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) + 2),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) + 2),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) + 2),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) + 2),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 s5 = Comput ((ProgramPart ((Initialize s) +* (stop (if>=0 (a,k1,I))))),((Initialize s) +* (stop (if>=0 (a,k1,I)))),2);
set i = (a,k1) >=0_goto 2;
set j = goto ((card I) + 1);
set SAl = Start-At (((card I) + 2),SCMPDS);
I2: s +* (Initialize (stop (if>=0 (a,k1,I)))) = (Initialize s) +* (stop (if>=0 (a,k1,I))) by COMPOS_1:125;
A1: if>=0 (a,k1,I) = ((a,k1) >=0_goto 2) ';' ((goto ((card I) + 1)) ';' I) by SCMPDS_4:52;
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 I2, FUNCT_4:12
.= s . (DataLoc ((s . a),k1)) by A2, I2, FUNCT_4:12 ;
A4: IC ((Initialize s) +* (stop (if>=0 (a,k1,I)))) = 0 by I2, FUNCT_4:26, SCMPDS_5:18;
A5: Initialize (stop (if>=0 (a,k1,I))) c= (Initialize s) +* (stop (if>=0 (a,k1,I))) by I2, 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;
A7: 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)))),2) by A5, AMI_1:81, COMPOS_1:132;
A8: 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 2),((Initialize s) +* (stop (if>=0 (a,k1,I))))) by A1, Th22, I2 ;
assume s . (DataLoc ((s . a),k1)) < 0 ; :: thesis: IExec ((if>=0 (a,k1,I)),s) = s +* (Start-At (((card I) + 2),SCMPDS))
then A9: IC (Comput ((ProgramPart ((Initialize s) +* (stop (if>=0 (a,k1,I))))),((Initialize s) +* (stop (if>=0 (a,k1,I)))),1)) = succ (IC ((Initialize s) +* (stop (if>=0 (a,k1,I))))) by A8, A3, SCMPDS_2:69
.= 0 + 1 by A4 ;
A10: 1 in dom (if>=0 (a,k1,I)) by Lm9;
then 1 in dom (stop (if>=0 (a,k1,I))) by Th18;
then A11: (Comput ((ProgramPart ((Initialize s) +* (stop (if>=0 (a,k1,I))))),((Initialize s) +* (stop (if>=0 (a,k1,I)))),1)) . 1 = (stop (if>=0 (a,k1,I))) . 1 by A6, GRFUNC_1:8
.= (if>=0 (a,k1,I)) . 1 by A10, Th19
.= goto ((card I) + 1) by Lm10 ;
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;
T: 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;
A12: Comput ((ProgramPart ((Initialize s) +* (stop (if>=0 (a,k1,I))))),((Initialize s) +* (stop (if>=0 (a,k1,I)))),(1 + 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)))),1))) by EXTPRO_1:4
.= Exec ((goto ((card I) + 1)),(Comput ((ProgramPart ((Initialize s) +* (stop (if>=0 (a,k1,I))))),((Initialize s) +* (stop (if>=0 (a,k1,I)))),1))) by A9, A11, Y, T ;
then A13: IC (Comput ((ProgramPart ((Initialize s) +* (stop (if>=0 (a,k1,I))))),((Initialize s) +* (stop (if>=0 (a,k1,I)))),2)) = ICplusConst ((Comput ((ProgramPart ((Initialize s) +* (stop (if>=0 (a,k1,I))))),((Initialize s) +* (stop (if>=0 (a,k1,I)))),1)),((card I) + 1)) by SCMPDS_2:66
.= ((card I) + 1) + 1 by A9, Th23
.= (card I) + (1 + 1) ;
Y: (ProgramPart (Comput ((ProgramPart ((Initialize s) +* (stop (if>=0 (a,k1,I))))),((Initialize s) +* (stop (if>=0 (a,k1,I)))),2))) /. (IC (Comput ((ProgramPart ((Initialize s) +* (stop (if>=0 (a,k1,I))))),((Initialize s) +* (stop (if>=0 (a,k1,I)))),2))) = (Comput ((ProgramPart ((Initialize s) +* (stop (if>=0 (a,k1,I))))),((Initialize s) +* (stop (if>=0 (a,k1,I)))),2)) . (IC (Comput ((ProgramPart ((Initialize s) +* (stop (if>=0 (a,k1,I))))),((Initialize s) +* (stop (if>=0 (a,k1,I)))),2))) 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)))),2)) by AMI_1:123;
A14: card (if>=0 (a,k1,I)) = (card I) + 2 by Lm8;
then (card I) + 2 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)))),2)) . ((card I) + 2) = (stop (if>=0 (a,k1,I))) . ((card I) + 2) by A7, GRFUNC_1:8
.= halt SCMPDS by A14, Th25 ;
then A15: 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)))),2))) = halt SCMPDS by A13, Y, TX;
then A16: ProgramPart ((Initialize s) +* (stop (if>=0 (a,k1,I)))) halts_on (Initialize s) +* (stop (if>=0 (a,k1,I))) by EXTPRO_1:30;
A17: CurInstr ((ProgramPart ((Initialize s) +* (stop (if>=0 (a,k1,I))))),((Initialize s) +* (stop (if>=0 (a,k1,I))))) = (a,k1) >=0_goto 2 by A1, Th22, I2;

now

let l be Element of NAT ; :: thesis: ( l < 1 + 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)))),b₁))) <> halt SCMPDS )
assume l < 1 + 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)))),b₁))) <> halt SCMPDS
then A18: l <= 1 by NAT_1:13;
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;
Y: (ProgramPart (Comput ((ProgramPart ((Initialize s) +* (stop (if>=0 (a,k1,I))))),((Initialize s) +* (stop (if>=0 (a,k1,I)))),l))) /. (IC (Comput ((ProgramPart ((Initialize s) +* (stop (if>=0 (a,k1,I))))),((Initialize s) +* (stop (if>=0 (a,k1,I)))),l))) = (Comput ((ProgramPart ((Initialize s) +* (stop (if>=0 (a,k1,I))))),((Initialize s) +* (stop (if>=0 (a,k1,I)))),l)) . (IC (Comput ((ProgramPart ((Initialize s) +* (stop (if>=0 (a,k1,I))))),((Initialize s) +* (stop (if>=0 (a,k1,I)))),l))) 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)))),l)) by AMI_1:123;

per cases ( l = 0 or l = 1 ) by A18, NAT_1:26;

suppose l = 0 ; :: 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)))),b₁))) <> halt SCMPDS

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 A17, Th31; :: thesis: verum

end;

suppose 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)))),b₁))) <> halt SCMPDS

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 A9, A11, Y, TX, SCMPDS_2:85; :: thesis: verum

end;

end;

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
2 <= l ;
then LifeSpan ((ProgramPart ((Initialize s) +* (stop (if>=0 (a,k1,I))))),((Initialize s) +* (stop (if>=0 (a,k1,I))))) = 2 by A15, A16, EXTPRO_1:def 14;
then A19: Comput ((ProgramPart ((Initialize s) +* (stop (if>=0 (a,k1,I))))),((Initialize s) +* (stop (if>=0 (a,k1,I)))),2) = Result ((ProgramPart ((Initialize s) +* (stop (if>=0 (a,k1,I))))),((Initialize s) +* (stop (if>=0 (a,k1,I))))) by A16, EXTPRO_1:23;
A20: dom (ProgramPart s) = NAT by COMPOS_1:34;

A21: now

A22: dom (Start-At (((card I) + 2),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) + 2),SCMPDS))) . b₁ )
assume A24: x in dom (IExec ((if>=0 (a,k1,I)),s)) ; :: thesis: (IExec ((if>=0 (a,k1,I)),s)) . b₁ = (s +* (Start-At (((card I) + 2),SCMPDS))) . b₁

per cases ( x is Int_position or x = IC SCMPDS or x is Element of NAT ) by A24, SCMPDS_4:20;

suppose A25: x is Int_position ; :: thesis: (IExec ((if>=0 (a,k1,I)),s)) . b₁ = (s +* (Start-At (((card I) + 2),SCMPDS))) . b₁

then x <> IC SCMPDS by SCMPDS_2:52;
then A26: not x in dom (Start-At (((card I) + 2),SCMPDS)) by A22, TARSKI:def 1;
not x in dom (s | NAT) by A20, A25, 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)))),2)) . x by A19, FUNCT_4:12
.= (Comput ((ProgramPart ((Initialize s) +* (stop (if>=0 (a,k1,I))))),((Initialize s) +* (stop (if>=0 (a,k1,I)))),1)) . x by A12, A25, SCMPDS_2:66
.= ((Initialize s) +* (stop (if>=0 (a,k1,I)))) . x by A8, A25, SCMPDS_2:69
.= s . x by A25, SCMPDS_5:19
.= (s +* (Start-At (((card I) + 2),SCMPDS))) . x by A26, FUNCT_4:12 ;
:: thesis: verum

end;

suppose A27: x = IC SCMPDS ; :: thesis: (IExec ((if>=0 (a,k1,I)),s)) . b₁ = (s +* (Start-At (((card I) + 2),SCMPDS))) . b₁

not x in dom (s | NAT) by A20, A27, COMPOS_1:3;
hence (IExec ((if>=0 (a,k1,I)),s)) . x = (card I) + 2 by A13, A19, A27, FUNCT_4:12
.= (s +* (Start-At (((card I) + 2),SCMPDS))) . x by A27, 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) + 2),SCMPDS))) . b₁

hence (IExec ((if>=0 (a,k1,I)),s)) . x = (s +* (Start-At (((card I) + 2),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) + 2),SCMPDS))) by PARTFUN1:def 4 ;
hence IExec ((if>=0 (a,k1,I)),s) = s +* (Start-At (((card I) + 2),SCMPDS)) by A21, FUNCT_1:9; :: thesis: verum