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_add_fn_max {α : Type u_1} {β : Type u_2} [] [] (f : αβ) (a : α) (b : α) :
f (min a b) + f (max a b) = f a + f b
theorem fn_min_mul_fn_max {α : Type u_1} {β : Type u_2} [] [] (f : αβ) (a : α) (b : α) :
f (min a b) * f (max a b) = f a * f b
theorem fn_max_add_fn_min {α : Type u_1} {β : Type u_2} [] [] (f : αβ) (a : α) (b : α) :
f (max a b) + f (min a b) = f a + f b
theorem fn_max_mul_fn_min {α : Type u_1} {β : Type u_2} [] [] (f : αβ) (a : α) (b : α) :
f (max a b) * f (min a b) = f a * f b
@[simp]
theorem min_add_max {α : Type u_1} [] [] (a : α) (b : α) :
min a b + max a b = a + b
@[simp]
theorem min_mul_max {α : Type u_1} [] [] (a : α) (b : α) :
min a b * max a b = a * b
@[simp]
theorem max_add_min {α : Type u_1} [] [] (a : α) (b : α) :
max a b + min a b = a + b
@[simp]
theorem max_mul_min {α : Type u_1} [] [] (a : α) (b : α) :
max a b * min a b = a * b
theorem min_add_add_left {α : Type u_1} [] [Add α] [CovariantClass α α (fun (x x_1 : α) => x + x_1) fun (x x_1 : α) => x x_1] (a : α) (b : α) (c : α) :
min (a + b) (a + c) = a + min b c
theorem min_mul_mul_left {α : Type u_1} [] [Mul α] [CovariantClass α α (fun (x x_1 : α) => x * x_1) fun (x x_1 : α) => x x_1] (a : α) (b : α) (c : α) :
min (a * b) (a * c) = a * min b c
theorem max_add_add_left {α : Type u_1} [] [Add α] [CovariantClass α α (fun (x x_1 : α) => x + x_1) fun (x x_1 : α) => x x_1] (a : α) (b : α) (c : α) :
max (a + b) (a + c) = a + max b c
theorem max_mul_mul_left {α : Type u_1} [] [Mul α] [CovariantClass α α (fun (x x_1 : α) => x * x_1) fun (x x_1 : α) => x x_1] (a : α) (b : α) (c : α) :
max (a * b) (a * c) = a * max b c
theorem min_add_add_right {α : Type u_1} [] [Add α] [CovariantClass α α (Function.swap fun (x x_1 : α) => x + x_1) fun (x x_1 : α) => x x_1] (a : α) (b : α) (c : α) :
min (a + c) (b + c) = min a b + c
theorem min_mul_mul_right {α : Type u_1} [] [Mul α] [CovariantClass α α (Function.swap fun (x x_1 : α) => x * x_1) fun (x x_1 : α) => x x_1] (a : α) (b : α) (c : α) :
min (a * c) (b * c) = min a b * c
theorem max_add_add_right {α : Type u_1} [] [Add α] [CovariantClass α α (Function.swap fun (x x_1 : α) => x + x_1) fun (x x_1 : α) => x x_1] (a : α) (b : α) (c : α) :
max (a + c) (b + c) = max a b + c
theorem max_mul_mul_right {α : Type u_1} [] [Mul α] [CovariantClass α α (Function.swap fun (x x_1 : α) => x * x_1) fun (x x_1 : α) => x x_1] (a : α) (b : α) (c : α) :
max (a * c) (b * c) = max a b * c
theorem lt_or_lt_of_add_lt_add {α : Type u_1} [] [Add α] [CovariantClass α α (fun (x x_1 : α) => x + x_1) fun (x x_1 : α) => x x_1] [CovariantClass α α (Function.swap fun (x x_1 : α) => x + x_1) fun (x x_1 : α) => x x_1] {a₁ : α} {a₂ : α} {b₁ : α} {b₂ : α} :
a₁ + b₁ < a₂ + b₂a₁ < a₂ b₁ < b₂
theorem lt_or_lt_of_mul_lt_mul {α : Type u_1} [] [Mul α] [CovariantClass α α (fun (x x_1 : α) => x * x_1) fun (x x_1 : α) => x x_1] [CovariantClass α α (Function.swap fun (x x_1 : α) => x * x_1) fun (x x_1 : α) => x x_1] {a₁ : α} {a₂ : α} {b₁ : α} {b₂ : α} :
a₁ * b₁ < a₂ * b₂a₁ < a₂ b₁ < b₂
theorem le_or_lt_of_add_le_add {α : Type u_1} [] [Add α] [CovariantClass α α (fun (x x_1 : α) => x + x_1) fun (x x_1 : α) => x x_1] [CovariantClass α α (Function.swap fun (x x_1 : α) => x + x_1) fun (x x_1 : α) => x < x_1] {a₁ : α} {a₂ : α} {b₁ : α} {b₂ : α} :
a₁ + b₁ a₂ + b₂a₁ a₂ b₁ < b₂
theorem le_or_lt_of_mul_le_mul {α : Type u_1} [] [Mul α] [CovariantClass α α (fun (x x_1 : α) => x * x_1) fun (x x_1 : α) => x x_1] [CovariantClass α α (Function.swap fun (x x_1 : α) => x * x_1) fun (x x_1 : α) => x < x_1] {a₁ : α} {a₂ : α} {b₁ : α} {b₂ : α} :
a₁ * b₁ a₂ * b₂a₁ a₂ b₁ < b₂
theorem lt_or_le_of_add_le_add {α : Type u_1} [] [Add α] [CovariantClass α α (fun (x x_1 : α) => x + x_1) fun (x x_1 : α) => x < x_1] [CovariantClass α α (Function.swap fun (x x_1 : α) => x + x_1) fun (x x_1 : α) => x x_1] {a₁ : α} {a₂ : α} {b₁ : α} {b₂ : α} :
a₁ + b₁ a₂ + b₂a₁ < a₂ b₁ b₂
theorem lt_or_le_of_mul_le_mul {α : Type u_1} [] [Mul α] [CovariantClass α α (fun (x x_1 : α) => x * x_1) fun (x x_1 : α) => x < x_1] [CovariantClass α α (Function.swap fun (x x_1 : α) => x * x_1) fun (x x_1 : α) => x x_1] {a₁ : α} {a₂ : α} {b₁ : α} {b₂ : α} :
a₁ * b₁ a₂ * b₂a₁ < a₂ b₁ b₂
theorem le_or_le_of_add_le_add {α : Type u_1} [] [Add α] [CovariantClass α α (fun (x x_1 : α) => x + x_1) fun (x x_1 : α) => x < x_1] [CovariantClass α α (Function.swap fun (x x_1 : α) => x + x_1) fun (x x_1 : α) => x < x_1] {a₁ : α} {a₂ : α} {b₁ : α} {b₂ : α} :
a₁ + b₁ a₂ + b₂a₁ a₂ b₁ b₂
theorem le_or_le_of_mul_le_mul {α : Type u_1} [] [Mul α] [CovariantClass α α (fun (x x_1 : α) => x * x_1) fun (x x_1 : α) => x < x_1] [CovariantClass α α (Function.swap fun (x x_1 : α) => x * x_1) fun (x x_1 : α) => x < x_1] {a₁ : α} {a₂ : α} {b₁ : α} {b₂ : α} :
a₁ * b₁ a₂ * b₂a₁ a₂ b₁ b₂
theorem add_lt_add_iff_of_le_of_le {α : Type u_1} [] [Add α] [CovariantClass α α (fun (x x_1 : α) => x + x_1) fun (x x_1 : α) => x x_1] [CovariantClass α α (Function.swap fun (x x_1 : α) => x + x_1) fun (x x_1 : α) => x x_1] [CovariantClass α α (fun (x x_1 : α) => x + x_1) fun (x x_1 : α) => x < x_1] [CovariantClass α α (Function.swap fun (x x_1 : α) => x + x_1) fun (x x_1 : α) => x < x_1] {a₁ : α} {a₂ : α} {b₁ : α} {b₂ : α} (ha : a₁ a₂) (hb : b₁ b₂) :
a₁ + b₁ < a₂ + b₂ a₁ < a₂ b₁ < b₂
theorem mul_lt_mul_iff_of_le_of_le {α : Type u_1} [] [Mul α] [CovariantClass α α (fun (x x_1 : α) => x * x_1) fun (x x_1 : α) => x x_1] [CovariantClass α α (Function.swap fun (x x_1 : α) => x * x_1) fun (x x_1 : α) => x x_1] [CovariantClass α α (fun (x x_1 : α) => x * x_1) fun (x x_1 : α) => x < x_1] [CovariantClass α α (Function.swap fun (x x_1 : α) => x * x_1) fun (x x_1 : α) => x < x_1] {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_add_of_nonneg_right {α : Type u_1} [] [] [CovariantClass α α (fun (x x_1 : α) => x + x_1) fun (x x_1 : α) => x x_1] {a : α} {b : α} (hb : 0 b) :
min a b a + b
theorem min_le_mul_of_one_le_right {α : Type u_1} [] [] [CovariantClass α α (fun (x x_1 : α) => x * x_1) fun (x x_1 : α) => x x_1] {a : α} {b : α} (hb : 1 b) :
min a b a * b
theorem min_le_add_of_nonneg_left {α : Type u_1} [] [] [CovariantClass α α (Function.swap fun (x x_1 : α) => x + x_1) fun (x x_1 : α) => x x_1] {a : α} {b : α} (ha : 0 a) :
min a b a + b
theorem min_le_mul_of_one_le_left {α : Type u_1} [] [] [CovariantClass α α (Function.swap fun (x x_1 : α) => x * x_1) fun (x x_1 : α) => x x_1] {a : α} {b : α} (ha : 1 a) :
min a b a * b
theorem max_le_add_of_nonneg {α : Type u_1} [] [] [CovariantClass α α (fun (x x_1 : α) => x + x_1) fun (x x_1 : α) => x x_1] [CovariantClass α α (Function.swap fun (x x_1 : α) => x + x_1) fun (x x_1 : α) => x x_1] {a : α} {b : α} (ha : 0 a) (hb : 0 b) :
max a b a + b
theorem max_le_mul_of_one_le {α : Type u_1} [] [] [CovariantClass α α (fun (x x_1 : α) => x * x_1) fun (x x_1 : α) => x x_1] [CovariantClass α α (Function.swap fun (x x_1 : α) => x * x_1) fun (x x_1 : α) => x x_1] {a : α} {b : α} (ha : 1 a) (hb : 1 b) :
max a b a * b