Theorem: Matroid.insert_indep_iff (Matroid/Closure.lean)

State before:
α : Type u_1
ι : Type u_2
M : Matroid α
F I J X Y B C R : Set α
e f x y : α
⊢ M.Indep (insert e I) ↔ M.Indep I ∧ (e ∉ I → e ∈ M.E \ M.cl I)
Tactic:
by_cases hI : M.Indep I
State after:
case pos
α : Type u_1
ι : Type u_2
M : Matroid α
F I J X Y B C R : Set α
e f x y : α
hI : M.Indep I
⊢ M.Indep (insert e I) ↔ M.Indep I ∧ (e ∉ I → e ∈ M.E \ M.cl I)

case neg
α : Type u_1
ι : Type u_2
M : Matroid α
F I J X Y B C R : Set α
e f x y : α
hI : ¬M.Indep I
⊢ M.Indep (insert e I) ↔ M.Indep I ∧ (e ∉ I → e ∈ M.E \ M.cl I)

State before:
case neg
α : Type u_1
ι : Type u_2
M : Matroid α
F I J X Y B C R : Set α
e f x y : α
hI : ¬M.Indep I
⊢ M.Indep (insert e I) ↔ M.Indep I ∧ (e ∉ I → e ∈ M.E \ M.cl I)
Tactic:
simp [hI, show ¬ M.Indep (insert e I) from fun h ↦ hI <| h.subset <| subset_insert _ _]
State after:
no goals

State before:
case pos
α : Type u_1
ι : Type u_2
M : Matroid α
F I J X Y B C R : Set α
e f x y : α
hI : M.Indep I
⊢ M.Indep (insert e I) ↔ M.Indep I ∧ (e ∉ I → e ∈ M.E \ M.cl I)
Tactic:
rw [hI.insert_indep_iff, and_iff_right hI, or_iff_not_imp_right]
State after:
no goals