let s1, s2 be State of SCMPDS ; :: thesis: for I being Program of SCMPDS st I is_closed_on s1 & I is_halting_on s1 & DataPart s1 = DataPart s2 holds
( LifeSpan (ProgramPart ((Initialize s1) +* (stop I))),((Initialize s1) +* (stop I)) = LifeSpan (ProgramPart ((Initialize s2) +* (stop I))),((Initialize s2) +* (stop I)) & Result (ProgramPart ((Initialize s1) +* (stop I))),((Initialize s1) +* (stop I)), Result (ProgramPart ((Initialize s2) +* (stop I))),((Initialize s2) +* (stop I)) equal_outside NAT )
let I be Program of SCMPDS ; :: thesis: ( I is_closed_on s1 & I is_halting_on s1 & DataPart s1 = DataPart s2 implies ( LifeSpan (ProgramPart ((Initialize s1) +* (stop I))),((Initialize s1) +* (stop I)) = LifeSpan (ProgramPart ((Initialize s2) +* (stop I))),((Initialize s2) +* (stop I)) & Result (ProgramPart ((Initialize s1) +* (stop I))),((Initialize s1) +* (stop I)), Result (ProgramPart ((Initialize s2) +* (stop I))),((Initialize s2) +* (stop I)) equal_outside NAT ) )
assume A1: I is_closed_on s1 ; :: thesis: ( not I is_halting_on s1 or not DataPart s1 = DataPart s2 or ( LifeSpan (ProgramPart ((Initialize s1) +* (stop I))),((Initialize s1) +* (stop I)) = LifeSpan (ProgramPart ((Initialize s2) +* (stop I))),((Initialize s2) +* (stop I)) & Result (ProgramPart ((Initialize s1) +* (stop I))),((Initialize s1) +* (stop I)), Result (ProgramPart ((Initialize s2) +* (stop I))),((Initialize s2) +* (stop I)) equal_outside NAT ) )
set ss1 = (Initialize s1) +* (stop I);
set ss2 = (Initialize s2) +* (stop I);
assume A2: I is_halting_on s1 ; :: thesis: ( not DataPart s1 = DataPart s2 or ( LifeSpan (ProgramPart ((Initialize s1) +* (stop I))),((Initialize s1) +* (stop I)) = LifeSpan (ProgramPart ((Initialize s2) +* (stop I))),((Initialize s2) +* (stop I)) & Result (ProgramPart ((Initialize s1) +* (stop I))),((Initialize s1) +* (stop I)), Result (ProgramPart ((Initialize s2) +* (stop I))),((Initialize s2) +* (stop I)) equal_outside NAT ) )
then A3: ProgramPart ((Initialize s1) +* (stop I)) halts_on (Initialize s1) +* (stop I) by SCMPDS_6:def 3;
then A4: Result (ProgramPart ((Initialize s1) +* (stop I))),((Initialize s1) +* (stop I)) = Comput (ProgramPart ((Initialize s1) +* (stop I))),((Initialize s1) +* (stop I)),(LifeSpan (ProgramPart ((Initialize s1) +* (stop I))),((Initialize s1) +* (stop I))) by AMI_1:122;
assume A5: DataPart s1 = DataPart s2 ; :: thesis: ( LifeSpan (ProgramPart ((Initialize s1) +* (stop I))),((Initialize s1) +* (stop I)) = LifeSpan (ProgramPart ((Initialize s2) +* (stop I))),((Initialize s2) +* (stop I)) & Result (ProgramPart ((Initialize s1) +* (stop I))),((Initialize s1) +* (stop I)), Result (ProgramPart ((Initialize s2) +* (stop I))),((Initialize s2) +* (stop I)) equal_outside NAT )
then I is_halting_on s2 by A1, A2, SCMPDS_6:37;
then A6: ProgramPart ((Initialize s2) +* (stop I)) halts_on (Initialize s2) +* (stop I) by SCMPDS_6:def 3;

A7: now

let l be Element of NAT ; :: thesis: ( CurInstr (ProgramPart ((Initialize s2) +* (stop I))),(Comput (ProgramPart ((Initialize s2) +* (stop I))),((Initialize s2) +* (stop I)),l) = halt SCMPDS implies LifeSpan (ProgramPart ((Initialize s1) +* (stop I))),((Initialize s1) +* (stop I)) <= l )
assume A8: CurInstr (ProgramPart ((Initialize s2) +* (stop I))),(Comput (ProgramPart ((Initialize s2) +* (stop I))),((Initialize s2) +* (stop I)),l) = halt SCMPDS ; :: thesis: LifeSpan (ProgramPart ((Initialize s1) +* (stop I))),((Initialize s1) +* (stop I)) <= l
CurInstr (ProgramPart ((Initialize s1) +* (stop I))),(Comput (ProgramPart ((Initialize s1) +* (stop I))),((Initialize s1) +* (stop I)),l) = CurInstr (ProgramPart ((Initialize s2) +* (stop I))),(Comput (ProgramPart ((Initialize s2) +* (stop I))),((Initialize s2) +* (stop I)),l) by A1, A5, Th25;
hence LifeSpan (ProgramPart ((Initialize s1) +* (stop I))),((Initialize s1) +* (stop I)) <= l by A3, A8, AMI_1:def 46; :: thesis: verum

end;

CurInstr (ProgramPart ((Initialize s2) +* (stop I))),(Comput (ProgramPart ((Initialize s2) +* (stop I))),((Initialize s2) +* (stop I)),(LifeSpan (ProgramPart ((Initialize s1) +* (stop I))),((Initialize s1) +* (stop I)))) = CurInstr (ProgramPart ((Initialize s1) +* (stop I))),(Comput (ProgramPart ((Initialize s1) +* (stop I))),((Initialize s1) +* (stop I)),(LifeSpan (ProgramPart ((Initialize s1) +* (stop I))),((Initialize s1) +* (stop I)))) by A1, A5, Th25
.= halt SCMPDS by A3, AMI_1:def 46 ;
hence LifeSpan (ProgramPart ((Initialize s1) +* (stop I))),((Initialize s1) +* (stop I)) = LifeSpan (ProgramPart ((Initialize s2) +* (stop I))),((Initialize s2) +* (stop I)) by A7, A6, AMI_1:def 46; :: thesis: Result (ProgramPart ((Initialize s1) +* (stop I))),((Initialize s1) +* (stop I)), Result (ProgramPart ((Initialize s2) +* (stop I))),((Initialize s2) +* (stop I)) equal_outside NAT
then Result (ProgramPart ((Initialize s2) +* (stop I))),((Initialize s2) +* (stop I)) = Comput (ProgramPart ((Initialize s2) +* (stop I))),((Initialize s2) +* (stop I)),(LifeSpan (ProgramPart ((Initialize s1) +* (stop I))),((Initialize s1) +* (stop I))) by A6, AMI_1:122;
hence Result (ProgramPart ((Initialize s1) +* (stop I))),((Initialize s1) +* (stop I)), Result (ProgramPart ((Initialize s2) +* (stop I))),((Initialize s2) +* (stop I)) equal_outside NAT by A1, A5, A4, Th25; :: thesis: verum