let s be State of ; :: thesis: for I being Program of
for a, c being Int_position
for i being Integer
for n being Element of NAT st s . (DataLoc (s . a),i) <= 0 holds
IExec (for-down a,i,n,I),s = s +* (Start-At (inspos ((card I) + 3)))
let I be Program of ; :: thesis: for a, c being Int_position
for i being Integer
for n being Element of NAT st s . (DataLoc (s . a),i) <= 0 holds
IExec (for-down a,i,n,I),s = s +* (Start-At (inspos ((card I) + 3)))
let a, c be Int_position ; :: thesis: for i being Integer
for n being Element of NAT st s . (DataLoc (s . a),i) <= 0 holds
IExec (for-down a,i,n,I),s = s +* (Start-At (inspos ((card I) + 3)))
let i be Integer; :: thesis: for n being Element of NAT st s . (DataLoc (s . a),i) <= 0 holds
IExec (for-down a,i,n,I),s = s +* (Start-At (inspos ((card I) + 3)))
let n be Element of NAT ; :: thesis: ( s . (DataLoc (s . a),i) <= 0 implies IExec (for-down a,i,n,I),s = s +* (Start-At (inspos ((card I) + 3))) )
set d1 = DataLoc (s . a),i;
set FOR = for-down a,i,n,I;
set pFOR = stop (for-down a,i,n,I);
set iFOR = Initialized (stop (for-down a,i,n,I));
set s3 = s +* (Initialized (stop (for-down a,i,n,I)));
set s4 = Computation (s +* (Initialized (stop (for-down a,i,n,I)))),1;
set i1 = a,i <=0_goto ((card I) + 3);
set i2 = AddTo a,i,(- n);
set i3 = goto (- ((card I) + 2));
set SAl = Start-At (inspos ((card I) + 3));
A1: IC (s +* (Initialized (stop (for-down a,i,n,I)))) = inspos 0 by SCMPDS_6:21;
A2: not DataLoc (s . a),i in dom (Initialized (stop (for-down a,i,n,I))) by SCMPDS_4:31;
Initialized (stop (for-down a,i,n,I)) c= s +* (Initialized (stop (for-down a,i,n,I))) by FUNCT_4:26;
then A3: stop (for-down a,i,n,I) c= Computation (s +* (Initialized (stop (for-down a,i,n,I)))),1 by AMI_1:81, SCMPDS_4:57;
not a in dom (Initialized (stop (for-down a,i,n,I))) by SCMPDS_4:31;
then A4: (s +* (Initialized (stop (for-down a,i,n,I)))) . (DataLoc ((s +* (Initialized (stop (for-down a,i,n,I)))) . a),i) = (s +* (Initialized (stop (for-down a,i,n,I)))) . (DataLoc (s . a),i) by FUNCT_4:12
.= s . (DataLoc (s . a),i) by A2, FUNCT_4:12 ;
A5: for-down a,i,n,I = (a,i <=0_goto ((card I) + 3)) ';' ((I ';' (AddTo a,i,(- n))) ';' (goto (- ((card I) + 2)))) by Th15;
A6: Computation (s +* (Initialized (stop (for-down a,i,n,I)))),(0 + 1) = Following (Computation (s +* (Initialized (stop (for-down a,i,n,I)))),0 ) by AMI_1:14
.= Following (s +* (Initialized (stop (for-down a,i,n,I)))) by AMI_1:13
.= Exec (a,i <=0_goto ((card I) + 3)),(s +* (Initialized (stop (for-down a,i,n,I)))) by A5, SCMPDS_6:22 ;
assume s . (DataLoc (s . a),i) <= 0 ; :: thesis: IExec (for-down a,i,n,I),s = s +* (Start-At (inspos ((card I) + 3)))
then A7: IC (Computation (s +* (Initialized (stop (for-down a,i,n,I)))),1) = ICplusConst (s +* (Initialized (stop (for-down a,i,n,I)))),((card I) + 3) by A6, A4, SCMPDS_2:68
.= inspos (0 + ((card I) + 3)) by A1, SCMPDS_6:23 ;
A8: card (for-down a,i,n,I) = (card I) + 3 by Th60;
then inspos ((card I) + 3) in dom (stop (for-down a,i,n,I)) by SCMPDS_6:25;
then (Computation (s +* (Initialized (stop (for-down a,i,n,I)))),1) . (inspos ((card I) + 3)) = (stop (for-down a,i,n,I)) . (inspos ((card I) + 3)) by A3, GRFUNC_1:8
.= halt SCMPDS by A8, SCMPDS_6:25 ;
then A9: CurInstr (Computation (s +* (Initialized (stop (for-down a,i,n,I)))),1) = halt SCMPDS by A7;
then A10: ProgramPart (s +* (Initialized (stop (for-down a,i,n,I)))) halts_on s +* (Initialized (stop (for-down a,i,n,I))) by AMI_1:146;
A11: CurInstr (s +* (Initialized (stop (for-down a,i,n,I)))) = a,i <=0_goto ((card I) + 3) by A5, SCMPDS_6:22;

now

let l be Element of NAT ; :: thesis: ( l < 0 + 1 implies CurInstr (Computation (s +* (Initialized (stop (for-down a,i,n,I)))),l) <> halt SCMPDS )
assume l < 0 + 1 ; :: thesis: CurInstr (Computation (s +* (Initialized (stop (for-down a,i,n,I)))),l) <> halt SCMPDS
then l = 0 by NAT_1:13;
then CurInstr (Computation (s +* (Initialized (stop (for-down a,i,n,I)))),l) = CurInstr (s +* (Initialized (stop (for-down a,i,n,I)))) by AMI_1:13;
hence CurInstr (Computation (s +* (Initialized (stop (for-down a,i,n,I)))),l) <> halt SCMPDS by A11, SCMPDS_6:30; :: thesis: verum

end;

then for l being Element of NAT st CurInstr (Computation (s +* (Initialized (stop (for-down a,i,n,I)))),l) = halt SCMPDS holds
1 <= l ;
then LifeSpan (s +* (Initialized (stop (for-down a,i,n,I)))) = 1 by A9, A10, AMI_1:def 46;
then A12: Computation (s +* (Initialized (stop (for-down a,i,n,I)))),1 = Result (s +* (Initialized (stop (for-down a,i,n,I)))) by A10, AMI_1:122;
A13: dom (s | NAT ) = NAT by SCMPDS_6:1;

A14: now

let x be set ; :: thesis: ( x in dom (IExec (for-down a,i,n,I),s) implies (IExec (for-down a,i,n,I),s) . b₁ = (s +* (Start-At (inspos ((card I) + 3)))) . b₁ )
A15: dom (Start-At (inspos ((card I) + 3))) = {(IC SCMPDS )} by FUNCOP_1:19;
assume A16: x in dom (IExec (for-down a,i,n,I),s) ; :: thesis: (IExec (for-down a,i,n,I),s) . b₁ = (s +* (Start-At (inspos ((card I) + 3)))) . b₁

per cases ( x is Int_position or x = IC SCMPDS or x is Instruction-Location of SCMPDS ) by A16, SCMPDS_4:20;

suppose A17: x is Int_position ; :: thesis: (IExec (for-down a,i,n,I),s) . b₁ = (s +* (Start-At (inspos ((card I) + 3)))) . b₁

then x <> IC SCMPDS by SCMPDS_2:52;
then A18: not x in dom (Start-At (inspos ((card I) + 3))) by A15, TARSKI:def 1;

now

assume x in dom (s | NAT ) ; :: thesis: contradiction
then reconsider l = x as Instruction-Location of SCMPDS by A13, AMI_1:def 4;
l = x ;
hence contradiction by A17, SCMPDS_2:53; :: thesis: verum

end;

hence (IExec (for-down a,i,n,I),s) . x = (Computation (s +* (Initialized (stop (for-down a,i,n,I)))),1) . x by A12, FUNCT_4:12
.= (s +* (Initialized (stop (for-down a,i,n,I)))) . x by A6, A17, SCMPDS_2:68
.= s . x by A17, SCMPDS_5:19
.= (s +* (Start-At (inspos ((card I) + 3)))) . x by A18, FUNCT_4:12 ;
:: thesis: verum

end;

suppose A19: x = IC SCMPDS ; :: thesis: (IExec (for-down a,i,n,I),s) . b₁ = (s +* (Start-At (inspos ((card I) + 3)))) . b₁

now

assume x in dom (s | NAT ) ; :: thesis: contradiction
then reconsider l = x as Instruction-Location of SCMPDS by A13, AMI_1:def 4;
l = x ;
hence contradiction by A19, AMI_1:48; :: thesis: verum

end;

hence (IExec (for-down a,i,n,I),s) . x = inspos ((card I) + 3) by A7, A12, A19, FUNCT_4:12
.= (s +* (Start-At (inspos ((card I) + 3)))) . x by A19, Th12 ;
:: thesis: verum

end;

suppose x is Instruction-Location of SCMPDS ; :: thesis: (IExec (for-down a,i,n,I),s) . b₁ = (s +* (Start-At (inspos ((card I) + 3)))) . b₁

hence (IExec (for-down a,i,n,I),s) . x = (s +* (Start-At (inspos ((card I) + 3)))) . x by SCMPDS_6:27; :: thesis: verum

end;

end;

end;

dom (IExec (for-down a,i,n,I),s) = the carrier of SCMPDS by AMI_1:79
.= dom (s +* (Start-At (inspos ((card I) + 3)))) by AMI_1:79 ;
hence IExec (for-down a,i,n,I),s = s +* (Start-At (inspos ((card I) + 3))) by A14, FUNCT_1:9; :: thesis: verum