Quellcode-Bibliothek

^© Kompilation durch diese Firma

[Weder Korrektheit noch Funktionsfähigkeit der Software werden zugesichert.]

Datei: NaryFunctions.v Sprache: Coq

Original von: Coq^©

(************************************************************************)
(*         *   The Coq Proof Assistant / The Coq Development Team       *)
(*  v      *   INRIA, CNRS and contributors - Copyright 1999-2018       *)
(* <O___,, *       (see CREDITS file for the list of authors)           *)
(*   \VV/  **************************************************************)
(*    //   *    This file is distributed under the terms of the         *)
(*         *     GNU Lesser General Public License Version 2.1          *)
(*         *     (see LICENSE file for the text of the license)         *)
(************************************************************************)
(*          Pierre Letouzey, Jerome Vouillon, PPS, Paris 7, 2008        *)
(************************************************************************)

Local Open Scope type_scope.

Require Import List.

(** * Generic dependently-typed operators about [n]-ary functions *)

(** The type of [n]-ary function: [nfun A n B] is
    [A -> ... -> A -> B] with [n] occurrences of [A] in this type. *)

Fixpoint nfun A n B :=
match n with
  | O => B
  | S n => A -> (nfun A n B)
end.

Notation " A ^^ n --> B " := (nfun A n B)
(at level 50, n at next level) : type_scope.

(** [napply_cst _ _ a n f] iterates [n] times the application of a
    particular constant [a] to the [n]-ary function [f]. *)

Fixpoint napply_cst (A B:Type)(a:A) n : (A^^n-->B) -> B :=
match n return (A^^n-->B) -> B with
  | O => fun x => x
  | S n => fun x => napply_cst _ _ a n (x a)
end.

(** A generic transformation from an n-ary function to another one.*)

Fixpoint nfun_to_nfun (A B C:Type)(f:B -> C) n :
(A^^n-->B) -> (A^^n-->C) :=
match n return (A^^n-->B) -> (A^^n-->C) with
  | O => f
  | S n => fun g a => nfun_to_nfun _ _ _ f n (g a)
end.

(** [napply_except_last _ _ n f] expects [n] arguments of type [A],
    applies [n-1] of them to [f] and discard the last one. *)

Definition napply_except_last (A B:Type) :=
nfun_to_nfun A B (A->B) (fun b a => b).

(** [napply_then_last _ _ a n f] expects [n] arguments of type [A],
    applies them to [f] and then apply [a] to the result. *)

Definition napply_then_last (A B:Type)(a:A) :=
nfun_to_nfun A (A->B) B (fun fab => fab a).

(** [napply_discard _ b n] expects [n] arguments, discards then,
    and returns [b]. *)

Fixpoint napply_discard (A B:Type)(b:B) n : A^^n-->B :=
match n return A^^n-->B with
  | O => b
  | S n => fun _ => napply_discard _ _ b n
end.

(** A fold function *)

Fixpoint nfold A B (f:A->B->B)(b:B) n : (A^^n-->B) :=
match n return (A^^n-->B) with
  | O => b
  | S n => fun a => (nfold _ _ f (f a b) n)
end.

(** [n]-ary products : [nprod A n] is [A*...*A*unit],
  with [n] occurrences of [A] in this type. *)

Fixpoint nprod A n : Type := match n with
| O => unit
| S n => (A * nprod A n)%type
end.

Notation "A ^ n" := (nprod A n) : type_scope.

(** [n]-ary curryfication / uncurryfication *)

Fixpoint ncurry (A B:Type) n : (A^n -> B) -> (A^^n-->B) :=
  match n return (A^n -> B) -> (A^^n-->B) with
  | O => fun x => x tt
  | S n => fun f a => ncurry _ _ n (fun p => f (a,p))
  end.

Fixpoint nuncurry (A B:Type) n : (A^^n-->B) -> (A^n -> B) :=
match n return (A^^n-->B) -> (A^n -> B) with
  | O => fun x _ => x
  | S n => fun f p => let (x,p) := p in nuncurry _ _ n (f x) p
end.

(** Earlier functions can also be defined via [ncurry/nuncurry].
    For instance : *)

Definition nfun_to_nfun_bis A B C (f:B->C) n :
(A^^n-->B) -> (A^^n-->C) :=
fun anb => ncurry _ _ n (fun an => f ((nuncurry _ _ n anb) an)).

(** We can also us it to obtain another [fold] function,
    equivalent to the previous one, but with a nicer expansion
    (see for instance Int31.iszero). *)

Fixpoint nfold_bis A B (f:A->B->B)(b:B) n : (A^^n-->B) :=
match n return (A^^n-->B) with
  | O => b
  | S n => fun a =>
      nfun_to_nfun_bis _ _ _ (f a) n (nfold_bis _ _ f b n)
end.

(** From [nprod] to [list] *)

Fixpoint nprod_to_list (A:Type) n : A^n -> list A :=
match n with
  | O => fun _ => nil
  | S n => fun p => let (x,p) := p in x::(nprod_to_list _ n p)
end.

(** From [list] to [nprod] *)

Fixpoint nprod_of_list (A:Type)(l:list A) : A^(length l) :=
match l return A^(length l) with
  | nil => tt
  | x::l => (x, nprod_of_list _ l)
end.

(** This gives an additional way to write the fold *)

Definition nfold_list (A B:Type)(f:A->B->B)(b:B) n : (A^^n-->B) :=
  ncurry _ _ n (fun p => fold_right f b (nprod_to_list _ _ p)).

¤ Dauer der Verarbeitung: 0.22 Sekunden (vorverarbeitet) ¤

Download des Quellennavigators
Download des sprechenden Kalenders
in der Quellcodebibliothek suchen

Haftungshinweis

Die Informationen auf dieser Webseite wurden nach bestem Wissen sorgfältig zusammengestellt. Es wird jedoch weder Vollständigkeit, noch Richtigkeit, noch Qualität der bereit gestellten Informationen zugesichert.

Bemerkung:

Die farbliche Syntaxdarstellung ist noch experimentell.