| import Mathlib |
| import OAI.Computability.MinUncut.Search.InnerParameters |
|
|
| namespace OAI |
|
|
| noncomputable section |
| open scoped BigOperators |
| namespace MinUncut.Inner |
|
|
| lemma uniformWeight_sum {Ξ : Type*} [Fintype Ξ] (F : Ξ → ℝ) : |
| (∑ ξ, (Fintype.card Ξ:ℝ)⁻¹*F ξ) = 𝔼 ξ,F ξ := by |
| rw [← Finset.mul_sum,Fintype.expect_eq_sum_div_card] |
| ring |
|
|
| attribute [local irreducible] firstError secondError thirdError atomProbability |
|
|
| theorem InnerParameters.decode_expect {J : ℝ} (hJ : 1 ≤ J) (P : InnerParameters J) |
| {Ξ : Type*} [Fintype Ξ] [Nonempty Ξ] (V A : Ξ → Type*) |
| [∀ ξ, AddCommGroup (V ξ)] [∀ ξ, Module F₂ (V ξ)] [∀ ξ, AddTorsor (V ξ) (A ξ)] |
| [∀ ξ, Fintype (A ξ)] (f : ∀ ξ, FoldedProof (A ξ)) |
| (h1 : (𝔼 ξ,firstError (m := P.m) (n := P.n) (f ξ) (sourceSigma J) (sourceEta J)) ≤ J*(10*sourceSigma J)) |
| (h2 : (𝔼 ξ,secondError (m := P.m) (n := P.n) (f ξ) |
| ((sourceEta J)^2/(P.m:ℝ)) (sourceSigma J) (sourceEta J)) ≤ J*(10*(sourceEta J+sourceEta J))) |
| (h3 : (𝔼 ξ,thirdError (m := P.m) (n := P.n) (f ξ) (sourceSigma J) (sourceEta J)) ≤ |
| J/(Fintype.card (Code P.m P.n):ℝ)) : |
| (P.p:ℝ) ≤ 𝔼 ξ,atomProbability (m := P.m) (n := P.n) (f ξ) (sourceSigma J) (sourceEta J) P.θ := by |
| let w := fun _ : Ξ => (Fintype.card Ξ:ℝ)⁻¹ |
| have hw (ξ : Ξ) : 0≤w ξ := by dsimp [w] |
| have hw1 : ∑ ξ,w ξ=1 := by |
| simpa only [mul_one,Fintype.expect_const] using (uniformWeight_sum (fun _ : Ξ => (1:ℝ))) |
| have hh1 : (∑ ξ,w ξ*firstError (m := P.m) (n := P.n) (f ξ) (sourceSigma J) (sourceEta J)) ≤ J*(10*sourceSigma J) := by |
| simpa only [w,uniformWeight_sum] using h1 |
| have hh2 : (∑ ξ,w ξ*secondError (m := P.m) (n := P.n) (f ξ) ((sourceEta J)^2/(P.m:ℝ)) (sourceSigma J) (sourceEta J)) ≤ J*(10*(sourceEta J+sourceEta J)) := by |
| simpa only [w,uniformWeight_sum] using h2 |
| have hh3 : (∑ ξ,w ξ*thirdError (m := P.m) (n := P.n) (f ξ) (sourceSigma J) (sourceEta J)) ≤ J/(Fintype.card (Code P.m P.n):ℝ) := by |
| simpa only [w,uniformWeight_sum] using h3 |
| simpa only [w,uniformWeight_sum] using P.decode hJ V A w hw hw1 f hh1 hh2 hh3 |
| end MinUncut.Inner |
|
|
| open scoped BigOperators |
| namespace MinUncut.Outer |
| open MinUncut.Inner OuterSmoothness |
| attribute [local instance] Classical.propDecidable BinaryFourier.dualFintype |
| attribute [local irreducible] firstError secondError thirdError atomProbability |
| variable {Name I S : Type*} [Fintype I] [Fintype S] |
|
|
| abbrev EdgeSamples (I S : Type*) [Fintype I] (k : ℕ) := |
| FixedSets I k × (I → S) × (I → Fin 3) |
|
|
| def edgeSampleSecond (equations : S → Equation Name) {k : ℕ} (e : EdgeSamples I S k) : |
| I → SecondQuestion Name := secondQuestion (fun i => equations (e.2.1 i)) (hiddenSet e.1.val) e.2.2 |
|
|
| lemma edgeSample_expect (equations : S → Equation Name) (k : ℕ) (F : EdgeStatistic Name I) : |
| (𝔼 e : EdgeSamples I S k, F (fun i => equations (e.2.1 i)) (hiddenSet e.1.val) e.2.2) = |
| edgeMean equations k F := by |
| rw [expect_pair] |
| simp_rw [expect_pair] |
| rfl |
|
|
| theorem ProofFamily.atomTotal_lower [Nonempty S] |
| (f : ProofFamily Name I) (equations : S → Equation Name) {J : ℝ} (hJ : 1≤J) |
| (P : InnerParameters J) {k : ℕ} (hk : k≤Fintype.card I) |
| (h1 : edgeMean equations k (fun U h pos => firstError (m := P.m) (n := P.n) |
| (f.second (secondQuestion U h pos)) (sourceSigma J) (sourceEta J)) ≤ J*(10*sourceSigma J)) |
| (h2 : edgeMean equations k (fun U h pos => secondError (m := P.m) (n := P.n) |
| (f.second (secondQuestion U h pos)) ((sourceEta J)^2/(P.m:ℝ)) (sourceSigma J) (sourceEta J)) ≤ |
| J*(10*(sourceEta J+sourceEta J))) |
| (h3 : edgeMean equations k (fun U h pos => thirdError (m := P.m) (n := P.n) |
| (f.second (secondQuestion U h pos)) (sourceSigma J) (sourceEta J)) ≤ |
| J/(Fintype.card (Code P.m P.n):ℝ)) : |
| (P.p:ℝ) ≤ edgeMean equations k (f.atomTotal (m := P.m) (n := P.n) (sourceSigma J) (sourceEta J) P.θ) := by |
| let : Nonempty (FixedSets I k) := fixedSets_nonempty hk |
| let A := fun e : EdgeSamples I S k => SecondAlphabet (edgeSampleSecond equations e) |
| rw [← edgeSample_expect equations k (f.atomTotal (m := P.m) (n := P.n) (sourceSigma J) (sourceEta J) P.θ)] |
| change (P.p:ℝ) ≤ 𝔼 e : EdgeSamples I S k,atomProbability (m := P.m) (n := P.n) |
| (f.second (edgeSampleSecond equations e)) (sourceSigma J) (sourceEta J) P.θ |
| apply P.decode_expect hJ A A (fun e => f.second (edgeSampleSecond equations e)) |
| · simp only [A,edgeSampleSecond] |
| erw [edgeSample_expect equations k (fun U h pos => firstError (m := P.m) (n := P.n) |
| (f.second (secondQuestion U h pos)) (sourceSigma J) (sourceEta J))] |
| exact h1 |
| · simp only [A,edgeSampleSecond] |
| erw [edgeSample_expect equations k (fun U h pos => secondError (m := P.m) (n := P.n) |
| (f.second (secondQuestion U h pos)) ((sourceEta J)^2/(P.m:ℝ)) (sourceSigma J) (sourceEta J))] |
| exact h2 |
| · simp only [A,edgeSampleSecond] |
| erw [edgeSample_expect equations k (fun U h pos => thirdError (m := P.m) (n := P.n) |
| (f.second (secondQuestion U h pos)) (sourceSigma J) (sourceEta J))] |
| exact h3 |
| end MinUncut.Outer |
|
|
| end |
|
|
| end OAI |
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|
