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{-# OPTIONS --without-K --safe #-}
open import Level using (Level; 0ℓ)
open import Data.Hypergraph.Label using (HypergraphLabel)
module Data.Hypergraph {c ℓ : Level} (HL : HypergraphLabel) where
import Data.List.Relation.Binary.Pointwise as PW
import Function.Reasoning as →-Reasoning
import Relation.Binary.PropositionalEquality as ≡
import Data.Hypergraph.Edge HL as Hyperedge
import Data.List.Relation.Binary.Permutation.Propositional as List-↭
open import Data.List using (List; map)
open import Data.List.Relation.Binary.Permutation.Propositional using (_↭_; ↭-refl; ↭-sym; ↭-trans)
open import Data.Nat using (ℕ)
open import Data.String using (String; unlines)
open import Function using (_∘_; _⟶ₛ_; Func)
open import Relation.Binary using (Setoid)
module Edge = Hyperedge
open Edge using (Edge; Edgeₛ)
-- A hypergraph is a list of edges
record Hypergraph (v : ℕ) : Set c where
constructor mkHypergraph
field
edges : List (Edge v)
module _ {v : ℕ} where
show : Hypergraph v → String
show (mkHypergraph e) = unlines (map Edge.show e)
-- an equivalence relation on hypergraphs
record _≈_ (H H′ : Hypergraph v) : Set ℓ where
constructor mk≈
module H = Hypergraph H
module H′ = Hypergraph H′
field
↭-edges : H.edges ↭ H′.edges
infixr 4 _≈_
≈-refl : {H : Hypergraph v} → H ≈ H
≈-refl = mk≈ ↭-refl
≈-sym : {H H′ : Hypergraph v} → H ≈ H′ → H′ ≈ H
≈-sym (mk≈ ≡n) = mk≈ (↭-sym ≡n)
≈-trans : {H H′ H″ : Hypergraph v} → H ≈ H′ → H′ ≈ H″ → H ≈ H″
≈-trans (mk≈ ≡n₁) (mk≈ ≡n₂) = mk≈ (↭-trans ≡n₁ ≡n₂)
-- The setoid of labeled hypergraphs with v nodes
Hypergraphₛ : ℕ → Setoid c ℓ
Hypergraphₛ v = record
{ Carrier = Hypergraph v
; _≈_ = _≈_
; isEquivalence = record
{ refl = ≈-refl
; sym = ≈-sym
; trans = ≈-trans
}
}
open Func
List∘Edgeₛ : (n : ℕ) → Setoid 0ℓ 0ℓ
List∘Edgeₛ = PW.setoid ∘ Edgeₛ
mkHypergraphₛ : {n : ℕ} → List∘Edgeₛ n ⟶ₛ Hypergraphₛ n
mkHypergraphₛ .to = mkHypergraph
mkHypergraphₛ {n} .cong ≋-edges = ≋-edges
|> PW.map Edge.≈⇒≡
|> PW.Pointwise-≡⇒≡
|> ≡.cong mkHypergraph
|> Setoid.reflexive (Hypergraphₛ n)
where
open →-Reasoning
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