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(************************************************************************)
(* * The Coq Proof Assistant / The Coq Development Team *)
(* v * Copyright INRIA, CNRS and contributors *)
(* <O___,, * (see version control and CREDITS file for authors & dates) *)
(* \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 [S n] arguments of type [A],
applies [n] of them to [f] and discards the last one. *)
Fixpoint napply_except_last (A B:Type) (n : nat) (f : A^^n-->B) {struct n} : A^^S n-->B.
Proof.
destruct n.
- exact (fun _ => f).
- exact (fun arg => napply_except_last A B n (f arg)).
Defined.
(** [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)).
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