File: Mathlib/Algebra/Order/Group/Lattice.lean

full_name: mul_sup
code: @[to_additive]
lemma mul_sup (a b c : α) : c * (a ⊔ b) = c * a ⊔ c * b 
goal: c * (a ⊔ b) = c * a ⊔ c * b

full_name: add_sup
code: @[to_additive]
lemma mul_sup (a b c : α) : c * (a ⊔ b) = c * a ⊔ c * b 
goal: c + a ⊔ b = (c + a) ⊔ (c + b)

full_name: sup_mul
code: @[to_additive]
lemma sup_mul (a b c : α) : (a ⊔ b) * c = a * c ⊔ b * c 
goal: (a ⊔ b) * c = a * c ⊔ b * c

full_name: sup_add
code: @[to_additive]
lemma sup_mul (a b c : α) : (a ⊔ b) * c = a * c ⊔ b * c 
goal: a ⊔ b + c = (a + c) ⊔ (b + c)

full_name: mul_inf
code: @[to_additive]
lemma mul_inf (a b c : α) : c * (a ⊓ b) = c * a ⊓ c * b 
goal: c * (a ⊓ b) = c * a ⊓ c * b

full_name: add_inf
code: @[to_additive]
lemma mul_inf (a b c : α) : c * (a ⊓ b) = c * a ⊓ c * b 
goal: c + a ⊓ b = (c + a) ⊓ (c + b)

full_name: inf_add
code: @[to_additive]
lemma inf_mul (a b c : α) : (a ⊓ b) * c = a * c ⊓ b * c 
goal: a ⊓ b + c = (a + c) ⊓ (b + c)

full_name: inf_mul
code: @[to_additive]
lemma inf_mul (a b c : α) : (a ⊓ b) * c = a * c ⊓ b * c 
goal: (a ⊓ b) * c = a * c ⊓ b * c

full_name: inv_sup
code: @[to_additive] lemma inv_sup (a b : α) : (a ⊔ b)⁻¹ = a⁻¹ ⊓ b⁻¹ 
goal: (a ⊔ b)⁻¹ = a⁻¹ ⊓ b⁻¹

full_name: neg_sup
code: @[to_additive] lemma inv_sup (a b : α) : (a ⊔ b)⁻¹ = a⁻¹ ⊓ b⁻¹ 
goal: -(a ⊔ b) = -a ⊓ -b

full_name: inv_inf
code: @[to_additive] lemma inv_inf (a b : α) : (a ⊓ b)⁻¹ = a⁻¹ ⊔ b⁻¹ 
goal: (a ⊓ b)⁻¹ = a⁻¹ ⊔ b⁻¹

full_name: neg_inf
code: @[to_additive] lemma inv_inf (a b : α) : (a ⊓ b)⁻¹ = a⁻¹ ⊔ b⁻¹ 
goal: -(a ⊓ b) = -a ⊔ -b

full_name: sub_sup
code: @[to_additive]
lemma div_sup (a b c : α) : c / (a ⊔ b) = c / a ⊓ c / b 
goal: c - a ⊔ b = (c - a) ⊓ (c - b)

full_name: div_sup
code: @[to_additive]
lemma div_sup (a b c : α) : c / (a ⊔ b) = c / a ⊓ c / b 
goal: c / (a ⊔ b) = c / a ⊓ c / b

full_name: sup_sub
code: @[to_additive]
lemma sup_div (a b c : α) : (a ⊔ b) / c = a / c ⊔ b / c 
goal: a ⊔ b - c = (a - c) ⊔ (b - c)

full_name: sup_div
code: @[to_additive]
lemma sup_div (a b c : α) : (a ⊔ b) / c = a / c ⊔ b / c 
goal: (a ⊔ b) / c = a / c ⊔ b / c

full_name: div_inf
code: @[to_additive]
lemma div_inf (a b c : α) : c / (a ⊓ b) = c / a ⊔ c / b 
goal: c / (a ⊓ b) = c / a ⊔ c / b

full_name: sub_inf
code: @[to_additive]
lemma div_inf (a b c : α) : c / (a ⊓ b) = c / a ⊔ c / b 
goal: c - a ⊓ b = (c - a) ⊔ (c - b)

full_name: inf_div
code: @[to_additive]
lemma inf_div (a b c : α) : (a ⊓ b) / c = a / c ⊓ b / c 
goal: (a ⊓ b) / c = a / c ⊓ b / c

full_name: inf_sub
code: @[to_additive]
lemma inf_div (a b c : α) : (a ⊓ b) / c = a / c ⊓ b / c 
goal: a ⊓ b - c = (a - c) ⊓ (b - c)

full_name: pow_two_semiclosed
code: @[to_additive]
lemma pow_two_semiclosed
    [CovariantClass α α (swap (· * ·)) (· ≤ ·)] {a : α} (ha : 1 ≤ a ^ 2) : 1 ≤ a 
goal: 1 ≤ a

full_name: nsmul_two_semiclosed
code: @[to_additive]
lemma pow_two_semiclosed
    [CovariantClass α α (swap (· * ·)) (· ≤ ·)] {a : α} (ha : 1 ≤ a ^ 2) : 1 ≤ a 
goal: 0 ≤ a

full_name: inf_add_sup
code: @[to_additive]
lemma inf_mul_sup [CovariantClass α α (· * ·) (· ≤ ·)] (a b : α) : (a ⊓ b) * (a ⊔ b) = a * b 
goal: a ⊓ b + a ⊔ b = a + b

full_name: inf_mul_sup
code: @[to_additive]
lemma inf_mul_sup [CovariantClass α α (· * ·) (· ≤ ·)] (a b : α) : (a ⊓ b) * (a ⊔ b) = a * b 
goal: (a ⊓ b) * (a ⊔ b) = a * b

full_name: AddCommGroup.toDistribLattice
code: /-- Every lattice ordered commutative group is a distributive lattice. -/
-- Non-comm case needs cancellation law https://ncatlab.org/nlab/show/distributive+lattice
@[to_additive "Every lattice ordered commutative additive group is a distributive lattice"]
def CommGroup.toDistribLattice (α : Type*) [Lattice α] [CommGroup α]
    [CovariantClass α α (· * ·) (· ≤ ·)] : DistribLattice α 
goal: DistribLattice α

full_name: CommGroup.toDistribLattice
code: /-- Every lattice ordered commutative group is a distributive lattice. -/
-- Non-comm case needs cancellation law https://ncatlab.org/nlab/show/distributive+lattice
@[to_additive "Every lattice ordered commutative additive group is a distributive lattice"]
def CommGroup.toDistribLattice (α : Type*) [Lattice α] [CommGroup α]
    [CovariantClass α α (· * ·) (· ≤ ·)] : DistribLattice α 
goal: DistribLattice α