openAI-math-dataset / lean /OAI /Probability /CriticalPercolation /Criticality /FiniteInterpolationUniform.lean
| import Mathlib | |
| import OAI.Probability.CriticalPercolation.Criticality.FiniteInterpolation | |
| import OAI.Probability.CriticalPercolation.Criticality.InterpolationUniform | |
| namespace OAI | |
| section | |
| section | |
| noncomputable section | |
| namespace CriticalPercolation.BinaryLaw | |
| open scoped Classical | |
| open FiniteProduct BondGraph | |
| variable {V E : Type*} [Fintype V] [Fintype E] | |
| theorem finite_interpolation_uniform (G : BondGraph V E) (d : ℕ) (x : ℕ → V) | |
| (hd : 2≤d) {p : ℝ} (hp : p ∈ Set.Ioo 0 1) (w : E → Prop → ℝ) | |
| (h0 : ∀ e, w e False = 1-p) (h1 : ∀ e, w e True = p) | |
| {c ε : ℝ} (hc : 0<c) (hε : 0<ε) (hεp : ε≤p) (hεq : ε≤1-p) | |
| (hconn : ∀ j k, j<k → k≤d → k-j<d → | |
| c ≤ prob w (fun ω => (G.openGraph ω).Reachable (x j) (x k))) | |
| (s : ℕ) (hδ : 0 < prob w (endpointEvent G (x 0) (x d) s)) : | |
| ∃ j<d, 0 < prob w (endpointEvent G (x j) (x (j+1)) s) ∧ | |
| -Real.log (prob w (endpointEvent G (x j) (x (j+1)) s)) ≤ | |
| interpolationConstant c ε * | |
| (Real.log (1/prob w (endpointEvent G (x 0) (x d) s))+Nat.clog 2 d+1) := by | |
| have hw := homogeneous_normalized w p h0 h1 | |
| have hn := fun e b => (homogeneous_positive w hp h0 h1 e b).le | |
| have hc1 : c≤1 := (hconn 0 1 (by omega) (by omega) (by omega)).trans (prob_le_one w hw hn _) | |
| obtain ⟨j,hj,hpos,hbound⟩ := finite_interpolation_index G d x hd hp w h0 h1 hc hconn s hδ | |
| exact ⟨j,hj,hpos,hbound.trans | |
| (interpolation_uniform_algebra hp hc hc1 hε hεp hεq hδ (prob_le_one w hw hn _) _)⟩ | |
| theorem finite_interpolation_edge (G : BondGraph V E) (d : ℕ) (x : ℕ → V) | |
| (hd : 2≤d) (path : Fin d → E) | |
| (hpath : ∀ j : Fin d, G.ends (path j) = s(x j,x (j+1))) | |
| {p : ℝ} (hp : p ∈ Set.Ioo 0 1) (w : E → Prop → ℝ) | |
| (h0 : ∀ e, w e False = 1-p) (h1 : ∀ e, w e True = p) | |
| {c ε : ℝ} (hc : 0<c) (hε : 0<ε) (hεp : ε≤p) (hεq : ε≤1-p) | |
| (hconn : ∀ j k, j<k → k≤d → k-j<d → | |
| c ≤ prob w (fun ω => (G.openGraph ω).Reachable (x j) (x k))) | |
| (s : ℕ) (hδ : 0 < prob w (endpointEvent G (x 0) (x d) s)) : | |
| ∃ j : Fin d, G.ends (path j) = s(x j,x (j+1)) ∧ | |
| 0 < prob w (endpointEvent G (x j) (x (j+1)) s) ∧ | |
| -Real.log (prob w (endpointEvent G (x j) (x (j+1)) s)) ≤ | |
| interpolationConstant c ε * | |
| (Real.log (1/prob w (endpointEvent G (x 0) (x d) s))+Nat.clog 2 d+1) := by | |
| obtain ⟨j,hj,hpos,hbound⟩ := finite_interpolation_uniform G d x hd hp w h0 h1 hc hε hεp hεq hconn s hδ | |
| exact ⟨⟨j,hj⟩,hpath ⟨j,hj⟩,hpos,hbound⟩ | |
| end CriticalPercolation.BinaryLaw | |
| end | |
| end | |
| end | |
| end OAI | |