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.