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 (inspos ((card I) + 2)))
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 (inspos ((card I) + 2)))
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 (inspos ((card I) + 2)))
let k1 be Integer; :: thesis: ( s . (DataLoc (s . a),k1) > 0 implies IExec (if<=0 a,k1,I),s = s +* (Start-At (inspos ((card I) + 2))) )
set b = DataLoc (s . a),k1;
assume A1: s . (DataLoc (s . a),k1) > 0 ; :: thesis: IExec (if<=0 a,k1,I),s = s +* (Start-At (inspos ((card I) + 2)))
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 = Computation (s +* (Initialized (stop (if<=0 a,k1,I)))),1;
set s5 = Computation (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 (inspos ((card I) + 2));
A2: if<=0 a,k1,I = (a,k1 <=0_goto 2) ';' ((goto ((card I) + 1)) ';' I) by SCMPDS_4:52;
A3: IC (s +* (Initialized (stop (if<=0 a,k1,I)))) = inspos 0 by FUNCT_4:26, SCMPDS_5:18;
A4: CurInstr (s +* (Initialized (stop (if<=0 a,k1,I)))) = a,k1 <=0_goto 2 by A2, Th22;
A5: Computation (s +* (Initialized (stop (if<=0 a,k1,I)))),(0 + 1) = Following (Computation (s +* (Initialized (stop (if<=0 a,k1,I)))),0 ) by AMI_1:14
.= Following (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 A2, Th22 ;
A6: ( not DataLoc (s . a),k1 in dom (Initialized (stop (if<=0 a,k1,I))) & DataLoc (s . a),k1 in dom s ) by SCMPDS_2:49, SCMPDS_4:31;
( not a in dom (Initialized (stop (if<=0 a,k1,I))) & a in dom s ) by SCMPDS_2:49, SCMPDS_4:31;
then A7: (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 A6, FUNCT_4:12 ;
A8: Initialized (stop (if<=0 a,k1,I)) c= s +* (Initialized (stop (if<=0 a,k1,I))) by FUNCT_4:26;
then A9: stop (if<=0 a,k1,I) c= Computation (s +* (Initialized (stop (if<=0 a,k1,I)))),1 by AMI_1:81, SCMPDS_4:57;
A10: inspos 1 in dom (if<=0 a,k1,I) by Lm10;
then A11: inspos 1 in dom (stop (if<=0 a,k1,I)) by Th18;
A12: IC (Computation (s +* (Initialized (stop (if<=0 a,k1,I)))),1) = Next (IC (s +* (Initialized (stop (if<=0 a,k1,I))))) by A1, A5, A7, SCMPDS_2:68
.= inspos (0 + 1) by A3 ;
A13: (Computation (s +* (Initialized (stop (if<=0 a,k1,I)))),1) . (inspos 1) = (stop (if<=0 a,k1,I)) . (inspos 1) by A9, A11, GRFUNC_1:8
.= (if<=0 a,k1,I) . (inspos 1) by A10, Th19
.= goto ((card I) + 1) by Lm11 ;
A14: Computation (s +* (Initialized (stop (if<=0 a,k1,I)))),(1 + 1) = Following (Computation (s +* (Initialized (stop (if<=0 a,k1,I)))),1) by AMI_1:14
.= Exec (goto ((card I) + 1)),(Computation (s +* (Initialized (stop (if<=0 a,k1,I)))),1) by A12, A13 ;
A15: stop (if<=0 a,k1,I) c= Computation (s +* (Initialized (stop (if<=0 a,k1,I)))),2 by A8, AMI_1:81, SCMPDS_4:57;
A16: card (if<=0 a,k1,I) = (card I) + 2 by Lm9;
then A17: inspos ((card I) + 2) in dom (stop (if<=0 a,k1,I)) by Th25;
A18: IC (Computation (s +* (Initialized (stop (if<=0 a,k1,I)))),2) = ICplusConst (Computation (s +* (Initialized (stop (if<=0 a,k1,I)))),1),((card I) + 1) by A14, SCMPDS_2:66
.= inspos (((card I) + 1) + 1) by A12, Th23
.= inspos ((card I) + (1 + 1)) ;
(Computation (s +* (Initialized (stop (if<=0 a,k1,I)))),2) . (inspos ((card I) + 2)) = (stop (if<=0 a,k1,I)) . (inspos ((card I) + 2)) by A15, A17, GRFUNC_1:8
.= halt SCMPDS by A16, Th25 ;
then A19: CurInstr (Computation (s +* (Initialized (stop (if<=0 a,k1,I)))),2) = halt SCMPDS by A18;
then A20: s +* (Initialized (stop (if<=0 a,k1,I))) is halting by AMI_1:def 20;

now

let l be Element of NAT ; :: thesis: ( l < 1 + 1 implies CurInstr (Computation (s +* (Initialized (stop (if<=0 a,k1,I)))),b₁) <> halt SCMPDS )
assume l < 1 + 1 ; :: thesis: CurInstr (Computation (s +* (Initialized (stop (if<=0 a,k1,I)))),b₁) <> halt SCMPDS
then A21: l <= 1 by NAT_1:13;

per cases ( l = 0 or l = 1 ) by A21, NAT_1:26;

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

then CurInstr (Computation (s +* (Initialized (stop (if<=0 a,k1,I)))),l) = CurInstr (s +* (Initialized (stop (if<=0 a,k1,I)))) by AMI_1:13;
hence CurInstr (Computation (s +* (Initialized (stop (if<=0 a,k1,I)))),l) <> halt SCMPDS by A4, Th30; :: thesis: verum

end;

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

hence CurInstr (Computation (s +* (Initialized (stop (if<=0 a,k1,I)))),l) <> halt SCMPDS by A12, A13, SCMPDS_2:85; :: thesis: verum

end;

end;

end;

then for l being Element of NAT st CurInstr (Computation (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 A19, A20, AMI_1:def 46;
then A22: Computation (s +* (Initialized (stop (if<=0 a,k1,I)))),2 = Result (s +* (Initialized (stop (if<=0 a,k1,I)))) by A20, AMI_1:122;
A23: dom (IExec (if<=0 a,k1,I),s) = the carrier of SCMPDS by AMI_1:79
.= dom (s +* (Start-At (inspos ((card I) + 2)))) by AMI_1:79 ;
A24: dom (s | NAT ) = NAT by Th1;

now

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 (inspos ((card I) + 2)))) . b₁ )
A25: IExec (if<=0 a,k1,I),s = (Result (s +* (Initialized (stop (if<=0 a,k1,I))))) +* (s | NAT ) by SCMPDS_4:def 8;
assume A26: x in dom (IExec (if<=0 a,k1,I),s) ; :: thesis: (IExec (if<=0 a,k1,I),s) . b₁ = (s +* (Start-At (inspos ((card I) + 2)))) . b₁
A27: dom (Start-At (inspos ((card I) + 2))) = {(IC SCMPDS )} by FUNCOP_1:19;

per cases ( x is Int_position or x = IC SCMPDS or x is Instruction-Location of SCMPDS ) by A26, SCMPDS_4:20;

suppose A28: x is Int_position ; :: thesis: (IExec (if<=0 a,k1,I),s) . b₁ = (s +* (Start-At (inspos ((card I) + 2)))) . b₁

then x <> IC SCMPDS by SCMPDS_2:52;
then A29: not x in dom (Start-At (inspos ((card I) + 2))) by A27, TARSKI:def 1;

now

assume x in dom (s | NAT ) ; :: thesis: contradiction
then reconsider l = x as Instruction-Location of SCMPDS by A24, AMI_1:def 4;
l = x ;
hence contradiction by A28, SCMPDS_2:53; :: thesis: verum

end;

hence (IExec (if<=0 a,k1,I),s) . x = (Computation (s +* (Initialized (stop (if<=0 a,k1,I)))),2) . x by A22, A25, FUNCT_4:12
.= (Computation (s +* (Initialized (stop (if<=0 a,k1,I)))),1) . x by A14, A28, SCMPDS_2:66
.= (s +* (Initialized (stop (if<=0 a,k1,I)))) . x by A5, A28, SCMPDS_2:68
.= s . x by A28, SCMPDS_5:19
.= (s +* (Start-At (inspos ((card I) + 2)))) . x by A29, FUNCT_4:12 ;
:: thesis: verum

end;

suppose A30: x = IC SCMPDS ; :: thesis: (IExec (if<=0 a,k1,I),s) . b₁ = (s +* (Start-At (inspos ((card I) + 2)))) . b₁

now

assume x in dom (s | NAT ) ; :: thesis: contradiction
then reconsider l = x as Instruction-Location of SCMPDS by A24, AMI_1:def 4;
l = x ;
hence contradiction by A30, AMI_1:48; :: thesis: verum

end;

hence (IExec (if<=0 a,k1,I),s) . x = inspos ((card I) + 2) by A22, A25, A18, A30, FUNCT_4:12
.= (s +* (Start-At (inspos ((card I) + 2)))) . x by A30, Lm8 ;
:: thesis: verum

end;

suppose x is Instruction-Location of SCMPDS ; :: thesis: (IExec (if<=0 a,k1,I),s) . b₁ = (s +* (Start-At (inspos ((card I) + 2)))) . b₁

hence (IExec (if<=0 a,k1,I),s) . x = (s +* (Start-At (inspos ((card I) + 2)))) . x by Th27; :: thesis: verum

end;

end;

end;

hence IExec (if<=0 a,k1,I),s = s +* (Start-At (inspos ((card I) + 2))) by A23, FUNCT_1:9; :: thesis: verum