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August 18, 2016 13:19
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Functor in Coq
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Require Import Coq.Program.Basics. | |
Require Import Coq.Program.Syntax. | |
Require Import Coq.Init.Datatypes. | |
Require Import Coq.Unicode.Utf8. | |
Open Local Scope program_scope. | |
Open Local Scope list_scope. | |
Open Local Scope type_scope. | |
Class Functor (φ : Type → Type) := { | |
fmap : forall {α β}, (α → β) → φ α → φ β; | |
(** Functor laws **) | |
fmap_identity : forall α (x : φ α), fmap id x = x; | |
fmap_composition : forall α β γ (f : β → γ) (g : α → β) (x : φ α), | |
fmap (f ∘ g) x = (fmap f ∘ fmap g) x | |
}. | |
Instance list_functor : Functor list := { | |
fmap α β := fix fmap f xs := match xs with | |
| [] => [] | |
| x :: xs => (f x) :: (fmap f xs) | |
end | |
}. | |
Proof. | |
(** fmap_identity **) | |
intros. | |
induction x as [| x xs]. | |
reflexivity. | |
rewrite IHxs. reflexivity. | |
(** fmap_composition **) | |
intros. | |
induction x as [| x xs]. | |
reflexivity. | |
rewrite IHxs. reflexivity. | |
Qed. |
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