| import Mathlib |
| import OAI.Geometry.Cannon.Energy.RecordData |
|
|
| namespace OAI |
|
|
| section |
| section |
| section |
| noncomputable section |
| section |
| section |
| section |
| namespace CannonApprox |
|
|
| open Set Metric Filter Topology MeasureTheory CannonNode CannonMagnification CannonPacking CannonAnnular |
|
|
| structure FlowSetup {X Γ : Type*} [MetricSpace X] [MeasurableSpace X] |
| {action : Γ → X ≃ₜ X} {ι : ℕ → Type*} [∀ n, Fintype (ι n)] |
| {p : ∀ n, ι n → X} {r : ℕ → ℝ} (R : RecordData action ι p r) |
| (ell L CE LE : ℝ) (Q : ℕ) where |
| ε : ℝ |
| εpos : 0 < ε |
| local_domain : closedBall R.x₀ (2*ε) ⊆ R.Ω |
| c₀ : ℝ |
| A : ℝ |
| B : ℝ |
| cPat : ℝ |
| H : ℝ |
| core : 2*L+2 < c₀ |
| inner : A=4*L*c₀+20 |
| outer : B=A+10 |
| pattern : cPat=2*CE*(2*L)/(A-2)^2 |
| pattern_pos : 0 < cPat |
| margin : 12*cPat*(B+2)^2*LE < c₀ |
| separation : max (4*(B+1)) (max (4*R.Cμ*c₀) 2) < H |
| ρ : ℝ |
| ρpos : 0 < ρ |
| small : (B+2)*ρ ≤ ε |
| cutoff : ρ ≤ ε/2 |
| n : ℕ |
| npos : 0 < n |
| decay : radius ell (160*(Q+1)*n) < 1/4 |
| S : SourceConstruction R.f R.x₀ ε ell c₀ A B H ρ Q n |
| energy : ∀ g, ∑ i, (S.len g i)^2 ≤ R.E * (R.ν univ : ℝ) |
| good : ∀ N : ℕ, ∃ g ≥ N, (1/2:ℝ) ≤ S.flow.goodMass g |
|
|
| variable {X Γ : Type*} [MetricSpace X] [CompactSpace X] [LocallyPathConnectedSpace X] |
| [MeasurableSpace X] [BorelSpace X] |
| {action : Γ → X ≃ₜ X} {ι : ℕ → Type*} [∀ n, Fintype (ι n)] |
| {p : ∀ n, ι n → X} {r : ℕ → ℝ} |
|
|
| theorem RecordData.make_flow (R : RecordData action ι p r) |
| (Q : ℕ) (CP CE LE ell L d₀ : ℝ) |
| (hpack : PolynomialPacking (X:=X) Q CP) (hCP : 0 ≤ CP) |
| (hCE : 0 < CE) (hLE : 0 < LE) (hell : 0 < ell) (hL : 1 ≤ L) |
| (hdc : d₀ ≤ 1/(9*LE)) |
| (M : ℕ → ℝ) (hMnon : ∀ j, 0 ≤ M j) |
| (hM : ∀ lam : ℝ, 1 < lam → ∃ C : ℝ, 0 < C ∧ ∀ j, M j ≤ C*lam^j) |
| (δ : ℕ → Type*) [∀ j, Fintype (δ j)] (ξ : ∀ j, δ j → X) |
| (hξ : ∀ j i i', i ≠ i' → radius ell j ≤ dist (ξ j i) (ξ j i')) |
| (ω : ∀ j, δ j → ℝ) (hω : ∀ j, MacroscopicAdmissible (ξ j) (radius ell j) d₀ (ω j)) |
| (hωmass : ∀ j, (∑ i, (ω j i)^2) ≤ M j) |
| (h : X → ℕ → X ≃ₜ X) |
| (hexp : ∀ x k, Expansion (h x k) x (radius ell k) CE LE) |
| (hpaths : ∀ x u v : X, ∀ s : ℝ, 0 < s → dist u x ≤ s → dist v x ≤ s → |
| JoinedIn (ball x (L*s)) u v) : |
| Nonempty (FlowSetup R ell L CE LE Q) := by |
| classical |
| obtain ⟨e,he,heΩ⟩ := Metric.isOpen_iff.mp R.isOpen R.x₀ R.x₀mem |
| let ε := e/4 |
| have hε : 0 < ε := by dsimp [ε]; positivity |
| have hloc : closedBall R.x₀ (2*ε) ⊆ R.Ω := |
| (closedBall_subset_ball (by dsimp [ε]; linarith only [he])).trans heΩ |
| have hKΩ : closedBall R.x₀ ε ⊆ R.Ω := |
| (closedBall_subset_closedBall (by linarith only [hε])).trans hloc |
| obtain ⟨s₀,hs₀,hmeasure⟩ := R.measure_bound (closedBall R.x₀ ε) isClosed_closedBall.isCompact hKΩ |
| obtain ⟨c₀,A,B,cPat,H,hcore,hinner,houter,hpattern,hcPat,hmargin,hsep⟩ := |
| exists_flow_constants hL hCE hLE R.Cμ_pos.le |
| have hc₀ : 2 ≤ c₀ := by linarith only [hcore,hL] |
| have hc₀pos : 0 < c₀ := by linarith only [hc₀] |
| have hA : 1 ≤ A := by rw [hinner]; nlinarith only [hL,hc₀] |
| have hAB : A+3 ≤ B := by rw [houter]; linarith |
| have hB : 0 ≤ B+1 := by linarith only [hA,hAB] |
| have hB₂ : 0 < B+2 := by linarith only [hB] |
| have hH : 0 < H := lt_trans (by norm_num : (0:ℝ) < 2) |
| ((le_max_right _ _).trans_lt ((le_max_right _ _).trans_lt hsep)) |
| have hLc : L*c₀ ≤ A := by rw [hinner]; nlinarith only [mul_pos (zero_lt_one.trans_le hL) hc₀pos] |
| let ρ := min (s₀/c₀) (min (ε/(B+2)) (ε/2)) |
| have hρ : 0 < ρ := lt_min (div_pos hs₀ hc₀pos) |
| (lt_min (div_pos hε hB₂) (by positivity)) |
| have hρs : c₀*ρ ≤ s₀ := by |
| have hh := (le_div_iff₀ hc₀pos).mp (min_le_left (s₀/c₀) (min (ε/(B+2)) (ε/2))) |
| simpa only [mul_comm,ρ] using hh |
| have hρB : (B+2)*ρ ≤ ε := by |
| have hmin : ρ ≤ ε/(B+2) := (min_le_right _ _).trans (min_le_left _ _) |
| have hh := (le_div_iff₀ hB₂).mp hmin |
| simpa only [mul_comm,ρ] using hh |
| have hρcut : ρ ≤ ε/2 := (min_le_right _ _).trans (min_le_right _ _) |
| obtain ⟨n,hn,hθ,⟨S⟩⟩ := exists_source_construction Q CP A B H c₀ CE LE d₀ ell L |
| hpack hCP hA hAB hH hc₀ hCE.le hLE hdc hell hL hLc M hMnon hM δ ξ hξ ω hω hωmass |
| h hexp hpaths R.Ω R.f R.continuous R.unit R.level R.diameter R.diameter_pos |
| (fun t ht => ⟨(R.level_properties t ht).2.1.isPreconnected, |
| (R.level_properties t ht).2.2.1,(R.level_properties t ht).2.2.2⟩) |
| R.level_hits R.x₀ ε ρ hε hρ hloc R.variation |
| have hE : ∀ g, ∑ i, (S.len g i)^2 ≤ R.E*(R.ν univ : ℝ) := by |
| apply S.level_energy R.ν.toMeasure R.Cμ R.E R.E_nonneg hB hc₀pos.le hθ |
| (lt_of_le_of_lt (le_max_left _ _) hsep) ?_ ?_ ?_ |
| · have hh := lt_of_le_of_lt (le_max_left (4*R.Cμ*c₀) 2) |
| (lt_of_le_of_lt (le_max_right (4*(B+1)) _) hsep) |
| nlinarith only [hh,mul_pos R.Cμ_pos hc₀pos] |
| · intro x hx s hs hsr |
| have hh := hmeasure (c₀*s) (mul_pos hc₀pos hs) |
| ((mul_le_mul_of_nonneg_left hsr hc₀pos.le).trans hρs) x hx |
| exact R.continuous.mono hh.1 |
| · intro x hx s hs hsr y hy z hz |
| have hh := hmeasure (c₀*s) (mul_pos hc₀pos hs) |
| ((mul_le_mul_of_nonneg_left hsr hc₀pos.le).trans hρs) x hx |
| have hb := (isCompact_closedBall x (c₀*s)).image_of_continuousOn (R.continuous.mono hh.1) |
| have hyl := csInf_le hb.bddBelow (mem_image_of_mem R.f hy) |
| have hyu := le_csSup hb.bddAbove (mem_image_of_mem R.f hy) |
| have hzl := csInf_le hb.bddBelow (mem_image_of_mem R.f hz) |
| have hzu := le_csSup hb.bddAbove (mem_image_of_mem R.f hz) |
| have hd : |R.f y-R.f z| ≤ CannonCrossEnergy.oscillation R.f (closedBall x (c₀*s)) := by |
| unfold CannonCrossEnergy.oscillation |
| exact abs_le.mpr ⟨by linarith only [hyl,hzu],by linarith only [hzl,hyu]⟩ |
| have hsq := pow_le_pow_left₀ (abs_nonneg _) hd 2 |
| change (R.f y-R.f z)^2 ≤ R.E*(R.ν (closedBall x (R.Cμ*(c₀*s))) : ℝ) |
| simpa only [sq_abs] using hsq.trans hh.2 |
| exact ⟨{ |
| ε := ε, εpos := hε, local_domain := hloc, c₀ := c₀, A := A, B := B, |
| cPat := cPat, H := H, core := hcore, inner := hinner, outer := houter, |
| pattern := hpattern, pattern_pos := hcPat, margin := hmargin, separation := hsep, |
| ρ := ρ, ρpos := hρ, small := hρB, cutoff := hρcut, n := n, npos := hn, decay := hθ, |
| S := S, energy := hE, good := S.arbitrarily_late_good hell hn _ hE }⟩ |
|
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| end CannonApprox |
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| end |
|
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| section |
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|
| section |
|
|
| section |
|
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| open Set Topology |
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| namespace CannonRelease |
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| open Set Topology |
|
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| universe v |
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| variable {G : Type v} [Group G] [TopologicalSpace G] [DiscreteTopology G] |
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| variable (D : CayleyData G) |
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| variable (hthin : D.ThinTriangles) |
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| local instance flowSetupRaySetoid1 : Setoid (BasedRay D) := raySetoid D |
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| end CannonRelease |
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| namespace CannonRelease |
|
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| open Set Topology |
|
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| universe v |
|
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| variable {G : Type v} [Group G] [TopologicalSpace G] [DiscreteTopology G] |
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| variable (D : CayleyData G) (hthin : D.ThinTriangles) |
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| local instance flowSetupRaySetoid2 : Setoid (BasedRay D) := raySetoid D |
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| end CannonRelease |
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| namespace CannonRelease |
|
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| open CannonGraph CannonVisual |
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| universe v |
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| variable {G : Type v} [Group G] [TopologicalSpace G] [DiscreteTopology G] |
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| variable (D : CayleyData G) {δ : ℕ} (hδ : Thin D.graph δ) |
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| local instance flowSetupRaySetoid3 : Setoid (BasedRay D) := raySetoid D |
|
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| end CannonRelease |
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| namespace CannonRelease |
|
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| open CannonGraph CannonVisual Filter Topology |
| open scoped NNReal |
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| universe v |
|
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| variable {G : Type v} [Group G] [TopologicalSpace G] [DiscreteTopology G] |
|
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| variable (D : CayleyData G) {δ : ℕ} (hδ : Thin D.graph δ) |
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| local instance flowSetupRaySetoid4 : Setoid (BasedRay D) := raySetoid D |
|
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| end CannonRelease |
|
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| namespace CannonRelease |
|
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| open CannonGraph CannonVisual Filter Topology |
| open scoped NNReal |
|
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| universe v |
|
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| variable {G : Type v} [Group G] [TopologicalSpace G] [DiscreteTopology G] |
|
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| variable (D : CayleyData G) {δ : ℕ} (hδ : Thin D.graph δ) |
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| local instance flowSetupRaySetoid5 : Setoid (BasedRay D) := raySetoid D |
|
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| end CannonRelease |
|
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| namespace CannonRelease |
|
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| open CannonGraph CannonVisual Filter Topology |
| open scoped NNReal |
|
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| universe v |
|
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| variable {G : Type v} [Group G] [TopologicalSpace G] [DiscreteTopology G] |
|
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| variable (D : CayleyData G) {δ : ℕ} (hδ : Thin D.graph δ) |
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| local instance flowSetupRaySetoid6 : Setoid (BasedRay D) := raySetoid D |
|
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| end CannonRelease |
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| namespace CannonRelease |
|
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| open CannonGraph CannonVisual Filter Topology |
| open scoped NNReal |
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| universe v |
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| variable {G : Type v} [Group G] [TopologicalSpace G] [DiscreteTopology G] |
|
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| variable (D : CayleyData G) {δ : ℕ} (hδ : Thin D.graph δ) |
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| local instance flowSetupRaySetoid7 : Setoid (BasedRay D) := raySetoid D |
|
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| end CannonRelease |
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| namespace CannonRelease |
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| open CannonGraph CannonVisual Filter Topology |
| open scoped NNReal |
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| universe v |
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| variable {G : Type v} [Group G] [TopologicalSpace G] [DiscreteTopology G] |
|
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| variable (D : CayleyData G) {δ : ℕ} (hδ : Thin D.graph δ) |
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| local instance flowSetupRaySetoid8 : Setoid (BasedRay D) := raySetoid D |
|
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| end CannonRelease |
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| namespace CannonRelease |
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| open CannonGraph CannonVisual Filter Topology |
| open scoped NNReal |
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| universe v |
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| variable {G : Type v} [Group G] [TopologicalSpace G] [DiscreteTopology G] |
|
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| variable (D : CayleyData G) {δ : ℕ} (hδ : Thin D.graph δ) |
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| local instance flowSetupRaySetoid9 : Setoid (BasedRay D) := raySetoid D |
|
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| end CannonRelease |
|
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| namespace CannonRelease |
|
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| open CannonGraph CannonVisual Filter Topology |
| open scoped NNReal |
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| universe v |
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| variable {G : Type v} [Group G] [TopologicalSpace G] [DiscreteTopology G] |
|
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| variable (D : CayleyData G) {δ : ℕ} (hδ : Thin D.graph δ) |
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| local instance flowSetupRaySetoid10 : Setoid (BasedRay D) := raySetoid D |
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| end CannonRelease |
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| namespace CannonRelease |
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| open CannonGraph CannonVisual Filter Topology |
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| universe v |
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| variable {G : Type v} [Group G] [TopologicalSpace G] [DiscreteTopology G] |
|
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| variable (D : CayleyData G) {δ : ℕ} (hδ : Thin D.graph δ) |
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| local instance flowSetupRaySetoid11 : Setoid (BasedRay D) := raySetoid D |
|
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| end CannonRelease |
|
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| namespace CannonRelease |
|
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| open CannonGraph CannonVisual Filter Topology |
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| universe v |
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| variable {G : Type v} [Group G] [TopologicalSpace G] [DiscreteTopology G] |
|
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| variable (D : CayleyData G) {δ : ℕ} (hδ : Thin D.graph δ) |
|
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| local instance flowSetupRaySetoid12 : Setoid (BasedRay D) := raySetoid D |
|
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| end CannonRelease |
|
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| namespace CannonRelease |
|
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| open CannonGraph CannonVisual Filter Topology Set |
|
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| universe v |
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| variable {G : Type v} [Group G] [TopologicalSpace G] [DiscreteTopology G] |
|
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| variable (D : CayleyData G) {δ : ℕ} (hδ : Thin D.graph δ) |
|
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| local instance flowSetupRaySetoid13 : Setoid (BasedRay D) := raySetoid D |
|
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| end CannonRelease |
|
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| namespace CannonRelease |
|
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| open CannonGraph CannonVisual Filter Topology Set |
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| universe v |
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| variable {G : Type v} [Group G] [TopologicalSpace G] [DiscreteTopology G] |
|
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| variable (D : CayleyData G) {δ : ℕ} (hδ : Thin D.graph δ) |
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| local instance flowSetupRaySetoid14 : Setoid (BasedRay D) := raySetoid D |
|
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| end CannonRelease |
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| namespace CannonRelease |
|
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| open CannonGraph CannonVisual Set |
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| universe v |
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| variable {G : Type v} [Group G] [TopologicalSpace G] [DiscreteTopology G] |
|
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| variable (D : CayleyData G) {δ : ℕ} (hδ : Thin D.graph δ) |
|
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| local instance flowSetupRaySetoid15 : Setoid (BasedRay D) := raySetoid D |
|
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| end CannonRelease |
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| namespace CannonRelease |
|
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| open CannonGraph CannonVisual Set |
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| universe v |
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| variable {G : Type v} [Group G] [TopologicalSpace G] [DiscreteTopology G] |
|
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| variable (D : CayleyData G) {δ : ℕ} (hδ : Thin D.graph δ) |
|
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| local instance flowSetupRaySetoid16 : Setoid (BasedRay D) := raySetoid D |
|
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| end CannonRelease |
|
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| namespace CannonRelease |
|
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| open CannonGraph CannonVisual Set |
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| universe v |
|
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| variable {G : Type v} [Group G] [TopologicalSpace G] [DiscreteTopology G] |
|
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| variable (D : CayleyData G) {δ : ℕ} (hδ : Thin D.graph δ) |
|
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| local instance flowSetupRaySetoid17 : Setoid (BasedRay D) := raySetoid D |
|
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| end CannonRelease |
|
|
| open Set Metric Topology |
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| namespace CannonRelease |
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| open CannonGraph CannonVisual CannonPaths Set Metric |
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| universe v |
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| variable {G : Type v} [Group G] [TopologicalSpace G] [DiscreteTopology G] |
|
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| variable (D : CayleyData G) {δ : ℕ} (hδ : Thin D.graph δ) |
|
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| local instance flowSetupRaySetoid18 : Setoid (BasedRay D) := raySetoid D |
|
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| end CannonRelease |
|
|
| open Set Topology Filter Metric unitInterval |
| open scoped InnerProductSpace |
|
|
| end |
| end |
| end |
| end |
| end |
| end |
| end |
| end |
| end |
|
|
| end OAI |
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|
