| import Mathlib |
| import OAI.Probability.CriticalPercolation.Entropy.BinarySwitches |
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| namespace OAI |
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| section |
| section |
| namespace CriticalPercolation.BinaryLaw |
| open scoped Classical |
| open FiniteProduct BondGraph LazyDFS BinaryInterpolation InterpolationForest |
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| variable {V E : Type*} (G : BondGraph V E) |
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| def HasSide (L : List (V × Bool)) (ω : Set E) (b : Bool) (v : V) : Prop := |
| ∃ z ∈ L, z.2=b ∧ (G.openGraph ω).Reachable z.1 v |
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| lemma hasSide_mono_reachable (L : List (V × Bool)) (ω : Set E) (b : Bool) {u v : V} |
| (hu : HasSide G L ω b u) (hr : (G.openGraph ω).Reachable u v) : |
| HasSide G L ω b v := by |
| obtain ⟨z,hz,hb,hrz⟩ := hu |
| exact ⟨z,hz,hb,hrz.trans hr⟩ |
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| lemma hasSide_unique (L : List (V × Bool)) (ω : Set E) (hL : Consistent G L ω) |
| {b c : Bool} {v : V} (hb : HasSide G L ω b v) (hc : HasSide G L ω c v) : b=c := by |
| obtain ⟨z,hz,hzb,hzv⟩ := hb |
| obtain ⟨t,ht,htc,htv⟩ := hc |
| exact hzb.symm.trans ((hL z hz t ht (hzv.trans htv.symm)).trans htc) |
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|
| section Output |
| variable [Fintype V] [Fintype E] [DecidableEq V] [DecidableEq E] |
| variable (order : List (Dart V E)) (d : ℕ) (x : ℕ → V) |
| variable (p : ℝ) (w : E → Prop → ℝ) (r₀ : (E → Prop) → ℝ) |
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| lemma switched_side_adj (n : ℕ) (h : History E n) (ω : E → Prop) |
| (hr : density G order d x p w r₀ n h ω ≠ 0) |
| (hL : Consistent G (anchors d x n h) ω) |
| (S : Finset ℕ) (hS : ∀ i ∈ S, 2 ≤ i ∧ i < n+2) {b : Bool} {u v : V} |
| (hu : HasSide G (anchors d x n h) ω b u) |
| (huv : (G.openGraph (switched n h ω S)).Adj u v) : |
| HasSide G (anchors d x n h) ω b v := by |
| obtain ⟨⟨e,he,hev⟩,_⟩ := (SimpleGraph.fromEdgeSet_adj _).mp huv |
| rcases he with he|⟨i,hi,him⟩ |
| · exact hasSide_mono_reachable G _ ω b hu (prop_openEdge_reachable G ω he hev) |
| · obtain ⟨_,a,c,heac,ha,hc⟩ := node_mark_geometry G order d x p w r₀ n h ω hr i |
| (hS i hi).1 (hS i hi).2 e him |
| have hip := nodeParent_lt G d x ω n h i (hS i hi).1 |
| have hcol := node_colour_parent G order d x p w r₀ n h ω hr i (hS i hi).1 (hS i hi).2 |
| have hA : HasSide G (anchors d x n h) ω (node d x n h i).2 a := |
| ⟨_,node_mem d x n h i (hS i hi).2,rfl,ha⟩ |
| have hC : HasSide G (anchors d x n h) ω (node d x n h i).2 c := |
| ⟨_,node_mem d x n h _ (hip.trans (hS i hi).2),hcol.symm,hc⟩ |
| rcases Sym2.eq_iff.mp (hev.symm.trans heac) with ⟨hua,hvc⟩|⟨huc,hva⟩ |
| · subst u; subst v |
| exact (hasSide_unique G _ ω hL hu hA).symm ▸ hC |
| · subst u; subst v |
| exact (hasSide_unique G _ ω hL hu hC).symm ▸ hA |
|
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| lemma switched_side_reachable (n : ℕ) (h : History E n) (ω : E → Prop) |
| (hr : density G order d x p w r₀ n h ω ≠ 0) |
| (hL : Consistent G (anchors d x n h) ω) |
| (S : Finset ℕ) (hS : ∀ i ∈ S, 2 ≤ i ∧ i < n+2) {b : Bool} {u v : V} |
| (hu : HasSide G (anchors d x n h) ω b u) |
| (huv : (G.openGraph (switched n h ω S)).Reachable u v) : |
| HasSide G (anchors d x n h) ω b v := by |
| rw [SimpleGraph.reachable_iff_reflTransGen] at huv |
| induction huv with |
| | refl => exact hu |
| | tail _ hstep ih => exact switched_side_adj G order d x p w r₀ n h ω hr hL S hS ih hstep |
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| theorem switched_terminal_distinct (s n : ℕ) (h : History E n) (ω : E → Prop) |
| (hr : density G order d x p w (seed G w (x 0) (x d) s) n h ω ≠ 0) |
| (S : Finset ℕ) (hS : ∀ i ∈ S, 2 ≤ i ∧ i < n+2) : |
| ¬(G.openGraph (switched n h ω S)).Reachable |
| (x (interval d n h).1) (x (interval d n h).2) := by |
| intro hreach |
| have hL := density_consistent G order d x p w s n h ω hr |
| have he := endpoint_labels d x n h |
| have hl : HasSide G (anchors d x n h) ω true (x (interval d n h).1) := |
| ⟨_,he.1,rfl,.refl _⟩ |
| have hr' : HasSide G (anchors d x n h) ω false (x (interval d n h).2) := |
| ⟨_,he.2,rfl,.refl _⟩ |
| have hh := switched_side_reachable G order d x p w _ n h ω hr hL S hS hl hreach |
| cases hasSide_unique G _ ω hL hh hr' |
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| omit [Fintype V] [Fintype E] [DecidableEq V] [DecidableEq E] in |
| lemma active_index_bounds (d : ℕ) (x : ℕ → V) (n : ℕ) (h : History E n) (ω : Set E) : |
| ∀ i ∈ activeNodes G d x n h ω, 2 ≤ i ∧ i < n+2 := by |
| intro i hi |
| exact ⟨((historyForest G d x ω n h).mem_active.mp hi).2,active_lt G d x n h ω i hi⟩ |
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| lemma active_lineage_reachable (n : ℕ) (h : History E n) (ω : E → Prop) |
| (hr : density G order d x p w r₀ n h ω ≠ 0) |
| (u : ℕ) (hu : u = owner G d x n h ω (x (interval d n h).1) ∨ |
| u = owner G d x n h ω (x (interval d n h).2)) |
| (i : ℕ) (hi : i ∈ (historyForest G d x ω n h).lineage u) (hin : i < n+2) : |
| (G.openGraph (switched n h ω (activeNodes G d x n h ω))).Reachable |
| (node d x n h i).1 (node d x n h ((historyForest G d x ω n h).root i)).1 := by |
| induction i using Nat.strong_induction_on with |
| | h i ih => |
| by_cases hib : i < 2 |
| · rw [Forest.root_base _ hib] |
| · have hi2 : 2 ≤ i := by omega |
| have hia : i ∈ activeNodes G d x n h ω := by |
| apply (Forest.mem_active _).mpr |
| exact ⟨hu.elim (fun e => Or.inl (e ▸ hi)) (fun e => Or.inr (e ▸ hi)),hi2⟩ |
| have hm := active_nonhold G order d x p w r₀ n h ω hr i hia |
| obtain ⟨e,he⟩ := Option.ne_none_iff_exists'.mp hm |
| obtain ⟨_,a,b,heab,ha,hb⟩ := node_mark_geometry G order d x p w r₀ n h ω hr i hi2 hin e he |
| have hmon := G.openGraph_mono (switched_mono n h ω (activeNodes G d x n h ω)) |
| have heopen : switched n h ω (activeNodes G d x n h ω) e := Or.inr ⟨i,hia,he⟩ |
| have hparent := (ha.mono hmon).trans |
| ((prop_openEdge_reachable G _ heopen heab).trans (hb.symm.mono hmon)) |
| rw [Forest.root_step _ hi2] |
| exact hparent.trans (ih _ (nodeParent_lt G d x ω n h i hi2) |
| ((historyForest G d x ω n h).parent_mem_lineage hi2 hi) |
| ((nodeParent_lt G d x ω n h i hi2).trans hin)) |
|
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| theorem switched_terminal_bases (s n : ℕ) (h : History E n) (ω : E → Prop) |
| (hr : density G order d x p w (seed G w (x 0) (x d) s) n h ω ≠ 0) : |
| let η := switched n h ω (activeNodes G d x n h ω) |
| (G.openGraph η).Reachable (x (interval d n h).1) (x 0) ∧ |
| (G.openGraph η).Reachable (x (interval d n h).2) (x d) := by |
| have he := endpoint_labels d x n h |
| have hrts := terminal_roots G order d x p w s n h ω hr |
| have hb := node_bases d x n h |
| have hmon := G.openGraph_mono (switched_mono n h ω (activeNodes G d x n h ω)) |
| constructor |
| · have hanc := active_lineage_reachable G order d x p w _ n h ω hr _ (Or.inl rfl) _ |
| ((historyForest G d x ω n h).self_mem_lineage _) (owner_lt G d x n h ω _ ⟨true,he.1⟩) |
| rw [hrts.1,hb.1] at hanc |
| exact ((owner_reachable G d x n h ω _ ⟨true,he.1⟩).mono hmon).trans hanc |
| · have hanc := active_lineage_reachable G order d x p w _ n h ω hr _ (Or.inr rfl) _ |
| ((historyForest G d x ω n h).self_mem_lineage _) (owner_lt G d x n h ω _ ⟨false,he.2⟩) |
| rw [hrts.2,hb.2] at hanc |
| exact ((owner_reachable G d x n h ω _ ⟨false,he.2⟩).mono hmon).trans hanc |
|
|
| lemma density_seed_support (n : ℕ) (h : History E n) (ω : E → Prop) |
| (hr : density G order d x p w r₀ n h ω ≠ 0) : r₀ ω ≠ 0 := by |
| induction n with |
| | zero => exact hr |
| | succ n ih => exact ih h.1 (density_step_support G order d x p w r₀ n h ω hr).1 |
|
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| theorem switched_endpointEvent (s n : ℕ) (h : History E n) (ω : E → Prop) |
| (hr : density G order d x p w (seed G w (x 0) (x d) s) n h ω ≠ 0) : |
| endpointEvent G (x (interval d n h).1) (x (interval d n h).2) s |
| (switched n h ω (activeNodes G d x n h ω)) := by |
| have hseed := density_seed_support G order d x p w _ n h ω hr |
| have ho : endpointEvent G (x 0) (x d) s ω := by |
| by_contra ho |
| simp only [seed,ite_eq_right ho,mul_zero] at hseed |
| exact hseed rfl |
| have hb := switched_terminal_bases G order d x p w s n h ω hr |
| have hmon := G.openGraph_mono (switched_mono n h ω (activeNodes G d x n h ω)) |
| refine ⟨switched_terminal_distinct G order d x p w s n h ω hr _ |
| (active_index_bounds G d x n h ω),?_,?_⟩ |
| · apply ho.2.1.trans |
| apply Finset.card_le_card |
| intro v hv |
| have hv' : (G.openGraph ω).Reachable (x 0) v := (Set.mem_toFinset (s := G.cluster ω (x 0)) (a := v)).mp hv |
| exact (Set.mem_toFinset (s := G.cluster (switched n h ω (activeNodes G d x n h ω)) (x (interval d n h).1)) (a := v)).mpr (hb.1.trans (hv'.mono hmon)) |
| · apply ho.2.2.trans |
| apply Finset.card_le_card |
| intro v hv |
| have hv' : (G.openGraph ω).Reachable (x d) v := (Set.mem_toFinset (s := G.cluster ω (x d)) (a := v)).mp hv |
| exact (Set.mem_toFinset (s := G.cluster (switched n h ω (activeNodes G d x n h ω)) (x (interval d n h).2)) (a := v)).mpr (hb.2.trans (hv'.mono hmon)) |
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|
| end Output |
| end CriticalPercolation.BinaryLaw |
| end |
| end |
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| end OAI |
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