Documentation

Mathlib.Algebra.Order.Ring.Basic

Basic lemmas about ordered rings #

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theorem IsSquare.nonneg {R : Type u_3} [Semiring R] [LinearOrder R] [IsRightCancelAdd R] [ZeroLEOneClass R] [ExistsAddOfLE R] [PosMulMono R] [AddLeftStrictMono R] {x : R} (h : IsSquare x) :
0 ≤ x
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theorem MonoidHom.map_neg_one {M : Type u_2} {R : Type u_3} [Ring R] [Monoid M] [LinearOrder M] [MulLeftMono M] (f : R →* M) :
f (-1) = 1
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@[simp]
theorem MonoidHom.map_neg {M : Type u_2} {R : Type u_3} [Ring R] [Monoid M] [LinearOrder M] [MulLeftMono M] (f : R →* M) (x : R) :
f (-x) = f x
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theorem MonoidHom.map_sub_swap {M : Type u_2} {R : Type u_3} [Ring R] [Monoid M] [LinearOrder M] [MulLeftMono M] (f : R →* M) (x y : R) :
f (x - y) = f (y - x)
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theorem pow_add_pow_le {R : Type u_3} [OrderedSemiring R] {x y : R} {n : ℕ} (hx : 0 ≤ x) (hy : 0 ≤ y) (hn : n ≠ 0) :
x ^ n + y ^ n ≤ (x + y) ^ n
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@[deprecated mul_le_one₀ (since := "2024-09-28")]
theorem mul_le_one {M₀ : Type u_1} [MonoidWithZero M₀] [Preorder M₀] {a b : M₀} [ZeroLEOneClass M₀] [PosMulMono M₀] [MulPosMono M₀] (ha : a ≤ 1) (hb₀ : 0 ≤ b) (hb : b ≤ 1) :
a * b ≤ 1

Alias of mul_le_one₀.

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@[deprecated pow_le_one₀ (since := "2024-09-28")]
theorem pow_le_one {M₀ : Type u_1} [MonoidWithZero M₀] [Preorder M₀] {a : M₀} [ZeroLEOneClass M₀] [PosMulMono M₀] [MulPosMono M₀] {n : ℕ} :
0 ≤ a → a ≤ 1 → a ^ n ≤ 1

Alias of pow_le_one₀.

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@[deprecated pow_lt_one₀ (since := "2024-09-28")]
theorem pow_lt_one {M₀ : Type u_1} [MonoidWithZero M₀] [Preorder M₀] {a : M₀} [ZeroLEOneClass M₀] [PosMulMono M₀] [MulPosMono M₀] (h₀ : 0 ≤ a) (h₁ : a < 1) {n : ℕ} :
n ≠ 0 → a ^ n < 1

Alias of pow_lt_one₀.

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@[deprecated one_le_pow₀ (since := "2024-09-28")]
theorem one_le_pow_of_one_le {M₀ : Type u_1} [MonoidWithZero M₀] [Preorder M₀] {a : M₀} [ZeroLEOneClass M₀] [PosMulMono M₀] [MulPosMono M₀] (ha : 1 ≤ a) {n : ℕ} :
1 ≤ a ^ n

Alias of one_le_pow₀.

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@[deprecated one_lt_pow₀ (since := "2024-09-28")]
theorem one_lt_pow {M₀ : Type u_1} [MonoidWithZero M₀] [Preorder M₀] {a : M₀} [ZeroLEOneClass M₀] [PosMulMono M₀] [MulPosMono M₀] (ha : 1 < a) {n : ℕ} :
n ≠ 0 → 1 < a ^ n

Alias of one_lt_pow₀.

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@[deprecated pow_right_mono₀ (since := "2024-10-04")]
theorem pow_right_mono {M₀ : Type u_1} [MonoidWithZero M₀] [Preorder M₀] {a : M₀} [ZeroLEOneClass M₀] [PosMulMono M₀] [MulPosMono M₀] (h : 1 ≤ a) :
Monotone fun (x : ℕ) => a ^ x

Alias of pow_right_mono₀.

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@[deprecated pow_le_pow_right₀ (since := "2024-10-04")]
theorem pow_le_pow_right {M₀ : Type u_1} [MonoidWithZero M₀] [Preorder M₀] {a : M₀} {m n : ℕ} [ZeroLEOneClass M₀] [PosMulMono M₀] [MulPosMono M₀] (ha : 1 ≤ a) (hmn : m ≤ n) :
a ^ m ≤ a ^ n

Alias of pow_le_pow_right₀.

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@[deprecated pow_le_pow_left₀ (since := "2024-11-13")]
theorem pow_le_pow_left {R : Type u_3} [OrderedSemiring R] {a b : R} (ha : 0 ≤ a) (hab : a ≤ b) (n : ℕ) :
a ^ n ≤ b ^ n
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theorem pow_add_pow_le' {R : Type u_3} [OrderedSemiring R] {a b : R} {n : ℕ} (ha : 0 ≤ a) (hb : 0 ≤ b) :
a ^ n + b ^ n ≤ 2 * (a + b) ^ n
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@[reducible, inline]
abbrev OrderedRing.toStrictOrderedRing (α : Type u_4) [OrderedRing α] [NoZeroDivisors α] [Nontrivial α] :
StrictOrderedRing α

Turn an ordered domain into a strict ordered ring.

Equations
Instances For
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    @[deprecated pow_lt_pow_left₀ (since := "2024-11-13")]
    theorem pow_lt_pow_left {R : Type u_3} [StrictOrderedSemiring R] {x y : R} (h : x < y) (hx : 0 ≤ x) {n : ℕ} :
    n ≠ 0 → x ^ n < y ^ n
    source
    @[deprecated pow_left_strictMonoOn₀ (since := "2024-11-13")]
    theorem pow_left_strictMonoOn {R : Type u_3} [StrictOrderedSemiring R] {n : ℕ} (hn : n ≠ 0) :
    StrictMonoOn (fun (x : R) => x ^ n) {a : R | 0 ≤ a}
    source
    @[deprecated pow_right_strictMono₀ (since := "2024-11-13")]
    theorem pow_right_strictMono {R : Type u_3} [StrictOrderedSemiring R] {a : R} (h : 1 < a) :
    StrictMono fun (x : ℕ) => a ^ x
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    @[deprecated pow_lt_pow_right₀ (since := "2024-11-13")]
    theorem pow_lt_pow_right {R : Type u_3} [StrictOrderedSemiring R] {a : R} {n m : ℕ} (h : 1 < a) (hmn : m < n) :
    a ^ m < a ^ n
    source
    @[deprecated pow_lt_pow_iff_right₀ (since := "2024-11-13")]
    theorem pow_lt_pow_iff_right {R : Type u_3} [StrictOrderedSemiring R] {a : R} {n m : ℕ} (h : 1 < a) :
    a ^ n < a ^ m ↔ n < m
    source
    @[deprecated pow_le_pow_iff_right₀ (since := "2024-11-13")]
    theorem pow_le_pow_iff_right {R : Type u_3} [StrictOrderedSemiring R] {a : R} {n m : ℕ} (h : 1 < a) :
    a ^ n ≤ a ^ m ↔ n ≤ m
    source
    @[deprecated lt_self_pow₀ (since := "2024-11-13")]
    theorem lt_self_pow {R : Type u_3} [StrictOrderedSemiring R] {a : R} {m : ℕ} (h : 1 < a) (hm : 1 < m) :
    a < a ^ m
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    @[deprecated pow_right_strictAnti₀ (since := "2024-11-13")]
    theorem pow_right_strictAnti {R : Type u_3} [StrictOrderedSemiring R] {a : R} (h₀ : 0 < a) (h₁ : a < 1) :
    StrictAnti fun (x : ℕ) => a ^ x
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    @[deprecated pow_lt_pow_iff_right_of_lt_one₀ (since := "2024-11-13")]
    theorem pow_lt_pow_iff_right_of_lt_one {R : Type u_3} [StrictOrderedSemiring R] {a : R} {n m : ℕ} (h₀ : 0 < a) (h₁ : a < 1) :
    a ^ m < a ^ n ↔ n < m
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    @[deprecated pow_lt_pow_right_of_lt_one₀ (since := "2024-11-13")]
    theorem pow_lt_pow_right_of_lt_one {R : Type u_3} [StrictOrderedSemiring R] {a : R} {n m : ℕ} (h₀ : 0 < a) (h₁ : a < 1) (hmn : m < n) :
    a ^ n < a ^ m
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    @[deprecated pow_lt_self_of_lt_one₀ (since := "2024-11-13")]
    theorem pow_lt_self_of_lt_one {R : Type u_3} [StrictOrderedSemiring R] {a : R} {n : ℕ} (h₀ : 0 < a) (h₁ : a < 1) (hn : 1 < n) :
    a ^ n < a
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    theorem sq_pos_of_neg {R : Type u_3} [StrictOrderedRing R] {a : R} (ha : a < 0) :
    0 < a ^ 2
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    @[deprecated pow_le_pow_iff_left₀ (since := "2024-11-12")]
    theorem pow_le_pow_iff_left {R : Type u_3} [LinearOrderedSemiring R] {a b : R} {n : ℕ} (ha : 0 ≤ a) (hb : 0 ≤ b) (hn : n ≠ 0) :
    a ^ n ≤ b ^ n ↔ a ≤ b
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    @[deprecated pow_lt_pow_iff_left₀ (since := "2024-11-12")]
    theorem pow_lt_pow_iff_left {R : Type u_3} [LinearOrderedSemiring R] {a b : R} {n : ℕ} (ha : 0 ≤ a) (hb : 0 ≤ b) (hn : n ≠ 0) :
    a ^ n < b ^ n ↔ a < b
    source
    @[deprecated pow_left_inj₀ (since := "2024-11-12")]
    theorem pow_left_inj {R : Type u_3} [LinearOrderedSemiring R] {a b : R} {n : ℕ} (ha : 0 ≤ a) (hb : 0 ≤ b) (hn : n ≠ 0) :
    a ^ n = b ^ n ↔ a = b
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    @[deprecated pow_right_injective₀ (since := "2024-11-12")]
    theorem pow_right_injective {R : Type u_3} [LinearOrderedSemiring R] {a : R} (ha₀ : 0 < a) (ha₁ : a ≠ 1) :
    Function.Injective fun (x : ℕ) => a ^ x
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    @[deprecated pow_right_inj₀ (since := "2024-11-12")]
    theorem pow_right_inj {R : Type u_3} [LinearOrderedSemiring R] {a : R} {m n : ℕ} (ha₀ : 0 < a) (ha₁ : a ≠ 1) :
    a ^ m = a ^ n ↔ m = n
    source
    @[deprecated sq_le_one_iff₀ (since := "2024-11-12")]
    theorem sq_le_one_iff {R : Type u_3} [LinearOrderedSemiring R] {a : R} (ha : 0 ≤ a) :
    a ^ 2 ≤ 1 ↔ a ≤ 1
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    @[deprecated sq_lt_one_iff₀ (since := "2024-11-12")]
    theorem sq_lt_one_iff {R : Type u_3} [LinearOrderedSemiring R] {a : R} (ha : 0 ≤ a) :
    a ^ 2 < 1 ↔ a < 1
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    @[deprecated one_le_sq_iff₀ (since := "2024-11-12")]
    theorem one_le_sq_iff {R : Type u_3} [LinearOrderedSemiring R] {a : R} (ha : 0 ≤ a) :
    1 ≤ a ^ 2 ↔ 1 ≤ a
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    @[deprecated one_lt_sq_iff₀ (since := "2024-11-12")]
    theorem one_lt_sq_iff {R : Type u_3} [LinearOrderedSemiring R] {a : R} (ha : 0 ≤ a) :
    1 < a ^ 2 ↔ 1 < a
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    @[deprecated lt_of_pow_lt_pow_left₀ (since := "2024-11-12")]
    theorem lt_of_pow_lt_pow_left {R : Type u_3} [LinearOrderedSemiring R] {a b : R} (n : ℕ) (hb : 0 ≤ b) (h : a ^ n < b ^ n) :
    a < b
    source
    @[deprecated le_of_pow_le_pow_left₀ (since := "2024-11-12")]
    theorem le_of_pow_le_pow_left {R : Type u_3} [LinearOrderedSemiring R] {a b : R} {n : ℕ} (hn : n ≠ 0) (hb : 0 ≤ b) (h : a ^ n ≤ b ^ n) :
    a ≤ b
    source
    @[deprecated sq_eq_sq₀ (since := "2024-11-12")]
    theorem sq_eq_sq {R : Type u_3} [LinearOrderedSemiring R] {a b : R} (ha : 0 ≤ a) (hb : 0 ≤ b) :
    a ^ 2 = b ^ 2 ↔ a = b
    source
    @[deprecated lt_of_mul_self_lt_mul_self₀ (since := "2024-11-12")]
    theorem lt_of_mul_self_lt_mul_self {R : Type u_3} [LinearOrderedSemiring R] {a b : R} (hb : 0 ≤ b) :
    a * a < b * b → a < b

    Lemmas for canonically linear ordered semirings or linear ordered rings #

    The slightly unusual typeclass assumptions [LinearOrderedSemiring R] [ExistsAddOfLE R] cover two more familiar settings:

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    theorem add_sq_le {R : Type u_3} [LinearOrderedSemiring R] {a b : R} [ExistsAddOfLE R] :
    (a + b) ^ 2 ≤ 2 * (a ^ 2 + b ^ 2)
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    theorem add_pow_le {R : Type u_3} [LinearOrderedSemiring R] {a b : R} [ExistsAddOfLE R] (ha : 0 ≤ a) (hb : 0 ≤ b) (n : ℕ) :
    (a + b) ^ n ≤ 2 ^ (n - 1) * (a ^ n + b ^ n)
    source
    theorem Even.add_pow_le {R : Type u_3} [LinearOrderedSemiring R] {a b : R} {n : ℕ} [ExistsAddOfLE R] (hn : Even n) :
    (a + b) ^ n ≤ 2 ^ (n - 1) * (a ^ n + b ^ n)
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    theorem Even.pow_nonneg {R : Type u_3} [LinearOrderedSemiring R] {n : ℕ} [ExistsAddOfLE R] (hn : Even n) (a : R) :
    0 ≤ a ^ n
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    theorem Even.pow_pos {R : Type u_3} [LinearOrderedSemiring R] {a : R} {n : ℕ} [ExistsAddOfLE R] (hn : Even n) (ha : a ≠ 0) :
    0 < a ^ n
    source
    theorem Even.pow_pos_iff {R : Type u_3} [LinearOrderedSemiring R] {a : R} {n : ℕ} [ExistsAddOfLE R] (hn : Even n) (h₀ : n ≠ 0) :
    0 < a ^ n ↔ a ≠ 0
    source
    theorem Odd.pow_neg_iff {R : Type u_3} [LinearOrderedSemiring R] {a : R} {n : ℕ} [ExistsAddOfLE R] (hn : Odd n) :
    a ^ n < 0 ↔ a < 0
    source
    theorem Odd.pow_nonneg_iff {R : Type u_3} [LinearOrderedSemiring R] {a : R} {n : ℕ} [ExistsAddOfLE R] (hn : Odd n) :
    0 ≤ a ^ n ↔ 0 ≤ a
    source
    theorem Odd.pow_nonpos_iff {R : Type u_3} [LinearOrderedSemiring R] {a : R} {n : ℕ} [ExistsAddOfLE R] (hn : Odd n) :
    a ^ n ≤ 0 ↔ a ≤ 0
    source
    theorem Odd.pow_pos_iff {R : Type u_3} [LinearOrderedSemiring R] {a : R} {n : ℕ} [ExistsAddOfLE R] (hn : Odd n) :
    0 < a ^ n ↔ 0 < a
    source
    theorem Odd.pow_nonpos {R : Type u_3} [LinearOrderedSemiring R] {a : R} {n : ℕ} [ExistsAddOfLE R] (hn : Odd n) :
    a ≤ 0 → a ^ n ≤ 0

    Alias of the reverse direction of Odd.pow_nonpos_iff.

    source
    theorem Odd.pow_neg {R : Type u_3} [LinearOrderedSemiring R] {a : R} {n : ℕ} [ExistsAddOfLE R] (hn : Odd n) :
    a < 0 → a ^ n < 0

    Alias of the reverse direction of Odd.pow_neg_iff.

    source
    theorem Odd.strictMono_pow {R : Type u_3} [LinearOrderedSemiring R] {n : ℕ} [ExistsAddOfLE R] (hn : Odd n) :
    StrictMono fun (a : R) => a ^ n
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    theorem sq_pos_iff {R : Type u_3} [LinearOrderedSemiring R] [ExistsAddOfLE R] {a : R} :
    0 < a ^ 2 ↔ a ≠ 0
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    theorem sq_pos_of_ne_zero {R : Type u_3} [LinearOrderedSemiring R] [ExistsAddOfLE R] {a : R} :
    a ≠ 0 → 0 < a ^ 2

    Alias of the reverse direction of sq_pos_iff.

    source
    theorem pow_two_pos_of_ne_zero {R : Type u_3} [LinearOrderedSemiring R] [ExistsAddOfLE R] {a : R} :
    a ≠ 0 → 0 < a ^ 2

    Alias of the reverse direction of sq_pos_iff.

    Alias of the reverse direction of sq_pos_iff.

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    theorem pow_four_le_pow_two_of_pow_two_le {R : Type u_3} [LinearOrderedSemiring R] {a b : R} [ExistsAddOfLE R] (h : a ^ 2 ≤ b) :
    a ^ 4 ≤ b ^ 2