3 files changed, 378 insertions, 377 deletions
diff --git a/Data/Hypergraph/Base.agda b/Data/Hypergraph/Base.agda
index 0771a78..e43cbce 100644
--- a/Data/Hypergraph/Base.agda
+++ b/Data/Hypergraph/Base.agda
@@ -1,386 +1,25 @@
 {-# OPTIONS --without-K --safe #-}
 
-module Data.Hypergraph.Base where
+open import Data.Hypergraph.Label using (HypergraphLabel)
 
-open import Data.Castable using (IsCastable)
-open import Data.Fin using (Fin)
+module Data.Hypergraph.Base (HL : HypergraphLabel) where
 
-open import Relation.Binary
-  using
-    ( Rel
-    ; IsDecTotalOrder
-    ; IsStrictTotalOrder
-    ; Tri
-    ; Trichotomous
-    ; _Respectsˡ_
-    ; _Respectsʳ_
-    ; _Respects_
-    )
-open import Relation.Binary.Bundles using (DecTotalOrder; StrictTotalOrder)
-open import Relation.Binary.Structures using (IsEquivalence)
-open import Relation.Nullary using (¬_)
-open import Data.Nat.Base using (ℕ; _<_)
-open import Data.Product.Base using (_×_; _,_; proj₁; proj₂)
-open import Data.Nat.Properties using (<-irrefl; <-trans; <-resp₂-≡; <-cmp)
-open import Level using (Level; suc; _⊔_; 0ℓ)
-open import Data.String using (String; _<+>_; unlines)
-open import Data.Fin.Show using () renaming (show to showFin)
-open import Data.List.Show using () renaming (show to showList)
-open import Data.List.Base using (map)
-open import Data.Vec.Show using () renaming (show to showVec)
+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.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
+import Data.List.Sort.MergeSort as MergeSort
 
-record HypergraphLabel {ℓ : Level} : Set (suc ℓ) where
+record Hypergraph (v : ℕ) : Set where
   field
-    Label : ℕ → Set ℓ
-    showLabel : (n : ℕ) → Label n → String
-    isCastable : IsCastable Label
-    -- _[_≈_] : (n : ℕ) → Rel (Label n) ℓ
-    _[_≤_] : (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
-      }
+    edges : List (Edge v)
 
-  open DTOP using (<-strictTotalOrder) renaming (_<_ to <)
-  _[_<_] : (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
+sortHypergraph : {v : ℕ} → Hypergraph v → Hypergraph v
+sortHypergraph {v} H = record { edges = sort edges }
+  where
+    open Hypergraph H
+    open MergeSort decTotalOrder using (sort)
 
-module Edge (HL : HypergraphLabel) where
-
-  module HL = HypergraphLabel HL
-  open HL using (Label; cast; cast-is-id; cast-trans)
-  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
-    eta-equality
-    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 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
-    where
-      open import Data.Vec.Relation.Binary.Pointwise.Inductive using (≡⇒Pointwise-≡)
-
-  <-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 import Data.Vec.Relation.Binary.Pointwise.Inductive using (≡⇒Pointwise-≡)
-      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 import Data.Vec.Relation.Binary.Pointwise.Inductive using (≡⇒Pointwise-≡)
-      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 import Data.Vec.Relation.Binary.Pointwise.Inductive using (≡⇒Pointwise-≡)
-              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 import Data.Vec.Relation.Binary.Pointwise.Inductive using (≡⇒Pointwise-≡; Pointwise-≡⇒≡)
-              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 import Data.Vec.Relation.Binary.Pointwise.Inductive using (≡⇒Pointwise-≡)
-              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 = record
-              { refl = ≈-Edge-refl
-              ; sym = ≈-Edge-sym
-              ; trans = ≈-Edge-trans
-              }
-          ; 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}
-      }
-
-  open module STOP′ {v} = STOP (strictTotalOrder {v}) using (decTotalOrder) public
-
-  showEdge : {v : ℕ} → Edge v → String
-  showEdge record { arity = a ; label = l ; ports = p} = HL.showLabel a l <+> showVec showFin p
-
-module HypergraphList (HL : HypergraphLabel) where
-
-  open import Data.List.Base using (List)
-  open Edge HL using (Edge)
-
-  record Hypergraph (v : ℕ) : Set where
-    field
-      edges : List (Edge v)
-
-  sortHypergraph : {v : ℕ} → Hypergraph v → Hypergraph v
-  sortHypergraph {v} H = record { edges = sort edges }
-    where
-      open Hypergraph H
-      open import Data.List.Sort.MergeSort (Edge.decTotalOrder HL) using (sort)
-
-  showHypergraph : {v : ℕ} → Hypergraph v → String
-  showHypergraph record { edges = e} = unlines (map (Edge.showEdge HL) e)
+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..3c2a3a1
--- /dev/null
+++ b/Data/Hypergraph/Edge.agda
@@ -0,0 +1,326 @@
+{-# 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 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 = record
+            { refl = ≈-Edge-refl
+            ; sym = ≈-Edge-sym
+            ; trans = ≈-Edge-trans
+            }
+        ; 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
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