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 IsIF = Initialized (stop (if>=0 a,k1,I));
set s3 = s +* (Initialized (stop (if>=0 a,k1,I)));
set s4 = Comput (ProgramPart (s +* (Initialized (stop (if>=0 a,k1,I))))),(s +* (Initialized (stop (if>=0 a,k1,I)))),1;
set s5 = Comput (ProgramPart (s +* (Initialized (stop (if>=0 a,k1,I))))),(s +* (Initialized (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 ;
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 (Initialized (stop (if>=0 a,k1,I))) by SCMPDS_4:31;
not a in dom (Initialized (stop (if>=0 a,k1,I))) by SCMPDS_4:31;
then A3: (s +* (Initialized (stop (if>=0 a,k1,I)))) . (DataLoc ((s +* (Initialized (stop (if>=0 a,k1,I)))) . a),k1) = (s +* (Initialized (stop (if>=0 a,k1,I)))) . (DataLoc (s . a),k1) by FUNCT_4:12
.= s . (DataLoc (s . a),k1) by A2, FUNCT_4:12 ;
A4: IC (s +* (Initialized (stop (if>=0 a,k1,I)))) = 0 by FUNCT_4:26, SCMPDS_5:18;
A5: Initialized (stop (if>=0 a,k1,I)) c= s +* (Initialized (stop (if>=0 a,k1,I))) by FUNCT_4:26;
then A6: stop (if>=0 a,k1,I) c= Comput (ProgramPart (s +* (Initialized (stop (if>=0 a,k1,I))))),(s +* (Initialized (stop (if>=0 a,k1,I)))),1 by AMI_1:81, SCMPDS_4:57;
A7: stop (if>=0 a,k1,I) c= Comput (ProgramPart (s +* (Initialized (stop (if>=0 a,k1,I))))),(s +* (Initialized (stop (if>=0 a,k1,I)))),2 by A5, AMI_1:81, SCMPDS_4:57;
A8: Comput (ProgramPart (s +* (Initialized (stop (if>=0 a,k1,I))))),(s +* (Initialized (stop (if>=0 a,k1,I)))),(0 + 1) = Following (ProgramPart (s +* (Initialized (stop (if>=0 a,k1,I))))),(Comput (ProgramPart (s +* (Initialized (stop (if>=0 a,k1,I))))),(s +* (Initialized (stop (if>=0 a,k1,I)))),0 ) by AMI_1:14
.= Following (ProgramPart (s +* (Initialized (stop (if>=0 a,k1,I))))),(s +* (Initialized (stop (if>=0 a,k1,I)))) by AMI_1:13
.= Exec (a,k1 >=0_goto 2),(s +* (Initialized (stop (if>=0 a,k1,I)))) by A1, Th22 ;
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 (s +* (Initialized (stop (if>=0 a,k1,I))))),(s +* (Initialized (stop (if>=0 a,k1,I)))),1) = succ (IC (s +* (Initialized (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 (s +* (Initialized (stop (if>=0 a,k1,I))))),(s +* (Initialized (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 (s +* (Initialized (stop (if>=0 a,k1,I))))),(s +* (Initialized (stop (if>=0 a,k1,I)))),1)) /. (IC (Comput (ProgramPart (s +* (Initialized (stop (if>=0 a,k1,I))))),(s +* (Initialized (stop (if>=0 a,k1,I)))),1)) = (Comput (ProgramPart (s +* (Initialized (stop (if>=0 a,k1,I))))),(s +* (Initialized (stop (if>=0 a,k1,I)))),1) . (IC (Comput (ProgramPart (s +* (Initialized (stop (if>=0 a,k1,I))))),(s +* (Initialized (stop (if>=0 a,k1,I)))),1)) by AMI_1:150;
T: ProgramPart (s +* (Initialized (stop (if>=0 a,k1,I)))) = ProgramPart (Comput (ProgramPart (s +* (Initialized (stop (if>=0 a,k1,I))))),(s +* (Initialized (stop (if>=0 a,k1,I)))),1) by AMI_1:144;
A12: Comput (ProgramPart (s +* (Initialized (stop (if>=0 a,k1,I))))),(s +* (Initialized (stop (if>=0 a,k1,I)))),(1 + 1) = Following (ProgramPart (s +* (Initialized (stop (if>=0 a,k1,I))))),(Comput (ProgramPart (s +* (Initialized (stop (if>=0 a,k1,I))))),(s +* (Initialized (stop (if>=0 a,k1,I)))),1) by AMI_1:14
.= Exec (goto ((card I) + 1)),(Comput (ProgramPart (s +* (Initialized (stop (if>=0 a,k1,I))))),(s +* (Initialized (stop (if>=0 a,k1,I)))),1) by A9, A11, Y, T ;
then A13: IC (Comput (ProgramPart (s +* (Initialized (stop (if>=0 a,k1,I))))),(s +* (Initialized (stop (if>=0 a,k1,I)))),2) = ICplusConst (Comput (ProgramPart (s +* (Initialized (stop (if>=0 a,k1,I))))),(s +* (Initialized (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 (s +* (Initialized (stop (if>=0 a,k1,I))))),(s +* (Initialized (stop (if>=0 a,k1,I)))),2)) /. (IC (Comput (ProgramPart (s +* (Initialized (stop (if>=0 a,k1,I))))),(s +* (Initialized (stop (if>=0 a,k1,I)))),2)) = (Comput (ProgramPart (s +* (Initialized (stop (if>=0 a,k1,I))))),(s +* (Initialized (stop (if>=0 a,k1,I)))),2) . (IC (Comput (ProgramPart (s +* (Initialized (stop (if>=0 a,k1,I))))),(s +* (Initialized (stop (if>=0 a,k1,I)))),2)) by AMI_1:150;
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 (s +* (Initialized (stop (if>=0 a,k1,I))))),(s +* (Initialized (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 (Comput (ProgramPart (s +* (Initialized (stop (if>=0 a,k1,I))))),(s +* (Initialized (stop (if>=0 a,k1,I)))),2)),(Comput (ProgramPart (s +* (Initialized (stop (if>=0 a,k1,I))))),(s +* (Initialized (stop (if>=0 a,k1,I)))),2) = halt SCMPDS by A13, Y;
then A16: ProgramPart (s +* (Initialized (stop (if>=0 a,k1,I)))) halts_on s +* (Initialized (stop (if>=0 a,k1,I))) by AMI_1:146;
A17: CurInstr (ProgramPart (s +* (Initialized (stop (if>=0 a,k1,I))))),(s +* (Initialized (stop (if>=0 a,k1,I)))) = a,k1 >=0_goto 2 by A1, Th22;

now

let l be Element of NAT ; :: thesis: ( l < 1 + 1 implies CurInstr (ProgramPart (Comput (ProgramPart (s +* (Initialized (stop (if>=0 a,k1,I))))),(s +* (Initialized (stop (if>=0 a,k1,I)))),b₁)),(Comput (ProgramPart (s +* (Initialized (stop (if>=0 a,k1,I))))),(s +* (Initialized (stop (if>=0 a,k1,I)))),b₁) <> halt SCMPDS )
assume l < 1 + 1 ; :: thesis: CurInstr (ProgramPart (Comput (ProgramPart (s +* (Initialized (stop (if>=0 a,k1,I))))),(s +* (Initialized (stop (if>=0 a,k1,I)))),b₁)),(Comput (ProgramPart (s +* (Initialized (stop (if>=0 a,k1,I))))),(s +* (Initialized (stop (if>=0 a,k1,I)))),b₁) <> halt SCMPDS
then A18: l <= 1 by NAT_1:13;
u: Comput (ProgramPart (s +* (Initialized (stop (if>=0 a,k1,I))))),(s +* (Initialized (stop (if>=0 a,k1,I)))),0 = s +* (Initialized (stop (if>=0 a,k1,I))) by AMI_1:13;
Y: (ProgramPart (Comput (ProgramPart (s +* (Initialized (stop (if>=0 a,k1,I))))),(s +* (Initialized (stop (if>=0 a,k1,I)))),l)) /. (IC (Comput (ProgramPart (s +* (Initialized (stop (if>=0 a,k1,I))))),(s +* (Initialized (stop (if>=0 a,k1,I)))),l)) = (Comput (ProgramPart (s +* (Initialized (stop (if>=0 a,k1,I))))),(s +* (Initialized (stop (if>=0 a,k1,I)))),l) . (IC (Comput (ProgramPart (s +* (Initialized (stop (if>=0 a,k1,I))))),(s +* (Initialized (stop (if>=0 a,k1,I)))),l)) by AMI_1:150;

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

suppose l = 0 ; :: thesis: CurInstr (ProgramPart (Comput (ProgramPart (s +* (Initialized (stop (if>=0 a,k1,I))))),(s +* (Initialized (stop (if>=0 a,k1,I)))),b₁)),(Comput (ProgramPart (s +* (Initialized (stop (if>=0 a,k1,I))))),(s +* (Initialized (stop (if>=0 a,k1,I)))),b₁) <> halt SCMPDS

then CurInstr (ProgramPart (Comput (ProgramPart (s +* (Initialized (stop (if>=0 a,k1,I))))),(s +* (Initialized (stop (if>=0 a,k1,I)))),l)),(Comput (ProgramPart (s +* (Initialized (stop (if>=0 a,k1,I))))),(s +* (Initialized (stop (if>=0 a,k1,I)))),l) = CurInstr (ProgramPart (s +* (Initialized (stop (if>=0 a,k1,I))))),(s +* (Initialized (stop (if>=0 a,k1,I)))) by u;
hence CurInstr (ProgramPart (Comput (ProgramPart (s +* (Initialized (stop (if>=0 a,k1,I))))),(s +* (Initialized (stop (if>=0 a,k1,I)))),l)),(Comput (ProgramPart (s +* (Initialized (stop (if>=0 a,k1,I))))),(s +* (Initialized (stop (if>=0 a,k1,I)))),l) <> halt SCMPDS by A17, Th31; :: thesis: verum

end;

suppose l = 1 ; :: thesis: CurInstr (ProgramPart (Comput (ProgramPart (s +* (Initialized (stop (if>=0 a,k1,I))))),(s +* (Initialized (stop (if>=0 a,k1,I)))),b₁)),(Comput (ProgramPart (s +* (Initialized (stop (if>=0 a,k1,I))))),(s +* (Initialized (stop (if>=0 a,k1,I)))),b₁) <> halt SCMPDS

hence CurInstr (ProgramPart (Comput (ProgramPart (s +* (Initialized (stop (if>=0 a,k1,I))))),(s +* (Initialized (stop (if>=0 a,k1,I)))),l)),(Comput (ProgramPart (s +* (Initialized (stop (if>=0 a,k1,I))))),(s +* (Initialized (stop (if>=0 a,k1,I)))),l) <> halt SCMPDS by A9, A11, SCMPDS_2:85, Y; :: thesis: verum

end;

end;

end;

then for l being Element of NAT st CurInstr (ProgramPart (Comput (ProgramPart (s +* (Initialized (stop (if>=0 a,k1,I))))),(s +* (Initialized (stop (if>=0 a,k1,I)))),l)),(Comput (ProgramPart (s +* (Initialized (stop (if>=0 a,k1,I))))),(s +* (Initialized (stop (if>=0 a,k1,I)))),l) = halt SCMPDS holds
2 <= l ;
then LifeSpan (s +* (Initialized (stop (if>=0 a,k1,I)))) = 2 by A15, A16, AMI_1:def 46;
then A19: Comput (ProgramPart (s +* (Initialized (stop (if>=0 a,k1,I))))),(s +* (Initialized (stop (if>=0 a,k1,I)))),2 = Result (s +* (Initialized (stop (if>=0 a,k1,I)))) by A16, AMI_1:122;
A20: dom (s | NAT ) = NAT by Th1;

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 )
A23: IExec (if>=0 a,k1,I),s = (Result (s +* (Initialized (stop (if>=0 a,k1,I))))) +* (s | NAT ) by SCMPDS_4:def 8;
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 (s +* (Initialized (stop (if>=0 a,k1,I))))),(s +* (Initialized (stop (if>=0 a,k1,I)))),2) . x by A19, A23, FUNCT_4:12
.= (Comput (ProgramPart (s +* (Initialized (stop (if>=0 a,k1,I))))),(s +* (Initialized (stop (if>=0 a,k1,I)))),1) . x by A12, A25, SCMPDS_2:66
.= (s +* (Initialized (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, AMI_1:48;
hence (IExec (if>=0 a,k1,I),s) . x = (card I) + 2 by A13, A19, A23, A27, FUNCT_4:12
.= (s +* (Start-At ((card I) + 2),SCMPDS )) . x by A27, Lm7 ;
:: 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