# Library mathcomp.ssreflect.ssrfun

From mathcomp Require Import ssreflect.
From Coq Require Export ssrfun.
From mathcomp Require Export ssrnotations.
#[export] Set Warnings "-overwriting-delimiting-key".
because there is some Set Warnings "overwriting-delimiting-key". somewhere in the above
Set Implicit Arguments.
v8.20 additions The notations below are already defined in Coq.ssr.ssrfun, but we redefine them in different notation scopes (mostly fun_scope -> function_scope, in order to replace the former with the latter).
Open Scope function_scope.
Notation "f ^~ y" := (fun x ⇒ f x y) : function_scope.
Notation "@^~ x" := (fun f ⇒ f x) : function_scope.
Notation "[ 'fun' : T => E ]" := (SimplFun (fun _ : T ⇒ E)) : function_scope.
Notation "[ 'fun' x => E ]" := (SimplFun (fun x ⇒ E)) : function_scope.
Notation "[ 'fun' x y => E ]" := (fun x ⇒ [fun y ⇒ E]) : function_scope.
Notation "[ 'fun' x : T => E ]" := (SimplFun (fun x : T ⇒ E))
(only parsing) : function_scope.
Notation "[ 'fun' x y : T => E ]" := (fun x : T ⇒ [fun y : T ⇒ E])
(only parsing) : function_scope.
Notation "[ 'fun' ( x : T ) y => E ]" := (fun x : T ⇒ [fun y ⇒ E])
(only parsing) : function_scope.
Notation "[ 'fun' x ( y : T ) => E ]" := (fun x ⇒ [fun y : T ⇒ E])
(only parsing) : function_scope.
Notation "[ 'fun' ( x : T ) ( y : U ) => E ]" := (fun x : T ⇒ [fun y : U ⇒ E])
(only parsing) : function_scope.
Notation "f1 =1 f2" := (eqfun f1 f2) : type_scope.
Notation "f1 =1 f2 :> A" := (f1 =1 (f2 : A)) : type_scope.
Notation "f1 =2 f2" := (eqrel f1 f2) : type_scope.
Notation "f1 =2 f2 :> A" := (f1 =2 (f2 : A)) : type_scope.
Notation "f1 \o f2" := (comp f1 f2) : function_scope.
Notation "f1 \; f2" := (catcomp f1 f2) : function_scope.
Notation "[ 'eta' f ]" := (fun x ⇒ f x) : function_scope.
Notation "'fun' => E" := (fun _ ⇒ E) : function_scope.
Notation "@ 'id' T" := (fun x : T ⇒ x)
(at level 10, T at level 8, only parsing) : function_scope.
Notation "@ 'sval'" := (@proj1_sig) (at level 10, only parsing) :
function_scope.
not yet backported
#[export] Set Warnings "-hiding-delimiting-key".
Delimit Scope function_scope with FUN.
Close Scope fun_scope.
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.
Lemma inj_omap {aT rT : Type} (f : aT → rT) :
injective f → injective (omap f).
Lemma omap_id {T : Type} (x : option T) : omap id x = x.
Lemma eq_omap {aT rT : Type} (f g : aT → rT) : f =1 g → omap f =1 omap g.
Lemma omapK {aT rT : Type} (f : aT → rT) (g : rT → aT) :
cancel f g → cancel (omap f) (omap g).
Definition idempotent_op (S : Type) (op : S → S → S) := ∀ x, op x x = x.
#[deprecated(since="mathcomp 2.3.0", note="use `idempotent_op` instead")]
Notation idempotent:= idempotent_op (only parsing).
Definition idempotent_fun (U : Type) (f : U → U) := f \o f =1 f.