let N be non empty with_non-empty_elements set ; :: thesis: for S being non empty halting IC-Ins-separated AMI-Struct of NAT ,N
for p being finite PartFunc of NAT ,the Instructions of S
for s being State of S
for k being Element of NAT st IC (Comput p,s,k) in dom p & p . (IC (Comput p,s,k)) = halt S holds
Result p,s = Comput p,s,k
let S be non empty halting IC-Ins-separated AMI-Struct of NAT ,N; :: thesis: for p being finite PartFunc of NAT ,the Instructions of S
for s being State of S
for k being Element of NAT st IC (Comput p,s,k) in dom p & p . (IC (Comput p,s,k)) = halt S holds
Result p,s = Comput p,s,k
let p be finite PartFunc of NAT ,the Instructions of S; :: thesis: for s being State of S
for k being Element of NAT st IC (Comput p,s,k) in dom p & p . (IC (Comput p,s,k)) = halt S holds
Result p,s = Comput p,s,k
let s be State of S; :: thesis: for k being Element of NAT st IC (Comput p,s,k) in dom p & p . (IC (Comput p,s,k)) = halt S holds
Result p,s = Comput p,s,k
let k be Element of NAT ; :: thesis: ( IC (Comput p,s,k) in dom p & p . (IC (Comput p,s,k)) = halt S implies Result p,s = Comput p,s,k )
assume Z1: IC (Comput p,s,k) in dom p ; :: thesis: ( not p . (IC (Comput p,s,k)) = halt S or Result p,s = Comput p,s,k )
assume Z2: p . (IC (Comput p,s,k)) = halt S ; :: thesis: Result p,s = Comput p,s,k
then p /. (IC (Comput p,s,k)) = halt S by Z1, PARTFUN1:def 8;
then A1: CurInstr p,(Comput p,s,k) = halt S by Z1;
then p halts_on s by Z1, Z2, Def8;
hence Result p,s = Comput p,s,k by A1, Def10; :: thesis: verum