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dd14c26 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 | 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
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