# Library Coq.ZArith.Zmin

THIS FILE IS DEPRECATED. Use Zminmax instead.
Require Import BinInt Zorder Zminmax.
Open Local Scope Z_scope.
Zmin is now Zminmax.Zmin. 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 the minimum on binary integer numbers

Definition Zmin_case := Z.min_case.
Definition Zmin_case_strong := Z.min_case_strong.
Lemma Zmin_spec : forall x y,
x <= y /\ Zmin x y = x \/ x > y /\ Zmin x y = y.

# Greatest lower bound properties of min

Definition Zle_min_l : forall n m, Zmin n m <= n := Z.le_min_l.
Definition Zle_min_r : forall n m, Zmin n m <= m := Z.le_min_r.
Definition Zmin_glb : forall n m p, p <= n -> p <= m -> p <= Zmin n m
:= Z.min_glb.
Definition Zmin_glb_lt : forall n m p, p < n -> p < m -> p < Zmin n m
:= Z.min_glb_lt.

# Compatibility with order

Definition Zle_min_compat_r : forall n m p, n <= m -> Zmin n p <= Zmin m p
:= Z.min_le_compat_r.
Definition Zle_min_compat_l : forall n m p, n <= m -> Zmin p n <= Zmin p m
:= Z.min_le_compat_l.

# Semi-lattice properties of min

Definition Zmin_idempotent : forall n, Zmin n n = n := Z.min_id.
Notation Zmin_n_n := Zmin_idempotent (only parsing).
Definition Zmin_comm : forall n m, Zmin n m = Zmin m n := Z.min_comm.
Definition Zmin_assoc : forall n m p, Zmin n (Zmin m p) = Zmin (Zmin n m) p
:= Z.min_assoc.

# Additional properties of min

Lemma Zmin_irreducible_inf : forall n m, { Zmin n m = n } + { Zmin n m = m }.
Lemma Zmin_irreducible : forall n m, Zmin n m = n \/ Zmin n m = m.
Notation Zmin_or := Zmin_irreducible (only parsing).
Lemma Zmin_le_prime_inf : forall n m p, Zmin n m <= p -> { n <= p } + { m <= p }.

# Operations preserving min

Definition Zsucc_min_distr :
forall n m, Zsucc (Zmin n m) = Zmin (Zsucc n) (Zsucc m)
:= Z.succ_min_distr.
Notation Zmin_SS := Z.succ_min_distr (only parsing).
Definition Zplus_min_distr_r :
forall n m p, Zmin (n + p) (m + p) = Zmin n m + p
:= Z.plus_min_distr_r.
Notation Zmin_plus := Z.plus_min_distr_r (only parsing).

# Minimum and Zpos

Definition Zpos_min : forall p q, Zpos (Pmin p q) = Zmin (Zpos p) (Zpos q)
:= Z.pos_min.