openAI-math-dataset / lean /OAI /Probability /CriticalPercolation /GraphGeometry /GraphApproximateNeighborhoods.lean
| import Mathlib | |
| import OAI.Probability.CriticalPercolation.GraphTopology.GraphNeighborhoodCompact | |
| import OAI.Probability.CriticalPercolation.Growth.SparseDoubling | |
| namespace OAI | |
| section | |
| noncomputable section | |
| open Set Function Topology | |
| open scoped Classical Pointwise | |
| namespace CriticalPercolation.GraphTopology | |
| variable {X : Type*} (Γ : SimpleGraph X) | |
| def displacement (o : X) (r : ℕ) : Set (Γ ≃g Γ) := | |
| {g | g o∈reachBall Γ o r} | |
| lemma reachBall_mono (o : X) : Monotone (reachBall Γ o) := by | |
| intro r s hrs x hx | |
| obtain ⟨p,hp⟩ := hx | |
| exact ⟨p,hp.trans hrs⟩ | |
| lemma iso_reachBall_image (g : Γ ≃g Γ) (o : X) (r : ℕ) : | |
| g '' reachBall Γ o r=reachBall Γ (g o) r := by | |
| apply Set.Subset.antisymm | |
| · rintro x ⟨y,hy,rfl⟩ | |
| exact iso_reachBall Γ g hy | |
| · intro x hx | |
| refine ⟨g.symm x,?_,g.apply_symm_apply x⟩ | |
| simpa using iso_reachBall Γ g.symm hx | |
| lemma reachBall_ncard_eq (ht : ∀ u v, ∃ g : Γ ≃g Γ, g u=v) (o x : X) (r : ℕ) : | |
| (reachBall Γ x r).ncard=(reachBall Γ o r).ncard := by | |
| obtain ⟨g,rfl⟩ := ht o x | |
| rw [←iso_reachBall_image Γ g o r,Set.ncard_image_of_injective _ g.injective] | |
| lemma one_mem_displacement (o : X) (r : ℕ) : 1∈displacement Γ o r := | |
| reachBall_self Γ o r | |
| lemma displacement_inv {o : X} {r : ℕ} {g : Γ ≃g Γ} : | |
| g⁻¹∈displacement Γ o r ↔ g∈displacement Γ o r := by | |
| constructor | |
| · intro h | |
| change g o∈reachBall Γ o r | |
| simpa using inverse_displacement Γ g.symm h | |
| · exact inverse_displacement Γ g | |
| lemma displacement_mul_subset (o : X) (r s : ℕ) : | |
| displacement Γ o r * displacement Γ o s ⊆ displacement Γ o (r+s) := by | |
| rintro g ⟨a,ha,b,hb,rfl⟩ | |
| exact reachBall_add Γ ha (iso_reachBall Γ a hb) | |
| lemma reachBall_packing (hl : ∀ x, (Γ.neighborSet x).Finite) | |
| (ht : ∀ u v, ∃ g : Γ ≃g Γ, g u=v) (o : X) (r K : ℕ) | |
| (hd : (reachBall Γ o (5*r)).ncard≤K*(reachBall Γ o r).ncard) : | |
| ∃ F : Finset X, (↑F : Set X)⊆reachBall Γ o (4*r) ∧ F.card≤K ∧ | |
| ∀ u∈reachBall Γ o (4*r), ∃ v∈F, u∈reachBall Γ v (2*r) := by | |
| let A := (reachBall_finite Γ hl o (4*r)).toFinset | |
| let C : Finset (Finset X) := A.powerset.filter | |
| (fun F => (F : Set X).PairwiseDisjoint (fun v => reachBall Γ v r)) | |
| obtain ⟨F,hFmax⟩ := C.exists_maximal <| Finset.filter_nonempty_iff.mpr | |
| ⟨∅,Finset.empty_mem_powerset _,by simp⟩ | |
| simp only [C,Finset.mem_filter,Finset.mem_powerset] at hFmax | |
| obtain ⟨hFA,hF⟩ := hFmax.1 | |
| have hFo : (F : Set X)⊆reachBall Γ o (4*r) := by | |
| intro u hu | |
| exact (reachBall_finite Γ hl o (4*r)).mem_toFinset.mp (hFA hu) | |
| have hsub : (⋃ u∈(F : Set X), reachBall Γ u r) ⊆ reachBall Γ o (5*r) := by | |
| intro v hv | |
| obtain ⟨u,hu,hv⟩ := Set.mem_iUnion₂.mp hv | |
| simpa only [show 4*r+r=5*r by omega] using reachBall_add Γ (hFo hu) hv | |
| have hcount : F.card*(reachBall Γ o r).ncard≤(reachBall Γ o (5*r)).ncard := by | |
| calc | |
| F.card*(reachBall Γ o r).ncard = ∑ u∈F, (reachBall Γ u r).ncard := by | |
| symm | |
| apply Finset.sum_const_nat | |
| intro u _ | |
| exact reachBall_ncard_eq Γ ht o u r | |
| _ = (⋃ u∈(F : Set X), reachBall Γ u r).ncard := by | |
| rw [(F.finite_toSet).ncard_biUnion (fun u _ => reachBall_finite Γ hl u r) hF, | |
| finsum_mem_coe_finset] | |
| _ ≤ (reachBall Γ o (5*r)).ncard := Set.ncard_le_ncard hsub (reachBall_finite Γ hl o _) | |
| have hpos : 0<(reachBall Γ o r).ncard := | |
| (Set.ncard_pos (reachBall_finite Γ hl o r)).mpr ⟨o,reachBall_self Γ o r⟩ | |
| refine ⟨F,hFo,le_of_mul_le_mul_right (hcount.trans hd) hpos,?_⟩ | |
| intro u hu | |
| by_cases huF : u∈F | |
| · exact ⟨u,huF,reachBall_self Γ u _⟩ | |
| by_cases! H : ∀ v∈F, Disjoint (reachBall Γ u r) (reachBall Γ v r) | |
| · refine False.elim (hFmax.not_gt ?_ (Finset.ssubset_insert huF)) | |
| rw [Finset.insert_subset_iff,Finset.coe_insert] | |
| exact ⟨⟨(reachBall_finite Γ hl o (4*r)).mem_toFinset.mpr hu,hFA⟩, | |
| hF.insert (fun v hv _ => H v hv)⟩ | |
| · simp_rw [Set.not_disjoint_iff] at H | |
| obtain ⟨v,hv,w,hwu,hwv⟩ := H | |
| refine ⟨v,hv,?_⟩ | |
| simpa only [show r+r=2*r by omega] using | |
| reachBall_add Γ hwv (reachBall_symm Γ hwu) | |
| lemma displacement_cover_of_doubling (hl : ∀ x, (Γ.neighborSet x).Finite) | |
| (ht : ∀ u v, ∃ g : Γ ≃g Γ, g u=v) (o : X) (r K : ℕ) | |
| (hd : (reachBall Γ o (5*r)).ncard≤K*(reachBall Γ o r).ncard) : | |
| CovBySMul (Γ ≃g Γ) (K : ℝ) (displacement Γ o (4*r)) (displacement Γ o (2*r)) := by | |
| obtain ⟨F,hFo,hFK,hcover⟩ := reachBall_packing Γ hl ht o r K hd | |
| have hex : ∀ v : F, ∃ φ : Γ ≃g Γ, φ o=v.val := fun v => ht o v.val | |
| choose φ hφ using hex | |
| let H := Finset.univ.image φ | |
| have hH : H.card≤K := (Finset.card_image_le).trans (by simpa using hFK) | |
| refine ⟨H,by exact_mod_cast hH,?_⟩ | |
| intro ψ hψ | |
| obtain ⟨v,hv,huv⟩ := hcover (ψ o) hψ | |
| let c := φ ⟨v,hv⟩ | |
| have hc : c∈H := Finset.mem_image.mpr ⟨⟨v,hv⟩,Finset.mem_univ _,rfl⟩ | |
| have hco : c o=v := hφ ⟨v,hv⟩ | |
| have hrem : c⁻¹*ψ∈displacement Γ o (2*r) := by | |
| have hh := iso_reachBall Γ c.symm huv | |
| rw [←hco] at hh | |
| change c.symm (ψ o)∈reachBall Γ o (2*r) | |
| simpa using hh | |
| change ψ∈(H : Set (Γ ≃g Γ))*displacement Γ o (2*r) | |
| exact Set.mem_mul.mpr ⟨c,hc,c⁻¹*ψ,hrem,by simp⟩ | |
| lemma displacement_approximate_of_doubling (hl : ∀ x, (Γ.neighborSet x).Finite) | |
| (ht : ∀ u v, ∃ g : Γ ≃g Γ, g u=v) (o : X) (r K : ℕ) | |
| (hd : (reachBall Γ o (5*r)).ncard≤K*(reachBall Γ o r).ncard) : | |
| IsApproximateSubgroup (K : ℝ) (displacement Γ o (2*r)) where | |
| one_mem := one_mem_displacement Γ o _ | |
| inv_eq_self := by | |
| ext φ | |
| change φ⁻¹∈displacement Γ o (2*r) ↔ φ∈displacement Γ o (2*r) | |
| exact displacement_inv Γ | |
| sq_covBySMul := by | |
| apply (displacement_cover_of_doubling Γ hl ht o r K hd).subset_left | |
| rw [pow_two] | |
| simpa only [show 2*r+2*r=4*r by omega] using displacement_mul_subset Γ o (2*r) (2*r) | |
| theorem compact_open_approximate_scales | |
| [TopologicalSpace X] [DiscreteTopology X] | |
| (hc : Γ.Connected) (hl : ∀ x, (Γ.neighborSet x).Finite) | |
| (ht : ∀ u v, ∃ g : Γ ≃g Γ, g u=v) (o : X) (d : ℕ) | |
| (hpoly : ∀ R : ℕ, ∃ n≥R, (reachBall Γ o n).ncard≤n^d) : | |
| let _ : TopologicalSpace (Γ ≃g Γ) := autTopology Γ | |
| ∀ R : ℕ, ∃ n≥R, IsCompact (displacement Γ o (2*n)) ∧ | |
| IsOpen (displacement Γ o (2*n)) ∧ | |
| IsApproximateSubgroup (5^(d+1) : ℝ) (displacement Γ o (2*n)) := by | |
| let _ : TopologicalSpace (Γ ≃g Γ) := autTopology Γ | |
| dsimp only | |
| intro R | |
| obtain ⟨n,hn,hd⟩ := sparse_polynomial_doubling_scales | |
| (fun n => (reachBall Γ o n).ncard) | |
| (fun r s hrs => Set.ncard_le_ncard (reachBall_mono Γ o hrs) (reachBall_finite Γ hl o s)) | |
| (fun n => (Set.ncard_pos (reachBall_finite Γ hl o n)).mpr ⟨o,reachBall_self Γ o n⟩) | |
| d hpoly R | |
| refine ⟨n,hn,compact_displacement Γ hc hl o _,open_displacement Γ o _,?_⟩ | |
| exact_mod_cast displacement_approximate_of_doubling Γ hl ht o n (5^(d+1)) hd | |
| end CriticalPercolation.GraphTopology | |
| end | |
| end | |
| end OAI | |