| import Mathlib |
| import OAI.Probability.CriticalPercolation.Gluing.FreshCollar |
| import OAI.Probability.CriticalPercolation.Gluing.SourceGraph |
|
|
| namespace OAI |
|
|
| section |
| section |
| noncomputable section |
| open Set Function MeasureTheory ProbabilityTheory |
| open scoped Classical BigOperators ENNReal |
|
|
| namespace CriticalPercolation.BondGraph |
| open BernoulliProduct |
|
|
| variable {V E : Type*} [Countable V] [Countable E] (G : BondGraph V E) |
|
|
| omit [Countable V] [Countable E] in |
| lemma withSource_edge_cases (A : Set V) (e : E⊕A) (v w : Option V) |
| (he : (G.withSource A).ends e=s(v,w)) : |
| (∃ b x y, e=Sum.inl b ∧ v=some x ∧ w=some y ∧ G.ends b=s(x,y)) ∨ |
| (∃ a : A, e=Sum.inr a ∧ ((v=none ∧ w=some a.val) ∨ (w=none ∧ v=some a.val))) := by |
| cases e with |
| | inl e => |
| obtain ⟨⟨x,y⟩,hxy⟩ := Sym2.mk_surjective (G.ends e) |
| change s(x,y)=G.ends e at hxy |
| have hh : s(some x,some y)=s(v,w) := by simpa [withSource,← hxy] using he |
| rcases Sym2.eq_iff.mp hh with hh|hh |
| · exact Or.inl ⟨e,x,y,rfl,hh.1.symm,hh.2.symm,hxy.symm⟩ |
| · exact Or.inl ⟨e,y,x,rfl,hh.2.symm,hh.1.symm,by simpa only [Sym2.eq_swap] using hxy.symm⟩ |
| | inr a => |
| right |
| refine ⟨a,rfl,?_⟩ |
| have hh : s(none,some a.val)=s(v,w) := he |
| rcases Sym2.eq_iff.mp hh with hh|hh |
| · exact Or.inl ⟨hh.1.symm,hh.2.symm⟩ |
| · exact Or.inr ⟨hh.1.symm,hh.2.symm⟩ |
|
|
| theorem withSource_transfer_from_geometry (hlocal : G.LocallyFinite) |
| (A Input T : Set V) (hA : A.Finite) (q : (E⊕A) → unitInterval) |
| (π : V → ℤ) (H : ℤ) (J : Finset ℤ) (hJ : J.Nonempty) |
| (hH : ∀ j∈J, j<H) (hAlevel : ∀ a∈A, H ≤ π a) |
| (hInput : ∀ j∈J, ∀ v∈Input, π v<j) |
| (hcross : ∀ j∈J, ∀ e v w, G.ends e=s(v,w) → j ≤ π v → π w<j → π v=j) |
| (p : unitInterval) (hp : (p : ℝ)<1) |
| (hq : ∀ j∈J, ∀ e v w, G.ends e=s(v,w) → π v=j → π w<j → (q (Sum.inl e) : ℝ) ≤ p) |
| (D k M N : ℕ) (hD : ∀ j∈J, ∀ v, π v=j → (G.incidentEdges {v}).ncard ≤ D) |
| (hM : 0 < M) (hk : M*N ≤ k) |
| (C : V → V → Prop) (hrefl : ∀ v, C v v) (hsymm : ∀ {v w}, C v w → C w v) |
| (hfinite : ∀ v, {w | C v w}.Finite) (hbound : ∀ v, {w | C v w}.ncard ≤ M) |
| (F : ℤ → V → Set E) (B : ℤ → V → Set V) |
| (hF : ∀ j∈J, ∀ v, F j v ⊆ (G.inducedEdges {w | j ≤ π w})ᶜ) |
| (hconnect : ∀ j∈J, ∀ v, π v=j → ∀ b∈B j v, (G.openGraph (F j v)).Reachable v b) |
| (hdis : ∀ j∈J, ∀ v w, ¬C v w → Disjoint (F j v) (F j w)) |
| (s₀ K η lam : ℝ) (hK : 0 ≤ K) (hη : 0 ≤ η) (hlam : 0 < lam) |
| (hs : ∀ j∈J, ∀ v, π v=j → s₀ ≤ (BernoulliProduct.law (fun e => q (Sum.inl e))).real (allOpen (F j v))) |
| (hN : K ≤ N*s₀) |
| (hmean : ∀ j∈J, ∀ v, π v=j → |
| 1-η ≤ (BernoulliProduct.law (fun e => q (Sum.inl e))).real (G.seedHitEvent (B j v) T)) : |
| (BernoulliProduct.law q).real ((G.withSource A).hitEvent none (some '' Input)\ |
| (G.withSource A).hitEvent none (some '' T)) ≤ |
| 1/((J.card : ℝ)*(1-(p : ℝ))^(D*k))+Real.exp (-K)+(K+1)*η/lam+lam := by |
| let π' : Option V → ℤ := Option.elim' H π |
| let C' : Option V → Option V → Prop := fun v w => match v,w with |
| | none,none => True |
| | some v,some w => C v w |
| | _,_ => False |
| let F' : ℤ → Option V → Set (E⊕A) := fun j => Option.elim' ∅ (fun v => Sum.inl '' F j v) |
| let B' : ℤ → Option V → Set (Option V) := fun j => Option.elim' ∅ (fun v => some '' B j v) |
| have hvA (j : ℤ) (hj : j∈J) (v : V) (hv : π v=j) : v∉A := by |
| intro hva |
| have := hAlevel v hva |
| have := hH j hj |
| omega |
| have hcross' (j : ℤ) (hj : j∈J) (e : E⊕A) (v w : Option V) |
| (he : (G.withSource A).ends e=s(v,w)) (hv : j ≤ π' v) (hw : π' w<j) : π' v=j := by |
| rcases G.withSource_edge_cases A e v w he with ⟨b,x,y,rfl,rfl,rfl,hb⟩|⟨a,rfl,hh⟩ |
| · exact hcross j hj b x y hb hv hw |
| · obtain hh|hh := hh |
| · obtain ⟨rfl,rfl⟩ := hh |
| have := hAlevel a a.property |
| have := hH j hj |
| change π a<j at hw |
| omega |
| · obtain ⟨rfl,rfl⟩ := hh |
| change H<j at hw |
| have := hH j hj |
| omega |
| have hs' (j : ℤ) (hj : j∈J) (v : V) (hv : π v=j) : |
| s₀ ≤ (BernoulliProduct.law q).real (allOpen (Sum.inl '' F j v)) := by |
| have hm : Measurable (fun ω : Set (E⊕A) => Sum.inl ⁻¹' ω : Set (E⊕A) → Set E) := |
| Measurable.of_eval (fun edge => measurable_set_mem (Sum.inl edge)) |
| have lawproj := BernoulliProduct.law_map_preimage q Sum.inl Sum.inl_injective |
| have heq : allOpen (Sum.inl '' F j v)=(fun ω : Set (E⊕A) => Sum.inl ⁻¹' ω) ⁻¹' allOpen (F j v) := by |
| ext ω |
| simp only [allOpen,mem_ofPred_eq,mem_preimage] |
| exact Set.image_subset_iff |
| rw [heq,measureReal_def,← Measure.map_apply hm ((blockSigma_le _) _ (measurable_allOpen_block (F j v))),lawproj] |
| exact hs j hj v hv |
| apply (G.withSource A).fresh_collar_transfer_from_geometry (G.withSource_local hlocal hA) |
| q π' none (some '' Input) (some '' T) J hJ (fun j hj => (hH j hj).le) |
| (by rintro j hj _ ⟨v,hv,rfl⟩; exact hInput j hj v hv) hcross' p hp |
| (by |
| intro j hj e v w he hv hw |
| rcases G.withSource_edge_cases A e v w he with ⟨b,x,y,rfl,rfl,rfl,hb⟩|⟨a,rfl,hh⟩ |
| · exact hq j hj b x y hb hv hw |
| · obtain hh|hh := hh |
| · obtain ⟨rfl,rfl⟩ := hh |
| change H=j at hv |
| have := hH j hj |
| omega |
| · obtain ⟨rfl,rfl⟩ := hh |
| change H<j at hw |
| have := hH j hj |
| omega) |
| D k M N |
| (by |
| intro j hj v hv |
| cases v with |
| | none => have := hH j hj; change H=j at hv; omega |
| | some v => |
| rw [G.withSource_incident A (hvA j hj v hv),Set.ncard_image_of_injective _ Sum.inl_injective] |
| exact hD j hj v hv) |
| hM hk C' |
| (by intro v; cases v; trivial; exact hrefl _) |
| (by intro v w h; cases v <;> cases w <;> simp_all only [C']) |
| (by |
| intro v |
| cases v with |
| | none => |
| have he : {w | C' none w}={none} := by ext w; cases w <;> simp [C'] |
| rw [he] |
| exact Set.finite_singleton _ |
| | some v => |
| have he : {w | C' (some v) w}=some '' {w | C v w} := by ext w; cases w <;> simp [C'] |
| rw [he] |
| exact (hfinite v).image _) |
| (by |
| intro v |
| cases v with |
| | none => |
| have he : {w | C' none w}={none} := by ext w; cases w <;> simp [C'] |
| rw [he,Set.ncard_singleton] |
| omega |
| | some v => |
| have he : {w | C' (some v) w}=some '' {w | C v w} := by ext w; cases w <;> simp [C'] |
| rw [he,Set.ncard_image_of_injective _ (Option.some_injective V)] |
| exact hbound v) |
| F' B' |
| (by |
| intro j hj v |
| cases v with |
| | none => exact Set.empty_subset _ |
| | some v => |
| rintro e ⟨b,hb,rfl⟩ he |
| apply hF j hj v hb |
| intro z hz |
| exact he (some z) (by simpa [withSource] using hz)) |
| (by |
| intro j hj v hv b hb |
| cases v with |
| | none => have := hH j hj; change H=j at hv; omega |
| | some v => |
| obtain ⟨z,hz,rfl⟩ := hb |
| exact G.withSource_trial_reachable A (F j v) (hconnect j hj v hv z hz)) |
| (by |
| intro j hj v w h |
| cases v <;> cases w |
| · simp [F'] |
| · simp [F'] |
| · simp [F'] |
| · exact Set.disjoint_image_of_injective Sum.inl_injective (hdis j hj _ _ h)) |
| s₀ K η lam hK hη hlam |
| (by |
| intro j hj v hv |
| cases v with |
| | none => have := hH j hj; change H=j at hv; omega |
| | some v => exact hs' j hj v hv) |
| hN |
| (by |
| intro j hj v hv |
| cases v with |
| | none => have := hH j hj; change H=j at hv; omega |
| | some v => exact (hmean j hj v hv).trans (G.withSource_seedHit A (B j v) T q)) |
|
|
| end CriticalPercolation.BondGraph |
| end |
| end |
| end |
|
|
| end OAI |
|
|
