/-- Given `M : Matroid α`, the `IndepMatroid α` whose independent sets are
  the subsets of `M.E` that are disjoint from some base of `M` -/
@[simps] def dualIndepMatroid (M : Matroid α) : IndepMatroid α where
  E := M.E
  Indep I := I ⊆ M.E ∧ ∃ B, M.Base B ∧ Disjoint I B
  indep_empty := ⟨empty_subset M.E, M.exists_base.imp (fun B hB ↦ ⟨hB, empty_disjoint _⟩)⟩
  indep_subset := by
    rintro I J ⟨hJE, B, hB, hJB⟩ hIJ
    exact ⟨hIJ.trans hJE, ⟨B, hB, disjoint_of_subset_left hIJ hJB⟩⟩
  indep_aug := by
    rintro I X ⟨hIE, B, hB, hIB⟩ hI_not_max hX_max
    have hXE := hX_max.1.1
    have hB' := (base_compl_iff_mem_maximals_disjoint_base hXE).mpr hX_max

    set B' := M.E \ X with hX
    have hI := (not_iff_not.mpr (base_compl_iff_mem_maximals_disjoint_base)).mpr hI_not_max
    obtain ⟨B'', hB'', hB''₁, hB''₂⟩ := (hB'.indep.diff I).exists_base_subset_union_base hB
    rw [← compl_subset_compl, ← hIB.sdiff_eq_right, ← union_diff_distrib, diff_eq, compl_inter,
      compl_compl, union_subset_iff, compl_subset_compl] at hB''₂

    have hssu := (subset_inter (hB''₂.2) hIE).ssubset_of_ne
      (by { rintro rfl; apply hI; convert hB''; simp [hB''.subset_ground] })

    obtain ⟨e, ⟨(heB'' : e ∉ _), heE⟩, heI⟩ := exists_of_ssubset hssu
    use e
    simp_rw [mem_diff, insert_subset_iff, and_iff_left heI, and_iff_right heE, and_iff_right hIE]
    refine' ⟨by_contra (fun heX ↦ heB'' (hB''₁ ⟨_, heI⟩)), ⟨B'', hB'', _⟩⟩
    · rw [hX]; exact ⟨heE, heX⟩
    rw [← union_singleton, disjoint_union_left, disjoint_singleton_left, and_iff_left heB'']
    exact disjoint_of_subset_left hB''₂.2 disjoint_compl_left
  indep_maximal := by
    rintro X - I'⟨hI'E, B, hB, hI'B⟩ hI'X
    obtain ⟨I, hI⟩ :=  M.exists_basis (M.E \ X)
    obtain ⟨B', hB', hIB', hB'IB⟩ := hI.indep.exists_base_subset_union_base hB
    refine' ⟨(X \ B') ∩ M.E,
      ⟨_,subset_inter (subset_diff.mpr _) hI'E, (inter_subset_left _ _).trans (diff_subset _ _)⟩, _⟩
    · simp only [inter_subset_right, true_and]
      exact ⟨B', hB', disjoint_of_subset_left (inter_subset_left _ _) disjoint_sdiff_left⟩
    · rw [and_iff_right hI'X]
      refine' disjoint_of_subset_right hB'IB _
      rw [disjoint_union_right, and_iff_left hI'B]
      exact disjoint_of_subset hI'X hI.subset disjoint_sdiff_right
    simp only [mem_setOf_eq, subset_inter_iff, and_imp, forall_exists_index]
    intros J hJE B'' hB'' hdj _ hJX hssJ
    rw [and_iff_left hJE]
    rw [diff_eq, inter_right_comm, ← diff_eq, diff_subset_iff] at hssJ

    have hI' : (B'' ∩ X) ∪ (B' \ X) ⊆ B'
    · rw [union_subset_iff, and_iff_left (diff_subset _ _),
        ← inter_eq_self_of_subset_left hB''.subset_ground, inter_right_comm, inter_assoc]

      calc _ ⊆ _ := inter_subset_inter_right _ hssJ
           _ ⊆ _ := by rw [inter_distrib_left, hdj.symm.inter_eq, union_empty]
           _ ⊆ _ := inter_subset_right _ _

    obtain ⟨B₁,hB₁,hI'B₁,hB₁I⟩ := (hB'.indep.subset hI').exists_base_subset_union_base hB''
    rw [union_comm, ← union_assoc, union_eq_self_of_subset_right (inter_subset_left _ _)] at hB₁I

    have : B₁ = B'
    · refine hB₁.eq_of_subset_indep hB'.indep (fun e he ↦ ?_)
      refine (hB₁I he).elim (fun heB'' ↦ ?_) (fun h ↦ h.1)
      refine (em (e ∈ X)).elim (fun heX ↦ hI' (Or.inl ⟨heB'', heX⟩)) (fun heX ↦ hIB' ?_)
      

Proof state:
case this
α : Type u_1
M✝ : Matroid α
I✝ B✝ X✝ : Set α
M : Matroid α
X I' : Set α
hI'E : I' ⊆ M.E
B : Set α
hB : M.Base B
hI'B : Disjoint I' B
hI'X : I' ⊆ X
I : Set α
hI : M.Basis I (M.E \ X)
B' : Set α
hB' : M.Base B'
hIB' : I ⊆ B'
hB'IB : B' ⊆ I ∪ B
J : Set α
hJE : J ⊆ M.E
B'' : Set α
hB'' : M.Base B''
hdj : Disjoint J B''
a✝ : I' ⊆ J
hJX : J ⊆ X
hssJ : X ∩ M.E ⊆ B' ∪ J
hI' : B'' ∩ X ∪ B' \ X ⊆ B'
B₁ : Set α
hB₁ : M.Base B₁
hI'B₁ : B'' ∩ X ∪ B' \ X ⊆ B₁
hB₁I : B₁ ⊆ B'' ∪ B' \ X
e : α
he : e ∈ B₁
heB'' : e ∈ B''
heX : e ∉ X
⊢ e ∈ I

Next tactic:
refine hI.mem_of_insert_indep ⟨hB₁.subset_ground he, heX⟩
        (hB₁.indep.subset (insert_subset he ?_))