Theorems · Theorem · combinatorics
Matroid.isCircuit_iff_dep_forall_sdiff_singleton_indep
∀ {α : Type u_1} {M : Matroid α} {C : Set α}, M.IsCircuit C ↔ M.Dep C ∧ ∀ e ∈ C, M.Indep (C \ {e})- Defined in
- Mathlib.Combinatorics.Matroid.Circuit
- Cited by
- 3 results in Mathlib
- Foundations
- Depth 63 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites12
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.Eproof · cited by 550
- Matroid.Indepstatement and proof · cited by 367
- Matroid.IsCircuitstatement · cited by 108
- Matroid.Depstatement and proof · cited by 76
- Matroid.Indep.subsetproof · cited by 42
- Matroid.IsCircuit.subset_groundproof · cited by 15
- Matroid.not_indep_iffproof · cited by 11
- Matroid.Dep.subset_groundproof · cited by 7
- Set.minimal_iff_forall_sdiff_singletonproof · cited by 3
- Matroid.isCircuit_iff_minimal_not_indepproof · cited by 1
Cited by3
Results whose statement or proof uses this declaration.
- Matroid.Indep.insert_isCircuit_of_forallproof · cited by 2
- Matroid.IsNonloop.closure_eq_closure_iff_isCircuit_of_neproof · cited by 1
- Matroid.isCircuit_iff_dep_forall_diff_singleton_indepproof · cited by 0