openAI-math-dataset / lean /OAI /Probability /CriticalPercolation /Gluing /SourceRelaySeparation.lean
| import Mathlib | |
| import OAI.Probability.CriticalPercolation.Gluing.RelayScales | |
| namespace OAI | |
| section | |
| section | |
| noncomputable section | |
| open Set Function MeasureTheory ProbabilityTheory | |
| namespace CriticalPercolation.BondGraph | |
| variable {V E : Type*} (G : BondGraph V E) | |
| lemma source_outside_relay_collar (t z : V → ℝ) (σ e ρ P : ℝ) (L R m i : ℕ) | |
| (he : 0<e) (hρ : 0 ≤ ρ) (hi : i ≤ L) (hPi : 1 ≤ P+i) | |
| (heL : (L+3)*e<1/100) (hr : (R+m+1)*ρ<e/10) | |
| (ht : ∀ ⦃u v⦄, G.fullGraph.Adj u v → |t u-t v| ≤ ρ) | |
| (hz : ∀ ⦃u v⦄, G.fullGraph.Adj u v → |z u-z v| ≤ ρ) | |
| {a : V} (ha : t a ≤ m*ρ-P) : | |
| a∉G.vertexCollar (corridorRelay t z σ e ρ L i) R := by | |
| intro hin | |
| obtain ⟨hρe,_,_,_,hlast⟩ := relay_coordinate_scale L R m e ρ he hρ heL hr | |
| have hs := G.vertexCollar_mono (corridorRelay_subset t z σ e ρ hρe L i hi) R hin | |
| have hb := (G.relay_collar_bounds t z σ e ρ hρ ht hz R i hs).1 | |
| have hi' : (i:ℝ) ≤ L := by exact_mod_cast hi | |
| have hile : (i+1)*e ≤ (L+1)*e := mul_le_mul_of_nonneg_right (by linarith) he.le | |
| have hb' := (abs_le.mp hb).1 | |
| nlinarith | |
| end CriticalPercolation.BondGraph | |
| end | |
| end | |
| end | |
| end OAI | |