let N be non empty with_zero set ; for S being non empty with_non-empty_values IC-Ins-separated halting AMI-Struct over N
for P being Instruction-Sequence of S
for s being State of S
for k being Nat st CurInstr (P,(Comput (P,s,k))) = halt S holds
Comput (P,s,(LifeSpan (P,s))) = Comput (P,s,k)
let S be non empty with_non-empty_values IC-Ins-separated halting AMI-Struct over N; for P being Instruction-Sequence of S
for s being State of S
for k being Nat st CurInstr (P,(Comput (P,s,k))) = halt S holds
Comput (P,s,(LifeSpan (P,s))) = Comput (P,s,k)
let P be Instruction-Sequence of S; for s being State of S
for k being Nat st CurInstr (P,(Comput (P,s,k))) = halt S holds
Comput (P,s,(LifeSpan (P,s))) = Comput (P,s,k)
let s be State of S; for k being Nat st CurInstr (P,(Comput (P,s,k))) = halt S holds
Comput (P,s,(LifeSpan (P,s))) = Comput (P,s,k)
let k be Nat; ( CurInstr (P,(Comput (P,s,k))) = halt S implies Comput (P,s,(LifeSpan (P,s))) = Comput (P,s,k) )
assume A1:
CurInstr (P,(Comput (P,s,k))) = halt S
; Comput (P,s,(LifeSpan (P,s))) = Comput (P,s,k)
A2:
dom P = NAT
by PARTFUN1:def 2;
A3:
P halts_on s
by A2, A1;
set Ls = LifeSpan (P,s);
A4:
CurInstr (P,(Comput (P,s,(LifeSpan (P,s))))) = halt S
by A3, Def15;
LifeSpan (P,s) <= k
by A1, A3, Def15;
hence
Comput (P,s,(LifeSpan (P,s))) = Comput (P,s,k)
by A4, Th5; verum