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) + 1)))
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) + 1)))
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) + 1)))
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) + 1))) )
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) + 1)))
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 i = a,k1 <=0_goto ((card I) + 1);
set SAl = Start-At (inspos ((card I) + 1));
A2: IC (s +* (Initialized (stop (if>0 a,k1,I)))) = inspos 0 by FUNCT_4:26, SCMPDS_5:18;
A3: CurInstr (s +* (Initialized (stop (if>0 a,k1,I)))) = a,k1 <=0_goto ((card I) + 1) by Th22;
A4: 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 ((card I) + 1)),(s +* (Initialized (stop (if>0 a,k1,I)))) by Th22 ;
A5: dom (s | NAT ) = NAT by Th1;
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 ;
Initialized (stop (if>0 a,k1,I)) c= s +* (Initialized (stop (if>0 a,k1,I))) by FUNCT_4:26;
then A8: stop (if>0 a,k1,I) c= Computation (s +* (Initialized (stop (if>0 a,k1,I)))),1 by AMI_1:81, SCMPDS_4:57;
A9: card (if>0 a,k1,I) = (card I) + 1 by Th15;
then A10: inspos ((card I) + 1) in dom (stop (if>0 a,k1,I)) by Th25;
A11: IC (Computation (s +* (Initialized (stop (if>0 a,k1,I)))),1) = ICplusConst (s +* (Initialized (stop (if>0 a,k1,I)))),((card I) + 1) by A1, A4, A7, SCMPDS_2:68
.= inspos (0 + ((card I) + 1)) by A2, Th23 ;
(Computation (s +* (Initialized (stop (if>0 a,k1,I)))),1) . (inspos ((card I) + 1)) = (stop (if>0 a,k1,I)) . (inspos ((card I) + 1)) by A8, A10, GRFUNC_1:8
.= halt SCMPDS by A9, Th25 ;
then A12: CurInstr (Computation (s +* (Initialized (stop (if>0 a,k1,I)))),1) = halt SCMPDS by A11;
then A13: 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 implies CurInstr (Computation (s +* (Initialized (stop (if>0 a,k1,I)))),l) <> halt SCMPDS )
assume l < 1 ; :: thesis: CurInstr (Computation (s +* (Initialized (stop (if>0 a,k1,I)))),l) <> halt SCMPDS
then l < 1 + 0 ;
then l <= 0 by NAT_1:13;
then l = 0 ;
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 A3, Th30; :: thesis: verum

end;

then for l being Element of NAT st CurInstr (Computation (s +* (Initialized (stop (if>0 a,k1,I)))),l) = halt SCMPDS holds
1 <= l ;
then LifeSpan (s +* (Initialized (stop (if>0 a,k1,I)))) = 1 by A12, A13, AMI_1:def 46;
then A14: Computation (s +* (Initialized (stop (if>0 a,k1,I)))),1 = Result (s +* (Initialized (stop (if>0 a,k1,I)))) by A13, AMI_1:122;
A15: dom (IExec (if>0 a,k1,I),s) = the carrier of SCMPDS by AMI_1:79
.= dom (s +* (Start-At (inspos ((card I) + 1)))) by AMI_1:79 ;

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) + 1)))) . b₁ )
A16: IExec (if>0 a,k1,I),s = (Result (s +* (Initialized (stop (if>0 a,k1,I))))) +* (s | NAT ) by SCMPDS_4:def 8;
assume A17: 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) + 1)))) . b₁
A18: dom (Start-At (inspos ((card I) + 1))) = {(IC SCMPDS )} by FUNCOP_1:19;

per cases ( x is Int_position or x = IC SCMPDS or x is Instruction-Location of SCMPDS ) by A17, SCMPDS_4:20;

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

then x <> IC SCMPDS by SCMPDS_2:52;
then A20: not x in dom (Start-At (inspos ((card I) + 1))) by A18, TARSKI:def 1;

now

assume x in dom (s | NAT ) ; :: thesis: contradiction
then reconsider l = x as Instruction-Location of SCMPDS by A5, AMI_1:def 4;
l = x ;
hence contradiction by A19, SCMPDS_2:53; :: thesis: verum

end;

hence (IExec (if>0 a,k1,I),s) . x = (Computation (s +* (Initialized (stop (if>0 a,k1,I)))),1) . x by A14, A16, FUNCT_4:12
.= (s +* (Initialized (stop (if>0 a,k1,I)))) . x by A4, A19, SCMPDS_2:68
.= s . x by A19, SCMPDS_5:19
.= (s +* (Start-At (inspos ((card I) + 1)))) . x by A20, FUNCT_4:12 ;
:: thesis: verum

end;

suppose A21: x = IC SCMPDS ; :: thesis: (IExec (if>0 a,k1,I),s) . b₁ = (s +* (Start-At (inspos ((card I) + 1)))) . b₁

now

assume x in dom (s | NAT ) ; :: thesis: contradiction
then reconsider l = x as Instruction-Location of SCMPDS by A5, AMI_1:def 4;
l = x ;
hence contradiction by A21, AMI_1:48; :: thesis: verum

end;

hence (IExec (if>0 a,k1,I),s) . x = inspos ((card I) + 1) by A14, A16, A11, A21, FUNCT_4:12
.= (s +* (Start-At (inspos ((card I) + 1)))) . x by A21, 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) + 1)))) . b₁

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

end;

end;

end;

hence IExec (if>0 a,k1,I),s = s +* (Start-At (inspos ((card I) + 1))) by A15, FUNCT_1:9; :: thesis: verum