l3lab/ntp-mathlib huggingface dataset:
instance ofBddAugment_finiteRk (E : Set α) Indep indep_empty indep_subset indep_aug
    indep_bdd subset_ground : FiniteRk (IndepMatroid.ofBddAugment
      E Indep indep_empty indep_subset indep_aug indep_bdd subset_ground).matroid := by
  rw [IndepMatroid.ofBddAugment]
  

Proof state:
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 → Indep J → encard I < encard J → ∃ e ∈ J, e ∉ I ∧ Indep (insert e I)
indep_bdd : ∃ n, ∀ (I : Set α), Indep I → encard I ≤ ↑n
subset_ground : ∀ (I : Set α), Indep I → I ⊆ E
⊢ FiniteRk
    (IndepMatroid.matroid
      (IndepMatroid.ofBdd E Indep indep_empty indep_subset
        (_ :
          ∀ ⦃I B : Set α⦄,
            Indep I →
              I ∉ maximals (fun x x_1 => x ⊆ x_1) {I | Indep I} →
                B ∈ maximals (fun x x_1 => x ⊆ x_1) {I | Indep I} → ∃ x ∈ B \ I, Indep (insert x I))
        subset_ground indep_bdd))

Next tactic:
infer_instance
/-- If `E` is finite, then any collection of subsets of `E` satisfying
  the usual independence axioms determines a matroid -/
protected def ofFinite {E : Set α} (hE : E.Finite) (Indep : Set α → Prop)
    (indep_empty : Indep ∅)
    (indep_subset : ∀ ⦃I J⦄, Indep J → I ⊆ J → Indep I)
    (indep_aug :
      ∀ ⦃I J⦄, Indep I → Indep J → I.ncard < J.ncard → ∃ e ∈ J, e ∉ I ∧ Indep (insert e I))
    (subset_ground : ∀ ⦃I⦄, Indep I → I ⊆ E) : IndepMatroid α :=
  IndepMatroid.ofBddAugment (E := E) (Indep := Indep) (indep_empty := indep_empty)
    (indep_subset := indep_subset)
    (indep_aug := by
      

Proof state:
α : Type u_1
I B X E : Set α
hE : Set.Finite E
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 → Indep J → ncard I < ncard J → ∃ e ∈ J, e ∉ I ∧ Indep (insert e I)
subset_ground : ∀ ⦃I : Set α⦄, Indep I → I ⊆ E
⊢ ∀ ⦃I J : Set α⦄, Indep I → Indep J → encard I < encard J → ∃ e ∈ J, e ∉ I ∧ Indep (insert e I)

Next tactic:
refine fun {I J} hI hJ hIJ ↦ indep_aug hI hJ ?_
/-- If `E` is finite, then any collection of subsets of `E` satisfying
  the usual independence axioms determines a matroid -/
protected def ofFinite {E : Set α} (hE : E.Finite) (Indep : Set α → Prop)
    (indep_empty : Indep ∅)
    (indep_subset : ∀ ⦃I J⦄, Indep J → I ⊆ J → Indep I)
    (indep_aug :
      ∀ ⦃I J⦄, Indep I → Indep J → I.ncard < J.ncard → ∃ e ∈ J, e ∉ I ∧ Indep (insert e I))
    (subset_ground : ∀ ⦃I⦄, Indep I → I ⊆ E) : IndepMatroid α :=
  IndepMatroid.ofBddAugment (E := E) (Indep := Indep) (indep_empty := indep_empty)
    (indep_subset := indep_subset)
    (indep_aug := by
      refine fun {I J} hI hJ hIJ ↦ indep_aug hI hJ ?_
      

Proof state:
α : Type u_1
I✝ B X E : Set α
hE : Set.Finite E
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 → Indep J → ncard I < ncard J → ∃ e ∈ J, e ∉ I ∧ Indep (insert e I)
subset_ground : ∀ ⦃I : Set α⦄, Indep I → I ⊆ E
I J : Set α
hI : Indep I
hJ : Indep J
hIJ : encard I < encard J
⊢ ncard I < ncard J

Next tactic:
rwa [← Nat.cast_lt (α := ℕ∞), (hE.subset (subset_ground hJ)).cast_ncard_eq,
        (hE.subset (subset_ground hI)).cast_ncard_eq]
/-- If `E` is finite, then any collection of subsets of `E` satisfying
  the usual independence axioms determines a matroid -/
protected def ofFinite {E : Set α} (hE : E.Finite) (Indep : Set α → Prop)
    (indep_empty : Indep ∅)
    (indep_subset : ∀ ⦃I J⦄, Indep J → I ⊆ J → Indep I)
    (indep_aug :
      ∀ ⦃I J⦄, Indep I → Indep J → I.ncard < J.ncard → ∃ e ∈ J, e ∉ I ∧ Indep (insert e I))
    (subset_ground : ∀ ⦃I⦄, Indep I → I ⊆ E) : IndepMatroid α :=
  IndepMatroid.ofBddAugment (E := E) (Indep := Indep) (indep_empty := indep_empty)
    (indep_subset := indep_subset)
    (indep_aug := by
      refine fun {I J} hI hJ hIJ ↦ indep_aug hI hJ ?_
      rwa [← Nat.cast_lt (α := ℕ∞), (hE.subset (subset_ground hJ)).cast_ncard_eq,
        (hE.subset (subset_ground hI)).cast_ncard_eq] )
    (indep_bdd := ⟨E.ncard, fun I hI ↦ by
      

Proof state:
α : Type u_1
I✝ B X E : Set α
hE : Set.Finite E
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 → Indep J → ncard I < ncard J → ∃ e ∈ J, e ∉ I ∧ Indep (insert e I)
subset_ground : ∀ ⦃I : Set α⦄, Indep I → I ⊆ E
I : Set α
hI : Indep I
⊢ encard I ≤ ↑(ncard E)

Next tactic:
rw [hE.cast_ncard_eq]
/-- If `E` is finite, then any collection of subsets of `E` satisfying
  the usual independence axioms determines a matroid -/
protected def ofFinite {E : Set α} (hE : E.Finite) (Indep : Set α → Prop)
    (indep_empty : Indep ∅)
    (indep_subset : ∀ ⦃I J⦄, Indep J → I ⊆ J → Indep I)
    (indep_aug :
      ∀ ⦃I J⦄, Indep I → Indep J → I.ncard < J.ncard → ∃ e ∈ J, e ∉ I ∧ Indep (insert e I))
    (subset_ground : ∀ ⦃I⦄, Indep I → I ⊆ E) : IndepMatroid α :=
  IndepMatroid.ofBddAugment (E := E) (Indep := Indep) (indep_empty := indep_empty)
    (indep_subset := indep_subset)
    (indep_aug := by
      refine fun {I J} hI hJ hIJ ↦ indep_aug hI hJ ?_
      rwa [← Nat.cast_lt (α := ℕ∞), (hE.subset (subset_ground hJ)).cast_ncard_eq,
        (hE.subset (subset_ground hI)).cast_ncard_eq] )
    (indep_bdd := ⟨E.ncard, fun I hI ↦ by
      rw [hE.cast_ncard_eq]
      

Proof state:
α : Type u_1
I✝ B X E : Set α
hE : Set.Finite E
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 → Indep J → ncard I < ncard J → ∃ e ∈ J, e ∉ I ∧ Indep (insert e I)
subset_ground : ∀ ⦃I : Set α⦄, Indep I → I ⊆ E
I : Set α
hI : Indep I
⊢ encard I ≤ encard E

Next tactic:
exact encard_le_of_subset <| subset_ground hI
/-- An independence predicate on `Finset α` that obeys the finite matroid axioms determines a
  finitary matroid on `α`. -/
protected def ofFinset [DecidableEq α] (E : Set α) (Indep : Finset α → Prop)
    (indep_empty : Indep ∅)
    (indep_subset : ∀ ⦃I J⦄, Indep J → I ⊆ J → Indep I)
    (indep_aug : ∀ ⦃I J⦄, Indep I → Indep J → I.card < J.card → ∃ e ∈ J, e ∉ I ∧ Indep (insert e I))
    (subset_ground : ∀ ⦃I⦄, Indep I → (I : Set α) ⊆ E) : IndepMatroid α :=
  IndepMatroid.ofFinitary
    (E := E)
    (Indep := (fun I ↦ (∀ (J : Finset α), (J : Set α) ⊆ I → Indep J)))
    (indep_empty := by 

Proof state:
α : Type u_1
I B X : Set α
inst✝ : DecidableEq α
E : Set α
Indep : Finset α → Prop
indep_empty : Indep ∅
indep_subset : ∀ ⦃I J : Finset α⦄, Indep J → I ⊆ J → Indep I
indep_aug : ∀ ⦃I J : Finset α⦄, Indep I → Indep J → Finset.card I < Finset.card J → ∃ e ∈ J, e ∉ I ∧ Indep (insert e I)
subset_ground : ∀ ⦃I : Finset α⦄, Indep I → ↑I ⊆ E
⊢ (fun I => ∀ (J : Finset α), ↑J ⊆ I → Indep J) ∅

Next tactic:
simpa [subset_empty_iff]
/-- An independence predicate on `Finset α` that obeys the finite matroid axioms determines a
  finitary matroid on `α`. -/
protected def ofFinset [DecidableEq α] (E : Set α) (Indep : Finset α → Prop)
    (indep_empty : Indep ∅)
    (indep_subset : ∀ ⦃I J⦄, Indep J → I ⊆ J → Indep I)
    (indep_aug : ∀ ⦃I J⦄, Indep I → Indep J → I.card < J.card → ∃ e ∈ J, e ∉ I ∧ Indep (insert e I))
    (subset_ground : ∀ ⦃I⦄, Indep I → (I : Set α) ⊆ E) : IndepMatroid α :=
  IndepMatroid.ofFinitary
    (E := E)
    (Indep := (fun I ↦ (∀ (J : Finset α), (J : Set α) ⊆ I → Indep J)))
    (indep_empty := by simpa [subset_empty_iff])
    (indep_subset := ( fun I J hJ hIJ K hKI ↦ hJ _ (hKI.trans hIJ) ))
    (indep_aug := by
      

Proof state:
α : Type u_1
I B X : Set α
inst✝ : DecidableEq α
E : Set α
Indep : Finset α → Prop
indep_empty : Indep ∅
indep_subset : ∀ ⦃I J : Finset α⦄, Indep J → I ⊆ J → Indep I
indep_aug : ∀ ⦃I J : Finset α⦄, Indep I → Indep J → Finset.card I < Finset.card J → ∃ e ∈ J, e ∉ I ∧ Indep (insert e I)
subset_ground : ∀ ⦃I : Finset α⦄, Indep I → ↑I ⊆ E
⊢ ∀ ⦃I J : Set α⦄,
    (fun I => ∀ (J : Finset α), ↑J ⊆ I → Indep J) I →
      Set.Finite I →
        (fun I => ∀ (J : Finset α), ↑J ⊆ I → Indep J) J →
          Set.Finite J → ncard I < ncard J → ∃ e ∈ J, e ∉ I ∧ (fun I => ∀ (J : Finset α), ↑J ⊆ I → Indep J) (insert e I)

Next tactic:
intro I J hI hIfin hJ hJfin hIJ
/-- An independence predicate on `Finset α` that obeys the finite matroid axioms determines a
  finitary matroid on `α`. -/
protected def ofFinset [DecidableEq α] (E : Set α) (Indep : Finset α → Prop)
    (indep_empty : Indep ∅)
    (indep_subset : ∀ ⦃I J⦄, Indep J → I ⊆ J → Indep I)
    (indep_aug : ∀ ⦃I J⦄, Indep I → Indep J → I.card < J.card → ∃ e ∈ J, e ∉ I ∧ Indep (insert e I))
    (subset_ground : ∀ ⦃I⦄, Indep I → (I : Set α) ⊆ E) : IndepMatroid α :=
  IndepMatroid.ofFinitary
    (E := E)
    (Indep := (fun I ↦ (∀ (J : Finset α), (J : Set α) ⊆ I → Indep J)))
    (indep_empty := by simpa [subset_empty_iff])
    (indep_subset := ( fun I J hJ hIJ K hKI ↦ hJ _ (hKI.trans hIJ) ))
    (indep_aug := by
      intro I J hI hIfin hJ hJfin hIJ
      

Proof state:
α : Type u_1
I✝ B X : Set α
inst✝ : DecidableEq α
E : Set α
Indep : Finset α → Prop
indep_empty : Indep ∅
indep_subset : ∀ ⦃I J : Finset α⦄, Indep J → I ⊆ J → Indep I
indep_aug : ∀ ⦃I J : Finset α⦄, Indep I → Indep J → Finset.card I < Finset.card J → ∃ e ∈ J, e ∉ I ∧ Indep (insert e I)
subset_ground : ∀ ⦃I : Finset α⦄, Indep I → ↑I ⊆ E
I J : Set α
hI : ∀ (J : Finset α), ↑J ⊆ I → Indep J
hIfin : Set.Finite I
hJ : ∀ (J_1 : Finset α), ↑J_1 ⊆ J → Indep J_1
hJfin : Set.Finite J
hIJ : ncard I < ncard J
⊢ ∃ e ∈ J, e ∉ I ∧ (fun I => ∀ (J : Finset α), ↑J ⊆ I → Indep J) (insert e I)

Next tactic:
rw [ncard_eq_toFinset_card _ hIfin, ncard_eq_toFinset_card _ hJfin] at hIJ
/-- An independence predicate on `Finset α` that obeys the finite matroid axioms determines a
  finitary matroid on `α`. -/
protected def ofFinset [DecidableEq α] (E : Set α) (Indep : Finset α → Prop)
    (indep_empty : Indep ∅)
    (indep_subset : ∀ ⦃I J⦄, Indep J → I ⊆ J → Indep I)
    (indep_aug : ∀ ⦃I J⦄, Indep I → Indep J → I.card < J.card → ∃ e ∈ J, e ∉ I ∧ Indep (insert e I))
    (subset_ground : ∀ ⦃I⦄, Indep I → (I : Set α) ⊆ E) : IndepMatroid α :=
  IndepMatroid.ofFinitary
    (E := E)
    (Indep := (fun I ↦ (∀ (J : Finset α), (J : Set α) ⊆ I → Indep J)))
    (indep_empty := by simpa [subset_empty_iff])
    (indep_subset := ( fun I J hJ hIJ K hKI ↦ hJ _ (hKI.trans hIJ) ))
    (indep_aug := by
      intro I J hI hIfin hJ hJfin hIJ
      rw [ncard_eq_toFinset_card _ hIfin, ncard_eq_toFinset_card _ hJfin] at hIJ
      

Proof state:
α : Type u_1
I✝ B X : Set α
inst✝ : DecidableEq α
E : Set α
Indep : Finset α → Prop
indep_empty : Indep ∅
indep_subset : ∀ ⦃I J : Finset α⦄, Indep J → I ⊆ J → Indep I
indep_aug : ∀ ⦃I J : Finset α⦄, Indep I → Indep J → Finset.card I < Finset.card J → ∃ e ∈ J, e ∉ I ∧ Indep (insert e I)
subset_ground : ∀ ⦃I : Finset α⦄, Indep I → ↑I ⊆ E
I J : Set α
hI : ∀ (J : Finset α), ↑J ⊆ I → Indep J
hIfin : Set.Finite I
hJ : ∀ (J_1 : Finset α), ↑J_1 ⊆ J → Indep J_1
hJfin : Set.Finite J
hIJ : Finset.card (Finite.toFinset hIfin) < Finset.card (Finite.toFinset hJfin)
⊢ ∃ e ∈ J, e ∉ I ∧ (fun I => ∀ (J : Finset α), ↑J ⊆ I → Indep J) (insert e I)

Next tactic:
have aug := indep_aug (hI _ (by simp [Subset.rfl])) (hJ _ (by simp [Subset.rfl])) hIJ
/-- An independence predicate on `Finset α` that obeys the finite matroid axioms determines a
  finitary matroid on `α`. -/
protected def ofFinset [DecidableEq α] (E : Set α) (Indep : Finset α → Prop)
    (indep_empty : Indep ∅)
    (indep_subset : ∀ ⦃I J⦄, Indep J → I ⊆ J → Indep I)
    (indep_aug : ∀ ⦃I J⦄, Indep I → Indep J → I.card < J.card → ∃ e ∈ J, e ∉ I ∧ Indep (insert e I))
    (subset_ground : ∀ ⦃I⦄, Indep I → (I : Set α) ⊆ E) : IndepMatroid α :=
  IndepMatroid.ofFinitary
    (E := E)
    (Indep := (fun I ↦ (∀ (J : Finset α), (J : Set α) ⊆ I → Indep J)))
    (indep_empty := by simpa [subset_empty_iff])
    (indep_subset := ( fun I J hJ hIJ K hKI ↦ hJ _ (hKI.trans hIJ) ))
    (indep_aug := by
      intro I J hI hIfin hJ hJfin hIJ
      rw [ncard_eq_toFinset_card _ hIfin, ncard_eq_toFinset_card _ hJfin] at hIJ
      have aug := indep_aug (hI _ (by 

Proof state:
α : Type u_1
I✝ B X : Set α
inst✝ : DecidableEq α
E : Set α
Indep : Finset α → Prop
indep_empty : Indep ∅
indep_subset : ∀ ⦃I J : Finset α⦄, Indep J → I ⊆ J → Indep I
indep_aug : ∀ ⦃I J : Finset α⦄, Indep I → Indep J → Finset.card I < Finset.card J → ∃ e ∈ J, e ∉ I ∧ Indep (insert e I)
subset_ground : ∀ ⦃I : Finset α⦄, Indep I → ↑I ⊆ E
I J : Set α
hI : ∀ (J : Finset α), ↑J ⊆ I → Indep J
hIfin : Set.Finite I
hJ : ∀ (J_1 : Finset α), ↑J_1 ⊆ J → Indep J_1
hJfin : Set.Finite J
hIJ : Finset.card (Finite.toFinset hIfin) < Finset.card (Finite.toFinset hJfin)
⊢ ↑(Finite.toFinset hIfin) ⊆ I

Next tactic:
simp [Subset.rfl]
/-- An independence predicate on `Finset α` that obeys the finite matroid axioms determines a
  finitary matroid on `α`. -/
protected def ofFinset [DecidableEq α] (E : Set α) (Indep : Finset α → Prop)
    (indep_empty : Indep ∅)
    (indep_subset : ∀ ⦃I J⦄, Indep J → I ⊆ J → Indep I)
    (indep_aug : ∀ ⦃I J⦄, Indep I → Indep J → I.card < J.card → ∃ e ∈ J, e ∉ I ∧ Indep (insert e I))
    (subset_ground : ∀ ⦃I⦄, Indep I → (I : Set α) ⊆ E) : IndepMatroid α :=
  IndepMatroid.ofFinitary
    (E := E)
    (Indep := (fun I ↦ (∀ (J : Finset α), (J : Set α) ⊆ I → Indep J)))
    (indep_empty := by simpa [subset_empty_iff])
    (indep_subset := ( fun I J hJ hIJ K hKI ↦ hJ _ (hKI.trans hIJ) ))
    (indep_aug := by
      intro I J hI hIfin hJ hJfin hIJ
      rw [ncard_eq_toFinset_card _ hIfin, ncard_eq_toFinset_card _ hJfin] at hIJ
      have aug := indep_aug (hI _ (by simp [Subset.rfl])) (hJ _ (by 

Proof state:
α : Type u_1
I✝ B X : Set α
inst✝ : DecidableEq α
E : Set α
Indep : Finset α → Prop
indep_empty : Indep ∅
indep_subset : ∀ ⦃I J : Finset α⦄, Indep J → I ⊆ J → Indep I
indep_aug : ∀ ⦃I J : Finset α⦄, Indep I → Indep J → Finset.card I < Finset.card J → ∃ e ∈ J, e ∉ I ∧ Indep (insert e I)
subset_ground : ∀ ⦃I : Finset α⦄, Indep I → ↑I ⊆ E
I J : Set α
hI : ∀ (J : Finset α), ↑J ⊆ I → Indep J
hIfin : Set.Finite I
hJ : ∀ (J_1 : Finset α), ↑J_1 ⊆ J → Indep J_1
hJfin : Set.Finite J
hIJ : Finset.card (Finite.toFinset hIfin) < Finset.card (Finite.toFinset hJfin)
⊢ ↑(Finite.toFinset hJfin) ⊆ J

Next tactic:
simp [Subset.rfl]
/-- An independence predicate on `Finset α` that obeys the finite matroid axioms determines a
  finitary matroid on `α`. -/
protected def ofFinset [DecidableEq α] (E : Set α) (Indep : Finset α → Prop)
    (indep_empty : Indep ∅)
    (indep_subset : ∀ ⦃I J⦄, Indep J → I ⊆ J → Indep I)
    (indep_aug : ∀ ⦃I J⦄, Indep I → Indep J → I.card < J.card → ∃ e ∈ J, e ∉ I ∧ Indep (insert e I))
    (subset_ground : ∀ ⦃I⦄, Indep I → (I : Set α) ⊆ E) : IndepMatroid α :=
  IndepMatroid.ofFinitary
    (E := E)
    (Indep := (fun I ↦ (∀ (J : Finset α), (J : Set α) ⊆ I → Indep J)))
    (indep_empty := by simpa [subset_empty_iff])
    (indep_subset := ( fun I J hJ hIJ K hKI ↦ hJ _ (hKI.trans hIJ) ))
    (indep_aug := by
      intro I J hI hIfin hJ hJfin hIJ
      rw [ncard_eq_toFinset_card _ hIfin, ncard_eq_toFinset_card _ hJfin] at hIJ
      have aug := indep_aug (hI _ (by simp [Subset.rfl])) (hJ _ (by simp [Subset.rfl])) hIJ
      

Proof state:
α : Type u_1
I✝ B X : Set α
inst✝ : DecidableEq α
E : Set α
Indep : Finset α → Prop
indep_empty : Indep ∅
indep_subset : ∀ ⦃I J : Finset α⦄, Indep J → I ⊆ J → Indep I
indep_aug : ∀ ⦃I J : Finset α⦄, Indep I → Indep J → Finset.card I < Finset.card J → ∃ e ∈ J, e ∉ I ∧ Indep (insert e I)
subset_ground : ∀ ⦃I : Finset α⦄, Indep I → ↑I ⊆ E
I J : Set α
hI : ∀ (J : Finset α), ↑J ⊆ I → Indep J
hIfin : Set.Finite I
hJ : ∀ (J_1 : Finset α), ↑J_1 ⊆ J → Indep J_1
hJfin : Set.Finite J
hIJ : Finset.card (Finite.toFinset hIfin) < Finset.card (Finite.toFinset hJfin)
aug : ∃ e ∈ Finite.toFinset hJfin, e ∉ Finite.toFinset hIfin ∧ Indep (insert e (Finite.toFinset hIfin))
⊢ ∃ e ∈ J, e ∉ I ∧ (fun I => ∀ (J : Finset α), ↑J ⊆ I → Indep J) (insert e I)

Next tactic:
simp only [Finite.mem_toFinset] at aug
/-- An independence predicate on `Finset α` that obeys the finite matroid axioms determines a
  finitary matroid on `α`. -/
protected def ofFinset [DecidableEq α] (E : Set α) (Indep : Finset α → Prop)
    (indep_empty : Indep ∅)
    (indep_subset : ∀ ⦃I J⦄, Indep J → I ⊆ J → Indep I)
    (indep_aug : ∀ ⦃I J⦄, Indep I → Indep J → I.card < J.card → ∃ e ∈ J, e ∉ I ∧ Indep (insert e I))
    (subset_ground : ∀ ⦃I⦄, Indep I → (I : Set α) ⊆ E) : IndepMatroid α :=
  IndepMatroid.ofFinitary
    (E := E)
    (Indep := (fun I ↦ (∀ (J : Finset α), (J : Set α) ⊆ I → Indep J)))
    (indep_empty := by simpa [subset_empty_iff])
    (indep_subset := ( fun I J hJ hIJ K hKI ↦ hJ _ (hKI.trans hIJ) ))
    (indep_aug := by
      intro I J hI hIfin hJ hJfin hIJ
      rw [ncard_eq_toFinset_card _ hIfin, ncard_eq_toFinset_card _ hJfin] at hIJ
      have aug := indep_aug (hI _ (by simp [Subset.rfl])) (hJ _ (by simp [Subset.rfl])) hIJ
      simp only [Finite.mem_toFinset] at aug
      

Proof state:
α : Type u_1
I✝ B X : Set α
inst✝ : DecidableEq α
E : Set α
Indep : Finset α → Prop
indep_empty : Indep ∅
indep_subset : ∀ ⦃I J : Finset α⦄, Indep J → I ⊆ J → Indep I
indep_aug : ∀ ⦃I J : Finset α⦄, Indep I → Indep J → Finset.card I < Finset.card J → ∃ e ∈ J, e ∉ I ∧ Indep (insert e I)
subset_ground : ∀ ⦃I : Finset α⦄, Indep I → ↑I ⊆ E
I J : Set α
hI : ∀ (J : Finset α), ↑J ⊆ I → Indep J
hIfin : Set.Finite I
hJ : ∀ (J_1 : Finset α), ↑J_1 ⊆ J → Indep J_1
hJfin : Set.Finite J
hIJ : Finset.card (Finite.toFinset hIfin) < Finset.card (Finite.toFinset hJfin)
aug : ∃ e ∈ J, e ∉ I ∧ Indep (insert e (Finite.toFinset hIfin))
⊢ ∃ e ∈ J, e ∉ I ∧ (fun I => ∀ (J : Finset α), ↑J ⊆ I → Indep J) (insert e I)

Next tactic:
obtain ⟨e, heJ, heI, hi⟩ := aug
/-- An independence predicate on `Finset α` that obeys the finite matroid axioms determines a
  finitary matroid on `α`. -/
protected def ofFinset [DecidableEq α] (E : Set α) (Indep : Finset α → Prop)
    (indep_empty : Indep ∅)
    (indep_subset : ∀ ⦃I J⦄, Indep J → I ⊆ J → Indep I)
    (indep_aug : ∀ ⦃I J⦄, Indep I → Indep J → I.card < J.card → ∃ e ∈ J, e ∉ I ∧ Indep (insert e I))
    (subset_ground : ∀ ⦃I⦄, Indep I → (I : Set α) ⊆ E) : IndepMatroid α :=
  IndepMatroid.ofFinitary
    (E := E)
    (Indep := (fun I ↦ (∀ (J : Finset α), (J : Set α) ⊆ I → Indep J)))
    (indep_empty := by simpa [subset_empty_iff])
    (indep_subset := ( fun I J hJ hIJ K hKI ↦ hJ _ (hKI.trans hIJ) ))
    (indep_aug := by
      intro I J hI hIfin hJ hJfin hIJ
      rw [ncard_eq_toFinset_card _ hIfin, ncard_eq_toFinset_card _ hJfin] at hIJ
      have aug := indep_aug (hI _ (by simp [Subset.rfl])) (hJ _ (by simp [Subset.rfl])) hIJ
      simp only [Finite.mem_toFinset] at aug
      obtain ⟨e, heJ, heI, hi⟩ := aug
      

Proof state:
case intro.intro.intro
α : Type u_1
I✝ B X : Set α
inst✝ : DecidableEq α
E : Set α
Indep : Finset α → Prop
indep_empty : Indep ∅
indep_subset : ∀ ⦃I J : Finset α⦄, Indep J → I ⊆ J → Indep I
indep_aug : ∀ ⦃I J : Finset α⦄, Indep I → Indep J → Finset.card I < Finset.card J → ∃ e ∈ J, e ∉ I ∧ Indep (insert e I)
subset_ground : ∀ ⦃I : Finset α⦄, Indep I → ↑I ⊆ E
I J : Set α
hI : ∀ (J : Finset α), ↑J ⊆ I → Indep J
hIfin : Set.Finite I
hJ : ∀ (J_1 : Finset α), ↑J_1 ⊆ J → Indep J_1
hJfin : Set.Finite J
hIJ : Finset.card (Finite.toFinset hIfin) < Finset.card (Finite.toFinset hJfin)
e : α
heJ : e ∈ J
heI : e ∉ I
hi : Indep (insert e (Finite.toFinset hIfin))
⊢ ∃ e ∈ J, e ∉ I ∧ (fun I => ∀ (J : Finset α), ↑J ⊆ I → Indep J) (insert e I)

Next tactic:
exact ⟨e, heJ, heI, fun K hK ↦ indep_subset hi <| Finset.coe_subset.1 (by simpa)⟩
/-- An independence predicate on `Finset α` that obeys the finite matroid axioms determines a
  finitary matroid on `α`. -/
protected def ofFinset [DecidableEq α] (E : Set α) (Indep : Finset α → Prop)
    (indep_empty : Indep ∅)
    (indep_subset : ∀ ⦃I J⦄, Indep J → I ⊆ J → Indep I)
    (indep_aug : ∀ ⦃I J⦄, Indep I → Indep J → I.card < J.card → ∃ e ∈ J, e ∉ I ∧ Indep (insert e I))
    (subset_ground : ∀ ⦃I⦄, Indep I → (I : Set α) ⊆ E) : IndepMatroid α :=
  IndepMatroid.ofFinitary
    (E := E)
    (Indep := (fun I ↦ (∀ (J : Finset α), (J : Set α) ⊆ I → Indep J)))
    (indep_empty := by simpa [subset_empty_iff])
    (indep_subset := ( fun I J hJ hIJ K hKI ↦ hJ _ (hKI.trans hIJ) ))
    (indep_aug := by
      intro I J hI hIfin hJ hJfin hIJ
      rw [ncard_eq_toFinset_card _ hIfin, ncard_eq_toFinset_card _ hJfin] at hIJ
      have aug := indep_aug (hI _ (by simp [Subset.rfl])) (hJ _ (by simp [Subset.rfl])) hIJ
      simp only [Finite.mem_toFinset] at aug
      obtain ⟨e, heJ, heI, hi⟩ := aug
      exact ⟨e, heJ, heI, fun K hK ↦ indep_subset hi <| Finset.coe_subset.1 (by 

Proof state:
α : Type u_1
I✝ B X : Set α
inst✝ : DecidableEq α
E : Set α
Indep : Finset α → Prop
indep_empty : Indep ∅
indep_subset : ∀ ⦃I J : Finset α⦄, Indep J → I ⊆ J → Indep I
indep_aug : ∀ ⦃I J : Finset α⦄, Indep I → Indep J → Finset.card I < Finset.card J → ∃ e ∈ J, e ∉ I ∧ Indep (insert e I)
subset_ground : ∀ ⦃I : Finset α⦄, Indep I → ↑I ⊆ E
I J : Set α
hI : ∀ (J : Finset α), ↑J ⊆ I → Indep J
hIfin : Set.Finite I
hJ : ∀ (J_1 : Finset α), ↑J_1 ⊆ J → Indep J_1
hJfin : Set.Finite J
hIJ : Finset.card (Finite.toFinset hIfin) < Finset.card (Finite.toFinset hJfin)
e : α
heJ : e ∈ J
heI : e ∉ I
hi : Indep (insert e (Finite.toFinset hIfin))
K : Finset α
hK : ↑K ⊆ insert e I
⊢ ↑K ⊆ ↑(insert e (Finite.toFinset hIfin))

Next tactic:
simpa
/-- An independence predicate on `Finset α` that obeys the finite matroid axioms determines a
  finitary matroid on `α`. -/
protected def ofFinset [DecidableEq α] (E : Set α) (Indep : Finset α → Prop)
    (indep_empty : Indep ∅)
    (indep_subset : ∀ ⦃I J⦄, Indep J → I ⊆ J → Indep I)
    (indep_aug : ∀ ⦃I J⦄, Indep I → Indep J → I.card < J.card → ∃ e ∈ J, e ∉ I ∧ Indep (insert e I))
    (subset_ground : ∀ ⦃I⦄, Indep I → (I : Set α) ⊆ E) : IndepMatroid α :=
  IndepMatroid.ofFinitary
    (E := E)
    (Indep := (fun I ↦ (∀ (J : Finset α), (J : Set α) ⊆ I → Indep J)))
    (indep_empty := by simpa [subset_empty_iff])
    (indep_subset := ( fun I J hJ hIJ K hKI ↦ hJ _ (hKI.trans hIJ) ))
    (indep_aug := by
      intro I J hI hIfin hJ hJfin hIJ
      rw [ncard_eq_toFinset_card _ hIfin, ncard_eq_toFinset_card _ hJfin] at hIJ
      have aug := indep_aug (hI _ (by simp [Subset.rfl])) (hJ _ (by simp [Subset.rfl])) hIJ
      simp only [Finite.mem_toFinset] at aug
      obtain ⟨e, heJ, heI, hi⟩ := aug
      exact ⟨e, heJ, heI, fun K hK ↦ indep_subset hi <| Finset.coe_subset.1 (by simpa)⟩ )
    (indep_compact := fun I h J hJ ↦ h _ hJ J.finite_toSet _ Subset.rfl )
    (subset_ground := fun I hI x hxI ↦ by 

Proof state:
α : Type u_1
I✝ B X : Set α
inst✝ : DecidableEq α
E : Set α
Indep : Finset α → Prop
indep_empty : Indep ∅
indep_subset : ∀ ⦃I J : Finset α⦄, Indep J → I ⊆ J → Indep I
indep_aug : ∀ ⦃I J : Finset α⦄, Indep I → Indep J → Finset.card I < Finset.card J → ∃ e ∈ J, e ∉ I ∧ Indep (insert e I)
subset_ground : ∀ ⦃I : Finset α⦄, Indep I → ↑I ⊆ E
I : Set α
hI : (fun I => ∀ (J : Finset α), ↑J ⊆ I → Indep J) I
x : α
hxI : x ∈ I
⊢ x ∈ E

Next tactic:
simpa using subset_ground <| hI {x} (by simpa)