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(* 02110-1301 USA                                                     *)
Set Asymmetric Patterns.
(* Contribution to the Coq Library   V6.3 (July 1999)                    *)
(****************************************************************************)
(*                 The Calculus of Inductive Constructions                  *)
(*                                                                          *)
(*                                Projet Coq                                *)
(*                                                                          *)
(*                     INRIA                        ENS-CNRS                *)
(*              Rocquencourt                        Lyon                    *)
(*                                                                          *)
(*                                Coq V5.10                                 *)
(*                              Nov 25th 1994                               *)
(*                                                                          *)
(****************************************************************************)
(*                                Sorting.v                                 *)
(****************************************************************************)
(* These sorting examples have first been compiled by P. Manoury using the  *)
(* ProPre tool to build recursive functions by cases                        *)
(* == Quelques fonctions utiles == *)
(* Inductive Set bool := true : bool | false : bool. *)
Definition si (X : Set) (b : bool) (x y : X) :=
match b with
| true => x
| _ => y
end.
(* Inductive Set nat : O:nat | S:nat->nat. *)
Fixpoint egal_nat (n : nat) : nat -> bool :=
fun m : nat =>
match n, m with
| O, O => true
| S n, S m => egal_nat n m
| _, _ => false
end.
Fixpoint inf_egal (n : nat) : nat -> bool :=
fun m : nat =>
match n, m with
| O, m => true
| S n, O => false
| S n, S m => inf_egal n m
end.
Inductive list (X : Set) : Set :=
| Nil : list X
| Cons : X -> list X -> list X.
Definition cdr (X : Set) (l : list X) :=
match l with
| Nil => Nil X
| Cons _ xs => xs
end.
Fixpoint length (X : Set) (l : list X) {struct l} : nat :=
match l with
| Nil => 0
| Cons _ xs => S (length X xs)
end.
(* append x y = yx *)
Fixpoint append (X : Set) (ys xs : list X) {struct xs} :
list X :=
match xs with
| Nil => ys
| Cons x xs' => Cons X x (append X ys xs')
end.
(* == Proprietes d'un tri == *)
Fixpoint sorted (l : list nat) : Prop :=
match l with
| Cons n (Cons m x as l) => inf_egal n m = true /\ sorted l
| _ => True
end.
Fixpoint nocc (n : nat) (l : list nat) {struct l} : nat :=
match l with
| Nil => 0
| Cons m x => si nat (egal_nat n m) (S (nocc n x)) (nocc n x)
end.
(* == Un lemme utile == *)
(* -- Sur les booleens *)
(* -- Sur le predicat "sorted" *)
(* == Tri par insertion *)
Fixpoint ins (n : nat) (l : list nat) {struct l} :
list nat :=
match l with
| Nil => Cons nat n (Nil nat)
| Cons m x =>
si (list nat) (inf_egal n m) (Cons nat n (Cons nat m x))
(Cons nat m (ins n x))
end.
Fixpoint tri_ins (l : list nat) : list nat :=
match l with
| Nil => Nil nat
| Cons n x => ins n (tri_ins x)
end.
(* -- Preuve de correction *)
(* -- 1 : Le resultat de "tri_ins" est une liste sorted *)

Theorem sorted_ins_Cons :
forall (x : list nat) (n m : nat),
inf_egal n m = false ->
sorted (Cons nat m x) -> sorted (Cons nat m (ins n x)).