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-rw-r--r--DecorationFunctor/Graph.agda88
1 files changed, 35 insertions, 53 deletions
diff --git a/DecorationFunctor/Graph.agda b/DecorationFunctor/Graph.agda
index d8a0187..8b62430 100644
--- a/DecorationFunctor/Graph.agda
+++ b/DecorationFunctor/Graph.agda
@@ -4,62 +4,48 @@ module DecorationFunctor.Graph where
import Categories.Morphism as Morphism
+open import Level using (0ℓ)
+
open import Categories.Category.BinaryProducts using (module BinaryProducts)
open import Categories.Category.Cartesian using (Cartesian)
-open import Categories.Category.Cocartesian using (Cocartesian; module BinaryCoproducts)
-open import Categories.Category.Core using (Category)
-open import Categories.Category.Instance.Nat using (Nat-Cocartesian)
open import Categories.Category.Instance.Nat using (Nat)
open import Categories.Category.Instance.Setoids using (Setoids)
-open import Categories.Category.Instance.SingletonSet using (SingletonSetoid)
open import Categories.Category.Monoidal.Instance.Setoids using (Setoids-Cartesian)
open import Categories.Category.Product using (_⁂_)
open import Categories.Functor using () renaming (_∘F_ to _∘′_)
open import Categories.Functor.Core using (Functor)
open import Categories.Functor.Monoidal.Symmetric using (module Lax)
open import Categories.NaturalTransformation using (NaturalTransformation; ntHelper)
-
open import Category.Cocomplete.Finitely.Bundle using (FinitelyCocompleteCategory)
-open import Category.Instance.Setoids.SymmetricMonoidal using (Setoids-×)
open import Category.Instance.Nat.FinitelyCocomplete using (Nat-FinitelyCocomplete)
-
+open import Category.Instance.Setoids.SymmetricMonoidal {0ℓ} {0ℓ} using (Setoids-×; ×-symmetric′)
open import Data.Empty using (⊥-elim)
-open import Data.Fin using (#_)
-open import Data.Fin.Base using (Fin; splitAt; join; zero; suc; _↑ˡ_; _↑ʳ_)
+open import Data.Fin using (#_; Fin; splitAt; join; zero; suc; _↑ˡ_; _↑ʳ_)
open import Data.Fin.Patterns using (0F; 1F)
open import Data.Fin.Properties using (splitAt-join; join-splitAt; splitAt-↑ˡ; splitAt⁻¹-↑ˡ)
open import Data.Nat using (ℕ; _+_)
open import Data.Product.Base using (_,_)
open import Data.Product.Relation.Binary.Pointwise.NonDependent using (×-setoid)
+open import Data.Setoid.Unit {0ℓ} {0ℓ} using (⊤ₛ)
open import Data.Sum.Base using (_⊎_; map; inj₁; inj₂; swap) renaming ([_,_]′ to [_,_])
open import Data.Sum.Properties using (map-map; [,]-map; [,]-∘; [-,]-cong; [,-]-cong; map-cong; swap-involutive)
open import Data.Unit using (tt)
-open import Data.Unit.Properties using () renaming (≡-setoid to ⊤-setoid)
-
-open import Function.Base using (_∘_; id; const; case_of_)
-open import Function.Bundles using (Func; Inverse; _↔_; mk↔)
+open import Function using (_∘_; id; const; Func; Inverse; _↔_; mk↔)
open import Function.Construct.Composition using (_↔-∘_)
+open import Function.Construct.Constant using () renaming (function to Const)
open import Function.Construct.Identity using (↔-id)
open import Function.Construct.Symmetry using (↔-sym)
-
-open import Level using (0ℓ; lift)
-
-open import Relation.Binary.Bundles using (Setoid)
-open import Relation.Binary.PropositionalEquality using (_≗_)
-open import Relation.Binary.PropositionalEquality.Core using (_≡_; erefl; refl; sym; trans; cong)
+open import Relation.Binary using (Setoid)
+open import Relation.Binary.PropositionalEquality using (_≗_; _≡_; erefl; refl; sym; trans; cong)
open import Relation.Binary.PropositionalEquality.Properties using (isEquivalence; module ≡-Reasoning)
open import Relation.Nullary.Negation.Core using (¬_)
open Cartesian (Setoids-Cartesian {0ℓ} {0ℓ}) using (products)
-open Cocartesian Nat-Cocartesian using (coproducts)
open FinitelyCocompleteCategory Nat-FinitelyCocomplete
- using ()
- renaming (symmetricMonoidalCategory to Nat-smc)
-open Morphism (Setoids 0ℓ 0ℓ) using (_≅_)
+ using (-+-)
+ renaming (symmetricMonoidalCategory to Nat-smc; +-assoc to Nat-+-assoc)
open Lax using (SymmetricMonoidalFunctor)
-
open BinaryProducts products using (-×-)
-open BinaryCoproducts coproducts using (-+-) renaming (+-assoc to Nat-+-assoc)
record Graph (v : ℕ) : Set where
@@ -182,18 +168,15 @@ F = record
; F-resp-≈ = λ f≗g → F-resp-≈ f≗g
}
-empty-graph : Graph 0
-empty-graph = record
+discrete : {n : ℕ} → Graph n
+discrete = record
{ e = 0
; s = λ ()
; t = λ ()
}
-ε : Func (SingletonSetoid {0ℓ} {0ℓ}) (Graph-setoid 0)
-ε = record
- { to = const empty-graph
- ; cong = const Graph-same-refl
- }
+ε : Func ⊤ₛ (Graph-setoid 0)
+ε = Const ⊤ₛ (Graph-setoid 0) discrete
together : Graph n → Graph m → Graph (n + m)
together {n} {m} G₁ G₂ = record
@@ -521,7 +504,7 @@ associativity {X} {Y} {Z} G₁ G₂ G₃ = record
(G₁+[G₂+G₃].t ∘ [ [ _↑ˡ G₂.e + G₃.e , (G₁.e ↑ʳ_) ∘ (_↑ˡ G₃.e) ] ∘ splitAt G₁.e , (G₁.e ↑ʳ_) ∘ (G₂.e ↑ʳ_) ] ∘ splitAt _) x ≡⟨⟩
(G₁+[G₂+G₃].t ∘ to (+-assoc-↔ G₁.e G₂.e G₃.e)) x ∎
-unitaryˡ : Graph-same (together empty-graph G) G
+unitaryˡ : Graph-same (together (discrete {0}) G) G
unitaryˡ = Graph-same-refl
n+0↔n : ∀ n → Fin (n + 0) ↔ Fin n
@@ -544,7 +527,7 @@ n+0↔n n = record
unitaryʳ
: {G : Graph n}
- → Graph-same (map-nodes ([ (λ x → x) , (λ ()) ] ∘ splitAt n) (together G empty-graph)) G
+ → Graph-same (map-nodes ([ (λ x → x) , (λ ()) ] ∘ splitAt n) (together G discrete)) G
unitaryʳ {n} {G} = record
{ ↔e = e+0↔e
; ≗s = ≗s+0
@@ -637,28 +620,27 @@ braiding {n} {m} {G₁} {G₂} = record
≡⟨ (cong (join m n ∘ map G₂.t G₁.t) ∘ splitAt-join G₂.e G₁.e ∘ swap ∘ splitAt G₁.e) x ⟨
(join m n ∘ map G₂.t G₁.t ∘ splitAt G₂.e ∘ join G₂.e G₁.e ∘ swap ∘ splitAt G₁.e) x ∎
-graph : SymmetricMonoidalFunctor Nat-smc (Setoids-× {0ℓ})
-graph = record
- { F = F
- ; isBraidedMonoidal = record
- { isMonoidal = record
- { ε = ε
- ; ⊗-homo = ⊗-homomorphism
- ; associativity = λ { {x} {y} {z} {(G₁ , G₂) , G₃} → associativity G₁ G₂ G₃ }
- ; unitaryˡ = unitaryˡ
- ; unitaryʳ = unitaryʳ
- }
- ; braiding-compat = λ { {x} {y} {G₁ , G₂} → braiding {G₁ = G₁} {G₂ = G₂} }
- }
- }
+opaque
+ unfolding ×-symmetric′
+ graph : SymmetricMonoidalFunctor Nat-smc Setoids-×
+ graph = record
+ { F = F
+ ; isBraidedMonoidal = record
+ { isMonoidal = record
+ { ε = ε
+ ; ⊗-homo = ⊗-homomorphism
+ ; associativity = λ { {x} {y} {z} {(G₁ , G₂) , G₃} → associativity G₁ G₂ G₃ }
+ ; unitaryˡ = unitaryˡ
+ ; unitaryʳ = unitaryʳ
+ }
+ ; braiding-compat = λ { {x} {y} {G₁ , G₂} → braiding {G₁ = G₁} {G₂ = G₂} }
+ }
+ }
module F = SymmetricMonoidalFunctor graph
-and-gate : Func (SingletonSetoid {0ℓ} {0ℓ}) (F.₀ 3)
-and-gate = record
- { to = λ { (lift tt) → and-graph }
- ; cong = λ { (lift tt) → Graph-same-refl }
- }
+and-gate : Func ⊤ₛ (Graph-setoid 3)
+and-gate = Const ⊤ₛ (Graph-setoid 3) and-graph
where
and-graph : Graph 3
and-graph = record
generated by cgit v1.2.3 (git 2.39.1) at 2025-12-14 21:51:44 +0000