let P1, P2 be Instruction-Sequence of SCMPDS; :: thesis: for s being 0 -started State of SCMPDS

for I being parahalting Program of st stop I c= P1 & stop I c= P2 holds

( LifeSpan (P1,s) = LifeSpan (P2,s) & Result (P1,s) = Result (P2,s) )

let s be 0 -started State of SCMPDS; :: thesis: for I being parahalting Program of st stop I c= P1 & stop I c= P2 holds

( LifeSpan (P1,s) = LifeSpan (P2,s) & Result (P1,s) = Result (P2,s) )

let I be parahalting Program of ; :: thesis: ( stop I c= P1 & stop I c= P2 implies ( LifeSpan (P1,s) = LifeSpan (P2,s) & Result (P1,s) = Result (P2,s) ) )

set SI = stop I;

assume that

A1: stop I c= P1 and

A2: stop I c= P2 ; :: thesis: ( LifeSpan (P1,s) = LifeSpan (P2,s) & Result (P1,s) = Result (P2,s) )

A3: P2 halts_on s by A2, SCMPDS_4:def 7;

A4: P1 halts_on s by A1, SCMPDS_4:def 7;

.= halt SCMPDS by A4, EXTPRO_1:def 15 ;

hence A7: LifeSpan (P1,s) = LifeSpan (P2,s) by A5, A3, EXTPRO_1:def 15; :: thesis: Result (P1,s) = Result (P2,s)

P2 halts_on s by A2, SCMPDS_4:def 7;

then A8: Result (P2,s) = Comput (P2,s,(LifeSpan (P1,s))) by A7, EXTPRO_1:23;

P1 halts_on s by A1, SCMPDS_4:def 7;

then Result (P1,s) = Comput (P1,s,(LifeSpan (P1,s))) by EXTPRO_1:23;

hence Result (P1,s) = Result (P2,s) by A1, A2, A8, Th5; :: thesis: verum

for I being parahalting Program of st stop I c= P1 & stop I c= P2 holds

( LifeSpan (P1,s) = LifeSpan (P2,s) & Result (P1,s) = Result (P2,s) )

let s be 0 -started State of SCMPDS; :: thesis: for I being parahalting Program of st stop I c= P1 & stop I c= P2 holds

( LifeSpan (P1,s) = LifeSpan (P2,s) & Result (P1,s) = Result (P2,s) )

let I be parahalting Program of ; :: thesis: ( stop I c= P1 & stop I c= P2 implies ( LifeSpan (P1,s) = LifeSpan (P2,s) & Result (P1,s) = Result (P2,s) ) )

set SI = stop I;

assume that

A1: stop I c= P1 and

A2: stop I c= P2 ; :: thesis: ( LifeSpan (P1,s) = LifeSpan (P2,s) & Result (P1,s) = Result (P2,s) )

A3: P2 halts_on s by A2, SCMPDS_4:def 7;

A4: P1 halts_on s by A1, SCMPDS_4:def 7;

A5: now :: thesis: for l being Nat st CurInstr (P2,(Comput (P2,s,l))) = halt SCMPDS holds

LifeSpan (P1,s) <= l

CurInstr (P2,(Comput (P2,s,(LifeSpan (P1,s))))) =
CurInstr (P1,(Comput (P1,s,(LifeSpan (P1,s)))))
by A1, A2, Th5
LifeSpan (P1,s) <= l

let l be Nat; :: thesis: ( CurInstr (P2,(Comput (P2,s,l))) = halt SCMPDS implies LifeSpan (P1,s) <= l )

assume A6: CurInstr (P2,(Comput (P2,s,l))) = halt SCMPDS ; :: thesis: LifeSpan (P1,s) <= l

CurInstr (P1,(Comput (P1,s,l))) = CurInstr (P2,(Comput (P2,s,l))) by A1, A2, Th5;

hence LifeSpan (P1,s) <= l by A4, A6, EXTPRO_1:def 15; :: thesis: verum

end;assume A6: CurInstr (P2,(Comput (P2,s,l))) = halt SCMPDS ; :: thesis: LifeSpan (P1,s) <= l

CurInstr (P1,(Comput (P1,s,l))) = CurInstr (P2,(Comput (P2,s,l))) by A1, A2, Th5;

hence LifeSpan (P1,s) <= l by A4, A6, EXTPRO_1:def 15; :: thesis: verum

.= halt SCMPDS by A4, EXTPRO_1:def 15 ;

hence A7: LifeSpan (P1,s) = LifeSpan (P2,s) by A5, A3, EXTPRO_1:def 15; :: thesis: Result (P1,s) = Result (P2,s)

P2 halts_on s by A2, SCMPDS_4:def 7;

then A8: Result (P2,s) = Comput (P2,s,(LifeSpan (P1,s))) by A7, EXTPRO_1:23;

P1 halts_on s by A1, SCMPDS_4:def 7;

then Result (P1,s) = Comput (P1,s,(LifeSpan (P1,s))) by EXTPRO_1:23;

hence Result (P1,s) = Result (P2,s) by A1, A2, A8, Th5; :: thesis: verum