statement stringlengths 1 2.93k | proof stringlengths 0 19.2k | type stringclasses 13
values | symbolic_name stringlengths 1 131 | library stringlengths 4 62 | filename stringlengths 20 95 | imports listlengths 0 10 | deps listlengths 0 64 | docstring stringlengths 0 4.95k | source_url stringclasses 1
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unop_mul [Mul α] (x y : αᵐᵒᵖ) : unop (x * y) = unop y * unop x | rfl | lemma | MulOpposite.unop_mul | Algebra | Mathlib/Algebra/Opposites.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
op_inv [Inv α] (x : α) : op x⁻¹ = (op x)⁻¹ | rfl | lemma | MulOpposite.op_inv | Algebra | Mathlib/Algebra/Opposites.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
unop_inv [Inv α] (x : αᵐᵒᵖ) : unop x⁻¹ = (unop x)⁻¹ | rfl | lemma | MulOpposite.unop_inv | Algebra | Mathlib/Algebra/Opposites.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
op_sub [Sub α] (x y : α) : op (x - y) = op x - op y | rfl | lemma | MulOpposite.op_sub | Algebra | Mathlib/Algebra/Opposites.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
unop_sub [Sub α] (x y : αᵐᵒᵖ) : unop (x - y) = unop x - unop y | rfl | lemma | MulOpposite.unop_sub | Algebra | Mathlib/Algebra/Opposites.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
op_smul [SMul α β] (a : α) (b : β) : op (a • b) = a • op b | rfl | lemma | MulOpposite.op_smul | Algebra | Mathlib/Algebra/Opposites.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
unop_smul [SMul α β] (a : α) (b : βᵐᵒᵖ) : unop (a • b) = a • unop b | rfl | lemma | MulOpposite.unop_smul | Algebra | Mathlib/Algebra/Opposites.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
unop_eq_zero_iff [Zero α] (a : αᵐᵒᵖ) : a.unop = (0 : α) ↔ a = (0 : αᵐᵒᵖ) | unop_injective.eq_iff' rfl | theorem | MulOpposite.unop_eq_zero_iff | Algebra | Mathlib/Algebra/Opposites.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
op_eq_zero_iff [Zero α] (a : α) : op a = (0 : αᵐᵒᵖ) ↔ a = (0 : α) | op_injective.eq_iff' rfl | theorem | MulOpposite.op_eq_zero_iff | Algebra | Mathlib/Algebra/Opposites.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
unop_ne_zero_iff [Zero α] (a : αᵐᵒᵖ) : a.unop ≠ (0 : α) ↔ a ≠ (0 : αᵐᵒᵖ) | not_congr <| unop_eq_zero_iff a | theorem | MulOpposite.unop_ne_zero_iff | Algebra | Mathlib/Algebra/Opposites.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
op_ne_zero_iff [Zero α] (a : α) : op a ≠ (0 : αᵐᵒᵖ) ↔ a ≠ (0 : α) | not_congr <| op_eq_zero_iff a | theorem | MulOpposite.op_ne_zero_iff | Algebra | Mathlib/Algebra/Opposites.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
unop_eq_one_iff [One α] (a : αᵐᵒᵖ) : a.unop = 1 ↔ a = 1 | unop_injective.eq_iff' rfl | theorem | MulOpposite.unop_eq_one_iff | Algebra | Mathlib/Algebra/Opposites.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
op_eq_one_iff [One α] (a : α) : op a = 1 ↔ a = 1 | op_injective.eq_iff | lemma | MulOpposite.op_eq_one_iff | Algebra | Mathlib/Algebra/Opposites.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
instOne [One α] : One αᵃᵒᵖ | where one := op 1 | instance | AddOpposite.instOne | Algebra | Mathlib/Algebra/Opposites.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
unop_one [One α] : unop 1 = (1 : α) | rfl | lemma | AddOpposite.unop_one | Algebra | Mathlib/Algebra/Opposites.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
op_eq_one_iff [One α] {a : α} : op a = 1 ↔ a = 1 | op_injective.eq_iff | lemma | AddOpposite.op_eq_one_iff | Algebra | Mathlib/Algebra/Opposites.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
unop_eq_one_iff [One α] {a : αᵃᵒᵖ} : unop a = 1 ↔ a = 1 | unop_injective.eq_iff | lemma | AddOpposite.unop_eq_one_iff | Algebra | Mathlib/Algebra/Opposites.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
instMul [Mul α] : Mul αᵃᵒᵖ | where mul a b := op (unop a * unop b) | instance | AddOpposite.instMul | Algebra | Mathlib/Algebra/Opposites.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
op_mul [Mul α] (a b : α) : op (a * b) = op a * op b | rfl | lemma | AddOpposite.op_mul | Algebra | Mathlib/Algebra/Opposites.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
unop_mul [Mul α] (a b : αᵃᵒᵖ) : unop (a * b) = unop a * unop b | rfl | lemma | AddOpposite.unop_mul | Algebra | Mathlib/Algebra/Opposites.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
instInv [Inv α] : Inv αᵃᵒᵖ | where inv a := op (unop a)⁻¹ | instance | AddOpposite.instInv | Algebra | Mathlib/Algebra/Opposites.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
instInvolutiveInv [InvolutiveInv α] : InvolutiveInv αᵃᵒᵖ | where
inv_inv _ := unop_injective <| inv_inv _ | instance | AddOpposite.instInvolutiveInv | Algebra | Mathlib/Algebra/Opposites.lean | [] | [
"InvolutiveInv",
"inv_inv"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
op_inv [Inv α] (a : α) : op a⁻¹ = (op a)⁻¹ | rfl | lemma | AddOpposite.op_inv | Algebra | Mathlib/Algebra/Opposites.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
unop_inv [Inv α] (a : αᵃᵒᵖ) : unop a⁻¹ = (unop a)⁻¹ | rfl | lemma | AddOpposite.unop_inv | Algebra | Mathlib/Algebra/Opposites.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
instDiv [Div α] : Div αᵃᵒᵖ | where div a b := op (unop a / unop b) | instance | AddOpposite.instDiv | Algebra | Mathlib/Algebra/Opposites.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
op_div [Div α] (a b : α) : op (a / b) = op a / op b | rfl | lemma | AddOpposite.op_div | Algebra | Mathlib/Algebra/Opposites.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
unop_div [Div α] (a b : αᵃᵒᵖ) : unop (a / b) = unop a / unop b | rfl | lemma | AddOpposite.unop_div | Algebra | Mathlib/Algebra/Opposites.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
SemigroupPEmpty : Semigroup PEmpty.{u + 1} | where
mul x _ := by cases x
mul_assoc x y z := by cases x | instance | SemigroupPEmpty | Algebra | Mathlib/Algebra/PEmptyInstances.lean | [] | [
"Semigroup",
"mul_assoc"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
discrim [Ring R] (a b c : R) : R | b ^ 2 - 4 * a * c | def | discrim | Algebra | Mathlib/Algebra/QuadraticDiscriminant.lean | [] | [
"Ring"
] | Discriminant of a quadratic | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
discrim_neg [Ring R] (a b c : R) : discrim (-a) (-b) (-c) = discrim a b c | by
simp [discrim] | lemma | discrim_neg | Algebra | Mathlib/Algebra/QuadraticDiscriminant.lean | [] | [
"Ring",
"discrim"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
discrim_eq_sq_of_quadratic_eq_zero {x : R} (h : a * (x * x) + b * x + c = 0) :
discrim a b c = (2 * a * x + b) ^ 2 | by
rw [discrim]
linear_combination -4 * a * h | lemma | discrim_eq_sq_of_quadratic_eq_zero | Algebra | Mathlib/Algebra/QuadraticDiscriminant.lean | [] | [
"discrim"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
quadratic_eq_zero_iff_discrim_eq_sq [NeZero (2 : R)] [NoZeroDivisors R]
(ha : a ≠ 0) (x : R) :
a * (x * x) + b * x + c = 0 ↔ discrim a b c = (2 * a * x + b) ^ 2 | by
refine ⟨discrim_eq_sq_of_quadratic_eq_zero, fun h ↦ ?_⟩
rw [discrim] at h
have ha : 2 * 2 * a ≠ 0 := mul_ne_zero (mul_ne_zero (NeZero.ne _) (NeZero.ne _)) ha
apply mul_left_cancel₀ ha
linear_combination -h | theorem | quadratic_eq_zero_iff_discrim_eq_sq | Algebra | Mathlib/Algebra/QuadraticDiscriminant.lean | [] | [
"NoZeroDivisors",
"discrim",
"mul_left_cancel₀",
"mul_ne_zero"
] | A quadratic has roots if and only if its discriminant equals some square. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
quadratic_ne_zero_of_discrim_ne_sq (h : ∀ s : R, discrim a b c ≠ s ^ 2) (x : R) :
a * (x * x) + b * x + c ≠ 0 | mt discrim_eq_sq_of_quadratic_eq_zero (h _) | theorem | quadratic_ne_zero_of_discrim_ne_sq | Algebra | Mathlib/Algebra/QuadraticDiscriminant.lean | [] | [
"discrim",
"discrim_eq_sq_of_quadratic_eq_zero"
] | A quadratic has no root if its discriminant has no square root. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
quadratic_eq_zero_iff (ha : a ≠ 0) {s : K} (h : discrim a b c = s * s) (x : K) :
a * (x * x) + b * x + c = 0 ↔ x = (-b + s) / (2 * a) ∨ x = (-b - s) / (2 * a) | by
rw [quadratic_eq_zero_iff_discrim_eq_sq ha, h, sq, mul_self_eq_mul_self_iff]
field_simp
grind | theorem | quadratic_eq_zero_iff | Algebra | Mathlib/Algebra/QuadraticDiscriminant.lean | [] | [
"discrim",
"mul_self_eq_mul_self_iff",
"quadratic_eq_zero_iff_discrim_eq_sq"
] | Roots of a quadratic equation. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
exists_quadratic_eq_zero (ha : a ≠ 0) (h : ∃ s, discrim a b c = s * s) :
∃ x, a * (x * x) + b * x + c = 0 | by
rcases h with ⟨s, hs⟩
use (-b + s) / (2 * a)
rw [quadratic_eq_zero_iff ha hs]
simp | theorem | exists_quadratic_eq_zero | Algebra | Mathlib/Algebra/QuadraticDiscriminant.lean | [] | [
"discrim",
"quadratic_eq_zero_iff"
] | A quadratic has roots if its discriminant has square roots | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
quadratic_eq_zero_iff_of_discrim_eq_zero (ha : a ≠ 0) (h : discrim a b c = 0) (x : K) :
a * (x * x) + b * x + c = 0 ↔ x = -b / (2 * a) | by
have : discrim a b c = 0 * 0 := by rw [h, mul_zero]
rw [quadratic_eq_zero_iff ha this, add_zero, sub_zero, or_self_iff] | theorem | quadratic_eq_zero_iff_of_discrim_eq_zero | Algebra | Mathlib/Algebra/QuadraticDiscriminant.lean | [] | [
"discrim",
"quadratic_eq_zero_iff"
] | Root of a quadratic when its discriminant equals zero | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
discrim_eq_zero_of_existsUnique (ha : a ≠ 0) (h : ∃! x, a * (x * x) + b * x + c = 0) :
discrim a b c = 0 | by
simp_rw [quadratic_eq_zero_iff_discrim_eq_sq ha] at h
generalize discrim a b c = d at h
obtain ⟨x, rfl, hx⟩ := h
specialize hx (-(x + b / a))
grind | theorem | discrim_eq_zero_of_existsUnique | Algebra | Mathlib/Algebra/QuadraticDiscriminant.lean | [] | [
"discrim",
"quadratic_eq_zero_iff_discrim_eq_sq"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
discrim_eq_zero_iff (ha : a ≠ 0) :
discrim a b c = 0 ↔ (∃! x, a * (x * x) + b * x + c = 0) | by
refine ⟨fun hd => ?_, discrim_eq_zero_of_existsUnique ha⟩
simp_rw [quadratic_eq_zero_iff_of_discrim_eq_zero ha hd, existsUnique_eq] | theorem | discrim_eq_zero_iff | Algebra | Mathlib/Algebra/QuadraticDiscriminant.lean | [] | [
"discrim",
"discrim_eq_zero_of_existsUnique",
"existsUnique_eq",
"quadratic_eq_zero_iff_of_discrim_eq_zero"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
discrim_le_zero (h : ∀ x : K, 0 ≤ a * (x * x) + b * x + c) : discrim a b c ≤ 0 | by
rw [discrim, sq]
obtain ha | rfl | ha : a < 0 ∨ a = 0 ∨ 0 < a := lt_trichotomy a 0
-- if a < 0
· have : Tendsto (fun x => (a * x + b) * x + c) atTop atBot :=
tendsto_atBot_add_const_right _ c <|
(tendsto_atBot_add_const_right _ b (tendsto_id.const_mul_atTop_of_neg ha)).atBot_mul_atTop₀
... | theorem | discrim_le_zero | Algebra | Mathlib/Algebra/QuadraticDiscriminant.lean | [] | [
"discrim",
"eq_or_ne",
"lt_trichotomy",
"mul_assoc",
"mul_div_cancel₀",
"zero_le_four"
] | If a polynomial of degree 2 is always nonnegative, then its discriminant is nonpositive | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
discrim_le_zero_of_nonpos (h : ∀ x : K, a * (x * x) + b * x + c ≤ 0) : discrim a b c ≤ 0 | discrim_neg a b c ▸ discrim_le_zero <| by simpa only [neg_mul, ← neg_add, neg_nonneg] | lemma | discrim_le_zero_of_nonpos | Algebra | Mathlib/Algebra/QuadraticDiscriminant.lean | [] | [
"discrim",
"discrim_le_zero",
"discrim_neg",
"neg_mul"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
discrim_lt_zero (ha : a ≠ 0) (h : ∀ x : K, 0 < a * (x * x) + b * x + c) :
discrim a b c < 0 | by
have : ∀ x : K, 0 ≤ a * (x * x) + b * x + c := fun x => le_of_lt (h x)
refine lt_of_le_of_ne (discrim_le_zero this) fun h' ↦ ?_
have := h (-b / (2 * a))
have : a * (-b / (2 * a)) * (-b / (2 * a)) + b * (-b / (2 * a)) + c = 0 := by
rw [mul_assoc, quadratic_eq_zero_iff_of_discrim_eq_zero ha h' (-b / (2 * a... | theorem | discrim_lt_zero | Algebra | Mathlib/Algebra/QuadraticDiscriminant.lean | [] | [
"discrim",
"discrim_le_zero",
"le_of_lt",
"lt_of_le_of_ne",
"mul_assoc",
"quadratic_eq_zero_iff_of_discrim_eq_zero"
] | If a polynomial of degree 2 is always positive, then its discriminant is negative,
at least when the coefficient of the quadratic term is nonzero. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
discrim_lt_zero_of_neg (ha : a ≠ 0) (h : ∀ x : K, a * (x * x) + b * x + c < 0) :
discrim a b c < 0 | discrim_neg a b c ▸ discrim_lt_zero (neg_ne_zero.2 ha) <| by
simpa only [neg_mul, ← neg_add, neg_pos] | lemma | discrim_lt_zero_of_neg | Algebra | Mathlib/Algebra/QuadraticDiscriminant.lean | [] | [
"discrim",
"discrim_lt_zero",
"discrim_neg",
"neg_mul"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Shelf (α : Type u) where
/-- The action of the `Shelf` over `α` -/
act : α → α → α
/-- A verification that `act` is self-distributive -/
self_distrib : ∀ {x y z : α}, act x (act y z) = act (act x y) (act x z) | class | Shelf | Algebra | Mathlib/Algebra/Quandle.lean | [] | [] | A *Shelf* is a structure with a self-distributive binary operation.
The binary operation is regarded as a left action of the type on itself. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
UnitalShelf (α : Type u) extends Shelf α, One α where
one_act : ∀ a : α, act 1 a = a
act_one : ∀ a : α, act a 1 = a | class | UnitalShelf | Algebra | Mathlib/Algebra/Quandle.lean | [] | [
"Shelf"
] | A *unital shelf* is a shelf equipped with an element `1` such that, for all elements `x`,
we have both `x ◃ 1` and `1 ◃ x` equal `x`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
ShelfHom (S₁ : Type*) (S₂ : Type*) [Shelf S₁] [Shelf S₂] where
/-- The function under the Shelf Homomorphism -/
toFun : S₁ → S₂
/-- The homomorphism property of a Shelf Homomorphism -/
map_act' : ∀ {x y : S₁}, toFun (Shelf.act x y) = Shelf.act (toFun x) (toFun y) | structure | ShelfHom | Algebra | Mathlib/Algebra/Quandle.lean | [] | [
"Shelf"
] | The type of homomorphisms between shelves.
This is also the notion of rack and quandle homomorphisms. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Rack (α : Type u) extends Shelf α where
/-- The inverse actions of the elements -/
invAct : α → α → α
/-- Proof of left inverse -/
left_inv : ∀ x, Function.LeftInverse (invAct x) (act x)
/-- Proof of right inverse -/
right_inv : ∀ x, Function.RightInverse (invAct x) (act x)
/-- Action of a Shelf -/
scoped[... | class | Rack | Algebra | Mathlib/Algebra/Quandle.lean | [] | [
"Shelf",
"ShelfHom"
] | A *rack* is an automorphic set (a set with an action on itself by
bijections) that is self-distributive. It is a shelf such that each
element's action is invertible.
The notations `x ◃ y` and `x ◃⁻¹ y` denote the action and the
inverse action, respectively, and they are right associative. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
act_act_self_eq (x y : S) : (x ◃ y) ◃ x = x ◃ y | by
have h : (x ◃ y) ◃ x = (x ◃ y) ◃ (x ◃ 1) := by rw [act_one]
rw [h, ← Shelf.self_distrib, act_one] | lemma | UnitalShelf.act_act_self_eq | Algebra | Mathlib/Algebra/Quandle.lean | [] | [] | A monoid is *graphic* if, for all `x` and `y`, the *graphic identity*
`(x * y) * x = x * y` holds. For a unital shelf, this graphic
identity holds. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
act_idem (x : S) : (x ◃ x) = x | by rw [← act_one x, ← Shelf.self_distrib, act_one] | lemma | UnitalShelf.act_idem | Algebra | Mathlib/Algebra/Quandle.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
act_self_act_eq (x y : S) : x ◃ (x ◃ y) = x ◃ y | by
have h : x ◃ (x ◃ y) = (x ◃ 1) ◃ (x ◃ y) := by rw [act_one]
rw [h, ← Shelf.self_distrib, one_act] | lemma | UnitalShelf.act_self_act_eq | Algebra | Mathlib/Algebra/Quandle.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
assoc (x y z : S) : (x ◃ y) ◃ z = x ◃ y ◃ z | by
rw [self_distrib, self_distrib, act_act_self_eq, act_self_act_eq] | lemma | UnitalShelf.assoc | Algebra | Mathlib/Algebra/Quandle.lean | [] | [] | The associativity of a unital shelf comes for free. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
act' (x : R) : R ≃ R | where
toFun := Shelf.act x
invFun := invAct x
left_inv := left_inv x
right_inv := right_inv x | def | Rack.act' | Algebra | Mathlib/Algebra/Quandle.lean | [] | [] | A rack acts on itself by equivalences. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
act'_apply (x y : R) : act' x y = x ◃ y | rfl | theorem | Rack.act'_apply | Algebra | Mathlib/Algebra/Quandle.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
act'_symm_apply (x y : R) : (act' x).symm y = x ◃⁻¹ y | rfl | theorem | Rack.act'_symm_apply | Algebra | Mathlib/Algebra/Quandle.lean | [] | [
"symm"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
invAct_apply (x y : R) : (act' x)⁻¹ y = x ◃⁻¹ y | rfl | theorem | Rack.invAct_apply | Algebra | Mathlib/Algebra/Quandle.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
invAct_act_eq (x y : R) : x ◃⁻¹ x ◃ y = y | left_inv x y | theorem | Rack.invAct_act_eq | Algebra | Mathlib/Algebra/Quandle.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
act_invAct_eq (x y : R) : x ◃ x ◃⁻¹ y = y | right_inv x y | theorem | Rack.act_invAct_eq | Algebra | Mathlib/Algebra/Quandle.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
left_cancel (x : R) {y y' : R} : x ◃ y = x ◃ y' ↔ y = y' | by
constructor
· apply (act' x).injective
rintro rfl
rfl | theorem | Rack.left_cancel | Algebra | Mathlib/Algebra/Quandle.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
left_cancel_inv (x : R) {y y' : R} : x ◃⁻¹ y = x ◃⁻¹ y' ↔ y = y' | by
constructor
· apply (act' x).symm.injective
rintro rfl
rfl | theorem | Rack.left_cancel_inv | Algebra | Mathlib/Algebra/Quandle.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
self_distrib_inv {x y z : R} : x ◃⁻¹ y ◃⁻¹ z = (x ◃⁻¹ y) ◃⁻¹ x ◃⁻¹ z | by
rw [← left_cancel (x ◃⁻¹ y), right_inv, ← left_cancel x, right_inv, self_distrib]
repeat' rw [right_inv] | theorem | Rack.self_distrib_inv | Algebra | Mathlib/Algebra/Quandle.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
ad_conj {R : Type*} [Rack R] (x y : R) : act' (x ◃ y) = act' x * act' y * (act' x)⁻¹ | by
rw [eq_mul_inv_iff_mul_eq]; ext z
apply self_distrib.symm | theorem | Rack.ad_conj | Algebra | Mathlib/Algebra/Quandle.lean | [] | [
"Rack",
"eq_mul_inv_iff_mul_eq"
] | The *adjoint action* of a rack on itself is `op'`, and the adjoint
action of `x ◃ y` is the conjugate of the action of `y` by the action
of `x`. It is another way to understand the self-distributivity axiom.
This is used in the natural rack homomorphism `toConj` from `R` to
`Conj (R ≃ R)` defined by `op'`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
oppositeRack : Rack Rᵐᵒᵖ | where
act x y := op (invAct (unop x) (unop y))
self_distrib := by
intro x y z
induction x
induction y
induction z
simp only [op_inj, unop_op]
rw [self_distrib_inv]
invAct x y := op (Shelf.act (unop x) (unop y))
left_inv := MulOpposite.rec' fun x => MulOpposite.rec' fun y => by simp
rig... | instance | Rack.oppositeRack | Algebra | Mathlib/Algebra/Quandle.lean | [] | [
"MulOpposite.rec'",
"Rack"
] | The opposite rack, swapping the roles of `◃` and `◃⁻¹`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
op_act_op_eq {x y : R} : op x ◃ op y = op (x ◃⁻¹ y) | rfl | theorem | Rack.op_act_op_eq | Algebra | Mathlib/Algebra/Quandle.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
op_invAct_op_eq {x y : R} : op x ◃⁻¹ op y = op (x ◃ y) | rfl | theorem | Rack.op_invAct_op_eq | Algebra | Mathlib/Algebra/Quandle.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
self_act_act_eq {x y : R} : (x ◃ x) ◃ y = x ◃ y | by rw [← right_inv x y, ← self_distrib] | theorem | Rack.self_act_act_eq | Algebra | Mathlib/Algebra/Quandle.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
self_invAct_invAct_eq {x y : R} : (x ◃⁻¹ x) ◃⁻¹ y = x ◃⁻¹ y | by
have h := @self_act_act_eq _ _ (op x) (op y)
simpa using h | theorem | Rack.self_invAct_invAct_eq | Algebra | Mathlib/Algebra/Quandle.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
self_act_invAct_eq {x y : R} : (x ◃ x) ◃⁻¹ y = x ◃⁻¹ y | by
rw [← left_cancel (x ◃ x)]
rw [right_inv]
rw [self_act_act_eq]
rw [right_inv] | theorem | Rack.self_act_invAct_eq | Algebra | Mathlib/Algebra/Quandle.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
self_invAct_act_eq {x y : R} : (x ◃⁻¹ x) ◃ y = x ◃ y | by
have h := @self_act_invAct_eq _ _ (op x) (op y)
simpa using h | theorem | Rack.self_invAct_act_eq | Algebra | Mathlib/Algebra/Quandle.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
self_act_eq_iff_eq {x y : R} : x ◃ x = y ◃ y ↔ x = y | by
constructor; swap
· rintro rfl; rfl
intro h
trans (x ◃ x) ◃⁻¹ x ◃ x
· rw [← left_cancel (x ◃ x), right_inv, self_act_act_eq]
· rw [h, ← left_cancel (y ◃ y), right_inv, self_act_act_eq] | theorem | Rack.self_act_eq_iff_eq | Algebra | Mathlib/Algebra/Quandle.lean | [] | [
"trans"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
self_invAct_eq_iff_eq {x y : R} : x ◃⁻¹ x = y ◃⁻¹ y ↔ x = y | by
have h := @self_act_eq_iff_eq _ _ (op x) (op y)
simpa using h | theorem | Rack.self_invAct_eq_iff_eq | Algebra | Mathlib/Algebra/Quandle.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
selfApplyEquiv (R : Type*) [Rack R] : R ≃ R | where
toFun x := x ◃ x
invFun x := x ◃⁻¹ x
left_inv x := by simp
right_inv x := by simp | def | Rack.selfApplyEquiv | Algebra | Mathlib/Algebra/Quandle.lean | [] | [
"Rack"
] | The map `x ↦ x ◃ x` is a bijection. (This has applications for the
regular isotopy version of the Reidemeister I move for knot diagrams.) | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
IsInvolutory (R : Type*) [Rack R] : Prop | ∀ x : R, Function.Involutive (Shelf.act x) | def | Rack.IsInvolutory | Algebra | Mathlib/Algebra/Quandle.lean | [] | [
"Function.Involutive",
"Rack"
] | An involutory rack is one for which `Rack.oppositeRack R x` is an involution for every x. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
involutory_invAct_eq_act {R : Type*} [Rack R] (h : IsInvolutory R) (x y : R) :
x ◃⁻¹ y = x ◃ y | by
rw [← left_cancel x, right_inv, h x] | theorem | Rack.involutory_invAct_eq_act | Algebra | Mathlib/Algebra/Quandle.lean | [] | [
"Rack"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
IsAbelian (R : Type*) [Rack R] : Prop | ∀ x y z w : R, (x ◃ y) ◃ z ◃ w = (x ◃ z) ◃ y ◃ w | def | Rack.IsAbelian | Algebra | Mathlib/Algebra/Quandle.lean | [] | [
"Rack"
] | An abelian rack is one for which the mediality axiom holds. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
assoc_iff_id {R : Type*} [Rack R] {x y z : R} : x ◃ y ◃ z = (x ◃ y) ◃ z ↔ x ◃ z = z | by
rw [self_distrib]
rw [left_cancel] | theorem | Rack.assoc_iff_id | Algebra | Mathlib/Algebra/Quandle.lean | [] | [
"Rack"
] | Associative racks are uninteresting. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
toFun_eq_coe (f : S₁ →◃ S₂) : f.toFun = f | rfl | theorem | ShelfHom.toFun_eq_coe | Algebra | Mathlib/Algebra/Quandle.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
map_act (f : S₁ →◃ S₂) {x y : S₁} : f (x ◃ y) = f x ◃ f y | map_act' f | theorem | ShelfHom.map_act | Algebra | Mathlib/Algebra/Quandle.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
id (S : Type*) [Shelf S] : S →◃ S | where
toFun := fun x => x
map_act' := by simp | def | ShelfHom.id | Algebra | Mathlib/Algebra/Quandle.lean | [] | [
"Shelf"
] | The identity homomorphism | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
inhabited (S : Type*) [Shelf S] : Inhabited (S →◃ S) | ⟨id S⟩ | instance | ShelfHom.inhabited | Algebra | Mathlib/Algebra/Quandle.lean | [] | [
"Shelf"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
comp (g : S₂ →◃ S₃) (f : S₁ →◃ S₂) : S₁ →◃ S₃ | where
toFun := g.toFun ∘ f.toFun
map_act' := by simp | def | ShelfHom.comp | Algebra | Mathlib/Algebra/Quandle.lean | [] | [] | The composition of shelf homomorphisms | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
comp_apply (g : S₂ →◃ S₃) (f : S₁ →◃ S₂) (x : S₁) : (g.comp f) x = g (f x) | rfl | theorem | ShelfHom.comp_apply | Algebra | Mathlib/Algebra/Quandle.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Quandle (α : Type*) extends Rack α where
/-- The fixing property of a Quandle -/
fix : ∀ {x : α}, act x x = x | class | Quandle | Algebra | Mathlib/Algebra/Quandle.lean | [] | [
"Rack"
] | A quandle is a rack such that each automorphism fixes its corresponding element. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
fix_inv {x : Q} : x ◃⁻¹ x = x | by
rw [← left_cancel x]
simp | theorem | Quandle.fix_inv | Algebra | Mathlib/Algebra/Quandle.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
oppositeQuandle : Quandle Qᵐᵒᵖ | where
fix := by
intro x
induction x
simp | instance | Quandle.oppositeQuandle | Algebra | Mathlib/Algebra/Quandle.lean | [] | [
"Quandle"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Conj (G : Type*) | G | abbrev | Quandle.Conj | Algebra | Mathlib/Algebra/Quandle.lean | [] | [] | The conjugation quandle of a group. Each element of the group acts by
the corresponding inner automorphism. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
Conj.quandle (G : Type*) [Group G] : Quandle (Conj G) | where
act x := @MulAut.conj G _ x
self_distrib := by
intro x y z
dsimp only [MulAut.conj_apply]
simp [mul_assoc]
invAct x := (@MulAut.conj G _ x).symm
left_inv x y := by
simp [mul_assoc]
right_inv x y := by
simp [mul_assoc]
fix := by simp | instance | Quandle.Conj.quandle | Algebra | Mathlib/Algebra/Quandle.lean | [] | [
"Group",
"MulAut.conj",
"MulAut.conj_apply",
"Quandle",
"mul_assoc",
"symm"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
conj_act_eq_conj {G : Type*} [Group G] (x y : Conj G) :
x ◃ y = ((x : G) * (y : G) * (x : G)⁻¹ : G) | rfl | theorem | Quandle.conj_act_eq_conj | Algebra | Mathlib/Algebra/Quandle.lean | [] | [
"Group"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
conj_swap {G : Type*} [Group G] (x y : Conj G) : x ◃ y = y ↔ y ◃ x = x | by
grind [eq_mul_inv_iff_mul_eq] | theorem | Quandle.conj_swap | Algebra | Mathlib/Algebra/Quandle.lean | [] | [
"Group",
"eq_mul_inv_iff_mul_eq"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Conj.map {G : Type*} {H : Type*} [Group G] [Group H] (f : G →* H) : Conj G →◃ Conj H | where
toFun := f
map_act' := by simp | def | Quandle.Conj.map | Algebra | Mathlib/Algebra/Quandle.lean | [] | [
"Group"
] | `Conj` is functorial | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
Dihedral (n : ℕ) | ZMod n | def | Quandle.Dihedral | Algebra | Mathlib/Algebra/Quandle.lean | [] | [
"ZMod"
] | The dihedral quandle. This is the conjugation quandle of the dihedral group restricted to flips.
Used for Fox n-colorings of knots. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
dihedralAct (n : ℕ) (a : ZMod n) : ZMod n → ZMod n | fun b => 2 * a - b | def | Quandle.dihedralAct | Algebra | Mathlib/Algebra/Quandle.lean | [] | [
"ZMod"
] | The operation for the dihedral quandle. It does not need to be an equivalence
because it is an involution (see `dihedralAct.inv`). | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
dihedralAct.inv (n : ℕ) (a : ZMod n) : Function.Involutive (dihedralAct n a) | by
intro b
dsimp only [dihedralAct]
simp | theorem | Quandle.dihedralAct.inv | Algebra | Mathlib/Algebra/Quandle.lean | [] | [
"Function.Involutive",
"ZMod"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
toConj (R : Type*) [Rack R] : R →◃ Quandle.Conj (R ≃ R) | where
toFun := act'
map_act' := by
intro x y
exact ad_conj x y | def | Rack.toConj | Algebra | Mathlib/Algebra/Quandle.lean | [] | [
"Quandle.Conj",
"Rack"
] | This is the natural rack homomorphism to the conjugation quandle of the group `R ≃ R`
that acts on the rack. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
PreEnvelGroup (R : Type u) : Type u
| unit : PreEnvelGroup R
| incl (x : R) : PreEnvelGroup R
| mul (a b : PreEnvelGroup R) : PreEnvelGroup R
| inv (a : PreEnvelGroup R) : PreEnvelGroup R | inductive | Rack.PreEnvelGroup | Algebra | Mathlib/Algebra/Quandle.lean | [] | [] | Free generators of the enveloping group. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
PreEnvelGroup.inhabited (R : Type u) : Inhabited (PreEnvelGroup R) | ⟨PreEnvelGroup.unit⟩ | instance | Rack.PreEnvelGroup.inhabited | Algebra | Mathlib/Algebra/Quandle.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
PreEnvelGroupRel' (R : Type u) [Rack R] : PreEnvelGroup R → PreEnvelGroup R → Type u
| refl {a : PreEnvelGroup R} : PreEnvelGroupRel' R a a
| symm {a b : PreEnvelGroup R} (hab : PreEnvelGroupRel' R a b) : PreEnvelGroupRel' R b a
| trans {a b c : PreEnvelGroup R} (hab : PreEnvelGroupRel' R a b)
(hbc : PreEnvel... | inductive | Rack.PreEnvelGroupRel' | Algebra | Mathlib/Algebra/Quandle.lean | [] | [
"Rack",
"inv_mul_cancel",
"mul_one",
"one_mul",
"refl",
"symm",
"trans"
] | Relations for the enveloping group. This is a type-valued relation because
`toEnvelGroup.mapAux.well_def` inducts on it to show `toEnvelGroup.map`
is well-defined. The relation `PreEnvelGroupRel` is the `Prop`-valued version,
which is used to define `EnvelGroup` itself. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
PreEnvelGroupRel'.inhabited (R : Type u) [Rack R] :
Inhabited (PreEnvelGroupRel' R unit unit) | ⟨PreEnvelGroupRel'.refl⟩ | instance | Rack.PreEnvelGroupRel'.inhabited | Algebra | Mathlib/Algebra/Quandle.lean | [] | [
"Rack"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
PreEnvelGroupRel (R : Type u) [Rack R] : PreEnvelGroup R → PreEnvelGroup R → Prop
| rel {a b : PreEnvelGroup R} (r : PreEnvelGroupRel' R a b) : PreEnvelGroupRel R a b | inductive | Rack.PreEnvelGroupRel | Algebra | Mathlib/Algebra/Quandle.lean | [] | [
"Rack"
] | The `PreEnvelGroupRel` relation as a `Prop`. Used as the relation for `PreEnvelGroup.setoid`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
PreEnvelGroupRel'.rel {R : Type u} [Rack R] {a b : PreEnvelGroup R} :
PreEnvelGroupRel' R a b → PreEnvelGroupRel R a b | PreEnvelGroupRel.rel | theorem | Rack.PreEnvelGroupRel'.rel | Algebra | Mathlib/Algebra/Quandle.lean | [] | [
"Rack"
] | A quick way to convert a `PreEnvelGroupRel'` to a `PreEnvelGroupRel`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
PreEnvelGroupRel.refl {R : Type u} [Rack R] {a : PreEnvelGroup R} :
PreEnvelGroupRel R a a | PreEnvelGroupRel.rel PreEnvelGroupRel'.refl | theorem | Rack.PreEnvelGroupRel.refl | Algebra | Mathlib/Algebra/Quandle.lean | [] | [
"Rack"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
PreEnvelGroupRel.symm {R : Type u} [Rack R] {a b : PreEnvelGroup R} :
PreEnvelGroupRel R a b → PreEnvelGroupRel R b a | | ⟨r⟩ => r.symm.rel | theorem | Rack.PreEnvelGroupRel.symm | Algebra | Mathlib/Algebra/Quandle.lean | [] | [
"Rack"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
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