Require Export FunctionProperties.
Require Export Ensembles.
Require Export EnsemblesSpec.
Definition inverse_image {X Y:Type} (f:X->Y) (T:Ensemble Y) : Ensemble X :=
[ x:X | In T (f x) ].
Hint Unfold inverse_image : sets.
Hint Resolve @inverse_image_increasing : sets.
Hint Rewrite @inverse_image_empty @inverse_image_full
@inverse_image_intersection @inverse_image_union
@inverse_image_complement @inverse_image_composition : sets.

Lemma in_inverse_image
{X Y : Type}
(f : X -> Y)
(T : Ensemble Y)
(x : X) :
In T (f x) <-> In (inverse_image f T) x.