Theorem: Matroid.er_insert_eq_add_one (Matroid/Rank.lean)

State before:
α : Type u_1
ι : Type u_2
M✝ N : Matroid α
I B X✝ X' Y Y' Z R : Set α
n : ℕ∞
e f : α
M : Matroid α
X : Set α
he : e ∈ M.E \ M.cl X
⊢ M.er (insert e X) = M.er X + 1
Tactic:
obtain ⟨I, hI⟩ := M.exists_basis' X
State after:
case intro
α : Type u_1
ι : Type u_2
M✝ N : Matroid α
I✝ B X✝ X' Y Y' Z R : Set α
n : ℕ∞
e f : α
M : Matroid α
X : Set α
he : e ∈ M.E \ M.cl X
I : Set α
hI : M.Basis' I X
⊢ M.er (insert e X) = M.er X + 1

State before:
case intro
α : Type u_1
ι : Type u_2
M✝ N : Matroid α
I✝ B X✝ X' Y Y' Z R : Set α
n : ℕ∞
e f : α
M : Matroid α
X : Set α
he : e ∈ M.E \ M.cl X
I : Set α
hI : M.Basis' I X
⊢ M.er (insert e X) = M.er X + 1
Tactic:
rw [← hI.cl_eq_cl] at he
State after:
case intro
α : Type u_1
ι : Type u_2
M✝ N : Matroid α
I✝ B X✝ X' Y Y' Z R : Set α
n : ℕ∞
e f : α
M : Matroid α
X I : Set α
he : e ∈ M.E \ M.cl I
hI : M.Basis' I X
⊢ M.er (insert e X) = M.er X + 1

State before:
case intro
α : Type u_1
ι : Type u_2
M✝ N : Matroid α
I✝ B X✝ X' Y Y' Z R : Set α
n : ℕ∞
e f : α
M : Matroid α
X I : Set α
he : e ∈ M.E \ M.cl I
hI : M.Basis' I X
⊢ M.er (insert e X) = M.er X + 1
Tactic:
rw [← er_cl_eq, ← cl_insert_cl_eq_cl_insert, ← hI.cl_eq_cl, hI.er_eq_encard,
  cl_insert_cl_eq_cl_insert, er_cl_eq, Indep.er, encard_insert_of_not_mem]
State after:
case intro
α : Type u_1
ι : Type u_2
M✝ N : Matroid α
I✝ B X✝ X' Y Y' Z R : Set α
n : ℕ∞
e f : α
M : Matroid α
X I : Set α
he : e ∈ M.E \ M.cl I
hI : M.Basis' I X
⊢ e ∉ I

case intro
α : Type u_1
ι : Type u_2
M✝ N : Matroid α
I✝ B X✝ X' Y Y' Z R : Set α
n : ℕ∞
e f : α
M : Matroid α
X I : Set α
he : e ∈ M.E \ M.cl I
hI : M.Basis' I X
⊢ M.Indep (insert e I)

State before:
case intro
α : Type u_1
ι : Type u_2
M✝ N : Matroid α
I✝ B X✝ X' Y Y' Z R : Set α
n : ℕ∞
e f : α
M : Matroid α
X I : Set α
he : e ∈ M.E \ M.cl I
hI : M.Basis' I X
⊢ M.Indep (insert e I)
Tactic:
rw [hI.indep.insert_indep_iff]
State after:
case intro
α : Type u_1
ι : Type u_2
M✝ N : Matroid α
I✝ B X✝ X' Y Y' Z R : Set α
n : ℕ∞
e f : α
M : Matroid α
X I : Set α
he : e ∈ M.E \ M.cl I
hI : M.Basis' I X
⊢ e ∈ M.E \ M.cl I ∨ e ∈ I

State before:
case intro
α : Type u_1
ι : Type u_2
M✝ N : Matroid α
I✝ B X✝ X' Y Y' Z R : Set α
n : ℕ∞
e f : α
M : Matroid α
X I : Set α
he : e ∈ M.E \ M.cl I
hI : M.Basis' I X
⊢ e ∈ M.E \ M.cl I ∨ e ∈ I
Tactic:
exact Or.inl he
State after:
no goals

State before:
case intro
α : Type u_1
ι : Type u_2
M✝ N : Matroid α
I✝ B X✝ X' Y Y' Z R : Set α
n : ℕ∞
e f : α
M : Matroid α
X I : Set α
he : e ∈ M.E \ M.cl I
hI : M.Basis' I X
⊢ e ∉ I
Tactic:
exact fun heI ↦ he.2 (M.subset_cl I hI.indep.subset_ground heI)
State after:
no goals