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Mathlib.Algebra.Group.WithOne.Defs

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 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 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 algebra.group.with_one.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.

def WithZero (α : Type u_1) :

Type u_1

Add an extra element 0 to a type

Instances For

def WithOne (α : Type u_1) :

Type u_1

Add an extra element 1 to a type

Instances For

instance WithOne.instReprWithZero {α : Type u} [Repr α] :

Repr (WithZero α)

instance WithZero.instReprWithZero {α : Type u} [Repr α] :

Repr (WithZero α)

abbrev WithZero.instReprWithZero.match_1 {α : Type u_1} (motive : WithZero α → Sort u_2) :

(o : WithZero α) → (Unit → motive none) → ((a : α) → motive (some a)) → motive o

Instances For

instance WithOne.instReprWithOne {α : Type u} [Repr α] :

Repr (WithOne α)

instance WithZero.monad :

instance WithOne.monad :

instance WithZero.zero {α : Type u} :

Zero (WithZero α)

instance WithOne.one {α : Type u} :

One (WithOne α)

instance WithZero.add {α : Type u} [Add α] :

Add (WithZero α)

instance WithOne.mul {α : Type u} [Mul α] :

Mul (WithOne α)

instance WithZero.neg {α : Type u} [Neg α] :

Neg (WithZero α)

instance WithOne.inv {α : Type u} [Inv α] :

Inv (WithOne α)

theorem WithZero.negZeroClass.proof_1 {α : Type u_1} [Neg α] :

-0 = -0

instance WithZero.negZeroClass {α : Type u} [Neg α] :

NegZeroClass (WithZero α)

instance WithOne.invOneClass {α : Type u} [Inv α] :

InvOneClass (WithOne α)

instance WithZero.inhabited {α : Type u} :

Inhabited (WithZero α)

instance WithOne.inhabited {α : Type u} :

Inhabited (WithOne α)

instance WithZero.nontrivial {α : Type u} [Nonempty α] :

Nontrivial (WithZero α)

instance WithOne.nontrivial {α : Type u} [Nonempty α] :

Nontrivial (WithOne α)

def WithZero.coe {α : Type u} :

α → WithZero α

The canonical map from α into WithZero α

Instances For

def WithOne.coe {α : Type u} :

α → WithOne α

The canonical map from α into WithOne α

Instances For

instance WithZero.coeTC {α : Type u} :

CoeTC α (WithZero α)

instance WithOne.coeTC {α : Type u} :

CoeTC α (WithOne α)

def WithZero.recZeroCoe {α : Type u} {C : WithZero α → Sort 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.

Instances For

def WithOne.recOneCoe {α : Type u} {C : WithOne α → Sort 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.

Instances For

def WithZero.unzero {α : Type u} {x : WithZero α} (hx : x ≠ 0) :

α

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

Instances For

def WithOne.unone {α : Type u} {x : WithOne α} (hx : x ≠ 1) :

α

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

Instances For

@[simp]

theorem WithZero.unzero_coe {α : Type u} {x : α} (hx : ↑x ≠ 0) :

WithZero.unzero hx = x

@[simp]

theorem WithOne.unone_coe {α : Type u} {x : α} (hx : ↑x ≠ 1) :

WithOne.unone hx = x

@[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, ↑a = x

theorem WithOne.ne_one_iff_exists {α : Type u} {x : WithOne α} :

x ≠ 1 ↔ ∃ a, ↑a = x

instance WithZero.canLift {α : Type u} :

CanLift (WithZero α) α WithZero.coe fun a => a ≠ 0

theorem WithZero.canLift.proof_1 {α : Type u_1} :

CanLift (WithZero α) α WithZero.coe fun a => a ≠ 0

instance WithOne.canLift {α : Type u} :

CanLift (WithOne α) α WithOne.coe fun a => a ≠ 1

@[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 α → Prop} (x : WithZero α) :

P 0 → ((a : α) → P ↑a) → P x

theorem WithOne.cases_on {α : Type u} {P : WithOne α → Prop} (x : WithOne α) :

P 1 → ((a : α) → P ↑a) → P x

theorem WithZero.addZeroClass.proof_1 {α : Type u_1} [Add α] (a : Option α) :

Option.liftOrGet (fun x x_1 => x + x_1) none a = a

theorem WithZero.addZeroClass.proof_2 {α : Type u_1} [Add α] (a : Option α) :

Option.liftOrGet (fun x x_1 => x + x_1) a none = a

instance WithZero.addZeroClass {α : Type u} [Add α] :

AddZeroClass (WithZero α)

instance WithOne.mulOneClass {α : Type u} [Mul α] :

MulOneClass (WithOne α)

theorem WithZero.addMonoid.proof_5 {α : Type u_1} [AddSemigroup α] :

∀ (n : ℕ) (x : WithZero α), nsmulRec (n + 1) x = nsmulRec (n + 1) x

theorem WithZero.addMonoid.proof_3 {α : Type u_1} [AddSemigroup α] (a : WithZero α) :

a + 0 = a

theorem WithZero.addMonoid.proof_4 {α : Type u_1} [AddSemigroup α] :

∀ (x : WithZero α), nsmulRec 0 x = nsmulRec 0 x

theorem WithZero.addMonoid.proof_2 {α : Type u_1} [AddSemigroup α] (a : WithZero α) :

0 + a = a

theorem WithZero.addMonoid.proof_1 {α : Type u_1} [AddSemigroup α] (a : Option α) (b : Option α) (c : Option α) :

Option.liftOrGet (fun x x_1 => x + x_1) (Option.liftOrGet (fun x x_1 => x + x_1) a b) c = Option.liftOrGet (fun x x_1 => x + x_1) a (Option.liftOrGet (fun x x_1 => x + x_1) b c)

instance WithZero.addMonoid {α : Type u} [AddSemigroup α] :

AddMonoid (WithZero α)

instance WithOne.monoid {α : Type u} [Semigroup α] :

Monoid (WithOne α)

theorem WithZero.addCommMonoid.proof_5 {α : Type u_1} [AddCommSemigroup α] (a : Option α) (b : Option α) :

Option.liftOrGet (fun x x_1 => x + x_1) a b = Option.liftOrGet (fun x x_1 => x + x_1) b a

instance WithZero.addCommMonoid {α : Type u} [AddCommSemigroup α] :

AddCommMonoid (WithZero α)

theorem WithZero.addCommMonoid.proof_2 {α : Type u_1} [AddCommSemigroup α] (a : WithZero α) :

a + 0 = a

theorem WithZero.addCommMonoid.proof_4 {α : Type u_1} [AddCommSemigroup α] (n : ℕ) (x : WithZero α) :

AddMonoid.nsmul (n + 1) x = x + AddMonoid.nsmul n x

theorem WithZero.addCommMonoid.proof_3 {α : Type u_1} [AddCommSemigroup α] (x : WithZero α) :

AddMonoid.nsmul 0 x = 0

theorem WithZero.addCommMonoid.proof_1 {α : Type u_1} [AddCommSemigroup α] (a : WithZero α) :

0 + a = a

instance WithOne.commMonoid {α : Type u} [CommSemigroup α] :

CommMonoid (WithOne α)

@[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

@[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

instance WithZero.one {α : Type u} [one : One α] :

One (WithZero α)

@[simp]

theorem WithZero.coe_one {α : Type u} [One α] :

↑1 = 1

instance WithZero.mulZeroClass {α : Type u} [Mul α] :

MulZeroClass (WithZero α)

@[simp]

theorem WithZero.coe_mul {α : Type u} [Mul α] {a : α} {b : α} :

↑(a * b) = ↑a * ↑b

instance WithZero.noZeroDivisors {α : Type u} [Mul α] :

NoZeroDivisors (WithZero α)

instance WithZero.semigroupWithZero {α : Type u} [Semigroup α] :

SemigroupWithZero (WithZero α)

instance WithZero.commSemigroup {α : Type u} [CommSemigroup α] :

CommSemigroup (WithZero α)

instance WithZero.mulZeroOneClass {α : Type u} [MulOneClass α] :

MulZeroOneClass (WithZero α)

instance WithZero.pow {α : Type u} [One α] [Pow α ℕ] :

Pow (WithZero α) ℕ

@[simp]

theorem WithZero.coe_pow {α : Type u} [One α] [Pow α ℕ] {a : α} (n : ℕ) :

↑(a ^ n) = ↑a ^ n

instance WithZero.monoidWithZero {α : Type u} [Monoid α] :

MonoidWithZero (WithZero α)

instance WithZero.commMonoidWithZero {α : Type u} [CommMonoid α] :

CommMonoidWithZero (WithZero α)

instance WithZero.inv {α : Type u} [Inv α] :

Inv (WithZero α)

Given an inverse operation on α there is an inverse operation on WithZero α sending 0 to 0.

@[simp]

theorem WithZero.coe_inv {α : Type u} [Inv α] (a : α) :

↑a⁻¹ = (↑a)⁻¹

@[simp]

theorem WithZero.inv_zero {α : Type u} [Inv α] :

instance WithZero.invOneClass {α : Type u} [InvOneClass α] :

InvOneClass (WithZero α)

instance WithZero.div {α : Type u} [Div α] :

Div (WithZero α)

theorem WithZero.coe_div {α : Type u} [Div α] (a : α) (b : α) :

↑(a / b) = ↑a / ↑b

instance WithZero.instPowWithZeroInt {α : Type u} [One α] [Pow α ℤ] :

Pow (WithZero α) ℤ

@[simp]

theorem WithZero.coe_zpow {α : Type u} [DivInvMonoid α] {a : α} (n : ℤ) :

↑(a ^ n) = ↑a ^ n

instance WithZero.divInvMonoid {α : Type u} [DivInvMonoid α] :

DivInvMonoid (WithZero α)

instance WithZero.divInvOneMonoid {α : Type u} [DivInvOneMonoid α] :

DivInvOneMonoid (WithZero α)

instance WithZero.groupWithZero {α : Type u} [Group α] :

GroupWithZero (WithZero α)

if G is a group then WithZero G is a group with zero.

instance WithZero.commGroupWithZero {α : Type u} [CommGroup α] :

CommGroupWithZero (WithZero α)

instance WithZero.addMonoidWithOne {α : Type u} [AddMonoidWithOne α] :

AddMonoidWithOne (WithZero α)

instance WithZero.semiring {α : Type u} [Semiring α] :

Semiring (WithZero α)