let s be State of SCM+FSA ; :: thesis:
set D = Int-Locations \/ FinSeq-Locations ;
set A = NAT ;
let I be InitHalting keepInt0_1 Program of SCM+FSA ; :: thesis:
let J be InitHalting Program of SCM+FSA ; :: thesis:
set ps = s | NAT ;
set s1 = s +* (Initialized I);
set s2 = s +* (Initialized (I ';' J));
set s3 = (Computation (s +* (Initialized I)),(LifeSpan (s +* (Initialized I)))) +* (Initialized J);
set m1 = LifeSpan (s +* (Initialized I));
set m3 = LifeSpan ((Computation (s +* (Initialized I)),(LifeSpan (s +* (Initialized I)))) +* (Initialized J));
A1: Initialized (I ';' J) c= s +* (Initialized (I ';' J)) by FUNCT_4:26;
s +* (Initialized (I ';' J)) = s +* ((I ';' J) +* (((intloc 0 ) .--> 1) +* (Start-At (insloc 0 )))) by FUNCT_4:15
.= (s +* (I ';' J)) +* (((intloc 0 ) .--> 1) +* (Start-At (insloc 0 ))) by FUNCT_4:15 ;
then A2: ((intloc 0 ) .--> 1) +* (Start-At (insloc 0 )) c= s +* (Initialized (I ';' J)) by FUNCT_4:26;
A3: (s +* (Initialized I)) +* (I ';' J) = s +* ((Initialized I) +* (I ';' J)) by FUNCT_4:15
.= s +* (Initialized (I ';' J)) by SCMFSA6A:58 ;
A4: s +* (Initialized I) is halting by Th5, FUNCT_4:26;
A5: Initialized J c= (Computation (s +* (Initialized I)),(LifeSpan (s +* (Initialized I)))) +* (Initialized J) by FUNCT_4:26;
A6: (Computation (s +* (Initialized I)),(LifeSpan (s +* (Initialized I)))) +* (Initialized J),((Computation (s +* (Initialized I)),(LifeSpan (s +* (Initialized I)))) +* (s | NAT )) +* (Initialized J) equal_outside dom (s | NAT ) by FUNCT_7:31, FUNCT_7:106;
then A7: ((Computation (s +* (Initialized I)),(LifeSpan (s +* (Initialized I)))) +* (s | NAT )) +* (Initialized J),(Computation (s +* (Initialized I)),(LifeSpan (s +* (Initialized I)))) +* (Initialized J) equal_outside dom (s | NAT ) by FUNCT_7:28;
A8: ( IExec I,s = (Result (s +* (Initialized I))) +* (s | NAT ) & Initialized J c= (Result (s +* (Initialized I))) +* (Initialized J) ) by FUNCT_4:26, SCMFSA6B:def 1;
A9: dom (s | NAT ) = (dom s) /\ NAT by RELAT_1:90
.= (((Int-Locations \/ FinSeq-Locations ) \/ {(IC SCM+FSA )}) \/ NAT ) /\ NAT by AMI_1:79, SCMFSA_2:8
.= NAT by XBOOLE_1:21 ;
A10: Initialized J c= (IExec I,s) +* (Initialized J) by FUNCT_4:26;
A11: ( Initialized J c= (IExec I,s) +* (Initialized J) & Initialized J c= (Computation (s +* (Initialized I)),(LifeSpan (s +* (Initialized I)))) +* (Initialized J) ) by FUNCT_4:26;
A12: Initialized J c= (Computation (s +* (Initialized I)),(LifeSpan (s +* (Initialized I)))) +* (Initialized J) by FUNCT_4:26;
A13: Initialized I c= s +* (Initialized I) by FUNCT_4:26;
then A14: (Computation (s +* (Initialized I)),(LifeSpan (s +* (Initialized I)))) +* (Initialized J) = (Result (s +* (Initialized I))) +* (Initialized J) by Th5, AMI_1:122;
Result (s +* (Initialized I)) = Computation (s +* (Initialized I)),(LifeSpan (s +* (Initialized I))) by A13, Th5, AMI_1:122;
then A15: IC (Result ((Result (s +* (Initialized I))) +* (Initialized J))) = IC (Result ((IExec I,s) +* (Initialized J))) by A9, A6, A8, A10, Th15, AMI_1:121;
A16: Initialized I c= (s +* (Initialized (I ';' J))) +* I by FUNCT_4:26, SCMFSA6A:52;
( s +* (Initialized I),s +* (Initialized (I ';' J)) equal_outside NAT & s +* (Initialized (I ';' J)),(s +* (Initialized (I ';' J))) +* I equal_outside NAT ) by AMI_1:120, SCMFSA6A:53;
then s +* (Initialized I),(s +* (Initialized (I ';' J))) +* I equal_outside NAT by FUNCT_7:29;
then A17: LifeSpan ((s +* (Initialized (I ';' J))) +* I) = LifeSpan (s +* (Initialized I)) by A13, A16, Th15;
then A18: ProgramPart (Relocated J,(card I)) c= Computation (s +* (Initialized (I ';' J))),((LifeSpan (s +* (Initialized I))) + 1) by A1, Th25;
(s +* (Initialized (I ';' J))) +* (I +* (((intloc 0 ) .--> 1) +* (Start-At (insloc 0 )))) = ((s +* (Initialized (I ';' J))) +* I) +* (((intloc 0 ) .--> 1) +* (Start-At (insloc 0 ))) by FUNCT_4:15
.= ((s +* (Initialized (I ';' J))) +* (((intloc 0 ) .--> 1) +* (Start-At (insloc 0 )))) +* I by Th19
.= (s +* (Initialized (I ';' J))) +* I by A2, FUNCT_4:79 ;
then I +* (((intloc 0 ) .--> 1) +* (Start-At (insloc 0 ))) c= (s +* (Initialized (I ';' J))) +* I by FUNCT_4:26;
then Initialized I c= (s +* (Initialized (I ';' J))) +* I by FUNCT_4:15;
then (s +* (Initialized (I ';' J))) +* I is halting by Th5;
then DataPart (Computation ((s +* (Initialized (I ';' J))) +* I),(LifeSpan (s +* (Initialized I)))) = DataPart (Computation (((s +* (Initialized (I ';' J))) +* I) +* (I ';' J)),(LifeSpan (s +* (Initialized I)))) by A16, A17, Th18, SCMFSA6A:39
.= DataPart (Computation ((s +* (Initialized (I ';' J))) +* (I +* (I ';' J))),(LifeSpan (s +* (Initialized I)))) by FUNCT_4:15
.= DataPart (Computation ((s +* (Initialized (I ';' J))) +* (I ';' J)),(LifeSpan (s +* (Initialized I)))) by SCMFSA6A:57
.= DataPart (Computation (s +* ((Initialized (I ';' J)) +* (I ';' J))),(LifeSpan (s +* (Initialized I)))) by FUNCT_4:15
.= DataPart (Computation (s +* (Initialized (I ';' J))),(LifeSpan (s +* (Initialized I)))) by LATTICE2:8, SCMFSA6A:26
.= DataPart (Computation (s +* (Initialized I)),(LifeSpan (s +* (Initialized I)))) by A13, A4, A3, Th18, SCMFSA6A:39 ;
then A19: DataPart ((Computation ((s +* (Initialized (I ';' J))) +* I),(LifeSpan (s +* (Initialized I)))) +* (Initialized J)) = (DataPart (Computation (s +* (Initialized I)),(LifeSpan (s +* (Initialized I))))) +* (DataPart (Initialized J)) by FUNCT_4:75
.= DataPart ((Computation (s +* (Initialized I)),(LifeSpan (s +* (Initialized I)))) +* (Initialized J)) by FUNCT_4:75 ;
A20: ( IC (Computation (s +* (Initialized (I ';' J))),((LifeSpan (s +* (Initialized I))) + 1)) = insloc (card I) & DataPart (Computation (s +* (Initialized (I ';' J))),((LifeSpan (s +* (Initialized I))) + 1)) = DataPart ((Computation ((s +* (Initialized (I ';' J))) +* I),(LifeSpan (s +* (Initialized I)))) +* (Initialized J)) ) by A1, A17, Th25;
then A21: DataPart (Computation (Computation (s +* (Initialized (I ';' J))),((LifeSpan (s +* (Initialized I))) + 1)),(LifeSpan ((Computation (s +* (Initialized I)),(LifeSpan (s +* (Initialized I)))) +* (Initialized J)))) = DataPart (Computation ((Computation (s +* (Initialized I)),(LifeSpan (s +* (Initialized I)))) +* (Initialized J)),(LifeSpan ((Computation (s +* (Initialized I)),(LifeSpan (s +* (Initialized I)))) +* (Initialized J)))) by A18, A5, A19, Th12;
A22: IC (Computation (Computation (s +* (Initialized (I ';' J))),((LifeSpan (s +* (Initialized I))) + 1)),(LifeSpan ((Computation (s +* (Initialized I)),(LifeSpan (s +* (Initialized I)))) +* (Initialized J)))) = (IC (Computation ((Computation (s +* (Initialized I)),(LifeSpan (s +* (Initialized I)))) +* (Initialized J)),(LifeSpan ((Computation (s +* (Initialized I)),(LifeSpan (s +* (Initialized I)))) +* (Initialized J))))) + (card I) by A20, A18, A5, A19, Th12;
A23: IExec (I ';' J),s = (Result (s +* (Initialized (I ';' J)))) +* (s | NAT ) by SCMFSA6B:def 1
.= (Computation (s +* (Initialized (I ';' J))),(LifeSpan (s +* (Initialized (I ';' J))))) +* (s | NAT ) by A1, Th5, AMI_1:122
.= (Computation (s +* (Initialized (I ';' J))),(((LifeSpan (s +* (Initialized I))) + 1) + (LifeSpan ((Computation (s +* (Initialized I)),(LifeSpan (s +* (Initialized I)))) +* (Initialized J))))) +* (s | NAT ) by A14, Th28 ;
IExec I,s = (Result (s +* (Initialized I))) +* (s | NAT ) by SCMFSA6B:def 1
.= (Computation (s +* (Initialized I)),(LifeSpan (s +* (Initialized I)))) +* (s | NAT ) by A13, Th5, AMI_1:122 ;
then Result ((IExec I,s) +* (Initialized J)), Result ((Computation (s +* (Initialized I)),(LifeSpan (s +* (Initialized I)))) +* (Initialized J)) equal_outside NAT by A9, A7, A11, Th15;
then A24: (Result ((IExec I,s) +* (Initialized J))) +* (s | NAT ) = (Result ((Computation (s +* (Initialized I)),(LifeSpan (s +* (Initialized I)))) +* (Initialized J))) +* (s | NAT ) by A9, FUNCT_7:108;
(IExec I,s) | NAT = ((Result (s +* (Initialized I))) +* (s | NAT )) | NAT by SCMFSA6B:def 1
.= s | NAT by CARD_3:99 ;
then A25: IExec J,(IExec I,s) = (Result ((IExec I,s) +* (Initialized J))) +* (s | NAT ) by SCMFSA6B:def 1
.= (Computation ((Computation (s +* (Initialized I)),(LifeSpan (s +* (Initialized I)))) +* (Initialized J)),(LifeSpan ((Computation (s +* (Initialized I)),(LifeSpan (s +* (Initialized I)))) +* (Initialized J)))) +* (s | NAT ) by A12, A24, Th5, AMI_1:122 ;
A26: dom (s | NAT ) misses Int-Locations \/ FinSeq-Locations by A9, SCMFSA_2:13, SCMFSA_2:14, XBOOLE_1:70;
then A27: DataPart (IExec (I ';' J),s) = DataPart (Computation (s +* (Initialized (I ';' J))),(((LifeSpan (s +* (Initialized I))) + 1) + (LifeSpan ((Computation (s +* (Initialized I)),(LifeSpan (s +* (Initialized I)))) +* (Initialized J))))) by A23, FUNCT_4:76, SCMFSA_2:127
.= DataPart (Computation ((Computation (s +* (Initialized I)),(LifeSpan (s +* (Initialized I)))) +* (Initialized J)),(LifeSpan ((Computation (s +* (Initialized I)),(LifeSpan (s +* (Initialized I)))) +* (Initialized J)))) by A21, AMI_1:51
.= DataPart (IExec J,(IExec I,s)) by A25, A26, FUNCT_4:76, SCMFSA_2:127 ;
A28: IC (IExec (I ';' J),s) = IC (Result (s +* (Initialized (I ';' J)))) by SCMFSA8A:7
.= IC (Computation (s +* (Initialized (I ';' J))),(LifeSpan (s +* (Initialized (I ';' J))))) by A1, Th5, AMI_1:122
.= IC (Computation (s +* (Initialized (I ';' J))),(((LifeSpan (s +* (Initialized I))) + 1) + (LifeSpan ((Computation (s +* (Initialized I)),(LifeSpan (s +* (Initialized I)))) +* (Initialized J))))) by A14, Th28
.= (IC (Computation ((Computation (s +* (Initialized I)),(LifeSpan (s +* (Initialized I)))) +* (Initialized J)),(LifeSpan ((Computation (s +* (Initialized I)),(LifeSpan (s +* (Initialized I)))) +* (Initialized J))))) + (card I) by A22, AMI_1:51
.= (IC (Result ((Computation (s +* (Initialized I)),(LifeSpan (s +* (Initialized I)))) +* (Initialized J)))) + (card I) by A12, Th5, AMI_1:122
.= (IC (Result ((Result (s +* (Initialized I))) +* (Initialized J)))) + (card I) by A13, Th5, AMI_1:122
.= (IC (IExec J,(IExec I,s))) + (card I) by A15, SCMFSA8A:7 ;

hereby :: thesis:

reconsider l = (IC (IExec J,(IExec I,s))) + (card I) as Instruction-Location of SCM+FSA ;
A29: dom (Start-At l) = {(IC SCM+FSA )} by FUNCOP_1:19;

A30: now

let x be set ; :: thesis:
assume A31: x in dom (IExec (I ';' J),s) ; :: thesis:

per cases by A31, SCMFSA6A:35;

suppose A32: x is Int-Location ; :: thesis:

then x <> IC SCM+FSA by SCMFSA_2:81;
then A33: not x in dom (Start-At l) by A29, TARSKI:def 1;
(IExec (I ';' J),s) . x = (IExec J,(IExec I,s)) . x by A27, A32, SCMFSA6A:38;
hence (IExec (I ';' J),s) . x = ((IExec J,(IExec I,s)) +* (Start-At ((IC (IExec J,(IExec I,s))) + (card I)))) . x by A33, FUNCT_4:12; :: thesis:

end;

suppose A34: x is FinSeq-Location ; :: thesis:

then x <> IC SCM+FSA by SCMFSA_2:82;
then A35: not x in dom (Start-At l) by A29, TARSKI:def 1;
(IExec (I ';' J),s) . x = (IExec J,(IExec I,s)) . x by A27, A34, SCMFSA6A:38;
hence (IExec (I ';' J),s) . x = ((IExec J,(IExec I,s)) +* (Start-At ((IC (IExec J,(IExec I,s))) + (card I)))) . x by A35, FUNCT_4:12; :: thesis:

end;

suppose A36: x = IC SCM+FSA ; :: thesis:

then x in {(IC SCM+FSA )} by TARSKI:def 1;
then A37: x in dom (Start-At l) by FUNCOP_1:19;
thus (IExec (I ';' J),s) . x = (Start-At l) . (IC SCM+FSA ) by A28, A36, FUNCOP_1:87
.= ((IExec J,(IExec I,s)) +* (Start-At ((IC (IExec J,(IExec I,s))) + (card I)))) . x by A36, A37, FUNCT_4:14 ; :: thesis:

end;

suppose A38: x is Instruction-Location of SCM+FSA ; :: thesis:

then x <> IC SCM+FSA by AMI_1:48;
then A39: not x in dom (Start-At l) by A29, TARSKI:def 1;
(IExec (I ';' J),s) | NAT = s | NAT by A23, CARD_3:99
.= (IExec J,(IExec I,s)) | NAT by A25, CARD_3:99 ;
then (IExec (I ';' J),s) . x = (IExec J,(IExec I,s)) . x by A38, SCMFSA6A:36;
hence (IExec (I ';' J),s) . x = ((IExec J,(IExec I,s)) +* (Start-At ((IC (IExec J,(IExec I,s))) + (card I)))) . x by A39, FUNCT_4:12; :: thesis:

end;

dom (IExec (I ';' J),s) = the carrier of SCM+FSA by AMI_1:79
.= dom ((IExec J,(IExec I,s)) +* (Start-At ((IC (IExec J,(IExec I,s))) + (card I)))) by AMI_1:79 ;
hence IExec (I ';' J),s = (IExec J,(IExec I,s)) +* (Start-At ((IC (IExec J,(IExec I,s))) + (card I))) by A30, FUNCT_1:9; :: thesis:

end;