let i, j be Element of NAT ; :: thesis: for N being non empty with_non-empty_elements set st i <= j holds
for S being non empty IC-Ins-separated halting AMI-Struct of N
for P being NAT -defined the Instructions of b₂ -valued Function
for s being State of S st P halts_at IC (Comput (P,s,i)) holds
P halts_at IC (Comput (P,s,j))
let N be non empty with_non-empty_elements set ; :: thesis: ( i <= j implies for S being non empty IC-Ins-separated halting AMI-Struct of N
for P being NAT -defined the Instructions of b₁ -valued Function
for s being State of S st P halts_at IC (Comput (P,s,i)) holds
P halts_at IC (Comput (P,s,j)) )
assume A1: i <= j ; :: thesis: for S being non empty IC-Ins-separated halting AMI-Struct of N
for P being NAT -defined the Instructions of b₁ -valued Function
for s being State of S st P halts_at IC (Comput (P,s,i)) holds
P halts_at IC (Comput (P,s,j))
let S be non empty IC-Ins-separated halting AMI-Struct of N; :: thesis: for P being NAT -defined the Instructions of S -valued Function
for s being State of S st P halts_at IC (Comput (P,s,i)) holds
P halts_at IC (Comput (P,s,j))
let p be NAT -defined the Instructions of S -valued Function; :: thesis: for s being State of S st p halts_at IC (Comput (p,s,i)) holds
p halts_at IC (Comput (p,s,j))
let s be State of S; :: thesis: ( p halts_at IC (Comput (p,s,i)) implies p halts_at IC (Comput (p,s,j)) )
assume that
A2: IC (Comput (p,s,i)) in dom p and
A3: p . (IC (Comput (p,s,i))) = halt S ; :: according to COMPOS_1:def 6 :: thesis: p halts_at IC (Comput (p,s,j))
A4: CurInstr (p,(Comput (p,s,i))) = halt S by A2, A3, PARTFUN1:def 6;
hence IC (Comput (p,s,j)) in dom p by A2, A1, Th6; :: according to COMPOS_1:def 6 :: thesis: p . (IC (Comput (p,s,j))) = halt S
thus p . (IC (Comput (p,s,j))) = halt S by A1, A3, A4, Th6; :: thesis: verum