:: SCPQSORT semantic presentation

begin


registration
let I, J be shiftable Program of SCMPDS;
let a be Int_position;
let k1 be Integer;
cluster if>0 (a,k1,I,J) ->  shiftable ;
correctness
coherence
if>0 (a,k1,I,J) is shiftable ;
proof
set i = (a,k1) <=0_goto ((card I) + 2);
set G = Goto ((card J) + 1);
 (I ';' (Goto ((card J) + 1))) ';' J = (I ';' (Load (goto ((card J) + 1)))) ';' J by SCMPDS_6:def_1;
then reconsider IJ = (I ';' (Goto ((card J) + 1))) ';' J as shiftable Program of SCMPDS ;
if>0 (a,k1,I,J) = ((((a,k1) <=0_goto ((card I) + 2)) ';' I) ';' (Goto ((card J) + 1))) ';' J by SCMPDS_6:def_5
.= (((a,k1) <=0_goto ((card I) + 2)) ';' (I ';' (Goto ((card J) + 1)))) ';' J by SCMPDS_4:14
.= ((a,k1) <=0_goto ((card I) + 2)) ';' IJ by SCMPDS_4:14
.= (Load ((a,k1) <=0_goto ((card I) + 2))) ';' IJ by SCMPDS_4:def_2 ;
hence  if>0 (a,k1,I,J) is shiftable ; ::_thesis: verum
end;
end;


theorem Th1: :: SCPQSORT:1
for P being Instruction-Sequence of SCMPDS
for s being 0 -started State of SCMPDS
for I being halt-free shiftable Program of SCMPDS
for J being shiftable Program of SCMPDS
for a, b being Int_position
for k1 being Integer st s . (DataLoc ((s . a),k1)) > 0 & I is_closed_on s,P & I is_halting_on s,P holds
(IExec ((if>0 (a,k1,I,J)),P,s)) . b = (IExec (I,P,s)) . b
proof
let P be Instruction-Sequence of SCMPDS; ::_thesis: for s being 0 -started State of SCMPDS
for I being halt-free shiftable Program of SCMPDS
for J being shiftable Program of SCMPDS
for a, b being Int_position
for k1 being Integer st s . (DataLoc ((s . a),k1)) > 0 & I is_closed_on s,P & I is_halting_on s,P holds
(IExec ((if>0 (a,k1,I,J)),P,s)) . b = (IExec (I,P,s)) . b
let s be 0 -started State of SCMPDS; ::_thesis: for I being halt-free shiftable Program of SCMPDS
for J being shiftable Program of SCMPDS
for a, b being Int_position
for k1 being Integer st s . (DataLoc ((s . a),k1)) > 0 & I is_closed_on s,P & I is_halting_on s,P holds
(IExec ((if>0 (a,k1,I,J)),P,s)) . b = (IExec (I,P,s)) . b
let I be halt-free shiftable Program of SCMPDS; ::_thesis: for J being shiftable Program of SCMPDS
for a, b being Int_position
for k1 being Integer st s . (DataLoc ((s . a),k1)) > 0 & I is_closed_on s,P & I is_halting_on s,P holds
(IExec ((if>0 (a,k1,I,J)),P,s)) . b = (IExec (I,P,s)) . b
let J be shiftable Program of SCMPDS; ::_thesis: for a, b being Int_position
for k1 being Integer st s . (DataLoc ((s . a),k1)) > 0 & I is_closed_on s,P & I is_halting_on s,P holds
(IExec ((if>0 (a,k1,I,J)),P,s)) . b = (IExec (I,P,s)) . b
let a, b be Int_position; ::_thesis: for k1 being Integer st s . (DataLoc ((s . a),k1)) > 0 & I is_closed_on s,P & I is_halting_on s,P holds
(IExec ((if>0 (a,k1,I,J)),P,s)) . b = (IExec (I,P,s)) . b
let k1 be Integer; ::_thesis: ( s . (DataLoc ((s . a),k1)) > 0 & I is_closed_on s,P & I is_halting_on s,P implies (IExec ((if>0 (a,k1,I,J)),P,s)) . b = (IExec (I,P,s)) . b )
assume that
A1: s . (DataLoc ((s . a),k1)) > 0 and
A2: I is_closed_on s,P and
A3: I is_halting_on s,P ; ::_thesis: (IExec ((if>0 (a,k1,I,J)),P,s)) . b = (IExec (I,P,s)) . b
A4: not b in dom (Start-At ((((card I) + (card J)) + 2),SCMPDS)) by SCMPDS_4:18;
 IExec ((if>0 (a,k1,I,J)),P,s) = (IExec (I,P,s)) +* (Start-At ((((card I) + (card J)) + 2),SCMPDS)) by A1, A2, A3, SCMPDS_6:70;
hence  (IExec ((if>0 (a,k1,I,J)),P,s)) . b = (IExec (I,P,s)) . b by A4, FUNCT_4:11; ::_thesis: verum
end;

set A = NAT ;

set D = SCM-Data-Loc ;

Lm2:  for a being Int_position
for i being Integer
for I being Program of SCMPDS holds while>0 (a,i,I) = ((a,i) <=0_goto ((card I) + 2)) ';' (I ';' (goto (- ((card I) + 1))))
proof
let a be Int_position; ::_thesis: for i being Integer
for I being Program of SCMPDS holds while>0 (a,i,I) = ((a,i) <=0_goto ((card I) + 2)) ';' (I ';' (goto (- ((card I) + 1))))
let i be Integer; ::_thesis: for I being Program of SCMPDS holds while>0 (a,i,I) = ((a,i) <=0_goto ((card I) + 2)) ';' (I ';' (goto (- ((card I) + 1))))
let I be Program of SCMPDS; ::_thesis: while>0 (a,i,I) = ((a,i) <=0_goto ((card I) + 2)) ';' (I ';' (goto (- ((card I) + 1))))
set i1 = (a,i) <=0_goto ((card I) + 2);
set i2 = goto (- ((card I) + 1));
thus  while>0 (a,i,I) = (((a,i) <=0_goto ((card I) + 2)) ';' I) ';' (goto (- ((card I) + 1))) by SCMPDS_8:def_3
.= ((a,i) <=0_goto ((card I) + 2)) ';' (I ';' (goto (- ((card I) + 1)))) by SCMPDS_4:15 ; ::_thesis: verum
end;

Lm3:  for I being Program of SCMPDS
for a being Int_position
for i being Integer holds Shift (I,1) c= while>0 (a,i,I)
proof
let I be Program of SCMPDS; ::_thesis: for a being Int_position
for i being Integer holds Shift (I,1) c= while>0 (a,i,I)
let a be Int_position; ::_thesis: for i being Integer holds Shift (I,1) c= while>0 (a,i,I)
let i be Integer; ::_thesis: Shift (I,1) c= while>0 (a,i,I)
set i1 = (a,i) <=0_goto ((card I) + 2);
set i2 = goto (- ((card I) + 1));
A1: while>0 (a,i,I) = (((a,i) <=0_goto ((card I) + 2)) ';' I) ';' (goto (- ((card I) + 1))) by SCMPDS_8:def_3
.= (((a,i) <=0_goto ((card I) + 2)) ';' I) ';' (Load (goto (- ((card I) + 1)))) by SCMPDS_4:def_3
.= ((Load ((a,i) <=0_goto ((card I) + 2))) ';' I) ';' (Load (goto (- ((card I) + 1)))) by SCMPDS_4:def_2 ;
 card (Load ((a,i) <=0_goto ((card I) + 2))) = 1 by COMPOS_1:54;
hence  Shift (I,1) c= while>0 (a,i,I) by A1, SCMPDS_7:3; ::_thesis: verum
end;

theorem Th2: :: SCPQSORT:2
for P being Instruction-Sequence of SCMPDS
for s, sm being State of SCMPDS
for I being halt-free shiftable Program of SCMPDS
for a being Int_position
for i being Integer
for m being Element of NAT st I is_closed_on s,P & I is_halting_on s,P & s . (DataLoc ((s . a),i)) > 0 & m = (LifeSpan ((P +* (stop I)),(Initialize s))) + 2 & sm = Comput ((P +* (stop (while>0 (a,i,I)))),(Initialize s),m) holds
( DataPart sm = DataPart (IExec (I,P,(Initialize s))) & Initialize sm = sm )
proof
let P be Instruction-Sequence of SCMPDS; ::_thesis: for s, sm being State of SCMPDS
for I being halt-free shiftable Program of SCMPDS
for a being Int_position
for i being Integer
for m being Element of NAT st I is_closed_on s,P & I is_halting_on s,P & s . (DataLoc ((s . a),i)) > 0 & m = (LifeSpan ((P +* (stop I)),(Initialize s))) + 2 & sm = Comput ((P +* (stop (while>0 (a,i,I)))),(Initialize s),m) holds
( DataPart sm = DataPart (IExec (I,P,(Initialize s))) & Initialize sm = sm )
let s, sm be State of SCMPDS; ::_thesis: for I being halt-free shiftable Program of SCMPDS
for a being Int_position
for i being Integer
for m being Element of NAT st I is_closed_on s,P & I is_halting_on s,P & s . (DataLoc ((s . a),i)) > 0 & m = (LifeSpan ((P +* (stop I)),(Initialize s))) + 2 & sm = Comput ((P +* (stop (while>0 (a,i,I)))),(Initialize s),m) holds
( DataPart sm = DataPart (IExec (I,P,(Initialize s))) & Initialize sm = sm )
let I be halt-free shiftable Program of SCMPDS; ::_thesis: for a being Int_position
for i being Integer
for m being Element of NAT st I is_closed_on s,P & I is_halting_on s,P & s . (DataLoc ((s . a),i)) > 0 & m = (LifeSpan ((P +* (stop I)),(Initialize s))) + 2 & sm = Comput ((P +* (stop (while>0 (a,i,I)))),(Initialize s),m) holds
( DataPart sm = DataPart (IExec (I,P,(Initialize s))) & Initialize sm = sm )
let a be Int_position; ::_thesis: for i being Integer
for m being Element of NAT st I is_closed_on s,P & I is_halting_on s,P & s . (DataLoc ((s . a),i)) > 0 & m = (LifeSpan ((P +* (stop I)),(Initialize s))) + 2 & sm = Comput ((P +* (stop (while>0 (a,i,I)))),(Initialize s),m) holds
( DataPart sm = DataPart (IExec (I,P,(Initialize s))) & Initialize sm = sm )
let i be Integer; ::_thesis: for m being Element of NAT st I is_closed_on s,P & I is_halting_on s,P & s . (DataLoc ((s . a),i)) > 0 & m = (LifeSpan ((P +* (stop I)),(Initialize s))) + 2 & sm = Comput ((P +* (stop (while>0 (a,i,I)))),(Initialize s),m) holds
( DataPart sm = DataPart (IExec (I,P,(Initialize s))) & Initialize sm = sm )
let m be Element of NAT ; ::_thesis: ( I is_closed_on s,P & I is_halting_on s,P & s . (DataLoc ((s . a),i)) > 0 & m = (LifeSpan ((P +* (stop I)),(Initialize s))) + 2 & sm = Comput ((P +* (stop (while>0 (a,i,I)))),(Initialize s),m) implies ( DataPart sm = DataPart (IExec (I,P,(Initialize s))) & Initialize sm = sm ) )
set b = DataLoc ((s . a),i);
set WH = while>0 (a,i,I);
set iWH = stop (while>0 (a,i,I));
set IsI = stop I;
set i1 = (a,i) <=0_goto ((card I) + 2);
set i2 = goto (- ((card I) + 1));
set s2 = Initialize s;
set P2 = P +* (stop I);
set s3 = Initialize s;
set P3 = P +* (stop (while>0 (a,i,I)));
set s4 = Comput ((P +* (stop (while>0 (a,i,I)))),(Initialize s),1);
set P4 = P +* (stop (while>0 (a,i,I)));
assume that
A4: I is_closed_on s,P and
A5: I is_halting_on s,P and
A6: s . (DataLoc ((s . a),i)) > 0 and
A7: m = (LifeSpan ((P +* (stop I)),(Initialize s))) + 2 and
A8: sm = Comput ((P +* (stop (while>0 (a,i,I)))),(Initialize s),m) ; ::_thesis: ( DataPart sm = DataPart (IExec (I,P,(Initialize s))) & Initialize sm = sm )
A9: stop I c= P +* (stop I) by FUNCT_4:25;
A10: stop (while>0 (a,i,I)) c= P +* (stop (while>0 (a,i,I))) by FUNCT_4:25;
 while>0 (a,i,I) c= stop (while>0 (a,i,I)) by AFINSQ_1:74;
then A11: while>0 (a,i,I) c= P +* (stop (while>0 (a,i,I))) by A10, XBOOLE_1:1;
 Shift (I,1) c= while>0 (a,i,I) by Lm3;
then A12: Shift (I,1) c= P +* (stop (while>0 (a,i,I))) by A11, XBOOLE_1:1;
A13: while>0 (a,i,I) = ((a,i) <=0_goto ((card I) + 2)) ';' (I ';' (goto (- ((card I) + 1)))) by Lm2;
A14: Comput ((P +* (stop (while>0 (a,i,I)))),(Initialize s),(0 + 1)) = Following ((P +* (stop (while>0 (a,i,I)))),(Comput ((P +* (stop (while>0 (a,i,I)))),(Initialize s),0))) by EXTPRO_1:3
.= Following ((P +* (stop (while>0 (a,i,I)))),(Initialize s))
.= Exec (((a,i) <=0_goto ((card I) + 2)),(Initialize s)) by A13, SCMPDS_6:11 ;
set m2 = LifeSpan ((P +* (stop I)),(Initialize s));
set s5 = Comput ((P +* (stop (while>0 (a,i,I)))),(Comput ((P +* (stop (while>0 (a,i,I)))),(Initialize s),1)),(LifeSpan ((P +* (stop I)),(Initialize s))));
set P5 = P +* (stop (while>0 (a,i,I)));
set l1 = (card I) + 1;
A15: IC (Initialize s) = 0 by MEMSTR_0:def_11;
TT: not DataLoc ((s . a),i) in dom (Start-At (0,SCMPDS)) by SCMPDS_4:18;
 not a in dom (Start-At (0,SCMPDS)) by SCMPDS_4:18;
then A16: (Initialize s) . (DataLoc (((Initialize s) . a),i)) = (Initialize s) . (DataLoc ((s . a),i)) by FUNCT_4:11
.= s . (DataLoc ((s . a),i)) by TT, FUNCT_4:11 ;
set m3 = (LifeSpan ((P +* (stop I)),(Initialize s))) + 1;
set s6 = Comput ((P +* (stop (while>0 (a,i,I)))),(Initialize s),((LifeSpan ((P +* (stop I)),(Initialize s))) + 1));
set P6 = P +* (stop (while>0 (a,i,I)));
A17: Comput ((P +* (stop (while>0 (a,i,I)))),(Initialize s),((LifeSpan ((P +* (stop I)),(Initialize s))) + 1)) = Comput ((P +* (stop (while>0 (a,i,I)))),(Comput ((P +* (stop (while>0 (a,i,I)))),(Initialize s),1)),(LifeSpan ((P +* (stop I)),(Initialize s)))) by EXTPRO_1:4;
set s7 = Comput ((P +* (stop (while>0 (a,i,I)))),(Initialize s),(((LifeSpan ((P +* (stop I)),(Initialize s))) + 1) + 1));
 (card I) + 1 < (card I) + 2 by XREAL_1:6;
then A18: (card I) + 1 in dom (while>0 (a,i,I)) by SCMPDS_8:18;
A20: I is_closed_on Initialize s,P +* (stop I) by A4, SCMPDS_6:24;
 for x being Int_position holds (Initialize s) . x = (Comput ((P +* (stop (while>0 (a,i,I)))),(Initialize s),1)) . x by A14, SCMPDS_2:56;
then A22: DataPart (Initialize s) = DataPart (Comput ((P +* (stop (while>0 (a,i,I)))),(Initialize s),1)) by SCMPDS_4:8;
A23: P +* (stop I) halts_on Initialize s by A5, SCMPDS_6:def_3;
 (P +* (stop I)) +* (stop I) halts_on Initialize (Initialize s) by A5, SCMPDS_6:def_3;
then A25: I is_halting_on Initialize s,P +* (stop I) by SCMPDS_6:def_3;
A26: IC (Comput ((P +* (stop (while>0 (a,i,I)))),(Initialize s),1)) = (Comput ((P +* (stop (while>0 (a,i,I)))),(Initialize s),1)) . (IC )
.= succ 0 by A6, A15, A14, A16, SCMPDS_2:56
.= 0 + 1 ;
then  DataPart (Comput ((P +* (stop I)),(Initialize s),(LifeSpan ((P +* (stop I)),(Initialize s))))) = DataPart (Comput ((P +* (stop (while>0 (a,i,I)))),(Comput ((P +* (stop (while>0 (a,i,I)))),(Initialize s),1)),(LifeSpan ((P +* (stop I)),(Initialize s))))) by A9, A25, A20, A22, A12, SCMPDS_7:18;
then A27: DataPart (Comput ((P +* (stop (while>0 (a,i,I)))),(Comput ((P +* (stop (while>0 (a,i,I)))),(Initialize s),1)),(LifeSpan ((P +* (stop I)),(Initialize s))))) = DataPart (Result ((P +* (stop I)),(Initialize s))) by A23, EXTPRO_1:23
.= DataPart (Result ((P +* (stop I)),(Initialize s)))
.= DataPart (IExec (I,P,(Initialize s))) by SCMPDS_4:def_5 ;
A28: IC (Comput ((P +* (stop (while>0 (a,i,I)))),(Comput ((P +* (stop (while>0 (a,i,I)))),(Initialize s),1)),(LifeSpan ((P +* (stop I)),(Initialize s))))) = (card I) + 1 by A9, A25, A20, A26, A22, A12, SCMPDS_7:18;
then A29: CurInstr ((P +* (stop (while>0 (a,i,I)))),(Comput ((P +* (stop (while>0 (a,i,I)))),(Initialize s),((LifeSpan ((P +* (stop I)),(Initialize s))) + 1)))) = (P +* (stop (while>0 (a,i,I)))) . ((card I) + 1) by A17, PBOOLE:143
.= (P +* (stop (while>0 (a,i,I)))) . ((card I) + 1)
.= (P +* (stop (while>0 (a,i,I)))) . ((card I) + 1)
.= (while>0 (a,i,I)) . ((card I) + 1) by A18, A11, GRFUNC_1:2
.= goto (- ((card I) + 1)) by SCMPDS_8:19 ;
A30: Comput ((P +* (stop (while>0 (a,i,I)))),(Initialize s),(((LifeSpan ((P +* (stop I)),(Initialize s))) + 1) + 1)) = Following ((P +* (stop (while>0 (a,i,I)))),(Comput ((P +* (stop (while>0 (a,i,I)))),(Initialize s),((LifeSpan ((P +* (stop I)),(Initialize s))) + 1)))) by EXTPRO_1:3
.= Exec ((goto (- ((card I) + 1))),(Comput ((P +* (stop (while>0 (a,i,I)))),(Initialize s),((LifeSpan ((P +* (stop I)),(Initialize s))) + 1)))) by A29 ;
now__::_thesis:_for_x_being_Int_position_holds_sm_._x_=_(Comput_((P_+*_(stop_(while>0_(a,i,I)))),(Comput_((P_+*_(stop_(while>0_(a,i,I)))),(Initialize_s),1)),(LifeSpan_((P_+*_(stop_I)),(Initialize_s)))))_._x
let x be Int_position; ::_thesis: sm . x = (Comput ((P +* (stop (while>0 (a,i,I)))),(Comput ((P +* (stop (while>0 (a,i,I)))),(Initialize s),1)),(LifeSpan ((P +* (stop I)),(Initialize s))))) . x
thus sm . x = (Comput ((P +* (stop (while>0 (a,i,I)))),(Initialize s),((LifeSpan ((P +* (stop I)),(Initialize s))) + 1))) . x by A7, A8, A30, SCMPDS_2:54
.= (Comput ((P +* (stop (while>0 (a,i,I)))),(Comput ((P +* (stop (while>0 (a,i,I)))),(Initialize s),1)),(LifeSpan ((P +* (stop I)),(Initialize s))))) . x by EXTPRO_1:4 ; ::_thesis: verum
end;
hence  DataPart sm = DataPart (IExec (I,P,(Initialize s))) by A27, SCMPDS_4:8; ::_thesis: Initialize sm = sm
IC (Comput ((P +* (stop (while>0 (a,i,I)))),(Initialize s),(((LifeSpan ((P +* (stop I)),(Initialize s))) + 1) + 1))) = (Comput ((P +* (stop (while>0 (a,i,I)))),(Initialize s),(((LifeSpan ((P +* (stop I)),(Initialize s))) + 1) + 1))) . (IC )
.= ICplusConst ((Comput ((P +* (stop (while>0 (a,i,I)))),(Initialize s),((LifeSpan ((P +* (stop I)),(Initialize s))) + 1))),(0 - ((card I) + 1))) by A30, SCMPDS_2:54
.= 0 by A28, A17, SCMPDS_7:1 ;
hence  Initialize sm = sm by A7, A8, MEMSTR_0:46; ::_thesis: verum
end;

theorem Th3: :: SCPQSORT:3
for P being Instruction-Sequence of SCMPDS
for s being State of SCMPDS
for I being Program of SCMPDS st ( for t being 0 -started State of SCMPDS
for Q being Instruction-Sequence of SCMPDS st DataPart t = DataPart s holds
I is_halting_on t,Q ) holds
I is_closed_on s,P
proof
let P be Instruction-Sequence of SCMPDS; ::_thesis: for s being State of SCMPDS
for I being Program of SCMPDS st ( for t being 0 -started State of SCMPDS
for Q being Instruction-Sequence of SCMPDS st DataPart t = DataPart s holds
I is_halting_on t,Q ) holds
I is_closed_on s,P
let s be State of SCMPDS; ::_thesis: for I being Program of SCMPDS st ( for t being 0 -started State of SCMPDS
for Q being Instruction-Sequence of SCMPDS st DataPart t = DataPart s holds
I is_halting_on t,Q ) holds
I is_closed_on s,P
let I be Program of SCMPDS; ::_thesis: ( ( for t being 0 -started State of SCMPDS
for Q being Instruction-Sequence of SCMPDS st DataPart t = DataPart s holds
I is_halting_on t,Q ) implies I is_closed_on s,P )
assume A1: for t being 0 -started State of SCMPDS
for Q being Instruction-Sequence of SCMPDS st DataPart t = DataPart s holds
I is_halting_on t,Q ; ::_thesis: I is_closed_on s,P
set pI = stop I;
set sI = Initialize s;
set PI = P +* (stop I);
defpred S1[ Nat] means  not IC (Comput ((P +* (stop I)),(Initialize s),$1)) in dom (stop I);
A2: for a being Int_position holds s . a = (Initialize s) . a
proof
let a be Int_position; ::_thesis: s . a = (Initialize s) . a
 not a in dom (Start-At (0,SCMPDS)) by SCMPDS_4:18;
hence  s . a = (Initialize s) . a by FUNCT_4:11; ::_thesis: verum
end;
assume  not I is_closed_on s,P ; ::_thesis: contradiction
then  ex k being Element of NAT st S1[k] by SCMPDS_6:def_2;
then A4: ex k being Nat st S1[k] ;
consider n being Nat such that
A5: S1[n] and
A6: for m being Nat st S1[m] holds
n <= m from NAT_1:sch_5(A4);
reconsider n = n as Element of NAT by ORDINAL1:def_12;
set s2 = Comput ((P +* (stop I)),(Initialize s),n);
set P2 = P +* (stop I);
set Ig = ((IC (Comput ((P +* (stop I)),(Initialize s),n))),(succ (IC (Comput ((P +* (stop I)),(Initialize s),n))))) --> ((goto 1),(goto (- 1)));
reconsider P0 = (P +* (stop I)) +* (((IC (Comput ((P +* (stop I)),(Initialize s),n))),(succ (IC (Comput ((P +* (stop I)),(Initialize s),n))))) --> ((goto 1),(goto (- 1)))) as Instruction-Sequence of SCMPDS ;
reconsider P1 = (P +* (stop I)) +* (((IC (Comput ((P +* (stop I)),(Initialize s),n))),(succ (IC (Comput ((P +* (stop I)),(Initialize s),n))))) --> ((goto 1),(goto (- 1)))) as Instruction-Sequence of SCMPDS ;
reconsider P3 = (P +* (stop I)) +* ((IC (Comput ((P +* (stop I)),(Initialize s),n))),(goto 1)) as Instruction-Sequence of SCMPDS ;
reconsider P4 = P3 +* ((succ (IC (Comput ((P +* (stop I)),(Initialize s),n)))),(goto (- 1))) as Instruction-Sequence of SCMPDS ;
U: P0 = P4 by FUNCT_7:139;
A13: for m being Element of NAT st m < n holds
IC (Comput ((P +* (stop I)),(Initialize s),m)) in dom (stop I) by A6;
A17: stop I c= P +* (stop I) by FUNCT_4:25;
 stop I c= P3 by A5, FUNCT_4:25, FUNCT_7:89;
then A18: stop I c= P0 by U, A5, AFINSQ_1:73, FUNCT_7:89;
then XX: Comput (P0,(Initialize s),n) = Comput ((P +* (stop I)),(Initialize s),n) by A17, A13, SCMPDS_4:21;
DataPart (Initialize s) = DataPart (Initialize s)
.= DataPart s by A2, SCMPDS_4:8 ;
then  I is_halting_on Initialize s,P0 by A1;
then A20: P0 +* (stop I) halts_on Initialize (Initialize s) by SCMPDS_6:def_3;
A21: not P0 halts_on Comput ((P +* (stop I)),(Initialize s),n) by SCMPDS_4:20;
 P0 +* (stop I) = P0 by A18, FUNCT_4:98;
hence  contradiction by A21, XX, A20, EXTPRO_1:22; ::_thesis: verum
end;

theorem Th4: :: SCPQSORT:4
for i1, i2, i3, i4 being Instruction of SCMPDS holds card (((i1 ';' i2) ';' i3) ';' i4) = 4
proof
let x1, x2, x3, x4 be Instruction of SCMPDS; ::_thesis: card (((x1 ';' x2) ';' x3) ';' x4) = 4
thus  card (((x1 ';' x2) ';' x3) ';' x4) = (card ((x1 ';' x2) ';' x3)) + 1 by SCMP_GCD:4
.= ((card (x1 ';' x2)) + 1) + 1 by SCMP_GCD:4
.= (2 + 1) + 1 by SCMP_GCD:5
.= 4 ; ::_thesis: verum
end;

theorem Th5: :: SCPQSORT:5
for P being Instruction-Sequence of SCMPDS
for s being 0 -started State of SCMPDS
for I being halt-free shiftable Program of SCMPDS
for a, x, y being Int_position
for i, c being Integer st s . x >= c + (s . (DataLoc ((s . a),i))) & ( for t being 0 -started State of SCMPDS
for Q being Instruction-Sequence of SCMPDS st t . x >= c + (t . (DataLoc ((s . a),i))) & t . y = s . y & t . a = s . a & t . (DataLoc ((s . a),i)) > 0 holds
( (IExec (I,Q,t)) . a = t . a & I is_closed_on t,Q & I is_halting_on t,Q & (IExec (I,Q,t)) . (DataLoc ((s . a),i)) < t . (DataLoc ((s . a),i)) & (IExec (I,Q,t)) . x >= c + ((IExec (I,Q,t)) . (DataLoc ((s . a),i))) & (IExec (I,Q,t)) . y = t . y ) ) holds
( while>0 (a,i,I) is_closed_on s,P & while>0 (a,i,I) is_halting_on s,P & ( s . (DataLoc ((s . a),i)) > 0 implies IExec ((while>0 (a,i,I)),P,s) = IExec ((while>0 (a,i,I)),P,(Initialize (IExec (I,P,s)))) ) )
proof
let P be Instruction-Sequence of SCMPDS; ::_thesis: for s being 0 -started State of SCMPDS
for I being halt-free shiftable Program of SCMPDS
for a, x, y being Int_position
for i, c being Integer st s . x >= c + (s . (DataLoc ((s . a),i))) & ( for t being 0 -started State of SCMPDS
for Q being Instruction-Sequence of SCMPDS st t . x >= c + (t . (DataLoc ((s . a),i))) & t . y = s . y & t . a = s . a & t . (DataLoc ((s . a),i)) > 0 holds
( (IExec (I,Q,t)) . a = t . a & I is_closed_on t,Q & I is_halting_on t,Q & (IExec (I,Q,t)) . (DataLoc ((s . a),i)) < t . (DataLoc ((s . a),i)) & (IExec (I,Q,t)) . x >= c + ((IExec (I,Q,t)) . (DataLoc ((s . a),i))) & (IExec (I,Q,t)) . y = t . y ) ) holds
( while>0 (a,i,I) is_closed_on s,P & while>0 (a,i,I) is_halting_on s,P & ( s . (DataLoc ((s . a),i)) > 0 implies IExec ((while>0 (a,i,I)),P,s) = IExec ((while>0 (a,i,I)),P,(Initialize (IExec (I,P,s)))) ) )
let s be 0 -started State of SCMPDS; ::_thesis: for I being halt-free shiftable Program of SCMPDS
for a, x, y being Int_position
for i, c being Integer st s . x >= c + (s . (DataLoc ((s . a),i))) & ( for t being 0 -started State of SCMPDS
for Q being Instruction-Sequence of SCMPDS st t . x >= c + (t . (DataLoc ((s . a),i))) & t . y = s . y & t . a = s . a & t . (DataLoc ((s . a),i)) > 0 holds
( (IExec (I,Q,t)) . a = t . a & I is_closed_on t,Q & I is_halting_on t,Q & (IExec (I,Q,t)) . (DataLoc ((s . a),i)) < t . (DataLoc ((s . a),i)) & (IExec (I,Q,t)) . x >= c + ((IExec (I,Q,t)) . (DataLoc ((s . a),i))) & (IExec (I,Q,t)) . y = t . y ) ) holds
( while>0 (a,i,I) is_closed_on s,P & while>0 (a,i,I) is_halting_on s,P & ( s . (DataLoc ((s . a),i)) > 0 implies IExec ((while>0 (a,i,I)),P,s) = IExec ((while>0 (a,i,I)),P,(Initialize (IExec (I,P,s)))) ) )
let I be halt-free shiftable Program of SCMPDS; ::_thesis: for a, x, y being Int_position
for i, c being Integer st s . x >= c + (s . (DataLoc ((s . a),i))) & ( for t being 0 -started State of SCMPDS
for Q being Instruction-Sequence of SCMPDS st t . x >= c + (t . (DataLoc ((s . a),i))) & t . y = s . y & t . a = s . a & t . (DataLoc ((s . a),i)) > 0 holds
( (IExec (I,Q,t)) . a = t . a & I is_closed_on t,Q & I is_halting_on t,Q & (IExec (I,Q,t)) . (DataLoc ((s . a),i)) < t . (DataLoc ((s . a),i)) & (IExec (I,Q,t)) . x >= c + ((IExec (I,Q,t)) . (DataLoc ((s . a),i))) & (IExec (I,Q,t)) . y = t . y ) ) holds
( while>0 (a,i,I) is_closed_on s,P & while>0 (a,i,I) is_halting_on s,P & ( s . (DataLoc ((s . a),i)) > 0 implies IExec ((while>0 (a,i,I)),P,s) = IExec ((while>0 (a,i,I)),P,(Initialize (IExec (I,P,s)))) ) )
let a, x1, y1 be Int_position; ::_thesis: for i, c being Integer st s . x1 >= c + (s . (DataLoc ((s . a),i))) & ( for t being 0 -started State of SCMPDS
for Q being Instruction-Sequence of SCMPDS st t . x1 >= c + (t . (DataLoc ((s . a),i))) & t . y1 = s . y1 & t . a = s . a & t . (DataLoc ((s . a),i)) > 0 holds
( (IExec (I,Q,t)) . a = t . a & I is_closed_on t,Q & I is_halting_on t,Q & (IExec (I,Q,t)) . (DataLoc ((s . a),i)) < t . (DataLoc ((s . a),i)) & (IExec (I,Q,t)) . x1 >= c + ((IExec (I,Q,t)) . (DataLoc ((s . a),i))) & (IExec (I,Q,t)) . y1 = t . y1 ) ) holds
( while>0 (a,i,I) is_closed_on s,P & while>0 (a,i,I) is_halting_on s,P & ( s . (DataLoc ((s . a),i)) > 0 implies IExec ((while>0 (a,i,I)),P,s) = IExec ((while>0 (a,i,I)),P,(Initialize (IExec (I,P,s)))) ) )
let i, c be Integer; ::_thesis: ( s . x1 >= c + (s . (DataLoc ((s . a),i))) & ( for t being 0 -started State of SCMPDS
for Q being Instruction-Sequence of SCMPDS st t . x1 >= c + (t . (DataLoc ((s . a),i))) & t . y1 = s . y1 & t . a = s . a & t . (DataLoc ((s . a),i)) > 0 holds
( (IExec (I,Q,t)) . a = t . a & I is_closed_on t,Q & I is_halting_on t,Q & (IExec (I,Q,t)) . (DataLoc ((s . a),i)) < t . (DataLoc ((s . a),i)) & (IExec (I,Q,t)) . x1 >= c + ((IExec (I,Q,t)) . (DataLoc ((s . a),i))) & (IExec (I,Q,t)) . y1 = t . y1 ) ) implies ( while>0 (a,i,I) is_closed_on s,P & while>0 (a,i,I) is_halting_on s,P & ( s . (DataLoc ((s . a),i)) > 0 implies IExec ((while>0 (a,i,I)),P,s) = IExec ((while>0 (a,i,I)),P,(Initialize (IExec (I,P,s)))) ) ) )
set b = DataLoc ((s . a),i);
assume  s . x1 >= c + (s . (DataLoc ((s . a),i))) ; ::_thesis: ( ex t being 0 -started State of SCMPDS ex Q being Instruction-Sequence of SCMPDS st
( t . x1 >= c + (t . (DataLoc ((s . a),i))) & t . y1 = s . y1 & t . a = s . a & t . (DataLoc ((s . a),i)) > 0 & not ( (IExec (I,Q,t)) . a = t . a & I is_closed_on t,Q & I is_halting_on t,Q & (IExec (I,Q,t)) . (DataLoc ((s . a),i)) < t . (DataLoc ((s . a),i)) & (IExec (I,Q,t)) . x1 >= c + ((IExec (I,Q,t)) . (DataLoc ((s . a),i))) & (IExec (I,Q,t)) . y1 = t . y1 ) ) or ( while>0 (a,i,I) is_closed_on s,P & while>0 (a,i,I) is_halting_on s,P & ( s . (DataLoc ((s . a),i)) > 0 implies IExec ((while>0 (a,i,I)),P,s) = IExec ((while>0 (a,i,I)),P,(Initialize (IExec (I,P,s)))) ) ) )
then A2: for x being Int_position st x in {x1} holds
s . x >= c + (s . (DataLoc ((s . a),i))) by TARSKI:def_1;
assume A3: for t being 0 -started State of SCMPDS
for Q being Instruction-Sequence of SCMPDS st t . x1 >= c + (t . (DataLoc ((s . a),i))) & t . y1 = s . y1 & t . a = s . a & t . (DataLoc ((s . a),i)) > 0 holds
( (IExec (I,Q,t)) . a = t . a & I is_closed_on t,Q & I is_halting_on t,Q & (IExec (I,Q,t)) . (DataLoc ((s . a),i)) < t . (DataLoc ((s . a),i)) & (IExec (I,Q,t)) . x1 >= c + ((IExec (I,Q,t)) . (DataLoc ((s . a),i))) & (IExec (I,Q,t)) . y1 = t . y1 ) ; ::_thesis: ( while>0 (a,i,I) is_closed_on s,P & while>0 (a,i,I) is_halting_on s,P & ( s . (DataLoc ((s . a),i)) > 0 implies IExec ((while>0 (a,i,I)),P,s) = IExec ((while>0 (a,i,I)),P,(Initialize (IExec (I,P,s)))) ) )
now__::_thesis:_for_t_being_0_-started_State_of_SCMPDS
for_Q_being_Instruction-Sequence_of_SCMPDS_st_(_for_x_being_Int_position_st_x_in_{x1}_holds_
t_._x_>=_c_+_(t_._(DataLoc_((s_._a),i)))_)_&_(_for_x_being_Int_position_st_x_in_{y1}_holds_
t_._x_=_s_._x_)_&_t_._a_=_s_._a_&_t_._(DataLoc_((s_._a),i))_>_0_holds_
(_(IExec_(I,Q,t))_._a_=_t_._a_&_I_is_closed_on_t,Q_&_I_is_halting_on_t,Q_&_(IExec_(I,Q,t))_._(DataLoc_((s_._a),i))_<_t_._(DataLoc_((s_._a),i))_&_(_for_x_being_Int_position_st_x_in_{x1}_holds_
(IExec_(I,Q,t))_._x_>=_c_+_((IExec_(I,Q,t))_._(DataLoc_((s_._a),i)))_)_&_(_for_x_being_Int_position_st_x_in_{y1}_holds_
(IExec_(I,Q,t))_._x_=_t_._x_)_)
let t be 0 -started State of SCMPDS; ::_thesis: for Q being Instruction-Sequence of SCMPDS st ( for x being Int_position st x in {x1} holds
t . x >= c + (t . (DataLoc ((s . a),i))) ) & ( for x being Int_position st x in {y1} holds
t . x = s . x ) & t . a = s . a & t . (DataLoc ((s . a),i)) > 0 holds
( (IExec (I,Q,t)) . a = t . a & I is_closed_on t,Q & I is_halting_on t,Q & (IExec (I,Q,t)) . (DataLoc ((s . a),i)) < t . (DataLoc ((s . a),i)) & ( for x being Int_position st x in {x1} holds
(IExec (I,Q,t)) . x >= c + ((IExec (I,Q,t)) . (DataLoc ((s . a),i))) ) & ( for x being Int_position st x in {y1} holds
(IExec (I,Q,t)) . x = t . x ) )
let Q be Instruction-Sequence of SCMPDS; ::_thesis: ( ( for x being Int_position st x in {x1} holds
t . x >= c + (t . (DataLoc ((s . a),i))) ) & ( for x being Int_position st x in {y1} holds
t . x = s . x ) & t . a = s . a & t . (DataLoc ((s . a),i)) > 0 implies ( (IExec (I,Q,t)) . a = t . a & I is_closed_on t,Q & I is_halting_on t,Q & (IExec (I,Q,t)) . (DataLoc ((s . a),i)) < t . (DataLoc ((s . a),i)) & ( for x being Int_position st x in {x1} holds
(IExec (I,Q,t)) . x >= c + ((IExec (I,Q,t)) . (DataLoc ((s . a),i))) ) & ( for x being Int_position st x in {y1} holds
(IExec (I,Q,t)) . x = t . x ) ) )
assume that
A4: for x being Int_position st x in {x1} holds
t . x >= c + (t . (DataLoc ((s . a),i))) and
A5: for x being Int_position st x in {y1} holds
t . x = s . x and
A6: t . a = s . a and
A7: t . (DataLoc ((s . a),i)) > 0 ; ::_thesis: ( (IExec (I,Q,t)) . a = t . a & I is_closed_on t,Q & I is_halting_on t,Q & (IExec (I,Q,t)) . (DataLoc ((s . a),i)) < t . (DataLoc ((s . a),i)) & ( for x being Int_position st x in {x1} holds
(IExec (I,Q,t)) . x >= c + ((IExec (I,Q,t)) . (DataLoc ((s . a),i))) ) & ( for x being Int_position st x in {y1} holds
(IExec (I,Q,t)) . x = t . x ) )
 y1 in {y1} by TARSKI:def_1;
then A8: t . y1 = s . y1 by A5;
 x1 in {x1} by TARSKI:def_1;
then A9: t . x1 >= c + (t . (DataLoc ((s . a),i))) by A4;
hence  ( (IExec (I,Q,t)) . a = t . a & I is_closed_on t,Q & I is_halting_on t,Q & (IExec (I,Q,t)) . (DataLoc ((s . a),i)) < t . (DataLoc ((s . a),i)) ) by A3, A6, A7, A8; ::_thesis: ( ( for x being Int_position st x in {x1} holds
(IExec (I,Q,t)) . x >= c + ((IExec (I,Q,t)) . (DataLoc ((s . a),i))) ) & ( for x being Int_position st x in {y1} holds
(IExec (I,Q,t)) . x = t . x ) )
hereby  ::_thesis: for x being Int_position st x in {y1} holds
(IExec (I,Q,t)) . x = t . x
let x be Int_position; ::_thesis: ( x in {x1} implies (IExec (I,Q,t)) . x >= c + ((IExec (I,Q,t)) . (DataLoc ((s . a),i))) )
assume A10: x in {x1} ; ::_thesis: (IExec (I,Q,t)) . x >= c + ((IExec (I,Q,t)) . (DataLoc ((s . a),i)))
 (IExec (I,Q,t)) . x1 >= c + ((IExec (I,Q,t)) . (DataLoc ((s . a),i))) by A3, A6, A7, A9, A8;
hence  (IExec (I,Q,t)) . x >= c + ((IExec (I,Q,t)) . (DataLoc ((s . a),i))) by A10, TARSKI:def_1; ::_thesis: verum
end;
let x be Int_position; ::_thesis: ( x in {y1} implies (IExec (I,Q,t)) . x = t . x )
assume A11: x in {y1} ; ::_thesis: (IExec (I,Q,t)) . x = t . x
hence (IExec (I,Q,t)) . x = (IExec (I,Q,t)) . y1 by TARSKI:def_1
.= t . y1 by A3, A6, A7, A9, A8
.= t . x by A11, TARSKI:def_1 ;
::_thesis: verum
end;
hence  ( while>0 (a,i,I) is_closed_on s,P & while>0 (a,i,I) is_halting_on s,P & ( s . (DataLoc ((s . a),i)) > 0 implies IExec ((while>0 (a,i,I)),P,s) = IExec ((while>0 (a,i,I)),P,(Initialize (IExec (I,P,s)))) ) ) by A2, SCMPDS_8:27; ::_thesis: verum
end;

theorem Th6: :: SCPQSORT:6
for P being Instruction-Sequence of SCMPDS
for s being 0 -started State of SCMPDS
for I being halt-free shiftable Program of SCMPDS
for a, x, y being Int_position
for i, c being Integer st s . x >= c & ( for t being 0 -started State of SCMPDS
for Q being Instruction-Sequence of SCMPDS st t . x >= c & t . y = s . y & t . a = s . a & t . (DataLoc ((s . a),i)) > 0 holds
( (IExec (I,Q,t)) . a = t . a & I is_closed_on t,Q & I is_halting_on t,Q & (IExec (I,Q,t)) . (DataLoc ((s . a),i)) < t . (DataLoc ((s . a),i)) & (IExec (I,Q,t)) . x >= c & (IExec (I,Q,t)) . y = t . y ) ) holds
( while>0 (a,i,I) is_closed_on s,P & while>0 (a,i,I) is_halting_on s,P & ( s . (DataLoc ((s . a),i)) > 0 implies IExec ((while>0 (a,i,I)),P,s) = IExec ((while>0 (a,i,I)),P,(Initialize (IExec (I,P,s)))) ) )
proof
let P be Instruction-Sequence of SCMPDS; ::_thesis: for s being 0 -started State of SCMPDS
for I being halt-free shiftable Program of SCMPDS
for a, x, y being Int_position
for i, c being Integer st s . x >= c & ( for t being 0 -started State of SCMPDS
for Q being Instruction-Sequence of SCMPDS st t . x >= c & t . y = s . y & t . a = s . a & t . (DataLoc ((s . a),i)) > 0 holds
( (IExec (I,Q,t)) . a = t . a & I is_closed_on t,Q & I is_halting_on t,Q & (IExec (I,Q,t)) . (DataLoc ((s . a),i)) < t . (DataLoc ((s . a),i)) & (IExec (I,Q,t)) . x >= c & (IExec (I,Q,t)) . y = t . y ) ) holds
( while>0 (a,i,I) is_closed_on s,P & while>0 (a,i,I) is_halting_on s,P & ( s . (DataLoc ((s . a),i)) > 0 implies IExec ((while>0 (a,i,I)),P,s) = IExec ((while>0 (a,i,I)),P,(Initialize (IExec (I,P,s)))) ) )
let s be 0 -started State of SCMPDS; ::_thesis: for I being halt-free shiftable Program of SCMPDS
for a, x, y being Int_position
for i, c being Integer st s . x >= c & ( for t being 0 -started State of SCMPDS
for Q being Instruction-Sequence of SCMPDS st t . x >= c & t . y = s . y & t . a = s . a & t . (DataLoc ((s . a),i)) > 0 holds
( (IExec (I,Q,t)) . a = t . a & I is_closed_on t,Q & I is_halting_on t,Q & (IExec (I,Q,t)) . (DataLoc ((s . a),i)) < t . (DataLoc ((s . a),i)) & (IExec (I,Q,t)) . x >= c & (IExec (I,Q,t)) . y = t . y ) ) holds
( while>0 (a,i,I) is_closed_on s,P & while>0 (a,i,I) is_halting_on s,P & ( s . (DataLoc ((s . a),i)) > 0 implies IExec ((while>0 (a,i,I)),P,s) = IExec ((while>0 (a,i,I)),P,(Initialize (IExec (I,P,s)))) ) )
let I be halt-free shiftable Program of SCMPDS; ::_thesis: for a, x, y being Int_position
for i, c being Integer st s . x >= c & ( for t being 0 -started State of SCMPDS
for Q being Instruction-Sequence of SCMPDS st t . x >= c & t . y = s . y & t . a = s . a & t . (DataLoc ((s . a),i)) > 0 holds
( (IExec (I,Q,t)) . a = t . a & I is_closed_on t,Q & I is_halting_on t,Q & (IExec (I,Q,t)) . (DataLoc ((s . a),i)) < t . (DataLoc ((s . a),i)) & (IExec (I,Q,t)) . x >= c & (IExec (I,Q,t)) . y = t . y ) ) holds
( while>0 (a,i,I) is_closed_on s,P & while>0 (a,i,I) is_halting_on s,P & ( s . (DataLoc ((s . a),i)) > 0 implies IExec ((while>0 (a,i,I)),P,s) = IExec ((while>0 (a,i,I)),P,(Initialize (IExec (I,P,s)))) ) )
let a, x, y be Int_position; ::_thesis: for i, c being Integer st s . x >= c & ( for t being 0 -started State of SCMPDS
for Q being Instruction-Sequence of SCMPDS st t . x >= c & t . y = s . y & t . a = s . a & t . (DataLoc ((s . a),i)) > 0 holds
( (IExec (I,Q,t)) . a = t . a & I is_closed_on t,Q & I is_halting_on t,Q & (IExec (I,Q,t)) . (DataLoc ((s . a),i)) < t . (DataLoc ((s . a),i)) & (IExec (I,Q,t)) . x >= c & (IExec (I,Q,t)) . y = t . y ) ) holds
( while>0 (a,i,I) is_closed_on s,P & while>0 (a,i,I) is_halting_on s,P & ( s . (DataLoc ((s . a),i)) > 0 implies IExec ((while>0 (a,i,I)),P,s) = IExec ((while>0 (a,i,I)),P,(Initialize (IExec (I,P,s)))) ) )
let i, c be Integer; ::_thesis: ( s . x >= c & ( for t being 0 -started State of SCMPDS
for Q being Instruction-Sequence of SCMPDS st t . x >= c & t . y = s . y & t . a = s . a & t . (DataLoc ((s . a),i)) > 0 holds
( (IExec (I,Q,t)) . a = t . a & I is_closed_on t,Q & I is_halting_on t,Q & (IExec (I,Q,t)) . (DataLoc ((s . a),i)) < t . (DataLoc ((s . a),i)) & (IExec (I,Q,t)) . x >= c & (IExec (I,Q,t)) . y = t . y ) ) implies ( while>0 (a,i,I) is_closed_on s,P & while>0 (a,i,I) is_halting_on s,P & ( s . (DataLoc ((s . a),i)) > 0 implies IExec ((while>0 (a,i,I)),P,s) = IExec ((while>0 (a,i,I)),P,(Initialize (IExec (I,P,s)))) ) ) )
set b = DataLoc ((s . a),i);
defpred S1[ set ] means  ex t being State of SCMPDS st
( t = $1 & t . x >= c & t . y = s . y );
consider f being Function of (product (the_Values_of SCMPDS)),NAT such that
A2: for s being State of SCMPDS holds
( ( s . (DataLoc ((s . a),i)) <= 0 implies f . s = 0 ) & ( s . (DataLoc ((s . a),i)) > 0 implies f . s = s . (DataLoc ((s . a),i)) ) ) by SCMPDS_8:5;
deffunc H1( State of SCMPDS) ->  Element of NAT = f . $1;
A5: for t being 0 -started State of SCMPDS st S1[t] & H1(t) = 0 holds
t . (DataLoc ((s . a),i)) <= 0 by A2;
assume A6: s . x >= c ; ::_thesis: ( ex t being 0 -started State of SCMPDS ex Q being Instruction-Sequence of SCMPDS st
( t . x >= c & t . y = s . y & t . a = s . a & t . (DataLoc ((s . a),i)) > 0 & not ( (IExec (I,Q,t)) . a = t . a & I is_closed_on t,Q & I is_halting_on t,Q & (IExec (I,Q,t)) . (DataLoc ((s . a),i)) < t . (DataLoc ((s . a),i)) & (IExec (I,Q,t)) . x >= c & (IExec (I,Q,t)) . y = t . y ) ) or ( while>0 (a,i,I) is_closed_on s,P & while>0 (a,i,I) is_halting_on s,P & ( s . (DataLoc ((s . a),i)) > 0 implies IExec ((while>0 (a,i,I)),P,s) = IExec ((while>0 (a,i,I)),P,(Initialize (IExec (I,P,s)))) ) ) )
A7: S1[s] by A6;
assume A8: for t being 0 -started State of SCMPDS
for Q being Instruction-Sequence of SCMPDS st t . x >= c & t . y = s . y & t . a = s . a & t . (DataLoc ((s . a),i)) > 0 holds
( (IExec (I,Q,t)) . a = t . a & I is_closed_on t,Q & I is_halting_on t,Q & (IExec (I,Q,t)) . (DataLoc ((s . a),i)) < t . (DataLoc ((s . a),i)) & (IExec (I,Q,t)) . x >= c & (IExec (I,Q,t)) . y = t . y ) ; ::_thesis: ( while>0 (a,i,I) is_closed_on s,P & while>0 (a,i,I) is_halting_on s,P & ( s . (DataLoc ((s . a),i)) > 0 implies IExec ((while>0 (a,i,I)),P,s) = IExec ((while>0 (a,i,I)),P,(Initialize (IExec (I,P,s)))) ) )
A9: now__::_thesis:_for_t_being_0_-started_State_of_SCMPDS
for_Q_being_Instruction-Sequence_of_SCMPDS_st_S1[t]_&_t_._a_=_s_._a_&_t_._(DataLoc_((s_._a),i))_>_0_holds_
(_(IExec_(I,Q,t))_._a_=_t_._a_&_I_is_closed_on_t,Q_&_I_is_halting_on_t,Q_&_H1(_Initialize_(IExec_(I,Q,t)))_<_H1(t)_&_S1[_Initialize_(IExec_(I,Q,t))]_)
let t be 0 -started State of SCMPDS; ::_thesis: for Q being Instruction-Sequence of SCMPDS st S1[t] & t . a = s . a & t . (DataLoc ((s . a),i)) > 0 holds
( (IExec (I,Q,t)) . a = t . a & I is_closed_on t,Q & I is_halting_on t,Q & H1( Initialize (IExec (I,Q,t))) < H1(t) & S1[ Initialize (IExec (I,Q,t))] )
let Q be Instruction-Sequence of SCMPDS; ::_thesis: ( S1[t] & t . a = s . a & t . (DataLoc ((s . a),i)) > 0 implies ( (IExec (I,Q,t)) . a = t . a & I is_closed_on t,Q & I is_halting_on t,Q & H1( Initialize (IExec (I,Q,t))) < H1(t) & S1[ Initialize (IExec (I,Q,t))] ) )
assume that
A10: S1[t] and
A11: t . a = s . a and
A12: t . (DataLoc ((s . a),i)) > 0 ; ::_thesis: ( (IExec (I,Q,t)) . a = t . a & I is_closed_on t,Q & I is_halting_on t,Q & H1( Initialize (IExec (I,Q,t))) < H1(t) & S1[ Initialize (IExec (I,Q,t))] )
set It = IExec (I,Q,t);
set t2 = Initialize (IExec (I,Q,t));
thus  ( (IExec (I,Q,t)) . a = t . a & I is_closed_on t,Q & I is_halting_on t,Q ) by A8, A11, A12, A10; ::_thesis: ( H1( Initialize (IExec (I,Q,t))) < H1(t) & S1[ Initialize (IExec (I,Q,t))] )
thus  H1( Initialize (IExec (I,Q,t))) < H1(t) ::_thesis: S1[ Initialize (IExec (I,Q,t))]
proof
A15: H1(t) = t . (DataLoc ((s . a),i)) by A2, A12
.= t . (DataLoc ((s . a),i)) ;
assume A16: H1( Initialize (IExec (I,Q,t))) >= H1(t) ; ::_thesis: contradiction
then  (Initialize (IExec (I,Q,t))) . (DataLoc ((s . a),i)) > 0 by A2, A12, A15;
then H1( Initialize (IExec (I,Q,t))) = (Initialize (IExec (I,Q,t))) . (DataLoc ((s . a),i)) by A2
.= (IExec (I,Q,t)) . (DataLoc ((s . a),i)) by SCMPDS_5:15 ;
hence  contradiction by A8, A11, A12, A10, A16, A15; ::_thesis: verum
end;
thus  S1[ Initialize (IExec (I,Q,t))] ::_thesis: verum
proof
take v = Initialize (IExec (I,Q,t)); ::_thesis: ( v = Initialize (IExec (I,Q,t)) & v . x >= c & v . y = s . y )
thus  v = Initialize (IExec (I,Q,t)) ; ::_thesis: ( v . x >= c & v . y = s . y )
 (IExec (I,Q,t)) . x >= c by A8, A11, A12, A10;
hence  v . x >= c by SCMPDS_5:15; ::_thesis: v . y = s . y
 (IExec (I,Q,t)) . y = t . y by A8, A11, A12, A10;
hence  v . y = s . y by A10, SCMPDS_5:15; ::_thesis: verum
end;
end;
 ( while>0 (a,i,I) is_closed_on s,P & while>0 (a,i,I) is_halting_on s,P ) from SCMPDS_8:sch_3(A5, A7, A9);
hence  ( while>0 (a,i,I) is_closed_on s,P & while>0 (a,i,I) is_halting_on s,P ) ; ::_thesis: ( s . (DataLoc ((s . a),i)) > 0 implies IExec ((while>0 (a,i,I)),P,s) = IExec ((while>0 (a,i,I)),P,(Initialize (IExec (I,P,s)))) )
assume A17: s . (DataLoc ((s . a),i)) > 0 ; ::_thesis: IExec ((while>0 (a,i,I)),P,s) = IExec ((while>0 (a,i,I)),P,(Initialize (IExec (I,P,s))))
 IExec ((while>0 (a,i,I)),P,s) = IExec ((while>0 (a,i,I)),P,(Initialize (IExec (I,P,s)))) from SCMPDS_8:sch_4(A17, A5, A7, A9);
hence  IExec ((while>0 (a,i,I)),P,s) = IExec ((while>0 (a,i,I)),P,(Initialize (IExec (I,P,s)))) ; ::_thesis: verum
end;

theorem Th7: :: SCPQSORT:7
for P being Instruction-Sequence of SCMPDS
for s being 0 -started State of SCMPDS
for I being halt-free shiftable Program of SCMPDS
for a, x1, x2, x3, x4 being Int_position
for i, c, md being Integer st s . x4 = ((s . x3) - c) + (s . x1) & md <= (s . x3) - c & ( for t being 0 -started State of SCMPDS
for Q being Instruction-Sequence of SCMPDS st t . x4 = ((t . x3) - c) + (t . x1) & md <= (t . x3) - c & t . x2 = s . x2 & t . a = s . a & t . (DataLoc ((s . a),i)) > 0 holds
( (IExec (I,Q,t)) . a = t . a & I is_closed_on t,Q & I is_halting_on t,Q & (IExec (I,Q,t)) . (DataLoc ((s . a),i)) < t . (DataLoc ((s . a),i)) & (IExec (I,Q,t)) . x4 = (((IExec (I,Q,t)) . x3) - c) + ((IExec (I,Q,t)) . x1) & md <= ((IExec (I,Q,t)) . x3) - c & (IExec (I,Q,t)) . x2 = t . x2 ) ) holds
( while>0 (a,i,I) is_closed_on s,P & while>0 (a,i,I) is_halting_on s,P & ( s . (DataLoc ((s . a),i)) > 0 implies IExec ((while>0 (a,i,I)),P,s) = IExec ((while>0 (a,i,I)),P,(Initialize (IExec (I,P,s)))) ) )
proof
let P be Instruction-Sequence of SCMPDS; ::_thesis: for s being 0 -started State of SCMPDS
for I being halt-free shiftable Program of SCMPDS
for a, x1, x2, x3, x4 being Int_position
for i, c, md being Integer st s . x4 = ((s . x3) - c) + (s . x1) & md <= (s . x3) - c & ( for t being 0 -started State of SCMPDS
for Q being Instruction-Sequence of SCMPDS st t . x4 = ((t . x3) - c) + (t . x1) & md <= (t . x3) - c & t . x2 = s . x2 & t . a = s . a & t . (DataLoc ((s . a),i)) > 0 holds
( (IExec (I,Q,t)) . a = t . a & I is_closed_on t,Q & I is_halting_on t,Q & (IExec (I,Q,t)) . (DataLoc ((s . a),i)) < t . (DataLoc ((s . a),i)) & (IExec (I,Q,t)) . x4 = (((IExec (I,Q,t)) . x3) - c) + ((IExec (I,Q,t)) . x1) & md <= ((IExec (I,Q,t)) . x3) - c & (IExec (I,Q,t)) . x2 = t . x2 ) ) holds
( while>0 (a,i,I) is_closed_on s,P & while>0 (a,i,I) is_halting_on s,P & ( s . (DataLoc ((s . a),i)) > 0 implies IExec ((while>0 (a,i,I)),P,s) = IExec ((while>0 (a,i,I)),P,(Initialize (IExec (I,P,s)))) ) )
let s be 0 -started State of SCMPDS; ::_thesis: for I being halt-free shiftable Program of SCMPDS
for a, x1, x2, x3, x4 being Int_position
for i, c, md being Integer st s . x4 = ((s . x3) - c) + (s . x1) & md <= (s . x3) - c & ( for t being 0 -started State of SCMPDS
for Q being Instruction-Sequence of SCMPDS st t . x4 = ((t . x3) - c) + (t . x1) & md <= (t . x3) - c & t . x2 = s . x2 & t . a = s . a & t . (DataLoc ((s . a),i)) > 0 holds
( (IExec (I,Q,t)) . a = t . a & I is_closed_on t,Q & I is_halting_on t,Q & (IExec (I,Q,t)) . (DataLoc ((s . a),i)) < t . (DataLoc ((s . a),i)) & (IExec (I,Q,t)) . x4 = (((IExec (I,Q,t)) . x3) - c) + ((IExec (I,Q,t)) . x1) & md <= ((IExec (I,Q,t)) . x3) - c & (IExec (I,Q,t)) . x2 = t . x2 ) ) holds
( while>0 (a,i,I) is_closed_on s,P & while>0 (a,i,I) is_halting_on s,P & ( s . (DataLoc ((s . a),i)) > 0 implies IExec ((while>0 (a,i,I)),P,s) = IExec ((while>0 (a,i,I)),P,(Initialize (IExec (I,P,s)))) ) )
let I be halt-free shiftable Program of SCMPDS; ::_thesis: for a, x1, x2, x3, x4 being Int_position
for i, c, md being Integer st s . x4 = ((s . x3) - c) + (s . x1) & md <= (s . x3) - c & ( for t being 0 -started State of SCMPDS
for Q being Instruction-Sequence of SCMPDS st t . x4 = ((t . x3) - c) + (t . x1) & md <= (t . x3) - c & t . x2 = s . x2 & t . a = s . a & t . (DataLoc ((s . a),i)) > 0 holds
( (IExec (I,Q,t)) . a = t . a & I is_closed_on t,Q & I is_halting_on t,Q & (IExec (I,Q,t)) . (DataLoc ((s . a),i)) < t . (DataLoc ((s . a),i)) & (IExec (I,Q,t)) . x4 = (((IExec (I,Q,t)) . x3) - c) + ((IExec (I,Q,t)) . x1) & md <= ((IExec (I,Q,t)) . x3) - c & (IExec (I,Q,t)) . x2 = t . x2 ) ) holds
( while>0 (a,i,I) is_closed_on s,P & while>0 (a,i,I) is_halting_on s,P & ( s . (DataLoc ((s . a),i)) > 0 implies IExec ((while>0 (a,i,I)),P,s) = IExec ((while>0 (a,i,I)),P,(Initialize (IExec (I,P,s)))) ) )
let a, x1, x2, x3, x4 be Int_position; ::_thesis: for i, c, md being Integer st s . x4 = ((s . x3) - c) + (s . x1) & md <= (s . x3) - c & ( for t being 0 -started State of SCMPDS
for Q being Instruction-Sequence of SCMPDS st t . x4 = ((t . x3) - c) + (t . x1) & md <= (t . x3) - c & t . x2 = s . x2 & t . a = s . a & t . (DataLoc ((s . a),i)) > 0 holds
( (IExec (I,Q,t)) . a = t . a & I is_closed_on t,Q & I is_halting_on t,Q & (IExec (I,Q,t)) . (DataLoc ((s . a),i)) < t . (DataLoc ((s . a),i)) & (IExec (I,Q,t)) . x4 = (((IExec (I,Q,t)) . x3) - c) + ((IExec (I,Q,t)) . x1) & md <= ((IExec (I,Q,t)) . x3) - c & (IExec (I,Q,t)) . x2 = t . x2 ) ) holds
( while>0 (a,i,I) is_closed_on s,P & while>0 (a,i,I) is_halting_on s,P & ( s . (DataLoc ((s . a),i)) > 0 implies IExec ((while>0 (a,i,I)),P,s) = IExec ((while>0 (a,i,I)),P,(Initialize (IExec (I,P,s)))) ) )
let i, c, md be Integer; ::_thesis: ( s . x4 = ((s . x3) - c) + (s . x1) & md <= (s . x3) - c & ( for t being 0 -started State of SCMPDS
for Q being Instruction-Sequence of SCMPDS st t . x4 = ((t . x3) - c) + (t . x1) & md <= (t . x3) - c & t . x2 = s . x2 & t . a = s . a & t . (DataLoc ((s . a),i)) > 0 holds
( (IExec (I,Q,t)) . a = t . a & I is_closed_on t,Q & I is_halting_on t,Q & (IExec (I,Q,t)) . (DataLoc ((s . a),i)) < t . (DataLoc ((s . a),i)) & (IExec (I,Q,t)) . x4 = (((IExec (I,Q,t)) . x3) - c) + ((IExec (I,Q,t)) . x1) & md <= ((IExec (I,Q,t)) . x3) - c & (IExec (I,Q,t)) . x2 = t . x2 ) ) implies ( while>0 (a,i,I) is_closed_on s,P & while>0 (a,i,I) is_halting_on s,P & ( s . (DataLoc ((s . a),i)) > 0 implies IExec ((while>0 (a,i,I)),P,s) = IExec ((while>0 (a,i,I)),P,(Initialize (IExec (I,P,s)))) ) ) )
set b = DataLoc ((s . a),i);
defpred S1[ set ] means  ex t being State of SCMPDS st
( t = $1 & t . x4 = ((t . x3) - c) + (t . x1) & md <= (t . x3) - c & t . x2 = s . x2 );
assume that
A2: s . x4 = ((s . x3) - c) + (s . x1) and
A3: md <= (s . x3) - c ; ::_thesis: ( ex t being 0 -started State of SCMPDS ex Q being Instruction-Sequence of SCMPDS st
( t . x4 = ((t . x3) - c) + (t . x1) & md <= (t . x3) - c & t . x2 = s . x2 & t . a = s . a & t . (DataLoc ((s . a),i)) > 0 & not ( (IExec (I,Q,t)) . a = t . a & I is_closed_on t,Q & I is_halting_on t,Q & (IExec (I,Q,t)) . (DataLoc ((s . a),i)) < t . (DataLoc ((s . a),i)) & (IExec (I,Q,t)) . x4 = (((IExec (I,Q,t)) . x3) - c) + ((IExec (I,Q,t)) . x1) & md <= ((IExec (I,Q,t)) . x3) - c & (IExec (I,Q,t)) . x2 = t . x2 ) ) or ( while>0 (a,i,I) is_closed_on s,P & while>0 (a,i,I) is_halting_on s,P & ( s . (DataLoc ((s . a),i)) > 0 implies IExec ((while>0 (a,i,I)),P,s) = IExec ((while>0 (a,i,I)),P,(Initialize (IExec (I,P,s)))) ) ) )
consider f being Function of (product (the_Values_of SCMPDS)),NAT such that
A4: for s being State of SCMPDS holds
( ( s . (DataLoc ((s . a),i)) <= 0 implies f . s = 0 ) & ( s . (DataLoc ((s . a),i)) > 0 implies f . s = s . (DataLoc ((s . a),i)) ) ) by SCMPDS_8:5;
deffunc H1( State of SCMPDS) ->  Element of NAT = f . $1;
A7: for t being 0 -started State of SCMPDS st S1[t] & H1(t) = 0 holds
t . (DataLoc ((s . a),i)) <= 0 by A4;
assume A8: for t being 0 -started State of SCMPDS
for Q being Instruction-Sequence of SCMPDS st t . x4 = ((t . x3) - c) + (t . x1) & md <= (t . x3) - c & t . x2 = s . x2 & t . a = s . a & t . (DataLoc ((s . a),i)) > 0 holds
( (IExec (I,Q,t)) . a = t . a & I is_closed_on t,Q & I is_halting_on t,Q & (IExec (I,Q,t)) . (DataLoc ((s . a),i)) < t . (DataLoc ((s . a),i)) & (IExec (I,Q,t)) . x4 = (((IExec (I,Q,t)) . x3) - c) + ((IExec (I,Q,t)) . x1) & md <= ((IExec (I,Q,t)) . x3) - c & (IExec (I,Q,t)) . x2 = t . x2 ) ; ::_thesis: ( while>0 (a,i,I) is_closed_on s,P & while>0 (a,i,I) is_halting_on s,P & ( s . (DataLoc ((s . a),i)) > 0 implies IExec ((while>0 (a,i,I)),P,s) = IExec ((while>0 (a,i,I)),P,(Initialize (IExec (I,P,s)))) ) )
A9: now__::_thesis:_for_t_being_0_-started_State_of_SCMPDS
for_Q_being_Instruction-Sequence_of_SCMPDS_st_S1[t]_&_t_._a_=_s_._a_&_t_._(DataLoc_((s_._a),i))_>_0_holds_
(_(IExec_(I,Q,t))_._a_=_t_._a_&_I_is_closed_on_t,Q_&_I_is_halting_on_t,Q_&_H1(_Initialize_(IExec_(I,Q,t)))_<_H1(t)_&_S1[_Initialize_(IExec_(I,Q,t))]_)
let t be 0 -started State of SCMPDS; ::_thesis: for Q being Instruction-Sequence of SCMPDS st S1[t] & t . a = s . a & t . (DataLoc ((s . a),i)) > 0 holds
( (IExec (I,Q,t)) . a = t . a & I is_closed_on t,Q & I is_halting_on t,Q & H1( Initialize (IExec (I,Q,t))) < H1(t) & S1[ Initialize (IExec (I,Q,t))] )
let Q be Instruction-Sequence of SCMPDS; ::_thesis: ( S1[t] & t . a = s . a & t . (DataLoc ((s . a),i)) > 0 implies ( (IExec (I,Q,t)) . a = t . a & I is_closed_on t,Q & I is_halting_on t,Q & H1( Initialize (IExec (I,Q,t))) < H1(t) & S1[ Initialize (IExec (I,Q,t))] ) )
assume that
A10: S1[t] and
A11: t . a = s . a and
A12: t . (DataLoc ((s . a),i)) > 0 ; ::_thesis: ( (IExec (I,Q,t)) . a = t . a & I is_closed_on t,Q & I is_halting_on t,Q & H1( Initialize (IExec (I,Q,t))) < H1(t) & S1[ Initialize (IExec (I,Q,t))] )
set It = IExec (I,Q,t);
set t2 = Initialize (IExec (I,Q,t));
consider v being State of SCMPDS such that
A13: v = t and
A14: v . x4 = ((v . x3) - c) + (v . x1) and
A15: md <= (v . x3) - c and
A16: v . x2 = s . x2 by A10;
A17: t . x2 = s . x2 by A13, A16;
A18: t . x4 = ((t . x3) - c) + (t . x1) by A13, A14;
A19: md <= (t . x3) - c by A13, A15;
thus  ( (IExec (I,Q,t)) . a = t . a & I is_closed_on t,Q & I is_halting_on t,Q ) by A8, A11, A12, A18, A16, A13, A15; ::_thesis: ( H1( Initialize (IExec (I,Q,t))) < H1(t) & S1[ Initialize (IExec (I,Q,t))] )
thus  H1( Initialize (IExec (I,Q,t))) < H1(t) ::_thesis: S1[ Initialize (IExec (I,Q,t))]
proof
A20: H1(t) = t . (DataLoc ((s . a),i)) by A4, A12
.= t . (DataLoc ((s . a),i)) ;
assume A21: H1( Initialize (IExec (I,Q,t))) >= H1(t) ; ::_thesis: contradiction
then  (Initialize (IExec (I,Q,t))) . (DataLoc ((s . a),i)) > 0 by A4, A12, A20;
then H1( Initialize (IExec (I,Q,t))) = (Initialize (IExec (I,Q,t))) . (DataLoc ((s . a),i)) by A4
.= (IExec (I,Q,t)) . (DataLoc ((s . a),i)) by SCMPDS_5:15 ;
hence  contradiction by A8, A11, A12, A18, A19, A13, A16, A21, A20; ::_thesis: verum
end;
thus  S1[ Initialize (IExec (I,Q,t))] ::_thesis: verum
proof
take v = Initialize (IExec (I,Q,t)); ::_thesis: ( v = Initialize (IExec (I,Q,t)) & v . x4 = ((v . x3) - c) + (v . x1) & md <= (v . x3) - c & v . x2 = s . x2 )
thus  v = Initialize (IExec (I,Q,t)) ; ::_thesis: ( v . x4 = ((v . x3) - c) + (v . x1) & md <= (v . x3) - c & v . x2 = s . x2 )
 (IExec (I,Q,t)) . x4 = (((IExec (I,Q,t)) . x3) - c) + ((IExec (I,Q,t)) . x1) by A8, A11, A12, A18, A19, A17;
then  v . x4 = (((IExec (I,Q,t)) . x3) - c) + ((IExec (I,Q,t)) . x1) by SCMPDS_5:15;
then  v . x4 = ((v . x3) - c) + ((IExec (I,Q,t)) . x1) by SCMPDS_5:15;
hence  v . x4 = ((v . x3) - c) + (v . x1) by SCMPDS_5:15; ::_thesis: ( md <= (v . x3) - c & v . x2 = s . x2 )
 md <= ((IExec (I,Q,t)) . x3) - c by A8, A11, A12, A18, A19, A17;
hence  md <= (v . x3) - c by SCMPDS_5:15; ::_thesis: v . x2 = s . x2
 (IExec (I,Q,t)) . x2 = t . x2 by A8, A11, A12, A18, A19, A13, A16;
hence  v . x2 = s . x2 by A13, A16, SCMPDS_5:15; ::_thesis: verum
end;
end;
A22: S1[s] by A2, A3;
 ( while>0 (a,i,I) is_closed_on s,P & while>0 (a,i,I) is_halting_on s,P ) from SCMPDS_8:sch_3(A7, A22, A9);
hence  ( while>0 (a,i,I) is_closed_on s,P & while>0 (a,i,I) is_halting_on s,P ) ; ::_thesis: ( s . (DataLoc ((s . a),i)) > 0 implies IExec ((while>0 (a,i,I)),P,s) = IExec ((while>0 (a,i,I)),P,(Initialize (IExec (I,P,s)))) )
assume A23: s . (DataLoc ((s . a),i)) > 0 ; ::_thesis: IExec ((while>0 (a,i,I)),P,s) = IExec ((while>0 (a,i,I)),P,(Initialize (IExec (I,P,s))))
 IExec ((while>0 (a,i,I)),P,s) = IExec ((while>0 (a,i,I)),P,(Initialize (IExec (I,P,s)))) from SCMPDS_8:sch_4(A23, A7, A22, A9);
hence  IExec ((while>0 (a,i,I)),P,s) = IExec ((while>0 (a,i,I)),P,(Initialize (IExec (I,P,s)))) ; ::_thesis: verum
end;

theorem Th8: :: SCPQSORT:8
for f being FinSequence of INT
for m, k1, k, n being Element of NAT st k1 = k - 1 & f is_non_decreasing_on m,k1 & f is_non_decreasing_on k + 1,n & ( for i being Element of NAT st m <= i & i < k holds
f . i <= f . k ) & ( for i being Element of NAT st k < i & i <= n holds
f . k <= f . i ) holds
f is_non_decreasing_on m,n
proof
let f be FinSequence of INT ; ::_thesis: for m, k1, k, n being Element of NAT st k1 = k - 1 & f is_non_decreasing_on m,k1 & f is_non_decreasing_on k + 1,n & ( for i being Element of NAT st m <= i & i < k holds
f . i <= f . k ) & ( for i being Element of NAT st k < i & i <= n holds
f . k <= f . i ) holds
f is_non_decreasing_on m,n
let m, k1, k, n be Element of NAT ; ::_thesis: ( k1 = k - 1 & f is_non_decreasing_on m,k1 & f is_non_decreasing_on k + 1,n & ( for i being Element of NAT st m <= i & i < k holds
f . i <= f . k ) & ( for i being Element of NAT st k < i & i <= n holds
f . k <= f . i ) implies f is_non_decreasing_on m,n )
assume A1: k1 = k - 1 ; ::_thesis: ( not f is_non_decreasing_on m,k1 or not f is_non_decreasing_on k + 1,n or ex i being Element of NAT st
( m <= i & i < k & not f . i <= f . k ) or ex i being Element of NAT st
( k < i & i <= n & not f . k <= f . i ) or f is_non_decreasing_on m,n )
assume A2: f is_non_decreasing_on m,k1 ; ::_thesis: ( not f is_non_decreasing_on k + 1,n or ex i being Element of NAT st
( m <= i & i < k & not f . i <= f . k ) or ex i being Element of NAT st
( k < i & i <= n & not f . k <= f . i ) or f is_non_decreasing_on m,n )
assume A3: f is_non_decreasing_on k + 1,n ; ::_thesis: ( ex i being Element of NAT st
( m <= i & i < k & not f . i <= f . k ) or ex i being Element of NAT st
( k < i & i <= n & not f . k <= f . i ) or f is_non_decreasing_on m,n )
assume A4: for i being Element of NAT st m <= i & i < k holds
f . i <= f . k ; ::_thesis: ( ex i being Element of NAT st
( k < i & i <= n & not f . k <= f . i ) or f is_non_decreasing_on m,n )
assume A5: for i being Element of NAT st k < i & i <= n holds
f . k <= f . i ; ::_thesis: f is_non_decreasing_on m,n
now__::_thesis:_for_i,_j_being_Element_of_NAT_st_m_<=_i_&_i_<=_j_&_j_<=_n_holds_
f_._i_<=_f_._j
let i, j be Element of NAT ; ::_thesis: ( m <= i & i <= j & j <= n implies f . b1 <= f . b2 )
assume that
A6: m <= i and
A7: i <= j and
A8: j <= n ; ::_thesis: f . b1 <= f . b2
percases ( j < k or j = k or j > k ) by XXREAL_0:1;
suppose j < k ; ::_thesis: f . b1 <= f . b2
then  j + 1 <= k by INT_1:7;
then  j <= k1 by A1, XREAL_1:19;
hence  f . i <= f . j by A2, A6, A7, GRAPH_2:def_12; ::_thesis: verum
end;
supposeA9: j = k ; ::_thesis: f . b1 <= f . b2
percases ( i = j or i <> j ) ;
suppose i = j ; ::_thesis: f . b1 <= f . b2
hence  f . i <= f . j ; ::_thesis: verum
end;
suppose i <> j ; ::_thesis: f . b1 <= f . b2
then  i < j by A7, XXREAL_0:1;
hence  f . i <= f . j by A4, A6, A9; ::_thesis: verum
end;
end;
end;
supposeA10: j > k ; ::_thesis: f . b1 <= f . b2
percases ( i < k or i = k or i > k ) by XXREAL_0:1;
suppose i < k ; ::_thesis: f . b1 <= f . b2
then A11: f . k >= f . i by A4, A6;
 f . j >= f . k by A5, A8, A10;
hence  f . i <= f . j by A11, XXREAL_0:2; ::_thesis: verum
end;
supposeA12: i = k ; ::_thesis: f . b1 <= f . b2
percases ( i = j or i <> j ) ;
suppose i = j ; ::_thesis: f . b1 <= f . b2
hence  f . i <= f . j ; ::_thesis: verum
end;
suppose i <> j ; ::_thesis: f . b1 <= f . b2
then  i < j by A7, XXREAL_0:1;
hence  f . i <= f . j by A5, A8, A12; ::_thesis: verum
end;
end;
end;
suppose i > k ; ::_thesis: f . b1 <= f . b2
then  i >= k + 1 by INT_1:7;
hence  f . i <= f . j by A3, A7, A8, GRAPH_2:def_12; ::_thesis: verum
end;
end;
end;
end;
end;
hence  f is_non_decreasing_on m,n by GRAPH_2:def_12; ::_thesis: verum
end;

Lm4:  for P being Instruction-Sequence of SCMPDS
for s being 0 -started State of SCMPDS
for I being halt-free shiftable Program of SCMPDS
for a being Int_position
for i, c being Integer
for f, g being FinSequence of INT
for m, n, m1 being Element of NAT st s . a = c & len f = n & len g = n & f is_FinSequence_on s,m & g is_FinSequence_on IExec ((while>0 (a,i,I)),P,s),m & 1 = s . (DataLoc (c,i)) & m1 = (m + n) + 1 & m + 1 = s . (intpos m1) & m + n = s . (intpos (m1 + 1)) & ( for t being 0 -started State of SCMPDS
for Q being Instruction-Sequence of SCMPDS
for f1, f2 being FinSequence of INT
for k1, k2, y1, yn being Element of NAT st t . a = c & (2 * k1) + 1 = t . (DataLoc (c,i)) & k2 = ((m + n) + (2 * k1)) + 1 & m + y1 = t . (intpos k2) & m + yn = t . (intpos (k2 + 1)) & ( ( 1 <= y1 & yn <= n ) or y1 >= yn ) holds
( I is_closed_on t,Q & I is_halting_on t,Q & (IExec (I,Q,t)) . a = t . a & ( for j being Element of NAT st 1 <= j & j < (2 * k1) + 1 holds
(IExec (I,Q,t)) . (intpos ((m + n) + j)) = t . (intpos ((m + n) + j)) ) & ( y1 >= yn implies ( (IExec (I,Q,t)) . (DataLoc (c,i)) = (2 * k1) - 1 & ( for j being Element of NAT st 1 <= j & j <= n holds
(IExec (I,Q,t)) . (intpos (m + j)) = t . (intpos (m + j)) ) ) ) & ( y1 < yn implies ( (IExec (I,Q,t)) . (DataLoc (c,i)) = (2 * k1) + 3 & ( for j being Element of NAT st ( ( 1 <= j & j < y1 ) or ( yn < j & j <= n ) ) holds
(IExec (I,Q,t)) . (intpos (m + j)) = t . (intpos (m + j)) ) & ex ym being Element of NAT st
( y1 <= ym & ym <= yn & m + y1 = (IExec (I,Q,t)) . (intpos k2) & (m + ym) - 1 = (IExec (I,Q,t)) . (intpos (k2 + 1)) & (m + ym) + 1 = (IExec (I,Q,t)) . (intpos (k2 + 2)) & m + yn = (IExec (I,Q,t)) . (intpos (k2 + 3)) & ( for j being Element of NAT st y1 <= j & j < ym holds
(IExec (I,Q,t)) . (intpos (m + j)) <= (IExec (I,Q,t)) . (intpos (m + ym)) ) & ( for j being Element of NAT st ym < j & j <= yn holds
(IExec (I,Q,t)) . (intpos (m + j)) >= (IExec (I,Q,t)) . (intpos (m + ym)) ) ) ) ) & ( f1 is_FinSequence_on t,m & f2 is_FinSequence_on IExec (I,Q,t),m & len f1 = n & len f2 = n implies f1,f2 are_fiberwise_equipotent ) ) ) holds
( while>0 (a,i,I) is_halting_on s,P & f,g are_fiberwise_equipotent & g is_non_decreasing_on 1,n )
proof
let P be Instruction-Sequence of SCMPDS; ::_thesis: for s being 0 -started State of SCMPDS
for I being halt-free shiftable Program of SCMPDS
for a being Int_position
for i, c being Integer
for f, g being FinSequence of INT
for m, n, m1 being Element of NAT st s . a = c & len f = n & len g = n & f is_FinSequence_on s,m & g is_FinSequence_on IExec ((while>0 (a,i,I)),P,s),m & 1 = s . (DataLoc (c,i)) & m1 = (m + n) + 1 & m + 1 = s . (intpos m1) & m + n = s . (intpos (m1 + 1)) & ( for t being 0 -started State of SCMPDS
for Q being Instruction-Sequence of SCMPDS
for f1, f2 being FinSequence of INT
for k1, k2, y1, yn being Element of NAT st t . a = c & (2 * k1) + 1 = t . (DataLoc (c,i)) & k2 = ((m + n) + (2 * k1)) + 1 & m + y1 = t . (intpos k2) & m + yn = t . (intpos (k2 + 1)) & ( ( 1 <= y1 & yn <= n ) or y1 >= yn ) holds
( I is_closed_on t,Q & I is_halting_on t,Q & (IExec (I,Q,t)) . a = t . a & ( for j being Element of NAT st 1 <= j & j < (2 * k1) + 1 holds
(IExec (I,Q,t)) . (intpos ((m + n) + j)) = t . (intpos ((m + n) + j)) ) & ( y1 >= yn implies ( (IExec (I,Q,t)) . (DataLoc (c,i)) = (2 * k1) - 1 & ( for j being Element of NAT st 1 <= j & j <= n holds
(IExec (I,Q,t)) . (intpos (m + j)) = t . (intpos (m + j)) ) ) ) & ( y1 < yn implies ( (IExec (I,Q,t)) . (DataLoc (c,i)) = (2 * k1) + 3 & ( for j being Element of NAT st ( ( 1 <= j & j < y1 ) or ( yn < j & j <= n ) ) holds
(IExec (I,Q,t)) . (intpos (m + j)) = t . (intpos (m + j)) ) & ex ym being Element of NAT st
( y1 <= ym & ym <= yn & m + y1 = (IExec (I,Q,t)) . (intpos k2) & (m + ym) - 1 = (IExec (I,Q,t)) . (intpos (k2 + 1)) & (m + ym) + 1 = (IExec (I,Q,t)) . (intpos (k2 + 2)) & m + yn = (IExec (I,Q,t)) . (intpos (k2 + 3)) & ( for j being Element of NAT st y1 <= j & j < ym holds
(IExec (I,Q,t)) . (intpos (m + j)) <= (IExec (I,Q,t)) . (intpos (m + ym)) ) & ( for j being Element of NAT st ym < j & j <= yn holds
(IExec (I,Q,t)) . (intpos (m + j)) >= (IExec (I,Q,t)) . (intpos (m + ym)) ) ) ) ) & ( f1 is_FinSequence_on t,m & f2 is_FinSequence_on IExec (I,Q,t),m & len f1 = n & len f2 = n implies f1,f2 are_fiberwise_equipotent ) ) ) holds
( while>0 (a,i,I) is_halting_on s,P & f,g are_fiberwise_equipotent & g is_non_decreasing_on 1,n )
let s be 0 -started State of SCMPDS; ::_thesis: for I being halt-free shiftable Program of SCMPDS
for a being Int_position
for i, c being Integer
for f, g being FinSequence of INT
for m, n, m1 being Element of NAT st s . a = c & len f = n & len g = n & f is_FinSequence_on s,m & g is_FinSequence_on IExec ((while>0 (a,i,I)),P,s),m & 1 = s . (DataLoc (c,i)) & m1 = (m + n) + 1 & m + 1 = s . (intpos m1) & m + n = s . (intpos (m1 + 1)) & ( for t being 0 -started State of SCMPDS
for Q being Instruction-Sequence of SCMPDS
for f1, f2 being FinSequence of INT
for k1, k2, y1, yn being Element of NAT st t . a = c & (2 * k1) + 1 = t . (DataLoc (c,i)) & k2 = ((m + n) + (2 * k1)) + 1 & m + y1 = t . (intpos k2) & m + yn = t . (intpos (k2 + 1)) & ( ( 1 <= y1 & yn <= n ) or y1 >= yn ) holds
( I is_closed_on t,Q & I is_halting_on t,Q & (IExec (I,Q,t)) . a = t . a & ( for j being Element of NAT st 1 <= j & j < (2 * k1) + 1 holds
(IExec (I,Q,t)) . (intpos ((m + n) + j)) = t . (intpos ((m + n) + j)) ) & ( y1 >= yn implies ( (IExec (I,Q,t)) . (DataLoc (c,i)) = (2 * k1) - 1 & ( for j being Element of NAT st 1 <= j & j <= n holds
(IExec (I,Q,t)) . (intpos (m + j)) = t . (intpos (m + j)) ) ) ) & ( y1 < yn implies ( (IExec (I,Q,t)) . (DataLoc (c,i)) = (2 * k1) + 3 & ( for j being Element of NAT st ( ( 1 <= j & j < y1 ) or ( yn < j & j <= n ) ) holds
(IExec (I,Q,t)) . (intpos (m + j)) = t . (intpos (m + j)) ) & ex ym being Element of NAT st
( y1 <= ym & ym <= yn & m + y1 = (IExec (I,Q,t)) . (intpos k2) & (m + ym) - 1 = (IExec (I,Q,t)) . (intpos (k2 + 1)) & (m + ym) + 1 = (IExec (I,Q,t)) . (intpos (k2 + 2)) & m + yn = (IExec (I,Q,t)) . (intpos (k2 + 3)) & ( for j being Element of NAT st y1 <= j & j < ym holds
(IExec (I,Q,t)) . (intpos (m + j)) <= (IExec (I,Q,t)) . (intpos (m + ym)) ) & ( for j being Element of NAT st ym < j & j <= yn holds
(IExec (I,Q,t)) . (intpos (m + j)) >= (IExec (I,Q,t)) . (intpos (m + ym)) ) ) ) ) & ( f1 is_FinSequence_on t,m & f2 is_FinSequence_on IExec (I,Q,t),m & len f1 = n & len f2 = n implies f1,f2 are_fiberwise_equipotent ) ) ) holds
( while>0 (a,i,I) is_halting_on s,P & f,g are_fiberwise_equipotent & g is_non_decreasing_on 1,n )
let I be halt-free shiftable Program of SCMPDS; ::_thesis: for a being Int_position
for i, c being Integer
for f, g being FinSequence of INT
for m, n, m1 being Element of NAT st s . a = c & len f = n & len g = n & f is_FinSequence_on s,m & g is_FinSequence_on IExec ((while>0 (a,i,I)),P,s),m & 1 = s . (DataLoc (c,i)) & m1 = (m + n) + 1 & m + 1 = s . (intpos m1) & m + n = s . (intpos (m1 + 1)) & ( for t being 0 -started State of SCMPDS
for Q being Instruction-Sequence of SCMPDS
for f1, f2 being FinSequence of INT
for k1, k2, y1, yn being Element of NAT st t . a = c & (2 * k1) + 1 = t . (DataLoc (c,i)) & k2 = ((m + n) + (2 * k1)) + 1 & m + y1 = t . (intpos k2) & m + yn = t . (intpos (k2 + 1)) & ( ( 1 <= y1 & yn <= n ) or y1 >= yn ) holds
( I is_closed_on t,Q & I is_halting_on t,Q & (IExec (I,Q,t)) . a = t . a & ( for j being Element of NAT st 1 <= j & j < (2 * k1) + 1 holds
(IExec (I,Q,t)) . (intpos ((m + n) + j)) = t . (intpos ((m + n) + j)) ) & ( y1 >= yn implies ( (IExec (I,Q,t)) . (DataLoc (c,i)) = (2 * k1) - 1 & ( for j being Element of NAT st 1 <= j & j <= n holds
(IExec (I,Q,t)) . (intpos (m + j)) = t . (intpos (m + j)) ) ) ) & ( y1 < yn implies ( (IExec (I,Q,t)) . (DataLoc (c,i)) = (2 * k1) + 3 & ( for j being Element of NAT st ( ( 1 <= j & j < y1 ) or ( yn < j & j <= n ) ) holds
(IExec (I,Q,t)) . (intpos (m + j)) = t . (intpos (m + j)) ) & ex ym being Element of NAT st
( y1 <= ym & ym <= yn & m + y1 = (IExec (I,Q,t)) . (intpos k2) & (m + ym) - 1 = (IExec (I,Q,t)) . (intpos (k2 + 1)) & (m + ym) + 1 = (IExec (I,Q,t)) . (intpos (k2 + 2)) & m + yn = (IExec (I,Q,t)) . (intpos (k2 + 3)) & ( for j being Element of NAT st y1 <= j & j < ym holds
(IExec (I,Q,t)) . (intpos (m + j)) <= (IExec (I,Q,t)) . (intpos (m + ym)) ) & ( for j being Element of NAT st ym < j & j <= yn holds
(IExec (I,Q,t)) . (intpos (m + j)) >= (IExec (I,Q,t)) . (intpos (m + ym)) ) ) ) ) & ( f1 is_FinSequence_on t,m & f2 is_FinSequence_on IExec (I,Q,t),m & len f1 = n & len f2 = n implies f1,f2 are_fiberwise_equipotent ) ) ) holds
( while>0 (a,i,I) is_halting_on s,P & f,g are_fiberwise_equipotent & g is_non_decreasing_on 1,n )
let a be Int_position; ::_thesis: for i, c being Integer
for f, g being FinSequence of INT
for m, n, m1 being Element of NAT st s . a = c & len f = n & len g = n & f is_FinSequence_on s,m & g is_FinSequence_on IExec ((while>0 (a,i,I)),P,s),m & 1 = s . (DataLoc (c,i)) & m1 = (m + n) + 1 & m + 1 = s . (intpos m1) & m + n = s . (intpos (m1 + 1)) & ( for t being 0 -started State of SCMPDS
for Q being Instruction-Sequence of SCMPDS
for f1, f2 being FinSequence of INT
for k1, k2, y1, yn being Element of NAT st t . a = c & (2 * k1) + 1 = t . (DataLoc (c,i)) & k2 = ((m + n) + (2 * k1)) + 1 & m + y1 = t . (intpos k2) & m + yn = t . (intpos (k2 + 1)) & ( ( 1 <= y1 & yn <= n ) or y1 >= yn ) holds
( I is_closed_on t,Q & I is_halting_on t,Q & (IExec (I,Q,t)) . a = t . a & ( for j being Element of NAT st 1 <= j & j < (2 * k1) + 1 holds
(IExec (I,Q,t)) . (intpos ((m + n) + j)) = t . (intpos ((m + n) + j)) ) & ( y1 >= yn implies ( (IExec (I,Q,t)) . (DataLoc (c,i)) = (2 * k1) - 1 & ( for j being Element of NAT st 1 <= j & j <= n holds
(IExec (I,Q,t)) . (intpos (m + j)) = t . (intpos (m + j)) ) ) ) & ( y1 < yn implies ( (IExec (I,Q,t)) . (DataLoc (c,i)) = (2 * k1) + 3 & ( for j being Element of NAT st ( ( 1 <= j & j < y1 ) or ( yn < j & j <= n ) ) holds
(IExec (I,Q,t)) . (intpos (m + j)) = t . (intpos (m + j)) ) & ex ym being Element of NAT st
( y1 <= ym & ym <= yn & m + y1 = (IExec (I,Q,t)) . (intpos k2) & (m + ym) - 1 = (IExec (I,Q,t)) . (intpos (k2 + 1)) & (m + ym) + 1 = (IExec (I,Q,t)) . (intpos (k2 + 2)) & m + yn = (IExec (I,Q,t)) . (intpos (k2 + 3)) & ( for j being Element of NAT st y1 <= j & j < ym holds
(IExec (I,Q,t)) . (intpos (m + j)) <= (IExec (I,Q,t)) . (intpos (m + ym)) ) & ( for j being Element of NAT st ym < j & j <= yn holds
(IExec (I,Q,t)) . (intpos (m + j)) >= (IExec (I,Q,t)) . (intpos (m + ym)) ) ) ) ) & ( f1 is_FinSequence_on t,m & f2 is_FinSequence_on IExec (I,Q,t),m & len f1 = n & len f2 = n implies f1,f2 are_fiberwise_equipotent ) ) ) holds
( while>0 (a,i,I) is_halting_on s,P & f,g are_fiberwise_equipotent & g is_non_decreasing_on 1,n )
let i, c be Integer; ::_thesis: for f, g being FinSequence of INT
for m, n, m1 being Element of NAT st s . a = c & len f = n & len g = n & f is_FinSequence_on s,m & g is_FinSequence_on IExec ((while>0 (a,i,I)),P,s),m & 1 = s . (DataLoc (c,i)) & m1 = (m + n) + 1 & m + 1 = s . (intpos m1) & m + n = s . (intpos (m1 + 1)) & ( for t being 0 -started State of SCMPDS
for Q being Instruction-Sequence of SCMPDS
for f1, f2 being FinSequence of INT
for k1, k2, y1, yn being Element of NAT st t . a = c & (2 * k1) + 1 = t . (DataLoc (c,i)) & k2 = ((m + n) + (2 * k1)) + 1 & m + y1 = t . (intpos k2) & m + yn = t . (intpos (k2 + 1)) & ( ( 1 <= y1 & yn <= n ) or y1 >= yn ) holds
( I is_closed_on t,Q & I is_halting_on t,Q & (IExec (I,Q,t)) . a = t . a & ( for j being Element of NAT st 1 <= j & j < (2 * k1) + 1 holds
(IExec (I,Q,t)) . (intpos ((m + n) + j)) = t . (intpos ((m + n) + j)) ) & ( y1 >= yn implies ( (IExec (I,Q,t)) . (DataLoc (c,i)) = (2 * k1) - 1 & ( for j being Element of NAT st 1 <= j & j <= n holds
(IExec (I,Q,t)) . (intpos (m + j)) = t . (intpos (m + j)) ) ) ) & ( y1 < yn implies ( (IExec (I,Q,t)) . (DataLoc (c,i)) = (2 * k1) + 3 & ( for j being Element of NAT st ( ( 1 <= j & j < y1 ) or ( yn < j & j <= n ) ) holds
(IExec (I,Q,t)) . (intpos (m + j)) = t . (intpos (m + j)) ) & ex ym being Element of NAT st
( y1 <= ym & ym <= yn & m + y1 = (IExec (I,Q,t)) . (intpos k2) & (m + ym) - 1 = (IExec (I,Q,t)) . (intpos (k2 + 1)) & (m + ym) + 1 = (IExec (I,Q,t)) . (intpos (k2 + 2)) & m + yn = (IExec (I,Q,t)) . (intpos (k2 + 3)) & ( for j being Element of NAT st y1 <= j & j < ym holds
(IExec (I,Q,t)) . (intpos (m + j)) <= (IExec (I,Q,t)) . (intpos (m + ym)) ) & ( for j being Element of NAT st ym < j & j <= yn holds
(IExec (I,Q,t)) . (intpos (m + j)) >= (IExec (I,Q,t)) . (intpos (m + ym)) ) ) ) ) & ( f1 is_FinSequence_on t,m & f2 is_FinSequence_on IExec (I,Q,t),m & len f1 = n & len f2 = n implies f1,f2 are_fiberwise_equipotent ) ) ) holds
( while>0 (a,i,I) is_halting_on s,P & f,g are_fiberwise_equipotent & g is_non_decreasing_on 1,n )
let f, g be FinSequence of INT ; ::_thesis: for m, n, m1 being Element of NAT st s . a = c & len f = n & len g = n & f is_FinSequence_on s,m & g is_FinSequence_on IExec ((while>0 (a,i,I)),P,s),m & 1 = s . (DataLoc (c,i)) & m1 = (m + n) + 1 & m + 1 = s . (intpos m1) & m + n = s . (intpos (m1 + 1)) & ( for t being 0 -started State of SCMPDS
for Q being Instruction-Sequence of SCMPDS
for f1, f2 being FinSequence of INT
for k1, k2, y1, yn being Element of NAT st t . a = c & (2 * k1) + 1 = t . (DataLoc (c,i)) & k2 = ((m + n) + (2 * k1)) + 1 & m + y1 = t . (intpos k2) & m + yn = t . (intpos (k2 + 1)) & ( ( 1 <= y1 & yn <= n ) or y1 >= yn ) holds
( I is_closed_on t,Q & I is_halting_on t,Q & (IExec (I,Q,t)) . a = t . a & ( for j being Element of NAT st 1 <= j & j < (2 * k1) + 1 holds
(IExec (I,Q,t)) . (intpos ((m + n) + j)) = t . (intpos ((m + n) + j)) ) & ( y1 >= yn implies ( (IExec (I,Q,t)) . (DataLoc (c,i)) = (2 * k1) - 1 & ( for j being Element of NAT st 1 <= j & j <= n holds
(IExec (I,Q,t)) . (intpos (m + j)) = t . (intpos (m + j)) ) ) ) & ( y1 < yn implies ( (IExec (I,Q,t)) . (DataLoc (c,i)) = (2 * k1) + 3 & ( for j being Element of NAT st ( ( 1 <= j & j < y1 ) or ( yn < j & j <= n ) ) holds
(IExec (I,Q,t)) . (intpos (m + j)) = t . (intpos (m + j)) ) & ex ym being Element of NAT st
( y1 <= ym & ym <= yn & m + y1 = (IExec (I,Q,t)) . (intpos k2) & (m + ym) - 1 = (IExec (I,Q,t)) . (intpos (k2 + 1)) & (m + ym) + 1 = (IExec (I,Q,t)) . (intpos (k2 + 2)) & m + yn = (IExec (I,Q,t)) . (intpos (k2 + 3)) & ( for j being Element of NAT st y1 <= j & j < ym holds
(IExec (I,Q,t)) . (intpos (m + j)) <= (IExec (I,Q,t)) . (intpos (m + ym)) ) & ( for j being Element of NAT st ym < j & j <= yn holds
(IExec (I,Q,t)) . (intpos (m + j)) >= (IExec (I,Q,t)) . (intpos (m + ym)) ) ) ) ) & ( f1 is_FinSequence_on t,m & f2 is_FinSequence_on IExec (I,Q,t),m & len f1 = n & len f2 = n implies f1,f2 are_fiberwise_equipotent ) ) ) holds
( while>0 (a,i,I) is_halting_on s,P & f,g are_fiberwise_equipotent & g is_non_decreasing_on 1,n )
let m, n, m1 be Element of NAT ; ::_thesis: ( s . a = c & len f = n & len g = n & f is_FinSequence_on s,m & g is_FinSequence_on IExec ((while>0 (a,i,I)),P,s),m & 1 = s . (DataLoc (c,i)) & m1 = (m + n) + 1 & m + 1 = s . (intpos m1) & m + n = s . (intpos (m1 + 1)) & ( for t being 0 -started State of SCMPDS
for Q being Instruction-Sequence of SCMPDS
for f1, f2 being FinSequence of INT
for k1, k2, y1, yn being Element of NAT st t . a = c & (2 * k1) + 1 = t . (DataLoc (c,i)) & k2 = ((m + n) + (2 * k1)) + 1 & m + y1 = t . (intpos k2) & m + yn = t . (intpos (k2 + 1)) & ( ( 1 <= y1 & yn <= n ) or y1 >= yn ) holds
( I is_closed_on t,Q & I is_halting_on t,Q & (IExec (I,Q,t)) . a = t . a & ( for j being Element of NAT st 1 <= j & j < (2 * k1) + 1 holds
(IExec (I,Q,t)) . (intpos ((m + n) + j)) = t . (intpos ((m + n) + j)) ) & ( y1 >= yn implies ( (IExec (I,Q,t)) . (DataLoc (c,i)) = (2 * k1) - 1 & ( for j being Element of NAT st 1 <= j & j <= n holds
(IExec (I,Q,t)) . (intpos (m + j)) = t . (intpos (m + j)) ) ) ) & ( y1 < yn implies ( (IExec (I,Q,t)) . (DataLoc (c,i)) = (2 * k1) + 3 & ( for j being Element of NAT st ( ( 1 <= j & j < y1 ) or ( yn < j & j <= n ) ) holds
(IExec (I,Q,t)) . (intpos (m + j)) = t . (intpos (m + j)) ) & ex ym being Element of NAT st
( y1 <= ym & ym <= yn & m + y1 = (IExec (I,Q,t)) . (intpos k2) & (m + ym) - 1 = (IExec (I,Q,t)) . (intpos (k2 + 1)) & (m + ym) + 1 = (IExec (I,Q,t)) . (intpos (k2 + 2)) & m + yn = (IExec (I,Q,t)) . (intpos (k2 + 3)) & ( for j being Element of NAT st y1 <= j & j < ym holds
(IExec (I,Q,t)) . (intpos (m + j)) <= (IExec (I,Q,t)) . (intpos (m + ym)) ) & ( for j being Element of NAT st ym < j & j <= yn holds
(IExec (I,Q,t)) . (intpos (m + j)) >= (IExec (I,Q,t)) . (intpos (m + ym)) ) ) ) ) & ( f1 is_FinSequence_on t,m & f2 is_FinSequence_on IExec (I,Q,t),m & len f1 = n & len f2 = n implies f1,f2 are_fiberwise_equipotent ) ) ) implies ( while>0 (a,i,I) is_halting_on s,P & f,g are_fiberwise_equipotent & g is_non_decreasing_on 1,n ) )
I: Initialize s = s by MEMSTR_0:44;
set b = DataLoc (c,i);
assume that
A2: s . a = c and
A3: len f = n and
A4: len g = n ; ::_thesis: ( not f is_FinSequence_on s,m or not g is_FinSequence_on IExec ((while>0 (a,i,I)),P,s),m or not 1 = s . (DataLoc (c,i)) or not m1 = (m + n) + 1 or not m + 1 = s . (intpos m1) or not m + n = s . (intpos (m1 + 1)) or ex t being 0 -started State of SCMPDS ex Q being Instruction-Sequence of SCMPDS ex f1, f2 being FinSequence of INT ex k1, k2, y1, yn being Element of NAT st
( t . a = c & (2 * k1) + 1 = t . (DataLoc (c,i)) & k2 = ((m + n) + (2 * k1)) + 1 & m + y1 = t . (intpos k2) & m + yn = t . (intpos (k2 + 1)) & ( ( 1 <= y1 & yn <= n ) or y1 >= yn ) & not ( I is_closed_on t,Q & I is_halting_on t,Q & (IExec (I,Q,t)) . a = t . a & ( for j being Element of NAT st 1 <= j & j < (2 * k1) + 1 holds
(IExec (I,Q,t)) . (intpos ((m + n) + j)) = t . (intpos ((m + n) + j)) ) & ( y1 >= yn implies ( (IExec (I,Q,t)) . (DataLoc (c,i)) = (2 * k1) - 1 & ( for j being Element of NAT st 1 <= j & j <= n holds
(IExec (I,Q,t)) . (intpos (m + j)) = t . (intpos (m + j)) ) ) ) & ( y1 < yn implies ( (IExec (I,Q,t)) . (DataLoc (c,i)) = (2 * k1) + 3 & ( for j being Element of NAT st ( ( 1 <= j & j < y1 ) or ( yn < j & j <= n ) ) holds
(IExec (I,Q,t)) . (intpos (m + j)) = t . (intpos (m + j)) ) & ex ym being Element of NAT st
( y1 <= ym & ym <= yn & m + y1 = (IExec (I,Q,t)) . (intpos k2) & (m + ym) - 1 = (IExec (I,Q,t)) . (intpos (k2 + 1)) & (m + ym) + 1 = (IExec (I,Q,t)) . (intpos (k2 + 2)) & m + yn = (IExec (I,Q,t)) . (intpos (k2 + 3)) & ( for j being Element of NAT st y1 <= j & j < ym holds
(IExec (I,Q,t)) . (intpos (m + j)) <= (IExec (I,Q,t)) . (intpos (m + ym)) ) & ( for j being Element of NAT st ym < j & j <= yn holds
(IExec (I,Q,t)) . (intpos (m + j)) >= (IExec (I,Q,t)) . (intpos (m + ym)) ) ) ) ) & ( f1 is_FinSequence_on t,m & f2 is_FinSequence_on IExec (I,Q,t),m & len f1 = n & len f2 = n implies f1,f2 are_fiberwise_equipotent ) ) ) or ( while>0 (a,i,I) is_halting_on s,P & f,g are_fiberwise_equipotent & g is_non_decreasing_on 1,n ) )
set WH = while>0 (a,i,I);
set sWH = stop (while>0 (a,i,I));
assume A5: f is_FinSequence_on s,m ; ::_thesis: ( not g is_FinSequence_on IExec ((while>0 (a,i,I)),P,s),m or not 1 = s . (DataLoc (c,i)) or not m1 = (m + n) + 1 or not m + 1 = s . (intpos m1) or not m + n = s . (intpos (m1 + 1)) or ex t being 0 -started State of SCMPDS ex Q being Instruction-Sequence of SCMPDS ex f1, f2 being FinSequence of INT ex k1, k2, y1, yn being Element of NAT st
( t . a = c & (2 * k1) + 1 = t . (DataLoc (c,i)) & k2 = ((m + n) + (2 * k1)) + 1 & m + y1 = t . (intpos k2) & m + yn = t . (intpos (k2 + 1)) & ( ( 1 <= y1 & yn <= n ) or y1 >= yn ) & not ( I is_closed_on t,Q & I is_halting_on t,Q & (IExec (I,Q,t)) . a = t . a & ( for j being Element of NAT st 1 <= j & j < (2 * k1) + 1 holds
(IExec (I,Q,t)) . (intpos ((m + n) + j)) = t . (intpos ((m + n) + j)) ) & ( y1 >= yn implies ( (IExec (I,Q,t)) . (DataLoc (c,i)) = (2 * k1) - 1 & ( for j being Element of NAT st 1 <= j & j <= n holds
(IExec (I,Q,t)) . (intpos (m + j)) = t . (intpos (m + j)) ) ) ) & ( y1 < yn implies ( (IExec (I,Q,t)) . (DataLoc (c,i)) = (2 * k1) + 3 & ( for j being Element of NAT st ( ( 1 <= j & j < y1 ) or ( yn < j & j <= n ) ) holds
(IExec (I,Q,t)) . (intpos (m + j)) = t . (intpos (m + j)) ) & ex ym being Element of NAT st
( y1 <= ym & ym <= yn & m + y1 = (IExec (I,Q,t)) . (intpos k2) & (m + ym) - 1 = (IExec (I,Q,t)) . (intpos (k2 + 1)) & (m + ym) + 1 = (IExec (I,Q,t)) . (intpos (k2 + 2)) & m + yn = (IExec (I,Q,t)) . (intpos (k2 + 3)) & ( for j being Element of NAT st y1 <= j & j < ym holds
(IExec (I,Q,t)) . (intpos (m + j)) <= (IExec (I,Q,t)) . (intpos (m + ym)) ) & ( for j being Element of NAT st ym < j & j <= yn holds
(IExec (I,Q,t)) . (intpos (m + j)) >= (IExec (I,Q,t)) . (intpos (m + ym)) ) ) ) ) & ( f1 is_FinSequence_on t,m & f2 is_FinSequence_on IExec (I,Q,t),m & len f1 = n & len f2 = n implies f1,f2 are_fiberwise_equipotent ) ) ) or ( while>0 (a,i,I) is_halting_on s,P & f,g are_fiberwise_equipotent & g is_non_decreasing_on 1,n ) )
defpred S1[ Element of NAT ] means  for t being 0 -started State of SCMPDS
for Q being Instruction-Sequence of SCMPDS
for f1 being FinSequence of INT
for k1, k2, y1, yn being Element of NAT st t . a = c & (2 * k1) + 1 = t . (DataLoc (c,i)) & k2 = ((m + n) + (2 * k1)) + 1 & ( ( 1 <= y1 & yn <= n ) or y1 >= yn ) & m + y1 = t . (intpos k2) & m + yn = t . (intpos (k2 + 1)) & yn - y1 <= $1 & f1 is_FinSequence_on t,m & len f1 = n holds
ex k being Element of NAT ex f2 being FinSequence of INT st
( Initialize (Comput ((Q +* (stop (while>0 (a,i,I)))),t,k)) = Comput ((Q +* (stop (while>0 (a,i,I)))),t,k) & f2 is_FinSequence_on Comput ((Q +* (stop (while>0 (a,i,I)))),t,k),m & len f2 = n & f1,f2 are_fiberwise_equipotent & f2 is_non_decreasing_on y1,yn & ( for j being Element of NAT st y1 < yn & ( ( 1 <= j & j < y1 ) or ( yn < j & j <= n ) ) holds
f2 . j = t . (intpos (m + j)) ) & ( for j being Element of NAT st y1 >= yn & 1 <= j & j <= n holds
f2 . j = t . (intpos (m + j)) ) & ( for j being Element of NAT st 1 <= j & j < (2 * k1) + 1 holds
(Comput ((Q +* (stop (while>0 (a,i,I)))),t,k)) . (intpos ((m + n) + j)) = t . (intpos ((m + n) + j)) ) & (Comput ((Q +* (stop (while>0 (a,i,I)))),t,k)) . (DataLoc (c,i)) = (t . (DataLoc (c,i))) - 2 & (Comput ((Q +* (stop (while>0 (a,i,I)))),t,k)) . a = c );
assume A6: g is_FinSequence_on IExec ((while>0 (a,i,I)),P,s),m ; ::_thesis: ( not 1 = s . (DataLoc (c,i)) or not m1 = (m + n) + 1 or not m + 1 = s . (intpos m1) or not m + n = s . (intpos (m1 + 1)) or ex t being 0 -started State of SCMPDS ex Q being Instruction-Sequence of SCMPDS ex f1, f2 being FinSequence of INT ex k1, k2, y1, yn being Element of NAT st
( t . a = c & (2 * k1) + 1 = t . (DataLoc (c,i)) & k2 = ((m + n) + (2 * k1)) + 1 & m + y1 = t . (intpos k2) & m + yn = t . (intpos (k2 + 1)) & ( ( 1 <= y1 & yn <= n ) or y1 >= yn ) & not ( I is_closed_on t,Q & I is_halting_on t,Q & (IExec (I,Q,t)) . a = t . a & ( for j being Element of NAT st 1 <= j & j < (2 * k1) + 1 holds
(IExec (I,Q,t)) . (intpos ((m + n) + j)) = t . (intpos ((m + n) + j)) ) & ( y1 >= yn implies ( (IExec (I,Q,t)) . (DataLoc (c,i)) = (2 * k1) - 1 & ( for j being Element of NAT st 1 <= j & j <= n holds
(IExec (I,Q,t)) . (intpos (m + j)) = t . (intpos (m + j)) ) ) ) & ( y1 < yn implies ( (IExec (I,Q,t)) . (DataLoc (c,i)) = (2 * k1) + 3 & ( for j being Element of NAT st ( ( 1 <= j & j < y1 ) or ( yn < j & j <= n ) ) holds
(IExec (I,Q,t)) . (intpos (m + j)) = t . (intpos (m + j)) ) & ex ym being Element of NAT st
( y1 <= ym & ym <= yn & m + y1 = (IExec (I,Q,t)) . (intpos k2) & (m + ym) - 1 = (IExec (I,Q,t)) . (intpos (k2 + 1)) & (m + ym) + 1 = (IExec (I,Q,t)) . (intpos (k2 + 2)) & m + yn = (IExec (I,Q,t)) . (intpos (k2 + 3)) & ( for j being Element of NAT st y1 <= j & j < ym holds
(IExec (I,Q,t)) . (intpos (m + j)) <= (IExec (I,Q,t)) . (intpos (m + ym)) ) & ( for j being Element of NAT st ym < j & j <= yn holds
(IExec (I,Q,t)) . (intpos (m + j)) >= (IExec (I,Q,t)) . (intpos (m + ym)) ) ) ) ) & ( f1 is_FinSequence_on t,m & f2 is_FinSequence_on IExec (I,Q,t),m & len f1 = n & len f2 = n implies f1,f2 are_fiberwise_equipotent ) ) ) or ( while>0 (a,i,I) is_halting_on s,P & f,g are_fiberwise_equipotent & g is_non_decreasing_on 1,n ) )
assume A7: 1 = s . (DataLoc (c,i)) ; ::_thesis: ( not m1 = (m + n) + 1 or not m + 1 = s . (intpos m1) or not m + n = s . (intpos (m1 + 1)) or ex t being 0 -started State of SCMPDS ex Q being Instruction-Sequence of SCMPDS ex f1, f2 being FinSequence of INT ex k1, k2, y1, yn being Element of NAT st
( t . a = c & (2 * k1) + 1 = t . (DataLoc (c,i)) & k2 = ((m + n) + (2 * k1)) + 1 & m + y1 = t . (intpos k2) & m + yn = t . (intpos (k2 + 1)) & ( ( 1 <= y1 & yn <= n ) or y1 >= yn ) & not ( I is_closed_on t,Q & I is_halting_on t,Q & (IExec (I,Q,t)) . a = t . a & ( for j being Element of NAT st 1 <= j & j < (2 * k1) + 1 holds
(IExec (I,Q,t)) . (intpos ((m + n) + j)) = t . (intpos ((m + n) + j)) ) & ( y1 >= yn implies ( (IExec (I,Q,t)) . (DataLoc (c,i)) = (2 * k1) - 1 & ( for j being Element of NAT st 1 <= j & j <= n holds
(IExec (I,Q,t)) . (intpos (m + j)) = t . (intpos (m + j)) ) ) ) & ( y1 < yn implies ( (IExec (I,Q,t)) . (DataLoc (c,i)) = (2 * k1) + 3 & ( for j being Element of NAT st ( ( 1 <= j & j < y1 ) or ( yn < j & j <= n ) ) holds
(IExec (I,Q,t)) . (intpos (m + j)) = t . (intpos (m + j)) ) & ex ym being Element of NAT st
( y1 <= ym & ym <= yn & m + y1 = (IExec (I,Q,t)) . (intpos k2) & (m + ym) - 1 = (IExec (I,Q,t)) . (intpos (k2 + 1)) & (m + ym) + 1 = (IExec (I,Q,t)) . (intpos (k2 + 2)) & m + yn = (IExec (I,Q,t)) . (intpos (k2 + 3)) & ( for j being Element of NAT st y1 <= j & j < ym holds
(IExec (I,Q,t)) . (intpos (m + j)) <= (IExec (I,Q,t)) . (intpos (m + ym)) ) & ( for j being Element of NAT st ym < j & j <= yn holds
(IExec (I,Q,t)) . (intpos (m + j)) >= (IExec (I,Q,t)) . (intpos (m + ym)) ) ) ) ) & ( f1 is_FinSequence_on t,m & f2 is_FinSequence_on IExec (I,Q,t),m & len f1 = n & len f2 = n implies f1,f2 are_fiberwise_equipotent ) ) ) or ( while>0 (a,i,I) is_halting_on s,P & f,g are_fiberwise_equipotent & g is_non_decreasing_on 1,n ) )
assume that
A8: m1 = (m + n) + 1 and
A9: m + 1 = s . (intpos m1) and
A10: m + n = s . (intpos (m1 + 1)) ; ::_thesis: ( ex t being 0 -started State of SCMPDS ex Q being Instruction-Sequence of SCMPDS ex f1, f2 being FinSequence of INT ex k1, k2, y1, yn being Element of NAT st
( t . a = c & (2 * k1) + 1 = t . (DataLoc (c,i)) & k2 = ((m + n) + (2 * k1)) + 1 & m + y1 = t . (intpos k2) & m + yn = t . (intpos (k2 + 1)) & ( ( 1 <= y1 & yn <= n ) or y1 >= yn ) & not ( I is_closed_on t,Q & I is_halting_on t,Q & (IExec (I,Q,t)) . a = t . a & ( for j being Element of NAT st 1 <= j & j < (2 * k1) + 1 holds
(IExec (I,Q,t)) . (intpos ((m + n) + j)) = t . (intpos ((m + n) + j)) ) & ( y1 >= yn implies ( (IExec (I,Q,t)) . (DataLoc (c,i)) = (2 * k1) - 1 & ( for j being Element of NAT st 1 <= j & j <= n holds
(IExec (I,Q,t)) . (intpos (m + j)) = t . (intpos (m + j)) ) ) ) & ( y1 < yn implies ( (IExec (I,Q,t)) . (DataLoc (c,i)) = (2 * k1) + 3 & ( for j being Element of NAT st ( ( 1 <= j & j < y1 ) or ( yn < j & j <= n ) ) holds
(IExec (I,Q,t)) . (intpos (m + j)) = t . (intpos (m + j)) ) & ex ym being Element of NAT st
( y1 <= ym & ym <= yn & m + y1 = (IExec (I,Q,t)) . (intpos k2) & (m + ym) - 1 = (IExec (I,Q,t)) . (intpos (k2 + 1)) & (m + ym) + 1 = (IExec (I,Q,t)) . (intpos (k2 + 2)) & m + yn = (IExec (I,Q,t)) . (intpos (k2 + 3)) & ( for j being Element of NAT st y1 <= j & j < ym holds
(IExec (I,Q,t)) . (intpos (m + j)) <= (IExec (I,Q,t)) . (intpos (m + ym)) ) & ( for j being Element of NAT st ym < j & j <= yn holds
(IExec (I,Q,t)) . (intpos (m + j)) >= (IExec (I,Q,t)) . (intpos (m + ym)) ) ) ) ) & ( f1 is_FinSequence_on t,m & f2 is_FinSequence_on IExec (I,Q,t),m & len f1 = n & len f2 = n implies f1,f2 are_fiberwise_equipotent ) ) ) or ( while>0 (a,i,I) is_halting_on s,P & f,g are_fiberwise_equipotent & g is_non_decreasing_on 1,n ) )
A11: m1 = ((m + n) + (2 * 0)) + 1 by A8;
assume A12: for t being 0 -started State of SCMPDS
for Q being Instruction-Sequence of SCMPDS
for f1, f2 being FinSequence of INT
for k1, k2, y1, yn being Element of NAT st t . a = c & (2 * k1) + 1 = t . (DataLoc (c,i)) & k2 = ((m + n) + (2 * k1)) + 1 & m + y1 = t . (intpos k2) & m + yn = t . (intpos (k2 + 1)) & ( ( 1 <= y1 & yn <= n ) or y1 >= yn ) holds
( I is_closed_on t,Q & I is_halting_on t,Q & (IExec (I,Q,t)) . a = t . a & ( for j being Element of NAT st 1 <= j & j < (2 * k1) + 1 holds
(IExec (I,Q,t)) . (intpos ((m + n) + j)) = t . (intpos ((m + n) + j)) ) & ( y1 >= yn implies ( (IExec (I,Q,t)) . (DataLoc (c,i)) = (2 * k1) - 1 & ( for j being Element of NAT st 1 <= j & j <= n holds
(IExec (I,Q,t)) . (intpos (m + j)) = t . (intpos (m + j)) ) ) ) & ( y1 < yn implies ( (IExec (I,Q,t)) . (DataLoc (c,i)) = (2 * k1) + 3 & ( for j being Element of NAT st ( ( 1 <= j & j < y1 ) or ( yn < j & j <= n ) ) holds
(IExec (I,Q,t)) . (intpos (m + j)) = t . (intpos (m + j)) ) & ex ym being Element of NAT st
( y1 <= ym & ym <= yn & m + y1 = (IExec (I,Q,t)) . (intpos k2) & (m + ym) - 1 = (IExec (I,Q,t)) . (intpos (k2 + 1)) & (m + ym) + 1 = (IExec (I,Q,t)) . (intpos (k2 + 2)) & m + yn = (IExec (I,Q,t)) . (intpos (k2 + 3)) & ( for j being Element of NAT st y1 <= j & j < ym holds
(IExec (I,Q,t)) . (intpos (m + j)) <= (IExec (I,Q,t)) . (intpos (m + ym)) ) & ( for j being Element of NAT st ym < j & j <= yn holds
(IExec (I,Q,t)) . (intpos (m + j)) >= (IExec (I,Q,t)) . (intpos (m + ym)) ) ) ) ) & ( f1 is_FinSequence_on t,m & f2 is_FinSequence_on IExec (I,Q,t),m & len f1 = n & len f2 = n implies f1,f2 are_fiberwise_equipotent ) ) ; ::_thesis: ( while>0 (a,i,I) is_halting_on s,P & f,g are_fiberwise_equipotent & g is_non_decreasing_on 1,n )
A13: S1[ 0 ]
proof
let t be 0 -started State of SCMPDS; ::_thesis: for Q being Instruction-Sequence of SCMPDS
for f1 being FinSequence of INT
for k1, k2, y1, yn being Element of NAT st t . a = c & (2 * k1) + 1 = t . (DataLoc (c,i)) & k2 = ((m + n) + (2 * k1)) + 1 & ( ( 1 <= y1 & yn <= n ) or y1 >= yn ) & m + y1 = t . (intpos k2) & m + yn = t . (intpos (k2 + 1)) & yn - y1 <= 0 & f1 is_FinSequence_on t,m & len f1 = n holds
ex k being Element of NAT ex f2 being FinSequence of INT st
( Initialize (Comput ((Q +* (stop (while>0 (a,i,I)))),t,k)) = Comput ((Q +* (stop (while>0 (a,i,I)))),t,k) & f2 is_FinSequence_on Comput ((Q +* (stop (while>0 (a,i,I)))),t,k),m & len f2 = n & f1,f2 are_fiberwise_equipotent & f2 is_non_decreasing_on y1,yn & ( for j being Element of NAT st y1 < yn & ( ( 1 <= j & j < y1 ) or ( yn < j & j <= n ) ) holds
f2 . j = t . (intpos (m + j)) ) & ( for j being Element of NAT st y1 >= yn & 1 <= j & j <= n holds
f2 . j = t . (intpos (m + j)) ) & ( for j being Element of NAT st 1 <= j & j < (2 * k1) + 1 holds
(Comput ((Q +* (stop (while>0 (a,i,I)))),t,k)) . (intpos ((m + n) + j)) = t . (intpos ((m + n) + j)) ) & (Comput ((Q +* (stop (while>0 (a,i,I)))),t,k)) . (DataLoc (c,i)) = (t . (DataLoc (c,i))) - 2 & (Comput ((Q +* (stop (while>0 (a,i,I)))),t,k)) . a = c )
let Q be Instruction-Sequence of SCMPDS; ::_thesis: for f1 being FinSequence of INT
for k1, k2, y1, yn being Element of NAT st t . a = c & (2 * k1) + 1 = t . (DataLoc (c,i)) & k2 = ((m + n) + (2 * k1)) + 1 & ( ( 1 <= y1 & yn <= n ) or y1 >= yn ) & m + y1 = t . (intpos k2) & m + yn = t . (intpos (k2 + 1)) & yn - y1 <= 0 & f1 is_FinSequence_on t,m & len f1 = n holds
ex k being Element of NAT ex f2 being FinSequence of INT st
( Initialize (Comput ((Q +* (stop (while>0 (a,i,I)))),t,k)) = Comput ((Q +* (stop (while>0 (a,i,I)))),t,k) & f2 is_FinSequence_on Comput ((Q +* (stop (while>0 (a,i,I)))),t,k),m & len f2 = n & f1,f2 are_fiberwise_equipotent & f2 is_non_decreasing_on y1,yn & ( for j being Element of NAT st y1 < yn & ( ( 1 <= j & j < y1 ) or ( yn < j & j <= n ) ) holds
f2 . j = t . (intpos (m + j)) ) & ( for j being Element of NAT st y1 >= yn & 1 <= j & j <= n holds
f2 . j = t . (intpos (m + j)) ) & ( for j being Element of NAT st 1 <= j & j < (2 * k1) + 1 holds
(Comput ((Q +* (stop (while>0 (a,i,I)))),t,k)) . (intpos ((m + n) + j)) = t . (intpos ((m + n) + j)) ) & (Comput ((Q +* (stop (while>0 (a,i,I)))),t,k)) . (DataLoc (c,i)) = (t . (DataLoc (c,i))) - 2 & (Comput ((Q +* (stop (while>0 (a,i,I)))),t,k)) . a = c )
let f1 be FinSequence of INT ; ::_thesis: for k1, k2, y1, yn being Element of NAT st t . a = c & (2 * k1) + 1 = t . (DataLoc (c,i)) & k2 = ((m + n) + (2 * k1)) + 1 & ( ( 1 <= y1 & yn <= n ) or y1 >= yn ) & m + y1 = t . (intpos k2) & m + yn = t . (intpos (k2 + 1)) & yn - y1 <= 0 & f1 is_FinSequence_on t,m & len f1 = n holds
ex k being Element of NAT ex f2 being FinSequence of INT st
( Initialize (Comput ((Q +* (stop (while>0 (a,i,I)))),t,k)) = Comput ((Q +* (stop (while>0 (a,i,I)))),t,k) & f2 is_FinSequence_on Comput ((Q +* (stop (while>0 (a,i,I)))),t,k),m & len f2 = n & f1,f2 are_fiberwise_equipotent & f2 is_non_decreasing_on y1,yn & ( for j being Element of NAT st y1 < yn & ( ( 1 <= j & j < y1 ) or ( yn < j & j <= n ) ) holds
f2 . j = t . (intpos (m + j)) ) & ( for j being Element of NAT st y1 >= yn & 1 <= j & j <= n holds
f2 . j = t . (intpos (m + j)) ) & ( for j being Element of NAT st 1 <= j & j < (2 * k1) + 1 holds
(Comput ((Q +* (stop (while>0 (a,i,I)))),t,k)) . (intpos ((m + n) + j)) = t . (intpos ((m + n) + j)) ) & (Comput ((Q +* (stop (while>0 (a,i,I)))),t,k)) . (DataLoc (c,i)) = (t . (DataLoc (c,i))) - 2 & (Comput ((Q +* (stop (while>0 (a,i,I)))),t,k)) . a = c )
let k1, k2, y1, yn be Element of NAT ; ::_thesis: ( t . a = c & (2 * k1) + 1 = t . (DataLoc (c,i)) & k2 = ((m + n) + (2 * k1)) + 1 & ( ( 1 <= y1 & yn <= n ) or y1 >= yn ) & m + y1 = t . (intpos k2) & m + yn = t . (intpos (k2 + 1)) & yn - y1 <= 0 & f1 is_FinSequence_on t,m & len f1 = n implies ex k being Element of NAT ex f2 being FinSequence of INT st
( Initialize (Comput ((Q +* (stop (while>0 (a,i,I)))),t,k)) = Comput ((Q +* (stop (while>0 (a,i,I)))),t,k) & f2 is_FinSequence_on Comput ((Q +* (stop (while>0 (a,i,I)))),t,k),m & len f2 = n & f1,f2 are_fiberwise_equipotent & f2 is_non_decreasing_on y1,yn & ( for j being Element of NAT st y1 < yn & ( ( 1 <= j & j < y1 ) or ( yn < j & j <= n ) ) holds
f2 . j = t . (intpos (m + j)) ) & ( for j being Element of NAT st y1 >= yn & 1 <= j & j <= n holds
f2 . j = t . (intpos (m + j)) ) & ( for j being Element of NAT st 1 <= j & j < (2 * k1) + 1 holds
(Comput ((Q +* (stop (while>0 (a,i,I)))),t,k)) . (intpos ((m + n) + j)) = t . (intpos ((m + n) + j)) ) & (Comput ((Q +* (stop (while>0 (a,i,I)))),t,k)) . (DataLoc (c,i)) = (t . (DataLoc (c,i))) - 2 & (Comput ((Q +* (stop (while>0 (a,i,I)))),t,k)) . a = c ) )
T: Initialize t = t by MEMSTR_0:44;
assume that
A14: t . a = c and
A15: (2 * k1) + 1 = t . (DataLoc (c,i)) and
A16: k2 = ((m + n) + (2 * k1)) + 1 and
A17: ( ( 1 <= y1 & yn <= n ) or y1 >= yn ) and
A18: m + y1 = t . (intpos k2) and
A19: m + yn = t . (intpos (k2 + 1)) and
A20: yn - y1 <= 0 and
A21: f1 is_FinSequence_on t,m and
A22: len f1 = n ; ::_thesis: ex k being Element of NAT ex f2 being FinSequence of INT st
( Initialize (Comput ((Q +* (stop (while>0 (a,i,I)))),t,k)) = Comput ((Q +* (stop (while>0 (a,i,I)))),t,k) & f2 is_FinSequence_on Comput ((Q +* (stop (while>0 (a,i,I)))),t,k),m & len f2 = n & f1,f2 are_fiberwise_equipotent & f2 is_non_decreasing_on y1,yn & ( for j being Element of NAT st y1 < yn & ( ( 1 <= j & j < y1 ) or ( yn < j & j <= n ) ) holds
f2 . j = t . (intpos (m + j)) ) & ( for j being Element of NAT st y1 >= yn & 1 <= j & j <= n holds
f2 . j = t . (intpos (m + j)) ) & ( for j being Element of NAT st 1 <= j & j < (2 * k1) + 1 holds
(Comput ((Q +* (stop (while>0 (a,i,I)))),t,k)) . (intpos ((m + n) + j)) = t . (intpos ((m + n) + j)) ) & (Comput ((Q +* (stop (while>0 (a,i,I)))),t,k)) . (DataLoc (c,i)) = (t . (DataLoc (c,i))) - 2 & (Comput ((Q +* (stop (while>0 (a,i,I)))),t,k)) . a = c )
A23: I is_halting_on t,Q by A12, A14, A15, A16, A17, A18, A19;
take k = (LifeSpan ((Q +* (stop I)),t)) + 2; ::_thesis: ex f2 being FinSequence of INT st
( Initialize (Comput ((Q +* (stop (while>0 (a,i,I)))),t,k)) = Comput ((Q +* (stop (while>0 (a,i,I)))),t,k) & f2 is_FinSequence_on Comput ((Q +* (stop (while>0 (a,i,I)))),t,k),m & len f2 = n & f1,f2 are_fiberwise_equipotent & f2 is_non_decreasing_on y1,yn & ( for j being Element of NAT st y1 < yn & ( ( 1 <= j & j < y1 ) or ( yn < j & j <= n ) ) holds
f2 . j = t . (intpos (m + j)) ) & ( for j being Element of NAT st y1 >= yn & 1 <= j & j <= n holds
f2 . j = t . (intpos (m + j)) ) & ( for j being Element of NAT st 1 <= j & j < (2 * k1) + 1 holds
(Comput ((Q +* (stop (while>0 (a,i,I)))),t,k)) . (intpos ((m + n) + j)) = t . (intpos ((m + n) + j)) ) & (Comput ((Q +* (stop (while>0 (a,i,I)))),t,k)) . (DataLoc (c,i)) = (t . (DataLoc (c,i))) - 2 & (Comput ((Q +* (stop (while>0 (a,i,I)))),t,k)) . a = c )
set tk = Comput ((Q +* (stop (while>0 (a,i,I)))),t,k);
A24: I is_closed_on t,Q by A12, A14, A15, A16, A17, A18, A19;
then A25: DataPart (Comput ((Q +* (stop (while>0 (a,i,I)))),t,k)) = DataPart (IExec (I,Q,t)) by A14, A15, A23, Th2, T;
consider f2 being FinSequence of INT  such that
A26: len f2 = n and
A27: for i being Element of NAT st 1 <= i & i <= len f2 holds
f2 . i = (IExec (I,Q,t)) . (intpos (m + i)) by SCPISORT:1;
take  f2 ; ::_thesis: ( Initialize (Comput ((Q +* (stop (while>0 (a,i,I)))),t,k)) = Comput ((Q +* (stop (while>0 (a,i,I)))),t,k) & f2 is_FinSequence_on Comput ((Q +* (stop (while>0 (a,i,I)))),t,k),m & len f2 = n & f1,f2 are_fiberwise_equipotent & f2 is_non_decreasing_on y1,yn & ( for j being Element of NAT st y1 < yn & ( ( 1 <= j & j < y1 ) or ( yn < j & j <= n ) ) holds
f2 . j = t . (intpos (m + j)) ) & ( for j being Element of NAT st y1 >= yn & 1 <= j & j <= n holds
f2 . j = t . (intpos (m + j)) ) & ( for j being Element of NAT st 1 <= j & j < (2 * k1) + 1 holds
(Comput ((Q +* (stop (while>0 (a,i,I)))),t,k)) . (intpos ((m + n) + j)) = t . (intpos ((m + n) + j)) ) & (Comput ((Q +* (stop (while>0 (a,i,I)))),t,k)) . (DataLoc (c,i)) = (t . (DataLoc (c,i))) - 2 & (Comput ((Q +* (stop (while>0 (a,i,I)))),t,k)) . a = c )
thus  Initialize (Comput ((Q +* (stop (while>0 (a,i,I)))),t,k)) = Comput ((Q +* (stop (while>0 (a,i,I)))),t,k) by A14, A15, A24, A23, Th2, T; ::_thesis: ( f2 is_FinSequence_on Comput ((Q +* (stop (while>0 (a,i,I)))),t,k),m & len f2 = n & f1,f2 are_fiberwise_equipotent & f2 is_non_decreasing_on y1,yn & ( for j being Element of NAT st y1 < yn & ( ( 1 <= j & j < y1 ) or ( yn < j & j <= n ) ) holds
f2 . j = t . (intpos (m + j)) ) & ( for j being Element of NAT st y1 >= yn & 1 <= j & j <= n holds
f2 . j = t . (intpos (m + j)) ) & ( for j being Element of NAT st 1 <= j & j < (2 * k1) + 1 holds
(Comput ((Q +* (stop (while>0 (a,i,I)))),t,k)) . (intpos ((m + n) + j)) = t . (intpos ((m + n) + j)) ) & (Comput ((Q +* (stop (while>0 (a,i,I)))),t,k)) . (DataLoc (c,i)) = (t . (DataLoc (c,i))) - 2 & (Comput ((Q +* (stop (while>0 (a,i,I)))),t,k)) . a = c )
now__::_thesis:_for_i_being_Element_of_NAT_st_1_<=_i_&_i_<=_len_f2_holds_
f2_._i_=_(Comput_((Q_+*_(stop_(while>0_(a,i,I)))),t,k))_._(intpos_(m_+_i))
let i be Element of NAT ; ::_thesis: ( 1 <= i & i <= len f2 implies f2 . i = (Comput ((Q +* (stop (while>0 (a,i,I)))),t,k)) . (intpos (m + i)) )
assume that
A28: 1 <= i and
A29: i <= len f2 ; ::_thesis: f2 . i = (Comput ((Q +* (stop (while>0 (a,i,I)))),t,k)) . (intpos (m + i))
thus f2 . i = (IExec (I,Q,t)) . (intpos (m + i)) by A27, A28, A29
.= (Comput ((Q +* (stop (while>0 (a,i,I)))),t,k)) . (intpos (m + i)) by A25, SCMPDS_4:8 ; ::_thesis: verum
end;
hence  f2 is_FinSequence_on Comput ((Q +* (stop (while>0 (a,i,I)))),t,k),m by SCPISORT:def_1; ::_thesis: ( len f2 = n & f1,f2 are_fiberwise_equipotent & f2 is_non_decreasing_on y1,yn & ( for j being Element of NAT st y1 < yn & ( ( 1 <= j & j < y1 ) or ( yn < j & j <= n ) ) holds
f2 . j = t . (intpos (m + j)) ) & ( for j being Element of NAT st y1 >= yn & 1 <= j & j <= n holds
f2 . j = t . (intpos (m + j)) ) & ( for j being Element of NAT st 1 <= j & j < (2 * k1) + 1 holds
(Comput ((Q +* (stop (while>0 (a,i,I)))),t,k)) . (intpos ((m + n) + j)) = t . (intpos ((m + n) + j)) ) & (Comput ((Q +* (stop (while>0 (a,i,I)))),t,k)) . (DataLoc (c,i)) = (t . (DataLoc (c,i))) - 2 & (Comput ((Q +* (stop (while>0 (a,i,I)))),t,k)) . a = c )
thus  len f2 = n by A26; ::_thesis: ( f1,f2 are_fiberwise_equipotent & f2 is_non_decreasing_on y1,yn & ( for j being Element of NAT st y1 < yn & ( ( 1 <= j & j < y1 ) or ( yn < j & j <= n ) ) holds
f2 . j = t . (intpos (m + j)) ) & ( for j being Element of NAT st y1 >= yn & 1 <= j & j <= n holds
f2 . j = t . (intpos (m + j)) ) & ( for j being Element of NAT st 1 <= j & j < (2 * k1) + 1 holds
(Comput ((Q +* (stop (while>0 (a,i,I)))),t,k)) . (intpos ((m + n) + j)) = t . (intpos ((m + n) + j)) ) & (Comput ((Q +* (stop (while>0 (a,i,I)))),t,k)) . (DataLoc (c,i)) = (t . (DataLoc (c,i))) - 2 & (Comput ((Q +* (stop (while>0 (a,i,I)))),t,k)) . a = c )
 f2 is_FinSequence_on IExec (I,Q,t),m by A27, SCPISORT:def_1;
hence  f1,f2 are_fiberwise_equipotent by A12, A14, A15, A16, A17, A18, A19, A21, A22, A26; ::_thesis: ( f2 is_non_decreasing_on y1,yn & ( for j being Element of NAT st y1 < yn & ( ( 1 <= j & j < y1 ) or ( yn < j & j <= n ) ) holds
f2 . j = t . (intpos (m + j)) ) & ( for j being Element of NAT st y1 >= yn & 1 <= j & j <= n holds
f2 . j = t . (intpos (m + j)) ) & ( for j being Element of NAT st 1 <= j & j < (2 * k1) + 1 holds
(Comput ((Q +* (stop (while>0 (a,i,I)))),t,k)) . (intpos ((m + n) + j)) = t . (intpos ((m + n) + j)) ) & (Comput ((Q +* (stop (while>0 (a,i,I)))),t,k)) . (DataLoc (c,i)) = (t . (DataLoc (c,i))) - 2 & (Comput ((Q +* (stop (while>0 (a,i,I)))),t,k)) . a = c )
thus  f2 is_non_decreasing_on y1,yn by A20, GRAPH_2:63, XREAL_1:50; ::_thesis: ( ( for j being Element of NAT st y1 < yn & ( ( 1 <= j & j < y1 ) or ( yn < j & j <= n ) ) holds
f2 . j = t . (intpos (m + j)) ) & ( for j being Element of NAT st y1 >= yn & 1 <= j & j <= n holds
f2 . j = t . (intpos (m + j)) ) & ( for j being Element of NAT st 1 <= j & j < (2 * k1) + 1 holds
(Comput ((Q +* (stop (while>0 (a,i,I)))),t,k)) . (intpos ((m + n) + j)) = t . (intpos ((m + n) + j)) ) & (Comput ((Q +* (stop (while>0 (a,i,I)))),t,k)) . (DataLoc (c,i)) = (t . (DataLoc (c,i))) - 2 & (Comput ((Q +* (stop (while>0 (a,i,I)))),t,k)) . a = c )
thus  for j being Element of NAT st y1 < yn & ( ( 1 <= j & j < y1 ) or ( yn < j & j <= n ) ) holds
f2 . j = t . (intpos (m + j)) by A20, XREAL_1:50; ::_thesis: ( ( for j being Element of NAT st y1 >= yn & 1 <= j & j <= n holds
f2 . j = t . (intpos (m + j)) ) & ( for j being Element of NAT st 1 <= j & j < (2 * k1) + 1 holds
(Comput ((Q +* (stop (while>0 (a,i,I)))),t,k)) . (intpos ((m + n) + j)) = t . (intpos ((m + n) + j)) ) & (Comput ((Q +* (stop (while>0 (a,i,I)))),t,k)) . (DataLoc (c,i)) = (t . (DataLoc (c,i))) - 2 & (Comput ((Q +* (stop (while>0 (a,i,I)))),t,k)) . a = c )
hereby  ::_thesis: ( ( for j being Element of NAT st 1 <= j & j < (2 * k1) + 1 holds
(Comput ((Q +* (stop (while>0 (a,i,I)))),t,k)) . (intpos ((m + n) + j)) = t . (intpos ((m + n) + j)) ) & (Comput ((Q +* (stop (while>0 (a,i,I)))),t,k)) . (DataLoc (c,i)) = (t . (DataLoc (c,i))) - 2 & (Comput ((Q +* (stop (while>0 (a,i,I)))),t,k)) . a = c )
let j be Element of NAT ; ::_thesis: ( y1 >= yn & 1 <= j & j <= n implies f2 . j = t . (intpos (m + j)) )
assume that
A30: y1 >= yn and
A31: 1 <= j and
A32: j <= n ; ::_thesis: f2 . j = t . (intpos (m + j))
thus f2 . j = (IExec (I,Q,t)) . (intpos (m + j)) by A26, A27, A31, A32
.= t . (intpos (m + j)) by A12, A14, A15, A16, A18, A19, A30, A31, A32 ; ::_thesis: verum
end;
hereby  ::_thesis: ( (Comput ((Q +* (stop (while>0 (a,i,I)))),t,k)) . (DataLoc (c,i)) = (t . (DataLoc (c,i))) - 2 & (Comput ((Q +* (stop (while>0 (a,i,I)))),t,k)) . a = c )
let j be Element of NAT ; ::_thesis: ( 1 <= j & j < (2 * k1) + 1 implies (Comput ((Q +* (stop (while>0 (a,i,I)))),t,k)) . (intpos ((m + n) + j)) = t . (intpos ((m + n) + j)) )
assume that
A33: 1 <= j and
A34: j < (2 * k1) + 1 ; ::_thesis: (Comput ((Q +* (stop (while>0 (a,i,I)))),t,k)) . (intpos ((m + n) + j)) = t . (intpos ((m + n) + j))
thus (Comput ((Q +* (stop (while>0 (a,i,I)))),t,k)) . (intpos ((m + n) + j)) = (IExec (I,Q,t)) . (intpos ((m + n) + j)) by A25, SCMPDS_4:8
.= t . (intpos ((m + n) + j)) by A12, A14, A15, A16, A17, A18, A19, A33, A34 ; ::_thesis: verum
end;
 ( y1 >= yn implies ( (IExec (I,Q,t)) . (DataLoc (c,i)) = (2 * k1) - 1 & ( for j being Element of NAT st 1 <= j & j <= n holds
(IExec (I,Q,t)) . (intpos (m + j)) = t . (intpos (m + j)) ) ) ) by A12, A14, A15, A16, A18, A19;
hence  (Comput ((Q +* (stop (while>0 (a,i,I)))),t,k)) . (DataLoc (c,i)) = (t . (DataLoc (c,i))) - 2 by A15, A20, A25, SCMPDS_4:8, XREAL_1:50; ::_thesis: (Comput ((Q +* (stop (while>0 (a,i,I)))),t,k)) . a = c
 (IExec (I,Q,t)) . a = t . a by A12, A14, A15, A16, A17, A18, A19;
hence  (Comput ((Q +* (stop (while>0 (a,i,I)))),t,k)) . a = c by A14, A25, SCMPDS_4:8; ::_thesis: verum
end;
A35: now__::_thesis:_for_k_being_Element_of_NAT_st_S1[k]_holds_
S1[k_+_1]
let k be Element of NAT ; ::_thesis: ( S1[k] implies S1[k + 1] )
assume A36: S1[k] ; ::_thesis: S1[k + 1]
 S1[k + 1]
proof
let t be 0 -started State of SCMPDS; ::_thesis: for Q being Instruction-Sequence of SCMPDS
for f1 being FinSequence of INT
for k1, k2, y1, yn being Element of NAT st t . a = c & (2 * k1) + 1 = t . (DataLoc (c,i)) & k2 = ((m + n) + (2 * k1)) + 1 & ( ( 1 <= y1 & yn <= n ) or y1 >= yn ) & m + y1 = t . (intpos k2) & m + yn = t . (intpos (k2 + 1)) & yn - y1 <= k + 1 & f1 is_FinSequence_on t,m & len f1 = n holds
ex k being Element of NAT ex f2 being FinSequence of INT st
( Initialize (Comput ((Q +* (stop (while>0 (a,i,I)))),t,k)) = Comput ((Q +* (stop (while>0 (a,i,I)))),t,k) & f2 is_FinSequence_on Comput ((Q +* (stop (while>0 (a,i,I)))),t,k),m & len f2 = n & f1,f2 are_fiberwise_equipotent & f2 is_non_decreasing_on y1,yn & ( for j being Element of NAT st y1 < yn & ( ( 1 <= j & j < y1 ) or ( yn < j & j <= n ) ) holds
f2 . j = t . (intpos (m + j)) ) & ( for j being Element of NAT st y1 >= yn & 1 <= j & j <= n holds
f2 . j = t . (intpos (m + j)) ) & ( for j being Element of NAT st 1 <= j & j < (2 * k1) + 1 holds
(Comput ((Q +* (stop (while>0 (a,i,I)))),t,k)) . (intpos ((m + n) + j)) = t . (intpos ((m + n) + j)) ) & (Comput ((Q +* (stop (while>0 (a,i,I)))),t,k)) . (DataLoc (c,i)) = (t . (DataLoc (c,i))) - 2 & (Comput ((Q +* (stop (while>0 (a,i,I)))),t,k)) . a = c )
let Q be Instruction-Sequence of SCMPDS; ::_thesis: for f1 being FinSequence of INT
for k1, k2, y1, yn being Element of NAT st t . a = c & (2 * k1) + 1 = t . (DataLoc (c,i)) & k2 = ((m + n) + (2 * k1)) + 1 & ( ( 1 <= y1 & yn <= n ) or y1 >= yn ) & m + y1 = t . (intpos k2) & m + yn = t . (intpos (k2 + 1)) & yn - y1 <= k + 1 & f1 is_FinSequence_on t,m & len f1 = n holds
ex k being Element of NAT ex f2 being FinSequence of INT st
( Initialize (Comput ((Q +* (stop (while>0 (a,i,I)))),t,k)) = Comput ((Q +* (stop (while>0 (a,i,I)))),t,k) & f2 is_FinSequence_on Comput ((Q +* (stop (while>0 (a,i,I)))),t,k),m & len f2 = n & f1,f2 are_fiberwise_equipotent & f2 is_non_decreasing_on y1,yn & ( for j being Element of NAT st y1 < yn & ( ( 1 <= j & j < y1 ) or ( yn < j & j <= n ) ) holds
f2 . j = t . (intpos (m + j)) ) & ( for j being Element of NAT st y1 >= yn & 1 <= j & j <= n holds
f2 . j = t . (intpos (m + j)) ) & ( for j being Element of NAT st 1 <= j & j < (2 * k1) + 1 holds
(Comput ((Q +* (stop (while>0 (a,i,I)))),t,k)) . (intpos ((m + n) + j)) = t . (intpos ((m + n) + j)) ) & (Comput ((Q +* (stop (while>0 (a,i,I)))),t,k)) . (DataLoc (c,i)) = (t . (DataLoc (c,i))) - 2 & (Comput ((Q +* (stop (while>0 (a,i,I)))),t,k)) . a = c )
let f1 be FinSequence of INT ; ::_thesis: for k1, k2, y1, yn being Element of NAT st t . a = c & (2 * k1) + 1 = t . (DataLoc (c,i)) & k2 = ((m + n) + (2 * k1)) + 1 & ( ( 1 <= y1 & yn <= n ) or y1 >= yn ) & m + y1 = t . (intpos k2) & m + yn = t . (intpos (k2 + 1)) & yn - y1 <= k + 1 & f1 is_FinSequence_on t,m & len f1 = n holds
ex k being Element of NAT ex f2 being FinSequence of INT st
( Initialize (Comput ((Q +* (stop (while>0 (a,i,I)))),t,k)) = Comput ((Q +* (stop (while>0 (a,i,I)))),t,k) & f2 is_FinSequence_on Comput ((Q +* (stop (while>0 (a,i,I)))),t,k),m & len f2 = n & f1,f2 are_fiberwise_equipotent & f2 is_non_decreasing_on y1,yn & ( for j being Element of NAT st y1 < yn & ( ( 1 <= j & j < y1 ) or ( yn < j & j <= n ) ) holds
f2 . j = t . (intpos (m + j)) ) & ( for j being Element of NAT st y1 >= yn & 1 <= j & j <= n holds
f2 . j = t . (intpos (m + j)) ) & ( for j being Element of NAT st 1 <= j & j < (2 * k1) + 1 holds
(Comput ((Q +* (stop (while>0 (a,i,I)))),t,k)) . (intpos ((m + n) + j)) = t . (intpos ((m + n) + j)) ) & (Comput ((Q +* (stop (while>0 (a,i,I)))),t,k)) . (DataLoc (c,i)) = (t . (DataLoc (c,i))) - 2 & (Comput ((Q +* (stop (while>0 (a,i,I)))),t,k)) . a = c )
let k1, k2, y1, yn be Element of NAT ; ::_thesis: ( t . a = c & (2 * k1) + 1 = t . (DataLoc (c,i)) & k2 = ((m + n) + (2 * k1)) + 1 & ( ( 1 <= y1 & yn <= n ) or y1 >= yn ) & m + y1 = t . (intpos k2) & m + yn = t . (intpos (k2 + 1)) & yn - y1 <= k + 1 & f1 is_FinSequence_on t,m & len f1 = n implies ex k being Element of NAT ex f2 being FinSequence of INT st
( Initialize (Comput ((Q +* (stop (while>0 (a,i,I)))),t,k)) = Comput ((Q +* (stop (while>0 (a,i,I)))),t,k) & f2 is_FinSequence_on Comput ((Q +* (stop (while>0 (a,i,I)))),t,k),m & len f2 = n & f1,f2 are_fiberwise_equipotent & f2 is_non_decreasing_on y1,yn & ( for j being Element of NAT st y1 < yn & ( ( 1 <= j & j < y1 ) or ( yn < j & j <= n ) ) holds
f2 . j = t . (intpos (m + j)) ) & ( for j being Element of NAT st y1 >= yn & 1 <= j & j <= n holds
f2 . j = t . (intpos (m + j)) ) & ( for j being Element of NAT st 1 <= j & j < (2 * k1) + 1 holds
(Comput ((Q +* (stop (while>0 (a,i,I)))),t,k)) . (intpos ((m + n) + j)) = t . (intpos ((m + n) + j)) ) & (Comput ((Q +* (stop (while>0 (a,i,I)))),t,k)) . (DataLoc (c,i)) = (t . (DataLoc (c,i))) - 2 & (Comput ((Q +* (stop (while>0 (a,i,I)))),t,k)) . a = c ) )
T: Initialize t = t by MEMSTR_0:44;
assume that
A37: t . a = c and
A38: (2 * k1) + 1 = t . (DataLoc (c,i)) and
A39: k2 = ((m + n) + (2 * k1)) + 1 and
A40: ( ( 1 <= y1 & yn <= n ) or y1 >= yn ) and
A41: m + y1 = t . (intpos k2) and
A42: m + yn = t . (intpos (k2 + 1)) and
A43: yn - y1 <= k + 1 and
A44: f1 is_FinSequence_on t,m and
A45: len f1 = n ; ::_thesis: ex k being Element of NAT ex f2 being FinSequence of INT st
( Initialize (Comput ((Q +* (stop (while>0 (a,i,I)))),t,k)) = Comput ((Q +* (stop (while>0 (a,i,I)))),t,k) & f2 is_FinSequence_on Comput ((Q +* (stop (while>0 (a,i,I)))),t,k),m & len f2 = n & f1,f2 are_fiberwise_equipotent & f2 is_non_decreasing_on y1,yn & ( for j being Element of NAT st y1 < yn & ( ( 1 <= j & j < y1 ) or ( yn < j & j <= n ) ) holds
f2 . j = t . (intpos (m + j)) ) & ( for j being Element of NAT st y1 >= yn & 1 <= j & j <= n holds
f2 . j = t . (intpos (m + j)) ) & ( for j being Element of NAT st 1 <= j & j < (2 * k1) + 1 holds
(Comput ((Q +* (stop (while>0 (a,i,I)))),t,k)) . (intpos ((m + n) + j)) = t . (intpos ((m + n) + j)) ) & (Comput ((Q +* (stop (while>0 (a,i,I)))),t,k)) . (DataLoc (c,i)) = (t . (DataLoc (c,i))) - 2 & (Comput ((Q +* (stop (while>0 (a,i,I)))),t,k)) . a = c )
percases ( yn - y1 <= 0 or yn - y1 > 0 ) ;
suppose yn - y1 <= 0 ; ::_thesis: ex k being Element of NAT ex f2 being FinSequence of INT st
( Initialize (Comput ((Q +* (stop (while>0 (a,i,I)))),t,k)) = Comput ((Q +* (stop (while>0 (a,i,I)))),t,k) & f2 is_FinSequence_on Comput ((Q +* (stop (while>0 (a,i,I)))),t,k),m & len f2 = n & f1,f2 are_fiberwise_equipotent & f2 is_non_decreasing_on y1,yn & ( for j being Element of NAT st y1 < yn & ( ( 1 <= j & j < y1 ) or ( yn < j & j <= n ) ) holds
f2 . j = t . (intpos (m + j)) ) & ( for j being Element of NAT st y1 >= yn & 1 <= j & j <= n holds
f2 . j = t . (intpos (m + j)) ) & ( for j being Element of NAT st 1 <= j & j < (2 * k1) + 1 holds
(Comput ((Q +* (stop (while>0 (a,i,I)))),t,k)) . (intpos ((m + n) + j)) = t . (intpos ((m + n) + j)) ) & (Comput ((Q +* (stop (while>0 (a,i,I)))),t,k)) . (DataLoc (c,i)) = (t . (DataLoc (c,i))) - 2 & (Comput ((Q +* (stop (while>0 (a,i,I)))),t,k)) . a = c )
hence  ex kk being Element of NAT ex f2 being FinSequence of INT st
( Initialize (Comput ((Q +* (stop (while>0 (a,i,I)))),t,kk)) = Comput ((Q +* (stop (while>0 (a,i,I)))),t,kk) & f2 is_FinSequence_on Comput ((Q +* (stop (while>0 (a,i,I)))),t,kk),m & len f2 = n & f1,f2 are_fiberwise_equipotent & f2 is_non_decreasing_on y1,yn & ( for j being Element of NAT st y1 < yn & ( ( 1 <= j & j < y1 ) or ( yn < j & j <= n ) ) holds
f2 . j = t . (intpos (m + j)) ) & ( for j being Element of NAT st y1 >= yn & 1 <= j & j <= n holds
f2 . j = t . (intpos (m + j)) ) & ( for j being Element of NAT st 1 <= j & j < (2 * k1) + 1 holds
(Comput ((Q +* (stop (while>0 (a,i,I)))),t,kk)) . (intpos ((m + n) + j)) = t . (intpos ((m + n) + j)) ) & (Comput ((Q +* (stop (while>0 (a,i,I)))),t,kk)) . (DataLoc (c,i)) = (t . (DataLoc (c,i))) - 2 & (Comput ((Q +* (stop (while>0 (a,i,I)))),t,kk)) . a = c ) by A13, A37, A38, A39, A40, A41, A42, A44, A45; ::_thesis: verum
end;
supposeA46: yn - y1 > 0 ; ::_thesis: ex k being Element of NAT ex f2 being FinSequence of INT st
( Initialize (Comput ((Q +* (stop (while>0 (a,i,I)))),t,k)) = Comput ((Q +* (stop (while>0 (a,i,I)))),t,k) & f2 is_FinSequence_on Comput ((Q +* (stop (while>0 (a,i,I)))),t,k),m & len f2 = n & f1,f2 are_fiberwise_equipotent & f2 is_non_decreasing_on y1,yn & ( for j being Element of NAT st y1 < yn & ( ( 1 <= j & j < y1 ) or ( yn < j & j <= n ) ) holds
f2 . j = t . (intpos (m + j)) ) & ( for j being Element of NAT st y1 >= yn & 1 <= j & j <= n holds
f2 . j = t . (intpos (m + j)) ) & ( for j being Element of NAT st 1 <= j & j < (2 * k1) + 1 holds
(Comput ((Q +* (stop (while>0 (a,i,I)))),t,k)) . (intpos ((m + n) + j)) = t . (intpos ((m + n) + j)) ) & (Comput ((Q +* (stop (while>0 (a,i,I)))),t,k)) . (DataLoc (c,i)) = (t . (DataLoc (c,i))) - 2 & (Comput ((Q +* (stop (while>0 (a,i,I)))),t,k)) . a = c )
set m1 = (LifeSpan ((Q +* (stop I)),t)) + 2;
set t1 = Comput ((Q +* (stop (while>0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 2));
set Q1 = Q +* (stop (while>0 (a,i,I)));
A47: I is_halting_on t,Q by A12, A37, A38, A39, A40, A41, A42;
 y1 - 1 >= 0 by A40, A46, XREAL_1:47, XREAL_1:48;
then reconsider y0 = y1 - 1 as Element of NAT by INT_1:3;
set jj = (2 * k1) + 1;
A48: (yn - y1) - 1 <= (k + 1) - 1 by A43, XREAL_1:9;
A49: yn <= y1 + (k + 1) by A43, XREAL_1:20;
consider f2 being FinSequence of INT  such that
A50: len f2 = n and
A51: for i being Element of NAT st 1 <= i & i <= len f2 holds
f2 . i = (Comput ((Q +* (stop (while>0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 2))) . (intpos (m + i)) by SCPISORT:1;
set It = IExec (I,Q,t);
A53: ( y1 < yn implies ( (IExec (I,Q,t)) . (DataLoc (c,i)) = (2 * k1) + 3 & ( for j being Element of NAT st ( ( 1 <= j & j < y1 ) or ( yn < j & j <= n ) ) holds
(IExec (I,Q,t)) . (intpos (m + j)) = t . (intpos (m + j)) ) & ex ym being Element of NAT st
( y1 <= ym & ym <= yn & m + y1 = (IExec (I,Q,t)) . (intpos k2) & (m + ym) - 1 = (IExec (I,Q,t)) . (intpos (k2 + 1)) & (m + ym) + 1 = (IExec (I,Q,t)) . (intpos (k2 + 2)) & m + yn = (IExec (I,Q,t)) . (intpos (k2 + 3)) & ( for j being Element of NAT st y1 <= j & j < ym holds
(IExec (I,Q,t)) . (intpos (m + j)) <= (IExec (I,Q,t)) . (intpos (m + ym)) ) & ( for j being Element of NAT st ym < j & j <= yn holds
(IExec (I,Q,t)) . (intpos (m + j)) >= (IExec (I,Q,t)) . (intpos (m + ym)) ) ) ) ) by A12, A37, A38, A39, A40, A41, A42;
then consider ym being Element of NAT  such that
A54: y1 <= ym and
A55: ym <= yn and
A56: m + y1 = (IExec (I,Q,t)) . (intpos k2) and
A57: (m + ym) - 1 = (IExec (I,Q,t)) . (intpos (k2 + 1)) and
A58: (m + ym) + 1 = (IExec (I,Q,t)) . (intpos (k2 + 2)) and
A59: m + yn = (IExec (I,Q,t)) . (intpos (k2 + 3)) and
A60: for j being Element of NAT st y1 <= j & j < ym holds
(IExec (I,Q,t)) . (intpos (m + j)) <= (IExec (I,Q,t)) . (intpos (m + ym)) and
A61: for j being Element of NAT st ym < j & j <= yn holds
(IExec (I,Q,t)) . (intpos (m + j)) >= (IExec (I,Q,t)) . (intpos (m + ym)) by A46, XREAL_1:47;
A62: I is_closed_on t,Q by A12, A37, A38, A39, A40, A41, A42;
then A63: DataPart (Comput ((Q +* (stop (while>0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 2))) = DataPart (IExec (I,Q,t)) by A37, A38, A47, Th2, T;
then A64: (Comput ((Q +* (stop (while>0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 2))) . (DataLoc (c,i)) = (2 * k1) + 3 by A46, A53, SCMPDS_4:8, XREAL_1:47;
A65: ym >= 1 by A40, A46, A54, XREAL_1:47, XXREAL_0:2;
then reconsider yc = ym - 1 as Element of NAT by INT_1:3, XREAL_1:48;
A66: yc <= yn by A55, XREAL_1:146, XXREAL_0:2;
then A67: yc <= n by A40, A46, XREAL_1:47, XXREAL_0:2;
A68: (Comput ((Q +* (stop (while>0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 2))) . (DataLoc (c,i)) = (2 * (k1 + 1)) + 1 by A46, A53, A63, SCMPDS_4:8, XREAL_1:47;
 (IExec (I,Q,t)) . a = t . a by A12, A37, A38, A39, A40, A41, A42;
then A69: (Comput ((Q +* (stop (while>0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 2))) . a = c by A37, A63, SCMPDS_4:8;
set k3 = ((m + n) + (2 * (k1 + 1))) + 1;
set yd = ym + 1;
A70: ym + 1 > ym by XREAL_1:29;
then A71: ym + 1 >= y1 by A54, XXREAL_0:2;
then A72: ym + 1 >= 1 by A40, A46, XREAL_1:47, XXREAL_0:2;
A73: (Comput ((Q +* (stop (while>0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 2))) . (intpos ((((m + n) + (2 * (k1 + 1))) + 1) + 1)) = m + yn by A39, A63, A59, SCMPDS_4:8;
 ym + (1 + k) >= y1 + (1 + k) by A54, XREAL_1:6;
then  yn <= (ym + 1) + k by A49, XXREAL_0:2;
then A74: yn - (ym + 1) <= k by XREAL_1:20;
AA: (Comput ((Q +* (stop (while>0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 2))) . (intpos (((m + n) + (2 * (k1 + 1))) + 1)) = m + (ym + 1) by A39, A63, A58, SCMPDS_4:8;
X1: (Initialize (Comput ((Q +* (stop (while>0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 2)))) . a = (Comput ((Q +* (stop (while>0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 2))) . a by SCMPDS_5:15;
X2: (Initialize (Comput ((Q +* (stop (while>0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 2)))) . (DataLoc (c,i)) = (Comput ((Q +* (stop (while>0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 2))) . (DataLoc (c,i)) by SCMPDS_5:15;
X3: (Initialize (Comput ((Q +* (stop (while>0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 2)))) . (intpos (((m + n) + (2 * (k1 + 1))) + 1)) = (Comput ((Q +* (stop (while>0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 2))) . (intpos (((m + n) + (2 * (k1 + 1))) + 1)) by SCMPDS_5:15;
X4: (Initialize (Comput ((Q +* (stop (while>0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 2)))) . (intpos ((((m + n) + (2 * (k1 + 1))) + 1) + 1)) = (Comput ((Q +* (stop (while>0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 2))) . (intpos ((((m + n) + (2 * (k1 + 1))) + 1) + 1)) by SCMPDS_5:15;
 f2 is_FinSequence_on Initialize (Comput ((Q +* (stop (while>0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 2))),m
proof
let i be Element of NAT ; :: according to SCPISORT:def_1 ::_thesis: ( not 1 <= i or not i <= len f2 or f2 . i = (Initialize (Comput ((Q +* (stop (while>0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 2)))) . (intpos (m + i)) )
assume  ( 1 <= i & i <= len f2 ) ; ::_thesis: f2 . i = (Initialize (Comput ((Q +* (stop (while>0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 2)))) . (intpos (m + i))
then  f2 . i = (Comput ((Q +* (stop (while>0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 2))) . (intpos (m + i)) by A51;
hence  f2 . i = (Initialize (Comput ((Q +* (stop (while>0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 2)))) . (intpos (m + i)) by SCMPDS_5:15; ::_thesis: verum
end;
then consider kl being Element of NAT , f3 being FinSequence of INT  such that
W1: Initialize (Comput (((Q +* (stop (while>0 (a,i,I)))) +* (stop (while>0 (a,i,I)))),(Initialize (Comput ((Q +* (stop (while>0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 2)))),kl)) = Comput (((Q +* (stop (while>0 (a,i,I)))) +* (stop (while>0 (a,i,I)))),(Initialize (Comput ((Q +* (stop (while>0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 2)))),kl) and
W2: f3 is_FinSequence_on Comput (((Q +* (stop (while>0 (a,i,I)))) +* (stop (while>0 (a,i,I)))),(Initialize (Comput ((Q +* (stop (while>0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 2)))),kl),m and
W3: len f3 = n and
W4: f2,f3 are_fiberwise_equipotent and
W5: f3 is_non_decreasing_on ym + 1,yn and
W6: for j being Element of NAT st ym + 1 < yn & ( ( 1 <= j & j < ym + 1 ) or ( yn < j & j <= n ) ) holds
f3 . j = (Initialize (Comput ((Q +* (stop (while>0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 2)))) . (intpos (m + j)) and
W7: for j being Element of NAT st ym + 1 >= yn & 1 <= j & j <= n holds
f3 . j = (Initialize (Comput ((Q +* (stop (while>0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 2)))) . (intpos (m + j)) and
W8: for j being Element of NAT st 1 <= j & j < (2 * (k1 + 1)) + 1 holds
(Comput (((Q +* (stop (while>0 (a,i,I)))) +* (stop (while>0 (a,i,I)))),(Initialize (Comput ((Q +* (stop (while>0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 2)))),kl)) . (intpos ((m + n) + j)) = (Initialize (Comput ((Q +* (stop (while>0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 2)))) . (intpos ((m + n) + j)) and
W9: (Comput (((Q +* (stop (while>0 (a,i,I)))) +* (stop (while>0 (a,i,I)))),(Initialize (Comput ((Q +* (stop (while>0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 2)))),kl)) . (DataLoc (c,i)) = ((Initialize (Comput ((Q +* (stop (while>0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 2)))) . (DataLoc (c,i))) - 2 and
W10: (Comput (((Q +* (stop (while>0 (a,i,I)))) +* (stop (while>0 (a,i,I)))),(Initialize (Comput ((Q +* (stop (while>0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 2)))),kl)) . a = c by A36, A72, A74, A50, A40, A46, A69, X1, A68, X2, AA, X3, A73, X4, XREAL_1:47;
A75: Initialize (Comput (((Q +* (stop (while>0 (a,i,I)))) +* (stop (while>0 (a,i,I)))),(Initialize (Comput ((Q +* (stop (while>0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 2)))),kl)) = Comput (((Q +* (stop (while>0 (a,i,I)))) +* (stop (while>0 (a,i,I)))),(Initialize (Comput ((Q +* (stop (while>0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 2)))),kl) by W1;
A76: f3 is_FinSequence_on Comput (((Q +* (stop (while>0 (a,i,I)))) +* (stop (while>0 (a,i,I)))),(Initialize (Comput ((Q +* (stop (while>0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 2)))),kl),m by W2;
A77: len f3 = n by W3;
A78: f2,f3 are_fiberwise_equipotent by W4;
A79: f3 is_non_decreasing_on ym + 1,yn by W5;
A80: for j being Element of NAT st ym + 1 < yn & ( ( 1 <= j & j < ym + 1 ) or ( yn < j & j <= n ) ) holds
f3 . j = (Comput ((Q +* (stop (while>0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 2))) . (intpos (m + j))
proof
let j be Element of NAT ; ::_thesis: ( ym + 1 < yn & ( ( 1 <= j & j < ym + 1 ) or ( yn < j & j <= n ) ) implies f3 . j = (Comput ((Q +* (stop (while>0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 2))) . (intpos (m + j)) )
assume  ( ym + 1 < yn & ( ( 1 <= j & j < ym + 1 ) or ( yn < j & j <= n ) ) ) ; ::_thesis: f3 . j = (Comput ((Q +* (stop (while>0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 2))) . (intpos (m + j))
then  f3 . j = (Initialize (Comput ((Q +* (stop (while>0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 2)))) . (intpos (m + j)) by W6;
hence  f3 . j = (Comput ((Q +* (stop (while>0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 2))) . (intpos (m + j)) by SCMPDS_5:15; ::_thesis: verum
end;
A81: for j being Element of NAT st ym + 1 >= yn & 1 <= j & j <= n holds
f3 . j = (Comput ((Q +* (stop (while>0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 2))) . (intpos (m + j))
proof
let j be Element of NAT ; ::_thesis: ( ym + 1 >= yn & 1 <= j & j <= n implies f3 . j = (Comput ((Q +* (stop (while>0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 2))) . (intpos (m + j)) )
assume  ( ym + 1 >= yn & 1 <= j & j <= n ) ; ::_thesis: f3 . j = (Comput ((Q +* (stop (while>0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 2))) . (intpos (m + j))
then  f3 . j = (Initialize (Comput ((Q +* (stop (while>0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 2)))) . (intpos (m + j)) by W7;
hence  f3 . j = (Comput ((Q +* (stop (while>0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 2))) . (intpos (m + j)) by SCMPDS_5:15; ::_thesis: verum
end;
A82: for j being Element of NAT st 1 <= j & j < (2 * (k1 + 1)) + 1 holds
(Comput (((Q +* (stop (while>0 (a,i,I)))) +* (stop (while>0 (a,i,I)))),(Initialize (Comput ((Q +* (stop (while>0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 2)))),kl)) . (intpos ((m + n) + j)) = (Comput ((Q +* (stop (while>0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 2))) . (intpos ((m + n) + j))
proof
let j be Element of NAT ; ::_thesis: ( 1 <= j & j < (2 * (k1 + 1)) + 1 implies (Comput (((Q +* (stop (while>0 (a,i,I)))) +* (stop (while>0 (a,i,I)))),(Initialize (Comput ((Q +* (stop (while>0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 2)))),kl)) . (intpos ((m + n) + j)) = (Comput ((Q +* (stop (while>0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 2))) . (intpos ((m + n) + j)) )
assume  ( 1 <= j & j < (2 * (k1 + 1)) + 1 ) ; ::_thesis: (Comput (((Q +* (stop (while>0 (a,i,I)))) +* (stop (while>0 (a,i,I)))),(Initialize (Comput ((Q +* (stop (while>0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 2)))),kl)) . (intpos ((m + n) + j)) = (Comput ((Q +* (stop (while>0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 2))) . (intpos ((m + n) + j))
then  (Comput (((Q +* (stop (while>0 (a,i,I)))) +* (stop (while>0 (a,i,I)))),(Initialize (Comput ((Q +* (stop (while>0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 2)))),kl)) . (intpos ((m + n) + j)) = (Initialize (Comput ((Q +* (stop (while>0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 2)))) . (intpos ((m + n) + j)) by W8;
hence  (Comput (((Q +* (stop (while>0 (a,i,I)))) +* (stop (while>0 (a,i,I)))),(Initialize (Comput ((Q +* (stop (while>0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 2)))),kl)) . (intpos ((m + n) + j)) = (Comput ((Q +* (stop (while>0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 2))) . (intpos ((m + n) + j)) by SCMPDS_5:15; ::_thesis: verum
end;
set t2 = Comput (((Q +* (stop (while>0 (a,i,I)))) +* (stop (while>0 (a,i,I)))),(Initialize (Comput ((Q +* (stop (while>0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 2)))),kl);
set Q2 = (Q +* (stop (while>0 (a,i,I)))) +* (stop (while>0 (a,i,I)));
A83: (Comput (((Q +* (stop (while>0 (a,i,I)))) +* (stop (while>0 (a,i,I)))),(Initialize (Comput ((Q +* (stop (while>0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 2)))),kl)) . (DataLoc (c,i)) = ((Comput ((Q +* (stop (while>0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 2))) . (DataLoc (c,i))) - 2 by W9, X2;
A84: (Comput (((Q +* (stop (while>0 (a,i,I)))) +* (stop (while>0 (a,i,I)))),(Initialize (Comput ((Q +* (stop (while>0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 2)))),kl)) . a = c by W10;
A86: (2 * k1) + 3 = (2 * (k1 + 1)) + 1 ;
then  (2 * k1) + 1 < (2 * (k1 + 1)) + 1 by XREAL_1:6;
then A87: (Comput (((Q +* (stop (while>0 (a,i,I)))) +* (stop (while>0 (a,i,I)))),(Initialize (Comput ((Q +* (stop (while>0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 2)))),kl)) . (intpos k2) = (Comput ((Q +* (stop (while>0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 2))) . (intpos ((m + n) + ((2 * k1) + 1))) by A39, A82, NAT_1:11
.= m + y1 by A39, A63, A56, SCMPDS_4:8 ;
A88: ym <= n by A40, A46, A55, XREAL_1:47, XXREAL_0:2;
A89: now__::_thesis:_f3_._ym_=_(Comput_((Q_+*_(stop_(while>0_(a,i,I)))),t,((LifeSpan_((Q_+*_(stop_I)),t))_+_2)))_._(intpos_(m_+_ym))
percases ( ym + 1 < yn or ym + 1 >= yn ) ;
suppose ym + 1 < yn ; ::_thesis: f3 . ym = (Comput ((Q +* (stop (while>0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 2))) . (intpos (m + ym))
hence  f3 . ym = (Comput ((Q +* (stop (while>0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 2))) . (intpos (m + ym)) by A70, A80, A65; ::_thesis: verum
end;
suppose ym + 1 >= yn ; ::_thesis: f3 . ym = (Comput ((Q +* (stop (while>0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 2))) . (intpos (m + ym))
hence  f3 . ym = (Comput ((Q +* (stop (while>0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 2))) . (intpos (m + ym)) by A81, A65, A88; ::_thesis: verum
end;
end;
end;
A90: now__::_thesis:_for_i_being_Element_of_NAT_st_1_<=_i_&_i_<=_ym_holds_
f3_._i_=_f2_._i
let i be Element of NAT ; ::_thesis: ( 1 <= i & i <= ym implies f3 . i = f2 . i )
assume that
A91: 1 <= i and
A92: i <= ym ; ::_thesis: f3 . i = f2 . i
A93: i <= n by A88, A92, XXREAL_0:2;
A94: i < ym + 1 by A70, A92, XXREAL_0:2;
now__::_thesis:_f3_._i_=_(Comput_((Q_+*_(stop_(while>0_(a,i,I)))),t,((LifeSpan_((Q_+*_(stop_I)),t))_+_2)))_._(intpos_(m_+_i))
percases ( ym + 1 < yn or ym + 1 >= yn ) ;
suppose ym + 1 < yn ; ::_thesis: f3 . i = (Comput ((Q +* (stop (while>0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 2))) . (intpos (m + i))
hence  f3 . i = (Comput ((Q +* (stop (while>0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 2))) . (intpos (m + i)) by A80, A91, A94; ::_thesis: verum
end;
suppose ym + 1 >= yn ; ::_thesis: f3 . i = (Comput ((Q +* (stop (while>0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 2))) . (intpos (m + i))
hence  f3 . i = (Comput ((Q +* (stop (while>0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 2))) . (intpos (m + i)) by A81, A91, A93; ::_thesis: verum
end;
end;
end;
hence  f3 . i = f2 . i by A50, A51, A91, A93; ::_thesis: verum
end;
 yc <= yn - 1 by A55, XREAL_1:9;
then  yc - y1 <= (yn - 1) - y1 by XREAL_1:9;
then A95: yc - y1 <= k by A48, XXREAL_0:2;
A96: yc < ym + 1 by A70, XREAL_1:146, XXREAL_0:2;
set jj = (2 * k1) + 2;
 (2 * k1) + 2 >= 2 by NAT_1:11;
then A97: (2 * k1) + 2 >= 1 by XXREAL_0:2;
 (2 * k1) + 2 < (2 * (k1 + 1)) + 1 by A86, XREAL_1:6;
then AA: (Comput (((Q +* (stop (while>0 (a,i,I)))) +* (stop (while>0 (a,i,I)))),(Initialize (Comput ((Q +* (stop (while>0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 2)))),kl)) . (intpos (k2 + 1)) = (Comput ((Q +* (stop (while>0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 2))) . (intpos ((m + n) + ((2 * k1) + 2))) by A39, A82, A97
.= m + yc by A39, A63, A57, SCMPDS_4:8 ;
YY: ( ( 1 <= y1 & yc <= n ) or y1 >= yc ) by A46, A67, A40, XREAL_1:47;
X1: (Initialize (Comput (((Q +* (stop (while>0 (a,i,I)))) +* (stop (while>0 (a,i,I)))),(Initialize (Comput ((Q +* (stop (while>0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 2)))),kl))) . a = (Comput (((Q +* (stop (while>0 (a,i,I)))) +* (stop (while>0 (a,i,I)))),(Initialize (Comput ((Q +* (stop (while>0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 2)))),kl)) . a by SCMPDS_5:15;
X2: (Initialize (Comput (((Q +* (stop (while>0 (a,i,I)))) +* (stop (while>0 (a,i,I)))),(Initialize (Comput ((Q +* (stop (while>0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 2)))),kl))) . (DataLoc (c,i)) = (Comput (((Q +* (stop (while>0 (a,i,I)))) +* (stop (while>0 (a,i,I)))),(Initialize (Comput ((Q +* (stop (while>0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 2)))),kl)) . (DataLoc (c,i)) by SCMPDS_5:15;
X3: (Initialize (Comput (((Q +* (stop (while>0 (a,i,I)))) +* (stop (while>0 (a,i,I)))),(Initialize (Comput ((Q +* (stop (while>0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 2)))),kl))) . (intpos k2) = (Comput (((Q +* (stop (while>0 (a,i,I)))) +* (stop (while>0 (a,i,I)))),(Initialize (Comput ((Q +* (stop (while>0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 2)))),kl)) . (intpos k2) by SCMPDS_5:15;
X4: (Initialize (Comput (((Q +* (stop (while>0 (a,i,I)))) +* (stop (while>0 (a,i,I)))),(Initialize (Comput ((Q +* (stop (while>0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 2)))),kl))) . (intpos (k2 + 1)) = (Comput (((Q +* (stop (while>0 (a,i,I)))) +* (stop (while>0 (a,i,I)))),(Initialize (Comput ((Q +* (stop (while>0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 2)))),kl)) . (intpos (k2 + 1)) by SCMPDS_5:15;
 f3 is_FinSequence_on Initialize (Comput (((Q +* (stop (while>0 (a,i,I)))) +* (stop (while>0 (a,i,I)))),(Initialize (Comput ((Q +* (stop (while>0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 2)))),kl)),m
proof
let i be Element of NAT ; :: according to SCPISORT:def_1 ::_thesis: ( not 1 <= i or not i <= len f3 or f3 . i = (Initialize (Comput (((Q +* (stop (while>0 (a,i,I)))) +* (stop (while>0 (a,i,I)))),(Initialize (Comput ((Q +* (stop (while>0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 2)))),kl))) . (intpos (m + i)) )
assume  ( 1 <= i & i <= len f3 ) ; ::_thesis: f3 . i = (Initialize (Comput (((Q +* (stop (while>0 (a,i,I)))) +* (stop (while>0 (a,i,I)))),(Initialize (Comput ((Q +* (stop (while>0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 2)))),kl))) . (intpos (m + i))
then  f3 . i = (Comput (((Q +* (stop (while>0 (a,i,I)))) +* (stop (while>0 (a,i,I)))),(Initialize (Comput ((Q +* (stop (while>0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 2)))),kl)) . (intpos (m + i)) by A76, SCPISORT:def_1;
hence  f3 . i = (Initialize (Comput (((Q +* (stop (while>0 (a,i,I)))) +* (stop (while>0 (a,i,I)))),(Initialize (Comput ((Q +* (stop (while>0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 2)))),kl))) . (intpos (m + i)) by SCMPDS_5:15; ::_thesis: verum
end;
then consider km being Element of NAT , f4 being FinSequence of INT  such that
W1: Initialize (Comput ((((Q +* (stop (while>0 (a,i,I)))) +* (stop (while>0 (a,i,I)))) +* (stop (while>0 (a,i,I)))),(Initialize (Comput (((Q +* (stop (while>0 (a,i,I)))) +* (stop (while>0 (a,i,I)))),(Initialize (Comput ((Q +* (stop (while>0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 2)))),kl))),km)) = Comput ((((Q +* (stop (while>0 (a,i,I)))) +* (stop (while>0 (a,i,I)))) +* (stop (while>0 (a,i,I)))),(Initialize (Comput (((Q +* (stop (while>0 (a,i,I)))) +* (stop (while>0 (a,i,I)))),(Initialize (Comput ((Q +* (stop (while>0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 2)))),kl))),km) and
W2: f4 is_FinSequence_on Comput ((((Q +* (stop (while>0 (a,i,I)))) +* (stop (while>0 (a,i,I)))) +* (stop (while>0 (a,i,I)))),(Initialize (Comput (((Q +* (stop (while>0 (a,i,I)))) +* (stop (while>0 (a,i,I)))),(Initialize (Comput ((Q +* (stop (while>0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 2)))),kl))),km),m and
W3: len f4 = n and
W4: f3,f4 are_fiberwise_equipotent and
W5: f4 is_non_decreasing_on y1,yc and
W6: for j being Element of NAT st y1 < yc & ( ( 1 <= j & j < y1 ) or ( yc < j & j <= n ) ) holds
f4 . j = (Initialize (Comput (((Q +* (stop (while>0 (a,i,I)))) +* (stop (while>0 (a,i,I)))),(Initialize (Comput ((Q +* (stop (while>0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 2)))),kl))) . (intpos (m + j)) and
W7: for j being Element of NAT st y1 >= yc & 1 <= j & j <= n holds
f4 . j = (Initialize (Comput (((Q +* (stop (while>0 (a,i,I)))) +* (stop (while>0 (a,i,I)))),(Initialize (Comput ((Q +* (stop (while>0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 2)))),kl))) . (intpos (m + j)) and
W8: for j being Element of NAT st 1 <= j & j < (2 * k1) + 1 holds
(Comput ((((Q +* (stop (while>0 (a,i,I)))) +* (stop (while>0 (a,i,I)))) +* (stop (while>0 (a,i,I)))),(Initialize (Comput (((Q +* (stop (while>0 (a,i,I)))) +* (stop (while>0 (a,i,I)))),(Initialize (Comput ((Q +* (stop (while>0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 2)))),kl))),km)) . (intpos ((m + n) + j)) = (Initialize (Comput (((Q +* (stop (while>0 (a,i,I)))) +* (stop (while>0 (a,i,I)))),(Initialize (Comput ((Q +* (stop (while>0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 2)))),kl))) . (intpos ((m + n) + j)) and
W9: (Comput ((((Q +* (stop (while>0 (a,i,I)))) +* (stop (while>0 (a,i,I)))) +* (stop (while>0 (a,i,I)))),(Initialize (Comput (((Q +* (stop (while>0 (a,i,I)))) +* (stop (while>0 (a,i,I)))),(Initialize (Comput ((Q +* (stop (while>0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 2)))),kl))),km)) . (DataLoc (c,i)) = ((Initialize (Comput (((Q +* (stop (while>0 (a,i,I)))) +* (stop (while>0 (a,i,I)))),(Initialize (Comput ((Q +* (stop (while>0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 2)))),kl))) . (DataLoc (c,i))) - 2 and
W10: (Comput ((((Q +* (stop (while>0 (a,i,I)))) +* (stop (while>0 (a,i,I)))) +* (stop (while>0 (a,i,I)))),(Initialize (Comput (((Q +* (stop (while>0 (a,i,I)))) +* (stop (while>0 (a,i,I)))),(Initialize (Comput ((Q +* (stop (while>0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 2)))),kl))),km)) . a = c by A36, A95, A77, YY, A39, X1, A84, X2, A64, A83, X3, A87, X4, AA;
A98: Initialize (Comput ((((Q +* (stop (while>0 (a,i,I)))) +* (stop (while>0 (a,i,I)))) +* (stop (while>0 (a,i,I)))),(Initialize (Comput (((Q +* (stop (while>0 (a,i,I)))) +* (stop (while>0 (a,i,I)))),(Initialize (Comput ((Q +* (stop (while>0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 2)))),kl))),km)) = Comput ((((Q +* (stop (while>0 (a,i,I)))) +* (stop (while>0 (a,i,I)))) +* (stop (while>0 (a,i,I)))),(Initialize (Comput (((Q +* (stop (while>0 (a,i,I)))) +* (stop (while>0 (a,i,I)))),(Initialize (Comput ((Q +* (stop (while>0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 2)))),kl))),km) by W1;
A99: f4 is_FinSequence_on Comput ((((Q +* (stop (while>0 (a,i,I)))) +* (stop (while>0 (a,i,I)))) +* (stop (while>0 (a,i,I)))),(Initialize (Comput (((Q +* (stop (while>0 (a,i,I)))) +* (stop (while>0 (a,i,I)))),(Initialize (Comput ((Q +* (stop (while>0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 2)))),kl))),km),m by W2;
A100: len f4 = n by W3;
A101: f3,f4 are_fiberwise_equipotent by W4;
A102: f4 is_non_decreasing_on y1,yc by W5;
A103: for j being Element of NAT st y1 < yc & ( ( 1 <= j & j < y1 ) or ( yc < j & j <= n ) ) holds
f4 . j = (Comput (((Q +* (stop (while>0 (a,i,I)))) +* (stop (while>0 (a,i,I)))),(Initialize (Comput ((Q +* (stop (while>0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 2)))),kl)) . (intpos (m + j))
proof
let j be Element of NAT ; ::_thesis: ( y1 < yc & ( ( 1 <= j & j < y1 ) or ( yc < j & j <= n ) ) implies f4 . j = (Comput (((Q +* (stop (while>0 (a,i,I)))) +* (stop (while>0 (a,i,I)))),(Initialize (Comput ((Q +* (stop (while>0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 2)))),kl)) . (intpos (m + j)) )
assume  ( y1 < yc & ( ( 1 <= j & j < y1 ) or ( yc < j & j <= n ) ) ) ; ::_thesis: f4 . j = (Comput (((Q +* (stop (while>0 (a,i,I)))) +* (stop (while>0 (a,i,I)))),(Initialize (Comput ((Q +* (stop (while>0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 2)))),kl)) . (intpos (m + j))
then  f4 . j = (Initialize (Comput (((Q +* (stop (while>0 (a,i,I)))) +* (stop (while>0 (a,i,I)))),(Initialize (Comput ((Q +* (stop (while>0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 2)))),kl))) . (intpos (m + j)) by W6;
hence  f4 . j = (Comput (((Q +* (stop (while>0 (a,i,I)))) +* (stop (while>0 (a,i,I)))),(Initialize (Comput ((Q +* (stop (while>0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 2)))),kl)) . (intpos (m + j)) by SCMPDS_5:15; ::_thesis: verum
end;
A104: for j being Element of NAT st y1 >= yc & 1 <= j & j <= n holds
f4 . j = (Comput (((Q +* (stop (while>0 (a,i,I)))) +* (stop (while>0 (a,i,I)))),(Initialize (Comput ((Q +* (stop (while>0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 2)))),kl)) . (intpos (m + j))
proof
let j be Element of NAT ; ::_thesis: ( y1 >= yc & 1 <= j & j <= n implies f4 . j = (Comput (((Q +* (stop (while>0 (a,i,I)))) +* (stop (while>0 (a,i,I)))),(Initialize (Comput ((Q +* (stop (while>0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 2)))),kl)) . (intpos (m + j)) )
assume  ( y1 >= yc & 1 <= j & j <= n ) ; ::_thesis: f4 . j = (Comput (((Q +* (stop (while>0 (a,i,I)))) +* (stop (while>0 (a,i,I)))),(Initialize (Comput ((Q +* (stop (while>0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 2)))),kl)) . (intpos (m + j))
then  f4 . j = (Initialize (Comput (((Q +* (stop (while>0 (a,i,I)))) +* (stop (while>0 (a,i,I)))),(Initialize (Comput ((Q +* (stop (while>0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 2)))),kl))) . (intpos (m + j)) by W7;
hence  f4 . j = (Comput (((Q +* (stop (while>0 (a,i,I)))) +* (stop (while>0 (a,i,I)))),(Initialize (Comput ((Q +* (stop (while>0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 2)))),kl)) . (intpos (m + j)) by SCMPDS_5:15; ::_thesis: verum
end;
A105: for j being Element of NAT st 1 <= j & j < (2 * k1) + 1 holds
(Comput ((((Q +* (stop (while>0 (a,i,I)))) +* (stop (while>0 (a,i,I)))) +* (stop (while>0 (a,i,I)))),(Initialize (Comput (((Q +* (stop (while>0 (a,i,I)))) +* (stop (while>0 (a,i,I)))),(Initialize (Comput ((Q +* (stop (while>0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 2)))),kl))),km)) . (intpos ((m + n) + j)) = (Comput (((Q +* (stop (while>0 (a,i,I)))) +* (stop (while>0 (a,i,I)))),(Initialize (Comput ((Q +* (stop (while>0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 2)))),kl)) . (intpos ((m + n) + j))
proof
let j be Element of NAT ; ::_thesis: ( 1 <= j & j < (2 * k1) + 1 implies (Comput ((((Q +* (stop (while>0 (a,i,I)))) +* (stop (while>0 (a,i,I)))) +* (stop (while>0 (a,i,I)))),(Initialize (Comput (((Q +* (stop (while>0 (a,i,I)))) +* (stop (while>0 (a,i,I)))),(Initialize (Comput ((Q +* (stop (while>0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 2)))),kl))),km)) . (intpos ((m + n) + j)) = (Comput (((Q +* (stop (while>0 (a,i,I)))) +* (stop (while>0 (a,i,I)))),(Initialize (Comput ((Q +* (stop (while>0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 2)))),kl)) . (intpos ((m + n) + j)) )
assume  ( 1 <= j & j < (2 * k1) + 1 ) ; ::_thesis: (Comput ((((Q +* (stop (while>0 (a,i,I)))) +* (stop (while>0 (a,i,I)))) +* (stop (while>0 (a,i,I)))),(Initialize (Comput (((Q +* (stop (while>0 (a,i,I)))) +* (stop (while>0 (a,i,I)))),(Initialize (Comput ((Q +* (stop (while>0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 2)))),kl))),km)) . (intpos ((m + n) + j)) = (Comput (((Q +* (stop (while>0 (a,i,I)))) +* (stop (while>0 (a,i,I)))),(Initialize (Comput ((Q +* (stop (while>0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 2)))),kl)) . (intpos ((m + n) + j))
then  (Comput ((((Q +* (stop (while>0 (a,i,I)))) +* (stop (while>0 (a,i,I)))) +* (stop (while>0 (a,i,I)))),(Initialize (Comput (((Q +* (stop (while>0 (a,i,I)))) +* (stop (while>0 (a,i,I)))),(Initialize (Comput ((Q +* (stop (while>0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 2)))),kl))),km)) . (intpos ((m + n) + j)) = (Initialize (Comput (((Q +* (stop (while>0 (a,i,I)))) +* (stop (while>0 (a,i,I)))),(Initialize (Comput ((Q +* (stop (while>0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 2)))),kl))) . (intpos ((m + n) + j)) by W8;
hence  (Comput ((((Q +* (stop (while>0 (a,i,I)))) +* (stop (while>0 (a,i,I)))) +* (stop (while>0 (a,i,I)))),(Initialize (Comput (((Q +* (stop (while>0 (a,i,I)))) +* (stop (while>0 (a,i,I)))),(Initialize (Comput ((Q +* (stop (while>0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 2)))),kl))),km)) . (intpos ((m + n) + j)) = (Comput (((Q +* (stop (while>0 (a,i,I)))) +* (stop (while>0 (a,i,I)))),(Initialize (Comput ((Q +* (stop (while>0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 2)))),kl)) . (intpos ((m + n) + j)) by SCMPDS_5:15; ::_thesis: verum
end;
A106: (Comput ((((Q +* (stop (while>0 (a,i,I)))) +* (stop (while>0 (a,i,I)))) +* (stop (while>0 (a,i,I)))),(Initialize (Comput (((Q +* (stop (while>0 (a,i,I)))) +* (stop (while>0 (a,i,I)))),(Initialize (Comput ((Q +* (stop (while>0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 2)))),kl))),km)) . (DataLoc (c,i)) = ((Comput (((Q +* (stop (while>0 (a,i,I)))) +* (stop (while>0 (a,i,I)))),(Initialize (Comput ((Q +* (stop (while>0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 2)))),kl)) . (DataLoc (c,i))) - 2 by W9, X2;
A107: (Comput ((((Q +* (stop (while>0 (a,i,I)))) +* (stop (while>0 (a,i,I)))) +* (stop (while>0 (a,i,I)))),(Initialize (Comput (((Q +* (stop (while>0 (a,i,I)))) +* (stop (while>0 (a,i,I)))),(Initialize (Comput ((Q +* (stop (while>0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 2)))),kl))),km)) . a = c by W10;
A108: now__::_thesis:_for_i_being_Element_of_NAT_st_yc_<_i_&_i_<=_len_f4_holds_
f4_._i_=_f3_._i
let i be Element of NAT ; ::_thesis: ( yc < i & i <= len f4 implies f4 . i = f3 . i )
assume that
A109: yc < i and
A110: i <= len f4 ; ::_thesis: f4 . i = f3 . i
A111: 1 + 0 <= i by A109, INT_1:7;
now__::_thesis:_f4_._i_=_(Comput_(((Q_+*_(stop_(while>0_(a,i,I))))_+*_(stop_(while>0_(a,i,I)))),(Initialize_(Comput_((Q_+*_(stop_(while>0_(a,i,I)))),t,((LifeSpan_((Q_+*_(stop_I)),t))_+_2)))),kl))_._(intpos_(m_+_i))
percases ( y1 < yc or y1 >= yc ) ;
suppose y1 < yc ; ::_thesis: f4 . i = (Comput (((Q +* (stop (while>0 (a,i,I)))) +* (stop (while>0 (a,i,I)))),(Initialize (Comput ((Q +* (stop (while>0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 2)))),kl)) . (intpos (m + i))
hence  f4 . i = (Comput (((Q +* (stop (while>0 (a,i,I)))) +* (stop (while>0 (a,i,I)))),(Initialize (Comput ((Q +* (stop (while>0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 2)))),kl)) . (intpos (m + i)) by A100, A103, A109, A110; ::_thesis: verum
end;
suppose y1 >= yc ; ::_thesis: f4 . i = (Comput (((Q +* (stop (while>0 (a,i,I)))) +* (stop (while>0 (a,i,I)))),(Initialize (Comput ((Q +* (stop (while>0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 2)))),kl)) . (intpos (m + i))
hence  f4 . i = (Comput (((Q +* (stop (while>0 (a,i,I)))) +* (stop (while>0 (a,i,I)))),(Initialize (Comput ((Q +* (stop (while>0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 2)))),kl)) . (intpos (m + i)) by A100, A104, A110, A111; ::_thesis: verum
end;
end;
end;
hence  f4 . i = f3 . i by A76, A77, A100, A110, A111, SCPISORT:def_1; ::_thesis: verum
end;
then  f4 . ym = f3 . ym by A100, A88, XREAL_1:146;
then A112: f4 . ym = (IExec (I,Q,t)) . (intpos (m + ym)) by A63, A89, SCMPDS_4:8;
A113: now__::_thesis:_for_i_being_Element_of_NAT_st_yn_<_i_&_i_<=_len_f3_holds_
f3_._i_=_f2_._i
let i be Element of NAT ; ::_thesis: ( yn < i & i <= len f3 implies f3 . i = f2 . i )
assume that
A114: yn < i and
A115: i <= len f3 ; ::_thesis: f3 . i = f2 . i
A116: 1 + 0 <= i by A114, INT_1:7;
now__::_thesis:_f3_._i_=_(Comput_((Q_+*_(stop_(while>0_(a,i,I)))),t,((LifeSpan_((Q_+*_(stop_I)),t))_+_2)))_._(intpos_(m_+_i))
percases ( ym + 1 < yn or ym + 1 >= yn ) ;
suppose ym + 1 < yn ; ::_thesis: f3 . i = (Comput ((Q +* (stop (while>0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 2))) . (intpos (m + i))
hence  f3 . i = (Comput ((Q +* (stop (while>0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 2))) . (intpos (m + i)) by A77, A80, A114, A115; ::_thesis: verum
end;
suppose ym + 1 >= yn ; ::_thesis: f3 . i = (Comput ((Q +* (stop (while>0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 2))) . (intpos (m + i))
hence  f3 . i = (Comput ((Q +* (stop (while>0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 2))) . (intpos (m + i)) by A77, A81, A115, A116; ::_thesis: verum
end;
end;
end;
hence  f3 . i = f2 . i by A50, A51, A77, A115, A116; ::_thesis: verum
end;
A117: now__::_thesis:_for_i_being_Element_of_NAT_st_ym_<_i_&_i_<=_yn_holds_
f4_._ym_<=_f4_._i
let i be Element of NAT ; ::_thesis: ( ym < i & i <= yn implies f4 . ym <= f4 . i )
assume that
A118: ym < i and
A119: i <= yn ; ::_thesis: f4 . ym <= f4 . i
consider j being Element of NAT  such that
A120: ym < j and
A121: j <= yn and
A122: f3 . i = f2 . j by A40, A46, A55, A77, A78, A90, A113, A118, A119, RFINSEQ:32, XREAL_1:47;
A123: yc < i by A118, XREAL_1:146, XXREAL_0:2;
A124: 1 <= j by A65, A120, XXREAL_0:2;
A125: j <= len f2 by A40, A46, A50, A121, XREAL_1:47, XXREAL_0:2;
 i <= len f4 by A40, A46, A100, A119, XREAL_1:47, XXREAL_0:2;
then f4 . i = f2 . j by A108, A123, A122
.= (Comput ((Q +* (stop (while>0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 2))) . (intpos (m + j)) by A51, A124, A125
.= (IExec (I,Q,t)) . (intpos (m + j)) by A63, SCMPDS_4:8 ;
hence  f4 . ym <= f4 . i by A61, A112, A120, A121; ::_thesis: verum
end;
A126: yn > y1 by A46, XREAL_1:47;
A127: now__::_thesis:_for_i_being_Element_of_NAT_st_1_<=_i_&_i_<=_y0_holds_
f4_._i_=_f3_._i
let i be Element of NAT ; ::_thesis: ( 1 <= i & i <= y0 implies f4 . i = f3 . i )
assume that
A128: 1 <= i and
A129: i <= y0 ; ::_thesis: f4 . i = f3 . i
 i - 1 < y1 - 1 by A129, XREAL_1:146, XXREAL_0:2;
then A130: i < y1 by XREAL_1:9;
 y1 <= n by A40, A126, XXREAL_0:2;
then A131: i <= n by A130, XXREAL_0:2;
now__::_thesis:_f4_._i_=_(Comput_(((Q_+*_(stop_(while>0_(a,i,I))))_+*_(stop_(while>0_(a,i,I)))),(Initialize_(Comput_((Q_+*_(stop_(while>0_(a,i,I)))),t,((LifeSpan_((Q_+*_(stop_I)),t))_+_2)))),kl))_._(intpos_(m_+_i))
percases ( y1 < yc or y1 >= yc ) ;
suppose y1 < yc ; ::_thesis: f4 . i = (Comput (((Q +* (stop (while>0 (a,i,I)))) +* (stop (while>0 (a,i,I)))),(Initialize (Comput ((Q +* (stop (while>0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 2)))),kl)) . (intpos (m + i))
hence  f4 . i = (Comput (((Q +* (stop (while>0 (a,i,I)))) +* (stop (while>0 (a,i,I)))),(Initialize (Comput ((Q +* (stop (while>0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 2)))),kl)) . (intpos (m + i)) by A103, A128, A130; ::_thesis: verum
end;
suppose y1 >= yc ; ::_thesis: f4 . i = (Comput (((Q +* (stop (while>0 (a,i,I)))) +* (stop (while>0 (a,i,I)))),(Initialize (Comput ((Q +* (stop (while>0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 2)))),kl)) . (intpos (m + i))
hence  f4 . i = (Comput (((Q +* (stop (while>0 (a,i,I)))) +* (stop (while>0 (a,i,I)))),(Initialize (Comput ((Q +* (stop (while>0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 2)))),kl)) . (intpos (m + i)) by A104, A128, A131; ::_thesis: verum
end;
end;
end;
hence  f4 . i = f3 . i by A76, A77, A128, A131, SCPISORT:def_1; ::_thesis: verum
end;
A132: y0 <= yc by A54, XREAL_1:9;
A133: now__::_thesis:_for_i_being_Element_of_NAT_st_y1_<=_i_&_i_<_ym_holds_
f4_._i_<=_f4_._ym
let i be Element of NAT ; ::_thesis: ( y1 <= i & i < ym implies f4 . i <= f4 . ym )
assume that
A134: y1 <= i and
A135: i < ym ; ::_thesis: f4 . i <= f4 . ym
 i + 1 <= ym by A135, INT_1:7;
then A136: i <= yc by XREAL_1:19;
 y0 < i by A134, XREAL_1:146, XXREAL_0:2;
then consider j being Element of NAT  such that
A137: y0 < j and
A138: j <= yc and
A139: f4 . i = f3 . j by A67, A100, A101, A132, A127, A108, A136, RFINSEQ:32;
A140: 1 + 0 <= j by A137, INT_1:7;
A141: j <= n by A67, A138, XXREAL_0:2;
A142: j < ym + 1 by A96, A138, XXREAL_0:2;
now__::_thesis:_f3_._j_=_(Comput_((Q_+*_(stop_(while>0_(a,i,I)))),t,((LifeSpan_((Q_+*_(stop_I)),t))_+_2)))_._(intpos_(m_+_j))
percases ( ym + 1 < yn or ym + 1 >= yn ) ;
suppose ym + 1 < yn ; ::_thesis: f3 . j = (Comput ((Q +* (stop (while>0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 2))) . (intpos (m + j))
hence  f3 . j = (Comput ((Q +* (stop (while>0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 2))) . (intpos (m + j)) by A80, A140, A142; ::_thesis: verum
end;
suppose ym + 1 >= yn ; ::_thesis: f3 . j = (Comput ((Q +* (stop (while>0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 2))) . (intpos (m + j))
hence  f3 . j = (Comput ((Q +* (stop (while>0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 2))) . (intpos (m + j)) by A81, A140, A141; ::_thesis: verum
end;
end;
end;
then A143: f4 . i = (IExec (I,Q,t)) . (intpos (m + j)) by A63, A139, SCMPDS_4:8;
A144: j < ym by A138, XREAL_1:146, XXREAL_0:2;
 (y1 - 1) + 1 <= j by A137, INT_1:7;
hence  f4 . i <= f4 . ym by A60, A112, A144, A143; ::_thesis: verum
end;
take mm = ((LifeSpan ((Q +* (stop I)),t)) + 2) + (kl + km); ::_thesis: ex f2 being FinSequence of INT st
( Initialize (Comput ((Q +* (stop (while>0 (a,i,I)))),t,mm)) = Comput ((Q +* (stop (while>0 (a,i,I)))),t,mm) & f2 is_FinSequence_on Comput ((Q +* (stop (while>0 (a,i,I)))),t,mm),m & len f2 = n & f1,f2 are_fiberwise_equipotent & f2 is_non_decreasing_on y1,yn & ( for j being Element of NAT st y1 < yn & ( ( 1 <= j & j < y1 ) or ( yn < j & j <= n ) ) holds
f2 . j = t . (intpos (m + j)) ) & ( for j being Element of NAT st y1 >= yn & 1 <= j & j <= n holds
f2 . j = t . (intpos (m + j)) ) & ( for j being Element of NAT st 1 <= j & j < (2 * k1) + 1 holds
(Comput ((Q +* (stop (while>0 (a,i,I)))),t,mm)) . (intpos ((m + n) + j)) = t . (intpos ((m + n) + j)) ) & (Comput ((Q +* (stop (while>0 (a,i,I)))),t,mm)) . (DataLoc (c,i)) = (t . (DataLoc (c,i))) - 2 & (Comput ((Q +* (stop (while>0 (a,i,I)))),t,mm)) . a = c )
set tm = Comput ((Q +* (stop (while>0 (a,i,I)))),t,mm);
take  f4 ; ::_thesis: ( Initialize (Comput ((Q +* (stop (while>0 (a,i,I)))),t,mm)) = Comput ((Q +* (stop (while>0 (a,i,I)))),t,mm) & f4 is_FinSequence_on Comput ((Q +* (stop (while>0 (a,i,I)))),t,mm),m & len f4 = n & f1,f4 are_fiberwise_equipotent & f4 is_non_decreasing_on y1,yn & ( for j being Element of NAT st y1 < yn & ( ( 1 <= j & j < y1 ) or ( yn < j & j <= n ) ) holds
f4 . j = t . (intpos (m + j)) ) & ( for j being Element of NAT st y1 >= yn & 1 <= j & j <= n holds
f4 . j = t . (intpos (m + j)) ) & ( for j being Element of NAT st 1 <= j & j < (2 * k1) + 1 holds
(Comput ((Q +* (stop (while>0 (a,i,I)))),t,mm)) . (intpos ((m + n) + j)) = t . (intpos ((m + n) + j)) ) & (Comput ((Q +* (stop (while>0 (a,i,I)))),t,mm)) . (DataLoc (c,i)) = (t . (DataLoc (c,i))) - 2 & (Comput ((Q +* (stop (while>0 (a,i,I)))),t,mm)) . a = c )
 Initialize (Comput ((Q +* (stop (while>0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 2))) = Comput ((Q +* (stop (while>0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 2)) by A37, A38, A62, A47, Th2, T;
then A146: Comput ((Q +* (stop (while>0 (a,i,I)))),t,mm) = Comput ((Q +* (stop (while>0 (a,i,I)))),(Initialize (Comput ((Q +* (stop (while>0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 2)))),(kl + km)) by EXTPRO_1:4
.= Comput (((Q +* (stop (while>0 (a,i,I)))) +* (stop (while>0 (a,i,I)))),(Initialize (Comput ((Q +* (stop (while>0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 2)))),(kl + km))
.= Comput ((((Q +* (stop (while>0 (a,i,I)))) +* (stop (while>0 (a,i,I)))) +* (stop (while>0 (a,i,I)))),(Initialize (Comput (((Q +* (stop (while>0 (a,i,I)))) +* (stop (while>0 (a,i,I)))),(Initialize (Comput ((Q +* (stop (while>0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 2)))),kl))),km) by A75, EXTPRO_1:4 ;
hence  Initialize (Comput ((Q +* (stop (while>0 (a,i,I)))),t,mm)) = Comput ((Q +* (stop (while>0 (a,i,I)))),t,mm) by A98; ::_thesis: ( f4 is_FinSequence_on Comput ((Q +* (stop (while>0 (a,i,I)))),t,mm),m & len f4 = n & f1,f4 are_fiberwise_equipotent & f4 is_non_decreasing_on y1,yn & ( for j being Element of NAT st y1 < yn & ( ( 1 <= j & j < y1 ) or ( yn < j & j <= n ) ) holds
f4 . j = t . (intpos (m + j)) ) & ( for j being Element of NAT st y1 >= yn & 1 <= j & j <= n holds
f4 . j = t . (intpos (m + j)) ) & ( for j being Element of NAT st 1 <= j & j < (2 * k1) + 1 holds
(Comput ((Q +* (stop (while>0 (a,i,I)))),t,mm)) . (intpos ((m + n) + j)) = t . (intpos ((m + n) + j)) ) & (Comput ((Q +* (stop (while>0 (a,i,I)))),t,mm)) . (DataLoc (c,i)) = (t . (DataLoc (c,i))) - 2 & (Comput ((Q +* (stop (while>0 (a,i,I)))),t,mm)) . a = c )
thus  f4 is_FinSequence_on Comput ((Q +* (stop (while>0 (a,i,I)))),t,mm),m by A99, A146; ::_thesis: ( len f4 = n & f1,f4 are_fiberwise_equipotent & f4 is_non_decreasing_on y1,yn & ( for j being Element of NAT st y1 < yn & ( ( 1 <= j & j < y1 ) or ( yn < j & j <= n ) ) holds
f4 . j = t . (intpos (m + j)) ) & ( for j being Element of NAT st y1 >= yn & 1 <= j & j <= n holds
f4 . j = t . (intpos (m + j)) ) & ( for j being Element of NAT st 1 <= j & j < (2 * k1) + 1 holds
(Comput ((Q +* (stop (while>0 (a,i,I)))),t,mm)) . (intpos ((m + n) + j)) = t . (intpos ((m + n) + j)) ) & (Comput ((Q +* (stop (while>0 (a,i,I)))),t,mm)) . (DataLoc (c,i)) = (t . (DataLoc (c,i))) - 2 & (Comput ((Q +* (stop (while>0 (a,i,I)))),t,mm)) . a = c )
thus  len f4 = n by A100; ::_thesis: ( f1,f4 are_fiberwise_equipotent & f4 is_non_decreasing_on y1,yn & ( for j being Element of NAT st y1 < yn & ( ( 1 <= j & j < y1 ) or ( yn < j & j <= n ) ) holds
f4 . j = t . (intpos (m + j)) ) & ( for j being Element of NAT st y1 >= yn & 1 <= j & j <= n holds
f4 . j = t . (intpos (m + j)) ) & ( for j being Element of NAT st 1 <= j & j < (2 * k1) + 1 holds
(Comput ((Q +* (stop (while>0 (a,i,I)))),t,mm)) . (intpos ((m + n) + j)) = t . (intpos ((m + n) + j)) ) & (Comput ((Q +* (stop (while>0 (a,i,I)))),t,mm)) . (DataLoc (c,i)) = (t . (DataLoc (c,i))) - 2 & (Comput ((Q +* (stop (while>0 (a,i,I)))),t,mm)) . a = c )
now__::_thesis:_for_i_being_Element_of_NAT_st_1_<=_i_&_i_<=_len_f2_holds_
f2_._i_=_(IExec_(I,Q,t))_._(intpos_(m_+_i))
let i be Element of NAT ; ::_thesis: ( 1 <= i & i <= len f2 implies f2 . i = (IExec (I,Q,t)) . (intpos (m + i)) )
assume that
A147: 1 <= i and
A148: i <= len f2 ; ::_thesis: f2 . i = (IExec (I,Q,t)) . (intpos (m + i))
thus f2 . i = (Comput ((Q +* (stop (while>0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 2))) . (intpos (m + i)) by A51, A147, A148
.= (IExec (I,Q,t)) . (intpos (m + i)) by A63, SCMPDS_4:8 ; ::_thesis: verum
end;
then  f2 is_FinSequence_on IExec (I,Q,t),m by SCPISORT:def_1;
then  f1,f2 are_fiberwise_equipotent by A12, A37, A38, A39, A40, A41, A42, A44, A45, A50;
then  f1,f3 are_fiberwise_equipotent by A78, CLASSES1:76;
hence  f1,f4 are_fiberwise_equipotent by A101, CLASSES1:76; ::_thesis: ( f4 is_non_decreasing_on y1,yn & ( for j being Element of NAT st y1 < yn & ( ( 1 <= j & j < y1 ) or ( yn < j & j <= n ) ) holds
f4 . j = t . (intpos (m + j)) ) & ( for j being Element of NAT st y1 >= yn & 1 <= j & j <= n holds
f4 . j = t . (intpos (m + j)) ) & ( for j being Element of NAT st 1 <= j & j < (2 * k1) + 1 holds
(Comput ((Q +* (stop (while>0 (a,i,I)))),t,mm)) . (intpos ((m + n) + j)) = t . (intpos ((m + n) + j)) ) & (Comput ((Q +* (stop (while>0 (a,i,I)))),t,mm)) . (DataLoc (c,i)) = (t . (DataLoc (c,i))) - 2 & (Comput ((Q +* (stop (while>0 (a,i,I)))),t,mm)) . a = c )
A149: now__::_thesis:_for_j_being_Element_of_NAT_st_ym_+_1_<=_j_&_j_<=_yn_holds_
f4_._j_=_f3_._j
let j be Element of NAT ; ::_thesis: ( ym + 1 <= j & j <= yn implies f4 . j = f3 . j )
assume that
A150: ym + 1 <= j and
A151: j <= yn ; ::_thesis: f4 . j = f3 . j
A152: 1 <= j by A72, A150, XXREAL_0:2;
A153: j <= n by A40, A46, A151, XREAL_1:47, XXREAL_0:2;
A154: yc < j by A96, A150, XXREAL_0:2;
now__::_thesis:_f4_._j_=_(Comput_(((Q_+*_(stop_(while>0_(a,i,I))))_+*_(stop_(while>0_(a,i,I)))),(Initialize_(Comput_((Q_+*_(stop_(while>0_(a,i,I)))),t,((LifeSpan_((Q_+*_(stop_I)),t))_+_2)))),kl))_._(intpos_(m_+_j))
percases ( y1 < yc or y1 >= yc ) ;
suppose y1 < yc ; ::_thesis: f4 . j = (Comput (((Q +* (stop (while>0 (a,i,I)))) +* (stop (while>0 (a,i,I)))),(Initialize (Comput ((Q +* (stop (while>0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 2)))),kl)) . (intpos (m + j))
hence  f4 . j = (Comput (((Q +* (stop (while>0 (a,i,I)))) +* (stop (while>0 (a,i,I)))),(Initialize (Comput ((Q +* (stop (while>0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 2)))),kl)) . (intpos (m + j)) by A103, A153, A154; ::_thesis: verum
end;
suppose y1 >= yc ; ::_thesis: f4 . j = (Comput (((Q +* (stop (while>0 (a,i,I)))) +* (stop (while>0 (a,i,I)))),(Initialize (Comput ((Q +* (stop (while>0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 2)))),kl)) . (intpos (m + j))
hence  f4 . j = (Comput (((Q +* (stop (while>0 (a,i,I)))) +* (stop (while>0 (a,i,I)))),(Initialize (Comput ((Q +* (stop (while>0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 2)))),kl)) . (intpos (m + j)) by A104, A153, A152; ::_thesis: verum
end;
end;
end;
hence  f4 . j = f3 . j by A76, A77, A153, A152, SCPISORT:def_1; ::_thesis: verum
end;
now__::_thesis:_for_i,_j_being_Element_of_NAT_st_ym_+_1_<=_i_&_i_<=_j_&_j_<=_yn_holds_
f4_._i_<=_f4_._j
let i, j be Element of NAT ; ::_thesis: ( ym + 1 <= i & i <= j & j <= yn implies f4 . i <= f4 . j )
assume that
A155: ym + 1 <= i and
A156: i <= j and
A157: j <= yn ; ::_thesis: f4 . i <= f4 . j
 ym + 1 <= j by A155, A156, XXREAL_0:2;
then A158: f4 . j = f3 . j by A149, A157;
 i <= yn by A156, A157, XXREAL_0:2;
then  f4 . i = f3 . i by A149, A155;
hence  f4 . i <= f4 . j by A79, A155, A156, A157, A158, GRAPH_2:def_12; ::_thesis: verum
end;
then  f4 is_non_decreasing_on ym + 1,yn by GRAPH_2:def_12;
hence  f4 is_non_decreasing_on y1,yn by A102, A133, A117, Th8; ::_thesis: ( ( for j being Element of NAT st y1 < yn & ( ( 1 <= j & j < y1 ) or ( yn < j & j <= n ) ) holds
f4 . j = t . (intpos (m + j)) ) & ( for j being Element of NAT st y1 >= yn & 1 <= j & j <= n holds
f4 . j = t . (intpos (m + j)) ) & ( for j being Element of NAT st 1 <= j & j < (2 * k1) + 1 holds
(Comput ((Q +* (stop (while>0 (a,i,I)))),t,mm)) . (intpos ((m + n) + j)) = t . (intpos ((m + n) + j)) ) & (Comput ((Q +* (stop (while>0 (a,i,I)))),t,mm)) . (DataLoc (c,i)) = (t . (DataLoc (c,i))) - 2 & (Comput ((Q +* (stop (while>0 (a,i,I)))),t,mm)) . a = c )
thus  for j being Element of NAT st y1 < yn & ( ( 1 <= j & j < y1 ) or ( yn < j & j <= n ) ) holds
f4 . j = t . (intpos (m + j)) ::_thesis: ( ( for j being Element of NAT st y1 >= yn & 1 <= j & j <= n holds
f4 . j = t . (intpos (m + j)) ) & ( for j being Element of NAT st 1 <= j & j < (2 * k1) + 1 holds
(Comput ((Q +* (stop (while>0 (a,i,I)))),t,mm)) . (intpos ((m + n) + j)) = t . (intpos ((m + n) + j)) ) & (Comput ((Q +* (stop (while>0 (a,i,I)))),t,mm)) . (DataLoc (c,i)) = (t . (DataLoc (c,i))) - 2 & (Comput ((Q +* (stop (while>0 (a,i,I)))),t,mm)) . a = c )
proof
let j be Element of NAT ; ::_thesis: ( y1 < yn & ( ( 1 <= j & j < y1 ) or ( yn < j & j <= n ) ) implies f4 . j = t . (intpos (m + j)) )
assume that
A159: y1 < yn and
A160: ( ( 1 <= j & j < y1 ) or ( yn < j & j <= n ) ) ; ::_thesis: f4 . j = t . (intpos (m + j))
A161: ( 1 <= j & j <= n )
proof
percases ( ( 1 <= j & j < y1 ) or ( yn < j & j <= n ) ) by A160;
supposeA162: ( 1 <= j & j < y1 ) ; ::_thesis: ( 1 <= j & j <= n )
then  j < yn by A159, XXREAL_0:2;
hence  ( 1 <= j & j <= n ) by A40, A46, A162, XREAL_1:47, XXREAL_0:2; ::_thesis: verum
end;
supposeA163: ( yn < j & j <= n ) ; ::_thesis: ( 1 <= j & j <= n )
then  y1 < j by A159, XXREAL_0:2;
hence  ( 1 <= j & j <= n ) by A40, A46, A163, XREAL_1:47, XXREAL_0:2; ::_thesis: verum
end;
end;
end;
A164: ( ( 1 <= j & j < ym + 1 ) or ( yn < j & j <= n ) )
proof
percases ( ( 1 <= j & j < y1 ) or ( yn < j & j <= n ) ) by A160;
suppose ( 1 <= j & j < y1 ) ; ::_thesis: ( ( 1 <= j & j < ym + 1 ) or ( yn < j & j <= n ) )
hence  ( ( 1 <= j & j < ym + 1 ) or ( yn < j & j <= n ) ) by A71, XXREAL_0:2; ::_thesis: verum
end;
suppose ( yn < j & j <= n ) ; ::_thesis: ( ( 1 <= j & j < ym + 1 ) or ( yn < j & j <= n ) )
hence  ( ( 1 <= j & j < ym + 1 ) or ( yn < j & j <= n ) ) ; ::_thesis: verum
end;
end;
end;
A165: now__::_thesis:_f3_._j_=_(Comput_((Q_+*_(stop_(while>0_(a,i,I)))),t,((LifeSpan_((Q_+*_(stop_I)),t))_+_2)))_._(intpos_(m_+_j))
percases ( ym + 1 < yn or ym + 1 >= yn ) ;
suppose ym + 1 < yn ; ::_thesis: f3 . j = (Comput ((Q +* (stop (while>0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 2))) . (intpos (m + j))
hence  f3 . j = (Comput ((Q +* (stop (while>0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 2))) . (intpos (m + j)) by A80, A164; ::_thesis: verum
end;
suppose ym + 1 >= yn ; ::_thesis: f3 . j = (Comput ((Q +* (stop (while>0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 2))) . (intpos (m + j))
hence  f3 . j = (Comput ((Q +* (stop (while>0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 2))) . (intpos (m + j)) by A81, A161; ::_thesis: verum
end;
end;
end;
A166: ( ( 1 <= j & j < y1 ) or ( yc < j & j <= n ) )
proof
percases ( ( 1 <= j & j < y1 ) or ( yn < j & j <= n ) ) by A160;
suppose ( 1 <= j & j < y1 ) ; ::_thesis: ( ( 1 <= j & j < y1 ) or ( yc < j & j <= n ) )
hence  ( ( 1 <= j & j < y1 ) or ( yc < j & j <= n ) ) ; ::_thesis: verum
end;
suppose ( yn < j & j <= n ) ; ::_thesis: ( ( 1 <= j & j < y1 ) or ( yc < j & j <= n ) )
hence  ( ( 1 <= j & j < y1 ) or ( yc < j & j <= n ) ) by A66, XXREAL_0:2; ::_thesis: verum
end;
end;
end;
now__::_thesis:_f4_._j_=_(Comput_(((Q_+*_(stop_(while>0_(a,i,I))))_+*_(stop_(while>0_(a,i,I)))),(Initialize_(Comput_((Q_+*_(stop_(while>0_(a,i,I)))),t,((LifeSpan_((Q_+*_(stop_I)),t))_+_2)))),kl))_._(intpos_(m_+_j))
percases ( y1 < yc or y1 >= yc ) ;
suppose y1 < yc ; ::_thesis: f4 . j = (Comput (((Q +* (stop (while>0 (a,i,I)))) +* (stop (while>0 (a,i,I)))),(Initialize (Comput ((Q +* (stop (while>0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 2)))),kl)) . (intpos (m + j))
hence  f4 . j = (Comput (((Q +* (stop (while>0 (a,i,I)))) +* (stop (while>0 (a,i,I)))),(Initialize (Comput ((Q +* (stop (while>0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 2)))),kl)) . (intpos (m + j)) by A103, A166; ::_thesis: verum
end;
suppose y1 >= yc ; ::_thesis: f4 . j = (Comput (((Q +* (stop (while>0 (a,i,I)))) +* (stop (while>0 (a,i,I)))),(Initialize (Comput ((Q +* (stop (while>0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 2)))),kl)) . (intpos (m + j))
hence  f4 . j = (Comput (((Q +* (stop (while>0 (a,i,I)))) +* (stop (while>0 (a,i,I)))),(Initialize (Comput ((Q +* (stop (while>0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 2)))),kl)) . (intpos (m + j)) by A104, A161; ::_thesis: verum
end;
end;
end;
hence f4 . j = f3 . j by A76, A77, A161, SCPISORT:def_1
.= (IExec (I,Q,t)) . (intpos (m + j)) by A63, A165, SCMPDS_4:8
.= t . (intpos (m + j)) by A12, A37, A38, A39, A40, A41, A42, A159, A160 ;
::_thesis: verum
end;
thus  for j being Element of NAT st y1 >= yn & 1 <= j & j <= n holds
f4 . j = t . (intpos (m + j)) by A46, XREAL_1:47; ::_thesis: ( ( for j being Element of NAT st 1 <= j & j < (2 * k1) + 1 holds
(Comput ((Q +* (stop (while>0 (a,i,I)))),t,mm)) . (intpos ((m + n) + j)) = t . (intpos ((m + n) + j)) ) & (Comput ((Q +* (stop (while>0 (a,i,I)))),t,mm)) . (DataLoc (c,i)) = (t . (DataLoc (c,i))) - 2 & (Comput ((Q +* (stop (while>0 (a,i,I)))),t,mm)) . a = c )
hereby  ::_thesis: ( (Comput ((Q +* (stop (while>0 (a,i,I)))),t,mm)) . (DataLoc (c,i)) = (t . (DataLoc (c,i))) - 2 & (Comput ((Q +* (stop (while>0 (a,i,I)))),t,mm)) . a = c )
let j be Element of NAT ; ::_thesis: ( 1 <= j & j < (2 * k1) + 1 implies (Comput ((Q +* (stop (while>0 (a,i,I)))),t,mm)) . (intpos ((m + n) + j)) = t . (intpos ((m + n) + j)) )
assume that
A167: 1 <= j and
A168: j < (2 * k1) + 1 ; ::_thesis: (Comput ((Q +* (stop (while>0 (a,i,I)))),t,mm)) . (intpos ((m + n) + j)) = t . (intpos ((m + n) + j))
 (2 * k1) + 1 < (2 * (k1 + 1)) + 1 by A86, XREAL_1:6;
then A169: j < (2 * (k1 + 1)) + 1 by A168, XXREAL_0:2;
thus (Comput ((Q +* (stop (while>0 (a,i,I)))),t,mm)) . (intpos ((m + n) + j)) = (Comput (((Q +* (stop (while>0 (a,i,I)))) +* (stop (while>0 (a,i,I)))),(Initialize (Comput ((Q +* (stop (while>0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 2)))),kl)) . (intpos ((m + n) + j)) by A105, A146, A167, A168
.= (Comput ((Q +* (stop (while>0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 2))) . (intpos ((m + n) + j)) by A82, A167, A169
.= (IExec (I,Q,t)) . (intpos ((m + n) + j)) by A63, SCMPDS_4:8
.= t . (intpos ((m + n) + j)) by A12, A37, A38, A39, A40, A41, A42, A167, A168 ; ::_thesis: verum
end;
thus  (Comput ((Q +* (stop (while>0 (a,i,I)))),t,mm)) . (DataLoc (c,i)) = (t . (DataLoc (c,i))) - 2 by A38, A64, A83, A106, A146; ::_thesis: (Comput ((Q +* (stop (while>0 (a,i,I)))),t,mm)) . a = c
thus  (Comput ((Q +* (stop (while>0 (a,i,I)))),t,mm)) . a = c by A107, A146; ::_thesis: verum
end;
end;
end;
hence  S1[k + 1] ; ::_thesis: verum
end;
A170: for k being Element of NAT holds S1[k] from NAT_1:sch_1(A13, A35);
 ex k being Element of NAT ex f2 being FinSequence of INT st
( Initialize (Comput ((P +* (stop (while>0 (a,i,I)))),s,k)) = Comput ((P +* (stop (while>0 (a,i,I)))),s,k) & f2 is_FinSequence_on Comput ((P +* (stop (while>0 (a,i,I)))),s,k),m & len f2 = n & f,f2 are_fiberwise_equipotent & f2 is_non_decreasing_on 1,n & ( for j being Element of NAT st 1 < n & ( ( 1 <= j & j < 1 ) or ( n < j & j <= n ) ) holds
f2 . j = s . (intpos (m + j)) ) & ( for j being Element of NAT st 1 >= n & 1 <= j & j <= n holds
f2 . j = s . (intpos (m + j)) ) & ( for j being Element of NAT st 1 <= j & j < (2 * 0) + 1 holds
(Comput ((P +* (stop (while>0 (a,i,I)))),s,k)) . (intpos ((m + n) + j)) = s . (intpos ((m + n) + j)) ) & (Comput ((P +* (stop (while>0 (a,i,I)))),s,k)) . (DataLoc (c,i)) = (s . (DataLoc (c,i))) - 2 & (Comput ((P +* (stop (while>0 (a,i,I)))),s,k)) . a = c )
proof
percases ( n - 1 <= 0 or n - 1 > 0 ) ;
suppose n - 1 <= 0 ; ::_thesis: ex k being Element of NAT ex f2 being FinSequence of INT st
( Initialize (Comput ((P +* (stop (while>0 (a,i,I)))),s,k)) = Comput ((P +* (stop (while>0 (a,i,I)))),s,k) & f2 is_FinSequence_on Comput ((P +* (stop (while>0 (a,i,I)))),s,k),m & len f2 = n & f,f2 are_fiberwise_equipotent & f2 is_non_decreasing_on 1,n & ( for j being Element of NAT st 1 < n & ( ( 1 <= j & j < 1 ) or ( n < j & j <= n ) ) holds
f2 . j = s . (intpos (m + j)) ) & ( for j being Element of NAT st 1 >= n & 1 <= j & j <= n holds
f2 . j = s . (intpos (m + j)) ) & ( for j being Element of NAT st 1 <= j & j < (2 * 0) + 1 holds
(Comput ((P +* (stop (while>0 (a,i,I)))),s,k)) . (intpos ((m + n) + j)) = s . (intpos ((m + n) + j)) ) & (Comput ((P +* (stop (while>0 (a,i,I)))),s,k)) . (DataLoc (c,i)) = (s . (DataLoc (c,i))) - 2 & (Comput ((P +* (stop (while>0 (a,i,I)))),s,k)) . a = c )
hence  ex k being Element of NAT ex f2 being FinSequence of INT st
( Initialize (Comput ((P +* (stop (while>0 (a,i,I)))),s,k)) = Comput ((P +* (stop (while>0 (a,i,I)))),s,k) & f2 is_FinSequence_on Comput ((P +* (stop (while>0 (a,i,I)))),s,k),m & len f2 = n & f,f2 are_fiberwise_equipotent & f2 is_non_decreasing_on 1,n & ( for j being Element of NAT st 1 < n & ( ( 1 <= j & j < 1 ) or ( n < j & j <= n ) ) holds
f2 . j = s . (intpos (m + j)) ) & ( for j being Element of NAT st 1 >= n & 1 <= j & j <= n holds
f2 . j = s . (intpos (m + j)) ) & ( for j being Element of NAT st 1 <= j & j < (2 * 0) + 1 holds
(Comput ((P +* (stop (while>0 (a,i,I)))),s,k)) . (intpos ((m + n) + j)) = s . (intpos ((m + n) + j)) ) & (Comput ((P +* (stop (while>0 (a,i,I)))),s,k)) . (DataLoc (c,i)) = (s . (DataLoc (c,i))) - 2 & (Comput ((P +* (stop (while>0 (a,i,I)))),s,k)) . a = c ) by A2, A3, A5, A7, A9, A10, A13, A11; ::_thesis: verum
end;
suppose n - 1 > 0 ; ::_thesis: ex k being Element of NAT ex f2 being FinSequence of INT st
( Initialize (Comput ((P +* (stop (while>0 (a,i,I)))),s,k)) = Comput ((P +* (stop (while>0 (a,i,I)))),s,k) & f2 is_FinSequence_on Comput ((P +* (stop (while>0 (a,i,I)))),s,k),m & len f2 = n & f,f2 are_fiberwise_equipotent & f2 is_non_decreasing_on 1,n & ( for j being Element of NAT st 1 < n & ( ( 1 <= j & j < 1 ) or ( n < j & j <= n ) ) holds
f2 . j = s . (intpos (m + j)) ) & ( for j being Element of NAT st 1 >= n & 1 <= j & j <= n holds
f2 . j = s . (intpos (m + j)) ) & ( for j being Element of NAT st 1 <= j & j < (2 * 0) + 1 holds
(Comput ((P +* (stop (while>0 (a,i,I)))),s,k)) . (intpos ((m + n) + j)) = s . (intpos ((m + n) + j)) ) & (Comput ((P +* (stop (while>0 (a,i,I)))),s,k)) . (DataLoc (c,i)) = (s . (DataLoc (c,i))) - 2 & (Comput ((P +* (stop (while>0 (a,i,I)))),s,k)) . a = c )
then reconsider nn = n - 1 as Element of NAT by INT_1:3;
 S1[nn] by A170;
hence  ex k being Element of NAT ex f2 being FinSequence of INT st
( Initialize (Comput ((P +* (stop (while>0 (a,i,I)))),s,k)) = Comput ((P +* (stop (while>0 (a,i,I)))),s,k) & f2 is_FinSequence_on Comput ((P +* (stop (while>0 (a,i,I)))),s,k),m & len f2 = n & f,f2 are_fiberwise_equipotent & f2 is_non_decreasing_on 1,n & ( for j being Element of NAT st 1 < n & ( ( 1 <= j & j < 1 ) or ( n < j & j <= n ) ) holds
f2 . j = s . (intpos (m + j)) ) & ( for j being Element of NAT st 1 >= n & 1 <= j & j <= n holds
f2 . j = s . (intpos (m + j)) ) & ( for j being Element of NAT st 1 <= j & j < (2 * 0) + 1 holds
(Comput ((P +* (stop (while>0 (a,i,I)))),s,k)) . (intpos ((m + n) + j)) = s . (intpos ((m + n) + j)) ) & (Comput ((P +* (stop (while>0 (a,i,I)))),s,k)) . (DataLoc (c,i)) = (s . (DataLoc (c,i))) - 2 & (Comput ((P +* (stop (while>0 (a,i,I)))),s,k)) . a = c ) by A2, A3, A5, A7, A9, A10, A11; ::_thesis: verum
end;
end;
end;
then consider k being Element of NAT , f2 being FinSequence of INT  such that
A171: Initialize (Comput ((P +* (stop (while>0 (a,i,I)))),s,k)) = Comput ((P +* (stop (while>0 (a,i,I)))),s,k) and
A172: f2 is_FinSequence_on Comput ((P +* (stop (while>0 (a,i,I)))),s,k),m and
A173: len f2 = n and
A174: f,f2 are_fiberwise_equipotent and
A175: f2 is_non_decreasing_on 1,n and
A176: (Comput ((P +* (stop (while>0 (a,i,I)))),s,k)) . (DataLoc (c,i)) = (s . (DataLoc (c,i))) - 2 and
A177: (Comput ((P +* (stop (while>0 (a,i,I)))),s,k)) . a = c ;
set sk = Comput ((P +* (stop (while>0 (a,i,I)))),s,k);
set s1 = Initialize (Comput ((P +* (stop (while>0 (a,i,I)))),s,k));
set P1 = (P +* (stop (while>0 (a,i,I)))) +* (stop (while>0 (a,i,I)));
set s2 = Comput (((P +* (stop (while>0 (a,i,I)))) +* (stop (while>0 (a,i,I)))),(Initialize (Comput ((P +* (stop (while>0 (a,i,I)))),s,k))),1);
set P2 = (P +* (stop (while>0 (a,i,I)))) +* (stop (while>0 (a,i,I)));
A179: IC (Initialize (Comput ((P +* (stop (while>0 (a,i,I)))),s,k))) = 0 by MEMSTR_0:def_11;
set i1 = (a,i) <=0_goto ((card I) + 2);
set i2 = goto (- ((card I) + 1));
A180: card (while>0 (a,i,I)) = (card I) + 2 by SCMPDS_8:17;
then A181: (card I) + 2 in dom (stop (while>0 (a,i,I))) by COMPOS_1:64;
A182: dom g = Seg n by A4, FINSEQ_1:def_3;
 stop (while>0 (a,i,I)) c= (P +* (stop (while>0 (a,i,I)))) +* (stop (while>0 (a,i,I))) by FUNCT_4:25;
then A183: ((P +* (stop (while>0 (a,i,I)))) +* (stop (while>0 (a,i,I)))) . ((card I) + 2) = (stop (while>0 (a,i,I))) . ((card I) + 2) by A181, GRFUNC_1:2
.= halt SCMPDS by A180, COMPOS_1:64 ;
A184: while>0 (a,i,I) = ((a,i) <=0_goto ((card I) + 2)) ';' (I ';' (goto (- ((card I) + 1)))) by Lm2;
A185: Comput (((P +* (stop (while>0 (a,i,I)))) +* (stop (while>0 (a,i,I)))),(Initialize (Comput ((P +* (stop (while>0 (a,i,I)))),s,k))),(0 + 1)) = Following (((P +* (stop (while>0 (a,i,I)))) +* (stop (while>0 (a,i,I)))),(Comput (((P +* (stop (while>0 (a,i,I)))) +* (stop (while>0 (a,i,I)))),(Initialize (Comput ((P +* (stop (while>0 (a,i,I)))),s,k))),0))) by EXTPRO_1:3
.= Following (((P +* (stop (while>0 (a,i,I)))) +* (stop (while>0 (a,i,I)))),(Initialize (Comput ((P +* (stop (while>0 (a,i,I)))),s,k))))
.= Exec (((a,i) <=0_goto ((card I) + 2)),(Initialize (Comput ((P +* (stop (while>0 (a,i,I)))),s,k)))) by A184, SCMPDS_6:11 ;
IC (Comput (((P +* (stop (while>0 (a,i,I)))) +* (stop (while>0 (a,i,I)))),(Initialize (Comput ((P +* (stop (while>0 (a,i,I)))),s,k))),1)) = (Comput (((P +* (stop (while>0 (a,i,I)))) +* (stop (while>0 (a,i,I)))),(Initialize (Comput ((P +* (stop (while>0 (a,i,I)))),s,k))),1)) . (IC )
.= ICplusConst ((Initialize (Comput ((P +* (stop (while>0 (a,i,I)))),s,k))),((card I) + 2)) by A7, A171, A176, A177, A185, SCMPDS_2:56
.= 0 + ((card I) + 2) by A179, SCMPDS_6:12 ;
then A187: CurInstr (((P +* (stop (while>0 (a,i,I)))) +* (stop (while>0 (a,i,I)))),(Comput (((P +* (stop (while>0 (a,i,I)))) +* (stop (while>0 (a,i,I)))),(Initialize (Comput ((P +* (stop (while>0 (a,i,I)))),s,k))),1))) = halt SCMPDS by A183, PBOOLE:143;
A188: Comput (((P +* (stop (while>0 (a,i,I)))) +* (stop (while>0 (a,i,I)))),(Initialize (Comput ((P +* (stop (while>0 (a,i,I)))),s,k))),1) = Comput ((P +* (stop (while>0 (a,i,I)))),s,(k + 1)) by A171, EXTPRO_1:4;
A190: P +* (stop (while>0 (a,i,I))) halts_on s by A187, A188, EXTPRO_1:29;
hence  while>0 (a,i,I) is_halting_on s,P by I, SCMPDS_6:def_3; ::_thesis: ( f,g are_fiberwise_equipotent & g is_non_decreasing_on 1,n )
A191: Result ((P +* (stop (while>0 (a,i,I)))),s) = Comput (((P +* (stop (while>0 (a,i,I)))) +* (stop (while>0 (a,i,I)))),(Initialize (Comput ((P +* (stop (while>0 (a,i,I)))),s,k))),1) by A187, A188, A190, EXTPRO_1:def_9;
now__::_thesis:_for_i_being_Nat_st_i_in_dom_g_holds_
g_._i_=_f2_._i
let i be Nat; ::_thesis: ( i in dom g implies g . i = f2 . i )
reconsider a = i as Element of NAT by ORDINAL1:def_12;
set xi = intpos (m + a);
assume A194: i in dom g ; ::_thesis: g . i = f2 . i
then A195: 1 <= i by A182, FINSEQ_1:1;
A196: i <= n by A182, A194, FINSEQ_1:1;
 IExec ((while>0 (a,i,I)),P,s) = Comput (((P +* (stop (while>0 (a,i,I)))) +* (stop (while>0 (a,i,I)))),(Initialize (Comput ((P +* (stop (while>0 (a,i,I)))),s,k))),1) by A191, SCMPDS_4:def_5;
hence g . i = (Comput (((P +* (stop (while>0 (a,i,I)))) +* (stop (while>0 (a,i,I)))),(Initialize (Comput ((P +* (stop (while>0 (a,i,I)))),s,k))),1)) . (intpos (m + a)) by A4, A6, A195, A196, SCPISORT:def_1
.= (Comput (((P +* (stop (while>0 (a,i,I)))) +* (stop (while>0 (a,i,I)))),(Initialize (Comput ((P +* (stop (while>0 (a,i,I)))),s,k))),1)) . (intpos (m + a))
.= (Initialize (Comput ((P +* (stop (while>0 (a,i,I)))),s,k))) . (intpos (m + a)) by A185, SCMPDS_2:56
.= f2 . i by A171, A172, A173, A195, A196, SCPISORT:def_1 ;
::_thesis: verum
end;
hence  ( f,g are_fiberwise_equipotent & g is_non_decreasing_on 1,n ) by A4, A173, A174, A175, FINSEQ_2:9; ::_thesis: verum
end;

Lm5:  for P being Instruction-Sequence of SCMPDS
for s being 0 -started State of SCMPDS
for I being halt-free shiftable Program of SCMPDS
for a being Int_position
for i, c being Integer
for m, n, m1 being Element of NAT st s . a = c & 1 = s . (DataLoc (c,i)) & m1 = (m + n) + 1 & m + 1 = s . (intpos m1) & m + n = s . (intpos (m1 + 1)) & ( for t being 0 -started State of SCMPDS
for Q being Instruction-Sequence of SCMPDS
for f1, f2 being FinSequence of INT
for k1, k2, y1, yn being Element of NAT st t . a = c & (2 * k1) + 1 = t . (DataLoc (c,i)) & k2 = ((m + n) + (2 * k1)) + 1 & m + y1 = t . (intpos k2) & m + yn = t . (intpos (k2 + 1)) & ( ( 1 <= y1 & yn <= n ) or y1 >= yn ) holds
( I is_closed_on t,Q & I is_halting_on t,Q & (IExec (I,Q,t)) . a = t . a & ( for j being Element of NAT st 1 <= j & j < (2 * k1) + 1 holds
(IExec (I,Q,t)) . (intpos ((m + n) + j)) = t . (intpos ((m + n) + j)) ) & ( y1 >= yn implies ( (IExec (I,Q,t)) . (DataLoc (c,i)) = (2 * k1) - 1 & ( for j being Element of NAT st 1 <= j & j <= n holds
(IExec (I,Q,t)) . (intpos (m + j)) = t . (intpos (m + j)) ) ) ) & ( y1 < yn implies ( (IExec (I,Q,t)) . (DataLoc (c,i)) = (2 * k1) + 3 & ( for j being Element of NAT st ( ( 1 <= j & j < y1 ) or ( yn < j & j <= n ) ) holds
(IExec (I,Q,t)) . (intpos (m + j)) = t . (intpos (m + j)) ) & ex ym being Element of NAT st
( y1 <= ym & ym <= yn & m + y1 = (IExec (I,Q,t)) . (intpos k2) & (m + ym) - 1 = (IExec (I,Q,t)) . (intpos (k2 + 1)) & (m + ym) + 1 = (IExec (I,Q,t)) . (intpos (k2 + 2)) & m + yn = (IExec (I,Q,t)) . (intpos (k2 + 3)) & ( for j being Element of NAT st y1 <= j & j < ym holds
(IExec (I,Q,t)) . (intpos (m + j)) <= (IExec (I,Q,t)) . (intpos (m + ym)) ) & ( for j being Element of NAT st ym < j & j <= yn holds
(IExec (I,Q,t)) . (intpos (m + j)) >= (IExec (I,Q,t)) . (intpos (m + ym)) ) ) ) ) & ( f1 is_FinSequence_on t,m & f2 is_FinSequence_on IExec (I,Q,t),m & len f1 = n & len f2 = n implies f1,f2 are_fiberwise_equipotent ) ) ) holds
while>0 (a,i,I) is_halting_on s,P
proof
let P be Instruction-Sequence of SCMPDS; ::_thesis: for s being 0 -started State of SCMPDS
for I being halt-free shiftable Program of SCMPDS
for a being Int_position
for i, c being Integer
for m, n, m1 being Element of NAT st s . a = c & 1 = s . (DataLoc (c,i)) & m1 = (m + n) + 1 & m + 1 = s . (intpos m1) & m + n = s . (intpos (m1 + 1)) & ( for t being 0 -started State of SCMPDS
for Q being Instruction-Sequence of SCMPDS
for f1, f2 being FinSequence of INT
for k1, k2, y1, yn being Element of NAT st t . a = c & (2 * k1) + 1 = t . (DataLoc (c,i)) & k2 = ((m + n) + (2 * k1)) + 1 & m + y1 = t . (intpos k2) & m + yn = t . (intpos (k2 + 1)) & ( ( 1 <= y1 & yn <= n ) or y1 >= yn ) holds
( I is_closed_on t,Q & I is_halting_on t,Q & (IExec (I,Q,t)) . a = t . a & ( for j being Element of NAT st 1 <= j & j < (2 * k1) + 1 holds
(IExec (I,Q,t)) . (intpos ((m + n) + j)) = t . (intpos ((m + n) + j)) ) & ( y1 >= yn implies ( (IExec (I,Q,t)) . (DataLoc (c,i)) = (2 * k1) - 1 & ( for j being Element of NAT st 1 <= j & j <= n holds
(IExec (I,Q,t)) . (intpos (m + j)) = t . (intpos (m + j)) ) ) ) & ( y1 < yn implies ( (IExec (I,Q,t)) . (DataLoc (c,i)) = (2 * k1) + 3 & ( for j being Element of NAT st ( ( 1 <= j & j < y1 ) or ( yn < j & j <= n ) ) holds
(IExec (I,Q,t)) . (intpos (m + j)) = t . (intpos (m + j)) ) & ex ym being Element of NAT st
( y1 <= ym & ym <= yn & m + y1 = (IExec (I,Q,t)) . (intpos k2) & (m + ym) - 1 = (IExec (I,Q,t)) . (intpos (k2 + 1)) & (m + ym) + 1 = (IExec (I,Q,t)) . (intpos (k2 + 2)) & m + yn = (IExec (I,Q,t)) . (intpos (k2 + 3)) & ( for j being Element of NAT st y1 <= j & j < ym holds
(IExec (I,Q,t)) . (intpos (m + j)) <= (IExec (I,Q,t)) . (intpos (m + ym)) ) & ( for j being Element of NAT st ym < j & j <= yn holds
(IExec (I,Q,t)) . (intpos (m + j)) >= (IExec (I,Q,t)) . (intpos (m + ym)) ) ) ) ) & ( f1 is_FinSequence_on t,m & f2 is_FinSequence_on IExec (I,Q,t),m & len f1 = n & len f2 = n implies f1,f2 are_fiberwise_equipotent ) ) ) holds
while>0 (a,i,I) is_halting_on s,P
let s be 0 -started State of SCMPDS; ::_thesis: for I being halt-free shiftable Program of SCMPDS
for a being Int_position
for i, c being Integer
for m, n, m1 being Element of NAT st s . a = c & 1 = s . (DataLoc (c,i)) & m1 = (m + n) + 1 & m + 1 = s . (intpos m1) & m + n = s . (intpos (m1 + 1)) & ( for t being 0 -started State of SCMPDS
for Q being Instruction-Sequence of SCMPDS
for f1, f2 being FinSequence of INT
for k1, k2, y1, yn being Element of NAT st t . a = c & (2 * k1) + 1 = t . (DataLoc (c,i)) & k2 = ((m + n) + (2 * k1)) + 1 & m + y1 = t . (intpos k2) & m + yn = t . (intpos (k2 + 1)) & ( ( 1 <= y1 & yn <= n ) or y1 >= yn ) holds
( I is_closed_on t,Q & I is_halting_on t,Q & (IExec (I,Q,t)) . a = t . a & ( for j being Element of NAT st 1 <= j & j < (2 * k1) + 1 holds
(IExec (I,Q,t)) . (intpos ((m + n) + j)) = t . (intpos ((m + n) + j)) ) & ( y1 >= yn implies ( (IExec (I,Q,t)) . (DataLoc (c,i)) = (2 * k1) - 1 & ( for j being Element of NAT st 1 <= j & j <= n holds
(IExec (I,Q,t)) . (intpos (m + j)) = t . (intpos (m + j)) ) ) ) & ( y1 < yn implies ( (IExec (I,Q,t)) . (DataLoc (c,i)) = (2 * k1) + 3 & ( for j being Element of NAT st ( ( 1 <= j & j < y1 ) or ( yn < j & j <= n ) ) holds
(IExec (I,Q,t)) . (intpos (m + j)) = t . (intpos (m + j)) ) & ex ym being Element of NAT st
( y1 <= ym & ym <= yn & m + y1 = (IExec (I,Q,t)) . (intpos k2) & (m + ym) - 1 = (IExec (I,Q,t)) . (intpos (k2 + 1)) & (m + ym) + 1 = (IExec (I,Q,t)) . (intpos (k2 + 2)) & m + yn = (IExec (I,Q,t)) . (intpos (k2 + 3)) & ( for j being Element of NAT st y1 <= j & j < ym holds
(IExec (I,Q,t)) . (intpos (m + j)) <= (IExec (I,Q,t)) . (intpos (m + ym)) ) & ( for j being Element of NAT st ym < j & j <= yn holds
(IExec (I,Q,t)) . (intpos (m + j)) >= (IExec (I,Q,t)) . (intpos (m + ym)) ) ) ) ) & ( f1 is_FinSequence_on t,m & f2 is_FinSequence_on IExec (I,Q,t),m & len f1 = n & len f2 = n implies f1,f2 are_fiberwise_equipotent ) ) ) holds
while>0 (a,i,I) is_halting_on s,P
let I be halt-free shiftable Program of SCMPDS; ::_thesis: for a being Int_position
for i, c being Integer
for m, n, m1 being Element of NAT st s . a = c & 1 = s . (DataLoc (c,i)) & m1 = (m + n) + 1 & m + 1 = s . (intpos m1) & m + n = s . (intpos (m1 + 1)) & ( for t being 0 -started State of SCMPDS
for Q being Instruction-Sequence of SCMPDS
for f1, f2 being FinSequence of INT
for k1, k2, y1, yn being Element of NAT st t . a = c & (2 * k1) + 1 = t . (DataLoc (c,i)) & k2 = ((m + n) + (2 * k1)) + 1 & m + y1 = t . (intpos k2) & m + yn = t . (intpos (k2 + 1)) & ( ( 1 <= y1 & yn <= n ) or y1 >= yn ) holds
( I is_closed_on t,Q & I is_halting_on t,Q & (IExec (I,Q,t)) . a = t . a & ( for j being Element of NAT st 1 <= j & j < (2 * k1) + 1 holds
(IExec (I,Q,t)) . (intpos ((m + n) + j)) = t . (intpos ((m + n) + j)) ) & ( y1 >= yn implies ( (IExec (I,Q,t)) . (DataLoc (c,i)) = (2 * k1) - 1 & ( for j being Element of NAT st 1 <= j & j <= n holds
(IExec (I,Q,t)) . (intpos (m + j)) = t . (intpos (m + j)) ) ) ) & ( y1 < yn implies ( (IExec (I,Q,t)) . (DataLoc (c,i)) = (2 * k1) + 3 & ( for j being Element of NAT st ( ( 1 <= j & j < y1 ) or ( yn < j & j <= n ) ) holds
(IExec (I,Q,t)) . (intpos (m + j)) = t . (intpos (m + j)) ) & ex ym being Element of NAT st
( y1 <= ym & ym <= yn & m + y1 = (IExec (I,Q,t)) . (intpos k2) & (m + ym) - 1 = (IExec (I,Q,t)) . (intpos (k2 + 1)) & (m + ym) + 1 = (IExec (I,Q,t)) . (intpos (k2 + 2)) & m + yn = (IExec (I,Q,t)) . (intpos (k2 + 3)) & ( for j being Element of NAT st y1 <= j & j < ym holds
(IExec (I,Q,t)) . (intpos (m + j)) <= (IExec (I,Q,t)) . (intpos (m + ym)) ) & ( for j being Element of NAT st ym < j & j <= yn holds
(IExec (I,Q,t)) . (intpos (m + j)) >= (IExec (I,Q,t)) . (intpos (m + ym)) ) ) ) ) & ( f1 is_FinSequence_on t,m & f2 is_FinSequence_on IExec (I,Q,t),m & len f1 = n & len f2 = n implies f1,f2 are_fiberwise_equipotent ) ) ) holds
while>0 (a,i,I) is_halting_on s,P
let a be Int_position; ::_thesis: for i, c being Integer
for m, n, m1 being Element of NAT st s . a = c & 1 = s . (DataLoc (c,i)) & m1 = (m + n) + 1 & m + 1 = s . (intpos m1) & m + n = s . (intpos (m1 + 1)) & ( for t being 0 -started State of SCMPDS
for Q being Instruction-Sequence of SCMPDS
for f1, f2 being FinSequence of INT
for k1, k2, y1, yn being Element of NAT st t . a = c & (2 * k1) + 1 = t . (DataLoc (c,i)) & k2 = ((m + n) + (2 * k1)) + 1 & m + y1 = t . (intpos k2) & m + yn = t . (intpos (k2 + 1)) & ( ( 1 <= y1 & yn <= n ) or y1 >= yn ) holds
( I is_closed_on t,Q & I is_halting_on t,Q & (IExec (I,Q,t)) . a = t . a & ( for j being Element of NAT st 1 <= j & j < (2 * k1) + 1 holds
(IExec (I,Q,t)) . (intpos ((m + n) + j)) = t . (intpos ((m + n) + j)) ) & ( y1 >= yn implies ( (IExec (I,Q,t)) . (DataLoc (c,i)) = (2 * k1) - 1 & ( for j being Element of NAT st 1 <= j & j <= n holds
(IExec (I,Q,t)) . (intpos (m + j)) = t . (intpos (m + j)) ) ) ) & ( y1 < yn implies ( (IExec (I,Q,t)) . (DataLoc (c,i)) = (2 * k1) + 3 & ( for j being Element of NAT st ( ( 1 <= j & j < y1 ) or ( yn < j & j <= n ) ) holds
(IExec (I,Q,t)) . (intpos (m + j)) = t . (intpos (m + j)) ) & ex ym being Element of NAT st
( y1 <= ym & ym <= yn & m + y1 = (IExec (I,Q,t)) . (intpos k2) & (m + ym) - 1 = (IExec (I,Q,t)) . (intpos (k2 + 1)) & (m + ym) + 1 = (IExec (I,Q,t)) . (intpos (k2 + 2)) & m + yn = (IExec (I,Q,t)) . (intpos (k2 + 3)) & ( for j being Element of NAT st y1 <= j & j < ym holds
(IExec (I,Q,t)) . (intpos (m + j)) <= (IExec (I,Q,t)) . (intpos (m + ym)) ) & ( for j being Element of NAT st ym < j & j <= yn holds
(IExec (I,Q,t)) . (intpos (m + j)) >= (IExec (I,Q,t)) . (intpos (m + ym)) ) ) ) ) & ( f1 is_FinSequence_on t,m & f2 is_FinSequence_on IExec (I,Q,t),m & len f1 = n & len f2 = n implies f1,f2 are_fiberwise_equipotent ) ) ) holds
while>0 (a,i,I) is_halting_on s,P
let i, c be Integer; ::_thesis: for m, n, m1 being Element of NAT st s . a = c & 1 = s . (DataLoc (c,i)) & m1 = (m + n) + 1 & m + 1 = s . (intpos m1) & m + n = s . (intpos (m1 + 1)) & ( for t being 0 -started State of SCMPDS
for Q being Instruction-Sequence of SCMPDS
for f1, f2 being FinSequence of INT
for k1, k2, y1, yn being Element of NAT st t . a = c & (2 * k1) + 1 = t . (DataLoc (c,i)) & k2 = ((m + n) + (2 * k1)) + 1 & m + y1 = t . (intpos k2) & m + yn = t . (intpos (k2 + 1)) & ( ( 1 <= y1 & yn <= n ) or y1 >= yn ) holds
( I is_closed_on t,Q & I is_halting_on t,Q & (IExec (I,Q,t)) . a = t . a & ( for j being Element of NAT st 1 <= j & j < (2 * k1) + 1 holds
(IExec (I,Q,t)) . (intpos ((m + n) + j)) = t . (intpos ((m + n) + j)) ) & ( y1 >= yn implies ( (IExec (I,Q,t)) . (DataLoc (c,i)) = (2 * k1) - 1 & ( for j being Element of NAT st 1 <= j & j <= n holds
(IExec (I,Q,t)) . (intpos (m + j)) = t . (intpos (m + j)) ) ) ) & ( y1 < yn implies ( (IExec (I,Q,t)) . (DataLoc (c,i)) = (2 * k1) + 3 & ( for j being Element of NAT st ( ( 1 <= j & j < y1 ) or ( yn < j & j <= n ) ) holds
(IExec (I,Q,t)) . (intpos (m + j)) = t . (intpos (m + j)) ) & ex ym being Element of NAT st
( y1 <= ym & ym <= yn & m + y1 = (IExec (I,Q,t)) . (intpos k2) & (m + ym) - 1 = (IExec (I,Q,t)) . (intpos (k2 + 1)) & (m + ym) + 1 = (IExec (I,Q,t)) . (intpos (k2 + 2)) & m + yn = (IExec (I,Q,t)) . (intpos (k2 + 3)) & ( for j being Element of NAT st y1 <= j & j < ym holds
(IExec (I,Q,t)) . (intpos (m + j)) <= (IExec (I,Q,t)) . (intpos (m + ym)) ) & ( for j being Element of NAT st ym < j & j <= yn holds
(IExec (I,Q,t)) . (intpos (m + j)) >= (IExec (I,Q,t)) . (intpos (m + ym)) ) ) ) ) & ( f1 is_FinSequence_on t,m & f2 is_FinSequence_on IExec (I,Q,t),m & len f1 = n & len f2 = n implies f1,f2 are_fiberwise_equipotent ) ) ) holds
while>0 (a,i,I) is_halting_on s,P
let m, n, m1 be Element of NAT ; ::_thesis: ( s . a = c & 1 = s . (DataLoc (c,i)) & m1 = (m + n) + 1 & m + 1 = s . (intpos m1) & m + n = s . (intpos (m1 + 1)) & ( for t being 0 -started State of SCMPDS
for Q being Instruction-Sequence of SCMPDS
for f1, f2 being FinSequence of INT
for k1, k2, y1, yn being Element of NAT st t . a = c & (2 * k1) + 1 = t . (DataLoc (c,i)) & k2 = ((m + n) + (2 * k1)) + 1 & m + y1 = t . (intpos k2) & m + yn = t . (intpos (k2 + 1)) & ( ( 1 <= y1 & yn <= n ) or y1 >= yn ) holds
( I is_closed_on t,Q & I is_halting_on t,Q & (IExec (I,Q,t)) . a = t . a & ( for j being Element of NAT st 1 <= j & j < (2 * k1) + 1 holds
(IExec (I,Q,t)) . (intpos ((m + n) + j)) = t . (intpos ((m + n) + j)) ) & ( y1 >= yn implies ( (IExec (I,Q,t)) . (DataLoc (c,i)) = (2 * k1) - 1 & ( for j being Element of NAT st 1 <= j & j <= n holds
(IExec (I,Q,t)) . (intpos (m + j)) = t . (intpos (m + j)) ) ) ) & ( y1 < yn implies ( (IExec (I,Q,t)) . (DataLoc (c,i)) = (2 * k1) + 3 & ( for j being Element of NAT st ( ( 1 <= j & j < y1 ) or ( yn < j & j <= n ) ) holds
(IExec (I,Q,t)) . (intpos (m + j)) = t . (intpos (m + j)) ) & ex ym being Element of NAT st
( y1 <= ym & ym <= yn & m + y1 = (IExec (I,Q,t)) . (intpos k2) & (m + ym) - 1 = (IExec (I,Q,t)) . (intpos (k2 + 1)) & (m + ym) + 1 = (IExec (I,Q,t)) . (intpos (k2 + 2)) & m + yn = (IExec (I,Q,t)) . (intpos (k2 + 3)) & ( for j being Element of NAT st y1 <= j & j < ym holds
(IExec (I,Q,t)) . (intpos (m + j)) <= (IExec (I,Q,t)) . (intpos (m + ym)) ) & ( for j being Element of NAT st ym < j & j <= yn holds
(IExec (I,Q,t)) . (intpos (m + j)) >= (IExec (I,Q,t)) . (intpos (m + ym)) ) ) ) ) & ( f1 is_FinSequence_on t,m & f2 is_FinSequence_on IExec (I,Q,t),m & len f1 = n & len f2 = n implies f1,f2 are_fiberwise_equipotent ) ) ) implies while>0 (a,i,I) is_halting_on s,P )
set b = DataLoc (c,i);
assume A2: s . a = c ; ::_thesis: ( not 1 = s . (DataLoc (c,i)) or not m1 = (m + n) + 1 or not m + 1 = s . (intpos m1) or not m + n = s . (intpos (m1 + 1)) or ex t being 0 -started State of SCMPDS ex Q being Instruction-Sequence of SCMPDS ex f1, f2 being FinSequence of INT ex k1, k2, y1, yn being Element of NAT st
( t . a = c & (2 * k1) + 1 = t . (DataLoc (c,i)) & k2 = ((m + n) + (2 * k1)) + 1 & m + y1 = t . (intpos k2) & m + yn = t . (intpos (k2 + 1)) & ( ( 1 <= y1 & yn <= n ) or y1 >= yn ) & not ( I is_closed_on t,Q & I is_halting_on t,Q & (IExec (I,Q,t)) . a = t . a & ( for j being Element of NAT st 1 <= j & j < (2 * k1) + 1 holds
(IExec (I,Q,t)) . (intpos ((m + n) + j)) = t . (intpos ((m + n) + j)) ) & ( y1 >= yn implies ( (IExec (I,Q,t)) . (DataLoc (c,i)) = (2 * k1) - 1 & ( for j being Element of NAT st 1 <= j & j <= n holds
(IExec (I,Q,t)) . (intpos (m + j)) = t . (intpos (m + j)) ) ) ) & ( y1 < yn implies ( (IExec (I,Q,t)) . (DataLoc (c,i)) = (2 * k1) + 3 & ( for j being Element of NAT st ( ( 1 <= j & j < y1 ) or ( yn < j & j <= n ) ) holds
(IExec (I,Q,t)) . (intpos (m + j)) = t . (intpos (m + j)) ) & ex ym being Element of NAT st
( y1 <= ym & ym <= yn & m + y1 = (IExec (I,Q,t)) . (intpos k2) & (m + ym) - 1 = (IExec (I,Q,t)) . (intpos (k2 + 1)) & (m + ym) + 1 = (IExec (I,Q,t)) . (intpos (k2 + 2)) & m + yn = (IExec (I,Q,t)) . (intpos (k2 + 3)) & ( for j being Element of NAT st y1 <= j & j < ym holds
(IExec (I,Q,t)) . (intpos (m + j)) <= (IExec (I,Q,t)) . (intpos (m + ym)) ) & ( for j being Element of NAT st ym < j & j <= yn holds
(IExec (I,Q,t)) . (intpos (m + j)) >= (IExec (I,Q,t)) . (intpos (m + ym)) ) ) ) ) & ( f1 is_FinSequence_on t,m & f2 is_FinSequence_on IExec (I,Q,t),m & len f1 = n & len f2 = n implies f1,f2 are_fiberwise_equipotent ) ) ) or while>0 (a,i,I) is_halting_on s,P )
consider f being FinSequence of INT  such that
A3: len f = n and
A4: for i being Element of NAT st 1 <= i & i <= len f holds
f . i = s . (intpos (m + i)) by SCPISORT:1;
A5: f is_FinSequence_on s,m by A4, SCPISORT:def_1;
set ss = IExec ((while>0 (a,i,I)),P,s);
assume A6: 1 = s . (DataLoc (c,i)) ; ::_thesis: ( not m1 = (m + n) + 1 or not m + 1 = s . (intpos m1) or not m + n = s . (intpos (m1 + 1)) or ex t being 0 -started State of SCMPDS ex Q being Instruction-Sequence of SCMPDS ex f1, f2 being FinSequence of INT ex k1, k2, y1, yn being Element of NAT st
( t . a = c & (2 * k1) + 1 = t . (DataLoc (c,i)) & k2 = ((m + n) + (2 * k1)) + 1 & m + y1 = t . (intpos k2) & m + yn = t . (intpos (k2 + 1)) & ( ( 1 <= y1 & yn <= n ) or y1 >= yn ) & not ( I is_closed_on t,Q & I is_halting_on t,Q & (IExec (I,Q,t)) . a = t . a & ( for j being Element of NAT st 1 <= j & j < (2 * k1) + 1 holds
(IExec (I,Q,t)) . (intpos ((m + n) + j)) = t . (intpos ((m + n) + j)) ) & ( y1 >= yn implies ( (IExec (I,Q,t)) . (DataLoc (c,i)) = (2 * k1) - 1 & ( for j being Element of NAT st 1 <= j & j <= n holds
(IExec (I,Q,t)) . (intpos (m + j)) = t . (intpos (m + j)) ) ) ) & ( y1 < yn implies ( (IExec (I,Q,t)) . (DataLoc (c,i)) = (2 * k1) + 3 & ( for j being Element of NAT st ( ( 1 <= j & j < y1 ) or ( yn < j & j <= n ) ) holds
(IExec (I,Q,t)) . (intpos (m + j)) = t . (intpos (m + j)) ) & ex ym being Element of NAT st
( y1 <= ym & ym <= yn & m + y1 = (IExec (I,Q,t)) . (intpos k2) & (m + ym) - 1 = (IExec (I,Q,t)) . (intpos (k2 + 1)) & (m + ym) + 1 = (IExec (I,Q,t)) . (intpos (k2 + 2)) & m + yn = (IExec (I,Q,t)) . (intpos (k2 + 3)) & ( for j being Element of NAT st y1 <= j & j < ym holds
(IExec (I,Q,t)) . (intpos (m + j)) <= (IExec (I,Q,t)) . (intpos (m + ym)) ) & ( for j being Element of NAT st ym < j & j <= yn holds
(IExec (I,Q,t)) . (intpos (m + j)) >= (IExec (I,Q,t)) . (intpos (m + ym)) ) ) ) ) & ( f1 is_FinSequence_on t,m & f2 is_FinSequence_on IExec (I,Q,t),m & len f1 = n & len f2 = n implies f1,f2 are_fiberwise_equipotent ) ) ) or while>0 (a,i,I) is_halting_on s,P )
consider g being FinSequence of INT  such that
A7: len g = n and
A8: for i being Element of NAT st 1 <= i & i <= len g holds
g . i = (IExec ((while>0 (a,i,I)),P,s)) . (intpos (m + i)) by SCPISORT:1;
A9: g is_FinSequence_on IExec ((while>0 (a,i,I)),P,s),m by A8, SCPISORT:def_1;
assume that
A10: m1 = (m + n) + 1 and
A11: m + 1 = s . (intpos m1) and
A12: m + n = s . (intpos (m1 + 1)) ; ::_thesis: ( ex t being 0 -started State of SCMPDS ex Q being Instruction-Sequence of SCMPDS ex f1, f2 being FinSequence of INT ex k1, k2, y1, yn being Element of NAT st
( t . a = c & (2 * k1) + 1 = t . (DataLoc (c,i)) & k2 = ((m + n) + (2 * k1)) + 1 & m + y1 = t . (intpos k2) & m + yn = t . (intpos (k2 + 1)) & ( ( 1 <= y1 & yn <= n ) or y1 >= yn ) & not ( I is_closed_on t,Q & I is_halting_on t,Q & (IExec (I,Q,t)) . a = t . a & ( for j being Element of NAT st 1 <= j & j < (2 * k1) + 1 holds
(IExec (I,Q,t)) . (intpos ((m + n) + j)) = t . (intpos ((m + n) + j)) ) & ( y1 >= yn implies ( (IExec (I,Q,t)) . (DataLoc (c,i)) = (2 * k1) - 1 & ( for j being Element of NAT st 1 <= j & j <= n holds
(IExec (I,Q,t)) . (intpos (m + j)) = t . (intpos (m + j)) ) ) ) & ( y1 < yn implies ( (IExec (I,Q,t)) . (DataLoc (c,i)) = (2 * k1) + 3 & ( for j being Element of NAT st ( ( 1 <= j & j < y1 ) or ( yn < j & j <= n ) ) holds
(IExec (I,Q,t)) . (intpos (m + j)) = t . (intpos (m + j)) ) & ex ym being Element of NAT st
( y1 <= ym & ym <= yn & m + y1 = (IExec (I,Q,t)) . (intpos k2) & (m + ym) - 1 = (IExec (I,Q,t)) . (intpos (k2 + 1)) & (m + ym) + 1 = (IExec (I,Q,t)) . (intpos (k2 + 2)) & m + yn = (IExec (I,Q,t)) . (intpos (k2 + 3)) & ( for j being Element of NAT st y1 <= j & j < ym holds
(IExec (I,Q,t)) . (intpos (m + j)) <= (IExec (I,Q,t)) . (intpos (m + ym)) ) & ( for j being Element of NAT st ym < j & j <= yn holds
(IExec (I,Q,t)) . (intpos (m + j)) >= (IExec (I,Q,t)) . (intpos (m + ym)) ) ) ) ) & ( f1 is_FinSequence_on t,m & f2 is_FinSequence_on IExec (I,Q,t),m & len f1 = n & len f2 = n implies f1,f2 are_fiberwise_equipotent ) ) ) or while>0 (a,i,I) is_halting_on s,P )
assume  for t being 0 -started State of SCMPDS
for Q being Instruction-Sequence of SCMPDS
for f1, f2 being FinSequence of INT
for k1, k2, y1, yn being Element of NAT st t . a = c & (2 * k1) + 1 = t . (DataLoc (c,i)) & k2 = ((m + n) + (2 * k1)) + 1 & m + y1 = t . (intpos k2) & m + yn = t . (intpos (k2 + 1)) & ( ( 1 <= y1 & yn <= n ) or y1 >= yn ) holds
( I is_closed_on t,Q & I is_halting_on t,Q & (IExec (I,Q,t)) . a = t . a & ( for j being Element of NAT st 1 <= j & j < (2 * k1) + 1 holds
(IExec (I,Q,t)) . (intpos ((m + n) + j)) = t . (intpos ((m + n) + j)) ) & ( y1 >= yn implies ( (IExec (I,Q,t)) . (DataLoc (c,i)) = (2 * k1) - 1 & ( for j being Element of NAT st 1 <= j & j <= n holds
(IExec (I,Q,t)) . (intpos (m + j)) = t . (intpos (m + j)) ) ) ) & ( y1 < yn implies ( (IExec (I,Q,t)) . (DataLoc (c,i)) = (2 * k1) + 3 & ( for j being Element of NAT st ( ( 1 <= j & j < y1 ) or ( yn < j & j <= n ) ) holds
(IExec (I,Q,t)) . (intpos (m + j)) = t . (intpos (m + j)) ) & ex ym being Element of NAT st
( y1 <= ym & ym <= yn & m + y1 = (IExec (I,Q,t)) . (intpos k2) & (m + ym) - 1 = (IExec (I,Q,t)) . (intpos (k2 + 1)) & (m + ym) + 1 = (IExec (I,Q,t)) . (intpos (k2 + 2)) & m + yn = (IExec (I,Q,t)) . (intpos (k2 + 3)) & ( for j being Element of NAT st y1 <= j & j < ym holds
(IExec (I,Q,t)) . (intpos (m + j)) <= (IExec (I,Q,t)) . (intpos (m + ym)) ) & ( for j being Element of NAT st ym < j & j <= yn holds
(IExec (I,Q,t)) . (intpos (m + j)) >= (IExec (I,Q,t)) . (intpos (m + ym)) ) ) ) ) & ( f1 is_FinSequence_on t,m & f2 is_FinSequence_on IExec (I,Q,t),m & len f1 = n & len f2 = n implies f1,f2 are_fiberwise_equipotent ) ) ; ::_thesis: while>0 (a,i,I) is_halting_on s,P
hence  while>0 (a,i,I) is_halting_on s,P by A2, A6, A10, A11, A12, A3, A5, A7, A9, Lm4; ::_thesis: verum
end;

Lm6:  for P being Instruction-Sequence of SCMPDS
for s being 0 -started State of SCMPDS
for I being halt-free shiftable Program of SCMPDS
for a being Int_position
for i, c being Integer
for m, n, m1 being Element of NAT st s . a = c & 1 = s . (DataLoc (c,i)) & m1 = (m + n) + 1 & m + 1 = s . (intpos m1) & m + n = s . (intpos (m1 + 1)) & ( for t being 0 -started State of SCMPDS
for Q being Instruction-Sequence of SCMPDS
for f1, f2 being FinSequence of INT
for k1, k2, y1, yn being Element of NAT st t . a = c & (2 * k1) + 1 = t . (DataLoc (c,i)) & k2 = ((m + n) + (2 * k1)) + 1 & m + y1 = t . (intpos k2) & m + yn = t . (intpos (k2 + 1)) & ( ( 1 <= y1 & yn <= n ) or y1 >= yn ) holds
( I is_closed_on t,Q & I is_halting_on t,Q & (IExec (I,Q,t)) . a = t . a & ( for j being Element of NAT st 1 <= j & j < (2 * k1) + 1 holds
(IExec (I,Q,t)) . (intpos ((m + n) + j)) = t . (intpos ((m + n) + j)) ) & ( y1 >= yn implies ( (IExec (I,Q,t)) . (DataLoc (c,i)) = (2 * k1) - 1 & ( for j being Element of NAT st 1 <= j & j <= n holds
(IExec (I,Q,t)) . (intpos (m + j)) = t . (intpos (m + j)) ) ) ) & ( y1 < yn implies ( (IExec (I,Q,t)) . (DataLoc (c,i)) = (2 * k1) + 3 & ( for j being Element of NAT st ( ( 1 <= j & j < y1 ) or ( yn < j & j <= n ) ) holds
(IExec (I,Q,t)) . (intpos (m + j)) = t . (intpos (m + j)) ) & ex ym being Element of NAT st
( y1 <= ym & ym <= yn & m + y1 = (IExec (I,Q,t)) . (intpos k2) & (m + ym) - 1 = (IExec (I,Q,t)) . (intpos (k2 + 1)) & (m + ym) + 1 = (IExec (I,Q,t)) . (intpos (k2 + 2)) & m + yn = (IExec (I,Q,t)) . (intpos (k2 + 3)) & ( for j being Element of NAT st y1 <= j & j < ym holds
(IExec (I,Q,t)) . (intpos (m + j)) <= (IExec (I,Q,t)) . (intpos (m + ym)) ) & ( for j being Element of NAT st ym < j & j <= yn holds
(IExec (I,Q,t)) . (intpos (m + j)) >= (IExec (I,Q,t)) . (intpos (m + ym)) ) ) ) ) & ( f1 is_FinSequence_on t,m & f2 is_FinSequence_on IExec (I,Q,t),m & len f1 = n & len f2 = n implies f1,f2 are_fiberwise_equipotent ) ) ) holds
( while>0 (a,i,I) is_halting_on s,P & while>0 (a,i,I) is_closed_on s,P )
proof
let P be Instruction-Sequence of SCMPDS; ::_thesis: for s being 0 -started State of SCMPDS
for I being halt-free shiftable Program of SCMPDS
for a being Int_position
for i, c being Integer
for m, n, m1 being Element of NAT st s . a = c & 1 = s . (DataLoc (c,i)) & m1 = (m + n) + 1 & m + 1 = s . (intpos m1) & m + n = s . (intpos (m1 + 1)) & ( for t being 0 -started State of SCMPDS
for Q being Instruction-Sequence of SCMPDS
for f1, f2 being FinSequence of INT
for k1, k2, y1, yn being Element of NAT st t . a = c & (2 * k1) + 1 = t . (DataLoc (c,i)) & k2 = ((m + n) + (2 * k1)) + 1 & m + y1 = t . (intpos k2) & m + yn = t . (intpos (k2 + 1)) & ( ( 1 <= y1 & yn <= n ) or y1 >= yn ) holds
( I is_closed_on t,Q & I is_halting_on t,Q & (IExec (I,Q,t)) . a = t . a & ( for j being Element of NAT st 1 <= j & j < (2 * k1) + 1 holds
(IExec (I,Q,t)) . (intpos ((m + n) + j)) = t . (intpos ((m + n) + j)) ) & ( y1 >= yn implies ( (IExec (I,Q,t)) . (DataLoc (c,i)) = (2 * k1) - 1 & ( for j being Element of NAT st 1 <= j & j <= n holds
(IExec (I,Q,t)) . (intpos (m + j)) = t . (intpos (m + j)) ) ) ) & ( y1 < yn implies ( (IExec (I,Q,t)) . (DataLoc (c,i)) = (2 * k1) + 3 & ( for j being Element of NAT st ( ( 1 <= j & j < y1 ) or ( yn < j & j <= n ) ) holds
(IExec (I,Q,t)) . (intpos (m + j)) = t . (intpos (m + j)) ) & ex ym being Element of NAT st
( y1 <= ym & ym <= yn & m + y1 = (IExec (I,Q,t)) . (intpos k2) & (m + ym) - 1 = (IExec (I,Q,t)) . (intpos (k2 + 1)) & (m + ym) + 1 = (IExec (I,Q,t)) . (intpos (k2 + 2)) & m + yn = (IExec (I,Q,t)) . (intpos (k2 + 3)) & ( for j being Element of NAT st y1 <= j & j < ym holds
(IExec (I,Q,t)) . (intpos (m + j)) <= (IExec (I,Q,t)) . (intpos (m + ym)) ) & ( for j being Element of NAT st ym < j & j <= yn holds
(IExec (I,Q,t)) . (intpos (m + j)) >= (IExec (I,Q,t)) . (intpos (m + ym)) ) ) ) ) & ( f1 is_FinSequence_on t,m & f2 is_FinSequence_on IExec (I,Q,t),m & len f1 = n & len f2 = n implies f1,f2 are_fiberwise_equipotent ) ) ) holds
( while>0 (a,i,I) is_halting_on s,P & while>0 (a,i,I) is_closed_on s,P )
let s be 0 -started State of SCMPDS; ::_thesis: for I being halt-free shiftable Program of SCMPDS
for a being Int_position
for i, c being Integer
for m, n, m1 being Element of NAT st s . a = c & 1 = s . (DataLoc (c,i)) & m1 = (m + n) + 1 & m + 1 = s . (intpos m1) & m + n = s . (intpos (m1 + 1)) & ( for t being 0 -started State of SCMPDS
for Q being Instruction-Sequence of SCMPDS
for f1, f2 being FinSequence of INT
for k1, k2, y1, yn being Element of NAT st t . a = c & (2 * k1) + 1 = t . (DataLoc (c,i)) & k2 = ((m + n) + (2 * k1)) + 1 & m + y1 = t . (intpos k2) & m + yn = t . (intpos (k2 + 1)) & ( ( 1 <= y1 & yn <= n ) or y1 >= yn ) holds
( I is_closed_on t,Q & I is_halting_on t,Q & (IExec (I,Q,t)) . a = t . a & ( for j being Element of NAT st 1 <= j & j < (2 * k1) + 1 holds
(IExec (I,Q,t)) . (intpos ((m + n) + j)) = t . (intpos ((m + n) + j)) ) & ( y1 >= yn implies ( (IExec (I,Q,t)) . (DataLoc (c,i)) = (2 * k1) - 1 & ( for j being Element of NAT st 1 <= j & j <= n holds
(IExec (I,Q,t)) . (intpos (m + j)) = t . (intpos (m + j)) ) ) ) & ( y1 < yn implies ( (IExec (I,Q,t)) . (DataLoc (c,i)) = (2 * k1) + 3 & ( for j being Element of NAT st ( ( 1 <= j & j < y1 ) or ( yn < j & j <= n ) ) holds
(IExec (I,Q,t)) . (intpos (m + j)) = t . (intpos (m + j)) ) & ex ym being Element of NAT st
( y1 <= ym & ym <= yn & m + y1 = (IExec (I,Q,t)) . (intpos k2) & (m + ym) - 1 = (IExec (I,Q,t)) . (intpos (k2 + 1)) & (m + ym) + 1 = (IExec (I,Q,t)) . (intpos (k2 + 2)) & m + yn = (IExec (I,Q,t)) . (intpos (k2 + 3)) & ( for j being Element of NAT st y1 <= j & j < ym holds
(IExec (I,Q,t)) . (intpos (m + j)) <= (IExec (I,Q,t)) . (intpos (m + ym)) ) & ( for j being Element of NAT st ym < j & j <= yn holds
(IExec (I,Q,t)) . (intpos (m + j)) >= (IExec (I,Q,t)) . (intpos (m + ym)) ) ) ) ) & ( f1 is_FinSequence_on t,m & f2 is_FinSequence_on IExec (I,Q,t),m & len f1 = n & len f2 = n implies f1,f2 are_fiberwise_equipotent ) ) ) holds
( while>0 (a,i,I) is_halting_on s,P & while>0 (a,i,I) is_closed_on s,P )
let I be halt-free shiftable Program of SCMPDS; ::_thesis: for a being Int_position
for i, c being Integer
for m, n, m1 being Element of NAT st s . a = c & 1 = s . (DataLoc (c,i)) & m1 = (m + n) + 1 & m + 1 = s . (intpos m1) & m + n = s . (intpos (m1 + 1)) & ( for t being 0 -started State of SCMPDS
for Q being Instruction-Sequence of SCMPDS
for f1, f2 being FinSequence of INT
for k1, k2, y1, yn being Element of NAT st t . a = c & (2 * k1) + 1 = t . (DataLoc (c,i)) & k2 = ((m + n) + (2 * k1)) + 1 & m + y1 = t . (intpos k2) & m + yn = t . (intpos (k2 + 1)) & ( ( 1 <= y1 & yn <= n ) or y1 >= yn ) holds
( I is_closed_on t,Q & I is_halting_on t,Q & (IExec (I,Q,t)) . a = t . a & ( for j being Element of NAT st 1 <= j & j < (2 * k1) + 1 holds
(IExec (I,Q,t)) . (intpos ((m + n) + j)) = t . (intpos ((m + n) + j)) ) & ( y1 >= yn implies ( (IExec (I,Q,t)) . (DataLoc (c,i)) = (2 * k1) - 1 & ( for j being Element of NAT st 1 <= j & j <= n holds
(IExec (I,Q,t)) . (intpos (m + j)) = t . (intpos (m + j)) ) ) ) & ( y1 < yn implies ( (IExec (I,Q,t)) . (DataLoc (c,i)) = (2 * k1) + 3 & ( for j being Element of NAT st ( ( 1 <= j & j < y1 ) or ( yn < j & j <= n ) ) holds
(IExec (I,Q,t)) . (intpos (m + j)) = t . (intpos (m + j)) ) & ex ym being Element of NAT st
( y1 <= ym & ym <= yn & m + y1 = (IExec (I,Q,t)) . (intpos k2) & (m + ym) - 1 = (IExec (I,Q,t)) . (intpos (k2 + 1)) & (m + ym) + 1 = (IExec (I,Q,t)) . (intpos (k2 + 2)) & m + yn = (IExec (I,Q,t)) . (intpos (k2 + 3)) & ( for j being Element of NAT st y1 <= j & j < ym holds
(IExec (I,Q,t)) . (intpos (m + j)) <= (IExec (I,Q,t)) . (intpos (m + ym)) ) & ( for j being Element of NAT st ym < j & j <= yn holds
(IExec (I,Q,t)) . (intpos (m + j)) >= (IExec (I,Q,t)) . (intpos (m + ym)) ) ) ) ) & ( f1 is_FinSequence_on t,m & f2 is_FinSequence_on IExec (I,Q,t),m & len f1 = n & len f2 = n implies f1,f2 are_fiberwise_equipotent ) ) ) holds
( while>0 (a,i,I) is_halting_on s,P & while>0 (a,i,I) is_closed_on s,P )
let a be Int_position; ::_thesis: for i, c being Integer
for m, n, m1 being Element of NAT st s . a = c & 1 = s . (DataLoc (c,i)) & m1 = (m + n) + 1 & m + 1 = s . (intpos m1) & m + n = s . (intpos (m1 + 1)) & ( for t being 0 -started State of SCMPDS
for Q being Instruction-Sequence of SCMPDS
for f1, f2 being FinSequence of INT
for k1, k2, y1, yn being Element of NAT st t . a = c & (2 * k1) + 1 = t . (DataLoc (c,i)) & k2 = ((m + n) + (2 * k1)) + 1 & m + y1 = t . (intpos k2) & m + yn = t . (intpos (k2 + 1)) & ( ( 1 <= y1 & yn <= n ) or y1 >= yn ) holds
( I is_closed_on t,Q & I is_halting_on t,Q & (IExec (I,Q,t)) . a = t . a & ( for j being Element of NAT st 1 <= j & j < (2 * k1) + 1 holds
(IExec (I,Q,t)) . (intpos ((m + n) + j)) = t . (intpos ((m + n) + j)) ) & ( y1 >= yn implies ( (IExec (I,Q,t)) . (DataLoc (c,i)) = (2 * k1) - 1 & ( for j being Element of NAT st 1 <= j & j <= n holds
(IExec (I,Q,t)) . (intpos (m + j)) = t . (intpos (m + j)) ) ) ) & ( y1 < yn implies ( (IExec (I,Q,t)) . (DataLoc (c,i)) = (2 * k1) + 3 & ( for j being Element of NAT st ( ( 1 <= j & j < y1 ) or ( yn < j & j <= n ) ) holds
(IExec (I,Q,t)) . (intpos (m + j)) = t . (intpos (m + j)) ) & ex ym being Element of NAT st
( y1 <= ym & ym <= yn & m + y1 = (IExec (I,Q,t)) . (intpos k2) & (m + ym) - 1 = (IExec (I,Q,t)) . (intpos (k2 + 1)) & (m + ym) + 1 = (IExec (I,Q,t)) . (intpos (k2 + 2)) & m + yn = (IExec (I,Q,t)) . (intpos (k2 + 3)) & ( for j being Element of NAT st y1 <= j & j < ym holds
(IExec (I,Q,t)) . (intpos (m + j)) <= (IExec (I,Q,t)) . (intpos (m + ym)) ) & ( for j being Element of NAT st ym < j & j <= yn holds
(IExec (I,Q,t)) . (intpos (m + j)) >= (IExec (I,Q,t)) . (intpos (m + ym)) ) ) ) ) & ( f1 is_FinSequence_on t,m & f2 is_FinSequence_on IExec (I,Q,t),m & len f1 = n & len f2 = n implies f1,f2 are_fiberwise_equipotent ) ) ) holds
( while>0 (a,i,I) is_halting_on s,P & while>0 (a,i,I) is_closed_on s,P )
let i, c be Integer; ::_thesis: for m, n, m1 being Element of NAT st s . a = c & 1 = s . (DataLoc (c,i)) & m1 = (m + n) + 1 & m + 1 = s . (intpos m1) & m + n = s . (intpos (m1 + 1)) & ( for t being 0 -started State of SCMPDS
for Q being Instruction-Sequence of SCMPDS
for f1, f2 being FinSequence of INT
for k1, k2, y1, yn being Element of NAT st t . a = c & (2 * k1) + 1 = t . (DataLoc (c,i)) & k2 = ((m + n) + (2 * k1)) + 1 & m + y1 = t . (intpos k2) & m + yn = t . (intpos (k2 + 1)) & ( ( 1 <= y1 & yn <= n ) or y1 >= yn ) holds
( I is_closed_on t,Q & I is_halting_on t,Q & (IExec (I,Q,t)) . a = t . a & ( for j being Element of NAT st 1 <= j & j < (2 * k1) + 1 holds
(IExec (I,Q,t)) . (intpos ((m + n) + j)) = t . (intpos ((m + n) + j)) ) & ( y1 >= yn implies ( (IExec (I,Q,t)) . (DataLoc (c,i)) = (2 * k1) - 1 & ( for j being Element of NAT st 1 <= j & j <= n holds
(IExec (I,Q,t)) . (intpos (m + j)) = t . (intpos (m + j)) ) ) ) & ( y1 < yn implies ( (IExec (I,Q,t)) . (DataLoc (c,i)) = (2 * k1) + 3 & ( for j being Element of NAT st ( ( 1 <= j & j < y1 ) or ( yn < j & j <= n ) ) holds
(IExec (I,Q,t)) . (intpos (m + j)) = t . (intpos (m + j)) ) & ex ym being Element of NAT st
( y1 <= ym & ym <= yn & m + y1 = (IExec (I,Q,t)) . (intpos k2) & (m + ym) - 1 = (IExec (I,Q,t)) . (intpos (k2 + 1)) & (m + ym) + 1 = (IExec (I,Q,t)) . (intpos (k2 + 2)) & m + yn = (IExec (I,Q,t)) . (intpos (k2 + 3)) & ( for j being Element of NAT st y1 <= j & j < ym holds
(IExec (I,Q,t)) . (intpos (m + j)) <= (IExec (I,Q,t)) . (intpos (m + ym)) ) & ( for j being Element of NAT st ym < j & j <= yn holds
(IExec (I,Q,t)) . (intpos (m + j)) >= (IExec (I,Q,t)) . (intpos (m + ym)) ) ) ) ) & ( f1 is_FinSequence_on t,m & f2 is_FinSequence_on IExec (I,Q,t),m & len f1 = n & len f2 = n implies f1,f2 are_fiberwise_equipotent ) ) ) holds
( while>0 (a,i,I) is_halting_on s,P & while>0 (a,i,I) is_closed_on s,P )
let m, n, m1 be Element of NAT ; ::_thesis: ( s . a = c & 1 = s . (DataLoc (c,i)) & m1 = (m + n) + 1 & m + 1 = s . (intpos m1) & m + n = s . (intpos (m1 + 1)) & ( for t being 0 -started State of SCMPDS
for Q being Instruction-Sequence of SCMPDS
for f1, f2 being FinSequence of INT
for k1, k2, y1, yn being Element of NAT st t . a = c & (2 * k1) + 1 = t . (DataLoc (c,i)) & k2 = ((m + n) + (2 * k1)) + 1 & m + y1 = t . (intpos k2) & m + yn = t . (intpos (k2 + 1)) & ( ( 1 <= y1 & yn <= n ) or y1 >= yn ) holds
( I is_closed_on t,Q & I is_halting_on t,Q & (IExec (I,Q,t)) . a = t . a & ( for j being Element of NAT st 1 <= j & j < (2 * k1) + 1 holds
(IExec (I,Q,t)) . (intpos ((m + n) + j)) = t . (intpos ((m + n) + j)) ) & ( y1 >= yn implies ( (IExec (I,Q,t)) . (DataLoc (c,i)) = (2 * k1) - 1 & ( for j being Element of NAT st 1 <= j & j <= n holds
(IExec (I,Q,t)) . (intpos (m + j)) = t . (intpos (m + j)) ) ) ) & ( y1 < yn implies ( (IExec (I,Q,t)) . (DataLoc (c,i)) = (2 * k1) + 3 & ( for j being Element of NAT st ( ( 1 <= j & j < y1 ) or ( yn < j & j <= n ) ) holds
(IExec (I,Q,t)) . (intpos (m + j)) = t . (intpos (m + j)) ) & ex ym being Element of NAT st
( y1 <= ym & ym <= yn & m + y1 = (IExec (I,Q,t)) . (intpos k2) & (m + ym) - 1 = (IExec (I,Q,t)) . (intpos (k2 + 1)) & (m + ym) + 1 = (IExec (I,Q,t)) . (intpos (k2 + 2)) & m + yn = (IExec (I,Q,t)) . (intpos (k2 + 3)) & ( for j being Element of NAT st y1 <= j & j < ym holds
(IExec (I,Q,t)) . (intpos (m + j)) <= (IExec (I,Q,t)) . (intpos (m + ym)) ) & ( for j being Element of NAT st ym < j & j <= yn holds
(IExec (I,Q,t)) . (intpos (m + j)) >= (IExec (I,Q,t)) . (intpos (m + ym)) ) ) ) ) & ( f1 is_FinSequence_on t,m & f2 is_FinSequence_on IExec (I,Q,t),m & len f1 = n & len f2 = n implies f1,f2 are_fiberwise_equipotent ) ) ) implies ( while>0 (a,i,I) is_halting_on s,P & while>0 (a,i,I) is_closed_on s,P ) )
set b = DataLoc (c,i);
assume A2: s . a = c ; ::_thesis: ( not 1 = s . (DataLoc (c,i)) or not m1 = (m + n) + 1 or not m + 1 = s . (intpos m1) or not m + n = s . (intpos (m1 + 1)) or ex t being 0 -started State of SCMPDS ex Q being Instruction-Sequence of SCMPDS ex f1, f2 being FinSequence of INT ex k1, k2, y1, yn being Element of NAT st
( t . a = c & (2 * k1) + 1 = t . (DataLoc (c,i)) & k2 = ((m + n) + (2 * k1)) + 1 & m + y1 = t . (intpos k2) & m + yn = t . (intpos (k2 + 1)) & ( ( 1 <= y1 & yn <= n ) or y1 >= yn ) & not ( I is_closed_on t,Q & I is_halting_on t,Q & (IExec (I,Q,t)) . a = t . a & ( for j being Element of NAT st 1 <= j & j < (2 * k1) + 1 holds
(IExec (I,Q,t)) . (intpos ((m + n) + j)) = t . (intpos ((m + n) + j)) ) & ( y1 >= yn implies ( (IExec (I,Q,t)) . (DataLoc (c,i)) = (2 * k1) - 1 & ( for j being Element of NAT st 1 <= j & j <= n holds
(IExec (I,Q,t)) . (intpos (m + j)) = t . (intpos (m + j)) ) ) ) & ( y1 < yn implies ( (IExec (I,Q,t)) . (DataLoc (c,i)) = (2 * k1) + 3 & ( for j being Element of NAT st ( ( 1 <= j & j < y1 ) or ( yn < j & j <= n ) ) holds
(IExec (I,Q,t)) . (intpos (m + j)) = t . (intpos (m + j)) ) & ex ym being Element of NAT st
( y1 <= ym & ym <= yn & m + y1 = (IExec (I,Q,t)) . (intpos k2) & (m + ym) - 1 = (IExec (I,Q,t)) . (intpos (k2 + 1)) & (m + ym) + 1 = (IExec (I,Q,t)) . (intpos (k2 + 2)) & m + yn = (IExec (I,Q,t)) . (intpos (k2 + 3)) & ( for j being Element of NAT st y1 <= j & j < ym holds
(IExec (I,Q,t)) . (intpos (m + j)) <= (IExec (I,Q,t)) . (intpos (m + ym)) ) & ( for j being Element of NAT st ym < j & j <= yn holds
(IExec (I,Q,t)) . (intpos (m + j)) >= (IExec (I,Q,t)) . (intpos (m + ym)) ) ) ) ) & ( f1 is_FinSequence_on t,m & f2 is_FinSequence_on IExec (I,Q,t),m & len f1 = n & len f2 = n implies f1,f2 are_fiberwise_equipotent ) ) ) or ( while>0 (a,i,I) is_halting_on s,P & while>0 (a,i,I) is_closed_on s,P ) )
assume A3: 1 = s . (DataLoc (c,i)) ; ::_thesis: ( not m1 = (m + n) + 1 or not m + 1 = s . (intpos m1) or not m + n = s . (intpos (m1 + 1)) or ex t being 0 -started State of SCMPDS ex Q being Instruction-Sequence of SCMPDS ex f1, f2 being FinSequence of INT ex k1, k2, y1, yn being Element of NAT st
( t . a = c & (2 * k1) + 1 = t . (DataLoc (c,i)) & k2 = ((m + n) + (2 * k1)) + 1 & m + y1 = t . (intpos k2) & m + yn = t . (intpos (k2 + 1)) & ( ( 1 <= y1 & yn <= n ) or y1 >= yn ) & not ( I is_closed_on t,Q & I is_halting_on t,Q & (IExec (I,Q,t)) . a = t . a & ( for j being Element of NAT st 1 <= j & j < (2 * k1) + 1 holds
(IExec (I,Q,t)) . (intpos ((m + n) + j)) = t . (intpos ((m + n) + j)) ) & ( y1 >= yn implies ( (IExec (I,Q,t)) . (DataLoc (c,i)) = (2 * k1) - 1 & ( for j being Element of NAT st 1 <= j & j <= n holds
(IExec (I,Q,t)) . (intpos (m + j)) = t . (intpos (m + j)) ) ) ) & ( y1 < yn implies ( (IExec (I,Q,t)) . (DataLoc (c,i)) = (2 * k1) + 3 & ( for j being Element of NAT st ( ( 1 <= j & j < y1 ) or ( yn < j & j <= n ) ) holds
(IExec (I,Q,t)) . (intpos (m + j)) = t . (intpos (m + j)) ) & ex ym being Element of NAT st
( y1 <= ym & ym <= yn & m + y1 = (IExec (I,Q,t)) . (intpos k2) & (m + ym) - 1 = (IExec (I,Q,t)) . (intpos (k2 + 1)) & (m + ym) + 1 = (IExec (I,Q,t)) . (intpos (k2 + 2)) & m + yn = (IExec (I,Q,t)) . (intpos (k2 + 3)) & ( for j being Element of NAT st y1 <= j & j < ym holds
(IExec (I,Q,t)) . (intpos (m + j)) <= (IExec (I,Q,t)) . (intpos (m + ym)) ) & ( for j being Element of NAT st ym < j & j <= yn holds
(IExec (I,Q,t)) . (intpos (m + j)) >= (IExec (I,Q,t)) . (intpos (m + ym)) ) ) ) ) & ( f1 is_FinSequence_on t,m & f2 is_FinSequence_on IExec (I,Q,t),m & len f1 = n & len f2 = n implies f1,f2 are_fiberwise_equipotent ) ) ) or ( while>0 (a,i,I) is_halting_on s,P & while>0 (a,i,I) is_closed_on s,P ) )
assume that
A4: m1 = (m + n) + 1 and
A5: m + 1 = s . (intpos m1) and
A6: m + n = s . (intpos (m1 + 1)) ; ::_thesis: ( ex t being 0 -started State of SCMPDS ex Q being Instruction-Sequence of SCMPDS ex f1, f2 being FinSequence of INT ex k1, k2, y1, yn being Element of NAT st
( t . a = c & (2 * k1) + 1 = t . (DataLoc (c,i)) & k2 = ((m + n) + (2 * k1)) + 1 & m + y1 = t . (intpos k2) & m + yn = t . (intpos (k2 + 1)) & ( ( 1 <= y1 & yn <= n ) or y1 >= yn ) & not ( I is_closed_on t,Q & I is_halting_on t,Q & (IExec (I,Q,t)) . a = t . a & ( for j being Element of NAT st 1 <= j & j < (2 * k1) + 1 holds
(IExec (I,Q,t)) . (intpos ((m + n) + j)) = t . (intpos ((m + n) + j)) ) & ( y1 >= yn implies ( (IExec (I,Q,t)) . (DataLoc (c,i)) = (2 * k1) - 1 & ( for j being Element of NAT st 1 <= j & j <= n holds
(IExec (I,Q,t)) . (intpos (m + j)) = t . (intpos (m + j)) ) ) ) & ( y1 < yn implies ( (IExec (I,Q,t)) . (DataLoc (c,i)) = (2 * k1) + 3 & ( for j being Element of NAT st ( ( 1 <= j & j < y1 ) or ( yn < j & j <= n ) ) holds
(IExec (I,Q,t)) . (intpos (m + j)) = t . (intpos (m + j)) ) & ex ym being Element of NAT st
( y1 <= ym & ym <= yn & m + y1 = (IExec (I,Q,t)) . (intpos k2) & (m + ym) - 1 = (IExec (I,Q,t)) . (intpos (k2 + 1)) & (m + ym) + 1 = (IExec (I,Q,t)) . (intpos (k2 + 2)) & m + yn = (IExec (I,Q,t)) . (intpos (k2 + 3)) & ( for j being Element of NAT st y1 <= j & j < ym holds
(IExec (I,Q,t)) . (intpos (m + j)) <= (IExec (I,Q,t)) . (intpos (m + ym)) ) & ( for j being Element of NAT st ym < j & j <= yn holds
(IExec (I,Q,t)) . (intpos (m + j)) >= (IExec (I,Q,t)) . (intpos (m + ym)) ) ) ) ) & ( f1 is_FinSequence_on t,m & f2 is_FinSequence_on IExec (I,Q,t),m & len f1 = n & len f2 = n implies f1,f2 are_fiberwise_equipotent ) ) ) or ( while>0 (a,i,I) is_halting_on s,P & while>0 (a,i,I) is_closed_on s,P ) )
assume A7: for t being 0 -started State of SCMPDS
for Q being Instruction-Sequence of SCMPDS
for f1, f2 being FinSequence of INT
for k1, k2, y1, yn being Element of NAT st t . a = c & (2 * k1) + 1 = t . (DataLoc (c,i)) & k2 = ((m + n) + (2 * k1)) + 1 & m + y1 = t . (intpos k2) & m + yn = t . (intpos (k2 + 1)) & ( ( 1 <= y1 & yn <= n ) or y1 >= yn ) holds
( I is_closed_on t,Q & I is_halting_on t,Q & (IExec (I,Q,t)) . a = t . a & ( for j being Element of NAT st 1 <= j & j < (2 * k1) + 1 holds
(IExec (I,Q,t)) . (intpos ((m + n) + j)) = t . (intpos ((m + n) + j)) ) & ( y1 >= yn implies ( (IExec (I,Q,t)) . (DataLoc (c,i)) = (2 * k1) - 1 & ( for j being Element of NAT st 1 <= j & j <= n holds
(IExec (I,Q,t)) . (intpos (m + j)) = t . (intpos (m + j)) ) ) ) & ( y1 < yn implies ( (IExec (I,Q,t)) . (DataLoc (c,i)) = (2 * k1) + 3 & ( for j being Element of NAT st ( ( 1 <= j & j < y1 ) or ( yn < j & j <= n ) ) holds
(IExec (I,Q,t)) . (intpos (m + j)) = t . (intpos (m + j)) ) & ex ym being Element of NAT st
( y1 <= ym & ym <= yn & m + y1 = (IExec (I,Q,t)) . (intpos k2) & (m + ym) - 1 = (IExec (I,Q,t)) . (intpos (k2 + 1)) & (m + ym) + 1 = (IExec (I,Q,t)) . (intpos (k2 + 2)) & m + yn = (IExec (I,Q,t)) . (intpos (k2 + 3)) & ( for j being Element of NAT st y1 <= j & j < ym holds
(IExec (I,Q,t)) . (intpos (m + j)) <= (IExec (I,Q,t)) . (intpos (m + ym)) ) & ( for j being Element of NAT st ym < j & j <= yn holds
(IExec (I,Q,t)) . (intpos (m + j)) >= (IExec (I,Q,t)) . (intpos (m + ym)) ) ) ) ) & ( f1 is_FinSequence_on t,m & f2 is_FinSequence_on IExec (I,Q,t),m & len f1 = n & len f2 = n implies f1,f2 are_fiberwise_equipotent ) ) ; ::_thesis: ( while>0 (a,i,I) is_halting_on s,P & while>0 (a,i,I) is_closed_on s,P )
A8: now__::_thesis:_for_v_being_0_-started_State_of_SCMPDS
for_V_being_Instruction-Sequence_of_SCMPDS_st_DataPart_v_=_DataPart_s_holds_
while>0_(a,i,I)_is_halting_on_v,V
let v be 0 -started State of SCMPDS; ::_thesis: for V being Instruction-Sequence of SCMPDS st DataPart v = DataPart s holds
while>0 (a,i,I) is_halting_on v,V
let V be Instruction-Sequence of SCMPDS; ::_thesis: ( DataPart v = DataPart s implies while>0 (a,i,I) is_halting_on v,V )
assume A9: DataPart v = DataPart s ; ::_thesis: while>0 (a,i,I) is_halting_on v,V
then A10: 1 = v . (DataLoc (c,i)) by A3, SCMPDS_4:8;
A11: m + n = v . (intpos (m1 + 1)) by A6, A9, SCMPDS_4:8;
A12: m + 1 = v . (intpos m1) by A5, A9, SCMPDS_4:8;
 v . a = c by A2, A9, SCMPDS_4:8;
hence  while>0 (a,i,I) is_halting_on v,V by A4, A7, A10, A12, A11, Lm5; ::_thesis: verum
end;
hence  while>0 (a,i,I) is_halting_on s,P ; ::_thesis: while>0 (a,i,I) is_closed_on s,P
thus  while>0 (a,i,I) is_closed_on s,P by A8, Th3; ::_thesis: verum
end;

begin

definition
func Partition  ->  Program of SCMPDS equals :: SCPQSORT:def 1
((((((((GBP,5) := (GBP,4)) ';' (SubFrom (GBP,5,GBP,2))) ';' ((GBP,3) := (GBP,2))) ';' (AddTo (GBP,3,1))) ';' (while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))))))) ';' ((GBP,6) := ((intpos 4),0))) ';' (((intpos 4),0) := ((intpos 2),0))) ';' (((intpos 2),0) := (GBP,6));
coherence
 ((((((((GBP,5) := (GBP,4)) ';' (SubFrom (GBP,5,GBP,2))) ';' ((GBP,3) := (GBP,2))) ';' (AddTo (GBP,3,1))) ';' (while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))))))) ';' ((GBP,6) := ((intpos 4),0))) ';' (((intpos 4),0) := ((intpos 2),0))) ';' (((intpos 2),0) := (GBP,6)) is Program of SCMPDS ;
end;

:: deftheorem  defines Partition SCPQSORT:def_1_:_
 Partition = ((((((((GBP,5) := (GBP,4)) ';' (SubFrom (GBP,5,GBP,2))) ';' ((GBP,3) := (GBP,2))) ';' (AddTo (GBP,3,1))) ';' (while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))))))) ';' ((GBP,6) := ((intpos 4),0))) ';' (((intpos 4),0) := ((intpos 2),0))) ';' (((intpos 2),0) := (GBP,6));

begin

definition
let n, p0 be Element of NAT ;
func QuickSort (n,p0) ->  Program of SCMPDS equals :: SCPQSORT:def 2
((((GBP := 0) ';' (SBP := 1)) ';' ((SBP,(p0 + n)) := (p0 + 1))) ';' ((SBP,((p0 + n) + 1)) := (p0 + n))) ';' (while>0 (GBP,1,((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Partition) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2)))))))));
coherence
 ((((GBP := 0) ';' (SBP := 1)) ';' ((SBP,(p0 + n)) := (p0 + 1))) ';' ((SBP,((p0 + n) + 1)) := (p0 + n))) ';' (while>0 (GBP,1,((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Partition) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))))) is Program of SCMPDS ;
end;

:: deftheorem  defines QuickSort SCPQSORT:def_2_:_
 for n, p0 being Element of NAT holds QuickSort (n,p0) = ((((GBP := 0) ';' (SBP := 1)) ';' ((SBP,(p0 + n)) := (p0 + 1))) ';' ((SBP,((p0 + n) + 1)) := (p0 + n))) ';' (while>0 (GBP,1,((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Partition) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2)))))))));

set i1 = (GBP,7) := (GBP,5);

set i2 = AddTo (GBP,5,(- 1));

set i3 = (GBP,6) := ((intpos 4),0);

set i4 = SubFrom (GBP,6,(intpos 2),0);

set i5 = AddTo (GBP,4,(- 1));

set i6 = AddTo (GBP,7,(- 1));

set i7 = Load ((GBP,5) := 0);

set IF1 = if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0)));

set WB1 = (((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))));

set WH1 = while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))));

set j1 = (GBP,5) := (GBP,7);

set j2 = AddTo (GBP,7,(- 1));

set j3 = (GBP,6) := ((intpos 2),0);

set j4 = SubFrom (GBP,6,(intpos 3),0);

set j5 = AddTo (GBP,3,1);

set j6 = AddTo (GBP,5,(- 1));

set j7 = Load ((GBP,7) := 0);

set IF2 = if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)));

set WB2 = (((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))));

set WH2 = while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))));

set k1 = (GBP,5) := (GBP,4);

set k2 = SubFrom (GBP,5,GBP,2);

set k3 = (GBP,3) := (GBP,2);

set k4 = AddTo (GBP,3,1);

set K4 = ((((GBP,5) := (GBP,4)) ';' (SubFrom (GBP,5,GBP,2))) ';' ((GBP,3) := (GBP,2))) ';' (AddTo (GBP,3,1));

set k5 = (GBP,6) := ((intpos 4),0);

set k6 = ((intpos 4),0) := ((intpos 3),0);

set k7 = ((intpos 3),0) := (GBP,6);

set k8 = AddTo (GBP,5,(- 2));

set k9 = AddTo (GBP,3,1);

set k0 = AddTo (GBP,4,(- 1));

set IF3 = if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))));

set WB3 = ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))));

set WH3 = while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))));

set j8 = (GBP,6) := ((intpos 4),0);

set j9 = ((intpos 4),0) := ((intpos 2),0);

set j0 = ((intpos 2),0) := (GBP,6);

set a1 = intpos 1;

set a2 = intpos 2;

set a3 = intpos 3;

set a4 = intpos 4;

set a5 = intpos 5;

set a6 = intpos 6;

set a7 = intpos 7;

Lm7:  card ((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))) = 9
proof
thus  card ((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))) = (card (((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0)))) + (card (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))) by AFINSQ_1:17
.= 4 + (card (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))) by Th4
.= 4 + (((card ((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1))))) + (card (Load ((GBP,5) := 0)))) + 2) by SCMPDS_6:65
.= 4 + ((2 + (card (Load ((GBP,5) := 0)))) + 2) by SCMP_GCD:5
.= 4 + ((2 + 1) + 2) by COMPOS_1:54
.= 9 ; ::_thesis: verum
end;

Lm8:  for P being Instruction-Sequence of SCMPDS
for s being 0 -started State of SCMPDS
for md, me being Element of NAT st s . (intpos 2) = md & s . (intpos 4) = me & md >= 8 & me >= 8 & s . GBP = 0 holds
( (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),P,s)) . GBP = 0 & (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),P,s)) . (intpos 1) = s . (intpos 1) & (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),P,s)) . (intpos 2) = s . (intpos 2) & (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),P,s)) . (intpos 3) = s . (intpos 3) & ( for i being Element of NAT st i >= 8 holds
(IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),P,s)) . (intpos i) = s . (intpos i) ) & ( s . (intpos md) < s . (intpos me) implies ( (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),P,s)) . (intpos 5) = (s . (intpos 5)) - 1 & (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),P,s)) . (intpos 4) = (s . (intpos 4)) - 1 & (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),P,s)) . (intpos 7) = (s . (intpos 5)) - 1 ) ) & ( s . (intpos md) >= s . (intpos me) implies ( (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),P,s)) . (intpos 5) = 0 & (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),P,s)) . (intpos 4) = s . (intpos 4) & (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),P,s)) . (intpos 7) = s . (intpos 5) ) ) )
proof
let P be Instruction-Sequence of SCMPDS; ::_thesis: for s being 0 -started State of SCMPDS
for md, me being Element of NAT st s . (intpos 2) = md & s . (intpos 4) = me & md >= 8 & me >= 8 & s . GBP = 0 holds
( (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),P,s)) . GBP = 0 & (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),P,s)) . (intpos 1) = s . (intpos 1) & (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),P,s)) . (intpos 2) = s . (intpos 2) & (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),P,s)) . (intpos 3) = s . (intpos 3) & ( for i being Element of NAT st i >= 8 holds
(IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),P,s)) . (intpos i) = s . (intpos i) ) & ( s . (intpos md) < s . (intpos me) implies ( (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),P,s)) . (intpos 5) = (s . (intpos 5)) - 1 & (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),P,s)) . (intpos 4) = (s . (intpos 4)) - 1 & (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),P,s)) . (intpos 7) = (s . (intpos 5)) - 1 ) ) & ( s . (intpos md) >= s . (intpos me) implies ( (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),P,s)) . (intpos 5) = 0 & (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),P,s)) . (intpos 4) = s . (intpos 4) & (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),P,s)) . (intpos 7) = s . (intpos 5) ) ) )
set a = GBP ;
let s be 0 -started State of SCMPDS; ::_thesis: for md, me being Element of NAT st s . (intpos 2) = md & s . (intpos 4) = me & md >= 8 & me >= 8 & s . GBP = 0 holds
( (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),P,s)) . GBP = 0 & (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),P,s)) . (intpos 1) = s . (intpos 1) & (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),P,s)) . (intpos 2) = s . (intpos 2) & (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),P,s)) . (intpos 3) = s . (intpos 3) & ( for i being Element of NAT st i >= 8 holds
(IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),P,s)) . (intpos i) = s . (intpos i) ) & ( s . (intpos md) < s . (intpos me) implies ( (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),P,s)) . (intpos 5) = (s . (intpos 5)) - 1 & (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),P,s)) . (intpos 4) = (s . (intpos 4)) - 1 & (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),P,s)) . (intpos 7) = (s . (intpos 5)) - 1 ) ) & ( s . (intpos md) >= s . (intpos me) implies ( (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),P,s)) . (intpos 5) = 0 & (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),P,s)) . (intpos 4) = s . (intpos 4) & (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),P,s)) . (intpos 7) = s . (intpos 5) ) ) )
let md, me be Element of NAT ; ::_thesis: ( s . (intpos 2) = md & s . (intpos 4) = me & md >= 8 & me >= 8 & s . GBP = 0 implies ( (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),P,s)) . GBP = 0 & (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),P,s)) . (intpos 1) = s . (intpos 1) & (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),P,s)) . (intpos 2) = s . (intpos 2) & (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),P,s)) . (intpos 3) = s . (intpos 3) & ( for i being Element of NAT st i >= 8 holds
(IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),P,s)) . (intpos i) = s . (intpos i) ) & ( s . (intpos md) < s . (intpos me) implies ( (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),P,s)) . (intpos 5) = (s . (intpos 5)) - 1 & (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),P,s)) . (intpos 4) = (s . (intpos 4)) - 1 & (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),P,s)) . (intpos 7) = (s . (intpos 5)) - 1 ) ) & ( s . (intpos md) >= s . (intpos me) implies ( (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),P,s)) . (intpos 5) = 0 & (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),P,s)) . (intpos 4) = s . (intpos 4) & (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),P,s)) . (intpos 7) = s . (intpos 5) ) ) ) )
assume that
A1: s . (intpos 2) = md and
A2: s . (intpos 4) = me and
A3: md >= 8 and
A4: me >= 8 and
A5: s . GBP = 0 ; ::_thesis: ( (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),P,s)) . GBP = 0 & (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),P,s)) . (intpos 1) = s . (intpos 1) & (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),P,s)) . (intpos 2) = s . (intpos 2) & (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),P,s)) . (intpos 3) = s . (intpos 3) & ( for i being Element of NAT st i >= 8 holds
(IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),P,s)) . (intpos i) = s . (intpos i) ) & ( s . (intpos md) < s . (intpos me) implies ( (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),P,s)) . (intpos 5) = (s . (intpos 5)) - 1 & (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),P,s)) . (intpos 4) = (s . (intpos 4)) - 1 & (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),P,s)) . (intpos 7) = (s . (intpos 5)) - 1 ) ) & ( s . (intpos md) >= s . (intpos me) implies ( (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),P,s)) . (intpos 5) = 0 & (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),P,s)) . (intpos 4) = s . (intpos 4) & (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),P,s)) . (intpos 7) = s . (intpos 5) ) ) )
set t0 = s;
set Q0 = P;
set t1 = IExec ((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))),P,s);
set Q1 = P;
set t2 = IExec (((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))),P,s);
set t3 = IExec ((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))),P,s);
set Q2 = P;
set t4 = Exec (((GBP,7) := (GBP,5)),s);
set Q4 = P;
A6: s . GBP = 0 by A5;
A7: DataLoc ((s . GBP),7) = intpos (0 + 7) by A5, SCMP_GCD:1;
then A8: (Exec (((GBP,7) := (GBP,5)),s)) . GBP = 0 by A6, AMI_3:10, SCMPDS_2:47;
then A9: DataLoc (((Exec (((GBP,7) := (GBP,5)),s)) . GBP),5) = intpos (0 + 5) by SCMP_GCD:1;
A10: (Exec (((GBP,7) := (GBP,5)),s)) . (intpos 4) = me by A7, A2, AMI_3:10, SCMPDS_2:47;
A11: (IExec ((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))),P,s)) . (intpos 4) = (Exec ((AddTo (GBP,5,(- 1))),(Exec (((GBP,7) := (GBP,5)),s)))) . (intpos 4) by SCMPDS_5:42
.= me by A10, A9, AMI_3:10, SCMPDS_2:48 ;
then A12: DataLoc (((IExec ((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))),P,s)) . (intpos 4)),0) = intpos (me + 0) by SCMP_GCD:1;
A13: (IExec ((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))),P,s)) . GBP = (Exec ((AddTo (GBP,5,(- 1))),(Exec (((GBP,7) := (GBP,5)),s)))) . GBP by SCMPDS_5:42
.= 0 by A8, A9, AMI_3:10, SCMPDS_2:48 ;
then A14: DataLoc (((IExec ((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))),P,s)) . GBP),6) = intpos (0 + 6) by SCMP_GCD:1;
A15: (Exec (((GBP,7) := (GBP,5)),s)) . (intpos 2) = md by A7, A1, AMI_3:10, SCMPDS_2:47;
A16: (IExec ((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))),P,s)) . (intpos 2) = (Exec ((AddTo (GBP,5,(- 1))),(Exec (((GBP,7) := (GBP,5)),s)))) . (intpos 2) by SCMPDS_5:42
.= md by A15, A9, AMI_3:10, SCMPDS_2:48 ;
A18: (Exec (((GBP,7) := (GBP,5)),s)) . (intpos 5) = s . (intpos 5) by A7, AMI_3:10, SCMPDS_2:47;
set t01 = Initialize (Initialize (IExec ((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))),P,s)));
set ii7 = (GBP,5) := 0;
set t5 = Exec ((AddTo (GBP,4,(- 1))),(Initialize (Initialize (IExec ((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))),P,s)))));
A19: (IExec (((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))),P,s)) . (intpos 2) = (Exec (((GBP,6) := ((intpos 4),0)),(IExec ((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))),P,s)))) . (intpos 2) by SCMPDS_5:41
.= md by A16, A14, AMI_3:10, SCMPDS_2:47 ;
A20: (IExec (((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))),P,s)) . GBP = (Exec (((GBP,6) := ((intpos 4),0)),(IExec ((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))),P,s)))) . GBP by SCMPDS_5:41
.= 0 by A13, A14, AMI_3:10, SCMPDS_2:47 ;
then A21: DataLoc (((IExec (((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))),P,s)) . GBP),6) = intpos (0 + 6) by SCMP_GCD:1;
 DataLoc ((s . GBP),5) = intpos (0 + 5) by A6, SCMP_GCD:1;
then A22: (Exec (((GBP,7) := (GBP,5)),s)) . (intpos 7) = s . (intpos 5) by A7, SCMPDS_2:47;
A23: (IExec ((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))),P,s)) . (intpos 7) = (Exec ((AddTo (GBP,5,(- 1))),(Exec (((GBP,7) := (GBP,5)),s)))) . (intpos 7) by SCMPDS_5:42
.= s . (intpos 5) by A22, A9, AMI_3:10, SCMPDS_2:48 ;
A24: (IExec (((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))),P,s)) . (intpos 7) = (Exec (((GBP,6) := ((intpos 4),0)),(IExec ((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))),P,s)))) . (intpos 7) by SCMPDS_5:41
.= s . (intpos 5) by A23, A14, AMI_3:10, SCMPDS_2:47 ;
(IExec ((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))),P,s)) . (intpos 7) = (Exec ((SubFrom (GBP,6,(intpos 2),0)),(IExec (((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))),P,s)))) . (intpos 7) by SCMPDS_5:41
.= s . (intpos 5) by A24, A21, AMI_3:10, SCMPDS_2:50 ;
then A25: (Initialize (Initialize (IExec ((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))),P,s)))) . (intpos 7) = s . (intpos 5) by SCMPDS_5:15;
(IExec ((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))),P,s)) . (intpos 2) = (Exec ((SubFrom (GBP,6,(intpos 2),0)),(IExec (((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))),P,s)))) . (intpos 2) by SCMPDS_5:41
.= md by A19, A21, AMI_3:10, SCMPDS_2:50 ;
then A26: (Initialize (Initialize (IExec ((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))),P,s)))) . (intpos 2) = s . (intpos 2) by A1, SCMPDS_5:15;
A27: now__::_thesis:_for_i_being_Element_of_NAT_st_i_>=_8_holds_
(Exec_(((GBP,7)_:=_(GBP,5)),s))_._(intpos_i)_=_s_._(intpos_i)
let i be Element of NAT ; ::_thesis: ( i >= 8 implies (Exec (((GBP,7) := (GBP,5)),s)) . (intpos i) = s . (intpos i) )
assume  i >= 8 ; ::_thesis: (Exec (((GBP,7) := (GBP,5)),s)) . (intpos i) = s . (intpos i)
then  i > 7 by XXREAL_0:2;
hence (Exec (((GBP,7) := (GBP,5)),s)) . (intpos i) = s . (intpos i) by A7, AMI_3:10, SCMPDS_2:47
.= s . (intpos i) ;
::_thesis: verum
end;
A28: now__::_thesis:_for_i_being_Element_of_NAT_st_i_>=_8_holds_
(IExec_((((GBP,7)_:=_(GBP,5))_';'_(AddTo_(GBP,5,(-_1)))),P,s))_._(intpos_i)_=_s_._(intpos_i)
let i be Element of NAT ; ::_thesis: ( i >= 8 implies (IExec ((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))),P,s)) . (intpos i) = s . (intpos i) )
assume A29: i >= 8 ; ::_thesis: (IExec ((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))),P,s)) . (intpos i) = s . (intpos i)
then A30: i > 5 by XXREAL_0:2;
thus (IExec ((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))),P,s)) . (intpos i) = (Exec ((AddTo (GBP,5,(- 1))),(Exec (((GBP,7) := (GBP,5)),s)))) . (intpos i) by SCMPDS_5:42
.= (Exec (((GBP,7) := (GBP,5)),s)) . (intpos i) by A9, A30, AMI_3:10, SCMPDS_2:48
.= s . (intpos i) by A27, A29 ; ::_thesis: verum
end;
A31: now__::_thesis:_for_i_being_Element_of_NAT_st_i_>=_8_holds_
(IExec_(((((GBP,7)_:=_(GBP,5))_';'_(AddTo_(GBP,5,(-_1))))_';'_((GBP,6)_:=_((intpos_4),0))),P,s))_._(intpos_i)_=_s_._(intpos_i)
let i be Element of NAT ; ::_thesis: ( i >= 8 implies (IExec (((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))),P,s)) . (intpos i) = s . (intpos i) )
assume A32: i >= 8 ; ::_thesis: (IExec (((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))),P,s)) . (intpos i) = s . (intpos i)
then A33: i > 6 by XXREAL_0:2;
thus (IExec (((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))),P,s)) . (intpos i) = (Exec (((GBP,6) := ((intpos 4),0)),(IExec ((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))),P,s)))) . (intpos i) by SCMPDS_5:41
.= (IExec ((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))),P,s)) . (intpos i) by A14, A33, AMI_3:10, SCMPDS_2:47
.= s . (intpos i) by A28, A32 ; ::_thesis: verum
end;
A34: now__::_thesis:_for_i_being_Element_of_NAT_st_i_>=_8_holds_
(IExec_((((((GBP,7)_:=_(GBP,5))_';'_(AddTo_(GBP,5,(-_1))))_';'_((GBP,6)_:=_((intpos_4),0)))_';'_(SubFrom_(GBP,6,(intpos_2),0))),P,s))_._(intpos_i)_=_s_._(intpos_i)
let i be Element of NAT ; ::_thesis: ( i >= 8 implies (IExec ((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))),P,s)) . (intpos i) = s . (intpos i) )
assume A35: i >= 8 ; ::_thesis: (IExec ((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))),P,s)) . (intpos i) = s . (intpos i)
then A36: i > 6 by XXREAL_0:2;
thus (IExec ((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))),P,s)) . (intpos i) = (Exec ((SubFrom (GBP,6,(intpos 2),0)),(IExec (((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))),P,s)))) . (intpos i) by SCMPDS_5:41
.= (IExec (((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))),P,s)) . (intpos i) by A21, A36, AMI_3:10, SCMPDS_2:50
.= s . (intpos i) by A31, A35 ; ::_thesis: verum
end;
A37: (Exec (((GBP,7) := (GBP,5)),s)) . (intpos 3) = s . (intpos 3) by A7, AMI_3:10, SCMPDS_2:47;
A38: (IExec ((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))),P,s)) . (intpos 3) = (Exec ((AddTo (GBP,5,(- 1))),(Exec (((GBP,7) := (GBP,5)),s)))) . (intpos 3) by SCMPDS_5:42
.= s . (intpos 3) by A37, A9, AMI_3:10, SCMPDS_2:48 ;
A39: (Exec (((GBP,7) := (GBP,5)),s)) . (intpos 1) = s . (intpos 1) by A7, AMI_3:10, SCMPDS_2:47;
A40: (IExec ((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))),P,s)) . (intpos 1) = (Exec ((AddTo (GBP,5,(- 1))),(Exec (((GBP,7) := (GBP,5)),s)))) . (intpos 1) by SCMPDS_5:42
.= s . (intpos 1) by A39, A9, AMI_3:10, SCMPDS_2:48 ;
A41: (IExec (((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))),P,s)) . (intpos 1) = (Exec (((GBP,6) := ((intpos 4),0)),(IExec ((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))),P,s)))) . (intpos 1) by SCMPDS_5:41
.= s . (intpos 1) by A40, A14, AMI_3:10, SCMPDS_2:47 ;
(IExec ((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))),P,s)) . (intpos 1) = (Exec ((SubFrom (GBP,6,(intpos 2),0)),(IExec (((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))),P,s)))) . (intpos 1) by SCMPDS_5:41
.= s . (intpos 1) by A41, A21, AMI_3:10, SCMPDS_2:50 ;
then A42: (Initialize (Initialize (IExec ((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))),P,s)))) . (intpos 1) = s . (intpos 1) by SCMPDS_5:15;
A43: (IExec (((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))),P,s)) . (intpos 3) = (Exec (((GBP,6) := ((intpos 4),0)),(IExec ((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))),P,s)))) . (intpos 3) by SCMPDS_5:41
.= s . (intpos 3) by A38, A14, AMI_3:10, SCMPDS_2:47 ;
(IExec ((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))),P,s)) . (intpos 3) = (Exec ((SubFrom (GBP,6,(intpos 2),0)),(IExec (((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))),P,s)))) . (intpos 3) by SCMPDS_5:41
.= s . (intpos 3) by A43, A21, AMI_3:10, SCMPDS_2:50 ;
then A44: (Initialize (Initialize (IExec ((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))),P,s)))) . (intpos 3) = s . (intpos 3) by SCMPDS_5:15;
A45: (IExec ((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))),P,s)) . GBP = (Exec ((SubFrom (GBP,6,(intpos 2),0)),(IExec (((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))),P,s)))) . GBP by SCMPDS_5:41
.= 0 by A20, A21, AMI_3:10, SCMPDS_2:50 ;
then A46: (Initialize (Initialize (IExec ((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))),P,s)))) . GBP = 0 by SCMPDS_5:15;
then A47: DataLoc (((Initialize (Initialize (IExec ((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))),P,s)))) . GBP),5) = intpos (0 + 5) by SCMP_GCD:1;
 intpos 3 <> DataLoc (((Initialize (Initialize (IExec ((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))),P,s)))) . GBP),4) by A46, AMI_3:10, SCMP_GCD:1;
then A48: (Exec ((AddTo (GBP,4,(- 1))),(Initialize (Initialize (IExec ((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))),P,s)))))) . (intpos 3) = s . (intpos 3) by A44, SCMPDS_2:48;
 intpos 2 <> DataLoc (((Initialize (Initialize (IExec ((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))),P,s)))) . GBP),4) by A46, AMI_3:10, SCMP_GCD:1;
then A49: (Exec ((AddTo (GBP,4,(- 1))),(Initialize (Initialize (IExec ((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))),P,s)))))) . (intpos 2) = s . (intpos 2) by A26, SCMPDS_2:48;
 GBP <> DataLoc (((Initialize (Initialize (IExec ((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))),P,s)))) . GBP),4) by A46, AMI_3:10, SCMP_GCD:1;
then A50: (Exec ((AddTo (GBP,4,(- 1))),(Initialize (Initialize (IExec ((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))),P,s)))))) . GBP = 0 by A46, SCMPDS_2:48;
then A51: DataLoc (((Exec ((AddTo (GBP,4,(- 1))),(Initialize (Initialize (IExec ((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))),P,s)))))) . GBP),7) = intpos (0 + 7) by SCMP_GCD:1;
A52: (IExec (((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))),P,s)) . (intpos 6) = (Exec (((GBP,6) := ((intpos 4),0)),(IExec ((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))),P,s)))) . (intpos 6) by SCMPDS_5:41
.= (IExec ((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))),P,s)) . (intpos me) by A14, A12, SCMPDS_2:47
.= s . (intpos me) by A4, A28 ;
(IExec ((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))),P,s)) . (intpos 6) = (Exec ((SubFrom (GBP,6,(intpos 2),0)),(IExec (((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))),P,s)))) . (intpos 6) by SCMPDS_5:41
.= ((IExec (((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))),P,s)) . (intpos 6)) - ((IExec (((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))),P,s)) . (DataLoc (((IExec (((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))),P,s)) . (intpos 2)),0))) by A21, SCMPDS_2:50
.= ((IExec (((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))),P,s)) . (intpos 6)) - ((IExec (((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))),P,s)) . (intpos (md + 0))) by A19, SCMP_GCD:1
.= (s . (intpos me)) - (s . (intpos md)) by A3, A52, A31 ;
then A53: (IExec ((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))),P,s)) . (DataLoc (((IExec ((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))),P,s)) . GBP),6)) = (s . (intpos me)) - (s . (intpos md)) by A45, SCMP_GCD:1;
 intpos 1 <> DataLoc (((Initialize (Initialize (IExec ((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))),P,s)))) . GBP),4) by A46, AMI_3:10, SCMP_GCD:1;
then A54: (Exec ((AddTo (GBP,4,(- 1))),(Initialize (Initialize (IExec ((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))),P,s)))))) . (intpos 1) = s . (intpos 1) by A42, SCMPDS_2:48;
A56: now__::_thesis:_(_(IExec_((if>0_(GBP,6,((AddTo_(GBP,4,(-_1)))_';'_(AddTo_(GBP,7,(-_1)))),(Load_((GBP,5)_:=_0)))),P,(Initialize_(IExec_((((((GBP,7)_:=_(GBP,5))_';'_(AddTo_(GBP,5,(-_1))))_';'_((GBP,6)_:=_((intpos_4),0)))_';'_(SubFrom_(GBP,6,(intpos_2),0))),P,s)))))_._GBP_=_0_&_(IExec_((if>0_(GBP,6,((AddTo_(GBP,4,(-_1)))_';'_(AddTo_(GBP,7,(-_1)))),(Load_((GBP,5)_:=_0)))),P,(Initialize_(IExec_((((((GBP,7)_:=_(GBP,5))_';'_(AddTo_(GBP,5,(-_1))))_';'_((GBP,6)_:=_((intpos_4),0)))_';'_(SubFrom_(GBP,6,(intpos_2),0))),P,s)))))_._(intpos_1)_=_s_._(intpos_1)_&_(IExec_((if>0_(GBP,6,((AddTo_(GBP,4,(-_1)))_';'_(AddTo_(GBP,7,(-_1)))),(Load_((GBP,5)_:=_0)))),P,(Initialize_(IExec_((((((GBP,7)_:=_(GBP,5))_';'_(AddTo_(GBP,5,(-_1))))_';'_((GBP,6)_:=_((intpos_4),0)))_';'_(SubFrom_(GBP,6,(intpos_2),0))),P,s)))))_._(intpos_2)_=_s_._(intpos_2)_&_(IExec_((if>0_(GBP,6,((AddTo_(GBP,4,(-_1)))_';'_(AddTo_(GBP,7,(-_1)))),(Load_((GBP,5)_:=_0)))),P,(Initialize_(IExec_((((((GBP,7)_:=_(GBP,5))_';'_(AddTo_(GBP,5,(-_1))))_';'_((GBP,6)_:=_((intpos_4),0)))_';'_(SubFrom_(GBP,6,(intpos_2),0))),P,s)))))_._(intpos_3)_=_s_._(intpos_3)_&_(_for_i_being_Element_of_NAT_st_i_>=_8_holds_
(IExec_(((((((GBP,7)_:=_(GBP,5))_';'_(AddTo_(GBP,5,(-_1))))_';'_((GBP,6)_:=_((intpos_4),0)))_';'_(SubFrom_(GBP,6,(intpos_2),0)))_';'_(if>0_(GBP,6,((AddTo_(GBP,4,(-_1)))_';'_(AddTo_(GBP,7,(-_1)))),(Load_((GBP,5)_:=_0))))),P,s))_._(intpos_i)_=_s_._(intpos_i)_)_)
percases ( (Initialize (Initialize (IExec ((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))),P,s)))) . (DataLoc (((Initialize (Initialize (IExec ((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))),P,s)))) . GBP),6)) <= 0 or (Initialize (Initialize (IExec ((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))),P,s)))) . (DataLoc (((Initialize (Initialize (IExec ((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))),P,s)))) . GBP),6)) > 0 ) ;
supposeA57: (Initialize (Initialize (IExec ((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))),P,s)))) . (DataLoc (((Initialize (Initialize (IExec ((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))),P,s)))) . GBP),6)) <= 0 ; ::_thesis: ( (IExec ((if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0)))),P,(Initialize (IExec ((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))),P,s))))) . GBP = 0 & (IExec ((if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0)))),P,(Initialize (IExec ((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))),P,s))))) . (intpos 1) = s . (intpos 1) & (IExec ((if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0)))),P,(Initialize (IExec ((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))),P,s))))) . (intpos 2) = s . (intpos 2) & (IExec ((if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0)))),P,(Initialize (IExec ((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))),P,s))))) . (intpos 3) = s . (intpos 3) & ( for i being Element of NAT st i >= 8 holds
(IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),P,s)) . (intpos i) = s . (intpos i) ) )
hence (IExec ((if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0)))),P,(Initialize (IExec ((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))),P,s))))) . GBP = (IExec ((Load ((GBP,5) := 0)),P,(Initialize (IExec ((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))),P,s))))) . GBP by SCMPDS_6:74
.= (Exec (((GBP,5) := 0),(Initialize (Initialize (IExec ((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))),P,s)))))) . GBP by SCMPDS_5:40
.= 0 by A46, A47, AMI_3:10, SCMPDS_2:46 ;
::_thesis: ( (IExec ((if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0)))),P,(Initialize (IExec ((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))),P,s))))) . (intpos 1) = s . (intpos 1) & (IExec ((if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0)))),P,(Initialize (IExec ((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))),P,s))))) . (intpos 2) = s . (intpos 2) & (IExec ((if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0)))),P,(Initialize (IExec ((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))),P,s))))) . (intpos 3) = s . (intpos 3) & ( for i being Element of NAT st i >= 8 holds
(IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),P,s)) . (intpos i) = s . (intpos i) ) )
thus (IExec ((if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0)))),P,(Initialize (IExec ((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))),P,s))))) . (intpos 1) = (IExec ((Load ((GBP,5) := 0)),P,(Initialize (IExec ((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))),P,s))))) . (intpos 1) by A57, SCMPDS_6:74
.= (Exec (((GBP,5) := 0),(Initialize (Initialize (IExec ((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))),P,s)))))) . (intpos 1) by SCMPDS_5:40
.= s . (intpos 1) by A42, A47, AMI_3:10, SCMPDS_2:46 ; ::_thesis: ( (IExec ((if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0)))),P,(Initialize (IExec ((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))),P,s))))) . (intpos 2) = s . (intpos 2) & (IExec ((if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0)))),P,(Initialize (IExec ((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))),P,s))))) . (intpos 3) = s . (intpos 3) & ( for i being Element of NAT st i >= 8 holds
(IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),P,s)) . (intpos i) = s . (intpos i) ) )
thus (IExec ((if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0)))),P,(Initialize (IExec ((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))),P,s))))) . (intpos 2) = (IExec ((Load ((GBP,5) := 0)),P,(Initialize (IExec ((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))),P,s))))) . (intpos 2) by A57, SCMPDS_6:74
.= (Exec (((GBP,5) := 0),(Initialize (Initialize (IExec ((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))),P,s)))))) . (intpos 2) by SCMPDS_5:40
.= s . (intpos 2) by A26, A47, AMI_3:10, SCMPDS_2:46 ; ::_thesis: ( (IExec ((if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0)))),P,(Initialize (IExec ((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))),P,s))))) . (intpos 3) = s . (intpos 3) & ( for i being Element of NAT st i >= 8 holds
(IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),P,s)) . (intpos i) = s . (intpos i) ) )
thus (IExec ((if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0)))),P,(Initialize (IExec ((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))),P,s))))) . (intpos 3) = (IExec ((Load ((GBP,5) := 0)),P,(Initialize (IExec ((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))),P,s))))) . (intpos 3) by A57, SCMPDS_6:74
.= (Exec (((GBP,5) := 0),(Initialize (Initialize (IExec ((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))),P,s)))))) . (intpos 3) by SCMPDS_5:40
.= s . (intpos 3) by A44, A47, AMI_3:10, SCMPDS_2:46 ; ::_thesis: for i being Element of NAT st i >= 8 holds
(IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),P,s)) . (intpos i) = s . (intpos i)
let i be Element of NAT ; ::_thesis: ( i >= 8 implies (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),P,s)) . (intpos i) = s . (intpos i) )
assume A58: i >= 8 ; ::_thesis: (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),P,s)) . (intpos i) = s . (intpos i)
then A59: i > 5 by XXREAL_0:2;
thus (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),P,s)) . (intpos i) = (IExec ((if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0)))),P,(Initialize (IExec ((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))),P,s))))) . (intpos i) by SCMPDS_5:35
.= (IExec ((Load ((GBP,5) := 0)),P,(Initialize (IExec ((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))),P,s))))) . (intpos i) by A57, SCMPDS_6:74
.= (Exec (((GBP,5) := 0),(Initialize (Initialize (IExec ((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))),P,s)))))) . (intpos i) by SCMPDS_5:40
.= (Initialize (Initialize (IExec ((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))),P,s)))) . (intpos i) by A47, A59, AMI_3:10, SCMPDS_2:46
.= (IExec ((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))),P,s)) . (intpos i) by SCMPDS_5:15
.= s . (intpos i) by A34, A58 ; ::_thesis: verum
end;
supposeA60: (Initialize (Initialize (IExec ((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))),P,s)))) . (DataLoc (((Initialize (Initialize (IExec ((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))),P,s)))) . GBP),6)) > 0 ; ::_thesis: ( (IExec ((if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0)))),P,(Initialize (IExec ((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))),P,s))))) . GBP = 0 & (IExec ((if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0)))),P,(Initialize (IExec ((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))),P,s))))) . (intpos 1) = s . (intpos 1) & (IExec ((if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0)))),P,(Initialize (IExec ((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))),P,s))))) . (intpos 2) = s . (intpos 2) & (IExec ((if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0)))),P,(Initialize (IExec ((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))),P,s))))) . (intpos 3) = s . (intpos 3) & ( for i being Element of NAT st i >= 8 holds
(IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),P,s)) . (intpos i) = s . (intpos i) ) )
thus (IExec ((if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0)))),P,(Initialize (IExec ((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))),P,s))))) . GBP = (IExec (((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),P,(Initialize (IExec ((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))),P,s))))) . GBP by A60, SCMPDS_6:73
.= (Exec ((AddTo (GBP,7,(- 1))),(Exec ((AddTo (GBP,4,(- 1))),(Initialize (Initialize (IExec ((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))),P,s)))))))) . GBP by SCMPDS_5:42
.= 0 by A50, A51, AMI_3:10, SCMPDS_2:48 ; ::_thesis: ( (IExec ((if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0)))),P,(Initialize (IExec ((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))),P,s))))) . (intpos 1) = s . (intpos 1) & (IExec ((if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0)))),P,(Initialize (IExec ((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))),P,s))))) . (intpos 2) = s . (intpos 2) & (IExec ((if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0)))),P,(Initialize (IExec ((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))),P,s))))) . (intpos 3) = s . (intpos 3) & ( for i being Element of NAT st i >= 8 holds
(IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),P,s)) . (intpos i) = s . (intpos i) ) )
thus (IExec ((if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0)))),P,(Initialize (IExec ((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))),P,s))))) . (intpos 1) = (IExec (((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),P,(Initialize (IExec ((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))),P,s))))) . (intpos 1) by A60, SCMPDS_6:73
.= (Exec ((AddTo (GBP,7,(- 1))),(Exec ((AddTo (GBP,4,(- 1))),(Initialize (Initialize (IExec ((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))),P,s)))))))) . (intpos 1) by SCMPDS_5:42
.= s . (intpos 1) by A54, A51, AMI_3:10, SCMPDS_2:48 ; ::_thesis: ( (IExec ((if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0)))),P,(Initialize (IExec ((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))),P,s))))) . (intpos 2) = s . (intpos 2) & (IExec ((if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0)))),P,(Initialize (IExec ((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))),P,s))))) . (intpos 3) = s . (intpos 3) & ( for i being Element of NAT st i >= 8 holds
(IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),P,s)) . (intpos i) = s . (intpos i) ) )
thus (IExec ((if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0)))),P,(Initialize (IExec ((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))),P,s))))) . (intpos 2) = (IExec (((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),P,(Initialize (IExec ((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))),P,s))))) . (intpos 2) by A60, SCMPDS_6:73
.= (Exec ((AddTo (GBP,7,(- 1))),(Exec ((AddTo (GBP,4,(- 1))),(Initialize (Initialize (IExec ((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))),P,s)))))))) . (intpos 2) by SCMPDS_5:42
.= s . (intpos 2) by A49, A51, AMI_3:10, SCMPDS_2:48 ; ::_thesis: ( (IExec ((if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0)))),P,(Initialize (IExec ((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))),P,s))))) . (intpos 3) = s . (intpos 3) & ( for i being Element of NAT st i >= 8 holds
(IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),P,s)) . (intpos i) = s . (intpos i) ) )
thus (IExec ((if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0)))),P,(Initialize (IExec ((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))),P,s))))) . (intpos 3) = (IExec (((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),P,(Initialize (IExec ((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))),P,s))))) . (intpos 3) by A60, SCMPDS_6:73
.= (Exec ((AddTo (GBP,7,(- 1))),(Exec ((AddTo (GBP,4,(- 1))),(Initialize (Initialize (IExec ((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))),P,s)))))))) . (intpos 3) by SCMPDS_5:42
.= s . (intpos 3) by A48, A51, AMI_3:10, SCMPDS_2:48 ; ::_thesis: for i being Element of NAT st i >= 8 holds
(IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),P,s)) . (intpos i) = s . (intpos i)
let i be Element of NAT ; ::_thesis: ( i >= 8 implies (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),P,s)) . (intpos i) = s . (intpos i) )
assume A61: i >= 8 ; ::_thesis: (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),P,s)) . (intpos i) = s . (intpos i)
then A62: i > 7 by XXREAL_0:2;
 i > 4 by A61, XXREAL_0:2;
then A63: intpos i <> DataLoc (((Initialize (Initialize (IExec ((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))),P,s)))) . GBP),4) by A46, AMI_3:10, SCMP_GCD:1;
thus (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),P,s)) . (intpos i) = (IExec ((if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0)))),P,(Initialize (IExec ((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))),P,s))))) . (intpos i) by SCMPDS_5:35
.= (IExec (((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),P,(Initialize (IExec ((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))),P,s))))) . (intpos i) by A60, SCMPDS_6:73
.= (Exec ((AddTo (GBP,7,(- 1))),(Exec ((AddTo (GBP,4,(- 1))),(Initialize (Initialize (IExec ((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))),P,s)))))))) . (intpos i) by SCMPDS_5:42
.= (Exec ((AddTo (GBP,4,(- 1))),(Initialize (Initialize (IExec ((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))),P,s)))))) . (intpos i) by A51, A62, AMI_3:10, SCMPDS_2:48
.= (Initialize (Initialize (IExec ((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))),P,s)))) . (intpos i) by A63, SCMPDS_2:48
.= (IExec ((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))),P,s)) . (intpos i) by SCMPDS_5:15
.= s . (intpos i) by A34, A61 ; ::_thesis: verum
end;
end;
end;
hence  ( (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),P,s)) . GBP = 0 & (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),P,s)) . (intpos 1) = s . (intpos 1) & (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),P,s)) . (intpos 2) = s . (intpos 2) & (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),P,s)) . (intpos 3) = s . (intpos 3) ) by SCMPDS_5:35; ::_thesis: ( ( for i being Element of NAT st i >= 8 holds
(IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),P,s)) . (intpos i) = s . (intpos i) ) & ( s . (intpos md) < s . (intpos me) implies ( (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),P,s)) . (intpos 5) = (s . (intpos 5)) - 1 & (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),P,s)) . (intpos 4) = (s . (intpos 4)) - 1 & (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),P,s)) . (intpos 7) = (s . (intpos 5)) - 1 ) ) & ( s . (intpos md) >= s . (intpos me) implies ( (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),P,s)) . (intpos 5) = 0 & (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),P,s)) . (intpos 4) = s . (intpos 4) & (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),P,s)) . (intpos 7) = s . (intpos 5) ) ) )
thus  for i being Element of NAT st i >= 8 holds
(IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),P,s)) . (intpos i) = s . (intpos i) by A56; ::_thesis: ( ( s . (intpos md) < s . (intpos me) implies ( (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),P,s)) . (intpos 5) = (s . (intpos 5)) - 1 & (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),P,s)) . (intpos 4) = (s . (intpos 4)) - 1 & (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),P,s)) . (intpos 7) = (s . (intpos 5)) - 1 ) ) & ( s . (intpos md) >= s . (intpos me) implies ( (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),P,s)) . (intpos 5) = 0 & (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),P,s)) . (intpos 4) = s . (intpos 4) & (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),P,s)) . (intpos 7) = s . (intpos 5) ) ) )
A64: (IExec ((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))),P,s)) . (intpos 5) = (Exec ((AddTo (GBP,5,(- 1))),(Exec (((GBP,7) := (GBP,5)),s)))) . (intpos 5) by SCMPDS_5:42
.= ((Exec (((GBP,7) := (GBP,5)),s)) . (intpos 5)) + (- 1) by A9, SCMPDS_2:48
.= (s . (intpos 5)) - 1 by A18 ;
A65: (IExec (((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))),P,s)) . (intpos 5) = (Exec (((GBP,6) := ((intpos 4),0)),(IExec ((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))),P,s)))) . (intpos 5) by SCMPDS_5:41
.= (s . (intpos 5)) - 1 by A64, A14, AMI_3:10, SCMPDS_2:47 ;
(IExec ((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))),P,s)) . (intpos 5) = (Exec ((SubFrom (GBP,6,(intpos 2),0)),(IExec (((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))),P,s)))) . (intpos 5) by SCMPDS_5:41
.= (s . (intpos 5)) - 1 by A65, A21, AMI_3:10, SCMPDS_2:50 ;
then A66: (Initialize (Initialize (IExec ((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))),P,s)))) . (intpos 5) = (s . (intpos 5)) - 1 by SCMPDS_5:15;
A67: (IExec (((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))),P,s)) . (intpos 4) = (Exec (((GBP,6) := ((intpos 4),0)),(IExec ((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))),P,s)))) . (intpos 4) by SCMPDS_5:41
.= me by A11, A14, AMI_3:10, SCMPDS_2:47 ;
(IExec ((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))),P,s)) . (intpos 4) = (Exec ((SubFrom (GBP,6,(intpos 2),0)),(IExec (((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))),P,s)))) . (intpos 4) by SCMPDS_5:41
.= me by A67, A21, AMI_3:10, SCMPDS_2:50 ;
then A68: (Initialize (Initialize (IExec ((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))),P,s)))) . (intpos 4) = s . (intpos 4) by A2, SCMPDS_5:15;
A69: DataLoc (((Initialize (Initialize (IExec ((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))),P,s)))) . GBP),4) = intpos (0 + 4) by A46, SCMP_GCD:1;
C71: (Initialize (Initialize (IExec ((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))),P,s)))) . (DataLoc (((Initialize (Initialize (IExec ((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))),P,s)))) . GBP),6)) = (Initialize (Initialize (IExec ((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))),P,s)))) . (DataLoc (((IExec ((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))),P,s)) . GBP),6)) by SCMPDS_5:15
.= (IExec ((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))),P,s)) . (DataLoc (((IExec ((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))),P,s)) . GBP),6)) by SCMPDS_5:15 ;
hereby  ::_thesis: ( s . (intpos md) >= s . (intpos me) implies ( (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),P,s)) . (intpos 5) = 0 & (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),P,s)) . (intpos 4) = s . (intpos 4) & (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),P,s)) . (intpos 7) = s . (intpos 5) ) )
A70: intpos 5 <> DataLoc (((Initialize (Initialize (IExec ((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))),P,s)))) . GBP),4) by A46, AMI_3:10, SCMP_GCD:1;
assume  s . (intpos md) < s . (intpos me) ; ::_thesis: ( (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),P,s)) . (intpos 5) = (s . (intpos 5)) - 1 & (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),P,s)) . (intpos 4) = (s . (intpos 4)) - 1 & (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),P,s)) . (intpos 7) = (s . (intpos 5)) - 1 )
then B71: 0 < (s . (intpos me)) - (s . (intpos md)) by XREAL_1:50;
C71: (Initialize (Initialize (IExec ((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))),P,s)))) . (DataLoc (((Initialize (Initialize (IExec ((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))),P,s)))) . GBP),6)) = (Initialize (Initialize (IExec ((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))),P,s)))) . (DataLoc (((IExec ((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))),P,s)) . GBP),6)) by SCMPDS_5:15
.= (IExec ((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))),P,s)) . (DataLoc (((IExec ((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))),P,s)) . GBP),6)) by SCMPDS_5:15 ;
thus (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),P,s)) . (intpos 5) = (IExec ((if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0)))),P,(Initialize (Initialize (IExec ((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))),P,s)))))) . (intpos 5) by SCMPDS_5:35
.= (IExec (((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),P,(Initialize (Initialize (IExec ((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))),P,s)))))) . (intpos 5) by A53, B71, C71, SCMPDS_6:73
.= (Exec ((AddTo (GBP,7,(- 1))),(Exec ((AddTo (GBP,4,(- 1))),(Initialize (Initialize (IExec ((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))),P,s)))))))) . (intpos 5) by SCMPDS_5:42
.= (Exec ((AddTo (GBP,4,(- 1))),(Initialize (Initialize (IExec ((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))),P,s)))))) . (intpos 5) by A51, AMI_3:10, SCMPDS_2:48
.= (s . (intpos 5)) - 1 by A66, A70, SCMPDS_2:48 ; ::_thesis: ( (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),P,s)) . (intpos 4) = (s . (intpos 4)) - 1 & (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),P,s)) . (intpos 7) = (s . (intpos 5)) - 1 )
thus (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),P,s)) . (intpos 4) = (IExec ((if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0)))),P,(Initialize (Initialize (Initialize (IExec ((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))),P,s))))))) . (intpos 4) by SCMPDS_5:35
.= (IExec (((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),P,(Initialize (Initialize (Initialize (IExec ((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))),P,s))))))) . (intpos 4) by A53, C71, B71, SCMPDS_6:73
.= (Exec ((AddTo (GBP,7,(- 1))),(Exec ((AddTo (GBP,4,(- 1))),(Initialize (Initialize (IExec ((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))),P,s)))))))) . (intpos 4) by SCMPDS_5:42
.= (Exec ((AddTo (GBP,4,(- 1))),(Initialize (Initialize (IExec ((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))),P,s)))))) . (intpos 4) by A51, AMI_3:10, SCMPDS_2:48
.= ((Initialize (Initialize (IExec ((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))),P,s)))) . (intpos 4)) + (- 1) by A69, SCMPDS_2:48
.= (s . (intpos 4)) - 1 by A68 ; ::_thesis: (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),P,s)) . (intpos 7) = (s . (intpos 5)) - 1
A72: intpos 7 <> DataLoc (((Initialize (Initialize (IExec ((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))),P,s)))) . GBP),4) by A46, AMI_3:10, SCMP_GCD:1;
thus (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),P,s)) . (intpos 7) = (IExec ((if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0)))),P,(Initialize (IExec ((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))),P,s))))) . (intpos 7) by SCMPDS_5:35
.= (IExec (((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),P,(Initialize (IExec ((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))),P,s))))) . (intpos 7) by A53, C71, B71, SCMPDS_6:73
.= (Exec ((AddTo (GBP,7,(- 1))),(Exec ((AddTo (GBP,4,(- 1))),(Initialize (Initialize (IExec ((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))),P,s)))))))) . (intpos 7) by SCMPDS_5:42
.= ((Exec ((AddTo (GBP,4,(- 1))),(Initialize (Initialize (IExec ((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))),P,s)))))) . (intpos 7)) + (- 1) by A51, SCMPDS_2:48
.= ((Exec ((AddTo (GBP,4,(- 1))),(Initialize (Initialize (IExec ((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))),P,s)))))) . (intpos 7)) - 1
.= (s . (intpos 5)) - 1 by A25, A72, SCMPDS_2:48 ; ::_thesis: verum
end;
assume  s . (intpos md) >= s . (intpos me) ; ::_thesis: ( (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),P,s)) . (intpos 5) = 0 & (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),P,s)) . (intpos 4) = s . (intpos 4) & (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),P,s)) . (intpos 7) = s . (intpos 5) )
then B73: (s . (intpos me)) - (s . (intpos md)) <= 0 by XREAL_1:47;
thus (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),P,s)) . (intpos 5) = (IExec ((if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0)))),P,(Initialize (IExec ((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))),P,s))))) . (intpos 5) by SCMPDS_5:35
.= (IExec ((Load ((GBP,5) := 0)),P,(Initialize (IExec ((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))),P,s))))) . (intpos 5) by A53, C71, B73, SCMPDS_6:74
.= (Exec (((GBP,5) := 0),(Initialize (Initialize (IExec ((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))),P,s)))))) . (intpos 5) by SCMPDS_5:40
.= 0 by A47, SCMPDS_2:46 ; ::_thesis: ( (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),P,s)) . (intpos 4) = s . (intpos 4) & (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),P,s)) . (intpos 7) = s . (intpos 5) )
thus (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),P,s)) . (intpos 4) = (IExec ((if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0)))),P,(Initialize (IExec ((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))),P,s))))) . (intpos 4) by SCMPDS_5:35
.= (IExec ((Load ((GBP,5) := 0)),P,(Initialize (IExec ((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))),P,s))))) . (intpos 4) by A53, C71, B73, SCMPDS_6:74
.= (Exec (((GBP,5) := 0),(Initialize (Initialize (IExec ((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))),P,s)))))) . (intpos 4) by SCMPDS_5:40
.= s . (intpos 4) by A47, A68, AMI_3:10, SCMPDS_2:46 ; ::_thesis: (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),P,s)) . (intpos 7) = s . (intpos 5)
thus (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),P,s)) . (intpos 7) = (IExec ((if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0)))),P,(Initialize (IExec ((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))),P,s))))) . (intpos 7) by SCMPDS_5:35
.= (IExec ((Load ((GBP,5) := 0)),P,(Initialize (IExec ((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))),P,s))))) . (intpos 7) by A53, C71, B73, SCMPDS_6:74
.= (Exec (((GBP,5) := 0),(Initialize (Initialize (IExec ((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))),P,s)))))) . (intpos 7) by SCMPDS_5:40
.= s . (intpos 5) by A25, A47, AMI_3:10, SCMPDS_2:46 ; ::_thesis: verum
end;

Lm9:  for P being Instruction-Sequence of SCMPDS
for s being 0 -started State of SCMPDS
for m4, md being Element of NAT st s . GBP = 0 & s . (intpos 5) > 0 & s . (intpos 4) = m4 + (s . (intpos 5)) & m4 >= 8 & s . (intpos 2) = md & md >= 8 holds
( (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s)) . GBP = 0 & (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s)) . (intpos 1) = s . (intpos 1) & (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s)) . (intpos 5) = 0 & (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s)) . (intpos 2) = s . (intpos 2) & (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s)) . (intpos 3) = s . (intpos 3) & ( for i being Element of NAT st i >= 8 holds
(IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s)) . (intpos i) = s . (intpos i) ) & ex mE being Element of NAT st
( mE = (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s)) . (intpos 7) & (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s)) . (intpos 4) = m4 + mE & mE <= s . (intpos 5) & ( for i being Element of NAT st m4 + mE < i & i <= s . (intpos 4) holds
(IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s)) . (intpos md) < (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s)) . (intpos i) ) & ( mE = 0 or (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s)) . (intpos md) >= (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s)) . (intpos (m4 + mE)) ) ) )
proof
let P be Instruction-Sequence of SCMPDS; ::_thesis: for s being 0 -started State of SCMPDS
for m4, md being Element of NAT st s . GBP = 0 & s . (intpos 5) > 0 & s . (intpos 4) = m4 + (s . (intpos 5)) & m4 >= 8 & s . (intpos 2) = md & md >= 8 holds
( (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s)) . GBP = 0 & (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s)) . (intpos 1) = s . (intpos 1) & (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s)) . (intpos 5) = 0 & (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s)) . (intpos 2) = s . (intpos 2) & (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s)) . (intpos 3) = s . (intpos 3) & ( for i being Element of NAT st i >= 8 holds
(IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s)) . (intpos i) = s . (intpos i) ) & ex mE being Element of NAT st
( mE = (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s)) . (intpos 7) & (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s)) . (intpos 4) = m4 + mE & mE <= s . (intpos 5) & ( for i being Element of NAT st m4 + mE < i & i <= s . (intpos 4) holds
(IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s)) . (intpos md) < (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s)) . (intpos i) ) & ( mE = 0 or (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s)) . (intpos md) >= (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s)) . (intpos (m4 + mE)) ) ) )
set a = GBP ;
let s be 0 -started State of SCMPDS; ::_thesis: for m4, md being Element of NAT st s . GBP = 0 & s . (intpos 5) > 0 & s . (intpos 4) = m4 + (s . (intpos 5)) & m4 >= 8 & s . (intpos 2) = md & md >= 8 holds
( (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s)) . GBP = 0 & (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s)) . (intpos 1) = s . (intpos 1) & (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s)) . (intpos 5) = 0 & (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s)) . (intpos 2) = s . (intpos 2) & (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s)) . (intpos 3) = s . (intpos 3) & ( for i being Element of NAT st i >= 8 holds
(IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s)) . (intpos i) = s . (intpos i) ) & ex mE being Element of NAT st
( mE = (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s)) . (intpos 7) & (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s)) . (intpos 4) = m4 + mE & mE <= s . (intpos 5) & ( for i being Element of NAT st m4 + mE < i & i <= s . (intpos 4) holds
(IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s)) . (intpos md) < (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s)) . (intpos i) ) & ( mE = 0 or (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s)) . (intpos md) >= (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s)) . (intpos (m4 + mE)) ) ) )
let m4, md be Element of NAT ; ::_thesis: ( s . GBP = 0 & s . (intpos 5) > 0 & s . (intpos 4) = m4 + (s . (intpos 5)) & m4 >= 8 & s . (intpos 2) = md & md >= 8 implies ( (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s)) . GBP = 0 & (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s)) . (intpos 1) = s . (intpos 1) & (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s)) . (intpos 5) = 0 & (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s)) . (intpos 2) = s . (intpos 2) & (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s)) . (intpos 3) = s . (intpos 3) & ( for i being Element of NAT st i >= 8 holds
(IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s)) . (intpos i) = s . (intpos i) ) & ex mE being Element of NAT st
( mE = (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s)) . (intpos 7) & (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s)) . (intpos 4) = m4 + mE & mE <= s . (intpos 5) & ( for i being Element of NAT st m4 + mE < i & i <= s . (intpos 4) holds
(IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s)) . (intpos md) < (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s)) . (intpos i) ) & ( mE = 0 or (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s)) . (intpos md) >= (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s)) . (intpos (m4 + mE)) ) ) ) )
assume that
A1: s . GBP = 0 and
A2: s . (intpos 5) > 0 and
A3: s . (intpos 4) = m4 + (s . (intpos 5)) and
A4: m4 >= 8 and
A5: s . (intpos 2) = md and
A6: md >= 8 ; ::_thesis: ( (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s)) . GBP = 0 & (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s)) . (intpos 1) = s . (intpos 1) & (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s)) . (intpos 5) = 0 & (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s)) . (intpos 2) = s . (intpos 2) & (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s)) . (intpos 3) = s . (intpos 3) & ( for i being Element of NAT st i >= 8 holds
(IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s)) . (intpos i) = s . (intpos i) ) & ex mE being Element of NAT st
( mE = (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s)) . (intpos 7) & (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s)) . (intpos 4) = m4 + mE & mE <= s . (intpos 5) & ( for i being Element of NAT st m4 + mE < i & i <= s . (intpos 4) holds
(IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s)) . (intpos md) < (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s)) . (intpos i) ) & ( mE = 0 or (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s)) . (intpos md) >= (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s)) . (intpos (m4 + mE)) ) ) )
 s . (intpos 5) >= 1 + 0 by A2, INT_1:7;
then reconsider m5 = (s . (intpos 5)) - 1 as Element of NAT by INT_1:3, XREAL_1:48;
A7: s . (intpos 5) = m5 + 1 ;
defpred S1[ Element of NAT ] means  for t being 0 -started State of SCMPDS
for Q being Instruction-Sequence of SCMPDS st t . GBP = 0 & t . (intpos 5) = $1 + 1 & t . (intpos 4) = m4 + (t . (intpos 5)) & t . (intpos 2) = md holds
( (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . GBP = 0 & (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 1) = t . (intpos 1) & (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 5) = 0 & (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 2) = t . (intpos 2) & (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 3) = t . (intpos 3) & ( for i being Element of NAT st i >= 8 holds
(IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos i) = t . (intpos i) ) & ex mE being Element of NAT st
( mE = (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 7) & (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 4) = m4 + mE & mE <= t . (intpos 5) & ( for i being Element of NAT st m4 + mE < i & i <= t . (intpos 4) holds
(IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos md) < (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos i) ) & ( mE = 0 or (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos md) >= (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos (m4 + mE)) ) ) );
A8: now__::_thesis:_for_k_being_Element_of_NAT_st_S1[k]_holds_
S1[k_+_1]
let k be Element of NAT ; ::_thesis: ( S1[k] implies S1[k + 1] )
assume A9: S1[k] ; ::_thesis: S1[k + 1]
thus  S1[k + 1] ::_thesis: verum
proof
set me = m4 + ((k + 1) + 1);
let t be 0 -started State of SCMPDS; ::_thesis: for Q being Instruction-Sequence of SCMPDS st t . GBP = 0 & t . (intpos 5) = (k + 1) + 1 & t . (intpos 4) = m4 + (t . (intpos 5)) & t . (intpos 2) = md holds
( (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . GBP = 0 & (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 1) = t . (intpos 1) & (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 5) = 0 & (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 2) = t . (intpos 2) & (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 3) = t . (intpos 3) & ( for i being Element of NAT st i >= 8 holds
(IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos i) = t . (intpos i) ) & ex mE being Element of NAT st
( mE = (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 7) & (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 4) = m4 + mE & mE <= t . (intpos 5) & ( for i being Element of NAT st m4 + mE < i & i <= t . (intpos 4) holds
(IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos md) < (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos i) ) & ( mE = 0 or (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos md) >= (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos (m4 + mE)) ) ) )
let Q be Instruction-Sequence of SCMPDS; ::_thesis: ( t . GBP = 0 & t . (intpos 5) = (k + 1) + 1 & t . (intpos 4) = m4 + (t . (intpos 5)) & t . (intpos 2) = md implies ( (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . GBP = 0 & (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 1) = t . (intpos 1) & (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 5) = 0 & (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 2) = t . (intpos 2) & (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 3) = t . (intpos 3) & ( for i being Element of NAT st i >= 8 holds
(IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos i) = t . (intpos i) ) & ex mE being Element of NAT st
( mE = (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 7) & (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 4) = m4 + mE & mE <= t . (intpos 5) & ( for i being Element of NAT st m4 + mE < i & i <= t . (intpos 4) holds
(IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos md) < (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos i) ) & ( mE = 0 or (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos md) >= (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos (m4 + mE)) ) ) ) )
set b = DataLoc ((t . GBP),5);
assume that
A10: t . GBP = 0 and
A11: t . (intpos 5) = (k + 1) + 1 and
A12: t . (intpos 4) = m4 + (t . (intpos 5)) and
A13: t . (intpos 2) = md ; ::_thesis: ( (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . GBP = 0 & (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 1) = t . (intpos 1) & (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 5) = 0 & (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 2) = t . (intpos 2) & (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 3) = t . (intpos 3) & ( for i being Element of NAT st i >= 8 holds
(IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos i) = t . (intpos i) ) & ex mE being Element of NAT st
( mE = (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 7) & (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 4) = m4 + mE & mE <= t . (intpos 5) & ( for i being Element of NAT st m4 + mE < i & i <= t . (intpos 4) holds
(IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos md) < (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos i) ) & ( mE = 0 or (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos md) >= (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos (m4 + mE)) ) ) )
set It = IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),Q,t);
set IT = Q;
set It0 = Initialize (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),Q,t));
A23: m4 + ((k + 1) + 1) >= m4 by NAT_1:11;
then A24: m4 + ((k + 1) + 1) >= 8 by A4, XXREAL_0:2;
then A25: (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),Q,t)) . (intpos 1) = t . (intpos 1) by A6, A10, A11, A12, A13, Lm8;
A26: ( t . (intpos md) >= t . (intpos (m4 + ((k + 1) + 1))) implies ( (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),Q,t)) . (intpos 5) = 0 & (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),Q,t)) . (intpos 4) = t . (intpos 4) & (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),Q,t)) . (intpos 7) = t . (intpos 5) ) ) by A6, A10, A11, A12, A13, A24, Lm8;
A27: DataLoc ((t . GBP),5) = intpos (0 + 5) by A10, SCMP_GCD:1;
A28: now__::_thesis:_for_v_being_0_-started_State_of_SCMPDS
for_V_being_Instruction-Sequence_of_SCMPDS_st_v_._(intpos_4)_>=_m4_+_(v_._(DataLoc_((t_._GBP),5)))_&_v_._(intpos_2)_=_t_._(intpos_2)_&_v_._GBP_=_t_._GBP_&_v_._(DataLoc_((t_._GBP),5))_>_0_holds_
(_(IExec_(((((((GBP,7)_:=_(GBP,5))_';'_(AddTo_(GBP,5,(-_1))))_';'_((GBP,6)_:=_((intpos_4),0)))_';'_(SubFrom_(GBP,6,(intpos_2),0)))_';'_(if>0_(GBP,6,((AddTo_(GBP,4,(-_1)))_';'_(AddTo_(GBP,7,(-_1)))),(Load_((GBP,5)_:=_0))))),V,v))_._GBP_=_v_._GBP_&_(((((GBP,7)_:=_(GBP,5))_';'_(AddTo_(GBP,5,(-_1))))_';'_((GBP,6)_:=_((intpos_4),0)))_';'_(SubFrom_(GBP,6,(intpos_2),0)))_';'_(if>0_(GBP,6,((AddTo_(GBP,4,(-_1)))_';'_(AddTo_(GBP,7,(-_1)))),(Load_((GBP,5)_:=_0))))_is_closed_on_v,V_&_(((((GBP,7)_:=_(GBP,5))_';'_(AddTo_(GBP,5,(-_1))))_';'_((GBP,6)_:=_((intpos_4),0)))_';'_(SubFrom_(GBP,6,(intpos_2),0)))_';'_(if>0_(GBP,6,((AddTo_(GBP,4,(-_1)))_';'_(AddTo_(GBP,7,(-_1)))),(Load_((GBP,5)_:=_0))))_is_halting_on_v,V_&_(IExec_(((((((GBP,7)_:=_(GBP,5))_';'_(AddTo_(GBP,5,(-_1))))_';'_((GBP,6)_:=_((intpos_4),0)))_';'_(SubFrom_(GBP,6,(intpos_2),0)))_';'_(if>0_(GBP,6,((AddTo_(GBP,4,(-_1)))_';'_(AddTo_(GBP,7,(-_1)))),(Load_((GBP,5)_:=_0))))),V,v))_._(DataLoc_((t_._GBP),5))_<_v_._(DataLoc_((t_._GBP),5))_&_(IExec_(((((((GBP,7)_:=_(GBP,5))_';'_(AddTo_(GBP,5,(-_1))))_';'_((GBP,6)_:=_((intpos_4),0)))_';'_(SubFrom_(GBP,6,(intpos_2),0)))_';'_(if>0_(GBP,6,((AddTo_(GBP,4,(-_1)))_';'_(AddTo_(GBP,7,(-_1)))),(Load_((GBP,5)_:=_0))))),V,v))_._(intpos_4)_>=_m4_+_((IExec_(((((((GBP,7)_:=_(GBP,5))_';'_(AddTo_(GBP,5,(-_1))))_';'_((GBP,6)_:=_((intpos_4),0)))_';'_(SubFrom_(GBP,6,(intpos_2),0)))_';'_(if>0_(GBP,6,((AddTo_(GBP,4,(-_1)))_';'_(AddTo_(GBP,7,(-_1)))),(Load_((GBP,5)_:=_0))))),V,v))_._(DataLoc_((t_._GBP),5)))_&_(IExec_(((((((GBP,7)_:=_(GBP,5))_';'_(AddTo_(GBP,5,(-_1))))_';'_((GBP,6)_:=_((intpos_4),0)))_';'_(SubFrom_(GBP,6,(intpos_2),0)))_';'_(if>0_(GBP,6,((AddTo_(GBP,4,(-_1)))_';'_(AddTo_(GBP,7,(-_1)))),(Load_((GBP,5)_:=_0))))),V,v))_._(intpos_2)_=_v_._(intpos_2)_)
let v be 0 -started State of SCMPDS; ::_thesis: for V being Instruction-Sequence of SCMPDS st v . (intpos 4) >= m4 + (v . (DataLoc ((t . GBP),5))) & v . (intpos 2) = t . (intpos 2) & v . GBP = t . GBP & v . (DataLoc ((t . GBP),5)) > 0 holds
( (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . GBP = v . GBP & (((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0)))) is_closed_on v,V & (((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0)))) is_halting_on v,V & (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . (DataLoc ((t . GBP),5)) < v . (DataLoc ((t . GBP),5)) & (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . (intpos 4) >= m4 + ((IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . (DataLoc ((t . GBP),5))) & (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . (intpos 2) = v . (intpos 2) )
let V be Instruction-Sequence of SCMPDS; ::_thesis: ( v . (intpos 4) >= m4 + (v . (DataLoc ((t . GBP),5))) & v . (intpos 2) = t . (intpos 2) & v . GBP = t . GBP & v . (DataLoc ((t . GBP),5)) > 0 implies ( (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . GBP = v . GBP & (((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0)))) is_closed_on v,V & (((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0)))) is_halting_on v,V & (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . (DataLoc ((t . GBP),5)) < v . (DataLoc ((t . GBP),5)) & (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . (intpos 4) >= m4 + ((IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . (DataLoc ((t . GBP),5))) & (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . (intpos 2) = v . (intpos 2) ) )
assume that
A29: v . (intpos 4) >= m4 + (v . (DataLoc ((t . GBP),5))) and
A30: v . (intpos 2) = t . (intpos 2) and
A31: v . GBP = t . GBP and
A32: v . (DataLoc ((t . GBP),5)) > 0 ; ::_thesis: ( (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . GBP = v . GBP & (((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0)))) is_closed_on v,V & (((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0)))) is_halting_on v,V & (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . (DataLoc ((t . GBP),5)) < v . (DataLoc ((t . GBP),5)) & (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . (intpos 4) >= m4 + ((IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . (DataLoc ((t . GBP),5))) & (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . (intpos 2) = v . (intpos 2) )
A33: m4 + (v . (DataLoc ((t . GBP),5))) > m4 + 0 by A32, XREAL_1:6;
then reconsider ME = v . (intpos 4) as Element of NAT by A29, INT_1:3;
 v . (intpos 4) >= m4 by A29, A33, XXREAL_0:2;
then A41: ME >= 8 by A4, XXREAL_0:2;
then A42: ( v . (intpos md) < v . (intpos ME) implies ( (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . (intpos 5) = (v . (intpos 5)) - 1 & (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . (intpos 4) = (v . (intpos 4)) - 1 & (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . (intpos 7) = (v . (intpos 5)) - 1 ) ) by A6, A10, A13, A30, A31, Lm8;
thus  (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . GBP = v . GBP by A6, A10, A13, A30, A31, A41, Lm8; ::_thesis: ( (((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0)))) is_closed_on v,V & (((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0)))) is_halting_on v,V & (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . (DataLoc ((t . GBP),5)) < v . (DataLoc ((t . GBP),5)) & (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . (intpos 4) >= m4 + ((IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . (DataLoc ((t . GBP),5))) & (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . (intpos 2) = v . (intpos 2) )
thus  ( (((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0)))) is_closed_on v,V & (((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0)))) is_halting_on v,V ) by SCMPDS_6:20, SCMPDS_6:21; ::_thesis: ( (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . (DataLoc ((t . GBP),5)) < v . (DataLoc ((t . GBP),5)) & (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . (intpos 4) >= m4 + ((IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . (DataLoc ((t . GBP),5))) & (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . (intpos 2) = v . (intpos 2) )
A43: ( v . (intpos md) >= v . (intpos ME) implies ( (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . (intpos 5) = 0 & (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . (intpos 4) = v . (intpos 4) & (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . (intpos 7) = v . (intpos 5) ) ) by A6, A10, A13, A30, A31, A41, Lm8;
hereby  ::_thesis: (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . (intpos 2) = v . (intpos 2)
percases ( v . (intpos md) < v . (intpos ME) or v . (intpos md) >= v . (intpos ME) ) ;
supposeA44: v . (intpos md) < v . (intpos ME) ; ::_thesis: ( (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . (DataLoc ((t . GBP),5)) < v . (DataLoc ((t . GBP),5)) & (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . (intpos 4) >= m4 + ((IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . (DataLoc ((t . GBP),5))) )
hence  (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . (DataLoc ((t . GBP),5)) < v . (DataLoc ((t . GBP),5)) by A27, A42, XREAL_1:146; ::_thesis: (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . (intpos 4) >= m4 + ((IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . (DataLoc ((t . GBP),5)))
 (m4 + (v . (DataLoc ((t . GBP),5)))) - 1 = m4 + ((IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . (DataLoc ((t . GBP),5))) by A27, A42, A44;
hence  (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . (intpos 4) >= m4 + ((IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . (DataLoc ((t . GBP),5))) by A29, A42, A44, XREAL_1:9; ::_thesis: verum
end;
supposeA45: v . (intpos md) >= v . (intpos ME) ; ::_thesis: ( (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . (DataLoc ((t . GBP),5)) < v . (DataLoc ((t . GBP),5)) & (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . (intpos 4) >= m4 + ((IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . (DataLoc ((t . GBP),5))) )
hence  (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . (DataLoc ((t . GBP),5)) < v . (DataLoc ((t . GBP),5)) by A6, A10, A13, A27, A30, A31, A32, A41, Lm8; ::_thesis: (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . (intpos 4) >= m4 + ((IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . (DataLoc ((t . GBP),5)))
 m4 + ((IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . (DataLoc ((t . GBP),5))) < m4 + (v . (DataLoc ((t . GBP),5))) by A27, A32, A43, A45, XREAL_1:6;
hence  (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . (intpos 4) >= m4 + ((IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . (DataLoc ((t . GBP),5))) by A29, A43, A45, XXREAL_0:2; ::_thesis: verum
end;
end;
end;
thus  (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . (intpos 2) = v . (intpos 2) by A6, A10, A13, A30, A31, A41, Lm8; ::_thesis: verum
end;
A46: (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),Q,t)) . (intpos 3) = t . (intpos 3) by A6, A10, A11, A12, A13, A24, Lm8;
A47: ( t . (intpos md) < t . (intpos (m4 + ((k + 1) + 1))) implies ( (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),Q,t)) . (intpos 5) = (t . (intpos 5)) - 1 & (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),Q,t)) . (intpos 4) = (t . (intpos 4)) - 1 & (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),Q,t)) . (intpos 7) = (t . (intpos 5)) - 1 ) ) by A6, A10, A11, A12, A13, A24, Lm8;
A48: (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),Q,t)) . GBP = 0 by A6, A10, A11, A12, A13, A24, Lm8;
then B48: (Initialize (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),Q,t))) . GBP = 0 by SCMPDS_5:15;
A49: DataLoc (((IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),Q,t)) . GBP),5) = intpos (0 + 5) by A48, SCMP_GCD:1;
A50: (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),Q,t)) . (intpos 2) = t . (intpos 2) by A6, A10, A11, A12, A13, A24, Lm8;
then B50: (Initialize (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),Q,t))) . (intpos 2) = t . (intpos 2) by SCMPDS_5:15;
percases ( t . (intpos md) < t . (intpos (m4 + ((k + 1) + 1))) or t . (intpos md) >= t . (intpos (m4 + ((k + 1) + 1))) ) ;
supposeA51: t . (intpos md) < t . (intpos (m4 + ((k + 1) + 1))) ; ::_thesis: ( (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . GBP = 0 & (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 1) = t . (intpos 1) & (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 5) = 0 & (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 2) = t . (intpos 2) & (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 3) = t . (intpos 3) & ( for i being Element of NAT st i >= 8 holds
(IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos i) = t . (intpos i) ) & ex mE being Element of NAT st
( mE = (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 7) & (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 4) = m4 + mE & mE <= t . (intpos 5) & ( for i being Element of NAT st m4 + mE < i & i <= t . (intpos 4) holds
(IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos md) < (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos i) ) & ( mE = 0 or (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos md) >= (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos (m4 + mE)) ) ) )
then A52: (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),Q,t)) . (intpos 4) = m4 + ((IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),Q,t)) . (intpos 5)) by A12, A47;
B52: (Initialize (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),Q,t))) . (intpos 4) = m4 + ((IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),Q,t)) . (intpos 5)) by A12, A47, A51, SCMPDS_5:15
.= m4 + ((Initialize (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),Q,t))) . (intpos 5)) by SCMPDS_5:15 ;
YY: (Initialize (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),Q,t))) . (intpos 5) = (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),Q,t)) . (intpos 5) by SCMPDS_5:15;
Y1: (Initialize (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),Q,t))) . (intpos 1) = (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),Q,t)) . (intpos 1) by SCMPDS_5:15;
Y3: (Initialize (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),Q,t))) . (intpos 3) = (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),Q,t)) . (intpos 3) by SCMPDS_5:15;
Y4: (Initialize (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),Q,t))) . (intpos 4) = (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),Q,t)) . (intpos 4) by SCMPDS_5:15;
 (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),Q,t)) . (intpos 5) = k + 1 by A11, A47, A51;
then consider mE being Element of NAT  such that
A53: mE = (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,(Initialize (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),Q,t))))) . (intpos 7) and
A54: (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,(Initialize (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),Q,t))))) . (intpos 4) = m4 + mE and
A55: mE <= (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),Q,t)) . (intpos 5) and
A56: for i being Element of NAT st m4 + mE < i & i <= (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),Q,t)) . (intpos 4) holds
(IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,(Initialize (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),Q,t))))) . (intpos md) < (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,(Initialize (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),Q,t))))) . (intpos i) and
A57: ( mE = 0 or (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,(Initialize (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),Q,t))))) . (intpos md) >= (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,(Initialize (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),Q,t))))) . (intpos (m4 + mE)) ) by A9, A12, A13, A47, A26, B48, B50, B52, YY;
A58: IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t) = IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,(Initialize (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),Q,t)))) by A11, A12, A27, A28, Th5;
hence  (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . GBP = 0 by A9, A11, A12, A13, A47, A26, A52, YY, B50, B48, Y4; ::_thesis: ( (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 1) = t . (intpos 1) & (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 5) = 0 & (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 2) = t . (intpos 2) & (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 3) = t . (intpos 3) & ( for i being Element of NAT st i >= 8 holds
(IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos i) = t . (intpos i) ) & ex mE being Element of NAT st
( mE = (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 7) & (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 4) = m4 + mE & mE <= t . (intpos 5) & ( for i being Element of NAT st m4 + mE < i & i <= t . (intpos 4) holds
(IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos md) < (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos i) ) & ( mE = 0 or (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos md) >= (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos (m4 + mE)) ) ) )
thus  (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 1) = t . (intpos 1) by A9, A11, A12, A13, A25, A47, A26, A52, A58, YY, B50, B48, Y1, Y4; ::_thesis: ( (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 5) = 0 & (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 2) = t . (intpos 2) & (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 3) = t . (intpos 3) & ( for i being Element of NAT st i >= 8 holds
(IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos i) = t . (intpos i) ) & ex mE being Element of NAT st
( mE = (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 7) & (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 4) = m4 + mE & mE <= t . (intpos 5) & ( for i being Element of NAT st m4 + mE < i & i <= t . (intpos 4) holds
(IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos md) < (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos i) ) & ( mE = 0 or (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos md) >= (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos (m4 + mE)) ) ) )
thus  (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 5) = 0 by A9, A11, A12, A13, A47, A26, A52, A58, YY, B50, B48, Y4; ::_thesis: ( (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 2) = t . (intpos 2) & (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 3) = t . (intpos 3) & ( for i being Element of NAT st i >= 8 holds
(IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos i) = t . (intpos i) ) & ex mE being Element of NAT st
( mE = (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 7) & (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 4) = m4 + mE & mE <= t . (intpos 5) & ( for i being Element of NAT st m4 + mE < i & i <= t . (intpos 4) holds
(IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos md) < (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos i) ) & ( mE = 0 or (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos md) >= (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos (m4 + mE)) ) ) )
thus  (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 2) = t . (intpos 2) by A9, A11, A12, A13, A47, A26, A52, A58, YY, B50, B48, Y4; ::_thesis: ( (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 3) = t . (intpos 3) & ( for i being Element of NAT st i >= 8 holds
(IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos i) = t . (intpos i) ) & ex mE being Element of NAT st
( mE = (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 7) & (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 4) = m4 + mE & mE <= t . (intpos 5) & ( for i being Element of NAT st m4 + mE < i & i <= t . (intpos 4) holds
(IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos md) < (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos i) ) & ( mE = 0 or (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos md) >= (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos (m4 + mE)) ) ) )
thus  (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 3) = t . (intpos 3) by A9, A11, A12, A13, A46, A47, A26, A52, A58, YY, B50, B48, Y3, Y4; ::_thesis: ( ( for i being Element of NAT st i >= 8 holds
(IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos i) = t . (intpos i) ) & ex mE being Element of NAT st
( mE = (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 7) & (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 4) = m4 + mE & mE <= t . (intpos 5) & ( for i being Element of NAT st m4 + mE < i & i <= t . (intpos 4) holds
(IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos md) < (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos i) ) & ( mE = 0 or (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos md) >= (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos (m4 + mE)) ) ) )
A59: now__::_thesis:_for_i_being_Element_of_NAT_st_i_>=_8_holds_
(IExec_((while>0_(GBP,5,((((((GBP,7)_:=_(GBP,5))_';'_(AddTo_(GBP,5,(-_1))))_';'_((GBP,6)_:=_((intpos_4),0)))_';'_(SubFrom_(GBP,6,(intpos_2),0)))_';'_(if>0_(GBP,6,((AddTo_(GBP,4,(-_1)))_';'_(AddTo_(GBP,7,(-_1)))),(Load_((GBP,5)_:=_0))))))),Q,t))_._(intpos_i)_=_t_._(intpos_i)
let i be Element of NAT ; ::_thesis: ( i >= 8 implies (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos i) = t . (intpos i) )
assume A61: i >= 8 ; ::_thesis: (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos i) = t . (intpos i)
hence (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos i) = (Initialize (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),Q,t))) . (intpos i) by A9, A11, A12, A13, A47, A26, A52, A58, YY, B50, B48, Y4
.= (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),Q,t)) . (intpos i) by SCMPDS_5:15
.= t . (intpos i) by A6, A10, A11, A12, A13, A24, A61, Lm8 ;
::_thesis: verum
end;
hence  for i being Element of NAT st i >= 8 holds
(IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos i) = t . (intpos i) ; ::_thesis: ex mE being Element of NAT st
( mE = (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 7) & (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 4) = m4 + mE & mE <= t . (intpos 5) & ( for i being Element of NAT st m4 + mE < i & i <= t . (intpos 4) holds
(IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos md) < (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos i) ) & ( mE = 0 or (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos md) >= (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos (m4 + mE)) ) )
take  mE ; ::_thesis: ( mE = (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 7) & (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 4) = m4 + mE & mE <= t . (intpos 5) & ( for i being Element of NAT st m4 + mE < i & i <= t . (intpos 4) holds
(IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos md) < (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos i) ) & ( mE = 0 or (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos md) >= (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos (m4 + mE)) ) )
thus  mE = (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 7) by A11, A12, A27, A28, A53, Th5; ::_thesis: ( (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 4) = m4 + mE & mE <= t . (intpos 5) & ( for i being Element of NAT st m4 + mE < i & i <= t . (intpos 4) holds
(IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos md) < (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos i) ) & ( mE = 0 or (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos md) >= (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos (m4 + mE)) ) )
thus  (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 4) = m4 + mE by A11, A12, A27, A28, A54, Th5; ::_thesis: ( mE <= t . (intpos 5) & ( for i being Element of NAT st m4 + mE < i & i <= t . (intpos 4) holds
(IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos md) < (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos i) ) & ( mE = 0 or (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos md) >= (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos (m4 + mE)) ) )
thus  mE <= t . (intpos 5) by A47, A51, A55, XREAL_1:146, XXREAL_0:2; ::_thesis: ( ( for i being Element of NAT st m4 + mE < i & i <= t . (intpos 4) holds
(IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos md) < (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos i) ) & ( mE = 0 or (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos md) >= (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos (m4 + mE)) ) )
A62: (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),Q,t)) . (intpos 4) = (t . (intpos 4)) - 1 by A6, A10, A11, A12, A13, A24, A51, Lm8;
hereby  ::_thesis: ( mE = 0 or (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos md) >= (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos (m4 + mE)) )
let i be Element of NAT ; ::_thesis: ( m4 + mE < i & i <= t . (intpos 4) implies (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos md) < (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos b1) )
assume that
A63: m4 + mE < i and
A64: i <= t . (intpos 4) ; ::_thesis: (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos md) < (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos b1)
percases ( i = t . (intpos 4) or i <> t . (intpos 4) ) ;
supposeA65: i = t . (intpos 4) ; ::_thesis: (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos md) < (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos b1)
 (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos (m4 + ((k + 1) + 1))) = t . (intpos (m4 + ((k + 1) + 1))) by A4, A23, A59, XXREAL_0:2;
hence  (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos md) < (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos i) by A6, A11, A12, A51, A59, A65; ::_thesis: verum
end;
suppose i <> t . (intpos 4) ; ::_thesis: (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos md) < (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos b1)
then  i < t . (intpos 4) by A64, XXREAL_0:1;
then  i + 1 <= t . (intpos 4) by INT_1:7;
then  i <= (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),Q,t)) . (intpos 4) by A62, XREAL_1:19;
hence  (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos md) < (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos i) by A58, A56, A63; ::_thesis: verum
end;
end;
end;
thus  ( mE = 0 or (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos md) >= (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos (m4 + mE)) ) by A58, A57; ::_thesis: verum
end;
supposeA66: t . (intpos md) >= t . (intpos (m4 + ((k + 1) + 1))) ; ::_thesis: ( (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . GBP = 0 & (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 1) = t . (intpos 1) & (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 5) = 0 & (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 2) = t . (intpos 2) & (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 3) = t . (intpos 3) & ( for i being Element of NAT st i >= 8 holds
(IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos i) = t . (intpos i) ) & ex mE being Element of NAT st
( mE = (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 7) & (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 4) = m4 + mE & mE <= t . (intpos 5) & ( for i being Element of NAT st m4 + mE < i & i <= t . (intpos 4) holds
(IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos md) < (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos i) ) & ( mE = 0 or (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos md) >= (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos (m4 + mE)) ) ) )
A67: now__::_thesis:_for_x_being_Int_position_holds_(IExec_((while>0_(GBP,5,((((((GBP,7)_:=_(GBP,5))_';'_(AddTo_(GBP,5,(-_1))))_';'_((GBP,6)_:=_((intpos_4),0)))_';'_(SubFrom_(GBP,6,(intpos_2),0)))_';'_(if>0_(GBP,6,((AddTo_(GBP,4,(-_1)))_';'_(AddTo_(GBP,7,(-_1)))),(Load_((GBP,5)_:=_0))))))),Q,t))_._x_=_(IExec_(((((((GBP,7)_:=_(GBP,5))_';'_(AddTo_(GBP,5,(-_1))))_';'_((GBP,6)_:=_((intpos_4),0)))_';'_(SubFrom_(GBP,6,(intpos_2),0)))_';'_(if>0_(GBP,6,((AddTo_(GBP,4,(-_1)))_';'_(AddTo_(GBP,7,(-_1)))),(Load_((GBP,5)_:=_0))))),Q,t))_._x
X1: (Initialize (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),Q,t))) . GBP = (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),Q,t)) . GBP by SCMPDS_5:15;
X2: (Initialize (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),Q,t))) . (DataLoc (((IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),Q,t)) . GBP),5)) = (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),Q,t)) . (DataLoc (((IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),Q,t)) . GBP),5)) by SCMPDS_5:15;
let x be Int_position; ::_thesis: (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . x = (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),Q,t)) . x
thus (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . x = (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,(Initialize (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),Q,t))))) . x by A11, A12, A27, A28, Th5
.= (Initialize (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),Q,t))) . x by A26, A49, A66, X1, X2, SCMPDS_8:23
.= (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),Q,t)) . x by SCMPDS_5:15 ; ::_thesis: verum
end;
hence  (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . GBP = 0 by A48; ::_thesis: ( (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 1) = t . (intpos 1) & (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 5) = 0 & (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 2) = t . (intpos 2) & (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 3) = t . (intpos 3) & ( for i being Element of NAT st i >= 8 holds
(IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos i) = t . (intpos i) ) & ex mE being Element of NAT st
( mE = (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 7) & (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 4) = m4 + mE & mE <= t . (intpos 5) & ( for i being Element of NAT st m4 + mE < i & i <= t . (intpos 4) holds
(IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos md) < (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos i) ) & ( mE = 0 or (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos md) >= (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos (m4 + mE)) ) ) )
thus  (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 1) = t . (intpos 1) by A25, A67; ::_thesis: ( (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 5) = 0 & (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 2) = t . (intpos 2) & (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 3) = t . (intpos 3) & ( for i being Element of NAT st i >= 8 holds
(IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos i) = t . (intpos i) ) & ex mE being Element of NAT st
( mE = (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 7) & (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 4) = m4 + mE & mE <= t . (intpos 5) & ( for i being Element of NAT st m4 + mE < i & i <= t . (intpos 4) holds
(IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos md) < (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos i) ) & ( mE = 0 or (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos md) >= (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos (m4 + mE)) ) ) )
thus  (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 5) = 0 by A26, A66, A67; ::_thesis: ( (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 2) = t . (intpos 2) & (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 3) = t . (intpos 3) & ( for i being Element of NAT st i >= 8 holds
(IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos i) = t . (intpos i) ) & ex mE being Element of NAT st
( mE = (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 7) & (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 4) = m4 + mE & mE <= t . (intpos 5) & ( for i being Element of NAT st m4 + mE < i & i <= t . (intpos 4) holds
(IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos md) < (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos i) ) & ( mE = 0 or (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos md) >= (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos (m4 + mE)) ) ) )
thus  (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 2) = t . (intpos 2) by A50, A67; ::_thesis: ( (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 3) = t . (intpos 3) & ( for i being Element of NAT st i >= 8 holds
(IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos i) = t . (intpos i) ) & ex mE being Element of NAT st
( mE = (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 7) & (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 4) = m4 + mE & mE <= t . (intpos 5) & ( for i being Element of NAT st m4 + mE < i & i <= t . (intpos 4) holds
(IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos md) < (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos i) ) & ( mE = 0 or (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos md) >= (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos (m4 + mE)) ) ) )
thus  (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 3) = t . (intpos 3) by A46, A67; ::_thesis: ( ( for i being Element of NAT st i >= 8 holds
(IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos i) = t . (intpos i) ) & ex mE being Element of NAT st
( mE = (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 7) & (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 4) = m4 + mE & mE <= t . (intpos 5) & ( for i being Element of NAT st m4 + mE < i & i <= t . (intpos 4) holds
(IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos md) < (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos i) ) & ( mE = 0 or (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos md) >= (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos (m4 + mE)) ) ) )
A68: now__::_thesis:_for_i_being_Element_of_NAT_st_i_>=_8_holds_
(IExec_((while>0_(GBP,5,((((((GBP,7)_:=_(GBP,5))_';'_(AddTo_(GBP,5,(-_1))))_';'_((GBP,6)_:=_((intpos_4),0)))_';'_(SubFrom_(GBP,6,(intpos_2),0)))_';'_(if>0_(GBP,6,((AddTo_(GBP,4,(-_1)))_';'_(AddTo_(GBP,7,(-_1)))),(Load_((GBP,5)_:=_0))))))),Q,t))_._(intpos_i)_=_t_._(intpos_i)
let i be Element of NAT ; ::_thesis: ( i >= 8 implies (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos i) = t . (intpos i) )
assume A70: i >= 8 ; ::_thesis: (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos i) = t . (intpos i)
thus (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos i) = (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),Q,t)) . (intpos i) by A67
.= t . (intpos i) by A6, A10, A11, A12, A13, A24, A70, Lm8 ; ::_thesis: verum
end;
hence  for i being Element of NAT st i >= 8 holds
(IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos i) = t . (intpos i) ; ::_thesis: ex mE being Element of NAT st
( mE = (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 7) & (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 4) = m4 + mE & mE <= t . (intpos 5) & ( for i being Element of NAT st m4 + mE < i & i <= t . (intpos 4) holds
(IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos md) < (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos i) ) & ( mE = 0 or (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos md) >= (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos (m4 + mE)) ) )
take ME = (k + 1) + 1; ::_thesis: ( ME = (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 7) & (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 4) = m4 + ME & ME <= t . (intpos 5) & ( for i being Element of NAT st m4 + ME < i & i <= t . (intpos 4) holds
(IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos md) < (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos i) ) & ( ME = 0 or (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos md) >= (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos (m4 + ME)) ) )
thus  (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 7) = ME by A11, A26, A66, A67; ::_thesis: ( (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 4) = m4 + ME & ME <= t . (intpos 5) & ( for i being Element of NAT st m4 + ME < i & i <= t . (intpos 4) holds
(IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos md) < (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos i) ) & ( ME = 0 or (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos md) >= (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos (m4 + ME)) ) )
thus  (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 4) = m4 + ME by A11, A12, A26, A66, A67; ::_thesis: ( ME <= t . (intpos 5) & ( for i being Element of NAT st m4 + ME < i & i <= t . (intpos 4) holds
(IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos md) < (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos i) ) & ( ME = 0 or (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos md) >= (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos (m4 + ME)) ) )
thus  ME <= t . (intpos 5) by A11; ::_thesis: ( ( for i being Element of NAT st m4 + ME < i & i <= t . (intpos 4) holds
(IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos md) < (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos i) ) & ( ME = 0 or (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos md) >= (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos (m4 + ME)) ) )
thus  for i being Element of NAT st m4 + ME < i & i <= t . (intpos 4) holds
(IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos md) < (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos i) by A11, A12; ::_thesis: ( ME = 0 or (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos md) >= (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos (m4 + ME)) )
 (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos (m4 + ((k + 1) + 1))) = t . (intpos (m4 + ((k + 1) + 1))) by A4, A23, A68, XXREAL_0:2;
hence  ( ME = 0 or (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos md) >= (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos (m4 + ME)) ) by A6, A66, A68; ::_thesis: verum
end;
end;
end;
end;
A118: S1[ 0 ]
proof
set me = m4 + 1;
let t be 0 -started State of SCMPDS; ::_thesis: for Q being Instruction-Sequence of SCMPDS st t . GBP = 0 & t . (intpos 5) = 0 + 1 & t . (intpos 4) = m4 + (t . (intpos 5)) & t . (intpos 2) = md holds
( (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . GBP = 0 & (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 1) = t . (intpos 1) & (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 5) = 0 & (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 2) = t . (intpos 2) & (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 3) = t . (intpos 3) & ( for i being Element of NAT st i >= 8 holds
(IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos i) = t . (intpos i) ) & ex mE being Element of NAT st
( mE = (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 7) & (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 4) = m4 + mE & mE <= t . (intpos 5) & ( for i being Element of NAT st m4 + mE < i & i <= t . (intpos 4) holds
(IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos md) < (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos i) ) & ( mE = 0 or (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos md) >= (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos (m4 + mE)) ) ) )
let Q be Instruction-Sequence of SCMPDS; ::_thesis: ( t . GBP = 0 & t . (intpos 5) = 0 + 1 & t . (intpos 4) = m4 + (t . (intpos 5)) & t . (intpos 2) = md implies ( (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . GBP = 0 & (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 1) = t . (intpos 1) & (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 5) = 0 & (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 2) = t . (intpos 2) & (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 3) = t . (intpos 3) & ( for i being Element of NAT st i >= 8 holds
(IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos i) = t . (intpos i) ) & ex mE being Element of NAT st
( mE = (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 7) & (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 4) = m4 + mE & mE <= t . (intpos 5) & ( for i being Element of NAT st m4 + mE < i & i <= t . (intpos 4) holds
(IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos md) < (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos i) ) & ( mE = 0 or (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos md) >= (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos (m4 + mE)) ) ) ) )
set It = IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),Q,t);
set IT = Q;
set b = DataLoc ((t . GBP),5);
assume that
A80: t . GBP = 0 and
A81: t . (intpos 5) = 0 + 1 and
A82: t . (intpos 4) = m4 + (t . (intpos 5)) and
A83: t . (intpos 2) = md ; ::_thesis: ( (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . GBP = 0 & (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 1) = t . (intpos 1) & (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 5) = 0 & (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 2) = t . (intpos 2) & (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 3) = t . (intpos 3) & ( for i being Element of NAT st i >= 8 holds
(IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos i) = t . (intpos i) ) & ex mE being Element of NAT st
( mE = (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 7) & (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 4) = m4 + mE & mE <= t . (intpos 5) & ( for i being Element of NAT st m4 + mE < i & i <= t . (intpos 4) holds
(IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos md) < (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos i) ) & ( mE = 0 or (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos md) >= (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos (m4 + mE)) ) ) )
A84: m4 + 1 >= m4 by NAT_1:11;
then A85: m4 + 1 >= 8 by A4, XXREAL_0:2;
then A86: ( t . (intpos md) < t . (intpos (m4 + 1)) implies ( (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),Q,t)) . (intpos 5) = (t . (intpos 5)) - 1 & (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),Q,t)) . (intpos 4) = (t . (intpos 4)) - 1 & (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),Q,t)) . (intpos 7) = (t . (intpos 5)) - 1 ) ) by A6, A80, A81, A82, A83, Lm8;
A87: ( t . (intpos md) >= t . (intpos (m4 + 1)) implies ( (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),Q,t)) . (intpos 5) = 0 & (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),Q,t)) . (intpos 4) = t . (intpos 4) & (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),Q,t)) . (intpos 7) = t . (intpos 5) ) ) by A6, A80, A81, A82, A83, A85, Lm8;
A88: DataLoc ((t . GBP),5) = intpos (0 + 5) by A80, SCMP_GCD:1;
A89: now__::_thesis:_for_v_being_0_-started_State_of_SCMPDS
for_V_being_Instruction-Sequence_of_SCMPDS_st_v_._(intpos_4)_>=_m4_+_(v_._(DataLoc_((t_._GBP),5)))_&_v_._(intpos_2)_=_t_._(intpos_2)_&_v_._GBP_=_t_._GBP_&_v_._(DataLoc_((t_._GBP),5))_>_0_holds_
(_(IExec_(((((((GBP,7)_:=_(GBP,5))_';'_(AddTo_(GBP,5,(-_1))))_';'_((GBP,6)_:=_((intpos_4),0)))_';'_(SubFrom_(GBP,6,(intpos_2),0)))_';'_(if>0_(GBP,6,((AddTo_(GBP,4,(-_1)))_';'_(AddTo_(GBP,7,(-_1)))),(Load_((GBP,5)_:=_0))))),V,v))_._GBP_=_v_._GBP_&_(((((GBP,7)_:=_(GBP,5))_';'_(AddTo_(GBP,5,(-_1))))_';'_((GBP,6)_:=_((intpos_4),0)))_';'_(SubFrom_(GBP,6,(intpos_2),0)))_';'_(if>0_(GBP,6,((AddTo_(GBP,4,(-_1)))_';'_(AddTo_(GBP,7,(-_1)))),(Load_((GBP,5)_:=_0))))_is_closed_on_v,V_&_(((((GBP,7)_:=_(GBP,5))_';'_(AddTo_(GBP,5,(-_1))))_';'_((GBP,6)_:=_((intpos_4),0)))_';'_(SubFrom_(GBP,6,(intpos_2),0)))_';'_(if>0_(GBP,6,((AddTo_(GBP,4,(-_1)))_';'_(AddTo_(GBP,7,(-_1)))),(Load_((GBP,5)_:=_0))))_is_halting_on_v,V_&_(IExec_(((((((GBP,7)_:=_(GBP,5))_';'_(AddTo_(GBP,5,(-_1))))_';'_((GBP,6)_:=_((intpos_4),0)))_';'_(SubFrom_(GBP,6,(intpos_2),0)))_';'_(if>0_(GBP,6,((AddTo_(GBP,4,(-_1)))_';'_(AddTo_(GBP,7,(-_1)))),(Load_((GBP,5)_:=_0))))),V,v))_._(DataLoc_((t_._GBP),5))_<_v_._(DataLoc_((t_._GBP),5))_&_(IExec_(((((((GBP,7)_:=_(GBP,5))_';'_(AddTo_(GBP,5,(-_1))))_';'_((GBP,6)_:=_((intpos_4),0)))_';'_(SubFrom_(GBP,6,(intpos_2),0)))_';'_(if>0_(GBP,6,((AddTo_(GBP,4,(-_1)))_';'_(AddTo_(GBP,7,(-_1)))),(Load_((GBP,5)_:=_0))))),V,v))_._(intpos_4)_>=_m4_+_((IExec_(((((((GBP,7)_:=_(GBP,5))_';'_(AddTo_(GBP,5,(-_1))))_';'_((GBP,6)_:=_((intpos_4),0)))_';'_(SubFrom_(GBP,6,(intpos_2),0)))_';'_(if>0_(GBP,6,((AddTo_(GBP,4,(-_1)))_';'_(AddTo_(GBP,7,(-_1)))),(Load_((GBP,5)_:=_0))))),V,v))_._(DataLoc_((t_._GBP),5)))_&_(IExec_(((((((GBP,7)_:=_(GBP,5))_';'_(AddTo_(GBP,5,(-_1))))_';'_((GBP,6)_:=_((intpos_4),0)))_';'_(SubFrom_(GBP,6,(intpos_2),0)))_';'_(if>0_(GBP,6,((AddTo_(GBP,4,(-_1)))_';'_(AddTo_(GBP,7,(-_1)))),(Load_((GBP,5)_:=_0))))),V,v))_._(intpos_2)_=_v_._(intpos_2)_)
let v be 0 -started State of SCMPDS; ::_thesis: for V being Instruction-Sequence of SCMPDS st v . (intpos 4) >= m4 + (v . (DataLoc ((t . GBP),5))) & v . (intpos 2) = t . (intpos 2) & v . GBP = t . GBP & v . (DataLoc ((t . GBP),5)) > 0 holds
( (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . GBP = v . GBP & (((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0)))) is_closed_on v,V & (((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0)))) is_halting_on v,V & (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . (DataLoc ((t . GBP),5)) < v . (DataLoc ((t . GBP),5)) & (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . (intpos 4) >= m4 + ((IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . (DataLoc ((t . GBP),5))) & (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . (intpos 2) = v . (intpos 2) )
let V be Instruction-Sequence of SCMPDS; ::_thesis: ( v . (intpos 4) >= m4 + (v . (DataLoc ((t . GBP),5))) & v . (intpos 2) = t . (intpos 2) & v . GBP = t . GBP & v . (DataLoc ((t . GBP),5)) > 0 implies ( (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . GBP = v . GBP & (((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0)))) is_closed_on v,V & (((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0)))) is_halting_on v,V & (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . (DataLoc ((t . GBP),5)) < v . (DataLoc ((t . GBP),5)) & (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . (intpos 4) >= m4 + ((IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . (DataLoc ((t . GBP),5))) & (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . (intpos 2) = v . (intpos 2) ) )
assume that
A90: v . (intpos 4) >= m4 + (v . (DataLoc ((t . GBP),5))) and
A91: v . (intpos 2) = t . (intpos 2) and
A92: v . GBP = t . GBP and
A93: v . (DataLoc ((t . GBP),5)) > 0 ; ::_thesis: ( (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . GBP = v . GBP & (((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0)))) is_closed_on v,V & (((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0)))) is_halting_on v,V & (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . (DataLoc ((t . GBP),5)) < v . (DataLoc ((t . GBP),5)) & (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . (intpos 4) >= m4 + ((IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . (DataLoc ((t . GBP),5))) & (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . (intpos 2) = v . (intpos 2) )
A94: m4 + (v . (DataLoc ((t . GBP),5))) > m4 + 0 by A93, XREAL_1:6;
then reconsider ME = v . (intpos 4) as Element of NAT by A90, INT_1:3;
 v . (intpos 4) >= m4 by A90, A94, XXREAL_0:2;
then A102: ME >= 8 by A4, XXREAL_0:2;
then A103: ( v . (intpos md) < v . (intpos ME) implies ( (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . (intpos 5) = (v . (intpos 5)) - 1 & (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . (intpos 4) = (v . (intpos 4)) - 1 & (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . (intpos 7) = (v . (intpos 5)) - 1 ) ) by A6, A80, A83, A91, A92, Lm8;
thus  (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . GBP = v . GBP by A6, A80, A83, A91, A92, A102, Lm8; ::_thesis: ( (((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0)))) is_closed_on v,V & (((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0)))) is_halting_on v,V & (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . (DataLoc ((t . GBP),5)) < v . (DataLoc ((t . GBP),5)) & (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . (intpos 4) >= m4 + ((IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . (DataLoc ((t . GBP),5))) & (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . (intpos 2) = v . (intpos 2) )
thus  ( (((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0)))) is_closed_on v,V & (((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0)))) is_halting_on v,V ) by SCMPDS_6:20, SCMPDS_6:21; ::_thesis: ( (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . (DataLoc ((t . GBP),5)) < v . (DataLoc ((t . GBP),5)) & (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . (intpos 4) >= m4 + ((IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . (DataLoc ((t . GBP),5))) & (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . (intpos 2) = v . (intpos 2) )
A104: ( v . (intpos md) >= v . (intpos ME) implies ( (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . (intpos 5) = 0 & (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . (intpos 4) = v . (intpos 4) & (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . (intpos 7) = v . (intpos 5) ) ) by A6, A80, A83, A91, A92, A102, Lm8;
hereby  ::_thesis: (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . (intpos 2) = v . (intpos 2)
percases ( v . (intpos md) < v . (intpos ME) or v . (intpos md) >= v . (intpos ME) ) ;
supposeA105: v . (intpos md) < v . (intpos ME) ; ::_thesis: ( (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . (DataLoc ((t . GBP),5)) < v . (DataLoc ((t . GBP),5)) & (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . (intpos 4) >= m4 + ((IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . (DataLoc ((t . GBP),5))) )
hence  (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . (DataLoc ((t . GBP),5)) < v . (DataLoc ((t . GBP),5)) by A88, A103, XREAL_1:146; ::_thesis: (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . (intpos 4) >= m4 + ((IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . (DataLoc ((t . GBP),5)))
 (m4 + (v . (DataLoc ((t . GBP),5)))) - 1 = m4 + ((IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . (DataLoc ((t . GBP),5))) by A88, A103, A105;
hence  (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . (intpos 4) >= m4 + ((IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . (DataLoc ((t . GBP),5))) by A90, A103, A105, XREAL_1:9; ::_thesis: verum
end;
supposeA106: v . (intpos md) >= v . (intpos ME) ; ::_thesis: ( (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . (DataLoc ((t . GBP),5)) < v . (DataLoc ((t . GBP),5)) & (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . (intpos 4) >= m4 + ((IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . (DataLoc ((t . GBP),5))) )
hence  (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . (DataLoc ((t . GBP),5)) < v . (DataLoc ((t . GBP),5)) by A6, A80, A83, A88, A91, A92, A93, A102, Lm8; ::_thesis: (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . (intpos 4) >= m4 + ((IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . (DataLoc ((t . GBP),5)))
 m4 + ((IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . (DataLoc ((t . GBP),5))) < m4 + (v . (DataLoc ((t . GBP),5))) by A88, A93, A104, A106, XREAL_1:6;
hence  (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . (intpos 4) >= m4 + ((IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . (DataLoc ((t . GBP),5))) by A90, A104, A106, XXREAL_0:2; ::_thesis: verum
end;
end;
end;
thus  (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . (intpos 2) = v . (intpos 2) by A6, A80, A83, A91, A92, A102, Lm8; ::_thesis: verum
end;
A107: (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),Q,t)) . GBP = 0 by A6, A80, A81, A82, A83, A85, Lm8;
then A108: (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),Q,t)) . (DataLoc (((IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),Q,t)) . GBP),5)) = 0 by A81, A86, A87, SCMP_GCD:1;
A109: now__::_thesis:_for_x_being_Int_position_holds_(IExec_((while>0_(GBP,5,((((((GBP,7)_:=_(GBP,5))_';'_(AddTo_(GBP,5,(-_1))))_';'_((GBP,6)_:=_((intpos_4),0)))_';'_(SubFrom_(GBP,6,(intpos_2),0)))_';'_(if>0_(GBP,6,((AddTo_(GBP,4,(-_1)))_';'_(AddTo_(GBP,7,(-_1)))),(Load_((GBP,5)_:=_0))))))),Q,t))_._x_=_(IExec_(((((((GBP,7)_:=_(GBP,5))_';'_(AddTo_(GBP,5,(-_1))))_';'_((GBP,6)_:=_((intpos_4),0)))_';'_(SubFrom_(GBP,6,(intpos_2),0)))_';'_(if>0_(GBP,6,((AddTo_(GBP,4,(-_1)))_';'_(AddTo_(GBP,7,(-_1)))),(Load_((GBP,5)_:=_0))))),Q,t))_._x
X1: (Initialize (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),Q,t))) . GBP = (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),Q,t)) . GBP by SCMPDS_5:15;
X2: (Initialize (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),Q,t))) . (DataLoc (((IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),Q,t)) . GBP),5)) = (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),Q,t)) . (DataLoc (((IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),Q,t)) . GBP),5)) by SCMPDS_5:15;
let x be Int_position; ::_thesis: (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . x = (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),Q,t)) . x
thus (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . x = (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,(Initialize (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),Q,t))))) . x by A81, A82, A88, A89, Th5
.= (Initialize (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),Q,t))) . x by A108, X1, X2, SCMPDS_8:23
.= (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),Q,t)) . x by SCMPDS_5:15 ; ::_thesis: verum
end;
hence  (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . GBP = 0 by A107; ::_thesis: ( (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 1) = t . (intpos 1) & (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 5) = 0 & (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 2) = t . (intpos 2) & (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 3) = t . (intpos 3) & ( for i being Element of NAT st i >= 8 holds
(IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos i) = t . (intpos i) ) & ex mE being Element of NAT st
( mE = (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 7) & (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 4) = m4 + mE & mE <= t . (intpos 5) & ( for i being Element of NAT st m4 + mE < i & i <= t . (intpos 4) holds
(IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos md) < (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos i) ) & ( mE = 0 or (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos md) >= (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos (m4 + mE)) ) ) )
 (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),Q,t)) . (intpos 1) = t . (intpos 1) by A6, A80, A81, A82, A83, A85, Lm8;
hence  (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 1) = t . (intpos 1) by A109; ::_thesis: ( (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 5) = 0 & (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 2) = t . (intpos 2) & (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 3) = t . (intpos 3) & ( for i being Element of NAT st i >= 8 holds
(IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos i) = t . (intpos i) ) & ex mE being Element of NAT st
( mE = (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 7) & (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 4) = m4 + mE & mE <= t . (intpos 5) & ( for i being Element of NAT st m4 + mE < i & i <= t . (intpos 4) holds
(IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos md) < (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos i) ) & ( mE = 0 or (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos md) >= (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos (m4 + mE)) ) ) )
thus  (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 5) = 0 by A81, A86, A87, A109; ::_thesis: ( (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 2) = t . (intpos 2) & (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 3) = t . (intpos 3) & ( for i being Element of NAT st i >= 8 holds
(IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos i) = t . (intpos i) ) & ex mE being Element of NAT st
( mE = (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 7) & (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 4) = m4 + mE & mE <= t . (intpos 5) & ( for i being Element of NAT st m4 + mE < i & i <= t . (intpos 4) holds
(IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos md) < (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos i) ) & ( mE = 0 or (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos md) >= (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos (m4 + mE)) ) ) )
 (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),Q,t)) . (intpos 2) = t . (intpos 2) by A6, A80, A81, A82, A83, A85, Lm8;
hence  (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 2) = t . (intpos 2) by A109; ::_thesis: ( (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 3) = t . (intpos 3) & ( for i being Element of NAT st i >= 8 holds
(IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos i) = t . (intpos i) ) & ex mE being Element of NAT st
( mE = (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 7) & (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 4) = m4 + mE & mE <= t . (intpos 5) & ( for i being Element of NAT st m4 + mE < i & i <= t . (intpos 4) holds
(IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos md) < (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos i) ) & ( mE = 0 or (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos md) >= (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos (m4 + mE)) ) ) )
 (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),Q,t)) . (intpos 3) = t . (intpos 3) by A6, A80, A81, A82, A83, A85, Lm8;
hence  (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 3) = t . (intpos 3) by A109; ::_thesis: ( ( for i being Element of NAT st i >= 8 holds
(IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos i) = t . (intpos i) ) & ex mE being Element of NAT st
( mE = (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 7) & (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 4) = m4 + mE & mE <= t . (intpos 5) & ( for i being Element of NAT st m4 + mE < i & i <= t . (intpos 4) holds
(IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos md) < (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos i) ) & ( mE = 0 or (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos md) >= (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos (m4 + mE)) ) ) )
A110: now__::_thesis:_for_i_being_Element_of_NAT_st_i_>=_8_holds_
(IExec_((while>0_(GBP,5,((((((GBP,7)_:=_(GBP,5))_';'_(AddTo_(GBP,5,(-_1))))_';'_((GBP,6)_:=_((intpos_4),0)))_';'_(SubFrom_(GBP,6,(intpos_2),0)))_';'_(if>0_(GBP,6,((AddTo_(GBP,4,(-_1)))_';'_(AddTo_(GBP,7,(-_1)))),(Load_((GBP,5)_:=_0))))))),Q,t))_._(intpos_i)_=_t_._(intpos_i)
let i be Element of NAT ; ::_thesis: ( i >= 8 implies (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos i) = t . (intpos i) )
assume A111: i >= 8 ; ::_thesis: (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos i) = t . (intpos i)
thus (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos i) = (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),Q,t)) . (intpos i) by A109
.= t . (intpos i) by A6, A80, A81, A82, A83, A85, A111, Lm8 ; ::_thesis: verum
end;
hence  for i being Element of NAT st i >= 8 holds
(IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos i) = t . (intpos i) ; ::_thesis: ex mE being Element of NAT st
( mE = (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 7) & (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 4) = m4 + mE & mE <= t . (intpos 5) & ( for i being Element of NAT st m4 + mE < i & i <= t . (intpos 4) holds
(IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos md) < (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos i) ) & ( mE = 0 or (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos md) >= (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos (m4 + mE)) ) )
A113: (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos (m4 + 1)) = t . (intpos (m4 + 1)) by A4, A84, A110, XXREAL_0:2;
percases ( t . (intpos md) < t . (intpos (m4 + 1)) or t . (intpos md) >= t . (intpos (m4 + 1)) ) ;
supposeA114: t . (intpos md) < t . (intpos (m4 + 1)) ; ::_thesis: ex mE being Element of NAT st
( mE = (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 7) & (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 4) = m4 + mE & mE <= t . (intpos 5) & ( for i being Element of NAT st m4 + mE < i & i <= t . (intpos 4) holds
(IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos md) < (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos i) ) & ( mE = 0 or (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos md) >= (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos (m4 + mE)) ) )
take ME = 0 ; ::_thesis: ( ME = (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 7) & (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 4) = m4 + ME & ME <= t . (intpos 5) & ( for i being Element of NAT st m4 + ME < i & i <= t . (intpos 4) holds
(IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos md) < (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos i) ) & ( ME = 0 or (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos md) >= (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos (m4 + ME)) ) )
thus  (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 7) = ME by A81, A86, A109, A114; ::_thesis: ( (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 4) = m4 + ME & ME <= t . (intpos 5) & ( for i being Element of NAT st m4 + ME < i & i <= t . (intpos 4) holds
(IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos md) < (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos i) ) & ( ME = 0 or (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos md) >= (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos (m4 + ME)) ) )
thus  (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 4) = m4 + ME by A81, A82, A86, A109, A114; ::_thesis: ( ME <= t . (intpos 5) & ( for i being Element of NAT st m4 + ME < i & i <= t . (intpos 4) holds
(IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos md) < (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos i) ) & ( ME = 0 or (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos md) >= (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos (m4 + ME)) ) )
thus  ME <= t . (intpos 5) by A81; ::_thesis: ( ( for i being Element of NAT st m4 + ME < i & i <= t . (intpos 4) holds
(IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos md) < (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos i) ) & ( ME = 0 or (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos md) >= (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos (m4 + ME)) ) )
hereby  ::_thesis: ( ME = 0 or (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos md) >= (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos (m4 + ME)) )
let i be Element of NAT ; ::_thesis: ( m4 + ME < i & i <= t . (intpos 4) implies (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos md) < (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos i) )
assume that
A115: m4 + ME < i and
A116: i <= t . (intpos 4) ; ::_thesis: (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos md) < (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos i)
 m4 + 1 <= i by A115, INT_1:7;
then  i = m4 + 1 by A81, A82, A116, XXREAL_0:1;
hence  (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos md) < (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos i) by A6, A110, A113, A114; ::_thesis: verum
end;
thus  ( ME = 0 or (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos md) >= (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos (m4 + ME)) ) ; ::_thesis: verum
end;
supposeA117: t . (intpos md) >= t . (intpos (m4 + 1)) ; ::_thesis: ex mE being Element of NAT st
( mE = (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 7) & (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 4) = m4 + mE & mE <= t . (intpos 5) & ( for i being Element of NAT st m4 + mE < i & i <= t . (intpos 4) holds
(IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos md) < (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos i) ) & ( mE = 0 or (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos md) >= (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos (m4 + mE)) ) )
take ME = 1; ::_thesis: ( ME = (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 7) & (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 4) = m4 + ME & ME <= t . (intpos 5) & ( for i being Element of NAT st m4 + ME < i & i <= t . (intpos 4) holds
(IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos md) < (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos i) ) & ( ME = 0 or (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos md) >= (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos (m4 + ME)) ) )
thus  (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 7) = ME by A81, A87, A109, A117; ::_thesis: ( (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 4) = m4 + ME & ME <= t . (intpos 5) & ( for i being Element of NAT st m4 + ME < i & i <= t . (intpos 4) holds
(IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos md) < (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos i) ) & ( ME = 0 or (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos md) >= (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos (m4 + ME)) ) )
thus  (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 4) = m4 + ME by A81, A82, A87, A109, A117; ::_thesis: ( ME <= t . (intpos 5) & ( for i being Element of NAT st m4 + ME < i & i <= t . (intpos 4) holds
(IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos md) < (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos i) ) & ( ME = 0 or (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos md) >= (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos (m4 + ME)) ) )
thus  ME <= t . (intpos 5) by A81; ::_thesis: ( ( for i being Element of NAT st m4 + ME < i & i <= t . (intpos 4) holds
(IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos md) < (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos i) ) & ( ME = 0 or (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos md) >= (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos (m4 + ME)) ) )
thus  for i being Element of NAT st m4 + ME < i & i <= t . (intpos 4) holds
(IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos md) < (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos i) by A81, A82; ::_thesis: ( ME = 0 or (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos md) >= (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos (m4 + ME)) )
thus  ( ME = 0 or (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos md) >= (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos (m4 + ME)) ) by A6, A110, A113, A117; ::_thesis: verum
end;
end;
end;
 for k being Element of NAT holds S1[k] from NAT_1:sch_1(A118, A8);
hence  ( (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s)) . GBP = 0 & (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s)) . (intpos 1) = s . (intpos 1) & (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s)) . (intpos 5) = 0 & (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s)) . (intpos 2) = s . (intpos 2) & (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s)) . (intpos 3) = s . (intpos 3) & ( for i being Element of NAT st i >= 8 holds
(IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s)) . (intpos i) = s . (intpos i) ) & ex mE being Element of NAT st
( mE = (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s)) . (intpos 7) & (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s)) . (intpos 4) = m4 + mE & mE <= s . (intpos 5) & ( for i being Element of NAT st m4 + mE < i & i <= s . (intpos 4) holds
(IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s)) . (intpos md) < (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s)) . (intpos i) ) & ( mE = 0 or (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s)) . (intpos md) >= (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s)) . (intpos (m4 + mE)) ) ) ) by A1, A3, A5, A7; ::_thesis: verum
end;

Lm10:  for P being Instruction-Sequence of SCMPDS
for s being 0 -started State of SCMPDS
for m4, md being Element of NAT st s . GBP = 0 & s . (intpos 4) = m4 + (s . (intpos 5)) & m4 >= 8 & s . (intpos 2) = md & md >= 8 holds
( while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0)))))) is_closed_on s,P & while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0)))))) is_halting_on s,P )
proof
let P be Instruction-Sequence of SCMPDS; ::_thesis: for s being 0 -started State of SCMPDS
for m4, md being Element of NAT st s . GBP = 0 & s . (intpos 4) = m4 + (s . (intpos 5)) & m4 >= 8 & s . (intpos 2) = md & md >= 8 holds
( while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0)))))) is_closed_on s,P & while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0)))))) is_halting_on s,P )
set a = GBP ;
let s be 0 -started State of SCMPDS; ::_thesis: for m4, md being Element of NAT st s . GBP = 0 & s . (intpos 4) = m4 + (s . (intpos 5)) & m4 >= 8 & s . (intpos 2) = md & md >= 8 holds
( while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0)))))) is_closed_on s,P & while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0)))))) is_halting_on s,P )
let m4, md be Element of NAT ; ::_thesis: ( s . GBP = 0 & s . (intpos 4) = m4 + (s . (intpos 5)) & m4 >= 8 & s . (intpos 2) = md & md >= 8 implies ( while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0)))))) is_closed_on s,P & while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0)))))) is_halting_on s,P ) )
assume that
A1: s . GBP = 0 and
A2: s . (intpos 4) = m4 + (s . (intpos 5)) and
A3: m4 >= 8 and
A4: s . (intpos 2) = md and
A5: md >= 8 ; ::_thesis: ( while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0)))))) is_closed_on s,P & while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0)))))) is_halting_on s,P )
set b = DataLoc ((s . GBP),5);
A6: DataLoc ((s . GBP),5) = intpos (0 + 5) by A1, SCMP_GCD:1;
now__::_thesis:_for_v_being_0_-started_State_of_SCMPDS
for_V_being_Instruction-Sequence_of_SCMPDS_st_v_._(intpos_4)_>=_m4_+_(v_._(DataLoc_((s_._GBP),5)))_&_v_._(intpos_2)_=_s_._(intpos_2)_&_v_._GBP_=_s_._GBP_&_v_._(DataLoc_((s_._GBP),5))_>_0_holds_
(_(IExec_(((((((GBP,7)_:=_(GBP,5))_';'_(AddTo_(GBP,5,(-_1))))_';'_((GBP,6)_:=_((intpos_4),0)))_';'_(SubFrom_(GBP,6,(intpos_2),0)))_';'_(if>0_(GBP,6,((AddTo_(GBP,4,(-_1)))_';'_(AddTo_(GBP,7,(-_1)))),(Load_((GBP,5)_:=_0))))),V,v))_._GBP_=_v_._GBP_&_(((((GBP,7)_:=_(GBP,5))_';'_(AddTo_(GBP,5,(-_1))))_';'_((GBP,6)_:=_((intpos_4),0)))_';'_(SubFrom_(GBP,6,(intpos_2),0)))_';'_(if>0_(GBP,6,((AddTo_(GBP,4,(-_1)))_';'_(AddTo_(GBP,7,(-_1)))),(Load_((GBP,5)_:=_0))))_is_closed_on_v,V_&_(((((GBP,7)_:=_(GBP,5))_';'_(AddTo_(GBP,5,(-_1))))_';'_((GBP,6)_:=_((intpos_4),0)))_';'_(SubFrom_(GBP,6,(intpos_2),0)))_';'_(if>0_(GBP,6,((AddTo_(GBP,4,(-_1)))_';'_(AddTo_(GBP,7,(-_1)))),(Load_((GBP,5)_:=_0))))_is_halting_on_v,V_&_(IExec_(((((((GBP,7)_:=_(GBP,5))_';'_(AddTo_(GBP,5,(-_1))))_';'_((GBP,6)_:=_((intpos_4),0)))_';'_(SubFrom_(GBP,6,(intpos_2),0)))_';'_(if>0_(GBP,6,((AddTo_(GBP,4,(-_1)))_';'_(AddTo_(GBP,7,(-_1)))),(Load_((GBP,5)_:=_0))))),V,v))_._(DataLoc_((s_._GBP),5))_<_v_._(DataLoc_((s_._GBP),5))_&_(IExec_(((((((GBP,7)_:=_(GBP,5))_';'_(AddTo_(GBP,5,(-_1))))_';'_((GBP,6)_:=_((intpos_4),0)))_';'_(SubFrom_(GBP,6,(intpos_2),0)))_';'_(if>0_(GBP,6,((AddTo_(GBP,4,(-_1)))_';'_(AddTo_(GBP,7,(-_1)))),(Load_((GBP,5)_:=_0))))),V,v))_._(intpos_4)_>=_m4_+_((IExec_(((((((GBP,7)_:=_(GBP,5))_';'_(AddTo_(GBP,5,(-_1))))_';'_((GBP,6)_:=_((intpos_4),0)))_';'_(SubFrom_(GBP,6,(intpos_2),0)))_';'_(if>0_(GBP,6,((AddTo_(GBP,4,(-_1)))_';'_(AddTo_(GBP,7,(-_1)))),(Load_((GBP,5)_:=_0))))),V,v))_._(DataLoc_((s_._GBP),5)))_&_(IExec_(((((((GBP,7)_:=_(GBP,5))_';'_(AddTo_(GBP,5,(-_1))))_';'_((GBP,6)_:=_((intpos_4),0)))_';'_(SubFrom_(GBP,6,(intpos_2),0)))_';'_(if>0_(GBP,6,((AddTo_(GBP,4,(-_1)))_';'_(AddTo_(GBP,7,(-_1)))),(Load_((GBP,5)_:=_0))))),V,v))_._(intpos_2)_=_v_._(intpos_2)_)
let v be 0 -started State of SCMPDS; ::_thesis: for V being Instruction-Sequence of SCMPDS st v . (intpos 4) >= m4 + (v . (DataLoc ((s . GBP),5))) & v . (intpos 2) = s . (intpos 2) & v . GBP = s . GBP & v . (DataLoc ((s . GBP),5)) > 0 holds
( (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . GBP = v . GBP & (((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0)))) is_closed_on v,V & (((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0)))) is_halting_on v,V & (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . (DataLoc ((s . GBP),5)) < v . (DataLoc ((s . GBP),5)) & (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . (intpos 4) >= m4 + ((IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . (DataLoc ((s . GBP),5))) & (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . (intpos 2) = v . (intpos 2) )
let V be Instruction-Sequence of SCMPDS; ::_thesis: ( v . (intpos 4) >= m4 + (v . (DataLoc ((s . GBP),5))) & v . (intpos 2) = s . (intpos 2) & v . GBP = s . GBP & v . (DataLoc ((s . GBP),5)) > 0 implies ( (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . GBP = v . GBP & (((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0)))) is_closed_on v,V & (((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0)))) is_halting_on v,V & (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . (DataLoc ((s . GBP),5)) < v . (DataLoc ((s . GBP),5)) & (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . (intpos 4) >= m4 + ((IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . (DataLoc ((s . GBP),5))) & (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . (intpos 2) = v . (intpos 2) ) )
assume that
A7: v . (intpos 4) >= m4 + (v . (DataLoc ((s . GBP),5))) and
A8: v . (intpos 2) = s . (intpos 2) and
A9: v . GBP = s . GBP and
A10: v . (DataLoc ((s . GBP),5)) > 0 ; ::_thesis: ( (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . GBP = v . GBP & (((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0)))) is_closed_on v,V & (((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0)))) is_halting_on v,V & (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . (DataLoc ((s . GBP),5)) < v . (DataLoc ((s . GBP),5)) & (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . (intpos 4) >= m4 + ((IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . (DataLoc ((s . GBP),5))) & (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . (intpos 2) = v . (intpos 2) )
A11: m4 + (v . (DataLoc ((s . GBP),5))) > m4 + 0 by A10, XREAL_1:6;
then reconsider ME = v . (intpos 4) as Element of NAT by A7, INT_1:3;
 v . (intpos 4) >= m4 by A7, A11, XXREAL_0:2;
then A19: ME >= 8 by A3, XXREAL_0:2;
then A20: ( v . (intpos md) < v . (intpos ME) implies ( (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . (intpos 5) = (v . (intpos 5)) - 1 & (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . (intpos 4) = (v . (intpos 4)) - 1 & (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . (intpos 7) = (v . (intpos 5)) - 1 ) ) by A1, A4, A5, A8, A9, Lm8;
thus  (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . GBP = v . GBP by A1, A4, A5, A8, A9, A19, Lm8; ::_thesis: ( (((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0)))) is_closed_on v,V & (((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0)))) is_halting_on v,V & (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . (DataLoc ((s . GBP),5)) < v . (DataLoc ((s . GBP),5)) & (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . (intpos 4) >= m4 + ((IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . (DataLoc ((s . GBP),5))) & (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . (intpos 2) = v . (intpos 2) )
thus  ( (((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0)))) is_closed_on v,V & (((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0)))) is_halting_on v,V ) by SCMPDS_6:20, SCMPDS_6:21; ::_thesis: ( (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . (DataLoc ((s . GBP),5)) < v . (DataLoc ((s . GBP),5)) & (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . (intpos 4) >= m4 + ((IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . (DataLoc ((s . GBP),5))) & (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . (intpos 2) = v . (intpos 2) )
A21: ( v . (intpos md) >= v . (intpos ME) implies ( (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . (intpos 5) = 0 & (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . (intpos 4) = v . (intpos 4) & (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . (intpos 7) = v . (intpos 5) ) ) by A1, A4, A5, A8, A9, A19, Lm8;
hereby  ::_thesis: (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . (intpos 2) = v . (intpos 2)
percases ( v . (intpos md) < v . (intpos ME) or v . (intpos md) >= v . (intpos ME) ) ;
supposeA22: v . (intpos md) < v . (intpos ME) ; ::_thesis: ( (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . (DataLoc ((s . GBP),5)) < v . (DataLoc ((s . GBP),5)) & (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . (intpos 4) >= m4 + ((IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . (DataLoc ((s . GBP),5))) )
hence  (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . (DataLoc ((s . GBP),5)) < v . (DataLoc ((s . GBP),5)) by A6, A20, XREAL_1:146; ::_thesis: (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . (intpos 4) >= m4 + ((IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . (DataLoc ((s . GBP),5)))
 (m4 + (v . (DataLoc ((s . GBP),5)))) - 1 = m4 + ((IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . (DataLoc ((s . GBP),5))) by A6, A20, A22;
hence  (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . (intpos 4) >= m4 + ((IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . (DataLoc ((s . GBP),5))) by A7, A20, A22, XREAL_1:9; ::_thesis: verum
end;
supposeA23: v . (intpos md) >= v . (intpos ME) ; ::_thesis: ( (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . (DataLoc ((s . GBP),5)) < v . (DataLoc ((s . GBP),5)) & (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . (intpos 4) >= m4 + ((IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . (DataLoc ((s . GBP),5))) )
hence  (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . (DataLoc ((s . GBP),5)) < v . (DataLoc ((s . GBP),5)) by A1, A4, A5, A6, A8, A9, A10, A19, Lm8; ::_thesis: (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . (intpos 4) >= m4 + ((IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . (DataLoc ((s . GBP),5)))
 m4 + ((IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . (DataLoc ((s . GBP),5))) < m4 + (v . (DataLoc ((s . GBP),5))) by A6, A10, A21, A23, XREAL_1:6;
hence  (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . (intpos 4) >= m4 + ((IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . (DataLoc ((s . GBP),5))) by A7, A21, A23, XXREAL_0:2; ::_thesis: verum
end;
end;
end;
thus  (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . (intpos 2) = v . (intpos 2) by A1, A4, A5, A8, A9, A19, Lm8; ::_thesis: verum
end;
hence  ( while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0)))))) is_closed_on s,P & while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0)))))) is_halting_on s,P ) by A2, A6, Th5; ::_thesis: verum
end;

Lm11:  card (while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) = 11
proof
thus  card (while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) = 9 + 2 by Lm7, SCMPDS_8:17
.= 11 ; ::_thesis: verum
end;

Lm12:  card ((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))) = 9
proof
thus  card ((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))) = (card (((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0)))) + (card (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))) by AFINSQ_1:17
.= 4 + (card (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))) by Th4
.= 4 + (((card ((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1))))) + (card (Load ((GBP,7) := 0)))) + 2) by SCMPDS_6:65
.= 4 + ((2 + (card (Load ((GBP,7) := 0)))) + 2) by SCMP_GCD:5
.= 4 + ((2 + 1) + 2) by COMPOS_1:54
.= 9 ; ::_thesis: verum
end;

Lm13:  card (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))) = 11
proof
thus  card (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))) = 9 + 2 by Lm12, SCMPDS_8:17
.= 11 ; ::_thesis: verum
end;

Lm14:  for P being Instruction-Sequence of SCMPDS
for s being 0 -started State of SCMPDS
for md, me being Element of NAT st s . (intpos 2) = md & s . (intpos 3) = me & md >= 8 & me >= 8 & s . GBP = 0 holds
( (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),P,s)) . GBP = 0 & (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),P,s)) . (intpos 1) = s . (intpos 1) & (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),P,s)) . (intpos 2) = s . (intpos 2) & (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),P,s)) . (intpos 4) = s . (intpos 4) & ( for i being Element of NAT st i >= 8 holds
(IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),P,s)) . (intpos i) = s . (intpos i) ) & ( s . (intpos md) > s . (intpos me) implies ( (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),P,s)) . (intpos 7) = (s . (intpos 7)) - 1 & (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),P,s)) . (intpos 3) = (s . (intpos 3)) + 1 & (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),P,s)) . (intpos 5) = (s . (intpos 7)) - 1 ) ) & ( s . (intpos md) <= s . (intpos me) implies ( (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),P,s)) . (intpos 7) = 0 & (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),P,s)) . (intpos 3) = s . (intpos 3) & (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),P,s)) . (intpos 5) = s . (intpos 7) ) ) )
proof
let P be Instruction-Sequence of SCMPDS; ::_thesis: for s being 0 -started State of SCMPDS
for md, me being Element of NAT st s . (intpos 2) = md & s . (intpos 3) = me & md >= 8 & me >= 8 & s . GBP = 0 holds
( (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),P,s)) . GBP = 0 & (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),P,s)) . (intpos 1) = s . (intpos 1) & (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),P,s)) . (intpos 2) = s . (intpos 2) & (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),P,s)) . (intpos 4) = s . (intpos 4) & ( for i being Element of NAT st i >= 8 holds
(IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),P,s)) . (intpos i) = s . (intpos i) ) & ( s . (intpos md) > s . (intpos me) implies ( (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),P,s)) . (intpos 7) = (s . (intpos 7)) - 1 & (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),P,s)) . (intpos 3) = (s . (intpos 3)) + 1 & (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),P,s)) . (intpos 5) = (s . (intpos 7)) - 1 ) ) & ( s . (intpos md) <= s . (intpos me) implies ( (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),P,s)) . (intpos 7) = 0 & (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),P,s)) . (intpos 3) = s . (intpos 3) & (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),P,s)) . (intpos 5) = s . (intpos 7) ) ) )
set a = GBP ;
let s be 0 -started State of SCMPDS; ::_thesis: for md, me being Element of NAT st s . (intpos 2) = md & s . (intpos 3) = me & md >= 8 & me >= 8 & s . GBP = 0 holds
( (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),P,s)) . GBP = 0 & (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),P,s)) . (intpos 1) = s . (intpos 1) & (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),P,s)) . (intpos 2) = s . (intpos 2) & (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),P,s)) . (intpos 4) = s . (intpos 4) & ( for i being Element of NAT st i >= 8 holds
(IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),P,s)) . (intpos i) = s . (intpos i) ) & ( s . (intpos md) > s . (intpos me) implies ( (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),P,s)) . (intpos 7) = (s . (intpos 7)) - 1 & (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),P,s)) . (intpos 3) = (s . (intpos 3)) + 1 & (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),P,s)) . (intpos 5) = (s . (intpos 7)) - 1 ) ) & ( s . (intpos md) <= s . (intpos me) implies ( (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),P,s)) . (intpos 7) = 0 & (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),P,s)) . (intpos 3) = s . (intpos 3) & (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),P,s)) . (intpos 5) = s . (intpos 7) ) ) )
let md, me be Element of NAT ; ::_thesis: ( s . (intpos 2) = md & s . (intpos 3) = me & md >= 8 & me >= 8 & s . GBP = 0 implies ( (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),P,s)) . GBP = 0 & (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),P,s)) . (intpos 1) = s . (intpos 1) & (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),P,s)) . (intpos 2) = s . (intpos 2) & (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),P,s)) . (intpos 4) = s . (intpos 4) & ( for i being Element of NAT st i >= 8 holds
(IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),P,s)) . (intpos i) = s . (intpos i) ) & ( s . (intpos md) > s . (intpos me) implies ( (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),P,s)) . (intpos 7) = (s . (intpos 7)) - 1 & (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),P,s)) . (intpos 3) = (s . (intpos 3)) + 1 & (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),P,s)) . (intpos 5) = (s . (intpos 7)) - 1 ) ) & ( s . (intpos md) <= s . (intpos me) implies ( (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),P,s)) . (intpos 7) = 0 & (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),P,s)) . (intpos 3) = s . (intpos 3) & (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),P,s)) . (intpos 5) = s . (intpos 7) ) ) ) )
assume that
A1: s . (intpos 2) = md and
A2: s . (intpos 3) = me and
A3: md >= 8 and
A4: me >= 8 and
A5: s . GBP = 0 ; ::_thesis: ( (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),P,s)) . GBP = 0 & (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),P,s)) . (intpos 1) = s . (intpos 1) & (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),P,s)) . (intpos 2) = s . (intpos 2) & (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),P,s)) . (intpos 4) = s . (intpos 4) & ( for i being Element of NAT st i >= 8 holds
(IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),P,s)) . (intpos i) = s . (intpos i) ) & ( s . (intpos md) > s . (intpos me) implies ( (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),P,s)) . (intpos 7) = (s . (intpos 7)) - 1 & (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),P,s)) . (intpos 3) = (s . (intpos 3)) + 1 & (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),P,s)) . (intpos 5) = (s . (intpos 7)) - 1 ) ) & ( s . (intpos md) <= s . (intpos me) implies ( (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),P,s)) . (intpos 7) = 0 & (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),P,s)) . (intpos 3) = s . (intpos 3) & (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),P,s)) . (intpos 5) = s . (intpos 7) ) ) )
set t0 = s;
set t1 = IExec ((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))),P,s);
set Q1 = P;
set t2 = IExec (((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))),P,s);
set t3 = IExec ((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))),P,s);
set t4 = Exec (((GBP,5) := (GBP,7)),s);
A7: s . GBP = 0 by A5;
A8: DataLoc ((s . GBP),5) = intpos (0 + 5) by A5, SCMP_GCD:1;
then A9: (Exec (((GBP,5) := (GBP,7)),s)) . GBP = 0 by A5, AMI_3:10, SCMPDS_2:47;
then A10: DataLoc (((Exec (((GBP,5) := (GBP,7)),s)) . GBP),7) = intpos (0 + 7) by SCMP_GCD:1;
A11: (Exec (((GBP,5) := (GBP,7)),s)) . (intpos 2) = md by A8, A1, AMI_3:10, SCMPDS_2:47;
A12: (IExec ((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))),P,s)) . (intpos 2) = (Exec ((AddTo (GBP,7,(- 1))),(Exec (((GBP,5) := (GBP,7)),s)))) . (intpos 2) by SCMPDS_5:42
.= md by A11, A10, AMI_3:10, SCMPDS_2:48 ;
then A13: DataLoc (((IExec ((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))),P,s)) . (intpos 2)),0) = intpos (md + 0) by SCMP_GCD:1;
A14: (IExec ((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))),P,s)) . GBP = (Exec ((AddTo (GBP,7,(- 1))),(Exec (((GBP,5) := (GBP,7)),s)))) . GBP by SCMPDS_5:42
.= 0 by A9, A10, AMI_3:10, SCMPDS_2:48 ;
then A15: DataLoc (((IExec ((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))),P,s)) . GBP),6) = intpos (0 + 6) by SCMP_GCD:1;
A16: (Exec (((GBP,5) := (GBP,7)),s)) . (intpos 3) = me by A8, A2, AMI_3:10, SCMPDS_2:47;
A17: (IExec ((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))),P,s)) . (intpos 3) = (Exec ((AddTo (GBP,7,(- 1))),(Exec (((GBP,5) := (GBP,7)),s)))) . (intpos 3) by SCMPDS_5:42
.= me by A16, A10, AMI_3:10, SCMPDS_2:48 ;
A19: (Exec (((GBP,5) := (GBP,7)),s)) . (intpos 7) = s . (intpos 7) by A8, AMI_3:10, SCMPDS_2:47;
set t01 = Initialize (IExec ((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))),P,s));
set jj7 = (GBP,7) := 0;
set t5 = Exec ((AddTo (GBP,3,1)),(Initialize (IExec ((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))),P,s))));
A20: (IExec (((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))),P,s)) . (intpos 3) = (Exec (((GBP,6) := ((intpos 2),0)),(IExec ((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))),P,s)))) . (intpos 3) by SCMPDS_5:41
.= me by A17, A15, AMI_3:10, SCMPDS_2:47 ;
A21: (IExec (((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))),P,s)) . GBP = (Exec (((GBP,6) := ((intpos 2),0)),(IExec ((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))),P,s)))) . GBP by SCMPDS_5:41
.= 0 by A14, A15, AMI_3:10, SCMPDS_2:47 ;
then A22: DataLoc (((IExec (((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))),P,s)) . GBP),6) = intpos (0 + 6) by SCMP_GCD:1;
A23: (Exec (((GBP,5) := (GBP,7)),s)) . (intpos 4) = s . (intpos 4) by A8, AMI_3:10, SCMPDS_2:47;
A24: (IExec ((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))),P,s)) . (intpos 4) = (Exec ((AddTo (GBP,7,(- 1))),(Exec (((GBP,5) := (GBP,7)),s)))) . (intpos 4) by SCMPDS_5:42
.= s . (intpos 4) by A23, A10, AMI_3:10, SCMPDS_2:48 ;
A25: (Exec (((GBP,5) := (GBP,7)),s)) . (intpos 1) = s . (intpos 1) by A8, AMI_3:10, SCMPDS_2:47;
A26: (IExec ((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))),P,s)) . (intpos 1) = (Exec ((AddTo (GBP,7,(- 1))),(Exec (((GBP,5) := (GBP,7)),s)))) . (intpos 1) by SCMPDS_5:42
.= s . (intpos 1) by A25, A10, AMI_3:10, SCMPDS_2:48 ;
A27: (IExec (((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))),P,s)) . (intpos 1) = (Exec (((GBP,6) := ((intpos 2),0)),(IExec ((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))),P,s)))) . (intpos 1) by SCMPDS_5:41
.= s . (intpos 1) by A26, A15, AMI_3:10, SCMPDS_2:47 ;
(IExec ((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))),P,s)) . (intpos 1) = (Exec ((SubFrom (GBP,6,(intpos 3),0)),(IExec (((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))),P,s)))) . (intpos 1) by SCMPDS_5:41
.= s . (intpos 1) by A27, A22, AMI_3:10, SCMPDS_2:50 ;
then A28: (Initialize (IExec ((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))),P,s))) . (intpos 1) = s . (intpos 1) by SCMPDS_5:15;
A29: (IExec (((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))),P,s)) . (intpos 4) = (Exec (((GBP,6) := ((intpos 2),0)),(IExec ((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))),P,s)))) . (intpos 4) by SCMPDS_5:41
.= s . (intpos 4) by A24, A15, AMI_3:10, SCMPDS_2:47 ;
(IExec ((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))),P,s)) . (intpos 4) = (Exec ((SubFrom (GBP,6,(intpos 3),0)),(IExec (((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))),P,s)))) . (intpos 4) by SCMPDS_5:41
.= s . (intpos 4) by A29, A22, AMI_3:10, SCMPDS_2:50 ;
then A30: (Initialize (IExec ((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))),P,s))) . (intpos 4) = s . (intpos 4) by SCMPDS_5:15;
A31: (IExec ((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))),P,s)) . GBP = (Exec ((SubFrom (GBP,6,(intpos 3),0)),(IExec (((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))),P,s)))) . GBP by SCMPDS_5:41
.= 0 by A21, A22, AMI_3:10, SCMPDS_2:50 ;
then A32: (Initialize (IExec ((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))),P,s))) . GBP = 0 by SCMPDS_5:15;
then A33: DataLoc (((Initialize (IExec ((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))),P,s))) . GBP),7) = intpos (0 + 7) by SCMP_GCD:1;
 DataLoc ((s . GBP),7) = intpos (0 + 7) by A7, SCMP_GCD:1;
then A34: (Exec (((GBP,5) := (GBP,7)),s)) . (intpos 5) = s . (intpos 7) by A8, SCMPDS_2:47;
A35: (IExec ((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))),P,s)) . (intpos 5) = (Exec ((AddTo (GBP,7,(- 1))),(Exec (((GBP,5) := (GBP,7)),s)))) . (intpos 5) by SCMPDS_5:42
.= s . (intpos 7) by A34, A10, AMI_3:10, SCMPDS_2:48 ;
A36: (IExec (((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))),P,s)) . (intpos 5) = (Exec (((GBP,6) := ((intpos 2),0)),(IExec ((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))),P,s)))) . (intpos 5) by SCMPDS_5:41
.= s . (intpos 7) by A35, A15, AMI_3:10, SCMPDS_2:47 ;
(IExec ((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))),P,s)) . (intpos 5) = (Exec ((SubFrom (GBP,6,(intpos 3),0)),(IExec (((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))),P,s)))) . (intpos 5) by SCMPDS_5:41
.= s . (intpos 7) by A36, A22, AMI_3:10, SCMPDS_2:50 ;
then A37: (Initialize (IExec ((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))),P,s))) . (intpos 5) = s . (intpos 7) by SCMPDS_5:15;
A38: (IExec (((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))),P,s)) . (intpos 2) = (Exec (((GBP,6) := ((intpos 2),0)),(IExec ((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))),P,s)))) . (intpos 2) by SCMPDS_5:41
.= md by A12, A15, AMI_3:10, SCMPDS_2:47 ;
(IExec ((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))),P,s)) . (intpos 2) = (Exec ((SubFrom (GBP,6,(intpos 3),0)),(IExec (((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))),P,s)))) . (intpos 2) by SCMPDS_5:41
.= md by A38, A22, AMI_3:10, SCMPDS_2:50 ;
then A39: (Initialize (IExec ((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))),P,s))) . (intpos 2) = s . (intpos 2) by A1, SCMPDS_5:15;
 intpos 4 <> DataLoc (((Initialize (IExec ((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))),P,s))) . GBP),3) by A32, AMI_3:10, SCMP_GCD:1;
then A40: (Exec ((AddTo (GBP,3,1)),(Initialize (IExec ((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))),P,s))))) . (intpos 4) = s . (intpos 4) by A30, SCMPDS_2:48;
 intpos 2 <> DataLoc (((Initialize (IExec ((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))),P,s))) . GBP),3) by A32, AMI_3:10, SCMP_GCD:1;
then A41: (Exec ((AddTo (GBP,3,1)),(Initialize (IExec ((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))),P,s))))) . (intpos 2) = s . (intpos 2) by A39, SCMPDS_2:48;
 GBP <> DataLoc (((Initialize (IExec ((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))),P,s))) . GBP),3) by A32, AMI_3:10, SCMP_GCD:1;
then A42: (Exec ((AddTo (GBP,3,1)),(Initialize (IExec ((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))),P,s))))) . GBP = 0 by A32, SCMPDS_2:48;
then A43: DataLoc (((Exec ((AddTo (GBP,3,1)),(Initialize (IExec ((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))),P,s))))) . GBP),5) = intpos (0 + 5) by SCMP_GCD:1;
A44: now__::_thesis:_for_i_being_Element_of_NAT_st_i_>=_8_holds_
(Exec_(((GBP,5)_:=_(GBP,7)),s))_._(intpos_i)_=_s_._(intpos_i)
let i be Element of NAT ; ::_thesis: ( i >= 8 implies (Exec (((GBP,5) := (GBP,7)),s)) . (intpos i) = s . (intpos i) )
assume  i >= 8 ; ::_thesis: (Exec (((GBP,5) := (GBP,7)),s)) . (intpos i) = s . (intpos i)
then  i > 5 by XXREAL_0:2;
hence (Exec (((GBP,5) := (GBP,7)),s)) . (intpos i) = s . (intpos i) by A8, AMI_3:10, SCMPDS_2:47
.= s . (intpos i) ;
::_thesis: verum
end;
A45: now__::_thesis:_for_i_being_Element_of_NAT_st_i_>=_8_holds_
(IExec_((((GBP,5)_:=_(GBP,7))_';'_(AddTo_(GBP,7,(-_1)))),P,s))_._(intpos_i)_=_s_._(intpos_i)
let i be Element of NAT ; ::_thesis: ( i >= 8 implies (IExec ((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))),P,s)) . (intpos i) = s . (intpos i) )
assume A46: i >= 8 ; ::_thesis: (IExec ((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))),P,s)) . (intpos i) = s . (intpos i)
then A47: i > 7 by XXREAL_0:2;
thus (IExec ((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))),P,s)) . (intpos i) = (Exec ((AddTo (GBP,7,(- 1))),(Exec (((GBP,5) := (GBP,7)),s)))) . (intpos i) by SCMPDS_5:42
.= (Exec (((GBP,5) := (GBP,7)),s)) . (intpos i) by A10, A47, AMI_3:10, SCMPDS_2:48
.= s . (intpos i) by A44, A46 ; ::_thesis: verum
end;
A48: now__::_thesis:_for_i_being_Element_of_NAT_st_i_>=_8_holds_
(IExec_(((((GBP,5)_:=_(GBP,7))_';'_(AddTo_(GBP,7,(-_1))))_';'_((GBP,6)_:=_((intpos_2),0))),P,s))_._(intpos_i)_=_s_._(intpos_i)
let i be Element of NAT ; ::_thesis: ( i >= 8 implies (IExec (((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))),P,s)) . (intpos i) = s . (intpos i) )
assume A49: i >= 8 ; ::_thesis: (IExec (((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))),P,s)) . (intpos i) = s . (intpos i)
then A50: i > 6 by XXREAL_0:2;
thus (IExec (((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))),P,s)) . (intpos i) = (Exec (((GBP,6) := ((intpos 2),0)),(IExec ((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))),P,s)))) . (intpos i) by SCMPDS_5:41
.= (IExec ((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))),P,s)) . (intpos i) by A15, A50, AMI_3:10, SCMPDS_2:47
.= s . (intpos i) by A45, A49 ; ::_thesis: verum
end;
A51: now__::_thesis:_for_i_being_Element_of_NAT_st_i_>=_8_holds_
(IExec_((((((GBP,5)_:=_(GBP,7))_';'_(AddTo_(GBP,7,(-_1))))_';'_((GBP,6)_:=_((intpos_2),0)))_';'_(SubFrom_(GBP,6,(intpos_3),0))),P,s))_._(intpos_i)_=_s_._(intpos_i)
let i be Element of NAT ; ::_thesis: ( i >= 8 implies (IExec ((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))),P,s)) . (intpos i) = s . (intpos i) )
assume A52: i >= 8 ; ::_thesis: (IExec ((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))),P,s)) . (intpos i) = s . (intpos i)
then A53: i > 6 by XXREAL_0:2;
thus (IExec ((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))),P,s)) . (intpos i) = (Exec ((SubFrom (GBP,6,(intpos 3),0)),(IExec (((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))),P,s)))) . (intpos i) by SCMPDS_5:41
.= (IExec (((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))),P,s)) . (intpos i) by A22, A53, AMI_3:10, SCMPDS_2:50
.= s . (intpos i) by A48, A52 ; ::_thesis: verum
end;
A54: (IExec (((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))),P,s)) . (intpos 6) = (Exec (((GBP,6) := ((intpos 2),0)),(IExec ((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))),P,s)))) . (intpos 6) by SCMPDS_5:41
.= (IExec ((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))),P,s)) . (intpos md) by A15, A13, SCMPDS_2:47
.= s . (intpos md) by A3, A45 ;
(IExec ((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))),P,s)) . (intpos 6) = (Exec ((SubFrom (GBP,6,(intpos 3),0)),(IExec (((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))),P,s)))) . (intpos 6) by SCMPDS_5:41
.= ((IExec (((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))),P,s)) . (intpos 6)) - ((IExec (((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))),P,s)) . (DataLoc (((IExec (((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))),P,s)) . (intpos 3)),0))) by A22, SCMPDS_2:50
.= ((IExec (((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))),P,s)) . (intpos 6)) - ((IExec (((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))),P,s)) . (intpos (me + 0))) by A20, SCMP_GCD:1
.= (s . (intpos md)) - (s . (intpos me)) by A4, A54, A48 ;
then A55: (IExec ((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))),P,s)) . (DataLoc (((IExec ((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))),P,s)) . GBP),6)) = (s . (intpos md)) - (s . (intpos me)) by A31, SCMP_GCD:1;
 intpos 1 <> DataLoc (((Initialize (IExec ((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))),P,s))) . GBP),3) by A32, AMI_3:10, SCMP_GCD:1;
then A56: (Exec ((AddTo (GBP,3,1)),(Initialize (IExec ((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))),P,s))))) . (intpos 1) = s . (intpos 1) by A28, SCMPDS_2:48;
A57: now__::_thesis:_(_(IExec_((if>0_(GBP,6,((AddTo_(GBP,3,1))_';'_(AddTo_(GBP,5,(-_1)))),(Load_((GBP,7)_:=_0)))),P,(Initialize_(IExec_((((((GBP,5)_:=_(GBP,7))_';'_(AddTo_(GBP,7,(-_1))))_';'_((GBP,6)_:=_((intpos_2),0)))_';'_(SubFrom_(GBP,6,(intpos_3),0))),P,s)))))_._GBP_=_0_&_(IExec_((if>0_(GBP,6,((AddTo_(GBP,3,1))_';'_(AddTo_(GBP,5,(-_1)))),(Load_((GBP,7)_:=_0)))),P,(Initialize_(IExec_((((((GBP,5)_:=_(GBP,7))_';'_(AddTo_(GBP,7,(-_1))))_';'_((GBP,6)_:=_((intpos_2),0)))_';'_(SubFrom_(GBP,6,(intpos_3),0))),P,s)))))_._(intpos_1)_=_s_._(intpos_1)_&_(IExec_((if>0_(GBP,6,((AddTo_(GBP,3,1))_';'_(AddTo_(GBP,5,(-_1)))),(Load_((GBP,7)_:=_0)))),P,(Initialize_(IExec_((((((GBP,5)_:=_(GBP,7))_';'_(AddTo_(GBP,7,(-_1))))_';'_((GBP,6)_:=_((intpos_2),0)))_';'_(SubFrom_(GBP,6,(intpos_3),0))),P,s)))))_._(intpos_2)_=_s_._(intpos_2)_&_(IExec_((if>0_(GBP,6,((AddTo_(GBP,3,1))_';'_(AddTo_(GBP,5,(-_1)))),(Load_((GBP,7)_:=_0)))),P,(Initialize_(IExec_((((((GBP,5)_:=_(GBP,7))_';'_(AddTo_(GBP,7,(-_1))))_';'_((GBP,6)_:=_((intpos_2),0)))_';'_(SubFrom_(GBP,6,(intpos_3),0))),P,s)))))_._(intpos_4)_=_s_._(intpos_4)_&_(_for_i_being_Element_of_NAT_st_i_>=_8_holds_
(IExec_(((((((GBP,5)_:=_(GBP,7))_';'_(AddTo_(GBP,7,(-_1))))_';'_((GBP,6)_:=_((intpos_2),0)))_';'_(SubFrom_(GBP,6,(intpos_3),0)))_';'_(if>0_(GBP,6,((AddTo_(GBP,3,1))_';'_(AddTo_(GBP,5,(-_1)))),(Load_((GBP,7)_:=_0))))),P,s))_._(intpos_i)_=_s_._(intpos_i)_)_)
percases ( (Initialize (IExec ((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))),P,s))) . (DataLoc (((Initialize (IExec ((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))),P,s))) . GBP),6)) <= 0 or (Initialize (IExec ((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))),P,s))) . (DataLoc (((Initialize (IExec ((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))),P,s))) . GBP),6)) > 0 ) ;
supposeA58: (Initialize (IExec ((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))),P,s))) . (DataLoc (((Initialize (IExec ((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))),P,s))) . GBP),6)) <= 0 ; ::_thesis: ( (IExec ((if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))),P,(Initialize (IExec ((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))),P,s))))) . GBP = 0 & (IExec ((if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))),P,(Initialize (IExec ((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))),P,s))))) . (intpos 1) = s . (intpos 1) & (IExec ((if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))),P,(Initialize (IExec ((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))),P,s))))) . (intpos 2) = s . (intpos 2) & (IExec ((if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))),P,(Initialize (IExec ((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))),P,s))))) . (intpos 4) = s . (intpos 4) & ( for i being Element of NAT st i >= 8 holds
(IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),P,s)) . (intpos i) = s . (intpos i) ) )
hence (IExec ((if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))),P,(Initialize (IExec ((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))),P,s))))) . GBP = (IExec ((Load ((GBP,7) := 0)),P,(Initialize (IExec ((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))),P,s))))) . GBP by SCMPDS_6:74
.= (Exec (((GBP,7) := 0),(Initialize (IExec ((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))),P,s))))) . GBP by SCMPDS_5:40
.= 0 by A32, A33, AMI_3:10, SCMPDS_2:46 ;
::_thesis: ( (IExec ((if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))),P,(Initialize (IExec ((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))),P,s))))) . (intpos 1) = s . (intpos 1) & (IExec ((if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))),P,(Initialize (IExec ((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))),P,s))))) . (intpos 2) = s . (intpos 2) & (IExec ((if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))),P,(Initialize (IExec ((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))),P,s))))) . (intpos 4) = s . (intpos 4) & ( for i being Element of NAT st i >= 8 holds
(IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),P,s)) . (intpos i) = s . (intpos i) ) )
thus (IExec ((if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))),P,(Initialize (IExec ((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))),P,s))))) . (intpos 1) = (IExec ((Load ((GBP,7) := 0)),P,(Initialize (IExec ((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))),P,s))))) . (intpos 1) by A58, SCMPDS_6:74
.= (Exec (((GBP,7) := 0),(Initialize (IExec ((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))),P,s))))) . (intpos 1) by SCMPDS_5:40
.= s . (intpos 1) by A28, A33, AMI_3:10, SCMPDS_2:46 ; ::_thesis: ( (IExec ((if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))),P,(Initialize (IExec ((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))),P,s))))) . (intpos 2) = s . (intpos 2) & (IExec ((if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))),P,(Initialize (IExec ((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))),P,s))))) . (intpos 4) = s . (intpos 4) & ( for i being Element of NAT st i >= 8 holds
(IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),P,s)) . (intpos i) = s . (intpos i) ) )
thus (IExec ((if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))),P,(Initialize (IExec ((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))),P,s))))) . (intpos 2) = (IExec ((Load ((GBP,7) := 0)),P,(Initialize (IExec ((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))),P,s))))) . (intpos 2) by A58, SCMPDS_6:74
.= (Exec (((GBP,7) := 0),(Initialize (IExec ((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))),P,s))))) . (intpos 2) by SCMPDS_5:40
.= s . (intpos 2) by A39, A33, AMI_3:10, SCMPDS_2:46 ; ::_thesis: ( (IExec ((if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))),P,(Initialize (IExec ((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))),P,s))))) . (intpos 4) = s . (intpos 4) & ( for i being Element of NAT st i >= 8 holds
(IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),P,s)) . (intpos i) = s . (intpos i) ) )
thus (IExec ((if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))),P,(Initialize (IExec ((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))),P,s))))) . (intpos 4) = (IExec ((Load ((GBP,7) := 0)),P,(Initialize (IExec ((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))),P,s))))) . (intpos 4) by A58, SCMPDS_6:74
.= (Exec (((GBP,7) := 0),(Initialize (IExec ((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))),P,s))))) . (intpos 4) by SCMPDS_5:40
.= s . (intpos 4) by A30, A33, AMI_3:10, SCMPDS_2:46 ; ::_thesis: for i being Element of NAT st i >= 8 holds
(IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),P,s)) . (intpos i) = s . (intpos i)
let i be Element of NAT ; ::_thesis: ( i >= 8 implies (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),P,s)) . (intpos i) = s . (intpos i) )
assume A59: i >= 8 ; ::_thesis: (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),P,s)) . (intpos i) = s . (intpos i)
then A60: i > 7 by XXREAL_0:2;
thus (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),P,s)) . (intpos i) = (IExec ((if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))),P,(Initialize (IExec ((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))),P,s))))) . (intpos i) by SCMPDS_5:35
.= (IExec ((Load ((GBP,7) := 0)),P,(Initialize (IExec ((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))),P,s))))) . (intpos i) by A58, SCMPDS_6:74
.= (Exec (((GBP,7) := 0),(Initialize (IExec ((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))),P,s))))) . (intpos i) by SCMPDS_5:40
.= (Initialize (IExec ((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))),P,s))) . (intpos i) by A33, A60, AMI_3:10, SCMPDS_2:46
.= (IExec ((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))),P,s)) . (intpos i) by SCMPDS_5:15
.= s . (intpos i) by A51, A59 ; ::_thesis: verum
end;
supposeA61: (Initialize (IExec ((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))),P,s))) . (DataLoc (((Initialize (IExec ((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))),P,s))) . GBP),6)) > 0 ; ::_thesis: ( (IExec ((if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))),P,(Initialize (IExec ((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))),P,s))))) . GBP = 0 & (IExec ((if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))),P,(Initialize (IExec ((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))),P,s))))) . (intpos 1) = s . (intpos 1) & (IExec ((if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))),P,(Initialize (IExec ((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))),P,s))))) . (intpos 2) = s . (intpos 2) & (IExec ((if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))),P,(Initialize (IExec ((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))),P,s))))) . (intpos 4) = s . (intpos 4) & ( for i being Element of NAT st i >= 8 holds
(IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),P,s)) . (intpos i) = s . (intpos i) ) )
hence (IExec ((if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))),P,(Initialize (IExec ((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))),P,s))))) . GBP = (IExec (((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),P,(Initialize (IExec ((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))),P,s))))) . GBP by SCMPDS_6:73
.= (Exec ((AddTo (GBP,5,(- 1))),(Exec ((AddTo (GBP,3,1)),(Initialize (IExec ((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))),P,s))))))) . GBP by SCMPDS_5:42
.= 0 by A42, A43, AMI_3:10, SCMPDS_2:48 ;
::_thesis: ( (IExec ((if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))),P,(Initialize (IExec ((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))),P,s))))) . (intpos 1) = s . (intpos 1) & (IExec ((if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))),P,(Initialize (IExec ((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))),P,s))))) . (intpos 2) = s . (intpos 2) & (IExec ((if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))),P,(Initialize (IExec ((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))),P,s))))) . (intpos 4) = s . (intpos 4) & ( for i being Element of NAT st i >= 8 holds
(IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),P,s)) . (intpos i) = s . (intpos i) ) )
thus (IExec ((if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))),P,(Initialize (IExec ((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))),P,s))))) . (intpos 1) = (IExec (((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),P,(Initialize (IExec ((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))),P,s))))) . (intpos 1) by A61, SCMPDS_6:73
.= (Exec ((AddTo (GBP,5,(- 1))),(Exec ((AddTo (GBP,3,1)),(Initialize (IExec ((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))),P,s))))))) . (intpos 1) by SCMPDS_5:42
.= s . (intpos 1) by A56, A43, AMI_3:10, SCMPDS_2:48 ; ::_thesis: ( (IExec ((if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))),P,(Initialize (IExec ((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))),P,s))))) . (intpos 2) = s . (intpos 2) & (IExec ((if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))),P,(Initialize (IExec ((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))),P,s))))) . (intpos 4) = s . (intpos 4) & ( for i being Element of NAT st i >= 8 holds
(IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),P,s)) . (intpos i) = s . (intpos i) ) )
thus (IExec ((if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))),P,(Initialize (IExec ((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))),P,s))))) . (intpos 2) = (IExec (((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),P,(Initialize (IExec ((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))),P,s))))) . (intpos 2) by A61, SCMPDS_6:73
.= (Exec ((AddTo (GBP,5,(- 1))),(Exec ((AddTo (GBP,3,1)),(Initialize (IExec ((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))),P,s))))))) . (intpos 2) by SCMPDS_5:42
.= s . (intpos 2) by A41, A43, AMI_3:10, SCMPDS_2:48 ; ::_thesis: ( (IExec ((if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))),P,(Initialize (IExec ((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))),P,s))))) . (intpos 4) = s . (intpos 4) & ( for i being Element of NAT st i >= 8 holds
(IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),P,s)) . (intpos i) = s . (intpos i) ) )
thus (IExec ((if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))),P,(Initialize (IExec ((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))),P,s))))) . (intpos 4) = (IExec (((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),P,(Initialize (IExec ((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))),P,s))))) . (intpos 4) by A61, SCMPDS_6:73
.= (Exec ((AddTo (GBP,5,(- 1))),(Exec ((AddTo (GBP,3,1)),(Initialize (IExec ((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))),P,s))))))) . (intpos 4) by SCMPDS_5:42
.= s . (intpos 4) by A40, A43, AMI_3:10, SCMPDS_2:48 ; ::_thesis: for i being Element of NAT st i >= 8 holds
(IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),P,s)) . (intpos i) = s . (intpos i)
let i be Element of NAT ; ::_thesis: ( i >= 8 implies (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),P,s)) . (intpos i) = s . (intpos i) )
assume A62: i >= 8 ; ::_thesis: (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),P,s)) . (intpos i) = s . (intpos i)
then A63: i > 5 by XXREAL_0:2;
 i > 3 by A62, XXREAL_0:2;
then A64: intpos i <> DataLoc (((Initialize (IExec ((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))),P,s))) . GBP),3) by A32, AMI_3:10, SCMP_GCD:1;
thus (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),P,s)) . (intpos i) = (IExec ((if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))),P,(Initialize (IExec ((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))),P,s))))) . (intpos i) by SCMPDS_5:35
.= (IExec (((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),P,(Initialize (IExec ((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))),P,s))))) . (intpos i) by A61, SCMPDS_6:73
.= (Exec ((AddTo (GBP,5,(- 1))),(Exec ((AddTo (GBP,3,1)),(Initialize (IExec ((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))),P,s))))))) . (intpos i) by SCMPDS_5:42
.= (Exec ((AddTo (GBP,3,1)),(Initialize (IExec ((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))),P,s))))) . (intpos i) by A43, A63, AMI_3:10, SCMPDS_2:48
.= (Initialize (IExec ((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))),P,s))) . (intpos i) by A64, SCMPDS_2:48
.= (IExec ((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))),P,s)) . (intpos i) by SCMPDS_5:15
.= s . (intpos i) by A51, A62 ; ::_thesis: verum
end;
end;
end;
hence  ( (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),P,s)) . GBP = 0 & (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),P,s)) . (intpos 1) = s . (intpos 1) & (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),P,s)) . (intpos 2) = s . (intpos 2) & (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),P,s)) . (intpos 4) = s . (intpos 4) ) by SCMPDS_5:35; ::_thesis: ( ( for i being Element of NAT st i >= 8 holds
(IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),P,s)) . (intpos i) = s . (intpos i) ) & ( s . (intpos md) > s . (intpos me) implies ( (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),P,s)) . (intpos 7) = (s . (intpos 7)) - 1 & (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),P,s)) . (intpos 3) = (s . (intpos 3)) + 1 & (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),P,s)) . (intpos 5) = (s . (intpos 7)) - 1 ) ) & ( s . (intpos md) <= s . (intpos me) implies ( (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),P,s)) . (intpos 7) = 0 & (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),P,s)) . (intpos 3) = s . (intpos 3) & (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),P,s)) . (intpos 5) = s . (intpos 7) ) ) )
thus  for i being Element of NAT st i >= 8 holds
(IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),P,s)) . (intpos i) = s . (intpos i) by A57; ::_thesis: ( ( s . (intpos md) > s . (intpos me) implies ( (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),P,s)) . (intpos 7) = (s . (intpos 7)) - 1 & (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),P,s)) . (intpos 3) = (s . (intpos 3)) + 1 & (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),P,s)) . (intpos 5) = (s . (intpos 7)) - 1 ) ) & ( s . (intpos md) <= s . (intpos me) implies ( (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),P,s)) . (intpos 7) = 0 & (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),P,s)) . (intpos 3) = s . (intpos 3) & (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),P,s)) . (intpos 5) = s . (intpos 7) ) ) )
A65: (IExec ((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))),P,s)) . (intpos 7) = (Exec ((AddTo (GBP,7,(- 1))),(Exec (((GBP,5) := (GBP,7)),s)))) . (intpos 7) by SCMPDS_5:42
.= ((Exec (((GBP,5) := (GBP,7)),s)) . (intpos 7)) + (- 1) by A10, SCMPDS_2:48
.= (s . (intpos 7)) - 1 by A19 ;
A66: (IExec (((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))),P,s)) . (intpos 7) = (Exec (((GBP,6) := ((intpos 2),0)),(IExec ((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))),P,s)))) . (intpos 7) by SCMPDS_5:41
.= (s . (intpos 7)) - 1 by A65, A15, AMI_3:10, SCMPDS_2:47 ;
(IExec ((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))),P,s)) . (intpos 7) = (Exec ((SubFrom (GBP,6,(intpos 3),0)),(IExec (((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))),P,s)))) . (intpos 7) by SCMPDS_5:41
.= (s . (intpos 7)) - 1 by A66, A22, AMI_3:10, SCMPDS_2:50 ;
then A67: (Initialize (IExec ((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))),P,s))) . (intpos 7) = (s . (intpos 7)) - 1 by SCMPDS_5:15;
(IExec ((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))),P,s)) . (intpos 3) = (Exec ((SubFrom (GBP,6,(intpos 3),0)),(IExec (((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))),P,s)))) . (intpos 3) by SCMPDS_5:41
.= me by A20, A22, AMI_3:10, SCMPDS_2:50 ;
then A68: (Initialize (IExec ((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))),P,s))) . (intpos 3) = s . (intpos 3) by A2, SCMPDS_5:15;
A69: DataLoc (((Initialize (IExec ((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))),P,s))) . GBP),3) = intpos (0 + 3) by A32, SCMP_GCD:1;
XX: (Initialize (IExec ((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))),P,s))) . (DataLoc (((Initialize (IExec ((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))),P,s))) . GBP),6)) = (IExec ((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))),P,s)) . (DataLoc (((Initialize (IExec ((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))),P,s))) . GBP),6)) by SCMPDS_5:15
.= (IExec ((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))),P,s)) . (DataLoc (((IExec ((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))),P,s)) . GBP),6)) by SCMPDS_5:15 ;
hereby  ::_thesis: ( s . (intpos md) <= s . (intpos me) implies ( (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),P,s)) . (intpos 7) = 0 & (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),P,s)) . (intpos 3) = s . (intpos 3) & (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),P,s)) . (intpos 5) = s . (intpos 7) ) )
A70: intpos 7 <> DataLoc (((Initialize (IExec ((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))),P,s))) . GBP),3) by A32, AMI_3:10, SCMP_GCD:1;
assume  s . (intpos md) > s . (intpos me) ; ::_thesis: ( (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),P,s)) . (intpos 7) = (s . (intpos 7)) - 1 & (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),P,s)) . (intpos 3) = (s . (intpos 3)) + 1 & (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),P,s)) . (intpos 5) = (s . (intpos 7)) - 1 )
then B71: (s . (intpos md)) - (s . (intpos me)) > 0 by XREAL_1:50;
thus (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),P,s)) . (intpos 7) = (IExec ((if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))),P,(Initialize (IExec ((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))),P,s))))) . (intpos 7) by SCMPDS_5:35
.= (IExec (((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),P,(Initialize (IExec ((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))),P,s))))) . (intpos 7) by A55, XX, B71, SCMPDS_6:73
.= (Exec ((AddTo (GBP,5,(- 1))),(Exec ((AddTo (GBP,3,1)),(Initialize (IExec ((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))),P,s))))))) . (intpos 7) by SCMPDS_5:42
.= (Exec ((AddTo (GBP,3,1)),(Initialize (IExec ((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))),P,s))))) . (intpos 7) by A43, AMI_3:10, SCMPDS_2:48
.= (s . (intpos 7)) - 1 by A67, A70, SCMPDS_2:48 ; ::_thesis: ( (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),P,s)) . (intpos 3) = (s . (intpos 3)) + 1 & (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),P,s)) . (intpos 5) = (s . (intpos 7)) - 1 )
thus (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),P,s)) . (intpos 3) = (IExec ((if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))),P,(Initialize (IExec ((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))),P,s))))) . (intpos 3) by SCMPDS_5:35
.= (IExec (((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),P,(Initialize (IExec ((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))),P,s))))) . (intpos 3) by A55, XX, B71, SCMPDS_6:73
.= (Exec ((AddTo (GBP,5,(- 1))),(Exec ((AddTo (GBP,3,1)),(Initialize (IExec ((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))),P,s))))))) . (intpos 3) by SCMPDS_5:42
.= (Exec ((AddTo (GBP,3,1)),(Initialize (IExec ((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))),P,s))))) . (intpos 3) by A43, AMI_3:10, SCMPDS_2:48
.= (s . (intpos 3)) + 1 by A69, A68, SCMPDS_2:48 ; ::_thesis: (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),P,s)) . (intpos 5) = (s . (intpos 7)) - 1
A72: intpos 5 <> DataLoc (((Initialize (IExec ((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))),P,s))) . GBP),3) by A32, AMI_3:10, SCMP_GCD:1;
thus (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),P,s)) . (intpos 5) = (IExec ((if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))),P,(Initialize (IExec ((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))),P,s))))) . (intpos 5) by SCMPDS_5:35
.= (IExec (((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),P,(Initialize (IExec ((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))),P,s))))) . (intpos 5) by A55, XX, B71, SCMPDS_6:73
.= (Exec ((AddTo (GBP,5,(- 1))),(Exec ((AddTo (GBP,3,1)),(Initialize (IExec ((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))),P,s))))))) . (intpos 5) by SCMPDS_5:42
.= ((Exec ((AddTo (GBP,3,1)),(Initialize (IExec ((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))),P,s))))) . (intpos 5)) + (- 1) by A43, SCMPDS_2:48
.= ((Exec ((AddTo (GBP,3,1)),(Initialize (IExec ((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))),P,s))))) . (intpos 5)) - 1
.= (s . (intpos 7)) - 1 by A37, A72, SCMPDS_2:48 ; ::_thesis: verum
end;
hereby  ::_thesis: verum
assume  s . (intpos md) <= s . (intpos me) ; ::_thesis: ( (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),P,s)) . (intpos 7) = 0 & (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),P,s)) . (intpos 3) = s . (intpos 3) & (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),P,s)) . (intpos 5) = s . (intpos 7) )
then B73: 0 >= (s . (intpos md)) - (s . (intpos me)) by XREAL_1:47;
thus (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),P,s)) . (intpos 7) = (IExec ((if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))),P,(Initialize (IExec ((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))),P,s))))) . (intpos 7) by SCMPDS_5:35
.= (IExec ((Load ((GBP,7) := 0)),P,(Initialize (IExec ((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))),P,s))))) . (intpos 7) by A55, XX, B73, SCMPDS_6:74
.= (Exec (((GBP,7) := 0),(Initialize (IExec ((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))),P,s))))) . (intpos 7) by SCMPDS_5:40
.= 0 by A33, SCMPDS_2:46 ; ::_thesis: ( (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),P,s)) . (intpos 3) = s . (intpos 3) & (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),P,s)) . (intpos 5) = s . (intpos 7) )
thus (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),P,s)) . (intpos 3) = (IExec ((if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))),P,(Initialize (IExec ((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))),P,s))))) . (intpos 3) by SCMPDS_5:35
.= (IExec ((Load ((GBP,7) := 0)),P,(Initialize (IExec ((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))),P,s))))) . (intpos 3) by A55, XX, B73, SCMPDS_6:74
.= (Exec (((GBP,7) := 0),(Initialize (IExec ((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))),P,s))))) . (intpos 3) by SCMPDS_5:40
.= s . (intpos 3) by A33, A68, AMI_3:10, SCMPDS_2:46 ; ::_thesis: (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),P,s)) . (intpos 5) = s . (intpos 7)
thus (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),P,s)) . (intpos 5) = (IExec ((if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))),P,(Initialize (IExec ((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))),P,s))))) . (intpos 5) by SCMPDS_5:35
.= (IExec ((Load ((GBP,7) := 0)),P,(Initialize (IExec ((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))),P,s))))) . (intpos 5) by A55, XX, B73, SCMPDS_6:74
.= (Exec (((GBP,7) := 0),(Initialize (IExec ((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))),P,s))))) . (intpos 5) by SCMPDS_5:40
.= s . (intpos 7) by A37, A33, AMI_3:10, SCMPDS_2:46 ; ::_thesis: verum
end;
end;

Lm15:  for P being Instruction-Sequence of SCMPDS
for s being 0 -started State of SCMPDS
for m3, md being Element of NAT st s . GBP = 0 & s . (intpos 7) > 0 & (s . (intpos 3)) + (s . (intpos 7)) = m3 & s . (intpos 3) >= 8 & s . (intpos 2) = md & md >= 8 holds
( (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),P,s)) . GBP = 0 & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),P,s)) . (intpos 1) = s . (intpos 1) & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),P,s)) . (intpos 7) = 0 & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),P,s)) . (intpos 2) = s . (intpos 2) & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),P,s)) . (intpos 4) = s . (intpos 4) & ( for i being Element of NAT st i >= 8 holds
(IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),P,s)) . (intpos i) = s . (intpos i) ) & ex m5, mE3 being Element of NAT st
( m5 = (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),P,s)) . (intpos 5) & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),P,s)) . (intpos 3) = mE3 & mE3 + m5 = m3 & m5 <= s . (intpos 7) & ( for i being Element of NAT st s . (intpos 3) <= i & i < mE3 holds
(IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),P,s)) . (intpos md) > (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),P,s)) . (intpos i) ) & ( m5 = 0 or (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),P,s)) . (intpos md) <= (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),P,s)) . (intpos mE3) ) ) )
proof
let P be Instruction-Sequence of SCMPDS; ::_thesis: for s being 0 -started State of SCMPDS
for m3, md being Element of NAT st s . GBP = 0 & s . (intpos 7) > 0 & (s . (intpos 3)) + (s . (intpos 7)) = m3 & s . (intpos 3) >= 8 & s . (intpos 2) = md & md >= 8 holds
( (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),P,s)) . GBP = 0 & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),P,s)) . (intpos 1) = s . (intpos 1) & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),P,s)) . (intpos 7) = 0 & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),P,s)) . (intpos 2) = s . (intpos 2) & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),P,s)) . (intpos 4) = s . (intpos 4) & ( for i being Element of NAT st i >= 8 holds
(IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),P,s)) . (intpos i) = s . (intpos i) ) & ex m5, mE3 being Element of NAT st
( m5 = (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),P,s)) . (intpos 5) & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),P,s)) . (intpos 3) = mE3 & mE3 + m5 = m3 & m5 <= s . (intpos 7) & ( for i being Element of NAT st s . (intpos 3) <= i & i < mE3 holds
(IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),P,s)) . (intpos md) > (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),P,s)) . (intpos i) ) & ( m5 = 0 or (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),P,s)) . (intpos md) <= (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),P,s)) . (intpos mE3) ) ) )
set a = GBP ;
let s be 0 -started State of SCMPDS; ::_thesis: for m3, md being Element of NAT st s . GBP = 0 & s . (intpos 7) > 0 & (s . (intpos 3)) + (s . (intpos 7)) = m3 & s . (intpos 3) >= 8 & s . (intpos 2) = md & md >= 8 holds
( (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),P,s)) . GBP = 0 & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),P,s)) . (intpos 1) = s . (intpos 1) & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),P,s)) . (intpos 7) = 0 & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),P,s)) . (intpos 2) = s . (intpos 2) & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),P,s)) . (intpos 4) = s . (intpos 4) & ( for i being Element of NAT st i >= 8 holds
(IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),P,s)) . (intpos i) = s . (intpos i) ) & ex m5, mE3 being Element of NAT st
( m5 = (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),P,s)) . (intpos 5) & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),P,s)) . (intpos 3) = mE3 & mE3 + m5 = m3 & m5 <= s . (intpos 7) & ( for i being Element of NAT st s . (intpos 3) <= i & i < mE3 holds
(IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),P,s)) . (intpos md) > (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),P,s)) . (intpos i) ) & ( m5 = 0 or (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),P,s)) . (intpos md) <= (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),P,s)) . (intpos mE3) ) ) )
let m3, md be Element of NAT ; ::_thesis: ( s . GBP = 0 & s . (intpos 7) > 0 & (s . (intpos 3)) + (s . (intpos 7)) = m3 & s . (intpos 3) >= 8 & s . (intpos 2) = md & md >= 8 implies ( (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),P,s)) . GBP = 0 & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),P,s)) . (intpos 1) = s . (intpos 1) & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),P,s)) . (intpos 7) = 0 & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),P,s)) . (intpos 2) = s . (intpos 2) & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),P,s)) . (intpos 4) = s . (intpos 4) & ( for i being Element of NAT st i >= 8 holds
(IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),P,s)) . (intpos i) = s . (intpos i) ) & ex m5, mE3 being Element of NAT st
( m5 = (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),P,s)) . (intpos 5) & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),P,s)) . (intpos 3) = mE3 & mE3 + m5 = m3 & m5 <= s . (intpos 7) & ( for i being Element of NAT st s . (intpos 3) <= i & i < mE3 holds
(IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),P,s)) . (intpos md) > (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),P,s)) . (intpos i) ) & ( m5 = 0 or (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),P,s)) . (intpos md) <= (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),P,s)) . (intpos mE3) ) ) ) )
assume that
A1: s . GBP = 0 and
A2: s . (intpos 7) > 0 and
A3: (s . (intpos 3)) + (s . (intpos 7)) = m3 and
A4: s . (intpos 3) >= 8 and
A5: s . (intpos 2) = md and
A6: md >= 8 ; ::_thesis: ( (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),P,s)) . GBP = 0 & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),P,s)) . (intpos 1) = s . (intpos 1) & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),P,s)) . (intpos 7) = 0 & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),P,s)) . (intpos 2) = s . (intpos 2) & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),P,s)) . (intpos 4) = s . (intpos 4) & ( for i being Element of NAT st i >= 8 holds
(IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),P,s)) . (intpos i) = s . (intpos i) ) & ex m5, mE3 being Element of NAT st
( m5 = (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),P,s)) . (intpos 5) & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),P,s)) . (intpos 3) = mE3 & mE3 + m5 = m3 & m5 <= s . (intpos 7) & ( for i being Element of NAT st s . (intpos 3) <= i & i < mE3 holds
(IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),P,s)) . (intpos md) > (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),P,s)) . (intpos i) ) & ( m5 = 0 or (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),P,s)) . (intpos md) <= (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),P,s)) . (intpos mE3) ) ) )
 s . (intpos 7) >= 1 + 0 by A2, INT_1:7;
then reconsider m7 = (s . (intpos 7)) - 1 as Element of NAT by INT_1:3, XREAL_1:48;
A7: s . (intpos 7) = m7 + 1 ;
defpred S1[ Element of NAT ] means  for t being 0 -started State of SCMPDS
for Q being Instruction-Sequence of SCMPDS st t . GBP = 0 & t . (intpos 7) = $1 + 1 & (t . (intpos 3)) + (t . (intpos 7)) = m3 & t . (intpos 3) >= 8 & t . (intpos 2) = md holds
( (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . GBP = 0 & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos 1) = t . (intpos 1) & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos 7) = 0 & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos 2) = t . (intpos 2) & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos 4) = t . (intpos 4) & ( for i being Element of NAT st i >= 8 holds
(IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos i) = t . (intpos i) ) & ex m5, mE3 being Element of NAT st
( m5 = (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos 5) & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos 3) = mE3 & mE3 + m5 = m3 & m5 <= t . (intpos 7) & ( for i being Element of NAT st t . (intpos 3) <= i & i < mE3 holds
(IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos md) > (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos i) ) & ( m5 = 0 or (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos md) <= (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos mE3) ) ) );
A8: now__::_thesis:_for_k_being_Element_of_NAT_st_S1[k]_holds_
S1[k_+_1]
let k be Element of NAT ; ::_thesis: ( S1[k] implies S1[k + 1] )
assume A9: S1[k] ; ::_thesis: S1[k + 1]
now__::_thesis:_for_t_being_0_-started_State_of_SCMPDS
for_Q_being_Instruction-Sequence_of_SCMPDS_st_t_._GBP_=_0_&_t_._(intpos_7)_=_(k_+_1)_+_1_&_(t_._(intpos_3))_+_(t_._(intpos_7))_=_m3_&_t_._(intpos_3)_>=_8_&_t_._(intpos_2)_=_md_holds_
(_(IExec_((while>0_(GBP,7,((((((GBP,5)_:=_(GBP,7))_';'_(AddTo_(GBP,7,(-_1))))_';'_((GBP,6)_:=_((intpos_2),0)))_';'_(SubFrom_(GBP,6,(intpos_3),0)))_';'_(if>0_(GBP,6,((AddTo_(GBP,3,1))_';'_(AddTo_(GBP,5,(-_1)))),(Load_((GBP,7)_:=_0))))))),Q,t))_._GBP_=_0_&_(IExec_((while>0_(GBP,7,((((((GBP,5)_:=_(GBP,7))_';'_(AddTo_(GBP,7,(-_1))))_';'_((GBP,6)_:=_((intpos_2),0)))_';'_(SubFrom_(GBP,6,(intpos_3),0)))_';'_(if>0_(GBP,6,((AddTo_(GBP,3,1))_';'_(AddTo_(GBP,5,(-_1)))),(Load_((GBP,7)_:=_0))))))),Q,t))_._(intpos_1)_=_t_._(intpos_1)_&_(IExec_((while>0_(GBP,7,((((((GBP,5)_:=_(GBP,7))_';'_(AddTo_(GBP,7,(-_1))))_';'_((GBP,6)_:=_((intpos_2),0)))_';'_(SubFrom_(GBP,6,(intpos_3),0)))_';'_(if>0_(GBP,6,((AddTo_(GBP,3,1))_';'_(AddTo_(GBP,5,(-_1)))),(Load_((GBP,7)_:=_0))))))),Q,t))_._(intpos_7)_=_0_&_(IExec_((while>0_(GBP,7,((((((GBP,5)_:=_(GBP,7))_';'_(AddTo_(GBP,7,(-_1))))_';'_((GBP,6)_:=_((intpos_2),0)))_';'_(SubFrom_(GBP,6,(intpos_3),0)))_';'_(if>0_(GBP,6,((AddTo_(GBP,3,1))_';'_(AddTo_(GBP,5,(-_1)))),(Load_((GBP,7)_:=_0))))))),Q,t))_._(intpos_2)_=_t_._(intpos_2)_&_(IExec_((while>0_(GBP,7,((((((GBP,5)_:=_(GBP,7))_';'_(AddTo_(GBP,7,(-_1))))_';'_((GBP,6)_:=_((intpos_2),0)))_';'_(SubFrom_(GBP,6,(intpos_3),0)))_';'_(if>0_(GBP,6,((AddTo_(GBP,3,1))_';'_(AddTo_(GBP,5,(-_1)))),(Load_((GBP,7)_:=_0))))))),Q,t))_._(intpos_4)_=_t_._(intpos_4)_&_(_for_i_being_Element_of_NAT_st_i_>=_8_holds_
(IExec_((while>0_(GBP,7,((((((GBP,5)_:=_(GBP,7))_';'_(AddTo_(GBP,7,(-_1))))_';'_((GBP,6)_:=_((intpos_2),0)))_';'_(SubFrom_(GBP,6,(intpos_3),0)))_';'_(if>0_(GBP,6,((AddTo_(GBP,3,1))_';'_(AddTo_(GBP,5,(-_1)))),(Load_((GBP,7)_:=_0))))))),Q,t))_._(intpos_i)_=_t_._(intpos_i)_)_&_ex_m5,_mE3_being_Element_of_NAT_st_
(_m5_=_(IExec_((while>0_(GBP,7,((((((GBP,5)_:=_(GBP,7))_';'_(AddTo_(GBP,7,(-_1))))_';'_((GBP,6)_:=_((intpos_2),0)))_';'_(SubFrom_(GBP,6,(intpos_3),0)))_';'_(if>0_(GBP,6,((AddTo_(GBP,3,1))_';'_(AddTo_(GBP,5,(-_1)))),(Load_((GBP,7)_:=_0))))))),Q,t))_._(intpos_5)_&_(IExec_((while>0_(GBP,7,((((((GBP,5)_:=_(GBP,7))_';'_(AddTo_(GBP,7,(-_1))))_';'_((GBP,6)_:=_((intpos_2),0)))_';'_(SubFrom_(GBP,6,(intpos_3),0)))_';'_(if>0_(GBP,6,((AddTo_(GBP,3,1))_';'_(AddTo_(GBP,5,(-_1)))),(Load_((GBP,7)_:=_0))))))),Q,t))_._(intpos_3)_=_mE3_&_mE3_+_m5_=_m3_&_m5_<=_t_._(intpos_7)_&_(_for_i_being_Element_of_NAT_st_t_._(intpos_3)_<=_i_&_i_<_mE3_holds_
(IExec_((while>0_(GBP,7,((((((GBP,5)_:=_(GBP,7))_';'_(AddTo_(GBP,7,(-_1))))_';'_((GBP,6)_:=_((intpos_2),0)))_';'_(SubFrom_(GBP,6,(intpos_3),0)))_';'_(if>0_(GBP,6,((AddTo_(GBP,3,1))_';'_(AddTo_(GBP,5,(-_1)))),(Load_((GBP,7)_:=_0))))))),Q,t))_._(intpos_md)_>_(IExec_((while>0_(GBP,7,((((((GBP,5)_:=_(GBP,7))_';'_(AddTo_(GBP,7,(-_1))))_';'_((GBP,6)_:=_((intpos_2),0)))_';'_(SubFrom_(GBP,6,(intpos_3),0)))_';'_(if>0_(GBP,6,((AddTo_(GBP,3,1))_';'_(AddTo_(GBP,5,(-_1)))),(Load_((GBP,7)_:=_0))))))),Q,t))_._(intpos_i)_)_&_(_m5_=_0_or_(IExec_((while>0_(GBP,7,((((((GBP,5)_:=_(GBP,7))_';'_(AddTo_(GBP,7,(-_1))))_';'_((GBP,6)_:=_((intpos_2),0)))_';'_(SubFrom_(GBP,6,(intpos_3),0)))_';'_(if>0_(GBP,6,((AddTo_(GBP,3,1))_';'_(AddTo_(GBP,5,(-_1)))),(Load_((GBP,7)_:=_0))))))),Q,t))_._(intpos_md)_<=_(IExec_((while>0_(GBP,7,((((((GBP,5)_:=_(GBP,7))_';'_(AddTo_(GBP,7,(-_1))))_';'_((GBP,6)_:=_((intpos_2),0)))_';'_(SubFrom_(GBP,6,(intpos_3),0)))_';'_(if>0_(GBP,6,((AddTo_(GBP,3,1))_';'_(AddTo_(GBP,5,(-_1)))),(Load_((GBP,7)_:=_0))))))),Q,t))_._(intpos_mE3)_)_)_)
let t be 0 -started State of SCMPDS; ::_thesis: for Q being Instruction-Sequence of SCMPDS st t . GBP = 0 & t . (intpos 7) = (k + 1) + 1 & (t . (intpos 3)) + (t . (intpos 7)) = m3 & t . (intpos 3) >= 8 & t . (intpos 2) = md holds
( (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),b3,b2)) . GBP = 0 & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),b3,b2)) . (intpos 1) = b2 . (intpos 1) & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),b3,b2)) . (intpos 7) = 0 & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),b3,b2)) . (intpos 2) = b2 . (intpos 2) & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),b3,b2)) . (intpos 4) = b2 . (intpos 4) & ( for i being Element of NAT st b4 >= 8 holds
(IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),b3,i)) . (intpos b4) = i . (intpos b4) ) & ex m5, mE3 being Element of NAT st
( b4 = (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),mE3,m5)) . (intpos 5) & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),mE3,m5)) . (intpos 3) = b5 & b5 + b4 = m3 & b4 <= m5 . (intpos 7) & ( for i being Element of NAT st m5 . (intpos 3) <= b6 & b6 < b5 holds
(IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),mE3,m5)) . (intpos md) > (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),mE3,m5)) . (intpos b6) ) & ( b4 = 0 or (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),mE3,m5)) . (intpos md) <= (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),mE3,m5)) . (intpos b5) ) ) )
let Q be Instruction-Sequence of SCMPDS; ::_thesis: ( t . GBP = 0 & t . (intpos 7) = (k + 1) + 1 & (t . (intpos 3)) + (t . (intpos 7)) = m3 & t . (intpos 3) >= 8 & t . (intpos 2) = md implies ( (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),b2,b1)) . GBP = 0 & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),b2,b1)) . (intpos 1) = b1 . (intpos 1) & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),b2,b1)) . (intpos 7) = 0 & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),b2,b1)) . (intpos 2) = b1 . (intpos 2) & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),b2,b1)) . (intpos 4) = b1 . (intpos 4) & ( for i being Element of NAT st b3 >= 8 holds
(IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),b2,i)) . (intpos b3) = i . (intpos b3) ) & ex m5, mE3 being Element of NAT st
( b3 = (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),mE3,m5)) . (intpos 5) & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),mE3,m5)) . (intpos 3) = b4 & b4 + b3 = m3 & b3 <= m5 . (intpos 7) & ( for i being Element of NAT st m5 . (intpos 3) <= b5 & b5 < b4 holds
(IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),mE3,m5)) . (intpos md) > (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),mE3,m5)) . (intpos b5) ) & ( b3 = 0 or (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),mE3,m5)) . (intpos md) <= (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),mE3,m5)) . (intpos b4) ) ) ) )
set b = DataLoc ((t . GBP),7);
assume that
A10: t . GBP = 0 and
A11: t . (intpos 7) = (k + 1) + 1 and
A12: (t . (intpos 3)) + (t . (intpos 7)) = m3 and
A13: t . (intpos 3) >= 8 and
A14: t . (intpos 2) = md ; ::_thesis: ( (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),b2,b1)) . GBP = 0 & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),b2,b1)) . (intpos 1) = b1 . (intpos 1) & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),b2,b1)) . (intpos 7) = 0 & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),b2,b1)) . (intpos 2) = b1 . (intpos 2) & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),b2,b1)) . (intpos 4) = b1 . (intpos 4) & ( for i being Element of NAT st b3 >= 8 holds
(IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),b2,i)) . (intpos b3) = i . (intpos b3) ) & ex m5, mE3 being Element of NAT st
( b3 = (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),mE3,m5)) . (intpos 5) & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),mE3,m5)) . (intpos 3) = b4 & b4 + b3 = m3 & b3 <= m5 . (intpos 7) & ( for i being Element of NAT st m5 . (intpos 3) <= b5 & b5 < b4 holds
(IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),mE3,m5)) . (intpos md) > (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),mE3,m5)) . (intpos b5) ) & ( b3 = 0 or (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),mE3,m5)) . (intpos md) <= (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),mE3,m5)) . (intpos b4) ) ) )
reconsider me = t . (intpos 3) as Element of NAT by A13, INT_1:3;
set It = IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),Q,t);
set IT = Q;
A24: ( t . (intpos md) > t . (intpos me) implies ( (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),Q,t)) . (intpos 7) = (t . (intpos 7)) - 1 & (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),Q,t)) . (intpos 3) = (t . (intpos 3)) + 1 & (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),Q,t)) . (intpos 5) = (t . (intpos 7)) - 1 ) ) by A6, A10, A13, A14, Lm14;
A25: DataLoc ((t . GBP),7) = intpos (0 + 7) by A10, SCMP_GCD:1;
A26: now__::_thesis:_for_v_being_0_-started_State_of_SCMPDS
for_V_being_Instruction-Sequence_of_SCMPDS_st_v_._(intpos_3)_>=_8_&_v_._(intpos_2)_=_t_._(intpos_2)_&_v_._GBP_=_t_._GBP_&_v_._(DataLoc_((t_._GBP),7))_>_0_holds_
(_(IExec_(((((((GBP,5)_:=_(GBP,7))_';'_(AddTo_(GBP,7,(-_1))))_';'_((GBP,6)_:=_((intpos_2),0)))_';'_(SubFrom_(GBP,6,(intpos_3),0)))_';'_(if>0_(GBP,6,((AddTo_(GBP,3,1))_';'_(AddTo_(GBP,5,(-_1)))),(Load_((GBP,7)_:=_0))))),V,v))_._GBP_=_v_._GBP_&_(((((GBP,5)_:=_(GBP,7))_';'_(AddTo_(GBP,7,(-_1))))_';'_((GBP,6)_:=_((intpos_2),0)))_';'_(SubFrom_(GBP,6,(intpos_3),0)))_';'_(if>0_(GBP,6,((AddTo_(GBP,3,1))_';'_(AddTo_(GBP,5,(-_1)))),(Load_((GBP,7)_:=_0))))_is_closed_on_v,V_&_(((((GBP,5)_:=_(GBP,7))_';'_(AddTo_(GBP,7,(-_1))))_';'_((GBP,6)_:=_((intpos_2),0)))_';'_(SubFrom_(GBP,6,(intpos_3),0)))_';'_(if>0_(GBP,6,((AddTo_(GBP,3,1))_';'_(AddTo_(GBP,5,(-_1)))),(Load_((GBP,7)_:=_0))))_is_halting_on_v,V_&_(IExec_(((((((GBP,5)_:=_(GBP,7))_';'_(AddTo_(GBP,7,(-_1))))_';'_((GBP,6)_:=_((intpos_2),0)))_';'_(SubFrom_(GBP,6,(intpos_3),0)))_';'_(if>0_(GBP,6,((AddTo_(GBP,3,1))_';'_(AddTo_(GBP,5,(-_1)))),(Load_((GBP,7)_:=_0))))),V,v))_._(DataLoc_((t_._GBP),7))_<_v_._(DataLoc_((t_._GBP),7))_&_(IExec_(((((((GBP,5)_:=_(GBP,7))_';'_(AddTo_(GBP,7,(-_1))))_';'_((GBP,6)_:=_((intpos_2),0)))_';'_(SubFrom_(GBP,6,(intpos_3),0)))_';'_(if>0_(GBP,6,((AddTo_(GBP,3,1))_';'_(AddTo_(GBP,5,(-_1)))),(Load_((GBP,7)_:=_0))))),V,v))_._(intpos_3)_>=_8_&_(IExec_(((((((GBP,5)_:=_(GBP,7))_';'_(AddTo_(GBP,7,(-_1))))_';'_((GBP,6)_:=_((intpos_2),0)))_';'_(SubFrom_(GBP,6,(intpos_3),0)))_';'_(if>0_(GBP,6,((AddTo_(GBP,3,1))_';'_(AddTo_(GBP,5,(-_1)))),(Load_((GBP,7)_:=_0))))),V,v))_._(intpos_2)_=_v_._(intpos_2)_)
let v be 0 -started State of SCMPDS; ::_thesis: for V being Instruction-Sequence of SCMPDS st v . (intpos 3) >= 8 & v . (intpos 2) = t . (intpos 2) & v . GBP = t . GBP & v . (DataLoc ((t . GBP),7)) > 0 holds
( (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),V,v)) . GBP = v . GBP & (((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))) is_closed_on v,V & (((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))) is_halting_on v,V & (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),V,v)) . (DataLoc ((t . GBP),7)) < v . (DataLoc ((t . GBP),7)) & (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),V,v)) . (intpos 3) >= 8 & (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),V,v)) . (intpos 2) = v . (intpos 2) )
let V be Instruction-Sequence of SCMPDS; ::_thesis: ( v . (intpos 3) >= 8 & v . (intpos 2) = t . (intpos 2) & v . GBP = t . GBP & v . (DataLoc ((t . GBP),7)) > 0 implies ( (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),V,v)) . GBP = v . GBP & (((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))) is_closed_on v,V & (((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))) is_halting_on v,V & (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),V,v)) . (DataLoc ((t . GBP),7)) < v . (DataLoc ((t . GBP),7)) & (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),V,v)) . (intpos 3) >= 8 & (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),V,v)) . (intpos 2) = v . (intpos 2) ) )
assume that
A27: v . (intpos 3) >= 8 and
A28: v . (intpos 2) = t . (intpos 2) and
A29: v . GBP = t . GBP and
A30: v . (DataLoc ((t . GBP),7)) > 0 ; ::_thesis: ( (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),V,v)) . GBP = v . GBP & (((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))) is_closed_on v,V & (((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))) is_halting_on v,V & (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),V,v)) . (DataLoc ((t . GBP),7)) < v . (DataLoc ((t . GBP),7)) & (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),V,v)) . (intpos 3) >= 8 & (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),V,v)) . (intpos 2) = v . (intpos 2) )
reconsider ME = v . (intpos 3) as Element of NAT by A27, INT_1:3;
A38: ME = v . (intpos 3) ;
hence  (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),V,v)) . GBP = v . GBP by A6, A10, A14, A27, A28, A29, Lm14; ::_thesis: ( (((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))) is_closed_on v,V & (((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))) is_halting_on v,V & (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),V,v)) . (DataLoc ((t . GBP),7)) < v . (DataLoc ((t . GBP),7)) & (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),V,v)) . (intpos 3) >= 8 & (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),V,v)) . (intpos 2) = v . (intpos 2) )
thus  ( (((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))) is_closed_on v,V & (((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))) is_halting_on v,V ) by SCMPDS_6:20, SCMPDS_6:21; ::_thesis: ( (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),V,v)) . (DataLoc ((t . GBP),7)) < v . (DataLoc ((t . GBP),7)) & (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),V,v)) . (intpos 3) >= 8 & (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),V,v)) . (intpos 2) = v . (intpos 2) )
A39: ( v . (intpos md) > v . (intpos ME) implies ( (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),V,v)) . (intpos 7) = (v . (intpos 7)) - 1 & (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),V,v)) . (intpos 3) = (v . (intpos 3)) + 1 & (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),V,v)) . (intpos 5) = (v . (intpos 7)) - 1 ) ) by A6, A10, A14, A27, A28, A29, Lm14;
hereby  ::_thesis: (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),V,v)) . (intpos 2) = v . (intpos 2)
percases ( v . (intpos md) > v . (intpos ME) or v . (intpos md) <= v . (intpos ME) ) ;
supposeA40: v . (intpos md) > v . (intpos ME) ; ::_thesis: ( (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),V,v)) . (DataLoc ((t . GBP),7)) < v . (DataLoc ((t . GBP),7)) & (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),V,v)) . (intpos 3) >= 8 )
hence  (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),V,v)) . (DataLoc ((t . GBP),7)) < v . (DataLoc ((t . GBP),7)) by A25, A39, XREAL_1:146; ::_thesis: (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),V,v)) . (intpos 3) >= 8
A41: (v . (intpos 3)) + 1 > v . (intpos 3) by XREAL_1:29;
 (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),V,v)) . (intpos 3) = (v . (intpos 3)) + 1 by A6, A10, A14, A27, A28, A29, A40, Lm14;
hence  (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),V,v)) . (intpos 3) >= 8 by A27, A41, XXREAL_0:2; ::_thesis: verum
end;
supposeA42: v . (intpos md) <= v . (intpos ME) ; ::_thesis: ( (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),V,v)) . (DataLoc ((t . GBP),7)) < v . (DataLoc ((t . GBP),7)) & (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),V,v)) . (intpos 3) >= 8 )
hence  (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),V,v)) . (DataLoc ((t . GBP),7)) < v . (DataLoc ((t . GBP),7)) by A6, A10, A14, A25, A27, A28, A29, A30, Lm14; ::_thesis: (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),V,v)) . (intpos 3) >= 8
thus  (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),V,v)) . (intpos 3) >= 8 by A6, A10, A14, A27, A28, A29, A42, Lm14; ::_thesis: verum
end;
end;
end;
thus  (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),V,v)) . (intpos 2) = v . (intpos 2) by A6, A10, A14, A27, A28, A29, A38, Lm14; ::_thesis: verum
end;
A43: ( t . (intpos md) <= t . (intpos me) implies ( (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),Q,t)) . (intpos 7) = 0 & (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),Q,t)) . (intpos 3) = t . (intpos 3) & (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),Q,t)) . (intpos 5) = t . (intpos 7) ) ) by A6, A10, A13, A14, Lm14;
A44: me = t . (intpos 3) ;
then A45: (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),Q,t)) . GBP = 0 by A6, A10, A13, A14, Lm14;
then A46: DataLoc (((IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),Q,t)) . GBP),7) = intpos (0 + 7) by SCMP_GCD:1;
A47: (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),Q,t)) . (intpos 1) = t . (intpos 1) by A6, A10, A13, A14, A44, Lm14;
A48: (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),Q,t)) . (intpos 4) = t . (intpos 4) by A6, A10, A13, A14, A44, Lm14;
A49: (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),Q,t)) . (intpos 2) = t . (intpos 2) by A6, A10, A13, A14, A44, Lm14;
B47: (Initialize (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),Q,t))) . (intpos 1) = (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),Q,t)) . (intpos 1) by SCMPDS_5:15;
B48: (Initialize (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),Q,t))) . (intpos 4) = (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),Q,t)) . (intpos 4) by SCMPDS_5:15;
B49: (Initialize (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),Q,t))) . (intpos 2) = (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),Q,t)) . (intpos 2) by SCMPDS_5:15;
percases ( t . (intpos md) > t . (intpos me) or t . (intpos md) <= t . (intpos me) ) ;
supposeA50: t . (intpos md) > t . (intpos me) ; ::_thesis: ( (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),b2,b1)) . GBP = 0 & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),b2,b1)) . (intpos 1) = b1 . (intpos 1) & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),b2,b1)) . (intpos 7) = 0 & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),b2,b1)) . (intpos 2) = b1 . (intpos 2) & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),b2,b1)) . (intpos 4) = b1 . (intpos 4) & ( for i being Element of NAT st b3 >= 8 holds
(IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),b2,i)) . (intpos b3) = i . (intpos b3) ) & ex m5, mE3 being Element of NAT st
( b3 = (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),mE3,m5)) . (intpos 5) & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),mE3,m5)) . (intpos 3) = b4 & b4 + b3 = m3 & b3 <= m5 . (intpos 7) & ( for i being Element of NAT st m5 . (intpos 3) <= b5 & b5 < b4 holds
(IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),mE3,m5)) . (intpos md) > (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),mE3,m5)) . (intpos b5) ) & ( b3 = 0 or (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),mE3,m5)) . (intpos md) <= (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),mE3,m5)) . (intpos b4) ) ) )
then  t . (intpos 3) < (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),Q,t)) . (intpos 3) by A24, XREAL_1:29;
then A51: (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),Q,t)) . (intpos 3) >= 8 by A13, XXREAL_0:2;
A52: ((IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),Q,t)) . (intpos 3)) + ((IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),Q,t)) . (intpos 7)) = m3 by A12, A24, A50;
X1: (Initialize (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),Q,t))) . GBP = (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),Q,t)) . GBP by SCMPDS_5:15;
X2: (Initialize (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),Q,t))) . (intpos 3) = (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),Q,t)) . (intpos 3) by SCMPDS_5:15;
X3: (Initialize (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),Q,t))) . (intpos 7) = (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),Q,t)) . (intpos 7) by SCMPDS_5:15;
consider m5, mE3 being Element of NAT  such that
A53: m5 = (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,(Initialize (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),Q,t))))) . (intpos 5) and
A54: (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,(Initialize (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),Q,t))))) . (intpos 3) = mE3 and
A55: mE3 + m5 = m3 and
A56: m5 <= (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),Q,t)) . (intpos 7) and
A57: for i being Element of NAT st (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),Q,t)) . (intpos 3) <= i & i < mE3 holds
(IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,(Initialize (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),Q,t))))) . (intpos md) > (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,(Initialize (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),Q,t))))) . (intpos i) and
A58: ( m5 = 0 or (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,(Initialize (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),Q,t))))) . (intpos md) <= (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,(Initialize (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),Q,t))))) . (intpos mE3) ) by A9, A11, A12, A14, A45, A49, B49, A24, A43, A51, A52, X1, X2, X3;
A59: IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t) = IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,(Initialize (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),Q,t)))) by A11, A13, A25, A26, Th6;
hence  (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . GBP = 0 by A9, A11, A12, A14, A45, A49, B49, A24, A43, A51, A52, X1, X2, X3; ::_thesis: ( (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos 1) = t . (intpos 1) & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos 7) = 0 & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos 2) = t . (intpos 2) & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos 4) = t . (intpos 4) & ( for i being Element of NAT st i >= 8 holds
(IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos i) = t . (intpos i) ) & ex m5, mE3 being Element of NAT st
( m5 = (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos 5) & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos 3) = mE3 & mE3 + m5 = m3 & m5 <= t . (intpos 7) & ( for i being Element of NAT st t . (intpos 3) <= i & i < mE3 holds
(IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos md) > (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos i) ) & ( m5 = 0 or (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos md) <= (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos mE3) ) ) )
thus  (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos 1) = t . (intpos 1) by A9, A11, A12, A14, A45, A47, B47, A49, B49, A24, A43, A51, A52, A59, X1, X2, X3; ::_thesis: ( (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos 7) = 0 & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos 2) = t . (intpos 2) & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos 4) = t . (intpos 4) & ( for i being Element of NAT st i >= 8 holds
(IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos i) = t . (intpos i) ) & ex m5, mE3 being Element of NAT st
( m5 = (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos 5) & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos 3) = mE3 & mE3 + m5 = m3 & m5 <= t . (intpos 7) & ( for i being Element of NAT st t . (intpos 3) <= i & i < mE3 holds
(IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos md) > (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos i) ) & ( m5 = 0 or (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos md) <= (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos mE3) ) ) )
thus  (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos 7) = 0 by A9, A11, A12, A14, A45, A49, B49, A24, A43, A51, A52, A59, X1, X2, X3; ::_thesis: ( (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos 2) = t . (intpos 2) & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos 4) = t . (intpos 4) & ( for i being Element of NAT st i >= 8 holds
(IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos i) = t . (intpos i) ) & ex m5, mE3 being Element of NAT st
( m5 = (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos 5) & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos 3) = mE3 & mE3 + m5 = m3 & m5 <= t . (intpos 7) & ( for i being Element of NAT st t . (intpos 3) <= i & i < mE3 holds
(IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos md) > (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos i) ) & ( m5 = 0 or (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos md) <= (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos mE3) ) ) )
thus  (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos 2) = t . (intpos 2) by A9, A11, A12, A14, A45, A49, B49, A24, A43, A51, A52, A59, X1, X2, X3; ::_thesis: ( (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos 4) = t . (intpos 4) & ( for i being Element of NAT st i >= 8 holds
(IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos i) = t . (intpos i) ) & ex m5, mE3 being Element of NAT st
( m5 = (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos 5) & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos 3) = mE3 & mE3 + m5 = m3 & m5 <= t . (intpos 7) & ( for i being Element of NAT st t . (intpos 3) <= i & i < mE3 holds
(IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos md) > (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos i) ) & ( m5 = 0 or (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos md) <= (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos mE3) ) ) )
thus  (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos 4) = t . (intpos 4) by A9, A11, A12, A14, A45, A49, B49, A48, B48, A24, A43, A51, A52, A59, X1, X2, X3; ::_thesis: ( ( for i being Element of NAT st i >= 8 holds
(IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos i) = t . (intpos i) ) & ex m5, mE3 being Element of NAT st
( m5 = (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos 5) & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos 3) = mE3 & mE3 + m5 = m3 & m5 <= t . (intpos 7) & ( for i being Element of NAT st t . (intpos 3) <= i & i < mE3 holds
(IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos md) > (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos i) ) & ( m5 = 0 or (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos md) <= (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos mE3) ) ) )
A60: now__::_thesis:_for_i_being_Element_of_NAT_st_i_>=_8_holds_
(IExec_((while>0_(GBP,7,((((((GBP,5)_:=_(GBP,7))_';'_(AddTo_(GBP,7,(-_1))))_';'_((GBP,6)_:=_((intpos_2),0)))_';'_(SubFrom_(GBP,6,(intpos_3),0)))_';'_(if>0_(GBP,6,((AddTo_(GBP,3,1))_';'_(AddTo_(GBP,5,(-_1)))),(Load_((GBP,7)_:=_0))))))),Q,t))_._(intpos_i)_=_t_._(intpos_i)
let i be Element of NAT ; ::_thesis: ( i >= 8 implies (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos i) = t . (intpos i) )
assume A62: i >= 8 ; ::_thesis: (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos i) = t . (intpos i)
 (Initialize (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),Q,t))) . (intpos i) = (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),Q,t)) . (intpos i) by SCMPDS_5:15;
hence (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos i) = (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),Q,t)) . (intpos i) by A9, A11, A12, A14, A45, A49, B49, A24, A43, A51, A52, A59, X1, X2, X3, A62
.= t . (intpos i) by A6, A10, A13, A14, A44, A62, Lm14 ;
::_thesis: verum
end;
hence  for i being Element of NAT st i >= 8 holds
(IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos i) = t . (intpos i) ; ::_thesis: ex m5, mE3 being Element of NAT st
( m5 = (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos 5) & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos 3) = mE3 & mE3 + m5 = m3 & m5 <= t . (intpos 7) & ( for i being Element of NAT st t . (intpos 3) <= i & i < mE3 holds
(IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos md) > (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos i) ) & ( m5 = 0 or (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos md) <= (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos mE3) ) )
take m5 = m5; ::_thesis: ex mE3 being Element of NAT st
( m5 = (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos 5) & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos 3) = mE3 & mE3 + m5 = m3 & m5 <= t . (intpos 7) & ( for i being Element of NAT st t . (intpos 3) <= i & i < mE3 holds
(IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos md) > (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos i) ) & ( m5 = 0 or (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos md) <= (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos mE3) ) )
take mE3 = mE3; ::_thesis: ( m5 = (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos 5) & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos 3) = mE3 & mE3 + m5 = m3 & m5 <= t . (intpos 7) & ( for i being Element of NAT st t . (intpos 3) <= i & i < mE3 holds
(IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos md) > (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos i) ) & ( m5 = 0 or (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos md) <= (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos mE3) ) )
thus  m5 = (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos 5) by A11, A13, A25, A26, A53, Th6; ::_thesis: ( (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos 3) = mE3 & mE3 + m5 = m3 & m5 <= t . (intpos 7) & ( for i being Element of NAT st t . (intpos 3) <= i & i < mE3 holds
(IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos md) > (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos i) ) & ( m5 = 0 or (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos md) <= (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos mE3) ) )
thus  (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos 3) = mE3 by A11, A13, A25, A26, A54, Th6; ::_thesis: ( mE3 + m5 = m3 & m5 <= t . (intpos 7) & ( for i being Element of NAT st t . (intpos 3) <= i & i < mE3 holds
(IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos md) > (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos i) ) & ( m5 = 0 or (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos md) <= (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos mE3) ) )
thus  mE3 + m5 = m3 by A55; ::_thesis: ( m5 <= t . (intpos 7) & ( for i being Element of NAT st t . (intpos 3) <= i & i < mE3 holds
(IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos md) > (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos i) ) & ( m5 = 0 or (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos md) <= (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos mE3) ) )
thus  m5 <= t . (intpos 7) by A24, A50, A56, XREAL_1:146, XXREAL_0:2; ::_thesis: ( ( for i being Element of NAT st t . (intpos 3) <= i & i < mE3 holds
(IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos md) > (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos i) ) & ( m5 = 0 or (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos md) <= (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos mE3) ) )
A63: (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),Q,t)) . (intpos 3) = (t . (intpos 3)) + 1 by A6, A10, A13, A14, A50, Lm14;
hereby  ::_thesis: ( m5 = 0 or (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos md) <= (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos mE3) )
let i be Element of NAT ; ::_thesis: ( t . (intpos 3) <= i & i < mE3 implies (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos md) > (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos b1) )
assume that
A64: t . (intpos 3) <= i and
A65: i < mE3 ; ::_thesis: (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos md) > (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos b1)
percases ( i = t . (intpos 3) or i <> t . (intpos 3) ) ;
supposeA66: i = t . (intpos 3) ; ::_thesis: (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos md) > (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos b1)
 (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos me) = t . (intpos me) by A13, A60;
hence  (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos md) > (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos i) by A6, A50, A60, A66; ::_thesis: verum
end;
suppose i <> t . (intpos 3) ; ::_thesis: (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos md) > (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos b1)
then  t . (intpos 3) < i by A64, XXREAL_0:1;
then  (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),Q,t)) . (intpos 3) <= i by A63, INT_1:7;
hence  (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos md) > (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos i) by A59, A57, A65; ::_thesis: verum
end;
end;
end;
thus  ( m5 = 0 or (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos md) <= (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos mE3) ) by A59, A58; ::_thesis: verum
end;
supposeA67: t . (intpos md) <= t . (intpos me) ; ::_thesis: ( (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),b2,b1)) . GBP = 0 & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),b2,b1)) . (intpos 1) = b1 . (intpos 1) & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),b2,b1)) . (intpos 7) = 0 & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),b2,b1)) . (intpos 2) = b1 . (intpos 2) & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),b2,b1)) . (intpos 4) = b1 . (intpos 4) & ( for i being Element of NAT st b3 >= 8 holds
(IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),b2,i)) . (intpos b3) = i . (intpos b3) ) & ex m5 being Element of NAT ex mE3 being Element of NAT st
( (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),mE3,m5)) . (intpos 5) = b3 & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),mE3,m5)) . (intpos 3) = b4 & b4 + b3 = m3 & b3 <= m5 . (intpos 7) & ( for i being Element of NAT st m5 . (intpos 3) <= b5 & b5 < b4 holds
(IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),mE3,m5)) . (intpos md) > (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),mE3,m5)) . (intpos b5) ) & ( b3 = 0 or (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),mE3,m5)) . (intpos md) <= (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),mE3,m5)) . (intpos b4) ) ) )
A68: now__::_thesis:_for_x_being_Int_position_holds_(IExec_((while>0_(GBP,7,((((((GBP,5)_:=_(GBP,7))_';'_(AddTo_(GBP,7,(-_1))))_';'_((GBP,6)_:=_((intpos_2),0)))_';'_(SubFrom_(GBP,6,(intpos_3),0)))_';'_(if>0_(GBP,6,((AddTo_(GBP,3,1))_';'_(AddTo_(GBP,5,(-_1)))),(Load_((GBP,7)_:=_0))))))),Q,t))_._x_=_(IExec_(((((((GBP,5)_:=_(GBP,7))_';'_(AddTo_(GBP,7,(-_1))))_';'_((GBP,6)_:=_((intpos_2),0)))_';'_(SubFrom_(GBP,6,(intpos_3),0)))_';'_(if>0_(GBP,6,((AddTo_(GBP,3,1))_';'_(AddTo_(GBP,5,(-_1)))),(Load_((GBP,7)_:=_0))))),Q,t))_._x
let x be Int_position; ::_thesis: (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . x = (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),Q,t)) . x
X1: (Initialize (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),Q,t))) . GBP = (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),Q,t)) . GBP by SCMPDS_5:15;
X2: (Initialize (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),Q,t))) . (DataLoc (((IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),Q,t)) . GBP),7)) = (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),Q,t)) . (DataLoc (((IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),Q,t)) . GBP),7)) by SCMPDS_5:15;
thus (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . x = (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,(Initialize (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),Q,t))))) . x by A11, A13, A25, A26, Th6
.= (Initialize (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),Q,t))) . x by A43, A46, A67, X1, X2, SCMPDS_8:23
.= (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),Q,t)) . x by SCMPDS_5:15 ; ::_thesis: verum
end;
hence  (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . GBP = 0 by A45; ::_thesis: ( (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos 1) = t . (intpos 1) & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos 7) = 0 & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos 2) = t . (intpos 2) & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos 4) = t . (intpos 4) & ( for i being Element of NAT st i >= 8 holds
(IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos i) = t . (intpos i) ) & ex m5 being Element of NAT ex mE3 being Element of NAT st
( (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos 5) = m5 & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos 3) = mE3 & mE3 + m5 = m3 & m5 <= t . (intpos 7) & ( for i being Element of NAT st t . (intpos 3) <= i & i < mE3 holds
(IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos md) > (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos i) ) & ( m5 = 0 or (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos md) <= (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos mE3) ) ) )
thus  (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos 1) = t . (intpos 1) by A47, A68; ::_thesis: ( (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos 7) = 0 & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos 2) = t . (intpos 2) & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos 4) = t . (intpos 4) & ( for i being Element of NAT st i >= 8 holds
(IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos i) = t . (intpos i) ) & ex m5 being Element of NAT ex mE3 being Element of NAT st
( (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos 5) = m5 & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos 3) = mE3 & mE3 + m5 = m3 & m5 <= t . (intpos 7) & ( for i being Element of NAT st t . (intpos 3) <= i & i < mE3 holds
(IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos md) > (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos i) ) & ( m5 = 0 or (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos md) <= (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos mE3) ) ) )
thus  (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos 7) = 0 by A43, A67, A68; ::_thesis: ( (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos 2) = t . (intpos 2) & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos 4) = t . (intpos 4) & ( for i being Element of NAT st i >= 8 holds
(IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos i) = t . (intpos i) ) & ex m5 being Element of NAT ex mE3 being Element of NAT st
( (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos 5) = m5 & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos 3) = mE3 & mE3 + m5 = m3 & m5 <= t . (intpos 7) & ( for i being Element of NAT st t . (intpos 3) <= i & i < mE3 holds
(IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos md) > (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos i) ) & ( m5 = 0 or (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos md) <= (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos mE3) ) ) )
thus  (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos 2) = t . (intpos 2) by A49, A68; ::_thesis: ( (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos 4) = t . (intpos 4) & ( for i being Element of NAT st i >= 8 holds
(IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos i) = t . (intpos i) ) & ex m5 being Element of NAT ex mE3 being Element of NAT st
( (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos 5) = m5 & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos 3) = mE3 & mE3 + m5 = m3 & m5 <= t . (intpos 7) & ( for i being Element of NAT st t . (intpos 3) <= i & i < mE3 holds
(IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos md) > (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos i) ) & ( m5 = 0 or (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos md) <= (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos mE3) ) ) )
thus  (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos 4) = t . (intpos 4) by A48, A68; ::_thesis: ( ( for i being Element of NAT st i >= 8 holds
(IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos i) = t . (intpos i) ) & ex m5 being Element of NAT ex mE3 being Element of NAT st
( (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos 5) = m5 & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos 3) = mE3 & mE3 + m5 = m3 & m5 <= t . (intpos 7) & ( for i being Element of NAT st t . (intpos 3) <= i & i < mE3 holds
(IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos md) > (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos i) ) & ( m5 = 0 or (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos md) <= (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos mE3) ) ) )
A69: now__::_thesis:_for_i_being_Element_of_NAT_st_i_>=_8_holds_
(IExec_((while>0_(GBP,7,((((((GBP,5)_:=_(GBP,7))_';'_(AddTo_(GBP,7,(-_1))))_';'_((GBP,6)_:=_((intpos_2),0)))_';'_(SubFrom_(GBP,6,(intpos_3),0)))_';'_(if>0_(GBP,6,((AddTo_(GBP,3,1))_';'_(AddTo_(GBP,5,(-_1)))),(Load_((GBP,7)_:=_0))))))),Q,t))_._(intpos_i)_=_t_._(intpos_i)
let i be Element of NAT ; ::_thesis: ( i >= 8 implies (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos i) = t . (intpos i) )
assume A71: i >= 8 ; ::_thesis: (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos i) = t . (intpos i)
thus (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos i) = (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),Q,t)) . (intpos i) by A68
.= t . (intpos i) by A6, A10, A13, A14, A44, A71, Lm14 ; ::_thesis: verum
end;
hence  for i being Element of NAT st i >= 8 holds
(IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos i) = t . (intpos i) ; ::_thesis: ex m5 being Element of NAT ex mE3 being Element of NAT st
( (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos 5) = m5 & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos 3) = mE3 & mE3 + m5 = m3 & m5 <= t . (intpos 7) & ( for i being Element of NAT st t . (intpos 3) <= i & i < mE3 holds
(IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos md) > (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos i) ) & ( m5 = 0 or (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos md) <= (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos mE3) ) )
take m5 = (k + 1) + 1; ::_thesis: ex mE3 being Element of NAT st
( (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos 5) = m5 & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos 3) = mE3 & mE3 + m5 = m3 & m5 <= t . (intpos 7) & ( for i being Element of NAT st t . (intpos 3) <= i & i < mE3 holds
(IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos md) > (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos i) ) & ( m5 = 0 or (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos md) <= (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos mE3) ) )
take mE3 = me; ::_thesis: ( (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos 5) = m5 & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos 3) = mE3 & mE3 + m5 = m3 & m5 <= t . (intpos 7) & ( for i being Element of NAT st t . (intpos 3) <= i & i < mE3 holds
(IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos md) > (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos i) ) & ( m5 = 0 or (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos md) <= (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos mE3) ) )
thus  (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos 5) = m5 by A11, A43, A67, A68; ::_thesis: ( (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos 3) = mE3 & mE3 + m5 = m3 & m5 <= t . (intpos 7) & ( for i being Element of NAT st t . (intpos 3) <= i & i < mE3 holds
(IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos md) > (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos i) ) & ( m5 = 0 or (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos md) <= (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos mE3) ) )
thus  (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos 3) = mE3 by A43, A67, A68; ::_thesis: ( mE3 + m5 = m3 & m5 <= t . (intpos 7) & ( for i being Element of NAT st t . (intpos 3) <= i & i < mE3 holds
(IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos md) > (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos i) ) & ( m5 = 0 or (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos md) <= (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos mE3) ) )
thus  mE3 + m5 = m3 by A11, A12; ::_thesis: ( m5 <= t . (intpos 7) & ( for i being Element of NAT st t . (intpos 3) <= i & i < mE3 holds
(IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos md) > (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos i) ) & ( m5 = 0 or (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos md) <= (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos mE3) ) )
thus  m5 <= t . (intpos 7) by A11; ::_thesis: ( ( for i being Element of NAT st t . (intpos 3) <= i & i < mE3 holds
(IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos md) > (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos i) ) & ( m5 = 0 or (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos md) <= (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos mE3) ) )
thus  for i being Element of NAT st t . (intpos 3) <= i & i < mE3 holds
(IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos md) > (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos i) ; ::_thesis: ( m5 = 0 or (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos md) <= (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos mE3) )
 (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos me) = t . (intpos me) by A13, A69;
hence  ( m5 = 0 or (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos md) <= (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos mE3) ) by A6, A67, A69; ::_thesis: verum
end;
end;
end;
hence  S1[k + 1] ; ::_thesis: verum
end;
now__::_thesis:_for_t_being_0_-started_State_of_SCMPDS
for_Q_being_Instruction-Sequence_of_SCMPDS_st_t_._GBP_=_0_&_t_._(intpos_7)_=_0_+_1_&_(t_._(intpos_3))_+_(t_._(intpos_7))_=_m3_&_t_._(intpos_3)_>=_8_&_t_._(intpos_2)_=_md_holds_
(_(IExec_((while>0_(GBP,7,((((((GBP,5)_:=_(GBP,7))_';'_(AddTo_(GBP,7,(-_1))))_';'_((GBP,6)_:=_((intpos_2),0)))_';'_(SubFrom_(GBP,6,(intpos_3),0)))_';'_(if>0_(GBP,6,((AddTo_(GBP,3,1))_';'_(AddTo_(GBP,5,(-_1)))),(Load_((GBP,7)_:=_0))))))),Q,t))_._GBP_=_0_&_(IExec_((while>0_(GBP,7,((((((GBP,5)_:=_(GBP,7))_';'_(AddTo_(GBP,7,(-_1))))_';'_((GBP,6)_:=_((intpos_2),0)))_';'_(SubFrom_(GBP,6,(intpos_3),0)))_';'_(if>0_(GBP,6,((AddTo_(GBP,3,1))_';'_(AddTo_(GBP,5,(-_1)))),(Load_((GBP,7)_:=_0))))))),Q,t))_._(intpos_1)_=_t_._(intpos_1)_&_(IExec_((while>0_(GBP,7,((((((GBP,5)_:=_(GBP,7))_';'_(AddTo_(GBP,7,(-_1))))_';'_((GBP,6)_:=_((intpos_2),0)))_';'_(SubFrom_(GBP,6,(intpos_3),0)))_';'_(if>0_(GBP,6,((AddTo_(GBP,3,1))_';'_(AddTo_(GBP,5,(-_1)))),(Load_((GBP,7)_:=_0))))))),Q,t))_._(intpos_7)_=_0_&_(IExec_((while>0_(GBP,7,((((((GBP,5)_:=_(GBP,7))_';'_(AddTo_(GBP,7,(-_1))))_';'_((GBP,6)_:=_((intpos_2),0)))_';'_(SubFrom_(GBP,6,(intpos_3),0)))_';'_(if>0_(GBP,6,((AddTo_(GBP,3,1))_';'_(AddTo_(GBP,5,(-_1)))),(Load_((GBP,7)_:=_0))))))),Q,t))_._(intpos_2)_=_t_._(intpos_2)_&_(IExec_((while>0_(GBP,7,((((((GBP,5)_:=_(GBP,7))_';'_(AddTo_(GBP,7,(-_1))))_';'_((GBP,6)_:=_((intpos_2),0)))_';'_(SubFrom_(GBP,6,(intpos_3),0)))_';'_(if>0_(GBP,6,((AddTo_(GBP,3,1))_';'_(AddTo_(GBP,5,(-_1)))),(Load_((GBP,7)_:=_0))))))),Q,t))_._(intpos_4)_=_t_._(intpos_4)_&_(_for_i_being_Element_of_NAT_st_i_>=_8_holds_
(IExec_((while>0_(GBP,7,((((((GBP,5)_:=_(GBP,7))_';'_(AddTo_(GBP,7,(-_1))))_';'_((GBP,6)_:=_((intpos_2),0)))_';'_(SubFrom_(GBP,6,(intpos_3),0)))_';'_(if>0_(GBP,6,((AddTo_(GBP,3,1))_';'_(AddTo_(GBP,5,(-_1)))),(Load_((GBP,7)_:=_0))))))),Q,t))_._(intpos_i)_=_t_._(intpos_i)_)_&_ex_m5_being_Element_of_NAT_ex_mE3_being_Element_of_NAT_st_
(_(IExec_((while>0_(GBP,7,((((((GBP,5)_:=_(GBP,7))_';'_(AddTo_(GBP,7,(-_1))))_';'_((GBP,6)_:=_((intpos_2),0)))_';'_(SubFrom_(GBP,6,(intpos_3),0)))_';'_(if>0_(GBP,6,((AddTo_(GBP,3,1))_';'_(AddTo_(GBP,5,(-_1)))),(Load_((GBP,7)_:=_0))))))),Q,t))_._(intpos_5)_=_m5_&_(IExec_((while>0_(GBP,7,((((((GBP,5)_:=_(GBP,7))_';'_(AddTo_(GBP,7,(-_1))))_';'_((GBP,6)_:=_((intpos_2),0)))_';'_(SubFrom_(GBP,6,(intpos_3),0)))_';'_(if>0_(GBP,6,((AddTo_(GBP,3,1))_';'_(AddTo_(GBP,5,(-_1)))),(Load_((GBP,7)_:=_0))))))),Q,t))_._(intpos_3)_=_mE3_&_mE3_+_m5_=_m3_&_m5_<=_t_._(intpos_7)_&_(_for_i_being_Element_of_NAT_st_t_._(intpos_3)_<=_i_&_i_<_mE3_holds_
(IExec_((while>0_(GBP,7,((((((GBP,5)_:=_(GBP,7))_';'_(AddTo_(GBP,7,(-_1))))_';'_((GBP,6)_:=_((intpos_2),0)))_';'_(SubFrom_(GBP,6,(intpos_3),0)))_';'_(if>0_(GBP,6,((AddTo_(GBP,3,1))_';'_(AddTo_(GBP,5,(-_1)))),(Load_((GBP,7)_:=_0))))))),Q,t))_._(intpos_md)_>_(IExec_((while>0_(GBP,7,((((((GBP,5)_:=_(GBP,7))_';'_(AddTo_(GBP,7,(-_1))))_';'_((GBP,6)_:=_((intpos_2),0)))_';'_(SubFrom_(GBP,6,(intpos_3),0)))_';'_(if>0_(GBP,6,((AddTo_(GBP,3,1))_';'_(AddTo_(GBP,5,(-_1)))),(Load_((GBP,7)_:=_0))))))),Q,t))_._(intpos_i)_)_&_(_m5_=_0_or_(IExec_((while>0_(GBP,7,((((((GBP,5)_:=_(GBP,7))_';'_(AddTo_(GBP,7,(-_1))))_';'_((GBP,6)_:=_((intpos_2),0)))_';'_(SubFrom_(GBP,6,(intpos_3),0)))_';'_(if>0_(GBP,6,((AddTo_(GBP,3,1))_';'_(AddTo_(GBP,5,(-_1)))),(Load_((GBP,7)_:=_0))))))),Q,t))_._(intpos_md)_<=_(IExec_((while>0_(GBP,7,((((((GBP,5)_:=_(GBP,7))_';'_(AddTo_(GBP,7,(-_1))))_';'_((GBP,6)_:=_((intpos_2),0)))_';'_(SubFrom_(GBP,6,(intpos_3),0)))_';'_(if>0_(GBP,6,((AddTo_(GBP,3,1))_';'_(AddTo_(GBP,5,(-_1)))),(Load_((GBP,7)_:=_0))))))),Q,t))_._(intpos_mE3)_)_)_)
let t be 0 -started State of SCMPDS; ::_thesis: for Q being Instruction-Sequence of SCMPDS st t . GBP = 0 & t . (intpos 7) = 0 + 1 & (t . (intpos 3)) + (t . (intpos 7)) = m3 & t . (intpos 3) >= 8 & t . (intpos 2) = md holds
( (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . GBP = 0 & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos 1) = t . (intpos 1) & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos 7) = 0 & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos 2) = t . (intpos 2) & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos 4) = t . (intpos 4) & ( for i being Element of NAT st i >= 8 holds
(IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos i) = t . (intpos i) ) & ex m5 being Element of NAT ex mE3 being Element of NAT st
( (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos 5) = m5 & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos 3) = mE3 & mE3 + m5 = m3 & m5 <= t . (intpos 7) & ( for i being Element of NAT st t . (intpos 3) <= i & i < mE3 holds
(IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos md) > (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos i) ) & ( m5 = 0 or (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos md) <= (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos mE3) ) ) )
let Q be Instruction-Sequence of SCMPDS; ::_thesis: ( t . GBP = 0 & t . (intpos 7) = 0 + 1 & (t . (intpos 3)) + (t . (intpos 7)) = m3 & t . (intpos 3) >= 8 & t . (intpos 2) = md implies ( (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . GBP = 0 & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos 1) = t . (intpos 1) & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos 7) = 0 & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos 2) = t . (intpos 2) & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos 4) = t . (intpos 4) & ( for i being Element of NAT st i >= 8 holds
(IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos i) = t . (intpos i) ) & ex m5 being Element of NAT ex mE3 being Element of NAT st
( (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos 5) = m5 & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos 3) = mE3 & mE3 + m5 = m3 & m5 <= t . (intpos 7) & ( for i being Element of NAT st t . (intpos 3) <= i & i < mE3 holds
(IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos md) > (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos i) ) & ( m5 = 0 or (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos md) <= (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos mE3) ) ) ) )
set b = DataLoc ((t . GBP),7);
assume that
A72: t . GBP = 0 and
A73: t . (intpos 7) = 0 + 1 and
A74: (t . (intpos 3)) + (t . (intpos 7)) = m3 and
A75: t . (intpos 3) >= 8 and
A76: t . (intpos 2) = md ; ::_thesis: ( (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . GBP = 0 & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos 1) = t . (intpos 1) & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos 7) = 0 & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos 2) = t . (intpos 2) & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos 4) = t . (intpos 4) & ( for i being Element of NAT st i >= 8 holds
(IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos i) = t . (intpos i) ) & ex m5 being Element of NAT ex mE3 being Element of NAT st
( (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos 5) = m5 & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos 3) = mE3 & mE3 + m5 = m3 & m5 <= t . (intpos 7) & ( for i being Element of NAT st t . (intpos 3) <= i & i < mE3 holds
(IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos md) > (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos i) ) & ( m5 = 0 or (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos md) <= (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos mE3) ) ) )
reconsider me = t . (intpos 3) as Element of NAT by A75, INT_1:3;
set It = IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),Q,t);
set IT = Q;
A86: ( t . (intpos md) > t . (intpos me) implies ( (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),Q,t)) . (intpos 7) = (t . (intpos 7)) - 1 & (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),Q,t)) . (intpos 3) = (t . (intpos 3)) + 1 & (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),Q,t)) . (intpos 5) = (t . (intpos 7)) - 1 ) ) by A6, A72, A75, A76, Lm14;
A87: DataLoc ((t . GBP),7) = intpos (0 + 7) by A72, SCMP_GCD:1;
A88: now__::_thesis:_for_v_being_0_-started_State_of_SCMPDS
for_V_being_Instruction-Sequence_of_SCMPDS_st_v_._(intpos_3)_>=_8_&_v_._(intpos_2)_=_t_._(intpos_2)_&_v_._GBP_=_t_._GBP_&_v_._(DataLoc_((t_._GBP),7))_>_0_holds_
(_(IExec_(((((((GBP,5)_:=_(GBP,7))_';'_(AddTo_(GBP,7,(-_1))))_';'_((GBP,6)_:=_((intpos_2),0)))_';'_(SubFrom_(GBP,6,(intpos_3),0)))_';'_(if>0_(GBP,6,((AddTo_(GBP,3,1))_';'_(AddTo_(GBP,5,(-_1)))),(Load_((GBP,7)_:=_0))))),V,v))_._GBP_=_v_._GBP_&_(((((GBP,5)_:=_(GBP,7))_';'_(AddTo_(GBP,7,(-_1))))_';'_((GBP,6)_:=_((intpos_2),0)))_';'_(SubFrom_(GBP,6,(intpos_3),0)))_';'_(if>0_(GBP,6,((AddTo_(GBP,3,1))_';'_(AddTo_(GBP,5,(-_1)))),(Load_((GBP,7)_:=_0))))_is_closed_on_v,V_&_(((((GBP,5)_:=_(GBP,7))_';'_(AddTo_(GBP,7,(-_1))))_';'_((GBP,6)_:=_((intpos_2),0)))_';'_(SubFrom_(GBP,6,(intpos_3),0)))_';'_(if>0_(GBP,6,((AddTo_(GBP,3,1))_';'_(AddTo_(GBP,5,(-_1)))),(Load_((GBP,7)_:=_0))))_is_halting_on_v,V_&_(IExec_(((((((GBP,5)_:=_(GBP,7))_';'_(AddTo_(GBP,7,(-_1))))_';'_((GBP,6)_:=_((intpos_2),0)))_';'_(SubFrom_(GBP,6,(intpos_3),0)))_';'_(if>0_(GBP,6,((AddTo_(GBP,3,1))_';'_(AddTo_(GBP,5,(-_1)))),(Load_((GBP,7)_:=_0))))),V,v))_._(DataLoc_((t_._GBP),7))_<_v_._(DataLoc_((t_._GBP),7))_&_(IExec_(((((((GBP,5)_:=_(GBP,7))_';'_(AddTo_(GBP,7,(-_1))))_';'_((GBP,6)_:=_((intpos_2),0)))_';'_(SubFrom_(GBP,6,(intpos_3),0)))_';'_(if>0_(GBP,6,((AddTo_(GBP,3,1))_';'_(AddTo_(GBP,5,(-_1)))),(Load_((GBP,7)_:=_0))))),V,v))_._(intpos_3)_>=_8_&_(IExec_(((((((GBP,5)_:=_(GBP,7))_';'_(AddTo_(GBP,7,(-_1))))_';'_((GBP,6)_:=_((intpos_2),0)))_';'_(SubFrom_(GBP,6,(intpos_3),0)))_';'_(if>0_(GBP,6,((AddTo_(GBP,3,1))_';'_(AddTo_(GBP,5,(-_1)))),(Load_((GBP,7)_:=_0))))),V,v))_._(intpos_2)_=_v_._(intpos_2)_)
let v be 0 -started State of SCMPDS; ::_thesis: for V being Instruction-Sequence of SCMPDS st v . (intpos 3) >= 8 & v . (intpos 2) = t . (intpos 2) & v . GBP = t . GBP & v . (DataLoc ((t . GBP),7)) > 0 holds
( (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),V,v)) . GBP = v . GBP & (((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))) is_closed_on v,V & (((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))) is_halting_on v,V & (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),V,v)) . (DataLoc ((t . GBP),7)) < v . (DataLoc ((t . GBP),7)) & (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),V,v)) . (intpos 3) >= 8 & (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),V,v)) . (intpos 2) = v . (intpos 2) )
let V be Instruction-Sequence of SCMPDS; ::_thesis: ( v . (intpos 3) >= 8 & v . (intpos 2) = t . (intpos 2) & v . GBP = t . GBP & v . (DataLoc ((t . GBP),7)) > 0 implies ( (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),V,v)) . GBP = v . GBP & (((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))) is_closed_on v,V & (((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))) is_halting_on v,V & (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),V,v)) . (DataLoc ((t . GBP),7)) < v . (DataLoc ((t . GBP),7)) & (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),V,v)) . (intpos 3) >= 8 & (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),V,v)) . (intpos 2) = v . (intpos 2) ) )
assume that
A89: v . (intpos 3) >= 8 and
A90: v . (intpos 2) = t . (intpos 2) and
A91: v . GBP = t . GBP and
A92: v . (DataLoc ((t . GBP),7)) > 0 ; ::_thesis: ( (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),V,v)) . GBP = v . GBP & (((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))) is_closed_on v,V & (((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))) is_halting_on v,V & (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),V,v)) . (DataLoc ((t . GBP),7)) < v . (DataLoc ((t . GBP),7)) & (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),V,v)) . (intpos 3) >= 8 & (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),V,v)) . (intpos 2) = v . (intpos 2) )
reconsider ME = v . (intpos 3) as Element of NAT by A89, INT_1:3;
A93: ME = v . (intpos 3) ;
A96: v . (intpos ME) = v . (intpos ME) ;
thus  (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),V,v)) . GBP = v . GBP by A6, A72, A76, A89, A90, A91, Lm14, A96; ::_thesis: ( (((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))) is_closed_on v,V & (((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))) is_halting_on v,V & (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),V,v)) . (DataLoc ((t . GBP),7)) < v . (DataLoc ((t . GBP),7)) & (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),V,v)) . (intpos 3) >= 8 & (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),V,v)) . (intpos 2) = v . (intpos 2) )
thus  ( (((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))) is_closed_on v,V & (((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))) is_halting_on v,V ) by SCMPDS_6:20, SCMPDS_6:21; ::_thesis: ( (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),V,v)) . (DataLoc ((t . GBP),7)) < v . (DataLoc ((t . GBP),7)) & (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),V,v)) . (intpos 3) >= 8 & (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),V,v)) . (intpos 2) = v . (intpos 2) )
A101: ( v . (intpos md) > v . (intpos ME) implies ( (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),V,v)) . (intpos 7) = (v . (intpos 7)) - 1 & (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),V,v)) . (intpos 3) = (v . (intpos 3)) + 1 & (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),V,v)) . (intpos 5) = (v . (intpos 7)) - 1 ) ) by A6, A72, A76, A89, A90, A91, Lm14;
hereby  ::_thesis: (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),V,v)) . (intpos 2) = v . (intpos 2)
percases ( v . (intpos md) > v . (intpos ME) or v . (intpos md) <= v . (intpos ME) ) ;
supposeA102: v . (intpos md) > v . (intpos ME) ; ::_thesis: ( (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),V,v)) . (DataLoc ((t . GBP),7)) < v . (DataLoc ((t . GBP),7)) & (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),V,v)) . (intpos 3) >= 8 )
hence  (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),V,v)) . (DataLoc ((t . GBP),7)) < v . (DataLoc ((t . GBP),7)) by A87, A101, XREAL_1:146; ::_thesis: (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),V,v)) . (intpos 3) >= 8
A103: (v . (intpos 3)) + 1 > v . (intpos 3) by XREAL_1:29;
 (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),V,v)) . (intpos 3) = (v . (intpos 3)) + 1 by A6, A72, A76, A89, A90, A91, A102, Lm14;
hence  (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),V,v)) . (intpos 3) >= 8 by A89, A103, XXREAL_0:2; ::_thesis: verum
end;
supposeA104: v . (intpos md) <= v . (intpos ME) ; ::_thesis: ( (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),V,v)) . (DataLoc ((t . GBP),7)) < v . (DataLoc ((t . GBP),7)) & (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),V,v)) . (intpos 3) >= 8 )
hence  (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),V,v)) . (DataLoc ((t . GBP),7)) < v . (DataLoc ((t . GBP),7)) by A6, A72, A76, A87, A89, A90, A91, A92, Lm14; ::_thesis: (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),V,v)) . (intpos 3) >= 8
thus  (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),V,v)) . (intpos 3) >= 8 by A6, A72, A76, A89, A90, A91, A104, Lm14; ::_thesis: verum
end;
end;
end;
thus  (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),V,v)) . (intpos 2) = v . (intpos 2) by A6, A72, A76, A89, A90, A91, A93, Lm14; ::_thesis: verum
end;
A105: ( t . (intpos md) <= t . (intpos me) implies ( (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),Q,t)) . (intpos 7) = 0 & (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),Q,t)) . (intpos 3) = t . (intpos 3) & (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),Q,t)) . (intpos 5) = t . (intpos 7) ) ) by A6, A72, A75, A76, Lm14;
A106: me = t . (intpos 3) ;
then A107: (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),Q,t)) . GBP = 0 by A6, A72, A75, A76, Lm14;
then A108: (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),Q,t)) . (DataLoc (((IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),Q,t)) . GBP),7)) = 0 by A73, A86, A105, SCMP_GCD:1;
A109: now__::_thesis:_for_x_being_Int_position_holds_(IExec_((while>0_(GBP,7,((((((GBP,5)_:=_(GBP,7))_';'_(AddTo_(GBP,7,(-_1))))_';'_((GBP,6)_:=_((intpos_2),0)))_';'_(SubFrom_(GBP,6,(intpos_3),0)))_';'_(if>0_(GBP,6,((AddTo_(GBP,3,1))_';'_(AddTo_(GBP,5,(-_1)))),(Load_((GBP,7)_:=_0))))))),Q,t))_._x_=_(IExec_(((((((GBP,5)_:=_(GBP,7))_';'_(AddTo_(GBP,7,(-_1))))_';'_((GBP,6)_:=_((intpos_2),0)))_';'_(SubFrom_(GBP,6,(intpos_3),0)))_';'_(if>0_(GBP,6,((AddTo_(GBP,3,1))_';'_(AddTo_(GBP,5,(-_1)))),(Load_((GBP,7)_:=_0))))),Q,t))_._x
let x be Int_position; ::_thesis: (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . x = (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),Q,t)) . x
X1: (Initialize (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),Q,t))) . GBP = (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),Q,t)) . GBP by SCMPDS_5:15;
X2: (Initialize (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),Q,t))) . (DataLoc (((IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),Q,t)) . GBP),7)) = (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),Q,t)) . (DataLoc (((IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),Q,t)) . GBP),7)) by SCMPDS_5:15;
thus (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . x = (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,(Initialize (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),Q,t))))) . x by A73, A75, A87, A88, Th6
.= (Initialize (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),Q,t))) . x by A108, X1, X2, SCMPDS_8:23
.= (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),Q,t)) . x by SCMPDS_5:15 ; ::_thesis: verum
end;
hence  (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . GBP = 0 by A107; ::_thesis: ( (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos 1) = t . (intpos 1) & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos 7) = 0 & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos 2) = t . (intpos 2) & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos 4) = t . (intpos 4) & ( for i being Element of NAT st i >= 8 holds
(IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos i) = t . (intpos i) ) & ex m5 being Element of NAT ex mE3 being Element of NAT st
( (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos 5) = m5 & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos 3) = mE3 & mE3 + m5 = m3 & m5 <= t . (intpos 7) & ( for i being Element of NAT st t . (intpos 3) <= i & i < mE3 holds
(IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos md) > (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos i) ) & ( m5 = 0 or (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos md) <= (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos mE3) ) ) )
 (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),Q,t)) . (intpos 1) = t . (intpos 1) by A6, A72, A75, A76, A106, Lm14;
hence  (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos 1) = t . (intpos 1) by A109; ::_thesis: ( (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos 7) = 0 & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos 2) = t . (intpos 2) & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos 4) = t . (intpos 4) & ( for i being Element of NAT st i >= 8 holds
(IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos i) = t . (intpos i) ) & ex m5 being Element of NAT ex mE3 being Element of NAT st
( (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos 5) = m5 & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos 3) = mE3 & mE3 + m5 = m3 & m5 <= t . (intpos 7) & ( for i being Element of NAT st t . (intpos 3) <= i & i < mE3 holds
(IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos md) > (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos i) ) & ( m5 = 0 or (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos md) <= (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos mE3) ) ) )
thus  (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos 7) = 0 by A73, A86, A105, A109; ::_thesis: ( (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos 2) = t . (intpos 2) & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos 4) = t . (intpos 4) & ( for i being Element of NAT st i >= 8 holds
(IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos i) = t . (intpos i) ) & ex m5 being Element of NAT ex mE3 being Element of NAT st
( (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos 5) = m5 & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos 3) = mE3 & mE3 + m5 = m3 & m5 <= t . (intpos 7) & ( for i being Element of NAT st t . (intpos 3) <= i & i < mE3 holds
(IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos md) > (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos i) ) & ( m5 = 0 or (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos md) <= (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos mE3) ) ) )
 (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),Q,t)) . (intpos 2) = t . (intpos 2) by A6, A72, A75, A76, A106, Lm14;
hence  (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos 2) = t . (intpos 2) by A109; ::_thesis: ( (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos 4) = t . (intpos 4) & ( for i being Element of NAT st i >= 8 holds
(IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos i) = t . (intpos i) ) & ex m5 being Element of NAT ex mE3 being Element of NAT st
( (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos 5) = m5 & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos 3) = mE3 & mE3 + m5 = m3 & m5 <= t . (intpos 7) & ( for i being Element of NAT st t . (intpos 3) <= i & i < mE3 holds
(IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos md) > (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos i) ) & ( m5 = 0 or (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos md) <= (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos mE3) ) ) )
 (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),Q,t)) . (intpos 4) = t . (intpos 4) by A6, A72, A75, A76, A106, Lm14;
hence  (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos 4) = t . (intpos 4) by A109; ::_thesis: ( ( for i being Element of NAT st i >= 8 holds
(IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos i) = t . (intpos i) ) & ex m5 being Element of NAT ex mE3 being Element of NAT st
( (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos 5) = m5 & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos 3) = mE3 & mE3 + m5 = m3 & m5 <= t . (intpos 7) & ( for i being Element of NAT st t . (intpos 3) <= i & i < mE3 holds
(IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos md) > (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos i) ) & ( m5 = 0 or (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos md) <= (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos mE3) ) ) )
A110: now__::_thesis:_for_i_being_Element_of_NAT_st_i_>=_8_holds_
(IExec_((while>0_(GBP,7,((((((GBP,5)_:=_(GBP,7))_';'_(AddTo_(GBP,7,(-_1))))_';'_((GBP,6)_:=_((intpos_2),0)))_';'_(SubFrom_(GBP,6,(intpos_3),0)))_';'_(if>0_(GBP,6,((AddTo_(GBP,3,1))_';'_(AddTo_(GBP,5,(-_1)))),(Load_((GBP,7)_:=_0))))))),Q,t))_._(intpos_i)_=_t_._(intpos_i)
let i be Element of NAT ; ::_thesis: ( i >= 8 implies (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos i) = t . (intpos i) )
assume A111: i >= 8 ; ::_thesis: (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos i) = t . (intpos i)
thus (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos i) = (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),Q,t)) . (intpos i) by A109
.= t . (intpos i) by A6, A72, A75, A76, A106, A111, Lm14 ; ::_thesis: verum
end;
hence  for i being Element of NAT st i >= 8 holds
(IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos i) = t . (intpos i) ; ::_thesis: ex m5 being Element of NAT ex mE3 being Element of NAT st
( (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos 5) = m5 & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos 3) = mE3 & mE3 + m5 = m3 & m5 <= t . (intpos 7) & ( for i being Element of NAT st t . (intpos 3) <= i & i < mE3 holds
(IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos md) > (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos i) ) & ( m5 = 0 or (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos md) <= (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos mE3) ) )
A113: (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos me) = t . (intpos me) by A75, A110;
hereby  ::_thesis: verum
percases ( t . (intpos md) > t . (intpos me) or t . (intpos md) <= t . (intpos me) ) ;
supposeA114: t . (intpos md) > t . (intpos me) ; ::_thesis: ex m5 being Element of NAT ex mE3 being Element of NAT st
( (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos 5) = m5 & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos 3) = mE3 & mE3 + m5 = m3 & m5 <= t . (intpos 7) & ( for i being Element of NAT st t . (intpos 3) <= i & i < mE3 holds
(IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos md) > (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos i) ) & ( m5 = 0 or (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos md) <= (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos mE3) ) )
take m5 = 0 ; ::_thesis: ex mE3 being Element of NAT st
( (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos 5) = m5 & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos 3) = mE3 & mE3 + m5 = m3 & m5 <= t . (intpos 7) & ( for i being Element of NAT st t . (intpos 3) <= i & i < mE3 holds
(IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos md) > (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos i) ) & ( m5 = 0 or (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos md) <= (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos mE3) ) )
take mE3 = m3; ::_thesis: ( (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos 5) = m5 & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos 3) = mE3 & mE3 + m5 = m3 & m5 <= t . (intpos 7) & ( for i being Element of NAT st t . (intpos 3) <= i & i < mE3 holds
(IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos md) > (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos i) ) & ( m5 = 0 or (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos md) <= (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos mE3) ) )
thus  (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos 5) = m5 by A73, A86, A109, A114; ::_thesis: ( (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos 3) = mE3 & mE3 + m5 = m3 & m5 <= t . (intpos 7) & ( for i being Element of NAT st t . (intpos 3) <= i & i < mE3 holds
(IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos md) > (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos i) ) & ( m5 = 0 or (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos md) <= (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos mE3) ) )
thus  (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos 3) = mE3 by A73, A74, A86, A109, A114; ::_thesis: ( mE3 + m5 = m3 & m5 <= t . (intpos 7) & ( for i being Element of NAT st t . (intpos 3) <= i & i < mE3 holds
(IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos md) > (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos i) ) & ( m5 = 0 or (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos md) <= (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos mE3) ) )
thus  mE3 + m5 = m3 ; ::_thesis: ( m5 <= t . (intpos 7) & ( for i being Element of NAT st t . (intpos 3) <= i & i < mE3 holds
(IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos md) > (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos i) ) & ( m5 = 0 or (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos md) <= (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos mE3) ) )
thus  m5 <= t . (intpos 7) by A73; ::_thesis: ( ( for i being Element of NAT st t . (intpos 3) <= i & i < mE3 holds
(IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos md) > (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos i) ) & ( m5 = 0 or (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos md) <= (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos mE3) ) )
hereby  ::_thesis: ( m5 = 0 or (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos md) <= (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos mE3) )
let i be Element of NAT ; ::_thesis: ( t . (intpos 3) <= i & i < mE3 implies (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos md) > (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos i) )
assume that
A115: t . (intpos 3) <= i and
A116: i < mE3 ; ::_thesis: (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos md) > (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos i)
 i <= me by A73, A74, A116, NAT_1:13;
then  i = t . (intpos 3) by A115, XXREAL_0:1;
hence  (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos md) > (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos i) by A6, A110, A113, A114; ::_thesis: verum
end;
thus  ( m5 = 0 or (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos md) <= (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos mE3) ) ; ::_thesis: verum
end;
supposeA117: t . (intpos md) <= t . (intpos me) ; ::_thesis: ex m5 being Element of NAT ex mE3 being Element of NAT st
( (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos 5) = m5 & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos 3) = mE3 & mE3 + m5 = m3 & m5 <= t . (intpos 7) & ( for i being Element of NAT st t . (intpos 3) <= i & i < mE3 holds
(IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos md) > (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos i) ) & ( m5 = 0 or (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos md) <= (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos mE3) ) )
take m5 = 1; ::_thesis: ex mE3 being Element of NAT st
( (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos 5) = m5 & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos 3) = mE3 & mE3 + m5 = m3 & m5 <= t . (intpos 7) & ( for i being Element of NAT st t . (intpos 3) <= i & i < mE3 holds
(IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos md) > (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos i) ) & ( m5 = 0 or (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos md) <= (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos mE3) ) )
take mE3 = me; ::_thesis: ( (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos 5) = m5 & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos 3) = mE3 & mE3 + m5 = m3 & m5 <= t . (intpos 7) & ( for i being Element of NAT st t . (intpos 3) <= i & i < mE3 holds
(IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos md) > (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos i) ) & ( m5 = 0 or (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos md) <= (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos mE3) ) )
thus  (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos 5) = m5 by A73, A105, A109, A117; ::_thesis: ( (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos 3) = mE3 & mE3 + m5 = m3 & m5 <= t . (intpos 7) & ( for i being Element of NAT st t . (intpos 3) <= i & i < mE3 holds
(IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos md) > (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos i) ) & ( m5 = 0 or (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos md) <= (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos mE3) ) )
thus  (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos 3) = mE3 by A105, A109, A117; ::_thesis: ( mE3 + m5 = m3 & m5 <= t . (intpos 7) & ( for i being Element of NAT st t . (intpos 3) <= i & i < mE3 holds
(IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos md) > (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos i) ) & ( m5 = 0 or (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos md) <= (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos mE3) ) )
thus  mE3 + m5 = m3 by A73, A74; ::_thesis: ( m5 <= t . (intpos 7) & ( for i being Element of NAT st t . (intpos 3) <= i & i < mE3 holds
(IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos md) > (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos i) ) & ( m5 = 0 or (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos md) <= (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos mE3) ) )
thus  m5 <= t . (intpos 7) by A73; ::_thesis: ( ( for i being Element of NAT st t . (intpos 3) <= i & i < mE3 holds
(IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos md) > (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos i) ) & ( m5 = 0 or (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos md) <= (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos mE3) ) )
thus  for i being Element of NAT st t . (intpos 3) <= i & i < mE3 holds
(IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos md) > (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos i) ; ::_thesis: ( m5 = 0 or (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos md) <= (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos mE3) )
thus  ( m5 = 0 or (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos md) <= (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos mE3) ) by A6, A110, A113, A117; ::_thesis: verum
end;
end;
end;
end;
then A118: S1[ 0 ] ;
 for k being Element of NAT holds S1[k] from NAT_1:sch_1(A118, A8);
hence  ( (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),P,s)) . GBP = 0 & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),P,s)) . (intpos 1) = s . (intpos 1) & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),P,s)) . (intpos 7) = 0 & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),P,s)) . (intpos 2) = s . (intpos 2) & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),P,s)) . (intpos 4) = s . (intpos 4) & ( for i being Element of NAT st i >= 8 holds
(IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),P,s)) . (intpos i) = s . (intpos i) ) & ex m5, mE3 being Element of NAT st
( m5 = (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),P,s)) . (intpos 5) & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),P,s)) . (intpos 3) = mE3 & mE3 + m5 = m3 & m5 <= s . (intpos 7) & ( for i being Element of NAT st s . (intpos 3) <= i & i < mE3 holds
(IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),P,s)) . (intpos md) > (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),P,s)) . (intpos i) ) & ( m5 = 0 or (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),P,s)) . (intpos md) <= (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),P,s)) . (intpos mE3) ) ) ) by A1, A3, A4, A5, A7; ::_thesis: verum
end;

Lm16:  for P being Instruction-Sequence of SCMPDS
for s being 0 -started State of SCMPDS
for md being Element of NAT st s . GBP = 0 & s . (intpos 3) >= 8 & s . (intpos 2) = md & md >= 8 holds
( while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))) is_closed_on s,P & while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))) is_halting_on s,P )
proof
let P be Instruction-Sequence of SCMPDS; ::_thesis: for s being 0 -started State of SCMPDS
for md being Element of NAT st s . GBP = 0 & s . (intpos 3) >= 8 & s . (intpos 2) = md & md >= 8 holds
( while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))) is_closed_on s,P & while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))) is_halting_on s,P )
set a = GBP ;
let s be 0 -started State of SCMPDS; ::_thesis: for md being Element of NAT st s . GBP = 0 & s . (intpos 3) >= 8 & s . (intpos 2) = md & md >= 8 holds
( while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))) is_closed_on s,P & while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))) is_halting_on s,P )
let md be Element of NAT ; ::_thesis: ( s . GBP = 0 & s . (intpos 3) >= 8 & s . (intpos 2) = md & md >= 8 implies ( while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))) is_closed_on s,P & while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))) is_halting_on s,P ) )
set b = DataLoc ((s . GBP),7);
assume that
A1: s . GBP = 0 and
A2: s . (intpos 3) >= 8 and
A3: s . (intpos 2) = md and
A4: md >= 8 ; ::_thesis: ( while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))) is_closed_on s,P & while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))) is_halting_on s,P )
A5: DataLoc ((s . GBP),7) = intpos (0 + 7) by A1, SCMP_GCD:1;
now__::_thesis:_for_v_being_0_-started_State_of_SCMPDS
for_V_being_Instruction-Sequence_of_SCMPDS_st_v_._(intpos_3)_>=_8_&_v_._(intpos_2)_=_s_._(intpos_2)_&_v_._GBP_=_s_._GBP_&_v_._(DataLoc_((s_._GBP),7))_>_0_holds_
(_(IExec_(((((((GBP,5)_:=_(GBP,7))_';'_(AddTo_(GBP,7,(-_1))))_';'_((GBP,6)_:=_((intpos_2),0)))_';'_(SubFrom_(GBP,6,(intpos_3),0)))_';'_(if>0_(GBP,6,((AddTo_(GBP,3,1))_';'_(AddTo_(GBP,5,(-_1)))),(Load_((GBP,7)_:=_0))))),V,v))_._GBP_=_v_._GBP_&_(((((GBP,5)_:=_(GBP,7))_';'_(AddTo_(GBP,7,(-_1))))_';'_((GBP,6)_:=_((intpos_2),0)))_';'_(SubFrom_(GBP,6,(intpos_3),0)))_';'_(if>0_(GBP,6,((AddTo_(GBP,3,1))_';'_(AddTo_(GBP,5,(-_1)))),(Load_((GBP,7)_:=_0))))_is_closed_on_v,V_&_(((((GBP,5)_:=_(GBP,7))_';'_(AddTo_(GBP,7,(-_1))))_';'_((GBP,6)_:=_((intpos_2),0)))_';'_(SubFrom_(GBP,6,(intpos_3),0)))_';'_(if>0_(GBP,6,((AddTo_(GBP,3,1))_';'_(AddTo_(GBP,5,(-_1)))),(Load_((GBP,7)_:=_0))))_is_halting_on_v,V_&_(IExec_(((((((GBP,5)_:=_(GBP,7))_';'_(AddTo_(GBP,7,(-_1))))_';'_((GBP,6)_:=_((intpos_2),0)))_';'_(SubFrom_(GBP,6,(intpos_3),0)))_';'_(if>0_(GBP,6,((AddTo_(GBP,3,1))_';'_(AddTo_(GBP,5,(-_1)))),(Load_((GBP,7)_:=_0))))),V,v))_._(DataLoc_((s_._GBP),7))_<_v_._(DataLoc_((s_._GBP),7))_&_(IExec_(((((((GBP,5)_:=_(GBP,7))_';'_(AddTo_(GBP,7,(-_1))))_';'_((GBP,6)_:=_((intpos_2),0)))_';'_(SubFrom_(GBP,6,(intpos_3),0)))_';'_(if>0_(GBP,6,((AddTo_(GBP,3,1))_';'_(AddTo_(GBP,5,(-_1)))),(Load_((GBP,7)_:=_0))))),V,v))_._(intpos_3)_>=_8_&_(IExec_(((((((GBP,5)_:=_(GBP,7))_';'_(AddTo_(GBP,7,(-_1))))_';'_((GBP,6)_:=_((intpos_2),0)))_';'_(SubFrom_(GBP,6,(intpos_3),0)))_';'_(if>0_(GBP,6,((AddTo_(GBP,3,1))_';'_(AddTo_(GBP,5,(-_1)))),(Load_((GBP,7)_:=_0))))),V,v))_._(intpos_2)_=_v_._(intpos_2)_)
let v be 0 -started State of SCMPDS; ::_thesis: for V being Instruction-Sequence of SCMPDS st v . (intpos 3) >= 8 & v . (intpos 2) = s . (intpos 2) & v . GBP = s . GBP & v . (DataLoc ((s . GBP),7)) > 0 holds
( (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),V,v)) . GBP = v . GBP & (((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))) is_closed_on v,V & (((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))) is_halting_on v,V & (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),V,v)) . (DataLoc ((s . GBP),7)) < v . (DataLoc ((s . GBP),7)) & (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),V,v)) . (intpos 3) >= 8 & (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),V,v)) . (intpos 2) = v . (intpos 2) )
let V be Instruction-Sequence of SCMPDS; ::_thesis: ( v . (intpos 3) >= 8 & v . (intpos 2) = s . (intpos 2) & v . GBP = s . GBP & v . (DataLoc ((s . GBP),7)) > 0 implies ( (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),V,v)) . GBP = v . GBP & (((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))) is_closed_on v,V & (((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))) is_halting_on v,V & (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),V,v)) . (DataLoc ((s . GBP),7)) < v . (DataLoc ((s . GBP),7)) & (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),V,v)) . (intpos 3) >= 8 & (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),V,v)) . (intpos 2) = v . (intpos 2) ) )
assume that
A6: v . (intpos 3) >= 8 and
A7: v . (intpos 2) = s . (intpos 2) and
A8: v . GBP = s . GBP and
A9: v . (DataLoc ((s . GBP),7)) > 0 ; ::_thesis: ( (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),V,v)) . GBP = v . GBP & (((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))) is_closed_on v,V & (((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))) is_halting_on v,V & (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),V,v)) . (DataLoc ((s . GBP),7)) < v . (DataLoc ((s . GBP),7)) & (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),V,v)) . (intpos 3) >= 8 & (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),V,v)) . (intpos 2) = v . (intpos 2) )
reconsider ME = v . (intpos 3) as Element of NAT by A6, INT_1:3;
A17: ME = v . (intpos 3) ;
hence  (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),V,v)) . GBP = v . GBP by A1, A3, A4, A6, A7, A8, Lm14; ::_thesis: ( (((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))) is_closed_on v,V & (((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))) is_halting_on v,V & (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),V,v)) . (DataLoc ((s . GBP),7)) < v . (DataLoc ((s . GBP),7)) & (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),V,v)) . (intpos 3) >= 8 & (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),V,v)) . (intpos 2) = v . (intpos 2) )
thus  ( (((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))) is_closed_on v,V & (((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))) is_halting_on v,V ) by SCMPDS_6:20, SCMPDS_6:21; ::_thesis: ( (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),V,v)) . (DataLoc ((s . GBP),7)) < v . (DataLoc ((s . GBP),7)) & (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),V,v)) . (intpos 3) >= 8 & (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),V,v)) . (intpos 2) = v . (intpos 2) )
A18: ( v . (intpos md) > v . (intpos ME) implies ( (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),V,v)) . (intpos 7) = (v . (intpos 7)) - 1 & (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),V,v)) . (intpos 3) = (v . (intpos 3)) + 1 & (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),V,v)) . (intpos 5) = (v . (intpos 7)) - 1 ) ) by A1, A3, A4, A6, A7, A8, Lm14;
hereby  ::_thesis: (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),V,v)) . (intpos 2) = v . (intpos 2)
percases ( v . (intpos md) > v . (intpos ME) or v . (intpos md) <= v . (intpos ME) ) ;
supposeA19: v . (intpos md) > v . (intpos ME) ; ::_thesis: ( (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),V,v)) . (DataLoc ((s . GBP),7)) < v . (DataLoc ((s . GBP),7)) & (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),V,v)) . (intpos 3) >= 8 )
hence  (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),V,v)) . (DataLoc ((s . GBP),7)) < v . (DataLoc ((s . GBP),7)) by A5, A18, XREAL_1:146; ::_thesis: (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),V,v)) . (intpos 3) >= 8
A20: (v . (intpos 3)) + 1 > v . (intpos 3) by XREAL_1:29;
 (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),V,v)) . (intpos 3) = (v . (intpos 3)) + 1 by A1, A3, A4, A6, A7, A8, A19, Lm14;
hence  (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),V,v)) . (intpos 3) >= 8 by A6, A20, XXREAL_0:2; ::_thesis: verum
end;
supposeA21: v . (intpos md) <= v . (intpos ME) ; ::_thesis: ( (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),V,v)) . (DataLoc ((s . GBP),7)) < v . (DataLoc ((s . GBP),7)) & (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),V,v)) . (intpos 3) >= 8 )
hence  (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),V,v)) . (DataLoc ((s . GBP),7)) < v . (DataLoc ((s . GBP),7)) by A1, A3, A4, A5, A6, A7, A8, A9, Lm14; ::_thesis: (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),V,v)) . (intpos 3) >= 8
thus  (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),V,v)) . (intpos 3) >= 8 by A1, A3, A4, A6, A7, A8, A21, Lm14; ::_thesis: verum
end;
end;
end;
thus  (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),V,v)) . (intpos 2) = v . (intpos 2) by A1, A3, A4, A6, A7, A8, A17, Lm14; ::_thesis: verum
end;
hence  ( while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))) is_closed_on s,P & while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))) is_halting_on s,P ) by A2, Th6; ::_thesis: verum
end;

Lm17:  card (((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))) = 29
proof
thus  card (((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))) = (card ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))))) + (card (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))) by AFINSQ_1:17
.= (11 + 11) + (card (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))) by Lm11, Lm13, AFINSQ_1:17
.= 22 + ((card (((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))) + 1) by SCMPDS_6:75
.= 22 + (((card ((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1)))) + 1) + 1) by SCMP_GCD:4
.= 22 + ((((card (((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2))))) + 1) + 1) + 1) by SCMP_GCD:4
.= 22 + (((((card ((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6)))) + 1) + 1) + 1) + 1) by SCMP_GCD:4
.= 22 + ((((((card (((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0)))) + 1) + 1) + 1) + 1) + 1) by SCMP_GCD:4
.= 22 + (((((2 + 1) + 1) + 1) + 1) + 1) by SCMP_GCD:5
.= 29 ; ::_thesis: verum
end;

Lm18:  card (while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))) = 31
proof
thus  card (while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))) = 29 + 2 by Lm17, SCMPDS_8:17
.= 31 ; ::_thesis: verum
end;

begin

theorem Th12: :: SCPQSORT:9
card Partition = 38
proof
thus  card Partition = (card ((((((((GBP,5) := (GBP,4)) ';' (SubFrom (GBP,5,GBP,2))) ';' ((GBP,3) := (GBP,2))) ';' (AddTo (GBP,3,1))) ';' (while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))))))) ';' ((GBP,6) := ((intpos 4),0))) ';' (((intpos 4),0) := ((intpos 2),0)))) + 1 by SCMP_GCD:4
.= ((card (((((((GBP,5) := (GBP,4)) ';' (SubFrom (GBP,5,GBP,2))) ';' ((GBP,3) := (GBP,2))) ';' (AddTo (GBP,3,1))) ';' (while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))))))) ';' ((GBP,6) := ((intpos 4),0)))) + 1) + 1 by SCMP_GCD:4
.= (((card ((((((GBP,5) := (GBP,4)) ';' (SubFrom (GBP,5,GBP,2))) ';' ((GBP,3) := (GBP,2))) ';' (AddTo (GBP,3,1))) ';' (while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))))) + 1) + 1) + 1 by SCMP_GCD:4
.= ((((card (((((GBP,5) := (GBP,4)) ';' (SubFrom (GBP,5,GBP,2))) ';' ((GBP,3) := (GBP,2))) ';' (AddTo (GBP,3,1)))) + (card (while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))))) + 1) + 1) + 1 by AFINSQ_1:17
.= (((4 + 31) + 1) + 1) + 1 by Lm18, Th4
.= 38 ; ::_thesis: verum
end;

Lm19:  for P being Instruction-Sequence of SCMPDS
for s being 0 -started State of SCMPDS
for m3, m4 being Element of NAT st s . GBP = 0 & s . (intpos 5) > 0 & s . (intpos 3) = m3 & s . (intpos 4) = m4 & m3 > 6 & m4 > 6 holds
( (IExec ((if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))),P,(Initialize s))) . GBP = 0 & (IExec ((if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))),P,(Initialize s))) . (intpos 1) = s . (intpos 1) & (IExec ((if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))),P,(Initialize s))) . (intpos 2) = s . (intpos 2) & (IExec ((if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))),P,(Initialize s))) . (intpos m3) = s . (intpos m4) & (IExec ((if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))),P,(Initialize s))) . (intpos m4) = s . (intpos m3) & (IExec ((if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))),P,(Initialize s))) . (intpos 3) = (s . (intpos 3)) + 1 & (IExec ((if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))),P,(Initialize s))) . (intpos 4) = (s . (intpos 4)) - 1 & (IExec ((if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))),P,(Initialize s))) . (intpos 5) = (s . (intpos 5)) - 2 & ( for i being Element of NAT st i >= 8 & i <> m3 & i <> m4 holds
(IExec ((if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))),P,(Initialize s))) . (intpos i) = s . (intpos i) ) )
proof
let P be Instruction-Sequence of SCMPDS; ::_thesis: for s being 0 -started State of SCMPDS
for m3, m4 being Element of NAT st s . GBP = 0 & s . (intpos 5) > 0 & s . (intpos 3) = m3 & s . (intpos 4) = m4 & m3 > 6 & m4 > 6 holds
( (IExec ((if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))),P,(Initialize s))) . GBP = 0 & (IExec ((if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))),P,(Initialize s))) . (intpos 1) = s . (intpos 1) & (IExec ((if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))),P,(Initialize s))) . (intpos 2) = s . (intpos 2) & (IExec ((if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))),P,(Initialize s))) . (intpos m3) = s . (intpos m4) & (IExec ((if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))),P,(Initialize s))) . (intpos m4) = s . (intpos m3) & (IExec ((if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))),P,(Initialize s))) . (intpos 3) = (s . (intpos 3)) + 1 & (IExec ((if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))),P,(Initialize s))) . (intpos 4) = (s . (intpos 4)) - 1 & (IExec ((if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))),P,(Initialize s))) . (intpos 5) = (s . (intpos 5)) - 2 & ( for i being Element of NAT st i >= 8 & i <> m3 & i <> m4 holds
(IExec ((if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))),P,(Initialize s))) . (intpos i) = s . (intpos i) ) )
set a = GBP ;
let s be 0 -started State of SCMPDS; ::_thesis: for m3, m4 being Element of NAT st s . GBP = 0 & s . (intpos 5) > 0 & s . (intpos 3) = m3 & s . (intpos 4) = m4 & m3 > 6 & m4 > 6 holds
( (IExec ((if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))),P,(Initialize s))) . GBP = 0 & (IExec ((if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))),P,(Initialize s))) . (intpos 1) = s . (intpos 1) & (IExec ((if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))),P,(Initialize s))) . (intpos 2) = s . (intpos 2) & (IExec ((if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))),P,(Initialize s))) . (intpos m3) = s . (intpos m4) & (IExec ((if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))),P,(Initialize s))) . (intpos m4) = s . (intpos m3) & (IExec ((if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))),P,(Initialize s))) . (intpos 3) = (s . (intpos 3)) + 1 & (IExec ((if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))),P,(Initialize s))) . (intpos 4) = (s . (intpos 4)) - 1 & (IExec ((if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))),P,(Initialize s))) . (intpos 5) = (s . (intpos 5)) - 2 & ( for i being Element of NAT st i >= 8 & i <> m3 & i <> m4 holds
(IExec ((if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))),P,(Initialize s))) . (intpos i) = s . (intpos i) ) )
let m3, m4 be Element of NAT ; ::_thesis: ( s . GBP = 0 & s . (intpos 5) > 0 & s . (intpos 3) = m3 & s . (intpos 4) = m4 & m3 > 6 & m4 > 6 implies ( (IExec ((if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))),P,(Initialize s))) . GBP = 0 & (IExec ((if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))),P,(Initialize s))) . (intpos 1) = s . (intpos 1) & (IExec ((if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))),P,(Initialize s))) . (intpos 2) = s . (intpos 2) & (IExec ((if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))),P,(Initialize s))) . (intpos m3) = s . (intpos m4) & (IExec ((if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))),P,(Initialize s))) . (intpos m4) = s . (intpos m3) & (IExec ((if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))),P,(Initialize s))) . (intpos 3) = (s . (intpos 3)) + 1 & (IExec ((if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))),P,(Initialize s))) . (intpos 4) = (s . (intpos 4)) - 1 & (IExec ((if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))),P,(Initialize s))) . (intpos 5) = (s . (intpos 5)) - 2 & ( for i being Element of NAT st i >= 8 & i <> m3 & i <> m4 holds
(IExec ((if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))),P,(Initialize s))) . (intpos i) = s . (intpos i) ) ) )
assume that
A1: s . GBP = 0 and
A2: s . (intpos 5) > 0 and
A3: s . (intpos 3) = m3 and
A4: s . (intpos 4) = m4 and
A5: m3 > 6 and
A6: m4 > 6 ; ::_thesis: ( (IExec ((if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))),P,(Initialize s))) . GBP = 0 & (IExec ((if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))),P,(Initialize s))) . (intpos 1) = s . (intpos 1) & (IExec ((if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))),P,(Initialize s))) . (intpos 2) = s . (intpos 2) & (IExec ((if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))),P,(Initialize s))) . (intpos m3) = s . (intpos m4) & (IExec ((if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))),P,(Initialize s))) . (intpos m4) = s . (intpos m3) & (IExec ((if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))),P,(Initialize s))) . (intpos 3) = (s . (intpos 3)) + 1 & (IExec ((if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))),P,(Initialize s))) . (intpos 4) = (s . (intpos 4)) - 1 & (IExec ((if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))),P,(Initialize s))) . (intpos 5) = (s . (intpos 5)) - 2 & ( for i being Element of NAT st i >= 8 & i <> m3 & i <> m4 holds
(IExec ((if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))),P,(Initialize s))) . (intpos i) = s . (intpos i) ) )
A7: m4 > 3 by A6, XXREAL_0:2;
set x = intpos m3;
set y = intpos m4;
set t0 = Initialize s;
set t1 = IExec ((((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))),P,(Initialize s));
set t2 = IExec (((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))),P,(Initialize s));
set t3 = IExec ((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))),P,(Initialize s));
set t4 = IExec (((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))),P,(Initialize s));
set t5 = IExec ((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))),P,(Initialize s));
set t6 = Exec (((GBP,6) := ((intpos 4),0)),(Initialize s));
A8: (Initialize s) . GBP = 0 by A1, SCMPDS_5:15;
then A9: DataLoc (((Initialize s) . GBP),6) = intpos (0 + 6) by SCMP_GCD:1;
then A10: (Exec (((GBP,6) := ((intpos 4),0)),(Initialize s))) . GBP = 0 by A8, AMI_3:10, SCMPDS_2:47;
A11: (Initialize s) . (intpos 4) = m4 by A4, SCMPDS_5:15;
then A12: (Exec (((GBP,6) := ((intpos 4),0)),(Initialize s))) . (intpos 4) = m4 by A9, AMI_3:10, SCMPDS_2:47;
then A13: DataLoc (((Exec (((GBP,6) := ((intpos 4),0)),(Initialize s))) . (intpos 4)),0) = intpos (m4 + 0) by SCMP_GCD:1;
A14: (IExec ((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))),P,(Initialize s))) . GBP = (Exec ((((intpos 4),0) := ((intpos 3),0)),(Exec (((GBP,6) := ((intpos 4),0)),(Initialize s))))) . GBP by SCMPDS_5:42
.= 0 by A6, A10, A13, AMI_3:10, SCMPDS_2:47 ;
A15: (Exec (((GBP,6) := ((intpos 4),0)),(Initialize s))) . (intpos 6) = (Initialize s) . (DataLoc (((Initialize s) . (intpos 4)),0)) by A9, SCMPDS_2:47
.= (Initialize s) . (intpos (m4 + 0)) by A11, SCMP_GCD:1
.= s . (intpos m4) by SCMPDS_5:15 ;
(IExec ((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))),P,(Initialize s))) . (intpos 6) = (Exec ((((intpos 4),0) := ((intpos 3),0)),(Exec (((GBP,6) := ((intpos 4),0)),(Initialize s))))) . (intpos 6) by SCMPDS_5:42
.= s . (intpos m4) by A6, A15, A13, AMI_3:10, SCMPDS_2:47 ;
then A16: (IExec ((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))),P,(Initialize s))) . (DataLoc (((IExec ((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))),P,(Initialize s))) . GBP),6)) = s . (intpos m4) by A14, SCMP_GCD:1;
 (Initialize s) . (intpos 3) = m3 by A3, SCMPDS_5:15;
then A17: (Exec (((GBP,6) := ((intpos 4),0)),(Initialize s))) . (intpos 3) = m3 by A9, AMI_3:10, SCMPDS_2:47;
A18: (IExec ((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))),P,(Initialize s))) . (intpos 3) = (Exec ((((intpos 4),0) := ((intpos 3),0)),(Exec (((GBP,6) := ((intpos 4),0)),(Initialize s))))) . (intpos 3) by SCMPDS_5:42
.= m3 by A7, A17, A13, AMI_3:10, SCMPDS_2:47 ;
then A19: DataLoc (((IExec ((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))),P,(Initialize s))) . (intpos 3)),0) = intpos (m3 + 0) by SCMP_GCD:1;
A20: (Initialize s) . (intpos m3) = s . (intpos m3) by SCMPDS_5:15;
A21: (IExec ((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))),P,(Initialize s))) . (intpos m4) = (Exec ((((intpos 4),0) := ((intpos 3),0)),(Exec (((GBP,6) := ((intpos 4),0)),(Initialize s))))) . (intpos m4) by SCMPDS_5:42
.= (Exec (((GBP,6) := ((intpos 4),0)),(Initialize s))) . (DataLoc (((Exec (((GBP,6) := ((intpos 4),0)),(Initialize s))) . (intpos 3)),0)) by A13, SCMPDS_2:47
.= (Exec (((GBP,6) := ((intpos 4),0)),(Initialize s))) . (intpos (m3 + 0)) by A17, SCMP_GCD:1
.= s . (intpos m3) by A5, A20, A9, AMI_3:10, SCMPDS_2:47 ;
A22: now__::_thesis:_(IExec_(((((GBP,6)_:=_((intpos_4),0))_';'_(((intpos_4),0)_:=_((intpos_3),0)))_';'_(((intpos_3),0)_:=_(GBP,6))),P,(Initialize_s)))_._(intpos_m4)_=_s_._(intpos_m3)
percases ( intpos m4 <> DataLoc (((IExec ((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))),P,(Initialize s))) . (intpos 3)),0) or intpos m4 = DataLoc (((IExec ((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))),P,(Initialize s))) . (intpos 3)),0) ) ;
supposeA23: intpos m4 <> DataLoc (((IExec ((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))),P,(Initialize s))) . (intpos 3)),0) ; ::_thesis: (IExec (((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))),P,(Initialize s))) . (intpos m4) = s . (intpos m3)
thus (IExec (((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))),P,(Initialize s))) . (intpos m4) = (Exec ((((intpos 3),0) := (GBP,6)),(IExec ((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))),P,(Initialize s))))) . (intpos m4) by SCMPDS_5:41
.= s . (intpos m3) by A21, A23, SCMPDS_2:47 ; ::_thesis: verum
end;
supposeA24: intpos m4 = DataLoc (((IExec ((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))),P,(Initialize s))) . (intpos 3)),0) ; ::_thesis: (IExec (((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))),P,(Initialize s))) . (intpos m4) = s . (intpos m3)
thus (IExec (((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))),P,(Initialize s))) . (intpos m4) = (Exec ((((intpos 3),0) := (GBP,6)),(IExec ((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))),P,(Initialize s))))) . (intpos m4) by SCMPDS_5:41
.= s . (intpos m3) by A19, A16, A24, SCMPDS_2:47 ; ::_thesis: verum
end;
end;
end;
 (Initialize s) . (intpos 2) = s . (intpos 2) by SCMPDS_5:15;
then A25: (Exec (((GBP,6) := ((intpos 4),0)),(Initialize s))) . (intpos 2) = s . (intpos 2) by A9, AMI_3:10, SCMPDS_2:47;
A26: m4 > 2 by A6, XXREAL_0:2;
A27: m3 > 5 by A5, XXREAL_0:2;
A28: (IExec (((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))),P,(Initialize s))) . GBP = (Exec ((((intpos 3),0) := (GBP,6)),(IExec ((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))),P,(Initialize s))))) . GBP by SCMPDS_5:41
.= 0 by A5, A14, A19, AMI_3:10, SCMPDS_2:47 ;
then A29: DataLoc (((IExec (((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))),P,(Initialize s))) . GBP),5) = intpos (0 + 5) by SCMP_GCD:1;
A30: (IExec ((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))),P,(Initialize s))) . GBP = (Exec ((AddTo (GBP,5,(- 2))),(IExec (((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))),P,(Initialize s))))) . GBP by SCMPDS_5:41
.= 0 by A28, A29, AMI_3:10, SCMPDS_2:48 ;
then A31: GBP <> DataLoc (((IExec ((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))),P,(Initialize s))) . GBP),3) by AMI_3:10, SCMP_GCD:1;
A32: m3 > 3 by A5, XXREAL_0:2;
then A33: intpos m3 <> DataLoc (((IExec ((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))),P,(Initialize s))) . GBP),3) by A30, AMI_3:10, SCMP_GCD:1;
A34: (IExec (((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))),P,(Initialize s))) . (intpos 3) = (Exec ((((intpos 3),0) := (GBP,6)),(IExec ((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))),P,(Initialize s))))) . (intpos 3) by SCMPDS_5:41
.= m3 by A32, A18, A19, AMI_3:10, SCMPDS_2:47 ;
A35: (IExec ((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))),P,(Initialize s))) . (intpos 3) = (Exec ((AddTo (GBP,5,(- 2))),(IExec (((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))),P,(Initialize s))))) . (intpos 3) by SCMPDS_5:41
.= m3 by A34, A29, AMI_3:10, SCMPDS_2:48 ;
A36: DataLoc (((IExec ((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))),P,(Initialize s))) . GBP),3) = intpos (0 + 3) by A30, SCMP_GCD:1;
A37: (IExec (((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))),P,(Initialize s))) . (intpos 3) = (Exec ((AddTo (GBP,3,1)),(IExec ((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))),P,(Initialize s))))) . (intpos 3) by SCMPDS_5:41
.= m3 + 1 by A35, A36, SCMPDS_2:48 ;
A38: m3 > 2 by A5, XXREAL_0:2;
A39: m4 > 5 by A6, XXREAL_0:2;
A40: (IExec ((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))),P,(Initialize s))) . (intpos 2) = (Exec ((((intpos 4),0) := ((intpos 3),0)),(Exec (((GBP,6) := ((intpos 4),0)),(Initialize s))))) . (intpos 2) by SCMPDS_5:42
.= s . (intpos 2) by A25, A13, A26, AMI_3:10, SCMPDS_2:47 ;
A41: (IExec (((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))),P,(Initialize s))) . (intpos m3) = (Exec ((((intpos 3),0) := (GBP,6)),(IExec ((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))),P,(Initialize s))))) . (intpos m3) by SCMPDS_5:41
.= s . (intpos m4) by A19, A16, SCMPDS_2:47 ;
A42: (IExec ((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))),P,(Initialize s))) . (intpos m3) = (Exec ((AddTo (GBP,5,(- 2))),(IExec (((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))),P,(Initialize s))))) . (intpos m3) by SCMPDS_5:41
.= s . (intpos m4) by A27, A41, A29, AMI_3:10, SCMPDS_2:48 ;
A43: (IExec (((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))),P,(Initialize s))) . (intpos m3) = (Exec ((AddTo (GBP,3,1)),(IExec ((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))),P,(Initialize s))))) . (intpos m3) by SCMPDS_5:41
.= s . (intpos m4) by A42, A33, SCMPDS_2:48 ;
A44: (IExec ((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))),P,(Initialize s))) . (intpos m4) = (Exec ((AddTo (GBP,5,(- 2))),(IExec (((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))),P,(Initialize s))))) . (intpos m4) by SCMPDS_5:41
.= s . (intpos m3) by A39, A22, A29, AMI_3:10, SCMPDS_2:48 ;
A45: intpos m4 <> DataLoc (((IExec ((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))),P,(Initialize s))) . GBP),3) by A7, A30, AMI_3:10, SCMP_GCD:1;
A46: (IExec (((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))),P,(Initialize s))) . (intpos m4) = (Exec ((AddTo (GBP,3,1)),(IExec ((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))),P,(Initialize s))))) . (intpos m4) by SCMPDS_5:41
.= s . (intpos m3) by A44, A45, SCMPDS_2:48 ;
A47: (IExec (((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))),P,(Initialize s))) . (intpos 2) = (Exec ((((intpos 3),0) := (GBP,6)),(IExec ((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))),P,(Initialize s))))) . (intpos 2) by SCMPDS_5:41
.= s . (intpos 2) by A40, A19, A38, AMI_3:10, SCMPDS_2:47 ;
A48: (IExec ((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))),P,(Initialize s))) . (intpos 2) = (Exec ((AddTo (GBP,5,(- 2))),(IExec (((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))),P,(Initialize s))))) . (intpos 2) by SCMPDS_5:41
.= s . (intpos 2) by A47, A29, AMI_3:10, SCMPDS_2:48 ;
A49: intpos 2 <> DataLoc (((IExec ((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))),P,(Initialize s))) . GBP),3) by A30, AMI_3:10, SCMP_GCD:1;
A50: (IExec (((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))),P,(Initialize s))) . (intpos 2) = (Exec ((AddTo (GBP,3,1)),(IExec ((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))),P,(Initialize s))))) . (intpos 2) by SCMPDS_5:41
.= s . (intpos 2) by A48, A49, SCMPDS_2:48 ;
A51: m4 > 1 by A6, XXREAL_0:2;
A52: DataLoc ((s . GBP),5) = intpos (0 + 5) by A1, SCMP_GCD:1;
 (Initialize s) . (intpos 5) = s . (intpos 5) by SCMPDS_5:15;
then A53: (Exec (((GBP,6) := ((intpos 4),0)),(Initialize s))) . (intpos 5) = s . (intpos 5) by A9, AMI_3:10, SCMPDS_2:47;
A54: (IExec ((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))),P,(Initialize s))) . (intpos 5) = (Exec ((((intpos 4),0) := ((intpos 3),0)),(Exec (((GBP,6) := ((intpos 4),0)),(Initialize s))))) . (intpos 5) by SCMPDS_5:42
.= s . (intpos 5) by A39, A53, A13, AMI_3:10, SCMPDS_2:47 ;
A55: (IExec (((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))),P,(Initialize s))) . (intpos 5) = (Exec ((((intpos 3),0) := (GBP,6)),(IExec ((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))),P,(Initialize s))))) . (intpos 5) by SCMPDS_5:41
.= s . (intpos 5) by A27, A54, A19, AMI_3:10, SCMPDS_2:47 ;
A56: (IExec ((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))),P,(Initialize s))) . (intpos 5) = (Exec ((AddTo (GBP,5,(- 2))),(IExec (((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))),P,(Initialize s))))) . (intpos 5) by SCMPDS_5:41
.= ((IExec (((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))),P,(Initialize s))) . (intpos 5)) + (- 2) by A29, SCMPDS_2:48
.= (s . (intpos 5)) - 2 by A55 ;
A57: intpos 5 <> DataLoc (((IExec ((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))),P,(Initialize s))) . GBP),3) by A30, AMI_3:10, SCMP_GCD:1;
A58: (IExec (((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))),P,(Initialize s))) . (intpos 5) = (Exec ((AddTo (GBP,3,1)),(IExec ((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))),P,(Initialize s))))) . (intpos 5) by SCMPDS_5:41
.= (s . (intpos 5)) - 2 by A56, A57, SCMPDS_2:48 ;
 (Initialize s) . (intpos 1) = s . (intpos 1) by SCMPDS_5:15;
then A59: (Exec (((GBP,6) := ((intpos 4),0)),(Initialize s))) . (intpos 1) = s . (intpos 1) by A9, AMI_3:10, SCMPDS_2:47;
A60: m3 > 1 by A5, XXREAL_0:2;
A61: (IExec (((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))),P,(Initialize s))) . GBP = (Exec ((AddTo (GBP,3,1)),(IExec ((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))),P,(Initialize s))))) . GBP by SCMPDS_5:41
.= 0 by A30, A31, SCMPDS_2:48 ;
then A62: GBP <> DataLoc (((IExec (((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))),P,(Initialize s))) . GBP),4) by AMI_3:10, SCMP_GCD:1;
A63: m4 > 4 by A6, XXREAL_0:2;
then A64: intpos m4 <> DataLoc (((IExec (((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))),P,(Initialize s))) . GBP),4) by A61, AMI_3:10, SCMP_GCD:1;
A65: m3 > 4 by A5, XXREAL_0:2;
then A66: intpos m3 <> DataLoc (((IExec (((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))),P,(Initialize s))) . GBP),4) by A61, AMI_3:10, SCMP_GCD:1;
A67: intpos 4 <> DataLoc (((IExec ((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))),P,(Initialize s))) . GBP),3) by A30, AMI_3:10, SCMP_GCD:1;
A68: (IExec ((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))),P,(Initialize s))) . (intpos 4) = (Exec ((((intpos 4),0) := ((intpos 3),0)),(Exec (((GBP,6) := ((intpos 4),0)),(Initialize s))))) . (intpos 4) by SCMPDS_5:42
.= m4 by A63, A12, A13, AMI_3:10, SCMPDS_2:47 ;
A69: (IExec (((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))),P,(Initialize s))) . (intpos 4) = (Exec ((((intpos 3),0) := (GBP,6)),(IExec ((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))),P,(Initialize s))))) . (intpos 4) by SCMPDS_5:41
.= m4 by A65, A68, A19, AMI_3:10, SCMPDS_2:47 ;
A70: (IExec ((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))),P,(Initialize s))) . (intpos 4) = (Exec ((AddTo (GBP,5,(- 2))),(IExec (((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))),P,(Initialize s))))) . (intpos 4) by SCMPDS_5:41
.= m4 by A69, A29, AMI_3:10, SCMPDS_2:48 ;
A71: (IExec (((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))),P,(Initialize s))) . (intpos 4) = (Exec ((AddTo (GBP,3,1)),(IExec ((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))),P,(Initialize s))))) . (intpos 4) by SCMPDS_5:41
.= m4 by A70, A67, SCMPDS_2:48 ;
A72: intpos 3 <> DataLoc (((IExec (((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))),P,(Initialize s))) . GBP),4) by A61, AMI_3:10, SCMP_GCD:1;
A73: (IExec ((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))),P,(Initialize s))) . (intpos 1) = (Exec ((((intpos 4),0) := ((intpos 3),0)),(Exec (((GBP,6) := ((intpos 4),0)),(Initialize s))))) . (intpos 1) by SCMPDS_5:42
.= s . (intpos 1) by A59, A13, A51, AMI_3:10, SCMPDS_2:47 ;
A74: (IExec (((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))),P,(Initialize s))) . (intpos 1) = (Exec ((((intpos 3),0) := (GBP,6)),(IExec ((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))),P,(Initialize s))))) . (intpos 1) by SCMPDS_5:41
.= s . (intpos 1) by A73, A19, A60, AMI_3:10, SCMPDS_2:47 ;
A75: (IExec ((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))),P,(Initialize s))) . (intpos 1) = (Exec ((AddTo (GBP,5,(- 2))),(IExec (((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))),P,(Initialize s))))) . (intpos 1) by SCMPDS_5:41
.= s . (intpos 1) by A74, A29, AMI_3:10, SCMPDS_2:48 ;
A76: intpos 1 <> DataLoc (((IExec ((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))),P,(Initialize s))) . GBP),3) by A30, AMI_3:10, SCMP_GCD:1;
A77: (IExec (((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))),P,(Initialize s))) . (intpos 1) = (Exec ((AddTo (GBP,3,1)),(IExec ((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))),P,(Initialize s))))) . (intpos 1) by SCMPDS_5:41
.= s . (intpos 1) by A75, A76, SCMPDS_2:48 ;
A78: intpos 1 <> DataLoc (((IExec (((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))),P,(Initialize s))) . GBP),4) by A61, AMI_3:10, SCMP_GCD:1;
(IExec ((((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))),P,(Initialize s))) . GBP = (Exec ((AddTo (GBP,4,(- 1))),(IExec (((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))),P,(Initialize s))))) . GBP by SCMPDS_5:41
.= 0 by A61, A62, SCMPDS_2:48 ;
hence  (IExec ((if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))),P,(Initialize s))) . GBP = 0 by A2, A52, SCMPDS_6:83; ::_thesis: ( (IExec ((if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))),P,(Initialize s))) . (intpos 1) = s . (intpos 1) & (IExec ((if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))),P,(Initialize s))) . (intpos 2) = s . (intpos 2) & (IExec ((if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))),P,(Initialize s))) . (intpos m3) = s . (intpos m4) & (IExec ((if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))),P,(Initialize s))) . (intpos m4) = s . (intpos m3) & (IExec ((if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))),P,(Initialize s))) . (intpos 3) = (s . (intpos 3)) + 1 & (IExec ((if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))),P,(Initialize s))) . (intpos 4) = (s . (intpos 4)) - 1 & (IExec ((if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))),P,(Initialize s))) . (intpos 5) = (s . (intpos 5)) - 2 & ( for i being Element of NAT st i >= 8 & i <> m3 & i <> m4 holds
(IExec ((if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))),P,(Initialize s))) . (intpos i) = s . (intpos i) ) )
(IExec ((((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))),P,(Initialize s))) . (intpos 1) = (Exec ((AddTo (GBP,4,(- 1))),(IExec (((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))),P,(Initialize s))))) . (intpos 1) by SCMPDS_5:41
.= s . (intpos 1) by A77, A78, SCMPDS_2:48 ;
hence  (IExec ((if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))),P,(Initialize s))) . (intpos 1) = s . (intpos 1) by A2, A52, SCMPDS_6:83; ::_thesis: ( (IExec ((if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))),P,(Initialize s))) . (intpos 2) = s . (intpos 2) & (IExec ((if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))),P,(Initialize s))) . (intpos m3) = s . (intpos m4) & (IExec ((if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))),P,(Initialize s))) . (intpos m4) = s . (intpos m3) & (IExec ((if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))),P,(Initialize s))) . (intpos 3) = (s . (intpos 3)) + 1 & (IExec ((if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))),P,(Initialize s))) . (intpos 4) = (s . (intpos 4)) - 1 & (IExec ((if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))),P,(Initialize s))) . (intpos 5) = (s . (intpos 5)) - 2 & ( for i being Element of NAT st i >= 8 & i <> m3 & i <> m4 holds
(IExec ((if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))),P,(Initialize s))) . (intpos i) = s . (intpos i) ) )
A79: intpos 2 <> DataLoc (((IExec (((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))),P,(Initialize s))) . GBP),4) by A61, AMI_3:10, SCMP_GCD:1;
(IExec ((((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))),P,(Initialize s))) . (intpos 2) = (Exec ((AddTo (GBP,4,(- 1))),(IExec (((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))),P,(Initialize s))))) . (intpos 2) by SCMPDS_5:41
.= s . (intpos 2) by A50, A79, SCMPDS_2:48 ;
hence  (IExec ((if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))),P,(Initialize s))) . (intpos 2) = s . (intpos 2) by A2, A52, SCMPDS_6:83; ::_thesis: ( (IExec ((if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))),P,(Initialize s))) . (intpos m3) = s . (intpos m4) & (IExec ((if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))),P,(Initialize s))) . (intpos m4) = s . (intpos m3) & (IExec ((if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))),P,(Initialize s))) . (intpos 3) = (s . (intpos 3)) + 1 & (IExec ((if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))),P,(Initialize s))) . (intpos 4) = (s . (intpos 4)) - 1 & (IExec ((if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))),P,(Initialize s))) . (intpos 5) = (s . (intpos 5)) - 2 & ( for i being Element of NAT st i >= 8 & i <> m3 & i <> m4 holds
(IExec ((if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))),P,(Initialize s))) . (intpos i) = s . (intpos i) ) )
(IExec ((((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))),P,(Initialize s))) . (intpos m3) = (Exec ((AddTo (GBP,4,(- 1))),(IExec (((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))),P,(Initialize s))))) . (intpos m3) by SCMPDS_5:41
.= s . (intpos m4) by A43, A66, SCMPDS_2:48 ;
hence  (IExec ((if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))),P,(Initialize s))) . (intpos m3) = s . (intpos m4) by A2, A52, SCMPDS_6:83; ::_thesis: ( (IExec ((if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))),P,(Initialize s))) . (intpos m4) = s . (intpos m3) & (IExec ((if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))),P,(Initialize s))) . (intpos 3) = (s . (intpos 3)) + 1 & (IExec ((if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))),P,(Initialize s))) . (intpos 4) = (s . (intpos 4)) - 1 & (IExec ((if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))),P,(Initialize s))) . (intpos 5) = (s . (intpos 5)) - 2 & ( for i being Element of NAT st i >= 8 & i <> m3 & i <> m4 holds
(IExec ((if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))),P,(Initialize s))) . (intpos i) = s . (intpos i) ) )
(IExec ((((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))),P,(Initialize s))) . (intpos m4) = (Exec ((AddTo (GBP,4,(- 1))),(IExec (((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))),P,(Initialize s))))) . (intpos m4) by SCMPDS_5:41
.= s . (intpos m3) by A46, A64, SCMPDS_2:48 ;
hence  (IExec ((if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))),P,(Initialize s))) . (intpos m4) = s . (intpos m3) by A2, A52, SCMPDS_6:83; ::_thesis: ( (IExec ((if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))),P,(Initialize s))) . (intpos 3) = (s . (intpos 3)) + 1 & (IExec ((if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))),P,(Initialize s))) . (intpos 4) = (s . (intpos 4)) - 1 & (IExec ((if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))),P,(Initialize s))) . (intpos 5) = (s . (intpos 5)) - 2 & ( for i being Element of NAT st i >= 8 & i <> m3 & i <> m4 holds
(IExec ((if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))),P,(Initialize s))) . (intpos i) = s . (intpos i) ) )
A80: DataLoc (((IExec (((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))),P,(Initialize s))) . GBP),4) = intpos (0 + 4) by A61, SCMP_GCD:1;
(IExec ((((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))),P,(Initialize s))) . (intpos 3) = (Exec ((AddTo (GBP,4,(- 1))),(IExec (((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))),P,(Initialize s))))) . (intpos 3) by SCMPDS_5:41
.= m3 + 1 by A37, A72, SCMPDS_2:48 ;
hence  (IExec ((if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))),P,(Initialize s))) . (intpos 3) = (s . (intpos 3)) + 1 by A2, A3, A52, SCMPDS_6:83; ::_thesis: ( (IExec ((if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))),P,(Initialize s))) . (intpos 4) = (s . (intpos 4)) - 1 & (IExec ((if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))),P,(Initialize s))) . (intpos 5) = (s . (intpos 5)) - 2 & ( for i being Element of NAT st i >= 8 & i <> m3 & i <> m4 holds
(IExec ((if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))),P,(Initialize s))) . (intpos i) = s . (intpos i) ) )
(IExec ((((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))),P,(Initialize s))) . (intpos 4) = (Exec ((AddTo (GBP,4,(- 1))),(IExec (((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))),P,(Initialize s))))) . (intpos 4) by SCMPDS_5:41
.= ((IExec (((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))),P,(Initialize s))) . (intpos 4)) + (- 1) by A80, SCMPDS_2:48
.= (s . (intpos 4)) - 1 by A4, A71 ;
hence  (IExec ((if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))),P,(Initialize s))) . (intpos 4) = (s . (intpos 4)) - 1 by A2, A52, SCMPDS_6:83; ::_thesis: ( (IExec ((if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))),P,(Initialize s))) . (intpos 5) = (s . (intpos 5)) - 2 & ( for i being Element of NAT st i >= 8 & i <> m3 & i <> m4 holds
(IExec ((if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))),P,(Initialize s))) . (intpos i) = s . (intpos i) ) )
A81: intpos 5 <> DataLoc (((IExec (((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))),P,(Initialize s))) . GBP),4) by A61, AMI_3:10, SCMP_GCD:1;
(IExec ((((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))),P,(Initialize s))) . (intpos 5) = (Exec ((AddTo (GBP,4,(- 1))),(IExec (((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))),P,(Initialize s))))) . (intpos 5) by SCMPDS_5:41
.= (s . (intpos 5)) - 2 by A58, A81, SCMPDS_2:48 ;
hence  (IExec ((if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))),P,(Initialize s))) . (intpos 5) = (s . (intpos 5)) - 2 by A2, A52, SCMPDS_6:83; ::_thesis: for i being Element of NAT st i >= 8 & i <> m3 & i <> m4 holds
(IExec ((if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))),P,(Initialize s))) . (intpos i) = s . (intpos i)
A82: now__::_thesis:_for_i_being_Element_of_NAT_st_i_>=_8_&_i_<>_m3_&_i_<>_m4_holds_
(Exec_(((GBP,6)_:=_((intpos_4),0)),(Initialize_s)))_._(intpos_i)_=_s_._(intpos_i)
let i be Element of NAT ; ::_thesis: ( i >= 8 & i <> m3 & i <> m4 implies (Exec (((GBP,6) := ((intpos 4),0)),(Initialize s))) . (intpos i) = s . (intpos i) )
assume that
A83: i >= 8 and
 i <> m3 and
 i <> m4 ; ::_thesis: (Exec (((GBP,6) := ((intpos 4),0)),(Initialize s))) . (intpos i) = s . (intpos i)
 i > 6 by A83, XXREAL_0:2;
hence (Exec (((GBP,6) := ((intpos 4),0)),(Initialize s))) . (intpos i) = (Initialize s) . (intpos i) by A9, AMI_3:10, SCMPDS_2:47
.= s . (intpos i) by SCMPDS_5:15 ;
::_thesis: verum
end;
A84: now__::_thesis:_for_i_being_Element_of_NAT_st_i_>=_8_&_i_<>_m3_&_i_<>_m4_holds_
(IExec_((((GBP,6)_:=_((intpos_4),0))_';'_(((intpos_4),0)_:=_((intpos_3),0))),P,(Initialize_s)))_._(intpos_i)_=_s_._(intpos_i)
let i be Element of NAT ; ::_thesis: ( i >= 8 & i <> m3 & i <> m4 implies (IExec ((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))),P,(Initialize s))) . (intpos i) = s . (intpos i) )
assume that
A85: i >= 8 and
A86: i <> m3 and
A87: i <> m4 ; ::_thesis: (IExec ((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))),P,(Initialize s))) . (intpos i) = s . (intpos i)
thus (IExec ((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))),P,(Initialize s))) . (intpos i) = (Exec ((((intpos 4),0) := ((intpos 3),0)),(Exec (((GBP,6) := ((intpos 4),0)),(Initialize s))))) . (intpos i) by SCMPDS_5:42
.= (Exec (((GBP,6) := ((intpos 4),0)),(Initialize s))) . (intpos i) by A13, A87, AMI_3:10, SCMPDS_2:47
.= s . (intpos i) by A82, A85, A86, A87 ; ::_thesis: verum
end;
A88: now__::_thesis:_for_i_being_Element_of_NAT_st_i_>=_8_&_i_<>_m3_&_i_<>_m4_holds_
(IExec_(((((GBP,6)_:=_((intpos_4),0))_';'_(((intpos_4),0)_:=_((intpos_3),0)))_';'_(((intpos_3),0)_:=_(GBP,6))),P,(Initialize_s)))_._(intpos_i)_=_s_._(intpos_i)
let i be Element of NAT ; ::_thesis: ( i >= 8 & i <> m3 & i <> m4 implies (IExec (((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))),P,(Initialize s))) . (intpos i) = s . (intpos i) )
assume that
A89: i >= 8 and
A90: i <> m3 and
A91: i <> m4 ; ::_thesis: (IExec (((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))),P,(Initialize s))) . (intpos i) = s . (intpos i)
thus (IExec (((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))),P,(Initialize s))) . (intpos i) = (Exec ((((intpos 3),0) := (GBP,6)),(IExec ((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))),P,(Initialize s))))) . (intpos i) by SCMPDS_5:41
.= (IExec ((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))),P,(Initialize s))) . (intpos i) by A19, A90, AMI_3:10, SCMPDS_2:47
.= s . (intpos i) by A84, A89, A90, A91 ; ::_thesis: verum
end;
A92: now__::_thesis:_for_i_being_Element_of_NAT_st_i_>=_8_&_i_<>_m3_&_i_<>_m4_holds_
(IExec_((((((GBP,6)_:=_((intpos_4),0))_';'_(((intpos_4),0)_:=_((intpos_3),0)))_';'_(((intpos_3),0)_:=_(GBP,6)))_';'_(AddTo_(GBP,5,(-_2)))),P,(Initialize_s)))_._(intpos_i)_=_s_._(intpos_i)
let i be Element of NAT ; ::_thesis: ( i >= 8 & i <> m3 & i <> m4 implies (IExec ((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))),P,(Initialize s))) . (intpos i) = s . (intpos i) )
assume that
A93: i >= 8 and
A94: i <> m3 and
A95: i <> m4 ; ::_thesis: (IExec ((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))),P,(Initialize s))) . (intpos i) = s . (intpos i)
A96: i > 5 by A93, XXREAL_0:2;
thus (IExec ((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))),P,(Initialize s))) . (intpos i) = (Exec ((AddTo (GBP,5,(- 2))),(IExec (((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))),P,(Initialize s))))) . (intpos i) by SCMPDS_5:41
.= (IExec (((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))),P,(Initialize s))) . (intpos i) by A29, A96, AMI_3:10, SCMPDS_2:48
.= s . (intpos i) by A88, A93, A94, A95 ; ::_thesis: verum
end;
A97: now__::_thesis:_for_i_being_Element_of_NAT_st_i_>=_8_&_i_<>_m3_&_i_<>_m4_holds_
(IExec_(((((((GBP,6)_:=_((intpos_4),0))_';'_(((intpos_4),0)_:=_((intpos_3),0)))_';'_(((intpos_3),0)_:=_(GBP,6)))_';'_(AddTo_(GBP,5,(-_2))))_';'_(AddTo_(GBP,3,1))),P,(Initialize_s)))_._(intpos_i)_=_s_._(intpos_i)
let i be Element of NAT ; ::_thesis: ( i >= 8 & i <> m3 & i <> m4 implies (IExec (((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))),P,(Initialize s))) . (intpos i) = s . (intpos i) )
assume that
A98: i >= 8 and
A99: i <> m3 and
A100: i <> m4 ; ::_thesis: (IExec (((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))),P,(Initialize s))) . (intpos i) = s . (intpos i)
 i > 3 by A98, XXREAL_0:2;
then A101: intpos i <> DataLoc (((IExec ((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))),P,(Initialize s))) . GBP),3) by A30, AMI_3:10, SCMP_GCD:1;
thus (IExec (((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))),P,(Initialize s))) . (intpos i) = (Exec ((AddTo (GBP,3,1)),(IExec ((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))),P,(Initialize s))))) . (intpos i) by SCMPDS_5:41
.= (IExec ((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))),P,(Initialize s))) . (intpos i) by A101, SCMPDS_2:48
.= s . (intpos i) by A92, A98, A99, A100 ; ::_thesis: verum
end;
A102: now__::_thesis:_for_i_being_Element_of_NAT_st_i_>=_8_&_i_<>_m3_&_i_<>_m4_holds_
(IExec_((((((((GBP,6)_:=_((intpos_4),0))_';'_(((intpos_4),0)_:=_((intpos_3),0)))_';'_(((intpos_3),0)_:=_(GBP,6)))_';'_(AddTo_(GBP,5,(-_2))))_';'_(AddTo_(GBP,3,1)))_';'_(AddTo_(GBP,4,(-_1)))),P,(Initialize_s)))_._(intpos_i)_=_s_._(intpos_i)
let i be Element of NAT ; ::_thesis: ( i >= 8 & i <> m3 & i <> m4 implies (IExec ((((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))),P,(Initialize s))) . (intpos i) = s . (intpos i) )
assume that
A103: i >= 8 and
A104: i <> m3 and
A105: i <> m4 ; ::_thesis: (IExec ((((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))),P,(Initialize s))) . (intpos i) = s . (intpos i)
 i > 4 by A103, XXREAL_0:2;
then A106: intpos i <> DataLoc (((IExec (((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))),P,(Initialize s))) . GBP),4) by A61, AMI_3:10, SCMP_GCD:1;
thus (IExec ((((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))),P,(Initialize s))) . (intpos i) = (Exec ((AddTo (GBP,4,(- 1))),(IExec (((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))),P,(Initialize s))))) . (intpos i) by SCMPDS_5:41
.= (IExec (((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))),P,(Initialize s))) . (intpos i) by A106, SCMPDS_2:48
.= s . (intpos i) by A97, A103, A104, A105 ; ::_thesis: verum
end;
hereby  ::_thesis: verum
let i be Element of NAT ; ::_thesis: ( i >= 8 & i <> m3 & i <> m4 implies (IExec ((if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))),P,(Initialize s))) . (intpos i) = s . (intpos i) )
assume that
A107: i >= 8 and
A108: i <> m3 and
A109: i <> m4 ; ::_thesis: (IExec ((if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))),P,(Initialize s))) . (intpos i) = s . (intpos i)
thus (IExec ((if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))),P,(Initialize s))) . (intpos i) = (IExec ((((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))),P,(Initialize s))) . (intpos i) by A2, A52, SCMPDS_6:83
.= s . (intpos i) by A102, A107, A108, A109 ; ::_thesis: verum
end;
end;

Lm20:  for P being Instruction-Sequence of SCMPDS
for s being 0 -started State of SCMPDS
for md, m3 being Element of NAT st s . GBP = 0 & s . (intpos 5) > 0 & s . (intpos 4) = m3 + (s . (intpos 5)) & m3 = (s . (intpos 3)) - 1 & s . (intpos 2) = md & md >= 8 & md <= m3 holds
( ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))) is_closed_on s,P & ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))) is_halting_on s,P & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . GBP = 0 & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 1) = s . (intpos 1) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 2) = md & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 3) >= s . (intpos 3) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 4) <= s . (intpos 4) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 4) >= m3 & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 5) < s . (intpos 5) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 5) >= - 1 & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 4) = (((IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 3)) - 1) + ((IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 5)) & ex n1, n2 being Element of NAT st
( n1 = ((IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 3)) - 1 & n2 = ((IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 4)) + 1 & ( for i being Element of NAT st i >= 8 & i <> n1 & i <> n2 holds
(IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos i) = s . (intpos i) ) & ( ( (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos n1) = s . (intpos n1) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos n2) = s . (intpos n2) ) or ( n1 >= s . (intpos 3) & n2 <= s . (intpos 4) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos n1) = s . (intpos n2) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos n2) = s . (intpos n1) ) ) & ( for i being Element of NAT st s . (intpos 3) <= i & i <= n1 holds
(IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos md) >= (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos i) ) & ( for i being Element of NAT st n2 <= i & i <= s . (intpos 4) holds
(IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos md) <= (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos i) ) ) )
proof
let P be Instruction-Sequence of SCMPDS; ::_thesis: for s being 0 -started State of SCMPDS
for md, m3 being Element of NAT st s . GBP = 0 & s . (intpos 5) > 0 & s . (intpos 4) = m3 + (s . (intpos 5)) & m3 = (s . (intpos 3)) - 1 & s . (intpos 2) = md & md >= 8 & md <= m3 holds
( ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))) is_closed_on s,P & ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))) is_halting_on s,P & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . GBP = 0 & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 1) = s . (intpos 1) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 2) = md & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 3) >= s . (intpos 3) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 4) <= s . (intpos 4) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 4) >= m3 & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 5) < s . (intpos 5) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 5) >= - 1 & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 4) = (((IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 3)) - 1) + ((IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 5)) & ex n1, n2 being Element of NAT st
( n1 = ((IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 3)) - 1 & n2 = ((IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 4)) + 1 & ( for i being Element of NAT st i >= 8 & i <> n1 & i <> n2 holds
(IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos i) = s . (intpos i) ) & ( ( (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos n1) = s . (intpos n1) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos n2) = s . (intpos n2) ) or ( n1 >= s . (intpos 3) & n2 <= s . (intpos 4) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos n1) = s . (intpos n2) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos n2) = s . (intpos n1) ) ) & ( for i being Element of NAT st s . (intpos 3) <= i & i <= n1 holds
(IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos md) >= (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos i) ) & ( for i being Element of NAT st n2 <= i & i <= s . (intpos 4) holds
(IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos md) <= (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos i) ) ) )
set a = GBP ;
let s be 0 -started State of SCMPDS; ::_thesis: for md, m3 being Element of NAT st s . GBP = 0 & s . (intpos 5) > 0 & s . (intpos 4) = m3 + (s . (intpos 5)) & m3 = (s . (intpos 3)) - 1 & s . (intpos 2) = md & md >= 8 & md <= m3 holds
( ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))) is_closed_on s,P & ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))) is_halting_on s,P & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . GBP = 0 & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 1) = s . (intpos 1) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 2) = md & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 3) >= s . (intpos 3) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 4) <= s . (intpos 4) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 4) >= m3 & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 5) < s . (intpos 5) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 5) >= - 1 & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 4) = (((IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 3)) - 1) + ((IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 5)) & ex n1, n2 being Element of NAT st
( n1 = ((IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 3)) - 1 & n2 = ((IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 4)) + 1 & ( for i being Element of NAT st i >= 8 & i <> n1 & i <> n2 holds
(IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos i) = s . (intpos i) ) & ( ( (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos n1) = s . (intpos n1) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos n2) = s . (intpos n2) ) or ( n1 >= s . (intpos 3) & n2 <= s . (intpos 4) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos n1) = s . (intpos n2) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos n2) = s . (intpos n1) ) ) & ( for i being Element of NAT st s . (intpos 3) <= i & i <= n1 holds
(IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos md) >= (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos i) ) & ( for i being Element of NAT st n2 <= i & i <= s . (intpos 4) holds
(IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos md) <= (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos i) ) ) )
let md, m3 be Element of NAT ; ::_thesis: ( s . GBP = 0 & s . (intpos 5) > 0 & s . (intpos 4) = m3 + (s . (intpos 5)) & m3 = (s . (intpos 3)) - 1 & s . (intpos 2) = md & md >= 8 & md <= m3 implies ( ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))) is_closed_on s,P & ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))) is_halting_on s,P & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . GBP = 0 & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 1) = s . (intpos 1) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 2) = md & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 3) >= s . (intpos 3) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 4) <= s . (intpos 4) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 4) >= m3 & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 5) < s . (intpos 5) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 5) >= - 1 & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 4) = (((IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 3)) - 1) + ((IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 5)) & ex n1, n2 being Element of NAT st
( n1 = ((IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 3)) - 1 & n2 = ((IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 4)) + 1 & ( for i being Element of NAT st i >= 8 & i <> n1 & i <> n2 holds
(IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos i) = s . (intpos i) ) & ( ( (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos n1) = s . (intpos n1) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos n2) = s . (intpos n2) ) or ( n1 >= s . (intpos 3) & n2 <= s . (intpos 4) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos n1) = s . (intpos n2) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos n2) = s . (intpos n1) ) ) & ( for i being Element of NAT st s . (intpos 3) <= i & i <= n1 holds
(IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos md) >= (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos i) ) & ( for i being Element of NAT st n2 <= i & i <= s . (intpos 4) holds
(IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos md) <= (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos i) ) ) ) )
I: Initialize s = s by MEMSTR_0:44;
assume that
A1: s . GBP = 0 and
A2: s . (intpos 5) > 0 and
A3: s . (intpos 4) = m3 + (s . (intpos 5)) and
A4: m3 = (s . (intpos 3)) - 1 and
A5: s . (intpos 2) = md and
A6: md >= 8 and
A7: md <= m3 ; ::_thesis: ( ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))) is_closed_on s,P & ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))) is_halting_on s,P & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . GBP = 0 & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 1) = s . (intpos 1) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 2) = md & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 3) >= s . (intpos 3) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 4) <= s . (intpos 4) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 4) >= m3 & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 5) < s . (intpos 5) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 5) >= - 1 & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 4) = (((IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 3)) - 1) + ((IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 5)) & ex n1, n2 being Element of NAT st
( n1 = ((IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 3)) - 1 & n2 = ((IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 4)) + 1 & ( for i being Element of NAT st i >= 8 & i <> n1 & i <> n2 holds
(IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos i) = s . (intpos i) ) & ( ( (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos n1) = s . (intpos n1) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos n2) = s . (intpos n2) ) or ( n1 >= s . (intpos 3) & n2 <= s . (intpos 4) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos n1) = s . (intpos n2) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos n2) = s . (intpos n1) ) ) & ( for i being Element of NAT st s . (intpos 3) <= i & i <= n1 holds
(IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos md) >= (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos i) ) & ( for i being Element of NAT st n2 <= i & i <= s . (intpos 4) holds
(IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos md) <= (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos i) ) ) )
set s1 = IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s);
A8: (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s)) . GBP = (Initialize (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s))) . GBP by SCMPDS_5:15;
A9: (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s)) . (intpos 1) = (Initialize (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s))) . (intpos 1) by SCMPDS_5:15;
A10: (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s)) . (intpos 2) = (Initialize (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s))) . (intpos 2) by SCMPDS_5:15;
A11: (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s)) . (intpos 3) = (Initialize (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s))) . (intpos 3) by SCMPDS_5:15;
A12: (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s)) . (intpos 4) = (Initialize (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s))) . (intpos 4) by SCMPDS_5:15;
A13: (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s)) . (intpos 7) = (Initialize (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s))) . (intpos 7) by SCMPDS_5:15;
A14: m3 >= 8 by A6, A7, XXREAL_0:2;
then consider mE being Element of NAT  such that
A15: mE = (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s)) . (intpos 7) and
A16: (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s)) . (intpos 4) = m3 + mE and
A17: mE <= s . (intpos 5) and
A18: for i being Element of NAT st m3 + mE < i & i <= s . (intpos 4) holds
(IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s)) . (intpos md) < (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s)) . (intpos i) and
A19: ( mE = 0 or (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s)) . (intpos md) >= (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s)) . (intpos (m3 + mE)) ) by A1, A2, A3, A5, A6, Lm9;
A20: (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s)) . GBP = 0 by A1, A2, A3, A5, A6, A14, Lm9;
A21: (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s)) . (intpos 1) = s . (intpos 1) by A1, A2, A3, A5, A6, A14, Lm9;
A22: while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0)))))) is_halting_on s,P by A1, A3, A5, A6, A14, Lm10;
set s2 = IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),P,(Initialize (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s))));
set sm = (m3 + 1) + mE;
set s12 = IExec (((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))),P,s);
set s3 = IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s);
set P12 = P;
set P3 = P;
set m4 = m3 + mE;
A23: (IExec (((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))),P,s)) . GBP = (Initialize (IExec (((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))),P,s))) . GBP by SCMPDS_5:15;
A24: (IExec (((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))),P,s)) . (intpos 1) = (Initialize (IExec (((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))),P,s))) . (intpos 1) by SCMPDS_5:15;
A25: (IExec (((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))),P,s)) . (intpos 2) = (Initialize (IExec (((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))),P,s))) . (intpos 2) by SCMPDS_5:15;
A26: (IExec (((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))),P,s)) . (intpos 3) = (Initialize (IExec (((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))),P,s))) . (intpos 3) by SCMPDS_5:15;
A27: (IExec (((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))),P,s)) . (intpos 4) = (Initialize (IExec (((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))),P,s))) . (intpos 4) by SCMPDS_5:15;
A28: (IExec (((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))),P,s)) . (intpos 5) = (Initialize (IExec (((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))),P,s))) . (intpos 5) by SCMPDS_5:15;
A29: ((IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s)) . (intpos 3)) + ((IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s)) . (intpos 7)) = (m3 + 1) + mE by A1, A2, A3, A4, A5, A6, A14, A15, Lm9;
A30: m3 + mE >= 8 by A14, NAT_1:12;
A31: (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s)) . (intpos 2) = s . (intpos 2) by A1, A2, A3, A5, A6, A14, Lm9;
A32: (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s)) . (intpos 3) = s . (intpos 3) by A1, A2, A3, A5, A6, A14, Lm9;
 (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s)) . (intpos 3) = m3 + 1 by A1, A2, A3, A4, A5, A6, A14, Lm9;
then  m3 < (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s)) . (intpos 3) by XREAL_1:29;
then A33: (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s)) . (intpos 3) >= 8 by A14, XXREAL_0:2;
A34: (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s)) . (intpos 5) = 0 by A1, A2, A3, A5, A6, A14, Lm9;
A35: m3 + mE <= m3 + (s . (intpos 5)) by A17, XREAL_1:6;
A36: while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0)))))) is_closed_on s,P by A1, A3, A5, A6, A14, Lm10;
percases ( mE <= 0 or mE > 0 ) ;
supposeA38: mE <= 0 ; ::_thesis: ( ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))) is_closed_on s,P & ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))) is_halting_on s,P & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . GBP = 0 & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 1) = s . (intpos 1) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 2) = md & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 3) >= s . (intpos 3) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 4) <= s . (intpos 4) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 4) >= m3 & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 5) < s . (intpos 5) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 5) >= - 1 & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 4) = (((IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 3)) - 1) + ((IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 5)) & ex n1, n2 being Element of NAT st
( n1 = ((IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 3)) - 1 & n2 = ((IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 4)) + 1 & ( for i being Element of NAT st i >= 8 & i <> n1 & i <> n2 holds
(IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos i) = s . (intpos i) ) & ( ( (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos n1) = s . (intpos n1) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos n2) = s . (intpos n2) ) or ( n1 >= s . (intpos 3) & n2 <= s . (intpos 4) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos n1) = s . (intpos n2) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos n2) = s . (intpos n1) ) ) & ( for i being Element of NAT st s . (intpos 3) <= i & i <= n1 holds
(IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos md) >= (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos i) ) & ( for i being Element of NAT st n2 <= i & i <= s . (intpos 4) holds
(IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos md) <= (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos i) ) ) )
X1: (Initialize (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s))) . GBP = (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s)) . GBP by SCMPDS_5:15;
X2: (Initialize (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s))) . (DataLoc (((IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s)) . GBP),7)) = (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s)) . (DataLoc (((IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s)) . GBP),7)) by SCMPDS_5:15;
X3: (Initialize (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s))) . (intpos 5) = (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s)) . (intpos 5) by SCMPDS_5:15;
X5: (Initialize (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s))) . (intpos 1) = (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s)) . (intpos 1) by SCMPDS_5:15;
X6: (Initialize (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s))) . (intpos 2) = (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s)) . (intpos 2) by SCMPDS_5:15;
X7: (Initialize (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s))) . (intpos 3) = (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s)) . (intpos 3) by SCMPDS_5:15;
X8: (Initialize (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s))) . (intpos 4) = (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s)) . (intpos 4) by SCMPDS_5:15;
A39: DataLoc (((IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s)) . GBP),7) = intpos (0 + 7) by A20, SCMP_GCD:1;
then A40: while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))) is_halting_on IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s),P by A15, A38, SCMPDS_8:20;
A41: while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))) is_closed_on IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s),P by A15, A38, A39, SCMPDS_8:20;
then A42: (IExec (((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))),P,s)) . (intpos 5) = (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),P,(Initialize (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s))))) . (intpos 5) by A36, A22, A40, SCMPDS_7:30
.= (Initialize (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s))) . (intpos 5) by A15, A38, A39, X1, X2, SCMPDS_8:23
.= 0 by A34, X3 ;
A43: (IExec (((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))),P,s)) . GBP = (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),P,(Initialize (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s))))) . GBP by A36, A22, A41, A40, SCMPDS_7:30
.= (Initialize (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s))) . GBP by A15, A38, A39, X1, X2, SCMPDS_8:23
.= 0 by A20, X1 ;
then A44: DataLoc (((IExec (((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))),P,s)) . GBP),5) = intpos (0 + 5) by SCMP_GCD:1;
A45: (while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))) is_halting_on s,P by A36, A22, A41, A40, I, SCMPDS_7:24;
A46: (while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))) is_closed_on s,P by A36, A22, A41, A40, I, SCMPDS_7:24;
hence  ( ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))) is_closed_on s,P & ((while>0 (GBP,5,((((((GBP