let s be State of SCMPDS; :: thesis: for I being Program of SCMPDS
for a being Int_position
for k1 being Integer st s . (DataLoc ((s . a),k1)) <> 0 holds
IExec ((if=0 (a,k1,I)),s) = s +* (Start-At (((card I) + 1),SCMPDS))
let I be Program of SCMPDS; :: thesis: for a being Int_position
for k1 being Integer st s . (DataLoc ((s . a),k1)) <> 0 holds
IExec ((if=0 (a,k1,I)),s) = s +* (Start-At (((card I) + 1),SCMPDS))
let a be Int_position ; :: thesis: for k1 being Integer st s . (DataLoc ((s . a),k1)) <> 0 holds
IExec ((if=0 (a,k1,I)),s) = s +* (Start-At (((card I) + 1),SCMPDS))
let k1 be Integer; :: thesis: ( s . (DataLoc ((s . a),k1)) <> 0 implies IExec ((if=0 (a,k1,I)),s) = s +* (Start-At (((card I) + 1),SCMPDS)) )
set b = DataLoc ((s . a),k1);
set IF = if=0 (a,k1,I);
set pIF = stop (if=0 (a,k1,I));
set s3 = (Initialize s) +* (stop (if=0 (a,k1,I)));
set s4 = Comput ((ProgramPart ((Initialize s) +* (stop (if=0 (a,k1,I))))),((Initialize s) +* (stop (if=0 (a,k1,I)))),1);
set i = (a,k1) <>0_goto ((card I) + 1);
set SAl = Start-At (((card I) + 1),SCMPDS);
I1: (Initialize s) +* (stop (if=0 (a,k1,I))) = s +* (Initialize (stop (if=0 (a,k1,I)))) by COMPOS_1:125;
A1: IC ((Initialize s) +* (stop (if=0 (a,k1,I)))) = 0 by I1, FUNCT_4:26, SCMPDS_5:18;
A2: not DataLoc ((s . a),k1) in dom (Initialize (stop (if=0 (a,k1,I)))) by SCMPDS_4:31;
not a in dom (Initialize (stop (if=0 (a,k1,I)))) by SCMPDS_4:31;
then A3: ((Initialize s) +* (stop (if=0 (a,k1,I)))) . (DataLoc ((((Initialize s) +* (stop (if=0 (a,k1,I)))) . a),k1)) = ((Initialize s) +* (stop (if=0 (a,k1,I)))) . (DataLoc ((s . a),k1)) by I1, FUNCT_4:12
.= s . (DataLoc ((s . a),k1)) by A2, I1, FUNCT_4:12 ;
A4: Comput ((ProgramPart ((Initialize s) +* (stop (if=0 (a,k1,I))))),((Initialize s) +* (stop (if=0 (a,k1,I)))),(0 + 1)) = Following ((ProgramPart ((Initialize s) +* (stop (if=0 (a,k1,I))))),(Comput ((ProgramPart ((Initialize s) +* (stop (if=0 (a,k1,I))))),((Initialize s) +* (stop (if=0 (a,k1,I)))),0))) by EXTPRO_1:4
.= Following ((ProgramPart ((Initialize s) +* (stop (if=0 (a,k1,I))))),((Initialize s) +* (stop (if=0 (a,k1,I))))) by EXTPRO_1:3
.= Exec (((a,k1) <>0_goto ((card I) + 1)),((Initialize s) +* (stop (if=0 (a,k1,I))))) by Th22, I1 ;
assume s . (DataLoc ((s . a),k1)) <> 0 ; :: thesis: IExec ((if=0 (a,k1,I)),s) = s +* (Start-At (((card I) + 1),SCMPDS))
then A5: IC (Comput ((ProgramPart ((Initialize s) +* (stop (if=0 (a,k1,I))))),((Initialize s) +* (stop (if=0 (a,k1,I)))),1)) = ICplusConst (((Initialize s) +* (stop (if=0 (a,k1,I)))),((card I) + 1)) by A4, A3, SCMPDS_2:67
.= 0 + ((card I) + 1) by A1, Th23 ;
Initialize (stop (if=0 (a,k1,I))) c= (Initialize s) +* (stop (if=0 (a,k1,I))) by I1, FUNCT_4:26;
then A6: stop (if=0 (a,k1,I)) c= Comput ((ProgramPart ((Initialize s) +* (stop (if=0 (a,k1,I))))),((Initialize s) +* (stop (if=0 (a,k1,I)))),1) by AMI_1:81, COMPOS_1:132;
Y: (ProgramPart (Comput ((ProgramPart ((Initialize s) +* (stop (if=0 (a,k1,I))))),((Initialize s) +* (stop (if=0 (a,k1,I)))),1))) /. (IC (Comput ((ProgramPart ((Initialize s) +* (stop (if=0 (a,k1,I))))),((Initialize s) +* (stop (if=0 (a,k1,I)))),1))) = (Comput ((ProgramPart ((Initialize s) +* (stop (if=0 (a,k1,I))))),((Initialize s) +* (stop (if=0 (a,k1,I)))),1)) . (IC (Comput ((ProgramPart ((Initialize s) +* (stop (if=0 (a,k1,I))))),((Initialize s) +* (stop (if=0 (a,k1,I)))),1))) by COMPOS_1:38;
TX: ProgramPart ((Initialize s) +* (stop (if=0 (a,k1,I)))) = ProgramPart (Comput ((ProgramPart ((Initialize s) +* (stop (if=0 (a,k1,I))))),((Initialize s) +* (stop (if=0 (a,k1,I)))),1)) by AMI_1:123;
A7: card (if=0 (a,k1,I)) = (card I) + 1 by Th15;
then (card I) + 1 in dom (stop (if=0 (a,k1,I))) by Th25;
then (Comput ((ProgramPart ((Initialize s) +* (stop (if=0 (a,k1,I))))),((Initialize s) +* (stop (if=0 (a,k1,I)))),1)) . ((card I) + 1) = (stop (if=0 (a,k1,I))) . ((card I) + 1) by A6, GRFUNC_1:8
.= halt SCMPDS by A7, Th25 ;
then A8: CurInstr ((ProgramPart ((Initialize s) +* (stop (if=0 (a,k1,I))))),(Comput ((ProgramPart ((Initialize s) +* (stop (if=0 (a,k1,I))))),((Initialize s) +* (stop (if=0 (a,k1,I)))),1))) = halt SCMPDS by A5, Y, TX;
then A9: ProgramPart ((Initialize s) +* (stop (if=0 (a,k1,I)))) halts_on (Initialize s) +* (stop (if=0 (a,k1,I))) by EXTPRO_1:30;
A10: CurInstr ((ProgramPart ((Initialize s) +* (stop (if=0 (a,k1,I))))),((Initialize s) +* (stop (if=0 (a,k1,I))))) = (a,k1) <>0_goto ((card I) + 1) by Th22, I1;

now

let l be Element of NAT ; :: thesis: ( l < 1 implies CurInstr ((ProgramPart ((Initialize s) +* (stop (if=0 (a,k1,I))))),(Comput ((ProgramPart ((Initialize s) +* (stop (if=0 (a,k1,I))))),((Initialize s) +* (stop (if=0 (a,k1,I)))),l))) <> halt SCMPDS )
u: Comput ((ProgramPart ((Initialize s) +* (stop (if=0 (a,k1,I))))),((Initialize s) +* (stop (if=0 (a,k1,I)))),0) = (Initialize s) +* (stop (if=0 (a,k1,I))) by EXTPRO_1:3;
assume l < 1 ; :: thesis: CurInstr ((ProgramPart ((Initialize s) +* (stop (if=0 (a,k1,I))))),(Comput ((ProgramPart ((Initialize s) +* (stop (if=0 (a,k1,I))))),((Initialize s) +* (stop (if=0 (a,k1,I)))),l))) <> halt SCMPDS
then l < 1 + 0 ;
then l = 0 by NAT_1:13;
then CurInstr ((ProgramPart ((Initialize s) +* (stop (if=0 (a,k1,I))))),(Comput ((ProgramPart ((Initialize s) +* (stop (if=0 (a,k1,I))))),((Initialize s) +* (stop (if=0 (a,k1,I)))),l))) = CurInstr ((ProgramPart ((Initialize s) +* (stop (if=0 (a,k1,I))))),((Initialize s) +* (stop (if=0 (a,k1,I))))) by u;
hence CurInstr ((ProgramPart ((Initialize s) +* (stop (if=0 (a,k1,I))))),(Comput ((ProgramPart ((Initialize s) +* (stop (if=0 (a,k1,I))))),((Initialize s) +* (stop (if=0 (a,k1,I)))),l))) <> halt SCMPDS by A10, Th29; :: thesis: verum

end;

then for l being Element of NAT st CurInstr ((ProgramPart ((Initialize s) +* (stop (if=0 (a,k1,I))))),(Comput ((ProgramPart ((Initialize s) +* (stop (if=0 (a,k1,I))))),((Initialize s) +* (stop (if=0 (a,k1,I)))),l))) = halt SCMPDS holds
1 <= l ;
then LifeSpan ((ProgramPart ((Initialize s) +* (stop (if=0 (a,k1,I))))),((Initialize s) +* (stop (if=0 (a,k1,I))))) = 1 by A8, A9, EXTPRO_1:def 14;
then A11: Comput ((ProgramPart ((Initialize s) +* (stop (if=0 (a,k1,I))))),((Initialize s) +* (stop (if=0 (a,k1,I)))),1) = Result ((ProgramPart ((Initialize s) +* (stop (if=0 (a,k1,I))))),((Initialize s) +* (stop (if=0 (a,k1,I))))) by A9, EXTPRO_1:23;
A12: dom (ProgramPart s) = NAT by COMPOS_1:34;

A13: now

A14: dom (Start-At (((card I) + 1),SCMPDS)) = {(IC SCMPDS)} by FUNCOP_1:19;
let x be set ; :: thesis: ( x in dom (IExec ((if=0 (a,k1,I)),s)) implies (IExec ((if=0 (a,k1,I)),s)) . b₁ = (s +* (Start-At (((card I) + 1),SCMPDS))) . b₁ )
assume A16: x in dom (IExec ((if=0 (a,k1,I)),s)) ; :: thesis: (IExec ((if=0 (a,k1,I)),s)) . b₁ = (s +* (Start-At (((card I) + 1),SCMPDS))) . b₁

per cases ( x is Int_position or x = IC SCMPDS or x is Element of NAT ) by A16, SCMPDS_4:20;

suppose A17: x is Int_position ; :: thesis: (IExec ((if=0 (a,k1,I)),s)) . b₁ = (s +* (Start-At (((card I) + 1),SCMPDS))) . b₁

then x <> IC SCMPDS by SCMPDS_2:52;
then A18: not x in dom (Start-At (((card I) + 1),SCMPDS)) by A14, TARSKI:def 1;
not x in dom (s | NAT) by A12, A17, SCMPDS_2:53;
hence (IExec ((if=0 (a,k1,I)),s)) . x = (Comput ((ProgramPart ((Initialize s) +* (stop (if=0 (a,k1,I))))),((Initialize s) +* (stop (if=0 (a,k1,I)))),1)) . x by A11, FUNCT_4:12
.= ((Initialize s) +* (stop (if=0 (a,k1,I)))) . x by A4, A17, SCMPDS_2:67
.= s . x by A17, SCMPDS_5:19
.= (s +* (Start-At (((card I) + 1),SCMPDS))) . x by A18, FUNCT_4:12 ;
:: thesis: verum

end;

suppose A19: x = IC SCMPDS ; :: thesis: (IExec ((if=0 (a,k1,I)),s)) . b₁ = (s +* (Start-At (((card I) + 1),SCMPDS))) . b₁

not x in dom (s | NAT) by A12, A19, COMPOS_1:3;
hence (IExec ((if=0 (a,k1,I)),s)) . x = (card I) + 1 by A5, A11, A19, FUNCT_4:12
.= (s +* (Start-At (((card I) + 1),SCMPDS))) . x by A19, FUNCT_4:121 ;
:: thesis: verum

end;

suppose x is Element of NAT ; :: thesis: (IExec ((if=0 (a,k1,I)),s)) . b₁ = (s +* (Start-At (((card I) + 1),SCMPDS))) . b₁

hence (IExec ((if=0 (a,k1,I)),s)) . x = (s +* (Start-At (((card I) + 1),SCMPDS))) . x by Th27; :: thesis: verum

end;

end;

end;

dom (IExec ((if=0 (a,k1,I)),s)) = the carrier of SCMPDS by PARTFUN1:def 4
.= dom (s +* (Start-At (((card I) + 1),SCMPDS))) by PARTFUN1:def 4 ;
hence IExec ((if=0 (a,k1,I)),s) = s +* (Start-At (((card I) + 1),SCMPDS)) by A13, FUNCT_1:9; :: thesis: verum