Quellcode-Bibliothek

^© Kompilation durch diese Firma

[Weder Korrektheit noch Funktionsfähigkeit der Software werden zugesichert.]

Datei: Qpower.v Sprache: Coq

Original von: Coq^©

(************************************************************************)
(*         *   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)         *)
(************************************************************************)

Require Import Zpow_facts Qfield Qreduction.

Lemma Qpower_positive_1 : forall n, Qpower_positive 1 n == 1.
Proof.
induction n;
simpl; try rewrite IHn; reflexivity.
Qed.

Lemma Qpower_1 : forall n, 1^n == 1.
Proof.
  intros [|n|n]; simpl; try rewrite Qpower_positive_1; reflexivity.
Qed.

Lemma Qpower_positive_0 : forall n, Qpower_positive 0 n == 0.
Proof.
induction n;
simpl; try rewrite IHn; reflexivity.
Qed.

Lemma Qpower_0 : forall n, (n<>0)%Z -> 0^n == 0.
Proof.
  intros [|n|n] Hn; try (elim Hn; reflexivity); simpl;
  rewrite Qpower_positive_0; reflexivity.
Qed.

Lemma Qpower_not_0_positive : forall a n, ~a==0 -> ~Qpower_positive a n == 0.
Proof.
intros a n X H.
apply X; clear X.
induction n; simpl in *; try assumption;
destruct (Qmult_integral _ _ H);
try destruct (Qmult_integral _ _ H0); auto.
Qed.

Lemma Qpower_pos_positive : forall p n, 0 <= p -> 0 <= Qpower_positive p n.
Proof.
intros p n Hp.
induction n; simpl; repeat apply Qmult_le_0_compat;assumption.
Qed.

Lemma Qpower_pos : forall p n, 0 <= p -> 0 <= p^n.
Proof.
  intros p [|n|n] Hp; simpl; try discriminate;
  try apply Qinv_le_0_compat;  apply Qpower_pos_positive; assumption.
Qed.

Lemma Qmult_power_positive  : forall a b n, Qpower_positive (a*b) n == (Qpower_positive a n)*(Qpower_positive b n).
Proof.
induction n;
simpl; repeat rewrite IHn; ring.
Qed.

Lemma Qmult_power : forall a b n, (a*b)^n == a^n*b^n.
Proof.
  intros a b [|n|n]; simpl;
  try rewrite Qmult_power_positive;
  try rewrite Qinv_mult_distr;
  reflexivity.
Qed.

Lemma Qinv_power_positive  : forall a n, Qpower_positive (/a) n == /(Qpower_positive a n).
Proof.
induction n;
simpl; repeat (rewrite IHn || rewrite Qinv_mult_distr); reflexivity.
Qed.

Lemma Qinv_power : forall a n, (/a)^n == /a^n.
Proof.
  intros a [|n|n]; simpl;
  try rewrite Qinv_power_positive;
  reflexivity.
Qed.

Lemma Qdiv_power : forall a b n, (a/b)^n == (a^n/b^n).
Proof.
unfold Qdiv.
intros a b n.
rewrite Qmult_power.
rewrite Qinv_power.
reflexivity.
Qed.

Lemma Qinv_power_n : forall n p, (1#p)^n == /(inject_Z (Zpos p))^n.
Proof.
intros n p.
rewrite Qmake_Qdiv.
rewrite Qdiv_power.
rewrite Qpower_1.
unfold Qdiv.
ring.
Qed.

Lemma Qpower_plus_positive : forall a n m, Qpower_positive a (n+m) == (Qpower_positive a n)*(Qpower_positive a m).
Proof.
intros a n m.
unfold Qpower_positive.
apply pow_pos_add.
apply Q_Setoid.
apply Qmult_comp.
apply Qmult_assoc.
Qed.

Lemma Qpower_opp : forall a n, a^(-n) == /a^n.
Proof.
intros a [|n|n]; simpl; try reflexivity.
symmetry; apply Qinv_involutive.
Qed.

Lemma Qpower_minus_positive : forall a (n m:positive),
  (m < n)%positive ->
  Qpower_positive a (n-m)%positive == (Qpower_positive a n)/(Qpower_positive a m).
Proof.
intros a n m H.
destruct (Qeq_dec a 0) as [EQ|NEQ].
- now rewrite EQ, !Qpower_positive_0.
- rewrite <- (Qdiv_mult_l (Qpower_positive a (n - m)) (Qpower_positive a m)) by
   (now apply Qpower_not_0_positive).
  f_equiv.
  rewrite <- Qpower_plus_positive.
  now rewrite Pos.sub_add.
Qed.

Lemma Qpower_plus : forall a n m, ~a==0 -> a^(n+m) == a^n*a^m.
Proof.
intros a [|n|n] [|m|m] H; simpl; try ring;
try rewrite Qpower_plus_positive;
try apply Qinv_mult_distr; try reflexivity;
rewrite ?Z.pos_sub_spec;
case Pos.compare_spec; intros H0; simpl; subst;
try rewrite Qpower_minus_positive;
try (field; try split; apply Qpower_not_0_positive);
assumption.
Qed.

Lemma Qpower_plus' : forall a n m, (n+m <> 0)%Z -> a^(n+m) == a^n*a^m.
Proof.
intros a n m H.
destruct (Qeq_dec a 0)as [X|X].
rewrite X.
rewrite Qpower_0 by assumption.
destruct n; destruct m; try (elim H; reflexivity);
simpl; repeat rewrite Qpower_positive_0; ring_simplify;
reflexivity.
apply Qpower_plus.
assumption.
Qed.

Lemma Qpower_mult_positive : forall a n m,
  Qpower_positive a (n*m) == Qpower_positive (Qpower_positive a n) m.
Proof.
intros a n m.
induction n using Pos.peano_ind.
reflexivity.
rewrite Pos.mul_succ_l.
rewrite <- Pos.add_1_l.
do 2 rewrite Qpower_plus_positive.
rewrite IHn.
rewrite Qmult_power_positive.
reflexivity.
Qed.

Lemma Qpower_mult : forall a n m, a^(n*m) ==  (a^n)^m.
Proof.
intros a [|n|n] [|m|m]; simpl;
try rewrite Qpower_positive_1;
try rewrite Qpower_mult_positive;
try rewrite Qinv_power_positive;
try rewrite Qinv_involutive;
try reflexivity.
Qed.

Lemma Zpower_Qpower : forall (a n:Z), (0<=n)%Z -> inject_Z (a^n) == (inject_Z a)^n.
Proof.
intros a [|n|n] H;[reflexivity| |elim H; reflexivity].
induction n using Pos.peano_ind.
replace (a^1)%Z with a by ring.
ring.
rewrite Pos2Z.inj_succ.
unfold Z.succ.
rewrite Zpower_exp; auto with *; try discriminate.
rewrite Qpower_plus' by discriminate.
rewrite <- IHn by discriminate.
replace (a^Zpos n*a^1)%Z with (a^Zpos n*a)%Z by ring.
ring_simplify.
reflexivity.
Qed.

Lemma Qsqr_nonneg : forall a, 0 <= a^2.
Proof.
intros a.
destruct (Qlt_le_dec 0 a) as [A|A].
apply (Qmult_le_0_compat a a);
(apply Qlt_le_weak; assumption).
setoid_replace (a^2) with ((-a)*(-a)) by ring.
rewrite Qle_minus_iff in A.
setoid_replace (0+ - a) with (-a) in A by ring.
apply Qmult_le_0_compat; assumption.
Qed.

Theorem Qpower_decomp p x y :
  Qpower_positive (x#y) p = x ^ Zpos p # (y ^ p).
Proof.
induction p; intros; simpl Qpower_positive; rewrite ?IHp.
- (* xI *)
  unfold Qmult, Qnum, Qden. f_equal.
  + now rewrite <- Z.pow_twice_r, <- Z.pow_succ_r.
  + apply Pos2Z.inj; rewrite !Pos2Z.inj_mul, !Pos2Z.inj_pow.
    now rewrite <- Z.pow_twice_r, <- Z.pow_succ_r.
- (* xO *)
  unfold Qmult, Qnum, Qden. f_equal.
  + now rewrite <- Z.pow_twice_r.
  + apply Pos2Z.inj; rewrite !Pos2Z.inj_mul, !Pos2Z.inj_pow.
    now rewrite <- Z.pow_twice_r.
- (* xO *)
  now rewrite Z.pow_1_r, Pos.pow_1_r.
Qed.

¤ Dauer der Verarbeitung: 0.19 Sekunden (vorverarbeitet) ¤

Download des Quellennavigators
Download des sprechenden Kalenders
in der Quellcodebibliothek suchen

Haftungshinweis

Die Informationen auf dieser Webseite wurden nach bestem Wissen sorgfältig zusammengestellt. Es wird jedoch weder Vollständigkeit, noch Richtigkeit, noch Qualität der bereit gestellten Informationen zugesichert.

Bemerkung:

Die farbliche Syntaxdarstellung ist noch experimentell.