import Mathlib import OAI.Probability.SKGap.Gaussian.GOEScalarTail namespace OAI section noncomputable section namespace SKGap open Matrix MeasureTheory ProbabilityTheory Real Set open scoped BigOperators MatrixOrder Matrix.Norms.Frobenius theorem compact_conditional_uniform_tail {α : Type*} [TopologicalSpace α] [CompactSpace α] {j A : ℝ} (hj : 0 < j) (hj1 : j < 1) (hA : 0 < A) (hsub : sqrt j*A < 1) (b e s q κ : α → ℝ) (hb : Continuous b) (he : Continuous e) (hs : Continuous s) (hq : Continuous q) (hk : Continuous κ) (hb0 : ∀ x,0 < b x) (hb1 : ∀ x,b x ≤ 1) (he0 : ∀ x,0 ≤ e x) (hs0 : ∀ x,0 < s x) (hq0 : ∀ x,0 < q x) (hkq : ∀ x,κ x^2=1-j*q x/s x) : ∃ δ ρ : ℝ, 0 < δ ∧ δ < 1 ∧ 0 < ρ ∧ ∀ U : ℝ, 0 ≤ U → ∀ ε τ : ℝ, 0 < ε → 0 < τ → ∃ c : ℝ, ∃ N : ℕ, 0 < c ∧ 0 < N ∧ ∀ n, N ≤ n → ∀ a : Fin n → ℝ, (∀ i,0 ≤ a i) → (∀ i,a i ≤ A) → 0 < j/(n:ℝ)*∑ i,a i → augmentedGramTolerance (j/(n:ℝ)*∑ i,a i) ε < ρ/2 → ∀ p : Fin 3 → EuclideanSpace ℝ (Fin n), ∀ u : EuclideanSpace ℝ (Fin n), (∀ r,‖p r‖ ≤ U) → ‖u‖ ≤ U → u.ofLp⬝ᵥu.ofLp=1 → ∀ x (C E : Matrix ((Fin 3) ⊕ Unit) ((Fin 3) ⊕ Unit) ℝ), C.IsHermitian → E.IsHermitian → τ*‖E‖ < ρ/2 → ‖diagonalAugmentedGram a p-augmentedGram (ν := Unit) (limitingGram (b x) (e x) (s x))‖ < ρ/2 → ‖C-conditionalCoefficient j ((2*j*q x+2*(j*b x)*(κ x-1))/q x) j ((κ x-1)*√(j*b x)/√(q x))‖ < ρ/2 → (Measure.pi (fun _ : MatrixCoordinates (Fin n) => gaussianReal 0 1)).real {g | ¬(conditionalBase j a g- pathDiagonal a 1*conditionalUpdateColumns j a p u g* (C+matrixBilinear (goeMatrix (j/(n:ℝ)) g) u u • E)* (conditionalUpdateColumns j a p u g)ᵀ*pathDiagonal a 1- (δ*((1-sqrt j*A)^2/4)) • (1 : Matrix (Fin n) (Fin n) ℝ)).PosDef} ≤ 101*exp (-c*(n:ℝ)) := by obtain ⟨δ,ρ,hδ,hδ1,hρ,hcrit⟩ := compact_conditional_rank_criterion hj hj1 b e s q κ hb he hs hq hk hb0 hb1 he0 hs0 hq0 hkq refine ⟨δ,ρ,hδ,hδ1,hρ,?_⟩ intro U hU ε τ hε hτ obtain ⟨c₀,N,hc₀,hN,hgram⟩ := conditional_augmented_gram_tail hj hA hsub hU hε let c₁ := τ^2/(π^2*(2*(U+1)^2)^2*j) have hc₁ : 0 < c₁ := by dsimp [c₁]; positivity refine ⟨min c₀ c₁,N,lt_min hc₀ hc₁,hN,?_⟩ intro n hn a ha haA hB heps p u hp hu hunit x C E hC hE hEbound hdiag hcoef let μ := Measure.pi (fun _ : MatrixCoordinates (Fin n) => gaussianReal 0 1) let badG := {g | ¬bilinearPathGood j A a g ∨ augmentedGramTolerance (j/(n:ℝ)*∑ i,a i) ε < ‖actualAugmentedGram j a p u g-diagonalAugmentedGram a p‖} let badS := {g | τ ≤ |matrixBilinear (goeMatrix (j/(n:ℝ)) g) u u|} have hnpos := hN.trans_le hn have hgprob : μ.real badG ≤ 99*exp (-c₀*(n:ℝ)) := hgram n hn a ha haA hB p u hp hu hunit have hsprob : μ.real badS ≤ 2*exp (-c₁*(n:ℝ)) := by have hh := goe_scalar_tail hnpos hj hU hτ.le u hu convert hh using 1 congr 2 dsimp [c₁] ring have hsubsets : {g | ¬(conditionalBase j a g- pathDiagonal a 1*conditionalUpdateColumns j a p u g* (C+matrixBilinear (goeMatrix (j/(n:ℝ)) g) u u • E)* (conditionalUpdateColumns j a p u g)ᵀ*pathDiagonal a 1- (δ*((1-sqrt j*A)^2/4)) • (1 : Matrix (Fin n) (Fin n) ℝ)).PosDef} ⊆ badG ∪ badS := by intro g hg by_contra hbad have hgG : g ∉ badG := fun hh => hbad (Or.inl hh) have hgS : g ∉ badS := fun hh => hbad (Or.inr hh) simp only [badG,mem_ofPred_eq,not_or,not_lt] at hgG have hgS' : |matrixBilinear (goeMatrix (j/(n:ℝ)) g) u u| < τ := lt_of_not_ge hgS let D := pathDiagonal a 1 let S := conditionalBase j a g let V := conditionalUpdateColumns j a p u g let Cg := C+matrixBilinear (goeMatrix (j/(n:ℝ)) g) u u • E let lo := (1-sqrt j*A)^2/4 have hmargin : (S-lo • (1 : Matrix (Fin n) (Fin n) ℝ)).PosDef := bilinearPathGood_margin a g (not_not.mp hgG.1) have hlo : 0 ≤ lo := by dsimp [lo]; positivity have hSpos : S.PosDef := by have hh := hmargin.add_posSemidef ((Matrix.PosSemidef.one : (1 : Matrix (Fin n) (Fin n) ℝ).PosSemidef).smul hlo) convert hh using 1; abel have hD : D.IsHermitian := isHermitian_diagonal_iff.mpr (fun i => isSelfAdjoint_iff.mpr (star_trivial _)) have hCg : Cg.IsHermitian := hC.add (hE.smul (R := ℝ) (isSelfAdjoint_iff.mpr (star_trivial _))) have hG : ‖Vᵀ*D*S⁻¹*D*V-augmentedGram (ν := Unit) (limitingGram (b x) (e x) (s x))‖ < ρ := by have hh := norm_sub_le_norm_sub_add_norm_sub (actualAugmentedGram j a p u g) (diagonalAugmentedGram a p) (augmentedGram (ν := Unit) (limitingGram (b x) (e x) (s x))) have heq : Vᵀ*D*S⁻¹*D*V=actualAugmentedGram j a p u g := by simp only [V,D,S,conditionalUpdateColumns,conditionalBase,actualAugmentedGram, exactG,Fintype.card_fin,Matrix.mul_assoc] rw [heq] exact hh.trans_lt (by linarith [hgG.2]) have hCgclose : ‖Cg-conditionalCoefficient j ((2*j*q x+2*(j*b x)*(κ x-1))/q x) j ((κ x-1)*√(j*b x)/√(q x))‖ < ρ := by have hn : ‖matrixBilinear (goeMatrix (j/(n:ℝ)) g) u u • E‖ ≤ τ*‖E‖ := by rw [norm_smul,Real.norm_eq_abs] exact mul_le_mul_of_nonneg_right hgS'.le (norm_nonneg _) have hh := norm_add_le (C-conditionalCoefficient j ((2*j*q x+2*(j*b x)*(κ x-1))/q x) j ((κ x-1)*√(j*b x)/√(q x))) (matrixBilinear (goeMatrix (j/(n:ℝ)) g) u u • E) have heq : Cg-conditionalCoefficient j ((2*j*q x+2*(j*b x)*(κ x-1))/q x) j ((κ x-1)*√(j*b x)/√(q x))= (C-conditionalCoefficient j ((2*j*q x+2*(j*b x)*(κ x-1))/q x) j ((κ x-1)*√(j*b x)/√(q x)))+matrixBilinear (goeMatrix (j/(n:ℝ)) g) u u • E := by dsimp [Cg]; abel rw [heq] exact hh.trans_lt (by linarith) exact hg (hcrit x (Fin n) S D V Cg lo hSpos hD hCg hmargin.posSemidef hG hCgclose) apply (measureReal_mono hsubsets (measure_ne_top _ _)).trans apply (measureReal_union_le badG badS).trans apply (add_le_add hgprob hsprob).trans have he₀ : exp (-c₀*(n:ℝ)) ≤ exp (-min c₀ c₁*(n:ℝ)) := by apply exp_le_exp.mpr; simpa only [neg_mul] using neg_le_neg (mul_le_mul_of_nonneg_right (min_le_left c₀ c₁) (Nat.cast_nonneg n : (0:ℝ) ≤ n)) have he₁ : exp (-c₁*(n:ℝ)) ≤ exp (-min c₀ c₁*(n:ℝ)) := by apply exp_le_exp.mpr; simpa only [neg_mul] using neg_le_neg (mul_le_mul_of_nonneg_right (min_le_right c₀ c₁) (Nat.cast_nonneg n : (0:ℝ) ≤ n)) nlinarith end SKGap end end end OAI