openAI-math-dataset / lean /OAI /Probability /CriticalPercolation /Exploration /ReachableMemPhaseClosed.lean
| import Mathlib | |
| import OAI.Probability.CriticalPercolation.Exploration.PhaseFrontier | |
| namespace OAI | |
| section | |
| section | |
| open Set MeasureTheory ProbabilityTheory | |
| open Set | |
| open scoped BigOperators | |
| open scoped Classical | |
| namespace CriticalPercolation.LazyDFS | |
| open scoped Classical | |
| variable {V E : Type*} [DecidableEq V] [DecidableEq E] | |
| omit [DecidableEq V] [DecidableEq E] in | |
| lemma reachable_mem_of_phase_closed (H : SimpleGraph V) (A : Set V) (S : Finset V) | |
| (hc : ∀ v ∈ S, v ∉ A → ∀ t, H.Adj v t → t ∈ S) {z t : V} | |
| (ha : ∀ a ∈ A, ¬H.Reachable z a) (ht : H.Reachable z t) (hz : z ∈ S) : t ∈ S := by | |
| have hh : ∀ {u v : V}, H.Walk u v → H.Reachable z u → u ∈ S → v ∈ S := by | |
| intro u v path | |
| induction path with | |
| | nil => exact fun _ h => h | |
| | @cons u v y h path ih => | |
| intro hzu hu | |
| exact ih (hzu.trans h.reachable) (hc u hu (fun hA => ha u hA hzu) v h) | |
| exact ht.elim (fun path => hh path (.refl _) hz) | |
| lemma opened_phaseFrontier (G : BondGraph V E) (order : List (Dart V E)) | |
| (ho : Exhaustive G order) (st : State V E) (f : E) (z : V) : | |
| PhaseFrontier G (st.reached : Set V) (st.opened f z) (push G order z) := by | |
| intro v hv hvA e t het _ | |
| rcases Finset.mem_insert.mp hv with rfl | hv | |
| · exact Or.inr (List.mem_filter.mpr | |
| ⟨ho v e t het,by simpa only [decide_eq_true_eq,true_and] using het⟩) | |
| · exact (hvA hv).elim | |
| variable [Fintype V] | |
| theorem phase_hits_of_avoiding_connection (G : BondGraph V E) (A : Set V) | |
| (order : List (Dart V E)) (ho : Exhaustive G order) (B : Set V) | |
| (fuel : ℕ) (st : State V E) (young : List (Dart V E)) (ω : E → Bool) | |
| (hf : PhaseFrontier G A st young) (hc : ReadConsistent G st ω) | |
| (hr : rank order st young ≤ fuel) (H : SimpleGraph V) | |
| (hH : H ≤ G.openGraph {e | ω e = true}) (z : V) (hz : z ∈ st.reached) | |
| (ha : ∀ a ∈ A, ¬H.Reachable z a) (ht : ∃ t ∈ B, H.Reachable z t) : | |
| ∃ t ∈ ((compile G order B fuel st young).run ω).reached, t ∈ B := by | |
| rcases phase_success_or_closed G A order ho B fuel st young ω hf hc hr with h | h | |
| · exact h | |
| · obtain ⟨t,htB,hzt⟩ := ht | |
| have hclosed : ∀ v ∈ ((compile G order B fuel st young).run ω).reached, | |
| v ∉ A → ∀ u, H.Adj v u → u ∈ ((compile G order B fuel st young).run ω).reached := by | |
| intro v hv hvA u hvu | |
| exact h v hv hvA u (hH hvu) | |
| exact ⟨t,reachable_mem_of_phase_closed H A _ hclosed ha hzt | |
| ((compile_mono G order B fuel st young ω).1 hz),htB⟩ | |
| theorem phase_no_older_reads (G : BondGraph V E) (A : Set V) | |
| (order : List (Dart V E)) (ho : Exhaustive G order) (B : Set V) | |
| (fuel : ℕ) (st : State V E) (young older : List (Dart V E)) (ω : E → Bool) | |
| (hf : PhaseFrontier G A st young) (hc : ReadConsistent G st ω) | |
| (hr : rank order st young ≤ fuel) (H : SimpleGraph V) | |
| (hH : H ≤ G.openGraph {e | ω e = true}) (z : V) (hz : z ∈ st.reached) | |
| (ha : ∀ a ∈ A, ¬H.Reachable z a) (ht : ∃ t ∈ B, H.Reachable z t) : | |
| (compile G order B fuel st (young++older)).reads ω = | |
| (compile G order B fuel st young).reads ω := | |
| (compile_append_of_success G order B fuel st young older ω | |
| (phase_hits_of_avoiding_connection G A order ho B fuel st young ω hf hc hr H hH z hz ha ht)).2 | |
| end CriticalPercolation.LazyDFS | |
| end | |
| end | |
| end OAI | |