| import OAI.Geometry.HeilbronnTriangle.DigitColumnLaw |
| import OAI.Geometry.HeilbronnTriangle.BadTripleAverage |
| import OAI.Geometry.HeilbronnTriangle.FiniteFieldLabels |
|
|
| namespace OAI |
|
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|
|
| noncomputable section |
|
|
| namespace Problem355.DigitBadTripleAverage |
|
|
| open scoped BigOperators |
| open DigitColumnLaw |
|
|
| def labelTripleEquiv (Λ : Type*) : (Fin 3 → Λ) ≃ Λ × Λ × Λ where |
| toFun l := (l 0, l 1, l 2) |
| invFun t := ![t.1, t.2.1, t.2.2] |
| left_inv l := by funext i; fin_cases i <;> rfl |
| right_inv t := rfl |
|
|
| theorem uniform_label_average {Λ : Type*} [Fintype Λ] |
| (F : (Fin 3 → Λ) → ℝ) : |
| (∑ l : Fin 3 → Λ, uniformWeight (Fin 3 → Λ) l * F l) = |
| (∑ a : Λ, ∑ b : Λ, ∑ c : Λ, F ![a, b, c]) / |
| (Fintype.card Λ : ℝ) ^ 3 := by |
| classical |
| calc |
| _ = ∑ t : Λ × Λ × Λ, |
| F ![t.1, t.2.1, t.2.2] / (Fintype.card Λ : ℝ) ^ 3 := by |
| apply Fintype.sum_equiv (labelTripleEquiv Λ) |
| intro l |
| change uniformWeight (Fin 3 → Λ) l * F l = |
| F ![l 0, l 1, l 2] / (Fintype.card Λ : ℝ) ^ 3 |
| have hl : (![l 0, l 1, l 2] : Fin 3 → Λ) = l := by |
| funext i; fin_cases i <;> rfl |
| simp [uniformWeight, hl, div_eq_mul_inv, mul_comm] |
| _ = _ := by simp only [Fintype.sum_prod_type, Finset.sum_div] |
|
|
| lemma injective_labels {Λ : Type*} {a b c : Λ} |
| (hab : a ≠ b) (hac : a ≠ c) (hbc : b ≠ c) : |
| Function.Injective (![a, b, c] : Fin 3 → Λ) := by |
| intro i j hij |
| fin_cases i <;> fin_cases j <;> simp_all |
|
|
| theorem expectation_le_six {Λ X : Type*} |
| [Fintype Λ] [Nonempty Λ] [Fintype X] {k : ℕ} |
| (D probability : (Fin 3 → Λ) → Array X k → ℝ) |
| (A : ℝ) (hA : 0 ≤ A) |
| (hmoment : ∀ labels, |
| (∑ f : Array X k, uniformWeight (Array X k) f * D labels f) ≤ 2) |
| (hzero : ∀ labels f, Function.Injective labels → probability labels f = 0) |
| (hbad : ∀ labels f, ¬ Function.Injective labels → |
| probability labels f ≤ A * D labels f) : |
| (∑ s : Fin 3 → Latent Λ X k, |
| ConditionalSamples.productWeight (uniformWeight (Latent Λ X k)) s * |
| probability (tripleEquiv s).1 (tripleEquiv s).2) ≤ |
| 6 * A / (Fintype.card Λ : ℝ) := by |
| classical |
| rw [expectation_triple, uniform_label_average] |
| apply Sampling.bad_triple_average_le_six |
| (fun _ _ _ f => uniformWeight (Array X k) f) |
| (fun a b c f => D ![a, b, c] f) |
| (fun a b c f => probability ![a, b, c] f) A hA |
| · intro a b c f |
| exact uniformWeight_nonneg _ f |
| · intro a b c |
| exact hmoment ![a, b, c] |
| · intro a b c f hab hac hbc |
| exact hzero _ f (injective_labels hab hac hbc) |
| · intro a b c f hrep |
| apply hbad |
| intro hinj |
| rcases hrep with hab | hac | hbc |
| · have h := hinj (show (![a, b, c] : Fin 3 → Λ) 0 = ![a, b, c] 1 from hab) |
| exact (by decide : (0 : Fin 3) ≠ 1) h |
| · have h := hinj (show (![a, b, c] : Fin 3 → Λ) 0 = ![a, b, c] 2 from hac) |
| exact (by decide : (0 : Fin 3) ≠ 2) h |
| · have h := hinj (show (![a, b, c] : Fin 3 → Λ) 1 = ![a, b, c] 2 from hbc) |
| exact (by decide : (1 : Fin 3) ≠ 2) h |
|
|
| theorem latent_expectation_le_six {Λ X : Type*} |
| [Fintype Λ] [Nonempty Λ] [Fintype X] {k : ℕ} |
| (D probability : (Fin 3 → Latent Λ X k) → ℝ) |
| (A : ℝ) (hA : 0 ≤ A) |
| (hmoment : ∀ labels, |
| (∑ f : Array X k, uniformWeight (Array X k) f * |
| D (tripleEquiv.symm (labels, f))) ≤ 2) |
| (hzero : ∀ s, Function.Injective (fun i => (s i).1) → probability s = 0) |
| (hbad : ∀ s, ¬ Function.Injective (fun i => (s i).1) → |
| probability s ≤ A * D s) : |
| (∑ s : Fin 3 → Latent Λ X k, |
| ConditionalSamples.productWeight (uniformWeight (Latent Λ X k)) s * |
| probability s) ≤ 6 * A / (Fintype.card Λ : ℝ) := by |
| have h := expectation_le_six |
| (fun labels f => D (tripleEquiv.symm (labels, f))) |
| (fun labels f => probability (tripleEquiv.symm (labels, f))) A hA hmoment |
| (fun labels f hi => hzero _ hi) (fun labels f hi => hbad _ hi) |
| simpa only [Prod.mk.eta, Equiv.symm_apply_apply] using h |
|
|
| theorem expectation_le_six_prime_labels {r : ℕ} [Fact r.Prime] |
| [Fintype (FiniteFieldLabels.Label r)] {X : Type*} [Fintype X] {k : ℕ} |
| (D probability : (Fin 3 → FiniteFieldLabels.Label r) → Array X k → ℝ) |
| (A : ℝ) (hA : 0 ≤ A) |
| (hmoment : ∀ labels, |
| (∑ f : Array X k, uniformWeight (Array X k) f * D labels f) ≤ 2) |
| (hzero : ∀ labels f, Function.Injective labels → probability labels f = 0) |
| (hbad : ∀ labels f, ¬ Function.Injective labels → |
| probability labels f ≤ A * D labels f) : |
| (∑ s : Fin 3 → Latent (FiniteFieldLabels.Label r) X k, |
| ConditionalSamples.productWeight |
| (uniformWeight (Latent (FiniteFieldLabels.Label r) X k)) s * |
| probability (tripleEquiv s).1 (tripleEquiv s).2) ≤ |
| 6 * A / (r : ℝ) ^ 41 := by |
| have hc : Fintype.card (FiniteFieldLabels.Label r) = r ^ 41 := by |
| rw [← Nat.card_eq_fintype_card, FiniteFieldLabels.card_label] |
| rfl |
| simpa only [hc, Nat.cast_pow] using |
| expectation_le_six D probability A hA hmoment hzero hbad |
|
|
| theorem expectation_le_of_lifting_bound {r : ℕ} [Fact r.Prime] |
| [Fintype (FiniteFieldLabels.Label r)] {X : Type*} [Fintype X] {k : ℕ} |
| (D probability : (Fin 3 → FiniteFieldLabels.Label r) → Array X k → ℝ) |
| (C R h N : ℝ) (hC : 0 ≤ C) (hh : 0 ≤ h) (hN : 0 < N) |
| (hmoment : ∀ labels, |
| (∑ f : Array X k, uniformWeight (Array X k) f * D labels f) ≤ 2) |
| (hzero : ∀ labels f, Function.Injective labels → probability labels f = 0) |
| (hbad : ∀ labels f, ¬ Function.Injective labels → |
| probability labels f ≤ (64 / 21 : ℝ) * C * R ^ 2 * h * D labels f / N ^ 3) : |
| (∑ s : Fin 3 → Latent (FiniteFieldLabels.Label r) X k, |
| ConditionalSamples.productWeight |
| (uniformWeight (Latent (FiniteFieldLabels.Label r) X k)) s * |
| probability (tripleEquiv s).1 (tripleEquiv s).2) ≤ |
| (128 / 7 : ℝ) * C * R ^ 2 * h / (N ^ 3 * (r : ℝ) ^ 41) := by |
| have hbound := expectation_le_six_prime_labels D probability |
| ((64 / 21 : ℝ) * C * R ^ 2 * h / N ^ 3) (by positivity) |
| hmoment hzero (by |
| intro labels f hi |
| convert hbad labels f hi using 1; ring) |
| convert hbound using 1; ring |
|
|
| end Problem355.DigitBadTripleAverage |
|
|
| namespace Problem355.DigitColumnLaw |
|
|
| open scoped BigOperators |
|
|
| def labelsEquiv (Λ : Type*) : (Fin 3 → Λ) ≃ Λ × Λ × Λ where |
| toFun l := (l 0, l 1, l 2) |
| invFun t := ![t.1, t.2.1, t.2.2] |
| left_inv l := by funext i; fin_cases i <;> rfl |
| right_inv t := rfl |
|
|
| theorem expectation_labels_le_of_repeated |
| {Λ : Type*} [Fintype Λ] [DecidableEq Λ] [Nonempty Λ] |
| (F : (Fin 3 → Λ) → ℝ) (M : ℝ) (hM : 0 ≤ M) |
| (hbound : ∀ l, F l ≤ M) |
| (hzero : ∀ l, l 0 ≠ l 1 → l 0 ≠ l 2 → l 1 ≠ l 2 → F l = 0) : |
| (∑ l : Fin 3 → Λ, uniformWeight (Fin 3 → Λ) l * F l) ≤ |
| 3 * M / (Fintype.card Λ : ℝ) := by |
| have h := Sampling.average_repeated_labels_le |
| (fun a b c => F ![a, b, c]) M hM |
| (fun a b c => hbound _) (fun a b c hab hac hbc => hzero _ hab hac hbc) |
| have hs : (∑ l : Fin 3 → Λ, F l) = ∑ a, ∑ b, ∑ c, F ![a, b, c] := by |
| calc |
| _ = ∑ t : Λ × Λ × Λ, F ((labelsEquiv Λ).symm t) := |
| (labelsEquiv Λ).symm.sum_comp F |>.symm |
| _ = _ := by simp only [Fintype.sum_prod_type]; rfl |
| have he : (∑ l : Fin 3 → Λ, uniformWeight (Fin 3 → Λ) l * F l) = |
| (∑ l : Fin 3 → Λ, F l) / (Fintype.card Λ : ℝ) ^ 3 := by |
| unfold uniformWeight |
| rw [← Finset.mul_sum] |
| simp only [Fintype.card_fun, Fintype.card_fin, Nat.cast_pow] |
| ring |
| rw [he, hs] |
| exact h |
|
|
| theorem expectation_triple_le_of_repeated |
| {Λ X : Type*} [Fintype Λ] [DecidableEq Λ] [Nonempty Λ] [Fintype X] {k : ℕ} |
| (F : (Fin 3 → Λ) → Array X k → ℝ) (M : ℝ) (hM : 0 ≤ M) |
| (hbound : ∀ l, (∑ f : Array X k, uniformWeight (Array X k) f * F l f) ≤ M) |
| (hzero : ∀ l f, l 0 ≠ l 1 → l 0 ≠ l 2 → l 1 ≠ l 2 → F l f = 0) : |
| (∑ s : Fin 3 → Latent Λ X k, |
| ConditionalSamples.productWeight (uniformWeight (Latent Λ X k)) s * |
| F (tripleEquiv s).1 (tripleEquiv s).2) ≤ |
| 3 * M / (Fintype.card Λ : ℝ) := by |
| rw [expectation_triple] |
| apply expectation_labels_le_of_repeated _ M hM hbound |
| intro l hab hac hbc |
| simp only [hzero l _ hab hac hbc, mul_zero, Finset.sum_const_zero] |
|
|
| theorem bad_triple_expectation_le |
| {Λ X : Type*} [Fintype Λ] [DecidableEq Λ] [Nonempty Λ] [Fintype X] {k : ℕ} |
| (D : (Fin 3 → Λ) → Array X k → ℝ) |
| (p : (Fin 3 → Latent Λ X k) → ℝ) (A : ℝ) (hA : 0 ≤ A) |
| (hmoment : ∀ l, (∑ f : Array X k, uniformWeight (Array X k) f * D l f) ≤ 2) |
| (hbound : ∀ l f, p (tripleEquiv.symm (l, f)) ≤ A * D l f) |
| (hzero : ∀ s, (s 0).1 ≠ (s 1).1 → (s 0).1 ≠ (s 2).1 → |
| (s 1).1 ≠ (s 2).1 → p s = 0) : |
| (∑ s : Fin 3 → Latent Λ X k, |
| ConditionalSamples.productWeight (uniformWeight (Latent Λ X k)) s * p s) ≤ |
| 6 * A / (Fintype.card Λ : ℝ) := by |
| have h := expectation_triple_le_of_repeated |
| (fun l f => p (tripleEquiv.symm (l, f))) (A * 2) (by positivity) |
| (by |
| intro l |
| calc |
| _ ≤ ∑ f : Array X k, uniformWeight (Array X k) f * (A * D l f) := |
| Finset.sum_le_sum fun f _ => mul_le_mul_of_nonneg_left |
| (hbound l f) (uniformWeight_nonneg _ f) |
| _ = A * ∑ f : Array X k, uniformWeight (Array X k) f * D l f := by |
| rw [Finset.mul_sum] |
| apply Finset.sum_congr rfl |
| intro f _ |
| ring |
| _ ≤ A * 2 := mul_le_mul_of_nonneg_left (hmoment l) hA) |
| (by intro l f hab hac hbc; exact hzero _ hab hac hbc) |
| simpa only [Prod.mk.eta, Equiv.symm_apply_apply, |
| show 3 * (A * 2) = 6 * A by ring] using h |
|
|
| end Problem355.DigitColumnLaw |
|
|
| end |
|
|
| end OAI |
|
|
