set J3 = (Goto 0) ';' (Stop SCM+FSA);
set J = Stop SCM+FSA;
let a be Int-Location ; :: thesis:
let I be Program of SCM+FSA; :: thesis:
let s be State of SCM+FSA; :: thesis:
set s1 = s +* ((while>0 (a,I)) +* (Start-At (0,SCM+FSA)));
set sI = s +* (I +* (Start-At (0,SCM+FSA)));
set life = LifeSpan ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(s +* (I +* (Start-At (0,SCM+FSA)))));
set sK1 = Comput ((ProgramPart (s +* ((while>0 (a,I)) +* (Start-At (0,SCM+FSA))))),(s +* ((while>0 (a,I)) +* (Start-At (0,SCM+FSA)))),(1 + (LifeSpan ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(s +* (I +* (Start-At (0,SCM+FSA))))))));
set sK2 = Comput ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(s +* (I +* (Start-At (0,SCM+FSA)))),(LifeSpan ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(s +* (I +* (Start-At (0,SCM+FSA)))))));
set I1 = I ';' (Goto 0);
set i = a >0_goto ((card (Stop SCM+FSA)) + 3);
reconsider n = IC (Comput ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(s +* (I +* (Start-At (0,SCM+FSA)))),(LifeSpan ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(s +* (I +* (Start-At (0,SCM+FSA)))))))) as Element of NAT ;
set Mi = ((a >0_goto ((card (Stop SCM+FSA)) + 3)) ';' (Stop SCM+FSA)) ';' (Goto ((card (I ';' (Goto 0))) + 1));
set J2 = (I ';' (Goto 0)) ';' (Stop SCM+FSA);
while>0 (a,I) c= (while>0 (a,I)) +* (Start-At (0,SCM+FSA)) by SCMFSA8A:9;
then A1: dom (while>0 (a,I)) c= dom ((while>0 (a,I)) +* (Start-At (0,SCM+FSA))) by GRFUNC_1:8;
I c= I +* (Start-At (0,SCM+FSA)) by SCMFSA8A:9;
then A2: dom I c= dom (I +* (Start-At (0,SCM+FSA))) by GRFUNC_1:8;
assume I is_closed_on s ; :: thesis:
then A3: n in dom I by SCMFSA7B:def 7;
then n < card I by AFINSQ_1:70;
then A4: n + 4 < (card I) + 6 by XREAL_1:10;
Y: (ProgramPart (Comput ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(s +* (I +* (Start-At (0,SCM+FSA)))),(LifeSpan ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(s +* (I +* (Start-At (0,SCM+FSA))))))))) /. (IC (Comput ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(s +* (I +* (Start-At (0,SCM+FSA)))),(LifeSpan ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(s +* (I +* (Start-At (0,SCM+FSA))))))))) = (Comput ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(s +* (I +* (Start-At (0,SCM+FSA)))),(LifeSpan ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(s +* (I +* (Start-At (0,SCM+FSA)))))))) . (IC (Comput ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(s +* (I +* (Start-At (0,SCM+FSA)))),(LifeSpan ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(s +* (I +* (Start-At (0,SCM+FSA))))))))) by COMPOS_1:38;
TX: ProgramPart (Comput ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(s +* (I +* (Start-At (0,SCM+FSA)))),(LifeSpan ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(s +* (I +* (Start-At (0,SCM+FSA)))))))) = ProgramPart (s +* (I +* (Start-At (0,SCM+FSA)))) by AMI_1:123;
assume I is_halting_on s ; :: thesis:
then A5: ProgramPart (s +* (I +* (Start-At (0,SCM+FSA)))) halts_on s +* (I +* (Start-At (0,SCM+FSA))) by SCMFSA7B:def 8;
CurInstr ((ProgramPart (Comput ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(s +* (I +* (Start-At (0,SCM+FSA)))),(LifeSpan ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(s +* (I +* (Start-At (0,SCM+FSA))))))))),(Comput ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(s +* (I +* (Start-At (0,SCM+FSA)))),(LifeSpan ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(s +* (I +* (Start-At (0,SCM+FSA))))))))) = (Comput ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(s +* (I +* (Start-At (0,SCM+FSA)))),(LifeSpan ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(s +* (I +* (Start-At (0,SCM+FSA)))))))) . n by Y
.= (s +* (I +* (Start-At (0,SCM+FSA)))) . n by AMI_1:54
.= (I +* (Start-At (0,SCM+FSA))) . n by A3, A2, FUNCT_4:14
.= I . n by A3, COMPOS_1:145 ;
then A6: I . n = halt SCM+FSA by A5, TX, EXTPRO_1:def 14;
A7: (I ';' (Goto 0)) ';' (Stop SCM+FSA) = I ';' ((Goto 0) ';' (Stop SCM+FSA)) by SCMFSA6A:67;
then dom ((I ';' (Goto 0)) ';' (Stop SCM+FSA)) = (dom (Directed I)) \/ (dom (ProgramPart (Relocated (((Goto 0) ';' (Stop SCM+FSA)),(card I))))) by FUNCT_4:def 1
.= (dom I) \/ (dom (ProgramPart (Relocated (((Goto 0) ';' (Stop SCM+FSA)),(card I))))) by FUNCT_4:105 ;
then A8: n in dom ((I ';' (Goto 0)) ';' (Stop SCM+FSA)) by A3, XBOOLE_0:def 3;
then n + 4 in { (il + 4) where il is Element of NAT : il in dom ((I ';' (Goto 0)) ';' (Stop SCM+FSA)) } ;
then A9: n + 4 in dom (Shift (((I ';' (Goto 0)) ';' (Stop SCM+FSA)),4)) by VALUED_1:def 12;
then A10: (Shift (((I ';' (Goto 0)) ';' (Stop SCM+FSA)),4)) /. (n + 4) = (Shift (((I ';' (Goto 0)) ';' (Stop SCM+FSA)),4)) . (n + 4) by PARTFUN1:def 8
.= ((I ';' (Goto 0)) ';' (Stop SCM+FSA)) . n by A8, VALUED_1:def 12
.= (Directed I) . n by A3, A7, SCMFSA8A:28
.= goto (card I) by A3, A6, SCMFSA8A:30 ;
set f = ((card I) + 4) .--> (goto 0);
assume A11: IC (Comput ((ProgramPart (s +* ((while>0 (a,I)) +* (Start-At (0,SCM+FSA))))),(s +* ((while>0 (a,I)) +* (Start-At (0,SCM+FSA)))),(1 + (LifeSpan ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(s +* (I +* (Start-At (0,SCM+FSA))))))))) = (IC (Comput ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(s +* (I +* (Start-At (0,SCM+FSA)))),(LifeSpan ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(s +* (I +* (Start-At (0,SCM+FSA))))))))) + 4 ; :: thesis:
( dom (((card I) + 4) .--> (goto 0)) = {((card I) + 4)} & n + 4 <> (card I) + 4 ) by A3, FUNCOP_1:19;
then A12: not n + 4 in dom (((card I) + 4) .--> (goto 0)) by TARSKI:def 1;
A13: card (((a >0_goto ((card (Stop SCM+FSA)) + 3)) ';' (Stop SCM+FSA)) ';' (Goto ((card (I ';' (Goto 0))) + 1))) = (card ((a >0_goto ((card (Stop SCM+FSA)) + 3)) ';' (Stop SCM+FSA))) + (card (Goto ((card (I ';' (Goto 0))) + 1))) by SCMFSA6A:61
.= (card ((a >0_goto ((card (Stop SCM+FSA)) + 3)) ';' (Stop SCM+FSA))) + 1 by SCMFSA8A:29
.= ((card (Macro (a >0_goto ((card (Stop SCM+FSA)) + 3)))) + (card (Stop SCM+FSA))) + 1 by SCMFSA6A:61
.= (2 + 1) + 1 by LL, COMPOS_1:150
.= 3 + 1 ;
then n + 4 >= card (((a >0_goto ((card (Stop SCM+FSA)) + 3)) ';' (Stop SCM+FSA)) ';' (Goto ((card (I ';' (Goto 0))) + 1))) by NAT_1:11;
then A14: not n + 4 in dom (((a >0_goto ((card (Stop SCM+FSA)) + 3)) ';' (Stop SCM+FSA)) ';' (Goto ((card (I ';' (Goto 0))) + 1))) by AFINSQ_1:70;
card (while>0 (a,I)) = (card I) + 6 by Th5;
then A15: n + 4 in dom (while>0 (a,I)) by A4, AFINSQ_1:70;
A16: dom (while>0 (a,I)) = (dom (if>0 (a,(I ';' (Goto 0)),(Stop SCM+FSA)))) \/ (dom (((card I) + 4) .--> (goto 0))) by FUNCT_4:def 1;
then A17: n + 4 in dom (if>0 (a,(I ';' (Goto 0)),(Stop SCM+FSA))) by A12, A15, XBOOLE_0:def 3;
A18: if>0 (a,(I ';' (Goto 0)),(Stop SCM+FSA)) = ((((a >0_goto ((card (Stop SCM+FSA)) + 3)) ';' (Stop SCM+FSA)) ';' (Goto ((card (I ';' (Goto 0))) + 1))) ';' (I ';' (Goto 0))) ';' (Stop SCM+FSA) by SCMFSA8B:def 2
.= (((a >0_goto ((card (Stop SCM+FSA)) + 3)) ';' (Stop SCM+FSA)) ';' (Goto ((card (I ';' (Goto 0))) + 1))) ';' ((I ';' (Goto 0)) ';' (Stop SCM+FSA)) by SCMFSA6A:67
.= (Directed (((a >0_goto ((card (Stop SCM+FSA)) + 3)) ';' (Stop SCM+FSA)) ';' (Goto ((card (I ';' (Goto 0))) + 1)))) +* (ProgramPart (Relocated (((I ';' (Goto 0)) ';' (Stop SCM+FSA)),4))) by A13 ;
then A19: dom (if>0 (a,(I ';' (Goto 0)),(Stop SCM+FSA))) = (dom (Directed (((a >0_goto ((card (Stop SCM+FSA)) + 3)) ';' (Stop SCM+FSA)) ';' (Goto ((card (I ';' (Goto 0))) + 1))))) \/ (dom (ProgramPart (Relocated (((I ';' (Goto 0)) ';' (Stop SCM+FSA)),4)))) by FUNCT_4:def 1;
then dom (if>0 (a,(I ';' (Goto 0)),(Stop SCM+FSA))) = (dom (((a >0_goto ((card (Stop SCM+FSA)) + 3)) ';' (Stop SCM+FSA)) ';' (Goto ((card (I ';' (Goto 0))) + 1)))) \/ (dom (ProgramPart (Relocated (((I ';' (Goto 0)) ';' (Stop SCM+FSA)),4)))) by FUNCT_4:105;
then A20: n + 4 in dom (ProgramPart (Relocated (((I ';' (Goto 0)) ';' (Stop SCM+FSA)),4))) by A17, A14, XBOOLE_0:def 3;
Z: (ProgramPart (Comput ((ProgramPart (s +* ((while>0 (a,I)) +* (Start-At (0,SCM+FSA))))),(s +* ((while>0 (a,I)) +* (Start-At (0,SCM+FSA)))),(1 + (LifeSpan ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(s +* (I +* (Start-At (0,SCM+FSA)))))))))) /. (IC (Comput ((ProgramPart (s +* ((while>0 (a,I)) +* (Start-At (0,SCM+FSA))))),(s +* ((while>0 (a,I)) +* (Start-At (0,SCM+FSA)))),(1 + (LifeSpan ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(s +* (I +* (Start-At (0,SCM+FSA)))))))))) = (Comput ((ProgramPart (s +* ((while>0 (a,I)) +* (Start-At (0,SCM+FSA))))),(s +* ((while>0 (a,I)) +* (Start-At (0,SCM+FSA)))),(1 + (LifeSpan ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(s +* (I +* (Start-At (0,SCM+FSA))))))))) . (IC (Comput ((ProgramPart (s +* ((while>0 (a,I)) +* (Start-At (0,SCM+FSA))))),(s +* ((while>0 (a,I)) +* (Start-At (0,SCM+FSA)))),(1 + (LifeSpan ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(s +* (I +* (Start-At (0,SCM+FSA)))))))))) by COMPOS_1:38;
(Comput ((ProgramPart (s +* ((while>0 (a,I)) +* (Start-At (0,SCM+FSA))))),(s +* ((while>0 (a,I)) +* (Start-At (0,SCM+FSA)))),(1 + (LifeSpan ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(s +* (I +* (Start-At (0,SCM+FSA))))))))) . (n + 4) = (s +* ((while>0 (a,I)) +* (Start-At (0,SCM+FSA)))) . (n + 4) by AMI_1:54
.= ((while>0 (a,I)) +* (Start-At (0,SCM+FSA))) . (n + 4) by A15, A1, FUNCT_4:14
.= ((if>0 (a,(I ';' (Goto 0)),(Stop SCM+FSA))) +* (((card I) + 4) .--> (goto 0))) . (n + 4) by A15, COMPOS_1:145
.= ((Directed (((a >0_goto ((card (Stop SCM+FSA)) + 3)) ';' (Stop SCM+FSA)) ';' (Goto ((card (I ';' (Goto 0))) + 1)))) +* (ProgramPart (Relocated (((I ';' (Goto 0)) ';' (Stop SCM+FSA)),4)))) . (n + 4) by A12, A15, A16, A18, FUNCT_4:def 1
.= (ProgramPart (Relocated (((I ';' (Goto 0)) ';' (Stop SCM+FSA)),4))) . (n + 4) by A17, A19, A20, FUNCT_4:def 1
.= (Reloc ((ProgramPart ((I ';' (Goto 0)) ';' (Stop SCM+FSA))),4)) . (n + 4) by COMPOS_1:116
.= (Reloc (((I ';' (Goto 0)) ';' (Stop SCM+FSA)),4)) . (n + 4) by RELAT_1:209
.= IncAddr ((goto (card I)),4) by A9, A10, SCMFSA_4:24
.= goto ((card I) + 4) by SCMFSA_4:14 ;
hence CurInstr ((ProgramPart (Comput ((ProgramPart (s +* ((while>0 (a,I)) +* (Start-At (0,SCM+FSA))))),(s +* ((while>0 (a,I)) +* (Start-At (0,SCM+FSA)))),(1 + (LifeSpan ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(s +* (I +* (Start-At (0,SCM+FSA)))))))))),(Comput ((ProgramPart (s +* ((while>0 (a,I)) +* (Start-At (0,SCM+FSA))))),(s +* ((while>0 (a,I)) +* (Start-At (0,SCM+FSA)))),(1 + (LifeSpan ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(s +* (I +* (Start-At (0,SCM+FSA)))))))))) = goto ((card I) + 4) by A11, Z; :: thesis: