Documentation

Mathlib.Algebra.Group.WithOne.Defs

Adjoining a zero/one to semigroups and related algebraic structures #

This file contains different results about adjoining an element to an algebraic structure which then behaves like a zero or a one. An example is adjoining a one to a semigroup to obtain a monoid. That this provides an example of an adjunction is proved in Mathlib.Algebra.Category.MonCat.Adjunctions.

Another result says that adjoining to a group an element zero gives a GroupWithZero. For more information about these structures (which are not that standard in informal mathematics, see Mathlib.Algebra.GroupWithZero.Basic)

Porting notes #

In Lean 3, we use id here and there to get correct types of proofs. This is required because WithOne and WithZero are marked as irreducible at the end of Mathlib.Algebra.Group.WithOne.Defs, so proofs that use Option α instead of WithOne α no longer typecheck. In Lean 4, both types are plain defs, so we don't need these ids.

TODO #

WithOne.coe_mul and WithZero.coe_mul have inconsistent use of implicit parameters

def WithZero (α : Type u_1) :
Type u_1

Add an extra element 0 to a type

Equations
def WithOne (α : Type u_1) :
Type u_1

Add an extra element 1 to a type

Equations
instance WithOne.instReprWithZero {α : Type u} [Repr α] :
Equations
instance WithZero.instRepr {α : Type u} [Repr α] :
Equations
instance WithOne.instRepr {α : Type u} [Repr α] :
Equations
instance WithZero.zero {α : Type u} :
Equations
  • WithZero.zero = { zero := none }
instance WithOne.one {α : Type u} :
Equations
  • WithOne.one = { one := none }
instance WithZero.add {α : Type u} [Add α] :
Equations
instance WithOne.mul {α : Type u} [Mul α] :
Equations
instance WithZero.neg {α : Type u} [Neg α] :
Equations
instance WithOne.inv {α : Type u} [Inv α] :
Equations
instance WithZero.negZeroClass {α : Type u} [Neg α] :
Equations
instance WithOne.invOneClass {α : Type u} [Inv α] :
Equations
instance WithZero.inhabited {α : Type u} :
Equations
  • WithZero.inhabited = { default := 0 }
instance WithOne.inhabited {α : Type u} :
Equations
  • WithOne.inhabited = { default := 1 }
instance WithZero.nontrivial {α : Type u} [Nonempty α] :
Equations
  • =
instance WithOne.nontrivial {α : Type u} [Nonempty α] :
Equations
  • =
def WithZero.coe {α : Type u} :
αWithZero α

The canonical map from α into WithZero α

Equations
  • WithZero.coe = some
def WithOne.coe {α : Type u} :
αWithOne α

The canonical map from α into WithOne α

Equations
  • WithOne.coe = some
instance WithZero.coeTC {α : Type u} :
CoeTC α (WithZero α)
Equations
  • WithZero.coeTC = { coe := WithZero.coe }
instance WithOne.coeTC {α : Type u} :
CoeTC α (WithOne α)
Equations
  • WithOne.coeTC = { coe := WithOne.coe }
def WithZero.recZeroCoe {α : Type u} {C : WithZero αSort u_1} (h₁ : C 0) (h₂ : (a : α) → C a) (n : WithZero α) :
C n

Recursor for WithZero using the preferred forms 0 and ↑a.

Equations
def WithOne.recOneCoe {α : Type u} {C : WithOne αSort u_1} (h₁ : C 1) (h₂ : (a : α) → C a) (n : WithOne α) :
C n

Recursor for WithOne using the preferred forms 1 and ↑a.

Equations
@[simp]
theorem WithZero.recZeroCoe_zero {α : Type u} {C : WithZero αSort u_1} (h₁ : C 0) (h₂ : (a : α) → C a) :
WithZero.recZeroCoe h₁ h₂ 0 = h₁
@[simp]
theorem WithOne.recOneCoe_one {α : Type u} {C : WithOne αSort u_1} (h₁ : C 1) (h₂ : (a : α) → C a) :
WithOne.recOneCoe h₁ h₂ 1 = h₁
@[simp]
theorem WithZero.recZeroCoe_coe {α : Type u} {C : WithZero αSort u_1} (h₁ : C 0) (h₂ : (a : α) → C a) (a : α) :
WithZero.recZeroCoe h₁ h₂ a = h₂ a
@[simp]
theorem WithOne.recOneCoe_coe {α : Type u} {C : WithOne αSort u_1} (h₁ : C 1) (h₂ : (a : α) → C a) (a : α) :
WithOne.recOneCoe h₁ h₂ a = h₂ a
def WithZero.unzero {α : Type u} {x : WithZero α} :
x 0α

Deconstruct an x : WithZero α to the underlying value in α, given a proof that x ≠ 0.

Equations
def WithOne.unone {α : Type u} {x : WithOne α} :
x 1α

Deconstruct an x : WithOne α to the underlying value in α, given a proof that x ≠ 1.

Equations
@[simp]
theorem WithZero.unzero_coe {α : Type u} {x : α} (hx : x 0) :
@[simp]
theorem WithOne.unone_coe {α : Type u} {x : α} (hx : x 1) :
@[simp]
theorem WithZero.coe_unzero {α : Type u} {x : WithZero α} (hx : x 0) :
(WithZero.unzero hx) = x
@[simp]
theorem WithOne.coe_unone {α : Type u} {x : WithOne α} (hx : x 1) :
(WithOne.unone hx) = x
@[simp]
theorem WithZero.coe_ne_zero {α : Type u} {a : α} :
a 0
@[simp]
theorem WithOne.coe_ne_one {α : Type u} {a : α} :
a 1
@[simp]
theorem WithZero.zero_ne_coe {α : Type u} {a : α} :
0 a
@[simp]
theorem WithOne.one_ne_coe {α : Type u} {a : α} :
1 a
theorem WithZero.ne_zero_iff_exists {α : Type u} {x : WithZero α} :
x 0 ∃ (a : α), a = x
theorem WithOne.ne_one_iff_exists {α : Type u} {x : WithOne α} :
x 1 ∃ (a : α), a = x
instance WithZero.canLift {α : Type u} :
CanLift (WithZero α) α WithZero.coe fun (a : WithZero α) => a 0
Equations
  • =
instance WithOne.canLift {α : Type u} :
CanLift (WithOne α) α WithOne.coe fun (a : WithOne α) => a 1
Equations
  • =
@[simp]
theorem WithZero.coe_inj {α : Type u} {a : α} {b : α} :
a = b a = b
@[simp]
theorem WithOne.coe_inj {α : Type u} {a : α} {b : α} :
a = b a = b
theorem WithZero.cases_on {α : Type u} {P : WithZero αProp} (x : WithZero α) :
P 0(∀ (a : α), P a)P x
theorem WithOne.cases_on {α : Type u} {P : WithOne αProp} (x : WithOne α) :
P 1(∀ (a : α), P a)P x
instance WithZero.addZeroClass {α : Type u} [Add α] :
Equations
instance WithOne.mulOneClass {α : Type u} [Mul α] :
Equations
@[simp]
theorem WithZero.coe_add {α : Type u} [Add α] (a : α) (b : α) :
(a + b) = a + b
@[simp]
theorem WithOne.coe_mul {α : Type u} [Mul α] (a : α) (b : α) :
(a * b) = a * b
Equations
instance WithOne.monoid {α : Type u} [Semigroup α] :
Equations
  • WithOne.monoid = Monoid.mk npowRecAuto
Equations
Equations
@[simp]
theorem WithZero.coe_neg {α : Type u} [Neg α] (a : α) :
(-a) = -a
@[simp]
theorem WithOne.coe_inv {α : Type u} [Inv α] (a : α) :
a⁻¹ = (↑a)⁻¹