Library mathcomp.ssreflect.ssrfun

From mathcomp Require Import ssreflect.
From Coq Require Export ssrfun.
From mathcomp Require Export ssrnotations.

Definition injective2 (rT aT1 aT2 : Type) (f : aT1 → aT2 → rT) :=
  ∀ (x1 x2 : aT1) (y1 y2 : aT2), f x1 y1 = f x2 y2 → (x1 = x2) × (y1 = y2).

Arguments injective2 [rT aT1 aT2] f.

Set Implicit Arguments.

Definition olift aT rT (f : aT → rT) := Some \o f.

Lemma obindEapp {aT rT} (f : aT → option rT) : obind f = oapp f None.

Lemma omapEbind {aT rT} (f : aT → rT) : omap f = obind (olift f).

Lemma omapEapp {aT rT} (f : aT → rT) : omap f = oapp (olift f) None.

Lemma oappEmap {aT rT} (f : aT → rT) (y0 : rT) x :
  oapp f y0 x = odflt y0 (omap f x).

Lemma omap_comp aT rT sT (f : aT → rT) (g : rT → sT) :
  omap (g \o f) =1 omap g \o omap f.

Lemma oapp_comp aT rT sT (f : aT → rT) (g : rT → sT) x :
  oapp (g \o f) x =1 (@oapp _ _)^~ x g \o omap f.

Lemma oapp_comp_f {aT rT sT} (f : aT → rT) (g : rT → sT) (x : rT) :
  oapp (g \o f) (g x) =1 g \o oapp f x.

Lemma olift_comp aT rT sT (f : aT → rT) (g : rT → sT) :
  olift (g \o f) = olift g \o f.

Lemma compA {A B C D : Type} (f : B → A) (g : C → B) (h : D → C) :
  f \o (g \o h) = (f \o g) \o h.

Lemma ocan_comp [A B C : Type] [f : B → option A] [h : C → option B]
    [f' : A → B] [h' : B → C] :
  ocancel f f' → ocancel h h' → ocancel (obind f \o h) (h' \o f').