:: SCPISORT semantic presentation

begin


definition
let f be FinSequence of INT ;
let s be State of SCMPDS;
let m be Element of NAT ;
predf is_FinSequence_on s,m means :Def1: :: SCPISORT:def 1
 for i being Element of NAT st 1 <= i & i <= len f holds
f . i = s . (intpos (m + i));
end;

:: deftheorem Def1 defines is_FinSequence_on SCPISORT:def_1_:_
 for f being FinSequence of INT
for s being State of SCMPDS
for m being Element of NAT holds
( f is_FinSequence_on s,m iff for i being Element of NAT st 1 <= i & i <= len f holds
f . i = s . (intpos (m + i)) );

theorem Th1: :: SCPISORT:1
for s being State of SCMPDS
for n, m being Element of NAT ex f being FinSequence of INT st
( len f = n & ( for i being Element of NAT st 1 <= i & i <= len f holds
f . i = s . (intpos (m + i)) ) )
proof
let s be State of SCMPDS; ::_thesis: for n, m being Element of NAT ex f being FinSequence of INT st
( len f = n & ( for i being Element of NAT st 1 <= i & i <= len f holds
f . i = s . (intpos (m + i)) ) )
let n, m be Element of NAT ; ::_thesis: ex f being FinSequence of INT st
( len f = n & ( for i being Element of NAT st 1 <= i & i <= len f holds
f . i = s . (intpos (m + i)) ) )
deffunc H1( Nat) ->  set = s . (intpos (m + $1));
consider f being FinSequence such that
A1: ( len f = n & ( for i being Nat st i in dom f holds
f . i = H1(i) ) ) from FINSEQ_1:sch_2();
now__::_thesis:_for_i_being_Nat_st_i_in_dom_f_holds_
f_._i_in_INT
let i be Nat; ::_thesis: ( i in dom f implies f . i in INT )
reconsider a = i as Element of NAT by ORDINAL1:def_12;
assume  i in dom f ; ::_thesis: f . i in INT
then  f . i = s . (intpos (m + a)) by A1;
hence  f . i in INT by INT_1:def_2; ::_thesis: verum
end;
then reconsider f = f as FinSequence of INT by FINSEQ_2:12;
take  f ; ::_thesis: ( len f = n & ( for i being Element of NAT st 1 <= i & i <= len f holds
f . i = s . (intpos (m + i)) ) )
thus  len f = n by A1; ::_thesis: for i being Element of NAT st 1 <= i & i <= len f holds
f . i = s . (intpos (m + i))
hereby  ::_thesis: verum
let i be Element of NAT ; ::_thesis: ( 1 <= i & i <= len f implies f . i = s . (intpos (m + i)) )
assume  ( 1 <= i & i <= len f ) ; ::_thesis: f . i = s . (intpos (m + i))
then  i in dom f by FINSEQ_3:25;
hence  f . i = s . (intpos (m + i)) by A1; ::_thesis: verum
end;
end;

theorem :: SCPISORT:2
for s being State of SCMPDS
for n, m being Element of NAT ex f being FinSequence of INT st
( len f = n & f is_FinSequence_on s,m )
proof
let s be State of SCMPDS; ::_thesis: for n, m being Element of NAT ex f being FinSequence of INT st
( len f = n & f is_FinSequence_on s,m )
let n, m be Element of NAT ; ::_thesis: ex f being FinSequence of INT st
( len f = n & f is_FinSequence_on s,m )
consider f being FinSequence of INT  such that
A1: len f = n and
A2: for i being Element of NAT st 1 <= i & i <= len f holds
f . i = s . (intpos (m + i)) by Th1;
take  f ; ::_thesis: ( len f = n & f is_FinSequence_on s,m )
thus  len f = n by A1; ::_thesis: f is_FinSequence_on s,m
thus  f is_FinSequence_on s,m by A2, Def1; ::_thesis: verum
end;

theorem Th3: :: SCPISORT:3
for f, g being FinSequence of INT
for m, n being Element of NAT st 1 <= n & n <= len f & 1 <= m & m <= len f & len f = len g & f . m = g . n & f . n = g . m & ( for k being Element of NAT st k <> m & k <> n & 1 <= k & k <= len f holds
f . k = g . k ) holds
f,g are_fiberwise_equipotent
proof
let f, g be FinSequence of INT ; ::_thesis: for m, n being Element of NAT st 1 <= n & n <= len f & 1 <= m & m <= len f & len f = len g & f . m = g . n & f . n = g . m & ( for k being Element of NAT st k <> m & k <> n & 1 <= k & k <= len f holds
f . k = g . k ) holds
f,g are_fiberwise_equipotent
let m, n be Element of NAT ; ::_thesis: ( 1 <= n & n <= len f & 1 <= m & m <= len f & len f = len g & f . m = g . n & f . n = g . m & ( for k being Element of NAT st k <> m & k <> n & 1 <= k & k <= len f holds
f . k = g . k ) implies f,g are_fiberwise_equipotent )
assume that
A1: ( 1 <= n & n <= len f ) and
A2: ( 1 <= m & m <= len f ) and
A3: len f = len g and
A4: ( f . m = g . n & f . n = g . m ) and
A5: for k being Element of NAT st k <> m & k <> n & 1 <= k & k <= len f holds
f . k = g . k ; ::_thesis: f,g are_fiberwise_equipotent
A6: m in Seg (len f) by A2, FINSEQ_1:1;
A7: Seg (len f) = dom f by FINSEQ_1:def_3;
A8: now__::_thesis:_for_k_being_set_st_k_<>_m_&_k_<>_n_&_k_in_dom_f_holds_
f_._k_=_g_._k
let k be set ; ::_thesis: ( k <> m & k <> n & k in dom f implies f . k = g . k )
assume that
A9: ( k <> m & k <> n ) and
A10: k in dom f ; ::_thesis: f . k = g . k
reconsider i = k as Element of NAT by A10;
 ( 1 <= i & i <= len f ) by A7, A10, FINSEQ_1:1;
hence  f . k = g . k by A5, A9; ::_thesis: verum
end;
 ( n in dom f & dom f = dom g ) by A1, A3, A7, FINSEQ_1:1, FINSEQ_1:def_3;
hence  f,g are_fiberwise_equipotent by A4, A7, A6, A8, RFINSEQ:28; ::_thesis: verum
end;

set A = NAT ;

set D = SCM-Data-Loc ;

theorem Th4: :: SCPISORT:4
for s1, s2 being State of SCMPDS st ( for a being Int_position holds s1 . a = s2 . a ) holds
Initialize s1 = Initialize s2
proof
let s1, s2 be State of SCMPDS; ::_thesis: ( ( for a being Int_position holds s1 . a = s2 . a ) implies Initialize s1 = Initialize s2 )
assume  for a being Int_position holds s1 . a = s2 . a ; ::_thesis: Initialize s1 = Initialize s2
then  DataPart s1 = DataPart s2 by SCMPDS_4:8;
hence  Initialize s1 = Initialize s2 by MEMSTR_0:80; ::_thesis: verum
end;


theorem Th5: :: SCPISORT:5
for P being Instruction-Sequence of SCMPDS
for s being State of SCMPDS
for I being halt-free Program of SCMPDS
for j being shiftable parahalting Instruction of SCMPDS st I is_closed_on s,P & I is_halting_on s,P holds
( I ';' j is_closed_on s,P & I ';' j is_halting_on s,P )
proof
let P be Instruction-Sequence of SCMPDS; ::_thesis: for s being State of SCMPDS
for I being halt-free Program of SCMPDS
for j being shiftable parahalting Instruction of SCMPDS st I is_closed_on s,P & I is_halting_on s,P holds
( I ';' j is_closed_on s,P & I ';' j is_halting_on s,P )
let s be State of SCMPDS; ::_thesis: for I being halt-free Program of SCMPDS
for j being shiftable parahalting Instruction of SCMPDS st I is_closed_on s,P & I is_halting_on s,P holds
( I ';' j is_closed_on s,P & I ';' j is_halting_on s,P )
let I be halt-free Program of SCMPDS; ::_thesis: for j being shiftable parahalting Instruction of SCMPDS st I is_closed_on s,P & I is_halting_on s,P holds
( I ';' j is_closed_on s,P & I ';' j is_halting_on s,P )
let j be shiftable parahalting Instruction of SCMPDS; ::_thesis: ( I is_closed_on s,P & I is_halting_on s,P implies ( I ';' j is_closed_on s,P & I ';' j is_halting_on s,P ) )
set Mj = Load j;
A1: ( Load j is_closed_on IExec (I,P,(Initialize s)),P & Load j is_halting_on IExec (I,P,(Initialize s)),P ) by SCMPDS_6:20, SCMPDS_6:21;
assume  ( I is_closed_on s,P & I is_halting_on s,P ) ; ::_thesis: ( I ';' j is_closed_on s,P & I ';' j is_halting_on s,P )
then  ( I ';' (Load j) is_closed_on s,P & I ';' (Load j) is_halting_on s,P ) by A1, SCMPDS_7:24;
hence  ( I ';' j is_closed_on s,P & I ';' j is_halting_on s,P ) by SCMPDS_4:def_3; ::_thesis: verum
end;

theorem :: SCPISORT:6
for P being Instruction-Sequence of SCMPDS
for s being 0 -started State of SCMPDS
for I being halt-free Program of SCMPDS
for J being parahalting shiftable Program of SCMPDS
for a being Int_position st I is_closed_on s,P & I is_halting_on s,P holds
(IExec ((I ';' J),P,s)) . a = (IExec (J,P,(Initialize (IExec (I,P,s))))) . a
proof
let P be Instruction-Sequence of SCMPDS; ::_thesis: for s being 0 -started State of SCMPDS
for I being halt-free Program of SCMPDS
for J being parahalting shiftable Program of SCMPDS
for a being Int_position st I is_closed_on s,P & I is_halting_on s,P holds
(IExec ((I ';' J),P,s)) . a = (IExec (J,P,(Initialize (IExec (I,P,s))))) . a
let s be 0 -started State of SCMPDS; ::_thesis: for I being halt-free Program of SCMPDS
for J being parahalting shiftable Program of SCMPDS
for a being Int_position st I is_closed_on s,P & I is_halting_on s,P holds
(IExec ((I ';' J),P,s)) . a = (IExec (J,P,(Initialize (IExec (I,P,s))))) . a
let I be halt-free Program of SCMPDS; ::_thesis: for J being parahalting shiftable Program of SCMPDS
for a being Int_position st I is_closed_on s,P & I is_halting_on s,P holds
(IExec ((I ';' J),P,s)) . a = (IExec (J,P,(Initialize (IExec (I,P,s))))) . a
let J be parahalting shiftable Program of SCMPDS; ::_thesis: for a being Int_position st I is_closed_on s,P & I is_halting_on s,P holds
(IExec ((I ';' J),P,s)) . a = (IExec (J,P,(Initialize (IExec (I,P,s))))) . a
let a be Int_position; ::_thesis: ( I is_closed_on s,P & I is_halting_on s,P implies (IExec ((I ';' J),P,s)) . a = (IExec (J,P,(Initialize (IExec (I,P,s))))) . a )
 ( J is_closed_on IExec (I,P,s),P & J is_halting_on IExec (I,P,s),P ) by SCMPDS_6:20, SCMPDS_6:21;
hence  ( I is_closed_on s,P & I is_halting_on s,P implies (IExec ((I ';' J),P,s)) . a = (IExec (J,P,(Initialize (IExec (I,P,s))))) . a ) by SCMPDS_7:30; ::_thesis: verum
end;

theorem :: SCPISORT:7
for P being Instruction-Sequence of SCMPDS
for s being 0 -started State of SCMPDS
for I being halt-free parahalting Program of SCMPDS
for J being shiftable Program of SCMPDS
for a being Int_position st J is_closed_on IExec (I,P,s),P & J is_halting_on IExec (I,P,s),P holds
(IExec ((I ';' J),P,s)) . a = (IExec (J,P,(Initialize (IExec (I,P,s))))) . a
proof
let P be Instruction-Sequence of SCMPDS; ::_thesis: for s being 0 -started State of SCMPDS
for I being halt-free parahalting Program of SCMPDS
for J being shiftable Program of SCMPDS
for a being Int_position st J is_closed_on IExec (I,P,s),P & J is_halting_on IExec (I,P,s),P holds
(IExec ((I ';' J),P,s)) . a = (IExec (J,P,(Initialize (IExec (I,P,s))))) . a
let s be 0 -started State of SCMPDS; ::_thesis: for I being halt-free parahalting Program of SCMPDS
for J being shiftable Program of SCMPDS
for a being Int_position st J is_closed_on IExec (I,P,s),P & J is_halting_on IExec (I,P,s),P holds
(IExec ((I ';' J),P,s)) . a = (IExec (J,P,(Initialize (IExec (I,P,s))))) . a
let I be halt-free parahalting Program of SCMPDS; ::_thesis: for J being shiftable Program of SCMPDS
for a being Int_position st J is_closed_on IExec (I,P,s),P & J is_halting_on IExec (I,P,s),P holds
(IExec ((I ';' J),P,s)) . a = (IExec (J,P,(Initialize (IExec (I,P,s))))) . a
let J be shiftable Program of SCMPDS; ::_thesis: for a being Int_position st J is_closed_on IExec (I,P,s),P & J is_halting_on IExec (I,P,s),P holds
(IExec ((I ';' J),P,s)) . a = (IExec (J,P,(Initialize (IExec (I,P,s))))) . a
let a be Int_position; ::_thesis: ( J is_closed_on IExec (I,P,s),P & J is_halting_on IExec (I,P,s),P implies (IExec ((I ';' J),P,s)) . a = (IExec (J,P,(Initialize (IExec (I,P,s))))) . a )
 ( I is_closed_on s,P & I is_halting_on s,P ) by SCMPDS_6:20, SCMPDS_6:21;
hence  ( J is_closed_on IExec (I,P,s),P & J is_halting_on IExec (I,P,s),P implies (IExec ((I ';' J),P,s)) . a = (IExec (J,P,(Initialize (IExec (I,P,s))))) . a ) by SCMPDS_7:30; ::_thesis: verum
end;

theorem :: SCPISORT:8
for P being Instruction-Sequence of SCMPDS
for s being State of SCMPDS
for I being Program of SCMPDS
for J being parahalting shiftable Program of SCMPDS st I is_closed_on s,P & I is_halting_on s,P holds
( I ';' J is_closed_on s,P & I ';' J 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 J being parahalting shiftable Program of SCMPDS st I is_closed_on s,P & I is_halting_on s,P holds
( I ';' J is_closed_on s,P & I ';' J is_halting_on s,P )
let s be State of SCMPDS; ::_thesis: for I being Program of SCMPDS
for J being parahalting shiftable Program of SCMPDS st I is_closed_on s,P & I is_halting_on s,P holds
( I ';' J is_closed_on s,P & I ';' J is_halting_on s,P )
let I be Program of SCMPDS; ::_thesis: for J being parahalting shiftable Program of SCMPDS st I is_closed_on s,P & I is_halting_on s,P holds
( I ';' J is_closed_on s,P & I ';' J is_halting_on s,P )
let J be parahalting shiftable Program of SCMPDS; ::_thesis: ( I is_closed_on s,P & I is_halting_on s,P implies ( I ';' J is_closed_on s,P & I ';' J is_halting_on s,P ) )
A1: ( J is_closed_on IExec (I,P,(Initialize s)),P & J is_halting_on IExec (I,P,(Initialize s)),P ) by SCMPDS_6:20, SCMPDS_6:21;
assume  ( I is_closed_on s,P & I is_halting_on s,P ) ; ::_thesis: ( I ';' J is_closed_on s,P & I ';' J is_halting_on s,P )
hence  ( I ';' J is_closed_on s,P & I ';' J is_halting_on s,P ) by A1, SCMPDS_7:24; ::_thesis: verum
end;

theorem :: SCPISORT:9
for P being Instruction-Sequence of SCMPDS
for s being State of SCMPDS
for I being parahalting Program of SCMPDS
for J being shiftable Program of SCMPDS st J is_closed_on IExec (I,P,(Initialize s)),P & J is_halting_on IExec (I,P,(Initialize s)),P holds
( I ';' J is_closed_on s,P & I ';' J is_halting_on s,P )
proof
let P be Instruction-Sequence of SCMPDS; ::_thesis: for s being State of SCMPDS
for I being parahalting Program of SCMPDS
for J being shiftable Program of SCMPDS st J is_closed_on IExec (I,P,(Initialize s)),P & J is_halting_on IExec (I,P,(Initialize s)),P holds
( I ';' J is_closed_on s,P & I ';' J is_halting_on s,P )
let s be State of SCMPDS; ::_thesis: for I being parahalting Program of SCMPDS
for J being shiftable Program of SCMPDS st J is_closed_on IExec (I,P,(Initialize s)),P & J is_halting_on IExec (I,P,(Initialize s)),P holds
( I ';' J is_closed_on s,P & I ';' J is_halting_on s,P )
let I be parahalting Program of SCMPDS; ::_thesis: for J being shiftable Program of SCMPDS st J is_closed_on IExec (I,P,(Initialize s)),P & J is_halting_on IExec (I,P,(Initialize s)),P holds
( I ';' J is_closed_on s,P & I ';' J is_halting_on s,P )
let J be shiftable Program of SCMPDS; ::_thesis: ( J is_closed_on IExec (I,P,(Initialize s)),P & J is_halting_on IExec (I,P,(Initialize s)),P implies ( I ';' J is_closed_on s,P & I ';' J is_halting_on s,P ) )
A1: ( I is_closed_on s,P & I is_halting_on s,P ) by SCMPDS_6:20, SCMPDS_6:21;
assume  ( J is_closed_on IExec (I,P,(Initialize s)),P & J is_halting_on IExec (I,P,(Initialize s)),P ) ; ::_thesis: ( I ';' J is_closed_on s,P & I ';' J is_halting_on s,P )
hence  ( I ';' J is_closed_on s,P & I ';' J is_halting_on s,P ) by A1, SCMPDS_7:24; ::_thesis: verum
end;

theorem :: SCPISORT:10
for P being Instruction-Sequence of SCMPDS
for s being State of SCMPDS
for I being Program of SCMPDS
for j being shiftable parahalting Instruction of SCMPDS st I is_closed_on s,P & I is_halting_on s,P holds
( I ';' j is_closed_on s,P & I ';' j 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 j being shiftable parahalting Instruction of SCMPDS st I is_closed_on s,P & I is_halting_on s,P holds
( I ';' j is_closed_on s,P & I ';' j is_halting_on s,P )
let s be State of SCMPDS; ::_thesis: for I being Program of SCMPDS
for j being shiftable parahalting Instruction of SCMPDS st I is_closed_on s,P & I is_halting_on s,P holds
( I ';' j is_closed_on s,P & I ';' j is_halting_on s,P )
let I be Program of SCMPDS; ::_thesis: for j being shiftable parahalting Instruction of SCMPDS st I is_closed_on s,P & I is_halting_on s,P holds
( I ';' j is_closed_on s,P & I ';' j is_halting_on s,P )
let j be shiftable parahalting Instruction of SCMPDS; ::_thesis: ( I is_closed_on s,P & I is_halting_on s,P implies ( I ';' j is_closed_on s,P & I ';' j is_halting_on s,P ) )
A1: ( Load j is_closed_on IExec (I,P,(Initialize s)),P & Load j is_halting_on IExec (I,P,(Initialize s)),P ) by SCMPDS_6:20, SCMPDS_6:21;
assume  ( I is_closed_on s,P & I is_halting_on s,P ) ; ::_thesis: ( I ';' j is_closed_on s,P & I ';' j is_halting_on s,P )
then  ( I ';' (Load j) is_closed_on s,P & I ';' (Load j) is_halting_on s,P ) by A1, SCMPDS_7:24;
hence  ( I ';' j is_closed_on s,P & I ';' j is_halting_on s,P ) by SCMPDS_4:def_3; ::_thesis: verum
end;

Lm1:  for a being Int_position
for i being Integer
for n being Element of NAT
for I being Program of SCMPDS holds card (stop (for-down (a,i,n,I))) = (card I) + 4
proof
let a be Int_position; ::_thesis: for i being Integer
for n being Element of NAT
for I being Program of SCMPDS holds card (stop (for-down (a,i,n,I))) = (card I) + 4
let i be Integer; ::_thesis: for n being Element of NAT
for I being Program of SCMPDS holds card (stop (for-down (a,i,n,I))) = (card I) + 4
let n be Element of NAT ; ::_thesis: for I being Program of SCMPDS holds card (stop (for-down (a,i,n,I))) = (card I) + 4
let I be Program of SCMPDS; ::_thesis: card (stop (for-down (a,i,n,I))) = (card I) + 4
thus  card (stop (for-down (a,i,n,I))) = (card (for-down (a,i,n,I))) + 1 by COMPOS_1:55
.= ((card I) + 3) + 1 by SCMPDS_7:41
.= (card I) + 4 ; ::_thesis: verum
end;

Lm2:  for a being Int_position
for i being Integer
for n being Element of NAT
for I being Program of SCMPDS holds for-down (a,i,n,I) = ((a,i) <=0_goto ((card I) + 3)) ';' ((I ';' (AddTo (a,i,(- n)))) ';' (goto (- ((card I) + 2))))
proof
let a be Int_position; ::_thesis: for i being Integer
for n being Element of NAT
for I being Program of SCMPDS holds for-down (a,i,n,I) = ((a,i) <=0_goto ((card I) + 3)) ';' ((I ';' (AddTo (a,i,(- n)))) ';' (goto (- ((card I) + 2))))
let i be Integer; ::_thesis: for n being Element of NAT
for I being Program of SCMPDS holds for-down (a,i,n,I) = ((a,i) <=0_goto ((card I) + 3)) ';' ((I ';' (AddTo (a,i,(- n)))) ';' (goto (- ((card I) + 2))))
let n be Element of NAT ; ::_thesis: for I being Program of SCMPDS holds for-down (a,i,n,I) = ((a,i) <=0_goto ((card I) + 3)) ';' ((I ';' (AddTo (a,i,(- n)))) ';' (goto (- ((card I) + 2))))
let I be Program of SCMPDS; ::_thesis: for-down (a,i,n,I) = ((a,i) <=0_goto ((card I) + 3)) ';' ((I ';' (AddTo (a,i,(- n)))) ';' (goto (- ((card I) + 2))))
set i1 = (a,i) <=0_goto ((card I) + 3);
set i2 = AddTo (a,i,(- n));
set i3 = goto (- ((card I) + 2));
thus  for-down (a,i,n,I) = ((((a,i) <=0_goto ((card I) + 3)) ';' I) ';' (AddTo (a,i,(- n)))) ';' (goto (- ((card I) + 2))) by SCMPDS_7:def_2
.= ((a,i) <=0_goto ((card I) + 3)) ';' ((I ';' (AddTo (a,i,(- n)))) ';' (goto (- ((card I) + 2)))) by SCMPDS_7:2 ; ::_thesis: verum
end;

Lm3:  for I being Program of SCMPDS
for a being Int_position
for i being Integer
for n being Element of NAT holds Shift ((I ';' (AddTo (a,i,(- n)))),1) c= for-down (a,i,n,I)
proof
let I be Program of SCMPDS; ::_thesis: for a being Int_position
for i being Integer
for n being Element of NAT holds Shift ((I ';' (AddTo (a,i,(- n)))),1) c= for-down (a,i,n,I)
let a be Int_position; ::_thesis: for i being Integer
for n being Element of NAT holds Shift ((I ';' (AddTo (a,i,(- n)))),1) c= for-down (a,i,n,I)
let i be Integer; ::_thesis: for n being Element of NAT holds Shift ((I ';' (AddTo (a,i,(- n)))),1) c= for-down (a,i,n,I)
let n be Element of NAT ; ::_thesis: Shift ((I ';' (AddTo (a,i,(- n)))),1) c= for-down (a,i,n,I)
set i1 = (a,i) <=0_goto ((card I) + 3);
set i2 = AddTo (a,i,(- n));
set i3 = goto (- ((card I) + 2));
A1: for-down (a,i,n,I) = ((((a,i) <=0_goto ((card I) + 3)) ';' I) ';' (AddTo (a,i,(- n)))) ';' (goto (- ((card I) + 2))) by SCMPDS_7:def_2
.= (((a,i) <=0_goto ((card I) + 3)) ';' (I ';' (AddTo (a,i,(- n))))) ';' (goto (- ((card I) + 2))) by SCMPDS_4:15
.= ((Load ((a,i) <=0_goto ((card I) + 3))) ';' (I ';' (AddTo (a,i,(- n))))) ';' (goto (- ((card I) + 2))) by SCMPDS_4:def_2
.= ((Load ((a,i) <=0_goto ((card I) + 3))) ';' (I ';' (AddTo (a,i,(- n))))) ';' (Load (goto (- ((card I) + 2)))) by SCMPDS_4:def_3 ;
 card (Load ((a,i) <=0_goto ((card I) + 3))) = 1 by COMPOS_1:54;
hence  Shift ((I ';' (AddTo (a,i,(- n)))),1) c= for-down (a,i,n,I) by A1, SCMPDS_7:3; ::_thesis: verum
end;

begin

scheme :: SCPISORT:sch 1
ForDownHalt{ P1[ set ], F1() ->  0 -started State of SCMPDS, F2() ->  Instruction-Sequence of SCMPDS, F3() ->  halt-free shiftable Program of SCMPDS, F4() ->  Int_position, F5() ->  Integer, F6() ->  Element of NAT } :
 ( for-down (F4(),F5(),F6(),F3()) is_closed_on F1(),F2() & for-down (F4(),F5(),F6(),F3()) is_halting_on F1(),F2() )
provided
A1: F6() > 0 and
A2: P1[F1()] and
A3: for t being 0 -started State of SCMPDS
for Q being Instruction-Sequence of SCMPDS st P1[t] & t . F4() = F1() . F4() & t . (DataLoc ((F1() . F4()),F5())) > 0 holds
( (IExec ((F3() ';' (AddTo (F4(),F5(),(- F6())))),Q,t)) . F4() = t . F4() & (IExec ((F3() ';' (AddTo (F4(),F5(),(- F6())))),Q,t)) . (DataLoc ((F1() . F4()),F5())) = (t . (DataLoc ((F1() . F4()),F5()))) - F6() & F3() is_closed_on t,Q & F3() is_halting_on t,Q & P1[ Initialize (IExec ((F3() ';' (AddTo (F4(),F5(),(- F6())))),Q,t))] )
proof
set i1 = (F4(),F5()) <=0_goto ((card F3()) + 3);
set J = F3() ';' (AddTo (F4(),F5(),(- F6())));
set i3 = goto (- ((card F3()) + 2));
set b = DataLoc ((F1() . F4()),F5());
set FOR = for-down (F4(),F5(),F6(),F3());
set pFOR = stop (for-down (F4(),F5(),F6(),F3()));
set pJ = stop (F3() ';' (AddTo (F4(),F5(),(- F6()))));
defpred S1[ Nat] means  for t being 0 -started State of SCMPDS
for Q being Instruction-Sequence of SCMPDS st t . (DataLoc ((F1() . F4()),F5())) <= $1 & P1[t] & t . F4() = F1() . F4() holds
( for-down (F4(),F5(),F6(),F3()) is_closed_on t,Q & for-down (F4(),F5(),F6(),F3()) is_halting_on t,Q );
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_t_._(DataLoc_((F1()_._F4()),F5()))_<=_k_+_1_&_P1[t]_&_t_._F4()_=_F1()_._F4()_holds_
(_for-down_(F4(),F5(),F6(),F3())_is_closed_on_t,Q_&_for-down_(F4(),F5(),F6(),F3())_is_halting_on_t,Q_)
let t be 0 -started State of SCMPDS; ::_thesis: for Q being Instruction-Sequence of SCMPDS st t . (DataLoc ((F1() . F4()),F5())) <= k + 1 & P1[t] & t . F4() = F1() . F4() holds
( for-down (F4(),F5(),F6(),F3()) is_closed_on b2,b3 & for-down (F4(),F5(),F6(),F3()) is_halting_on b2,b3 )
let Q be Instruction-Sequence of SCMPDS; ::_thesis: ( t . (DataLoc ((F1() . F4()),F5())) <= k + 1 & P1[t] & t . F4() = F1() . F4() implies ( for-down (F4(),F5(),F6(),F3()) is_closed_on b1,b2 & for-down (F4(),F5(),F6(),F3()) is_halting_on b1,b2 ) )
A6: Initialize t = t by MEMSTR_0:44;
assume A7: t . (DataLoc ((F1() . F4()),F5())) <= k + 1 ; ::_thesis: ( P1[t] & t . F4() = F1() . F4() implies ( for-down (F4(),F5(),F6(),F3()) is_closed_on b1,b2 & for-down (F4(),F5(),F6(),F3()) is_halting_on b1,b2 ) )
assume A8: P1[t] ; ::_thesis: ( t . F4() = F1() . F4() implies ( for-down (F4(),F5(),F6(),F3()) is_closed_on b1,b2 & for-down (F4(),F5(),F6(),F3()) is_halting_on b1,b2 ) )
assume A9: t . F4() = F1() . F4() ; ::_thesis: ( for-down (F4(),F5(),F6(),F3()) is_closed_on b1,b2 & for-down (F4(),F5(),F6(),F3()) is_halting_on b1,b2 )
percases ( t . (DataLoc ((F1() . F4()),F5())) <= 0 or t . (DataLoc ((F1() . F4()),F5())) > 0 ) ;
suppose t . (DataLoc ((F1() . F4()),F5())) <= 0 ; ::_thesis: ( for-down (F4(),F5(),F6(),F3()) is_closed_on b1,b2 & for-down (F4(),F5(),F6(),F3()) is_halting_on b1,b2 )
hence  ( for-down (F4(),F5(),F6(),F3()) is_closed_on t,Q & for-down (F4(),F5(),F6(),F3()) is_halting_on t,Q ) by A9, SCMPDS_7:44; ::_thesis: verum
end;
supposeA10: t . (DataLoc ((F1() . F4()),F5())) > 0 ; ::_thesis: ( for-down (F4(),F5(),F6(),F3()) is_closed_on b1,b2 & for-down (F4(),F5(),F6(),F3()) is_halting_on b1,b2 )
A11: 0 in dom (stop (for-down (F4(),F5(),F6(),F3()))) by COMPOS_1:36;
 - (- F6()) > 0 by A1;
then  - F6() < 0 ;
then  - F6() <= - 1 by INT_1:8;
then A12: (- F6()) + (t . (DataLoc ((F1() . F4()),F5()))) <= (- 1) + (t . (DataLoc ((F1() . F4()),F5()))) by XREAL_1:6;
 (t . (DataLoc ((F1() . F4()),F5()))) - 1 <= k by A7, XREAL_1:20;
then A13: (- F6()) + (t . (DataLoc ((F1() . F4()),F5()))) <= k by A12, XXREAL_0:2;
set Q2 = Q +* (stop (F3() ';' (AddTo (F4(),F5(),(- F6())))));
set Q3 = Q +* (stop (for-down (F4(),F5(),F6(),F3())));
set t4 = Comput ((Q +* (stop (for-down (F4(),F5(),F6(),F3())))),t,1);
set Q4 = Q +* (stop (for-down (F4(),F5(),F6(),F3())));
set Jt = IExec ((F3() ';' (AddTo (F4(),F5(),(- F6())))),Q,t);
A14: stop (F3() ';' (AddTo (F4(),F5(),(- F6())))) c= Q +* (stop (F3() ';' (AddTo (F4(),F5(),(- F6()))))) by FUNCT_4:25;
A15: for-down (F4(),F5(),F6(),F3()) = ((F4(),F5()) <=0_goto ((card F3()) + 3)) ';' ((F3() ';' (AddTo (F4(),F5(),(- F6())))) ';' (goto (- ((card F3()) + 2)))) by Lm2;
A16: Comput ((Q +* (stop (for-down (F4(),F5(),F6(),F3())))),t,(0 + 1)) = Following ((Q +* (stop (for-down (F4(),F5(),F6(),F3())))),(Comput ((Q +* (stop (for-down (F4(),F5(),F6(),F3())))),t,0))) by EXTPRO_1:3
.= Following ((Q +* (stop (for-down (F4(),F5(),F6(),F3())))),t) by EXTPRO_1:2
.= Exec (((F4(),F5()) <=0_goto ((card F3()) + 3)),t) by A15, A6, SCMPDS_6:11 ;
 for x being Int_position holds t . x = (Comput ((Q +* (stop (for-down (F4(),F5(),F6(),F3())))),t,1)) . x by A16, SCMPDS_2:56;
then A17: DataPart t = DataPart (Comput ((Q +* (stop (for-down (F4(),F5(),F6(),F3())))),t,1)) by SCMPDS_4:8;
A18: (IExec ((F3() ';' (AddTo (F4(),F5(),(- F6())))),Q,t)) . (DataLoc ((F1() . F4()),F5())) = (t . (DataLoc ((F1() . F4()),F5()))) - F6() by A3, A8, A9, A10;
A19: (IExec ((F3() ';' (AddTo (F4(),F5(),(- F6())))),Q,t)) . F4() = t . F4() by A3, A8, A9, A10;
set m2 = LifeSpan ((Q +* (stop (F3() ';' (AddTo (F4(),F5(),(- F6())))))),t);
set t5 = Comput ((Q +* (stop (for-down (F4(),F5(),F6(),F3())))),(Comput ((Q +* (stop (for-down (F4(),F5(),F6(),F3())))),t,1)),(LifeSpan ((Q +* (stop (F3() ';' (AddTo (F4(),F5(),(- F6())))))),t)));
set Q5 = Q +* (stop (for-down (F4(),F5(),F6(),F3())));
set l1 = (card (F3() ';' (AddTo (F4(),F5(),(- F6()))))) + 1;
A20: card (F3() ';' (AddTo (F4(),F5(),(- F6())))) = (card F3()) + 1 by SCMP_GCD:4;
A21: t . (DataLoc ((t . F4()),F5())) = t . (DataLoc ((F1() . F4()),F5())) by A9;
A22: IC t = 0 by A6, MEMSTR_0:47;
A23: ( F3() is_closed_on t,Q & F3() is_halting_on t,Q ) by A3, A8, A9, A10;
then A24: F3() ';' (AddTo (F4(),F5(),(- F6()))) is_closed_on t,Q by Th5;
then A25: F3() ';' (AddTo (F4(),F5(),(- F6()))) is_closed_on t,Q +* (stop (F3() ';' (AddTo (F4(),F5(),(- F6()))))) by A6, SCMPDS_6:24;
 (card F3()) + 2 < (card F3()) + 3 by XREAL_1:6;
then A26: (card (F3() ';' (AddTo (F4(),F5(),(- F6()))))) + 1 in dom (for-down (F4(),F5(),F6(),F3())) by A20, SCMPDS_7:42;
set m3 = (LifeSpan ((Q +* (stop (F3() ';' (AddTo (F4(),F5(),(- F6())))))),t)) + 1;
set t6 = Comput ((Q +* (stop (for-down (F4(),F5(),F6(),F3())))),t,((LifeSpan ((Q +* (stop (F3() ';' (AddTo (F4(),F5(),(- F6())))))),t)) + 1));
set Q6 = Q +* (stop (for-down (F4(),F5(),F6(),F3())));
A27: Comput ((Q +* (stop (for-down (F4(),F5(),F6(),F3())))),t,((LifeSpan ((Q +* (stop (F3() ';' (AddTo (F4(),F5(),(- F6())))))),t)) + 1)) = Comput ((Q +* (stop (for-down (F4(),F5(),F6(),F3())))),(Comput ((Q +* (stop (for-down (F4(),F5(),F6(),F3())))),t,1)),(LifeSpan ((Q +* (stop (F3() ';' (AddTo (F4(),F5(),(- F6())))))),t))) by EXTPRO_1:4;
A28: F3() ';' (AddTo (F4(),F5(),(- F6()))) is_halting_on t,Q by A23, Th5;
then A29: Q +* (stop (F3() ';' (AddTo (F4(),F5(),(- F6()))))) halts_on t by A6, SCMPDS_6:def_3;
 (Q +* (stop (F3() ';' (AddTo (F4(),F5(),(- F6())))))) +* (stop (F3() ';' (AddTo (F4(),F5(),(- F6()))))) halts_on t by A28, A6, SCMPDS_6:def_3;
then A30: F3() ';' (AddTo (F4(),F5(),(- F6()))) is_halting_on t,Q +* (stop (F3() ';' (AddTo (F4(),F5(),(- F6()))))) by A6, SCMPDS_6:def_3;
set m4 = ((LifeSpan ((Q +* (stop (F3() ';' (AddTo (F4(),F5(),(- F6())))))),t)) + 1) + 1;
set t7 = Comput ((Q +* (stop (for-down (F4(),F5(),F6(),F3())))),t,(((LifeSpan ((Q +* (stop (F3() ';' (AddTo (F4(),F5(),(- F6())))))),t)) + 1) + 1));
set Q7 = Q +* (stop (for-down (F4(),F5(),F6(),F3())));
A31: stop (for-down (F4(),F5(),F6(),F3())) c= Q +* (stop (for-down (F4(),F5(),F6(),F3()))) by FUNCT_4:25;
A32: for-down (F4(),F5(),F6(),F3()) c= stop (for-down (F4(),F5(),F6(),F3())) by AFINSQ_1:74;
then A33: for-down (F4(),F5(),F6(),F3()) c= Q +* (stop (for-down (F4(),F5(),F6(),F3()))) by A31, XBOOLE_1:1;
 Shift ((F3() ';' (AddTo (F4(),F5(),(- F6())))),1) c= for-down (F4(),F5(),F6(),F3()) by Lm3;
then  Shift ((F3() ';' (AddTo (F4(),F5(),(- F6())))),1) c= stop (for-down (F4(),F5(),F6(),F3())) by A32, XBOOLE_1:1;
then A34: Shift ((F3() ';' (AddTo (F4(),F5(),(- F6())))),1) c= Q +* (stop (for-down (F4(),F5(),F6(),F3()))) by A31, XBOOLE_1:1;
A35: IC (Comput ((Q +* (stop (for-down (F4(),F5(),F6(),F3())))),t,1)) = succ (IC t) by A10, A16, A21, SCMPDS_2:56
.= 0 + 1 by A22 ;
then A36: DataPart (Comput ((Q +* (stop (F3() ';' (AddTo (F4(),F5(),(- F6())))))),t,(LifeSpan ((Q +* (stop (F3() ';' (AddTo (F4(),F5(),(- F6())))))),t)))) = DataPart (Comput ((Q +* (stop (for-down (F4(),F5(),F6(),F3())))),(Comput ((Q +* (stop (for-down (F4(),F5(),F6(),F3())))),t,1)),(LifeSpan ((Q +* (stop (F3() ';' (AddTo (F4(),F5(),(- F6())))))),t)))) by A14, A30, A25, A17, A34, SCMPDS_7:18;
then A37: DataPart (Comput ((Q +* (stop (for-down (F4(),F5(),F6(),F3())))),(Comput ((Q +* (stop (for-down (F4(),F5(),F6(),F3())))),t,1)),(LifeSpan ((Q +* (stop (F3() ';' (AddTo (F4(),F5(),(- F6())))))),t)))) = DataPart (Result ((Q +* (stop (F3() ';' (AddTo (F4(),F5(),(- F6())))))),t)) by A29, EXTPRO_1:23
.= DataPart (IExec ((F3() ';' (AddTo (F4(),F5(),(- F6())))),Q,t)) by SCMPDS_4:def_5 ;
A38: IC (Comput ((Q +* (stop (for-down (F4(),F5(),F6(),F3())))),(Comput ((Q +* (stop (for-down (F4(),F5(),F6(),F3())))),t,1)),(LifeSpan ((Q +* (stop (F3() ';' (AddTo (F4(),F5(),(- F6())))))),t)))) = (card (F3() ';' (AddTo (F4(),F5(),(- F6()))))) + 1 by A14, A30, A25, A35, A17, A34, SCMPDS_7:18;
then A39: CurInstr ((Q +* (stop (for-down (F4(),F5(),F6(),F3())))),(Comput ((Q +* (stop (for-down (F4(),F5(),F6(),F3())))),t,((LifeSpan ((Q +* (stop (F3() ';' (AddTo (F4(),F5(),(- F6())))))),t)) + 1)))) = (Q +* (stop (for-down (F4(),F5(),F6(),F3())))) . ((card (F3() ';' (AddTo (F4(),F5(),(- F6()))))) + 1) by A27, PBOOLE:143
.= (for-down (F4(),F5(),F6(),F3())) . ((card (F3() ';' (AddTo (F4(),F5(),(- F6()))))) + 1) by A26, A33, GRFUNC_1:2
.= goto (- ((card F3()) + 2)) by A20, SCMPDS_7:43 ;
A40: Comput ((Q +* (stop (for-down (F4(),F5(),F6(),F3())))),t,(((LifeSpan ((Q +* (stop (F3() ';' (AddTo (F4(),F5(),(- F6())))))),t)) + 1) + 1)) = Following ((Q +* (stop (for-down (F4(),F5(),F6(),F3())))),(Comput ((Q +* (stop (for-down (F4(),F5(),F6(),F3())))),t,((LifeSpan ((Q +* (stop (F3() ';' (AddTo (F4(),F5(),(- F6())))))),t)) + 1)))) by EXTPRO_1:3
.= Exec ((goto (- ((card F3()) + 2))),(Comput ((Q +* (stop (for-down (F4(),F5(),F6(),F3())))),t,((LifeSpan ((Q +* (stop (F3() ';' (AddTo (F4(),F5(),(- F6())))))),t)) + 1)))) by A39 ;
IC (Comput ((Q +* (stop (for-down (F4(),F5(),F6(),F3())))),t,(((LifeSpan ((Q +* (stop (F3() ';' (AddTo (F4(),F5(),(- F6())))))),t)) + 1) + 1))) = ICplusConst ((Comput ((Q +* (stop (for-down (F4(),F5(),F6(),F3())))),t,((LifeSpan ((Q +* (stop (F3() ';' (AddTo (F4(),F5(),(- F6())))))),t)) + 1))),(0 - ((card F3()) + 2))) by A40, SCMPDS_2:54
.= 0 by A20, A38, A27, SCMPDS_7:1 ;
then A41: Initialize (Comput ((Q +* (stop (for-down (F4(),F5(),F6(),F3())))),t,(((LifeSpan ((Q +* (stop (F3() ';' (AddTo (F4(),F5(),(- F6())))))),t)) + 1) + 1))) = Comput ((Q +* (stop (for-down (F4(),F5(),F6(),F3())))),t,(((LifeSpan ((Q +* (stop (F3() ';' (AddTo (F4(),F5(),(- F6())))))),t)) + 1) + 1)) by MEMSTR_0:46;
 InsCode (goto (- ((card F3()) + 2))) = 14 by SCMPDS_2:12;
then  InsCode (goto (- ((card F3()) + 2))) in {0,4,5,6,14} by ENUMSET1:def_3;
then  Comput ((Q +* (stop (for-down (F4(),F5(),F6(),F3())))),t,(((LifeSpan ((Q +* (stop (F3() ';' (AddTo (F4(),F5(),(- F6())))))),t)) + 1) + 1)) = Initialize (Comput ((Q +* (stop (for-down (F4(),F5(),F6(),F3())))),t,((LifeSpan ((Q +* (stop (F3() ';' (AddTo (F4(),F5(),(- F6())))))),t)) + 1))) by A40, A41, SCMPDS_8:3
.= Initialize (IExec ((F3() ';' (AddTo (F4(),F5(),(- F6())))),Q,t)) by A37, A27, MEMSTR_0:80 ;
then A42: P1[ Comput ((Q +* (stop (for-down (F4(),F5(),F6(),F3())))),t,(((LifeSpan ((Q +* (stop (F3() ';' (AddTo (F4(),F5(),(- F6())))))),t)) + 1) + 1))] by A3, A8, A9, A10;
A43: (Q +* (stop (for-down (F4(),F5(),F6(),F3())))) +* (stop (for-down (F4(),F5(),F6(),F3()))) = Q +* (stop (for-down (F4(),F5(),F6(),F3()))) ;
(Comput ((Q +* (stop (for-down (F4(),F5(),F6(),F3())))),(Comput ((Q +* (stop (for-down (F4(),F5(),F6(),F3())))),t,1)),(LifeSpan ((Q +* (stop (F3() ';' (AddTo (F4(),F5(),(- F6())))))),t)))) . (DataLoc ((F1() . F4()),F5())) = (Comput ((Q +* (stop (F3() ';' (AddTo (F4(),F5(),(- F6())))))),t,(LifeSpan ((Q +* (stop (F3() ';' (AddTo (F4(),F5(),(- F6())))))),t)))) . (DataLoc ((F1() . F4()),F5())) by A36, SCMPDS_4:8
.= (Result ((Q +* (stop (F3() ';' (AddTo (F4(),F5(),(- F6())))))),t)) . (DataLoc ((F1() . F4()),F5())) by A29, EXTPRO_1:23
.= (t . (DataLoc ((F1() . F4()),F5()))) - F6() by A18, SCMPDS_4:def_5 ;
then A44: (Comput ((Q +* (stop (for-down (F4(),F5(),F6(),F3())))),t,(((LifeSpan ((Q +* (stop (F3() ';' (AddTo (F4(),F5(),(- F6())))))),t)) + 1) + 1))) . (DataLoc ((F1() . F4()),F5())) = (- F6()) + (t . (DataLoc ((F1() . F4()),F5()))) by A27, A40, SCMPDS_2:54;
(Comput ((Q +* (stop (for-down (F4(),F5(),F6(),F3())))),(Comput ((Q +* (stop (for-down (F4(),F5(),F6(),F3())))),t,1)),(LifeSpan ((Q +* (stop (F3() ';' (AddTo (F4(),F5(),(- F6())))))),t)))) . F4() = (Comput ((Q +* (stop (F3() ';' (AddTo (F4(),F5(),(- F6())))))),t,(LifeSpan ((Q +* (stop (F3() ';' (AddTo (F4(),F5(),(- F6())))))),t)))) . F4() by A36, SCMPDS_4:8
.= (Result ((Q +* (stop (F3() ';' (AddTo (F4(),F5(),(- F6())))))),t)) . F4() by A29, EXTPRO_1:23
.= F1() . F4() by A9, A19, SCMPDS_4:def_5 ;
then A45: (Comput ((Q +* (stop (for-down (F4(),F5(),F6(),F3())))),t,(((LifeSpan ((Q +* (stop (F3() ';' (AddTo (F4(),F5(),(- F6())))))),t)) + 1) + 1))) . F4() = F1() . F4() by A27, A40, SCMPDS_2:54;
then A46: for-down (F4(),F5(),F6(),F3()) is_closed_on Comput ((Q +* (stop (for-down (F4(),F5(),F6(),F3())))),t,(((LifeSpan ((Q +* (stop (F3() ';' (AddTo (F4(),F5(),(- F6())))))),t)) + 1) + 1)),Q +* (stop (for-down (F4(),F5(),F6(),F3()))) by A5, A42, A44, A13, A41;
now__::_thesis:_for_k_being_Element_of_NAT_holds_IC_(Comput_((Q_+*_(stop_(for-down_(F4(),F5(),F6(),F3())))),t,k))_in_dom_(stop_(for-down_(F4(),F5(),F6(),F3())))
let k be Element of NAT ; ::_thesis: IC (Comput ((Q +* (stop (for-down (F4(),F5(),F6(),F3())))),t,b1)) in dom (stop (for-down (F4(),F5(),F6(),F3())))
percases ( k < ((LifeSpan ((Q +* (stop (F3() ';' (AddTo (F4(),F5(),(- F6())))))),t)) + 1) + 1 or k >= ((LifeSpan ((Q +* (stop (F3() ';' (AddTo (F4(),F5(),(- F6())))))),t)) + 1) + 1 ) ;
suppose k < ((LifeSpan ((Q +* (stop (F3() ';' (AddTo (F4(),F5(),(- F6())))))),t)) + 1) + 1 ; ::_thesis: IC (Comput ((Q +* (stop (for-down (F4(),F5(),F6(),F3())))),t,b1)) in dom (stop (for-down (F4(),F5(),F6(),F3())))
then A47: k <= (LifeSpan ((Q +* (stop (F3() ';' (AddTo (F4(),F5(),(- F6())))))),t)) + 1 by INT_1:7;
hereby  ::_thesis: verum
percases ( k <= LifeSpan ((Q +* (stop (F3() ';' (AddTo (F4(),F5(),(- F6())))))),t) or k = (LifeSpan ((Q +* (stop (F3() ';' (AddTo (F4(),F5(),(- F6())))))),t)) + 1 ) by A47, NAT_1:8;
supposeA48: k <= LifeSpan ((Q +* (stop (F3() ';' (AddTo (F4(),F5(),(- F6())))))),t) ; ::_thesis: IC (Comput ((Q +* (stop (for-down (F4(),F5(),F6(),F3())))),t,k)) in dom (stop (for-down (F4(),F5(),F6(),F3())))
percases ( k = 0 or k <> 0 ) ;
suppose k = 0 ; ::_thesis: IC (Comput ((Q +* (stop (for-down (F4(),F5(),F6(),F3())))),t,k)) in dom (stop (for-down (F4(),F5(),F6(),F3())))
hence  IC (Comput ((Q +* (stop (for-down (F4(),F5(),F6(),F3())))),t,k)) in dom (stop (for-down (F4(),F5(),F6(),F3()))) by A11, A22, EXTPRO_1:2; ::_thesis: verum
end;
suppose k <> 0 ; ::_thesis: IC (Comput ((Q +* (stop (for-down (F4(),F5(),F6(),F3())))),t,k)) in dom (stop (for-down (F4(),F5(),F6(),F3())))
then consider kn being Nat such that
A49: k = kn + 1 by NAT_1:6;
reconsider kn = kn as Element of NAT by ORDINAL1:def_12;
reconsider lm = IC (Comput ((Q +* (stop (F3() ';' (AddTo (F4(),F5(),(- F6())))))),t,kn)) as Element of NAT ;
 kn < k by A49, XREAL_1:29;
then  kn < LifeSpan ((Q +* (stop (F3() ';' (AddTo (F4(),F5(),(- F6())))))),t) by A48, XXREAL_0:2;
then  (IC (Comput ((Q +* (stop (F3() ';' (AddTo (F4(),F5(),(- F6())))))),t,kn))) + 1 = IC (Comput ((Q +* (stop (for-down (F4(),F5(),F6(),F3())))),(Comput ((Q +* (stop (for-down (F4(),F5(),F6(),F3())))),t,1)),kn)) by A14, A30, A25, A35, A17, A34, SCMPDS_7:16;
then A50: IC (Comput ((Q +* (stop (for-down (F4(),F5(),F6(),F3())))),t,k)) = lm + 1 by A49, EXTPRO_1:4;
 IC (Comput ((Q +* (stop (F3() ';' (AddTo (F4(),F5(),(- F6())))))),t,kn)) in dom (stop (F3() ';' (AddTo (F4(),F5(),(- F6()))))) by A24, A6, SCMPDS_6:def_2;
then  lm < card (stop (F3() ';' (AddTo (F4(),F5(),(- F6()))))) by AFINSQ_1:66;
then  lm < (card (F3() ';' (AddTo (F4(),F5(),(- F6()))))) + 1 by COMPOS_1:55;
then A51: lm + 1 <= (card (F3() ';' (AddTo (F4(),F5(),(- F6()))))) + 1 by INT_1:7;
 (card F3()) + 2 < (card F3()) + 4 by XREAL_1:6;
then  lm + 1 < (card F3()) + 4 by A20, A51, XXREAL_0:2;
then  lm + 1 < card (stop (for-down (F4(),F5(),F6(),F3()))) by Lm1;
hence  IC (Comput ((Q +* (stop (for-down (F4(),F5(),F6(),F3())))),t,k)) in dom (stop (for-down (F4(),F5(),F6(),F3()))) by A50, AFINSQ_1:66; ::_thesis: verum
end;
end;
end;
supposeA52: k = (LifeSpan ((Q +* (stop (F3() ';' (AddTo (F4(),F5(),(- F6())))))),t)) + 1 ; ::_thesis: IC (Comput ((Q +* (stop (for-down (F4(),F5(),F6(),F3())))),t,k)) in dom (stop (for-down (F4(),F5(),F6(),F3())))
 (card (F3() ';' (AddTo (F4(),F5(),(- F6()))))) + 1 in dom (stop (for-down (F4(),F5(),F6(),F3()))) by A26, COMPOS_1:62;
hence  IC (Comput ((Q +* (stop (for-down (F4(),F5(),F6(),F3())))),t,k)) in dom (stop (for-down (F4(),F5(),F6(),F3()))) by A14, A30, A25, A35, A17, A34, A27, A52, SCMPDS_7:18; ::_thesis: verum
end;
end;
end;
end;
suppose k >= ((LifeSpan ((Q +* (stop (F3() ';' (AddTo (F4(),F5(),(- F6())))))),t)) + 1) + 1 ; ::_thesis: IC (Comput ((Q +* (stop (for-down (F4(),F5(),F6(),F3())))),t,b1)) in dom (stop (for-down (F4(),F5(),F6(),F3())))
then consider nn being Nat such that
A53: k = (((LifeSpan ((Q +* (stop (F3() ';' (AddTo (F4(),F5(),(- F6())))))),t)) + 1) + 1) + nn by NAT_1:10;
reconsider nn = nn as Element of NAT by ORDINAL1:def_12;
Comput ((Q +* (stop (for-down (F4(),F5(),F6(),F3())))),t,k) = Comput ((Q +* (stop (for-down (F4(),F5(),F6(),F3())))),(Comput ((Q +* (stop (for-down (F4(),F5(),F6(),F3())))),t,(((LifeSpan ((Q +* (stop (F3() ';' (AddTo (F4(),F5(),(- F6())))))),t)) + 1) + 1))),nn) by A53, EXTPRO_1:4
.= Comput (((Q +* (stop (for-down (F4(),F5(),F6(),F3())))) +* (stop (for-down (F4(),F5(),F6(),F3())))),(Comput ((Q +* (stop (for-down (F4(),F5(),F6(),F3())))),t,(((LifeSpan ((Q +* (stop (F3() ';' (AddTo (F4(),F5(),(- F6())))))),t)) + 1) + 1))),nn) ;
hence  IC (Comput ((Q +* (stop (for-down (F4(),F5(),F6(),F3())))),t,k)) in dom (stop (for-down (F4(),F5(),F6(),F3()))) by A46, A41, SCMPDS_6:def_2; ::_thesis: verum
end;
end;
end;
hence  for-down (F4(),F5(),F6(),F3()) is_closed_on t,Q by A6, SCMPDS_6:def_2; ::_thesis: for-down (F4(),F5(),F6(),F3()) is_halting_on t,Q
 for-down (F4(),F5(),F6(),F3()) is_halting_on Comput ((Q +* (stop (for-down (F4(),F5(),F6(),F3())))),t,(((LifeSpan ((Q +* (stop (F3() ';' (AddTo (F4(),F5(),(- F6())))))),t)) + 1) + 1)),Q +* (stop (for-down (F4(),F5(),F6(),F3()))) by A5, A45, A42, A44, A13, A41;
then  Q +* (stop (for-down (F4(),F5(),F6(),F3()))) halts_on Comput ((Q +* (stop (for-down (F4(),F5(),F6(),F3())))),t,(((LifeSpan ((Q +* (stop (F3() ';' (AddTo (F4(),F5(),(- F6())))))),t)) + 1) + 1)) by A43, A41, SCMPDS_6:def_3;
then  Q +* (stop (for-down (F4(),F5(),F6(),F3()))) halts_on t by EXTPRO_1:22;
hence  for-down (F4(),F5(),F6(),F3()) is_halting_on t,Q by A6, SCMPDS_6:def_3; ::_thesis: verum
end;
end;
end;
hence  S1[k + 1] ; ::_thesis: verum
end;
A54: S1[ 0 ] by SCMPDS_7:44;
A55: for k being Element of NAT holds S1[k] from NAT_1:sch_1(A54, A4);
percases ( F1() . (DataLoc ((F1() . F4()),F5())) <= 0 or F1() . (DataLoc ((F1() . F4()),F5())) > 0 ) ;
suppose F1() . (DataLoc ((F1() . F4()),F5())) <= 0 ; ::_thesis: ( for-down (F4(),F5(),F6(),F3()) is_closed_on F1(),F2() & for-down (F4(),F5(),F6(),F3()) is_halting_on F1(),F2() )
hence  ( for-down (F4(),F5(),F6(),F3()) is_closed_on F1(),F2() & for-down (F4(),F5(),F6(),F3()) is_halting_on F1(),F2() ) by SCMPDS_7:44; ::_thesis: verum
end;
suppose F1() . (DataLoc ((F1() . F4()),F5())) > 0 ; ::_thesis: ( for-down (F4(),F5(),F6(),F3()) is_closed_on F1(),F2() & for-down (F4(),F5(),F6(),F3()) is_halting_on F1(),F2() )
then reconsider m = F1() . (DataLoc ((F1() . F4()),F5())) as Element of NAT by INT_1:3;
 S1[m] by A55;
hence  ( for-down (F4(),F5(),F6(),F3()) is_closed_on F1(),F2() & for-down (F4(),F5(),F6(),F3()) is_halting_on F1(),F2() ) by A2; ::_thesis: verum
end;
end;
end;

scheme :: SCPISORT:sch 2
ForDownExec{ P1[ set ], F1() ->  0 -started State of SCMPDS, F2() ->  Instruction-Sequence of SCMPDS, F3() ->  halt-free shiftable Program of SCMPDS, F4() ->  Int_position, F5() ->  Integer, F6() ->  Element of NAT } :
 IExec ((for-down (F4(),F5(),F6(),F3())),F2(),F1()) = IExec ((for-down (F4(),F5(),F6(),F3())),F2(),(Initialize (IExec ((F3() ';' (AddTo (F4(),F5(),(- F6())))),F2(),F1()))))
provided
A1: F6() > 0 and
A2: F1() . (DataLoc ((F1() . F4()),F5())) > 0 and
A3: P1[F1()] and
A4: for t being 0 -started State of SCMPDS
for Q being Instruction-Sequence of SCMPDS st P1[t] & t . F4() = F1() . F4() & t . (DataLoc ((F1() . F4()),F5())) > 0 holds
( (IExec ((F3() ';' (AddTo (F4(),F5(),(- F6())))),Q,t)) . F4() = t . F4() & (IExec ((F3() ';' (AddTo (F4(),F5(),(- F6())))),Q,t)) . (DataLoc ((F1() . F4()),F5())) = (t . (DataLoc ((F1() . F4()),F5()))) - F6() & F3() is_closed_on t,Q & F3() is_halting_on t,Q & P1[ Initialize (IExec ((F3() ';' (AddTo (F4(),F5(),(- F6())))),Q,t))] )
proof
set i1 = (F4(),F5()) <=0_goto ((card F3()) + 3);
set J = F3() ';' (AddTo (F4(),F5(),(- F6())));
set i3 = goto (- ((card F3()) + 2));
set FOR = for-down (F4(),F5(),F6(),F3());
set pFOR = stop (for-down (F4(),F5(),F6(),F3()));
set P1 = F2() +* (stop (for-down (F4(),F5(),F6(),F3())));
set PJ = F2() +* (stop (F3() ';' (AddTo (F4(),F5(),(- F6())))));
set mJ = LifeSpan ((F2() +* (stop (F3() ';' (AddTo (F4(),F5(),(- F6())))))),F1());
set m1 = (LifeSpan ((F2() +* (stop (F3() ';' (AddTo (F4(),F5(),(- F6())))))),F1())) + 2;
set s2 = Initialize (IExec ((F3() ';' (AddTo (F4(),F5(),(- F6())))),F2(),F1()));
set m2 = LifeSpan ((F2() +* (stop (for-down (F4(),F5(),F6(),F3())))),(Initialize (IExec ((F3() ';' (AddTo (F4(),F5(),(- F6())))),F2(),F1()))));
A5: stop (F3() ';' (AddTo (F4(),F5(),(- F6())))) c= F2() +* (stop (F3() ';' (AddTo (F4(),F5(),(- F6()))))) by FUNCT_4:25;
A6: Initialize F1() = F1() by MEMSTR_0:44;
A7: ( F3() is_closed_on F1(),F2() & F3() is_halting_on F1(),F2() ) by A2, A3, A4;
then  F3() ';' (AddTo (F4(),F5(),(- F6()))) is_closed_on F1(),F2() by Th5;
then A8: F3() ';' (AddTo (F4(),F5(),(- F6()))) is_closed_on F1(),F2() +* (stop (F3() ';' (AddTo (F4(),F5(),(- F6()))))) by A6, SCMPDS_6:24;
A9: F3() ';' (AddTo (F4(),F5(),(- F6()))) is_halting_on F1(),F2() by A7, Th5;
then A10: F2() +* (stop (F3() ';' (AddTo (F4(),F5(),(- F6()))))) halts_on F1() by A6, SCMPDS_6:def_3;
 (F2() +* (stop (F3() ';' (AddTo (F4(),F5(),(- F6())))))) +* (stop (F3() ';' (AddTo (F4(),F5(),(- F6()))))) halts_on Initialize F1() by A9, SCMPDS_6:def_3;
then A11: F3() ';' (AddTo (F4(),F5(),(- F6()))) is_halting_on F1(),F2() +* (stop (F3() ';' (AddTo (F4(),F5(),(- F6()))))) by SCMPDS_6:def_3;
A12: for-down (F4(),F5(),F6(),F3()) = ((F4(),F5()) <=0_goto ((card F3()) + 3)) ';' ((F3() ';' (AddTo (F4(),F5(),(- F6())))) ';' (goto (- ((card F3()) + 2)))) by Lm2;
A13: Comput ((F2() +* (stop (for-down (F4(),F5(),F6(),F3())))),F1(),(0 + 1)) = Following ((F2() +* (stop (for-down (F4(),F5(),F6(),F3())))),(Comput ((F2() +* (stop (for-down (F4(),F5(),F6(),F3())))),F1(),0))) by EXTPRO_1:3
.= Following ((F2() +* (stop (for-down (F4(),F5(),F6(),F3())))),F1()) by EXTPRO_1:2
.= Exec (((F4(),F5()) <=0_goto ((card F3()) + 3)),F1()) by A12, A6, SCMPDS_6:11 ;
set m3 = (LifeSpan ((F2() +* (stop (F3() ';' (AddTo (F4(),F5(),(- F6())))))),F1())) + 1;
set s4 = Comput ((F2() +* (stop (for-down (F4(),F5(),F6(),F3())))),F1(),1);
set P4 = F2() +* (stop (for-down (F4(),F5(),F6(),F3())));
set b = DataLoc ((F1() . F4()),F5());
A14: card (F3() ';' (AddTo (F4(),F5(),(- F6())))) = (card F3()) + 1 by SCMP_GCD:4;
set s6 = Comput ((F2() +* (stop (for-down (F4(),F5(),F6(),F3())))),F1(),((LifeSpan ((F2() +* (stop (F3() ';' (AddTo (F4(),F5(),(- F6())))))),F1())) + 1));
set P6 = F2() +* (stop (for-down (F4(),F5(),F6(),F3())));
set s5 = Comput ((F2() +* (stop (for-down (F4(),F5(),F6(),F3())))),(Comput ((F2() +* (stop (for-down (F4(),F5(),F6(),F3())))),F1(),1)),(LifeSpan ((F2() +* (stop (F3() ';' (AddTo (F4(),F5(),(- F6())))))),F1())));
set l1 = (card (F3() ';' (AddTo (F4(),F5(),(- F6()))))) + 1;
set P5 = F2() +* (stop (for-down (F4(),F5(),F6(),F3())));
 (card F3()) + 2 < (card F3()) + 3 by XREAL_1:6;
then A15: (card (F3() ';' (AddTo (F4(),F5(),(- F6()))))) + 1 in dom (for-down (F4(),F5(),F6(),F3())) by A14, SCMPDS_7:42;
A16: Comput ((F2() +* (stop (for-down (F4(),F5(),F6(),F3())))),F1(),((LifeSpan ((F2() +* (stop (F3() ';' (AddTo (F4(),F5(),(- F6())))))),F1())) + 1)) = Comput ((F2() +* (stop (for-down (F4(),F5(),F6(),F3())))),(Comput ((F2() +* (stop (for-down (F4(),F5(),F6(),F3())))),F1(),1)),(LifeSpan ((F2() +* (stop (F3() ';' (AddTo (F4(),F5(),(- F6())))))),F1()))) by EXTPRO_1:4;
 for x being Int_position holds F1() . x = (Comput ((F2() +* (stop (for-down (F4(),F5(),F6(),F3())))),F1(),1)) . x by A13, SCMPDS_2:56;
then A17: DataPart F1() = DataPart (Comput ((F2() +* (stop (for-down (F4(),F5(),F6(),F3())))),F1(),1)) by SCMPDS_4:8;
A18: IC F1() = 0 by MEMSTR_0:def_12;
A19: stop (for-down (F4(),F5(),F6(),F3())) c= F2() +* (stop (for-down (F4(),F5(),F6(),F3()))) by FUNCT_4:25;
A20: for-down (F4(),F5(),F6(),F3()) c= stop (for-down (F4(),F5(),F6(),F3())) by AFINSQ_1:74;
then A21: for-down (F4(),F5(),F6(),F3()) c= F2() +* (stop (for-down (F4(),F5(),F6(),F3()))) by A19, XBOOLE_1:1;
 Shift ((F3() ';' (AddTo (F4(),F5(),(- F6())))),1) c= for-down (F4(),F5(),F6(),F3()) by Lm3;
then  Shift ((F3() ';' (AddTo (F4(),F5(),(- F6())))),1) c= stop (for-down (F4(),F5(),F6(),F3())) by A20, XBOOLE_1:1;
then A22: Shift ((F3() ';' (AddTo (F4(),F5(),(- F6())))),1) c= F2() +* (stop (for-down (F4(),F5(),F6(),F3()))) by A19, XBOOLE_1:1;
set m0 = LifeSpan ((F2() +* (stop (for-down (F4(),F5(),F6(),F3())))),F1());
set m4 = ((LifeSpan ((F2() +* (stop (F3() ';' (AddTo (F4(),F5(),(- F6())))))),F1())) + 1) + 1;
set s7 = Comput ((F2() +* (stop (for-down (F4(),F5(),F6(),F3())))),F1(),(((LifeSpan ((F2() +* (stop (F3() ';' (AddTo (F4(),F5(),(- F6())))))),F1())) + 1) + 1));
A23: IC (Comput ((F2() +* (stop (for-down (F4(),F5(),F6(),F3())))),F1(),1)) = succ (IC F1()) by A2, A13, SCMPDS_2:56
.= 0 + 1 by A18 ;
then A24: IC (Comput ((F2() +* (stop (for-down (F4(),F5(),F6(),F3())))),(Comput ((F2() +* (stop (for-down (F4(),F5(),F6(),F3())))),F1(),1)),(LifeSpan ((F2() +* (stop (F3() ';' (AddTo (F4(),F5(),(- F6())))))),F1())))) = (card (F3() ';' (AddTo (F4(),F5(),(- F6()))))) + 1 by A5, A11, A8, A17, A22, SCMPDS_7:18;
then A25: CurInstr ((F2() +* (stop (for-down (F4(),F5(),F6(),F3())))),(Comput ((F2() +* (stop (for-down (F4(),F5(),F6(),F3())))),F1(),((LifeSpan ((F2() +* (stop (F3() ';' (AddTo (F4(),F5(),(- F6())))))),F1())) + 1)))) = (F2() +* (stop (for-down (F4(),F5(),F6(),F3())))) . ((card (F3() ';' (AddTo (F4(),F5(),(- F6()))))) + 1) by A16, PBOOLE:143
.= (for-down (F4(),F5(),F6(),F3())) . ((card (F3() ';' (AddTo (F4(),F5(),(- F6()))))) + 1) by A15, A21, GRFUNC_1:2
.= goto (- ((card F3()) + 2)) by A14, SCMPDS_7:43 ;
A26: Comput ((F2() +* (stop (for-down (F4(),F5(),F6(),F3())))),F1(),(((LifeSpan ((F2() +* (stop (F3() ';' (AddTo (F4(),F5(),(- F6())))))),F1())) + 1) + 1)) = Following ((F2() +* (stop (for-down (F4(),F5(),F6(),F3())))),(Comput ((F2() +* (stop (for-down (F4(),F5(),F6(),F3())))),F1(),((LifeSpan ((F2() +* (stop (F3() ';' (AddTo (F4(),F5(),(- F6())))))),F1())) + 1)))) by EXTPRO_1:3
.= Exec ((goto (- ((card F3()) + 2))),(Comput ((F2() +* (stop (for-down (F4(),F5(),F6(),F3())))),F1(),((LifeSpan ((F2() +* (stop (F3() ';' (AddTo (F4(),F5(),(- F6())))))),F1())) + 1)))) by A25 ;
A27: DataPart (Comput ((F2() +* (stop (F3() ';' (AddTo (F4(),F5(),(- F6())))))),F1(),(LifeSpan ((F2() +* (stop (F3() ';' (AddTo (F4(),F5(),(- F6())))))),F1())))) = DataPart (Comput ((F2() +* (stop (for-down (F4(),F5(),F6(),F3())))),(Comput ((F2() +* (stop (for-down (F4(),F5(),F6(),F3())))),F1(),1)),(LifeSpan ((F2() +* (stop (F3() ';' (AddTo (F4(),F5(),(- F6())))))),F1())))) by A5, A11, A8, A23, A17, A22, SCMPDS_7:18;
now__::_thesis:_for_x_being_Int_position_holds_(Comput_((F2()_+*_(stop_(for-down_(F4(),F5(),F6(),F3())))),F1(),(((LifeSpan_((F2()_+*_(stop_(F3()_';'_(AddTo_(F4(),F5(),(-_F6())))))),F1()))_+_1)_+_1)))_._x_=_(Initialize_(IExec_((F3()_';'_(AddTo_(F4(),F5(),(-_F6())))),F2(),F1())))_._x
let x be Int_position; ::_thesis: (Comput ((F2() +* (stop (for-down (F4(),F5(),F6(),F3())))),F1(),(((LifeSpan ((F2() +* (stop (F3() ';' (AddTo (F4(),F5(),(- F6())))))),F1())) + 1) + 1))) . x = (Initialize (IExec ((F3() ';' (AddTo (F4(),F5(),(- F6())))),F2(),F1()))) . x
 not x in dom (Start-At (0,SCMPDS)) by SCMPDS_4:18;
then A28: (Initialize (IExec ((F3() ';' (AddTo (F4(),F5(),(- F6())))),F2(),F1()))) . x = (IExec ((F3() ';' (AddTo (F4(),F5(),(- F6())))),F2(),F1())) . x by FUNCT_4:11;
(Comput ((F2() +* (stop (for-down (F4(),F5(),F6(),F3())))),(Comput ((F2() +* (stop (for-down (F4(),F5(),F6(),F3())))),F1(),1)),(LifeSpan ((F2() +* (stop (F3() ';' (AddTo (F4(),F5(),(- F6())))))),F1())))) . x = (Comput ((F2() +* (stop (F3() ';' (AddTo (F4(),F5(),(- F6())))))),F1(),(LifeSpan ((F2() +* (stop (F3() ';' (AddTo (F4(),F5(),(- F6())))))),F1())))) . x by A27, SCMPDS_4:8
.= (Result ((F2() +* (stop (F3() ';' (AddTo (F4(),F5(),(- F6())))))),F1())) . x by A10, EXTPRO_1:23
.= (IExec ((F3() ';' (AddTo (F4(),F5(),(- F6())))),F2(),F1())) . x by SCMPDS_4:def_5 ;
hence  (Comput ((F2() +* (stop (for-down (F4(),F5(),F6(),F3())))),F1(),(((LifeSpan ((F2() +* (stop (F3() ';' (AddTo (F4(),F5(),(- F6())))))),F1())) + 1) + 1))) . x = (Initialize (IExec ((F3() ';' (AddTo (F4(),F5(),(- F6())))),F2(),F1()))) . x by A16, A26, A28, SCMPDS_2:54; ::_thesis: verum
end;
then A29: DataPart (Comput ((F2() +* (stop (for-down (F4(),F5(),F6(),F3())))),F1(),(((LifeSpan ((F2() +* (stop (F3() ';' (AddTo (F4(),F5(),(- F6())))))),F1())) + 1) + 1))) = DataPart (Initialize (IExec ((F3() ';' (AddTo (F4(),F5(),(- F6())))),F2(),F1()))) by SCMPDS_4:8;
A30: for t being 0 -started State of SCMPDS
for Q being Instruction-Sequence of SCMPDS st P1[t] & t . F4() = F1() . F4() & t . (DataLoc ((F1() . F4()),F5())) > 0 holds
( (IExec ((F3() ';' (AddTo (F4(),F5(),(- F6())))),Q,t)) . F4() = t . F4() & (IExec ((F3() ';' (AddTo (F4(),F5(),(- F6())))),Q,t)) . (DataLoc ((F1() . F4()),F5())) = (t . (DataLoc ((F1() . F4()),F5()))) - F6() & F3() is_closed_on t,Q & F3() is_halting_on t,Q & P1[ Initialize (IExec ((F3() ';' (AddTo (F4(),F5(),(- F6())))),Q,t))] ) by A4;
A31: P1[F1()] by A3;
 ( for-down (F4(),F5(),F6(),F3()) is_closed_on F1(),F2() & for-down (F4(),F5(),F6(),F3()) is_halting_on F1(),F2() ) from SCPISORT:sch_1(A1, A31, A30);
then A32: F2() +* (stop (for-down (F4(),F5(),F6(),F3()))) halts_on F1() by A6, SCMPDS_6:def_3;
set Es = IExec ((F3() ';' (AddTo (F4(),F5(),(- F6())))),F2(),F1());
set bj = DataLoc (((Initialize (IExec ((F3() ';' (AddTo (F4(),F5(),(- F6())))),F2(),F1()))) . F4()),F5());
(Initialize (IExec ((F3() ';' (AddTo (F4(),F5(),(- F6())))),F2(),F1()))) . F4() = (IExec ((F3() ';' (AddTo (F4(),F5(),(- F6())))),F2(),F1())) . F4() by SCMPDS_5:15
.= F1() . F4() by A2, A3, A4 ;
then A33: for t being 0 -started State of SCMPDS
for Q being Instruction-Sequence of SCMPDS st P1[t] & t . F4() = (Initialize (IExec ((F3() ';' (AddTo (F4(),F5(),(- F6())))),F2(),F1()))) . F4() & t . (DataLoc (((Initialize (IExec ((F3() ';' (AddTo (F4(),F5(),(- F6())))),F2(),F1()))) . F4()),F5())) > 0 holds
( (IExec ((F3() ';' (AddTo (F4(),F5(),(- F6())))),Q,t)) . F4() = t . F4() & (IExec ((F3() ';' (AddTo (F4(),F5(),(- F6())))),Q,t)) . (DataLoc (((Initialize (IExec ((F3() ';' (AddTo (F4(),F5(),(- F6())))),F2(),F1()))) . F4()),F5())) = (t . (DataLoc (((Initialize (IExec ((F3() ';' (AddTo (F4(),F5(),(- F6())))),F2(),F1()))) . F4()),F5()))) - F6() & F3() is_closed_on t,Q & F3() is_halting_on t,Q & P1[ Initialize (IExec ((F3() ';' (AddTo (F4(),F5(),(- F6())))),Q,t))] ) by A4;
A34: P1[ Initialize (IExec ((F3() ';' (AddTo (F4(),F5(),(- F6())))),F2(),F1()))] by A2, A3, A4;
 ( for-down (F4(),F5(),F6(),F3()) is_closed_on Initialize (IExec ((F3() ';' (AddTo (F4(),F5(),(- F6())))),F2(),F1())),F2() & for-down (F4(),F5(),F6(),F3()) is_halting_on Initialize (IExec ((F3() ';' (AddTo (F4(),F5(),(- F6())))),F2(),F1())),F2() ) from SCPISORT:sch_1(A1, A34, A33);
then A35: F2() +* (stop (for-down (F4(),F5(),F6(),F3()))) halts_on Initialize (Initialize (IExec ((F3() ';' (AddTo (F4(),F5(),(- F6())))),F2(),F1()))) by SCMPDS_6:def_3;
IC (Comput ((F2() +* (stop (for-down (F4(),F5(),F6(),F3())))),F1(),(((LifeSpan ((F2() +* (stop (F3() ';' (AddTo (F4(),F5(),(- F6())))))),F1())) + 1) + 1))) = ICplusConst ((Comput ((F2() +* (stop (for-down (F4(),F5(),F6(),F3())))),F1(),((LifeSpan ((F2() +* (stop (F3() ';' (AddTo (F4(),F5(),(- F6())))))),F1())) + 1))),(0 - ((card F3()) + 2))) by A26, SCMPDS_2:54
.= 0 by A14, A24, A16, SCMPDS_7:1 ;
then A36: IC (Initialize (IExec ((F3() ';' (AddTo (F4(),F5(),(- F6())))),F2(),F1()))) = IC (Comput ((F2() +* (stop (for-down (F4(),F5(),F6(),F3())))),F1(),((LifeSpan ((F2() +* (stop (F3() ';' (AddTo (F4(),F5(),(- F6())))))),F1())) + 2))) by MEMSTR_0:47;
A37: Comput ((F2() +* (stop (for-down (F4(),F5(),F6(),F3())))),F1(),((LifeSpan ((F2() +* (stop (F3() ';' (AddTo (F4(),F5(),(- F6())))))),F1())) + 2)) = Initialize (IExec ((F3() ';' (AddTo (F4(),F5(),(- F6())))),F2(),F1())) by A29, A36, MEMSTR_0:78;
then  CurInstr ((F2() +* (stop (for-down (F4(),F5(),F6(),F3())))),(Comput ((F2() +* (stop (for-down (F4(),F5(),F6(),F3())))),F1(),((LifeSpan ((F2() +* (stop (F3() ';' (AddTo (F4(),F5(),(- F6())))))),F1())) + 2)))) = (F4(),F5()) <=0_goto ((card F3()) + 3) by A12, SCMPDS_6:11;
then  LifeSpan ((F2() +* (stop (for-down (F4(),F5(),F6(),F3())))),F1()) > (LifeSpan ((F2() +* (stop (F3() ';' (AddTo (F4(),F5(),(- F6())))))),F1())) + 2 by A32, EXTPRO_1:36, SCMPDS_6:17;
then consider nn being Nat such that
A38: LifeSpan ((F2() +* (stop (for-down (F4(),F5(),F6(),F3())))),F1()) = ((LifeSpan ((F2() +* (stop (F3() ';' (AddTo (F4(),F5(),(- F6())))))),F1())) + 2) + nn by NAT_1:10;
reconsider nn = nn as Element of NAT by ORDINAL1:def_12;
 Comput ((F2() +* (stop (for-down (F4(),F5(),F6(),F3())))),F1(),(((LifeSpan ((F2() +* (stop (F3() ';' (AddTo (F4(),F5(),(- F6())))))),F1())) + 2) + (LifeSpan ((F2() +* (stop (for-down (F4(),F5(),F6(),F3())))),(Initialize (IExec ((F3() ';' (AddTo (F4(),F5(),(- F6())))),F2(),F1()))))))) = Comput ((F2() +* (stop (for-down (F4(),F5(),F6(),F3())))),(Initialize (IExec ((F3() ';' (AddTo (F4(),F5(),(- F6())))),F2(),F1()))),(LifeSpan ((F2() +* (stop (for-down (F4(),F5(),F6(),F3())))),(Initialize (IExec ((F3() ';' (AddTo (F4(),F5(),(- F6())))),F2(),F1())))))) by A37, EXTPRO_1:4;
then  CurInstr ((F2() +* (stop (for-down (F4(),F5(),F6(),F3())))),(Comput ((F2() +* (stop (for-down (F4(),F5(),F6(),F3())))),F1(),(((LifeSpan ((F2() +* (stop (F3() ';' (AddTo (F4(),F5(),(- F6())))))),F1())) + 2) + (LifeSpan ((F2() +* (stop (for-down (F4(),F5(),F6(),F3())))),(Initialize (IExec ((F3() ';' (AddTo (F4(),F5(),(- F6())))),F2(),F1()))))))))) = halt SCMPDS by A35, EXTPRO_1:def_15;
then  ((LifeSpan ((F2() +* (stop (F3() ';' (AddTo (F4(),F5(),(- F6())))))),F1())) + 2) + (LifeSpan ((F2() +* (stop (for-down (F4(),F5(),F6(),F3())))),(Initialize (IExec ((F3() ';' (AddTo (F4(),F5(),(- F6())))),F2(),F1()))))) >= LifeSpan ((F2() +* (stop (for-down (F4(),F5(),F6(),F3())))),F1()) by A32, EXTPRO_1:def_15;
then A39: LifeSpan ((F2() +* (stop (for-down (F4(),F5(),F6(),F3())))),(Initialize (IExec ((F3() ';' (AddTo (F4(),F5(),(- F6())))),F2(),F1())))) >= nn by A38, XREAL_1:6;
A40: Comput ((F2() +* (stop (for-down (F4(),F5(),F6(),F3())))),F1(),(LifeSpan ((F2() +* (stop (for-down (F4(),F5(),F6(),F3())))),F1()))) = Comput ((F2() +* (stop (for-down (F4(),F5(),F6(),F3())))),(Initialize (IExec ((F3() ';' (AddTo (F4(),F5(),(- F6())))),F2(),F1()))),nn) by A37, A38, EXTPRO_1:4;
then  CurInstr ((F2() +* (stop (for-down (F4(),F5(),F6(),F3())))),(Comput ((F2() +* (stop (for-down (F4(),F5(),F6(),F3())))),(Initialize (IExec ((F3() ';' (AddTo (F4(),F5(),(- F6())))),F2(),F1()))),nn))) = halt SCMPDS by A32, EXTPRO_1:def_15;
then  nn >= LifeSpan ((F2() +* (stop (for-down (F4(),F5(),F6(),F3())))),(Initialize (IExec ((F3() ';' (AddTo (F4(),F5(),(- F6())))),F2(),F1())))) by A35, EXTPRO_1:def_15;
then  nn = LifeSpan ((F2() +* (stop (for-down (F4(),F5(),F6(),F3())))),(Initialize (IExec ((F3() ';' (AddTo (F4(),F5(),(- F6())))),F2(),F1())))) by A39, XXREAL_0:1;
then  Result ((F2() +* (stop (for-down (F4(),F5(),F6(),F3())))),F1()) = Comput ((F2() +* (stop (for-down (F4(),F5(),F6(),F3())))),(Initialize (IExec ((F3() ';' (AddTo (F4(),F5(),(- F6())))),F2(),F1()))),(LifeSpan ((F2() +* (stop (for-down (F4(),F5(),F6(),F3())))),(Initialize (IExec ((F3() ';' (AddTo (F4(),F5(),(- F6())))),F2(),F1())))))) by A32, A40, EXTPRO_1:23;
hence  IExec ((for-down (F4(),F5(),F6(),F3())),F2(),F1()) = Comput ((F2() +* (stop (for-down (F4(),F5(),F6(),F3())))),(Initialize (IExec ((F3() ';' (AddTo (F4(),F5(),(- F6())))),F2(),F1()))),(LifeSpan ((F2() +* (stop (for-down (F4(),F5(),F6(),F3())))),(Initialize (IExec ((F3() ';' (AddTo (F4(),F5(),(- F6())))),F2(),F1())))))) by SCMPDS_4:def_5
.= Result ((F2() +* (stop (for-down (F4(),F5(),F6(),F3())))),(Initialize (IExec ((F3() ';' (AddTo (F4(),F5(),(- F6())))),F2(),F1())))) by A35, EXTPRO_1:23
.= IExec ((for-down (F4(),F5(),F6(),F3())),F2(),(Initialize (IExec ((F3() ';' (AddTo (F4(),F5(),(- F6())))),F2(),F1())))) by SCMPDS_4:def_5 ;
::_thesis: verum
end;

scheme :: SCPISORT:sch 3
ForDownEnd{ P1[ set ], F1() ->  0 -started State of SCMPDS, F2() ->  halt-free shiftable Program of SCMPDS, F3() ->  Instruction-Sequence of SCMPDS, F4() ->  Int_position, F5() ->  Integer, F6() ->  Element of NAT } :
 ( (IExec ((for-down (F4(),F5(),F6(),F2())),F3(),F1())) . (DataLoc ((F1() . F4()),F5())) <= 0 & P1[ Initialize (IExec ((for-down (F4(),F5(),F6(),F2())),F3(),F1()))] )
provided
A1: F6() > 0 and
A2: P1[F1()] and
A3: for Q being Instruction-Sequence of SCMPDS
for t being 0 -started State of SCMPDS st P1[t] & t . F4() = F1() . F4() & t . (DataLoc ((F1() . F4()),F5())) > 0 holds
( (IExec ((F2() ';' (AddTo (F4(),F5(),(- F6())))),Q,t)) . F4() = t . F4() & (IExec ((F2() ';' (AddTo (F4(),F5(),(- F6())))),Q,t)) . (DataLoc ((F1() . F4()),F5())) = (t . (DataLoc ((F1() . F4()),F5()))) - F6() & F2() is_closed_on t,Q & F2() is_halting_on t,Q & P1[ Initialize (IExec ((F2() ';' (AddTo (F4(),F5(),(- F6())))),Q,t))] )
proof
set b = DataLoc ((F1() . F4()),F5());
set FR = for-down (F4(),F5(),F6(),F2());
defpred S1[ Nat] means  for t being 0 -started State of SCMPDS
for Q being Instruction-Sequence of SCMPDS st t . (DataLoc ((F1() . F4()),F5())) <= $1 & t . F4() = F1() . F4() & P1[t] holds
( (IExec ((for-down (F4(),F5(),F6(),F2())),Q,t)) . (DataLoc ((F1() . F4()),F5())) <= 0 & P1[ Initialize (IExec ((for-down (F4(),F5(),F6(),F2())),Q,t))] );
A4: S1[ 0 ]
proof
let t be 0 -started State of SCMPDS; ::_thesis: for Q being Instruction-Sequence of SCMPDS st t . (DataLoc ((F1() . F4()),F5())) <= 0 & t . F4() = F1() . F4() & P1[t] holds
( (IExec ((for-down (F4(),F5(),F6(),F2())),Q,t)) . (DataLoc ((F1() . F4()),F5())) <= 0 & P1[ Initialize (IExec ((for-down (F4(),F5(),F6(),F2())),Q,t))] )
let Q be Instruction-Sequence of SCMPDS; ::_thesis: ( t . (DataLoc ((F1() . F4()),F5())) <= 0 & t . F4() = F1() . F4() & P1[t] implies ( (IExec ((for-down (F4(),F5(),F6(),F2())),Q,t)) . (DataLoc ((F1() . F4()),F5())) <= 0 & P1[ Initialize (IExec ((for-down (F4(),F5(),F6(),F2())),Q,t))] ) )
assume that
A5: ( t . (DataLoc ((F1() . F4()),F5())) <= 0 & t . F4() = F1() . F4() ) and
A6: P1[t] ; ::_thesis: ( (IExec ((for-down (F4(),F5(),F6(),F2())),Q,t)) . (DataLoc ((F1() . F4()),F5())) <= 0 & P1[ Initialize (IExec ((for-down (F4(),F5(),F6(),F2())),Q,t))] )
A7: Initialize t = t by MEMSTR_0:44;
hence  (IExec ((for-down (F4(),F5(),F6(),F2())),Q,t)) . (DataLoc ((F1() . F4()),F5())) <= 0 by A5, SCMPDS_7:47; ::_thesis: P1[ Initialize (IExec ((for-down (F4(),F5(),F6(),F2())),Q,t))]
 for x being Int_position holds (IExec ((for-down (F4(),F5(),F6(),F2())),Q,t)) . x = t . x by A5, A7, SCMPDS_7:47;
hence  P1[ Initialize (IExec ((for-down (F4(),F5(),F6(),F2())),Q,t))] by A6, Th4, A7; ::_thesis: verum
end;
A8: now__::_thesis:_for_k_being_Element_of_NAT_st_S1[k]_holds_
S1[k_+_1]
let k be Element of NAT ; ::_thesis: ( S1[k] implies S1[k + 1] )
assume A9: S1[k] ; ::_thesis: S1[k + 1]
thus  S1[k + 1] ::_thesis: verum
proof
let u be 0 -started State of SCMPDS; ::_thesis: for Q being Instruction-Sequence of SCMPDS st u . (DataLoc ((F1() . F4()),F5())) <= k + 1 & u . F4() = F1() . F4() & P1[u] holds
( (IExec ((for-down (F4(),F5(),F6(),F2())),Q,u)) . (DataLoc ((F1() . F4()),F5())) <= 0 & P1[ Initialize (IExec ((for-down (F4(),F5(),F6(),F2())),Q,u))] )
let U be Instruction-Sequence of SCMPDS; ::_thesis: ( u . (DataLoc ((F1() . F4()),F5())) <= k + 1 & u . F4() = F1() . F4() & P1[u] implies ( (IExec ((for-down (F4(),F5(),F6(),F2())),U,u)) . (DataLoc ((F1() . F4()),F5())) <= 0 & P1[ Initialize (IExec ((for-down (F4(),F5(),F6(),F2())),U,u))] ) )
assume that
A10: u . (DataLoc ((F1() . F4()),F5())) <= k + 1 and
A11: u . F4() = F1() . F4() and
A12: P1[u] ; ::_thesis: ( (IExec ((for-down (F4(),F5(),F6(),F2())),U,u)) . (DataLoc ((F1() . F4()),F5())) <= 0 & P1[ Initialize (IExec ((for-down (F4(),F5(),F6(),F2())),U,u))] )
percases ( u . (DataLoc ((F1() . F4()),F5())) <= 0 or u . (DataLoc ((F1() . F4()),F5())) > 0 ) ;
suppose u . (DataLoc ((F1() . F4()),F5())) <= 0 ; ::_thesis: ( (IExec ((for-down (F4(),F5(),F6(),F2())),U,u)) . (DataLoc ((F1() . F4()),F5())) <= 0 & P1[ Initialize (IExec ((for-down (F4(),F5(),F6(),F2())),U,u))] )
hence  ( (IExec ((for-down (F4(),F5(),F6(),F2())),U,u)) . (DataLoc ((F1() . F4()),F5())) <= 0 & P1[ Initialize (IExec ((for-down (F4(),F5(),F6(),F2())),U,u))] ) by A4, A11, A12; ::_thesis: verum
end;
supposeA13: u . (DataLoc ((F1() . F4()),F5())) > 0 ; ::_thesis: ( (IExec ((for-down (F4(),F5(),F6(),F2())),U,u)) . (DataLoc ((F1() . F4()),F5())) <= 0 & P1[ Initialize (IExec ((for-down (F4(),F5(),F6(),F2())),U,u))] )
set Ad = AddTo (F4(),F5(),(- F6()));
set Iu = IExec ((F2() ';' (AddTo (F4(),F5(),(- F6())))),U,u);
A14: ( (IExec ((F2() ';' (AddTo (F4(),F5(),(- F6())))),U,u)) . F4() = F1() . F4() & P1[ Initialize (IExec ((F2() ';' (AddTo (F4(),F5(),(- F6())))),U,u))] ) by A3, A11, A12, A13;
 (IExec ((F2() ';' (AddTo (F4(),F5(),(- F6())))),U,u)) . (DataLoc ((F1() . F4()),F5())) = (u . (DataLoc ((F1() . F4()),F5()))) - F6() by A3, A11, A12, A13;
then  ((IExec ((F2() ';' (AddTo (F4(),F5(),(- F6())))),U,u)) . (DataLoc ((F1() . F4()),F5()))) + 1 <= u . (DataLoc ((F1() . F4()),F5())) by A1, INT_1:7, XREAL_1:44;
then  ((IExec ((F2() ';' (AddTo (F4(),F5(),(- F6())))),U,u)) . (DataLoc ((F1() . F4()),F5()))) + 1 <= k + 1 by A10, XXREAL_0:2;
then A15: (IExec ((F2() ';' (AddTo (F4(),F5(),(- F6())))),U,u)) . (DataLoc ((F1() . F4()),F5())) <= k by XREAL_1:6;
A16: P1[u] by A12;
A17: for t being 0 -started State of SCMPDS
for Q being Instruction-Sequence of SCMPDS st P1[t] & t . F4() = u . F4() & t . (DataLoc ((u . F4()),F5())) > 0 holds
( (IExec ((F2() ';' (AddTo (F4(),F5(),(- F6())))),Q,t)) . F4() = t . F4() & (IExec ((F2() ';' (AddTo (F4(),F5(),(- F6())))),Q,t)) . (DataLoc ((u . F4()),F5())) = (t . (DataLoc ((u . F4()),F5()))) - F6() & F2() is_closed_on t,Q & F2() is_halting_on t,Q & P1[ Initialize (IExec ((F2() ';' (AddTo (F4(),F5(),(- F6())))),Q,t))] ) by A3, A11;
A18: u . (DataLoc ((u . F4()),F5())) > 0 by A11, A13;
A19: (Initialize (IExec ((F2() ';' (AddTo (F4(),F5(),(- F6())))),U,u))) . (DataLoc ((F1() . F4()),F5())) = (IExec ((F2() ';' (AddTo (F4(),F5(),(- F6())))),U,u)) . (DataLoc ((F1() . F4()),F5())) by SCMPDS_5:15;
A20: (Initialize (IExec ((F2() ';' (AddTo (F4(),F5(),(- F6())))),U,u))) . F4() = (IExec ((F2() ';' (AddTo (F4(),F5(),(- F6())))),U,u)) . F4() by SCMPDS_5:15;
 IExec ((for-down (F4(),F5(),F6(),F2())),U,u) = IExec ((for-down (F4(),F5(),F6(),F2())),U,(Initialize (IExec ((F2() ';' (AddTo (F4(),F5(),(- F6())))),U,u)))) from SCPISORT:sch_2(A1, A18, A16, A17);
hence  ( (IExec ((for-down (F4(),F5(),F6(),F2())),U,u)) . (DataLoc ((F1() . F4()),F5())) <= 0 & P1[ Initialize (IExec ((for-down (F4(),F5(),F6(),F2())),U,u))] ) by A9, A15, A14, A19, A20; ::_thesis: verum
end;
end;
end;
end;
A21: for k being Element of NAT holds S1[k] from NAT_1:sch_1(A4, A8);
percases ( F1() . (DataLoc ((F1() . F4()),F5())) > 0 or F1() . (DataLoc ((F1() . F4()),F5())) <= 0 ) ;
suppose F1() . (DataLoc ((F1() . F4()),F5())) > 0 ; ::_thesis: ( (IExec ((for-down (F4(),F5(),F6(),F2())),F3(),F1())) . (DataLoc ((F1() . F4()),F5())) <= 0 & P1[ Initialize (IExec ((for-down (F4(),F5(),F6(),F2())),F3(),F1()))] )
then reconsider m = F1() . (DataLoc ((F1() . F4()),F5())) as Element of NAT by INT_1:3;
 S1[m] by A21;
hence  ( (IExec ((for-down (F4(),F5(),F6(),F2())),F3(),F1())) . (DataLoc ((F1() . F4()),F5())) <= 0 & P1[ Initialize (IExec ((for-down (F4(),F5(),F6(),F2())),F3(),F1()))] ) by A2; ::_thesis: verum
end;
suppose F1() . (DataLoc ((F1() . F4()),F5())) <= 0 ; ::_thesis: ( (IExec ((for-down (F4(),F5(),F6(),F2())),F3(),F1())) . (DataLoc ((F1() . F4()),F5())) <= 0 & P1[ Initialize (IExec ((for-down (F4(),F5(),F6(),F2())),F3(),F1()))] )
hence  ( (IExec ((for-down (F4(),F5(),F6(),F2())),F3(),F1())) . (DataLoc ((F1() . F4()),F5())) <= 0 & P1[ Initialize (IExec ((for-down (F4(),F5(),F6(),F2())),F3(),F1()))] ) by A2, A4; ::_thesis: verum
end;
end;
end;

theorem Th11: :: SCPISORT:11
for P being Instruction-Sequence of SCMPDS
for s being 0 -started State of SCMPDS
for I being halt-free shiftable Program of SCMPDS
for a, x, y being Int_position
for i, c being Integer
for n being Element of NAT st n > 0 & s . x >= (s . y) + c & ( for t being 0 -started State of SCMPDS
for Q being Instruction-Sequence of SCMPDS st t . x >= (t . y) + c & t . a = s . a & t . (DataLoc ((s . a),i)) > 0 holds
( (IExec ((I ';' (AddTo (a,i,(- n)))),Q,t)) . a = t . a & (IExec ((I ';' (AddTo (a,i,(- n)))),Q,t)) . (DataLoc ((s . a),i)) = (t . (DataLoc ((s . a),i))) - n & I is_closed_on t,Q & I is_halting_on t,Q & (IExec ((I ';' (AddTo (a,i,(- n)))),Q,t)) . x >= ((IExec ((I ';' (AddTo (a,i,(- n)))),Q,t)) . y) + c ) ) holds
( for-down (a,i,n,I) is_closed_on s,P & for-down (a,i,n,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, x, y being Int_position
for i, c being Integer
for n being Element of NAT st n > 0 & s . x >= (s . y) + c & ( for t being 0 -started State of SCMPDS
for Q being Instruction-Sequence of SCMPDS st t . x >= (t . y) + c & t . a = s . a & t . (DataLoc ((s . a),i)) > 0 holds
( (IExec ((I ';' (AddTo (a,i,(- n)))),Q,t)) . a = t . a & (IExec ((I ';' (AddTo (a,i,(- n)))),Q,t)) . (DataLoc ((s . a),i)) = (t . (DataLoc ((s . a),i))) - n & I is_closed_on t,Q & I is_halting_on t,Q & (IExec ((I ';' (AddTo (a,i,(- n)))),Q,t)) . x >= ((IExec ((I ';' (AddTo (a,i,(- n)))),Q,t)) . y) + c ) ) holds
( for-down (a,i,n,I) is_closed_on s,P & for-down (a,i,n,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, x, y being Int_position
for i, c being Integer
for n being Element of NAT st n > 0 & s . x >= (s . y) + c & ( for t being 0 -started State of SCMPDS
for Q being Instruction-Sequence of SCMPDS st t . x >= (t . y) + c & t . a = s . a & t . (DataLoc ((s . a),i)) > 0 holds
( (IExec ((I ';' (AddTo (a,i,(- n)))),Q,t)) . a = t . a & (IExec ((I ';' (AddTo (a,i,(- n)))),Q,t)) . (DataLoc ((s . a),i)) = (t . (DataLoc ((s . a),i))) - n & I is_closed_on t,Q & I is_halting_on t,Q & (IExec ((I ';' (AddTo (a,i,(- n)))),Q,t)) . x >= ((IExec ((I ';' (AddTo (a,i,(- n)))),Q,t)) . y) + c ) ) holds
( for-down (a,i,n,I) is_closed_on s,P & for-down (a,i,n,I) is_halting_on s,P )
let I be halt-free shiftable Program of SCMPDS; ::_thesis: for a, x, y being Int_position
for i, c being Integer
for n being Element of NAT st n > 0 & s . x >= (s . y) + c & ( for t being 0 -started State of SCMPDS
for Q being Instruction-Sequence of SCMPDS st t . x >= (t . y) + c & t . a = s . a & t . (DataLoc ((s . a),i)) > 0 holds
( (IExec ((I ';' (AddTo (a,i,(- n)))),Q,t)) . a = t . a & (IExec ((I ';' (AddTo (a,i,(- n)))),Q,t)) . (DataLoc ((s . a),i)) = (t . (DataLoc ((s . a),i))) - n & I is_closed_on t,Q & I is_halting_on t,Q & (IExec ((I ';' (AddTo (a,i,(- n)))),Q,t)) . x >= ((IExec ((I ';' (AddTo (a,i,(- n)))),Q,t)) . y) + c ) ) holds
( for-down (a,i,n,I) is_closed_on s,P & for-down (a,i,n,I) is_halting_on s,P )
let a, x, y be Int_position; ::_thesis: for i, c being Integer
for n being Element of NAT st n > 0 & s . x >= (s . y) + c & ( for t being 0 -started State of SCMPDS
for Q being Instruction-Sequence of SCMPDS st t . x >= (t . y) + c & t . a = s . a & t . (DataLoc ((s . a),i)) > 0 holds
( (IExec ((I ';' (AddTo (a,i,(- n)))),Q,t)) . a = t . a & (IExec ((I ';' (AddTo (a,i,(- n)))),Q,t)) . (DataLoc ((s . a),i)) = (t . (DataLoc ((s . a),i))) - n & I is_closed_on t,Q & I is_halting_on t,Q & (IExec ((I ';' (AddTo (a,i,(- n)))),Q,t)) . x >= ((IExec ((I ';' (AddTo (a,i,(- n)))),Q,t)) . y) + c ) ) holds
( for-down (a,i,n,I) is_closed_on s,P & for-down (a,i,n,I) is_halting_on s,P )
let i, c be Integer; ::_thesis: for n being Element of NAT st n > 0 & s . x >= (s . y) + c & ( for t being 0 -started State of SCMPDS
for Q being Instruction-Sequence of SCMPDS st t . x >= (t . y) + c & t . a = s . a & t . (DataLoc ((s . a),i)) > 0 holds
( (IExec ((I ';' (AddTo (a,i,(- n)))),Q,t)) . a = t . a & (IExec ((I ';' (AddTo (a,i,(- n)))),Q,t)) . (DataLoc ((s . a),i)) = (t . (DataLoc ((s . a),i))) - n & I is_closed_on t,Q & I is_halting_on t,Q & (IExec ((I ';' (AddTo (a,i,(- n)))),Q,t)) . x >= ((IExec ((I ';' (AddTo (a,i,(- n)))),Q,t)) . y) + c ) ) holds
( for-down (a,i,n,I) is_closed_on s,P & for-down (a,i,n,I) is_halting_on s,P )
let n be Element of NAT ; ::_thesis: ( n > 0 & s . x >= (s . y) + c & ( for t being 0 -started State of SCMPDS
for Q being Instruction-Sequence of SCMPDS st t . x >= (t . y) + c & t . a = s . a & t . (DataLoc ((s . a),i)) > 0 holds
( (IExec ((I ';' (AddTo (a,i,(- n)))),Q,t)) . a = t . a & (IExec ((I ';' (AddTo (a,i,(- n)))),Q,t)) . (DataLoc ((s . a),i)) = (t . (DataLoc ((s . a),i))) - n & I is_closed_on t,Q & I is_halting_on t,Q & (IExec ((I ';' (AddTo (a,i,(- n)))),Q,t)) . x >= ((IExec ((I ';' (AddTo (a,i,(- n)))),Q,t)) . y) + c ) ) implies ( for-down (a,i,n,I) is_closed_on s,P & for-down (a,i,n,I) is_halting_on s,P ) )
set b = DataLoc ((s . a),i);
set J = I ';' (AddTo (a,i,(- n)));
assume A1: n > 0 ; ::_thesis: ( not s . x >= (s . y) + c or ex t being 0 -started State of SCMPDS ex Q being Instruction-Sequence of SCMPDS st
( t . x >= (t . y) + c & t . a = s . a & t . (DataLoc ((s . a),i)) > 0 & not ( (IExec ((I ';' (AddTo (a,i,(- n)))),Q,t)) . a = t . a & (IExec ((I ';' (AddTo (a,i,(- n)))),Q,t)) . (DataLoc ((s . a),i)) = (t . (DataLoc ((s . a),i))) - n & I is_closed_on t,Q & I is_halting_on t,Q & (IExec ((I ';' (AddTo (a,i,(- n)))),Q,t)) . x >= ((IExec ((I ';' (AddTo (a,i,(- n)))),Q,t)) . y) + c ) ) or ( for-down (a,i,n,I) is_closed_on s,P & for-down (a,i,n,I) is_halting_on s,P ) )
defpred S1[ set ] means  ex t being State of SCMPDS st
( t = $1 & t . x >= (t . y) + c );
assume A2: s . x >= (s . y) + c ; ::_thesis: ( ex t being 0 -started State of SCMPDS ex Q being Instruction-Sequence of SCMPDS st
( t . x >= (t . y) + c & t . a = s . a & t . (DataLoc ((s . a),i)) > 0 & not ( (IExec ((I ';' (AddTo (a,i,(- n)))),Q,t)) . a = t . a & (IExec ((I ';' (AddTo (a,i,(- n)))),Q,t)) . (DataLoc ((s . a),i)) = (t . (DataLoc ((s . a),i))) - n & I is_closed_on t,Q & I is_halting_on t,Q & (IExec ((I ';' (AddTo (a,i,(- n)))),Q,t)) . x >= ((IExec ((I ';' (AddTo (a,i,(- n)))),Q,t)) . y) + c ) ) or ( for-down (a,i,n,I) is_closed_on s,P & for-down (a,i,n,I) is_halting_on s,P ) )
A3: S1[s] by A2;
assume A4: for t being 0 -started State of SCMPDS
for Q being Instruction-Sequence of SCMPDS st t . x >= (t . y) + c & t . a = s . a & t . (DataLoc ((s . a),i)) > 0 holds
( (IExec ((I ';' (AddTo (a,i,(- n)))),Q,t)) . a = t . a & (IExec ((I ';' (AddTo (a,i,(- n)))),Q,t)) . (DataLoc ((s . a),i)) = (t . (DataLoc ((s . a),i))) - n & I is_closed_on t,Q & I is_halting_on t,Q & (IExec ((I ';' (AddTo (a,i,(- n)))),Q,t)) . x >= ((IExec ((I ';' (AddTo (a,i,(- n)))),Q,t)) . y) + c ) ; ::_thesis: ( for-down (a,i,n,I) is_closed_on s,P & for-down (a,i,n,I) is_halting_on s,P )
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_';'_(AddTo_(a,i,(-_n)))),Q,t))_._a_=_t_._a_&_(IExec_((I_';'_(AddTo_(a,i,(-_n)))),Q,t))_._(DataLoc_((s_._a),i))_=_(t_._(DataLoc_((s_._a),i)))_-_n_&_I_is_closed_on_t,Q_&_I_is_halting_on_t,Q_&_S1[_Initialize_(IExec_((I_';'_(AddTo_(a,i,(-_n)))),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 ';' (AddTo (a,i,(- n)))),Q,t)) . a = t . a & (IExec ((I ';' (AddTo (a,i,(- n)))),Q,t)) . (DataLoc ((s . a),i)) = (t . (DataLoc ((s . a),i))) - n & I is_closed_on t,Q & I is_halting_on t,Q & S1[ Initialize (IExec ((I ';' (AddTo (a,i,(- n)))),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 ';' (AddTo (a,i,(- n)))),Q,t)) . a = t . a & (IExec ((I ';' (AddTo (a,i,(- n)))),Q,t)) . (DataLoc ((s . a),i)) = (t . (DataLoc ((s . a),i))) - n & I is_closed_on t,Q & I is_halting_on t,Q & S1[ Initialize (IExec ((I ';' (AddTo (a,i,(- n)))),Q,t))] ) )
assume that
A6: S1[t] and
A7: ( t . a = s . a & t . (DataLoc ((s . a),i)) > 0 ) ; ::_thesis: ( (IExec ((I ';' (AddTo (a,i,(- n)))),Q,t)) . a = t . a & (IExec ((I ';' (AddTo (a,i,(- n)))),Q,t)) . (DataLoc ((s . a),i)) = (t . (DataLoc ((s . a),i))) - n & I is_closed_on t,Q & I is_halting_on t,Q & S1[ Initialize (IExec ((I ';' (AddTo (a,i,(- n)))),Q,t))] )
consider v being State of SCMPDS such that
A8: v = t and
A9: v . x >= (v . y) + c by A6;
thus  ( (IExec ((I ';' (AddTo (a,i,(- n)))),Q,t)) . a = t . a & (IExec ((I ';' (AddTo (a,i,(- n)))),Q,t)) . (DataLoc ((s . a),i)) = (t . (DataLoc ((s . a),i))) - n & I is_closed_on t,Q & I is_halting_on t,Q ) by A4, A7, A8, A9; ::_thesis: S1[ Initialize (IExec ((I ';' (AddTo (a,i,(- n)))),Q,t))]
thus  S1[ Initialize (IExec ((I ';' (AddTo (a,i,(- n)))),Q,t))] ::_thesis: verum
proof
take v = Initialize (IExec ((I ';' (AddTo (a,i,(- n)))),Q,t)); ::_thesis: ( v = Initialize (IExec ((I ';' (AddTo (a,i,(- n)))),Q,t)) & v . x >= (v . y) + c )
thus  v = Initialize (IExec ((I ';' (AddTo (a,i,(- n)))),Q,t)) ; ::_thesis: v . x >= (v . y) + c
 v . x = (IExec ((I ';' (AddTo (a,i,(- n)))),Q,t)) . x by SCMPDS_5:15;
then  v . x >= ((IExec ((I ';' (AddTo (a,i,(- n)))),Q,t)) . y) + c by A4, A7, A8, A9;
hence  v . x >= (v . y) + c by SCMPDS_5:15; ::_thesis: verum
end;
end;
 ( for-down (a,i,n,I) is_closed_on s,P & for-down (a,i,n,I) is_halting_on s,P ) from SCPISORT:sch_1(A1, A3, A5);
hence  ( for-down (a,i,n,I) is_closed_on s,P & for-down (a,i,n,I) is_halting_on s,P ) ; ::_thesis: verum
end;

theorem Th12: :: SCPISORT:12
for P being Instruction-Sequence of SCMPDS
for s being 0 -started State of SCMPDS
for I being halt-free shiftable Program of SCMPDS
for a, x, y being Int_position
for i, c being Integer
for n being Element of NAT st n > 0 & s . x >= (s . y) + c & s . (DataLoc ((s . a),i)) > 0 & ( for t being 0 -started State of SCMPDS
for Q being Instruction-Sequence of SCMPDS st t . x >= (t . y) + c & t . a = s . a & t . (DataLoc ((s . a),i)) > 0 holds
( (IExec ((I ';' (AddTo (a,i,(- n)))),Q,t)) . a = t . a & (IExec ((I ';' (AddTo (a,i,(- n)))),Q,t)) . (DataLoc ((s . a),i)) = (t . (DataLoc ((s . a),i))) - n & I is_closed_on t,Q & I is_halting_on t,Q & (IExec ((I ';' (AddTo (a,i,(- n)))),Q,t)) . x >= ((IExec ((I ';' (AddTo (a,i,(- n)))),Q,t)) . y) + c ) ) holds
IExec ((for-down (a,i,n,I)),P,s) = IExec ((for-down (a,i,n,I)),P,(Initialize (IExec ((I ';' (AddTo (a,i,(- n)))),P,s))))
proof
let P be Instruction-Sequence of SCMPDS; ::_thesis: for s being 0 -started State of SCMPDS
for I being halt-free shiftable Program of SCMPDS
for a, x, y being Int_position
for i, c being Integer
for n being Element of NAT st n > 0 & s . x >= (s . y) + c & s . (DataLoc ((s . a),i)) > 0 & ( for t being 0 -started State of SCMPDS
for Q being Instruction-Sequence of SCMPDS st t . x >= (t . y) + c & t . a = s . a & t . (DataLoc ((s . a),i)) > 0 holds
( (IExec ((I ';' (AddTo (a,i,(- n)))),Q,t)) . a = t . a & (IExec ((I ';' (AddTo (a,i,(- n)))),Q,t)) . (DataLoc ((s . a),i)) = (t . (DataLoc ((s . a),i))) - n & I is_closed_on t,Q & I is_halting_on t,Q & (IExec ((I ';' (AddTo (a,i,(- n)))),Q,t)) . x >= ((IExec ((I ';' (AddTo (a,i,(- n)))),Q,t)) . y) + c ) ) holds
IExec ((for-down (a,i,n,I)),P,s) = IExec ((for-down (a,i,n,I)),P,(Initialize (IExec ((I ';' (AddTo (a,i,(- n)))),P,s))))
let s be 0 -started State of SCMPDS; ::_thesis: for I being halt-free shiftable Program of SCMPDS
for a, x, y being Int_position
for i, c being Integer
for n being Element of NAT st n > 0 & s . x >= (s . y) + c & s . (DataLoc ((s . a),i)) > 0 & ( for t being 0 -started State of SCMPDS
for Q being Instruction-Sequence of SCMPDS st t . x >= (t . y) + c & t . a = s . a & t . (DataLoc ((s . a),i)) > 0 holds
( (IExec ((I ';' (AddTo (a,i,(- n)))),Q,t)) . a = t . a & (IExec ((I ';' (AddTo (a,i,(- n)))),Q,t)) . (DataLoc ((s . a),i)) = (t . (DataLoc ((s . a),i))) - n & I is_closed_on t,Q & I is_halting_on t,Q & (IExec ((I ';' (AddTo (a,i,(- n)))),Q,t)) . x >= ((IExec ((I ';' (AddTo (a,i,(- n)))),Q,t)) . y) + c ) ) holds
IExec ((for-down (a,i,n,I)),P,s) = IExec ((for-down (a,i,n,I)),P,(Initialize (IExec ((I ';' (AddTo (a,i,(- n)))),P,s))))
let I be halt-free shiftable Program of SCMPDS; ::_thesis: for a, x, y being Int_position
for i, c being Integer
for n being Element of NAT st n > 0 & s . x >= (s . y) + c & s . (DataLoc ((s . a),i)) > 0 & ( for t being 0 -started State of SCMPDS
for Q being Instruction-Sequence of SCMPDS st t . x >= (t . y) + c & t . a = s . a & t . (DataLoc ((s . a),i)) > 0 holds
( (IExec ((I ';' (AddTo (a,i,(- n)))),Q,t)) . a = t . a & (IExec ((I ';' (AddTo (a,i,(- n)))),Q,t)) . (DataLoc ((s . a),i)) = (t . (DataLoc ((s . a),i))) - n & I is_closed_on t,Q & I is_halting_on t,Q & (IExec ((I ';' (AddTo (a,i,(- n)))),Q,t)) . x >= ((IExec ((I ';' (AddTo (a,i,(- n)))),Q,t)) . y) + c ) ) holds
IExec ((for-down (a,i,n,I)),P,s) = IExec ((for-down (a,i,n,I)),P,(Initialize (IExec ((I ';' (AddTo (a,i,(- n)))),P,s))))
let a, x, y be Int_position; ::_thesis: for i, c being Integer
for n being Element of NAT st n > 0 & s . x >= (s . y) + c & s . (DataLoc ((s . a),i)) > 0 & ( for t being 0 -started State of SCMPDS
for Q being Instruction-Sequence of SCMPDS st t . x >= (t . y) + c & t . a = s . a & t . (DataLoc ((s . a),i)) > 0 holds
( (IExec ((I ';' (AddTo (a,i,(- n)))),Q,t)) . a = t . a & (IExec ((I ';' (AddTo (a,i,(- n)))),Q,t)) . (DataLoc ((s . a),i)) = (t . (DataLoc ((s . a),i))) - n & I is_closed_on t,Q & I is_halting_on t,Q & (IExec ((I ';' (AddTo (a,i,(- n)))),Q,t)) . x >= ((IExec ((I ';' (AddTo (a,i,(- n)))),Q,t)) . y) + c ) ) holds
IExec ((for-down (a,i,n,I)),P,s) = IExec ((for-down (a,i,n,I)),P,(Initialize (IExec ((I ';' (AddTo (a,i,(- n)))),P,s))))
let i, c be Integer; ::_thesis: for n being Element of NAT st n > 0 & s . x >= (s . y) + c & s . (DataLoc ((s . a),i)) > 0 & ( for t being 0 -started State of SCMPDS
for Q being Instruction-Sequence of SCMPDS st t . x >= (t . y) + c & t . a = s . a & t . (DataLoc ((s . a),i)) > 0 holds
( (IExec ((I ';' (AddTo (a,i,(- n)))),Q,t)) . a = t . a & (IExec ((I ';' (AddTo (a,i,(- n)))),Q,t)) . (DataLoc ((s . a),i)) = (t . (DataLoc ((s . a),i))) - n & I is_closed_on t,Q & I is_halting_on t,Q & (IExec ((I ';' (AddTo (a,i,(- n)))),Q,t)) . x >= ((IExec ((I ';' (AddTo (a,i,(- n)))),Q,t)) . y) + c ) ) holds
IExec ((for-down (a,i,n,I)),P,s) = IExec ((for-down (a,i,n,I)),P,(Initialize (IExec ((I ';' (AddTo (a,i,(- n)))),P,s))))
let n be Element of NAT ; ::_thesis: ( n > 0 & s . x >= (s . y) + c & s . (DataLoc ((s . a),i)) > 0 & ( for t being 0 -started State of SCMPDS
for Q being Instruction-Sequence of SCMPDS st t . x >= (t . y) + c & t . a = s . a & t . (DataLoc ((s . a),i)) > 0 holds
( (IExec ((I ';' (AddTo (a,i,(- n)))),Q,t)) . a = t . a & (IExec ((I ';' (AddTo (a,i,(- n)))),Q,t)) . (DataLoc ((s . a),i)) = (t . (DataLoc ((s . a),i))) - n & I is_closed_on t,Q & I is_halting_on t,Q & (IExec ((I ';' (AddTo (a,i,(- n)))),Q,t)) . x >= ((IExec ((I ';' (AddTo (a,i,(- n)))),Q,t)) . y) + c ) ) implies IExec ((for-down (a,i,n,I)),P,s) = IExec ((for-down (a,i,n,I)),P,(Initialize (IExec ((I ';' (AddTo (a,i,(- n)))),P,s)))) )
set b = DataLoc ((s . a),i);
set J = I ';' (AddTo (a,i,(- n)));
assume A1: n > 0 ; ::_thesis: ( not s . x >= (s . y) + c or not s . (DataLoc ((s . a),i)) > 0 or ex t being 0 -started State of SCMPDS ex Q being Instruction-Sequence of SCMPDS st
( t . x >= (t . y) + c & t . a = s . a & t . (DataLoc ((s . a),i)) > 0 & not ( (IExec ((I ';' (AddTo (a,i,(- n)))),Q,t)) . a = t . a & (IExec ((I ';' (AddTo (a,i,(- n)))),Q,t)) . (DataLoc ((s . a),i)) = (t . (DataLoc ((s . a),i))) - n & I is_closed_on t,Q & I is_halting_on t,Q & (IExec ((I ';' (AddTo (a,i,(- n)))),Q,t)) . x >= ((IExec ((I ';' (AddTo (a,i,(- n)))),Q,t)) . y) + c ) ) or IExec ((for-down (a,i,n,I)),P,s) = IExec ((for-down (a,i,n,I)),P,(Initialize (IExec ((I ';' (AddTo (a,i,(- n)))),P,s)))) )
defpred S1[ set ] means  ex t being State of SCMPDS st
( t = $1 & t . x >= (t . y) + c );
assume A2: s . x >= (s . y) + c ; ::_thesis: ( not s . (DataLoc ((s . a),i)) > 0 or ex t being 0 -started State of SCMPDS ex Q being Instruction-Sequence of SCMPDS st
( t . x >= (t . y) + c & t . a = s . a & t . (DataLoc ((s . a),i)) > 0 & not ( (IExec ((I ';' (AddTo (a,i,(- n)))),Q,t)) . a = t . a & (IExec ((I ';' (AddTo (a,i,(- n)))),Q,t)) . (DataLoc ((s . a),i)) = (t . (DataLoc ((s . a),i))) - n & I is_closed_on t,Q & I is_halting_on t,Q & (IExec ((I ';' (AddTo (a,i,(- n)))),Q,t)) . x >= ((IExec ((I ';' (AddTo (a,i,(- n)))),Q,t)) . y) + c ) ) or IExec ((for-down (a,i,n,I)),P,s) = IExec ((for-down (a,i,n,I)),P,(Initialize (IExec ((I ';' (AddTo (a,i,(- n)))),P,s)))) )
A3: S1[s] by A2;
assume A4: s . (DataLoc ((s . a),i)) > 0 ; ::_thesis: ( ex t being 0 -started State of SCMPDS ex Q being Instruction-Sequence of SCMPDS st
( t . x >= (t . y) + c & t . a = s . a & t . (DataLoc ((s . a),i)) > 0 & not ( (IExec ((I ';' (AddTo (a,i,(- n)))),Q,t)) . a = t . a & (IExec ((I ';' (AddTo (a,i,(- n)))),Q,t)) . (DataLoc ((s . a),i)) = (t . (DataLoc ((s . a),i))) - n & I is_closed_on t,Q & I is_halting_on t,Q & (IExec ((I ';' (AddTo (a,i,(- n)))),Q,t)) . x >= ((IExec ((I ';' (AddTo (a,i,(- n)))),Q,t)) . y) + c ) ) or IExec ((for-down (a,i,n,I)),P,s) = IExec ((for-down (a,i,n,I)),P,(Initialize (IExec ((I ';' (AddTo (a,i,(- n)))),P,s)))) )
assume A5: for t being 0 -started State of SCMPDS
for Q being Instruction-Sequence of SCMPDS st t . x >= (t . y) + c & t . a = s . a & t . (DataLoc ((s . a),i)) > 0 holds
( (IExec ((I ';' (AddTo (a,i,(- n)))),Q,t)) . a = t . a & (IExec ((I ';' (AddTo (a,i,(- n)))),Q,t)) . (DataLoc ((s . a),i)) = (t . (DataLoc ((s . a),i))) - n & I is_closed_on t,Q & I is_halting_on t,Q & (IExec ((I ';' (AddTo (a,i,(- n)))),Q,t)) . x >= ((IExec ((I ';' (AddTo (a,i,(- n)))),Q,t)) . y) + c ) ; ::_thesis: IExec ((for-down (a,i,n,I)),P,s) = IExec ((for-down (a,i,n,I)),P,(Initialize (IExec ((I ';' (AddTo (a,i,(- n)))),P,s))))
A6: 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_';'_(AddTo_(a,i,(-_n)))),Q,t))_._a_=_t_._a_&_(IExec_((I_';'_(AddTo_(a,i,(-_n)))),Q,t))_._(DataLoc_((s_._a),i))_=_(t_._(DataLoc_((s_._a),i)))_-_n_&_I_is_closed_on_t,Q_&_I_is_halting_on_t,Q_&_S1[_Initialize_(IExec_((I_';'_(AddTo_(a,i,(-_n)))),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 ';' (AddTo (a,i,(- n)))),Q,t)) . a = t . a & (IExec ((I ';' (AddTo (a,i,(- n)))),Q,t)) . (DataLoc ((s . a),i)) = (t . (DataLoc ((s . a),i))) - n & I is_closed_on t,Q & I is_halting_on t,Q & S1[ Initialize (IExec ((I ';' (AddTo (a,i,(- n)))),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 ';' (AddTo (a,i,(- n)))),Q,t)) . a = t . a & (IExec ((I ';' (AddTo (a,i,(- n)))),Q,t)) . (DataLoc ((s . a),i)) = (t . (DataLoc ((s . a),i))) - n & I is_closed_on t,Q & I is_halting_on t,Q & S1[ Initialize (IExec ((I ';' (AddTo (a,i,(- n)))),Q,t))] ) )
assume that
A7: S1[t] and
A8: ( t . a = s . a & t . (DataLoc ((s . a),i)) > 0 ) ; ::_thesis: ( (IExec ((I ';' (AddTo (a,i,(- n)))),Q,t)) . a = t . a & (IExec ((I ';' (AddTo (a,i,(- n)))),Q,t)) . (DataLoc ((s . a),i)) = (t . (DataLoc ((s . a),i))) - n & I is_closed_on t,Q & I is_halting_on t,Q & S1[ Initialize (IExec ((I ';' (AddTo (a,i,(- n)))),Q,t))] )
consider v being State of SCMPDS such that
A9: v = t and
A10: v . x >= (v . y) + c by A7;
thus  ( (IExec ((I ';' (AddTo (a,i,(- n)))),Q,t)) . a = t . a & (IExec ((I ';' (AddTo (a,i,(- n)))),Q,t)) . (DataLoc ((s . a),i)) = (t . (DataLoc ((s . a),i))) - n & I is_closed_on t,Q & I is_halting_on t,Q ) by A5, A8, A9, A10; ::_thesis: S1[ Initialize (IExec ((I ';' (AddTo (a,i,(- n)))),Q,t))]
thus  S1[ Initialize (IExec ((I ';' (AddTo (a,i,(- n)))),Q,t))] ::_thesis: verum
proof
take v = Initialize (IExec ((I ';' (AddTo (a,i,(- n)))),Q,t)); ::_thesis: ( v = Initialize (IExec ((I ';' (AddTo (a,i,(- n)))),Q,t)) & v . x >= (v . y) + c )
thus  v = Initialize (IExec ((I ';' (AddTo (a,i,(- n)))),Q,t)) ; ::_thesis: v . x >= (v . y) + c
 v . x = (IExec ((I ';' (AddTo (a,i,(- n)))),Q,t)) . x by SCMPDS_5:15;
then  v . x >= ((IExec ((I ';' (AddTo (a,i,(- n)))),Q,t)) . y) + c by A5, A8, A9, A10;
hence  v . x >= (v . y) + c by SCMPDS_5:15; ::_thesis: verum
end;
end;
 IExec ((for-down (a,i,n,I)),P,s) = IExec ((for-down (a,i,n,I)),P,(Initialize (IExec ((I ';' (AddTo (a,i,(- n)))),P,s)))) from SCPISORT:sch_2(A1, A4, A3, A6);
hence  IExec ((for-down (a,i,n,I)),P,s) = IExec ((for-down (a,i,n,I)),P,(Initialize (IExec ((I ';' (AddTo (a,i,(- n)))),P,s)))) ; ::_thesis: verum
end;

theorem :: SCPISORT: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 n being Element of NAT st s . (DataLoc ((s . a),i)) > 0 & n > 0 & a <> DataLoc ((s . a),i) & ( for t being 0 -started State of SCMPDS
for Q being Instruction-Sequence of SCMPDS st t . a = s . a 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 ) ) holds
( for-down (a,i,n,I) is_closed_on s,P & for-down (a,i,n,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 n being Element of NAT st s . (DataLoc ((s . a),i)) > 0 & n > 0 & a <> DataLoc ((s . a),i) & ( for t being 0 -started State of SCMPDS
for Q being Instruction-Sequence of SCMPDS st t . a = s . a 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 ) ) holds
( for-down (a,i,n,I) is_closed_on s,P & for-down (a,i,n,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 n being Element of NAT st s . (DataLoc ((s . a),i)) > 0 & n > 0 & a <> DataLoc ((s . a),i) & ( for t being 0 -started State of SCMPDS
for Q being Instruction-Sequence of SCMPDS st t . a = s . a 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 ) ) holds
( for-down (a,i,n,I) is_closed_on s,P & for-down (a,i,n,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 n being Element of NAT st s . (DataLoc ((s . a),i)) > 0 & n > 0 & a <> DataLoc ((s . a),i) & ( for t being 0 -started State of SCMPDS
for Q being Instruction-Sequence of SCMPDS st t . a = s . a 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 ) ) holds
( for-down (a,i,n,I) is_closed_on s,P & for-down (a,i,n,I) is_halting_on s,P )
let a be Int_position; ::_thesis: for i being Integer
for n being Element of NAT st s . (DataLoc ((s . a),i)) > 0 & n > 0 & a <> DataLoc ((s . a),i) & ( for t being 0 -started State of SCMPDS
for Q being Instruction-Sequence of SCMPDS st t . a = s . a 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 ) ) holds
( for-down (a,i,n,I) is_closed_on s,P & for-down (a,i,n,I) is_halting_on s,P )
let i be Integer; ::_thesis: for n being Element of NAT st s . (DataLoc ((s . a),i)) > 0 & n > 0 & a <> DataLoc ((s . a),i) & ( for t being 0 -started State of SCMPDS
for Q being Instruction-Sequence of SCMPDS st t . a = s . a 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 ) ) holds
( for-down (a,i,n,I) is_closed_on s,P & for-down (a,i,n,I) is_halting_on s,P )
let n be Element of NAT ; ::_thesis: ( s . (DataLoc ((s . a),i)) > 0 & n > 0 & a <> DataLoc ((s . a),i) & ( for t being 0 -started State of SCMPDS
for Q being Instruction-Sequence of SCMPDS st t . a = s . a 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 ) ) implies ( for-down (a,i,n,I) is_closed_on s,P & for-down (a,i,n,I) is_halting_on s,P ) )
assume that
A1: ( s . (DataLoc ((s . a),i)) > 0 & n > 0 & a <> DataLoc ((s . a),i) ) and
A2: for t being 0 -started State of SCMPDS
for Q being Instruction-Sequence of SCMPDS st t . a = s . a 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 ) ; ::_thesis: ( for-down (a,i,n,I) is_closed_on s,P & for-down (a,i,n,I) is_halting_on s,P )
 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 = s . x ) & t . a = s . a 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 y being Int_position st y in {} holds
(IExec (I,Q,t)) . y = t . y ) ) by A2;
hence  ( for-down (a,i,n,I) is_closed_on s,P & for-down (a,i,n,I) is_halting_on s,P ) by A1, SCMPDS_7:48; ::_thesis: verum
end;

begin

definition
let n, p0 be Element of NAT ;
func insert-sort (n,p0) ->  Program of SCMPDS equals :: SCPISORT:def 2
((((GBP := 0) ';' ((GBP,1) := 0)) ';' ((GBP,2) := (n - 1))) ';' ((GBP,3) := p0)) ';' (for-down (GBP,2,1,(((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))) ';' (while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))))))));
coherence
 ((((GBP := 0) ';' ((GBP,1) := 0)) ';' ((GBP,2) := (n - 1))) ';' ((GBP,3) := p0)) ';' (for-down (GBP,2,1,(((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))) ';' (while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0)))))))))) is Program of SCMPDS ;
end;

:: deftheorem  defines insert-sort SCPISORT:def_2_:_
 for n, p0 being Element of NAT holds insert-sort (n,p0) = ((((GBP := 0) ';' ((GBP,1) := 0)) ';' ((GBP,2) := (n - 1))) ';' ((GBP,3) := p0)) ';' (for-down (GBP,2,1,(((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))) ';' (while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))))))));

set j1 = AddTo (GBP,3,1);

set j2 = (GBP,4) := (GBP,3);

set j3 = AddTo (GBP,1,1);

set j4 = (GBP,6) := (GBP,1);

set k1 = (GBP,5) := ((intpos 4),(- 1));

set k2 = SubFrom (GBP,5,(intpos 4),0);

set k3 = (GBP,5) := ((intpos 4),(- 1));

set k4 = ((intpos 4),(- 1)) := ((intpos 4),0);

set k5 = ((intpos 4),0) := (GBP,5);

set k6 = AddTo (GBP,4,(- 1));

set k7 = AddTo (GBP,6,(- 1));

set FA = Load ((GBP,6) := 0);

set TR = (((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)));

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

set B1 = (((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))));

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

set J4 = (((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1));

set B2 = ((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))) ';' (while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0)))))));

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

Lm4:  card ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))) = 10
proof
thus  card ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))) = (card (((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0)))) + (card (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))) by AFINSQ_1:17
.= 2 + (card (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))) by SCMP_GCD:5
.= 2 + (((card ((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1))))) + (card (Load ((GBP,6) := 0)))) + 2) by SCMPDS_6:65
.= 2 + ((((card (((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1))))) + 1) + (card (Load ((GBP,6) := 0)))) + 2) by SCMP_GCD:4
.= 2 + (((((card ((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5)))) + 1) + 1) + (card (Load ((GBP,6) := 0)))) + 2) by SCMP_GCD:4
.= 2 + ((((((card (((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0)))) + 1) + 1) + 1) + (card (Load ((GBP,6) := 0)))) + 2) by SCMP_GCD:4
.= 2 + (((((2 + 1) + 1) + 1) + (card (Load ((GBP,6) := 0)))) + 2) by SCMP_GCD:5
.= 2 + (((((2 + 1) + 1) + 1) + 1) + 2) by COMPOS_1:54
.= 10 ; ::_thesis: verum
end;

Lm5:  card (((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))) ';' (while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0)))))))) = 16
proof
thus  card (((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))) ';' (while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0)))))))) = (card ((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1)))) + (card (while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0)))))))) by AFINSQ_1:17
.= ((card (((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1)))) + 1) + (card (while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0)))))))) by SCMP_GCD:4
.= (((card ((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3)))) + 1) + 1) + (card (while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0)))))))) by SCMP_GCD:4
.= ((2 + 1) + 1) + (card (while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0)))))))) by SCMP_GCD:5
.= ((2 + 1) + 1) + (10 + 2) by Lm4, SCMPDS_8:17
.= 16 ; ::_thesis: verum
end;

set a1 = intpos 1;

set a2 = intpos 2;

set a3 = intpos 3;

set a4 = intpos 4;

set a5 = intpos 5;

set a6 = intpos 6;

Lm6:  for P being Instruction-Sequence of SCMPDS
for s being 0 -started State of SCMPDS st s . (intpos 4) >= 7 + (s . (intpos 6)) & s . GBP = 0 & s . (intpos 6) > 0 holds
( (IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))),P,s)) . GBP = 0 & (IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))),P,s)) . (intpos 1) = s . (intpos 1) )
proof
let P be Instruction-Sequence of SCMPDS; ::_thesis: for s being 0 -started State of SCMPDS st s . (intpos 4) >= 7 + (s . (intpos 6)) & s . GBP = 0 & s . (intpos 6) > 0 holds
( (IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))),P,s)) . GBP = 0 & (IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))),P,s)) . (intpos 1) = s . (intpos 1) )
let s be 0 -started State of SCMPDS; ::_thesis: ( s . (intpos 4) >= 7 + (s . (intpos 6)) & s . GBP = 0 & s . (intpos 6) > 0 implies ( (IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))),P,s)) . GBP = 0 & (IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))),P,s)) . (intpos 1) = s . (intpos 1) ) )
set a = GBP ;
set x = DataLoc ((s . (intpos 4)),(- 1));
set y = DataLoc ((s . (intpos 4)),0);
assume that
A1: s . (intpos 4) >= 7 + (s . (intpos 6)) and
A2: s . GBP = 0 and
A3: s . (intpos 6) > 0 ; ::_thesis: ( (IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))),P,s)) . GBP = 0 & (IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))),P,s)) . (intpos 1) = s . (intpos 1) )
A4: 7 + (s . (intpos 6)) > 7 + 0 by A3, XREAL_1:6;
set t0 = s;
set t1 = Exec (((GBP,5) := ((intpos 4),(- 1))),s);
set t2 = IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s);
set Q2 = P;
A5: DataLoc ((s . GBP),5) = intpos (0 + 5) by A2, SCMP_GCD:1;
then A6: (Exec (((GBP,5) := ((intpos 4),(- 1))),s)) . GBP = 0 by A2, AMI_3:10, SCMPDS_2:47;
then A7: DataLoc (((Exec (((GBP,5) := ((intpos 4),(- 1))),s)) . GBP),5) = intpos (0 + 5) by SCMP_GCD:1;
A8: (Exec (((GBP,5) := ((intpos 4),(- 1))),s)) . (intpos 4) = s . (intpos 4) by A5, AMI_3:10, SCMPDS_2:47;
A9: s . (intpos 4) >= 1 + (6 + (s . (intpos 6))) by A1;
then A10: (s . (intpos 4)) - 1 >= 6 + (s . (intpos 6)) by XREAL_1:19;
set Fi = (GBP,6) := 0;
set t02 = Initialize (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s));
set Q02 = P;
set t3 = IExec ((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))),P,(Initialize (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s))));
set t4 = IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))),P,(Initialize (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s))));
set t5 = IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))),P,(Initialize (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s))));
set t6 = Exec (((GBP,5) := ((intpos 4),(- 1))),(Initialize (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s))));
(IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s)) . GBP = (Exec ((SubFrom (GBP,5,(intpos 4),0)),(Exec (((GBP,5) := ((intpos 4),(- 1))),s)))) . GBP by SCMPDS_5:42
.= 0 by A6, A7, AMI_3:10, SCMPDS_2:50 ;
then A11: (Initialize (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s))) . GBP = 0 by SCMPDS_5:15;
then A12: DataLoc (((Initialize (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s))) . GBP),5) = intpos (0 + 5) by SCMP_GCD:1;
then A13: (Exec (((GBP,5) := ((intpos 4),(- 1))),(Initialize (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s))))) . GBP = 0 by A11, AMI_3:10, SCMPDS_2:47;
(IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s)) . (intpos 4) = (Exec ((SubFrom (GBP,5,(intpos 4),0)),(Exec (((GBP,5) := ((intpos 4),(- 1))),s)))) . (intpos 4) by SCMPDS_5:42
.= s . (intpos 4) by A8, A7, AMI_3:10, SCMPDS_2:50 ;
then  (Initialize (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s))) . (intpos 4) = s . (intpos 4) by SCMPDS_5:15;
then A14: (Exec (((GBP,5) := ((intpos 4),(- 1))),(Initialize (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s))))) . (intpos 4) = s . (intpos 4) by A12, AMI_3:10, SCMPDS_2:47;
A15: 6 + (s . (intpos 6)) > 6 + 0 by A3, XREAL_1:6;
then  0 <> abs (((Exec (((GBP,5) := ((intpos 4),(- 1))),(Initialize (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s))))) . (intpos 4)) + (- 1)) by A10, A14, ABSVALUE:def_1;
then A16: GBP <> DataLoc (((Exec (((GBP,5) := ((intpos 4),(- 1))),(Initialize (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s))))) . (intpos 4)),(- 1)) by XTUPLE_0:1;
 (s . (intpos 4)) - 1 > 0 by A3, A9, XREAL_1:19;
then A17: abs (((Exec (((GBP,5) := ((intpos 4),(- 1))),(Initialize (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s))))) . (intpos 4)) + (- 1)) = (s . (intpos 4)) - 1 by A14, ABSVALUE:def_1;
then  4 <> abs (((Exec (((GBP,5) := ((intpos 4),(- 1))),(Initialize (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s))))) . (intpos 4)) + (- 1)) by A10, A15, XXREAL_0:2;
then A18: intpos 4 <> DataLoc (((Exec (((GBP,5) := ((intpos 4),(- 1))),(Initialize (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s))))) . (intpos 4)),(- 1)) by XTUPLE_0:1;
A19: (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))),P,(Initialize (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s))))) . (intpos 4) = (Exec ((((intpos 4),(- 1)) := ((intpos 4),0)),(Exec (((GBP,5) := ((intpos 4),(- 1))),(Initialize (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s))))))) . (intpos 4) by SCMPDS_5:42
.= s . (intpos 4) by A14, A18, SCMPDS_2:47 ;
then  0 <> abs (((IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))),P,(Initialize (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s))))) . (intpos 4)) + 0) by A1, A4, ABSVALUE:def_1;
then A20: GBP <> DataLoc (((IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))),P,(Initialize (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s))))) . (intpos 4)),0) by XTUPLE_0:1;
A21: (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))),P,(Initialize (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s))))) . GBP = (Exec ((((intpos 4),(- 1)) := ((intpos 4),0)),(Exec (((GBP,5) := ((intpos 4),(- 1))),(Initialize (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s))))))) . GBP by SCMPDS_5:42
.= 0 by A13, A16, SCMPDS_2:47 ;
A22: (IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))),P,(Initialize (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s))))) . GBP = (Exec ((((intpos 4),0) := (GBP,5)),(IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))),P,(Initialize (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s))))))) . GBP by SCMPDS_5:41
.= 0 by A21, A20, SCMPDS_2:47 ;
then A23: GBP <> DataLoc (((IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))),P,(Initialize (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s))))) . GBP),4) by AMI_3:10, SCMP_GCD:1;
A24: (Exec (((GBP,5) := ((intpos 4),(- 1))),s)) . (intpos 1) = s . (intpos 1) by A5, AMI_3:10, SCMPDS_2:47;
(IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s)) . (intpos 1) = (Exec ((SubFrom (GBP,5,(intpos 4),0)),(Exec (((GBP,5) := ((intpos 4),(- 1))),s)))) . (intpos 1) by SCMPDS_5:42
.= s . (intpos 1) by A24, A7, AMI_3:10, SCMPDS_2:50 ;
then A25: (Initialize (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s))) . (intpos 1) = s . (intpos 1) by SCMPDS_5:15;
then A26: (Exec (((GBP,5) := ((intpos 4),(- 1))),(Initialize (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s))))) . (intpos 1) = s . (intpos 1) by A12, AMI_3:10, SCMPDS_2:47;
A27: (IExec ((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))),P,(Initialize (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s))))) . GBP = (Exec ((AddTo (GBP,4,(- 1))),(IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))),P,(Initialize (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s))))))) . GBP by SCMPDS_5:41
.= 0 by A22, A23, SCMPDS_2:48 ;
then A28: DataLoc (((IExec ((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))),P,(Initialize (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s))))) . GBP),6) = intpos (0 + 6) by SCMP_GCD:1;
A29: DataLoc (((Initialize (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s))) . GBP),6) = intpos (0 + 6) by A11, SCMP_GCD:1;
now__::_thesis:_(IExec_((if>0_(GBP,5,((((((GBP,5)_:=_((intpos_4),(-_1)))_';'_(((intpos_4),(-_1))_:=_((intpos_4),0)))_';'_(((intpos_4),0)_:=_(GBP,5)))_';'_(AddTo_(GBP,4,(-_1))))_';'_(AddTo_(GBP,6,(-_1)))),(Load_((GBP,6)_:=_0)))),P,(Initialize_(IExec_((((GBP,5)_:=_((intpos_4),(-_1)))_';'_(SubFrom_(GBP,5,(intpos_4),0))),P,s)))))_._GBP_=_0
percases ( (Initialize (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s))) . (DataLoc (((Initialize (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s))) . GBP),5)) <= 0 or (Initialize (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s))) . (DataLoc (((Initialize (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s))) . GBP),5)) > 0 ) ;
suppose (Initialize (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s))) . (DataLoc (((Initialize (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s))) . GBP),5)) <= 0 ; ::_thesis: (IExec ((if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0)))),P,(Initialize (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s))))) . GBP = 0
hence (IExec ((if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0)))),P,(Initialize (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s))))) . GBP = (IExec ((Load ((GBP,6) := 0)),P,(Initialize (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s))))) . GBP by SCMPDS_6:74
.= (Exec (((GBP,6) := 0),(Initialize (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s))))) . GBP by SCMPDS_5:40
.= 0 by A11, A29, AMI_3:10, SCMPDS_2:46 ;
::_thesis: verum
end;
suppose (Initialize (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s))) . (DataLoc (((Initialize (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s))) . GBP),5)) > 0 ; ::_thesis: (IExec ((if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0)))),P,(Initialize (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s))))) . GBP = 0
hence (IExec ((if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0)))),P,(Initialize (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s))))) . GBP = (IExec (((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),P,(Initialize (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s))))) . GBP by SCMPDS_6:73
.= (Exec ((AddTo (GBP,6,(- 1))),(IExec ((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))),P,(Initialize (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s))))))) . GBP by SCMPDS_5:41
.= 0 by A27, A28, AMI_3:10, SCMPDS_2:48 ;
::_thesis: verum
end;
end;
end;
hence  (IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))),P,s)) . GBP = 0 by SCMPDS_5:35; ::_thesis: (IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))),P,s)) . (intpos 1) = s . (intpos 1)
A30: intpos 1 <> DataLoc (((IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))),P,(Initialize (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s))))) . GBP),4) by A22, AMI_3:10, SCMP_GCD:1;
 abs (((IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))),P,(Initialize (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s))))) . (intpos 4)) + 0) = s . (intpos 4) by A1, A4, A19, ABSVALUE:def_1;
then  1 <> abs (((IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))),P,(Initialize (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s))))) . (intpos 4)) + 0) by A1, A4, XXREAL_0:2;
then A31: intpos 1 <> DataLoc (((IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))),P,(Initialize (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s))))) . (intpos 4)),0) by XTUPLE_0:1;
 1 <> abs (((Exec (((GBP,5) := ((intpos 4),(- 1))),(Initialize (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s))))) . (intpos 4)) + (- 1)) by A10, A15, A17, XXREAL_0:2;
then A32: intpos 1 <> DataLoc (((Exec (((GBP,5) := ((intpos 4),(- 1))),(Initialize (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s))))) . (intpos 4)),(- 1)) by XTUPLE_0:1;
A33: (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))),P,(Initialize (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s))))) . (intpos 1) = (Exec ((((intpos 4),(- 1)) := ((intpos 4),0)),(Exec (((GBP,5) := ((intpos 4),(- 1))),(Initialize (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s))))))) . (intpos 1) by SCMPDS_5:42
.= s . (intpos 1) by A26, A32, SCMPDS_2:47 ;
A34: (IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))),P,(Initialize (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s))))) . (intpos 1) = (Exec ((((intpos 4),0) := (GBP,5)),(IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))),P,(Initialize (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s))))))) . (intpos 1) by SCMPDS_5:41
.= s . (intpos 1) by A33, A31, SCMPDS_2:47 ;
A35: (IExec ((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))),P,(Initialize (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s))))) . (intpos 1) = (Exec ((AddTo (GBP,4,(- 1))),(IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))),P,(Initialize (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s))))))) . (intpos 1) by SCMPDS_5:41
.= s . (intpos 1) by A34, A30, SCMPDS_2:48 ;
now__::_thesis:_(IExec_((if>0_(GBP,5,((((((GBP,5)_:=_((intpos_4),(-_1)))_';'_(((intpos_4),(-_1))_:=_((intpos_4),0)))_';'_(((intpos_4),0)_:=_(GBP,5)))_';'_(AddTo_(GBP,4,(-_1))))_';'_(AddTo_(GBP,6,(-_1)))),(Load_((GBP,6)_:=_0)))),P,(Initialize_(IExec_((((GBP,5)_:=_((intpos_4),(-_1)))_';'_(SubFrom_(GBP,5,(intpos_4),0))),P,s)))))_._(intpos_1)_=_s_._(intpos_1)
percases ( (Initialize (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s))) . (DataLoc (((Initialize (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s))) . GBP),5)) <= 0 or (Initialize (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s))) . (DataLoc (((Initialize (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s))) . GBP),5)) > 0 ) ;
suppose (Initialize (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s))) . (DataLoc (((Initialize (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s))) . GBP),5)) <= 0 ; ::_thesis: (IExec ((if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0)))),P,(Initialize (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s))))) . (intpos 1) = s . (intpos 1)
hence (IExec ((if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0)))),P,(Initialize (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s))))) . (intpos 1) = (IExec ((Load ((GBP,6) := 0)),P,(Initialize (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s))))) . (intpos 1) by SCMPDS_6:74
.= (Exec (((GBP,6) := 0),(Initialize (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s))))) . (intpos 1) by SCMPDS_5:40
.= s . (intpos 1) by A25, A29, AMI_3:10, SCMPDS_2:46 ;
::_thesis: verum
end;
suppose (Initialize (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s))) . (DataLoc (((Initialize (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s))) . GBP),5)) > 0 ; ::_thesis: (IExec ((if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0)))),P,(Initialize (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s))))) . (intpos 1) = s . (intpos 1)
hence (IExec ((if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0)))),P,(Initialize (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s))))) . (intpos 1) = (IExec (((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),P,(Initialize (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s))))) . (intpos 1) by SCMPDS_6:73
.= (Exec ((AddTo (GBP,6,(- 1))),(IExec ((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))),P,(Initialize (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s))))))) . (intpos 1) by SCMPDS_5:41
.= s . (intpos 1) by A35, A28, AMI_3:10, SCMPDS_2:48 ;
::_thesis: verum
end;
end;
end;
hence  (IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))),P,s)) . (intpos 1) = s . (intpos 1) by SCMPDS_5:35; ::_thesis: verum
end;

Lm7:  for P being Instruction-Sequence of SCMPDS
for s being 0 -started State of SCMPDS st s . (intpos 4) >= 7 + (s . (intpos 6)) & s . GBP = 0 & s . (intpos 6) > 0 holds
( (IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))),P,s)) . (intpos 2) = s . (intpos 2) & (IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))),P,s)) . (intpos 3) = s . (intpos 3) & (IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))),P,s)) . (intpos 6) < s . (intpos 6) & (IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))),P,s)) . (intpos 4) >= 7 + ((IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))),P,s)) . (intpos 6)) & ( for i being Nat st i >= 7 & i <> (s . (intpos 4)) - 1 & i <> s . (intpos 4) holds
(IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))),P,s)) . (intpos i) = s . (intpos i) ) & ( s . (DataLoc ((s . (intpos 4)),(- 1))) > s . (DataLoc ((s . (intpos 4)),0)) implies ( (IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))),P,s)) . (DataLoc ((s . (intpos 4)),(- 1))) = s . (DataLoc ((s . (intpos 4)),0)) & (IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))),P,s)) . (DataLoc ((s . (intpos 4)),0)) = s . (DataLoc ((s . (intpos 4)),(- 1))) & (IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))),P,s)) . (intpos 6) = (s . (intpos 6)) - 1 & (IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))),P,s)) . (intpos 4) = (s . (intpos 4)) - 1 ) ) & ( s . (DataLoc ((s . (intpos 4)),(- 1))) <= s . (DataLoc ((s . (intpos 4)),0)) implies ( (IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))),P,s)) . (DataLoc ((s . (intpos 4)),(- 1))) = s . (DataLoc ((s . (intpos 4)),(- 1))) & (IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))),P,s)) . (DataLoc ((s . (intpos 4)),0)) = s . (DataLoc ((s . (intpos 4)),0)) & (IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))),P,s)) . (intpos 6) = 0 ) ) )
proof
let P be Instruction-Sequence of SCMPDS; ::_thesis: for s being 0 -started State of SCMPDS st s . (intpos 4) >= 7 + (s . (intpos 6)) & s . GBP = 0 & s . (intpos 6) > 0 holds
( (IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))),P,s)) . (intpos 2) = s . (intpos 2) & (IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))),P,s)) . (intpos 3) = s . (intpos 3) & (IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))),P,s)) . (intpos 6) < s . (intpos 6) & (IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))),P,s)) . (intpos 4) >= 7 + ((IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))),P,s)) . (intpos 6)) & ( for i being Nat st i >= 7 & i <> (s . (intpos 4)) - 1 & i <> s . (intpos 4) holds
(IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))),P,s)) . (intpos i) = s . (intpos i) ) & ( s . (DataLoc ((s . (intpos 4)),(- 1))) > s . (DataLoc ((s . (intpos 4)),0)) implies ( (IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))),P,s)) . (DataLoc ((s . (intpos 4)),(- 1))) = s . (DataLoc ((s . (intpos 4)),0)) & (IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))),P,s)) . (DataLoc ((s . (intpos 4)),0)) = s . (DataLoc ((s . (intpos 4)),(- 1))) & (IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))),P,s)) . (intpos 6) = (s . (intpos 6)) - 1 & (IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))),P,s)) . (intpos 4) = (s . (intpos 4)) - 1 ) ) & ( s . (DataLoc ((s . (intpos 4)),(- 1))) <= s . (DataLoc ((s . (intpos 4)),0)) implies ( (IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))),P,s)) . (DataLoc ((s . (intpos 4)),(- 1))) = s . (DataLoc ((s . (intpos 4)),(- 1))) & (IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))),P,s)) . (DataLoc ((s . (intpos 4)),0)) = s . (DataLoc ((s . (intpos 4)),0)) & (IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))),P,s)) . (intpos 6) = 0 ) ) )
let s be 0 -started State of SCMPDS; ::_thesis: ( s . (intpos 4) >= 7 + (s . (intpos 6)) & s . GBP = 0 & s . (intpos 6) > 0 implies ( (IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))),P,s)) . (intpos 2) = s . (intpos 2) & (IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))),P,s)) . (intpos 3) = s . (intpos 3) & (IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))),P,s)) . (intpos 6) < s . (intpos 6) & (IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))),P,s)) . (intpos 4) >= 7 + ((IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))),P,s)) . (intpos 6)) & ( for i being Nat st i >= 7 & i <> (s . (intpos 4)) - 1 & i <> s . (intpos 4) holds
(IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))),P,s)) . (intpos i) = s . (intpos i) ) & ( s . (DataLoc ((s . (intpos 4)),(- 1))) > s . (DataLoc ((s . (intpos 4)),0)) implies ( (IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))),P,s)) . (DataLoc ((s . (intpos 4)),(- 1))) = s . (DataLoc ((s . (intpos 4)),0)) & (IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))),P,s)) . (DataLoc ((s . (intpos 4)),0)) = s . (DataLoc ((s . (intpos 4)),(- 1))) & (IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))),P,s)) . (intpos 6) = (s . (intpos 6)) - 1 & (IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))),P,s)) . (intpos 4) = (s . (intpos 4)) - 1 ) ) & ( s . (DataLoc ((s . (intpos 4)),(- 1))) <= s . (DataLoc ((s . (intpos 4)),0)) implies ( (IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))),P,s)) . (DataLoc ((s . (intpos 4)),(- 1))) = s . (DataLoc ((s . (intpos 4)),(- 1))) & (IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))),P,s)) . (DataLoc ((s . (intpos 4)),0)) = s . (DataLoc ((s . (intpos 4)),0)) & (IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))),P,s)) . (intpos 6) = 0 ) ) ) )
set a = GBP ;
set x = DataLoc ((s . (intpos 4)),(- 1));
set y = DataLoc ((s . (intpos 4)),0);
assume that
A1: s . (intpos 4) >= 7 + (s . (intpos 6)) and
A2: s . GBP = 0 and
A3: s . (intpos 6) > 0 ; ::_thesis: ( (IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))),P,s)) . (intpos 2) = s . (intpos 2) & (IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))),P,s)) . (intpos 3) = s . (intpos 3) & (IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))),P,s)) . (intpos 6) < s . (intpos 6) & (IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))),P,s)) . (intpos 4) >= 7 + ((IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))),P,s)) . (intpos 6)) & ( for i being Nat st i >= 7 & i <> (s . (intpos 4)) - 1 & i <> s . (intpos 4) holds
(IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))),P,s)) . (intpos i) = s . (intpos i) ) & ( s . (DataLoc ((s . (intpos 4)),(- 1))) > s . (DataLoc ((s . (intpos 4)),0)) implies ( (IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))),P,s)) . (DataLoc ((s . (intpos 4)),(- 1))) = s . (DataLoc ((s . (intpos 4)),0)) & (IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))),P,s)) . (DataLoc ((s . (intpos 4)),0)) = s . (DataLoc ((s . (intpos 4)),(- 1))) & (IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))),P,s)) . (intpos 6) = (s . (intpos 6)) - 1 & (IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))),P,s)) . (intpos 4) = (s . (intpos 4)) - 1 ) ) & ( s . (DataLoc ((s . (intpos 4)),(- 1))) <= s . (DataLoc ((s . (intpos 4)),0)) implies ( (IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))),P,s)) . (DataLoc ((s . (intpos 4)),(- 1))) = s . (DataLoc ((s . (intpos 4)),(- 1))) & (IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))),P,s)) . (DataLoc ((s . (intpos 4)),0)) = s . (DataLoc ((s . (intpos 4)),0)) & (IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))),P,s)) . (intpos 6) = 0 ) ) )
set t0 = s;
set t1 = Exec (((GBP,5) := ((intpos 4),(- 1))),s);
set t2 = IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s);
set Q2 = P;
A4: 7 + (s . (intpos 6)) > 7 + 0 by A3, XREAL_1:6;
then A5: abs (s . (intpos 4)) = s . (intpos 4) by A1, ABSVALUE:def_1;
set Fi = (GBP,6) := 0;
set t02 = Initialize (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s));
set Q02 = P;
set t3 = IExec ((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))),P,(Initialize (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s))));
set t4 = IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))),P,(Initialize (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s))));
set t5 = IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))),P,(Initialize (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s))));
set t6 = Exec (((GBP,5) := ((intpos 4),(- 1))),(Initialize (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s))));
A6: DataLoc ((s . GBP),5) = intpos (0 + 5) by A2, SCMP_GCD:1;
then A7: (Exec (((GBP,5) := ((intpos 4),(- 1))),s)) . GBP = 0 by A2, AMI_3:10, SCMPDS_2:47;
then A8: DataLoc (((Exec (((GBP,5) := ((intpos 4),(- 1))),s)) . GBP),5) = intpos (0 + 5) by SCMP_GCD:1;
then A9: abs (((Exec (((GBP,5) := ((intpos 4),(- 1))),s)) . GBP) + 5) = 0 + 5 by XTUPLE_0:1;
then  abs ((s . (intpos 4)) + 0) <> abs (((Exec (((GBP,5) := ((intpos 4),(- 1))),s)) . GBP) + 5) by A1, A4, A5, XXREAL_0:2;
then A10: DataLoc ((s . (intpos 4)),0) <> DataLoc (((Exec (((GBP,5) := ((intpos 4),(- 1))),s)) . GBP),5) by XTUPLE_0:1;
A11: abs ((s . GBP) + 5) = 0 + 5 by A6, XTUPLE_0:1;
then  abs ((s . (intpos 4)) + 0) <> abs ((s . GBP) + 5) by A1, A4, A5, XXREAL_0:2;
then  ( s . (DataLoc ((s . (intpos 4)),0)) = s . (DataLoc ((s . (intpos 4)),0)) & DataLoc ((s . (intpos 4)),0) <> DataLoc ((s . GBP),5) ) by XTUPLE_0:1;
then A12: (Exec (((GBP,5) := ((intpos 4),(- 1))),s)) . (DataLoc ((s . (intpos 4)),0)) = s . (DataLoc ((s . (intpos 4)),0)) by SCMPDS_2:47;
A13: (Exec (((GBP,5) := ((intpos 4),(- 1))),s)) . (intpos 5) = s . (DataLoc ((s . (intpos 4)),(- 1))) by A6, SCMPDS_2:47;
A14: (Exec (((GBP,5) := ((intpos 4),(- 1))),s)) . (intpos 4) = s . (intpos 4) by A6, AMI_3:10, SCMPDS_2:47;
(IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s)) . (DataLoc ((s . (intpos 4)),0)) = (Exec ((SubFrom (GBP,5,(intpos 4),0)),(Exec (((GBP,5) := ((intpos 4),(- 1))),s)))) . (DataLoc ((s . (intpos 4)),0)) by SCMPDS_5:42
.= s . (DataLoc ((s . (intpos 4)),0)) by A12, A10, SCMPDS_2:50 ;
then A15: (Initialize (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s))) . (DataLoc ((s . (intpos 4)),0)) = s . (DataLoc ((s . (intpos 4)),0)) by SCMPDS_5:15;
A16: (Initialize (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s))) . GBP = (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s)) . GBP by SCMPDS_5:15
.= (Exec ((SubFrom (GBP,5,(intpos 4),0)),(Exec (((GBP,5) := ((intpos 4),(- 1))),s)))) . GBP by SCMPDS_5:42
.= 0 by A7, A8, AMI_3:10, SCMPDS_2:50 ;
A17: DataLoc (((Initialize (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s))) . GBP),5) = intpos (0 + 5) by A16, SCMP_GCD:1;
then A18: (Exec (((GBP,5) := ((intpos 4),(- 1))),(Initialize (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s))))) . GBP = 0 by A16, AMI_3:10, SCMPDS_2:47;
 abs (((Initialize (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s))) . GBP) + 5) = 0 + 5 by A17, XTUPLE_0:1;
then  abs ((s . (intpos 4)) + 0) <> abs (((Initialize (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s))) . GBP) + 5) by A1, A4, A5, XXREAL_0:2;
then  DataLoc ((s . (intpos 4)),0) <> DataLoc (((Initialize (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s))) . GBP),5) by XTUPLE_0:1;
then A19: (Exec (((GBP,5) := ((intpos 4),(- 1))),(Initialize (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s))))) . (DataLoc ((s . (intpos 4)),0)) = s . (DataLoc ((s . (intpos 4)),0)) by A15, SCMPDS_2:47;
(IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s)) . (intpos 4) = (Exec ((SubFrom (GBP,5,(intpos 4),0)),(Exec (((GBP,5) := ((intpos 4),(- 1))),s)))) . (intpos 4) by SCMPDS_5:42
.= s . (intpos 4) by A14, A8, AMI_3:10, SCMPDS_2:50 ;
then A20: (Initialize (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s))) . (intpos 4) = s . (intpos 4) by SCMPDS_5:15;
then A21: (Exec (((GBP,5) := ((intpos 4),(- 1))),(Initialize (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s))))) . (intpos 4) = s . (intpos 4) by A17, AMI_3:10, SCMPDS_2:47;
then A22: (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))),P,(Initialize (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s))))) . (DataLoc ((s . (intpos 4)),(- 1))) = (Exec ((((intpos 4),(- 1)) := ((intpos 4),0)),(Exec (((GBP,5) := ((intpos 4),(- 1))),(Initialize (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s))))))) . (DataLoc (((Exec (((GBP,5) := ((intpos 4),(- 1))),(Initialize (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s))))) . (intpos 4)),(- 1))) by SCMPDS_5:42
.= s . (DataLoc ((s . (intpos 4)),0)) by A21, A19, SCMPDS_2:47 ;
A23: (Exec (((GBP,5) := ((intpos 4),(- 1))),s)) . (intpos 3) = s . (intpos 3) by A6, AMI_3:10, SCMPDS_2:47;
(IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s)) . (intpos 3) = (Exec ((SubFrom (GBP,5,(intpos 4),0)),(Exec (((GBP,5) := ((intpos 4),(- 1))),s)))) . (intpos 3) by SCMPDS_5:42
.= s . (intpos 3) by A23, A8, AMI_3:10, SCMPDS_2:50 ;
then A24: (Initialize (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s))) . (intpos 3) = s . (intpos 3) by SCMPDS_5:15;
then A25: (Exec (((GBP,5) := ((intpos 4),(- 1))),(Initialize (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s))))) . (intpos 3) = s . (intpos 3) by A17, AMI_3:10, SCMPDS_2:47;
A26: s . (intpos 4) >= 1 + (6 + (s . (intpos 6))) by A1;
then A27: (s . (intpos 4)) - 1 >= 6 + (s . (intpos 6)) by XREAL_1:19;
A28: (Exec (((GBP,5) := ((intpos 4),(- 1))),s)) . (intpos 2) = s . (intpos 2) by A6, AMI_3:10, SCMPDS_2:47;
(IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s)) . (intpos 2) = (Exec ((SubFrom (GBP,5,(intpos 4),0)),(Exec (((GBP,5) := ((intpos 4),(- 1))),s)))) . (intpos 2) by SCMPDS_5:42
.= s . (intpos 2) by A28, A8, AMI_3:10, SCMPDS_2:50 ;
then A29: (Initialize (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s))) . (intpos 2) = s . (intpos 2) by SCMPDS_5:15;
then A30: (Exec (((GBP,5) := ((intpos 4),(- 1))),(Initialize (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s))))) . (intpos 2) = s . (intpos 2) by A17, AMI_3:10, SCMPDS_2:47;
A31: (Exec (((GBP,5) := ((intpos 4),(- 1))),s)) . (intpos 6) = s . (intpos 6) by A6, AMI_3:10, SCMPDS_2:47;
(IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s)) . (intpos 6) = (Exec ((SubFrom (GBP,5,(intpos 4),0)),(Exec (((GBP,5) := ((intpos 4),(- 1))),s)))) . (intpos 6) by SCMPDS_5:42
.= s . (intpos 6) by A31, A8, AMI_3:10, SCMPDS_2:50 ;
then  (Initialize (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s))) . (intpos 6) = s . (intpos 6) by SCMPDS_5:15;
then A32: (Exec (((GBP,5) := ((intpos 4),(- 1))),(Initialize (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s))))) . (intpos 6) = s . (intpos 6) by A17, AMI_3:10, SCMPDS_2:47;
A33: DataLoc (((Initialize (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s))) . GBP),6) = intpos (0 + 6) by A16, SCMP_GCD:1;
A34: now__::_thesis:_(_(Initialize_(IExec_((((GBP,5)_:=_((intpos_4),(-_1)))_';'_(SubFrom_(GBP,5,(intpos_4),0))),P,s)))_._(DataLoc_(((Initialize_(IExec_((((GBP,5)_:=_((intpos_4),(-_1)))_';'_(SubFrom_(GBP,5,(intpos_4),0))),P,s)))_._GBP),5))_<=_0_implies_(IExec_(((((GBP,5)_:=_((intpos_4),(-_1)))_';'_(SubFrom_(GBP,5,(intpos_4),0)))_';'_(if>0_(GBP,5,((((((GBP,5)_:=_((intpos_4),(-_1)))_';'_(((intpos_4),(-_1))_:=_((intpos_4),0)))_';'_(((intpos_4),0)_:=_(GBP,5)))_';'_(AddTo_(GBP,4,(-_1))))_';'_(AddTo_(GBP,6,(-_1)))),(Load_((GBP,6)_:=_0))))),P,s))_._(intpos_6)_=_0_)
assume  (Initialize (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s))) . (DataLoc (((Initialize (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s))) . GBP),5)) <= 0 ; ::_thesis: (IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))),P,s)) . (intpos 6) = 0
then (IExec ((if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0)))),P,(Initialize (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s))))) . (intpos 6) = (IExec ((Load ((GBP,6) := 0)),P,(Initialize (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s))))) . (intpos 6) by SCMPDS_6:74
.= (Exec (((GBP,6) := 0),(Initialize (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s))))) . (intpos 6) by SCMPDS_5:40
.= 0 by A33, SCMPDS_2:46 ;
hence  (IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))),P,s)) . (intpos 6) = 0 by SCMPDS_5:35; ::_thesis: verum
end;
A35: 6 + (s . (intpos 6)) > 6 + 0 by A3, XREAL_1:6;
then  0 <> abs (((Exec (((GBP,5) := ((intpos 4),(- 1))),(Initialize (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s))))) . (intpos 4)) + (- 1)) by A27, A21, ABSVALUE:def_1;
then A36: GBP <> DataLoc (((Exec (((GBP,5) := ((intpos 4),(- 1))),(Initialize (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s))))) . (intpos 4)),(- 1)) by XTUPLE_0:1;
A37: (s . (intpos 4)) - 1 > 0 by A3, A26, XREAL_1:19;
then A38: abs (((Exec (((GBP,5) := ((intpos 4),(- 1))),(Initialize (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s))))) . (intpos 4)) + (- 1)) = (s . (intpos 4)) - 1 by A21, ABSVALUE:def_1;
then  4 <> abs (((Exec (((GBP,5) := ((intpos 4),(- 1))),(Initialize (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s))))) . (intpos 4)) + (- 1)) by A27, A35, XXREAL_0:2;
then A39: intpos 4 <> DataLoc (((Exec (((GBP,5) := ((intpos 4),(- 1))),(Initialize (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s))))) . (intpos 4)),(- 1)) by XTUPLE_0:1;
A40: (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))),P,(Initialize (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s))))) . (intpos 4) = (Exec ((((intpos 4),(- 1)) := ((intpos 4),0)),(Exec (((GBP,5) := ((intpos 4),(- 1))),(Initialize (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s))))))) . (intpos 4) by SCMPDS_5:42
.= s . (intpos 4) by A21, A39, SCMPDS_2:47 ;
then A41: abs (((IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))),P,(Initialize (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s))))) . (intpos 4)) + 0) = s . (intpos 4) by A1, A4, ABSVALUE:def_1;
then  4 <> abs (((IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))),P,(Initialize (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s))))) . (intpos 4)) + 0) by A1, A4, XXREAL_0:2;
then A42: intpos 4 <> DataLoc (((IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))),P,(Initialize (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s))))) . (intpos 4)),0) by XTUPLE_0:1;
 3 <> abs (((Exec (((GBP,5) := ((intpos 4),(- 1))),(Initialize (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s))))) . (intpos 4)) + (- 1)) by A27, A35, A38, XXREAL_0:2;
then A43: intpos 3 <> DataLoc (((Exec (((GBP,5) := ((intpos 4),(- 1))),(Initialize (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s))))) . (intpos 4)),(- 1)) by XTUPLE_0:1;
 3 <> abs (((IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))),P,(Initialize (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s))))) . (intpos 4)) + 0) by A1, A4, A41, XXREAL_0:2;
then A44: intpos 3 <> DataLoc (((IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))),P,(Initialize (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s))))) . (intpos 4)),0) by XTUPLE_0:1;
A45: (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))),P,(Initialize (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s))))) . (intpos 3) = (Exec ((((intpos 4),(- 1)) := ((intpos 4),0)),(Exec (((GBP,5) := ((intpos 4),(- 1))),(Initialize (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s))))))) . (intpos 3) by SCMPDS_5:42
.= s . (intpos 3) by A25, A43, SCMPDS_2:47 ;
A46: (IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))),P,(Initialize (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s))))) . (intpos 3) = (Exec ((((intpos 4),0) := (GBP,5)),(IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))),P,(Initialize (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s))))))) . (intpos 3) by SCMPDS_5:41
.= s . (intpos 3) by A45, A44, SCMPDS_2:47 ;
 2 <> abs (((IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))),P,(Initialize (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s))))) . (intpos 4)) + 0) by A1, A4, A41, XXREAL_0:2;
then A47: intpos 2 <> DataLoc (((IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))),P,(Initialize (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s))))) . (intpos 4)),0) by XTUPLE_0:1;
A48: (IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))),P,(Initialize (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s))))) . (intpos 4) = (Exec ((((intpos 4),0) := (GBP,5)),(IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))),P,(Initialize (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s))))))) . (intpos 4) by SCMPDS_5:41
.= s . (intpos 4) by A40, A42, SCMPDS_2:47 ;
A49: (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))),P,(Initialize (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s))))) . GBP = (Exec ((((intpos 4),(- 1)) := ((intpos 4),0)),(Exec (((GBP,5) := ((intpos 4),(- 1))),(Initialize (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s))))))) . GBP by SCMPDS_5:42
.= 0 by A18, A36, SCMPDS_2:47 ;
A50: (2 * (abs ((s . (intpos 4)) + (- 1)))) + 1 = (2 * ((s . (intpos 4)) - 1)) + 1 by A27, A35, ABSVALUE:def_1;
then  abs ((s . (intpos 4)) + (- 1)) <> abs (((Exec (((GBP,5) := ((intpos 4),(- 1))),s)) . GBP) + 5) by A3, A27, A9, XREAL_1:6;
then A51: DataLoc ((s . (intpos 4)),(- 1)) <> DataLoc (((Exec (((GBP,5) := ((intpos 4),(- 1))),s)) . GBP),5) by XTUPLE_0:1;
 0 <> abs (((IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))),P,(Initialize (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s))))) . (intpos 4)) + 0) by A1, A4, A40, ABSVALUE:def_1;
then A52: GBP <> DataLoc (((IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))),P,(Initialize (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s))))) . (intpos 4)),0) by XTUPLE_0:1;
A53: (IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))),P,(Initialize (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s))))) . GBP = (Exec ((((intpos 4),0) := (GBP,5)),(IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))),P,(Initialize (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s))))))) . GBP by SCMPDS_5:41
.= 0 by A49, A52, SCMPDS_2:47 ;
then A54: GBP <> DataLoc (((IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))),P,(Initialize (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s))))) . GBP),4) by AMI_3:10, SCMP_GCD:1;
 2 <> abs (((Exec (((GBP,5) := ((intpos 4),(- 1))),(Initialize (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s))))) . (intpos 4)) + (- 1)) by A27, A35, A38, XXREAL_0:2;
then A55: intpos 2 <> DataLoc (((Exec (((GBP,5) := ((intpos 4),(- 1))),(Initialize (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s))))) . (intpos 4)),(- 1)) by XTUPLE_0:1;
A56: (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))),P,(Initialize (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s))))) . (intpos 2) = (Exec ((((intpos 4),(- 1)) := ((intpos 4),0)),(Exec (((GBP,5) := ((intpos 4),(- 1))),(Initialize (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s))))))) . (intpos 2) by SCMPDS_5:42
.= s . (intpos 2) by A30, A55, SCMPDS_2:47 ;
A57: (IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))),P,(Initialize (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s))))) . (intpos 2) = (Exec ((((intpos 4),0) := (GBP,5)),(IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))),P,(Initialize (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s))))))) . (intpos 2) by SCMPDS_5:41
.= s . (intpos 2) by A56, A47, SCMPDS_2:47 ;
A58: intpos 2 <> DataLoc (((IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))),P,(Initialize (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s))))) . GBP),4) by A53, AMI_3:10, SCMP_GCD:1;
(IExec ((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))),P,(Initialize (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s))))) . GBP = (Exec ((AddTo (GBP,4,(- 1))),(IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))),P,(Initialize (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s))))))) . GBP by SCMPDS_5:41
.= 0 by A53, A54, SCMPDS_2:48 ;
then A59: DataLoc (((IExec ((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))),P,(Initialize (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s))))) . GBP),6) = intpos (0 + 6) by SCMP_GCD:1;
then A60: abs (((IExec ((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))),P,(Initialize (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s))))) . GBP) + 6) = 0 + 6 by XTUPLE_0:1;
A61: DataLoc (((IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))),P,(Initialize (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s))))) . GBP),4) = intpos (0 + 4) by A53, SCMP_GCD:1;
then A62: abs (((IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))),P,(Initialize (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s))))) . GBP) + 4) = 0 + 4 by XTUPLE_0:1;
then  abs ((s . (intpos 4)) + (- 1)) <> abs (((IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))),P,(Initialize (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s))))) . GBP) + 4) by A27, A35, A50, XXREAL_0:2;
then A63: DataLoc ((s . (intpos 4)),(- 1)) <> DataLoc (((IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))),P,(Initialize (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s))))) . GBP),4) by XTUPLE_0:1;
A64: (IExec ((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))),P,(Initialize (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s))))) . (intpos 2) = (Exec ((AddTo (GBP,4,(- 1))),(IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))),P,(Initialize (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s))))))) . (intpos 2) by SCMPDS_5:41
.= s . (intpos 2) by A57, A58, SCMPDS_2:48 ;
now__::_thesis:_(IExec_((if>0_(GBP,5,((((((GBP,5)_:=_((intpos_4),(-_1)))_';'_(((intpos_4),(-_1))_:=_((intpos_4),0)))_';'_(((intpos_4),0)_:=_(GBP,5)))_';'_(AddTo_(GBP,4,(-_1))))_';'_(AddTo_(GBP,6,(-_1)))),(Load_((GBP,6)_:=_0)))),P,(Initialize_(IExec_((((GBP,5)_:=_((intpos_4),(-_1)))_';'_(SubFrom_(GBP,5,(intpos_4),0))),P,s)))))_._(intpos_2)_=_s_._(intpos_2)
percases ( (Initialize (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s))) . (DataLoc (((Initialize (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s))) . GBP),5)) <= 0 or (Initialize (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s))) . (DataLoc (((Initialize (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s))) . GBP),5)) > 0 ) ;
suppose (Initialize (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s))) . (DataLoc (((Initialize (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s))) . GBP),5)) <= 0 ; ::_thesis: (IExec ((if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0)))),P,(Initialize (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s))))) . (intpos 2) = s . (intpos 2)
hence (IExec ((if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0)))),P,(Initialize (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s))))) . (intpos 2) = (IExec ((Load ((GBP,6) := 0)),P,(Initialize (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s))))) . (intpos 2) by SCMPDS_6:74
.= (Exec (((GBP,6) := 0),(Initialize (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s))))) . (intpos 2) by SCMPDS_5:40
.= s . (intpos 2) by A29, A33, AMI_3:10, SCMPDS_2:46 ;
::_thesis: verum
end;
suppose (Initialize (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s))) . (DataLoc (((Initialize (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s))) . GBP),5)) > 0 ; ::_thesis: (IExec ((if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0)))),P,(Initialize (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s))))) . (intpos 2) = s . (intpos 2)
hence (IExec ((if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0)))),P,(Initialize (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s))))) . (intpos 2) = (IExec (((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),P,(Initialize (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s))))) . (intpos 2) by SCMPDS_6:73
.= (Exec ((AddTo (GBP,6,(- 1))),(IExec ((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))),P,(Initialize (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s))))))) . (intpos 2) by SCMPDS_5:41
.= s . (intpos 2) by A64, A59, AMI_3:10, SCMPDS_2:48 ;
::_thesis: verum
end;
end;
end;
hence  (IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))),P,s)) . (intpos 2) = s . (intpos 2) by SCMPDS_5:35; ::_thesis: ( (IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))),P,s)) . (intpos 3) = s . (intpos 3) & (IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))),P,s)) . (intpos 6) < s . (intpos 6) & (IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))),P,s)) . (intpos 4) >= 7 + ((IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))),P,s)) . (intpos 6)) & ( for i being Nat st i >= 7 & i <> (s . (intpos 4)) - 1 & i <> s . (intpos 4) holds
(IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))),P,s)) . (intpos i) = s . (intpos i) ) & ( s . (DataLoc ((s . (intpos 4)),(- 1))) > s . (DataLoc ((s . (intpos 4)),0)) implies ( (IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))),P,s)) . (DataLoc ((s . (intpos 4)),(- 1))) = s . (DataLoc ((s . (intpos 4)),0)) & (IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))),P,s)) . (DataLoc ((s . (intpos 4)),0)) = s . (DataLoc ((s . (intpos 4)),(- 1))) & (IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))),P,s)) . (intpos 6) = (s . (intpos 6)) - 1 & (IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))),P,s)) . (intpos 4) = (s . (intpos 4)) - 1 ) ) & ( s . (DataLoc ((s . (intpos 4)),(- 1))) <= s . (DataLoc ((s . (intpos 4)),0)) implies ( (IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))),P,s)) . (DataLoc ((s . (intpos 4)),(- 1))) = s . (DataLoc ((s . (intpos 4)),(- 1))) & (IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))),P,s)) . (DataLoc ((s . (intpos 4)),0)) = s . (DataLoc ((s . (intpos 4)),0)) & (IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))),P,s)) . (intpos 6) = 0 ) ) )
A65: intpos 3 <> DataLoc (((IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))),P,(Initialize (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s))))) . GBP),4) by A53, AMI_3:10, SCMP_GCD:1;
A66: (IExec ((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))),P,(Initialize (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s))))) . (intpos 3) = (Exec ((AddTo (GBP,4,(- 1))),(IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))),P,(Initialize (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s))))))) . (intpos 3) by SCMPDS_5:41
.= s . (intpos 3) by A46, A65, SCMPDS_2:48 ;
now__::_thesis:_(IExec_((if>0_(GBP,5,((((((GBP,5)_:=_((intpos_4),(-_1)))_';'_(((intpos_4),(-_1))_:=_((intpos_4),0)))_';'_(((intpos_4),0)_:=_(GBP,5)))_';'_(AddTo_(GBP,4,(-_1))))_';'_(AddTo_(GBP,6,(-_1)))),(Load_((GBP,6)_:=_0)))),P,(Initialize_(IExec_((((GBP,5)_:=_((intpos_4),(-_1)))_';'_(SubFrom_(GBP,5,(intpos_4),0))),P,s)))))_._(intpos_3)_=_s_._(intpos_3)
percases ( (Initialize (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s))) . (DataLoc (((Initialize (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s))) . GBP),5)) <= 0 or (Initialize (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s))) . (DataLoc (((Initialize (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s))) . GBP),5)) > 0 ) ;
suppose (Initialize (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s))) . (DataLoc (((Initialize (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s))) . GBP),5)) <= 0 ; ::_thesis: (IExec ((if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0)))),P,(Initialize (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s))))) . (intpos 3) = s . (intpos 3)
hence (IExec ((if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0)))),P,(Initialize (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s))))) . (intpos 3) = (IExec ((Load ((GBP,6) := 0)),P,(Initialize (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s))))) . (intpos 3) by SCMPDS_6:74
.= (Exec (((GBP,6) := 0),(Initialize (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s))))) . (intpos 3) by SCMPDS_5:40
.= s . (intpos 3) by A24, A33, AMI_3:10, SCMPDS_2:46 ;
::_thesis: verum
end;
suppose (Initialize (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s))) . (DataLoc (((Initialize (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s))) . GBP),5)) > 0 ; ::_thesis: (IExec ((if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0)))),P,(Initialize (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s))))) . (intpos 3) = s . (intpos 3)
hence (IExec ((if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0)))),P,(Initialize (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s))))) . (intpos 3) = (IExec (((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),P,(Initialize (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s))))) . (intpos 3) by SCMPDS_6:73
.= (Exec ((AddTo (GBP,6,(- 1))),(IExec ((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))),P,(Initialize (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s))))))) . (intpos 3) by SCMPDS_5:41
.= s . (intpos 3) by A66, A59, AMI_3:10, SCMPDS_2:48 ;
::_thesis: verum
end;
end;
end;
hence  (IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))),P,s)) . (intpos 3) = s . (intpos 3) by SCMPDS_5:35; ::_thesis: ( (IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))),P,s)) . (intpos 6) < s . (intpos 6) & (IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))),P,s)) . (intpos 4) >= 7 + ((IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))),P,s)) . (intpos 6)) & ( for i being Nat st i >= 7 & i <> (s . (intpos 4)) - 1 & i <> s . (intpos 4) holds
(IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))),P,s)) . (intpos i) = s . (intpos i) ) & ( s . (DataLoc ((s . (intpos 4)),(- 1))) > s . (DataLoc ((s . (intpos 4)),0)) implies ( (IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))),P,s)) . (DataLoc ((s . (intpos 4)),(- 1))) = s . (DataLoc ((s . (intpos 4)),0)) & (IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))),P,s)) . (DataLoc ((s . (intpos 4)),0)) = s . (DataLoc ((s . (intpos 4)),(- 1))) & (IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))),P,s)) . (intpos 6) = (s . (intpos 6)) - 1 & (IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))),P,s)) . (intpos 4) = (s . (intpos 4)) - 1 ) ) & ( s . (DataLoc ((s . (intpos 4)),(- 1))) <= s . (DataLoc ((s . (intpos 4)),0)) implies ( (IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))),P,s)) . (DataLoc ((s . (intpos 4)),(- 1))) = s . (DataLoc ((s . (intpos 4)),(- 1))) & (IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))),P,s)) . (DataLoc ((s . (intpos 4)),0)) = s . (DataLoc ((s . (intpos 4)),0)) & (IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))),P,s)) . (intpos 6) = 0 ) ) )
A67: intpos 6 <> DataLoc (((IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))),P,(Initialize (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s))))) . GBP),4) by A53, AMI_3:10, SCMP_GCD:1;
 6 <> abs (((Exec (((GBP,5) := ((intpos 4),(- 1))),(Initialize (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s))))) . (intpos 4)) + (- 1)) by A26, A35, A38, XREAL_1:19;
then A68: intpos 6 <> DataLoc (((Exec (((GBP,5) := ((intpos 4),(- 1))),(Initialize (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s))))) . (intpos 4)),(- 1)) by XTUPLE_0:1;
A69: (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))),P,(Initialize (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s))))) . (intpos 6) = (Exec ((((intpos 4),(- 1)) := ((intpos 4),0)),(Exec (((GBP,5) := ((intpos 4),(- 1))),(Initialize (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s))))))) . (intpos 6) by SCMPDS_5:42
.= s . (intpos 6) by A32, A68, SCMPDS_2:47 ;
 6 <> abs (((IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))),P,(Initialize (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s))))) . (intpos 4)) + 0) by A1, A4, A41, XXREAL_0:2;
then A70: intpos 6 <> DataLoc (((IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))),P,(Initialize (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s))))) . (intpos 4)),0) by XTUPLE_0:1;
A71: (IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))),P,(Initialize (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s))))) . (intpos 6) = (Exec ((((intpos 4),0) := (GBP,5)),(IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))),P,(Initialize (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s))))))) . (intpos 6) by SCMPDS_5:41
.= s . (intpos 6) by A69, A70, SCMPDS_2:47 ;
A72: (IExec ((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))),P,(Initialize (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s))))) . (intpos 6) = (Exec ((AddTo (GBP,4,(- 1))),(IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))),P,(Initialize (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s))))))) . (intpos 6) by SCMPDS_5:41
.= s . (intpos 6) by A71, A67, SCMPDS_2:48 ;
A73: (IExec ((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))),P,(Initialize (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s))))) . (intpos 4) = (Exec ((AddTo (GBP,4,(- 1))),(IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))),P,(Initialize (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s))))))) . (intpos 4) by SCMPDS_5:41
.= ((IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))),P,(Initialize (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s))))) . (intpos 4)) + (- 1) by A61, SCMPDS_2:48
.= (s . (intpos 4)) - 1 by A48 ;
A74: now__::_thesis:_(_(Initialize_(IExec_((((GBP,5)_:=_((intpos_4),(-_1)))_';'_(SubFrom_(GBP,5,(intpos_4),0))),P,s)))_._(DataLoc_(((Initialize_(IExec_((((GBP,5)_:=_((intpos_4),(-_1)))_';'_(SubFrom_(GBP,5,(intpos_4),0))),P,s)))_._GBP),5))_>_0_implies_(_(IExec_(((((GBP,5)_:=_((intpos_4),(-_1)))_';'_(SubFrom_(GBP,5,(intpos_4),0)))_';'_(if>0_(GBP,5,((((((GBP,5)_:=_((intpos_4),(-_1)))_';'_(((intpos_4),(-_1))_:=_((intpos_4),0)))_';'_(((intpos_4),0)_:=_(GBP,5)))_';'_(AddTo_(GBP,4,(-_1))))_';'_(AddTo_(GBP,6,(-_1)))),(Load_((GBP,6)_:=_0))))),P,s))_._(intpos_6)_=_(s_._(intpos_6))_-_1_&_(IExec_(((((GBP,5)_:=_((intpos_4),(-_1)))_';'_(SubFrom_(GBP,5,(intpos_4),0)))_';'_(if>0_(GBP,5,((((((GBP,5)_:=_((intpos_4),(-_1)))_';'_(((intpos_4),(-_1))_:=_((intpos_4),0)))_';'_(((intpos_4),0)_:=_(GBP,5)))_';'_(AddTo_(GBP,4,(-_1))))_';'_(AddTo_(GBP,6,(-_1)))),(Load_((GBP,6)_:=_0))))),P,s))_._(intpos_4)_=_(s_._(intpos_4))_-_1_)_)
assume A75: (Initialize (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s))) . (DataLoc (((Initialize (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s))) . GBP),5)) > 0 ; ::_thesis: ( (IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))),P,s)) . (intpos 6) = (s . (intpos 6)) - 1 & (IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))),P,s)) . (intpos 4) = (s . (intpos 4)) - 1 )
then (IExec ((if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0)))),P,(Initialize (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s))))) . (intpos 6) = (IExec (((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),P,(Initialize (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s))))) . (intpos 6) by SCMPDS_6:73
.= (Exec ((AddTo (GBP,6,(- 1))),(IExec ((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))),P,(Initialize (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s))))))) . (intpos 6) by SCMPDS_5:41
.= (s . (intpos 6)) + (- 1) by A72, A59, SCMPDS_2:48
.= (s . (intpos 6)) - 1 ;
hence  (IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))),P,s)) . (intpos 6) = (s . (intpos 6)) - 1 by SCMPDS_5:35; ::_thesis: (IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))),P,s)) . (intpos 4) = (s . (intpos 4)) - 1
(IExec ((if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0)))),P,(Initialize (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s))))) . (intpos 4) = (IExec (((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),P,(Initialize (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s))))) . (intpos 4) by A75, SCMPDS_6:73
.= (Exec ((AddTo (GBP,6,(- 1))),(IExec ((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))),P,(Initialize (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s))))))) . (intpos 4) by SCMPDS_5:41
.= (s . (intpos 4)) - 1 by A73, A59, AMI_3:10, SCMPDS_2:48 ;
hence  (IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))),P,s)) . (intpos 4) = (s . (intpos 4)) - 1 by SCMPDS_5:35; ::_thesis: verum
end;
hereby  ::_thesis: ( (IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))),P,s)) . (intpos 4) >= 7 + ((IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))),P,s)) . (intpos 6)) & ( for i being Nat st i >= 7 & i <> (s . (intpos 4)) - 1 & i <> s . (intpos 4) holds
(IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))),P,s)) . (intpos i) = s . (intpos i) ) & ( s . (DataLoc ((s . (intpos 4)),(- 1))) > s . (DataLoc ((s . (intpos 4)),0)) implies ( (IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))),P,s)) . (DataLoc ((s . (intpos 4)),(- 1))) = s . (DataLoc ((s . (intpos 4)),0)) & (IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))),P,s)) . (DataLoc ((s . (intpos 4)),0)) = s . (DataLoc ((s . (intpos 4)),(- 1))) & (IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))),P,s)) . (intpos 6) = (s . (intpos 6)) - 1 & (IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))),P,s)) . (intpos 4) = (s . (intpos 4)) - 1 ) ) & ( s . (DataLoc ((s . (intpos 4)),(- 1))) <= s . (DataLoc ((s . (intpos 4)),0)) implies ( (IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))),P,s)) . (DataLoc ((s . (intpos 4)),(- 1))) = s . (DataLoc ((s . (intpos 4)),(- 1))) & (IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))),P,s)) . (DataLoc ((s . (intpos 4)),0)) = s . (DataLoc ((s . (intpos 4)),0)) & (IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))),P,s)) . (intpos 6) = 0 ) ) )
percases ( (Initialize (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s))) . (DataLoc (((Initialize (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s))) . GBP),5)) <= 0 or (Initialize (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s))) . (DataLoc (((Initialize (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s))) . GBP),5)) > 0 ) ;
suppose (Initialize (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s))) . (DataLoc (((Initialize (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s))) . GBP),5)) <= 0 ; ::_thesis: (IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))),P,s)) . (intpos 6) < s . (intpos 6)
hence  (IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))),P,s)) . (intpos 6) < s . (intpos 6) by A3, A34; ::_thesis: verum
end;
suppose (Initialize (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s))) . (DataLoc (((Initialize (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s))) . GBP),5)) > 0 ; ::_thesis: (IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))),P,s)) . (intpos 6) < s . (intpos 6)
hence  (IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))),P,s)) . (intpos 6) < s . (intpos 6) by A74, XREAL_1:146; ::_thesis: verum
end;
end;
end;
hereby  ::_thesis: ( ( for i being Nat st i >= 7 & i <> (s . (intpos 4)) - 1 & i <> s . (intpos 4) holds
(IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))),P,s)) . (intpos i) = s . (intpos i) ) & ( s . (DataLoc ((s . (intpos 4)),(- 1))) > s . (DataLoc ((s . (intpos 4)),0)) implies ( (IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))),P,s)) . (DataLoc ((s . (intpos 4)),(- 1))) = s . (DataLoc ((s . (intpos 4)),0)) & (IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))),P,s)) . (DataLoc ((s . (intpos 4)),0)) = s . (DataLoc ((s . (intpos 4)),(- 1))) & (IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))),P,s)) . (intpos 6) = (s . (intpos 6)) - 1 & (IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))),P,s)) . (intpos 4) = (s . (intpos 4)) - 1 ) ) & ( s . (DataLoc ((s . (intpos 4)),(- 1))) <= s . (DataLoc ((s . (intpos 4)),0)) implies ( (IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))),P,s)) . (DataLoc ((s . (intpos 4)),(- 1))) = s . (DataLoc ((s . (intpos 4)),(- 1))) & (IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))),P,s)) . (DataLoc ((s . (intpos 4)),0)) = s . (DataLoc ((s . (intpos 4)),0)) & (IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))),P,s)) . (intpos 6) = 0 ) ) )
percases ( (Initialize (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s))) . (DataLoc (((Initialize (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s))) . GBP),5)) <= 0 or (Initialize (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s))) . (DataLoc (((Initialize (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s))) . GBP),5)) > 0 ) ;
supposeA76: (Initialize (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s))) . (DataLoc (((Initialize (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s))) . GBP),5)) <= 0 ; ::_thesis: (IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))),P,s)) . (intpos 4) >= 7 + ((IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))),P,s)) . (intpos 6))
then (IExec ((if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0)))),P,(Initialize (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s))))) . (intpos 4) = (IExec ((Load ((GBP,6) := 0)),P,(Initialize (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s))))) . (intpos 4) by SCMPDS_6:74
.= (Exec (((GBP,6) := 0),(Initialize (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s))))) . (intpos 4) by SCMPDS_5:40
.= s . (intpos 4) by A20, A33, AMI_3:10, SCMPDS_2:46 ;
then  (IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))),P,s)) . (intpos 4) = s . (intpos 4) by SCMPDS_5:35;
hence  (IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))),P,s)) . (intpos 4) >= 7 + ((IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))),P,s)) . (intpos 6)) by A1, A4, A34, A76, XXREAL_0:2; ::_thesis: verum
end;
supposeA77: (Initialize (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s))) . (DataLoc (((Initialize (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s))) . GBP),5)) > 0 ; ::_thesis: (IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))),P,s)) . (intpos 4) >= 7 + ((IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))),P,s)) . (intpos 6))
 (s . (intpos 4)) - 1 >= (7 + (s . (intpos 6))) - 1 by A1, XREAL_1:9;
hence  (IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))),P,s)) . (intpos 4) >= 7 + ((IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))),P,s)) . (intpos 6)) by A74, A77; ::_thesis: verum
end;
end;
end;
A78: now__::_thesis:_for_i_being_Element_of_NAT_st_i_>=_7_&_i_<>_(s_._(intpos_4))_-_1_&_i_<>_s_._(intpos_4)_holds_
(Exec_(((GBP,5)_:=_((intpos_4),(-_1))),s))_._(intpos_i)_=_s_._(intpos_i)
let i be Element of NAT ; ::_thesis: ( i >= 7 & i <> (s . (intpos 4)) - 1 & i <> s . (intpos 4) implies (Exec (((GBP,5) := ((intpos 4),(- 1))),s)) . (intpos i) = s . (intpos i) )
assume that
A79: i >= 7 and
 i <> (s . (intpos 4)) - 1 and
 i <> s . (intpos 4) ; ::_thesis: (Exec (((GBP,5) := ((intpos 4),(- 1))),s)) . (intpos i) = s . (intpos i)
 i > 5 by A79, XXREAL_0:2;
hence  (Exec (((GBP,5) := ((intpos 4),(- 1))),s)) . (intpos i) = s . (intpos i) by A6, AMI_3:10, SCMPDS_2:47; ::_thesis: verum
end;
A80: now__::_thesis:_for_i_being_Element_of_NAT_st_i_>=_7_&_i_<>_(s_._(intpos_4))_-_1_&_i_<>_s_._(intpos_4)_holds_
(IExec_((((GBP,5)_:=_((intpos_4),(-_1)))_';'_(SubFrom_(GBP,5,(intpos_4),0))),P,s))_._(intpos_i)_=_s_._(intpos_i)
let i be Element of NAT ; ::_thesis: ( i >= 7 & i <> (s . (intpos 4)) - 1 & i <> s . (intpos 4) implies (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s)) . (intpos i) = s . (intpos i) )
assume that
A81: i >= 7 and
A82: ( i <> (s . (intpos 4)) - 1 & i <> s . (intpos 4) ) ; ::_thesis: (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s)) . (intpos i) = s . (intpos i)
A83: i > 5 by A81, XXREAL_0:2;
thus (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s)) . (intpos i) = (Exec ((SubFrom (GBP,5,(intpos 4),0)),(Exec (((GBP,5) := ((intpos 4),(- 1))),s)))) . (intpos i) by SCMPDS_5:42
.= (Exec (((GBP,5) := ((intpos 4),(- 1))),s)) . (intpos i) by A8, A83, AMI_3:10, SCMPDS_2:50
.= s . (intpos i) by A78, A81, A82 ; ::_thesis: verum
end;
A84: now__::_thesis:_for_i_being_Element_of_NAT_st_i_>=_7_&_i_<>_(s_._(intpos_4))_-_1_&_i_<>_s_._(intpos_4)_holds_
(Exec_(((GBP,5)_:=_((intpos_4),(-_1))),(Initialize_(IExec_((((GBP,5)_:=_((intpos_4),(-_1)))_';'_(SubFrom_(GBP,5,(intpos_4),0))),P,s)))))_._(intpos_i)_=_s_._(intpos_i)
let i be Element of NAT ; ::_thesis: ( i >= 7 & i <> (s . (intpos 4)) - 1 & i <> s . (intpos 4) implies (Exec (((GBP,5) := ((intpos 4),(- 1))),(Initialize (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s))))) . (intpos i) = s . (intpos i) )
assume that
A85: i >= 7 and
A86: ( i <> (s . (intpos 4)) - 1 & i <> s . (intpos 4) ) ; ::_thesis: (Exec (((GBP,5) := ((intpos 4),(- 1))),(Initialize (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s))))) . (intpos i) = s . (intpos i)
 i > 5 by A85, XXREAL_0:2;
hence (Exec (((GBP,5) := ((intpos 4),(- 1))),(Initialize (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s))))) . (intpos i) = (Initialize (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s))) . (intpos i) by A17, AMI_3:10, SCMPDS_2:47
.= (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s)) . (intpos i) by SCMPDS_5:15
.= s . (intpos i) by A80, A85, A86 ;
::_thesis: verum
end;
A87: now__::_thesis:_for_i_being_Element_of_NAT_st_i_>=_7_&_i_<>_(s_._(intpos_4))_-_1_&_i_<>_s_._(intpos_4)_holds_
(IExec_((((GBP,5)_:=_((intpos_4),(-_1)))_';'_(((intpos_4),(-_1))_:=_((intpos_4),0))),P,(Initialize_(IExec_((((GBP,5)_:=_((intpos_4),(-_1)))_';'_(SubFrom_(GBP,5,(intpos_4),0))),P,s)))))_._(intpos_i)_=_s_._(intpos_i)
let i be Element of NAT ; ::_thesis: ( i >= 7 & i <> (s . (intpos 4)) - 1 & i <> s . (intpos 4) implies (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))),P,(Initialize (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s))))) . (intpos i) = s . (intpos i) )
assume that
A88: i >= 7 and
A89: i <> (s . (intpos 4)) - 1 and
A90: i <> s . (intpos 4) ; ::_thesis: (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))),P,(Initialize (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s))))) . (intpos i) = s . (intpos i)
A91: intpos i <> DataLoc (((Exec (((GBP,5) := ((intpos 4),(- 1))),(Initialize (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s))))) . (intpos 4)),(- 1))
proof
assume  intpos i = DataLoc (((Exec (((GBP,5) := ((intpos 4),(- 1))),(Initialize (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s))))) . (intpos 4)),(- 1)) ; ::_thesis: contradiction
then  i = abs (((Exec (((GBP,5) := ((intpos 4),(- 1))),(Initialize (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s))))) . (intpos 4)) + (- 1)) by XTUPLE_0:1;
hence  contradiction by A37, A21, A89, ABSVALUE:def_1; ::_thesis: verum
end;
thus (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))),P,(Initialize (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s))))) . (intpos i) = (Exec ((((intpos 4),(- 1)) := ((intpos 4),0)),(Exec (((GBP,5) := ((intpos 4),(- 1))),(Initialize (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s))))))) . (intpos i) by SCMPDS_5:42
.= (Exec (((GBP,5) := ((intpos 4),(- 1))),(Initialize (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s))))) . (intpos i) by A91, SCMPDS_2:47
.= s . (intpos i) by A84, A88, A89, A90 ; ::_thesis: verum
end;
A92: now__::_thesis:_for_i_being_Element_of_NAT_st_i_>=_7_&_i_<>_(s_._(intpos_4))_-_1_&_i_<>_s_._(intpos_4)_holds_
(IExec_(((((GBP,5)_:=_((intpos_4),(-_1)))_';'_(((intpos_4),(-_1))_:=_((intpos_4),0)))_';'_(((intpos_4),0)_:=_(GBP,5))),P,(Initialize_(IExec_((((GBP,5)_:=_((intpos_4),(-_1)))_';'_(SubFrom_(GBP,5,(intpos_4),0))),P,s)))))_._(intpos_i)_=_s_._(intpos_i)
let i be Element of NAT ; ::_thesis: ( i >= 7 & i <> (s . (intpos 4)) - 1 & i <> s . (intpos 4) implies (IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))),P,(Initialize (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s))))) . (intpos i) = s . (intpos i) )
assume that
A93: ( i >= 7 & i <> (s . (intpos 4)) - 1 ) and
A94: i <> s . (intpos 4) ; ::_thesis: (IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))),P,(Initialize (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s))))) . (intpos i) = s . (intpos i)
A95: intpos i <> DataLoc (((IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))),P,(Initialize (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s))))) . (intpos 4)),0)
proof
assume  intpos i = DataLoc (((IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))),P,(Initialize (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s))))) . (intpos 4)),0) ; ::_thesis: contradiction
then  i = abs (((IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))),P,(Initialize (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s))))) . (intpos 4)) + 0) by XTUPLE_0:1;
hence  contradiction by A1, A4, A40, A94, ABSVALUE:def_1; ::_thesis: verum
end;
thus (IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))),P,(Initialize (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s))))) . (intpos i) = (Exec ((((intpos 4),0) := (GBP,5)),(IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))),P,(Initialize (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s))))))) . (intpos i) by SCMPDS_5:41
.= (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))),P,(Initialize (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s))))) . (intpos i) by A95, SCMPDS_2:47
.= s . (intpos i) by A87, A93, A94 ; ::_thesis: verum
end;
A96: now__::_thesis:_for_i_being_Element_of_NAT_st_i_>=_7_&_i_<>_(s_._(intpos_4))_-_1_&_i_<>_s_._(intpos_4)_holds_
(IExec_((((((GBP,5)_:=_((intpos_4),(-_1)))_';'_(((intpos_4),(-_1))_:=_((intpos_4),0)))_';'_(((intpos_4),0)_:=_(GBP,5)))_';'_(AddTo_(GBP,4,(-_1)))),P,(Initialize_(IExec_((((GBP,5)_:=_((intpos_4),(-_1)))_';'_(SubFrom_(GBP,5,(intpos_4),0))),P,s)))))_._(intpos_i)_=_s_._(intpos_i)
let i be Element of NAT ; ::_thesis: ( i >= 7 & i <> (s . (intpos 4)) - 1 & i <> s . (intpos 4) implies (IExec ((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))),P,(Initialize (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s))))) . (intpos i) = s . (intpos i) )
assume that
A97: i >= 7 and
A98: ( i <> (s . (intpos 4)) - 1 & i <> s . (intpos 4) ) ; ::_thesis: (IExec ((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))),P,(Initialize (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s))))) . (intpos i) = s . (intpos i)
 i > 4 by A97, XXREAL_0:2;
then A99: intpos i <> DataLoc (((IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))),P,(Initialize (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s))))) . GBP),4) by A53, AMI_3:10, SCMP_GCD:1;
thus (IExec ((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))),P,(Initialize (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s))))) . (intpos i) = (Exec ((AddTo (GBP,4,(- 1))),(IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))),P,(Initialize (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s))))))) . (intpos i) by SCMPDS_5:41
.= (IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))),P,(Initialize (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s))))) . (intpos i) by A99, SCMPDS_2:48
.= s . (intpos i) by A92, A97, A98 ; ::_thesis: verum
end;
hereby  ::_thesis: ( ( s . (DataLoc ((s . (intpos 4)),(- 1))) > s . (DataLoc ((s . (intpos 4)),0)) implies ( (IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))),P,s)) . (DataLoc ((s . (intpos 4)),(- 1))) = s . (DataLoc ((s . (intpos 4)),0)) & (IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))),P,s)) . (DataLoc ((s . (intpos 4)),0)) = s . (DataLoc ((s . (intpos 4)),(- 1))) & (IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))),P,s)) . (intpos 6) = (s . (intpos 6)) - 1 & (IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))),P,s)) . (intpos 4) = (s . (intpos 4)) - 1 ) ) & ( s . (DataLoc ((s . (intpos 4)),(- 1))) <= s . (DataLoc ((s . (intpos 4)),0)) implies ( (IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))),P,s)) . (DataLoc ((s . (intpos 4)),(- 1))) = s . (DataLoc ((s . (intpos 4)),(- 1))) & (IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))),P,s)) . (DataLoc ((s . (intpos 4)),0)) = s . (DataLoc ((s . (intpos 4)),0)) & (IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))),P,s)) . (intpos 6) = 0 ) ) )
let i be Nat; ::_thesis: ( i >= 7 & i <> (s . (intpos 4)) - 1 & i <> s . (intpos 4) implies (IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))),P,s)) . (intpos b1) = s . (intpos b1) )
A100: i in NAT by ORDINAL1:def_12;
set xi = intpos i;
assume that
A101: i >= 7 and
A102: ( i <> (s . (intpos 4)) - 1 & i <> s . (intpos 4) ) ; ::_thesis: (IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))),P,s)) . (intpos b1) = s . (intpos b1)
A103: i > 6 by A101, XXREAL_0:2;
percases ( (Initialize (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s))) . (DataLoc (((Initialize (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s))) . GBP),5)) <= 0 or (Initialize (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s))) . (DataLoc (((Initialize (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s))) . GBP),5)) > 0 ) ;
suppose (Initialize (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s))) . (DataLoc (((Initialize (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s))) . GBP),5)) <= 0 ; ::_thesis: (IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))),P,s)) . (intpos b1) = s . (intpos b1)
then (IExec ((if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0)))),P,(Initialize (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s))))) . (intpos i) = (IExec ((Load ((GBP,6) := 0)),P,(Initialize (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s))))) . (intpos i) by SCMPDS_6:74
.= (Exec (((GBP,6) := 0),(Initialize (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s))))) . (intpos i) by SCMPDS_5:40
.= (Initialize (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s))) . (intpos i) by A33, A103, AMI_3:10, SCMPDS_2:46
.= (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s)) . (intpos i) by SCMPDS_5:15
.= s . (intpos i) by A80, A100, A101, A102 ;
hence  (IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))),P,s)) . (intpos i) = s . (intpos i) by SCMPDS_5:35; ::_thesis: verum
end;
suppose (Initialize (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s))) . (DataLoc (((Initialize (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s))) . GBP),5)) > 0 ; ::_thesis: (IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))),P,s)) . (intpos b1) = s . (intpos b1)
then (IExec ((if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0)))),P,(Initialize (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s))))) . (intpos i) = (IExec (((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),P,(Initialize (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s))))) . (intpos i) by SCMPDS_6:73
.= (Exec ((AddTo (GBP,6,(- 1))),(IExec ((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))),P,(Initialize (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s))))))) . (intpos i) by SCMPDS_5:41
.= (IExec ((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))),P,(Initialize (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s))))) . (intpos i) by A59, A103, AMI_3:10, SCMPDS_2:48
.= s . (intpos i) by A96, A100, A101, A102 ;
hence  (IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))),P,s)) . (intpos i) = s . (intpos i) by SCMPDS_5:35; ::_thesis: verum
end;
end;
end;
A104: (Initialize (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s))) . (intpos 5) = (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s)) . (intpos 5) by SCMPDS_5:15
.= (Exec ((SubFrom (GBP,5,(intpos 4),0)),(Exec (((GBP,5) := ((intpos 4),(- 1))),s)))) . (intpos 5) by SCMPDS_5:42
.= (s . (DataLoc ((s . (intpos 4)),(- 1)))) - (s . (DataLoc ((s . (intpos 4)),0))) by A14, A13, A12, A8, SCMPDS_2:50 ;
then A105: (Initialize (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s))) . (DataLoc (((Initialize (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s))) . GBP),5)) = (s . (DataLoc ((s . (intpos 4)),(- 1)))) - (s . (DataLoc ((s . (intpos 4)),0))) by A16, SCMP_GCD:1;
 abs ((s . (intpos 4)) + (- 1)) <> abs ((s . GBP) + 5) by A3, A27, A50, A11, XREAL_1:6;
then  DataLoc ((s . (intpos 4)),(- 1)) <> DataLoc ((s . GBP),5) by XTUPLE_0:1;
then A106: (Exec (((GBP,5) := ((intpos 4),(- 1))),s)) . (DataLoc ((s . (intpos 4)),(- 1))) = s . (DataLoc ((s . (intpos 4)),(- 1))) by SCMPDS_2:47;
(IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s)) . (DataLoc ((s . (intpos 4)),(- 1))) = (Exec ((SubFrom (GBP,5,(intpos 4),0)),(Exec (((GBP,5) := ((intpos 4),(- 1))),s)))) . (DataLoc ((s . (intpos 4)),(- 1))) by SCMPDS_5:42
.= s . (DataLoc ((s . (intpos 4)),(- 1))) by A106, A51, SCMPDS_2:50 ;
then A107: (Initialize (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s))) . (DataLoc ((s . (intpos 4)),(- 1))) = s . (DataLoc ((s . (intpos 4)),(- 1))) by SCMPDS_5:15;
then A108: (Exec (((GBP,5) := ((intpos 4),(- 1))),(Initialize (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s))))) . (intpos 5) = s . (DataLoc ((s . (intpos 4)),(- 1))) by A20, A17, SCMPDS_2:47;
 5 <> abs (((Exec (((GBP,5) := ((intpos 4),(- 1))),(Initialize (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s))))) . (intpos 4)) + (- 1)) by A27, A35, A38, XXREAL_0:2;
then A109: intpos 5 <> DataLoc (((Exec (((GBP,5) := ((intpos 4),(- 1))),(Initialize (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s))))) . (intpos 4)),(- 1)) by XTUPLE_0:1;
A110: (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))),P,(Initialize (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s))))) . (intpos 5) = (Exec ((((intpos 4),(- 1)) := ((intpos 4),0)),(Exec (((GBP,5) := ((intpos 4),(- 1))),(Initialize (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s))))))) . (intpos 5) by SCMPDS_5:42
.= s . (DataLoc ((s . (intpos 4)),(- 1))) by A108, A109, SCMPDS_2:47 ;
 abs ((s . (intpos 4)) + 0) <> abs (((IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))),P,(Initialize (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s))))) . GBP) + 4) by A1, A4, A5, A62, XXREAL_0:2;
then A111: DataLoc ((s . (intpos 4)),0) <> DataLoc (((IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))),P,(Initialize (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s))))) . GBP),4) by XTUPLE_0:1;
A112: (IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))),P,(Initialize (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s))))) . (DataLoc ((s . (intpos 4)),0)) = (Exec ((((intpos 4),0) := (GBP,5)),(IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))),P,(Initialize (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s))))))) . (DataLoc (((IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))),P,(Initialize (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s))))) . (intpos 4)),0)) by A40, SCMPDS_5:41
.= (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))),P,(Initialize (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s))))) . (DataLoc (((IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))),P,(Initialize (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s))))) . GBP),5)) by SCMPDS_2:47
.= s . (DataLoc ((s . (intpos 4)),(- 1))) by A49, A110, SCMP_GCD:1 ;
A113: (IExec ((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))),P,(Initialize (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s))))) . (DataLoc ((s . (intpos 4)),0)) = (Exec ((AddTo (GBP,4,(- 1))),(IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))),P,(Initialize (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s))))))) . (DataLoc ((s . (intpos 4)),0)) by SCMPDS_5:41
.= s . (DataLoc ((s . (intpos 4)),(- 1))) by A112, A111, SCMPDS_2:48 ;
 abs ((s . (intpos 4)) + (- 1)) <> abs (((IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))),P,(Initialize (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s))))) . (intpos 4)) + 0) by A50, A41;
then A114: DataLoc ((s . (intpos 4)),(- 1)) <> DataLoc (((IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))),P,(Initialize (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s))))) . (intpos 4)),0) by XTUPLE_0:1;
A115: (IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))),P,(Initialize (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s))))) . (DataLoc ((s . (intpos 4)),(- 1))) = (Exec ((((intpos 4),0) := (GBP,5)),(IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))),P,(Initialize (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s))))))) . (DataLoc ((s . (intpos 4)),(- 1))) by SCMPDS_5:41
.= s . (DataLoc ((s . (intpos 4)),0)) by A22, A114, SCMPDS_2:47 ;
A116: (IExec ((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))),P,(Initialize (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s))))) . (DataLoc ((s . (intpos 4)),(- 1))) = (Exec ((AddTo (GBP,4,(- 1))),(IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))),P,(Initialize (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s))))))) . (DataLoc ((s . (intpos 4)),(- 1))) by SCMPDS_5:41
.= s . (DataLoc ((s . (intpos 4)),0)) by A115, A63, SCMPDS_2:48 ;
hereby  ::_thesis: ( s . (DataLoc ((s . (intpos 4)),(- 1))) <= s . (DataLoc ((s . (intpos 4)),0)) implies ( (IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))),P,s)) . (DataLoc ((s . (intpos 4)),(- 1))) = s . (DataLoc ((s . (intpos 4)),(- 1))) & (IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))),P,s)) . (DataLoc ((s . (intpos 4)),0)) = s . (DataLoc ((s . (intpos 4)),0)) & (IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))),P,s)) . (intpos 6) = 0 ) )
A117: DataLoc ((s . (intpos 4)),(- 1)) <> DataLoc (((IExec ((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))),P,(Initialize (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s))))) . GBP),6) by A27, A35, A50, A59, XTUPLE_0:1;
assume  s . (DataLoc ((s . (intpos 4)),(- 1))) > s . (DataLoc ((s . (intpos 4)),0)) ; ::_thesis: ( (IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))),P,s)) . (DataLoc ((s . (intpos 4)),(- 1))) = s . (DataLoc ((s . (intpos 4)),0)) & (IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))),P,s)) . (DataLoc ((s . (intpos 4)),0)) = s . (DataLoc ((s . (intpos 4)),(- 1))) & (IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))),P,s)) . (intpos 6) = (s . (intpos 6)) - 1 & (IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))),P,s)) . (intpos 4) = (s . (intpos 4)) - 1 )
then A118: (s . (DataLoc ((s . (intpos 4)),(- 1)))) - (s . (DataLoc ((s . (intpos 4)),0))) > (s . (DataLoc ((s . (intpos 4)),0))) - (s . (DataLoc ((s . (intpos 4)),0))) by XREAL_1:9;
then (IExec ((if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0)))),P,(Initialize (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s))))) . (DataLoc ((s . (intpos 4)),(- 1))) = (IExec (((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),P,(Initialize (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s))))) . (DataLoc ((s . (intpos 4)),(- 1))) by A105, SCMPDS_6:73
.= (Exec ((AddTo (GBP,6,(- 1))),(IExec ((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))),P,(Initialize (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s))))))) . (DataLoc ((s . (intpos 4)),(- 1))) by SCMPDS_5:41
.= s . (DataLoc ((s . (intpos 4)),0)) by A116, A117, SCMPDS_2:48 ;
hence  (IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))),P,s)) . (DataLoc ((s . (intpos 4)),(- 1))) = s . (DataLoc ((s . (intpos 4)),0)) by SCMPDS_5:35; ::_thesis: ( (IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))),P,s)) . (DataLoc ((s . (intpos 4)),0)) = s . (DataLoc ((s . (intpos 4)),(- 1))) & (IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))),P,s)) . (intpos 6) = (s . (intpos 6)) - 1 & (IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))),P,s)) . (intpos 4) = (s . (intpos 4)) - 1 )
 abs ((s . (intpos 4)) + 0) <> abs (((IExec ((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))),P,(Initialize (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s))))) . GBP) + 6) by A1, A4, A5, A60, XXREAL_0:2;
then A119: DataLoc ((s . (intpos 4)),0) <> DataLoc (((IExec ((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))),P,(Initialize (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s))))) . GBP),6) by XTUPLE_0:1;
(IExec ((if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0)))),P,(Initialize (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s))))) . (DataLoc ((s . (intpos 4)),0)) = (IExec (((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),P,(Initialize (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s))))) . (DataLoc ((s . (intpos 4)),0)) by A105, A118, SCMPDS_6:73
.= (Exec ((AddTo (GBP,6,(- 1))),(IExec ((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))),P,(Initialize (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s))))))) . (DataLoc ((s . (intpos 4)),0)) by SCMPDS_5:41
.= s . (DataLoc ((s . (intpos 4)),(- 1))) by A113, A119, SCMPDS_2:48 ;
hence  (IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))),P,s)) . (DataLoc ((s . (intpos 4)),0)) = s . (DataLoc ((s . (intpos 4)),(- 1))) by SCMPDS_5:35; ::_thesis: ( (IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))),P,s)) . (intpos 6) = (s . (intpos 6)) - 1 & (IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))),P,s)) . (intpos 4) = (s . (intpos 4)) - 1 )
thus  ( (IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))),P,s)) . (intpos 6) = (s . (intpos 6)) - 1 & (IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))),P,s)) . (intpos 4) = (s . (intpos 4)) - 1 ) by A16, A104, A74, A118, SCMP_GCD:1; ::_thesis: verum
end;
A120: abs (((Initialize (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s))) . GBP) + 6) = 0 + 6 by A33, XTUPLE_0:1;
hereby  ::_thesis: verum
A121: DataLoc ((s . (intpos 4)),(- 1)) <> DataLoc (((Initialize (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s))) . GBP),6) by A27, A35, A50, A33, XTUPLE_0:1;
assume  s . (DataLoc ((s . (intpos 4)),(- 1))) <= s . (DataLoc ((s . (intpos 4)),0)) ; ::_thesis: ( (IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))),P,s)) . (DataLoc ((s . (intpos 4)),(- 1))) = s . (DataLoc ((s . (intpos 4)),(- 1))) & (IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))),P,s)) . (DataLoc ((s . (intpos 4)),0)) = s . (DataLoc ((s . (intpos 4)),0)) & (IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))),P,s)) . (intpos 6) = 0 )
then A122: (s . (DataLoc ((s . (intpos 4)),(- 1)))) - (s . (DataLoc ((s . (intpos 4)),0))) <= (s . (DataLoc ((s . (intpos 4)),0))) - (s . (DataLoc ((s . (intpos 4)),0))) by XREAL_1:9;
then (IExec ((if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0)))),P,(Initialize (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s))))) . (DataLoc ((s . (intpos 4)),(- 1))) = (IExec ((Load ((GBP,6) := 0)),P,(Initialize (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s))))) . (DataLoc ((s . (intpos 4)),(- 1))) by A105, SCMPDS_6:74
.= (Exec (((GBP,6) := 0),(Initialize (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s))))) . (DataLoc ((s . (intpos 4)),(- 1))) by SCMPDS_5:40
.= s . (DataLoc ((s . (intpos 4)),(- 1))) by A107, A121, SCMPDS_2:46 ;
hence  (IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))),P,s)) . (DataLoc ((s . (intpos 4)),(- 1))) = s . (DataLoc ((s . (intpos 4)),(- 1))) by SCMPDS_5:35; ::_thesis: ( (IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))),P,s)) . (DataLoc ((s . (intpos 4)),0)) = s . (DataLoc ((s . (intpos 4)),0)) & (IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))),P,s)) . (intpos 6) = 0 )
 abs ((s . (intpos 4)) + 0) <> abs (((Initialize (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s))) . GBP) + 6) by A1, A4, A5, A120, XXREAL_0:2;
then A123: DataLoc ((s . (intpos 4)),0) <> DataLoc (((Initialize (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s))) . GBP),6) by XTUPLE_0:1;
(IExec ((if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0)))),P,(Initialize (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s))))) . (DataLoc ((s . (intpos 4)),0)) = (IExec ((Load ((GBP,6) := 0)),P,(Initialize (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s))))) . (DataLoc ((s . (intpos 4)),0)) by A105, A122, SCMPDS_6:74
.= (Exec (((GBP,6) := 0),(Initialize (IExec ((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))),P,s))))) . (DataLoc ((s . (intpos 4)),0)) by SCMPDS_5:40
.= s . (DataLoc ((s . (intpos 4)),0)) by A15, A123, SCMPDS_2:46 ;
hence  (IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))),P,s)) . (DataLoc ((s . (intpos 4)),0)) = s . (DataLoc ((s . (intpos 4)),0)) by SCMPDS_5:35; ::_thesis: (IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))),P,s)) . (intpos 6) = 0
thus  (IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))),P,s)) . (intpos 6) = 0 by A16, A104, A34, A122, SCMP_GCD:1; ::_thesis: verum
end;
end;

Lm8:  for P being Instruction-Sequence of SCMPDS
for s being 0 -started State of SCMPDS st s . (intpos 4) >= 7 + (s . (DataLoc ((s . GBP),6))) & s . GBP = 0 holds
( while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0)))))) is_closed_on s,P & while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0)))))) is_halting_on s,P )
proof
let P be Instruction-Sequence of SCMPDS; ::_thesis: for s being 0 -started State of SCMPDS st s . (intpos 4) >= 7 + (s . (DataLoc ((s . GBP),6))) & s . GBP = 0 holds
( while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0)))))) is_closed_on s,P & while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0)))))) is_halting_on s,P )
let s be 0 -started State of SCMPDS; ::_thesis: ( s . (intpos 4) >= 7 + (s . (DataLoc ((s . GBP),6))) & s . GBP = 0 implies ( while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0)))))) is_closed_on s,P & while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0)))))) is_halting_on s,P ) )
set a = GBP ;
set b = DataLoc ((s . GBP),6);
assume that
A1: s . (intpos 4) >= 7 + (s . (DataLoc ((s . GBP),6))) and
A2: s . GBP = 0 ; ::_thesis: ( while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0)))))) is_closed_on s,P & while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0)))))) is_halting_on s,P )
A3: DataLoc ((s . GBP),6) = intpos (0 + 6) by A2, SCMP_GCD:1;
A4: 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_{(intpos_4)}_holds_
t_._x_>=_7_+_(t_._(DataLoc_((s_._GBP),6)))_)_&_t_._GBP_=_s_._GBP_&_t_._(DataLoc_((s_._GBP),6))_>_0_holds_
(_(IExec_(((((GBP,5)_:=_((intpos_4),(-_1)))_';'_(SubFrom_(GBP,5,(intpos_4),0)))_';'_(if>0_(GBP,5,((((((GBP,5)_:=_((intpos_4),(-_1)))_';'_(((intpos_4),(-_1))_:=_((intpos_4),0)))_';'_(((intpos_4),0)_:=_(GBP,5)))_';'_(AddTo_(GBP,4,(-_1))))_';'_(AddTo_(GBP,6,(-_1)))),(Load_((GBP,6)_:=_0))))),Q,t))_._GBP_=_t_._GBP_&_(((GBP,5)_:=_((intpos_4),(-_1)))_';'_(SubFrom_(GBP,5,(intpos_4),0)))_';'_(if>0_(GBP,5,((((((GBP,5)_:=_((intpos_4),(-_1)))_';'_(((intpos_4),(-_1))_:=_((intpos_4),0)))_';'_(((intpos_4),0)_:=_(GBP,5)))_';'_(AddTo_(GBP,4,(-_1))))_';'_(AddTo_(GBP,6,(-_1)))),(Load_((GBP,6)_:=_0))))_is_closed_on_t,Q_&_(((GBP,5)_:=_((intpos_4),(-_1)))_';'_(SubFrom_(GBP,5,(intpos_4),0)))_';'_(if>0_(GBP,5,((((((GBP,5)_:=_((intpos_4),(-_1)))_';'_(((intpos_4),(-_1))_:=_((intpos_4),0)))_';'_(((intpos_4),0)_:=_(GBP,5)))_';'_(AddTo_(GBP,4,(-_1))))_';'_(AddTo_(GBP,6,(-_1)))),(Load_((GBP,6)_:=_0))))_is_halting_on_t,Q_&_(IExec_(((((GBP,5)_:=_((intpos_4),(-_1)))_';'_(SubFrom_(GBP,5,(intpos_4),0)))_';'_(if>0_(GBP,5,((((((GBP,5)_:=_((intpos_4),(-_1)))_';'_(((intpos_4),(-_1))_:=_((intpos_4),0)))_';'_(((intpos_4),0)_:=_(GBP,5)))_';'_(AddTo_(GBP,4,(-_1))))_';'_(AddTo_(GBP,6,(-_1)))),(Load_((GBP,6)_:=_0))))),Q,t))_._(DataLoc_((s_._GBP),6))_<_t_._(DataLoc_((s_._GBP),6))_&_(_for_x_being_Int_position_st_x_in_{(intpos_4)}_holds_
(IExec_(((((GBP,5)_:=_((intpos_4),(-_1)))_';'_(SubFrom_(GBP,5,(intpos_4),0)))_';'_(if>0_(GBP,5,((((((GBP,5)_:=_((intpos_4),(-_1)))_';'_(((intpos_4),(-_1))_:=_((intpos_4),0)))_';'_(((intpos_4),0)_:=_(GBP,5)))_';'_(AddTo_(GBP,4,(-_1))))_';'_(AddTo_(GBP,6,(-_1)))),(Load_((GBP,6)_:=_0))))),Q,t))_._x_>=_7_+_((IExec_(((((GBP,5)_:=_((intpos_4),(-_1)))_';'_(SubFrom_(GBP,5,(intpos_4),0)))_';'_(if>0_(GBP,5,((((((GBP,5)_:=_((intpos_4),(-_1)))_';'_(((intpos_4),(-_1))_:=_((intpos_4),0)))_';'_(((intpos_4),0)_:=_(GBP,5)))_';'_(AddTo_(GBP,4,(-_1))))_';'_(AddTo_(GBP,6,(-_1)))),(Load_((GBP,6)_:=_0))))),Q,t))_._(DataLoc_((s_._GBP),6)))_)_)
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 {(intpos 4)} holds
t . x >= 7 + (t . (DataLoc ((s . GBP),6))) ) & t . GBP = s . GBP & t . (DataLoc ((s . GBP),6)) > 0 holds
( (IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))),Q,t)) . GBP = t . GBP & (((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0)))) is_closed_on t,Q & (((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0)))) is_halting_on t,Q & (IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))),Q,t)) . (DataLoc ((s . GBP),6)) < t . (DataLoc ((s . GBP),6)) & ( for x being Int_position st x in {(intpos 4)} holds
(IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))),Q,t)) . x >= 7 + ((IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))),Q,t)) . (DataLoc ((s . GBP),6))) ) )
let Q be Instruction-Sequence of SCMPDS; ::_thesis: ( ( for x being Int_position st x in {(intpos 4)} holds
t . x >= 7 + (t . (DataLoc ((s . GBP),6))) ) & t . GBP = s . GBP & t . (DataLoc ((s . GBP),6)) > 0 implies ( (IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))),Q,t)) . GBP = t . GBP & (((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0)))) is_closed_on t,Q & (((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0)))) is_halting_on t,Q & (IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))),Q,t)) . (DataLoc ((s . GBP),6)) < t . (DataLoc ((s . GBP),6)) & ( for x being Int_position st x in {(intpos 4)} holds
(IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))),Q,t)) . x >= 7 + ((IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))),Q,t)) . (DataLoc ((s . GBP),6))) ) ) )
assume that
A5: for x being Int_position st x in {(intpos 4)} holds
t . x >= 7 + (t . (DataLoc ((s . GBP),6))) and
A6: ( t . GBP = s . GBP & t . (DataLoc ((s . GBP),6)) > 0 ) ; ::_thesis: ( (IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))),Q,t)) . GBP = t . GBP & (((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0)))) is_closed_on t,Q & (((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0)))) is_halting_on t,Q & (IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))),Q,t)) . (DataLoc ((s . GBP),6)) < t . (DataLoc ((s . GBP),6)) & ( for x being Int_position st x in {(intpos 4)} holds
(IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))),Q,t)) . x >= 7 + ((IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))),Q,t)) . (DataLoc ((s . GBP),6))) ) )
set Bt = IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))),Q,t);
A7: intpos 4 in {(intpos 4)} by TARSKI:def_1;
then A8: t . (intpos 4) >= 7 + (t . (intpos 6)) by A3, A5;
hence  (IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))),Q,t)) . GBP = t . GBP by A2, A3, A6, Lm6; ::_thesis: ( (((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0)))) is_closed_on t,Q & (((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0)))) is_halting_on t,Q & (IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))),Q,t)) . (DataLoc ((s . GBP),6)) < t . (DataLoc ((s . GBP),6)) & ( for x being Int_position st x in {(intpos 4)} holds
(IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))),Q,t)) . x >= 7 + ((IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))),Q,t)) . (DataLoc ((s . GBP),6))) ) )
thus  ( (((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0)))) is_closed_on t,Q & (((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0)))) is_halting_on t,Q ) by SCMPDS_6:20, SCMPDS_6:21; ::_thesis: ( (IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))),Q,t)) . (DataLoc ((s . GBP),6)) < t . (DataLoc ((s . GBP),6)) & ( for x being Int_position st x in {(intpos 4)} holds
(IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))),Q,t)) . x >= 7 + ((IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))),Q,t)) . (DataLoc ((s . GBP),6))) ) )
thus  (IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))),Q,t)) . (DataLoc ((s . GBP),6)) < t . (DataLoc ((s . GBP),6)) by A2, A3, A6, A8, Lm7; ::_thesis: for x being Int_position st x in {(intpos 4)} holds
(IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))),Q,t)) . x >= 7 + ((IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))),Q,t)) . (DataLoc ((s . GBP),6)))
 t . (intpos 4) >= 7 + (t . (DataLoc ((s . GBP),6))) by A5, A7;
then  (IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))),Q,t)) . (intpos 4) >= 7 + ((IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))),Q,t)) . (intpos 6)) by A2, A3, A6, Lm7;
hence  for x being Int_position st x in {(intpos 4)} holds
(IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))),Q,t)) . x >= 7 + ((IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))),Q,t)) . (DataLoc ((s . GBP),6))) by A3, TARSKI:def_1; ::_thesis: verum
end;
 for x being Int_position st x in {(intpos 4)} holds
s . x >= 7 + (s . (DataLoc ((s . GBP),6))) by A1, TARSKI:def_1;
hence  ( while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0)))))) is_closed_on s,P & while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0)))))) is_halting_on s,P ) by A4, SCMPDS_8:29; ::_thesis: verum
end;

Lm9:  for P being Instruction-Sequence of SCMPDS
for s being 0 -started State of SCMPDS st s . (intpos 4) >= 7 + (s . (DataLoc ((s . GBP),6))) & s . GBP = 0 holds
( (IExec ((while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))))),P,s)) . GBP = 0 & (IExec ((while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))))),P,s)) . (intpos 1) = s . (intpos 1) & (IExec ((while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))))),P,s)) . (intpos 2) = s . (intpos 2) & (IExec ((while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))))),P,s)) . (intpos 3) = s . (intpos 3) )
proof
let P be Instruction-Sequence of SCMPDS; ::_thesis: for s being 0 -started State of SCMPDS st s . (intpos 4) >= 7 + (s . (DataLoc ((s . GBP),6))) & s . GBP = 0 holds
( (IExec ((while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))))),P,s)) . GBP = 0 & (IExec ((while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))))),P,s)) . (intpos 1) = s . (intpos 1) & (IExec ((while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))))),P,s)) . (intpos 2) = s . (intpos 2) & (IExec ((while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))))),P,s)) . (intpos 3) = s . (intpos 3) )
let s be 0 -started State of SCMPDS; ::_thesis: ( s . (intpos 4) >= 7 + (s . (DataLoc ((s . GBP),6))) & s . GBP = 0 implies ( (IExec ((while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))))),P,s)) . GBP = 0 & (IExec ((while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))))),P,s)) . (intpos 1) = s . (intpos 1) & (IExec ((while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))))),P,s)) . (intpos 2) = s . (intpos 2) & (IExec ((while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))))),P,s)) . (intpos 3) = s . (intpos 3) ) )
set b = DataLoc ((s . GBP),6);
set a = GBP ;
assume that
A1: s . (intpos 4) >= 7 + (s . (DataLoc ((s . GBP),6))) and
A2: s . GBP = 0 ; ::_thesis: ( (IExec ((while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))))),P,s)) . GBP = 0 & (IExec ((while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))))),P,s)) . (intpos 1) = s . (intpos 1) & (IExec ((while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))))),P,s)) . (intpos 2) = s . (intpos 2) & (IExec ((while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))))),P,s)) . (intpos 3) = s . (intpos 3) )
A3: DataLoc ((s . GBP),6) = intpos (0 + 6) by A2, SCMP_GCD:1;
defpred S1[ Nat] means  for t being 0 -started State of SCMPDS
for Q being Instruction-Sequence of SCMPDS st t . (intpos 6) <= $1 & t . (intpos 4) >= 7 + (t . (intpos 6)) & t . GBP = 0 holds
( (IExec ((while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))))),Q,t)) . GBP = 0 & (IExec ((while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))))),Q,t)) . (intpos 1) = t . (intpos 1) & (IExec ((while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))))),Q,t)) . (intpos 2) = t . (intpos 2) & (IExec ((while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))))),Q,t)) . (intpos 3) = t . (intpos 3) );
A4: S1[ 0 ]
proof
let t be 0 -started State of SCMPDS; ::_thesis: for Q being Instruction-Sequence of SCMPDS st t . (intpos 6) <= 0 & t . (intpos 4) >= 7 + (t . (intpos 6)) & t . GBP = 0 holds
( (IExec ((while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))))),Q,t)) . GBP = 0 & (IExec ((while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))))),Q,t)) . (intpos 1) = t . (intpos 1) & (IExec ((while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))))),Q,t)) . (intpos 2) = t . (intpos 2) & (IExec ((while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))))),Q,t)) . (intpos 3) = t . (intpos 3) )
let Q be Instruction-Sequence of SCMPDS; ::_thesis: ( t . (intpos 6) <= 0 & t . (intpos 4) >= 7 + (t . (intpos 6)) & t . GBP = 0 implies ( (IExec ((while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))))),Q,t)) . GBP = 0 & (IExec ((while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))))),Q,t)) . (intpos 1) = t . (intpos 1) & (IExec ((while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))))),Q,t)) . (intpos 2) = t . (intpos 2) & (IExec ((while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))))),Q,t)) . (intpos 3) = t . (intpos 3) ) )
assume that
A5: t . (intpos 6) <= 0 and
 t . (intpos 4) >= 7 + (t . (intpos 6)) and
A6: t . GBP = 0 ; ::_thesis: ( (IExec ((while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))))),Q,t)) . GBP = 0 & (IExec ((while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))))),Q,t)) . (intpos 1) = t . (intpos 1) & (IExec ((while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))))),Q,t)) . (intpos 2) = t . (intpos 2) & (IExec ((while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))))),Q,t)) . (intpos 3) = t . (intpos 3) )
A7: DataLoc ((t . GBP),6) = intpos (0 + 6) by A6, SCMP_GCD:1;
hence  (IExec ((while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))))),Q,t)) . GBP = 0 by A5, A6, SCMPDS_8:23; ::_thesis: ( (IExec ((while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))))),Q,t)) . (intpos 1) = t . (intpos 1) & (IExec ((while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))))),Q,t)) . (intpos 2) = t . (intpos 2) & (IExec ((while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))))),Q,t)) . (intpos 3) = t . (intpos 3) )
thus  ( (IExec ((while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))))),Q,t)) . (intpos 1) = t . (intpos 1) & (IExec ((while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))))),Q,t)) . (intpos 2) = t . (intpos 2) & (IExec ((while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))))),Q,t)) . (intpos 3) = t . (intpos 3) ) by A5, A7, SCMPDS_8:23; ::_thesis: verum
end;
A8: 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 A9: S1[k] ; ::_thesis: S1[k + 1]
thus  S1[k + 1] ::_thesis: verum
proof
let t be 0 -started State of SCMPDS; ::_thesis: for Q being Instruction-Sequence of SCMPDS st t . (intpos 6) <= k + 1 & t . (intpos 4) >= 7 + (t . (intpos 6)) & t . GBP = 0 holds
( (IExec ((while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))))),Q,t)) . GBP = 0 & (IExec ((while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))))),Q,t)) . (intpos 1) = t . (intpos 1) & (IExec ((while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))))),Q,t)) . (intpos 2) = t . (intpos 2) & (IExec ((while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))))),Q,t)) . (intpos 3) = t . (intpos 3) )
let Q be Instruction-Sequence of SCMPDS; ::_thesis: ( t . (intpos 6) <= k + 1 & t . (intpos 4) >= 7 + (t . (intpos 6)) & t . GBP = 0 implies ( (IExec ((while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))))),Q,t)) . GBP = 0 & (IExec ((while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))))),Q,t)) . (intpos 1) = t . (intpos 1) & (IExec ((while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))))),Q,t)) . (intpos 2) = t . (intpos 2) & (IExec ((while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))))),Q,t)) . (intpos 3) = t . (intpos 3) ) )
set bt = DataLoc ((t . GBP),6);
assume that
A10: t . (intpos 6) <= k + 1 and
A11: t . (intpos 4) >= 7 + (t . (intpos 6)) and
A12: t . GBP = 0 ; ::_thesis: ( (IExec ((while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))))),Q,t)) . GBP = 0 & (IExec ((while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))))),Q,t)) . (intpos 1) = t . (intpos 1) & (IExec ((while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))))),Q,t)) . (intpos 2) = t . (intpos 2) & (IExec ((while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))))),Q,t)) . (intpos 3) = t . (intpos 3) )
A13: DataLoc ((t . GBP),6) = intpos (0 + 6) by A12, SCMP_GCD:1;
percases ( t . (DataLoc ((t . GBP),6)) <= 0 or t . (DataLoc ((t . GBP),6)) > 0 ) ;
suppose t . (DataLoc ((t . GBP),6)) <= 0 ; ::_thesis: ( (IExec ((while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))))),Q,t)) . GBP = 0 & (IExec ((while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))))),Q,t)) . (intpos 1) = t . (intpos 1) & (IExec ((while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))))),Q,t)) . (intpos 2) = t . (intpos 2) & (IExec ((while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))))),Q,t)) . (intpos 3) = t . (intpos 3) )
hence  ( (IExec ((while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))))),Q,t)) . GBP = 0 & (IExec ((while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))))),Q,t)) . (intpos 1) = t . (intpos 1) & (IExec ((while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))))),Q,t)) . (intpos 2) = t . (intpos 2) & (IExec ((while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))))),Q,t)) . (intpos 3) = t . (intpos 3) ) by A12, SCMPDS_8:23; ::_thesis: verum
end;
supposeA14: t . (DataLoc ((t . GBP),6)) > 0 ; ::_thesis: ( (IExec ((while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))))),Q,t)) . GBP = 0 & (IExec ((while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))))),Q,t)) . (intpos 1) = t . (intpos 1) & (IExec ((while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))))),Q,t)) . (intpos 2) = t . (intpos 2) & (IExec ((while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))))),Q,t)) . (intpos 3) = t . (intpos 3) )
A15: for v being 0 -started State of SCMPDS
for Q being Instruction-Sequence of SCMPDS st ( for x being Int_position st x in {(intpos 4)} holds
v . x >= 7 + (v . (DataLoc ((t . GBP),6))) ) & v . GBP = t . GBP & v . (DataLoc ((t . GBP),6)) > 0 holds
( (IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))),Q,v)) . GBP = v . GBP & (((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0)))) is_closed_on v,Q & (((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0)))) is_halting_on v,Q & (IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))),Q,v)) . (DataLoc ((t . GBP),6)) < v . (DataLoc ((t . GBP),6)) & ( for x being Int_position st x in {(intpos 4)} holds
(IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))),Q,v)) . x >= 7 + ((IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))),Q,v)) . (DataLoc ((t . GBP),6))) ) )
proof
let v be 0 -started State of SCMPDS; ::_thesis: for Q being Instruction-Sequence of SCMPDS st ( for x being Int_position st x in {(intpos 4)} holds
v . x >= 7 + (v . (DataLoc ((t . GBP),6))) ) & v . GBP = t . GBP & v . (DataLoc ((t . GBP),6)) > 0 holds
( (IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))),Q,v)) . GBP = v . GBP & (((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0)))) is_closed_on v,Q & (((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0)))) is_halting_on v,Q & (IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))),Q,v)) . (DataLoc ((t . GBP),6)) < v . (DataLoc ((t . GBP),6)) & ( for x being Int_position st x in {(intpos 4)} holds
(IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))),Q,v)) . x >= 7 + ((IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))),Q,v)) . (DataLoc ((t . GBP),6))) ) )
let V be Instruction-Sequence of SCMPDS; ::_thesis: ( ( for x being Int_position st x in {(intpos 4)} holds
v . x >= 7 + (v . (DataLoc ((t . GBP),6))) ) & v . GBP = t . GBP & v . (DataLoc ((t . GBP),6)) > 0 implies ( (IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))),V,v)) . GBP = v . GBP & (((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0)))) is_closed_on v,V & (((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0)))) is_halting_on v,V & (IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))),V,v)) . (DataLoc ((t . GBP),6)) < v . (DataLoc ((t . GBP),6)) & ( for x being Int_position st x in {(intpos 4)} holds
(IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))),V,v)) . x >= 7 + ((IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))),V,v)) . (DataLoc ((t . GBP),6))) ) ) )
A16: Initialize v = v by MEMSTR_0:44;
assume that
A17: for x being Int_position st x in {(intpos 4)} holds
v . x >= 7 + (v . (DataLoc ((t . GBP),6))) and
A18: ( v . GBP = t . GBP & v . (DataLoc ((t . GBP),6)) > 0 ) ; ::_thesis: ( (IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))),V,v)) . GBP = v . GBP & (((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0)))) is_closed_on v,V & (((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0)))) is_halting_on v,V & (IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))),V,v)) . (DataLoc ((t . GBP),6)) < v . (DataLoc ((t . GBP),6)) & ( for x being Int_position st x in {(intpos 4)} holds
(IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))),V,v)) . x >= 7 + ((IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))),V,v)) . (DataLoc ((t . GBP),6))) ) )
set Iv = IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))),V,(Initialize (Initialize v)));
A19: v . GBP = (Initialize v) . GBP by SCMPDS_5:15;
A20: v . (intpos 4) = (Initialize v) . (intpos 4) by SCMPDS_5:15;
A21: v . (intpos 6) = (Initialize v) . (intpos 6) by SCMPDS_5:15;
A22: intpos 4 in {(intpos 4)} by TARSKI:def_1;
then A23: v . (intpos 4) >= 7 + (v . (intpos 6)) by A13, A17;
hence  (IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))),V,v)) . GBP = v . GBP by A12, A13, A18, Lm6; ::_thesis: ( (((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0)))) is_closed_on v,V & (((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0)))) is_halting_on v,V & (IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))),V,v)) . (DataLoc ((t . GBP),6)) < v . (DataLoc ((t . GBP),6)) & ( for x being Int_position st x in {(intpos 4)} holds
(IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))),V,v)) . x >= 7 + ((IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))),V,v)) . (DataLoc ((t . GBP),6))) ) )
thus  ( (((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0)))) is_closed_on v,V & (((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0)))) is_halting_on v,V ) by SCMPDS_6:20, SCMPDS_6:21; ::_thesis: ( (IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))),V,v)) . (DataLoc ((t . GBP),6)) < v . (DataLoc ((t . GBP),6)) & ( for x being Int_position st x in {(intpos 4)} holds
(IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))),V,v)) . x >= 7 + ((IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))),V,v)) . (DataLoc ((t . GBP),6))) ) )
thus  (IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))),V,v)) . (DataLoc ((t . GBP),6)) < v . (DataLoc ((t . GBP),6)) by A12, A13, A18, A23, Lm7; ::_thesis: for x being Int_position st x in {(intpos 4)} holds
(IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))),V,v)) . x >= 7 + ((IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))),V,v)) . (DataLoc ((t . GBP),6)))
 v . (intpos 4) >= 7 + (v . (DataLoc ((t . GBP),6))) by A17, A22;
then  (IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))),V,(Initialize (Initialize v)))) . (intpos 4) >= 7 + ((IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))),V,(Initialize (Initialize v)))) . (intpos 6)) by A12, A13, A18, Lm7, A19, A20, A21;
hence  for x being Int_position st x in {(intpos 4)} holds
(IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))),V,v)) . x >= 7 + ((IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))),V,v)) . (DataLoc ((t . GBP),6))) by A13, A16, TARSKI:def_1; ::_thesis: verum
end;
set It = IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))),Q,t);
set IT = Q;
A24: (IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))),Q,t)) . (intpos 1) = (Initialize (IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))),Q,t))) . (intpos 1) by SCMPDS_5:15;
A25: (IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))),Q,t)) . (intpos 6) = (Initialize (IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))),Q,t))) . (intpos 6) by SCMPDS_5:15;
A26: (IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))),Q,t)) . (intpos 4) = (Initialize (IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))),Q,t))) . (intpos 4) by SCMPDS_5:15;
A27: (IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))),Q,t)) . GBP = (Initialize (IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))),Q,t))) . GBP by SCMPDS_5:15;
A28: (IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))),Q,t)) . (intpos 2) = (Initialize (IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))),Q,t))) . (intpos 2) by SCMPDS_5:15;
A29: (IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))),Q,t)) . (intpos 3) = (Initialize (IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))),Q,t))) . (intpos 3) by SCMPDS_5:15;
 (IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))),Q,t)) . (intpos 6) < t . (intpos 6) by A11, A12, A13, A14, Lm7;
then  ((IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))),Q,t)) . (intpos 6)) + 1 <= t . (intpos 6) by INT_1:7;
then  ((IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))),Q,t)) . (intpos 6)) + 1 <= k + 1 by A10, XXREAL_0:2;
then A30: (IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))),Q,t)) . (intpos 6) <= k by XREAL_1:6;
A31: ( (IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))),Q,t)) . GBP = 0 & (IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))),Q,t)) . (intpos 4) >= 7 + ((IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))),Q,t)) . (intpos 6)) ) by A11, A12, A13, A14, Lm6, Lm7;
then A32: (IExec ((while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))))),Q,(Initialize (IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))),Q,t))))) . (intpos 1) = (IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))),Q,t)) . (intpos 1) by A9, A30, A24, A25, A26, A27;
A33: (IExec ((while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))))),Q,(Initialize (IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))),Q,t))))) . (intpos 3) = (IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))),Q,t)) . (intpos 3) by A9, A31, A30, A25, A26, A27, A29;
A34: (IExec ((while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))))),Q,(Initialize (IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))),Q,t))))) . (intpos 2) = (IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))),Q,t)) . (intpos 2) by A9, A31, A30, A25, A26, A27, A28;
A35: for x being Int_position st x in {(intpos 4)} holds
t . x >= 7 + (t . (DataLoc ((t . GBP),6))) by A11, A13, TARSKI:def_1;
 (IExec ((while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))))),Q,(Initialize (IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))),Q,t))))) . GBP = 0 by A9, A31, A30, A25, A26, A27;
hence  (IExec ((while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))))),Q,t)) . GBP = 0 by A14, A35, A15, SCMPDS_8:29; ::_thesis: ( (IExec ((while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))))),Q,t)) . (intpos 1) = t . (intpos 1) & (IExec ((while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))))),Q,t)) . (intpos 2) = t . (intpos 2) & (IExec ((while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))))),Q,t)) . (intpos 3) = t . (intpos 3) )
 (IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))),Q,t)) . (intpos 1) = t . (intpos 1) by A11, A12, A13, A14, Lm6;
hence  (IExec ((while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))))),Q,t)) . (intpos 1) = t . (intpos 1) by A14, A35, A15, A32, SCMPDS_8:29; ::_thesis: ( (IExec ((while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))))),Q,t)) . (intpos 2) = t . (intpos 2) & (IExec ((while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))))),Q,t)) . (intpos 3) = t . (intpos 3) )
 (IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))),Q,t)) . (intpos 2) = t . (intpos 2) by A11, A12, A13, A14, Lm7;
hence  (IExec ((while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))))),Q,t)) . (intpos 2) = t . (intpos 2) by A14, A35, A15, A34, SCMPDS_8:29; ::_thesis: (IExec ((while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))))),Q,t)) . (intpos 3) = t . (intpos 3)
 (IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))),Q,t)) . (intpos 3) = t . (intpos 3) by A11, A12, A13, A14, Lm7;
hence  (IExec ((while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))))),Q,t)) . (intpos 3) = t . (intpos 3) by A14, A35, A15, A33, SCMPDS_8:29; ::_thesis: verum
end;
end;
end;
end;
percases ( s . (intpos 6) <= 0 or s . (intpos 6) > 0 ) ;
suppose s . (intpos 6) <= 0 ; ::_thesis: ( (IExec ((while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))))),P,s)) . GBP = 0 & (IExec ((while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))))),P,s)) . (intpos 1) = s . (intpos 1) & (IExec ((while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))))),P,s)) . (intpos 2) = s . (intpos 2) & (IExec ((while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))))),P,s)) . (intpos 3) = s . (intpos 3) )
hence  ( (IExec ((while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))))),P,s)) . GBP = 0 & (IExec ((while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))))),P,s)) . (intpos 1) = s . (intpos 1) & (IExec ((while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))))),P,s)) . (intpos 2) = s . (intpos 2) & (IExec ((while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))))),P,s)) . (intpos 3) = s . (intpos 3) ) by A2, A3, SCMPDS_8:23; ::_thesis: verum
end;
suppose s . (intpos 6) > 0 ; ::_thesis: ( (IExec ((while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))))),P,s)) . GBP = 0 & (IExec ((while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))))),P,s)) . (intpos 1) = s . (intpos 1) & (IExec ((while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))))),P,s)) . (intpos 2) = s . (intpos 2) & (IExec ((while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))))),P,s)) . (intpos 3) = s . (intpos 3) )
then reconsider m = s . (intpos 6) as Element of NAT by INT_1:3;
 for k being Element of NAT holds S1[k] from NAT_1:sch_1(A4, A8);
then  S1[m] ;
hence  ( (IExec ((while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))))),P,s)) . GBP = 0 & (IExec ((while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))))),P,s)) . (intpos 1) = s . (intpos 1) & (IExec ((while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))))),P,s)) . (intpos 2) = s . (intpos 2) & (IExec ((while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))))),P,s)) . (intpos 3) = s . (intpos 3) ) by A1, A2, A3; ::_thesis: verum
end;
end;
end;

Lm10:  for P being Instruction-Sequence of SCMPDS
for s being 0 -started State of SCMPDS st s . GBP = 0 holds
( (IExec (((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))),P,s)) . GBP = 0 & (IExec (((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))),P,s)) . (intpos 1) = (s . (intpos 1)) + 1 & (IExec (((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))),P,s)) . (intpos 2) = s . (intpos 2) & (IExec (((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))),P,s)) . (intpos 3) = (s . (intpos 3)) + 1 & (IExec (((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))),P,s)) . (intpos 4) = (s . (intpos 3)) + 1 & (IExec (((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))),P,s)) . (intpos 6) = (s . (intpos 1)) + 1 & ( for i being Element of NAT st i >= 7 holds
(IExec (((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))),P,s)) . (intpos i) = s . (intpos i) ) )
proof
let P be Instruction-Sequence of SCMPDS; ::_thesis: for s being 0 -started State of SCMPDS st s . GBP = 0 holds
( (IExec (((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))),P,s)) . GBP = 0 & (IExec (((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))),P,s)) . (intpos 1) = (s . (intpos 1)) + 1 & (IExec (((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))),P,s)) . (intpos 2) = s . (intpos 2) & (IExec (((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))),P,s)) . (intpos 3) = (s . (intpos 3)) + 1 & (IExec (((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))),P,s)) . (intpos 4) = (s . (intpos 3)) + 1 & (IExec (((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))),P,s)) . (intpos 6) = (s . (intpos 1)) + 1 & ( for i being Element of NAT st i >= 7 holds
(IExec (((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))),P,s)) . (intpos i) = s . (intpos i) ) )
set a = GBP ;
let s be 0 -started State of SCMPDS; ::_thesis: ( s . GBP = 0 implies ( (IExec (((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))),P,s)) . GBP = 0 & (IExec (((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))),P,s)) . (intpos 1) = (s . (intpos 1)) + 1 & (IExec (((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))),P,s)) . (intpos 2) = s . (intpos 2) & (IExec (((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))),P,s)) . (intpos 3) = (s . (intpos 3)) + 1 & (IExec (((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))),P,s)) . (intpos 4) = (s . (intpos 3)) + 1 & (IExec (((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))),P,s)) . (intpos 6) = (s . (intpos 1)) + 1 & ( for i being Element of NAT st i >= 7 holds
(IExec (((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))),P,s)) . (intpos i) = s . (intpos i) ) ) )
A1: Initialize s = s by MEMSTR_0:44;
set t0 = Initialize s;
set t1 = IExec (((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))),P,s);
set t2 = IExec ((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))),P,s);
set t3 = IExec (((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))),P,s);
set t4 = Exec ((AddTo (GBP,3,1)),s);
assume A2: s . GBP = 0 ; ::_thesis: ( (IExec (((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))),P,s)) . GBP = 0 & (IExec (((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))),P,s)) . (intpos 1) = (s . (intpos 1)) + 1 & (IExec (((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))),P,s)) . (intpos 2) = s . (intpos 2) & (IExec (((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))),P,s)) . (intpos 3) = (s . (intpos 3)) + 1 & (IExec (((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))),P,s)) . (intpos 4) = (s . (intpos 3)) + 1 & (IExec (((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))),P,s)) . (intpos 6) = (s . (intpos 1)) + 1 & ( for i being Element of NAT st i >= 7 holds
(IExec (((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))),P,s)) . (intpos i) = s . (intpos i) ) )
then A3: (Initialize s) . GBP = 0 by SCMPDS_5:15;
then A4: DataLoc (((Initialize s) . GBP),3) = intpos (0 + 3) by SCMP_GCD:1;
then A5: (Exec ((AddTo (GBP,3,1)),s)) . GBP = 0 by A3, A2, AMI_3:10, SCMPDS_2:48;
then A6: DataLoc (((Exec ((AddTo (GBP,3,1)),s)) . GBP),4) = intpos (0 + 4) by SCMP_GCD:1;
A7: (Exec ((AddTo (GBP,3,1)),s)) . (intpos 2) = s . (intpos 2) by A4, A2, A3, AMI_3:10, SCMPDS_2:48;
A8: (IExec (((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))),P,s)) . (intpos 2) = (Exec (((GBP,4) := (GBP,3)),(Exec ((AddTo (GBP,3,1)),s)))) . (intpos 2) by SCMPDS_5:42
.= s . (intpos 2) by A7, A6, AMI_3:10, SCMPDS_2:47 ;
A9: (Exec ((AddTo (GBP,3,1)),s)) . (intpos 1) = s . (intpos 1) by A4, A2, A3, AMI_3:10, SCMPDS_2:48;
A10: (IExec (((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))),P,s)) . (intpos 1) = (Exec (((GBP,4) := (GBP,3)),(Exec ((AddTo (GBP,3,1)),s)))) . (intpos 1) by SCMPDS_5:42
.= s . (intpos 1) by A9, A6, AMI_3:10, SCMPDS_2:47 ;
A11: (Exec ((AddTo (GBP,3,1)),s)) . (intpos 3) = (s . (intpos 3)) + 1 by A4, A1, SCMPDS_2:48;
A12: (IExec (((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))),P,s)) . GBP = (Exec (((GBP,4) := (GBP,3)),(Exec ((AddTo (GBP,3,1)),s)))) . GBP by SCMPDS_5:42
.= 0 by A5, A6, AMI_3:10, SCMPDS_2:47 ;
then A13: DataLoc (((IExec (((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))),P,s)) . GBP),1) = intpos (0 + 1) by SCMP_GCD:1;
A14: DataLoc (((Exec ((AddTo (GBP,3,1)),s)) . GBP),3) = intpos (0 + 3) by A5, SCMP_GCD:1;
A15: (IExec (((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))),P,s)) . (intpos 4) = (Exec (((GBP,4) := (GBP,3)),(Exec ((AddTo (GBP,3,1)),s)))) . (intpos 4) by SCMPDS_5:42
.= (s . (intpos 3)) + 1 by A11, A6, A14, SCMPDS_2:47 ;
A16: (IExec ((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))),P,s)) . (intpos 4) = (Exec ((AddTo (GBP,1,1)),(IExec (((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))),P,s)))) . (intpos 4) by SCMPDS_5:41
.= (s . (intpos 3)) + 1 by A15, A13, AMI_3:10, SCMPDS_2:48 ;
A17: (IExec ((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))),P,s)) . (intpos 2) = (Exec ((AddTo (GBP,1,1)),(IExec (((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))),P,s)))) . (intpos 2) by SCMPDS_5:41
.= s . (intpos 2) by A8, A13, AMI_3:10, SCMPDS_2:48 ;
A18: (IExec ((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))),P,s)) . (intpos 1) = (Exec ((AddTo (GBP,1,1)),(IExec (((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))),P,s)))) . (intpos 1) by SCMPDS_5:41
.= (s . (intpos 1)) + 1 by A10, A13, SCMPDS_2:48 ;
A19: (IExec (((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))),P,s)) . (intpos 3) = (Exec (((GBP,4) := (GBP,3)),(Exec ((AddTo (GBP,3,1)),s)))) . (intpos 3) by SCMPDS_5:42
.= (s . (intpos 3)) + 1 by A11, A6, AMI_3:10, SCMPDS_2:47 ;
A20: (IExec ((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))),P,s)) . (intpos 3) = (Exec ((AddTo (GBP,1,1)),(IExec (((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))),P,s)))) . (intpos 3) by SCMPDS_5:41
.= (s . (intpos 3)) + 1 by A19, A13, AMI_3:10, SCMPDS_2:48 ;
A21: (IExec ((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))),P,s)) . GBP = (Exec ((AddTo (GBP,1,1)),(IExec (((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))),P,s)))) . GBP by SCMPDS_5:41
.= 0 by A12, A13, AMI_3:10, SCMPDS_2:48 ;
then A22: DataLoc (((IExec ((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))),P,s)) . GBP),6) = intpos (0 + 6) by SCMP_GCD:1;
thus (IExec (((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))),P,s)) . GBP = (Exec (((GBP,6) := (GBP,1)),(IExec ((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))),P,s)))) . GBP by SCMPDS_5:41
.= 0 by A21, A22, AMI_3:10, SCMPDS_2:47 ; ::_thesis: ( (IExec (((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))),P,s)) . (intpos 1) = (s . (intpos 1)) + 1 & (IExec (((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))),P,s)) . (intpos 2) = s . (intpos 2) & (IExec (((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))),P,s)) . (intpos 3) = (s . (intpos 3)) + 1 & (IExec (((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))),P,s)) . (intpos 4) = (s . (intpos 3)) + 1 & (IExec (((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))),P,s)) . (intpos 6) = (s . (intpos 1)) + 1 & ( for i being Element of NAT st i >= 7 holds
(IExec (((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))),P,s)) . (intpos i) = s . (intpos i) ) )
thus (IExec (((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))),P,s)) . (intpos 1) = (Exec (((GBP,6) := (GBP,1)),(IExec ((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))),P,s)))) . (intpos 1) by SCMPDS_5:41
.= (s . (intpos 1)) + 1 by A18, A22, AMI_3:10, SCMPDS_2:47 ; ::_thesis: ( (IExec (((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))),P,s)) . (intpos 2) = s . (intpos 2) & (IExec (((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))),P,s)) . (intpos 3) = (s . (intpos 3)) + 1 & (IExec (((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))),P,s)) . (intpos 4) = (s . (intpos 3)) + 1 & (IExec (((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))),P,s)) . (intpos 6) = (s . (intpos 1)) + 1 & ( for i being Element of NAT st i >= 7 holds
(IExec (((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))),P,s)) . (intpos i) = s . (intpos i) ) )
thus (IExec (((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))),P,s)) . (intpos 2) = (Exec (((GBP,6) := (GBP,1)),(IExec ((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))),P,s)))) . (intpos 2) by SCMPDS_5:41
.= s . (intpos 2) by A17, A22, AMI_3:10, SCMPDS_2:47 ; ::_thesis: ( (IExec (((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))),P,s)) . (intpos 3) = (s . (intpos 3)) + 1 & (IExec (((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))),P,s)) . (intpos 4) = (s . (intpos 3)) + 1 & (IExec (((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))),P,s)) . (intpos 6) = (s . (intpos 1)) + 1 & ( for i being Element of NAT st i >= 7 holds
(IExec (((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))),P,s)) . (intpos i) = s . (intpos i) ) )
A23: DataLoc (((IExec ((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))),P,s)) . GBP),1) = intpos (0 + 1) by A21, SCMP_GCD:1;
thus (IExec (((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))),P,s)) . (intpos 3) = (Exec (((GBP,6) := (GBP,1)),(IExec ((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))),P,s)))) . (intpos 3) by SCMPDS_5:41
.= (s . (intpos 3)) + 1 by A20, A22, AMI_3:10, SCMPDS_2:47 ; ::_thesis: ( (IExec (((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))),P,s)) . (intpos 4) = (s . (intpos 3)) + 1 & (IExec (((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))),P,s)) . (intpos 6) = (s . (intpos 1)) + 1 & ( for i being Element of NAT st i >= 7 holds
(IExec (((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))),P,s)) . (intpos i) = s . (intpos i) ) )
thus (IExec (((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))),P,s)) . (intpos 4) = (Exec (((GBP,6) := (GBP,1)),(IExec ((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))),P,s)))) . (intpos 4) by SCMPDS_5:41
.= (s . (intpos 3)) + 1 by A16, A22, AMI_3:10, SCMPDS_2:47 ; ::_thesis: ( (IExec (((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))),P,s)) . (intpos 6) = (s . (intpos 1)) + 1 & ( for i being Element of NAT st i >= 7 holds
(IExec (((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))),P,s)) . (intpos i) = s . (intpos i) ) )
thus (IExec (((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))),P,s)) . (intpos 6) = (Exec (((GBP,6) := (GBP,1)),(IExec ((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))),P,s)))) . (intpos 6) by SCMPDS_5:41
.= (s . (intpos 1)) + 1 by A18, A22, A23, SCMPDS_2:47 ; ::_thesis: for i being Element of NAT st i >= 7 holds
(IExec (((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))),P,s)) . (intpos i) = s . (intpos i)
A24: now__::_thesis:_for_i_being_Element_of_NAT_st_i_>=_7_holds_
(Exec_((AddTo_(GBP,3,1)),s))_._(intpos_i)_=_s_._(intpos_i)
let i be Element of NAT ; ::_thesis: ( i >= 7 implies (Exec ((AddTo (GBP,3,1)),s)) . (intpos i) = s . (intpos i) )
assume  i >= 7 ; ::_thesis: (Exec ((AddTo (GBP,3,1)),s)) . (intpos i) = s . (intpos i)
then  i > 3 by XXREAL_0:2;
hence (Exec ((AddTo (GBP,3,1)),s)) . (intpos i) = (Initialize s) . (intpos i) by A4, A1, AMI_3:10, SCMPDS_2:48
.= s . (intpos i) by SCMPDS_5:15 ;
::_thesis: verum
end;
A25: now__::_thesis:_for_i_being_Element_of_NAT_st_i_>=_7_holds_
(IExec_(((AddTo_(GBP,3,1))_';'_((GBP,4)_:=_(GBP,3))),P,s))_._(intpos_i)_=_s_._(intpos_i)
let i be Element of NAT ; ::_thesis: ( i >= 7 implies (IExec (((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))),P,s)) . (intpos i) = s . (intpos i) )
assume A26: i >= 7 ; ::_thesis: (IExec (((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))),P,s)) . (intpos i) = s . (intpos i)
then A27: i > 4 by XXREAL_0:2;
thus (IExec (((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))),P,s)) . (intpos i) = (Exec (((GBP,4) := (GBP,3)),(Exec ((AddTo (GBP,3,1)),s)))) . (intpos i) by SCMPDS_5:42
.= (Exec ((AddTo (GBP,3,1)),s)) . (intpos i) by A6, A27, AMI_3:10, SCMPDS_2:47
.= s . (intpos i) by A24, A26 ; ::_thesis: verum
end;
A28: now__::_thesis:_for_i_being_Element_of_NAT_st_i_>=_7_holds_
(IExec_((((AddTo_(GBP,3,1))_';'_((GBP,4)_:=_(GBP,3)))_';'_(AddTo_(GBP,1,1))),P,s))_._(intpos_i)_=_s_._(intpos_i)
let i be Element of NAT ; ::_thesis: ( i >= 7 implies (IExec ((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))),P,s)) . (intpos i) = s . (intpos i) )
assume A29: i >= 7 ; ::_thesis: (IExec ((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))),P,s)) . (intpos i) = s . (intpos i)
then A30: i > 1 by XXREAL_0:2;
thus (IExec ((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))),P,s)) . (intpos i) = (Exec ((AddTo (GBP,1,1)),(IExec (((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))),P,s)))) . (intpos i) by SCMPDS_5:41
.= (IExec (((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))),P,s)) . (intpos i) by A13, A30, AMI_3:10, SCMPDS_2:48
.= s . (intpos i) by A25, A29 ; ::_thesis: verum
end;
hereby  ::_thesis: verum
let i be Element of NAT ; ::_thesis: ( i >= 7 implies (IExec (((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))),P,s)) . (intpos i) = s . (intpos i) )
assume A31: i >= 7 ; ::_thesis: (IExec (((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))),P,s)) . (intpos i) = s . (intpos i)
then A32: i > 6 by XXREAL_0:2;
thus (IExec (((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))),P,s)) . (intpos i) = (Exec (((GBP,6) := (GBP,1)),(IExec ((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))),P,s)))) . (intpos i) by SCMPDS_5:41
.= (IExec ((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))),P,s)) . (intpos i) by A22, A32, AMI_3:10, SCMPDS_2:47
.= s . (intpos i) by A28, A31 ; ::_thesis: verum
end;
end;

set jf = AddTo (GBP,2,(- 1));

set B3 = (((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))) ';' (while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0)))))))) ';' (AddTo (GBP,2,(- 1)));

Lm11:  for P being Instruction-Sequence of SCMPDS
for s being 0 -started State of SCMPDS st s . (intpos 3) >= (s . (intpos 1)) + 7 & s . GBP = 0 holds
( (IExec (((((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))) ';' (while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0)))))))) ';' (AddTo (GBP,2,(- 1)))),P,s)) . GBP = 0 & (IExec (((((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))) ';' (while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0)))))))) ';' (AddTo (GBP,2,(- 1)))),P,s)) . (intpos 2) = (s . (intpos 2)) - 1 & (IExec (((((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))) ';' (while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0)))))))) ';' (AddTo (GBP,2,(- 1)))),P,s)) . (intpos 3) = (s . (intpos 3)) + 1 & (IExec (((((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))) ';' (while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0)))))))) ';' (AddTo (GBP,2,(- 1)))),P,s)) . (intpos 1) = (s . (intpos 1)) + 1 & ( for i being Element of NAT st i <> 2 holds
(IExec (((((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))) ';' (while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0)))))))) ';' (AddTo (GBP,2,(- 1)))),P,s)) . (intpos i) = (IExec ((while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))))),P,(Initialize (IExec (((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))),P,s))))) . (intpos i) ) )
proof
let P be Instruction-Sequence of SCMPDS; ::_thesis: for s being 0 -started State of SCMPDS st s . (intpos 3) >= (s . (intpos 1)) + 7 & s . GBP = 0 holds
( (IExec (((((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))) ';' (while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0)))))))) ';' (AddTo (GBP,2,(- 1)))),P,s)) . GBP = 0 & (IExec (((((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))) ';' (while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0)))))))) ';' (AddTo (GBP,2,(- 1)))),P,s)) . (intpos 2) = (s . (intpos 2)) - 1 & (IExec (((((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))) ';' (while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0)))))))) ';' (AddTo (GBP,2,(- 1)))),P,s)) . (intpos 3) = (s . (intpos 3)) + 1 & (IExec (((((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))) ';' (while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0)))))))) ';' (AddTo (GBP,2,(- 1)))),P,s)) . (intpos 1) = (s . (intpos 1)) + 1 & ( for i being Element of NAT st i <> 2 holds
(IExec (((((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))) ';' (while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0)))))))) ';' (AddTo (GBP,2,(- 1)))),P,s)) . (intpos i) = (IExec ((while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))))),P,(Initialize (IExec (((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))),P,s))))) . (intpos i) ) )
set a = GBP ;
let s be 0 -started State of SCMPDS; ::_thesis: ( s . (intpos 3) >= (s . (intpos 1)) + 7 & s . GBP = 0 implies ( (IExec (((((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))) ';' (while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0)))))))) ';' (AddTo (GBP,2,(- 1)))),P,s)) . GBP = 0 & (IExec (((((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))) ';' (while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0)))))))) ';' (AddTo (GBP,2,(- 1)))),P,s)) . (intpos 2) = (s . (intpos 2)) - 1 & (IExec (((((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))) ';' (while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0)))))))) ';' (AddTo (GBP,2,(- 1)))),P,s)) . (intpos 3) = (s . (intpos 3)) + 1 & (IExec (((((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))) ';' (while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0)))))))) ';' (AddTo (GBP,2,(- 1)))),P,s)) . (intpos 1) = (s . (intpos 1)) + 1 & ( for i being Element of NAT st i <> 2 holds
(IExec (((((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))) ';' (while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0)))))))) ';' (AddTo (GBP,2,(- 1)))),P,s)) . (intpos i) = (IExec ((while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))))),P,(Initialize (IExec (((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))),P,s))))) . (intpos i) ) ) )
set s1 = IExec (((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))),P,s);
set Bt = IExec ((((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))) ';' (while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0)))))))),P,s);
set P1 = P;
A1: Initialize s = s by MEMSTR_0:44;
A2: (IExec (((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))),P,s)) . GBP = (Initialize (IExec (((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))),P,s))) . GBP by SCMPDS_5:15;
A3: (IExec (((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))),P,s)) . (intpos 4) = (Initialize (IExec (((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))),P,s))) . (intpos 4) by SCMPDS_5:15;
A4: (IExec (((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))),P,s)) . (intpos 6) = (Initialize (IExec (((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))),P,s))) . (intpos 6) by SCMPDS_5:15;
A5: (IExec (((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))),P,s)) . (intpos 3) = (Initialize (IExec (((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))),P,s))) . (intpos 3) by SCMPDS_5:15;
A6: (IExec (((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))),P,s)) . (intpos 1) = (Initialize (IExec (((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))),P,s))) . (intpos 1) by SCMPDS_5:15;
A7: (IExec (((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))),P,s)) . (intpos 2) = (Initialize (IExec (((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))),P,s))) . (intpos 2) by SCMPDS_5:15;
assume that
A8: s . (intpos 3) >= (s . (intpos 1)) + 7 and
A9: s . GBP = 0 ; ::_thesis: ( (IExec (((((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))) ';' (while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0)))))))) ';' (AddTo (GBP,2,(- 1)))),P,s)) . GBP = 0 & (IExec (((((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))) ';' (while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0)))))))) ';' (AddTo (GBP,2,(- 1)))),P,s)) . (intpos 2) = (s . (intpos 2)) - 1 & (IExec (((((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))) ';' (while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0)))))))) ';' (AddTo (GBP,2,(- 1)))),P,s)) . (intpos 3) = (s . (intpos 3)) + 1 & (IExec (((((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))) ';' (while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0)))))))) ';' (AddTo (GBP,2,(- 1)))),P,s)) . (intpos 1) = (s . (intpos 1)) + 1 & ( for i being Element of NAT st i <> 2 holds
(IExec (((((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))) ';' (while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0)))))))) ';' (AddTo (GBP,2,(- 1)))),P,s)) . (intpos i) = (IExec ((while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))))),P,(Initialize (IExec (((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))),P,s))))) . (intpos i) ) )
A10: (IExec (((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))),P,s)) . (intpos 1) = (s . (intpos 1)) + 1 by A9, Lm10;
A11: (IExec (((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))),P,s)) . (intpos 3) = (s . (intpos 3)) + 1 by A9, Lm10;
A12: (IExec (((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))),P,s)) . (intpos 2) = s . (intpos 2) by A9, Lm10;
 ( (IExec (((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))),P,s)) . (intpos 6) = (s . (intpos 1)) + 1 & (s . (intpos 3)) + 1 >= (7 + (s . (intpos 1))) + 1 ) by A8, A9, Lm10, XREAL_1:6;
then A13: (IExec (((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))),P,s)) . (intpos 4) >= 7 + ((IExec (((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))),P,s)) . (intpos 6)) by A9, Lm10;
A14: (IExec (((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))),P,s)) . GBP = 0 by A9, Lm10;
then A15: DataLoc (((IExec (((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))),P,s)) . GBP),6) = intpos (0 + 6) by SCMP_GCD:1;
then  ( while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0)))))) is_closed_on Initialize (IExec (((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))),P,s)),P & while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0)))))) is_halting_on Initialize (IExec (((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))),P,s)),P ) by A14, A13, Lm8, A2, A3, A4;
then A16: ( while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0)))))) is_closed_on IExec (((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))),P,s),P & while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0)))))) is_halting_on IExec (((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))),P,s),P ) by SCMPDS_6:125, SCMPDS_6:126;
A17: ( (((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1)) is_closed_on s,P & (((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1)) is_halting_on s,P ) by SCMPDS_6:20, SCMPDS_6:21;
then A18: (IExec ((((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))) ';' (while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0)))))))),P,s)) . GBP = (IExec ((while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))))),P,(Initialize (IExec (((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))),P,s))))) . GBP by A16, SCMPDS_7:30
.= 0 by A14, A15, A13, Lm9, A2, A3, A4 ;
then A19: DataLoc (((IExec ((((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))) ';' (while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0)))))))),P,s)) . GBP),2) = intpos (0 + 2) by SCMP_GCD:1;
A20: ( ((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))) ';' (while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))))) is_closed_on s,P & ((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))) ';' (while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))))) is_halting_on s,P ) by A16, A17, A1, SCMPDS_7:24;
hence (IExec (((((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))) ';' (while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0)))))))) ';' (AddTo (GBP,2,(- 1)))),P,s)) . GBP = (Exec ((AddTo (GBP,2,(- 1))),(IExec ((((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))) ';' (while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0)))))))),P,s)))) . GBP by SCMPDS_7:31
.= 0 by A18, A19, AMI_3:10, SCMPDS_2:48 ;
::_thesis: ( (IExec (((((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))) ';' (while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0)))))))) ';' (AddTo (GBP,2,(- 1)))),P,s)) . (intpos 2) = (s . (intpos 2)) - 1 & (IExec (((((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))) ';' (while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0)))))))) ';' (AddTo (GBP,2,(- 1)))),P,s)) . (intpos 3) = (s . (intpos 3)) + 1 & (IExec (((((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))) ';' (while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0)))))))) ';' (AddTo (GBP,2,(- 1)))),P,s)) . (intpos 1) = (s . (intpos 1)) + 1 & ( for i being Element of NAT st i <> 2 holds
(IExec (((((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))) ';' (while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0)))))))) ';' (AddTo (GBP,2,(- 1)))),P,s)) . (intpos i) = (IExec ((while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))))),P,(Initialize (IExec (((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))),P,s))))) . (intpos i) ) )
thus (IExec (((((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))) ';' (while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0)))))))) ';' (AddTo (GBP,2,(- 1)))),P,s)) . (intpos 2) = (Exec ((AddTo (GBP,2,(- 1))),(IExec ((((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))) ';' (while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0)))))))),P,s)))) . (intpos 2) by A20, SCMPDS_7:31
.= ((IExec ((((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))) ';' (while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0)))))))),P,s)) . (intpos 2)) + (- 1) by A19, SCMPDS_2:48
.= ((IExec ((((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))) ';' (while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0)))))))),P,s)) . (intpos 2)) - 1
.= ((IExec ((while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))))),P,(Initialize (IExec (((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))),P,s))))) . (intpos 2)) - 1 by A16, A17, SCMPDS_7:30
.= (s . (intpos 2)) - 1 by A14, A12, A15, A13, Lm9, A2, A3, A4, A7 ; ::_thesis: ( (IExec (((((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))) ';' (while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0)))))))) ';' (AddTo (GBP,2,(- 1)))),P,s)) . (intpos 3) = (s . (intpos 3)) + 1 & (IExec (((((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))) ';' (while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0)))))))) ';' (AddTo (GBP,2,(- 1)))),P,s)) . (intpos 1) = (s . (intpos 1)) + 1 & ( for i being Element of NAT st i <> 2 holds
(IExec (((((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))) ';' (while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0)))))))) ';' (AddTo (GBP,2,(- 1)))),P,s)) . (intpos i) = (IExec ((while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))))),P,(Initialize (IExec (((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))),P,s))))) . (intpos i) ) )
thus (IExec (((((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))) ';' (while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0)))))))) ';' (AddTo (GBP,2,(- 1)))),P,s)) . (intpos 3) = (Exec ((AddTo (GBP,2,(- 1))),(IExec ((((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))) ';' (while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0)))))))),P,s)))) . (intpos 3) by A20, SCMPDS_7:31
.= (IExec ((((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))) ';' (while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0)))))))),P,s)) . (intpos 3) by A19, AMI_3:10, SCMPDS_2:48
.= (IExec ((while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))))),P,(Initialize (IExec (((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))),P,s))))) . (intpos 3) by A16, A17, SCMPDS_7:30
.= (s . (intpos 3)) + 1 by A14, A11, A15, A13, Lm9, A2, A3, A4, A5 ; ::_thesis: ( (IExec (((((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))) ';' (while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0)))))))) ';' (AddTo (GBP,2,(- 1)))),P,s)) . (intpos 1) = (s . (intpos 1)) + 1 & ( for i being Element of NAT st i <> 2 holds
(IExec (((((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))) ';' (while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0)))))))) ';' (AddTo (GBP,2,(- 1)))),P,s)) . (intpos i) = (IExec ((while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))))),P,(Initialize (IExec (((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))),P,s))))) . (intpos i) ) )
thus (IExec (((((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))) ';' (while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0)))))))) ';' (AddTo (GBP,2,(- 1)))),P,s)) . (intpos 1) = (Exec ((AddTo (GBP,2,(- 1))),(IExec ((((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))) ';' (while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0)))))))),P,s)))) . (intpos 1) by A20, SCMPDS_7:31
.= (IExec ((((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))) ';' (while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0)))))))),P,s)) . (intpos 1) by A19, AMI_3:10, SCMPDS_2:48
.= (IExec ((while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))))),P,(Initialize (IExec (((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))),P,s))))) . (intpos 1) by A16, A17, SCMPDS_7:30
.= (s . (intpos 1)) + 1 by A14, A10, A15, A13, Lm9, A2, A3, A4, A6 ; ::_thesis: for i being Element of NAT st i <> 2 holds
(IExec (((((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))) ';' (while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0)))))))) ';' (AddTo (GBP,2,(- 1)))),P,s)) . (intpos i) = (IExec ((while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))))),P,(Initialize (IExec (((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))),P,s))))) . (intpos i)
hereby  ::_thesis: verum
let i be Element of NAT ; ::_thesis: ( i <> 2 implies (IExec (((((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))) ';' (while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0)))))))) ';' (AddTo (GBP,2,(- 1)))),P,s)) . (intpos i) = (IExec ((while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))))),P,(Initialize (IExec (((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))),P,s))))) . (intpos i) )
assume A21: i <> 2 ; ::_thesis: (IExec (((((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))) ';' (while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0)))))))) ';' (AddTo (GBP,2,(- 1)))),P,s)) . (intpos i) = (IExec ((while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))))),P,(Initialize (IExec (((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))),P,s))))) . (intpos i)
thus (IExec (((((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))) ';' (while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0)))))))) ';' (AddTo (GBP,2,(- 1)))),P,s)) . (intpos i) = (Exec ((AddTo (GBP,2,(- 1))),(IExec ((((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))) ';' (while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0)))))))),P,s)))) . (intpos i) by A20, SCMPDS_7:31
.= (IExec ((((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))) ';' (while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0)))))))),P,s)) . (intpos i) by A19, A21, AMI_3:10, SCMPDS_2:48
.= (IExec ((while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))))),P,(Initialize (IExec (((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))),P,s))))) . (intpos i) by A16, A17, SCMPDS_7:30 ; ::_thesis: verum
end;
end;

Lm12:  for P being Instruction-Sequence of SCMPDS
for s being 0 -started State of SCMPDS st s . (intpos 3) >= (s . (intpos 1)) + 7 & s . GBP = 0 holds
( for-down (GBP,2,1,(((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))) ';' (while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))))))) is_closed_on s,P & for-down (GBP,2,1,(((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))) ';' (while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))))))) is_halting_on s,P )
proof
let P be Instruction-Sequence of SCMPDS; ::_thesis: for s being 0 -started State of SCMPDS st s . (intpos 3) >= (s . (intpos 1)) + 7 & s . GBP = 0 holds
( for-down (GBP,2,1,(((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))) ';' (while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))))))) is_closed_on s,P & for-down (GBP,2,1,(((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))) ';' (while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))))))) is_halting_on s,P )
let s be 0 -started State of SCMPDS; ::_thesis: ( s . (intpos 3) >= (s . (intpos 1)) + 7 & s . GBP = 0 implies ( for-down (GBP,2,1,(((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))) ';' (while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))))))) is_closed_on s,P & for-down (GBP,2,1,(((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))) ';' (while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))))))) is_halting_on s,P ) )
set a = GBP ;
set b = DataLoc ((s . GBP),2);
assume that
A1: s . (intpos 3) >= (s . (intpos 1)) + 7 and
A2: s . GBP = 0 ; ::_thesis: ( for-down (GBP,2,1,(((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))) ';' (while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))))))) is_closed_on s,P & for-down (GBP,2,1,(((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))) ';' (while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))))))) is_halting_on s,P )
A3: DataLoc ((s . GBP),2) = intpos (0 + 2) by A2, SCMP_GCD:1;
now__::_thesis:_for_t_being_0_-started_State_of_SCMPDS
for_Q_being_Instruction-Sequence_of_SCMPDS_st_t_._(intpos_3)_>=_(t_._(intpos_1))_+_7_&_t_._GBP_=_s_._GBP_&_t_._(DataLoc_((s_._GBP),2))_>_0_holds_
(_(IExec_(((((((AddTo_(GBP,3,1))_';'_((GBP,4)_:=_(GBP,3)))_';'_(AddTo_(GBP,1,1)))_';'_((GBP,6)_:=_(GBP,1)))_';'_(while>0_(GBP,6,((((GBP,5)_:=_((intpos_4),(-_1)))_';'_(SubFrom_(GBP,5,(intpos_4),0)))_';'_(if>0_(GBP,5,((((((GBP,5)_:=_((intpos_4),(-_1)))_';'_(((intpos_4),(-_1))_:=_((intpos_4),0)))_';'_(((intpos_4),0)_:=_(GBP,5)))_';'_(AddTo_(GBP,4,(-_1))))_';'_(AddTo_(GBP,6,(-_1)))),(Load_((GBP,6)_:=_0))))))))_';'_(AddTo_(GBP,2,(-_1)))),Q,t))_._GBP_=_t_._GBP_&_(IExec_(((((((AddTo_(GBP,3,1))_';'_((GBP,4)_:=_(GBP,3)))_';'_(AddTo_(GBP,1,1)))_';'_((GBP,6)_:=_(GBP,1)))_';'_(while>0_(GBP,6,((((GBP,5)_:=_((intpos_4),(-_1)))_';'_(SubFrom_(GBP,5,(intpos_4),0)))_';'_(if>0_(GBP,5,((((((GBP,5)_:=_((intpos_4),(-_1)))_';'_(((intpos_4),(-_1))_:=_((intpos_4),0)))_';'_(((intpos_4),0)_:=_(GBP,5)))_';'_(AddTo_(GBP,4,(-_1))))_';'_(AddTo_(GBP,6,(-_1)))),(Load_((GBP,6)_:=_0))))))))_';'_(AddTo_(GBP,2,(-_1)))),Q,t))_._(DataLoc_((s_._GBP),2))_=_(t_._(DataLoc_((s_._GBP),2)))_-_1_&_((((AddTo_(GBP,3,1))_';'_((GBP,4)_:=_(GBP,3)))_';'_(AddTo_(GBP,1,1)))_';'_((GBP,6)_:=_(GBP,1)))_';'_(while>0_(GBP,6,((((GBP,5)_:=_((intpos_4),(-_1)))_';'_(SubFrom_(GBP,5,(intpos_4),0)))_';'_(if>0_(GBP,5,((((((GBP,5)_:=_((intpos_4),(-_1)))_';'_(((intpos_4),(-_1))_:=_((intpos_4),0)))_';'_(((intpos_4),0)_:=_(GBP,5)))_';'_(AddTo_(GBP,4,(-_1))))_';'_(AddTo_(GBP,6,(-_1)))),(Load_((GBP,6)_:=_0)))))))_is_closed_on_t,Q_&_((((AddTo_(GBP,3,1))_';'_((GBP,4)_:=_(GBP,3)))_';'_(AddTo_(GBP,1,1)))_';'_((GBP,6)_:=_(GBP,1)))_';'_(while>0_(GBP,6,((((GBP,5)_:=_((intpos_4),(-_1)))_';'_(SubFrom_(GBP,5,(intpos_4),0)))_';'_(if>0_(GBP,5,((((((GBP,5)_:=_((intpos_4),(-_1)))_';'_(((intpos_4),(-_1))_:=_((intpos_4),0)))_';'_(((intpos_4),0)_:=_(GBP,5)))_';'_(AddTo_(GBP,4,(-_1))))_';'_(AddTo_(GBP,6,(-_1)))),(Load_((GBP,6)_:=_0)))))))_is_halting_on_t,Q_&_(IExec_(((((((AddTo_(GBP,3,1))_';'_((GBP,4)_:=_(GBP,3)))_';'_(AddTo_(GBP,1,1)))_';'_((GBP,6)_:=_(GBP,1)))_';'_(while>0_(GBP,6,((((GBP,5)_:=_((intpos_4),(-_1)))_';'_(SubFrom_(GBP,5,(intpos_4),0)))_';'_(if>0_(GBP,5,((((((GBP,5)_:=_((intpos_4),(-_1)))_';'_(((intpos_4),(-_1))_:=_((intpos_4),0)))_';'_(((intpos_4),0)_:=_(GBP,5)))_';'_(AddTo_(GBP,4,(-_1))))_';'_(AddTo_(GBP,6,(-_1)))),(Load_((GBP,6)_:=_0))))))))_';'_(AddTo_(GBP,2,(-_1)))),Q,t))_._(intpos_3)_>=_((IExec_(((((((AddTo_(GBP,3,1))_';'_((GBP,4)_:=_(GBP,3)))_';'_(AddTo_(GBP,1,1)))_';'_((GBP,6)_:=_(GBP,1)))_';'_(while>0_(GBP,6,((((GBP,5)_:=_((intpos_4),(-_1)))_';'_(SubFrom_(GBP,5,(intpos_4),0)))_';'_(if>0_(GBP,5,((((((GBP,5)_:=_((intpos_4),(-_1)))_';'_(((intpos_4),(-_1))_:=_((intpos_4),0)))_';'_(((intpos_4),0)_:=_(GBP,5)))_';'_(AddTo_(GBP,4,(-_1))))_';'_(AddTo_(GBP,6,(-_1)))),(Load_((GBP,6)_:=_0))))))))_';'_(AddTo_(GBP,2,(-_1)))),Q,t))_._(intpos_1))_+_7_)
let t be 0 -started State of SCMPDS; ::_thesis: for Q being Instruction-Sequence of SCMPDS st t . (intpos 3) >= (t . (intpos 1)) + 7 & t . GBP = s . GBP & t . (DataLoc ((s . GBP),2)) > 0 holds
( (IExec (((((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))) ';' (while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0)))))))) ';' (AddTo (GBP,2,(- 1)))),Q,t)) . GBP = t . GBP & (IExec (((((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))) ';' (while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0)))))))) ';' (AddTo (GBP,2,(- 1)))),Q,t)) . (DataLoc ((s . GBP),2)) = (t . (DataLoc ((s . GBP),2))) - 1 & ((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))) ';' (while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))))) is_closed_on t,Q & ((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))) ';' (while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))))) is_halting_on t,Q & (IExec (((((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))) ';' (while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0)))))))) ';' (AddTo (GBP,2,(- 1)))),Q,t)) . (intpos 3) >= ((IExec (((((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))) ';' (while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0)))))))) ';' (AddTo (GBP,2,(- 1)))),Q,t)) . (intpos 1)) + 7 )
let Q be Instruction-Sequence of SCMPDS; ::_thesis: ( t . (intpos 3) >= (t . (intpos 1)) + 7 & t . GBP = s . GBP & t . (DataLoc ((s . GBP),2)) > 0 implies ( (IExec (((((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))) ';' (while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0)))))))) ';' (AddTo (GBP,2,(- 1)))),Q,t)) . GBP = t . GBP & (IExec (((((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))) ';' (while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0)))))))) ';' (AddTo (GBP,2,(- 1)))),Q,t)) . (DataLoc ((s . GBP),2)) = (t . (DataLoc ((s . GBP),2))) - 1 & ((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))) ';' (while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))))) is_closed_on t,Q & ((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))) ';' (while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))))) is_halting_on t,Q & (IExec (((((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))) ';' (while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0)))))))) ';' (AddTo (GBP,2,(- 1)))),Q,t)) . (intpos 3) >= ((IExec (((((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))) ';' (while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0)))))))) ';' (AddTo (GBP,2,(- 1)))),Q,t)) . (intpos 1)) + 7 ) )
A4: Initialize t = t by MEMSTR_0:44;
assume that
A5: t . (intpos 3) >= (t . (intpos 1)) + 7 and
A6: t . GBP = s . GBP and
 t . (DataLoc ((s . GBP),2)) > 0 ; ::_thesis: ( (IExec (((((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))) ';' (while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0)))))))) ';' (AddTo (GBP,2,(- 1)))),Q,t)) . GBP = t . GBP & (IExec (((((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))) ';' (while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0)))))))) ';' (AddTo (GBP,2,(- 1)))),Q,t)) . (DataLoc ((s . GBP),2)) = (t . (DataLoc ((s . GBP),2))) - 1 & ((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))) ';' (while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))))) is_closed_on t,Q & ((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))) ';' (while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))))) is_halting_on t,Q & (IExec (((((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))) ';' (while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0)))))))) ';' (AddTo (GBP,2,(- 1)))),Q,t)) . (intpos 3) >= ((IExec (((((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))) ';' (while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0)))))))) ';' (AddTo (GBP,2,(- 1)))),Q,t)) . (intpos 1)) + 7 )
set t1 = IExec (((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))),Q,t);
set Q1 = Q;
A7: (IExec (((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))),Q,t)) . GBP = (Initialize (IExec (((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))),Q,t))) . GBP by SCMPDS_5:15;
A8: (IExec (((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))),Q,t)) . (intpos 4) = (Initialize (IExec (((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))),Q,t))) . (intpos 4) by SCMPDS_5:15;
A9: (IExec (((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))),Q,t)) . (intpos 6) = (Initialize (IExec (((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))),Q,t))) . (intpos 6) by SCMPDS_5:15;
A10: (t . (intpos 3)) + 1 >= (7 + (t . (intpos 1))) + 1 by A5, XREAL_1:6;
thus  (IExec (((((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))) ';' (while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0)))))))) ';' (AddTo (GBP,2,(- 1)))),Q,t)) . GBP = t . GBP by A2, A5, A6, Lm11; ::_thesis: ( (IExec (((((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))) ';' (while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0)))))))) ';' (AddTo (GBP,2,(- 1)))),Q,t)) . (DataLoc ((s . GBP),2)) = (t . (DataLoc ((s . GBP),2))) - 1 & ((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))) ';' (while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))))) is_closed_on t,Q & ((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))) ';' (while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))))) is_halting_on t,Q & (IExec (((((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))) ';' (while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0)))))))) ';' (AddTo (GBP,2,(- 1)))),Q,t)) . (intpos 3) >= ((IExec (((((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))) ';' (while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0)))))))) ';' (AddTo (GBP,2,(- 1)))),Q,t)) . (intpos 1)) + 7 )
thus  (IExec (((((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))) ';' (while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0)))))))) ';' (AddTo (GBP,2,(- 1)))),Q,t)) . (DataLoc ((s . GBP),2)) = (t . (DataLoc ((s . GBP),2))) - 1 by A2, A3, A5, A6, Lm11; ::_thesis: ( ((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))) ';' (while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))))) is_closed_on t,Q & ((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))) ';' (while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))))) is_halting_on t,Q & (IExec (((((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))) ';' (while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0)))))))) ';' (AddTo (GBP,2,(- 1)))),Q,t)) . (intpos 3) >= ((IExec (((((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))) ';' (while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0)))))))) ';' (AddTo (GBP,2,(- 1)))),Q,t)) . (intpos 1)) + 7 )
A11: ( (((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1)) is_closed_on t,Q & (((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1)) is_halting_on t,Q ) by SCMPDS_6:20, SCMPDS_6:21;
 (IExec (((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))),Q,t)) . (intpos 6) = (t . (intpos 1)) + 1 by A2, A6, Lm10;
then A12: (IExec (((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))),Q,t)) . (intpos 4) >= 7 + ((IExec (((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))),Q,t)) . (intpos 6)) by A2, A6, A10, Lm10;
A13: (IExec (((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))),Q,t)) . GBP = 0 by A2, A6, Lm10;
then  DataLoc (((IExec (((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))),Q,t)) . GBP),6) = intpos (0 + 6) by SCMP_GCD:1;
then  ( while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0)))))) is_closed_on Initialize (IExec (((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))),Q,t)),Q & while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0)))))) is_halting_on Initialize (IExec (((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))),Q,t)),Q ) by A13, A12, Lm8, A8, A9, A7;
then  ( while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0)))))) is_closed_on IExec (((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))),Q,t),Q & while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0)))))) is_halting_on IExec (((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))),Q,t),Q ) by SCMPDS_6:125, SCMPDS_6:126;
hence  ( ((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))) ';' (while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))))) is_closed_on t,Q & ((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))) ';' (while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))))) is_halting_on t,Q ) by A11, A4, SCMPDS_7:24; ::_thesis: (IExec (((((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))) ';' (while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0)))))))) ';' (AddTo (GBP,2,(- 1)))),Q,t)) . (intpos 3) >= ((IExec (((((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))) ';' (while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0)))))))) ';' (AddTo (GBP,2,(- 1)))),Q,t)) . (intpos 1)) + 7
 (IExec (((((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))) ';' (while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0)))))))) ';' (AddTo (GBP,2,(- 1)))),Q,t)) . (intpos 1) = (t . (intpos 1)) + 1 by A2, A5, A6, Lm11;
hence  (IExec (((((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))) ';' (while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0)))))))) ';' (AddTo (GBP,2,(- 1)))),Q,t)) . (intpos 3) >= ((IExec (((((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))) ';' (while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0)))))))) ';' (AddTo (GBP,2,(- 1)))),Q,t)) . (intpos 1)) + 7 by A2, A5, A6, A10, Lm11; ::_thesis: verum
end;
hence  ( for-down (GBP,2,1,(((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))) ';' (while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))))))) is_closed_on s,P & for-down (GBP,2,1,(((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))) ';' (while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))))))) is_halting_on s,P ) by A1, Th11; ::_thesis: verum
end;

Lm13:  for P being Instruction-Sequence of SCMPDS
for s being 0 -started State of SCMPDS st s . (intpos 3) >= (s . (intpos 1)) + 7 & s . GBP = 0 & s . (intpos 2) > 0 holds
IExec ((for-down (GBP,2,1,(((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))) ';' (while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0)))))))))),P,s) = IExec ((for-down (GBP,2,1,(((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))) ';' (while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0)))))))))),P,(Initialize (IExec (((((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))) ';' (while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0)))))))) ';' (AddTo (GBP,2,(- 1)))),P,s))))
proof
let P be Instruction-Sequence of SCMPDS; ::_thesis: for s being 0 -started State of SCMPDS st s . (intpos 3) >= (s . (intpos 1)) + 7 & s . GBP = 0 & s . (intpos 2) > 0 holds
IExec ((for-down (GBP,2,1,(((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))) ';' (while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0)))))))))),P,s) = IExec ((for-down (GBP,2,1,(((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))) ';' (while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0)))))))))),P,(Initialize (IExec (((((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))) ';' (while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0)))))))) ';' (AddTo (GBP,2,(- 1)))),P,s))))
let s be 0 -started State of SCMPDS; ::_thesis: ( s . (intpos 3) >= (s . (intpos 1)) + 7 & s . GBP = 0 & s . (intpos 2) > 0 implies IExec ((for-down (GBP,2,1,(((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))) ';' (while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0)))))))))),P,s) = IExec ((for-down (GBP,2,1,(((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))) ';' (while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0)))))))))),P,(Initialize (IExec (((((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))) ';' (while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0)))))))) ';' (AddTo (GBP,2,(- 1)))),P,s)))) )
set a = GBP ;
set b = DataLoc ((s . GBP),2);
assume that
A1: s . (intpos 3) >= (s . (intpos 1)) + 7 and
A2: s . GBP = 0 and
A3: s . (intpos 2) > 0 ; ::_thesis: IExec ((for-down (GBP,2,1,(((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))) ';' (while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0)))))))))),P,s) = IExec ((for-down (GBP,2,1,(((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))) ';' (while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0)))))))))),P,(Initialize (IExec (((((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))) ';' (while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0)))))))) ';' (AddTo (GBP,2,(- 1)))),P,s))))
A4: DataLoc ((s . GBP),2) = intpos (0 + 2) by A2, SCMP_GCD:1;
now__::_thesis:_for_t_being_0_-started_State_of_SCMPDS
for_Q_being_Instruction-Sequence_of_SCMPDS_st_t_._(intpos_3)_>=_(t_._(intpos_1))_+_7_&_t_._GBP_=_s_._GBP_&_t_._(DataLoc_((s_._GBP),2))_>_0_holds_
(_(IExec_(((((((AddTo_(GBP,3,1))_';'_((GBP,4)_:=_(GBP,3)))_';'_(AddTo_(GBP,1,1)))_';'_((GBP,6)_:=_(GBP,1)))_';'_(while>0_(GBP,6,((((GBP,5)_:=_((intpos_4),(-_1)))_';'_(SubFrom_(GBP,5,(intpos_4),0)))_';'_(if>0_(GBP,5,((((((GBP,5)_:=_((intpos_4),(-_1)))_';'_(((intpos_4),(-_1))_:=_((intpos_4),0)))_';'_(((intpos_4),0)_:=_(GBP,5)))_';'_(AddTo_(GBP,4,(-_1))))_';'_(AddTo_(GBP,6,(-_1)))),(Load_((GBP,6)_:=_0))))))))_';'_(AddTo_(GBP,2,(-_1)))),Q,t))_._GBP_=_t_._GBP_&_(IExec_(((((((AddTo_(GBP,3,1))_';'_((GBP,4)_:=_(GBP,3)))_';'_(AddTo_(GBP,1,1)))_';'_((GBP,6)_:=_(GBP,1)))_';'_(while>0_(GBP,6,((((GBP,5)_:=_((intpos_4),(-_1)))_';'_(SubFrom_(GBP,5,(intpos_4),0)))_';'_(if>0_(GBP,5,((((((GBP,5)_:=_((intpos_4),(-_1)))_';'_(((intpos_4),(-_1))_:=_((intpos_4),0)))_';'_(((intpos_4),0)_:=_(GBP,5)))_';'_(AddTo_(GBP,4,(-_1))))_';'_(AddTo_(GBP,6,(-_1)))),(Load_((GBP,6)_:=_0))))))))_';'_(AddTo_(GBP,2,(-_1)))),Q,t))_._(DataLoc_((s_._GBP),2))_=_(t_._(DataLoc_((s_._GBP),2)))_-_1_&_((((AddTo_(GBP,3,1))_';'_((GBP,4)_:=_(GBP,3)))_';'_(AddTo_(GBP,1,1)))_';'_((GBP,6)_:=_(GBP,1)))_';'_(while>0_(GBP,6,((((GBP,5)_:=_((intpos_4),(-_1)))_';'_(SubFrom_(GBP,5,(intpos_4),0)))_';'_(if>0_(GBP,5,((((((GBP,5)_:=_((intpos_4),(-_1)))_';'_(((intpos_4),(-_1))_:=_((intpos_4),0)))_';'_(((intpos_4),0)_:=_(GBP,5)))_';'_(AddTo_(GBP,4,(-_1))))_';'_(AddTo_(GBP,6,(-_1)))),(Load_((GBP,6)_:=_0)))))))_is_closed_on_t,Q_&_((((AddTo_(GBP,3,1))_';'_((GBP,4)_:=_(GBP,3)))_';'_(AddTo_(GBP,1,1)))_';'_((GBP,6)_:=_(GBP,1)))_';'_(while>0_(GBP,6,((((GBP,5)_:=_((intpos_4),(-_1)))_';'_(SubFrom_(GBP,5,(intpos_4),0)))_';'_(if>0_(GBP,5,((((((GBP,5)_:=_((intpos_4),(-_1)))_';'_(((intpos_4),(-_1))_:=_((intpos_4),0)))_';'_(((intpos_4),0)_:=_(GBP,5)))_';'_(AddTo_(GBP,4,(-_1))))_';'_(AddTo_(GBP,6,(-_1)))),(Load_((GBP,6)_:=_0)))))))_is_halting_on_t,Q_&_(IExec_(((((((AddTo_(GBP,3,1))_';'_((GBP,4)_:=_(GBP,3)))_';'_(AddTo_(GBP,1,1)))_';'_((GBP,6)_:=_(GBP,1)))_';'_(while>0_(GBP,6,((((GBP,5)_:=_((intpos_4),(-_1)))_';'_(SubFrom_(GBP,5,(intpos_4),0)))_';'_(if>0_(GBP,5,((((((GBP,5)_:=_((intpos_4),(-_1)))_';'_(((intpos_4),(-_1))_:=_((intpos_4),0)))_';'_(((intpos_4),0)_:=_(GBP,5)))_';'_(AddTo_(GBP,4,(-_1))))_';'_(AddTo_(GBP,6,(-_1)))),(Load_((GBP,6)_:=_0))))))))_';'_(AddTo_(GBP,2,(-_1)))),Q,t))_._(intpos_3)_>=_((IExec_(((((((AddTo_(GBP,3,1))_';'_((GBP,4)_:=_(GBP,3)))_';'_(AddTo_(GBP,1,1)))_';'_((GBP,6)_:=_(GBP,1)))_';'_(while>0_(GBP,6,((((GBP,5)_:=_((intpos_4),(-_1)))_';'_(SubFrom_(GBP,5,(intpos_4),0)))_';'_(if>0_(GBP,5,((((((GBP,5)_:=_((intpos_4),(-_1)))_';'_(((intpos_4),(-_1))_:=_((intpos_4),0)))_';'_(((intpos_4),0)_:=_(GBP,5)))_';'_(AddTo_(GBP,4,(-_1))))_';'_(AddTo_(GBP,6,(-_1)))),(Load_((GBP,6)_:=_0))))))))_';'_(AddTo_(GBP,2,(-_1)))),Q,t))_._(intpos_1))_+_7_)
let t be 0 -started State of SCMPDS; ::_thesis: for Q being Instruction-Sequence of SCMPDS st t . (intpos 3) >= (t . (intpos 1)) + 7 & t . GBP = s . GBP & t . (DataLoc ((s . GBP),2)) > 0 holds
( (IExec (((((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))) ';' (while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0)))))))) ';' (AddTo (GBP,2,(- 1)))),Q,t)) . GBP = t . GBP & (IExec (((((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))) ';' (while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0)))))))) ';' (AddTo (GBP,2,(- 1)))),Q,t)) . (DataLoc ((s . GBP),2)) = (t . (DataLoc ((s . GBP),2))) - 1 & ((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))) ';' (while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))))) is_closed_on t,Q & ((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))) ';' (while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))))) is_halting_on t,Q & (IExec (((((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))) ';' (while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0)))))))) ';' (AddTo (GBP,2,(- 1)))),Q,t)) . (intpos 3) >= ((IExec (((((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))) ';' (while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0)))))))) ';' (AddTo (GBP,2,(- 1)))),Q,t)) . (intpos 1)) + 7 )
let Q be Instruction-Sequence of SCMPDS; ::_thesis: ( t . (intpos 3) >= (t . (intpos 1)) + 7 & t . GBP = s . GBP & t . (DataLoc ((s . GBP),2)) > 0 implies ( (IExec (((((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))) ';' (while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0)))))))) ';' (AddTo (GBP,2,(- 1)))),Q,t)) . GBP = t . GBP & (IExec (((((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))) ';' (while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0)))))))) ';' (AddTo (GBP,2,(- 1)))),Q,t)) . (DataLoc ((s . GBP),2)) = (t . (DataLoc ((s . GBP),2))) - 1 & ((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))) ';' (while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))))) is_closed_on t,Q & ((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))) ';' (while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))))) is_halting_on t,Q & (IExec (((((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))) ';' (while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0)))))))) ';' (AddTo (GBP,2,(- 1)))),Q,t)) . (intpos 3) >= ((IExec (((((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))) ';' (while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0)))))))) ';' (AddTo (GBP,2,(- 1)))),Q,t)) . (intpos 1)) + 7 ) )
A5: Initialize t = t by MEMSTR_0:44;
assume that
A6: t . (intpos 3) >= (t . (intpos 1)) + 7 and
A7: t . GBP = s . GBP and
 t . (DataLoc ((s . GBP),2)) > 0 ; ::_thesis: ( (IExec (((((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))) ';' (while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0)))))))) ';' (AddTo (GBP,2,(- 1)))),Q,t)) . GBP = t . GBP & (IExec (((((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))) ';' (while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0)))))))) ';' (AddTo (GBP,2,(- 1)))),Q,t)) . (DataLoc ((s . GBP),2)) = (t . (DataLoc ((s . GBP),2))) - 1 & ((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))) ';' (while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))))) is_closed_on t,Q & ((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))) ';' (while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))))) is_halting_on t,Q & (IExec (((((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))) ';' (while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0)))))))) ';' (AddTo (GBP,2,(- 1)))),Q,t)) . (intpos 3) >= ((IExec (((((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))) ';' (while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0)))))))) ';' (AddTo (GBP,2,(- 1)))),Q,t)) . (intpos 1)) + 7 )
set t1 = IExec (((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))),Q,t);
set Q1 = Q;
A8: (t . (intpos 3)) + 1 >= (7 + (t . (intpos 1))) + 1 by A6, XREAL_1:6;
thus  (IExec (((((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))) ';' (while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0)))))))) ';' (AddTo (GBP,2,(- 1)))),Q,t)) . GBP = t . GBP by A2, A6, A7, Lm11; ::_thesis: ( (IExec (((((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))) ';' (while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0)))))))) ';' (AddTo (GBP,2,(- 1)))),Q,t)) . (DataLoc ((s . GBP),2)) = (t . (DataLoc ((s . GBP),2))) - 1 & ((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))) ';' (while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))))) is_closed_on t,Q & ((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))) ';' (while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))))) is_halting_on t,Q & (IExec (((((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))) ';' (while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0)))))))) ';' (AddTo (GBP,2,(- 1)))),Q,t)) . (intpos 3) >= ((IExec (((((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))) ';' (while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0)))))))) ';' (AddTo (GBP,2,(- 1)))),Q,t)) . (intpos 1)) + 7 )
thus  (IExec (((((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))) ';' (while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0)))))))) ';' (AddTo (GBP,2,(- 1)))),Q,t)) . (DataLoc ((s . GBP),2)) = (t . (DataLoc ((s . GBP),2))) - 1 by A2, A4, A6, A7, Lm11; ::_thesis: ( ((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))) ';' (while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))))) is_closed_on t,Q & ((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))) ';' (while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))))) is_halting_on t,Q & (IExec (((((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))) ';' (while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0)))))))) ';' (AddTo (GBP,2,(- 1)))),Q,t)) . (intpos 3) >= ((IExec (((((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))) ';' (while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0)))))))) ';' (AddTo (GBP,2,(- 1)))),Q,t)) . (intpos 1)) + 7 )
A9: ( (((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1)) is_closed_on t,Q & (((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1)) is_halting_on t,Q ) by SCMPDS_6:20, SCMPDS_6:21;
 (IExec (((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))),Q,t)) . (intpos 6) = (t . (intpos 1)) + 1 by A2, A7, Lm10;
then A10: (IExec (((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))),Q,t)) . (intpos 4) >= 7 + ((IExec (((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))),Q,t)) . (intpos 6)) by A2, A7, A8, Lm10;
A11: (IExec (((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))),Q,t)) . GBP = (Initialize (IExec (((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))),Q,t))) . GBP by SCMPDS_5:15;
A12: (IExec (((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))),Q,t)) . (intpos 4) = (Initialize (IExec (((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))),Q,t))) . (intpos 4) by SCMPDS_5:15;
A13: (IExec (((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))),Q,t)) . (intpos 6) = (Initialize (IExec (((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))),Q,t))) . (intpos 6) by SCMPDS_5:15;
A14: (IExec (((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))),Q,t)) . GBP = 0 by A2, A7, Lm10;
then  DataLoc (((IExec (((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))),Q,t)) . GBP),6) = intpos (0 + 6) by SCMP_GCD:1;
then  ( while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0)))))) is_closed_on Initialize (IExec (((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))),Q,t)),Q & while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0)))))) is_halting_on Initialize (IExec (((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))),Q,t)),Q ) by A14, A10, Lm8, A11, A12, A13;
then  ( while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0)))))) is_closed_on IExec (((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))),Q,t),Q & while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0)))))) is_halting_on IExec (((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))),Q,t),Q ) by SCMPDS_6:125, SCMPDS_6:126;
hence  ( ((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))) ';' (while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))))) is_closed_on t,Q & ((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))) ';' (while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))))) is_halting_on t,Q ) by A9, A5, SCMPDS_7:24; ::_thesis: (IExec (((((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))) ';' (while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0)))))))) ';' (AddTo (GBP,2,(- 1)))),Q,t)) . (intpos 3) >= ((IExec (((((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))) ';' (while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0)))))))) ';' (AddTo (GBP,2,(- 1)))),Q,t)) . (intpos 1)) + 7
 (IExec (((((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))) ';' (while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0)))))))) ';' (AddTo (GBP,2,(- 1)))),Q,t)) . (intpos 1) = (t . (intpos 1)) + 1 by A2, A6, A7, Lm11;
hence  (IExec (((((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))) ';' (while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0)))))))) ';' (AddTo (GBP,2,(- 1)))),Q,t)) . (intpos 3) >= ((IExec (((((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))) ';' (while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0)))))))) ';' (AddTo (GBP,2,(- 1)))),Q,t)) . (intpos 1)) + 7 by A2, A6, A7, A8, Lm11; ::_thesis: verum
end;
hence  IExec ((for-down (GBP,2,1,(((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))) ';' (while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0)))))))))),P,s) = IExec ((for-down (GBP,2,1,(((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))) ';' (while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0)))))))))),P,(Initialize (IExec (((((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))) ';' (while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0)))))))) ';' (AddTo (GBP,2,(- 1)))),P,s)))) by A1, A3, A4, Th12; ::_thesis: verum
end;

begin

theorem :: SCPISORT:14
for n, p0 being Element of NAT holds card (insert-sort (n,p0)) = 23
proof
let n, p0 be Element of NAT ; ::_thesis: card (insert-sort (n,p0)) = 23
set i1 = GBP := 0;
set i2 = (GBP,1) := 0;
set i3 = (GBP,2) := (n - 1);
set i4 = (GBP,3) := p0;
thus  card (insert-sort (n,p0)) = (card ((((GBP := 0) ';' ((GBP,1) := 0)) ';' ((GBP,2) := (n - 1))) ';' ((GBP,3) := p0))) + (card (for-down (GBP,2,1,(((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))) ';' (while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))))))))) by AFINSQ_1:17
.= ((card (((GBP := 0) ';' ((GBP,1) := 0)) ';' ((GBP,2) := (n - 1)))) + 1) + (card (for-down (GBP,2,1,(((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))) ';' (while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))))))))) by SCMP_GCD:4
.= (((card ((GBP := 0) ';' ((GBP,1) := 0))) + 1) + 1) + (card (for-down (GBP,2,1,(((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))) ';' (while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))))))))) by SCMP_GCD:4
.= ((2 + 1) + 1) + (card (for-down (GBP,2,1,(((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))) ';' (while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))))))))) by SCMP_GCD:5
.= 4 + ((card (((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))) ';' (while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))))))) + 3) by SCMPDS_7:41
.= 23 by Lm5 ; ::_thesis: verum
end;

theorem :: SCPISORT:15
for p0, n being Element of NAT st p0 >= 7 holds
insert-sort (n,p0) is parahalting
proof
let p0, n be Element of NAT ; ::_thesis: ( p0 >= 7 implies insert-sort (n,p0) is parahalting )
set a = GBP ;
set i1 = GBP := 0;
set i2 = (GBP,1) := 0;
set i3 = (GBP,2) := (n - 1);
set i4 = (GBP,3) := p0;
set I = (((GBP := 0) ';' ((GBP,1) := 0)) ';' ((GBP,2) := (n - 1))) ';' ((GBP,3) := p0);
assume A1: p0 >= 7 ; ::_thesis: insert-sort (n,p0) is parahalting
now__::_thesis:_for_s_being_State_of_SCMPDS
for_P_being_Instruction-Sequence_of_SCMPDS_holds_insert-sort_(n,p0)_is_halting_on_s,P
let s be State of SCMPDS; ::_thesis: for P being Instruction-Sequence of SCMPDS holds insert-sort (n,p0) is_halting_on s,P
let P be Instruction-Sequence of SCMPDS; ::_thesis: insert-sort (n,p0) is_halting_on s,P
set s1 = IExec (((((GBP := 0) ';' ((GBP,1) := 0)) ';' ((GBP,2) := (n - 1))) ';' ((GBP,3) := p0)),P,(Initialize s));
set s2 = IExec ((((GBP := 0) ';' ((GBP,1) := 0)) ';' ((GBP,2) := (n - 1))),P,(Initialize s));
set P1 = P;
set s3 = IExec (((GBP := 0) ';' ((GBP,1) := 0)),P,(Initialize s));
set s4 = Exec ((GBP := 0),(Initialize s));
A2: (IExec (((((GBP := 0) ';' ((GBP,1) := 0)) ';' ((GBP,2) := (n - 1))) ';' ((GBP,3) := p0)),P,(Initialize s))) . GBP = (Initialize (IExec (((((GBP := 0) ';' ((GBP,1) := 0)) ';' ((GBP,2) := (n - 1))) ';' ((GBP,3) := p0)),P,(Initialize s)))) . GBP by SCMPDS_5:15;
A3: (IExec (((((GBP := 0) ';' ((GBP,1) := 0)) ';' ((GBP,2) := (n - 1))) ';' ((GBP,3) := p0)),P,(Initialize s))) . (intpos 1) = (Initialize (IExec (((((GBP := 0) ';' ((GBP,1) := 0)) ';' ((GBP,2) := (n - 1))) ';' ((GBP,3) := p0)),P,(Initialize s)))) . (intpos 1) by SCMPDS_5:15;
A4: (IExec (((((GBP := 0) ';' ((GBP,1) := 0)) ';' ((GBP,2) := (n - 1))) ';' ((GBP,3) := p0)),P,(Initialize s))) . (intpos 3) = (Initialize (IExec (((((GBP := 0) ';' ((GBP,1) := 0)) ';' ((GBP,2) := (n - 1))) ';' ((GBP,3) := p0)),P,(Initialize s)))) . (intpos 3) by SCMPDS_5:15;
A5: ( (((GBP := 0) ';' ((GBP,1) := 0)) ';' ((GBP,2) := (n - 1))) ';' ((GBP,3) := p0) is_closed_on s,P & (((GBP := 0) ';' ((GBP,1) := 0)) ';' ((GBP,2) := (n - 1))) ';' ((GBP,3) := p0) is_halting_on s,P ) by SCMPDS_6:20, SCMPDS_6:21;
A6: (Exec ((GBP := 0),(Initialize s))) . GBP = 0 by SCMPDS_2:45;
then A7: DataLoc (((Exec ((GBP := 0),(Initialize s))) . GBP),1) = intpos (0 + 1) by SCMP_GCD:1;
A8: (IExec (((GBP := 0) ';' ((GBP,1) := 0)),P,(Initialize s))) . GBP = (Exec (((GBP,1) := 0),(Exec ((GBP := 0),(Initialize s))))) . GBP by SCMPDS_5:42
.= 0 by A6, A7, AMI_3:10, SCMPDS_2:46 ;
then A9: DataLoc (((IExec (((GBP := 0) ';' ((GBP,1) := 0)),P,(Initialize s))) . GBP),2) = intpos (0 + 2) by SCMP_GCD:1;
A10: (IExec (((GBP := 0) ';' ((GBP,1) := 0)),P,(Initialize s))) . (intpos 1) = (Exec (((GBP,1) := 0),(Exec ((GBP := 0),(Initialize s))))) . (intpos 1) by SCMPDS_5:42
.= 0 by A7, SCMPDS_2:46 ;
A11: (IExec ((((GBP := 0) ';' ((GBP,1) := 0)) ';' ((GBP,2) := (n - 1))),P,(Initialize s))) . (intpos 1) = (Exec (((GBP,2) := (n - 1)),(IExec (((GBP := 0) ';' ((GBP,1) := 0)),P,(Initialize s))))) . (intpos 1) by SCMPDS_5:41
.= 0 by A10, A9, AMI_3:10, SCMPDS_2:46 ;
A12: (IExec ((((GBP := 0) ';' ((GBP,1) := 0)) ';' ((GBP,2) := (n - 1))),P,(Initialize s))) . GBP = (Exec (((GBP,2) := (n - 1)),(IExec (((GBP := 0) ';' ((GBP,1) := 0)),P,(Initialize s))))) . GBP by SCMPDS_5:41
.= 0 by A8, A9, AMI_3:10, SCMPDS_2:46 ;
then A13: DataLoc (((IExec ((((GBP := 0) ';' ((GBP,1) := 0)) ';' ((GBP,2) := (n - 1))),P,(Initialize s))) . GBP),3) = intpos (0 + 3) by SCMP_GCD:1;
A14: (IExec (((((GBP := 0) ';' ((GBP,1) := 0)) ';' ((GBP,2) := (n - 1))) ';' ((GBP,3) := p0)),P,(Initialize s))) . (intpos 3) = (Exec (((GBP,3) := p0),(IExec ((((GBP := 0) ';' ((GBP,1) := 0)) ';' ((GBP,2) := (n - 1))),P,(Initialize s))))) . (intpos 3) by SCMPDS_5:41
.= p0 by A13, SCMPDS_2:46 ;
(IExec (((((GBP := 0) ';' ((GBP,1) := 0)) ';' ((GBP,2) := (n - 1))) ';' ((GBP,3) := p0)),P,(Initialize s))) . (intpos 1) = (Exec (((GBP,3) := p0),(IExec ((((GBP := 0) ';' ((GBP,1) := 0)) ';' ((GBP,2) := (n - 1))),P,(Initialize s))))) . (intpos 1) by SCMPDS_5:41
.= 0 by A11, A13, AMI_3:10, SCMPDS_2:46 ;
then A15: (IExec (((((GBP := 0) ';' ((GBP,1) := 0)) ';' ((GBP,2) := (n - 1))) ';' ((GBP,3) := p0)),P,(Initialize s))) . (intpos 3) >= ((IExec (((((GBP := 0) ';' ((GBP,1) := 0)) ';' ((GBP,2) := (n - 1))) ';' ((GBP,3) := p0)),P,(Initialize s))) . (intpos 1)) + 7 by A1, A14;
(IExec (((((GBP := 0) ';' ((GBP,1) := 0)) ';' ((GBP,2) := (n - 1))) ';' ((GBP,3) := p0)),P,(Initialize s))) . GBP = (Exec (((GBP,3) := p0),(IExec ((((GBP := 0) ';' ((GBP,1) := 0)) ';' ((GBP,2) := (n - 1))),P,(Initialize s))))) . GBP by SCMPDS_5:41
.= 0 by A12, A13, AMI_3:10, SCMPDS_2:46 ;
then  ( for-down (GBP,2,1,(((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))) ';' (while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))))))) is_closed_on Initialize (IExec (((((GBP := 0) ';' ((GBP,1) := 0)) ';' ((GBP,2) := (n - 1))) ';' ((GBP,3) := p0)),P,(Initialize s))),P & for-down (GBP,2,1,(((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))) ';' (while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))))))) is_halting_on Initialize (IExec (((((GBP := 0) ';' ((GBP,1) := 0)) ';' ((GBP,2) := (n - 1))) ';' ((GBP,3) := p0)),P,(Initialize s))),P ) by A15, Lm12, A2, A3, A4;
then  ( for-down (GBP,2,1,(((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))) ';' (while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))))))) is_closed_on IExec (((((GBP := 0) ';' ((GBP,1) := 0)) ';' ((GBP,2) := (n - 1))) ';' ((GBP,3) := p0)),P,(Initialize s)),P & for-down (GBP,2,1,(((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))) ';' (while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))))))) is_halting_on IExec (((((GBP := 0) ';' ((GBP,1) := 0)) ';' ((GBP,2) := (n - 1))) ';' ((GBP,3) := p0)),P,(Initialize s)),P ) by SCMPDS_6:125, SCMPDS_6:126;
hence  insert-sort (n,p0) is_halting_on s,P by A5, SCMPDS_7:24; ::_thesis: verum
end;
hence  insert-sort (n,p0) is parahalting by SCMPDS_6:21; ::_thesis: verum
end;

Lm14:  for P being Instruction-Sequence of SCMPDS
for s being 0 -started State of SCMPDS st s . (intpos 4) >= 7 + (s . (intpos 6)) & s . GBP = 0 & s . (intpos 6) > 0 holds
IExec ((while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))))),P,s) = IExec ((while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))))),P,(Initialize (IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))),P,s))))
proof
let P be Instruction-Sequence of SCMPDS; ::_thesis: for s being 0 -started State of SCMPDS st s . (intpos 4) >= 7 + (s . (intpos 6)) & s . GBP = 0 & s . (intpos 6) > 0 holds
IExec ((while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))))),P,s) = IExec ((while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))))),P,(Initialize (IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))),P,s))))
let s be 0 -started State of SCMPDS; ::_thesis: ( s . (intpos 4) >= 7 + (s . (intpos 6)) & s . GBP = 0 & s . (intpos 6) > 0 implies IExec ((while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))))),P,s) = IExec ((while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))))),P,(Initialize (IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))),P,s)))) )
set a = GBP ;
set b = DataLoc ((s . GBP),6);
assume that
A1: s . (intpos 4) >= 7 + (s . (intpos 6)) and
A2: s . GBP = 0 and
A3: s . (intpos 6) > 0 ; ::_thesis: IExec ((while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))))),P,s) = IExec ((while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))))),P,(Initialize (IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))),P,s))))
A4: DataLoc ((s . GBP),6) = intpos (0 + 6) by A2, SCMP_GCD:1;
A5: 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_{(intpos_4)}_holds_
t_._x_>=_7_+_(t_._(DataLoc_((s_._GBP),6)))_)_&_t_._GBP_=_s_._GBP_&_t_._(DataLoc_((s_._GBP),6))_>_0_holds_
(_(IExec_(((((GBP,5)_:=_((intpos_4),(-_1)))_';'_(SubFrom_(GBP,5,(intpos_4),0)))_';'_(if>0_(GBP,5,((((((GBP,5)_:=_((intpos_4),(-_1)))_';'_(((intpos_4),(-_1))_:=_((intpos_4),0)))_';'_(((intpos_4),0)_:=_(GBP,5)))_';'_(AddTo_(GBP,4,(-_1))))_';'_(AddTo_(GBP,6,(-_1)))),(Load_((GBP,6)_:=_0))))),Q,t))_._GBP_=_t_._GBP_&_(((GBP,5)_:=_((intpos_4),(-_1)))_';'_(SubFrom_(GBP,5,(intpos_4),0)))_';'_(if>0_(GBP,5,((((((GBP,5)_:=_((intpos_4),(-_1)))_';'_(((intpos_4),(-_1))_:=_((intpos_4),0)))_';'_(((intpos_4),0)_:=_(GBP,5)))_';'_(AddTo_(GBP,4,(-_1))))_';'_(AddTo_(GBP,6,(-_1)))),(Load_((GBP,6)_:=_0))))_is_closed_on_t,Q_&_(((GBP,5)_:=_((intpos_4),(-_1)))_';'_(SubFrom_(GBP,5,(intpos_4),0)))_';'_(if>0_(GBP,5,((((((GBP,5)_:=_((intpos_4),(-_1)))_';'_(((intpos_4),(-_1))_:=_((intpos_4),0)))_';'_(((intpos_4),0)_:=_(GBP,5)))_';'_(AddTo_(GBP,4,(-_1))))_';'_(AddTo_(GBP,6,(-_1)))),(Load_((GBP,6)_:=_0))))_is_halting_on_t,Q_&_(IExec_(((((GBP,5)_:=_((intpos_4),(-_1)))_';'_(SubFrom_(GBP,5,(intpos_4),0)))_';'_(if>0_(GBP,5,((((((GBP,5)_:=_((intpos_4),(-_1)))_';'_(((intpos_4),(-_1))_:=_((intpos_4),0)))_';'_(((intpos_4),0)_:=_(GBP,5)))_';'_(AddTo_(GBP,4,(-_1))))_';'_(AddTo_(GBP,6,(-_1)))),(Load_((GBP,6)_:=_0))))),Q,t))_._(DataLoc_((s_._GBP),6))_<_t_._(DataLoc_((s_._GBP),6))_&_(_for_x_being_Int_position_st_x_in_{(intpos_4)}_holds_
(IExec_(((((GBP,5)_:=_((intpos_4),(-_1)))_';'_(SubFrom_(GBP,5,(intpos_4),0)))_';'_(if>0_(GBP,5,((((((GBP,5)_:=_((intpos_4),(-_1)))_';'_(((intpos_4),(-_1))_:=_((intpos_4),0)))_';'_(((intpos_4),0)_:=_(GBP,5)))_';'_(AddTo_(GBP,4,(-_1))))_';'_(AddTo_(GBP,6,(-_1)))),(Load_((GBP,6)_:=_0))))),Q,t))_._x_>=_7_+_((IExec_(((((GBP,5)_:=_((intpos_4),(-_1)))_';'_(SubFrom_(GBP,5,(intpos_4),0)))_';'_(if>0_(GBP,5,((((((GBP,5)_:=_((intpos_4),(-_1)))_';'_(((intpos_4),(-_1))_:=_((intpos_4),0)))_';'_(((intpos_4),0)_:=_(GBP,5)))_';'_(AddTo_(GBP,4,(-_1))))_';'_(AddTo_(GBP,6,(-_1)))),(Load_((GBP,6)_:=_0))))),Q,t))_._(DataLoc_((s_._GBP),6)))_)_)
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 {(intpos 4)} holds
t . x >= 7 + (t . (DataLoc ((s . GBP),6))) ) & t . GBP = s . GBP & t . (DataLoc ((s . GBP),6)) > 0 holds
( (IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))),Q,t)) . GBP = t . GBP & (((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0)))) is_closed_on t,Q & (((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0)))) is_halting_on t,Q & (IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))),Q,t)) . (DataLoc ((s . GBP),6)) < t . (DataLoc ((s . GBP),6)) & ( for x being Int_position st x in {(intpos 4)} holds
(IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))),Q,t)) . x >= 7 + ((IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))),Q,t)) . (DataLoc ((s . GBP),6))) ) )
let Q be Instruction-Sequence of SCMPDS; ::_thesis: ( ( for x being Int_position st x in {(intpos 4)} holds
t . x >= 7 + (t . (DataLoc ((s . GBP),6))) ) & t . GBP = s . GBP & t . (DataLoc ((s . GBP),6)) > 0 implies ( (IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))),Q,t)) . GBP = t . GBP & (((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0)))) is_closed_on t,Q & (((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0)))) is_halting_on t,Q & (IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))),Q,t)) . (DataLoc ((s . GBP),6)) < t . (DataLoc ((s . GBP),6)) & ( for x being Int_position st x in {(intpos 4)} holds
(IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))),Q,t)) . x >= 7 + ((IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))),Q,t)) . (DataLoc ((s . GBP),6))) ) ) )
A6: Initialize t = t by MEMSTR_0:44;
assume that
A7: for x being Int_position st x in {(intpos 4)} holds
t . x >= 7 + (t . (DataLoc ((s . GBP),6))) and
A8: ( t . GBP = s . GBP & t . (DataLoc ((s . GBP),6)) > 0 ) ; ::_thesis: ( (IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))),Q,t)) . GBP = t . GBP & (((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0)))) is_closed_on t,Q & (((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0)))) is_halting_on t,Q & (IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))),Q,t)) . (DataLoc ((s . GBP),6)) < t . (DataLoc ((s . GBP),6)) & ( for x being Int_position st x in {(intpos 4)} holds
(IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))),Q,t)) . x >= 7 + ((IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))),Q,t)) . (DataLoc ((s . GBP),6))) ) )
set Bt = IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))),Q,(Initialize t));
A9: intpos 4 in {(intpos 4)} by TARSKI:def_1;
then A10: t . (intpos 4) >= 7 + (t . (intpos 6)) by A4, A7;
hence  (IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))),Q,t)) . GBP = t . GBP by A2, A4, A8, Lm6; ::_thesis: ( (((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0)))) is_closed_on t,Q & (((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0)))) is_halting_on t,Q & (IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))),Q,t)) . (DataLoc ((s . GBP),6)) < t . (DataLoc ((s . GBP),6)) & ( for x being Int_position st x in {(intpos 4)} holds
(IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))),Q,t)) . x >= 7 + ((IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))),Q,t)) . (DataLoc ((s . GBP),6))) ) )
thus  ( (((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0)))) is_closed_on t,Q & (((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0)))) is_halting_on t,Q ) by SCMPDS_6:20, SCMPDS_6:21; ::_thesis: ( (IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))),Q,t)) . (DataLoc ((s . GBP),6)) < t . (DataLoc ((s . GBP),6)) & ( for x being Int_position st x in {(intpos 4)} holds
(IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))),Q,t)) . x >= 7 + ((IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))),Q,t)) . (DataLoc ((s . GBP),6))) ) )
thus  (IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))),Q,t)) . (DataLoc ((s . GBP),6)) < t . (DataLoc ((s . GBP),6)) by A2, A4, A8, A10, Lm7; ::_thesis: for x being Int_position st x in {(intpos 4)} holds
(IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))),Q,t)) . x >= 7 + ((IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))),Q,t)) . (DataLoc ((s . GBP),6)))
 t . (intpos 4) >= 7 + (t . (DataLoc ((s . GBP),6))) by A7, A9;
then  (IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))),Q,(Initialize t))) . (intpos 4) >= 7 + ((IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))),Q,(Initialize t))) . (intpos 6)) by A2, A4, A8, Lm7, A6;
hence  for x being Int_position st x in {(intpos 4)} holds
(IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))),Q,t)) . x >= 7 + ((IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))),Q,t)) . (DataLoc ((s . GBP),6))) by A4, A6, TARSKI:def_1; ::_thesis: verum
end;
 for x being Int_position st x in {(intpos 4)} holds
s . x >= 7 + (s . (DataLoc ((s . GBP),6))) by A1, A4, TARSKI:def_1;
hence  IExec ((while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))))),P,s) = IExec ((while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))))),P,(Initialize (IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))),P,s)))) by A3, A4, A5, SCMPDS_8:29; ::_thesis: verum
end;

theorem Th16: :: SCPISORT:16
for P being Instruction-Sequence of SCMPDS
for s being 0 -started State of SCMPDS
for f, g being FinSequence of INT
for k0, k being Element of NAT st s . (intpos 4) >= 7 + (s . (intpos 6)) & s . GBP = 0 & k = s . (intpos 6) & k0 = ((s . (intpos 4)) - (s . (intpos 6))) - 1 & f is_FinSequence_on s,k0 & g is_FinSequence_on IExec ((while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))))),P,s),k0 & len f = len g & len f > k & f is_non_decreasing_on 1,k holds
( f,g are_fiberwise_equipotent & g is_non_decreasing_on 1,k + 1 & ( for i being Element of NAT st i > k + 1 & i <= len f holds
f . i = g . i ) & ( for i being Element of NAT st 1 <= i & i <= k + 1 holds
ex j being Element of NAT st
( 1 <= j & j <= k + 1 & g . i = f . j ) ) )
proof
let P be Instruction-Sequence of SCMPDS; ::_thesis: for s being 0 -started State of SCMPDS
for f, g being FinSequence of INT
for k0, k being Element of NAT st s . (intpos 4) >= 7 + (s . (intpos 6)) & s . GBP = 0 & k = s . (intpos 6) & k0 = ((s . (intpos 4)) - (s . (intpos 6))) - 1 & f is_FinSequence_on s,k0 & g is_FinSequence_on IExec ((while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))))),P,s),k0 & len f = len g & len f > k & f is_non_decreasing_on 1,k holds
( f,g are_fiberwise_equipotent & g is_non_decreasing_on 1,k + 1 & ( for i being Element of NAT st i > k + 1 & i <= len f holds
f . i = g . i ) & ( for i being Element of NAT st 1 <= i & i <= k + 1 holds
ex j being Element of NAT st
( 1 <= j & j <= k + 1 & g . i = f . j ) ) )
set a = GBP ;
let s be 0 -started State of SCMPDS; ::_thesis: for f, g being FinSequence of INT
for k0, k being Element of NAT st s . (intpos 4) >= 7 + (s . (intpos 6)) & s . GBP = 0 & k = s . (intpos 6) & k0 = ((s . (intpos 4)) - (s . (intpos 6))) - 1 & f is_FinSequence_on s,k0 & g is_FinSequence_on IExec ((while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))))),P,s),k0 & len f = len g & len f > k & f is_non_decreasing_on 1,k holds
( f,g are_fiberwise_equipotent & g is_non_decreasing_on 1,k + 1 & ( for i being Element of NAT st i > k + 1 & i <= len f holds
f . i = g . i ) & ( for i being Element of NAT st 1 <= i & i <= k + 1 holds
ex j being Element of NAT st
( 1 <= j & j <= k + 1 & g . i = f . j ) ) )
let f, g be FinSequence of INT ; ::_thesis: for k0, k being Element of NAT st s . (intpos 4) >= 7 + (s . (intpos 6)) & s . GBP = 0 & k = s . (intpos 6) & k0 = ((s . (intpos 4)) - (s . (intpos 6))) - 1 & f is_FinSequence_on s,k0 & g is_FinSequence_on IExec ((while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))))),P,s),k0 & len f = len g & len f > k & f is_non_decreasing_on 1,k holds
( f,g are_fiberwise_equipotent & g is_non_decreasing_on 1,k + 1 & ( for i being Element of NAT st i > k + 1 & i <= len f holds
f . i = g . i ) & ( for i being Element of NAT st 1 <= i & i <= k + 1 holds
ex j being Element of NAT st
( 1 <= j & j <= k + 1 & g . i = f . j ) ) )
let m, n be Element of NAT ; ::_thesis: ( s . (intpos 4) >= 7 + (s . (intpos 6)) & s . GBP = 0 & n = s . (intpos 6) & m = ((s . (intpos 4)) - (s . (intpos 6))) - 1 & f is_FinSequence_on s,m & g is_FinSequence_on IExec ((while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))))),P,s),m & len f = len g & len f > n & f is_non_decreasing_on 1,n implies ( f,g are_fiberwise_equipotent & g is_non_decreasing_on 1,n + 1 & ( for i being Element of NAT st i > n + 1 & i <= len f holds
f . i = g . i ) & ( for i being Element of NAT st 1 <= i & i <= n + 1 holds
ex j being Element of NAT st
( 1 <= j & j <= n + 1 & g . i = f . j ) ) ) )
assume A1: ( s . (intpos 4) >= 7 + (s . (intpos 6)) & s . GBP = 0 & n = s . (intpos 6) & m = ((s . (intpos 4)) - (s . (intpos 6))) - 1 ) ; ::_thesis: ( not f is_FinSequence_on s,m or not g is_FinSequence_on IExec ((while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))))),P,s),m or not len f = len g or not len f > n or not f is_non_decreasing_on 1,n or ( f,g are_fiberwise_equipotent & g is_non_decreasing_on 1,n + 1 & ( for i being Element of NAT st i > n + 1 & i <= len f holds
f . i = g . i ) & ( for i being Element of NAT st 1 <= i & i <= n + 1 holds
ex j being Element of NAT st
( 1 <= j & j <= n + 1 & g . i = f . j ) ) ) )
defpred S1[ Element of NAT ] means  for t being 0 -started State of SCMPDS
for Q being Instruction-Sequence of SCMPDS
for f1, f2 being FinSequence of INT st t . (intpos 4) >= 7 + (t . (intpos 6)) & t . GBP = 0 & $1 = t . (intpos 6) & m = ((t . (intpos 4)) - (t . (intpos 6))) - 1 & f1 is_FinSequence_on t,m & f2 is_FinSequence_on IExec ((while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))))),Q,t),m & len f1 = len f2 & len f1 > $1 & f1 is_non_decreasing_on 1,$1 holds
( f1,f2 are_fiberwise_equipotent & f2 is_non_decreasing_on 1,$1 + 1 & ( for i being Element of NAT st i > $1 + 1 & i <= len f1 holds
f1 . i = f2 . i ) & ( for i being Element of NAT st 1 <= i & i <= $1 + 1 holds
ex j being Element of NAT st
( 1 <= j & j <= $1 + 1 & f2 . i = f1 . j ) ) );
assume A2: ( f is_FinSequence_on s,m & g is_FinSequence_on IExec ((while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))))),P,s),m ) ; ::_thesis: ( not len f = len g or not len f > n or not f is_non_decreasing_on 1,n or ( f,g are_fiberwise_equipotent & g is_non_decreasing_on 1,n + 1 & ( for i being Element of NAT st i > n + 1 & i <= len f holds
f . i = g . i ) & ( for i being Element of NAT st 1 <= i & i <= n + 1 holds
ex j being Element of NAT st
( 1 <= j & j <= n + 1 & g . i = f . j ) ) ) )
A3: now__::_thesis:_for_k_being_Element_of_NAT_st_S1[k]_holds_
S1[k_+_1]
let k be Element of NAT ; ::_thesis: ( S1[k] implies S1[k + 1] )
assume A4: S1[k] ; ::_thesis: S1[k + 1]
thus  S1[k + 1] ::_thesis: verum
proof
let t be 0 -started State of SCMPDS; ::_thesis: for Q being Instruction-Sequence of SCMPDS
for f1, f2 being FinSequence of INT st t . (intpos 4) >= 7 + (t . (intpos 6)) & t . GBP = 0 & k + 1 = t . (intpos 6) & m = ((t . (intpos 4)) - (t . (intpos 6))) - 1 & f1 is_FinSequence_on t,m & f2 is_FinSequence_on IExec ((while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))))),Q,t),m & len f1 = len f2 & len f1 > k + 1 & f1 is_non_decreasing_on 1,k + 1 holds
( f1,f2 are_fiberwise_equipotent & f2 is_non_decreasing_on 1,(k + 1) + 1 & ( for i being Element of NAT st i > (k + 1) + 1 & i <= len f1 holds
f1 . i = f2 . i ) & ( for i being Element of NAT st 1 <= i & i <= (k + 1) + 1 holds
ex j being Element of NAT st
( 1 <= j & j <= (k + 1) + 1 & f2 . i = f1 . j ) ) )
let Q be Instruction-Sequence of SCMPDS; ::_thesis: for f1, f2 being FinSequence of INT st t . (intpos 4) >= 7 + (t . (intpos 6)) & t . GBP = 0 & k + 1 = t . (intpos 6) & m = ((t . (intpos 4)) - (t . (intpos 6))) - 1 & f1 is_FinSequence_on t,m & f2 is_FinSequence_on IExec ((while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))))),Q,t),m & len f1 = len f2 & len f1 > k + 1 & f1 is_non_decreasing_on 1,k + 1 holds
( f1,f2 are_fiberwise_equipotent & f2 is_non_decreasing_on 1,(k + 1) + 1 & ( for i being Element of NAT st i > (k + 1) + 1 & i <= len f1 holds
f1 . i = f2 . i ) & ( for i being Element of NAT st 1 <= i & i <= (k + 1) + 1 holds
ex j being Element of NAT st
( 1 <= j & j <= (k + 1) + 1 & f2 . i = f1 . j ) ) )
let f1, f2 be FinSequence of INT ; ::_thesis: ( t . (intpos 4) >= 7 + (t . (intpos 6)) & t . GBP = 0 & k + 1 = t . (intpos 6) & m = ((t . (intpos 4)) - (t . (intpos 6))) - 1 & f1 is_FinSequence_on t,m & f2 is_FinSequence_on IExec ((while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))))),Q,t),m & len f1 = len f2 & len f1 > k + 1 & f1 is_non_decreasing_on 1,k + 1 implies ( f1,f2 are_fiberwise_equipotent & f2 is_non_decreasing_on 1,(k + 1) + 1 & ( for i being Element of NAT st i > (k + 1) + 1 & i <= len f1 holds
f1 . i = f2 . i ) & ( for i being Element of NAT st 1 <= i & i <= (k + 1) + 1 holds
ex j being Element of NAT st
( 1 <= j & j <= (k + 1) + 1 & f2 . i = f1 . j ) ) ) )
A5: Initialize t = t by MEMSTR_0:44;
assume that
A6: t . (intpos 4) >= 7 + (t . (intpos 6)) and
A7: t . GBP = 0 and
A8: k + 1 = t . (intpos 6) and
A9: m = ((t . (intpos 4)) - (t . (intpos 6))) - 1 and
A10: f1 is_FinSequence_on t,m and
A11: f2 is_FinSequence_on IExec ((while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))))),Q,t),m and
A12: len f1 = len f2 and
A13: len f1 > k + 1 and
A14: f1 is_non_decreasing_on 1,k + 1 ; ::_thesis: ( f1,f2 are_fiberwise_equipotent & f2 is_non_decreasing_on 1,(k + 1) + 1 & ( for i being Element of NAT st i > (k + 1) + 1 & i <= len f1 holds
f1 . i = f2 . i ) & ( for i being Element of NAT st 1 <= i & i <= (k + 1) + 1 holds
ex j being Element of NAT st
( 1 <= j & j <= (k + 1) + 1 & f2 . i = f1 . j ) ) )
set Bt = IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))),Q,t);
set x = DataLoc ((t . (intpos 4)),(- 1));
set y = DataLoc ((t . (intpos 4)),0);
A15: (IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))),Q,t)) . GBP = 0 by A6, A7, A8, Lm6;
 (m + 1) + (k + 1) >= 7 + (t . (intpos 6)) by A6, A8, A9;
then A16: m + 1 >= 7 by A8, XREAL_1:6;
A17: DataLoc ((t . (intpos 4)),(- 1)) = DataLoc (m,(k + 1)) by A8, A9
.= intpos (m + (k + 1)) by SCMP_GCD:1 ;
A18: (IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))),Q,t)) . (intpos 6) = (Initialize (IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))),Q,t))) . (intpos 6) by SCMPDS_5:15;
A19: (IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))),Q,t)) . (intpos 4) = (Initialize (IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))),Q,t))) . (intpos 4) by SCMPDS_5:15;
A20: (Initialize (IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))),Q,t))) . GBP = (IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))),Q,t)) . GBP by SCMPDS_5:15;
A21: ( t . (DataLoc ((t . (intpos 4)),(- 1))) > t . (DataLoc ((t . (intpos 4)),0)) implies ( (IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))),Q,t)) . (DataLoc ((t . (intpos 4)),(- 1))) = t . (DataLoc ((t . (intpos 4)),0)) & (IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))),Q,t)) . (DataLoc ((t . (intpos 4)),0)) = t . (DataLoc ((t . (intpos 4)),(- 1))) & (IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))),Q,t)) . (intpos 6) = (t . (intpos 6)) - 1 & (IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))),Q,t)) . (intpos 4) = (t . (intpos 4)) - 1 ) ) by A6, A7, A8, Lm7;
A22: ( t . (DataLoc ((t . (intpos 4)),(- 1))) <= t . (DataLoc ((t . (intpos 4)),0)) implies ( (IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))),Q,t)) . (DataLoc ((t . (intpos 4)),(- 1))) = t . (DataLoc ((t . (intpos 4)),(- 1))) & (IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))),Q,t)) . (DataLoc ((t . (intpos 4)),0)) = t . (DataLoc ((t . (intpos 4)),0)) & (IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))),Q,t)) . (intpos 6) = 0 ) ) by A6, A7, A8, Lm7;
A23: DataLoc ((t . (intpos 4)),0) = intpos (m + (k + 2)) by A8, A9, SCMP_GCD:1;
A24: (IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))),Q,t)) . (intpos 4) >= 7 + ((IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))),Q,t)) . (intpos 6)) by A6, A7, A8, Lm7;
percases ( t . (DataLoc ((t . (intpos 4)),(- 1))) > t . (DataLoc ((t . (intpos 4)),0)) or t . (DataLoc ((t . (intpos 4)),(- 1))) <= t . (DataLoc ((t . (intpos 4)),0)) ) ;
supposeA25: t . (DataLoc ((t . (intpos 4)),(- 1))) > t . (DataLoc ((t . (intpos 4)),0)) ; ::_thesis: ( f1,f2 are_fiberwise_equipotent & f2 is_non_decreasing_on 1,(k + 1) + 1 & ( for i being Element of NAT st i > (k + 1) + 1 & i <= len f1 holds
f1 . i = f2 . i ) & ( for i being Element of NAT st 1 <= i & i <= (k + 1) + 1 holds
ex j being Element of NAT st
( 1 <= j & j <= (k + 1) + 1 & f2 . i = f1 . j ) ) )
now__::_thesis:_for_i_being_Element_of_NAT_st_1_<=_i_&_i_<=_len_f2_holds_
f2_._i_=_(IExec_((while>0_(GBP,6,((((GBP,5)_:=_((intpos_4),(-_1)))_';'_(SubFrom_(GBP,5,(intpos_4),0)))_';'_(if>0_(GBP,5,((((((GBP,5)_:=_((intpos_4),(-_1)))_';'_(((intpos_4),(-_1))_:=_((intpos_4),0)))_';'_(((intpos_4),0)_:=_(GBP,5)))_';'_(AddTo_(GBP,4,(-_1))))_';'_(AddTo_(GBP,6,(-_1)))),(Load_((GBP,6)_:=_0))))))),Q,(Initialize_(IExec_(((((GBP,5)_:=_((intpos_4),(-_1)))_';'_(SubFrom_(GBP,5,(intpos_4),0)))_';'_(if>0_(GBP,5,((((((GBP,5)_:=_((intpos_4),(-_1)))_';'_(((intpos_4),(-_1))_:=_((intpos_4),0)))_';'_(((intpos_4),0)_:=_(GBP,5)))_';'_(AddTo_(GBP,4,(-_1))))_';'_(AddTo_(GBP,6,(-_1)))),(Load_((GBP,6)_:=_0))))),Q,t)))))_._(intpos_(m_+_i))
let i be Element of NAT ; ::_thesis: ( 1 <= i & i <= len f2 implies f2 . i = (IExec ((while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))))),Q,(Initialize (IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))),Q,t))))) . (intpos (m + i)) )
assume  ( 1 <= i & i <= len f2 ) ; ::_thesis: f2 . i = (IExec ((while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))))),Q,(Initialize (IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))),Q,t))))) . (intpos (m + i))
hence f2 . i = (IExec ((while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))))),Q,t)) . (intpos (m + i)) by A11, Def1
.= (IExec ((while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))))),Q,(Initialize (IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))),Q,t))))) . (intpos (m + i)) by A6, A7, A8, Lm14 ;
::_thesis: verum
end;
then A26: f2 is_FinSequence_on IExec ((while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))))),Q,(Initialize (IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))),Q,t)))),m by Def1;
A27: k + 1 < k + 2 by XREAL_1:6;
consider h being FinSequence of INT  such that
A28: len h = len f1 and
A29: for i being Element of NAT st 1 <= i & i <= len h holds
h . i = (IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))),Q,t)) . (intpos (m + i)) by Th1;
 k + 1 > k by XREAL_1:29;
then A30: len h > k by A13, A28, XXREAL_0:2;
A31: now__::_thesis:_for_i_being_Element_of_NAT_st_i_<>_k_+_1_&_i_<>_k_+_2_&_1_<=_i_&_i_<=_len_f1_holds_
h_._i_=_f1_._i
let i be Element of NAT ; ::_thesis: ( i <> k + 1 & i <> k + 2 & 1 <= i & i <= len f1 implies h . i = f1 . i )
assume that
A32: ( i <> k + 1 & i <> k + 2 ) and
A33: 1 <= i and
A34: i <= len f1 ; ::_thesis: h . i = f1 . i
A35: ( m + i <> (t . (intpos 4)) - 1 & m + i <> t . (intpos 4) ) by A8, A9, A32;
 m + i >= m + 1 by A33, XREAL_1:6;
then A36: m + i >= 7 by A16, XXREAL_0:2;
thus h . i = (IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))),Q,t)) . (intpos (m + i)) by A28, A29, A33, A34
.= t . (intpos (m + i)) by A6, A7, A8, A36, A35, Lm7
.= f1 . i by A10, A33, A34, Def1 ; ::_thesis: verum
end;
now__::_thesis:_for_i,_j_being_Element_of_NAT_st_1_<=_i_&_i_<=_j_&_j_<=_k_holds_
h_._i_<=_h_._j
let i, j be Element of NAT ; ::_thesis: ( 1 <= i & i <= j & j <= k implies h . i <= h . j )
assume that
A37: 1 <= i and
A38: i <= j and
A39: j <= k ; ::_thesis: h . i <= h . j
A40: j <= len f1 by A28, A30, A39, XXREAL_0:2;
then A41: i <= len f1 by A38, XXREAL_0:2;
A42: k < k + 1 by XREAL_1:29;
then A43: j < k + 1 by A39, XXREAL_0:2;
 k + 1 < (k + 1) + 1 by XREAL_1:29;
then A44: k < (k + 1) + 1 by A42, XXREAL_0:2;
 j >= 1 by A37, A38, XXREAL_0:2;
then A45: h . j = f1 . j by A31, A39, A42, A44, A40;
 j < k + 2 by A39, A44, XXREAL_0:2;
then  h . i = f1 . i by A31, A37, A38, A43, A41;
hence  h . i <= h . j by A14, A37, A38, A43, A45, GRAPH_2:def_12; ::_thesis: verum
end;
then A46: h is_non_decreasing_on 1,k by GRAPH_2:def_12;
A47: len f1 >= (k + 1) + 1 by A13, INT_1:7;
A48: 1 <= k + 1 by NAT_1:11;
then A49: 1 <= k + 2 by A27, XXREAL_0:2;
then A50: h . (k + 2) = t . (DataLoc ((t . (intpos 4)),(- 1))) by A23, A21, A25, A28, A29, A47;
then A51: h . (k + 2) = f1 . (k + 1) by A10, A13, A17, A48, Def1;
A52: (((IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))),Q,t)) . (intpos 4)) - ((IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))),Q,t)) . (intpos 6))) - 1 = m by A9, A21, A25;
A53: h . (k + 1) = t . (DataLoc ((t . (intpos 4)),0)) by A13, A17, A21, A25, A28, A29, NAT_1:11;
then  h . (k + 1) = f1 . (k + 2) by A10, A23, A49, A47, Def1;
then A54: f1,h are_fiberwise_equipotent by A13, A28, A31, A48, A49, A47, A51, Th3;
A55: h is_FinSequence_on Initialize (IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))),Q,t)),m
proof
let i be Element of NAT ; :: according to SCPISORT:def_1 ::_thesis: ( 1 <= i & i <= len h implies h . i = (Initialize (IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))),Q,t))) . (intpos (m + i)) )
assume  ( 1 <= i & i <= len h ) ; ::_thesis: h . i = (Initialize (IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))),Q,t))) . (intpos (m + i))
then  h . i = (IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))),Q,t)) . (intpos (m + i)) by A29;
hence  h . i = (Initialize (IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))),Q,t))) . (intpos (m + i)) by SCMPDS_5:15; ::_thesis: verum
end;
 h,f2 are_fiberwise_equipotent by A4, A8, A12, A15, A24, A21, A25, A28, A52, A26, A30, A46, A55, A18, A19, A20;
hence  f1,f2 are_fiberwise_equipotent by A54, CLASSES1:76; ::_thesis: ( f2 is_non_decreasing_on 1,(k + 1) + 1 & ( for i being Element of NAT st i > (k + 1) + 1 & i <= len f1 holds
f1 . i = f2 . i ) & ( for i being Element of NAT st 1 <= i & i <= (k + 1) + 1 holds
ex j being Element of NAT st
( 1 <= j & j <= (k + 1) + 1 & f2 . i = f1 . j ) ) )
A56: f2 is_non_decreasing_on 1,k + 1 by A4, A8, A12, A15, A24, A21, A25, A28, A52, A55, A18, A19, A20, A26, A30, A46;
now__::_thesis:_for_i,_j_being_Element_of_NAT_st_1_<=_i_&_i_<=_j_&_j_<=_(k_+_1)_+_1_holds_
f2_._i_<=_f2_._j
let i, j be Element of NAT ; ::_thesis: ( 1 <= i & i <= j & j <= (k + 1) + 1 implies f2 . b1 <= f2 . b2 )
assume that
A57: 1 <= i and
A58: i <= j and
A59: j <= (k + 1) + 1 ; ::_thesis: f2 . b1 <= f2 . b2
percases ( j <= k + 1 or j = (k + 1) + 1 ) by A59, NAT_1:8;
suppose j <= k + 1 ; ::_thesis: f2 . b1 <= f2 . b2
hence  f2 . i <= f2 . j by A56, A57, A58, GRAPH_2:def_12; ::_thesis: verum
end;
supposeA60: j = (k + 1) + 1 ; ::_thesis: f2 . b1 <= f2 . b2
hereby  ::_thesis: verum
percases ( i = j or i <> j ) ;
suppose i = j ; ::_thesis: f2 . i <= f2 . j
hence  f2 . i <= f2 . j ; ::_thesis: verum
end;
suppose i <> j ; ::_thesis: f2 . i <= f2 . j
then  i < j by A58, XXREAL_0:1;
then  i <= k + 1 by A60, NAT_1:13;
then consider mm being Element of NAT  such that
A61: 1 <= mm and
A62: mm <= k + 1 and
A63: f2 . i = h . mm by A4, A8, A12, A15, A24, A21, A25, A28, A52, A55, A26, A30, A46, A57, A18, A19, A20;
A64: f2 . j = h . (k + 2) by A4, A8, A12, A15, A24, A21, A25, A28, A52, A55, A26, A30, A46, A27, A47, A60, A18, A19, A20;
hereby  ::_thesis: verum
percases ( mm = k + 1 or mm <> k + 1 ) ;
suppose mm = k + 1 ; ::_thesis: f2 . i <= f2 . j
hence  f2 . i <= f2 . j by A13, A17, A21, A22, A28, A29, A50, A61, A63, A64; ::_thesis: verum
end;
supposeA65: mm <> k + 1 ; ::_thesis: f2 . i <= f2 . j
 mm < k + 2 by A27, A62, XXREAL_0:2;
then  mm < len h by A28, A47, XXREAL_0:2;
then A66: h . mm = f1 . mm by A28, A31, A27, A61, A62, A65;
 f2 . j = f1 . (k + 1) by A10, A13, A17, A48, A50, A64, Def1;
hence  f2 . i <= f2 . j by A14, A61, A62, A63, A66, GRAPH_2:def_12; ::_thesis: verum
end;
end;
end;
end;
end;
end;
end;
end;
end;
hence  f2 is_non_decreasing_on 1,(k + 1) + 1 by GRAPH_2:def_12; ::_thesis: ( ( for i being Element of NAT st i > (k + 1) + 1 & i <= len f1 holds
f1 . i = f2 . i ) & ( for i being Element of NAT st 1 <= i & i <= (k + 1) + 1 holds
ex j being Element of NAT st
( 1 <= j & j <= (k + 1) + 1 & f2 . i = f1 . j ) ) )
hereby  ::_thesis: for i being Element of NAT st 1 <= i & i <= (k + 1) + 1 holds
ex j being Element of NAT st
( 1 <= j & j <= (k + 1) + 1 & f2 . i = f1 . j )
let i be Element of NAT ; ::_thesis: ( i > (k + 1) + 1 & i <= len f1 implies f2 . i = f1 . i )
assume that
A67: i > (k + 1) + 1 and
A68: i <= len f1 ; ::_thesis: f2 . i = f1 . i
A69: k + 1 < (k + 1) + 1 by XREAL_1:29;
then A70: i > k + 1 by A67, XXREAL_0:2;
 1 <= k + 1 by NAT_1:11;
then A71: 1 <= i by A70, XXREAL_0:2;
thus f2 . i = h . i by A4, A8, A12, A15, A24, A21, A25, A28, A52, A55, A26, A30, A46, A68, A70, A18, A19, A20
.= f1 . i by A31, A67, A68, A69, A71 ; ::_thesis: verum
end;
hereby  ::_thesis: verum
let i be Element of NAT ; ::_thesis: ( 1 <= i & i <= (k + 1) + 1 implies ex j being Element of NAT st
( 1 <= b2 & b2 <= (k + 1) + 1 & f2 . j = f1 . b2 ) )
assume that
A72: 1 <= i and
A73: i <= (k + 1) + 1 ; ::_thesis: ex j being Element of NAT st
( 1 <= b2 & b2 <= (k + 1) + 1 & f2 . j = f1 . b2 )
percases ( i = (k + 1) + 1 or i <> (k + 1) + 1 ) ;
supposeA74: i = (k + 1) + 1 ; ::_thesis: ex j being Element of NAT st
( 1 <= b2 & b2 <= (k + 1) + 1 & f2 . j = f1 . b2 )
take j = k + 1; ::_thesis: ( 1 <= j & j <= (k + 1) + 1 & f2 . i = f1 . j )
thus  1 <= j by NAT_1:11; ::_thesis: ( j <= (k + 1) + 1 & f2 . i = f1 . j )
thus  j <= (k + 1) + 1 by NAT_1:11; ::_thesis: f2 . i = f1 . j
thus  f2 . i = f1 . j by A4, A8, A12, A15, A24, A21, A25, A28, A52, A55, A26, A30, A46, A27, A47, A51, A74, A18, A19, A20; ::_thesis: verum
end;
suppose i <> (k + 1) + 1 ; ::_thesis: ex j being Element of NAT st
( 1 <= b2 & b2 <= (k + 1) + 1 & f2 . j = f1 . b2 )
then  i < (k + 1) + 1 by A73, XXREAL_0:1;
then  i <= k + 1 by NAT_1:13;
then consider mm being Element of NAT  such that
A75: 1 <= mm and
A76: mm <= k + 1 and
A77: f2 . i = h . mm by A4, A8, A12, A15, A24, A21, A25, A28, A52, A55, A26, A30, A46, A72, A18, A19, A20;
hereby  ::_thesis: verum
A78: k + 2 = (k + 1) + 1 ;
percases ( mm = k + 1 or mm <> k + 1 ) ;
supposeA79: mm = k + 1 ; ::_thesis: ex j being Element of NAT st
( 1 <= j & j <= (k + 1) + 1 & f2 . i = f1 . j )
take j = k + 2; ::_thesis: ( 1 <= j & j <= (k + 1) + 1 & f2 . i = f1 . j )
thus  1 <= j by A78, NAT_1:11; ::_thesis: ( j <= (k + 1) + 1 & f2 . i = f1 . j )
thus  j <= (k + 1) + 1 ; ::_thesis: f2 . i = f1 . j
thus  f2 . i = f1 . j by A10, A23, A49, A47, A53, A77, A79, Def1; ::_thesis: verum
end;
supposeA80: mm <> k + 1 ; ::_thesis: ex mm being Element of NAT st
( 1 <= mm & mm <= (k + 1) + 1 & f2 . i = f1 . mm )
take mm = mm; ::_thesis: ( 1 <= mm & mm <= (k + 1) + 1 & f2 . i = f1 . mm )
thus  1 <= mm by A75; ::_thesis: ( mm <= (k + 1) + 1 & f2 . i = f1 . mm )
thus  mm <= (k + 1) + 1 by A27, A76, XXREAL_0:2; ::_thesis: f2 . i = f1 . mm
 mm < k + 2 by A27, A76, XXREAL_0:2;
then  mm < len f1 by A47, XXREAL_0:2;
hence  f2 . i = f1 . mm by A31, A27, A75, A76, A77, A80; ::_thesis: verum
end;
end;
end;
end;
end;
end;
end;
supposeA81: t . (DataLoc ((t . (intpos 4)),(- 1))) <= t . (DataLoc ((t . (intpos 4)),0)) ; ::_thesis: ( f1,f2 are_fiberwise_equipotent & f2 is_non_decreasing_on 1,(k + 1) + 1 & ( for i being Element of NAT st i > (k + 1) + 1 & i <= len f1 holds
f1 . i = f2 . i ) & ( for i being Element of NAT st 1 <= i & i <= (k + 1) + 1 holds
ex j being Element of NAT st
( 1 <= j & j <= (k + 1) + 1 & f2 . i = f1 . j ) ) )
A82: now__::_thesis:_for_i_being_Nat_st_i_>=_1_&_i_<=_len_f1_holds_
f1_._i_=_f2_._i
let i be Nat; ::_thesis: ( i >= 1 & i <= len f1 implies f1 . b1 = f2 . b1 )
assume that
A83: i >= 1 and
A84: i <= len f1 ; ::_thesis: f1 . b1 = f2 . b1
A85: i in NAT by ORDINAL1:def_12;
then A86: f1 . i = t . (intpos (m + i)) by A10, A83, A84, Def1;
 (IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))),Q,t)) . (DataLoc (((IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))),Q,t)) . GBP),6)) = 0 by A15, A22, A81, SCMP_GCD:1;
then A87: (Initialize (IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))),Q,t))) . (DataLoc (((IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))),Q,t)) . GBP),6)) = 0 by SCMPDS_5:15;
 m + i >= m + 1 by A83, XREAL_1:6;
then A88: m + i >= 7 by A16, XXREAL_0:2;
A89: (Initialize (IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))),Q,t))) . (intpos (m + i)) = (IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))),Q,t)) . (intpos (m + i)) by SCMPDS_5:15;
A90: (Initialize (IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))),Q,t))) . (DataLoc (((Initialize (IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))),Q,t))) . GBP),6)) <= 0 by A87, SCMPDS_5:15;
percases ( m + i = (t . (intpos 4)) - 1 or m + i = t . (intpos 4) or ( m + i <> (t . (intpos 4)) - 1 & m + i <> t . (intpos 4) ) ) ;
supposeA91: m + i = (t . (intpos 4)) - 1 ; ::_thesis: f1 . b1 = f2 . b1
hence f1 . i = (IExec ((while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))))),Q,(Initialize (IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))),Q,t))))) . (DataLoc ((t . (intpos 4)),(- 1))) by A8, A9, A17, A22, A81, A86, A89, A90, SCMPDS_8:23
.= (IExec ((while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))))),Q,(Initialize t))) . (DataLoc ((t . (intpos 4)),(- 1))) by A6, A7, A8, Lm14, A5
.= f2 . i by A8, A9, A11, A12, A13, A17, A83, A91, Def1, A5 ;
::_thesis: verum
end;
supposeA92: m + i = t . (intpos 4) ; ::_thesis: f1 . b1 = f2 . b1
hence f1 . i = (IExec ((while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))))),Q,(Initialize (IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))),Q,t))))) . (DataLoc ((t . (intpos 4)),0)) by A8, A9, A23, A22, A81, A86, A89, A90, SCMPDS_8:23
.= (IExec ((while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))))),Q,(Initialize t))) . (DataLoc ((t . (intpos 4)),0)) by A6, A7, A8, Lm14, A5
.= f2 . i by A8, A9, A11, A12, A23, A83, A84, A92, Def1, A5 ;
::_thesis: verum
end;
suppose ( m + i <> (t . (intpos 4)) - 1 & m + i <> t . (intpos 4) ) ; ::_thesis: f1 . b1 = f2 . b1
hence f1 . i = (IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))),Q,t)) . (intpos (m + i)) by A6, A7, A8, A88, A86, Lm7
.= (IExec ((while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))))),Q,(Initialize (IExec (((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))),Q,t))))) . (intpos (m + i)) by A89, A90, SCMPDS_8:23
.= (IExec ((while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))))),Q,(Initialize t))) . (intpos (m + i)) by A6, A7, A8, Lm14, A5
.= f2 . i by A11, A12, A85, A83, A84, Def1, A5 ;
::_thesis: verum
end;
end;
end;
then A93: f1 = f2 by A12, FINSEQ_1:14;
thus  f1,f2 are_fiberwise_equipotent by A12, A82, FINSEQ_1:14; ::_thesis: ( f2 is_non_decreasing_on 1,(k + 1) + 1 & ( for i being Element of NAT st i > (k + 1) + 1 & i <= len f1 holds
f1 . i = f2 . i ) & ( for i being Element of NAT st 1 <= i & i <= (k + 1) + 1 holds
ex j being Element of NAT st
( 1 <= j & j <= (k + 1) + 1 & f2 . i = f1 . j ) ) )
now__::_thesis:_for_i,_j_being_Element_of_NAT_st_1_<=_i_&_i_<=_j_&_j_<=_(k_+_1)_+_1_holds_
f1_._i_<=_f1_._j
let i, j be Element of NAT ; ::_thesis: ( 1 <= i & i <= j & j <= (k + 1) + 1 implies f1 . b1 <= f1 . b2 )
assume that
A94: 1 <= i and
A95: i <= j and
A96: j <= (k + 1) + 1 ; ::_thesis: f1 . b1 <= f1 . b2
percases ( j <= k + 1 or j = (k + 1) + 1 ) by A96, NAT_1:8;
suppose j <= k + 1 ; ::_thesis: f1 . b1 <= f1 . b2
hence  f1 . i <= f1 . j by A14, A94, A95, GRAPH_2:def_12; ::_thesis: verum
end;
supposeA97: j = (k + 1) + 1 ; ::_thesis: f1 . b1 <= f1 . b2
hereby  ::_thesis: verum
percases ( i = j or i <> j ) ;
suppose i = j ; ::_thesis: f1 . i <= f1 . j
hence  f1 . i <= f1 . j ; ::_thesis: verum
end;
suppose i <> j ; ::_thesis: f1 . i <= f1 . j
then  i < j by A95, XXREAL_0:1;
then  i <= k + 1 by A97, NAT_1:13;
then A98: f1 . i <= f1 . (k + 1) by A14, A94, GRAPH_2:def_12;
 1 <= k + 1 by NAT_1:11;
then A99: f1 . (k + 1) = t . (DataLoc ((t . (intpos 4)),(- 1))) by A10, A13, A17, Def1;
 ( 1 <= (k + 1) + 1 & j <= len f1 ) by A13, A97, INT_1:7, NAT_1:11;
then  f1 . j = t . (DataLoc ((t . (intpos 4)),0)) by A10, A23, A97, Def1;
hence  f1 . i <= f1 . j by A81, A98, A99, XXREAL_0:2; ::_thesis: verum
end;
end;
end;
end;
end;
end;
hence  f2 is_non_decreasing_on 1,(k + 1) + 1 by A93, GRAPH_2:def_12; ::_thesis: ( ( for i being Element of NAT st i > (k + 1) + 1 & i <= len f1 holds
f1 . i = f2 . i ) & ( for i being Element of NAT st 1 <= i & i <= (k + 1) + 1 holds
ex j being Element of NAT st
( 1 <= j & j <= (k + 1) + 1 & f2 . i = f1 . j ) ) )
thus  for i being Element of NAT st i > (k + 1) + 1 & i <= len f1 holds
f1 . i = f2 . i by A12, A82, FINSEQ_1:14; ::_thesis: for i being Element of NAT st 1 <= i & i <= (k + 1) + 1 holds
ex j being Element of NAT st
( 1 <= j & j <= (k + 1) + 1 & f2 . i = f1 . j )
thus  for i being Element of NAT st 1 <= i & i <= (k + 1) + 1 holds
ex j being Element of NAT st
( 1 <= j & j <= (k + 1) + 1 & f2 . i = f1 . j ) by A93; ::_thesis: verum
end;
end;
end;
end;
A100: S1[ 0 ]
proof
let t be 0 -started State of SCMPDS; ::_thesis: for Q being Instruction-Sequence of SCMPDS
for f1, f2 being FinSequence of INT st t . (intpos 4) >= 7 + (t . (intpos 6)) & t . GBP = 0 & 0 = t . (intpos 6) & m = ((t . (intpos 4)) - (t . (intpos 6))) - 1 & f1 is_FinSequence_on t,m & f2 is_FinSequence_on IExec ((while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))))),Q,t),m & len f1 = len f2 & len f1 > 0 & f1 is_non_decreasing_on 1, 0 holds
( f1,f2 are_fiberwise_equipotent & f2 is_non_decreasing_on 1,0 + 1 & ( for i being Element of NAT st i > 0 + 1 & i <= len f1 holds
f1 . i = f2 . i ) & ( for i being Element of NAT st 1 <= i & i <= 0 + 1 holds
ex j being Element of NAT st
( 1 <= j & j <= 0 + 1 & f2 . i = f1 . j ) ) )
let Q be Instruction-Sequence of SCMPDS; ::_thesis: for f1, f2 being FinSequence of INT st t . (intpos 4) >= 7 + (t . (intpos 6)) & t . GBP = 0 & 0 = t . (intpos 6) & m = ((t . (intpos 4)) - (t . (intpos 6))) - 1 & f1 is_FinSequence_on t,m & f2 is_FinSequence_on IExec ((while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))))),Q,t),m & len f1 = len f2 & len f1 > 0 & f1 is_non_decreasing_on 1, 0 holds
( f1,f2 are_fiberwise_equipotent & f2 is_non_decreasing_on 1,0 + 1 & ( for i being Element of NAT st i > 0 + 1 & i <= len f1 holds
f1 . i = f2 . i ) & ( for i being Element of NAT st 1 <= i & i <= 0 + 1 holds
ex j being Element of NAT st
( 1 <= j & j <= 0 + 1 & f2 . i = f1 . j ) ) )
let f1, f2 be FinSequence of INT ; ::_thesis: ( t . (intpos 4) >= 7 + (t . (intpos 6)) & t . GBP = 0 & 0 = t . (intpos 6) & m = ((t . (intpos 4)) - (t . (intpos 6))) - 1 & f1 is_FinSequence_on t,m & f2 is_FinSequence_on IExec ((while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))))),Q,t),m & len f1 = len f2 & len f1 > 0 & f1 is_non_decreasing_on 1, 0 implies ( f1,f2 are_fiberwise_equipotent & f2 is_non_decreasing_on 1,0 + 1 & ( for i being Element of NAT st i > 0 + 1 & i <= len f1 holds
f1 . i = f2 . i ) & ( for i being Element of NAT st 1 <= i & i <= 0 + 1 holds
ex j being Element of NAT st
( 1 <= j & j <= 0 + 1 & f2 . i = f1 . j ) ) ) )
A101: Initialize t = t by MEMSTR_0:44;
assume that
 t . (intpos 4) >= 7 + (t . (intpos 6)) and
A102: ( t . GBP = 0 & 0 = t . (intpos 6) ) and
 m = ((t . (intpos 4)) - (t . (intpos 6))) - 1 and
A103: f1 is_FinSequence_on t,m and
A104: f2 is_FinSequence_on IExec ((while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))))),Q,t),m and
A105: len f1 = len f2 and
 len f1 > 0 and
 f1 is_non_decreasing_on 1, 0 ; ::_thesis: ( f1,f2 are_fiberwise_equipotent & f2 is_non_decreasing_on 1,0 + 1 & ( for i being Element of NAT st i > 0 + 1 & i <= len f1 holds
f1 . i = f2 . i ) & ( for i being Element of NAT st 1 <= i & i <= 0 + 1 holds
ex j being Element of NAT st
( 1 <= j & j <= 0 + 1 & f2 . i = f1 . j ) ) )
A106: t . (DataLoc ((t . GBP),6)) = 0 by A102, SCMP_GCD:1;
A107: now__::_thesis:_for_i_being_Nat_st_1_<=_i_&_i_<=_len_f1_holds_
f1_._i_=_f2_._i
let i be Nat; ::_thesis: ( 1 <= i & i <= len f1 implies f1 . i = f2 . i )
assume A108: ( 1 <= i & i <= len f1 ) ; ::_thesis: f1 . i = f2 . i
A109: i in NAT by ORDINAL1:def_12;
hence f1 . i = t . (intpos (m + i)) by A103, A108, Def1
.= (IExec ((while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))))),Q,(Initialize t))) . (intpos (m + i)) by A106, A101, SCMPDS_8:23
.= f2 . i by A104, A105, A109, A108, Def1, A101 ;
::_thesis: verum
end;
hence  f1,f2 are_fiberwise_equipotent by A105, FINSEQ_1:14; ::_thesis: ( f2 is_non_decreasing_on 1,0 + 1 & ( for i being Element of NAT st i > 0 + 1 & i <= len f1 holds
f1 . i = f2 . i ) & ( for i being Element of NAT st 1 <= i & i <= 0 + 1 holds
ex j being Element of NAT st
( 1 <= j & j <= 0 + 1 & f2 . i = f1 . j ) ) )
thus  f2 is_non_decreasing_on 1,0 + 1 by GRAPH_2:63; ::_thesis: ( ( for i being Element of NAT st i > 0 + 1 & i <= len f1 holds
f1 . i = f2 . i ) & ( for i being Element of NAT st 1 <= i & i <= 0 + 1 holds
ex j being Element of NAT st
( 1 <= j & j <= 0 + 1 & f2 . i = f1 . j ) ) )
thus  for i being Element of NAT st i > 0 + 1 & i <= len f1 holds
f1 . i = f2 . i by A107; ::_thesis: for i being Element of NAT st 1 <= i & i <= 0 + 1 holds
ex j being Element of NAT st
( 1 <= j & j <= 0 + 1 & f2 . i = f1 . j )
 f1 = f2 by A105, A107, FINSEQ_1:14;
hence  for i being Element of NAT st 1 <= i & i <= 0 + 1 holds
ex j being Element of NAT st
( 1 <= j & j <= 0 + 1 & f2 . i = f1 . j ) ; ::_thesis: verum
end;
A110: for k being Element of NAT holds S1[k] from NAT_1:sch_1(A100, A3);
assume  ( len f = len g & len f > n & f is_non_decreasing_on 1,n ) ; ::_thesis: ( f,g are_fiberwise_equipotent & g is_non_decreasing_on 1,n + 1 & ( for i being Element of NAT st i > n + 1 & i <= len f holds
f . i = g . i ) & ( for i being Element of NAT st 1 <= i & i <= n + 1 holds
ex j being Element of NAT st
( 1 <= j & j <= n + 1 & g . i = f . j ) ) )
hence  ( f,g are_fiberwise_equipotent & g is_non_decreasing_on 1,n + 1 & ( for i being Element of NAT st i > n + 1 & i <= len f holds
f . i = g . i ) & ( for i being Element of NAT st 1 <= i & i <= n + 1 holds
ex j being Element of NAT st
( 1 <= j & j <= n + 1 & g . i = f . j ) ) ) by A1, A2, A110; ::_thesis: verum
end;

Lm15:  for P being Instruction-Sequence of SCMPDS
for s being 0 -started State of SCMPDS
for f, g being FinSequence of INT
for p0, n being Element of NAT st s . GBP = 0 & s . (intpos 2) = n - 1 & s . (intpos 3) = p0 + 1 & s . (intpos 1) = 0 & p0 >= 6 & f is_FinSequence_on s,p0 & g is_FinSequence_on IExec ((for-down (GBP,2,1,(((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))) ';' (while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0)))))))))),P,s),p0 & len f = n & len g = n holds
( f,g are_fiberwise_equipotent & g is_non_decreasing_on 1,n )
proof
let P be Instruction-Sequence of SCMPDS; ::_thesis: for s being 0 -started State of SCMPDS
for f, g being FinSequence of INT
for p0, n being Element of NAT st s . GBP = 0 & s . (intpos 2) = n - 1 & s . (intpos 3) = p0 + 1 & s . (intpos 1) = 0 & p0 >= 6 & f is_FinSequence_on s,p0 & g is_FinSequence_on IExec ((for-down (GBP,2,1,(((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))) ';' (while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0)))))))))),P,s),p0 & len f = n & len g = n holds
( f,g are_fiberwise_equipotent & g is_non_decreasing_on 1,n )
set a = GBP ;
let s be 0 -started State of SCMPDS; ::_thesis: for f, g being FinSequence of INT
for p0, n being Element of NAT st s . GBP = 0 & s . (intpos 2) = n - 1 & s . (intpos 3) = p0 + 1 & s . (intpos 1) = 0 & p0 >= 6 & f is_FinSequence_on s,p0 & g is_FinSequence_on IExec ((for-down (GBP,2,1,(((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))) ';' (while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0)))))))))),P,s),p0 & len f = n & len g = n holds
( f,g are_fiberwise_equipotent & g is_non_decreasing_on 1,n )
let f, g be FinSequence of INT ; ::_thesis: for p0, n being Element of NAT st s . GBP = 0 & s . (intpos 2) = n - 1 & s . (intpos 3) = p0 + 1 & s . (intpos 1) = 0 & p0 >= 6 & f is_FinSequence_on s,p0 & g is_FinSequence_on IExec ((for-down (GBP,2,1,(((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))) ';' (while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0)))))))))),P,s),p0 & len f = n & len g = n holds
( f,g are_fiberwise_equipotent & g is_non_decreasing_on 1,n )
let p0, n be Element of NAT ; ::_thesis: ( s . GBP = 0 & s . (intpos 2) = n - 1 & s . (intpos 3) = p0 + 1 & s . (intpos 1) = 0 & p0 >= 6 & f is_FinSequence_on s,p0 & g is_FinSequence_on IExec ((for-down (GBP,2,1,(((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))) ';' (while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0)))))))))),P,s),p0 & len f = n & len g = n implies ( f,g are_fiberwise_equipotent & g is_non_decreasing_on 1,n ) )
assume that
A1: s . GBP = 0 and
A2: s . (intpos 2) = n - 1 and
A3: s . (intpos 3) = p0 + 1 and
A4: s . (intpos 1) = 0 and
A5: p0 >= 6 and
A6: ( f is_FinSequence_on s,p0 & g is_FinSequence_on IExec ((for-down (GBP,2,1,(((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))) ';' (while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0)))))))))),P,s),p0 ) and
A7: len f = n and
A8: len g = n ; ::_thesis: ( f,g are_fiberwise_equipotent & g is_non_decreasing_on 1,n )
percases ( n = 0 or n <> 0 ) ;
supposeA9: n = 0 ; ::_thesis: ( f,g are_fiberwise_equipotent & g is_non_decreasing_on 1,n )
then  g = {} by A8;
hence  f,g are_fiberwise_equipotent by A7, A9; ::_thesis: g is_non_decreasing_on 1,n
thus  g is_non_decreasing_on 1,n by A9, GRAPH_2:63; ::_thesis: verum
end;
suppose n <> 0 ; ::_thesis: ( f,g are_fiberwise_equipotent & g is_non_decreasing_on 1,n )
then  n >= 1 + 0 by INT_1:7;
then  n - 1 >= 0 by XREAL_1:19;
then reconsider n1 = n - 1 as Element of NAT by INT_1:3;
defpred S1[ Nat] means  for t being 0 -started State of SCMPDS
for Q being Instruction-Sequence of SCMPDS
for f1, f2 being FinSequence of INT
for m being Element of NAT st t . GBP = 0 & (t . (intpos 2)) + (t . (intpos 1)) = n - 1 & t . (intpos 2) = $1 & m = n - (t . (intpos 2)) & p0 = ((t . (intpos 3)) - (t . (intpos 1))) - 1 & f1 is_FinSequence_on t,p0 & f2 is_FinSequence_on IExec ((for-down (GBP,2,1,(((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))) ';' (while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0)))))))))),Q,t),p0 & f1 is_non_decreasing_on 1,m & len f1 = n & len f2 = n holds
( f1,f2 are_fiberwise_equipotent & f2 is_non_decreasing_on 1,n );
A10: ( (s . (intpos 2)) + (s . (intpos 1)) = (n - 1) + 0 & 1 = n - (s . (intpos 2)) ) by A2, A4;
A11: now__::_thesis:_for_k_being_Element_of_NAT_st_S1[k]_holds_
S1[k_+_1]
let k be Element of NAT ; ::_thesis: ( S1[k] implies S1[k + 1] )
assume A12: S1[k] ; ::_thesis: S1[k + 1]
now__::_thesis:_for_t_being_0_-started_State_of_SCMPDS
for_f1,_f2_being_FinSequence_of_INT_
for_m_being_Element_of_NAT_
for_Q_being_Instruction-Sequence_of_SCMPDS_st_t_._GBP_=_0_&_(t_._(intpos_2))_+_(t_._(intpos_1))_=_n_-_1_&_t_._(intpos_2)_=_k_+_1_&_m_=_n_-_(t_._(intpos_2))_&_p0_=_((t_._(intpos_3))_-_(t_._(intpos_1)))_-_1_&_f1_is_FinSequence_on_t,p0_&_f2_is_FinSequence_on_IExec_((for-down_(GBP,2,1,(((((AddTo_(GBP,3,1))_';'_((GBP,4)_:=_(GBP,3)))_';'_(AddTo_(GBP,1,1)))_';'_((GBP,6)_:=_(GBP,1)))_';'_(while>0_(GBP,6,((((GBP,5)_:=_((intpos_4),(-_1)))_';'_(SubFrom_(GBP,5,(intpos_4),0)))_';'_(if>0_(GBP,5,((((((GBP,5)_:=_((intpos_4),(-_1)))_';'_(((intpos_4),(-_1))_:=_((intpos_4),0)))_';'_(((intpos_4),0)_:=_(GBP,5)))_';'_(AddTo_(GBP,4,(-_1))))_';'_(AddTo_(GBP,6,(-_1)))),(Load_((GBP,6)_:=_0)))))))))),Q,t),p0_&_f1_is_non_decreasing_on_1,m_&_len_f1_=_n_&_len_f2_=_n_holds_
(_f1,f2_are_fiberwise_equipotent_&_f2_is_non_decreasing_on_1,n_)
let t be 0 -started State of SCMPDS; ::_thesis: for f1, f2 being FinSequence of INT
for m being Element of NAT
for Q being Instruction-Sequence of SCMPDS st t . GBP = 0 & (t . (intpos 2)) + (t . (intpos 1)) = n - 1 & t . (intpos 2) = k + 1 & m = n - (t . (intpos 2)) & p0 = ((t . (intpos 3)) - (t . (intpos 1))) - 1 & f1 is_FinSequence_on t,p0 & f2 is_FinSequence_on IExec ((for-down (GBP,2,1,(((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))) ';' (while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0)))))))))),Q,t),p0 & f1 is_non_decreasing_on 1,m & len f1 = n & len f2 = n holds
( f1,f2 are_fiberwise_equipotent & f2 is_non_decreasing_on 1,n )
let f1, f2 be FinSequence of INT ; ::_thesis: for m being Element of NAT
for Q being Instruction-Sequence of SCMPDS st t . GBP = 0 & (t . (intpos 2)) + (t . (intpos 1)) = n - 1 & t . (intpos 2) = k + 1 & m = n - (t . (intpos 2)) & p0 = ((t . (intpos 3)) - (t . (intpos 1))) - 1 & f1 is_FinSequence_on t,p0 & f2 is_FinSequence_on IExec ((for-down (GBP,2,1,(((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))) ';' (while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0)))))))))),Q,t),p0 & f1 is_non_decreasing_on 1,m & len f1 = n & len f2 = n holds
( f1,f2 are_fiberwise_equipotent & f2 is_non_decreasing_on 1,n )
let m be Element of NAT ; ::_thesis: for Q being Instruction-Sequence of SCMPDS st t . GBP = 0 & (t . (intpos 2)) + (t . (intpos 1)) = n - 1 & t . (intpos 2) = k + 1 & m = n - (t . (intpos 2)) & p0 = ((t . (intpos 3)) - (t . (intpos 1))) - 1 & f1 is_FinSequence_on t,p0 & f2 is_FinSequence_on IExec ((for-down (GBP,2,1,(((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))) ';' (while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0)))))))))),Q,t),p0 & f1 is_non_decreasing_on 1,m & len f1 = n & len f2 = n holds
( f1,f2 are_fiberwise_equipotent & f2 is_non_decreasing_on 1,n )
let Q be Instruction-Sequence of SCMPDS; ::_thesis: ( t . GBP = 0 & (t . (intpos 2)) + (t . (intpos 1)) = n - 1 & t . (intpos 2) = k + 1 & m = n - (t . (intpos 2)) & p0 = ((t . (intpos 3)) - (t . (intpos 1))) - 1 & f1 is_FinSequence_on t,p0 & f2 is_FinSequence_on IExec ((for-down (GBP,2,1,(((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))) ';' (while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0)))))))))),Q,t),p0 & f1 is_non_decreasing_on 1,m & len f1 = n & len f2 = n implies ( f1,f2 are_fiberwise_equipotent & f2 is_non_decreasing_on 1,n ) )
assume that
A13: t . GBP = 0 and
A14: (t . (intpos 2)) + (t . (intpos 1)) = n - 1 and
A15: t . (intpos 2) = k + 1 and
A16: m = n - (t . (intpos 2)) and
A17: p0 = ((t . (intpos 3)) - (t . (intpos 1))) - 1 and
A18: f1 is_FinSequence_on t,p0 and
A19: f2 is_FinSequence_on IExec ((for-down (GBP,2,1,(((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))) ';' (while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0)))))))))),Q,t),p0 and
A20: ( f1 is_non_decreasing_on 1,m & len f1 = n ) and
A21: len f2 = n ; ::_thesis: ( f1,f2 are_fiberwise_equipotent & f2 is_non_decreasing_on 1,n )
set t1 = IExec (((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))),Q,t);
set Bt = IExec (((((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))) ';' (while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0)))))))) ';' (AddTo (GBP,2,(- 1)))),Q,t);
set Q1 = Q;
A22: (IExec (((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))),Q,t)) . GBP = (Initialize (IExec (((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))),Q,t))) . GBP by SCMPDS_5:15;
A23: (IExec (((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))),Q,t)) . (intpos 4) = (Initialize (IExec (((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))),Q,t))) . (intpos 4) by SCMPDS_5:15;
A24: (IExec (((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))),Q,t)) . (intpos 6) = (Initialize (IExec (((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))),Q,t))) . (intpos 6) by SCMPDS_5:15;
A25: (IExec (((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))),Q,t)) . (intpos 4) = (t . (intpos 3)) + 1 by A13, Lm10;
 p0 + ((t . (intpos 1)) + 1) = t . (intpos 3) by A17;
then  t . (intpos 3) >= 6 + ((t . (intpos 1)) + 1) by A5, XREAL_1:6;
then A26: t . (intpos 3) >= (6 + 1) + (t . (intpos 1)) ;
then A27: (IExec (((((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))) ';' (while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0)))))))) ';' (AddTo (GBP,2,(- 1)))),Q,t)) . GBP = 0 by A13, Lm11;
A28: (IExec (((((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))) ';' (while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0)))))))) ';' (AddTo (GBP,2,(- 1)))),Q,t)) . (intpos 2) = (t . (intpos 2)) - 1 by A13, A26, Lm11;
then A29: n - ((IExec (((((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))) ';' (while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0)))))))) ';' (AddTo (GBP,2,(- 1)))),Q,t)) . (intpos 2)) = m + 1 by A16;
A30: (IExec (((((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))) ';' (while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0)))))))) ';' (AddTo (GBP,2,(- 1)))),Q,t)) . (intpos 1) = (t . (intpos 1)) + 1 by A13, A26, Lm11;
then A31: ((IExec (((((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))) ';' (while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0)))))))) ';' (AddTo (GBP,2,(- 1)))),Q,t)) . (intpos 2)) + ((IExec (((((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))) ';' (while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0)))))))) ';' (AddTo (GBP,2,(- 1)))),Q,t)) . (intpos 1)) = n - 1 by A14, A28;
 (IExec (((((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))) ';' (while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0)))))))) ';' (AddTo (GBP,2,(- 1)))),Q,t)) . (intpos 3) = (t . (intpos 3)) + 1 by A13, A26, Lm11;
then A32: (((IExec (((((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))) ';' (while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0)))))))) ';' (AddTo (GBP,2,(- 1)))),Q,t)) . (intpos 3)) - ((IExec (((((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))) ';' (while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0)))))))) ';' (AddTo (GBP,2,(- 1)))),Q,t)) . (intpos 1))) - 1 = p0 by A17, A30;
A33: (IExec (((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))),Q,t)) . (intpos 6) = (t . (intpos 1)) + 1 by A13, Lm10;
then A34: p0 = (((IExec (((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))),Q,t)) . (intpos 4)) - ((IExec (((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))),Q,t)) . (intpos 6))) - 1 by A17, A25;
now__::_thesis:_for_i_being_Element_of_NAT_st_1_<=_i_&_i_<=_len_f2_holds_
f2_._i_=_(IExec_((for-down_(GBP,2,1,(((((AddTo_(GBP,3,1))_';'_((GBP,4)_:=_(GBP,3)))_';'_(AddTo_(GBP,1,1)))_';'_((GBP,6)_:=_(GBP,1)))_';'_(while>0_(GBP,6,((((GBP,5)_:=_((intpos_4),(-_1)))_';'_(SubFrom_(GBP,5,(intpos_4),0)))_';'_(if>0_(GBP,5,((((((GBP,5)_:=_((intpos_4),(-_1)))_';'_(((intpos_4),(-_1))_:=_((intpos_4),0)))_';'_(((intpos_4),0)_:=_(GBP,5)))_';'_(AddTo_(GBP,4,(-_1))))_';'_(AddTo_(GBP,6,(-_1)))),(Load_((GBP,6)_:=_0)))))))))),Q,(Initialize_(IExec_(((((((AddTo_(GBP,3,1))_';'_((GBP,4)_:=_(GBP,3)))_';'_(AddTo_(GBP,1,1)))_';'_((GBP,6)_:=_(GBP,1)))_';'_(while>0_(GBP,6,((((GBP,5)_:=_((intpos_4),(-_1)))_';'_(SubFrom_(GBP,5,(intpos_4),0)))_';'_(if>0_(GBP,5,((((((GBP,5)_:=_((intpos_4),(-_1)))_';'_(((intpos_4),(-_1))_:=_((intpos_4),0)))_';'_(((intpos_4),0)_:=_(GBP,5)))_';'_(AddTo_(GBP,4,(-_1))))_';'_(AddTo_(GBP,6,(-_1)))),(Load_((GBP,6)_:=_0))))))))_';'_(AddTo_(GBP,2,(-_1)))),Q,t)))))_._(intpos_(p0_+_i))
let i be Element of NAT ; ::_thesis: ( 1 <= i & i <= len f2 implies f2 . i = (IExec ((for-down (GBP,2,1,(((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))) ';' (while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0)))))))))),Q,(Initialize (IExec (((((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))) ';' (while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0)))))))) ';' (AddTo (GBP,2,(- 1)))),Q,t))))) . (intpos (p0 + i)) )
assume  ( 1 <= i & i <= len f2 ) ; ::_thesis: f2 . i = (IExec ((for-down (GBP,2,1,(((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))) ';' (while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0)))))))))),Q,(Initialize (IExec (((((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))) ';' (while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0)))))))) ';' (AddTo (GBP,2,(- 1)))),Q,t))))) . (intpos (p0 + i))
hence f2 . i = (IExec ((for-down (GBP,2,1,(((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))) ';' (while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0)))))))))),Q,t)) . (intpos (p0 + i)) by A19, Def1
.= (IExec ((for-down (GBP,2,1,(((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))) ';' (while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0)))))))))),Q,(Initialize (IExec (((((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))) ';' (while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0)))))))) ';' (AddTo (GBP,2,(- 1)))),Q,t))))) . (intpos (p0 + i)) by A13, A15, A26, Lm13 ;
::_thesis: verum
end;
then A35: f2 is_FinSequence_on IExec ((for-down (GBP,2,1,(((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))) ';' (while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0)))))))))),Q,(Initialize (IExec (((((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))) ';' (while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0)))))))) ';' (AddTo (GBP,2,(- 1)))),Q,t)))),p0 by Def1;
A36: now__::_thesis:_for_i_being_Element_of_NAT_st_1_<=_i_&_i_<=_len_f1_holds_
f1_._i_=_(IExec_(((((AddTo_(GBP,3,1))_';'_((GBP,4)_:=_(GBP,3)))_';'_(AddTo_(GBP,1,1)))_';'_((GBP,6)_:=_(GBP,1))),Q,t))_._(intpos_(p0_+_i))
A37: p0 + 1 >= 6 + 1 by A5, XREAL_1:6;
let i be Element of NAT ; ::_thesis: ( 1 <= i & i <= len f1 implies f1 . i = (IExec (((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))),Q,t)) . (intpos (p0 + i)) )
assume that
A38: 1 <= i and
A39: i <= len f1 ; ::_thesis: f1 . i = (IExec (((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))),Q,t)) . (intpos (p0 + i))
 p0 + 1 <= p0 + i by A38, XREAL_1:6;
then A40: p0 + i >= 7 by A37, XXREAL_0:2;
thus f1 . i = t . (intpos (p0 + i)) by A18, A38, A39, Def1
.= (IExec (((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))),Q,t)) . (intpos (p0 + i)) by A13, A40, Lm10 ; ::_thesis: verum
end;
 (IExec (((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))),Q,t)) . (intpos 4) = p0 + (((IExec (((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))),Q,t)) . (intpos 6)) + 1) by A17, A25, A33;
then  (IExec (((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))),Q,t)) . (intpos 4) >= 6 + (((IExec (((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))),Q,t)) . (intpos 6)) + 1) by A5, XREAL_1:6;
then A41: (IExec (((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))),Q,t)) . (intpos 4) >= (6 + 1) + ((IExec (((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))),Q,t)) . (intpos 6)) ;
 m + (k + 1) = n by A15, A16;
then A42: n > 0 + m by XREAL_1:6;
consider h being FinSequence of INT  such that
A43: len h = n and
A44: for i being Element of NAT st 1 <= i & i <= len h holds
h . i = (IExec ((while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))))),Q,(Initialize (IExec (((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))),Q,t))))) . (intpos (p0 + i)) by Th1;
A45: h is_FinSequence_on IExec ((while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))))),Q,(Initialize (IExec (((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))),Q,t)))),p0 by A44, Def1;
A46: now__::_thesis:_for_i_being_Element_of_NAT_st_1_<=_i_&_i_<=_len_h_holds_
h_._i_=_(IExec_(((((((AddTo_(GBP,3,1))_';'_((GBP,4)_:=_(GBP,3)))_';'_(AddTo_(GBP,1,1)))_';'_((GBP,6)_:=_(GBP,1)))_';'_(while>0_(GBP,6,((((GBP,5)_:=_((intpos_4),(-_1)))_';'_(SubFrom_(GBP,5,(intpos_4),0)))_';'_(if>0_(GBP,5,((((((GBP,5)_:=_((intpos_4),(-_1)))_';'_(((intpos_4),(-_1))_:=_((intpos_4),0)))_';'_(((intpos_4),0)_:=_(GBP,5)))_';'_(AddTo_(GBP,4,(-_1))))_';'_(AddTo_(GBP,6,(-_1)))),(Load_((GBP,6)_:=_0))))))))_';'_(AddTo_(GBP,2,(-_1)))),Q,t))_._(intpos_(p0_+_i))
A47: p0 + 1 >= 6 + 1 by A5, XREAL_1:6;
let i be Element of NAT ; ::_thesis: ( 1 <= i & i <= len h implies h . i = (IExec (((((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))) ';' (while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0)))))))) ';' (AddTo (GBP,2,(- 1)))),Q,t)) . (intpos (p0 + i)) )
assume that
A48: 1 <= i and
A49: i <= len h ; ::_thesis: h . i = (IExec (((((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))) ';' (while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0)))))))) ';' (AddTo (GBP,2,(- 1)))),Q,t)) . (intpos (p0 + i))
 p0 + 1 <= p0 + i by A48, XREAL_1:6;
then  p0 + i >= 7 by A47, XXREAL_0:2;
then A50: p0 + i > 2 by XXREAL_0:2;
thus h . i = (IExec ((while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))))),Q,(Initialize (IExec (((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))),Q,t))))) . (intpos (p0 + i)) by A44, A48, A49
.= (IExec (((((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))) ';' (while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0)))))))) ';' (AddTo (GBP,2,(- 1)))),Q,t)) . (intpos (p0 + i)) by A13, A26, A50, Lm11 ; ::_thesis: verum
end;
A51: f1 is_FinSequence_on Initialize (IExec (((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))),Q,t)),p0
proof
let i be Element of NAT ; :: according to SCPISORT:def_1 ::_thesis: ( 1 <= i & i <= len f1 implies f1 . i = (Initialize (IExec (((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))),Q,t))) . (intpos (p0 + i)) )
assume  ( 1 <= i & i <= len f1 ) ; ::_thesis: f1 . i = (Initialize (IExec (((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))),Q,t))) . (intpos (p0 + i))
then  f1 . i = (IExec (((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))),Q,t)) . (intpos (p0 + i)) by A36;
hence  f1 . i = (Initialize (IExec (((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))),Q,t))) . (intpos (p0 + i)) by SCMPDS_5:15; ::_thesis: verum
end;
A52: (IExec (((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))),Q,t)) . GBP = 0 by A13, Lm10;
then A53: f1,h are_fiberwise_equipotent by A14, A16, A17, A20, A25, A43, A34, A41, A45, A42, Th16, A23, A24, A22, A51;
A54: h is_non_decreasing_on 1,m + 1 by A14, A16, A17, A20, A25, A43, A34, A41, A45, A42, Th16, A23, A24, A22, A51, A52;
A55: (Initialize (IExec (((((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))) ';' (while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0)))))))) ';' (AddTo (GBP,2,(- 1)))),Q,t))) . GBP = (IExec (((((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))) ';' (while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0)))))))) ';' (AddTo (GBP,2,(- 1)))),Q,t)) . GBP by SCMPDS_5:15;
A56: (Initialize (IExec (((((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))) ';' (while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0)))))))) ';' (AddTo (GBP,2,(- 1)))),Q,t))) . (intpos 1) = (IExec (((((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))) ';' (while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0)))))))) ';' (AddTo (GBP,2,(- 1)))),Q,t)) . (intpos 1) by SCMPDS_5:15;
A57: (Initialize (IExec (((((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))) ';' (while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0)))))))) ';' (AddTo (GBP,2,(- 1)))),Q,t))) . (intpos 2) = (IExec (((((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))) ';' (while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0)))))))) ';' (AddTo (GBP,2,(- 1)))),Q,t)) . (intpos 2) by SCMPDS_5:15;
A58: (Initialize (IExec (((((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))) ';' (while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0)))))))) ';' (AddTo (GBP,2,(- 1)))),Q,t))) . (intpos 3) = (IExec (((((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))) ';' (while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0)))))))) ';' (AddTo (GBP,2,(- 1)))),Q,t)) . (intpos 3) by SCMPDS_5:15;
A59: h is_FinSequence_on Initialize (IExec (((((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))) ';' (while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0)))))))) ';' (AddTo (GBP,2,(- 1)))),Q,t)),p0
proof
let i be Element of NAT ; :: according to SCPISORT:def_1 ::_thesis: ( 1 <= i & i <= len h implies h . i = (Initialize (IExec (((((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))) ';' (while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0)))))))) ';' (AddTo (GBP,2,(- 1)))),Q,t))) . (intpos (p0 + i)) )
assume A60: ( 1 <= i & i <= len h ) ; ::_thesis: h . i = (Initialize (IExec (((((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))) ';' (while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0)))))))) ';' (AddTo (GBP,2,(- 1)))),Q,t))) . (intpos (p0 + i))
thus h . i = (IExec (((((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))) ';' (while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0)))))))) ';' (AddTo (GBP,2,(- 1)))),Q,t)) . (intpos (p0 + i)) by A60, A46
.= (Initialize (IExec (((((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))) ';' (while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0)))))))) ';' (AddTo (GBP,2,(- 1)))),Q,t))) . (intpos (p0 + i)) by SCMPDS_5:15 ; ::_thesis: verum
end;
then  h,f2 are_fiberwise_equipotent by A12, A15, A16, A21, A43, A27, A31, A29, A32, A35, A54, A55, A56, A57, A58;
hence  f1,f2 are_fiberwise_equipotent by A53, CLASSES1:76; ::_thesis: f2 is_non_decreasing_on 1,n
thus  f2 is_non_decreasing_on 1,n by A12, A15, A16, A21, A43, A54, A27, A31, A29, A32, A35, A55, A56, A57, A58, A59; ::_thesis: verum
end;
hence  S1[k + 1] ; ::_thesis: verum
end;
A61: S1[ 0 ]
proof
let t be 0 -started State of SCMPDS; ::_thesis: for Q being Instruction-Sequence of SCMPDS
for f1, f2 being FinSequence of INT
for m being Element of NAT st t . GBP = 0 & (t . (intpos 2)) + (t . (intpos 1)) = n - 1 & t . (intpos 2) = 0 & m = n - (t . (intpos 2)) & p0 = ((t . (intpos 3)) - (t . (intpos 1))) - 1 & f1 is_FinSequence_on t,p0 & f2 is_FinSequence_on IExec ((for-down (GBP,2,1,(((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))) ';' (while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0)))))))))),Q,t),p0 & f1 is_non_decreasing_on 1,m & len f1 = n & len f2 = n holds
( f1,f2 are_fiberwise_equipotent & f2 is_non_decreasing_on 1,n )
let Q be Instruction-Sequence of SCMPDS; ::_thesis: for f1, f2 being FinSequence of INT
for m being Element of NAT st t . GBP = 0 & (t . (intpos 2)) + (t . (intpos 1)) = n - 1 & t . (intpos 2) = 0 & m = n - (t . (intpos 2)) & p0 = ((t . (intpos 3)) - (t . (intpos 1))) - 1 & f1 is_FinSequence_on t,p0 & f2 is_FinSequence_on IExec ((for-down (GBP,2,1,(((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))) ';' (while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0)))))))))),Q,t),p0 & f1 is_non_decreasing_on 1,m & len f1 = n & len f2 = n holds
( f1,f2 are_fiberwise_equipotent & f2 is_non_decreasing_on 1,n )
let f1, f2 be FinSequence of INT ; ::_thesis: for m being Element of NAT st t . GBP = 0 & (t . (intpos 2)) + (t . (intpos 1)) = n - 1 & t . (intpos 2) = 0 & m = n - (t . (intpos 2)) & p0 = ((t . (intpos 3)) - (t . (intpos 1))) - 1 & f1 is_FinSequence_on t,p0 & f2 is_FinSequence_on IExec ((for-down (GBP,2,1,(((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))) ';' (while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0)))))))))),Q,t),p0 & f1 is_non_decreasing_on 1,m & len f1 = n & len f2 = n holds
( f1,f2 are_fiberwise_equipotent & f2 is_non_decreasing_on 1,n )
let m be Element of NAT ; ::_thesis: ( t . GBP = 0 & (t . (intpos 2)) + (t . (intpos 1)) = n - 1 & t . (intpos 2) = 0 & m = n - (t . (intpos 2)) & p0 = ((t . (intpos 3)) - (t . (intpos 1))) - 1 & f1 is_FinSequence_on t,p0 & f2 is_FinSequence_on IExec ((for-down (GBP,2,1,(((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))) ';' (while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0)))))))))),Q,t),p0 & f1 is_non_decreasing_on 1,m & len f1 = n & len f2 = n implies ( f1,f2 are_fiberwise_equipotent & f2 is_non_decreasing_on 1,n ) )
A62: Initialize t = t by MEMSTR_0:44;
assume that
A63: t . GBP = 0 and
 (t . (intpos 2)) + (t . (intpos 1)) = n - 1 and
A64: t . (intpos 2) = 0 and
A65: m = n - (t . (intpos 2)) and
 p0 = ((t . (intpos 3)) - (t . (intpos 1))) - 1 and
A66: f1 is_FinSequence_on t,p0 and
A67: f2 is_FinSequence_on IExec ((for-down (GBP,2,1,(((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))) ';' (while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0)))))))))),Q,t),p0 and
A68: f1 is_non_decreasing_on 1,m and
A69: ( len f1 = n & len f2 = n ) ; ::_thesis: ( f1,f2 are_fiberwise_equipotent & f2 is_non_decreasing_on 1,n )
A70: t . (DataLoc ((t . GBP),2)) = 0 by A63, A64, SCMP_GCD:1;
A71: now__::_thesis:_for_i_being_Nat_st_1_<=_i_&_i_<=_len_f2_holds_
f2_._i_=_f1_._i
let i be Nat; ::_thesis: ( 1 <= i & i <= len f2 implies f2 . i = f1 . i )
assume A72: ( 1 <= i & i <= len f2 ) ; ::_thesis: f2 . i = f1 . i
A73: i in NAT by ORDINAL1:def_12;
hence f2 . i = (IExec ((for-down (GBP,2,1,(((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))) ';' (while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0)))))))))),Q,(Initialize t))) . (intpos (p0 + i)) by A67, A72, Def1, A62
.= t . (intpos (p0 + i)) by A70, SCMPDS_7:47
.= f1 . i by A66, A69, A73, A72, Def1 ;
::_thesis: verum
end;
hence  f1,f2 are_fiberwise_equipotent by A69, FINSEQ_1:14; ::_thesis: f2 is_non_decreasing_on 1,n
thus  f2 is_non_decreasing_on 1,n by A64, A65, A68, A69, A71, FINSEQ_1:14; ::_thesis: verum
end;
 for k being Element of NAT holds S1[k] from NAT_1:sch_1(A61, A11);
then A74: S1[n1] ;
 ( p0 = ((s . (intpos 3)) - (s . (intpos 1))) - 1 & f is_non_decreasing_on 1,1 ) by A3, A4, GRAPH_2:63;
hence  ( f,g are_fiberwise_equipotent & g is_non_decreasing_on 1,n ) by A1, A6, A7, A8, A10, A74; ::_thesis: verum
end;
end;
end;

theorem :: SCPISORT:17
for P being Instruction-Sequence of SCMPDS
for s being 0 -started State of SCMPDS
for f, g being FinSequence of INT
for p0, n being Element of NAT st p0 >= 6 & len f = n & len g = n & f is_FinSequence_on s,p0 & g is_FinSequence_on IExec ((insert-sort (n,(p0 + 1))),P,s),p0 holds
( f,g are_fiberwise_equipotent & g is_non_decreasing_on 1,n )
proof
let P be Instruction-Sequence of SCMPDS; ::_thesis: for s being 0 -started State of SCMPDS
for f, g being FinSequence of INT
for p0, n being Element of NAT st p0 >= 6 & len f = n & len g = n & f is_FinSequence_on s,p0 & g is_FinSequence_on IExec ((insert-sort (n,(p0 + 1))),P,s),p0 holds
( f,g are_fiberwise_equipotent & g is_non_decreasing_on 1,n )
set a = GBP ;
let s be 0 -started State of SCMPDS; ::_thesis: for f, g being FinSequence of INT
for p0, n being Element of NAT st p0 >= 6 & len f = n & len g = n & f is_FinSequence_on s,p0 & g is_FinSequence_on IExec ((insert-sort (n,(p0 + 1))),P,s),p0 holds
( f,g are_fiberwise_equipotent & g is_non_decreasing_on 1,n )
let f, g be FinSequence of INT ; ::_thesis: for p0, n being Element of NAT st p0 >= 6 & len f = n & len g = n & f is_FinSequence_on s,p0 & g is_FinSequence_on IExec ((insert-sort (n,(p0 + 1))),P,s),p0 holds
( f,g are_fiberwise_equipotent & g is_non_decreasing_on 1,n )
let p0, n be Element of NAT ; ::_thesis: ( p0 >= 6 & len f = n & len g = n & f is_FinSequence_on s,p0 & g is_FinSequence_on IExec ((insert-sort (n,(p0 + 1))),P,s),p0 implies ( f,g are_fiberwise_equipotent & g is_non_decreasing_on 1,n ) )
assume that
A1: p0 >= 6 and
A2: ( len f = n & len g = n ) and
A3: f is_FinSequence_on s,p0 and
A4: g is_FinSequence_on IExec ((insert-sort (n,(p0 + 1))),P,s),p0 ; ::_thesis: ( f,g are_fiberwise_equipotent & g is_non_decreasing_on 1,n )
A5: p0 + 1 >= 6 + 1 by A1, XREAL_1:6;
set i1 = GBP := 0;
set i2 = (GBP,1) := 0;
set i3 = (GBP,2) := (n - 1);
set i4 = (GBP,3) := (p0 + 1);
set I4 = (((GBP := 0) ';' ((GBP,1) := 0)) ';' ((GBP,2) := (n - 1))) ';' ((GBP,3) := (p0 + 1));
set t1 = IExec (((((GBP := 0) ';' ((GBP,1) := 0)) ';' ((GBP,2) := (n - 1))) ';' ((GBP,3) := (p0 + 1))),P,s);
set t2 = IExec ((((GBP := 0) ';' ((GBP,1) := 0)) ';' ((GBP,2) := (n - 1))),P,s);
set t3 = IExec (((GBP := 0) ';' ((GBP,1) := 0)),P,s);
set t4 = Exec ((GBP := 0),s);
A6: (Exec ((GBP := 0),s)) . GBP = 0 by SCMPDS_2:45;
then A7: DataLoc (((Exec ((GBP := 0),s)) . GBP),1) = intpos (0 + 1) by SCMP_GCD:1;
A8: (IExec (((GBP := 0) ';' ((GBP,1) := 0)),P,s)) . GBP = (Exec (((GBP,1) := 0),(Exec ((GBP := 0),s)))) . GBP by SCMPDS_5:42
.= 0 by A6, A7, AMI_3:10, SCMPDS_2:46 ;
then A9: DataLoc (((IExec (((GBP := 0) ';' ((GBP,1) := 0)),P,s)) . GBP),2) = intpos (0 + 2) by SCMP_GCD:1;
A10: (IExec ((((GBP := 0) ';' ((GBP,1) := 0)) ';' ((GBP,2) := (n - 1))),P,s)) . GBP = (Exec (((GBP,2) := (n - 1)),(IExec (((GBP := 0) ';' ((GBP,1) := 0)),P,s)))) . GBP by SCMPDS_5:41
.= 0 by A8, A9, AMI_3:10, SCMPDS_2:46 ;
then A11: DataLoc (((IExec ((((GBP := 0) ';' ((GBP,1) := 0)) ';' ((GBP,2) := (n - 1))),P,s)) . GBP),3) = intpos (0 + 3) by SCMP_GCD:1;
A12: now__::_thesis:_for_i_being_Element_of_NAT_st_i_>_3_holds_
(IExec_(((GBP_:=_0)_';'_((GBP,1)_:=_0)),P,s))_._(intpos_i)_=_s_._(intpos_i)
let i be Element of NAT ; ::_thesis: ( i > 3 implies (IExec (((GBP := 0) ';' ((GBP,1) := 0)),P,s)) . (intpos i) = s . (intpos i) )
assume A13: i > 3 ; ::_thesis: (IExec (((GBP := 0) ';' ((GBP,1) := 0)),P,s)) . (intpos i) = s . (intpos i)
then A14: i > 1 by XXREAL_0:2;
thus (IExec (((GBP := 0) ';' ((GBP,1) := 0)),P,s)) . (intpos i) = (Exec (((GBP,1) := 0),(Exec ((GBP := 0),s)))) . (intpos i) by SCMPDS_5:42
.= (Exec ((GBP := 0),s)) . (intpos i) by A7, A14, AMI_3:10, SCMPDS_2:46
.= s . (intpos i) by A13, AMI_3:10, SCMPDS_2:45 ; ::_thesis: verum
end;
A15: now__::_thesis:_for_i_being_Element_of_NAT_st_i_>_3_holds_
(IExec_((((GBP_:=_0)_';'_((GBP,1)_:=_0))_';'_((GBP,2)_:=_(n_-_1))),P,s))_._(intpos_i)_=_s_._(intpos_i)
let i be Element of NAT ; ::_thesis: ( i > 3 implies (IExec ((((GBP := 0) ';' ((GBP,1) := 0)) ';' ((GBP,2) := (n - 1))),P,s)) . (intpos i) = s . (intpos i) )
assume A16: i > 3 ; ::_thesis: (IExec ((((GBP := 0) ';' ((GBP,1) := 0)) ';' ((GBP,2) := (n - 1))),P,s)) . (intpos i) = s . (intpos i)
then A17: i > 2 by XXREAL_0:2;
thus (IExec ((((GBP := 0) ';' ((GBP,1) := 0)) ';' ((GBP,2) := (n - 1))),P,s)) . (intpos i) = (Exec (((GBP,2) := (n - 1)),(IExec (((GBP := 0) ';' ((GBP,1) := 0)),P,s)))) . (intpos i) by SCMPDS_5:41
.= (IExec (((GBP := 0) ';' ((GBP,1) := 0)),P,s)) . (intpos i) by A9, A17, AMI_3:10, SCMPDS_2:46
.= s . (intpos i) by A12, A16 ; ::_thesis: verum
end;
now__::_thesis:_for_i_being_Element_of_NAT_st_1_<=_i_&_i_<=_len_f_holds_
(Initialize_(IExec_(((((GBP_:=_0)_';'_((GBP,1)_:=_0))_';'_((GBP,2)_:=_(n_-_1)))_';'_((GBP,3)_:=_(p0_+_1))),P,s)))_._(intpos_(p0_+_i))_=_f_._i
let i be Element of NAT ; ::_thesis: ( 1 <= i & i <= len f implies (Initialize (IExec (((((GBP := 0) ';' ((GBP,1) := 0)) ';' ((GBP,2) := (n - 1))) ';' ((GBP,3) := (p0 + 1))),P,s))) . (intpos (p0 + i)) = f . i )
assume that
A18: 1 <= i and
A19: i <= len f ; ::_thesis: (Initialize (IExec (((((GBP := 0) ';' ((GBP,1) := 0)) ';' ((GBP,2) := (n - 1))) ';' ((GBP,3) := (p0 + 1))),P,s))) . (intpos (p0 + i)) = f . i
set pi = p0 + i;
 p0 + i >= p0 + 1 by A18, XREAL_1:6;
then  p0 + i >= 7 by A5, XXREAL_0:2;
then A20: p0 + i > 3 by XXREAL_0:2;
thus (Initialize (IExec (((((GBP := 0) ';' ((GBP,1) := 0)) ';' ((GBP,2) := (n - 1))) ';' ((GBP,3) := (p0 + 1))),P,s))) . (intpos (p0 + i)) = (IExec (((((GBP := 0) ';' ((GBP,1) := 0)) ';' ((GBP,2) := (n - 1))) ';' ((GBP,3) := (p0 + 1))),P,s)) . (intpos (p0 + i)) by SCMPDS_5:15
.= (Exec (((GBP,3) := (p0 + 1)),(IExec ((((GBP := 0) ';' ((GBP,1) := 0)) ';' ((GBP,2) := (n - 1))),P,s)))) . (intpos (p0 + i)) by SCMPDS_5:41
.= (IExec ((((GBP := 0) ';' ((GBP,1) := 0)) ';' ((GBP,2) := (n - 1))),P,s)) . (intpos (p0 + i)) by A11, A20, AMI_3:10, SCMPDS_2:46
.= s . (intpos (p0 + i)) by A15, A20
.= f . i by A3, A18, A19, Def1 ; ::_thesis: verum
end;
then A21: f is_FinSequence_on Initialize (IExec (((((GBP := 0) ';' ((GBP,1) := 0)) ';' ((GBP,2) := (n - 1))) ';' ((GBP,3) := (p0 + 1))),P,s)),p0 by Def1;
A22: (IExec (((GBP := 0) ';' ((GBP,1) := 0)),P,s)) . (intpos 1) = (Exec (((GBP,1) := 0),(Exec ((GBP := 0),s)))) . (intpos 1) by SCMPDS_5:42
.= 0 by A7, SCMPDS_2:46 ;
A23: (IExec ((((GBP := 0) ';' ((GBP,1) := 0)) ';' ((GBP,2) := (n - 1))),P,s)) . (intpos 1) = (Exec (((GBP,2) := (n - 1)),(IExec (((GBP := 0) ';' ((GBP,1) := 0)),P,s)))) . (intpos 1) by SCMPDS_5:41
.= 0 by A22, A9, AMI_3:10, SCMPDS_2:46 ;
A24: ( (((GBP := 0) ';' ((GBP,1) := 0)) ';' ((GBP,2) := (n - 1))) ';' ((GBP,3) := (p0 + 1)) is_closed_on s,P & (((GBP := 0) ';' ((GBP,1) := 0)) ';' ((GBP,2) := (n - 1))) ';' ((GBP,3) := (p0 + 1)) is_halting_on s,P ) by SCMPDS_6:20, SCMPDS_6:21;
A25: (IExec (((((GBP := 0) ';' ((GBP,1) := 0)) ';' ((GBP,2) := (n - 1))) ';' ((GBP,3) := (p0 + 1))),P,s)) . GBP = (Exec (((GBP,3) := (p0 + 1)),(IExec ((((GBP := 0) ';' ((GBP,1) := 0)) ';' ((GBP,2) := (n - 1))),P,s)))) . GBP by SCMPDS_5:41
.= 0 by A10, A11, AMI_3:10, SCMPDS_2:46 ;
A26: (IExec ((((GBP := 0) ';' ((GBP,1) := 0)) ';' ((GBP,2) := (n - 1))),P,s)) . (intpos 2) = (Exec (((GBP,2) := (n - 1)),(IExec (((GBP := 0) ';' ((GBP,1) := 0)),P,s)))) . (intpos 2) by SCMPDS_5:41
.= n - 1 by A9, SCMPDS_2:46 ;
A27: (IExec (((((GBP := 0) ';' ((GBP,1) := 0)) ';' ((GBP,2) := (n - 1))) ';' ((GBP,3) := (p0 + 1))),P,s)) . (intpos 3) = (Exec (((GBP,3) := (p0 + 1)),(IExec ((((GBP := 0) ';' ((GBP,1) := 0)) ';' ((GBP,2) := (n - 1))),P,s)))) . (intpos 3) by SCMPDS_5:41
.= p0 + 1 by A11, SCMPDS_2:46 ;
A28: (IExec (((((GBP := 0) ';' ((GBP,1) := 0)) ';' ((GBP,2) := (n - 1))) ';' ((GBP,3) := (p0 + 1))),P,s)) . (intpos 3) = (Initialize (IExec (((((GBP := 0) ';' ((GBP,1) := 0)) ';' ((GBP,2) := (n - 1))) ';' ((GBP,3) := (p0 + 1))),P,s))) . (intpos 3) by SCMPDS_5:15;
A29: (IExec (((((GBP := 0) ';' ((GBP,1) := 0)) ';' ((GBP,2) := (n - 1))) ';' ((GBP,3) := (p0 + 1))),P,s)) . (intpos 1) = (Initialize (IExec (((((GBP := 0) ';' ((GBP,1) := 0)) ';' ((GBP,2) := (n - 1))) ';' ((GBP,3) := (p0 + 1))),P,s))) . (intpos 1) by SCMPDS_5:15;
A30: (IExec (((((GBP := 0) ';' ((GBP,1) := 0)) ';' ((GBP,2) := (n - 1))) ';' ((GBP,3) := (p0 + 1))),P,s)) . GBP = (Initialize (IExec (((((GBP := 0) ';' ((GBP,1) := 0)) ';' ((GBP,2) := (n - 1))) ';' ((GBP,3) := (p0 + 1))),P,s))) . GBP by SCMPDS_5:15;
A31: (IExec (((((GBP := 0) ';' ((GBP,1) := 0)) ';' ((GBP,2) := (n - 1))) ';' ((GBP,3) := (p0 + 1))),P,s)) . (intpos 2) = (Initialize (IExec (((((GBP := 0) ';' ((GBP,1) := 0)) ';' ((GBP,2) := (n - 1))) ';' ((GBP,3) := (p0 + 1))),P,s))) . (intpos 2) by SCMPDS_5:15;
A32: (IExec (((((GBP := 0) ';' ((GBP,1) := 0)) ';' ((GBP,2) := (n - 1))) ';' ((GBP,3) := (p0 + 1))),P,s)) . (intpos 1) = (Exec (((GBP,3) := (p0 + 1)),(IExec ((((GBP := 0) ';' ((GBP,1) := 0)) ';' ((GBP,2) := (n - 1))),P,s)))) . (intpos 1) by SCMPDS_5:41
.= 0 by A23, A11, AMI_3:10, SCMPDS_2:46 ;
then  (IExec (((((GBP := 0) ';' ((GBP,1) := 0)) ';' ((GBP,2) := (n - 1))) ';' ((GBP,3) := (p0 + 1))),P,s)) . (intpos 3) >= ((IExec (((((GBP := 0) ';' ((GBP,1) := 0)) ';' ((GBP,2) := (n - 1))) ';' ((GBP,3) := (p0 + 1))),P,s)) . (intpos 1)) + 7 by A27, A5;
then  ( for-down (GBP,2,1,(((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))) ';' (while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))))))) is_closed_on Initialize (IExec (((((GBP := 0) ';' ((GBP,1) := 0)) ';' ((GBP,2) := (n - 1))) ';' ((GBP,3) := (p0 + 1))),P,s)),P & for-down (GBP,2,1,(((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))) ';' (while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))))))) is_halting_on Initialize (IExec (((((GBP := 0) ';' ((GBP,1) := 0)) ';' ((GBP,2) := (n - 1))) ';' ((GBP,3) := (p0 + 1))),P,s)),P ) by A25, Lm12, A28, A29, A30;
then A33: ( for-down (GBP,2,1,(((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))) ';' (while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))))))) is_closed_on IExec (((((GBP := 0) ';' ((GBP,1) := 0)) ';' ((GBP,2) := (n - 1))) ';' ((GBP,3) := (p0 + 1))),P,s),P & for-down (GBP,2,1,(((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))) ';' (while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))))))) is_halting_on IExec (((((GBP := 0) ';' ((GBP,1) := 0)) ';' ((GBP,2) := (n - 1))) ';' ((GBP,3) := (p0 + 1))),P,s),P ) by SCMPDS_6:125, SCMPDS_6:126;
now__::_thesis:_for_i_being_Element_of_NAT_st_1_<=_i_&_i_<=_len_g_holds_
g_._i_=_(IExec_((for-down_(GBP,2,1,(((((AddTo_(GBP,3,1))_';'_((GBP,4)_:=_(GBP,3)))_';'_(AddTo_(GBP,1,1)))_';'_((GBP,6)_:=_(GBP,1)))_';'_(while>0_(GBP,6,((((GBP,5)_:=_((intpos_4),(-_1)))_';'_(SubFrom_(GBP,5,(intpos_4),0)))_';'_(if>0_(GBP,5,((((((GBP,5)_:=_((intpos_4),(-_1)))_';'_(((intpos_4),(-_1))_:=_((intpos_4),0)))_';'_(((intpos_4),0)_:=_(GBP,5)))_';'_(AddTo_(GBP,4,(-_1))))_';'_(AddTo_(GBP,6,(-_1)))),(Load_((GBP,6)_:=_0)))))))))),P,(Initialize_(IExec_(((((GBP_:=_0)_';'_((GBP,1)_:=_0))_';'_((GBP,2)_:=_(n_-_1)))_';'_((GBP,3)_:=_(p0_+_1))),P,s)))))_._(intpos_(p0_+_i))
let i be Element of NAT ; ::_thesis: ( 1 <= i & i <= len g implies g . i = (IExec ((for-down (GBP,2,1,(((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))) ';' (while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0)))))))))),P,(Initialize (IExec (((((GBP := 0) ';' ((GBP,1) := 0)) ';' ((GBP,2) := (n - 1))) ';' ((GBP,3) := (p0 + 1))),P,s))))) . (intpos (p0 + i)) )
assume  ( 1 <= i & i <= len g ) ; ::_thesis: g . i = (IExec ((for-down (GBP,2,1,(((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))) ';' (while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0)))))))))),P,(Initialize (IExec (((((GBP := 0) ';' ((GBP,1) := 0)) ';' ((GBP,2) := (n - 1))) ';' ((GBP,3) := (p0 + 1))),P,s))))) . (intpos (p0 + i))
hence g . i = (IExec ((((((GBP := 0) ';' ((GBP,1) := 0)) ';' ((GBP,2) := (n - 1))) ';' ((GBP,3) := (p0 + 1))) ';' (for-down (GBP,2,1,(((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))) ';' (while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0))))))))))),P,s)) . (intpos (p0 + i)) by A4, Def1
.= (IExec ((for-down (GBP,2,1,(((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))) ';' (while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0)))))))))),P,(Initialize (IExec (((((GBP := 0) ';' ((GBP,1) := 0)) ';' ((GBP,2) := (n - 1))) ';' ((GBP,3) := (p0 + 1))),P,s))))) . (intpos (p0 + i)) by A24, A33, SCMPDS_7:30 ;
::_thesis: verum
end;
then A34: g is_FinSequence_on IExec ((for-down (GBP,2,1,(((((AddTo (GBP,3,1)) ';' ((GBP,4) := (GBP,3))) ';' (AddTo (GBP,1,1))) ';' ((GBP,6) := (GBP,1))) ';' (while>0 (GBP,6,((((GBP,5) := ((intpos 4),(- 1))) ';' (SubFrom (GBP,5,(intpos 4),0))) ';' (if>0 (GBP,5,((((((GBP,5) := ((intpos 4),(- 1))) ';' (((intpos 4),(- 1)) := ((intpos 4),0))) ';' (((intpos 4),0) := (GBP,5))) ';' (AddTo (GBP,4,(- 1)))) ';' (AddTo (GBP,6,(- 1)))),(Load ((GBP,6) := 0)))))))))),P,(Initialize (IExec (((((GBP := 0) ';' ((GBP,1) := 0)) ';' ((GBP,2) := (n - 1))) ';' ((GBP,3) := (p0 + 1))),P,s)))),p0 by Def1;
(IExec (((((GBP := 0) ';' ((GBP,1) := 0)) ';' ((GBP,2) := (n - 1))) ';' ((GBP,3) := (p0 + 1))),P,s)) . (intpos 2) = (Exec (((GBP,3) := (p0 + 1)),(IExec ((((GBP := 0) ';' ((GBP,1) := 0)) ';' ((GBP,2) := (n - 1))),P,s)))) . (intpos 2) by SCMPDS_5:41
.= n - 1 by A26, A11, AMI_3:10, SCMPDS_2:46 ;
hence  ( f,g are_fiberwise_equipotent & g is_non_decreasing_on 1,n ) by A1, A2, A25, A32, A27, A21, A34, Lm15, A28, A29, A30, A31; ::_thesis: verum
end;