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.