mathlib documentation

algebra.group_with_zero.defs

Typeclasses for groups with an adjoined zero element #

This file provides just the typeclass definitions, and the projection lemmas that expose their members.

Main definitions #

@[instance]
def mul_zero_class.to_has_mul (M₀ : Type u_4) [self : mul_zero_class M₀] :
has_mul M₀
@[class]
structure mul_zero_class (M₀ : Type u_4) :
Type u_4
  • mul : M₀ → M₀ → M₀
  • zero : M₀
  • zero_mul : ∀ (a : M₀), 0 * a = 0
  • mul_zero : ∀ (a : M₀), a * 0 = 0

Typeclass for expressing that a type M₀ with multiplication and a zero satisfies 0 * a = 0 and a * 0 = 0 for all a : M₀.

Instances
@[instance]
def mul_zero_class.to_has_zero (M₀ : Type u_4) [self : mul_zero_class M₀] :
@[simp]
theorem zero_mul {M₀ : Type u_1} [mul_zero_class M₀] (a : M₀) :
0 * a = 0
@[simp]
theorem mul_zero {M₀ : Type u_1} [mul_zero_class M₀] (a : M₀) :
a * 0 = 0
@[class]
structure semigroup_with_zero (S₀ : Type u_4) :
Type u_4
  • mul : S₀ → S₀ → S₀
  • mul_assoc : ∀ (a b c : S₀), (a * b) * c = a * b * c
  • zero : S₀
  • zero_mul : ∀ (a : S₀), 0 * a = 0
  • mul_zero : ∀ (a : S₀), a * 0 = 0

A type S₀ is a "semigroup with zero” if it is a semigroup with zero element, and 0 is left and right absorbing.

Instances
@[instance]
def semigroup_with_zero.to_mul_zero_class (S₀ : Type u_4) [self : semigroup_with_zero S₀] :
@[instance]
def semigroup_with_zero.to_semigroup (S₀ : Type u_4) [self : semigroup_with_zero S₀] :
@[class]
structure mul_zero_one_class (M₀ : Type u_4) :
Type u_4
  • one : M₀
  • mul : M₀ → M₀ → M₀
  • one_mul : ∀ (a : M₀), 1 * a = a
  • mul_one : ∀ (a : M₀), a * 1 = a
  • zero : M₀
  • zero_mul : ∀ (a : M₀), 0 * a = 0
  • mul_zero : ∀ (a : M₀), a * 0 = 0

A typeclass for non-associative monoids with zero elements.

Instances
@[instance]
def mul_zero_one_class.to_mul_zero_class (M₀ : Type u_4) [self : mul_zero_one_class M₀] :
@[instance]
def mul_zero_one_class.to_mul_one_class (M₀ : Type u_4) [self : mul_zero_one_class M₀] :
@[instance]
def monoid_with_zero.to_mul_zero_one_class (M₀ : Type u_4) [self : monoid_with_zero M₀] :
@[instance]
def monoid_with_zero.to_monoid (M₀ : Type u_4) [self : monoid_with_zero M₀] :
monoid M₀
@[class]
structure monoid_with_zero (M₀ : Type u_4) :
Type u_4
  • mul : M₀ → M₀ → M₀
  • mul_assoc : ∀ (a b c : M₀), (a * b) * c = a * b * c
  • one : M₀
  • one_mul : ∀ (a : M₀), 1 * a = a
  • mul_one : ∀ (a : M₀), a * 1 = a
  • npow : M₀ → M₀
  • npow_zero' : (∀ (x : M₀), monoid_with_zero.npow 0 x = 1) . "try_refl_tac"
  • npow_succ' : (∀ (n : ) (x : M₀), monoid_with_zero.npow n.succ x = x * monoid_with_zero.npow n x) . "try_refl_tac"
  • zero : M₀
  • zero_mul : ∀ (a : M₀), 0 * a = 0
  • mul_zero : ∀ (a : M₀), a * 0 = 0

A type M₀ is a “monoid with zero” if it is a monoid with zero element, and 0 is left and right absorbing.

Instances
@[protected, instance]
Equations
@[instance]
@[class]
structure cancel_monoid_with_zero (M₀ : Type u_4) :
Type u_4
  • mul : M₀ → M₀ → M₀
  • mul_assoc : ∀ (a b c : M₀), (a * b) * c = a * b * c
  • one : M₀
  • one_mul : ∀ (a : M₀), 1 * a = a
  • mul_one : ∀ (a : M₀), a * 1 = a
  • npow : M₀ → M₀
  • npow_zero' : (∀ (x : M₀), cancel_monoid_with_zero.npow 0 x = 1) . "try_refl_tac"
  • npow_succ' : (∀ (n : ) (x : M₀), cancel_monoid_with_zero.npow n.succ x = x * cancel_monoid_with_zero.npow n x) . "try_refl_tac"
  • zero : M₀
  • zero_mul : ∀ (a : M₀), 0 * a = 0
  • mul_zero : ∀ (a : M₀), a * 0 = 0
  • mul_left_cancel_of_ne_zero : ∀ {a b c : M₀}, a 0a * b = a * cb = c
  • mul_right_cancel_of_ne_zero : ∀ {a b c : M₀}, b 0a * b = c * ba = c

A type M is a cancel_monoid_with_zero if it is a monoid with zero element, 0 is left and right absorbing, and left/right multiplication by a non-zero element is injective.

Instances
theorem mul_left_cancel₀ {M₀ : Type u_1} [cancel_monoid_with_zero M₀] {a b c : M₀} (ha : a 0) (h : a * b = a * c) :
b = c
theorem mul_right_cancel₀ {M₀ : Type u_1} [cancel_monoid_with_zero M₀] {a b c : M₀} (hb : b 0) (h : a * b = c * b) :
a = c
theorem mul_right_injective₀ {M₀ : Type u_1} [cancel_monoid_with_zero M₀] {a : M₀} (ha : a 0) :
theorem mul_left_injective₀ {M₀ : Type u_1} [cancel_monoid_with_zero M₀] {b : M₀} (hb : b 0) :
function.injective (λ (a : M₀), a * b)
@[instance]
def comm_monoid_with_zero.to_comm_monoid (M₀ : Type u_4) [self : comm_monoid_with_zero M₀] :
@[instance]
@[class]
structure comm_monoid_with_zero (M₀ : Type u_4) :
Type u_4

A type M is a commutative “monoid with zero” if it is a commutative monoid with zero element, and 0 is left and right absorbing.

Instances
@[class]
structure comm_cancel_monoid_with_zero (M₀ : Type u_4) :
Type u_4
  • mul : M₀ → M₀ → M₀
  • mul_assoc : ∀ (a b c : M₀), (a * b) * c = a * b * c
  • one : M₀
  • one_mul : ∀ (a : M₀), 1 * a = a
  • mul_one : ∀ (a : M₀), a * 1 = a
  • npow : M₀ → M₀
  • npow_zero' : (∀ (x : M₀), comm_cancel_monoid_with_zero.npow 0 x = 1) . "try_refl_tac"
  • npow_succ' : (∀ (n : ) (x : M₀), comm_cancel_monoid_with_zero.npow n.succ x = x * comm_cancel_monoid_with_zero.npow n x) . "try_refl_tac"
  • mul_comm : ∀ (a b : M₀), a * b = b * a
  • zero : M₀
  • zero_mul : ∀ (a : M₀), 0 * a = 0
  • mul_zero : ∀ (a : M₀), a * 0 = 0
  • mul_left_cancel_of_ne_zero : ∀ {a b c : M₀}, a 0a * b = a * cb = c
  • mul_right_cancel_of_ne_zero : ∀ {a b c : M₀}, b 0a * b = c * ba = c

A type M is a comm_cancel_monoid_with_zero if it is a commutative monoid with zero element, 0 is left and right absorbing, and left/right multiplication by a non-zero element is injective.

Instances
@[instance]
def group_with_zero.to_nontrivial (G₀ : Type u) [self : group_with_zero G₀] :
@[instance]
def group_with_zero.to_monoid_with_zero (G₀ : Type u) [self : group_with_zero G₀] :
@[class]
structure group_with_zero (G₀ : Type u) :
Type u

A type G₀ is a “group with zero” if it is a monoid with zero element (distinct from 1) such that every nonzero element is invertible. The type is required to come with an “inverse” function, and the inverse of 0 must be 0.

Examples include division rings and the ordered monoids that are the target of valuations in general valuation theory.

Instances
@[instance]
def group_with_zero.to_div_inv_monoid (G₀ : Type u) [self : group_with_zero G₀] :
@[simp]
theorem inv_zero {G₀ : Type u} [group_with_zero G₀] :
0⁻¹ = 0
@[simp]
theorem mul_inv_cancel {G₀ : Type u} [group_with_zero G₀] {a : G₀} (h : a 0) :
a * a⁻¹ = 1
@[class]
structure comm_group_with_zero (G₀ : Type u_4) :
Type u_4

A type G₀ is a commutative “group with zero” if it is a commutative monoid with zero element (distinct from 1) such that every nonzero element is invertible. The type is required to come with an “inverse” function, and the inverse of 0 must be 0.

Instances
@[instance]