import Mathlib import OAI.Analysis.TracialSplitting.PreparedBlocks import OAI.Analysis.JiangSu.DimensionDrop.IteratedSpectrum import OAI.Analysis.JiangSu.DimensionDrop.SortingNetwork namespace OAI noncomputable section namespace JiangSuConstruction open scoped CStarAlgebra ComplexOrder Kronecker theorem continuous_matching_control_two_scales (p n : ℕ) {δ r : ℝ} (hr : 0 (x i:ℝ)≤t).card≤ (Finset.univ.filter fun i => (y i:ℝ)≤t+δ).card) (hyx : ∀ t : ℝ,(Finset.univ.filter fun i => (y i:ℝ)≤t).card≤ (Finset.univ.filter fun i => (x i:ℝ)≤t+δ).card) (f : Interval → Mat p) {η ηs : ℝ} (hη : 0≤η) (hηs : 0≤ηs) (hf : ∀ s t,dist s t≤δ → ‖f s-f t‖≤η) (hfs : ∀ s t,dist s t≤r → ‖f s-f t‖≤ηs) : ‖unitaryConjugation (continuousMatchingUnitary p n r (x,y)) (blockDiagonalHom (f ∘ x))-blockDiagonalHom (f ∘ y)‖≤ 6*((sortingNetwork n).length:ℝ)*ηs+η := by let ux := networkUnitary p n r (sortingNetwork n) x let uy := networkUnitary p n r (sortingNetwork n) y let X := blockDiagonalHom (f ∘ x) let Y := blockDiagonalHom (f ∘ y) let SX := blockDiagonalHom (f ∘ (x ∘ Tuple.sort x)) let SY := blockDiagonalHom (f ∘ (y ∘ Tuple.sort y)) have hx : ‖unitaryConjugation ux X-SX‖≤3*((sortingNetwork n).length:ℝ)*ηs := continuous_sort_unitary_control p n hr f hηs hfs x have hy : ‖unitaryConjugation uy Y-SY‖≤3*((sortingNetwork n).length:ℝ)*ηs := continuous_sort_unitary_control p n hr f hηs hfs y have hs : ‖SX-SY‖≤η := by change ‖blockDiagonalHom _-blockDiagonalHom _‖≤_ rw [←map_sub] apply (NonUnitalStarAlgHom.norm_apply_le (blockDiagonalHom : (Fin n → Mat p) →⋆ₐ[ℂ] TensorMat p n) _).trans apply (pi_norm_le_iff_of_nonneg hη).mpr intro i exact hf _ _ (interval_sorted_displacement x y δ hxy hyx i) have hcancel : unitaryConjugation uy⁻¹ (unitaryConjugation uy Y)=Y := by rw [←unitaryConjugation_mul,inv_mul_cancel,unitaryConjugation_one] have hb : ‖unitaryConjugation ux X-unitaryConjugation uy Y‖≤ 6*((sortingNetwork n).length:ℝ)*ηs+η := by have h₁ := norm_sub_le_norm_sub_add_norm_sub (unitaryConjugation ux X) SX SY have h₂ := norm_sub_le_norm_sub_add_norm_sub (unitaryConjugation ux X) SY (unitaryConjugation uy Y) rw [norm_sub_rev SY] at h₂ linarith change ‖unitaryConjugation (uy⁻¹*ux) X-Y‖≤_ rw [unitaryConjugation_mul,←hcancel,←map_sub] exact (NonUnitalStarAlgHom.norm_apply_le (unitaryConjugation uy⁻¹) _).trans hb lemma filter_card_equiv {I J : Type*} [Fintype I] [Fintype J] (e : I ≃ J) (P : J → Prop) [DecidablePred P] : (Finset.univ.filter fun i : I => P (e i)).card= (Finset.univ.filter P).card := by classical rw [←Fintype.card_subtype,←Fintype.card_subtype] exact Fintype.card_congr (e.subtypeEquiv (fun _ => Iff.rfl)) abbrev iteratedSize (n k : ℕ) := Fintype.card (IteratedIndex n k) def iteratedTuple (n k : ℕ) (s : Interval) : Fin (iteratedSize n k) → Interval := fun j => iteratedFamily n k ((Fintype.equivFin (IteratedIndex n k)).symm j) s lemma iteratedTuple_continuous (n k : ℕ) : Continuous (iteratedTuple n k) := by apply continuous_pi intro j exact (iteratedFamily n k _).continuous lemma iteratedTuple_shifted_counts (n : ℕ) {δ : ℝ} (hδ : 0<δ) : ∃ k : ℕ,∀ l,k≤l → ∀ s t : Interval,∀ v : ℝ, (Finset.univ.filter fun i => (iteratedTuple n l s i:ℝ)≤v).card≤ (Finset.univ.filter fun i => (iteratedTuple n l t i:ℝ)≤v+δ).card := by classical obtain ⟨k,hk⟩ := iterated_spectrum_shifted_counts n hδ refine ⟨k,?_⟩ intro l hl s t v change (Finset.univ.filter fun i => (iteratedFamily n l ((Fintype.equivFin (IteratedIndex n l)).symm i) s:ℝ)≤v).card≤_ rw [filter_card_equiv (Fintype.equivFin (IteratedIndex n l)).symm (fun i => (iteratedFamily n l i s:ℝ)≤v)] change _≤(Finset.univ.filter fun i => (iteratedFamily n l ((Fintype.equivFin (IteratedIndex n l)).symm i) t:ℝ)≤v+δ).card rw [filter_card_equiv (Fintype.equivFin (IteratedIndex n l)).symm (fun i => (iteratedFamily n l i t:ℝ)≤v+δ)] exact hk l hl s t v lemma finite_continuous_modulus {p : ℕ} (F : Finset C(Interval,Mat p)) {η : ℝ} (hη : 0<η) : ∃ δ : ℝ,0<δ ∧ ∀ f∈F,∀ s t : Interval,dist s t≤δ → ‖f s-f t‖≤η := by classical let g : Interval → (F → Mat p) := fun s f => f.val s have hc : Continuous g := by apply continuous_pi; intro f; exact f.val.continuous have hu := CompactSpace.uniformContinuous_of_continuous hc obtain ⟨δ,hδ,hd⟩ := Metric.uniformContinuous_iff.mp hu (η/2) (by positivity) refine ⟨δ/2,by positivity,?_⟩ intro f hf s t hst have hh : dist (g s) (g t)<η/2 := hd (by linarith) have hi := dist_le_pi_dist (g s) (g t) (⟨f,hf⟩:F) have hh' : dist (g s (⟨f,hf⟩:F)) (g t (⟨f,hf⟩:F))≤η := hi.trans (le_trans hh.le (by linarith)) simpa only [g,dist_eq_norm] using hh' theorem actual_iterated_continuous_matching (n p : ℕ) (F : Finset C(Interval,Mat p)) {ε : ℝ} (hε : 0<ε) : ∃ k : ℕ,∀ l,k≤l → ∃ U : C(Interval×Interval,unitary (TensorMat p (iteratedSize n l))), ∀ f∈F,∀ s t : Interval, ‖unitaryConjugation (U (s,t)) (blockDiagonalHom (f ∘ iteratedTuple n l s))- blockDiagonalHom (f ∘ iteratedTuple n l t)‖<ε := by obtain ⟨δ,hδ,hF⟩ := finite_continuous_modulus F (show 0<ε/3 by positivity) obtain ⟨k,hk⟩ := iteratedTuple_shifted_counts n hδ refine ⟨k,?_⟩ intro l hl let L : ℝ := ((sortingNetwork (iteratedSize n l)).length:ℝ) have hL : 0≤L := Nat.cast_nonneg _ let ηs := ε/(3*(6*L+1)) have hηs : 0<ηs := by dsimp [ηs]; positivity obtain ⟨r,hr,hFs⟩ := finite_continuous_modulus F hηs let U : C(Interval×Interval,unitary (TensorMat p (iteratedSize n l))) := (continuousMatchingUnitary p (iteratedSize n l) r).comp ⟨fun z => (iteratedTuple n l z.1,iteratedTuple n l z.2), ((iteratedTuple_continuous n l).comp continuous_fst).prodMk ((iteratedTuple_continuous n l).comp continuous_snd)⟩ refine ⟨U,?_⟩ intro f hf s t have hh := continuous_matching_control_two_scales p (iteratedSize n l) hr (iteratedTuple n l s) (iteratedTuple n l t) (hk l hl s t) (hk l hl t s) f (show 0≤ε/3 by positivity) hηs.le (hF f hf) (hFs f hf) change ‖unitaryConjugation (continuousMatchingUnitary p (iteratedSize n l) r (iteratedTuple n l s,iteratedTuple n l t)) _-_‖<ε apply hh.trans_lt change 6*L*ηs+ε/3<ε have heq : ηs*(3*(6*L+1))=ε := by dsimp [ηs] exact div_mul_cancel₀ _ (by positivity) nlinarith end JiangSuConstruction end end OAI