/-- An independence predicate satisfying the finite matroid axioms determines a matroid,
  provided independence is determined by its behaviour on finite sets.
  This fundamentally needs choice, since it can be used to prove that every vector space
  has a basis. -/
@[simps E] protected def ofFinitary (E : Set α) (Indep : Set α → Prop)
    (indep_empty : Indep ∅)
    (indep_subset : ∀ ⦃I J⦄, Indep J → I ⊆ J → Indep I)
    (indep_aug : ∀ ⦃I J⦄, Indep I → I.Finite → Indep J → J.Finite → I.ncard < J.ncard →
      ∃ e ∈ J, e ∉ I ∧ Indep (insert e I))
    (indep_compact : ∀ I, (∀ J, J ⊆ I → J.Finite → Indep J) → Indep I)
    (subset_ground : ∀ I, Indep I → I ⊆ E) : IndepMatroid α :=
  have htofin : ∀ I e, Indep I → ¬ Indep (insert e I) →
    ∃ I₀, I₀ ⊆ I ∧ I₀.Finite ∧ ¬ Indep (insert e I₀) := by
      by_contra h; push_neg at h
      obtain ⟨I, e, -, hIe, h⟩ := h
      refine hIe <| indep_compact _ fun J hJss hJfin ↦ ?_
      exact indep_subset (h (J \ {e}) (by rwa [diff_subset_iff]) (hJfin.diff _)) (by simp)
  IndepMatroid.mk
  (E := E)
  (Indep := Indep)
  (indep_empty := indep_empty)
  (indep_subset := indep_subset)
  (indep_aug := by
    intro I B hI hImax hBmax
    simp only [mem_maximals_iff, mem_setOf_eq, not_and, not_forall, exists_prop,
      exists_and_left, iff_true_intro hI, true_imp_iff] at hImax hBmax
    obtain ⟨I', hI', hII', hne⟩ := hImax
    obtain ⟨e, heI', heI⟩ := exists_of_ssubset (hII'.ssubset_of_ne hne)
    have hins : Indep (insert e I) := indep_subset hI' (insert_subset heI' hII')
    obtain (heB | heB) := em (e ∈ B)
    · exact ⟨e, ⟨heB, heI⟩, hins⟩
    by_contra hcon; push_neg at hcon

    have heBdep : ¬Indep (insert e B) :=
      fun hi ↦ heB <| insert_eq_self.1 (hBmax.2 hi (subset_insert _ _)).symm

    -- There is a finite subset `B₀` of `B` so that `B₀ + e` is dependent
    obtain ⟨B₀, hB₀B, hB₀fin, hB₀e⟩ := htofin B e hBmax.1  heBdep
    have hB₀ := indep_subset hBmax.1 hB₀B

    -- There is a finite subset `I₀` of `I` so that `I₀` doesn't extend into `B₀`
    have hexI₀ : ∃ I₀, I₀ ⊆ I ∧ I₀.Finite ∧ ∀ x, x ∈ B₀ \ I₀ → ¬Indep (insert x I₀)
    · have hchoose : ∀ (b : ↑(B₀ \ I)), ∃ Ib, Ib ⊆ I ∧ Ib.Finite ∧ ¬Indep (insert (b : α) Ib)
      · rintro ⟨b, hb⟩; exact htofin I b hI (hcon b ⟨hB₀B hb.1, hb.2⟩)
      choose! f hf using hchoose
      have _ := finite_coe_iff.2 (hB₀fin.diff I)
      refine ⟨iUnion f ∪ (B₀ ∩ I),
        union_subset (iUnion_subset (fun i ↦ (hf i).1)) (inter_subset_right _ _),
        (finite_iUnion <| fun i ↦ (hf i).2.1).union (hB₀fin.subset (inter_subset_left _ _)),
        fun x ⟨hxB₀, hxn⟩ hi ↦ ?_⟩
      have hxI : x ∉ I := fun hxI ↦ hxn <| Or.inr ⟨hxB₀, hxI⟩
      refine (hf ⟨x, ⟨hxB₀, hxI⟩⟩).2.2 (indep_subset hi <| insert_subset_insert ?_)
      apply subset_union_of_subset_left
      apply subset_iUnion

    obtain ⟨I₀, hI₀I, hI₀fin, hI₀⟩ := hexI₀

    set E₀ := insert e (I₀ ∪ B₀)
    have hE₀fin : E₀.Finite := (hI₀fin.union hB₀fin).insert e

    -- Extend `B₀` to a maximal independent subset of `I₀ ∪ B₀ + e`
    obtain ⟨J, ⟨hB₀J, hJ, hJss⟩, hJmax⟩ := Finite.exists_maximal_wrt (f := id)
      (s := {J | B₀ ⊆ J ∧ Indep J ∧ J ⊆ E₀})
      (hE₀fin.finite_subsets.subset (by simp))
      ⟨B₀, Subset.rfl, hB₀, (subset_union_right _ _).trans (subset_insert _ _)⟩

    have heI₀ : e ∉ I₀ := not_mem_subset hI₀I heI
    have heI₀i : Indep (insert e I₀) := indep_subset hins (insert_subset_insert hI₀I)

    have heJ : e ∉ J := fun heJ ↦ hB₀e (indep_subset hJ <| insert_subset heJ hB₀J)

    have hJfin := hE₀fin.subset hJss

    -- We have `|I₀ + e| ≤ |J|`, since otherwise we could extend the maximal set `J`
    have hcard : (insert e I₀).ncard ≤ J.ncard
    · refine not_lt.1 fun hlt ↦ ?_
      obtain ⟨f, hfI, hfJ, hfi⟩ := indep_aug hJ hJfin heI₀i (hI₀fin.insert e) hlt
      have hfE₀ : f ∈ E₀ := mem_of_mem_of_subset hfI (insert_subset_insert (subset_union_left _ _))
      refine hfJ (insert_eq_self.1 <| Eq.symm (hJmax _
        ⟨hB₀J.trans <| subset_insert _ _,hfi,insert_subset hfE₀ hJss⟩ (subset_insert _ _)))

    -- But this means `|I₀| < |J|`, and extending `I₀` into `J` gives a contradiction
    rw [ncard_insert_of_not_mem heI₀ hI₀fin, ← Nat.lt_iff_add_one_le] at hcard

    obtain ⟨f, hfJ, hfI₀, hfi⟩ := indep_aug (indep_subset hI hI₀I) hI₀fin hJ hJfin hcard
    

Proof state:
case intro.intro.intro.intro.intro.inr.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
α : Type u_1
I✝ B✝ X E : Set α
Indep : Set α → Prop
indep_empty : Indep ∅
indep_subset : ∀ ⦃I J : Set α⦄, Indep J → I ⊆ J → Indep I
indep_aug :
  ∀ ⦃I J : Set α⦄,
    Indep I → Set.Finite I → Indep J → Set.Finite J → ncard I < ncard J → ∃ e ∈ J, e ∉ I ∧ Indep (insert e I)
indep_compact : ∀ (I : Set α), (∀ J ⊆ I, Set.Finite J → Indep J) → Indep I
subset_ground : ∀ (I : Set α), Indep I → I ⊆ E
htofin : ∀ (I : Set α) (e : α), Indep I → ¬Indep (insert e I) → ∃ I₀ ⊆ I, Set.Finite I₀ ∧ ¬Indep (insert e I₀)
I B : Set α
hI : Indep I
hBmax : Indep B ∧ ∀ ⦃y : Set α⦄, Indep y → B ⊆ y → B = y
I' : Set α
hI' : Indep I'
hII' : I ⊆ I'
hne : ¬I = I'
e : α
heI' : e ∈ I'
heI : e ∉ I
hins : Indep (insert e I)
heB : e ∉ B
hcon : ∀ x ∈ B \ I, ¬Indep (insert x I)
heBdep : ¬Indep (insert e B)
B₀ : Set α
hB₀B : B₀ ⊆ B
hB₀fin : Set.Finite B₀
hB₀e : ¬Indep (insert e B₀)
hB₀ : Indep B₀
I₀ : Set α
hI₀I : I₀ ⊆ I
hI₀fin : Set.Finite I₀
hI₀ : ∀ x ∈ B₀ \ I₀, ¬Indep (insert x I₀)
E₀ : Set α := insert e (I₀ ∪ B₀)
hE₀fin : Set.Finite E₀
J : Set α
hJmax : ∀ a' ∈ {J | B₀ ⊆ J ∧ Indep J ∧ J ⊆ E₀}, id J ≤ id a' → id J = id a'
hB₀J : B₀ ⊆ J
hJ : Indep J
hJss : J ⊆ E₀
heI₀ : e ∉ I₀
heI₀i : Indep (insert e I₀)
heJ : e ∉ J
hJfin : Set.Finite J
hcard : ncard I₀ < ncard J
f : α
hfJ : f ∈ J
hfI₀ : f ∉ I₀
hfi : Indep (insert f I₀)
⊢ False

Next tactic:
exact hI₀ f ⟨Or.elim (hJss hfJ) (fun hfe ↦ (heJ <| hfe ▸ hfJ).elim) (by aesop), hfI₀⟩ hfi