mathlib documentation

algebra.order.field.power

Lemmas about powers in ordered fields. #

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Integer powers #

theorem zpow_le_of_le {α : Type u_1} [linear_ordered_semifield α] {a : α} {m n : } (ha : 1 a) (h : m n) :
a ^ m a ^ n
theorem zpow_le_one_of_nonpos {α : Type u_1} [linear_ordered_semifield α] {a : α} {n : } (ha : 1 a) (hn : n 0) :
a ^ n 1
theorem one_le_zpow_of_nonneg {α : Type u_1} [linear_ordered_semifield α] {a : α} {n : } (ha : 1 a) (hn : 0 n) :
1 a ^ n
@[protected]
theorem nat.zpow_pos_of_pos {α : Type u_1} [linear_ordered_semifield α] {a : } (h : 0 < a) (n : ) :
0 < a ^ n
theorem nat.zpow_ne_zero_of_pos {α : Type u_1} [linear_ordered_semifield α] {a : } (h : 0 < a) (n : ) :
a ^ n 0
theorem one_lt_zpow {α : Type u_1} [linear_ordered_semifield α] {a : α} (ha : 1 < a) (n : ) :
0 < n 1 < a ^ n
theorem zpow_strict_mono {α : Type u_1} [linear_ordered_semifield α] {a : α} (hx : 1 < a) :
theorem zpow_strict_anti {α : Type u_1} [linear_ordered_semifield α] {a : α} (h₀ : 0 < a) (h₁ : a < 1) :
@[simp]
theorem zpow_lt_iff_lt {α : Type u_1} [linear_ordered_semifield α] {a : α} {m n : } (hx : 1 < a) :
a ^ m < a ^ n m < n
@[simp]
theorem zpow_le_iff_le {α : Type u_1} [linear_ordered_semifield α] {a : α} {m n : } (hx : 1 < a) :
a ^ m a ^ n m n
@[simp]
theorem div_pow_le {α : Type u_1} [linear_ordered_semifield α] {a b : α} (ha : 0 a) (hb : 1 b) (k : ) :
a / b ^ k a
theorem zpow_injective {α : Type u_1} [linear_ordered_semifield α] {a : α} (h₀ : 0 < a) (h₁ : a 1) :
@[simp]
theorem zpow_inj {α : Type u_1} [linear_ordered_semifield α] {a : α} {m n : } (h₀ : 0 < a) (h₁ : a 1) :
a ^ m = a ^ n m = n
theorem zpow_le_max_of_min_le {α : Type u_1} [linear_ordered_semifield α] {x : α} (hx : 1 x) {a b c : } (h : linear_order.min a b c) :
x ^ -c linear_order.max (x ^ -a) (x ^ -b)
theorem zpow_le_max_iff_min_le {α : Type u_1} [linear_ordered_semifield α] {x : α} (hx : 1 < x) {a b c : } :

Lemmas about powers to numerals. #

theorem zpow_bit0_nonneg {α : Type u_1} [linear_ordered_field α] (a : α) (n : ) :
0 a ^ bit0 n
theorem zpow_two_nonneg {α : Type u_1} [linear_ordered_field α] (a : α) :
0 a ^ 2
theorem zpow_bit0_pos {α : Type u_1} [linear_ordered_field α] {a : α} (h : a 0) (n : ) :
0 < a ^ bit0 n
theorem zpow_two_pos_of_ne_zero {α : Type u_1} [linear_ordered_field α] {a : α} (h : a 0) :
0 < a ^ 2
@[simp]
theorem zpow_bit0_pos_iff {α : Type u_1} [linear_ordered_field α] {a : α} {n : } (hn : n 0) :
0 < a ^ bit0 n a 0
@[simp]
theorem zpow_bit1_neg_iff {α : Type u_1} [linear_ordered_field α] {a : α} {n : } :
a ^ bit1 n < 0 a < 0
@[simp]
theorem zpow_bit1_nonneg_iff {α : Type u_1} [linear_ordered_field α] {a : α} {n : } :
0 a ^ bit1 n 0 a
@[simp]
theorem zpow_bit1_nonpos_iff {α : Type u_1} [linear_ordered_field α] {a : α} {n : } :
a ^ bit1 n 0 a 0
@[simp]
theorem zpow_bit1_pos_iff {α : Type u_1} [linear_ordered_field α] {a : α} {n : } :
0 < a ^ bit1 n 0 < a
@[protected]
theorem even.zpow_nonneg {α : Type u_1} [linear_ordered_field α] {n : } (hn : even n) (a : α) :
0 a ^ n
theorem even.zpow_pos_iff {α : Type u_1} [linear_ordered_field α] {a : α} {n : } (hn : even n) (h : n 0) :
0 < a ^ n a 0
theorem odd.zpow_neg_iff {α : Type u_1} [linear_ordered_field α] {a : α} {n : } (hn : odd n) :
a ^ n < 0 a < 0
@[protected]
theorem odd.zpow_nonneg_iff {α : Type u_1} [linear_ordered_field α] {a : α} {n : } (hn : odd n) :
0 a ^ n 0 a
theorem odd.zpow_nonpos_iff {α : Type u_1} [linear_ordered_field α] {a : α} {n : } (hn : odd n) :
a ^ n 0 a 0
theorem odd.zpow_pos_iff {α : Type u_1} [linear_ordered_field α] {a : α} {n : } (hn : odd n) :
0 < a ^ n 0 < a
theorem even.zpow_pos {α : Type u_1} [linear_ordered_field α] {a : α} {n : } (hn : even n) (h : n 0) :
a 0 0 < a ^ n

Alias of the reverse direction of even.zpow_pos_iff.

theorem odd.zpow_neg {α : Type u_1} [linear_ordered_field α] {a : α} {n : } (hn : odd n) :
a < 0 a ^ n < 0

Alias of the reverse direction of odd.zpow_neg_iff.

theorem odd.zpow_nonpos {α : Type u_1} [linear_ordered_field α] {a : α} {n : } (hn : odd n) :
a 0 a ^ n 0

Alias of the reverse direction of odd.zpow_nonpos_iff.

theorem even.zpow_abs {α : Type u_1} [linear_ordered_field α] {p : } (hp : even p) (a : α) :
|a| ^ p = a ^ p
@[simp]
theorem zpow_bit0_abs {α : Type u_1} [linear_ordered_field α] (a : α) (p : ) :
|a| ^ bit0 p = a ^ bit0 p

Miscellaneous lemmmas #

theorem nat.cast_le_pow_sub_div_sub {α : Type u_1} [linear_ordered_field α] {a : α} (H : 1 < a) (n : ) :
n (a ^ n - 1) / (a - 1)

Bernoulli's inequality reformulated to estimate (n : α).

theorem nat.cast_le_pow_div_sub {α : Type u_1} [linear_ordered_field α] {a : α} (H : 1 < a) (n : ) :
n a ^ n / (a - 1)

For any a > 1 and a natural n we have n ≤ a ^ n / (a - 1). See also nat.cast_le_pow_sub_div_sub for a stronger inequality with a ^ n - 1 in the numerator.