openAI-math-dataset / lean /OAI /Probability /MatroidSecretary /Sampling /UniformPermutationImage.lean
| import Mathlib.Data.Fintype.Perm | |
| import Mathlib.Logic.Equiv.Fintype | |
| import Mathlib.Data.Finset.Powerset | |
| import Mathlib.Algebra.BigOperators.Group.Finset.Basic | |
| import Mathlib.Basic.Real.Basic | |
| import Mathlib.Tactic | |
| namespace OAI | |
| /-! Counting core for the secretary random-prefix law: a uniformly random | |
| permutation sends each fixed finite set uniformly onto sets of the same size. | |
| No conditioning on the complete secretary seed is asserted here. | |
| Source: `sections/secretary.tex`, equation `eq:secretary-prefix-mask`, | |
| lines 54--66 of the pinned manuscript; SHA256 | |
| `3554fe0f7296782edf63cc7f1305c377e7a7b9d8f8948b9510ea4ad02a702d63`. | |
| These are supporting counting lemmas, not the complete secretary endpoint. | |
| The empty ambient type and empty/full fixed subsets are included. -/ | |
| namespace MatroidProphet.Secretary | |
| open Finset | |
| variable {α : Type*} [Fintype α] [DecidableEq α] | |
| /-- Equal-cardinality target sets have equally many permutation preimages. -/ | |
| theorem permutation_image_count_eq (s t u : Finset α) (htu : t.card = u.card) : | |
| (∑ σ : Equiv.Perm α, if s.map σ.toEmbedding = t then (1 : ℝ) else 0) = | |
| ∑ σ : Equiv.Perm α, if s.map σ.toEmbedding = u then (1 : ℝ) else 0 := by | |
| classical | |
| obtain ⟨ρ, hρ⟩ := Equiv.Perm.exists_map_finset_eq t u htu | |
| let e : Equiv.Perm α ≃ Equiv.Perm α := | |
| { toFun := fun σ => σ.trans ρ | |
| invFun := fun σ => σ.trans ρ.symm | |
| left_inv := fun σ => by ext a; simp | |
| right_inv := fun σ => by ext a; simp } | |
| have he (σ : Equiv.Perm α) : | |
| s.map (e σ).toEmbedding = u ↔ s.map σ.toEmbedding = t := by | |
| have hmap : s.map (e σ).toEmbedding = (s.map σ.toEmbedding).map ρ.toEmbedding := by | |
| simp [e, Finset.map_map, Function.Embedding.trans] | |
| rw [hmap, ← hρ] | |
| exact Finset.map_injective ρ.toEmbedding |>.eq_iff | |
| calc | |
| _ = ∑ σ : Equiv.Perm α, if s.map (e σ).toEmbedding = u then (1 : ℝ) else 0 := by | |
| apply Finset.sum_congr rfl | |
| intro σ _ | |
| simp only [he] | |
| _ = _ := e.sum_comp (fun σ => if s.map σ.toEmbedding = u then (1 : ℝ) else 0) | |
| /-- Exact probability numerator for a prescribed image of a fixed set. -/ | |
| theorem permutation_image_count (s t : Finset α) (hst : s.card = t.card) : | |
| (∑ σ : Equiv.Perm α, if s.map σ.toEmbedding = t then (1 : ℝ) else 0) = | |
| (Fintype.card (Equiv.Perm α) : ℝ) / (Fintype.card α).choose s.card := by | |
| classical | |
| let C : ℝ := ∑ σ : Equiv.Perm α, if s.map σ.toEmbedding = t then (1 : ℝ) else 0 | |
| let T := (Finset.univ : Finset α).powersetCard s.card | |
| have hT (u : Finset α) : u ∈ T ↔ u.card = s.card := by | |
| simp [T] | |
| have hsum : (Nat.choose (Fintype.card α) s.card : ℝ) * C = | |
| Fintype.card (Equiv.Perm α) := by | |
| calc | |
| _ = ∑ u ∈ T, C := by simp [T, Finset.card_powersetCard] | |
| _ = ∑ u ∈ T, ∑ σ : Equiv.Perm α, | |
| if s.map σ.toEmbedding = u then (1 : ℝ) else 0 := by | |
| apply Finset.sum_congr rfl | |
| intro u hu | |
| exact permutation_image_count_eq s t u (hst.symm.trans (hT u |>.mp hu).symm) | |
| _ = ∑ σ : Equiv.Perm α, ∑ u ∈ T, | |
| if s.map σ.toEmbedding = u then (1 : ℝ) else 0 := by | |
| rw [Finset.sum_comm] | |
| _ = ∑ _ : Equiv.Perm α, (1 : ℝ) := by | |
| apply Finset.sum_congr rfl | |
| intro σ _ | |
| have hmem : s.map σ.toEmbedding ∈ T := (hT _).2 (Finset.card_map _) | |
| simp [hmem] | |
| _ = _ := by simp | |
| have hc : (Nat.choose (Fintype.card α) s.card : ℝ) ≠ 0 := by | |
| exact_mod_cast (Nat.choose_pos (Finset.card_le_univ s)).ne' | |
| apply (eq_div_iff hc).2 | |
| simpa only [mul_comm] using hsum | |
| /-- A prescribed image has probability the reciprocal of the number of | |
| sets of that size, and probability zero if its cardinality is different. -/ | |
| theorem uniform_permutation_image_atom (s t : Finset α) : | |
| (∑ σ : Equiv.Perm α, if s.map σ.toEmbedding = t then (1 : ℝ) else 0) / | |
| Fintype.card (Equiv.Perm α) = | |
| if s.card = t.card then ((Fintype.card α).choose s.card : ℝ)⁻¹ else 0 := by | |
| classical | |
| by_cases hst : s.card = t.card | |
| · rw [ite_eq_left hst, permutation_image_count s t hst] | |
| have hc : (Fintype.card (Equiv.Perm α) : ℝ) ≠ 0 := by | |
| exact_mod_cast (Fintype.card_ne_zero : Fintype.card (Equiv.Perm α) ≠ 0) | |
| field_simp | |
| · rw [ite_eq_right hst] | |
| have hne (σ : Equiv.Perm α) : s.map σ.toEmbedding ≠ t := by | |
| intro h | |
| apply hst | |
| simpa using congrArg Finset.card h | |
| simp [hne] | |
| /-- Averaging over all permutations equals averaging over all possible | |
| images of the fixed set. This form can be mixed over a random prefix length. -/ | |
| theorem uniform_permutation_image_average (s : Finset α) (f : Finset α → ℝ) : | |
| (∑ σ : Equiv.Perm α, f (s.map σ.toEmbedding)) / | |
| Fintype.card (Equiv.Perm α) = | |
| (∑ t ∈ (Finset.univ : Finset α).powersetCard s.card, f t) / | |
| (Fintype.card α).choose s.card := by | |
| classical | |
| let T := (Finset.univ : Finset α).powersetCard s.card | |
| have hT (t : Finset α) : t ∈ T ↔ t.card = s.card := by simp [T] | |
| calc | |
| _ = (∑ σ : Equiv.Perm α, ∑ t ∈ T, | |
| (if s.map σ.toEmbedding = t then (1 : ℝ) else 0) * f t) / | |
| Fintype.card (Equiv.Perm α) := by | |
| congr 1 | |
| apply Finset.sum_congr rfl | |
| intro σ _ | |
| have hmem : s.map σ.toEmbedding ∈ T := (hT _).2 (Finset.card_map _) | |
| simp [hmem] | |
| _ = ∑ t ∈ T, ((∑ σ : Equiv.Perm α, | |
| if s.map σ.toEmbedding = t then (1 : ℝ) else 0) / | |
| Fintype.card (Equiv.Perm α)) * f t := by | |
| rw [Finset.sum_comm, Finset.sum_div] | |
| apply Finset.sum_congr rfl | |
| intro t _ | |
| rw [← Finset.sum_mul] | |
| ring | |
| _ = ∑ t ∈ T, (((Fintype.card α).choose s.card : ℝ)⁻¹ * f t) := by | |
| apply Finset.sum_congr rfl | |
| intro t ht | |
| rw [uniform_permutation_image_atom, ite_eq_left ((hT t).1 ht).symm] | |
| _ = _ := by | |
| rw [← Finset.mul_sum] | |
| change _ = (∑ t ∈ T, f t) / _ | |
| rw [div_eq_mul_inv, mul_comm] | |
| end MatroidProphet.Secretary | |
| end OAI | |