let s be State of SCM+FSA ; :: thesis: for I being InitClosed Program of SCM+FSA st ProgramPart (s +* (Initialized I)) halts_on s +* (Initialized I) holds
for J being Program of SCM+FSA
for k being Element of NAT st k <= LifeSpan (ProgramPart (s +* (Initialized I))),(s +* (Initialized I)) holds
Comput (ProgramPart (s +* (Initialized I))),(s +* (Initialized I)),k, Comput (ProgramPart (s +* (Initialized (I ';' J)))),(s +* (Initialized (I ';' J))),k equal_outside NAT
let I be InitClosed Program of SCM+FSA ; :: thesis: ( ProgramPart (s +* (Initialized I)) halts_on s +* (Initialized I) implies for J being Program of SCM+FSA
for k being Element of NAT st k <= LifeSpan (ProgramPart (s +* (Initialized I))),(s +* (Initialized I)) holds
Comput (ProgramPart (s +* (Initialized I))),(s +* (Initialized I)),k, Comput (ProgramPart (s +* (Initialized (I ';' J)))),(s +* (Initialized (I ';' J))),k equal_outside NAT )
assume A1: ProgramPart (s +* (Initialized I)) halts_on s +* (Initialized I) ; :: thesis: for J being Program of SCM+FSA
for k being Element of NAT st k <= LifeSpan (ProgramPart (s +* (Initialized I))),(s +* (Initialized I)) holds
Comput (ProgramPart (s +* (Initialized I))),(s +* (Initialized I)),k, Comput (ProgramPart (s +* (Initialized (I ';' J)))),(s +* (Initialized (I ';' J))),k equal_outside NAT
set s1 = s +* (Initialized I);
let J be Program of SCM+FSA ; :: thesis: for k being Element of NAT st k <= LifeSpan (ProgramPart (s +* (Initialized I))),(s +* (Initialized I)) holds
Comput (ProgramPart (s +* (Initialized I))),(s +* (Initialized I)),k, Comput (ProgramPart (s +* (Initialized (I ';' J)))),(s +* (Initialized (I ';' J))),k equal_outside NAT
set s2 = s +* (Initialized (I ';' J));
A2: Initialized I c= s +* (Initialized I) by FUNCT_4:26;
defpred S1[ Nat] means ( $1 <= LifeSpan (ProgramPart (s +* (Initialized I))),(s +* (Initialized I)) implies Comput (ProgramPart (s +* (Initialized I))),(s +* (Initialized I)),$1, Comput (ProgramPart (s +* (Initialized (I ';' J)))),(s +* (Initialized (I ';' J))),$1 equal_outside NAT );
A3: Initialized (I ';' J) c= s +* (Initialized (I ';' J)) by FUNCT_4:26;
A4: for m being Element of NAT st S1[m] holds
S1[m + 1]
proof
dom (I ';' J) = (dom (Directed I)) \/ (dom (ProgramPart (Relocated J,(card I)))) by FUNCT_4:def 1
.= (dom I) \/ (dom (ProgramPart (Relocated J,(card I)))) by FUNCT_4:105 ;
then A5: dom I c= dom (I ';' J) by XBOOLE_1:7;
set sx = s +* (Initialized (I ';' J));
let m be Element of NAT ; :: thesis: ( S1[m] implies S1[m + 1] )
assume A6: ( m <= LifeSpan (ProgramPart (s +* (Initialized I))),(s +* (Initialized I)) implies Comput (ProgramPart (s +* (Initialized I))),(s +* (Initialized I)),m, Comput (ProgramPart (s +* (Initialized (I ';' J)))),(s +* (Initialized (I ';' J))),m equal_outside NAT ) ; :: thesis: S1[m + 1]
assume A7: m + 1 <= LifeSpan (ProgramPart (s +* (Initialized I))),(s +* (Initialized I)) ; :: thesis: Comput (ProgramPart (s +* (Initialized I))),(s +* (Initialized I)),(m + 1), Comput (ProgramPart (s +* (Initialized (I ';' J)))),(s +* (Initialized (I ';' J))),(m + 1) equal_outside NAT
then A8: IC (Comput (ProgramPart (s +* (Initialized I))),(s +* (Initialized I)),m) = IC (Comput (ProgramPart (s +* (Initialized (I ';' J)))),(s +* (Initialized (I ';' J))),m) by A6, COMPOS_1:24, NAT_1:13;
A9: I ';' J c= Comput (ProgramPart (s +* (Initialized (I ';' J)))),(s +* (Initialized (I ';' J))),m by A3, Th13, AMI_1:81;
T: ProgramPart (s +* (Initialized I)) = ProgramPart (Comput (ProgramPart (s +* (Initialized I))),(s +* (Initialized I)),m) by AMI_1:123;
A10: Comput (ProgramPart (s +* (Initialized I))),(s +* (Initialized I)),(m + 1) = Following (ProgramPart (s +* (Initialized I))),(Comput (ProgramPart (s +* (Initialized I))),(s +* (Initialized I)),m) by AMI_1:14
.= Exec (CurInstr (ProgramPart (Comput (ProgramPart (s +* (Initialized I))),(s +* (Initialized I)),m)),(Comput (ProgramPart (s +* (Initialized I))),(s +* (Initialized I)),m)),(Comput (ProgramPart (s +* (Initialized I))),(s +* (Initialized I)),m) by T ;
T: ProgramPart (s +* (Initialized (I ';' J))) = ProgramPart (Comput (ProgramPart (s +* (Initialized (I ';' J)))),(s +* (Initialized (I ';' J))),m) by AMI_1:123;
A11: Comput (ProgramPart (s +* (Initialized (I ';' J)))),(s +* (Initialized (I ';' J))),(m + 1) = Following (ProgramPart (s +* (Initialized (I ';' J)))),(Comput (ProgramPart (s +* (Initialized (I ';' J)))),(s +* (Initialized (I ';' J))),m) by AMI_1:14
.= Exec (CurInstr (ProgramPart (Comput (ProgramPart (s +* (Initialized (I ';' J)))),(s +* (Initialized (I ';' J))),m)),(Comput (ProgramPart (s +* (Initialized (I ';' J)))),(s +* (Initialized (I ';' J))),m)),(Comput (ProgramPart (s +* (Initialized (I ';' J)))),(s +* (Initialized (I ';' J))),m) by T ;
A12: IC (Comput (ProgramPart (s +* (Initialized I))),(s +* (Initialized I)),m) in dom I by A2, Def1;
Y: (ProgramPart (Comput (ProgramPart (s +* (Initialized I))),(s +* (Initialized I)),m)) /. (IC (Comput (ProgramPart (s +* (Initialized I))),(s +* (Initialized I)),m)) = (Comput (ProgramPart (s +* (Initialized I))),(s +* (Initialized I)),m) . (IC (Comput (ProgramPart (s +* (Initialized I))),(s +* (Initialized I)),m)) by COMPOS_1:38;
TX: ProgramPart (s +* (Initialized I)) = ProgramPart (Comput (ProgramPart (s +* (Initialized I))),(s +* (Initialized I)),m) by AMI_1:123;
I c= Comput (ProgramPart (s +* (Initialized I))),(s +* (Initialized I)),m by A2, Th13, AMI_1:81;
then A13: CurInstr (ProgramPart (s +* (Initialized I))),(Comput (ProgramPart (s +* (Initialized I))),(s +* (Initialized I)),m) = I . (IC (Comput (ProgramPart (s +* (Initialized I))),(s +* (Initialized I)),m)) by A12, Y, TX, GRFUNC_1:8;
Y: (ProgramPart (Comput (ProgramPart (s +* (Initialized (I ';' J)))),(s +* (Initialized (I ';' J))),m)) /. (IC (Comput (ProgramPart (s +* (Initialized (I ';' J)))),(s +* (Initialized (I ';' J))),m)) = (Comput (ProgramPart (s +* (Initialized (I ';' J)))),(s +* (Initialized (I ';' J))),m) . (IC (Comput (ProgramPart (s +* (Initialized (I ';' J)))),(s +* (Initialized (I ';' J))),m)) by COMPOS_1:38;
m < LifeSpan (ProgramPart (s +* (Initialized I))),(s +* (Initialized I)) by A7, NAT_1:13;
then I . (IC (Comput (ProgramPart (s +* (Initialized I))),(s +* (Initialized I)),m)) <> halt SCM+FSA by A1, A13, AMI_1:def 46;
then CurInstr (ProgramPart (Comput (ProgramPart (s +* (Initialized I))),(s +* (Initialized I)),m)),(Comput (ProgramPart (s +* (Initialized I))),(s +* (Initialized I)),m) = (I ';' J) . (IC (Comput (ProgramPart (s +* (Initialized I))),(s +* (Initialized I)),m)) by A12, A13, TX, SCMFSA6A:54
.= CurInstr (ProgramPart (Comput (ProgramPart (s +* (Initialized (I ';' J)))),(s +* (Initialized (I ';' J))),m)),(Comput (ProgramPart (s +* (Initialized (I ';' J)))),(s +* (Initialized (I ';' J))),m) by A8, A12, A9, A5, Y, GRFUNC_1:8 ;
hence Comput (ProgramPart (s +* (Initialized I))),(s +* (Initialized I)),(m + 1), Comput (ProgramPart (s +* (Initialized (I ';' J)))),(s +* (Initialized (I ';' J))),(m + 1) equal_outside NAT by A6, A7, A10, A11, NAT_1:13, SCMFSA6A:32; :: thesis: verum
end;
A14: ( (s +* (((intloc 0 ) .--> 1) +* (Start-At 0 ,SCM+FSA ))) +* I,s +* (((intloc 0 ) .--> 1) +* (Start-At 0 ,SCM+FSA )) equal_outside NAT & s +* (((intloc 0 ) .--> 1) +* (Start-At 0 ,SCM+FSA )),(s +* (((intloc 0 ) .--> 1) +* (Start-At 0 ,SCM+FSA ))) +* (I ';' J) equal_outside NAT ) by FUNCT_7:28, FUNCT_7:132;
A15: s +* (Initialized (I ';' J)) = s +* ((I ';' J) +* (((intloc 0 ) .--> 1) +* (Start-At 0 ,SCM+FSA ))) by FUNCT_4:15
.= (s +* (I ';' J)) +* (((intloc 0 ) .--> 1) +* (Start-At 0 ,SCM+FSA )) by FUNCT_4:15
.= (s +* (((intloc 0 ) .--> 1) +* (Start-At 0 ,SCM+FSA ))) +* (I ';' J) by Th19 ;
A16: ( Comput (ProgramPart (s +* (Initialized I))),(s +* (Initialized I)),0 = s +* (Initialized I) & Comput (ProgramPart (s +* (Initialized (I ';' J)))),(s +* (Initialized (I ';' J))),0 = s +* (Initialized (I ';' J)) ) by AMI_1:13;
s +* (Initialized I) = s +* (I +* (((intloc 0 ) .--> 1) +* (Start-At 0 ,SCM+FSA ))) by FUNCT_4:15
.= (s +* I) +* (((intloc 0 ) .--> 1) +* (Start-At 0 ,SCM+FSA )) by FUNCT_4:15
.= (s +* (((intloc 0 ) .--> 1) +* (Start-At 0 ,SCM+FSA ))) +* I by Th19 ;
then A17: S1[ 0 ] by A15, A14, A16, FUNCT_7:29;
thus for k being Element of NAT holds S1[k] from NAT_1:sch 1(A17, A4); :: thesis: verum