let P1, P2 be Instruction-Sequence of SCMPDS; 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; 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 ; ( 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
; ( 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 for l being Nat st CurInstr (P2,(Comput (P2,s,l))) = halt SCMPDS holds
LifeSpan (P1,s) <= llet l be
Nat;
( CurInstr (P2,(Comput (P2,s,l))) = halt SCMPDS implies LifeSpan (P1,s) <= l )assume A6:
CurInstr (
P2,
(Comput (P2,s,l)))
= halt SCMPDS
;
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;
verum end;
CurInstr (P2,(Comput (P2,s,(LifeSpan (P1,s))))) =
CurInstr (P1,(Comput (P1,s,(LifeSpan (P1,s)))))
by A1, A2, Th5
.=
halt SCMPDS
by A4, EXTPRO_1:def 15
;
hence A7:
LifeSpan (P1,s) = LifeSpan (P2,s)
by A5, A3, EXTPRO_1:def 15; 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; verum