ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- le_or_lt_of_add_le_add
@[to_additive]
theorem le_or_lt_of_mul_le_mul [CovariantClass α α (· * ·) (· ≤ ·)]
    [CovariantClass α α (Function.swap (· * ·)) (· < ·)] {a₁ a₂ b₁ b₂ : α} :
    a₁ * b₁ ≤ a₂ * b₂ → a₁ ≤ a₂ ∨ b₁ < b₂ 
Arguments: {α : Type u_1} [LinearOrder α] [Add α] [CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] [CovariantClass α α (Function.swap fun x x_1 => x + x_1) fun x x_1 => x < x_1] {a₁ a₂ b₁ b₂ : α}
Goal: a₁ + b₁ ≤ a₂ + b₂ → a₁ ≤ a₂ ∨ b₁ < b₂
-- le_or_lt_of_mul_le_mul
@[to_additive]
theorem le_or_lt_of_mul_le_mul [CovariantClass α α (· * ·) (· ≤ ·)]
    [CovariantClass α α (Function.swap (· * ·)) (· < ·)] {a₁ a₂ b₁ b₂ : α} :
    a₁ * b₁ ≤ a₂ * b₂ → a₁ ≤ a₂ ∨ b₁ < b₂ 
Arguments: {α : Type u_1} [LinearOrder α] [Mul α] [CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1] [CovariantClass α α (Function.swap fun x x_1 => x * x_1) fun x x_1 => x < x_1] {a₁ a₂ b₁ b₂ : α}
Goal: a₁ * b₁ ≤ a₂ * b₂ → a₁ ≤ a₂ ∨ b₁ < b₂
-- lt_or_le_of_mul_le_mul
@[to_additive]
theorem lt_or_le_of_mul_le_mul [CovariantClass α α (· * ·) (· < ·)]
    [CovariantClass α α (Function.swap (· * ·)) (· ≤ ·)] {a₁ a₂ b₁ b₂ : α} :
    a₁ * b₁ ≤ a₂ * b₂ → a₁ < a₂ ∨ b₁ ≤ b₂ 
Arguments: {α : Type u_1} [LinearOrder α] [Mul α] [CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x < x_1] [CovariantClass α α (Function.swap fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1] {a₁ a₂ b₁ b₂ : α}
Goal: a₁ * b₁ ≤ a₂ * b₂ → a₁ < a₂ ∨ b₁ ≤ b₂
-- lt_or_le_of_add_le_add
@[to_additive]
theorem lt_or_le_of_mul_le_mul [CovariantClass α α (· * ·) (· < ·)]
    [CovariantClass α α (Function.swap (· * ·)) (· ≤ ·)] {a₁ a₂ b₁ b₂ : α} :
    a₁ * b₁ ≤ a₂ * b₂ → a₁ < a₂ ∨ b₁ ≤ b₂ 
Arguments: {α : Type u_1} [LinearOrder α] [Add α] [CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x < x_1] [CovariantClass α α (Function.swap fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] {a₁ a₂ b₁ b₂ : α}
Goal: a₁ + b₁ ≤ a₂ + b₂ → a₁ < a₂ ∨ b₁ ≤ b₂
-- le_or_le_of_mul_le_mul
@[to_additive]
theorem le_or_le_of_mul_le_mul [CovariantClass α α (· * ·) (· < ·)]
    [CovariantClass α α (Function.swap (· * ·)) (· < ·)] {a₁ a₂ b₁ b₂ : α} :
    a₁ * b₁ ≤ a₂ * b₂ → a₁ ≤ a₂ ∨ b₁ ≤ b₂ 
Arguments: {α : Type u_1} [LinearOrder α] [Mul α] [CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x < x_1] [CovariantClass α α (Function.swap fun x x_1 => x * x_1) fun x x_1 => x < x_1] {a₁ a₂ b₁ b₂ : α}
Goal: a₁ * b₁ ≤ a₂ * b₂ → a₁ ≤ a₂ ∨ b₁ ≤ b₂
-- le_or_le_of_add_le_add
@[to_additive]
theorem le_or_le_of_mul_le_mul [CovariantClass α α (· * ·) (· < ·)]
    [CovariantClass α α (Function.swap (· * ·)) (· < ·)] {a₁ a₂ b₁ b₂ : α} :
    a₁ * b₁ ≤ a₂ * b₂ → a₁ ≤ a₂ ∨ b₁ ≤ b₂ 
Arguments: {α : Type u_1} [LinearOrder α] [Add α] [CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x < x_1] [CovariantClass α α (Function.swap fun x x_1 => x + x_1) fun x x_1 => x < x_1] {a₁ a₂ b₁ b₂ : α}
Goal: a₁ + b₁ ≤ a₂ + b₂ → a₁ ≤ a₂ ∨ b₁ ≤ b₂
-- mul_lt_mul_iff_of_le_of_le
@[to_additive]
theorem mul_lt_mul_iff_of_le_of_le [CovariantClass α α (· * ·) (· ≤ ·)]
    [CovariantClass α α (Function.swap (· * ·)) (· ≤ ·)] [CovariantClass α α (· * ·) (· < ·)]
    [CovariantClass α α (Function.swap (· * ·)) (· < ·)] {a₁ a₂ b₁ b₂ : α} (ha : a₁ ≤ a₂)
    (hb : b₁ ≤ b₂) : a₁ * b₁ < a₂ * b₂ ↔ a₁ < a₂ ∨ b₁ < b₂ 
Arguments: {α : Type u_1} [LinearOrder α] [Mul α] [CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1] [CovariantClass α α (Function.swap fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1] [CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x < x_1] [CovariantClass α α (Function.swap fun x x_1 => x * x_1) fun x x_1 => x < x_1] {a₁ a₂ b₁ b₂ : α} (ha : a₁ ≤ a₂) (hb : b₁ ≤ b₂)
Goal: a₁ * b₁ < a₂ * b₂ ↔ a₁ < a₂ ∨ b₁ < b₂
-- add_lt_add_iff_of_le_of_le
@[to_additive]
theorem mul_lt_mul_iff_of_le_of_le [CovariantClass α α (· * ·) (· ≤ ·)]
    [CovariantClass α α (Function.swap (· * ·)) (· ≤ ·)] [CovariantClass α α (· * ·) (· < ·)]
    [CovariantClass α α (Function.swap (· * ·)) (· < ·)] {a₁ a₂ b₁ b₂ : α} (ha : a₁ ≤ a₂)
    (hb : b₁ ≤ b₂) : a₁ * b₁ < a₂ * b₂ ↔ a₁ < a₂ ∨ b₁ < b₂ 
Arguments: {α : Type u_1} [LinearOrder α] [Add α] [CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] [CovariantClass α α (Function.swap fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] [CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x < x_1] [CovariantClass α α (Function.swap fun x x_1 => x + x_1) fun x x_1 => x < x_1] {a₁ a₂ b₁ b₂ : α} (ha : a₁ ≤ a₂) (hb : b₁ ≤ b₂)
Goal: a₁ + b₁ < a₂ + b₂ ↔ a₁ < a₂ ∨ b₁ < b₂
-- min_le_mul_of_one_le_right
@[to_additive]
theorem min_le_mul_of_one_le_right [CovariantClass α α (· * ·) (· ≤ ·)] {a b : α} (hb : 1 ≤ b) :
    min a b ≤ a * b 
Arguments: {α : Type u_1} [LinearOrder α] [MulOneClass α] [CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1] {a b : α} (hb : 1 ≤ b)
Goal: min a b ≤ a * b
-- min_le_add_of_nonneg_right
@[to_additive]
theorem min_le_mul_of_one_le_right [CovariantClass α α (· * ·) (· ≤ ·)] {a b : α} (hb : 1 ≤ b) :
    min a b ≤ a * b 
Arguments: {α : Type u_1} [LinearOrder α] [AddZeroClass α] [CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] {a b : α} (hb : 0 ≤ b)
Goal: min a b ≤ a + b
-- min_le_mul_of_one_le_left
@[to_additive]
theorem min_le_mul_of_one_le_left [CovariantClass α α (Function.swap (· * ·)) (· ≤ ·)] {a b : α}
    (ha : 1 ≤ a) : min a b ≤ a * b 
Arguments: {α : Type u_1} [LinearOrder α] [MulOneClass α] [CovariantClass α α (Function.swap fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1] {a b : α} (ha : 1 ≤ a)
Goal: min a b ≤ a * b
-- min_le_add_of_nonneg_left
@[to_additive]
theorem min_le_mul_of_one_le_left [CovariantClass α α (Function.swap (· * ·)) (· ≤ ·)] {a b : α}
    (ha : 1 ≤ a) : min a b ≤ a * b 
Arguments: {α : Type u_1} [LinearOrder α] [AddZeroClass α] [CovariantClass α α (Function.swap fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] {a b : α} (ha : 0 ≤ a)
Goal: min a b ≤ a + b
-- max_le_mul_of_one_le
@[to_additive]
theorem max_le_mul_of_one_le [CovariantClass α α (· * ·) (· ≤ ·)]
    [CovariantClass α α (Function.swap (· * ·)) (· ≤ ·)] {a b : α} (ha : 1 ≤ a) (hb : 1 ≤ b) :
    max a b ≤ a * b 
Arguments: {α : Type u_1} [LinearOrder α] [MulOneClass α] [CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1] [CovariantClass α α (Function.swap fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1] {a b : α} (ha : 1 ≤ a) (hb : 1 ≤ b)
Goal: max a b ≤ a * b
-- max_le_add_of_nonneg
@[to_additive]
theorem max_le_mul_of_one_le [CovariantClass α α (· * ·) (· ≤ ·)]
    [CovariantClass α α (Function.swap (· * ·)) (· ≤ ·)] {a b : α} (ha : 1 ≤ a) (hb : 1 ≤ b) :
    max a b ≤ a * b 
Arguments: {α : Type u_1} [LinearOrder α] [AddZeroClass α] [CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] [CovariantClass α α (Function.swap fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] {a b : α} (ha : 0 ≤ a) (hb : 0 ≤ b)
Goal: max a b ≤ a + b
-- CovariantClass.elim
class CovariantClass : Prop where
    protected elim : Covariant M N μ r
Arguments: {M : Type u_1} {N : Type u_2} {μ : M → N → N} {r : N → N → Prop} [self : CovariantClass M N μ r]
Goal: Covariant M N μ r
-- ContravariantClass.elim
class ContravariantClass : Prop where
    protected elim : Contravariant M N μ r
Arguments: {M : Type u_1} {N : Type u_2} {μ : M → N → N} {r : N → N → Prop} [self : ContravariantClass M N μ r]
Goal: Contravariant M N μ r
-- rel_iff_cov
theorem rel_iff_cov [CovariantClass M N μ r] [ContravariantClass M N μ r] (m : M) {a b : N} :
    r (μ m a) (μ m b) ↔ r a b 
Arguments: (M : Type u_1) (N : Type u_2) (μ : M → N → N) (r : N → N → Prop) [CovariantClass M N μ r] [ContravariantClass M N μ r] (m : M) {a b : N}
Goal: r (μ m a) (μ m b) ↔ r a b
-- Covariant.flip
theorem Covariant.flip (h : Covariant M N μ r) : Covariant M N μ (flip r) 
Arguments: {M : Type u_1} {N : Type u_2} {μ : M → N → N} {r : N → N → Prop} (h : Covariant M N μ r)
Goal: Covariant M N μ (flip r)
-- Contravariant.flip
theorem Contravariant.flip (h : Contravariant M N μ r) : Contravariant M N μ (flip r) 
Arguments: {M : Type u_1} {N : Type u_2} {μ : M → N → N} {r : N → N → Prop} (h : Contravariant M N μ r)
Goal: Contravariant M N μ (flip r)
-- act_rel_act_of_rel
theorem act_rel_act_of_rel (m : M) {a b : N} (ab : r a b) : r (μ m a) (μ m b) 
Arguments: {M : Type u_1} {N : Type u_2} {μ : M → N → N} {r : N → N → Prop} [CovariantClass M N μ r] (m : M) {a b : N} (ab : r a b)
Goal: r (μ m a) (μ m b)
-- Group.covariant_iff_contravariant
@[to_additive]
theorem Group.covariant_iff_contravariant [Group N] :
    Covariant N N (· * ·) r ↔ Contravariant N N (· * ·) r 
Arguments: {N : Type u_2} {r : N → N → Prop} [Group N]
Goal: Covariant N N (fun x x_1 => x * x_1) r ↔ Contravariant N N (fun x x_1 => x * x_1) r
-- AddGroup.covariant_iff_contravariant
@[to_additive]
theorem Group.covariant_iff_contravariant [Group N] :
    Covariant N N (· * ·) r ↔ Contravariant N N (· * ·) r 
Arguments: {N : Type u_2} {r : N → N → Prop} [AddGroup N]
Goal: Covariant N N (fun x x_1 => x + x_1) r ↔ Contravariant N N (fun x x_1 => x + x_1) r
-- AddGroup.covariant_swap_iff_contravariant_swap
@[to_additive]
theorem Group.covariant_swap_iff_contravariant_swap [Group N] :
    Covariant N N (swap (· * ·)) r ↔ Contravariant N N (swap (· * ·)) r 
Arguments: {N : Type u_2} {r : N → N → Prop} [AddGroup N]
Goal: Covariant N N (Function.swap fun x x_1 => x + x_1) r ↔ Contravariant N N (Function.swap fun x x_1 => x + x_1) r
-- Group.covariant_swap_iff_contravariant_swap
@[to_additive]
theorem Group.covariant_swap_iff_contravariant_swap [Group N] :
    Covariant N N (swap (· * ·)) r ↔ Contravariant N N (swap (· * ·)) r 
Arguments: {N : Type u_2} {r : N → N → Prop} [Group N]
Goal: Covariant N N (Function.swap fun x x_1 => x * x_1) r ↔ Contravariant N N (Function.swap fun x x_1 => x * x_1) r
-- act_rel_of_rel_of_act_rel
theorem act_rel_of_rel_of_act_rel (ab : r a b) (rl : r (μ m b) c) : r (μ m a) c 
Arguments: {M : Type u_1} {N : Type u_2} {μ : M → N → N} {r : N → N → Prop} [CovariantClass M N μ r] [IsTrans N r] (m : M) {a b c : N} (ab : r a b) (rl : r (μ m b) c)
Goal: r (μ m a) c
-- rel_act_of_rel_of_rel_act
theorem rel_act_of_rel_of_rel_act (ab : r a b) (rr : r c (μ m a)) : r c (μ m b) 
Arguments: {M : Type u_1} {N : Type u_2} {μ : M → N → N} {r : N → N → Prop} [CovariantClass M N μ r] [IsTrans N r] (m : M) {a b c : N} (ab : r a b) (rr : r c (μ m a))
Goal: r c (μ m b)
-- act_rel_act_of_rel_of_rel
theorem act_rel_act_of_rel_of_rel (ab : r a b) (cd : r c d) : r (mu a c) (mu b d) 
Arguments: {N : Type u_2} {r : N → N → Prop} {mu : N → N → N} [IsTrans N r] [i : CovariantClass N N mu r] [i' : CovariantClass N N (Function.swap mu) r] {a b c d : N} (ab : r a b) (cd : r c d)
Goal: r (mu a c) (mu b d)
-- rel_of_act_rel_act
theorem rel_of_act_rel_act (m : M) {a b : N} (ab : r (μ m a) (μ m b)) : r a b 
Arguments: {M : Type u_1} {N : Type u_2} {μ : M → N → N} {r : N → N → Prop} [ContravariantClass M N μ r] (m : M) {a b : N} (ab : r (μ m a) (μ m b))
Goal: r a b
-- act_rel_of_act_rel_of_rel_act_rel
theorem act_rel_of_act_rel_of_rel_act_rel (ab : r (μ m a) b) (rl : r (μ m b) (μ m c)) :
    r (μ m a) c 
Arguments: {M : Type u_1} {N : Type u_2} {μ : M → N → N} {r : N → N → Prop} [ContravariantClass M N μ r] [IsTrans N r] (m : M) {a b c : N} (ab : r (μ m a) b) (rl : r (μ m b) (μ m c))
Goal: r (μ m a) c
-- rel_act_of_act_rel_act_of_rel_act
theorem rel_act_of_act_rel_act_of_rel_act (ab : r (μ m a) (μ m b)) (rr : r b (μ m c)) :
    r a (μ m c) 
Arguments: {M : Type u_1} {N : Type u_2} {μ : M → N → N} {r : N → N → Prop} [ContravariantClass M N μ r] [IsTrans N r] (m : M) {a b c : N} (ab : r (μ m a) (μ m b)) (rr : r b (μ m c))
Goal: r a (μ m c)
-- Covariant.monotone_of_const
theorem Covariant.monotone_of_const [CovariantClass M N μ (· ≤ ·)] (m : M) : Monotone (μ m) 
Arguments: {M : Type u_1} {N : Type u_2} {μ : M → N → N} [Preorder N] [CovariantClass M N μ fun x x_1 => x ≤ x_1] (m : M)
Goal: Monotone (μ m)
-- Monotone.covariant_of_const
theorem Monotone.covariant_of_const [CovariantClass M N μ (· ≤ ·)] (hf : Monotone f) (m : M) :
    Monotone (f <| μ m ·) 
Arguments: {M : Type u_1} {N : Type u_2} {μ : M → N → N} {α : Type u_3} [Preorder α] [Preorder N] {f : N → α} [CovariantClass M N μ fun x x_1 => x ≤ x_1] (hf : Monotone f) (m : M)
Goal: Monotone fun x => f (μ m x)
-- Monotone.covariant_of_const'
theorem Monotone.covariant_of_const' {μ : N → N → N} [CovariantClass N N (swap μ) (· ≤ ·)]
    (hf : Monotone f) (m : N) : Monotone (f <| μ · m) 
Arguments: {N : Type u_2} {α : Type u_3} [Preorder α] [Preorder N] {f : N → α} {μ : N → N → N} [CovariantClass N N (Function.swap μ) fun x x_1 => x ≤ x_1] (hf : Monotone f) (m : N)
Goal: Monotone fun x => f (μ x m)
-- Antitone.covariant_of_const
theorem Antitone.covariant_of_const [CovariantClass M N μ (· ≤ ·)] (hf : Antitone f) (m : M) :
    Antitone (f <| μ m ·) 
Arguments: {M : Type u_1} {N : Type u_2} {μ : M → N → N} {α : Type u_3} [Preorder α] [Preorder N] {f : N → α} [CovariantClass M N μ fun x x_1 => x ≤ x_1] (hf : Antitone f) (m : M)
Goal: Antitone fun x => f (μ m x)
-- Antitone.covariant_of_const'
theorem Antitone.covariant_of_const' {μ : N → N → N} [CovariantClass N N (swap μ) (· ≤ ·)]
    (hf : Antitone f) (m : N) : Antitone (f <| μ · m) 
Arguments: {N : Type u_2} {α : Type u_3} [Preorder α] [Preorder N] {f : N → α} {μ : N → N → N} [CovariantClass N N (Function.swap μ) fun x x_1 => x ≤ x_1] (hf : Antitone f) (m : N)
Goal: Antitone fun x => f (μ x m)
-- covariant_le_of_covariant_lt
theorem covariant_le_of_covariant_lt [PartialOrder N] :
    Covariant M N μ (· < ·) → Covariant M N μ (· ≤ ·) 
Arguments: (M : Type u_1) (N : Type u_2) (μ : M → N → N) [PartialOrder N]
Goal: (Covariant M N μ fun x x_1 => x < x_1) → Covariant M N μ fun x x_1 => x ≤ x_1
-- covariantClass_le_of_lt
theorem covariantClass_le_of_lt [PartialOrder N] [CovariantClass M N μ (· < ·)] :
    CovariantClass M N μ (· ≤ ·) 
Arguments: (M : Type u_1) (N : Type u_2) (μ : M → N → N) [PartialOrder N] [CovariantClass M N μ fun x x_1 => x < x_1]
Goal: CovariantClass M N μ fun x x_1 => x ≤ x_1
-- contravariant_le_iff_contravariant_lt_and_eq
theorem contravariant_le_iff_contravariant_lt_and_eq [PartialOrder N] :
    Contravariant M N μ (· ≤ ·) ↔ Contravariant M N μ (· < ·) ∧ Contravariant M N μ (· = ·) 
Arguments: (M : Type u_1) (N : Type u_2) (μ : M → N → N) [PartialOrder N]
Goal: (Contravariant M N μ fun x x_1 => x ≤ x_1) ↔
  (Contravariant M N μ fun x x_1 => x < x_1) ∧ Contravariant M N μ fun x x_1 => x = x_1
-- contravariant_lt_of_contravariant_le
theorem contravariant_lt_of_contravariant_le [PartialOrder N] :
    Contravariant M N μ (· ≤ ·) → Contravariant M N μ (· < ·) 
Arguments: (M : Type u_1) (N : Type u_2) (μ : M → N → N) [PartialOrder N]
Goal: (Contravariant M N μ fun x x_1 => x ≤ x_1) → Contravariant M N μ fun x x_1 => x < x_1
-- covariant_le_iff_contravariant_lt
theorem covariant_le_iff_contravariant_lt [LinearOrder N] :
    Covariant M N μ (· ≤ ·) ↔ Contravariant M N μ (· < ·) 
Arguments: (M : Type u_1) (N : Type u_2) (μ : M → N → N) [LinearOrder N]
Goal: (Covariant M N μ fun x x_1 => x ≤ x_1) ↔ Contravariant M N μ fun x x_1 => x < x_1
-- covariant_lt_iff_contravariant_le
theorem covariant_lt_iff_contravariant_le [LinearOrder N] :
    Covariant M N μ (· < ·) ↔ Contravariant M N μ (· ≤ ·) 
Arguments: (M : Type u_1) (N : Type u_2) (μ : M → N → N) [LinearOrder N]
Goal: (Covariant M N μ fun x x_1 => x < x_1) ↔ Contravariant M N μ fun x x_1 => x ≤ x_1
-- covariant_flip_iff
theorem covariant_flip_iff [IsSymmOp N N mu] :
    Covariant N N (flip mu) r ↔ Covariant N N mu r 
Arguments: (N : Type u_2) (r : N → N → Prop) (mu : N → N → N) [IsSymmOp N N mu]
Goal: Covariant N N (flip mu) r ↔ Covariant N N mu r
-- contravariant_flip_iff
theorem contravariant_flip_iff [IsSymmOp N N mu] :
    Contravariant N N (flip mu) r ↔ Contravariant N N mu r 
Arguments: (N : Type u_2) (r : N → N → Prop) (mu : N → N → N) [IsSymmOp N N mu]
Goal: Contravariant N N (flip mu) r ↔ Contravariant N N mu r
-- covariant_lt_of_covariant_le_of_contravariant_eq
theorem covariant_lt_of_covariant_le_of_contravariant_eq [ContravariantClass M N μ (· = ·)]
    [PartialOrder N] [CovariantClass M N μ (· ≤ ·)] : CovariantClass M N μ (· < ·) 
Arguments: (M : Type u_1) (N : Type u_2) (μ : M → N → N) [ContravariantClass M N μ fun x x_1 => x = x_1] [PartialOrder N] [CovariantClass M N μ fun x x_1 => x ≤ x_1]
Goal: CovariantClass M N μ fun x x_1 => x < x_1
-- contravariant_le_of_contravariant_eq_and_lt
theorem contravariant_le_of_contravariant_eq_and_lt [PartialOrder N]
    [ContravariantClass M N μ (· = ·)] [ContravariantClass M N μ (· < ·)] :
    ContravariantClass M N μ (· ≤ ·) 
Arguments: (M : Type u_1) (N : Type u_2) (μ : M → N → N) [PartialOrder N] [ContravariantClass M N μ fun x x_1 => x = x_1] [ContravariantClass M N μ fun x x_1 => x < x_1]
Goal: ContravariantClass M N μ fun x x_1 => x ≤ x_1
-- WithZero.toMulBot_zero
@[simp]
theorem toMulBot_zero : toMulBot (0 : WithZero (Multiplicative α)) = Multiplicative.ofAdd ⊥ 
Arguments: {α : Type u} [Add α]
Goal: WithZero.toMulBot 0 = Multiplicative.ofAdd ⊥
-- WithZero.toMulBot_coe
@[simp]
theorem toMulBot_coe (x : Multiplicative α) :
    toMulBot ↑x = Multiplicative.ofAdd (↑(Multiplicative.toAdd x) : WithBot α) 
Arguments: {α : Type u} [Add α] (x : Multiplicative α)
Goal: WithZero.toMulBot ↑x = Multiplicative.ofAdd ↑(Multiplicative.toAdd x)
-- WithZero.toMulBot_symm_bot
@[simp]
theorem toMulBot_symm_bot : toMulBot.symm (Multiplicative.ofAdd (⊥ : WithBot α)) = 0 
Arguments: {α : Type u} [Add α]
Goal: WithZero.toMulBot.symm (Multiplicative.ofAdd ⊥) = 0
-- WithZero.toMulBot_coe_ofAdd
@[simp]
theorem toMulBot_coe_ofAdd (x : α) :
    toMulBot.symm (Multiplicative.ofAdd (x : WithBot α)) = Multiplicative.ofAdd x 
Arguments: {α : Type u} [Add α] (x : α)
Goal: WithZero.toMulBot.symm (Multiplicative.ofAdd ↑x) = ↑(Multiplicative.ofAdd x)
-- WithZero.toMulBot_strictMono
theorem toMulBot_strictMono : StrictMono (@toMulBot α _) 
Arguments: {α : Type u} [Add α] [Preorder α]
Goal: StrictMono ⇑WithZero.toMulBot
-- WithZero.toMulBot_le
@[simp]
theorem toMulBot_le : toMulBot a ≤ toMulBot b ↔ a ≤ b 
Arguments: {α : Type u} [Add α] [Preorder α] (a b : WithZero (Multiplicative α))
Goal: WithZero.toMulBot a ≤ WithZero.toMulBot b ↔ a ≤ b
-- WithZero.toMulBot_lt
@[simp]
theorem toMulBot_lt : toMulBot a < toMulBot b ↔ a < b 
Arguments: {α : Type u} [Add α] [Preorder α] (a b : WithZero (Multiplicative α))
Goal: WithZero.toMulBot a < WithZero.toMulBot b ↔ a < b
-- OrderedAddCommMonoid.add_le_add_left
class OrderedAddCommMonoid (α : Type*) extends AddCommMonoid α, PartialOrder α where
  protected add_le_add_left : ∀ a b : α, a ≤ b → ∀ c, c + a ≤ c + b
Arguments: {α : Type u_3} [self : OrderedAddCommMonoid α] (a b : α)
Goal: a ≤ b → ∀ (c : α), c + a ≤ c + b
-- OrderedCommMonoid.mul_le_mul_left
@[to_additive]
class OrderedCommMonoid (α : Type*) extends CommMonoid α, PartialOrder α where
  protected mul_le_mul_left : ∀ a b : α, a ≤ b → ∀ c, c * a ≤ c * b
Arguments: {α : Type u_3} [self : OrderedCommMonoid α] (a b : α)
Goal: a ≤ b → ∀ (c : α), c * a ≤ c * b
-- OrderedCommMonoid.toCovariantClassRight
@[to_additive]
theorem OrderedCommMonoid.toCovariantClassRight (M : Type*) [OrderedCommMonoid M] :
    CovariantClass M M (swap (· * ·)) (· ≤ ·) 
Arguments: (M : Type u_3) [OrderedCommMonoid M]
Goal: CovariantClass M M (Function.swap fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1
-- OrderedAddCommMonoid.toCovariantClassRight
@[to_additive]
theorem OrderedCommMonoid.toCovariantClassRight (M : Type*) [OrderedCommMonoid M] :
    CovariantClass M M (swap (· * ·)) (· ≤ ·) 
Arguments: (M : Type u_3) [OrderedAddCommMonoid M]
Goal: CovariantClass M M (Function.swap fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1
-- OrderedCancelAddCommMonoid.le_of_add_le_add_left
class OrderedCancelAddCommMonoid (α : Type*) extends OrderedAddCommMonoid α where
  protected le_of_add_le_add_left : ∀ a b c : α, a + b ≤ a + c → b ≤ c
Arguments: {α : Type u_3} [self : OrderedCancelAddCommMonoid α] (a b c : α)
Goal: a + b ≤ a + c → b ≤ c
-- OrderedCancelCommMonoid.le_of_mul_le_mul_left
@[to_additive OrderedCancelAddCommMonoid]
class OrderedCancelCommMonoid (α : Type*) extends OrderedCommMonoid α where
  protected le_of_mul_le_mul_left : ∀ a b c : α, a * b ≤ a * c → b ≤ c
Arguments: {α : Type u_3} [self : OrderedCancelCommMonoid α] (a b c : α)
Goal: a * b ≤ a * c → b ≤ c
-- OrderedCancelAddCommMonoid.toContravariantClassRight
@[to_additive]
theorem OrderedCancelCommMonoid.toContravariantClassRight :
    ContravariantClass α α (swap (· * ·)) (· < ·) 
Arguments: {α : Type u_1} [OrderedCancelAddCommMonoid α]
Goal: ContravariantClass α α (Function.swap fun x x_1 => x + x_1) fun x x_1 => x < x_1
-- OrderedCancelCommMonoid.toContravariantClassRight
@[to_additive]
theorem OrderedCancelCommMonoid.toContravariantClassRight :
    ContravariantClass α α (swap (· * ·)) (· < ·) 
Arguments: {α : Type u_1} [OrderedCancelCommMonoid α]
Goal: ContravariantClass α α (Function.swap fun x x_1 => x * x_1) fun x x_1 => x < x_1
-- one_le_mul_self_iff
@[to_additive (attr := simp)]
theorem one_le_mul_self_iff : 1 ≤ a * a ↔ 1 ≤ a 
Arguments: {α : Type u_1} [LinearOrderedCommMonoid α] {a : α}
Goal: 1 ≤ a * a ↔ 1 ≤ a
-- nonneg_add_self_iff
@[to_additive (attr := simp)]
theorem one_le_mul_self_iff : 1 ≤ a * a ↔ 1 ≤ a 
Arguments: {α : Type u_1} [LinearOrderedAddCommMonoid α] {a : α}
Goal: 0 ≤ a + a ↔ 0 ≤ a
-- one_lt_mul_self_iff
@[to_additive (attr := simp)]
theorem one_lt_mul_self_iff : 1 < a * a ↔ 1 < a 
Arguments: {α : Type u_1} [LinearOrderedCommMonoid α] {a : α}
Goal: 1 < a * a ↔ 1 < a
-- pos_add_self_iff
@[to_additive (attr := simp)]
theorem one_lt_mul_self_iff : 1 < a * a ↔ 1 < a 
Arguments: {α : Type u_1} [LinearOrderedAddCommMonoid α] {a : α}
Goal: 0 < a + a ↔ 0 < a
-- add_self_nonpos_iff
@[to_additive (attr := simp)]
theorem mul_self_le_one_iff : a * a ≤ 1 ↔ a ≤ 1 
Arguments: {α : Type u_1} [LinearOrderedAddCommMonoid α] {a : α}
Goal: a + a ≤ 0 ↔ a ≤ 0
-- mul_self_le_one_iff
@[to_additive (attr := simp)]
theorem mul_self_le_one_iff : a * a ≤ 1 ↔ a ≤ 1 
Arguments: {α : Type u_1} [LinearOrderedCommMonoid α] {a : α}
Goal: a * a ≤ 1 ↔ a ≤ 1
-- add_self_neg_iff
@[to_additive (attr := simp)]
theorem mul_self_lt_one_iff : a * a < 1 ↔ a < 1 
Arguments: {α : Type u_1} [LinearOrderedAddCommMonoid α] {a : α}
Goal: a + a < 0 ↔ a < 0
-- mul_self_lt_one_iff
@[to_additive (attr := simp)]
theorem mul_self_lt_one_iff : a * a < 1 ↔ a < 1 
Arguments: {α : Type u_1} [LinearOrderedCommMonoid α] {a : α}
Goal: a * a < 1 ↔ a < 1
-- lt_add_one
lemma lt_add_one [One α] [AddZeroClass α] [PartialOrder α] [ZeroLEOneClass α]
    [NeZero (1 : α)] [CovariantClass α α (·+·) (·<·)] (a : α) : a < a + 1 
Arguments: {α : Type u_1} [One α] [AddZeroClass α] [PartialOrder α] [ZeroLEOneClass α] [NeZero 1] [CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x < x_1] (a : α)
Goal: a < a + 1
-- lt_one_add
lemma lt_one_add [One α] [AddZeroClass α] [PartialOrder α] [ZeroLEOneClass α]
    [NeZero (1 : α)] [CovariantClass α α (swap (·+·)) (·<·)] (a : α) : a < 1 + a 
Arguments: {α : Type u_1} [One α] [AddZeroClass α] [PartialOrder α] [ZeroLEOneClass α] [NeZero 1] [CovariantClass α α (Function.swap fun x x_1 => x + x_1) fun x x_1 => x < x_1] (a : α)
Goal: a < 1 + a
-- zero_le_two
lemma zero_le_two [Preorder α] [ZeroLEOneClass α] [CovariantClass α α (·+·) (·≤·)] :
    (0 : α) ≤ 2 
Arguments: {α : Type u_1} [AddMonoidWithOne α] [Preorder α] [ZeroLEOneClass α] [CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1]
Goal: 0 ≤ 2
-- zero_le_three
lemma zero_le_three [Preorder α] [ZeroLEOneClass α] [CovariantClass α α (·+·) (·≤·)] :
    (0 : α) ≤ 3 
Arguments: {α : Type u_1} [AddMonoidWithOne α] [Preorder α] [ZeroLEOneClass α] [CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1]
Goal: 0 ≤ 3
-- zero_le_four
lemma zero_le_four [Preorder α] [ZeroLEOneClass α] [CovariantClass α α (·+·) (·≤·)] :
    (0 : α) ≤ 4 
Arguments: {α : Type u_1} [AddMonoidWithOne α] [Preorder α] [ZeroLEOneClass α] [CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1]
Goal: 0 ≤ 4
-- one_le_two
lemma one_le_two [LE α] [ZeroLEOneClass α] [CovariantClass α α (·+·) (·≤·)] :
    (1 : α) ≤ 2 
Arguments: {α : Type u_1} [AddMonoidWithOne α] [LE α] [ZeroLEOneClass α] [CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1]
Goal: 1 ≤ 2