Zmin.v

Built with Alectryon, running Coq+SerAPI v8.10.0+0.7.0. Coq sources are in this panel; goals and messages will appear in the other. Bubbles () indicate interactive fragments: hover for details, tap to reveal contents. Use Ctrl+↑ Ctrl+↓ to navigate, Ctrl+🖱️ to focus.

(************************************************************************)
(*         *   The Coq Proof Assistant / The Coq Development Team       *)
(*  v      *   INRIA, CNRS and contributors - Copyright 1999-2018       *)
(* <O___,, *       (see CREDITS file for the list of authors)           *)
(*   \VV/  **************************************************************)
(*    //   *    This file is distributed under the terms of the         *)
(*         *     GNU Lesser General Public License Version 2.1          *)
(*         *     (see LICENSE file for the text of the license)         *)
(************************************************************************)

THIS FILE IS DEPRECATED.

Require Import BinInt Zcompare Zorder.

Local Open Scope Z_scope.

Definition Z.min is now BinInt.Z.min.

Exact compatibility

Notation Zle_min_compat_r := Z.min_le_compat_r (only parsing).
Notation Zle_min_compat_l := Z.min_le_compat_l (only parsing).
Notation Zmin_idempotent := Z.min_id (only parsing).
Notation Zmin_n_n := Z.min_id (only parsing).
Notation Zmin_irreducible_inf := Z.min_dec (only parsing).
Notation Zmin_SS := Z.succ_min_distr (only parsing).
Notation Zplus_min_distr_r := Z.add_min_distr_r (only parsing).
Notation Zmin_plus := Z.add_min_distr_r (only parsing).
Notation Zpos_min := Pos2Z.inj_min (only parsing).

Slightly different lemmas

Lemma Zmin_spec x y :
  x <= y /\ Z.min x y = x  \/  x > y /\ Z.min x y = y.x, y:Z
x <= y /\ Z.min x y = x \/ x > y /\ Z.min x y = y
Proof.x, y:Z
x <= y /\ Z.min x y = x \/ x > y /\ Z.min x y = y
 Z.swap_greater.x, y:Z
x <= y /\ Z.min x y = x \/ y < x /\ Z.min x y = y rewrite Z.min_comm.x, y:Z
x <= y /\ Z.min y x = x \/ y < x /\ Z.min y x = y destruct (Z.min_spec y x); auto.
Qed.

Lemma Zmin_irreducible n m : Z.min n m = n \/ Z.min n m = m.n, m:Z
Z.min n m = n \/ Z.min n m = m
Proof.n, m:Z
Z.min n m = n \/ Z.min n m = m destruct (Z.min_dec n m); auto. Qed.

Notation Zmin_or := Zmin_irreducible (only parsing).

Lemma Zmin_le_prime_inf n m p : Z.min n m <= p -> {n <= p} + {m <= p}.n, m, p:Z
Z.min n m <= p -> {n <= p} + {m <= p}
Proof.n, m, p:Z
Z.min n m <= p -> {n <= p} + {m <= p} apply Z.min_case; auto. Qed.

Lemma Zpos_min_1 p : Z.min 1 (Zpos p) = 1.p:positive
Z.min 1 (Z.pos p) = 1
Proof.p:positive
Z.min 1 (Z.pos p) = 1
 now destruct p.
Qed.