File: Mathlib/Algebra/Order/Monoid/Unbundled/MinMax.lean

full_name: fn_min_add_fn_max
code: @[to_additive]
lemma fn_min_mul_fn_max  (f : α → β) (a b : α) : f (min a b) * f (max a b) = f a * f b 
goal: f (min a b) + f (max a b) = f a + f b

full_name: fn_min_mul_fn_max
code: @[to_additive]
lemma fn_min_mul_fn_max  (f : α → β) (a b : α) : f (min a b) * f (max a b) = f a * f b 
goal: f (min a b) * f (max a b) = f a * f b

full_name: fn_max_mul_fn_min
code: @[to_additive]
lemma fn_max_mul_fn_min (f : α → β) (a b : α) : f (max a b) * f (min a b) = f a * f b 
goal: f (max a b) * f (min a b) = f a * f b

full_name: fn_max_add_fn_min
code: @[to_additive]
lemma fn_max_mul_fn_min (f : α → β) (a b : α) : f (max a b) * f (min a b) = f a * f b 
goal: f (max a b) + f (min a b) = f a + f b

full_name: min_add_max
code: @[to_additive (attr := simp)]
lemma min_mul_max (a b : α) : min a b * max a b = a * b 
goal: min a b + max a b = a + b

full_name: min_mul_max
code: @[to_additive (attr := simp)]
lemma min_mul_max (a b : α) : min a b * max a b = a * b 
goal: min a b * max a b = a * b

full_name: max_add_min
code: @[to_additive (attr := simp)]
lemma max_mul_min (a b : α) : max a b * min a b = a * b 
goal: max a b + min a b = a + b

full_name: max_mul_min
code: @[to_additive (attr := simp)]
lemma max_mul_min (a b : α) : max a b * min a b = a * b 
goal: max a b * min a b = a * b

full_name: min_mul_mul_left
code: @[to_additive]
theorem min_mul_mul_left (a b c : α) : min (a * b) (a * c) = a * min b c 
goal: min (a * b) (a * c) = a * min b c

full_name: min_add_add_left
code: @[to_additive]
theorem min_mul_mul_left (a b c : α) : min (a * b) (a * c) = a * min b c 
goal: min (a + b) (a + c) = a + min b c

full_name: max_mul_mul_left
code: @[to_additive]
theorem max_mul_mul_left (a b c : α) : max (a * b) (a * c) = a * max b c 
goal: max (a * b) (a * c) = a * max b c

full_name: max_add_add_left
code: @[to_additive]
theorem max_mul_mul_left (a b c : α) : max (a * b) (a * c) = a * max b c 
goal: max (a + b) (a + c) = a + max b c

full_name: min_add_add_right
code: @[to_additive]
theorem min_mul_mul_right (a b c : α) : min (a * c) (b * c) = min a b * c 
goal: min (a + c) (b + c) = min a b + c

full_name: min_mul_mul_right
code: @[to_additive]
theorem min_mul_mul_right (a b c : α) : min (a * c) (b * c) = min a b * c 
goal: min (a * c) (b * c) = min a b * c

full_name: max_mul_mul_right
code: @[to_additive]
theorem max_mul_mul_right (a b c : α) : max (a * c) (b * c) = max a b * c 
goal: max (a * c) (b * c) = max a b * c

full_name: max_add_add_right
code: @[to_additive]
theorem max_mul_mul_right (a b c : α) : max (a * c) (b * c) = max a b * c 
goal: max (a + c) (b + c) = max a b + c

full_name: lt_or_lt_of_add_lt_add
code: @[to_additive]
theorem lt_or_lt_of_mul_lt_mul [CovariantClass α α (· * ·) (· ≤ ·)]
    [CovariantClass α α (Function.swap (· * ·)) (· ≤ ·)] {a₁ a₂ b₁ b₂ : α} :
    a₁ * b₁ < a₂ * b₂ → a₁ < a₂ ∨ b₁ < b₂ 
goal: a₁ + b₁ < a₂ + b₂ → a₁ < a₂ ∨ b₁ < b₂

full_name: lt_or_lt_of_mul_lt_mul
code: @[to_additive]
theorem lt_or_lt_of_mul_lt_mul [CovariantClass α α (· * ·) (· ≤ ·)]
    [CovariantClass α α (Function.swap (· * ·)) (· ≤ ·)] {a₁ a₂ b₁ b₂ : α} :
    a₁ * b₁ < a₂ * b₂ → a₁ < a₂ ∨ b₁ < b₂ 
goal: a₁ * b₁ < a₂ * b₂ → a₁ < a₂ ∨ b₁ < b₂

full_name: le_or_lt_of_add_le_add
code: @[to_additive]
theorem le_or_lt_of_mul_le_mul [CovariantClass α α (· * ·) (· ≤ ·)]
    [CovariantClass α α (Function.swap (· * ·)) (· < ·)] {a₁ a₂ b₁ b₂ : α} :
    a₁ * b₁ ≤ a₂ * b₂ → a₁ ≤ a₂ ∨ b₁ < b₂ 
goal: a₁ + b₁ ≤ a₂ + b₂ → a₁ ≤ a₂ ∨ b₁ < b₂

full_name: le_or_lt_of_mul_le_mul
code: @[to_additive]
theorem le_or_lt_of_mul_le_mul [CovariantClass α α (· * ·) (· ≤ ·)]
    [CovariantClass α α (Function.swap (· * ·)) (· < ·)] {a₁ a₂ b₁ b₂ : α} :
    a₁ * b₁ ≤ a₂ * b₂ → a₁ ≤ a₂ ∨ b₁ < b₂ 
goal: a₁ * b₁ ≤ a₂ * b₂ → a₁ ≤ a₂ ∨ b₁ < b₂

full_name: lt_or_le_of_mul_le_mul
code: @[to_additive]
theorem lt_or_le_of_mul_le_mul [CovariantClass α α (· * ·) (· < ·)]
    [CovariantClass α α (Function.swap (· * ·)) (· ≤ ·)] {a₁ a₂ b₁ b₂ : α} :
    a₁ * b₁ ≤ a₂ * b₂ → a₁ < a₂ ∨ b₁ ≤ b₂ 
goal: a₁ * b₁ ≤ a₂ * b₂ → a₁ < a₂ ∨ b₁ ≤ b₂

full_name: lt_or_le_of_add_le_add
code: @[to_additive]
theorem lt_or_le_of_mul_le_mul [CovariantClass α α (· * ·) (· < ·)]
    [CovariantClass α α (Function.swap (· * ·)) (· ≤ ·)] {a₁ a₂ b₁ b₂ : α} :
    a₁ * b₁ ≤ a₂ * b₂ → a₁ < a₂ ∨ b₁ ≤ b₂ 
goal: a₁ + b₁ ≤ a₂ + b₂ → a₁ < a₂ ∨ b₁ ≤ b₂

full_name: le_or_le_of_mul_le_mul
code: @[to_additive]
theorem le_or_le_of_mul_le_mul [CovariantClass α α (· * ·) (· < ·)]
    [CovariantClass α α (Function.swap (· * ·)) (· < ·)] {a₁ a₂ b₁ b₂ : α} :
    a₁ * b₁ ≤ a₂ * b₂ → a₁ ≤ a₂ ∨ b₁ ≤ b₂ 
goal: a₁ * b₁ ≤ a₂ * b₂ → a₁ ≤ a₂ ∨ b₁ ≤ b₂

full_name: le_or_le_of_add_le_add
code: @[to_additive]
theorem le_or_le_of_mul_le_mul [CovariantClass α α (· * ·) (· < ·)]
    [CovariantClass α α (Function.swap (· * ·)) (· < ·)] {a₁ a₂ b₁ b₂ : α} :
    a₁ * b₁ ≤ a₂ * b₂ → a₁ ≤ a₂ ∨ b₁ ≤ b₂ 
goal: a₁ + b₁ ≤ a₂ + b₂ → a₁ ≤ a₂ ∨ b₁ ≤ b₂

full_name: mul_lt_mul_iff_of_le_of_le
code: @[to_additive]
theorem mul_lt_mul_iff_of_le_of_le [CovariantClass α α (· * ·) (· ≤ ·)]
    [CovariantClass α α (Function.swap (· * ·)) (· ≤ ·)] [CovariantClass α α (· * ·) (· < ·)]
    [CovariantClass α α (Function.swap (· * ·)) (· < ·)] {a₁ a₂ b₁ b₂ : α} (ha : a₁ ≤ a₂)
    (hb : b₁ ≤ b₂) : a₁ * b₁ < a₂ * b₂ ↔ a₁ < a₂ ∨ b₁ < b₂ 
goal: a₁ * b₁ < a₂ * b₂ ↔ a₁ < a₂ ∨ b₁ < b₂

full_name: add_lt_add_iff_of_le_of_le
code: @[to_additive]
theorem mul_lt_mul_iff_of_le_of_le [CovariantClass α α (· * ·) (· ≤ ·)]
    [CovariantClass α α (Function.swap (· * ·)) (· ≤ ·)] [CovariantClass α α (· * ·) (· < ·)]
    [CovariantClass α α (Function.swap (· * ·)) (· < ·)] {a₁ a₂ b₁ b₂ : α} (ha : a₁ ≤ a₂)
    (hb : b₁ ≤ b₂) : a₁ * b₁ < a₂ * b₂ ↔ a₁ < a₂ ∨ b₁ < b₂ 
goal: a₁ + b₁ < a₂ + b₂ ↔ a₁ < a₂ ∨ b₁ < b₂

full_name: min_le_mul_of_one_le_right
code: @[to_additive]
theorem min_le_mul_of_one_le_right [CovariantClass α α (· * ·) (· ≤ ·)] {a b : α} (hb : 1 ≤ b) :
    min a b ≤ a * b 
goal: min a b ≤ a * b

full_name: min_le_add_of_nonneg_right
code: @[to_additive]
theorem min_le_mul_of_one_le_right [CovariantClass α α (· * ·) (· ≤ ·)] {a b : α} (hb : 1 ≤ b) :
    min a b ≤ a * b 
goal: min a b ≤ a + b

full_name: min_le_mul_of_one_le_left
code: @[to_additive]
theorem min_le_mul_of_one_le_left [CovariantClass α α (Function.swap (· * ·)) (· ≤ ·)] {a b : α}
    (ha : 1 ≤ a) : min a b ≤ a * b 
goal: min a b ≤ a * b

full_name: min_le_add_of_nonneg_left
code: @[to_additive]
theorem min_le_mul_of_one_le_left [CovariantClass α α (Function.swap (· * ·)) (· ≤ ·)] {a b : α}
    (ha : 1 ≤ a) : min a b ≤ a * b 
goal: min a b ≤ a + b

full_name: max_le_mul_of_one_le
code: @[to_additive]
theorem max_le_mul_of_one_le [CovariantClass α α (· * ·) (· ≤ ·)]
    [CovariantClass α α (Function.swap (· * ·)) (· ≤ ·)] {a b : α} (ha : 1 ≤ a) (hb : 1 ≤ b) :
    max a b ≤ a * b 
goal: max a b ≤ a * b

full_name: max_le_add_of_nonneg
code: @[to_additive]
theorem max_le_mul_of_one_le [CovariantClass α α (· * ·) (· ≤ ·)]
    [CovariantClass α α (Function.swap (· * ·)) (· ≤ ·)] {a b : α} (ha : 1 ≤ a) (hb : 1 ≤ b) :
    max a b ≤ a * b 
goal: max a b ≤ a + b