Documentation

Mathlib.Algebra.Group.ULift

`ULift` instances for groups and monoids #

This file defines instances for group, monoid, semigroup and related structures on ULift types.

(Recall ULift α is just a "copy" of a type α in a higher universe.)

We also provide MulEquiv.ulift : ULift R ≃* R (and its additive analogue).

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

One (ULift.{u_1, u} α)

Equations

ULift.one = { one := { down := 1 } }

instance ULift.zero {α : Type u} [Zero α] :

Zero (ULift.{u_1, u} α)

Equations

ULift.zero = { zero := { down := 0 } }

@[simp]

theorem ULift.one_down {α : Type u} [One α] :

down 1 = 1

@[simp]

theorem ULift.zero_down {α : Type u} [Zero α] :

down 0 = 0

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

Mul (ULift.{u_1, u} α)

Equations

ULift.mul = { mul := fun (f g : ULift.{?u.3, ?u.4} α) => { down := f.down * g.down } }

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

Add (ULift.{u_1, u} α)

Equations

ULift.add = { add := fun (f g : ULift.{?u.3, ?u.4} α) => { down := f.down + g.down } }

@[simp]

theorem ULift.mul_down {α : Type u} {x y : ULift.{w, u} α} [Mul α] :

(x * y).down = x.down * y.down

@[simp]

theorem ULift.add_down {α : Type u} {x y : ULift.{w, u} α} [Add α] :

(x + y).down = x.down + y.down

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

Div (ULift.{u_1, u} α)

Equations

ULift.div = { div := fun (f g : ULift.{?u.3, ?u.4} α) => { down := f.down / g.down } }

instance ULift.sub {α : Type u} [Sub α] :

Sub (ULift.{u_1, u} α)

Equations

ULift.sub = { sub := fun (f g : ULift.{?u.3, ?u.4} α) => { down := f.down - g.down } }

@[simp]

theorem ULift.div_down {α : Type u} {x y : ULift.{w, u} α} [Div α] :

(x / y).down = x.down / y.down

@[simp]

theorem ULift.sub_down {α : Type u} {x y : ULift.{w, u} α} [Sub α] :

(x - y).down = x.down - y.down

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

Inv (ULift.{u_1, u} α)

Equations

ULift.inv = { inv := fun (f : ULift.{?u.3, ?u.4} α) => { down := f.down⁻¹ } }

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

Neg (ULift.{u_1, u} α)

Equations

ULift.neg = { neg := fun (f : ULift.{?u.3, ?u.4} α) => { down := -f.down } }

@[simp]

theorem ULift.inv_down {α : Type u} {x : ULift.{w, u} α} [Inv α] :

x⁻¹.down = x.down⁻¹

@[simp]

theorem ULift.neg_down {α : Type u} {x : ULift.{w, u} α} [Neg α] :

(-x).down = -x.down

instance ULift.smul {α : Type u} {β : Type v} [SMul α β] :

SMul α (ULift.{u_1, v} β)

Equations

ULift.smul = { smul := fun (n : α) (x : ULift.{?u.4, ?u.5} β) => { down := n • x.down } }

instance ULift.vadd {α : Type u} {β : Type v} [VAdd α β] :

VAdd α (ULift.{u_1, v} β)

Equations

ULift.vadd = { vadd := fun (n : α) (x : ULift.{?u.4, ?u.5} β) => { down := n +ᵥ x.down } }

@[simp]

theorem ULift.smul_down {α : Type u} {β : Type v} [SMul α β] (a : α) (b : ULift.{w, v} β) :

(a • b).down = a • b.down

@[simp]

theorem ULift.vadd_down {α : Type u} {β : Type v} [VAdd α β] (a : α) (b : ULift.{w, v} β) :

(a +ᵥ b).down = a +ᵥ b.down

instance ULift.pow {α : Type u} {β : Type v} [Pow α β] :

Pow (ULift.{u_1, u} α) β

Equations

ULift.pow = { pow := fun (x : ULift.{?u.4, ?u.6} α) (n : β) => { down := x.down ^ n } }

@[simp]

theorem ULift.pow_down {α : Type u} {β : Type v} [Pow α β] (a : ULift.{w, u} α) (b : β) :

(a ^ b).down = a.down ^ b

def MulEquiv.ulift {α : Type u} [Mul α] :

ULift.{u_1, u} α ≃* α

The multiplicative equivalence between ULift α and α.

Equations

MulEquiv.ulift = { toEquiv := Equiv.ulift, map_mul' := ⋯ }

Instances For

def AddEquiv.ulift {α : Type u} [Add α] :

ULift.{u_1, u} α ≃+ α

The additive equivalence between ULift α and α.

Equations

AddEquiv.ulift = { toEquiv := Equiv.ulift, map_add' := ⋯ }

Instances For

instance ULift.semigroup {α : Type u} [Semigroup α] :

Semigroup (ULift.{u_1, u} α)

Equations

ULift.semigroup = Function.Injective.semigroup ⇑MulEquiv.ulift ⋯ ⋯

instance ULift.addSemigroup {α : Type u} [AddSemigroup α] :

AddSemigroup (ULift.{u_1, u} α)

Equations

ULift.addSemigroup = Function.Injective.addSemigroup ⇑AddEquiv.ulift ⋯ ⋯

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

CommSemigroup (ULift.{u_1, u} α)

Equations

ULift.commSemigroup = Function.Injective.commSemigroup ⇑Equiv.ulift ⋯ ⋯

instance ULift.addCommSemigroup {α : Type u} [AddCommSemigroup α] :

AddCommSemigroup (ULift.{u_1, u} α)

Equations

ULift.addCommSemigroup = Function.Injective.addCommSemigroup ⇑Equiv.ulift ⋯ ⋯

instance ULift.mulOneClass {α : Type u} [MulOneClass α] :

MulOneClass (ULift.{u_1, u} α)

Equations

ULift.mulOneClass = Function.Injective.mulOneClass ⇑Equiv.ulift ⋯ ⋯ ⋯

instance ULift.addZeroClass {α : Type u} [AddZeroClass α] :

AddZeroClass (ULift.{u_1, u} α)

Equations

ULift.addZeroClass = Function.Injective.addZeroClass ⇑Equiv.ulift ⋯ ⋯ ⋯

instance ULift.monoid {α : Type u} [Monoid α] :

Monoid (ULift.{u_1, u} α)

Equations

ULift.monoid = Function.Injective.monoid ⇑Equiv.ulift ⋯ ⋯ ⋯ ⋯

instance ULift.addMonoid {α : Type u} [AddMonoid α] :

AddMonoid (ULift.{u_1, u} α)

Equations

ULift.addMonoid = Function.Injective.addMonoid ⇑Equiv.ulift ⋯ ⋯ ⋯ ⋯

instance ULift.commMonoid {α : Type u} [CommMonoid α] :

CommMonoid (ULift.{u_1, u} α)

Equations

ULift.commMonoid = Function.Injective.commMonoid ⇑Equiv.ulift ⋯ ⋯ ⋯ ⋯

instance ULift.addCommMonoid {α : Type u} [AddCommMonoid α] :

AddCommMonoid (ULift.{u_1, u} α)

Equations

ULift.addCommMonoid = Function.Injective.addCommMonoid ⇑Equiv.ulift ⋯ ⋯ ⋯ ⋯

instance ULift.instNatCast {α : Type u} [NatCast α] :

NatCast (ULift.{u_1, u} α)

Equations

ULift.instNatCast = { natCast := fun (x : ℕ) => { down := ↑x } }

instance ULift.instIntCast {α : Type u} [IntCast α] :

IntCast (ULift.{u_1, u} α)

Equations

ULift.instIntCast = { intCast := fun (x : ℤ) => { down := ↑x } }

@[simp]

theorem ULift.up_natCast {α : Type u} [NatCast α] (n : ℕ) :

{ down := ↑n } = ↑n

@[simp]

theorem ULift.up_ofNat {α : Type u} [NatCast α] (n : ℕ) [n.AtLeastTwo] :

{ down := OfNat.ofNat n } = OfNat.ofNat n

@[simp]

theorem ULift.up_intCast {α : Type u} [IntCast α] (n : ℤ) :

{ down := ↑n } = ↑n

@[simp]

theorem ULift.down_natCast {α : Type u} [NatCast α] (n : ℕ) :

(↑n).down = ↑n

@[simp]

theorem ULift.down_ofNat {α : Type u} [NatCast α] (n : ℕ) [n.AtLeastTwo] :

(OfNat.ofNat n).down = OfNat.ofNat n

@[simp]

theorem ULift.down_intCast {α : Type u} [IntCast α] (n : ℤ) :

(↑n).down = ↑n

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

AddMonoidWithOne (ULift.{u_1, u} α)

Equations

ULift.addMonoidWithOne = { natCast := fun (x : ℕ) => { down := ↑x }, toAddMonoid := ULift.addMonoid, toOne := ULift.one, natCast_zero := ⋯, natCast_succ := ⋯ }

instance ULift.addCommMonoidWithOne {α : Type u} [AddCommMonoidWithOne α] :

AddCommMonoidWithOne (ULift.{u_1, u} α)

Equations

ULift.addCommMonoidWithOne = { toAddMonoidWithOne := ULift.addMonoidWithOne, add_comm := ⋯ }

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

DivInvMonoid (ULift.{u_1, u} α)

Equations

ULift.divInvMonoid = Function.Injective.divInvMonoid ⇑Equiv.ulift ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯

instance ULift.subNegAddMonoid {α : Type u} [SubNegMonoid α] :

SubNegMonoid (ULift.{u_1, u} α)

Equations

ULift.subNegAddMonoid = Function.Injective.subNegMonoid ⇑Equiv.ulift ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯

instance ULift.group {α : Type u} [Group α] :

Group (ULift.{u_1, u} α)

Equations

ULift.group = Function.Injective.group ⇑Equiv.ulift ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯

instance ULift.addGroup {α : Type u} [AddGroup α] :

AddGroup (ULift.{u_1, u} α)

Equations

ULift.addGroup = Function.Injective.addGroup ⇑Equiv.ulift ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯

instance ULift.commGroup {α : Type u} [CommGroup α] :

CommGroup (ULift.{u_1, u} α)

Equations

ULift.commGroup = Function.Injective.commGroup ⇑Equiv.ulift ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯

instance ULift.addCommGroup {α : Type u} [AddCommGroup α] :

AddCommGroup (ULift.{u_1, u} α)

Equations

ULift.addCommGroup = Function.Injective.addCommGroup ⇑Equiv.ulift ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯

instance ULift.addGroupWithOne {α : Type u} [AddGroupWithOne α] :

AddGroupWithOne (ULift.{u_1, u} α)

Equations

One or more equations did not get rendered due to their size.

instance ULift.addCommGroupWithOne {α : Type u} [AddCommGroupWithOne α] :

AddCommGroupWithOne (ULift.{u_1, u} α)

Equations

One or more equations did not get rendered due to their size.

instance ULift.leftCancelSemigroup {α : Type u} [LeftCancelSemigroup α] :

LeftCancelSemigroup (ULift.{u_1, u} α)

Equations

ULift.leftCancelSemigroup = Function.Injective.leftCancelSemigroup ⇑Equiv.ulift ⋯ ⋯

instance ULift.addLeftCancelSemigroup {α : Type u} [AddLeftCancelSemigroup α] :

AddLeftCancelSemigroup (ULift.{u_1, u} α)

Equations

ULift.addLeftCancelSemigroup = Function.Injective.addLeftCancelSemigroup ⇑Equiv.ulift ⋯ ⋯

instance ULift.rightCancelSemigroup {α : Type u} [RightCancelSemigroup α] :

RightCancelSemigroup (ULift.{u_1, u} α)

Equations

ULift.rightCancelSemigroup = Function.Injective.rightCancelSemigroup ⇑Equiv.ulift ⋯ ⋯

instance ULift.addRightCancelSemigroup {α : Type u} [AddRightCancelSemigroup α] :

AddRightCancelSemigroup (ULift.{u_1, u} α)

Equations

ULift.addRightCancelSemigroup = Function.Injective.addRightCancelSemigroup ⇑Equiv.ulift ⋯ ⋯

instance ULift.leftCancelMonoid {α : Type u} [LeftCancelMonoid α] :

LeftCancelMonoid (ULift.{u_1, u} α)

Equations

ULift.leftCancelMonoid = Function.Injective.leftCancelMonoid ⇑Equiv.ulift ⋯ ⋯ ⋯ ⋯

instance ULift.addLeftCancelMonoid {α : Type u} [AddLeftCancelMonoid α] :

AddLeftCancelMonoid (ULift.{u_1, u} α)

Equations

ULift.addLeftCancelMonoid = Function.Injective.addLeftCancelMonoid ⇑Equiv.ulift ⋯ ⋯ ⋯ ⋯

instance ULift.rightCancelMonoid {α : Type u} [RightCancelMonoid α] :

RightCancelMonoid (ULift.{u_1, u} α)

Equations

ULift.rightCancelMonoid = Function.Injective.rightCancelMonoid ⇑Equiv.ulift ⋯ ⋯ ⋯ ⋯

instance ULift.addRightCancelMonoid {α : Type u} [AddRightCancelMonoid α] :

AddRightCancelMonoid (ULift.{u_1, u} α)

Equations

ULift.addRightCancelMonoid = Function.Injective.addRightCancelMonoid ⇑Equiv.ulift ⋯ ⋯ ⋯ ⋯

instance ULift.cancelMonoid {α : Type u} [CancelMonoid α] :

CancelMonoid (ULift.{u_1, u} α)

Equations

ULift.cancelMonoid = Function.Injective.cancelMonoid ⇑Equiv.ulift ⋯ ⋯ ⋯ ⋯

instance ULift.addCancelMonoid {α : Type u} [AddCancelMonoid α] :

AddCancelMonoid (ULift.{u_1, u} α)

Equations

ULift.addCancelMonoid = Function.Injective.addCancelMonoid ⇑Equiv.ulift ⋯ ⋯ ⋯ ⋯

instance ULift.cancelCommMonoid {α : Type u} [CancelCommMonoid α] :

CancelCommMonoid (ULift.{u_1, u} α)

Equations

ULift.cancelCommMonoid = Function.Injective.cancelCommMonoid ⇑Equiv.ulift ⋯ ⋯ ⋯ ⋯

instance ULift.addCancelCommMonoid {α : Type u} [AddCancelCommMonoid α] :

AddCancelCommMonoid (ULift.{u_1, u} α)

Equations

ULift.addCancelCommMonoid = Function.Injective.addCancelCommMonoid ⇑Equiv.ulift ⋯ ⋯ ⋯ ⋯

instance ULift.nontrivial {α : Type u} [Nontrivial α] :

Nontrivial (ULift.{u_1, u} α)