openAI-math-dataset / lean /OAI /Probability /CriticalPercolation /Exploration /GrowthHistoryGeometry.lean
| import Mathlib | |
| import OAI.Probability.CriticalPercolation.GraphGeometry.LatticeGrowth | |
| import OAI.Probability.CriticalPercolation.Gluing.CorridorGraph | |
| namespace OAI | |
| section | |
| section | |
| noncomputable section | |
| open Set Function | |
| open scoped Classical | |
| namespace CriticalPercolation.Corridor | |
| open BondGraph Boundary | |
| variable {V E : Type*} (G : BondGraph V E) (ξ : V → Point) (j h : ℝ) | |
| def revealed (S : GrowthState) : Set E := | |
| boxEdges G ξ j h (0,0)∪⋃ e∈S.seen, corridorEdges G ξ j h e | |
| lemma revealed_mono {S T : GrowthState} (hST : S.seen⊆T.seen) : | |
| revealed G ξ j h S⊆revealed G ξ j h T := by | |
| apply Set.union_subset_union_right | |
| intro b hb | |
| obtain ⟨e,he,hb⟩ := Set.mem_iUnion₂.mp hb | |
| exact Set.mem_iUnion₂.mpr ⟨e,hST he,hb⟩ | |
| lemma boxEdges_subset_corridor_source (e : Edge) : | |
| boxEdges G ξ j h (source e)⊆corridorEdges G ξ j h e := by | |
| intro b hb v hv | |
| exact Or.inl (Or.inl (hb v hv)) | |
| lemma boxEdges_subset_corridor_target (e : Edge) : | |
| boxEdges G ξ j h (target e)⊆corridorEdges G ξ j h e := by | |
| intro b hb v hv | |
| exact Or.inl (Or.inr (hb v hv)) | |
| lemma good_box_revealed {S : GrowthState} (hs : S.Covered) {v : Site} (hv : v∈S.good) : | |
| boxEdges G ξ j h v⊆revealed G ξ j h S := by | |
| rcases hs v hv with rfl|⟨e,he,rfl⟩ | |
| · exact Set.subset_union_left | |
| · intro b hb | |
| exact Or.inr (Set.mem_iUnion₂.mpr ⟨e,he,boxEdges_subset_corridor_target G ξ j h e hb⟩) | |
| lemma revealed_initial : revealed G ξ j h GrowthState.initial=boxEdges G ξ j h (0,0) := by | |
| simp [revealed,GrowthState.initial] | |
| lemma revealed_advance (n : ℕ) (S : GrowthState) (b : Bool) : | |
| revealed G ξ j h (S.advance n b)=revealed G ξ j h S∪ | |
| ⋃ e∈(incoming n S.good).toFinset, corridorEdges G ξ j h e := by | |
| ext e | |
| constructor | |
| · rintro (he|he) | |
| · exact Or.inl (Or.inl he) | |
| · obtain ⟨f,hf,he⟩ := Set.mem_iUnion₂.mp he | |
| rcases Finset.mem_union.mp hf with hf|hf | |
| · exact Or.inl (Or.inr (Set.mem_iUnion₂.mpr ⟨f,hf,he⟩)) | |
| · exact Or.inr (Set.mem_iUnion₂.mpr ⟨f,hf,he⟩) | |
| · rintro ((he|he)|he) | |
| · exact Or.inl he | |
| · obtain ⟨f,hf,he⟩ := Set.mem_iUnion₂.mp he | |
| exact Or.inr (Set.mem_iUnion₂.mpr ⟨f,Finset.mem_union.mpr (Or.inl hf),he⟩) | |
| · obtain ⟨f,hf,he⟩ := Set.mem_iUnion₂.mp he | |
| exact Or.inr (Set.mem_iUnion₂.mpr ⟨f,Finset.mem_union.mpr (Or.inr hf),he⟩) | |
| lemma revealed_incoming (n : ℕ) (S : GrowthState) (b : Bool) {e : Edge} | |
| (he : incoming n S.good=some e) : | |
| revealed G ξ j h (S.advance n b)=revealed G ξ j h S∪corridorEdges G ξ j h e := by | |
| rw [revealed_advance,he] | |
| simp | |
| lemma later_revealed_intersection (hh : 0≤h) (hhj : h≤j) (hsep : 2*j+h<1) | |
| (n : ℕ) (S T : GrowthState) (hc : S.Covered) (hv : T.Valid n) | |
| {e : Edge} (hp : source e∈S.good) (ht : n ≤ gridSlot (target e)) : | |
| revealed G ξ j h T∩corridorEdges G ξ j h e⊆revealed G ξ j h S := by | |
| intro b hb | |
| rcases hb.1 with hb0|hbT | |
| · exact Or.inl hb0 | |
| · have hh' := T.seen_intersection n hv ht | |
| have hh'' := history_intersection_subset G ξ j h hh hhj hsep (↑T.seen) hh'.1 hh'.2 | |
| exact good_box_revealed G ξ j h hc hp (hh'' ⟨hbT,hb.2⟩) | |
| lemma outgoing_unrevealed (hh : 0≤h) (hhj : h≤j) (hsep : 2*j+h<1) | |
| (n : ℕ) (S : GrowthState) (hv : S.Valid n) (hn : 0<n) | |
| {e : Edge} (he : source e=slotSite n) : | |
| Disjoint (revealed G ξ j h S) (corridorEdges G ξ j h e) := by | |
| apply Disjoint.sup_left | |
| · apply edge_box_disjoint G ξ j h hh hhj hsep | |
| intro hroot | |
| have he1 := parent_slot_lt e (he ▸ slotSite_quadrant n) | |
| have he2 : gridSlot (source e)=n := by rw [he,gridSlot_slotSite] | |
| rcases hroot with hh|hh | |
| · have he0 : gridSlot (source e)=0 := by rw [← hh]; rfl | |
| omega | |
| · have he0 : gridSlot (target e)=0 := by rw [← hh]; rfl | |
| omega | |
| · exact history_disjoint G ξ j h hh hhj hsep (↑S.seen) | |
| (fun f hf => S.outgoing_nonincident n hv he hf) | |
| end CriticalPercolation.Corridor | |
| end | |
| end | |
| end | |
| end OAI | |