openAI-math-dataset / lean /OAI /Combinatorics /BalancedRyser /Probability /CoverCountingExposure.lean
| import OAI.Combinatorics.BalancedRyser.Deletion.Deletion | |
| import OAI.Combinatorics.BalancedRyser.Probability.AtypicalLines | |
| import OAI.Combinatorics.BalancedRyser.Probability.PermutationHistoryLaw | |
| namespace OAI | |
| namespace Balanced.CoverCountingExposure | |
| noncomputable section | |
| open Finset PermutationHistoryLaw | |
| open scoped BigOperators | |
| attribute [local instance] Classical.propDecidable | |
| variable {F : Type*} [Field F] [Fintype F] [DecidableEq F] | |
| def Marked (U : Finset (Point F)) (ε : ℝ) (l : Line F) : Prop := | |
| ε < |(affineIntersectionCount U l : ℝ) / (Fintype.card F : ℝ) - | |
| (U.card : ℝ) / (Fintype.card F : ℝ) ^ 2| | |
| omit [Field F] in | |
| theorem order_sq_le_card_unused (I : Finset (Line F)) | |
| (hI : I.card ≤ Fintype.card F) (σ : Equiv.Perm (Line F)) : | |
| (Fintype.card F : ℝ) ^ 2 ≤ ((unused I σ).card : ℝ) := by | |
| have hcard := Finset.card_sdiff_add_card_eq_card | |
| (Finset.subset_univ (I.map σ.toEmbedding)) | |
| have hcast := congrArg (fun n : ℕ => (n : ℝ)) hcard | |
| have hIR : (I.card : ℝ) ≤ Fintype.card F := by exact_mod_cast hI | |
| simp only [Finset.card_map, Finset.card_univ, Fintype.card_prod, | |
| Fintype.card_option, Nat.cast_add, Nat.cast_mul, Nat.cast_one] at hcast | |
| change (Fintype.card F : ℝ) ^ 2 ≤ | |
| ((Finset.univ \ I.map σ.toEmbedding).card : ℝ) | |
| nlinarith | |
| theorem conditional_mark_le (I : Finset (Line F)) | |
| (hI : I.card ≤ Fintype.card F) (σ : Equiv.Perm (Line F)) | |
| (U : Finset (Point F)) {ε : ℝ} (hε : 0 < ε) : | |
| (𝔼 l ∈ unused I σ, if Marked U ε l then (1 : ℝ) else 0) ≤ | |
| 1 / (4 * ε ^ 2 * (Fintype.card F : ℝ)) := by | |
| have hq : (0 : ℝ) < Fintype.card F := by exact_mod_cast Fintype.card_pos (α := F) | |
| have hden := order_sq_le_card_unused I hI σ | |
| have hdenpos : (0 : ℝ) < (unused I σ).card := | |
| (sq_pos_of_pos hq).trans_le hden | |
| have hcount : (((unused I σ).filter (Marked U ε)).card : ℝ) ≤ | |
| (Fintype.card F : ℝ) / (4 * ε ^ 2) := by | |
| apply le_trans _ (AtypicalLines.affine_card_marked_lines_le U ε hε) | |
| exact Nat.cast_le.mpr (Finset.card_le_card | |
| (Finset.filter_subset_filter (Marked U ε) (Finset.subset_univ (unused I σ)))) | |
| rw [Finset.expect_eq_sum_div_card] | |
| simp only [Finset.sum_boole] | |
| calc | |
| (((unused I σ).filter (Marked U ε)).card : ℝ) / ((unused I σ).card : ℝ) ≤ | |
| ((Fintype.card F : ℝ) / (4 * ε ^ 2)) / ((unused I σ).card : ℝ) := | |
| div_le_div_of_nonneg_right hcount hdenpos.le | |
| _ ≤ ((Fintype.card F : ℝ) / (4 * ε ^ 2)) / (Fintype.card F : ℝ) ^ 2 := | |
| div_le_div_of_nonneg_left (by positivity) (sq_pos_of_pos hq) hden | |
| _ = _ := by field_simp | |
| theorem past_event_and_mark_le | |
| (I : Finset (Line F)) (hI : I.card ≤ Fintype.card F) | |
| (i : Line F) (hi : i ∉ I) | |
| (A : Equiv.Perm (Line F) → Prop) | |
| (U : Equiv.Perm (Line F) → Finset (Point F)) | |
| (hA : ∀ σ τ, σ ∈ histories I τ → (A σ ↔ A τ)) | |
| (hU : ∀ σ τ, σ ∈ histories I τ → U σ = U τ) | |
| {ε : ℝ} (hε : 0 < ε) : | |
| (𝔼 σ : Equiv.Perm (Line F), | |
| if A σ ∧ Marked (U σ) ε (σ i) then (1 : ℝ) else 0) ≤ | |
| (1 / (4 * ε ^ 2 * (Fintype.card F : ℝ))) * | |
| (𝔼 σ : Equiv.Perm (Line F), if A σ then (1 : ℝ) else 0) := by | |
| rw [expect_past_next I i hi | |
| (fun σ l => if A σ ∧ Marked (U σ) ε l then (1 : ℝ) else 0) (by | |
| intro σ τ hστ | |
| funext l | |
| simp only [hA σ τ hστ, hU σ τ hστ])] | |
| rw [Finset.mul_expect] | |
| apply Finset.expect_le_expect | |
| intro σ _ | |
| by_cases h : A σ | |
| · simp only [h, true_and, ite_true, mul_one] | |
| exact conditional_mark_le I hI σ (U σ) hε | |
| · simp only [h, false_and, ite_false, Finset.expect_const_zero, mul_zero, le_refl] | |
| def pastUncovered (D : Finset (Line F)) (I : Finset (Line F)) | |
| (σ : Equiv.Perm (Line F)) : Finset (Point F) := | |
| uncoveredBy (D ∪ I.map σ.toEmbedding) | |
| theorem pastUncovered_eq_of_history (D I : Finset (Line F)) | |
| (σ τ : Equiv.Perm (Line F)) (h : σ ∈ histories I τ) : | |
| pastUncovered D I σ = pastUncovered D I τ := by | |
| simp only [pastUncovered, map_eq_of_history I σ τ h] | |
| theorem pastUncovered_eq_of_larger_history (D J I : Finset (Line F)) | |
| (hJI : J ⊆ I) (σ τ : Equiv.Perm (Line F)) (h : σ ∈ histories I τ) : | |
| pastUncovered D J σ = pastUncovered D J τ := by | |
| apply pastUncovered_eq_of_history D J σ τ | |
| apply (mem_histories J τ σ).mpr | |
| intro j hj | |
| exact (mem_histories I τ σ).mp h j (hJI hj) | |
| theorem earlier_cover_mark_iff (D J I : Finset (Line F)) | |
| (hJI : J ⊆ I) (j : Line F) (hj : j ∈ I) | |
| (σ τ : Equiv.Perm (Line F)) (h : σ ∈ histories I τ) (ε : ℝ) : | |
| Marked (pastUncovered D J σ) ε (σ j) ↔ | |
| Marked (pastUncovered D J τ) ε (τ j) := by | |
| rw [pastUncovered_eq_of_larger_history D J I hJI σ τ h, | |
| (mem_histories I τ σ).mp h j hj] | |
| theorem earlier_deletion_mark_iff (D J I : Finset (Line F)) | |
| (hJI : J ⊆ I) (l : Line F) | |
| (σ τ : Equiv.Perm (Line F)) (h : σ ∈ histories I τ) (ε : ℝ) : | |
| Marked (pastUncovered D J σ) ε l ↔ | |
| Marked (pastUncovered D J τ) ε l := by | |
| rw [pastUncovered_eq_of_larger_history D J I hJI σ τ h] | |
| def priorMixedMarks {δ κ : Type*} (sD : Finset δ) (sC : Finset κ) | |
| (pastD₁ : δ → Finset (Line F)) (pastC₁ : δ → Finset (Line F)) | |
| (targetD : δ → Line F) | |
| (pastD₂ : κ → Finset (Line F)) (pastC₂ : κ → Finset (Line F)) | |
| (positionC : κ → Line F) (ε : ℝ) (σ : Equiv.Perm (Line F)) : Prop := | |
| (∀ j ∈ sD, Marked (pastUncovered (pastD₁ j) (pastC₁ j) σ) ε (targetD j)) ∧ | |
| ∀ j ∈ sC, Marked (pastUncovered (pastD₂ j) (pastC₂ j) σ) ε (σ (positionC j)) | |
| theorem priorMixedMarks_iff_of_history {δ κ : Type*} | |
| (sD : Finset δ) (sC : Finset κ) | |
| (pastD₁ : δ → Finset (Line F)) (pastC₁ : δ → Finset (Line F)) | |
| (targetD : δ → Line F) | |
| (pastD₂ : κ → Finset (Line F)) (pastC₂ : κ → Finset (Line F)) | |
| (positionC : κ → Line F) (I : Finset (Line F)) | |
| (hD : ∀ j ∈ sD, pastC₁ j ⊆ I) | |
| (hC : ∀ j ∈ sC, pastC₂ j ⊆ I ∧ positionC j ∈ I) | |
| (σ τ : Equiv.Perm (Line F)) (h : σ ∈ histories I τ) (ε : ℝ) : | |
| priorMixedMarks sD sC pastD₁ pastC₁ targetD pastD₂ pastC₂ positionC ε σ ↔ | |
| priorMixedMarks sD sC pastD₁ pastC₁ targetD pastD₂ pastC₂ positionC ε τ := by | |
| unfold priorMixedMarks | |
| constructor | |
| · rintro ⟨h₁, h₂⟩ | |
| constructor | |
| · intro j hj | |
| exact (earlier_deletion_mark_iff _ _ I (hD j hj) _ σ τ h ε).mp (h₁ j hj) | |
| · intro j hj | |
| exact (earlier_cover_mark_iff _ _ I (hC j hj).1 _ (hC j hj).2 σ τ h ε).mp (h₂ j hj) | |
| · rintro ⟨h₁, h₂⟩ | |
| constructor | |
| · intro j hj | |
| exact (earlier_deletion_mark_iff _ _ I (hD j hj) _ σ τ h ε).mpr (h₁ j hj) | |
| · intro j hj | |
| exact (earlier_cover_mark_iff _ _ I (hC j hj).1 _ (hC j hj).2 σ τ h ε).mpr (h₂ j hj) | |
| theorem past_event_and_actual_mark_le | |
| (D I : Finset (Line F)) (hI : I.card ≤ Fintype.card F) | |
| (i : Line F) (hi : i ∉ I) | |
| (A : Equiv.Perm (Line F) → Prop) | |
| (hA : ∀ σ τ, σ ∈ histories I τ → (A σ ↔ A τ)) | |
| {ε : ℝ} (hε : 0 < ε) : | |
| (𝔼 σ : Equiv.Perm (Line F), | |
| if A σ ∧ Marked (pastUncovered D I σ) ε (σ i) then (1 : ℝ) else 0) ≤ | |
| (1 / (4 * ε ^ 2 * (Fintype.card F : ℝ))) * | |
| (𝔼 σ : Equiv.Perm (Line F), if A σ then (1 : ℝ) else 0) := | |
| past_event_and_mark_le I hI i hi A (pastUncovered D I) hA | |
| (pastUncovered_eq_of_history D I) hε | |
| theorem prior_mixed_marks_and_actual_mark_le {δ κ : Type*} | |
| (sD : Finset δ) (sC : Finset κ) | |
| (pastD₁ : δ → Finset (Line F)) (pastC₁ : δ → Finset (Line F)) | |
| (targetD : δ → Line F) | |
| (pastD₂ : κ → Finset (Line F)) (pastC₂ : κ → Finset (Line F)) | |
| (positionC : κ → Line F) | |
| (D I : Finset (Line F)) (hI : I.card ≤ Fintype.card F) | |
| (i : Line F) (hi : i ∉ I) | |
| (hD : ∀ j ∈ sD, pastC₁ j ⊆ I) | |
| (hC : ∀ j ∈ sC, pastC₂ j ⊆ I ∧ positionC j ∈ I) | |
| {ε : ℝ} (hε : 0 < ε) : | |
| let A := priorMixedMarks sD sC pastD₁ pastC₁ targetD pastD₂ pastC₂ positionC ε | |
| (𝔼 σ : Equiv.Perm (Line F), | |
| if A σ ∧ Marked (pastUncovered D I σ) ε (σ i) then (1 : ℝ) else 0) ≤ | |
| (1 / (4 * ε ^ 2 * (Fintype.card F : ℝ))) * | |
| (𝔼 σ : Equiv.Perm (Line F), if A σ then (1 : ℝ) else 0) := by | |
| apply past_event_and_actual_mark_le D I hI i hi _ _ hε | |
| intro σ τ hστ | |
| exact priorMixedMarks_iff_of_history sD sC pastD₁ pastC₁ targetD pastD₂ pastC₂ | |
| positionC I hD hC σ τ hστ ε | |
| theorem measureReal_past_event_and_actual_mark_le | |
| [MeasurableSpace (Equiv.Perm (Line F))] | |
| [MeasurableSingletonClass (Equiv.Perm (Line F))] | |
| (D I : Finset (Line F)) (hI : I.card ≤ Fintype.card F) | |
| (i : Line F) (hi : i ∉ I) | |
| (A : Equiv.Perm (Line F) → Prop) | |
| (hA : ∀ σ τ, σ ∈ histories I τ → (A σ ↔ A τ)) | |
| {ε : ℝ} (hε : 0 < ε) : | |
| (PMF.uniformOfFintype (Equiv.Perm (Line F))).toMeasure.real | |
| {σ | A σ ∧ Marked (pastUncovered D I σ) ε (σ i)} ≤ | |
| (1 / (4 * ε ^ 2 * (Fintype.card F : ℝ))) * | |
| (PMF.uniformOfFintype (Equiv.Perm (Line F))).toMeasure.real {σ | A σ} := by | |
| simpa only [uniform_measureReal_eq_expect] using | |
| past_event_and_actual_mark_le D I hI i hi A hA hε | |
| theorem measure_actual_mark_inter_past_le | |
| [MeasurableSpace (Equiv.Perm (Line F))] | |
| [MeasurableSingletonClass (Equiv.Perm (Line F))] | |
| (D I : Finset (Line F)) (hI : I.card ≤ Fintype.card F) | |
| (i : Line F) (hi : i ∉ I) | |
| (A : Equiv.Perm (Line F) → Prop) | |
| (hA : ∀ σ τ, σ ∈ histories I τ → (A σ ↔ A τ)) | |
| {ε : ℝ} (hε : 0 < ε) : | |
| (PMF.uniformOfFintype (Equiv.Perm (Line F))).toMeasure | |
| ({σ | Marked (pastUncovered D I σ) ε (σ i)} ∩ {σ | A σ}) ≤ | |
| ENNReal.ofReal (1 / (4 * ε ^ 2 * (Fintype.card F : ℝ))) * | |
| (PMF.uniformOfFintype (Equiv.Perm (Line F))).toMeasure {σ | A σ} := by | |
| apply (ENNReal.toReal_le_toReal | |
| (MeasureTheory.measure_ne_top _ _) | |
| (ENNReal.mul_ne_top ENNReal.ofReal_ne_top (MeasureTheory.measure_ne_top _ _))).mp | |
| rw [ENNReal.toReal_mul, ENNReal.toReal_ofReal (by positivity)] | |
| have h := measureReal_past_event_and_actual_mark_le D I hI i hi A hA hε | |
| simpa only [MeasureTheory.Measure.real, Set.ofPred_and, Set.inter_comm] using h | |
| omit [Field F] in | |
| theorem witness_marginal (J : Finset (Line F)) (g : Finset (Line F) → ℝ) : | |
| (𝔼 σ : Equiv.Perm (Line F), g (J.map σ.toEmbedding)) = | |
| 𝔼 C ∈ (Finset.univ : Finset (Line F)).powersetCard J.card, g C := | |
| PermutationSubsetLaw.expect_image J g | |
| omit [Field F] in | |
| theorem witness_and_order_law (J : Finset (Line F)) | |
| (g : Finset (Line F) → Equiv.Perm (Line F) → ℝ) : | |
| (𝔼 σ : Equiv.Perm (Line F), g (J.map σ.toEmbedding) σ) = | |
| 𝔼 C ∈ (Finset.univ : Finset (Line F)).powersetCard J.card, | |
| 𝔼 σ ∈ Finset.univ.filter (fun σ : Equiv.Perm (Line F) => | |
| J.map σ.toEmbedding = C), g C σ := by | |
| rw [expect_by_image J (fun σ => g (J.map σ.toEmbedding) σ)] | |
| apply Finset.expect_congr rfl | |
| intro C _ | |
| apply Finset.expect_congr rfl | |
| intro σ hσ | |
| rw [(Finset.mem_filter.mp hσ).2] | |
| theorem independent_outer_past_event_and_actual_mark_le | |
| {Ω : Type*} (S : Finset Ω) (w : Ω → ℝ) (hw : ∀ ω ∈ S, 0 ≤ w ω) | |
| (D : Ω → Finset (Line F)) (I : Finset (Line F)) | |
| (hI : I.card ≤ Fintype.card F) (i : Line F) (hi : i ∉ I) | |
| (A : Ω → Equiv.Perm (Line F) → Prop) | |
| (hA : ∀ ω ∈ S, ∀ σ τ, σ ∈ histories I τ → (A ω σ ↔ A ω τ)) | |
| {ε : ℝ} (hε : 0 < ε) : | |
| (∑ ω ∈ S, w ω * (𝔼 σ : Equiv.Perm (Line F), | |
| if A ω σ ∧ Marked (pastUncovered (D ω) I σ) ε (σ i) then (1 : ℝ) else 0)) ≤ | |
| (1 / (4 * ε ^ 2 * (Fintype.card F : ℝ))) * | |
| ∑ ω ∈ S, w ω * | |
| (𝔼 σ : Equiv.Perm (Line F), if A ω σ then (1 : ℝ) else 0) := by | |
| calc | |
| _ ≤ ∑ ω ∈ S, w ω * ((1 / (4 * ε ^ 2 * (Fintype.card F : ℝ))) * | |
| (𝔼 σ : Equiv.Perm (Line F), if A ω σ then (1 : ℝ) else 0)) := by | |
| apply Finset.sum_le_sum | |
| intro ω hω | |
| exact mul_le_mul_of_nonneg_left | |
| (past_event_and_actual_mark_le (D ω) I hI i hi (A ω) (hA ω hω) hε) (hw ω hω) | |
| _ = _ := by | |
| rw [Finset.mul_sum] | |
| apply Finset.sum_congr rfl | |
| intro ω _ | |
| ring | |
| end | |
| end Balanced.CoverCountingExposure | |
| end OAI | |