# Library Coq.Arith.Min

``` 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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