openAI-math-dataset / lean /OAI /Probability /CriticalPercolation /GraphGeometry /EllipseUniform.lean
| import Mathlib | |
| import OAI.Probability.CriticalPercolation.ProductMeasure.BitMeasure | |
| import OAI.Probability.CriticalPercolation.GraphGeometry.EuclideanEllipse | |
| import OAI.Probability.CriticalPercolation.GraphGeometry.MovingStrips | |
| import OAI.Probability.CriticalPercolation.Gluing.EllipseMoves | |
| import OAI.Probability.CriticalPercolation.GraphGeometry.EllipseOrigin | |
| import OAI.Probability.CriticalPercolation.Gluing.ApproximateMoveEndgame | |
| import OAI.Probability.CriticalPercolation.Modular.MoveTransport | |
| namespace OAI | |
| section | |
| section | |
| noncomputable section | |
| open Set Function MeasureTheory ProbabilityTheory Filter | |
| open scoped Classical Topology ENNReal | |
| namespace CriticalPercolation.BondGraph | |
| variable {V E : Type*} (G : BondGraph V E) | |
| lemma ellipse_aut_differences (pi : V → ℂ) (R W : ℝ) (c d e : Circle) (phi : G.Aut) | |
| (hphi : ∀ x y, pi (phi.vertex x)-pi (phi.vertex y)=pi x-pi y) : | |
| (∀ x y, ellipseTime (EuclideanEllipse.normalize R W c ∘ pi) d e (phi.vertex x)- | |
| ellipseTime (EuclideanEllipse.normalize R W c ∘ pi) d e (phi.vertex y)= | |
| ellipseTime (EuclideanEllipse.normalize R W c ∘ pi) d e x- | |
| ellipseTime (EuclideanEllipse.normalize R W c ∘ pi) d e y) ∧ | |
| (∀ x y, ellipseHeight (EuclideanEllipse.normalize R W c ∘ pi) d e (phi.vertex x)- | |
| ellipseHeight (EuclideanEllipse.normalize R W c ∘ pi) d e (phi.vertex y)= | |
| ellipseHeight (EuclideanEllipse.normalize R W c ∘ pi) d e x- | |
| ellipseHeight (EuclideanEllipse.normalize R W c ∘ pi) d e y) := by | |
| have hPsi : ∀ x y, | |
| EuclideanEllipse.normalize R W c (pi (phi.vertex x))-EuclideanEllipse.normalize R W c (pi (phi.vertex y))= | |
| EuclideanEllipse.normalize R W c (pi x)-EuclideanEllipse.normalize R W c (pi y) := by | |
| intro x y | |
| rw [←EuclideanEllipse.normalize_sub,←EuclideanEllipse.normalize_sub,hphi] | |
| constructor | |
| · intro x y | |
| rw [(ellipse_coordinates_sub _ d e (phi.vertex x) (phi.vertex y)).1, | |
| (ellipse_coordinates_sub _ d e x y).1] | |
| exact congrArg (fun z : ℂ => EllipseLinear.time (d : ℂ).re (d : ℂ).im (e : ℂ).re (e : ℂ).im z.re z.im) (hPsi x y) | |
| · intro x y | |
| rw [(ellipse_coordinates_sub _ d e (phi.vertex x) (phi.vertex y)).2, | |
| (ellipse_coordinates_sub _ d e x y).2] | |
| exact congrArg (fun z : ℂ => EllipseLinear.height (d : ℂ).re (d : ℂ).im (e : ℂ).re (e : ℂ).im z.re z.im) (hPsi x y) | |
| theorem ellipse_uniform_moves [Countable V] [Countable E] | |
| (hc : G.fullGraph.Connected) (hl : G.LocallyFinite) | |
| (pi : V → ℂ) (o : V) (c : ℕ) (D : ℝ) (pc : unitInterval) | |
| (hD : 0≤D) (hpc : 0<pc) (hpo : pi o=0) | |
| (hpi : ∀ ⦃x y⦄, G.fullGraph.Adj x y → ‖pi x-pi y‖≤D) | |
| (hdense : ∀ y, ∃ phi : G.Aut, phi.vertex o∈G.vertexBall y c ∧ | |
| ∀ x y, pi (phi.vertex x)-pi (phi.vertex y)=pi x-pi y) | |
| (hone : BernoulliProduct.law (fun _ : E => pc) G.percolates=1) | |
| (hbox : ∀ U W : ℝ, ∀ᵐ ω ∂BernoulliProduct.law (fun _ : E => pc), ∀ x, | |
| (G.cluster (ω∩G.inducedEdges {x | |(pi x).re|≤U ∧ |(pi x).im|≤W}) x).Finite) | |
| (hnull : ∀ a b : ℝ, a^2+b^2=1 → ∀ w : ℝ, | |
| BernoulliProduct.law (fun _ : E => pc) | |
| (G.stripPercolates (normalCoordinate (fun x => (pi x).re) (fun x => (pi x).im) a b) w)=0) | |
| (eps eta : ℝ) (heps : 0<eps) (heps' : eps<1/10) (heta : 0<eta) : | |
| ∃ m : ℕ, ∀ rho : ℝ, 0<rho → ∃ t z : V → ℝ, ∃ p : unitInterval, | |
| (pc : ℝ)/2≤p ∧ p<pc ∧ | |
| (∀ ⦃x y⦄, G.fullGraph.Adj x y → |t x-t y|≤rho) ∧ | |
| (∀ ⦃x y⦄, G.fullGraph.Adj x y → |z x-z y|≤rho) ∧ | |
| ∀ y s, |s|=1 → 1-eta≤(BernoulliProduct.law (fun _ : E => p)).real | |
| (G.coordinateMove t z m eps y s) := by | |
| obtain ⟨m,hm⟩ := G.ellipse_origin_moves hc hl pi o D pc hD hpo hpi hone hbox hnull eps eta heps heps' heta | |
| refine ⟨m+c,?_⟩ | |
| intro rho hrho | |
| let r := min rho (eps/(8*((c : ℝ)+1))) | |
| have hr : 0<r := lt_min hrho (div_pos heps (by positivity)) | |
| have hrrho : r≤rho := min_le_left _ _ | |
| have hrs : r≤eps/(8*((c : ℝ)+1)) := min_le_right _ _ | |
| have hmesh : (c : ℝ)*r≤eps/8 := by | |
| have hh := (le_div_iff₀ (by positivity : 0<8*((c : ℝ)+1))).mp hrs | |
| nlinarith | |
| obtain ⟨R,W,a,b,d,ht,hz,hmoves⟩ := hm r hr | |
| let t := ellipseTime (EuclideanEllipse.normalize R W a ∘ pi) b d | |
| let z := ellipseHeight (EuclideanEllipse.normalize R W a ∘ pi) b d | |
| have htdense : ∀ y, ∃ phi : G.Aut, phi.vertex o∈G.vertexBall y c ∧ | |
| (∀ x y, t (phi.vertex x)-t (phi.vertex y)=t x-t y) ∧ | |
| (∀ x y, z (phi.vertex x)-z (phi.vertex y)=z x-z y) := by | |
| intro y | |
| obtain ⟨phi,hphi,hphiP⟩ := hdense y | |
| exact ⟨phi,hphi,G.ellipse_aut_differences pi R W a b d phi hphiP⟩ | |
| obtain ⟨p,hpl,hpu,hp⟩ := G.uniform_moves_of_origin t z o m c eps eta r heps' hr.le hmesh ht hz htdense pc hpc hmoves | |
| exact ⟨t,z,p,hpl,hpu,fun {_ _} h => (ht h).trans hrrho,fun {_ _} h => (hz h).trans hrrho,hp⟩ | |
| theorem ellipse_critical_contradiction [Countable V] [Countable E] | |
| (hc : G.fullGraph.Connected) (hl : G.LocallyFinite) (hq : G.QuasiTransitive) | |
| (pi : V → ℂ) (o : V) (c : ℕ) (D : ℝ) | |
| (hD : 0≤D) (hpc : 0<(G.criticalProbability : ℝ)) (hpc1 : (G.criticalProbability : ℝ)<1) | |
| (hpo : pi o=0) | |
| (hpi : ∀ ⦃x y⦄, G.fullGraph.Adj x y → ‖pi x-pi y‖≤D) | |
| (hdense : ∀ y, ∃ phi : G.Aut, phi.vertex o∈G.vertexBall y c ∧ | |
| ∀ x y, pi (phi.vertex x)-pi (phi.vertex y)=pi x-pi y) | |
| (hone : BernoulliProduct.law (fun _ : E => G.criticalProbability) G.percolates=1) | |
| (hbox : ∀ U W : ℝ, ∀ᵐ ω ∂BernoulliProduct.law (fun _ : E => G.criticalProbability), ∀ x, | |
| (G.cluster (ω∩G.inducedEdges {x | |(pi x).re|≤U ∧ |(pi x).im|≤W}) x).Finite) | |
| (hnull : ∀ a b : ℝ, a^2+b^2=1 → ∀ w : ℝ, | |
| BernoulliProduct.law (fun _ : E => G.criticalProbability) | |
| (G.stripPercolates (normalCoordinate (fun x => (pi x).re) (fun x => (pi x).im) a b) w)=0) : False := by | |
| apply G.approximate_moves_contradiction hc hl hq hpc hpc1 | |
| intro eps eta he he' heta _heta' | |
| exact G.ellipse_uniform_moves hc hl pi o c D G.criticalProbability hD hpc hpo hpi hdense hone hbox hnull eps eta he he' heta | |
| end CriticalPercolation.BondGraph | |
| end | |
| end | |
| end | |
| end OAI | |