let s be State of SCM+FSA ; :: thesis: for I being InitClosed Program of SCM+FSA st ProgramPart (s +* I) halts_on s +* I & Directed I c= s & ((intloc 0 ) .--> 1) +* (Start-At 0 ,SCM+FSA ) c= s holds
IC (Comput (ProgramPart s),s,((LifeSpan (ProgramPart (s +* I)),(s +* I)) + 1)) = card I
set A = NAT ;
let I be InitClosed Program of SCM+FSA ; :: thesis: ( ProgramPart (s +* I) halts_on s +* I & Directed I c= s & ((intloc 0 ) .--> 1) +* (Start-At 0 ,SCM+FSA ) c= s implies IC (Comput (ProgramPart s),s,((LifeSpan (ProgramPart (s +* I)),(s +* I)) + 1)) = card I )
assume that
A1: ProgramPart (s +* I) halts_on s +* I and
A2: Directed I c= s and
A3: ((intloc 0 ) .--> 1) +* (Start-At 0 ,SCM+FSA ) c= s ; :: thesis: IC (Comput (ProgramPart s),s,((LifeSpan (ProgramPart (s +* I)),(s +* I)) + 1)) = card I
set sISA0 = s +* (I +* (((intloc 0 ) .--> 1) +* (Start-At 0 ,SCM+FSA )));
A4: Initialized I c= s +* (I +* (((intloc 0 ) .--> 1) +* (Start-At 0 ,SCM+FSA ))) by A3, Th20;
set IAt = I +* (Start-At 0 ,SCM+FSA );
dom I misses dom (Start-At 0 ,SCM+FSA ) by SF_MASTR:64;
then I c= I +* (Start-At 0 ,SCM+FSA ) by FUNCT_4:33;
then A5: dom I c= dom (I +* (Start-At 0 ,SCM+FSA )) by GRFUNC_1:8;
A6: s +* (I +* (((intloc 0 ) .--> 1) +* (Start-At 0 ,SCM+FSA ))) = s +* I by A3, Th20;
reconsider sISA0 = s +* (I +* (((intloc 0 ) .--> 1) +* (Start-At 0 ,SCM+FSA ))) as State of SCM+FSA ;
set m = LifeSpan (ProgramPart sISA0),sISA0;
set l1 = IC (Comput (ProgramPart sISA0),sISA0,(LifeSpan (ProgramPart sISA0),sISA0));
A7: IC (Comput (ProgramPart sISA0),sISA0,(LifeSpan (ProgramPart sISA0),sISA0)) in dom I by A4, Def1;
set s2 = sISA0 +* (Directed I);
A8: now
set s1 = sISA0 +* (I ';' I);
let k be Element of NAT ; :: thesis: ( k <= LifeSpan (ProgramPart sISA0),sISA0 implies Comput (ProgramPart sISA0),sISA0,k, Comput (ProgramPart (sISA0 +* (Directed I))),(sISA0 +* (Directed I)),k equal_outside NAT )
defpred S1[ Nat] means ( $1 <= k implies Comput (ProgramPart (sISA0 +* (I ';' I))),(sISA0 +* (I ';' I)),$1, Comput (ProgramPart (sISA0 +* (Directed I))),(sISA0 +* (Directed I)),$1 equal_outside NAT );
assume A9: k <= LifeSpan (ProgramPart sISA0),sISA0 ; :: thesis: Comput (ProgramPart sISA0),sISA0,k, Comput (ProgramPart (sISA0 +* (Directed I))),(sISA0 +* (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 Comput (ProgramPart (sISA0 +* (I ';' I))),(sISA0 +* (I ';' I)),n, Comput (ProgramPart (sISA0 +* (Directed I))),(sISA0 +* (Directed I)),n equal_outside NAT ) ; :: thesis: S1[n + 1]
T: ProgramPart (sISA0 +* (Directed I)) = ProgramPart (Comput (ProgramPart (sISA0 +* (Directed I))),(sISA0 +* (Directed I)),n) by AMI_1:123;
A12: Comput (ProgramPart (sISA0 +* (Directed I))),(sISA0 +* (Directed I)),(n + 1) = Following (ProgramPart (sISA0 +* (Directed I))),(Comput (ProgramPart (sISA0 +* (Directed I))),(sISA0 +* (Directed I)),n) by AMI_1:14
.= Exec (CurInstr (ProgramPart (Comput (ProgramPart (sISA0 +* (Directed I))),(sISA0 +* (Directed I)),n)),(Comput (ProgramPart (sISA0 +* (Directed I))),(sISA0 +* (Directed I)),n)),(Comput (ProgramPart (sISA0 +* (Directed I))),(sISA0 +* (Directed I)),n) by T ;
T: ProgramPart (sISA0 +* (I ';' I)) = ProgramPart (Comput (ProgramPart (sISA0 +* (I ';' I))),(sISA0 +* (I ';' I)),n) by AMI_1:123;
A13: Comput (ProgramPart (sISA0 +* (I ';' I))),(sISA0 +* (I ';' I)),(n + 1) = Following (ProgramPart (sISA0 +* (I ';' I))),(Comput (ProgramPart (sISA0 +* (I ';' I))),(sISA0 +* (I ';' I)),n) by AMI_1:14
.= Exec (CurInstr (ProgramPart (Comput (ProgramPart (sISA0 +* (I ';' I))),(sISA0 +* (I ';' I)),n)),(Comput (ProgramPart (sISA0 +* (I ';' I))),(sISA0 +* (I ';' I)),n)),(Comput (ProgramPart (sISA0 +* (I ';' I))),(sISA0 +* (I ';' I)),n) by T ;
A14: n <= n + 1 by NAT_1:12;
assume A15: n + 1 <= k ; :: thesis: Comput (ProgramPart (sISA0 +* (I ';' I))),(sISA0 +* (I ';' I)),(n + 1), Comput (ProgramPart (sISA0 +* (Directed I))),(sISA0 +* (Directed I)),(n + 1) equal_outside NAT
then A16: IC (Comput (ProgramPart (sISA0 +* (I ';' I))),(sISA0 +* (I ';' I)),n) = IC (Comput (ProgramPart (sISA0 +* (Directed I))),(sISA0 +* (Directed I)),n) by A11, A14, COMPOS_1:24, XXREAL_0:2;
n <= k by A15, A14, XXREAL_0:2;
then n <= LifeSpan (ProgramPart sISA0),sISA0 by A9, XXREAL_0:2;
then IC (Comput (ProgramPart sISA0),sISA0,n) = IC (Comput (ProgramPart (sISA0 +* (I ';' I))),(sISA0 +* (I ';' I)),n) by A1, A4, A6, Th18, COMPOS_1:24;
then A17: IC (Comput (ProgramPart (sISA0 +* (I ';' I))),(sISA0 +* (I ';' I)),n) in dom I by A4, Def1;
then A18: IC (Comput (ProgramPart (sISA0 +* (Directed I))),(sISA0 +* (Directed I)),n) in dom (Directed I) by A16, FUNCT_4:105;
Y: (ProgramPart (Comput (ProgramPart (sISA0 +* (Directed I))),(sISA0 +* (Directed I)),n)) /. (IC (Comput (ProgramPart (sISA0 +* (Directed I))),(sISA0 +* (Directed I)),n)) = (Comput (ProgramPart (sISA0 +* (Directed I))),(sISA0 +* (Directed I)),n) . (IC (Comput (ProgramPart (sISA0 +* (Directed I))),(sISA0 +* (Directed I)),n)) by COMPOS_1:38;
Z: (ProgramPart (Comput (ProgramPart (sISA0 +* (I ';' I))),(sISA0 +* (I ';' I)),n)) /. (IC (Comput (ProgramPart (sISA0 +* (I ';' I))),(sISA0 +* (I ';' I)),n)) = (Comput (ProgramPart (sISA0 +* (I ';' I))),(sISA0 +* (I ';' I)),n) . (IC (Comput (ProgramPart (sISA0 +* (I ';' I))),(sISA0 +* (I ';' I)),n)) by COMPOS_1:38;
A19: CurInstr (ProgramPart (Comput (ProgramPart (sISA0 +* (Directed I))),(sISA0 +* (Directed I)),n)),(Comput (ProgramPart (sISA0 +* (Directed I))),(sISA0 +* (Directed I)),n) = (sISA0 +* (Directed I)) . (IC (Comput (ProgramPart (sISA0 +* (Directed I))),(sISA0 +* (Directed I)),n)) by Y, AMI_1:54
.= (Directed I) . (IC (Comput (ProgramPart (sISA0 +* (Directed I))),(sISA0 +* (Directed I)),n)) by A18, FUNCT_4:14 ;
( dom I c= dom (I ';' I) & CurInstr (ProgramPart (Comput (ProgramPart (sISA0 +* (I ';' I))),(sISA0 +* (I ';' I)),n)),(Comput (ProgramPart (sISA0 +* (I ';' I))),(sISA0 +* (I ';' I)),n) = (sISA0 +* (I ';' I)) . (IC (Comput (ProgramPart (sISA0 +* (I ';' I))),(sISA0 +* (I ';' I)),n)) ) by Z, AMI_1:54, SCMFSA6A:56;
then ( Directed I c= I ';' I & CurInstr (ProgramPart (Comput (ProgramPart (sISA0 +* (I ';' I))),(sISA0 +* (I ';' I)),n)),(Comput (ProgramPart (sISA0 +* (I ';' I))),(sISA0 +* (I ';' I)),n) = (I ';' I) . (IC (Comput (ProgramPart (sISA0 +* (I ';' I))),(sISA0 +* (I ';' I)),n)) ) by A17, FUNCT_4:14, SCMFSA6A:55;
then CurInstr (ProgramPart (Comput (ProgramPart (sISA0 +* (I ';' I))),(sISA0 +* (I ';' I)),n)),(Comput (ProgramPart (sISA0 +* (I ';' I))),(sISA0 +* (I ';' I)),n) = (Directed I) . (IC (Comput (ProgramPart (sISA0 +* (I ';' I))),(sISA0 +* (I ';' I)),n)) by A16, A18, GRFUNC_1:8;
hence Comput (ProgramPart (sISA0 +* (I ';' I))),(sISA0 +* (I ';' I)),(n + 1), Comput (ProgramPart (sISA0 +* (Directed I))),(sISA0 +* (Directed I)),(n + 1) equal_outside NAT by A11, A15, A14, A16, A19, A13, A12, SCMFSA6A:32, XXREAL_0:2; :: thesis: verum
end;
( Comput (ProgramPart (sISA0 +* (I ';' I))),(sISA0 +* (I ';' I)),0 = sISA0 +* (I ';' I) & Comput (ProgramPart (sISA0 +* (Directed I))),(sISA0 +* (Directed I)),0 = sISA0 +* (Directed I) ) by AMI_1:13;
then Comput (ProgramPart (sISA0 +* (Directed I))),(sISA0 +* (Directed I)),0 , Comput (ProgramPart (sISA0 +* (I ';' I))),(sISA0 +* (I ';' I)),0 equal_outside NAT by FUNCT_7:107, FUNCT_7:133;
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: Comput (ProgramPart (sISA0 +* (I ';' I))),(sISA0 +* (I ';' I)),k, Comput (ProgramPart (sISA0 +* (Directed I))),(sISA0 +* (Directed I)),k equal_outside NAT ;
Comput (ProgramPart sISA0),sISA0,k, Comput (ProgramPart (sISA0 +* (I ';' I))),(sISA0 +* (I ';' I)),k equal_outside NAT by A1, A4, A6, A9, Th18;
hence Comput (ProgramPart sISA0),sISA0,k, Comput (ProgramPart (sISA0 +* (Directed I))),(sISA0 +* (Directed I)),k equal_outside NAT by A21, FUNCT_7:29; :: thesis: verum
end;
then A22: IC (Comput (ProgramPart sISA0),sISA0,(LifeSpan (ProgramPart sISA0),sISA0)) = IC (Comput (ProgramPart (sISA0 +* (Directed I))),(sISA0 +* (Directed I)),(LifeSpan (ProgramPart sISA0),sISA0)) by COMPOS_1:24;
Y: (ProgramPart (Comput (ProgramPart sISA0),sISA0,(LifeSpan (ProgramPart sISA0),sISA0))) /. (IC (Comput (ProgramPart sISA0),sISA0,(LifeSpan (ProgramPart sISA0),sISA0))) = (Comput (ProgramPart sISA0),sISA0,(LifeSpan (ProgramPart sISA0),sISA0)) . (IC (Comput (ProgramPart sISA0),sISA0,(LifeSpan (ProgramPart sISA0),sISA0))) by COMPOS_1:38;
TX: ProgramPart sISA0 = ProgramPart (Comput (ProgramPart sISA0),sISA0,(LifeSpan (ProgramPart sISA0),sISA0)) by AMI_1:123;
I +* (Start-At 0 ,SCM+FSA ) c= Initialized I by Th6;
then I +* (Start-At 0 ,SCM+FSA ) c= sISA0 by A4, XBOOLE_1:1;
then sISA0 . (IC (Comput (ProgramPart sISA0),sISA0,(LifeSpan (ProgramPart sISA0),sISA0))) = (I +* (Start-At 0 ,SCM+FSA )) . (IC (Comput (ProgramPart sISA0),sISA0,(LifeSpan (ProgramPart sISA0),sISA0))) by A7, A5, GRFUNC_1:8;
then A23: I . (IC (Comput (ProgramPart sISA0),sISA0,(LifeSpan (ProgramPart sISA0),sISA0))) = sISA0 . (IC (Comput (ProgramPart sISA0),sISA0,(LifeSpan (ProgramPart sISA0),sISA0))) by A7, SCMFSA6B:7
.= CurInstr (ProgramPart sISA0),(Comput (ProgramPart sISA0),sISA0,(LifeSpan (ProgramPart sISA0),sISA0)) by Y, TX, AMI_1:54
.= halt SCM+FSA by A1, A6, AMI_1:def 46 ;
IC (Comput (ProgramPart (sISA0 +* (Directed I))),(sISA0 +* (Directed I)),(LifeSpan (ProgramPart sISA0),sISA0)) in dom I by A8, A7, COMPOS_1:24;
then IC (Comput (ProgramPart (sISA0 +* (Directed I))),(sISA0 +* (Directed I)),(LifeSpan (ProgramPart sISA0),sISA0)) in dom (Directed I) by FUNCT_4:105;
then A24: (sISA0 +* (Directed I)) . (IC (Comput (ProgramPart sISA0),sISA0,(LifeSpan (ProgramPart sISA0),sISA0))) = (Directed I) . (IC (Comput (ProgramPart sISA0),sISA0,(LifeSpan (ProgramPart sISA0),sISA0))) by A22, FUNCT_4:14
.= goto (card I) by A7, A23, FUNCT_4:112 ;
Y: (ProgramPart (Comput (ProgramPart (sISA0 +* (Directed I))),(sISA0 +* (Directed I)),(LifeSpan (ProgramPart sISA0),sISA0))) /. (IC (Comput (ProgramPart (sISA0 +* (Directed I))),(sISA0 +* (Directed I)),(LifeSpan (ProgramPart sISA0),sISA0))) = (Comput (ProgramPart (sISA0 +* (Directed I))),(sISA0 +* (Directed I)),(LifeSpan (ProgramPart sISA0),sISA0)) . (IC (Comput (ProgramPart (sISA0 +* (Directed I))),(sISA0 +* (Directed I)),(LifeSpan (ProgramPart sISA0),sISA0))) by COMPOS_1:38;
T: ProgramPart (sISA0 +* (Directed I)) = ProgramPart (Comput (ProgramPart (sISA0 +* (Directed I))),(sISA0 +* (Directed I)),(LifeSpan (ProgramPart sISA0),sISA0)) by AMI_1:123;
Comput (ProgramPart (sISA0 +* (Directed I))),(sISA0 +* (Directed I)),((LifeSpan (ProgramPart sISA0),sISA0) + 1) = Following (ProgramPart (sISA0 +* (Directed I))),(Comput (ProgramPart (sISA0 +* (Directed I))),(sISA0 +* (Directed I)),(LifeSpan (ProgramPart sISA0),sISA0)) by AMI_1:14
.= Exec (goto (card I)),(Comput (ProgramPart (sISA0 +* (Directed I))),(sISA0 +* (Directed I)),(LifeSpan (ProgramPart sISA0),sISA0)) by A22, A24, Y, T, AMI_1:54 ;
then A25: IC (Comput (ProgramPart (sISA0 +* (Directed I))),(sISA0 +* (Directed I)),((LifeSpan (ProgramPart sISA0),sISA0) + 1)) = card I by SCMFSA_2:95;
X: s +* (I +* (((intloc 0 ) .--> 1) +* (Start-At 0 ,SCM+FSA ))) = s +* I by A3, Th20;
sISA0 +* (Directed I) = s +* (Directed I) by A3, Th20
.= s by A2, FUNCT_4:79 ;
hence IC (Comput (ProgramPart s),s,((LifeSpan (ProgramPart (s +* I)),(s +* I)) + 1)) = card I by A25, X; :: thesis: verum