ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- StrictMonoOn.add
@[to_additive add "The sum of two strictly monotone functions is strictly monotone."]
theorem StrictMonoOn.mul' [CovariantClass α α (· * ·) (· < ·)]
    [CovariantClass α α (swap (· * ·)) (· < ·)] (hf : StrictMonoOn f s) (hg : StrictMonoOn g s) :
    StrictMonoOn (fun x => f x * g x) s 
Arguments: {α : Type u_1} {β : Type u_2} [Add α] [Preorder α] [Preorder β] {f g : β → α} {s : Set β} [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] (hf : StrictMonoOn f s) (hg : StrictMonoOn g s)
Goal: StrictMonoOn (fun x => f x + g x) s
-- StrictAnti.add
@[to_additive add "The sum of two strictly antitone functions is strictly antitone."]
theorem StrictAnti.mul' [CovariantClass α α (· * ·) (· < ·)]
    [CovariantClass α α (swap (· * ·)) (· < ·)] (hf : StrictAnti f) (hg : StrictAnti g) :
    StrictAnti fun x => f x * g x 
Arguments: {α : Type u_1} {β : Type u_2} [Add α] [Preorder α] [Preorder β] {f g : β → 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] (hf : StrictAnti f) (hg : StrictAnti g)
Goal: StrictAnti fun x => f x + g x
-- StrictAnti.mul'
@[to_additive add "The sum of two strictly antitone functions is strictly antitone."]
theorem StrictAnti.mul' [CovariantClass α α (· * ·) (· < ·)]
    [CovariantClass α α (swap (· * ·)) (· < ·)] (hf : StrictAnti f) (hg : StrictAnti g) :
    StrictAnti fun x => f x * g x 
Arguments: {α : Type u_1} {β : Type u_2} [Mul α] [Preorder α] [Preorder β] {f g : β → 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] (hf : StrictAnti f) (hg : StrictAnti g)
Goal: StrictAnti fun x => f x * g x
-- StrictAntiOn.mul'
@[to_additive add "The sum of two strictly antitone functions is strictly antitone."]
theorem StrictAntiOn.mul' [CovariantClass α α (· * ·) (· < ·)]
    [CovariantClass α α (swap (· * ·)) (· < ·)] (hf : StrictAntiOn f s) (hg : StrictAntiOn g s) :
    StrictAntiOn (fun x => f x * g x) s 
Arguments: {α : Type u_1} {β : Type u_2} [Mul α] [Preorder α] [Preorder β] {f g : β → α} {s : Set β} [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] (hf : StrictAntiOn f s) (hg : StrictAntiOn g s)
Goal: StrictAntiOn (fun x => f x * g x) s
-- StrictAntiOn.add
@[to_additive add "The sum of two strictly antitone functions is strictly antitone."]
theorem StrictAntiOn.mul' [CovariantClass α α (· * ·) (· < ·)]
    [CovariantClass α α (swap (· * ·)) (· < ·)] (hf : StrictAntiOn f s) (hg : StrictAntiOn g s) :
    StrictAntiOn (fun x => f x * g x) s 
Arguments: {α : Type u_1} {β : Type u_2} [Add α] [Preorder α] [Preorder β] {f g : β → α} {s : Set β} [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] (hf : StrictAntiOn f s) (hg : StrictAntiOn g s)
Goal: StrictAntiOn (fun x => f x + g x) s
-- Monotone.add_strictMono
@[to_additive add_strictMono "The sum of a monotone function and a strictly monotone function is
strictly monotone."]
theorem Monotone.mul_strictMono' [CovariantClass α α (· * ·) (· < ·)]
    [CovariantClass α α (swap (· * ·)) (· ≤ ·)] {f g : β → α} (hf : Monotone f)
    (hg : StrictMono g) :
    StrictMono fun x => f x * g x 
Arguments: {α : Type u_1} {β : Type u_2} [Add α] [Preorder α] [Preorder β] [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] {f g : β → α} (hf : Monotone f) (hg : StrictMono g)
Goal: StrictMono fun x => f x + g x
-- Monotone.mul_strictMono'
@[to_additive add_strictMono "The sum of a monotone function and a strictly monotone function is
strictly monotone."]
theorem Monotone.mul_strictMono' [CovariantClass α α (· * ·) (· < ·)]
    [CovariantClass α α (swap (· * ·)) (· ≤ ·)] {f g : β → α} (hf : Monotone f)
    (hg : StrictMono g) :
    StrictMono fun x => f x * g x 
Arguments: {α : Type u_1} {β : Type u_2} [Mul α] [Preorder α] [Preorder β] [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] {f g : β → α} (hf : Monotone f) (hg : StrictMono g)
Goal: StrictMono fun x => f x * g x
-- MonotoneOn.add_strictMono
@[to_additive add_strictMono "The sum of a monotone function and a strictly monotone function is
strictly monotone."]
theorem MonotoneOn.mul_strictMono' [CovariantClass α α (· * ·) (· < ·)]
    [CovariantClass α α (swap (· * ·)) (· ≤ ·)] {f g : β → α} (hf : MonotoneOn f s)
    (hg : StrictMonoOn g s) : StrictMonoOn (fun x => f x * g x) s 
Arguments: {α : Type u_1} {β : Type u_2} [Add α] [Preorder α] [Preorder β] {s : Set β} [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] {f g : β → α} (hf : MonotoneOn f s) (hg : StrictMonoOn g s)
Goal: StrictMonoOn (fun x => f x + g x) s
-- MonotoneOn.mul_strictMono'
@[to_additive add_strictMono "The sum of a monotone function and a strictly monotone function is
strictly monotone."]
theorem MonotoneOn.mul_strictMono' [CovariantClass α α (· * ·) (· < ·)]
    [CovariantClass α α (swap (· * ·)) (· ≤ ·)] {f g : β → α} (hf : MonotoneOn f s)
    (hg : StrictMonoOn g s) : StrictMonoOn (fun x => f x * g x) s 
Arguments: {α : Type u_1} {β : Type u_2} [Mul α] [Preorder α] [Preorder β] {s : Set β} [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] {f g : β → α} (hf : MonotoneOn f s) (hg : StrictMonoOn g s)
Goal: StrictMonoOn (fun x => f x * g x) s
-- Antitone.mul_strictAnti'
@[to_additive add_strictAnti "The sum of an antitone function and a strictly antitone function is
strictly antitone."]
theorem Antitone.mul_strictAnti' [CovariantClass α α (· * ·) (· < ·)]
    [CovariantClass α α (swap (· * ·)) (· ≤ ·)] {f g : β → α} (hf : Antitone f)
    (hg : StrictAnti g) :
    StrictAnti fun x => f x * g x 
Arguments: {α : Type u_1} {β : Type u_2} [Mul α] [Preorder α] [Preorder β] [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] {f g : β → α} (hf : Antitone f) (hg : StrictAnti g)
Goal: StrictAnti fun x => f x * g x
-- Antitone.add_strictAnti
@[to_additive add_strictAnti "The sum of an antitone function and a strictly antitone function is
strictly antitone."]
theorem Antitone.mul_strictAnti' [CovariantClass α α (· * ·) (· < ·)]
    [CovariantClass α α (swap (· * ·)) (· ≤ ·)] {f g : β → α} (hf : Antitone f)
    (hg : StrictAnti g) :
    StrictAnti fun x => f x * g x 
Arguments: {α : Type u_1} {β : Type u_2} [Add α] [Preorder α] [Preorder β] [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] {f g : β → α} (hf : Antitone f) (hg : StrictAnti g)
Goal: StrictAnti fun x => f x + g x
-- AntitoneOn.add_strictAnti
@[to_additive add_strictAnti "The sum of an antitone function and a strictly antitone function is
strictly antitone."]
theorem AntitoneOn.mul_strictAnti' [CovariantClass α α (· * ·) (· < ·)]
    [CovariantClass α α (swap (· * ·)) (· ≤ ·)] {f g : β → α} (hf : AntitoneOn f s)
    (hg : StrictAntiOn g s) :
    StrictAntiOn (fun x => f x * g x) s 
Arguments: {α : Type u_1} {β : Type u_2} [Add α] [Preorder α] [Preorder β] {s : Set β} [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] {f g : β → α} (hf : AntitoneOn f s) (hg : StrictAntiOn g s)
Goal: StrictAntiOn (fun x => f x + g x) s
-- AntitoneOn.mul_strictAnti'
@[to_additive add_strictAnti "The sum of an antitone function and a strictly antitone function is
strictly antitone."]
theorem AntitoneOn.mul_strictAnti' [CovariantClass α α (· * ·) (· < ·)]
    [CovariantClass α α (swap (· * ·)) (· ≤ ·)] {f g : β → α} (hf : AntitoneOn f s)
    (hg : StrictAntiOn g s) :
    StrictAntiOn (fun x => f x * g x) s 
Arguments: {α : Type u_1} {β : Type u_2} [Mul α] [Preorder α] [Preorder β] {s : Set β} [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] {f g : β → α} (hf : AntitoneOn f s) (hg : StrictAntiOn g s)
Goal: StrictAntiOn (fun x => f x * g x) s
-- StrictMono.add_monotone
@[to_additive add_monotone "The sum of a strictly monotone function and a monotone function is
strictly monotone."]
theorem StrictMono.mul_monotone' (hf : StrictMono f) (hg : Monotone g) :
    StrictMono fun x => f x * g x 
Arguments: {α : Type u_1} {β : Type u_2} [Add α] [Preorder α] [Preorder β] {f g : β → 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] (hf : StrictMono f) (hg : Monotone g)
Goal: StrictMono fun x => f x + g x
-- StrictMono.mul_monotone'
@[to_additive add_monotone "The sum of a strictly monotone function and a monotone function is
strictly monotone."]
theorem StrictMono.mul_monotone' (hf : StrictMono f) (hg : Monotone g) :
    StrictMono fun x => f x * g x 
Arguments: {α : Type u_1} {β : Type u_2} [Mul α] [Preorder α] [Preorder β] {f g : β → 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] (hf : StrictMono f) (hg : Monotone g)
Goal: StrictMono fun x => f x * g x
-- StrictMonoOn.mul_monotone'
@[to_additive add_monotone "The sum of a strictly monotone function and a monotone function is
strictly monotone."]
theorem StrictMonoOn.mul_monotone' (hf : StrictMonoOn f s) (hg : MonotoneOn g s) :
    StrictMonoOn (fun x => f x * g x) s 
Arguments: {α : Type u_1} {β : Type u_2} [Mul α] [Preorder α] [Preorder β] {f g : β → α} {s : Set β} [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] (hf : StrictMonoOn f s) (hg : MonotoneOn g s)
Goal: StrictMonoOn (fun x => f x * g x) s
-- StrictMonoOn.add_monotone
@[to_additive add_monotone "The sum of a strictly monotone function and a monotone function is
strictly monotone."]
theorem StrictMonoOn.mul_monotone' (hf : StrictMonoOn f s) (hg : MonotoneOn g s) :
    StrictMonoOn (fun x => f x * g x) s 
Arguments: {α : Type u_1} {β : Type u_2} [Add α] [Preorder α] [Preorder β] {f g : β → α} {s : Set β} [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] (hf : StrictMonoOn f s) (hg : MonotoneOn g s)
Goal: StrictMonoOn (fun x => f x + g x) s
-- StrictAnti.add_antitone
@[to_additive add_antitone "The sum of a strictly antitone function and an antitone function is
strictly antitone."]
theorem StrictAnti.mul_antitone' (hf : StrictAnti f) (hg : Antitone g) :
    StrictAnti fun x => f x * g x 
Arguments: {α : Type u_1} {β : Type u_2} [Add α] [Preorder α] [Preorder β] {f g : β → 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] (hf : StrictAnti f) (hg : Antitone g)
Goal: StrictAnti fun x => f x + g x
-- StrictAnti.mul_antitone'
@[to_additive add_antitone "The sum of a strictly antitone function and an antitone function is
strictly antitone."]
theorem StrictAnti.mul_antitone' (hf : StrictAnti f) (hg : Antitone g) :
    StrictAnti fun x => f x * g x 
Arguments: {α : Type u_1} {β : Type u_2} [Mul α] [Preorder α] [Preorder β] {f g : β → 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] (hf : StrictAnti f) (hg : Antitone g)
Goal: StrictAnti fun x => f x * g x
-- StrictAntiOn.mul_antitone'
@[to_additive add_antitone "The sum of a strictly antitone function and an antitone function is
strictly antitone."]
theorem StrictAntiOn.mul_antitone' (hf : StrictAntiOn f s) (hg : AntitoneOn g s) :
    StrictAntiOn (fun x => f x * g x) s 
Arguments: {α : Type u_1} {β : Type u_2} [Mul α] [Preorder α] [Preorder β] {f g : β → α} {s : Set β} [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] (hf : StrictAntiOn f s) (hg : AntitoneOn g s)
Goal: StrictAntiOn (fun x => f x * g x) s
-- StrictAntiOn.add_antitone
@[to_additive add_antitone "The sum of a strictly antitone function and an antitone function is
strictly antitone."]
theorem StrictAntiOn.mul_antitone' (hf : StrictAntiOn f s) (hg : AntitoneOn g s) :
    StrictAntiOn (fun x => f x * g x) s 
Arguments: {α : Type u_1} {β : Type u_2} [Add α] [Preorder α] [Preorder β] {f g : β → α} {s : Set β} [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] (hf : StrictAntiOn f s) (hg : AntitoneOn g s)
Goal: StrictAntiOn (fun x => f x + g x) s
-- cmp_add_left
@[to_additive (attr := simp) cmp_add_left]
theorem cmp_mul_left' {α : Type*} [Mul α] [LinearOrder α] [CovariantClass α α (· * ·) (· < ·)]
    (a b c : α) :
    cmp (a * b) (a * c) = cmp b c 
Arguments: {α : Type u_3} [Add α] [LinearOrder α] [CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x < x_1] (a b c : α)
Goal: cmp (a + b) (a + c) = cmp b c
-- cmp_mul_left'
@[to_additive (attr := simp) cmp_add_left]
theorem cmp_mul_left' {α : Type*} [Mul α] [LinearOrder α] [CovariantClass α α (· * ·) (· < ·)]
    (a b c : α) :
    cmp (a * b) (a * c) = cmp b c 
Arguments: {α : Type u_3} [Mul α] [LinearOrder α] [CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x < x_1] (a b c : α)
Goal: cmp (a * b) (a * c) = cmp b c
-- cmp_add_right
@[to_additive (attr := simp) cmp_add_right]
theorem cmp_mul_right' {α : Type*} [Mul α] [LinearOrder α]
    [CovariantClass α α (swap (· * ·)) (· < ·)] (a b c : α) :
    cmp (a * c) (b * c) = cmp a b 
Arguments: {α : Type u_3} [Add α] [LinearOrder α] [CovariantClass α α (Function.swap fun x x_1 => x + x_1) fun x x_1 => x < x_1] (a b c : α)
Goal: cmp (a + c) (b + c) = cmp a b
-- cmp_mul_right'
@[to_additive (attr := simp) cmp_add_right]
theorem cmp_mul_right' {α : Type*} [Mul α] [LinearOrder α]
    [CovariantClass α α (swap (· * ·)) (· < ·)] (a b c : α) :
    cmp (a * c) (b * c) = cmp a b 
Arguments: {α : Type u_3} [Mul α] [LinearOrder α] [CovariantClass α α (Function.swap fun x x_1 => x * x_1) fun x x_1 => x < x_1] (a b c : α)
Goal: cmp (a * c) (b * c) = cmp a b
-- Contravariant.AddLECancellable
@[to_additive]
theorem Contravariant.MulLECancellable [Mul α] [LE α] [ContravariantClass α α (· * ·) (· ≤ ·)]
    {a : α} :
    MulLECancellable a 
Arguments: {α : Type u_1} [Add α] [LE α] [ContravariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] {a : α}
Goal: AddLECancellable a
-- Contravariant.MulLECancellable
@[to_additive]
theorem Contravariant.MulLECancellable [Mul α] [LE α] [ContravariantClass α α (· * ·) (· ≤ ·)]
    {a : α} :
    MulLECancellable a 
Arguments: {α : Type u_1} [Mul α] [LE α] [ContravariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1] {a : α}
Goal: MulLECancellable a
-- addLECancellable_zero
@[to_additive]
theorem mulLECancellable_one [Monoid α] [LE α] : MulLECancellable (1 : α) 
Arguments: {α : Type u_1} [AddMonoid α] [LE α]
Goal: AddLECancellable 0
-- mulLECancellable_one
@[to_additive]
theorem mulLECancellable_one [Monoid α] [LE α] : MulLECancellable (1 : α) 
Arguments: {α : Type u_1} [Monoid α] [LE α]
Goal: MulLECancellable 1
-- AddLECancellable.Injective
@[to_additive]
protected theorem Injective [Mul α] [PartialOrder α] {a : α} (ha : MulLECancellable a) :
    Injective (a * ·) 
Arguments: {α : Type u_1} [Add α] [PartialOrder α] {a : α} (ha : AddLECancellable a)
Goal: Function.Injective fun x => a + x
-- MulLECancellable.Injective
@[to_additive]
protected theorem Injective [Mul α] [PartialOrder α] {a : α} (ha : MulLECancellable a) :
    Injective (a * ·) 
Arguments: {α : Type u_1} [Mul α] [PartialOrder α] {a : α} (ha : MulLECancellable a)
Goal: Function.Injective fun x => a * x
-- AddLECancellable.inj
@[to_additive]
protected theorem inj [Mul α] [PartialOrder α] {a b c : α} (ha : MulLECancellable a) :
    a * b = a * c ↔ b = c 
Arguments: {α : Type u_1} [Add α] [PartialOrder α] {a b c : α} (ha : AddLECancellable a)
Goal: a + b = a + c ↔ b = c
-- MulLECancellable.inj
@[to_additive]
protected theorem inj [Mul α] [PartialOrder α] {a b c : α} (ha : MulLECancellable a) :
    a * b = a * c ↔ b = c 
Arguments: {α : Type u_1} [Mul α] [PartialOrder α] {a b c : α} (ha : MulLECancellable a)
Goal: a * b = a * c ↔ b = c
-- AddLECancellable.injective_left
@[to_additive]
protected theorem injective_left [Mul α] [i : IsSymmOp α α (· * ·)] [PartialOrder α] {a : α}
    (ha : MulLECancellable a) :
    Injective (· * a) 
Arguments: {α : Type u_1} [Add α] [i : IsSymmOp α α fun x x_1 => x + x_1] [PartialOrder α] {a : α} (ha : AddLECancellable a)
Goal: Function.Injective fun x => x + a
-- MulLECancellable.injective_left
@[to_additive]
protected theorem injective_left [Mul α] [i : IsSymmOp α α (· * ·)] [PartialOrder α] {a : α}
    (ha : MulLECancellable a) :
    Injective (· * a) 
Arguments: {α : Type u_1} [Mul α] [i : IsSymmOp α α fun x x_1 => x * x_1] [PartialOrder α] {a : α} (ha : MulLECancellable a)
Goal: Function.Injective fun x => x * a
-- AddLECancellable.inj_left
@[to_additive]
protected theorem inj_left [Mul α] [IsSymmOp α α (· * ·)] [PartialOrder α] {a b c : α}
    (hc : MulLECancellable c) :
    a * c = b * c ↔ a = b 
Arguments: {α : Type u_1} [Add α] [IsSymmOp α α fun x x_1 => x + x_1] [PartialOrder α] {a b c : α} (hc : AddLECancellable c)
Goal: a + c = b + c ↔ a = b
-- MulLECancellable.inj_left
@[to_additive]
protected theorem inj_left [Mul α] [IsSymmOp α α (· * ·)] [PartialOrder α] {a b c : α}
    (hc : MulLECancellable c) :
    a * c = b * c ↔ a = b 
Arguments: {α : Type u_1} [Mul α] [IsSymmOp α α fun x x_1 => x * x_1] [PartialOrder α] {a b c : α} (hc : MulLECancellable c)
Goal: a * c = b * c ↔ a = b
-- MulLECancellable.mul_le_mul_iff_left
@[to_additive]
protected theorem mul_le_mul_iff_left [Mul α] [CovariantClass α α (· * ·) (· ≤ ·)] {a b c : α}
    (ha : MulLECancellable a) : a * b ≤ a * c ↔ b ≤ c 
Arguments: {α : Type u_1} [LE α] [Mul α] [CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1] {a b c : α} (ha : MulLECancellable a)
Goal: a * b ≤ a * c ↔ b ≤ c
-- AddLECancellable.add_le_add_iff_left
@[to_additive]
protected theorem mul_le_mul_iff_left [Mul α] [CovariantClass α α (· * ·) (· ≤ ·)] {a b c : α}
    (ha : MulLECancellable a) : a * b ≤ a * c ↔ b ≤ c 
Arguments: {α : Type u_1} [LE α] [Add α] [CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] {a b c : α} (ha : AddLECancellable a)
Goal: a + b ≤ a + c ↔ b ≤ c
-- AddLECancellable.add_le_add_iff_right
@[to_additive]
protected theorem mul_le_mul_iff_right [Mul α] [i : IsSymmOp α α (· * ·)]
    [CovariantClass α α (· * ·) (· ≤ ·)] {a b c : α} (ha : MulLECancellable a) :
    b * a ≤ c * a ↔ b ≤ c 
Arguments: {α : Type u_1} [LE α] [Add α] [i : IsSymmOp α α fun x x_1 => x + x_1] [CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] {a b c : α} (ha : AddLECancellable a)
Goal: b + a ≤ c + a ↔ b ≤ c
-- MulLECancellable.mul_le_mul_iff_right
@[to_additive]
protected theorem mul_le_mul_iff_right [Mul α] [i : IsSymmOp α α (· * ·)]
    [CovariantClass α α (· * ·) (· ≤ ·)] {a b c : α} (ha : MulLECancellable a) :
    b * a ≤ c * a ↔ b ≤ c 
Arguments: {α : Type u_1} [LE α] [Mul α] [i : IsSymmOp α α fun x x_1 => x * x_1] [CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1] {a b c : α} (ha : MulLECancellable a)
Goal: b * a ≤ c * a ↔ b ≤ c
-- MulLECancellable.le_mul_iff_one_le_right
@[to_additive]
protected theorem le_mul_iff_one_le_right [MulOneClass α] [CovariantClass α α (· * ·) (· ≤ ·)]
    {a b : α} (ha : MulLECancellable a) :
    a ≤ a * b ↔ 1 ≤ b 
Arguments: {α : Type u_1} [LE α] [MulOneClass α] [CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1] {a b : α} (ha : MulLECancellable a)
Goal: a ≤ a * b ↔ 1 ≤ b
-- AddLECancellable.le_add_iff_nonneg_right
@[to_additive]
protected theorem le_mul_iff_one_le_right [MulOneClass α] [CovariantClass α α (· * ·) (· ≤ ·)]
    {a b : α} (ha : MulLECancellable a) :
    a ≤ a * b ↔ 1 ≤ b 
Arguments: {α : Type u_1} [LE α] [AddZeroClass α] [CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] {a b : α} (ha : AddLECancellable a)
Goal: a ≤ a + b ↔ 0 ≤ b
-- MulLECancellable.mul_le_iff_le_one_right
@[to_additive]
protected theorem mul_le_iff_le_one_right [MulOneClass α] [CovariantClass α α (· * ·) (· ≤ ·)]
    {a b : α} (ha : MulLECancellable a) :
    a * b ≤ a ↔ b ≤ 1 
Arguments: {α : Type u_1} [LE α] [MulOneClass α] [CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1] {a b : α} (ha : MulLECancellable a)
Goal: a * b ≤ a ↔ b ≤ 1
-- AddLECancellable.add_le_iff_nonpos_right
@[to_additive]
protected theorem mul_le_iff_le_one_right [MulOneClass α] [CovariantClass α α (· * ·) (· ≤ ·)]
    {a b : α} (ha : MulLECancellable a) :
    a * b ≤ a ↔ b ≤ 1 
Arguments: {α : Type u_1} [LE α] [AddZeroClass α] [CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] {a b : α} (ha : AddLECancellable a)
Goal: a + b ≤ a ↔ b ≤ 0
-- AddLECancellable.le_add_iff_nonneg_left
@[to_additive]
protected theorem le_mul_iff_one_le_left [MulOneClass α] [i : IsSymmOp α α (· * ·)]
    [CovariantClass α α (· * ·) (· ≤ ·)] {a b : α} (ha : MulLECancellable a) :
    a ≤ b * a ↔ 1 ≤ b 
Arguments: {α : Type u_1} [LE α] [AddZeroClass α] [i : IsSymmOp α α fun x x_1 => x + x_1] [CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] {a b : α} (ha : AddLECancellable a)
Goal: a ≤ b + a ↔ 0 ≤ b
-- MulLECancellable.le_mul_iff_one_le_left
@[to_additive]
protected theorem le_mul_iff_one_le_left [MulOneClass α] [i : IsSymmOp α α (· * ·)]
    [CovariantClass α α (· * ·) (· ≤ ·)] {a b : α} (ha : MulLECancellable a) :
    a ≤ b * a ↔ 1 ≤ b 
Arguments: {α : Type u_1} [LE α] [MulOneClass α] [i : IsSymmOp α α fun x x_1 => x * x_1] [CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1] {a b : α} (ha : MulLECancellable a)
Goal: a ≤ b * a ↔ 1 ≤ b
-- MulLECancellable.mul_le_iff_le_one_left
@[to_additive]
protected theorem mul_le_iff_le_one_left [MulOneClass α] [i : IsSymmOp α α (· * ·)]
    [CovariantClass α α (· * ·) (· ≤ ·)] {a b : α} (ha : MulLECancellable a) :
    b * a ≤ a ↔ b ≤ 1 
Arguments: {α : Type u_1} [LE α] [MulOneClass α] [i : IsSymmOp α α fun x x_1 => x * x_1] [CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1] {a b : α} (ha : MulLECancellable a)
Goal: b * a ≤ a ↔ b ≤ 1
-- AddLECancellable.add_le_iff_nonpos_left
@[to_additive]
protected theorem mul_le_iff_le_one_left [MulOneClass α] [i : IsSymmOp α α (· * ·)]
    [CovariantClass α α (· * ·) (· ≤ ·)] {a b : α} (ha : MulLECancellable a) :
    b * a ≤ a ↔ b ≤ 1 
Arguments: {α : Type u_1} [LE α] [AddZeroClass α] [i : IsSymmOp α α fun x x_1 => x + x_1] [CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] {a b : α} (ha : AddLECancellable a)
Goal: b + a ≤ a ↔ b ≤ 0
-- WithTop.coe_zero
@[to_additive (attr := simp, norm_cast)]
theorem coe_one : ((1 : α) : WithTop α) = 1 
Arguments: {α : Type u} [Zero α]
Goal: ↑0 = 0
-- WithTop.coe_one
@[to_additive (attr := simp, norm_cast)]
theorem coe_one : ((1 : α) : WithTop α) = 1 
Arguments: {α : Type u} [One α]
Goal: ↑1 = 1
-- WithTop.coe_eq_zero
@[to_additive (attr := simp, norm_cast)]
lemma coe_eq_one : (a : WithTop α) = 1 ↔ a = 1 
Arguments: {α : Type u} [Zero α] {a : α}
Goal: ↑a = 0 ↔ a = 0
-- WithTop.coe_eq_one
@[to_additive (attr := simp, norm_cast)]
lemma coe_eq_one : (a : WithTop α) = 1 ↔ a = 1 
Arguments: {α : Type u} [One α] {a : α}
Goal: ↑a = 1 ↔ a = 1
-- WithTop.one_eq_coe
@[to_additive (attr := simp, norm_cast)]
lemma one_eq_coe : 1 = (a : WithTop α) ↔ a = 1 
Arguments: {α : Type u} [One α] {a : α}
Goal: 1 = ↑a ↔ a = 1
-- WithTop.zero_eq_coe
@[to_additive (attr := simp, norm_cast)]
lemma one_eq_coe : 1 = (a : WithTop α) ↔ a = 1 
Arguments: {α : Type u} [Zero α] {a : α}
Goal: 0 = ↑a ↔ a = 0