ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- OrderEmbedding.addLeft_apply
@[to_additive (attr := simps!)
      "The order embedding sending `b` to `a + b`, for some fixed `a`.
       See also `OrderIso.addLeft` when working in an additive ordered group."]
def OrderEmbedding.mulLeft {α : Type*} [Mul α] [LinearOrder α]
    [CovariantClass α α (· * ·) (· < ·)] (m : α) : α ↪o α 
Arguments: {α : Type u_2} [Add α] [LinearOrder α] [CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x < x_1] (m n : α)
Goal: (OrderEmbedding.addLeft m) n = m + n
-- OrderEmbedding.mulLeft_apply
@[to_additive (attr := simps!)
      "The order embedding sending `b` to `a + b`, for some fixed `a`.
       See also `OrderIso.addLeft` when working in an additive ordered group."]
def OrderEmbedding.mulLeft {α : Type*} [Mul α] [LinearOrder α]
    [CovariantClass α α (· * ·) (· < ·)] (m : α) : α ↪o α 
Arguments: {α : Type u_2} [Mul α] [LinearOrder α] [CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x < x_1] (m n : α)
Goal: (OrderEmbedding.mulLeft m) n = m * n
-- OrderEmbedding.addRight_apply
@[to_additive (attr := simps!)
      "The order embedding sending `b` to `b + a`, for some fixed `a`.
       See also `OrderIso.addRight` when working in an additive ordered group."]
def OrderEmbedding.mulRight {α : Type*} [Mul α] [LinearOrder α]
    [CovariantClass α α (swap (· * ·)) (· < ·)] (m : α) : α ↪o α 
Arguments: {α : Type u_2} [Add α] [LinearOrder α] [CovariantClass α α (Function.swap fun x x_1 => x + x_1) fun x x_1 => x < x_1] (m n : α)
Goal: (OrderEmbedding.addRight m) n = n + m
-- OrderEmbedding.mulRight_apply
@[to_additive (attr := simps!)
      "The order embedding sending `b` to `b + a`, for some fixed `a`.
       See also `OrderIso.addRight` when working in an additive ordered group."]
def OrderEmbedding.mulRight {α : Type*} [Mul α] [LinearOrder α]
    [CovariantClass α α (swap (· * ·)) (· < ·)] (m : α) : α ↪o α 
Arguments: {α : Type u_2} [Mul α] [LinearOrder α] [CovariantClass α α (Function.swap fun x x_1 => x * x_1) fun x x_1 => x < x_1] (m n : α)
Goal: (OrderEmbedding.mulRight m) n = n * m
-- OrderedSub.tsub_le_iff_right
class OrderedSub (α : Type*) [LE α] [Add α] [Sub α] : Prop where
    tsub_le_iff_right : ∀ a b c : α, a - b ≤ c ↔ a ≤ c + b
Arguments: {α : Type u_3} [LE α] [Add α] [Sub α] [self : OrderedSub α] (a b c : α)
Goal: a - b ≤ c ↔ a ≤ c + b
-- tsub_le_iff_right
@[simp]
theorem tsub_le_iff_right [LE α] [Add α] [Sub α] [OrderedSub α] {a b c : α} :
    a - b ≤ c ↔ a ≤ c + b 
Arguments: {α : Type u_1} [LE α] [Add α] [Sub α] [OrderedSub α] {a b c : α}
Goal: a - b ≤ c ↔ a ≤ c + b
-- add_tsub_le_right
theorem add_tsub_le_right : a + b - b ≤ a 
Arguments: {α : Type u_1} [Preorder α] [Add α] [Sub α] [OrderedSub α] {a b : α}
Goal: a + b - b ≤ a
-- le_tsub_add
theorem le_tsub_add : b ≤ b - a + a 
Arguments: {α : Type u_1} [Preorder α] [Add α] [Sub α] [OrderedSub α] {a b : α}
Goal: b ≤ b - a + a
-- tsub_le_iff_left
theorem tsub_le_iff_left : a - b ≤ c ↔ a ≤ b + c 
Arguments: {α : Type u_1} [Preorder α] [AddCommSemigroup α] [Sub α] [OrderedSub α] {a b c : α}
Goal: a - b ≤ c ↔ a ≤ b + c
-- le_add_tsub
theorem le_add_tsub : a ≤ b + (a - b) 
Arguments: {α : Type u_1} [Preorder α] [AddCommSemigroup α] [Sub α] [OrderedSub α] {a b : α}
Goal: a ≤ b + (a - b)
-- add_tsub_le_left
theorem add_tsub_le_left : a + b - a ≤ b 
Arguments: {α : Type u_1} [Preorder α] [AddCommSemigroup α] [Sub α] [OrderedSub α] {a b : α}
Goal: a + b - a ≤ b
-- tsub_le_tsub_right
@[gcongr] theorem tsub_le_tsub_right (h : a ≤ b) (c : α) : a - c ≤ b - c 
Arguments: {α : Type u_1} [Preorder α] [AddCommSemigroup α] [Sub α] [OrderedSub α] {a b : α} (h : a ≤ b) (c : α)
Goal: a - c ≤ b - c
-- tsub_le_iff_tsub_le
theorem tsub_le_iff_tsub_le : a - b ≤ c ↔ a - c ≤ b 
Arguments: {α : Type u_1} [Preorder α] [AddCommSemigroup α] [Sub α] [OrderedSub α] {a b c : α}
Goal: a - b ≤ c ↔ a - c ≤ b
-- tsub_tsub_le
theorem tsub_tsub_le : b - (b - a) ≤ a 
Arguments: {α : Type u_1} [Preorder α] [AddCommSemigroup α] [Sub α] [OrderedSub α] {a b : α}
Goal: b - (b - a) ≤ a
-- tsub_le_tsub_left
@[gcongr] theorem tsub_le_tsub_left (h : a ≤ b) (c : α) : c - b ≤ c - a 
Arguments: {α : Type u_1} [Preorder α] [AddCommSemigroup α] [Sub α] [OrderedSub α] {a b : α} [CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] (h : a ≤ b) (c : α)
Goal: c - b ≤ c - a
-- tsub_le_tsub
@[gcongr] theorem tsub_le_tsub (hab : a ≤ b) (hcd : c ≤ d) : a - d ≤ b - c 
Arguments: {α : Type u_1} [Preorder α] [AddCommSemigroup α] [Sub α] [OrderedSub α] {a b c d : α} [CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] (hab : a ≤ b) (hcd : c ≤ d)
Goal: a - d ≤ b - c
-- antitone_const_tsub
theorem antitone_const_tsub : Antitone fun x => c - x 
Arguments: {α : Type u_1} [Preorder α] [AddCommSemigroup α] [Sub α] [OrderedSub α] {c : α} [CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1]
Goal: Antitone fun x => c - x
-- add_tsub_le_assoc
theorem add_tsub_le_assoc : a + b - c ≤ a + (b - c) 
Arguments: {α : Type u_1} [Preorder α] [AddCommSemigroup α] [Sub α] [OrderedSub α] {a b c : α} [CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1]
Goal: a + b - c ≤ a + (b - c)
-- add_tsub_le_tsub_add
theorem add_tsub_le_tsub_add : a + b - c ≤ a - c + b 
Arguments: {α : Type u_1} [Preorder α] [AddCommSemigroup α] [Sub α] [OrderedSub α] {a b c : α} [CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1]
Goal: a + b - c ≤ a - c + b
-- add_le_add_add_tsub
theorem add_le_add_add_tsub : a + b ≤ a + c + (b - c) 
Arguments: {α : Type u_1} [Preorder α] [AddCommSemigroup α] [Sub α] [OrderedSub α] {a b c : α} [CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1]
Goal: a + b ≤ a + c + (b - c)
-- le_tsub_add_add
theorem le_tsub_add_add : a + b ≤ a - c + (b + c) 
Arguments: {α : Type u_1} [Preorder α] [AddCommSemigroup α] [Sub α] [OrderedSub α] {a b c : α} [CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1]
Goal: a + b ≤ a - c + (b + c)
-- tsub_le_tsub_add_tsub
theorem tsub_le_tsub_add_tsub : a - c ≤ a - b + (b - c) 
Arguments: {α : Type u_1} [Preorder α] [AddCommSemigroup α] [Sub α] [OrderedSub α] {a b c : α} [CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1]
Goal: a - c ≤ a - b + (b - c)
-- tsub_tsub_tsub_le_tsub
theorem tsub_tsub_tsub_le_tsub : c - a - (c - b) ≤ b - a 
Arguments: {α : Type u_1} [Preorder α] [AddCommSemigroup α] [Sub α] [OrderedSub α] {a b c : α} [CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1]
Goal: c - a - (c - b) ≤ b - a
-- tsub_tsub_le_tsub_add
theorem tsub_tsub_le_tsub_add {a b c : α} : a - (b - c) ≤ a - b + c 
Arguments: {α : Type u_1} [Preorder α] [AddCommSemigroup α] [Sub α] [OrderedSub α] [CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] {a b c : α}
Goal: a - (b - c) ≤ a - b + c
-- add_tsub_add_le_tsub_add_tsub
theorem add_tsub_add_le_tsub_add_tsub : a + b - (c + d) ≤ a - c + (b - d) 
Arguments: {α : Type u_1} [Preorder α] [AddCommSemigroup α] [Sub α] [OrderedSub α] {a b c d : α} [CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1]
Goal: a + b - (c + d) ≤ a - c + (b - d)
-- add_tsub_add_le_tsub_left
theorem add_tsub_add_le_tsub_left : a + b - (a + c) ≤ b - c 
Arguments: {α : Type u_1} [Preorder α] [AddCommSemigroup α] [Sub α] [OrderedSub α] {a b c : α} [CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1]
Goal: a + b - (a + c) ≤ b - c
-- add_tsub_add_le_tsub_right
theorem add_tsub_add_le_tsub_right : a + c - (b + c) ≤ a - b 
Arguments: {α : Type u_1} [Preorder α] [AddCommSemigroup α] [Sub α] [OrderedSub α] {a b c : α} [CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1]
Goal: a + c - (b + c) ≤ a - b
-- AddLECancellable.le_add_tsub_swap
protected theorem le_add_tsub_swap (hb : AddLECancellable b) : a ≤ b + a - b 
Arguments: {α : Type u_1} [Preorder α] [AddCommSemigroup α] [Sub α] [OrderedSub α] {a b : α} (hb : AddLECancellable b)
Goal: a ≤ b + a - b
-- AddLECancellable.le_add_tsub
protected theorem le_add_tsub (hb : AddLECancellable b) : a ≤ a + b - b 
Arguments: {α : Type u_1} [Preorder α] [AddCommSemigroup α] [Sub α] [OrderedSub α] {a b : α} (hb : AddLECancellable b)
Goal: a ≤ a + b - b
-- AddLECancellable.le_tsub_of_add_le_left
protected theorem le_tsub_of_add_le_left (ha : AddLECancellable a) (h : a + b ≤ c) : b ≤ c - a 
Arguments: {α : Type u_1} [Preorder α] [AddCommSemigroup α] [Sub α] [OrderedSub α] {a b c : α} (ha : AddLECancellable a) (h : a + b ≤ c)
Goal: b ≤ c - a
-- AddLECancellable.le_tsub_of_add_le_right
protected theorem le_tsub_of_add_le_right (hb : AddLECancellable b) (h : a + b ≤ c) : a ≤ c - b 
Arguments: {α : Type u_1} [Preorder α] [AddCommSemigroup α] [Sub α] [OrderedSub α] {a b c : α} (hb : AddLECancellable b) (h : a + b ≤ c)
Goal: a ≤ c - b
-- le_add_tsub_swap
theorem le_add_tsub_swap : a ≤ b + a - b 
Arguments: {α : Type u_1} [Preorder α] [AddCommSemigroup α] [Sub α] [OrderedSub α] {a b : α} [ContravariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1]
Goal: a ≤ b + a - b
-- le_add_tsub'
theorem le_add_tsub' : a ≤ a + b - b 
Arguments: {α : Type u_1} [Preorder α] [AddCommSemigroup α] [Sub α] [OrderedSub α] {a b : α} [ContravariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1]
Goal: a ≤ a + b - b
-- le_tsub_of_add_le_left
theorem le_tsub_of_add_le_left (h : a + b ≤ c) : b ≤ c - a 
Arguments: {α : Type u_1} [Preorder α] [AddCommSemigroup α] [Sub α] [OrderedSub α] {a b c : α} [ContravariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] (h : a + b ≤ c)
Goal: b ≤ c - a
-- le_tsub_of_add_le_right
theorem le_tsub_of_add_le_right (h : a + b ≤ c) : a ≤ c - b 
Arguments: {α : Type u_1} [Preorder α] [AddCommSemigroup α] [Sub α] [OrderedSub α] {a b c : α} [ContravariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] (h : a + b ≤ c)
Goal: a ≤ c - b
-- tsub_nonpos
theorem tsub_nonpos : a - b ≤ 0 ↔ a ≤ b 
Arguments: {α : Type u_1} [Preorder α] [AddCommMonoid α] [Sub α] [OrderedSub α] {a b : α}
Goal: a - b ≤ 0 ↔ a ≤ b
-- tsub_tsub
theorem tsub_tsub (b a c : α) : b - a - c = b - (a + c) 
Arguments: {α : Type u_1} [PartialOrder α] [AddCommSemigroup α] [Sub α] [OrderedSub α] (b a c : α)
Goal: b - a - c = b - (a + c)
-- tsub_add_eq_tsub_tsub
theorem tsub_add_eq_tsub_tsub (a b c : α) : a - (b + c) = a - b - c 
Arguments: {α : Type u_1} [PartialOrder α] [AddCommSemigroup α] [Sub α] [OrderedSub α] (a b c : α)
Goal: a - (b + c) = a - b - c
-- tsub_add_eq_tsub_tsub_swap
theorem tsub_add_eq_tsub_tsub_swap (a b c : α) : a - (b + c) = a - c - b 
Arguments: {α : Type u_1} [PartialOrder α] [AddCommSemigroup α] [Sub α] [OrderedSub α] (a b c : α)
Goal: a - (b + c) = a - c - b
-- tsub_right_comm
theorem tsub_right_comm : a - b - c = a - c - b 
Arguments: {α : Type u_1} [PartialOrder α] [AddCommSemigroup α] [Sub α] [OrderedSub α] {a b c : α}
Goal: a - b - c = a - c - b
-- AddLECancellable.tsub_eq_of_eq_add
protected theorem tsub_eq_of_eq_add (hb : AddLECancellable b) (h : a = c + b) : a - b = c 
Arguments: {α : Type u_1} [PartialOrder α] [AddCommSemigroup α] [Sub α] [OrderedSub α] {a b c : α} (hb : AddLECancellable b) (h : a = c + b)
Goal: a - b = c
-- AddLECancellable.eq_tsub_of_add_eq
protected theorem eq_tsub_of_add_eq (hc : AddLECancellable c) (h : a + c = b) : a = b - c 
Arguments: {α : Type u_1} [PartialOrder α] [AddCommSemigroup α] [Sub α] [OrderedSub α] {a b c : α} (hc : AddLECancellable c) (h : a + c = b)
Goal: a = b - c
-- AddLECancellable.tsub_eq_of_eq_add_rev
protected theorem tsub_eq_of_eq_add_rev (hb : AddLECancellable b) (h : a = b + c) : a - b = c 
Arguments: {α : Type u_1} [PartialOrder α] [AddCommSemigroup α] [Sub α] [OrderedSub α] {a b c : α} (hb : AddLECancellable b) (h : a = b + c)
Goal: a - b = c
-- AddLECancellable.add_tsub_cancel_right
@[simp]
protected theorem add_tsub_cancel_right (hb : AddLECancellable b) : a + b - b = a 
Arguments: {α : Type u_1} [PartialOrder α] [AddCommSemigroup α] [Sub α] [OrderedSub α] {a b : α} (hb : AddLECancellable b)
Goal: a + b - b = a
-- AddLECancellable.add_tsub_cancel_left
@[simp]
protected theorem add_tsub_cancel_left (ha : AddLECancellable a) : a + b - a = b 
Arguments: {α : Type u_1} [PartialOrder α] [AddCommSemigroup α] [Sub α] [OrderedSub α] {a b : α} (ha : AddLECancellable a)
Goal: a + b - a = b
-- AddLECancellable.lt_add_of_tsub_lt_left
protected theorem lt_add_of_tsub_lt_left (hb : AddLECancellable b) (h : a - b < c) : a < b + c 
Arguments: {α : Type u_1} [PartialOrder α] [AddCommSemigroup α] [Sub α] [OrderedSub α] {a b c : α} (hb : AddLECancellable b) (h : a - b < c)
Goal: a < b + c
-- AddLECancellable.lt_add_of_tsub_lt_right
protected theorem lt_add_of_tsub_lt_right (hc : AddLECancellable c) (h : a - c < b) :
    a < b + c 
Arguments: {α : Type u_1} [PartialOrder α] [AddCommSemigroup α] [Sub α] [OrderedSub α] {a b c : α} (hc : AddLECancellable c) (h : a - c < b)
Goal: a < b + c
-- AddLECancellable.lt_tsub_of_add_lt_right
protected theorem lt_tsub_of_add_lt_right (hc : AddLECancellable c) (h : a + c < b) : a < b - c 
Arguments: {α : Type u_1} [PartialOrder α] [AddCommSemigroup α] [Sub α] [OrderedSub α] {a b c : α} (hc : AddLECancellable c) (h : a + c < b)
Goal: a < b - c
-- AddLECancellable.lt_tsub_of_add_lt_left
protected theorem lt_tsub_of_add_lt_left (ha : AddLECancellable a) (h : a + c < b) : c < b - a 
Arguments: {α : Type u_1} [PartialOrder α] [AddCommSemigroup α] [Sub α] [OrderedSub α] {a b c : α} (ha : AddLECancellable a) (h : a + c < b)
Goal: c < b - a
-- tsub_eq_of_eq_add
theorem tsub_eq_of_eq_add (h : a = c + b) : a - b = c 
Arguments: {α : Type u_1} [PartialOrder α] [AddCommSemigroup α] [Sub α] [OrderedSub α] {a b c : α} [ContravariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] (h : a = c + b)
Goal: a - b = c
-- eq_tsub_of_add_eq
theorem eq_tsub_of_add_eq (h : a + c = b) : a = b - c 
Arguments: {α : Type u_1} [PartialOrder α] [AddCommSemigroup α] [Sub α] [OrderedSub α] {a b c : α} [ContravariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] (h : a + c = b)
Goal: a = b - c
-- tsub_eq_of_eq_add_rev
theorem tsub_eq_of_eq_add_rev (h : a = b + c) : a - b = c 
Arguments: {α : Type u_1} [PartialOrder α] [AddCommSemigroup α] [Sub α] [OrderedSub α] {a b c : α} [ContravariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] (h : a = b + c)
Goal: a - b = c
-- add_tsub_cancel_right
@[simp]
theorem add_tsub_cancel_right (a b : α) : a + b - b = a 
Arguments: {α : Type u_1} [PartialOrder α] [AddCommSemigroup α] [Sub α] [OrderedSub α] [ContravariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] (a b : α)
Goal: a + b - b = a
-- add_tsub_cancel_left
@[simp]
theorem add_tsub_cancel_left (a b : α) : a + b - a = b 
Arguments: {α : Type u_1} [PartialOrder α] [AddCommSemigroup α] [Sub α] [OrderedSub α] [ContravariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] (a b : α)
Goal: a + b - a = b
-- lt_add_of_tsub_lt_left
theorem lt_add_of_tsub_lt_left (h : a - b < c) : a < b + c 
Arguments: {α : Type u_1} [PartialOrder α] [AddCommSemigroup α] [Sub α] [OrderedSub α] {a b c : α} [ContravariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] (h : a - b < c)
Goal: a < b + c
-- lt_add_of_tsub_lt_right
theorem lt_add_of_tsub_lt_right (h : a - c < b) : a < b + c 
Arguments: {α : Type u_1} [PartialOrder α] [AddCommSemigroup α] [Sub α] [OrderedSub α] {a b c : α} [ContravariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] (h : a - c < b)
Goal: a < b + c
-- lt_tsub_of_add_lt_left
theorem lt_tsub_of_add_lt_left : a + c < b → c < b - a 
Arguments: {α : Type u_1} [PartialOrder α] [AddCommSemigroup α] [Sub α] [OrderedSub α] {a b c : α} [ContravariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1]
Goal: a + c < b → c < b - a
-- lt_tsub_of_add_lt_right
theorem lt_tsub_of_add_lt_right : a + c < b → a < b - c 
Arguments: {α : Type u_1} [PartialOrder α] [AddCommSemigroup α] [Sub α] [OrderedSub α] {a b c : α} [ContravariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1]
Goal: a + c < b → a < b - c
-- add_tsub_add_eq_tsub_right
theorem add_tsub_add_eq_tsub_right (a c b : α) : a + c - (b + c) = a - b 
Arguments: {α : Type u_1} [PartialOrder α] [AddCommSemigroup α] [Sub α] [OrderedSub α] [CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] [ContravariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] (a c b : α)
Goal: a + c - (b + c) = a - b
-- add_tsub_add_eq_tsub_left
theorem add_tsub_add_eq_tsub_left (a b c : α) : a + b - (a + c) = b - c 
Arguments: {α : Type u_1} [PartialOrder α] [AddCommSemigroup α] [Sub α] [OrderedSub α] [CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] [ContravariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] (a b c : α)
Goal: a + b - (a + c) = b - c
-- lt_of_tsub_lt_tsub_right
theorem lt_of_tsub_lt_tsub_right (h : a - c < b - c) : a < b 
Arguments: {α : Type u_1} {a b c : α} [LinearOrder α] [AddCommSemigroup α] [Sub α] [OrderedSub α] (h : a - c < b - c)
Goal: a < b
-- lt_tsub_iff_right
theorem lt_tsub_iff_right : a < b - c ↔ a + c < b 
Arguments: {α : Type u_1} {a b c : α} [LinearOrder α] [AddCommSemigroup α] [Sub α] [OrderedSub α]
Goal: a < b - c ↔ a + c < b
-- lt_tsub_iff_left
theorem lt_tsub_iff_left : a < b - c ↔ c + a < b 
Arguments: {α : Type u_1} {a b c : α} [LinearOrder α] [AddCommSemigroup α] [Sub α] [OrderedSub α]
Goal: a < b - c ↔ c + a < b
-- lt_tsub_comm
theorem lt_tsub_comm : a < b - c ↔ c < b - a 
Arguments: {α : Type u_1} {a b c : α} [LinearOrder α] [AddCommSemigroup α] [Sub α] [OrderedSub α]
Goal: a < b - c ↔ c < b - a
-- lt_of_tsub_lt_tsub_left
theorem lt_of_tsub_lt_tsub_left (h : a - b < a - c) : c < b 
Arguments: {α : Type u_1} {a b c : α} [LinearOrder α] [AddCommSemigroup α] [Sub α] [OrderedSub α] [CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] (h : a - b < a - c)
Goal: c < b
-- tsub_zero
@[simp]
theorem tsub_zero (a : α) : a - 0 = a 
Arguments: {α : Type u_1} [PartialOrder α] [AddCommMonoid α] [Sub α] [OrderedSub α] (a : α)
Goal: a - 0 = a
-- ExistsMulOfLE.exists_mul_of_le
class ExistsMulOfLE (α : Type u) [Mul α] [LE α] : Prop where
    exists_mul_of_le : ∀ {a b : α}, a ≤ b → ∃ c : α, b = a * c
Arguments: {α : Type u} [Mul α] [LE α] [self : ExistsMulOfLE α] {a b : α}
Goal: a ≤ b → ∃ c, b = a * c
-- ExistsAddOfLE.exists_add_of_le
class ExistsAddOfLE (α : Type u) [Add α] [LE α] : Prop where
    exists_add_of_le : ∀ {a b : α}, a ≤ b → ∃ c : α, b = a + c
Arguments: {α : Type u} [Add α] [LE α] [self : ExistsAddOfLE α] {a b : α}
Goal: a ≤ b → ∃ c, b = a + c
-- exists_pos_add_of_lt'
@[to_additive]
theorem exists_one_lt_mul_of_lt' (h : a < b) : ∃ c, 1 < c ∧ a * c = b 
Arguments: {α : Type u} [AddZeroClass α] [Preorder α] [ContravariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x < x_1] [ExistsAddOfLE α] {a b : α} (h : a < b)
Goal: ∃ c, 0 < c ∧ a + c = b
-- exists_one_lt_mul_of_lt'
@[to_additive]
theorem exists_one_lt_mul_of_lt' (h : a < b) : ∃ c, 1 < c ∧ a * c = b 
Arguments: {α : Type u} [MulOneClass α] [Preorder α] [ContravariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x < x_1] [ExistsMulOfLE α] {a b : α} (h : a < b)
Goal: ∃ c, 1 < c ∧ a * c = b
-- le_of_forall_pos_le_add
@[to_additive]
theorem le_of_forall_one_lt_le_mul (h : ∀ ε : α, 1 < ε → a ≤ b * ε) : a ≤ b 
Arguments: {α : Type u} [LinearOrder α] [DenselyOrdered α] [AddMonoid α] [ExistsAddOfLE α] [CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x < x_1] [ContravariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x < x_1] {a b : α} (h : ∀ (ε : α), 0 < ε → a ≤ b + ε)
Goal: a ≤ b
-- le_of_forall_one_lt_le_mul
@[to_additive]
theorem le_of_forall_one_lt_le_mul (h : ∀ ε : α, 1 < ε → a ≤ b * ε) : a ≤ b 
Arguments: {α : Type u} [LinearOrder α] [DenselyOrdered α] [Monoid α] [ExistsMulOfLE α] [CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x < x_1] [ContravariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x < x_1] {a b : α} (h : ∀ (ε : α), 1 < ε → a ≤ b * ε)
Goal: a ≤ b
-- le_of_forall_pos_lt_add'
@[to_additive]
theorem le_of_forall_one_lt_lt_mul' (h : ∀ ε : α, 1 < ε → a < b * ε) : a ≤ b 
Arguments: {α : Type u} [LinearOrder α] [DenselyOrdered α] [AddMonoid α] [ExistsAddOfLE α] [CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x < x_1] [ContravariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x < x_1] {a b : α} (h : ∀ (ε : α), 0 < ε → a < b + ε)
Goal: a ≤ b
-- le_of_forall_one_lt_lt_mul'
@[to_additive]
theorem le_of_forall_one_lt_lt_mul' (h : ∀ ε : α, 1 < ε → a < b * ε) : a ≤ b 
Arguments: {α : Type u} [LinearOrder α] [DenselyOrdered α] [Monoid α] [ExistsMulOfLE α] [CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x < x_1] [ContravariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x < x_1] {a b : α} (h : ∀ (ε : α), 1 < ε → a < b * ε)
Goal: a ≤ b
-- le_iff_forall_pos_lt_add'
@[to_additive]
theorem le_iff_forall_one_lt_lt_mul' : a ≤ b ↔ ∀ ε, 1 < ε → a < b * ε 
Arguments: {α : Type u} [LinearOrder α] [DenselyOrdered α] [AddMonoid α] [ExistsAddOfLE α] [CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x < x_1] [ContravariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x < x_1] {a b : α}
Goal: a ≤ b ↔ ∀ (ε : α), 0 < ε → a < b + ε
-- le_iff_forall_one_lt_lt_mul'
@[to_additive]
theorem le_iff_forall_one_lt_lt_mul' : a ≤ b ↔ ∀ ε, 1 < ε → a < b * ε 
Arguments: {α : Type u} [LinearOrder α] [DenselyOrdered α] [Monoid α] [ExistsMulOfLE α] [CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x < x_1] [ContravariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x < x_1] {a b : α}
Goal: a ≤ b ↔ ∀ (ε : α), 1 < ε → a < b * ε
-- Polynomial.natDegree_pos_of_aeval_root
theorem natDegree_pos_of_aeval_root [Algebra R S] {p : R[X]} (hp : p ≠ 0) {z : S}
    (hz : aeval z p = 0) (inj : ∀ x : R, algebraMap R S x = 0 → x = 0) : 0 < p.natDegree 
Arguments: {R : Type u} {S : Type v} [CommRing R] [Semiring S] [Algebra R S] {p : R[X]} (hp : p ≠ 0) {z : S} (hz : (Polynomial.aeval z) p = 0) (inj : ∀ (x : R), (algebraMap R S) x = 0 → x = 0)
Goal: 0 < p.natDegree
-- Polynomial.degree_pos_of_aeval_root
theorem degree_pos_of_aeval_root [Algebra R S] {p : R[X]} (hp : p ≠ 0) {z : S} (hz : aeval z p = 0)
    (inj : ∀ x : R, algebraMap R S x = 0 → x = 0) : 0 < p.degree 
Arguments: {R : Type u} {S : Type v} [CommRing R] [Semiring S] [Algebra R S] {p : R[X]} (hp : p ≠ 0) {z : S} (hz : (Polynomial.aeval z) p = 0) (inj : ∀ (x : R), (algebraMap R S) x = 0 → x = 0)
Goal: 0 < p.degree
-- Polynomial.modByMonic_eq_of_dvd_sub
theorem modByMonic_eq_of_dvd_sub (hq : q.Monic) {p₁ p₂ : R[X]} (h : q ∣ p₁ - p₂) :
    p₁ %ₘ q = p₂ %ₘ q 
Arguments: {R : Type u} [CommRing R] {q : R[X]} (hq : q.Monic) {p₁ p₂ : R[X]} (h : q ∣ p₁ - p₂)
Goal: p₁ %ₘ q = p₂ %ₘ q
-- Polynomial.add_modByMonic
theorem add_modByMonic (p₁ p₂ : R[X]) : (p₁ + p₂) %ₘ q = p₁ %ₘ q + p₂ %ₘ q 
Arguments: {R : Type u} [CommRing R] {q : R[X]} (p₁ p₂ : R[X])
Goal: (p₁ + p₂) %ₘ q = p₁ %ₘ q + p₂ %ₘ q
-- Polynomial.smul_modByMonic
theorem smul_modByMonic (c : R) (p : R[X]) : c • p %ₘ q = c • (p %ₘ q) 
Arguments: {R : Type u} [CommRing R] {q : R[X]} (c : R) (p : R[X])
Goal: c • p %ₘ q = c • (p %ₘ q)
-- Polynomial.modByMonicHom_apply
@[simps]
def modByMonicHom (q : R[X]) : R[X] →ₗ[R] R[X] 
Arguments: {R : Type u} [CommRing R] (q p : R[X])
Goal: q.modByMonicHom p = p %ₘ q
-- Polynomial.neg_modByMonic
theorem neg_modByMonic (p mod : R[X]) : (-p) %ₘ mod = - (p %ₘ mod) 
Arguments: {R : Type u} [CommRing R] (p mod : R[X])
Goal: -p %ₘ mod = -(p %ₘ mod)
-- Polynomial.sub_modByMonic
theorem sub_modByMonic (a b mod : R[X]) : (a - b) %ₘ mod = a %ₘ mod - b %ₘ mod 
Arguments: {R : Type u} [CommRing R] (a b mod : R[X])
Goal: (a - b) %ₘ mod = a %ₘ mod - b %ₘ mod
-- Polynomial.aeval_modByMonic_eq_self_of_root
theorem aeval_modByMonic_eq_self_of_root [Algebra R S] {p q : R[X]} (hq : q.Monic) {x : S}
    (hx : aeval x q = 0) : aeval x (p %ₘ q) = aeval x p 
Arguments: {R : Type u} {S : Type v} [CommRing R] [Ring S] [Algebra R S] {p q : R[X]} (hq : q.Monic) {x : S} (hx : (Polynomial.aeval x) q = 0)
Goal: (Polynomial.aeval x) (p %ₘ q) = (Polynomial.aeval x) p
-- Polynomial.natDegree_mul
theorem natDegree_mul (hp : p ≠ 0) (hq : q ≠ 0) : (p*q).natDegree = p.natDegree + q.natDegree 
Arguments: {R : Type u} [Semiring R] [NoZeroDivisors R] {p q : R[X]} (hp : p ≠ 0) (hq : q ≠ 0)
Goal: (p * q).natDegree = p.natDegree + q.natDegree
-- Polynomial.trailingDegree_mul
theorem trailingDegree_mul : (p * q).trailingDegree = p.trailingDegree + q.trailingDegree 
Arguments: {R : Type u} [Semiring R] [NoZeroDivisors R] {p q : R[X]}
Goal: (p * q).trailingDegree = p.trailingDegree + q.trailingDegree