openAI-math-dataset / lean /OAI /NumberTheory /PiExponent /Analysis /FiniteDeterminantCollision.lean
| import Mathlib | |
| import OAI.NumberTheory.PiExponent.Analysis.CollisionGeometricSum | |
| namespace OAI | |
| open scoped BigOperators | |
| namespace PiExponent.FiniteDeterminantCollision | |
| theorem det_finite_row_sum {ι κ δ : Type*} [Fintype ι] [DecidableEq ι] | |
| [Fintype δ] (group : ι → κ) (B : ι → δ → ℂ) (C : κ → δ → ι → ℂ) : | |
| Matrix.det (fun i j => ∑ d, B i d * C (group i) d j) = | |
| ∑ f : ι → δ, (∏ i, B i (f i)) * | |
| Matrix.det (fun i j => C (group i) (f i) j) := by | |
| classical | |
| have hm := (Matrix.detRowAlternating : (ι → ℂ) [⋀^ι]→ₗ[ℂ] ℂ).toMultilinearMap.map_sum | |
| (fun i d => B i d • C (group i) d) | |
| calc | |
| _ = Matrix.det (fun i => ∑ d, B i d • C (group i) d) := by | |
| congr 1 | |
| funext i j | |
| simp | |
| _ = ∑ f : ι → δ, Matrix.det (fun i => B i (f i) • C (group i) (f i)) := hm | |
| _ = _ := by | |
| apply Finset.sum_congr rfl | |
| intro f _ | |
| exact Matrix.det_mul_column (fun i => B i (f i)) (fun i j => C (group i) (f i) j) | |
| theorem norm_det_le_factorial_mul_pow {ι : Type*} [Fintype ι] [DecidableEq ι] | |
| (A : Matrix ι ι ℂ) {D : ℝ} (hD : 0 ≤ D) (hA : ∀ i j, ‖A i j‖ ≤ D) : | |
| ‖A.det‖ ≤ (Fintype.card ι).factorial * D ^ Fintype.card ι := by | |
| rw [Matrix.det_apply] | |
| calc | |
| _ ≤ ∑ σ : Equiv.Perm ι, ‖σ.sign • ∏ i, A (σ i) i‖ := norm_sum_le _ _ | |
| _ = ∑ σ : Equiv.Perm ι, ∏ i, ‖A (σ i) i‖ := by | |
| apply Finset.sum_congr rfl | |
| intro σ _ | |
| rcases Int.units_eq_one_or σ.sign with hs | hs <;> simp [hs, norm_prod] | |
| _ ≤ ∑ _σ : Equiv.Perm ι, D ^ Fintype.card ι := by | |
| apply Finset.sum_le_sum | |
| intro σ _ | |
| calc | |
| _ ≤ ∏ _index : ι, |D| := Finset.prod_le_prod₀ | |
| (fun index _ => norm_nonneg _) | |
| (fun index _ => (hA (σ index) index).trans (le_abs_self D)) | |
| _ = D ^ Fintype.card ι := by simp [abs_of_nonneg hD] | |
| _ = _ := by simp [Fintype.card_perm] | |
| theorem finite_collision_bound | |
| {ι κ : Type*} [Fintype ι] [DecidableEq ι] [Fintype κ] [DecidableEq κ] | |
| (group : ι → κ) (B : ι → ℕ → ℂ) (C : κ → ℕ → ι → ℂ) | |
| (D s : ℝ) (L : ι → ℝ) (hD : 0 ≤ D) (hs0 : 0 ≤ s) (hs1 : s < 1) | |
| (hL : ∀ i, 0 ≤ L i) | |
| (hC : ∀ a d j, ‖C a d j‖ ≤ D) | |
| (hB : ∀ i d, ‖B i d‖ ≤ L i * (s * s) ^ d) (N : ℕ) : | |
| ‖Matrix.det (fun i j => ∑ d : Fin N, B i d * C (group i) d j)‖ ≤ | |
| ((Fintype.card ι).factorial * D ^ Fintype.card ι * (∏ i, L i)) * | |
| s ^ (∑ a, (Collision.multiplicity Finset.univ group a).choose 2) * | |
| ((1 - s)⁻¹) ^ Fintype.card ι := by | |
| classical | |
| let good := (Finset.univ : Finset (ι → Fin N)).filter | |
| (fun f => Function.Injective (fun i => (group i, (f i : ℕ)))) | |
| let Q := (Fintype.card ι).factorial * D ^ Fintype.card ι * (∏ i, L i) | |
| have hQ : 0 ≤ Q := | |
| mul_nonneg (mul_nonneg (Nat.cast_nonneg _) (pow_nonneg hD _)) | |
| (Finset.prod_nonneg fun i _ => hL i) | |
| have hexpand : Matrix.det (fun i j => ∑ d : Fin N, B i d * C (group i) d j) = | |
| ∑ f ∈ good, (∏ i, B i (f i)) * Matrix.det (fun i j => C (group i) (f i) j) := by | |
| rw [det_finite_row_sum group (fun i (d : Fin N) => B i d) | |
| (fun a (d : Fin N) j => C a d j)] | |
| symm | |
| apply Finset.sum_subset (Finset.filter_subset _ _) | |
| intro f _ hf | |
| have hn : ¬ Function.Injective (fun i => (group i, (f i : ℕ))) := by | |
| simpa [good] using hf | |
| obtain ⟨i, j, hij, hne⟩ := Function.not_injective_iff.mp hn | |
| have hg : group i = group j := congrArg Prod.fst hij | |
| have hd : (f i : ℕ) = (f j : ℕ) := congrArg Prod.snd hij | |
| have hz : Matrix.det (fun r c => C (group r) (f r) c) = 0 := by | |
| apply Matrix.det_zero_of_row_eq hne | |
| funext c | |
| simp only [hg, hd] | |
| simp [hz] | |
| have hterm (f : ι → Fin N) : | |
| ‖(∏ i, B i (f i)) * Matrix.det (fun i j => C (group i) (f i) j)‖ ≤ | |
| Q * (s * s) ^ (∑ i, (f i : ℕ)) := by | |
| have hbp : (∏ i, ‖B i (f i)‖) ≤ ∏ i, L i * (s * s) ^ (f i : ℕ) := | |
| Finset.prod_le_prod₀ (fun i _ => norm_nonneg _) (fun i _ => hB i (f i)) | |
| have hcp := norm_det_le_factorial_mul_pow | |
| (fun i j => C (group i) (f i) j) hD (fun i j => hC _ _ _) | |
| rw [norm_mul, norm_prod] | |
| calc | |
| _ ≤ (∏ i, L i * (s * s) ^ (f i : ℕ)) * | |
| ((Fintype.card ι).factorial * D ^ Fintype.card ι) := | |
| mul_le_mul hbp hcp (norm_nonneg _) (Finset.prod_nonneg fun i _ => | |
| mul_nonneg (hL i) (pow_nonneg (mul_self_nonneg s) _)) | |
| _ = _ := by | |
| rw [Finset.prod_mul_distrib, Finset.prod_pow_eq_pow_sum] | |
| dsimp [Q] | |
| ring | |
| rw [hexpand] | |
| calc | |
| _ ≤ ∑ f ∈ good, | |
| ‖(∏ i, B i (f i)) * Matrix.det (fun i j => C (group i) (f i) j)‖ := | |
| norm_sum_le _ _ | |
| _ ≤ ∑ f ∈ good, Q * (s * s) ^ (∑ i, (f i : ℕ)) := | |
| Finset.sum_le_sum (fun f _ => hterm f) | |
| _ = Q * ∑ f ∈ good, (s * s) ^ (∑ i, (f i : ℕ)) := by rw [Finset.mul_sum] | |
| _ ≤ Q * (s ^ (∑ a, (Collision.multiplicity Finset.univ group a).choose 2) * | |
| ((1 - s)⁻¹) ^ Fintype.card ι) := | |
| mul_le_mul_of_nonneg_left (Collision.collision_geometric_sum_le group hs0 hs1 N) hQ | |
| _ = _ := by dsimp [Q]; ring | |
| end PiExponent.FiniteDeterminantCollision | |
| end OAI | |