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Prove that predecessors imply edges

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Signed-off-by: Danila Fedorin <danila.fedorin@gmail.com>
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Danila Fedorin 2024-05-09 23:18:51 -07:00
parent 41ada43047
commit e0248397b7
1 changed files with 6 additions and 1 deletions
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Language/Graphs.agda
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@ -6,7 +6,7 @@ open import Data.Fin as Fin using (Fin; suc; zero)
open import Data.Fin.Properties as FinProp using (suc-injective) open import Data.Fin.Properties as FinProp using (suc-injective)
open import Data.List as List using (List; []; _∷_) open import Data.List as List using (List; []; _∷_)
open import Data.List.Membership.Propositional as ListMem using () open import Data.List.Membership.Propositional as ListMem using ()
open import Data.List.Membership.Propositional.Properties as ListMemProp using (∈-filter⁺) open import Data.List.Membership.Propositional.Properties as ListMemProp using (∈-filter⁺; ∈-filter⁻)
open import Data.List.Relation.Unary.All using (All; []; _∷_) open import Data.List.Relation.Unary.All using (All; []; _∷_)
open import Data.List.Relation.Unary.Any as RelAny using () open import Data.List.Relation.Unary.Any as RelAny using ()
open import Data.Nat as Nat using (ℕ; suc) open import Data.Nat as Nat using (ℕ; suc)
@ -167,3 +167,8 @@ module _ (g : Graph) where
edge⇒predecessor {idx₁} {idx₂} idx₁,idx₂∈es = edge⇒predecessor {idx₁} {idx₂} idx₁,idx₂∈es =
∈-filter⁺ (λ idx' → (idx' , idx₂) ∈? (Graph.edges g)) ∈-filter⁺ (λ idx' → (idx' , idx₂) ∈? (Graph.edges g))
(indices-complete idx₁) idx₁,idx₂∈es (indices-complete idx₁) idx₁,idx₂∈es
predecessor⇒edge : ∀ {idx₁ idx₂ : Graph.Index g} → idx₁ ListMem.∈ (predecessors idx₂) →
(idx₁ , idx₂) ListMem.∈ (Graph.edges g)
predecessor⇒edge {idx₁} {idx₂} idx₁∈pred =
proj₂ (∈-filter⁻ (λ idx' → (idx' , idx₂) ∈? (Graph.edges g)) {v = idx₁} {xs = indices} idx₁∈pred )