# Library Coq.ZArith.Zmax

THIS FILE IS DEPRECATED. Use Zminmax instead.
Require Export BinInt Zorder Zminmax.
Open Local Scope Z_scope.
Zmax is now Zminmax.Zmax. Code that do things like unfold Zmin.Zmin will have to be adapted, and neither a Definition or a Notation here can help much.

# Characterization of maximum on binary integer numbers

Definition Zmax_case := Z.max_case.
Definition Zmax_case_strong := Z.max_case_strong.
Lemma Zmax_spec : forall x y,
x >= y /\ Zmax x y = x \/ x < y /\ Zmax x y = y.
Lemma Zmax_left : forall n m, n>=m -> Zmax n m = n.
Definition Zmax_right : forall n m, n<=m -> Zmax n m = m := Zmax_r.

# Least upper bound properties of max

Definition Zle_max_l : forall n m, n <= Zmax n m := Z.le_max_l.
Definition Zle_max_r : forall n m, m <= Zmax n m := Z.le_max_r.
Definition Zmax_lub : forall n m p, n <= p -> m <= p -> Zmax n m <= p
:= Z.max_lub.
Definition Zmax_lub_lt : forall n m p:Z, n < p -> m < p -> Zmax n m < p
:= Z.max_lub_lt.

# Compatibility with order

Definition Zle_max_compat_r : forall n m p, n <= m -> Zmax n p <= Zmax m p
:= Z.max_le_compat_r.
Definition Zle_max_compat_l : forall n m p, n <= m -> Zmax p n <= Zmax p m
:= Z.max_le_compat_l.

# Semi-lattice properties of max

Definition Zmax_idempotent : forall n, Zmax n n = n := Z.max_id.
Definition Zmax_comm : forall n m, Zmax n m = Zmax m n := Z.max_comm.
Definition Zmax_assoc : forall n m p, Zmax n (Zmax m p) = Zmax (Zmax n m) p
:= Z.max_assoc.

# Additional properties of max

Lemma Zmax_irreducible_dec : forall n m, {Zmax n m = n} + {Zmax n m = m}.
Definition Zmax_le_prime : forall n m p, p <= Zmax n m -> p <= n \/ p <= m
:= Z.max_le.

# Operations preserving max

Definition Zsucc_max_distr :
forall n m:Z, Zsucc (Zmax n m) = Zmax (Zsucc n) (Zsucc m)
:= Z.succ_max_distr.
Definition Zplus_max_distr_l : forall n m p:Z, Zmax (p + n) (p + m) = p + Zmax n m
:= Z.plus_max_distr_l.
Definition Zplus_max_distr_r : forall n m p:Z, Zmax (n + p) (m + p) = Zmax n m + p
:= Z.plus_max_distr_r.

# Maximum and Zpos

Definition Zpos_max : forall p q, Zpos (Pmax p q) = Zmax (Zpos p) (Zpos q)
:= Z.pos_max.
Definition Zpos_max_1 : forall p, Zmax 1 (Zpos p) = Zpos p
:= Z.pos_max_1.

# Characterization of Pminus in term of Zminus and Zmax

Definition Zpos_minus :
forall p q, Zpos (Pminus p q) = Zmax 1 (Zpos p - Zpos q)
:= Zpos_minus.