Theorems · Theorem · combinatorics
Matroid.mem_closure_iff_exists_isCircuit
∀ {α : Type u_1} {M : Matroid α} {X : Set α} {e : α},
e ∉ X → (e ∈ M.closure X ↔ ∃ C ⊆ insert e X, M.IsCircuit C ∧ e ∈ C)- Defined in
- Mathlib.Combinatorics.Matroid.Circuit
- Cited by
- 4 results in Mathlib
- Foundations
- Depth 92 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites8
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Setstatement and proof · cited by 53,352
- Matroidstatement and proof · cited by 1,258
- Matroid.closurestatement and proof · cited by 272
- Matroid.IsCircuitstatement and proof · cited by 108
- Set.mem_of_mem_of_subsetproof · cited by 36
- Matroid.closure_subset_closureproof · cited by 24
- Matroid.IsCircuit.mem_closure_sdiff_singleton_of_memproof · cited by 7
- Matroid.exists_isCircuit_of_mem_closureproof · cited by 2
Cited by4
Results whose statement or proof uses this declaration.
- Matroid.closure_union_eq_of_subset_coloopsproof · cited by 4
- Matroid.isColoop_tfaeproof · cited by 4
- Matroid.IsCircuit.strong_multi_elimination_insertproof · cited by 1
- Matroid.restrict_isColoop_iffproof · cited by 1