| import Mathlib |
| import OAI.Geometry.Cannon.Energy.PathCosts |
| import OAI.Geometry.Cannon.ScalarLimits.LocalOscillation |
|
|
| namespace OAI |
|
|
| section |
| section |
| section |
| noncomputable section |
| section |
| section |
| section |
| section |
| section |
| open Set Topology |
| open Set Metric Topology |
| open Set Topology Filter Metric unitInterval |
| open scoped InnerProductSpace |
| open Set unitInterval |
|
|
| namespace CannonApprox |
|
|
| open CannonGrid CannonRegularization CannonLevel CannonCells CannonBands |
|
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| variable {ι : Type*} [Fintype ι] |
|
|
| local instance : PathConnectedSpace (Strip height (1 / 3) (2 / 3)) := |
| strip_pathConnected _ _ (by norm_num) (by norm_num) (by norm_num) |
|
|
| def rawFunction (C : ι → Set CannonGrid.Square) (ρ : ι → ℝ) : CannonGrid.Square → ℝ := |
| rawExtended height (1/3) (2/3) C ρ |
|
|
| def Admissible (C : ι → Set CannonGrid.Square) (ρ : ι → ℝ) : Prop := |
| ∀ u : Strip height (1/3) (2/3), height u.val = 1/3 → |
| ∀ v : Strip height (1/3) (2/3), height v.val = 2/3 → |
| ∀ p : Path u v, 1 ≤ cost (fun i => Subtype.val ⁻¹' C i) ρ (range p) |
|
|
| theorem rawFunction_properties (C : ι → Set CannonGrid.Square) (ρ : ι → ℝ) |
| (hρ : ∀ i, 0 ≤ ρ i) (hcross : Admissible C ρ) : |
| (∀ z, rawFunction C ρ z ∈ Icc 0 1) ∧ |
| (∀ z ∈ topBand, rawFunction C ρ z = 0) ∧ |
| (∀ z ∈ bottomBand, rawFunction C ρ z = 1) ∧ |
| (∀ u v, ∀ p : Path u v, |rawFunction C ρ u-rawFunction C ρ v| ≤ cost C ρ (range p)) := by |
| exact rawExtended_properties height (by unfold height; fun_prop) (1/3) (2/3) (by norm_num) C ρ hρ |
| (strip_top_nonempty _ _ (by norm_num) (by norm_num) (by norm_num)) hcross |
|
|
| def initial (C : ι → Set CannonGrid.Square) (ρ : ι → ℝ) (n : ℕ) (hn : 0 < n) : |
| CannonGrid.Square → ℝ := interpolate (3*n) (by omega) (rawFunction C ρ) |
|
|
| def scalar (C : ι → Set CannonGrid.Square) (ρ : ι → ℝ) (n : ℕ) (hn : 0 < n) : |
| CannonGrid.Square → ℝ := drain (flood (initial C ρ n hn) topBand) bottomBand |
|
|
| lemma initial_properties (C : ι → Set CannonGrid.Square) (ρ : ι → ℝ) |
| (hρ : ∀ i, 0 ≤ ρ i) (hcross : Admissible C ρ) (n : ℕ) (hn : 0 < n) : |
| Continuous (initial C ρ n hn) ∧ (∀ z, initial C ρ n hn z ∈ Icc 0 1) ∧ |
| (∀ z ∈ topBand, initial C ρ n hn z = 0) ∧ |
| (∀ z ∈ bottomBand, initial C ρ n hn z = 1) := by |
| obtain ⟨hrange,htop,hbottom,_⟩ := rawFunction_properties C ρ hρ hcross |
| have hbands := interpolate_preserves_bands n hn (rawFunction C ρ) htop hbottom |
| exact ⟨continuous_interpolate _ _ _, interpolate_bounds _ _ _ hrange,hbands⟩ |
|
|
| theorem scalar_properties (C : ι → Set CannonGrid.Square) (ρ : ι → ℝ) |
| (hρ : ∀ i, 0 ≤ ρ i) (hcross : Admissible C ρ) (n : ℕ) (hn : 0 < n) : |
| Continuous (scalar C ρ n hn) ∧ (∀ z, scalar C ρ n hn z ∈ Icc 0 1) ∧ |
| (∀ z ∈ topBand, scalar C ρ n hn z = 0) ∧ |
| (∀ z ∈ bottomBand, scalar C ρ n hn z = 1) := by |
| obtain ⟨hf,hfI,hT,hB⟩ := initial_properties C ρ hρ hcross n hn |
| have hF := continuous_flood topBand_pathConnected.nonempty hf (fun z => (hfI z).2) |
| have hFk : ∀ z, 0 ≤ flood (initial C ρ n hn) topBand z := fun z => |
| (hfI z).1.trans (le_flood topBand_pathConnected.nonempty (fun w => (hfI w).2) z) |
| exact ⟨continuous_drain bottomBand_pathConnected.nonempty hF hFk, |
| regularized_values topBand_pathConnected.nonempty bottomBand_pathConnected.nonempty |
| (fun z => (hfI z).1) (fun z => (hfI z).2) hT hB⟩ |
|
|
| theorem scalar_level (C : ι → Set CannonGrid.Square) (ρ : ι → ℝ) |
| (hρ : ∀ i, 0 ≤ ρ i) (hcross : Admissible C ρ) (n : ℕ) (hn : 0 < n) |
| {t : ℝ} (ht : t ∈ Ioo (0:ℝ) 1) : |
| ∃ A : Set CannonGrid.Square, IsCompact A ∧ IsConnected A ∧ A ⊆ centralRectangle ∧ |
| (∀ z ∈ A, scalar C ρ n hn z = t) ∧ (∃ z ∈ A, z.1 = 0) ∧ (∃ z ∈ A, z.1 = 1) ∧ |
| (∀ {x y : CannonGrid.Square} (p : Path x y), scalar C ρ n hn x < t → |
| t < scalar C ρ n hn y → ∃ s, p s ∈ A) := by |
| obtain ⟨hf,hfI,hT,hB⟩ := initial_properties C ρ hρ hcross n hn |
| exact rectangle_separating_level hf (fun z => (hfI z).1) (fun z => (hfI z).2) hT hB ht.1 ht.2 |
|
|
| theorem exists_mesh {X : Type*} [MetricSpace X] (e : CannonGrid.Square → X) |
| (he : Continuous e) {r : ℝ} (hr : 0 < r) : |
| ∃ 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 := |
| exists_metric_mesh e he hr |
|
|
| theorem scalar_local_oscillation {X : Type*} [MetricSpace X] |
| (e : CannonGrid.Square → X) (he : Topology.IsEmbedding e) |
| (L J r : ℝ) (hL : 1 ≤ L) (hJ : 0 < J) (hr : 0 < r) |
| (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) |
| (x : X) (hchart : Metric.ball x (L*(L*J+1)*r) ⊆ range e) |
| (q : ι → X) (ρ : ι → ℝ) (hρ : ∀ i, 0 ≤ ρ i) |
| (hcross : Admissible (fun i => e ⁻¹' Metric.ball (q i) (2*r)) ρ) |
| (n : ℕ) (hn : 0 < n) |
| (hmesh : ∀ 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) (ha : dist x (e a) ≤ J*r) (hb : dist x (e b) ≤ J*r) : |
| let C := fun i => e ⁻¹' Metric.ball (q i) (2*r) |
| |scalar C ρ n hn a-scalar C ρ n hn b| ≤ |
| ∑ i ∈ Finset.univ.filter (fun i => dist (q i) x ≤ (L*(L*J+1)+3)*r), ρ i := by |
| let C := fun i => e ⁻¹' Metric.ball (q i) (2*r) |
| obtain ⟨hrawI,_,_,hraw⟩ := rawFunction_properties C ρ hρ hcross |
| exact CannonScalarLocal.local_oscillation e he L J r hL hJ hr hpaths x hchart q ρ hρ |
| (rawFunction C ρ) hrawI hraw (3*n) (by omega) hmesh topBand bottomBand |
| topBand_pathConnected.nonempty bottomBand_pathConnected.nonempty a b ha hb |
|
|
| end CannonApprox |
|
|
| namespace CannonRectangle |
|
|
| open CannonGrid CannonCells CannonBands |
|
|
| abbrev Middle := Strip height (1/3) (2/3) |
|
|
| def normalize (z : Middle) : CannonGrid.Square := |
| (z.val.1, ⟨3*(z.val.2:ℝ)-1, by |
| have hz := z.property |
| change (z.val.2:ℝ) ∈ Icc (1/3:ℝ) (2/3) at hz |
| constructor <;> linarith [hz.1,hz.2]⟩) |
|
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| lemma continuous_normalize : Continuous normalize := by |
| unfold normalize |
| fun_prop |
|
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| lemma normalize_injective : Function.Injective normalize := by |
| intro x y h |
| apply Subtype.ext |
| apply Prod.ext |
| · simpa only [normalize] using congrArg Prod.fst h |
| · apply Subtype.ext |
| have hh := congrArg (fun z : CannonGrid.Square => (z.2:ℝ)) h |
| change 3*(x.val.2:ℝ)-1 = 3*(y.val.2:ℝ)-1 at hh |
| linarith |
|
|
| def leftRight : Set (Set CannonGrid.Square) := |
| {A | ∃ u v : Middle, u.val.1 = 0 ∧ v.val.1 = 1 ∧ |
| ∃ p : Path u v, A = Subtype.val '' range p} |
|
|
| def topBottom : Set (Set CannonGrid.Square) := |
| {A | ∃ u v : Middle, height u.val = 1/3 ∧ height v.val = 2/3 ∧ |
| ∃ p : Path u v, A = Subtype.val '' range p} |
|
|
| lemma leftRight_nonempty : leftRight.Nonempty := by |
| let u : Middle := ⟨(0,⟨1/2,by norm_num⟩),by norm_num [height]⟩ |
| let v : Middle := ⟨(1,⟨1/2,by norm_num⟩),by norm_num [height]⟩ |
| let : PathConnectedSpace Middle := strip_pathConnected _ _ (by norm_num) (by norm_num) (by norm_num) |
| exact ⟨Subtype.val '' range (PathConnectedSpace.somePath u v), |
| u,v,rfl,rfl,PathConnectedSpace.somePath u v,rfl⟩ |
|
|
| lemma leftRight_members_nonempty (A : Set CannonGrid.Square) (hA : A ∈ leftRight) : |
| A.Nonempty := by |
| obtain ⟨u,v,_,_,p,rfl⟩ := hA |
| exact ⟨(p 0).val,⟨p 0,⟨0,rfl⟩,rfl⟩⟩ |
|
|
| lemma crossings_intersect (A : Set CannonGrid.Square) (hA : A ∈ leftRight) |
| (B : Set CannonGrid.Square) (hB : B ∈ topBottom) : (A ∩ B).Nonempty := by |
| obtain ⟨u,v,hu,hv,p,rfl⟩ := hA |
| obtain ⟨w,z,hw,hz,q,rfl⟩ := hB |
| have hp0 : (normalize (p 0)).1 = 0 := by simpa [normalize] using hu |
| have hp1 : (normalize (p 1)).1 = 1 := by simpa [normalize] using hv |
| have hq0 : (normalize (q.symm 0)).2 = 1 := by |
| apply Subtype.ext |
| change 3*((q.symm 0).val.2:ℝ)-1 = 1 |
| simpa only [Path.source] using show 3*(z.val.2:ℝ)-1 = 1 by |
| change (z.val.2:ℝ) = 2/3 at hz |
| linarith |
| have hq1 : (normalize (q.symm 1)).2 = 0 := by |
| apply Subtype.ext |
| change 3*((q.symm 1).val.2:ℝ)-1 = 0 |
| simpa only [Path.target] using show 3*(w.val.2:ℝ)-1 = 0 by |
| change (w.val.2:ℝ) = 1/3 at hw |
| linarith |
| obtain ⟨s,t,hst⟩ := CannonPlanar.paths_cross |
| (continuous_normalize.comp p.continuous) (continuous_normalize.comp q.symm.continuous) |
| hp0 hp1 hq0 hq1 |
| have he := normalize_injective hst |
| refine ⟨(p s).val,⟨p s,⟨s,rfl⟩,rfl⟩,?_⟩ |
| refine ⟨q.symm t,?_,congrArg Subtype.val he.symm⟩ |
| exact (q.symm_range ▸ mem_range_self t) |
|
|
| theorem exists_dual_weights {ι : Type*} [Fintype ι] |
| (C : ι → Set CannonGrid.Square) (hcover : ∀ x, ∃ i, x ∈ C i) : |
| 0 < modulus C leftRight ∧ ∃ ρ : ι → ℝ, (∀ i, 0 ≤ ρ i) ∧ |
| CannonApprox.Admissible C ρ ∧ (∑ i, (ρ i)^2) = (modulus C leftRight)⁻¹ := by |
| obtain ⟨hpos,_⟩ := modulus_minimum C hcover leftRight_nonempty leftRight_members_nonempty |
| obtain ⟨ρ,hρ,he⟩ := transverse_weights C hcover leftRight_nonempty |
| leftRight_members_nonempty crossings_intersect |
| obtain ⟨hn,hadm⟩ := (admissible_iff C topBottom ρ).mp hρ |
| refine ⟨hpos,ρ,hn,?_,?_⟩ |
| · intro u hu v hv p |
| rw [cost_preimage_image] |
| exact hadm _ ⟨u,v,hu,hv,p,rfl⟩ |
| · simpa only [EuclideanSpace.real_norm_sq_eq] using he |
|
|
| end CannonRectangle |
|
|
| end |
| end |
| end |
| end |
| end |
| end |
| end |
| end |
| end |
|
|
| end OAI |
|
|
