| import Mathlib |
| import OAI.Geometry.Cannon.Flows.EscapingLevelEnergyObstruction |
| import OAI.Geometry.Cannon.Flows.FarDiam |
|
|
| namespace OAI |
|
|
| section |
| section |
| section |
| noncomputable section |
| section |
| section |
| section |
| section |
| section |
| section |
| namespace CannonPairs |
|
|
| open Set Metric Filter Topology CannonApprox CannonCrossEnergyLocal |
|
|
| lemma interval_avoiding_short_range {u v l h : ℝ} (huv : u < v) |
| (hshort : h-l < (v-u)/4) : |
| ∃ a d e : ℝ, u < a ∧ a < d ∧ d < v ∧ 0 < e ∧ |
| ∀ t ∈ Icc a d, ∀ z ∈ Icc l h, e ≤ |t-z| := by |
| by_cases hl : u+(v-u)/4 < l |
| · refine ⟨u+(v-u)/12,u+(v-u)/6,(v-u)/12,by linarith,by linarith, |
| by linarith,by linarith,?_⟩ |
| intro t ht z hz |
| have hh : (v-u)/12 ≤ z-t := by linarith only [hl,ht.2,hz.1] |
| exact hh.trans (by rw [abs_sub_comm]; exact le_abs_self _) |
| · have hl' := le_of_not_gt hl |
| refine ⟨u+2*(v-u)/3,u+5*(v-u)/6,(v-u)/12,by linarith,by linarith, |
| by linarith,by linarith,?_⟩ |
| intro t ht z hz |
| exact (show (v-u)/12 ≤ t-z by linarith only [huv,hshort,hl',ht.1,hz.2]).trans |
| (le_abs_self _) |
|
|
| lemma cells_into_puncture_exterior {X : Type*} [MetricSpace X] |
| (ι : ℕ → Type*) (p : ∀ n, ι n → X) (r : ℕ → ℝ) |
| (hrlim : Tendsto r atTop (𝓝 0)) (b c : X) {ρ ε : ℝ} (hρ : 0 < ρ) (hε : 0 < ε) : |
| ∀ᶠ n in atTop, ∀ i, |
| (ball (p n i) (2*r n) ∩ {z | ρ ≤ dist z b ∧ ε ≤ dist z c}).Nonempty → |
| closedBall (p n i) (2*r n) ⊆ {z | ρ/2 ≤ dist z b ∧ ε/2 ≤ dist z c} := by |
| have ht : ∀ᶠ n in atTop, 4*r n < min ρ ε/2 := by |
| have hh : Tendsto (fun n => 4*r n) atTop (𝓝 0) := by |
| simpa only [mul_zero] using (tendsto_const_nhds.mul hrlim : |
| Tendsto (fun n => 4*r n) atTop (𝓝 (4*0))) |
| exact hh.eventually_lt_const (div_pos (lt_min hρ hε) (by norm_num)) |
| filter_upwards [ht] with n hn |
| intro i hi z hz |
| obtain ⟨t,hti,htK⟩ := hi |
| have htz := dist_triangle t (p n i) z |
| have hti' := mem_ball.mp hti |
| have hzi := mem_closedBall.mp hz |
| rw [dist_comm (p n i) z] at htz |
| have hbtri := dist_triangle t z b |
| have hctri := dist_triangle t z c |
| have hbmin := min_le_left ρ ε |
| have hcmin := min_le_right ρ ε |
| obtain ⟨htb,htc⟩ := htK |
| constructor <;> linarith |
|
|
| theorem path_forces_opposite_oscillation {X : Type*} [MetricSpace X] [CompactSpace X] |
| (ι : ℕ → Type*) [∀ n, Fintype (ι n)] (p : ∀ n, ι n → X) |
| (r : ℕ → ℝ) (hr : ∀ n, 0 < r n) (hrlim : Tendsto r atTop (𝓝 0)) |
| (hpnet : ∀ n x, ∃ i, dist (p n i) x ≤ r n) |
| (f w : ℕ → X → ℝ) (W V : X → ℝ) (b c : X) |
| (ρ ε : ℝ) (hρ : 0 < ρ) (hε : 0 < ε) |
| (v₀ v₁ : X) (γ : Path v₀ v₁) (hvar : W v₀ < W v₁) |
| (hP : ∀ x ∈ range γ, ρ < dist x b) |
| (q : ℝ) (hq : 0 < q) |
| (hgap : ∀ x ∈ range γ, ∀ z ∈ closedBall b ρ, q ≤ |V x-V z|) |
| (hW : ContinuousOn W (closedBall c ε)) |
| (hflim : TendstoUniformlyOn f W atTop (closedBall c ε)) |
| (hwlimP : TendstoUniformlyOn w V atTop (range γ)) |
| (hwlimB : TendstoUniformlyOn w V atTop (closedBall b ρ)) |
| (hfcont : ∀ᶠ n in atTop, |
| ContinuousOn (f n) {z | ρ/2 ≤ dist z b ∧ ε/2 ≤ dist z c}) |
| (hwcont : ∀ᶠ n in atTop, |
| ContinuousOn (w n) {z | ρ/2 ≤ dist z b ∧ ε/2 ≤ dist z c}) |
| (hfe : Tendsto (fun n => meshEnergy (p n) (r n) 2 (f n) |
| {z | ρ ≤ dist z b ∧ ε ≤ dist z c}) atTop (𝓝 0)) |
| (hwe : Tendsto (fun n => meshEnergy (p n) (r n) 2 (w n) |
| {z | ρ ≤ dist z b ∧ ε ≤ dist z c}) atTop (𝓝 0)) |
| (hlevels : ∀ a d : ℝ, W v₀ < a → a < d → d < W v₁ → |
| ∀ᶠ n in atTop, ∀ t ∈ Icc a d, ∃ D : Set X, |
| IsCompact D ∧ IsConnected D ∧ (D ∩ range γ).Nonempty ∧ |
| (∀ x ∈ D, f n x = t) ∧ ∃ z ∈ D, dist z b < ρ) : |
| (W v₁-W v₀)/4 ≤ CannonCrossEnergy.oscillation W (closedBall c ε) := by |
| by_contra hn |
| have hshort : sSup (W '' closedBall c ε)-sInf (W '' closedBall c ε) < |
| (W v₁-W v₀)/4 := lt_of_not_ge hn |
| obtain ⟨a,d,e,hva,had,hdv,he,hegap⟩ := interval_avoiding_short_range hvar hshort |
| have hCc : IsCompact (closedBall c ε) := isClosed_closedBall.isCompact |
| have havoid : ∀ᶠ n in atTop, ∀ t ∈ Icc a d, ∀ z ∈ closedBall c ε, f n z ≠ t := by |
| filter_upwards [Metric.tendstoUniformlyOn_iff.mp hflim e he] with n hn |
| intro t ht z hz hzt |
| have hwr := CannonCrossEnergyLocal.mem_rangeInterval hW hCc hz |
| have hg := hegap t ht (W z) hwr |
| have hh := hn z hz |
| rw [Real.dist_eq,hzt,abs_sub_comm] at hh |
| exact (not_lt_of_ge hg) hh |
| have hwgap : ∀ᶠ n in atTop, ∀ x ∈ range γ, ∀ z, dist z b = ρ → |
| q/2 ≤ |w n x-w n z| := by |
| filter_upwards [Metric.tendstoUniformlyOn_iff.mp hwlimP (q/4) (by positivity), |
| Metric.tendstoUniformlyOn_iff.mp hwlimB (q/4) (by positivity)] with n hnP hnB |
| intro x hx z hz |
| have hzB : z ∈ closedBall b ρ := mem_closedBall.mpr hz.le |
| have hxerr := hnP x hx |
| have hzerr := hnB z hzB |
| rw [Real.dist_eq] at hxerr hzerr |
| have hg := hgap x hx z hzB |
| have ha := abs_sub_le (V x) (w n x) (V z) |
| have hb := abs_sub_le (w n x) (w n z) (V z) |
| rw [abs_sub_comm (w n z) (V z)] at hb |
| linarith |
| apply escaping_level_energy_obstruction ι p r hr hpnet f w |
| (fun _ => {z | ρ/2 ≤ dist z b ∧ ε/2 ≤ dist z c}) b c ρ ε (range γ) hP a d (q/2) |
| had (by positivity) (Eventually.of_forall (by |
| intro n z hz |
| constructor <;> linarith [hz.1,hz.2])) |
| (cells_into_puncture_exterior ι p r hrlim b c hρ hε) hfcont hwcont hwgap _ hfe hwe |
| filter_upwards [hlevels a d hva had hdv,havoid] with n hn hnA |
| intro t ht |
| obtain ⟨D,hD,hDconn,hDP,hDlev,hDesc⟩ := hn t ht |
| refine ⟨D,hD,hDconn,hDP,hDlev,hDesc,?_⟩ |
| intro z hz |
| by_contra hnz |
| exact hnA t ht z (mem_closedBall.mpr (lt_of_not_ge hnz).le) (hDlev z hz) |
|
|
| end CannonPairs |
|
|
| namespace CannonPairs |
|
|
| open Set Metric Filter Topology CannonApprox CannonCrossEnergyLocal |
|
|
| lemma separated_paths_avoid_short_range {X : Type*} [TopologicalSpace X] |
| (V : X → ℝ) (P₀ P₁ E : Set X) (hP₀ : IsCompact P₀) (hne : P₀.Nonempty) |
| (hE : IsCompact E) (hVP : ContinuousOn V P₀) (hVE : ContinuousOn V E) |
| (δ : ℝ) (hδ : 0 < δ) |
| (hsep : ∀ x ∈ P₀, ∀ y ∈ P₁, V x+δ ≤ V y) |
| (hshort : CannonCrossEnergy.oscillation V E < δ/4) : |
| (∀ x ∈ P₀, ∀ z ∈ E, δ/4 ≤ |V x-V z|) ∨ |
| (∀ x ∈ P₁, ∀ z ∈ E, δ/4 ≤ |V x-V z|) := by |
| let u₀ := sSup (V '' P₀) |
| have hbound₀ : ∀ x ∈ P₀, V x ≤ u₀ := by |
| intro x hx |
| exact le_csSup (hP₀.image_of_continuousOn hVP).bddAbove (mem_image_of_mem V hx) |
| have hbound₁ : ∀ y ∈ P₁, u₀+δ ≤ V y := by |
| intro y hy |
| have hh : u₀ ≤ V y-δ := csSup_le (hne.image V) (by |
| rintro _ ⟨x,hx,rfl⟩ |
| linarith only [hsep x hx y hy]) |
| linarith only [hh] |
| by_cases hhi : sSup (V '' E) ≤ u₀+δ/2 |
| · right |
| intro x hx z hz |
| have hzx := (CannonCrossEnergyLocal.mem_rangeInterval hVE hE hz).2 |
| have hxx := hbound₁ x hx |
| exact (show δ/4 ≤ V x-V z by linarith only [hhi,hzx,hxx,hδ]).trans (le_abs_self _) |
| · left |
| have hhi' := lt_of_not_ge hhi |
| intro x hx z hz |
| have hzx := (CannonCrossEnergyLocal.mem_rangeInterval hVE hE hz).1 |
| have hxx := hbound₀ x hx |
| dsimp [CannonCrossEnergy.oscillation] at hshort |
| have hh : δ/4 ≤ V z-V x := by linarith only [hhi',hzx,hxx,hshort] |
| exact hh.trans (by rw [abs_sub_comm]; exact le_abs_self _) |
|
|
| theorem two_path_pair_alternative {X : Type*} [MetricSpace X] [CompactSpace X] |
| (ι : ℕ → Type*) [∀ n, Fintype (ι n)] (p : ∀ n, ι n → X) |
| (r : ℕ → ℝ) (hr : ∀ n, 0 < r n) (hrlim : Tendsto r atTop (𝓝 0)) |
| (hpnet : ∀ n x, ∃ i, dist (p n i) x ≤ r n) |
| (f w : ℕ → X → ℝ) (W V : X → ℝ) (b c : X) |
| (ρ ε : ℝ) (hρ : 0 < ρ) (hε : 0 < ε) |
| (v₀ v₁ : Bool → X) (γ : ∀ s, Path (v₀ s) (v₁ s)) |
| (gap δ : ℝ) (hgap : 0 < gap) (hδ : 0 < δ) |
| (hvar : ∀ s, gap ≤ W (v₁ s)-W (v₀ s)) |
| (hP : ∀ s, ∀ x ∈ range (γ s), ρ < dist x b) |
| (hsep : ∀ x ∈ range (γ false), ∀ y ∈ range (γ true), V x+δ ≤ V y) |
| (hVP : ContinuousOn V (range (γ false))) |
| (hVE : ContinuousOn V (closedBall b ρ)) |
| (hW : ContinuousOn W (closedBall c ε)) |
| (hflim : TendstoUniformlyOn f W atTop (closedBall c ε)) |
| (hwlimP : ∀ s, TendstoUniformlyOn w V atTop (range (γ s))) |
| (hwlimB : TendstoUniformlyOn w V atTop (closedBall b ρ)) |
| (hfcont : ∀ᶠ n in atTop, |
| ContinuousOn (f n) {z | ρ/2 ≤ dist z b ∧ ε/2 ≤ dist z c}) |
| (hwcont : ∀ᶠ n in atTop, |
| ContinuousOn (w n) {z | ρ/2 ≤ dist z b ∧ ε/2 ≤ dist z c}) |
| (hfe : Tendsto (fun n => meshEnergy (p n) (r n) 2 (f n) |
| {z | ρ ≤ dist z b ∧ ε ≤ dist z c}) atTop (𝓝 0)) |
| (hwe : Tendsto (fun n => meshEnergy (p n) (r n) 2 (w n) |
| {z | ρ ≤ dist z b ∧ ε ≤ dist z c}) atTop (𝓝 0)) |
| (hlevels : ∀ s, ∀ a d : ℝ, W (v₀ s) < a → a < d → d < W (v₁ s) → |
| ∀ᶠ n in atTop, ∀ t ∈ Icc a d, ∃ D : Set X, |
| IsCompact D ∧ IsConnected D ∧ (D ∩ range (γ s)).Nonempty ∧ |
| (∀ x ∈ D, f n x = t) ∧ ∃ z ∈ D, dist z b < ρ) : |
| gap/4 ≤ CannonCrossEnergy.oscillation W (closedBall c ε) ∨ |
| δ/4 ≤ CannonCrossEnergy.oscillation V (closedBall b ρ) := by |
| by_cases hv : δ/4 ≤ CannonCrossEnergy.oscillation V (closedBall b ρ) |
| · exact Or.inr hv |
| left |
| have hor := separated_paths_avoid_short_range V (range (γ false)) (range (γ true)) |
| (closedBall b ρ) (isCompact_range (γ false).continuous) (range_nonempty _) |
| isClosed_closedBall.isCompact hVP hVE δ hδ hsep (lt_of_not_ge hv) |
| obtain ⟨s,hs⟩ : ∃ s : Bool, ∀ x ∈ range (γ s), ∀ z ∈ closedBall b ρ, |
| δ/4 ≤ |V x-V z| := by |
| rcases hor with h | h |
| · exact ⟨false,h⟩ |
| · exact ⟨true,h⟩ |
| have hh := path_forces_opposite_oscillation ι p r hr hrlim hpnet f w W V b c ρ ε |
| hρ hε (v₀ s) (v₁ s) (γ s) (by linarith only [hgap,hvar s]) (hP s) (δ/4) |
| (by positivity) hs hW hflim (hwlimP s) hwlimB hfcont hwcont hfe hwe (hlevels s) |
| linarith only [hh,hvar s] |
|
|
| end CannonPairs |
|
|
| end |
|
|
| namespace CannonPairs |
|
|
| open Set Metric Filter Topology CannonMagnification CannonApprox |
|
|
| lemma uniformly_on_preimage {X : Type*} [MetricSpace X] (h : X ≃ₜ X) |
| (H : ℕ → X → ℝ) (f : X → ℝ) (Ω T : Set X) |
| (hlim : ∀ K : Set X, IsCompact K → K ⊆ Ω → TendstoUniformlyOn H f atTop K) |
| (hT : IsCompact T) (hTΩ : h.symm '' T ⊆ Ω) : |
| TendstoUniformlyOn (fun n => H n ∘ h.symm) (f ∘ h.symm) atTop T := |
| ((hlim (h.symm '' T) (hT.image h.symm.continuous) hTΩ).comp h.symm).mono |
| (fun _ hx => mem_image_of_mem _ hx) |
|
|
| lemma continuousOn_preimage {X : Type*} [TopologicalSpace X] (h : X ≃ₜ X) |
| (f : X → ℝ) (Ω T : Set X) (hf : ContinuousOn f Ω) (hTΩ : h.symm '' T ⊆ Ω) : |
| ContinuousOn (f ∘ h.symm) T := |
| hf.comp h.symm.continuous.continuousOn (fun _ hx => hTΩ (mem_image_of_mem _ hx)) |
|
|
| lemma puncture_exterior_domain {X : Type*} [MetricSpace X] |
| (h : X ≃ₜ X) (x p : X) (R U A ε : ℝ) (Ω : Set X) |
| (hR : 0 < R) (hA : 0 < A) (hroom : 2*U/A ≤ ε) |
| (hΩ : closedBall x ε ⊆ Ω) |
| (hupper : ∀ y, dist (h y) p*denominator x R y ≤ U*R) : |
| h.symm '' {z | A*R/2 ≤ dist z p} ⊆ Ω := by |
| rintro y ⟨z,hz,rfl⟩ |
| apply hΩ |
| rw [mem_closedBall,dist_comm] |
| have hD : 0 ≤ denominator x R (h.symm z) := hR.le.trans (le_max_left _ _) |
| have hm := mul_le_mul_of_nonneg_right hz hD |
| have hu := hupper (h.symm z) |
| rw [h.apply_symm_apply] at hu |
| have hd : denominator x R (h.symm z) ≤ 2*U/A := by |
| apply (le_div_iff₀ hA).mpr |
| have hh : R*(A*denominator x R (h.symm z)) ≤ R*(2*U) := by |
| nlinarith only [hm,hu] |
| simpa only [mul_comm A] using (mul_le_mul_iff_right₀ hR).mp hh |
| exact (le_max_right _ _).trans (hd.trans hroom) |
|
|
| lemma pattern_puncture_distance {X : Type*} [MetricSpace X] |
| (h : X ≃ₜ X) (x p : X) (R V B cutoff : ℝ) |
| (hR : 0 < R) (hV : 0 < V) (hB : 1 ≤ B) (hsmall : B*R ≤ cutoff) |
| (hlower : ∀ y, denominator x R y ≤ cutoff → |
| R ≤ V*dist (h y) p*denominator x R y) |
| (T : Set X) (hT : h.symm '' T ⊆ closedBall x (B*R)) : |
| ∀ z ∈ T, 1/(V*B) ≤ dist z p := by |
| intro z hz |
| have hd := mem_closedBall.mp (hT (mem_image_of_mem _ hz)) |
| have hD : denominator x R (h.symm z) ≤ B*R := by |
| apply max_le (by nlinarith only [hB,hR]) |
| simpa only [dist_comm] using hd |
| have hh := hlower (h.symm z) (hD.trans hsmall) |
| rw [h.apply_symm_apply] at hh |
| have hm := mul_le_mul_of_nonneg_left hD (mul_nonneg hV.le (dist_nonneg (x:=z) (y:=p))) |
| have hh' : R*1 ≤ R*(V*B*dist z p) := by nlinarith only [hh,hm] |
| have hz' := (mul_le_mul_iff_right₀ hR).mp hh' |
| exact (div_le_iff₀ (mul_pos hV (zero_lt_one.trans_le hB))).mpr (by nlinarith only [hz']) |
|
|
| lemma wide_image_enters {X : Type*} [MetricSpace X] |
| (h : X ≃ₜ X) (x p : X) (R U A d : ℝ) (hR : 0 < R) |
| (hd : 0 < d) (hA : 3*U/d < A) |
| (hupper : ∀ y, dist (h y) p*denominator x R y ≤ U*R) |
| (K : Set X) (hK : d ≤ diam K) : |
| ∃ z ∈ h '' K, dist z p < A*R := by |
| obtain ⟨y,hy,hyfar⟩ := exists_far_of_diam K d hd hK x |
| have hm := mul_le_mul_of_nonneg_left (hyfar.trans (le_max_right R (dist x y))) |
| (dist_nonneg (x:=h y) (y:=p)) |
| change dist (h y) p*(d/3) ≤ dist (h y) p*denominator x R y at hm |
| have hh : dist (h y) p ≤ (3*U/d)*R := by |
| apply (mul_le_mul_iff_left₀ (show 0 < d/3 by positivity)).mp |
| calc |
| _ ≤ U*R := hm.trans (hupper y) |
| _ = ((3*U/d)*R)*(d/3) := by field_simp |
| exact ⟨h y,mem_image_of_mem h hy,hh.trans_lt (mul_lt_mul_of_pos_right hA hR)⟩ |
|
|
| theorem transformed_level_paths {X : Type*} [MetricSpace X] [CompactSpace X] |
| (Ω : Set X) (H : ℕ → X → ℝ) (f : X → ℝ) |
| (hfunit : ∀ x ∈ Ω, f x ∈ Icc (0:ℝ) 1) |
| (hlim : ∀ K : Set X, IsCompact K → K ⊆ Ω → TendstoUniformlyOn H f atTop K) |
| (d : ℝ) (hd : 0 < d) |
| (hlevels : ∀ n t, t ∈ Ioo (0:ℝ) 1 → ∃ K : Set X, |
| IsCompact K ∧ IsConnected K ∧ d ≤ diam K ∧ (∀ y ∈ K, H n y=t) ∧ |
| ∀ u v, ∀ γ : Path u v, range γ ⊆ Ω → H n u < t → t < H n v → |
| (K ∩ range γ).Nonempty) |
| (h : X ≃ₜ X) (x p : X) (R U A : ℝ) (hR : 0 < R) (hA : 3*U/d < A) |
| (hupper : ∀ y, dist (h y) p*denominator x R y ≤ U*R) |
| (v₀ v₁ : X) (γ : Path v₀ v₁) (hγ : h.symm '' range γ ⊆ Ω) |
| (a b : ℝ) (ha : f (h.symm v₀) < a) (_hab : a < b) (hb : b < f (h.symm v₁)) : |
| ∀ᶠ n in atTop, ∀ t ∈ Icc a b, ∃ D : Set X, |
| IsCompact D ∧ IsConnected D ∧ (D ∩ range γ).Nonempty ∧ |
| (∀ y ∈ D, (H n ∘ h.symm) y=t) ∧ ∃ y ∈ D, dist y p < A*R := by |
| let π : Path (h.symm v₀) (h.symm v₁) := γ.map h.symm.continuous |
| have hπ : range π ⊆ Ω := by |
| rintro y ⟨s,rfl⟩ |
| exact hγ (mem_image_of_mem _ (mem_range_self s)) |
| have huv₀ : 0 ≤ f (h.symm v₀) := (hfunit _ (hπ π.source_mem_range)).1 |
| have huv₁ : f (h.symm v₁) ≤ 1 := (hfunit _ (hπ π.target_mem_range)).2 |
| have hconv := hlim (range π) (isCompact_range π.continuous) hπ |
| have ht₀ : ∀ᶠ n in atTop, H n (h.symm v₀) < a := |
| (hconv.tendsto_at π.source_mem_range).eventually_lt_const ha |
| have ht₁ : ∀ᶠ n in atTop, b < H n (h.symm v₁) := |
| (hconv.tendsto_at π.target_mem_range).eventually_const_lt hb |
| filter_upwards [ht₀,ht₁] with n hn₀ hn₁ |
| intro t ht |
| have htt : t ∈ Ioo (0:ℝ) 1 := ⟨huv₀.trans_lt (ha.trans_le ht.1),ht.2.trans_lt (hb.trans_le huv₁)⟩ |
| obtain ⟨K,hK,hKc,hKd,hKt,hhit⟩ := hlevels n t htt |
| obtain ⟨z,hz,⟨s,hs⟩⟩ := hhit _ _ π hπ (hn₀.trans_le ht.1) (ht.2.trans_lt hn₁) |
| refine ⟨h '' K,hK.image h.continuous,hKc.image h h.continuous.continuousOn,?_,?_, |
| wide_image_enters h x p R U A d hR hd hA hupper K hKd⟩ |
| · refine ⟨γ s,?_,mem_range_self s⟩ |
| refine ⟨z,hz,?_⟩ |
| rw [← hs] |
| exact h.apply_symm_apply (γ s) |
| · rintro y ⟨z,hz,rfl⟩ |
| simpa only [Function.comp_apply,h.symm_apply_apply] using hKt z hz |
|
|
| end CannonPairs |
|
|
| open Set Filter Metric Topology CannonMagnification CannonApprox |
|
|
| namespace CannonPairs |
|
|
| theorem fixed_source_pair {X : Type*} [MetricSpace X] [CompactSpace X] |
| (ι : ℕ → Type*) [∀ n, Fintype (ι n)] (p : ∀ n, ι n → X) |
| (r : ℕ → ℝ) (hr : ∀ n, 0 < r n) (hrlim : Tendsto r atTop (𝓝 0)) |
| (hpnet : ∀ n x, ∃ i, dist (p n i) x ≤ r n) |
| (Ω : Set X) (H : ℕ → X → ℝ) (f : X → ℝ) |
| (hf : ContinuousOn f Ω) (hH : ∀ n, ContinuousOn (H n) Ω) |
| (hfunit : ∀ x ∈ Ω, f x ∈ Icc (0:ℝ) 1) |
| (hlim : ∀ K : Set X, IsCompact K → K ⊆ Ω → TendstoUniformlyOn H f atTop K) |
| (d : ℝ) (hd : 0 < d) |
| (hlevels : ∀ n t, t ∈ Ioo (0:ℝ) 1 → ∃ K : Set X, |
| IsCompact K ∧ IsConnected K ∧ d ≤ diam K ∧ (∀ y ∈ K, H n y=t) ∧ |
| ∀ u v, ∀ γ : Path u v, range γ ⊆ Ω → H n u < t → t < H n v → |
| (K ∩ range γ).Nonempty) |
| (hi hj : X ≃ₜ X) (xi xj pi pj : X) (R U A ε : ℝ) |
| (hR : 0 < R) (hA : 0 < A) (hwide : 3*U/d < A) |
| (hroom : 2*U/A ≤ ε) (hΩi : closedBall xi ε ⊆ Ω) (hΩj : closedBall xj ε ⊆ Ω) |
| (hui : ∀ y, dist (hi y) pi*denominator xi R y ≤ U*R) |
| (huj : ∀ y, dist (hj y) pj*denominator xj R y ≤ U*R) |
| (hfar : 2*A*R ≤ dist pi pj) |
| (v₀ v₁ : Bool → X) (γ : ∀ s, Path (v₀ s) (v₁ s)) |
| (hP : ∀ s z, z ∈ range (γ s) → A*R < dist z pi ∧ A*R < dist z pj) |
| (gap δ : ℝ) (hgap : 0 < gap) (hδ : 0 < δ) |
| (hvar : ∀ s, gap ≤ f (hi.symm (v₁ s))-f (hi.symm (v₀ s))) |
| (hsep : ∀ x ∈ range (γ false), ∀ y ∈ range (γ true), |
| f (hj.symm x)+δ ≤ f (hj.symm y)) |
| (hei : ∀ K : Set X, IsCompact K → hi.symm '' K ⊆ Ω → |
| Tendsto (fun n => meshEnergy (p n) (r n) 2 (H n ∘ hi.symm) K) atTop (𝓝 0)) |
| (hej : ∀ K : Set X, IsCompact K → hj.symm '' K ⊆ Ω → |
| Tendsto (fun n => meshEnergy (p n) (r n) 2 (H n ∘ hj.symm) K) atTop (𝓝 0)) : |
| gap/4 ≤ CannonCrossEnergy.oscillation f (hi.symm '' closedBall pj (A*R)) ∨ |
| δ/4 ≤ CannonCrossEnergy.oscillation f (hj.symm '' closedBall pi (A*R)) := by |
| have hρ : 0 < A*R := mul_pos hA hR |
| have hDi := puncture_exterior_domain hi xi pi R U A ε Ω hR hA hroom hΩi hui |
| have hDj := puncture_exterior_domain hj xj pj R U A ε Ω hR hA hroom hΩj huj |
| have hbi : hi.symm '' closedBall pj (A*R) ⊆ Ω := by |
| apply Subset.trans (image_mono ?_) hDi |
| intro z hz |
| have hz' := mem_closedBall.mp hz |
| have htri := dist_triangle pi z pj |
| rw [dist_comm pi z] at htri |
| change A*R/2 ≤ dist z pi |
| linarith only [hfar,htri,hz',hρ] |
| have hbj : hj.symm '' closedBall pi (A*R) ⊆ Ω := by |
| apply Subset.trans (image_mono ?_) hDj |
| intro z hz |
| have hz' := mem_closedBall.mp hz |
| have htri := dist_triangle pi z pj |
| rw [dist_comm pi z] at htri |
| change A*R/2 ≤ dist z pj |
| linarith only [hfar,htri,hz',hρ] |
| have hpathi (s : Bool) : hi.symm '' range (γ s) ⊆ Ω := by |
| apply Subset.trans (image_mono ?_) hDi |
| intro z hz |
| change A*R/2 ≤ dist z pi |
| linarith only [(hP s z hz).1,hρ] |
| have hpathj (s : Bool) : hj.symm '' range (γ s) ⊆ Ω := by |
| apply Subset.trans (image_mono ?_) hDj |
| intro z hz |
| change A*R/2 ≤ dist z pj |
| linarith only [(hP s z hz).2,hρ] |
| let T : Set X := {z | A*R/2 ≤ dist z pi ∧ A*R/2 ≤ dist z pj} |
| let K : Set X := {z | A*R ≤ dist z pi ∧ A*R ≤ dist z pj} |
| have hTi : hi.symm '' T ⊆ Ω := |
| (image_mono (fun _ hz => hz.1)).trans hDi |
| have hTj : hj.symm '' T ⊆ Ω := |
| (image_mono (fun _ hz => hz.2)).trans hDj |
| have hKT : K ⊆ T := by |
| intro z hz |
| exact ⟨(by linarith only [hρ] : A*R/2 ≤ A*R).trans hz.1, |
| (by linarith only [hρ] : A*R/2 ≤ A*R).trans hz.2⟩ |
| have hK : IsCompact K := by |
| apply IsClosed.isCompact |
| exact (isClosed_le continuous_const (continuous_id.dist continuous_const)).inter |
| (isClosed_le continuous_const (continuous_id.dist continuous_const)) |
| have hh := two_path_pair_alternative ι p r hr hrlim hpnet |
| (fun n => H n ∘ hi.symm) (fun n => H n ∘ hj.symm) |
| (f ∘ hi.symm) (f ∘ hj.symm) pi pj (A*R) (A*R) hρ hρ v₀ v₁ γ gap δ |
| hgap hδ hvar (fun s z hz => (hP s z hz).1) hsep |
| (continuousOn_preimage hj f Ω _ hf (hpathj false)) |
| (continuousOn_preimage hj f Ω _ hf hbj) |
| (continuousOn_preimage hi f Ω _ hf hbi) |
| (uniformly_on_preimage hi H f Ω _ hlim isClosed_closedBall.isCompact hbi) |
| (fun s => uniformly_on_preimage hj H f Ω _ hlim (isCompact_range (γ s).continuous) (hpathj s)) |
| (uniformly_on_preimage hj H f Ω _ hlim isClosed_closedBall.isCompact hbj) |
| (Eventually.of_forall (fun n => continuousOn_preimage hi (H n) Ω T (hH n) hTi)) |
| (Eventually.of_forall (fun n => continuousOn_preimage hj (H n) Ω T (hH n) hTj)) |
| (hei K hK ((image_mono hKT).trans hTi)) |
| (hej K hK ((image_mono hKT).trans hTj)) |
| (fun s a b ha hab hb => transformed_level_paths Ω H f hfunit hlim d hd hlevels |
| hi xi pi R U A hR hwide hui (v₀ s) (v₁ s) (γ s) (hpathi s) a b ha hab hb) |
| simpa only [CannonCrossEnergy.oscillation, image_comp] using hh |
|
|
| end CannonPairs |
|
|
| end |
|
|
| end |
| end |
| end |
| end |
| end |
| end |
| end |
| end |
|
|
| end OAI |
|
|
