openAI-math-dataset / lean /OAI /Combinatorics /BalancedRyser /Probability /ConditionalOrderLaw.lean
| import OAI.Combinatorics.BalancedRyser.Probability.ConditionalOrderVariance | |
| namespace OAI | |
| open scoped BigOperators ENNReal | |
| namespace Balanced.ConditionalOrderLaw | |
| theorem filter_uniform_eq {Ω : Type*} [Fintype Ω] [Nonempty Ω] | |
| (s : Finset Ω) (hs : s.Nonempty) | |
| (h : ∃ a ∈ (s : Set Ω), a ∈ (PMF.uniformOfFintype Ω).support) : | |
| (PMF.uniformOfFintype Ω).filter (s : Set Ω) h = PMF.uniformOfFinset s hs := by | |
| classical | |
| have hn : (Fintype.card Ω : ℝ≥0∞) ≠ 0 := by exact_mod_cast Fintype.card_ne_zero | |
| have hs0 : (s.card : ℝ≥0∞) ≠ 0 := by exact_mod_cast hs.card_pos.ne' | |
| have hsum : (∑' a : Ω, (s : Set Ω).indicator (PMF.uniformOfFintype Ω) a) = | |
| (s.card : ℝ≥0∞) * (Fintype.card Ω : ℝ≥0∞)⁻¹ := by | |
| simp [tsum_fintype, Set.indicator, Finset.sum_ite_mem] | |
| ext a | |
| rw [PMF.filter_apply, hsum, PMF.uniformOfFinset_apply] | |
| by_cases ha : a ∈ s | |
| · simp only [Set.indicator_of_mem ha, PMF.uniformOfFintype_apply, ite_eq_left ha] | |
| rw [ENNReal.mul_inv (Or.inl hs0) (Or.inl (ENNReal.natCast_ne_top _)), inv_inv] | |
| calc | |
| _ = (s.card : ℝ≥0∞)⁻¹ * | |
| ((Fintype.card Ω : ℝ≥0∞)⁻¹ * Fintype.card Ω) := by ac_rfl | |
| _ = _ := by rw [ENNReal.inv_mul_cancel hn (ENNReal.natCast_ne_top _), mul_one] | |
| · simp [ha] | |
| open ConditionalOrderVariance MeasureTheory | |
| variable {F : Type*} [Field F] [Fintype F] [DecidableEq F] | |
| omit [Field F] [Fintype F] in | |
| theorem target_event_supported (U V : Finset (Line F)) (i x : U) : | |
| ∃ p ∈ {p : OrderPair U V | p.1 i = x}, | |
| p ∈ (PMF.uniformOfFintype (OrderPair U V)).support := by | |
| exact ⟨(Equiv.swap i x, Equiv.refl V), by simp, by simp⟩ | |
| omit [Field F] [Fintype F] in | |
| theorem conditioned_eq_filter (U V : Finset (Line F)) (i x : U) : | |
| conditionedOrderLaw U V i x = | |
| (PMF.uniformOfFintype (OrderPair U V)).filter | |
| {p | p.1 i = x} (target_event_supported U V i x) := by | |
| classical | |
| let s := (Finset.univ.filter (fun σ : Equiv.Perm U => σ i = x)) ×ˢ | |
| (Finset.univ : Finset (Equiv.Perm V)) | |
| have hs : s.Nonempty := ⟨(Equiv.swap i x, Equiv.refl V), by simp [s]⟩ | |
| have hset : (s : Set (OrderPair U V)) = {p | p.1 i = x} := by | |
| ext p | |
| simp [s] | |
| have hsupport : ∃ p ∈ (s : Set (OrderPair U V)), | |
| p ∈ (PMF.uniformOfFintype (OrderPair U V)).support := by | |
| rw [hset] | |
| exact target_event_supported U V i x | |
| change PMF.uniformOfFinset s hs = _ | |
| have h := (filter_uniform_eq s hs hsupport).symm | |
| simpa only [hset] using h | |
| omit [Field F] in | |
| theorem target_event_probability (U V : Finset (Line F)) | |
| [MeasurableSpace (OrderPair U V)] [MeasurableSingletonClass (OrderPair U V)] | |
| (i x : U) : | |
| (PMF.uniformOfFintype (OrderPair U V)).toMeasure.real {p | p.1 i = x} = | |
| 1 / (U.card : ℝ) := by | |
| classical | |
| rw [← integral_indicator_one (Set.to_countable _).measurableSet, | |
| PMF.uniformOfFintype, WithoutReplacement.integral_uniformOfFinset, | |
| ← Finset.univ_product_univ, Finset.expect_product] | |
| simp only [Set.indicator, Set.mem_ofPred_eq, Pi.one_apply, Fintype.expect_const] | |
| simpa only [Fintype.card_coe] using PermutationSubsetLaw.target_probability i x | |
| omit [Field F] in | |
| theorem target_event_probability_pos (U V : Finset (Line F)) | |
| [MeasurableSpace (OrderPair U V)] [MeasurableSingletonClass (OrderPair U V)] | |
| (i x : U) : | |
| 0 < (PMF.uniformOfFintype (OrderPair U V)).toMeasure.real {p | p.1 i = x} := by | |
| rw [target_event_probability] | |
| have hU : (0 : ℝ) < U.card := by | |
| exact_mod_cast (Finset.card_pos.mpr ⟨i.val, i.property⟩) | |
| positivity | |
| omit [Field F] [Fintype F] [DecidableEq F] in | |
| theorem uniform_pair_atom {α β : Type*} [Fintype α] [Nonempty α] | |
| [Fintype β] [Nonempty β] (a : α) (b : β) : | |
| PMF.uniformOfFintype (α × β) (a, b) = | |
| PMF.uniformOfFintype α a * PMF.uniformOfFintype β b := by | |
| simp only [PMF.uniformOfFintype_apply, Fintype.card_prod, Nat.cast_mul] | |
| exact ENNReal.mul_inv (Or.inl (by exact_mod_cast Fintype.card_ne_zero)) | |
| (Or.inl (ENNReal.natCast_ne_top _)) | |
| end Balanced.ConditionalOrderLaw | |
| end OAI | |