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Theorems · Theorem · combinatorics

Matroid.dual_dual

∀ {α : Type u_1} (M : Matroid α), M✶✶ = M
Defined in
Mathlib.Combinatorics.Matroid.Dual
Cited by
22 results in Mathlib
Foundations
Depth 72 from the axioms · uses propext, Classical.choice, Quot.sound
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Around this declaration

Dashed lines are statement dependencies; solid lines are citations in proofs.

Matroid.dual_contract · cited by 9Matroid.dual_contractMatroid.dual_delete · cited by 5Matroid.dual_deleteMatroid.contract_empty · cited by 3Matroid.contract_emptyMatroid.dual_involutive · cited by 2Matroid.dual_involutiveMatroid.dual_isLoop_iff_isColoop · cited by 2Matroid.dual_isLoop_iff_i…Matroid.IsLoop.dual_isColoop · cited by 2IsLoop.dual_isColoopMatroid.dual_coindep_iff · cited by 2Matroid.dual_coindep_iffMatroid.dual_coloops · cited by 2Matroid.dual_coloopsMatroid.IsBase.inter_isBasis_iff_compl_inter_isBasis_dual · cited by 1IsBase.inter_isBasis_iff_…Matroid.IsCircuit.isCocircuit · cited by 1IsCircuit.isCocircuitMatroid.freeOn_dual_eq · cited by 1Matroid.freeOn_dual_eqMatroid.IsLoop.not_isColoop · cited by 1IsLoop.not_isColoopMatroid.eRk_dual_add_eRank · cited by 1Matroid.eRk_dual_add_eRankMatroid.Indep.contract_isBase_iff · cited by 1Indep.contract_isBase_iffMatroid.rankPos_iff_exists_isCocircuit · cited by 0Matroid.rankPos_iff_exist…Set · cited by 53352SetMatroid · cited by 1258MatroidMatroid.E · cited by 550Matroid.EMatroid.IsBase · cited by 239Matroid.IsBaseMatroid.dual · cited by 79Matroid.dualSet.sdiff_sdiff_cancel_left · cited by 21Set.sdiff_sdiff_cancel_le…Matroid.dual_ground · cited by 9Matroid.dual_groundMatroid.dual_isBase_iff · cited by 9Matroid.dual_isBase_iffMatroid.ext_isBase · cited by 7Matroid.ext_isBaseMatroid.dual_dualCITED BYCITES

Cites9

Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.

Cited by22

Results whose statement or proof uses this declaration.