4 files changed, 447 insertions, 0 deletions
diff --git a/Data/Hypergraph/Base.agda b/Data/Hypergraph/Base.agda
new file mode 100644
index 0000000..6988cf0
--- /dev/null
+++ b/Data/Hypergraph/Base.agda
@@ -0,0 +1,26 @@
+{-# OPTIONS --without-K --safe #-}
+
+open import Data.Hypergraph.Label using (HypergraphLabel)
+
+module Data.Hypergraph.Base (HL : HypergraphLabel) where
+
+open import Data.Hypergraph.Edge HL using (Edge; decTotalOrder; showEdge)
+open import Data.List.Base using (List; map)
+open import Data.Nat.Base using (ℕ)
+open import Data.String using (String; unlines)
+
+import Data.List.Sort as Sort
+
+record Hypergraph (v : ℕ) : Set where
+  constructor mkHypergraph
+  field
+    edges : List (Edge v)
+
+sortHypergraph : {v : ℕ} → Hypergraph v → Hypergraph v
+sortHypergraph {v} H = record { edges = sort edges }
+  where
+    open Hypergraph H
+    open Sort decTotalOrder using (sort)
+
+showHypergraph : {v : ℕ} → Hypergraph v → String
+showHypergraph record { edges = e} = unlines (map showEdge e)
diff --git a/Data/Hypergraph/Edge.agda b/Data/Hypergraph/Edge.agda
new file mode 100644
index 0000000..13b9278
--- /dev/null
+++ b/Data/Hypergraph/Edge.agda
@@ -0,0 +1,335 @@
+{-# OPTIONS --without-K --safe #-}
+
+open import Data.Hypergraph.Label using (HypergraphLabel)
+
+module Data.Hypergraph.Edge (HL : HypergraphLabel) where
+
+
+open import Relation.Binary using (Rel; IsStrictTotalOrder; Tri; Trichotomous; _Respects_)
+open import Data.Castable using (IsCastable)
+open import Data.Fin using (Fin)
+open import Data.Fin.Show using () renaming (show to showFin)
+open import Data.Nat.Base using (ℕ; _<_)
+open import Data.Nat.Properties using (<-irrefl; <-trans; <-resp₂-≡; <-cmp)
+open import Data.Product.Base using (_,_; proj₁; proj₂)
+open import Data.String using (String; _<+>_)
+open import Data.Vec.Relation.Binary.Pointwise.Inductive using (≡⇒Pointwise-≡; Pointwise-≡⇒≡)
+open import Data.Vec.Show using () renaming (show to showVec)
+open import Level using (0ℓ)
+open import Relation.Binary.Bundles using (DecTotalOrder; StrictTotalOrder)
+open import Relation.Binary.Structures using (IsEquivalence)
+open import Relation.Nullary using (¬_)
+
+import Data.Fin.Base as Fin
+import Data.Fin.Properties as FinProp
+import Data.Vec.Base as VecBase
+import Data.Vec.Relation.Binary.Equality.Cast as VecCast
+import Data.Vec.Relation.Binary.Lex.Strict as Lex
+import Relation.Binary.PropositionalEquality as ≡
+import Relation.Binary.Properties.DecTotalOrder as DTOP
+import Relation.Binary.Properties.StrictTotalOrder as STOP
+
+module HL = HypergraphLabel HL
+open HL using (Label; cast; cast-is-id)
+open VecBase using (Vec)
+
+record Edge (v : ℕ) : Set where
+  field
+    {arity} : ℕ
+    label : Label arity
+    ports : Vec (Fin v) arity
+
+open ≡ using (_≡_)
+open VecCast using (_≈[_]_)
+
+record ≈-Edge {n : ℕ} (E E′ : Edge n) : Set where
+  module E = Edge E
+  module E′ = Edge E′
+  field
+    ≡arity : E.arity ≡ E′.arity
+    ≡label : cast ≡arity E.label ≡ E′.label
+    ≡ports : E.ports ≈[ ≡arity ] E′.ports
+
+≈-Edge-refl : {v : ℕ} {x : Edge v} → ≈-Edge x x
+≈-Edge-refl {_} {x} = record
+    { ≡arity = ≡.refl
+    ; ≡label = HL.≈-reflexive ≡.refl
+    ; ≡ports = VecCast.≈-reflexive ≡.refl
+    }
+  where
+    open Edge x using (arity; label)
+    open DecTotalOrder (HL.decTotalOrder arity) using (module Eq)
+
+≈-Edge-sym : {v : ℕ} {x y : Edge v} → ≈-Edge x y → ≈-Edge y x
+≈-Edge-sym {_} {x} {y} x≈y = record
+    { ≡arity = ≡.sym ≡arity
+    ; ≡label = HL.≈-sym ≡label
+    ; ≡ports = VecCast.≈-sym ≡ports
+    }
+  where
+    open ≈-Edge x≈y
+    open DecTotalOrder (HL.decTotalOrder E.arity) using (module Eq)
+
+≈-Edge-trans : {v : ℕ} {i j k : Edge v} → ≈-Edge i j → ≈-Edge j k → ≈-Edge i k
+≈-Edge-trans {_} {i} {j} {k} i≈j j≈k = record
+    { ≡arity = ≡.trans i≈j.≡arity j≈k.≡arity
+    ; ≡label = HL.≈-trans i≈j.≡label j≈k.≡label
+    ; ≡ports = VecCast.≈-trans i≈j.≡ports j≈k.≡ports
+    }
+  where
+    module i≈j = ≈-Edge i≈j
+    module j≈k = ≈-Edge j≈k
+
+open import Relation.Binary using (IsEquivalence)
+≈-Edge-IsEquivalence : {v : ℕ} → IsEquivalence (≈-Edge {v})
+≈-Edge-IsEquivalence = record
+    { refl = ≈-Edge-refl
+    ; sym = ≈-Edge-sym
+    ; trans = ≈-Edge-trans
+    }
+
+open HL using (_[_<_])
+_<<_ : {v a : ℕ} → Rel (Vec (Fin v) a) 0ℓ
+_<<_ {v} = Lex.Lex-< _≡_ (Fin._<_ {v})
+data <-Edge {v : ℕ} : Edge v → Edge v → Set where
+  <-arity
+      : {x y : Edge v}
+      → Edge.arity x < Edge.arity y
+      → <-Edge x y
+  <-label
+      : {x y : Edge v}
+        (≡a : Edge.arity x ≡ Edge.arity y)
+      → Edge.arity y [ cast ≡a (Edge.label x) < Edge.label y ]
+      → <-Edge x y
+  <-ports
+      : {x y : Edge v}
+        (≡a : Edge.arity x ≡ Edge.arity y)
+        (≡l : Edge.label x HL.≈[ ≡a ] Edge.label y)
+      → VecBase.cast ≡a (Edge.ports x) << Edge.ports y
+      → <-Edge x y
+
+<-Edge-irrefl : {v : ℕ} {x y : Edge v} → ≈-Edge x y → ¬ <-Edge x y
+<-Edge-irrefl record { ≡arity = ≡a } (<-arity n<m) = <-irrefl ≡a n<m
+<-Edge-irrefl record { ≡label = ≡l } (<-label _ (_ , x≉y)) = x≉y ≡l
+<-Edge-irrefl record { ≡ports = ≡p } (<-ports ≡.refl ≡l x<y)
+    = Lex.<-irrefl FinProp.<-irrefl (≡⇒Pointwise-≡ ≡p) x<y
+
+<-Edge-trans : {v : ℕ} {i j k : Edge v} → <-Edge i j → <-Edge j k → <-Edge i k
+<-Edge-trans (<-arity i<j) (<-arity j<k) = <-arity (<-trans i<j j<k)
+<-Edge-trans (<-arity i<j) (<-label ≡.refl j<k) = <-arity i<j
+<-Edge-trans (<-arity i<j) (<-ports ≡.refl _ j<k) = <-arity i<j
+<-Edge-trans (<-label ≡.refl i<j) (<-arity j<k) = <-arity j<k
+<-Edge-trans {_} {i} (<-label ≡.refl i<j) (<-label ≡.refl j<k)
+    = <-label ≡.refl (<-label-trans i<j (<-respˡ-≈ (HL.≈-reflexive ≡.refl) j<k))
+  where
+    open DTOP (HL.decTotalOrder (Edge.arity i)) using (<-respˡ-≈) renaming (<-trans to <-label-trans)
+<-Edge-trans {k = k} (<-label ≡.refl i<j) (<-ports ≡.refl ≡.refl _)
+    = <-label ≡.refl (<-respʳ-≈ (≡.sym (HL.≈-reflexive ≡.refl)) i<j)
+  where
+    open DTOP (HL.decTotalOrder (Edge.arity k)) using (<-respʳ-≈)
+<-Edge-trans (<-ports ≡.refl _ _) (<-arity j<k) = <-arity j<k
+<-Edge-trans {k = k} (<-ports ≡.refl ≡.refl _) (<-label ≡.refl j<k)
+    = <-label ≡.refl (<-respˡ-≈ (≡.cong (cast _) (HL.≈-reflexive ≡.refl)) j<k)
+  where
+    open DTOP (HL.decTotalOrder (Edge.arity k)) using (<-respˡ-≈)
+<-Edge-trans {j = j} (<-ports ≡.refl ≡l₁ i<j) (<-ports ≡.refl ≡l₂ j<k)
+  rewrite (VecCast.cast-is-id ≡.refl (Edge.ports j))
+    = <-ports ≡.refl
+        (HL.≈-trans ≡l₁ ≡l₂)
+        (Lex.<-trans ≡-isPartialEquivalence FinProp.<-resp₂-≡ FinProp.<-trans i<j j<k)
+  where
+    open IsEquivalence ≡.isEquivalence using () renaming (isPartialEquivalence to ≡-isPartialEquivalence)
+
+<-Edge-respˡ-≈ : {v : ℕ} {y : Edge v} → (λ x → <-Edge x y) Respects ≈-Edge
+<-Edge-respˡ-≈ ≈x (<-arity x₁<y) = <-arity (proj₂ <-resp₂-≡ ≡arity x₁<y)
+  where
+    open ≈-Edge ≈x using (≡arity)
+<-Edge-respˡ-≈ {_} {y} record { ≡arity = ≡.refl ; ≡label = ≡.refl } (<-label ≡.refl x₁<y)
+    = <-label ≡.refl (<-respˡ-≈ (≡.sym (HL.≈-reflexive ≡.refl)) x₁<y)
+  where
+    module y = Edge y
+    open DTOP (HL.decTotalOrder y.arity) using (<-respˡ-≈)
+<-Edge-respˡ-≈ record { ≡arity = ≡.refl ; ≡label = ≡.refl; ≡ports = ≡.refl} (<-ports ≡.refl ≡.refl x₁<y)
+    = <-ports
+        ≡.refl
+        (≡.cong (cast _) (HL.≈-reflexive ≡.refl))
+        (Lex.<-respectsˡ
+            ≡-isPartialEquivalence
+            FinProp.<-respˡ-≡
+            (≡⇒Pointwise-≡ (≡.sym (VecCast.≈-reflexive ≡.refl)))
+            x₁<y)
+  where
+    open IsEquivalence ≡.isEquivalence using () renaming (isPartialEquivalence to ≡-isPartialEquivalence)
+
+<-Edge-respʳ-≈ : {v : ℕ} {x : Edge v} → <-Edge x Respects ≈-Edge
+<-Edge-respʳ-≈ record { ≡arity = ≡a } (<-arity x<y₁) = <-arity (proj₁ <-resp₂-≡ ≡a x<y₁)
+<-Edge-respʳ-≈ {_} {x} record { ≡arity = ≡.refl ; ≡label = ≡.refl } (<-label ≡.refl x<y₁)
+    = <-label ≡.refl (<-respʳ-≈ (≡.sym (HL.≈-reflexive ≡.refl)) x<y₁)
+  where
+    module x = Edge x
+    open DTOP (HL.decTotalOrder x.arity) using (<-respʳ-≈)
+<-Edge-respʳ-≈ record { ≡arity = ≡.refl ; ≡label = ≡.refl; ≡ports = ≡.refl} (<-ports ≡.refl ≡.refl x<y₁)
+    = <-ports
+        ≡.refl
+        (≡.cong (cast _) (≡.sym (HL.≈-reflexive ≡.refl)))
+        (Lex.<-respectsʳ
+            ≡-isPartialEquivalence
+            FinProp.<-respʳ-≡
+            (≡⇒Pointwise-≡ (≡.sym (VecCast.≈-reflexive ≡.refl)))
+            x<y₁)
+  where
+    open IsEquivalence ≡.isEquivalence using () renaming (isPartialEquivalence to ≡-isPartialEquivalence)
+
+open Tri
+open ≈-Edge
+tri : {v : ℕ} → Trichotomous (≈-Edge {v}) (<-Edge {v})
+tri x y with <-cmp x.arity y.arity
+  where
+    module x = Edge x
+    module y = Edge y
+tri x y | tri< x<y x≢y y≮x = tri< (<-arity x<y) (λ x≡y → x≢y (≡arity x≡y)) ¬y<x
+  where
+    ¬y<x :  ¬ <-Edge y x
+    ¬y<x (<-arity y<x) = y≮x y<x
+    ¬y<x (<-label ≡a _) = x≢y (≡.sym ≡a)
+    ¬y<x (<-ports ≡a _ _) = x≢y (≡.sym ≡a)
+tri x y | tri≈ x≮y ≡.refl y≮x = compare-label
+  where
+    module x = Edge x
+    module y = Edge y
+    open StrictTotalOrder (HL.strictTotalOrder x.arity) using (compare)
+    import Relation.Binary.Properties.DecTotalOrder
+    open DTOP (HL.decTotalOrder x.arity) using (<-respˡ-≈)
+    compare-label : Tri (<-Edge x y) (≈-Edge x y) (<-Edge y x)
+    compare-label with compare x.label y.label
+    ... | tri< x<y x≢y y≮x′ = tri<
+            (<-label ≡.refl (<-respˡ-≈ (≡.sym (HL.≈-reflexive ≡.refl)) x<y))
+            (λ x≡y → x≢y (≡.trans (≡.sym (HL.≈-reflexive ≡.refl)) (≡label x≡y)))
+            ¬y<x
+      where
+        ¬y<x :  ¬ <-Edge y x
+        ¬y<x (<-arity y<x) = y≮x y<x
+        ¬y<x (<-label _ y<x) = y≮x′ (<-respˡ-≈ (HL.≈-reflexive ≡.refl) y<x)
+        ¬y<x (<-ports _ ≡l _) = x≢y (≡.trans (≡.sym ≡l) (cast-is-id ≡.refl y.label))
+    ... | tri≈ x≮y′ x≡y′ y≮x′ = compare-ports
+      where
+        compare-ports : Tri (<-Edge x y) (≈-Edge x y) (<-Edge y x)
+        compare-ports with Lex.<-cmp ≡.sym FinProp.<-cmp x.ports y.ports
+        ... | tri< x<y x≢y y≮x″ =
+                tri<
+                  (<-ports ≡.refl
+                    (HL.≈-reflexive x≡y′)
+                    (Lex.<-respectsˡ
+                      ≡-isPartialEquivalence
+                      FinProp.<-respˡ-≡
+                      (≡⇒Pointwise-≡ (≡.sym (VecCast.≈-reflexive ≡.refl)))
+                      x<y))
+                  (λ x≡y → x≢y (≡⇒Pointwise-≡ (≡.trans (≡.sym (VecCast.≈-reflexive ≡.refl)) (≡ports x≡y))))
+                  ¬y<x
+          where
+            open IsEquivalence ≡.isEquivalence using () renaming (isPartialEquivalence to ≡-isPartialEquivalence)
+            ¬y<x :  ¬ <-Edge y x
+            ¬y<x (<-arity y<x) = y≮x y<x
+            ¬y<x (<-label _ y<x) = y≮x′ (<-respˡ-≈ (HL.≈-reflexive ≡.refl) y<x)
+            ¬y<x (<-ports _ _ y<x) =
+                y≮x″
+                  (Lex.<-respectsˡ
+                    ≡-isPartialEquivalence
+                    FinProp.<-respˡ-≡
+                    (≡⇒Pointwise-≡ (VecCast.≈-reflexive ≡.refl))
+                    y<x)
+        ... | tri≈ x≮y″ x≡y″ y≮x″ = tri≈
+                ¬x<y
+                (record { ≡arity = ≡.refl ; ≡label = HL.≈-reflexive x≡y′ ; ≡ports = VecCast.≈-reflexive (Pointwise-≡⇒≡ x≡y″) })
+                ¬y<x
+          where
+            open IsEquivalence ≡.isEquivalence using () renaming (isPartialEquivalence to ≡-isPartialEquivalence)
+            ¬x<y : ¬ <-Edge x y
+            ¬x<y (<-arity x<y) = x≮y x<y
+            ¬x<y (<-label _ x<y) = x≮y′ (<-respˡ-≈ (HL.≈-reflexive ≡.refl) x<y)
+            ¬x<y (<-ports _ _ x<y) =
+                x≮y″
+                  (Lex.<-respectsˡ
+                    ≡-isPartialEquivalence
+                    FinProp.<-respˡ-≡
+                    (≡⇒Pointwise-≡ (VecCast.≈-reflexive ≡.refl))
+                    x<y)
+            ¬y<x : ¬ <-Edge y x
+            ¬y<x (<-arity y<x) = y≮x y<x
+            ¬y<x (<-label _ y<x) = y≮x′ (<-respˡ-≈ (HL.≈-reflexive ≡.refl) y<x)
+            ¬y<x (<-ports _ _ y<x) =
+                y≮x″
+                  (Lex.<-respectsˡ
+                    ≡-isPartialEquivalence
+                    FinProp.<-respˡ-≡
+                    (≡⇒Pointwise-≡ (VecCast.≈-reflexive ≡.refl))
+                    y<x)
+
+        ... | tri> x≮y″ x≢y y<x =
+                tri>
+                  ¬x<y
+                  (λ x≡y → x≢y (≡⇒Pointwise-≡ (≡.trans (≡.sym (VecCast.≈-reflexive ≡.refl)) (≡ports x≡y))))
+                  (<-ports
+                    ≡.refl
+                    (HL.≈-sym (HL.≈-reflexive x≡y′))
+                    (Lex.<-respectsˡ
+                      ≡-isPartialEquivalence
+                      FinProp.<-respˡ-≡
+                      (≡⇒Pointwise-≡ (≡.sym (VecCast.≈-reflexive ≡.refl)))
+                      y<x))
+          where
+            open IsEquivalence ≡.isEquivalence using () renaming (isPartialEquivalence to ≡-isPartialEquivalence)
+            ¬x<y : ¬ <-Edge x y
+            ¬x<y (<-arity x<y) = x≮y x<y
+            ¬x<y (<-label _ x<y) = x≮y′ (<-respˡ-≈ (HL.≈-reflexive ≡.refl) x<y)
+            ¬x<y (<-ports _ _ x<y) =
+                x≮y″
+                  (Lex.<-respectsˡ
+                    ≡-isPartialEquivalence
+                    FinProp.<-respˡ-≡
+                    (≡⇒Pointwise-≡ (VecCast.≈-reflexive ≡.refl))
+                    x<y)
+    ... | tri> x≮y′ x≢y y<x = tri>
+            ¬x<y
+            (λ x≡y → x≢y (≡.trans (≡.sym (HL.≈-reflexive ≡.refl)) (≡label x≡y)))
+            (<-label ≡.refl (<-respˡ-≈ (≡.sym (HL.≈-reflexive ≡.refl)) y<x))
+      where
+        ¬x<y : ¬ <-Edge x y
+        ¬x<y (<-arity x<y) = x≮y x<y
+        ¬x<y (<-label ≡a x<y) = x≮y′ (<-respˡ-≈ (HL.≈-reflexive ≡.refl) x<y)
+        ¬x<y (<-ports _ ≡l _) = x≢y (≡.trans (≡.sym (HL.≈-reflexive ≡.refl)) ≡l)
+tri x y | tri> x≮y x≢y y<x = tri> ¬x<y (λ x≡y → x≢y (≡arity x≡y)) (<-arity y<x)
+  where
+    ¬x<y :  ¬ <-Edge x y
+    ¬x<y (<-arity x<y) = x≮y x<y
+    ¬x<y (<-label ≡a x<y) = x≢y ≡a
+    ¬x<y (<-ports ≡a _ _) = x≢y ≡a
+
+isStrictTotalOrder : {v : ℕ} → IsStrictTotalOrder (≈-Edge {v}) (<-Edge {v})
+isStrictTotalOrder = record
+    { isStrictPartialOrder = record
+        { isEquivalence = ≈-Edge-IsEquivalence
+        ; irrefl = <-Edge-irrefl
+        ; trans = <-Edge-trans
+        ; <-resp-≈ = <-Edge-respʳ-≈ , <-Edge-respˡ-≈
+        }
+    ; compare = tri
+    }
+
+strictTotalOrder : {v : ℕ} → StrictTotalOrder 0ℓ 0ℓ 0ℓ
+strictTotalOrder {v} = record
+    { Carrier = Edge v
+    ; _≈_ = ≈-Edge {v}
+    ; _<_ = <-Edge {v}
+    ; isStrictTotalOrder = isStrictTotalOrder {v}
+    }
+
+showEdge : {v : ℕ} → Edge v → String
+showEdge record { arity = a ; label = l ; ports = p} = HL.showLabel a l <+> showVec showFin p
+
+open module STOP′ {v} = STOP (strictTotalOrder {v}) using (decTotalOrder) public
+
+≈-Edge⇒≡ : {v : ℕ} {x y : Edge v} → ≈-Edge x y → x ≡ y
+≈-Edge⇒≡ {v} {record { label = l ; ports = p }} record { ≡arity = ≡.refl ; ≡label = ≡.refl ; ≡ports = ≡.refl }
+  rewrite cast-is-id ≡.refl l
+  rewrite VecCast.cast-is-id ≡.refl p = ≡.refl
diff --git a/Data/Hypergraph/Label.agda b/Data/Hypergraph/Label.agda
new file mode 100644
index 0000000..c23d33a
--- /dev/null
+++ b/Data/Hypergraph/Label.agda
@@ -0,0 +1,36 @@
+{-# OPTIONS --without-K --safe #-}
+module Data.Hypergraph.Label where
+
+open import Data.Castable using (IsCastable)
+open import Data.Nat.Base using (ℕ)
+open import Data.String using (String)
+open import Level using (Level; suc)
+open import Relation.Binary using (Rel; IsDecTotalOrder)
+open import Relation.Binary.Bundles using (DecTotalOrder; StrictTotalOrder)
+open import Relation.Binary.Properties.DecTotalOrder using (<-strictTotalOrder; _<_)
+open import Relation.Binary.PropositionalEquality using (_≡_)
+
+record HypergraphLabel {ℓ : Level} : Set (suc ℓ) where
+
+  field
+    Label : ℕ → Set ℓ
+    showLabel : (n : ℕ) → Label n → String
+    isCastable : IsCastable Label
+    _[_≤_] : (n : ℕ) → Rel (Label n) ℓ
+    isDecTotalOrder : (n : ℕ) → IsDecTotalOrder _≡_ (n [_≤_])
+
+  decTotalOrder : (n : ℕ) → DecTotalOrder ℓ ℓ ℓ
+  decTotalOrder n = record
+      { Carrier = Label n
+      ; _≈_ = _≡_
+      ; _≤_ = n [_≤_]
+      ; isDecTotalOrder = isDecTotalOrder n
+      }
+
+  _[_<_] : (n : ℕ) → Rel (Label n) ℓ
+  _[_<_] n =  _<_ (decTotalOrder n)
+
+  strictTotalOrder : (n : ℕ) → StrictTotalOrder ℓ ℓ ℓ
+  strictTotalOrder n = <-strictTotalOrder (decTotalOrder n)
+
+  open IsCastable isCastable public
diff --git a/Data/Hypergraph/Setoid.agda b/Data/Hypergraph/Setoid.agda
new file mode 100644
index 0000000..c39977d
--- /dev/null
+++ b/Data/Hypergraph/Setoid.agda
@@ -0,0 +1,50 @@
+{-# OPTIONS --without-K --safe #-}
+
+open import Data.Hypergraph.Label using (HypergraphLabel)
+
+module Data.Hypergraph.Setoid (HL : HypergraphLabel) where
+
+open import Data.Hypergraph.Edge HL using (decTotalOrder)
+open import Data.Hypergraph.Base HL using (Hypergraph; sortHypergraph)
+open import Data.Nat.Base using (ℕ)
+open import Level using (0ℓ)
+open import Relation.Binary.Bundles using (Setoid)
+open import Relation.Binary.PropositionalEquality as ≡ using (_≡_)
+
+record ≈-Hypergraph {v : ℕ} (H H′ : Hypergraph v) : Set where
+  module H = Hypergraph H
+  module H′ = Hypergraph H′
+  field
+    ≡sorted : sortHypergraph H ≡ sortHypergraph H′
+  open Hypergraph using (edges)
+  ≡edges : edges (sortHypergraph H) ≡ edges (sortHypergraph H′)
+  ≡edges = ≡.cong edges ≡sorted
+  open import Data.List.Relation.Binary.Permutation.Propositional using (_↭_; ↭-reflexive; ↭-sym; ↭-trans)
+  open import Data.List.Sort decTotalOrder using (sort-↭)
+  ↭edges : H.edges ↭ H′.edges
+  ↭edges = ↭-trans (↭-sym (sort-↭ H.edges)) (↭-trans (↭-reflexive ≡edges) (sort-↭ H′.edges))
+
+≈-refl : {v : ℕ} {H : Hypergraph v} → ≈-Hypergraph H H
+≈-refl = record { ≡sorted = ≡.refl }
+
+≈-sym : {v : ℕ} {H H′ : Hypergraph v} → ≈-Hypergraph H H′ → ≈-Hypergraph H′ H
+≈-sym ≈H = record { ≡sorted = ≡.sym ≡sorted }
+  where
+    open ≈-Hypergraph ≈H 
+
+≈-trans : {v : ℕ} {H H′ H″ : Hypergraph v} → ≈-Hypergraph H H′ → ≈-Hypergraph H′ H″ → ≈-Hypergraph H H″
+≈-trans ≈H₁ ≈H₂ = record { ≡sorted = ≡.trans ≈H₁.≡sorted ≈H₂.≡sorted }
+  where
+    module ≈H₁ = ≈-Hypergraph ≈H₁
+    module ≈H₂ = ≈-Hypergraph ≈H₂
+
+Hypergraph-Setoid : (v : ℕ) → Setoid 0ℓ 0ℓ
+Hypergraph-Setoid v = record
+    { Carrier = Hypergraph v
+    ; _≈_ = ≈-Hypergraph
+    ; isEquivalence = record
+        { refl = ≈-refl
+        ; sym = ≈-sym
+        ; trans = ≈-trans
+        }
+    }