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97c3405 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 | 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
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