Require Import Arith.
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.
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 -> Set), 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.
Lemma max_case2 : forall n m (P:nat -> Prop), 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.