l3lab/ntp-mathlib huggingface dataset:
/-- Transfer `Group` across an `Equiv` -/
@[to_additive (attr := reducible) "Transfer `AddGroup` across an `Equiv`"]
protected def group [Group β] : Group α := by
  let one := e.one
  let mul := e.mul
  let inv := e.Inv
  let div := e.div
  let npow := e.pow ℕ
  let zpow := e.pow ℤ
  apply e.injective.group _ <;> intros <;> 

Proof state:
case npow
α : Type u
β : Type v
e : α ≃ β
inst✝ : Group β
one : One α := Equiv.one e
mul : Mul α := Equiv.mul e
inv : Inv α := Equiv.Inv e
div : Div α := Equiv.div e
npow : Pow α ℕ := Equiv.pow e ℕ
zpow : Pow α ℤ := Equiv.pow e ℤ
x✝ : α
n✝ : ℕ
⊢ e (x✝ ^ n✝) = e x✝ ^ n✝

Next tactic:
exact e.apply_symm_apply _
/-- Transfer `Group` across an `Equiv` -/
@[to_additive (attr := reducible) "Transfer `AddGroup` across an `Equiv`"]
protected def group [Group β] : Group α := by
  let one := e.one
  let mul := e.mul
  let inv := e.Inv
  let div := e.div
  let npow := e.pow ℕ
  let zpow := e.pow ℤ
  apply e.injective.group _ <;> intros <;> 

Proof state:
case zpow
α : Type u
β : Type v
e : α ≃ β
inst✝ : Group β
one : One α := Equiv.one e
mul : Mul α := Equiv.mul e
inv : Inv α := Equiv.Inv e
div : Div α := Equiv.div e
npow : Pow α ℕ := Equiv.pow e ℕ
zpow : Pow α ℤ := Equiv.pow e ℤ
x✝ : α
n✝ : ℤ
⊢ e (x✝ ^ n✝) = e x✝ ^ n✝

Next tactic:
exact e.apply_symm_apply _
/-- Transfer `CommGroup` across an `Equiv` -/
@[to_additive (attr := reducible) "Transfer `AddCommGroup` across an `Equiv`"]
protected def commGroup [CommGroup β] : CommGroup α := by
  

Proof state:
α : Type u
β : Type v
e : α ≃ β
inst✝ : CommGroup β
⊢ CommGroup α

Next tactic:
let one := e.one
/-- Transfer `CommGroup` across an `Equiv` -/
@[to_additive (attr := reducible) "Transfer `AddCommGroup` across an `Equiv`"]
protected def commGroup [CommGroup β] : CommGroup α := by
  let one := e.one
  

Proof state:
α : Type u
β : Type v
e : α ≃ β
inst✝ : CommGroup β
one : One α := Equiv.one e
⊢ CommGroup α

Next tactic:
let mul := e.mul
/-- Transfer `CommGroup` across an `Equiv` -/
@[to_additive (attr := reducible) "Transfer `AddCommGroup` across an `Equiv`"]
protected def commGroup [CommGroup β] : CommGroup α := by
  let one := e.one
  let mul := e.mul
  

Proof state:
α : Type u
β : Type v
e : α ≃ β
inst✝ : CommGroup β
one : One α := Equiv.one e
mul : Mul α := Equiv.mul e
⊢ CommGroup α

Next tactic:
let inv := e.Inv
/-- Transfer `CommGroup` across an `Equiv` -/
@[to_additive (attr := reducible) "Transfer `AddCommGroup` across an `Equiv`"]
protected def commGroup [CommGroup β] : CommGroup α := by
  let one := e.one
  let mul := e.mul
  let inv := e.Inv
  

Proof state:
α : Type u
β : Type v
e : α ≃ β
inst✝ : CommGroup β
one : One α := Equiv.one e
mul : Mul α := Equiv.mul e
inv : Inv α := Equiv.Inv e
⊢ CommGroup α

Next tactic:
let div := e.div
/-- Transfer `CommGroup` across an `Equiv` -/
@[to_additive (attr := reducible) "Transfer `AddCommGroup` across an `Equiv`"]
protected def commGroup [CommGroup β] : CommGroup α := by
  let one := e.one
  let mul := e.mul
  let inv := e.Inv
  let div := e.div
  

Proof state:
α : Type u
β : Type v
e : α ≃ β
inst✝ : CommGroup β
one : One α := Equiv.one e
mul : Mul α := Equiv.mul e
inv : Inv α := Equiv.Inv e
div : Div α := Equiv.div e
⊢ CommGroup α

Next tactic:
let npow := e.pow ℕ
/-- Transfer `CommGroup` across an `Equiv` -/
@[to_additive (attr := reducible) "Transfer `AddCommGroup` across an `Equiv`"]
protected def commGroup [CommGroup β] : CommGroup α := by
  let one := e.one
  let mul := e.mul
  let inv := e.Inv
  let div := e.div
  let npow := e.pow ℕ
  

Proof state:
α : Type u
β : Type v
e : α ≃ β
inst✝ : CommGroup β
one : One α := Equiv.one e
mul : Mul α := Equiv.mul e
inv : Inv α := Equiv.Inv e
div : Div α := Equiv.div e
npow : Pow α ℕ := Equiv.pow e ℕ
⊢ CommGroup α

Next tactic:
let zpow := e.pow ℤ
/-- Transfer `CommGroup` across an `Equiv` -/
@[to_additive (attr := reducible) "Transfer `AddCommGroup` across an `Equiv`"]
protected def commGroup [CommGroup β] : CommGroup α := by
  let one := e.one
  let mul := e.mul
  let inv := e.Inv
  let div := e.div
  let npow := e.pow ℕ
  let zpow := e.pow ℤ
  

Proof state:
α : Type u
β : Type v
e : α ≃ β
inst✝ : CommGroup β
one : One α := Equiv.one e
mul : Mul α := Equiv.mul e
inv : Inv α := Equiv.Inv e
div : Div α := Equiv.div e
npow : Pow α ℕ := Equiv.pow e ℕ
zpow : Pow α ℤ := Equiv.pow e ℤ
⊢ CommGroup α

Next tactic:
apply e.injective.commGroup _
/-- Transfer `CommGroup` across an `Equiv` -/
@[to_additive (attr := reducible) "Transfer `AddCommGroup` across an `Equiv`"]
protected def commGroup [CommGroup β] : CommGroup α := by
  let one := e.one
  let mul := e.mul
  let inv := e.Inv
  let div := e.div
  let npow := e.pow ℕ
  let zpow := e.pow ℤ
  apply e.injective.commGroup _ <;> 

Proof state:
case one
α : Type u
β : Type v
e : α ≃ β
inst✝ : CommGroup β
one : One α := Equiv.one e
mul : Mul α := Equiv.mul e
inv : Inv α := Equiv.Inv e
div : Div α := Equiv.div e
npow : Pow α ℕ := Equiv.pow e ℕ
zpow : Pow α ℤ := Equiv.pow e ℤ
⊢ e 1 = 1

Next tactic:
intros
/-- Transfer `CommGroup` across an `Equiv` -/
@[to_additive (attr := reducible) "Transfer `AddCommGroup` across an `Equiv`"]
protected def commGroup [CommGroup β] : CommGroup α := by
  let one := e.one
  let mul := e.mul
  let inv := e.Inv
  let div := e.div
  let npow := e.pow ℕ
  let zpow := e.pow ℤ
  apply e.injective.commGroup _ <;> 

Proof state:
case mul
α : Type u
β : Type v
e : α ≃ β
inst✝ : CommGroup β
one : One α := Equiv.one e
mul : Mul α := Equiv.mul e
inv : Inv α := Equiv.Inv e
div : Div α := Equiv.div e
npow : Pow α ℕ := Equiv.pow e ℕ
zpow : Pow α ℤ := Equiv.pow e ℤ
⊢ ∀ (x y : α), e (x * y) = e x * e y

Next tactic:
intros
/-- Transfer `CommGroup` across an `Equiv` -/
@[to_additive (attr := reducible) "Transfer `AddCommGroup` across an `Equiv`"]
protected def commGroup [CommGroup β] : CommGroup α := by
  let one := e.one
  let mul := e.mul
  let inv := e.Inv
  let div := e.div
  let npow := e.pow ℕ
  let zpow := e.pow ℤ
  apply e.injective.commGroup _ <;> 

Proof state:
case inv
α : Type u
β : Type v
e : α ≃ β
inst✝ : CommGroup β
one : One α := Equiv.one e
mul : Mul α := Equiv.mul e
inv : Inv α := Equiv.Inv e
div : Div α := Equiv.div e
npow : Pow α ℕ := Equiv.pow e ℕ
zpow : Pow α ℤ := Equiv.pow e ℤ
⊢ ∀ (x : α), e x⁻¹ = (e x)⁻¹

Next tactic:
intros
/-- Transfer `CommGroup` across an `Equiv` -/
@[to_additive (attr := reducible) "Transfer `AddCommGroup` across an `Equiv`"]
protected def commGroup [CommGroup β] : CommGroup α := by
  let one := e.one
  let mul := e.mul
  let inv := e.Inv
  let div := e.div
  let npow := e.pow ℕ
  let zpow := e.pow ℤ
  apply e.injective.commGroup _ <;> 

Proof state:
case div
α : Type u
β : Type v
e : α ≃ β
inst✝ : CommGroup β
one : One α := Equiv.one e
mul : Mul α := Equiv.mul e
inv : Inv α := Equiv.Inv e
div : Div α := Equiv.div e
npow : Pow α ℕ := Equiv.pow e ℕ
zpow : Pow α ℤ := Equiv.pow e ℤ
⊢ ∀ (x y : α), e (x / y) = e x / e y

Next tactic:
intros
/-- Transfer `CommGroup` across an `Equiv` -/
@[to_additive (attr := reducible) "Transfer `AddCommGroup` across an `Equiv`"]
protected def commGroup [CommGroup β] : CommGroup α := by
  let one := e.one
  let mul := e.mul
  let inv := e.Inv
  let div := e.div
  let npow := e.pow ℕ
  let zpow := e.pow ℤ
  apply e.injective.commGroup _ <;> 

Proof state:
case npow
α : Type u
β : Type v
e : α ≃ β
inst✝ : CommGroup β
one : One α := Equiv.one e
mul : Mul α := Equiv.mul e
inv : Inv α := Equiv.Inv e
div : Div α := Equiv.div e
npow : Pow α ℕ := Equiv.pow e ℕ
zpow : Pow α ℤ := Equiv.pow e ℤ
⊢ ∀ (x : α) (n : ℕ), e (x ^ n) = e x ^ n

Next tactic:
intros
/-- Transfer `CommGroup` across an `Equiv` -/
@[to_additive (attr := reducible) "Transfer `AddCommGroup` across an `Equiv`"]
protected def commGroup [CommGroup β] : CommGroup α := by
  let one := e.one
  let mul := e.mul
  let inv := e.Inv
  let div := e.div
  let npow := e.pow ℕ
  let zpow := e.pow ℤ
  apply e.injective.commGroup _ <;> 

Proof state:
case zpow
α : Type u
β : Type v
e : α ≃ β
inst✝ : CommGroup β
one : One α := Equiv.one e
mul : Mul α := Equiv.mul e
inv : Inv α := Equiv.Inv e
div : Div α := Equiv.div e
npow : Pow α ℕ := Equiv.pow e ℕ
zpow : Pow α ℤ := Equiv.pow e ℤ
⊢ ∀ (x : α) (n : ℤ), e (x ^ n) = e x ^ n

Next tactic:
intros
/-- Transfer `CommGroup` across an `Equiv` -/
@[to_additive (attr := reducible) "Transfer `AddCommGroup` across an `Equiv`"]
protected def commGroup [CommGroup β] : CommGroup α := by
  let one := e.one
  let mul := e.mul
  let inv := e.Inv
  let div := e.div
  let npow := e.pow ℕ
  let zpow := e.pow ℤ
  apply e.injective.commGroup _ <;> intros <;> 

Proof state:
case one
α : Type u
β : Type v
e : α ≃ β
inst✝ : CommGroup β
one : One α := Equiv.one e
mul : Mul α := Equiv.mul e
inv : Inv α := Equiv.Inv e
div : Div α := Equiv.div e
npow : Pow α ℕ := Equiv.pow e ℕ
zpow : Pow α ℤ := Equiv.pow e ℤ
⊢ e 1 = 1

Next tactic:
exact e.apply_symm_apply _
/-- Transfer `CommGroup` across an `Equiv` -/
@[to_additive (attr := reducible) "Transfer `AddCommGroup` across an `Equiv`"]
protected def commGroup [CommGroup β] : CommGroup α := by
  let one := e.one
  let mul := e.mul
  let inv := e.Inv
  let div := e.div
  let npow := e.pow ℕ
  let zpow := e.pow ℤ
  apply e.injective.commGroup _ <;> intros <;> 

Proof state:
case mul
α : Type u
β : Type v
e : α ≃ β
inst✝ : CommGroup β
one : One α := Equiv.one e
mul : Mul α := Equiv.mul e
inv : Inv α := Equiv.Inv e
div : Div α := Equiv.div e
npow : Pow α ℕ := Equiv.pow e ℕ
zpow : Pow α ℤ := Equiv.pow e ℤ
x✝ y✝ : α
⊢ e (x✝ * y✝) = e x✝ * e y✝

Next tactic:
exact e.apply_symm_apply _
/-- Transfer `CommGroup` across an `Equiv` -/
@[to_additive (attr := reducible) "Transfer `AddCommGroup` across an `Equiv`"]
protected def commGroup [CommGroup β] : CommGroup α := by
  let one := e.one
  let mul := e.mul
  let inv := e.Inv
  let div := e.div
  let npow := e.pow ℕ
  let zpow := e.pow ℤ
  apply e.injective.commGroup _ <;> intros <;> 

Proof state:
case inv
α : Type u
β : Type v
e : α ≃ β
inst✝ : CommGroup β
one : One α := Equiv.one e
mul : Mul α := Equiv.mul e
inv : Inv α := Equiv.Inv e
div : Div α := Equiv.div e
npow : Pow α ℕ := Equiv.pow e ℕ
zpow : Pow α ℤ := Equiv.pow e ℤ
x✝ : α
⊢ e x✝⁻¹ = (e x✝)⁻¹

Next tactic:
exact e.apply_symm_apply _
/-- Transfer `CommGroup` across an `Equiv` -/
@[to_additive (attr := reducible) "Transfer `AddCommGroup` across an `Equiv`"]
protected def commGroup [CommGroup β] : CommGroup α := by
  let one := e.one
  let mul := e.mul
  let inv := e.Inv
  let div := e.div
  let npow := e.pow ℕ
  let zpow := e.pow ℤ
  apply e.injective.commGroup _ <;> intros <;> 

Proof state:
case div
α : Type u
β : Type v
e : α ≃ β
inst✝ : CommGroup β
one : One α := Equiv.one e
mul : Mul α := Equiv.mul e
inv : Inv α := Equiv.Inv e
div : Div α := Equiv.div e
npow : Pow α ℕ := Equiv.pow e ℕ
zpow : Pow α ℤ := Equiv.pow e ℤ
x✝ y✝ : α
⊢ e (x✝ / y✝) = e x✝ / e y✝

Next tactic:
exact e.apply_symm_apply _
/-- Transfer `CommGroup` across an `Equiv` -/
@[to_additive (attr := reducible) "Transfer `AddCommGroup` across an `Equiv`"]
protected def commGroup [CommGroup β] : CommGroup α := by
  let one := e.one
  let mul := e.mul
  let inv := e.Inv
  let div := e.div
  let npow := e.pow ℕ
  let zpow := e.pow ℤ
  apply e.injective.commGroup _ <;> intros <;> 

Proof state:
case npow
α : Type u
β : Type v
e : α ≃ β
inst✝ : CommGroup β
one : One α := Equiv.one e
mul : Mul α := Equiv.mul e
inv : Inv α := Equiv.Inv e
div : Div α := Equiv.div e
npow : Pow α ℕ := Equiv.pow e ℕ
zpow : Pow α ℤ := Equiv.pow e ℤ
x✝ : α
n✝ : ℕ
⊢ e (x✝ ^ n✝) = e x✝ ^ n✝

Next tactic:
exact e.apply_symm_apply _
/-- Transfer `CommGroup` across an `Equiv` -/
@[to_additive (attr := reducible) "Transfer `AddCommGroup` across an `Equiv`"]
protected def commGroup [CommGroup β] : CommGroup α := by
  let one := e.one
  let mul := e.mul
  let inv := e.Inv
  let div := e.div
  let npow := e.pow ℕ
  let zpow := e.pow ℤ
  apply e.injective.commGroup _ <;> intros <;> 

Proof state:
case zpow
α : Type u
β : Type v
e : α ≃ β
inst✝ : CommGroup β
one : One α := Equiv.one e
mul : Mul α := Equiv.mul e
inv : Inv α := Equiv.Inv e
div : Div α := Equiv.div e
npow : Pow α ℕ := Equiv.pow e ℕ
zpow : Pow α ℤ := Equiv.pow e ℤ
x✝ : α
n✝ : ℤ
⊢ e (x✝ ^ n✝) = e x✝ ^ n✝

Next tactic:
exact e.apply_symm_apply _
/-- Transfer `NonUnitalNonAssocSemiring` across an `Equiv` -/
@[reducible]
protected def nonUnitalNonAssocSemiring [NonUnitalNonAssocSemiring β] :
    NonUnitalNonAssocSemiring α := by
  

Proof state:
α : Type u
β : Type v
e : α ≃ β
inst✝ : NonUnitalNonAssocSemiring β
⊢ NonUnitalNonAssocSemiring α

Next tactic:
let zero := e.zero
/-- Transfer `NonUnitalNonAssocSemiring` across an `Equiv` -/
@[reducible]
protected def nonUnitalNonAssocSemiring [NonUnitalNonAssocSemiring β] :
    NonUnitalNonAssocSemiring α := by
  let zero := e.zero
  

Proof state:
α : Type u
β : Type v
e : α ≃ β
inst✝ : NonUnitalNonAssocSemiring β
zero : Zero α := Equiv.zero e
⊢ NonUnitalNonAssocSemiring α

Next tactic:
let add := e.add
/-- Transfer `NonUnitalNonAssocSemiring` across an `Equiv` -/
@[reducible]
protected def nonUnitalNonAssocSemiring [NonUnitalNonAssocSemiring β] :
    NonUnitalNonAssocSemiring α := by
  let zero := e.zero
  let add := e.add
  

Proof state:
α : Type u
β : Type v
e : α ≃ β
inst✝ : NonUnitalNonAssocSemiring β
zero : Zero α := Equiv.zero e
add : Add α := Equiv.add e
⊢ NonUnitalNonAssocSemiring α

Next tactic:
let mul := e.mul
/-- Transfer `NonUnitalNonAssocSemiring` across an `Equiv` -/
@[reducible]
protected def nonUnitalNonAssocSemiring [NonUnitalNonAssocSemiring β] :
    NonUnitalNonAssocSemiring α := by
  let zero := e.zero
  let add := e.add
  let mul := e.mul
  

Proof state:
α : Type u
β : Type v
e : α ≃ β
inst✝ : NonUnitalNonAssocSemiring β
zero : Zero α := Equiv.zero e
add : Add α := Equiv.add e
mul : Mul α := Equiv.mul e
⊢ NonUnitalNonAssocSemiring α

Next tactic:
let nsmul := e.smul ℕ
/-- Transfer `NonUnitalNonAssocSemiring` across an `Equiv` -/
@[reducible]
protected def nonUnitalNonAssocSemiring [NonUnitalNonAssocSemiring β] :
    NonUnitalNonAssocSemiring α := by
  let zero := e.zero
  let add := e.add
  let mul := e.mul
  let nsmul := e.smul ℕ
  

Proof state:
α : Type u
β : Type v
e : α ≃ β
inst✝ : NonUnitalNonAssocSemiring β
zero : Zero α := Equiv.zero e
add : Add α := Equiv.add e
mul : Mul α := Equiv.mul e
nsmul : SMul ℕ α := Equiv.smul e ℕ
⊢ NonUnitalNonAssocSemiring α

Next tactic:
apply e.injective.nonUnitalNonAssocSemiring _
/-- Transfer `NonUnitalNonAssocSemiring` across an `Equiv` -/
@[reducible]
protected def nonUnitalNonAssocSemiring [NonUnitalNonAssocSemiring β] :
    NonUnitalNonAssocSemiring α := by
  let zero := e.zero
  let add := e.add
  let mul := e.mul
  let nsmul := e.smul ℕ
  apply e.injective.nonUnitalNonAssocSemiring _ <;> 

Proof state:
case zero
α : Type u
β : Type v
e : α ≃ β
inst✝ : NonUnitalNonAssocSemiring β
zero : Zero α := Equiv.zero e
add : Add α := Equiv.add e
mul : Mul α := Equiv.mul e
nsmul : SMul ℕ α := Equiv.smul e ℕ
⊢ e 0 = 0

Next tactic:
intros
/-- Transfer `NonUnitalNonAssocSemiring` across an `Equiv` -/
@[reducible]
protected def nonUnitalNonAssocSemiring [NonUnitalNonAssocSemiring β] :
    NonUnitalNonAssocSemiring α := by
  let zero := e.zero
  let add := e.add
  let mul := e.mul
  let nsmul := e.smul ℕ
  apply e.injective.nonUnitalNonAssocSemiring _ <;> 

Proof state:
case add
α : Type u
β : Type v
e : α ≃ β
inst✝ : NonUnitalNonAssocSemiring β
zero : Zero α := Equiv.zero e
add : Add α := Equiv.add e
mul : Mul α := Equiv.mul e
nsmul : SMul ℕ α := Equiv.smul e ℕ
⊢ ∀ (x y : α), e (x + y) = e x + e y

Next tactic:
intros
/-- Transfer `NonUnitalNonAssocSemiring` across an `Equiv` -/
@[reducible]
protected def nonUnitalNonAssocSemiring [NonUnitalNonAssocSemiring β] :
    NonUnitalNonAssocSemiring α := by
  let zero := e.zero
  let add := e.add
  let mul := e.mul
  let nsmul := e.smul ℕ
  apply e.injective.nonUnitalNonAssocSemiring _ <;> 

Proof state:
case mul
α : Type u
β : Type v
e : α ≃ β
inst✝ : NonUnitalNonAssocSemiring β
zero : Zero α := Equiv.zero e
add : Add α := Equiv.add e
mul : Mul α := Equiv.mul e
nsmul : SMul ℕ α := Equiv.smul e ℕ
⊢ ∀ (x y : α), e (x * y) = e x * e y

Next tactic:
intros
/-- Transfer `NonUnitalNonAssocSemiring` across an `Equiv` -/
@[reducible]
protected def nonUnitalNonAssocSemiring [NonUnitalNonAssocSemiring β] :
    NonUnitalNonAssocSemiring α := by
  let zero := e.zero
  let add := e.add
  let mul := e.mul
  let nsmul := e.smul ℕ
  apply e.injective.nonUnitalNonAssocSemiring _ <;> 

Proof state:
case nsmul
α : Type u
β : Type v
e : α ≃ β
inst✝ : NonUnitalNonAssocSemiring β
zero : Zero α := Equiv.zero e
add : Add α := Equiv.add e
mul : Mul α := Equiv.mul e
nsmul : SMul ℕ α := Equiv.smul e ℕ
⊢ ∀ (x : α) (n : ℕ), e (n • x) = n • e x

Next tactic:
intros
/-- Transfer `NonUnitalNonAssocSemiring` across an `Equiv` -/
@[reducible]
protected def nonUnitalNonAssocSemiring [NonUnitalNonAssocSemiring β] :
    NonUnitalNonAssocSemiring α := by
  let zero := e.zero
  let add := e.add
  let mul := e.mul
  let nsmul := e.smul ℕ
  apply e.injective.nonUnitalNonAssocSemiring _ <;> intros <;> 

Proof state:
case zero
α : Type u
β : Type v
e : α ≃ β
inst✝ : NonUnitalNonAssocSemiring β
zero : Zero α := Equiv.zero e
add : Add α := Equiv.add e
mul : Mul α := Equiv.mul e
nsmul : SMul ℕ α := Equiv.smul e ℕ
⊢ e 0 = 0

Next tactic:
exact e.apply_symm_apply _
/-- Transfer `NonUnitalNonAssocSemiring` across an `Equiv` -/
@[reducible]
protected def nonUnitalNonAssocSemiring [NonUnitalNonAssocSemiring β] :
    NonUnitalNonAssocSemiring α := by
  let zero := e.zero
  let add := e.add
  let mul := e.mul
  let nsmul := e.smul ℕ
  apply e.injective.nonUnitalNonAssocSemiring _ <;> intros <;> 

Proof state:
case add
α : Type u
β : Type v
e : α ≃ β
inst✝ : NonUnitalNonAssocSemiring β
zero : Zero α := Equiv.zero e
add : Add α := Equiv.add e
mul : Mul α := Equiv.mul e
nsmul : SMul ℕ α := Equiv.smul e ℕ
x✝ y✝ : α
⊢ e (x✝ + y✝) = e x✝ + e y✝

Next tactic:
exact e.apply_symm_apply _
/-- Transfer `NonUnitalNonAssocSemiring` across an `Equiv` -/
@[reducible]
protected def nonUnitalNonAssocSemiring [NonUnitalNonAssocSemiring β] :
    NonUnitalNonAssocSemiring α := by
  let zero := e.zero
  let add := e.add
  let mul := e.mul
  let nsmul := e.smul ℕ
  apply e.injective.nonUnitalNonAssocSemiring _ <;> intros <;> 

Proof state:
case mul
α : Type u
β : Type v
e : α ≃ β
inst✝ : NonUnitalNonAssocSemiring β
zero : Zero α := Equiv.zero e
add : Add α := Equiv.add e
mul : Mul α := Equiv.mul e
nsmul : SMul ℕ α := Equiv.smul e ℕ
x✝ y✝ : α
⊢ e (x✝ * y✝) = e x✝ * e y✝

Next tactic:
exact e.apply_symm_apply _
/-- Transfer `NonUnitalNonAssocSemiring` across an `Equiv` -/
@[reducible]
protected def nonUnitalNonAssocSemiring [NonUnitalNonAssocSemiring β] :
    NonUnitalNonAssocSemiring α := by
  let zero := e.zero
  let add := e.add
  let mul := e.mul
  let nsmul := e.smul ℕ
  apply e.injective.nonUnitalNonAssocSemiring _ <;> intros <;> 

Proof state:
case nsmul
α : Type u
β : Type v
e : α ≃ β
inst✝ : NonUnitalNonAssocSemiring β
zero : Zero α := Equiv.zero e
add : Add α := Equiv.add e
mul : Mul α := Equiv.mul e
nsmul : SMul ℕ α := Equiv.smul e ℕ
x✝ : α
n✝ : ℕ
⊢ e (n✝ • x✝) = n✝ • e x✝

Next tactic:
exact e.apply_symm_apply _
/-- Transfer `NonUnitalSemiring` across an `Equiv` -/
@[reducible]
protected def nonUnitalSemiring [NonUnitalSemiring β] : NonUnitalSemiring α := by
  

Proof state:
α : Type u
β : Type v
e : α ≃ β
inst✝ : NonUnitalSemiring β
⊢ NonUnitalSemiring α

Next tactic:
let zero := e.zero
/-- Transfer `NonUnitalSemiring` across an `Equiv` -/
@[reducible]
protected def nonUnitalSemiring [NonUnitalSemiring β] : NonUnitalSemiring α := by
  let zero := e.zero
  

Proof state:
α : Type u
β : Type v
e : α ≃ β
inst✝ : NonUnitalSemiring β
zero : Zero α := Equiv.zero e
⊢ NonUnitalSemiring α

Next tactic:
let add := e.add
/-- Transfer `NonUnitalSemiring` across an `Equiv` -/
@[reducible]
protected def nonUnitalSemiring [NonUnitalSemiring β] : NonUnitalSemiring α := by
  let zero := e.zero
  let add := e.add
  

Proof state:
α : Type u
β : Type v
e : α ≃ β
inst✝ : NonUnitalSemiring β
zero : Zero α := Equiv.zero e
add : Add α := Equiv.add e
⊢ NonUnitalSemiring α

Next tactic:
let mul := e.mul
/-- Transfer `NonUnitalSemiring` across an `Equiv` -/
@[reducible]
protected def nonUnitalSemiring [NonUnitalSemiring β] : NonUnitalSemiring α := by
  let zero := e.zero
  let add := e.add
  let mul := e.mul
  

Proof state:
α : Type u
β : Type v
e : α ≃ β
inst✝ : NonUnitalSemiring β
zero : Zero α := Equiv.zero e
add : Add α := Equiv.add e
mul : Mul α := Equiv.mul e
⊢ NonUnitalSemiring α

Next tactic:
let nsmul := e.smul ℕ
/-- Transfer `NonUnitalSemiring` across an `Equiv` -/
@[reducible]
protected def nonUnitalSemiring [NonUnitalSemiring β] : NonUnitalSemiring α := by
  let zero := e.zero
  let add := e.add
  let mul := e.mul
  let nsmul := e.smul ℕ
  

Proof state:
α : Type u
β : Type v
e : α ≃ β
inst✝ : NonUnitalSemiring β
zero : Zero α := Equiv.zero e
add : Add α := Equiv.add e
mul : Mul α := Equiv.mul e
nsmul : SMul ℕ α := Equiv.smul e ℕ
⊢ NonUnitalSemiring α

Next tactic:
apply e.injective.nonUnitalSemiring _
/-- Transfer `NonUnitalSemiring` across an `Equiv` -/
@[reducible]
protected def nonUnitalSemiring [NonUnitalSemiring β] : NonUnitalSemiring α := by
  let zero := e.zero
  let add := e.add
  let mul := e.mul
  let nsmul := e.smul ℕ
  apply e.injective.nonUnitalSemiring _ <;> 

Proof state:
case zero
α : Type u
β : Type v
e : α ≃ β
inst✝ : NonUnitalSemiring β
zero : Zero α := Equiv.zero e
add : Add α := Equiv.add e
mul : Mul α := Equiv.mul e
nsmul : SMul ℕ α := Equiv.smul e ℕ
⊢ e 0 = 0

Next tactic:
intros
/-- Transfer `NonUnitalSemiring` across an `Equiv` -/
@[reducible]
protected def nonUnitalSemiring [NonUnitalSemiring β] : NonUnitalSemiring α := by
  let zero := e.zero
  let add := e.add
  let mul := e.mul
  let nsmul := e.smul ℕ
  apply e.injective.nonUnitalSemiring _ <;> 

Proof state:
case add
α : Type u
β : Type v
e : α ≃ β
inst✝ : NonUnitalSemiring β
zero : Zero α := Equiv.zero e
add : Add α := Equiv.add e
mul : Mul α := Equiv.mul e
nsmul : SMul ℕ α := Equiv.smul e ℕ
⊢ ∀ (x y : α), e (x + y) = e x + e y

Next tactic:
intros
/-- Transfer `NonUnitalSemiring` across an `Equiv` -/
@[reducible]
protected def nonUnitalSemiring [NonUnitalSemiring β] : NonUnitalSemiring α := by
  let zero := e.zero
  let add := e.add
  let mul := e.mul
  let nsmul := e.smul ℕ
  apply e.injective.nonUnitalSemiring _ <;> 

Proof state:
case mul
α : Type u
β : Type v
e : α ≃ β
inst✝ : NonUnitalSemiring β
zero : Zero α := Equiv.zero e
add : Add α := Equiv.add e
mul : Mul α := Equiv.mul e
nsmul : SMul ℕ α := Equiv.smul e ℕ
⊢ ∀ (x y : α), e (x * y) = e x * e y

Next tactic:
intros
/-- Transfer `NonUnitalSemiring` across an `Equiv` -/
@[reducible]
protected def nonUnitalSemiring [NonUnitalSemiring β] : NonUnitalSemiring α := by
  let zero := e.zero
  let add := e.add
  let mul := e.mul
  let nsmul := e.smul ℕ
  apply e.injective.nonUnitalSemiring _ <;> 

Proof state:
case nsmul
α : Type u
β : Type v
e : α ≃ β
inst✝ : NonUnitalSemiring β
zero : Zero α := Equiv.zero e
add : Add α := Equiv.add e
mul : Mul α := Equiv.mul e
nsmul : SMul ℕ α := Equiv.smul e ℕ
⊢ ∀ (x : α) (n : ℕ), e (n • x) = n • e x

Next tactic:
intros
/-- Transfer `NonUnitalSemiring` across an `Equiv` -/
@[reducible]
protected def nonUnitalSemiring [NonUnitalSemiring β] : NonUnitalSemiring α := by
  let zero := e.zero
  let add := e.add
  let mul := e.mul
  let nsmul := e.smul ℕ
  apply e.injective.nonUnitalSemiring _ <;> intros <;> 

Proof state:
case zero
α : Type u
β : Type v
e : α ≃ β
inst✝ : NonUnitalSemiring β
zero : Zero α := Equiv.zero e
add : Add α := Equiv.add e
mul : Mul α := Equiv.mul e
nsmul : SMul ℕ α := Equiv.smul e ℕ
⊢ e 0 = 0

Next tactic:
exact e.apply_symm_apply _
/-- Transfer `NonUnitalSemiring` across an `Equiv` -/
@[reducible]
protected def nonUnitalSemiring [NonUnitalSemiring β] : NonUnitalSemiring α := by
  let zero := e.zero
  let add := e.add
  let mul := e.mul
  let nsmul := e.smul ℕ
  apply e.injective.nonUnitalSemiring _ <;> intros <;> 

Proof state:
case add
α : Type u
β : Type v
e : α ≃ β
inst✝ : NonUnitalSemiring β
zero : Zero α := Equiv.zero e
add : Add α := Equiv.add e
mul : Mul α := Equiv.mul e
nsmul : SMul ℕ α := Equiv.smul e ℕ
x✝ y✝ : α
⊢ e (x✝ + y✝) = e x✝ + e y✝

Next tactic:
exact e.apply_symm_apply _