let s be State of SCM+FSA; :: thesis: for I, J being Program of SCM+FSA st I is_pseudo-closed_on s holds
for k being Element of NAT st k <= pseudo-LifeSpan (s,I) holds
Comput ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(s +* (I +* (Start-At (0,SCM+FSA)))),k), Comput ((ProgramPart (s +* ((I ';' J) +* (Start-At (0,SCM+FSA))))),(s +* ((I ';' J) +* (Start-At (0,SCM+FSA)))),k) equal_outside NAT
let I, J be Program of SCM+FSA; :: thesis: ( I is_pseudo-closed_on s implies for k being Element of NAT st k <= pseudo-LifeSpan (s,I) holds
Comput ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(s +* (I +* (Start-At (0,SCM+FSA)))),k), Comput ((ProgramPart (s +* ((I ';' J) +* (Start-At (0,SCM+FSA))))),(s +* ((I ';' J) +* (Start-At (0,SCM+FSA)))),k) equal_outside NAT )
set s1 = s +* (I +* (Start-At (0,SCM+FSA)));
set s2 = s +* ((I ';' J) +* (Start-At (0,SCM+FSA)));
set I1 = I +* (Start-At (0,SCM+FSA));
set I2 = (I ';' J) +* (Start-At (0,SCM+FSA));
defpred S₁[ Nat] means ( $1 <= pseudo-LifeSpan (s,I) implies Comput ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(s +* (I +* (Start-At (0,SCM+FSA)))),$1), Comput ((ProgramPart (s +* ((I ';' J) +* (Start-At (0,SCM+FSA))))),(s +* ((I ';' J) +* (Start-At (0,SCM+FSA)))),$1) equal_outside NAT );
assume A1: I is_pseudo-closed_on s ; :: thesis: for k being Element of NAT st k <= pseudo-LifeSpan (s,I) holds
Comput ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(s +* (I +* (Start-At (0,SCM+FSA)))),k), Comput ((ProgramPart (s +* ((I ';' J) +* (Start-At (0,SCM+FSA))))),(s +* ((I ';' J) +* (Start-At (0,SCM+FSA)))),k) equal_outside NAT

A2: now

let k be Element of NAT ; :: thesis: ( S₁[k] implies S₁[k + 1] )
assume A3: S₁[k] ; :: thesis: S₁[k + 1]
thus S₁[k + 1] :: thesis: verum

proof

T: ProgramPart (s +* ((I ';' J) +* (Start-At (0,SCM+FSA)))) = ProgramPart (Comput ((ProgramPart (s +* ((I ';' J) +* (Start-At (0,SCM+FSA))))),(s +* ((I ';' J) +* (Start-At (0,SCM+FSA)))),k)) by AMI_1:123;
A4: Comput ((ProgramPart (s +* ((I ';' J) +* (Start-At (0,SCM+FSA))))),(s +* ((I ';' J) +* (Start-At (0,SCM+FSA)))),(k + 1)) = Following ((ProgramPart (s +* ((I ';' J) +* (Start-At (0,SCM+FSA))))),(Comput ((ProgramPart (s +* ((I ';' J) +* (Start-At (0,SCM+FSA))))),(s +* ((I ';' J) +* (Start-At (0,SCM+FSA)))),k))) by EXTPRO_1:4
.= Exec ((CurInstr ((ProgramPart (Comput ((ProgramPart (s +* ((I ';' J) +* (Start-At (0,SCM+FSA))))),(s +* ((I ';' J) +* (Start-At (0,SCM+FSA)))),k))),(Comput ((ProgramPart (s +* ((I ';' J) +* (Start-At (0,SCM+FSA))))),(s +* ((I ';' J) +* (Start-At (0,SCM+FSA)))),k)))),(Comput ((ProgramPart (s +* ((I ';' J) +* (Start-At (0,SCM+FSA))))),(s +* ((I ';' J) +* (Start-At (0,SCM+FSA)))),k))) by T ;
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)))),k)) by AMI_1:123;
A5: Comput ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(s +* (I +* (Start-At (0,SCM+FSA)))),(k + 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)))),k))) by EXTPRO_1:4
.= Exec ((CurInstr ((ProgramPart (Comput ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(s +* (I +* (Start-At (0,SCM+FSA)))),k))),(Comput ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(s +* (I +* (Start-At (0,SCM+FSA)))),k)))),(Comput ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(s +* (I +* (Start-At (0,SCM+FSA)))),k))) by T ;
A6: dom I c= dom (I ';' J) by SCMFSA6A:56;
( I ';' J c= (I ';' J) +* (Start-At (0,SCM+FSA)) & (I ';' J) +* (Start-At (0,SCM+FSA)) c= s +* ((I ';' J) +* (Start-At (0,SCM+FSA))) ) by Th9, FUNCT_4:26;
then I ';' J c= s +* ((I ';' J) +* (Start-At (0,SCM+FSA))) by XBOOLE_1:1;
then A7: I ';' J c= Comput ((ProgramPart (s +* ((I ';' J) +* (Start-At (0,SCM+FSA))))),(s +* ((I ';' J) +* (Start-At (0,SCM+FSA)))),k) by AMI_1:81;
A8: k + 0 < k + 1 by XREAL_1:8;
assume A9: k + 1 <= pseudo-LifeSpan (s,I) ; :: thesis: Comput ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(s +* (I +* (Start-At (0,SCM+FSA)))),(k + 1)), Comput ((ProgramPart (s +* ((I ';' J) +* (Start-At (0,SCM+FSA))))),(s +* ((I ';' J) +* (Start-At (0,SCM+FSA)))),(k + 1)) equal_outside NAT
then A10: k < pseudo-LifeSpan (s,I) by A8, XXREAL_0:2;
then A11: IC (Comput ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(s +* (I +* (Start-At (0,SCM+FSA)))),k)) in dom I by A1, Th31;
Y: (ProgramPart (Comput ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(s +* (I +* (Start-At (0,SCM+FSA)))),k))) /. (IC (Comput ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(s +* (I +* (Start-At (0,SCM+FSA)))),k))) = (Comput ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(s +* (I +* (Start-At (0,SCM+FSA)))),k)) . (IC (Comput ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(s +* (I +* (Start-At (0,SCM+FSA)))),k))) by COMPOS_1:38;
Z: (ProgramPart (Comput ((ProgramPart (s +* ((I ';' J) +* (Start-At (0,SCM+FSA))))),(s +* ((I ';' J) +* (Start-At (0,SCM+FSA)))),k))) /. (IC (Comput ((ProgramPart (s +* ((I ';' J) +* (Start-At (0,SCM+FSA))))),(s +* ((I ';' J) +* (Start-At (0,SCM+FSA)))),k))) = (Comput ((ProgramPart (s +* ((I ';' J) +* (Start-At (0,SCM+FSA))))),(s +* ((I ';' J) +* (Start-At (0,SCM+FSA)))),k)) . (IC (Comput ((ProgramPart (s +* ((I ';' J) +* (Start-At (0,SCM+FSA))))),(s +* ((I ';' J) +* (Start-At (0,SCM+FSA)))),k))) by COMPOS_1:38;
( I c= I +* (Start-At (0,SCM+FSA)) & I +* (Start-At (0,SCM+FSA)) c= s +* (I +* (Start-At (0,SCM+FSA))) ) by Th9, FUNCT_4:26;
then I c= s +* (I +* (Start-At (0,SCM+FSA))) by XBOOLE_1:1;
then I c= Comput ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(s +* (I +* (Start-At (0,SCM+FSA)))),k) by AMI_1:81;
then A12: CurInstr ((ProgramPart (Comput ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(s +* (I +* (Start-At (0,SCM+FSA)))),k))),(Comput ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(s +* (I +* (Start-At (0,SCM+FSA)))),k))) = I . (IC (Comput ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(s +* (I +* (Start-At (0,SCM+FSA)))),k))) by A11, Y, GRFUNC_1:8;
then I . (IC (Comput ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(s +* (I +* (Start-At (0,SCM+FSA)))),k))) <> halt SCM+FSA by A1, A10, Th31;
then CurInstr ((ProgramPart (Comput ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(s +* (I +* (Start-At (0,SCM+FSA)))),k))),(Comput ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(s +* (I +* (Start-At (0,SCM+FSA)))),k))) = (I ';' J) . (IC (Comput ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(s +* (I +* (Start-At (0,SCM+FSA)))),k))) by A11, A12, SCMFSA6A:54
.= (Comput ((ProgramPart (s +* ((I ';' J) +* (Start-At (0,SCM+FSA))))),(s +* ((I ';' J) +* (Start-At (0,SCM+FSA)))),k)) . (IC (Comput ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(s +* (I +* (Start-At (0,SCM+FSA)))),k))) by A7, A11, A6, GRFUNC_1:8
.= CurInstr ((ProgramPart (Comput ((ProgramPart (s +* ((I ';' J) +* (Start-At (0,SCM+FSA))))),(s +* ((I ';' J) +* (Start-At (0,SCM+FSA)))),k))),(Comput ((ProgramPart (s +* ((I ';' J) +* (Start-At (0,SCM+FSA))))),(s +* ((I ';' J) +* (Start-At (0,SCM+FSA)))),k))) by A3, A9, A8, Z, COMPOS_1:24, XXREAL_0:2 ;
hence Comput ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(s +* (I +* (Start-At (0,SCM+FSA)))),(k + 1)), Comput ((ProgramPart (s +* ((I ';' J) +* (Start-At (0,SCM+FSA))))),(s +* ((I ';' J) +* (Start-At (0,SCM+FSA)))),(k + 1)) equal_outside NAT by A3, A9, A8, A5, A4, SCMFSA6A:32, XXREAL_0:2; :: thesis: verum

end;

end;

A13: S₁[ 0 ]

proof

assume 0 <= pseudo-LifeSpan (s,I) ; :: thesis: Comput ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(s +* (I +* (Start-At (0,SCM+FSA)))),0), Comput ((ProgramPart (s +* ((I ';' J) +* (Start-At (0,SCM+FSA))))),(s +* ((I ';' J) +* (Start-At (0,SCM+FSA)))),0) equal_outside NAT
s +* (I +* (Start-At (0,SCM+FSA))),s +* ((I ';' J) +* (Start-At (0,SCM+FSA))) equal_outside NAT by Th14;
then Comput ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(s +* (I +* (Start-At (0,SCM+FSA)))),0),s +* ((I ';' J) +* (Start-At (0,SCM+FSA))) 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 +* ((I ';' J) +* (Start-At (0,SCM+FSA))))),(s +* ((I ';' J) +* (Start-At (0,SCM+FSA)))),0) equal_outside NAT by EXTPRO_1:3; :: thesis: verum

end;

thus for k being Element of NAT holds S₁[k] from NAT_1:sch 1(A13, A2); :: thesis: verum