let P be Instruction-Sequence of SCMPDS; :: thesis: for s being 0 -started State of SCMPDS
for I being halt-free shiftable Program of
for J being shiftable Program of
for a being Int_position
for k1 being Integer st s . (DataLoc ((s . a),k1)) < 0 & I is_closed_on s,P & I is_halting_on s,P holds
IExec ((if<0 (a,k1,I,J)),P,s) = (IExec (I,P,s)) +* (Start-At ((((card I) + (card J)) + 2),SCMPDS))
let s be 0 -started State of SCMPDS; :: thesis: for I being halt-free shiftable Program of
for J being shiftable Program of
for a being Int_position
for k1 being Integer st s . (DataLoc ((s . a),k1)) < 0 & I is_closed_on s,P & I is_halting_on s,P holds
IExec ((if<0 (a,k1,I,J)),P,s) = (IExec (I,P,s)) +* (Start-At ((((card I) + (card J)) + 2),SCMPDS))
let I be halt-free shiftable Program of ; :: thesis: for J being shiftable Program of
for a being Int_position
for k1 being Integer st s . (DataLoc ((s . a),k1)) < 0 & I is_closed_on s,P & I is_halting_on s,P holds
IExec ((if<0 (a,k1,I,J)),P,s) = (IExec (I,P,s)) +* (Start-At ((((card I) + (card J)) + 2),SCMPDS))
let J be shiftable Program of ; :: thesis: for a being Int_position
for k1 being Integer st s . (DataLoc ((s . a),k1)) < 0 & I is_closed_on s,P & I is_halting_on s,P holds
IExec ((if<0 (a,k1,I,J)),P,s) = (IExec (I,P,s)) +* (Start-At ((((card I) + (card J)) + 2),SCMPDS))
let a be Int_position; :: thesis: for k1 being Integer st s . (DataLoc ((s . a),k1)) < 0 & I is_closed_on s,P & I is_halting_on s,P holds
IExec ((if<0 (a,k1,I,J)),P,s) = (IExec (I,P,s)) +* (Start-At ((((card I) + (card J)) + 2),SCMPDS))
let k1 be Integer; :: thesis: ( s . (DataLoc ((s . a),k1)) < 0 & I is_closed_on s,P & I is_halting_on s,P implies IExec ((if<0 (a,k1,I,J)),P,s) = (IExec (I,P,s)) +* (Start-At ((((card I) + (card J)) + 2),SCMPDS)) )
set b = DataLoc ((s . a),k1);
set G = Goto ((card J) + 1);
set I2 = (I ';' (Goto ((card J) + 1))) ';' J;
set IF = if<0 (a,k1,I,J);
set pIF = stop (if<0 (a,k1,I,J));
set pI2 = stop ((I ';' (Goto ((card J) + 1))) ';' J);
set P2 = P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J));
set P3 = P +* (stop (if<0 (a,k1,I,J)));
set s4 = Comput ((P +* (stop (if<0 (a,k1,I,J)))),s,1);
set P4 = P +* (stop (if<0 (a,k1,I,J)));
set i = (a,k1) >=0_goto ((card I) + 2);
set SAl = Start-At ((((card I) + (card J)) + 2),SCMPDS);
A1: Initialize s = s by MEMSTR_0:44;
then A2: IC s = 0 by MEMSTR_0:47;
A3: if<0 (a,k1,I,J) = (((a,k1) >=0_goto ((card I) + 2)) ';' (I ';' (Goto ((card J) + 1)))) ';' J by SCMPDS_4:14
.= ((a,k1) >=0_goto ((card I) + 2)) ';' ((I ';' (Goto ((card J) + 1))) ';' J) by SCMPDS_4:14 ;
then A4: Shift ((stop ((I ';' (Goto ((card J) + 1))) ';' J)),1) c= P +* (stop (if<0 (a,k1,I,J))) by Lm6;
A5: Comput ((P +* (stop (if<0 (a,k1,I,J)))),s,(0 + 1)) = Following ((P +* (stop (if<0 (a,k1,I,J)))),(Comput ((P +* (stop (if<0 (a,k1,I,J)))),s,0))) by EXTPRO_1:3
.= Following ((P +* (stop (if<0 (a,k1,I,J)))),s) by EXTPRO_1:2
.= Exec (((a,k1) >=0_goto ((card I) + 2)),s) by A3, Th3, A1 ;
assume s . (DataLoc ((s . a),k1)) < 0 ; :: thesis: ( not I is_closed_on s,P or not I is_halting_on s,P or IExec ((if<0 (a,k1,I,J)),P,s) = (IExec (I,P,s)) +* (Start-At ((((card I) + (card J)) + 2),SCMPDS)) )
then A6: IC (Comput ((P +* (stop (if<0 (a,k1,I,J)))),s,1)) = (IC s) + 1 by A5, SCMPDS_2:57
.= 0 + 1 by A2 ;
for a being Int_position holds s . a = (Comput ((P +* (stop (if<0 (a,k1,I,J)))),s,1)) . a by A5, SCMPDS_2:57;
then A7: DataPart s = DataPart (Comput ((P +* (stop (if<0 (a,k1,I,J)))),s,1)) by SCMPDS_4:8;
assume A8: I is_closed_on s,P ; :: thesis: ( not I is_halting_on s,P or IExec ((if<0 (a,k1,I,J)),P,s) = (IExec (I,P,s)) +* (Start-At ((((card I) + (card J)) + 2),SCMPDS)) )
assume A9: I is_halting_on s,P ; :: thesis: IExec ((if<0 (a,k1,I,J)),P,s) = (IExec (I,P,s)) +* (Start-At ((((card I) + (card J)) + 2),SCMPDS))
then (I ';' (Goto ((card J) + 1))) ';' J is_halting_on s,P by A8, Th21;
then A10: P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J)) halts_on s by A1;
(I ';' (Goto ((card J) + 1))) ';' J is_closed_on s,P by A8, A9, Th21;
then A11: ( Start-At (0,SCMPDS) c= s & (I ';' (Goto ((card J) + 1))) ';' J is_closed_on s,P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J)) ) by A1, FUNCT_4:25;
A12: stop ((I ';' (Goto ((card J) + 1))) ';' J) c= P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J)) by FUNCT_4:25;
A13: Comput ((P +* (stop (if<0 (a,k1,I,J)))),s,((LifeSpan ((P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J))),s)) + 1)) = Comput ((P +* (stop (if<0 (a,k1,I,J)))),(Comput ((P +* (stop (if<0 (a,k1,I,J)))),s,1)),(LifeSpan ((P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J))),s))) by EXTPRO_1:4;
A14: CurInstr ((P +* (stop (if<0 (a,k1,I,J)))),(Comput ((P +* (stop (if<0 (a,k1,I,J)))),s,((LifeSpan ((P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J))),s)) + 1)))) = CurInstr ((P +* (stop (if<0 (a,k1,I,J)))),(Comput ((P +* (stop (if<0 (a,k1,I,J)))),(Comput ((P +* (stop (if<0 (a,k1,I,J)))),s,1)),(LifeSpan ((P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J))),s))))) by A13
.= CurInstr ((P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J))),(Comput ((P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J))),s,(LifeSpan ((P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J))),s))))) by A11, A4, A6, A7, Th22, A12
.= halt SCMPDS by A10, EXTPRO_1:def 15 ;
then A15: P +* (stop (if<0 (a,k1,I,J))) halts_on s by EXTPRO_1:29;
A16: CurInstr ((P +* (stop (if<0 (a,k1,I,J)))),s) = (a,k1) >=0_goto ((card I) + 2) by A3, Th3, A1;

now :: thesis: for l being Nat st l < (LifeSpan ((P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J))),s)) + 1 holds
CurInstr ((P +* (stop (if<0 (a,k1,I,J)))),(Comput ((P +* (stop (if<0 (a,k1,I,J)))),s,l))) <> halt SCMPDS

let l be Nat; :: thesis: ( l < (LifeSpan ((P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J))),s)) + 1 implies CurInstr ((P +* (stop (if<0 (a,k1,I,J)))),(Comput ((P +* (stop (if<0 (a,k1,I,J)))),s,b₁))) <> halt SCMPDS )
assume A17: l < (LifeSpan ((P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J))),s)) + 1 ; :: thesis: CurInstr ((P +* (stop (if<0 (a,k1,I,J)))),(Comput ((P +* (stop (if<0 (a,k1,I,J)))),s,b₁))) <> halt SCMPDS
A18: Comput ((P +* (stop (if<0 (a,k1,I,J)))),s,0) = s by EXTPRO_1:2;

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

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

then CurInstr ((P +* (stop (if<0 (a,k1,I,J)))),(Comput ((P +* (stop (if<0 (a,k1,I,J)))),s,l))) = CurInstr ((P +* (stop (if<0 (a,k1,I,J)))),s) by A18;
hence CurInstr ((P +* (stop (if<0 (a,k1,I,J)))),(Comput ((P +* (stop (if<0 (a,k1,I,J)))),s,l))) <> halt SCMPDS by A16; :: thesis: verum

end;

suppose l <> 0 ; :: thesis: not CurInstr ((P +* (stop (if<0 (a,k1,I,J)))),(Comput ((P +* (stop (if<0 (a,k1,I,J)))),s,b₁))) = halt SCMPDS

then consider n being Nat such that
A19: l = n + 1 by NAT_1:6;
reconsider n = n as Nat ;
A20: n < LifeSpan ((P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J))),s) by A17, A19, XREAL_1:6;
assume A21: CurInstr ((P +* (stop (if<0 (a,k1,I,J)))),(Comput ((P +* (stop (if<0 (a,k1,I,J)))),s,l))) = halt SCMPDS ; :: thesis: contradiction
A22: Comput ((P +* (stop (if<0 (a,k1,I,J)))),s,(n + 1)) = Comput ((P +* (stop (if<0 (a,k1,I,J)))),(Comput ((P +* (stop (if<0 (a,k1,I,J)))),s,1)),n) by EXTPRO_1:4;
CurInstr ((P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J))),(Comput ((P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J))),s,n))) = CurInstr ((P +* (stop (if<0 (a,k1,I,J)))),(Comput ((P +* (stop (if<0 (a,k1,I,J)))),(Comput ((P +* (stop (if<0 (a,k1,I,J)))),s,1)),n))) by A11, A4, A6, A7, Th22, A12
.= halt SCMPDS by A19, A21, A22 ;
hence contradiction by A10, A20, EXTPRO_1:def 15; :: thesis: verum

end;

end;

end;

then for l being Nat st CurInstr ((P +* (stop (if<0 (a,k1,I,J)))),(Comput ((P +* (stop (if<0 (a,k1,I,J)))),s,l))) = halt SCMPDS holds
(LifeSpan ((P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J))),s)) + 1 <= l ;
then A23: LifeSpan ((P +* (stop (if<0 (a,k1,I,J)))),s) = (LifeSpan ((P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J))),s)) + 1 by A14, A15, EXTPRO_1:def 15;
A24: DataPart (Result ((P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J))),s)) = DataPart (Comput ((P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J))),s,(LifeSpan ((P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J))),s)))) by A10, EXTPRO_1:23
.= DataPart (Comput ((P +* (stop (if<0 (a,k1,I,J)))),(Comput ((P +* (stop (if<0 (a,k1,I,J)))),s,1)),(LifeSpan ((P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J))),s)))) by A11, A4, A6, A7, Th22, A12
.= DataPart (Comput ((P +* (stop (if<0 (a,k1,I,J)))),s,((LifeSpan ((P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J))),s)) + 1))) by EXTPRO_1:4
.= DataPart (Result ((P +* (stop (if<0 (a,k1,I,J)))),s)) by A15, A23, EXTPRO_1:23 ;

A25: now :: thesis: for x being object st x in dom (IExec ((if<0 (a,k1,I,J)),P,s)) holds
(IExec ((if<0 (a,k1,I,J)),P,s)) . x = ((IExec (((I ';' (Goto ((card J) + 1))) ';' J),P,s)) +* (Start-At ((((card I) + (card J)) + 2),SCMPDS))) . x

let x be object ; :: thesis: ( x in dom (IExec ((if<0 (a,k1,I,J)),P,s)) implies (IExec ((if<0 (a,k1,I,J)),P,s)) . b₁ = ((IExec (((I ';' (Goto ((card J) + 1))) ';' J),P,s)) +* (Start-At ((((card I) + (card J)) + 2),SCMPDS))) . b₁ )
A26: dom (Start-At ((((card I) + (card J)) + 2),SCMPDS)) = {(IC )} by FUNCOP_1:13;
assume A27: x in dom (IExec ((if<0 (a,k1,I,J)),P,s)) ; :: thesis: (IExec ((if<0 (a,k1,I,J)),P,s)) . b₁ = ((IExec (((I ';' (Goto ((card J) + 1))) ';' J),P,s)) +* (Start-At ((((card I) + (card J)) + 2),SCMPDS))) . b₁

per cases ( x is Int_position or x = IC ) by A27, SCMPDS_4:6;

suppose A28: x is Int_position ; :: thesis: (IExec ((if<0 (a,k1,I,J)),P,s)) . b₁ = ((IExec (((I ';' (Goto ((card J) + 1))) ';' J),P,s)) +* (Start-At ((((card I) + (card J)) + 2),SCMPDS))) . b₁

then x <> IC by SCMPDS_2:43;
then A29: not x in dom (Start-At ((((card I) + (card J)) + 2),SCMPDS)) by A26, TARSKI:def 1;
thus (IExec ((if<0 (a,k1,I,J)),P,s)) . x = (Result ((P +* (stop (if<0 (a,k1,I,J)))),s)) . x
.= (Result ((P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J))),s)) . x by A24, A28, SCMPDS_4:8
.= (IExec (((I ';' (Goto ((card J) + 1))) ';' J),P,s)) . x
.= ((IExec (((I ';' (Goto ((card J) + 1))) ';' J),P,s)) +* (Start-At ((((card I) + (card J)) + 2),SCMPDS))) . x by A29, FUNCT_4:11 ; :: thesis: verum

end;

suppose A30: x = IC ; :: thesis: (IExec ((if<0 (a,k1,I,J)),P,s)) . b₁ = ((IExec (((I ';' (Goto ((card J) + 1))) ';' J),P,s)) +* (Start-At ((((card I) + (card J)) + 2),SCMPDS))) . b₁

A31: IC (Result ((P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J))),s)) = IC (IExec (((I ';' (Goto ((card J) + 1))) ';' J),P,s))
.= ((card I) + (card J)) + 1 by A8, A9, Th23 ;
A32: x in dom (Start-At ((((card I) + (card J)) + 2),SCMPDS)) by A26, A30, TARSKI:def 1;
thus (IExec ((if<0 (a,k1,I,J)),P,s)) . x = (Result ((P +* (stop (if<0 (a,k1,I,J)))),s)) . x
.= (Comput ((P +* (stop (if<0 (a,k1,I,J)))),s,((LifeSpan ((P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J))),s)) + 1))) . x by A15, A23, EXTPRO_1:23
.= IC (Comput ((P +* (stop (if<0 (a,k1,I,J)))),(Comput ((P +* (stop (if<0 (a,k1,I,J)))),s,1)),(LifeSpan ((P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J))),s)))) by A30, EXTPRO_1:4
.= (IC (Comput ((P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J))),s,(LifeSpan ((P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J))),s))))) + 1 by A11, A4, A6, A7, Th22, A12
.= (IC (Result ((P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J))),s))) + 1 by A10, EXTPRO_1:23
.= IC (Start-At (((((card I) + (card J)) + 1) + 1),SCMPDS)) by A31, FUNCOP_1:72
.= ((IExec (((I ';' (Goto ((card J) + 1))) ';' J),P,s)) +* (Start-At ((((card I) + (card J)) + 2),SCMPDS))) . x by A30, A32, FUNCT_4:13 ; :: thesis: verum

end;

end;

end;

dom (IExec ((if<0 (a,k1,I,J)),P,s)) = the carrier of SCMPDS by PARTFUN1:def 2
.= dom ((IExec (((I ';' (Goto ((card J) + 1))) ';' J),P,s)) +* (Start-At ((((card I) + (card J)) + 2),SCMPDS))) by PARTFUN1:def 2 ;
hence IExec ((if<0 (a,k1,I,J)),P,s) = (IExec (((I ';' (Goto ((card J) + 1))) ';' J),P,s)) +* (Start-At ((((card I) + (card J)) + 2),SCMPDS)) by A25, FUNCT_1:2
.= ((IExec (I,P,s)) +* (Start-At ((((card I) + (card J)) + 1),SCMPDS))) +* (Start-At ((((card I) + (card J)) + 2),SCMPDS)) by A8, A9, Th24
.= (IExec (I,P,s)) +* (Start-At ((((card I) + (card J)) + 2),SCMPDS)) by MEMSTR_0:36 ;
:: thesis: verum