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(************************************************************************)
(* 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 *)
(************************************************************************)
(*i $Id$ i*)
(** THIS IS A OLD CONTRIB. IT IS NO LONGER MAINTAINED ***)
Require Import Le.
Parameter List_Dom : Set.
Definition A := List_Dom.
Inductive list : Set :=
| nil : list
| cons : A -> list -> list.
Fixpoint app (l m:list) {struct l} : list :=
match l return list with
| nil => m
| cons a l1 => cons a (app l1 m)
end.
Lemma app_nil_end : forall l:list, l = app l nil.
Proof.
intro l; elim l; simpl in |- *; auto.
simple induction 1; auto.
Qed.
Hint Resolve app_nil_end: list v62.
Lemma app_ass : forall l m n:list, app (app l m) n = app l (app m n).
Proof.
intros l m n; elim l; simpl in |- *; auto with list.
simple induction 1; auto with list.
Qed.
Hint Resolve app_ass: list v62.
Lemma ass_app : forall l m n:list, app l (app m n) = app (app l m) n.
Proof.
auto with list.
Qed.
Hint Resolve ass_app: list v62.
Definition tail (l:list) : list :=
match l return list with
| cons _ m => m
| _ => nil
end.
Lemma nil_cons : forall (a:A) (m:list), nil <> cons a m.
intros; discriminate.
Qed.
(****************************************)
(* Length of lists *)
(****************************************)
Fixpoint length (l:list) : nat :=
match l return nat with
| cons _ m => S (length m)
| _ => 0
end.
(******************************)
(* Length order of lists *)
(******************************)
Section length_order.
Definition lel (l m:list) := length l <= length m.
Hint Unfold lel: list.
Variables a b : A.
Variables l m n : list.
Lemma lel_refl : lel l l.
Proof.
unfold lel in |- *; auto with list.
Qed.
Lemma lel_trans : lel l m -> lel m n -> lel l n.
Proof.
unfold lel in |- *; intros.
apply le_trans with (length m); auto with list.
Qed.
Lemma lel_cons_cons : lel l m -> lel (cons a l) (cons b m).
Proof.
unfold lel in |- *; simpl in |- *; auto with list arith.
Qed.
Lemma lel_cons : lel l m -> lel l (cons b m).
Proof.
unfold lel in |- *; simpl in |- *; auto with list arith.
Qed.
Lemma lel_tail : lel (cons a l) (cons b m) -> lel l m.
Proof.
unfold lel in |- *; simpl in |- *; auto with list arith.
Qed.
Lemma lel_nil : forall l':list, lel l' nil -> nil = l'.
Proof.
intro l'; elim l'; auto with list arith.
intros a' y H H0.
(* <list>nil=(cons a' y)
============================
H0 : (lel (cons a' y) nil)
H : (lel y nil)->(<list>nil=y)
y : list
a' : A
l' : list *)
absurd (S (length y) <= 0); auto with list arith.
Qed.
End length_order.
Hint Resolve lel_refl lel_cons_cons lel_cons lel_nil lel_nil nil_cons: list
v62.
Fixpoint In (a:A) (l:list) {struct l} : Prop :=
match l with
| nil => False
| cons b m => b = a \/ In a m
end.
Lemma in_eq : forall (a:A) (l:list), In a (cons a l).
Proof.
simpl in |- *; auto with list.
Qed.
Hint Resolve in_eq: list v62.
Lemma in_cons : forall (a b:A) (l:list), In b l -> In b (cons a l).
Proof.
simpl in |- *; auto with list.
Qed.
Hint Resolve in_cons: list v62.
Lemma in_app_or : forall (l m:list) (a:A), In a (app l m) -> In a l \/ In a m.
Proof.
intros l m a.
elim l; simpl in |- *; auto with list.
intros a0 y H H0.
(* ((<A>a0=a)\/(In a y))\/(In a m)
============================
H0 : (<A>a0=a)\/(In a (app y m))
H : (In a (app y m))->((In a y)\/(In a m))
y : list
a0 : A
a : A
m : list
l : list *)
elim H0; auto with list.
intro H1.
(* ((<A>a0=a)\/(In a y))\/(In a m)
============================
H1 : (In a (app y m)) *)
elim (H H1); auto with list.
Qed.
Hint Immediate in_app_or: list v62.
Lemma in_or_app : forall (l m:list) (a:A), In a l \/ In a m -> In a (app l m).
Proof.
intros l m a.
elim l; simpl in |- *; intro H.
(* 1 (In a m)
============================
H : False\/(In a m)
a : A
m : list
l : list *)
elim H; auto with list; intro H0.
(* (In a m)
============================
H0 : False *)
elim H0. (* subProof completed *)
intros y H0 H1.
(* 2 (<A>H=a)\/(In a (app y m))
============================
H1 : ((<A>H=a)\/(In a y))\/(In a m)
H0 : ((In a y)\/(In a m))->(In a (app y m))
y : list *)
elim H1; auto 4 with list.
intro H2.
(* (<A>H=a)\/(In a (app y m))
============================
H2 : (<A>H=a)\/(In a y) *)
elim H2; auto with list.
Qed.
Hint Resolve in_or_app: list v62.
Definition incl (l m:list) := forall a:A, In a l -> In a m.
Hint Unfold incl: list v62.
Lemma incl_refl : forall l:list, incl l l.
Proof.
auto with list.
Qed.
Hint Resolve incl_refl: list v62.
Lemma incl_tl : forall (a:A) (l m:list), incl l m -> incl l (cons a m).
Proof.
auto with list.
Qed.
Hint Immediate incl_tl: list v62.
Lemma incl_tran : forall l m n:list, incl l m -> incl m n -> incl l n.
Proof.
auto with list.
Qed.
Lemma incl_appl : forall l m n:list, incl l n -> incl l (app n m).
Proof.
auto with list.
Qed.
Hint Immediate incl_appl: list v62.
Lemma incl_appr : forall l m n:list, incl l n -> incl l (app m n).
Proof.
auto with list.
Qed.
Hint Immediate incl_appr: list v62.
Lemma incl_cons :
forall (a:A) (l m:list), In a m -> incl l m -> incl (cons a l) m.
Proof.
unfold incl in |- *; simpl in |- *; intros a l m H H0 a0 H1.
(* (In a0 m)
============================
H1 : (<A>a=a0)\/(In a0 l)
a0 : A
H0 : (a:A)(In a l)->(In a m)
H : (In a m)
m : list
l : list
a : A *)
elim H1.
(* 1 (<A>a=a0)->(In a0 m) *)
elim H1; auto with list; intro H2.
(* (<A>a=a0)->(In a0 m)
============================
H2 : <A>a=a0 *)
elim H2; auto with list. (* solves subgoal *)
(* 2 (In a0 l)->(In a0 m) *)
auto with list.
Qed.
Hint Resolve incl_cons: list v62.
Lemma incl_app : forall l m n:list, incl l n -> incl m n -> incl (app l m) n.
Proof.
unfold incl in |- *; simpl in |- *; intros l m n H H0 a H1.
(* (In a n)
============================
H1 : (In a (app l m))
a : A
H0 : (a:A)(In a m)->(In a n)
H : (a:A)(In a l)->(In a n)
n : list
m : list
l : list *)
elim (in_app_or l m a); auto with list.
Qed.
Hint Resolve incl_app: list v62.
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