# Library Coq.Numbers.Natural.Abstract.NMaxMin

Require Import NAxioms NSub GenericMinMax.

# Properties of minimum and maximum specific to natural numbers

Module Type NMaxMinProp (Import N : NAxiomsMiniSig').
Include NSubProp N.
Zero
Lemma max_0_l : forall n, max 0 n == n.
Lemma max_0_r : forall n, max n 0 == n.
Lemma min_0_l : forall n, min 0 n == 0.
Lemma min_0_r : forall n, min n 0 == 0.
The following results are concrete instances of max_monotone and similar lemmas.
Succ
Lemma succ_max_distr : forall n m, S (max n m) == max (S n) (S m).
Lemma succ_min_distr : forall n m, S (min n m) == min (S n) (S m).
Add
Lemma add_max_distr_l : forall n m p, max (p + n) (p + m) == p + max n m.
Lemma add_max_distr_r : forall n m p, max (n + p) (m + p) == max n m + p.
Lemma add_min_distr_l : forall n m p, min (p + n) (p + m) == p + min n m.
Lemma add_min_distr_r : forall n m p, min (n + p) (m + p) == min n m + p.
Mul
Lemma mul_max_distr_l : forall n m p, max (p * n) (p * m) == p * max n m.
Lemma mul_max_distr_r : forall n m p, max (n * p) (m * p) == max n m * p.
Lemma mul_min_distr_l : forall n m p, min (p * n) (p * m) == p * min n m.
Lemma mul_min_distr_r : forall n m p, min (n * p) (m * p) == min n m * p.
Sub
Lemma sub_max_distr_l : forall n m p, max (p - n) (p - m) == p - min n m.
Lemma sub_max_distr_r : forall n m p, max (n - p) (m - p) == max n m - p.
Lemma sub_min_distr_l : forall n m p, min (p - n) (p - m) == p - max n m.
Lemma sub_min_distr_r : forall n m p, min (n - p) (m - p) == min n m - p.
End NMaxMinProp.