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[Merged by Bors] - chore: missing simps lemmas for Multiplicative defs #7016

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[Merged by Bors] - chore: missing simps lemmas for Multiplicative defs #7016

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chore: missing simps lemmas for `Multiplicative` defs
eric-wieser Sep 7, 2023
6936b9a
remove ?
eric-wieser Sep 7, 2023
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67 changes: 53 additions & 14 deletions Mathlib/Algebra/Hom/Equiv/TypeTags.lean
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Original file line number Diff line number Diff line change
Expand Up @@ -16,56 +16,93 @@ import Mathlib.Algebra.Group.TypeTags
variable {G H : Type*}

/-- Reinterpret `G ≃+ H` as `Multiplicative G ≃* Multiplicative H`. -/
@[simps]
def AddEquiv.toMultiplicative [AddZeroClass G] [AddZeroClass H] :
G ≃+ H ≃ (Multiplicative G ≃* Multiplicative H) where
toFun f :=
⟨⟨AddMonoidHom.toMultiplicative f.toAddMonoidHom,
AddMonoidHom.toMultiplicative f.symm.toAddMonoidHom, f.1.3, f.1.4⟩, f.2⟩
invFun f := ⟨⟨f.toMonoidHom, f.symm.toMonoidHom, f.1.3, f.1.4⟩, f.2⟩
{ toFun := AddMonoidHom.toMultiplicative f.toAddMonoidHom
invFun := AddMonoidHom.toMultiplicative f.symm.toAddMonoidHom
left_inv := f.left_inv
right_inv := f.right_inv
map_mul' := f.map_add }
invFun f :=
{ toFun := AddMonoidHom.toMultiplicative.symm f.toMonoidHom
invFun := AddMonoidHom.toMultiplicative.symm f.symm.toMonoidHom
left_inv := f.left_inv
right_inv := f.right_inv
map_add' := f.map_mul }
left_inv x := by ext; rfl
right_inv x := by ext; rfl
#align add_equiv.to_multiplicative AddEquiv.toMultiplicative

/-- Reinterpret `G ≃* H` as `Additive G ≃+ Additive H`. -/
@[simps]
def MulEquiv.toAdditive [MulOneClass G] [MulOneClass H] :
G ≃* H ≃ (Additive G ≃+ Additive H) where
toFun f := ⟨⟨MonoidHom.toAdditive f.toMonoidHom,
MonoidHom.toAdditive f.symm.toMonoidHom, f.1.3, f.1.4⟩, f.2⟩
invFun f := ⟨⟨f.toAddMonoidHom, f.symm.toAddMonoidHom, f.1.3, f.1.4⟩, f.2⟩
toFun f :=
{ toFun := MonoidHom.toAdditive f.toMonoidHom
invFun := MonoidHom.toAdditive f.symm.toMonoidHom
left_inv := f.left_inv
right_inv := f.right_inv
map_add' := f.map_mul }
invFun f :=
{ toFun := MonoidHom.toAdditive.symm f.toAddMonoidHom
invFun := MonoidHom.toAdditive.symm f.symm.toAddMonoidHom
left_inv := f.left_inv
right_inv := f.right_inv
map_mul' := f.map_add }
left_inv x := by ext; rfl
right_inv x := by ext; rfl
#align mul_equiv.to_additive MulEquiv.toAdditive

/-- Reinterpret `Additive G ≃+ H` as `G ≃* Multiplicative H`. -/
@[simps]
def AddEquiv.toMultiplicative' [MulOneClass G] [AddZeroClass H] :
Additive G ≃+ H ≃ (G ≃* Multiplicative H) where
toFun f :=
⟨⟨AddMonoidHom.toMultiplicative' f.toAddMonoidHom,
AddMonoidHom.toMultiplicative'' f.symm.toAddMonoidHom, f.1.3, f.1.4⟩, f.2⟩
invFun f := ⟨⟨f.toMonoidHom, f.symm.toMonoidHom, f.1.3, f.1.4⟩, f.2⟩
{ toFun := AddMonoidHom.toMultiplicative' f.toAddMonoidHom
invFun := AddMonoidHom.toMultiplicative'' f.symm.toAddMonoidHom
left_inv := f.left_inv
right_inv := f.right_inv
map_mul' := f.map_add }
invFun f :=
{ toFun := AddMonoidHom.toMultiplicative'.symm f.toMonoidHom
invFun := AddMonoidHom.toMultiplicative''.symm f.symm.toMonoidHom
left_inv := f.left_inv
right_inv := f.right_inv
map_add' := f.map_mul }
left_inv x := by ext; rfl
right_inv x := by ext; rfl
#align add_equiv.to_multiplicative' AddEquiv.toMultiplicative'

/-- Reinterpret `G ≃* Multiplicative H` as `Additive G ≃+ H` as. -/
def MulEquiv.toAdditive' [MulOneClass G] [AddZeroClass H] :
abbrev MulEquiv.toAdditive' [MulOneClass G] [AddZeroClass H] :
G ≃* Multiplicative H ≃ (Additive G ≃+ H) :=
AddEquiv.toMultiplicative'.symm
#align mul_equiv.to_additive' MulEquiv.toAdditive'

/-- Reinterpret `G ≃+ Additive H` as `Multiplicative G ≃* H`. -/
@[simps]
def AddEquiv.toMultiplicative'' [AddZeroClass G] [MulOneClass H] :
G ≃+ Additive H ≃ (Multiplicative G ≃* H) where
toFun f :=
⟨⟨AddMonoidHom.toMultiplicative'' f.toAddMonoidHom,
AddMonoidHom.toMultiplicative' f.symm.toAddMonoidHom, f.1.3, f.1.4⟩, f.2⟩
invFun f := ⟨⟨f.toMonoidHom, f.symm.toMonoidHom, f.1.3, f.1.4⟩, f.2⟩
{ toFun := AddMonoidHom.toMultiplicative'' f.toAddMonoidHom
invFun := AddMonoidHom.toMultiplicative' f.symm.toAddMonoidHom
left_inv := f.left_inv
right_inv := f.right_inv
map_mul' := f.map_add }
invFun f :=
{ toFun := AddMonoidHom.toMultiplicative''.symm f.toMonoidHom
invFun := AddMonoidHom.toMultiplicative'.symm f.symm.toMonoidHom
left_inv := f.left_inv
right_inv := f.right_inv
map_add' := f.map_mul }
left_inv x := by ext; rfl
right_inv x := by ext; rfl
#align add_equiv.to_multiplicative'' AddEquiv.toMultiplicative''

/-- Reinterpret `Multiplicative G ≃* H` as `G ≃+ Additive H` as. -/
def MulEquiv.toAdditive'' [AddZeroClass G] [MulOneClass H] :
abbrev MulEquiv.toAdditive'' [AddZeroClass G] [MulOneClass H] :
Multiplicative G ≃* H ≃ (G ≃+ Additive H) :=
AddEquiv.toMultiplicative''.symm
#align mul_equiv.to_additive'' MulEquiv.toAdditive''
Expand All @@ -75,11 +112,13 @@ section
variable (G) (H)

/-- `Additive (Multiplicative G)` is just `G`. -/
@[simps!]
def AddEquiv.additiveMultiplicative [AddZeroClass G] : Additive (Multiplicative G) ≃+ G :=
MulEquiv.toAdditive' (MulEquiv.refl (Multiplicative G))
#align add_equiv.additive_multiplicative AddEquiv.additiveMultiplicative

/-- `Multiplicative (Additive H)` is just `H`. -/
@[simps!]
def MulEquiv.multiplicativeAdditive [MulOneClass H] : Multiplicative (Additive H) ≃* H :=
AddEquiv.toMultiplicative'' (AddEquiv.refl (Additive H))
#align mul_equiv.multiplicative_additive MulEquiv.multiplicativeAdditive
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