| import Mathlib |
| import OAI.Geometry.Cannon.Boundary.BoundaryHomeomorph |
|
|
| namespace OAI |
|
|
| section |
| section |
| section |
| section |
| section |
| section |
| noncomputable section |
| noncomputable section |
| open Set Topology |
|
|
| noncomputable section |
| namespace CannonGraph |
| open SimpleGraph |
|
|
| variable {V : Type*} {G : SimpleGraph V} |
|
|
| lemma product2_base_upper (hG : G.Connected) (o v x y : V) : |
| product2 (G := G) o x y ≤ product2 (G := G) v x y + 2*G.dist o v := by |
| have h₁ := product2_add_dist hG o x y |
| have h₂ := product2_add_dist hG v x y |
| have h₃ := @Connected.dist_triangle _ G o v x hG |
| have h₄ := @Connected.dist_triangle _ G o v y hG |
| omega |
|
|
| lemma product2_replace_endpoints (hG : G.Connected) {δ : ℕ} (hδ : Thin G δ) |
| (o a b a' b' : V) |
| (ha : product2 (G := G) o a b + 2*(8*δ+6) < product2 (G := G) o a a') |
| (hb : product2 (G := G) o a b + 2*(8*δ+6) < product2 (G := G) o b b') : |
| product2 (G := G) o a b ≤ product2 (G := G) o a' b' + 2*(8*δ+6) ∧ |
| product2 (G := G) o a' b' ≤ product2 (G := G) o a b + 2*(8*δ+6) := by |
| have h₁ := CannonVisual.product2_four hG hδ o a' a b b' |
| have h₂ := CannonVisual.product2_four hG hδ o a a' b' b |
| rw [product2_comm (o := o) (x := a') (y := a)] at h₁ |
| rw [product2_comm (o := o) (x := b') (y := b)] at h₂ |
| omega |
|
|
| lemma product2_base_cross (hG : G.Connected) (o v a b c d : V) : |
| product2 (G := G) o a b + product2 (G := G) o c d + |
| product2 (G := G) v a d + product2 (G := G) v b c = |
| product2 (G := G) o a d + product2 (G := G) o b c + |
| product2 (G := G) v a b + product2 (G := G) v c d := by |
| have h₁ := product2_add_dist hG o a b |
| have h₂ := product2_add_dist hG o c d |
| have h₃ := product2_add_dist hG o a d |
| have h₄ := product2_add_dist hG o b c |
| have h₅ := product2_add_dist hG v a b |
| have h₆ := product2_add_dist hG v c d |
| have h₇ := product2_add_dist hG v a d |
| have h₈ := product2_add_dist hG v b c |
| omega |
|
|
| end CannonGraph |
|
|
| namespace CannonRelease |
| open CannonGraph CannonVisual Filter Topology |
| open scoped NNReal |
|
|
| universe v |
| variable {G : Type v} [Group G] [TopologicalSpace G] [DiscreteTopology G] |
| variable (D : CayleyData G) {δ : ℕ} (hδ : Thin D.graph δ) |
| local instance : Setoid (BasedRay D) := raySetoid D |
|
|
| omit [TopologicalSpace G] [DiscreteTopology G] in |
| lemma product2_mul (g o x y : G) : |
| product2 (G := D.graph) (g*o) (g*x) (g*y) = product2 (G := D.graph) o x y := by |
| simp only [product2, D.dist_mul_left] |
|
|
| omit [TopologicalSpace G] [DiscreteTopology G] in |
| include hδ in |
| lemma product2_ray_bounded {r s : BasedRay D} (h : ¬ (raySetoid D).r r s) : |
| ∃ C : ℕ, ∀ n, product2 (G := D.graph) 1 (r.val n) (s.val n) ≤ C := by |
| have hn : ∃ n, 2*δ < D.graph.dist (r.val n) (s.val n) := by |
| by_contra hn |
| push Not at hn |
| exact h ⟨2*δ,hn⟩ |
| obtain ⟨k,hk⟩ := hn |
| exact ⟨2*k+2*δ,product2_le_of_separated D.connected hδ r s hk⟩ |
|
|
| lemma translated_mixed_product_large (g : G) (r : BasedRay D) (C : ℕ) : |
| ∀ᶠ n : ℕ in atTop, C < product2 (G := D.graph) 1 (g*r.val n) |
| ((translateRay D ⟨δ,hδ⟩ g r).val n) := by |
| let r' := translateRay D ⟨δ,hδ⟩ g r |
| filter_upwards [Filter.eventually_ge_atTop (C+4*D.graph.dist 1 g+4*δ+1)] with n hn |
| have hr := translateRay_uniform D hδ g r n |
| have hp := product2_add_dist D.connected 1 (g*r.val n) (r'.val n) |
| have ht := @SimpleGraph.Connected.dist_triangle _ D.graph g 1 (g*r.val n) D.connected |
| have hg : D.graph.dist g (g*r.val n) = n := by |
| simpa only [mul_one, r.property.1, Nat.dist_zero_left] using |
| (D.dist_mul_left g 1 (r.val n)).trans (by simpa only [r.property.1, Nat.dist_zero_left] using r.property.2 0 n) |
| have hr' : D.graph.dist 1 (r'.val n) = n := by |
| simpa only [r'.property.1, Nat.dist_zero_left] using r'.property.2 0 n |
| rw [hg, SimpleGraph.dist_comm (u := g) (v := 1)] at ht |
| dsimp only [CayleyData.graph] at hr' hp |
| rw [hr', SimpleGraph.dist_comm (u := g*r.val n) (v := r'.val n)] at hp |
| change D.graph.dist (r'.val n) (g*r.val n) ≤ _ at hr |
| change C < product2 (G := D.graph) 1 (g*r.val n) (r'.val n) |
| dsimp only [CayleyData.graph] at * |
| omega |
|
|
| lemma translated_product_independent_error (g : G) (r s : BasedRay D) |
| (h : ¬ (raySetoid D).r r s) : |
| ∀ᶠ n : ℕ in atTop, |
| product2 (G := D.graph) 1 (g*r.val n) (g*s.val n) ≤ |
| product2 (G := D.graph) 1 ((translateRay D ⟨δ,hδ⟩ g r).val n) |
| ((translateRay D ⟨δ,hδ⟩ g s).val n) + 2*(8*δ+6) ∧ |
| product2 (G := D.graph) 1 ((translateRay D ⟨δ,hδ⟩ g r).val n) |
| ((translateRay D ⟨δ,hδ⟩ g s).val n) ≤ |
| product2 (G := D.graph) 1 (g*r.val n) (g*s.val n) + 2*(8*δ+6) := by |
| obtain ⟨C,hC⟩ := product2_ray_bounded D hδ h |
| let B := C + 2*D.graph.dist 1 g + 2*(8*δ+6) |
| filter_upwards [translated_mixed_product_large D hδ g r B, |
| translated_mixed_product_large D hδ g s B] with n hr hs |
| have hp := product2_base_upper D.connected 1 g (g*r.val n) (g*s.val n) |
| have he : product2 (G := D.graph) g (g*r.val n) (g*s.val n) = |
| product2 (G := D.graph) 1 (r.val n) (s.val n) := by |
| simpa only [mul_one] using product2_mul D g 1 (r.val n) (s.val n) |
| dsimp only [CayleyData.graph] at hp he |
| rw [he] at hp |
| have hbound := hC n |
| dsimp only [CayleyData.graph] at hbound |
| apply product2_replace_endpoints D.connected hδ |
| · dsimp [B] at hr |
| dsimp only [CayleyData.graph] at * |
| omega |
| · dsimp [B] at hs |
| dsimp only [CayleyData.graph] at * |
| omega |
|
|
| end CannonRelease |
|
|
| namespace CannonRelease |
| open CannonGraph CannonVisual Filter Topology |
| open scoped NNReal |
|
|
| universe v |
| variable {G : Type v} [Group G] [TopologicalSpace G] [DiscreteTopology G] |
| variable (D : CayleyData G) {δ : ℕ} (hδ : Thin D.graph δ) |
| local instance : Setoid (BasedRay D) := raySetoid D |
|
|
| omit [TopologicalSpace G] [DiscreteTopology G] in |
| lemma translated_product2_cross (g a b c d : G) : |
| product2 (G := D.graph) 1 (g*a) (g*b) + product2 (G := D.graph) 1 (g*c) (g*d) + |
| product2 (G := D.graph) 1 a d + product2 (G := D.graph) 1 b c = |
| product2 (G := D.graph) 1 (g*a) (g*d) + product2 (G := D.graph) 1 (g*b) (g*c) + |
| product2 (G := D.graph) 1 a b + product2 (G := D.graph) 1 c d := by |
| have hm (x y : G) : product2 (G := D.graph) g (g*x) (g*y) = product2 (G := D.graph) 1 x y := by |
| simpa only [mul_one] using product2_mul D g 1 x y |
| have h := product2_base_cross D.connected 1 g (g*a) (g*b) (g*c) (g*d) |
| dsimp only [CayleyData.graph] at h hm ⊢ |
| simpa only [hm] using h |
|
|
| lemma translated_preDist_crossratio (g : G) (a b c d : BasedRay D) |
| (hab : ¬ (raySetoid D).r a b) (hcd : ¬ (raySetoid D).r c d) |
| (had : ¬ (raySetoid D).r a d) (hbc : ¬ (raySetoid D).r b c) : |
| (preDist (visualExponent δ) (translateRay D ⟨δ,hδ⟩ g a) (translateRay D ⟨δ,hδ⟩ g b) : ℝ) * |
| preDist (visualExponent δ) (translateRay D ⟨δ,hδ⟩ g c) (translateRay D ⟨δ,hδ⟩ g d) * |
| preDist (visualExponent δ) a d * preDist (visualExponent δ) b c ≤ |
| Real.exp (visualExponent δ * (8*(8*δ+6) : ℕ)) * |
| ((preDist (visualExponent δ) (translateRay D ⟨δ,hδ⟩ g a) (translateRay D ⟨δ,hδ⟩ g d) : ℝ) * |
| preDist (visualExponent δ) (translateRay D ⟨δ,hδ⟩ g b) (translateRay D ⟨δ,hδ⟩ g c) * |
| preDist (visualExponent δ) a b * preDist (visualExponent δ) c d) := by |
| let a' := translateRay D ⟨δ,hδ⟩ g a |
| let b' := translateRay D ⟨δ,hδ⟩ g b |
| let c' := translateRay D ⟨δ,hδ⟩ g c |
| let d' := translateRay D ⟨δ,hδ⟩ g d |
| have hlim (r s : BasedRay D) : Tendsto (fun n => (decay (visualExponent δ) r s n : ℝ)) atTop |
| (𝓝 (preDist (visualExponent δ) r s : ℝ)) := |
| (NNReal.continuous_coe.tendsto _).comp (tendsto_decay D.connected (visualExponent_pos δ).le r s) |
| apply le_of_tendsto_of_tendsto |
| ((((hlim a' b').mul (hlim c' d')).mul (hlim a d)).mul (hlim b c)) |
| (((((hlim a' d').mul (hlim b' c')).mul (hlim a b)).mul (hlim c d)).const_mul _) |
| filter_upwards [translated_product_independent_error D hδ g a b hab, |
| translated_product_independent_error D hδ g c d hcd, |
| translated_product_independent_error D hδ g a d had, |
| translated_product_independent_error D hδ g b c hbc] with n hab' hcd' had' hbc' |
| have hident := translated_product2_cross D g (a.val n) (b.val n) (c.val n) (d.val n) |
| have hsum : product2 (G := D.graph) 1 (a'.val n) (d'.val n) + |
| product2 (G := D.graph) 1 (b'.val n) (c'.val n) + |
| product2 (G := D.graph) 1 (a.val n) (b.val n) + |
| product2 (G := D.graph) 1 (c.val n) (d.val n) ≤ |
| product2 (G := D.graph) 1 (a'.val n) (b'.val n) + |
| product2 (G := D.graph) 1 (c'.val n) (d'.val n) + |
| product2 (G := D.graph) 1 (a.val n) (d.val n) + |
| product2 (G := D.graph) 1 (b.val n) (c.val n) + 8*(8*δ+6) := by |
| dsimp only [a',b',c',d'] |
| omega |
| have hsum' : (product2 (G := D.graph) 1 (a'.val n) (d'.val n) : ℝ) + |
| product2 (G := D.graph) 1 (b'.val n) (c'.val n) + |
| product2 (G := D.graph) 1 (a.val n) (b.val n) + |
| product2 (G := D.graph) 1 (c.val n) (d.val n) ≤ |
| (product2 (G := D.graph) 1 (a'.val n) (b'.val n) : ℝ) + |
| product2 (G := D.graph) 1 (c'.val n) (d'.val n) + |
| product2 (G := D.graph) 1 (a.val n) (d.val n) + |
| product2 (G := D.graph) 1 (b.val n) (c.val n) + (8*(8*δ+6) : ℕ) := by |
| exact_mod_cast hsum |
| change Real.exp _ * Real.exp _ * Real.exp _ * Real.exp _ ≤ |
| Real.exp _ * (Real.exp _ * Real.exp _ * Real.exp _ * Real.exp _) |
| repeat rw [← Real.exp_add] |
| apply Real.exp_le_exp.mpr |
| have he := visualExponent_pos δ |
| dsimp only [CayleyData.graph] at * |
| nlinarith |
|
|
| end CannonRelease |
|
|
| namespace CannonReal |
| lemma mul_four_le {a b c d A B C D : ℝ} (ha : a ≤ A) (hb : b ≤ B) |
| (hc : c ≤ C) (hd : d ≤ D) (hb₀ : 0 ≤ b) (hd₀ : 0 ≤ d) |
| (hc₀ : 0 ≤ c) (hA : 0 ≤ A) (hB : 0 ≤ B) (hC : 0 ≤ C) : |
| a*b*c*d ≤ A*B*C*D := by |
| have h₁ := mul_le_mul ha hb hb₀ hA |
| have h₂ := mul_le_mul hc hd hd₀ hC |
| simpa only [mul_assoc] using mul_le_mul h₁ h₂ (mul_nonneg hc₀ hd₀) (mul_nonneg hA hB) |
| end CannonReal |
|
|
| namespace CannonRelease |
| open CannonGraph CannonVisual Filter Topology |
| open scoped NNReal |
|
|
| universe v |
| variable {G : Type v} [Group G] [TopologicalSpace G] [DiscreteTopology G] |
| variable (D : CayleyData G) {δ : ℕ} (hδ : Thin D.graph δ) |
| local instance : Setoid (BasedRay D) := raySetoid D |
|
|
| theorem boundary_uniform_cross_bound (g : G) (x y z w : Boundary D) |
| (hxy : x ≠ y) (hzw : z ≠ w) (hxw : x ≠ w) (hyz : y ≠ z) : |
| let _ := visualMetricSpace D hδ |
| (@dist (Boundary D) (visualMetricSpace D hδ).toDist) (boundaryTranslate D ⟨δ,hδ⟩ g x) (boundaryTranslate D ⟨δ,hδ⟩ g y) * |
| (@dist (Boundary D) (visualMetricSpace D hδ).toDist) (boundaryTranslate D ⟨δ,hδ⟩ g z) (boundaryTranslate D ⟨δ,hδ⟩ g w) * |
| (@dist (Boundary D) (visualMetricSpace D hδ).toDist) x w * (@dist (Boundary D) (visualMetricSpace D hδ).toDist) y z ≤ |
| (16*Real.exp (visualExponent δ * (8*(8*δ+6):ℕ))) * |
| ((@dist (Boundary D) (visualMetricSpace D hδ).toDist) (boundaryTranslate D ⟨δ,hδ⟩ g x) (boundaryTranslate D ⟨δ,hδ⟩ g w) * |
| (@dist (Boundary D) (visualMetricSpace D hδ).toDist) (boundaryTranslate D ⟨δ,hδ⟩ g y) (boundaryTranslate D ⟨δ,hδ⟩ g z) * |
| (@dist (Boundary D) (visualMetricSpace D hδ).toDist) x y * (@dist (Boundary D) (visualMetricSpace D hδ).toDist) z w) := by |
| let _ := visualMetricSpace D hδ |
| induction x,y using Quotient.inductionOn₂ with | h a b => |
| induction z,w using Quotient.inductionOn₂ with | h c d => |
| let a' := translateRay D ⟨δ,hδ⟩ g a |
| let b' := translateRay D ⟨δ,hδ⟩ g b |
| let c' := translateRay D ⟨δ,hδ⟩ g c |
| let d' := translateRay D ⟨δ,hδ⟩ g d |
| have hab : ¬ (raySetoid D).r a b := fun h => hxy (Quotient.sound h) |
| have hcd : ¬ (raySetoid D).r c d := fun h => hzw (Quotient.sound h) |
| have had : ¬ (raySetoid D).r a d := fun h => hxw (Quotient.sound h) |
| have hbc : ¬ (raySetoid D).r b c := fun h => hyz (Quotient.sound h) |
| have huv := translated_preDist_crossratio D hδ g a b c d hab hcd had hbc |
| have hlow := CannonReal.mul_four_le |
| (visualMetric_comparison D hδ a' b').1 (visualMetric_comparison D hδ c' d').1 |
| (visualMetric_comparison D hδ a d).1 (visualMetric_comparison D hδ b c).1 |
| dist_nonneg dist_nonneg dist_nonneg (NNReal.coe_nonneg _) (NNReal.coe_nonneg _) (NNReal.coe_nonneg _) |
| have hupp := CannonReal.mul_four_le |
| (visualMetric_comparison D hδ a' d').2 (visualMetric_comparison D hδ b' c').2 |
| (visualMetric_comparison D hδ a b).2 (visualMetric_comparison D hδ c d).2 |
| (NNReal.coe_nonneg _) (NNReal.coe_nonneg _) (NNReal.coe_nonneg _) |
| (mul_nonneg (by norm_num) dist_nonneg) (mul_nonneg (by norm_num) dist_nonneg) |
| (mul_nonneg (by norm_num) dist_nonneg) |
| have hfinal := (hlow.trans huv).trans (mul_le_mul_of_nonneg_left hupp |
| (Real.exp_pos (visualExponent δ * (8*(8*δ+6):ℕ))).le) |
| change (@dist (Boundary D) (visualMetricSpace D hδ).toDist) (Quotient.mk' a' : Boundary D) (Quotient.mk' b' : Boundary D) * (@dist (Boundary D) (visualMetricSpace D hδ).toDist) (Quotient.mk' c' : Boundary D) (Quotient.mk' d' : Boundary D) * |
| (@dist (Boundary D) (visualMetricSpace D hδ).toDist) (Quotient.mk' a : Boundary D) (Quotient.mk' d : Boundary D) * (@dist (Boundary D) (visualMetricSpace D hδ).toDist) (Quotient.mk' b : Boundary D) (Quotient.mk' c : Boundary D) ≤ |
| (16*Real.exp (visualExponent δ * (8*(8*δ+6):ℕ))) * |
| ((@dist (Boundary D) (visualMetricSpace D hδ).toDist) (Quotient.mk' a' : Boundary D) (Quotient.mk' d' : Boundary D) * (@dist (Boundary D) (visualMetricSpace D hδ).toDist) (Quotient.mk' b' : Boundary D) (Quotient.mk' c' : Boundary D) * |
| (@dist (Boundary D) (visualMetricSpace D hδ).toDist) (Quotient.mk' a : Boundary D) (Quotient.mk' b : Boundary D) * (@dist (Boundary D) (visualMetricSpace D hδ).toDist) (Quotient.mk' c : Boundary D) (Quotient.mk' d : Boundary D)) |
| exact hfinal.trans_eq (by ring) |
|
|
| end CannonRelease |
|
|
| namespace CannonGraph |
| open SimpleGraph |
|
|
| variable {V : Type*} {G : SimpleGraph V} {o : V} |
|
|
| lemma product2_prefix (hG : G.Connected) {δ : ℕ} (hδ : Thin G δ) |
| (r s : CannonVisual.Ray G o) {k n : ℕ} (hkn : k ≤ n) : |
| product2 (G := G) o (r.val k) (s.val n) ≤ |
| min (2*k) (product2 (G := G) o (r.val n) (s.val n)) ∧ |
| min (2*k) (product2 (G := G) o (r.val n) (s.val n)) ≤ |
| product2 (G := G) o (r.val k) (s.val n) + (8*δ+6) := by |
| have hr (j : ℕ) : G.dist o (r.val j) = j := by |
| simpa only [r.property.1, Nat.dist_zero_left] using r.property.2 0 j |
| have hs (j : ℕ) : G.dist o (s.val j) = j := by |
| simpa only [s.property.1, Nat.dist_zero_left] using s.property.2 0 j |
| have h₁ := product2_add_dist hG o (r.val k) (s.val n) |
| have h₂ := product2_add_dist hG o (r.val n) (s.val n) |
| have ht := @Connected.dist_triangle _ G (r.val n) (r.val k) (s.val n) hG |
| rw [r.property.2, Nat.dist_eq_sub_of_le_right hkn] at ht |
| rw [hr, hs] at h₁ h₂ |
| have hup := product2_le_twice_left hG o (r.val k) (s.val n) |
| rw [hr] at hup |
| have hself : product2 (G := G) o (r.val k) (r.val n) = 2*k := by |
| unfold product2 |
| rw [hr,hr,r.property.2,Nat.dist_eq_sub_of_le hkn] |
| omega |
| have hlow := product2_hyperbolic hG hδ o (r.val k) (r.val n) (s.val n) |
| rw [hself] at hlow |
| constructor < |
|
|
| lemma product2_base_change (hG : G.Connected) (o v y z : V) : |
| product2 (G := G) v y z + product2 (G := G) o v y + product2 (G := G) o v z = |
| product2 (G := G) o y z + 2*G.dist o v := by |
| have h₁ := product2_add_dist hG v y z |
| have h₂ := product2_add_dist hG o v y |
| have h₃ := product2_add_dist hG o v z |
| have h₄ := product2_add_dist hG o y z |
| omega |
|
|
| end CannonGraph |
|
|
| namespace CannonRelease |
| open CannonGraph CannonVisual Filter Topology |
| open scoped NNReal |
|
|
| universe v |
| variable {G : Type v} [Group G] [TopologicalSpace G] [DiscreteTopology G] |
| variable (D : CayleyData G) {δ : ℕ} (hδ : Thin D.graph δ) |
| local instance : Setoid (BasedRay D) := raySetoid D |
|
|
| lemma expansion_product_sums (x y z : BasedRay D) (k : ℕ) |
| (hyz : ¬ (raySetoid D).r y z) : |
| let g := (x.val k)⁻¹ |
| ∀ᶠ n : ℕ in atTop, |
| product2 (G := D.graph) 1 ((translateRay D ⟨δ,hδ⟩ g y).val n) |
| ((translateRay D ⟨δ,hδ⟩ g z).val n) + |
| min (2*k) (product2 (G := D.graph) 1 (x.val n) (y.val n)) + |
| min (2*k) (product2 (G := D.graph) 1 (x.val n) (z.val n)) ≤ |
| product2 (G := D.graph) 1 (y.val n) (z.val n) + 2*k + 4*(8*δ+6) ∧ |
| product2 (G := D.graph) 1 (y.val n) (z.val n) + 2*k ≤ |
| product2 (G := D.graph) 1 ((translateRay D ⟨δ,hδ⟩ g y).val n) |
| ((translateRay D ⟨δ,hδ⟩ g z).val n) + |
| min (2*k) (product2 (G := D.graph) 1 (x.val n) (y.val n)) + |
| min (2*k) (product2 (G := D.graph) 1 (x.val n) (z.val n)) + 2*(8*δ+6) := by |
| let g := (x.val k)⁻¹ |
| filter_upwards [translated_product_independent_error D hδ g y z hyz, |
| Filter.eventually_ge_atTop k] with n hn hkn |
| have hy := product2_prefix D.connected hδ x y hkn |
| have hz := product2_prefix D.connected hδ x z hkn |
| have hid := product2_base_change D.connected 1 (x.val k) (y.val n) (z.val n) |
| have hd : D.graph.dist 1 (x.val k) = k := by |
| simpa only [x.property.1, Nat.dist_zero_left] using x.property.2 0 k |
| have he : product2 (G := D.graph) 1 (g*y.val n) (g*z.val n) = |
| product2 (G := D.graph) (x.val k) (y.val n) (z.val n) := by |
| simpa only [g,inv_mul_cancel] using product2_mul D ((x.val k)⁻¹) (x.val k) (y.val n) (z.val n) |
| dsimp only [CayleyData.graph] at * |
| rw [hd] at hid |
| rw [he] at hn |
| dsimp only [g] at * |
| constructor < |
|
|
| end CannonRelease |
|
|
| namespace CannonReal |
| lemma exp_neg_nat_min (ε : ℝ) (hε : 0 ≤ ε) (a b : ℕ) : |
| Real.exp (-ε * (min a b : ℕ)) = |
| max (Real.exp (-ε * (a : ℝ))) (Real.exp (-ε * (b : ℝ))) := by |
| rcases le_total a b with hab | hba |
| · rw [min_eq_left hab, max_eq_left] |
| apply Real.exp_le_exp.mpr |
| exact mul_le_mul_of_nonpos_left (by exact_mod_cast hab) (neg_nonpos.mpr hε) |
| · rw [min_eq_right hba, max_eq_right] |
| apply Real.exp_le_exp.mpr |
| exact mul_le_mul_of_nonpos_left (by exact_mod_cast hba) (neg_nonpos.mpr hε) |
|
|
| lemma exp_expansion (ε : ℝ) (hε : 0 ≤ ε) (E A B C K k : ℕ) |
| (h₁ : E + min (2*k) A + min (2*k) B ≤ C+2*k+4*K) |
| (h₂ : C+2*k ≤ E+min (2*k) A+min (2*k) B+2*K) : |
| Real.exp (-ε*(E:ℝ)) * max (Real.exp (-ε*(2*k:ℕ))) (Real.exp (-ε*(A:ℝ))) * |
| max (Real.exp (-ε*(2*k:ℕ))) (Real.exp (-ε*(B:ℝ))) ≤ |
| Real.exp (ε*(2*K:ℕ)) * Real.exp (-ε*(2*k:ℕ)) * Real.exp (-ε*(C:ℝ)) ∧ |
| Real.exp (-ε*(2*k:ℕ)) * Real.exp (-ε*(C:ℝ)) ≤ |
| Real.exp (ε*(4*K:ℕ)) * (Real.exp (-ε*(E:ℝ)) * |
| max (Real.exp (-ε*(2*k:ℕ))) (Real.exp (-ε*(A:ℝ))) * |
| max (Real.exp (-ε*(2*k:ℕ))) (Real.exp (-ε*(B:ℝ)))) := by |
| rw [← exp_neg_nat_min ε hε, ← exp_neg_nat_min ε hε] |
| have h₁' : (E:ℝ) + (min (2*k) A : ℕ) + (min (2*k) B : ℕ) ≤ |
| (C:ℝ) + (2*k:ℕ) + (4*K:ℕ) := by exact_mod_cast h₁ |
| have h₂' : (C:ℝ)+(2*k:ℕ) ≤ (E:ℝ)+(min (2*k) A : ℕ)+(min (2*k) B : ℕ)+(2*K:ℕ) := by |
| exact_mod_cast h₂ |
| constructor < |
| · apply Real.exp_le_exp.mpr |
| nlinarith |
| · apply Real.exp_le_exp.mpr |
| nlinarith |
| end CannonReal |
|
|
| namespace CannonVisual |
| noncomputable def visualScale (δ k : ℕ) : ℝ := Real.exp (-visualExponent δ * (2*k:ℕ)) |
| lemma visualScale_pos (δ k : ℕ) : 0 < visualScale δ k := Real.exp_pos _ |
| end CannonVisual |
|
|
| namespace CannonRelease |
| open CannonGraph CannonVisual Filter Topology |
| open scoped NNReal |
|
|
| universe v |
| variable {G : Type v} [Group G] [TopologicalSpace G] [DiscreteTopology G] |
| variable (D : CayleyData G) {δ : ℕ} (hδ : Thin D.graph δ) |
| local instance : Setoid (BasedRay D) := raySetoid D |
|
|
| lemma preDist_expansion (x y z : BasedRay D) (k : ℕ) |
| (hyz : ¬ (raySetoid D).r y z) : |
| let g := (x.val k)⁻¹ |
| let R := visualScale δ k |
| let E := (preDist (visualExponent δ) (translateRay D ⟨δ,hδ⟩ g y) |
| (translateRay D ⟨δ,hδ⟩ g z) : ℝ) |
| let A := max R (preDist (visualExponent δ) x y : ℝ) |
| let B := max R (preDist (visualExponent δ) x z : ℝ) |
| E*A*B ≤ Real.exp (visualExponent δ * (2*(8*δ+6):ℕ))*R*(preDist (visualExponent δ) y z : ℝ) ∧ |
| R*(preDist (visualExponent δ) y z : ℝ) ≤ Real.exp (visualExponent δ * (4*(8*δ+6):ℕ))*(E*A*B) := by |
| let g := (x.val k)⁻¹ |
| let y' := translateRay D ⟨δ,hδ⟩ g y |
| let z' := translateRay D ⟨δ,hδ⟩ g z |
| have hlim (r s : BasedRay D) : Tendsto (fun n => (decay (visualExponent δ) r s n : ℝ)) atTop |
| (𝓝 (preDist (visualExponent δ) r s : ℝ)) := |
| (NNReal.continuous_coe.tendsto _).comp (tendsto_decay D.connected (visualExponent_pos δ).le r s) |
| have he : ∀ᶠ n : ℕ in atTop, |
| (decay (visualExponent δ) y' z' n : ℝ) * |
| max (visualScale δ k) (decay (visualExponent δ) x y n : ℝ) * |
| max (visualScale δ k) (decay (visualExponent δ) x z n : ℝ) ≤ |
| Real.exp (visualExponent δ * (2*(8*δ+6):ℕ))*visualScale δ k*(decay (visualExponent δ) y z n : ℝ) ∧ |
| visualScale δ k*(decay (visualExponent δ) y z n : ℝ) ≤ |
| Real.exp (visualExponent δ * (4*(8*δ+6):ℕ)) * |
| ((decay (visualExponent δ) y' z' n : ℝ) * |
| max (visualScale δ k) (decay (visualExponent δ) x y n : ℝ) * |
| max (visualScale δ k) (decay (visualExponent δ) x z n : ℝ)) := by |
| filter_upwards [expansion_product_sums D hδ x y z k hyz] with n hn |
| exact CannonReal.exp_expansion (visualExponent δ) (visualExponent_pos δ).le _ _ _ _ _ _ hn.1 hn.2 |
| have hf := ((hlim y' z').mul ((tendsto_const_nhds (x := visualScale δ k)).max (hlim x y))).mul |
| ((tendsto_const_nhds (x := visualScale δ k)).max (hlim x z)) |
| constructor |
| · exact le_of_tendsto_of_tendsto hf ((hlim y z).const_mul _) (he.mono fun _ h => h.1) |
| · exact le_of_tendsto_of_tendsto ((hlim y z).const_mul _) (hf.const_mul _) (he.mono fun _ h => h.2) |
|
|
| end CannonRelease |
|
|
| namespace CannonReal |
| lemma max_le_twice_max {R d p : ℝ} (hR : 0 ≤ R) (hp : p ≤ 2*d) : |
| max R p ≤ 2*max R d := by |
| apply max_le |
| · have := le_max_left R d |
| nlinarith |
| · exact hp.trans (mul_le_mul_of_nonneg_left (le_max_right R d) (by norm_num)) |
|
|
| lemma mul_three_le {a b c A B C : ℝ} (ha : a ≤ A) (hb : b ≤ B) (hc : c ≤ C) |
| (hb₀ : 0 ≤ b) (hc₀ : 0 ≤ c) (hA : 0 ≤ A) (hB : 0 ≤ B) : |
| a*b*c ≤ A*B*C := |
| mul_le_mul (mul_le_mul ha hb hb₀ hA) hc hc₀ (mul_nonneg hA hB) |
| end CannonReal |
|
|
| namespace CannonRelease |
| open CannonGraph CannonVisual Filter Topology |
| open scoped NNReal |
|
|
| universe v |
| variable {G : Type v} [Group G] [TopologicalSpace G] [DiscreteTopology G] |
| variable (D : CayleyData G) {δ : ℕ} (hδ : Thin D.graph δ) |
| local instance : Setoid (BasedRay D) := raySetoid D |
|
|
| theorem visual_expansion_ray (r : BasedRay D) (k : ℕ) (y z : Boundary D) : |
| let _ := visualMetricSpace D hδ |
| let g := (r.val k)⁻¹ |
| let x : Boundary D := Quotient.mk' r |
| let R := visualScale δ k |
| let E := (@dist (Boundary D) (visualMetricSpace D hδ).toDist) (boundaryTranslate D ⟨δ,hδ⟩ g y) (boundaryTranslate D ⟨δ,hδ⟩ g z) |
| let A := max R ((@dist (Boundary D) (visualMetricSpace D hδ).toDist) x y) |
| let B := max R ((@dist (Boundary D) (visualMetricSpace D hδ).toDist) x z) |
| E*A*B ≤ (2*Real.exp (visualExponent δ * (2*(8*δ+6):ℕ)))*R*(@dist (Boundary D) (visualMetricSpace D hδ).toDist) y z ∧ |
| R*(@dist (Boundary D) (visualMetricSpace D hδ).toDist) y z ≤ (8*Real.exp (visualExponent δ * (4*(8*δ+6):ℕ)))*(E*A*B) := by |
| let _ := visualMetricSpace D hδ |
| by_cases hyz : y = z |
| · subst z |
| simp only [dist_self,zero_mul,mul_zero,le_refl,and_self] |
| induction y,z using Quotient.inductionOn₂ with | h a b => |
| let g := (r.val k)⁻¹ |
| let a' := translateRay D ⟨δ,hδ⟩ g a |
| let b' := translateRay D ⟨δ,hδ⟩ g b |
| let R := visualScale δ k |
| have hR : 0 ≤ R := (visualScale_pos δ k).le |
| have hab : ¬ (raySetoid D).r a b := fun h => hyz (Quotient.sound h) |
| have hh := preDist_expansion D hδ r a b k hab |
| have hc (s t : BasedRay D) := visualMetric_comparison D hδ s t |
| have hm (s : BasedRay D) : max R (preDist (visualExponent δ) r s : ℝ) ≤ |
| 2*max R ((@dist (Boundary D) (visualMetricSpace D hδ).toDist) (Quotient.mk' r : Boundary D) (Quotient.mk' s : Boundary D)) := |
| CannonReal.max_le_twice_max hR (hc r s).2 |
| have hn₁ : 0 ≤ max R ((@dist (Boundary D) (visualMetricSpace D hδ).toDist) (Quotient.mk' r : Boundary D) (Quotient.mk' a : Boundary D)) := hR.trans (le_max_left _ _) |
| have hn₂ : 0 ≤ max R ((@dist (Boundary D) (visualMetricSpace D hδ).toDist) (Quotient.mk' r : Boundary D) (Quotient.mk' b : Boundary D)) := hR.trans (le_max_left _ _) |
| have hp₁ : 0 ≤ max R (preDist (visualExponent δ) r a : ℝ) := hR.trans (le_max_left _ _) |
| have hp₂ : 0 ≤ max R (preDist (visualExponent δ) r b : ℝ) := hR.trans (le_max_left _ _) |
| have hlo := CannonReal.mul_three_le (hc a' b').1 |
| (max_le_max_left R (hc r a).1) (max_le_max_left R (hc r b).1) |
| hn₁ hn₂ (NNReal.coe_nonneg _) hp₁ |
| have hup := CannonReal.mul_three_le (hc a' b').2 (hm a) (hm b) |
| hp₁ hp₂ (mul_nonneg (by norm_num) dist_nonneg) (mul_nonneg (by norm_num) hn₁) |
| change (@dist (Boundary D) (visualMetricSpace D hδ).toDist) (Quotient.mk' a' : Boundary D) (Quotient.mk' b' : Boundary D) * |
| max R ((@dist (Boundary D) (visualMetricSpace D hδ).toDist) (Quotient.mk' r : Boundary D) (Quotient.mk' a : Boundary D)) * max R ((@dist (Boundary D) (visualMetricSpace D hδ).toDist) (Quotient.mk' r : Boundary D) (Quotient.mk' b : Boundary D)) ≤ |
| (2*Real.exp (visualExponent δ * (2*(8*δ+6):ℕ)))*R*(@dist (Boundary D) (visualMetricSpace D hδ).toDist) (Quotient.mk' a : Boundary D) (Quotient.mk' b : Boundary D) ∧ |
| R*(@dist (Boundary D) (visualMetricSpace D hδ).toDist) (Quotient.mk' a : Boundary D) (Quotient.mk' b : Boundary D) ≤ |
| (8*Real.exp (visualExponent δ * (4*(8*δ+6):ℕ)))* |
| ((@dist (Boundary D) (visualMetricSpace D hδ).toDist) (Quotient.mk' a' : Boundary D) (Quotient.mk' b' : Boundary D) * |
| max R ((@dist (Boundary D) (visualMetricSpace D hδ).toDist) (Quotient.mk' r : Boundary D) (Quotient.mk' a : Boundary D)) * max R ((@dist (Boundary D) (visualMetricSpace D hδ).toDist) (Quotient.mk' r : Boundary D) (Quotient.mk' b : Boundary D))) |
| constructor |
| · have h := (hlo.trans hh.1).trans (mul_le_mul_of_nonneg_left (hc a b).2 |
| (mul_nonneg (Real.exp_pos _).le hR)) |
| exact h.trans_eq (by ring) |
| · have h := (mul_le_mul_of_nonneg_left (hc a b).1 hR).trans hh.2 |
| exact (h.trans (mul_le_mul_of_nonneg_left hup (Real.exp_pos _).le)).trans_eq (by ring) |
|
|
| end CannonRelease |
|
|
| namespace CannonGraph |
| open SimpleGraph |
|
|
| variable {V : Type*} {G : SimpleGraph V} {o : V} |
|
|
| lemma product2_replace_one (hG : G.Connected) {δ : ℕ} (hδ : Thin G δ) |
| (o v a b : V) (hab : 2*G.dist o v < product2 (G := G) o a b) : |
| product2 (G := G) o v a ≤ product2 (G := G) o v b + (8*δ+6) ∧ |
| product2 (G := G) o v b ≤ product2 (G := G) o v a + (8*δ+6) := by |
| have h₁ := product2_hyperbolic hG hδ o v a b |
| have h₂ := product2_hyperbolic hG hδ o v b a |
| rw [product2_comm (o := o) (x := b) (y := a)] at h₂ |
| have h₃ := product2_le_twice_left hG o v a |
| have h₄ := product2_le_twice_left hG o v b |
| constructor < |
|
|
| lemma product2_complement (hG : G.Connected) (o v a : V) : |
| product2 (G := G) v o a + product2 (G := G) o v a = 2*G.dist o v := by |
| have h₁ := product2_add_dist hG v o a |
| have h₂ := product2_add_dist hG o v a |
| rw [SimpleGraph.dist_comm (u := v) (v := o)] at h₁ |
| omega |
|
|
| lemma vertex_close_to_ray (hG : G.Connected) {δ : ℕ} (hδ : Thin G δ) |
| (s : CannonVisual.Ray G o) (v : V) (n C : ℕ) |
| (hn : G.dist o v ≤ n) |
| (hprod : 2*G.dist o v ≤ product2 (G := G) o v (s.val n) + C) : |
| G.dist v (s.val (G.dist o v)) ≤ 2*C+4*δ+2 := by |
| let w := G.dist o v |
| have hs (i : ℕ) : G.dist o (s.val i) = i := by |
| simpa only [s.property.1,Nat.dist_zero_left] using s.property.2 0 i |
| by_cases hw : w ≤ C+δ+1 |
| · have h := @Connected.dist_triangle _ G v o (s.val w) hG |
| rw [SimpleGraph.dist_comm (u := v) (v := o),hs] at h |
| change G.dist v (s.val w) ≤ _ |
| dsimp [w] at * |
| omega |
| obtain ⟨p,hp⟩ := hG.exists_walk_length_eq_dist o v |
| let q : G.Walk o (s.val n) := (raySegment s.val s.property.2 n).copy s.property.1 rfl |
| have hq : q.length = G.dist o (s.val n) := by simp only [q,Walk.length_copy,raySegment_length,hs] |
| let m := w-(C+δ+1) |
| have hm : m ≤ p.length := by dsimp [m,w] |
| have hm' : m ≤ q.length := by rw [hq,hs] |
| have had := product2_add_dist hG o v (s.val n) |
| have hclose := fellow_travel_below_product hG hδ p q hp hq hm hm' (by dsimp [m,w] at * |
| have hpm := getVert_end_dist hG p hp hm |
| have hqm : q.getVert m = s.val m := by |
| simp only [q,Walk.getVert_copy,raySegment_getVert s.val s.property.2 (show m ≤ n by dsimp [m,w] |
| rw [hqm] at hclose |
| have h₁ := @Connected.dist_triangle _ G v (p.getVert m) (s.val m) hG |
| have h₂ := @Connected.dist_triangle _ G v (s.val m) (s.val w) hG |
| rw [SimpleGraph.dist_comm (u := v) (v := p.getVert m)] at h₁ |
| rw [s.property.2,Nat.dist_eq_sub_of_le (show m ≤ w by dsimp [m] |
| change G.dist v (s.val w) ≤ _ |
| dsimp [m,w] at * |
| omega |
|
|
| end CannonGraph |
|
|
| namespace CannonRelease |
| open CannonGraph CannonVisual Filter Topology |
| open scoped NNReal |
|
|
| universe v |
| variable {G : Type v} [Group G] [TopologicalSpace G] [DiscreteTopology G] |
| variable (D : CayleyData G) {δ : ℕ} (hδ : Thin D.graph δ) |
|
|
| include hδ in |
|
|
| theorem coarse_prefix_of_distinct_rays (r s : BasedRay D) |
| (hrs : ¬ (raySetoid D).r r s) : |
| ∃ B : ℕ, ∀ v : G, ∃ t : BasedRay D, |
| D.graph.dist v (t.val (D.graph.dist 1 v)) ≤ B := by |
| obtain ⟨C,hC⟩ := product2_ray_bounded D hδ hrs |
| refine ⟨2*(C+2*(8*δ+6))+4*δ+2,?_⟩ |
| intro v |
| let r' := translateRay D ⟨δ,hδ⟩ v r |
| let s' := translateRay D ⟨δ,hδ⟩ v s |
| have he := (translated_mixed_product_large D hδ v r (2*D.graph.dist 1 v)).and |
| ((translated_mixed_product_large D hδ v s (2*D.graph.dist 1 v)).and |
| (Filter.eventually_ge_atTop (D.graph.dist 1 v))) |
| obtain ⟨n,hr,hs,hn⟩ := he.exists |
| have hrb := product2_replace_one D.connected hδ 1 v (v*r.val n) (r'.val n) hr |
| have hsb := product2_replace_one D.connected hδ 1 v (v*s.val n) (s'.val n) hs |
| have hmin := product2_hyperbolic D.connected hδ v (v*r.val n) 1 (v*s.val n) |
| have hp₁ := product2_complement D.connected 1 v (v*r.val n) |
| have hp₂ := product2_complement D.connected 1 v (v*s.val n) |
| have heq : product2 (G := D.graph) v (v*r.val n) (v*s.val n) = |
| product2 (G := D.graph) 1 (r.val n) (s.val n) := by |
| simpa only [mul_one] using product2_mul D v 1 (r.val n) (s.val n) |
| have hb := hC n |
| dsimp only [CayleyData.graph] at * |
| rw [heq,product2_comm (o := v) (x := v*r.val n) (y := 1)] at hmin |
| have hposs : 2*(SimpleGraph.mulCayley ↑D.generators).dist 1 v ≤ |
| product2 (G := D.graph) 1 v (r'.val n) + (C+2*(8*δ+6)) ∨ |
| 2*(SimpleGraph.mulCayley ↑D.generators).dist 1 v ≤ |
| product2 (G := D.graph) 1 v (s'.val n) + (C+2*(8*δ+6)) := by |
| dsimp only [CayleyData.graph] |
| omega |
| rcases hposs with hr' | hs' |
| · exact ⟨r',vertex_close_to_ray D.connected hδ r' v n (C+2*(8*δ+6)) hn hr'⟩ |
| · exact ⟨s',vertex_close_to_ray D.connected hδ s' v n (C+2*(8*δ+6)) hn hs'⟩ |
|
|
| end CannonRelease |
|
|
| namespace CannonVisual |
| lemma visualScale_eq_pow (δ k : ℕ) : |
| visualScale δ k = (Real.exp (-(2*visualExponent δ)))^k := by |
| rw [← Real.exp_nat_mul] |
| unfold visualScale |
| congr 1 |
| push_cast |
| ring |
|
|
| lemma visualScale_near (δ : ℕ) {r : ℝ} (hr : 0 < r) (hr₁ : r ≤ 1) : |
| ∃ k : ℕ, r ≤ visualScale δ k ∧ |
| visualScale δ k ≤ Real.exp (2*visualExponent δ)*r := by |
| have he := visualExponent_pos δ |
| have hq : Real.exp (-(2*visualExponent δ)) < 1 := by |
| rw [Real.exp_lt_one_iff] |
| linarith |
| obtain ⟨k,hk₁,hk₂⟩ := exists_nat_pow_near_of_lt_one hr hr₁ (Real.exp_pos _) hq |
| refine ⟨k,by simpa [visualScale_eq_pow] using hk₂,?_⟩ |
| rw [pow_succ] at hk₁ |
| have h := mul_lt_mul_of_pos_right hk₁ (Real.exp_pos (2*visualExponent δ)) |
| have hex : Real.exp (-(2*visualExponent δ))*Real.exp (2*visualExponent δ) = 1 := by |
| rw [← Real.exp_add,neg_add_cancel,Real.exp_zero] |
| rw [mul_assoc,hex,mul_one] at h |
| simpa only [visualScale_eq_pow,mul_comm] using h.le |
|
|
| lemma visualScale_tendsto_zero (δ : ℕ) : |
| Filter.Tendsto (visualScale δ) Filter.atTop (nhds 0) := by |
| change Filter.Tendsto (fun k => visualScale δ k) Filter.atTop (nhds 0) |
| simpa only [visualScale_eq_pow] using tendsto_pow_atTop_nhds_zero_of_lt_one |
| (Real.exp_pos (-(2*visualExponent δ))).le |
| (show Real.exp (-(2*visualExponent δ)) < 1 by |
| rw [Real.exp_lt_one_iff] |
| end CannonVisual |
|
|
| end |
| end |
| end |
| end |
| end |
| end |
| end |
| end |
| end |
|
|
| end OAI |
|
|
