import Mathlib import OAI.Geometry.Cannon.ScalarLimits.OscillationPairwise import OAI.Geometry.Cannon.Energy.Extend namespace OAI section section section noncomputable section section section section section section open Set Topology open Set Topology Filter Metric unitInterval open scoped InnerProductSpace open Set Metric Topology open Set Topology unitInterval open Set Filter Topology open Set MeasureTheory open scoped BigOperators open Set unitInterval namespace CannonApprox open CannonGrid CannonChart CannonPacking CannonCrossEnergy CannonScalarEnergy CannonAnnular Metric theorem exists_scalar_annular_constant {X : Type*} [MetricSpace X] [LocallyPathConnectedSpace X] (Q : ℕ) (CP C c L : ℝ) (hpack : PolynomialPacking (X := X) Q CP) (hCP : 0 ≤ CP) (hC : 0 ≤ C) (hc : 0 < c) (hL : 1 ≤ L) (hpaths : ∀ (o u v : X) (s : ℝ), 0 < s → dist o u ≤ s → dist o v ≤ s → JoinedIn {z | dist o z < L*s} u v) : ∃ E : ℝ, 0 ≤ E ∧ ∀ (e : CannonGrid.Square → X), Topology.IsEmbedding e → ∀ (d₁ : ℝ), (∀ A : Set CannonGrid.Square, IsCompact A → (∃ x ∈ A, x.1 = 0) → (∃ x ∈ A, x.1 = 1) → d₁ ≤ diam (e '' A)) → ∀ {β γ : Type*} [Fintype β] [Fintype γ] (ξ : β → X) (p : γ → X) (h : X → X), Continuous h → ∀ (o : X) (R r d₀ s : ℝ), 0 < R → R ≤ 1 → 4*R < d₁ → 0 < r → r ≤ R/2 → (L*(L*2+1)+3)*r ≤ R/4 → max (C+2) (max c 1)*(r/R) ≤ 1 → 0 < s → L*s ≤ R/2 → Metric.ball o (4*R) ⊆ range e → (∀ i j, i ≠ j → r/R ≤ dist (ξ i) (ξ j)) → (∀ i j, i ≠ j → r ≤ dist (p i) (p j)) → (∀ y : X, ∃ j, dist (p j) y ≤ r) → (∀ u v, dist u o ≤ 3*R → dist v o ≤ 3*R → dist (h u) (h v) ≤ C/R * dist u v) → (∀ u v, dist u o ≤ 3*R → dist v o ≤ 3*R → c/R * dist u v ≤ dist (h u) (h v)) → (∀ u v, dist u o ≤ 2*R → dist v o ≤ 2*R → R ≤ dist u v → d₀ ≤ dist (h u) (h v)) → ∀ (ω : β → ℝ), MacroscopicAdmissible ξ (r/R) d₀ ω → ∀ (ρ : γ → ℝ), (∀ j, 0 ≤ ρ j) → Admissible (fun j => e ⁻¹' Metric.ball (p j) (2*r)) ρ → ∀ (n : ℕ) (hn : 0 < n), (∀ z : CannonGrid.Square, ∀ i : Index (3*n), weight (3*n) z i ≠ 0 → dist (e z) (e (vertex (3*n) (by omega) i)) < r) → ∀ a b : CannonGrid.Square, dist o (e a) ≤ s → dist o (e b) ≤ s → let f := scalar (fun j => e ⁻¹' Metric.ball (p j) (2*r)) ρ n hn (f b-f a)^2 ≤ E*(∑ j, (ω j)^2)* (∑ j ∈ Finset.univ.filter (fun j => R/4 ≤ dist (p j) o ∧ dist (p j) o ≤ 4*R), (ρ j)^2) := by classical let A := L*(L*2+1)+3 have hLp : 0 < L := zero_lt_one.trans_le hL have hA : 0 < A := by dsimp [A]; positivity have hAone : 1 ≤ A := by dsimp [A]; nlinarith [sq_nonneg L] obtain ⟨E,hE,hbound⟩ := exists_packed_annular_constant Q CP C c A hpack hCP hC hc hA refine ⟨E,hE,?_⟩ intro e he d₁ hd₁ β γ _ _ ξ p h hh o R r d₀ s hR hRone hdR hr hrR hAr hscaleT hs hsR hchart hξsep hpsep hpnet hzoom hlow hlarge ω hω ρ hρ hadm n hn hmesh a b ha hb let Cells := fun j => e ⁻¹' Metric.ball (p j) (2*r) let f := scalar Cells ρ n hn have hf := (scalar_properties Cells ρ hρ hadm n hn).1 have hfI := (scalar_properties Cells ρ hρ hadm n hn).2.1 let α := {j : γ // R/2 ≤ dist (p j) o ∧ dist (p j) o ≤ 3*R} let q : α → X := fun i => p i.val have hq (i : α) : R/2 ≤ dist (q i) o ∧ dist (q i) o ≤ 3*R := i.property have hqsep (i j : α) (hne : i ≠ j) : r ≤ dist (q i) (q j) := hpsep i.val j.val (fun heq => hne (Subtype.ext heq)) have hnet (y : X) (hy : R ≤ dist y o) (hy' : dist y o ≤ 2*R) : ∃ i : α, dist (q i) y ≤ r := by obtain ⟨j,hj⟩ := hpnet y have ht₁ := dist_triangle y (p j) o have ht₂ := dist_triangle (p j) y o rw [dist_comm y (p j)] at ht₁ have hmem : R/2 ≤ dist (p j) o ∧ dist (p j) o ≤ 3*R := by constructor <;> linarith exact ⟨⟨j,hmem⟩,hj⟩ have hchartq (i : α) : Metric.ball (q i) (L*(L*2+1)*r) ⊆ range e := by intro y hy apply hchart change dist y o < 4*R have ht := dist_triangle y (q i) o have hhq := (hq i).2 change dist y (q i) < L*(L*2+1)*r at hy change A*r ≤ R/4 at hAr dsimp [A] at hAr nlinarith have hinner : 2 < L*(L*2+1) := by nlinarith [sq_nonneg (L-1)] have hclosedq (i : α) : Metric.closedBall (q i) (2*r) ⊆ range e := by intro y hy apply hchartq i change dist y (q i) < L*(L*2+1)*r exact (show dist y (q i) ≤ 2*r from hy).trans_lt (mul_lt_mul_of_pos_right hinner hr) let K (i : α) := e ⁻¹' Metric.closedBall (q i) (2*r) have hK (i : α) : IsCompact (K i) := (isClosed_closedBall.preimage he.continuous).isCompact have hKn (i : α) : (K i).Nonempty := by obtain ⟨x,hx⟩ := hclosedq i (Metric.mem_closedBall_self (by positivity : 0 ≤ 2*r)) refine ⟨x,?_⟩ change e x ∈ Metric.closedBall (q i) (2*r) rw [hx] exact Metric.mem_closedBall_self (by positivity) let lo := fun i => sInf (f '' K i) let hi := fun i => sSup (f '' K i) have hlu (i : α) : lo i ≤ hi i := csInf_le_csSup ((hKn i).image f) ((hK i).image hf).bddBelow ((hK i).image hf).bddAbove have hrange (i : α) (z : X) (hz : dist z (q i) ≤ 2*r) : extend e f z ∈ Icc (lo i) (hi i) := by obtain ⟨x,rfl⟩ := hclosedq i hz rw [extend_apply e he.injective] exact mem_rangeInterval hf (hK i) hz have hlocal (i : α) : hi i-lo i ≤ ∑ j ∈ Finset.univ.filter (fun j => dist (p j) (q i) ≤ A*r), ρ j := by apply oscillation_le_of_pairwise hf (hK i) (Finset.sum_nonneg (fun j _ => hρ j)) intro x hx y hy exact scalar_local_oscillation e he L 2 r hL (by norm_num) hr hpaths (q i) (hchartq i) p ρ hρ hadm n hn hmesh x y (by simpa only [K,mem_preimage,mem_closedBall,dist_comm] using hx) (by simpa only [K,mem_preimage,mem_closedBall,dist_comm] using hy) have hscaleS : max A 1*r ≤ 1 := by rw [max_eq_left hAone] exact hAr.trans ((div_le_self hR.le (by norm_num : (1:ℝ) ≤ 4)).trans hRone) have hordered (a b : CannonGrid.Square) (ha : dist o (e a) ≤ s) (hb : dist o (e b) ≤ s) (hab : f a ≤ f b) : (f b-f a)^2 ≤ E*(∑ j, (ω j)^2)* (∑ j ∈ Finset.univ.filter (fun j => R/4 ≤ dist (p j) o ∧ dist (p j) o ≤ 4*R), (ρ j)^2) := by obtain ⟨π,hπ⟩ := hpaths o (e a) (e b) s hs ha hb have hπr (v : unitInterval) : π v ∈ range e := by apply hchart change dist (π v) o < 4*R have hv := hπ v change dist o (π v) < L*s at hv rw [dist_comm] at hv linarith have hlev : ∀ t : ℝ, ∃ S : Set X, t ∈ Ioo (f a) (f b) → IsPreconnected S ∧ 4*R < diam S ∧ (∃ z ∈ S, dist z o ≤ R/2) ∧ ∀ z ∈ S, extend e f z = t := by intro t by_cases ht : t ∈ Ioo (f a) (f b) · have ht01 : t ∈ Ioo (0:ℝ) 1 := ⟨(hfI a).1.trans_lt ht.1,ht.2.trans_le (hfI b).2⟩ obtain ⟨S,hS,hcon,_,hd,hval,hhit⟩ := scalar_ambient_level e he Cells ρ hρ hadm n hn d₁ hd₁ t ht01 obtain ⟨v,hv⟩ := hhit π hπr ht.1 ht.2 refine ⟨S,fun _ => ⟨hcon.isPreconnected,hdR.trans_le hd,⟨π v,hv,?_⟩,hval⟩⟩ have hpv := hπ v change dist o (π v) < L*s at hpv simpa only [dist_comm] using hpv.le.trans hsR · exact ⟨∅,fun htt => (ht htt).elim⟩ choose Clevel hlevel using hlev exact hbound q ξ p h hh (extend e f) lo hi o R r d₀ (f a) (f b) hR hr hrR hAr hscaleT hscaleS hab hlu hq hqsep hξsep hpsep hnet hzoom hlow hlarge hrange ω hω ρ hρ hlocal Clevel hlevel rcases le_total (f a) (f b) with hab | hab · exact hordered a b ha hb hab · change (f b-f a)^2 ≤ _ have hsquare : (f b-f a)^2 = (f a-f b)^2 := by ring rw [hsquare] exact hordered b a hb ha hab end CannonApprox end end end end end end end end end end OAI