ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- one_le_two'
lemma one_le_two' [LE α] [ZeroLEOneClass α] [CovariantClass α α (swap (·+·)) (·≤·)] :
    (1 : α) ≤ 2 
Arguments: {α : Type u_1} [AddMonoidWithOne α] [LE α] [ZeroLEOneClass α] [CovariantClass α α (Function.swap fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1]
Goal: 1 ≤ 2
-- zero_lt_two
@[simp] lemma zero_lt_two : (0 : α) < 2 
Arguments: {α : Type u_1} [AddMonoidWithOne α] [PartialOrder α] [ZeroLEOneClass α] [NeZero 1] [CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1]
Goal: 0 < 2
-- zero_lt_three
@[simp] lemma zero_lt_three : (0 : α) < 3 
Arguments: {α : Type u_1} [AddMonoidWithOne α] [PartialOrder α] [ZeroLEOneClass α] [NeZero 1] [CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1]
Goal: 0 < 3
-- zero_lt_four
@[simp] lemma zero_lt_four : (0 : α) < 4 
Arguments: {α : Type u_1} [AddMonoidWithOne α] [PartialOrder α] [ZeroLEOneClass α] [NeZero 1] [CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1]
Goal: 0 < 4
-- zero_lt_two'
lemma zero_lt_two' : (0 : α) < 2 
Arguments: (α : Type u_1) [AddMonoidWithOne α] [PartialOrder α] [ZeroLEOneClass α] [NeZero 1] [CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1]
Goal: 0 < 2
-- zero_lt_three'
lemma zero_lt_three' : (0 : α) < 3 
Arguments: (α : Type u_1) [AddMonoidWithOne α] [PartialOrder α] [ZeroLEOneClass α] [NeZero 1] [CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1]
Goal: 0 < 3
-- zero_lt_four'
lemma zero_lt_four' : (0 : α) < 4 
Arguments: (α : Type u_1) [AddMonoidWithOne α] [PartialOrder α] [ZeroLEOneClass α] [NeZero 1] [CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1]
Goal: 0 < 4
-- one_lt_two
lemma one_lt_two [CovariantClass α α (·+·) (·<·)] : (1 : α) < 2 
Arguments: {α : Type u_1} [AddMonoidWithOne α] [PartialOrder α] [ZeroLEOneClass α] [NeZero 1] [CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x < x_1]
Goal: 1 < 2
-- nsmul_le_nsmul_right
@[to_additive (attr := mono, gcongr) nsmul_le_nsmul_right]
theorem pow_le_pow_left' [CovariantClass M M (swap (· * ·)) (· ≤ ·)] {a b : M} (hab : a ≤ b) :
    ∀ i : ℕ, a ^ i ≤ b ^ i
  
Arguments: {M : Type u_3} [AddMonoid M] [Preorder M] [CovariantClass M M (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] [CovariantClass M M (Function.swap fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] {a b : M} (hab : a ≤ b) (i : ℕ)
Goal: i • a ≤ i • b
-- pow_le_pow_left'
@[to_additive (attr := mono, gcongr) nsmul_le_nsmul_right]
theorem pow_le_pow_left' [CovariantClass M M (swap (· * ·)) (· ≤ ·)] {a b : M} (hab : a ≤ b) :
    ∀ i : ℕ, a ^ i ≤ b ^ i
  
Arguments: {M : Type u_3} [Monoid M] [Preorder M] [CovariantClass M M (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1] [CovariantClass M M (Function.swap fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1] {a b : M} (hab : a ≤ b) (i : ℕ)
Goal: a ^ i ≤ b ^ i
-- one_le_pow_of_one_le'
@[to_additive nsmul_nonneg]
theorem one_le_pow_of_one_le' {a : M} (H : 1 ≤ a) : ∀ n : ℕ, 1 ≤ a ^ n
  
Arguments: {M : Type u_3} [Monoid M] [Preorder M] [CovariantClass M M (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1] {a : M} (H : 1 ≤ a) (n : ℕ)
Goal: 1 ≤ a ^ n
-- nsmul_nonneg
@[to_additive nsmul_nonneg]
theorem one_le_pow_of_one_le' {a : M} (H : 1 ≤ a) : ∀ n : ℕ, 1 ≤ a ^ n
  
Arguments: {M : Type u_3} [AddMonoid M] [Preorder M] [CovariantClass M M (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] {a : M} (H : 0 ≤ a) (n : ℕ)
Goal: 0 ≤ n • a
-- pow_le_one'
@[to_additive nsmul_nonpos]
theorem pow_le_one' {a : M} (H : a ≤ 1) (n : ℕ) : a ^ n ≤ 1 
Arguments: {M : Type u_3} [Monoid M] [Preorder M] [CovariantClass M M (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1] {a : M} (H : a ≤ 1) (n : ℕ)
Goal: a ^ n ≤ 1
-- nsmul_nonpos
@[to_additive nsmul_nonpos]
theorem pow_le_one' {a : M} (H : a ≤ 1) (n : ℕ) : a ^ n ≤ 1 
Arguments: {M : Type u_3} [AddMonoid M] [Preorder M] [CovariantClass M M (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] {a : M} (H : a ≤ 0) (n : ℕ)
Goal: n • a ≤ 0
-- pow_le_pow_right'
@[to_additive (attr := gcongr) nsmul_le_nsmul_left]
theorem pow_le_pow_right' {a : M} {n m : ℕ} (ha : 1 ≤ a) (h : n ≤ m) : a ^ n ≤ a ^ m 
Arguments: {M : Type u_3} [Monoid M] [Preorder M] [CovariantClass M M (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1] {a : M} {n m : ℕ} (ha : 1 ≤ a) (h : n ≤ m)
Goal: a ^ n ≤ a ^ m
-- nsmul_le_nsmul_left
@[to_additive (attr := gcongr) nsmul_le_nsmul_left]
theorem pow_le_pow_right' {a : M} {n m : ℕ} (ha : 1 ≤ a) (h : n ≤ m) : a ^ n ≤ a ^ m 
Arguments: {M : Type u_3} [AddMonoid M] [Preorder M] [CovariantClass M M (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] {a : M} {n m : ℕ} (ha : 0 ≤ a) (h : n ≤ m)
Goal: n • a ≤ m • a
-- pow_le_pow_right_of_le_one'
@[to_additive nsmul_le_nsmul_left_of_nonpos]
theorem pow_le_pow_right_of_le_one' {a : M} {n m : ℕ} (ha : a ≤ 1) (h : n ≤ m) : a ^ m ≤ a ^ n 
Arguments: {M : Type u_3} [Monoid M] [Preorder M] [CovariantClass M M (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1] {a : M} {n m : ℕ} (ha : a ≤ 1) (h : n ≤ m)
Goal: a ^ m ≤ a ^ n
-- nsmul_le_nsmul_left_of_nonpos
@[to_additive nsmul_le_nsmul_left_of_nonpos]
theorem pow_le_pow_right_of_le_one' {a : M} {n m : ℕ} (ha : a ≤ 1) (h : n ≤ m) : a ^ m ≤ a ^ n 
Arguments: {M : Type u_3} [AddMonoid M] [Preorder M] [CovariantClass M M (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] {a : M} {n m : ℕ} (ha : a ≤ 0) (h : n ≤ m)
Goal: m • a ≤ n • a
-- nsmul_pos
@[to_additive nsmul_pos]
theorem one_lt_pow' {a : M} (ha : 1 < a) {k : ℕ} (hk : k ≠ 0) : 1 < a ^ k 
Arguments: {M : Type u_3} [AddMonoid M] [Preorder M] [CovariantClass M M (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] {a : M} (ha : 0 < a) {k : ℕ} (hk : k ≠ 0)
Goal: 0 < k • a
-- one_lt_pow'
@[to_additive nsmul_pos]
theorem one_lt_pow' {a : M} (ha : 1 < a) {k : ℕ} (hk : k ≠ 0) : 1 < a ^ k 
Arguments: {M : Type u_3} [Monoid M] [Preorder M] [CovariantClass M M (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1] {a : M} (ha : 1 < a) {k : ℕ} (hk : k ≠ 0)
Goal: 1 < a ^ k
-- nsmul_neg
@[to_additive nsmul_neg]
theorem pow_lt_one' {a : M} (ha : a < 1) {k : ℕ} (hk : k ≠ 0) : a ^ k < 1 
Arguments: {M : Type u_3} [AddMonoid M] [Preorder M] [CovariantClass M M (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] {a : M} (ha : a < 0) {k : ℕ} (hk : k ≠ 0)
Goal: k • a < 0
-- pow_lt_one'
@[to_additive nsmul_neg]
theorem pow_lt_one' {a : M} (ha : a < 1) {k : ℕ} (hk : k ≠ 0) : a ^ k < 1 
Arguments: {M : Type u_3} [Monoid M] [Preorder M] [CovariantClass M M (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1] {a : M} (ha : a < 1) {k : ℕ} (hk : k ≠ 0)
Goal: a ^ k < 1
-- pow_lt_pow_right'
@[to_additive (attr := gcongr) nsmul_lt_nsmul_left]
theorem pow_lt_pow_right' [CovariantClass M M (· * ·) (· < ·)] {a : M} {n m : ℕ} (ha : 1 < a)
    (h : n < m) : a ^ n < a ^ m 
Arguments: {M : Type u_3} [Monoid M] [Preorder M] [CovariantClass M M (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1] [CovariantClass M M (fun x x_1 => x * x_1) fun x x_1 => x < x_1] {a : M} {n m : ℕ} (ha : 1 < a) (h : n < m)
Goal: a ^ n < a ^ m
-- nsmul_lt_nsmul_left
@[to_additive (attr := gcongr) nsmul_lt_nsmul_left]
theorem pow_lt_pow_right' [CovariantClass M M (· * ·) (· < ·)] {a : M} {n m : ℕ} (ha : 1 < a)
    (h : n < m) : a ^ n < a ^ m 
Arguments: {M : Type u_3} [AddMonoid M] [Preorder M] [CovariantClass M M (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] [CovariantClass M M (fun x x_1 => x + x_1) fun x x_1 => x < x_1] {a : M} {n m : ℕ} (ha : 0 < a) (h : n < m)
Goal: n • a < m • a
-- nsmul_left_strictMono
@[to_additive nsmul_left_strictMono]
theorem pow_right_strictMono' [CovariantClass M M (· * ·) (· < ·)] {a : M} (ha : 1 < a) :
    StrictMono ((a ^ ·) : ℕ → M) 
Arguments: {M : Type u_3} [AddMonoid M] [Preorder M] [CovariantClass M M (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] [CovariantClass M M (fun x x_1 => x + x_1) fun x x_1 => x < x_1] {a : M} (ha : 0 < a)
Goal: StrictMono fun x => x • a
-- pow_right_strictMono'
@[to_additive nsmul_left_strictMono]
theorem pow_right_strictMono' [CovariantClass M M (· * ·) (· < ·)] {a : M} (ha : 1 < a) :
    StrictMono ((a ^ ·) : ℕ → M) 
Arguments: {M : Type u_3} [Monoid M] [Preorder M] [CovariantClass M M (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1] [CovariantClass M M (fun x x_1 => x * x_1) fun x x_1 => x < x_1] {a : M} (ha : 1 < a)
Goal: StrictMono fun x => a ^ x
-- Left.pow_nonneg
@[to_additive Left.pow_nonneg]
theorem Left.one_le_pow_of_le (hx : 1 ≤ x) : ∀ {n : ℕ}, 1 ≤ x ^ n
  
Arguments: {M : Type u_3} [AddMonoid M] [Preorder M] [CovariantClass M M (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] {x : M} (hx : 0 ≤ x) {n : ℕ}
Goal: 0 ≤ n • x
-- Left.one_le_pow_of_le
@[to_additive Left.pow_nonneg]
theorem Left.one_le_pow_of_le (hx : 1 ≤ x) : ∀ {n : ℕ}, 1 ≤ x ^ n
  
Arguments: {M : Type u_3} [Monoid M] [Preorder M] [CovariantClass M M (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1] {x : M} (hx : 1 ≤ x) {n : ℕ}
Goal: 1 ≤ x ^ n
-- Left.pow_le_one_of_le
@[to_additive Left.pow_nonpos]
theorem Left.pow_le_one_of_le (hx : x ≤ 1) : ∀ {n : ℕ}, x ^ n ≤ 1
  
Arguments: {M : Type u_3} [Monoid M] [Preorder M] [CovariantClass M M (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1] {x : M} (hx : x ≤ 1) {n : ℕ}
Goal: x ^ n ≤ 1
-- Left.pow_nonpos
@[to_additive Left.pow_nonpos]
theorem Left.pow_le_one_of_le (hx : x ≤ 1) : ∀ {n : ℕ}, x ^ n ≤ 1
  
Arguments: {M : Type u_3} [AddMonoid M] [Preorder M] [CovariantClass M M (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] {x : M} (hx : x ≤ 0) {n : ℕ}
Goal: n • x ≤ 0
-- Right.one_le_pow_of_le
@[to_additive Right.pow_nonneg]
theorem Right.one_le_pow_of_le (hx : 1 ≤ x) : ∀ {n : ℕ}, 1 ≤ x ^ n
  
Arguments: {M : Type u_3} [Monoid M] [Preorder M] [CovariantClass M M (Function.swap fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1] {x : M} (hx : 1 ≤ x) {n : ℕ}
Goal: 1 ≤ x ^ n
-- Right.pow_nonneg
@[to_additive Right.pow_nonneg]
theorem Right.one_le_pow_of_le (hx : 1 ≤ x) : ∀ {n : ℕ}, 1 ≤ x ^ n
  
Arguments: {M : Type u_3} [AddMonoid M] [Preorder M] [CovariantClass M M (Function.swap fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] {x : M} (hx : 0 ≤ x) {n : ℕ}
Goal: 0 ≤ n • x
-- Right.pow_le_one_of_le
@[to_additive Right.pow_nonpos]
theorem Right.pow_le_one_of_le (hx : x ≤ 1) : ∀ {n : ℕ}, x ^ n ≤ 1
  
Arguments: {M : Type u_3} [Monoid M] [Preorder M] [CovariantClass M M (Function.swap fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1] {x : M} (hx : x ≤ 1) {n : ℕ}
Goal: x ^ n ≤ 1
-- Right.pow_nonpos
@[to_additive Right.pow_nonpos]
theorem Right.pow_le_one_of_le (hx : x ≤ 1) : ∀ {n : ℕ}, x ^ n ≤ 1
  
Arguments: {M : Type u_3} [AddMonoid M] [Preorder M] [CovariantClass M M (Function.swap fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] {x : M} (hx : x ≤ 0) {n : ℕ}
Goal: n • x ≤ 0
-- StrictMono.pow_const
@[to_additive StrictMono.const_nsmul]
theorem StrictMono.pow_const (hf : StrictMono f) : ∀ {n : ℕ}, n ≠ 0 → StrictMono (f · ^ n)
  
Arguments: {β : Type u_1} {M : Type u_3} [Monoid M] [Preorder M] [Preorder β] [CovariantClass M M (fun x x_1 => x * x_1) fun x x_1 => x < x_1] [CovariantClass M M (Function.swap fun x x_1 => x * x_1) fun x x_1 => x < x_1] {f : β → M} (hf : StrictMono f) {n : ℕ}
Goal: n ≠ 0 → StrictMono fun x => f x ^ n
-- StrictMono.const_nsmul
@[to_additive StrictMono.const_nsmul]
theorem StrictMono.pow_const (hf : StrictMono f) : ∀ {n : ℕ}, n ≠ 0 → StrictMono (f · ^ n)
  
Arguments: {β : Type u_1} {M : Type u_3} [AddMonoid M] [Preorder M] [Preorder β] [CovariantClass M M (fun x x_1 => x + x_1) fun x x_1 => x < x_1] [CovariantClass M M (Function.swap fun x x_1 => x + x_1) fun x x_1 => x < x_1] {f : β → M} (hf : StrictMono f) {n : ℕ}
Goal: n ≠ 0 → StrictMono fun x => n • f x
-- pow_left_strictMono
@[to_additive nsmul_right_strictMono]  -- Porting note: nolint to_additive_doc
theorem pow_left_strictMono (hn : n ≠ 0) : StrictMono (· ^ n : M → M) 
Arguments: {M : Type u_3} [Monoid M] [Preorder M] [CovariantClass M M (fun x x_1 => x * x_1) fun x x_1 => x < x_1] [CovariantClass M M (Function.swap fun x x_1 => x * x_1) fun x x_1 => x < x_1] {n : ℕ} (hn : n ≠ 0)
Goal: StrictMono fun x => x ^ n
-- nsmul_right_strictMono
@[to_additive nsmul_right_strictMono]  -- Porting note: nolint to_additive_doc
theorem pow_left_strictMono (hn : n ≠ 0) : StrictMono (· ^ n : M → M) 
Arguments: {M : Type u_3} [AddMonoid M] [Preorder M] [CovariantClass M M (fun x x_1 => x + x_1) fun x x_1 => x < x_1] [CovariantClass M M (Function.swap fun x x_1 => x + x_1) fun x x_1 => x < x_1] {n : ℕ} (hn : n ≠ 0)
Goal: StrictMono fun x => n • x
-- nsmul_lt_nsmul_right
@[to_additive (attr := mono, gcongr) nsmul_lt_nsmul_right]
lemma pow_lt_pow_left' (hn : n ≠ 0) {a b : M} (hab : a < b) : a ^ n < b ^ n 
Arguments: {M : Type u_3} [AddMonoid M] [Preorder M] [CovariantClass M M (fun x x_1 => x + x_1) fun x x_1 => x < x_1] [CovariantClass M M (Function.swap fun x x_1 => x + x_1) fun x x_1 => x < x_1] {n : ℕ} (hn : n ≠ 0) {a b : M} (hab : a < b)
Goal: n • a < n • b
-- pow_lt_pow_left'
@[to_additive (attr := mono, gcongr) nsmul_lt_nsmul_right]
lemma pow_lt_pow_left' (hn : n ≠ 0) {a b : M} (hab : a < b) : a ^ n < b ^ n 
Arguments: {M : Type u_3} [Monoid M] [Preorder M] [CovariantClass M M (fun x x_1 => x * x_1) fun x x_1 => x < x_1] [CovariantClass M M (Function.swap fun x x_1 => x * x_1) fun x x_1 => x < x_1] {n : ℕ} (hn : n ≠ 0) {a b : M} (hab : a < b)
Goal: a ^ n < b ^ n
-- Monotone.const_nsmul
@[to_additive Monotone.const_nsmul]
theorem Monotone.pow_const {f : β → M} (hf : Monotone f) : ∀ n : ℕ, Monotone fun a => f a ^ n
  
Arguments: {β : Type u_1} {M : Type u_3} [AddMonoid M] [Preorder M] [Preorder β] [CovariantClass M M (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] [CovariantClass M M (Function.swap fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] {f : β → M} (hf : Monotone f) (n : ℕ)
Goal: Monotone fun a => n • f a
-- Monotone.pow_const
@[to_additive Monotone.const_nsmul]
theorem Monotone.pow_const {f : β → M} (hf : Monotone f) : ∀ n : ℕ, Monotone fun a => f a ^ n
  
Arguments: {β : Type u_1} {M : Type u_3} [Monoid M] [Preorder M] [Preorder β] [CovariantClass M M (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1] [CovariantClass M M (Function.swap fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1] {f : β → M} (hf : Monotone f) (n : ℕ)
Goal: Monotone fun a => f a ^ n
-- nsmul_right_mono
@[to_additive nsmul_right_mono]
theorem pow_left_mono (n : ℕ) : Monotone fun a : M => a ^ n 
Arguments: {M : Type u_3} [AddMonoid M] [Preorder M] [CovariantClass M M (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] [CovariantClass M M (Function.swap fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] (n : ℕ)
Goal: Monotone fun a => n • a
-- pow_left_mono
@[to_additive nsmul_right_mono]
theorem pow_left_mono (n : ℕ) : Monotone fun a : M => a ^ n 
Arguments: {M : Type u_3} [Monoid M] [Preorder M] [CovariantClass M M (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1] [CovariantClass M M (Function.swap fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1] (n : ℕ)
Goal: Monotone fun a => a ^ n
-- Left.pow_neg
@[to_additive Left.pow_neg]
theorem Left.pow_lt_one_of_lt [CovariantClass M M (· * ·) (· < ·)] {n : ℕ} {x : M} (hn : 0 < n)
    (h : x < 1) : x ^ n < 1 
Arguments: {M : Type u_3} [AddMonoid M] [Preorder M] [CovariantClass M M (fun x x_1 => x + x_1) fun x x_1 => x < x_1] {n : ℕ} {x : M} (hn : 0 < n) (h : x < 0)
Goal: n • x < 0
-- Left.pow_lt_one_of_lt
@[to_additive Left.pow_neg]
theorem Left.pow_lt_one_of_lt [CovariantClass M M (· * ·) (· < ·)] {n : ℕ} {x : M} (hn : 0 < n)
    (h : x < 1) : x ^ n < 1 
Arguments: {M : Type u_3} [Monoid M] [Preorder M] [CovariantClass M M (fun x x_1 => x * x_1) fun x x_1 => x < x_1] {n : ℕ} {x : M} (hn : 0 < n) (h : x < 1)
Goal: x ^ n < 1
-- Right.pow_lt_one_of_lt
@[to_additive Right.pow_neg]
theorem Right.pow_lt_one_of_lt [CovariantClass M M (swap (· * ·)) (· < ·)] {n : ℕ} {x : M}
    (hn : 0 < n) (h : x < 1) : x ^ n < 1 
Arguments: {M : Type u_3} [Monoid M] [Preorder M] [CovariantClass M M (Function.swap fun x x_1 => x * x_1) fun x x_1 => x < x_1] {n : ℕ} {x : M} (hn : 0 < n) (h : x < 1)
Goal: x ^ n < 1
-- Right.pow_neg
@[to_additive Right.pow_neg]
theorem Right.pow_lt_one_of_lt [CovariantClass M M (swap (· * ·)) (· < ·)] {n : ℕ} {x : M}
    (hn : 0 < n) (h : x < 1) : x ^ n < 1 
Arguments: {M : Type u_3} [AddMonoid M] [Preorder M] [CovariantClass M M (Function.swap fun x x_1 => x + x_1) fun x x_1 => x < x_1] {n : ℕ} {x : M} (hn : 0 < n) (h : x < 0)
Goal: n • x < 0
-- nsmul_nonneg_iff
@[to_additive nsmul_nonneg_iff]
theorem one_le_pow_iff {x : M} {n : ℕ} (hn : n ≠ 0) : 1 ≤ x ^ n ↔ 1 ≤ x 
Arguments: {M : Type u_3} [AddMonoid M] [LinearOrder M] [CovariantClass M M (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] {x : M} {n : ℕ} (hn : n ≠ 0)
Goal: 0 ≤ n • x ↔ 0 ≤ x
-- one_le_pow_iff
@[to_additive nsmul_nonneg_iff]
theorem one_le_pow_iff {x : M} {n : ℕ} (hn : n ≠ 0) : 1 ≤ x ^ n ↔ 1 ≤ x 
Arguments: {M : Type u_3} [Monoid M] [LinearOrder M] [CovariantClass M M (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1] {x : M} {n : ℕ} (hn : n ≠ 0)
Goal: 1 ≤ x ^ n ↔ 1 ≤ x
-- nsmul_nonpos_iff
@[to_additive]
theorem pow_le_one_iff {x : M} {n : ℕ} (hn : n ≠ 0) : x ^ n ≤ 1 ↔ x ≤ 1 
Arguments: {M : Type u_3} [AddMonoid M] [LinearOrder M] [CovariantClass M M (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] {x : M} {n : ℕ} (hn : n ≠ 0)
Goal: n • x ≤ 0 ↔ x ≤ 0
-- pow_le_one_iff
@[to_additive]
theorem pow_le_one_iff {x : M} {n : ℕ} (hn : n ≠ 0) : x ^ n ≤ 1 ↔ x ≤ 1 
Arguments: {M : Type u_3} [Monoid M] [LinearOrder M] [CovariantClass M M (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1] {x : M} {n : ℕ} (hn : n ≠ 0)
Goal: x ^ n ≤ 1 ↔ x ≤ 1
-- one_lt_pow_iff
@[to_additive nsmul_pos_iff]
theorem one_lt_pow_iff {x : M} {n : ℕ} (hn : n ≠ 0) : 1 < x ^ n ↔ 1 < x 
Arguments: {M : Type u_3} [Monoid M] [LinearOrder M] [CovariantClass M M (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1] {x : M} {n : ℕ} (hn : n ≠ 0)
Goal: 1 < x ^ n ↔ 1 < x
-- nsmul_pos_iff
@[to_additive nsmul_pos_iff]
theorem one_lt_pow_iff {x : M} {n : ℕ} (hn : n ≠ 0) : 1 < x ^ n ↔ 1 < x 
Arguments: {M : Type u_3} [AddMonoid M] [LinearOrder M] [CovariantClass M M (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] {x : M} {n : ℕ} (hn : n ≠ 0)
Goal: 0 < n • x ↔ 0 < x
-- pow_lt_one_iff
@[to_additive]
theorem pow_lt_one_iff {x : M} {n : ℕ} (hn : n ≠ 0) : x ^ n < 1 ↔ x < 1 
Arguments: {M : Type u_3} [Monoid M] [LinearOrder M] [CovariantClass M M (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1] {x : M} {n : ℕ} (hn : n ≠ 0)
Goal: x ^ n < 1 ↔ x < 1
-- nsmul_neg_iff
@[to_additive]
theorem pow_lt_one_iff {x : M} {n : ℕ} (hn : n ≠ 0) : x ^ n < 1 ↔ x < 1 
Arguments: {M : Type u_3} [AddMonoid M] [LinearOrder M] [CovariantClass M M (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] {x : M} {n : ℕ} (hn : n ≠ 0)
Goal: n • x < 0 ↔ x < 0
-- pow_eq_one_iff
@[to_additive]
theorem pow_eq_one_iff {x : M} {n : ℕ} (hn : n ≠ 0) : x ^ n = 1 ↔ x = 1 
Arguments: {M : Type u_3} [Monoid M] [LinearOrder M] [CovariantClass M M (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1] {x : M} {n : ℕ} (hn : n ≠ 0)
Goal: x ^ n = 1 ↔ x = 1
-- nsmul_eq_zero_iff
@[to_additive]
theorem pow_eq_one_iff {x : M} {n : ℕ} (hn : n ≠ 0) : x ^ n = 1 ↔ x = 1 
Arguments: {M : Type u_3} [AddMonoid M] [LinearOrder M] [CovariantClass M M (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] {x : M} {n : ℕ} (hn : n ≠ 0)
Goal: n • x = 0 ↔ x = 0
-- nsmul_le_nsmul_iff_left
@[to_additive nsmul_le_nsmul_iff_left]
theorem pow_le_pow_iff_right' (ha : 1 < a) : a ^ m ≤ a ^ n ↔ m ≤ n 
Arguments: {M : Type u_3} [AddMonoid M] [LinearOrder M] [CovariantClass M M (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] [CovariantClass M M (fun x x_1 => x + x_1) fun x x_1 => x < x_1] {a : M} {m n : ℕ} (ha : 0 < a)
Goal: m • a ≤ n • a ↔ m ≤ n
-- pow_le_pow_iff_right'
@[to_additive nsmul_le_nsmul_iff_left]
theorem pow_le_pow_iff_right' (ha : 1 < a) : a ^ m ≤ a ^ n ↔ m ≤ n 
Arguments: {M : Type u_3} [Monoid M] [LinearOrder M] [CovariantClass M M (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1] [CovariantClass M M (fun x x_1 => x * x_1) fun x x_1 => x < x_1] {a : M} {m n : ℕ} (ha : 1 < a)
Goal: a ^ m ≤ a ^ n ↔ m ≤ n
-- nsmul_lt_nsmul_iff_left
@[to_additive nsmul_lt_nsmul_iff_left]
theorem pow_lt_pow_iff_right' (ha : 1 < a) : a ^ m < a ^ n ↔ m < n 
Arguments: {M : Type u_3} [AddMonoid M] [LinearOrder M] [CovariantClass M M (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] [CovariantClass M M (fun x x_1 => x + x_1) fun x x_1 => x < x_1] {a : M} {m n : ℕ} (ha : 0 < a)
Goal: m • a < n • a ↔ m < n
-- pow_lt_pow_iff_right'
@[to_additive nsmul_lt_nsmul_iff_left]
theorem pow_lt_pow_iff_right' (ha : 1 < a) : a ^ m < a ^ n ↔ m < n 
Arguments: {M : Type u_3} [Monoid M] [LinearOrder M] [CovariantClass M M (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1] [CovariantClass M M (fun x x_1 => x * x_1) fun x x_1 => x < x_1] {a : M} {m n : ℕ} (ha : 1 < a)
Goal: a ^ m < a ^ n ↔ m < n
-- lt_of_nsmul_lt_nsmul_right
@[to_additive lt_of_nsmul_lt_nsmul_right]
theorem lt_of_pow_lt_pow_left' {a b : M} (n : ℕ) : a ^ n < b ^ n → a < b 
Arguments: {M : Type u_3} [AddMonoid M] [LinearOrder M] [CovariantClass M M (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] [CovariantClass M M (Function.swap fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] {a b : M} (n : ℕ)
Goal: n • a < n • b → a < b
-- lt_of_pow_lt_pow_left'
@[to_additive lt_of_nsmul_lt_nsmul_right]
theorem lt_of_pow_lt_pow_left' {a b : M} (n : ℕ) : a ^ n < b ^ n → a < b 
Arguments: {M : Type u_3} [Monoid M] [LinearOrder M] [CovariantClass M M (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1] [CovariantClass M M (Function.swap fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1] {a b : M} (n : ℕ)
Goal: a ^ n < b ^ n → a < b
-- min_lt_of_add_lt_two_nsmul
@[to_additive min_lt_of_add_lt_two_nsmul]
theorem min_lt_of_mul_lt_sq {a b c : M} (h : a * b < c ^ 2) : min a b < c 
Arguments: {M : Type u_3} [AddMonoid M] [LinearOrder M] [CovariantClass M M (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] [CovariantClass M M (Function.swap fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] {a b c : M} (h : a + b < 2 • c)
Goal: min a b < c
-- min_lt_of_mul_lt_sq
@[to_additive min_lt_of_add_lt_two_nsmul]
theorem min_lt_of_mul_lt_sq {a b c : M} (h : a * b < c ^ 2) : min a b < c 
Arguments: {M : Type u_3} [Monoid M] [LinearOrder M] [CovariantClass M M (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1] [CovariantClass M M (Function.swap fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1] {a b c : M} (h : a * b < c ^ 2)
Goal: min a b < c
-- lt_max_of_sq_lt_mul
@[to_additive lt_max_of_two_nsmul_lt_add]
theorem lt_max_of_sq_lt_mul {a b c : M} (h : a ^ 2 < b * c) : a < max b c 
Arguments: {M : Type u_3} [Monoid M] [LinearOrder M] [CovariantClass M M (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1] [CovariantClass M M (Function.swap fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1] {a b c : M} (h : a ^ 2 < b * c)
Goal: a < max b c
-- lt_max_of_two_nsmul_lt_add
@[to_additive lt_max_of_two_nsmul_lt_add]
theorem lt_max_of_sq_lt_mul {a b c : M} (h : a ^ 2 < b * c) : a < max b c 
Arguments: {M : Type u_3} [AddMonoid M] [LinearOrder M] [CovariantClass M M (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] [CovariantClass M M (Function.swap fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] {a b c : M} (h : 2 • a < b + c)
Goal: a < max b c
-- le_of_nsmul_le_nsmul_right
@[to_additive le_of_nsmul_le_nsmul_right]
theorem le_of_pow_le_pow_left' {a b : M} {n : ℕ} (hn : n ≠ 0) : a ^ n ≤ b ^ n → a ≤ b 
Arguments: {M : Type u_3} [AddMonoid M] [LinearOrder M] [CovariantClass M M (fun x x_1 => x + x_1) fun x x_1 => x < x_1] [CovariantClass M M (Function.swap fun x x_1 => x + x_1) fun x x_1 => x < x_1] {a b : M} {n : ℕ} (hn : n ≠ 0)
Goal: n • a ≤ n • b → a ≤ b
-- le_of_pow_le_pow_left'
@[to_additive le_of_nsmul_le_nsmul_right]
theorem le_of_pow_le_pow_left' {a b : M} {n : ℕ} (hn : n ≠ 0) : a ^ n ≤ b ^ n → a ≤ b 
Arguments: {M : Type u_3} [Monoid M] [LinearOrder M] [CovariantClass M M (fun x x_1 => x * x_1) fun x x_1 => x < x_1] [CovariantClass M M (Function.swap fun x x_1 => x * x_1) fun x x_1 => x < x_1] {a b : M} {n : ℕ} (hn : n ≠ 0)
Goal: a ^ n ≤ b ^ n → a ≤ b
-- min_le_of_add_le_two_nsmul
@[to_additive min_le_of_add_le_two_nsmul]
theorem min_le_of_mul_le_sq {a b c : M} (h : a * b ≤ c ^ 2) : min a b ≤ c 
Arguments: {M : Type u_3} [AddMonoid M] [LinearOrder M] [CovariantClass M M (fun x x_1 => x + x_1) fun x x_1 => x < x_1] [CovariantClass M M (Function.swap fun x x_1 => x + x_1) fun x x_1 => x < x_1] {a b c : M} (h : a + b ≤ 2 • c)
Goal: min a b ≤ c
-- min_le_of_mul_le_sq
@[to_additive min_le_of_add_le_two_nsmul]
theorem min_le_of_mul_le_sq {a b c : M} (h : a * b ≤ c ^ 2) : min a b ≤ c 
Arguments: {M : Type u_3} [Monoid M] [LinearOrder M] [CovariantClass M M (fun x x_1 => x * x_1) fun x x_1 => x < x_1] [CovariantClass M M (Function.swap fun x x_1 => x * x_1) fun x x_1 => x < x_1] {a b c : M} (h : a * b ≤ c ^ 2)
Goal: min a b ≤ c