Theorems · Definition · combinatorics
Matroid.contract
{α : Type u_1} → Matroid α → Set α → Matroid αThe contraction M ／ C is the matroid on M.E \ C in which a set I ⊆ M.E \ C is independent
if and only if I ∪ J is independent, where J is an arbitrarily chosen basis for C.
It is also equal by definition to (M✶ ＼ C)✶; see Matroid.IsBasis.contract_indep_iff for
a proof that its independent sets are the claimed ones.
- Cited by
- 103 results in Mathlib
- Foundations
- Depth 79 from the axioms, rests on 796 definitions · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites4
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.deleteproof · cited by 86
- Matroid.dualproof · cited by 79
Cited by104
Results whose statement or proof uses this declaration.
- Matroid.IsMinorproof · cited by 19
- Matroid.contract_groundstatement · cited by 12
- Matroid.dual_contractstatement · cited by 9
- Matroid.dual_delete_dualstatement · cited by 9
- Matroid.Indep.contract_indep_iffstatement · cited by 9
- Matroid.contract_inter_ground_eqstatement and proof · cited by 7
- Matroid.dual_deletestatement and proof · cited by 5
- Matroid.IsBasis'.contract_eq_contract_deletestatement and proof · cited by 5
- Matroid.contract_contractstatement and proof · cited by 5
- Matroid.Indep.contract_dep_iffstatement and proof · cited by 4
- Matroid.IsBasis'.contract_indep_iffstatement · cited by 4
- Matroid.IsCircuit.contract_isCircuitstatement and proof · cited by 4