openAI-math-dataset / lean /OAI /Probability /CriticalPercolation /Clusters /ThreeClusterAvoidanceSecond.lean
| import Mathlib | |
| import OAI.Probability.CriticalPercolation.Model | |
| import OAI.Probability.CriticalPercolation.GraphGeometry.IsolateThree | |
| import OAI.Probability.CriticalPercolation.ProductMeasure.ProbDisjoint | |
| namespace OAI | |
| section | |
| section | |
| open Set MeasureTheory ProbabilityTheory | |
| open Set | |
| open scoped BigOperators | |
| namespace CriticalPercolation.BondGraph | |
| open FiniteProduct | |
| open scoped Classical | |
| variable {V E : Type*} (G : BondGraph V E) [Fintype V] [Fintype E] | |
| lemma threeCluster_avoidance_second (w : E → Prop → ℝ) | |
| (hw : ∀ e, ∑ z, w e z = 1) (hp : ∀ e z, 0 ≤ w e z) (hc : ∀ e, 0 < w e False) | |
| (a s t b : V) (hat : t ≠ a) (hst : t ≠ s) : | |
| prob w (fun ω => G.joinedPair t a s ω ∧ (G.openGraph ω).Reachable t b) * | |
| prob w (G.separateThree a s t) ≤ | |
| prob w (G.joinedPair t a s) * | |
| prob w (fun ω => G.separateThree a s t ω ∧ (G.openGraph ω).Reachable t b) := by | |
| have h := G.threeCluster_avoidance w hw hp hc t a s b hat hst | |
| rw [prob_congr w (G.separateThree_rotate a s t), | |
| prob_congr w (fun ω => and_congr_left (fun _ => G.separateThree_rotate a s t ω))] at h | |
| exact h | |
| omit [Fintype V] in | |
| lemma threeCluster_event_bounds (w : E → Prop → ℝ) (hp : ∀ e z, 0 ≤ w e z) | |
| (a s t b : V) : | |
| prob w (fun ω => G.separateThree a s t ω ∧ (G.openGraph ω).Reachable t b) ≤ | |
| prob w (G.separateThree a s t) ∧ | |
| prob w (fun ω => G.separateThree a s t ω ∧ (G.openGraph ω).Reachable t b) ≤ | |
| prob w (G.separateThree a s t) + prob w (G.joinedPair a s t) - | |
| prob w (fun ω => G.separateThree a s t ω ∧ (G.openGraph ω).Reachable a b) - | |
| prob w (fun ω => G.joinedPair a s t ω ∧ (G.openGraph ω).Reachable a b) := by | |
| constructor | |
| · exact prob_mono w hp (fun _ h => h.1) | |
| · have h1 := prob_disjoint_le w hp | |
| (fun ω => G.separateThree a s t ω ∧ (G.openGraph ω).Reachable t b) | |
| (fun ω => G.separateThree a s t ω ∧ (G.openGraph ω).Reachable a b) | |
| (G.separateThree a s t) (fun _ h => h.1) (fun _ h => h.1) | |
| (fun _ h => h.1.1.1.2 (h.2.2.trans h.1.2.symm)) | |
| have h2 := prob_mono w hp (P := fun ω => G.joinedPair a s t ω ∧ (G.openGraph ω).Reachable a b) | |
| (Q := G.joinedPair a s t) (fun _ h => h.1) | |
| linarith | |
| end CriticalPercolation.BondGraph | |
| end | |
| end | |
| end OAI | |