Documentation

Mathlib.Algebra.Order.Field.Power

Lemmas about powers in ordered fields. #

Integer powers #

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theorem zpow_le_of_le {α : Type u_1} [LinearOrderedSemifield α] {a : α} {m : ℤ} {n : ℤ} (ha : 1 ≤ a) (h : m ≤ n) :
a ^ m ≤ a ^ n
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theorem zpow_le_one_of_nonpos {α : Type u_1} [LinearOrderedSemifield α] {a : α} {n : ℤ} (ha : 1 ≤ a) (hn : n ≤ 0) :
a ^ n ≤ 1
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theorem one_le_zpow_of_nonneg {α : Type u_1} [LinearOrderedSemifield α] {a : α} {n : ℤ} (ha : 1 ≤ a) (hn : 0 ≤ n) :
1 ≤ a ^ n
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theorem Nat.zpow_pos_of_pos {α : Type u_1} [LinearOrderedSemifield α] {a : ℕ} (h : 0 < a) (n : ℤ) :
0 < ↑a ^ n
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theorem Nat.zpow_ne_zero_of_pos {α : Type u_1} [LinearOrderedSemifield α] {a : ℕ} (h : 0 < a) (n : ℤ) :
↑a ^ n ≠ 0
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theorem one_lt_zpow {α : Type u_1} [LinearOrderedSemifield α] {a : α} (ha : 1 < a) (n : ℤ) :
0 < n → 1 < a ^ n
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theorem zpow_strictMono {α : Type u_1} [LinearOrderedSemifield α] {a : α} (hx : 1 < a) :
StrictMono fun (x : ℤ) => a ^ x
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theorem zpow_strictAnti {α : Type u_1} [LinearOrderedSemifield α] {a : α} (h₀ : 0 < a) (h₁ : a < 1) :
StrictAnti fun (x : ℤ) => a ^ x
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@[simp]
theorem zpow_lt_iff_lt {α : Type u_1} [LinearOrderedSemifield α] {a : α} {m : ℤ} {n : ℤ} (hx : 1 < a) :
a ^ m < a ^ n ↔ m < n
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theorem GCongr.zpow_lt_of_lt {α : Type u_1} [LinearOrderedSemifield α] {a : α} {m : ℤ} {n : ℤ} (hx : 1 < a) :
m < n → a ^ m < a ^ n

Alias of the reverse direction of zpow_lt_iff_lt.

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@[deprecated GCongr.zpow_lt_of_lt]
theorem zpow_lt_of_lt {α : Type u_1} [LinearOrderedSemifield α] {a : α} {m : ℤ} {n : ℤ} (hx : 1 < a) :
m < n → a ^ m < a ^ n

Alias of the reverse direction of zpow_lt_iff_lt.

Alias of the reverse direction of zpow_lt_iff_lt.

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@[simp]
theorem zpow_le_iff_le {α : Type u_1} [LinearOrderedSemifield α] {a : α} {m : ℤ} {n : ℤ} (hx : 1 < a) :
a ^ m ≤ a ^ n ↔ m ≤ n
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@[simp]
theorem div_pow_le {α : Type u_1} [LinearOrderedSemifield α] {a : α} {b : α} (ha : 0 ≤ a) (hb : 1 ≤ b) (k : ℕ) :
a / b ^ k ≤ a
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theorem zpow_injective {α : Type u_1} [LinearOrderedSemifield α] {a : α} (h₀ : 0 < a) (h₁ : a ≠ 1) :
Function.Injective fun (x : ℤ) => a ^ x
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@[simp]
theorem zpow_inj {α : Type u_1} [LinearOrderedSemifield α] {a : α} {m : ℤ} {n : ℤ} (h₀ : 0 < a) (h₁ : a ≠ 1) :
a ^ m = a ^ n ↔ m = n
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theorem zpow_le_max_of_min_le {α : Type u_1} [LinearOrderedSemifield α] {x : α} (hx : 1 ≤ x) {a : ℤ} {b : ℤ} {c : ℤ} (h : min a b ≤ c) :
x ^ (-c) ≤ max (x ^ (-a)) (x ^ (-b))
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theorem zpow_le_max_iff_min_le {α : Type u_1} [LinearOrderedSemifield α] {x : α} (hx : 1 < x) {a : ℤ} {b : ℤ} {c : ℤ} :
x ^ (-c) ≤ max (x ^ (-a)) (x ^ (-b)) ↔ min a b ≤ c
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theorem Even.zpow_nonneg {α : Type u_1} [LinearOrderedField α] {n : ℤ} (hn : Even n) (a : α) :
0 ≤ a ^ n
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theorem zpow_two_nonneg {α : Type u_1} [LinearOrderedField α] (a : α) :
0 ≤ a ^ 2
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theorem zpow_neg_two_nonneg {α : Type u_1} [LinearOrderedField α] (a : α) :
0 ≤ a ^ (-2)
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theorem Even.zpow_pos {α : Type u_1} [LinearOrderedField α] {a : α} {n : ℤ} (hn : Even n) (ha : a ≠ 0) :
0 < a ^ n
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theorem zpow_two_pos_of_ne_zero {α : Type u_1} [LinearOrderedField α] {a : α} (ha : a ≠ 0) :
0 < a ^ 2
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theorem Even.zpow_pos_iff {α : Type u_1} [LinearOrderedField α] {a : α} {n : ℤ} (hn : Even n) (h : n ≠ 0) :
0 < a ^ n ↔ a ≠ 0
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theorem Odd.zpow_neg_iff {α : Type u_1} [LinearOrderedField α] {a : α} {n : ℤ} (hn : Odd n) :
a ^ n < 0 ↔ a < 0
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theorem Odd.zpow_nonneg_iff {α : Type u_1} [LinearOrderedField α] {a : α} {n : ℤ} (hn : Odd n) :
0 ≤ a ^ n ↔ 0 ≤ a
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theorem Odd.zpow_nonpos_iff {α : Type u_1} [LinearOrderedField α] {a : α} {n : ℤ} (hn : Odd n) :
a ^ n ≤ 0 ↔ a ≤ 0
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theorem Odd.zpow_pos_iff {α : Type u_1} [LinearOrderedField α] {a : α} {n : ℤ} (hn : Odd n) :
0 < a ^ n ↔ 0 < a
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theorem Odd.zpow_neg {α : Type u_1} [LinearOrderedField α] {a : α} {n : ℤ} (hn : Odd n) :
a < 0 → a ^ n < 0

Alias of the reverse direction of Odd.zpow_neg_iff.

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theorem Odd.zpow_nonpos {α : Type u_1} [LinearOrderedField α] {a : α} {n : ℤ} (hn : Odd n) :
a ≤ 0 → a ^ n ≤ 0

Alias of the reverse direction of Odd.zpow_nonpos_iff.

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

Bernoulli's inequality #

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theorem Nat.cast_le_pow_sub_div_sub {α : Type u_1} [LinearOrderedField α] {a : α} (H : 1 < a) (n : ℕ) :
↑n ≤ (a ^ n - 1) / (a - 1)

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

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theorem Nat.cast_le_pow_div_sub {α : Type u_1} [LinearOrderedField α] {a : α} (H : 1 < a) (n : 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.

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def Mathlib.Meta.Positivity.evalZPow :
Mathlib.Meta.Positivity.PositivityExt

The positivity extension which identifies expressions of the form a ^ (b : ℤ), such that positivity successfully recognises both a and b.

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