# Library Coq.Numbers.Natural.Abstract.NPow

Properties of the power function
Require Import Bool NAxioms NSub NParity NZPow.
Derived properties of power, specialized on natural numbers
Module Type NPowProp
(Import A : NAxiomsSig')
(Import B : NSubProp A)
(Import C : NParityProp A B).
Module Import Private_NZPow := Nop <+ NZPowProp A A B.
Ltac auto' := trivial; try rewrite <- neq_0_lt_0; auto using le_0_l.
Ltac wrap l := intros; apply l; auto'.
Lemma pow_succ_r' : forall a b, a ^( S b ) == a * a ^ b.
Power and basic constants
Lemma pow_0_l : forall a, a ~=0 -> 0^ a == 0.
Definition pow_1_r : forall a, a ^1 == a
:= pow_1_r.
Lemma pow_1_l : forall a, 1^ a == 1.
Definition pow_2_r : forall a, a ^2 == a * a
:= pow_2_r.
Power and addition, multiplication
Lemma pow_add_r : forall a b c, a ^( b + c ) == a ^ b * a ^ c.
Lemma pow_mul_l : forall a b c, ( a * b )^ c == a ^ c * b ^ c.
Lemma pow_mul_r : forall a b c, a ^( b * c ) == ( a ^ b )^ c.
Power and nullity
Lemma pow_eq_0 : forall a b, b ~=0 -> a ^ b == 0 -> a == 0.
Lemma pow_nonzero : forall a b, a ~=0 -> a ^ b ~= 0.
Lemma pow_eq_0_iff : forall a b, a ^ b == 0 <-> b ~=0 /\ a ==0.
Monotonicity
Lemma pow_lt_mono_l : forall a b c, c ~=0 -> a < b -> a ^ c < b ^ c.
Lemma pow_le_mono_l : forall a b c, a <= b -> a ^ c <= b ^ c.
Lemma pow_gt_1 : forall a b, 1< a -> b ~=0 -> 1< a ^ b.
Lemma pow_lt_mono_r : forall a b c, 1< a -> b < c -> a ^ b < a ^ c.
NB: since 0^0 > 0^1, the following result isn't valid with a=0
Lemma pow_le_mono_r : forall a b c, a ~=0 -> b <= c -> a ^ b <= a ^ c.
Lemma pow_le_mono : forall a b c d, a ~=0 -> a <= c -> b <= d ->
a ^ b <= c ^ d.
Definition pow_lt_mono : forall a b c d, 0< a < c -> 0< b < d ->
a ^ b < c ^ d
:= pow_lt_mono.
Injectivity
Lemma pow_inj_l : forall a b c, c ~=0 -> a ^ c == b ^ c -> a == b.
Lemma pow_inj_r : forall a b c, 1< a -> a ^ b == a ^ c -> b == c.
Monotonicity results, both ways
Lemma pow_lt_mono_l_iff : forall a b c, c ~=0 ->
(a < b <-> a ^ c < b ^ c).
Lemma pow_le_mono_l_iff : forall a b c, c ~=0 ->
(a <= b <-> a ^ c <= b ^ c).
Lemma pow_lt_mono_r_iff : forall a b c, 1< a ->
(b < c <-> a ^ b < a ^ c).
Lemma pow_le_mono_r_iff : forall a b c, 1< a ->
(b <= c <-> a ^ b <= a ^ c).
For any a>1, the a^x function is above the identity function
Lemma pow_gt_lin_r : forall a b, 1< a -> b < a ^ b.
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, c ~=0 ->
a ^ c + b ^ c <= ( a + b )^ c.
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, c ~=0 ->
( a + b )^ c <= 2^( pred c ) * ( a ^ c + b ^ c ).
Power and parity
Lemma even_pow : forall a b, b ~=0 -> even (a ^ b) = even a.
Lemma odd_pow : forall a b, b ~=0 -> odd (a ^ b) = odd a.
End NPowProp.