| import OAI.Combinatorics.ContingencyTables.Machines.FiniteWordMachine |
|
|
| namespace OAI |
|
|
| /-! # Every-path output properties of the finite-word machine -/ |
|
|
| namespace ContingencyTables.FiniteWordExecution |
| open MatchingFPRAS MatchingFPRAS.PhysicalStream Turing |
| open scoped BigOperators Classical |
|
|
| variable {n : List Symbol → ℕ} {f : List Symbol×List Bool → List Symbol} |
|
|
| /-- A property verified for every finite source word holds on every actual |
| machine tape once the fixed execution bound has elapsed. -/ |
| theorem machine_all (N : TreeTyped.Realizer n) (F : TreeTyped.Realizer f) |
| (as : List Symbol) (ha : ∀ a∈as,a≠0) |
| (hout : ∀ bs,∀ a∈f (as,bs),a≠6 ∧ a≠7) |
| (P : Tape Symbol → Prop) |
| (hf : ∀ bits : Bits (n as),P (Tape.mk₁ (f (as,List.ofFn bits)))) |
| (t : ℕ) (ht : machineTime N F as≤t) : |
| ∀ bits : Bits t,P (run (machine N F) as bits).tape := by |
| let g : Tape Symbol → ℚ := fun tape => if P tape then 0 else 1 |
| have hm := (machine_law N F as ha hout t ht).2 g |
| have hz : average (fun bits : Bits t => g (run (machine N F) as bits).tape)=0 := by |
| rw [hm] |
| have he : (fun bits : Bits (n as) => g (Tape.mk₁ (f (as,List.ofFn bits))))= |
| (fun _ => (0:ℚ)) := by |
| funext bits |
| exact ite_eq_left (hf bits) |
| rw [he,average_const] |
| have hpow : (2:ℚ)^t≠0 := pow_ne_zero _ (by norm_num) |
| have hs : ∑ bits : Bits t,g (run (machine N F) as bits).tape=0 := by |
| have h := congrArg (fun a : ℚ => a*(2:ℚ)^t) hz |
| simpa only [average,div_mul_cancel₀ _ hpow,zero_mul] using h |
| have hnonneg (bits : Bits t) : 0≤g (run (machine N F) as bits).tape := by |
| dsimp only [g] |
| split_ifs <;> norm_num |
| intro bits |
| have hb := (Finset.sum_eq_zero_iff_of_nonneg (fun b _ => hnonneg b)).mp hs bits (Finset.mem_univ bits) |
| by_contra hn |
| have hbad : (1:ℚ)=0 := by simpa only [g,ite_eq_right hn] using hb |
| norm_num at hbad |
|
|
| end ContingencyTables.FiniteWordExecution |
|
|
| end OAI |
|
|
