loopless_iff_forall_isCircuit — Mathlib

English

M.Indep(I \ K) ↔ M.Indep(I) when K ⊆ M.coloops (same as previous form).

Русский

Nезависимость I \ K эквивалентна независимости I, если K ⊆ M.coloops.

LaTeX

$$$$ K \subseteq M.coloops \Rightarrow M.Indep(I \setminus K) \iff M.Indep(I). $$$$

Lean4

theorem loopless_iff_forall_isCircuit : M.Loopless ↔ ∀ C, M.IsCircuit C → C.Nontrivial :=
  by
  suffices (∃ x ∈ M.E, M.IsLoop x) ↔ ∃ x, M.IsCircuit x ∧ x.Subsingleton by
    simpa [loopless_iff_forall_not_isLoop, ← not_iff_not (a := ∀ _, _)]
  refine ⟨fun ⟨e, _, he⟩ ↦ ⟨{ e }, he.isCircuit, by simp⟩, fun ⟨C, hC, hCs⟩ ↦ ?_⟩
  obtain (rfl | ⟨e, rfl⟩) := hCs.eq_empty_or_singleton
  · simpa using hC.nonempty
  exact ⟨e, (singleton_isCircuit.1 hC).mem_ground, singleton_isCircuit.1 hC⟩

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