openAI-math-dataset / lean /OAI /GroupTheory /PolycyclicRecognition /Analysis /BoxInteriorSlack.lean
| import OAI.GroupTheory.PolycyclicRecognition.Algebra.UniformInteriorSlack | |
| import OAI.GroupTheory.PolycyclicRecognition.Analysis.BoxWordExpansion | |
| namespace OAI | |
| /-! # Interior slack directly from membership in rescaled boxes -/ | |
| noncomputable section | |
| namespace PolycyclicRecognition | |
| open Module LieModule Filter | |
| open scoped Topology BigOperators | |
| variable {I L G A Z : Type*} [Fintype I] [DecidableEq I] | |
| [LieRing L] [LieAlgebra ℝ L] [FiniteDimensional ℝ L] | |
| [Group G] [NormedRing A] [NormedAlgebra ℝ A] [CompleteSpace A] | |
| attribute [local instance 100] LieRing.ofAssociativeRing | |
| /-- The actual logarithms of an enlarged suffix and an interior family | |
| satisfy the uniform conjugation conclusion. The finite word expansions | |
| needed by interior slack are constructed here from the two box bounds. -/ | |
| theorem rescaled_box_conjugation_tendstoUniformly | |
| [Nonempty (Fin (finrank ℝ L))] | |
| (X : I → L) {m k₀ c₀ : ℕ} | |
| (hk : lowerCentralSeries ℝ L L k₀ = ⊥) | |
| (hnil : lowerCentralSeries ℝ L L (m + 1) = ⊥) | |
| (b₀ : Basis (Fin (finrank ℝ L)) ℝ L) (s : I → ℝ) | |
| (β : ℝ → WordBasisSelections X (boundedBracketWords k₀)) | |
| (hβ : ∀ r, IsMaximizingColumnBasis | |
| (wordListCoordinates X (boundedBracketWords k₀) b₀) | |
| (fun w => bracketWordExponent s w) (β r).val) | |
| (Exp : L → G) (hinj : Function.Injective Exp) | |
| (hneg : ∀ x, Exp (-x) = (Exp x)⁻¹) | |
| (a : DynkinCoefficients c₀ 2) | |
| (hm : ∀ x y, Exp (finiteDynkinPolynomial a ![x, y]) = Exp x * Exp y) | |
| (j : L →ₗ⁅ℝ⁆ A) (ρ : G →* Aˣ) (hρ : Function.Injective ρ) | |
| (hmatch : ∀ x, ρ (Exp x) = algebraExpUnit (j x)) | |
| {δ c lo hi : ℝ} (hδ : 0 < δ) (hc : 0 ≤ c) (hhi : 0 ≤ hi) | |
| (hcost : ∀ w : boundedBracketWords (I := I) k₀, | |
| lo ≤ bracketWordExponent s w ∧ bracketWordExponent s w ≤ hi) | |
| (hgap : (m : ℝ) * hi < c * lo) | |
| (ε : ℝ → ℝ) (hε : Tendsto ε atTop (𝓝 0)) | |
| (k : ℝ → L) (u : ℝ → Z → L) | |
| (hkbox : ∀ᶠ r : ℝ in atTop, | |
| ‖(exponentialRescaleBasis | |
| (WordBasisSelections.basis X (boundedBracketWords k₀) (β r)) | |
| (fun i => (1 + δ) * bracketWordExponent s ((β r).val i)) r).equivFun (k r)‖ ≤ | |
| Real.exp (r * ε r)) | |
| (hubox : ∀ᶠ r : ℝ in atTop, ∀ z, | |
| ‖(exponentialRescaleBasis | |
| (WordBasisSelections.basis X (boundedBracketWords k₀) (β r)) | |
| (fun i => (1 - c * δ) * bracketWordExponent s ((β r).val i)) r).equivFun (u r z)‖ ≤ | |
| Real.exp (r * ε r)) : | |
| let b := fun r => exponentialRescaleBasis | |
| (WordBasisSelections.basis X (boundedBracketWords k₀) (β r)) | |
| (fun i => bracketWordExponent s ((β r).val i)) r | |
| TendstoUniformly (fun r z => dynkinProduct a (basisBracket (b r)) | |
| (dynkinProduct a (basisBracket (b r)) ((b r).equivFun (k r)) | |
| ((b r).equivFun (u r z))) (-((b r).equivFun (k r)))) (fun _ => 0) atTop := by | |
| classical | |
| let S := boundedBracketWords (I := I) k₀ | |
| let cost : S → ℝ := fun w => bracketWordExponent s w | |
| let bk := fun r => exponentialRescaleBasis (WordBasisSelections.basis X S (β r)) | |
| (fun i => (1 + δ) * cost ((β r).val i)) r | |
| let bu := fun r => exponentialRescaleBasis (WordBasisSelections.basis X S (β r)) | |
| (fun i => (1 - c * δ) * cost ((β r).val i)) r | |
| have hek (r : ℝ) := exists_word_expansion_of_rescaled_norm_le X S (β r) | |
| (fun w => (1 + δ) * cost w) r (norm_nonneg ((bk r).equivFun (k r))) (k r) le_rfl | |
| have heu (r : ℝ) (z : Z) := exists_word_expansion_of_rescaled_norm_le X S (β r) | |
| (fun w => (1 - c * δ) * cost w) r (norm_nonneg ((bu r).equivFun (u r z))) (u r z) le_rfl | |
| choose ak hak hkeq using hek | |
| choose au hau hueq using heu | |
| have hkcoeff : ∀ᶠ r : ℝ in atTop, ∀ w, | |
| |ak r w| ≤ (finrank ℝ L : ℝ) * Real.exp (r * ((1 + δ) * cost w + ε r)) := by | |
| filter_upwards [hkbox] with r hr w | |
| apply (hak r w).trans | |
| calc | |
| _ ≤ (finrank ℝ L : ℝ) * Real.exp (r * ε r) * | |
| Real.exp (r * ((1 + δ) * cost w)) := by gcongr | |
| _ = _ := by rw [mul_assoc, ← Real.exp_add]; congr 2; ring | |
| have hucoeff : ∀ᶠ r : ℝ in atTop, ∀ z w, | |
| |au r z w| ≤ (finrank ℝ L : ℝ) * Real.exp (r * ((1 - c * δ) * cost w + ε r)) := by | |
| filter_upwards [hubox] with r hr z w | |
| apply (hau r z w).trans | |
| calc | |
| _ ≤ (finrank ℝ L : ℝ) * Real.exp (r * ε r) * | |
| Real.exp (r * ((1 - c * δ) * cost w)) := by | |
| gcongr | |
| exact hr z | |
| _ = _ := by rw [mul_assoc, ← Real.exp_add]; congr 2; ring | |
| have h := rescaled_word_conjugation_tendstoUniformly X hk hnil b₀ s β hβ | |
| Exp hinj hneg a hm j ρ hρ hmatch hδ hc hhi (Nat.cast_nonneg _) (Nat.cast_nonneg _) | |
| hcost hgap ε hε ak au hkcoeff hucoeff | |
| dsimp only [S] at hkeq hueq | |
| simpa only [hkeq, hueq] using h | |
| end PolycyclicRecognition | |
| end | |
| end OAI | |