let s be State of SCMPDS ; :: thesis: for I being parahalting Program of SCMPDS
for k being Element of NAT st Initialize I c= s & k <= LifeSpan (ProgramPart ((Initialize s) +* (stop I))),((Initialize s) +* (stop I)) holds
Comput (ProgramPart s),s,k, Comput (ProgramPart ((Initialize s) +* (stop I))),((Initialize s) +* (stop I)),k equal_outside NAT
let I be parahalting Program of SCMPDS ; :: thesis: for k being Element of NAT st Initialize I c= s & k <= LifeSpan (ProgramPart ((Initialize s) +* (stop I))),((Initialize s) +* (stop I)) holds
Comput (ProgramPart s),s,k, Comput (ProgramPart ((Initialize s) +* (stop I))),((Initialize s) +* (stop I)),k equal_outside NAT
let k be Element of NAT ; :: thesis: ( Initialize I c= s & k <= LifeSpan (ProgramPart ((Initialize s) +* (stop I))),((Initialize s) +* (stop I)) implies Comput (ProgramPart s),s,k, Comput (ProgramPart ((Initialize s) +* (stop I))),((Initialize s) +* (stop I)),k equal_outside NAT )
set m = LifeSpan (ProgramPart ((Initialize s) +* (stop I))),((Initialize s) +* (stop I));
assume that
A1: Initialize I c= s and
A2: k <= LifeSpan (ProgramPart ((Initialize s) +* (stop I))),((Initialize s) +* (stop I)) ; :: thesis: Comput (ProgramPart s),s,k, Comput (ProgramPart ((Initialize s) +* (stop I))),((Initialize s) +* (stop I)),k equal_outside NAT
set s2 = (Initialize s) +* (stop I);
defpred S₁[ Element of NAT ] means ( $1 <= LifeSpan (ProgramPart ((Initialize s) +* (stop I))),((Initialize s) +* (stop I)) implies Comput (ProgramPart s),s,$1, Comput (ProgramPart ((Initialize s) +* (stop I))),((Initialize s) +* (stop I)),$1 equal_outside NAT );
I2: s +* (Initialize I) = (Initialize s) +* I by SCMPDS_4:5;
A3: (Initialize s) +* (stop I) = s +* (stop I) by A1, SCMPDS_4:34;
A4: s = (Initialize s) +* I by A1, I2, 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 ((Initialize s) +* (stop I)) = ProgramPart (Comput (ProgramPart ((Initialize s) +* (stop I))),((Initialize s) +* (stop I)),k) by AMI_1:123;
A7: Comput (ProgramPart ((Initialize s) +* (stop I))),((Initialize s) +* (stop I)),(k + 1) = Following (ProgramPart ((Initialize s) +* (stop I))),(Comput (ProgramPart ((Initialize s) +* (stop I))),((Initialize s) +* (stop I)),k) by AMI_1:14
.= Exec (CurInstr (ProgramPart (Comput (ProgramPart ((Initialize s) +* (stop I))),((Initialize s) +* (stop I)),k)),(Comput (ProgramPart ((Initialize s) +* (stop I))),((Initialize s) +* (stop I)),k)),(Comput (ProgramPart ((Initialize s) +* (stop I))),((Initialize s) +* (stop I)),k) by T ;
T: ProgramPart s = ProgramPart (Comput (ProgramPart s),s,k) by AMI_1:123;
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 (ProgramPart ((Initialize s) +* (stop I))),((Initialize s) +* (stop I)) ; :: thesis: Comput (ProgramPart s),s,(k + 1), Comput (ProgramPart ((Initialize s) +* (stop I))),((Initialize s) +* (stop I)),(k + 1) equal_outside NAT
then k < LifeSpan (ProgramPart ((Initialize s) +* (stop I))),((Initialize s) +* (stop I)) by A9, XXREAL_0:2;
then A11: IC (Comput (ProgramPart ((Initialize s) +* (stop I))),((Initialize s) +* (stop I)),k) in dom I by Th28;
then A12: IC (Comput (ProgramPart ((Initialize s) +* (stop I))),((Initialize s) +* (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 COMPOS_1:38;
Z: (ProgramPart (Comput (ProgramPart ((Initialize s) +* (stop I))),((Initialize s) +* (stop I)),k)) /. (IC (Comput (ProgramPart ((Initialize s) +* (stop I))),((Initialize s) +* (stop I)),k)) = (Comput (ProgramPart ((Initialize s) +* (stop I))),((Initialize s) +* (stop I)),k) . (IC (Comput (ProgramPart ((Initialize s) +* (stop I))),((Initialize s) +* (stop I)),k)) by COMPOS_1:38;
IC (Comput (ProgramPart s),s,k) = IC (Comput (ProgramPart ((Initialize s) +* (stop I))),((Initialize s) +* (stop I)),k) by A6, A10, A9, COMPOS_1:24, XXREAL_0:2;
then CurInstr (ProgramPart (Comput (ProgramPart s),s,k)),(Comput (ProgramPart s),s,k) = s . (IC (Comput (ProgramPart ((Initialize s) +* (stop I))),((Initialize s) +* (stop I)),k)) by Y, AMI_1:54
.= I . (IC (Comput (ProgramPart ((Initialize s) +* (stop I))),((Initialize s) +* (stop I)),k)) by A4, A11, FUNCT_4:14
.= (stop I) . (IC (Comput (ProgramPart ((Initialize s) +* (stop I))),((Initialize s) +* (stop I)),k)) by A11, SCMPDS_4:37
.= ((Initialize s) +* (stop I)) . (IC (Comput (ProgramPart ((Initialize s) +* (stop I))),((Initialize s) +* (stop I)),k)) by A12, FUNCT_4:14
.= CurInstr (ProgramPart (Comput (ProgramPart ((Initialize s) +* (stop I))),((Initialize s) +* (stop I)),k)),(Comput (ProgramPart ((Initialize s) +* (stop I))),((Initialize s) +* (stop I)),k) by Z, AMI_1:54 ;
hence Comput (ProgramPart s),s,(k + 1), Comput (ProgramPart ((Initialize s) +* (stop I))),((Initialize s) +* (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 (ProgramPart ((Initialize s) +* (stop I))),((Initialize s) +* (stop I)) ; :: thesis: Comput (ProgramPart s),s,0 , Comput (ProgramPart ((Initialize s) +* (stop I))),((Initialize s) +* (stop I)),0 equal_outside NAT
A14: Comput (ProgramPart ((Initialize s) +* (stop I))),((Initialize s) +* (stop I)),0 = (Initialize s) +* (stop I) by AMI_1:13;
Comput (ProgramPart s),s,0 = s by AMI_1:13;
hence Comput (ProgramPart s),s,0 , Comput (ProgramPart ((Initialize s) +* (stop I))),((Initialize s) +* (stop I)),0 equal_outside NAT by A3, A14, FUNCT_7:132; :: 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 ((Initialize s) +* (stop I))),((Initialize s) +* (stop I)),k equal_outside NAT by A2; :: thesis: verum