:: The Construction and Computation of While-loop Programs for SCMPDS
:: by JingChao Chen
::
:: Copyright (c) 2000-2021 Association of Mizar Users

set A = NAT ;

set D = SCM-Data-Loc ;

theorem :: SCMPDS_8:1
for a being Int_position ex i being Nat st a = intpos i
proof end;

theorem :: SCMPDS_8:2
canceled;

::$CT theorem Th2: :: SCMPDS_8:3 for t being State of SCMPDS for i being Instruction of SCMPDS st InsCode i in {0,4,5,6,14} holds Initialize t = Initialize (Exec (i,t)) proof end; theorem :: SCMPDS_8:4 canceled; ::$CT
theorem Th3: :: SCMPDS_8:5
for a being Int_position ex f being Function of ,NAT st
for s being State of SCMPDS holds
( ( s . a <= 0 implies f . s = 0 ) & ( s . a > 0 implies f . s = s . a ) )
proof end;

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

:: deftheorem SCMPDS_8:def 1 :
canceled;

:: deftheorem defines while<0 SCMPDS_8:def 2 :
for a being Int_position
for i being Integer
for I being Program of SCMPDS holds while<0 (a,i,I) = (((a,i) >=0_goto ((card I) + 2)) ';' I) ';' (goto (- ((card I) + 1)));

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

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

theorem Th4: :: SCMPDS_8:6
for a being Int_position
for i being Integer
for I being Program of SCMPDS holds card (while<0 (a,i,I)) = (card I) + 2
proof end;

Lm1: for a being Int_position
for i being Integer
for I being Program of SCMPDS holds card (stop (while<0 (a,i,I))) = (card I) + 3

proof end;

theorem Th5: :: SCMPDS_8:7
for a being Int_position
for i being Integer
for m being Nat
for I being Program of SCMPDS holds
( m < (card I) + 2 iff m in dom (while<0 (a,i,I)) )
proof end;

theorem Th6: :: SCMPDS_8:8
for a being Int_position
for i being Integer
for I being Program of SCMPDS holds
( (while<0 (a,i,I)) . 0 = (a,i) >=0_goto ((card I) + 2) & (while<0 (a,i,I)) . ((card I) + 1) = goto (- ((card I) + 1)) )
proof end;

theorem Th7: :: SCMPDS_8:9
for P being Instruction-Sequence of SCMPDS
for s being State of SCMPDS
for I being Program of SCMPDS
for a being Int_position
for i being Integer st s . (DataLoc ((s . a),i)) >= 0 holds
( while<0 (a,i,I) is_closed_on s,P & while<0 (a,i,I) is_halting_on s,P )
proof end;

theorem Th8: :: SCMPDS_8:10
for P being Instruction-Sequence of SCMPDS
for s being 0 -started State of SCMPDS
for I being Program of SCMPDS
for a, c being Int_position
for i being Integer st s . (DataLoc ((s . a),i)) >= 0 holds
IExec ((while<0 (a,i,I)),P,s) = s +* (Start-At (((card I) + 2),SCMPDS))
proof end;

theorem :: SCMPDS_8:11
for P being Instruction-Sequence of SCMPDS
for s being 0 -started State of SCMPDS
for I being Program of SCMPDS
for a being Int_position
for i being Integer st s . (DataLoc ((s . a),i)) >= 0 holds
IC (IExec ((while<0 (a,i,I)),P,s)) = (card I) + 2
proof end;

theorem :: SCMPDS_8:12
for P being Instruction-Sequence of SCMPDS
for s being 0 -started State of SCMPDS
for I being Program of SCMPDS
for a, b being Int_position
for i being Integer st s . (DataLoc ((s . a),i)) >= 0 holds
(IExec ((while<0 (a,i,I)),P,s)) . b = s . b
proof end;

Lm2: for I being Program of SCMPDS
for a being Int_position
for i being Integer holds Shift (I,1) c= while<0 (a,i,I)

proof end;

scheme :: SCMPDS_8:sch 1
WhileLHalt{ F1( State of SCMPDS) -> Nat, F2() -> 0 -started State of SCMPDS, F3() -> Instruction-Sequence of SCMPDS, F4() -> halt-free shiftable Program of SCMPDS, F5() -> Int_position, F6() -> Integer, P1[ State of SCMPDS] } :
( while<0 (F5(),F6(),F4()) is_closed_on F2(),F3() & while<0 (F5(),F6(),F4()) is_halting_on F2(),F3() )
provided
A1: for t being 0 -started State of SCMPDS st P1[t] & F1(t) = 0 holds
t . (DataLoc ((F2() . F5()),F6())) >= 0 and
A2: P1[F2()] and
A3: for t being 0 -started State of SCMPDS
for Q being Instruction-Sequence of SCMPDS st P1[t] & t . F5() = F2() . F5() & t . (DataLoc ((F2() . F5()),F6())) < 0 holds
( (IExec (F4(),Q,t)) . F5() = t . F5() & F4() is_closed_on t,Q & F4() is_halting_on t,Q & F1((Initialize (IExec (F4(),Q,t)))) < F1(t) & P1[ Initialize (IExec (F4(),Q,t))] )
proof end;

scheme :: SCMPDS_8:sch 2
WhileLExec{ F1( State of SCMPDS) -> Nat, F2() -> 0 -started State of SCMPDS, F3() -> Instruction-Sequence of SCMPDS, F4() -> halt-free shiftable Program of SCMPDS, F5() -> Int_position, F6() -> Integer, P1[ State of SCMPDS] } :
IExec ((while<0 (F5(),F6(),F4())),F3(),F2()) = IExec ((while<0 (F5(),F6(),F4())),F3(),(Initialize (IExec (F4(),F3(),F2()))))
provided
A1: F2() . (DataLoc ((F2() . F5()),F6())) < 0 and
A2: for t being 0 -started State of SCMPDS st P1[t] & F1(t) = 0 holds
t . (DataLoc ((F2() . F5()),F6())) >= 0 and
A3: P1[F2()] and
A4: for t being 0 -started State of SCMPDS
for Q being Instruction-Sequence of SCMPDS st P1[t] & t . F5() = F2() . F5() & t . (DataLoc ((F2() . F5()),F6())) < 0 holds
( (IExec (F4(),Q,t)) . F5() = t . F5() & F4() is_closed_on t,Q & F4() is_halting_on t,Q & F1((Initialize (IExec (F4(),Q,t)))) < F1(t) & P1[ Initialize (IExec (F4(),Q,t))] )
proof end;

theorem :: SCMPDS_8:13
for P being Instruction-Sequence of SCMPDS
for s being 0 -started State of SCMPDS
for I being halt-free shiftable Program of SCMPDS
for a being Int_position
for i being Integer
for X being set
for f being Function of ,NAT st ( for t being 0 -started State of SCMPDS st f . t = 0 holds
t . (DataLoc ((s . a),i)) >= 0 ) & ( for t being 0 -started State of SCMPDS
for Q being Instruction-Sequence of SCMPDS st ( for x being Int_position st x in X holds
t . x = s . x ) & t . a = s . a & t . (DataLoc ((s . a),i)) < 0 holds
( (IExec (I,Q,t)) . a = t . a & f . (Initialize (IExec (I,Q,t))) < f . t & I is_closed_on t,Q & I is_halting_on t,Q & ( for x being Int_position st x in X holds
(IExec (I,Q,t)) . x = t . x ) ) ) holds
( while<0 (a,i,I) is_closed_on s,P & while<0 (a,i,I) is_halting_on s,P )
proof end;

theorem :: SCMPDS_8:14
for P being Instruction-Sequence of SCMPDS
for s being 0 -started State of SCMPDS
for I being halt-free shiftable Program of SCMPDS
for a being Int_position
for i being Integer
for X being set
for f being Function of ,NAT st s . (DataLoc ((s . a),i)) < 0 & ( for t being 0 -started State of SCMPDS st f . t = 0 holds
t . (DataLoc ((s . a),i)) >= 0 ) & ( for t being 0 -started State of SCMPDS
for Q being Instruction-Sequence of SCMPDS st ( for x being Int_position st x in X holds
t . x = s . x ) & t . a = s . a & t . (DataLoc ((s . a),i)) < 0 holds
( (IExec (I,Q,t)) . a = t . a & I is_closed_on t,Q & I is_halting_on t,Q & f . (Initialize (IExec (I,Q,t))) < f . t & ( for x being Int_position st x in X holds
(IExec (I,Q,t)) . x = t . x ) ) ) holds
IExec ((while<0 (a,i,I)),P,s) = IExec ((while<0 (a,i,I)),P,(Initialize (IExec (I,P,s))))
proof end;

theorem Th13: :: SCMPDS_8:15
for P being Instruction-Sequence of SCMPDS
for s being 0 -started State of SCMPDS
for I being halt-free shiftable Program of SCMPDS
for a being Int_position
for i being Integer
for X being set st ( for t being 0 -started State of SCMPDS
for Q being Instruction-Sequence of SCMPDS st ( for x being Int_position st x in X holds
t . x = s . x ) & t . a = s . a & t . (DataLoc ((s . a),i)) < 0 holds
( (IExec (I,Q,t)) . a = t . a & (IExec (I,Q,t)) . (DataLoc ((s . a),i)) > t . (DataLoc ((s . a),i)) & I is_closed_on t,Q & I is_halting_on t,Q & ( for x being Int_position st x in X holds
(IExec (I,Q,t)) . x = t . x ) ) ) holds
( while<0 (a,i,I) is_closed_on s,P & while<0 (a,i,I) is_halting_on s,P )
proof end;

theorem :: SCMPDS_8:16
for P being Instruction-Sequence of SCMPDS
for s being 0 -started State of SCMPDS
for I being halt-free shiftable Program of SCMPDS
for a being Int_position
for i being Integer
for X being set st s . (DataLoc ((s . a),i)) < 0 & ( for t being 0 -started State of SCMPDS
for Q being Instruction-Sequence of SCMPDS st ( for x being Int_position st x in X holds
t . x = s . x ) & t . a = s . a & t . (DataLoc ((s . a),i)) < 0 holds
( (IExec (I,Q,t)) . a = t . a & (IExec (I,Q,t)) . (DataLoc ((s . a),i)) > t . (DataLoc ((s . a),i)) & I is_closed_on t,Q & I is_halting_on t,Q & ( for x being Int_position st x in X holds
(IExec (I,Q,t)) . x = t . x ) ) ) holds
IExec ((while<0 (a,i,I)),P,s) = IExec ((while<0 (a,i,I)),P,(Initialize (IExec (I,P,s))))
proof end;

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

:: deftheorem defines while>0 SCMPDS_8:def 3 :
for a being Int_position
for i being Integer
for I being Program of SCMPDS holds while>0 (a,i,I) = (((a,i) <=0_goto ((card I) + 2)) ';' I) ';' (goto (- ((card I) + 1)));

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

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

theorem Th15: :: SCMPDS_8:17
for a being Int_position
for i being Integer
for I being Program of SCMPDS holds card (while>0 (a,i,I)) = (card I) + 2
proof end;

Lm3: for a being Int_position
for i being Integer
for I being Program of SCMPDS holds card (stop (while>0 (a,i,I))) = (card I) + 3

proof end;

theorem Th16: :: SCMPDS_8:18
for a being Int_position
for i being Integer
for m being Nat
for I being Program of SCMPDS holds
( m < (card I) + 2 iff m in dom (while>0 (a,i,I)) )
proof end;

theorem Th17: :: SCMPDS_8:19
for a being Int_position
for i being Integer
for I being Program of SCMPDS holds
( (while>0 (a,i,I)) . 0 = (a,i) <=0_goto ((card I) + 2) & (while>0 (a,i,I)) . ((card I) + 1) = goto (- ((card I) + 1)) )
proof end;

theorem Th18: :: SCMPDS_8:20
for P being Instruction-Sequence of SCMPDS
for s being State of SCMPDS
for I being Program of SCMPDS
for a being Int_position
for i being Integer st s . (DataLoc ((s . a),i)) <= 0 holds
( while>0 (a,i,I) is_closed_on s,P & while>0 (a,i,I) is_halting_on s,P )
proof end;

theorem Th19: :: SCMPDS_8:21
for P being Instruction-Sequence of SCMPDS
for s being 0 -started State of SCMPDS
for I being Program of SCMPDS
for a, c being Int_position
for i being Integer st s . (DataLoc ((s . a),i)) <= 0 holds
IExec ((while>0 (a,i,I)),P,s) = s +* (Start-At (((card I) + 2),SCMPDS))
proof end;

theorem :: SCMPDS_8:22
for P being Instruction-Sequence of SCMPDS
for s being 0 -started State of SCMPDS
for I being Program of SCMPDS
for a being Int_position
for i being Integer st s . (DataLoc ((s . a),i)) <= 0 holds
IC (IExec ((while>0 (a,i,I)),P,s)) = (card I) + 2
proof end;

theorem :: SCMPDS_8:23
for P being Instruction-Sequence of SCMPDS
for s being 0 -started State of SCMPDS
for I being Program of SCMPDS
for a, b being Int_position
for i being Integer st s . (DataLoc ((s . a),i)) <= 0 holds
(IExec ((while>0 (a,i,I)),P,s)) . b = s . b
proof end;

Lm4: for I being Program of SCMPDS
for a being Int_position
for i being Integer holds Shift (I,1) c= while>0 (a,i,I)

proof end;

scheme :: SCMPDS_8:sch 3
WhileGHalt{ F1( State of SCMPDS) -> Nat, F2() -> 0 -started State of SCMPDS, F3() -> Instruction-Sequence of SCMPDS, F4() -> halt-free shiftable Program of SCMPDS, F5() -> Int_position, F6() -> Integer, P1[ State of SCMPDS] } :
( while>0 (F5(),F6(),F4()) is_closed_on F2(),F3() & while>0 (F5(),F6(),F4()) is_halting_on F2(),F3() )
provided
A1: for t being 0 -started State of SCMPDS st P1[t] & F1(t) = 0 holds
t . (DataLoc ((F2() . F5()),F6())) <= 0 and
A2: P1[F2()] and
A3: for t being 0 -started State of SCMPDS
for Q being Instruction-Sequence of SCMPDS st P1[t] & t . F5() = F2() . F5() & t . (DataLoc ((F2() . F5()),F6())) > 0 holds
( (IExec (F4(),Q,t)) . F5() = t . F5() & F4() is_closed_on t,Q & F4() is_halting_on t,Q & F1((Initialize (IExec (F4(),Q,t)))) < F1(t) & P1[ Initialize (IExec (F4(),Q,t))] )
proof end;

scheme :: SCMPDS_8:sch 4
WhileGExec{ F1( State of SCMPDS) -> Nat, F2() -> 0 -started State of SCMPDS, F3() -> Instruction-Sequence of SCMPDS, F4() -> halt-free shiftable Program of SCMPDS, F5() -> Int_position, F6() -> Integer, P1[ State of SCMPDS] } :
IExec ((while>0 (F5(),F6(),F4())),F3(),F2()) = IExec ((while>0 (F5(),F6(),F4())),F3(),(Initialize (IExec (F4(),F3(),F2()))))
provided
A1: F2() . (DataLoc ((F2() . F5()),F6())) > 0 and
A2: for t being 0 -started State of SCMPDS st P1[t] & F1(t) = 0 holds
t . (DataLoc ((F2() . F5()),F6())) <= 0 and
A3: P1[F2()] and
A4: for t being 0 -started State of SCMPDS
for Q being Instruction-Sequence of SCMPDS st P1[t] & t . F5() = F2() . F5() & t . (DataLoc ((F2() . F5()),F6())) > 0 holds
( (IExec (F4(),Q,t)) . F5() = t . F5() & F4() is_closed_on t,Q & F4() is_halting_on t,Q & F1((Initialize (IExec (F4(),Q,t)))) < F1(t) & P1[ Initialize (IExec (F4(),Q,t))] )
proof end;

theorem Th22: :: SCMPDS_8:24
for P being Instruction-Sequence of SCMPDS
for s being 0 -started State of SCMPDS
for I being halt-free shiftable Program of SCMPDS
for a being Int_position
for i, c being Integer
for X, Y being set
for f being Function of ,NAT st ( for t being 0 -started State of SCMPDS st f . t = 0 holds
t . (DataLoc ((s . a),i)) <= 0 ) & ( for x being Int_position st x in X holds
s . x >= c + (s . (DataLoc ((s . a),i))) ) & ( for t being 0 -started State of SCMPDS
for Q being Instruction-Sequence of SCMPDS st ( for x being Int_position st x in X holds
t . x >= c + (t . (DataLoc ((s . a),i))) ) & ( for x being Int_position st x in Y holds
t . x = s . x ) & t . a = s . a & t . (DataLoc ((s . a),i)) > 0 holds
( (IExec (I,Q,t)) . a = t . a & I is_closed_on t,Q & I is_halting_on t,Q & f . (Initialize (IExec (I,Q,t))) < f . t & ( for x being Int_position st x in X holds
(IExec (I,Q,t)) . x >= c + ((IExec (I,Q,t)) . (DataLoc ((s . a),i))) ) & ( for x being Int_position st x in Y holds
(IExec (I,Q,t)) . x = t . x ) ) ) holds
( while>0 (a,i,I) is_closed_on s,P & while>0 (a,i,I) is_halting_on s,P )
proof end;

theorem Th23: :: SCMPDS_8:25
for P being Instruction-Sequence of SCMPDS
for s being 0 -started State of SCMPDS
for I being halt-free shiftable Program of SCMPDS
for a being Int_position
for i, c being Integer
for X, Y being set
for f being Function of ,NAT st s . (DataLoc ((s . a),i)) > 0 & ( for t being 0 -started State of SCMPDS st f . t = 0 holds
t . (DataLoc ((s . a),i)) <= 0 ) & ( for x being Int_position st x in X holds
s . x >= c + (s . (DataLoc ((s . a),i))) ) & ( for t being 0 -started State of SCMPDS
for Q being Instruction-Sequence of SCMPDS st ( for x being Int_position st x in X holds
t . x >= c + (t . (DataLoc ((s . a),i))) ) & ( for x being Int_position st x in Y holds
t . x = s . x ) & t . a = s . a & t . (DataLoc ((s . a),i)) > 0 holds
( (IExec (I,Q,t)) . a = t . a & I is_closed_on t,Q & I is_halting_on t,Q & f . (Initialize (IExec (I,Q,t))) < f . t & ( for x being Int_position st x in X holds
(IExec (I,Q,t)) . x >= c + ((IExec (I,Q,t)) . (DataLoc ((s . a),i))) ) & ( for x being Int_position st x in Y holds
(IExec (I,Q,t)) . x = t . x ) ) ) holds
IExec ((while>0 (a,i,I)),P,s) = IExec ((while>0 (a,i,I)),P,(Initialize (IExec (I,P,s))))
proof end;

theorem :: SCMPDS_8:26
for P being Instruction-Sequence of SCMPDS
for s being 0 -started State of SCMPDS
for I being halt-free shiftable Program of SCMPDS
for a being Int_position
for i being Integer
for X being set
for f being Function of ,NAT st ( for t being 0 -started State of SCMPDS st f . t = 0 holds
t . (DataLoc ((s . a),i)) <= 0 ) & ( for t being 0 -started State of SCMPDS
for Q being Instruction-Sequence of SCMPDS st ( for x being Int_position st x in X holds
t . x = s . x ) & t . a = s . a & t . (DataLoc ((s . a),i)) > 0 holds
( (IExec (I,Q,t)) . a = t . a & I is_closed_on t,Q & I is_halting_on t,Q & f . (Initialize (IExec (I,Q,t))) < f . t & ( for x being Int_position st x in X holds
(IExec (I,Q,t)) . x = t . x ) ) ) holds
( while>0 (a,i,I) is_closed_on s,P & while>0 (a,i,I) is_halting_on s,P & ( s . (DataLoc ((s . a),i)) > 0 implies IExec ((while>0 (a,i,I)),P,s) = IExec ((while>0 (a,i,I)),P,(Initialize (IExec (I,P,s)))) ) )
proof end;

theorem Th25: :: SCMPDS_8:27
for P being Instruction-Sequence of SCMPDS
for s being 0 -started State of SCMPDS
for I being halt-free shiftable Program of SCMPDS
for a being Int_position
for i, c being Integer
for X, Y being set st ( for x being Int_position st x in X holds
s . x >= c + (s . (DataLoc ((s . a),i))) ) & ( for t being 0 -started State of SCMPDS
for Q being Instruction-Sequence of SCMPDS st ( for x being Int_position st x in X holds
t . x >= c + (t . (DataLoc ((s . a),i))) ) & ( for x being Int_position st x in Y holds
t . x = s . x ) & t . a = s . a & t . (DataLoc ((s . a),i)) > 0 holds
( (IExec (I,Q,t)) . a = t . a & I is_closed_on t,Q & I is_halting_on t,Q & (IExec (I,Q,t)) . (DataLoc ((s . a),i)) < t . (DataLoc ((s . a),i)) & ( for x being Int_position st x in X holds
(IExec (I,Q,t)) . x >= c + ((IExec (I,Q,t)) . (DataLoc ((s . a),i))) ) & ( for x being Int_position st x in Y holds
(IExec (I,Q,t)) . x = t . x ) ) ) holds
( while>0 (a,i,I) is_closed_on s,P & while>0 (a,i,I) is_halting_on s,P & ( s . (DataLoc ((s . a),i)) > 0 implies IExec ((while>0 (a,i,I)),P,s) = IExec ((while>0 (a,i,I)),P,(Initialize (IExec (I,P,s)))) ) )
proof end;

theorem :: SCMPDS_8:28
for P being Instruction-Sequence of SCMPDS
for s being 0 -started State of SCMPDS
for I being halt-free shiftable Program of SCMPDS
for a being Int_position
for i being Integer
for X being set st ( for t being 0 -started State of SCMPDS
for Q being Instruction-Sequence of SCMPDS st ( for x being Int_position st x in X holds
t . x = s . x ) & t . a = s . a & t . (DataLoc ((s . a),i)) > 0 holds
( (IExec (I,Q,t)) . a = t . a & I is_closed_on t,Q & I is_halting_on t,Q & (IExec (I,Q,t)) . (DataLoc ((s . a),i)) < t . (DataLoc ((s . a),i)) & ( for x being Int_position st x in X holds
(IExec (I,Q,t)) . x = t . x ) ) ) holds
( while>0 (a,i,I) is_closed_on s,P & while>0 (a,i,I) is_halting_on s,P & ( s . (DataLoc ((s . a),i)) > 0 implies IExec ((while>0 (a,i,I)),P,s) = IExec ((while>0 (a,i,I)),P,(Initialize (IExec (I,P,s)))) ) )
proof end;

theorem :: SCMPDS_8:29
for P being Instruction-Sequence of SCMPDS
for s being 0 -started State of SCMPDS
for I being halt-free shiftable Program of SCMPDS
for a being Int_position
for i, c being Integer
for X being set st ( for x being Int_position st x in X holds
s . x >= c + (s . (DataLoc ((s . a),i))) ) & ( for t being 0 -started State of SCMPDS
for Q being Instruction-Sequence of SCMPDS st ( for x being Int_position st x in X holds
t . x >= c + (t . (DataLoc ((s . a),i))) ) & t . a = s . a & t . (DataLoc ((s . a),i)) > 0 holds
( (IExec (I,Q,t)) . a = t . a & I is_closed_on t,Q & I is_halting_on t,Q & (IExec (I,Q,t)) . (DataLoc ((s . a),i)) < t . (DataLoc ((s . a),i)) & ( for x being Int_position st x in X holds
(IExec (I,Q,t)) . x >= c + ((IExec (I,Q,t)) . (DataLoc ((s . a),i))) ) ) ) holds
( while>0 (a,i,I) is_closed_on s,P & while>0 (a,i,I) is_halting_on s,P & ( s . (DataLoc ((s . a),i)) > 0 implies IExec ((while>0 (a,i,I)),P,s) = IExec ((while>0 (a,i,I)),P,(Initialize (IExec (I,P,s)))) ) )
proof end;