let N be non empty with_non-empty_elements set ; :: thesis: for S being non empty stored-program IC-Ins-separated definite halting AMI-Struct of N
for P being the Instructions of b₁ -valued ManySortedSet of NAT
for s being State of S
for k being Element of NAT st P . (IC (Comput P,s,k)) = halt S holds
Result P,s = Comput P,s,k
let S be non empty stored-program IC-Ins-separated definite halting AMI-Struct of N; :: thesis: for P being the Instructions of S -valued ManySortedSet of NAT
for s being State of S
for k being Element of NAT st P . (IC (Comput P,s,k)) = halt S holds
Result P,s = Comput P,s,k
let P be the Instructions of S -valued ManySortedSet of NAT ; :: thesis: for s being State of S
for k being Element of NAT st 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 P . (IC (Comput P,s,k)) = halt S holds
Result P,s = Comput P,s,k
let k be Element of NAT ; :: thesis: ( P . (IC (Comput P,s,k)) = halt S implies Result P,s = Comput P,s,k )
D: dom P = NAT by PARTFUN1:def 4;
X: IC (Comput P,s,k) in dom P by D;
then Y: P /. (IC (Comput P,s,k)) = P . (IC (Comput P,s,k)) by PARTFUN1:def 8;
assume P . (IC (Comput P,s,k)) = halt S ; :: thesis: Result P,s = Comput P,s,k
then A1: CurInstr P,(Comput P,s,k) = halt S by Y;
then CurInstr P,(Comput P,s,k) = halt S ;
then P halts_on s by Def20, X;
hence Result P,s = Comput P,s,k by A1, Def22; :: thesis: verum