From Coq Require Export Image.
From Coq Require Export Program.Basics.
Arguments injective {U} {V}.
Definition surjective {X Y:Type} (f:X->Y) :=
forall y:Y, exists x:X, f x = y.
Definition bijective {X Y:Type} (f:X->Y) :=
injective f /\ surjective f.
Inductive invertible {X Y:Type} (f:X->Y) : Prop :=
| intro_invertible: forall g:Y->X,
(forall x:X, g (f x) = x) -> (forall y:Y, f (g y) = y) ->
invertible f.
Definition function_inverse {X Y:Type} (f:X->Y)
(i:invertible f) : { g:Y->X | (forall x:X, g (f x) = x) /\
(forall y:Y, f (g y) = y) }
:=
(constructive_definite_description _
(unique_inverse f i)).

Lemma bijective_impl_invertible: forall {X Y:Type} (f:X->Y),
bijective f -> invertible f.