/
TransferInstance.lean
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/
TransferInstance.lean
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/-
Copyright (c) 2018 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Algebra.Algebra.Equiv
import Mathlib.Algebra.Field.Basic
import Mathlib.Logic.Equiv.Defs
import Mathlib.Logic.Small.Defs
#align_import logic.equiv.transfer_instance from "leanprover-community/mathlib"@"ec1c7d810034d4202b0dd239112d1792be9f6fdc"
/-!
# Transfer algebraic structures across `Equiv`s
In this file we prove theorems of the following form: if `β` has a
group structure and `α ≃ β` then `α` has a group structure, and
similarly for monoids, semigroups, rings, integral domains, fields and
so on.
Note that most of these constructions can also be obtained using the `transport` tactic.
### Implementation details
When adding new definitions that transfer type-classes across an equivalence, please mark them
`@[reducible]`. See note [reducible non-instances].
## Tags
equiv, group, ring, field, module, algebra
-/
universe u v
variable {α : Type u} {β : Type v}
namespace Equiv
section Instances
variable (e : α ≃ β)
/-- Transfer `One` across an `Equiv` -/
@[to_additive (attr := reducible) "Transfer `Zero` across an `Equiv`"]
protected def one [One β] : One α :=
⟨e.symm 1⟩
#align equiv.has_one Equiv.one
#align equiv.has_zero Equiv.zero
@[to_additive]
theorem one_def [One β] :
letI := e.one
1 = e.symm 1 :=
rfl
#align equiv.one_def Equiv.one_def
#align equiv.zero_def Equiv.zero_def
@[to_additive]
noncomputable instance [Small.{v} α] [One α] : One (Shrink.{v} α) :=
(equivShrink α).symm.one
/-- Transfer `Mul` across an `Equiv` -/
@[to_additive (attr := reducible) "Transfer `Add` across an `Equiv`"]
protected def mul [Mul β] : Mul α :=
⟨fun x y => e.symm (e x * e y)⟩
#align equiv.has_mul Equiv.mul
#align equiv.has_add Equiv.add
@[to_additive]
theorem mul_def [Mul β] (x y : α) :
letI := Equiv.mul e
x * y = e.symm (e x * e y) :=
rfl
#align equiv.mul_def Equiv.mul_def
#align equiv.add_def Equiv.add_def
@[to_additive]
noncomputable instance [Small.{v} α] [Mul α] : Mul (Shrink.{v} α) :=
(equivShrink α).symm.mul
/-- Transfer `Div` across an `Equiv` -/
@[to_additive (attr := reducible) "Transfer `Sub` across an `Equiv`"]
protected def div [Div β] : Div α :=
⟨fun x y => e.symm (e x / e y)⟩
#align equiv.has_div Equiv.div
#align equiv.has_sub Equiv.sub
@[to_additive]
theorem div_def [Div β] (x y : α) :
letI := Equiv.div e
x / y = e.symm (e x / e y) :=
rfl
#align equiv.div_def Equiv.div_def
#align equiv.sub_def Equiv.sub_def
@[to_additive]
noncomputable instance [Small.{v} α] [Div α] : Div (Shrink.{v} α) :=
(equivShrink α).symm.div
-- Porting note: this should be called `inv`,
-- but we already have an `Equiv.inv` (which perhaps should move to `Perm.inv`?)
/-- Transfer `Inv` across an `Equiv` -/
@[to_additive (attr := reducible) "Transfer `Neg` across an `Equiv`"]
protected def Inv [Inv β] : Inv α :=
⟨fun x => e.symm (e x)⁻¹⟩
#align equiv.has_inv Equiv.Inv
#align equiv.has_neg Equiv.Neg
@[to_additive]
theorem inv_def [Inv β] (x : α) :
letI := Equiv.Inv e
x⁻¹ = e.symm (e x)⁻¹ :=
rfl
#align equiv.inv_def Equiv.inv_def
#align equiv.neg_def Equiv.neg_def
@[to_additive]
noncomputable instance [Small.{v} α] [Inv α] : Inv (Shrink.{v} α) :=
(equivShrink α).symm.Inv
/-- Transfer `SMul` across an `Equiv` -/
@[reducible]
protected def smul (R : Type*) [SMul R β] : SMul R α :=
⟨fun r x => e.symm (r • e x)⟩
#align equiv.has_smul Equiv.smul
theorem smul_def {R : Type*} [SMul R β] (r : R) (x : α) :
letI := e.smul R
r • x = e.symm (r • e x) :=
rfl
#align equiv.smul_def Equiv.smul_def
noncomputable instance [Small.{v} α] (R : Type*) [SMul R α] : SMul R (Shrink.{v} α) :=
(equivShrink α).symm.smul R
/-- Transfer `Pow` across an `Equiv` -/
@[to_additive (attr := reducible) existing smul]
protected def pow (N : Type*) [Pow β N] : Pow α N :=
⟨fun x n => e.symm (e x ^ n)⟩
#align equiv.has_pow Equiv.pow
theorem pow_def {N : Type*} [Pow β N] (n : N) (x : α) :
letI := e.pow N
x ^ n = e.symm (e x ^ n) :=
rfl
#align equiv.pow_def Equiv.pow_def
noncomputable instance [Small.{v} α] (N : Type*) [Pow α N] : Pow (Shrink.{v} α) N :=
(equivShrink α).symm.pow N
/-- An equivalence `e : α ≃ β` gives a multiplicative equivalence `α ≃* β` where
the multiplicative structure on `α` is the one obtained by transporting a multiplicative structure
on `β` back along `e`. -/
@[to_additive "An equivalence `e : α ≃ β` gives an additive equivalence `α ≃+ β` where
the additive structure on `α` is the one obtained by transporting an additive structure
on `β` back along `e`."]
def mulEquiv (e : α ≃ β) [Mul β] :
let mul := Equiv.mul e
α ≃* β := by
intros
exact
{ e with
map_mul' := fun x y => by
apply e.symm.injective
simp [mul_def] }
#align equiv.mul_equiv Equiv.mulEquiv
#align equiv.add_equiv Equiv.addEquiv
@[to_additive (attr := simp)]
theorem mulEquiv_apply (e : α ≃ β) [Mul β] (a : α) : (mulEquiv e) a = e a :=
rfl
#align equiv.mul_equiv_apply Equiv.mulEquiv_apply
#align equiv.add_equiv_apply Equiv.addEquiv_apply
@[to_additive]
theorem mulEquiv_symm_apply (e : α ≃ β) [Mul β] (b : β) :
letI := Equiv.mul e
(mulEquiv e).symm b = e.symm b :=
rfl
#align equiv.mul_equiv_symm_apply Equiv.mulEquiv_symm_apply
#align equiv.add_equiv_symm_apply Equiv.addEquiv_symm_apply
/-- Shrink `α` to a smaller universe preserves multiplication. -/
@[to_additive "Shrink `α` to a smaller universe preserves addition."]
noncomputable def _root_.Shrink.mulEquiv [Small.{v} α] [Mul α] : Shrink.{v} α ≃* α :=
(equivShrink α).symm.mulEquiv
/-- An equivalence `e : α ≃ β` gives a ring equivalence `α ≃+* β`
where the ring structure on `α` is
the one obtained by transporting a ring structure on `β` back along `e`.
-/
def ringEquiv (e : α ≃ β) [Add β] [Mul β] : by
let add := Equiv.add e
let mul := Equiv.mul e
exact α ≃+* β := by
intros
exact
{ e with
map_add' := fun x y => by
apply e.symm.injective
simp [add_def]
map_mul' := fun x y => by
apply e.symm.injective
simp [mul_def] }
#align equiv.ring_equiv Equiv.ringEquiv
@[simp]
theorem ringEquiv_apply (e : α ≃ β) [Add β] [Mul β] (a : α) : (ringEquiv e) a = e a :=
rfl
#align equiv.ring_equiv_apply Equiv.ringEquiv_apply
theorem ringEquiv_symm_apply (e : α ≃ β) [Add β] [Mul β] (b : β) : by
letI := Equiv.add e
letI := Equiv.mul e
exact (ringEquiv e).symm b = e.symm b := rfl
#align equiv.ring_equiv_symm_apply Equiv.ringEquiv_symm_apply
variable (α) in
/-- Shrink `α` to a smaller universe preserves ring structure. -/
noncomputable def _root_.Shrink.ringEquiv [Small.{v} α] [Ring α] : Shrink.{v} α ≃+* α :=
(equivShrink α).symm.ringEquiv
/-- Transfer `Semigroup` across an `Equiv` -/
@[to_additive (attr := reducible) "Transfer `add_semigroup` across an `Equiv`"]
protected def semigroup [Semigroup β] : Semigroup α := by
let mul := e.mul
apply e.injective.semigroup _; intros; exact e.apply_symm_apply _
#align equiv.semigroup Equiv.semigroup
#align equiv.add_semigroup Equiv.addSemigroup
@[to_additive]
noncomputable instance [Small.{v} α] [Semigroup α] : Semigroup (Shrink.{v} α) :=
(equivShrink α).symm.semigroup
/-- Transfer `SemigroupWithZero` across an `Equiv` -/
@[reducible]
protected def semigroupWithZero [SemigroupWithZero β] : SemigroupWithZero α := by
let mul := e.mul
let zero := e.zero
apply e.injective.semigroupWithZero _ <;> intros <;> exact e.apply_symm_apply _
#align equiv.semigroup_with_zero Equiv.semigroupWithZero
@[to_additive]
noncomputable instance [Small.{v} α] [SemigroupWithZero α] : SemigroupWithZero (Shrink.{v} α) :=
(equivShrink α).symm.semigroupWithZero
/-- Transfer `CommSemigroup` across an `Equiv` -/
@[to_additive (attr := reducible) "Transfer `AddCommSemigroup` across an `Equiv`"]
protected def commSemigroup [CommSemigroup β] : CommSemigroup α := by
let mul := e.mul
apply e.injective.commSemigroup _; intros; exact e.apply_symm_apply _
#align equiv.comm_semigroup Equiv.commSemigroup
#align equiv.add_comm_semigroup Equiv.addCommSemigroup
@[to_additive]
noncomputable instance [Small.{v} α] [CommSemigroup α] : CommSemigroup (Shrink.{v} α) :=
(equivShrink α).symm.commSemigroup
/-- Transfer `MulZeroClass` across an `Equiv` -/
@[reducible]
protected def mulZeroClass [MulZeroClass β] : MulZeroClass α := by
let zero := e.zero
let mul := e.mul
apply e.injective.mulZeroClass _ <;> intros <;> exact e.apply_symm_apply _
#align equiv.mul_zero_class Equiv.mulZeroClass
noncomputable instance [Small.{v} α] [MulZeroClass α] : MulZeroClass (Shrink.{v} α) :=
(equivShrink α).symm.mulZeroClass
/-- Transfer `MulOneClass` across an `Equiv` -/
@[to_additive (attr := reducible) "Transfer `AddZeroClass` across an `Equiv`"]
protected def mulOneClass [MulOneClass β] : MulOneClass α := by
let one := e.one
let mul := e.mul
apply e.injective.mulOneClass _ <;> intros <;> exact e.apply_symm_apply _
#align equiv.mul_one_class Equiv.mulOneClass
#align equiv.add_zero_class Equiv.addZeroClass
@[to_additive]
noncomputable instance [Small.{v} α] [MulOneClass α] : MulOneClass (Shrink.{v} α) :=
(equivShrink α).symm.mulOneClass
/-- Transfer `MulZeroOneClass` across an `Equiv` -/
@[reducible]
protected def mulZeroOneClass [MulZeroOneClass β] : MulZeroOneClass α := by
let zero := e.zero
let one := e.one
let mul := e.mul
apply e.injective.mulZeroOneClass _ <;> intros <;> exact e.apply_symm_apply _
#align equiv.mul_zero_one_class Equiv.mulZeroOneClass
noncomputable instance [Small.{v} α] [MulZeroOneClass α] : MulZeroOneClass (Shrink.{v} α) :=
(equivShrink α).symm.mulZeroOneClass
/-- Transfer `Monoid` across an `Equiv` -/
@[to_additive (attr := reducible) "Transfer `AddMonoid` across an `Equiv`"]
protected def monoid [Monoid β] : Monoid α := by
let one := e.one
let mul := e.mul
let pow := e.pow ℕ
apply e.injective.monoid _ <;> intros <;> exact e.apply_symm_apply _
#align equiv.monoid Equiv.monoid
#align equiv.add_monoid Equiv.addMonoid
@[to_additive]
noncomputable instance [Small.{v} α] [Monoid α] : Monoid (Shrink.{v} α) :=
(equivShrink α).symm.monoid
/-- Transfer `CommMonoid` across an `Equiv` -/
@[to_additive (attr := reducible) "Transfer `AddCommMonoid` across an `Equiv`"]
protected def commMonoid [CommMonoid β] : CommMonoid α := by
let one := e.one
let mul := e.mul
let pow := e.pow ℕ
apply e.injective.commMonoid _ <;> intros <;> exact e.apply_symm_apply _
#align equiv.comm_monoid Equiv.commMonoid
#align equiv.add_comm_monoid Equiv.addCommMonoid
@[to_additive]
noncomputable instance [Small.{v} α] [CommMonoid α] : CommMonoid (Shrink.{v} α) :=
(equivShrink α).symm.commMonoid
/-- 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 <;> exact e.apply_symm_apply _
#align equiv.group Equiv.group
#align equiv.add_group Equiv.addGroup
@[to_additive]
noncomputable instance [Small.{v} α] [Group α] : Group (Shrink.{v} α) :=
(equivShrink α).symm.group
/-- 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 <;> exact e.apply_symm_apply _
#align equiv.comm_group Equiv.commGroup
#align equiv.add_comm_group Equiv.addCommGroup
@[to_additive]
noncomputable instance [Small.{v} α] [CommGroup α] : CommGroup (Shrink.{v} α) :=
(equivShrink α).symm.commGroup
/-- 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 <;> exact e.apply_symm_apply _
#align equiv.non_unital_non_assoc_semiring Equiv.nonUnitalNonAssocSemiring
noncomputable instance [Small.{v} α] [NonUnitalNonAssocSemiring α] :
NonUnitalNonAssocSemiring (Shrink.{v} α) :=
(equivShrink α).symm.nonUnitalNonAssocSemiring
/-- 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 <;> exact e.apply_symm_apply _
#align equiv.non_unital_semiring Equiv.nonUnitalSemiring
noncomputable instance [Small.{v} α] [NonUnitalSemiring α] : NonUnitalSemiring (Shrink.{v} α) :=
(equivShrink α).symm.nonUnitalSemiring
/-- Transfer `AddMonoidWithOne` across an `Equiv` -/
@[reducible]
protected def addMonoidWithOne [AddMonoidWithOne β] : AddMonoidWithOne α :=
{ e.addMonoid, e.one with
natCast := fun n => e.symm n
natCast_zero := e.injective (by simp [zero_def])
natCast_succ := fun n => e.injective (by simp [add_def, one_def]) }
#align equiv.add_monoid_with_one Equiv.addMonoidWithOne
noncomputable instance [Small.{v} α] [AddMonoidWithOne α] : AddMonoidWithOne (Shrink.{v} α) :=
(equivShrink α).symm.addMonoidWithOne
/-- Transfer `AddGroupWithOne` across an `Equiv` -/
@[reducible]
protected def addGroupWithOne [AddGroupWithOne β] : AddGroupWithOne α :=
{ e.addMonoidWithOne,
e.addGroup with
intCast := fun n => e.symm n
intCast_ofNat := fun n => by simp only [Int.cast_ofNat]; rfl
intCast_negSucc := fun n =>
congr_arg e.symm <| (Int.cast_negSucc _).trans <| congr_arg _ (e.apply_symm_apply _).symm }
#align equiv.add_group_with_one Equiv.addGroupWithOne
noncomputable instance [Small.{v} α] [AddGroupWithOne α] : AddGroupWithOne (Shrink.{v} α) :=
(equivShrink α).symm.addGroupWithOne
/-- Transfer `NonAssocSemiring` across an `Equiv` -/
@[reducible]
protected def nonAssocSemiring [NonAssocSemiring β] : NonAssocSemiring α := by
let mul := e.mul
let add_monoid_with_one := e.addMonoidWithOne
apply e.injective.nonAssocSemiring _ <;> intros <;> exact e.apply_symm_apply _
#align equiv.non_assoc_semiring Equiv.nonAssocSemiring
noncomputable instance [Small.{v} α] [NonAssocSemiring α] : NonAssocSemiring (Shrink.{v} α) :=
(equivShrink α).symm.nonAssocSemiring
/-- Transfer `Semiring` across an `Equiv` -/
@[reducible]
protected def semiring [Semiring β] : Semiring α := by
let mul := e.mul
let add_monoid_with_one := e.addMonoidWithOne
let npow := e.pow ℕ
apply e.injective.semiring _ <;> intros <;> exact e.apply_symm_apply _
#align equiv.semiring Equiv.semiring
noncomputable instance [Small.{v} α] [Semiring α] : Semiring (Shrink.{v} α) :=
(equivShrink α).symm.semiring
/-- Transfer `NonUnitalCommSemiring` across an `Equiv` -/
@[reducible]
protected def nonUnitalCommSemiring [NonUnitalCommSemiring β] : NonUnitalCommSemiring α := by
let zero := e.zero
let add := e.add
let mul := e.mul
let nsmul := e.smul ℕ
apply e.injective.nonUnitalCommSemiring _ <;> intros <;> exact e.apply_symm_apply _
#align equiv.non_unital_comm_semiring Equiv.nonUnitalCommSemiring
noncomputable instance [Small.{v} α] [NonUnitalCommSemiring α] :
NonUnitalCommSemiring (Shrink.{v} α) :=
(equivShrink α).symm.nonUnitalCommSemiring
/-- Transfer `CommSemiring` across an `Equiv` -/
@[reducible]
protected def commSemiring [CommSemiring β] : CommSemiring α := by
let mul := e.mul
let add_monoid_with_one := e.addMonoidWithOne
let npow := e.pow ℕ
apply e.injective.commSemiring _ <;> intros <;> exact e.apply_symm_apply _
#align equiv.comm_semiring Equiv.commSemiring
noncomputable instance [Small.{v} α] [CommSemiring α] : CommSemiring (Shrink.{v} α) :=
(equivShrink α).symm.commSemiring
/-- Transfer `NonUnitalNonAssocRing` across an `Equiv` -/
@[reducible]
protected def nonUnitalNonAssocRing [NonUnitalNonAssocRing β] : NonUnitalNonAssocRing α := by
let zero := e.zero
let add := e.add
let mul := e.mul
let neg := e.Neg
let sub := e.sub
let nsmul := e.smul ℕ
let zsmul := e.smul ℤ
apply e.injective.nonUnitalNonAssocRing _ <;> intros <;> exact e.apply_symm_apply _
#align equiv.non_unital_non_assoc_ring Equiv.nonUnitalNonAssocRing
noncomputable instance [Small.{v} α] [NonUnitalNonAssocRing α] :
NonUnitalNonAssocRing (Shrink.{v} α) :=
(equivShrink α).symm.nonUnitalNonAssocRing
/-- Transfer `NonUnitalRing` across an `Equiv` -/
@[reducible]
protected def nonUnitalRing [NonUnitalRing β] : NonUnitalRing α := by
let zero := e.zero
let add := e.add
let mul := e.mul
let neg := e.Neg
let sub := e.sub
let nsmul := e.smul ℕ
let zsmul := e.smul ℤ
apply e.injective.nonUnitalRing _ <;> intros <;> exact e.apply_symm_apply _
#align equiv.non_unital_ring Equiv.nonUnitalRing
noncomputable instance [Small.{v} α] [NonUnitalRing α] : NonUnitalRing (Shrink.{v} α) :=
(equivShrink α).symm.nonUnitalRing
/-- Transfer `NonAssocRing` across an `Equiv` -/
@[reducible]
protected def nonAssocRing [NonAssocRing β] : NonAssocRing α := by
let add_group_with_one := e.addGroupWithOne
let mul := e.mul
apply e.injective.nonAssocRing _ <;> intros <;> exact e.apply_symm_apply _
#align equiv.non_assoc_ring Equiv.nonAssocRing
noncomputable instance [Small.{v} α] [NonAssocRing α] : NonAssocRing (Shrink.{v} α) :=
(equivShrink α).symm.nonAssocRing
/-- Transfer `Ring` across an `Equiv` -/
@[reducible]
protected def ring [Ring β] : Ring α := by
let mul := e.mul
let add_group_with_one := e.addGroupWithOne
let npow := e.pow ℕ
apply e.injective.ring _ <;> intros <;> exact e.apply_symm_apply _
#align equiv.ring Equiv.ring
noncomputable instance [Small.{v} α] [Ring α] : Ring (Shrink.{v} α) :=
(equivShrink α).symm.ring
/-- Transfer `NonUnitalCommRing` across an `Equiv` -/
@[reducible]
protected def nonUnitalCommRing [NonUnitalCommRing β] : NonUnitalCommRing α := by
let zero := e.zero
let add := e.add
let mul := e.mul
let neg := e.Neg
let sub := e.sub
let nsmul := e.smul ℕ
let zsmul := e.smul ℤ
apply e.injective.nonUnitalCommRing _ <;> intros <;> exact e.apply_symm_apply _
#align equiv.non_unital_comm_ring Equiv.nonUnitalCommRing
noncomputable instance [Small.{v} α] [NonUnitalCommRing α] : NonUnitalCommRing (Shrink.{v} α) :=
(equivShrink α).symm.nonUnitalCommRing
/-- Transfer `CommRing` across an `Equiv` -/
@[reducible]
protected def commRing [CommRing β] : CommRing α := by
let mul := e.mul
let add_group_with_one := e.addGroupWithOne
let npow := e.pow ℕ
apply e.injective.commRing _ <;> intros <;> exact e.apply_symm_apply _
#align equiv.comm_ring Equiv.commRing
noncomputable instance [Small.{v} α] [CommRing α] : CommRing (Shrink.{v} α) :=
(equivShrink α).symm.commRing
/-- Transfer `Nontrivial` across an `Equiv` -/
@[reducible]
protected theorem nontrivial [Nontrivial β] : Nontrivial α :=
e.surjective.nontrivial
#align equiv.nontrivial Equiv.nontrivial
noncomputable instance [Small.{v} α] [Nontrivial α] : Nontrivial (Shrink.{v} α) :=
(equivShrink α).symm.nontrivial
/-- Transfer `IsDomain` across an `Equiv` -/
@[reducible]
protected theorem isDomain [Ring α] [Ring β] [IsDomain β] (e : α ≃+* β) : IsDomain α :=
Function.Injective.isDomain e.toRingHom e.injective
#align equiv.is_domain Equiv.isDomain
noncomputable instance [Small.{v} α] [Ring α] [IsDomain α] : IsDomain (Shrink.{v} α) :=
Equiv.isDomain (Shrink.ringEquiv α)
/-- Transfer `RatCast` across an `Equiv` -/
@[reducible]
protected def RatCast [RatCast β] : RatCast α where ratCast n := e.symm n
#align equiv.has_rat_cast Equiv.RatCast
noncomputable instance [Small.{v} α] [RatCast α] : RatCast (Shrink.{v} α) :=
(equivShrink α).symm.RatCast
/-- Transfer `DivisionRing` across an `Equiv` -/
@[reducible]
protected def divisionRing [DivisionRing β] : DivisionRing α := by
let add_group_with_one := e.addGroupWithOne
let inv := e.Inv
let div := e.div
let mul := e.mul
let npow := e.pow ℕ
let zpow := e.pow ℤ
let rat_cast := e.RatCast
let qsmul := e.smul ℚ
apply e.injective.divisionRing _ <;> intros <;> exact e.apply_symm_apply _
#align equiv.division_ring Equiv.divisionRing
noncomputable instance [Small.{v} α] [DivisionRing α] : DivisionRing (Shrink.{v} α) :=
(equivShrink α).symm.divisionRing
/-- Transfer `Field` across an `Equiv` -/
@[reducible]
protected def field [Field β] : Field α := by
let add_group_with_one := e.addGroupWithOne
let neg := e.Neg
let inv := e.Inv
let div := e.div
let mul := e.mul
let npow := e.pow ℕ
let zpow := e.pow ℤ
let rat_cast := e.RatCast
let qsmul := e.smul ℚ
apply e.injective.field _ <;> intros <;> exact e.apply_symm_apply _
#align equiv.field Equiv.field
noncomputable instance [Small.{v} α] [Field α] : Field (Shrink.{v} α) :=
(equivShrink α).symm.field
section R
variable (R : Type*)
section
variable [Monoid R]
/-- Transfer `MulAction` across an `Equiv` -/
@[reducible]
protected def mulAction (e : α ≃ β) [MulAction R β] : MulAction R α :=
{ e.smul R with
one_smul := by simp [smul_def]
mul_smul := by simp [smul_def, mul_smul] }
#align equiv.mul_action Equiv.mulAction
noncomputable instance [Small.{v} α] [MulAction R α] : MulAction R (Shrink.{v} α) :=
(equivShrink α).symm.mulAction R
/-- Transfer `DistribMulAction` across an `Equiv` -/
@[reducible]
protected def distribMulAction (e : α ≃ β) [AddCommMonoid β] :
letI := Equiv.addCommMonoid e
∀ [DistribMulAction R β], DistribMulAction R α := by
intros
letI := Equiv.addCommMonoid e
exact
({ Equiv.mulAction R e with
smul_zero := by simp [zero_def, smul_def]
smul_add := by simp [add_def, smul_def, smul_add] } :
DistribMulAction R α)
#align equiv.distrib_mul_action Equiv.distribMulAction
noncomputable instance [Small.{v} α] [AddCommMonoid α] [DistribMulAction R α] :
DistribMulAction R (Shrink.{v} α) :=
(equivShrink α).symm.distribMulAction R
end
section
variable [Semiring R]
/-- Transfer `Module` across an `Equiv` -/
@[reducible]
protected def module (e : α ≃ β) [AddCommMonoid β] :
let addCommMonoid := Equiv.addCommMonoid e
∀ [Module R β], Module R α := by
intros
exact
({ Equiv.distribMulAction R e with
zero_smul := by simp [smul_def, zero_smul, zero_def]
add_smul := by simp [add_def, smul_def, add_smul] } :
Module R α)
#align equiv.module Equiv.module
noncomputable instance [Small.{v} α] [AddCommMonoid α] [Module R α] : Module R (Shrink.{v} α) :=
(equivShrink α).symm.module R
/-- An equivalence `e : α ≃ β` gives a linear equivalence `α ≃ₗ[R] β`
where the `R`-module structure on `α` is
the one obtained by transporting an `R`-module structure on `β` back along `e`.
-/
def linearEquiv (e : α ≃ β) [AddCommMonoid β] [Module R β] : by
let addCommMonoid := Equiv.addCommMonoid e
let module := Equiv.module R e
exact α ≃ₗ[R] β := by
intros
exact
{ Equiv.addEquiv e with
map_smul' := fun r x => by
apply e.symm.injective
simp only [toFun_as_coe, RingHom.id_apply, EmbeddingLike.apply_eq_iff_eq]
exact Iff.mpr (apply_eq_iff_eq_symm_apply _) rfl }
#align equiv.linear_equiv Equiv.linearEquiv
variable (α) in
/-- Shrink `α` to a smaller universe preserves module structure. -/
@[simps!]
noncomputable def _root_.Shrink.linearEquiv [Small.{v} α] [AddCommMonoid α] [Module R α] :
Shrink.{v} α ≃ₗ[R] α :=
Equiv.linearEquiv _ (equivShrink α).symm
end
section
variable [CommSemiring R]
/-- Transfer `Algebra` across an `Equiv` -/
@[reducible]
protected def algebra (e : α ≃ β) [Semiring β] :
let semiring := Equiv.semiring e
∀ [Algebra R β], Algebra R α := by
intros
letI : Module R α := e.module R
fapply Algebra.ofModule
· intro r x y
show e.symm (e (e.symm (r • e x)) * e y) = e.symm (r • e.ringEquiv (x * y))
simp only [apply_symm_apply, Algebra.smul_mul_assoc, map_mul, ringEquiv_apply]
· intro r x y
show e.symm (e x * e (e.symm (r • e y))) = e.symm (r • e (e.symm (e x * e y)))
simp only [apply_symm_apply, Algebra.mul_smul_comm]
#align equiv.algebra Equiv.algebra
lemma algebraMap_def (e : α ≃ β) [Semiring β] [Algebra R β] (r : R) :
let semiring := Equiv.semiring e
let algebra := Equiv.algebra R e
(algebraMap R α) r = e.symm ((algebraMap R β) r) := by
intros
simp only [Algebra.algebraMap_eq_smul_one]
show e.symm (r • e 1) = e.symm (r • 1)
simp only [Equiv.one_def, apply_symm_apply]
noncomputable instance [Small.{v} α] [Semiring α] [Algebra R α] :
Algebra R (Shrink.{v} α) :=
(equivShrink α).symm.algebra _
/-- An equivalence `e : α ≃ β` gives an algebra equivalence `α ≃ₐ[R] β`
where the `R`-algebra structure on `α` is
the one obtained by transporting an `R`-algebra structure on `β` back along `e`.
-/
def algEquiv (e : α ≃ β) [Semiring β] [Algebra R β] : by
let semiring := Equiv.semiring e
let algebra := Equiv.algebra R e
exact α ≃ₐ[R] β := by
intros
exact
{ Equiv.ringEquiv e with
commutes' := fun r => by
apply e.symm.injective
simp only [RingEquiv.toEquiv_eq_coe, toFun_as_coe, EquivLike.coe_coe, ringEquiv_apply,
symm_apply_apply, algebraMap_def] }
#align equiv.alg_equiv Equiv.algEquiv
@[simp]
theorem algEquiv_apply (e : α ≃ β) [Semiring β] [Algebra R β] (a : α) : (algEquiv R e) a = e a :=
rfl
theorem algEquiv_symm_apply (e : α ≃ β) [Semiring β] [Algebra R β] (b : β) : by
letI := Equiv.semiring e
letI := Equiv.algebra R e
exact (algEquiv R e).symm b = e.symm b := by intros; rfl
variable (α) in
/-- Shrink `α` to a smaller universe preserves algebra structure. -/
@[simps!]
noncomputable def _root_.Shrink.algEquiv [Small.{v} α] [Semiring α] [Algebra R α] :
Shrink.{v} α ≃ₐ[R] α :=
Equiv.algEquiv _ (equivShrink α).symm
end
end R
end Instances
end Equiv
namespace Finite
attribute [-instance] Fin.instMulFin
/-- Any finite group in universe `u` is equivalent to some finite group in universe `0`. -/
lemma exists_type_zero_nonempty_mulEquiv (G : Type u) [Group G] [Finite G] :
∃ (G' : Type) (_ : Group G') (_ : Fintype G'), Nonempty (G ≃* G') := by
obtain ⟨n, ⟨e⟩⟩ := Finite.exists_equiv_fin G
letI groupH : Group (Fin n) := Equiv.group e.symm
exact ⟨Fin n, inferInstance, inferInstance, ⟨MulEquiv.symm <| Equiv.mulEquiv e.symm⟩⟩
end Finite