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e320e82 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 | 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
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