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(* -*- coq-prog-args: ("-emacs-U" "-nois") -*- *)
(************************************************************************)
(* v * The Coq Proof Assistant / The Coq Development Team *)
(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *)
(* \VV/ **************************************************************)
(* // * This file is distributed under the terms of the *)
(* * GNU Lesser General Public License Version 2.1 *)
(************************************************************************)
(* Standard functions and proofs about them.
* Author: Matthieu Sozeau
* Institution: LRI, CNRS UMR 8623 - UniversitÃcopyright Paris Sud
* 91405 Orsay, France *)
(* $Id: FSetAVL_prog.v 616 2007-08-08 12:28:10Z msozeau $ *)
Set Implicit Arguments.
Unset Strict Implicit.
Require Export Coq.Program.FunctionalExtensionality.
Notation " 'λ' x : T , y " := (fun x:T => y) (at level 100, x,T,y at level 10, no associativity) : program_scope.
Open Local Scope program_scope.
Definition id {A} := λ x : A, x.
Definition compose {A B C} (g : B -> C) (f : A -> B) := λ x : A , g (f x).
Hint Unfold compose.
Notation " g ∘ f " := (compose g f) (at level 50, left associativity) : program_scope.
Lemma compose_id_left : forall A B (f : A -> B), id ∘ f = f.
Proof.
intros.
unfold id, compose.
symmetry ; apply eta_expansion.
Qed.
Lemma compose_id_right : forall A B (f : A -> B), f ∘ id = f.
Proof.
intros.
unfold id, compose.
symmetry ; apply eta_expansion.
Qed.
Lemma compose_assoc : forall A B C D (f : A -> B) (g : B -> C) (h : C -> D),
h ∘ g ∘ f = h ∘ (g ∘ f).
Proof.
intros.
reflexivity.
Qed.
Hint Rewrite @compose_id_left @compose_id_right @compose_assoc : core.
Definition arrow (A B : Type) := A -> B.
Definition impl (A B : Prop) : Prop := A -> B.
(* Notation " f x " := (f x) (at level 100, x at level 200, only parsing) : program_scope. *)
Definition const {A B} (a : A) := fun x : B => a.
Definition flip {A B C} (f : A -> B -> C) x y := f y x.
Lemma flip_flip : forall A B C (f : A -> B -> C), flip (flip f) = f.
Proof.
unfold flip.
intros.
extensionality x ; extensionality y.
reflexivity.
Qed.
Definition apply {A B} (f : A -> B) (x : A) := f x.
(** Notations for the unit type and value. *)
Notation " () " := Datatypes.unit : type_scope.
Notation " () " := tt.
(** Set maximally inserted implicit arguments for standard definitions. *)
Implicit Arguments eq [[A]].
Implicit Arguments Some [[A]].
Implicit Arguments None [[A]].
Implicit Arguments inl [[A] [B]].
Implicit Arguments inr [[A] [B]].
Implicit Arguments left [[A] [B]].
Implicit Arguments right [[A] [B]].
(** Curryfication. *)
Definition curry {a b c} (f : a -> b -> c) (p : prod a b) : c :=
let (x, y) := p in f x y.
Definition uncurry {a b c} (f : prod a b -> c) (x : a) (y : b) : c :=
f (x, y).
Lemma uncurry_curry : forall a b c (f : a -> b -> c), uncurry (curry f) = f.
Proof.
simpl ; intros.
unfold uncurry, curry.
extensionality x ; extensionality y.
reflexivity.
Qed.
Lemma curry_uncurry : forall a b c (f : prod a b -> c), curry (uncurry f) = f.
Proof.
simpl ; intros.
unfold uncurry, curry.
extensionality x.
destruct x ; simpl ; reflexivity.
Qed.
(** n-ary exists ! *)
Notation " 'exists' x y , p" := (ex (fun x => (ex (fun y => p))))
(at level 200, x ident, y ident, right associativity) : type_scope.
Notation " 'exists' x y z , p" := (ex (fun x => (ex (fun y => (ex (fun z => p))))))
(at level 200, x ident, y ident, z ident, right associativity) : type_scope.
Notation " 'exists' x y z w , p" := (ex (fun x => (ex (fun y => (ex (fun z => (ex (fun w => p))))))))
(at level 200, x ident, y ident, z ident, w ident, right associativity) : type_scope.
Tactic Notation "exist" constr(x) := exists x.
Tactic Notation "exist" constr(x) constr(y) := exists x ; exists y.
Tactic Notation "exist" constr(x) constr(y) constr(z) := exists x ; exists y ; exists z.
Tactic Notation "exist" constr(x) constr(y) constr(z) constr(w) := exists x ; exists y ; exists z ; exists w.
(* Notation " 'Σ' x : T , p" := (sigT (fun x : T => p)) *)
(* (at level 200, x ident, y ident, right associativity) : program_scope. *)
(* Notation " 'Π' x : T , p " := (forall x : T, p) *)
(* (at level 200, x ident, right associativity) : program_scope. *)
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