openAI-math-dataset / lean /OAI /Probability /CriticalPercolation /Clusters /ThreeClusterAvoidanceGeneral.lean
| import Mathlib | |
| import OAI.Probability.CriticalPercolation.Model | |
| import OAI.Probability.CriticalPercolation.Clusters.ThreeClusterAvoidanceSecond | |
| import OAI.Probability.CriticalPercolation.Exploration.PivotalComparison | |
| import OAI.Probability.CriticalPercolation.GraphGeometry.IsolateThree | |
| import OAI.Probability.CriticalPercolation.ProductMeasure.ClosedThin | |
| 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_general (w : E → Prop → ℝ) | |
| (hw : ∀ e, ∑ z, w e z = 1) (hp : ∀ e z, 0 ≤ w e z) | |
| (a s t b : V) (has : a ≠ s) (hat : a ≠ t) : | |
| prob w (fun ω => G.joinedPair a s t ω ∧ (G.openGraph ω).Reachable a 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_mul_le_of_strict w hw hp _ _ _ _ (fun w' hw' hp' hc' => | |
| G.threeCluster_avoidance w' hw' hp' hc' a s t b has hat) | |
| lemma threeCluster_avoidance_second_general (w : E → Prop → ℝ) | |
| (hw : ∀ e, ∑ z, w e z = 1) (hp : ∀ e z, 0 ≤ w e z) | |
| (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_general w hw hp 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 | |
| theorem residual_converted_variance_general (w : E → Prop → ℝ) | |
| (hw : ∀ e, ∑ z, w e z = 1) (hp : ∀ e z, 0 ≤ w e z) | |
| (a s t b : V) (has : a ≠ s) (hat : a ≠ t) (hst : s ≠ t) | |
| {p : ℝ} (hparam : p ∈ Set.Ioo 0 1) : | |
| let u := prob w (G.separateThree a s t) | |
| let v := prob w (G.joinedPair a s t) | |
| let α := prob w (fun ω => G.separateThree a s t ω ∧ (G.openGraph ω).Reachable t b) | |
| let β := prob w (fun ω => G.separateThree a s t ω ∧ (G.openGraph ω).Reachable a b) | |
| let γ := prob w (fun ω => G.joinedPair a s t ω ∧ (G.openGraph ω).Reachable a b) | |
| let σ := prob w (fun ω => G.joinedPair t a s ω ∧ (G.openGraph ω).Reachable t b) | |
| p*(1-p)*α*(α+σ) ≤ | |
| p*(1-p)*(u+v)*(u+v-β-γ)/((1-p)*(u+v)+p*(u+v-β-γ)) * | |
| ((v+α-γ)/(u+v-β-γ)-v/(u+v))^2 := by | |
| dsimp only | |
| have h1 := G.threeCluster_avoidance_general w hw hp a s t b has hat | |
| have h2 := G.threeCluster_avoidance_second_general w hw hp a s t b hat.symm hst.symm | |
| have h3 := G.threeCluster_harris w hw hp a s t | |
| have h4 := G.threeCluster_event_bounds w hp a s t b | |
| apply DFS.converted_variance hparam (prob_nonneg w hp _) (prob_nonneg w hp _) | |
| (prob_nonneg w hp _) (prob_nonneg w hp _) (prob_nonneg w hp _) (prob_nonneg w hp _) | |
| h4.1 h4.2 _ h2 h3 | |
| simpa only [mul_comm] using h1 | |
| end CriticalPercolation.BondGraph | |
| end | |
| end | |
| end OAI | |