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82 lines
2.5 KiB
Plaintext
82 lines
2.5 KiB
Plaintext
\documentclass{article}
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% The following packages are needed because unicode
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% is translated (using the next set of packages) to
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% latex commands. You may need more packages if you
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% use more unicode characters:
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\usepackage{amssymb}
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\usepackage{bbm}
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\usepackage[greek,english]{babel}
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% This handles the translation of unicode to latex:
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\usepackage{ucs}
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\usepackage[utf8x]{inputenc}
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\usepackage{autofe}
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% Some characters that are not automatically defined
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% (you figure out by the latex compilation errors you get),
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% and you need to define:
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\DeclareUnicodeCharacter{8988}{\ensuremath{\ulcorner}}
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\DeclareUnicodeCharacter{8989}{\ensuremath{\urcorner}}
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\DeclareUnicodeCharacter{8803}{\ensuremath{\overline{\equiv}}}
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% Add more as you need them (shouldn’t happen often).
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% Using “\newenvironment” to redefine verbatim to
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% be called “code” doesn’t always work properly.
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% You can more reliably use:
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\usepackage{fancyvrb}
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\DefineVerbatimEnvironment
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{code}{Verbatim}
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{} % Add fancy options here if you like.
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\begin{document}
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\begin{code}
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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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\end{code}
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\end{document} |