internlm/Lean-Github huggingface dataset:
Theorem: Matroid.ModularCut.extIndep_aug_of_not_coloop
Proof state before:
α : Type u_1
M : Matroid α
I J B F₀ F F' X Y : Set α
e f : α
U✝ U : M.ModularCut
he : M.cl (M.E \ {e}) = M.E
hI : U.ExtIndep e I
hInmax :
  ¬(M.cl (I \ {e}) = M.cl (insert e M.E \ {e}) ∧ ((e ∈ I ↔ M.cl (insert e M.E \ {e}) ∈ U) → e ∉ insert e M.E) ∨
      (M.cl (I \ {e}) ⋖[M] M.cl (insert e M.E \ {e})) ∧ e ∈ I ∧ M.cl (insert e M.E \ {e}) ∈ U)
hBmax :
  M.cl (B \ {e}) = M.cl (insert e M.E \ {e}) ∧ ((e ∈ B ↔ M.cl (insert e M.E \ {e}) ∈ U) → e ∉ insert e M.E) ∨
    (M.cl (B \ {e}) ⋖[M] M.cl (insert e M.E \ {e})) ∧ e ∈ B ∧ M.cl (insert e M.E \ {e}) ∈ U
hcon : ∀ x ∈ B \ I, ¬U.ExtIndep e (insert x I)
hB : U.ExtIndep e B
hIeE : I \ {e} ⊆ M.E
hBeE : B \ {e} ⊆ M.E
hss : B \ {e} ⊆ (I ∪ B) \ {e}
hIBe : I ∪ B ⊆ insert e M.E
hIBe' : (I ∪ B) \ {e} ⊆ M.E
hImax : I ∈ maximals (fun x x_1 => x ⊆ x_1) {J | U.ExtIndep e J ∧ J ⊆ I ∪ B}
hrw : {J | U.ExtIndep e J} = {J | U.ExtIndep e J ∧ J ⊆ insert e M.E}
⊢ False
Tactic:
rw [U.maximal_extIndep_iff hIBe hI subset_union_left] at hImax
Proof state after:
α : Type u_1
M : Matroid α
I J B F₀ F F' X Y : Set α
e f : α
U✝ U : M.ModularCut
he : M.cl (M.E \ {e}) = M.E
hI : U.ExtIndep e I
hInmax :
  ¬(M.cl (I \ {e}) = M.cl (insert e M.E \ {e}) ∧ ((e ∈ I ↔ M.cl (insert e M.E \ {e}) ∈ U) → e ∉ insert e M.E) ∨
      (M.cl (I \ {e}) ⋖[M] M.cl (insert e M.E \ {e})) ∧ e ∈ I ∧ M.cl (insert e M.E \ {e}) ∈ U)
hBmax :
  M.cl (B \ {e}) = M.cl (insert e M.E \ {e}) ∧ ((e ∈ B ↔ M.cl (insert e M.E \ {e}) ∈ U) → e ∉ insert e M.E) ∨
    (M.cl (B \ {e}) ⋖[M] M.cl (insert e M.E \ {e})) ∧ e ∈ B ∧ M.cl (insert e M.E \ {e}) ∈ U
hcon : ∀ x ∈ B \ I, ¬U.ExtIndep e (insert x I)
hB : U.ExtIndep e B
hIeE : I \ {e} ⊆ M.E
hBeE : B \ {e} ⊆ M.E
hss : B \ {e} ⊆ (I ∪ B) \ {e}
hIBe : I ∪ B ⊆ insert e M.E
hIBe' : (I ∪ B) \ {e} ⊆ M.E
hImax :
  M.cl (I \ {e}) = M.cl ((I ∪ B) \ {e}) ∧ ((e ∈ I ↔ M.cl ((I ∪ B) \ {e}) ∈ U) → e ∉ I ∪ B) ∨
    (M.cl (I \ {e}) ⋖[M] M.cl ((I ∪ B) \ {e})) ∧ e ∈ I ∧ M.cl ((I ∪ B) \ {e}) ∈ U
hrw : {J | U.ExtIndep e J} = {J | U.ExtIndep e J ∧ J ⊆ insert e M.E}
⊢ False
Theorem: Matroid.ModularCut.extIndep_aug_of_not_coloop
Proof state before:
case inr
α : Type u_1
M : Matroid α
I J B F₀ F F' X Y : Set α
e f : α
U✝ U : M.ModularCut
he : M.cl (M.E \ {e}) = M.E
hI : U.ExtIndep e I
hInmax :
  ¬(M.cl (I \ {e}) = M.cl (insert e M.E \ {e}) ∧ ((e ∈ I ↔ M.cl (insert e M.E \ {e}) ∈ U) → e ∉ insert e M.E) ∨
      (M.cl (I \ {e}) ⋖[M] M.cl (insert e M.E \ {e})) ∧ e ∈ I ∧ M.cl (insert e M.E \ {e}) ∈ U)
hBmax :
  M.cl (B \ {e}) = M.cl (insert e M.E \ {e}) ∧ ((e ∈ B ↔ M.cl (insert e M.E \ {e}) ∈ U) → e ∉ insert e M.E) ∨
    (M.cl (B \ {e}) ⋖[M] M.cl (insert e M.E \ {e})) ∧ e ∈ B ∧ M.cl (insert e M.E \ {e}) ∈ U
hcon : ∀ x ∈ B \ I, ¬U.ExtIndep e (insert x I)
hB : U.ExtIndep e B
hIeE : I \ {e} ⊆ M.E
hBeE : B \ {e} ⊆ M.E
hss : B \ {e} ⊆ (I ∪ B) \ {e}
hIBe : I ∪ B ⊆ insert e M.E
hIBe' : (I ∪ B) \ {e} ⊆ M.E
hImax :
  M.cl (I \ {e}) = M.cl ((I ∪ B) \ {e}) ∧ ((e ∈ I ↔ M.cl ((I ∪ B) \ {e}) ∈ U) → e ∉ I ∪ B) ∨
    (M.cl (I \ {e}) ⋖[M] M.cl ((I ∪ B) \ {e})) ∧ e ∈ I ∧ M.cl ((I ∪ B) \ {e}) ∈ U
hrw : {J | U.ExtIndep e J} = {J | U.ExtIndep e J ∧ J ⊆ insert e M.E}
hU : M.E ∈ U
⊢ False
Tactic:
simp only [mem_singleton_iff, insert_diff_of_mem, he, not_false_eq_true, diff_singleton_eq_self,
  cl_ground, hU, iff_true, mem_insert_iff, or_false, not_true_eq_false, and_false, imp_false,
  and_true, not_or, not_and, not_not, mem_union, and_self_left,
  ← spanning_iff_cl hBeE, ← spanning_iff_cl hIeE, ← hyperplane_iff_covBy] at hBmax hInmax
Proof state after:
case inr
α : Type u_1
M : Matroid α
I J B F₀ F F' X Y : Set α
e f : α
U✝ U : M.ModularCut
he : M.cl (M.E \ {e}) = M.E
hI : U.ExtIndep e I
hcon : ∀ x ∈ B \ I, ¬U.ExtIndep e (insert x I)
hB : U.ExtIndep e B
hIeE : I \ {e} ⊆ M.E
hBeE : B \ {e} ⊆ M.E
hss : B \ {e} ⊆ (I ∪ B) \ {e}
hIBe : I ∪ B ⊆ insert e M.E
hIBe' : (I ∪ B) \ {e} ⊆ M.E
hImax :
  M.cl (I \ {e}) = M.cl ((I ∪ B) \ {e}) ∧ ((e ∈ I ↔ M.cl ((I ∪ B) \ {e}) ∈ U) → e ∉ I ∪ B) ∨
    (M.cl (I \ {e}) ⋖[M] M.cl ((I ∪ B) \ {e})) ∧ e ∈ I ∧ M.cl ((I ∪ B) \ {e}) ∈ U
hrw : {J | U.ExtIndep e J} = {J | U.ExtIndep e J ∧ J ⊆ insert e M.E}
hU : M.E ∈ U
hBmax : M.Spanning (B \ {e}) ∧ (e ∈ B → False ∧ e ∉ M.E) ∨ M.Hyperplane (M.cl (B \ {e})) ∧ e ∈ B
hInmax : (M.Spanning (I \ {e}) → ¬(e ∈ I → False ∧ e ∉ M.E)) ∧ (M.Hyperplane (M.cl (I \ {e})) → e ∉ I)
⊢ False
Theorem: Matroid.ModularCut.extIndep_aug_of_not_coloop
Proof state before:
case inr
α : Type u_1
M : Matroid α
I J B F₀ F F' X Y : Set α
e f : α
U✝ U : M.ModularCut
he : M.cl (M.E \ {e}) = M.E
hI : U.ExtIndep e I
hcon : ∀ x ∈ B \ I, ¬U.ExtIndep e (insert x I)
hB : U.ExtIndep e B
hIeE : I \ {e} ⊆ M.E
hBeE : B \ {e} ⊆ M.E
hss : B \ {e} ⊆ (I ∪ B) \ {e}
hIBe : I ∪ B ⊆ insert e M.E
hIBe' : (I ∪ B) \ {e} ⊆ M.E
hImax :
  M.cl (I \ {e}) = M.cl ((I ∪ B) \ {e}) ∧ ((e ∈ I ↔ M.cl ((I ∪ B) \ {e}) ∈ U) → e ∉ I ∪ B) ∨
    (M.cl (I \ {e}) ⋖[M] M.cl ((I ∪ B) \ {e})) ∧ e ∈ I ∧ M.cl ((I ∪ B) \ {e}) ∈ U
hrw : {J | U.ExtIndep e J} = {J | U.ExtIndep e J ∧ J ⊆ insert e M.E}
hU : M.E ∈ U
hBmax : M.Spanning (B \ {e}) ∧ (e ∈ B → False ∧ e ∉ M.E) ∨ M.Hyperplane (M.cl (B \ {e})) ∧ e ∈ B
hInmax : (M.Spanning (I \ {e}) → ¬(e ∈ I → False ∧ e ∉ M.E)) ∧ (M.Hyperplane (M.cl (I \ {e})) → e ∉ I)
⊢ False
Tactic:
obtain (hsp | hsp) := em <| M.Spanning ((I ∪ B) \ {e})
Proof state after:
case inr.inl
α : Type u_1
M : Matroid α
I J B F₀ F F' X Y : Set α
e f : α
U✝ U : M.ModularCut
he : M.cl (M.E \ {e}) = M.E
hI : U.ExtIndep e I
hcon : ∀ x ∈ B \ I, ¬U.ExtIndep e (insert x I)
hB : U.ExtIndep e B
hIeE : I \ {e} ⊆ M.E
hBeE : B \ {e} ⊆ M.E
hss : B \ {e} ⊆ (I ∪ B) \ {e}
hIBe : I ∪ B ⊆ insert e M.E
hIBe' : (I ∪ B) \ {e} ⊆ M.E
hImax :
  M.cl (I \ {e}) = M.cl ((I ∪ B) \ {e}) ∧ ((e ∈ I ↔ M.cl ((I ∪ B) \ {e}) ∈ U) → e ∉ I ∪ B) ∨
    (M.cl (I \ {e}) ⋖[M] M.cl ((I ∪ B) \ {e})) ∧ e ∈ I ∧ M.cl ((I ∪ B) \ {e}) ∈ U
hrw : {J | U.ExtIndep e J} = {J | U.ExtIndep e J ∧ J ⊆ insert e M.E}
hU : M.E ∈ U
hBmax : M.Spanning (B \ {e}) ∧ (e ∈ B → False ∧ e ∉ M.E) ∨ M.Hyperplane (M.cl (B \ {e})) ∧ e ∈ B
hInmax : (M.Spanning (I \ {e}) → ¬(e ∈ I → False ∧ e ∉ M.E)) ∧ (M.Hyperplane (M.cl (I \ {e})) → e ∉ I)
hsp : M.Spanning ((I ∪ B) \ {e})
⊢ False

case inr.inr
α : Type u_1
M : Matroid α
I J B F₀ F F' X Y : Set α
e f : α
U✝ U : M.ModularCut
he : M.cl (M.E \ {e}) = M.E
hI : U.ExtIndep e I
hcon : ∀ x ∈ B \ I, ¬U.ExtIndep e (insert x I)
hB : U.ExtIndep e B
hIeE : I \ {e} ⊆ M.E
hBeE : B \ {e} ⊆ M.E
hss : B \ {e} ⊆ (I ∪ B) \ {e}
hIBe : I ∪ B ⊆ insert e M.E
hIBe' : (I ∪ B) \ {e} ⊆ M.E
hImax :
  M.cl (I \ {e}) = M.cl ((I ∪ B) \ {e}) ∧ ((e ∈ I ↔ M.cl ((I ∪ B) \ {e}) ∈ U) → e ∉ I ∪ B) ∨
    (M.cl (I \ {e}) ⋖[M] M.cl ((I ∪ B) \ {e})) ∧ e ∈ I ∧ M.cl ((I ∪ B) \ {e}) ∈ U
hrw : {J | U.ExtIndep e J} = {J | U.ExtIndep e J ∧ J ⊆ insert e M.E}
hU : M.E ∈ U
hBmax : M.Spanning (B \ {e}) ∧ (e ∈ B → False ∧ e ∉ M.E) ∨ M.Hyperplane (M.cl (B \ {e})) ∧ e ∈ B
hInmax : (M.Spanning (I \ {e}) → ¬(e ∈ I → False ∧ e ∉ M.E)) ∧ (M.Hyperplane (M.cl (I \ {e})) → e ∉ I)
hsp : ¬M.Spanning ((I ∪ B) \ {e})
⊢ False
Theorem: Matroid.ModularCut.extIndep_aug_of_not_coloop
Proof state before:
case inr.inr
α : Type u_1
M : Matroid α
I J B F₀ F F' X Y : Set α
e f : α
U✝ U : M.ModularCut
he : M.cl (M.E \ {e}) = M.E
hI : U.ExtIndep e I
hcon : ∀ x ∈ B \ I, ¬U.ExtIndep e (insert x I)
hB : U.ExtIndep e B
hIeE : I \ {e} ⊆ M.E
hBeE : B \ {e} ⊆ M.E
hss : B \ {e} ⊆ (I ∪ B) \ {e}
hIBe : I ∪ B ⊆ insert e M.E
hIBe' : (I ∪ B) \ {e} ⊆ M.E
hImax :
  M.cl (I \ {e}) = M.cl ((I ∪ B) \ {e}) ∧ ((e ∈ I ↔ M.cl ((I ∪ B) \ {e}) ∈ U) → e ∉ I ∪ B) ∨
    (M.cl (I \ {e}) ⋖[M] M.cl ((I ∪ B) \ {e})) ∧ e ∈ I ∧ M.cl ((I ∪ B) \ {e}) ∈ U
hrw : {J | U.ExtIndep e J} = {J | U.ExtIndep e J ∧ J ⊆ insert e M.E}
hU : M.E ∈ U
hBmax : M.Spanning (B \ {e}) ∧ (e ∈ B → False ∧ e ∉ M.E) ∨ M.Hyperplane (M.cl (B \ {e})) ∧ e ∈ B
hInmax : (M.Spanning (I \ {e}) → ¬(e ∈ I → False ∧ e ∉ M.E)) ∧ (M.Hyperplane (M.cl (I \ {e})) → e ∉ I)
hsp : ¬M.Spanning ((I ∪ B) \ {e})
⊢ False
Tactic:
obtain (⟨hBsp, -⟩ | ⟨hBhp, heB⟩) := hBmax
Proof state after:
case inr.inr.inl.intro
α : Type u_1
M : Matroid α
I J B F₀ F F' X Y : Set α
e f : α
U✝ U : M.ModularCut
he : M.cl (M.E \ {e}) = M.E
hI : U.ExtIndep e I
hcon : ∀ x ∈ B \ I, ¬U.ExtIndep e (insert x I)
hB : U.ExtIndep e B
hIeE : I \ {e} ⊆ M.E
hBeE : B \ {e} ⊆ M.E
hss : B \ {e} ⊆ (I ∪ B) \ {e}
hIBe : I ∪ B ⊆ insert e M.E
hIBe' : (I ∪ B) \ {e} ⊆ M.E
hImax :
  M.cl (I \ {e}) = M.cl ((I ∪ B) \ {e}) ∧ ((e ∈ I ↔ M.cl ((I ∪ B) \ {e}) ∈ U) → e ∉ I ∪ B) ∨
    (M.cl (I \ {e}) ⋖[M] M.cl ((I ∪ B) \ {e})) ∧ e ∈ I ∧ M.cl ((I ∪ B) \ {e}) ∈ U
hrw : {J | U.ExtIndep e J} = {J | U.ExtIndep e J ∧ J ⊆ insert e M.E}
hU : M.E ∈ U
hInmax : (M.Spanning (I \ {e}) → ¬(e ∈ I → False ∧ e ∉ M.E)) ∧ (M.Hyperplane (M.cl (I \ {e})) → e ∉ I)
hsp : ¬M.Spanning ((I ∪ B) \ {e})
hBsp : M.Spanning (B \ {e})
⊢ False

case inr.inr.inr.intro
α : Type u_1
M : Matroid α
I J B F₀ F F' X Y : Set α
e f : α
U✝ U : M.ModularCut
he : M.cl (M.E \ {e}) = M.E
hI : U.ExtIndep e I
hcon : ∀ x ∈ B \ I, ¬U.ExtIndep e (insert x I)
hB : U.ExtIndep e B
hIeE : I \ {e} ⊆ M.E
hBeE : B \ {e} ⊆ M.E
hss : B \ {e} ⊆ (I ∪ B) \ {e}
hIBe : I ∪ B ⊆ insert e M.E
hIBe' : (I ∪ B) \ {e} ⊆ M.E
hImax :
  M.cl (I \ {e}) = M.cl ((I ∪ B) \ {e}) ∧ ((e ∈ I ↔ M.cl ((I ∪ B) \ {e}) ∈ U) → e ∉ I ∪ B) ∨
    (M.cl (I \ {e}) ⋖[M] M.cl ((I ∪ B) \ {e})) ∧ e ∈ I ∧ M.cl ((I ∪ B) \ {e}) ∈ U
hrw : {J | U.ExtIndep e J} = {J | U.ExtIndep e J ∧ J ⊆ insert e M.E}
hU : M.E ∈ U
hInmax : (M.Spanning (I \ {e}) → ¬(e ∈ I → False ∧ e ∉ M.E)) ∧ (M.Hyperplane (M.cl (I \ {e})) → e ∉ I)
hsp : ¬M.Spanning ((I ∪ B) \ {e})
hBhp : M.Hyperplane (M.cl (B \ {e}))
heB : e ∈ B
⊢ False
Theorem: Matroid.ModularCut.extIndep_aug_of_not_coloop
Proof state before:
case inr.inr.inr.intro
α : Type u_1
M : Matroid α
I J B F₀ F F' X Y : Set α
e f : α
U✝ U : M.ModularCut
he : M.cl (M.E \ {e}) = M.E
hI : U.ExtIndep e I
hcon : ∀ x ∈ B \ I, ¬U.ExtIndep e (insert x I)
hB : U.ExtIndep e B
hIeE : I \ {e} ⊆ M.E
hBeE : B \ {e} ⊆ M.E
hss : B \ {e} ⊆ (I ∪ B) \ {e}
hIBe : I ∪ B ⊆ insert e M.E
hIBe' : (I ∪ B) \ {e} ⊆ M.E
hImax :
  M.cl (I \ {e}) = M.cl ((I ∪ B) \ {e}) ∧ ((e ∈ I ↔ M.cl ((I ∪ B) \ {e}) ∈ U) → e ∉ I ∪ B) ∨
    (M.cl (I \ {e}) ⋖[M] M.cl ((I ∪ B) \ {e})) ∧ e ∈ I ∧ M.cl ((I ∪ B) \ {e}) ∈ U
hrw : {J | U.ExtIndep e J} = {J | U.ExtIndep e J ∧ J ⊆ insert e M.E}
hU : M.E ∈ U
hInmax : (M.Spanning (I \ {e}) → ¬(e ∈ I → False ∧ e ∉ M.E)) ∧ (M.Hyperplane (M.cl (I \ {e})) → e ∉ I)
hsp : ¬M.Spanning ((I ∪ B) \ {e})
hBhp : M.Hyperplane (M.cl (B \ {e}))
heB : e ∈ B
⊢ False
Tactic:
have hclcl : M.cl (B \ {e}) = M.cl ((I ∪ B) \ {e}) := by
  refine by_contra fun hne ↦ hsp <| ?_
  rw [← cl_spanning_iff hIBe']
  have hssu := (M.cl_subset_cl hss).ssubset_of_ne hne
  exact hBhp.spanning_of_ssuperset hssu
Proof state after:
case inr.inr.inr.intro
α : Type u_1
M : Matroid α
I J B F₀ F F' X Y : Set α
e f : α
U✝ U : M.ModularCut
he : M.cl (M.E \ {e}) = M.E
hI : U.ExtIndep e I
hcon : ∀ x ∈ B \ I, ¬U.ExtIndep e (insert x I)
hB : U.ExtIndep e B
hIeE : I \ {e} ⊆ M.E
hBeE : B \ {e} ⊆ M.E
hss : B \ {e} ⊆ (I ∪ B) \ {e}
hIBe : I ∪ B ⊆ insert e M.E
hIBe' : (I ∪ B) \ {e} ⊆ M.E
hImax :
  M.cl (I \ {e}) = M.cl ((I ∪ B) \ {e}) ∧ ((e ∈ I ↔ M.cl ((I ∪ B) \ {e}) ∈ U) → e ∉ I ∪ B) ∨
    (M.cl (I \ {e}) ⋖[M] M.cl ((I ∪ B) \ {e})) ∧ e ∈ I ∧ M.cl ((I ∪ B) \ {e}) ∈ U
hrw : {J | U.ExtIndep e J} = {J | U.ExtIndep e J ∧ J ⊆ insert e M.E}
hU : M.E ∈ U
hInmax : (M.Spanning (I \ {e}) → ¬(e ∈ I → False ∧ e ∉ M.E)) ∧ (M.Hyperplane (M.cl (I \ {e})) → e ∉ I)
hsp : ¬M.Spanning ((I ∪ B) \ {e})
hBhp : M.Hyperplane (M.cl (B \ {e}))
heB : e ∈ B
hclcl : M.cl (B \ {e}) = M.cl ((I ∪ B) \ {e})
⊢ False
Theorem: Matroid.ModularCut.extIndep_aug_of_not_coloop
Proof state before:
case inr.inr.inr.intro
α : Type u_1
M : Matroid α
I J B F₀ F F' X Y : Set α
e f : α
U✝ U : M.ModularCut
he : M.cl (M.E \ {e}) = M.E
hI : U.ExtIndep e I
hcon : ∀ x ∈ B \ I, ¬U.ExtIndep e (insert x I)
hB : U.ExtIndep e B
hIeE : I \ {e} ⊆ M.E
hBeE : B \ {e} ⊆ M.E
hss : B \ {e} ⊆ (I ∪ B) \ {e}
hIBe : I ∪ B ⊆ insert e M.E
hIBe' : (I ∪ B) \ {e} ⊆ M.E
hImax :
  M.cl (I \ {e}) = M.cl ((I ∪ B) \ {e}) ∧ ((e ∈ I ↔ M.cl ((I ∪ B) \ {e}) ∈ U) → e ∉ I ∪ B) ∨
    (M.cl (I \ {e}) ⋖[M] M.cl ((I ∪ B) \ {e})) ∧ e ∈ I ∧ M.cl ((I ∪ B) \ {e}) ∈ U
hrw : {J | U.ExtIndep e J} = {J | U.ExtIndep e J ∧ J ⊆ insert e M.E}
hU : M.E ∈ U
hInmax : (M.Spanning (I \ {e}) → ¬(e ∈ I → False ∧ e ∉ M.E)) ∧ (M.Hyperplane (M.cl (I \ {e})) → e ∉ I)
hsp : ¬M.Spanning ((I ∪ B) \ {e})
hBhp : M.Hyperplane (M.cl (B \ {e}))
heB : e ∈ B
hclcl : M.cl (B \ {e}) = M.cl ((I ∪ B) \ {e})
⊢ False
Tactic:
rw [extIndep_iff_of_mem heB] at hB
Proof state after:
case inr.inr.inr.intro
α : Type u_1
M : Matroid α
I J B F₀ F F' X Y : Set α
e f : α
U✝ U : M.ModularCut
he : M.cl (M.E \ {e}) = M.E
hI : U.ExtIndep e I
hcon : ∀ x ∈ B \ I, ¬U.ExtIndep e (insert x I)
hB : M.Indep (B \ {e}) ∧ M.cl (B \ {e}) ∉ U
hIeE : I \ {e} ⊆ M.E
hBeE : B \ {e} ⊆ M.E
hss : B \ {e} ⊆ (I ∪ B) \ {e}
hIBe : I ∪ B ⊆ insert e M.E
hIBe' : (I ∪ B) \ {e} ⊆ M.E
hImax :
  M.cl (I \ {e}) = M.cl ((I ∪ B) \ {e}) ∧ ((e ∈ I ↔ M.cl ((I ∪ B) \ {e}) ∈ U) → e ∉ I ∪ B) ∨
    (M.cl (I \ {e}) ⋖[M] M.cl ((I ∪ B) \ {e})) ∧ e ∈ I ∧ M.cl ((I ∪ B) \ {e}) ∈ U
hrw : {J | U.ExtIndep e J} = {J | U.ExtIndep e J ∧ J ⊆ insert e M.E}
hU : M.E ∈ U
hInmax : (M.Spanning (I \ {e}) → ¬(e ∈ I → False ∧ e ∉ M.E)) ∧ (M.Hyperplane (M.cl (I \ {e})) → e ∉ I)
hsp : ¬M.Spanning ((I ∪ B) \ {e})
hBhp : M.Hyperplane (M.cl (B \ {e}))
heB : e ∈ B
hclcl : M.cl (B \ {e}) = M.cl ((I ∪ B) \ {e})
⊢ False
Theorem: Matroid.ModularCut.extIndep_aug_of_not_coloop
Proof state before:
case inr.inr.inr.intro
α : Type u_1
M : Matroid α
I J B F₀ F F' X Y : Set α
e f : α
U✝ U : M.ModularCut
he : M.cl (M.E \ {e}) = M.E
hI : U.ExtIndep e I
hcon : ∀ x ∈ B \ I, ¬U.ExtIndep e (insert x I)
hB : M.Indep (B \ {e}) ∧ M.cl (B \ {e}) ∉ U
hIeE : I \ {e} ⊆ M.E
hBeE : B \ {e} ⊆ M.E
hss : B \ {e} ⊆ (I ∪ B) \ {e}
hIBe : I ∪ B ⊆ insert e M.E
hIBe' : (I ∪ B) \ {e} ⊆ M.E
hImax :
  M.cl (I \ {e}) = M.cl ((I ∪ B) \ {e}) ∧ ((e ∈ I ↔ M.cl ((I ∪ B) \ {e}) ∈ U) → e ∉ I ∪ B) ∨
    (M.cl (I \ {e}) ⋖[M] M.cl ((I ∪ B) \ {e})) ∧ e ∈ I ∧ M.cl ((I ∪ B) \ {e}) ∈ U
hrw : {J | U.ExtIndep e J} = {J | U.ExtIndep e J ∧ J ⊆ insert e M.E}
hU : M.E ∈ U
hInmax : (M.Spanning (I \ {e}) → ¬(e ∈ I → False ∧ e ∉ M.E)) ∧ (M.Hyperplane (M.cl (I \ {e})) → e ∉ I)
hsp : ¬M.Spanning ((I ∪ B) \ {e})
hBhp : M.Hyperplane (M.cl (B \ {e}))
heB : e ∈ B
hclcl : M.cl (B \ {e}) = M.cl ((I ∪ B) \ {e})
⊢ False
Tactic:
replace hImax := show M.cl (I \ {e}) = M.cl (B \ {e}) ∧ e ∈ I by
  simpa [heB, ← hclcl, hB.2] using hImax
Proof state after:
case inr.inr.inr.intro
α : Type u_1
M : Matroid α
I J B F₀ F F' X Y : Set α
e f : α
U✝ U : M.ModularCut
he : M.cl (M.E \ {e}) = M.E
hI : U.ExtIndep e I
hcon : ∀ x ∈ B \ I, ¬U.ExtIndep e (insert x I)
hB : M.Indep (B \ {e}) ∧ M.cl (B \ {e}) ∉ U
hIeE : I \ {e} ⊆ M.E
hBeE : B \ {e} ⊆ M.E
hss : B \ {e} ⊆ (I ∪ B) \ {e}
hIBe : I ∪ B ⊆ insert e M.E
hIBe' : (I ∪ B) \ {e} ⊆ M.E
hrw : {J | U.ExtIndep e J} = {J | U.ExtIndep e J ∧ J ⊆ insert e M.E}
hU : M.E ∈ U
hInmax : (M.Spanning (I \ {e}) → ¬(e ∈ I → False ∧ e ∉ M.E)) ∧ (M.Hyperplane (M.cl (I \ {e})) → e ∉ I)
hsp : ¬M.Spanning ((I ∪ B) \ {e})
hBhp : M.Hyperplane (M.cl (B \ {e}))
heB : e ∈ B
hclcl : M.cl (B \ {e}) = M.cl ((I ∪ B) \ {e})
hImax : M.cl (I \ {e}) = M.cl (B \ {e}) ∧ e ∈ I
⊢ False
Theorem: Matroid.ModularCut.extIndep_aug_of_not_coloop
Proof state before:
case inr.inr.inr.intro
α : Type u_1
M : Matroid α
I J B F₀ F F' X Y : Set α
e f : α
U✝ U : M.ModularCut
he : M.cl (M.E \ {e}) = M.E
hI : U.ExtIndep e I
hcon : ∀ x ∈ B \ I, ¬U.ExtIndep e (insert x I)
hB : M.Indep (B \ {e}) ∧ M.cl (B \ {e}) ∉ U
hIeE : I \ {e} ⊆ M.E
hBeE : B \ {e} ⊆ M.E
hss : B \ {e} ⊆ (I ∪ B) \ {e}
hIBe : I ∪ B ⊆ insert e M.E
hIBe' : (I ∪ B) \ {e} ⊆ M.E
hrw : {J | U.ExtIndep e J} = {J | U.ExtIndep e J ∧ J ⊆ insert e M.E}
hU : M.E ∈ U
hInmax : (M.Spanning (I \ {e}) → ¬(e ∈ I → False ∧ e ∉ M.E)) ∧ (M.Hyperplane (M.cl (I \ {e})) → e ∉ I)
hsp : ¬M.Spanning ((I ∪ B) \ {e})
hBhp : M.Hyperplane (M.cl (B \ {e}))
heB : e ∈ B
hclcl : M.cl (B \ {e}) = M.cl ((I ∪ B) \ {e})
hImax : M.cl (I \ {e}) = M.cl (B \ {e}) ∧ e ∈ I
⊢ False
Tactic:
replace hInmax := show ¬M.Hyperplane (M.cl (I \ {e})) by simpa [hImax.2] using hInmax
Proof state after:
case inr.inr.inr.intro
α : Type u_1
M : Matroid α
I J B F₀ F F' X Y : Set α
e f : α
U✝ U : M.ModularCut
he : M.cl (M.E \ {e}) = M.E
hI : U.ExtIndep e I
hcon : ∀ x ∈ B \ I, ¬U.ExtIndep e (insert x I)
hB : M.Indep (B \ {e}) ∧ M.cl (B \ {e}) ∉ U
hIeE : I \ {e} ⊆ M.E
hBeE : B \ {e} ⊆ M.E
hss : B \ {e} ⊆ (I ∪ B) \ {e}
hIBe : I ∪ B ⊆ insert e M.E
hIBe' : (I ∪ B) \ {e} ⊆ M.E
hrw : {J | U.ExtIndep e J} = {J | U.ExtIndep e J ∧ J ⊆ insert e M.E}
hU : M.E ∈ U
hsp : ¬M.Spanning ((I ∪ B) \ {e})
hBhp : M.Hyperplane (M.cl (B \ {e}))
heB : e ∈ B
hclcl : M.cl (B \ {e}) = M.cl ((I ∪ B) \ {e})
hImax : M.cl (I \ {e}) = M.cl (B \ {e}) ∧ e ∈ I
hInmax : ¬M.Hyperplane (M.cl (I \ {e}))
⊢ False
Theorem: Matroid.ModularCut.extIndep_aug_of_not_coloop
Proof state before:
case inr.inr.inr.intro
α : Type u_1
M : Matroid α
I J B F₀ F F' X Y : Set α
e f : α
U✝ U : M.ModularCut
he : M.cl (M.E \ {e}) = M.E
hI : U.ExtIndep e I
hcon : ∀ x ∈ B \ I, ¬U.ExtIndep e (insert x I)
hB : M.Indep (B \ {e}) ∧ M.cl (B \ {e}) ∉ U
hIeE : I \ {e} ⊆ M.E
hBeE : B \ {e} ⊆ M.E
hss : B \ {e} ⊆ (I ∪ B) \ {e}
hIBe : I ∪ B ⊆ insert e M.E
hIBe' : (I ∪ B) \ {e} ⊆ M.E
hrw : {J | U.ExtIndep e J} = {J | U.ExtIndep e J ∧ J ⊆ insert e M.E}
hU : M.E ∈ U
hsp : ¬M.Spanning ((I ∪ B) \ {e})
hBhp : M.Hyperplane (M.cl (B \ {e}))
heB : e ∈ B
hclcl : M.cl (B \ {e}) = M.cl ((I ∪ B) \ {e})
hImax : M.cl (I \ {e}) = M.cl (B \ {e}) ∧ e ∈ I
hInmax : ¬M.Hyperplane (M.cl (I \ {e}))
⊢ False
Tactic:
exact hInmax <| (hImax.1.symm ▸ hBhp)
Proof state after:
no goals
Theorem: Matroid.ModularCut.extIndep_aug_of_not_coloop
Proof state before:
α : Type u_1
M : Matroid α
I J B F₀ F F' X Y : Set α
e f : α
U✝ U : M.ModularCut
he : M.cl (M.E \ {e}) = M.E
hI : U.ExtIndep e I
hInmax : I ∉ maximals (fun x x_1 => x ⊆ x_1) {I | U.ExtIndep e I}
hBmax : B ∈ maximals (fun x x_1 => x ⊆ x_1) {I | U.ExtIndep e I}
hcon : ∀ x ∈ B \ I, ¬U.ExtIndep e (insert x I)
hB : U.ExtIndep e B
hIeE : I \ {e} ⊆ M.E
hBeE : B \ {e} ⊆ M.E
hss : B \ {e} ⊆ (I ∪ B) \ {e}
hIBe : I ∪ B ⊆ insert e M.E
⊢ (I ∪ B) \ {e} ⊆ M.E
Tactic:
rwa [diff_singleton_subset_iff]
Proof state after:
no goals
Theorem: Matroid.ModularCut.extIndep_aug_of_not_coloop
Proof state before:
α : Type u_1
M : Matroid α
I J B F₀ F F' X Y : Set α
e f : α
U✝ U : M.ModularCut
he : M.cl (M.E \ {e}) = M.E
hI : U.ExtIndep e I
hInmax : I ∉ maximals (fun x x_1 => x ⊆ x_1) {I | U.ExtIndep e I}
hBmax : B ∈ maximals (fun x x_1 => x ⊆ x_1) {I | U.ExtIndep e I}
hcon : ∀ x ∈ B \ I, ¬U.ExtIndep e (insert x I)
hB : U.ExtIndep e B
hIeE : I \ {e} ⊆ M.E
hBeE : B \ {e} ⊆ M.E
hss : B \ {e} ⊆ (I ∪ B) \ {e}
hIBe : I ∪ B ⊆ insert e M.E
hIBe' : (I ∪ B) \ {e} ⊆ M.E
⊢ I ∈ maximals (fun x x_1 => x ⊆ x_1) {J | U.ExtIndep e J ∧ J ⊆ I ∪ B}
Tactic:
rw [mem_maximals_iff_forall_insert (fun _ _ ht hst ↦ ⟨ht.1.subset hst, hst.trans ht.2⟩),
  and_iff_right hI, and_iff_right subset_union_left]
Proof state after:
α : Type u_1
M : Matroid α
I J B F₀ F F' X Y : Set α
e f : α
U✝ U : M.ModularCut
he : M.cl (M.E \ {e}) = M.E
hI : U.ExtIndep e I
hInmax : I ∉ maximals (fun x x_1 => x ⊆ x_1) {I | U.ExtIndep e I}
hBmax : B ∈ maximals (fun x x_1 => x ⊆ x_1) {I | U.ExtIndep e I}
hcon : ∀ x ∈ B \ I, ¬U.ExtIndep e (insert x I)
hB : U.ExtIndep e B
hIeE : I \ {e} ⊆ M.E
hBeE : B \ {e} ⊆ M.E
hss : B \ {e} ⊆ (I ∪ B) \ {e}
hIBe : I ∪ B ⊆ insert e M.E
hIBe' : (I ∪ B) \ {e} ⊆ M.E
⊢ ∀ x ∉ I, ¬(U.ExtIndep e (insert x I) ∧ insert x I ⊆ I ∪ B)
Theorem: Matroid.ModularCut.extIndep_aug_of_not_coloop
Proof state before:
α : Type u_1
M : Matroid α
I J B F₀ F F' X Y : Set α
e f : α
U✝ U : M.ModularCut
he : M.cl (M.E \ {e}) = M.E
hI : U.ExtIndep e I
hInmax : I ∉ maximals (fun x x_1 => x ⊆ x_1) {I | U.ExtIndep e I}
hBmax : B ∈ maximals (fun x x_1 => x ⊆ x_1) {I | U.ExtIndep e I}
hcon : ∀ x ∈ B \ I, ¬U.ExtIndep e (insert x I)
hB : U.ExtIndep e B
hIeE : I \ {e} ⊆ M.E
hBeE : B \ {e} ⊆ M.E
hss : B \ {e} ⊆ (I ∪ B) \ {e}
hIBe : I ∪ B ⊆ insert e M.E
hIBe' : (I ∪ B) \ {e} ⊆ M.E
⊢ ∀ x ∉ I, ¬(U.ExtIndep e (insert x I) ∧ insert x I ⊆ I ∪ B)
Tactic:
intro x hxI h'
Proof state after:
α : Type u_1
M : Matroid α
I J B F₀ F F' X Y : Set α
e f : α
U✝ U : M.ModularCut
he : M.cl (M.E \ {e}) = M.E
hI : U.ExtIndep e I
hInmax : I ∉ maximals (fun x x_1 => x ⊆ x_1) {I | U.ExtIndep e I}
hBmax : B ∈ maximals (fun x x_1 => x ⊆ x_1) {I | U.ExtIndep e I}
hcon : ∀ x ∈ B \ I, ¬U.ExtIndep e (insert x I)
hB : U.ExtIndep e B
hIeE : I \ {e} ⊆ M.E
hBeE : B \ {e} ⊆ M.E
hss : B \ {e} ⊆ (I ∪ B) \ {e}
hIBe : I ∪ B ⊆ insert e M.E
hIBe' : (I ∪ B) \ {e} ⊆ M.E
x : α
hxI : x ∉ I
h' : U.ExtIndep e (insert x I) ∧ insert x I ⊆ I ∪ B
⊢ False
Theorem: Matroid.ModularCut.extIndep_aug_of_not_coloop
Proof state before:
α : Type u_1
M : Matroid α
I J B F₀ F F' X Y : Set α
e f : α
U✝ U : M.ModularCut
he : M.cl (M.E \ {e}) = M.E
hI : U.ExtIndep e I
hInmax : I ∉ maximals (fun x x_1 => x ⊆ x_1) {I | U.ExtIndep e I}
hBmax : B ∈ maximals (fun x x_1 => x ⊆ x_1) {I | U.ExtIndep e I}
hcon : ∀ x ∈ B \ I, ¬U.ExtIndep e (insert x I)
hB : U.ExtIndep e B
hIeE : I \ {e} ⊆ M.E
hBeE : B \ {e} ⊆ M.E
hss : B \ {e} ⊆ (I ∪ B) \ {e}
hIBe : I ∪ B ⊆ insert e M.E
hIBe' : (I ∪ B) \ {e} ⊆ M.E
x : α
hxI : x ∉ I
h' : U.ExtIndep e (insert x I) ∧ insert x I ⊆ I ∪ B
⊢ False
Tactic:
rw [insert_subset_iff, mem_union, or_iff_right hxI] at h'
Proof state after:
α : Type u_1
M : Matroid α
I J B F₀ F F' X Y : Set α
e f : α
U✝ U : M.ModularCut
he : M.cl (M.E \ {e}) = M.E
hI : U.ExtIndep e I
hInmax : I ∉ maximals (fun x x_1 => x ⊆ x_1) {I | U.ExtIndep e I}
hBmax : B ∈ maximals (fun x x_1 => x ⊆ x_1) {I | U.ExtIndep e I}
hcon : ∀ x ∈ B \ I, ¬U.ExtIndep e (insert x I)
hB : U.ExtIndep e B
hIeE : I \ {e} ⊆ M.E
hBeE : B \ {e} ⊆ M.E
hss : B \ {e} ⊆ (I ∪ B) \ {e}
hIBe : I ∪ B ⊆ insert e M.E
hIBe' : (I ∪ B) \ {e} ⊆ M.E
x : α
hxI : x ∉ I
h' : U.ExtIndep e (insert x I) ∧ x ∈ B ∧ I ⊆ I ∪ B
⊢ False
Theorem: Matroid.ModularCut.extIndep_aug_of_not_coloop
Proof state before:
α : Type u_1
M : Matroid α
I J B F₀ F F' X Y : Set α
e f : α
U✝ U : M.ModularCut
he : M.cl (M.E \ {e}) = M.E
hI : U.ExtIndep e I
hInmax : I ∉ maximals (fun x x_1 => x ⊆ x_1) {I | U.ExtIndep e I}
hBmax : B ∈ maximals (fun x x_1 => x ⊆ x_1) {I | U.ExtIndep e I}
hcon : ∀ x ∈ B \ I, ¬U.ExtIndep e (insert x I)
hB : U.ExtIndep e B
hIeE : I \ {e} ⊆ M.E
hBeE : B \ {e} ⊆ M.E
hss : B \ {e} ⊆ (I ∪ B) \ {e}
hIBe : I ∪ B ⊆ insert e M.E
hIBe' : (I ∪ B) \ {e} ⊆ M.E
x : α
hxI : x ∉ I
h' : U.ExtIndep e (insert x I) ∧ x ∈ B ∧ I ⊆ I ∪ B
⊢ False
Tactic:
exact hcon x ⟨h'.2.1, hxI⟩ h'.1
Proof state after:
no goals
Theorem: Matroid.ModularCut.extIndep_aug_of_not_coloop
Proof state before:
α : Type u_1
M : Matroid α
I J B F₀ F F' X Y : Set α
e f : α
U✝ U : M.ModularCut
he : M.cl (M.E \ {e}) = M.E
hI : U.ExtIndep e I
hInmax : I ∉ maximals (fun x x_1 => x ⊆ x_1) {I | U.ExtIndep e I}
hBmax : B ∈ maximals (fun x x_1 => x ⊆ x_1) {I | U.ExtIndep e I}
hcon : ∀ x ∈ B \ I, ¬U.ExtIndep e (insert x I)
hB : U.ExtIndep e B
hIeE : I \ {e} ⊆ M.E
hBeE : B \ {e} ⊆ M.E
hss : B \ {e} ⊆ (I ∪ B) \ {e}
hIBe : I ∪ B ⊆ insert e M.E
hIBe' : (I ∪ B) \ {e} ⊆ M.E
hImax : I ∈ maximals (fun x x_1 => x ⊆ x_1) {J | U.ExtIndep e J ∧ J ⊆ I ∪ B}
⊢ {J | U.ExtIndep e J} = {J | U.ExtIndep e J ∧ J ⊆ insert e M.E}
Tactic:
simp only [ext_iff, mem_setOf_eq, iff_self_and]
Proof state after:
α : Type u_1
M : Matroid α
I J B F₀ F F' X Y : Set α
e f : α
U✝ U : M.ModularCut
he : M.cl (M.E \ {e}) = M.E
hI : U.ExtIndep e I
hInmax : I ∉ maximals (fun x x_1 => x ⊆ x_1) {I | U.ExtIndep e I}
hBmax : B ∈ maximals (fun x x_1 => x ⊆ x_1) {I | U.ExtIndep e I}
hcon : ∀ x ∈ B \ I, ¬U.ExtIndep e (insert x I)
hB : U.ExtIndep e B
hIeE : I \ {e} ⊆ M.E
hBeE : B \ {e} ⊆ M.E
hss : B \ {e} ⊆ (I ∪ B) \ {e}
hIBe : I ∪ B ⊆ insert e M.E
hIBe' : (I ∪ B) \ {e} ⊆ M.E
hImax : I ∈ maximals (fun x x_1 => x ⊆ x_1) {J | U.ExtIndep e J ∧ J ⊆ I ∪ B}
⊢ ∀ (x : Set α), U.ExtIndep e x → x ⊆ insert e M.E