:: SCMPDS_8 semantic presentation

begin

   set A = NAT ;

set D = SCM-Data-Loc ;

theorem :: SCMPDS_8:1
for a being Int_position ex i being Element of NAT st a = intpos i
proof
let a be Int_position; ::_thesis: ex i being Element of NAT st a = intpos i
 a in SCM-Data-Loc by AMI_2:def_16;
then consider x, y being set  such that
A1: x in {1} and
A2: y in NAT and
A3: a = [x,y] by ZFMISC_1:84;
reconsider k = y as Element of NAT by A2;
take  k ; ::_thesis: a = intpos k
thus  intpos k = dl. k by SCMP_GCD:def_1
.= a by A1, A3, TARSKI:def_1 ; ::_thesis: verum
end;

theorem :: SCMPDS_8:2
canceled;

theorem Th3: :: SCMPDS_8:3
for t being State of SCMPDS
for i being Instruction of SCMPDS st InsCode i in {0,4,5,6,14} holds
Initialize t = Initialize (Exec (i,t))
proof
let t be State of SCMPDS; ::_thesis: for i being Instruction of SCMPDS st InsCode i in {0,4,5,6,14} holds
Initialize t = Initialize (Exec (i,t))
let i be Instruction of SCMPDS; ::_thesis: ( InsCode i in {0,4,5,6,14} implies Initialize t = Initialize (Exec (i,t)) )
assume  InsCode i in {0,4,5,6,14} ; ::_thesis: Initialize t = Initialize (Exec (i,t))
then  DataPart (Exec (i,t)) = DataPart t by SCMPDS_7:7;
hence  Initialize t = Initialize (Exec (i,t)) by MEMSTR_0:80; ::_thesis: verum
end;

theorem :: SCMPDS_8:4
canceled;

theorem Th5: :: SCMPDS_8:5
for a being Int_position ex f being Function of (product (the_Values_of SCMPDS)),NAT st
for s being State of SCMPDS holds
( ( s . a <= 0 implies f . s = 0 ) & ( s . a > 0 implies f . s = s . a ) )
proof
let a be Int_position; ::_thesis: ex f being Function of (product (the_Values_of SCMPDS)),NAT st
for s being State of SCMPDS holds
( ( s . a <= 0 implies f . s = 0 ) & ( s . a > 0 implies f . s = s . a ) )
defpred S1[ set , set ] means  ex t being State of SCMPDS st
( t = $1 & ( t . a <= 0 implies $2 = 0 ) & ( t . a > 0 implies $2 = t . a ) );
A1: now__::_thesis:_for_s_being_Element_of_product_(the_Values_of_SCMPDS)_ex_y_being_Element_of_NAT_st_S1[s,y]
let s be Element of product (the_Values_of SCMPDS); ::_thesis: ex y being Element of NAT st S1[y,b2]
percases ( s . a <= 0 or s . a > 0 ) ;
supposeA2: s . a <= 0 ; ::_thesis: ex y being Element of NAT st S1[y,b2]
reconsider y = 0 as Element of NAT ;
take y = y; ::_thesis: S1[s,y]
thus  S1[s,y] by A2; ::_thesis: verum
end;
supposeA3: s . a > 0 ; ::_thesis: ex y being Element of NAT st S1[y,b2]
then reconsider y = s . a as Element of NAT by INT_1:3;
take y = y; ::_thesis: S1[s,y]
thus  S1[s,y] by A3; ::_thesis: verum
end;
end;
end;
consider f being Function of (product (the_Values_of SCMPDS)),NAT such that
A4: for s being Element of product (the_Values_of SCMPDS) holds S1[s,f . s] from FUNCT_2:sch_3(A1);
A5: for s being State of SCMPDS holds S1[s,f . s]
proof
let s be State of SCMPDS; ::_thesis: S1[s,f . s]
reconsider s = s as Element of product (the_Values_of SCMPDS) by CARD_3:107;
 S1[s,f . s] by A4;
hence  S1[s,f . s] ; ::_thesis: verum
end;
take  f ; ::_thesis: for s being State of SCMPDS holds
( ( s . a <= 0 implies f . s = 0 ) & ( s . a > 0 implies f . s = s . a ) )
hereby  ::_thesis: verum
let s be State of SCMPDS; ::_thesis: ( ( s . a <= 0 implies f . s = 0 ) & ( s . a > 0 implies f . s = s . a ) )
 S1[s,f . s] by A5;
hence  ( ( s . a <= 0 implies f . s = 0 ) & ( s . a > 0 implies f . s = s . a ) ) ; ::_thesis: verum
end;
end;

begin

definition
canceled;
let a be Int_position;
let i be Integer;
let I be Program of SCMPDS;
func while<0 (a,i,I) ->  Program of SCMPDS equals :: SCMPDS_8:def 2
(((a,i) >=0_goto ((card I) + 2)) ';' I) ';' (goto (- ((card I) + 1)));
coherence
 (((a,i) >=0_goto ((card I) + 2)) ';' I) ';' (goto (- ((card I) + 1))) is Program of SCMPDS ;
end;

:: deftheorem  SCMPDS_8:def_1_:_
canceled;

:: deftheorem  defines while<0 SCMPDS_8:def_2_:_
 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)));

registration
let I be shiftable Program of SCMPDS;
let a be Int_position;
let i be Integer;
cluster while<0 (a,i,I) ->  shiftable ;
correctness
coherence
while<0 (a,i,I) is shiftable ;
proof
set WHL = while<0 (a,i,I);
set i1 = (a,i) >=0_goto ((card I) + 2);
set PF = (Load ((a,i) >=0_goto ((card I) + 2))) ';' I;
A1: (Load ((a,i) >=0_goto ((card I) + 2))) ';' I = ((a,i) >=0_goto ((card I) + 2)) ';' I by SCMPDS_4:def_2;
then  card ((Load ((a,i) >=0_goto ((card I) + 2))) ';' I) = (card I) + 1 by SCMPDS_6:6;
then  (card ((Load ((a,i) >=0_goto ((card I) + 2))) ';' I)) + (- ((card I) + 1)) = 0 ;
hence  while<0 (a,i,I) is shiftable by A1, SCMPDS_4:23; ::_thesis: verum
end;
end;

registration
let I be halt-free Program of SCMPDS;
let a be Int_position;
let i be Integer;
cluster while<0 (a,i,I) ->  halt-free ;
correctness
coherence
while<0 (a,i,I) is halt-free ;
;
end;

theorem Th6: :: SCMPDS_8:6
for a being Int_position
for i being Integer
for I being Program of SCMPDS holds card (while<0 (a,i,I)) = (card I) + 2
proof
let a be Int_position; ::_thesis: for i being Integer
for I being Program of SCMPDS holds card (while<0 (a,i,I)) = (card I) + 2
let i be Integer; ::_thesis: for I being Program of SCMPDS holds card (while<0 (a,i,I)) = (card I) + 2
let I be Program of SCMPDS; ::_thesis: card (while<0 (a,i,I)) = (card I) + 2
set i1 = (a,i) >=0_goto ((card I) + 2);
set I4 = ((a,i) >=0_goto ((card I) + 2)) ';' I;
thus  card (while<0 (a,i,I)) = (card (((a,i) >=0_goto ((card I) + 2)) ';' I)) + 1 by SCMP_GCD:4
.= ((card I) + 1) + 1 by SCMPDS_6:6
.= (card I) + 2 ; ::_thesis: verum
end;

Lm1:  for a being Int_position
for i being Integer
for I being Program of SCMPDS holds card (stop (while<0 (a,i,I))) = (card I) + 3
proof
let a be Int_position; ::_thesis: for i being Integer
for I being Program of SCMPDS holds card (stop (while<0 (a,i,I))) = (card I) + 3
let i be Integer; ::_thesis: for I being Program of SCMPDS holds card (stop (while<0 (a,i,I))) = (card I) + 3
let I be Program of SCMPDS; ::_thesis: card (stop (while<0 (a,i,I))) = (card I) + 3
thus  card (stop (while<0 (a,i,I))) = (card (while<0 (a,i,I))) + 1 by COMPOS_1:55
.= ((card I) + 2) + 1 by Th6
.= (card I) + 3 ; ::_thesis: verum
end;

theorem Th7: :: SCMPDS_8:7
for a being Int_position
for i being Integer
for m being Element of NAT
for I being Program of SCMPDS holds
( m < (card I) + 2 iff m in dom (while<0 (a,i,I)) )
proof
let a be Int_position; ::_thesis: for i being Integer
for m being Element of NAT
for I being Program of SCMPDS holds
( m < (card I) + 2 iff m in dom (while<0 (a,i,I)) )
let i be Integer; ::_thesis: for m being Element of NAT
for I being Program of SCMPDS holds
( m < (card I) + 2 iff m in dom (while<0 (a,i,I)) )
let m be Element of NAT ; ::_thesis: for I being Program of SCMPDS holds
( m < (card I) + 2 iff m in dom (while<0 (a,i,I)) )
let I be Program of SCMPDS; ::_thesis: ( m < (card I) + 2 iff m in dom (while<0 (a,i,I)) )
 card (while<0 (a,i,I)) = (card I) + 2 by Th6;
hence  ( m < (card I) + 2 iff m in dom (while<0 (a,i,I)) ) by AFINSQ_1:66; ::_thesis: verum
end;

theorem Th8: :: SCMPDS_8:8
for a being Int_position
for i being Integer
for I being Program of SCMPDS holds
( (while<0 (a,i,I)) . 0 = (a,i) >=0_goto ((card I) + 2) & (while<0 (a,i,I)) . ((card I) + 1) = 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)) . 0 = (a,i) >=0_goto ((card I) + 2) & (while<0 (a,i,I)) . ((card I) + 1) = goto (- ((card I) + 1)) )
let i be Integer; ::_thesis: for I being Program of SCMPDS holds
( (while<0 (a,i,I)) . 0 = (a,i) >=0_goto ((card I) + 2) & (while<0 (a,i,I)) . ((card I) + 1) = goto (- ((card I) + 1)) )
let I be Program of SCMPDS; ::_thesis: ( (while<0 (a,i,I)) . 0 = (a,i) >=0_goto ((card I) + 2) & (while<0 (a,i,I)) . ((card I) + 1) = goto (- ((card I) + 1)) )
set i1 = (a,i) >=0_goto ((card I) + 2);
set i2 = goto (- ((card I) + 1));
set I4 = ((a,i) >=0_goto ((card I) + 2)) ';' I;
set J5 = I ';' (goto (- ((card I) + 1)));
set FLOOP = while<0 (a,i,I);
 while<0 (a,i,I) = ((a,i) >=0_goto ((card I) + 2)) ';' (I ';' (goto (- ((card I) + 1)))) by SCMPDS_4:15;
hence  (while<0 (a,i,I)) . 0 = (a,i) >=0_goto ((card I) + 2) by SCMPDS_6:7; ::_thesis: (while<0 (a,i,I)) . ((card I) + 1) = goto (- ((card I) + 1))
 card (((a,i) >=0_goto ((card I) + 2)) ';' I) = (card I) + 1 by SCMPDS_6:6;
hence  (while<0 (a,i,I)) . ((card I) + 1) = goto (- ((card I) + 1)) by SCMP_GCD:6; ::_thesis: verum
end;


theorem Th9: :: SCMPDS_8:9
for P being Instruction-Sequence of SCMPDS
for s being State of SCMPDS
for I being Program of SCMPDS
for a being Int_position
for i being Integer st s . (DataLoc ((s . a),i)) >= 0 holds
( while<0 (a,i,I) is_closed_on s,P & while<0 (a,i,I) is_halting_on s,P )
proof
let P be Instruction-Sequence of SCMPDS; ::_thesis: for s being State of SCMPDS
for I being Program of SCMPDS
for a being Int_position
for i being Integer st s . (DataLoc ((s . a),i)) >= 0 holds
( while<0 (a,i,I) is_closed_on s,P & while<0 (a,i,I) is_halting_on s,P )
let s be State of SCMPDS; ::_thesis: for I being Program of SCMPDS
for a being Int_position
for i being Integer st s . (DataLoc ((s . a),i)) >= 0 holds
( while<0 (a,i,I) is_closed_on s,P & while<0 (a,i,I) is_halting_on s,P )
let I be Program of SCMPDS; ::_thesis: for a being Int_position
for i being Integer st s . (DataLoc ((s . a),i)) >= 0 holds
( while<0 (a,i,I) is_closed_on s,P & while<0 (a,i,I) is_halting_on s,P )
let a be Int_position; ::_thesis: for i being Integer st s . (DataLoc ((s . a),i)) >= 0 holds
( while<0 (a,i,I) is_closed_on s,P & while<0 (a,i,I) is_halting_on s,P )
let i be Integer; ::_thesis: ( s . (DataLoc ((s . a),i)) >= 0 implies ( while<0 (a,i,I) is_closed_on s,P & while<0 (a,i,I) is_halting_on s,P ) )
set d1 = DataLoc ((s . a),i);
assume A1: s . (DataLoc ((s . a),i)) >= 0 ; ::_thesis: ( while<0 (a,i,I) is_closed_on s,P & while<0 (a,i,I) is_halting_on s,P )
set i1 = (a,i) >=0_goto ((card I) + 2);
set i2 = goto (- ((card I) + 1));
set WHL = while<0 (a,i,I);
set pWHL = stop (while<0 (a,i,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)));
A3: IC (Initialize s) = 0 by MEMSTR_0:def_11;
A4: 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 A5: (Initialize s) . (DataLoc (((Initialize s) . a),i)) = (Initialize s) . (DataLoc ((s . a),i)) by FUNCT_4:11
.= s . (DataLoc ((s . a),i)) by A4, FUNCT_4:11 ;
A6: while<0 (a,i,I) = ((a,i) >=0_goto ((card I) + 2)) ';' (I ';' (goto (- ((card I) + 1)))) by SCMPDS_4:15;
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 A6, SCMPDS_6:11 ;
then A7: IC (Comput ((P +* (stop (while<0 (a,i,I)))),(Initialize s),1)) = ICplusConst ((Initialize s),((card I) + 2)) by A1, A5, SCMPDS_2:57
.= 0 + ((card I) + 2) by A3, SCMPDS_6:12 ;
A8: card (while<0 (a,i,I)) = (card I) + 2 by Th6;
then A9: (card I) + 2 in dom (stop (while<0 (a,i,I))) by COMPOS_1:64;
 stop (while<0 (a,i,I)) c= P +* (stop (while<0 (a,i,I))) by FUNCT_4:25;
then (P +* (stop (while<0 (a,i,I)))) . ((card I) + 2) = (stop (while<0 (a,i,I))) . ((card I) + 2) by A9, GRFUNC_1:2
.= halt SCMPDS by A8, COMPOS_1:64 ;
then A11: CurInstr ((P +* (stop (while<0 (a,i,I)))),(Comput ((P +* (stop (while<0 (a,i,I)))),(Initialize s),1))) = halt SCMPDS by A7, PBOOLE:143;
now__::_thesis:_for_k_being_Element_of_NAT_holds_IC_(Comput_((P_+*_(stop_(while<0_(a,i,I)))),(Initialize_s),k))_in_dom_(stop_(while<0_(a,i,I)))
let k be Element of NAT ; ::_thesis: IC (Comput ((P +* (stop (while<0 (a,i,I)))),(Initialize s),b1)) in dom (stop (while<0 (a,i,I)))
percases ( 0 < k or k = 0 ) ;
suppose 0 < k ; ::_thesis: IC (Comput ((P +* (stop (while<0 (a,i,I)))),(Initialize s),b1)) in dom (stop (while<0 (a,i,I)))
then  1 + 0 <= k by INT_1:7;
hence  IC (Comput ((P +* (stop (while<0 (a,i,I)))),(Initialize s),k)) in dom (stop (while<0 (a,i,I))) by A9, A7, A11, EXTPRO_1:5; ::_thesis: verum
end;
suppose k = 0 ; ::_thesis: IC (Comput ((P +* (stop (while<0 (a,i,I)))),(Initialize s),b1)) in dom (stop (while<0 (a,i,I)))
then  Comput ((P +* (stop (while<0 (a,i,I)))),(Initialize s),k) = Initialize s by EXTPRO_1:2;
hence  IC (Comput ((P +* (stop (while<0 (a,i,I)))),(Initialize s),k)) in dom (stop (while<0 (a,i,I))) by A3, COMPOS_1:36; ::_thesis: verum
end;
end;
end;
hence  while<0 (a,i,I) is_closed_on s,P by SCMPDS_6:def_2; ::_thesis: while<0 (a,i,I) is_halting_on s,P
 P +* (stop (while<0 (a,i,I))) halts_on Initialize s by A11, EXTPRO_1:29;
hence  while<0 (a,i,I) is_halting_on s,P by SCMPDS_6:def_3; ::_thesis: verum
end;

theorem Th10: :: SCMPDS_8:10
for P being Instruction-Sequence of SCMPDS
for s being 0 -started State of SCMPDS
for I being Program of SCMPDS
for a, c being Int_position
for i being Integer st s . (DataLoc ((s . a),i)) >= 0 holds
IExec ((while<0 (a,i,I)),P,s) = s +* (Start-At (((card I) + 2),SCMPDS))
proof
let P be Instruction-Sequence of SCMPDS; ::_thesis: for s being 0 -started State of SCMPDS
for I being Program of SCMPDS
for a, c being Int_position
for i being Integer st s . (DataLoc ((s . a),i)) >= 0 holds
IExec ((while<0 (a,i,I)),P,s) = s +* (Start-At (((card I) + 2),SCMPDS))
let s be 0 -started State of SCMPDS; ::_thesis: for I being Program of SCMPDS
for a, c being Int_position
for i being Integer st s . (DataLoc ((s . a),i)) >= 0 holds
IExec ((while<0 (a,i,I)),P,s) = s +* (Start-At (((card I) + 2),SCMPDS))
let I be Program of SCMPDS; ::_thesis: for a, c being Int_position
for i being Integer st s . (DataLoc ((s . a),i)) >= 0 holds
IExec ((while<0 (a,i,I)),P,s) = s +* (Start-At (((card I) + 2),SCMPDS))
let a, c be Int_position; ::_thesis: for i being Integer st s . (DataLoc ((s . a),i)) >= 0 holds
IExec ((while<0 (a,i,I)),P,s) = s +* (Start-At (((card I) + 2),SCMPDS))
let i be Integer; ::_thesis: ( s . (DataLoc ((s . a),i)) >= 0 implies IExec ((while<0 (a,i,I)),P,s) = s +* (Start-At (((card I) + 2),SCMPDS)) )
set d1 = DataLoc ((s . a),i);
set WHL = while<0 (a,i,I);
set pWHL = stop (while<0 (a,i,I));
set P3 = P +* (stop (while<0 (a,i,I)));
set s4 = Comput ((P +* (stop (while<0 (a,i,I)))),s,1);
set P4 = P +* (stop (while<0 (a,i,I)));
set i1 = (a,i) >=0_goto ((card I) + 2);
set i2 = goto (- ((card I) + 1));
set SAl = Start-At (((card I) + 2),SCMPDS);
I: Initialize s = s by MEMSTR_0:44;
A2: IC s = 0 by MEMSTR_0:def_11;
A3: while<0 (a,i,I) = ((a,i) >=0_goto ((card I) + 2)) ';' (I ';' (goto (- ((card I) + 1)))) by SCMPDS_4:15;
A15: CurInstr ((P +* (stop (while<0 (a,i,I)))),s) = (a,i) >=0_goto ((card I) + 2) by A3, I, SCMPDS_6:11;
A6: Comput ((P +* (stop (while<0 (a,i,I)))),s,(0 + 1)) = Following ((P +* (stop (while<0 (a,i,I)))),(Comput ((P +* (stop (while<0 (a,i,I)))),s,0))) by EXTPRO_1:3
.= Following ((P +* (stop (while<0 (a,i,I)))),s)
.= Exec (((a,i) >=0_goto ((card I) + 2)),s) by A3, I, SCMPDS_6:11 ;
A7: stop (while<0 (a,i,I)) c= P +* (stop (while<0 (a,i,I))) by FUNCT_4:25;
A8: IExec ((while<0 (a,i,I)),P,s) = Result ((P +* (stop (while<0 (a,i,I)))),s) by SCMPDS_4:def_5;
assume  s . (DataLoc ((s . a),i)) >= 0 ; ::_thesis: IExec ((while<0 (a,i,I)),P,s) = s +* (Start-At (((card I) + 2),SCMPDS))
then A9: IC (Comput ((P +* (stop (while<0 (a,i,I)))),s,1)) = ICplusConst (s,((card I) + 2)) by A6, SCMPDS_2:57
.= 0 + ((card I) + 2) by A2, SCMPDS_6:12 ;
A12: card (while<0 (a,i,I)) = (card I) + 2 by Th6;
then  (card I) + 2 in dom (stop (while<0 (a,i,I))) by COMPOS_1:64;
then (P +* (stop (while<0 (a,i,I)))) . ((card I) + 2) = (stop (while<0 (a,i,I))) . ((card I) + 2) by A7, GRFUNC_1:2
.= halt SCMPDS by A12, COMPOS_1:64 ;
then A13: CurInstr ((P +* (stop (while<0 (a,i,I)))),(Comput ((P +* (stop (while<0 (a,i,I)))),s,1))) = halt SCMPDS by A9, PBOOLE:143;
then A14: P +* (stop (while<0 (a,i,I))) halts_on s by EXTPRO_1:29;
now__::_thesis:_for_l_being_Element_of_NAT_st_l_<_0_+_1_holds_
CurInstr_((P_+*_(stop_(while<0_(a,i,I)))),(Comput_((P_+*_(stop_(while<0_(a,i,I)))),s,l)))_<>_halt_SCMPDS
let l be Element of NAT ; ::_thesis: ( l < 0 + 1 implies CurInstr ((P +* (stop (while<0 (a,i,I)))),(Comput ((P +* (stop (while<0 (a,i,I)))),s,l))) <> halt SCMPDS )
A16: Comput ((P +* (stop (while<0 (a,i,I)))),s,0) = s ;
assume  l < 0 + 1 ; ::_thesis: CurInstr ((P +* (stop (while<0 (a,i,I)))),(Comput ((P +* (stop (while<0 (a,i,I)))),s,l))) <> halt SCMPDS
then  l = 0 by NAT_1:13;
hence  CurInstr ((P +* (stop (while<0 (a,i,I)))),(Comput ((P +* (stop (while<0 (a,i,I)))),s,l))) <> halt SCMPDS by A15, A16; ::_thesis: verum
end;
then  for l being Element of NAT st CurInstr ((P +* (stop (while<0 (a,i,I)))),(Comput ((P +* (stop (while<0 (a,i,I)))),s,l))) = halt SCMPDS holds
1 <= l ;
then B17: LifeSpan ((P +* (stop (while<0 (a,i,I)))),s) = 1 by A13, A14, EXTPRO_1:def_15;
A19: now__::_thesis:_for_x_being_set_st_x_in_dom_(IExec_((while<0_(a,i,I)),P,s))_holds_
(IExec_((while<0_(a,i,I)),P,s))_._x_=_(s_+*_(Start-At_(((card_I)_+_2),SCMPDS)))_._x
let x be set ; ::_thesis: ( x in dom (IExec ((while<0 (a,i,I)),P,s)) implies (IExec ((while<0 (a,i,I)),P,s)) . b1 = (s +* (Start-At (((card I) + 2),SCMPDS))) . b1 )
A20: dom (Start-At (((card I) + 2),SCMPDS)) = {(IC )} by FUNCOP_1:13;
assume A21: x in dom (IExec ((while<0 (a,i,I)),P,s)) ; ::_thesis: (IExec ((while<0 (a,i,I)),P,s)) . b1 = (s +* (Start-At (((card I) + 2),SCMPDS))) . b1
percases ( x is Int_position or x = IC ) by A21, SCMPDS_4:6;
supposeA22: x is Int_position ; ::_thesis: (IExec ((while<0 (a,i,I)),P,s)) . b1 = (s +* (Start-At (((card I) + 2),SCMPDS))) . b1
then  x <> IC by SCMPDS_2:43;
then A23: not x in dom (Start-At (((card I) + 2),SCMPDS)) by A20, TARSKI:def_1;
thus (IExec ((while<0 (a,i,I)),P,s)) . x = (Comput ((P +* (stop (while<0 (a,i,I)))),s,1)) . x by B17, A8, A14, EXTPRO_1:23
.= s . x by A6, A22, SCMPDS_2:57
.= (s +* (Start-At (((card I) + 2),SCMPDS))) . x by A23, FUNCT_4:11 ; ::_thesis: verum
end;
supposeA24: x = IC ; ::_thesis: (IExec ((while<0 (a,i,I)),P,s)) . b1 = (s +* (Start-At (((card I) + 2),SCMPDS))) . b1
hence (IExec ((while<0 (a,i,I)),P,s)) . x = (card I) + 2 by A9, B17, A8, A14, EXTPRO_1:23
.= (s +* (Start-At (((card I) + 2),SCMPDS))) . x by A24, FUNCT_4:113 ;
::_thesis: verum
end;
end;
end;
dom (IExec ((while<0 (a,i,I)),P,s)) = the carrier of SCMPDS by PARTFUN1:def_2
.= dom (s +* (Start-At (((card I) + 2),SCMPDS))) by PARTFUN1:def_2 ;
hence  IExec ((while<0 (a,i,I)),P,s) = s +* (Start-At (((card I) + 2),SCMPDS)) by A19, FUNCT_1:2; ::_thesis: verum
end;

theorem :: SCMPDS_8:11
for P being Instruction-Sequence of SCMPDS
for s being 0 -started State of SCMPDS
for I being Program of SCMPDS
for a being Int_position
for i being Integer st s . (DataLoc ((s . a),i)) >= 0 holds
IC (IExec ((while<0 (a,i,I)),P,s)) = (card I) + 2
proof
let P be Instruction-Sequence of SCMPDS; ::_thesis: for s being 0 -started State of SCMPDS
for I being Program of SCMPDS
for a being Int_position
for i being Integer st s . (DataLoc ((s . a),i)) >= 0 holds
IC (IExec ((while<0 (a,i,I)),P,s)) = (card I) + 2
let s be 0 -started State of SCMPDS; ::_thesis: for I being Program of SCMPDS
for a being Int_position
for i being Integer st s . (DataLoc ((s . a),i)) >= 0 holds
IC (IExec ((while<0 (a,i,I)),P,s)) = (card I) + 2
let I be Program of SCMPDS; ::_thesis: for a being Int_position
for i being Integer st s . (DataLoc ((s . a),i)) >= 0 holds
IC (IExec ((while<0 (a,i,I)),P,s)) = (card I) + 2
let a be Int_position; ::_thesis: for i being Integer st s . (DataLoc ((s . a),i)) >= 0 holds
IC (IExec ((while<0 (a,i,I)),P,s)) = (card I) + 2
let i be Integer; ::_thesis: ( s . (DataLoc ((s . a),i)) >= 0 implies IC (IExec ((while<0 (a,i,I)),P,s)) = (card I) + 2 )
assume  s . (DataLoc ((s . a),i)) >= 0 ; ::_thesis: IC (IExec ((while<0 (a,i,I)),P,s)) = (card I) + 2
then  IExec ((while<0 (a,i,I)),P,s) = s +* (Start-At (((card I) + 2),SCMPDS)) by Th10;
hence  IC (IExec ((while<0 (a,i,I)),P,s)) = (card I) + 2 by FUNCT_4:113; ::_thesis: verum
end;

theorem :: SCMPDS_8:12
for P being Instruction-Sequence of SCMPDS
for s being 0 -started State of SCMPDS
for I being Program of SCMPDS
for a, b being Int_position
for i being Integer st s . (DataLoc ((s . a),i)) >= 0 holds
(IExec ((while<0 (a,i,I)),P,s)) . b = s . b
proof
let P be Instruction-Sequence of SCMPDS; ::_thesis: for s being 0 -started State of SCMPDS
for I being Program of SCMPDS
for a, b being Int_position
for i being Integer st s . (DataLoc ((s . a),i)) >= 0 holds
(IExec ((while<0 (a,i,I)),P,s)) . b = s . b
let s be 0 -started State of SCMPDS; ::_thesis: for I being Program of SCMPDS
for a, b being Int_position
for i being Integer st s . (DataLoc ((s . a),i)) >= 0 holds
(IExec ((while<0 (a,i,I)),P,s)) . b = s . b
let I be Program of SCMPDS; ::_thesis: for a, b being Int_position
for i being Integer st s . (DataLoc ((s . a),i)) >= 0 holds
(IExec ((while<0 (a,i,I)),P,s)) . b = s . b
let a, b be Int_position; ::_thesis: for i being Integer st s . (DataLoc ((s . a),i)) >= 0 holds
(IExec ((while<0 (a,i,I)),P,s)) . b = s . b
let i be Integer; ::_thesis: ( s . (DataLoc ((s . a),i)) >= 0 implies (IExec ((while<0 (a,i,I)),P,s)) . b = s . b )
assume  s . (DataLoc ((s . a),i)) >= 0 ; ::_thesis: (IExec ((while<0 (a,i,I)),P,s)) . b = s . b
then A1: IExec ((while<0 (a,i,I)),P,s) = s +* (Start-At (((card I) + 2),SCMPDS)) by Th10;
 not b in dom (Start-At (((card I) + 2),SCMPDS)) by SCMPDS_4:18;
hence  (IExec ((while<0 (a,i,I)),P,s)) . b = s . b by A1, FUNCT_4:11; ::_thesis: verum
end;

Lm2:  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) ';' (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;

scheme :: SCMPDS_8:sch 1
WhileLHalt{ F1( State of SCMPDS) ->  Element of NAT , F2() ->  0 -started State of SCMPDS, F3() ->  Instruction-Sequence of SCMPDS, F4() ->  halt-free shiftable Program of SCMPDS, F5() ->  Int_position, F6() ->  Integer, P1[ State of SCMPDS] } :
 ( while<0 (F5(),F6(),F4()) is_closed_on F2(),F3() & while<0 (F5(),F6(),F4()) is_halting_on F2(),F3() )
provided
A2: for t being 0 -started State of SCMPDS st P1[t] & F1(t) = 0 holds
t . (DataLoc ((F2() . F5()),F6())) >= 0 and
A3: P1[F2()] and
A4: for t being 0 -started State of SCMPDS
for Q being Instruction-Sequence of SCMPDS st P1[t] & t . F5() = F2() . F5() & t . (DataLoc ((F2() . F5()),F6())) < 0 holds
( (IExec (F4(),Q,t)) . F5() = t . F5() & F4() is_closed_on t,Q & F4() is_halting_on t,Q & F1((Initialize (IExec (F4(),Q,t)))) < F1(t) & P1[ Initialize (IExec (F4(),Q,t))] )
proof
set i1 = (F5(),F6()) >=0_goto ((card F4()) + 2);
set i2 = goto (- ((card F4()) + 1));
set WHL = while<0 (F5(),F6(),F4());
set pWHL = stop (while<0 (F5(),F6(),F4()));
set pI = stop F4();
set b = DataLoc ((F2() . F5()),F6());
defpred S1[ Element of NAT ] means  for t being 0 -started State of SCMPDS
for Q being Instruction-Sequence of SCMPDS st F1(t) <= $1 & P1[t] & t . F5() = F2() . F5() holds
( while<0 (F5(),F6(),F4()) is_closed_on t,Q & while<0 (F5(),F6(),F4()) is_halting_on t,Q );
A5: for k being Element of NAT st S1[k] holds
S1[k + 1]
proof
let k be Element of NAT ; ::_thesis: ( S1[k] implies S1[k + 1] )
assume A6: S1[k] ; ::_thesis: S1[k + 1]
let t be 0 -started State of SCMPDS; ::_thesis: for Q being Instruction-Sequence of SCMPDS st F1(t) <= k + 1 & P1[t] & t . F5() = F2() . F5() holds
( while<0 (F5(),F6(),F4()) is_closed_on t,Q & while<0 (F5(),F6(),F4()) is_halting_on t,Q )
let Q be Instruction-Sequence of SCMPDS; ::_thesis: ( F1(t) <= k + 1 & P1[t] & t . F5() = F2() . F5() implies ( while<0 (F5(),F6(),F4()) is_closed_on t,Q & while<0 (F5(),F6(),F4()) is_halting_on t,Q ) )
T: Initialize t = t by MEMSTR_0:44;
assume A7: F1(t) <= k + 1 ; ::_thesis: ( not P1[t] or not t . F5() = F2() . F5() or ( while<0 (F5(),F6(),F4()) is_closed_on t,Q & while<0 (F5(),F6(),F4()) is_halting_on t,Q ) )
assume A8: P1[t] ; ::_thesis: ( not t . F5() = F2() . F5() or ( while<0 (F5(),F6(),F4()) is_closed_on t,Q & while<0 (F5(),F6(),F4()) is_halting_on t,Q ) )
assume A9: t . F5() = F2() . F5() ; ::_thesis: ( while<0 (F5(),F6(),F4()) is_closed_on t,Q & while<0 (F5(),F6(),F4()) is_halting_on t,Q )
percases ( t . (DataLoc ((F2() . F5()),F6())) >= 0 or t . (DataLoc ((F2() . F5()),F6())) < 0 ) ;
suppose t . (DataLoc ((F2() . F5()),F6())) >= 0 ; ::_thesis: ( while<0 (F5(),F6(),F4()) is_closed_on t,Q & while<0 (F5(),F6(),F4()) is_halting_on t,Q )
hence  ( while<0 (F5(),F6(),F4()) is_closed_on t,Q & while<0 (F5(),F6(),F4()) is_halting_on t,Q ) by A9, Th9; ::_thesis: verum
end;
supposeA10: t . (DataLoc ((F2() . F5()),F6())) < 0 ; ::_thesis: ( while<0 (F5(),F6(),F4()) is_closed_on t,Q & while<0 (F5(),F6(),F4()) is_halting_on t,Q )
A14: 0 in dom (stop (while<0 (F5(),F6(),F4()))) by COMPOS_1:36;
A17: while<0 (F5(),F6(),F4()) = ((F5(),F6()) >=0_goto ((card F4()) + 2)) ';' (F4() ';' (goto (- ((card F4()) + 1)))) by SCMPDS_4:15;
set Q2 = Q +* (stop F4());
set Q3 = Q +* (stop (while<0 (F5(),F6(),F4())));
set t4 = Comput ((Q +* (stop (while<0 (F5(),F6(),F4())))),t,1);
set Q4 = Q +* (stop (while<0 (F5(),F6(),F4())));
A20: stop F4() c= Q +* (stop F4()) by FUNCT_4:25;
set m2 = LifeSpan ((Q +* (stop F4())),t);
set t5 = Comput ((Q +* (stop (while<0 (F5(),F6(),F4())))),(Comput ((Q +* (stop (while<0 (F5(),F6(),F4())))),t,1)),(LifeSpan ((Q +* (stop F4())),t)));
set Q5 = Q +* (stop (while<0 (F5(),F6(),F4())));
set l1 = (card F4()) + 1;
A21: IC t = 0 by MEMSTR_0:def_11;
set m3 = (LifeSpan ((Q +* (stop F4())),t)) + 1;
set t6 = Comput ((Q +* (stop (while<0 (F5(),F6(),F4())))),t,((LifeSpan ((Q +* (stop F4())),t)) + 1));
set Q6 = Q +* (stop (while<0 (F5(),F6(),F4())));
set t7 = Comput ((Q +* (stop (while<0 (F5(),F6(),F4())))),t,(((LifeSpan ((Q +* (stop F4())),t)) + 1) + 1));
set Q7 = Q +* (stop (while<0 (F5(),F6(),F4())));
 (card F4()) + 1 < (card F4()) + 2 by XREAL_1:6;
then A22: (card F4()) + 1 in dom (while<0 (F5(),F6(),F4())) by Th7;
AA: stop (while<0 (F5(),F6(),F4())) c= Q +* (stop (while<0 (F5(),F6(),F4()))) by FUNCT_4:25;
 while<0 (F5(),F6(),F4()) c= stop (while<0 (F5(),F6(),F4())) by AFINSQ_1:74;
then A23: while<0 (F5(),F6(),F4()) c= Q +* (stop (while<0 (F5(),F6(),F4()))) by AA, XBOOLE_1:1;
 Shift (F4(),1) c= while<0 (F5(),F6(),F4()) by Lm2;
then A24: Shift (F4(),1) c= Q +* (stop (while<0 (F5(),F6(),F4()))) by A23, XBOOLE_1:1;
A25: Comput ((Q +* (stop (while<0 (F5(),F6(),F4())))),t,(0 + 1)) = Following ((Q +* (stop (while<0 (F5(),F6(),F4())))),(Comput ((Q +* (stop (while<0 (F5(),F6(),F4())))),t,0))) by EXTPRO_1:3
.= Following ((Q +* (stop (while<0 (F5(),F6(),F4())))),t)
.= Exec (((F5(),F6()) >=0_goto ((card F4()) + 2)),t) by A17, T, SCMPDS_6:11 ;
 for a being Int_position holds t . a = (Comput ((Q +* (stop (while<0 (F5(),F6(),F4())))),t,1)) . a by A25, SCMPDS_2:57;
then A27: DataPart t = DataPart (Comput ((Q +* (stop (while<0 (F5(),F6(),F4())))),t,1)) by SCMPDS_4:8;
 F4() is_halting_on t,Q by A4, A8, A9, A10;
then A28: Q +* (stop F4()) halts_on t by T, SCMPDS_6:def_3;
 (Q +* (stop F4())) +* (stop F4()) halts_on t by A28;
then A30: F4() is_halting_on t,Q +* (stop F4()) by T, SCMPDS_6:def_3;
A31: IExec (F4(),Q,t) = Result ((Q +* (stop F4())),t) by SCMPDS_4:def_5;
A32: P1[ Initialize (IExec (F4(),Q,t))] by A4, A8, A9, A10;
A33: F4() is_closed_on t,Q by A4, A8, A9, A10;
A34: F4() is_closed_on t,Q +* (stop F4()) by A4, A8, A9, A10;
A35: IC (Comput ((Q +* (stop (while<0 (F5(),F6(),F4())))),t,1)) = succ (IC t) by A10, A25, A9, SCMPDS_2:57
.= 0 + 1 by A21 ;
then A36: IC (Comput ((Q +* (stop (while<0 (F5(),F6(),F4())))),(Comput ((Q +* (stop (while<0 (F5(),F6(),F4())))),t,1)),(LifeSpan ((Q +* (stop F4())),t)))) = (card F4()) + 1 by A20, A30, A34, A27, A24, SCMPDS_7:18;
A37: (Q +* (stop (while<0 (F5(),F6(),F4())))) /. (IC (Comput ((Q +* (stop (while<0 (F5(),F6(),F4())))),t,((LifeSpan ((Q +* (stop F4())),t)) + 1)))) = (Q +* (stop (while<0 (F5(),F6(),F4())))) . (IC (Comput ((Q +* (stop (while<0 (F5(),F6(),F4())))),t,((LifeSpan ((Q +* (stop F4())),t)) + 1)))) by PBOOLE:143;
A38: Comput ((Q +* (stop (while<0 (F5(),F6(),F4())))),t,((LifeSpan ((Q +* (stop F4())),t)) + 1)) = Comput ((Q +* (stop (while<0 (F5(),F6(),F4())))),(Comput ((Q +* (stop (while<0 (F5(),F6(),F4())))),t,1)),(LifeSpan ((Q +* (stop F4())),t))) by EXTPRO_1:4;
then A39: CurInstr ((Q +* (stop (while<0 (F5(),F6(),F4())))),(Comput ((Q +* (stop (while<0 (F5(),F6(),F4())))),t,((LifeSpan ((Q +* (stop F4())),t)) + 1)))) = (Q +* (stop (while<0 (F5(),F6(),F4())))) . ((card F4()) + 1) by A20, A30, A34, A35, A27, A24, A37, SCMPDS_7:18
.= (while<0 (F5(),F6(),F4())) . ((card F4()) + 1) by A22, A23, GRFUNC_1:2
.= goto (- ((card F4()) + 1)) by Th8 ;
A41: Comput ((Q +* (stop (while<0 (F5(),F6(),F4())))),t,(((LifeSpan ((Q +* (stop F4())),t)) + 1) + 1)) = Following ((Q +* (stop (while<0 (F5(),F6(),F4())))),(Comput ((Q +* (stop (while<0 (F5(),F6(),F4())))),t,((LifeSpan ((Q +* (stop F4())),t)) + 1)))) by EXTPRO_1:3
.= Exec ((goto (- ((card F4()) + 1))),(Comput ((Q +* (stop (while<0 (F5(),F6(),F4())))),t,((LifeSpan ((Q +* (stop F4())),t)) + 1)))) by A39 ;
then  IC (Comput ((Q +* (stop (while<0 (F5(),F6(),F4())))),t,(((LifeSpan ((Q +* (stop F4())),t)) + 1) + 1))) = ICplusConst ((Comput ((Q +* (stop (while<0 (F5(),F6(),F4())))),t,((LifeSpan ((Q +* (stop F4())),t)) + 1))),(0 - ((card F4()) + 1))) by SCMPDS_2:54
.= 0 by A36, A38, SCMPDS_7:1 ;
then A42: Initialize (Comput ((Q +* (stop (while<0 (F5(),F6(),F4())))),t,(((LifeSpan ((Q +* (stop F4())),t)) + 1) + 1))) = Comput ((Q +* (stop (while<0 (F5(),F6(),F4())))),t,(((LifeSpan ((Q +* (stop F4())),t)) + 1) + 1)) by MEMSTR_0:46;
A43: DataPart (Comput ((Q +* (stop F4())),t,(LifeSpan ((Q +* (stop F4())),t)))) = DataPart (Comput ((Q +* (stop (while<0 (F5(),F6(),F4())))),(Comput ((Q +* (stop (while<0 (F5(),F6(),F4())))),t,1)),(LifeSpan ((Q +* (stop F4())),t)))) by A20, A30, A34, A35, A27, A24, SCMPDS_7:18;
then A44: DataPart (Comput ((Q +* (stop (while<0 (F5(),F6(),F4())))),(Comput ((Q +* (stop (while<0 (F5(),F6(),F4())))),t,1)),(LifeSpan ((Q +* (stop F4())),t)))) = DataPart (Result ((Q +* (stop F4())),t)) by A28, EXTPRO_1:23
.= DataPart (IExec (F4(),Q,t)) by SCMPDS_4:def_5 ;
 InsCode (goto (- ((card F4()) + 1))) = 14 by SCMPDS_2:12;
then  InsCode (goto (- ((card F4()) + 1))) in {0,4,5,6,14} by ENUMSET1:def_3;
then A45: Initialize (Comput ((Q +* (stop (while<0 (F5(),F6(),F4())))),t,(((LifeSpan ((Q +* (stop F4())),t)) + 1) + 1))) = Initialize (Comput ((Q +* (stop (while<0 (F5(),F6(),F4())))),t,((LifeSpan ((Q +* (stop F4())),t)) + 1))) by A41, Th3
.= Initialize (IExec (F4(),Q,t)) by A44, A38, MEMSTR_0:80 ;
A46: now__::_thesis:_not_F1((Initialize_(Comput_((Q_+*_(stop_(while<0_(F5(),F6(),F4())))),t,(((LifeSpan_((Q_+*_(stop_F4())),t))_+_1)_+_1)))))_>_k
 F1((Initialize (IExec (F4(),Q,t)))) < F1(t) by A4, A8, A9, A10;
then A47: F1((Initialize (Comput ((Q +* (stop (while<0 (F5(),F6(),F4())))),t,(((LifeSpan ((Q +* (stop F4())),t)) + 1) + 1))))) < k + 1 by A7, A45, XXREAL_0:2;
assume  F1((Initialize (Comput ((Q +* (stop (while<0 (F5(),F6(),F4())))),t,(((LifeSpan ((Q +* (stop F4())),t)) + 1) + 1))))) > k ; ::_thesis: contradiction
hence  contradiction by A47, INT_1:7; ::_thesis: verum
end;
A48: (Comput ((Q +* (stop (while<0 (F5(),F6(),F4())))),(Comput ((Q +* (stop (while<0 (F5(),F6(),F4())))),t,1)),(LifeSpan ((Q +* (stop F4())),t)))) . F5() = (Comput ((Q +* (stop F4())),t,(LifeSpan ((Q +* (stop F4())),t)))) . F5() by A43, SCMPDS_4:8
.= (Result ((Q +* (stop F4())),t)) . F5() by A28, EXTPRO_1:23
.= F2() . F5() by A9, A4, A8, A10, A31 ;
A50: (Comput ((Q +* (stop (while<0 (F5(),F6(),F4())))),t,(((LifeSpan ((Q +* (stop F4())),t)) + 1) + 1))) . F5() = (Comput ((Q +* (stop (while<0 (F5(),F6(),F4())))),t,((LifeSpan ((Q +* (stop F4())),t)) + 1))) . F5() by A41, SCMPDS_2:54
.= F2() . F5() by A48, EXTPRO_1:4 ;
then A51: while<0 (F5(),F6(),F4()) is_closed_on Comput ((Q +* (stop (while<0 (F5(),F6(),F4())))),t,(((LifeSpan ((Q +* (stop F4())),t)) + 1) + 1)),Q +* (stop (while<0 (F5(),F6(),F4()))) by A6, A32, A45, A46, A42;
now__::_thesis:_for_k_being_Element_of_NAT_holds_IC_(Comput_((Q_+*_(stop_(while<0_(F5(),F6(),F4())))),t,k))_in_dom_(stop_(while<0_(F5(),F6(),F4())))
let k be Element of NAT ; ::_thesis: IC (Comput ((Q +* (stop (while<0 (F5(),F6(),F4())))),t,b1)) in dom (stop (while<0 (F5(),F6(),F4())))
percases ( k < ((LifeSpan ((Q +* (stop F4())),t)) + 1) + 1 or k >= ((LifeSpan ((Q +* (stop F4())),t)) + 1) + 1 ) ;
suppose k < ((LifeSpan ((Q +* (stop F4())),t)) + 1) + 1 ; ::_thesis: IC (Comput ((Q +* (stop (while<0 (F5(),F6(),F4())))),t,b1)) in dom (stop (while<0 (F5(),F6(),F4())))
then A52: k <= (LifeSpan ((Q +* (stop F4())),t)) + 1 by INT_1:7;
hereby  ::_thesis: verum
percases ( k <= LifeSpan ((Q +* (stop F4())),t) or k = (LifeSpan ((Q +* (stop F4())),t)) + 1 ) by A52, NAT_1:8;
supposeA53: k <= LifeSpan ((Q +* (stop F4())),t) ; ::_thesis: IC (Comput ((Q +* (stop (while<0 (F5(),F6(),F4())))),t,k)) in dom (stop (while<0 (F5(),F6(),F4())))
hereby  ::_thesis: verum
percases ( k = 0 or k <> 0 ) ;
suppose k = 0 ; ::_thesis: IC (Comput ((Q +* (stop (while<0 (F5(),F6(),F4())))),t,k)) in dom (stop (while<0 (F5(),F6(),F4())))
hence  IC (Comput ((Q +* (stop (while<0 (F5(),F6(),F4())))),t,k)) in dom (stop (while<0 (F5(),F6(),F4()))) by A14, A21, EXTPRO_1:2; ::_thesis: verum
end;
suppose k <> 0 ; ::_thesis: IC (Comput ((Q +* (stop (while<0 (F5(),F6(),F4())))),t,k)) in dom (stop (while<0 (F5(),F6(),F4())))
then consider kn being Nat such that
A54: k = kn + 1 by NAT_1:6;
reconsider kn = kn as Element of NAT by ORDINAL1:def_12;
reconsider lm = IC (Comput ((Q +* (stop F4())),t,kn)) as Element of NAT ;
 kn < k by A54, XREAL_1:29;
then  kn < LifeSpan ((Q +* (stop F4())),t) by A53, XXREAL_0:2;
then  (IC (Comput ((Q +* (stop F4())),t,kn))) + 1 = IC (Comput ((Q +* (stop (while<0 (F5(),F6(),F4())))),(Comput ((Q +* (stop (while<0 (F5(),F6(),F4())))),t,1)),kn)) by A20, A30, A34, A35, A27, A24, SCMPDS_7:16;
then A56: IC (Comput ((Q +* (stop (while<0 (F5(),F6(),F4())))),t,k)) = lm + 1 by A54, EXTPRO_1:4;
 IC (Comput ((Q +* (stop F4())),t,kn)) in dom (stop F4()) by A33, T, SCMPDS_6:def_2;
then  lm < card (stop F4()) by AFINSQ_1:66;
then  lm < (card F4()) + 1 by COMPOS_1:55;
then A57: lm + 1 <= (card F4()) + 1 by INT_1:7;
 (card F4()) + 1 < (card F4()) + 3 by XREAL_1:6;
then  lm + 1 < (card F4()) + 3 by A57, XXREAL_0:2;
then  lm + 1 < card (stop (while<0 (F5(),F6(),F4()))) by Lm1;
hence  IC (Comput ((Q +* (stop (while<0 (F5(),F6(),F4())))),t,k)) in dom (stop (while<0 (F5(),F6(),F4()))) by A56, AFINSQ_1:66; ::_thesis: verum
end;
end;
end;
end;
supposeA58: k = (LifeSpan ((Q +* (stop F4())),t)) + 1 ; ::_thesis: IC (Comput ((Q +* (stop (while<0 (F5(),F6(),F4())))),t,k)) in dom (stop (while<0 (F5(),F6(),F4())))
 (card F4()) + 1 in dom (stop (while<0 (F5(),F6(),F4()))) by A22, COMPOS_1:62;
hence  IC (Comput ((Q +* (stop (while<0 (F5(),F6(),F4())))),t,k)) in dom (stop (while<0 (F5(),F6(),F4()))) by A20, A30, A34, A35, A27, A24, A38, A58, SCMPDS_7:18; ::_thesis: verum
end;
end;
end;
end;
suppose k >= ((LifeSpan ((Q +* (stop F4())),t)) + 1) + 1 ; ::_thesis: IC (Comput ((Q +* (stop (while<0 (F5(),F6(),F4())))),t,b1)) in dom (stop (while<0 (F5(),F6(),F4())))
then consider nn being Nat such that
A59: k = (((LifeSpan ((Q +* (stop F4())),t)) + 1) + 1) + nn by NAT_1:10;
reconsider nn = nn as Element of NAT by ORDINAL1:def_12;
 Comput ((Q +* (stop (while<0 (F5(),F6(),F4())))),t,k) = Comput (((Q +* (stop (while<0 (F5(),F6(),F4())))) +* (stop (while<0 (F5(),F6(),F4())))),(Comput ((Q +* (stop (while<0 (F5(),F6(),F4())))),t,(((LifeSpan ((Q +* (stop F4())),t)) + 1) + 1))),nn) by A59, EXTPRO_1:4;
hence  IC (Comput ((Q +* (stop (while<0 (F5(),F6(),F4())))),t,k)) in dom (stop (while<0 (F5(),F6(),F4()))) by A51, A42, SCMPDS_6:def_2; ::_thesis: verum
end;
end;
end;
hence  while<0 (F5(),F6(),F4()) is_closed_on t,Q by T, SCMPDS_6:def_2; ::_thesis: while<0 (F5(),F6(),F4()) is_halting_on t,Q
RR: Q +* (stop (while<0 (F5(),F6(),F4()))) = (Q +* (stop (while<0 (F5(),F6(),F4())))) +* (stop (while<0 (F5(),F6(),F4()))) ;
 while<0 (F5(),F6(),F4()) is_halting_on Comput ((Q +* (stop (while<0 (F5(),F6(),F4())))),t,(((LifeSpan ((Q +* (stop F4())),t)) + 1) + 1)),Q +* (stop (while<0 (F5(),F6(),F4()))) by A6, A32, A50, A45, A46, A42;
then  Q +* (stop (while<0 (F5(),F6(),F4()))) halts_on Comput ((Q +* (stop (while<0 (F5(),F6(),F4())))),t,(((LifeSpan ((Q +* (stop F4())),t)) + 1) + 1)) by A42, RR, SCMPDS_6:def_3;
then  Q +* (stop (while<0 (F5(),F6(),F4()))) halts_on t by EXTPRO_1:22;
hence  while<0 (F5(),F6(),F4()) is_halting_on t,Q by T, SCMPDS_6:def_3; ::_thesis: verum
end;
end;
end;
set n = F1(F2());
A60: S1[ 0 ]
proof
let t be 0 -started State of SCMPDS; ::_thesis: for Q being Instruction-Sequence of SCMPDS st F1(t) <= 0 & P1[t] & t . F5() = F2() . F5() holds
( while<0 (F5(),F6(),F4()) is_closed_on t,Q & while<0 (F5(),F6(),F4()) is_halting_on t,Q )
let Q be Instruction-Sequence of SCMPDS; ::_thesis: ( F1(t) <= 0 & P1[t] & t . F5() = F2() . F5() implies ( while<0 (F5(),F6(),F4()) is_closed_on t,Q & while<0 (F5(),F6(),F4()) is_halting_on t,Q ) )
assume that
A61: F1(t) <= 0 and
A62: P1[t] and
A63: t . F5() = F2() . F5() ; ::_thesis: ( while<0 (F5(),F6(),F4()) is_closed_on t,Q & while<0 (F5(),F6(),F4()) is_halting_on t,Q )
 F1(t) = 0 by A61;
then  t . (DataLoc ((F2() . F5()),F6())) >= 0 by A2, A62;
hence  ( while<0 (F5(),F6(),F4()) is_closed_on t,Q & while<0 (F5(),F6(),F4()) is_halting_on t,Q ) by A63, Th9; ::_thesis: verum
end;
 for k being Element of NAT holds S1[k] from NAT_1:sch_1(A60, A5);
then  S1[F1(F2())] ;
hence  ( while<0 (F5(),F6(),F4()) is_closed_on F2(),F3() & while<0 (F5(),F6(),F4()) is_halting_on F2(),F3() ) by A3; ::_thesis: verum
end;

scheme :: SCMPDS_8:sch 2
WhileLExec{ F1( State of SCMPDS) ->  Element of NAT , F2() ->  0 -started State of SCMPDS, F3() ->  Instruction-Sequence of SCMPDS, F4() ->  halt-free shiftable Program of SCMPDS, F5() ->  Int_position, F6() ->  Integer, P1[ State of SCMPDS] } :
 IExec ((while<0 (F5(),F6(),F4())),F3(),F2()) = IExec ((while<0 (F5(),F6(),F4())),F3(),(Initialize (IExec (F4(),F3(),F2()))))
provided
A2: F2() . (DataLoc ((F2() . F5()),F6())) < 0 and
A3: for t being 0 -started State of SCMPDS st P1[t] & F1(t) = 0 holds
t . (DataLoc ((F2() . F5()),F6())) >= 0 and
A4: P1[F2()] and
A5: for t being 0 -started State of SCMPDS
for Q being Instruction-Sequence of SCMPDS st P1[t] & t . F5() = F2() . F5() & t . (DataLoc ((F2() . F5()),F6())) < 0 holds
( (IExec (F4(),Q,t)) . F5() = t . F5() & F4() is_closed_on t,Q & F4() is_halting_on t,Q & F1((Initialize (IExec (F4(),Q,t)))) < F1(t) & P1[ Initialize (IExec (F4(),Q,t))] )
proof
II: Initialize F2() = F2() by MEMSTR_0:44;
set WHL = while<0 (F5(),F6(),F4());
set pWHL = stop (while<0 (F5(),F6(),F4()));
set P1 = F3() +* (stop (while<0 (F5(),F6(),F4())));
set PI = F3() +* (stop F4());
set m1 = (LifeSpan ((F3() +* (stop F4())),F2())) + 2;
set s2 = Initialize (IExec (F4(),F3(),F2()));
set P2 = F3() +* (stop (while<0 (F5(),F6(),F4())));
set m2 = LifeSpan ((F3() +* (stop (while<0 (F5(),F6(),F4())))),(Initialize (IExec (F4(),F3(),F2()))));
A9: P1[F2()] by A4;
A10: stop F4() c= F3() +* (stop F4()) by FUNCT_4:25;
A11: F4() is_closed_on F2(),F3() +* (stop F4()) by A2, A4, A5;
 F4() is_halting_on F2(),F3() by A2, A4, A5;
then A12: F3() +* (stop F4()) halts_on F2() by II, SCMPDS_6:def_3;
 (F3() +* (stop F4())) +* (stop F4()) halts_on F2() by A12;
then A14: F4() is_halting_on F2(),F3() +* (stop F4()) by II, SCMPDS_6:def_3;
set Es = IExec (F4(),F3(),F2());
set bj = DataLoc (((Initialize (IExec (F4(),F3(),F2()))) . F5()),F6());
set EP = F3();
A15: for t being 0 -started State of SCMPDS
for Q being Instruction-Sequence of SCMPDS st P1[t] & t . F5() = F2() . F5() & t . (DataLoc ((F2() . F5()),F6())) < 0 holds
( (IExec (F4(),Q,t)) . F5() = t . F5() & F4() is_closed_on t,Q & F4() is_halting_on t,Q & F1((Initialize (IExec (F4(),Q,t)))) < F1(t) & P1[ Initialize (IExec (F4(),Q,t))] ) by A5;
A16: for t being 0 -started State of SCMPDS st P1[t] & F1(t) = 0 holds
t . (DataLoc ((F2() . F5()),F6())) >= 0 by A3;
 ( while<0 (F5(),F6(),F4()) is_closed_on F2(),F3() & while<0 (F5(),F6(),F4()) is_halting_on F2(),F3() ) from SCMPDS_8:sch_1(A16, A9, A15);
then A17: F3() +* (stop (while<0 (F5(),F6(),F4()))) halts_on F2() by II, SCMPDS_6:def_3;
B18: (IExec (F4(),F3(),F2())) . F5() = (Initialize (IExec (F4(),F3(),F2()))) . F5() by SCMPDS_5:15;
A18: (IExec (F4(),F3(),F2())) . F5() = F2() . F5() by A2, A4, A5;
then A19: for t being 0 -started State of SCMPDS st P1[t] & F1(t) = 0 holds
t . (DataLoc (((Initialize (IExec (F4(),F3(),F2()))) . F5()),F6())) >= 0 by A3, B18;
A20: for t being 0 -started State of SCMPDS
for Q being Instruction-Sequence of SCMPDS st P1[t] & t . F5() = (Initialize (IExec (F4(),F3(),F2()))) . F5() & t . (DataLoc (((Initialize (IExec (F4(),F3(),F2()))) . F5()),F6())) < 0 holds
( (IExec (F4(),Q,t)) . F5() = t . F5() & F4() is_closed_on t,Q & F4() is_halting_on t,Q & F1((Initialize (IExec (F4(),Q,t)))) < F1(t) & P1[ Initialize (IExec (F4(),Q,t))] ) by A5, A18, B18;
A21: P1[ Initialize (IExec (F4(),F3(),F2()))] by A2, A4, A5;
 ( while<0 (F5(),F6(),F4()) is_closed_on Initialize (IExec (F4(),F3(),F2())),F3() & while<0 (F5(),F6(),F4()) is_halting_on Initialize (IExec (F4(),F3(),F2())),F3() ) from SCMPDS_8:sch_1(A19, A21, A20);
then A22: F3() +* (stop (while<0 (F5(),F6(),F4()))) halts_on Initialize (Initialize (IExec (F4(),F3(),F2()))) by SCMPDS_6:def_3;
set s4 = Comput ((F3() +* (stop (while<0 (F5(),F6(),F4())))),F2(),1);
set P4 = F3() +* (stop (while<0 (F5(),F6(),F4())));
set i1 = (F5(),F6()) >=0_goto ((card F4()) + 2);
set i2 = goto (- ((card F4()) + 1));
set b = DataLoc ((F2() . F5()),F6());
A23: while<0 (F5(),F6(),F4()) = ((F5(),F6()) >=0_goto ((card F4()) + 2)) ';' (F4() ';' (goto (- ((card F4()) + 1)))) by SCMPDS_4:15;
set mI = LifeSpan ((F3() +* (stop F4())),F2());
set s5 = Comput ((F3() +* (stop (while<0 (F5(),F6(),F4())))),(Comput ((F3() +* (stop (while<0 (F5(),F6(),F4())))),F2(),1)),(LifeSpan ((F3() +* (stop F4())),F2())));
set P5 = F3() +* (stop (while<0 (F5(),F6(),F4())));
set l1 = (card F4()) + 1;
A24: IC F2() = 0 by MEMSTR_0:def_11;
A25: Comput ((F3() +* (stop (while<0 (F5(),F6(),F4())))),F2(),(0 + 1)) = Following ((F3() +* (stop (while<0 (F5(),F6(),F4())))),(Comput ((F3() +* (stop (while<0 (F5(),F6(),F4())))),F2(),0))) by EXTPRO_1:3
.= Following ((F3() +* (stop (while<0 (F5(),F6(),F4())))),F2())
.= Exec (((F5(),F6()) >=0_goto ((card F4()) + 2)),F2()) by A23, II, SCMPDS_6:11 ;
 for a being Int_position holds F2() . a = (Comput ((F3() +* (stop (while<0 (F5(),F6(),F4())))),F2(),1)) . a by A25, SCMPDS_2:57;
then A27: DataPart F2() = DataPart (Comput ((F3() +* (stop (while<0 (F5(),F6(),F4())))),F2(),1)) by SCMPDS_4:8;
set m3 = (LifeSpan ((F3() +* (stop F4())),F2())) + 1;
set s6 = Comput ((F3() +* (stop (while<0 (F5(),F6(),F4())))),F2(),((LifeSpan ((F3() +* (stop F4())),F2())) + 1));
set P6 = F3() +* (stop (while<0 (F5(),F6(),F4())));
 (card F4()) + 1 < (card F4()) + 2 by XREAL_1:6;
then A28: (card F4()) + 1 in dom (while<0 (F5(),F6(),F4())) by Th7;
set m0 = LifeSpan ((F3() +* (stop (while<0 (F5(),F6(),F4())))),F2());
set s7 = Comput ((F3() +* (stop (while<0 (F5(),F6(),F4())))),F2(),(((LifeSpan ((F3() +* (stop F4())),F2())) + 1) + 1));
XX: while<0 (F5(),F6(),F4()) c= stop (while<0 (F5(),F6(),F4())) by AFINSQ_1:74;
 stop (while<0 (F5(),F6(),F4())) c= F3() +* (stop (while<0 (F5(),F6(),F4()))) by FUNCT_4:25;
then A33: while<0 (F5(),F6(),F4()) c= F3() +* (stop (while<0 (F5(),F6(),F4()))) by XX, XBOOLE_1:1;
 Shift (F4(),1) c= while<0 (F5(),F6(),F4()) by Lm2;
then A34: Shift (F4(),1) c= F3() +* (stop (while<0 (F5(),F6(),F4()))) by A33, XBOOLE_1:1;
A35: IC (Comput ((F3() +* (stop (while<0 (F5(),F6(),F4())))),F2(),1)) = succ (IC F2()) by A2, A25, SCMPDS_2:57
.= 0 + 1 by A24 ;
then A36: IC (Comput ((F3() +* (stop (while<0 (F5(),F6(),F4())))),(Comput ((F3() +* (stop (while<0 (F5(),F6(),F4())))),F2(),1)),(LifeSpan ((F3() +* (stop F4())),F2())))) = (card F4()) + 1 by A10, A14, A11, A27, A34, SCMPDS_7:18;
A37: (F3() +* (stop (while<0 (F5(),F6(),F4())))) /. (IC (Comput ((F3() +* (stop (while<0 (F5(),F6(),F4())))),F2(),((LifeSpan ((F3() +* (stop F4())),F2())) + 1)))) = (F3() +* (stop (while<0 (F5(),F6(),F4())))) . (IC (Comput ((F3() +* (stop (while<0 (F5(),F6(),F4())))),F2(),((LifeSpan ((F3() +* (stop F4())),F2())) + 1)))) by PBOOLE:143;
A38: Comput ((F3() +* (stop (while<0 (F5(),F6(),F4())))),F2(),((LifeSpan ((F3() +* (stop F4())),F2())) + 1)) = Comput ((F3() +* (stop (while<0 (F5(),F6(),F4())))),(Comput ((F3() +* (stop (while<0 (F5(),F6(),F4())))),F2(),1)),(LifeSpan ((F3() +* (stop F4())),F2()))) by EXTPRO_1:4;
then A39: CurInstr ((F3() +* (stop (while<0 (F5(),F6(),F4())))),(Comput ((F3() +* (stop (while<0 (F5(),F6(),F4())))),F2(),((LifeSpan ((F3() +* (stop F4())),F2())) + 1)))) = (F3() +* (stop (while<0 (F5(),F6(),F4())))) . ((card F4()) + 1) by A10, A14, A11, A35, A27, A34, A37, SCMPDS_7:18
.= (while<0 (F5(),F6(),F4())) . ((card F4()) + 1) by A28, A33, GRFUNC_1:2
.= goto (- ((card F4()) + 1)) by Th8 ;
A41: Comput ((F3() +* (stop (while<0 (F5(),F6(),F4())))),F2(),(((LifeSpan ((F3() +* (stop F4())),F2())) + 1) + 1)) = Following ((F3() +* (stop (while<0 (F5(),F6(),F4())))),(Comput ((F3() +* (stop (while<0 (F5(),F6(),F4())))),F2(),((LifeSpan ((F3() +* (stop F4())),F2())) + 1)))) by EXTPRO_1:3
.= Exec ((goto (- ((card F4()) + 1))),(Comput ((F3() +* (stop (while<0 (F5(),F6(),F4())))),F2(),((LifeSpan ((F3() +* (stop F4())),F2())) + 1)))) by A39 ;
then  IC (Comput ((F3() +* (stop (while<0 (F5(),F6(),F4())))),F2(),(((LifeSpan ((F3() +* (stop F4())),F2())) + 1) + 1))) = ICplusConst ((Comput ((F3() +* (stop (while<0 (F5(),F6(),F4())))),F2(),((LifeSpan ((F3() +* (stop F4())),F2())) + 1))),(0 - ((card F4()) + 1))) by SCMPDS_2:54
.= 0 by A36, A38, SCMPDS_7:1 ;
then A42: IC (Initialize (IExec (F4(),F3(),F2()))) = IC (Comput ((F3() +* (stop (while<0 (F5(),F6(),F4())))),F2(),((LifeSpan ((F3() +* (stop F4())),F2())) + 2))) by MEMSTR_0:def_11;
A43: DataPart (Comput ((F3() +* (stop F4())),F2(),(LifeSpan ((F3() +* (stop F4())),F2())))) = DataPart (Comput ((F3() +* (stop (while<0 (F5(),F6(),F4())))),(Comput ((F3() +* (stop (while<0 (F5(),F6(),F4())))),F2(),1)),(LifeSpan ((F3() +* (stop F4())),F2())))) by A10, A14, A11, A35, A27, A34, SCMPDS_7:18;
now__::_thesis:_for_x_being_Int_position_holds_(Comput_((F3()_+*_(stop_(while<0_(F5(),F6(),F4())))),F2(),(((LifeSpan_((F3()_+*_(stop_F4())),F2()))_+_1)_+_1)))_._x_=_(Initialize_(IExec_(F4(),F3(),F2())))_._x
let x be Int_position; ::_thesis: (Comput ((F3() +* (stop (while<0 (F5(),F6(),F4())))),F2(),(((LifeSpan ((F3() +* (stop F4())),F2())) + 1) + 1))) . x = (Initialize (IExec (F4(),F3(),F2()))) . x
A44: not x in dom (Start-At (0,SCMPDS)) by SCMPDS_4:18;
(Comput ((F3() +* (stop (while<0 (F5(),F6(),F4())))),(Comput ((F3() +* (stop (while<0 (F5(),F6(),F4())))),F2(),1)),(LifeSpan ((F3() +* (stop F4())),F2())))) . x = (Comput ((F3() +* (stop F4())),F2(),(LifeSpan ((F3() +* (stop F4())),F2())))) . x by A43, SCMPDS_4:8
.= (Result ((F3() +* (stop F4())),F2())) . x by A12, EXTPRO_1:23
.= (IExec (F4(),F3(),F2())) . x by SCMPDS_4:def_5 ;
hence (Comput ((F3() +* (stop (while<0 (F5(),F6(),F4())))),F2(),(((LifeSpan ((F3() +* (stop F4())),F2())) + 1) + 1))) . x = (IExec (F4(),F3(),F2())) . x by A38, A41, SCMPDS_2:54
.= (Initialize (IExec (F4(),F3(),F2()))) . x by A44, FUNCT_4:11 ;
::_thesis: verum
end;
then A46: DataPart (Comput ((F3() +* (stop (while<0 (F5(),F6(),F4())))),F2(),(((LifeSpan ((F3() +* (stop F4())),F2())) + 1) + 1))) = DataPart (Initialize (IExec (F4(),F3(),F2()))) by SCMPDS_4:8;
A48: Comput ((F3() +* (stop (while<0 (F5(),F6(),F4())))),F2(),((LifeSpan ((F3() +* (stop F4())),F2())) + 2)) = Initialize (IExec (F4(),F3(),F2())) by A46, A42, MEMSTR_0:78;
then  CurInstr ((F3() +* (stop (while<0 (F5(),F6(),F4())))),(Comput ((F3() +* (stop (while<0 (F5(),F6(),F4())))),F2(),((LifeSpan ((F3() +* (stop F4())),F2())) + 2)))) = (F5(),F6()) >=0_goto ((card F4()) + 2) by A23, SCMPDS_6:11;
then  LifeSpan ((F3() +* (stop (while<0 (F5(),F6(),F4())))),F2()) > (LifeSpan ((F3() +* (stop F4())),F2())) + 2 by A17, EXTPRO_1:36, SCMPDS_6:18;
then consider nn being Nat such that
A49: LifeSpan ((F3() +* (stop (while<0 (F5(),F6(),F4())))),F2()) = ((LifeSpan ((F3() +* (stop F4())),F2())) + 2) + nn by NAT_1:10;
reconsider nn = nn as Element of NAT by ORDINAL1:def_12;
 Comput ((F3() +* (stop (while<0 (F5(),F6(),F4())))),F2(),(((LifeSpan ((F3() +* (stop F4())),F2())) + 2) + (LifeSpan ((F3() +* (stop (while<0 (F5(),F6(),F4())))),(Initialize (IExec (F4(),F3(),F2()))))))) = Comput ((F3() +* (stop (while<0 (F5(),F6(),F4())))),(Initialize (IExec (F4(),F3(),F2()))),(LifeSpan ((F3() +* (stop (while<0 (F5(),F6(),F4())))),(Initialize (IExec (F4(),F3(),F2())))))) by A48, EXTPRO_1:4;
then  CurInstr ((F3() +* (stop (while<0 (F5(),F6(),F4())))),(Comput ((F3() +* (stop (while<0 (F5(),F6(),F4())))),F2(),(((LifeSpan ((F3() +* (stop F4())),F2())) + 2) + (LifeSpan ((F3() +* (stop (while<0 (F5(),F6(),F4())))),(Initialize (IExec (F4(),F3(),F2()))))))))) = halt SCMPDS by A22, EXTPRO_1:def_15;
then  ((LifeSpan ((F3() +* (stop F4())),F2())) + 2) + (LifeSpan ((F3() +* (stop (while<0 (F5(),F6(),F4())))),(Initialize (IExec (F4(),F3(),F2()))))) >= LifeSpan ((F3() +* (stop (while<0 (F5(),F6(),F4())))),F2()) by A17, EXTPRO_1:def_15;
then A51: LifeSpan ((F3() +* (stop (while<0 (F5(),F6(),F4())))),(Initialize (IExec (F4(),F3(),F2())))) >= nn by A49, XREAL_1:6;
A52: Comput ((F3() +* (stop (while<0 (F5(),F6(),F4())))),F2(),(LifeSpan ((F3() +* (stop (while<0 (F5(),F6(),F4())))),F2()))) = Comput ((F3() +* (stop (while<0 (F5(),F6(),F4())))),(Initialize (IExec (F4(),F3(),F2()))),nn) by A48, A49, EXTPRO_1:4;
then  CurInstr ((F3() +* (stop (while<0 (F5(),F6(),F4())))),(Comput ((F3() +* (stop (while<0 (F5(),F6(),F4())))),(Initialize (IExec (F4(),F3(),F2()))),nn))) = halt SCMPDS by A17, EXTPRO_1:def_15;
then  nn >= LifeSpan ((F3() +* (stop (while<0 (F5(),F6(),F4())))),(Initialize (IExec (F4(),F3(),F2())))) by A22, EXTPRO_1:def_15;
then  nn = LifeSpan ((F3() +* (stop (while<0 (F5(),F6(),F4())))),(Initialize (IExec (F4(),F3(),F2())))) by A51, XXREAL_0:1;
then  Result ((F3() +* (stop (while<0 (F5(),F6(),F4())))),F2()) = Comput ((F3() +* (stop (while<0 (F5(),F6(),F4())))),(Initialize (IExec (F4(),F3(),F2()))),(LifeSpan ((F3() +* (stop (while<0 (F5(),F6(),F4())))),(Initialize (IExec (F4(),F3(),F2())))))) by A17, A52, EXTPRO_1:23;
hence  IExec ((while<0 (F5(),F6(),F4())),F3(),F2()) = Comput ((F3() +* (stop (while<0 (F5(),F6(),F4())))),(Initialize (IExec (F4(),F3(),F2()))),(LifeSpan ((F3() +* (stop (while<0 (F5(),F6(),F4())))),(Initialize (IExec (F4(),F3(),F2())))))) by SCMPDS_4:def_5
.= Result ((F3() +* (stop (while<0 (F5(),F6(),F4())))),(Initialize (IExec (F4(),F3(),F2())))) by A22, EXTPRO_1:23
.= IExec ((while<0 (F5(),F6(),F4())),F3(),(Initialize (IExec (F4(),F3(),F2())))) by SCMPDS_4:def_5 ;
::_thesis: verum
end;

theorem :: SCMPDS_8:13
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 being Integer
for X being set
for f being Function of (product (the_Values_of SCMPDS)),NAT st ( for t being 0 -started State of SCMPDS st f . t = 0 holds
t . (DataLoc ((s . a),i)) >= 0 ) & ( for t being 0 -started State of SCMPDS
for Q being Instruction-Sequence of SCMPDS st ( for x being Int_position st x in X holds
t . x = s . x ) & t . a = s . a & t . (DataLoc ((s . a),i)) < 0 holds
( (IExec (I,Q,t)) . a = t . a & f . (Initialize (IExec (I,Q,t))) < f . t & I is_closed_on t,Q & I is_halting_on t,Q & ( for x being Int_position st x in X holds
(IExec (I,Q,t)) . x = t . x ) ) ) holds
( while<0 (a,i,I) is_closed_on s,P & 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 being Integer
for X being set
for f being Function of (product (the_Values_of SCMPDS)),NAT st ( for t being 0 -started State of SCMPDS st f . t = 0 holds
t . (DataLoc ((s . a),i)) >= 0 ) & ( for t being 0 -started State of SCMPDS
for Q being Instruction-Sequence of SCMPDS st ( for x being Int_position st x in X holds
t . x = s . x ) & t . a = s . a & t . (DataLoc ((s . a),i)) < 0 holds
( (IExec (I,Q,t)) . a = t . a & f . (Initialize (IExec (I,Q,t))) < f . t & I is_closed_on t,Q & I is_halting_on t,Q & ( for x being Int_position st x in X holds
(IExec (I,Q,t)) . x = t . x ) ) ) holds
( while<0 (a,i,I) is_closed_on s,P & 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 being Integer
for X being set
for f being Function of (product (the_Values_of SCMPDS)),NAT st ( for t being 0 -started State of SCMPDS st f . t = 0 holds
t . (DataLoc ((s . a),i)) >= 0 ) & ( for t being 0 -started State of SCMPDS
for Q being Instruction-Sequence of SCMPDS st ( for x being Int_position st x in X holds
t . x = s . x ) & t . a = s . a & t . (DataLoc ((s . a),i)) < 0 holds
( (IExec (I,Q,t)) . a = t . a & f . (Initialize (IExec (I,Q,t))) < f . t & I is_closed_on t,Q & I is_halting_on t,Q & ( for x being Int_position st x in X holds
(IExec (I,Q,t)) . x = t . x ) ) ) holds
( while<0 (a,i,I) is_closed_on s,P & 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 being Integer
for X being set
for f being Function of (product (the_Values_of SCMPDS)),NAT st ( for t being 0 -started State of SCMPDS st f . t = 0 holds
t . (DataLoc ((s . a),i)) >= 0 ) & ( for t being 0 -started State of SCMPDS
for Q being Instruction-Sequence of SCMPDS st ( for x being Int_position st x in X holds
t . x = s . x ) & t . a = s . a & t . (DataLoc ((s . a),i)) < 0 holds
( (IExec (I,Q,t)) . a = t . a & f . (Initialize (IExec (I,Q,t))) < f . t & I is_closed_on t,Q & I is_halting_on t,Q & ( for x being Int_position st x in X holds
(IExec (I,Q,t)) . x = t . x ) ) ) holds
( while<0 (a,i,I) is_closed_on s,P & while<0 (a,i,I) is_halting_on s,P )
let a be Int_position; ::_thesis: for i being Integer
for X being set
for f being Function of (product (the_Values_of SCMPDS)),NAT st ( for t being 0 -started State of SCMPDS st f . t = 0 holds
t . (DataLoc ((s . a),i)) >= 0 ) & ( for t being 0 -started State of SCMPDS
for Q being Instruction-Sequence of SCMPDS st ( for x being Int_position st x in X holds
t . x = s . x ) & t . a = s . a & t . (DataLoc ((s . a),i)) < 0 holds
( (IExec (I,Q,t)) . a = t . a & f . (Initialize (IExec (I,Q,t))) < f . t & I is_closed_on t,Q & I is_halting_on t,Q & ( for x being Int_position st x in X holds
(IExec (I,Q,t)) . x = t . x ) ) ) holds
( while<0 (a,i,I) is_closed_on s,P & while<0 (a,i,I) is_halting_on s,P )
let i be Integer; ::_thesis: for X being set
for f being Function of (product (the_Values_of SCMPDS)),NAT st ( for t being 0 -started State of SCMPDS st f . t = 0 holds
t . (DataLoc ((s . a),i)) >= 0 ) & ( for t being 0 -started State of SCMPDS
for Q being Instruction-Sequence of SCMPDS st ( for x being Int_position st x in X holds
t . x = s . x ) & t . a = s . a & t . (DataLoc ((s . a),i)) < 0 holds
( (IExec (I,Q,t)) . a = t . a & f . (Initialize (IExec (I,Q,t))) < f . t & I is_closed_on t,Q & I is_halting_on t,Q & ( for x being Int_position st x in X holds
(IExec (I,Q,t)) . x = t . x ) ) ) holds
( while<0 (a,i,I) is_closed_on s,P & while<0 (a,i,I) is_halting_on s,P )
let X be set ; ::_thesis: for f being Function of (product (the_Values_of SCMPDS)),NAT st ( for t being 0 -started State of SCMPDS st f . t = 0 holds
t . (DataLoc ((s . a),i)) >= 0 ) & ( for t being 0 -started State of SCMPDS
for Q being Instruction-Sequence of SCMPDS st ( for x being Int_position st x in X holds
t . x = s . x ) & t . a = s . a & t . (DataLoc ((s . a),i)) < 0 holds
( (IExec (I,Q,t)) . a = t . a & f . (Initialize (IExec (I,Q,t))) < f . t & I is_closed_on t,Q & I is_halting_on t,Q & ( for x being Int_position st x in X holds
(IExec (I,Q,t)) . x = t . x ) ) ) holds
( while<0 (a,i,I) is_closed_on s,P & while<0 (a,i,I) is_halting_on s,P )
let f be Function of (product (the_Values_of SCMPDS)),NAT; ::_thesis: ( ( for t being 0 -started State of SCMPDS st f . t = 0 holds
t . (DataLoc ((s . a),i)) >= 0 ) & ( for t being 0 -started State of SCMPDS
for Q being Instruction-Sequence of SCMPDS st ( for x being Int_position st x in X holds
t . x = s . x ) & t . a = s . a & t . (DataLoc ((s . a),i)) < 0 holds
( (IExec (I,Q,t)) . a = t . a & f . (Initialize (IExec (I,Q,t))) < f . t & I is_closed_on t,Q & I is_halting_on t,Q & ( for x being Int_position st x in X holds
(IExec (I,Q,t)) . x = t . x ) ) ) implies ( while<0 (a,i,I) is_closed_on s,P & while<0 (a,i,I) is_halting_on s,P ) )
set b = DataLoc ((s . a),i);
set WHL = while<0 (a,i,I);
set pWHL = stop (while<0 (a,i,I));
set pI = stop I;
set i1 = (a,i) >=0_goto ((card I) + 2);
set i2 = goto (- ((card I) + 1));
defpred S1[ Element of NAT ] means  for t being 0 -started State of SCMPDS
for Q being Instruction-Sequence of SCMPDS st f . t <= $1 & ( for x being Int_position st x in X holds
t . x = s . x ) & t . a = s . a holds
( while<0 (a,i,I) is_closed_on t,Q & while<0 (a,i,I) is_halting_on t,Q );
assume A2: for t being 0 -started State of SCMPDS st f . t = 0 holds
t . (DataLoc ((s . a),i)) >= 0 ; ::_thesis: ( ex t being 0 -started State of SCMPDS ex Q being Instruction-Sequence of SCMPDS st
( ( for x being Int_position st x in X holds
t . x = s . x ) & t . a = s . a & t . (DataLoc ((s . a),i)) < 0 & not ( (IExec (I,Q,t)) . a = t . a & f . (Initialize (IExec (I,Q,t))) < f . t & I is_closed_on t,Q & I is_halting_on t,Q & ( for x being Int_position st x in X holds
(IExec (I,Q,t)) . x = t . x ) ) ) or ( while<0 (a,i,I) is_closed_on s,P & while<0 (a,i,I) is_halting_on s,P ) )
assume A3: for t being 0 -started State of SCMPDS
for Q being Instruction-Sequence of SCMPDS st ( for x being Int_position st x in X holds
t . x = s . x ) & t . a = s . a & t . (DataLoc ((s . a),i)) < 0 holds
( (IExec (I,Q,t)) . a = t . a & f . (Initialize (IExec (I,Q,t))) < f . t & I is_closed_on t,Q & I is_halting_on t,Q & ( for x being Int_position st x in X holds
(IExec (I,Q,t)) . x = t . x ) ) ; ::_thesis: ( while<0 (a,i,I) is_closed_on s,P & while<0 (a,i,I) is_halting_on s,P )
A4: for k being Element of NAT st S1[k] holds
S1[k + 1]
proof
let k be Element of NAT ; ::_thesis: ( S1[k] implies S1[k + 1] )
assume A5: S1[k] ; ::_thesis: S1[k + 1]
now__::_thesis:_for_t_being_0_-started_State_of_SCMPDS
for_Q_being_Instruction-Sequence_of_SCMPDS_st_f_._t_<=_k_+_1_&_(_for_x_being_Int_position_st_x_in_X_holds_
t_._x_=_s_._x_)_&_t_._a_=_s_._a_holds_
(_while<0_(a,i,I)_is_closed_on_t,Q_&_while<0_(a,i,I)_is_halting_on_t,Q_)
let t be 0 -started State of SCMPDS; ::_thesis: for Q being Instruction-Sequence of SCMPDS st f . t <= k + 1 & ( for x being Int_position st x in X holds
t . x = s . x ) & t . a = s . a holds
( while<0 (a,i,I) is_closed_on b2,b3 & while<0 (a,i,I) is_halting_on b2,b3 )
let Q be Instruction-Sequence of SCMPDS; ::_thesis: ( f . t <= k + 1 & ( for x being Int_position st x in X holds
t . x = s . x ) & t . a = s . a implies ( while<0 (a,i,I) is_closed_on b1,b2 & while<0 (a,i,I) is_halting_on b1,b2 ) )
T: Initialize t = t by MEMSTR_0:44;
assume A6: f . t <= k + 1 ; ::_thesis: ( ( for x being Int_position st x in X holds
t . x = s . x ) & t . a = s . a implies ( while<0 (a,i,I) is_closed_on b1,b2 & while<0 (a,i,I) is_halting_on b1,b2 ) )
assume A7: for x being Int_position st x in X holds
t . x = s . x ; ::_thesis: ( t . a = s . a implies ( while<0 (a,i,I) is_closed_on b1,b2 & while<0 (a,i,I) is_halting_on b1,b2 ) )
assume A8: t . a = s . a ; ::_thesis: ( while<0 (a,i,I) is_closed_on b1,b2 & while<0 (a,i,I) is_halting_on b1,b2 )
percases ( t . (DataLoc ((s . a),i)) >= 0 or t . (DataLoc ((s . a),i)) < 0 ) ;
suppose t . (DataLoc ((s . a),i)) >= 0 ; ::_thesis: ( while<0 (a,i,I) is_closed_on b1,b2 & while<0 (a,i,I) is_halting_on b1,b2 )
hence  ( while<0 (a,i,I) is_closed_on t,Q & while<0 (a,i,I) is_halting_on t,Q ) by A8, Th9; ::_thesis: verum
end;
supposeA9: t . (DataLoc ((s . a),i)) < 0 ; ::_thesis: ( while<0 (a,i,I) is_closed_on b1,b2 & while<0 (a,i,I) is_halting_on b1,b2 )
A13: 0 in dom (stop (while<0 (a,i,I))) by COMPOS_1:36;
A15: while<0 (a,i,I) = ((a,i) >=0_goto ((card I) + 2)) ';' (I ';' (goto (- ((card I) + 1)))) by SCMPDS_4:15;
A16: f . (Initialize (IExec (I,Q,t))) < f . t by A3, A7, A8, A9;
set t2 = t;
set Q2 = Q +* (stop I);
set t3 = t;
set Q3 = Q +* (stop (while<0 (a,i,I)));
set t4 = Comput ((Q +* (stop (while<0 (a,i,I)))),t,1);
set Q4 = Q +* (stop (while<0 (a,i,I)));
A20: stop I c= Q +* (stop I) by FUNCT_4:25;
A21: Comput ((Q +* (stop (while<0 (a,i,I)))),t,(0 + 1)) = Following ((Q +* (stop (while<0 (a,i,I)))),(Comput ((Q +* (stop (while<0 (a,i,I)))),t,0))) by EXTPRO_1:3
.= Following ((Q +* (stop (while<0 (a,i,I)))),t)
.= Exec (((a,i) >=0_goto ((card I) + 2)),t) by A15, T, SCMPDS_6:11 ;
 for a being Int_position holds t . a = (Comput ((Q +* (stop (while<0 (a,i,I)))),t,1)) . a by A21, SCMPDS_2:57;
then A23: DataPart t = DataPart (Comput ((Q +* (stop (while<0 (a,i,I)))),t,1)) by SCMPDS_4:8;
XX: while<0 (a,i,I) c= stop (while<0 (a,i,I)) by AFINSQ_1:74;
 stop (while<0 (a,i,I)) c= Q +* (stop (while<0 (a,i,I))) by FUNCT_4:25;
then A24: while<0 (a,i,I) c= Q +* (stop (while<0 (a,i,I))) by XX, XBOOLE_1:1;
 Shift (I,1) c= while<0 (a,i,I) by Lm2;
then A25: Shift (I,1) c= Q +* (stop (while<0 (a,i,I))) by A24, XBOOLE_1:1;
set m2 = LifeSpan ((Q +* (stop I)),t);
set t5 = Comput ((Q +* (stop (while<0 (a,i,I)))),(Comput ((Q +* (stop (while<0 (a,i,I)))),t,1)),(LifeSpan ((Q +* (stop I)),t)));
set Q5 = Q +* (stop (while<0 (a,i,I)));
set l1 = (card I) + 1;
A26: IC t = 0 by MEMSTR_0:def_11;
set m3 = (LifeSpan ((Q +* (stop I)),t)) + 1;
set t6 = Comput ((Q +* (stop (while<0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 1));
set Q6 = Q +* (stop (while<0 (a,i,I)));
set t7 = Comput ((Q +* (stop (while<0 (a,i,I)))),t,(((LifeSpan ((Q +* (stop I)),t)) + 1) + 1));
set Q7 = Q +* (stop (while<0 (a,i,I)));
 (card I) + 1 < (card I) + 2 by XREAL_1:6;
then A27: (card I) + 1 in dom (while<0 (a,i,I)) by Th7;
A28: I is_closed_on t,Q by A3, A7, A8, A9;
A29: I is_closed_on t,Q +* (stop I) by A3, A7, A8, A9;
 I is_halting_on t,Q by A3, A7, A8, A9;
then A30: Q +* (stop I) halts_on t by T, SCMPDS_6:def_3;
 (Q +* (stop I)) +* (stop I) halts_on t by A30;
then A32: I is_halting_on t,Q +* (stop I) by T, SCMPDS_6:def_3;
A33: IC (Comput ((Q +* (stop (while<0 (a,i,I)))),t,1)) = succ (IC t) by A9, A21, A8, SCMPDS_2:57
.= 0 + 1 by A26 ;
then A34: IC (Comput ((Q +* (stop (while<0 (a,i,I)))),(Comput ((Q +* (stop (while<0 (a,i,I)))),t,1)),(LifeSpan ((Q +* (stop I)),t)))) = (card I) + 1 by A20, A32, A29, A23, A25, SCMPDS_7:18;
A35: (Q +* (stop (while<0 (a,i,I)))) /. (IC (Comput ((Q +* (stop (while<0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 1)))) = (Q +* (stop (while<0 (a,i,I)))) . (IC (Comput ((Q +* (stop (while<0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 1)))) by PBOOLE:143;
A36: Comput ((Q +* (stop (while<0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 1)) = Comput ((Q +* (stop (while<0 (a,i,I)))),(Comput ((Q +* (stop (while<0 (a,i,I)))),t,1)),(LifeSpan ((Q +* (stop I)),t))) by EXTPRO_1:4;
then A37: CurInstr ((Q +* (stop (while<0 (a,i,I)))),(Comput ((Q +* (stop (while<0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 1)))) = (Q +* (stop (while<0 (a,i,I)))) . ((card I) + 1) by A20, A32, A29, A33, A23, A25, A35, SCMPDS_7:18
.= (while<0 (a,i,I)) . ((card I) + 1) by A27, A24, GRFUNC_1:2
.= goto (- ((card I) + 1)) by Th8 ;
A38: DataPart (Comput ((Q +* (stop I)),t,(LifeSpan ((Q +* (stop I)),t)))) = DataPart (Comput ((Q +* (stop (while<0 (a,i,I)))),(Comput ((Q +* (stop (while<0 (a,i,I)))),t,1)),(LifeSpan ((Q +* (stop I)),t)))) by A20, A32, A29, A33, A23, A25, SCMPDS_7:18;
then A39: DataPart (Comput ((Q +* (stop (while<0 (a,i,I)))),(Comput ((Q +* (stop (while<0 (a,i,I)))),t,1)),(LifeSpan ((Q +* (stop I)),t)))) = DataPart (Result ((Q +* (stop I)),t)) by A30, EXTPRO_1:23
.= DataPart (IExec (I,Q,t)) by SCMPDS_4:def_5 ;
A41: Comput ((Q +* (stop (while<0 (a,i,I)))),t,(((LifeSpan ((Q +* (stop I)),t)) + 1) + 1)) = Following ((Q +* (stop (while<0 (a,i,I)))),(Comput ((Q +* (stop (while<0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 1)))) by EXTPRO_1:3
.= Exec ((goto (- ((card I) + 1))),(Comput ((Q +* (stop (while<0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 1)))) by A37 ;
then  IC (Comput ((Q +* (stop (while<0 (a,i,I)))),t,(((LifeSpan ((Q +* (stop I)),t)) + 1) + 1))) = ICplusConst ((Comput ((Q +* (stop (while<0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 1))),(0 - ((card I) + 1))) by SCMPDS_2:54
.= 0 by A34, A36, SCMPDS_7:1 ;
then A42: Initialize (Comput ((Q +* (stop (while<0 (a,i,I)))),t,(((LifeSpan ((Q +* (stop I)),t)) + 1) + 1))) = Comput ((Q +* (stop (while<0 (a,i,I)))),t,(((LifeSpan ((Q +* (stop I)),t)) + 1) + 1)) by MEMSTR_0:46;
A43: IExec (I,Q,t) = Result ((Q +* (stop I)),t) by SCMPDS_4:def_5;
A44: now__::_thesis:_for_x_being_Int_position_st_x_in_X_holds_
(Comput_((Q_+*_(stop_(while<0_(a,i,I)))),t,(((LifeSpan_((Q_+*_(stop_I)),t))_+_1)_+_1)))_._x_=_s_._x
let x be Int_position; ::_thesis: ( x in X implies (Comput ((Q +* (stop (while<0 (a,i,I)))),t,(((LifeSpan ((Q +* (stop I)),t)) + 1) + 1))) . x = s . x )
assume A45: x in X ; ::_thesis: (Comput ((Q +* (stop (while<0 (a,i,I)))),t,(((LifeSpan ((Q +* (stop I)),t)) + 1) + 1))) . x = s . x
(Comput ((Q +* (stop (while<0 (a,i,I)))),(Comput ((Q +* (stop (while<0 (a,i,I)))),t,1)),(LifeSpan ((Q +* (stop I)),t)))) . x = (Comput ((Q +* (stop I)),t,(LifeSpan ((Q +* (stop I)),t)))) . x by A38, SCMPDS_4:8
.= (Result ((Q +* (stop I)),t)) . x by A30, EXTPRO_1:23
.= (IExec (I,Q,t)) . x by SCMPDS_4:def_5
.= t . x by A3, A7, A8, A9, A45
.= s . x by A7, A45 ;
hence  (Comput ((Q +* (stop (while<0 (a,i,I)))),t,(((LifeSpan ((Q +* (stop I)),t)) + 1) + 1))) . x = s . x by A36, A41, SCMPDS_2:54; ::_thesis: verum
end;
 InsCode (goto (- ((card I) + 1))) = 14 by SCMPDS_2:12;
then  InsCode (goto (- ((card I) + 1))) in {0,4,5,6,14} by ENUMSET1:def_3;
then A47: Initialize (Comput ((Q +* (stop (while<0 (a,i,I)))),t,(((LifeSpan ((Q +* (stop I)),t)) + 1) + 1))) = Initialize (Comput ((Q +* (stop (while<0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 1))) by A41, Th3
.= Initialize (IExec (I,Q,t)) by A39, A36, MEMSTR_0:80 ;
A48: now__::_thesis:_not_f_._(Comput_((Q_+*_(stop_(while<0_(a,i,I)))),t,(((LifeSpan_((Q_+*_(stop_I)),t))_+_1)_+_1)))_>_k
assume A49: f . (Comput ((Q +* (stop (while<0 (a,i,I)))),t,(((LifeSpan ((Q +* (stop I)),t)) + 1) + 1))) > k ; ::_thesis: contradiction
 f . (Comput ((Q +* (stop (while<0 (a,i,I)))),t,(((LifeSpan ((Q +* (stop I)),t)) + 1) + 1))) < k + 1 by A6, A16, A47, A42, XXREAL_0:2;
hence  contradiction by A49, INT_1:7; ::_thesis: verum
end;
A50: (Comput ((Q +* (stop (while<0 (a,i,I)))),(Comput ((Q +* (stop (while<0 (a,i,I)))),t,1)),(LifeSpan ((Q +* (stop I)),t)))) . a = (Comput ((Q +* (stop I)),t,(LifeSpan ((Q +* (stop I)),t)))) . a by A38, SCMPDS_4:8
.= (Result ((Q +* (stop I)),t)) . a by A30, EXTPRO_1:23
.= s . a by A8, A3, A7, A9, A43 ;
A52: (Comput ((Q +* (stop (while<0 (a,i,I)))),t,(((LifeSpan ((Q +* (stop I)),t)) + 1) + 1))) . a = (Comput ((Q +* (stop (while<0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 1))) . a by A41, SCMPDS_2:54
.= s . a by A50, EXTPRO_1:4 ;
then A53: while<0 (a,i,I) is_closed_on Comput ((Q +* (stop (while<0 (a,i,I)))),t,(((LifeSpan ((Q +* (stop I)),t)) + 1) + 1)),Q +* (stop (while<0 (a,i,I))) by A5, A44, A48, A42;
now__::_thesis:_for_k_being_Element_of_NAT_holds_IC_(Comput_((Q_+*_(stop_(while<0_(a,i,I)))),t,k))_in_dom_(stop_(while<0_(a,i,I)))
let k be Element of NAT ; ::_thesis: IC (Comput ((Q +* (stop (while<0 (a,i,I)))),t,b1)) in dom (stop (while<0 (a,i,I)))
percases ( k < ((LifeSpan ((Q +* (stop I)),t)) + 1) + 1 or k >= ((LifeSpan ((Q +* (stop I)),t)) + 1) + 1 ) ;
suppose k < ((LifeSpan ((Q +* (stop I)),t)) + 1) + 1 ; ::_thesis: IC (Comput ((Q +* (stop (while<0 (a,i,I)))),t,b1)) in dom (stop (while<0 (a,i,I)))
then A54: k <= (LifeSpan ((Q +* (stop I)),t)) + 1 by INT_1:7;
hereby  ::_thesis: verum
percases ( k <= LifeSpan ((Q +* (stop I)),t) or k = (LifeSpan ((Q +* (stop I)),t)) + 1 ) by A54, NAT_1:8;
supposeA55: k <= LifeSpan ((Q +* (stop I)),t) ; ::_thesis: IC (Comput ((Q +* (stop (while<0 (a,i,I)))),t,k)) in dom (stop (while<0 (a,i,I)))
hereby  ::_thesis: verum
percases ( k = 0 or k <> 0 ) ;
suppose k = 0 ; ::_thesis: IC (Comput ((Q +* (stop (while<0 (a,i,I)))),t,k)) in dom (stop (while<0 (a,i,I)))
hence  IC (Comput ((Q +* (stop (while<0 (a,i,I)))),t,k)) in dom (stop (while<0 (a,i,I))) by A13, A26, EXTPRO_1:2; ::_thesis: verum
end;
suppose k <> 0 ; ::_thesis: IC (Comput ((Q +* (stop (while<0 (a,i,I)))),t,k)) in dom (stop (while<0 (a,i,I)))
then consider kn being Nat such that
A56: k = kn + 1 by NAT_1:6;
reconsider kn = kn as Element of NAT by ORDINAL1:def_12;
reconsider lm = IC (Comput ((Q +* (stop I)),t,kn)) as Element of NAT ;
 kn < k by A56, XREAL_1:29;
then  kn < LifeSpan ((Q +* (stop I)),t) by A55, XXREAL_0:2;
then  (IC (Comput ((Q +* (stop I)),t,kn))) + 1 = IC (Comput ((Q +* (stop (while<0 (a,i,I)))),(Comput ((Q +* (stop (while<0 (a,i,I)))),t,1)),kn)) by A20, A32, A29, A33, A23, A25, SCMPDS_7:16;
then A58: IC (Comput ((Q +* (stop (while<0 (a,i,I)))),t,k)) = lm + 1 by A56, EXTPRO_1:4;
 IC (Comput ((Q +* (stop I)),t,kn)) in dom (stop I) by A28, T, SCMPDS_6:def_2;
then  lm < card (stop I) by AFINSQ_1:66;
then  lm < (card I) + 1 by COMPOS_1:55;
then A59: lm + 1 <= (card I) + 1 by INT_1:7;
 (card I) + 1 < (card I) + 3 by XREAL_1:6;
then  lm + 1 < (card I) + 3 by A59, XXREAL_0:2;
then  lm + 1 < card (stop (while<0 (a,i,I))) by Lm1;
hence  IC (Comput ((Q +* (stop (while<0 (a,i,I)))),t,k)) in dom (stop (while<0 (a,i,I))) by A58, AFINSQ_1:66; ::_thesis: verum
end;
end;
end;
end;
supposeA60: k = (LifeSpan ((Q +* (stop I)),t)) + 1 ; ::_thesis: IC (Comput ((Q +* (stop (while<0 (a,i,I)))),t,k)) in dom (stop (while<0 (a,i,I)))
 (card I) + 1 in dom (stop (while<0 (a,i,I))) by A27, COMPOS_1:62;
hence  IC (Comput ((Q +* (stop (while<0 (a,i,I)))),t,k)) in dom (stop (while<0 (a,i,I))) by A20, A32, A29, A33, A23, A25, A36, A60, SCMPDS_7:18; ::_thesis: verum
end;
end;
end;
end;
suppose k >= ((LifeSpan ((Q +* (stop I)),t)) + 1) + 1 ; ::_thesis: IC (Comput ((Q +* (stop (while<0 (a,i,I)))),t,b1)) in dom (stop (while<0 (a,i,I)))
then consider nn being Nat such that
A61: k = (((LifeSpan ((Q +* (stop I)),t)) + 1) + 1) + nn by NAT_1:10;
reconsider nn = nn as Element of NAT by ORDINAL1:def_12;
 Comput ((Q +* (stop (while<0 (a,i,I)))),t,k) = Comput (((Q +* (stop (while<0 (a,i,I)))) +* (stop (while<0 (a,i,I)))),(Comput ((Q +* (stop (while<0 (a,i,I)))),t,(((LifeSpan ((Q +* (stop I)),t)) + 1) + 1))),nn) by A61, EXTPRO_1:4;
hence  IC (Comput ((Q +* (stop (while<0 (a,i,I)))),t,k)) in dom (stop (while<0 (a,i,I))) by A53, A42, SCMPDS_6:def_2; ::_thesis: verum
end;
end;
end;
hence  while<0 (a,i,I) is_closed_on t,Q by T, SCMPDS_6:def_2; ::_thesis: while<0 (a,i,I) is_halting_on t,Q
RR: Q +* (stop (while<0 (a,i,I))) = (Q +* (stop (while<0 (a,i,I)))) +* (stop (while<0 (a,i,I))) ;
 while<0 (a,i,I) is_halting_on Comput ((Q +* (stop (while<0 (a,i,I)))),t,(((LifeSpan ((Q +* (stop I)),t)) + 1) + 1)),Q +* (stop (while<0 (a,i,I))) by A5, A52, A44, A48, A42;
then  Q +* (stop (while<0 (a,i,I))) halts_on Comput ((Q +* (stop (while<0 (a,i,I)))),t,(((LifeSpan ((Q +* (stop I)),t)) + 1) + 1)) by A42, RR, SCMPDS_6:def_3;
then  Q +* (stop (while<0 (a,i,I))) halts_on t by EXTPRO_1:22;
hence  while<0 (a,i,I) is_halting_on t,Q by T, SCMPDS_6:def_3; ::_thesis: verum
end;
end;
end;
hence  S1[k + 1] ; ::_thesis: verum
end;
set n = f . s;
A62: S1[ 0 ]
proof
let t be 0 -started State of SCMPDS; ::_thesis: for Q being Instruction-Sequence of SCMPDS st f . t <= 0 & ( for x being Int_position st x in X holds
t . x = s . x ) & t . a = s . a holds
( while<0 (a,i,I) is_closed_on t,Q & while<0 (a,i,I) is_halting_on t,Q )
let Q be Instruction-Sequence of SCMPDS; ::_thesis: ( f . t <= 0 & ( for x being Int_position st x in X holds
t . x = s . x ) & t . a = s . a implies ( while<0 (a,i,I) is_closed_on t,Q & while<0 (a,i,I) is_halting_on t,Q ) )
assume  f . t <= 0 ; ::_thesis: ( ex x being Int_position st
( x in X & not t . x = s . x ) or not t . a = s . a or ( while<0 (a,i,I) is_closed_on t,Q & while<0 (a,i,I) is_halting_on t,Q ) )
then  f . t = 0 ;
then A63: t . (DataLoc ((s . a),i)) >= 0 by A2;
assume  for x being Int_position st x in X holds
t . x = s . x ; ::_thesis: ( not t . a = s . a or ( while<0 (a,i,I) is_closed_on t,Q & while<0 (a,i,I) is_halting_on t,Q ) )
assume  t . a = s . a ; ::_thesis: ( while<0 (a,i,I) is_closed_on t,Q & while<0 (a,i,I) is_halting_on t,Q )
hence  ( while<0 (a,i,I) is_closed_on t,Q & while<0 (a,i,I) is_halting_on t,Q ) by A63, Th9; ::_thesis: verum
end;
 for k being Element of NAT holds S1[k] from NAT_1:sch_1(A62, A4);
then A64: S1[f . s] ;
 for x being Int_position st x in X holds
s . x = s . x ;
hence  ( while<0 (a,i,I) is_closed_on s,P & while<0 (a,i,I) is_halting_on s,P ) by A64; ::_thesis: verum
end;

theorem :: SCMPDS_8:14
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 being Integer
for X being set
for f being Function of (product (the_Values_of SCMPDS)),NAT st s . (DataLoc ((s . a),i)) < 0 & ( for t being 0 -started State of SCMPDS st f . t = 0 holds
t . (DataLoc ((s . a),i)) >= 0 ) & ( for t being 0 -started State of SCMPDS
for Q being Instruction-Sequence of SCMPDS st ( for x being Int_position st x in X 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 & f . (Initialize (IExec (I,Q,t))) < f . t & ( for x being Int_position st x in X holds
(IExec (I,Q,t)) . x = t . x ) ) ) holds
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 being Int_position
for i being Integer
for X being set
for f being Function of (product (the_Values_of SCMPDS)),NAT st s . (DataLoc ((s . a),i)) < 0 & ( for t being 0 -started State of SCMPDS st f . t = 0 holds
t . (DataLoc ((s . a),i)) >= 0 ) & ( for t being 0 -started State of SCMPDS
for Q being Instruction-Sequence of SCMPDS st ( for x being Int_position st x in X 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 & f . (Initialize (IExec (I,Q,t))) < f . t & ( for x being Int_position st x in X holds
(IExec (I,Q,t)) . x = t . x ) ) ) holds
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 being Int_position
for i being Integer
for X being set
for f being Function of (product (the_Values_of SCMPDS)),NAT st s . (DataLoc ((s . a),i)) < 0 & ( for t being 0 -started State of SCMPDS st f . t = 0 holds
t . (DataLoc ((s . a),i)) >= 0 ) & ( for t being 0 -started State of SCMPDS
for Q being Instruction-Sequence of SCMPDS st ( for x being Int_position st x in X 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 & f . (Initialize (IExec (I,Q,t))) < f . t & ( for x being Int_position st x in X holds
(IExec (I,Q,t)) . x = t . x ) ) ) holds
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 being Int_position
for i being Integer
for X being set
for f being Function of (product (the_Values_of SCMPDS)),NAT st s . (DataLoc ((s . a),i)) < 0 & ( for t being 0 -started State of SCMPDS st f . t = 0 holds
t . (DataLoc ((s . a),i)) >= 0 ) & ( for t being 0 -started State of SCMPDS
for Q being Instruction-Sequence of SCMPDS st ( for x being Int_position st x in X 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 & f . (Initialize (IExec (I,Q,t))) < f . t & ( for x being Int_position st x in X holds
(IExec (I,Q,t)) . x = t . x ) ) ) holds
IExec ((while<0 (a,i,I)),P,s) = IExec ((while<0 (a,i,I)),P,(Initialize (IExec (I,P,s))))
let a be Int_position; ::_thesis: for i being Integer
for X being set
for f being Function of (product (the_Values_of SCMPDS)),NAT st s . (DataLoc ((s . a),i)) < 0 & ( for t being 0 -started State of SCMPDS st f . t = 0 holds
t . (DataLoc ((s . a),i)) >= 0 ) & ( for t being 0 -started State of SCMPDS
for Q being Instruction-Sequence of SCMPDS st ( for x being Int_position st x in X 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 & f . (Initialize (IExec (I,Q,t))) < f . t & ( for x being Int_position st x in X holds
(IExec (I,Q,t)) . x = t . x ) ) ) holds
IExec ((while<0 (a,i,I)),P,s) = IExec ((while<0 (a,i,I)),P,(Initialize (IExec (I,P,s))))
let i be Integer; ::_thesis: for X being set
for f being Function of (product (the_Values_of SCMPDS)),NAT st s . (DataLoc ((s . a),i)) < 0 & ( for t being 0 -started State of SCMPDS st f . t = 0 holds
t . (DataLoc ((s . a),i)) >= 0 ) & ( for t being 0 -started State of SCMPDS
for Q being Instruction-Sequence of SCMPDS st ( for x being Int_position st x in X 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 & f . (Initialize (IExec (I,Q,t))) < f . t & ( for x being Int_position st x in X holds
(IExec (I,Q,t)) . x = t . x ) ) ) holds
IExec ((while<0 (a,i,I)),P,s) = IExec ((while<0 (a,i,I)),P,(Initialize (IExec (I,P,s))))
let X be set ; ::_thesis: for f being Function of (product (the_Values_of SCMPDS)),NAT st s . (DataLoc ((s . a),i)) < 0 & ( for t being 0 -started State of SCMPDS st f . t = 0 holds
t . (DataLoc ((s . a),i)) >= 0 ) & ( for t being 0 -started State of SCMPDS
for Q being Instruction-Sequence of SCMPDS st ( for x being Int_position st x in X 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 & f . (Initialize (IExec (I,Q,t))) < f . t & ( for x being Int_position st x in X holds
(IExec (I,Q,t)) . x = t . x ) ) ) holds
IExec ((while<0 (a,i,I)),P,s) = IExec ((while<0 (a,i,I)),P,(Initialize (IExec (I,P,s))))
let f be Function of (product (the_Values_of SCMPDS)),NAT; ::_thesis: ( s . (DataLoc ((s . a),i)) < 0 & ( for t being 0 -started State of SCMPDS st f . t = 0 holds
t . (DataLoc ((s . a),i)) >= 0 ) & ( for t being 0 -started State of SCMPDS
for Q being Instruction-Sequence of SCMPDS st ( for x being Int_position st x in X 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 & f . (Initialize (IExec (I,Q,t))) < f . t & ( for x being Int_position st x in X holds
(IExec (I,Q,t)) . x = t . x ) ) ) 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);
deffunc H1( State of SCMPDS) ->  Element of NAT = f . $1;
defpred S1[ State of SCMPDS] means  for x being Int_position st x in X holds
$1 . x = s . x;
assume A2: s . (DataLoc ((s . a),i)) < 0 ; ::_thesis: ( ex t being 0 -started State of SCMPDS st
( f . t = 0 & not t . (DataLoc ((s . a),i)) >= 0 ) or ex t being 0 -started State of SCMPDS ex Q being Instruction-Sequence of SCMPDS st
( ( for x being Int_position st x in X holds
t . x = s . x ) & 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 & f . (Initialize (IExec (I,Q,t))) < f . t & ( for x being Int_position st x in X holds
(IExec (I,Q,t)) . x = t . x ) ) ) or IExec ((while<0 (a,i,I)),P,s) = IExec ((while<0 (a,i,I)),P,(Initialize (IExec (I,P,s)))) )
assume  for t being 0 -started State of SCMPDS st f . t = 0 holds
t . (DataLoc ((s . a),i)) >= 0 ; ::_thesis: ( ex t being 0 -started State of SCMPDS ex Q being Instruction-Sequence of SCMPDS st
( ( for x being Int_position st x in X holds
t . x = s . x ) & 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 & f . (Initialize (IExec (I,Q,t))) < f . t & ( for x being Int_position st x in X holds
(IExec (I,Q,t)) . x = t . x ) ) ) or IExec ((while<0 (a,i,I)),P,s) = IExec ((while<0 (a,i,I)),P,(Initialize (IExec (I,P,s)))) )
then A3: for t being 0 -started State of SCMPDS st S1[t] & H1(t) = 0 holds
t . (DataLoc ((s . a),i)) >= 0 ;
assume A4: for t being 0 -started State of SCMPDS
for Q being Instruction-Sequence of SCMPDS st ( for x being Int_position st x in X 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 & f . (Initialize (IExec (I,Q,t))) < f . t & ( for x being Int_position st x in X holds
(IExec (I,Q,t)) . x = t . x ) ) ; ::_thesis: IExec ((while<0 (a,i,I)),P,s) = IExec ((while<0 (a,i,I)),P,(Initialize (IExec (I,P,s))))
A5: 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))] ) )
set v = t;
assume that
A6: S1[t] and
A7: ( t . a = s . a & 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);
thus  ( (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) ) by A4, A7, A6; ::_thesis: S1[ Initialize (IExec (I,Q,t))]
thus  S1[ Initialize (IExec (I,Q,t))] ::_thesis: verum
proof
set v = Initialize (IExec (I,Q,t));
let x be Int_position; ::_thesis: ( x in X implies (Initialize (IExec (I,Q,t))) . x = s . x )
assume A9: x in X ; ::_thesis: (Initialize (IExec (I,Q,t))) . x = s . x
then  (IExec (I,Q,t)) . x = t . x by A4, A7, A6;
then  (Initialize (IExec (I,Q,t))) . x = t . x by SCMPDS_5:15;
hence  (Initialize (IExec (I,Q,t))) . x = s . x by A6, A9; ::_thesis: verum
end;
end;
A10: S1[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_2(A2, A3, A10, A5);
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 Th15: :: SCMPDS_8:15
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 being Integer
for X being set st ( for t being 0 -started State of SCMPDS
for Q being Instruction-Sequence of SCMPDS st ( for x being Int_position st x in X holds
t . x = s . x ) & t . a = s . a & t . (DataLoc ((s . a),i)) < 0 holds
( (IExec (I,Q,t)) . a = t . a & (IExec (I,Q,t)) . (DataLoc ((s . a),i)) > t . (DataLoc ((s . a),i)) & I is_closed_on t,Q & I is_halting_on t,Q & ( for x being Int_position st x in X holds
(IExec (I,Q,t)) . x = t . x ) ) ) holds
( while<0 (a,i,I) is_closed_on s,P & 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 being Integer
for X being set st ( for t being 0 -started State of SCMPDS
for Q being Instruction-Sequence of SCMPDS st ( for x being Int_position st x in X holds
t . x = s . x ) & t . a = s . a & t . (DataLoc ((s . a),i)) < 0 holds
( (IExec (I,Q,t)) . a = t . a & (IExec (I,Q,t)) . (DataLoc ((s . a),i)) > t . (DataLoc ((s . a),i)) & I is_closed_on t,Q & I is_halting_on t,Q & ( for x being Int_position st x in X holds
(IExec (I,Q,t)) . x = t . x ) ) ) holds
( while<0 (a,i,I) is_closed_on s,P & 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 being Integer
for X being set st ( for t being 0 -started State of SCMPDS
for Q being Instruction-Sequence of SCMPDS st ( for x being Int_position st x in X holds
t . x = s . x ) & t . a = s . a & t . (DataLoc ((s . a),i)) < 0 holds
( (IExec (I,Q,t)) . a = t . a & (IExec (I,Q,t)) . (DataLoc ((s . a),i)) > t . (DataLoc ((s . a),i)) & I is_closed_on t,Q & I is_halting_on t,Q & ( for x being Int_position st x in X holds
(IExec (I,Q,t)) . x = t . x ) ) ) holds
( while<0 (a,i,I) is_closed_on s,P & 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 being Integer
for X being set st ( for t being 0 -started State of SCMPDS
for Q being Instruction-Sequence of SCMPDS st ( for x being Int_position st x in X holds
t . x = s . x ) & t . a = s . a & t . (DataLoc ((s . a),i)) < 0 holds
( (IExec (I,Q,t)) . a = t . a & (IExec (I,Q,t)) . (DataLoc ((s . a),i)) > t . (DataLoc ((s . a),i)) & I is_closed_on t,Q & I is_halting_on t,Q & ( for x being Int_position st x in X holds
(IExec (I,Q,t)) . x = t . x ) ) ) holds
( while<0 (a,i,I) is_closed_on s,P & while<0 (a,i,I) is_halting_on s,P )
let a be Int_position; ::_thesis: for i being Integer
for X being set st ( for t being 0 -started State of SCMPDS
for Q being Instruction-Sequence of SCMPDS st ( for x being Int_position st x in X holds
t . x = s . x ) & t . a = s . a & t . (DataLoc ((s . a),i)) < 0 holds
( (IExec (I,Q,t)) . a = t . a & (IExec (I,Q,t)) . (DataLoc ((s . a),i)) > t . (DataLoc ((s . a),i)) & I is_closed_on t,Q & I is_halting_on t,Q & ( for x being Int_position st x in X holds
(IExec (I,Q,t)) . x = t . x ) ) ) holds
( while<0 (a,i,I) is_closed_on s,P & while<0 (a,i,I) is_halting_on s,P )
let i be Integer; ::_thesis: for X being set st ( for t being 0 -started State of SCMPDS
for Q being Instruction-Sequence of SCMPDS st ( for x being Int_position st x in X holds
t . x = s . x ) & t . a = s . a & t . (DataLoc ((s . a),i)) < 0 holds
( (IExec (I,Q,t)) . a = t . a & (IExec (I,Q,t)) . (DataLoc ((s . a),i)) > t . (DataLoc ((s . a),i)) & I is_closed_on t,Q & I is_halting_on t,Q & ( for x being Int_position st x in X holds
(IExec (I,Q,t)) . x = t . x ) ) ) holds
( while<0 (a,i,I) is_closed_on s,P & while<0 (a,i,I) is_halting_on s,P )
let X be set ; ::_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 X holds
t . x = s . x ) & t . a = s . a & t . (DataLoc ((s . a),i)) < 0 holds
( (IExec (I,Q,t)) . a = t . a & (IExec (I,Q,t)) . (DataLoc ((s . a),i)) > t . (DataLoc ((s . a),i)) & I is_closed_on t,Q & I is_halting_on t,Q & ( for x being Int_position st x in X holds
(IExec (I,Q,t)) . x = t . x ) ) ) implies ( while<0 (a,i,I) is_closed_on s,P & while<0 (a,i,I) is_halting_on s,P ) )
set b = DataLoc ((s . a),i);
set WHL = while<0 (a,i,I);
set pWHL = stop (while<0 (a,i,I));
set pI = stop I;
set i1 = (a,i) >=0_goto ((card I) + 2);
set i2 = goto (- ((card I) + 1));
defpred S1[ Element of NAT ] means  for t being 0 -started State of SCMPDS
for Q being Instruction-Sequence of SCMPDS st - (t . (DataLoc ((s . a),i))) <= $1 & ( for x being Int_position st x in X holds
t . x = s . x ) & t . a = s . a holds
( while<0 (a,i,I) is_closed_on t,Q & while<0 (a,i,I) is_halting_on t,Q );
assume A2: for t being 0 -started State of SCMPDS
for Q being Instruction-Sequence of SCMPDS st ( for x being Int_position st x in X holds
t . x = s . x ) & t . a = s . a & t . (DataLoc ((s . a),i)) < 0 holds
( (IExec (I,Q,t)) . a = t . a & (IExec (I,Q,t)) . (DataLoc ((s . a),i)) > t . (DataLoc ((s . a),i)) & I is_closed_on t,Q & I is_halting_on t,Q & ( for x being Int_position st x in X holds
(IExec (I,Q,t)) . x = t . x ) ) ; ::_thesis: ( while<0 (a,i,I) is_closed_on s,P & while<0 (a,i,I) is_halting_on s,P )
A3: for k being Element of NAT st S1[k] holds
S1[k + 1]
proof
let k be Element of NAT ; ::_thesis: ( S1[k] implies S1[k + 1] )
assume A4: 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_._(DataLoc_((s_._a),i)))_<=_k_+_1_&_(_for_x_being_Int_position_st_x_in_X_holds_
t_._x_=_s_._x_)_&_t_._a_=_s_._a_holds_
(_while<0_(a,i,I)_is_closed_on_t,Q_&_while<0_(a,i,I)_is_halting_on_t,Q_)
let t be 0 -started State of SCMPDS; ::_thesis: for Q being Instruction-Sequence of SCMPDS st - (t . (DataLoc ((s . a),i))) <= k + 1 & ( for x being Int_position st x in X holds
t . x = s . x ) & t . a = s . a holds
( while<0 (a,i,I) is_closed_on b2,b3 & while<0 (a,i,I) is_halting_on b2,b3 )
let Q be Instruction-Sequence of SCMPDS; ::_thesis: ( - (t . (DataLoc ((s . a),i))) <= k + 1 & ( for x being Int_position st x in X holds
t . x = s . x ) & t . a = s . a implies ( while<0 (a,i,I) is_closed_on b1,b2 & while<0 (a,i,I) is_halting_on b1,b2 ) )
T: Initialize t = t by MEMSTR_0:44;
assume A5: - (t . (DataLoc ((s . a),i))) <= k + 1 ; ::_thesis: ( ( for x being Int_position st x in X holds
t . x = s . x ) & t . a = s . a implies ( while<0 (a,i,I) is_closed_on b1,b2 & while<0 (a,i,I) is_halting_on b1,b2 ) )
assume A6: for x being Int_position st x in X holds
t . x = s . x ; ::_thesis: ( t . a = s . a implies ( while<0 (a,i,I) is_closed_on b1,b2 & while<0 (a,i,I) is_halting_on b1,b2 ) )
assume A7: t . a = s . a ; ::_thesis: ( while<0 (a,i,I) is_closed_on b1,b2 & while<0 (a,i,I) is_halting_on b1,b2 )
percases ( t . (DataLoc ((s . a),i)) >= 0 or t . (DataLoc ((s . a),i)) < 0 ) ;
suppose t . (DataLoc ((s . a),i)) >= 0 ; ::_thesis: ( while<0 (a,i,I) is_closed_on b1,b2 & while<0 (a,i,I) is_halting_on b1,b2 )
hence  ( while<0 (a,i,I) is_closed_on t,Q & while<0 (a,i,I) is_halting_on t,Q ) by A7, Th9; ::_thesis: verum
end;
supposeA8: t . (DataLoc ((s . a),i)) < 0 ; ::_thesis: ( while<0 (a,i,I) is_closed_on b1,b2 & while<0 (a,i,I) is_halting_on b1,b2 )
A12: (IExec (I,Q,(Initialize t))) . (DataLoc ((s . a),i)) > t . (DataLoc ((s . a),i)) by A2, A6, A7, A8, T;
A13: 0 in dom (stop (while<0 (a,i,I))) by COMPOS_1:36;
A15: not DataLoc ((s . a),i) in dom (Start-At (0,SCMPDS)) by SCMPDS_4:18;
A16: while<0 (a,i,I) = ((a,i) >=0_goto ((card I) + 2)) ';' (I ';' (goto (- ((card I) + 1)))) by SCMPDS_4:15;
set t2 = Initialize t;
set Q2 = Q +* (stop I);
set t3 = Initialize t;
set Q3 = Q +* (stop (while<0 (a,i,I)));
set t4 = Comput ((Q +* (stop (while<0 (a,i,I)))),(Initialize t),1);
set Q4 = Q +* (stop (while<0 (a,i,I)));
A19: stop I c= Q +* (stop I) by FUNCT_4:25;
A20: Comput ((Q +* (stop (while<0 (a,i,I)))),(Initialize t),(0 + 1)) = Following ((Q +* (stop (while<0 (a,i,I)))),(Comput ((Q +* (stop (while<0 (a,i,I)))),(Initialize t),0))) by EXTPRO_1:3
.= Following ((Q +* (stop (while<0 (a,i,I)))),(Initialize t))
.= Exec (((a,i) >=0_goto ((card I) + 2)),(Initialize t)) by A16, SCMPDS_6:11 ;
 for a being Int_position holds (Initialize t) . a = (Comput ((Q +* (stop (while<0 (a,i,I)))),(Initialize t),1)) . a by A20, SCMPDS_2:57;
then A22: DataPart (Initialize t) = DataPart (Comput ((Q +* (stop (while<0 (a,i,I)))),(Initialize t),1)) by SCMPDS_4:8;
XX: while<0 (a,i,I) c= stop (while<0 (a,i,I)) by AFINSQ_1:74;
 stop (while<0 (a,i,I)) c= Q +* (stop (while<0 (a,i,I))) by FUNCT_4:25;
then A23: while<0 (a,i,I) c= Q +* (stop (while<0 (a,i,I))) by XX, XBOOLE_1:1;
 Shift (I,1) c= while<0 (a,i,I) by Lm2;
then A24: Shift (I,1) c= Q +* (stop (while<0 (a,i,I))) by A23, XBOOLE_1:1;
set m2 = LifeSpan ((Q +* (stop I)),(Initialize t));
set t5 = Comput ((Q +* (stop (while<0 (a,i,I)))),(Comput ((Q +* (stop (while<0 (a,i,I)))),(Initialize t),1)),(LifeSpan ((Q +* (stop I)),(Initialize t))));
set Q5 = Q +* (stop (while<0 (a,i,I)));
set l1 = (card I) + 1;
A25: IC (Initialize t) = 0 by MEMSTR_0:def_11;
set m3 = (LifeSpan ((Q +* (stop I)),(Initialize t))) + 1;
set t6 = Comput ((Q +* (stop (while<0 (a,i,I)))),(Initialize t),((LifeSpan ((Q +* (stop I)),(Initialize t))) + 1));
set Q6 = Q +* (stop (while<0 (a,i,I)));
set t7 = Comput ((Q +* (stop (while<0 (a,i,I)))),(Initialize t),(((LifeSpan ((Q +* (stop I)),(Initialize t))) + 1) + 1));
set Q7 = Q +* (stop (while<0 (a,i,I)));
 (card I) + 1 < (card I) + 2 by XREAL_1:6;
then A26: (card I) + 1 in dom (while<0 (a,i,I)) by Th7;
A27: IExec (I,Q,(Initialize t)) = Result ((Q +* (stop I)),(Initialize t)) by SCMPDS_4:def_5;
A28: I is_closed_on t,Q by A2, A6, A7, A8;
then A29: I is_closed_on Initialize t,Q +* (stop I) by SCMPDS_6:24;
 I is_halting_on t,Q by A2, A6, A7, A8;
then A30: Q +* (stop I) halts_on Initialize t by SCMPDS_6:def_3;
 (Q +* (stop I)) +* (stop I) halts_on Initialize (Initialize t) by A30;
then A32: I is_halting_on Initialize t,Q +* (stop I) by SCMPDS_6:def_3;
 not a in dom (Start-At (0,SCMPDS)) by SCMPDS_4:18;
then (Initialize t) . (DataLoc (((Initialize t) . a),i)) = (Initialize t) . (DataLoc ((s . a),i)) by A7, FUNCT_4:11
.= t . (DataLoc ((s . a),i)) by A15, FUNCT_4:11 ;
then A33: IC (Comput ((Q +* (stop (while<0 (a,i,I)))),(Initialize t),1)) = succ (IC (Initialize t)) by A8, A20, SCMPDS_2:57
.= 0 + 1 by A25 ;
then A34: IC (Comput ((Q +* (stop (while<0 (a,i,I)))),(Comput ((Q +* (stop (while<0 (a,i,I)))),(Initialize t),1)),(LifeSpan ((Q +* (stop I)),(Initialize t))))) = (card I) + 1 by A19, A32, A29, A22, A24, SCMPDS_7:18;
A35: (Q +* (stop (while<0 (a,i,I)))) /. (IC (Comput ((Q +* (stop (while<0 (a,i,I)))),(Initialize t),((LifeSpan ((Q +* (stop I)),(Initialize t))) + 1)))) = (Q +* (stop (while<0 (a,i,I)))) . (IC (Comput ((Q +* (stop (while<0 (a,i,I)))),(Initialize t),((LifeSpan ((Q +* (stop I)),(Initialize t))) + 1)))) by PBOOLE:143;
A36: Comput ((Q +* (stop (while<0 (a,i,I)))),(Initialize t),((LifeSpan ((Q +* (stop I)),(Initialize t))) + 1)) = Comput ((Q +* (stop (while<0 (a,i,I)))),(Comput ((Q +* (stop (while<0 (a,i,I)))),(Initialize t),1)),(LifeSpan ((Q +* (stop I)),(Initialize t)))) by EXTPRO_1:4;
then A37: CurInstr ((Q +* (stop (while<0 (a,i,I)))),(Comput ((Q +* (stop (while<0 (a,i,I)))),(Initialize t),((LifeSpan ((Q +* (stop I)),(Initialize t))) + 1)))) = (Q +* (stop (while<0 (a,i,I)))) . ((card I) + 1) by A19, A32, A29, A33, A22, A24, A35, SCMPDS_7:18
.= (while<0 (a,i,I)) . ((card I) + 1) by A26, A23, GRFUNC_1:2
.= goto (- ((card I) + 1)) by Th8 ;
A38: DataPart (Comput ((Q +* (stop I)),(Initialize t),(LifeSpan ((Q +* (stop I)),(Initialize t))))) = DataPart (Comput ((Q +* (stop (while<0 (a,i,I)))),(Comput ((Q +* (stop (while<0 (a,i,I)))),(Initialize t),1)),(LifeSpan ((Q +* (stop I)),(Initialize t))))) by A19, A32, A29, A33, A22, A24, SCMPDS_7:18;
then A39: (Comput ((Q +* (stop (while<0 (a,i,I)))),(Comput ((Q +* (stop (while<0 (a,i,I)))),(Initialize t),1)),(LifeSpan ((Q +* (stop I)),(Initialize t))))) . a = (Comput ((Q +* (stop I)),(Initialize t),(LifeSpan ((Q +* (stop I)),(Initialize t))))) . a by SCMPDS_4:8
.= (Result ((Q +* (stop I)),(Initialize t))) . a by A30, EXTPRO_1:23
.= s . a by A7, A2, A6, A8, A27, T ;
A41: Comput ((Q +* (stop (while<0 (a,i,I)))),(Initialize t),(((LifeSpan ((Q +* (stop I)),(Initialize t))) + 1) + 1)) = Following ((Q +* (stop (while<0 (a,i,I)))),(Comput ((Q +* (stop (while<0 (a,i,I)))),(Initialize t),((LifeSpan ((Q +* (stop I)),(Initialize t))) + 1)))) by EXTPRO_1:3
.= Exec ((goto (- ((card I) + 1))),(Comput ((Q +* (stop (while<0 (a,i,I)))),(Initialize t),((LifeSpan ((Q +* (stop I)),(Initialize t))) + 1)))) by A37 ;
then  IC (Comput ((Q +* (stop (while<0 (a,i,I)))),(Initialize t),(((LifeSpan ((Q +* (stop I)),(Initialize t))) + 1) + 1))) = ICplusConst ((Comput ((Q +* (stop (while<0 (a,i,I)))),(Initialize t),((LifeSpan ((Q +* (stop I)),(Initialize t))) + 1))),(0 - ((card I) + 1))) by SCMPDS_2:54
.= 0 by A34, A36, SCMPDS_7:1 ;
then A42: Initialize (Comput ((Q +* (stop (while<0 (a,i,I)))),(Initialize t),(((LifeSpan ((Q +* (stop I)),(Initialize t))) + 1) + 1))) = Comput ((Q +* (stop (while<0 (a,i,I)))),(Initialize t),(((LifeSpan ((Q +* (stop I)),(Initialize t))) + 1) + 1)) by MEMSTR_0:46;
A43: now__::_thesis:_for_x_being_Int_position_st_x_in_X_holds_
(Comput_((Q_+*_(stop_(while<0_(a,i,I)))),(Initialize_t),(((LifeSpan_((Q_+*_(stop_I)),(Initialize_t)))_+_1)_+_1)))_._x_=_s_._x
let x be Int_position; ::_thesis: ( x in X implies (Comput ((Q +* (stop (while<0 (a,i,I)))),(Initialize t),(((LifeSpan ((Q +* (stop I)),(Initialize t))) + 1) + 1))) . x = s . x )
assume A44: x in X ; ::_thesis: (Comput ((Q +* (stop (while<0 (a,i,I)))),(Initialize t),(((LifeSpan ((Q +* (stop I)),(Initialize t))) + 1) + 1))) . x = s . x
(Comput ((Q +* (stop (while<0 (a,i,I)))),(Comput ((Q +* (stop (while<0 (a,i,I)))),(Initialize t),1)),(LifeSpan ((Q +* (stop I)),(Initialize t))))) . x = (Comput ((Q +* (stop I)),(Initialize t),(LifeSpan ((Q +* (stop I)),(Initialize t))))) . x by A38, SCMPDS_4:8
.= (Result ((Q +* (stop I)),(Initialize t))) . x by A30, EXTPRO_1:23
.= (IExec (I,Q,(Initialize t))) . x by SCMPDS_4:def_5
.= t . x by A2, A6, A7, A8, A44, T
.= s . x by A6, A44 ;
hence  (Comput ((Q +* (stop (while<0 (a,i,I)))),(Initialize t),(((LifeSpan ((Q +* (stop I)),(Initialize t))) + 1) + 1))) . x = s . x by A36, A41, SCMPDS_2:54; ::_thesis: verum
end;
(Comput ((Q +* (stop (while<0 (a,i,I)))),(Comput ((Q +* (stop (while<0 (a,i,I)))),(Initialize t),1)),(LifeSpan ((Q +* (stop I)),(Initialize t))))) . (DataLoc ((s . a),i)) = (Comput ((Q +* (stop I)),(Initialize t),(LifeSpan ((Q +* (stop I)),(Initialize t))))) . (DataLoc ((s . a),i)) by A38, SCMPDS_4:8
.= (Result ((Q +* (stop I)),(Initialize t))) . (DataLoc ((s . a),i)) by A30, EXTPRO_1:23
.= (IExec (I,Q,(Initialize t))) . (DataLoc ((s . a),i)) by SCMPDS_4:def_5 ;
then A46: (Comput ((Q +* (stop (while<0 (a,i,I)))),(Initialize t),(((LifeSpan ((Q +* (stop I)),(Initialize t))) + 1) + 1))) . (DataLoc ((s . a),i)) = (IExec (I,Q,(Initialize t))) . (DataLoc ((s . a),i)) by A36, A41, SCMPDS_2:54;
A47: now__::_thesis:_not_-_((Comput_((Q_+*_(stop_(while<0_(a,i,I)))),(Initialize_t),(((LifeSpan_((Q_+*_(stop_I)),(Initialize_t)))_+_1)_+_1)))_._(DataLoc_((s_._a),i)))_>_k
 - ((Comput ((Q +* (stop (while<0 (a,i,I)))),(Initialize t),(((LifeSpan ((Q +* (stop I)),(Initialize t))) + 1) + 1))) . (DataLoc ((s . a),i))) < - (t . (DataLoc ((s . a),i))) by A12, A46, XREAL_1:24;
then A48: - ((Comput ((Q +* (stop (while<0 (a,i,I)))),(Initialize t),(((LifeSpan ((Q +* (stop I)),(Initialize t))) + 1) + 1))) . (DataLoc ((s . a),i))) < k + 1 by A5, XXREAL_0:2;
assume  - ((Comput ((Q +* (stop (while<0 (a,i,I)))),(Initialize t),(((LifeSpan ((Q +* (stop I)),(Initialize t))) + 1) + 1))) . (DataLoc ((s . a),i))) > k ; ::_thesis: contradiction
hence  contradiction by A48, INT_1:7; ::_thesis: verum
end;
A50: (Comput ((Q +* (stop (while<0 (a,i,I)))),(Initialize t),(((LifeSpan ((Q +* (stop I)),(Initialize t))) + 1) + 1))) . a = (Comput ((Q +* (stop (while<0 (a,i,I)))),(Initialize t),((LifeSpan ((Q +* (stop I)),(Initialize t))) + 1))) . a by A41, SCMPDS_2:54
.= s . a by A39, EXTPRO_1:4 ;
then A51: while<0 (a,i,I) is_closed_on Comput ((Q +* (stop (while<0 (a,i,I)))),(Initialize t),(((LifeSpan ((Q +* (stop I)),(Initialize t))) + 1) + 1)),Q +* (stop (while<0 (a,i,I))) by A4, A43, A47, A42;
now__::_thesis:_for_k_being_Element_of_NAT_holds_IC_(Comput_((Q_+*_(stop_(while<0_(a,i,I)))),(Initialize_t),k))_in_dom_(stop_(while<0_(a,i,I)))
let k be Element of NAT ; ::_thesis: IC (Comput ((Q +* (stop (while<0 (a,i,I)))),(Initialize t),b1)) in dom (stop (while<0 (a,i,I)))
percases ( k < ((LifeSpan ((Q +* (stop I)),(Initialize t))) + 1) + 1 or k >= ((LifeSpan ((Q +* (stop I)),(Initialize t))) + 1) + 1 ) ;
suppose k < ((LifeSpan ((Q +* (stop I)),(Initialize t))) + 1) + 1 ; ::_thesis: IC (Comput ((Q +* (stop (while<0 (a,i,I)))),(Initialize t),b1)) in dom (stop (while<0 (a,i,I)))
then A52: k <= (LifeSpan ((Q +* (stop I)),(Initialize t))) + 1 by INT_1:7;
hereby  ::_thesis: verum
percases ( k <= LifeSpan ((Q +* (stop I)),(Initialize t)) or k = (LifeSpan ((Q +* (stop I)),(Initialize t))) + 1 ) by A52, NAT_1:8;
supposeA53: k <= LifeSpan ((Q +* (stop I)),(Initialize t)) ; ::_thesis: IC (Comput ((Q +* (stop (while<0 (a,i,I)))),(Initialize t),k)) in dom (stop (while<0 (a,i,I)))
hereby  ::_thesis: verum
percases ( k = 0 or k <> 0 ) ;
suppose k = 0 ; ::_thesis: IC (Comput ((Q +* (stop (while<0 (a,i,I)))),(Initialize t),k)) in dom (stop (while<0 (a,i,I)))
hence  IC (Comput ((Q +* (stop (while<0 (a,i,I)))),(Initialize t),k)) in dom (stop (while<0 (a,i,I))) by A13, A25, EXTPRO_1:2; ::_thesis: verum
end;
suppose k <> 0 ; ::_thesis: IC (Comput ((Q +* (stop (while<0 (a,i,I)))),(Initialize t),k)) in dom (stop (while<0 (a,i,I)))
then consider kn being Nat such that
A54: k = kn + 1 by NAT_1:6;
reconsider kn = kn as Element of NAT by ORDINAL1:def_12;
reconsider lm = IC (Comput ((Q +* (stop I)),(Initialize t),kn)) as Element of NAT ;
 kn < k by A54, XREAL_1:29;
then  kn < LifeSpan ((Q +* (stop I)),(Initialize t)) by A53, XXREAL_0:2;
then  (IC (Comput ((Q +* (stop I)),(Initialize t),kn))) + 1 = IC (Comput ((Q +* (stop (while<0 (a,i,I)))),(Comput ((Q +* (stop (while<0 (a,i,I)))),(Initialize t),1)),kn)) by A19, A32, A29, A33, A22, A24, SCMPDS_7:16;
then A56: IC (Comput ((Q +* (stop (while<0 (a,i,I)))),(Initialize t),k)) = lm + 1 by A54, EXTPRO_1:4;
 IC (Comput ((Q +* (stop I)),(Initialize t),kn)) in dom (stop I) by A28, SCMPDS_6:def_2;
then  lm < card (stop I) by AFINSQ_1:66;
then  lm < (card I) + 1 by COMPOS_1:55;
then A57: lm + 1 <= (card I) + 1 by INT_1:7;
 (card I) + 1 < (card I) + 3 by XREAL_1:6;
then  lm + 1 < (card I) + 3 by A57, XXREAL_0:2;
then  lm + 1 < card (stop (while<0 (a,i,I))) by Lm1;
hence  IC (Comput ((Q +* (stop (while<0 (a,i,I)))),(Initialize t),k)) in dom (stop (while<0 (a,i,I))) by A56, AFINSQ_1:66; ::_thesis: verum
end;
end;
end;
end;
supposeA58: k = (LifeSpan ((Q +* (stop I)),(Initialize t))) + 1 ; ::_thesis: IC (Comput ((Q +* (stop (while<0 (a,i,I)))),(Initialize t),k)) in dom (stop (while<0 (a,i,I)))
 (card I) + 1 in dom (stop (while<0 (a,i,I))) by A26, COMPOS_1:62;
hence  IC (Comput ((Q +* (stop (while<0 (a,i,I)))),(Initialize t),k)) in dom (stop (while<0 (a,i,I))) by A19, A32, A29, A33, A22, A24, A36, A58, SCMPDS_7:18; ::_thesis: verum
end;
end;
end;
end;
suppose k >= ((LifeSpan ((Q +* (stop I)),(Initialize t))) + 1) + 1 ; ::_thesis: IC (Comput ((Q +* (stop (while<0 (a,i,I)))),(Initialize t),b1)) in dom (stop (while<0 (a,i,I)))
then consider nn being Nat such that
A59: k = (((LifeSpan ((Q +* (stop I)),(Initialize t))) + 1) + 1) + nn by NAT_1:10;
reconsider nn = nn as Element of NAT by ORDINAL1:def_12;
 Comput ((Q +* (stop (while<0 (a,i,I)))),(Initialize t),k) = Comput (((Q +* (stop (while<0 (a,i,I)))) +* (stop (while<0 (a,i,I)))),(Initialize (Comput ((Q +* (stop (while<0 (a,i,I)))),(Initialize t),(((LifeSpan ((Q +* (stop I)),(Initialize t))) + 1) + 1)))),nn) by A42, A59, EXTPRO_1:4;
hence  IC (Comput ((Q +* (stop (while<0 (a,i,I)))),(Initialize t),k)) in dom (stop (while<0 (a,i,I))) by A51, SCMPDS_6:def_2; ::_thesis: verum
end;
end;
end;
hence  while<0 (a,i,I) is_closed_on t,Q by SCMPDS_6:def_2; ::_thesis: while<0 (a,i,I) is_halting_on t,Q
RR: Q +* (stop (while<0 (a,i,I))) = (Q +* (stop (while<0 (a,i,I)))) +* (stop (while<0 (a,i,I))) ;
 while<0 (a,i,I) is_halting_on Comput ((Q +* (stop (while<0 (a,i,I)))),(Initialize t),(((LifeSpan ((Q +* (stop I)),(Initialize t))) + 1) + 1)),Q +* (stop (while<0 (a,i,I))) by A4, A50, A43, A47, A42;
then  Q +* (stop (while<0 (a,i,I))) halts_on Comput ((Q +* (stop (while<0 (a,i,I)))),(Initialize t),(((LifeSpan ((Q +* (stop I)),(Initialize t))) + 1) + 1)) by A42, RR, SCMPDS_6:def_3;
then  Q +* (stop (while<0 (a,i,I))) halts_on Initialize t by EXTPRO_1:22;
hence  while<0 (a,i,I) is_halting_on t,Q by SCMPDS_6:def_3; ::_thesis: verum
end;
end;
end;
hence  S1[k + 1] ; ::_thesis: verum
end;
A60: S1[ 0 ]
proof
let t be 0 -started State of SCMPDS; ::_thesis: for Q being Instruction-Sequence of SCMPDS st - (t . (DataLoc ((s . a),i))) <= 0 & ( for x being Int_position st x in X holds
t . x = s . x ) & t . a = s . a holds
( while<0 (a,i,I) is_closed_on t,Q & while<0 (a,i,I) is_halting_on t,Q )
let Q be Instruction-Sequence of SCMPDS; ::_thesis: ( - (t . (DataLoc ((s . a),i))) <= 0 & ( for x being Int_position st x in X holds
t . x = s . x ) & t . a = s . a implies ( while<0 (a,i,I) is_closed_on t,Q & while<0 (a,i,I) is_halting_on t,Q ) )
assume  - (t . (DataLoc ((s . a),i))) <= 0 ; ::_thesis: ( ex x being Int_position st
( x in X & not t . x = s . x ) or not t . a = s . a or ( while<0 (a,i,I) is_closed_on t,Q & while<0 (a,i,I) is_halting_on t,Q ) )
then  - (t . (DataLoc ((s . a),i))) <= - 0 ;
then A61: t . (DataLoc ((s . a),i)) >= 0 by XREAL_1:24;
assume  for x being Int_position st x in X holds
t . x = s . x ; ::_thesis: ( not t . a = s . a or ( while<0 (a,i,I) is_closed_on t,Q & while<0 (a,i,I) is_halting_on t,Q ) )
assume  t . a = s . a ; ::_thesis: ( while<0 (a,i,I) is_closed_on t,Q & while<0 (a,i,I) is_halting_on t,Q )
hence  ( while<0 (a,i,I) is_closed_on t,Q & while<0 (a,i,I) is_halting_on t,Q ) by A61, Th9; ::_thesis: verum
end;
A62: for k being Element of NAT holds S1[k] from NAT_1:sch_1(A60, A3);
percases ( s . (DataLoc ((s . a),i)) >= 0 or s . (DataLoc ((s . a),i)) < 0 ) ;
suppose s . (DataLoc ((s . a),i)) >= 0 ; ::_thesis: ( while<0 (a,i,I) is_closed_on s,P & while<0 (a,i,I) is_halting_on s,P )
hence  ( while<0 (a,i,I) is_closed_on s,P & while<0 (a,i,I) is_halting_on s,P ) by Th9; ::_thesis: verum
end;
suppose s . (DataLoc ((s . a),i)) < 0 ; ::_thesis: ( while<0 (a,i,I) is_closed_on s,P & while<0 (a,i,I) is_halting_on s,P )
then reconsider n = - (s . (DataLoc ((s . a),i))) as Element of NAT by INT_1:3;
 ( S1[n] & ( for x being Int_position st x in X holds
s . x = s . x ) ) by A62;
hence  ( while<0 (a,i,I) is_closed_on s,P & while<0 (a,i,I) is_halting_on s,P ) ; ::_thesis: verum
end;
end;
end;

theorem :: SCMPDS_8:16
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 being Integer
for X being set st s . (DataLoc ((s . a),i)) < 0 & ( for t being 0 -started State of SCMPDS
for Q being Instruction-Sequence of SCMPDS st ( for x being Int_position st x in X holds
t . x = s . x ) & t . a = s . a & t . (DataLoc ((s . a),i)) < 0 holds
( (IExec (I,Q,t)) . a = t . a & (IExec (I,Q,t)) . (DataLoc ((s . a),i)) > t . (DataLoc ((s . a),i)) & I is_closed_on t,Q & I is_halting_on t,Q & ( for x being Int_position st x in X holds
(IExec (I,Q,t)) . x = t . x ) ) ) holds
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 being Int_position
for i being Integer
for X being set st s . (DataLoc ((s . a),i)) < 0 & ( for t being 0 -started State of SCMPDS
for Q being Instruction-Sequence of SCMPDS st ( for x being Int_position st x in X holds
t . x = s . x ) & t . a = s . a & t . (DataLoc ((s . a),i)) < 0 holds
( (IExec (I,Q,t)) . a = t . a & (IExec (I,Q,t)) . (DataLoc ((s . a),i)) > t . (DataLoc ((s . a),i)) & I is_closed_on t,Q & I is_halting_on t,Q & ( for x being Int_position st x in X holds
(IExec (I,Q,t)) . x = t . x ) ) ) holds
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 being Int_position
for i being Integer
for X being set st s . (DataLoc ((s . a),i)) < 0 & ( for t being 0 -started State of SCMPDS
for Q being Instruction-Sequence of SCMPDS st ( for x being Int_position st x in X holds
t . x = s . x ) & t . a = s . a & t . (DataLoc ((s . a),i)) < 0 holds
( (IExec (I,Q,t)) . a = t . a & (IExec (I,Q,t)) . (DataLoc ((s . a),i)) > t . (DataLoc ((s . a),i)) & I is_closed_on t,Q & I is_halting_on t,Q & ( for x being Int_position st x in X holds
(IExec (I,Q,t)) . x = t . x ) ) ) holds
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 being Int_position
for i being Integer
for X being set st s . (DataLoc ((s . a),i)) < 0 & ( for t being 0 -started State of SCMPDS
for Q being Instruction-Sequence of SCMPDS st ( for x being Int_position st x in X holds
t . x = s . x ) & t . a = s . a & t . (DataLoc ((s . a),i)) < 0 holds
( (IExec (I,Q,t)) . a = t . a & (IExec (I,Q,t)) . (DataLoc ((s . a),i)) > t . (DataLoc ((s . a),i)) & I is_closed_on t,Q & I is_halting_on t,Q & ( for x being Int_position st x in X holds
(IExec (I,Q,t)) . x = t . x ) ) ) holds
IExec ((while<0 (a,i,I)),P,s) = IExec ((while<0 (a,i,I)),P,(Initialize (IExec (I,P,s))))
let a be Int_position; ::_thesis: for i being Integer
for X being set st s . (DataLoc ((s . a),i)) < 0 & ( for t being 0 -started State of SCMPDS
for Q being Instruction-Sequence of SCMPDS st ( for x being Int_position st x in X holds
t . x = s . x ) & t . a = s . a & t . (DataLoc ((s . a),i)) < 0 holds
( (IExec (I,Q,t)) . a = t . a & (IExec (I,Q,t)) . (DataLoc ((s . a),i)) > t . (DataLoc ((s . a),i)) & I is_closed_on t,Q & I is_halting_on t,Q & ( for x being Int_position st x in X holds
(IExec (I,Q,t)) . x = t . x ) ) ) holds
IExec ((while<0 (a,i,I)),P,s) = IExec ((while<0 (a,i,I)),P,(Initialize (IExec (I,P,s))))
let i be Integer; ::_thesis: for X being set st s . (DataLoc ((s . a),i)) < 0 & ( for t being 0 -started State of SCMPDS
for Q being Instruction-Sequence of SCMPDS st ( for x being Int_position st x in X holds
t . x = s . x ) & t . a = s . a & t . (DataLoc ((s . a),i)) < 0 holds
( (IExec (I,Q,t)) . a = t . a & (IExec (I,Q,t)) . (DataLoc ((s . a),i)) > t . (DataLoc ((s . a),i)) & I is_closed_on t,Q & I is_halting_on t,Q & ( for x being Int_position st x in X holds
(IExec (I,Q,t)) . x = t . x ) ) ) holds
IExec ((while<0 (a,i,I)),P,s) = IExec ((while<0 (a,i,I)),P,(Initialize (IExec (I,P,s))))
let X be set ; ::_thesis: ( s . (DataLoc ((s . a),i)) < 0 & ( for t being 0 -started State of SCMPDS
for Q being Instruction-Sequence of SCMPDS st ( for x being Int_position st x in X holds
t . x = s . x ) & t . a = s . a & t . (DataLoc ((s . a),i)) < 0 holds
( (IExec (I,Q,t)) . a = t . a & (IExec (I,Q,t)) . (DataLoc ((s . a),i)) > t . (DataLoc ((s . a),i)) & I is_closed_on t,Q & I is_halting_on t,Q & ( for x being Int_position st x in X holds
(IExec (I,Q,t)) . x = t . x ) ) ) implies IExec ((while<0 (a,i,I)),P,s) = IExec ((while<0 (a,i,I)),P,(Initialize (IExec (I,P,s)))) )
I: Initialize s = s by MEMSTR_0:44;
set b = DataLoc ((s . a),i);
set WHL = while<0 (a,i,I);
set pWHL = stop (while<0 (a,i,I));
set P1 = P +* (stop (while<0 (a,i,I)));
set i1 = (a,i) >=0_goto ((card I) + 2);
set i2 = goto (- ((card I) + 1));
assume A2: s . (DataLoc ((s . a),i)) < 0 ; ::_thesis: ( ex t being 0 -started State of SCMPDS ex Q being Instruction-Sequence of SCMPDS st
( ( for x being Int_position st x in X holds
t . x = s . x ) & t . a = s . a & t . (DataLoc ((s . a),i)) < 0 & not ( (IExec (I,Q,t)) . a = t . a & (IExec (I,Q,t)) . (DataLoc ((s . a),i)) > t . (DataLoc ((s . a),i)) & I is_closed_on t,Q & I is_halting_on t,Q & ( for x being Int_position st x in X holds
(IExec (I,Q,t)) . x = t . x ) ) ) or IExec ((while<0 (a,i,I)),P,s) = IExec ((while<0 (a,i,I)),P,(Initialize (IExec (I,P,s)))) )
set Es = IExec (I,P,s);
set bj = DataLoc (((Initialize (IExec (I,P,s))) . a),i);
set EP = P;
set PI = P +* (stop I);
set m1 = (LifeSpan ((P +* (stop I)),s)) + 2;
set s2 = Initialize (IExec (I,P,s));
set P2 = P +* (stop (while<0 (a,i,I)));
set m2 = LifeSpan ((P +* (stop (while<0 (a,i,I)))),(Initialize (IExec (I,P,s))));
assume A6: for t being 0 -started State of SCMPDS
for Q being Instruction-Sequence of SCMPDS st ( for x being Int_position st x in X holds
t . x = s . x ) & t . a = s . a & t . (DataLoc ((s . a),i)) < 0 holds
( (IExec (I,Q,t)) . a = t . a & (IExec (I,Q,t)) . (DataLoc ((s . a),i)) > t . (DataLoc ((s . a),i)) & I is_closed_on t,Q & I is_halting_on t,Q & ( for x being Int_position st x in X holds
(IExec (I,Q,t)) . x = t . x ) ) ; ::_thesis: IExec ((while<0 (a,i,I)),P,s) = IExec ((while<0 (a,i,I)),P,(Initialize (IExec (I,P,s))))
then  while<0 (a,i,I) is_halting_on s,P by Th15;
then A7: P +* (stop (while<0 (a,i,I))) halts_on s by I, SCMPDS_6:def_3;
A8: stop I c= P +* (stop I) by FUNCT_4:25;
A9: for x being Int_position st x in X holds
s . x = s . x ;
then  I is_halting_on s,P by A2, A6;
then A10: P +* (stop I) halts_on s by I, SCMPDS_6:def_3;
 (P +* (stop I)) +* (stop I) halts_on s by A10;
then A12: I is_halting_on s,P +* (stop I) by I, SCMPDS_6:def_3;
A13: (Initialize (IExec (I,P,s))) . a = (IExec (I,P,s)) . a by SCMPDS_5:15
.= s . a by A2, A6, A9 ;
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_X_holds_
t_._x_=_(Initialize_(IExec_(I,P,s)))_._x_)_&_t_._a_=_(Initialize_(IExec_(I,P,s)))_._a_&_t_._(DataLoc_(((Initialize_(IExec_(I,P,s)))_._a),i))_<_0_holds_
(_(IExec_(I,Q,t))_._a_=_t_._a_&_(IExec_(I,Q,t))_._(DataLoc_(((Initialize_(IExec_(I,P,s)))_._a),i))_>_t_._(DataLoc_(((Initialize_(IExec_(I,P,s)))_._a),i))_&_I_is_closed_on_t,Q_&_I_is_halting_on_t,Q_&_(_for_x_being_Int_position_st_x_in_X_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 X holds
t . x = (Initialize (IExec (I,P,s))) . x ) & t . a = (Initialize (IExec (I,P,s))) . a & t . (DataLoc (((Initialize (IExec (I,P,s))) . a),i)) < 0 holds
( (IExec (I,Q,t)) . a = t . a & (IExec (I,Q,t)) . (DataLoc (((Initialize (IExec (I,P,s))) . a),i)) > t . (DataLoc (((Initialize (IExec (I,P,s))) . a),i)) & I is_closed_on t,Q & I is_halting_on t,Q & ( for x being Int_position st x in X holds
(IExec (I,Q,t)) . x = t . x ) )
let Q be Instruction-Sequence of SCMPDS; ::_thesis: ( ( for x being Int_position st x in X holds
t . x = (Initialize (IExec (I,P,s))) . x ) & t . a = (Initialize (IExec (I,P,s))) . a & t . (DataLoc (((Initialize (IExec (I,P,s))) . a),i)) < 0 implies ( (IExec (I,Q,t)) . a = t . a & (IExec (I,Q,t)) . (DataLoc (((Initialize (IExec (I,P,s))) . a),i)) > t . (DataLoc (((Initialize (IExec (I,P,s))) . a),i)) & I is_closed_on t,Q & I is_halting_on t,Q & ( for x being Int_position st x in X holds
(IExec (I,Q,t)) . x = t . x ) ) )
assume that
A14: for x being Int_position st x in X holds
t . x = (Initialize (IExec (I,P,s))) . x and
A15: ( t . a = (Initialize (IExec (I,P,s))) . a & t . (DataLoc (((Initialize (IExec (I,P,s))) . a),i)) < 0 ) ; ::_thesis: ( (IExec (I,Q,t)) . a = t . a & (IExec (I,Q,t)) . (DataLoc (((Initialize (IExec (I,P,s))) . a),i)) > t . (DataLoc (((Initialize (IExec (I,P,s))) . a),i)) & I is_closed_on t,Q & I is_halting_on t,Q & ( for x being Int_position st x in X holds
(IExec (I,Q,t)) . x = t . x ) )
A16: now__::_thesis:_for_x_being_Int_position_st_x_in_X_holds_
t_._x_=_s_._x
let x be Int_position; ::_thesis: ( x in X implies t . x = s . x )
assume A17: x in X ; ::_thesis: t . x = s . x
hence t . x = (Initialize (IExec (I,P,s))) . x by A14
.= (IExec (I,P,s)) . x by SCMPDS_5:15
.= s . x by A2, A6, A9, A17 ;
::_thesis: verum
end;
hence  (IExec (I,Q,t)) . a = t . a by A6, A13, A15; ::_thesis: ( (IExec (I,Q,t)) . (DataLoc (((Initialize (IExec (I,P,s))) . a),i)) > t . (DataLoc (((Initialize (IExec (I,P,s))) . a),i)) & I is_closed_on t,Q & I is_halting_on t,Q & ( for x being Int_position st x in X holds
(IExec (I,Q,t)) . x = t . x ) )
thus  (IExec (I,Q,t)) . (DataLoc (((Initialize (IExec (I,P,s))) . a),i)) > t . (DataLoc (((Initialize (IExec (I,P,s))) . a),i)) by A6, A13, A15, A16; ::_thesis: ( I is_closed_on t,Q & I is_halting_on t,Q & ( for x being Int_position st x in X holds
(IExec (I,Q,t)) . x = t . x ) )
thus  ( I is_closed_on t,Q & I is_halting_on t,Q & ( for x being Int_position st x in X holds
(IExec (I,Q,t)) . x = t . x ) ) by A6, A13, A15, A16; ::_thesis: verum
end;
then  while<0 (a,i,I) is_halting_on Initialize (IExec (I,P,s)),P by Th15;
then XX: P +* (stop (while<0 (a,i,I))) halts_on Initialize (Initialize (IExec (I,P,s))) by SCMPDS_6:def_3;
set m0 = LifeSpan ((P +* (stop (while<0 (a,i,I)))),s);
set s4 = Comput ((P +* (stop (while<0 (a,i,I)))),s,1);
set P4 = P +* (stop (while<0 (a,i,I)));
A21: IC s = 0 by MEMSTR_0:def_11;
A22: while<0 (a,i,I) = ((a,i) >=0_goto ((card I) + 2)) ';' (I ';' (goto (- ((card I) + 1)))) by SCMPDS_4:15;
A23: Comput ((P +* (stop (while<0 (a,i,I)))),s,(0 + 1)) = Following ((P +* (stop (while<0 (a,i,I)))),(Comput ((P +* (stop (while<0 (a,i,I)))),s,0))) by EXTPRO_1:3
.= Following ((P +* (stop (while<0 (a,i,I)))),s)
.= Exec (((a,i) >=0_goto ((card I) + 2)),s) by A22, I, SCMPDS_6:11 ;
A26: IC (Comput ((P +* (stop (while<0 (a,i,I)))),s,1)) = succ (IC s) by A2, A23, SCMPDS_2:57
.= 0 + 1 by A21 ;
set mI = LifeSpan ((P +* (stop I)),s);
set s5 = Comput ((P +* (stop (while<0 (a,i,I)))),(Comput ((P +* (stop (while<0 (a,i,I)))),s,1)),(LifeSpan ((P +* (stop I)),s)));
set P5 = P +* (stop (while<0 (a,i,I)));
set l1 = (card I) + 1;
 for a being Int_position holds s . a = (Comput ((P +* (stop (while<0 (a,i,I)))),s,1)) . a by A23, SCMPDS_2:57;
then A28: DataPart s = DataPart (Comput ((P +* (stop (while<0 (a,i,I)))),s,1)) by SCMPDS_4:8;
set m3 = (LifeSpan ((P +* (stop I)),s)) + 1;
set s6 = Comput ((P +* (stop (while<0 (a,i,I)))),s,((LifeSpan ((P +* (stop I)),s)) + 1));
set P6 = P +* (stop (while<0 (a,i,I)));
 (card I) + 1 < (card I) + 2 by XREAL_1:6;
then A29: (card I) + 1 in dom (while<0 (a,i,I)) by Th7;
set s7 = Comput ((P +* (stop (while<0 (a,i,I)))),s,(((LifeSpan ((P +* (stop I)),s)) + 1) + 1));
YY: while<0 (a,i,I) c= stop (while<0 (a,i,I)) by AFINSQ_1:74;
 stop (while<0 (a,i,I)) c= P +* (stop (while<0 (a,i,I))) by FUNCT_4:25;
then A31: while<0 (a,i,I) c= P +* (stop (while<0 (a,i,I))) by YY, XBOOLE_1:1;
 Shift (I,1) c= while<0 (a,i,I) by Lm2;
then A32: Shift (I,1) c= P +* (stop (while<0 (a,i,I))) by A31, XBOOLE_1:1;
A33: I is_closed_on s,P +* (stop I) by A2, A6, A9;
then A34: IC (Comput ((P +* (stop (while<0 (a,i,I)))),(Comput ((P +* (stop (while<0 (a,i,I)))),s,1)),(LifeSpan ((P +* (stop I)),s)))) = (card I) + 1 by A8, A12, A26, A28, A32, SCMPDS_7:18;
A35: (P +* (stop (while<0 (a,i,I)))) /. (IC (Comput ((P +* (stop (while<0 (a,i,I)))),s,((LifeSpan ((P +* (stop I)),s)) + 1)))) = (P +* (stop (while<0 (a,i,I)))) . (IC (Comput ((P +* (stop (while<0 (a,i,I)))),s,((LifeSpan ((P +* (stop I)),s)) + 1)))) by PBOOLE:143;
A36: Comput ((P +* (stop (while<0 (a,i,I)))),s,((LifeSpan ((P +* (stop I)),s)) + 1)) = Comput ((P +* (stop (while<0 (a,i,I)))),(Comput ((P +* (stop (while<0 (a,i,I)))),s,1)),(LifeSpan ((P +* (stop I)),s))) by EXTPRO_1:4;
then A37: CurInstr ((P +* (stop (while<0 (a,i,I)))),(Comput ((P +* (stop (while<0 (a,i,I)))),s,((LifeSpan ((P +* (stop I)),s)) + 1)))) = (P +* (stop (while<0 (a,i,I)))) . ((card I) + 1) by A8, A12, A33, A26, A28, A32, A35, SCMPDS_7:18
.= (while<0 (a,i,I)) . ((card I) + 1) by A29, A31, GRFUNC_1:2
.= goto (- ((card I) + 1)) by Th8 ;
A39: Comput ((P +* (stop (while<0 (a,i,I)))),s,(((LifeSpan ((P +* (stop I)),s)) + 1) + 1)) = Following ((P +* (stop (while<0 (a,i,I)))),(Comput ((P +* (stop (while<0 (a,i,I)))),s,((LifeSpan ((P +* (stop I)),s)) + 1)))) by EXTPRO_1:3
.= Exec ((goto (- ((card I) + 1))),(Comput ((P +* (stop (while<0 (a,i,I)))),s,((LifeSpan ((P +* (stop I)),s)) + 1)))) by A37 ;
then  IC (Comput ((P +* (stop (while<0 (a,i,I)))),s,(((LifeSpan ((P +* (stop I)),s)) + 1) + 1))) = ICplusConst ((Comput ((P +* (stop (while<0 (a,i,I)))),s,((LifeSpan ((P +* (stop I)),s)) + 1))),(0 - ((card I) + 1))) by SCMPDS_2:54
.= 0 by A34, A36, SCMPDS_7:1 ;
then A40: IC (Initialize (IExec (I,P,s))) = IC (Comput ((P +* (stop (while<0 (a,i,I)))),s,((LifeSpan ((P +* (stop I)),s)) + 2))) by MEMSTR_0:def_11;
A41: DataPart (Comput ((P +* (stop I)),s,(LifeSpan ((P +* (stop I)),s)))) = DataPart (Comput ((P +* (stop (while<0 (a,i,I)))),(Comput ((P +* (stop (while<0 (a,i,I)))),s,1)),(LifeSpan ((P +* (stop I)),s)))) by A8, A12, A33, A26, A28, A32, SCMPDS_7:18;
now__::_thesis:_for_x_being_Int_position_holds_(Comput_((P_+*_(stop_(while<0_(a,i,I)))),s,(((LifeSpan_((P_+*_(stop_I)),s))_+_1)_+_1)))_._x_=_(Initialize_(IExec_(I,P,s)))_._x
let x be Int_position; ::_thesis: (Comput ((P +* (stop (while<0 (a,i,I)))),s,(((LifeSpan ((P +* (stop I)),s)) + 1) + 1))) . x = (Initialize (IExec (I,P,s))) . x
A42: not x in dom (Start-At (0,SCMPDS)) by SCMPDS_4:18;
(Comput ((P +* (stop (while<0 (a,i,I)))),(Comput ((P +* (stop (while<0 (a,i,I)))),s,1)),(LifeSpan ((P +* (stop I)),s)))) . x = (Comput ((P +* (stop I)),s,(LifeSpan ((P +* (stop I)),s)))) . x by A41, SCMPDS_4:8
.= (Result ((P +* (stop I)),s)) . x by A10, EXTPRO_1:23
.= (IExec (I,P,s)) . x by SCMPDS_4:def_5 ;
hence (Comput ((P +* (stop (while<0 (a,i,I)))),s,(((LifeSpan ((P +* (stop I)),s)) + 1) + 1))) . x = (IExec (I,P,s)) . x by A36, A39, SCMPDS_2:54
.= (Initialize (IExec (I,P,s))) . x by A42, FUNCT_4:11 ;
::_thesis: verum
end;
then A44: DataPart (Comput ((P +* (stop (while<0 (a,i,I)))),s,(((LifeSpan ((P +* (stop I)),s)) + 1) + 1))) = DataPart (Initialize (IExec (I,P,s))) by SCMPDS_4:8;
A46: Comput ((P +* (stop (while<0 (a,i,I)))),s,((LifeSpan ((P +* (stop I)),s)) + 2)) = Initialize (IExec (I,P,s)) by A44, A40, MEMSTR_0:78;
then  CurInstr ((P +* (stop (while<0 (a,i,I)))),(Comput ((P +* (stop (while<0 (a,i,I)))),s,((LifeSpan ((P +* (stop I)),s)) + 2)))) = (a,i) >=0_goto ((card I) + 2) by A22, SCMPDS_6:11;
then  LifeSpan ((P +* (stop (while<0 (a,i,I)))),s) > (LifeSpan ((P +* (stop I)),s)) + 2 by A7, EXTPRO_1:36, SCMPDS_6:18;
then consider nn being Nat such that
A47: LifeSpan ((P +* (stop (while<0 (a,i,I)))),s) = ((LifeSpan ((P +* (stop I)),s)) + 2) + nn by NAT_1:10;
reconsider nn = nn as Element of NAT by ORDINAL1:def_12;
 Comput ((P +* (stop (while<0 (a,i,I)))),s,(((LifeSpan ((P +* (stop I)),s)) + 2) + (LifeSpan ((P +* (stop (while<0 (a,i,I)))),(Initialize (IExec (I,P,s))))))) = Comput ((P +* (stop (while<0 (a,i,I)))),(Initialize (IExec (I,P,s))),(LifeSpan ((P +* (stop (while<0 (a,i,I)))),(Initialize (IExec (I,P,s)))))) by A46, EXTPRO_1:4;
then  CurInstr ((P +* (stop (while<0 (a,i,I)))),(Comput ((P +* (stop (while<0 (a,i,I)))),s,(((LifeSpan ((P +* (stop I)),s)) + 2) + (LifeSpan ((P +* (stop (while<0 (a,i,I)))),(Initialize (IExec (I,P,s))))))))) = halt SCMPDS by XX, EXTPRO_1:def_15;
then  ((LifeSpan ((P +* (stop I)),s)) + 2) + (LifeSpan ((P +* (stop (while<0 (a,i,I)))),(Initialize (IExec (I,P,s))))) >= LifeSpan ((P +* (stop (while<0 (a,i,I)))),s) by A7, EXTPRO_1:def_15;
then A49: LifeSpan ((P +* (stop (while<0 (a,i,I)))),(Initialize (IExec (I,P,s)))) >= nn by A47, XREAL_1:6;
A50: Comput ((P +* (stop (while<0 (a,i,I)))),s,(LifeSpan ((P +* (stop (while<0 (a,i,I)))),s))) = Comput ((P +* (stop (while<0 (a,i,I)))),(Initialize (IExec (I,P,s))),nn) by A46, A47, EXTPRO_1:4;
then  CurInstr ((P +* (stop (while<0 (a,i,I)))),(Comput ((P +* (stop (while<0 (a,i,I)))),(Initialize (IExec (I,P,s))),nn))) = halt SCMPDS by A7, EXTPRO_1:def_15;
then  nn >= LifeSpan ((P +* (stop (while<0 (a,i,I)))),(Initialize (IExec (I,P,s)))) by XX, EXTPRO_1:def_15;
then  nn = LifeSpan ((P +* (stop (while<0 (a,i,I)))),(Initialize (IExec (I,P,s)))) by A49, XXREAL_0:1;
then  Result ((P +* (stop (while<0 (a,i,I)))),s) = Comput ((P +* (stop (while<0 (a,i,I)))),(Initialize (IExec (I,P,s))),(LifeSpan ((P +* (stop (while<0 (a,i,I)))),(Initialize (IExec (I,P,s)))))) by A7, A50, EXTPRO_1:23;
hence  IExec ((while<0 (a,i,I)),P,s) = Comput ((P +* (stop (while<0 (a,i,I)))),(Initialize (IExec (I,P,s))),(LifeSpan ((P +* (stop (while<0 (a,i,I)))),(Initialize (IExec (I,P,s)))))) by SCMPDS_4:def_5
.= Result ((P +* (stop (while<0 (a,i,I)))),(Initialize (IExec (I,P,s)))) by XX, EXTPRO_1:23
.= IExec ((while<0 (a,i,I)),P,(Initialize (IExec (I,P,s)))) by SCMPDS_4:def_5 ;
::_thesis: verum
end;

begin

definition
let a be Int_position;
let i be Integer;
let I be Program of SCMPDS;
func while>0 (a,i,I) ->  Program of SCMPDS equals :: SCMPDS_8:def 3
(((a,i) <=0_goto ((card I) + 2)) ';' I) ';' (goto (- ((card I) + 1)));
coherence
 (((a,i) <=0_goto ((card I) + 2)) ';' I) ';' (goto (- ((card I) + 1))) is Program of SCMPDS ;
end;

:: deftheorem  defines while>0 SCMPDS_8:def_3_:_
 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)));

registration
let I be shiftable Program of SCMPDS;
let a be Int_position;
let i be Integer;
cluster while>0 (a,i,I) ->  shiftable ;
correctness
coherence
while>0 (a,i,I) is shiftable ;
proof
set WHL = while>0 (a,i,I);
set i1 = (a,i) <=0_goto ((card I) + 2);
reconsider PF = (Load ((a,i) <=0_goto ((card I) + 2))) ';' I as shiftable Program of SCMPDS ;
A1: PF = ((a,i) <=0_goto ((card I) + 2)) ';' I by SCMPDS_4:def_2;
then  card PF = (card I) + 1 by SCMPDS_6:6;
then  (card PF) + (- ((card I) + 1)) = 0 ;
hence  while>0 (a,i,I) is shiftable by A1, SCMPDS_4:23; ::_thesis: verum
end;
end;

registration
let I be halt-free Program of SCMPDS;
let a be Int_position;
let i be Integer;
cluster while>0 (a,i,I) ->  halt-free ;
correctness
coherence
while>0 (a,i,I) is halt-free ;
;
end;

theorem Th17: :: SCMPDS_8:17
for a being Int_position
for i being Integer
for I being Program of SCMPDS holds card (while>0 (a,i,I)) = (card I) + 2
proof
let a be Int_position; ::_thesis: for i being Integer
for I being Program of SCMPDS holds card (while>0 (a,i,I)) = (card I) + 2
let i be Integer; ::_thesis: for I being Program of SCMPDS holds card (while>0 (a,i,I)) = (card I) + 2
let I be Program of SCMPDS; ::_thesis: card (while>0 (a,i,I)) = (card I) + 2
set i1 = (a,i) <=0_goto ((card I) + 2);
set I4 = ((a,i) <=0_goto ((card I) + 2)) ';' I;
thus  card (while>0 (a,i,I)) = (card (((a,i) <=0_goto ((card I) + 2)) ';' I)) + 1 by SCMP_GCD:4
.= ((card I) + 1) + 1 by SCMPDS_6:6
.= (card I) + 2 ; ::_thesis: verum
end;

Lm3:  for a being Int_position
for i being Integer
for I being Program of SCMPDS holds card (stop (while>0 (a,i,I))) = (card I) + 3
proof
let a be Int_position; ::_thesis: for i being Integer
for I being Program of SCMPDS holds card (stop (while>0 (a,i,I))) = (card I) + 3
let i be Integer; ::_thesis: for I being Program of SCMPDS holds card (stop (while>0 (a,i,I))) = (card I) + 3
let I be Program of SCMPDS; ::_thesis: card (stop (while>0 (a,i,I))) = (card I) + 3
thus  card (stop (while>0 (a,i,I))) = (card (while>0 (a,i,I))) + 1 by COMPOS_1:55
.= ((card I) + 2) + 1 by Th17
.= (card I) + 3 ; ::_thesis: verum
end;

theorem Th18: :: SCMPDS_8:18
for a being Int_position
for i being Integer
for m being Element of NAT
for I being Program of SCMPDS holds
( m < (card I) + 2 iff m in dom (while>0 (a,i,I)) )
proof
let a be Int_position; ::_thesis: for i being Integer
for m being Element of NAT
for I being Program of SCMPDS holds
( m < (card I) + 2 iff m in dom (while>0 (a,i,I)) )
let i be Integer; ::_thesis: for m being Element of NAT
for I being Program of SCMPDS holds
( m < (card I) + 2 iff m in dom (while>0 (a,i,I)) )
let m be Element of NAT ; ::_thesis: for I being Program of SCMPDS holds
( m < (card I) + 2 iff m in dom (while>0 (a,i,I)) )
let I be Program of SCMPDS; ::_thesis: ( m < (card I) + 2 iff m in dom (while>0 (a,i,I)) )
 card (while>0 (a,i,I)) = (card I) + 2 by Th17;
hence  ( m < (card I) + 2 iff m in dom (while>0 (a,i,I)) ) by AFINSQ_1:66; ::_thesis: verum
end;

theorem Th19: :: SCMPDS_8:19
for a being Int_position
for i being Integer
for I being Program of SCMPDS holds
( (while>0 (a,i,I)) . 0 = (a,i) <=0_goto ((card I) + 2) & (while>0 (a,i,I)) . ((card I) + 1) = 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)) . 0 = (a,i) <=0_goto ((card I) + 2) & (while>0 (a,i,I)) . ((card I) + 1) = goto (- ((card I) + 1)) )
let i be Integer; ::_thesis: for I being Program of SCMPDS holds
( (while>0 (a,i,I)) . 0 = (a,i) <=0_goto ((card I) + 2) & (while>0 (a,i,I)) . ((card I) + 1) = goto (- ((card I) + 1)) )
let I be Program of SCMPDS; ::_thesis: ( (while>0 (a,i,I)) . 0 = (a,i) <=0_goto ((card I) + 2) & (while>0 (a,i,I)) . ((card I) + 1) = goto (- ((card I) + 1)) )
set i1 = (a,i) <=0_goto ((card I) + 2);
set i2 = goto (- ((card I) + 1));
set I4 = ((a,i) <=0_goto ((card I) + 2)) ';' I;
set J5 = I ';' (goto (- ((card I) + 1)));
set WHL = while>0 (a,i,I);
 while>0 (a,i,I) = ((a,i) <=0_goto ((card I) + 2)) ';' (I ';' (goto (- ((card I) + 1)))) by SCMPDS_4:15;
hence  (while>0 (a,i,I)) . 0 = (a,i) <=0_goto ((card I) + 2) by SCMPDS_6:7; ::_thesis: (while>0 (a,i,I)) . ((card I) + 1) = goto (- ((card I) + 1))
 card (((a,i) <=0_goto ((card I) + 2)) ';' I) = (card I) + 1 by SCMPDS_6:6;
hence  (while>0 (a,i,I)) . ((card I) + 1) = goto (- ((card I) + 1)) by SCMP_GCD:6; ::_thesis: verum
end;

theorem Th20: :: SCMPDS_8:20
for P being Instruction-Sequence of SCMPDS
for s being State of SCMPDS
for I being Program of SCMPDS
for a being Int_position
for i being Integer st s . (DataLoc ((s . a),i)) <= 0 holds
( while>0 (a,i,I) is_closed_on s,P & while>0 (a,i,I) is_halting_on s,P )
proof
let P be Instruction-Sequence of SCMPDS; ::_thesis: for s being State of SCMPDS
for I being Program of SCMPDS
for a being Int_position
for i being Integer st s . (DataLoc ((s . a),i)) <= 0 holds
( while>0 (a,i,I) is_closed_on s,P & while>0 (a,i,I) is_halting_on s,P )
let s be State of SCMPDS; ::_thesis: for I being Program of SCMPDS
for a being Int_position
for i being Integer st s . (DataLoc ((s . a),i)) <= 0 holds
( while>0 (a,i,I) is_closed_on s,P & while>0 (a,i,I) is_halting_on s,P )
let I be Program of SCMPDS; ::_thesis: for a being Int_position
for i being Integer st s . (DataLoc ((s . a),i)) <= 0 holds
( while>0 (a,i,I) is_closed_on s,P & while>0 (a,i,I) is_halting_on s,P )
let a be Int_position; ::_thesis: for i being Integer st s . (DataLoc ((s . a),i)) <= 0 holds
( while>0 (a,i,I) is_closed_on s,P & while>0 (a,i,I) is_halting_on s,P )
let i be Integer; ::_thesis: ( s . (DataLoc ((s . a),i)) <= 0 implies ( while>0 (a,i,I) is_closed_on s,P & while>0 (a,i,I) is_halting_on s,P ) )
set d1 = DataLoc ((s . a),i);
assume A1: s . (DataLoc ((s . a),i)) <= 0 ; ::_thesis: ( while>0 (a,i,I) is_closed_on s,P & while>0 (a,i,I) is_halting_on s,P )
set i1 = (a,i) <=0_goto ((card I) + 2);
set i2 = goto (- ((card I) + 1));
set WHL = while>0 (a,i,I);
set pWHL = stop (while>0 (a,i,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)));
A3: IC (Initialize s) = 0 by MEMSTR_0:def_11;
A4: 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 A5: (Initialize s) . (DataLoc (((Initialize s) . a),i)) = (Initialize s) . (DataLoc ((s . a),i)) by FUNCT_4:11
.= s . (DataLoc ((s . a),i)) by A4, FUNCT_4:11 ;
A6: while>0 (a,i,I) = ((a,i) <=0_goto ((card I) + 2)) ';' (I ';' (goto (- ((card I) + 1)))) by SCMPDS_4:15;
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 A6, SCMPDS_6:11 ;
then A7: IC (Comput ((P +* (stop (while>0 (a,i,I)))),(Initialize s),1)) = ICplusConst ((Initialize s),((card I) + 2)) by A1, A5, SCMPDS_2:56
.= 0 + ((card I) + 2) by A3, SCMPDS_6:12 ;
A8: card (while>0 (a,i,I)) = (card I) + 2 by Th17;
then A9: (card I) + 2 in dom (stop (while>0 (a,i,I))) by COMPOS_1:64;
 stop (while>0 (a,i,I)) c= P +* (stop (while>0 (a,i,I))) by FUNCT_4:25;
then (P +* (stop (while>0 (a,i,I)))) . ((card I) + 2) = (stop (while>0 (a,i,I))) . ((card I) + 2) by A9, GRFUNC_1:2
.= halt SCMPDS by A8, COMPOS_1:64 ;
then A11: CurInstr ((P +* (stop (while>0 (a,i,I)))),(Comput ((P +* (stop (while>0 (a,i,I)))),(Initialize s),1))) = halt SCMPDS by A7, PBOOLE:143;
now__::_thesis:_for_k_being_Element_of_NAT_holds_IC_(Comput_((P_+*_(stop_(while>0_(a,i,I)))),(Initialize_s),k))_in_dom_(stop_(while>0_(a,i,I)))
let k be Element of NAT ; ::_thesis: IC (Comput ((P +* (stop (while>0 (a,i,I)))),(Initialize s),b1)) in dom (stop (while>0 (a,i,I)))
percases ( 0 < k or k = 0 ) ;
suppose 0 < k ; ::_thesis: IC (Comput ((P +* (stop (while>0 (a,i,I)))),(Initialize s),b1)) in dom (stop (while>0 (a,i,I)))
then  1 + 0 <= k by INT_1:7;
hence  IC (Comput ((P +* (stop (while>0 (a,i,I)))),(Initialize s),k)) in dom (stop (while>0 (a,i,I))) by A9, A7, A11, EXTPRO_1:5; ::_thesis: verum
end;
suppose k = 0 ; ::_thesis: IC (Comput ((P +* (stop (while>0 (a,i,I)))),(Initialize s),b1)) in dom (stop (while>0 (a,i,I)))
then  Comput ((P +* (stop (while>0 (a,i,I)))),(Initialize s),k) = Initialize s by EXTPRO_1:2;
hence  IC (Comput ((P +* (stop (while>0 (a,i,I)))),(Initialize s),k)) in dom (stop (while>0 (a,i,I))) by A3, COMPOS_1:36; ::_thesis: verum
end;
end;
end;
hence  while>0 (a,i,I) is_closed_on s,P by SCMPDS_6:def_2; ::_thesis: while>0 (a,i,I) is_halting_on s,P
 P +* (stop (while>0 (a,i,I))) halts_on Initialize s by A11, EXTPRO_1:29;
hence  while>0 (a,i,I) is_halting_on s,P by SCMPDS_6:def_3; ::_thesis: verum
end;

theorem Th21: :: SCMPDS_8:21
for P being Instruction-Sequence of SCMPDS
for s being 0 -started State of SCMPDS
for I being Program of SCMPDS
for a, c being Int_position
for i being Integer st s . (DataLoc ((s . a),i)) <= 0 holds
IExec ((while>0 (a,i,I)),P,s) = s +* (Start-At (((card I) + 2),SCMPDS))
proof
let P be Instruction-Sequence of SCMPDS; ::_thesis: for s being 0 -started State of SCMPDS
for I being Program of SCMPDS
for a, c being Int_position
for i being Integer st s . (DataLoc ((s . a),i)) <= 0 holds
IExec ((while>0 (a,i,I)),P,s) = s +* (Start-At (((card I) + 2),SCMPDS))
let s be 0 -started State of SCMPDS; ::_thesis: for I being Program of SCMPDS
for a, c being Int_position
for i being Integer st s . (DataLoc ((s . a),i)) <= 0 holds
IExec ((while>0 (a,i,I)),P,s) = s +* (Start-At (((card I) + 2),SCMPDS))
let I be Program of SCMPDS; ::_thesis: for a, c being Int_position
for i being Integer st s . (DataLoc ((s . a),i)) <= 0 holds
IExec ((while>0 (a,i,I)),P,s) = s +* (Start-At (((card I) + 2),SCMPDS))
let a, c be Int_position; ::_thesis: for i being Integer st s . (DataLoc ((s . a),i)) <= 0 holds
IExec ((while>0 (a,i,I)),P,s) = s +* (Start-At (((card I) + 2),SCMPDS))
let i be Integer; ::_thesis: ( s . (DataLoc ((s . a),i)) <= 0 implies IExec ((while>0 (a,i,I)),P,s) = s +* (Start-At (((card I) + 2),SCMPDS)) )
set d1 = DataLoc ((s . a),i);
set WHL = while>0 (a,i,I);
set pWHL = stop (while>0 (a,i,I));
set P3 = P +* (stop (while>0 (a,i,I)));
set s4 = Comput ((P +* (stop (while>0 (a,i,I)))),s,1);
set P4 = P +* (stop (while>0 (a,i,I)));
set i1 = (a,i) <=0_goto ((card I) + 2);
set i2 = goto (- ((card I) + 1));
set SAl = Start-At (((card I) + 2),SCMPDS);
I: Initialize s = s by MEMSTR_0:44;
A2: IC s = 0 by MEMSTR_0:def_11;
A3: while>0 (a,i,I) = ((a,i) <=0_goto ((card I) + 2)) ';' (I ';' (goto (- ((card I) + 1)))) by SCMPDS_4:15;
A6: Comput ((P +* (stop (while>0 (a,i,I)))),s,(0 + 1)) = Following ((P +* (stop (while>0 (a,i,I)))),(Comput ((P +* (stop (while>0 (a,i,I)))),s,0))) by EXTPRO_1:3
.= Following ((P +* (stop (while>0 (a,i,I)))),s)
.= Exec (((a,i) <=0_goto ((card I) + 2)),s) by A3, I, SCMPDS_6:11 ;
A7: stop (while>0 (a,i,I)) c= P +* (stop (while>0 (a,i,I))) by FUNCT_4:25;
A8: IExec ((while>0 (a,i,I)),P,s) = Result ((P +* (stop (while>0 (a,i,I)))),s) by SCMPDS_4:def_5;
assume  s . (DataLoc ((s . a),i)) <= 0 ; ::_thesis: IExec ((while>0 (a,i,I)),P,s) = s +* (Start-At (((card I) + 2),SCMPDS))
then A9: IC (Comput ((P +* (stop (while>0 (a,i,I)))),s,1)) = ICplusConst (s,((card I) + 2)) by A6, SCMPDS_2:56
.= 0 + ((card I) + 2) by A2, SCMPDS_6:12 ;
A12: card (while>0 (a,i,I)) = (card I) + 2 by Th17;
then  (card I) + 2 in dom (stop (while>0 (a,i,I))) by COMPOS_1:64;
then (P +* (stop (while>0 (a,i,I)))) . ((card I) + 2) = (stop (while>0 (a,i,I))) . ((card I) + 2) by A7, GRFUNC_1:2
.= halt SCMPDS by A12, COMPOS_1:64 ;
then A13: CurInstr ((P +* (stop (while>0 (a,i,I)))),(Comput ((P +* (stop (while>0 (a,i,I)))),s,1))) = halt SCMPDS by A9, PBOOLE:143;
then A14: P +* (stop (while>0 (a,i,I))) halts_on s by EXTPRO_1:29;
A15: CurInstr ((P +* (stop (while>0 (a,i,I)))),s) = (a,i) <=0_goto ((card I) + 2) by A3, I, SCMPDS_6:11;
now__::_thesis:_for_l_being_Element_of_NAT_st_l_<_0_+_1_holds_
CurInstr_((P_+*_(stop_(while>0_(a,i,I)))),(Comput_((P_+*_(stop_(while>0_(a,i,I)))),s,l)))_<>_halt_SCMPDS
let l be Element of NAT ; ::_thesis: ( l < 0 + 1 implies CurInstr ((P +* (stop (while>0 (a,i,I)))),(Comput ((P +* (stop (while>0 (a,i,I)))),s,l))) <> halt SCMPDS )
A16: Comput ((P +* (stop (while>0 (a,i,I)))),s,0) = s ;
assume  l < 0 + 1 ; ::_thesis: CurInstr ((P +* (stop (while>0 (a,i,I)))),(Comput ((P +* (stop (while>0 (a,i,I)))),s,l))) <> halt SCMPDS
then  l = 0 by NAT_1:13;
hence  CurInstr ((P +* (stop (while>0 (a,i,I)))),(Comput ((P +* (stop (while>0 (a,i,I)))),s,l))) <> halt SCMPDS by A15, A16; ::_thesis: verum
end;
then  for l being Element of NAT st CurInstr ((P +* (stop (while>0 (a,i,I)))),(Comput ((P +* (stop (while>0 (a,i,I)))),s,l))) = halt SCMPDS holds
1 <= l ;
then B17: LifeSpan ((P +* (stop (while>0 (a,i,I)))),s) = 1 by A13, A14, EXTPRO_1:def_15;
A19: now__::_thesis:_for_x_being_set_st_x_in_dom_(IExec_((while>0_(a,i,I)),P,s))_holds_
(IExec_((while>0_(a,i,I)),P,s))_._x_=_(s_+*_(Start-At_(((card_I)_+_2),SCMPDS)))_._x
let x be set ; ::_thesis: ( x in dom (IExec ((while>0 (a,i,I)),P,s)) implies (IExec ((while>0 (a,i,I)),P,s)) . b1 = (s +* (Start-At (((card I) + 2),SCMPDS))) . b1 )
A20: dom (Start-At (((card I) + 2),SCMPDS)) = {(IC )} by FUNCOP_1:13;
assume A21: x in dom (IExec ((while>0 (a,i,I)),P,s)) ; ::_thesis: (IExec ((while>0 (a,i,I)),P,s)) . b1 = (s +* (Start-At (((card I) + 2),SCMPDS))) . b1
percases ( x is Int_position or x = IC ) by A21, SCMPDS_4:6;
supposeA22: x is Int_position ; ::_thesis: (IExec ((while>0 (a,i,I)),P,s)) . b1 = (s +* (Start-At (((card I) + 2),SCMPDS))) . b1
then  x <> IC by SCMPDS_2:43;
then A23: not x in dom (Start-At (((card I) + 2),SCMPDS)) by A20, TARSKI:def_1;
thus (IExec ((while>0 (a,i,I)),P,s)) . x = (Comput ((P +* (stop (while>0 (a,i,I)))),s,1)) . x by A8, B17, A14, EXTPRO_1:23
.= s . x by A6, A22, SCMPDS_2:56
.= (s +* (Start-At (((card I) + 2),SCMPDS))) . x by A23, FUNCT_4:11 ; ::_thesis: verum
end;
supposeA24: x = IC ; ::_thesis: (IExec ((while>0 (a,i,I)),P,s)) . b1 = (s +* (Start-At (((card I) + 2),SCMPDS))) . b1
hence (IExec ((while>0 (a,i,I)),P,s)) . x = (card I) + 2 by A8, A9, B17, A14, EXTPRO_1:23
.= (s +* (Start-At (((card I) + 2),SCMPDS))) . x by A24, FUNCT_4:113 ;
::_thesis: verum
end;
end;
end;
dom (IExec ((while>0 (a,i,I)),P,s)) = the carrier of SCMPDS by PARTFUN1:def_2
.= dom (s +* (Start-At (((card I) + 2),SCMPDS))) by PARTFUN1:def_2 ;
hence  IExec ((while>0 (a,i,I)),P,s) = s +* (Start-At (((card I) + 2),SCMPDS)) by A19, FUNCT_1:2; ::_thesis: verum
end;

theorem :: SCMPDS_8:22
for P being Instruction-Sequence of SCMPDS
for s being 0 -started State of SCMPDS
for I being Program of SCMPDS
for a being Int_position
for i being Integer st s . (DataLoc ((s . a),i)) <= 0 holds
IC (IExec ((while>0 (a,i,I)),P,s)) = (card I) + 2
proof
let P be Instruction-Sequence of SCMPDS; ::_thesis: for s being 0 -started State of SCMPDS
for I being Program of SCMPDS
for a being Int_position
for i being Integer st s . (DataLoc ((s . a),i)) <= 0 holds
IC (IExec ((while>0 (a,i,I)),P,s)) = (card I) + 2
let s be 0 -started State of SCMPDS; ::_thesis: for I being Program of SCMPDS
for a being Int_position
for i being Integer st s . (DataLoc ((s . a),i)) <= 0 holds
IC (IExec ((while>0 (a,i,I)),P,s)) = (card I) + 2
let I be Program of SCMPDS; ::_thesis: for a being Int_position
for i being Integer st s . (DataLoc ((s . a),i)) <= 0 holds
IC (IExec ((while>0 (a,i,I)),P,s)) = (card I) + 2
let a be Int_position; ::_thesis: for i being Integer st s . (DataLoc ((s . a),i)) <= 0 holds
IC (IExec ((while>0 (a,i,I)),P,s)) = (card I) + 2
let i be Integer; ::_thesis: ( s . (DataLoc ((s . a),i)) <= 0 implies IC (IExec ((while>0 (a,i,I)),P,s)) = (card I) + 2 )
assume  s . (DataLoc ((s . a),i)) <= 0 ; ::_thesis: IC (IExec ((while>0 (a,i,I)),P,s)) = (card I) + 2
then  IExec ((while>0 (a,i,I)),P,s) = s +* (Start-At (((card I) + 2),SCMPDS)) by Th21;
hence  IC (IExec ((while>0 (a,i,I)),P,s)) = (card I) + 2 by FUNCT_4:113; ::_thesis: verum
end;

theorem :: SCMPDS_8:23
for P being Instruction-Sequence of SCMPDS
for s being 0 -started State of SCMPDS
for I being Program of SCMPDS
for a, b being Int_position
for i being Integer st s . (DataLoc ((s . a),i)) <= 0 holds
(IExec ((while>0 (a,i,I)),P,s)) . b = s . b
proof
let P be Instruction-Sequence of SCMPDS; ::_thesis: for s being 0 -started State of SCMPDS
for I being Program of SCMPDS
for a, b being Int_position
for i being Integer st s . (DataLoc ((s . a),i)) <= 0 holds
(IExec ((while>0 (a,i,I)),P,s)) . b = s . b
let s be 0 -started State of SCMPDS; ::_thesis: for I being Program of SCMPDS
for a, b being Int_position
for i being Integer st s . (DataLoc ((s . a),i)) <= 0 holds
(IExec ((while>0 (a,i,I)),P,s)) . b = s . b
let I be Program of SCMPDS; ::_thesis: for a, b being Int_position
for i being Integer st s . (DataLoc ((s . a),i)) <= 0 holds
(IExec ((while>0 (a,i,I)),P,s)) . b = s . b
let a, b be Int_position; ::_thesis: for i being Integer st s . (DataLoc ((s . a),i)) <= 0 holds
(IExec ((while>0 (a,i,I)),P,s)) . b = s . b
let i be Integer; ::_thesis: ( s . (DataLoc ((s . a),i)) <= 0 implies (IExec ((while>0 (a,i,I)),P,s)) . b = s . b )
assume  s . (DataLoc ((s . a),i)) <= 0 ; ::_thesis: (IExec ((while>0 (a,i,I)),P,s)) . b = s . b
then A1: IExec ((while>0 (a,i,I)),P,s) = s +* (Start-At (((card I) + 2),SCMPDS)) by Th21;
 not b in dom (Start-At (((card I) + 2),SCMPDS)) by SCMPDS_4:18;
hence  (IExec ((while>0 (a,i,I)),P,s)) . b = s . b by A1, FUNCT_4:11; ::_thesis: verum
end;

Lm4:  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) ';' (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;

scheme :: SCMPDS_8:sch 3
WhileGHalt{ F1( State of SCMPDS) ->  Element of NAT , F2() ->  0 -started State of SCMPDS, F3() ->  Instruction-Sequence of SCMPDS, F4() ->  halt-free shiftable Program of SCMPDS, F5() ->  Int_position, F6() ->  Integer, P1[ State of SCMPDS] } :
 ( while>0 (F5(),F6(),F4()) is_closed_on F2(),F3() & while>0 (F5(),F6(),F4()) is_halting_on F2(),F3() )
provided
A2: for t being 0 -started State of SCMPDS st P1[t] & F1(t) = 0 holds
t . (DataLoc ((F2() . F5()),F6())) <= 0 and
A3: P1[F2()] and
A4: for t being 0 -started State of SCMPDS
for Q being Instruction-Sequence of SCMPDS st P1[t] & t . F5() = F2() . F5() & t . (DataLoc ((F2() . F5()),F6())) > 0 holds
( (IExec (F4(),Q,t)) . F5() = t . F5() & F4() is_closed_on t,Q & F4() is_halting_on t,Q & F1((Initialize (IExec (F4(),Q,t)))) < F1(t) & P1[ Initialize (IExec (F4(),Q,t))] )
proof
set i1 = (F5(),F6()) <=0_goto ((card F4()) + 2);
set i2 = goto (- ((card F4()) + 1));
set WHL = while>0 (F5(),F6(),F4());
set pWHL = stop (while>0 (F5(),F6(),F4()));
set pI = stop F4();
set b = DataLoc ((F2() . F5()),F6());
defpred S1[ Element of NAT ] means  for t being 0 -started State of SCMPDS
for Q being Instruction-Sequence of SCMPDS st F1(t) <= $1 & P1[t] & t . F5() = F2() . F5() holds
( while>0 (F5(),F6(),F4()) is_closed_on t,Q & while>0 (F5(),F6(),F4()) is_halting_on t,Q );
A5: for k being Element of NAT st S1[k] holds
S1[k + 1]
proof
let k be Element of NAT ; ::_thesis: ( S1[k] implies S1[k + 1] )
assume A6: S1[k] ; ::_thesis: S1[k + 1]
now__::_thesis:_for_t_being_0_-started_State_of_SCMPDS
for_Q,_Q_being_Instruction-Sequence_of_SCMPDS_st_F1(t)_<=_k_+_1_&_P1[t]_&_t_._F5()_=_F2()_._F5()_holds_
(_while>0_(F5(),F6(),F4())_is_closed_on_t,Q_&_while>0_(F5(),F6(),F4())_is_halting_on_t,Q_)
let t be 0 -started State of SCMPDS; ::_thesis: for Q, Q being Instruction-Sequence of SCMPDS st F1(t) <= k + 1 & P1[t] & t . F5() = F2() . F5() holds
( while>0 (F5(),F6(),F4()) is_closed_on b3,b5 & while>0 (F5(),F6(),F4()) is_halting_on b3,b5 )
let Q, Q be Instruction-Sequence of SCMPDS; ::_thesis: ( F1(t) <= k + 1 & P1[t] & t . F5() = F2() . F5() implies ( while>0 (F5(),F6(),F4()) is_closed_on b1,b3 & while>0 (F5(),F6(),F4()) is_halting_on b1,b3 ) )
assume A7: F1(t) <= k + 1 ; ::_thesis: ( P1[t] & t . F5() = F2() . F5() implies ( while>0 (F5(),F6(),F4()) is_closed_on b1,b3 & while>0 (F5(),F6(),F4()) is_halting_on b1,b3 ) )
assume A8: P1[t] ; ::_thesis: ( t . F5() = F2() . F5() implies ( while>0 (F5(),F6(),F4()) is_closed_on b1,b3 & while>0 (F5(),F6(),F4()) is_halting_on b1,b3 ) )
assume A9: t . F5() = F2() . F5() ; ::_thesis: ( while>0 (F5(),F6(),F4()) is_closed_on b1,b3 & while>0 (F5(),F6(),F4()) is_halting_on b1,b3 )
percases ( t . (DataLoc ((F2() . F5()),F6())) <= 0 or t . (DataLoc ((F2() . F5()),F6())) > 0 ) ;
suppose t . (DataLoc ((F2() . F5()),F6())) <= 0 ; ::_thesis: ( while>0 (F5(),F6(),F4()) is_closed_on b1,b3 & while>0 (F5(),F6(),F4()) is_halting_on b1,b3 )
hence  ( while>0 (F5(),F6(),F4()) is_closed_on t,Q & while>0 (F5(),F6(),F4()) is_halting_on t,Q ) by A9, Th20; ::_thesis: verum
end;
supposeA10: t . (DataLoc ((F2() . F5()),F6())) > 0 ; ::_thesis: ( while>0 (F5(),F6(),F4()) is_closed_on b1,b3 & while>0 (F5(),F6(),F4()) is_halting_on b1,b3 )
A14: 0 in dom (stop (while>0 (F5(),F6(),F4()))) by COMPOS_1:36;
A17: while>0 (F5(),F6(),F4()) = ((F5(),F6()) <=0_goto ((card F4()) + 2)) ';' (F4() ';' (goto (- ((card F4()) + 1)))) by SCMPDS_4:15;
set t2 = t;
set Q2 = Q +* (stop F4());
set t3 = t;
set Q3 = Q +* (stop (while>0 (F5(),F6(),F4())));
set t4 = Comput ((Q +* (stop (while>0 (F5(),F6(),F4())))),t,1);
set Q4 = Q +* (stop (while>0 (F5(),F6(),F4())));
A20: stop F4() c= Q +* (stop F4()) by FUNCT_4:25;
set m2 = LifeSpan ((Q +* (stop F4())),t);
set t5 = Comput ((Q +* (stop (while>0 (F5(),F6(),F4())))),(Comput ((Q +* (stop (while>0 (F5(),F6(),F4())))),t,1)),(LifeSpan ((Q +* (stop F4())),t)));
set Q5 = Q +* (stop (while>0 (F5(),F6(),F4())));
set l1 = (card F4()) + 1;
I: Initialize t = t by MEMSTR_0:44;
A21: IC t = 0 by MEMSTR_0:def_11;
set m3 = (LifeSpan ((Q +* (stop F4())),t)) + 1;
set t6 = Comput ((Q +* (stop (while>0 (F5(),F6(),F4())))),t,((LifeSpan ((Q +* (stop F4())),t)) + 1));
set Q6 = Q +* (stop (while>0 (F5(),F6(),F4())));
set t7 = Comput ((Q +* (stop (while>0 (F5(),F6(),F4())))),t,(((LifeSpan ((Q +* (stop F4())),t)) + 1) + 1));
set Q7 = Q +* (stop (while>0 (F5(),F6(),F4())));
 (card F4()) + 1 < (card F4()) + 2 by XREAL_1:6;
then A22: (card F4()) + 1 in dom (while>0 (F5(),F6(),F4())) by Th18;
XX: while>0 (F5(),F6(),F4()) c= stop (while>0 (F5(),F6(),F4())) by AFINSQ_1:74;
 stop (while>0 (F5(),F6(),F4())) c= Q +* (stop (while>0 (F5(),F6(),F4()))) by FUNCT_4:25;
then A23: while>0 (F5(),F6(),F4()) c= Q +* (stop (while>0 (F5(),F6(),F4()))) by XX, XBOOLE_1:1;
 Shift (F4(),1) c= while>0 (F5(),F6(),F4()) by Lm4;
then A24: Shift (F4(),1) c= Q +* (stop (while>0 (F5(),F6(),F4()))) by A23, XBOOLE_1:1;
A25: Comput ((Q +* (stop (while>0 (F5(),F6(),F4())))),t,(0 + 1)) = Following ((Q +* (stop (while>0 (F5(),F6(),F4())))),(Comput ((Q +* (stop (while>0 (F5(),F6(),F4())))),t,0))) by EXTPRO_1:3
.= Following ((Q +* (stop (while>0 (F5(),F6(),F4())))),t)
.= Exec (((F5(),F6()) <=0_goto ((card F4()) + 2)),t) by A17, I, SCMPDS_6:11 ;
 for a being Int_position holds t . a = (Comput ((Q +* (stop (while>0 (F5(),F6(),F4())))),t,1)) . a by A25, SCMPDS_2:56;
then A27: DataPart t = DataPart (Comput ((Q +* (stop (while>0 (F5(),F6(),F4())))),t,1)) by SCMPDS_4:8;
 F4() is_halting_on t,Q by A4, A8, A9, A10;
then A28: Q +* (stop F4()) halts_on t by I, SCMPDS_6:def_3;
 (Q +* (stop F4())) +* (stop F4()) halts_on t by A28;
then A30: F4() is_halting_on t,Q +* (stop F4()) by I, SCMPDS_6:def_3;
A31: IExec (F4(),Q,t) = Result ((Q +* (stop F4())),t) by SCMPDS_4:def_5;
A32: P1[ Initialize (IExec (F4(),Q,t))] by A4, A8, A9, A10;
A33: F4() is_closed_on t,Q by A4, A8, A9, A10;
A34: F4() is_closed_on t,Q +* (stop F4()) by A4, A8, A9, A10;
A35: IC (Comput ((Q +* (stop (while>0 (F5(),F6(),F4())))),t,1)) = succ (IC t) by A10, A25, A9, SCMPDS_2:56
.= 0 + 1 by A21 ;
then A36: IC (Comput ((Q +* (stop (while>0 (F5(),F6(),F4())))),(Comput ((Q +* (stop (while>0 (F5(),F6(),F4())))),t,1)),(LifeSpan ((Q +* (stop F4())),t)))) = (card F4()) + 1 by A20, A30, A34, A27, A24, SCMPDS_7:18;
A37: (Q +* (stop (while>0 (F5(),F6(),F4())))) /. (IC (Comput ((Q +* (stop (while>0 (F5(),F6(),F4())))),t,((LifeSpan ((Q +* (stop F4())),t)) + 1)))) = (Q +* (stop (while>0 (F5(),F6(),F4())))) . (IC (Comput ((Q +* (stop (while>0 (F5(),F6(),F4())))),t,((LifeSpan ((Q +* (stop F4())),t)) + 1)))) by PBOOLE:143;
A38: Comput ((Q +* (stop (while>0 (F5(),F6(),F4())))),t,((LifeSpan ((Q +* (stop F4())),t)) + 1)) = Comput ((Q +* (stop (while>0 (F5(),F6(),F4())))),(Comput ((Q +* (stop (while>0 (F5(),F6(),F4())))),t,1)),(LifeSpan ((Q +* (stop F4())),t))) by EXTPRO_1:4;
then A39: CurInstr ((Q +* (stop (while>0 (F5(),F6(),F4())))),(Comput ((Q +* (stop (while>0 (F5(),F6(),F4())))),t,((LifeSpan ((Q +* (stop F4())),t)) + 1)))) = (Q +* (stop (while>0 (F5(),F6(),F4())))) . ((card F4()) + 1) by A20, A30, A34, A35, A27, A24, A37, SCMPDS_7:18
.= (while>0 (F5(),F6(),F4())) . ((card F4()) + 1) by A22, A23, GRFUNC_1:2
.= goto (- ((card F4()) + 1)) by Th19 ;
A41: Comput ((Q +* (stop (while>0 (F5(),F6(),F4())))),t,(((LifeSpan ((Q +* (stop F4())),t)) + 1) + 1)) = Following ((Q +* (stop (while>0 (F5(),F6(),F4())))),(Comput ((Q +* (stop (while>0 (F5(),F6(),F4())))),t,((LifeSpan ((Q +* (stop F4())),t)) + 1)))) by EXTPRO_1:3
.= Exec ((goto (- ((card F4()) + 1))),(Comput ((Q +* (stop (while>0 (F5(),F6(),F4())))),t,((LifeSpan ((Q +* (stop F4())),t)) + 1)))) by A39 ;
then  IC (Comput ((Q +* (stop (while>0 (F5(),F6(),F4())))),t,(((LifeSpan ((Q +* (stop F4())),t)) + 1) + 1))) = ICplusConst ((Comput ((Q +* (stop (while>0 (F5(),F6(),F4())))),t,((LifeSpan ((Q +* (stop F4())),t)) + 1))),(0 - ((card F4()) + 1))) by SCMPDS_2:54
.= 0 by A36, A38, SCMPDS_7:1 ;
then A42: Initialize (Comput ((Q +* (stop (while>0 (F5(),F6(),F4())))),t,(((LifeSpan ((Q +* (stop F4())),t)) + 1) + 1))) = Comput ((Q +* (stop (while>0 (F5(),F6(),F4())))),t,(((LifeSpan ((Q +* (stop F4())),t)) + 1) + 1)) by MEMSTR_0:46;
A43: DataPart (Comput ((Q +* (stop F4())),t,(LifeSpan ((Q +* (stop F4())),t)))) = DataPart (Comput ((Q +* (stop (while>0 (F5(),F6(),F4())))),(Comput ((Q +* (stop (while>0 (F5(),F6(),F4())))),t,1)),(LifeSpan ((Q +* (stop F4())),t)))) by A20, A30, A34, A35, A27, A24, SCMPDS_7:18;
then A44: DataPart (Comput ((Q +* (stop (while>0 (F5(),F6(),F4())))),(Comput ((Q +* (stop (while>0 (F5(),F6(),F4())))),t,1)),(LifeSpan ((Q +* (stop F4())),t)))) = DataPart (Result ((Q +* (stop F4())),t)) by A28, EXTPRO_1:23
.= DataPart (IExec (F4(),Q,t)) by SCMPDS_4:def_5 ;
 InsCode (goto (- ((card F4()) + 1))) = 14 by SCMPDS_2:12;
then  InsCode (goto (- ((card F4()) + 1))) in {0,4,5,6,14} by ENUMSET1:def_3;
then A45: Initialize (Comput ((Q +* (stop (while>0 (F5(),F6(),F4())))),t,(((LifeSpan ((Q +* (stop F4())),t)) + 1) + 1))) = Initialize (Comput ((Q +* (stop (while>0 (F5(),F6(),F4())))),t,((LifeSpan ((Q +* (stop F4())),t)) + 1))) by A41, Th3
.= Initialize (IExec (F4(),Q,t)) by A44, A38, MEMSTR_0:80 ;
A46: now__::_thesis:_not_F1((Initialize_(Comput_((Q_+*_(stop_(while>0_(F5(),F6(),F4())))),t,(((LifeSpan_((Q_+*_(stop_F4())),t))_+_1)_+_1)))))_>_k
 F1((Initialize (IExec (F4(),Q,t)))) < F1((Initialize t)) by A4, A8, A9, A10, I;
then A47: F1((Initialize (Comput ((Q +* (stop (while>0 (F5(),F6(),F4())))),t,(((LifeSpan ((Q +* (stop F4())),t)) + 1) + 1))))) < k + 1 by A7, A45, I, XXREAL_0:2;
assume  F1((Initialize (Comput ((Q +* (stop (while>0 (F5(),F6(),F4())))),t,(((LifeSpan ((Q +* (stop F4())),t)) + 1) + 1))))) > k ; ::_thesis: contradiction
hence  contradiction by A47, INT_1:7; ::_thesis: verum
end;
A48: (Comput ((Q +* (stop (while>0 (F5(),F6(),F4())))),(Comput ((Q +* (stop (while>0 (F5(),F6(),F4())))),t,1)),(LifeSpan ((Q +* (stop F4())),t)))) . F5() = (Comput ((Q +* (stop F4())),t,(LifeSpan ((Q +* (stop F4())),t)))) . F5() by A43, SCMPDS_4:8
.= (Result ((Q +* (stop F4())),t)) . F5() by A28, EXTPRO_1:23
.= F2() . F5() by A9, A4, A8, A10, A31 ;
A50: (Comput ((Q +* (stop (while>0 (F5(),F6(),F4())))),t,(((LifeSpan ((Q +* (stop F4())),t)) + 1) + 1))) . F5() = (Comput ((Q +* (stop (while>0 (F5(),F6(),F4())))),t,((LifeSpan ((Q +* (stop F4())),t)) + 1))) . F5() by A41, SCMPDS_2:54
.= F2() . F5() by A48, EXTPRO_1:4 ;
then A51: while>0 (F5(),F6(),F4()) is_closed_on Comput ((Q +* (stop (while>0 (F5(),F6(),F4())))),t,(((LifeSpan ((Q +* (stop F4())),t)) + 1) + 1)),Q +* (stop (while>0 (F5(),F6(),F4()))) by A6, A32, A45, A46, A42;
now__::_thesis:_for_k_being_Element_of_NAT_holds_IC_(Comput_((Q_+*_(stop_(while>0_(F5(),F6(),F4())))),t,k))_in_dom_(stop_(while>0_(F5(),F6(),F4())))
let k be Element of NAT ; ::_thesis: IC (Comput ((Q +* (stop (while>0 (F5(),F6(),F4())))),t,b1)) in dom (stop (while>0 (F5(),F6(),F4())))
percases ( k < ((LifeSpan ((Q +* (stop F4())),t)) + 1) + 1 or k >= ((LifeSpan ((Q +* (stop F4())),t)) + 1) + 1 ) ;
suppose k < ((LifeSpan ((Q +* (stop F4())),t)) + 1) + 1 ; ::_thesis: IC (Comput ((Q +* (stop (while>0 (F5(),F6(),F4())))),t,b1)) in dom (stop (while>0 (F5(),F6(),F4())))
then A52: k <= (LifeSpan ((Q +* (stop F4())),t)) + 1 by INT_1:7;
percases ( k <= LifeSpan ((Q +* (stop F4())),t) or k = (LifeSpan ((Q +* (stop F4())),t)) + 1 ) by A52, NAT_1:8;
supposeA53: k <= LifeSpan ((Q +* (stop F4())),t) ; ::_thesis: IC (Comput ((Q +* (stop (while>0 (F5(),F6(),F4())))),t,b1)) in dom (stop (while>0 (F5(),F6(),F4())))
hereby  ::_thesis: verum
percases ( k = 0 or k <> 0 ) ;
suppose k = 0 ; ::_thesis: IC (Comput ((Q +* (stop (while>0 (F5(),F6(),F4())))),t,k)) in dom (stop (while>0 (F5(),F6(),F4())))
hence  IC (Comput ((Q +* (stop (while>0 (F5(),F6(),F4())))),t,k)) in dom (stop (while>0 (F5(),F6(),F4()))) by A14, A21, EXTPRO_1:2; ::_thesis: verum
end;
suppose k <> 0 ; ::_thesis: IC (Comput ((Q +* (stop (while>0 (F5(),F6(),F4())))),t,k)) in dom (stop (while>0 (F5(),F6(),F4())))
then consider kn being Nat such that
A54: k = kn + 1 by NAT_1:6;
reconsider kn = kn as Element of NAT by ORDINAL1:def_12;
reconsider lm = IC (Comput ((Q +* (stop F4())),t,kn)) as Element of NAT ;
 kn < k by A54, XREAL_1:29;
then  kn < LifeSpan ((Q +* (stop F4())),t) by A53, XXREAL_0:2;
then  (IC (Comput ((Q +* (stop F4())),t,kn))) + 1 = IC (Comput ((Q +* (stop (while>0 (F5(),F6(),F4())))),(Comput ((Q +* (stop (while>0 (F5(),F6(),F4())))),t,1)),kn)) by A20, A30, A34, A35, A27, A24, SCMPDS_7:16;
then A56: IC (Comput ((Q +* (stop (while>0 (F5(),F6(),F4())))),t,k)) = lm + 1 by A54, EXTPRO_1:4;
 IC (Comput ((Q +* (stop F4())),t,kn)) in dom (stop F4()) by A33, I, SCMPDS_6:def_2;
then  lm < card (stop F4()) by AFINSQ_1:66;
then  lm < (card F4()) + 1 by COMPOS_1:55;
then A57: lm + 1 <= (card F4()) + 1 by INT_1:7;
 (card F4()) + 1 < (card F4()) + 3 by XREAL_1:6;
then  lm + 1 < (card F4()) + 3 by A57, XXREAL_0:2;
then  lm + 1 < card (stop (while>0 (F5(),F6(),F4()))) by Lm3;
hence  IC (Comput ((Q +* (stop (while>0 (F5(),F6(),F4())))),t,k)) in dom (stop (while>0 (F5(),F6(),F4()))) by A56, AFINSQ_1:66; ::_thesis: verum
end;
end;
end;
end;
supposeA58: k = (LifeSpan ((Q +* (stop F4())),t)) + 1 ; ::_thesis: IC (Comput ((Q +* (stop (while>0 (F5(),F6(),F4())))),t,b1)) in dom (stop (while>0 (F5(),F6(),F4())))
 (card F4()) + 1 in dom (stop (while>0 (F5(),F6(),F4()))) by A22, COMPOS_1:62;
hence  IC (Comput ((Q +* (stop (while>0 (F5(),F6(),F4())))),t,k)) in dom (stop (while>0 (F5(),F6(),F4()))) by A20, A30, A34, A35, A27, A24, A38, A58, SCMPDS_7:18; ::_thesis: verum
end;
end;
end;
suppose k >= ((LifeSpan ((Q +* (stop F4())),t)) + 1) + 1 ; ::_thesis: IC (Comput ((Q +* (stop (while>0 (F5(),F6(),F4())))),t,b1)) in dom (stop (while>0 (F5(),F6(),F4())))
then consider nn being Nat such that
A59: k = (((LifeSpan ((Q +* (stop F4())),t)) + 1) + 1) + nn by NAT_1:10;
A61: nn in NAT by ORDINAL1:def_12;
 Comput ((Q +* (stop (while>0 (F5(),F6(),F4())))),t,k) = Comput (((Q +* (stop (while>0 (F5(),F6(),F4())))) +* (stop (while>0 (F5(),F6(),F4())))),(Comput ((Q +* (stop (while>0 (F5(),F6(),F4())))),t,(((LifeSpan ((Q +* (stop F4())),t)) + 1) + 1))),nn) by A59, A61, EXTPRO_1:4;
hence  IC (Comput ((Q +* (stop (while>0 (F5(),F6(),F4())))),t,k)) in dom (stop (while>0 (F5(),F6(),F4()))) by A51, A61, A42, SCMPDS_6:def_2; ::_thesis: verum
end;
end;
end;
hence  while>0 (F5(),F6(),F4()) is_closed_on t,Q by I, SCMPDS_6:def_2; ::_thesis: while>0 (F5(),F6(),F4()) is_halting_on t,Q
RR: (Q +* (stop (while>0 (F5(),F6(),F4())))) +* (stop (while>0 (F5(),F6(),F4()))) = Q +* (stop (while>0 (F5(),F6(),F4()))) ;
 while>0 (F5(),F6(),F4()) is_halting_on Comput ((Q +* (stop (while>0 (F5(),F6(),F4())))),t,(((LifeSpan ((Q +* (stop F4())),t)) + 1) + 1)),Q +* (stop (while>0 (F5(),F6(),F4()))) by A6, A32, A50, A45, A46, A42;
then  Q +* (stop (while>0 (F5(),F6(),F4()))) halts_on Comput ((Q +* (stop (while>0 (F5(),F6(),F4())))),t,(((LifeSpan ((Q +* (stop F4())),t)) + 1) + 1)) by A42, RR, SCMPDS_6:def_3;
then  Q +* (stop (while>0 (F5(),F6(),F4()))) halts_on t by EXTPRO_1:22;
hence  while>0 (F5(),F6(),F4()) is_halting_on t,Q by I, SCMPDS_6:def_3; ::_thesis: verum
end;
end;
end;
hence  S1[k + 1] ; ::_thesis: verum
end;
set n = F1(F2());
A62: S1[ 0 ]
proof
let t be 0 -started State of SCMPDS; ::_thesis: for Q being Instruction-Sequence of SCMPDS st F1(t) <= 0 & P1[t] & t . F5() = F2() . F5() holds
( while>0 (F5(),F6(),F4()) is_closed_on t,Q & while>0 (F5(),F6(),F4()) is_halting_on t,Q )
let Q be Instruction-Sequence of SCMPDS; ::_thesis: ( F1(t) <= 0 & P1[t] & t . F5() = F2() . F5() implies ( while>0 (F5(),F6(),F4()) is_closed_on t,Q & while>0 (F5(),F6(),F4()) is_halting_on t,Q ) )
assume that
A63: F1(t) <= 0 and
A64: P1[t] and
A65: t . F5() = F2() . F5() ; ::_thesis: ( while>0 (F5(),F6(),F4()) is_closed_on t,Q & while>0 (F5(),F6(),F4()) is_halting_on t,Q )
 F1(t) = 0 by A63;
then  t . (DataLoc ((F2() . F5()),F6())) <= 0 by A2, A64;
hence  ( while>0 (F5(),F6(),F4()) is_closed_on t,Q & while>0 (F5(),F6(),F4()) is_halting_on t,Q ) by A65, Th20; ::_thesis: verum
end;
 for k being Element of NAT holds S1[k] from NAT_1:sch_1(A62, A5);
then  S1[F1(F2())] ;
hence  ( while>0 (F5(),F6(),F4()) is_closed_on F2(),F3() & while>0 (F5(),F6(),F4()) is_halting_on F2(),F3() ) by A3; ::_thesis: verum
end;

scheme :: SCMPDS_8:sch 4
WhileGExec{ F1( State of SCMPDS) ->  Element of NAT , F2() ->  0 -started State of SCMPDS, F3() ->  Instruction-Sequence of SCMPDS, F4() ->  halt-free shiftable Program of SCMPDS, F5() ->  Int_position, F6() ->  Integer, P1[ State of SCMPDS] } :
 IExec ((while>0 (F5(),F6(),F4())),F3(),F2()) = IExec ((while>0 (F5(),F6(),F4())),F3(),(Initialize (IExec (F4(),F3(),F2()))))
provided
A2: F2() . (DataLoc ((F2() . F5()),F6())) > 0 and
A3: for t being 0 -started State of SCMPDS st P1[t] & F1(t) = 0 holds
t . (DataLoc ((F2() . F5()),F6())) <= 0 and
A4: P1[F2()] and
A5: for t being 0 -started State of SCMPDS
for Q being Instruction-Sequence of SCMPDS st P1[t] & t . F5() = F2() . F5() & t . (DataLoc ((F2() . F5()),F6())) > 0 holds
( (IExec (F4(),Q,t)) . F5() = t . F5() & F4() is_closed_on t,Q & F4() is_halting_on t,Q & F1((Initialize (IExec (F4(),Q,t)))) < F1(t) & P1[ Initialize (IExec (F4(),Q,t))] )
proof
set WHL = while>0 (F5(),F6(),F4());
set pWHL = stop (while>0 (F5(),F6(),F4()));
set P1 = F3() +* (stop (while>0 (F5(),F6(),F4())));
set PI = F3() +* (stop F4());
set m1 = (LifeSpan ((F3() +* (stop F4())),F2())) + 2;
set s2 = Initialize (IExec (F4(),F3(),F2()));
set P2 = F3() +* (stop (while>0 (F5(),F6(),F4())));
set m2 = LifeSpan ((F3() +* (stop (while>0 (F5(),F6(),F4())))),(Initialize (IExec (F4(),F3(),F2()))));
I: Initialize F2() = F2() by MEMSTR_0:44;
A9: P1[F2()] by A4;
A10: stop F4() c= F3() +* (stop F4()) by FUNCT_4:25;
A11: F4() is_closed_on F2(),F3() +* (stop F4()) by A2, A4, A5;
 F4() is_halting_on F2(),F3() by A2, A4, A5;
then A12: F3() +* (stop F4()) halts_on F2() by I, SCMPDS_6:def_3;
 (F3() +* (stop F4())) +* (stop F4()) halts_on F2() by A12;
then A14: F4() is_halting_on F2(),F3() +* (stop F4()) by I, SCMPDS_6:def_3;
set Es = IExec (F4(),F3(),F2());
set bj = DataLoc (((Initialize (IExec (F4(),F3(),F2()))) . F5()),F6());
set EP = F3();
deffunc H1( State of SCMPDS) ->  Element of NAT = F1($1);
A15: for t being 0 -started State of SCMPDS
for Q being Instruction-Sequence of SCMPDS st P1[t] & t . F5() = F2() . F5() & t . (DataLoc ((F2() . F5()),F6())) > 0 holds
( (IExec (F4(),Q,t)) . F5() = t . F5() & F4() is_closed_on t,Q & F4() is_halting_on t,Q & H1( Initialize (IExec (F4(),Q,t))) < H1(t) & P1[ Initialize (IExec (F4(),Q,t))] ) by A5;
A16: for t being 0 -started State of SCMPDS st P1[t] & H1(t) = 0 holds
t . (DataLoc ((F2() . F5()),F6())) <= 0 by A3;
 ( while>0 (F5(),F6(),F4()) is_closed_on F2(),F3() & while>0 (F5(),F6(),F4()) is_halting_on F2(),F3() ) from SCMPDS_8:sch_3(A16, A9, A15);
then A17: F3() +* (stop (while>0 (F5(),F6(),F4()))) halts_on F2() by I, SCMPDS_6:def_3;
deffunc H2( State of SCMPDS) ->  Element of NAT = F1($1);
A18: (Initialize (IExec (F4(),F3(),F2()))) . F5() = (IExec (F4(),F3(),F2())) . F5() by SCMPDS_5:15
.= F2() . F5() by A2, A4, A5 ;
then A19: for t being 0 -started State of SCMPDS st P1[t] & H2(t) = 0 holds
t . (DataLoc (((Initialize (IExec (F4(),F3(),F2()))) . F5()),F6())) <= 0 by A3;
A20: for t being 0 -started State of SCMPDS
for Q being Instruction-Sequence of SCMPDS st P1[t] & t . F5() = (Initialize (IExec (F4(),F3(),F2()))) . F5() & t . (DataLoc (((Initialize (IExec (F4(),F3(),F2()))) . F5()),F6())) > 0 holds
( (IExec (F4(),Q,t)) . F5() = t . F5() & F4() is_closed_on t,Q & F4() is_halting_on t,Q & H2( Initialize (IExec (F4(),Q,t))) < H2(t) & P1[ Initialize (IExec (F4(),Q,t))] ) by A5, A18;
A21: P1[ Initialize (IExec (F4(),F3(),F2()))] by A2, A4, A5;
 ( while>0 (F5(),F6(),F4()) is_closed_on Initialize (IExec (F4(),F3(),F2())),F3() & while>0 (F5(),F6(),F4()) is_halting_on Initialize (IExec (F4(),F3(),F2())),F3() ) from SCMPDS_8:sch_3(A19, A21, A20);
then A22: F3() +* (stop (while>0 (F5(),F6(),F4()))) halts_on Initialize (Initialize (IExec (F4(),F3(),F2()))) by SCMPDS_6:def_3;
set s4 = Comput ((F3() +* (stop (while>0 (F5(),F6(),F4())))),F2(),1);
set P4 = F3() +* (stop (while>0 (F5(),F6(),F4())));
set i1 = (F5(),F6()) <=0_goto ((card F4()) + 2);
set i2 = goto (- ((card F4()) + 1));
set b = DataLoc ((F2() . F5()),F6());
A23: while>0 (F5(),F6(),F4()) = ((F5(),F6()) <=0_goto ((card F4()) + 2)) ';' (F4() ';' (goto (- ((card F4()) + 1)))) by SCMPDS_4:15;
set mI = LifeSpan ((F3() +* (stop F4())),F2());
set s5 = Comput ((F3() +* (stop (while>0 (F5(),F6(),F4())))),(Comput ((F3() +* (stop (while>0 (F5(),F6(),F4())))),F2(),1)),(LifeSpan ((F3() +* (stop F4())),F2())));
set P5 = F3() +* (stop (while>0 (F5(),F6(),F4())));
set l1 = (card F4()) + 1;
A24: IC F2() = 0 by MEMSTR_0:def_11;
A25: Comput ((F3() +* (stop (while>0 (F5(),F6(),F4())))),F2(),(0 + 1)) = Following ((F3() +* (stop (while>0 (F5(),F6(),F4())))),(Comput ((F3() +* (stop (while>0 (F5(),F6(),F4())))),F2(),0))) by EXTPRO_1:3
.= Following ((F3() +* (stop (while>0 (F5(),F6(),F4())))),F2())
.= Exec (((F5(),F6()) <=0_goto ((card F4()) + 2)),F2()) by A23, I, SCMPDS_6:11 ;
 for a being Int_position holds F2() . a = (Comput ((F3() +* (stop (while>0 (F5(),F6(),F4())))),F2(),1)) . a by A25, SCMPDS_2:56;
then A27: DataPart F2() = DataPart (Comput ((F3() +* (stop (while>0 (F5(),F6(),F4())))),F2(),1)) by SCMPDS_4:8;
set m3 = (LifeSpan ((F3() +* (stop F4())),F2())) + 1;
set s6 = Comput ((F3() +* (stop (while>0 (F5(),F6(),F4())))),F2(),((LifeSpan ((F3() +* (stop F4())),F2())) + 1));
set P6 = F3() +* (stop (while>0 (F5(),F6(),F4())));
 (card F4()) + 1 < (card F4()) + 2 by XREAL_1:6;
then A28: (card F4()) + 1 in dom (while>0 (F5(),F6(),F4())) by Th18;
set m0 = LifeSpan ((F3() +* (stop (while>0 (F5(),F6(),F4())))),F2());
set s7 = Comput ((F3() +* (stop (while>0 (F5(),F6(),F4())))),F2(),(((LifeSpan ((F3() +* (stop F4())),F2())) + 1) + 1));
XX: while>0 (F5(),F6(),F4()) c= stop (while>0 (F5(),F6(),F4())) by AFINSQ_1:74;
 stop (while>0 (F5(),F6(),F4())) c= F3() +* (stop (while>0 (F5(),F6(),F4()))) by FUNCT_4:25;
then A33: while>0 (F5(),F6(),F4()) c= F3() +* (stop (while>0 (F5(),F6(),F4()))) by XX, XBOOLE_1:1;
 Shift (F4(),1) c= while>0 (F5(),F6(),F4()) by Lm4;
then A34: Shift (F4(),1) c= F3() +* (stop (while>0 (F5(),F6(),F4()))) by A33, XBOOLE_1:1;
A35: IC (Comput ((F3() +* (stop (while>0 (F5(),F6(),F4())))),F2(),1)) = succ (IC F2()) by A2, A25, SCMPDS_2:56
.= 0 + 1 by A24 ;
then A36: IC (Comput ((F3() +* (stop (while>0 (F5(),F6(),F4())))),(Comput ((F3() +* (stop (while>0 (F5(),F6(),F4())))),F2(),1)),(LifeSpan ((F3() +* (stop F4())),F2())))) = (card F4()) + 1 by A10, A14, A11, A27, A34, SCMPDS_7:18;
A37: (F3() +* (stop (while>0 (F5(),F6(),F4())))) /. (IC (Comput ((F3() +* (stop (while>0 (F5(),F6(),F4())))),F2(),((LifeSpan ((F3() +* (stop F4())),F2())) + 1)))) = (F3() +* (stop (while>0 (F5(),F6(),F4())))) . (IC (Comput ((F3() +* (stop (while>0 (F5(),F6(),F4())))),F2(),((LifeSpan ((F3() +* (stop F4())),F2())) + 1)))) by PBOOLE:143;
A38: Comput ((F3() +* (stop (while>0 (F5(),F6(),F4())))),F2(),((LifeSpan ((F3() +* (stop F4())),F2())) + 1)) = Comput ((F3() +* (stop (while>0 (F5(),F6(),F4())))),(Comput ((F3() +* (stop (while>0 (F5(),F6(),F4())))),F2(),1)),(LifeSpan ((F3() +* (stop F4())),F2()))) by EXTPRO_1:4;
then A39: CurInstr ((F3() +* (stop (while>0 (F5(),F6(),F4())))),(Comput ((F3() +* (stop (while>0 (F5(),F6(),F4())))),F2(),((LifeSpan ((F3() +* (stop F4())),F2())) + 1)))) = (F3() +* (stop (while>0 (F5(),F6(),F4())))) . ((card F4()) + 1) by A10, A14, A11, A35, A27, A34, A37, SCMPDS_7:18
.= (while>0 (F5(),F6(),F4())) . ((card F4()) + 1) by A28, A33, GRFUNC_1:2
.= goto (- ((card F4()) + 1)) by Th19 ;
A41: Comput ((F3() +* (stop (while>0 (F5(),F6(),F4())))),F2(),(((LifeSpan ((F3() +* (stop F4())),F2())) + 1) + 1)) = Following ((F3() +* (stop (while>0 (F5(),F6(),F4())))),(Comput ((F3() +* (stop (while>0 (F5(),F6(),F4())))),F2(),((LifeSpan ((F3() +* (stop F4())),F2())) + 1)))) by EXTPRO_1:3
.= Exec ((goto (- ((card F4()) + 1))),(Comput ((F3() +* (stop (while>0 (F5(),F6(),F4())))),F2(),((LifeSpan ((F3() +* (stop F4())),F2())) + 1)))) by A39 ;
then  IC (Comput ((F3() +* (stop (while>0 (F5(),F6(),F4())))),F2(),(((LifeSpan ((F3() +* (stop F4())),F2())) + 1) + 1))) = ICplusConst ((Comput ((F3() +* (stop (while>0 (F5(),F6(),F4())))),F2(),((LifeSpan ((F3() +* (stop F4())),F2())) + 1))),(0 - ((card F4()) + 1))) by SCMPDS_2:54
.= 0 by A36, A38, SCMPDS_7:1 ;
then A42: IC (Initialize (IExec (F4(),F3(),F2()))) = IC (Comput ((F3() +* (stop (while>0 (F5(),F6(),F4())))),F2(),((LifeSpan ((F3() +* (stop F4())),F2())) + 2))) by MEMSTR_0:def_11;
A43: DataPart (Comput ((F3() +* (stop F4())),F2(),(LifeSpan ((F3() +* (stop F4())),F2())))) = DataPart (Comput ((F3() +* (stop (while>0 (F5(),F6(),F4())))),(Comput ((F3() +* (stop (while>0 (F5(),F6(),F4())))),F2(),1)),(LifeSpan ((F3() +* (stop F4())),F2())))) by A10, A14, A11, A35, A27, A34, SCMPDS_7:18;
now__::_thesis:_for_x_being_Int_position_holds_(Comput_((F3()_+*_(stop_(while>0_(F5(),F6(),F4())))),F2(),(((LifeSpan_((F3()_+*_(stop_F4())),F2()))_+_1)_+_1)))_._x_=_(Initialize_(IExec_(F4(),F3(),F2())))_._x
let x be Int_position; ::_thesis: (Comput ((F3() +* (stop (while>0 (F5(),F6(),F4())))),F2(),(((LifeSpan ((F3() +* (stop F4())),F2())) + 1) + 1))) . x = (Initialize (IExec (F4(),F3(),F2()))) . x
A44: not x in dom (Start-At (0,SCMPDS)) by SCMPDS_4:18;
(Comput ((F3() +* (stop (while>0 (F5(),F6(),F4())))),(Comput ((F3() +* (stop (while>0 (F5(),F6(),F4())))),F2(),1)),(LifeSpan ((F3() +* (stop F4())),F2())))) . x = (Comput ((F3() +* (stop F4())),F2(),(LifeSpan ((F3() +* (stop F4())),F2())))) . x by A43, SCMPDS_4:8
.= (Result ((F3() +* (stop F4())),F2())) . x by A12, EXTPRO_1:23
.= (IExec (F4(),F3(),F2())) . x by SCMPDS_4:def_5 ;
hence (Comput ((F3() +* (stop (while>0 (F5(),F6(),F4())))),F2(),(((LifeSpan ((F3() +* (stop F4())),F2())) + 1) + 1))) . x = (IExec (F4(),F3(),F2())) . x by A38, A41, SCMPDS_2:54
.= (Initialize (IExec (F4(),F3(),F2()))) . x by A44, FUNCT_4:11 ;
::_thesis: verum
end;
then A46: DataPart (Comput ((F3() +* (stop (while>0 (F5(),F6(),F4())))),F2(),(((LifeSpan ((F3() +* (stop F4())),F2())) + 1) + 1))) = DataPart (Initialize (IExec (F4(),F3(),F2()))) by SCMPDS_4:8;
A48: Comput ((F3() +* (stop (while>0 (F5(),F6(),F4())))),F2(),((LifeSpan ((F3() +* (stop F4())),F2())) + 2)) = Initialize (IExec (F4(),F3(),F2())) by A46, A42, MEMSTR_0:78;
then  CurInstr ((F3() +* (stop (while>0 (F5(),F6(),F4())))),(Comput ((F3() +* (stop (while>0 (F5(),F6(),F4())))),F2(),((LifeSpan ((F3() +* (stop F4())),F2())) + 2)))) = (F5(),F6()) <=0_goto ((card F4()) + 2) by A23, SCMPDS_6:11;
then  LifeSpan ((F3() +* (stop (while>0 (F5(),F6(),F4())))),F2()) > (LifeSpan ((F3() +* (stop F4())),F2())) + 2 by A17, EXTPRO_1:36, SCMPDS_6:17;
then consider nn being Nat such that
A49: LifeSpan ((F3() +* (stop (while>0 (F5(),F6(),F4())))),F2()) = ((LifeSpan ((F3() +* (stop F4())),F2())) + 2) + nn by NAT_1:10;
reconsider nn = nn as Element of NAT by ORDINAL1:def_12;
 Comput ((F3() +* (stop (while>0 (F5(),F6(),F4())))),F2(),(((LifeSpan ((F3() +* (stop F4())),F2())) + 2) + (LifeSpan ((F3() +* (stop (while>0 (F5(),F6(),F4())))),(Initialize (IExec (F4(),F3(),F2()))))))) = Comput ((F3() +* (stop (while>0 (F5(),F6(),F4())))),(Initialize (IExec (F4(),F3(),F2()))),(LifeSpan ((F3() +* (stop (while>0 (F5(),F6(),F4())))),(Initialize (IExec (F4(),F3(),F2())))))) by A48, EXTPRO_1:4;
then  CurInstr ((F3() +* (stop (while>0 (F5(),F6(),F4())))),(Comput ((F3() +* (stop (while>0 (F5(),F6(),F4())))),F2(),(((LifeSpan ((F3() +* (stop F4())),F2())) + 2) + (LifeSpan ((F3() +* (stop (while>0 (F5(),F6(),F4())))),(Initialize (IExec (F4(),F3(),F2()))))))))) = halt SCMPDS by A22, EXTPRO_1:def_15;
then  ((LifeSpan ((F3() +* (stop F4())),F2())) + 2) + (LifeSpan ((F3() +* (stop (while>0 (F5(),F6(),F4())))),(Initialize (IExec (F4(),F3(),F2()))))) >= LifeSpan ((F3() +* (stop (while>0 (F5(),F6(),F4())))),F2()) by A17, EXTPRO_1:def_15;
then A51: LifeSpan ((F3() +* (stop (while>0 (F5(),F6(),F4())))),(Initialize (IExec (F4(),F3(),F2())))) >= nn by A49, XREAL_1:6;
A52: Comput ((F3() +* (stop (while>0 (F5(),F6(),F4())))),F2(),(LifeSpan ((F3() +* (stop (while>0 (F5(),F6(),F4())))),F2()))) = Comput ((F3() +* (stop (while>0 (F5(),F6(),F4())))),(Initialize (IExec (F4(),F3(),F2()))),nn) by A48, A49, EXTPRO_1:4;
then  CurInstr ((F3() +* (stop (while>0 (F5(),F6(),F4())))),(Comput ((F3() +* (stop (while>0 (F5(),F6(),F4())))),(Initialize (IExec (F4(),F3(),F2()))),nn))) = halt SCMPDS by A17, EXTPRO_1:def_15;
then  nn >= LifeSpan ((F3() +* (stop (while>0 (F5(),F6(),F4())))),(Initialize (IExec (F4(),F3(),F2())))) by A22, EXTPRO_1:def_15;
then  nn = LifeSpan ((F3() +* (stop (while>0 (F5(),F6(),F4())))),(Initialize (IExec (F4(),F3(),F2())))) by A51, XXREAL_0:1;
then  Result ((F3() +* (stop (while>0 (F5(),F6(),F4())))),F2()) = Comput ((F3() +* (stop (while>0 (F5(),F6(),F4())))),(Initialize (IExec (F4(),F3(),F2()))),(LifeSpan ((F3() +* (stop (while>0 (F5(),F6(),F4())))),(Initialize (IExec (F4(),F3(),F2())))))) by A17, A52, EXTPRO_1:23;
hence  IExec ((while>0 (F5(),F6(),F4())),F3(),F2()) = Comput ((F3() +* (stop (while>0 (F5(),F6(),F4())))),(Initialize (IExec (F4(),F3(),F2()))),(LifeSpan ((F3() +* (stop (while>0 (F5(),F6(),F4())))),(Initialize (IExec (F4(),F3(),F2())))))) by SCMPDS_4:def_5
.= Result ((F3() +* (stop (while>0 (F5(),F6(),F4())))),(Initialize (IExec (F4(),F3(),F2())))) by A22, EXTPRO_1:23
.= IExec ((while>0 (F5(),F6(),F4())),F3(),(Initialize (IExec (F4(),F3(),F2())))) by SCMPDS_4:def_5 ;
::_thesis: verum
end;

theorem Th24: :: SCMPDS_8:24
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 X, Y being set
for f being Function of (product (the_Values_of SCMPDS)),NAT st ( for t being 0 -started State of SCMPDS st f . t = 0 holds
t . (DataLoc ((s . a),i)) <= 0 ) & ( for x being Int_position st x in X holds
s . x >= c + (s . (DataLoc ((s . a),i))) ) & ( for t being 0 -started State of SCMPDS
for Q being Instruction-Sequence of SCMPDS st ( for x being Int_position st x in X holds
t . x >= c + (t . (DataLoc ((s . a),i))) ) & ( for x being Int_position st x in Y 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 & f . (Initialize (IExec (I,Q,t))) < f . t & ( for x being Int_position st x in X holds
(IExec (I,Q,t)) . x >= c + ((IExec (I,Q,t)) . (DataLoc ((s . a),i))) ) & ( for x being Int_position st x in Y holds
(IExec (I,Q,t)) . x = t . x ) ) ) holds
( while>0 (a,i,I) is_closed_on s,P & 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 X, Y being set
for f being Function of (product (the_Values_of SCMPDS)),NAT st ( for t being 0 -started State of SCMPDS st f . t = 0 holds
t . (DataLoc ((s . a),i)) <= 0 ) & ( for x being Int_position st x in X holds
s . x >= c + (s . (DataLoc ((s . a),i))) ) & ( for t being 0 -started State of SCMPDS
for Q being Instruction-Sequence of SCMPDS st ( for x being Int_position st x in X holds
t . x >= c + (t . (DataLoc ((s . a),i))) ) & ( for x being Int_position st x in Y 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 & f . (Initialize (IExec (I,Q,t))) < f . t & ( for x being Int_position st x in X holds
(IExec (I,Q,t)) . x >= c + ((IExec (I,Q,t)) . (DataLoc ((s . a),i))) ) & ( for x being Int_position st x in Y holds
(IExec (I,Q,t)) . x = t . x ) ) ) holds
( while>0 (a,i,I) is_closed_on s,P & 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 X, Y being set
for f being Function of (product (the_Values_of SCMPDS)),NAT st ( for t being 0 -started State of SCMPDS st f . t = 0 holds
t . (DataLoc ((s . a),i)) <= 0 ) & ( for x being Int_position st x in X holds
s . x >= c + (s . (DataLoc ((s . a),i))) ) & ( for t being 0 -started State of SCMPDS
for Q being Instruction-Sequence of SCMPDS st ( for x being Int_position st x in X holds
t . x >= c + (t . (DataLoc ((s . a),i))) ) & ( for x being Int_position st x in Y 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 & f . (Initialize (IExec (I,Q,t))) < f . t & ( for x being Int_position st x in X holds
(IExec (I,Q,t)) . x >= c + ((IExec (I,Q,t)) . (DataLoc ((s . a),i))) ) & ( for x being Int_position st x in Y holds
(IExec (I,Q,t)) . x = t . x ) ) ) holds
( while>0 (a,i,I) is_closed_on s,P & 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 X, Y being set
for f being Function of (product (the_Values_of SCMPDS)),NAT st ( for t being 0 -started State of SCMPDS st f . t = 0 holds
t . (DataLoc ((s . a),i)) <= 0 ) & ( for x being Int_position st x in X holds
s . x >= c + (s . (DataLoc ((s . a),i))) ) & ( for t being 0 -started State of SCMPDS
for Q being Instruction-Sequence of SCMPDS st ( for x being Int_position st x in X holds
t . x >= c + (t . (DataLoc ((s . a),i))) ) & ( for x being Int_position st x in Y 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 & f . (Initialize (IExec (I,Q,t))) < f . t & ( for x being Int_position st x in X holds
(IExec (I,Q,t)) . x >= c + ((IExec (I,Q,t)) . (DataLoc ((s . a),i))) ) & ( for x being Int_position st x in Y holds
(IExec (I,Q,t)) . x = t . x ) ) ) holds
( while>0 (a,i,I) is_closed_on s,P & while>0 (a,i,I) is_halting_on s,P )
let a be Int_position; ::_thesis: for i, c being Integer
for X, Y being set
for f being Function of (product (the_Values_of SCMPDS)),NAT st ( for t being 0 -started State of SCMPDS st f . t = 0 holds
t . (DataLoc ((s . a),i)) <= 0 ) & ( for x being Int_position st x in X holds
s . x >= c + (s . (DataLoc ((s . a),i))) ) & ( for t being 0 -started State of SCMPDS
for Q being Instruction-Sequence of SCMPDS st ( for x being Int_position st x in X holds
t . x >= c + (t . (DataLoc ((s . a),i))) ) & ( for x being Int_position st x in Y 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 & f . (Initialize (IExec (I,Q,t))) < f . t & ( for x being Int_position st x in X holds
(IExec (I,Q,t)) . x >= c + ((IExec (I,Q,t)) . (DataLoc ((s . a),i))) ) & ( for x being Int_position st x in Y holds
(IExec (I,Q,t)) . x = t . x ) ) ) holds
( while>0 (a,i,I) is_closed_on s,P & while>0 (a,i,I) is_halting_on s,P )
let i, c be Integer; ::_thesis: for X, Y being set
for f being Function of (product (the_Values_of SCMPDS)),NAT st ( for t being 0 -started State of SCMPDS st f . t = 0 holds
t . (DataLoc ((s . a),i)) <= 0 ) & ( for x being Int_position st x in X holds
s . x >= c + (s . (DataLoc ((s . a),i))) ) & ( for t being 0 -started State of SCMPDS
for Q being Instruction-Sequence of SCMPDS st ( for x being Int_position st x in X holds
t . x >= c + (t . (DataLoc ((s . a),i))) ) & ( for x being Int_position st x in Y 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 & f . (Initialize (IExec (I,Q,t))) < f . t & ( for x being Int_position st x in X holds
(IExec (I,Q,t)) . x >= c + ((IExec (I,Q,t)) . (DataLoc ((s . a),i))) ) & ( for x being Int_position st x in Y holds
(IExec (I,Q,t)) . x = t . x ) ) ) holds
( while>0 (a,i,I) is_closed_on s,P & while>0 (a,i,I) is_halting_on s,P )
let X, Y be set ; ::_thesis: for f being Function of (product (the_Values_of SCMPDS)),NAT st ( for t being 0 -started State of SCMPDS st f . t = 0 holds
t . (DataLoc ((s . a),i)) <= 0 ) & ( for x being Int_position st x in X holds
s . x >= c + (s . (DataLoc ((s . a),i))) ) & ( for t being 0 -started State of SCMPDS
for Q being Instruction-Sequence of SCMPDS st ( for x being Int_position st x in X holds
t . x >= c + (t . (DataLoc ((s . a),i))) ) & ( for x being Int_position st x in Y 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 & f . (Initialize (IExec (I,Q,t))) < f . t & ( for x being Int_position st x in X holds
(IExec (I,Q,t)) . x >= c + ((IExec (I,Q,t)) . (DataLoc ((s . a),i))) ) & ( for x being Int_position st x in Y holds
(IExec (I,Q,t)) . x = t . x ) ) ) holds
( while>0 (a,i,I) is_closed_on s,P & while>0 (a,i,I) is_halting_on s,P )
let f be Function of (product (the_Values_of SCMPDS)),NAT; ::_thesis: ( ( for t being 0 -started State of SCMPDS st f . t = 0 holds
t . (DataLoc ((s . a),i)) <= 0 ) & ( for x being Int_position st x in X holds
s . x >= c + (s . (DataLoc ((s . a),i))) ) & ( for t being 0 -started State of SCMPDS
for Q being Instruction-Sequence of SCMPDS st ( for x being Int_position st x in X holds
t . x >= c + (t . (DataLoc ((s . a),i))) ) & ( for x being Int_position st x in Y 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 & f . (Initialize (IExec (I,Q,t))) < f . t & ( for x being Int_position st x in X holds
(IExec (I,Q,t)) . x >= c + ((IExec (I,Q,t)) . (DataLoc ((s . a),i))) ) & ( for x being Int_position st x in Y holds
(IExec (I,Q,t)) . x = t . x ) ) ) implies ( while>0 (a,i,I) is_closed_on s,P & while>0 (a,i,I) is_halting_on s,P ) )
set b = DataLoc ((s . a),i);
set WHL = while>0 (a,i,I);
set pWHL = stop (while>0 (a,i,I));
set pI = stop I;
set i1 = (a,i) <=0_goto ((card I) + 2);
set i2 = goto (- ((card I) + 1));
defpred S1[ Element of NAT ] means  for t being 0 -started State of SCMPDS
for Q being Instruction-Sequence of SCMPDS st f . t <= $1 & ( for x being Int_position st x in X holds
t . x >= c + (t . (DataLoc ((s . a),i))) ) & ( for x being Int_position st x in Y holds
t . x = s . x ) & t . a = s . a holds
( while>0 (a,i,I) is_closed_on t,Q & while>0 (a,i,I) is_halting_on t,Q );
assume A2: for t being 0 -started State of SCMPDS st f . t = 0 holds
t . (DataLoc ((s . a),i)) <= 0 ; ::_thesis: ( ex x being Int_position st
( x in X & not s . x >= c + (s . (DataLoc ((s . a),i))) ) or ex t being 0 -started State of SCMPDS ex Q being Instruction-Sequence of SCMPDS st
( ( for x being Int_position st x in X holds
t . x >= c + (t . (DataLoc ((s . a),i))) ) & ( for x being Int_position st x in Y holds
t . x = s . x ) & 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 & f . (Initialize (IExec (I,Q,t))) < f . t & ( for x being Int_position st x in X holds
(IExec (I,Q,t)) . x >= c + ((IExec (I,Q,t)) . (DataLoc ((s . a),i))) ) & ( for x being Int_position st x in Y holds
(IExec (I,Q,t)) . x = t . x ) ) ) or ( while>0 (a,i,I) is_closed_on s,P & while>0 (a,i,I) is_halting_on s,P ) )
assume A3: for x being Int_position st x in X holds
s . x >= c + (s . (DataLoc ((s . a),i))) ; ::_thesis: ( ex t being 0 -started State of SCMPDS ex Q being Instruction-Sequence of SCMPDS st
( ( for x being Int_position st x in X holds
t . x >= c + (t . (DataLoc ((s . a),i))) ) & ( for x being Int_position st x in Y holds
t . x = s . x ) & 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 & f . (Initialize (IExec (I,Q,t))) < f . t & ( for x being Int_position st x in X holds
(IExec (I,Q,t)) . x >= c + ((IExec (I,Q,t)) . (DataLoc ((s . a),i))) ) & ( for x being Int_position st x in Y holds
(IExec (I,Q,t)) . x = t . x ) ) ) or ( while>0 (a,i,I) is_closed_on s,P & while>0 (a,i,I) is_halting_on s,P ) )
assume A4: for t being 0 -started State of SCMPDS
for Q being Instruction-Sequence of SCMPDS st ( for x being Int_position st x in X holds
t . x >= c + (t . (DataLoc ((s . a),i))) ) & ( for x being Int_position st x in Y 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 & f . (Initialize (IExec (I,Q,t))) < f . t & ( for x being Int_position st x in X holds
(IExec (I,Q,t)) . x >= c + ((IExec (I,Q,t)) . (DataLoc ((s . a),i))) ) & ( for x being Int_position st x in Y holds
(IExec (I,Q,t)) . x = t . x ) ) ; ::_thesis: ( while>0 (a,i,I) is_closed_on s,P & while>0 (a,i,I) is_halting_on s,P )
A5: for k being Element of NAT st S1[k] holds
S1[k + 1]
proof
let k be Element of NAT ; ::_thesis: ( S1[k] implies S1[k + 1] )
assume A6: S1[k] ; ::_thesis: S1[k + 1]
now__::_thesis:_for_t_being_0_-started_State_of_SCMPDS
for_Q_being_Instruction-Sequence_of_SCMPDS_st_f_._t_<=_k_+_1_&_(_for_x_being_Int_position_st_x_in_X_holds_
t_._x_>=_c_+_(t_._(DataLoc_((s_._a),i)))_)_&_(_for_x_being_Int_position_st_x_in_Y_holds_
t_._x_=_s_._x_)_&_t_._a_=_s_._a_holds_
(_while>0_(a,i,I)_is_closed_on_t,Q_&_while>0_(a,i,I)_is_halting_on_t,Q_)
let t be 0 -started State of SCMPDS; ::_thesis: for Q being Instruction-Sequence of SCMPDS st f . t <= k + 1 & ( for x being Int_position st x in X holds
t . x >= c + (t . (DataLoc ((s . a),i))) ) & ( for x being Int_position st x in Y holds
t . x = s . x ) & t . a = s . a holds
( while>0 (a,i,I) is_closed_on b2,b3 & while>0 (a,i,I) is_halting_on b2,b3 )
let Q be Instruction-Sequence of SCMPDS; ::_thesis: ( f . t <= k + 1 & ( for x being Int_position st x in X holds
t . x >= c + (t . (DataLoc ((s . a),i))) ) & ( for x being Int_position st x in Y holds
t . x = s . x ) & t . a = s . a implies ( while>0 (a,i,I) is_closed_on b1,b2 & while>0 (a,i,I) is_halting_on b1,b2 ) )
T: Initialize t = t by MEMSTR_0:44;
assume A7: f . t <= k + 1 ; ::_thesis: ( ( for x being Int_position st x in X holds
t . x >= c + (t . (DataLoc ((s . a),i))) ) & ( for x being Int_position st x in Y holds
t . x = s . x ) & t . a = s . a implies ( while>0 (a,i,I) is_closed_on b1,b2 & while>0 (a,i,I) is_halting_on b1,b2 ) )
assume A8: for x being Int_position st x in X holds
t . x >= c + (t . (DataLoc ((s . a),i))) ; ::_thesis: ( ( for x being Int_position st x in Y holds
t . x = s . x ) & t . a = s . a implies ( while>0 (a,i,I) is_closed_on b1,b2 & while>0 (a,i,I) is_halting_on b1,b2 ) )
assume A9: for x being Int_position st x in Y holds
t . x = s . x ; ::_thesis: ( t . a = s . a implies ( while>0 (a,i,I) is_closed_on b1,b2 & while>0 (a,i,I) is_halting_on b1,b2 ) )
assume A10: t . a = s . a ; ::_thesis: ( while>0 (a,i,I) is_closed_on b1,b2 & while>0 (a,i,I) is_halting_on b1,b2 )
percases ( t . (DataLoc ((s . a),i)) <= 0 or t . (DataLoc ((s . a),i)) > 0 ) ;
suppose t . (DataLoc ((s . a),i)) <= 0 ; ::_thesis: ( while>0 (a,i,I) is_closed_on b1,b2 & while>0 (a,i,I) is_halting_on b1,b2 )
hence  ( while>0 (a,i,I) is_closed_on t,Q & while>0 (a,i,I) is_halting_on t,Q ) by A10, Th20; ::_thesis: verum
end;
supposeA11: t . (DataLoc ((s . a),i)) > 0 ; ::_thesis: ( while>0 (a,i,I) is_closed_on b1,b2 & while>0 (a,i,I) is_halting_on b1,b2 )
A15: 0 in dom (stop (while>0 (a,i,I))) by COMPOS_1:36;
A18: while>0 (a,i,I) = ((a,i) <=0_goto ((card I) + 2)) ';' (I ';' (goto (- ((card I) + 1)))) by SCMPDS_4:15;
set Q2 = Q +* (stop I);
set Q3 = Q +* (stop (while>0 (a,i,I)));
set t4 = Comput ((Q +* (stop (while>0 (a,i,I)))),t,1);
set Q4 = Q +* (stop (while>0 (a,i,I)));
A21: stop I c= Q +* (stop I) by FUNCT_4:25;
A22: Comput ((Q +* (stop (while>0 (a,i,I)))),t,(0 + 1)) = Following ((Q +* (stop (while>0 (a,i,I)))),(Comput ((Q +* (stop (while>0 (a,i,I)))),t,0))) by EXTPRO_1:3
.= Following ((Q +* (stop (while>0 (a,i,I)))),t)
.= Exec (((a,i) <=0_goto ((card I) + 2)),t) by A18, T, SCMPDS_6:11 ;
 for a being Int_position holds t . a = (Comput ((Q +* (stop (while>0 (a,i,I)))),t,1)) . a by A22, SCMPDS_2:56;
then A24: DataPart t = DataPart (Comput ((Q +* (stop (while>0 (a,i,I)))),t,1)) by SCMPDS_4:8;
XX: while>0 (a,i,I) c= stop (while>0 (a,i,I)) by AFINSQ_1:74;
 stop (while>0 (a,i,I)) c= Q +* (stop (while>0 (a,i,I))) by FUNCT_4:25;
then A25: while>0 (a,i,I) c= Q +* (stop (while>0 (a,i,I))) by XX, XBOOLE_1:1;
 Shift (I,1) c= while>0 (a,i,I) by Lm4;
then A26: Shift (I,1) c= Q +* (stop (while>0 (a,i,I))) by A25, XBOOLE_1:1;
A27: IExec (I,Q,t) = Result ((Q +* (stop I)),t) by SCMPDS_4:def_5;
set m2 = LifeSpan ((Q +* (stop I)),t);
set t5 = Comput ((Q +* (stop (while>0 (a,i,I)))),(Comput ((Q +* (stop (while>0 (a,i,I)))),t,1)),(LifeSpan ((Q +* (stop I)),t)));
set Q5 = Q +* (stop (while>0 (a,i,I)));
set l1 = (card I) + 1;
A28: IC t = 0 by MEMSTR_0:def_11;
set m3 = (LifeSpan ((Q +* (stop I)),t)) + 1;
set t6 = Comput ((Q +* (stop (while>0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 1));
set Q6 = Q +* (stop (while>0 (a,i,I)));
set t7 = Comput ((Q +* (stop (while>0 (a,i,I)))),t,(((LifeSpan ((Q +* (stop I)),t)) + 1) + 1));
set Q7 = Q +* (stop (while>0 (a,i,I)));
 (card I) + 1 < (card I) + 2 by XREAL_1:6;
then A29: (card I) + 1 in dom (while>0 (a,i,I)) by Th18;
A30: I is_closed_on t,Q by A4, A8, A9, A10, A11;
A31: I is_closed_on t,Q +* (stop I) by A4, A8, A9, A10, A11;
 I is_halting_on t,Q by A4, A8, A9, A10, A11;
then A32: Q +* (stop I) halts_on t by T, SCMPDS_6:def_3;
 (Q +* (stop I)) +* (stop I) halts_on t by A32;
then A34: I is_halting_on t,Q +* (stop I) by T, SCMPDS_6:def_3;
A35: IC (Comput ((Q +* (stop (while>0 (a,i,I)))),t,1)) = succ (IC t) by A11, A22, A10, SCMPDS_2:56
.= 0 + 1 by A28 ;
then A36: IC (Comput ((Q +* (stop (while>0 (a,i,I)))),(Comput ((Q +* (stop (while>0 (a,i,I)))),t,1)),(LifeSpan ((Q +* (stop I)),t)))) = (card I) + 1 by A21, A34, A31, A24, A26, SCMPDS_7:18;
A37: (Q +* (stop (while>0 (a,i,I)))) /. (IC (Comput ((Q +* (stop (while>0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 1)))) = (Q +* (stop (while>0 (a,i,I)))) . (IC (Comput ((Q +* (stop (while>0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 1)))) by PBOOLE:143;
A38: Comput ((Q +* (stop (while>0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 1)) = Comput ((Q +* (stop (while>0 (a,i,I)))),(Comput ((Q +* (stop (while>0 (a,i,I)))),t,1)),(LifeSpan ((Q +* (stop I)),t))) by EXTPRO_1:4;
then A39: CurInstr ((Q +* (stop (while>0 (a,i,I)))),(Comput ((Q +* (stop (while>0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 1)))) = (Q +* (stop (while>0 (a,i,I)))) . ((card I) + 1) by A21, A34, A31, A35, A24, A26, A37, SCMPDS_7:18
.= (while>0 (a,i,I)) . ((card I) + 1) by A29, A25, GRFUNC_1:2
.= goto (- ((card I) + 1)) by Th19 ;
A41: Comput ((Q +* (stop (while>0 (a,i,I)))),t,(((LifeSpan ((Q +* (stop I)),t)) + 1) + 1)) = Following ((Q +* (stop (while>0 (a,i,I)))),(Comput ((Q +* (stop (while>0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 1)))) by EXTPRO_1:3
.= Exec ((goto (- ((card I) + 1))),(Comput ((Q +* (stop (while>0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 1)))) by A39 ;
then  IC (Comput ((Q +* (stop (while>0 (a,i,I)))),t,(((LifeSpan ((Q +* (stop I)),t)) + 1) + 1))) = ICplusConst ((Comput ((Q +* (stop (while>0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 1))),(0 - ((card I) + 1))) by SCMPDS_2:54
.= 0 by A36, A38, SCMPDS_7:1 ;
then A42: Initialize (Comput ((Q +* (stop (while>0 (a,i,I)))),t,(((LifeSpan ((Q +* (stop I)),t)) + 1) + 1))) = Comput ((Q +* (stop (while>0 (a,i,I)))),t,(((LifeSpan ((Q +* (stop I)),t)) + 1) + 1)) by MEMSTR_0:46;
A43: DataPart (Comput ((Q +* (stop I)),t,(LifeSpan ((Q +* (stop I)),t)))) = DataPart (Comput ((Q +* (stop (while>0 (a,i,I)))),(Comput ((Q +* (stop (while>0 (a,i,I)))),t,1)),(LifeSpan ((Q +* (stop I)),t)))) by A21, A34, A31, A35, A24, A26, SCMPDS_7:18;
then A44: DataPart (Comput ((Q +* (stop (while>0 (a,i,I)))),(Comput ((Q +* (stop (while>0 (a,i,I)))),t,1)),(LifeSpan ((Q +* (stop I)),t)))) = DataPart (Result ((Q +* (stop I)),t)) by A32, EXTPRO_1:23
.= DataPart (IExec (I,Q,t)) by SCMPDS_4:def_5 ;
A45: now__::_thesis:_for_x_being_Int_position_st_x_in_Y_holds_
(Comput_((Q_+*_(stop_(while>0_(a,i,I)))),t,(((LifeSpan_((Q_+*_(stop_I)),t))_+_1)_+_1)))_._x_=_s_._x
let x be Int_position; ::_thesis: ( x in Y implies (Comput ((Q +* (stop (while>0 (a,i,I)))),t,(((LifeSpan ((Q +* (stop I)),t)) + 1) + 1))) . x = s . x )
assume A46: x in Y ; ::_thesis: (Comput ((Q +* (stop (while>0 (a,i,I)))),t,(((LifeSpan ((Q +* (stop I)),t)) + 1) + 1))) . x = s . x
thus (Comput ((Q +* (stop (while>0 (a,i,I)))),t,(((LifeSpan ((Q +* (stop I)),t)) + 1) + 1))) . x = (Comput ((Q +* (stop (while>0 (a,i,I)))),(Comput ((Q +* (stop (while>0 (a,i,I)))),t,1)),(LifeSpan ((Q +* (stop I)),t)))) . x by A38, A41, SCMPDS_2:54
.= (IExec (I,Q,t)) . x by A44, SCMPDS_3:3
.= t . x by A4, A8, A9, A10, A11, A46
.= s . x by A9, A46 ; ::_thesis: verum
end;
 InsCode (goto (- ((card I) + 1))) = 14 by SCMPDS_2:12;
then  InsCode (goto (- ((card I) + 1))) in {0,4,5,6,14} by ENUMSET1:def_3;
then A47: Initialize (Comput ((Q +* (stop (while>0 (a,i,I)))),t,(((LifeSpan ((Q +* (stop I)),t)) + 1) + 1))) = Initialize (Comput ((Q +* (stop (while>0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 1))) by A41, Th3
.= Initialize (IExec (I,Q,t)) by A44, A38, MEMSTR_0:80 ;
A48: now__::_thesis:_not_f_._(Initialize_(Comput_((Q_+*_(stop_(while>0_(a,i,I)))),t,(((LifeSpan_((Q_+*_(stop_I)),t))_+_1)_+_1))))_>_k
 f . (Initialize (IExec (I,Q,t))) < f . (Initialize t) by A4, A8, A9, A10, A11, T;
then A49: f . (Comput ((Q +* (stop (while>0 (a,i,I)))),t,(((LifeSpan ((Q +* (stop I)),t)) + 1) + 1))) < k + 1 by A7, A47, A42, T, XXREAL_0:2;
assume  f . (Initialize (Comput ((Q +* (stop (while>0 (a,i,I)))),t,(((LifeSpan ((Q +* (stop I)),t)) + 1) + 1)))) > k ; ::_thesis: contradiction
hence  contradiction by A49, A42, INT_1:7; ::_thesis: verum
end;
A50: (Comput ((Q +* (stop (while>0 (a,i,I)))),t,(((LifeSpan ((Q +* (stop I)),t)) + 1) + 1))) . (DataLoc ((s . a),i)) = (Comput ((Q +* (stop (while>0 (a,i,I)))),(Comput ((Q +* (stop (while>0 (a,i,I)))),t,1)),(LifeSpan ((Q +* (stop I)),t)))) . (DataLoc ((s . a),i)) by A38, A41, SCMPDS_2:54
.= (IExec (I,Q,t)) . (DataLoc ((s . a),i)) by A44, SCMPDS_3:3 ;
A51: now__::_thesis:_for_x_being_Int_position_st_x_in_X_holds_
(Comput_((Q_+*_(stop_(while>0_(a,i,I)))),t,(((LifeSpan_((Q_+*_(stop_I)),t))_+_1)_+_1)))_._x_>=_c_+_((Comput_((Q_+*_(stop_(while>0_(a,i,I)))),t,(((LifeSpan_((Q_+*_(stop_I)),t))_+_1)_+_1)))_._(DataLoc_((s_._a),i)))
let x be Int_position; ::_thesis: ( x in X implies (Comput ((Q +* (stop (while>0 (a,i,I)))),t,(((LifeSpan ((Q +* (stop I)),t)) + 1) + 1))) . x >= c + ((Comput ((Q +* (stop (while>0 (a,i,I)))),t,(((LifeSpan ((Q +* (stop I)),t)) + 1) + 1))) . (DataLoc ((s . a),i))) )
assume A52: x in X ; ::_thesis: (Comput ((Q +* (stop (while>0 (a,i,I)))),t,(((LifeSpan ((Q +* (stop I)),t)) + 1) + 1))) . x >= c + ((Comput ((Q +* (stop (while>0 (a,i,I)))),t,(((LifeSpan ((Q +* (stop I)),t)) + 1) + 1))) . (DataLoc ((s . a),i)))
(Comput ((Q +* (stop (while>0 (a,i,I)))),t,(((LifeSpan ((Q +* (stop I)),t)) + 1) + 1))) . x = (Comput ((Q +* (stop (while>0 (a,i,I)))),(Comput ((Q +* (stop (while>0 (a,i,I)))),t,1)),(LifeSpan ((Q +* (stop I)),t)))) . x by A38, A41, SCMPDS_2:54
.= (IExec (I,Q,t)) . x by A44, SCMPDS_3:3 ;
hence  (Comput ((Q +* (stop (while>0 (a,i,I)))),t,(((LifeSpan ((Q +* (stop I)),t)) + 1) + 1))) . x >= c + ((Comput ((Q +* (stop (while>0 (a,i,I)))),t,(((LifeSpan ((Q +* (stop I)),t)) + 1) + 1))) . (DataLoc ((s . a),i))) by A4, A8, A9, A10, A11, A50, A52; ::_thesis: verum
end;
A53: (Comput ((Q +* (stop (while>0 (a,i,I)))),(Comput ((Q +* (stop (while>0 (a,i,I)))),t,1)),(LifeSpan ((Q +* (stop I)),t)))) . a = (Comput ((Q +* (stop I)),t,(LifeSpan ((Q +* (stop I)),t)))) . a by A43, SCMPDS_4:8
.= (Result ((Q +* (stop I)),t)) . a by A32, EXTPRO_1:23
.= s . a by A10, A4, A8, A9, A11, A27 ;
A55: (Comput ((Q +* (stop (while>0 (a,i,I)))),t,(((LifeSpan ((Q +* (stop I)),t)) + 1) + 1))) . a = (Comput ((Q +* (stop (while>0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 1))) . a by A41, SCMPDS_2:54
.= s . a by A53, EXTPRO_1:4 ;
then A56: while>0 (a,i,I) is_closed_on Comput ((Q +* (stop (while>0 (a,i,I)))),t,(((LifeSpan ((Q +* (stop I)),t)) + 1) + 1)),Q +* (stop (while>0 (a,i,I))) by A6, A51, A45, A48, A42;
now__::_thesis:_for_k_being_Element_of_NAT_holds_IC_(Comput_((Q_+*_(stop_(while>0_(a,i,I)))),t,k))_in_dom_(stop_(while>0_(a,i,I)))
let k be Element of NAT ; ::_thesis: IC (Comput ((Q +* (stop (while>0 (a,i,I)))),t,b1)) in dom (stop (while>0 (a,i,I)))
percases ( k < ((LifeSpan ((Q +* (stop I)),t)) + 1) + 1 or k >= ((LifeSpan ((Q +* (stop I)),t)) + 1) + 1 ) ;
suppose k < ((LifeSpan ((Q +* (stop I)),t)) + 1) + 1 ; ::_thesis: IC (Comput ((Q +* (stop (while>0 (a,i,I)))),t,b1)) in dom (stop (while>0 (a,i,I)))
then A57: k <= (LifeSpan ((Q +* (stop I)),t)) + 1 by INT_1:7;
hereby  ::_thesis: verum
percases ( k <= LifeSpan ((Q +* (stop I)),t) or k = (LifeSpan ((Q +* (stop I)),t)) + 1 ) by A57, NAT_1:8;
supposeA58: k <= LifeSpan ((Q +* (stop I)),t) ; ::_thesis: IC (Comput ((Q +* (stop (while>0 (a,i,I)))),t,k)) in dom (stop (while>0 (a,i,I)))
hereby  ::_thesis: verum
percases ( k = 0 or k <> 0 ) ;
suppose k = 0 ; ::_thesis: IC (Comput ((Q +* (stop (while>0 (a,i,I)))),t,k)) in dom (stop (while>0 (a,i,I)))
hence  IC (Comput ((Q +* (stop (while>0 (a,i,I)))),t,k)) in dom (stop (while>0 (a,i,I))) by A15, A28, EXTPRO_1:2; ::_thesis: verum
end;
suppose k <> 0 ; ::_thesis: IC (Comput ((Q +* (stop (while>0 (a,i,I)))),t,k)) in dom (stop (while>0 (a,i,I)))
then consider kn being Nat such that
A59: k = kn + 1 by NAT_1:6;
reconsider kn = kn as Element of NAT by ORDINAL1:def_12;
reconsider lm = IC (Comput ((Q +* (stop I)),t,kn)) as Element of NAT ;
 kn < k by A59, XREAL_1:29;
then  kn < LifeSpan ((Q +* (stop I)),t) by A58, XXREAL_0:2;
then  (IC (Comput ((Q +* (stop I)),t,kn))) + 1 = IC (Comput ((Q +* (stop (while>0 (a,i,I)))),(Comput ((Q +* (stop (while>0 (a,i,I)))),t,1)),kn)) by A21, A34, A31, A35, A24, A26, SCMPDS_7:16;
then A61: IC (Comput ((Q +* (stop (while>0 (a,i,I)))),t,k)) = lm + 1 by A59, EXTPRO_1:4;
 IC (Comput ((Q +* (stop I)),t,kn)) in dom (stop I) by A30, T, SCMPDS_6:def_2;
then  lm < card (stop I) by AFINSQ_1:66;
then  lm < (card I) + 1 by COMPOS_1:55;
then A62: lm + 1 <= (card I) + 1 by INT_1:7;
 (card I) + 1 < (card I) + 3 by XREAL_1:6;
then  lm + 1 < (card I) + 3 by A62, XXREAL_0:2;
then  lm + 1 < card (stop (while>0 (a,i,I))) by Lm3;
hence  IC (Comput ((Q +* (stop (while>0 (a,i,I)))),t,k)) in dom (stop (while>0 (a,i,I))) by A61, AFINSQ_1:66; ::_thesis: verum
end;
end;
end;
end;
supposeA63: k = (LifeSpan ((Q +* (stop I)),t)) + 1 ; ::_thesis: IC (Comput ((Q +* (stop (while>0 (a,i,I)))),t,k)) in dom (stop (while>0 (a,i,I)))
 (card I) + 1 in dom (stop (while>0 (a,i,I))) by A29, COMPOS_1:62;
hence  IC (Comput ((Q +* (stop (while>0 (a,i,I)))),t,k)) in dom (stop (while>0 (a,i,I))) by A21, A34, A31, A35, A24, A26, A38, A63, SCMPDS_7:18; ::_thesis: verum
end;
end;
end;
end;
suppose k >= ((LifeSpan ((Q +* (stop I)),t)) + 1) + 1 ; ::_thesis: IC (Comput ((Q +* (stop (while>0 (a,i,I)))),t,b1)) in dom (stop (while>0 (a,i,I)))
then consider nn being Nat such that
A64: k = (((LifeSpan ((Q +* (stop I)),t)) + 1) + 1) + nn by NAT_1:10;
A66: nn in NAT by ORDINAL1:def_12;
 Comput ((Q +* (stop (while>0 (a,i,I)))),t,k) = Comput (((Q +* (stop (while>0 (a,i,I)))) +* (stop (while>0 (a,i,I)))),(Comput ((Q +* (stop (while>0 (a,i,I)))),t,(((LifeSpan ((Q +* (stop I)),t)) + 1) + 1))),nn) by A64, A66, EXTPRO_1:4;
hence  IC (Comput ((Q +* (stop (while>0 (a,i,I)))),t,k)) in dom (stop (while>0 (a,i,I))) by A56, A66, A42, SCMPDS_6:def_2; ::_thesis: verum
end;
end;
end;
hence  while>0 (a,i,I) is_closed_on t,Q by T, SCMPDS_6:def_2; ::_thesis: while>0 (a,i,I) is_halting_on t,Q
RR: (Q +* (stop (while>0 (a,i,I)))) +* (stop (while>0 (a,i,I))) = Q +* (stop (while>0 (a,i,I))) ;
 while>0 (a,i,I) is_halting_on Comput ((Q +* (stop (while>0 (a,i,I)))),t,(((LifeSpan ((Q +* (stop I)),t)) + 1) + 1)),Q +* (stop (while>0 (a,i,I))) by A6, A55, A51, A45, A48, A42;
then  Q +* (stop (while>0 (a,i,I))) halts_on Comput ((Q +* (stop (while>0 (a,i,I)))),t,(((LifeSpan ((Q +* (stop I)),t)) + 1) + 1)) by A42, RR, SCMPDS_6:def_3;
then  Q +* (stop (while>0 (a,i,I))) halts_on t by EXTPRO_1:22;
hence  while>0 (a,i,I) is_halting_on t,Q by T, SCMPDS_6:def_3; ::_thesis: verum
end;
end;
end;
hence  S1[k + 1] ; ::_thesis: verum
end;
set n = f . s;
A67: for x being Int_position st x in Y holds
s . x = s . x ;
A68: S1[ 0 ]
proof
let t be 0 -started State of SCMPDS; ::_thesis: for Q being Instruction-Sequence of SCMPDS st f . t <= 0 & ( for x being Int_position st x in X holds
t . x >= c + (t . (DataLoc ((s . a),i))) ) & ( for x being Int_position st x in Y holds
t . x = s . x ) & t . a = s . a holds
( while>0 (a,i,I) is_closed_on t,Q & while>0 (a,i,I) is_halting_on t,Q )
let Q be Instruction-Sequence of SCMPDS; ::_thesis: ( f . t <= 0 & ( for x being Int_position st x in X holds
t . x >= c + (t . (DataLoc ((s . a),i))) ) & ( for x being Int_position st x in Y holds
t . x = s . x ) & t . a = s . a implies ( while>0 (a,i,I) is_closed_on t,Q & while>0 (a,i,I) is_halting_on t,Q ) )
assume  f . t <= 0 ; ::_thesis: ( ex x being Int_position st
( x in X & not t . x >= c + (t . (DataLoc ((s . a),i))) ) or ex x being Int_position st
( x in Y & not t . x = s . x ) or not t . a = s . a or ( while>0 (a,i,I) is_closed_on t,Q & while>0 (a,i,I) is_halting_on t,Q ) )
then  f . t = 0 ;
then A69: t . (DataLoc ((s . a),i)) <= 0 by A2;
assume  for x being Int_position st x in X holds
t . x >= c + (t . (DataLoc ((s . a),i))) ; ::_thesis: ( ex x being Int_position st
( x in Y & not t . x = s . x ) or not t . a = s . a or ( while>0 (a,i,I) is_closed_on t,Q & while>0 (a,i,I) is_halting_on t,Q ) )
assume  for x being Int_position st x in Y holds
t . x = s . x ; ::_thesis: ( not t . a = s . a or ( while>0 (a,i,I) is_closed_on t,Q & while>0 (a,i,I) is_halting_on t,Q ) )
assume  t . a = s . a ; ::_thesis: ( while>0 (a,i,I) is_closed_on t,Q & while>0 (a,i,I) is_halting_on t,Q )
hence  ( while>0 (a,i,I) is_closed_on t,Q & while>0 (a,i,I) is_halting_on t,Q ) by A69, Th20; ::_thesis: verum
end;
 for k being Element of NAT holds S1[k] from NAT_1:sch_1(A68, A5);
then  S1[f . s] ;
hence  ( while>0 (a,i,I) is_closed_on s,P & while>0 (a,i,I) is_halting_on s,P ) by A3, A67; ::_thesis: verum
end;

theorem Th25: :: SCMPDS_8:25
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 X, Y being set
for f being Function of (product (the_Values_of SCMPDS)),NAT st s . (DataLoc ((s . a),i)) > 0 & ( for t being 0 -started State of SCMPDS st f . t = 0 holds
t . (DataLoc ((s . a),i)) <= 0 ) & ( for x being Int_position st x in X holds
s . x >= c + (s . (DataLoc ((s . a),i))) ) & ( for t being 0 -started State of SCMPDS
for Q being Instruction-Sequence of SCMPDS st ( for x being Int_position st x in X holds
t . x >= c + (t . (DataLoc ((s . a),i))) ) & ( for x being Int_position st x in Y 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 & f . (Initialize (IExec (I,Q,t))) < f . t & ( for x being Int_position st x in X holds
(IExec (I,Q,t)) . x >= c + ((IExec (I,Q,t)) . (DataLoc ((s . a),i))) ) & ( for x being Int_position st x in Y holds
(IExec (I,Q,t)) . x = t . x ) ) ) holds
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 being Int_position
for i, c being Integer
for X, Y being set
for f being Function of (product (the_Values_of SCMPDS)),NAT st s . (DataLoc ((s . a),i)) > 0 & ( for t being 0 -started State of SCMPDS st f . t = 0 holds
t . (DataLoc ((s . a),i)) <= 0 ) & ( for x being Int_position st x in X holds
s . x >= c + (s . (DataLoc ((s . a),i))) ) & ( for t being 0 -started State of SCMPDS
for Q being Instruction-Sequence of SCMPDS st ( for x being Int_position st x in X holds
t . x >= c + (t . (DataLoc ((s . a),i))) ) & ( for x being Int_position st x in Y 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 & f . (Initialize (IExec (I,Q,t))) < f . t & ( for x being Int_position st x in X holds
(IExec (I,Q,t)) . x >= c + ((IExec (I,Q,t)) . (DataLoc ((s . a),i))) ) & ( for x being Int_position st x in Y holds
(IExec (I,Q,t)) . x = t . x ) ) ) holds
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 being Int_position
for i, c being Integer
for X, Y being set
for f being Function of (product (the_Values_of SCMPDS)),NAT st s . (DataLoc ((s . a),i)) > 0 & ( for t being 0 -started State of SCMPDS st f . t = 0 holds
t . (DataLoc ((s . a),i)) <= 0 ) & ( for x being Int_position st x in X holds
s . x >= c + (s . (DataLoc ((s . a),i))) ) & ( for t being 0 -started State of SCMPDS
for Q being Instruction-Sequence of SCMPDS st ( for x being Int_position st x in X holds
t . x >= c + (t . (DataLoc ((s . a),i))) ) & ( for x being Int_position st x in Y 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 & f . (Initialize (IExec (I,Q,t))) < f . t & ( for x being Int_position st x in X holds
(IExec (I,Q,t)) . x >= c + ((IExec (I,Q,t)) . (DataLoc ((s . a),i))) ) & ( for x being Int_position st x in Y holds
(IExec (I,Q,t)) . x = t . x ) ) ) holds
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 being Int_position
for i, c being Integer
for X, Y being set
for f being Function of (product (the_Values_of SCMPDS)),NAT st s . (DataLoc ((s . a),i)) > 0 & ( for t being 0 -started State of SCMPDS st f . t = 0 holds
t . (DataLoc ((s . a),i)) <= 0 ) & ( for x being Int_position st x in X holds
s . x >= c + (s . (DataLoc ((s . a),i))) ) & ( for t being 0 -started State of SCMPDS
for Q being Instruction-Sequence of SCMPDS st ( for x being Int_position st x in X holds
t . x >= c + (t . (DataLoc ((s . a),i))) ) & ( for x being Int_position st x in Y 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 & f . (Initialize (IExec (I,Q,t))) < f . t & ( for x being Int_position st x in X holds
(IExec (I,Q,t)) . x >= c + ((IExec (I,Q,t)) . (DataLoc ((s . a),i))) ) & ( for x being Int_position st x in Y holds
(IExec (I,Q,t)) . x = t . x ) ) ) holds
IExec ((while>0 (a,i,I)),P,s) = IExec ((while>0 (a,i,I)),P,(Initialize (IExec (I,P,s))))
let a be Int_position; ::_thesis: for i, c being Integer
for X, Y being set
for f being Function of (product (the_Values_of SCMPDS)),NAT st s . (DataLoc ((s . a),i)) > 0 & ( for t being 0 -started State of SCMPDS st f . t = 0 holds
t . (DataLoc ((s . a),i)) <= 0 ) & ( for x being Int_position st x in X holds
s . x >= c + (s . (DataLoc ((s . a),i))) ) & ( for t being 0 -started State of SCMPDS
for Q being Instruction-Sequence of SCMPDS st ( for x being Int_position st x in X holds
t . x >= c + (t . (DataLoc ((s . a),i))) ) & ( for x being Int_position st x in Y 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 & f . (Initialize (IExec (I,Q,t))) < f . t & ( for x being Int_position st x in X holds
(IExec (I,Q,t)) . x >= c + ((IExec (I,Q,t)) . (DataLoc ((s . a),i))) ) & ( for x being Int_position st x in Y holds
(IExec (I,Q,t)) . x = t . x ) ) ) holds
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: for X, Y being set
for f being Function of (product (the_Values_of SCMPDS)),NAT st s . (DataLoc ((s . a),i)) > 0 & ( for t being 0 -started State of SCMPDS st f . t = 0 holds
t . (DataLoc ((s . a),i)) <= 0 ) & ( for x being Int_position st x in X holds
s . x >= c + (s . (DataLoc ((s . a),i))) ) & ( for t being 0 -started State of SCMPDS
for Q being Instruction-Sequence of SCMPDS st ( for x being Int_position st x in X holds
t . x >= c + (t . (DataLoc ((s . a),i))) ) & ( for x being Int_position st x in Y 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 & f . (Initialize (IExec (I,Q,t))) < f . t & ( for x being Int_position st x in X holds
(IExec (I,Q,t)) . x >= c + ((IExec (I,Q,t)) . (DataLoc ((s . a),i))) ) & ( for x being Int_position st x in Y holds
(IExec (I,Q,t)) . x = t . x ) ) ) holds
IExec ((while>0 (a,i,I)),P,s) = IExec ((while>0 (a,i,I)),P,(Initialize (IExec (I,P,s))))
let X, Y be set ; ::_thesis: for f being Function of (product (the_Values_of SCMPDS)),NAT st s . (DataLoc ((s . a),i)) > 0 & ( for t being 0 -started State of SCMPDS st f . t = 0 holds
t . (DataLoc ((s . a),i)) <= 0 ) & ( for x being Int_position st x in X holds
s . x >= c + (s . (DataLoc ((s . a),i))) ) & ( for t being 0 -started State of SCMPDS
for Q being Instruction-Sequence of SCMPDS st ( for x being Int_position st x in X holds
t . x >= c + (t . (DataLoc ((s . a),i))) ) & ( for x being Int_position st x in Y 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 & f . (Initialize (IExec (I,Q,t))) < f . t & ( for x being Int_position st x in X holds
(IExec (I,Q,t)) . x >= c + ((IExec (I,Q,t)) . (DataLoc ((s . a),i))) ) & ( for x being Int_position st x in Y holds
(IExec (I,Q,t)) . x = t . x ) ) ) holds
IExec ((while>0 (a,i,I)),P,s) = IExec ((while>0 (a,i,I)),P,(Initialize (IExec (I,P,s))))
let f be Function of (product (the_Values_of SCMPDS)),NAT; ::_thesis: ( s . (DataLoc ((s . a),i)) > 0 & ( for t being 0 -started State of SCMPDS st f . t = 0 holds
t . (DataLoc ((s . a),i)) <= 0 ) & ( for x being Int_position st x in X holds
s . x >= c + (s . (DataLoc ((s . a),i))) ) & ( for t being 0 -started State of SCMPDS
for Q being Instruction-Sequence of SCMPDS st ( for x being Int_position st x in X holds
t . x >= c + (t . (DataLoc ((s . a),i))) ) & ( for x being Int_position st x in Y 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 & f . (Initialize (IExec (I,Q,t))) < f . t & ( for x being Int_position st x in X holds
(IExec (I,Q,t)) . x >= c + ((IExec (I,Q,t)) . (DataLoc ((s . a),i))) ) & ( for x being Int_position st x in Y holds
(IExec (I,Q,t)) . x = t . x ) ) ) implies IExec ((while>0 (a,i,I)),P,s) = IExec ((while>0 (a,i,I)),P,(Initialize (IExec (I,P,s)))) )
I: Initialize s = s by MEMSTR_0:44;
set b = DataLoc ((s . a),i);
set WHL = while>0 (a,i,I);
set pWHL = stop (while>0 (a,i,I));
set P1 = P +* (stop (while>0 (a,i,I)));
set i1 = (a,i) <=0_goto ((card I) + 2);
set i2 = goto (- ((card I) + 1));
assume A2: s . (DataLoc ((s . a),i)) > 0 ; ::_thesis: ( ex t being 0 -started State of SCMPDS st
( f . t = 0 & not t . (DataLoc ((s . a),i)) <= 0 ) or ex x being Int_position st
( x in X & not s . x >= c + (s . (DataLoc ((s . a),i))) ) or ex t being 0 -started State of SCMPDS ex Q being Instruction-Sequence of SCMPDS st
( ( for x being Int_position st x in X holds
t . x >= c + (t . (DataLoc ((s . a),i))) ) & ( for x being Int_position st x in Y holds
t . x = s . x ) & 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 & f . (Initialize (IExec (I,Q,t))) < f . t & ( for x being Int_position st x in X holds
(IExec (I,Q,t)) . x >= c + ((IExec (I,Q,t)) . (DataLoc ((s . a),i))) ) & ( for x being Int_position st x in Y holds
(IExec (I,Q,t)) . x = t . x ) ) ) or IExec ((while>0 (a,i,I)),P,s) = IExec ((while>0 (a,i,I)),P,(Initialize (IExec (I,P,s)))) )
set s4 = Comput ((P +* (stop (while>0 (a,i,I)))),s,1);
set P4 = P +* (stop (while>0 (a,i,I)));
A3: IC s = 0 by MEMSTR_0:def_11;
A4: while>0 (a,i,I) = ((a,i) <=0_goto ((card I) + 2)) ';' (I ';' (goto (- ((card I) + 1)))) by SCMPDS_4:15;
A5: Comput ((P +* (stop (while>0 (a,i,I)))),s,(0 + 1)) = Following ((P +* (stop (while>0 (a,i,I)))),(Comput ((P +* (stop (while>0 (a,i,I)))),s,0))) by EXTPRO_1:3
.= Following ((P +* (stop (while>0 (a,i,I)))),s)
.= Exec (((a,i) <=0_goto ((card I) + 2)),s) by A4, I, SCMPDS_6:11 ;
set m0 = LifeSpan ((P +* (stop (while>0 (a,i,I)))),s);
set Es = IExec (I,P,s);
set bj = DataLoc (((Initialize (IExec (I,P,s))) . a),i);
set EP = P;
assume A9: for t being 0 -started State of SCMPDS st f . t = 0 holds
t . (DataLoc ((s . a),i)) <= 0 ; ::_thesis: ( ex x being Int_position st
( x in X & not s . x >= c + (s . (DataLoc ((s . a),i))) ) or ex t being 0 -started State of SCMPDS ex Q being Instruction-Sequence of SCMPDS st
( ( for x being Int_position st x in X holds
t . x >= c + (t . (DataLoc ((s . a),i))) ) & ( for x being Int_position st x in Y holds
t . x = s . x ) & 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 & f . (Initialize (IExec (I,Q,t))) < f . t & ( for x being Int_position st x in X holds
(IExec (I,Q,t)) . x >= c + ((IExec (I,Q,t)) . (DataLoc ((s . a),i))) ) & ( for x being Int_position st x in Y holds
(IExec (I,Q,t)) . x = t . x ) ) ) or IExec ((while>0 (a,i,I)),P,s) = IExec ((while>0 (a,i,I)),P,(Initialize (IExec (I,P,s)))) )
assume A10: for x being Int_position st x in X holds
s . x >= c + (s . (DataLoc ((s . a),i))) ; ::_thesis: ( ex t being 0 -started State of SCMPDS ex Q being Instruction-Sequence of SCMPDS st
( ( for x being Int_position st x in X holds
t . x >= c + (t . (DataLoc ((s . a),i))) ) & ( for x being Int_position st x in Y holds
t . x = s . x ) & 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 & f . (Initialize (IExec (I,Q,t))) < f . t & ( for x being Int_position st x in X holds
(IExec (I,Q,t)) . x >= c + ((IExec (I,Q,t)) . (DataLoc ((s . a),i))) ) & ( for x being Int_position st x in Y holds
(IExec (I,Q,t)) . x = t . x ) ) ) or IExec ((while>0 (a,i,I)),P,s) = IExec ((while>0 (a,i,I)),P,(Initialize (IExec (I,P,s)))) )
assume A11: for t being 0 -started State of SCMPDS
for Q being Instruction-Sequence of SCMPDS st ( for x being Int_position st x in X holds
t . x >= c + (t . (DataLoc ((s . a),i))) ) & ( for x being Int_position st x in Y 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 & f . (Initialize (IExec (I,Q,t))) < f . t & ( for x being Int_position st x in X holds
(IExec (I,Q,t)) . x >= c + ((IExec (I,Q,t)) . (DataLoc ((s . a),i))) ) & ( for x being Int_position st x in Y holds
(IExec (I,Q,t)) . x = t . x ) ) ; ::_thesis: IExec ((while>0 (a,i,I)),P,s) = IExec ((while>0 (a,i,I)),P,(Initialize (IExec (I,P,s))))
then  while>0 (a,i,I) is_halting_on s,P by A9, A10, Th24;
then A12: P +* (stop (while>0 (a,i,I))) halts_on s by I, SCMPDS_6:def_3;
A13: for x being Int_position st x in Y holds
s . x = s . x ;
A14: DataLoc (((Initialize (IExec (I,P,s))) . a),i) = DataLoc (((IExec (I,P,s)) . a),i) by SCMPDS_5:15
.= DataLoc ((s . a),i) by A2, A10, A11, A13 ;
set PI = P +* (stop I);
set m1 = (LifeSpan ((P +* (stop I)),s)) + 2;
set s2 = Initialize (IExec (I,P,s));
set P2 = P +* (stop (while>0 (a,i,I)));
set m2 = LifeSpan ((P +* (stop (while>0 (a,i,I)))),(Initialize (IExec (I,P,s))));
A17: stop I c= P +* (stop I) by FUNCT_4:25;
A18: (Initialize (IExec (I,P,s))) . a = (IExec (I,P,s)) . a by SCMPDS_5:15
.= s . a by A2, A10, A11, A13 ;
A19: 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_X_holds_
t_._x_>=_c_+_(t_._(DataLoc_(((Initialize_(IExec_(I,P,s)))_._a),i)))_)_&_(_for_x_being_Int_position_st_x_in_Y_holds_
t_._x_=_(Initialize_(IExec_(I,P,s)))_._x_)_&_t_._a_=_(Initialize_(IExec_(I,P,s)))_._a_&_t_._(DataLoc_(((Initialize_(IExec_(I,P,s)))_._a),i))_>_0_holds_
(_(Initialize_(IExec_(I,Q,t)))_._a_=_t_._a_&_I_is_closed_on_t,Q_&_I_is_halting_on_t,Q_&_f_._(Initialize_(IExec_(I,Q,t)))_<_f_._t_&_(_for_x_being_Int_position_st_x_in_X_holds_
(Initialize_(IExec_(I,Q,t)))_._x_>=_c_+_((Initialize_(IExec_(I,Q,t)))_._(DataLoc_(((Initialize_(IExec_(I,P,s)))_._a),i)))_)_&_(_for_x_being_Int_position_st_x_in_Y_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 X holds
t . x >= c + (t . (DataLoc (((Initialize (IExec (I,P,s))) . a),i))) ) & ( for x being Int_position st x in Y holds
t . x = (Initialize (IExec (I,P,s))) . x ) & t . a = (Initialize (IExec (I,P,s))) . a & t . (DataLoc (((Initialize (IExec (I,P,s))) . a),i)) > 0 holds
( (Initialize (IExec (I,Q,t))) . a = t . a & I is_closed_on t,Q & I is_halting_on t,Q & f . (Initialize (IExec (I,Q,t))) < f . t & ( for x being Int_position st x in X holds
(Initialize (IExec (I,Q,t))) . x >= c + ((Initialize (IExec (I,Q,t))) . (DataLoc (((Initialize (IExec (I,P,s))) . a),i))) ) & ( for x being Int_position st x in Y holds
(IExec (I,Q,t)) . x = t . x ) )
let Q be Instruction-Sequence of SCMPDS; ::_thesis: ( ( for x being Int_position st x in X holds
t . x >= c + (t . (DataLoc (((Initialize (IExec (I,P,s))) . a),i))) ) & ( for x being Int_position st x in Y holds
t . x = (Initialize (IExec (I,P,s))) . x ) & t . a = (Initialize (IExec (I,P,s))) . a & t . (DataLoc (((Initialize (IExec (I,P,s))) . a),i)) > 0 implies ( (Initialize (IExec (I,Q,t))) . a = t . a & I is_closed_on t,Q & I is_halting_on t,Q & f . (Initialize (IExec (I,Q,t))) < f . t & ( for x being Int_position st x in X holds
(Initialize (IExec (I,Q,t))) . x >= c + ((Initialize (IExec (I,Q,t))) . (DataLoc (((Initialize (IExec (I,P,s))) . a),i))) ) & ( for x being Int_position st x in Y holds
(IExec (I,Q,t)) . x = t . x ) ) )
assume that
A20: for x being Int_position st x in X holds
t . x >= c + (t . (DataLoc (((Initialize (IExec (I,P,s))) . a),i))) and
A21: for x being Int_position st x in Y holds
t . x = (Initialize (IExec (I,P,s))) . x and
A22: t . a = (Initialize (IExec (I,P,s))) . a and
A23: t . (DataLoc (((Initialize (IExec (I,P,s))) . a),i)) > 0 ; ::_thesis: ( (Initialize (IExec (I,Q,t))) . a = t . a & I is_closed_on t,Q & I is_halting_on t,Q & f . (Initialize (IExec (I,Q,t))) < f . t & ( for x being Int_position st x in X holds
(Initialize (IExec (I,Q,t))) . x >= c + ((Initialize (IExec (I,Q,t))) . (DataLoc (((Initialize (IExec (I,P,s))) . a),i))) ) & ( for x being Int_position st x in Y holds
(IExec (I,Q,t)) . x = t . x ) )
A24: now__::_thesis:_for_x_being_Int_position_st_x_in_Y_holds_
t_._x_=_s_._x
let x be Int_position; ::_thesis: ( x in Y implies t . x = s . x )
assume A25: x in Y ; ::_thesis: t . x = s . x
hence t . x = (Initialize (IExec (I,P,s))) . x by A21
.= (IExec (I,P,s)) . x by SCMPDS_5:15
.= s . x by A2, A10, A11, A13, A25 ;
::_thesis: verum
end;
thus (Initialize (IExec (I,Q,t))) . a = (IExec (I,Q,t)) . a by SCMPDS_5:15
.= t . a by A11, A18, A20, A22, A23, A24 ; ::_thesis: ( I is_closed_on t,Q & I is_halting_on t,Q & f . (Initialize (IExec (I,Q,t))) < f . t & ( for x being Int_position st x in X holds
(Initialize (IExec (I,Q,t))) . x >= c + ((Initialize (IExec (I,Q,t))) . (DataLoc (((Initialize (IExec (I,P,s))) . a),i))) ) & ( for x being Int_position st x in Y holds
(IExec (I,Q,t)) . x = t . x ) )
A26: t . a = (IExec (I,P,s)) . a by A22, SCMPDS_5:15
.= s . a by A2, A10, A11, A13 ;
hence  ( I is_closed_on t,Q & I is_halting_on t,Q ) by A11, A20, A22, A23, A24; ::_thesis: ( f . (Initialize (IExec (I,Q,t))) < f . t & ( for x being Int_position st x in X holds
(Initialize (IExec (I,Q,t))) . x >= c + ((Initialize (IExec (I,Q,t))) . (DataLoc (((Initialize (IExec (I,P,s))) . a),i))) ) & ( for x being Int_position st x in Y holds
(IExec (I,Q,t)) . x = t . x ) )
thus  f . (Initialize (IExec (I,Q,t))) < f . t by A11, A20, A22, A23, A24, A26; ::_thesis: ( ( for x being Int_position st x in X holds
(Initialize (IExec (I,Q,t))) . x >= c + ((Initialize (IExec (I,Q,t))) . (DataLoc (((Initialize (IExec (I,P,s))) . a),i))) ) & ( for x being Int_position st x in Y holds
(IExec (I,Q,t)) . x = t . x ) )
thus  for x being Int_position st x in X holds
(Initialize (IExec (I,Q,t))) . x >= c + ((Initialize (IExec (I,Q,t))) . (DataLoc (((Initialize (IExec (I,P,s))) . a),i))) ::_thesis: for x being Int_position st x in Y holds
(IExec (I,Q,t)) . x = t . x
proof
let x be Int_position; ::_thesis: ( x in X implies (Initialize (IExec (I,Q,t))) . x >= c + ((Initialize (IExec (I,Q,t))) . (DataLoc (((Initialize (IExec (I,P,s))) . a),i))) )
 ( (Initialize (IExec (I,Q,t))) . x = (IExec (I,Q,t)) . x & (Initialize (IExec (I,Q,t))) . (DataLoc (((Initialize (IExec (I,P,s))) . a),i)) = (IExec (I,Q,t)) . (DataLoc (((Initialize (IExec (I,P,s))) . a),i)) ) by SCMPDS_5:15;
hence  ( x in X implies (Initialize (IExec (I,Q,t))) . x >= c + ((Initialize (IExec (I,Q,t))) . (DataLoc (((Initialize (IExec (I,P,s))) . a),i))) ) by A11, A18, A20, A22, A23, A24; ::_thesis: verum
end;
thus  for x being Int_position st x in Y holds
(IExec (I,Q,t)) . x = t . x by A11, A20, A22, A23, A24, A26; ::_thesis: verum
end;
XX: for x being Int_position st x in X holds
(Initialize (IExec (I,P,s))) . x >= c + ((Initialize (IExec (I,P,s))) . (DataLoc (((Initialize (IExec (I,P,s))) . a),i)))
proof
let x be Int_position; ::_thesis: ( x in X implies (Initialize (IExec (I,P,s))) . x >= c + ((Initialize (IExec (I,P,s))) . (DataLoc (((Initialize (IExec (I,P,s))) . a),i))) )
F: (Initialize (IExec (I,P,s))) . x = (IExec (I,P,s)) . x by SCMPDS_5:15;
G: (Initialize (IExec (I,P,s))) . (DataLoc (((Initialize (IExec (I,P,s))) . a),i)) = (IExec (I,P,s)) . (DataLoc (((Initialize (IExec (I,P,s))) . a),i)) by SCMPDS_5:15;
assume  x in X ; ::_thesis: (Initialize (IExec (I,P,s))) . x >= c + ((Initialize (IExec (I,P,s))) . (DataLoc (((Initialize (IExec (I,P,s))) . a),i)))
hence  (Initialize (IExec (I,P,s))) . x >= c + ((Initialize (IExec (I,P,s))) . (DataLoc (((Initialize (IExec (I,P,s))) . a),i))) by A2, A10, A11, A13, A18, F, G; ::_thesis: verum
end;
 for t being 0 -started State of SCMPDS
for Q being Instruction-Sequence of SCMPDS st ( for x being Int_position st x in X holds
t . x >= c + (t . (DataLoc (((Initialize (IExec (I,P,s))) . a),i))) ) & ( for x being Int_position st x in Y holds
t . x = (Initialize (IExec (I,P,s))) . x ) & t . a = (Initialize (IExec (I,P,s))) . a & t . (DataLoc (((Initialize (IExec (I,P,s))) . a),i)) > 0 holds
( (IExec (I,Q,t)) . a = t . a & I is_closed_on t,Q & I is_halting_on t,Q & f . (Initialize (IExec (I,Q,t))) < f . t & ( for x being Int_position st x in X holds
(IExec (I,Q,t)) . x >= c + ((IExec (I,Q,t)) . (DataLoc (((Initialize (IExec (I,P,s))) . a),i))) ) & ( for x being Int_position st x in Y holds
(IExec (I,Q,t)) . x = t . x ) )
proof
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 X holds
t . x >= c + (t . (DataLoc (((Initialize (IExec (I,P,s))) . a),i))) ) & ( for x being Int_position st x in Y holds
t . x = (Initialize (IExec (I,P,s))) . x ) & t . a = (Initialize (IExec (I,P,s))) . a & t . (DataLoc (((Initialize (IExec (I,P,s))) . a),i)) > 0 holds
( (IExec (I,Q,t)) . a = t . a & I is_closed_on t,Q & I is_halting_on t,Q & f . (Initialize (IExec (I,Q,t))) < f . t & ( for x being Int_position st x in X holds
(IExec (I,Q,t)) . x >= c + ((IExec (I,Q,t)) . (DataLoc (((Initialize (IExec (I,P,s))) . a),i))) ) & ( for x being Int_position st x in Y holds
(IExec (I,Q,t)) . x = t . x ) )
let Q be Instruction-Sequence of SCMPDS; ::_thesis: ( ( for x being Int_position st x in X holds
t . x >= c + (t . (DataLoc (((Initialize (IExec (I,P,s))) . a),i))) ) & ( for x being Int_position st x in Y holds
t . x = (Initialize (IExec (I,P,s))) . x ) & t . a = (Initialize (IExec (I,P,s))) . a & t . (DataLoc (((Initialize (IExec (I,P,s))) . a),i)) > 0 implies ( (IExec (I,Q,t)) . a = t . a & I is_closed_on t,Q & I is_halting_on t,Q & f . (Initialize (IExec (I,Q,t))) < f . t & ( for x being Int_position st x in X holds
(IExec (I,Q,t)) . x >= c + ((IExec (I,Q,t)) . (DataLoc (((Initialize (IExec (I,P,s))) . a),i))) ) & ( for x being Int_position st x in Y holds
(IExec (I,Q,t)) . x = t . x ) ) )
assume that
Z1: for x being Int_position st x in X holds
t . x >= c + (t . (DataLoc (((Initialize (IExec (I,P,s))) . a),i))) and
Z2: for x being Int_position st x in Y holds
t . x = (Initialize (IExec (I,P,s))) . x and
Z3: t . a = (Initialize (IExec (I,P,s))) . a and
Z4: t . (DataLoc (((Initialize (IExec (I,P,s))) . a),i)) > 0 ; ::_thesis: ( (IExec (I,Q,t)) . a = t . a & I is_closed_on t,Q & I is_halting_on t,Q & f . (Initialize (IExec (I,Q,t))) < f . t & ( for x being Int_position st x in X holds
(IExec (I,Q,t)) . x >= c + ((IExec (I,Q,t)) . (DataLoc (((Initialize (IExec (I,P,s))) . a),i))) ) & ( for x being Int_position st x in Y holds
(IExec (I,Q,t)) . x = t . x ) )
thus (IExec (I,Q,t)) . a = (Initialize (IExec (I,Q,t))) . a by SCMPDS_5:15
.= t . a by A19, Z1, Z2, Z3, Z4 ; ::_thesis: ( I is_closed_on t,Q & I is_halting_on t,Q & f . (Initialize (IExec (I,Q,t))) < f . t & ( for x being Int_position st x in X holds
(IExec (I,Q,t)) . x >= c + ((IExec (I,Q,t)) . (DataLoc (((Initialize (IExec (I,P,s))) . a),i))) ) & ( for x being Int_position st x in Y holds
(IExec (I,Q,t)) . x = t . x ) )
thus  ( I is_closed_on t,Q & I is_halting_on t,Q ) by A19, Z1, Z2, Z3, Z4; ::_thesis: ( f . (Initialize (IExec (I,Q,t))) < f . t & ( for x being Int_position st x in X holds
(IExec (I,Q,t)) . x >= c + ((IExec (I,Q,t)) . (DataLoc (((Initialize (IExec (I,P,s))) . a),i))) ) & ( for x being Int_position st x in Y holds
(IExec (I,Q,t)) . x = t . x ) )
thus  f . (Initialize (IExec (I,Q,t))) < f . t by A19, Z1, Z2, Z3, Z4; ::_thesis: ( ( for x being Int_position st x in X holds
(IExec (I,Q,t)) . x >= c + ((IExec (I,Q,t)) . (DataLoc (((Initialize (IExec (I,P,s))) . a),i))) ) & ( for x being Int_position st x in Y holds
(IExec (I,Q,t)) . x = t . x ) )
thus  for x being Int_position st x in X holds
(IExec (I,Q,t)) . x >= c + ((IExec (I,Q,t)) . (DataLoc (((Initialize (IExec (I,P,s))) . a),i))) ::_thesis: for x being Int_position st x in Y holds
(IExec (I,Q,t)) . x = t . x
proof
let x be Int_position; ::_thesis: ( x in X implies (IExec (I,Q,t)) . x >= c + ((IExec (I,Q,t)) . (DataLoc (((Initialize (IExec (I,P,s))) . a),i))) )
 ( (IExec (I,Q,t)) . x = (Initialize (IExec (I,Q,t))) . x & (IExec (I,Q,t)) . (DataLoc (((Initialize (IExec (I,P,s))) . a),i)) = (Initialize (IExec (I,Q,t))) . (DataLoc (((Initialize (IExec (I,P,s))) . a),i)) ) by SCMPDS_5:15;
hence  ( x in X implies (IExec (I,Q,t)) . x >= c + ((IExec (I,Q,t)) . (DataLoc (((Initialize (IExec (I,P,s))) . a),i))) ) by A19, Z1, Z2, Z3, Z4; ::_thesis: verum
end;
thus  for x being Int_position st x in Y holds
(IExec (I,Q,t)) . x = t . x by A19, Z1, Z2, Z3, Z4; ::_thesis: verum
end;
then  while>0 (a,i,I) is_halting_on Initialize (IExec (I,P,s)),P by A9, A14, Th24, XX;
then B27: P +* (stop (while>0 (a,i,I))) halts_on Initialize (Initialize (IExec (I,P,s))) by SCMPDS_6:def_3;
 for a being Int_position holds s . a = (Comput ((P +* (stop (while>0 (a,i,I)))),s,1)) . a by A5, SCMPDS_2:56;
then A29: DataPart s = DataPart (Comput ((P +* (stop (while>0 (a,i,I)))),s,1)) by SCMPDS_4:8;
set mI = LifeSpan ((P +* (stop I)),s);
set s5 = Comput ((P +* (stop (while>0 (a,i,I)))),(Comput ((P +* (stop (while>0 (a,i,I)))),s,1)),(LifeSpan ((P +* (stop I)),s)));
set P5 = P +* (stop (while>0 (a,i,I)));
set l1 = (card I) + 1;
XX: while>0 (a,i,I) c= stop (while>0 (a,i,I)) by AFINSQ_1:74;
 stop (while>0 (a,i,I)) c= P +* (stop (while>0 (a,i,I))) by FUNCT_4:25;
then A30: while>0 (a,i,I) c= P +* (stop (while>0 (a,i,I))) by XX, XBOOLE_1:1;
set m3 = (LifeSpan ((P +* (stop I)),s)) + 1;
set s7 = Comput ((P +* (stop (while>0 (a,i,I)))),s,(((LifeSpan ((P +* (stop I)),s)) + 1) + 1));
set P7 = P +* (stop (while>0 (a,i,I)));
set s6 = Comput ((P +* (stop (while>0 (a,i,I)))),s,((LifeSpan ((P +* (stop I)),s)) + 1));
set P6 = P +* (stop (while>0 (a,i,I)));
 (card I) + 1 < (card I) + 2 by XREAL_1:6;
then A32: (card I) + 1 in dom (while>0 (a,i,I)) by Th18;
 Shift (I,1) c= while>0 (a,i,I) by Lm4;
then A33: Shift (I,1) c= P +* (stop (while>0 (a,i,I))) by A30, XBOOLE_1:1;
 I is_halting_on s,P by A2, A10, A11, A13;
then A34: P +* (stop I) halts_on s by I, SCMPDS_6:def_3;
 (P +* (stop I)) +* (stop I) halts_on s by A34;
then A36: I is_halting_on s,P +* (stop I) by I, SCMPDS_6:def_3;
A39: IC (Comput ((P +* (stop (while>0 (a,i,I)))),s,1)) = succ (IC s) by A2, A5, SCMPDS_2:56
.= 0 + 1 by A3 ;
A40: I is_closed_on s,P +* (stop I) by A2, A10, A11, A13;
then A41: IC (Comput ((P +* (stop (while>0 (a,i,I)))),(Comput ((P +* (stop (while>0 (a,i,I)))),s,1)),(LifeSpan ((P +* (stop I)),s)))) = (card I) + 1 by A17, A36, A39, A29, A33, SCMPDS_7:18;
A42: (P +* (stop (while>0 (a,i,I)))) /. (IC (Comput ((P +* (stop (while>0 (a,i,I)))),s,((LifeSpan ((P +* (stop I)),s)) + 1)))) = (P +* (stop (while>0 (a,i,I)))) . (IC (Comput ((P +* (stop (while>0 (a,i,I)))),s,((LifeSpan ((P +* (stop I)),s)) + 1)))) by PBOOLE:143;
A43: Comput ((P +* (stop (while>0 (a,i,I)))),s,((LifeSpan ((P +* (stop I)),s)) + 1)) = Comput ((P +* (stop (while>0 (a,i,I)))),(Comput ((P +* (stop (while>0 (a,i,I)))),s,1)),(LifeSpan ((P +* (stop I)),s))) by EXTPRO_1:4;
then A44: CurInstr ((P +* (stop (while>0 (a,i,I)))),(Comput ((P +* (stop (while>0 (a,i,I)))),s,((LifeSpan ((P +* (stop I)),s)) + 1)))) = (P +* (stop (while>0 (a,i,I)))) . ((card I) + 1) by A17, A36, A40, A39, A29, A33, A42, SCMPDS_7:18
.= (while>0 (a,i,I)) . ((card I) + 1) by A32, A30, GRFUNC_1:2
.= goto (- ((card I) + 1)) by Th19 ;
A46: Comput ((P +* (stop (while>0 (a,i,I)))),s,(((LifeSpan ((P +* (stop I)),s)) + 1) + 1)) = Following ((P +* (stop (while>0 (a,i,I)))),(Comput ((P +* (stop (while>0 (a,i,I)))),s,((LifeSpan ((P +* (stop I)),s)) + 1)))) by EXTPRO_1:3
.= Exec ((goto (- ((card I) + 1))),(Comput ((P +* (stop (while>0 (a,i,I)))),s,((LifeSpan ((P +* (stop I)),s)) + 1)))) by A44 ;
then  IC (Comput ((P +* (stop (while>0 (a,i,I)))),s,(((LifeSpan ((P +* (stop I)),s)) + 1) + 1))) = ICplusConst ((Comput ((P +* (stop (while>0 (a,i,I)))),s,((LifeSpan ((P +* (stop I)),s)) + 1))),(0 - ((card I) + 1))) by SCMPDS_2:54
.= 0 by A41, A43, SCMPDS_7:1 ;
then A47: IC (Initialize (IExec (I,P,s))) = IC (Comput ((P +* (stop (while>0 (a,i,I)))),s,((LifeSpan ((P +* (stop I)),s)) + 2))) by MEMSTR_0:def_11;
A48: DataPart (Comput ((P +* (stop I)),s,(LifeSpan ((P +* (stop I)),s)))) = DataPart (Comput ((P +* (stop (while>0 (a,i,I)))),(Comput ((P +* (stop (while>0 (a,i,I)))),s,1)),(LifeSpan ((P +* (stop I)),s)))) by A17, A36, A40, A39, A29, A33, SCMPDS_7:18;
now__::_thesis:_for_x_being_Int_position_holds_(Comput_((P_+*_(stop_(while>0_(a,i,I)))),s,(((LifeSpan_((P_+*_(stop_I)),s))_+_1)_+_1)))_._x_=_(Initialize_(IExec_(I,P,s)))_._x
let x be Int_position; ::_thesis: (Comput ((P +* (stop (while>0 (a,i,I)))),s,(((LifeSpan ((P +* (stop I)),s)) + 1) + 1))) . x = (Initialize (IExec (I,P,s))) . x
A49: not x in dom (Start-At (0,SCMPDS)) by SCMPDS_4:18;
(Comput ((P +* (stop (while>0 (a,i,I)))),(Comput ((P +* (stop (while>0 (a,i,I)))),s,1)),(LifeSpan ((P +* (stop I)),s)))) . x = (Comput ((P +* (stop I)),s,(LifeSpan ((P +* (stop I)),s)))) . x by A48, SCMPDS_4:8
.= (Result ((P +* (stop I)),s)) . x by A34, EXTPRO_1:23
.= (IExec (I,P,s)) . x by SCMPDS_4:def_5 ;
hence (Comput ((P +* (stop (while>0 (a,i,I)))),s,(((LifeSpan ((P +* (stop I)),s)) + 1) + 1))) . x = (IExec (I,P,s)) . x by A43, A46, SCMPDS_2:54
.= (Initialize (IExec (I,P,s))) . x by A49, FUNCT_4:11 ;
::_thesis: verum
end;
then A51: DataPart (Comput ((P +* (stop (while>0 (a,i,I)))),s,(((LifeSpan ((P +* (stop I)),s)) + 1) + 1))) = DataPart (Initialize (IExec (I,P,s))) by SCMPDS_4:8;
A53: Comput ((P +* (stop (while>0 (a,i,I)))),s,((LifeSpan ((P +* (stop I)),s)) + 2)) = Initialize (IExec (I,P,s)) by A51, A47, MEMSTR_0:78;
then  CurInstr ((P +* (stop (while>0 (a,i,I)))),(Comput ((P +* (stop (while>0 (a,i,I)))),s,((LifeSpan ((P +* (stop I)),s)) + 2)))) = (a,i) <=0_goto ((card I) + 2) by A4, SCMPDS_6:11;
then  LifeSpan ((P +* (stop (while>0 (a,i,I)))),s) > (LifeSpan ((P +* (stop I)),s)) + 2 by A12, EXTPRO_1:36, SCMPDS_6:17;
then consider nn being Nat such that
A54: LifeSpan ((P +* (stop (while>0 (a,i,I)))),s) = ((LifeSpan ((P +* (stop I)),s)) + 2) + nn by NAT_1:10;
reconsider nn = nn as Element of NAT by ORDINAL1:def_12;
 Comput ((P +* (stop (while>0 (a,i,I)))),s,(((LifeSpan ((P +* (stop I)),s)) + 2) + (LifeSpan ((P +* (stop (while>0 (a,i,I)))),(Initialize (IExec (I,P,s))))))) = Comput ((P +* (stop (while>0 (a,i,I)))),(Initialize (IExec (I,P,s))),(LifeSpan ((P +* (stop (while>0 (a,i,I)))),(Initialize (IExec (I,P,s)))))) by A53, EXTPRO_1:4;
then  CurInstr ((P +* (stop (while>0 (a,i,I)))),(Comput ((P +* (stop (while>0 (a,i,I)))),s,(((LifeSpan ((P +* (stop I)),s)) + 2) + (LifeSpan ((P +* (stop (while>0 (a,i,I)))),(Initialize (IExec (I,P,s))))))))) = halt SCMPDS by B27, EXTPRO_1:def_15;
then  ((LifeSpan ((P +* (stop I)),s)) + 2) + (LifeSpan ((P +* (stop (while>0 (a,i,I)))),(Initialize (IExec (I,P,s))))) >= LifeSpan ((P +* (stop (while>0 (a,i,I)))),s) by A12, EXTPRO_1:def_15;
then A56: LifeSpan ((P +* (stop (while>0 (a,i,I)))),(Initialize (IExec (I,P,s)))) >= nn by A54, XREAL_1:6;
A57: Comput ((P +* (stop (while>0 (a,i,I)))),s,(LifeSpan ((P +* (stop (while>0 (a,i,I)))),s))) = Comput ((P +* (stop (while>0 (a,i,I)))),(Initialize (IExec (I,P,s))),nn) by A53, A54, EXTPRO_1:4;
then  CurInstr ((P +* (stop (while>0 (a,i,I)))),(Comput ((P +* (stop (while>0 (a,i,I)))),(Initialize (IExec (I,P,s))),nn))) = halt SCMPDS by A12, EXTPRO_1:def_15;
then  nn >= LifeSpan ((P +* (stop (while>0 (a,i,I)))),(Initialize (IExec (I,P,s)))) by B27, EXTPRO_1:def_15;
then  nn = LifeSpan ((P +* (stop (while>0 (a,i,I)))),(Initialize (IExec (I,P,s)))) by A56, XXREAL_0:1;
then  Result ((P +* (stop (while>0 (a,i,I)))),s) = Comput ((P +* (stop (while>0 (a,i,I)))),(Initialize (IExec (I,P,s))),(LifeSpan ((P +* (stop (while>0 (a,i,I)))),(Initialize (IExec (I,P,s)))))) by A12, A57, EXTPRO_1:23;
hence  IExec ((while>0 (a,i,I)),P,s) = Comput ((P +* (stop (while>0 (a,i,I)))),(Initialize (IExec (I,P,s))),(LifeSpan ((P +* (stop (while>0 (a,i,I)))),(Initialize (IExec (I,P,s)))))) by SCMPDS_4:def_5
.= Result ((P +* (stop (while>0 (a,i,I)))),(Initialize (IExec (I,P,s)))) by B27, EXTPRO_1:23
.= IExec ((while>0 (a,i,I)),P,(Initialize (IExec (I,P,s)))) by SCMPDS_4:def_5 ;
::_thesis: verum
end;

theorem :: SCMPDS_8:26
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 being Integer
for X being set
for f being Function of (product (the_Values_of SCMPDS)),NAT st ( for t being 0 -started State of SCMPDS st f . t = 0 holds
t . (DataLoc ((s . a),i)) <= 0 ) & ( for t being 0 -started State of SCMPDS
for Q being Instruction-Sequence of SCMPDS st ( for x being Int_position st x in X 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 & f . (Initialize (IExec (I,Q,t))) < f . t & ( for x being Int_position st x in X holds
(IExec (I,Q,t)) . x = t . x ) ) ) 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 being Int_position
for i being Integer
for X being set
for f being Function of (product (the_Values_of SCMPDS)),NAT st ( for t being 0 -started State of SCMPDS st f . t = 0 holds
t . (DataLoc ((s . a),i)) <= 0 ) & ( for t being 0 -started State of SCMPDS
for Q being Instruction-Sequence of SCMPDS st ( for x being Int_position st x in X 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 & f . (Initialize (IExec (I,Q,t))) < f . t & ( for x being Int_position st x in X holds
(IExec (I,Q,t)) . x = t . x ) ) ) 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 being Int_position
for i being Integer
for X being set
for f being Function of (product (the_Values_of SCMPDS)),NAT st ( for t being 0 -started State of SCMPDS st f . t = 0 holds
t . (DataLoc ((s . a),i)) <= 0 ) & ( for t being 0 -started State of SCMPDS
for Q being Instruction-Sequence of SCMPDS st ( for x being Int_position st x in X 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 & f . (Initialize (IExec (I,Q,t))) < f . t & ( for x being Int_position st x in X holds
(IExec (I,Q,t)) . x = t . x ) ) ) 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 being Int_position
for i being Integer
for X being set
for f being Function of (product (the_Values_of SCMPDS)),NAT st ( for t being 0 -started State of SCMPDS st f . t = 0 holds
t . (DataLoc ((s . a),i)) <= 0 ) & ( for t being 0 -started State of SCMPDS
for Q being Instruction-Sequence of SCMPDS st ( for x being Int_position st x in X 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 & f . (Initialize (IExec (I,Q,t))) < f . t & ( for x being Int_position st x in X holds
(IExec (I,Q,t)) . x = t . x ) ) ) 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 be Int_position; ::_thesis: for i being Integer
for X being set
for f being Function of (product (the_Values_of SCMPDS)),NAT st ( for t being 0 -started State of SCMPDS st f . t = 0 holds
t . (DataLoc ((s . a),i)) <= 0 ) & ( for t being 0 -started State of SCMPDS
for Q being Instruction-Sequence of SCMPDS st ( for x being Int_position st x in X 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 & f . (Initialize (IExec (I,Q,t))) < f . t & ( for x being Int_position st x in X holds
(IExec (I,Q,t)) . x = t . x ) ) ) 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 Integer; ::_thesis: for X being set
for f being Function of (product (the_Values_of SCMPDS)),NAT st ( for t being 0 -started State of SCMPDS st f . t = 0 holds
t . (DataLoc ((s . a),i)) <= 0 ) & ( for t being 0 -started State of SCMPDS
for Q being Instruction-Sequence of SCMPDS st ( for x being Int_position st x in X 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 & f . (Initialize (IExec (I,Q,t))) < f . t & ( for x being Int_position st x in X holds
(IExec (I,Q,t)) . x = t . x ) ) ) 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 X be set ; ::_thesis: for f being Function of (product (the_Values_of SCMPDS)),NAT st ( for t being 0 -started State of SCMPDS st f . t = 0 holds
t . (DataLoc ((s . a),i)) <= 0 ) & ( for t being 0 -started State of SCMPDS
for Q being Instruction-Sequence of SCMPDS st ( for x being Int_position st x in X 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 & f . (Initialize (IExec (I,Q,t))) < f . t & ( for x being Int_position st x in X holds
(IExec (I,Q,t)) . x = t . x ) ) ) 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 f be Function of (product (the_Values_of SCMPDS)),NAT; ::_thesis: ( ( for t being 0 -started State of SCMPDS st f . t = 0 holds
t . (DataLoc ((s . a),i)) <= 0 ) & ( for t being 0 -started State of SCMPDS
for Q being Instruction-Sequence of SCMPDS st ( for x being Int_position st x in X 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 & f . (Initialize (IExec (I,Q,t))) < f . t & ( for x being Int_position st x in X holds
(IExec (I,Q,t)) . x = t . x ) ) ) 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 A2: for t being 0 -started State of SCMPDS st f . t = 0 holds
t . (DataLoc ((s . a),i)) <= 0 ; ::_thesis: ( ex t being 0 -started State of SCMPDS ex Q being Instruction-Sequence of SCMPDS st
( ( for x being Int_position st x in X holds
t . x = s . x ) & 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 & f . (Initialize (IExec (I,Q,t))) < f . t & ( for x being Int_position st x in X holds
(IExec (I,Q,t)) . x = t . x ) ) ) 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)))) ) ) )
assume A3: for t being 0 -started State of SCMPDS
for Q being Instruction-Sequence of SCMPDS st ( for x being Int_position st x in X 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 & f . (Initialize (IExec (I,Q,t))) < f . t & ( for x being Int_position st x in X holds
(IExec (I,Q,t)) . x = t . x ) ) ; ::_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)))) ) )
then A4: for t being 0 -started State of SCMPDS
for Q being Instruction-Sequence of SCMPDS st ( for x being Int_position st x in {} holds
t . x >= 0 + (t . (DataLoc ((s . a),i))) ) & ( for x being Int_position st x in X 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 & f . (Initialize (IExec (I,Q,t))) < f . t & ( for x being Int_position st x in {} holds
(IExec (I,Q,t)) . x >= 0 + ((IExec (I,Q,t)) . (DataLoc ((s . a),i))) ) & ( for x being Int_position st x in X holds
(IExec (I,Q,t)) . x = t . x ) ) ;
A5: for x being Int_position st x in {} holds
s . x >= 0 + (s . (DataLoc ((s . a),i))) ;
 for t being 0 -started State of SCMPDS
for Q being Instruction-Sequence of SCMPDS st ( for x being Int_position st x in {} holds
t . x >= 0 + (t . (DataLoc ((s . a),i))) ) & ( for x being Int_position st x in X 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 & f . (Initialize (IExec (I,Q,t))) < f . t & ( for x being Int_position st x in {} holds
(IExec (I,Q,t)) . x >= 0 + ((IExec (I,Q,t)) . (DataLoc ((s . a),i))) ) & ( for x being Int_position st x in X holds
(IExec (I,Q,t)) . x = t . x ) ) by A3;
hence  ( while>0 (a,i,I) is_closed_on s,P & while>0 (a,i,I) is_halting_on s,P ) by A2, A5, Th24; ::_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  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))))
hence  IExec ((while>0 (a,i,I)),P,s) = IExec ((while>0 (a,i,I)),P,(Initialize (IExec (I,P,s)))) by A2, A5, A4, Th25; ::_thesis: verum
end;

theorem Th27: :: SCMPDS_8:27
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 X, Y being set st ( for x being Int_position st x in X holds
s . x >= c + (s . (DataLoc ((s . a),i))) ) & ( for t being 0 -started State of SCMPDS
for Q being Instruction-Sequence of SCMPDS st ( for x being Int_position st x in X holds
t . x >= c + (t . (DataLoc ((s . a),i))) ) & ( for x being Int_position st x in Y 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 X holds
(IExec (I,Q,t)) . x >= c + ((IExec (I,Q,t)) . (DataLoc ((s . a),i))) ) & ( for x being Int_position st x in Y holds
(IExec (I,Q,t)) . x = t . x ) ) ) 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 being Int_position
for i, c being Integer
for X, Y being set st ( for x being Int_position st x in X holds
s . x >= c + (s . (DataLoc ((s . a),i))) ) & ( for t being 0 -started State of SCMPDS
for Q being Instruction-Sequence of SCMPDS st ( for x being Int_position st x in X holds
t . x >= c + (t . (DataLoc ((s . a),i))) ) & ( for x being Int_position st x in Y 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 X holds
(IExec (I,Q,t)) . x >= c + ((IExec (I,Q,t)) . (DataLoc ((s . a),i))) ) & ( for x being Int_position st x in Y holds
(IExec (I,Q,t)) . x = t . x ) ) ) 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 being Int_position
for i, c being Integer
for X, Y being set st ( for x being Int_position st x in X holds
s . x >= c + (s . (DataLoc ((s . a),i))) ) & ( for t being 0 -started State of SCMPDS
for Q being Instruction-Sequence of SCMPDS st ( for x being Int_position st x in X holds
t . x >= c + (t . (DataLoc ((s . a),i))) ) & ( for x being Int_position st x in Y 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 X holds
(IExec (I,Q,t)) . x >= c + ((IExec (I,Q,t)) . (DataLoc ((s . a),i))) ) & ( for x being Int_position st x in Y holds
(IExec (I,Q,t)) . x = t . x ) ) ) 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 being Int_position
for i, c being Integer
for X, Y being set st ( for x being Int_position st x in X holds
s . x >= c + (s . (DataLoc ((s . a),i))) ) & ( for t being 0 -started State of SCMPDS
for Q being Instruction-Sequence of SCMPDS st ( for x being Int_position st x in X holds
t . x >= c + (t . (DataLoc ((s . a),i))) ) & ( for x being Int_position st x in Y 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 X holds
(IExec (I,Q,t)) . x >= c + ((IExec (I,Q,t)) . (DataLoc ((s . a),i))) ) & ( for x being Int_position st x in Y holds
(IExec (I,Q,t)) . x = t . x ) ) ) 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 be Int_position; ::_thesis: for i, c being Integer
for X, Y being set st ( for x being Int_position st x in X holds
s . x >= c + (s . (DataLoc ((s . a),i))) ) & ( for t being 0 -started State of SCMPDS
for Q being Instruction-Sequence of SCMPDS st ( for x being Int_position st x in X holds
t . x >= c + (t . (DataLoc ((s . a),i))) ) & ( for x being Int_position st x in Y 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 X holds
(IExec (I,Q,t)) . x >= c + ((IExec (I,Q,t)) . (DataLoc ((s . a),i))) ) & ( for x being Int_position st x in Y holds
(IExec (I,Q,t)) . x = t . x ) ) ) 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: for X, Y being set st ( for x being Int_position st x in X holds
s . x >= c + (s . (DataLoc ((s . a),i))) ) & ( for t being 0 -started State of SCMPDS
for Q being Instruction-Sequence of SCMPDS st ( for x being Int_position st x in X holds
t . x >= c + (t . (DataLoc ((s . a),i))) ) & ( for x being Int_position st x in Y 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 X holds
(IExec (I,Q,t)) . x >= c + ((IExec (I,Q,t)) . (DataLoc ((s . a),i))) ) & ( for x being Int_position st x in Y holds
(IExec (I,Q,t)) . x = t . x ) ) ) 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 X, Y be set ; ::_thesis: ( ( for x being Int_position st x in X holds
s . x >= c + (s . (DataLoc ((s . a),i))) ) & ( for t being 0 -started State of SCMPDS
for Q being Instruction-Sequence of SCMPDS st ( for x being Int_position st x in X holds
t . x >= c + (t . (DataLoc ((s . a),i))) ) & ( for x being Int_position st x in Y 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 X holds
(IExec (I,Q,t)) . x >= c + ((IExec (I,Q,t)) . (DataLoc ((s . a),i))) ) & ( for x being Int_position st x in Y holds
(IExec (I,Q,t)) . x = t . x ) ) ) 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[ State of SCMPDS] means ( ( for x being Int_position st x in X holds
$1 . x >= c + ($1 . (DataLoc ((s . a),i))) ) & ( for x being Int_position st x in Y holds
$1 . x = s . x ) );
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 Th5;
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: for x being Int_position st x in X holds
s . x >= c + (s . (DataLoc ((s . a),i))) ; ::_thesis: ( ex t being 0 -started State of SCMPDS ex Q being Instruction-Sequence of SCMPDS st
( ( for x being Int_position st x in X holds
t . x >= c + (t . (DataLoc ((s . a),i))) ) & ( for x being Int_position st x in Y holds
t . x = s . x ) & 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)) & ( for x being Int_position st x in X holds
(IExec (I,Q,t)) . x >= c + ((IExec (I,Q,t)) . (DataLoc ((s . a),i))) ) & ( for x being Int_position st x in Y holds
(IExec (I,Q,t)) . x = t . x ) ) ) 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 ( for x being Int_position st x in X holds
t . x >= c + (t . (DataLoc ((s . a),i))) ) & ( for x being Int_position st x in Y 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 X holds
(IExec (I,Q,t)) . x >= c + ((IExec (I,Q,t)) . (DataLoc ((s . a),i))) ) & ( for x being Int_position st x in Y holds
(IExec (I,Q,t)) . x = t . x ) ) ; ::_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));
set t1 = t;
consider v being State of SCMPDS such that
A13: v = t and
A14: for x being Int_position st x in X holds
v . x >= c + (v . (DataLoc ((s . a),i))) and
A15: for x being Int_position st x in Y holds
v . x = s . x by A10;
thus  ( (IExec (I,Q,t)) . a = t . a & I is_closed_on t,Q & I is_halting_on t,Q ) by A8, A11, A12, A13, A15, A14; ::_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
A18: H1(t) = t . (DataLoc ((s . a),i)) by A2, A12;
assume A19: H1( Initialize (IExec (I,Q,t))) >= H1(t) ; ::_thesis: contradiction
H1( Initialize (IExec (I,Q,t))) = (Initialize (IExec (I,Q,t))) . (DataLoc ((s . a),i)) by A2, A12, A18, A19
.= (IExec (I,Q,t)) . (DataLoc ((s . a),i)) by SCMPDS_5:15 ;
hence  contradiction by A8, A11, A12, A10, A19, A18; ::_thesis: verum
end;
thus  S1[ Initialize (IExec (I,Q,t))] ::_thesis: verum
proof
set v = Initialize (IExec (I,Q,t));
hereby  ::_thesis: for x being Int_position st x in Y holds
(Initialize (IExec (I,Q,t))) . x = s . x
let x be Int_position; ::_thesis: ( x in X implies (Initialize (IExec (I,Q,t))) . x >= c + ((Initialize (IExec (I,Q,t))) . (DataLoc ((s . a),i))) )
assume  x in X ; ::_thesis: (Initialize (IExec (I,Q,t))) . x >= c + ((Initialize (IExec (I,Q,t))) . (DataLoc ((s . a),i)))
then  (IExec (I,Q,t)) . x >= c + ((IExec (I,Q,t)) . (DataLoc ((s . a),i))) by A8, A11, A12, A10;
then  (Initialize (IExec (I,Q,t))) . x >= c + ((IExec (I,Q,t)) . (DataLoc ((s . a),i))) by SCMPDS_5:15;
hence  (Initialize (IExec (I,Q,t))) . x >= c + ((Initialize (IExec (I,Q,t))) . (DataLoc ((s . a),i))) by SCMPDS_5:15; ::_thesis: verum
end;
hereby  ::_thesis: verum
let x be Int_position; ::_thesis: ( x in Y implies (Initialize (IExec (I,Q,t))) . x = s . x )
assume A20: x in Y ; ::_thesis: (Initialize (IExec (I,Q,t))) . x = s . x
then  (IExec (I,Q,t)) . x = t . x by A8, A11, A12, A10;
then  (Initialize (IExec (I,Q,t))) . x = t . x by SCMPDS_5:15;
hence  (Initialize (IExec (I,Q,t))) . x = s . x by A13, A15, A20; ::_thesis: verum
end;
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 A21: 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(A21, 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 :: SCMPDS_8:28
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 being Integer
for X being set st ( for t being 0 -started State of SCMPDS
for Q being Instruction-Sequence of SCMPDS st ( for x being Int_position st x in X 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 X holds
(IExec (I,Q,t)) . x = t . x ) ) ) 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 being Int_position
for i being Integer
for X being set st ( for t being 0 -started State of SCMPDS
for Q being Instruction-Sequence of SCMPDS st ( for x being Int_position st x in X 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 X holds
(IExec (I,Q,t)) . x = t . x ) ) ) 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 being Int_position
for i being Integer
for X being set st ( for t being 0 -started State of SCMPDS
for Q being Instruction-Sequence of SCMPDS st ( for x being Int_position st x in X 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 X holds
(IExec (I,Q,t)) . x = t . x ) ) ) 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 being Int_position
for i being Integer
for X being set st ( for t being 0 -started State of SCMPDS
for Q being Instruction-Sequence of SCMPDS st ( for x being Int_position st x in X 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 X holds
(IExec (I,Q,t)) . x = t . x ) ) ) 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 be Int_position; ::_thesis: for i being Integer
for X being set st ( for t being 0 -started State of SCMPDS
for Q being Instruction-Sequence of SCMPDS st ( for x being Int_position st x in X 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 X holds
(IExec (I,Q,t)) . x = t . x ) ) ) 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 Integer; ::_thesis: for X being set st ( for t being 0 -started State of SCMPDS
for Q being Instruction-Sequence of SCMPDS st ( for x being Int_position st x in X 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 X holds
(IExec (I,Q,t)) . x = t . x ) ) ) 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 X be set ; ::_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 X 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 X holds
(IExec (I,Q,t)) . x = t . x ) ) ) 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 A2: for t being 0 -started State of SCMPDS
for Q being Instruction-Sequence of SCMPDS st ( for x being Int_position st x in X 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 X holds
(IExec (I,Q,t)) . x = t . x ) ) ; ::_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)))) ) )
then A3: for t being 0 -started State of SCMPDS
for Q being Instruction-Sequence of SCMPDS st ( for x being Int_position st x in {} holds
t . x >= 0 + (t . (DataLoc ((s . a),i))) ) & ( for x being Int_position st x in X 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 {} holds
(IExec (I,Q,t)) . x >= 0 + ((IExec (I,Q,t)) . (DataLoc ((s . a),i))) ) & ( for x being Int_position st x in X holds
(IExec (I,Q,t)) . x = t . x ) ) ;
A4: for x being Int_position st x in {} holds
s . x >= 0 + (s . (DataLoc ((s . a),i))) ;
 for t being 0 -started State of SCMPDS
for Q being Instruction-Sequence of SCMPDS st ( for x being Int_position st x in {} holds
t . x >= 0 + (t . (DataLoc ((s . a),i))) ) & ( for x being Int_position st x in X 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 {} holds
(IExec (I,Q,t)) . x >= 0 + ((IExec (I,Q,t)) . (DataLoc ((s . a),i))) ) & ( for x being Int_position st x in X holds
(IExec (I,Q,t)) . x = t . x ) ) by A2;
hence  ( while>0 (a,i,I) is_closed_on s,P & while>0 (a,i,I) is_halting_on s,P ) by A4, Th27; ::_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  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))))
hence  IExec ((while>0 (a,i,I)),P,s) = IExec ((while>0 (a,i,I)),P,(Initialize (IExec (I,P,s)))) by A4, A3, Th27; ::_thesis: verum
end;

theorem :: SCMPDS_8:29
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 X being set st ( for x being Int_position st x in X holds
s . x >= c + (s . (DataLoc ((s . a),i))) ) & ( for t being 0 -started State of SCMPDS
for Q being Instruction-Sequence of SCMPDS st ( for x being Int_position st x in X holds
t . x >= c + (t . (DataLoc ((s . a),i))) ) & 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 X holds
(IExec (I,Q,t)) . x >= c + ((IExec (I,Q,t)) . (DataLoc ((s . a),i))) ) ) ) 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 being Int_position
for i, c being Integer
for X being set st ( for x being Int_position st x in X holds
s . x >= c + (s . (DataLoc ((s . a),i))) ) & ( for t being 0 -started State of SCMPDS
for Q being Instruction-Sequence of SCMPDS st ( for x being Int_position st x in X holds
t . x >= c + (t . (DataLoc ((s . a),i))) ) & 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 X holds
(IExec (I,Q,t)) . x >= c + ((IExec (I,Q,t)) . (DataLoc ((s . a),i))) ) ) ) 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 being Int_position
for i, c being Integer
for X being set st ( for x being Int_position st x in X holds
s . x >= c + (s . (DataLoc ((s . a),i))) ) & ( for t being 0 -started State of SCMPDS
for Q being Instruction-Sequence of SCMPDS st ( for x being Int_position st x in X holds
t . x >= c + (t . (DataLoc ((s . a),i))) ) & 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 X holds
(IExec (I,Q,t)) . x >= c + ((IExec (I,Q,t)) . (DataLoc ((s . a),i))) ) ) ) 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 being Int_position
for i, c being Integer
for X being set st ( for x being Int_position st x in X holds
s . x >= c + (s . (DataLoc ((s . a),i))) ) & ( for t being 0 -started State of SCMPDS
for Q being Instruction-Sequence of SCMPDS st ( for x being Int_position st x in X holds
t . x >= c + (t . (DataLoc ((s . a),i))) ) & 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 X holds
(IExec (I,Q,t)) . x >= c + ((IExec (I,Q,t)) . (DataLoc ((s . a),i))) ) ) ) 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 be Int_position; ::_thesis: for i, c being Integer
for X being set st ( for x being Int_position st x in X holds
s . x >= c + (s . (DataLoc ((s . a),i))) ) & ( for t being 0 -started State of SCMPDS
for Q being Instruction-Sequence of SCMPDS st ( for x being Int_position st x in X holds
t . x >= c + (t . (DataLoc ((s . a),i))) ) & 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 X holds
(IExec (I,Q,t)) . x >= c + ((IExec (I,Q,t)) . (DataLoc ((s . a),i))) ) ) ) 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: for X being set st ( for x being Int_position st x in X holds
s . x >= c + (s . (DataLoc ((s . a),i))) ) & ( for t being 0 -started State of SCMPDS
for Q being Instruction-Sequence of SCMPDS st ( for x being Int_position st x in X holds
t . x >= c + (t . (DataLoc ((s . a),i))) ) & 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 X holds
(IExec (I,Q,t)) . x >= c + ((IExec (I,Q,t)) . (DataLoc ((s . a),i))) ) ) ) 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 X be set ; ::_thesis: ( ( for x being Int_position st x in X holds
s . x >= c + (s . (DataLoc ((s . a),i))) ) & ( for t being 0 -started State of SCMPDS
for Q being Instruction-Sequence of SCMPDS st ( for x being Int_position st x in X holds
t . x >= c + (t . (DataLoc ((s . a),i))) ) & 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 X holds
(IExec (I,Q,t)) . x >= c + ((IExec (I,Q,t)) . (DataLoc ((s . a),i))) ) ) ) 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 A2: for x being Int_position st x in X holds
s . x >= c + (s . (DataLoc ((s . a),i))) ; ::_thesis: ( ex t being 0 -started State of SCMPDS ex Q being Instruction-Sequence of SCMPDS st
( ( for x being Int_position st x in X holds
t . x >= c + (t . (DataLoc ((s . a),i))) ) & 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)) & ( for x being Int_position st x in X holds
(IExec (I,Q,t)) . x >= c + ((IExec (I,Q,t)) . (DataLoc ((s . a),i))) ) ) ) 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)))) ) ) )
assume A3: for t being 0 -started State of SCMPDS
for Q being Instruction-Sequence of SCMPDS st ( for x being Int_position st x in X holds
t . x >= c + (t . (DataLoc ((s . a),i))) ) & 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 X holds
(IExec (I,Q,t)) . x >= c + ((IExec (I,Q,t)) . (DataLoc ((s . a),i))) ) ) ; ::_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)))) ) )
then  for t being 0 -started State of SCMPDS
for Q being Instruction-Sequence of SCMPDS st ( for x being Int_position st x in X holds
t . x >= c + (t . (DataLoc ((s . a),i))) ) & ( for x being Int_position st x in {} 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 X holds
(IExec (I,Q,t)) . x >= c + ((IExec (I,Q,t)) . (DataLoc ((s . a),i))) ) & ( for x being Int_position st x in {} holds
(IExec (I,Q,t)) . x = t . x ) ) ;
hence  ( while>0 (a,i,I) is_closed_on s,P & while>0 (a,i,I) is_halting_on s,P ) by A2, Th27; ::_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)))) )
 for t being 0 -started State of SCMPDS
for Q being Instruction-Sequence of SCMPDS st ( for x being Int_position st x in X holds
t . x >= c + (t . (DataLoc ((s . a),i))) ) & ( for x being Int_position st x in {} 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 X holds
(IExec (I,Q,t)) . x >= c + ((IExec (I,Q,t)) . (DataLoc ((s . a),i))) ) & ( for x being Int_position st x in {} holds
(IExec (I,Q,t)) . x = t . x ) ) by A3;
hence  ( 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, Th27; ::_thesis: verum
end;