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d5b7741 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 | import OAI.Combinatorics.ContingencyTables.Rooted.ComputedLargeForest
namespace OAI
/-! # The computed forest retains its literal adjacency-list code -/
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
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