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Diffstat (limited to 'DecorationFunctor/Hypergraph')
| -rw-r--r-- | DecorationFunctor/Hypergraph/Labeled.agda | 689 |
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diff --git a/DecorationFunctor/Hypergraph/Labeled.agda b/DecorationFunctor/Hypergraph/Labeled.agda new file mode 100644 index 0000000..31402b1 --- /dev/null +++ b/DecorationFunctor/Hypergraph/Labeled.agda @@ -0,0 +1,689 @@ +{-# OPTIONS --without-K --safe #-} + +open import Level using (Level) + +module DecorationFunctor.Hypergraph.Labeled {c ℓ : Level} where + +import Categories.Morphism as Morphism + +open import Categories.Category.BinaryProducts using (module BinaryProducts) +open import Categories.Category.Cartesian using (Cartesian) +open import Categories.Category.Core using (Category) +open import Categories.Category.Instance.Nat using (Nat) +open import Categories.Category.Instance.Setoids using (Setoids) +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.Nat.FinitelyCocomplete using (Nat-FinitelyCocomplete) +open import Category.Instance.Setoids.SymmetricMonoidal {c} {ℓ} using (Setoids-×; ×-symmetric′) +open import Data.Fin using (#_; Fin; splitAt; join; zero; suc; _↑ˡ_; _↑ʳ_; cast) +open import Data.Fin.Patterns using (0F; 1F; 2F) +open import Data.Fin.Properties + using + ( splitAt-join ; join-splitAt + ; cast-is-id ; cast-trans ; subst-is-cast + ; splitAt-↑ˡ ; splitAt-↑ʳ + ; splitAt⁻¹-↑ˡ ; ↑ˡ-injective + ) +open import Data.Nat using (ℕ; _+_) +open import Data.Product.Base using (_,_; Σ) +open import Data.Product.Relation.Binary.Pointwise.NonDependent using (_×ₛ_) +open import Data.Setoid.Unit {c} {ℓ} using (⊤ₛ) +open import Data.Sum.Base using (_⊎_; map; inj₁; inj₂; swap; map₂) renaming ([_,_]′ to [_,_]) +open import Data.Sum.Properties using (map-map; [,]-map; [,]-∘; [-,]-cong; [,-]-cong; [,]-cong; map-cong; swap-involutive) +open import Data.Unit.Properties using () renaming (≡-setoid to ⊤-setoid) +open import Function using (_∘_; id; const; Func; Inverse; _↔_; mk↔; _⟶ₛ_) +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 Relation.Binary using (Setoid) +open import Relation.Binary.PropositionalEquality using (_≗_; _≡_; erefl; refl; sym; trans; cong; cong₂; subst; cong-app) +open import Relation.Binary.PropositionalEquality.Properties using (isEquivalence; module ≡-Reasoning; dcong; dcong₂; subst-∘; subst-subst; sym-cong; subst-subst-sym; trans-cong; cong-∘; trans-reflʳ) +open import Relation.Nullary.Negation.Core using (¬_) + +open Cartesian (Setoids-Cartesian {c} {ℓ}) using (products) +open FinitelyCocompleteCategory Nat-FinitelyCocomplete + using (-+-) + renaming (symmetricMonoidalCategory to Nat-smc; +-assoc to Nat-+-assoc) +open import Category.Monoidal.Instance.Nat using (Nat,+,0) +open Morphism (Setoids c ℓ) using (_≅_) +open Lax using (SymmetricMonoidalFunctor) + +open BinaryProducts products using (-×-) + +open import Data.Circuit.Gate using (Gate; cast-gate; cast-gate-trans; cast-gate-is-id; subst-is-cast-gate) + +record Hypergraph (v : ℕ) : Set c where + + field + h : ℕ + a : Fin h → ℕ + j : (e : Fin h) → Fin (a e) → Fin v + l : (e : Fin h) → Gate (a e) + +record Hypergraph-same {n : ℕ} (H H′ : Hypergraph n) : Set ℓ where + + open Hypergraph H public + open Hypergraph H′ renaming (h to h′; a to a′; j to j′; l to l′) public + + field + ↔h : Fin h ↔ Fin h′ + + open Inverse ↔h public + + field + ≗a : a ≗ a′ ∘ to + ≗j : (e : Fin h) + (i : Fin (a e)) + → j e i + ≡ j′ (to e) (cast (≗a e) i) + ≗l : (e : Fin h) + → l e + ≡ cast-gate (sym (≗a e)) (l′ (to e)) + +private + + variable + n n′ m m′ o : ℕ + H H′ H″ H₁ H₁′ : Hypergraph n + H₂ H₂′ : Hypergraph m + H₃ : Hypergraph o + +Hypergraph-same-refl : Hypergraph-same H H +Hypergraph-same-refl {_} {H} = record + { ↔h = ↔-id (Fin h) + ; ≗a = cong a ∘ erefl + ; ≗j = λ e i → cong (j e) (sym (cast-is-id refl i)) + ; ≗l = λ { e → sym (cast-gate-is-id refl (l e)) } + } + where + open Hypergraph H + +sym-sym : {A : Set} {x y : A} (p : x ≡ y) → sym (sym p) ≡ p +sym-sym refl = refl + +Hypergraph-same-sym : Hypergraph-same H H′ → Hypergraph-same H′ H +Hypergraph-same-sym {V} {H} {H′} ≡H = record + { ↔h = ↔-sym ↔h + ; ≗a = sym ∘ a∘from≗a′ + ; ≗j = ≗j′ + ; ≗l = ≗l′ + } + where + open Hypergraph-same ≡H + open ≡-Reasoning + a∘from≗a′ : a ∘ from ≗ a′ + a∘from≗a′ x = begin + (a ∘ from) x ≡⟨ (≗a ∘ from) x ⟩ + (a′ ∘ to ∘ from) x ≡⟨ (cong a′ ∘ inverseˡ ∘ erefl ∘ from) x ⟩ + a′ x ∎ + id≗to∘from : id ≗ to ∘ from + id≗to∘from e = sym (inverseˡ refl) + ≗arity′ : a′ ≗ a ∘ from + ≗arity′ e = sym (a∘from≗a′ e) + ≗arity- : a′ ≗ a′ ∘ to ∘ from + ≗arity- e = cong a′ (id≗to∘from e) + + ≗j′ : (e : Fin h′) (i : Fin (a′ e)) → j′ e i ≡ j (from e) (cast (≗arity′ e) i) + ≗j′ e i = begin + j′ e i ≡⟨ dcong₂ j′ (id≗to∘from e) (subst-∘ (id≗to∘from e)) ⟩ + j′ (to (from e)) (subst Fin (cong a′ (id≗to∘from e)) i) ≡⟨ cong (j′ (to (from e))) (subst-is-cast (cong a′ (id≗to∘from e)) i) ⟩ + j′ (to (from e)) (cast (cong a′ (id≗to∘from e)) i) ≡⟨⟩ + j′ (to (from e)) (cast (trans (≗arity′ e) (≗a (from e))) i) ≡⟨ cong (j′ (to (from e))) (cast-trans (≗arity′ e) (≗a (from e)) i) ⟨ + j′ (to (from e)) (cast (≗a (from e)) (cast (≗arity′ e) i)) ≡⟨ ≗j (from e) (cast (≗arity′ e) i) ⟨ + j (from e) (cast (≗arity′ e) i) ∎ + + ≗l′ : (e : Fin h′) → l′ e ≡ cast-gate (sym (sym (a∘from≗a′ e))) (l (from e)) + ≗l′ e = begin + l′ e ≡⟨ dcong l′ (sym (id≗to∘from e)) ⟨ + subst (Gate ∘ a′) (sym (id≗to∘from e)) (l′ (to (from e))) ≡⟨ subst-∘ (sym (id≗to∘from e)) ⟩ + subst Gate (cong a′ (sym (id≗to∘from e))) (l′ (to (from e))) ≡⟨ subst-is-cast-gate (cong a′ (sym (id≗to∘from e))) (l′ (to (from e))) ⟩ + cast-gate _ (l′ (to (from e))) ≡⟨ cast-gate-trans _ (sym (sym (a∘from≗a′ e))) (l′ (to (from e))) ⟨ + cast-gate (sym (sym (a∘from≗a′ e))) (cast-gate _ (l′ (to (from e)))) ≡⟨ cong (cast-gate (sym (sym (a∘from≗a′ e)))) (≗l (from e)) ⟨ + cast-gate (sym (sym (a∘from≗a′ e))) (l (from e)) ∎ + +Hypergraph-same-trans : Hypergraph-same H H′ → Hypergraph-same H′ H″ → Hypergraph-same H H″ +Hypergraph-same-trans ≡H₁ ≡H₂ = record + { ↔h = ↔h ≡H₂ ↔-∘ ↔h ≡H₁ + ; ≗a = λ { x → trans (≗a ≡H₁ x) (≗a ≡H₂ (to (↔h ≡H₁) x)) } + ; ≗j = λ { e i → trans (≗j ≡H₁ e i) (≗j₂ e i) } + ; ≗l = λ { e → trans (≗l ≡H₁ e) (≗l₂ e) } + } + where + open Hypergraph-same + open Inverse + open ≡-Reasoning + ≗j₂ : (e : Fin (h ≡H₁)) + (i : Fin (a ≡H₁ e)) + → j ≡H₂ (to (↔h ≡H₁) e) (cast (≗a ≡H₁ e) i) + ≡ j′ ≡H₂ (to (↔h ≡H₂) (to (↔h ≡H₁) e)) (cast (trans (≗a ≡H₁ e) (≗a ≡H₂ (to (↔h ≡H₁) e))) i) + ≗j₂ e i = begin + j ≡H₂ (to (↔h ≡H₁) e) (cast (≗a ≡H₁ e) i) + ≡⟨ ≗j ≡H₂ (to (↔h ≡H₁) e) (cast (≗a ≡H₁ e) i) ⟩ + j′ ≡H₂ (to (↔h ≡H₂) (to (↔h ≡H₁) e)) (cast (≗a ≡H₂ (to (↔h ≡H₁) e)) (cast (≗a ≡H₁ e) i)) + ≡⟨ cong (j′ ≡H₂ (to (↔h ≡H₂) (to (↔h ≡H₁) e))) (cast-trans (≗a ≡H₁ e) (≗a ≡H₂ (to (↔h ≡H₁) e)) i) ⟩ + j′ ≡H₂ (to (↔h ≡H₂) (to (↔h ≡H₁) e)) (cast (trans (≗a ≡H₁ e) (≗a ≡H₂ (to (↔h ≡H₁) e))) i) ∎ + ≗l₂ : (e : Fin (h ≡H₁)) → cast-gate _ (l′ ≡H₁ (to ≡H₁ e)) ≡ cast-gate _ (l′ ≡H₂ (to ≡H₂ (to ≡H₁ e))) + ≗l₂ e = trans (cong (cast-gate _) (≗l ≡H₂ (to ≡H₁ e))) (cast-gate-trans _ (sym (≗a ≡H₁ e)) (l′ ≡H₂ (to ≡H₂ (to ≡H₁ e)))) + +Hypergraph-setoid : ℕ → Setoid c ℓ +Hypergraph-setoid p = record + { Carrier = Hypergraph p + ; _≈_ = Hypergraph-same + ; isEquivalence = record + { refl = Hypergraph-same-refl + ; sym = Hypergraph-same-sym + ; trans = Hypergraph-same-trans + } + } + +map-nodes : (Fin n → Fin m) → Hypergraph n → Hypergraph m +map-nodes f H = record + { h = h + ; a = a + ; j = λ e i → f (j e i) + ; l = l + } + where + open Hypergraph H + +Hypergraph-same-cong + : (f : Fin n → Fin m) + → Hypergraph-same H H′ + → Hypergraph-same (map-nodes f H) (map-nodes f H′) +Hypergraph-same-cong f ≡H = record + { ↔h = ↔h + ; ≗a = ≗a + ; ≗j = λ { e i → cong f (≗j e i) } + ; ≗l = ≗l + } + where + open Hypergraph-same ≡H + +Hypergraph-Func : (Fin n → Fin m) → Hypergraph-setoid n ⟶ₛ Hypergraph-setoid m +Hypergraph-Func f = record + { to = map-nodes f + ; cong = Hypergraph-same-cong f + } + +F-resp-≈ + : {f g : Fin n → Fin m} + → f ≗ g + → Hypergraph-same (map-nodes f H) (map-nodes g H) +F-resp-≈ {g = g} f≗g = record + { ↔h = ↔h + ; ≗a = ≗a + ; ≗j = λ { e i → trans (f≗g (j e i)) (cong g (≗j e i)) } + ; ≗l = ≗l + } + where + open Hypergraph-same Hypergraph-same-refl + +homomorphism + : (f : Fin n → Fin m) + → (g : Fin m → Fin o) + → Hypergraph-same (map-nodes (g ∘ f) H) (map-nodes g (map-nodes f H)) +homomorphism {n} {m} {o} {H} f g = record + { ↔h = ↔h + ; ≗a = ≗a + ; ≗j = λ e i → cong (g ∘ f) (≗j e i) + ; ≗l = ≗l + } + where + open Hypergraph-same Hypergraph-same-refl + +F : Functor Nat (Setoids c ℓ) +F = record + { F₀ = λ n → Hypergraph-setoid n + ; F₁ = Hypergraph-Func + ; identity = λ { {n} {H} → Hypergraph-same-refl {H = H} } + ; homomorphism = λ { {f = f} {g = g} → homomorphism f g } + ; F-resp-≈ = λ f≗g → F-resp-≈ f≗g + } + +-- monoidal structure + +discrete : {n : ℕ} → Hypergraph n +discrete = record + { h = 0 + ; a = λ () + ; j = λ () + ; l = λ () + } + +ε : ⊤ₛ ⟶ₛ Hypergraph-setoid 0 +ε = Const ⊤ₛ (Hypergraph-setoid 0) discrete + +module _ (H₁ : Hypergraph n) (H₂ : Hypergraph m) where + private + module H₁ = Hypergraph H₁ + module H₂ = Hypergraph H₂ + j+j : (e : Fin (H₁.h + H₂.h)) + → Fin ([ H₁.a , H₂.a ] (splitAt H₁.h e)) + → Fin (n + m) + j+j e i with splitAt H₁.h e + ... | inj₁ e₁ = H₁.j e₁ i ↑ˡ m + ... | inj₂ e₂ = n ↑ʳ H₂.j e₂ i + l+l : (e : Fin (H₁.h + H₂.h)) → Gate ([ H₁.a , H₂.a ] (splitAt H₁.h e)) + l+l e with splitAt H₁.h e + ... | inj₁ e₁ = H₁.l e₁ + ... | inj₂ e₂ = H₂.l e₂ + +together : Hypergraph n → Hypergraph m → Hypergraph (n + m) +together {n} {m} H₁ H₂ = record + { h = h H₁ + h H₂ + ; a = [ a H₁ , a H₂ ] ∘ splitAt (h H₁) + ; j = j+j H₁ H₂ + ; l = l+l H₁ H₂ + } + where + open Hypergraph + ++-resp-↔ + : {n n′ m m′ : ℕ} + → Fin n ↔ Fin n′ + → Fin m ↔ Fin m′ + → Fin (n + m) ↔ Fin (n′ + m′) ++-resp-↔ {n} {n′} {m} {m′} ↔n ↔m = record + { to = join n′ m′ ∘ map ↔n.to ↔m.to ∘ splitAt n + ; from = join n m ∘ map ↔n.from ↔m.from ∘ splitAt n′ + ; to-cong = cong (join n′ m′ ∘ map ↔n.to ↔m.to ∘ splitAt n) + ; from-cong = cong (join n m ∘ map ↔n.from ↔m.from ∘ splitAt n′) + ; inverse = (λ { refl → to∘from _ }) , λ { refl → from∘to _ } + } + where + module ↔n = Inverse ↔n + module ↔m = Inverse ↔m + open ≡-Reasoning + to∘from : join n′ m′ ∘ map ↔n.to ↔m.to ∘ splitAt n ∘ join n m ∘ map ↔n.from ↔m.from ∘ splitAt n′ ≗ id + to∘from x = begin + (join n′ m′ ∘ map ↔n.to ↔m.to ∘ splitAt n ∘ join n m ∘ map ↔n.from ↔m.from ∘ splitAt n′) x + ≡⟨ cong + (join n′ m′ ∘ map ↔n.to ↔m.to) + (splitAt-join n m (map ↔n.from ↔m.from (splitAt n′ x))) ⟩ + (join n′ m′ ∘ map ↔n.to ↔m.to ∘ map ↔n.from ↔m.from ∘ splitAt n′) x + ≡⟨ cong (join n′ m′) (map-map (splitAt n′ x)) ⟩ + (join n′ m′ ∘ map (↔n.to ∘ ↔n.from) (↔m.to ∘ ↔m.from) ∘ splitAt n′) x + ≡⟨ cong + (join n′ m′) + (map-cong + (λ _ → ↔n.inverseˡ refl) + (λ _ → ↔m.inverseˡ refl) + (splitAt n′ x)) ⟩ + (join n′ m′ ∘ map id id ∘ splitAt n′) x ≡⟨ [,]-map (splitAt n′ x) ⟩ + (join n′ m′ ∘ splitAt n′) x ≡⟨ join-splitAt n′ m′ x ⟩ + x ∎ + from∘to : join n m ∘ map ↔n.from ↔m.from ∘ splitAt n′ ∘ join n′ m′ ∘ map ↔n.to ↔m.to ∘ splitAt n ≗ id + from∘to x = begin + (join n m ∘ map ↔n.from ↔m.from ∘ splitAt n′ ∘ join n′ m′ ∘ map ↔n.to ↔m.to ∘ splitAt n) x + ≡⟨ cong + (join n m ∘ map ↔n.from ↔m.from) + (splitAt-join n′ m′ (map ↔n.to ↔m.to (splitAt n x))) ⟩ + (join n m ∘ map ↔n.from ↔m.from ∘ map ↔n.to ↔m.to ∘ splitAt n) x + ≡⟨ cong (join n m) (map-map (splitAt n x)) ⟩ + (join n m ∘ map (↔n.from ∘ ↔n.to) (↔m.from ∘ ↔m.to) ∘ splitAt n) x + ≡⟨ cong + (join n m) + (map-cong + (λ _ → ↔n.inverseʳ refl) + (λ _ → ↔m.inverseʳ refl) + (splitAt n x)) ⟩ + (join n m ∘ map id id ∘ splitAt n) x ≡⟨ [,]-map (splitAt n x) ⟩ + (join n m ∘ splitAt n) x ≡⟨ join-splitAt n m x ⟩ + x ∎ + +together-resp-same + : Hypergraph-same H₁ H₁′ + → Hypergraph-same H₂ H₂′ + → Hypergraph-same (together H₁ H₂) (together H₁′ H₂′) +together-resp-same {n} {H₁} {H₁′} {m} {H₂} {H₂′} ≡H₁ ≡H₂ = record + { ↔h = +-resp-↔ ≡H₁.↔h ≡H₂.↔h + ; ≗a = ≗a + ; ≗j = ≗j + ; ≗l = ≗l + } + where + module ≡H₁ = Hypergraph-same ≡H₁ + module ≡H₂ = Hypergraph-same ≡H₂ + module H₁+H₂ = Hypergraph (together H₁ H₂) + module H₁+H₂′ = Hypergraph (together H₁′ H₂′) + open ≡-Reasoning + open Inverse + open Hypergraph + ≗a : [ ≡H₁.a , ≡H₂.a ] ∘ splitAt ≡H₁.h + ≗ [ ≡H₁.a′ , ≡H₂.a′ ] ∘ splitAt ≡H₁.h′ + ∘ join ≡H₁.h′ ≡H₂.h′ ∘ map ≡H₁.to ≡H₂.to ∘ splitAt ≡H₁.h + ≗a e = begin + [ ≡H₁.a , ≡H₂.a ] (splitAt ≡H₁.h e) ≡⟨ [,]-cong ≡H₁.≗a ≡H₂.≗a (splitAt ≡H₁.h e) ⟩ + ([ ≡H₁.a′ ∘ ≡H₁.to , ≡H₂.a′ ∘ ≡H₂.to ] ∘ splitAt ≡H₁.h) e ≡⟨ [,]-map (splitAt ≡H₁.h e) ⟨ + ([ ≡H₁.a′ , ≡H₂.a′ ] ∘ map ≡H₁.to ≡H₂.to ∘ splitAt ≡H₁.h) e ≡⟨ (cong [ ≡H₁.a′ , ≡H₂.a′ ] ∘ splitAt-join ≡H₁.h′ ≡H₂.h′ ∘ map ≡H₁.to ≡H₂.to ∘ splitAt ≡H₁.h) e ⟨ + ([ ≡H₁.a′ , ≡H₂.a′ ] ∘ splitAt ≡H₁.h′ ∘ join ≡H₁.h′ ≡H₂.h′ ∘ map ≡H₁.to ≡H₂.to ∘ splitAt ≡H₁.h) e ∎ + ≗j : (e : Fin H₁+H₂.h) + (i : Fin (H₁+H₂.a e)) + → H₁+H₂.j e i + ≡ H₁+H₂′.j (to (+-resp-↔ ≡H₁.↔h ≡H₂.↔h) e) (cast (≗a e) i) + ≗j e i with splitAt ≡H₁.h e + ... | inj₁ e₁ rewrite splitAt-↑ˡ ≡H₁.h′ (≡H₁.to e₁) ≡H₂.h′ = cong (_↑ˡ m) (≡H₁.≗j e₁ i) + ... | inj₂ e₂ rewrite splitAt-↑ʳ ≡H₁.h′ ≡H₂.h′ (≡H₂.to e₂) = cong (n ↑ʳ_) (≡H₂.≗j e₂ i) + ≗l : (e : Fin H₁+H₂.h) → l+l H₁ H₂ e ≡ cast-gate (sym (≗a e)) (l+l H₁′ H₂′ (to (+-resp-↔ ≡H₁.↔h ≡H₂.↔h) e)) + ≗l e with splitAt ≡H₁.h e | .{≗a e} + ... | inj₁ e₁ rewrite splitAt-↑ˡ ≡H₁.h′ (≡H₁.to e₁) ≡H₂.h′ = ≡H₁.≗l e₁ + ... | inj₂ e₂ rewrite splitAt-↑ʳ ≡H₁.h′ ≡H₂.h′ (≡H₂.to e₂) = ≡H₂.≗l e₂ + +commute + : (f : Fin n → Fin n′) + → (g : Fin m → Fin m′) + → Hypergraph-same + (together (map-nodes f H₁) (map-nodes g H₂)) + (map-nodes ([ (_↑ˡ m′) ∘ f , (n′ ↑ʳ_) ∘ g ] ∘ splitAt n) (together H₁ H₂)) +commute {n} {n′} {m} {m′} {H₁} {H₂} f g = record + { ↔h = ≡H₁+H₂.↔h + ; ≗a = ≡H₁+H₂.≗a + ; ≗j = ≗j + ; ≗l = ≗l + } + where + module H₁ = Hypergraph H₁ + module H₂ = Hypergraph H₂ + module H₁+H₂ = Hypergraph (together H₁ H₂) + module ≡H₁+H₂ = Hypergraph-same {H = together H₁ H₂} Hypergraph-same-refl + open Hypergraph + open ≡-Reasoning + ≗j : (e : Fin (H₁.h + H₂.h)) + (i : Fin (([ H₁.a , H₂.a ] ∘ splitAt H₁.h) e)) + → j (together (map-nodes f H₁) (map-nodes g H₂)) e i + ≡ j (map-nodes ([ (_↑ˡ m′) ∘ f , (n′ ↑ʳ_) ∘ g ] ∘ splitAt n) (together H₁ H₂)) (≡H₁+H₂.to e) (cast refl i) + ≗j e i rewrite (cast-is-id refl i) with splitAt H₁.h e + ... | inj₁ e₁ rewrite splitAt-↑ˡ n (H₁.j e₁ i) m = refl + ... | inj₂ e₂ rewrite splitAt-↑ʳ n m (H₂.j e₂ i) = refl + ≗l : (e : Fin (H₁.h + H₂.h)) + → l+l (map-nodes f H₁) (map-nodes g H₂) e + ≡ cast-gate refl (l+l H₁ H₂ (≡H₁+H₂.to e)) + ≗l e rewrite cast-gate-is-id refl (l+l H₁ H₂ (≡H₁+H₂.to e)) with splitAt H₁.h e + ... | inj₁ e₁ = refl + ... | inj₂ e₁ = refl + +⊗-homomorphism : NaturalTransformation (-×- ∘′ (F ⁂ F)) (F ∘′ -+-) +⊗-homomorphism = record + { η = λ { (m , n) → η } + ; commute = λ { (f , g) {H₁ , H₂} → commute {H₁ = H₁} {H₂ = H₂} f g } + ; sym-commute = λ { (f , g) {H₁ , H₂} → Hypergraph-same-sym (commute {H₁ = H₁} {H₂ = H₂} f g) } + } + where + η : Hypergraph-setoid n ×ₛ Hypergraph-setoid m ⟶ₛ Hypergraph-setoid (n + m) + η = record + { to = λ { (H₁ , H₂) → together H₁ H₂ } + ; cong = λ { (≡H₁ , ≡H₂) → together-resp-same ≡H₁ ≡H₂ } + } + ++-assoc-↔ : ∀ (x y z : ℕ) → Fin (x + y + z) ↔ Fin (x + (y + z)) ++-assoc-↔ x y z = record + { to = to + ; from = from + ; to-cong = λ { refl → refl } + ; from-cong = λ { refl → refl } + ; inverse = (λ { refl → isoˡ _ }) , λ { refl → isoʳ _ } + } + where + module Nat = Morphism Nat + open Nat._≅_ (Nat-+-assoc {x} {y} {z}) + +associativity + : {X Y Z : ℕ} + → {H₁ : Hypergraph X} + → {H₂ : Hypergraph Y} + → {H₃ : Hypergraph Z} + → Hypergraph-same + (map-nodes (Inverse.to (+-assoc-↔ X Y Z)) (together (together H₁ H₂) H₃)) + (together H₁ (together H₂ H₃)) +associativity {X} {Y} {Z} {H₁} {H₂} {H₃} = record + { ↔h = ↔h + ; ≗a = ≗a + ; ≗j = ≗j + ; ≗l = ≗l + } + where + module H₁ = Hypergraph H₁ + module H₂ = Hypergraph H₂ + module H₃ = Hypergraph H₃ + open ≡-Reasoning + open Hypergraph + ↔h : Fin (H₁.h + H₂.h + H₃.h) ↔ Fin (H₁.h + (H₂.h + H₃.h)) + ↔h = +-assoc-↔ H₁.h H₂.h H₃.h + ≗a : (x : Fin (H₁.h + H₂.h + H₃.h)) + → [ [ H₁.a , H₂.a ] ∘ splitAt H₁.h , H₃.a ] (splitAt (H₁.h + H₂.h) x) + ≡ [ H₁.a , [ H₂.a , H₃.a ] ∘ splitAt H₂.h ] (splitAt H₁.h ([ [ _↑ˡ H₂.h + H₃.h , (H₁.h ↑ʳ_) ∘ (_↑ˡ H₃.h) ] ∘ splitAt H₁.h , (H₁.h ↑ʳ_) ∘ (H₂.h ↑ʳ_) ] (splitAt (H₁.h + H₂.h) x))) + ≗a x = begin + ([ [ H₁.a , H₂.a ] ∘ splitAt H₁.h , H₃.a ] ∘ splitAt (H₁.h + H₂.h)) x + ≡⟨⟩ + ([ [ H₁.a , [ H₂.a , H₃.a ] ∘ inj₁ ] ∘ splitAt H₁.h , H₃.a ] ∘ splitAt (H₁.h + H₂.h)) x + ≡⟨ [-,]-cong ([,-]-cong (cong [ H₂.a , H₃.a ] ∘ (λ i → splitAt-↑ˡ H₂.h i H₃.h)) ∘ splitAt H₁.h) (splitAt (H₁.h + H₂.h) x) ⟨ + ([ [ H₁.a , [ H₂.a , H₃.a ] ∘ splitAt H₂.h ∘ (_↑ˡ H₃.h) ] ∘ splitAt H₁.h , H₃.a ] ∘ splitAt (H₁.h + H₂.h)) x + ≡⟨ [-,]-cong ([,]-∘ [ H₁.a , [ H₂.a , H₃.a ] ∘ splitAt H₂.h ] ∘ splitAt H₁.h) (splitAt (H₁.h + H₂.h) x) ⟨ + ([ [ H₁.a , [ H₂.a , H₃.a ] ∘ splitAt H₂.h ] ∘ map₂ (_↑ˡ H₃.h) ∘ splitAt H₁.h , H₃.a ] ∘ splitAt (H₁.h + H₂.h)) x + ≡⟨⟩ + ([ [ H₁.a , [ H₂.a , H₃.a ] ∘ splitAt H₂.h ] ∘ map₂ (_↑ˡ H₃.h) ∘ splitAt H₁.h , [ H₂.a , H₃.a ] ∘ inj₂ ] ∘ splitAt (H₁.h + H₂.h)) x + ≡⟨ [,-]-cong (cong [ H₂.a , H₃.a ] ∘ splitAt-↑ʳ H₂.h H₃.h) (splitAt (H₁.h + H₂.h) x) ⟨ + ([ [ H₁.a , [ H₂.a , H₃.a ] ∘ splitAt H₂.h ] ∘ map₂ (_↑ˡ H₃.h) ∘ splitAt H₁.h , [ H₂.a , H₃.a ] ∘ splitAt H₂.h ∘ (H₂.h ↑ʳ_) ] ∘ splitAt (H₁.h + H₂.h)) x + ≡⟨⟩ + ([ [ H₁.a , [ H₂.a , H₃.a ] ∘ splitAt H₂.h ] ∘ map₂ (_↑ˡ H₃.h) ∘ splitAt H₁.h , [ H₁.a , [ H₂.a , H₃.a ] ∘ splitAt H₂.h ] ∘ inj₂ ∘ (H₂.h ↑ʳ_) ] ∘ splitAt (H₁.h + H₂.h)) x + ≡⟨ [,]-∘ [ H₁.a , [ H₂.a , H₃.a ] ∘ splitAt H₂.h ] (splitAt (H₁.h + H₂.h) x) ⟨ + ([ H₁.a , [ H₂.a , H₃.a ] ∘ splitAt H₂.h ] ∘ [ map₂ (_↑ˡ H₃.h) ∘ splitAt H₁.h , inj₂ ∘ (H₂.h ↑ʳ_) ] ∘ splitAt (H₁.h + H₂.h)) x + ≡⟨ cong [ H₁.a , [ H₂.a , H₃.a ] ∘ splitAt H₂.h ] ([,-]-cong (splitAt-↑ʳ H₁.h (H₂.h + H₃.h) ∘ (H₂.h ↑ʳ_)) (splitAt (H₁.h + H₂.h) x)) ⟨ + ([ H₁.a , [ H₂.a , H₃.a ] ∘ splitAt H₂.h ] ∘ [ map₂ (_↑ˡ H₃.h) ∘ splitAt H₁.h , splitAt H₁.h ∘ (H₁.h ↑ʳ_) ∘ (H₂.h ↑ʳ_) ] ∘ splitAt (H₁.h + H₂.h)) x + ≡⟨ cong [ H₁.a , [ H₂.a , H₃.a ] ∘ splitAt H₂.h ] ([-,]-cong (splitAt-join H₁.h (H₂.h + H₃.h) ∘ map₂ (_↑ˡ H₃.h) ∘ splitAt H₁.h) (splitAt (H₁.h + H₂.h) x)) ⟨ + ([ H₁.a , [ H₂.a , H₃.a ] ∘ splitAt H₂.h ] ∘ [ splitAt H₁.h ∘ join H₁.h (H₂.h + H₃.h) ∘ map₂ (_↑ˡ H₃.h) ∘ splitAt H₁.h , splitAt H₁.h ∘ (H₁.h ↑ʳ_) ∘ (H₂.h ↑ʳ_) ] ∘ splitAt (H₁.h + H₂.h)) x + ≡⟨ cong [ H₁.a , [ H₂.a , H₃.a ] ∘ splitAt H₂.h ] ([,]-∘ (splitAt H₁.h) (splitAt (H₁.h + H₂.h) x)) ⟨ + ([ H₁.a , [ H₂.a , H₃.a ] ∘ splitAt H₂.h ] ∘ splitAt H₁.h ∘ [ join H₁.h (H₂.h + H₃.h) ∘ map₂ (_↑ˡ H₃.h) ∘ splitAt H₁.h , (H₁.h ↑ʳ_) ∘ (H₂.h ↑ʳ_) ] ∘ splitAt (H₁.h + H₂.h)) x + ≡⟨⟩ + ([ H₁.a , [ H₂.a , H₃.a ] ∘ splitAt H₂.h ] ∘ splitAt H₁.h ∘ [ [ _↑ˡ H₂.h + H₃.h , H₁.h ↑ʳ_ ] ∘ [ inj₁ , inj₂ ∘ (_↑ˡ H₃.h) ] ∘ splitAt H₁.h , (H₁.h ↑ʳ_) ∘ (H₂.h ↑ʳ_) ] ∘ splitAt (H₁.h + H₂.h)) x + ≡⟨ cong ([ H₁.a , [ H₂.a , H₃.a ] ∘ splitAt H₂.h ] ∘ splitAt H₁.h) ([-,]-cong ([,]-∘ [ _↑ˡ H₂.h + H₃.h , H₁.h ↑ʳ_ ] ∘ splitAt H₁.h) (splitAt (H₁.h + H₂.h) x)) ⟩ + ([ H₁.a , [ H₂.a , H₃.a ] ∘ splitAt H₂.h ] ∘ splitAt H₁.h ∘ [ [ _↑ˡ H₂.h + H₃.h , (H₁.h ↑ʳ_) ∘ (_↑ˡ H₃.h) ] ∘ splitAt H₁.h , (H₁.h ↑ʳ_) ∘ (H₂.h ↑ʳ_) ] ∘ splitAt (H₁.h + H₂.h)) x ∎ + ≗j : (e : Fin (H₁.h + H₂.h + H₃.h)) + (i : Fin ([ [ H₁.a , H₂.a ] ∘ splitAt H₁.h , H₃.a ] (splitAt (H₁.h + H₂.h) e))) + → Inverse.to (+-assoc-↔ X Y Z) (j+j (together H₁ H₂) H₃ e i) + ≡ j+j H₁ (together H₂ H₃) (Inverse.to ↔h e) (cast (≗a e) i) + ≗j e i with splitAt (H₁.h + H₂.h) e + ≗j e i | inj₁ e₁₂ with splitAt H₁.h e₁₂ + ≗j e i | inj₁ e₁₂ | inj₁ e₁ + rewrite splitAt-↑ˡ H₁.h e₁ (H₂.h + H₃.h) + rewrite splitAt-↑ˡ (X + Y) (H₁.j e₁ i ↑ˡ Y) Z + rewrite splitAt-↑ˡ X (H₁.j e₁ i) Y = cong ((_↑ˡ Y + Z) ∘ H₁.j e₁) (sym (cast-is-id refl i)) + ≗j e i | inj₁ e₁₂ | inj₂ e₂ + rewrite splitAt-↑ʳ H₁.h (H₂.h + H₃.h) (e₂ ↑ˡ H₃.h) + rewrite splitAt-↑ˡ H₂.h e₂ H₃.h + rewrite splitAt-↑ˡ (X + Y) (X ↑ʳ H₂.j e₂ i) Z + rewrite splitAt-↑ʳ X Y (H₂.j e₂ i) = cong ((X ↑ʳ_) ∘ (_↑ˡ Z) ∘ H₂.j e₂) (sym (cast-is-id refl i)) + ≗j e i | inj₂ e₃ + rewrite splitAt-↑ʳ H₁.h (H₂.h + H₃.h) (H₂.h ↑ʳ e₃) + rewrite splitAt-↑ʳ H₂.h H₃.h e₃ + rewrite splitAt-↑ʳ (X + Y) Z (H₃.j e₃ i) = cong ((X ↑ʳ_) ∘ (Y ↑ʳ_) ∘ H₃.j e₃) (sym (cast-is-id refl i)) + ≗l : (e : Fin (H₁.h + H₂.h + H₃.h)) + → l (map-nodes (Inverse.to (+-assoc-↔ X Y Z)) (together (together H₁ H₂) H₃)) e + ≡ cast-gate (sym (≗a e)) (l (together H₁ (together H₂ H₃)) (Inverse.to ↔h e)) + ≗l e with splitAt (H₁.h + H₂.h) e + ≗l e | inj₁ e₁₂ with splitAt H₁.h e₁₂ + ≗l e | inj₁ e₁₂ | inj₁ e₁ + rewrite splitAt-↑ˡ H₁.h e₁ (H₂.h + H₃.h) = sym (cast-gate-is-id refl (H₁.l e₁)) + ≗l e | inj₁ e₁₂ | inj₂ e₂ + rewrite splitAt-↑ʳ H₁.h (H₂.h + H₃.h) (e₂ ↑ˡ H₃.h) + rewrite splitAt-↑ˡ H₂.h e₂ H₃.h = sym (cast-gate-is-id refl (H₂.l e₂)) + ≗l e | inj₂ e₃ + rewrite splitAt-↑ʳ H₁.h (H₂.h + H₃.h) (H₂.h ↑ʳ e₃) + rewrite splitAt-↑ʳ H₂.h H₃.h e₃ = sym (cast-gate-is-id refl (H₃.l e₃)) + +n+0↔n : ∀ n → Fin (n + 0) ↔ Fin n +n+0↔n n = record + { to = to + ; from = from + ; to-cong = λ { refl → refl } + ; from-cong = λ { refl → refl } + ; inverse = (λ { refl → to∘from _ }) , λ { refl → from∘to _ } + } + where + to : Fin (n + 0) → Fin n + to x with inj₁ x₁ ← splitAt n x = x₁ + from : Fin n → Fin (n + 0) + from x = x ↑ˡ 0 + from∘to : (x : Fin (n + 0)) → from (to x) ≡ x + from∘to x with inj₁ x₁ ← splitAt n x in eq = splitAt⁻¹-↑ˡ eq + to∘from : (x : Fin n) → to (from x) ≡ x + to∘from x rewrite splitAt-↑ˡ n x 0 = refl + +unitaryʳ : Hypergraph-same (map-nodes ([ id , (λ ()) ] ∘ splitAt n) (together H (discrete {0}))) H +unitaryʳ {n} {H} = record + { ↔h = h+0↔h + ; ≗a = ≗a + ; ≗j = ≗j + ; ≗l = ≗l + } + where + module H = Hypergraph H + module H+0 = Hypergraph (together H (discrete {0})) + h+0↔h : Fin H+0.h ↔ Fin H.h + h+0↔h = n+0↔n H.h + ≗a : (e : Fin (H.h + 0)) → [ H.a , (λ ()) ] (splitAt H.h e) ≡ H.a (Inverse.to h+0↔h e) + ≗a e with inj₁ e₁ ← splitAt H.h e in eq = refl + ≗j : (e : Fin (H.h + 0)) + (i : Fin ([ H.a , (λ ()) ] (splitAt H.h e))) + → [ (λ x → x) , (λ ()) ] (splitAt n (j+j H discrete e i)) + ≡ H.j (Inverse.to h+0↔h e) (cast (≗a e) i) + ≗j e i = ≗j-aux (splitAt H.h e) refl (j+j H discrete e) refl (≗a e) i + where + ≗j-aux + : (w : Fin H.h ⊎ Fin 0) + → (eq₁ : splitAt H.h e ≡ w) + → (w₁ : Fin ([ H.a , (λ ()) ] w) → Fin (n + 0)) + → j+j H discrete e ≡ subst (λ hole → Fin ([ H.a , (λ ()) ] hole) → Fin (n + 0)) (sym eq₁) w₁ + → (w₂ : [ H.a , (λ ()) ] w ≡ H.a (Inverse.to h+0↔h e)) + (i : Fin ([ H.a , (λ ()) ] w)) + → [ (λ x → x) , (λ ()) ] (splitAt n (w₁ i)) + ≡ H.j (Inverse.to h+0↔h e) (cast w₂ i) + ≗j-aux (inj₁ e₁) eq w₁ eq₁ w₂ i + with (inj₁ x) ← splitAt n (w₁ i) in eq₂ + rewrite eq = trans + (↑ˡ-injective 0 x (H.j e₁ i) (trans (splitAt⁻¹-↑ˡ eq₂) (sym (cong-app eq₁ i)))) + (cong (H.j e₁) (sym (cast-is-id refl i))) + ≗l : (e : Fin (H.h + 0)) + → l+l H discrete e + ≡ cast-gate (sym (≗a e)) (H.l (Inverse.to h+0↔h e)) + ≗l e with splitAt H.h e | {(≗a e)} + ... | inj₁ e₁ = sym (cast-gate-is-id refl (H.l e₁)) + ++-comm-↔ : ∀ (n m : ℕ) → Fin (n + m) ↔ Fin (m + n) ++-comm-↔ n m = record + { to = join m n ∘ swap ∘ splitAt n + ; from = join n m ∘ swap ∘ splitAt m + ; to-cong = λ { refl → refl } + ; from-cong = λ { refl → refl } + ; inverse = (λ { refl → to∘from _ }) , λ { refl → from∘to _ } + } + where + open ≡-Reasoning + to∘from : join m n ∘ swap ∘ splitAt n ∘ join n m ∘ swap ∘ splitAt m ≗ id + to∘from x = begin + (join m n ∘ swap ∘ splitAt n ∘ join n m ∘ swap ∘ splitAt m) x ≡⟨ (cong (join m n ∘ swap) ∘ splitAt-join n m ∘ swap ∘ splitAt m) x ⟩ + (join m n ∘ swap ∘ swap ∘ splitAt m) x ≡⟨ (cong (join m n) ∘ swap-involutive ∘ splitAt m) x ⟩ + (join m n ∘ splitAt m) x ≡⟨ join-splitAt m n x ⟩ + x ∎ + from∘to : join n m ∘ swap ∘ splitAt m ∘ join m n ∘ swap ∘ splitAt n ≗ id + from∘to x = begin + (join n m ∘ swap ∘ splitAt m ∘ join m n ∘ swap ∘ splitAt n) x ≡⟨ (cong (join n m ∘ swap) ∘ splitAt-join m n ∘ swap ∘ splitAt n) x ⟩ + (join n m ∘ swap ∘ swap ∘ splitAt n) x ≡⟨ (cong (join n m) ∘ swap-involutive ∘ splitAt n) x ⟩ + (join n m ∘ splitAt n) x ≡⟨ join-splitAt n m x ⟩ + x ∎ + +[,]∘swap : {A B C : Set} {f : A → C} {g : B → C} → [ f , g ] ∘ swap ≗ [ g , f ] +[,]∘swap (inj₁ x) = refl +[,]∘swap (inj₂ y) = refl + +braiding + : {n m : ℕ} + {H₁ : Hypergraph n} + {H₂ : Hypergraph m} + → Hypergraph-same (map-nodes ([ m ↑ʳ_ , _↑ˡ n ] ∘ splitAt n) (together H₁ H₂)) (together H₂ H₁) +braiding {n} {m} {H₁} {H₂} = record + { ↔h = +-comm-↔ H₁.h H₂.h + ; ≗a = ≗a + ; ≗j = ≗j + ; ≗l = ≗l + } + where + open ≡-Reasoning + module H₁ = Hypergraph H₁ + module H₂ = Hypergraph H₂ + ≗a : (e : Fin (H₁.h + H₂.h)) + → [ H₁.a , H₂.a ] (splitAt H₁.h e) + ≡ [ H₂.a , H₁.a ] (splitAt H₂.h (join H₂.h H₁.h (swap (splitAt H₁.h e)))) + ≗a e = begin + [ H₁.a , H₂.a ] (splitAt H₁.h e) ≡⟨ [,]∘swap (splitAt H₁.h e) ⟨ + [ H₂.a , H₁.a ] (swap (splitAt H₁.h e)) ≡⟨ cong [ H₂.a , H₁.a ] (splitAt-join H₂.h H₁.h (swap (splitAt H₁.h e))) ⟨ + [ H₂.a , H₁.a ] (splitAt H₂.h (join H₂.h H₁.h (swap (splitAt H₁.h e)))) ∎ + ≗j : (e : Fin (Hypergraph.h (map-nodes ([ m ↑ʳ_ , _↑ˡ n ] ∘ splitAt n) (together H₁ H₂)))) + (i : Fin (Hypergraph.a (map-nodes ([ m ↑ʳ_ , _↑ˡ n ] ∘ splitAt n) (together H₁ H₂)) e)) + → Hypergraph.j (map-nodes ([ _↑ʳ_ m , _↑ˡ n ] ∘ splitAt n) (together H₁ H₂)) e i + ≡ Hypergraph.j (together H₂ H₁) (Inverse.to (+-comm-↔ H₁.h H₂.h) e) (cast (≗a e) i) + ≗j e i with splitAt H₁.h e + ≗j e i | inj₁ e₁ + rewrite splitAt-↑ˡ n (H₁.j e₁ i) m + rewrite splitAt-↑ʳ H₂.h H₁.h e₁ = cong ((m ↑ʳ_) ∘ H₁.j e₁) (sym (cast-is-id refl i)) + ≗j e i | inj₂ e₂ + rewrite splitAt-↑ʳ n m (H₂.j e₂ i) + rewrite splitAt-↑ˡ H₂.h e₂ H₁.h = cong ((_↑ˡ n) ∘ H₂.j e₂) (sym (cast-is-id refl i)) + ≗l : (e : Fin (H₁.h + H₂.h)) + → l+l H₁ H₂ e + ≡ cast-gate (sym (≗a e)) (l+l H₂ H₁ (Inverse.to (+-comm-↔ H₁.h H₂.h) e)) + ≗l e with splitAt H₁.h e | .{≗a e} + ≗l e | inj₁ e₁ rewrite splitAt-↑ʳ H₂.h H₁.h e₁ = sym (cast-gate-is-id refl (H₁.l e₁)) + ≗l e | inj₂ e₂ rewrite splitAt-↑ˡ H₂.h e₂ H₁.h = sym (cast-gate-is-id refl (H₂.l e₂)) + +opaque + unfolding ×-symmetric′ + Circ : SymmetricMonoidalFunctor Nat,+,0 Setoids-× + Circ = record + { F = F + ; isBraidedMonoidal = record + { isMonoidal = record + { ε = ε + ; ⊗-homo = ntHelper record + { η = λ { (m , n) → η } + ; commute = λ { (f , g) {H₁ , H₂} → commute {H₁ = H₁} {H₂ = H₂} f g } + } + ; associativity = λ { {X} {Y} {Z} {(H₁ , H₂) , H₃} → associativity {X} {Y} {Z} {H₁} {H₂} {H₃} } + ; unitaryˡ = Hypergraph-same-refl + ; unitaryʳ = unitaryʳ + } + ; braiding-compat = λ { {X} {Y} {H₁ , H₂} → braiding {X} {Y} {H₁} {H₂} } + } + } + where + η : Hypergraph-setoid n ×ₛ Hypergraph-setoid m ⟶ₛ Hypergraph-setoid (n + m) + η = record + { to = λ { (H₁ , H₂) → together H₁ H₂ } + ; cong = λ { (≡H₁ , ≡H₂) → together-resp-same ≡H₁ ≡H₂ } + } + +module F = SymmetricMonoidalFunctor Circ + +open Gate + +and-gate : ⊤ₛ ⟶ₛ Hypergraph-setoid 3 +and-gate = Const ⊤ₛ (Hypergraph-setoid 3) and-graph + where + and-graph : Hypergraph 3 + and-graph = record + { h = 1 + ; a = λ { 0F → 3 } + ; j = λ { 0F → edge-0-nodes } + ; l = λ { 0F → AND } + } + where + edge-0-nodes : Fin 3 → Fin 3 + edge-0-nodes 0F = # 0 + edge-0-nodes 1F = # 1 + edge-0-nodes 2F = # 2 |