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 inverse_image_add
{X Y : Type}
(f : X -> Y)
(g : Y -> X)
(F : Family Y)
(T : Ensemble Y) :
(forall x, g (f x) = x) ->
(forall y, f (g y) = y) ->
inverse_image (inverse_image g) (Add F T) = Add (inverse_image (inverse_image g) F) (inverse_image f T).