let P be Instruction-Sequence of SCMPDS; :: thesis: for s being State of SCMPDS
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)),P,(Initialize s)) = s +* (Start-At (((card I) + 1),SCMPDS))
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)),P,(Initialize 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)),P,(Initialize 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)),P,(Initialize 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)),P,(Initialize 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;
set P3 = P +* (stop (if>0 (a,k1,I)));
set s4 = Comput ((P +* (stop (if>0 (a,k1,I)))),(Initialize s),1);
set P4 = P +* (stop (if>0 (a,k1,I)));
set i = (a,k1) <=0_goto ((card I) + 1);
set SAl = Start-At (((card I) + 1),SCMPDS);
A2: IC (Initialize s) = 0 by MEMSTR_0:47;
A3: not DataLoc ((s . a),k1) in dom (Start-At (0,SCMPDS)) by SCMPDS_4:18;
not a in dom (Start-At (0,SCMPDS)) by SCMPDS_4:18;
then A4: (Initialize s) . (DataLoc (((Initialize s) . a),k1)) = (Initialize s) . (DataLoc ((s . a),k1)) by FUNCT_4:11
.= s . (DataLoc ((s . a),k1)) by A3, FUNCT_4:11 ;
A5: Comput ((P +* (stop (if>0 (a,k1,I)))),(Initialize s),(0 + 1)) = Following ((P +* (stop (if>0 (a,k1,I)))),(Comput ((P +* (stop (if>0 (a,k1,I)))),(Initialize s),0))) by EXTPRO_1:3
.= Following ((P +* (stop (if>0 (a,k1,I)))),(Initialize s)) by EXTPRO_1:2
.= Exec (((a,k1) <=0_goto ((card I) + 1)),(Initialize s)) by Th22 ;
assume s . (DataLoc ((s . a),k1)) <= 0 ; :: thesis: IExec ((if>0 (a,k1,I)),P,(Initialize s)) = s +* (Start-At (((card I) + 1),SCMPDS))
then A6: IC (Comput ((P +* (stop (if>0 (a,k1,I)))),(Initialize s),1)) = ICplusConst ((Initialize s),((card I) + 1)) by A5, A4, SCMPDS_2:56
.= 0 + ((card I) + 1) by A2, Th23 ;
stop (if>0 (a,k1,I)) c= P +* (stop (if>0 (a,k1,I))) by FUNCT_4:25;
then A7: stop (if>0 (a,k1,I)) c= P +* (stop (if>0 (a,k1,I))) ;
A8: (P +* (stop (if>0 (a,k1,I)))) /. (IC (Comput ((P +* (stop (if>0 (a,k1,I)))),(Initialize s),1))) = (P +* (stop (if>0 (a,k1,I)))) . (IC (Comput ((P +* (stop (if>0 (a,k1,I)))),(Initialize s),1))) by PBOOLE:143;
A10: card (if>0 (a,k1,I)) = (card I) + 1 by Th15;
then (card I) + 1 in dom (stop (if>0 (a,k1,I))) by COMPOS_1:64;
then (P +* (stop (if>0 (a,k1,I)))) . ((card I) + 1) = (stop (if>0 (a,k1,I))) . ((card I) + 1) by A7, GRFUNC_1:2
.= halt SCMPDS by A10, COMPOS_1:64 ;
then A11: CurInstr ((P +* (stop (if>0 (a,k1,I)))),(Comput ((P +* (stop (if>0 (a,k1,I)))),(Initialize s),1))) = halt SCMPDS by A6, A8;
then A12: P +* (stop (if>0 (a,k1,I))) halts_on Initialize s by EXTPRO_1:29;
A13: CurInstr ((P +* (stop (if>0 (a,k1,I)))),(Initialize s)) = (a,k1) <=0_goto ((card I) + 1) by Th22;

now

let l be Element of NAT ; :: thesis: ( l < 1 implies CurInstr ((P +* (stop (if>0 (a,k1,I)))),(Comput ((P +* (stop (if>0 (a,k1,I)))),(Initialize s),l))) <> halt SCMPDS )
A14: Comput ((P +* (stop (if>0 (a,k1,I)))),(Initialize s),0) = Initialize s by EXTPRO_1:2;
assume l < 1 ; :: thesis: CurInstr ((P +* (stop (if>0 (a,k1,I)))),(Comput ((P +* (stop (if>0 (a,k1,I)))),(Initialize s),l))) <> halt SCMPDS
then l < 1 + 0 ;
then l = 0 by NAT_1:13;
then CurInstr ((P +* (stop (if>0 (a,k1,I)))),(Comput ((P +* (stop (if>0 (a,k1,I)))),(Initialize s),l))) = CurInstr ((P +* (stop (if>0 (a,k1,I)))),(Initialize s)) by A14;
hence CurInstr ((P +* (stop (if>0 (a,k1,I)))),(Comput ((P +* (stop (if>0 (a,k1,I)))),(Initialize s),l))) <> halt SCMPDS by A13; :: thesis: verum

end;

then for l being Element of NAT st CurInstr ((P +* (stop (if>0 (a,k1,I)))),(Comput ((P +* (stop (if>0 (a,k1,I)))),(Initialize s),l))) = halt SCMPDS holds
1 <= l ;
then LifeSpan ((P +* (stop (if>0 (a,k1,I)))),(Initialize s)) = 1 by A11, A12, EXTPRO_1:def 15;
then A15: Comput ((P +* (stop (if>0 (a,k1,I)))),(Initialize s),1) = Result ((P +* (stop (if>0 (a,k1,I)))),(Initialize s)) by A12, EXTPRO_1:23;

A17: now

A18: dom (Start-At (((card I) + 1),SCMPDS)) = {(IC )} by FUNCOP_1:13;
let x be set ; :: thesis: ( x in dom (IExec ((if>0 (a,k1,I)),P,(Initialize s))) implies (IExec ((if>0 (a,k1,I)),P,(Initialize s))) . b₁ = (s +* (Start-At (((card I) + 1),SCMPDS))) . b₁ )
assume A19: x in dom (IExec ((if>0 (a,k1,I)),P,(Initialize s))) ; :: thesis: (IExec ((if>0 (a,k1,I)),P,(Initialize s))) . b₁ = (s +* (Start-At (((card I) + 1),SCMPDS))) . b₁

per cases ( x is Int_position or x = IC ) by A19, SCMPDS_4:6;

suppose A20: x is Int_position ; :: thesis: (IExec ((if>0 (a,k1,I)),P,(Initialize s))) . b₁ = (s +* (Start-At (((card I) + 1),SCMPDS))) . b₁

then x <> IC by SCMPDS_2:43;
then A21: not x in dom (Start-At (((card I) + 1),SCMPDS)) by A18, TARSKI:def 1;
TT: not x in dom (Start-At (0,SCMPDS)) by A20, SCMPDS_4:18;
thus (IExec ((if>0 (a,k1,I)),P,(Initialize s))) . x = (Comput ((P +* (stop (if>0 (a,k1,I)))),(Initialize s),1)) . x by A15
.= (Initialize s) . x by A5, A20, SCMPDS_2:56
.= s . x by TT, FUNCT_4:11
.= (s +* (Start-At (((card I) + 1),SCMPDS))) . x by A21, FUNCT_4:11 ; :: thesis: verum

end;

suppose A22: x = IC ; :: thesis: (IExec ((if>0 (a,k1,I)),P,(Initialize s))) . b₁ = (s +* (Start-At (((card I) + 1),SCMPDS))) . b₁

hence (IExec ((if>0 (a,k1,I)),P,(Initialize s))) . x = (card I) + 1 by A6, A15
.= (s +* (Start-At (((card I) + 1),SCMPDS))) . x by A22, FUNCT_4:113 ;
:: thesis: verum

end;

end;

end;

dom (IExec ((if>0 (a,k1,I)),P,(Initialize s))) = the carrier of SCMPDS by PARTFUN1:def 2
.= dom (s +* (Start-At (((card I) + 1),SCMPDS))) by PARTFUN1:def 2 ;
hence IExec ((if>0 (a,k1,I)),P,(Initialize s)) = s +* (Start-At (((card I) + 1),SCMPDS)) by A17, FUNCT_1:2; :: thesis: verum