let s be State of ; :: thesis: for I being paraclosed Program of st ProgramPart (s +* I) halts_on s +* I & Directed I c= s & Start-At (insloc 0 ) c= s holds
IC (Computation s,((LifeSpan (s +* I)) + 1)) = insloc (card I)
set A = NAT ;
let I be paraclosed Program of ; :: thesis: ( ProgramPart (s +* I) halts_on s +* I & Directed I c= s & Start-At (insloc 0 ) c= s implies IC (Computation s,((LifeSpan (s +* I)) + 1)) = insloc (card I) )
assume that
A1: ProgramPart (s +* I) halts_on s +* I and
A2: Directed I c= s and
A3: Start-At (insloc 0 ) c= s ; :: thesis: IC (Computation s,((LifeSpan (s +* I)) + 1)) = insloc (card I)
set sISA0 = s +* (I +* (Start-At (insloc 0 )));
set s2 = (s +* (I +* (Start-At (insloc 0 )))) +* (Directed I);
set IAt = I +* (Start-At (insloc 0 ));
A4: dom (Directed I) = dom I by FUNCT_4:105;
set m = LifeSpan (s +* (I +* (Start-At (insloc 0 ))));
set l1 = IC (Computation (s +* (I +* (Start-At (insloc 0 )))),(LifeSpan (s +* (I +* (Start-At (insloc 0 ))))));
A5: I +* (Start-At (insloc 0 )) c= s +* (I +* (Start-At (insloc 0 ))) by FUNCT_4:26;
then A6: IC (Computation (s +* (I +* (Start-At (insloc 0 )))),(LifeSpan (s +* (I +* (Start-At (insloc 0 )))))) in dom I by Def2;
A7: s +* (I +* (Start-At (insloc 0 ))) = (s +* I) +* (Start-At (insloc 0 )) by FUNCT_4:15
.= (s +* (Start-At (insloc 0 ))) +* I by Th14
.= s +* I by A3, FUNCT_4:79 ;

A8: now

set s1 = (s +* (I +* (Start-At (insloc 0 )))) +* (I ';' I);
let k be Element of NAT ; :: thesis: ( k <= LifeSpan (s +* (I +* (Start-At (insloc 0 )))) implies Computation (s +* (I +* (Start-At (insloc 0 )))),k, Computation ((s +* (I +* (Start-At (insloc 0 )))) +* (Directed I)),k equal_outside NAT )
defpred S₁[ Element of NAT ] means ( $1 <= k implies Computation ((s +* (I +* (Start-At (insloc 0 )))) +* (I ';' I)),$1, Computation ((s +* (I +* (Start-At (insloc 0 )))) +* (Directed I)),$1 equal_outside NAT );
assume A9: k <= LifeSpan (s +* (I +* (Start-At (insloc 0 )))) ; :: thesis: Computation (s +* (I +* (Start-At (insloc 0 )))),k, Computation ((s +* (I +* (Start-At (insloc 0 )))) +* (Directed I)),k equal_outside NAT
A10: for n being Element of NAT st S₁[n] holds
S₁[n + 1]

proof

let n be Element of NAT ; :: thesis: ( S₁[n] implies S₁[n + 1] )
assume A11: ( n <= k implies Computation ((s +* (I +* (Start-At (insloc 0 )))) +* (I ';' I)),n, Computation ((s +* (I +* (Start-At (insloc 0 )))) +* (Directed I)),n equal_outside NAT ) ; :: thesis: S₁[n + 1]
A12: Computation ((s +* (I +* (Start-At (insloc 0 )))) +* (Directed I)),(n + 1) = Following (Computation ((s +* (I +* (Start-At (insloc 0 )))) +* (Directed I)),n) by AMI_1:14
.= Exec (CurInstr (Computation ((s +* (I +* (Start-At (insloc 0 )))) +* (Directed I)),n)),(Computation ((s +* (I +* (Start-At (insloc 0 )))) +* (Directed I)),n) ;
A13: Computation ((s +* (I +* (Start-At (insloc 0 )))) +* (I ';' I)),(n + 1) = Following (Computation ((s +* (I +* (Start-At (insloc 0 )))) +* (I ';' I)),n) by AMI_1:14
.= Exec (CurInstr (Computation ((s +* (I +* (Start-At (insloc 0 )))) +* (I ';' I)),n)),(Computation ((s +* (I +* (Start-At (insloc 0 )))) +* (I ';' I)),n) ;
A14: n <= n + 1 by NAT_1:12;
assume A15: n + 1 <= k ; :: thesis: Computation ((s +* (I +* (Start-At (insloc 0 )))) +* (I ';' I)),(n + 1), Computation ((s +* (I +* (Start-At (insloc 0 )))) +* (Directed I)),(n + 1) equal_outside NAT
then A16: IC (Computation ((s +* (I +* (Start-At (insloc 0 )))) +* (I ';' I)),n) = IC (Computation ((s +* (I +* (Start-At (insloc 0 )))) +* (Directed I)),n) by A11, A14, AMI_1:121, XXREAL_0:2;
n <= k by A15, A14, XXREAL_0:2;
then n <= LifeSpan (s +* (I +* (Start-At (insloc 0 )))) by A9, XXREAL_0:2;
then IC (Computation (s +* (I +* (Start-At (insloc 0 )))),n) = IC (Computation ((s +* (I +* (Start-At (insloc 0 )))) +* (I ';' I)),n) by A1, A5, A7, Th36, AMI_1:121;
then A17: IC (Computation ((s +* (I +* (Start-At (insloc 0 )))) +* (I ';' I)),n) in dom I by A5, Def2;
then A18: IC (Computation ((s +* (I +* (Start-At (insloc 0 )))) +* (Directed I)),n) in dom (Directed I) by A16, FUNCT_4:105;
A19: CurInstr (Computation ((s +* (I +* (Start-At (insloc 0 )))) +* (Directed I)),n) = ((s +* (I +* (Start-At (insloc 0 )))) +* (Directed I)) . (IC (Computation ((s +* (I +* (Start-At (insloc 0 )))) +* (Directed I)),n)) by AMI_1:54
.= (Directed I) . (IC (Computation ((s +* (I +* (Start-At (insloc 0 )))) +* (Directed I)),n)) by A18, FUNCT_4:14 ;
( dom I c= dom (I ';' I) & CurInstr (Computation ((s +* (I +* (Start-At (insloc 0 )))) +* (I ';' I)),n) = ((s +* (I +* (Start-At (insloc 0 )))) +* (I ';' I)) . (IC (Computation ((s +* (I +* (Start-At (insloc 0 )))) +* (I ';' I)),n)) ) by AMI_1:54, SCMFSA6A:56;
then ( Directed I c= I ';' I & CurInstr (Computation ((s +* (I +* (Start-At (insloc 0 )))) +* (I ';' I)),n) = (I ';' I) . (IC (Computation ((s +* (I +* (Start-At (insloc 0 )))) +* (I ';' I)),n)) ) by A17, FUNCT_4:14, SCMFSA6A:55;
then CurInstr (Computation ((s +* (I +* (Start-At (insloc 0 )))) +* (I ';' I)),n) = (Directed I) . (IC (Computation ((s +* (I +* (Start-At (insloc 0 )))) +* (I ';' I)),n)) by A16, A18, GRFUNC_1:8;
hence Computation ((s +* (I +* (Start-At (insloc 0 )))) +* (I ';' I)),(n + 1), Computation ((s +* (I +* (Start-At (insloc 0 )))) +* (Directed I)),(n + 1) equal_outside NAT by A11, A15, A14, A16, A19, A13, A12, SCMFSA6A:32, XXREAL_0:2; :: thesis: verum

end;

( Computation ((s +* (I +* (Start-At (insloc 0 )))) +* (I ';' I)),0 = (s +* (I +* (Start-At (insloc 0 )))) +* (I ';' I) & Computation ((s +* (I +* (Start-At (insloc 0 )))) +* (Directed I)),0 = (s +* (I +* (Start-At (insloc 0 )))) +* (Directed I) ) by AMI_1:13;
then Computation ((s +* (I +* (Start-At (insloc 0 )))) +* (Directed I)),0 , Computation ((s +* (I +* (Start-At (insloc 0 )))) +* (I ';' I)),0 equal_outside NAT by FUNCT_7:107, SCMFSA6A:42;
then A20: S₁[ 0 ] by FUNCT_7:28;
for n being Element of NAT holds S₁[n] from NAT_1:sch 1(A20, A10);
then A21: Computation ((s +* (I +* (Start-At (insloc 0 )))) +* (I ';' I)),k, Computation ((s +* (I +* (Start-At (insloc 0 )))) +* (Directed I)),k equal_outside NAT ;
Computation (s +* (I +* (Start-At (insloc 0 )))),k, Computation ((s +* (I +* (Start-At (insloc 0 )))) +* (I ';' I)),k equal_outside NAT by A1, A7, A9, Th36, FUNCT_4:26;
hence Computation (s +* (I +* (Start-At (insloc 0 )))),k, Computation ((s +* (I +* (Start-At (insloc 0 )))) +* (Directed I)),k equal_outside NAT by A21, FUNCT_7:29; :: thesis: verum

end;

then A22: IC (Computation (s +* (I +* (Start-At (insloc 0 )))),(LifeSpan (s +* (I +* (Start-At (insloc 0 )))))) = IC (Computation ((s +* (I +* (Start-At (insloc 0 )))) +* (Directed I)),(LifeSpan (s +* (I +* (Start-At (insloc 0 )))))) by AMI_1:121;
dom I misses dom (Start-At (insloc 0 )) by SF_MASTR:64;
then I c= I +* (Start-At (insloc 0 )) by FUNCT_4:33;
then dom I c= dom (I +* (Start-At (insloc 0 ))) by GRFUNC_1:8;
then (s +* (I +* (Start-At (insloc 0 )))) . (IC (Computation (s +* (I +* (Start-At (insloc 0 )))),(LifeSpan (s +* (I +* (Start-At (insloc 0 ))))))) = (I +* (Start-At (insloc 0 ))) . (IC (Computation (s +* (I +* (Start-At (insloc 0 )))),(LifeSpan (s +* (I +* (Start-At (insloc 0 ))))))) by A5, A6, GRFUNC_1:8;
then A23: I . (IC (Computation (s +* (I +* (Start-At (insloc 0 )))),(LifeSpan (s +* (I +* (Start-At (insloc 0 ))))))) = (s +* (I +* (Start-At (insloc 0 )))) . (IC (Computation (s +* (I +* (Start-At (insloc 0 )))),(LifeSpan (s +* (I +* (Start-At (insloc 0 ))))))) by A6, Th7
.= CurInstr (Computation (s +* (I +* (Start-At (insloc 0 )))),(LifeSpan (s +* (I +* (Start-At (insloc 0 )))))) by AMI_1:54
.= halt SCM+FSA by A1, A7, AMI_1:def 46 ;
IC (Computation ((s +* (I +* (Start-At (insloc 0 )))) +* (Directed I)),(LifeSpan (s +* (I +* (Start-At (insloc 0 )))))) in dom I by A8, A6, AMI_1:121;
then IC (Computation ((s +* (I +* (Start-At (insloc 0 )))) +* (Directed I)),(LifeSpan (s +* (I +* (Start-At (insloc 0 )))))) in dom (Directed I) by FUNCT_4:105;
then A24: ((s +* (I +* (Start-At (insloc 0 )))) +* (Directed I)) . (IC (Computation (s +* (I +* (Start-At (insloc 0 )))),(LifeSpan (s +* (I +* (Start-At (insloc 0 ))))))) = (Directed I) . (IC (Computation (s +* (I +* (Start-At (insloc 0 )))),(LifeSpan (s +* (I +* (Start-At (insloc 0 ))))))) by A22, FUNCT_4:14
.= goto (insloc (card I)) by A6, A23, FUNCT_4:112 ;
A25: (s +* (I +* (Start-At (insloc 0 )))) +* (Directed I) = ((s +* I) +* (Start-At (insloc 0 ))) +* (Directed I) by FUNCT_4:15
.= ((s +* (Start-At (insloc 0 ))) +* I) +* (Directed I) by Th14
.= (s +* I) +* (Directed I) by A3, FUNCT_4:79
.= s +* (I +* (Directed I)) by FUNCT_4:15
.= s +* (Directed I) by A4, FUNCT_4:20
.= s by A2, FUNCT_4:79 ;
Computation ((s +* (I +* (Start-At (insloc 0 )))) +* (Directed I)),((LifeSpan (s +* (I +* (Start-At (insloc 0 ))))) + 1) = Following (Computation ((s +* (I +* (Start-At (insloc 0 )))) +* (Directed I)),(LifeSpan (s +* (I +* (Start-At (insloc 0 )))))) by AMI_1:14
.= Exec (goto (insloc (card I))),(Computation ((s +* (I +* (Start-At (insloc 0 )))) +* (Directed I)),(LifeSpan (s +* (I +* (Start-At (insloc 0 )))))) by A22, A24, AMI_1:54 ;
hence IC (Computation s,((LifeSpan (s +* I)) + 1)) = insloc (card I) by A7, A25, SCMFSA_2:95; :: thesis: verum