| import Mathlib |
| import OAI.Geometry.Cannon.Boundary.WordBall |
|
|
| namespace OAI |
|
|
| section |
| section |
| section |
| noncomputable section |
| section |
| section |
| open Set Topology |
|
|
| namespace CannonRelease |
|
|
| open CannonGraph CannonVisual Filter Topology |
|
|
| 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 visualSynchronousPrefixSetoid : Setoid (BasedRay D) := raySetoid D |
|
|
| theorem visual_synchronous_prefix (C : ℝ) (_hC : 0 ≤ C) : |
| ∃ B : ℕ, ∀ (r s : BasedRay D) (j : ℕ), |
| boundaryDist D hδ (Quotient.mk' r) (Quotient.mk' s) ≤ C*visualScale δ j → |
| D.graph.dist (r.val j) (s.val j) ≤ B := by |
| have hε := visualExponent_pos δ |
| have hexp : Tendsto (fun n : ℕ => Real.exp (visualExponent δ*(n:ℝ))) atTop atTop := |
| Real.tendsto_exp_atTop.comp (tendsto_natCast_atTop_atTop.const_mul_atTop hε) |
| obtain ⟨A,hA⟩ := (hexp.eventually (eventually_gt_atTop (2*C))).exists |
| refine ⟨2*(A+(8*δ+6))+4*δ+2,fun r s j hj => ?_⟩ |
| have hc := (visualMetric_comparison D hδ r s).2 |
| have hp : (preDist (visualExponent δ) r s : ℝ) < |
| Real.exp (visualExponent δ*(A:ℝ)-2*visualExponent δ*j) := by |
| calc |
| _ ≤ 2*boundaryDist D hδ (Quotient.mk' r) (Quotient.mk' s) := hc |
| _ ≤ 2*(C*visualScale δ j) := mul_le_mul_of_nonneg_left hj (by norm_num) |
| _ < Real.exp (visualExponent δ*(A:ℝ))*visualScale δ j := by |
| nlinarith only [mul_lt_mul_of_pos_right hA (visualScale_pos δ j)] |
| _ = _ := by rw [visualScale,← Real.exp_add]; congr 1; push_cast; ring |
| have hlim : 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 hε.le r s) |
| obtain ⟨n,hn,hdec⟩ := ((eventually_ge_atTop j).and (hlim.eventually_lt_const hp)).exists |
| change Real.exp (-visualExponent δ*(product2 (G:=D.graph) 1 (r.val n) (s.val n):ℝ)) < _ at hdec |
| have hprod : 2*j ≤ product2 (G:=D.graph) 1 (r.val n) (s.val n)+A := by |
| have he := Real.exp_lt_exp.mp hdec |
| have he' : (2*j:ℕ) ≤ product2 (G:=D.graph) 1 (r.val n) (s.val n)+A := by |
| have hh : ((2*j:ℕ):ℝ) ≤ ((product2 (G:=D.graph) 1 (r.val n) (s.val n)+A:ℕ):ℝ) := by |
| push_cast |
| nlinarith only [he,hε] |
| exact_mod_cast hh |
| exact he' |
| have hpref := (product2_prefix D.connected hδ r s hn).2 |
| have hr : D.graph.dist 1 (r.val j) = j := by |
| simpa only [r.property.1,Nat.dist_zero_left] using r.property.2 0 j |
| dsimp only [CayleyData.graph] at * |
| have hv := vertex_close_to_ray D.connected hδ s (r.val j) n (A+(8*δ+6)) |
| (by simpa only [hr] using hn) (by rw [hr]; omega) |
| simpa only [hr] using hv |
| |
| theorem separated_ray_products {e : ℝ} (he : 0 < e) : |
| ∃ C : ℕ, ∀ (r s : BasedRay D), |
| e ≤ boundaryDist D hδ (Quotient.mk' r) (Quotient.mk' s) → |
| ∀ n, product2 (G:=D.graph) 1 (r.val n) (s.val n) ≤ C := by |
| have hε := visualExponent_pos δ |
| have ht : Tendsto (fun n : ℕ => Real.exp (-visualExponent δ*(n:ℝ))) atTop (𝓝 0) := by |
| apply Real.tendsto_exp_atBot.comp |
| simpa only [neg_mul,Function.comp_def] using tendsto_neg_atTop_atBot.comp |
| (tendsto_natCast_atTop_atTop.const_mul_atTop hε) |
| obtain ⟨C,hC⟩ := (ht.eventually_lt_const he).exists |
| refine ⟨C,fun r s hsep n => ?_⟩ |
| have hc := (visualMetric_comparison D hδ r s).1 |
| have hd := preDist_le_decay (visualExponent δ) r s n |
| have hd' : (preDist (visualExponent δ) r s : ℝ) ≤ |
| Real.exp (-visualExponent δ*(product2 (G:=D.graph) 1 (r.val n) (s.val n):ℝ)) := hd |
| by_contra hn |
| have hCN : (C:ℝ) ≤ product2 (G:=D.graph) 1 (r.val n) (s.val n) := by exact_mod_cast (lt_of_not_ge hn).le |
| have hm := Real.exp_le_exp.mpr (mul_le_mul_of_nonpos_left hCN (neg_nonpos.mpr hε.le)) |
| exact (not_lt_of_ge (hsep.trans (hc.trans hd'))) (hm.trans_lt hC) |
|
|
| theorem inverse_puncture_prefix {e : ℝ} (he : 0 < e) : |
| ∃ B : ℕ, ∀ (r z : BasedRay D) (k : ℕ), |
| e ≤ boundaryDist D hδ (Quotient.mk' r) (Quotient.mk' z) → |
| D.graph.dist (r.val k)⁻¹ |
| ((translateRay D ⟨δ,hδ⟩ (r.val k)⁻¹ z).val k) ≤ B := by |
| obtain ⟨C,hC⟩ := separated_ray_products D hδ he |
| refine ⟨2*(C+(8*δ+6))+4*δ+2,fun r z k hz => ?_⟩ |
| let g := (r.val k)⁻¹ |
| let p := translateRay D ⟨δ,hδ⟩ g z |
| have hk : D.graph.dist 1 (r.val k) = k := by |
| simpa only [r.property.1,Nat.dist_zero_left] using r.property.2 0 k |
| have hg : D.graph.dist 1 g = k := by |
| have hh := D.dist_mul_left g (r.val k) 1 |
| simpa [g,SimpleGraph.dist_comm,hk] using hh |
| obtain ⟨n,hn,hbig⟩ := ((eventually_ge_atTop k).and |
| (translated_mixed_product_large D hδ g z (2*D.graph.dist 1 g))).exists |
| have hrepl := product2_replace_one D.connected hδ 1 g (g*z.val n) (p.val n) hbig |
| have hpref := (product2_prefix D.connected hδ r z hn).1 |
| have hbound := hC r z hz n |
| have hcomp := product2_complement D.connected 1 (r.val k) (z.val n) |
| have hmul : product2 (G:=D.graph) 1 g (g*z.val n) = |
| product2 (G:=D.graph) (r.val k) 1 (z.val n) := by |
| have hh := product2_mul D g (r.val k) 1 (z.val n) |
| simpa [g] using hh |
| change product2 (G:=D.graph) (r.val k) 1 (z.val n) + |
| product2 (G:=D.graph) 1 (r.val k) (z.val n) = 2*D.graph.dist 1 (r.val k) at hcomp |
| rw [hk,← hmul] at hcomp |
| dsimp only [CayleyData.graph] at * |
| have hprod : 2*D.graph.dist 1 g ≤ product2 (G:=D.graph) 1 g (p.val n)+(C+(8*δ+6)) := by |
| dsimp only [CayleyData.graph] |
| rw [hg] |
| omega |
| have hh := vertex_close_to_ray D.connected hδ p g n (C+(8*δ+6)) (hg ▸ hn) hprod |
| simpa only [hg] using hh |
|
|
| end CannonRelease |
|
|
| namespace CannonRelease |
|
|
| open CannonGraph CannonVisual Filter Topology Finset |
|
|
| 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 visualSynchronousGraphBallSetoid : Setoid (BasedRay D) := raySetoid D |
|
|
| local instance visualSynchronousGraphBallDecidable : ∀ p : Prop, Decidable p := |
| Classical.propDecidable |
|
|
| omit [TopologicalSpace G] [DiscreteTopology G] in |
| lemma uniform_graph_ball_card (B : ℕ) : |
| ∃ N : ℕ, ∀ (x : G) (S : Finset G), |
| (∀ y ∈ S, D.graph.dist x y ≤ B) → S.card ≤ N := by |
| classical |
| let T := (finite_graph_ball D.connected (finite_cayley_neighbors D.generators) 1 B).toFinset |
| refine ⟨T.card,fun x S hS => ?_⟩ |
| apply card_le_card_of_injOn (fun y => x⁻¹*y) ?_ (by intro a _ b _ hab; exact mul_left_cancel hab) |
| intro y hy |
| change x⁻¹*y ∈ (finite_graph_ball D.connected (finite_cayley_neighbors D.generators) 1 B).toFinset |
| rw [Set.Finite.mem_toFinset] |
| have hh := D.dist_mul_left x 1 (x⁻¹*y) |
| simp only [mul_one,mul_inv_cancel_left] at hh |
| change D.graph.dist 1 (x⁻¹*y) ≤ B |
| rw [← hh] |
| exact hS y hy |
|
|
| theorem vertex_assignment_multiplicity {H : ℝ} (hH : 0 < H) : |
| ∃ N : ℕ, ∀ {ι : Type*} (S : Finset ι) (r : ι → BasedRay D) (k : ℕ), |
| (∀ i ∈ S, ∀ j ∈ S, i ≠ j → H*visualScale δ k ≤ |
| boundaryDist D hδ (Quotient.mk' (r i)) (Quotient.mk' (r j))) → |
| ∀ v : G, (S.filter (fun i => (r i).val k = v)).card ≤ N := by |
| classical |
| let _ := visualMetricSpace D hδ |
| obtain ⟨N,hN⟩ := relative_packing_bound D hδ hH |
| refine ⟨N,fun {ι} S r k hsep v => ?_⟩ |
| let T := S.filter (fun i => (r i).val k = v) |
| by_cases hT : T.Nonempty |
| · obtain ⟨i,hi⟩ := hT |
| let x : ι → Boundary D := fun j => Quotient.mk' (r j) |
| have hinj : Set.InjOn x (T : Set ι) := by |
| intro j hj j' hj' heq |
| by_contra hne |
| have hh := hsep j (mem_filter.mp hj).1 j' (mem_filter.mp hj').1 hne |
| change H*visualScale δ k ≤ dist (x j) (x j') at hh |
| rw [heq,dist_self] at hh |
| exact (not_le_of_gt (mul_pos hH (visualScale_pos δ k))) hh |
| have hh := hN (x i) (visualScale δ k) (visualScale_pos δ k) |
| (by rw [visualScale]; apply Real.exp_le_one_iff.mpr; exact mul_nonpos_of_nonpos_of_nonneg (neg_nonpos.mpr (visualExponent_pos δ).le) (Nat.cast_nonneg _)) (T.image x) (by |
| intro y hy |
| obtain ⟨j,hj,rfl⟩ := mem_image.mp hy |
| exact boundaryDist_le_of_prefix D hδ (r i) (r j) |
| ((mem_filter.mp hi).2.trans (mem_filter.mp hj).2.symm)) (by |
| intro y hy y' hy' hne |
| obtain ⟨j,hj,rfl⟩ := mem_image.mp hy |
| obtain ⟨j',hj',rfl⟩ := mem_image.mp hy' |
| exact hsep j (mem_filter.mp hj).1 j' (mem_filter.mp hj').1 |
| (fun he => hne (congrArg x he))) |
| rwa [card_image_of_injOn hinj] at hh |
| · change T.card ≤ _ |
| rw [Finset.not_nonempty_iff_eq_empty.mp hT,Finset.card_empty] |
| exact Nat.zero_le _ |
|
|
| theorem puncture_ball_multiplicity {e H C : ℝ} (he : 0 < e) (hH : 0 < H) (hC : 0 ≤ C) : |
| ∃ N : ℕ, ∀ {ι : Type*} (S : Finset ι) (r : ι → BasedRay D) (z : BasedRay D) (k : ℕ), |
| (∀ i ∈ S, e ≤ boundaryDist D hδ (Quotient.mk' (r i)) (Quotient.mk' z)) → |
| (∀ i ∈ S, ∀ j ∈ S, i ≠ j → H*visualScale δ k ≤ |
| boundaryDist D hδ (Quotient.mk' (r i)) (Quotient.mk' (r j))) → |
| ∀ b : Boundary D, (S.filter (fun i => boundaryDist D hδ |
| (boundaryTranslate D ⟨δ,hδ⟩ ((r i).val k)⁻¹ (Quotient.mk' z)) b ≤ C*visualScale δ k)).card ≤ N := by |
| classical |
| let _ := visualMetricSpace D hδ |
| obtain ⟨B,hB⟩ := inverse_puncture_prefix D hδ he |
| obtain ⟨B',hB'⟩ := visual_synchronous_prefix D hδ (2*C) (by positivity) |
| obtain ⟨Ng,hNg⟩ := uniform_graph_ball_card D (B+B') |
| obtain ⟨Nv,hNv⟩ := vertex_assignment_multiplicity D hδ hH |
| refine ⟨Nv*Ng,fun {ι} S r z k hsep hcent b => ?_⟩ |
| let p : ι → BasedRay D := fun i => translateRay D ⟨δ,hδ⟩ ((r i).val k)⁻¹ z |
| let T := S.filter (fun i => boundaryDist D hδ (Quotient.mk' (p i)) b ≤ C*visualScale δ k) |
| change T.card ≤ Nv*Ng |
| by_cases hT : T.Nonempty |
| · obtain ⟨i₀,hi₀⟩ := hT |
| have hnear (i : ι) (hi : i ∈ T) : |
| D.graph.dist ((r i).val k)⁻¹ ((p i₀).val k) ≤ B+B' := by |
| have h₁ := hB (r i) z k (hsep i (mem_filter.mp hi).1) |
| have hpd : boundaryDist D hδ (Quotient.mk' (p i)) (Quotient.mk' (p i₀)) ≤ |
| (2*C)*visualScale δ k := by |
| have hh := @dist_triangle (Boundary D) _ (Quotient.mk' (p i)) b (Quotient.mk' (p i₀)) |
| rw [@dist_comm (Boundary D) _ b (Quotient.mk' (p i₀))] at hh |
| change boundaryDist D hδ (Quotient.mk' (p i)) (Quotient.mk' (p i₀)) ≤ |
| boundaryDist D hδ (Quotient.mk' (p i)) b + boundaryDist D hδ (Quotient.mk' (p i₀)) b at hh |
| linarith only [hh,(mem_filter.mp hi).2,(mem_filter.mp hi₀).2] |
| have h₂ := hB' (p i) (p i₀) k hpd |
| have ht := @SimpleGraph.Connected.dist_triangle _ D.graph |
| ((r i).val k)⁻¹ ((p i).val k) ((p i₀).val k) D.connected |
| exact ht.trans (Nat.add_le_add h₁ h₂) |
| have hcard := hNg ((p i₀).val k) (T.image (fun i => ((r i).val k)⁻¹)) (by |
| intro y hy |
| obtain ⟨i,hi,rfl⟩ := mem_image.mp hy |
| rw [SimpleGraph.dist_comm] |
| exact hnear i hi) |
| have hfib : ∀ v ∈ T.image (fun i => ((r i).val k)⁻¹), |
| (T.filter (fun i => ((r i).val k)⁻¹ = v)).card ≤ Nv := by |
| intro v _ |
| have hh := hNv T r k (fun i hi j hj hij => hcent i (mem_filter.mp hi).1 |
| j (mem_filter.mp hj).1 hij) v⁻¹ |
| simpa only [inv_eq_iff_eq_inv] using hh |
| exact (card_le_mul_card_image T Nv hfib).trans (Nat.mul_le_mul_left Nv hcard) |
| · rw [Finset.not_nonempty_iff_eq_empty.mp hT,Finset.card_empty] |
| exact Nat.zero_le _ |
|
|
| end CannonRelease |
|
|
| namespace CannonGraph |
|
|
| open SimpleGraph CannonVisual |
|
|
| variable {V : Type*} {Γ : SimpleGraph V} {o : V} |
|
|
| lemma prefix_of_near_ray (hΓ : Γ.Connected) {δ : ℕ} (hδ : Thin Γ δ) |
| (p : Ray Γ o) (v w : V) (k l B : ℕ) (hl : l ≤ k) |
| (_hv : Γ.dist o v = k) (hw : Γ.dist o w = l) |
| (hwv : Γ.dist w v = k-l) (hvp : Γ.dist v (p.val k) ≤ B) : |
| Γ.dist w (p.val l) ≤ 2*B+4*δ+2 := by |
| have hp : Γ.dist o (p.val k) = k := by |
| simpa only [p.property.1,Nat.dist_zero_left] using p.property.2 0 k |
| have ht := @Connected.dist_triangle _ Γ w v (p.val k) hΓ |
| have hprod := product2_add_dist hΓ o w (p.val k) |
| have hlarge : 2*Γ.dist o w ≤ product2 (G:=Γ) o w (p.val k)+B := by |
| rw [hwv] at ht |
| rw [hw,hp] at hprod |
| rw [hw] |
| omega |
| have hh := vertex_close_to_ray hΓ hδ p w k B (by simpa only [hw] using hl) hlarge |
| simpa only [hw] using hh |
|
|
| end CannonGraph |
|
|
| namespace CannonRelease |
|
|
| open CannonGraph CannonVisual Finset |
|
|
| 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 visualSynchronousInverseRaySetoid : Setoid (BasedRay D) := raySetoid D |
|
|
| local instance visualSynchronousInverseRayDecidable : ∀ p : Prop, Decidable p := |
| Classical.propDecidable |
|
|
| omit [TopologicalSpace G] [DiscreteTopology G] in |
| lemma inverse_ray_prefix (r : BasedRay D) {k l : ℕ} (hl : l ≤ k) : |
| D.graph.dist 1 ((r.val k)⁻¹*r.val (k-l)) = l ∧ |
| D.graph.dist ((r.val k)⁻¹*r.val (k-l)) (r.val k)⁻¹ = k-l := by |
| have h₁ := D.dist_mul_left (r.val k) 1 ((r.val k)⁻¹*r.val (k-l)) |
| have h₂ := D.dist_mul_left (r.val k) ((r.val k)⁻¹*r.val (k-l)) (r.val k)⁻¹ |
| simp only [mul_one,mul_inv_cancel_left,mul_inv_cancel] at h₁ h₂ |
| have hkk := r.property.2 k (k-l) |
| have hk₀ := r.property.2 (k-l) 0 |
| rw [r.property.1,Nat.dist_zero_right] at hk₀ |
| dsimp only [CayleyData.graph] at h₁ h₂ hkk hk₀ ⊢ |
| constructor |
| · rw [← h₁,hkk,Nat.dist_eq_sub_of_le_right (Nat.sub_le _ _)] |
| omega |
| · rw [← h₂,hk₀] |
|
|
| theorem near_puncture_descendant_count {e H C : ℝ} |
| (he : 0 < e) (hH : 0 < H) (hC : 0 ≤ C) : |
| ∃ N : ℕ, ∀ {ι : Type*} (S : Finset ι) (r : ι → BasedRay D) |
| (z a b : BasedRay D) (k l : ℕ), l ≤ k → |
| (∀ i ∈ S, e ≤ boundaryDist D hδ (Quotient.mk' (r i)) (Quotient.mk' z)) → |
| (∀ i ∈ S, ∀ j ∈ S, i ≠ j → H*visualScale δ k ≤ |
| boundaryDist D hδ (Quotient.mk' (r i)) (Quotient.mk' (r j))) → |
| (∀ i ∈ S, boundaryDist D hδ (Quotient.mk' (r i)) (Quotient.mk' a) ≤ |
| C*visualScale δ (k-l)) → |
| (∀ i ∈ S, boundaryDist D hδ |
| (boundaryTranslate D ⟨δ,hδ⟩ ((r i).val k)⁻¹ (Quotient.mk' z)) |
| (Quotient.mk' b) ≤ visualScale δ l) → |
| S.card ≤ N := by |
| classical |
| obtain ⟨B,hB⟩ := inverse_puncture_prefix D hδ he |
| obtain ⟨B₁,hB₁⟩ := visual_synchronous_prefix D hδ 1 (by norm_num) |
| obtain ⟨B₂,hB₂⟩ := visual_synchronous_prefix D hδ C hC |
| obtain ⟨N₁,hN₁⟩ := uniform_graph_ball_card D (2*B+4*δ+2+B₁) |
| obtain ⟨N₂,hN₂⟩ := uniform_graph_ball_card D B₂ |
| obtain ⟨Nv,hNv⟩ := vertex_assignment_multiplicity D hδ hH |
| refine ⟨Nv*(N₁*N₂),fun {ι} S r z a b k l hl hsep hcent hanc hpunc => ?_⟩ |
| let u : ι → G := fun i => (r i).val (k-l) |
| let w : ι → G := fun i => ((r i).val k)⁻¹*u i |
| let p : ι → BasedRay D := fun i => translateRay D ⟨δ,hδ⟩ ((r i).val k)⁻¹ z |
| have hw (i : ι) (hi : i ∈ S) : |
| D.graph.dist (w i) (b.val l) ≤ 2*B+4*δ+2+B₁ := by |
| have hg : D.graph.dist 1 ((r i).val k)⁻¹ = k := by |
| have hh := D.dist_mul_left ((r i).val k) 1 ((r i).val k)⁻¹ |
| have hk := (r i).property.2 k 0 |
| rw [(r i).property.1,Nat.dist_zero_right] at hk |
| simpa only [mul_one,mul_inv_cancel,hk] using hh.symm |
| have hpre := inverse_ray_prefix D (r i) hl |
| have hnear := hB (r i) z k (hsep i hi) |
| have hh := prefix_of_near_ray D.connected hδ (p i) ((r i).val k)⁻¹ (w i) |
| k l B hl hg hpre.1 hpre.2 hnear |
| have hpi := hpunc i hi |
| have heq : boundaryTranslate D ⟨δ,hδ⟩ ((r i).val k)⁻¹ (Quotient.mk' z) = |
| Quotient.mk' (p i) := rfl |
| rw [heq] at hpi |
| have hsync := hB₁ (p i) b l (by simpa only [one_mul] using hpi) |
| exact (@SimpleGraph.Connected.dist_triangle _ D.graph (w i) ((p i).val l) (b.val l) |
| D.connected).trans (Nat.add_le_add hh hsync) |
| have hu (i : ι) (hi : i ∈ S) : D.graph.dist (u i) (a.val (k-l)) ≤ B₂ := |
| hB₂ (r i) a (k-l) (hanc i hi) |
| have hwcard : (S.image w).card ≤ N₁ := hN₁ (b.val l) (S.image w) (by |
| intro y hy |
| obtain ⟨i,hi,rfl⟩ := mem_image.mp hy |
| rw [SimpleGraph.dist_comm] |
| exact hw i hi) |
| have hucard : (S.image u).card ≤ N₂ := hN₂ (a.val (k-l)) (S.image u) (by |
| intro y hy |
| obtain ⟨i,hi,rfl⟩ := mem_image.mp hy |
| rw [SimpleGraph.dist_comm] |
| exact hu i hi) |
| let f : ι → G × G := fun i => (w i,u i) |
| have hfcard : (S.image f).card ≤ N₁*N₂ := by |
| have hsub : S.image f ⊆ (S.image w) ×ˢ (S.image u) := by |
| intro y hy |
| obtain ⟨i,hi,rfl⟩ := mem_image.mp hy |
| exact mem_product.mpr ⟨mem_image.mpr ⟨i,hi,rfl⟩,mem_image.mpr ⟨i,hi,rfl⟩⟩ |
| exact (Finset.card_le_card hsub).trans (by rw [card_product]; exact Nat.mul_le_mul hwcard hucard) |
| have hfib : ∀ y ∈ S.image f, (S.filter (fun i => f i = y)).card ≤ Nv := by |
| intro y _ |
| apply (Finset.card_le_card (show S.filter (fun i => f i = y) ⊆ |
| S.filter (fun i => (r i).val k = y.2*y.1⁻¹) from ?_)).trans (hNv S r k hcent _) |
| intro i hi |
| obtain ⟨hi,heq⟩ := mem_filter.mp hi |
| have hw := congrArg Prod.fst heq |
| have hu := congrArg Prod.snd heq |
| change w i = y.1 at hw |
| change u i = y.2 at hu |
| refine mem_filter.mpr ⟨hi,?_⟩ |
| rw [← hw,← hu] |
| simp only [w,mul_inv_rev,inv_inv,mul_inv_cancel_left] |
| exact (card_le_mul_card_image S Nv hfib).trans (Nat.mul_le_mul_left Nv hfcard) |
|
|
| end CannonRelease |
|
|
| end |
|
|
| end |
| end |
| end |
| end |
| end |
|
|
| end OAI |
|
|
