let s be State of SCM+FSA; :: thesis: for I being Program of SCM+FSA st I is_closed_on s & I is_halting_on s holds
for m being Element of NAT st m <= LifeSpan ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(s +* (I +* (Start-At (0,SCM+FSA))))) holds
Comput ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(s +* (I +* (Start-At (0,SCM+FSA)))),m), Comput ((ProgramPart (s +* ((loop I) +* (Start-At (0,SCM+FSA))))),(s +* ((loop I) +* (Start-At (0,SCM+FSA)))),m) equal_outside NAT
set A = NAT ;
let I be Program of SCM+FSA; :: thesis: ( I is_closed_on s & I is_halting_on s implies for m being Element of NAT st m <= LifeSpan ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(s +* (I +* (Start-At (0,SCM+FSA))))) holds
Comput ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(s +* (I +* (Start-At (0,SCM+FSA)))),m), Comput ((ProgramPart (s +* ((loop I) +* (Start-At (0,SCM+FSA))))),(s +* ((loop I) +* (Start-At (0,SCM+FSA)))),m) equal_outside NAT )
set s1 = s +* (I +* (Start-At (0,SCM+FSA)));
set s2 = s +* ((loop I) +* (Start-At (0,SCM+FSA)));
assume A1: I is_closed_on s ; :: thesis: ( not I is_halting_on s or for m being Element of NAT st m <= LifeSpan ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(s +* (I +* (Start-At (0,SCM+FSA))))) holds
Comput ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(s +* (I +* (Start-At (0,SCM+FSA)))),m), Comput ((ProgramPart (s +* ((loop I) +* (Start-At (0,SCM+FSA))))),(s +* ((loop I) +* (Start-At (0,SCM+FSA)))),m) equal_outside NAT )
defpred S₁[ Nat] means ( $1 <= LifeSpan ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(s +* (I +* (Start-At (0,SCM+FSA))))) implies Comput ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(s +* (I +* (Start-At (0,SCM+FSA)))),$1), Comput ((ProgramPart (s +* ((loop I) +* (Start-At (0,SCM+FSA))))),(s +* ((loop I) +* (Start-At (0,SCM+FSA)))),$1) equal_outside NAT );
assume I is_halting_on s ; :: thesis: for m being Element of NAT st m <= LifeSpan ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(s +* (I +* (Start-At (0,SCM+FSA))))) holds
Comput ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(s +* (I +* (Start-At (0,SCM+FSA)))),m), Comput ((ProgramPart (s +* ((loop I) +* (Start-At (0,SCM+FSA))))),(s +* ((loop I) +* (Start-At (0,SCM+FSA)))),m) equal_outside NAT
then A2: ProgramPart (s +* (I +* (Start-At (0,SCM+FSA)))) halts_on s +* (I +* (Start-At (0,SCM+FSA))) by SCMFSA7B:def 8;
A3: for m being Element of NAT st S₁[m] holds
S₁[m + 1]

proof

A4: loop I c= (loop I) +* (Start-At (0,SCM+FSA)) by SCMFSA8A:9;
A5: I c= I +* (Start-At (0,SCM+FSA)) by SCMFSA8A:9;
let m be Element of NAT ; :: thesis: ( S₁[m] implies S₁[m + 1] )
assume A6: ( m <= LifeSpan ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(s +* (I +* (Start-At (0,SCM+FSA))))) implies Comput ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(s +* (I +* (Start-At (0,SCM+FSA)))),m), Comput ((ProgramPart (s +* ((loop I) +* (Start-At (0,SCM+FSA))))),(s +* ((loop I) +* (Start-At (0,SCM+FSA)))),m) equal_outside NAT ) ; :: thesis: S₁[m + 1]
(loop I) +* (Start-At (0,SCM+FSA)) c= s +* ((loop I) +* (Start-At (0,SCM+FSA))) by FUNCT_4:26;
then loop I c= s +* ((loop I) +* (Start-At (0,SCM+FSA))) by A4, XBOOLE_1:1;
then A7: loop I c= Comput ((ProgramPart (s +* ((loop I) +* (Start-At (0,SCM+FSA))))),(s +* ((loop I) +* (Start-At (0,SCM+FSA)))),m) by AMI_1:81;
T: ProgramPart (s +* (I +* (Start-At (0,SCM+FSA)))) = ProgramPart (Comput ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(s +* (I +* (Start-At (0,SCM+FSA)))),m)) by AMI_1:123;
A8: Comput ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(s +* (I +* (Start-At (0,SCM+FSA)))),(m + 1)) = Following ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(Comput ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(s +* (I +* (Start-At (0,SCM+FSA)))),m))) by EXTPRO_1:4
.= Exec ((CurInstr ((ProgramPart (Comput ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(s +* (I +* (Start-At (0,SCM+FSA)))),m))),(Comput ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(s +* (I +* (Start-At (0,SCM+FSA)))),m)))),(Comput ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(s +* (I +* (Start-At (0,SCM+FSA)))),m))) by T ;
T: ProgramPart (s +* ((loop I) +* (Start-At (0,SCM+FSA)))) = ProgramPart (Comput ((ProgramPart (s +* ((loop I) +* (Start-At (0,SCM+FSA))))),(s +* ((loop I) +* (Start-At (0,SCM+FSA)))),m)) by AMI_1:123;
A9: Comput ((ProgramPart (s +* ((loop I) +* (Start-At (0,SCM+FSA))))),(s +* ((loop I) +* (Start-At (0,SCM+FSA)))),(m + 1)) = Following ((ProgramPart (s +* ((loop I) +* (Start-At (0,SCM+FSA))))),(Comput ((ProgramPart (s +* ((loop I) +* (Start-At (0,SCM+FSA))))),(s +* ((loop I) +* (Start-At (0,SCM+FSA)))),m))) by EXTPRO_1:4
.= Exec ((CurInstr ((ProgramPart (Comput ((ProgramPart (s +* ((loop I) +* (Start-At (0,SCM+FSA))))),(s +* ((loop I) +* (Start-At (0,SCM+FSA)))),m))),(Comput ((ProgramPart (s +* ((loop I) +* (Start-At (0,SCM+FSA))))),(s +* ((loop I) +* (Start-At (0,SCM+FSA)))),m)))),(Comput ((ProgramPart (s +* ((loop I) +* (Start-At (0,SCM+FSA))))),(s +* ((loop I) +* (Start-At (0,SCM+FSA)))),m))) by T ;
A10: IC (Comput ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(s +* (I +* (Start-At (0,SCM+FSA)))),m)) in dom I by A1, SCMFSA7B:def 7;
Y: (ProgramPart (Comput ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(s +* (I +* (Start-At (0,SCM+FSA)))),m))) /. (IC (Comput ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(s +* (I +* (Start-At (0,SCM+FSA)))),m))) = (Comput ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(s +* (I +* (Start-At (0,SCM+FSA)))),m)) . (IC (Comput ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(s +* (I +* (Start-At (0,SCM+FSA)))),m))) by COMPOS_1:38;
TX1: ProgramPart (s +* (I +* (Start-At (0,SCM+FSA)))) = ProgramPart (Comput ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(s +* (I +* (Start-At (0,SCM+FSA)))),m)) by AMI_1:123;
I +* (Start-At (0,SCM+FSA)) c= s +* (I +* (Start-At (0,SCM+FSA))) by FUNCT_4:26;
then I c= s +* (I +* (Start-At (0,SCM+FSA))) by A5, XBOOLE_1:1;
then I c= Comput ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(s +* (I +* (Start-At (0,SCM+FSA)))),m) by AMI_1:81;
then A11: CurInstr ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(Comput ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(s +* (I +* (Start-At (0,SCM+FSA)))),m))) = I . (IC (Comput ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(s +* (I +* (Start-At (0,SCM+FSA)))),m))) by A10, Y, TX1, GRFUNC_1:8;
assume A12: m + 1 <= LifeSpan ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(s +* (I +* (Start-At (0,SCM+FSA))))) ; :: thesis: Comput ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(s +* (I +* (Start-At (0,SCM+FSA)))),(m + 1)), Comput ((ProgramPart (s +* ((loop I) +* (Start-At (0,SCM+FSA))))),(s +* ((loop I) +* (Start-At (0,SCM+FSA)))),(m + 1)) equal_outside NAT
then m < LifeSpan ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(s +* (I +* (Start-At (0,SCM+FSA))))) by NAT_1:13;
then I . (IC (Comput ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(s +* (I +* (Start-At (0,SCM+FSA)))),m))) <> halt SCM+FSA by A2, A11, EXTPRO_1:def 14;
then A13: I . (IC (Comput ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(s +* (I +* (Start-At (0,SCM+FSA)))),m))) = (loop I) . (IC (Comput ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(s +* (I +* (Start-At (0,SCM+FSA)))),m))) by FUNCT_4:111;
Z: (ProgramPart (Comput ((ProgramPart (s +* ((loop I) +* (Start-At (0,SCM+FSA))))),(s +* ((loop I) +* (Start-At (0,SCM+FSA)))),m))) /. (IC (Comput ((ProgramPart (s +* ((loop I) +* (Start-At (0,SCM+FSA))))),(s +* ((loop I) +* (Start-At (0,SCM+FSA)))),m))) = (Comput ((ProgramPart (s +* ((loop I) +* (Start-At (0,SCM+FSA))))),(s +* ((loop I) +* (Start-At (0,SCM+FSA)))),m)) . (IC (Comput ((ProgramPart (s +* ((loop I) +* (Start-At (0,SCM+FSA))))),(s +* ((loop I) +* (Start-At (0,SCM+FSA)))),m))) by COMPOS_1:38;
IC (Comput ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(s +* (I +* (Start-At (0,SCM+FSA)))),m)) in dom (loop I) by A10, FUNCT_4:105;
then CurInstr ((ProgramPart (Comput ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(s +* (I +* (Start-At (0,SCM+FSA)))),m))),(Comput ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(s +* (I +* (Start-At (0,SCM+FSA)))),m))) = (Comput ((ProgramPart (s +* ((loop I) +* (Start-At (0,SCM+FSA))))),(s +* ((loop I) +* (Start-At (0,SCM+FSA)))),m)) . (IC (Comput ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(s +* (I +* (Start-At (0,SCM+FSA)))),m))) by A7, A11, A13, TX1, GRFUNC_1:8
.= CurInstr ((ProgramPart (Comput ((ProgramPart (s +* ((loop I) +* (Start-At (0,SCM+FSA))))),(s +* ((loop I) +* (Start-At (0,SCM+FSA)))),m))),(Comput ((ProgramPart (s +* ((loop I) +* (Start-At (0,SCM+FSA))))),(s +* ((loop I) +* (Start-At (0,SCM+FSA)))),m))) by A6, A12, Z, COMPOS_1:24, NAT_1:13 ;
hence Comput ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(s +* (I +* (Start-At (0,SCM+FSA)))),(m + 1)), Comput ((ProgramPart (s +* ((loop I) +* (Start-At (0,SCM+FSA))))),(s +* ((loop I) +* (Start-At (0,SCM+FSA)))),(m + 1)) equal_outside NAT by A6, A12, A8, A9, NAT_1:13, SCMFSA6A:32; :: thesis: verum

end;

A14: S₁[ 0 ]

proof

assume 0 <= LifeSpan ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(s +* (I +* (Start-At (0,SCM+FSA))))) ; :: thesis: Comput ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(s +* (I +* (Start-At (0,SCM+FSA)))),0), Comput ((ProgramPart (s +* ((loop I) +* (Start-At (0,SCM+FSA))))),(s +* ((loop I) +* (Start-At (0,SCM+FSA)))),0) equal_outside NAT
A15: s,s +* (loop I) equal_outside NAT by FUNCT_7:132;
s +* I,s equal_outside NAT by FUNCT_7:28, FUNCT_7:132;
then s +* I,s +* (loop I) equal_outside NAT by A15, FUNCT_7:29;
then (s +* I) +* (Start-At (0,SCM+FSA)),(s +* (loop I)) +* (Start-At (0,SCM+FSA)) equal_outside NAT by FUNCT_7:106;
then (s +* I) +* (Start-At (0,SCM+FSA)),s +* ((loop I) +* (Start-At (0,SCM+FSA))) equal_outside NAT by FUNCT_4:15;
then s +* (I +* (Start-At (0,SCM+FSA))),s +* ((loop I) +* (Start-At (0,SCM+FSA))) equal_outside NAT by FUNCT_4:15;
then s +* (I +* (Start-At (0,SCM+FSA))), Comput ((ProgramPart (s +* ((loop I) +* (Start-At (0,SCM+FSA))))),(s +* ((loop I) +* (Start-At (0,SCM+FSA)))),0) equal_outside NAT by EXTPRO_1:3;
hence Comput ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(s +* (I +* (Start-At (0,SCM+FSA)))),0), Comput ((ProgramPart (s +* ((loop I) +* (Start-At (0,SCM+FSA))))),(s +* ((loop I) +* (Start-At (0,SCM+FSA)))),0) equal_outside NAT by EXTPRO_1:3; :: thesis: verum

end;

thus for m being Element of NAT holds S₁[m] from NAT_1:sch 1(A14, A3); :: thesis: verum