# Library Coq.Numbers.Integer.Abstract.ZMaxMin

Require Import ZAxioms ZMulOrder GenericMinMax.

# Properties of minimum and maximum specific to integer numbers

Module Type ZMaxMinProp (Import Z : ZAxiomsMiniSig').
Include ZMulOrderProp Z.

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).

Pred

Lemma pred_max_distr : forall n m, P (max n m) == max (P n) (P m).

Lemma pred_min_distr : forall n m, P (min n m) == min (P n) (P m).

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.

Opp

Lemma opp_max_distr : forall n m, -(max n m) == min (-n) (-m).

Lemma opp_min_distr : forall n m, -(min n m) == max (-n) (-m).

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.

Mul

Lemma mul_max_distr_nonneg_l : forall n m p, 0 <= p ->
max (p * n) (p * m) == p * max n m.

Lemma mul_max_distr_nonneg_r : forall n m p, 0 <= p ->
max (n * p) (m * p) == max n m * p.

Lemma mul_min_distr_nonneg_l : forall n m p, 0 <= p ->
min (p * n) (p * m) == p * min n m.

Lemma mul_min_distr_nonneg_r : forall n m p, 0 <= p ->
min (n * p) (m * p) == min n m * p.

Lemma mul_max_distr_nonpos_l : forall n m p, p <= 0 ->
max (p * n) (p * m) == p * min n m.

Lemma mul_max_distr_nonpos_r : forall n m p, p <= 0 ->
max (n * p) (m * p) == min n m * p.

Lemma mul_min_distr_nonpos_l : forall n m p, p <= 0 ->
min (p * n) (p * m) == p * max n m.

Lemma mul_min_distr_nonpos_r : forall n m p, p <= 0 ->
min (n * p) (m * p) == max n m * p.

End ZMaxMinProp.