Lemmas about min and max in an ordered monoid.

Some lemmas about types that have an ordering and a binary operation, with no rules relating them.

theorem fn_min_mul_fn_max {α : Type u_1} {β : Type u_2} [LinearOrder α] [CommSemigroup β] (f : α → β) (a : α) (b : α) : f (a ⊓ b) * f (a ⊔ b) = f a * f b

theorem fn_min_add_fn_max {α : Type u_1} {β : Type u_2} [LinearOrder α] [AddCommSemigroup β] (f : α → β) (a : α) (b : α) : f (a ⊓ b) + f (a ⊔ b) = f a + f b

theorem fn_max_mul_fn_min {α : Type u_1} {β : Type u_2} [LinearOrder α] [CommSemigroup β] (f : α → β) (a : α) (b : α) : f (a ⊔ b) * f (a ⊓ b) = f a * f b

theorem fn_max_add_fn_min {α : Type u_1} {β : Type u_2} [LinearOrder α] [AddCommSemigroup β] (f : α → β) (a : α) (b : α) : f (a ⊔ b) + f (a ⊓ b) = f a + f b

@[simp]

theorem min_mul_max {α : Type u_1} [LinearOrder α] [CommSemigroup α] (a : α) (b : α) : (a ⊓ b) * (a ⊔ b) = a * b

@[simp]

theorem min_add_max {α : Type u_1} [LinearOrder α] [AddCommSemigroup α] (a : α) (b : α) : a ⊓ b + a ⊔ b = a + b

@[simp]

theorem max_mul_min {α : Type u_1} [LinearOrder α] [CommSemigroup α] (a : α) (b : α) : (a ⊔ b) * (a ⊓ b) = a * b

@[simp]

theorem max_add_min {α : Type u_1} [LinearOrder α] [AddCommSemigroup α] (a : α) (b : α) : a ⊔ b + a ⊓ b = a + b

theorem min_mul_mul_left {α : Type u_1} [LinearOrder α] [Mul α] [MulLeftMono α] (a : α) (b : α) (c : α) : a * b ⊓ a * c = a * (b ⊓ c)

theorem min_add_add_left {α : Type u_1} [LinearOrder α] [Add α] [AddLeftMono α] (a : α) (b : α) (c : α) : (a + b) ⊓ (a + c) = a + b ⊓ c

theorem max_mul_mul_left {α : Type u_1} [LinearOrder α] [Mul α] [MulLeftMono α] (a : α) (b : α) (c : α) : a * b ⊔ a * c = a * (b ⊔ c)

theorem max_add_add_left {α : Type u_1} [LinearOrder α] [Add α] [AddLeftMono α] (a : α) (b : α) (c : α) : (a + b) ⊔ (a + c) = a + b ⊔ c

theorem min_mul_mul_right {α : Type u_1} [LinearOrder α] [Mul α] [MulRightMono α] (a : α) (b : α) (c : α) : a * c ⊓ b * c = (a ⊓ b) * c

theorem min_add_add_right {α : Type u_1} [LinearOrder α] [Add α] [AddRightMono α] (a : α) (b : α) (c : α) : (a + c) ⊓ (b + c) = a ⊓ b + c

theorem max_mul_mul_right {α : Type u_1} [LinearOrder α] [Mul α] [MulRightMono α] (a : α) (b : α) (c : α) : a * c ⊔ b * c = (a ⊔ b) * c

theorem max_add_add_right {α : Type u_1} [LinearOrder α] [Add α] [AddRightMono α] (a : α) (b : α) (c : α) : (a + c) ⊔ (b + c) = a ⊔ b + c

theorem lt_or_lt_of_mul_lt_mul {α : Type u_1} [LinearOrder α] [Mul α] [MulLeftMono α] [MulRightMono α] {a₁ : α} {a₂ : α} {b₁ : α} {b₂ : α} : a₁ * b₁ < a₂ * b₂ → a₁ < a₂ ∨ b₁ < b₂

theorem lt_or_lt_of_add_lt_add {α : Type u_1} [LinearOrder α] [Add α] [AddLeftMono α] [AddRightMono α] {a₁ : α} {a₂ : α} {b₁ : α} {b₂ : α} : a₁ + b₁ < a₂ + b₂ → a₁ < a₂ ∨ b₁ < b₂

theorem le_or_lt_of_mul_le_mul {α : Type u_1} [LinearOrder α] [Mul α] [MulLeftMono α] [MulRightStrictMono α] {a₁ : α} {a₂ : α} {b₁ : α} {b₂ : α} : a₁ * b₁ ≤ a₂ * b₂ → a₁ ≤ a₂ ∨ b₁ < b₂

theorem le_or_lt_of_add_le_add {α : Type u_1} [LinearOrder α] [Add α] [AddLeftMono α] [AddRightStrictMono α] {a₁ : α} {a₂ : α} {b₁ : α} {b₂ : α} : a₁ + b₁ ≤ a₂ + b₂ → a₁ ≤ a₂ ∨ b₁ < b₂

theorem lt_or_le_of_mul_le_mul {α : Type u_1} [LinearOrder α] [Mul α] [MulLeftStrictMono α] [MulRightMono α] {a₁ : α} {a₂ : α} {b₁ : α} {b₂ : α} : a₁ * b₁ ≤ a₂ * b₂ → a₁ < a₂ ∨ b₁ ≤ b₂

theorem lt_or_le_of_add_le_add {α : Type u_1} [LinearOrder α] [Add α] [AddLeftStrictMono α] [AddRightMono α] {a₁ : α} {a₂ : α} {b₁ : α} {b₂ : α} : a₁ + b₁ ≤ a₂ + b₂ → a₁ < a₂ ∨ b₁ ≤ b₂

theorem le_or_le_of_mul_le_mul {α : Type u_1} [LinearOrder α] [Mul α] [MulLeftStrictMono α] [MulRightStrictMono α] {a₁ : α} {a₂ : α} {b₁ : α} {b₂ : α} : a₁ * b₁ ≤ a₂ * b₂ → a₁ ≤ a₂ ∨ b₁ ≤ b₂

theorem le_or_le_of_add_le_add {α : Type u_1} [LinearOrder α] [Add α] [AddLeftStrictMono α] [AddRightStrictMono α] {a₁ : α} {a₂ : α} {b₁ : α} {b₂ : α} : a₁ + b₁ ≤ a₂ + b₂ → a₁ ≤ a₂ ∨ b₁ ≤ b₂

theorem mul_lt_mul_iff_of_le_of_le {α : Type u_1} [LinearOrder α] [Mul α] [MulLeftMono α] [MulRightMono α] [MulLeftStrictMono α] [MulRightStrictMono α] {a₁ : α} {a₂ : α} {b₁ : α} {b₂ : α} (ha : a₁ ≤ a₂) (hb : b₁ ≤ b₂) : a₁ * b₁ < a₂ * b₂ ↔ a₁ < a₂ ∨ b₁ < b₂

theorem add_lt_add_iff_of_le_of_le {α : Type u_1} [LinearOrder α] [Add α] [AddLeftMono α] [AddRightMono α] [AddLeftStrictMono α] [AddRightStrictMono α] {a₁ : α} {a₂ : α} {b₁ : α} {b₂ : α} (ha : a₁ ≤ a₂) (hb : b₁ ≤ b₂) : a₁ + b₁ < a₂ + b₂ ↔ a₁ < a₂ ∨ b₁ < b₂

theorem min_le_mul_of_one_le_right {α : Type u_1} [LinearOrder α] [MulOneClass α] [MulLeftMono α] {a : α} {b : α} (hb : 1 ≤ b) : a ⊓ b ≤ a * b

theorem min_le_add_of_nonneg_right {α : Type u_1} [LinearOrder α] [AddZeroClass α] [AddLeftMono α] {a : α} {b : α} (hb : 0 ≤ b) : a ⊓ b ≤ a + b

theorem min_le_mul_of_one_le_left {α : Type u_1} [LinearOrder α] [MulOneClass α] [MulRightMono α] {a : α} {b : α} (ha : 1 ≤ a) : a ⊓ b ≤ a * b

theorem min_le_add_of_nonneg_left {α : Type u_1} [LinearOrder α] [AddZeroClass α] [AddRightMono α] {a : α} {b : α} (ha : 0 ≤ a) : a ⊓ b ≤ a + b

theorem max_le_mul_of_one_le {α : Type u_1} [LinearOrder α] [MulOneClass α] [MulLeftMono α] [MulRightMono α] {a : α} {b : α} (ha : 1 ≤ a) (hb : 1 ≤ b) : a ⊔ b ≤ a * b

theorem max_le_add_of_nonneg {α : Type u_1} [LinearOrder α] [AddZeroClass α] [AddLeftMono α] [AddRightMono α] {a : α} {b : α} (ha : 0 ≤ a) (hb : 0 ≤ b) : a ⊔ b ≤ a + b