| import OAI.Combinatorics.ContingencyTables.Rooted.ComputedLargeForest |
|
|
| namespace OAI |
|
|
| /-! |
|
|
| namespace ContingencyTables.ComputedLargeForest |
| open SimpleGraph MaximumForest ForestCapacity ForestGreedySemantics |
| open FiniteGraphRootProgram LargeEdgeCatalogProgram |
| open scoped Classical |
|
|
| variable {m n : ℕ} |
|
|
| private lemma code_congr {N : ℕ} (G H : SimpleGraph (Fin N)) |
| [DecidableRel G.Adj] [DecidableRel H.Adj] (h : G=H) : graphCode G=graphCode H := by |
| subst H |
| apply congrArg (@graphCode N G) |
| exact Subsingleton.elim _ _ |
|
|
| lemma computed_code (r : Fin m→ℕ) (c : Fin n→ℕ) (b : Fin m→Fin n→ℕ) |
| (W : ℕ) (hW : 1 ≤ W) : |
| graphCode ((computed r c b W).comap finSumFinEquiv.symm)= |
| ForestGreedyProgram.forest (NetworkVertexCode.data r c b,W) := by |
| obtain ⟨F,hF,_⟩ := iterate_optimal (graph b W) ⊥ (capacities b) |
| (catalog (NetworkVertexCode.data r c b,W)) (catalog_correct r c b W) |
| (capacity_positive b W hW) (MaximumForestExchange.bottom_extendible _ _) |
| ((m+n)*(m+n)) (by |
| have hb := unordered_pair_budget (n:=m+n) |
| omega) |
| have hzero (d : DecidableRel (⊥ : SimpleGraph (Fin (m+n))).Adj) : |
| @graphCode (m+n) ⊥ d=ForestGreedyProgram.emptyGraph (NetworkVertexCode.data r c b,W) := by |
| symm |
| simp only [ForestGreedyProgram.emptyGraph,vertexCount_code,DirectedBFSCode.range_eq_map, |
| ←List.ofFn_eq_map,List.map_ofFn,graphCode] |
| apply congrArg List.ofFn |
| funext v |
| simp [adjacencyRow] |
| have hout : ForestGreedyProgram.forest (NetworkVertexCode.data r c b,W)=graphCode F := by |
| unfold ForestGreedyProgram.forest ForestGreedyProgram.prepare |
| rw [←hzero (fun a b => Classical.propDecidable _),execute_iterate,graphCode_length] |
| exact congrArg Prod.fst hF |
| have hc : (computed r c b W).comap finSumFinEquiv.symm= |
| decoded (ForestGreedyProgram.forest (NetworkVertexCode.data r c b,W)) := by |
| ext u v |
| change (decoded (ForestGreedyProgram.forest (NetworkVertexCode.data r c b,W))).Adj |
| (vertexEquiv (finSumFinEquiv.symm u)) (vertexEquiv (finSumFinEquiv.symm v)) ↔ _ |
| simp only [vertexEquiv,Equiv.apply_symm_apply] |
| have he : (computed r c b W).comap finSumFinEquiv.symm=F := by |
| simpa only [hout,decoded_graphCode] using hc |
| exact (code_congr _ _ he).trans hout.symm |
|
|
| end ContingencyTables.ComputedLargeForest |
|
|
| end OAI |
|
|
