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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: MonoList.v 8642 2006-03-17 10:09:02Z notin $ 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 IT IS NO LONGER MAINTAINED ***)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. \ No newline at end of file
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