diff options
Diffstat (limited to 'Category/Equivalence')
| -rw-r--r-- | Category/Equivalence/Instance/Nat.agda | 28 |
1 files changed, 14 insertions, 14 deletions
diff --git a/Category/Equivalence/Instance/Nat.agda b/Category/Equivalence/Instance/Nat.agda index 7b73c36..46f5b1e 100644 --- a/Category/Equivalence/Instance/Nat.agda +++ b/Category/Equivalence/Instance/Nat.agda @@ -32,7 +32,7 @@ open import Data.Vec using (Vec) open import Data.Vec.Functional using (tail) open import Function using (Func; _⟨$⟩_; flip; _∘_; _⇔_; mk⇔) renaming (id to idf) open import Relation.Binary.PropositionalEquality as ≡ using (_≡_; _≗_; module ≡-Reasoning) -open import Relation.Nullary.Decidable using (⌊_⌋; Dec; yes; no; _×-dec_; isYes≗does; dec-true; dec-false; does-⇔) +open import Relation.Nullary.Decidable using (⌊_⌋; Dec; yes; no; _×?_; isYes≗does; dec-true; dec-false; does-⇔) open import Relation.Unary using (Pred; Decidable) module 𝔹-rig = Semiring 𝔹.semiring @@ -83,7 +83,7 @@ open Dec where open ≡-Reasoning -⌊A×B⌋ : {A B : Set} (A? : Dec A) (B? : Dec B) → ⌊ A? ⌋ ∧ ⌊ B? ⌋ ≡ ⌊ A? ×-dec B? ⌋ +⌊A×B⌋ : {A B : Set} (A? : Dec A) (B? : Dec B) → ⌊ A? ⌋ ∧ ⌊ B? ⌋ ≡ ⌊ A? ×? B? ⌋ ⌊A×B⌋ (yes a) (yes b) = ≡.refl ⌊A×B⌋ (yes a) (no ¬b) = ≡.refl ⌊A×B⌋ (no ¬a) B? = ≡.refl @@ -119,8 +119,8 @@ functional-index functional-index f i j = begin sum (λ k → ⌊ f k ≟ j ⌋ ∧ ⌊ f k ≟ i ⌋) ∨ identity i j ≡⟨ ≡.cong (_∨ identity i j) (sum-cong (λ k → ∧-comm (⌊ f k ≟ j ⌋) (⌊ f k ≟ i ⌋))) ⟩ sum (λ k → ⌊ f k ≟ i ⌋ ∧ ⌊ f k ≟ j ⌋) ∨ identity i j ≡⟨ ≡.cong₂ _∨_ (sum-cong (λ k → ⌊A×B⌋ (f k ≟ i) (f k ≟ j))) (≡.sym (graph-id i j)) ⟩ - sum (λ k → ⌊ f k ≟ i ×-dec f k ≟ j ⌋) ∨ ⌊ i ≟ j ⌋ ≡⟨ ≡.cong (_∨ ⌊ i ≟ j ⌋) (⌊Σ?⌋ (λ k → f k ≟ i ×-dec f k ≟ j)) ⟨ - ⌊ Σ? (λ k → f k ≟ i ×-dec f k ≟ j) ⌋ ∨ ⌊ i ≟ j ⌋ ≡⟨ A→B⇒A∨B≡B (func f) ⟩ + sum (λ k → ⌊ f k ≟ i ×? f k ≟ j ⌋) ∨ ⌊ i ≟ j ⌋ ≡⟨ ≡.cong (_∨ ⌊ i ≟ j ⌋) (⌊Σ?⌋ (λ k → f k ≟ i ×? f k ≟ j)) ⟨ + ⌊ Σ? (λ k → f k ≟ i ×? f k ≟ j) ⌋ ∨ ⌊ i ≟ j ⌋ ≡⟨ A→B⇒A∨B≡B (func f) ⟩ ⌊ i ≟ j ⌋ ≡⟨ graph-id i j ⟩ identity i j ∎ where @@ -135,10 +135,10 @@ entire-index {n} {m} f i j = begin identity i j ∨ sum (λ k → ⌊ f j ≟ k ⌋ ∧ ⌊ f i ≟ k ⌋) ≡⟨ ≡.cong (identity i j ∨_) (sum-cong (λ k → ∧-comm (⌊ f j ≟ k ⌋) (⌊ f i ≟ k ⌋))) ⟩ identity i j ∨ sum (λ k → ⌊ f i ≟ k ⌋ ∧ ⌊ f j ≟ k ⌋) ≡⟨ ≡.cong (_∨ sum (λ k → ⌊ f i ≟ k ⌋ ∧ ⌊ f j ≟ k ⌋)) (graph-id i j) ⟨ ⌊ i ≟ j ⌋ ∨ sum (λ k → ⌊ f i ≟ k ⌋ ∧ ⌊ f j ≟ k ⌋) ≡⟨ ≡.cong (⌊ i ≟ j ⌋ ∨_) (sum-cong (λ k → ⌊A×B⌋ (f i ≟ k) (f j ≟ k))) ⟩ - ⌊ i ≟ j ⌋ ∨ sum (λ k → ⌊ f i ≟ k ×-dec f j ≟ k ⌋) ≡⟨ ≡.cong (⌊ i ≟ j ⌋ ∨_) (⌊Σ?⌋ (λ k → f i ≟ k ×-dec f j ≟ k)) ⟨ - ⌊ i ≟ j ⌋ ∨ ⌊ Σ? (λ k → f i ≟ k ×-dec f j ≟ k) ⌋ ≡⟨ A→B⇒A∨B≡B (enti f) ⟩ - ⌊ Σ? (λ k → f i ≟ k ×-dec f j ≟ k) ⌋ ≡⟨ ⌊Σ?⌋ (λ k → f i ≟ k ×-dec f j ≟ k) ⟩ - sum (λ k → ⌊ f i ≟ k ×-dec f j ≟ k ⌋) ≡⟨ sum-cong (λ k → ⌊A×B⌋ (f i ≟ k) (f j ≟ k)) ⟨ + ⌊ i ≟ j ⌋ ∨ sum (λ k → ⌊ f i ≟ k ×? f j ≟ k ⌋) ≡⟨ ≡.cong (⌊ i ≟ j ⌋ ∨_) (⌊Σ?⌋ (λ k → f i ≟ k ×? f j ≟ k)) ⟨ + ⌊ i ≟ j ⌋ ∨ ⌊ Σ? (λ k → f i ≟ k ×? f j ≟ k) ⌋ ≡⟨ A→B⇒A∨B≡B (enti f) ⟩ + ⌊ Σ? (λ k → f i ≟ k ×? f j ≟ k) ⌋ ≡⟨ ⌊Σ?⌋ (λ k → f i ≟ k ×? f j ≟ k) ⟩ + sum (λ k → ⌊ f i ≟ k ×? f j ≟ k ⌋) ≡⟨ sum-cong (λ k → ⌊A×B⌋ (f i ≟ k) (f j ≟ k)) ⟨ sum (λ k → ⌊ f i ≟ k ⌋ ∧ ⌊ f j ≟ k ⌋) ≡⟨ sum-cong (λ k → ∧-comm (⌊ f i ≟ k ⌋) (⌊ f j ≟ k ⌋)) ⟩ sum (λ k → ⌊ f j ≟ k ⌋ ∧ ⌊ f i ≟ k ⌋) ∎ where @@ -209,12 +209,12 @@ homomorphism-index (j : Fin Z) → ⌊ (g (f i) ≟ j) ⌋ ≡ sum (λ k → ⌊ g k ≟ j ⌋ ∧ ⌊ f i ≟ k ⌋) homomorphism-index f g i j = begin - ⌊ g (f i) ≟ j ⌋ ≡⟨ isYes≗does (g (f i) ≟ j) ⟩ - does (g (f i) ≟ j) ≡⟨ does-⇔ (homo f g i j) (g (f i) ≟ j) (Σ? (λ k → g k ≟ j ×-dec f i ≟ k)) ⟩ - does (Σ? (λ k → g k ≟ j ×-dec f i ≟ k)) ≡⟨ isYes≗does (Σ? (λ k → g k ≟ j ×-dec f i ≟ k)) ⟨ - ⌊ Σ? (λ k → g k ≟ j ×-dec f i ≟ k) ⌋ ≡⟨ ⌊Σ?⌋ (λ k → g k ≟ j ×-dec f i ≟ k) ⟩ - sum (λ k → ⌊ g k ≟ j ×-dec f i ≟ k ⌋) ≡⟨ sum-cong (λ k → ⌊A×B⌋ (g k ≟ j) (f i ≟ k)) ⟨ - sum (λ k → ⌊ g k ≟ j ⌋ ∧ ⌊ f i ≟ k ⌋) ∎ + ⌊ g (f i) ≟ j ⌋ ≡⟨ isYes≗does (g (f i) ≟ j) ⟩ + does (g (f i) ≟ j) ≡⟨ does-⇔ (homo f g i j) (g (f i) ≟ j) (Σ? (λ k → g k ≟ j ×? f i ≟ k)) ⟩ + does (Σ? (λ k → g k ≟ j ×? f i ≟ k)) ≡⟨ isYes≗does (Σ? (λ k → g k ≟ j ×? f i ≟ k)) ⟨ + ⌊ Σ? (λ k → g k ≟ j ×? f i ≟ k) ⌋ ≡⟨ ⌊Σ?⌋ (λ k → g k ≟ j ×? f i ≟ k) ⟩ + sum (λ k → ⌊ g k ≟ j ×? f i ≟ k ⌋) ≡⟨ sum-cong (λ k → ⌊A×B⌋ (g k ≟ j) (f i ≟ k)) ⟨ + sum (λ k → ⌊ g k ≟ j ⌋ ∧ ⌊ f i ≟ k ⌋) ∎ where open ≡-Reasoning |