| import Mathlib |
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| namespace OAI |
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| section |
| section |
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| noncomputable section |
| open scoped BigOperators |
| namespace HyperbolicObstruction |
| namespace BernoulliCandidates |
| variable {V A : Type*} [Fintype V] [Fintype A] [DecidableEq A] |
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| / |
| of globally injective free values avoiding every pinned letter. This is only a |
| lower bound; the actual candidate set does not prohibit cross-field coincidences. -/ |
| lemma card_lower_of_embeddings (C : Finset (V → A)) (B : Finset A) |
| (hC : ∀ f : V ↪ {a : A // a ∉ B}, (fun v => (f v).val) ∈ C) : |
| ((Fintype.card A - B.card).descFactorial (Fintype.card V)) ≤ C.card := by |
| classical |
| have hi : Function.Injective (fun f : V ↪ {a : A // a ∉ B} => |
| (⟨fun v => (f v).val, hC f⟩ : C)) := by |
| intro f g h |
| apply DFunLike.ext |
| intro v |
| exact Subtype.ext (congrFun (congrArg Subtype.val h) v) |
| have hh := Fintype.card_le_of_injective _ hi |
| rw [Fintype.card_embedding_eq] at hh |
| have hb : Fintype.card {a : A // a ∉ B} = Fintype.card A - B.card := by |
| simp |
| simpa only [hb, Fintype.card_coe] using hh |
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| lemma real_descFactorial_lower (N b v : ℕ) (h : 2 * (b + v) ≤ N) : |
| ((N : ℝ) / 2) ^ v ≤ ((N - b).descFactorial v : ℝ) := by |
| have hb : b ≤ N := by omega |
| have hnv : v ≤ N - b + 1 := by omega |
| have hs : (N : ℝ) / 2 ≤ (N - b + 1 - v : ℕ) := by |
| rw [Nat.cast_sub hnv, Nat.cast_add, Nat.cast_sub hb] |
| push_cast |
| have h' : (2 : ℝ) * (b + v) ≤ N := by exact_mod_cast h |
| linarith |
| exact (pow_le_pow_left₀ (by positivity) hs v).trans |
| (by exact_mod_cast (N - b).pow_sub_le_descFactorial v) |
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| lemma card_lower_half (C : Finset (V → A)) (B : Finset A) |
| (hC : ∀ f : V ↪ {a : A // a ∉ B}, (fun v => (f v).val) ∈ C) |
| (h : 2 * (B.card + Fintype.card V) ≤ Fintype.card A) : |
| ((Fintype.card A : ℝ) / 2) ^ Fintype.card V ≤ C.card := by |
| exact (real_descFactorial_lower _ _ _ h).trans |
| (by exact_mod_cast card_lower_of_embeddings C B hC) |
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| end BernoulliCandidates |
| end HyperbolicObstruction |
| end |
| end |
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| section |
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| end |
| end |
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| end OAI |
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