openAI-math-dataset / lean /OAI /Combinatorics /BalancedRyser /Probability /ConditionalOrderVariance.lean
| import OAI.Combinatorics.BalancedRyser.Probability.PermutationSubsetLaw | |
| import OAI.Combinatorics.BalancedRyser.Probability.UncoveredSubsetVariance | |
| namespace OAI | |
| open scoped BigOperators | |
| namespace Balanced.ConditionalOrderVariance | |
| open WithoutReplacement PermutationSubsetLaw UncoveredSampling | |
| theorem variance_two_images_conditioned_target | |
| {α β : Type*} [Fintype α] [DecidableEq α] [Fintype β] [DecidableEq β] | |
| (I : Finset α) (J : Finset β) (i x : α) (hi : i ∉ I) | |
| (g : Finset α → Finset β → ℝ) : | |
| finiteVariance | |
| ((Finset.univ.filter (fun σ : Equiv.Perm α => σ i = x)) ×ˢ | |
| (Finset.univ : Finset (Equiv.Perm β))) | |
| (fun p => g (I.map p.1.toEmbedding) (J.map p.2.toEmbedding)) = | |
| finiteVariance ((Finset.univ.erase x).powersetCard I.card ×ˢ | |
| (Finset.univ : Finset β).powersetCard J.card) (fun p => g p.1 p.2) := by | |
| unfold finiteVariance | |
| simp_rw [Finset.expect_product] | |
| rw [expect_two_images_conditioned_target I J i x hi g] | |
| exact expect_two_images_conditioned_target I J i x hi | |
| (fun S T => (g S T - (𝔼 A ∈ (Finset.univ.erase x).powersetCard I.card, | |
| 𝔼 B ∈ (Finset.univ : Finset β).powersetCard J.card, g A B))^2) | |
| theorem conditioned_order_variance_le | |
| {α β : Type*} [Fintype α] [DecidableEq α] [Fintype β] [DecidableEq β] | |
| (I : Finset α) (J : Finset β) (i x : α) (hi : i ∉ I) | |
| (g : Finset α → Finset β → ℝ) {c d : ℝ} (hc : 0 ≤ c) (hd : 0 ≤ d) | |
| (hfirst : ∀ S : Finset α, S ⊆ Finset.univ.erase x → S.card = I.card → | |
| ∀ T : Finset β, T ⊆ Finset.univ → T.card = J.card → | |
| ∀ a ∈ S, ∀ b ∈ Finset.univ.erase x, b ∉ S → | |
| |g S T - g (insert b (S.erase a)) T| ≤ c) | |
| (hsecond : ∀ S : Finset α, S ⊆ Finset.univ.erase x → S.card = I.card → | |
| ∀ T : Finset β, T ⊆ Finset.univ → T.card = J.card → | |
| ∀ a ∈ T, ∀ b ∈ Finset.univ, b ∉ T → | |
| |g S T - g S (insert b (T.erase a))| ≤ d) : | |
| finiteVariance | |
| ((Finset.univ.filter (fun σ : Equiv.Perm α => σ i = x)) ×ˢ | |
| (Finset.univ : Finset (Equiv.Perm β))) | |
| (fun p => g (I.map p.1.toEmbedding) (J.map p.2.toEmbedding)) ≤ | |
| (I.card : ℝ) * c^2 + (J.card : ℝ) * d^2 := by | |
| rw [variance_two_images_conditioned_target I J i x hi] | |
| have hI : I ⊆ Finset.univ.erase i := by | |
| intro a ha | |
| exact Finset.mem_erase.mpr ⟨ne_of_mem_of_not_mem ha hi, Finset.mem_univ _⟩ | |
| have hm : I.card ≤ (Finset.univ.erase x).card := by | |
| simpa using Finset.card_le_card hI | |
| exact finiteVariance_powersetCard_product_le _ _ _ _ g hc hd hm | |
| (Finset.card_le_card (Finset.subset_univ J)) hfirst hsecond | |
| section Affine | |
| variable {F : Type*} [Field F] [Fintype F] [DecidableEq F] | |
| def populationLines (U : Finset (Line F)) (S : Finset U) : Finset (Line F) := | |
| S.map ⟨Subtype.val, Subtype.val_injective⟩ | |
| omit [Field F] [Fintype F] in | |
| @[simp] theorem populationLines_insert (U : Finset (Line F)) (S : Finset U) (a : U) : | |
| populationLines U (insert a S) = insert a.val (populationLines U S) := by | |
| simp [populationLines] | |
| omit [Field F] [Fintype F] in | |
| @[simp] theorem populationLines_erase (U : Finset (Line F)) (S : Finset U) (a : U) : | |
| populationLines U (S.erase a) = (populationLines U S).erase a.val := by | |
| simp [populationLines, Finset.map_erase] | |
| omit [Field F] [Fintype F] [DecidableEq F] in | |
| theorem mem_populationLines (U : Finset (Line F)) (S : Finset U) {a : U} | |
| (ha : a ∈ S) : a.val ∈ populationLines U S := | |
| Finset.mem_map.mpr ⟨a, ha, rfl⟩ | |
| theorem plane_conditioned_order_variance_le | |
| (U V : Finset (Line F)) (I : Finset U) (J : Finset V) | |
| (i x : U) (hi : i ∉ I) : | |
| finiteVariance | |
| ((Finset.univ.filter (fun σ : Equiv.Perm U => σ i = x)) ×ˢ | |
| (Finset.univ : Finset (Equiv.Perm V))) | |
| (fun p => planeFraction | |
| (populationLines U (I.map p.1.toEmbedding)) | |
| (populationLines V (J.map p.2.toEmbedding))) ≤ | |
| 4 * ((I.card : ℝ) + J.card) / (Fintype.card F : ℝ)^2 := by | |
| have hc : 0 ≤ 2 / (Fintype.card F : ℝ) := by positivity | |
| have h := conditioned_order_variance_le I J i x hi | |
| (fun S T => planeFraction (populationLines U S) (populationLines V T)) hc hc | |
| (by | |
| intro S _ _ T _ _ a ha b _ _ | |
| simpa using planeFraction_exchange_left (populationLines U S) | |
| (populationLines V T) (mem_populationLines U S ha) b.val) | |
| (by | |
| intro S _ _ T _ _ a ha b _ _ | |
| simpa using planeFraction_exchange_right (populationLines U S) | |
| (populationLines V T) (mem_populationLines V T ha) b.val) | |
| exact h.trans_eq (by ring) | |
| theorem target_conditioned_order_variance_le | |
| (U V : Finset (Line F)) (hUV : Disjoint U V) | |
| (I : Finset U) (J : Finset V) (i x : U) (hi : i ∉ I) : | |
| finiteVariance | |
| ((Finset.univ.filter (fun σ : Equiv.Perm U => σ i = x)) ×ˢ | |
| (Finset.univ : Finset (Equiv.Perm V))) | |
| (fun p => targetFraction x.val | |
| (populationLines U (I.map p.1.toEmbedding)) | |
| (populationLines V (J.map p.2.toEmbedding))) ≤ | |
| 4 * ((I.card : ℝ) + J.card) / (Fintype.card F : ℝ)^2 := by | |
| have hc : 0 ≤ 2 / (Fintype.card F : ℝ) := by positivity | |
| have hxV (a : V) : x.val ≠ a.val := by | |
| intro h | |
| exact Finset.disjoint_left.mp hUV x.property (h ▸ a.property) | |
| have h := conditioned_order_variance_le I J i x hi | |
| (fun S T => targetFraction x.val (populationLines U S) (populationLines V T)) hc hc | |
| (by | |
| intro S hS _ T _ _ a ha b hb _ | |
| have hxa : x.val ≠ a.val := by | |
| intro h | |
| exact (Finset.mem_erase.mp (hS ha)).1 (Subtype.ext h.symm) | |
| have hxb : x.val ≠ b.val := by | |
| intro h | |
| exact (Finset.mem_erase.mp hb).1 (Subtype.ext h.symm) | |
| simpa using targetFraction_exchange_left x.val (populationLines U S) | |
| (populationLines V T) (mem_populationLines U S ha) b.val hxa hxb) | |
| (by | |
| intro S _ _ T _ _ a ha b _ _ | |
| simpa using targetFraction_exchange_right x.val (populationLines U S) | |
| (populationLines V T) (mem_populationLines V T ha) b.val (hxV a) (hxV b)) | |
| exact h.trans_eq (by ring) | |
| abbrev OrderPair (U V : Finset (Line F)) := Equiv.Perm U × Equiv.Perm V | |
| noncomputable def conditionedOrderLaw (U V : Finset (Line F)) (i x : U) : | |
| PMF (OrderPair U V) := | |
| PMF.uniformOfFinset | |
| ((Finset.univ.filter (fun σ : Equiv.Perm U => σ i = x)) ×ˢ | |
| (Finset.univ : Finset (Equiv.Perm V))) | |
| ⟨(Equiv.swap i x, Equiv.refl V), by simp⟩ | |
| theorem plane_conditioned_order_measure_variance_le | |
| (U V : Finset (Line F)) | |
| [MeasurableSpace (OrderPair U V)] [MeasurableSingletonClass (OrderPair U V)] | |
| (I : Finset U) (J : Finset V) (i x : U) (hi : i ∉ I) : | |
| ProbabilityTheory.variance | |
| (fun p : OrderPair U V => planeFraction | |
| (populationLines U (I.map p.1.toEmbedding)) | |
| (populationLines V (J.map p.2.toEmbedding))) | |
| (conditionedOrderLaw U V i x).toMeasure ≤ | |
| 4 * ((I.card : ℝ) + J.card) / (Fintype.card F : ℝ)^2 := by | |
| erw [conditionedOrderLaw, variance_uniformOfFinset] | |
| exact plane_conditioned_order_variance_le U V I J i x hi | |
| theorem target_conditioned_order_measure_variance_le | |
| (U V : Finset (Line F)) | |
| [MeasurableSpace (OrderPair U V)] [MeasurableSingletonClass (OrderPair U V)] | |
| (hUV : Disjoint U V) (I : Finset U) (J : Finset V) (i x : U) (hi : i ∉ I) : | |
| ProbabilityTheory.variance | |
| (fun p : OrderPair U V => targetFraction x.val | |
| (populationLines U (I.map p.1.toEmbedding)) | |
| (populationLines V (J.map p.2.toEmbedding))) | |
| (conditionedOrderLaw U V i x).toMeasure ≤ | |
| 4 * ((I.card : ℝ) + J.card) / (Fintype.card F : ℝ)^2 := by | |
| erw [conditionedOrderLaw, variance_uniformOfFinset] | |
| exact target_conditioned_order_variance_le U V hUV I J i x hi | |
| end Affine | |
| end Balanced.ConditionalOrderVariance | |
| end OAI | |