let s be State of SCMPDS ; :: thesis: for I being parahalting Program of SCMPDS
for k being Element of NAT st Initialized I c= s & k <= LifeSpan (s +* (Initialized (stop I))) holds
Comput (ProgramPart s),s,k, Comput (ProgramPart (s +* (Initialized (stop I)))),(s +* (Initialized (stop I))),k equal_outside NAT
let I be parahalting Program of SCMPDS ; :: thesis: for k being Element of NAT st Initialized I c= s & k <= LifeSpan (s +* (Initialized (stop I))) holds
Comput (ProgramPart s),s,k, Comput (ProgramPart (s +* (Initialized (stop I)))),(s +* (Initialized (stop I))),k equal_outside NAT
let k be Element of NAT ; :: thesis: ( Initialized I c= s & k <= LifeSpan (s +* (Initialized (stop I))) implies Comput (ProgramPart s),s,k, Comput (ProgramPart (s +* (Initialized (stop I)))),(s +* (Initialized (stop I))),k equal_outside NAT )
set II = Initialized I;
set IsI = Initialized (stop I);
set m = LifeSpan (s +* (Initialized (stop I)));
assume that
A1: Initialized I c= s and
A2: k <= LifeSpan (s +* (Initialized (stop I))) ; :: thesis: Comput (ProgramPart s),s,k, Comput (ProgramPart (s +* (Initialized (stop I)))),(s +* (Initialized (stop I))),k equal_outside NAT
set s2 = s +* (Initialized (stop I));
defpred S1[ Element of NAT ] means ( $1 <= LifeSpan (s +* (Initialized (stop I))) implies Comput (ProgramPart s),s,$1, Comput (ProgramPart (s +* (Initialized (stop I)))),(s +* (Initialized (stop I))),$1 equal_outside NAT );
A3: s +* (Initialized (stop I)) = s +* (stop I) by A1, SCMPDS_4:34;
A4: s = s +* (Initialized I) by A1, FUNCT_4:79
.= s +* I by A1, SCMPDS_4:34 ;
A5: now
let k be Element of NAT ; :: thesis: ( S1[k] implies S1[k + 1] )
assume A6: S1[k] ; :: thesis: S1[k + 1]
now
T: ProgramPart (s +* (Initialized (stop I))) = ProgramPart (Comput (ProgramPart (s +* (Initialized (stop I)))),(s +* (Initialized (stop I))),k) by AMI_1:144;
A7: Comput (ProgramPart (s +* (Initialized (stop I)))),(s +* (Initialized (stop I))),(k + 1) = Following (ProgramPart (s +* (Initialized (stop I)))),(Comput (ProgramPart (s +* (Initialized (stop I)))),(s +* (Initialized (stop I))),k) by AMI_1:14
.= Exec (CurInstr (ProgramPart (Comput (ProgramPart (s +* (Initialized (stop I)))),(s +* (Initialized (stop I))),k)),(Comput (ProgramPart (s +* (Initialized (stop I)))),(s +* (Initialized (stop I))),k)),(Comput (ProgramPart (s +* (Initialized (stop I)))),(s +* (Initialized (stop I))),k) by T ;
T: ProgramPart s = ProgramPart (Comput (ProgramPart s),s,k) by AMI_1:144;
A8: Comput (ProgramPart s),s,(k + 1) = Following (ProgramPart s),(Comput (ProgramPart s),s,k) by AMI_1:14
.= Exec (CurInstr (ProgramPart (Comput (ProgramPart s),s,k)),(Comput (ProgramPart s),s,k)),(Comput (ProgramPart s),s,k) by T ;
A9: k < k + 1 by XREAL_1:31;
assume A10: k + 1 <= LifeSpan (s +* (Initialized (stop I))) ; :: thesis: Comput (ProgramPart s),s,(k + 1), Comput (ProgramPart (s +* (Initialized (stop I)))),(s +* (Initialized (stop I))),(k + 1) equal_outside NAT
then k < LifeSpan (s +* (Initialized (stop I))) by A9, XXREAL_0:2;
then A11: IC (Comput (ProgramPart (s +* (Initialized (stop I)))),(s +* (Initialized (stop I))),k) in dom I by Th28;
then A12: IC (Comput (ProgramPart (s +* (Initialized (stop I)))),(s +* (Initialized (stop I))),k) in dom (stop I) by FUNCT_4:13;
Y: (ProgramPart (Comput (ProgramPart s),s,k)) /. (IC (Comput (ProgramPart s),s,k)) = (Comput (ProgramPart s),s,k) . (IC (Comput (ProgramPart s),s,k)) by AMI_1:150;
Z: (ProgramPart (Comput (ProgramPart (s +* (Initialized (stop I)))),(s +* (Initialized (stop I))),k)) /. (IC (Comput (ProgramPart (s +* (Initialized (stop I)))),(s +* (Initialized (stop I))),k)) = (Comput (ProgramPart (s +* (Initialized (stop I)))),(s +* (Initialized (stop I))),k) . (IC (Comput (ProgramPart (s +* (Initialized (stop I)))),(s +* (Initialized (stop I))),k)) by AMI_1:150;
IC (Comput (ProgramPart s),s,k) = IC (Comput (ProgramPart (s +* (Initialized (stop I)))),(s +* (Initialized (stop I))),k) by A6, A10, A9, AMI_1:121, XXREAL_0:2;
then CurInstr (ProgramPart (Comput (ProgramPart s),s,k)),(Comput (ProgramPart s),s,k) = s . (IC (Comput (ProgramPart (s +* (Initialized (stop I)))),(s +* (Initialized (stop I))),k)) by AMI_1:54, Y
.= I . (IC (Comput (ProgramPart (s +* (Initialized (stop I)))),(s +* (Initialized (stop I))),k)) by A4, A11, FUNCT_4:14
.= (stop I) . (IC (Comput (ProgramPart (s +* (Initialized (stop I)))),(s +* (Initialized (stop I))),k)) by A11, SCMPDS_4:37
.= (s +* (Initialized (stop I))) . (IC (Comput (ProgramPart (s +* (Initialized (stop I)))),(s +* (Initialized (stop I))),k)) by A3, A12, FUNCT_4:14
.= CurInstr (ProgramPart (Comput (ProgramPart (s +* (Initialized (stop I)))),(s +* (Initialized (stop I))),k)),(Comput (ProgramPart (s +* (Initialized (stop I)))),(s +* (Initialized (stop I))),k) by AMI_1:54, Z ;
hence Comput (ProgramPart s),s,(k + 1), Comput (ProgramPart (s +* (Initialized (stop I)))),(s +* (Initialized (stop I))),(k + 1) equal_outside NAT by A6, A10, A9, A8, A7, SCMPDS_4:15, XXREAL_0:2; :: thesis: verum
end;
hence S1[k + 1] ; :: thesis: verum
end;
A13: S1[ 0 ]
proof
assume 0 <= LifeSpan (s +* (Initialized (stop I))) ; :: thesis: Comput (ProgramPart s),s,0 , Comput (ProgramPart (s +* (Initialized (stop I)))),(s +* (Initialized (stop I))),0 equal_outside NAT
A14: Comput (ProgramPart (s +* (Initialized (stop I)))),(s +* (Initialized (stop I))),0 = s +* (Initialized (stop I)) by AMI_1:13;
Comput (ProgramPart s),s,0 = s by AMI_1:13;
hence Comput (ProgramPart s),s,0 , Comput (ProgramPart (s +* (Initialized (stop I)))),(s +* (Initialized (stop I))),0 equal_outside NAT by A3, A14, AMI_1:120; :: thesis: verum
end;
for k being Element of NAT holds S1[k] from NAT_1:sch 1(A13, A5);
 hence Comput (ProgramPart s),s,k, Comput (ProgramPart (s +* (Initialized (stop I)))),(s +* (Initialized (stop I))),k equal_outside NAT by A2; :: thesis: verum