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Diffstat (limited to 'theories/Arith/Min.v')
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diff --git a/theories/Arith/Min.v b/theories/Arith/Min.v new file mode 100755 index 00000000..912e7ba3 --- /dev/null +++ b/theories/Arith/Min.v @@ -0,0 +1,83 @@ +(************************************************************************) +(* 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: Min.v,v 1.10.2.1 2004/07/16 19:31:00 herbelin Exp $ i*) + +Require Import Arith. + +Open Local Scope nat_scope. + +Implicit Types m n : nat. + +(** minimum of two natural numbers *) + +Fixpoint min n m {struct n} : nat := + match n, m with + | O, _ => 0 + | S n', O => 0 + | S n', S m' => S (min n' m') + end. + +(** Simplifications of [min] *) + +Lemma min_SS : forall n m, S (min n m) = min (S n) (S m). +Proof. +auto with arith. +Qed. + +Lemma min_comm : forall n m, min n m = min m n. +Proof. +induction n; induction m; simpl in |- *; auto with arith. +Qed. + +(** [min] and [le] *) + +Lemma min_l : forall n m, n <= m -> min n m = n. +Proof. +induction n; induction m; simpl in |- *; auto with arith. +Qed. + +Lemma min_r : forall n m, m <= n -> min n m = m. +Proof. +induction n; induction m; simpl in |- *; auto with arith. +Qed. + +Lemma le_min_l : forall n m, min n m <= n. +Proof. +induction n; intros; simpl in |- *; auto with arith. +elim m; intros; simpl in |- *; auto with arith. +Qed. + +Lemma le_min_r : forall n m, min n m <= m. +Proof. +induction n; simpl in |- *; auto with arith. +induction m; simpl in |- *; auto with arith. +Qed. +Hint Resolve min_l min_r le_min_l le_min_r: arith v62. + +(** [min n m] is equal to [n] or [m] *) + +Lemma min_dec : forall n m, {min n m = n} + {min n m = m}. +Proof. +induction n; induction m; simpl in |- *; auto with arith. +elim (IHn m); intro H; elim H; auto. +Qed. + +Lemma min_case : forall n m (P:nat -> Set), P n -> P m -> P (min n m). +Proof. +induction n; simpl in |- *; auto with arith. +induction m; intros; simpl in |- *; auto with arith. +pattern (min n m) in |- *; apply IHn; auto with arith. +Qed. + +Lemma min_case2 : forall n m (P:nat -> Prop), P n -> P m -> P (min n m). +Proof. +induction n; simpl in |- *; auto with arith. +induction m; intros; simpl in |- *; auto with arith. +pattern (min n m) in |- *; apply IHn; auto with arith. +Qed.
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