Library Coq.Numbers.NatInt.NZPow

Power Function

Interface of a power function, then its specification on naturals

Module Type Pow (Import A : Typ).
Parameters Inline pow : t -> t -> t.
End Pow.

Module Type PowNotation (A : Typ)(Import B : Pow A).
Infix "^" := pow.
End PowNotation.

Module Type Pow' (A : Typ) := Pow A <+ PowNotation A.

Module Type NZPowSpec (Import A : NZOrdAxiomsSig')(Import B : Pow' A).
Declare Instance pow_wd : Proper (eq ==>eq ==>eq) pow.
Axiom pow_0_r : forall a, a ^0 == 1.
Axiom pow_succ_r : forall a b, 0<=b -> a ^(succ b ) == a * a ^b.
Axiom pow_neg_r : forall a b, b <0 -> a ^b == 0.
End NZPowSpec.

The above pow_neg_r specification is useless (and trivially provable) for N. Having it here already allows deriving some slightly more general statements.

Module Type NZPow (A : NZOrdAxiomsSig) := Pow A <+ NZPowSpec A.
Module Type NZPow' (A : NZOrdAxiomsSig) := Pow' A <+ NZPowSpec A.

Derived properties of power

Module Type NZPowProp
(Import A : NZOrdAxiomsSig')
(Import B : NZPow' A)
(Import C : NZMulOrderProp A).

Hint Rewrite pow_0_r pow_succ_r : nz.

Power and basic constants

Lemma pow_0_l : forall a, 0<a -> 0^a == 0.
Proof.
intros a Ha.
destruct (lt_exists_pred _ _ Ha) as (a' & EQ & Ha').
rewrite EQ. now nzsimpl.
Qed.

Lemma pow_0_l' : forall a, a ~=0 -> 0^a == 0.
Proof.
intros a Ha.
destruct (lt_trichotomy a 0) as [LT|[EQ|GT]]; try order.
now rewrite pow_neg_r.
now apply pow_0_l.
Qed.

Lemma pow_1_r : forall a, a ^1 == a.
Proof.
intros. now nzsimpl'.
Qed.

Lemma pow_1_l : forall a, 0<=a -> 1^a == 1.
Proof.
apply le_ind; intros. solve_proper.
now nzsimpl.
now nzsimpl.
Qed.

Hint Rewrite pow_1_r pow_1_l : nz.

Lemma pow_2_r : forall a, a ^2 == a *a.
Proof.
intros. rewrite two_succ. nzsimpl; order'.
Qed.

Hint Rewrite pow_2_r : nz.

Power and nullity

Lemma pow_eq_0 : forall a b, 0<=b -> a ^b == 0 -> a == 0.
Proof.
intros a b Hb. apply le_ind with (4:=Hb).
solve_proper.
rewrite pow_0_r. order'.
clear b Hb. intros b Hb IH.
rewrite pow_succ_r by trivial.
intros H. apply eq_mul_0 in H. destruct H; trivial.
now apply IH.
Qed.

Lemma pow_nonzero : forall a b, a ~=0 -> 0<=b -> a ^b ~= 0.
Proof.
intros a b Ha Hb. contradict Ha. now apply pow_eq_0 with b.
Qed.

Lemma pow_eq_0_iff : forall a b, a ^b == 0 <-> b <0 \/ (0<b /\ a ==0).
Proof.
intros a b. split.
intros H.
destruct (lt_trichotomy b 0) as [Hb|[Hb|Hb]].
now left.
rewrite Hb, pow_0_r in H; order'.
right. split; trivial. apply pow_eq_0 with b; order.
intros [Hb|[Hb Ha]]. now rewrite pow_neg_r.
rewrite Ha. apply pow_0_l'. order.
Qed.

Power and addition, multiplication

Lemma pow_add_r : forall a b c, 0<=b -> 0<=c ->
  a ^(b +c ) == a ^b * a ^c.
Proof.
intros a b c Hb. apply le_ind with (4:=Hb). solve_proper.
now nzsimpl.
clear b Hb. intros b Hb IH Hc.
nzsimpl; trivial.
rewrite IH; trivial. apply mul_assoc.
now apply add_nonneg_nonneg.
Qed.

Lemma pow_mul_l : forall a b c,
  (a *b )^c == a ^c * b ^c.
Proof.
intros a b c.
destruct (lt_ge_cases c 0) as [Hc|Hc].
rewrite !(pow_neg_r _ _ Hc). now nzsimpl.
apply le_ind with (4:=Hc). solve_proper.
now nzsimpl.
clear c Hc. intros c Hc IH.
nzsimpl; trivial.
rewrite IH; trivial. apply mul_shuffle1.
Qed.

Lemma pow_mul_r : forall a b c, 0<=b -> 0<=c ->
  a ^(b *c ) == (a ^b )^c.
Proof.
intros a b c Hb. apply le_ind with (4:=Hb). solve_proper.
intros. now nzsimpl.
clear b Hb. intros b Hb IH Hc.
nzsimpl; trivial.
rewrite pow_add_r, IH, pow_mul_l; trivial. apply mul_comm.
now apply mul_nonneg_nonneg.
Qed.

Positivity

Lemma pow_nonneg : forall a b, 0<=a -> 0<=a ^b.
Proof.
intros a b Ha.
destruct (lt_ge_cases b 0) as [Hb|Hb].
now rewrite !(pow_neg_r _ _ Hb).
apply le_ind with (4:=Hb). solve_proper.
nzsimpl; order'.
clear b Hb. intros b Hb IH.
nzsimpl; trivial. now apply mul_nonneg_nonneg.
Qed.

Lemma pow_pos_nonneg : forall a b, 0<a -> 0<=b -> 0<a ^b.
Proof.
intros a b Ha Hb. apply le_ind with (4:=Hb). solve_proper.
nzsimpl; order'.
clear b Hb. intros b Hb IH.
nzsimpl; trivial. now apply mul_pos_pos.
Qed.

Monotonicity

Lemma pow_lt_mono_l : forall a b c, 0<c -> 0<=a <b -> a ^c < b ^c.
Proof.
intros a b c Hc. apply lt_ind with (4:=Hc). solve_proper.
intros (Ha,H). nzsimpl; trivial; order.
clear c Hc. intros c Hc IH (Ha,H).
nzsimpl; try order.
apply mul_lt_mono_nonneg; trivial.
apply pow_nonneg; try order.
apply IH. now split.
Qed.

Lemma pow_le_mono_l : forall a b c, 0<=a <=b -> a ^c <= b ^c.
Proof.
intros a b c (Ha,H).
destruct (lt_trichotomy c 0) as [Hc|[Hc|Hc]].
rewrite !(pow_neg_r _ _ Hc); now nzsimpl.
rewrite Hc; now nzsimpl.
apply lt_eq_cases in H. destruct H as [H|H]; [|now rewrite <- H].
apply lt_le_incl, pow_lt_mono_l; now try split.
Qed.

Lemma pow_gt_1 : forall a b, 1<a -> (0<b <-> 1<a ^b ).
Proof.
intros a b Ha. split; intros Hb.
rewrite <- (pow_1_l b) by order.
apply pow_lt_mono_l; try split; order'.
destruct (lt_trichotomy b 0) as [H|[H|H]]; trivial.
rewrite pow_neg_r in Hb; order'.
rewrite H, pow_0_r in Hb. order.
Qed.

Lemma pow_lt_mono_r : forall a b c, 1<a -> 0<=c -> b <c -> a ^b < a ^c.
Proof.
intros a b c Ha Hc H.
destruct (lt_ge_cases b 0) as [Hb|Hb].
rewrite pow_neg_r by trivial. apply pow_pos_nonneg; order'.
assert (H' : b<=c) by order.
destruct (le_exists_sub _ _ H') as (d & EQ & Hd).
rewrite EQ, pow_add_r; trivial. rewrite <- (mul_1_l (a^b)) at 1.
apply mul_lt_mono_pos_r.
apply pow_pos_nonneg; order'.
apply pow_gt_1; trivial.
apply lt_eq_cases in Hd; destruct Hd as [LT|EQ']; trivial.
rewrite <- EQ' in *. rewrite add_0_l in EQ. order.
Qed.

NB: since 0^0 > 0^1, the following result isn't valid with a=0

Lemma pow_le_mono_r : forall a b c, 0<a -> b <=c -> a ^b <= a ^c.
Proof.
intros a b c Ha H.
destruct (lt_ge_cases b 0) as [Hb|Hb].
rewrite (pow_neg_r _ _ Hb). apply pow_nonneg; order.
apply le_succ_l in Ha; rewrite <- one_succ in Ha.
apply lt_eq_cases in Ha; destruct Ha as [Ha|Ha]; [|rewrite <- Ha].
apply lt_eq_cases in H; destruct H as [H|H]; [|now rewrite <- H].
apply lt_le_incl, pow_lt_mono_r; order.
nzsimpl; order.
Qed.

Lemma pow_le_mono : forall a b c d, 0<a <=c -> b <=d ->
a ^b <= c ^d.
Proof.
intros. transitivity (a^d).
apply pow_le_mono_r; intuition order.
apply pow_le_mono_l; intuition order.
Qed.

Lemma pow_lt_mono : forall a b c d, 0<a <c -> 0<b <d ->
a ^b < c ^d.
Proof.
intros a b c d (Ha,Hac) (Hb,Hbd).
apply le_succ_l in Ha; rewrite <- one_succ in Ha.
apply lt_eq_cases in Ha; destruct Ha as [Ha|Ha]; [|rewrite <- Ha].
transitivity (a^d).
apply pow_lt_mono_r; intuition order.
apply pow_lt_mono_l; try split; order'.
nzsimpl; try order. apply pow_gt_1; order.
Qed.

Injectivity

Lemma pow_inj_l : forall a b c, 0<=a -> 0<=b -> 0<c ->
a ^c == b ^c -> a == b.
Proof.
intros a b c Ha Hb Hc EQ.
destruct (lt_trichotomy a b) as [LT|[EQ'|GT]]; trivial.
assert (a^c < b^c) by (apply pow_lt_mono_l; try split; trivial).
order.
assert (b^c < a^c) by (apply pow_lt_mono_l; try split; trivial).
order.
Qed.

Lemma pow_inj_r : forall a b c, 1<a -> 0<=b -> 0<=c ->
a ^b == a ^c -> b == c.
Proof.
intros a b c Ha Hb Hc EQ.
destruct (lt_trichotomy b c) as [LT|[EQ'|GT]]; trivial.
assert (a^b < a^c) by (apply pow_lt_mono_r; try split; trivial).
order.
assert (a^c < a^b) by (apply pow_lt_mono_r; try split; trivial).
order.
Qed.

Monotonicity results, both ways

Lemma pow_lt_mono_l_iff : forall a b c, 0<=a -> 0<=b -> 0<c ->
  (a <b <-> a ^c < b ^c ).
Proof.
intros a b c Ha Hb Hc.
split; intro LT.
apply pow_lt_mono_l; try split; trivial.
destruct (le_gt_cases b a) as [LE|GT]; trivial.
assert (b^c <= a^c) by (apply pow_le_mono_l; try split; order).
order.
Qed.

Lemma pow_le_mono_l_iff : forall a b c, 0<=a -> 0<=b -> 0<c ->
  (a <=b <-> a ^c <= b ^c ).
Proof.
intros a b c Ha Hb Hc.
split; intro LE.
apply pow_le_mono_l; try split; trivial.
destruct (le_gt_cases a b) as [LE'|GT]; trivial.
assert (b^c < a^c) by (apply pow_lt_mono_l; try split; trivial).
order.
Qed.

Lemma pow_lt_mono_r_iff : forall a b c, 1<a -> 0<=c ->
  (b <c <-> a ^b < a ^c ).
Proof.
intros a b c Ha Hc.
split; intro LT.
now apply pow_lt_mono_r.
destruct (le_gt_cases c b) as [LE|GT]; trivial.
assert (a^c <= a^b) by (apply pow_le_mono_r; order').
order.
Qed.

Lemma pow_le_mono_r_iff : forall a b c, 1<a -> 0<=c ->
  (b <=c <-> a ^b <= a ^c ).
Proof.
intros a b c Ha Hc.
split; intro LE.
apply pow_le_mono_r; order'.
destruct (le_gt_cases b c) as [LE'|GT]; trivial.
assert (a^c < a^b) by (apply pow_lt_mono_r; order').
order.
Qed.

For any a>1, the a^x function is above the identity function

Lemma pow_gt_lin_r : forall a b, 1<a -> 0<=b -> b < a ^b.
Proof.
intros a b Ha Hb. apply le_ind with (4:=Hb). solve_proper.
nzsimpl. order'.
clear b Hb. intros b Hb IH. nzsimpl; trivial.
rewrite <- !le_succ_l in *. rewrite <- two_succ in Ha.
transitivity (2*(S b)).
nzsimpl'. rewrite <- 2 succ_le_mono.
rewrite <- (add_0_l b) at 1. apply add_le_mono; order.
apply mul_le_mono_nonneg; trivial.
order'.
now apply lt_le_incl, lt_succ_r.
Qed.

Someday, we should say something about the full Newton formula. In the meantime, we can at least provide some inequalities about (a+b)^c.

Lemma pow_add_lower : forall a b c, 0<=a -> 0<=b -> 0<c ->
  a ^c + b ^c <= (a +b )^c.
Proof.
intros a b c Ha Hb Hc. apply lt_ind with (4:=Hc). solve_proper.
nzsimpl; order.
clear c Hc. intros c Hc IH.
assert (0<=c) by order'.
nzsimpl; trivial.
transitivity ((a+b)*(a^c + b^c)).
rewrite mul_add_distr_r, !mul_add_distr_l.
apply add_le_mono.
rewrite <- add_0_r at 1. apply add_le_mono_l.
  apply mul_nonneg_nonneg; trivial.
  apply pow_nonneg; trivial.
rewrite <- add_0_l at 1. apply add_le_mono_r.
  apply mul_nonneg_nonneg; trivial.
  apply pow_nonneg; trivial.
apply mul_le_mono_nonneg_l; trivial.
now apply add_nonneg_nonneg.
Qed.

This upper bound can also be seen as a convexity proof for x^c : image of (a+b)/2 is below the middle of the images of a and b

Lemma pow_add_upper : forall a b c, 0<=a -> 0<=b -> 0<c ->
  (a +b )^c <= 2^(pred c ) * (a ^c + b ^c ).
Proof.
assert (aux : forall a b c, 0<=a <=b -> 0<c ->
         (a + b ) * (a ^ c + b ^ c ) <= 2 * (a * a ^ c + b * b ^ c )).
  intros a b c (Ha,H) Hc.
  rewrite !mul_add_distr_l, !mul_add_distr_r. nzsimpl'.
  rewrite <- !add_assoc. apply add_le_mono_l.
  rewrite !add_assoc. apply add_le_mono_r.
  destruct (le_exists_sub _ _ H) as (d & EQ & Hd).
  rewrite EQ.
  rewrite 2 mul_add_distr_r.
  rewrite !add_assoc. apply add_le_mono_r.
  rewrite add_comm. apply add_le_mono_l.
  apply mul_le_mono_nonneg_l; trivial.
  apply pow_le_mono_l; try split; order.
intros a b c Ha Hb Hc. apply lt_ind with (4:=Hc). solve_proper.
nzsimpl; order.
clear c Hc. intros c Hc IH.
assert (0<=c) by order.
nzsimpl; trivial.
transitivity ((a+b)*(2^(pred c) * (a^c + b^c))).
apply mul_le_mono_nonneg_l; trivial.
now apply add_nonneg_nonneg.
rewrite mul_assoc. rewrite (mul_comm (a+b)).
assert (EQ : S (P c) == c) by (apply lt_succ_pred with 0; order').
assert (LE : 0 <= P c) by (now rewrite succ_le_mono, EQ, le_succ_l).
assert (EQ' : 2^c == 2^(P c) * 2) by (rewrite <- EQ at 1; nzsimpl'; order).
rewrite EQ', <- !mul_assoc.
apply mul_le_mono_nonneg_l.
apply pow_nonneg; order'.
destruct (le_gt_cases a b).
apply aux; try split; order'.
rewrite (add_comm a), (add_comm (a^c)), (add_comm (a*a^c)).
apply aux; try split; order'.
Qed.

End NZPowProp.