| import Mathlib.Logic.Equiv.Fintype | |
| namespace OAI | |
| /-! Two placements of the same finite graph differ by an ambient permutation. | |
| The statement is about arbitrary embeddings and therefore preserves isolated | |
| vertices as well as vertices incident to edges. -/ | |
| namespace LeanBlast.SecondKahnKalai | |
| theorem exists_perm_comp_embedding {α β : Type*} [Finite β] | |
| (f g : α ↪ β) : ∃ σ : Equiv.Perm β, ∀ a, σ (f a) = g a := by | |
| classical | |
| let : Finite α := Finite.of_injective f f.injective | |
| let : Fintype α := Fintype.ofFinite α | |
| let e : Set.range f ≃ Set.range g := f.toEquivRange.symm.trans g.toEquivRange | |
| refine ⟨e.extendSubtype, ?_⟩ | |
| intro a | |
| rw [e.extendSubtype_apply_of_mem (f a) ⟨a, rfl⟩] | |
| change (e (f.toEquivRange a) : β) = g a | |
| simp [e] | |
| end LeanBlast.SecondKahnKalai | |
| end OAI | |