-- Mathlib/Data/Matroid/Circuit.lean
/-- For an element `e` of a base `B`, the complement of the closure of `B \ {e}` is a cocircuit. -/
lemma IsBase.compl_closure_diff_singleton_isCocircuit (hB : M.IsBase B) (he : e ∈ B) :
    M.IsCocircuit (M.E \ M.closure (B \ {e}))

Proof state:
α : Type u_1
M : Matroid α
e : α
B : Set α
hB : M.IsBase B
he : e ∈ B
hB' : Minimal M.Spanning B
X : Set α
hX : ¬M.Spanning (M.E \ X)
hXss : X ⊆ M.E ∧ Disjoint X (M.closure (B \ {e}))
f : α
hf : f ∈ M.E \ X
fcl : f ∉ M.closure (B \ {e})
⊢ M.Spanning (M.E \ X)

Tactic:
suffices hsp : M.IsBase (insert f (B \ {e})) by
  refine hsp.spanning.superset <| insert_subset hf <|
    (M.subset_closure _ (diff_subset.trans hB.subset_ground)).trans ?_
  rw [subset_diff, and_iff_left hXss.2.symm]
  apply closure_subset_ground

Result:
α : Type u_1
M : Matroid α
e : α
B : Set α
hB : M.IsBase B
he : e ∈ B
hB' : Minimal M.Spanning B
X : Set α
hX : ¬M.Spanning (M.E \ X)
hXss : X ⊆ M.E ∧ Disjoint X (M.closure (B \ {e}))
f : α
hf : f ∈ M.E \ X
fcl : f ∉ M.closure (B \ {e})
⊢ M.IsBase (insert f (B \ {e}))

Proof state:
α : Type u_1
M : Matroid α
e : α
B : Set α
hB : M.IsBase B
he : e ∈ B
⊢ M.IsCocircuit (M.E \ M.closure (B \ {e}))

Tactic:
rw [isCocircuit_iff_minimal_compl_nonspanning, minimal_subset_iff,
  diff_diff_cancel_left (M.closure_subset_ground _),
  closure_spanning_iff (diff_subset.trans hB.subset_ground)]

Result:
α : Type u_1
M : Matroid α
e : α
B : Set α
hB : M.IsBase B
he : e ∈ B
⊢ ¬M.Spanning (B \ {e}) ∧
    ∀ ⦃t : Set α⦄, ¬M.Spanning (M.E \ t) → t ⊆ M.E \ M.closure (B \ {e}) → M.E \ M.closure (B \ {e}) = t

Proof state:
α : Type u_1
M : Matroid α
e : α
B : Set α
hB : M.IsBase B
he : e ∈ B
⊢ ¬M.Spanning (B \ {e}) ∧
    ∀ ⦃t : Set α⦄, ¬M.Spanning (M.E \ t) → t ⊆ M.E \ M.closure (B \ {e}) → M.E \ M.closure (B \ {e}) = t

Tactic:
have hB' := (isBase_iff_minimal_spanning.1 hB)

Result:
α : Type u_1
M : Matroid α
e : α
B : Set α
hB : M.IsBase B
he : e ∈ B
hB' : Minimal M.Spanning B
⊢ ¬M.Spanning (B \ {e}) ∧
    ∀ ⦃t : Set α⦄, ¬M.Spanning (M.E \ t) → t ⊆ M.E \ M.closure (B \ {e}) → M.E \ M.closure (B \ {e}) = t

Proof state:
α : Type u_1
M : Matroid α
e : α
B : Set α
hB : M.IsBase B
he : e ∈ B
hB' : Minimal M.Spanning B
⊢ ¬M.Spanning (B \ {e}) ∧
    ∀ ⦃t : Set α⦄, ¬M.Spanning (M.E \ t) → t ⊆ M.E \ M.closure (B \ {e}) → M.E \ M.closure (B \ {e}) = t

Tactic:
refine ⟨fun hsp ↦ hB'.not_mem_of_prop_diff_singleton hsp he, fun X hX hXss ↦ hXss.antisymm' ?_⟩

Result:
α : Type u_1
M : Matroid α
e : α
B : Set α
hB : M.IsBase B
he : e ∈ B
hB' : Minimal M.Spanning B
X : Set α
hX : ¬M.Spanning (M.E \ X)
hXss : X ⊆ M.E \ M.closure (B \ {e})
⊢ M.E \ M.closure (B \ {e}) ⊆ X

Proof state:
α : Type u_1
M : Matroid α
e : α
B : Set α
hB : M.IsBase B
he : e ∈ B
hB' : Minimal M.Spanning B
X : Set α
hX : ¬M.Spanning (M.E \ X)
hXss : X ⊆ M.E ∧ Disjoint X (M.closure (B \ {e}))
f : α
hf : f ∈ M.E \ X
fcl : f ∉ M.closure (B \ {e})
hsp : M.IsBase (insert f (B \ {e}))
⊢ M.closure (B \ {e}) ⊆ M.E

Tactic:
apply closure_subset_ground

Result:
no goals

Proof state:
α : Type u_1
M : Matroid α
e : α
B : Set α
hB : M.IsBase B
he : e ∈ B
hB' : Minimal M.Spanning B
X : Set α
hX : ¬M.Spanning (M.E \ X)
hXss : X ⊆ M.E \ M.closure (B \ {e})
⊢ M.E \ M.closure (B \ {e}) ⊆ X

Tactic:
rw [diff_subset_comm]

Result:
α : Type u_1
M : Matroid α
e : α
B : Set α
hB : M.IsBase B
he : e ∈ B
hB' : Minimal M.Spanning B
X : Set α
hX : ¬M.Spanning (M.E \ X)
hXss : X ⊆ M.E \ M.closure (B \ {e})
⊢ M.E \ X ⊆ M.closure (B \ {e})

Proof state:
α : Type u_1
M : Matroid α
e : α
B : Set α
hB : M.IsBase B
he : e ∈ B
hB' : Minimal M.Spanning B
X : Set α
hX : ¬M.Spanning (M.E \ X)
hXss : X ⊆ M.E ∧ Disjoint X (M.closure (B \ {e}))
f : α
hf : f ∈ M.E \ X
fcl : f ∉ M.closure (B \ {e})
hsp : M.IsBase (insert f (B \ {e}))
⊢ M.Spanning (M.E \ X)

Tactic:
refine hsp.spanning.superset <| insert_subset hf <|
  (M.subset_closure _ (diff_subset.trans hB.subset_ground)).trans ?_

Result:
α : Type u_1
M : Matroid α
e : α
B : Set α
hB : M.IsBase B
he : e ∈ B
hB' : Minimal M.Spanning B
X : Set α
hX : ¬M.Spanning (M.E \ X)
hXss : X ⊆ M.E ∧ Disjoint X (M.closure (B \ {e}))
f : α
hf : f ∈ M.E \ X
fcl : f ∉ M.closure (B \ {e})
hsp : M.IsBase (insert f (B \ {e}))
⊢ M.closure (B \ {e}) ⊆ M.E \ X

Proof state:
α : Type u_1
M : Matroid α
e : α
B : Set α
hB : M.IsBase B
he : e ∈ B
hB' : Minimal M.Spanning B
X : Set α
hX : ¬M.Spanning (M.E \ X)
hXss : X ⊆ M.E \ M.closure (B \ {e})
f : α
hf : f ∈ M.E \ X
fcl : f ∉ M.closure (B \ {e})
⊢ M.Spanning (M.E \ X)

Tactic:
rw [subset_diff] at hXss

Result:
α : Type u_1
M : Matroid α
e : α
B : Set α
hB : M.IsBase B
he : e ∈ B
hB' : Minimal M.Spanning B
X : Set α
hX : ¬M.Spanning (M.E \ X)
hXss : X ⊆ M.E ∧ Disjoint X (M.closure (B \ {e}))
f : α
hf : f ∈ M.E \ X
fcl : f ∉ M.closure (B \ {e})
⊢ M.Spanning (M.E \ X)

Proof state:
α : Type u_1
M : Matroid α
e : α
B : Set α
hB : M.IsBase B
he : e ∈ B
hB' : Minimal M.Spanning B
X : Set α
hX : ¬M.Spanning (M.E \ X)
hXss : X ⊆ M.E \ M.closure (B \ {e})
⊢ M.E \ X ⊆ M.closure (B \ {e})

Tactic:
refine fun f hf ↦ by_contra fun fcl ↦ hX ?_

Result:
α : Type u_1
M : Matroid α
e : α
B : Set α
hB : M.IsBase B
he : e ∈ B
hB' : Minimal M.Spanning B
X : Set α
hX : ¬M.Spanning (M.E \ X)
hXss : X ⊆ M.E \ M.closure (B \ {e})
f : α
hf : f ∈ M.E \ X
fcl : f ∉ M.closure (B \ {e})
⊢ M.Spanning (M.E \ X)

Proof state:
α : Type u_1
M : Matroid α
e : α
B : Set α
hB : M.IsBase B
he : e ∈ B
hB' : Minimal M.Spanning B
X : Set α
hX : ¬M.Spanning (M.E \ X)
hXss : X ⊆ M.E ∧ Disjoint X (M.closure (B \ {e}))
f : α
hf : f ∈ M.E \ X
fcl : f ∉ M.closure (B \ {e})
hsp : M.IsBase (insert f (B \ {e}))
⊢ M.closure (B \ {e}) ⊆ M.E \ X

Tactic:
rw [subset_diff, and_iff_left hXss.2.symm]

Result:
α : Type u_1
M : Matroid α
e : α
B : Set α
hB : M.IsBase B
he : e ∈ B
hB' : Minimal M.Spanning B
X : Set α
hX : ¬M.Spanning (M.E \ X)
hXss : X ⊆ M.E ∧ Disjoint X (M.closure (B \ {e}))
f : α
hf : f ∈ M.E \ X
fcl : f ∉ M.closure (B \ {e})
hsp : M.IsBase (insert f (B \ {e}))
⊢ M.closure (B \ {e}) ⊆ M.E

Proof state:
α : Type u_1
M : Matroid α
e : α
B : Set α
hB : M.IsBase B
he : e ∈ B
hB' : Minimal M.Spanning B
X : Set α
hX : ¬M.Spanning (M.E \ X)
hXss : X ⊆ M.E ∧ Disjoint X (M.closure (B \ {e}))
f : α
hf : f ∈ M.E \ X
fcl : f ∉ M.closure (B \ {e})
⊢ M.IsBase (insert f (B \ {e}))

Tactic:
exact hB.exchange_base_of_not_mem_closure he fcl

Result:
no goals