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 (insloc 0 )));
set sI = s +* (I +* (Start-At (insloc 0 )));
set life = LifeSpan (s +* (I +* (Start-At (insloc 0 ))));
set sK1 = Computation (s +* ((while=0 a,I) +* (Start-At (insloc 0 )))),(1 + (LifeSpan (s +* (I +* (Start-At (insloc 0 ))))));
set sK2 = Computation (s +* (I +* (Start-At (insloc 0 )))),(LifeSpan (s +* (I +* (Start-At (insloc 0 )))));
assume A1: I is_closed_on s ; :: thesis:
assume A2: I is_halting_on s ; :: thesis:
assume A3: IC (Computation (s +* ((while=0 a,I) +* (Start-At (insloc 0 )))),(1 + (LifeSpan (s +* (I +* (Start-At (insloc 0 ))))))) = (IC (Computation (s +* (I +* (Start-At (insloc 0 )))),(LifeSpan (s +* (I +* (Start-At (insloc 0 ))))))) + 4 ; :: thesis:
set I1 = I ';' (Goto (insloc 0 ));
set J = Stop SCM+FSA ;
set f = (insloc ((card I) + 4)) .--> (goto (insloc 0 ));
set i = a =0_goto (insloc ((card (Stop SCM+FSA )) + 3));
A4: dom ((insloc ((card I) + 4)) .--> (goto (insloc 0 ))) = {(insloc ((card I) + 4))} by FUNCOP_1:19;
reconsider n = IC (Computation (s +* (I +* (Start-At (insloc 0 )))),(LifeSpan (s +* (I +* (Start-At (insloc 0 )))))) as Element of NAT by ORDINAL1:def 13;
A6: insloc n in dom I by A1, SCMFSA7B:def 7;
then A7: n < card I by SCMFSA6A:15;
n + 4 <> (card I) + 4 by A6, SCMFSA6A:15;
then A8: not insloc (n + 4) in dom ((insloc ((card I) + 4)) .--> (goto (insloc 0 ))) by A4, TARSKI:def 1;
A9: n + 4 < (card I) + 6 by A7, XREAL_1:10;
card (while=0 a,I) = (card I) + 6 by Th4;
then A10: insloc (n + 4) in dom (while=0 a,I) by A9, SCMFSA6A:15;
A11: dom (while=0 a,I) = (dom (if=0 a,(I ';' (Goto (insloc 0 ))),(Stop SCM+FSA ))) \/ (dom ((insloc ((card I) + 4)) .--> (goto (insloc 0 )))) by FUNCT_4:def 1;
then A12: insloc (n + 4) in dom (if=0 a,(I ';' (Goto (insloc 0 ))),(Stop SCM+FSA )) by A8, A10, XBOOLE_0:def 3;
set Mi = ((a =0_goto (insloc ((card (Stop SCM+FSA )) + 3))) ';' (Stop SCM+FSA )) ';' (Goto (insloc ((card (I ';' (Goto (insloc 0 )))) + 1)));
set J2 = (I ';' (Goto (insloc 0 ))) ';' (Stop SCM+FSA );
set J3 = (Goto (insloc 0 )) ';' (Stop SCM+FSA );
A13: card (((a =0_goto (insloc ((card (Stop SCM+FSA )) + 3))) ';' (Stop SCM+FSA )) ';' (Goto (insloc ((card (I ';' (Goto (insloc 0 )))) + 1)))) = (card ((a =0_goto (insloc ((card (Stop SCM+FSA )) + 3))) ';' (Stop SCM+FSA ))) + (card (Goto (insloc ((card (I ';' (Goto (insloc 0 )))) + 1)))) by SCMFSA6A:61
.= (card ((a =0_goto (insloc ((card (Stop SCM+FSA )) + 3))) ';' (Stop SCM+FSA ))) + 1 by SCMFSA8A:29
.= ((card (Macro (a =0_goto (insloc ((card (Stop SCM+FSA )) + 3))))) + (card (Stop SCM+FSA ))) + 1 by SCMFSA6A:61
.= (2 + 1) + 1 by SCMFSA7B:6, SCMFSA8A:17
.= 3 + 1 ;
then n + 4 >= card (((a =0_goto (insloc ((card (Stop SCM+FSA )) + 3))) ';' (Stop SCM+FSA )) ';' (Goto (insloc ((card (I ';' (Goto (insloc 0 )))) + 1)))) by NAT_1:11;
then A14: not insloc (n + 4) in dom (((a =0_goto (insloc ((card (Stop SCM+FSA )) + 3))) ';' (Stop SCM+FSA )) ';' (Goto (insloc ((card (I ';' (Goto (insloc 0 )))) + 1)))) by SCMFSA6A:15;
A15: if=0 a,(I ';' (Goto (insloc 0 ))),(Stop SCM+FSA ) = ((((a =0_goto (insloc ((card (Stop SCM+FSA )) + 3))) ';' (Stop SCM+FSA )) ';' (Goto (insloc ((card (I ';' (Goto (insloc 0 )))) + 1)))) ';' (I ';' (Goto (insloc 0 )))) ';' (Stop SCM+FSA ) by SCMFSA8B:def 1
.= (((a =0_goto (insloc ((card (Stop SCM+FSA )) + 3))) ';' (Stop SCM+FSA )) ';' (Goto (insloc ((card (I ';' (Goto (insloc 0 )))) + 1)))) ';' ((I ';' (Goto (insloc 0 ))) ';' (Stop SCM+FSA )) by SCMFSA6A:67
.= (Directed (((a =0_goto (insloc ((card (Stop SCM+FSA )) + 3))) ';' (Stop SCM+FSA )) ';' (Goto (insloc ((card (I ';' (Goto (insloc 0 )))) + 1))))) +* (ProgramPart (Relocated ((I ';' (Goto (insloc 0 ))) ';' (Stop SCM+FSA )),4)) by A13 ;
then A16: dom (if=0 a,(I ';' (Goto (insloc 0 ))),(Stop SCM+FSA )) = (dom (Directed (((a =0_goto (insloc ((card (Stop SCM+FSA )) + 3))) ';' (Stop SCM+FSA )) ';' (Goto (insloc ((card (I ';' (Goto (insloc 0 )))) + 1)))))) \/ (dom (ProgramPart (Relocated ((I ';' (Goto (insloc 0 ))) ';' (Stop SCM+FSA )),4))) by FUNCT_4:def 1;
then dom (if=0 a,(I ';' (Goto (insloc 0 ))),(Stop SCM+FSA )) = (dom (((a =0_goto (insloc ((card (Stop SCM+FSA )) + 3))) ';' (Stop SCM+FSA )) ';' (Goto (insloc ((card (I ';' (Goto (insloc 0 )))) + 1))))) \/ (dom (ProgramPart (Relocated ((I ';' (Goto (insloc 0 ))) ';' (Stop SCM+FSA )),4))) by FUNCT_4:105;
then A17: insloc (n + 4) in dom (ProgramPart (Relocated ((I ';' (Goto (insloc 0 ))) ';' (Stop SCM+FSA )),4)) by A12, A14, XBOOLE_0:def 3;
while=0 a,I c= (while=0 a,I) +* (Start-At (insloc 0 )) by SCMFSA8A:9;
then A19: dom (while=0 a,I) c= dom ((while=0 a,I) +* (Start-At (insloc 0 ))) by GRFUNC_1:8;
I c= I +* (Start-At (insloc 0 )) by SCMFSA8A:9;
then A20: dom I c= dom (I +* (Start-At (insloc 0 ))) by GRFUNC_1:8;
A21: CurInstr (Computation (s +* (I +* (Start-At (insloc 0 )))),(LifeSpan (s +* (I +* (Start-At (insloc 0 )))))) = (Computation (s +* (I +* (Start-At (insloc 0 )))),(LifeSpan (s +* (I +* (Start-At (insloc 0 )))))) . (insloc n) by AMI_1:def 17
.= (s +* (I +* (Start-At (insloc 0 )))) . (insloc n) by AMI_1:54
.= (I +* (Start-At (insloc 0 ))) . (insloc n) by A6, A20, FUNCT_4:14
.= I . (insloc n) by A6, SCMFSA6B:7 ;
s +* (I +* (Start-At (insloc 0 ))) is halting by A2, SCMFSA7B:def 8;
then A22: I . (insloc n) = halt SCM+FSA by A21, AMI_1:def 46;
A23: (I ';' (Goto (insloc 0 ))) ';' (Stop SCM+FSA ) = I ';' ((Goto (insloc 0 )) ';' (Stop SCM+FSA )) by SCMFSA6A:67;
then dom ((I ';' (Goto (insloc 0 ))) ';' (Stop SCM+FSA )) = (dom (Directed I)) \/ (dom (ProgramPart (Relocated ((Goto (insloc 0 )) ';' (Stop SCM+FSA )),(card I)))) by FUNCT_4:def 1
.= (dom I) \/ (dom (ProgramPart (Relocated ((Goto (insloc 0 )) ';' (Stop SCM+FSA )),(card I)))) by FUNCT_4:105 ;
then A24: insloc n in dom ((I ';' (Goto (insloc 0 ))) ';' (Stop SCM+FSA )) by A6, XBOOLE_0:def 3;
then (insloc n) + 4 in { (il + 4) where il is Element of NAT : il in dom ((I ';' (Goto (insloc 0 ))) ';' (Stop SCM+FSA )) } ;
then A25: insloc (n + 4) in dom (Shift ((I ';' (Goto (insloc 0 ))) ';' (Stop SCM+FSA )),4) by VALUED_1:def 12;
then A26: pi (Shift ((I ';' (Goto (insloc 0 ))) ';' (Stop SCM+FSA )),4),(n + 4) = (Shift ((I ';' (Goto (insloc 0 ))) ';' (Stop SCM+FSA )),4) . ((insloc n) + 4) by AMI_1:def 47
.= ((I ';' (Goto (insloc 0 ))) ';' (Stop SCM+FSA )) . (insloc n) by A24, VALUED_1:def 12
.= (Directed I) . (insloc n) by A6, A23, SCMFSA8A:28
.= goto (insloc (card I)) by A6, A22, SCMFSA8A:30 ;
(Computation (s +* ((while=0 a,I) +* (Start-At (insloc 0 )))),(1 + (LifeSpan (s +* (I +* (Start-At (insloc 0 ))))))) . (insloc (n + 4)) = (s +* ((while=0 a,I) +* (Start-At (insloc 0 )))) . (insloc (n + 4)) by AMI_1:54
.= ((while=0 a,I) +* (Start-At (insloc 0 ))) . (insloc (n + 4)) by A10, A19, FUNCT_4:14
.= ((if=0 a,(I ';' (Goto (insloc 0 ))),(Stop SCM+FSA )) +* ((insloc ((card I) + 4)) .--> (goto (insloc 0 )))) . (insloc (n + 4)) by A10, SCMFSA6B:7
.= ((Directed (((a =0_goto (insloc ((card (Stop SCM+FSA )) + 3))) ';' (Stop SCM+FSA )) ';' (Goto (insloc ((card (I ';' (Goto (insloc 0 )))) + 1))))) +* (ProgramPart (Relocated ((I ';' (Goto (insloc 0 ))) ';' (Stop SCM+FSA )),4))) . (insloc (n + 4)) by A8, A10, A11, A15, FUNCT_4:def 1
.= (ProgramPart (Relocated ((I ';' (Goto (insloc 0 ))) ';' (Stop SCM+FSA )),4)) . (insloc (n + 4)) by A12, A16, A17, FUNCT_4:def 1
.= (IncAddr [(Shift (ProgramPart ((I ';' (Goto (insloc 0 ))) ';' (Stop SCM+FSA ))),4)],4) . (insloc (n + 4)) by SCMFSA_5:2
.= (IncAddr (Shift ((I ';' (Goto (insloc 0 ))) ';' (Stop SCM+FSA )),4),4) . (insloc (n + 4)) by AMI_1:105
.= IncAddr (goto (insloc (card I))),4 by A25, A26, SCMFSA_4:24
.= goto (insloc ((card I) + 4)) by SCMFSA_4:14 ;
hence CurInstr (Computation (s +* ((while=0 a,I) +* (Start-At (insloc 0 )))),(1 + (LifeSpan (s +* (I +* (Start-At (insloc 0 ))))))) = goto (insloc ((card I) + 4)) by A3, AMI_1:def 17; :: thesis: