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

Goal state before:
pp: case add
α : Type u_1
β : Type u_2
e : α ≃ β
inst✝ : NonUnitalNonAssocRing β
zero : Zero α := e.zero
add : Add α := e.add
mul : Mul α := e.mul
neg : Neg α := e.Neg
sub : Sub α := e.sub
nsmul : SMul ℕ α := Equiv.smul ℕ e
zsmul : SMul ℤ α := Equiv.smul ℤ e
x✝ y✝ : α
⊢ e (x✝ + y✝) = e x✝ + e y✝

Tactic:
exact e.apply_symm_apply _

Goal state after:
Module: Mathlib.Algebra.Ring.TransferInstance

Goal state before:
pp: case mul
α : Type u_1
β : Type u_2
e : α ≃ β
inst✝ : NonUnitalNonAssocRing β
zero : Zero α := e.zero
add : Add α := e.add
mul : Mul α := e.mul
neg : Neg α := e.Neg
sub : Sub α := e.sub
nsmul : SMul ℕ α := Equiv.smul ℕ e
zsmul : SMul ℤ α := Equiv.smul ℤ e
x✝ y✝ : α
⊢ e (x✝ * y✝) = e x✝ * e y✝

Tactic:
exact e.apply_symm_apply _

Goal state after:
Module: Mathlib.Algebra.Ring.TransferInstance

Goal state before:
pp: case neg
α : Type u_1
β : Type u_2
e : α ≃ β
inst✝ : NonUnitalNonAssocRing β
zero : Zero α := e.zero
add : Add α := e.add
mul : Mul α := e.mul
neg : Neg α := e.Neg
sub : Sub α := e.sub
nsmul : SMul ℕ α := Equiv.smul ℕ e
zsmul : SMul ℤ α := Equiv.smul ℤ e
x✝ : α
⊢ e (-x✝) = -e x✝

Tactic:
exact e.apply_symm_apply _

Goal state after:
Module: Mathlib.Algebra.Ring.TransferInstance

Goal state before:
pp: case sub
α : Type u_1
β : Type u_2
e : α ≃ β
inst✝ : NonUnitalNonAssocRing β
zero : Zero α := e.zero
add : Add α := e.add
mul : Mul α := e.mul
neg : Neg α := e.Neg
sub : Sub α := e.sub
nsmul : SMul ℕ α := Equiv.smul ℕ e
zsmul : SMul ℤ α := Equiv.smul ℤ e
x✝ y✝ : α
⊢ e (x✝ - y✝) = e x✝ - e y✝

Tactic:
exact e.apply_symm_apply _

Goal state after:
Module: Mathlib.Algebra.Ring.TransferInstance

Goal state before:
pp: case nsmul
α : Type u_1
β : Type u_2
e : α ≃ β
inst✝ : NonUnitalNonAssocRing β
zero : Zero α := e.zero
add : Add α := e.add
mul : Mul α := e.mul
neg : Neg α := e.Neg
sub : Sub α := e.sub
nsmul : SMul ℕ α := Equiv.smul ℕ e
zsmul : SMul ℤ α := Equiv.smul ℤ e
n✝ : ℕ
x✝ : α
⊢ e (n✝ • x✝) = n✝ • e x✝

Tactic:
exact e.apply_symm_apply _

Goal state after:
Module: Mathlib.Algebra.Ring.TransferInstance

Goal state before:
pp: case zsmul
α : Type u_1
β : Type u_2
e : α ≃ β
inst✝ : NonUnitalNonAssocRing β
zero : Zero α := e.zero
add : Add α := e.add
mul : Mul α := e.mul
neg : Neg α := e.Neg
sub : Sub α := e.sub
nsmul : SMul ℕ α := Equiv.smul ℕ e
zsmul : SMul ℤ α := Equiv.smul ℤ e
n✝ : ℤ
x✝ : α
⊢ e (n✝ • x✝) = n✝ • e x✝

Tactic:
exact e.apply_symm_apply _

Goal state after:
Module: Mathlib.Algebra.Ring.TransferInstance

Goal state before:
pp: case zero
α : Type u_1
β : Type u_2
e : α ≃ β
inst✝ : NonUnitalRing β
zero : Zero α := ⋯
add : Add α := ⋯
mul : Mul α := ⋯
neg : Neg α := ⋯
sub : Sub α := ⋯
nsmul : SMul ℕ α := ⋯
zsmul : SMul ℤ α := ⋯
⊢ e 0 = 0

Tactic:
exact e.apply_symm_apply _

Goal state after:
Module: Mathlib.Algebra.Ring.TransferInstance

Goal state before:
pp: case add
α : Type u_1
β : Type u_2
e : α ≃ β
inst✝ : NonUnitalRing β
zero : Zero α := e.zero
add : Add α := e.add
mul : Mul α := e.mul
neg : Neg α := e.Neg
sub : Sub α := e.sub
nsmul : SMul ℕ α := Equiv.smul ℕ e
zsmul : SMul ℤ α := Equiv.smul ℤ e
x✝ y✝ : α
⊢ e (x✝ + y✝) = e x✝ + e y✝

Tactic:
exact e.apply_symm_apply _

Goal state after:
Module: Mathlib.Algebra.Ring.TransferInstance

Goal state before:
pp: case mul
α : Type u_1
β : Type u_2
e : α ≃ β
inst✝ : NonUnitalRing β
zero : Zero α := e.zero
add : Add α := e.add
mul : Mul α := e.mul
neg : Neg α := e.Neg
sub : Sub α := e.sub
nsmul : SMul ℕ α := Equiv.smul ℕ e
zsmul : SMul ℤ α := Equiv.smul ℤ e
x✝ y✝ : α
⊢ e (x✝ * y✝) = e x✝ * e y✝

Tactic:
exact e.apply_symm_apply _

Goal state after:
Module: Mathlib.Algebra.Ring.TransferInstance

Goal state before:
pp: case neg
α : Type u_1
β : Type u_2
e : α ≃ β
inst✝ : NonUnitalRing β
zero : Zero α := e.zero
add : Add α := e.add
mul : Mul α := e.mul
neg : Neg α := e.Neg
sub : Sub α := e.sub
nsmul : SMul ℕ α := Equiv.smul ℕ e
zsmul : SMul ℤ α := Equiv.smul ℤ e
x✝ : α
⊢ e (-x✝) = -e x✝

Tactic:
exact e.apply_symm_apply _

Goal state after:
Module: Mathlib.Algebra.Ring.TransferInstance

Goal state before:
pp: case sub
α : Type u_1
β : Type u_2
e : α ≃ β
inst✝ : NonUnitalRing β
zero : Zero α := e.zero
add : Add α := e.add
mul : Mul α := e.mul
neg : Neg α := e.Neg
sub : Sub α := e.sub
nsmul : SMul ℕ α := Equiv.smul ℕ e
zsmul : SMul ℤ α := Equiv.smul ℤ e
x✝ y✝ : α
⊢ e (x✝ - y✝) = e x✝ - e y✝

Tactic:
exact e.apply_symm_apply _

Goal state after:
Module: Mathlib.Algebra.Ring.TransferInstance

Goal state before:
pp: case nsmul
α : Type u_1
β : Type u_2
e : α ≃ β
inst✝ : NonUnitalRing β
zero : Zero α := e.zero
add : Add α := e.add
mul : Mul α := e.mul
neg : Neg α := e.Neg
sub : Sub α := e.sub
nsmul : SMul ℕ α := Equiv.smul ℕ e
zsmul : SMul ℤ α := Equiv.smul ℤ e
n✝ : ℕ
x✝ : α
⊢ e (n✝ • x✝) = n✝ • e x✝

Tactic:
exact e.apply_symm_apply _

Goal state after:
Module: Mathlib.Algebra.Ring.TransferInstance

Goal state before:
pp: case zsmul
α : Type u_1
β : Type u_2
e : α ≃ β
inst✝ : NonUnitalRing β
zero : Zero α := e.zero
add : Add α := e.add
mul : Mul α := e.mul
neg : Neg α := e.Neg
sub : Sub α := e.sub
nsmul : SMul ℕ α := Equiv.smul ℕ e
zsmul : SMul ℤ α := Equiv.smul ℤ e
n✝ : ℤ
x✝ : α
⊢ e (n✝ • x✝) = n✝ • e x✝

Tactic:
exact e.apply_symm_apply _

Goal state after:
Module: Mathlib.Algebra.Ring.TransferInstance

Goal state before:
pp: case zero
α : Type u_1
β : Type u_2
e : α ≃ β
inst✝ : NonAssocRing β
add_group_with_one : AddGroupWithOne α := ⋯
mul : Mul α := ⋯
⊢ e 0 = 0

Tactic:
exact e.apply_symm_apply _

Goal state after:
Module: Mathlib.Algebra.Ring.TransferInstance

Goal state before:
pp: case one
α : Type u_1
β : Type u_2
e : α ≃ β
inst✝ : NonAssocRing β
add_group_with_one : AddGroupWithOne α := ⋯
mul : Mul α := ⋯
⊢ e 1 = 1

Tactic:
exact e.apply_symm_apply _

Goal state after:
Module: Mathlib.Algebra.Ring.TransferInstance

Goal state before:
pp: case add
α : Type u_1
β : Type u_2
e : α ≃ β
inst✝ : NonAssocRing β
add_group_with_one : AddGroupWithOne α := e.addGroupWithOne
mul : Mul α := e.mul
x✝ y✝ : α
⊢ e (x✝ + y✝) = e x✝ + e y✝

Tactic:
exact e.apply_symm_apply _

Goal state after:
Module: Mathlib.Algebra.Ring.TransferInstance

Goal state before:
pp: case mul
α : Type u_1
β : Type u_2
e : α ≃ β
inst✝ : NonAssocRing β
add_group_with_one : AddGroupWithOne α := e.addGroupWithOne
mul : Mul α := e.mul
x✝ y✝ : α
⊢ e (x✝ * y✝) = e x✝ * e y✝

Tactic:
exact e.apply_symm_apply _

Goal state after:
Module: Mathlib.Algebra.Ring.TransferInstance

Goal state before:
pp: case neg
α : Type u_1
β : Type u_2
e : α ≃ β
inst✝ : NonAssocRing β
add_group_with_one : AddGroupWithOne α := e.addGroupWithOne
mul : Mul α := e.mul
x✝ : α
⊢ e (-x✝) = -e x✝

Tactic:
exact e.apply_symm_apply _

Goal state after:
Module: Mathlib.Algebra.Ring.TransferInstance

Goal state before:
pp: case sub
α : Type u_1
β : Type u_2
e : α ≃ β
inst✝ : NonAssocRing β
add_group_with_one : AddGroupWithOne α := e.addGroupWithOne
mul : Mul α := e.mul
x✝ y✝ : α
⊢ e (x✝ - y✝) = e x✝ - e y✝

Tactic:
exact e.apply_symm_apply _

Goal state after:
Module: Mathlib.Algebra.Ring.TransferInstance

Goal state before:
pp: case nsmul
α : Type u_1
β : Type u_2
e : α ≃ β
inst✝ : NonAssocRing β
add_group_with_one : AddGroupWithOne α := e.addGroupWithOne
mul : Mul α := e.mul
n✝ : ℕ
x✝ : α
⊢ e (n✝ • x✝) = n✝ • e x✝

Tactic:
exact e.apply_symm_apply _

Goal state after:
Module: Mathlib.Algebra.Ring.TransferInstance

Goal state before:
pp: case zsmul
α : Type u_1
β : Type u_2
e : α ≃ β
inst✝ : NonAssocRing β
add_group_with_one : AddGroupWithOne α := e.addGroupWithOne
mul : Mul α := e.mul
n✝ : ℤ
x✝ : α
⊢ e (n✝ • x✝) = n✝ • e x✝

Tactic:
exact e.apply_symm_apply _

Goal state after:
Module: Mathlib.Algebra.Ring.TransferInstance

Goal state before:
pp: case natCast
α : Type u_1
β : Type u_2
e : α ≃ β
inst✝ : NonAssocRing β
add_group_with_one : AddGroupWithOne α := e.addGroupWithOne
mul : Mul α := e.mul
n✝ : ℕ
⊢ e ↑n✝ = ↑n✝

Tactic:
exact e.apply_symm_apply _

Goal state after:
Module: Mathlib.Algebra.Ring.TransferInstance

Goal state before:
pp: case intCast
α : Type u_1
β : Type u_2
e : α ≃ β
inst✝ : NonAssocRing β
add_group_with_one : AddGroupWithOne α := e.addGroupWithOne
mul : Mul α := e.mul
n✝ : ℤ
⊢ e ↑n✝ = ↑n✝

Tactic:
exact e.apply_symm_apply _

Goal state after:
Module: Mathlib.Algebra.Ring.TransferInstance

Goal state before:
pp: case zero
α : Type u_1
β : Type u_2
e : α ≃ β
inst✝ : Ring β
mul : Mul α := ⋯
add_group_with_one : AddGroupWithOne α := ⋯
npow : Pow α ℕ := ⋯
⊢ e 0 = 0

Tactic:
exact e.apply_symm_apply _

Goal state after:
Module: Mathlib.Algebra.Ring.TransferInstance

Goal state before:
pp: case one
α : Type u_1
β : Type u_2
e : α ≃ β
inst✝ : Ring β
mul : Mul α := ⋯
add_group_with_one : AddGroupWithOne α := ⋯
npow : Pow α ℕ := ⋯
⊢ e 1 = 1

Tactic:
exact e.apply_symm_apply _

Goal state after:
Module: Mathlib.Algebra.Ring.TransferInstance

Goal state before:
pp: case add
α : Type u_1
β : Type u_2
e : α ≃ β
inst✝ : Ring β
mul : Mul α := e.mul
add_group_with_one : AddGroupWithOne α := e.addGroupWithOne
npow : Pow α ℕ := Equiv.pow ℕ e
x✝ y✝ : α
⊢ e (x✝ + y✝) = e x✝ + e y✝

Tactic:
exact e.apply_symm_apply _

Goal state after:
Module: Mathlib.Algebra.Ring.TransferInstance

Goal state before:
pp: case mul
α : Type u_1
β : Type u_2
e : α ≃ β
inst✝ : Ring β
mul : Mul α := e.mul
add_group_with_one : AddGroupWithOne α := e.addGroupWithOne
npow : Pow α ℕ := Equiv.pow ℕ e
x✝ y✝ : α
⊢ e (x✝ * y✝) = e x✝ * e y✝

Tactic:
exact e.apply_symm_apply _

Goal state after:
Module: Mathlib.Algebra.Ring.TransferInstance

Goal state before:
pp: case neg
α : Type u_1
β : Type u_2
e : α ≃ β
inst✝ : Ring β
mul : Mul α := e.mul
add_group_with_one : AddGroupWithOne α := e.addGroupWithOne
npow : Pow α ℕ := Equiv.pow ℕ e
x✝ : α
⊢ e (-x✝) = -e x✝

Tactic:
exact e.apply_symm_apply _

Goal state after:
Module: Mathlib.Algebra.Ring.TransferInstance

Goal state before:
pp: case sub
α : Type u_1
β : Type u_2
e : α ≃ β
inst✝ : Ring β
mul : Mul α := e.mul
add_group_with_one : AddGroupWithOne α := e.addGroupWithOne
npow : Pow α ℕ := Equiv.pow ℕ e
x✝ y✝ : α
⊢ e (x✝ - y✝) = e x✝ - e y✝

Tactic:
exact e.apply_symm_apply _

Goal state after:
Module: Mathlib.Algebra.Ring.TransferInstance

Goal state before:
pp: case nsmul
α : Type u_1
β : Type u_2
e : α ≃ β
inst✝ : Ring β
mul : Mul α := e.mul
add_group_with_one : AddGroupWithOne α := e.addGroupWithOne
npow : Pow α ℕ := Equiv.pow ℕ e
n✝ : ℕ
x✝ : α
⊢ e (n✝ • x✝) = n✝ • e x✝

Tactic:
exact e.apply_symm_apply _

Goal state after:
Module: Mathlib.Algebra.Ring.TransferInstance

Goal state before:
pp: case zsmul
α : Type u_1
β : Type u_2
e : α ≃ β
inst✝ : Ring β
mul : Mul α := e.mul
add_group_with_one : AddGroupWithOne α := e.addGroupWithOne
npow : Pow α ℕ := Equiv.pow ℕ e
n✝ : ℤ
x✝ : α
⊢ e (n✝ • x✝) = n✝ • e x✝

Tactic:
exact e.apply_symm_apply _

Goal state after:
Module: Mathlib.Algebra.Ring.TransferInstance

Goal state before:
pp: case npow
α : Type u_1
β : Type u_2
e : α ≃ β
inst✝ : Ring β
mul : Mul α := e.mul
add_group_with_one : AddGroupWithOne α := e.addGroupWithOne
npow : Pow α ℕ := Equiv.pow ℕ e
x✝ : α
n✝ : ℕ
⊢ e (x✝ ^ n✝) = e x✝ ^ n✝

Tactic:
exact e.apply_symm_apply _

Goal state after:
Module: Mathlib.Algebra.Ring.TransferInstance

Goal state before:
pp: case natCast
α : Type u_1
β : Type u_2
e : α ≃ β
inst✝ : Ring β
mul : Mul α := e.mul
add_group_with_one : AddGroupWithOne α := e.addGroupWithOne
npow : Pow α ℕ := Equiv.pow ℕ e
n✝ : ℕ
⊢ e ↑n✝ = ↑n✝

Tactic:
exact e.apply_symm_apply _

Goal state after:
Module: Mathlib.Algebra.Ring.TransferInstance

Goal state before:
pp: case intCast
α : Type u_1
β : Type u_2
e : α ≃ β
inst✝ : Ring β
mul : Mul α := e.mul
add_group_with_one : AddGroupWithOne α := e.addGroupWithOne
npow : Pow α ℕ := Equiv.pow ℕ e
n✝ : ℤ
⊢ e ↑n✝ = ↑n✝

Tactic:
exact e.apply_symm_apply _

Goal state after:
Module: Mathlib.Algebra.Ring.TransferInstance

Goal state before:
pp: case zero
α : Type u_1
β : Type u_2
e : α ≃ β
inst✝ : NonUnitalCommRing β
zero : Zero α := ⋯
add : Add α := ⋯
mul : Mul α := ⋯
neg : Neg α := ⋯
sub : Sub α := ⋯
nsmul : SMul ℕ α := ⋯
zsmul : SMul ℤ α := ⋯
⊢ e 0 = 0

Tactic:
exact e.apply_symm_apply _

Goal state after:
Module: Mathlib.Algebra.Ring.TransferInstance

Goal state before:
pp: case add
α : Type u_1
β : Type u_2
e : α ≃ β
inst✝ : NonUnitalCommRing β
zero : Zero α := e.zero
add : Add α := e.add
mul : Mul α := e.mul
neg : Neg α := e.Neg
sub : Sub α := e.sub
nsmul : SMul ℕ α := Equiv.smul ℕ e
zsmul : SMul ℤ α := Equiv.smul ℤ e
x✝ y✝ : α
⊢ e (x✝ + y✝) = e x✝ + e y✝

Tactic:
exact e.apply_symm_apply _

Goal state after:
Module: Mathlib.Algebra.Ring.TransferInstance

Goal state before:
pp: case mul
α : Type u_1
β : Type u_2
e : α ≃ β
inst✝ : NonUnitalCommRing β
zero : Zero α := e.zero
add : Add α := e.add
mul : Mul α := e.mul
neg : Neg α := e.Neg
sub : Sub α := e.sub
nsmul : SMul ℕ α := Equiv.smul ℕ e
zsmul : SMul ℤ α := Equiv.smul ℤ e
x✝ y✝ : α
⊢ e (x✝ * y✝) = e x✝ * e y✝

Tactic:
exact e.apply_symm_apply _

Goal state after:
Module: Mathlib.Algebra.Ring.TransferInstance

Goal state before:
pp: case neg
α : Type u_1
β : Type u_2
e : α ≃ β
inst✝ : NonUnitalCommRing β
zero : Zero α := e.zero
add : Add α := e.add
mul : Mul α := e.mul
neg : Neg α := e.Neg
sub : Sub α := e.sub
nsmul : SMul ℕ α := Equiv.smul ℕ e
zsmul : SMul ℤ α := Equiv.smul ℤ e
x✝ : α
⊢ e (-x✝) = -e x✝

Tactic:
exact e.apply_symm_apply _

Goal state after:
Module: Mathlib.Algebra.Ring.TransferInstance

Goal state before:
pp: case sub
α : Type u_1
β : Type u_2
e : α ≃ β
inst✝ : NonUnitalCommRing β
zero : Zero α := e.zero
add : Add α := e.add
mul : Mul α := e.mul
neg : Neg α := e.Neg
sub : Sub α := e.sub
nsmul : SMul ℕ α := Equiv.smul ℕ e
zsmul : SMul ℤ α := Equiv.smul ℤ e
x✝ y✝ : α
⊢ e (x✝ - y✝) = e x✝ - e y✝

Tactic:
exact e.apply_symm_apply _

Goal state after:
Module: Mathlib.Algebra.Ring.TransferInstance

Goal state before:
pp: case nsmul
α : Type u_1
β : Type u_2
e : α ≃ β
inst✝ : NonUnitalCommRing β
zero : Zero α := e.zero
add : Add α := e.add
mul : Mul α := e.mul
neg : Neg α := e.Neg
sub : Sub α := e.sub
nsmul : SMul ℕ α := Equiv.smul ℕ e
zsmul : SMul ℤ α := Equiv.smul ℤ e
n✝ : ℕ
x✝ : α
⊢ e (n✝ • x✝) = n✝ • e x✝

Tactic:
exact e.apply_symm_apply _

Goal state after:
Module: Mathlib.Algebra.Ring.TransferInstance

Goal state before:
pp: case zsmul
α : Type u_1
β : Type u_2
e : α ≃ β
inst✝ : NonUnitalCommRing β
zero : Zero α := e.zero
add : Add α := e.add
mul : Mul α := e.mul
neg : Neg α := e.Neg
sub : Sub α := e.sub
nsmul : SMul ℕ α := Equiv.smul ℕ e
zsmul : SMul ℤ α := Equiv.smul ℤ e
n✝ : ℤ
x✝ : α
⊢ e (n✝ • x✝) = n✝ • e x✝

Tactic:
exact e.apply_symm_apply _

Goal state after:
Module: Mathlib.Algebra.Ring.TransferInstance

Goal state before:
pp: case zero
α : Type u_1
β : Type u_2
e : α ≃ β
inst✝ : CommRing β
mul : Mul α := ⋯
add_group_with_one : AddGroupWithOne α := ⋯
npow : Pow α ℕ := ⋯
⊢ e 0 = 0

Tactic:
exact e.apply_symm_apply _

Goal state after:
Module: Mathlib.Algebra.Ring.TransferInstance

Goal state before:
pp: case one
α : Type u_1
β : Type u_2
e : α ≃ β
inst✝ : CommRing β
mul : Mul α := ⋯
add_group_with_one : AddGroupWithOne α := ⋯
npow : Pow α ℕ := ⋯
⊢ e 1 = 1

Tactic:
exact e.apply_symm_apply _

Goal state after:
Module: Mathlib.Algebra.Ring.TransferInstance

Goal state before:
pp: case add
α : Type u_1
β : Type u_2
e : α ≃ β
inst✝ : CommRing β
mul : Mul α := e.mul
add_group_with_one : AddGroupWithOne α := e.addGroupWithOne
npow : Pow α ℕ := Equiv.pow ℕ e
x✝ y✝ : α
⊢ e (x✝ + y✝) = e x✝ + e y✝

Tactic:
exact e.apply_symm_apply _

Goal state after:
Module: Mathlib.Algebra.Ring.TransferInstance

Goal state before:
pp: case mul
α : Type u_1
β : Type u_2
e : α ≃ β
inst✝ : CommRing β
mul : Mul α := e.mul
add_group_with_one : AddGroupWithOne α := e.addGroupWithOne
npow : Pow α ℕ := Equiv.pow ℕ e
x✝ y✝ : α
⊢ e (x✝ * y✝) = e x✝ * e y✝

Tactic:
exact e.apply_symm_apply _

Goal state after:
Module: Mathlib.Algebra.Ring.TransferInstance

Goal state before:
pp: case neg
α : Type u_1
β : Type u_2
e : α ≃ β
inst✝ : CommRing β
mul : Mul α := e.mul
add_group_with_one : AddGroupWithOne α := e.addGroupWithOne
npow : Pow α ℕ := Equiv.pow ℕ e
x✝ : α
⊢ e (-x✝) = -e x✝

Tactic:
exact e.apply_symm_apply _

Goal state after:
Module: Mathlib.Algebra.Ring.TransferInstance

Goal state before:
pp: case sub
α : Type u_1
β : Type u_2
e : α ≃ β
inst✝ : CommRing β
mul : Mul α := e.mul
add_group_with_one : AddGroupWithOne α := e.addGroupWithOne
npow : Pow α ℕ := Equiv.pow ℕ e
x✝ y✝ : α
⊢ e (x✝ - y✝) = e x✝ - e y✝

Tactic:
exact e.apply_symm_apply _

Goal state after:
Module: Mathlib.Algebra.Ring.TransferInstance

Goal state before:
pp: case nsmul
α : Type u_1
β : Type u_2
e : α ≃ β
inst✝ : CommRing β
mul : Mul α := e.mul
add_group_with_one : AddGroupWithOne α := e.addGroupWithOne
npow : Pow α ℕ := Equiv.pow ℕ e
n✝ : ℕ
x✝ : α
⊢ e (n✝ • x✝) = n✝ • e x✝

Tactic:
exact e.apply_symm_apply _

Goal state after:
Module: Mathlib.Algebra.Ring.TransferInstance

Goal state before:
pp: case zsmul
α : Type u_1
β : Type u_2
e : α ≃ β
inst✝ : CommRing β
mul : Mul α := e.mul
add_group_with_one : AddGroupWithOne α := e.addGroupWithOne
npow : Pow α ℕ := Equiv.pow ℕ e
n✝ : ℤ
x✝ : α
⊢ e (n✝ • x✝) = n✝ • e x✝

Tactic:
exact e.apply_symm_apply _

Goal state after:
Module: Mathlib.Algebra.Ring.TransferInstance

Goal state before:
pp: case npow
α : Type u_1
β : Type u_2
e : α ≃ β
inst✝ : CommRing β
mul : Mul α := e.mul
add_group_with_one : AddGroupWithOne α := e.addGroupWithOne
npow : Pow α ℕ := Equiv.pow ℕ e
x✝ : α
n✝ : ℕ
⊢ e (x✝ ^ n✝) = e x✝ ^ n✝

Tactic:
exact e.apply_symm_apply _

Goal state after:
Module: Mathlib.Algebra.Ring.TransferInstance

Goal state before:
pp: case natCast
α : Type u_1
β : Type u_2
e : α ≃ β
inst✝ : CommRing β
mul : Mul α := e.mul
add_group_with_one : AddGroupWithOne α := e.addGroupWithOne
npow : Pow α ℕ := Equiv.pow ℕ e
n✝ : ℕ
⊢ e ↑n✝ = ↑n✝

Tactic:
exact e.apply_symm_apply _

Goal state after:
Module: Mathlib.Algebra.Ring.TransferInstance

Goal state before:
pp: case intCast
α : Type u_1
β : Type u_2
e : α ≃ β
inst✝ : CommRing β
mul : Mul α := e.mul
add_group_with_one : AddGroupWithOne α := e.addGroupWithOne
npow : Pow α ℕ := Equiv.pow ℕ e
n✝ : ℤ
⊢ e ↑n✝ = ↑n✝

Tactic:
exact e.apply_symm_apply _

Goal state after:
Module: Mathlib.Algebra.Algebra.Spectrum.Basic

Goal state before:
pp: R : Type u
A : Type v
inst✝² : CommSemiring R
inst✝¹ : Ring A
inst✝ : Algebra R A
a b : A
r : R
ha : r ∈ resolventSet R a
hb : r ∈ resolventSet R b
⊢ resolvent a r - resolvent b r = resolvent a r * (a - b) * resolvent b r

Tactic:
resolvent_eq ha,

Goal state after:
R : Type u
A : Type v
inst✝² : CommSemiring R
inst✝¹ : Ring A
inst✝ : Algebra R A
a b : A
r : R
ha : r ∈ resolventSet R a
hb : r ∈ resolventSet R b
⊢ ↑(IsUnit.unit ha)⁻¹ - resolvent b r = ↑(IsUnit.unit ha)⁻¹ * (a - b) * resolvent b r
Module: Mathlib.RingTheory.TensorProduct.Basic

Goal state before:
pp: case tmul
R : Type uR
A : Type uA
B : Type uB
inst✝⁴ : CommRing R
inst✝³ : Ring A
inst✝² : Ring B
inst✝¹ : Algebra R A
inst✝ : Algebra R B
x : A
y : B
⊢ x ⊗ₜ[R] y ∈ Subring.closure (Set.range ⇑includeLeft ∪ Set.range ⇑includeRight)

Tactic:
convert_to (Algebra.TensorProduct.includeLeftRingHom (R := R) x) *
      (Algebra.TensorProduct.includeRight y) ∈ _

Goal state after:
case e'_5
R : Type uR
A : Type uA
B : Type uB
inst✝⁴ : CommRing R
inst✝³ : Ring A
inst✝² : Ring B
inst✝¹ : Algebra R A
inst✝ : Algebra R B
x : A
y : B
⊢ x ⊗ₜ[R] y = includeLeftRingHom x * includeRight y

case tmul.convert_4
R : Type uR
A : Type uA
B : Type uB
inst✝⁴ : CommRing R
inst✝³ : Ring A
inst✝² : Ring B
inst✝¹ : Algebra R A
inst✝ : Algebra R B
x : A
y : B
⊢ includeLeftRingHom x * includeRight y ∈ Subring.closure (Set.range ⇑includeLeft ∪ Set.range ⇑includeRight)
Module: Mathlib.Algebra.Algebra.StrictPositivity

Goal state before:
pp: A : Type u_1
inst✝³ : Semiring A
inst✝² : StarRing A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
u a : A
hu : IsUnit u
⊢ IsStrictlyPositive (star u * a * u) ↔ IsStrictlyPositive a

Tactic:
simpa using hu.star.isStrictlyPositive_star_right_conjugate_iff

Goal state after:
Module: Mathlib.Algebra.Algebra.Subalgebra.Centralizer

Goal state before:
pp: case mp
R : Type u_1
inst✝⁵ : CommSemiring R
A : Type u_2
inst✝⁴ : Semiring A
inst✝³ : Algebra R A
B : Type u_3
inst✝² : Semiring B
inst✝¹ : Algebra R B
S : Set A
inst✝ : Module.Free R B
ℬ : Module.Basis (Module.Free.ChooseBasisIndex R B) R B := Module.Free.chooseBasis R B
b : Module.Free.ChooseBasisIndex R B →₀ A
hw : ∀ g ∈ ⇑includeLeft '' S, (g * b.sum fun i m ↦ m ⊗ₜ[R] ℬ i) = (b.sum fun i m ↦ m ⊗ₜ[R] ℬ i) * g
j : Module.Free.ChooseBasisIndex R B
hj : j ∈ b.support
x : A
⊢ x ∈ S → x * b j = b j * x

Tactic:
hx

Goal state after:
case mp
R : Type u_1
inst✝⁵ : CommSemiring R
A : Type u_2
inst✝⁴ : Semiring A
inst✝³ : Algebra R A
B : Type u_3
inst✝² : Semiring B
inst✝¹ : Algebra R B
S : Set A
inst✝ : Module.Free R B
ℬ : Module.Basis (Module.Free.ChooseBasisIndex R B) R B := Module.Free.chooseBasis R B
b : Module.Free.ChooseBasisIndex R B →₀ A
hw : ∀ g ∈ ⇑includeLeft '' S, (g * b.sum fun i m ↦ m ⊗ₜ[R] ℬ i) = (b.sum fun i m ↦ m ⊗ₜ[R] ℬ i) * g
j : Module.Free.ChooseBasisIndex R B
hj : j ∈ b.support
x : A
hx : x ∈ S
⊢ x * b j = b j * x
Module: Mathlib.Algebra.Order.Star.Basic

Goal state before:
pp: case neg
R : Type u_1
inst✝³ : NonUnitalSemiring R
inst✝² : PartialOrder R
inst✝¹ : StarRing R
inst✝ : StarOrderedRing R
x y : R
h : ¬x = y
⊢ star x < star y ↔ x < y

Tactic:
simpa [le_iff_lt_or_eq, h] using star_le_star_iff (x := x) (y := y)

Goal state after:
Module: Mathlib.Algebra.Order.Star.Basic

Goal state before:
pp: R : Type u_1
inst✝⁴ : NonUnitalSemiring R
inst✝³ : PartialOrder R
inst✝² : StarRing R
inst✝¹ : StarOrderedRing R
inst✝ : Nontrivial R
x : R
hx : IsRegular x
⊢ 0 < x * star x

Tactic:
by
  simpa using star_mul_self_pos hx.star

Goal state after:
Module: Mathlib.LinearAlgebra.Dimension.Constructions

Goal state before:
pp: R : Type u
M : Type v
inst✝² : Semiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
s : Submodule R M
lt : finrank R ↥s < finrank R M
⊢ s < ⊤

Tactic:
rw [← finrank_top R M] at lt
  exact lt_of_le_of_finrank_lt_finrank le_top lt

Goal state after:
Module: Mathlib.LinearAlgebra.Dimension.Finite

Goal state before:
pp: case hbc
R : Type u
M : Type v
inst✝⁴ : Semiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : StrongRankCondition R
ι : Type w
inst✝ : Module.Finite R M
v : ι → M
h : LinearIndependent R v
⊢ ↑(finrank R M) < ℵ₀

Tactic:
apply Cardinal.natCast_lt_aleph0

Goal state after:
Module: Mathlib.LinearAlgebra.Dimension.Finite

Goal state before:
pp: R : Type u
M : Type v
inst✝⁴ : Semiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : Nontrivial R
inst✝ : Subsingleton M
⊢ toNat 0 = 0

Tactic:
map_zero

Goal state after:
R : Type u
M : Type v
inst✝⁴ : Semiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : Nontrivial R
inst✝ : Subsingleton M
⊢ 0 = 0
Module: Mathlib.LinearAlgebra.Dimension.Finite

Goal state before:
pp: R : Type u
M : Type v
inst✝³ : Semiring R
inst✝² : AddCommMonoid M
inst✝¹ : Module R M
inst✝ : Free R M
h : ∀ (s : Finset M) (a : Basis (↑↑s) R M), False
s : Set M
b : Basis (↑s) R M
hs : s.Finite
⊢ Basis (↑↑hs.toFinset) R M

Tactic:
convert! b

Goal state after:
R : Type u
M : Type v
inst✝³ : Semiring R
inst✝² : AddCommMonoid M
inst✝¹ : Module R M
inst✝ : Free R M
h : ∀ (s : Finset M) (a : Basis (↑↑s) R M), False
s : Set M
b : Basis (↑s) R M
hs : s.Finite
⊢ ↑hs.toFinset = s