| 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 |
| |
