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March 13, 2026 10:08
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| import Mathlib | |
| import Mathlib.Algebra.Group.Pi.Basic | |
| import Mathlib.Algebra.Group.Subgroup.Ker | |
| import Mathlib.Data.List.Chain | |
| import Mathlib.Algebra.Group.Int.Defs | |
| import Mathlib.Algebra.BigOperators.Group.List.Defs | |
| namespace FreeGroup | |
| theorem map_injective' {α β : Type*} {f : α → β} | |
| (hf : Function.Injective f) : Function.Injective (map f) := by | |
| by_cases Nonempty α | |
| · have map_inv : Function.LeftInverse (map (Function.invFun f)) (map f) := by | |
| intro x | |
| have hgf : (Function.invFun f) ∘ f = id := by | |
| funext | |
| apply Function.leftInverse_invFun hf | |
| simp [map.comp, hgf] | |
| exact map_inv.injective | |
| · simp [not_nonempty_iff] at * | |
| intro x y _ | |
| exact Subsingleton.elim x y |
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