File size: 2,127 Bytes
7d9acbd | 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 | import Mathlib
namespace OAI
section
section
noncomputable section
open scoped BigOperators
namespace HyperbolicObstruction
namespace BernoulliCandidates
variable {V A : Type*} [Fintype V] [Fintype A] [DecidableEq A]
/-- A convenient subset of the actual fieldwise-injective candidates consists
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
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)
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)
end BernoulliCandidates
end HyperbolicObstruction
end
end
section
end
end
end OAI
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