openAI-math-dataset / lean /OAI /Probability /CriticalPercolation /Criticality /FiniteEdgeInterpolation.lean
| import Mathlib | |
| import OAI.Probability.CriticalPercolation.Model | |
| import OAI.Probability.CriticalPercolation.Criticality.InterpolationUniform | |
| import OAI.Probability.CriticalPercolation.GraphGeometry.RestrictVertices | |
| import OAI.Probability.CriticalPercolation.Modular.FiniteClusterTransport | |
| namespace OAI | |
| section | |
| section | |
| noncomputable section | |
| open Set MeasureTheory ProbabilityTheory | |
| open scoped Classical | |
| namespace CriticalPercolation.BondGraph | |
| variable {V E : Type*} (G : BondGraph V E) | |
| theorem finite_edge_interpolation_measure [Fintype E] | |
| (d : ℕ) (x : ℕ → V) (hd : 2≤d) (p : unitInterval) | |
| (hp : (p : ℝ) ∈ Set.Ioo 0 1) {c ε : ℝ} (hc : 0<c) (hε : 0<ε) | |
| (hεp : ε≤p) (hεq : ε≤1-(p:ℝ)) | |
| (hconn : ∀ j k, j<k → k≤d → k-j<d → | |
| c ≤ (G.law p).real {ω | (G.openGraph ω).Reachable (x j) (x k)}) | |
| (s : ℕ) (hδ : 0 < (G.law p).real (G.twoClusterEvent (x 0) (x d) s)) : | |
| ∃ j<d, 0 < (G.law p).real (G.twoClusterEvent (x j) (x (j+1)) s) ∧ | |
| -Real.log ((G.law p).real (G.twoClusterEvent (x j) (x (j+1)) s)) ≤ | |
| interpolationConstant c ε * | |
| (Real.log (1/(G.law p).real (G.twoClusterEvent (x 0) (x d) s))+Nat.clog 2 d+1) := by | |
| let U : Set V := Set.range (fun j : Fin (d+1) => x j.val) ∪ | |
| ⋃ e : E, {v | v ∈ G.ends e} | |
| have hUf : U.Finite := (Set.finite_range _).union | |
| (Set.finite_iUnion (fun e => by | |
| convert (G.ends e).toFinset.finite_toSet using 1 | |
| ext v | |
| exact Sym2.mem_toFinset.symm)) | |
| let : Fintype U := hUf.fintype | |
| have hU : ∀ e v, v∈G.ends e → v∈U := by | |
| intro e v hv | |
| exact Or.inr (Set.mem_iUnion.mpr ⟨e,hv⟩) | |
| let R := G.restrictVertices U hU | |
| let y (j : ℕ) : U := ⟨x (min j d),Or.inl ⟨⟨min j d,by omega⟩,rfl⟩⟩ | |
| have hy (j : ℕ) (hj : j≤d) : (y j).val=x j := by simp only [y,min_eq_left hj] | |
| have hreach (j k : ℕ) (hj : j≤d) (hk : k≤d) : | |
| (R.law p).real {ω | (R.openGraph ω).Reachable (y j) (y k)} = | |
| (G.law p).real {ω | (G.openGraph ω).Reachable (x j) (x k)} := by | |
| change (G.law p).real _ = _ | |
| congr 1 | |
| ext ω | |
| exact (G.restrictVertices_reachable U hU ω (y j) (y k)).trans (by simp only [hy j hj,hy k hk,Set.mem_ofPred_eq]) | |
| have htwo (j k : ℕ) (hj : j≤d) (hk : k≤d) : | |
| (R.law p).real (R.twoClusterEvent (y j) (y k) s) = | |
| (G.law p).real (G.twoClusterEvent (x j) (x k) s) := by | |
| change (G.law p).real _ = _ | |
| congr 1 | |
| ext ω | |
| exact (G.restrictVertices_twoClusterEvent U hU ω (y j) (y k) s).trans | |
| (by rw [hy j hj,hy k hk]) | |
| have hconn' : ∀ j k, j<k → k≤d → k-j<d → | |
| c ≤ (R.law p).real {ω | (R.openGraph ω).Reachable (y j) (y k)} := by | |
| intro j k hj hk hjk | |
| rw [hreach j k (by omega) hk] | |
| exact hconn j k hj hk hjk | |
| have hδ' : 0 < (R.law p).real (R.twoClusterEvent (y 0) (y d) s) := by | |
| rwa [htwo 0 d (by omega) le_rfl] | |
| obtain ⟨j,hj,hpos,hcost⟩ := R.finite_interpolation_measure d y hd p hp hc hε hεp hεq hconn' s hδ' | |
| rw [htwo j (j+1) (by omega) (by omega)] at hpos hcost | |
| rw [htwo 0 d (by omega) le_rfl] at hcost | |
| exact ⟨j,hj,hpos,hcost⟩ | |
| end CriticalPercolation.BondGraph | |
| end | |
| end | |
| end | |
| end OAI | |