Theorem: Matroid.modularSet_singleton (Matroid/Modular.lean)

State before:
ι : Type u_1
α : Type u_2
M✝ : Matroid α
B I J X X' Y Y' F : Set α
Xs Ys Js Is : ι → Set α
i j : ι
e f : α
M : Matroid α
he : e ∈ M.E
⊢ M.ModularSet {e}
Tactic:
refine modularSet_def.2 fun F hF ↦ ?_
State after:
ι : Type u_1
α : Type u_2
M✝ : Matroid α
B I J X X' Y Y' F✝ : Set α
Xs Ys Js Is : ι → Set α
i j : ι
e f : α
M : Matroid α
he : e ∈ M.E
F : Set α
hF : M.Flat F
⊢ M.ModularPair {e} F

State before:
ι : Type u_1
α : Type u_2
M✝ : Matroid α
B I J X X' Y Y' F✝ : Set α
Xs Ys Js Is : ι → Set α
i j : ι
e f : α
M : Matroid α
he : e ∈ M.E
F : Set α
hF : M.Flat F
⊢ M.ModularPair {e} F
Tactic:
by_cases heF : {e} ⊆ F
State after:
case pos
ι : Type u_1
α : Type u_2
M✝ : Matroid α
B I J X X' Y Y' F✝ : Set α
Xs Ys Js Is : ι → Set α
i j : ι
e f : α
M : Matroid α
he : e ∈ M.E
F : Set α
hF : M.Flat F
heF : {e} ⊆ F
⊢ M.ModularPair {e} F

case neg
ι : Type u_1
α : Type u_2
M✝ : Matroid α
B I J X X' Y Y' F✝ : Set α
Xs Ys Js Is : ι → Set α
i j : ι
e f : α
M : Matroid α
he : e ∈ M.E
F : Set α
hF : M.Flat F
heF : ¬{e} ⊆ F
⊢ M.ModularPair {e} F

State before:
case neg
ι : Type u_1
α : Type u_2
M✝ : Matroid α
B I J X X' Y Y' F✝ : Set α
Xs Ys Js Is : ι → Set α
i j : ι
e f : α
M : Matroid α
he : e ∈ M.E
F : Set α
hF : M.Flat F
heF : ¬{e} ⊆ F
⊢ M.ModularPair {e} F
Tactic:
rw [singleton_subset_iff, ← hF.cl] at heF
State after:
case neg
ι : Type u_1
α : Type u_2
M✝ : Matroid α
B I J X X' Y Y' F✝ : Set α
Xs Ys Js Is : ι → Set α
i j : ι
e f : α
M : Matroid α
he : e ∈ M.E
F : Set α
hF : M.Flat F
heF : e ∉ M.cl F
⊢ M.ModularPair {e} F

State before:
case neg
ι : Type u_1
α : Type u_2
M✝ : Matroid α
B I J X X' Y Y' F✝ : Set α
Xs Ys Js Is : ι → Set α
i j : ι
e f : α
M : Matroid α
he : e ∈ M.E
F : Set α
hF : M.Flat F
heF : e ∉ M.cl F
⊢ M.ModularPair {e} F
Tactic:
exact modularPair_singleton he hF.subset_ground heF
State after:
no goals

State before:
case pos
ι : Type u_1
α : Type u_2
M✝ : Matroid α
B I J X X' Y Y' F✝ : Set α
Xs Ys Js Is : ι → Set α
i j : ι
e f : α
M : Matroid α
he : e ∈ M.E
F : Set α
hF : M.Flat F
heF : {e} ⊆ F
⊢ M.ModularPair {e} F
Tactic:
apply modularPair_of_subset heF hF.subset_ground
State after:
no goals