diff options
Diffstat (limited to 'DecorationFunctor')
| -rw-r--r-- | DecorationFunctor/Hypergraph/Labeled.agda | 159 |
1 files changed, 58 insertions, 101 deletions
diff --git a/DecorationFunctor/Hypergraph/Labeled.agda b/DecorationFunctor/Hypergraph/Labeled.agda index d8f6e64..083ff80 100644 --- a/DecorationFunctor/Hypergraph/Labeled.agda +++ b/DecorationFunctor/Hypergraph/Labeled.agda @@ -26,7 +26,7 @@ open import Category.Instance.Nat.FinitelyCocomplete using (Nat-FinitelyCocomple open import Data.Empty using (⊥-elim) open import Data.Fin using (#_) open import Data.Fin.Base using (Fin; splitAt; join; zero; suc; _↑ˡ_; _↑ʳ_; Fin′; cast) -open import Data.Fin.Patterns using (0F; 1F) +open import Data.Fin.Patterns using (0F; 1F; 2F) open import Data.Fin.Permutation using (lift₀) open import Data.Fin.Properties using @@ -68,58 +68,20 @@ open Lax using (SymmetricMonoidalFunctor) open BinaryProducts products using (-×-) open BinaryCoproducts coproducts using (-+-) renaming (+-assoc to Nat-+-assoc) - -data Gate : ℕ → Set where - ZERO : Gate 1 - ONE : Gate 1 - NOT : Gate 2 - AND : Gate 3 - OR : Gate 3 - XOR : Gate 3 - NAND : Gate 3 - NOR : Gate 3 - XNOR : Gate 3 - -cast-gate : {e e′ : ℕ} → .(e ≡ e′) → Gate e → Gate e′ -cast-gate {1} {1} eq g = g -cast-gate {2} {2} eq g = g -cast-gate {3} {3} eq g = g - -cast-gate-trans - : {m n o : ℕ} - → .(eq₁ : m ≡ n) - .(eq₂ : n ≡ o) - (g : Gate m) - → cast-gate eq₂ (cast-gate eq₁ g) ≡ cast-gate (trans eq₁ eq₂) g -cast-gate-trans {1} {1} {1} eq₁ eq₂ g = refl -cast-gate-trans {2} {2} {2} eq₁ eq₂ g = refl -cast-gate-trans {3} {3} {3} eq₁ eq₂ g = refl - -cast-gate-is-id : {m : ℕ} .(eq : m ≡ m) (g : Gate m) → cast-gate eq g ≡ g -cast-gate-is-id {1} eq g = refl -cast-gate-is-id {2} eq g = refl -cast-gate-is-id {3} eq g = refl - -subst-is-cast-gate : {m n : ℕ} (eq : m ≡ n) (g : Gate m) → subst Gate eq g ≡ cast-gate eq g -subst-is-cast-gate refl g = sym (cast-gate-is-id refl g) +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 where field h : ℕ a : Fin h → ℕ - - arity : Fin h → ℕ - arity = ℕ.suc ∘ a - - field - j : (e : Fin h) → Fin (arity e) → Fin v - l : (e : Fin h) → Gate (arity e) + 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 where open Hypergraph H public - open Hypergraph H′ renaming (h to h′; a to a′; arity to arity′; j to j′; l to l′) public + open Hypergraph H′ renaming (h to h′; a to a′; j to j′; l to l′) public field ↔h : Fin h ↔ Fin h′ @@ -128,18 +90,13 @@ record Hypergraph-same {n : ℕ} (H H′ : Hypergraph n) : Set where field ≗a : a ≗ a′ ∘ to - - ≗arity : arity ≗ arity′ ∘ to - ≗arity e = cong ℕ.suc (≗a e) - - field ≗j : (e : Fin h) - (i : Fin (arity e)) + (i : Fin (a e)) → j e i - ≡ j′ (to e) (cast (≗arity e) i) + ≡ j′ (to e) (cast (≗a e) i) ≗l : (e : Fin h) → l e - ≡ cast-gate (sym (≗arity e)) (l′ (to e)) + ≡ cast-gate (sym (≗a e)) (l′ (to e)) private @@ -179,28 +136,28 @@ Hypergraph-same-sym {V} {H} {H′} ≡H = record a′ x ∎ id≗to∘from : id ≗ to ∘ from id≗to∘from e = sym (inverseˡ refl) - ≗arity′ : arity′ ≗ arity ∘ from - ≗arity′ e = cong ℕ.suc (sym (a∘from≗a′ e)) - ≗arity- : arity′ ≗ arity′ ∘ to ∘ from - ≗arity- e = cong arity′ (id≗to∘from e) + ≗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 (arity′ e)) → j′ e i ≡ j (from e) (cast (≗arity′ e) i) + ≗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 arity′ (id≗to∘from e)) i) ≡⟨ cong (j′ (to (from e))) (subst-is-cast (cong arity′ (id≗to∘from e)) i) ⟩ - j′ (to (from e)) (cast (cong arity′ (id≗to∘from e)) i) ≡⟨⟩ - j′ (to (from e)) (cast (trans (≗arity′ e) (≗arity (from e))) i) ≡⟨ cong (j′ (to (from e))) (cast-trans (≗arity′ e) (≗arity (from e)) i) ⟨ - j′ (to (from e)) (cast (≗arity (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 (cong ℕ.suc (sym (a∘from≗a′ e)))) (l (from e)) + 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 ∘ arity′) (sym (id≗to∘from e)) (l′ (to (from e))) ≡⟨ subst-∘ (sym (id≗to∘from e)) ⟩ - subst Gate (cong arity′ (sym (id≗to∘from e))) (l′ (to (from e))) ≡⟨ subst-is-cast-gate (cong arity′ (sym (id≗to∘from e))) (l′ (to (from e))) ⟩ - cast-gate _ (l′ (to (from e))) ≡⟨ cast-gate-trans _ (sym (cong ℕ.suc (sym (a∘from≗a′ e)))) (l′ (to (from e))) ⟨ - cast-gate (sym (cong ℕ.suc (sym (a∘from≗a′ e)))) (cast-gate _ (l′ (to (from e)))) ≡⟨ cong (cast-gate (sym (cong ℕ.suc (sym (a∘from≗a′ e))))) (≗l (from e)) ⟨ - cast-gate (sym (cong ℕ.suc (sym (a∘from≗a′ e)))) (l (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 @@ -214,17 +171,17 @@ Hypergraph-same-trans ≡H₁ ≡H₂ = record open Inverse open ≡-Reasoning ≗j₂ : (e : Fin (h ≡H₁)) - (i : Fin (arity ≡H₁ e)) - → j ≡H₂ (to (↔h ≡H₁) e) (cast (≗arity ≡H₁ e) i) - ≡ j′ ≡H₂ (to (↔h ≡H₂) (to (↔h ≡H₁) e)) (cast (trans (≗arity ≡H₁ e) (≗arity ≡H₂ (to (↔h ≡H₁) e))) i) + (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 (≗arity ≡H₁ e) i) - ≡⟨ ≗j ≡H₂ (to (↔h ≡H₁) e) (cast (≗arity ≡H₁ e) i) ⟩ - j′ ≡H₂ (to (↔h ≡H₂) (to (↔h ≡H₁) e)) (cast (≗arity ≡H₂ (to (↔h ≡H₁) e)) (cast (≗arity ≡H₁ e) i)) - ≡⟨ cong (j′ ≡H₂ (to (↔h ≡H₂) (to (↔h ≡H₁) e))) (cast-trans (≗arity ≡H₁ e) (≗arity ≡H₂ (to (↔h ≡H₁) e)) i) ⟩ - j′ ≡H₂ (to (↔h ≡H₂) (to (↔h ≡H₁) e)) (cast (trans (≗arity ≡H₁ e) (≗arity ≡H₂ (to (↔h ≡H₁) e))) i) ∎ + 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 (≗arity ≡H₁ e)) (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 0ℓ 0ℓ Hypergraph-setoid p = record @@ -322,12 +279,12 @@ module _ (H₁ : Hypergraph n) (H₂ : Hypergraph m) where module H₁ = Hypergraph H₁ module H₂ = Hypergraph H₂ j+j : (e : Fin (H₁.h + H₂.h)) - → Fin (ℕ.suc ([ H₁.a , H₂.a ] (splitAt H₁.h e))) + → 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 (ℕ.suc ([ H₁.a , H₂.a ] (splitAt H₁.h e))) + 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 @@ -421,17 +378,15 @@ together-resp-same {n} {H₁} {H₁′} {m} {H₂} {H₂′} ≡H₁ ≡H₂ = r ([ ≡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 ∎ - ≗arity : H₁+H₂.arity ≗ H₁+H₂′.arity ∘ join ≡H₁.h′ ≡H₂.h′ ∘ map ≡H₁.to ≡H₂.to ∘ splitAt ≡H₁.h - ≗arity = cong ℕ.suc ∘ ≗a ≗j : (e : Fin H₁+H₂.h) - (i : Fin (H₁+H₂.arity e)) + (i : Fin (H₁+H₂.a e)) → H₁+H₂.j e i - ≡ H₁+H₂′.j (to (+-resp-↔ ≡H₁.↔h ≡H₂.↔h) e) (cast (≗arity 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 (≗arity e)) (l+l H₁′ H₂′ (to (+-resp-↔ ≡H₁.↔h ≡H₂.↔h) e)) - ≗l e with splitAt ≡H₁.h e | .{≗arity e} + ≗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 @@ -455,7 +410,7 @@ commute {n} {n′} {m} {m′} {H₁} {H₂} f g = record open Hypergraph open ≡-Reasoning ≗j : (e : Fin (H₁.h + H₂.h)) - (i : Fin ((ℕ.suc ∘ [ H₁.a , H₂.a ] ∘ splitAt H₁.h) e)) + (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 @@ -545,9 +500,9 @@ associativity {X} {Y} {Z} {H₁} {H₂} {H₃} = record ≡⟨ 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 (ℕ.suc ([ [ H₁.a , H₂.a ] ∘ splitAt H₁.h , H₃.a ] (splitAt (H₁.h + H₂.h) e)))) + (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 (cong ℕ.suc (≗a 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 @@ -566,7 +521,7 @@ associativity {X} {Y} {Z} {H₁} {H₂} {H₃} = record 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 (cong ℕ.suc (≗a e))) (l (together H₁ (together H₂ H₃)) (Inverse.to ↔h 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 @@ -611,20 +566,20 @@ unitaryʳ {n} {H} = record ≗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 (ℕ.suc ([ H.a , (λ ()) ] (splitAt H.h e)))) + (i : Fin ([ H.a , (λ ()) ] (splitAt H.h e))) → [ (λ x → x) , (λ ()) ] (splitAt n (j+j H empty-hypergraph e i)) - ≡ H.j (Inverse.to h+0↔h e) (cast (cong ℕ.suc (≗a 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 empty-hypergraph e) refl (≗a e) i where ≗j-aux : (w : Fin H.h ⊎ Fin 0) → (eq₁ : splitAt H.h e ≡ w) - → (w₁ : Fin (ℕ.suc ([ H.a , (λ ()) ] w)) → Fin (n + 0)) - → j+j H empty-hypergraph e ≡ subst (λ hole → Fin (ℕ.suc ([ H.a , (λ ()) ] hole)) → Fin (n + 0)) (sym eq₁) w₁ + → (w₁ : Fin ([ H.a , (λ ()) ] w) → Fin (n + 0)) + → j+j H empty-hypergraph 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 (ℕ.suc ([ H.a , (λ ()) ] w))) + (i : Fin ([ H.a , (λ ()) ] w)) → [ (λ x → x) , (λ ()) ] (splitAt n (w₁ i)) - ≡ H.j (Inverse.to h+0↔h e) (cast (cong ℕ.suc w₂) 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 @@ -632,7 +587,7 @@ unitaryʳ {n} {H} = record (cong (H.j e₁) (sym (cast-is-id refl i))) ≗l : (e : Fin (H.h + 0)) → l+l H empty-hypergraph e - ≡ cast-gate (sym (cong ℕ.suc (≗a e))) (H.l (Inverse.to h+0↔h 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)) @@ -686,9 +641,9 @@ braiding {n} {m} {H₁} {H₂} = record [ 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.arity (map-nodes ([ m ↑ʳ_ , _↑ˡ n ] ∘ splitAt n) (together H₁ H₂)) e)) + (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 (cong ℕ.suc (≗a 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 @@ -698,7 +653,7 @@ braiding {n} {m} {H₁} {H₂} = record 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 (cong ℕ.suc (≗a e))) (l+l H₂ H₁ (Inverse.to (+-comm-↔ H₁.h H₂.h) 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)) @@ -729,6 +684,8 @@ hypergraph = record module F = SymmetricMonoidalFunctor hypergraph +open Gate + and-gate : Func (SingletonSetoid {0ℓ} {0ℓ}) (F.₀ 3) and-gate = record { to = λ { (lift tt) → and-graph } @@ -738,7 +695,7 @@ and-gate = record and-graph : Hypergraph 3 and-graph = record { h = 1 - ; a = λ { 0F → 2 } + ; a = λ { 0F → 3 } ; j = λ { 0F → edge-0-nodes } ; l = λ { 0F → AND } } |