-- Mathlib/Data/Matroid/Closure.lean
/-- For a nonempty collection of subsets of a given independent set,
the closure of the intersection is the intersection of the closure. -/
lemma Indep.closure_sInter_eq_biInter_closure_of_forall_subset {Js : Set (Set α)} (hI : M.Indep I)
    (hne : Js.Nonempty) (hIs : ∀ J ∈ Js, J ⊆ I) : M.closure (⋂₀ Js) = (⋂ J ∈ Js, M.closure J)

Proof state:
α : Type u_2
M : Matroid α
I : Set α
Js : Set (Set α)
hI : M.Indep I
hne : Js.Nonempty
hIs : ∀ J ∈ Js, J ⊆ I
hiX : ⋂₀ Js ⊆ I
hiI : M.Indep (⋂₀ Js)
e : α
he : ∀ i ∈ Js, e ∈ M.closure i
he' : M.Indep (insert e (⋂₀ Js))
heEI : e ∈ M.E \ I
J : Set α
hJI : M.IsBasis J (insert e I)
heJ : insert e (⋂₀ Js) ⊆ J
⊢ e ∈ M.closure I

Tactic:
exact (M.closure_subset_closure (hIs _ hne.some_mem)) (he _ hne.some_mem)

Result:
no goals

Proof state:
α : Type u_2
M : Matroid α
I : Set α
Js : Set (Set α)
hI : M.Indep I
hne : Js.Nonempty
hIs : ∀ J ∈ Js, J ⊆ I
⊢ (∀ i ∈ Js, M.closure (⋂₀ Js) ⊆ M.closure i) ∧ ⋂ J ∈ Js, M.closure J ⊆ M.closure (⋂₀ Js)

Tactic:
have hiX : ⋂₀ Js ⊆ I := (sInter_subset_of_mem hne.some_mem).trans (hIs _ hne.some_mem)

Result:
α : Type u_2
M : Matroid α
I : Set α
Js : Set (Set α)
hI : M.Indep I
hne : Js.Nonempty
hIs : ∀ J ∈ Js, J ⊆ I
hiX : ⋂₀ Js ⊆ I
⊢ (∀ i ∈ Js, M.closure (⋂₀ Js) ⊆ M.closure i) ∧ ⋂ J ∈ Js, M.closure J ⊆ M.closure (⋂₀ Js)

Proof state:
α : Type u_2
M : Matroid α
I : Set α
Js : Set (Set α)
hI : M.Indep I
hne : Js.Nonempty
hIs : ∀ J ∈ Js, J ⊆ I
hiX : ⋂₀ Js ⊆ I
hiI : M.Indep (⋂₀ Js)
e : α
he : ∀ i ∈ Js, e ∈ M.closure i
he' : M.Indep (insert e (⋂₀ Js))
heEI : e ∈ M.E \ I
J : Set α
hJI : M.IsBasis J (insert e I)
heJ : insert e (⋂₀ Js) ⊆ J
hIb : M.IsBasis I (insert e I)
f : α
hfIJ : f ∈ I \ J
hfb : M.IsBasis (insert f (J \ {e})) (insert e I)
⊢ J \ {e} ⊆ I

Tactic:
(rw [diff_subset_iff, singleton_union]; exact hJI.subset)

Result:
no goals

Proof state:
α : Type u_2
M : Matroid α
I : Set α
Js : Set (Set α)
hI : M.Indep I
hne : Js.Nonempty
hIs : ∀ J ∈ Js, J ⊆ I
hiX : ⋂₀ Js ⊆ I
hiI : M.Indep (⋂₀ Js)
⊢ (∀ i ∈ Js, M.closure (⋂₀ Js) ⊆ M.closure i) ∧ ⋂ J ∈ Js, M.closure J ⊆ M.closure (⋂₀ Js)

Tactic:
refine ⟨ fun X hX ↦ M.closure_subset_closure (sInter_subset_of_mem hX),
  fun e he ↦ by_contra fun he' ↦ ?_⟩

Result:
α : Type u_2
M : Matroid α
I : Set α
Js : Set (Set α)
hI : M.Indep I
hne : Js.Nonempty
hIs : ∀ J ∈ Js, J ⊆ I
hiX : ⋂₀ Js ⊆ I
hiI : M.Indep (⋂₀ Js)
e : α
he : e ∈ ⋂ J ∈ Js, M.closure J
he' : e ∉ M.closure (⋂₀ Js)
⊢ False

Proof state:
case intro.intro.intro.intro
α : Type u_2
M : Matroid α
Js : Set (Set α)
hne : Js.Nonempty
hiI : M.Indep (⋂₀ Js)
e : α
he : ∀ i ∈ Js, e ∈ M.closure i
he' : M.Indep (insert e (⋂₀ Js))
J : Set α
heJ : insert e (⋂₀ Js) ⊆ J
f : α
hI : M.Indep (insert f (J \ {e}))
hIs : ∀ J_1 ∈ Js, J_1 ⊆ insert f (J \ {e})
hiX : ⋂₀ Js ⊆ insert f (J \ {e})
heEI : e ∈ M.E \ insert f (J \ {e})
hJI : M.IsBasis J (insert e (insert f (J \ {e})))
hIb : M.IsBasis (insert f (J \ {e})) (insert e (insert f (J \ {e})))
hfIJ : f ∈ insert f (J \ {e}) \ J
hfb : M.IsBasis (insert f (J \ {e})) (insert e (insert f (J \ {e})))
X : Set α
hX' : X ∈ Js
hfX : f ∉ X
⊢ False

Tactic:
obtain (hd | heX) := ((hI.subset (hIs X hX')).mem_closure_iff).mp (he _ hX')

Result:
case intro.intro.intro.intro.inl
α : Type u_2
M : Matroid α
Js : Set (Set α)
hne : Js.Nonempty
hiI : M.Indep (⋂₀ Js)
e : α
he : ∀ i ∈ Js, e ∈ M.closure i
he' : M.Indep (insert e (⋂₀ Js))
J : Set α
heJ : insert e (⋂₀ Js) ⊆ J
f : α
hI : M.Indep (insert f (J \ {e}))
hIs : ∀ J_1 ∈ Js, J_1 ⊆ insert f (J \ {e})
hiX : ⋂₀ Js ⊆ insert f (J \ {e})
heEI : e ∈ M.E \ insert f (J \ {e})
hJI : M.IsBasis J (insert e (insert f (J \ {e})))
hIb : M.IsBasis (insert f (J \ {e})) (insert e (insert f (J \ {e})))
hfIJ : f ∈ insert f (J \ {e}) \ J
hfb : M.IsBasis (insert f (J \ {e})) (insert e (insert f (J \ {e})))
X : Set α
hX' : X ∈ Js
hfX : f ∉ X
hd : M.Dep (insert e X)
⊢ False

case intro.intro.intro.intro.inr
α : Type u_2
M : Matroid α
Js : Set (Set α)
hne : Js.Nonempty
hiI : M.Indep (⋂₀ Js)
e : α
he : ∀ i ∈ Js, e ∈ M.closure i
he' : M.Indep (insert e (⋂₀ Js))
J : Set α
heJ : insert e (⋂₀ Js) ⊆ J
f : α
hI : M.Indep (insert f (J \ {e}))
hIs : ∀ J_1 ∈ Js, J_1 ⊆ insert f (J \ {e})
hiX : ⋂₀ Js ⊆ insert f (J \ {e})
heEI : e ∈ M.E \ insert f (J \ {e})
hJI : M.IsBasis J (insert e (insert f (J \ {e})))
hIb : M.IsBasis (insert f (J \ {e})) (insert e (insert f (J \ {e})))
hfIJ : f ∈ insert f (J \ {e}) \ J
hfb : M.IsBasis (insert f (J \ {e})) (insert e (insert f (J \ {e})))
X : Set α
hX' : X ∈ Js
hfX : f ∉ X
heX : e ∈ X
⊢ False

Proof state:
α : Type u_2
M : Matroid α
I : Set α
Js : Set (Set α)
hI : M.Indep I
hne : Js.Nonempty
hIs : ∀ J ∈ Js, J ⊆ I
⊢ M.closure (⋂₀ Js) = ⋂ J ∈ Js, M.closure J

Tactic:
rw [subset_antisymm_iff, subset_iInter₂_iff]

Result:
α : Type u_2
M : Matroid α
I : Set α
Js : Set (Set α)
hI : M.Indep I
hne : Js.Nonempty
hIs : ∀ J ∈ Js, J ⊆ I
⊢ (∀ i ∈ Js, M.closure (⋂₀ Js) ⊆ M.closure i) ∧ ⋂ J ∈ Js, M.closure J ⊆ M.closure (⋂₀ Js)

Proof state:
case intro.intro.intro.intro.inl
α : Type u_2
M : Matroid α
Js : Set (Set α)
hne : Js.Nonempty
hiI : M.Indep (⋂₀ Js)
e : α
he : ∀ i ∈ Js, e ∈ M.closure i
he' : M.Indep (insert e (⋂₀ Js))
J : Set α
heJ : insert e (⋂₀ Js) ⊆ J
f : α
hI : M.Indep (insert f (J \ {e}))
hIs : ∀ J_1 ∈ Js, J_1 ⊆ insert f (J \ {e})
hiX : ⋂₀ Js ⊆ insert f (J \ {e})
heEI : e ∈ M.E \ insert f (J \ {e})
hJI : M.IsBasis J (insert e (insert f (J \ {e})))
hIb : M.IsBasis (insert f (J \ {e})) (insert e (insert f (J \ {e})))
hfIJ : f ∈ insert f (J \ {e}) \ J
hfb : M.IsBasis (insert f (J \ {e})) (insert e (insert f (J \ {e})))
X : Set α
hX' : X ∈ Js
hfX : f ∉ X
hd : M.Dep (insert e X)
⊢ X ⊆ J

Tactic:
specialize hIs _ hX'

Result:
case intro.intro.intro.intro.inl
α : Type u_2
M : Matroid α
Js : Set (Set α)
hne : Js.Nonempty
hiI : M.Indep (⋂₀ Js)
e : α
he : ∀ i ∈ Js, e ∈ M.closure i
he' : M.Indep (insert e (⋂₀ Js))
J : Set α
heJ : insert e (⋂₀ Js) ⊆ J
f : α
hI : M.Indep (insert f (J \ {e}))
hiX : ⋂₀ Js ⊆ insert f (J \ {e})
heEI : e ∈ M.E \ insert f (J \ {e})
hJI : M.IsBasis J (insert e (insert f (J \ {e})))
hIb : M.IsBasis (insert f (J \ {e})) (insert e (insert f (J \ {e})))
hfIJ : f ∈ insert f (J \ {e}) \ J
hfb : M.IsBasis (insert f (J \ {e})) (insert e (insert f (J \ {e})))
X : Set α
hX' : X ∈ Js
hfX : f ∉ X
hd : M.Dep (insert e X)
hIs : X ⊆ insert f (J \ {e})
⊢ X ⊆ J

Proof state:
case intro.intro.intro.intro.inl
α : Type u_2
M : Matroid α
Js : Set (Set α)
hne : Js.Nonempty
hiI : M.Indep (⋂₀ Js)
e : α
he : ∀ i ∈ Js, e ∈ M.closure i
he' : M.Indep (insert e (⋂₀ Js))
J : Set α
heJ : insert e (⋂₀ Js) ⊆ J
f : α
hI : M.Indep (insert f (J \ {e}))
hiX : ⋂₀ Js ⊆ insert f (J \ {e})
heEI : e ∈ M.E \ insert f (J \ {e})
hJI : M.IsBasis J (insert e (insert f (J \ {e})))
hIb : M.IsBasis (insert f (J \ {e})) (insert e (insert f (J \ {e})))
hfIJ : f ∈ insert f (J \ {e}) \ J
hfb : M.IsBasis (insert f (J \ {e})) (insert e (insert f (J \ {e})))
X : Set α
hX' : X ∈ Js
hfX : f ∉ X
hd : M.Dep (insert e X)
hIs : X ⊆ insert f (J \ {e})
⊢ X ⊆ J

Tactic:
rw [← singleton_union, ← diff_subset_iff, diff_singleton_eq_self hfX] at hIs

Result:
case intro.intro.intro.intro.inl
α : Type u_2
M : Matroid α
Js : Set (Set α)
hne : Js.Nonempty
hiI : M.Indep (⋂₀ Js)
e : α
he : ∀ i ∈ Js, e ∈ M.closure i
he' : M.Indep (insert e (⋂₀ Js))
J : Set α
heJ : insert e (⋂₀ Js) ⊆ J
f : α
hI : M.Indep (insert f (J \ {e}))
hiX : ⋂₀ Js ⊆ insert f (J \ {e})
heEI : e ∈ M.E \ insert f (J \ {e})
hJI : M.IsBasis J (insert e (insert f (J \ {e})))
hIb : M.IsBasis (insert f (J \ {e})) (insert e (insert f (J \ {e})))
hfIJ : f ∈ insert f (J \ {e}) \ J
hfb : M.IsBasis (insert f (J \ {e})) (insert e (insert f (J \ {e})))
X : Set α
hX' : X ∈ Js
hfX : f ∉ X
hd : M.Dep (insert e X)
hIs : X ⊆ J \ {e}
⊢ X ⊆ J