let P be Instruction-Sequence of SCM+FSA; :: thesis: for I, J being really-closed MacroInstruction of SCM+FSA
for a being read-write Int-Location
for s being State of SCM+FSA st s . a > 0 & I is_halting_on Initialized s,P holds
IExec ((if>0 (a,I,J)),P,s) = (IExec (I,P,s)) +* (Start-At ((((card I) + (card J)) + 3),SCM+FSA))
let I, J be really-closed MacroInstruction of SCM+FSA ; :: thesis: for a being read-write Int-Location
for s being State of SCM+FSA st s . a > 0 & I is_halting_on Initialized s,P holds
IExec ((if>0 (a,I,J)),P,s) = (IExec (I,P,s)) +* (Start-At ((((card I) + (card J)) + 3),SCM+FSA))
let a be read-write Int-Location; :: thesis: for s being State of SCM+FSA st s . a > 0 & I is_halting_on Initialized s,P holds
IExec ((if>0 (a,I,J)),P,s) = (IExec (I,P,s)) +* (Start-At ((((card I) + (card J)) + 3),SCM+FSA))
let s be State of SCM+FSA; :: thesis: ( s . a > 0 & I is_halting_on Initialized s,P implies IExec ((if>0 (a,I,J)),P,s) = (IExec (I,P,s)) +* (Start-At ((((card I) + (card J)) + 3),SCM+FSA)) )
set I1 = I ";" (Stop SCM+FSA);
set s1 = Initialized s;
set P1 = P +* (I ";" (Stop SCM+FSA));
set P3 = P +* (if>0 (a,I,J));
set s4 = Comput ((P +* (if>0 (a,I,J))),(Initialized s),1);
set i = a >0_goto ((card J) + 3);
A1: I ";" (Stop SCM+FSA) c= P +* (I ";" (Stop SCM+FSA)) by FUNCT_4:25;
A2: if>0 (a,I,J) = (((a >0_goto ((card J) + 3)) ";" J) ";" (Goto ((card I) + 1))) ";" (I ";" (Stop SCM+FSA)) by SCMFSA6A:25;
A3: 0 in dom (if>0 (a,I,J)) by AFINSQ_1:65;
A4: (P +* (if>0 (a,I,J))) . 0 = (if>0 (a,I,J)) . 0 by A3, FUNCT_4:13
.= a >0_goto ((card J) + 3) by Lm2 ;
A5: dom (Initialize ((intloc 0) .--> 1)) = {(intloc 0),(IC )} by SCMFSA_M:11;
( a <> intloc 0 & a <> IC ) by SCMFSA_2:56;
then A6: not a in dom (Initialize ((intloc 0) .--> 1)) by A5, TARSKI:def 2;
IC in dom (Initialize ((intloc 0) .--> 1)) by MEMSTR_0:48;
then A7: IC (Initialized s) = IC (Initialize ((intloc 0) .--> 1)) by FUNCT_4:13
.= 0 by MEMSTR_0:def 11 ;
A8: Comput ((P +* (if>0 (a,I,J))),(Initialized s),(0 + 1)) = Following ((P +* (if>0 (a,I,J))),(Comput ((P +* (if>0 (a,I,J))),(Initialized s),0))) by EXTPRO_1:3
.= Following ((P +* (if>0 (a,I,J))),(Initialized s))
.= Exec ((a >0_goto ((card J) + 3)),(Initialized s)) by A7, A4, PBOOLE:143 ;
A9: if>0 (a,I,J) c= P +* (if>0 (a,I,J)) by FUNCT_4:25;
assume s . a > 0 ; :: thesis: ( not I is_halting_on Initialized s,P or IExec ((if>0 (a,I,J)),P,s) = (IExec (I,P,s)) +* (Start-At ((((card I) + (card J)) + 3),SCM+FSA)) )
then (Initialized s) . a > 0 by A6, FUNCT_4:11;
then A10: IC (Comput ((P +* (if>0 (a,I,J))),(Initialized s),1)) = (card J) + 3 by A8, SCMFSA_2:71;
A11: for f being FinSeq-Location holds (Initialized s) . f = (Comput ((P +* (if>0 (a,I,J))),(Initialized s),1)) . f by A8, SCMFSA_2:71;
for a being Int-Location holds (Initialized s) . a = (Comput ((P +* (if>0 (a,I,J))),(Initialized s),1)) . a by A8, SCMFSA_2:71;
then A12: DataPart (Initialized s) = DataPart (Comput ((P +* (if>0 (a,I,J))),(Initialized s),1)) by A11, SCMFSA_M:2;
card (((a >0_goto ((card J) + 3)) ";" J) ";" (Goto ((card I) + 1))) = (card ((Macro (a >0_goto ((card J) + 3))) ";" J)) + (card (Goto ((card I) + 1))) by SCMFSA6A:21
.= (card ((Macro (a >0_goto ((card J) + 3))) ";" J)) + 1 by SCMFSA8A:15
.= ((card (Macro (a >0_goto ((card J) + 3)))) + (card J)) + 1 by SCMFSA6A:21
.= ((card J) + 2) + 1 by COMPOS_1:56
.= (card J) + (2 + 1) ;
then A13: Reloc ((I ";" (Stop SCM+FSA)),((card J) + 3)) c= if>0 (a,I,J) by A2, SCMFSA6A:38;
A14: Reloc ((I ";" (Stop SCM+FSA)),((card J) + 3)) c= P +* (if>0 (a,I,J)) by A13, A9, XBOOLE_1:1;
assume A15: I is_halting_on Initialized s,P ; :: thesis: IExec ((if>0 (a,I,J)),P,s) = (IExec (I,P,s)) +* (Start-At ((((card I) + (card J)) + 3),SCM+FSA))
then A16: P +* (I ";" (Stop SCM+FSA)) halts_on Initialized s by SCMFSA8A:34;
A17: CurInstr ((P +* (if>0 (a,I,J))),(Comput ((P +* (if>0 (a,I,J))),(Initialized s),((LifeSpan ((P +* (I ";" (Stop SCM+FSA))),(Initialized s))) + 1)))) = CurInstr ((P +* (if>0 (a,I,J))),(Comput ((P +* (if>0 (a,I,J))),(Comput ((P +* (if>0 (a,I,J))),(Initialized s),1)),(LifeSpan ((P +* (I ";" (Stop SCM+FSA))),(Initialized s)))))) by EXTPRO_1:4
.= IncAddr ((CurInstr ((P +* (I ";" (Stop SCM+FSA))),(Comput ((P +* (I ";" (Stop SCM+FSA))),(Initialized s),(LifeSpan ((P +* (I ";" (Stop SCM+FSA))),(Initialized s))))))),((card J) + 3)) by A10, A12, Th2, A14, A1
.= IncAddr ((halt SCM+FSA),((card J) + 3)) by A16, EXTPRO_1:def 15
.= halt SCM+FSA by COMPOS_0:4 ;
then A18: P +* (if>0 (a,I,J)) halts_on Initialized s by EXTPRO_1:29;
then for l being Nat st CurInstr ((P +* (if>0 (a,I,J))),(Comput ((P +* (if>0 (a,I,J))),(Initialized s),l))) = halt SCM+FSA holds
(LifeSpan ((P +* (I ";" (Stop SCM+FSA))),(Initialized s))) + 1 <= l ;
then A24: LifeSpan ((P +* (if>0 (a,I,J))),(Initialized s)) = (LifeSpan ((P +* (I ";" (Stop SCM+FSA))),(Initialized s))) + 1 by A17, A18, EXTPRO_1:def 15;
A25: DataPart (Result ((P +* (I ";" (Stop SCM+FSA))),(Initialized s))) = DataPart (Comput ((P +* (I ";" (Stop SCM+FSA))),(Initialized s),(LifeSpan ((P +* (I ";" (Stop SCM+FSA))),(Initialized s))))) by A15, EXTPRO_1:23, SCMFSA8A:34
.= DataPart (Comput ((P +* (if>0 (a,I,J))),(Comput ((P +* (if>0 (a,I,J))),(Initialized s),1)),(LifeSpan ((P +* (I ";" (Stop SCM+FSA))),(Initialized s))))) by A10, A12, Th2, A14, A1
.= DataPart (Comput ((P +* (if>0 (a,I,J))),(Initialized s),((LifeSpan ((P +* (I ";" (Stop SCM+FSA))),(Initialized s))) + 1))) by EXTPRO_1:4
.= DataPart (Result ((P +* (if>0 (a,I,J))),(Initialized s))) by A18, A24, EXTPRO_1:23 ;
dom (IExec ((if>0 (a,I,J)),P,s)) = the carrier of SCM+FSA by PARTFUN1:def 2
.= dom ((IExec ((I ";" (Stop SCM+FSA)),P,s)) +* (Start-At ((((card I) + (card J)) + 3),SCM+FSA))) by PARTFUN1:def 2 ;
hence IExec ((if>0 (a,I,J)),P,s) = (IExec ((I ";" (Stop SCM+FSA)),P,s)) +* (Start-At ((((card I) + (card J)) + 3),SCM+FSA)) by A26, FUNCT_1:2
.= ((IExec (I,P,s)) +* (Start-At ((card I),SCM+FSA))) +* (Start-At ((((card I) + (card J)) + 3),SCM+FSA)) by A15, SCMFSA8A:36
.= (IExec (I,P,s)) +* (Start-At ((((card I) + (card J)) + 3),SCM+FSA)) by FUNCT_4:114 ;
:: thesis: verum