let IL be non empty set ; :: thesis: for N being with_non-empty_elements set
for S being non empty stored-program halting IC-Ins-separated steady-programmed definite AMI-Struct of IL,N
for s being State of S st ex k being Element of NAT st s . (IC (Computation s,k)) = halt S holds
for i being Element of NAT holds Result s = Result (Computation s,i)
let N be with_non-empty_elements set ; :: thesis: for S being non empty stored-program halting IC-Ins-separated steady-programmed definite AMI-Struct of IL,N
for s being State of S st ex k being Element of NAT st s . (IC (Computation s,k)) = halt S holds
for i being Element of NAT holds Result s = Result (Computation s,i)
let S be non empty stored-program halting IC-Ins-separated steady-programmed definite AMI-Struct of IL,N; :: thesis: for s being State of S st ex k being Element of NAT st s . (IC (Computation s,k)) = halt S holds
for i being Element of NAT holds Result s = Result (Computation s,i)
let s be State of S; :: thesis: ( ex k being Element of NAT st s . (IC (Computation s,k)) = halt S implies for i being Element of NAT holds Result s = Result (Computation s,i) )
given k being Element of NAT such that A1: s . (IC (Computation s,k)) = halt S ; :: thesis: for i being Element of NAT holds Result s = Result (Computation s,i)
set s' = Computation s,k;
A2: CurInstr (Computation s,k) = halt S by A1, Th54;
let i be Element of NAT ; :: thesis: Result s = Result (Computation s,i)
now
per cases ( i <= k or i >= k ) ;
suppose i <= k ; :: thesis: Result s = Result (Computation s,i)
then consider j being Nat such that
A3: k = i + j by NAT_1:10;
reconsider j = j as Element of NAT by ORDINAL1:def 13;
A4: Computation s,k = Computation (Computation s,i),j by A3, Th51;
then A5: Computation s,i is halting by A2, Def20;
thus Result s = Computation s,k by A1, Th56
.= Result (Computation s,i) by A2, A4, A5, Def22 ; :: thesis: verum
end;
end;
end;
hence Result s = Result (Computation s,i) ; :: thesis: verum