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(************************************************************************)
(* v * The Coq Proof Assistant / The Coq Development Team *)
(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *)
(* \VV/ **************************************************************)
(* // * This file is distributed under the terms of the *)
(* * GNU Lesser General Public License Version 2.1 *)
(************************************************************************)
(*i $Id: Zmin.v,v 1.3.2.1 2004/07/16 19:31:21 herbelin Exp $ i*)
(** Binary Integers (Pierre Crégut (CNET, Lannion, France) *)
Require Import Arith.
Require Import BinInt.
Require Import Zcompare.
Require Import Zorder.
Open Local Scope Z_scope.
(**********************************************************************)
(** Minimum on binary integer numbers *)
Definition Zmin (n m:Z) :=
match n ?= m return Z with
| Eq => n
| Lt => n
| Gt => m
end.
(** Properties of minimum on binary integer numbers *)
Lemma Zmin_SS : forall n m:Z, Zsucc (Zmin n m) = Zmin (Zsucc n) (Zsucc m).
Proof.
intros n m; unfold Zmin in |- *; rewrite (Zcompare_succ_compat n m);
elim_compare n m; intros E; rewrite E; auto with arith.
Qed.
Lemma Zle_min_l : forall n m:Z, Zmin n m <= n.
Proof.
intros n m; unfold Zmin in |- *; elim_compare n m; intros E; rewrite E;
[ apply Zle_refl
| apply Zle_refl
| apply Zlt_le_weak; apply Zgt_lt; exact E ].
Qed.
Lemma Zle_min_r : forall n m:Z, Zmin n m <= m.
Proof.
intros n m; unfold Zmin in |- *; elim_compare n m; intros E; rewrite E;
[ unfold Zle in |- *; rewrite E; discriminate
| unfold Zle in |- *; rewrite E; discriminate
| apply Zle_refl ].
Qed.
Lemma Zmin_case : forall (n m:Z) (P:Z -> Set), P n -> P m -> P (Zmin n m).
Proof.
intros n m P H1 H2; unfold Zmin in |- *; case (n ?= m); auto with arith.
Qed.
Lemma Zmin_or : forall n m:Z, Zmin n m = n \/ Zmin n m = m.
Proof.
unfold Zmin in |- *; intros; elim (n ?= m); auto.
Qed.
Lemma Zmin_n_n : forall n:Z, Zmin n n = n.
Proof.
unfold Zmin in |- *; intros; elim (n ?= n); auto.
Qed.
Lemma Zmin_plus : forall n m p:Z, Zmin (n + p) (m + p) = Zmin n m + p.
Proof.
intros x y n; unfold Zmin in |- *.
rewrite (Zplus_comm x n); rewrite (Zplus_comm y n);
rewrite (Zcompare_plus_compat x y n).
case (x ?= y); apply Zplus_comm.
Qed.
(**********************************************************************)
(** Maximum of two binary integer numbers *)
Definition Zmax a b := match a ?= b with
| Lt => b
| _ => a
end.
(** Properties of maximum on binary integer numbers *)
Ltac CaseEq name :=
generalize (refl_equal name); pattern name at -1 in |- *; case name.
Theorem Zmax1 : forall a b, a <= Zmax a b.
Proof.
intros a b; unfold Zmax in |- *; CaseEq (a ?= b); simpl in |- *;
auto with zarith.
unfold Zle in |- *; intros H; rewrite H; red in |- *; intros; discriminate.
Qed.
Theorem Zmax2 : forall a b, b <= Zmax a b.
Proof.
intros a b; unfold Zmax in |- *; CaseEq (a ?= b); simpl in |- *;
auto with zarith.
intros H;
(case (Zle_or_lt b a); auto; unfold Zlt in |- *; rewrite H; intros;
discriminate).
intros H;
(case (Zle_or_lt b a); auto; unfold Zlt in |- *; rewrite H; intros;
discriminate).
Qed.
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