let i be Nat; :: thesis: for j being Element of NAT st i <= j holds
for N being non empty with_non-empty_elements set
for S being non empty stored-program halting IC-Ins-separated definite AMI-Struct of N
for p being NAT -defined the Instructions of b₃ -valued finite 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 j be Element of NAT ; :: thesis: ( i <= j implies for N being non empty with_non-empty_elements set
for S being non empty stored-program halting IC-Ins-separated definite AMI-Struct of N
for p being NAT -defined the Instructions of b₂ -valued finite 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 N being non empty with_non-empty_elements set
for S being non empty stored-program halting IC-Ins-separated definite AMI-Struct of N
for p being NAT -defined the Instructions of b₂ -valued finite 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: for S being non empty stored-program halting IC-Ins-separated definite AMI-Struct of N
for p being NAT -defined the Instructions of b₁ -valued finite 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 stored-program halting IC-Ins-separated definite AMI-Struct of N; :: thesis: for p being NAT -defined the Instructions of S -valued finite 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 finite 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
A3: IC (Comput p,s,i) in dom p and
A2: p . (IC (Comput p,s,i)) = halt S ; :: according to SCMNORM:def 15 :: thesis: p halts_at IC (Comput p,s,j)
X: CurInstr p,(Comput p,s,i) = halt S by A3, A2, PARTFUN1:def 8;
hence IC (Comput p,s,j) in dom p by A3, A1, Th7; :: according to SCMNORM:def 15 :: thesis: p . (IC (Comput p,s,j)) = halt S
thus p . (IC (Comput p,s,j)) = halt S by A1, A2, Th7, X; :: thesis: verum