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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*)
Require Import Le.
Open Local Scope nat_scope.
Implicit Types m n : nat.
(** * maximum of two natural numbers *)
Fixpoint max n m {struct n} : nat :=
match n, m with
| O, _ => m
| S n', O => n
| S n', S m' => S (max n' m')
end.
(** * Simplifications of [max] *)
Lemma max_SS : forall n m, S (max n m) = max (S n) (S m).
Proof.
auto with arith.
Qed.
Theorem max_assoc : forall m n p : nat, max m (max n p) = max (max m n) p.
Proof.
induction m; destruct n; destruct p; trivial.
simpl.
auto using IHm.
Qed.
Lemma max_comm : forall n m, max n m = max m n.
Proof.
induction n; induction m; simpl in |- *; auto with arith.
Qed.
(** * [max] and [le] *)
Lemma max_l : forall n m, m <= n -> max n m = n.
Proof.
induction n; induction m; simpl in |- *; auto with arith.
Qed.
Lemma max_r : forall n m, n <= m -> max n m = m.
Proof.
induction n; induction m; simpl in |- *; auto with arith.
Qed.
Lemma le_max_l : forall n m, n <= max n m.
Proof.
induction n; intros; simpl in |- *; auto with arith.
elim m; intros; simpl in |- *; auto with arith.
Qed.
Lemma le_max_r : forall n m, m <= max n m.
Proof.
induction n; simpl in |- *; auto with arith.
induction m; simpl in |- *; auto with arith.
Qed.
Hint Resolve max_r max_l le_max_l le_max_r: arith v62.
(** * [max n m] is equal to [n] or [m] *)
Lemma max_dec : forall n m, {max n m = n} + {max n m = m}.
Proof.
induction n; induction m; simpl in |- *; auto with arith.
elim (IHn m); intro H; elim H; auto.
Qed.
Lemma max_case : forall n m (P:nat -> Type), P n -> P m -> P (max n m).
Proof.
induction n; simpl in |- *; auto with arith.
induction m; intros; simpl in |- *; auto with arith.
pattern (max n m) in |- *; apply IHn; auto with arith.
Qed.
Notation max_case2 := max_case (only parsing).
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