mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Algebra.Field.TransferInstance

Goal state before:
pp: case add
α : Type u_1
β : Type u_2
e : α ≃ β
inst✝ : DivisionRing β
add_group_with_one : AddGroupWithOne α := ⋯
inv : Inv α := ⋯
div : Div α := ⋯
mul : Mul α := ⋯
npow : Pow α ℕ := ⋯
zpow : Pow α ℤ := ⋯
nnratCast : NNRatCast α := ⋯
ratCast : RatCast α := ⋯
nnqsmul : SMul ℚ≥0 α := ⋯
qsmul : SMul ℚ α := ⋯
⊢ ∀ (x y : α), e (x + y) = e x + e y

Tactic:
intros

Goal state after:
case add
α : Type u_1
β : Type u_2
e : α ≃ β
inst✝ : DivisionRing β
add_group_with_one : AddGroupWithOne α := e.addGroupWithOne
inv : Inv α := e.Inv
div : Div α := e.div
mul : Mul α := e.mul
npow : Pow α ℕ := Equiv.pow ℕ e
zpow : Pow α ℤ := Equiv.pow ℤ e
nnratCast : NNRatCast α := e.nnratCast
ratCast : RatCast α := e.ratCast
nnqsmul : SMul ℚ≥0 α := Equiv.smul ℚ≥0 e
qsmul : SMul ℚ α := Equiv.smul ℚ e
x✝ y✝ : α
⊢ e (x✝ + y✝) = e x✝ + e y✝
Module: Mathlib.Algebra.Field.TransferInstance

Goal state before:
pp: case mul
α : Type u_1
β : Type u_2
e : α ≃ β
inst✝ : DivisionRing β
add_group_with_one : AddGroupWithOne α := ⋯
inv : Inv α := ⋯
div : Div α := ⋯
mul : Mul α := ⋯
npow : Pow α ℕ := ⋯
zpow : Pow α ℤ := ⋯
nnratCast : NNRatCast α := ⋯
ratCast : RatCast α := ⋯
nnqsmul : SMul ℚ≥0 α := ⋯
qsmul : SMul ℚ α := ⋯
⊢ ∀ (x y : α), e (x * y) = e x * e y

Tactic:
intros

Goal state after:
case mul
α : Type u_1
β : Type u_2
e : α ≃ β
inst✝ : DivisionRing β
add_group_with_one : AddGroupWithOne α := e.addGroupWithOne
inv : Inv α := e.Inv
div : Div α := e.div
mul : Mul α := e.mul
npow : Pow α ℕ := Equiv.pow ℕ e
zpow : Pow α ℤ := Equiv.pow ℤ e
nnratCast : NNRatCast α := e.nnratCast
ratCast : RatCast α := e.ratCast
nnqsmul : SMul ℚ≥0 α := Equiv.smul ℚ≥0 e
qsmul : SMul ℚ α := Equiv.smul ℚ e
x✝ y✝ : α
⊢ e (x✝ * y✝) = e x✝ * e y✝
Module: Mathlib.Algebra.Field.TransferInstance

Goal state before:
pp: case neg
α : Type u_1
β : Type u_2
e : α ≃ β
inst✝ : DivisionRing β
add_group_with_one : AddGroupWithOne α := ⋯
inv : Inv α := ⋯
div : Div α := ⋯
mul : Mul α := ⋯
npow : Pow α ℕ := ⋯
zpow : Pow α ℤ := ⋯
nnratCast : NNRatCast α := ⋯
ratCast : RatCast α := ⋯
nnqsmul : SMul ℚ≥0 α := ⋯
qsmul : SMul ℚ α := ⋯
⊢ ∀ (x : α), e (-x) = -e x

Tactic:
intros

Goal state after:
case neg
α : Type u_1
β : Type u_2
e : α ≃ β
inst✝ : DivisionRing β
add_group_with_one : AddGroupWithOne α := e.addGroupWithOne
inv : Inv α := e.Inv
div : Div α := e.div
mul : Mul α := e.mul
npow : Pow α ℕ := Equiv.pow ℕ e
zpow : Pow α ℤ := Equiv.pow ℤ e
nnratCast : NNRatCast α := e.nnratCast
ratCast : RatCast α := e.ratCast
nnqsmul : SMul ℚ≥0 α := Equiv.smul ℚ≥0 e
qsmul : SMul ℚ α := Equiv.smul ℚ e
x✝ : α
⊢ e (-x✝) = -e x✝
Module: Mathlib.Algebra.Field.TransferInstance

Goal state before:
pp: case sub
α : Type u_1
β : Type u_2
e : α ≃ β
inst✝ : DivisionRing β
add_group_with_one : AddGroupWithOne α := ⋯
inv : Inv α := ⋯
div : Div α := ⋯
mul : Mul α := ⋯
npow : Pow α ℕ := ⋯
zpow : Pow α ℤ := ⋯
nnratCast : NNRatCast α := ⋯
ratCast : RatCast α := ⋯
nnqsmul : SMul ℚ≥0 α := ⋯
qsmul : SMul ℚ α := ⋯
⊢ ∀ (x y : α), e (x - y) = e x - e y

Tactic:
intros

Goal state after:
case sub
α : Type u_1
β : Type u_2
e : α ≃ β
inst✝ : DivisionRing β
add_group_with_one : AddGroupWithOne α := e.addGroupWithOne
inv : Inv α := e.Inv
div : Div α := e.div
mul : Mul α := e.mul
npow : Pow α ℕ := Equiv.pow ℕ e
zpow : Pow α ℤ := Equiv.pow ℤ e
nnratCast : NNRatCast α := e.nnratCast
ratCast : RatCast α := e.ratCast
nnqsmul : SMul ℚ≥0 α := Equiv.smul ℚ≥0 e
qsmul : SMul ℚ α := Equiv.smul ℚ e
x✝ y✝ : α
⊢ e (x✝ - y✝) = e x✝ - e y✝
Module: Mathlib.Algebra.Field.TransferInstance

Goal state before:
pp: case inv
α : Type u_1
β : Type u_2
e : α ≃ β
inst✝ : DivisionRing β
add_group_with_one : AddGroupWithOne α := ⋯
inv : Inv α := ⋯
div : Div α := ⋯
mul : Mul α := ⋯
npow : Pow α ℕ := ⋯
zpow : Pow α ℤ := ⋯
nnratCast : NNRatCast α := ⋯
ratCast : RatCast α := ⋯
nnqsmul : SMul ℚ≥0 α := ⋯
qsmul : SMul ℚ α := ⋯
⊢ ∀ (x : α), e x⁻¹ = (e x)⁻¹

Tactic:
intros

Goal state after:
case inv
α : Type u_1
β : Type u_2
e : α ≃ β
inst✝ : DivisionRing β
add_group_with_one : AddGroupWithOne α := e.addGroupWithOne
inv : Inv α := e.Inv
div : Div α := e.div
mul : Mul α := e.mul
npow : Pow α ℕ := Equiv.pow ℕ e
zpow : Pow α ℤ := Equiv.pow ℤ e
nnratCast : NNRatCast α := e.nnratCast
ratCast : RatCast α := e.ratCast
nnqsmul : SMul ℚ≥0 α := Equiv.smul ℚ≥0 e
qsmul : SMul ℚ α := Equiv.smul ℚ e
x✝ : α
⊢ e x✝⁻¹ = (e x✝)⁻¹
Module: Mathlib.Algebra.Field.TransferInstance

Goal state before:
pp: case div
α : Type u_1
β : Type u_2
e : α ≃ β
inst✝ : DivisionRing β
add_group_with_one : AddGroupWithOne α := ⋯
inv : Inv α := ⋯
div : Div α := ⋯
mul : Mul α := ⋯
npow : Pow α ℕ := ⋯
zpow : Pow α ℤ := ⋯
nnratCast : NNRatCast α := ⋯
ratCast : RatCast α := ⋯
nnqsmul : SMul ℚ≥0 α := ⋯
qsmul : SMul ℚ α := ⋯
⊢ ∀ (x y : α), e (x / y) = e x / e y

Tactic:
intros

Goal state after:
case div
α : Type u_1
β : Type u_2
e : α ≃ β
inst✝ : DivisionRing β
add_group_with_one : AddGroupWithOne α := e.addGroupWithOne
inv : Inv α := e.Inv
div : Div α := e.div
mul : Mul α := e.mul
npow : Pow α ℕ := Equiv.pow ℕ e
zpow : Pow α ℤ := Equiv.pow ℤ e
nnratCast : NNRatCast α := e.nnratCast
ratCast : RatCast α := e.ratCast
nnqsmul : SMul ℚ≥0 α := Equiv.smul ℚ≥0 e
qsmul : SMul ℚ α := Equiv.smul ℚ e
x✝ y✝ : α
⊢ e (x✝ / y✝) = e x✝ / e y✝
Module: Mathlib.Algebra.Field.TransferInstance

Goal state before:
pp: case nsmul
α : Type u_1
β : Type u_2
e : α ≃ β
inst✝ : DivisionRing β
add_group_with_one : AddGroupWithOne α := ⋯
inv : Inv α := ⋯
div : Div α := ⋯
mul : Mul α := ⋯
npow : Pow α ℕ := ⋯
zpow : Pow α ℤ := ⋯
nnratCast : NNRatCast α := ⋯
ratCast : RatCast α := ⋯
nnqsmul : SMul ℚ≥0 α := ⋯
qsmul : SMul ℚ α := ⋯
⊢ ∀ (n : ℕ) (x : α), e (n • x) = n • e x

Tactic:
intros

Goal state after:
case nsmul
α : Type u_1
β : Type u_2
e : α ≃ β
inst✝ : DivisionRing β
add_group_with_one : AddGroupWithOne α := e.addGroupWithOne
inv : Inv α := e.Inv
div : Div α := e.div
mul : Mul α := e.mul
npow : Pow α ℕ := Equiv.pow ℕ e
zpow : Pow α ℤ := Equiv.pow ℤ e
nnratCast : NNRatCast α := e.nnratCast
ratCast : RatCast α := e.ratCast
nnqsmul : SMul ℚ≥0 α := Equiv.smul ℚ≥0 e
qsmul : SMul ℚ α := Equiv.smul ℚ e
n✝ : ℕ
x✝ : α
⊢ e (n✝ • x✝) = n✝ • e x✝
Module: Mathlib.Algebra.Field.TransferInstance

Goal state before:
pp: case zsmul
α : Type u_1
β : Type u_2
e : α ≃ β
inst✝ : DivisionRing β
add_group_with_one : AddGroupWithOne α := ⋯
inv : Inv α := ⋯
div : Div α := ⋯
mul : Mul α := ⋯
npow : Pow α ℕ := ⋯
zpow : Pow α ℤ := ⋯
nnratCast : NNRatCast α := ⋯
ratCast : RatCast α := ⋯
nnqsmul : SMul ℚ≥0 α := ⋯
qsmul : SMul ℚ α := ⋯
⊢ ∀ (n : ℤ) (x : α), e (n • x) = n • e x

Tactic:
intros

Goal state after:
case zsmul
α : Type u_1
β : Type u_2
e : α ≃ β
inst✝ : DivisionRing β
add_group_with_one : AddGroupWithOne α := e.addGroupWithOne
inv : Inv α := e.Inv
div : Div α := e.div
mul : Mul α := e.mul
npow : Pow α ℕ := Equiv.pow ℕ e
zpow : Pow α ℤ := Equiv.pow ℤ e
nnratCast : NNRatCast α := e.nnratCast
ratCast : RatCast α := e.ratCast
nnqsmul : SMul ℚ≥0 α := Equiv.smul ℚ≥0 e
qsmul : SMul ℚ α := Equiv.smul ℚ e
n✝ : ℤ
x✝ : α
⊢ e (n✝ • x✝) = n✝ • e x✝
Module: Mathlib.Algebra.Field.TransferInstance

Goal state before:
pp: case nnqsmul
α : Type u_1
β : Type u_2
e : α ≃ β
inst✝ : DivisionRing β
add_group_with_one : AddGroupWithOne α := ⋯
inv : Inv α := ⋯
div : Div α := ⋯
mul : Mul α := ⋯
npow : Pow α ℕ := ⋯
zpow : Pow α ℤ := ⋯
nnratCast : NNRatCast α := ⋯
ratCast : RatCast α := ⋯
nnqsmul : SMul ℚ≥0 α := ⋯
qsmul : SMul ℚ α := ⋯
⊢ ∀ (q : ℚ≥0) (x : α), e (q • x) = q • e x

Tactic:
intros

Goal state after:
case nnqsmul
α : Type u_1
β : Type u_2
e : α ≃ β
inst✝ : DivisionRing β
add_group_with_one : AddGroupWithOne α := e.addGroupWithOne
inv : Inv α := e.Inv
div : Div α := e.div
mul : Mul α := e.mul
npow : Pow α ℕ := Equiv.pow ℕ e
zpow : Pow α ℤ := Equiv.pow ℤ e
nnratCast : NNRatCast α := e.nnratCast
ratCast : RatCast α := e.ratCast
nnqsmul : SMul ℚ≥0 α := Equiv.smul ℚ≥0 e
qsmul : SMul ℚ α := Equiv.smul ℚ e
q✝ : ℚ≥0
x✝ : α
⊢ e (q✝ • x✝) = q✝ • e x✝
Module: Mathlib.Algebra.Field.TransferInstance

Goal state before:
pp: case qsmul
α : Type u_1
β : Type u_2
e : α ≃ β
inst✝ : DivisionRing β
add_group_with_one : AddGroupWithOne α := ⋯
inv : Inv α := ⋯
div : Div α := ⋯
mul : Mul α := ⋯
npow : Pow α ℕ := ⋯
zpow : Pow α ℤ := ⋯
nnratCast : NNRatCast α := ⋯
ratCast : RatCast α := ⋯
nnqsmul : SMul ℚ≥0 α := ⋯
qsmul : SMul ℚ α := ⋯
⊢ ∀ (q : ℚ) (x : α), e (q • x) = q • e x

Tactic:
intros

Goal state after:
case qsmul
α : Type u_1
β : Type u_2
e : α ≃ β
inst✝ : DivisionRing β
add_group_with_one : AddGroupWithOne α := e.addGroupWithOne
inv : Inv α := e.Inv
div : Div α := e.div
mul : Mul α := e.mul
npow : Pow α ℕ := Equiv.pow ℕ e
zpow : Pow α ℤ := Equiv.pow ℤ e
nnratCast : NNRatCast α := e.nnratCast
ratCast : RatCast α := e.ratCast
nnqsmul : SMul ℚ≥0 α := Equiv.smul ℚ≥0 e
qsmul : SMul ℚ α := Equiv.smul ℚ e
q✝ : ℚ
x✝ : α
⊢ e (q✝ • x✝) = q✝ • e x✝
Module: Mathlib.Algebra.Field.TransferInstance

Goal state before:
pp: case npow
α : Type u_1
β : Type u_2
e : α ≃ β
inst✝ : DivisionRing β
add_group_with_one : AddGroupWithOne α := ⋯
inv : Inv α := ⋯
div : Div α := ⋯
mul : Mul α := ⋯
npow : Pow α ℕ := ⋯
zpow : Pow α ℤ := ⋯
nnratCast : NNRatCast α := ⋯
ratCast : RatCast α := ⋯
nnqsmul : SMul ℚ≥0 α := ⋯
qsmul : SMul ℚ α := ⋯
⊢ ∀ (x : α) (n : ℕ), e (x ^ n) = e x ^ n

Tactic:
intros

Goal state after:
case npow
α : Type u_1
β : Type u_2
e : α ≃ β
inst✝ : DivisionRing β
add_group_with_one : AddGroupWithOne α := e.addGroupWithOne
inv : Inv α := e.Inv
div : Div α := e.div
mul : Mul α := e.mul
npow : Pow α ℕ := Equiv.pow ℕ e
zpow : Pow α ℤ := Equiv.pow ℤ e
nnratCast : NNRatCast α := e.nnratCast
ratCast : RatCast α := e.ratCast
nnqsmul : SMul ℚ≥0 α := Equiv.smul ℚ≥0 e
qsmul : SMul ℚ α := Equiv.smul ℚ e
x✝ : α
n✝ : ℕ
⊢ e (x✝ ^ n✝) = e x✝ ^ n✝
Module: Mathlib.Algebra.Field.TransferInstance

Goal state before:
pp: case zpow
α : Type u_1
β : Type u_2
e : α ≃ β
inst✝ : DivisionRing β
add_group_with_one : AddGroupWithOne α := ⋯
inv : Inv α := ⋯
div : Div α := ⋯
mul : Mul α := ⋯
npow : Pow α ℕ := ⋯
zpow : Pow α ℤ := ⋯
nnratCast : NNRatCast α := ⋯
ratCast : RatCast α := ⋯
nnqsmul : SMul ℚ≥0 α := ⋯
qsmul : SMul ℚ α := ⋯
⊢ ∀ (x : α) (n : ℤ), e (x ^ n) = e x ^ n

Tactic:
intros

Goal state after:
case zpow
α : Type u_1
β : Type u_2
e : α ≃ β
inst✝ : DivisionRing β
add_group_with_one : AddGroupWithOne α := e.addGroupWithOne
inv : Inv α := e.Inv
div : Div α := e.div
mul : Mul α := e.mul
npow : Pow α ℕ := Equiv.pow ℕ e
zpow : Pow α ℤ := Equiv.pow ℤ e
nnratCast : NNRatCast α := e.nnratCast
ratCast : RatCast α := e.ratCast
nnqsmul : SMul ℚ≥0 α := Equiv.smul ℚ≥0 e
qsmul : SMul ℚ α := Equiv.smul ℚ e
x✝ : α
n✝ : ℤ
⊢ e (x✝ ^ n✝) = e x✝ ^ n✝
Module: Mathlib.Algebra.Field.TransferInstance

Goal state before:
pp: case natCast
α : Type u_1
β : Type u_2
e : α ≃ β
inst✝ : DivisionRing β
add_group_with_one : AddGroupWithOne α := ⋯
inv : Inv α := ⋯
div : Div α := ⋯
mul : Mul α := ⋯
npow : Pow α ℕ := ⋯
zpow : Pow α ℤ := ⋯
nnratCast : NNRatCast α := ⋯
ratCast : RatCast α := ⋯
nnqsmul : SMul ℚ≥0 α := ⋯
qsmul : SMul ℚ α := ⋯
⊢ ∀ (n : ℕ), e ↑n = ↑n

Tactic:
intros

Goal state after:
case natCast
α : Type u_1
β : Type u_2
e : α ≃ β
inst✝ : DivisionRing β
add_group_with_one : AddGroupWithOne α := e.addGroupWithOne
inv : Inv α := e.Inv
div : Div α := e.div
mul : Mul α := e.mul
npow : Pow α ℕ := Equiv.pow ℕ e
zpow : Pow α ℤ := Equiv.pow ℤ e
nnratCast : NNRatCast α := e.nnratCast
ratCast : RatCast α := e.ratCast
nnqsmul : SMul ℚ≥0 α := Equiv.smul ℚ≥0 e
qsmul : SMul ℚ α := Equiv.smul ℚ e
n✝ : ℕ
⊢ e ↑n✝ = ↑n✝
Module: Mathlib.Algebra.Field.TransferInstance

Goal state before:
pp: case intCast
α : Type u_1
β : Type u_2
e : α ≃ β
inst✝ : DivisionRing β
add_group_with_one : AddGroupWithOne α := ⋯
inv : Inv α := ⋯
div : Div α := ⋯
mul : Mul α := ⋯
npow : Pow α ℕ := ⋯
zpow : Pow α ℤ := ⋯
nnratCast : NNRatCast α := ⋯
ratCast : RatCast α := ⋯
nnqsmul : SMul ℚ≥0 α := ⋯
qsmul : SMul ℚ α := ⋯
⊢ ∀ (n : ℤ), e ↑n = ↑n

Tactic:
intros

Goal state after:
case intCast
α : Type u_1
β : Type u_2
e : α ≃ β
inst✝ : DivisionRing β
add_group_with_one : AddGroupWithOne α := e.addGroupWithOne
inv : Inv α := e.Inv
div : Div α := e.div
mul : Mul α := e.mul
npow : Pow α ℕ := Equiv.pow ℕ e
zpow : Pow α ℤ := Equiv.pow ℤ e
nnratCast : NNRatCast α := e.nnratCast
ratCast : RatCast α := e.ratCast
nnqsmul : SMul ℚ≥0 α := Equiv.smul ℚ≥0 e
qsmul : SMul ℚ α := Equiv.smul ℚ e
n✝ : ℤ
⊢ e ↑n✝ = ↑n✝
Module: Mathlib.Algebra.Field.TransferInstance

Goal state before:
pp: case nnratCast
α : Type u_1
β : Type u_2
e : α ≃ β
inst✝ : DivisionRing β
add_group_with_one : AddGroupWithOne α := ⋯
inv : Inv α := ⋯
div : Div α := ⋯
mul : Mul α := ⋯
npow : Pow α ℕ := ⋯
zpow : Pow α ℤ := ⋯
nnratCast : NNRatCast α := ⋯
ratCast : RatCast α := ⋯
nnqsmul : SMul ℚ≥0 α := ⋯
qsmul : SMul ℚ α := ⋯
⊢ ∀ (q : ℚ≥0), e ↑q = ↑q

Tactic:
intros

Goal state after:
case nnratCast
α : Type u_1
β : Type u_2
e : α ≃ β
inst✝ : DivisionRing β
add_group_with_one : AddGroupWithOne α := e.addGroupWithOne
inv : Inv α := e.Inv
div : Div α := e.div
mul : Mul α := e.mul
npow : Pow α ℕ := Equiv.pow ℕ e
zpow : Pow α ℤ := Equiv.pow ℤ e
nnratCast : NNRatCast α := e.nnratCast
ratCast : RatCast α := e.ratCast
nnqsmul : SMul ℚ≥0 α := Equiv.smul ℚ≥0 e
qsmul : SMul ℚ α := Equiv.smul ℚ e
q✝ : ℚ≥0
⊢ e ↑q✝ = ↑q✝
Module: Mathlib.Algebra.Field.TransferInstance

Goal state before:
pp: case ratCast
α : Type u_1
β : Type u_2
e : α ≃ β
inst✝ : DivisionRing β
add_group_with_one : AddGroupWithOne α := ⋯
inv : Inv α := ⋯
div : Div α := ⋯
mul : Mul α := ⋯
npow : Pow α ℕ := ⋯
zpow : Pow α ℤ := ⋯
nnratCast : NNRatCast α := ⋯
ratCast : RatCast α := ⋯
nnqsmul : SMul ℚ≥0 α := ⋯
qsmul : SMul ℚ α := ⋯
⊢ ∀ (q : ℚ), e ↑q = ↑q

Tactic:
intros

Goal state after:
case ratCast
α : Type u_1
β : Type u_2
e : α ≃ β
inst✝ : DivisionRing β
add_group_with_one : AddGroupWithOne α := e.addGroupWithOne
inv : Inv α := e.Inv
div : Div α := e.div
mul : Mul α := e.mul
npow : Pow α ℕ := Equiv.pow ℕ e
zpow : Pow α ℤ := Equiv.pow ℤ e
nnratCast : NNRatCast α := e.nnratCast
ratCast : RatCast α := e.ratCast
nnqsmul : SMul ℚ≥0 α := Equiv.smul ℚ≥0 e
qsmul : SMul ℚ α := Equiv.smul ℚ e
q✝ : ℚ
⊢ e ↑q✝ = ↑q✝
Module: Mathlib.Algebra.Field.TransferInstance

Goal state before:
pp: case zero
α : Type u_1
β : Type u_2
e : α ≃ β
inst✝ : Field β
add_group_with_one : AddGroupWithOne α := ⋯
neg : Neg α := ⋯
inv : Inv α := ⋯
div : Div α := ⋯
mul : Mul α := ⋯
npow : Pow α ℕ := ⋯
zpow : Pow α ℤ := ⋯
nnratCast : NNRatCast α := ⋯
ratCast : RatCast α := ⋯
nnqsmul : SMul ℚ≥0 α := ⋯
qsmul : SMul ℚ α := ⋯
⊢ e 0 = 0

Tactic:
intros

Goal state after:
case zero
α : Type u_1
β : Type u_2
e : α ≃ β
inst✝ : Field β
add_group_with_one : AddGroupWithOne α := ⋯
neg : Neg α := ⋯
inv : Inv α := ⋯
div : Div α := ⋯
mul : Mul α := ⋯
npow : Pow α ℕ := ⋯
zpow : Pow α ℤ := ⋯
nnratCast : NNRatCast α := ⋯
ratCast : RatCast α := ⋯
nnqsmul : SMul ℚ≥0 α := ⋯
qsmul : SMul ℚ α := ⋯
⊢ e 0 = 0
Module: Mathlib.Algebra.Field.TransferInstance

Goal state before:
pp: case one
α : Type u_1
β : Type u_2
e : α ≃ β
inst✝ : Field β
add_group_with_one : AddGroupWithOne α := ⋯
neg : Neg α := ⋯
inv : Inv α := ⋯
div : Div α := ⋯
mul : Mul α := ⋯
npow : Pow α ℕ := ⋯
zpow : Pow α ℤ := ⋯
nnratCast : NNRatCast α := ⋯
ratCast : RatCast α := ⋯
nnqsmul : SMul ℚ≥0 α := ⋯
qsmul : SMul ℚ α := ⋯
⊢ e 1 = 1

Tactic:
intros

Goal state after:
case one
α : Type u_1
β : Type u_2
e : α ≃ β
inst✝ : Field β
add_group_with_one : AddGroupWithOne α := ⋯
neg : Neg α := ⋯
inv : Inv α := ⋯
div : Div α := ⋯
mul : Mul α := ⋯
npow : Pow α ℕ := ⋯
zpow : Pow α ℤ := ⋯
nnratCast : NNRatCast α := ⋯
ratCast : RatCast α := ⋯
nnqsmul : SMul ℚ≥0 α := ⋯
qsmul : SMul ℚ α := ⋯
⊢ e 1 = 1
Module: Mathlib.Algebra.Field.TransferInstance

Goal state before:
pp: case add
α : Type u_1
β : Type u_2
e : α ≃ β
inst✝ : Field β
add_group_with_one : AddGroupWithOne α := ⋯
neg : Neg α := ⋯
inv : Inv α := ⋯
div : Div α := ⋯
mul : Mul α := ⋯
npow : Pow α ℕ := ⋯
zpow : Pow α ℤ := ⋯
nnratCast : NNRatCast α := ⋯
ratCast : RatCast α := ⋯
nnqsmul : SMul ℚ≥0 α := ⋯
qsmul : SMul ℚ α := ⋯
⊢ ∀ (x y : α), e (x + y) = e x + e y

Tactic:
intros

Goal state after:
case add
α : Type u_1
β : Type u_2
e : α ≃ β
inst✝ : Field β
add_group_with_one : AddGroupWithOne α := e.addGroupWithOne
neg : Neg α := e.Neg
inv : Inv α := e.Inv
div : Div α := e.div
mul : Mul α := e.mul
npow : Pow α ℕ := Equiv.pow ℕ e
zpow : Pow α ℤ := Equiv.pow ℤ e
nnratCast : NNRatCast α := e.nnratCast
ratCast : RatCast α := e.ratCast
nnqsmul : SMul ℚ≥0 α := Equiv.smul ℚ≥0 e
qsmul : SMul ℚ α := Equiv.smul ℚ e
x✝ y✝ : α
⊢ e (x✝ + y✝) = e x✝ + e y✝
Module: Mathlib.Algebra.Field.TransferInstance

Goal state before:
pp: case mul
α : Type u_1
β : Type u_2
e : α ≃ β
inst✝ : Field β
add_group_with_one : AddGroupWithOne α := ⋯
neg : Neg α := ⋯
inv : Inv α := ⋯
div : Div α := ⋯
mul : Mul α := ⋯
npow : Pow α ℕ := ⋯
zpow : Pow α ℤ := ⋯
nnratCast : NNRatCast α := ⋯
ratCast : RatCast α := ⋯
nnqsmul : SMul ℚ≥0 α := ⋯
qsmul : SMul ℚ α := ⋯
⊢ ∀ (x y : α), e (x * y) = e x * e y

Tactic:
intros

Goal state after:
case mul
α : Type u_1
β : Type u_2
e : α ≃ β
inst✝ : Field β
add_group_with_one : AddGroupWithOne α := e.addGroupWithOne
neg : Neg α := e.Neg
inv : Inv α := e.Inv
div : Div α := e.div
mul : Mul α := e.mul
npow : Pow α ℕ := Equiv.pow ℕ e
zpow : Pow α ℤ := Equiv.pow ℤ e
nnratCast : NNRatCast α := e.nnratCast
ratCast : RatCast α := e.ratCast
nnqsmul : SMul ℚ≥0 α := Equiv.smul ℚ≥0 e
qsmul : SMul ℚ α := Equiv.smul ℚ e
x✝ y✝ : α
⊢ e (x✝ * y✝) = e x✝ * e y✝
Module: Mathlib.Algebra.Field.TransferInstance

Goal state before:
pp: case neg
α : Type u_1
β : Type u_2
e : α ≃ β
inst✝ : Field β
add_group_with_one : AddGroupWithOne α := ⋯
neg : Neg α := ⋯
inv : Inv α := ⋯
div : Div α := ⋯
mul : Mul α := ⋯
npow : Pow α ℕ := ⋯
zpow : Pow α ℤ := ⋯
nnratCast : NNRatCast α := ⋯
ratCast : RatCast α := ⋯
nnqsmul : SMul ℚ≥0 α := ⋯
qsmul : SMul ℚ α := ⋯
⊢ ∀ (x : α), e (-x) = -e x

Tactic:
intros

Goal state after:
case neg
α : Type u_1
β : Type u_2
e : α ≃ β
inst✝ : Field β
add_group_with_one : AddGroupWithOne α := e.addGroupWithOne
neg : Neg α := e.Neg
inv : Inv α := e.Inv
div : Div α := e.div
mul : Mul α := e.mul
npow : Pow α ℕ := Equiv.pow ℕ e
zpow : Pow α ℤ := Equiv.pow ℤ e
nnratCast : NNRatCast α := e.nnratCast
ratCast : RatCast α := e.ratCast
nnqsmul : SMul ℚ≥0 α := Equiv.smul ℚ≥0 e
qsmul : SMul ℚ α := Equiv.smul ℚ e
x✝ : α
⊢ e (-x✝) = -e x✝
Module: Mathlib.Algebra.Field.TransferInstance

Goal state before:
pp: case sub
α : Type u_1
β : Type u_2
e : α ≃ β
inst✝ : Field β
add_group_with_one : AddGroupWithOne α := ⋯
neg : Neg α := ⋯
inv : Inv α := ⋯
div : Div α := ⋯
mul : Mul α := ⋯
npow : Pow α ℕ := ⋯
zpow : Pow α ℤ := ⋯
nnratCast : NNRatCast α := ⋯
ratCast : RatCast α := ⋯
nnqsmul : SMul ℚ≥0 α := ⋯
qsmul : SMul ℚ α := ⋯
⊢ ∀ (x y : α), e (x - y) = e x - e y

Tactic:
intros

Goal state after:
case sub
α : Type u_1
β : Type u_2
e : α ≃ β
inst✝ : Field β
add_group_with_one : AddGroupWithOne α := e.addGroupWithOne
neg : Neg α := e.Neg
inv : Inv α := e.Inv
div : Div α := e.div
mul : Mul α := e.mul
npow : Pow α ℕ := Equiv.pow ℕ e
zpow : Pow α ℤ := Equiv.pow ℤ e
nnratCast : NNRatCast α := e.nnratCast
ratCast : RatCast α := e.ratCast
nnqsmul : SMul ℚ≥0 α := Equiv.smul ℚ≥0 e
qsmul : SMul ℚ α := Equiv.smul ℚ e
x✝ y✝ : α
⊢ e (x✝ - y✝) = e x✝ - e y✝
Module: Mathlib.Algebra.Field.TransferInstance

Goal state before:
pp: case inv
α : Type u_1
β : Type u_2
e : α ≃ β
inst✝ : Field β
add_group_with_one : AddGroupWithOne α := ⋯
neg : Neg α := ⋯
inv : Inv α := ⋯
div : Div α := ⋯
mul : Mul α := ⋯
npow : Pow α ℕ := ⋯
zpow : Pow α ℤ := ⋯
nnratCast : NNRatCast α := ⋯
ratCast : RatCast α := ⋯
nnqsmul : SMul ℚ≥0 α := ⋯
qsmul : SMul ℚ α := ⋯
⊢ ∀ (x : α), e x⁻¹ = (e x)⁻¹

Tactic:
intros

Goal state after:
case inv
α : Type u_1
β : Type u_2
e : α ≃ β
inst✝ : Field β
add_group_with_one : AddGroupWithOne α := e.addGroupWithOne
neg : Neg α := e.Neg
inv : Inv α := e.Inv
div : Div α := e.div
mul : Mul α := e.mul
npow : Pow α ℕ := Equiv.pow ℕ e
zpow : Pow α ℤ := Equiv.pow ℤ e
nnratCast : NNRatCast α := e.nnratCast
ratCast : RatCast α := e.ratCast
nnqsmul : SMul ℚ≥0 α := Equiv.smul ℚ≥0 e
qsmul : SMul ℚ α := Equiv.smul ℚ e
x✝ : α
⊢ e x✝⁻¹ = (e x✝)⁻¹
Module: Mathlib.Algebra.Field.TransferInstance

Goal state before:
pp: case div
α : Type u_1
β : Type u_2
e : α ≃ β
inst✝ : Field β
add_group_with_one : AddGroupWithOne α := ⋯
neg : Neg α := ⋯
inv : Inv α := ⋯
div : Div α := ⋯
mul : Mul α := ⋯
npow : Pow α ℕ := ⋯
zpow : Pow α ℤ := ⋯
nnratCast : NNRatCast α := ⋯
ratCast : RatCast α := ⋯
nnqsmul : SMul ℚ≥0 α := ⋯
qsmul : SMul ℚ α := ⋯
⊢ ∀ (x y : α), e (x / y) = e x / e y

Tactic:
intros

Goal state after:
case div
α : Type u_1
β : Type u_2
e : α ≃ β
inst✝ : Field β
add_group_with_one : AddGroupWithOne α := e.addGroupWithOne
neg : Neg α := e.Neg
inv : Inv α := e.Inv
div : Div α := e.div
mul : Mul α := e.mul
npow : Pow α ℕ := Equiv.pow ℕ e
zpow : Pow α ℤ := Equiv.pow ℤ e
nnratCast : NNRatCast α := e.nnratCast
ratCast : RatCast α := e.ratCast
nnqsmul : SMul ℚ≥0 α := Equiv.smul ℚ≥0 e
qsmul : SMul ℚ α := Equiv.smul ℚ e
x✝ y✝ : α
⊢ e (x✝ / y✝) = e x✝ / e y✝
Module: Mathlib.Algebra.Field.TransferInstance

Goal state before:
pp: case nsmul
α : Type u_1
β : Type u_2
e : α ≃ β
inst✝ : Field β
add_group_with_one : AddGroupWithOne α := ⋯
neg : Neg α := ⋯
inv : Inv α := ⋯
div : Div α := ⋯
mul : Mul α := ⋯
npow : Pow α ℕ := ⋯
zpow : Pow α ℤ := ⋯
nnratCast : NNRatCast α := ⋯
ratCast : RatCast α := ⋯
nnqsmul : SMul ℚ≥0 α := ⋯
qsmul : SMul ℚ α := ⋯
⊢ ∀ (n : ℕ) (x : α), e (n • x) = n • e x

Tactic:
intros

Goal state after:
case nsmul
α : Type u_1
β : Type u_2
e : α ≃ β
inst✝ : Field β
add_group_with_one : AddGroupWithOne α := e.addGroupWithOne
neg : Neg α := e.Neg
inv : Inv α := e.Inv
div : Div α := e.div
mul : Mul α := e.mul
npow : Pow α ℕ := Equiv.pow ℕ e
zpow : Pow α ℤ := Equiv.pow ℤ e
nnratCast : NNRatCast α := e.nnratCast
ratCast : RatCast α := e.ratCast
nnqsmul : SMul ℚ≥0 α := Equiv.smul ℚ≥0 e
qsmul : SMul ℚ α := Equiv.smul ℚ e
n✝ : ℕ
x✝ : α
⊢ e (n✝ • x✝) = n✝ • e x✝
Module: Mathlib.Algebra.Field.TransferInstance

Goal state before:
pp: case zsmul
α : Type u_1
β : Type u_2
e : α ≃ β
inst✝ : Field β
add_group_with_one : AddGroupWithOne α := ⋯
neg : Neg α := ⋯
inv : Inv α := ⋯
div : Div α := ⋯
mul : Mul α := ⋯
npow : Pow α ℕ := ⋯
zpow : Pow α ℤ := ⋯
nnratCast : NNRatCast α := ⋯
ratCast : RatCast α := ⋯
nnqsmul : SMul ℚ≥0 α := ⋯
qsmul : SMul ℚ α := ⋯
⊢ ∀ (n : ℤ) (x : α), e (n • x) = n • e x

Tactic:
intros

Goal state after:
case zsmul
α : Type u_1
β : Type u_2
e : α ≃ β
inst✝ : Field β
add_group_with_one : AddGroupWithOne α := e.addGroupWithOne
neg : Neg α := e.Neg
inv : Inv α := e.Inv
div : Div α := e.div
mul : Mul α := e.mul
npow : Pow α ℕ := Equiv.pow ℕ e
zpow : Pow α ℤ := Equiv.pow ℤ e
nnratCast : NNRatCast α := e.nnratCast
ratCast : RatCast α := e.ratCast
nnqsmul : SMul ℚ≥0 α := Equiv.smul ℚ≥0 e
qsmul : SMul ℚ α := Equiv.smul ℚ e
n✝ : ℤ
x✝ : α
⊢ e (n✝ • x✝) = n✝ • e x✝
Module: Mathlib.Algebra.Field.TransferInstance

Goal state before:
pp: case nnqsmul
α : Type u_1
β : Type u_2
e : α ≃ β
inst✝ : Field β
add_group_with_one : AddGroupWithOne α := ⋯
neg : Neg α := ⋯
inv : Inv α := ⋯
div : Div α := ⋯
mul : Mul α := ⋯
npow : Pow α ℕ := ⋯
zpow : Pow α ℤ := ⋯
nnratCast : NNRatCast α := ⋯
ratCast : RatCast α := ⋯
nnqsmul : SMul ℚ≥0 α := ⋯
qsmul : SMul ℚ α := ⋯
⊢ ∀ (q : ℚ≥0) (x : α), e (q • x) = q • e x

Tactic:
intros

Goal state after:
case nnqsmul
α : Type u_1
β : Type u_2
e : α ≃ β
inst✝ : Field β
add_group_with_one : AddGroupWithOne α := e.addGroupWithOne
neg : Neg α := e.Neg
inv : Inv α := e.Inv
div : Div α := e.div
mul : Mul α := e.mul
npow : Pow α ℕ := Equiv.pow ℕ e
zpow : Pow α ℤ := Equiv.pow ℤ e
nnratCast : NNRatCast α := e.nnratCast
ratCast : RatCast α := e.ratCast
nnqsmul : SMul ℚ≥0 α := Equiv.smul ℚ≥0 e
qsmul : SMul ℚ α := Equiv.smul ℚ e
q✝ : ℚ≥0
x✝ : α
⊢ e (q✝ • x✝) = q✝ • e x✝