Timings for Qpower.v
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Require Import Zpow_facts Qfield Qreduction.
(** * Properties of Qpower_positive *)
(** ** Values of Qpower_positive for specific arguments *)
Lemma Qpower_positive_1 : forall n, Qpower_positive 1 n == 1.
induction n;
simpl; try rewrite IHn; reflexivity.
Lemma Qpower_positive_0 : forall n, Qpower_positive 0 n == 0.
induction n;
simpl; try rewrite IHn; reflexivity.
(** ** Relation of Qpower_positive to zero *)
Lemma Qpower_not_0_positive : forall a n, ~a==0 -> ~Qpower_positive a n == 0.
induction n; simpl in *; try assumption;
destruct (Qmult_integral _ _ H);
try destruct (Qmult_integral _ _ H0); auto.
Lemma Qpower_pos_positive : forall p n, 0 <= p -> 0 <= Qpower_positive p n.
induction n; simpl; repeat apply Qmult_le_0_compat;assumption.
(** ** Qpower_positive and multiplication, exponent subtraction *)
Lemma Qmult_power_positive : forall a b n, Qpower_positive (a*b) n == (Qpower_positive a n)*(Qpower_positive b n).
induction n;
simpl; repeat rewrite IHn; ring.
Lemma Qpower_plus_positive : forall a n m, Qpower_positive a (n+m) == (Qpower_positive a n)*(Qpower_positive a m).
(** ** Qpower_positive and inversion, division, exponent subtraction *)
Lemma Qinv_power_positive : forall a n, Qpower_positive (/a) n == /(Qpower_positive a n).
induction n;
simpl; repeat (rewrite IHn || rewrite Qinv_mult_distr); reflexivity.
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).
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).
rewrite <- Qpower_plus_positive.
(** ** Qpower and exponent multiplication *)
Lemma Qpower_mult_positive : forall a n m,
Qpower_positive a (n*m) == Qpower_positive (Qpower_positive a n) m.
induction n using Pos.peano_ind.
do 2 rewrite Qpower_plus_positive.
rewrite Qmult_power_positive.
(** ** Qpower_positive decomposition *)
Lemma Qpower_decomp_positive p x y :
Qpower_positive (x#y) p = x ^ Zpos p # (y ^ p).
induction p; intros; simpl Qpower_positive; rewrite ?IHp.
unfold Qmult, Qnum, Qden.
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.
unfold Qmult, Qnum, Qden.
now rewrite <- Z.pow_twice_r.
apply Pos2Z.inj; rewrite !Pos2Z.inj_mul, !Pos2Z.inj_pow.
now rewrite <- Z.pow_twice_r.
now rewrite Z.pow_1_r, Pos.pow_1_r.
(* This notation will be deprecated with a planned larger rework of Q lemma naming *)
Notation Qpower_decomp := Qpower_decomp_positive (only parsing).
(** * Properties of Qpower *)
(** ** Values of Qpower for specific arguments *)
Lemma Qpower_0 : forall n, (n<>0)%Z -> 0^n == 0.
intros [|n|n] Hn; try (elim Hn; reflexivity); simpl;
rewrite Qpower_positive_0; reflexivity.
Lemma Qpower_1 : forall n, 1^n == 1.
intros [|n|n]; simpl; try rewrite Qpower_positive_1; reflexivity.
Lemma Qpower_0_r: forall q:Q,
q^0 == 1.
Lemma Qpower_1_r: forall q:Q,
q^1 == q.
(** ** Relation of Qpower to zero *)
Lemma Qpower_not_0: forall (a : Q) (z : Z),
~ a == 0 -> ~ Qpower a z == 0.
intros a z H; destruct z.
apply Qpower_not_0_positive; assumption.
pose proof Qmult_inv_r (Qpower_positive a p) as H2.
specialize (H2 (Qpower_not_0_positive _ _ H)).
rewrite H1, Qmult_0_r in H2.
Lemma Qpower_0_le : forall (p : Q) (n : Z), 0 <= p -> 0 <= p^n.
intros p [|n|n] Hp; simpl; try discriminate;
try apply Qinv_le_0_compat; apply Qpower_pos_positive; assumption.
(* This notation will be deprecated with a planned larger rework of Q lemma naming *)
Notation Qpower_pos := Qpower_0_le (only parsing).
Lemma Qpower_0_lt: forall (a : Q) (z : Z), 0 < a -> 0 < Qpower a z.
pose proof Qpower_pos q z (Qlt_le_weak 0 q Hpos) as H1.
pose proof Qpower_not_0 q z as H2.
pose proof Qlt_not_eq 0 q Hpos as H3.
specialize (H2 (Qnot_eq_sym _ _ H3)); clear H3.
apply Qlt_leneq; split; assumption.
(** ** Relation of Qpower to 1 *)
Lemma Qpower_1_lt_pos:
forall (q : Q) (n : positive), (1<q)%Q -> (1 < q ^ (Z.pos n))%Q.
apply Qmult_lt_1_compat; assumption.
apply Qmult_lt_1_compat; assumption.
Lemma Qpower_1_lt:
forall (q : Q) (n : Z), (1<q)%Q -> (0<n)%Z -> (1 < q ^ n)%Q.
apply Qpower_1_lt_pos; assumption.
discriminate (Z.lt_trans _ _ _ Hn (Pos2Z.neg_is_neg p)).
Lemma Qpower_1_le_pos:
forall (q : Q) (n : positive), (1<=q)%Q -> (1 <= q ^ (Z.pos n))%Q.
apply Qmult_le_1_compat; assumption.
apply Qmult_le_1_compat; assumption.
Lemma Qpower_1_le:
forall (q : Q) (n : Z), (1<=q)%Q -> (0<=n)%Z -> (1 <= q ^ n)%Q.
apply Qpower_1_le_pos; assumption.
discriminate (Z.le_lt_trans _ _ _ Hn (Pos2Z.neg_is_neg p)).
(** ** Qpower and multiplication, exponent addition *)
Lemma Qmult_power : forall a b n, (a*b)^n == a^n*b^n.
intros a b [|n|n]; simpl;
try rewrite Qmult_power_positive;
try rewrite Qinv_mult_distr;
reflexivity.
Lemma Qpower_plus : forall a n m, ~a==0 -> a^(n+m) == a^n*a^m.
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.
Lemma Qpower_plus' : forall a n m, (n+m <> 0)%Z -> a^(n+m) == a^n*a^m.
destruct (Qeq_dec a 0)as [X|X].
rewrite Qpower_0 by assumption.
destruct n; destruct m; try (elim H; reflexivity);
simpl; repeat rewrite Qpower_positive_0; ring_simplify;
reflexivity.
(** ** Qpower and inversion, division, exponent subtraction *)
Lemma Qinv_power : forall a n, (/a)^n == /a^n.
intros a [|n|n]; simpl;
try rewrite Qinv_power_positive;
reflexivity.
Lemma Qdiv_power : forall a b n, (a/b)^n == (a^n/b^n).
Lemma Qinv_power_n : forall n p, (1#p)^n == /(inject_Z (Zpos p))^n.
Lemma Qpower_opp : forall a n, a^(-n) == /a^n.
intros a [|n|n]; simpl; try reflexivity.
symmetry; apply Qinv_involutive.
Lemma Qpower_minus: forall (a : Q) (n m : Z),
~ a == 0 -> a ^ (n - m) == a ^ n / a ^ m.
rewrite Qpower_plus by assumption.
apply Qpower_not_0; assumption.
Lemma Qpower_minus_pos: forall (a b : positive) (n m : Z),
(Z.pos a#b) ^ (n - m) == (Z.pos a#b) ^ n * (Z.pos b#a) ^ m.
rewrite Qpower_minus by discriminate.
rewrite <- (Qinv_pos b a), Qinv_power.
Lemma Qpower_minus_neg: forall (a b : positive) (n m : Z),
(Z.neg a#b) ^ (n - m) == (Z.neg a#b) ^ n * (Z.neg b#a) ^ m.
rewrite Qpower_minus by discriminate.
rewrite <- (Qinv_neg b a), Qinv_power.
(** ** Qpower and exponent multiplication *)
Lemma Qpower_mult : forall a n m, a^(n*m) == (a^n)^m.
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.
(** ** Qpower decomposition *)
Lemma Qpower_decomp_pos: forall (p : positive) (a : Z) (b : positive),
(a # b) ^ (Z.pos p) == a ^ (Z.pos p) # (b ^ p)%positive.
pose proof Qpower_decomp_positive p a b.
cbn; rewrite H; reflexivity.
Lemma Qpower_decomp_neg_pos: forall (p a b: positive),
(Z.pos a # b) ^ (Z.neg p) == (Z.pos b) ^ (Z.pos p) # (a ^ p)%positive.
rewrite <- Qinv_power_positive, Qinv_pos.
rewrite Qpower_decomp_positive.
Lemma Qpower_decomp_neg_neg: forall (p a b: positive),
(Z.neg a # b) ^ (Z.neg p) == (Z.neg b) ^ (Z.pos p) # (a ^ p)%positive.
rewrite <- Qinv_power_positive, Qinv_neg.
rewrite Qpower_decomp_positive.
(** ** Compatibility of Qpower with relational operators *)
Lemma Qpower_lt_compat_l:
forall (q : Q) (n m : Z), (n < m)%Z -> (1<q)%Q -> (q ^ n < q ^ m)%Q.
replace m with (n+(m-n))%Z by ring.
rewrite Qpower_plus, <- Qmult_1_r, <- Qmult_assoc.
rewrite Qmult_lt_l, Qmult_1_l.
exact (Qlt_trans 0 1 q ltac:(reflexivity) Hq).
rewrite Heqk; apply Z.lt_0_sub, Hnm.
Lemma Qpower_le_compat_l:
forall (q : Q) (n m : Z), (n <= m)%Z -> (1<=q)%Q -> (q ^ n <= q ^ m)%Q.
replace m with (n+(m-n))%Z by ring.
rewrite Qpower_plus, <- Qmult_1_r, <- Qmult_assoc.
rewrite Qmult_le_l, Qmult_1_l.
exact (Qlt_le_trans 0 1 q ltac:(reflexivity) Hq).
rewrite Heqk; apply Z.le_0_sub, Hnm.
Lemma Qpower_lt_compat_l_inv:
forall (q : Q) (n m : Z), (q ^ n < q ^ m)%Q -> (1<q)%Q -> (n < m)%Z.
destruct (Z_lt_le_dec n m) as [Hd|Hd].
pose proof Qpower_le_compat_l q m n Hd (Qlt_le_weak _ _ Hq) as Hnm'.
pose proof Qlt_le_trans _ _ _ Hnm Hnm' as Habsurd.
destruct (Qlt_irrefl _ Habsurd).
Lemma Qpower_le_compat_l_inv:
forall (q : Q) (n m : Z), (q ^ n <= q ^ m)%Q -> (1<q)%Q -> (n <= m)%Z.
destruct (Z_lt_le_dec m n) as [Hd|Hd].
pose proof Qpower_lt_compat_l q m n Hd Hq as Hnm'.
pose proof Qle_lt_trans _ _ _ Hnm Hnm' as Habsurd.
destruct (Qlt_irrefl _ Habsurd).
(** ** Qpower and inject_Z *)
Lemma Zpower_Qpower : forall (a n:Z), (0<=n)%Z -> inject_Z (a^n) == (inject_Z a)^n.
intros a [|n|n] H;[reflexivity| |elim H; reflexivity].
induction n using Pos.peano_ind.
replace (a^1)%Z with a by ring.
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.
Lemma Qsqr_nonneg : forall a, 0 <= a^2.
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.
(** ** Power of 2 positive upper bound *)
Lemma Qarchimedean_power2_pos : forall q : Q,
{p : positive | (q < Z.pos (2^p) # 1)%Q}.
destruct (Qarchimedean q) as [pexp Hpexp].
pose proof Pos.size_gt pexp as H1.
apply (Z.mul_lt_mono_pos_r (QDen q)) in H1; [|assumption].
apply (Z.lt_trans _ _ _ Hpexp H1).