Zmax.v

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THIS FILE IS DEPRECATED.

Require Export BinInt Zcompare Zorder.

Local Open Scope Z_scope.

Definition Z.max is now BinInt.Z.max.

Exact compatibility

Notation Zmax_right := Z.max_r (only parsing).
Notation Zle_max_compat_r := Z.max_le_compat_r (only parsing).
Notation Zle_max_compat_l := Z.max_le_compat_l (only parsing).
Notation Zmax_idempotent := Z.max_id (only parsing).
Notation Zmax_n_n := Z.max_id (only parsing).
Notation Zmax_irreducible_dec := Z.max_dec (only parsing).
Notation Zmax_le_prime := Z.max_le (only parsing).
Notation Zmax_SS := Z.succ_max_distr (only parsing).
Notation Zplus_max_distr_l := Z.add_max_distr_l (only parsing).
Notation Zplus_max_distr_r := Z.add_max_distr_r (only parsing).
Notation Zmax_plus := Z.add_max_distr_r (only parsing).
Notation Zmax1 := Z.le_max_l (only parsing).
Notation Zmax2 := Z.le_max_r (only parsing).
Notation Zmax_irreducible_inf := Z.max_dec (only parsing).
Notation Zmax_le_prime_inf := Z.max_le (only parsing).
Notation Zpos_max := Pos2Z.inj_max (only parsing).
Notation Zpos_minus := Pos2Z.inj_sub_max (only parsing).

Slightly different lemmas

Lemma Zmax_spec x y :
  x >= y /\ Z.max x y = x  \/ x < y /\ Z.max x y = y.x, y:Z
x >= y /\ Z.max x y = x \/ x < y /\ Z.max x y = y
Proof.x, y:Z
x >= y /\ Z.max x y = x \/ x < y /\ Z.max x y = y
 Z.swap_greater.x, y:Z
y <= x /\ Z.max x y = x \/ x < y /\ Z.max x y = y destruct (Z.max_spec x y); auto.
Qed.

Lemma Zmax_left n m : n>=m -> Z.max n m = n.n, m:Z
n >= m -> Z.max n m = n
Proof.n, m:Z
n >= m -> Z.max n m = n Z.swap_greater.n, m:Z
m <= n -> Z.max n m = n apply Z.max_l. Qed.

Lemma Zpos_max_1 p : Z.max 1 (Z.pos p) = Z.pos p.p:positive
Z.max 1 (Z.pos p) = Z.pos p
Proof.p:positive
Z.max 1 (Z.pos p) = Z.pos p
 now destruct p.
Qed.