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function.v
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function.v
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(**
This module defines basic properties for functions.
Copyright (C) 2018 Larry D. Lee Jr. <[email protected]>
This program is free software: you can redistribute it and/or modify
it under the terms of the GNU Lesser General Public License as
published by the Free Software Foundation, either version 3 of the
License, or (at your option) any later version.
This program is distributed in the hope that it will be useful,
but WITHOUT ANY WARRANTY; without even the implied warranty of
MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
GNU Lesser General Public License for more details.
You should have received a copy of the GNU Lesser General Public
License along with this program. If not, see
<https://www.gnu.org/licenses/>.
*)
(** Defines the injective predicate. *)
Definition is_injective (A B : Type) (f : A -> B) : Prop
:= forall x y : A, f x = f y -> x = y.
(** Defines the onto predicate. *)
Definition is_onto (A B : Type) (f : A -> B) : Prop
:= forall y : B, exists x : A, f x = y.
(** Defines the bijective predicate. *)
Definition is_bijective (A B : Type) (f : A -> B) : Prop
:= is_injective A B f /\ is_onto A B f.