Lemmas about subtraction in canonically ordered monoids

@[simp]

theorem add_tsub_cancel_of_le {α : Type u_1} [AddCommSemigroup α] [PartialOrder α] [ExistsAddOfLE α] [CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] [Sub α] [OrderedSub α] {a : α} {b : α} (h : a ≤ b) : a + (b - a) = b

theorem tsub_add_cancel_of_le {α : Type u_1} [AddCommSemigroup α] [PartialOrder α] [ExistsAddOfLE α] [CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] [Sub α] [OrderedSub α] {a : α} {b : α} (h : a ≤ b) : b - a + a = b

theorem add_le_of_le_tsub_right_of_le {α : Type u_1} [AddCommSemigroup α] [PartialOrder α] [ExistsAddOfLE α] [CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] [Sub α] [OrderedSub α] {a : α} {b : α} {c : α} (h : b ≤ c) (h2 : a ≤ c - b) : a + b ≤ c

theorem add_le_of_le_tsub_left_of_le {α : Type u_1} [AddCommSemigroup α] [PartialOrder α] [ExistsAddOfLE α] [CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] [Sub α] [OrderedSub α] {a : α} {b : α} {c : α} (h : a ≤ c) (h2 : b ≤ c - a) : a + b ≤ c

theorem tsub_le_tsub_iff_right {α : Type u_1} [AddCommSemigroup α] [PartialOrder α] [ExistsAddOfLE α] [CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] [Sub α] [OrderedSub α] {a : α} {b : α} {c : α} (h : c ≤ b) : a - c ≤ b - c ↔ a ≤ b

theorem tsub_left_inj {α : Type u_1} [AddCommSemigroup α] [PartialOrder α] [ExistsAddOfLE α] [CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] [Sub α] [OrderedSub α] {a : α} {b : α} {c : α} (h1 : c ≤ a) (h2 : c ≤ b) : a - c = b - c ↔ a = b

theorem tsub_inj_left {α : Type u_1} [AddCommSemigroup α] [PartialOrder α] [ExistsAddOfLE α] [CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] [Sub α] [OrderedSub α] {a : α} {b : α} {c : α} (h₁ : a ≤ b) (h₂ : a ≤ c) : b - a = c - a → b = c

theorem lt_of_tsub_lt_tsub_right_of_le {α : Type u_1} [AddCommSemigroup α] [PartialOrder α] [ExistsAddOfLE α] [CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] [Sub α] [OrderedSub α] {a : α} {b : α} {c : α} (h : c ≤ b) (h2 : a - c < b - c) : a < b

See lt_of_tsub_lt_tsub_right for a stronger statement in a linear order.

theorem tsub_add_tsub_cancel {α : Type u_1} [AddCommSemigroup α] [PartialOrder α] [ExistsAddOfLE α] [CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] [Sub α] [OrderedSub α] {a : α} {b : α} {c : α} (hab : b ≤ a) (hcb : c ≤ b) : a - b + (b - c) = a - c

theorem tsub_tsub_tsub_cancel_right {α : Type u_1} [AddCommSemigroup α] [PartialOrder α] [ExistsAddOfLE α] [CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] [Sub α] [OrderedSub α] {a : α} {b : α} {c : α} (h : c ≤ b) : a - c - (b - c) = a - b

Lemmas that assume that an element is AddLECancellable.

theorem AddLECancellable.eq_tsub_iff_add_eq_of_le {α : Type u_1} [AddCommSemigroup α] [PartialOrder α] [ExistsAddOfLE α] [CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] [Sub α] [OrderedSub α] {a : α} {b : α} {c : α} (hc : AddLECancellable c) (h : c ≤ b) : a = b - c ↔ a + c = b

theorem AddLECancellable.tsub_eq_iff_eq_add_of_le {α : Type u_1} [AddCommSemigroup α] [PartialOrder α] [ExistsAddOfLE α] [CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] [Sub α] [OrderedSub α] {a : α} {b : α} {c : α} (hb : AddLECancellable b) (h : b ≤ a) : a - b = c ↔ a = c + b

theorem AddLECancellable.add_tsub_assoc_of_le {α : Type u_1} [AddCommSemigroup α] [PartialOrder α] [ExistsAddOfLE α] [CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] [Sub α] [OrderedSub α] {b : α} {c : α} (hc : AddLECancellable c) (h : c ≤ b) (a : α) : a + b - c = a + (b - c)

theorem AddLECancellable.tsub_add_eq_add_tsub {α : Type u_1} [AddCommSemigroup α] [PartialOrder α] [ExistsAddOfLE α] [CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] [Sub α] [OrderedSub α] {a : α} {b : α} {c : α} (hb : AddLECancellable b) (h : b ≤ a) : a - b + c = a + c - b

theorem AddLECancellable.tsub_tsub_assoc {α : Type u_1} [AddCommSemigroup α] [PartialOrder α] [ExistsAddOfLE α] [CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] [Sub α] [OrderedSub α] {a : α} {b : α} {c : α} (hbc : AddLECancellable (b - c)) (h₁ : b ≤ a) (h₂ : c ≤ b) : a - (b - c) = a - b + c

theorem AddLECancellable.tsub_add_tsub_comm {α : Type u_1} [AddCommSemigroup α] [PartialOrder α] [ExistsAddOfLE α] [CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] [Sub α] [OrderedSub α] {a : α} {b : α} {c : α} {d : α} (hb : AddLECancellable b) (hd : AddLECancellable d) (hba : b ≤ a) (hdc : d ≤ c) : a - b + (c - d) = a + c - (b + d)

theorem AddLECancellable.le_tsub_iff_left {α : Type u_1} [AddCommSemigroup α] [PartialOrder α] [ExistsAddOfLE α] [CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] [Sub α] [OrderedSub α] {a : α} {b : α} {c : α} (ha : AddLECancellable a) (h : a ≤ c) : b ≤ c - a ↔ a + b ≤ c

theorem AddLECancellable.le_tsub_iff_right {α : Type u_1} [AddCommSemigroup α] [PartialOrder α] [ExistsAddOfLE α] [CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] [Sub α] [OrderedSub α] {a : α} {b : α} {c : α} (ha : AddLECancellable a) (h : a ≤ c) : b ≤ c - a ↔ b + a ≤ c

theorem AddLECancellable.tsub_lt_iff_left {α : Type u_1} [AddCommSemigroup α] [PartialOrder α] [ExistsAddOfLE α] [CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] [Sub α] [OrderedSub α] {a : α} {b : α} {c : α} (hb : AddLECancellable b) (hba : b ≤ a) : a - b < c ↔ a < b + c

theorem AddLECancellable.tsub_lt_iff_right {α : Type u_1} [AddCommSemigroup α] [PartialOrder α] [ExistsAddOfLE α] [CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] [Sub α] [OrderedSub α] {a : α} {b : α} {c : α} (hb : AddLECancellable b) (hba : b ≤ a) : a - b < c ↔ a < c + b

theorem AddLECancellable.tsub_lt_iff_tsub_lt {α : Type u_1} [AddCommSemigroup α] [PartialOrder α] [ExistsAddOfLE α] [CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] [Sub α] [OrderedSub α] {a : α} {b : α} {c : α} (hb : AddLECancellable b) (hc : AddLECancellable c) (h₁ : b ≤ a) (h₂ : c ≤ a) : a - b < c ↔ a - c < b

theorem AddLECancellable.le_tsub_iff_le_tsub {α : Type u_1} [AddCommSemigroup α] [PartialOrder α] [ExistsAddOfLE α] [CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] [Sub α] [OrderedSub α] {a : α} {b : α} {c : α} (ha : AddLECancellable a) (hc : AddLECancellable c) (h₁ : a ≤ b) (h₂ : c ≤ b) : a ≤ b - c ↔ c ≤ b - a

theorem AddLECancellable.lt_tsub_iff_right_of_le {α : Type u_1} [AddCommSemigroup α] [PartialOrder α] [ExistsAddOfLE α] [CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] [Sub α] [OrderedSub α] {a : α} {b : α} {c : α} (hc : AddLECancellable c) (h : c ≤ b) : a < b - c ↔ a + c < b

theorem AddLECancellable.lt_tsub_iff_left_of_le {α : Type u_1} [AddCommSemigroup α] [PartialOrder α] [ExistsAddOfLE α] [CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] [Sub α] [OrderedSub α] {a : α} {b : α} {c : α} (hc : AddLECancellable c) (h : c ≤ b) : a < b - c ↔ c + a < b

theorem AddLECancellable.tsub_inj_right {α : Type u_1} [AddCommSemigroup α] [PartialOrder α] [ExistsAddOfLE α] [CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] [Sub α] [OrderedSub α] {a : α} {b : α} {c : α} (hab : AddLECancellable (a - b)) (h₁ : b ≤ a) (h₂ : c ≤ a) (h₃ : a - b = a - c) : b = c

theorem AddLECancellable.lt_of_tsub_lt_tsub_left_of_le {α : Type u_1} [AddCommSemigroup α] [PartialOrder α] [ExistsAddOfLE α] [CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] [Sub α] [OrderedSub α] {a : α} {b : α} {c : α} [ContravariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x < x_1] (hb : AddLECancellable b) (hca : c ≤ a) (h : a - b < a - c) : c < b

theorem AddLECancellable.tsub_lt_tsub_left_of_le {α : Type u_1} [AddCommSemigroup α] [PartialOrder α] [ExistsAddOfLE α] [CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] [Sub α] [OrderedSub α] {a : α} {b : α} {c : α} (hab : AddLECancellable (a - b)) (h₁ : b ≤ a) (h : c < b) : a - b < a - c

theorem AddLECancellable.tsub_lt_tsub_right_of_le {α : Type u_1} [AddCommSemigroup α] [PartialOrder α] [ExistsAddOfLE α] [CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] [Sub α] [OrderedSub α] {a : α} {b : α} {c : α} (hc : AddLECancellable c) (h : c ≤ a) (h2 : a < b) : a - c < b - c

theorem AddLECancellable.tsub_lt_tsub_iff_left_of_le_of_le {α : Type u_1} [AddCommSemigroup α] [PartialOrder α] [ExistsAddOfLE α] [CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] [Sub α] [OrderedSub α] {a : α} {b : α} {c : α} [ContravariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x < x_1] (hb : AddLECancellable b) (hab : AddLECancellable (a - b)) (h₁ : b ≤ a) (h₂ : c ≤ a) : a - b < a - c ↔ c < b

@[simp]

theorem AddLECancellable.add_tsub_tsub_cancel {α : Type u_1} [AddCommSemigroup α] [PartialOrder α] [ExistsAddOfLE α] [CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] [Sub α] [OrderedSub α] {a : α} {b : α} {c : α} (hac : AddLECancellable (a - c)) (h : c ≤ a) : a + b - (a - c) = b + c

theorem AddLECancellable.tsub_tsub_cancel_of_le {α : Type u_1} [AddCommSemigroup α] [PartialOrder α] [ExistsAddOfLE α] [CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] [Sub α] [OrderedSub α] {a : α} {b : α} (hba : AddLECancellable (b - a)) (h : a ≤ b) : b - (b - a) = a

theorem AddLECancellable.tsub_tsub_tsub_cancel_left {α : Type u_1} [AddCommSemigroup α] [PartialOrder α] [ExistsAddOfLE α] [CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] [Sub α] [OrderedSub α] {a : α} {b : α} {c : α} (hab : AddLECancellable (a - b)) (h : b ≤ a) : a - c - (a - b) = b - c

Lemmas where addition is order-reflecting.

theorem eq_tsub_iff_add_eq_of_le {α : Type u_1} [AddCommSemigroup α] [PartialOrder α] [ExistsAddOfLE α] [CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] [Sub α] [OrderedSub α] {a : α} {b : α} {c : α} [ContravariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] (h : c ≤ b) : a = b - c ↔ a + c = b

theorem tsub_eq_iff_eq_add_of_le {α : Type u_1} [AddCommSemigroup α] [PartialOrder α] [ExistsAddOfLE α] [CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] [Sub α] [OrderedSub α] {a : α} {b : α} {c : α} [ContravariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] (h : b ≤ a) : a - b = c ↔ a = c + b

theorem add_tsub_assoc_of_le {α : Type u_1} [AddCommSemigroup α] [PartialOrder α] [ExistsAddOfLE α] [CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] [Sub α] [OrderedSub α] {b : α} {c : α} [ContravariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] (h : c ≤ b) (a : α) : a + b - c = a + (b - c)

See add_tsub_le_assoc for an inequality.

theorem tsub_add_eq_add_tsub {α : Type u_1} [AddCommSemigroup α] [PartialOrder α] [ExistsAddOfLE α] [CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] [Sub α] [OrderedSub α] {a : α} {b : α} {c : α} [ContravariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] (h : b ≤ a) : a - b + c = a + c - b

theorem tsub_tsub_assoc {α : Type u_1} [AddCommSemigroup α] [PartialOrder α] [ExistsAddOfLE α] [CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] [Sub α] [OrderedSub α] {a : α} {b : α} {c : α} [ContravariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] (h₁ : b ≤ a) (h₂ : c ≤ b) : a - (b - c) = a - b + c

theorem tsub_add_tsub_comm {α : Type u_1} [AddCommSemigroup α] [PartialOrder α] [ExistsAddOfLE α] [CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] [Sub α] [OrderedSub α] {a : α} {b : α} {c : α} {d : α} [ContravariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] (hba : b ≤ a) (hdc : d ≤ c) : a - b + (c - d) = a + c - (b + d)

theorem le_tsub_iff_left {α : Type u_1} [AddCommSemigroup α] [PartialOrder α] [ExistsAddOfLE α] [CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] [Sub α] [OrderedSub α] {a : α} {b : α} {c : α} [ContravariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] (h : a ≤ c) : b ≤ c - a ↔ a + b ≤ c

theorem le_tsub_iff_right {α : Type u_1} [AddCommSemigroup α] [PartialOrder α] [ExistsAddOfLE α] [CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] [Sub α] [OrderedSub α] {a : α} {b : α} {c : α} [ContravariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] (h : a ≤ c) : b ≤ c - a ↔ b + a ≤ c

theorem tsub_lt_iff_left {α : Type u_1} [AddCommSemigroup α] [PartialOrder α] [ExistsAddOfLE α] [CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] [Sub α] [OrderedSub α] {a : α} {b : α} {c : α} [ContravariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] (hbc : b ≤ a) : a - b < c ↔ a < b + c

theorem tsub_lt_iff_right {α : Type u_1} [AddCommSemigroup α] [PartialOrder α] [ExistsAddOfLE α] [CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] [Sub α] [OrderedSub α] {a : α} {b : α} {c : α} [ContravariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] (hbc : b ≤ a) : a - b < c ↔ a < c + b

theorem tsub_lt_iff_tsub_lt {α : Type u_1} [AddCommSemigroup α] [PartialOrder α] [ExistsAddOfLE α] [CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] [Sub α] [OrderedSub α] {a : α} {b : α} {c : α} [ContravariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] (h₁ : b ≤ a) (h₂ : c ≤ a) : a - b < c ↔ a - c < b

theorem le_tsub_iff_le_tsub {α : Type u_1} [AddCommSemigroup α] [PartialOrder α] [ExistsAddOfLE α] [CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] [Sub α] [OrderedSub α] {a : α} {b : α} {c : α} [ContravariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] (h₁ : a ≤ b) (h₂ : c ≤ b) : a ≤ b - c ↔ c ≤ b - a

theorem lt_tsub_iff_right_of_le {α : Type u_1} [AddCommSemigroup α] [PartialOrder α] [ExistsAddOfLE α] [CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] [Sub α] [OrderedSub α] {a : α} {b : α} {c : α} [ContravariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] (h : c ≤ b) : a < b - c ↔ a + c < b

See lt_tsub_iff_right for a stronger statement in a linear order.

theorem lt_tsub_iff_left_of_le {α : Type u_1} [AddCommSemigroup α] [PartialOrder α] [ExistsAddOfLE α] [CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] [Sub α] [OrderedSub α] {a : α} {b : α} {c : α} [ContravariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] (h : c ≤ b) : a < b - c ↔ c + a < b

See lt_tsub_iff_left for a stronger statement in a linear order.

theorem lt_of_tsub_lt_tsub_left_of_le {α : Type u_1} [AddCommSemigroup α] [PartialOrder α] [ExistsAddOfLE α] [CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] [Sub α] [OrderedSub α] {a : α} {b : α} {c : α} [ContravariantClass 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] (hca : c ≤ a) (h : a - b < a - c) : c < b

See lt_of_tsub_lt_tsub_left for a stronger statement in a linear order.

theorem tsub_lt_tsub_left_of_le {α : Type u_1} [AddCommSemigroup α] [PartialOrder α] [ExistsAddOfLE α] [CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] [Sub α] [OrderedSub α] {a : α} {b : α} {c : α} [ContravariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] : b ≤ a → c < b → a - b < a - c

theorem tsub_lt_tsub_right_of_le {α : Type u_1} [AddCommSemigroup α] [PartialOrder α] [ExistsAddOfLE α] [CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] [Sub α] [OrderedSub α] {a : α} {b : α} {c : α} [ContravariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] (h : c ≤ a) (h2 : a < b) : a - c < b - c

theorem tsub_inj_right {α : Type u_1} [AddCommSemigroup α] [PartialOrder α] [ExistsAddOfLE α] [CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] [Sub α] [OrderedSub α] {a : α} {b : α} {c : α} [ContravariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] (h₁ : b ≤ a) (h₂ : c ≤ a) (h₃ : a - b = a - c) : b = c

theorem tsub_lt_tsub_iff_left_of_le_of_le {α : Type u_1} [AddCommSemigroup α] [PartialOrder α] [ExistsAddOfLE α] [CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] [Sub α] [OrderedSub α] {a : α} {b : α} {c : α} [ContravariantClass 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] (h₁ : b ≤ a) (h₂ : c ≤ a) : a - b < a - c ↔ c < b

See tsub_lt_tsub_iff_left_of_le for a stronger statement in a linear order.

@[simp]

theorem add_tsub_tsub_cancel {α : Type u_1} [AddCommSemigroup α] [PartialOrder α] [ExistsAddOfLE α] [CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] [Sub α] [OrderedSub α] {a : α} {b : α} {c : α} [ContravariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] (h : c ≤ a) : a + b - (a - c) = b + c

theorem tsub_tsub_cancel_of_le {α : Type u_1} [AddCommSemigroup α] [PartialOrder α] [ExistsAddOfLE α] [CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] [Sub α] [OrderedSub α] {a : α} {b : α} [ContravariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] (h : a ≤ b) : b - (b - a) = a

See tsub_tsub_le for an inequality.

theorem tsub_tsub_tsub_cancel_left {α : Type u_1} [AddCommSemigroup α] [PartialOrder α] [ExistsAddOfLE α] [CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] [Sub α] [OrderedSub α] {a : α} {b : α} {c : α} [ContravariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] (h : b ≤ a) : a - c - (a - b) = b - c

Lemmas in a canonically ordered monoid.

theorem add_tsub_cancel_iff_le {α : Type u_1} [CanonicallyOrderedAddMonoid α] [Sub α] [OrderedSub α] {a : α} {b : α} : a + (b - a) = b ↔ a ≤ b

theorem tsub_add_cancel_iff_le {α : Type u_1} [CanonicallyOrderedAddMonoid α] [Sub α] [OrderedSub α] {a : α} {b : α} : b - a + a = b ↔ a ≤ b

@[simp]

theorem tsub_eq_zero_iff_le {α : Type u_1} [CanonicallyOrderedAddMonoid α] [Sub α] [OrderedSub α] {a : α} {b : α} : a - b = 0 ↔ a ≤ b

@[simp]

theorem tsub_eq_zero_of_le {α : Type u_1} [CanonicallyOrderedAddMonoid α] [Sub α] [OrderedSub α] {a : α} {b : α} : a ≤ b → a - b = 0

Alias of the reverse direction of tsub_eq_zero_iff_le.

theorem tsub_self {α : Type u_1} [CanonicallyOrderedAddMonoid α] [Sub α] [OrderedSub α] (a : α) : a - a = 0

theorem tsub_le_self {α : Type u_1} [CanonicallyOrderedAddMonoid α] [Sub α] [OrderedSub α] {a : α} {b : α} : a - b ≤ a

theorem zero_tsub {α : Type u_1} [CanonicallyOrderedAddMonoid α] [Sub α] [OrderedSub α] (a : α) : 0 - a = 0

theorem tsub_self_add {α : Type u_1} [CanonicallyOrderedAddMonoid α] [Sub α] [OrderedSub α] (a : α) (b : α) : a - (a + b) = 0

theorem tsub_pos_iff_not_le {α : Type u_1} [CanonicallyOrderedAddMonoid α] [Sub α] [OrderedSub α] {a : α} {b : α} : 0 < a - b ↔ ¬a ≤ b

theorem tsub_pos_of_lt {α : Type u_1} [CanonicallyOrderedAddMonoid α] [Sub α] [OrderedSub α] {a : α} {b : α} (h : a < b) : 0 < b - a

theorem tsub_lt_of_lt {α : Type u_1} [CanonicallyOrderedAddMonoid α] [Sub α] [OrderedSub α] {a : α} {b : α} {c : α} (h : a < b) : a - c < b

theorem AddLECancellable.tsub_le_tsub_iff_left {α : Type u_1} [CanonicallyOrderedAddMonoid α] [Sub α] [OrderedSub α] {a : α} {b : α} {c : α} (ha : AddLECancellable a) (hc : AddLECancellable c) (h : c ≤ a) : a - b ≤ a - c ↔ c ≤ b

theorem AddLECancellable.tsub_right_inj {α : Type u_1} [CanonicallyOrderedAddMonoid α] [Sub α] [OrderedSub α] {a : α} {b : α} {c : α} (ha : AddLECancellable a) (hb : AddLECancellable b) (hc : AddLECancellable c) (hba : b ≤ a) (hca : c ≤ a) : a - b = a - c ↔ b = c

Lemmas where addition is order-reflecting.

theorem tsub_le_tsub_iff_left {α : Type u_1} [CanonicallyOrderedAddMonoid α] [Sub α] [OrderedSub α] {a : α} {b : α} {c : α} [ContravariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] (h : c ≤ a) : a - b ≤ a - c ↔ c ≤ b

theorem tsub_right_inj {α : Type u_1} [CanonicallyOrderedAddMonoid α] [Sub α] [OrderedSub α] {a : α} {b : α} {c : α} [ContravariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] (hba : b ≤ a) (hca : c ≤ a) : a - b = a - c ↔ b = c

@[reducible]

def CanonicallyOrderedAddMonoid.toAddCancelCommMonoid (α : Type u_1) [CanonicallyOrderedAddMonoid α] [Sub α] [OrderedSub α] [ContravariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] : AddCancelCommMonoid α

A CanonicallyOrderedAddMonoid with ordered subtraction and order-reflecting addition is cancellative. This is not an instance as it would form a typeclass loop.

See note [reducible non-instances].

Equations

• CanonicallyOrderedAddMonoid.toAddCancelCommMonoid α = let src := inferInstance; AddCancelCommMonoid.mk (_ : ∀ (a b : α), a + b = b + a)

Lemmas in a linearly canonically ordered monoid.

@[simp]

theorem tsub_pos_iff_lt {α : Type u_1} [CanonicallyLinearOrderedAddMonoid α] [Sub α] [OrderedSub α] {a : α} {b : α} : 0 < a - b ↔ b < a

theorem tsub_eq_tsub_min {α : Type u_1} [CanonicallyLinearOrderedAddMonoid α] [Sub α] [OrderedSub α] (a : α) (b : α) : a - b = a - min a b

theorem AddLECancellable.lt_tsub_iff_right {α : Type u_1} [CanonicallyLinearOrderedAddMonoid α] [Sub α] [OrderedSub α] {a : α} {b : α} {c : α} (hc : AddLECancellable c) : a < b - c ↔ a + c < b

theorem AddLECancellable.lt_tsub_iff_left {α : Type u_1} [CanonicallyLinearOrderedAddMonoid α] [Sub α] [OrderedSub α] {a : α} {b : α} {c : α} (hc : AddLECancellable c) : a < b - c ↔ c + a < b

theorem AddLECancellable.tsub_lt_tsub_iff_right {α : Type u_1} [CanonicallyLinearOrderedAddMonoid α] [Sub α] [OrderedSub α] {a : α} {b : α} {c : α} (hc : AddLECancellable c) (h : c ≤ a) : a - c < b - c ↔ a < b

theorem AddLECancellable.tsub_lt_self {α : Type u_1} [CanonicallyLinearOrderedAddMonoid α] [Sub α] [OrderedSub α] {a : α} {b : α} (ha : AddLECancellable a) (h₁ : 0 < a) (h₂ : 0 < b) : a - b < a

theorem AddLECancellable.tsub_lt_self_iff {α : Type u_1} [CanonicallyLinearOrderedAddMonoid α] [Sub α] [OrderedSub α] {a : α} {b : α} (ha : AddLECancellable a) : a - b < a ↔ 0 < a ∧ 0 < b

theorem AddLECancellable.tsub_lt_tsub_iff_left_of_le {α : Type u_1} [CanonicallyLinearOrderedAddMonoid α] [Sub α] [OrderedSub α] {a : α} {b : α} {c : α} (ha : AddLECancellable a) (hb : AddLECancellable b) (h : b ≤ a) : a - b < a - c ↔ c < b

See lt_tsub_iff_left_of_le_of_le for a weaker statement in a partial order.

theorem tsub_lt_tsub_iff_right {α : Type u_1} [CanonicallyLinearOrderedAddMonoid α] [Sub α] [OrderedSub α] {a : α} {b : α} {c : α} [ContravariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] (h : c ≤ a) : a - c < b - c ↔ a < b

This lemma also holds for ENNReal, but we need a different proof for that.

theorem tsub_lt_self {α : Type u_1} [CanonicallyLinearOrderedAddMonoid α] [Sub α] [OrderedSub α] {a : α} {b : α} [ContravariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] : 0 < a → 0 < b → a - b < a

theorem tsub_lt_self_iff {α : Type u_1} [CanonicallyLinearOrderedAddMonoid α] [Sub α] [OrderedSub α] {a : α} {b : α} [ContravariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] : a - b < a ↔ 0 < a ∧ 0 < b

theorem tsub_lt_tsub_iff_left_of_le {α : Type u_1} [CanonicallyLinearOrderedAddMonoid α] [Sub α] [OrderedSub α] {a : α} {b : α} {c : α} [ContravariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] (h : b ≤ a) : a - b < a - c ↔ c < b

See lt_tsub_iff_left_of_le_of_le for a weaker statement in a partial order.

Lemmas about max and min.

theorem tsub_add_eq_max {α : Type u_1} [CanonicallyLinearOrderedAddMonoid α] [Sub α] [OrderedSub α] {a : α} {b : α} : a - b + b = max a b

theorem add_tsub_eq_max {α : Type u_1} [CanonicallyLinearOrderedAddMonoid α] [Sub α] [OrderedSub α] {a : α} {b : α} : a + (b - a) = max a b

theorem tsub_min {α : Type u_1} [CanonicallyLinearOrderedAddMonoid α] [Sub α] [OrderedSub α] {a : α} {b : α} : a - min a b = a - b

theorem tsub_add_min {α : Type u_1} [CanonicallyLinearOrderedAddMonoid α] [Sub α] [OrderedSub α] {a : α} {b : α} : a - b + min a b = a