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40 lines
1.4 KiB
Agda
40 lines
1.4 KiB
Agda
module NatCat where
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open import Relation.Binary.PropositionalEquality
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-- If you can show that a relation only ever has one inhabitant
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-- you get the category laws for free
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module
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EasyCategory
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(obj : Set)
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(_⟶_ : obj → obj → Set)
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(_∘_ : ∀ {x y z} → x ⟶ y → y ⟶ z → x ⟶ z)
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(id : ∀ x → x ⟶ x)
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(single-inhabitant : (x y : obj) (r s : x ⟶ y) → r ≡ s)
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where
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idʳ : ∀ x y (r : x ⟶ y) → r ∘ id y ≡ r
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idʳ x y r = single-inhabitant x y (r ∘ id y) r
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idˡ : ∀ x y (r : x ⟶ y) → id x ∘ r ≡ r
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idˡ x y r = single-inhabitant x y (id x ∘ r) r
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∘-assoc : ∀ w x y z (r : w ⟶ x) (s : x ⟶ y) (t : y ⟶ z) → (r ∘ s) ∘ t ≡ r ∘ (s ∘ t)
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∘-assoc w x y z r s t = single-inhabitant w z ((r ∘ s) ∘ t) (r ∘ (s ∘ t))
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open import Data.Nat
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same : (x y : ℕ) (r s : x ≤ y) → r ≡ s
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same .0 y z≤n z≤n = refl
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same .(suc m) .(suc n) (s≤s {m} {n} r) (s≤s s) = cong s≤s (same m n r s)
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≤-trans : ∀ x y z → x ≤ y → y ≤ z → x ≤ z
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≤-trans .0 y z z≤n s = z≤n
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≤-trans .(suc m) .(suc n) .(suc n₁) (s≤s {m} {n} r) (s≤s {.n} {n₁} s) = s≤s (≤-trans m n n₁ r s)
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≤-refl : ∀ x → x ≤ x
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≤-refl zero = z≤n
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≤-refl (suc x) = s≤s (≤-refl x)
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module Nat-EasyCategory = EasyCategory ℕ _≤_ (λ {x}{y}{z} → ≤-trans x y z) ≤-refl same
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