let s be State of SCMPDS; :: thesis: for I being halt-free shiftable Program of SCMPDS
for a being Int_position
for k1 being Integer st s . (DataLoc ((s . a),k1)) <= 0 & I is_closed_on s & I is_halting_on s holds
IExec ((if<=0 (a,k1,I)),s) = (IExec (I,s)) +* (Start-At (((card I) + 2),SCMPDS))
let I be halt-free shiftable Program of SCMPDS; :: thesis: for a being Int_position
for k1 being Integer st s . (DataLoc ((s . a),k1)) <= 0 & I is_closed_on s & I is_halting_on s holds
IExec ((if<=0 (a,k1,I)),s) = (IExec (I,s)) +* (Start-At (((card I) + 2),SCMPDS))
let a be Int_position ; :: thesis: for k1 being Integer st s . (DataLoc ((s . a),k1)) <= 0 & I is_closed_on s & I is_halting_on s holds
IExec ((if<=0 (a,k1,I)),s) = (IExec (I,s)) +* (Start-At (((card I) + 2),SCMPDS))
let k1 be Integer; :: thesis: ( s . (DataLoc ((s . a),k1)) <= 0 & I is_closed_on s & I is_halting_on s implies IExec ((if<=0 (a,k1,I)),s) = (IExec (I,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 pI = stop I;
set s2 = (Initialize s) +* (stop 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 i = (a,k1) <=0_goto 2;
set j = goto ((card I) + 1);
set SAl = Start-At (((card I) + 2),SCMPDS);
A1: s +* (Initialize (stop I)) = (Initialize s) +* (stop I) by COMPOS_1:125;
A2: s +* (Initialize (stop (if<=0 (a,k1,I)))) = (Initialize s) +* (stop (if<=0 (a,k1,I))) by COMPOS_1:125;
A3: if<=0 (a,k1,I) = ((a,k1) <=0_goto 2) ';' ((goto ((card I) + 1)) ';' I) by SCMPDS_4:52;
A4: 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 A3, Th22, A2 ;
A5: 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 A6: ((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 A2, FUNCT_4:12
.= s . (DataLoc ((s . a),k1)) by A5, A2, FUNCT_4:12 ;
A7: IC ((Initialize s) +* (stop (if<=0 (a,k1,I)))) = 0 by A2, FUNCT_4:26, SCMPDS_5:18;
A8: ( Initialize (stop I) c= (Initialize s) +* (stop I) & Shift ((stop I),2) c= Comput ((ProgramPart ((Initialize s) +* (stop (if<=0 (a,k1,I))))),((Initialize s) +* (stop (if<=0 (a,k1,I)))),1) ) by Lm7, A1, FUNCT_4:26;
A9: DataPart ((Initialize s) +* (stop I)) = DataPart ((Initialize s) +* (stop (if<=0 (a,k1,I)))) by COMPOS_1:138, FUNCT_7:134;

now

let a be Int_position ; :: thesis: ((Initialize s) +* (stop I)) . a = (Comput ((ProgramPart ((Initialize s) +* (stop (if<=0 (a,k1,I))))),((Initialize s) +* (stop (if<=0 (a,k1,I)))),1)) . a
thus ((Initialize s) +* (stop I)) . a = ((Initialize s) +* (stop (if<=0 (a,k1,I)))) . a by A9, SCMPDS_4:23
.= (Comput ((ProgramPart ((Initialize s) +* (stop (if<=0 (a,k1,I))))),((Initialize s) +* (stop (if<=0 (a,k1,I)))),1)) . a by A4, SCMPDS_2:68 ; :: thesis: verum

end;

then A10: DataPart ((Initialize s) +* (stop I)) = DataPart (Comput ((ProgramPart ((Initialize s) +* (stop (if<=0 (a,k1,I))))),((Initialize s) +* (stop (if<=0 (a,k1,I)))),1)) by SCMPDS_4:23;
assume s . (DataLoc ((s . a),k1)) <= 0 ; :: thesis: ( not I is_closed_on s or not I is_halting_on s or IExec ((if<=0 (a,k1,I)),s) = (IExec (I,s)) +* (Start-At (((card I) + 2),SCMPDS)) )
then A11: IC (Comput ((ProgramPart ((Initialize s) +* (stop (if<=0 (a,k1,I))))),((Initialize s) +* (stop (if<=0 (a,k1,I)))),1)) = ICplusConst (((Initialize s) +* (stop (if<=0 (a,k1,I)))),2) by A4, A6, SCMPDS_2:68
.= 0 + 2 by A7, Th23 ;
A12: dom (ProgramPart s) = NAT by COMPOS_1:34;
assume A13: I is_closed_on s ; :: thesis: ( not I is_halting_on s or IExec ((if<=0 (a,k1,I)),s) = (IExec (I,s)) +* (Start-At (((card I) + 2),SCMPDS)) )
then A14: I is_closed_on (Initialize s) +* (stop I) by Th38;
assume A15: I is_halting_on s ; :: thesis: IExec ((if<=0 (a,k1,I)),s) = (IExec (I,s)) +* (Start-At (((card I) + 2),SCMPDS))
then A16: ProgramPart ((Initialize s) +* (stop I)) halts_on (Initialize s) +* (stop I) by Def3;
A17: 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;
A18: Comput ((ProgramPart ((Initialize s) +* (stop (if<=0 (a,k1,I))))),((Initialize s) +* (stop (if<=0 (a,k1,I)))),((LifeSpan ((ProgramPart ((Initialize s) +* (stop I))),((Initialize s) +* (stop I)))) + 1)) = Comput ((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)),(LifeSpan ((ProgramPart ((Initialize s) +* (stop I))),((Initialize s) +* (stop I))))) by EXTPRO_1:5;
A19: ProgramPart (Comput ((ProgramPart ((Initialize s) +* (stop (if<=0 (a,k1,I))))),((Initialize s) +* (stop (if<=0 (a,k1,I)))),1)) = ProgramPart (Comput ((ProgramPart (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)),(LifeSpan ((ProgramPart ((Initialize s) +* (stop I))),((Initialize s) +* (stop I)))))) by AMI_1:123;
A20: ProgramPart ((Initialize s) +* (stop I)) = ProgramPart (Comput ((ProgramPart ((Initialize s) +* (stop I))),((Initialize s) +* (stop I)),(LifeSpan ((ProgramPart ((Initialize s) +* (stop I))),((Initialize s) +* (stop I)))))) by AMI_1:123;
A21: 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)))),((LifeSpan ((ProgramPart ((Initialize s) +* (stop I))),((Initialize s) +* (stop I)))) + 1)))) = CurInstr ((ProgramPart (Comput ((ProgramPart ((Initialize s) +* (stop (if<=0 (a,k1,I))))),((Initialize s) +* (stop (if<=0 (a,k1,I)))),1))),(Comput ((ProgramPart (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)),(LifeSpan ((ProgramPart ((Initialize s) +* (stop I))),((Initialize s) +* (stop I))))))) by A18, A17
.= CurInstr ((ProgramPart ((Initialize s) +* (stop I))),(Comput ((ProgramPart ((Initialize s) +* (stop I))),((Initialize s) +* (stop I)),(LifeSpan ((ProgramPart ((Initialize s) +* (stop I))),((Initialize s) +* (stop I))))))) by A14, A8, A11, A10, Th45, A20, A19
.= halt SCMPDS by A16, EXTPRO_1:def 14 ;
then A22: ProgramPart ((Initialize s) +* (stop (if<=0 (a,k1,I)))) halts_on (Initialize s) +* (stop (if<=0 (a,k1,I))) by EXTPRO_1:30;
A23: CurInstr ((ProgramPart ((Initialize s) +* (stop (if<=0 (a,k1,I))))),((Initialize s) +* (stop (if<=0 (a,k1,I))))) = (a,k1) <=0_goto 2 by A3, Th22, A2;

now

let l be Element of NAT ; :: thesis: ( l < (LifeSpan ((ProgramPart ((Initialize s) +* (stop I))),((Initialize s) +* (stop I)))) + 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 A24: l < (LifeSpan ((ProgramPart ((Initialize s) +* (stop I))),((Initialize s) +* (stop I)))) + 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
A25: 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;

per cases ( l = 0 or l <> 0 ) ;

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 A25;
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 A23, Th30; :: thesis: verum

end;

suppose l <> 0 ; :: thesis: not 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 consider n being Nat such that
A26: l = n + 1 by NAT_1:6;
reconsider n = n as Element of NAT by ORDINAL1:def 13;
A27: n < LifeSpan ((ProgramPart ((Initialize s) +* (stop I))),((Initialize s) +* (stop I))) by A24, A26, XREAL_1:8;
assume A28: 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 ; :: thesis: contradiction
A29: 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;
A30: Comput ((ProgramPart ((Initialize s) +* (stop (if<=0 (a,k1,I))))),((Initialize s) +* (stop (if<=0 (a,k1,I)))),(n + 1)) = Comput ((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)),n) by EXTPRO_1:5;
A31: ProgramPart ((Initialize s) +* (stop I)) = ProgramPart (Comput ((ProgramPart ((Initialize s) +* (stop I))),((Initialize s) +* (stop I)),n)) by AMI_1:123;
A32: ProgramPart (Comput ((ProgramPart ((Initialize s) +* (stop (if<=0 (a,k1,I))))),((Initialize s) +* (stop (if<=0 (a,k1,I)))),1)) = ProgramPart (Comput ((ProgramPart (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)),n)) by AMI_1:123;
CurInstr ((ProgramPart ((Initialize s) +* (stop I))),(Comput ((ProgramPart ((Initialize s) +* (stop I))),((Initialize s) +* (stop I)),n))) = CurInstr ((ProgramPart (Comput ((ProgramPart ((Initialize s) +* (stop (if<=0 (a,k1,I))))),((Initialize s) +* (stop (if<=0 (a,k1,I)))),1))),(Comput ((ProgramPart (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)),n))) by A14, A8, A11, A10, Th45, A31, A32
.= halt SCMPDS by A26, A28, A30, A29 ;
hence contradiction by A16, A27, EXTPRO_1:def 14; :: 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
(LifeSpan ((ProgramPart ((Initialize s) +* (stop I))),((Initialize s) +* (stop I)))) + 1 <= l ;
then A33: LifeSpan ((ProgramPart ((Initialize s) +* (stop (if<=0 (a,k1,I))))),((Initialize s) +* (stop (if<=0 (a,k1,I))))) = (LifeSpan ((ProgramPart ((Initialize s) +* (stop I))),((Initialize s) +* (stop I)))) + 1 by A21, A22, EXTPRO_1:def 14;
A34: 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;
A35: DataPart (Result ((ProgramPart ((Initialize s) +* (stop I))),((Initialize s) +* (stop I)))) = DataPart (Comput ((ProgramPart ((Initialize s) +* (stop I))),((Initialize s) +* (stop I)),(LifeSpan ((ProgramPart ((Initialize s) +* (stop I))),((Initialize s) +* (stop I)))))) by A16, EXTPRO_1:23
.= DataPart (Comput ((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)),(LifeSpan ((ProgramPart ((Initialize s) +* (stop I))),((Initialize s) +* (stop I)))))) by A14, A8, A11, A10, Th45, A34
.= DataPart (Comput ((ProgramPart ((Initialize s) +* (stop (if<=0 (a,k1,I))))),((Initialize s) +* (stop (if<=0 (a,k1,I)))),((LifeSpan ((ProgramPart ((Initialize s) +* (stop I))),((Initialize s) +* (stop I)))) + 1))) by EXTPRO_1:5
.= DataPart (Result ((ProgramPart ((Initialize s) +* (stop (if<=0 (a,k1,I))))),((Initialize s) +* (stop (if<=0 (a,k1,I)))))) by A22, A33, EXTPRO_1:23 ;

A36: 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 = ((IExec (I,s)) +* (Start-At (((card I) + 2),SCMPDS))) . b₁ )
A37: dom (Start-At (((card I) + 2),SCMPDS)) = {(IC )} by FUNCOP_1:19;
assume A38: x in dom (IExec ((if<=0 (a,k1,I)),s)) ; :: thesis: (IExec ((if<=0 (a,k1,I)),s)) . b₁ = ((IExec (I,s)) +* (Start-At (((card I) + 2),SCMPDS))) . b₁

per cases ( x is Int_position or x = IC or x is Element of NAT ) by A38, SCMPDS_4:20;

suppose A39: x is Int_position ; :: thesis: (IExec ((if<=0 (a,k1,I)),s)) . b₁ = ((IExec (I,s)) +* (Start-At (((card I) + 2),SCMPDS))) . b₁

then x <> IC by SCMPDS_2:52;
then A40: not x in dom (Start-At (((card I) + 2),SCMPDS)) by A37, TARSKI:def 1;
A41: not x in dom (s | NAT) by A12, A39, SCMPDS_2:53;
hence (IExec ((if<=0 (a,k1,I)),s)) . x = (Result ((ProgramPart ((Initialize s) +* (stop (if<=0 (a,k1,I))))),((Initialize s) +* (stop (if<=0 (a,k1,I)))))) . x by FUNCT_4:12
.= (Result ((ProgramPart ((Initialize s) +* (stop I))),((Initialize s) +* (stop I)))) . x by A35, A39, SCMPDS_4:23
.= (IExec (I,s)) . x by A41, FUNCT_4:12
.= ((IExec (I,s)) +* (Start-At (((card I) + 2),SCMPDS))) . x by A40, FUNCT_4:12 ;
:: thesis: verum

end;

suppose A42: x = IC ; :: thesis: (IExec ((if<=0 (a,k1,I)),s)) . b₁ = ((IExec (I,s)) +* (Start-At (((card I) + 2),SCMPDS))) . b₁

A43: not x in dom (s | NAT) by A12, A42, COMPOS_1:3;
then A44: IC (Result ((ProgramPart ((Initialize s) +* (stop I))),((Initialize s) +* (stop I)))) = IC (IExec (I,s)) by A42, FUNCT_4:12
.= card I by A13, A15, Th48 ;
A45: x in dom (Start-At (((card I) + 2),SCMPDS)) by A37, A42, TARSKI:def 1;
A46: 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;
thus (IExec ((if<=0 (a,k1,I)),s)) . x = (Result ((ProgramPart ((Initialize s) +* (stop (if<=0 (a,k1,I))))),((Initialize s) +* (stop (if<=0 (a,k1,I)))))) . x by A43, FUNCT_4:12
.= (Comput ((ProgramPart ((Initialize s) +* (stop (if<=0 (a,k1,I))))),((Initialize s) +* (stop (if<=0 (a,k1,I)))),((LifeSpan ((ProgramPart ((Initialize s) +* (stop I))),((Initialize s) +* (stop I)))) + 1))) . x by A22, A33, EXTPRO_1:23
.= IC (Comput ((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)),(LifeSpan ((ProgramPart ((Initialize s) +* (stop I))),((Initialize s) +* (stop I)))))) by A42, EXTPRO_1:5
.= (IC (Comput ((ProgramPart ((Initialize s) +* (stop I))),((Initialize s) +* (stop I)),(LifeSpan ((ProgramPart ((Initialize s) +* (stop I))),((Initialize s) +* (stop I))))))) + 2 by A14, A8, A11, A10, Th45, A46
.= (IC (Result ((ProgramPart ((Initialize s) +* (stop I))),((Initialize s) +* (stop I))))) + 2 by A16, EXTPRO_1:23
.= (Start-At (((card I) + 2),SCMPDS)) . (IC ) by A44, FUNCOP_1:87
.= ((IExec (I,s)) +* (Start-At (((card I) + 2),SCMPDS))) . x by A42, A45, FUNCT_4:14 ; :: thesis: verum

end;

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

hence (IExec ((if<=0 (a,k1,I)),s)) . x = ((IExec (I,s)) +* (Start-At (((card I) + 2),SCMPDS))) . x by Th26; :: thesis: verum

end;

end;

end;

dom (IExec ((if<=0 (a,k1,I)),s)) = the carrier of SCMPDS by PARTFUN1:def 4
.= dom ((IExec (I,s)) +* (Start-At (((card I) + 2),SCMPDS))) by PARTFUN1:def 4 ;
hence IExec ((if<=0 (a,k1,I)),s) = (IExec (I,s)) +* (Start-At (((card I) + 2),SCMPDS)) by A36, FUNCT_1:9; :: thesis: verum