Type tags that turn additive structures into multiplicative, and vice versa

We define two type tags:

• Additive α: turns any multiplicative structure on α into the corresponding additive structure on Additive α;
• Multiplicative α: turns any additive structure on α into the corresponding multiplicative structure on Multiplicative α.

We also define instances Additive.* and Multiplicative.* that actually transfer the structures.

See also

This file is similar to Order.Synonym.

Porting notes

• Since bundled morphism applications that rely on CoeFun currently don't work, they are ported as toFoo a rather than a.toFoo for now. (https://github.com/leanprover/lean4/issues/1910)

def Additive (α : Type u_1) : Type u_1

If α carries some multiplicative structure, then Additive α carries the corresponding additive structure.

Equations

• Additive α = α

def Multiplicative (α : Type u_1) : Type u_1

If α carries some additive structure, then Multiplicative α carries the corresponding multiplicative structure.

Equations

• Multiplicative α = α

def Additive.ofMul {α : Type u} : α ≃ Additive α

Reinterpret x : α as an element of Additive α.

Equations

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

def Additive.toMul {α : Type u} : Additive α ≃ α

Reinterpret x : Additive α as an element of α.

Equations

• Additive.toMul = Additive.ofMul.symm

@[simp]

theorem Additive.ofMul_symm_eq {α : Type u} : Additive.ofMul.symm = Additive.toMul

@[simp]

theorem Additive.toMul_symm_eq {α : Type u} : Additive.toMul.symm = Additive.ofMul

def Multiplicative.ofAdd {α : Type u} : α ≃ Multiplicative α

Reinterpret x : α as an element of Multiplicative α.

Equations

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

def Multiplicative.toAdd {α : Type u} : Multiplicative α ≃ α

Reinterpret x : Multiplicative α as an element of α.

Equations

• Multiplicative.toAdd = Multiplicative.ofAdd.symm

@[simp]

theorem Multiplicative.ofAdd_symm_eq {α : Type u} : Multiplicative.ofAdd.symm = Multiplicative.toAdd

@[simp]

theorem Multiplicative.toAdd_symm_eq {α : Type u} : Multiplicative.toAdd.symm = Multiplicative.ofAdd

@[simp]

theorem toAdd_ofAdd {α : Type u} (x : α) : Multiplicative.toAdd (Multiplicative.ofAdd x) = x

@[simp]

theorem ofAdd_toAdd {α : Type u} (x : Multiplicative α) : Multiplicative.ofAdd (Multiplicative.toAdd x) = x

@[simp]

theorem toMul_ofMul {α : Type u} (x : α) : Additive.toMul (Additive.ofMul x) = x

@[simp]

theorem ofMul_toMul {α : Type u} (x : Additive α) : Additive.ofMul (Additive.toMul x) = x

instance instSubsingletonAdditive {α : Type u} [Subsingleton α] : Subsingleton (Additive α)

Equations

• (_ : Subsingleton (Additive α)) = (_ : Subsingleton (Additive α))

instance instSubsingletonMultiplicative {α : Type u} [Subsingleton α] : Subsingleton (Multiplicative α)

Equations

• (_ : Subsingleton (Multiplicative α)) = (_ : Subsingleton (Multiplicative α))

instance instInhabitedAdditive {α : Type u} [Inhabited α] : Inhabited (Additive α)

Equations

• instInhabitedAdditive = { default := Additive.ofMul default }

instance instInhabitedMultiplicative {α : Type u} [Inhabited α] : Inhabited (Multiplicative α)

Equations

• instInhabitedMultiplicative = { default := Multiplicative.ofAdd default }

instance instUniqueAdditive {α : Type u} [Unique α] : Unique (Additive α)

Equations

• instUniqueAdditive = Equiv.unique Additive.toMul

instance instUniqueMultiplicative {α : Type u} [Unique α] : Unique (Multiplicative α)

Equations

• instUniqueMultiplicative = Equiv.unique Multiplicative.toAdd

instance instFiniteAdditive {α : Type u} [Finite α] : Finite (Additive α)

Equations

• (_ : Finite (Additive α)) = (_ : Finite (Additive α))

instance instFiniteMultiplicative {α : Type u} [Finite α] : Finite (Multiplicative α)

Equations

• (_ : Finite (Multiplicative α)) = (_ : Finite (Multiplicative α))

instance instInfiniteAdditive {α : Type u} [h : Infinite α] : Infinite (Additive α)

Equations

• (_ : Infinite (Additive α)) = h

instance instInfiniteMultiplicative {α : Type u} [h : Infinite α] : Infinite (Multiplicative α)

Equations

• (_ : Infinite (Multiplicative α)) = h

instance instDecidableEqMultiplicative {α : Type u} [h : DecidableEq α] : DecidableEq (Multiplicative α)

Equations

• instDecidableEqMultiplicative = h

instance instDecidableEqAdditive {α : Type u} [h : DecidableEq α] : DecidableEq (Additive α)

Equations

• instDecidableEqAdditive = h

instance Additive.instNontrivial {α : Type u} [Nontrivial α] : Nontrivial (Additive α)

Equations

• (_ : Nontrivial (Additive α)) = (_ : Nontrivial (Additive α))

instance Multiplicative.instNontrivial {α : Type u} [Nontrivial α] : Nontrivial (Multiplicative α)

Equations

• (_ : Nontrivial (Multiplicative α)) = (_ : Nontrivial (Multiplicative α))

instance Additive.add {α : Type u} [Mul α] : Add (Additive α)

Equations

• Additive.add = { add := fun (x y : Additive α) => Additive.ofMul (Additive.toMul x * Additive.toMul y) }

instance Multiplicative.mul {α : Type u} [Add α] : Mul (Multiplicative α)

Equations

• Multiplicative.mul = { mul := fun (x y : Multiplicative α) => Multiplicative.ofAdd (Multiplicative.toAdd x + Multiplicative.toAdd y) }

@[simp]

theorem ofAdd_add {α : Type u} [Add α] (x : α) (y : α) : Multiplicative.ofAdd (x + y) = Multiplicative.ofAdd x * Multiplicative.ofAdd y

@[simp]

theorem toAdd_mul {α : Type u} [Add α] (x : Multiplicative α) (y : Multiplicative α) : Multiplicative.toAdd (x * y) = Multiplicative.toAdd x + Multiplicative.toAdd y

@[simp]

theorem ofMul_mul {α : Type u} [Mul α] (x : α) (y : α) : Additive.ofMul (x * y) = Additive.ofMul x + Additive.ofMul y

@[simp]

theorem toMul_add {α : Type u} [Mul α] (x : Additive α) (y : Additive α) : Additive.toMul (x + y) = Additive.toMul x * Additive.toMul y

instance Additive.addSemigroup {α : Type u} [Semigroup α] : AddSemigroup (Additive α)

Equations

• Additive.addSemigroup = let src := Additive.add; AddSemigroup.mk (_ : ∀ (a b c : α), a * b * c = a * (b * c))

instance Multiplicative.semigroup {α : Type u} [AddSemigroup α] : Semigroup (Multiplicative α)

Equations

• Multiplicative.semigroup = let src := Multiplicative.mul; Semigroup.mk (_ : ∀ (a b c : α), a + b + c = a + (b + c))

instance Additive.addCommSemigroup {α : Type u} [CommSemigroup α] : AddCommSemigroup (Additive α)

Equations

• Additive.addCommSemigroup = let src := Additive.addSemigroup; AddCommSemigroup.mk (_ : ∀ (a b : α), a * b = b * a)

instance Multiplicative.commSemigroup {α : Type u} [AddCommSemigroup α] : CommSemigroup (Multiplicative α)

Equations

• Multiplicative.commSemigroup = let src := Multiplicative.semigroup; CommSemigroup.mk (_ : ∀ (a b : α), a + b = b + a)

instance Additive.isLeftCancelAdd {α : Type u} [Mul α] [IsLeftCancelMul α] : IsLeftCancelAdd (Additive α)

Equations

• (_ : IsLeftCancelAdd (Additive α)) = (_ : IsLeftCancelAdd (Additive α))

instance Multiplicative.isLeftCancelMul {α : Type u} [Add α] [IsLeftCancelAdd α] : IsLeftCancelMul (Multiplicative α)

Equations

• (_ : IsLeftCancelMul (Multiplicative α)) = (_ : IsLeftCancelMul (Multiplicative α))

instance Additive.isRightCancelAdd {α : Type u} [Mul α] [IsRightCancelMul α] : IsRightCancelAdd (Additive α)

Equations

• (_ : IsRightCancelAdd (Additive α)) = (_ : IsRightCancelAdd (Additive α))

instance Multiplicative.isRightCancelMul {α : Type u} [Add α] [IsRightCancelAdd α] : IsRightCancelMul (Multiplicative α)

Equations

• (_ : IsRightCancelMul (Multiplicative α)) = (_ : IsRightCancelMul (Multiplicative α))

instance Additive.isCancelAdd {α : Type u} [Mul α] [IsCancelMul α] : IsCancelAdd (Additive α)

Equations

• (_ : IsCancelAdd (Additive α)) = (_ : IsCancelAdd (Additive α))

instance Multiplicative.isCancelMul {α : Type u} [Add α] [IsCancelAdd α] : IsCancelMul (Multiplicative α)

Equations

• (_ : IsCancelMul (Multiplicative α)) = (_ : IsCancelMul (Multiplicative α))

instance Additive.addLeftCancelSemigroup {α : Type u} [LeftCancelSemigroup α] : AddLeftCancelSemigroup (Additive α)

Equations

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

instance Multiplicative.leftCancelSemigroup {α : Type u} [AddLeftCancelSemigroup α] : LeftCancelSemigroup (Multiplicative α)

Equations

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

instance Additive.addRightCancelSemigroup {α : Type u} [RightCancelSemigroup α] : AddRightCancelSemigroup (Additive α)

Equations

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

instance Multiplicative.rightCancelSemigroup {α : Type u} [AddRightCancelSemigroup α] : RightCancelSemigroup (Multiplicative α)

Equations

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

instance instZeroAdditive {α : Type u} [One α] : Zero (Additive α)

Equations

• instZeroAdditive = { zero := Additive.ofMul 1 }

@[simp]

theorem ofMul_one {α : Type u} [One α] : Additive.ofMul 1 = 0

@[simp]

theorem ofMul_eq_zero {A : Type u_1} [One A] {x : A} : Additive.ofMul x = 0 ↔ x = 1

@[simp]

theorem toMul_zero {α : Type u} [One α] : Additive.toMul 0 = 1

instance instOneMultiplicative {α : Type u} [Zero α] : One (Multiplicative α)

Equations

• instOneMultiplicative = { one := Multiplicative.ofAdd 0 }

@[simp]

theorem ofAdd_zero {α : Type u} [Zero α] : Multiplicative.ofAdd 0 = 1

@[simp]

theorem ofAdd_eq_one {A : Type u_1} [Zero A] {x : A} : Multiplicative.ofAdd x = 1 ↔ x = 0

@[simp]

theorem toAdd_one {α : Type u} [Zero α] : Multiplicative.toAdd 1 = 0

instance Additive.addZeroClass {α : Type u} [MulOneClass α] : AddZeroClass (Additive α)

Equations

• Additive.addZeroClass = AddZeroClass.mk (_ : ∀ (a : α), 1 * a = a) (_ : ∀ (a : α), a * 1 = a)

instance Multiplicative.mulOneClass {α : Type u} [AddZeroClass α] : MulOneClass (Multiplicative α)

Equations

• Multiplicative.mulOneClass = MulOneClass.mk (_ : ∀ (a : α), 0 + a = a) (_ : ∀ (a : α), a + 0 = a)

instance Additive.addMonoid {α : Type u} [h : Monoid α] : AddMonoid (Additive α)

Equations

• Additive.addMonoid = let src := Additive.addZeroClass; let src_1 := Additive.addSemigroup; AddMonoid.mk (_ : ∀ (a : Additive α), 0 + a = a) (_ : ∀ (a : Additive α), a + 0 = a) Monoid.npow

instance Multiplicative.monoid {α : Type u} [h : AddMonoid α] : Monoid (Multiplicative α)

Equations

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

@[simp]

theorem ofMul_pow {α : Type u} [Monoid α] (n : ℕ) (a : α) : Additive.ofMul (a ^ n) = n • Additive.ofMul a

@[simp]

theorem toMul_nsmul {α : Type u} [Monoid α] (n : ℕ) (a : Additive α) : Additive.toMul (n • a) = Additive.toMul a ^ n

@[simp]

theorem ofAdd_nsmul {α : Type u} [AddMonoid α] (n : ℕ) (a : α) : Multiplicative.ofAdd (n • a) = Multiplicative.ofAdd a ^ n

@[simp]

theorem toAdd_pow {α : Type u} [AddMonoid α] (a : Multiplicative α) (n : ℕ) : Multiplicative.toAdd (a ^ n) = n • Multiplicative.toAdd a

instance Additive.addLeftCancelMonoid {α : Type u} [LeftCancelMonoid α] : AddLeftCancelMonoid (Additive α)

Equations

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

instance Multiplicative.leftCancelMonoid {α : Type u} [AddLeftCancelMonoid α] : LeftCancelMonoid (Multiplicative α)

Equations

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

instance Additive.addRightCancelMonoid {α : Type u} [RightCancelMonoid α] : AddRightCancelMonoid (Additive α)

Equations

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

instance Multiplicative.rightCancelMonoid {α : Type u} [AddRightCancelMonoid α] : RightCancelMonoid (Multiplicative α)

Equations

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

instance Additive.addCommMonoid {α : Type u} [CommMonoid α] : AddCommMonoid (Additive α)

Equations

• Additive.addCommMonoid = let src := Additive.addMonoid; let src_1 := Additive.addCommSemigroup; AddCommMonoid.mk (_ : ∀ (a b : Additive α), a + b = b + a)

instance Multiplicative.commMonoid {α : Type u} [AddCommMonoid α] : CommMonoid (Multiplicative α)

Equations

• Multiplicative.commMonoid = let src := Multiplicative.monoid; let src_1 := Multiplicative.commSemigroup; CommMonoid.mk (_ : ∀ (a b : Multiplicative α), a * b = b * a)

instance Additive.neg {α : Type u} [Inv α] : Neg (Additive α)

Equations

• Additive.neg = { neg := fun (x : Additive α) => Multiplicative.ofAdd (Additive.toMul x)⁻¹ }

@[simp]

theorem ofMul_inv {α : Type u} [Inv α] (x : α) : Additive.ofMul x⁻¹ = -Additive.ofMul x

@[simp]

theorem toMul_neg {α : Type u} [Inv α] (x : Additive α) : Additive.toMul (-x) = (Additive.toMul x)⁻¹

instance Multiplicative.inv {α : Type u} [Neg α] : Inv (Multiplicative α)

Equations

• Multiplicative.inv = { inv := fun (x : Multiplicative α) => Additive.ofMul (-Multiplicative.toAdd x) }

@[simp]

theorem ofAdd_neg {α : Type u} [Neg α] (x : α) : Multiplicative.ofAdd (-x) = (Multiplicative.ofAdd x)⁻¹

@[simp]

theorem toAdd_inv {α : Type u} [Neg α] (x : Multiplicative α) : Multiplicative.toAdd x⁻¹ = -Multiplicative.toAdd x

instance Additive.sub {α : Type u} [Div α] : Sub (Additive α)

Equations

• Additive.sub = { sub := fun (x y : Additive α) => Additive.ofMul (Additive.toMul x / Additive.toMul y) }

instance Multiplicative.div {α : Type u} [Sub α] : Div (Multiplicative α)

Equations

• Multiplicative.div = { div := fun (x y : Multiplicative α) => Multiplicative.ofAdd (Multiplicative.toAdd x - Multiplicative.toAdd y) }

@[simp]

theorem ofAdd_sub {α : Type u} [Sub α] (x : α) (y : α) : Multiplicative.ofAdd (x - y) = Multiplicative.ofAdd x / Multiplicative.ofAdd y

@[simp]

theorem toAdd_div {α : Type u} [Sub α] (x : Multiplicative α) (y : Multiplicative α) : Multiplicative.toAdd (x / y) = Multiplicative.toAdd x - Multiplicative.toAdd y

@[simp]

theorem ofMul_div {α : Type u} [Div α] (x : α) (y : α) : Additive.ofMul (x / y) = Additive.ofMul x - Additive.ofMul y

@[simp]

theorem toMul_sub {α : Type u} [Div α] (x : Additive α) (y : Additive α) : Additive.toMul (x - y) = Additive.toMul x / Additive.toMul y

instance Additive.involutiveNeg {α : Type u} [InvolutiveInv α] : InvolutiveNeg (Additive α)

Equations

• Additive.involutiveNeg = let src := Additive.neg; InvolutiveNeg.mk (_ : ∀ (a : α), a⁻¹⁻¹ = a)

instance Multiplicative.involutiveInv {α : Type u} [InvolutiveNeg α] : InvolutiveInv (Multiplicative α)

Equations

• Multiplicative.involutiveInv = let src := Multiplicative.inv; InvolutiveInv.mk (_ : ∀ (a : α), - -a = a)

instance Additive.subNegMonoid {α : Type u} [DivInvMonoid α] : SubNegMonoid (Additive α)

Equations

• Additive.subNegMonoid = let src := Additive.neg; let src_1 := Additive.sub; let src_2 := Additive.addMonoid; SubNegMonoid.mk DivInvMonoid.zpow

instance Multiplicative.divInvMonoid {α : Type u} [SubNegMonoid α] : DivInvMonoid (Multiplicative α)

Equations

• Multiplicative.divInvMonoid = let src := Multiplicative.inv; let src_1 := Multiplicative.div; let src_2 := Multiplicative.monoid; DivInvMonoid.mk SubNegMonoid.zsmul

@[simp]

theorem ofMul_zpow {α : Type u} [DivInvMonoid α] (z : ℤ) (a : α) : Additive.ofMul (a ^ z) = z • Additive.ofMul a

@[simp]

theorem toMul_zsmul {α : Type u} [DivInvMonoid α] (z : ℤ) (a : Additive α) : Additive.toMul (z • a) = Additive.toMul a ^ z

@[simp]

theorem ofAdd_zsmul {α : Type u} [SubNegMonoid α] (z : ℤ) (a : α) : Multiplicative.ofAdd (z • a) = Multiplicative.ofAdd a ^ z

@[simp]

theorem toAdd_zpow {α : Type u} [SubNegMonoid α] (a : Multiplicative α) (z : ℤ) : Multiplicative.toAdd (a ^ z) = z • Multiplicative.toAdd a

instance Additive.subtractionMonoid {α : Type u} [DivisionMonoid α] : SubtractionMonoid (Additive α)

Equations

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

instance Multiplicative.divisionMonoid {α : Type u} [SubtractionMonoid α] : DivisionMonoid (Multiplicative α)

Equations

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

instance Additive.subtractionCommMonoid {α : Type u} [DivisionCommMonoid α] : SubtractionCommMonoid (Additive α)

Equations

• Additive.subtractionCommMonoid = let src := Additive.subtractionMonoid; let src_1 := Additive.addCommSemigroup; SubtractionCommMonoid.mk (_ : ∀ (a b : Additive α), a + b = b + a)

instance Multiplicative.divisionCommMonoid {α : Type u} [SubtractionCommMonoid α] : DivisionCommMonoid (Multiplicative α)

Equations

• Multiplicative.divisionCommMonoid = let src := Multiplicative.divisionMonoid; let src_1 := Multiplicative.commSemigroup; DivisionCommMonoid.mk (_ : ∀ (a b : Multiplicative α), a * b = b * a)

instance Additive.addGroup {α : Type u} [Group α] : AddGroup (Additive α)

Equations

• Additive.addGroup = let src := Additive.subNegMonoid; AddGroup.mk (_ : ∀ (a : α), a⁻¹ * a = 1)

instance Multiplicative.group {α : Type u} [AddGroup α] : Group (Multiplicative α)

Equations

• Multiplicative.group = let src := Multiplicative.divInvMonoid; Group.mk (_ : ∀ (a : α), -a + a = 0)

instance Additive.addCommGroup {α : Type u} [CommGroup α] : AddCommGroup (Additive α)

Equations

• Additive.addCommGroup = let src := Additive.addGroup; let src_1 := Additive.addCommMonoid; AddCommGroup.mk (_ : ∀ (a b : Additive α), a + b = b + a)

instance Multiplicative.commGroup {α : Type u} [AddCommGroup α] : CommGroup (Multiplicative α)

Equations

• Multiplicative.commGroup = let src := Multiplicative.group; let src_1 := Multiplicative.commMonoid; CommGroup.mk (_ : ∀ (a b : Multiplicative α), a * b = b * a)

@[simp]

theorem AddMonoidHom.toMultiplicative_apply_apply {α : Type u} {β : Type v} [AddZeroClass α] [AddZeroClass β] (f : α →+ β) (a : Multiplicative α) : (AddMonoidHom.toMultiplicative f) a = Multiplicative.ofAdd (f (Multiplicative.toAdd a))

@[simp]

theorem AddMonoidHom.toMultiplicative_symm_apply_apply {α : Type u} {β : Type v} [AddZeroClass α] [AddZeroClass β] (f : Multiplicative α →* Multiplicative β) (a : α) : (AddMonoidHom.toMultiplicative.symm f) a = Multiplicative.toAdd (f (Multiplicative.ofAdd a))

def AddMonoidHom.toMultiplicative {α : Type u} {β : Type v} [AddZeroClass α] [AddZeroClass β] : (α →+ β) ≃ (Multiplicative α →* Multiplicative β)

Reinterpret α →+ β as Multiplicative α →* Multiplicative β.

Equations

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

@[simp]

theorem MonoidHom.toAdditive_apply_apply {α : Type u} {β : Type v} [MulOneClass α] [MulOneClass β] (f : α →* β) (a : Additive α) : (MonoidHom.toAdditive f) a = Additive.ofMul (f (Additive.toMul a))

@[simp]

theorem MonoidHom.toAdditive_symm_apply_apply {α : Type u} {β : Type v} [MulOneClass α] [MulOneClass β] (f : Additive α →+ Additive β) (a : α) : (MonoidHom.toAdditive.symm f) a = Additive.toMul (f (Additive.ofMul a))

def MonoidHom.toAdditive {α : Type u} {β : Type v} [MulOneClass α] [MulOneClass β] : (α →* β) ≃ (Additive α →+ Additive β)

Reinterpret α →* β as Additive α →+ Additive β.

Equations

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

@[simp]

theorem AddMonoidHom.toMultiplicative'_apply_apply {α : Type u} {β : Type v} [MulOneClass α] [AddZeroClass β] (f : Additive α →+ β) (a : α) : (AddMonoidHom.toMultiplicative' f) a = Multiplicative.ofAdd (f (Additive.ofMul a))

@[simp]

theorem AddMonoidHom.toMultiplicative'_symm_apply_apply {α : Type u} {β : Type v} [MulOneClass α] [AddZeroClass β] (f : α →* Multiplicative β) (a : Additive α) : (AddMonoidHom.toMultiplicative'.symm f) a = Multiplicative.toAdd (f (Additive.toMul a))

def AddMonoidHom.toMultiplicative' {α : Type u} {β : Type v} [MulOneClass α] [AddZeroClass β] : (Additive α →+ β) ≃ (α →* Multiplicative β)

Reinterpret Additive α →+ β as α →* Multiplicative β.

Equations

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

@[simp]

theorem MonoidHom.toAdditive'_apply_apply {α : Type u} {β : Type v} [MulOneClass α] [AddZeroClass β] : ∀ (a : α →* Multiplicative β) (a_1 : Additive α), (MonoidHom.toAdditive' a) a_1 = Multiplicative.toAdd (a (Additive.toMul a_1))

@[simp]

theorem MonoidHom.toAdditive'_symm_apply_apply {α : Type u} {β : Type v} [MulOneClass α] [AddZeroClass β] : ∀ (a : Additive α →+ β) (a_1 : α), (MonoidHom.toAdditive'.symm a) a_1 = Multiplicative.ofAdd (a (Additive.ofMul a_1))

def MonoidHom.toAdditive' {α : Type u} {β : Type v} [MulOneClass α] [AddZeroClass β] : (α →* Multiplicative β) ≃ (Additive α →+ β)

Reinterpret α →* Multiplicative β as Additive α →+ β.

Equations

• MonoidHom.toAdditive' = AddMonoidHom.toMultiplicative'.symm

@[simp]

theorem AddMonoidHom.toMultiplicative''_symm_apply_apply {α : Type u} {β : Type v} [AddZeroClass α] [MulOneClass β] (f : Multiplicative α →* β) (a : α) : (AddMonoidHom.toMultiplicative''.symm f) a = Additive.ofMul (f (Multiplicative.ofAdd a))

@[simp]

theorem AddMonoidHom.toMultiplicative''_apply_apply {α : Type u} {β : Type v} [AddZeroClass α] [MulOneClass β] (f : α →+ Additive β) (a : Multiplicative α) : (AddMonoidHom.toMultiplicative'' f) a = Additive.toMul (f (Multiplicative.toAdd a))

def AddMonoidHom.toMultiplicative'' {α : Type u} {β : Type v} [AddZeroClass α] [MulOneClass β] : (α →+ Additive β) ≃ (Multiplicative α →* β)

Reinterpret α →+ Additive β as Multiplicative α →* β.

Equations

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

@[simp]

theorem MonoidHom.toAdditive''_symm_apply_apply {α : Type u} {β : Type v} [AddZeroClass α] [MulOneClass β] : ∀ (a : α →+ Additive β) (a_1 : Multiplicative α), (MonoidHom.toAdditive''.symm a) a_1 = Additive.toMul (a (Multiplicative.toAdd a_1))

@[simp]

theorem MonoidHom.toAdditive''_apply_apply {α : Type u} {β : Type v} [AddZeroClass α] [MulOneClass β] : ∀ (a : Multiplicative α →* β) (a_1 : α), (MonoidHom.toAdditive'' a) a_1 = Additive.ofMul (a (Multiplicative.ofAdd a_1))

def MonoidHom.toAdditive'' {α : Type u} {β : Type v} [AddZeroClass α] [MulOneClass β] : (Multiplicative α →* β) ≃ (α →+ Additive β)

Reinterpret Multiplicative α →* β as α →+ Additive β.

Equations

• MonoidHom.toAdditive'' = AddMonoidHom.toMultiplicative''.symm

instance Additive.coeToFun {α : Type u_1} {β : α → Sort u_2} [CoeFun α β] : CoeFun (Additive α) fun (a : Additive α) => β (Additive.toMul a)

If α has some multiplicative structure and coerces to a function, then Additive α should also coerce to the same function.

This allows Additive to be used on bundled function types with a multiplicative structure, which is often used for composition, without affecting the behavior of the function itself.

Equations

• Additive.coeToFun = { coe := fun (a : Additive α) => CoeFun.coe (Additive.toMul a) }

instance Multiplicative.coeToFun {α : Type u_1} {β : α → Sort u_2} [CoeFun α β] : CoeFun (Multiplicative α) fun (a : Multiplicative α) => β (Multiplicative.toAdd a)

If α has some additive structure and coerces to a function, then Multiplicative α should also coerce to the same function.

This allows Multiplicative to be used on bundled function types with an additive structure, which is often used for composition, without affecting the behavior of the function itself.

Equations

• Multiplicative.coeToFun = { coe := fun (a : Multiplicative α) => CoeFun.coe (Multiplicative.toAdd a) }