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[Merged by Bors] - feat: add FreeMonoid.lift_ofList #2193

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[Merged by Bors] - feat: add FreeMonoid.lift_ofList #2193

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8 changes: 6 additions & 2 deletions Mathlib/Algebra/FreeMonoid/Basic.lean
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Original file line number Diff line number Diff line change
Expand Up @@ -226,8 +226,7 @@ lemma prodAux_eq : ∀ l : List M, FreeMonoid.prodAux l = l.prod
/-- Equivalence between maps `α → M` and monoid homomorphisms `FreeMonoid α →* M`. -/
@[to_additive "Equivalence between maps `α → A` and additive monoid homomorphisms
`FreeAddMonoid α →+ A`."]
def lift : (α → M) ≃ (FreeMonoid α →* M)
where
def lift : (α → M) ≃ (FreeMonoid α →* M) where
toFun f :=
{ toFun := fun l ↦ prodAux ((toList l).map f)
map_one' := rfl
Expand All @@ -238,6 +237,11 @@ def lift : (α → M) ≃ (FreeMonoid α →* M)
#align free_monoid.lift FreeMonoid.lift
#align free_add_monoid.lift FreeAddMonoid.lift

-- porting note: new
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@[to_additive (attr := simp)]
theorem lift_ofList (f : α → M) (l : List α) : lift f (ofList l) = (l.map f).prod :=
prodAux_eq _

@[to_additive (attr := simp)]
theorem lift_symm_apply (f : FreeMonoid α →* M) : lift.symm f = f ∘ of := rfl
#align free_monoid.lift_symm_apply FreeMonoid.lift_symm_apply
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