repo_url
stringclasses 13
values | commit
stringclasses 13
values | file_path
stringlengths 15
79
| full_name
stringlengths 2
121
| status
stringclasses 2
values | state_pp
stringlengths 7
13.4k
| state_message
stringclasses 2
values | state_goals
stringlengths 42
31.4k
| tactic
stringlengths 3
4.55k
⌀ | result_type
stringclasses 6
values | result
stringlengths 4
14.1k
| result_message
stringlengths 0
13.4k
⌀ |
|---|---|---|---|---|---|---|---|---|---|---|---|
https://github.com/apnelson1/Matroid | 510262f56c0025be19502ab3457a788e2f6d2d1c | Matroid/Constructions/Uniform.lean | Matroid.unif_eq_freeOn | Status.PROVED | α : Type u_1
M : Matroid α
E I B X C : Set α
k : ℕ∞
a b a' b' : ℕ
h : b ≤ a
⊢ unif a b = freeOn univ | None | [Goal(assumptions=[Declaration(ident='α', lean_type='Type u_1'), Declaration(ident='M', lean_type='Matroid α'), Declaration(ident='E', lean_type='Set α'), Declaration(ident='I', lean_type='Set α'), Declaration(ident='B', lean_type='Set α'), Declaration(ident='X', lean_type='Set α'), Declaration(ident='C', lean_type='Set α'), Declaration(ident='k', lean_type='ℕ∞'), Declaration(ident='a', lean_type='ℕ'), Declaration(ident='b', lean_type='ℕ'), Declaration(ident="a'", lean_type='ℕ'), Declaration(ident="b'", lean_type='ℕ'), Declaration(ident='h', lean_type='b ≤ a')], conclusion='unif a b = freeOn univ')] | simpa [eq_freeOn_iff] | <class 'lean_dojo.interaction.dojo.ProofFinished'> | ProofFinished(tactic_state_id=1, message='') | |
https://github.com/apnelson1/Matroid | 510262f56c0025be19502ab3457a788e2f6d2d1c | Matroid/Closure.lean | Matroid.not_spanning_iff_closure | Status.PROVED | 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
S T : Set α
hS : autoParam (S ⊆ M.E) _auto✝
⊢ ¬M.Spanning S ↔ M.closure S ⊂ M.E | None | [Goal(assumptions=[Declaration(ident='α', lean_type='Type u_1'), Declaration(ident='ι', lean_type='Type u_2'), Declaration(ident='M', lean_type='Matroid α'), Declaration(ident='F', lean_type='Set α'), Declaration(ident='I', lean_type='Set α'), Declaration(ident='J', lean_type='Set α'), Declaration(ident='X', lean_type='Set α'), Declaration(ident='Y', lean_type='Set α'), Declaration(ident='B', lean_type='Set α'), Declaration(ident='C', lean_type='Set α'), Declaration(ident='R', lean_type='Set α'), Declaration(ident='e', lean_type='α'), Declaration(ident='f', lean_type='α'), Declaration(ident='x', lean_type='α'), Declaration(ident='y', lean_type='α'), Declaration(ident='S', lean_type='Set α'), Declaration(ident='T', lean_type='Set α'), Declaration(ident='hS', lean_type='autoParam (S ⊆ M.E) _auto✝')], conclusion='¬M.Spanning S ↔ M.closure S ⊂ M.E')] | rw [spanning_iff_closure, ssubset_iff_subset_ne, iff_and_self,
iff_true_intro (M.closure_subset_ground _)] | <class 'lean_dojo.interaction.dojo.TacticState'> | TacticState(pp='α : Type u_1\nι : Type u_2\nM : Matroid α\nF I J X Y B C R : Set α\ne f x y : α\nS T : Set α\nhS : autoParam (S ⊆ M.E) _auto✝\n⊢ ¬M.closure S = M.E → True', id=1, message='') | 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
S T : Set α
hS : autoParam (S ⊆ M.E) _auto✝
⊢ ¬M.closure S = M.E → True |
https://github.com/apnelson1/Matroid | 510262f56c0025be19502ab3457a788e2f6d2d1c | Matroid/Closure.lean | Matroid.not_spanning_iff_closure | Status.PROVED | 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
S T : Set α
hS : autoParam (S ⊆ M.E) _auto✝
⊢ ¬M.closure S = M.E → True | [Goal(assumptions=[Declaration(ident='α', lean_type='Type u_1'), Declaration(ident='ι', lean_type='Type u_2'), Declaration(ident='M', lean_type='Matroid α'), Declaration(ident='F', lean_type='Set α'), Declaration(ident='I', lean_type='Set α'), Declaration(ident='J', lean_type='Set α'), Declaration(ident='X', lean_type='Set α'), Declaration(ident='Y', lean_type='Set α'), Declaration(ident='B', lean_type='Set α'), Declaration(ident='C', lean_type='Set α'), Declaration(ident='R', lean_type='Set α'), Declaration(ident='e', lean_type='α'), Declaration(ident='f', lean_type='α'), Declaration(ident='x', lean_type='α'), Declaration(ident='y', lean_type='α'), Declaration(ident='S', lean_type='Set α'), Declaration(ident='T', lean_type='Set α'), Declaration(ident='hS', lean_type='autoParam (S ⊆ M.E) _auto✝')], conclusion='¬M.closure S = M.E → True')] | exact fun _ ↦ trivial | <class 'lean_dojo.interaction.dojo.ProofFinished'> | ProofFinished(tactic_state_id=2, message='') | ||
https://github.com/apnelson1/Matroid | 510262f56c0025be19502ab3457a788e2f6d2d1c | Matroid/Minor/Basic.lean | Matroid.delete_inter_ground_eq | Status.PROVED | α : Type u_1
M✝ M' N : Matroid α
e f : α
I J R✝ B X Y Z K D✝ D₁ D₂ R : Set α
M : Matroid α
D : Set α
⊢ M ＼ (D ∩ M.E) = M ＼ D | None | [Goal(assumptions=[Declaration(ident='α', lean_type='Type u_1'), Declaration(ident='M✝', lean_type='Matroid α'), Declaration(ident="M'", lean_type='Matroid α'), Declaration(ident='N', lean_type='Matroid α'), Declaration(ident='e', lean_type='α'), Declaration(ident='f', lean_type='α'), Declaration(ident='I', lean_type='Set α'), Declaration(ident='J', lean_type='Set α'), Declaration(ident='R✝', lean_type='Set α'), Declaration(ident='B', lean_type='Set α'), Declaration(ident='X', lean_type='Set α'), Declaration(ident='Y', lean_type='Set α'), Declaration(ident='Z', lean_type='Set α'), Declaration(ident='K', lean_type='Set α'), Declaration(ident='D✝', lean_type='Set α'), Declaration(ident='D₁', lean_type='Set α'), Declaration(ident='D₂', lean_type='Set α'), Declaration(ident='R', lean_type='Set α'), Declaration(ident='M', lean_type='Matroid α'), Declaration(ident='D', lean_type='Set α')], conclusion='M ＼ (D ∩ M.E) = M ＼ D')] | rw [← restrict_compl, ← restrict_compl, diff_inter_self_eq_diff] | <class 'lean_dojo.interaction.dojo.ProofFinished'> | ProofFinished(tactic_state_id=1, message='') | |
https://github.com/apnelson1/Matroid | 510262f56c0025be19502ab3457a788e2f6d2d1c | Matroid/ForMathlib/SetPartition.lean | Partition.setOf_rel_eq | Status.FAILED | α : Type u_1
inst✝² : CompleteLattice α
s✝¹ x y z : α
s✝ t : Set α
a b : α
P : Partition s✝
r : α → α → Prop
inst✝¹ : IsSymm α r
inst✝ : IsTrans α r
s : Set α
ht : t ∈ P
hx : x ∈ t
⊢ {y | P.Rel x y} = t | None | [Goal(assumptions=[Declaration(ident='α', lean_type='Type u_1'), Declaration(ident='inst✝²', lean_type='CompleteLattice α'), Declaration(ident='s✝¹', lean_type='α'), Declaration(ident='x', lean_type='α'), Declaration(ident='y', lean_type='α'), Declaration(ident='z', lean_type='α'), Declaration(ident='s✝', lean_type='Set α'), Declaration(ident='t', lean_type='Set α'), Declaration(ident='a', lean_type='α'), Declaration(ident='b', lean_type='α'), Declaration(ident='P', lean_type='Partition s✝'), Declaration(ident='r', lean_type='α → α → Prop'), Declaration(ident='inst✝¹', lean_type='IsSymm α r'), Declaration(ident='inst✝', lean_type='IsTrans α r'), Declaration(ident='s', lean_type='Set α'), Declaration(ident='ht', lean_type='t ∈ P'), Declaration(ident='hx', lean_type='x ∈ t')], conclusion='{y | P.Rel x y} = t')] | rwa [P.eq_of_mem_of_mem ht ht' hx hx'] | <class 'lean_dojo.interaction.dojo.LeanError'> | LeanError(error='no goals to be solved') | no goals to be solved |
https://github.com/apnelson1/Matroid | 510262f56c0025be19502ab3457a788e2f6d2d1c | Matroid/Rank.lean | Matroid.er_inter_add_er_union_le_er_add_er | Status.PROVED | 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 Y : Set α
⊢ M.er (X ∩ Y) + M.er (X ∪ Y) ≤ M.er X + M.er Y | None | [Goal(assumptions=[Declaration(ident='α', lean_type='Type u_1'), Declaration(ident='ι', lean_type='Type u_2'), Declaration(ident='M✝', lean_type='Matroid α'), Declaration(ident='N', lean_type='Matroid α'), Declaration(ident='I', lean_type='Set α'), Declaration(ident='B', lean_type='Set α'), Declaration(ident='X✝', lean_type='Set α'), Declaration(ident="X'", lean_type='Set α'), Declaration(ident='Y✝', lean_type='Set α'), Declaration(ident="Y'", lean_type='Set α'), Declaration(ident='Z', lean_type='Set α'), Declaration(ident='R', lean_type='Set α'), Declaration(ident='n', lean_type='ℕ∞'), Declaration(ident='e', lean_type='α'), Declaration(ident='f', lean_type='α'), Declaration(ident='M', lean_type='Matroid α'), Declaration(ident='X', lean_type='Set α'), Declaration(ident='Y', lean_type='Set α')], conclusion='M.er (X ∩ Y) + M.er (X ∪ Y) ≤ M.er X + M.er Y')] | obtain ⟨Ii, hIi⟩ := M.exists_basis' (X ∩ Y) | <class 'lean_dojo.interaction.dojo.TacticState'> | TacticState(pp="case intro\nα : Type u_1\nι : Type u_2\nM✝ N : Matroid α\nI B X✝ X' Y✝ Y' Z R : Set α\nn : ℕ∞\ne f : α\nM : Matroid α\nX Y Ii : Set α\nhIi : M.Basis' Ii (X ∩ Y)\n⊢ M.er (X ∩ Y) + M.er (X ∪ Y) ≤ M.er X + M.er Y", id=1, message='') | 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 Y Ii : Set α
hIi : M.Basis' Ii (X ∩ Y)
⊢ M.er (X ∩ Y) + M.er (X ∪ Y) ≤ M.er X + M.er Y |
https://github.com/apnelson1/Matroid | 510262f56c0025be19502ab3457a788e2f6d2d1c | Matroid/Rank.lean | Matroid.er_inter_add_er_union_le_er_add_er | Status.PROVED | 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 Y Ii : Set α
hIi : M.Basis' Ii (X ∩ Y)
⊢ M.er (X ∩ Y) + M.er (X ∪ Y) ≤ M.er X + M.er Y | [Goal(assumptions=[Declaration(ident='α', lean_type='Type u_1'), Declaration(ident='ι', lean_type='Type u_2'), Declaration(ident='M✝', lean_type='Matroid α'), Declaration(ident='N', lean_type='Matroid α'), Declaration(ident='I', lean_type='Set α'), Declaration(ident='B', lean_type='Set α'), Declaration(ident='X✝', lean_type='Set α'), Declaration(ident="X'", lean_type='Set α'), Declaration(ident='Y✝', lean_type='Set α'), Declaration(ident="Y'", lean_type='Set α'), Declaration(ident='Z', lean_type='Set α'), Declaration(ident='R', lean_type='Set α'), Declaration(ident='n', lean_type='ℕ∞'), Declaration(ident='e', lean_type='α'), Declaration(ident='f', lean_type='α'), Declaration(ident='M', lean_type='Matroid α'), Declaration(ident='X', lean_type='Set α'), Declaration(ident='Y', lean_type='Set α'), Declaration(ident='Ii', lean_type='Set α'), Declaration(ident='hIi', lean_type="M.Basis' Ii (X ∩ Y)")], conclusion='M.er (X ∩ Y) + M.er (X ∪ Y) ≤ M.er X + M.er Y')] | obtain ⟨IX, hIX, hIX'⟩ :=
hIi.indep.subset_basis'_of_subset (hIi.subset.trans inter_subset_left) | <class 'lean_dojo.interaction.dojo.TacticState'> | TacticState(pp="case intro.intro.intro\nα : Type u_1\nι : Type u_2\nM✝ N : Matroid α\nI B X✝ X' Y✝ Y' Z R : Set α\nn : ℕ∞\ne f : α\nM : Matroid α\nX Y Ii : Set α\nhIi : M.Basis' Ii (X ∩ Y)\nIX : Set α\nhIX : M.Basis' IX X\nhIX' : Ii ⊆ IX\n⊢ M.er (X ∩ Y) + M.er (X ∪ Y) ≤ M.er X + M.er Y", id=2, message='') | case intro.intro.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 Y Ii : Set α
hIi : M.Basis' Ii (X ∩ Y)
IX : Set α
hIX : M.Basis' IX X
hIX' : Ii ⊆ IX
⊢ M.er (X ∩ Y) + M.er (X ∪ Y) ≤ M.er X + M.er Y |
|
https://github.com/apnelson1/Matroid | 510262f56c0025be19502ab3457a788e2f6d2d1c | Matroid/Rank.lean | Matroid.er_inter_add_er_union_le_er_add_er | Status.PROVED | case intro.intro.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 Y Ii : Set α
hIi : M.Basis' Ii (X ∩ Y)
IX : Set α
hIX : M.Basis' IX X
hIX' : Ii ⊆ IX
⊢ M.er (X ∩ Y) + M.er (X ∪ Y) ≤ M.er X + M.er Y | [Goal(assumptions=[Declaration(ident='α', lean_type='Type u_1'), Declaration(ident='ι', lean_type='Type u_2'), Declaration(ident='M✝', lean_type='Matroid α'), Declaration(ident='N', lean_type='Matroid α'), Declaration(ident='I', lean_type='Set α'), Declaration(ident='B', lean_type='Set α'), Declaration(ident='X✝', lean_type='Set α'), Declaration(ident="X'", lean_type='Set α'), Declaration(ident='Y✝', lean_type='Set α'), Declaration(ident="Y'", lean_type='Set α'), Declaration(ident='Z', lean_type='Set α'), Declaration(ident='R', lean_type='Set α'), Declaration(ident='n', lean_type='ℕ∞'), Declaration(ident='e', lean_type='α'), Declaration(ident='f', lean_type='α'), Declaration(ident='M', lean_type='Matroid α'), Declaration(ident='X', lean_type='Set α'), Declaration(ident='Y', lean_type='Set α'), Declaration(ident='Ii', lean_type='Set α'), Declaration(ident='hIi', lean_type="M.Basis' Ii (X ∩ Y)"), Declaration(ident='IX', lean_type='Set α'), Declaration(ident='hIX', lean_type="M.Basis' IX X"), Declaration(ident="hIX'", lean_type='Ii ⊆ IX')], conclusion='M.er (X ∩ Y) + M.er (X ∪ Y) ≤ M.er X + M.er Y')] | obtain ⟨IY, hIY, hIY'⟩ :=
hIi.indep.subset_basis'_of_subset (hIi.subset.trans inter_subset_right) | <class 'lean_dojo.interaction.dojo.TacticState'> | TacticState(pp="case intro.intro.intro.intro.intro\nα : Type u_1\nι : Type u_2\nM✝ N : Matroid α\nI B X✝ X' Y✝ Y' Z R : Set α\nn : ℕ∞\ne f : α\nM : Matroid α\nX Y Ii : Set α\nhIi : M.Basis' Ii (X ∩ Y)\nIX : Set α\nhIX : M.Basis' IX X\nhIX' : Ii ⊆ IX\nIY : Set α\nhIY : M.Basis' IY Y\nhIY' : Ii ⊆ IY\n⊢ M.er (X ∩ Y) + M.er (X ∪ Y) ≤ M.er X + M.er Y", id=3, message='') | case intro.intro.intro.intro.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 Y Ii : Set α
hIi : M.Basis' Ii (X ∩ Y)
IX : Set α
hIX : M.Basis' IX X
hIX' : Ii ⊆ IX
IY : Set α
hIY : M.Basis' IY Y
hIY' : Ii ⊆ IY
⊢ M.er (X ∩ Y) + M.er (X ∪ Y) ≤ M.er X + M.er Y |
|
https://github.com/apnelson1/Matroid | 510262f56c0025be19502ab3457a788e2f6d2d1c | Matroid/Rank.lean | Matroid.er_inter_add_er_union_le_er_add_er | Status.PROVED | case intro.intro.intro.intro.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 Y Ii : Set α
hIi : M.Basis' Ii (X ∩ Y)
IX : Set α
hIX : M.Basis' IX X
hIX' : Ii ⊆ IX
IY : Set α
hIY : M.Basis' IY Y
hIY' : Ii ⊆ IY
⊢ M.er (X ∩ Y) + M.er (X ∪ Y) ≤ M.er X + M.er Y | [Goal(assumptions=[Declaration(ident='α', lean_type='Type u_1'), Declaration(ident='ι', lean_type='Type u_2'), Declaration(ident='M✝', lean_type='Matroid α'), Declaration(ident='N', lean_type='Matroid α'), Declaration(ident='I', lean_type='Set α'), Declaration(ident='B', lean_type='Set α'), Declaration(ident='X✝', lean_type='Set α'), Declaration(ident="X'", lean_type='Set α'), Declaration(ident='Y✝', lean_type='Set α'), Declaration(ident="Y'", lean_type='Set α'), Declaration(ident='Z', lean_type='Set α'), Declaration(ident='R', lean_type='Set α'), Declaration(ident='n', lean_type='ℕ∞'), Declaration(ident='e', lean_type='α'), Declaration(ident='f', lean_type='α'), Declaration(ident='M', lean_type='Matroid α'), Declaration(ident='X', lean_type='Set α'), Declaration(ident='Y', lean_type='Set α'), Declaration(ident='Ii', lean_type='Set α'), Declaration(ident='hIi', lean_type="M.Basis' Ii (X ∩ Y)"), Declaration(ident='IX', lean_type='Set α'), Declaration(ident='hIX', lean_type="M.Basis' IX X"), Declaration(ident="hIX'", lean_type='Ii ⊆ IX'), Declaration(ident='IY', lean_type='Set α'), Declaration(ident='hIY', lean_type="M.Basis' IY Y"), Declaration(ident="hIY'", lean_type='Ii ⊆ IY')], conclusion='M.er (X ∩ Y) + M.er (X ∪ Y) ≤ M.er X + M.er Y')] | rw [← hIX.er_eq_er_union, union_comm, ← hIY.er_eq_er_union, ← hIi.encard, ← hIX.encard,
← hIY.encard, union_comm, ← encard_union_add_encard_inter, add_comm] | <class 'lean_dojo.interaction.dojo.TacticState'> | TacticState(pp="case intro.intro.intro.intro.intro\nα : Type u_1\nι : Type u_2\nM✝ N : Matroid α\nI B X✝ X' Y✝ Y' Z R : Set α\nn : ℕ∞\ne f : α\nM : Matroid α\nX Y Ii : Set α\nhIi : M.Basis' Ii (X ∩ Y)\nIX : Set α\nhIX : M.Basis' IX X\nhIX' : Ii ⊆ IX\nIY : Set α\nhIY : M.Basis' IY Y\nhIY' : Ii ⊆ IY\n⊢ M.er (IX ∪ IY) + Ii.encard ≤ (IX ∪ IY).encard + (IX ∩ IY).encard", id=4, message='') | case intro.intro.intro.intro.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 Y Ii : Set α
hIi : M.Basis' Ii (X ∩ Y)
IX : Set α
hIX : M.Basis' IX X
hIX' : Ii ⊆ IX
IY : Set α
hIY : M.Basis' IY Y
hIY' : Ii ⊆ IY
⊢ M.er (IX ∪ IY) + Ii.encard ≤ (IX ∪ IY).encard + (IX ∩ IY).encard |
|
https://github.com/apnelson1/Matroid | 510262f56c0025be19502ab3457a788e2f6d2d1c | Matroid/Rank.lean | Matroid.er_inter_add_er_union_le_er_add_er | Status.PROVED | case intro.intro.intro.intro.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 Y Ii : Set α
hIi : M.Basis' Ii (X ∩ Y)
IX : Set α
hIX : M.Basis' IX X
hIX' : Ii ⊆ IX
IY : Set α
hIY : M.Basis' IY Y
hIY' : Ii ⊆ IY
⊢ M.er (IX ∪ IY) + Ii.encard ≤ (IX ∪ IY).encard + (IX ∩ IY).encard | [Goal(assumptions=[Declaration(ident='α', lean_type='Type u_1'), Declaration(ident='ι', lean_type='Type u_2'), Declaration(ident='M✝', lean_type='Matroid α'), Declaration(ident='N', lean_type='Matroid α'), Declaration(ident='I', lean_type='Set α'), Declaration(ident='B', lean_type='Set α'), Declaration(ident='X✝', lean_type='Set α'), Declaration(ident="X'", lean_type='Set α'), Declaration(ident='Y✝', lean_type='Set α'), Declaration(ident="Y'", lean_type='Set α'), Declaration(ident='Z', lean_type='Set α'), Declaration(ident='R', lean_type='Set α'), Declaration(ident='n', lean_type='ℕ∞'), Declaration(ident='e', lean_type='α'), Declaration(ident='f', lean_type='α'), Declaration(ident='M', lean_type='Matroid α'), Declaration(ident='X', lean_type='Set α'), Declaration(ident='Y', lean_type='Set α'), Declaration(ident='Ii', lean_type='Set α'), Declaration(ident='hIi', lean_type="M.Basis' Ii (X ∩ Y)"), Declaration(ident='IX', lean_type='Set α'), Declaration(ident='hIX', lean_type="M.Basis' IX X"), Declaration(ident="hIX'", lean_type='Ii ⊆ IX'), Declaration(ident='IY', lean_type='Set α'), Declaration(ident='hIY', lean_type="M.Basis' IY Y"), Declaration(ident="hIY'", lean_type='Ii ⊆ IY')], conclusion='M.er (IX ∪ IY) + Ii.encard ≤ (IX ∪ IY).encard + (IX ∩ IY).encard')] | exact add_le_add (er_le_encard _ _) (encard_mono (subset_inter hIX' hIY')) | <class 'lean_dojo.interaction.dojo.ProofFinished'> | ProofFinished(tactic_state_id=5, message='') | ||
https://github.com/apnelson1/Matroid | 510262f56c0025be19502ab3457a788e2f6d2d1c | Matroid/Minor/Basic.lean | Matroid.Minor.C_union_D_eq | Status.PROVED | α : Type u_1
M M' N : Matroid α
e f : α
I J R B X Y Z K : Set α
M₀ M₁ M₂ : Matroid α
h : N ≤m M
⊢ h.C ∪ h.D = M.E \ N.E | None | [Goal(assumptions=[Declaration(ident='α', lean_type='Type u_1'), Declaration(ident='M', lean_type='Matroid α'), Declaration(ident="M'", lean_type='Matroid α'), Declaration(ident='N', lean_type='Matroid α'), Declaration(ident='e', lean_type='α'), Declaration(ident='f', lean_type='α'), Declaration(ident='I', lean_type='Set α'), Declaration(ident='J', lean_type='Set α'), Declaration(ident='R', lean_type='Set α'), Declaration(ident='B', lean_type='Set α'), Declaration(ident='X', lean_type='Set α'), Declaration(ident='Y', lean_type='Set α'), Declaration(ident='Z', lean_type='Set α'), Declaration(ident='K', lean_type='Set α'), Declaration(ident='M₀', lean_type='Matroid α'), Declaration(ident='M₁', lean_type='Matroid α'), Declaration(ident='M₂', lean_type='Matroid α'), Declaration(ident='h', lean_type='N ≤m M')], conclusion='h.C ∪ h.D = M.E \\ N.E')] | simp only [h.eq_con_del, delete_ground, contract_ground, diff_diff] | <class 'lean_dojo.interaction.dojo.TacticState'> | TacticState(pp="α : Type u_1\nM M' N : Matroid α\ne f : α\nI J R B X Y Z K : Set α\nM₀ M₁ M₂ : Matroid α\nh : N ≤m M\n⊢ h.C ∪ h.D = M.E \\ (M.E \\ (h.C ∪ h.D))", id=1, message='') | α : Type u_1
M M' N : Matroid α
e f : α
I J R B X Y Z K : Set α
M₀ M₁ M₂ : Matroid α
h : N ≤m M
⊢ h.C ∪ h.D = M.E \ (M.E \ (h.C ∪ h.D)) |
https://github.com/apnelson1/Matroid | 510262f56c0025be19502ab3457a788e2f6d2d1c | Matroid/Minor/Basic.lean | Matroid.Minor.C_union_D_eq | Status.PROVED | α : Type u_1
M M' N : Matroid α
e f : α
I J R B X Y Z K : Set α
M₀ M₁ M₂ : Matroid α
h : N ≤m M
⊢ h.C ∪ h.D = M.E \ (M.E \ (h.C ∪ h.D)) | [Goal(assumptions=[Declaration(ident='α', lean_type='Type u_1'), Declaration(ident='M', lean_type='Matroid α'), Declaration(ident="M'", lean_type='Matroid α'), Declaration(ident='N', lean_type='Matroid α'), Declaration(ident='e', lean_type='α'), Declaration(ident='f', lean_type='α'), Declaration(ident='I', lean_type='Set α'), Declaration(ident='J', lean_type='Set α'), Declaration(ident='R', lean_type='Set α'), Declaration(ident='B', lean_type='Set α'), Declaration(ident='X', lean_type='Set α'), Declaration(ident='Y', lean_type='Set α'), Declaration(ident='Z', lean_type='Set α'), Declaration(ident='K', lean_type='Set α'), Declaration(ident='M₀', lean_type='Matroid α'), Declaration(ident='M₁', lean_type='Matroid α'), Declaration(ident='M₂', lean_type='Matroid α'), Declaration(ident='h', lean_type='N ≤m M')], conclusion='h.C ∪ h.D = M.E \\ (M.E \\ (h.C ∪ h.D))')] | rw [Set.diff_diff_cancel_left] | <class 'lean_dojo.interaction.dojo.TacticState'> | TacticState(pp="α : Type u_1\nM M' N : Matroid α\ne f : α\nI J R B X Y Z K : Set α\nM₀ M₁ M₂ : Matroid α\nh : N ≤m M\n⊢ h.C ∪ h.D ⊆ M.E", id=2, message='') | α : Type u_1
M M' N : Matroid α
e f : α
I J R B X Y Z K : Set α
M₀ M₁ M₂ : Matroid α
h : N ≤m M
⊢ h.C ∪ h.D ⊆ M.E |
|
https://github.com/apnelson1/Matroid | 510262f56c0025be19502ab3457a788e2f6d2d1c | Matroid/Minor/Basic.lean | Matroid.Minor.C_union_D_eq | Status.PROVED | α : Type u_1
M M' N : Matroid α
e f : α
I J R B X Y Z K : Set α
M₀ M₁ M₂ : Matroid α
h : N ≤m M
⊢ h.C ∪ h.D ⊆ M.E | [Goal(assumptions=[Declaration(ident='α', lean_type='Type u_1'), Declaration(ident='M', lean_type='Matroid α'), Declaration(ident="M'", lean_type='Matroid α'), Declaration(ident='N', lean_type='Matroid α'), Declaration(ident='e', lean_type='α'), Declaration(ident='f', lean_type='α'), Declaration(ident='I', lean_type='Set α'), Declaration(ident='J', lean_type='Set α'), Declaration(ident='R', lean_type='Set α'), Declaration(ident='B', lean_type='Set α'), Declaration(ident='X', lean_type='Set α'), Declaration(ident='Y', lean_type='Set α'), Declaration(ident='Z', lean_type='Set α'), Declaration(ident='K', lean_type='Set α'), Declaration(ident='M₀', lean_type='Matroid α'), Declaration(ident='M₁', lean_type='Matroid α'), Declaration(ident='M₂', lean_type='Matroid α'), Declaration(ident='h', lean_type='N ≤m M')], conclusion='h.C ∪ h.D ⊆ M.E')] | exact union_subset h.C_indep.subset_ground h.D_coindep.subset_ground | <class 'lean_dojo.interaction.dojo.ProofFinished'> | ProofFinished(tactic_state_id=3, message='') | ||
https://github.com/apnelson1/Matroid | 510262f56c0025be19502ab3457a788e2f6d2d1c | Matroid/Simple.lean | Matroid.closure_eq_self_of_subset_singleton | Status.FAILED | α : Type u_1
M N : Matroid α
e f g : α
I X P D : Set α
inst✝ : M.Simple
he : e ∈ M.E
hX : X ⊆ {e}
⊢ M.closure X = X | None | [Goal(assumptions=[Declaration(ident='α', lean_type='Type u_1'), Declaration(ident='M', lean_type='Matroid α'), Declaration(ident='N', lean_type='Matroid α'), Declaration(ident='e', lean_type='α'), Declaration(ident='f', lean_type='α'), Declaration(ident='g', lean_type='α'), Declaration(ident='I', lean_type='Set α'), Declaration(ident='X', lean_type='Set α'), Declaration(ident='P', lean_type='Set α'), Declaration(ident='D', lean_type='Set α'), Declaration(ident='inst✝', lean_type='M.Simple'), Declaration(ident='he', lean_type='e ∈ M.E'), Declaration(ident='hX', lean_type='X ⊆ {e}')], conclusion='M.closure X = X')] | obtain (rfl | rfl) := subset_singleton_iff_eq.1 hX | <class 'lean_dojo.interaction.dojo.TacticState'> | TacticState(pp='case inl\nα : Type u_1\nM N : Matroid α\ne f g : α\nI P D : Set α\ninst✝ : M.Simple\nhe : e ∈ M.E\nhX : ∅ ⊆ {e}\n⊢ M.closure ∅ = ∅\n\ncase inr\nα : Type u_1\nM N : Matroid α\ne f g : α\nI P D : Set α\ninst✝ : M.Simple\nhe : e ∈ M.E\nhX : {e} ⊆ {e}\n⊢ M.closure {e} = {e}', id=1, message='') | case inl
α : Type u_1
M N : Matroid α
e f g : α
I P D : Set α
inst✝ : M.Simple
he : e ∈ M.E
hX : ∅ ⊆ {e}
⊢ M.closure ∅ = ∅
case inr
α : Type u_1
M N : Matroid α
e f g : α
I P D : Set α
inst✝ : M.Simple
he : e ∈ M.E
hX : {e} ⊆ {e}
⊢ M.closure {e} = {e} |
https://github.com/apnelson1/Matroid | 510262f56c0025be19502ab3457a788e2f6d2d1c | Matroid/Simple.lean | Matroid.closure_eq_self_of_subset_singleton | Status.FAILED | case inl
α : Type u_1
M N : Matroid α
e f g : α
I P D : Set α
inst✝ : M.Simple
he : e ∈ M.E
hX : ∅ ⊆ {e}
⊢ M.closure ∅ = ∅
case inr
α : Type u_1
M N : Matroid α
e f g : α
I P D : Set α
inst✝ : M.Simple
he : e ∈ M.E
hX : {e} ⊆ {e}
⊢ M.closure {e} = {e} | [Goal(assumptions=[Declaration(ident='α', lean_type='Type u_1'), Declaration(ident='M', lean_type='Matroid α'), Declaration(ident='N', lean_type='Matroid α'), Declaration(ident='e', lean_type='α'), Declaration(ident='f', lean_type='α'), Declaration(ident='g', lean_type='α'), Declaration(ident='I', lean_type='Set α'), Declaration(ident='P', lean_type='Set α'), Declaration(ident='D', lean_type='Set α'), Declaration(ident='inst✝', lean_type='M.Simple'), Declaration(ident='he', lean_type='e ∈ M.E'), Declaration(ident='hX', lean_type='∅ ⊆ {e}')], conclusion='M.closure ∅ = ∅'), Goal(assumptions=[Declaration(ident='α', lean_type='Type u_1'), Declaration(ident='M', lean_type='Matroid α'), Declaration(ident='N', lean_type='Matroid α'), Declaration(ident='e', lean_type='α'), Declaration(ident='f', lean_type='α'), Declaration(ident='g', lean_type='α'), Declaration(ident='I', lean_type='Set α'), Declaration(ident='P', lean_type='Set α'), Declaration(ident='D', lean_type='Set α'), Declaration(ident='inst✝', lean_type='M.Simple'), Declaration(ident='he', lean_type='e ∈ M.E'), Declaration(ident='hX', lean_type='{e} ⊆ {e}')], conclusion='M.closure {e} = {e}')] | exact closure_singleton_eq he | <class 'lean_dojo.interaction.dojo.LeanError'> | LeanError(error='type mismatch\n Matroid.closure_singleton_eq he\nhas type\n M.closure {e} = {e} : Prop\nbut is expected to have type\n M.closure ∅ = ∅ : Prop') | type mismatch
Matroid.closure_singleton_eq he
has type
M.closure {e} = {e} : Prop
but is expected to have type
M.closure ∅ = ∅ : Prop |
|
https://github.com/apnelson1/Matroid | 510262f56c0025be19502ab3457a788e2f6d2d1c | Matroid/Rank.lean | Matroid.loopyOn_r_eq | Status.PROVED | 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
E✝ E X : Set α
⊢ (loopyOn E).r X = 0 | None | [Goal(assumptions=[Declaration(ident='α', lean_type='Type u_1'), Declaration(ident='ι', lean_type='Type u_2'), Declaration(ident='M', lean_type='Matroid α'), Declaration(ident='N', lean_type='Matroid α'), Declaration(ident='I', lean_type='Set α'), Declaration(ident='B', lean_type='Set α'), Declaration(ident='X✝', lean_type='Set α'), Declaration(ident="X'", lean_type='Set α'), Declaration(ident='Y', lean_type='Set α'), Declaration(ident="Y'", lean_type='Set α'), Declaration(ident='Z', lean_type='Set α'), Declaration(ident='R', lean_type='Set α'), Declaration(ident='n', lean_type='ℕ∞'), Declaration(ident='e', lean_type='α'), Declaration(ident='f', lean_type='α'), Declaration(ident='E✝', lean_type='Set α'), Declaration(ident='E', lean_type='Set α'), Declaration(ident='X', lean_type='Set α')], conclusion='(loopyOn E).r X = 0')] | rw [← er_toNat_eq_r, loopyOn_er_eq] | <class 'lean_dojo.interaction.dojo.TacticState'> | TacticState(pp="α : Type u_1\nι : Type u_2\nM N : Matroid α\nI B X✝ X' Y Y' Z R : Set α\nn : ℕ∞\ne f : α\nE✝ E X : Set α\n⊢ toNat 0 = 0", id=1, message='') | 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
E✝ E X : Set α
⊢ toNat 0 = 0 |
https://github.com/apnelson1/Matroid | 510262f56c0025be19502ab3457a788e2f6d2d1c | Matroid/Rank.lean | Matroid.loopyOn_r_eq | Status.PROVED | 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
E✝ E X : Set α
⊢ toNat 0 = 0 | [Goal(assumptions=[Declaration(ident='α', lean_type='Type u_1'), Declaration(ident='ι', lean_type='Type u_2'), Declaration(ident='M', lean_type='Matroid α'), Declaration(ident='N', lean_type='Matroid α'), Declaration(ident='I', lean_type='Set α'), Declaration(ident='B', lean_type='Set α'), Declaration(ident='X✝', lean_type='Set α'), Declaration(ident="X'", lean_type='Set α'), Declaration(ident='Y', lean_type='Set α'), Declaration(ident="Y'", lean_type='Set α'), Declaration(ident='Z', lean_type='Set α'), Declaration(ident='R', lean_type='Set α'), Declaration(ident='n', lean_type='ℕ∞'), Declaration(ident='e', lean_type='α'), Declaration(ident='f', lean_type='α'), Declaration(ident='E✝', lean_type='Set α'), Declaration(ident='E', lean_type='Set α'), Declaration(ident='X', lean_type='Set α')], conclusion='toNat 0 = 0')] | rfl | <class 'lean_dojo.interaction.dojo.ProofFinished'> | ProofFinished(tactic_state_id=2, message='') | ||
https://github.com/apnelson1/Matroid | 510262f56c0025be19502ab3457a788e2f6d2d1c | Matroid/Minor/Basic.lean | Matroid.restriction_iff_exists_eq_delete | Status.FAILED | α : Type u_1
M M' N : Matroid α
e f : α
I J R✝ B X Y Z K D D₁ D₂ R : Set α
⊢ N ≤r M ↔ ∃ D ⊆ M.E, N = M ＼ D | None | [Goal(assumptions=[Declaration(ident='α', lean_type='Type u_1'), Declaration(ident='M', lean_type='Matroid α'), Declaration(ident="M'", lean_type='Matroid α'), Declaration(ident='N', lean_type='Matroid α'), Declaration(ident='e', lean_type='α'), Declaration(ident='f', lean_type='α'), Declaration(ident='I', lean_type='Set α'), Declaration(ident='J', lean_type='Set α'), Declaration(ident='R✝', lean_type='Set α'), Declaration(ident='B', lean_type='Set α'), Declaration(ident='X', lean_type='Set α'), Declaration(ident='Y', lean_type='Set α'), Declaration(ident='Z', lean_type='Set α'), Declaration(ident='K', lean_type='Set α'), Declaration(ident='D', lean_type='Set α'), Declaration(ident='D₁', lean_type='Set α'), Declaration(ident='D₂', lean_type='Set α'), Declaration(ident='R', lean_type='Set α')], conclusion='N ≤r M ↔ ∃ D ⊆ M.E, N = M ＼ D')] | rintro ⟨D, -, rfl⟩ | <class 'lean_dojo.interaction.dojo.LeanError'> | LeanError(error="tactic 'introN' failed, insufficient number of binders\nα : Type u_1\nM M' N : Matroid α\ne f : α\nI J R✝ B X Y Z K D D₁ D₂ R : Set α\n⊢ N ≤r M ↔ ∃ D ⊆ M.E, N = M ＼ D") | tactic 'introN' failed, insufficient number of binders
α : Type u_1
M M' N : Matroid α
e f : α
I J R✝ B X Y Z K D D₁ D₂ R : Set α
⊢ N ≤r M ↔ ∃ D ⊆ M.E, N = M ＼ D |
https://github.com/apnelson1/Matroid | 510262f56c0025be19502ab3457a788e2f6d2d1c | Matroid/Flat.lean | Matroid.Flat.iInter_inter_ground | Status.FAILED | α : Type u_1
M : Matroid α
I F X Y F' F₀ F₁ F₂ P L H H₁ H₂ H' B C K : Set α
e f : α
ι : Type u_2
Fs : ι → Set α
hFs : ∀ (i : ι), M.Flat (Fs i)
⊢ M.Flat ((⋂ i, Fs i) ∩ M.E) | None | [Goal(assumptions=[Declaration(ident='α', lean_type='Type u_1'), Declaration(ident='M', lean_type='Matroid α'), Declaration(ident='I', lean_type='Set α'), Declaration(ident='F', lean_type='Set α'), Declaration(ident='X', lean_type='Set α'), Declaration(ident='Y', lean_type='Set α'), Declaration(ident="F'", lean_type='Set α'), Declaration(ident='F₀', lean_type='Set α'), Declaration(ident='F₁', lean_type='Set α'), Declaration(ident='F₂', lean_type='Set α'), Declaration(ident='P', lean_type='Set α'), Declaration(ident='L', lean_type='Set α'), Declaration(ident='H', lean_type='Set α'), Declaration(ident='H₁', lean_type='Set α'), Declaration(ident='H₂', lean_type='Set α'), Declaration(ident="H'", lean_type='Set α'), Declaration(ident='B', lean_type='Set α'), Declaration(ident='C', lean_type='Set α'), Declaration(ident='K', lean_type='Set α'), Declaration(ident='e', lean_type='α'), Declaration(ident='f', lean_type='α'), Declaration(ident='ι', lean_type='Type u_2'), Declaration(ident='Fs', lean_type='ι → Set α'), Declaration(ident='hFs', lean_type='∀ (i : ι), M.Flat (Fs i)')], conclusion='M.Flat ((⋂ i, Fs i) ∩ M.E)')] | obtain (hι | hι) := isEmpty_or_nonempty ι | <class 'lean_dojo.interaction.dojo.TacticState'> | TacticState(pp="case inl\nα : Type u_1\nM : Matroid α\nI F X Y F' F₀ F₁ F₂ P L H H₁ H₂ H' B C K : Set α\ne f : α\nι : Type u_2\nFs : ι → Set α\nhFs : ∀ (i : ι), M.Flat (Fs i)\nhι : IsEmpty ι\n⊢ M.Flat ((⋂ i, Fs i) ∩ M.E)\n\ncase inr\nα : Type u_1\nM : Matroid α\nI F X Y F' F₀ F₁ F₂ P L H H₁ H₂ H' B C K : Set α\ne f : α\nι : Type u_2\nFs : ι → Set α\nhFs : ∀ (i : ι), M.Flat (Fs i)\nhι : Nonempty ι\n⊢ M.Flat ((⋂ i, Fs i) ∩ M.E)", id=1, message='') | case inl
α : Type u_1
M : Matroid α
I F X Y F' F₀ F₁ F₂ P L H H₁ H₂ H' B C K : Set α
e f : α
ι : Type u_2
Fs : ι → Set α
hFs : ∀ (i : ι), M.Flat (Fs i)
hι : IsEmpty ι
⊢ M.Flat ((⋂ i, Fs i) ∩ M.E)
case inr
α : Type u_1
M : Matroid α
I F X Y F' F₀ F₁ F₂ P L H H₁ H₂ H' B C K : Set α
e f : α
ι : Type u_2
Fs : ι → Set α
hFs : ∀ (i : ι), M.Flat (Fs i)
hι : Nonempty ι
⊢ M.Flat ((⋂ i, Fs i) ∩ M.E) |
https://github.com/apnelson1/Matroid | 510262f56c0025be19502ab3457a788e2f6d2d1c | Matroid/Flat.lean | Matroid.Flat.iInter_inter_ground | Status.FAILED | case inl
α : Type u_1
M : Matroid α
I F X Y F' F₀ F₁ F₂ P L H H₁ H₂ H' B C K : Set α
e f : α
ι : Type u_2
Fs : ι → Set α
hFs : ∀ (i : ι), M.Flat (Fs i)
hι : IsEmpty ι
⊢ M.Flat ((⋂ i, Fs i) ∩ M.E)
case inr
α : Type u_1
M : Matroid α
I F X Y F' F₀ F₁ F₂ P L H H₁ H₂ H' B C K : Set α
e f : α
ι : Type u_2
Fs : ι → Set α
hFs : ∀ (i : ι), M.Flat (Fs i)
hι : Nonempty ι
⊢ M.Flat ((⋂ i, Fs i) ∩ M.E) | [Goal(assumptions=[Declaration(ident='α', lean_type='Type u_1'), Declaration(ident='M', lean_type='Matroid α'), Declaration(ident='I', lean_type='Set α'), Declaration(ident='F', lean_type='Set α'), Declaration(ident='X', lean_type='Set α'), Declaration(ident='Y', lean_type='Set α'), Declaration(ident="F'", lean_type='Set α'), Declaration(ident='F₀', lean_type='Set α'), Declaration(ident='F₁', lean_type='Set α'), Declaration(ident='F₂', lean_type='Set α'), Declaration(ident='P', lean_type='Set α'), Declaration(ident='L', lean_type='Set α'), Declaration(ident='H', lean_type='Set α'), Declaration(ident='H₁', lean_type='Set α'), Declaration(ident='H₂', lean_type='Set α'), Declaration(ident="H'", lean_type='Set α'), Declaration(ident='B', lean_type='Set α'), Declaration(ident='C', lean_type='Set α'), Declaration(ident='K', lean_type='Set α'), Declaration(ident='e', lean_type='α'), Declaration(ident='f', lean_type='α'), Declaration(ident='ι', lean_type='Type u_2'), Declaration(ident='Fs', lean_type='ι → Set α'), Declaration(ident='hFs', lean_type='∀ (i : ι), M.Flat (Fs i)'), Declaration(ident='hι', lean_type='IsEmpty ι')], conclusion='M.Flat ((⋂ i, Fs i) ∩ M.E)'), Goal(assumptions=[Declaration(ident='α', lean_type='Type u_1'), Declaration(ident='M', lean_type='Matroid α'), Declaration(ident='I', lean_type='Set α'), Declaration(ident='F', lean_type='Set α'), Declaration(ident='X', lean_type='Set α'), Declaration(ident='Y', lean_type='Set α'), Declaration(ident="F'", lean_type='Set α'), Declaration(ident='F₀', lean_type='Set α'), Declaration(ident='F₁', lean_type='Set α'), Declaration(ident='F₂', lean_type='Set α'), Declaration(ident='P', lean_type='Set α'), Declaration(ident='L', lean_type='Set α'), Declaration(ident='H', lean_type='Set α'), Declaration(ident='H₁', lean_type='Set α'), Declaration(ident='H₂', lean_type='Set α'), Declaration(ident="H'", lean_type='Set α'), Declaration(ident='B', lean_type='Set α'), Declaration(ident='C', lean_type='Set α'), Declaration(ident='K', lean_type='Set α'), Declaration(ident='e', lean_type='α'), Declaration(ident='f', lean_type='α'), Declaration(ident='ι', lean_type='Type u_2'), Declaration(ident='Fs', lean_type='ι → Set α'), Declaration(ident='hFs', lean_type='∀ (i : ι), M.Flat (Fs i)'), Declaration(ident='hι', lean_type='Nonempty ι')], conclusion='M.Flat ((⋂ i, Fs i) ∩ M.E)')] | exact (Flat.iInter hFs).inter M.ground_flat | <class 'lean_dojo.interaction.dojo.LeanError'> | LeanError(error='failed to synthesize\n Nonempty ι\nuse `set_option diagnostics true` to get diagnostic information') | failed to synthesize
Nonempty ι
use `set_option diagnostics true` to get diagnostic information |
|
https://github.com/apnelson1/Matroid | 510262f56c0025be19502ab3457a788e2f6d2d1c | Matroid/Minor/Rank.lean | Matroid.restrict_r_eq' | Status.PROVED | α : Type u_1
M✝ N : Matroid α
I J C D X✝ Y Z : Set α
e f : α
M : Matroid α
R X : Set α
⊢ (M ↾ R).r X = M.r (X ∩ R) | None | [Goal(assumptions=[Declaration(ident='α', lean_type='Type u_1'), Declaration(ident='M✝', lean_type='Matroid α'), Declaration(ident='N', lean_type='Matroid α'), Declaration(ident='I', lean_type='Set α'), Declaration(ident='J', lean_type='Set α'), Declaration(ident='C', lean_type='Set α'), Declaration(ident='D', lean_type='Set α'), Declaration(ident='X✝', lean_type='Set α'), Declaration(ident='Y', lean_type='Set α'), Declaration(ident='Z', lean_type='Set α'), Declaration(ident='e', lean_type='α'), Declaration(ident='f', lean_type='α'), Declaration(ident='M', lean_type='Matroid α'), Declaration(ident='R', lean_type='Set α'), Declaration(ident='X', lean_type='Set α')], conclusion='(M ↾ R).r X = M.r (X ∩ R)')] | rw [r, restrict_er_eq', r] | <class 'lean_dojo.interaction.dojo.ProofFinished'> | ProofFinished(tactic_state_id=1, message='') | |
https://github.com/apnelson1/Matroid | 510262f56c0025be19502ab3457a788e2f6d2d1c | Matroid/Minor/Basic.lean | Matroid.Minor.eq_of_ground_subset | Status.PROVED | α : Type u_1
M M' N : Matroid α
e f : α
I J R B X Y Z K : Set α
M₀ M₁ M₂ : Matroid α
h : N ≤m M
hE : M.E ⊆ N.E
⊢ M = N | None | [Goal(assumptions=[Declaration(ident='α', lean_type='Type u_1'), Declaration(ident='M', lean_type='Matroid α'), Declaration(ident="M'", lean_type='Matroid α'), Declaration(ident='N', lean_type='Matroid α'), Declaration(ident='e', lean_type='α'), Declaration(ident='f', lean_type='α'), Declaration(ident='I', lean_type='Set α'), Declaration(ident='J', lean_type='Set α'), Declaration(ident='R', lean_type='Set α'), Declaration(ident='B', lean_type='Set α'), Declaration(ident='X', lean_type='Set α'), Declaration(ident='Y', lean_type='Set α'), Declaration(ident='Z', lean_type='Set α'), Declaration(ident='K', lean_type='Set α'), Declaration(ident='M₀', lean_type='Matroid α'), Declaration(ident='M₁', lean_type='Matroid α'), Declaration(ident='M₂', lean_type='Matroid α'), Declaration(ident='h', lean_type='N ≤m M'), Declaration(ident='hE', lean_type='M.E ⊆ N.E')], conclusion='M = N')] | obtain ⟨C, D, -, -, -, rfl⟩ := h | <class 'lean_dojo.interaction.dojo.TacticState'> | TacticState(pp="case intro.intro.intro.intro.intro\nα : Type u_1\nM M' : Matroid α\ne f : α\nI J R B X Y Z K : Set α\nM₀ M₁ M₂ : Matroid α\nC D : Set α\nhE : M.E ⊆ (M ／ C ＼ D).E\n⊢ M = M ／ C ＼ D", id=1, message='') | case intro.intro.intro.intro.intro
α : Type u_1
M M' : Matroid α
e f : α
I J R B X Y Z K : Set α
M₀ M₁ M₂ : Matroid α
C D : Set α
hE : M.E ⊆ (M ／ C ＼ D).E
⊢ M = M ／ C ＼ D |
https://github.com/apnelson1/Matroid | 510262f56c0025be19502ab3457a788e2f6d2d1c | Matroid/Minor/Basic.lean | Matroid.Minor.eq_of_ground_subset | Status.PROVED | case intro.intro.intro.intro.intro
α : Type u_1
M M' : Matroid α
e f : α
I J R B X Y Z K : Set α
M₀ M₁ M₂ : Matroid α
C D : Set α
hE : M.E ⊆ (M ／ C ＼ D).E
⊢ M = M ／ C ＼ D | [Goal(assumptions=[Declaration(ident='α', lean_type='Type u_1'), Declaration(ident='M', lean_type='Matroid α'), Declaration(ident="M'", lean_type='Matroid α'), Declaration(ident='e', lean_type='α'), Declaration(ident='f', lean_type='α'), Declaration(ident='I', lean_type='Set α'), Declaration(ident='J', lean_type='Set α'), Declaration(ident='R', lean_type='Set α'), Declaration(ident='B', lean_type='Set α'), Declaration(ident='X', lean_type='Set α'), Declaration(ident='Y', lean_type='Set α'), Declaration(ident='Z', lean_type='Set α'), Declaration(ident='K', lean_type='Set α'), Declaration(ident='M₀', lean_type='Matroid α'), Declaration(ident='M₁', lean_type='Matroid α'), Declaration(ident='M₂', lean_type='Matroid α'), Declaration(ident='C', lean_type='Set α'), Declaration(ident='D', lean_type='Set α'), Declaration(ident='hE', lean_type='M.E ⊆ (M ／ C ＼ D).E')], conclusion='M = M ／ C ＼ D')] | rw [delete_ground, contract_ground, subset_diff, subset_diff] at hE | <class 'lean_dojo.interaction.dojo.TacticState'> | TacticState(pp="case intro.intro.intro.intro.intro\nα : Type u_1\nM M' : Matroid α\ne f : α\nI J R B X Y Z K : Set α\nM₀ M₁ M₂ : Matroid α\nC D : Set α\nhE : (M.E ⊆ M.E ∧ Disjoint M.E C) ∧ Disjoint M.E D\n⊢ M = M ／ C ＼ D", id=2, message='') | case intro.intro.intro.intro.intro
α : Type u_1
M M' : Matroid α
e f : α
I J R B X Y Z K : Set α
M₀ M₁ M₂ : Matroid α
C D : Set α
hE : (M.E ⊆ M.E ∧ Disjoint M.E C) ∧ Disjoint M.E D
⊢ M = M ／ C ＼ D |
|
https://github.com/apnelson1/Matroid | 510262f56c0025be19502ab3457a788e2f6d2d1c | Matroid/Minor/Basic.lean | Matroid.Minor.eq_of_ground_subset | Status.PROVED | case intro.intro.intro.intro.intro
α : Type u_1
M M' : Matroid α
e f : α
I J R B X Y Z K : Set α
M₀ M₁ M₂ : Matroid α
C D : Set α
hE : (M.E ⊆ M.E ∧ Disjoint M.E C) ∧ Disjoint M.E D
⊢ M = M ／ C ＼ D | [Goal(assumptions=[Declaration(ident='α', lean_type='Type u_1'), Declaration(ident='M', lean_type='Matroid α'), Declaration(ident="M'", lean_type='Matroid α'), Declaration(ident='e', lean_type='α'), Declaration(ident='f', lean_type='α'), Declaration(ident='I', lean_type='Set α'), Declaration(ident='J', lean_type='Set α'), Declaration(ident='R', lean_type='Set α'), Declaration(ident='B', lean_type='Set α'), Declaration(ident='X', lean_type='Set α'), Declaration(ident='Y', lean_type='Set α'), Declaration(ident='Z', lean_type='Set α'), Declaration(ident='K', lean_type='Set α'), Declaration(ident='M₀', lean_type='Matroid α'), Declaration(ident='M₁', lean_type='Matroid α'), Declaration(ident='M₂', lean_type='Matroid α'), Declaration(ident='C', lean_type='Set α'), Declaration(ident='D', lean_type='Set α'), Declaration(ident='hE', lean_type='(M.E ⊆ M.E ∧ Disjoint M.E C) ∧ Disjoint M.E D')], conclusion='M = M ／ C ＼ D')] | rw [← contract_inter_ground_eq, hE.1.2.symm.inter_eq, contract_empty, ← delete_inter_ground_eq,
hE.2.symm.inter_eq, delete_empty] | <class 'lean_dojo.interaction.dojo.ProofFinished'> | ProofFinished(tactic_state_id=3, message='') | ||
https://github.com/apnelson1/Matroid | 510262f56c0025be19502ab3457a788e2f6d2d1c | Matroid/Constructions/Uniform.lean | Matroid.unifOn_indep_iff | Status.PROVED | α : Type u_1
M : Matroid α
E I B X C : Set α
k : ℕ∞
⊢ (unifOn E k).Indep I ↔ I.encard ≤ k ∧ I ⊆ E | None | [Goal(assumptions=[Declaration(ident='α', lean_type='Type u_1'), Declaration(ident='M', lean_type='Matroid α'), Declaration(ident='E', lean_type='Set α'), Declaration(ident='I', lean_type='Set α'), Declaration(ident='B', lean_type='Set α'), Declaration(ident='X', lean_type='Set α'), Declaration(ident='C', lean_type='Set α'), Declaration(ident='k', lean_type='ℕ∞')], conclusion='(unifOn E k).Indep I ↔ I.encard ≤ k ∧ I ⊆ E')] | simp [unifOn, and_comm] | <class 'lean_dojo.interaction.dojo.ProofFinished'> | ProofFinished(tactic_state_id=1, message='') | |
https://github.com/apnelson1/Matroid | 510262f56c0025be19502ab3457a788e2f6d2d1c | Matroid/Minor/Basic.lean | Matroid.Minor.eq_con_del | Status.PROVED | α : Type u_1
M M' N : Matroid α
e f : α
I J R B X Y Z K : Set α
M₀ M₁ M₂ : Matroid α
h : N ≤m M
⊢ N = M ／ h.C ＼ h.D | None | [Goal(assumptions=[Declaration(ident='α', lean_type='Type u_1'), Declaration(ident='M', lean_type='Matroid α'), Declaration(ident="M'", lean_type='Matroid α'), Declaration(ident='N', lean_type='Matroid α'), Declaration(ident='e', lean_type='α'), Declaration(ident='f', lean_type='α'), Declaration(ident='I', lean_type='Set α'), Declaration(ident='J', lean_type='Set α'), Declaration(ident='R', lean_type='Set α'), Declaration(ident='B', lean_type='Set α'), Declaration(ident='X', lean_type='Set α'), Declaration(ident='Y', lean_type='Set α'), Declaration(ident='Z', lean_type='Set α'), Declaration(ident='K', lean_type='Set α'), Declaration(ident='M₀', lean_type='Matroid α'), Declaration(ident='M₁', lean_type='Matroid α'), Declaration(ident='M₂', lean_type='Matroid α'), Declaration(ident='h', lean_type='N ≤m M')], conclusion='N = M ／ h.C ＼ h.D')] | obtain ⟨-,-,-,h⟩ := h.exists_contract_indep_delete_coindep.choose_spec.choose_spec | <class 'lean_dojo.interaction.dojo.TacticState'> | TacticState(pp="case intro.intro.intro\nα : Type u_1\nM M' N : Matroid α\ne f : α\nI J R B X Y Z K : Set α\nM₀ M₁ M₂ : Matroid α\nh✝ : N ≤m M\nh : N = M ／ ⋯.choose ＼ ⋯.choose\n⊢ N = M ／ h✝.C ＼ h✝.D", id=1, message='') | case intro.intro.intro
α : Type u_1
M M' N : Matroid α
e f : α
I J R B X Y Z K : Set α
M₀ M₁ M₂ : Matroid α
h✝ : N ≤m M
h : N = M ／ ⋯.choose ＼ ⋯.choose
⊢ N = M ／ h✝.C ＼ h✝.D |
https://github.com/apnelson1/Matroid | 510262f56c0025be19502ab3457a788e2f6d2d1c | Matroid/Minor/Basic.lean | Matroid.Minor.eq_con_del | Status.PROVED | case intro.intro.intro
α : Type u_1
M M' N : Matroid α
e f : α
I J R B X Y Z K : Set α
M₀ M₁ M₂ : Matroid α
h✝ : N ≤m M
h : N = M ／ ⋯.choose ＼ ⋯.choose
⊢ N = M ／ h✝.C ＼ h✝.D | [Goal(assumptions=[Declaration(ident='α', lean_type='Type u_1'), Declaration(ident='M', lean_type='Matroid α'), Declaration(ident="M'", lean_type='Matroid α'), Declaration(ident='N', lean_type='Matroid α'), Declaration(ident='e', lean_type='α'), Declaration(ident='f', lean_type='α'), Declaration(ident='I', lean_type='Set α'), Declaration(ident='J', lean_type='Set α'), Declaration(ident='R', lean_type='Set α'), Declaration(ident='B', lean_type='Set α'), Declaration(ident='X', lean_type='Set α'), Declaration(ident='Y', lean_type='Set α'), Declaration(ident='Z', lean_type='Set α'), Declaration(ident='K', lean_type='Set α'), Declaration(ident='M₀', lean_type='Matroid α'), Declaration(ident='M₁', lean_type='Matroid α'), Declaration(ident='M₂', lean_type='Matroid α'), Declaration(ident='h✝', lean_type='N ≤m M'), Declaration(ident='h', lean_type='N = M ／ ⋯.choose ＼ ⋯.choose')], conclusion='N = M ／ h✝.C ＼ h✝.D')] | exact h | <class 'lean_dojo.interaction.dojo.ProofFinished'> | ProofFinished(tactic_state_id=2, message='') | ||
https://github.com/apnelson1/Matroid | 510262f56c0025be19502ab3457a788e2f6d2d1c | Matroid/Simple.lean | Matroid.IsSimplification.eq_self_iff | Status.PROVED | α : Type u_1
M N : Matroid α
e f g : α
I X P D : Set α
x y : α
h : N.IsSimplification M
⊢ N = M ↔ M.Simple | None | [Goal(assumptions=[Declaration(ident='α', lean_type='Type u_1'), Declaration(ident='M', lean_type='Matroid α'), Declaration(ident='N', lean_type='Matroid α'), Declaration(ident='e', lean_type='α'), Declaration(ident='f', lean_type='α'), Declaration(ident='g', lean_type='α'), Declaration(ident='I', lean_type='Set α'), Declaration(ident='X', lean_type='Set α'), Declaration(ident='P', lean_type='Set α'), Declaration(ident='D', lean_type='Set α'), Declaration(ident='x', lean_type='α'), Declaration(ident='y', lean_type='α'), Declaration(ident='h', lean_type='N.IsSimplification M')], conclusion='N = M ↔ M.Simple')] | refine ⟨fun h' ↦ h' ▸ h.simple, fun h' ↦ ?_⟩ | <class 'lean_dojo.interaction.dojo.TacticState'> | TacticState(pp="α : Type u_1\nM N : Matroid α\ne f g : α\nI X P D : Set α\nx y : α\nh : N.IsSimplification M\nh' : M.Simple\n⊢ N = M", id=1, message='') | α : Type u_1
M N : Matroid α
e f g : α
I X P D : Set α
x y : α
h : N.IsSimplification M
h' : M.Simple
⊢ N = M |
https://github.com/apnelson1/Matroid | 510262f56c0025be19502ab3457a788e2f6d2d1c | Matroid/Simple.lean | Matroid.IsSimplification.eq_self_iff | Status.PROVED | α : Type u_1
M N : Matroid α
e f g : α
I X P D : Set α
x y : α
h : N.IsSimplification M
h' : M.Simple
⊢ N = M | [Goal(assumptions=[Declaration(ident='α', lean_type='Type u_1'), Declaration(ident='M', lean_type='Matroid α'), Declaration(ident='N', lean_type='Matroid α'), Declaration(ident='e', lean_type='α'), Declaration(ident='f', lean_type='α'), Declaration(ident='g', lean_type='α'), Declaration(ident='I', lean_type='Set α'), Declaration(ident='X', lean_type='Set α'), Declaration(ident='P', lean_type='Set α'), Declaration(ident='D', lean_type='Set α'), Declaration(ident='x', lean_type='α'), Declaration(ident='y', lean_type='α'), Declaration(ident='h', lean_type='N.IsSimplification M'), Declaration(ident="h'", lean_type='M.Simple')], conclusion='N = M')] | obtain ⟨f, rfl⟩ := h | <class 'lean_dojo.interaction.dojo.TacticState'> | TacticState(pp="case intro\nα : Type u_1\nM : Matroid α\ne f✝ g : α\nI X P D : Set α\nx y : α\nh' : M.Simple\nf : M.parallelClasses.RepFun\n⊢ M ↾ ⇑f '' {e | M.Nonloop e} = M", id=2, message='') | case intro
α : Type u_1
M : Matroid α
e f✝ g : α
I X P D : Set α
x y : α
h' : M.Simple
f : M.parallelClasses.RepFun
⊢ M ↾ ⇑f '' {e | M.Nonloop e} = M |
|
https://github.com/apnelson1/Matroid | 510262f56c0025be19502ab3457a788e2f6d2d1c | Matroid/Simple.lean | Matroid.IsSimplification.eq_self_iff | Status.PROVED | case intro
α : Type u_1
M : Matroid α
e f✝ g : α
I X P D : Set α
x y : α
h' : M.Simple
f : M.parallelClasses.RepFun
⊢ M ↾ ⇑f '' {e | M.Nonloop e} = M | [Goal(assumptions=[Declaration(ident='α', lean_type='Type u_1'), Declaration(ident='M', lean_type='Matroid α'), Declaration(ident='e', lean_type='α'), Declaration(ident='f✝', lean_type='α'), Declaration(ident='g', lean_type='α'), Declaration(ident='I', lean_type='Set α'), Declaration(ident='X', lean_type='Set α'), Declaration(ident='P', lean_type='Set α'), Declaration(ident='D', lean_type='Set α'), Declaration(ident='x', lean_type='α'), Declaration(ident='y', lean_type='α'), Declaration(ident="h'", lean_type='M.Simple'), Declaration(ident='f', lean_type='M.parallelClasses.RepFun')], conclusion="M ↾ ⇑f '' {e | M.Nonloop e} = M")] | rw [restrict_eq_self_iff, f.coeFun_eq_id_of_eq_discrete M.parallelClasses_eq_discrete, image_id,
Loopless.ground_eq] | <class 'lean_dojo.interaction.dojo.ProofFinished'> | ProofFinished(tactic_state_id=3, message='') | ||
https://github.com/apnelson1/Matroid | 510262f56c0025be19502ab3457a788e2f6d2d1c | Matroid/Loop.lean | Matroid.Nonloop.closure_eq_closure_iff_eq_or_dep | Status.FAILED | α : Type u_1
β : Type u_2
M N : Matroid α
e f : α
B L L' I X Y Z F C K : Set α
he : M.Nonloop e
hf : M.Nonloop f
⊢ M.closure {e} = M.closure {f} ↔ e = f ∨ ¬M.Indep {e, f} | None | [Goal(assumptions=[Declaration(ident='α', lean_type='Type u_1'), Declaration(ident='β', lean_type='Type u_2'), Declaration(ident='M', lean_type='Matroid α'), Declaration(ident='N', lean_type='Matroid α'), Declaration(ident='e', lean_type='α'), Declaration(ident='f', lean_type='α'), Declaration(ident='B', lean_type='Set α'), Declaration(ident='L', lean_type='Set α'), Declaration(ident="L'", lean_type='Set α'), Declaration(ident='I', lean_type='Set α'), Declaration(ident='X', lean_type='Set α'), Declaration(ident='Y', lean_type='Set α'), Declaration(ident='Z', lean_type='Set α'), Declaration(ident='F', lean_type='Set α'), Declaration(ident='C', lean_type='Set α'), Declaration(ident='K', lean_type='Set α'), Declaration(ident='he', lean_type='M.Nonloop e'), Declaration(ident='hf', lean_type='M.Nonloop f')], conclusion='M.closure {e} = M.closure {f} ↔ e = f ∨ ¬M.Indep {e, f}')] | obtain (rfl | hne) := eq_or_ne e f | <class 'lean_dojo.interaction.dojo.TacticState'> | TacticState(pp="case inl\nα : Type u_1\nβ : Type u_2\nM N : Matroid α\ne : α\nB L L' I X Y Z F C K : Set α\nhe hf : M.Nonloop e\n⊢ M.closure {e} = M.closure {e} ↔ e = e ∨ ¬M.Indep {e, e}\n\ncase inr\nα : Type u_1\nβ : Type u_2\nM N : Matroid α\ne f : α\nB L L' I X Y Z F C K : Set α\nhe : M.Nonloop e\nhf : M.Nonloop f\nhne : e ≠ f\n⊢ M.closure {e} = M.closure {f} ↔ e = f ∨ ¬M.Indep {e, f}", id=1, message='') | case inl
α : Type u_1
β : Type u_2
M N : Matroid α
e : α
B L L' I X Y Z F C K : Set α
he hf : M.Nonloop e
⊢ M.closure {e} = M.closure {e} ↔ e = e ∨ ¬M.Indep {e, e}
case inr
α : Type u_1
β : Type u_2
M N : Matroid α
e f : α
B L L' I X Y Z F C K : Set α
he : M.Nonloop e
hf : M.Nonloop f
hne : e ≠ f
⊢ M.closure {e} = M.closure {f} ↔ e = f ∨ ¬M.Indep {e, f} |
https://github.com/apnelson1/Matroid | 510262f56c0025be19502ab3457a788e2f6d2d1c | Matroid/Loop.lean | Matroid.Nonloop.closure_eq_closure_iff_eq_or_dep | Status.FAILED | case inl
α : Type u_1
β : Type u_2
M N : Matroid α
e : α
B L L' I X Y Z F C K : Set α
he hf : M.Nonloop e
⊢ M.closure {e} = M.closure {e} ↔ e = e ∨ ¬M.Indep {e, e}
case inr
α : Type u_1
β : Type u_2
M N : Matroid α
e f : α
B L L' I X Y Z F C K : Set α
he : M.Nonloop e
hf : M.Nonloop f
hne : e ≠ f
⊢ M.closure {e} = M.closure {f} ↔ e = f ∨ ¬M.Indep {e, f} | [Goal(assumptions=[Declaration(ident='α', lean_type='Type u_1'), Declaration(ident='β', lean_type='Type u_2'), Declaration(ident='M', lean_type='Matroid α'), Declaration(ident='N', lean_type='Matroid α'), Declaration(ident='e', lean_type='α'), Declaration(ident='B', lean_type='Set α'), Declaration(ident='L', lean_type='Set α'), Declaration(ident="L'", lean_type='Set α'), Declaration(ident='I', lean_type='Set α'), Declaration(ident='X', lean_type='Set α'), Declaration(ident='Y', lean_type='Set α'), Declaration(ident='Z', lean_type='Set α'), Declaration(ident='F', lean_type='Set α'), Declaration(ident='C', lean_type='Set α'), Declaration(ident='K', lean_type='Set α'), Declaration(ident='he', lean_type='M.Nonloop e'), Declaration(ident='hf', lean_type='M.Nonloop e')], conclusion='M.closure {e} = M.closure {e} ↔ e = e ∨ ¬M.Indep {e, e}'), Goal(assumptions=[Declaration(ident='α', lean_type='Type u_1'), Declaration(ident='β', lean_type='Type u_2'), Declaration(ident='M', lean_type='Matroid α'), Declaration(ident='N', lean_type='Matroid α'), Declaration(ident='e', lean_type='α'), Declaration(ident='f', lean_type='α'), Declaration(ident='B', lean_type='Set α'), Declaration(ident='L', lean_type='Set α'), Declaration(ident="L'", lean_type='Set α'), Declaration(ident='I', lean_type='Set α'), Declaration(ident='X', lean_type='Set α'), Declaration(ident='Y', lean_type='Set α'), Declaration(ident='Z', lean_type='Set α'), Declaration(ident='F', lean_type='Set α'), Declaration(ident='C', lean_type='Set α'), Declaration(ident='K', lean_type='Set α'), Declaration(ident='he', lean_type='M.Nonloop e'), Declaration(ident='hf', lean_type='M.Nonloop f'), Declaration(ident='hne', lean_type='e ≠ f')], conclusion='M.closure {e} = M.closure {f} ↔ e = f ∨ ¬M.Indep {e, f}')] | simp_rw [he.closure_eq_closure_iff_circuit_of_ne hne, or_iff_right hne,
circuit_iff_dep_forall_diff_singleton_indep, dep_iff, insert_subset_iff, singleton_subset_iff,
and_iff_left hf.mem_ground, and_iff_left he.mem_ground, and_iff_left_iff_imp] | <class 'lean_dojo.interaction.dojo.LeanError'> | LeanError(error='simp made no progress') | simp made no progress |
|
https://github.com/apnelson1/Matroid | 510262f56c0025be19502ab3457a788e2f6d2d1c | Matroid/Flat.lean | Matroid.hyperplane_iff_maximal_proper_flat | Status.PROVED | α : Type u_1
M : Matroid α
I F X Y F' F₀ F₁ F₂ P L H H₁ H₂ H' B C K : Set α
e f : α
⊢ M.Hyperplane H ↔ M.Flat H ∧ H ⊂ M.E ∧ ∀ (F : Set α), H ⊂ F → M.Flat F → F = M.E | None | [Goal(assumptions=[Declaration(ident='α', lean_type='Type u_1'), Declaration(ident='M', lean_type='Matroid α'), Declaration(ident='I', lean_type='Set α'), Declaration(ident='F', lean_type='Set α'), Declaration(ident='X', lean_type='Set α'), Declaration(ident='Y', lean_type='Set α'), Declaration(ident="F'", lean_type='Set α'), Declaration(ident='F₀', lean_type='Set α'), Declaration(ident='F₁', lean_type='Set α'), Declaration(ident='F₂', lean_type='Set α'), Declaration(ident='P', lean_type='Set α'), Declaration(ident='L', lean_type='Set α'), Declaration(ident='H', lean_type='Set α'), Declaration(ident='H₁', lean_type='Set α'), Declaration(ident='H₂', lean_type='Set α'), Declaration(ident="H'", lean_type='Set α'), Declaration(ident='B', lean_type='Set α'), Declaration(ident='C', lean_type='Set α'), Declaration(ident='K', lean_type='Set α'), Declaration(ident='e', lean_type='α'), Declaration(ident='f', lean_type='α')], conclusion='M.Hyperplane H ↔ M.Flat H ∧ H ⊂ M.E ∧ ∀ (F : Set α), H ⊂ F → M.Flat F → F = M.E')] | rw [hyperplane_iff_covBy, covBy_iff, and_iff_right M.ground_flat, and_congr_right_iff,
and_congr_right_iff] | <class 'lean_dojo.interaction.dojo.TacticState'> | TacticState(pp="α : Type u_1\nM : Matroid α\nI F X Y F' F₀ F₁ F₂ P L H H₁ H₂ H' B C K : Set α\ne f : α\n⊢ M.Flat H →\n H ⊂ M.E →\n ((∀ (F : Set α), M.Flat F → H ⊆ F → F ⊆ M.E → F = H ∨ F = M.E) ↔ ∀ (F : Set α), H ⊂ F → M.Flat F → F = M.E)", id=1, message='') | α : Type u_1
M : Matroid α
I F X Y F' F₀ F₁ F₂ P L H H₁ H₂ H' B C K : Set α
e f : α
⊢ M.Flat H →
H ⊂ M.E →
((∀ (F : Set α), M.Flat F → H ⊆ F → F ⊆ M.E → F = H ∨ F = M.E) ↔ ∀ (F : Set α), H ⊂ F → M.Flat F → F = M.E) |
https://github.com/apnelson1/Matroid | 510262f56c0025be19502ab3457a788e2f6d2d1c | Matroid/Flat.lean | Matroid.hyperplane_iff_maximal_proper_flat | Status.PROVED | α : Type u_1
M : Matroid α
I F X Y F' F₀ F₁ F₂ P L H H₁ H₂ H' B C K : Set α
e f : α
⊢ M.Flat H →
H ⊂ M.E →
((∀ (F : Set α), M.Flat F → H ⊆ F → F ⊆ M.E → F = H ∨ F = M.E) ↔ ∀ (F : Set α), H ⊂ F → M.Flat F → F = M.E) | [Goal(assumptions=[Declaration(ident='α', lean_type='Type u_1'), Declaration(ident='M', lean_type='Matroid α'), Declaration(ident='I', lean_type='Set α'), Declaration(ident='F', lean_type='Set α'), Declaration(ident='X', lean_type='Set α'), Declaration(ident='Y', lean_type='Set α'), Declaration(ident="F'", lean_type='Set α'), Declaration(ident='F₀', lean_type='Set α'), Declaration(ident='F₁', lean_type='Set α'), Declaration(ident='F₂', lean_type='Set α'), Declaration(ident='P', lean_type='Set α'), Declaration(ident='L', lean_type='Set α'), Declaration(ident='H', lean_type='Set α'), Declaration(ident='H₁', lean_type='Set α'), Declaration(ident='H₂', lean_type='Set α'), Declaration(ident="H'", lean_type='Set α'), Declaration(ident='B', lean_type='Set α'), Declaration(ident='C', lean_type='Set α'), Declaration(ident='K', lean_type='Set α'), Declaration(ident='e', lean_type='α'), Declaration(ident='f', lean_type='α')], conclusion='M.Flat H →\n H ⊂ M.E →\n ((∀ (F : Set α), M.Flat F → H ⊆ F → F ⊆ M.E → F = H ∨ F = M.E) ↔ ∀ (F : Set α), H ⊂ F → M.Flat F → F = M.E)')] | simp_rw [or_iff_not_imp_left, ssubset_iff_subset_ne, and_imp] | <class 'lean_dojo.interaction.dojo.TacticState'> | TacticState(pp="α : Type u_1\nM : Matroid α\nI F X Y F' F₀ F₁ F₂ P L H H₁ H₂ H' B C K : Set α\ne f : α\n⊢ M.Flat H →\n H ⊆ M.E →\n H ≠ M.E →\n ((∀ (F : Set α), M.Flat F → H ⊆ F → F ⊆ M.E → ¬F = H → F = M.E) ↔\n ∀ (F : Set α), H ⊆ F → H ≠ F → M.Flat F → F = M.E)", id=2, message='') | α : Type u_1
M : Matroid α
I F X Y F' F₀ F₁ F₂ P L H H₁ H₂ H' B C K : Set α
e f : α
⊢ M.Flat H →
H ⊆ M.E →
H ≠ M.E →
((∀ (F : Set α), M.Flat F → H ⊆ F → F ⊆ M.E → ¬F = H → F = M.E) ↔
∀ (F : Set α), H ⊆ F → H ≠ F → M.Flat F → F = M.E) |
|
https://github.com/apnelson1/Matroid | 510262f56c0025be19502ab3457a788e2f6d2d1c | Matroid/Flat.lean | Matroid.hyperplane_iff_maximal_proper_flat | Status.PROVED | α : Type u_1
M : Matroid α
I F X Y F' F₀ F₁ F₂ P L H H₁ H₂ H' B C K : Set α
e f : α
⊢ M.Flat H →
H ⊆ M.E →
H ≠ M.E →
((∀ (F : Set α), M.Flat F → H ⊆ F → F ⊆ M.E → ¬F = H → F = M.E) ↔
∀ (F : Set α), H ⊆ F → H ≠ F → M.Flat F → F = M.E) | [Goal(assumptions=[Declaration(ident='α', lean_type='Type u_1'), Declaration(ident='M', lean_type='Matroid α'), Declaration(ident='I', lean_type='Set α'), Declaration(ident='F', lean_type='Set α'), Declaration(ident='X', lean_type='Set α'), Declaration(ident='Y', lean_type='Set α'), Declaration(ident="F'", lean_type='Set α'), Declaration(ident='F₀', lean_type='Set α'), Declaration(ident='F₁', lean_type='Set α'), Declaration(ident='F₂', lean_type='Set α'), Declaration(ident='P', lean_type='Set α'), Declaration(ident='L', lean_type='Set α'), Declaration(ident='H', lean_type='Set α'), Declaration(ident='H₁', lean_type='Set α'), Declaration(ident='H₂', lean_type='Set α'), Declaration(ident="H'", lean_type='Set α'), Declaration(ident='B', lean_type='Set α'), Declaration(ident='C', lean_type='Set α'), Declaration(ident='K', lean_type='Set α'), Declaration(ident='e', lean_type='α'), Declaration(ident='f', lean_type='α')], conclusion='M.Flat H →\n H ⊆ M.E →\n H ≠ M.E →\n ((∀ (F : Set α), M.Flat F → H ⊆ F → F ⊆ M.E → ¬F = H → F = M.E) ↔\n ∀ (F : Set α), H ⊆ F → H ≠ F → M.Flat F → F = M.E)')] | exact fun _ _ _ ↦
⟨fun h F hHF hne' hF ↦ h F hF hHF hF.subset_ground hne'.symm, fun h F hF hHF _ hne' ↦
h F hHF (Ne.symm hne') hF⟩ | <class 'lean_dojo.interaction.dojo.ProofFinished'> | ProofFinished(tactic_state_id=3, message='') | ||
https://github.com/apnelson1/Matroid | 510262f56c0025be19502ab3457a788e2f6d2d1c | Matroid/Constructions/Uniform.lean | Matroid.unif_isoMinor_unif_iff | Status.PROVED | α : Type u_1
M : Matroid α
E I B X C : Set α
k : ℕ∞
a b a' b' a₁ a₂ d₁ d₂ : ℕ
⊢ Nonempty (unif a₁ (a₁ + d₁) ≤i unif a₂ (a₂ + d₂)) ↔ a₁ ≤ a₂ ∧ d₁ ≤ d₂ | None | [Goal(assumptions=[Declaration(ident='α', lean_type='Type u_1'), Declaration(ident='M', lean_type='Matroid α'), Declaration(ident='E', lean_type='Set α'), Declaration(ident='I', lean_type='Set α'), Declaration(ident='B', lean_type='Set α'), Declaration(ident='X', lean_type='Set α'), Declaration(ident='C', lean_type='Set α'), Declaration(ident='k', lean_type='ℕ∞'), Declaration(ident='a', lean_type='ℕ'), Declaration(ident='b', lean_type='ℕ'), Declaration(ident="a'", lean_type='ℕ'), Declaration(ident="b'", lean_type='ℕ'), Declaration(ident='a₁', lean_type='ℕ'), Declaration(ident='a₂', lean_type='ℕ'), Declaration(ident='d₁', lean_type='ℕ'), Declaration(ident='d₂', lean_type='ℕ')], conclusion='Nonempty (unif a₁ (a₁ + d₁) ≤i unif a₂ (a₂ + d₂)) ↔ a₁ ≤ a₂ ∧ d₁ ≤ d₂')] | refine ⟨fun ⟨e⟩ ↦ ⟨by simpa using e.rk_le, by simpa using IsoMinor.rk_le e.dual⟩, fun h ↦ ?_⟩ | <class 'lean_dojo.interaction.dojo.TacticState'> | TacticState(pp="α : Type u_1\nM : Matroid α\nE I B X C : Set α\nk : ℕ∞\na b a' b' a₁ a₂ d₁ d₂ : ℕ\nh : a₁ ≤ a₂ ∧ d₁ ≤ d₂\n⊢ Nonempty (unif a₁ (a₁ + d₁) ≤i unif a₂ (a₂ + d₂))", id=1, message='') | α : Type u_1
M : Matroid α
E I B X C : Set α
k : ℕ∞
a b a' b' a₁ a₂ d₁ d₂ : ℕ
h : a₁ ≤ a₂ ∧ d₁ ≤ d₂
⊢ Nonempty (unif a₁ (a₁ + d₁) ≤i unif a₂ (a₂ + d₂)) |
https://github.com/apnelson1/Matroid | 510262f56c0025be19502ab3457a788e2f6d2d1c | Matroid/Constructions/Uniform.lean | Matroid.unif_isoMinor_unif_iff | Status.PROVED | α : Type u_1
M : Matroid α
E I B X C : Set α
k : ℕ∞
a b a' b' a₁ a₂ d₁ d₂ : ℕ
h : a₁ ≤ a₂ ∧ d₁ ≤ d₂
⊢ Nonempty (unif a₁ (a₁ + d₁) ≤i unif a₂ (a₂ + d₂)) | [Goal(assumptions=[Declaration(ident='α', lean_type='Type u_1'), Declaration(ident='M', lean_type='Matroid α'), Declaration(ident='E', lean_type='Set α'), Declaration(ident='I', lean_type='Set α'), Declaration(ident='B', lean_type='Set α'), Declaration(ident='X', lean_type='Set α'), Declaration(ident='C', lean_type='Set α'), Declaration(ident='k', lean_type='ℕ∞'), Declaration(ident='a', lean_type='ℕ'), Declaration(ident='b', lean_type='ℕ'), Declaration(ident="a'", lean_type='ℕ'), Declaration(ident="b'", lean_type='ℕ'), Declaration(ident='a₁', lean_type='ℕ'), Declaration(ident='a₂', lean_type='ℕ'), Declaration(ident='d₁', lean_type='ℕ'), Declaration(ident='d₂', lean_type='ℕ'), Declaration(ident='h', lean_type='a₁ ≤ a₂ ∧ d₁ ≤ d₂')], conclusion='Nonempty (unif a₁ (a₁ + d₁) ≤i unif a₂ (a₂ + d₂))')] | obtain ⟨j, rfl⟩ := Nat.exists_eq_add_of_le h.1 | <class 'lean_dojo.interaction.dojo.TacticState'> | TacticState(pp="case intro\nα : Type u_1\nM : Matroid α\nE I B X C : Set α\nk : ℕ∞\na b a' b' a₁ d₁ d₂ j : ℕ\nh : a₁ ≤ a₁ + j ∧ d₁ ≤ d₂\n⊢ Nonempty (unif a₁ (a₁ + d₁) ≤i unif (a₁ + j) (a₁ + j + d₂))", id=2, message='') | case intro
α : Type u_1
M : Matroid α
E I B X C : Set α
k : ℕ∞
a b a' b' a₁ d₁ d₂ j : ℕ
h : a₁ ≤ a₁ + j ∧ d₁ ≤ d₂
⊢ Nonempty (unif a₁ (a₁ + d₁) ≤i unif (a₁ + j) (a₁ + j + d₂)) |
|
https://github.com/apnelson1/Matroid | 510262f56c0025be19502ab3457a788e2f6d2d1c | Matroid/Constructions/Uniform.lean | Matroid.unif_isoMinor_unif_iff | Status.PROVED | case intro
α : Type u_1
M : Matroid α
E I B X C : Set α
k : ℕ∞
a b a' b' a₁ d₁ d₂ j : ℕ
h : a₁ ≤ a₁ + j ∧ d₁ ≤ d₂
⊢ Nonempty (unif a₁ (a₁ + d₁) ≤i unif (a₁ + j) (a₁ + j + d₂)) | [Goal(assumptions=[Declaration(ident='α', lean_type='Type u_1'), Declaration(ident='M', lean_type='Matroid α'), Declaration(ident='E', lean_type='Set α'), Declaration(ident='I', lean_type='Set α'), Declaration(ident='B', lean_type='Set α'), Declaration(ident='X', lean_type='Set α'), Declaration(ident='C', lean_type='Set α'), Declaration(ident='k', lean_type='ℕ∞'), Declaration(ident='a', lean_type='ℕ'), Declaration(ident='b', lean_type='ℕ'), Declaration(ident="a'", lean_type='ℕ'), Declaration(ident="b'", lean_type='ℕ'), Declaration(ident='a₁', lean_type='ℕ'), Declaration(ident='d₁', lean_type='ℕ'), Declaration(ident='d₂', lean_type='ℕ'), Declaration(ident='j', lean_type='ℕ'), Declaration(ident='h', lean_type='a₁ ≤ a₁ + j ∧ d₁ ≤ d₂')], conclusion='Nonempty (unif a₁ (a₁ + d₁) ≤i unif (a₁ + j) (a₁ + j + d₂))')] | exact ⟨(unif_isoMinor_contr a₁ (a₁ + d₁) j).trans (unif_isoRestr_unif _ (by linarith)).isoMinor⟩ | <class 'lean_dojo.interaction.dojo.ProofFinished'> | ProofFinished(tactic_state_id=3, message='') | ||
https://github.com/apnelson1/Matroid | 510262f56c0025be19502ab3457a788e2f6d2d1c | Matroid/Rank.lean | Matroid.r_le_rk | Status.PROVED | 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
inst✝ : M.FiniteRk
X : Set α
⊢ M.r X ≤ M.rk | None | [Goal(assumptions=[Declaration(ident='α', lean_type='Type u_1'), Declaration(ident='ι', lean_type='Type u_2'), Declaration(ident='M✝', lean_type='Matroid α'), Declaration(ident='N', lean_type='Matroid α'), Declaration(ident='I', lean_type='Set α'), Declaration(ident='B', lean_type='Set α'), Declaration(ident='X✝', lean_type='Set α'), Declaration(ident="X'", lean_type='Set α'), Declaration(ident='Y', lean_type='Set α'), Declaration(ident="Y'", lean_type='Set α'), Declaration(ident='Z', lean_type='Set α'), Declaration(ident='R', lean_type='Set α'), Declaration(ident='n', lean_type='ℕ∞'), Declaration(ident='e', lean_type='α'), Declaration(ident='f', lean_type='α'), Declaration(ident='M', lean_type='Matroid α'), Declaration(ident='inst✝', lean_type='M.FiniteRk'), Declaration(ident='X', lean_type='Set α')], conclusion='M.r X ≤ M.rk')] | rw [r_eq_r_inter_ground, rk_def] | <class 'lean_dojo.interaction.dojo.TacticState'> | TacticState(pp="α : Type u_1\nι : Type u_2\nM✝ N : Matroid α\nI B X✝ X' Y Y' Z R : Set α\nn : ℕ∞\ne f : α\nM : Matroid α\ninst✝ : M.FiniteRk\nX : Set α\n⊢ M.r (X ∩ M.E) ≤ M.r M.E", id=1, message='') | 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
inst✝ : M.FiniteRk
X : Set α
⊢ M.r (X ∩ M.E) ≤ M.r M.E |
https://github.com/apnelson1/Matroid | 510262f56c0025be19502ab3457a788e2f6d2d1c | Matroid/Rank.lean | Matroid.r_le_rk | Status.PROVED | 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
inst✝ : M.FiniteRk
X : Set α
⊢ M.r (X ∩ M.E) ≤ M.r M.E | [Goal(assumptions=[Declaration(ident='α', lean_type='Type u_1'), Declaration(ident='ι', lean_type='Type u_2'), Declaration(ident='M✝', lean_type='Matroid α'), Declaration(ident='N', lean_type='Matroid α'), Declaration(ident='I', lean_type='Set α'), Declaration(ident='B', lean_type='Set α'), Declaration(ident='X✝', lean_type='Set α'), Declaration(ident="X'", lean_type='Set α'), Declaration(ident='Y', lean_type='Set α'), Declaration(ident="Y'", lean_type='Set α'), Declaration(ident='Z', lean_type='Set α'), Declaration(ident='R', lean_type='Set α'), Declaration(ident='n', lean_type='ℕ∞'), Declaration(ident='e', lean_type='α'), Declaration(ident='f', lean_type='α'), Declaration(ident='M', lean_type='Matroid α'), Declaration(ident='inst✝', lean_type='M.FiniteRk'), Declaration(ident='X', lean_type='Set α')], conclusion='M.r (X ∩ M.E) ≤ M.r M.E')] | exact M.r_mono inter_subset_right | <class 'lean_dojo.interaction.dojo.ProofFinished'> | ProofFinished(tactic_state_id=2, message='') | ||
https://github.com/apnelson1/Matroid | 510262f56c0025be19502ab3457a788e2f6d2d1c | Matroid/Minor/Rank.lean | Matroid.relRank_add_of_subset_of_subset | Status.PROVED | α : Type u_1
M✝ N : Matroid α
I J C D X Y Z : Set α
e f : α
M : Matroid α
hXY : X ⊆ Y
hYZ : Y ⊆ Z
⊢ M.relRank X Y + M.relRank Y Z = M.relRank X Z | None | [Goal(assumptions=[Declaration(ident='α', lean_type='Type u_1'), Declaration(ident='M✝', lean_type='Matroid α'), Declaration(ident='N', lean_type='Matroid α'), Declaration(ident='I', lean_type='Set α'), Declaration(ident='J', lean_type='Set α'), Declaration(ident='C', lean_type='Set α'), Declaration(ident='D', lean_type='Set α'), Declaration(ident='X', lean_type='Set α'), Declaration(ident='Y', lean_type='Set α'), Declaration(ident='Z', lean_type='Set α'), Declaration(ident='e', lean_type='α'), Declaration(ident='f', lean_type='α'), Declaration(ident='M', lean_type='Matroid α'), Declaration(ident='hXY', lean_type='X ⊆ Y'), Declaration(ident='hYZ', lean_type='Y ⊆ Z')], conclusion='M.relRank X Y + M.relRank Y Z = M.relRank X Z')] | obtain ⟨I, hI⟩ := M.exists_basis' X | <class 'lean_dojo.interaction.dojo.TacticState'> | TacticState(pp="case intro\nα : Type u_1\nM✝ N : Matroid α\nI✝ J C D X Y Z : Set α\ne f : α\nM : Matroid α\nhXY : X ⊆ Y\nhYZ : Y ⊆ Z\nI : Set α\nhI : M.Basis' I X\n⊢ M.relRank X Y + M.relRank Y Z = M.relRank X Z", id=1, message='') | case intro
α : Type u_1
M✝ N : Matroid α
I✝ J C D X Y Z : Set α
e f : α
M : Matroid α
hXY : X ⊆ Y
hYZ : Y ⊆ Z
I : Set α
hI : M.Basis' I X
⊢ M.relRank X Y + M.relRank Y Z = M.relRank X Z |
https://github.com/apnelson1/Matroid | 510262f56c0025be19502ab3457a788e2f6d2d1c | Matroid/Minor/Rank.lean | Matroid.relRank_add_of_subset_of_subset | Status.PROVED | case intro
α : Type u_1
M✝ N : Matroid α
I✝ J C D X Y Z : Set α
e f : α
M : Matroid α
hXY : X ⊆ Y
hYZ : Y ⊆ Z
I : Set α
hI : M.Basis' I X
⊢ M.relRank X Y + M.relRank Y Z = M.relRank X Z | [Goal(assumptions=[Declaration(ident='α', lean_type='Type u_1'), Declaration(ident='M✝', lean_type='Matroid α'), Declaration(ident='N', lean_type='Matroid α'), Declaration(ident='I✝', lean_type='Set α'), Declaration(ident='J', lean_type='Set α'), Declaration(ident='C', lean_type='Set α'), Declaration(ident='D', lean_type='Set α'), Declaration(ident='X', lean_type='Set α'), Declaration(ident='Y', lean_type='Set α'), Declaration(ident='Z', lean_type='Set α'), Declaration(ident='e', lean_type='α'), Declaration(ident='f', lean_type='α'), Declaration(ident='M', lean_type='Matroid α'), Declaration(ident='hXY', lean_type='X ⊆ Y'), Declaration(ident='hYZ', lean_type='Y ⊆ Z'), Declaration(ident='I', lean_type='Set α'), Declaration(ident='hI', lean_type="M.Basis' I X")], conclusion='M.relRank X Y + M.relRank Y Z = M.relRank X Z')] | obtain ⟨J, hJ, hIJ⟩ := hI.indep.subset_basis'_of_subset (hI.subset.trans hXY) | <class 'lean_dojo.interaction.dojo.TacticState'> | TacticState(pp="case intro.intro.intro\nα : Type u_1\nM✝ N : Matroid α\nI✝ J✝ C D X Y Z : Set α\ne f : α\nM : Matroid α\nhXY : X ⊆ Y\nhYZ : Y ⊆ Z\nI : Set α\nhI : M.Basis' I X\nJ : Set α\nhJ : M.Basis' J Y\nhIJ : I ⊆ J\n⊢ M.relRank X Y + M.relRank Y Z = M.relRank X Z", id=2, message='') | case intro.intro.intro
α : Type u_1
M✝ N : Matroid α
I✝ J✝ C D X Y Z : Set α
e f : α
M : Matroid α
hXY : X ⊆ Y
hYZ : Y ⊆ Z
I : Set α
hI : M.Basis' I X
J : Set α
hJ : M.Basis' J Y
hIJ : I ⊆ J
⊢ M.relRank X Y + M.relRank Y Z = M.relRank X Z |
|
https://github.com/apnelson1/Matroid | 510262f56c0025be19502ab3457a788e2f6d2d1c | Matroid/Minor/Rank.lean | Matroid.relRank_add_of_subset_of_subset | Status.PROVED | case intro.intro.intro
α : Type u_1
M✝ N : Matroid α
I✝ J✝ C D X Y Z : Set α
e f : α
M : Matroid α
hXY : X ⊆ Y
hYZ : Y ⊆ Z
I : Set α
hI : M.Basis' I X
J : Set α
hJ : M.Basis' J Y
hIJ : I ⊆ J
⊢ M.relRank X Y + M.relRank Y Z = M.relRank X Z | [Goal(assumptions=[Declaration(ident='α', lean_type='Type u_1'), Declaration(ident='M✝', lean_type='Matroid α'), Declaration(ident='N', lean_type='Matroid α'), Declaration(ident='I✝', lean_type='Set α'), Declaration(ident='J✝', lean_type='Set α'), Declaration(ident='C', lean_type='Set α'), Declaration(ident='D', lean_type='Set α'), Declaration(ident='X', lean_type='Set α'), Declaration(ident='Y', lean_type='Set α'), Declaration(ident='Z', lean_type='Set α'), Declaration(ident='e', lean_type='α'), Declaration(ident='f', lean_type='α'), Declaration(ident='M', lean_type='Matroid α'), Declaration(ident='hXY', lean_type='X ⊆ Y'), Declaration(ident='hYZ', lean_type='Y ⊆ Z'), Declaration(ident='I', lean_type='Set α'), Declaration(ident='hI', lean_type="M.Basis' I X"), Declaration(ident='J', lean_type='Set α'), Declaration(ident='hJ', lean_type="M.Basis' J Y"), Declaration(ident='hIJ', lean_type='I ⊆ J')], conclusion='M.relRank X Y + M.relRank Y Z = M.relRank X Z')] | obtain ⟨K, hK, hJK⟩ := hJ.indep.subset_basis'_of_subset (hJ.subset.trans hYZ) | <class 'lean_dojo.interaction.dojo.TacticState'> | TacticState(pp="case intro.intro.intro.intro.intro\nα : Type u_1\nM✝ N : Matroid α\nI✝ J✝ C D X Y Z : Set α\ne f : α\nM : Matroid α\nhXY : X ⊆ Y\nhYZ : Y ⊆ Z\nI : Set α\nhI : M.Basis' I X\nJ : Set α\nhJ : M.Basis' J Y\nhIJ : I ⊆ J\nK : Set α\nhK : M.Basis' K Z\nhJK : J ⊆ K\n⊢ M.relRank X Y + M.relRank Y Z = M.relRank X Z", id=3, message='') | case intro.intro.intro.intro.intro
α : Type u_1
M✝ N : Matroid α
I✝ J✝ C D X Y Z : Set α
e f : α
M : Matroid α
hXY : X ⊆ Y
hYZ : Y ⊆ Z
I : Set α
hI : M.Basis' I X
J : Set α
hJ : M.Basis' J Y
hIJ : I ⊆ J
K : Set α
hK : M.Basis' K Z
hJK : J ⊆ K
⊢ M.relRank X Y + M.relRank Y Z = M.relRank X Z |
|
https://github.com/apnelson1/Matroid | 510262f56c0025be19502ab3457a788e2f6d2d1c | Matroid/Minor/Rank.lean | Matroid.relRank_add_of_subset_of_subset | Status.PROVED | case intro.intro.intro.intro.intro
α : Type u_1
M✝ N : Matroid α
I✝ J✝ C D X Y Z : Set α
e f : α
M : Matroid α
hXY : X ⊆ Y
hYZ : Y ⊆ Z
I : Set α
hI : M.Basis' I X
J : Set α
hJ : M.Basis' J Y
hIJ : I ⊆ J
K : Set α
hK : M.Basis' K Z
hJK : J ⊆ K
⊢ M.relRank X Y + M.relRank Y Z = M.relRank X Z | [Goal(assumptions=[Declaration(ident='α', lean_type='Type u_1'), Declaration(ident='M✝', lean_type='Matroid α'), Declaration(ident='N', lean_type='Matroid α'), Declaration(ident='I✝', lean_type='Set α'), Declaration(ident='J✝', lean_type='Set α'), Declaration(ident='C', lean_type='Set α'), Declaration(ident='D', lean_type='Set α'), Declaration(ident='X', lean_type='Set α'), Declaration(ident='Y', lean_type='Set α'), Declaration(ident='Z', lean_type='Set α'), Declaration(ident='e', lean_type='α'), Declaration(ident='f', lean_type='α'), Declaration(ident='M', lean_type='Matroid α'), Declaration(ident='hXY', lean_type='X ⊆ Y'), Declaration(ident='hYZ', lean_type='Y ⊆ Z'), Declaration(ident='I', lean_type='Set α'), Declaration(ident='hI', lean_type="M.Basis' I X"), Declaration(ident='J', lean_type='Set α'), Declaration(ident='hJ', lean_type="M.Basis' J Y"), Declaration(ident='hIJ', lean_type='I ⊆ J'), Declaration(ident='K', lean_type='Set α'), Declaration(ident='hK', lean_type="M.Basis' K Z"), Declaration(ident='hJK', lean_type='J ⊆ K')], conclusion='M.relRank X Y + M.relRank Y Z = M.relRank X Z')] | obtain rfl := hI.inter_eq_of_subset_indep hIJ hJ.indep | <class 'lean_dojo.interaction.dojo.TacticState'> | TacticState(pp="case intro.intro.intro.intro.intro\nα : Type u_1\nM✝ N : Matroid α\nI J✝ C D X Y Z : Set α\ne f : α\nM : Matroid α\nhXY : X ⊆ Y\nhYZ : Y ⊆ Z\nJ : Set α\nhJ : M.Basis' J Y\nK : Set α\nhK : M.Basis' K Z\nhJK : J ⊆ K\nhI : M.Basis' (J ∩ X) X\nhIJ : J ∩ X ⊆ J\n⊢ M.relRank X Y + M.relRank Y Z = M.relRank X Z", id=4, message='') | case intro.intro.intro.intro.intro
α : Type u_1
M✝ N : Matroid α
I J✝ C D X Y Z : Set α
e f : α
M : Matroid α
hXY : X ⊆ Y
hYZ : Y ⊆ Z
J : Set α
hJ : M.Basis' J Y
K : Set α
hK : M.Basis' K Z
hJK : J ⊆ K
hI : M.Basis' (J ∩ X) X
hIJ : J ∩ X ⊆ J
⊢ M.relRank X Y + M.relRank Y Z = M.relRank X Z |
|
https://github.com/apnelson1/Matroid | 510262f56c0025be19502ab3457a788e2f6d2d1c | Matroid/Minor/Rank.lean | Matroid.relRank_add_of_subset_of_subset | Status.PROVED | case intro.intro.intro.intro.intro
α : Type u_1
M✝ N : Matroid α
I J✝ C D X Y Z : Set α
e f : α
M : Matroid α
hXY : X ⊆ Y
hYZ : Y ⊆ Z
J : Set α
hJ : M.Basis' J Y
K : Set α
hK : M.Basis' K Z
hJK : J ⊆ K
hI : M.Basis' (J ∩ X) X
hIJ : J ∩ X ⊆ J
⊢ M.relRank X Y + M.relRank Y Z = M.relRank X Z | [Goal(assumptions=[Declaration(ident='α', lean_type='Type u_1'), Declaration(ident='M✝', lean_type='Matroid α'), Declaration(ident='N', lean_type='Matroid α'), Declaration(ident='I', lean_type='Set α'), Declaration(ident='J✝', lean_type='Set α'), Declaration(ident='C', lean_type='Set α'), Declaration(ident='D', lean_type='Set α'), Declaration(ident='X', lean_type='Set α'), Declaration(ident='Y', lean_type='Set α'), Declaration(ident='Z', lean_type='Set α'), Declaration(ident='e', lean_type='α'), Declaration(ident='f', lean_type='α'), Declaration(ident='M', lean_type='Matroid α'), Declaration(ident='hXY', lean_type='X ⊆ Y'), Declaration(ident='hYZ', lean_type='Y ⊆ Z'), Declaration(ident='J', lean_type='Set α'), Declaration(ident='hJ', lean_type="M.Basis' J Y"), Declaration(ident='K', lean_type='Set α'), Declaration(ident='hK', lean_type="M.Basis' K Z"), Declaration(ident='hJK', lean_type='J ⊆ K'), Declaration(ident='hI', lean_type="M.Basis' (J ∩ X) X"), Declaration(ident='hIJ', lean_type='J ∩ X ⊆ J')], conclusion='M.relRank X Y + M.relRank Y Z = M.relRank X Z')] | obtain rfl := hJ.inter_eq_of_subset_indep hJK hK.indep | <class 'lean_dojo.interaction.dojo.TacticState'> | TacticState(pp="case intro.intro.intro.intro.intro\nα : Type u_1\nM✝ N : Matroid α\nI J C D X Y Z : Set α\ne f : α\nM : Matroid α\nhXY : X ⊆ Y\nhYZ : Y ⊆ Z\nK : Set α\nhK : M.Basis' K Z\nhJ : M.Basis' (K ∩ Y) Y\nhJK : K ∩ Y ⊆ K\nhI : M.Basis' (K ∩ Y ∩ X) X\nhIJ : K ∩ Y ∩ X ⊆ K ∩ Y\n⊢ M.relRank X Y + M.relRank Y Z = M.relRank X Z", id=5, message='') | case intro.intro.intro.intro.intro
α : Type u_1
M✝ N : Matroid α
I J C D X Y Z : Set α
e f : α
M : Matroid α
hXY : X ⊆ Y
hYZ : Y ⊆ Z
K : Set α
hK : M.Basis' K Z
hJ : M.Basis' (K ∩ Y) Y
hJK : K ∩ Y ⊆ K
hI : M.Basis' (K ∩ Y ∩ X) X
hIJ : K ∩ Y ∩ X ⊆ K ∩ Y
⊢ M.relRank X Y + M.relRank Y Z = M.relRank X Z |
|
https://github.com/apnelson1/Matroid | 510262f56c0025be19502ab3457a788e2f6d2d1c | Matroid/Minor/Rank.lean | Matroid.relRank_add_of_subset_of_subset | Status.PROVED | case intro.intro.intro.intro.intro
α : Type u_1
M✝ N : Matroid α
I J C D X Y Z : Set α
e f : α
M : Matroid α
hXY : X ⊆ Y
hYZ : Y ⊆ Z
K : Set α
hK : M.Basis' K Z
hJ : M.Basis' (K ∩ Y) Y
hJK : K ∩ Y ⊆ K
hI : M.Basis' (K ∩ Y ∩ X) X
hIJ : K ∩ Y ∩ X ⊆ K ∩ Y
⊢ M.relRank X Y + M.relRank Y Z = M.relRank X Z | [Goal(assumptions=[Declaration(ident='α', lean_type='Type u_1'), Declaration(ident='M✝', lean_type='Matroid α'), Declaration(ident='N', lean_type='Matroid α'), Declaration(ident='I', lean_type='Set α'), Declaration(ident='J', lean_type='Set α'), Declaration(ident='C', lean_type='Set α'), Declaration(ident='D', lean_type='Set α'), Declaration(ident='X', lean_type='Set α'), Declaration(ident='Y', lean_type='Set α'), Declaration(ident='Z', lean_type='Set α'), Declaration(ident='e', lean_type='α'), Declaration(ident='f', lean_type='α'), Declaration(ident='M', lean_type='Matroid α'), Declaration(ident='hXY', lean_type='X ⊆ Y'), Declaration(ident='hYZ', lean_type='Y ⊆ Z'), Declaration(ident='K', lean_type='Set α'), Declaration(ident='hK', lean_type="M.Basis' K Z"), Declaration(ident='hJ', lean_type="M.Basis' (K ∩ Y) Y"), Declaration(ident='hJK', lean_type='K ∩ Y ⊆ K'), Declaration(ident='hI', lean_type="M.Basis' (K ∩ Y ∩ X) X"), Declaration(ident='hIJ', lean_type='K ∩ Y ∩ X ⊆ K ∩ Y')], conclusion='M.relRank X Y + M.relRank Y Z = M.relRank X Z')] | rw [hJ.relRank_eq_encard_diff_of_subset hXY hI, hK.relRank_eq_encard_diff_of_subset hYZ hJ,
hK.relRank_eq_encard_diff_of_subset (hXY.trans hYZ)
(by rwa [inter_assoc, inter_eq_self_of_subset_right hXY] at hI),
← encard_union_eq, diff_eq, diff_eq, inter_assoc, ← inter_union_distrib_left,
inter_union_distrib_right, union_compl_self, univ_inter, ← compl_inter,
inter_eq_self_of_subset_left hXY, diff_eq] | <class 'lean_dojo.interaction.dojo.TacticState'> | TacticState(pp="case intro.intro.intro.intro.intro\nα : Type u_1\nM✝ N : Matroid α\nI J C D X Y Z : Set α\ne f : α\nM : Matroid α\nhXY : X ⊆ Y\nhYZ : Y ⊆ Z\nK : Set α\nhK : M.Basis' K Z\nhJ : M.Basis' (K ∩ Y) Y\nhJK : K ∩ Y ⊆ K\nhI : M.Basis' (K ∩ Y ∩ X) X\nhIJ : K ∩ Y ∩ X ⊆ K ∩ Y\n⊢ Disjoint ((K ∩ Y) \\ X) (K \\ Y)", id=6, message='') | case intro.intro.intro.intro.intro
α : Type u_1
M✝ N : Matroid α
I J C D X Y Z : Set α
e f : α
M : Matroid α
hXY : X ⊆ Y
hYZ : Y ⊆ Z
K : Set α
hK : M.Basis' K Z
hJ : M.Basis' (K ∩ Y) Y
hJK : K ∩ Y ⊆ K
hI : M.Basis' (K ∩ Y ∩ X) X
hIJ : K ∩ Y ∩ X ⊆ K ∩ Y
⊢ Disjoint ((K ∩ Y) \ X) (K \ Y) |
|
https://github.com/apnelson1/Matroid | 510262f56c0025be19502ab3457a788e2f6d2d1c | Matroid/Minor/Rank.lean | Matroid.relRank_add_of_subset_of_subset | Status.PROVED | case intro.intro.intro.intro.intro
α : Type u_1
M✝ N : Matroid α
I J C D X Y Z : Set α
e f : α
M : Matroid α
hXY : X ⊆ Y
hYZ : Y ⊆ Z
K : Set α
hK : M.Basis' K Z
hJ : M.Basis' (K ∩ Y) Y
hJK : K ∩ Y ⊆ K
hI : M.Basis' (K ∩ Y ∩ X) X
hIJ : K ∩ Y ∩ X ⊆ K ∩ Y
⊢ Disjoint ((K ∩ Y) \ X) (K \ Y) | [Goal(assumptions=[Declaration(ident='α', lean_type='Type u_1'), Declaration(ident='M✝', lean_type='Matroid α'), Declaration(ident='N', lean_type='Matroid α'), Declaration(ident='I', lean_type='Set α'), Declaration(ident='J', lean_type='Set α'), Declaration(ident='C', lean_type='Set α'), Declaration(ident='D', lean_type='Set α'), Declaration(ident='X', lean_type='Set α'), Declaration(ident='Y', lean_type='Set α'), Declaration(ident='Z', lean_type='Set α'), Declaration(ident='e', lean_type='α'), Declaration(ident='f', lean_type='α'), Declaration(ident='M', lean_type='Matroid α'), Declaration(ident='hXY', lean_type='X ⊆ Y'), Declaration(ident='hYZ', lean_type='Y ⊆ Z'), Declaration(ident='K', lean_type='Set α'), Declaration(ident='hK', lean_type="M.Basis' K Z"), Declaration(ident='hJ', lean_type="M.Basis' (K ∩ Y) Y"), Declaration(ident='hJK', lean_type='K ∩ Y ⊆ K'), Declaration(ident='hI', lean_type="M.Basis' (K ∩ Y ∩ X) X"), Declaration(ident='hIJ', lean_type='K ∩ Y ∩ X ⊆ K ∩ Y')], conclusion='Disjoint ((K ∩ Y) \\ X) (K \\ Y)')] | exact disjoint_of_subset_left (diff_subset.trans inter_subset_right)
disjoint_sdiff_right | <class 'lean_dojo.interaction.dojo.ProofFinished'> | ProofFinished(tactic_state_id=7, message='') | ||
https://github.com/apnelson1/Matroid | 510262f56c0025be19502ab3457a788e2f6d2d1c | Matroid/ForMathlib/Card.lean | Fin.nonempty_embedding_iff_le_encard | Status.FAILED | α : Type u_1
β : Type u_2
s t : Set α
n : ℕ
⊢ Nonempty (Fin n ↪ ↑s) ↔ ↑n ≤ s.encard | None | [Goal(assumptions=[Declaration(ident='α', lean_type='Type u_1'), Declaration(ident='β', lean_type='Type u_2'), Declaration(ident='s', lean_type='Set α'), Declaration(ident='t', lean_type='Set α'), Declaration(ident='n', lean_type='ℕ')], conclusion='Nonempty (Fin n ↪ ↑s) ↔ ↑n ≤ s.encard')] | refine ⟨fun ⟨i⟩ ↦ ?_, fun h ↦ ?_⟩ | <class 'lean_dojo.interaction.dojo.TacticState'> | TacticState(pp='case refine_1\nα : Type u_1\nβ : Type u_2\ns t : Set α\nn : ℕ\nx✝ : Nonempty (Fin n ↪ ↑s)\ni : Fin n ↪ ↑s\n⊢ ↑n ≤ s.encard\n\ncase refine_2\nα : Type u_1\nβ : Type u_2\ns t : Set α\nn : ℕ\nh : ↑n ≤ s.encard\n⊢ Nonempty (Fin n ↪ ↑s)', id=1, message='') | case refine_1
α : Type u_1
β : Type u_2
s t : Set α
n : ℕ
x✝ : Nonempty (Fin n ↪ ↑s)
i : Fin n ↪ ↑s
⊢ ↑n ≤ s.encard
case refine_2
α : Type u_1
β : Type u_2
s t : Set α
n : ℕ
h : ↑n ≤ s.encard
⊢ Nonempty (Fin n ↪ ↑s) |
https://github.com/apnelson1/Matroid | 510262f56c0025be19502ab3457a788e2f6d2d1c | Matroid/ForMathlib/Card.lean | Fin.nonempty_embedding_iff_le_encard | Status.FAILED | case refine_1
α : Type u_1
β : Type u_2
s t : Set α
n : ℕ
x✝ : Nonempty (Fin n ↪ ↑s)
i : Fin n ↪ ↑s
⊢ ↑n ≤ s.encard
case refine_2
α : Type u_1
β : Type u_2
s t : Set α
n : ℕ
h : ↑n ≤ s.encard
⊢ Nonempty (Fin n ↪ ↑s) | [Goal(assumptions=[Declaration(ident='α', lean_type='Type u_1'), Declaration(ident='β', lean_type='Type u_2'), Declaration(ident='s', lean_type='Set α'), Declaration(ident='t', lean_type='Set α'), Declaration(ident='n', lean_type='ℕ'), Declaration(ident='x✝', lean_type='Nonempty (Fin n ↪ ↑s)'), Declaration(ident='i', lean_type='Fin n ↪ ↑s')], conclusion='↑n ≤ s.encard'), Goal(assumptions=[Declaration(ident='α', lean_type='Type u_1'), Declaration(ident='β', lean_type='Type u_2'), Declaration(ident='s', lean_type='Set α'), Declaration(ident='t', lean_type='Set α'), Declaration(ident='n', lean_type='ℕ'), Declaration(ident='h', lean_type='↑n ≤ s.encard')], conclusion='Nonempty (Fin n ↪ ↑s)')] | obtain ⟨t, hts, hcard⟩ := exists_subset_encard_eq h | <class 'lean_dojo.interaction.dojo.LeanError'> | LeanError(error="unknown identifier 'h'") | unknown identifier 'h' |
|
https://github.com/apnelson1/Matroid | 510262f56c0025be19502ab3457a788e2f6d2d1c | Matroid/Extension.lean | Matroid.ModularCut.closure_mem_of_mem | Status.PROVED | α : Type u_1
M : Matroid α
I J B F₀ F F' X Y : Set α
e f : α
U : M.ModularCut
hF : F ∈ U
⊢ M.closure F ∈ U | None | [Goal(assumptions=[Declaration(ident='α', lean_type='Type u_1'), Declaration(ident='M', lean_type='Matroid α'), Declaration(ident='I', lean_type='Set α'), Declaration(ident='J', lean_type='Set α'), Declaration(ident='B', lean_type='Set α'), Declaration(ident='F₀', lean_type='Set α'), Declaration(ident='F', lean_type='Set α'), Declaration(ident="F'", lean_type='Set α'), Declaration(ident='X', lean_type='Set α'), Declaration(ident='Y', lean_type='Set α'), Declaration(ident='e', lean_type='α'), Declaration(ident='f', lean_type='α'), Declaration(ident='U', lean_type='M.ModularCut'), Declaration(ident='hF', lean_type='F ∈ U')], conclusion='M.closure F ∈ U')] | rwa [(U.flat_of_mem hF).closure] | <class 'lean_dojo.interaction.dojo.ProofFinished'> | ProofFinished(tactic_state_id=1, message='') | |
https://github.com/apnelson1/Matroid | 510262f56c0025be19502ab3457a788e2f6d2d1c | Matroid/ForMathlib/Card.lean | Set.coe_le_encard_iff | Status.FAILED | α : Type u_1
β : Type u_2
s t : Set α
n : ℕ
⊢ ↑n ≤ s.encard ↔ s.Finite → n ≤ s.ncard | None | [Goal(assumptions=[Declaration(ident='α', lean_type='Type u_1'), Declaration(ident='β', lean_type='Type u_2'), Declaration(ident='s', lean_type='Set α'), Declaration(ident='t', lean_type='Set α'), Declaration(ident='n', lean_type='ℕ')], conclusion='↑n ≤ s.encard ↔ s.Finite → n ≤ s.ncard')] | obtain (hfin | hinf) := s.finite_or_infinite | <class 'lean_dojo.interaction.dojo.TacticState'> | TacticState(pp='case inl\nα : Type u_1\nβ : Type u_2\ns t : Set α\nn : ℕ\nhfin : s.Finite\n⊢ ↑n ≤ s.encard ↔ s.Finite → n ≤ s.ncard\n\ncase inr\nα : Type u_1\nβ : Type u_2\ns t : Set α\nn : ℕ\nhinf : s.Infinite\n⊢ ↑n ≤ s.encard ↔ s.Finite → n ≤ s.ncard', id=1, message='') | case inl
α : Type u_1
β : Type u_2
s t : Set α
n : ℕ
hfin : s.Finite
⊢ ↑n ≤ s.encard ↔ s.Finite → n ≤ s.ncard
case inr
α : Type u_1
β : Type u_2
s t : Set α
n : ℕ
hinf : s.Infinite
⊢ ↑n ≤ s.encard ↔ s.Finite → n ≤ s.ncard |
https://github.com/apnelson1/Matroid | 510262f56c0025be19502ab3457a788e2f6d2d1c | Matroid/ForMathlib/Card.lean | Set.coe_le_encard_iff | Status.FAILED | case inl
α : Type u_1
β : Type u_2
s t : Set α
n : ℕ
hfin : s.Finite
⊢ ↑n ≤ s.encard ↔ s.Finite → n ≤ s.ncard
case inr
α : Type u_1
β : Type u_2
s t : Set α
n : ℕ
hinf : s.Infinite
⊢ ↑n ≤ s.encard ↔ s.Finite → n ≤ s.ncard | [Goal(assumptions=[Declaration(ident='α', lean_type='Type u_1'), Declaration(ident='β', lean_type='Type u_2'), Declaration(ident='s', lean_type='Set α'), Declaration(ident='t', lean_type='Set α'), Declaration(ident='n', lean_type='ℕ'), Declaration(ident='hfin', lean_type='s.Finite')], conclusion='↑n ≤ s.encard ↔ s.Finite → n ≤ s.ncard'), Goal(assumptions=[Declaration(ident='α', lean_type='Type u_1'), Declaration(ident='β', lean_type='Type u_2'), Declaration(ident='s', lean_type='Set α'), Declaration(ident='t', lean_type='Set α'), Declaration(ident='n', lean_type='ℕ'), Declaration(ident='hinf', lean_type='s.Infinite')], conclusion='↑n ≤ s.encard ↔ s.Finite → n ≤ s.ncard')] | rw [hinf.encard_eq, iff_true_intro le_top, true_iff, iff_false_intro hinf, false_imp_iff] | <class 'lean_dojo.interaction.dojo.LeanError'> | LeanError(error="tactic 'rewrite' failed, equality or iff proof expected\n ?m.5635\ncase inl\nα : Type u_1\nβ : Type u_2\ns t : Set α\nn : ℕ\nhfin : s.Finite\n⊢ ↑n ≤ s.encard ↔ s.Finite → n ≤ s.ncard") | tactic 'rewrite' failed, equality or iff proof expected
?m.5635
case inl
α : Type u_1
β : Type u_2
s t : Set α
n : ℕ
hfin : s.Finite
⊢ ↑n ≤ s.encard ↔ s.Finite → n ≤ s.ncard |
|
https://github.com/apnelson1/Matroid | 510262f56c0025be19502ab3457a788e2f6d2d1c | Matroid/Extension.lean | Matroid.ModularCut.maximal_extIndep_iff | Status.FAILED | α : Type u_1
M : Matroid α
I J B F₀ F F' X Y : Set α
e f : α
U : M.ModularCut
hX : X ⊆ insert e M.E
hI : U.ExtIndep e I
hIX : I ⊆ X
⊢ I ∈ maximals (fun x x_1 => x ⊆ x_1) {J | U.ExtIndep e J ∧ J ⊆ X} ↔
M.closure (I \ {e}) = M.closure (X \ {e}) ∧ ((e ∈ I ↔ M.closure (X \ {e}) ∈ U) → e ∉ X) ∨
(M.closure (I \ {e}) ⋖[M] M.closure (X \ {e})) ∧ e ∈ I ∧ M.closure (X \ {e}) ∈ U | None | [Goal(assumptions=[Declaration(ident='α', lean_type='Type u_1'), Declaration(ident='M', lean_type='Matroid α'), Declaration(ident='I', lean_type='Set α'), Declaration(ident='J', lean_type='Set α'), Declaration(ident='B', lean_type='Set α'), Declaration(ident='F₀', lean_type='Set α'), Declaration(ident='F', lean_type='Set α'), Declaration(ident="F'", lean_type='Set α'), Declaration(ident='X', lean_type='Set α'), Declaration(ident='Y', lean_type='Set α'), Declaration(ident='e', lean_type='α'), Declaration(ident='f', lean_type='α'), Declaration(ident='U', lean_type='M.ModularCut'), Declaration(ident='hX', lean_type='X ⊆ insert e M.E'), Declaration(ident='hI', lean_type='U.ExtIndep e I'), Declaration(ident='hIX', lean_type='I ⊆ X')], conclusion='I ∈ maximals (fun x x_1 => x ⊆ x_1) {J | U.ExtIndep e J ∧ J ⊆ X} ↔\n M.closure (I \\ {e}) = M.closure (X \\ {e}) ∧ ((e ∈ I ↔ M.closure (X \\ {e}) ∈ U) → e ∉ X) ∨\n (M.closure (I \\ {e}) ⋖[M] M.closure (X \\ {e})) ∧ e ∈ I ∧ M.closure (X \\ {e}) ∈ U')] | have hss : I \ {e} ⊆ X \ {e} := diff_subset_diff_left hIX | <class 'lean_dojo.interaction.dojo.TacticState'> | TacticState(pp="α : Type u_1\nM : Matroid α\nI J B F₀ F F' X Y : Set α\ne f : α\nU : M.ModularCut\nhX : X ⊆ insert e M.E\nhI : U.ExtIndep e I\nhIX : I ⊆ X\nhss : I \\ {e} ⊆ X \\ {e}\n⊢ I ∈ maximals (fun x x_1 => x ⊆ x_1) {J | U.ExtIndep e J ∧ J ⊆ X} ↔\n M.closure (I \\ {e}) = M.closure (X \\ {e}) ∧ ((e ∈ I ↔ M.closure (X \\ {e}) ∈ U) → e ∉ X) ∨\n (M.closure (I \\ {e}) ⋖[M] M.closure (X \\ {e})) ∧ e ∈ I ∧ M.closure (X \\ {e}) ∈ U", id=1, message='') | α : Type u_1
M : Matroid α
I J B F₀ F F' X Y : Set α
e f : α
U : M.ModularCut
hX : X ⊆ insert e M.E
hI : U.ExtIndep e I
hIX : I ⊆ X
hss : I \ {e} ⊆ X \ {e}
⊢ I ∈ maximals (fun x x_1 => x ⊆ x_1) {J | U.ExtIndep e J ∧ J ⊆ X} ↔
M.closure (I \ {e}) = M.closure (X \ {e}) ∧ ((e ∈ I ↔ M.closure (X \ {e}) ∈ U) → e ∉ X) ∨
(M.closure (I \ {e}) ⋖[M] M.closure (X \ {e})) ∧ e ∈ I ∧ M.closure (X \ {e}) ∈ U |
https://github.com/apnelson1/Matroid | 510262f56c0025be19502ab3457a788e2f6d2d1c | Matroid/Extension.lean | Matroid.ModularCut.maximal_extIndep_iff | Status.FAILED | α : Type u_1
M : Matroid α
I J B F₀ F F' X Y : Set α
e f : α
U : M.ModularCut
hX : X ⊆ insert e M.E
hI : U.ExtIndep e I
hIX : I ⊆ X
hss : I \ {e} ⊆ X \ {e}
⊢ I ∈ maximals (fun x x_1 => x ⊆ x_1) {J | U.ExtIndep e J ∧ J ⊆ X} ↔
M.closure (I \ {e}) = M.closure (X \ {e}) ∧ ((e ∈ I ↔ M.closure (X \ {e}) ∈ U) → e ∉ X) ∨
(M.closure (I \ {e}) ⋖[M] M.closure (X \ {e})) ∧ e ∈ I ∧ M.closure (X \ {e}) ∈ U | [Goal(assumptions=[Declaration(ident='α', lean_type='Type u_1'), Declaration(ident='M', lean_type='Matroid α'), Declaration(ident='I', lean_type='Set α'), Declaration(ident='J', lean_type='Set α'), Declaration(ident='B', lean_type='Set α'), Declaration(ident='F₀', lean_type='Set α'), Declaration(ident='F', lean_type='Set α'), Declaration(ident="F'", lean_type='Set α'), Declaration(ident='X', lean_type='Set α'), Declaration(ident='Y', lean_type='Set α'), Declaration(ident='e', lean_type='α'), Declaration(ident='f', lean_type='α'), Declaration(ident='U', lean_type='M.ModularCut'), Declaration(ident='hX', lean_type='X ⊆ insert e M.E'), Declaration(ident='hI', lean_type='U.ExtIndep e I'), Declaration(ident='hIX', lean_type='I ⊆ X'), Declaration(ident='hss', lean_type='I \\ {e} ⊆ X \\ {e}')], conclusion='I ∈ maximals (fun x x_1 => x ⊆ x_1) {J | U.ExtIndep e J ∧ J ⊆ X} ↔\n M.closure (I \\ {e}) = M.closure (X \\ {e}) ∧ ((e ∈ I ↔ M.closure (X \\ {e}) ∈ U) → e ∉ X) ∨\n (M.closure (I \\ {e}) ⋖[M] M.closure (X \\ {e})) ∧ e ∈ I ∧ M.closure (X \\ {e}) ∈ U')] | have hX' : X \ {e} ⊆ M.E := by simpa | <class 'lean_dojo.interaction.dojo.TacticState'> | TacticState(pp="α : Type u_1\nM : Matroid α\nI J B F₀ F F' X Y : Set α\ne f : α\nU : M.ModularCut\nhX : X ⊆ insert e M.E\nhI : U.ExtIndep e I\nhIX : I ⊆ X\nhss : I \\ {e} ⊆ X \\ {e}\nhX' : X \\ {e} ⊆ M.E\n⊢ I ∈ maximals (fun x x_1 => x ⊆ x_1) {J | U.ExtIndep e J ∧ J ⊆ X} ↔\n M.closure (I \\ {e}) = M.closure (X \\ {e}) ∧ ((e ∈ I ↔ M.closure (X \\ {e}) ∈ U) → e ∉ X) ∨\n (M.closure (I \\ {e}) ⋖[M] M.closure (X \\ {e})) ∧ e ∈ I ∧ M.closure (X \\ {e}) ∈ U", id=2, message='') | α : Type u_1
M : Matroid α
I J B F₀ F F' X Y : Set α
e f : α
U : M.ModularCut
hX : X ⊆ insert e M.E
hI : U.ExtIndep e I
hIX : I ⊆ X
hss : I \ {e} ⊆ X \ {e}
hX' : X \ {e} ⊆ M.E
⊢ I ∈ maximals (fun x x_1 => x ⊆ x_1) {J | U.ExtIndep e J ∧ J ⊆ X} ↔
M.closure (I \ {e}) = M.closure (X \ {e}) ∧ ((e ∈ I ↔ M.closure (X \ {e}) ∈ U) → e ∉ X) ∨
(M.closure (I \ {e}) ⋖[M] M.closure (X \ {e})) ∧ e ∈ I ∧ M.closure (X \ {e}) ∈ U |
|
https://github.com/apnelson1/Matroid | 510262f56c0025be19502ab3457a788e2f6d2d1c | Matroid/Extension.lean | Matroid.ModularCut.maximal_extIndep_iff | Status.FAILED | α : Type u_1
M : Matroid α
I J B F₀ F F' X Y : Set α
e f : α
U : M.ModularCut
hX : X ⊆ insert e M.E
hI : U.ExtIndep e I
hIX : I ⊆ X
hss : I \ {e} ⊆ X \ {e}
hX' : X \ {e} ⊆ M.E
⊢ I ∈ maximals (fun x x_1 => x ⊆ x_1) {J | U.ExtIndep e J ∧ J ⊆ X} ↔
M.closure (I \ {e}) = M.closure (X \ {e}) ∧ ((e ∈ I ↔ M.closure (X \ {e}) ∈ U) → e ∉ X) ∨
(M.closure (I \ {e}) ⋖[M] M.closure (X \ {e})) ∧ e ∈ I ∧ M.closure (X \ {e}) ∈ U | [Goal(assumptions=[Declaration(ident='α', lean_type='Type u_1'), Declaration(ident='M', lean_type='Matroid α'), Declaration(ident='I', lean_type='Set α'), Declaration(ident='J', lean_type='Set α'), Declaration(ident='B', lean_type='Set α'), Declaration(ident='F₀', lean_type='Set α'), Declaration(ident='F', lean_type='Set α'), Declaration(ident="F'", lean_type='Set α'), Declaration(ident='X', lean_type='Set α'), Declaration(ident='Y', lean_type='Set α'), Declaration(ident='e', lean_type='α'), Declaration(ident='f', lean_type='α'), Declaration(ident='U', lean_type='M.ModularCut'), Declaration(ident='hX', lean_type='X ⊆ insert e M.E'), Declaration(ident='hI', lean_type='U.ExtIndep e I'), Declaration(ident='hIX', lean_type='I ⊆ X'), Declaration(ident='hss', lean_type='I \\ {e} ⊆ X \\ {e}'), Declaration(ident="hX'", lean_type='X \\ {e} ⊆ M.E')], conclusion='I ∈ maximals (fun x x_1 => x ⊆ x_1) {J | U.ExtIndep e J ∧ J ⊆ X} ↔\n M.closure (I \\ {e}) = M.closure (X \\ {e}) ∧ ((e ∈ I ↔ M.closure (X \\ {e}) ∈ U) → e ∉ X) ∨\n (M.closure (I \\ {e}) ⋖[M] M.closure (X \\ {e})) ∧ e ∈ I ∧ M.closure (X \\ {e}) ∈ U')] | rw [mem_maximals_iff_forall_insert (fun _ _ ht hst ↦ ⟨ht.1.subset hst, hst.trans ht.2⟩)] | <class 'lean_dojo.interaction.dojo.TacticState'> | TacticState(pp="α : Type u_1\nM : Matroid α\nI J B F₀ F F' X Y : Set α\ne f : α\nU : M.ModularCut\nhX : X ⊆ insert e M.E\nhI : U.ExtIndep e I\nhIX : I ⊆ X\nhss : I \\ {e} ⊆ X \\ {e}\nhX' : X \\ {e} ⊆ M.E\n⊢ ((U.ExtIndep e I ∧ I ⊆ X) ∧ ∀ x ∉ I, ¬(U.ExtIndep e (insert x I) ∧ insert x I ⊆ X)) ↔\n M.closure (I \\ {e}) = M.closure (X \\ {e}) ∧ ((e ∈ I ↔ M.closure (X \\ {e}) ∈ U) → e ∉ X) ∨\n (M.closure (I \\ {e}) ⋖[M] M.closure (X \\ {e})) ∧ e ∈ I ∧ M.closure (X \\ {e}) ∈ U", id=3, message='') | α : Type u_1
M : Matroid α
I J B F₀ F F' X Y : Set α
e f : α
U : M.ModularCut
hX : X ⊆ insert e M.E
hI : U.ExtIndep e I
hIX : I ⊆ X
hss : I \ {e} ⊆ X \ {e}
hX' : X \ {e} ⊆ M.E
⊢ ((U.ExtIndep e I ∧ I ⊆ X) ∧ ∀ x ∉ I, ¬(U.ExtIndep e (insert x I) ∧ insert x I ⊆ X)) ↔
M.closure (I \ {e}) = M.closure (X \ {e}) ∧ ((e ∈ I ↔ M.closure (X \ {e}) ∈ U) → e ∉ X) ∨
(M.closure (I \ {e}) ⋖[M] M.closure (X \ {e})) ∧ e ∈ I ∧ M.closure (X \ {e}) ∈ U |
|
https://github.com/apnelson1/Matroid | 510262f56c0025be19502ab3457a788e2f6d2d1c | Matroid/Extension.lean | Matroid.ModularCut.maximal_extIndep_iff | Status.FAILED | α : Type u_1
M : Matroid α
I J B F₀ F F' X Y : Set α
e f : α
U : M.ModularCut
hX : X ⊆ insert e M.E
hI : U.ExtIndep e I
hIX : I ⊆ X
hss : I \ {e} ⊆ X \ {e}
hX' : X \ {e} ⊆ M.E
⊢ ((U.ExtIndep e I ∧ I ⊆ X) ∧ ∀ x ∉ I, ¬(U.ExtIndep e (insert x I) ∧ insert x I ⊆ X)) ↔
M.closure (I \ {e}) = M.closure (X \ {e}) ∧ ((e ∈ I ↔ M.closure (X \ {e}) ∈ U) → e ∉ X) ∨
(M.closure (I \ {e}) ⋖[M] M.closure (X \ {e})) ∧ e ∈ I ∧ M.closure (X \ {e}) ∈ U | [Goal(assumptions=[Declaration(ident='α', lean_type='Type u_1'), Declaration(ident='M', lean_type='Matroid α'), Declaration(ident='I', lean_type='Set α'), Declaration(ident='J', lean_type='Set α'), Declaration(ident='B', lean_type='Set α'), Declaration(ident='F₀', lean_type='Set α'), Declaration(ident='F', lean_type='Set α'), Declaration(ident="F'", lean_type='Set α'), Declaration(ident='X', lean_type='Set α'), Declaration(ident='Y', lean_type='Set α'), Declaration(ident='e', lean_type='α'), Declaration(ident='f', lean_type='α'), Declaration(ident='U', lean_type='M.ModularCut'), Declaration(ident='hX', lean_type='X ⊆ insert e M.E'), Declaration(ident='hI', lean_type='U.ExtIndep e I'), Declaration(ident='hIX', lean_type='I ⊆ X'), Declaration(ident='hss', lean_type='I \\ {e} ⊆ X \\ {e}'), Declaration(ident="hX'", lean_type='X \\ {e} ⊆ M.E')], conclusion='((U.ExtIndep e I ∧ I ⊆ X) ∧ ∀ x ∉ I, ¬(U.ExtIndep e (insert x I) ∧ insert x I ⊆ X)) ↔\n M.closure (I \\ {e}) = M.closure (X \\ {e}) ∧ ((e ∈ I ↔ M.closure (X \\ {e}) ∈ U) → e ∉ X) ∨\n (M.closure (I \\ {e}) ⋖[M] M.closure (X \\ {e})) ∧ e ∈ I ∧ M.closure (X \\ {e}) ∈ U')] | simp only [hI, hIX, and_self, insert_subset_iff, and_true, not_and, true_and, imp_not_comm] | <class 'lean_dojo.interaction.dojo.TacticState'> | TacticState(pp="α : Type u_1\nM : Matroid α\nI J B F₀ F F' X Y : Set α\ne f : α\nU : M.ModularCut\nhX : X ⊆ insert e M.E\nhI : U.ExtIndep e I\nhIX : I ⊆ X\nhss : I \\ {e} ⊆ X \\ {e}\nhX' : X \\ {e} ⊆ M.E\n⊢ (∀ x ∉ I, x ∈ X → ¬U.ExtIndep e (insert x I)) ↔\n M.closure (I \\ {e}) = M.closure (X \\ {e}) ∧ (e ∈ X → ¬(e ∈ I ↔ M.closure (X \\ {e}) ∈ U)) ∨\n (M.closure (I \\ {e}) ⋖[M] M.closure (X \\ {e})) ∧ e ∈ I ∧ M.closure (X \\ {e}) ∈ U", id=4, message='') | α : Type u_1
M : Matroid α
I J B F₀ F F' X Y : Set α
e f : α
U : M.ModularCut
hX : X ⊆ insert e M.E
hI : U.ExtIndep e I
hIX : I ⊆ X
hss : I \ {e} ⊆ X \ {e}
hX' : X \ {e} ⊆ M.E
⊢ (∀ x ∉ I, x ∈ X → ¬U.ExtIndep e (insert x I)) ↔
M.closure (I \ {e}) = M.closure (X \ {e}) ∧ (e ∈ X → ¬(e ∈ I ↔ M.closure (X \ {e}) ∈ U)) ∨
(M.closure (I \ {e}) ⋖[M] M.closure (X \ {e})) ∧ e ∈ I ∧ M.closure (X \ {e}) ∈ U |
|
https://github.com/apnelson1/Matroid | 510262f56c0025be19502ab3457a788e2f6d2d1c | Matroid/Extension.lean | Matroid.ModularCut.maximal_extIndep_iff | Status.FAILED | α : Type u_1
M : Matroid α
I J B F₀ F F' X Y : Set α
e f : α
U : M.ModularCut
hX : X ⊆ insert e M.E
hI : U.ExtIndep e I
hIX : I ⊆ X
hss : I \ {e} ⊆ X \ {e}
hX' : X \ {e} ⊆ M.E
⊢ (∀ x ∉ I, x ∈ X → ¬U.ExtIndep e (insert x I)) ↔
M.closure (I \ {e}) = M.closure (X \ {e}) ∧ (e ∈ X → ¬(e ∈ I ↔ M.closure (X \ {e}) ∈ U)) ∨
(M.closure (I \ {e}) ⋖[M] M.closure (X \ {e})) ∧ e ∈ I ∧ M.closure (X \ {e}) ∈ U | [Goal(assumptions=[Declaration(ident='α', lean_type='Type u_1'), Declaration(ident='M', lean_type='Matroid α'), Declaration(ident='I', lean_type='Set α'), Declaration(ident='J', lean_type='Set α'), Declaration(ident='B', lean_type='Set α'), Declaration(ident='F₀', lean_type='Set α'), Declaration(ident='F', lean_type='Set α'), Declaration(ident="F'", lean_type='Set α'), Declaration(ident='X', lean_type='Set α'), Declaration(ident='Y', lean_type='Set α'), Declaration(ident='e', lean_type='α'), Declaration(ident='f', lean_type='α'), Declaration(ident='U', lean_type='M.ModularCut'), Declaration(ident='hX', lean_type='X ⊆ insert e M.E'), Declaration(ident='hI', lean_type='U.ExtIndep e I'), Declaration(ident='hIX', lean_type='I ⊆ X'), Declaration(ident='hss', lean_type='I \\ {e} ⊆ X \\ {e}'), Declaration(ident="hX'", lean_type='X \\ {e} ⊆ M.E')], conclusion='(∀ x ∉ I, x ∈ X → ¬U.ExtIndep e (insert x I)) ↔\n M.closure (I \\ {e}) = M.closure (X \\ {e}) ∧ (e ∈ X → ¬(e ∈ I ↔ M.closure (X \\ {e}) ∈ U)) ∨\n (M.closure (I \\ {e}) ⋖[M] M.closure (X \\ {e})) ∧ e ∈ I ∧ M.closure (X \\ {e}) ∈ U')] | refine ⟨fun h ↦ ?_, fun h x hxI hi hind ↦ ?_⟩ | <class 'lean_dojo.interaction.dojo.TacticState'> | TacticState(pp="case refine_1\nα : Type u_1\nM : Matroid α\nI J B F₀ F F' X Y : Set α\ne f : α\nU : M.ModularCut\nhX : X ⊆ insert e M.E\nhI : U.ExtIndep e I\nhIX : I ⊆ X\nhss : I \\ {e} ⊆ X \\ {e}\nhX' : X \\ {e} ⊆ M.E\nh : ∀ x ∉ I, x ∈ X → ¬U.ExtIndep e (insert x I)\n⊢ M.closure (I \\ {e}) = M.closure (X \\ {e}) ∧ (e ∈ X → ¬(e ∈ I ↔ M.closure (X \\ {e}) ∈ U)) ∨\n (M.closure (I \\ {e}) ⋖[M] M.closure (X \\ {e})) ∧ e ∈ I ∧ M.closure (X \\ {e}) ∈ U\n\ncase refine_2\nα : Type u_1\nM : Matroid α\nI J B F₀ F F' X Y : Set α\ne f : α\nU : M.ModularCut\nhX : X ⊆ insert e M.E\nhI : U.ExtIndep e I\nhIX : I ⊆ X\nhss : I \\ {e} ⊆ X \\ {e}\nhX' : X \\ {e} ⊆ M.E\nh :\n M.closure (I \\ {e}) = M.closure (X \\ {e}) ∧ (e ∈ X → ¬(e ∈ I ↔ M.closure (X \\ {e}) ∈ U)) ∨\n (M.closure (I \\ {e}) ⋖[M] M.closure (X \\ {e})) ∧ e ∈ I ∧ M.closure (X \\ {e}) ∈ U\nx : α\nhxI : x ∉ I\nhi : x ∈ X\nhind : U.ExtIndep e (insert x I)\n⊢ False", id=5, message='') | case refine_1
α : Type u_1
M : Matroid α
I J B F₀ F F' X Y : Set α
e f : α
U : M.ModularCut
hX : X ⊆ insert e M.E
hI : U.ExtIndep e I
hIX : I ⊆ X
hss : I \ {e} ⊆ X \ {e}
hX' : X \ {e} ⊆ M.E
h : ∀ x ∉ I, x ∈ X → ¬U.ExtIndep e (insert x I)
⊢ M.closure (I \ {e}) = M.closure (X \ {e}) ∧ (e ∈ X → ¬(e ∈ I ↔ M.closure (X \ {e}) ∈ U)) ∨
(M.closure (I \ {e}) ⋖[M] M.closure (X \ {e})) ∧ e ∈ I ∧ M.closure (X \ {e}) ∈ U
case refine_2
α : Type u_1
M : Matroid α
I J B F₀ F F' X Y : Set α
e f : α
U : M.ModularCut
hX : X ⊆ insert e M.E
hI : U.ExtIndep e I
hIX : I ⊆ X
hss : I \ {e} ⊆ X \ {e}
hX' : X \ {e} ⊆ M.E
h :
M.closure (I \ {e}) = M.closure (X \ {e}) ∧ (e ∈ X → ¬(e ∈ I ↔ M.closure (X \ {e}) ∈ U)) ∨
(M.closure (I \ {e}) ⋖[M] M.closure (X \ {e})) ∧ e ∈ I ∧ M.closure (X \ {e}) ∈ U
x : α
hxI : x ∉ I
hi : x ∈ X
hind : U.ExtIndep e (insert x I)
⊢ False |
|
https://github.com/apnelson1/Matroid | 510262f56c0025be19502ab3457a788e2f6d2d1c | Matroid/Extension.lean | Matroid.ModularCut.maximal_extIndep_iff | Status.FAILED | case refine_1
α : Type u_1
M : Matroid α
I J B F₀ F F' X Y : Set α
e f : α
U : M.ModularCut
hX : X ⊆ insert e M.E
hI : U.ExtIndep e I
hIX : I ⊆ X
hss : I \ {e} ⊆ X \ {e}
hX' : X \ {e} ⊆ M.E
h : ∀ x ∉ I, x ∈ X → ¬U.ExtIndep e (insert x I)
⊢ M.closure (I \ {e}) = M.closure (X \ {e}) ∧ (e ∈ X → ¬(e ∈ I ↔ M.closure (X \ {e}) ∈ U)) ∨
(M.closure (I \ {e}) ⋖[M] M.closure (X \ {e})) ∧ e ∈ I ∧ M.closure (X \ {e}) ∈ U
case refine_2
α : Type u_1
M : Matroid α
I J B F₀ F F' X Y : Set α
e f : α
U : M.ModularCut
hX : X ⊆ insert e M.E
hI : U.ExtIndep e I
hIX : I ⊆ X
hss : I \ {e} ⊆ X \ {e}
hX' : X \ {e} ⊆ M.E
h :
M.closure (I \ {e}) = M.closure (X \ {e}) ∧ (e ∈ X → ¬(e ∈ I ↔ M.closure (X \ {e}) ∈ U)) ∨
(M.closure (I \ {e}) ⋖[M] M.closure (X \ {e})) ∧ e ∈ I ∧ M.closure (X \ {e}) ∈ U
x : α
hxI : x ∉ I
hi : x ∈ X
hind : U.ExtIndep e (insert x I)
⊢ False | [Goal(assumptions=[Declaration(ident='α', lean_type='Type u_1'), Declaration(ident='M', lean_type='Matroid α'), Declaration(ident='I', lean_type='Set α'), Declaration(ident='J', lean_type='Set α'), Declaration(ident='B', lean_type='Set α'), Declaration(ident='F₀', lean_type='Set α'), Declaration(ident='F', lean_type='Set α'), Declaration(ident="F'", lean_type='Set α'), Declaration(ident='X', lean_type='Set α'), Declaration(ident='Y', lean_type='Set α'), Declaration(ident='e', lean_type='α'), Declaration(ident='f', lean_type='α'), Declaration(ident='U', lean_type='M.ModularCut'), Declaration(ident='hX', lean_type='X ⊆ insert e M.E'), Declaration(ident='hI', lean_type='U.ExtIndep e I'), Declaration(ident='hIX', lean_type='I ⊆ X'), Declaration(ident='hss', lean_type='I \\ {e} ⊆ X \\ {e}'), Declaration(ident="hX'", lean_type='X \\ {e} ⊆ M.E'), Declaration(ident='h', lean_type='∀ x ∉ I, x ∈ X → ¬U.ExtIndep e (insert x I)')], conclusion='M.closure (I \\ {e}) = M.closure (X \\ {e}) ∧ (e ∈ X → ¬(e ∈ I ↔ M.closure (X \\ {e}) ∈ U)) ∨\n (M.closure (I \\ {e}) ⋖[M] M.closure (X \\ {e})) ∧ e ∈ I ∧ M.closure (X \\ {e}) ∈ U'), Goal(assumptions=[Declaration(ident='α', lean_type='Type u_1'), Declaration(ident='M', lean_type='Matroid α'), Declaration(ident='I', lean_type='Set α'), Declaration(ident='J', lean_type='Set α'), Declaration(ident='B', lean_type='Set α'), Declaration(ident='F₀', lean_type='Set α'), Declaration(ident='F', lean_type='Set α'), Declaration(ident="F'", lean_type='Set α'), Declaration(ident='X', lean_type='Set α'), Declaration(ident='Y', lean_type='Set α'), Declaration(ident='e', lean_type='α'), Declaration(ident='f', lean_type='α'), Declaration(ident='U', lean_type='M.ModularCut'), Declaration(ident='hX', lean_type='X ⊆ insert e M.E'), Declaration(ident='hI', lean_type='U.ExtIndep e I'), Declaration(ident='hIX', lean_type='I ⊆ X'), Declaration(ident='hss', lean_type='I \\ {e} ⊆ X \\ {e}'), Declaration(ident="hX'", lean_type='X \\ {e} ⊆ M.E'), Declaration(ident='h', lean_type='M.closure (I \\ {e}) = M.closure (X \\ {e}) ∧ (e ∈ X → ¬(e ∈ I ↔ M.closure (X \\ {e}) ∈ U)) ∨\n (M.closure (I \\ {e}) ⋖[M] M.closure (X \\ {e})) ∧ e ∈ I ∧ M.closure (X \\ {e}) ∈ U'), Declaration(ident='x', lean_type='α'), Declaration(ident='hxI', lean_type='x ∉ I'), Declaration(ident='hi', lean_type='x ∈ X'), Declaration(ident='hind', lean_type='U.ExtIndep e (insert x I)')], conclusion='False')] | by_cases heI : e ∈ I | <class 'lean_dojo.interaction.dojo.TacticState'> | TacticState(pp="case pos\nα : Type u_1\nM : Matroid α\nI J B F₀ F F' X Y : Set α\ne f : α\nU : M.ModularCut\nhX : X ⊆ insert e M.E\nhI : U.ExtIndep e I\nhIX : I ⊆ X\nhss : I \\ {e} ⊆ X \\ {e}\nhX' : X \\ {e} ⊆ M.E\nh : ∀ x ∉ I, x ∈ X → ¬U.ExtIndep e (insert x I)\nheI : e ∈ I\n⊢ M.closure (I \\ {e}) = M.closure (X \\ {e}) ∧ (e ∈ X → ¬(e ∈ I ↔ M.closure (X \\ {e}) ∈ U)) ∨\n (M.closure (I \\ {e}) ⋖[M] M.closure (X \\ {e})) ∧ e ∈ I ∧ M.closure (X \\ {e}) ∈ U\n\ncase neg\nα : Type u_1\nM : Matroid α\nI J B F₀ F F' X Y : Set α\ne f : α\nU : M.ModularCut\nhX : X ⊆ insert e M.E\nhI : U.ExtIndep e I\nhIX : I ⊆ X\nhss : I \\ {e} ⊆ X \\ {e}\nhX' : X \\ {e} ⊆ M.E\nh : ∀ x ∉ I, x ∈ X → ¬U.ExtIndep e (insert x I)\nheI : e ∉ I\n⊢ M.closure (I \\ {e}) = M.closure (X \\ {e}) ∧ (e ∈ X → ¬(e ∈ I ↔ M.closure (X \\ {e}) ∈ U)) ∨\n (M.closure (I \\ {e}) ⋖[M] M.closure (X \\ {e})) ∧ e ∈ I ∧ M.closure (X \\ {e}) ∈ U\n\ncase refine_2\nα : Type u_1\nM : Matroid α\nI J B F₀ F F' X Y : Set α\ne f : α\nU : M.ModularCut\nhX : X ⊆ insert e M.E\nhI : U.ExtIndep e I\nhIX : I ⊆ X\nhss : I \\ {e} ⊆ X \\ {e}\nhX' : X \\ {e} ⊆ M.E\nh :\n M.closure (I \\ {e}) = M.closure (X \\ {e}) ∧ (e ∈ X → ¬(e ∈ I ↔ M.closure (X \\ {e}) ∈ U)) ∨\n (M.closure (I \\ {e}) ⋖[M] M.closure (X \\ {e})) ∧ e ∈ I ∧ M.closure (X \\ {e}) ∈ U\nx : α\nhxI : x ∉ I\nhi : x ∈ X\nhind : U.ExtIndep e (insert x I)\n⊢ False", id=6, message='') | case pos
α : Type u_1
M : Matroid α
I J B F₀ F F' X Y : Set α
e f : α
U : M.ModularCut
hX : X ⊆ insert e M.E
hI : U.ExtIndep e I
hIX : I ⊆ X
hss : I \ {e} ⊆ X \ {e}
hX' : X \ {e} ⊆ M.E
h : ∀ x ∉ I, x ∈ X → ¬U.ExtIndep e (insert x I)
heI : e ∈ I
⊢ M.closure (I \ {e}) = M.closure (X \ {e}) ∧ (e ∈ X → ¬(e ∈ I ↔ M.closure (X \ {e}) ∈ U)) ∨
(M.closure (I \ {e}) ⋖[M] M.closure (X \ {e})) ∧ e ∈ I ∧ M.closure (X \ {e}) ∈ U
case neg
α : Type u_1
M : Matroid α
I J B F₀ F F' X Y : Set α
e f : α
U : M.ModularCut
hX : X ⊆ insert e M.E
hI : U.ExtIndep e I
hIX : I ⊆ X
hss : I \ {e} ⊆ X \ {e}
hX' : X \ {e} ⊆ M.E
h : ∀ x ∉ I, x ∈ X → ¬U.ExtIndep e (insert x I)
heI : e ∉ I
⊢ M.closure (I \ {e}) = M.closure (X \ {e}) ∧ (e ∈ X → ¬(e ∈ I ↔ M.closure (X \ {e}) ∈ U)) ∨
(M.closure (I \ {e}) ⋖[M] M.closure (X \ {e})) ∧ e ∈ I ∧ M.closure (X \ {e}) ∈ U
case refine_2
α : Type u_1
M : Matroid α
I J B F₀ F F' X Y : Set α
e f : α
U : M.ModularCut
hX : X ⊆ insert e M.E
hI : U.ExtIndep e I
hIX : I ⊆ X
hss : I \ {e} ⊆ X \ {e}
hX' : X \ {e} ⊆ M.E
h :
M.closure (I \ {e}) = M.closure (X \ {e}) ∧ (e ∈ X → ¬(e ∈ I ↔ M.closure (X \ {e}) ∈ U)) ∨
(M.closure (I \ {e}) ⋖[M] M.closure (X \ {e})) ∧ e ∈ I ∧ M.closure (X \ {e}) ∈ U
x : α
hxI : x ∉ I
hi : x ∈ X
hind : U.ExtIndep e (insert x I)
⊢ False |
|
https://github.com/apnelson1/Matroid | 510262f56c0025be19502ab3457a788e2f6d2d1c | Matroid/Extension.lean | Matroid.ModularCut.maximal_extIndep_iff | Status.FAILED | case pos
α : Type u_1
M : Matroid α
I J B F₀ F F' X Y : Set α
e f : α
U : M.ModularCut
hX : X ⊆ insert e M.E
hI : U.ExtIndep e I
hIX : I ⊆ X
hss : I \ {e} ⊆ X \ {e}
hX' : X \ {e} ⊆ M.E
h : ∀ x ∉ I, x ∈ X → ¬U.ExtIndep e (insert x I)
heI : e ∈ I
⊢ M.closure (I \ {e}) = M.closure (X \ {e}) ∧ (e ∈ X → ¬(e ∈ I ↔ M.closure (X \ {e}) ∈ U)) ∨
(M.closure (I \ {e}) ⋖[M] M.closure (X \ {e})) ∧ e ∈ I ∧ M.closure (X \ {e}) ∈ U
case neg
α : Type u_1
M : Matroid α
I J B F₀ F F' X Y : Set α
e f : α
U : M.ModularCut
hX : X ⊆ insert e M.E
hI : U.ExtIndep e I
hIX : I ⊆ X
hss : I \ {e} ⊆ X \ {e}
hX' : X \ {e} ⊆ M.E
h : ∀ x ∉ I, x ∈ X → ¬U.ExtIndep e (insert x I)
heI : e ∉ I
⊢ M.closure (I \ {e}) = M.closure (X \ {e}) ∧ (e ∈ X → ¬(e ∈ I ↔ M.closure (X \ {e}) ∈ U)) ∨
(M.closure (I \ {e}) ⋖[M] M.closure (X \ {e})) ∧ e ∈ I ∧ M.closure (X \ {e}) ∈ U
case refine_2
α : Type u_1
M : Matroid α
I J B F₀ F F' X Y : Set α
e f : α
U : M.ModularCut
hX : X ⊆ insert e M.E
hI : U.ExtIndep e I
hIX : I ⊆ X
hss : I \ {e} ⊆ X \ {e}
hX' : X \ {e} ⊆ M.E
h :
M.closure (I \ {e}) = M.closure (X \ {e}) ∧ (e ∈ X → ¬(e ∈ I ↔ M.closure (X \ {e}) ∈ U)) ∨
(M.closure (I \ {e}) ⋖[M] M.closure (X \ {e})) ∧ e ∈ I ∧ M.closure (X \ {e}) ∈ U
x : α
hxI : x ∉ I
hi : x ∈ X
hind : U.ExtIndep e (insert x I)
⊢ False | [Goal(assumptions=[Declaration(ident='α', lean_type='Type u_1'), Declaration(ident='M', lean_type='Matroid α'), Declaration(ident='I', lean_type='Set α'), Declaration(ident='J', lean_type='Set α'), Declaration(ident='B', lean_type='Set α'), Declaration(ident='F₀', lean_type='Set α'), Declaration(ident='F', lean_type='Set α'), Declaration(ident="F'", lean_type='Set α'), Declaration(ident='X', lean_type='Set α'), Declaration(ident='Y', lean_type='Set α'), Declaration(ident='e', lean_type='α'), Declaration(ident='f', lean_type='α'), Declaration(ident='U', lean_type='M.ModularCut'), Declaration(ident='hX', lean_type='X ⊆ insert e M.E'), Declaration(ident='hI', lean_type='U.ExtIndep e I'), Declaration(ident='hIX', lean_type='I ⊆ X'), Declaration(ident='hss', lean_type='I \\ {e} ⊆ X \\ {e}'), Declaration(ident="hX'", lean_type='X \\ {e} ⊆ M.E'), Declaration(ident='h', lean_type='∀ x ∉ I, x ∈ X → ¬U.ExtIndep e (insert x I)'), Declaration(ident='heI', lean_type='e ∈ I')], conclusion='M.closure (I \\ {e}) = M.closure (X \\ {e}) ∧ (e ∈ X → ¬(e ∈ I ↔ M.closure (X \\ {e}) ∈ U)) ∨\n (M.closure (I \\ {e}) ⋖[M] M.closure (X \\ {e})) ∧ e ∈ I ∧ M.closure (X \\ {e}) ∈ U'), Goal(assumptions=[Declaration(ident='α', lean_type='Type u_1'), Declaration(ident='M', lean_type='Matroid α'), Declaration(ident='I', lean_type='Set α'), Declaration(ident='J', lean_type='Set α'), Declaration(ident='B', lean_type='Set α'), Declaration(ident='F₀', lean_type='Set α'), Declaration(ident='F', lean_type='Set α'), Declaration(ident="F'", lean_type='Set α'), Declaration(ident='X', lean_type='Set α'), Declaration(ident='Y', lean_type='Set α'), Declaration(ident='e', lean_type='α'), Declaration(ident='f', lean_type='α'), Declaration(ident='U', lean_type='M.ModularCut'), Declaration(ident='hX', lean_type='X ⊆ insert e M.E'), Declaration(ident='hI', lean_type='U.ExtIndep e I'), Declaration(ident='hIX', lean_type='I ⊆ X'), Declaration(ident='hss', lean_type='I \\ {e} ⊆ X \\ {e}'), Declaration(ident="hX'", lean_type='X \\ {e} ⊆ M.E'), Declaration(ident='h', lean_type='∀ x ∉ I, x ∈ X → ¬U.ExtIndep e (insert x I)'), Declaration(ident='heI', lean_type='e ∉ I')], conclusion='M.closure (I \\ {e}) = M.closure (X \\ {e}) ∧ (e ∈ X → ¬(e ∈ I ↔ M.closure (X \\ {e}) ∈ U)) ∨\n (M.closure (I \\ {e}) ⋖[M] M.closure (X \\ {e})) ∧ e ∈ I ∧ M.closure (X \\ {e}) ∈ U'), Goal(assumptions=[Declaration(ident='α', lean_type='Type u_1'), Declaration(ident='M', lean_type='Matroid α'), Declaration(ident='I', lean_type='Set α'), Declaration(ident='J', lean_type='Set α'), Declaration(ident='B', lean_type='Set α'), Declaration(ident='F₀', lean_type='Set α'), Declaration(ident='F', lean_type='Set α'), Declaration(ident="F'", lean_type='Set α'), Declaration(ident='X', lean_type='Set α'), Declaration(ident='Y', lean_type='Set α'), Declaration(ident='e', lean_type='α'), Declaration(ident='f', lean_type='α'), Declaration(ident='U', lean_type='M.ModularCut'), Declaration(ident='hX', lean_type='X ⊆ insert e M.E'), Declaration(ident='hI', lean_type='U.ExtIndep e I'), Declaration(ident='hIX', lean_type='I ⊆ X'), Declaration(ident='hss', lean_type='I \\ {e} ⊆ X \\ {e}'), Declaration(ident="hX'", lean_type='X \\ {e} ⊆ M.E'), Declaration(ident='h', lean_type='M.closure (I \\ {e}) = M.closure (X \\ {e}) ∧ (e ∈ X → ¬(e ∈ I ↔ M.closure (X \\ {e}) ∈ U)) ∨\n (M.closure (I \\ {e}) ⋖[M] M.closure (X \\ {e})) ∧ e ∈ I ∧ M.closure (X \\ {e}) ∈ U'), Declaration(ident='x', lean_type='α'), Declaration(ident='hxI', lean_type='x ∉ I'), Declaration(ident='hi', lean_type='x ∈ X'), Declaration(ident='hind', lean_type='U.ExtIndep e (insert x I)')], conclusion='False')] | simp only [heI, not_false_eq_true, diff_singleton_eq_self, false_iff, not_not, false_and,
and_false, or_false] at h | <class 'lean_dojo.interaction.dojo.TacticState'> | TacticState(pp="case pos\nα : Type u_1\nM : Matroid α\nI J B F₀ F F' X Y : Set α\ne f : α\nU : M.ModularCut\nhX : X ⊆ insert e M.E\nhI : U.ExtIndep e I\nhIX : I ⊆ X\nhss : I \\ {e} ⊆ X \\ {e}\nhX' : X \\ {e} ⊆ M.E\nh : ∀ x ∉ I, x ∈ X → ¬U.ExtIndep e (insert x I)\nheI : e ∈ I\n⊢ M.closure (I \\ {e}) = M.closure (X \\ {e}) ∧ (e ∈ X → ¬(e ∈ I ↔ M.closure (X \\ {e}) ∈ U)) ∨\n (M.closure (I \\ {e}) ⋖[M] M.closure (X \\ {e})) ∧ e ∈ I ∧ M.closure (X \\ {e}) ∈ U\n\ncase neg\nα : Type u_1\nM : Matroid α\nI J B F₀ F F' X Y : Set α\ne f : α\nU : M.ModularCut\nhX : X ⊆ insert e M.E\nhI : U.ExtIndep e I\nhIX : I ⊆ X\nhss : I \\ {e} ⊆ X \\ {e}\nhX' : X \\ {e} ⊆ M.E\nh : ∀ x ∉ I, x ∈ X → ¬U.ExtIndep e (insert x I)\nheI : e ∉ I\n⊢ M.closure (I \\ {e}) = M.closure (X \\ {e}) ∧ (e ∈ X → ¬(e ∈ I ↔ M.closure (X \\ {e}) ∈ U)) ∨\n (M.closure (I \\ {e}) ⋖[M] M.closure (X \\ {e})) ∧ e ∈ I ∧ M.closure (X \\ {e}) ∈ U\n\ncase refine_2\nα : Type u_1\nM : Matroid α\nI J B F₀ F F' X Y : Set α\ne f : α\nU : M.ModularCut\nhX : X ⊆ insert e M.E\nhI : U.ExtIndep e I\nhIX : I ⊆ X\nhss : I \\ {e} ⊆ X \\ {e}\nhX' : X \\ {e} ⊆ M.E\nh :\n M.closure (I \\ {e}) = M.closure (X \\ {e}) ∧ (e ∈ X → ¬(e ∈ I ↔ M.closure (X \\ {e}) ∈ U)) ∨\n (M.closure (I \\ {e}) ⋖[M] M.closure (X \\ {e})) ∧ e ∈ I ∧ M.closure (X \\ {e}) ∈ U\nx : α\nhxI : x ∉ I\nhi : x ∈ X\nhind : U.ExtIndep e (insert x I)\n⊢ False", id=7, message='') | case pos
α : Type u_1
M : Matroid α
I J B F₀ F F' X Y : Set α
e f : α
U : M.ModularCut
hX : X ⊆ insert e M.E
hI : U.ExtIndep e I
hIX : I ⊆ X
hss : I \ {e} ⊆ X \ {e}
hX' : X \ {e} ⊆ M.E
h : ∀ x ∉ I, x ∈ X → ¬U.ExtIndep e (insert x I)
heI : e ∈ I
⊢ M.closure (I \ {e}) = M.closure (X \ {e}) ∧ (e ∈ X → ¬(e ∈ I ↔ M.closure (X \ {e}) ∈ U)) ∨
(M.closure (I \ {e}) ⋖[M] M.closure (X \ {e})) ∧ e ∈ I ∧ M.closure (X \ {e}) ∈ U
case neg
α : Type u_1
M : Matroid α
I J B F₀ F F' X Y : Set α
e f : α
U : M.ModularCut
hX : X ⊆ insert e M.E
hI : U.ExtIndep e I
hIX : I ⊆ X
hss : I \ {e} ⊆ X \ {e}
hX' : X \ {e} ⊆ M.E
h : ∀ x ∉ I, x ∈ X → ¬U.ExtIndep e (insert x I)
heI : e ∉ I
⊢ M.closure (I \ {e}) = M.closure (X \ {e}) ∧ (e ∈ X → ¬(e ∈ I ↔ M.closure (X \ {e}) ∈ U)) ∨
(M.closure (I \ {e}) ⋖[M] M.closure (X \ {e})) ∧ e ∈ I ∧ M.closure (X \ {e}) ∈ U
case refine_2
α : Type u_1
M : Matroid α
I J B F₀ F F' X Y : Set α
e f : α
U : M.ModularCut
hX : X ⊆ insert e M.E
hI : U.ExtIndep e I
hIX : I ⊆ X
hss : I \ {e} ⊆ X \ {e}
hX' : X \ {e} ⊆ M.E
h :
M.closure (I \ {e}) = M.closure (X \ {e}) ∧ (e ∈ X → ¬(e ∈ I ↔ M.closure (X \ {e}) ∈ U)) ∨
(M.closure (I \ {e}) ⋖[M] M.closure (X \ {e})) ∧ e ∈ I ∧ M.closure (X \ {e}) ∈ U
x : α
hxI : x ∉ I
hi : x ∈ X
hind : U.ExtIndep e (insert x I)
⊢ False |
|
https://github.com/apnelson1/Matroid | 510262f56c0025be19502ab3457a788e2f6d2d1c | Matroid/Extension.lean | Matroid.ModularCut.maximal_extIndep_iff | Status.FAILED | case pos
α : Type u_1
M : Matroid α
I J B F₀ F F' X Y : Set α
e f : α
U : M.ModularCut
hX : X ⊆ insert e M.E
hI : U.ExtIndep e I
hIX : I ⊆ X
hss : I \ {e} ⊆ X \ {e}
hX' : X \ {e} ⊆ M.E
h : ∀ x ∉ I, x ∈ X → ¬U.ExtIndep e (insert x I)
heI : e ∈ I
⊢ M.closure (I \ {e}) = M.closure (X \ {e}) ∧ (e ∈ X → ¬(e ∈ I ↔ M.closure (X \ {e}) ∈ U)) ∨
(M.closure (I \ {e}) ⋖[M] M.closure (X \ {e})) ∧ e ∈ I ∧ M.closure (X \ {e}) ∈ U
case neg
α : Type u_1
M : Matroid α
I J B F₀ F F' X Y : Set α
e f : α
U : M.ModularCut
hX : X ⊆ insert e M.E
hI : U.ExtIndep e I
hIX : I ⊆ X
hss : I \ {e} ⊆ X \ {e}
hX' : X \ {e} ⊆ M.E
h : ∀ x ∉ I, x ∈ X → ¬U.ExtIndep e (insert x I)
heI : e ∉ I
⊢ M.closure (I \ {e}) = M.closure (X \ {e}) ∧ (e ∈ X → ¬(e ∈ I ↔ M.closure (X \ {e}) ∈ U)) ∨
(M.closure (I \ {e}) ⋖[M] M.closure (X \ {e})) ∧ e ∈ I ∧ M.closure (X \ {e}) ∈ U
case refine_2
α : Type u_1
M : Matroid α
I J B F₀ F F' X Y : Set α
e f : α
U : M.ModularCut
hX : X ⊆ insert e M.E
hI : U.ExtIndep e I
hIX : I ⊆ X
hss : I \ {e} ⊆ X \ {e}
hX' : X \ {e} ⊆ M.E
h :
M.closure (I \ {e}) = M.closure (X \ {e}) ∧ (e ∈ X → ¬(e ∈ I ↔ M.closure (X \ {e}) ∈ U)) ∨
(M.closure (I \ {e}) ⋖[M] M.closure (X \ {e})) ∧ e ∈ I ∧ M.closure (X \ {e}) ∈ U
x : α
hxI : x ∉ I
hi : x ∈ X
hind : U.ExtIndep e (insert x I)
⊢ False | [Goal(assumptions=[Declaration(ident='α', lean_type='Type u_1'), Declaration(ident='M', lean_type='Matroid α'), Declaration(ident='I', lean_type='Set α'), Declaration(ident='J', lean_type='Set α'), Declaration(ident='B', lean_type='Set α'), Declaration(ident='F₀', lean_type='Set α'), Declaration(ident='F', lean_type='Set α'), Declaration(ident="F'", lean_type='Set α'), Declaration(ident='X', lean_type='Set α'), Declaration(ident='Y', lean_type='Set α'), Declaration(ident='e', lean_type='α'), Declaration(ident='f', lean_type='α'), Declaration(ident='U', lean_type='M.ModularCut'), Declaration(ident='hX', lean_type='X ⊆ insert e M.E'), Declaration(ident='hI', lean_type='U.ExtIndep e I'), Declaration(ident='hIX', lean_type='I ⊆ X'), Declaration(ident='hss', lean_type='I \\ {e} ⊆ X \\ {e}'), Declaration(ident="hX'", lean_type='X \\ {e} ⊆ M.E'), Declaration(ident='h', lean_type='∀ x ∉ I, x ∈ X → ¬U.ExtIndep e (insert x I)'), Declaration(ident='heI', lean_type='e ∈ I')], conclusion='M.closure (I \\ {e}) = M.closure (X \\ {e}) ∧ (e ∈ X → ¬(e ∈ I ↔ M.closure (X \\ {e}) ∈ U)) ∨\n (M.closure (I \\ {e}) ⋖[M] M.closure (X \\ {e})) ∧ e ∈ I ∧ M.closure (X \\ {e}) ∈ U'), Goal(assumptions=[Declaration(ident='α', lean_type='Type u_1'), Declaration(ident='M', lean_type='Matroid α'), Declaration(ident='I', lean_type='Set α'), Declaration(ident='J', lean_type='Set α'), Declaration(ident='B', lean_type='Set α'), Declaration(ident='F₀', lean_type='Set α'), Declaration(ident='F', lean_type='Set α'), Declaration(ident="F'", lean_type='Set α'), Declaration(ident='X', lean_type='Set α'), Declaration(ident='Y', lean_type='Set α'), Declaration(ident='e', lean_type='α'), Declaration(ident='f', lean_type='α'), Declaration(ident='U', lean_type='M.ModularCut'), Declaration(ident='hX', lean_type='X ⊆ insert e M.E'), Declaration(ident='hI', lean_type='U.ExtIndep e I'), Declaration(ident='hIX', lean_type='I ⊆ X'), Declaration(ident='hss', lean_type='I \\ {e} ⊆ X \\ {e}'), Declaration(ident="hX'", lean_type='X \\ {e} ⊆ M.E'), Declaration(ident='h', lean_type='∀ x ∉ I, x ∈ X → ¬U.ExtIndep e (insert x I)'), Declaration(ident='heI', lean_type='e ∉ I')], conclusion='M.closure (I \\ {e}) = M.closure (X \\ {e}) ∧ (e ∈ X → ¬(e ∈ I ↔ M.closure (X \\ {e}) ∈ U)) ∨\n (M.closure (I \\ {e}) ⋖[M] M.closure (X \\ {e})) ∧ e ∈ I ∧ M.closure (X \\ {e}) ∈ U'), Goal(assumptions=[Declaration(ident='α', lean_type='Type u_1'), Declaration(ident='M', lean_type='Matroid α'), Declaration(ident='I', lean_type='Set α'), Declaration(ident='J', lean_type='Set α'), Declaration(ident='B', lean_type='Set α'), Declaration(ident='F₀', lean_type='Set α'), Declaration(ident='F', lean_type='Set α'), Declaration(ident="F'", lean_type='Set α'), Declaration(ident='X', lean_type='Set α'), Declaration(ident='Y', lean_type='Set α'), Declaration(ident='e', lean_type='α'), Declaration(ident='f', lean_type='α'), Declaration(ident='U', lean_type='M.ModularCut'), Declaration(ident='hX', lean_type='X ⊆ insert e M.E'), Declaration(ident='hI', lean_type='U.ExtIndep e I'), Declaration(ident='hIX', lean_type='I ⊆ X'), Declaration(ident='hss', lean_type='I \\ {e} ⊆ X \\ {e}'), Declaration(ident="hX'", lean_type='X \\ {e} ⊆ M.E'), Declaration(ident='h', lean_type='M.closure (I \\ {e}) = M.closure (X \\ {e}) ∧ (e ∈ X → ¬(e ∈ I ↔ M.closure (X \\ {e}) ∈ U)) ∨\n (M.closure (I \\ {e}) ⋖[M] M.closure (X \\ {e})) ∧ e ∈ I ∧ M.closure (X \\ {e}) ∈ U'), Declaration(ident='x', lean_type='α'), Declaration(ident='hxI', lean_type='x ∉ I'), Declaration(ident='hi', lean_type='x ∈ X'), Declaration(ident='hind', lean_type='U.ExtIndep e (insert x I)')], conclusion='False')] | obtain (rfl | hne) := eq_or_ne e x | <class 'lean_dojo.interaction.dojo.LeanError'> | LeanError(error="unknown identifier 'x'") | unknown identifier 'x' |
|
https://github.com/apnelson1/Matroid | 510262f56c0025be19502ab3457a788e2f6d2d1c | Matroid/Parallel.lean | Matroid.Parallel'.mem_closure | Status.PROVED | α : Type u_1
M N : Matroid α
e f g : α
I X P D : Set α
h : M.Parallel' e f
⊢ e ∈ M.closure {f} | None | [Goal(assumptions=[Declaration(ident='α', lean_type='Type u_1'), Declaration(ident='M', lean_type='Matroid α'), Declaration(ident='N', lean_type='Matroid α'), Declaration(ident='e', lean_type='α'), Declaration(ident='f', lean_type='α'), Declaration(ident='g', lean_type='α'), Declaration(ident='I', lean_type='Set α'), Declaration(ident='X', lean_type='Set α'), Declaration(ident='P', lean_type='Set α'), Declaration(ident='D', lean_type='Set α'), Declaration(ident='h', lean_type="M.Parallel' e f")], conclusion='e ∈ M.closure {f}')] | rw [← h.closure_eq_closure] | <class 'lean_dojo.interaction.dojo.TacticState'> | TacticState(pp="α : Type u_1\nM N : Matroid α\ne f g : α\nI X P D : Set α\nh : M.Parallel' e f\n⊢ e ∈ M.closure {e}", id=1, message='') | α : Type u_1
M N : Matroid α
e f g : α
I X P D : Set α
h : M.Parallel' e f
⊢ e ∈ M.closure {e} |
https://github.com/apnelson1/Matroid | 510262f56c0025be19502ab3457a788e2f6d2d1c | Matroid/Parallel.lean | Matroid.Parallel'.mem_closure | Status.PROVED | α : Type u_1
M N : Matroid α
e f g : α
I X P D : Set α
h : M.Parallel' e f
⊢ e ∈ M.closure {e} | [Goal(assumptions=[Declaration(ident='α', lean_type='Type u_1'), Declaration(ident='M', lean_type='Matroid α'), Declaration(ident='N', lean_type='Matroid α'), Declaration(ident='e', lean_type='α'), Declaration(ident='f', lean_type='α'), Declaration(ident='g', lean_type='α'), Declaration(ident='I', lean_type='Set α'), Declaration(ident='X', lean_type='Set α'), Declaration(ident='P', lean_type='Set α'), Declaration(ident='D', lean_type='Set α'), Declaration(ident='h', lean_type="M.Parallel' e f")], conclusion='e ∈ M.closure {e}')] | apply mem_closure_self _ _ h.mem_ground_left | <class 'lean_dojo.interaction.dojo.ProofFinished'> | ProofFinished(tactic_state_id=2, message='') | ||
https://github.com/apnelson1/Matroid | 510262f56c0025be19502ab3457a788e2f6d2d1c | Matroid/Flat.lean | Matroid.Hyperplane.inter_ssubset_left_of_ne | Status.PROVED | α : Type u_1
M : Matroid α
I F X Y F' F₀ F₁ F₂ P L H H₁ H₂ H' B C K : Set α
e f : α
h₁ : M.Hyperplane H₁
h₂ : M.Hyperplane H₂
hne : H₁ ≠ H₂
⊢ H₁ ∩ H₂ ⊂ H₁ | None | [Goal(assumptions=[Declaration(ident='α', lean_type='Type u_1'), Declaration(ident='M', lean_type='Matroid α'), Declaration(ident='I', lean_type='Set α'), Declaration(ident='F', lean_type='Set α'), Declaration(ident='X', lean_type='Set α'), Declaration(ident='Y', lean_type='Set α'), Declaration(ident="F'", lean_type='Set α'), Declaration(ident='F₀', lean_type='Set α'), Declaration(ident='F₁', lean_type='Set α'), Declaration(ident='F₂', lean_type='Set α'), Declaration(ident='P', lean_type='Set α'), Declaration(ident='L', lean_type='Set α'), Declaration(ident='H', lean_type='Set α'), Declaration(ident='H₁', lean_type='Set α'), Declaration(ident='H₂', lean_type='Set α'), Declaration(ident="H'", lean_type='Set α'), Declaration(ident='B', lean_type='Set α'), Declaration(ident='C', lean_type='Set α'), Declaration(ident='K', lean_type='Set α'), Declaration(ident='e', lean_type='α'), Declaration(ident='f', lean_type='α'), Declaration(ident='h₁', lean_type='M.Hyperplane H₁'), Declaration(ident='h₂', lean_type='M.Hyperplane H₂'), Declaration(ident='hne', lean_type='H₁ ≠ H₂')], conclusion='H₁ ∩ H₂ ⊂ H₁')] | refine inter_subset_left.ssubset_of_ne fun h_eq ↦ hne <| h₁.eq_of_subset h₂ ?_ | <class 'lean_dojo.interaction.dojo.TacticState'> | TacticState(pp="α : Type u_1\nM : Matroid α\nI F X Y F' F₀ F₁ F₂ P L H H₁ H₂ H' B C K : Set α\ne f : α\nh₁ : M.Hyperplane H₁\nh₂ : M.Hyperplane H₂\nhne : H₁ ≠ H₂\nh_eq : H₁ ∩ H₂ = H₁\n⊢ H₁ ⊆ H₂", id=1, message='') | α : Type u_1
M : Matroid α
I F X Y F' F₀ F₁ F₂ P L H H₁ H₂ H' B C K : Set α
e f : α
h₁ : M.Hyperplane H₁
h₂ : M.Hyperplane H₂
hne : H₁ ≠ H₂
h_eq : H₁ ∩ H₂ = H₁
⊢ H₁ ⊆ H₂ |
https://github.com/apnelson1/Matroid | 510262f56c0025be19502ab3457a788e2f6d2d1c | Matroid/Flat.lean | Matroid.Hyperplane.inter_ssubset_left_of_ne | Status.PROVED | α : Type u_1
M : Matroid α
I F X Y F' F₀ F₁ F₂ P L H H₁ H₂ H' B C K : Set α
e f : α
h₁ : M.Hyperplane H₁
h₂ : M.Hyperplane H₂
hne : H₁ ≠ H₂
h_eq : H₁ ∩ H₂ = H₁
⊢ H₁ ⊆ H₂ | [Goal(assumptions=[Declaration(ident='α', lean_type='Type u_1'), Declaration(ident='M', lean_type='Matroid α'), Declaration(ident='I', lean_type='Set α'), Declaration(ident='F', lean_type='Set α'), Declaration(ident='X', lean_type='Set α'), Declaration(ident='Y', lean_type='Set α'), Declaration(ident="F'", lean_type='Set α'), Declaration(ident='F₀', lean_type='Set α'), Declaration(ident='F₁', lean_type='Set α'), Declaration(ident='F₂', lean_type='Set α'), Declaration(ident='P', lean_type='Set α'), Declaration(ident='L', lean_type='Set α'), Declaration(ident='H', lean_type='Set α'), Declaration(ident='H₁', lean_type='Set α'), Declaration(ident='H₂', lean_type='Set α'), Declaration(ident="H'", lean_type='Set α'), Declaration(ident='B', lean_type='Set α'), Declaration(ident='C', lean_type='Set α'), Declaration(ident='K', lean_type='Set α'), Declaration(ident='e', lean_type='α'), Declaration(ident='f', lean_type='α'), Declaration(ident='h₁', lean_type='M.Hyperplane H₁'), Declaration(ident='h₂', lean_type='M.Hyperplane H₂'), Declaration(ident='hne', lean_type='H₁ ≠ H₂'), Declaration(ident='h_eq', lean_type='H₁ ∩ H₂ = H₁')], conclusion='H₁ ⊆ H₂')] | rw [← h_eq] | <class 'lean_dojo.interaction.dojo.TacticState'> | TacticState(pp="α : Type u_1\nM : Matroid α\nI F X Y F' F₀ F₁ F₂ P L H H₁ H₂ H' B C K : Set α\ne f : α\nh₁ : M.Hyperplane H₁\nh₂ : M.Hyperplane H₂\nhne : H₁ ≠ H₂\nh_eq : H₁ ∩ H₂ = H₁\n⊢ H₁ ∩ H₂ ⊆ H₂", id=2, message='') | α : Type u_1
M : Matroid α
I F X Y F' F₀ F₁ F₂ P L H H₁ H₂ H' B C K : Set α
e f : α
h₁ : M.Hyperplane H₁
h₂ : M.Hyperplane H₂
hne : H₁ ≠ H₂
h_eq : H₁ ∩ H₂ = H₁
⊢ H₁ ∩ H₂ ⊆ H₂ |
|
https://github.com/apnelson1/Matroid | 510262f56c0025be19502ab3457a788e2f6d2d1c | Matroid/Flat.lean | Matroid.Hyperplane.inter_ssubset_left_of_ne | Status.PROVED | α : Type u_1
M : Matroid α
I F X Y F' F₀ F₁ F₂ P L H H₁ H₂ H' B C K : Set α
e f : α
h₁ : M.Hyperplane H₁
h₂ : M.Hyperplane H₂
hne : H₁ ≠ H₂
h_eq : H₁ ∩ H₂ = H₁
⊢ H₁ ∩ H₂ ⊆ H₂ | [Goal(assumptions=[Declaration(ident='α', lean_type='Type u_1'), Declaration(ident='M', lean_type='Matroid α'), Declaration(ident='I', lean_type='Set α'), Declaration(ident='F', lean_type='Set α'), Declaration(ident='X', lean_type='Set α'), Declaration(ident='Y', lean_type='Set α'), Declaration(ident="F'", lean_type='Set α'), Declaration(ident='F₀', lean_type='Set α'), Declaration(ident='F₁', lean_type='Set α'), Declaration(ident='F₂', lean_type='Set α'), Declaration(ident='P', lean_type='Set α'), Declaration(ident='L', lean_type='Set α'), Declaration(ident='H', lean_type='Set α'), Declaration(ident='H₁', lean_type='Set α'), Declaration(ident='H₂', lean_type='Set α'), Declaration(ident="H'", lean_type='Set α'), Declaration(ident='B', lean_type='Set α'), Declaration(ident='C', lean_type='Set α'), Declaration(ident='K', lean_type='Set α'), Declaration(ident='e', lean_type='α'), Declaration(ident='f', lean_type='α'), Declaration(ident='h₁', lean_type='M.Hyperplane H₁'), Declaration(ident='h₂', lean_type='M.Hyperplane H₂'), Declaration(ident='hne', lean_type='H₁ ≠ H₂'), Declaration(ident='h_eq', lean_type='H₁ ∩ H₂ = H₁')], conclusion='H₁ ∩ H₂ ⊆ H₂')] | exact inter_subset_right | <class 'lean_dojo.interaction.dojo.ProofFinished'> | ProofFinished(tactic_state_id=3, message='') | ||
https://github.com/apnelson1/Matroid | 510262f56c0025be19502ab3457a788e2f6d2d1c | Matroid/Equiv.lean | Matroid.Iso.preimage_subset_iff | Status.PROVED | α : Type u_1
β : Type u_2
M : Matroid α
N : Matroid β
e : M ≂ N
X : Set ↑N.E
Y : Set ↑M.E
⊢ ⇑e ⁻¹' X ⊆ Y ↔ X ⊆ ⇑e '' Y | None | [Goal(assumptions=[Declaration(ident='α', lean_type='Type u_1'), Declaration(ident='β', lean_type='Type u_2'), Declaration(ident='M', lean_type='Matroid α'), Declaration(ident='N', lean_type='Matroid β'), Declaration(ident='e', lean_type='M ≂ N'), Declaration(ident='X', lean_type='Set ↑N.E'), Declaration(ident='Y', lean_type='Set ↑M.E')], conclusion="⇑e ⁻¹' X ⊆ Y ↔ X ⊆ ⇑e '' Y")] | rw [← e.image_symm_eq_preimage, image_subset_iff, e.preimage_symm_eq_image] | <class 'lean_dojo.interaction.dojo.ProofFinished'> | ProofFinished(tactic_state_id=1, message='') | |
https://github.com/apnelson1/Matroid | 510262f56c0025be19502ab3457a788e2f6d2d1c | Matroid/Flat.lean | Matroid.closure_covBy_iff | Status.PROVED | α : Type u_1
M : Matroid α
I F X Y F' F₀ F₁ F₂ P L H H₁ H₂ H' B C K : Set α
e f : α
⊢ M.closure X ⋖[M] F ↔ ∃ e ∈ M.E \ M.closure X, F = M.closure (insert e X) | None | [Goal(assumptions=[Declaration(ident='α', lean_type='Type u_1'), Declaration(ident='M', lean_type='Matroid α'), Declaration(ident='I', lean_type='Set α'), Declaration(ident='F', lean_type='Set α'), Declaration(ident='X', lean_type='Set α'), Declaration(ident='Y', lean_type='Set α'), Declaration(ident="F'", lean_type='Set α'), Declaration(ident='F₀', lean_type='Set α'), Declaration(ident='F₁', lean_type='Set α'), Declaration(ident='F₂', lean_type='Set α'), Declaration(ident='P', lean_type='Set α'), Declaration(ident='L', lean_type='Set α'), Declaration(ident='H', lean_type='Set α'), Declaration(ident='H₁', lean_type='Set α'), Declaration(ident='H₂', lean_type='Set α'), Declaration(ident="H'", lean_type='Set α'), Declaration(ident='B', lean_type='Set α'), Declaration(ident='C', lean_type='Set α'), Declaration(ident='K', lean_type='Set α'), Declaration(ident='e', lean_type='α'), Declaration(ident='f', lean_type='α')], conclusion='M.closure X ⋖[M] F ↔ ∃ e ∈ M.E \\ M.closure X, F = M.closure (insert e X)')] | simp_rw [(M.closure_flat X).covBy_iff_eq_closure_insert, closure_insert_closure_eq_closure_insert] | <class 'lean_dojo.interaction.dojo.ProofFinished'> | ProofFinished(tactic_state_id=1, message='') | |
https://github.com/apnelson1/Matroid | 510262f56c0025be19502ab3457a788e2f6d2d1c | Matroid/Loop.lean | Matroid.Coloop.insert_indep_of_indep | Status.PROVED | α : Type u_1
β : Type u_2
M N : Matroid α
e f : α
B L L' I X Y Z F C K : Set α
he : M.Coloop e
hI : M.Indep I
⊢ M.Indep (insert e I) | None | [Goal(assumptions=[Declaration(ident='α', lean_type='Type u_1'), Declaration(ident='β', lean_type='Type u_2'), Declaration(ident='M', lean_type='Matroid α'), Declaration(ident='N', lean_type='Matroid α'), Declaration(ident='e', lean_type='α'), Declaration(ident='f', lean_type='α'), Declaration(ident='B', lean_type='Set α'), Declaration(ident='L', lean_type='Set α'), Declaration(ident="L'", lean_type='Set α'), Declaration(ident='I', lean_type='Set α'), Declaration(ident='X', lean_type='Set α'), Declaration(ident='Y', lean_type='Set α'), Declaration(ident='Z', lean_type='Set α'), Declaration(ident='F', lean_type='Set α'), Declaration(ident='C', lean_type='Set α'), Declaration(ident='K', lean_type='Set α'), Declaration(ident='he', lean_type='M.Coloop e'), Declaration(ident='hI', lean_type='M.Indep I')], conclusion='M.Indep (insert e I)')] | refine (em (e ∈ I)).elim (fun h ↦ by rwa [insert_eq_of_mem h]) fun h ↦ ?_ | <class 'lean_dojo.interaction.dojo.TacticState'> | TacticState(pp="α : Type u_1\nβ : Type u_2\nM N : Matroid α\ne f : α\nB L L' I X Y Z F C K : Set α\nhe : M.Coloop e\nhI : M.Indep I\nh : e ∉ I\n⊢ M.Indep (insert e I)", id=1, message='') | α : Type u_1
β : Type u_2
M N : Matroid α
e f : α
B L L' I X Y Z F C K : Set α
he : M.Coloop e
hI : M.Indep I
h : e ∉ I
⊢ M.Indep (insert e I) |
https://github.com/apnelson1/Matroid | 510262f56c0025be19502ab3457a788e2f6d2d1c | Matroid/Loop.lean | Matroid.Coloop.insert_indep_of_indep | Status.PROVED | α : Type u_1
β : Type u_2
M N : Matroid α
e f : α
B L L' I X Y Z F C K : Set α
he : M.Coloop e
hI : M.Indep I
h : e ∉ I
⊢ M.Indep (insert e I) | [Goal(assumptions=[Declaration(ident='α', lean_type='Type u_1'), Declaration(ident='β', lean_type='Type u_2'), Declaration(ident='M', lean_type='Matroid α'), Declaration(ident='N', lean_type='Matroid α'), Declaration(ident='e', lean_type='α'), Declaration(ident='f', lean_type='α'), Declaration(ident='B', lean_type='Set α'), Declaration(ident='L', lean_type='Set α'), Declaration(ident="L'", lean_type='Set α'), Declaration(ident='I', lean_type='Set α'), Declaration(ident='X', lean_type='Set α'), Declaration(ident='Y', lean_type='Set α'), Declaration(ident='Z', lean_type='Set α'), Declaration(ident='F', lean_type='Set α'), Declaration(ident='C', lean_type='Set α'), Declaration(ident='K', lean_type='Set α'), Declaration(ident='he', lean_type='M.Coloop e'), Declaration(ident='hI', lean_type='M.Indep I'), Declaration(ident='h', lean_type='e ∉ I')], conclusion='M.Indep (insert e I)')] | rw [← hI.not_mem_closure_iff_of_not_mem h] | <class 'lean_dojo.interaction.dojo.TacticState'> | TacticState(pp="α : Type u_1\nβ : Type u_2\nM N : Matroid α\ne f : α\nB L L' I X Y Z F C K : Set α\nhe : M.Coloop e\nhI : M.Indep I\nh : e ∉ I\n⊢ e ∉ M.closure I", id=2, message='') | α : Type u_1
β : Type u_2
M N : Matroid α
e f : α
B L L' I X Y Z F C K : Set α
he : M.Coloop e
hI : M.Indep I
h : e ∉ I
⊢ e ∉ M.closure I |
|
https://github.com/apnelson1/Matroid | 510262f56c0025be19502ab3457a788e2f6d2d1c | Matroid/Loop.lean | Matroid.Coloop.insert_indep_of_indep | Status.PROVED | α : Type u_1
β : Type u_2
M N : Matroid α
e f : α
B L L' I X Y Z F C K : Set α
he : M.Coloop e
hI : M.Indep I
h : e ∉ I
⊢ e ∉ M.closure I | [Goal(assumptions=[Declaration(ident='α', lean_type='Type u_1'), Declaration(ident='β', lean_type='Type u_2'), Declaration(ident='M', lean_type='Matroid α'), Declaration(ident='N', lean_type='Matroid α'), Declaration(ident='e', lean_type='α'), Declaration(ident='f', lean_type='α'), Declaration(ident='B', lean_type='Set α'), Declaration(ident='L', lean_type='Set α'), Declaration(ident="L'", lean_type='Set α'), Declaration(ident='I', lean_type='Set α'), Declaration(ident='X', lean_type='Set α'), Declaration(ident='Y', lean_type='Set α'), Declaration(ident='Z', lean_type='Set α'), Declaration(ident='F', lean_type='Set α'), Declaration(ident='C', lean_type='Set α'), Declaration(ident='K', lean_type='Set α'), Declaration(ident='he', lean_type='M.Coloop e'), Declaration(ident='hI', lean_type='M.Indep I'), Declaration(ident='h', lean_type='e ∉ I')], conclusion='e ∉ M.closure I')] | exact he.not_mem_closure_of_not_mem h | <class 'lean_dojo.interaction.dojo.ProofFinished'> | ProofFinished(tactic_state_id=3, message='') | ||
https://github.com/apnelson1/Matroid | 510262f56c0025be19502ab3457a788e2f6d2d1c | Matroid/ForMathlib/MatroidMap.lean | Matroid.restrictSubtype_ground_base_iff | Status.PROVED | α : Type u_1
β : Type u_2
M : Matroid α
X : Set α
B : Set ↑M.E
⊢ (M.restrictSubtype M.E).Base B ↔ M.Base (Subtype.val '' B) | None | [Goal(assumptions=[Declaration(ident='α', lean_type='Type u_1'), Declaration(ident='β', lean_type='Type u_2'), Declaration(ident='M', lean_type='Matroid α'), Declaration(ident='X', lean_type='Set α'), Declaration(ident='B', lean_type='Set ↑M.E')], conclusion="(M.restrictSubtype M.E).Base B ↔ M.Base (Subtype.val '' B)")] | rw [restrictSubtype_base_iff, basis'_iff_basis, basis_ground_iff] | <class 'lean_dojo.interaction.dojo.ProofFinished'> | ProofFinished(tactic_state_id=1, message='') | |
https://github.com/apnelson1/Matroid | 510262f56c0025be19502ab3457a788e2f6d2d1c | Matroid/Circuit.lean | Matroid.girth_emptyOn | Status.PROVED | α : Type u_1
M : Matroid α
C C' I X K C₁ C₂ R : Set α
e f x y : α
E D : Set α
⊢ (emptyOn α).girth = ⊤ | None | [Goal(assumptions=[Declaration(ident='α', lean_type='Type u_1'), Declaration(ident='M', lean_type='Matroid α'), Declaration(ident='C', lean_type='Set α'), Declaration(ident="C'", lean_type='Set α'), Declaration(ident='I', lean_type='Set α'), Declaration(ident='X', lean_type='Set α'), Declaration(ident='K', lean_type='Set α'), Declaration(ident='C₁', lean_type='Set α'), Declaration(ident='C₂', lean_type='Set α'), Declaration(ident='R', lean_type='Set α'), Declaration(ident='e', lean_type='α'), Declaration(ident='f', lean_type='α'), Declaration(ident='x', lean_type='α'), Declaration(ident='y', lean_type='α'), Declaration(ident='E', lean_type='Set α'), Declaration(ident='D', lean_type='Set α')], conclusion='(emptyOn α).girth = ⊤')] | simp [girth] | <class 'lean_dojo.interaction.dojo.ProofFinished'> | ProofFinished(tactic_state_id=1, message='') | |
https://github.com/apnelson1/Matroid | 510262f56c0025be19502ab3457a788e2f6d2d1c | Matroid/ForMathlib/Other.lean | Set.Finite.encard_le_iff_nonempty_embedding | Status.FAILED | α : Type u_1
β : Type u_2
s✝ s₁ s₂ t✝ t' : Set α
f : α → β
s : Set α
t : Set β
hs : s.Finite
⊢ s.encard ≤ t.encard ↔ Nonempty (↑s ↪ ↑t) | None | [Goal(assumptions=[Declaration(ident='α', lean_type='Type u_1'), Declaration(ident='β', lean_type='Type u_2'), Declaration(ident='s✝', lean_type='Set α'), Declaration(ident='s₁', lean_type='Set α'), Declaration(ident='s₂', lean_type='Set α'), Declaration(ident='t✝', lean_type='Set α'), Declaration(ident="t'", lean_type='Set α'), Declaration(ident='f', lean_type='α → β'), Declaration(ident='s', lean_type='Set α'), Declaration(ident='t', lean_type='Set β'), Declaration(ident='hs', lean_type='s.Finite')], conclusion='s.encard ≤ t.encard ↔ Nonempty (↑s ↪ ↑t)')] | cases isEmpty_or_nonempty β | <class 'lean_dojo.interaction.dojo.TacticState'> | TacticState(pp="case inl\nα : Type u_1\nβ : Type u_2\ns✝ s₁ s₂ t✝ t' : Set α\nf : α → β\ns : Set α\nt : Set β\nhs : s.Finite\nh✝ : IsEmpty β\n⊢ s.encard ≤ t.encard ↔ Nonempty (↑s ↪ ↑t)\n\ncase inr\nα : Type u_1\nβ : Type u_2\ns✝ s₁ s₂ t✝ t' : Set α\nf : α → β\ns : Set α\nt : Set β\nhs : s.Finite\nh✝ : Nonempty β\n⊢ s.encard ≤ t.encard ↔ Nonempty (↑s ↪ ↑t)", id=1, message='') | case inl
α : Type u_1
β : Type u_2
s✝ s₁ s₂ t✝ t' : Set α
f : α → β
s : Set α
t : Set β
hs : s.Finite
h✝ : IsEmpty β
⊢ s.encard ≤ t.encard ↔ Nonempty (↑s ↪ ↑t)
case inr
α : Type u_1
β : Type u_2
s✝ s₁ s₂ t✝ t' : Set α
f : α → β
s : Set α
t : Set β
hs : s.Finite
h✝ : Nonempty β
⊢ s.encard ≤ t.encard ↔ Nonempty (↑s ↪ ↑t) |
https://github.com/apnelson1/Matroid | 510262f56c0025be19502ab3457a788e2f6d2d1c | Matroid/ForMathlib/Other.lean | Set.Finite.encard_le_iff_nonempty_embedding | Status.FAILED | case inl
α : Type u_1
β : Type u_2
s✝ s₁ s₂ t✝ t' : Set α
f : α → β
s : Set α
t : Set β
hs : s.Finite
h✝ : IsEmpty β
⊢ s.encard ≤ t.encard ↔ Nonempty (↑s ↪ ↑t)
case inr
α : Type u_1
β : Type u_2
s✝ s₁ s₂ t✝ t' : Set α
f : α → β
s : Set α
t : Set β
hs : s.Finite
h✝ : Nonempty β
⊢ s.encard ≤ t.encard ↔ Nonempty (↑s ↪ ↑t) | [Goal(assumptions=[Declaration(ident='α', lean_type='Type u_1'), Declaration(ident='β', lean_type='Type u_2'), Declaration(ident='s✝', lean_type='Set α'), Declaration(ident='s₁', lean_type='Set α'), Declaration(ident='s₂', lean_type='Set α'), Declaration(ident='t✝', lean_type='Set α'), Declaration(ident="t'", lean_type='Set α'), Declaration(ident='f', lean_type='α → β'), Declaration(ident='s', lean_type='Set α'), Declaration(ident='t', lean_type='Set β'), Declaration(ident='hs', lean_type='s.Finite'), Declaration(ident='h✝', lean_type='IsEmpty β')], conclusion='s.encard ≤ t.encard ↔ Nonempty (↑s ↪ ↑t)'), Goal(assumptions=[Declaration(ident='α', lean_type='Type u_1'), Declaration(ident='β', lean_type='Type u_2'), Declaration(ident='s✝', lean_type='Set α'), Declaration(ident='s₁', lean_type='Set α'), Declaration(ident='s₂', lean_type='Set α'), Declaration(ident='t✝', lean_type='Set α'), Declaration(ident="t'", lean_type='Set α'), Declaration(ident='f', lean_type='α → β'), Declaration(ident='s', lean_type='Set α'), Declaration(ident='t', lean_type='Set β'), Declaration(ident='hs', lean_type='s.Finite'), Declaration(ident='h✝', lean_type='Nonempty β')], conclusion='s.encard ≤ t.encard ↔ Nonempty (↑s ↪ ↑t)')] | refine ⟨fun h ↦ ?_, fun ⟨e⟩ ↦ e.enccard_le⟩ | <class 'lean_dojo.interaction.dojo.TacticState'> | TacticState(pp="case inl\nα : Type u_1\nβ : Type u_2\ns✝ s₁ s₂ t✝ t' : Set α\nf : α → β\ns : Set α\nt : Set β\nhs : s.Finite\nh✝ : IsEmpty β\nh : s.encard ≤ t.encard\n⊢ Nonempty (↑s ↪ ↑t)\n\ncase inr\nα : Type u_1\nβ : Type u_2\ns✝ s₁ s₂ t✝ t' : Set α\nf : α → β\ns : Set α\nt : Set β\nhs : s.Finite\nh✝ : Nonempty β\n⊢ s.encard ≤ t.encard ↔ Nonempty (↑s ↪ ↑t)", id=2, message='') | case inl
α : Type u_1
β : Type u_2
s✝ s₁ s₂ t✝ t' : Set α
f : α → β
s : Set α
t : Set β
hs : s.Finite
h✝ : IsEmpty β
h : s.encard ≤ t.encard
⊢ Nonempty (↑s ↪ ↑t)
case inr
α : Type u_1
β : Type u_2
s✝ s₁ s₂ t✝ t' : Set α
f : α → β
s : Set α
t : Set β
hs : s.Finite
h✝ : Nonempty β
⊢ s.encard ≤ t.encard ↔ Nonempty (↑s ↪ ↑t) |
|
https://github.com/apnelson1/Matroid | 510262f56c0025be19502ab3457a788e2f6d2d1c | Matroid/ForMathlib/Other.lean | Set.Finite.encard_le_iff_nonempty_embedding | Status.FAILED | case inl
α : Type u_1
β : Type u_2
s✝ s₁ s₂ t✝ t' : Set α
f : α → β
s : Set α
t : Set β
hs : s.Finite
h✝ : IsEmpty β
h : s.encard ≤ t.encard
⊢ Nonempty (↑s ↪ ↑t)
case inr
α : Type u_1
β : Type u_2
s✝ s₁ s₂ t✝ t' : Set α
f : α → β
s : Set α
t : Set β
hs : s.Finite
h✝ : Nonempty β
⊢ s.encard ≤ t.encard ↔ Nonempty (↑s ↪ ↑t) | [Goal(assumptions=[Declaration(ident='α', lean_type='Type u_1'), Declaration(ident='β', lean_type='Type u_2'), Declaration(ident='s✝', lean_type='Set α'), Declaration(ident='s₁', lean_type='Set α'), Declaration(ident='s₂', lean_type='Set α'), Declaration(ident='t✝', lean_type='Set α'), Declaration(ident="t'", lean_type='Set α'), Declaration(ident='f', lean_type='α → β'), Declaration(ident='s', lean_type='Set α'), Declaration(ident='t', lean_type='Set β'), Declaration(ident='hs', lean_type='s.Finite'), Declaration(ident='h✝', lean_type='IsEmpty β'), Declaration(ident='h', lean_type='s.encard ≤ t.encard')], conclusion='Nonempty (↑s ↪ ↑t)'), Goal(assumptions=[Declaration(ident='α', lean_type='Type u_1'), Declaration(ident='β', lean_type='Type u_2'), Declaration(ident='s✝', lean_type='Set α'), Declaration(ident='s₁', lean_type='Set α'), Declaration(ident='s₂', lean_type='Set α'), Declaration(ident='t✝', lean_type='Set α'), Declaration(ident="t'", lean_type='Set α'), Declaration(ident='f', lean_type='α → β'), Declaration(ident='s', lean_type='Set α'), Declaration(ident='t', lean_type='Set β'), Declaration(ident='hs', lean_type='s.Finite'), Declaration(ident='h✝', lean_type='Nonempty β')], conclusion='s.encard ≤ t.encard ↔ Nonempty (↑s ↪ ↑t)')] | obtain ⟨f, hst, hf⟩ := hs.exists_injOn_of_encard_le h | <class 'lean_dojo.interaction.dojo.LeanError'> | LeanError(error='rcases tactic failed: x✝ : ?m.27594 is not an inductive datatype') | rcases tactic failed: x✝ : ?m.27594 is not an inductive datatype |
|
https://github.com/apnelson1/Matroid | 510262f56c0025be19502ab3457a788e2f6d2d1c | Matroid/Circuit.lean | Matroid.finitary_iff_forall_circuit_finite | Status.FAILED | α : Type u_1
M : Matroid α
C C' I X K C₁ C₂ R : Set α
e f x y : α
⊢ M.Finitary ↔ ∀ (C : Set α), M.Circuit C → C.Finite | None | [Goal(assumptions=[Declaration(ident='α', lean_type='Type u_1'), Declaration(ident='M', lean_type='Matroid α'), Declaration(ident='C', lean_type='Set α'), Declaration(ident="C'", lean_type='Set α'), Declaration(ident='I', lean_type='Set α'), Declaration(ident='X', lean_type='Set α'), Declaration(ident='K', lean_type='Set α'), Declaration(ident='C₁', lean_type='Set α'), Declaration(ident='C₂', lean_type='Set α'), Declaration(ident='R', lean_type='Set α'), Declaration(ident='e', lean_type='α'), Declaration(ident='f', lean_type='α'), Declaration(ident='x', lean_type='α'), Declaration(ident='y', lean_type='α')], conclusion='M.Finitary ↔ ∀ (C : Set α), M.Circuit C → C.Finite')] | refine ⟨fun _ _ ↦ Circuit.finite, fun h ↦
⟨fun I hI ↦ indep_iff_not_dep.2 ⟨fun hd ↦ ?_,fun x hx ↦ ?_⟩⟩ ⟩ | <class 'lean_dojo.interaction.dojo.TacticState'> | TacticState(pp="case refine_1\nα : Type u_1\nM : Matroid α\nC C' I✝ X K C₁ C₂ R : Set α\ne f x y : α\nh : ∀ (C : Set α), M.Circuit C → C.Finite\nI : Set α\nhI : ∀ J ⊆ I, J.Finite → M.Indep J\nhd : M.Dep I\n⊢ False\n\ncase refine_2\nα : Type u_1\nM : Matroid α\nC C' I✝ X K C₁ C₂ R : Set α\ne f x✝ y : α\nh : ∀ (C : Set α), M.Circuit C → C.Finite\nI : Set α\nhI : ∀ J ⊆ I, J.Finite → M.Indep J\nx : α\nhx : x ∈ I\n⊢ x ∈ M.E", id=1, message='') | case refine_1
α : Type u_1
M : Matroid α
C C' I✝ X K C₁ C₂ R : Set α
e f x y : α
h : ∀ (C : Set α), M.Circuit C → C.Finite
I : Set α
hI : ∀ J ⊆ I, J.Finite → M.Indep J
hd : M.Dep I
⊢ False
case refine_2
α : Type u_1
M : Matroid α
C C' I✝ X K C₁ C₂ R : Set α
e f x✝ y : α
h : ∀ (C : Set α), M.Circuit C → C.Finite
I : Set α
hI : ∀ J ⊆ I, J.Finite → M.Indep J
x : α
hx : x ∈ I
⊢ x ∈ M.E |
https://github.com/apnelson1/Matroid | 510262f56c0025be19502ab3457a788e2f6d2d1c | Matroid/Circuit.lean | Matroid.finitary_iff_forall_circuit_finite | Status.FAILED | case refine_1
α : Type u_1
M : Matroid α
C C' I✝ X K C₁ C₂ R : Set α
e f x y : α
h : ∀ (C : Set α), M.Circuit C → C.Finite
I : Set α
hI : ∀ J ⊆ I, J.Finite → M.Indep J
hd : M.Dep I
⊢ False
case refine_2
α : Type u_1
M : Matroid α
C C' I✝ X K C₁ C₂ R : Set α
e f x✝ y : α
h : ∀ (C : Set α), M.Circuit C → C.Finite
I : Set α
hI : ∀ J ⊆ I, J.Finite → M.Indep J
x : α
hx : x ∈ I
⊢ x ∈ M.E | [Goal(assumptions=[Declaration(ident='α', lean_type='Type u_1'), Declaration(ident='M', lean_type='Matroid α'), Declaration(ident='C', lean_type='Set α'), Declaration(ident="C'", lean_type='Set α'), Declaration(ident='I✝', lean_type='Set α'), Declaration(ident='X', lean_type='Set α'), Declaration(ident='K', lean_type='Set α'), Declaration(ident='C₁', lean_type='Set α'), Declaration(ident='C₂', lean_type='Set α'), Declaration(ident='R', lean_type='Set α'), Declaration(ident='e', lean_type='α'), Declaration(ident='f', lean_type='α'), Declaration(ident='x', lean_type='α'), Declaration(ident='y', lean_type='α'), Declaration(ident='h', lean_type='∀ (C : Set α), M.Circuit C → C.Finite'), Declaration(ident='I', lean_type='Set α'), Declaration(ident='hI', lean_type='∀ J ⊆ I, J.Finite → M.Indep J'), Declaration(ident='hd', lean_type='M.Dep I')], conclusion='False'), Goal(assumptions=[Declaration(ident='α', lean_type='Type u_1'), Declaration(ident='M', lean_type='Matroid α'), Declaration(ident='C', lean_type='Set α'), Declaration(ident="C'", lean_type='Set α'), Declaration(ident='I✝', lean_type='Set α'), Declaration(ident='X', lean_type='Set α'), Declaration(ident='K', lean_type='Set α'), Declaration(ident='C₁', lean_type='Set α'), Declaration(ident='C₂', lean_type='Set α'), Declaration(ident='R', lean_type='Set α'), Declaration(ident='e', lean_type='α'), Declaration(ident='f', lean_type='α'), Declaration(ident='x✝', lean_type='α'), Declaration(ident='y', lean_type='α'), Declaration(ident='h', lean_type='∀ (C : Set α), M.Circuit C → C.Finite'), Declaration(ident='I', lean_type='Set α'), Declaration(ident='hI', lean_type='∀ J ⊆ I, J.Finite → M.Indep J'), Declaration(ident='x', lean_type='α'), Declaration(ident='hx', lean_type='x ∈ I')], conclusion='x ∈ M.E')] | simpa using (hI {x} (by simpa) (finite_singleton _)).subset_ground | <class 'lean_dojo.interaction.dojo.LeanError'> | LeanError(error='type mismatch\n h✝\nhas type\n x ∈ M.E : Prop\nbut is expected to have type\n False : Prop') | type mismatch
h✝
has type
x ∈ M.E : Prop
but is expected to have type
False : Prop |
|
https://github.com/apnelson1/Matroid | 510262f56c0025be19502ab3457a788e2f6d2d1c | Matroid/Minor/Basic.lean | Matroid.contract_loop_iff_mem_closure | Status.PROVED | α : Type u_1
M M' N : Matroid α
e f : α
I J R B X Y Z K C C₁ C₂ : Set α
⊢ (M ／ C).Loop e ↔ e ∈ M.closure C \ C | None | [Goal(assumptions=[Declaration(ident='α', lean_type='Type u_1'), Declaration(ident='M', lean_type='Matroid α'), Declaration(ident="M'", lean_type='Matroid α'), Declaration(ident='N', lean_type='Matroid α'), Declaration(ident='e', lean_type='α'), Declaration(ident='f', lean_type='α'), Declaration(ident='I', lean_type='Set α'), Declaration(ident='J', lean_type='Set α'), Declaration(ident='R', lean_type='Set α'), Declaration(ident='B', lean_type='Set α'), Declaration(ident='X', lean_type='Set α'), Declaration(ident='Y', lean_type='Set α'), Declaration(ident='Z', lean_type='Set α'), Declaration(ident='K', lean_type='Set α'), Declaration(ident='C', lean_type='Set α'), Declaration(ident='C₁', lean_type='Set α'), Declaration(ident='C₂', lean_type='Set α')], conclusion='(M ／ C).Loop e ↔ e ∈ M.closure C \\ C')] | obtain ⟨I, D, hI, -, -, hM⟩ := M.exists_eq_contract_indep_delete C | <class 'lean_dojo.interaction.dojo.TacticState'> | TacticState(pp="case intro.intro.intro.intro.intro\nα : Type u_1\nM M' N : Matroid α\ne f : α\nI✝ J R B X Y Z K C C₁ C₂ I D : Set α\nhI : M.Basis I (C ∩ M.E)\nhM : M ／ C = M ／ I ＼ D\n⊢ (M ／ C).Loop e ↔ e ∈ M.closure C \\ C", id=1, message='') | case intro.intro.intro.intro.intro
α : Type u_1
M M' N : Matroid α
e f : α
I✝ J R B X Y Z K C C₁ C₂ I D : Set α
hI : M.Basis I (C ∩ M.E)
hM : M ／ C = M ／ I ＼ D
⊢ (M ／ C).Loop e ↔ e ∈ M.closure C \ C |
https://github.com/apnelson1/Matroid | 510262f56c0025be19502ab3457a788e2f6d2d1c | Matroid/Minor/Basic.lean | Matroid.contract_loop_iff_mem_closure | Status.PROVED | case intro.intro.intro.intro.intro
α : Type u_1
M M' N : Matroid α
e f : α
I✝ J R B X Y Z K C C₁ C₂ I D : Set α
hI : M.Basis I (C ∩ M.E)
hM : M ／ C = M ／ I ＼ D
⊢ (M ／ C).Loop e ↔ e ∈ M.closure C \ C | [Goal(assumptions=[Declaration(ident='α', lean_type='Type u_1'), Declaration(ident='M', lean_type='Matroid α'), Declaration(ident="M'", lean_type='Matroid α'), Declaration(ident='N', lean_type='Matroid α'), Declaration(ident='e', lean_type='α'), Declaration(ident='f', lean_type='α'), Declaration(ident='I✝', lean_type='Set α'), Declaration(ident='J', lean_type='Set α'), Declaration(ident='R', lean_type='Set α'), Declaration(ident='B', lean_type='Set α'), Declaration(ident='X', lean_type='Set α'), Declaration(ident='Y', lean_type='Set α'), Declaration(ident='Z', lean_type='Set α'), Declaration(ident='K', lean_type='Set α'), Declaration(ident='C', lean_type='Set α'), Declaration(ident='C₁', lean_type='Set α'), Declaration(ident='C₂', lean_type='Set α'), Declaration(ident='I', lean_type='Set α'), Declaration(ident='D', lean_type='Set α'), Declaration(ident='hI', lean_type='M.Basis I (C ∩ M.E)'), Declaration(ident='hM', lean_type='M ／ C = M ／ I ＼ D')], conclusion='(M ／ C).Loop e ↔ e ∈ M.closure C \\ C')] | rw [hM, delete_loop_iff, ← singleton_dep, hI.indep.contract_dep_iff, disjoint_singleton_left,
singleton_union, hI.indep.insert_dep_iff, mem_diff, ← M.closure_inter_ground C, hI.closure_eq_closure,
and_comm (a := e ∉ I), and_self_right, ← mem_diff, ← mem_diff, diff_diff] | <class 'lean_dojo.interaction.dojo.TacticState'> | TacticState(pp="case intro.intro.intro.intro.intro\nα : Type u_1\nM M' N : Matroid α\ne f : α\nI✝ J R B X Y Z K C C₁ C₂ I D : Set α\nhI : M.Basis I (C ∩ M.E)\nhM : M ／ C = M ／ I ＼ D\n⊢ e ∈ M.closure (C ∩ M.E) \\ (I ∪ D) ↔ e ∈ M.closure (C ∩ M.E) \\ C", id=2, message='') | case intro.intro.intro.intro.intro
α : Type u_1
M M' N : Matroid α
e f : α
I✝ J R B X Y Z K C C₁ C₂ I D : Set α
hI : M.Basis I (C ∩ M.E)
hM : M ／ C = M ／ I ＼ D
⊢ e ∈ M.closure (C ∩ M.E) \ (I ∪ D) ↔ e ∈ M.closure (C ∩ M.E) \ C |
|
https://github.com/apnelson1/Matroid | 510262f56c0025be19502ab3457a788e2f6d2d1c | Matroid/Minor/Basic.lean | Matroid.contract_loop_iff_mem_closure | Status.PROVED | case intro.intro.intro.intro.intro
α : Type u_1
M M' N : Matroid α
e f : α
I✝ J R B X Y Z K C C₁ C₂ I D : Set α
hI : M.Basis I (C ∩ M.E)
hM : M ／ C = M ／ I ＼ D
⊢ e ∈ M.closure (C ∩ M.E) \ (I ∪ D) ↔ e ∈ M.closure (C ∩ M.E) \ C | [Goal(assumptions=[Declaration(ident='α', lean_type='Type u_1'), Declaration(ident='M', lean_type='Matroid α'), Declaration(ident="M'", lean_type='Matroid α'), Declaration(ident='N', lean_type='Matroid α'), Declaration(ident='e', lean_type='α'), Declaration(ident='f', lean_type='α'), Declaration(ident='I✝', lean_type='Set α'), Declaration(ident='J', lean_type='Set α'), Declaration(ident='R', lean_type='Set α'), Declaration(ident='B', lean_type='Set α'), Declaration(ident='X', lean_type='Set α'), Declaration(ident='Y', lean_type='Set α'), Declaration(ident='Z', lean_type='Set α'), Declaration(ident='K', lean_type='Set α'), Declaration(ident='C', lean_type='Set α'), Declaration(ident='C₁', lean_type='Set α'), Declaration(ident='C₂', lean_type='Set α'), Declaration(ident='I', lean_type='Set α'), Declaration(ident='D', lean_type='Set α'), Declaration(ident='hI', lean_type='M.Basis I (C ∩ M.E)'), Declaration(ident='hM', lean_type='M ／ C = M ／ I ＼ D')], conclusion='e ∈ M.closure (C ∩ M.E) \\ (I ∪ D) ↔ e ∈ M.closure (C ∩ M.E) \\ C')] | apply_fun Matroid.E at hM | <class 'lean_dojo.interaction.dojo.TacticState'> | TacticState(pp="case intro.intro.intro.intro.intro\nα : Type u_1\nM M' N : Matroid α\ne f : α\nI✝ J R B X Y Z K C C₁ C₂ I D : Set α\nhI : M.Basis I (C ∩ M.E)\nhM : (M ／ C).E = (M ／ I ＼ D).E\n⊢ e ∈ M.closure (C ∩ M.E) \\ (I ∪ D) ↔ e ∈ M.closure (C ∩ M.E) \\ C", id=3, message='') | case intro.intro.intro.intro.intro
α : Type u_1
M M' N : Matroid α
e f : α
I✝ J R B X Y Z K C C₁ C₂ I D : Set α
hI : M.Basis I (C ∩ M.E)
hM : (M ／ C).E = (M ／ I ＼ D).E
⊢ e ∈ M.closure (C ∩ M.E) \ (I ∪ D) ↔ e ∈ M.closure (C ∩ M.E) \ C |
|
https://github.com/apnelson1/Matroid | 510262f56c0025be19502ab3457a788e2f6d2d1c | Matroid/Minor/Basic.lean | Matroid.contract_loop_iff_mem_closure | Status.PROVED | case intro.intro.intro.intro.intro
α : Type u_1
M M' N : Matroid α
e f : α
I✝ J R B X Y Z K C C₁ C₂ I D : Set α
hI : M.Basis I (C ∩ M.E)
hM : (M ／ C).E = (M ／ I ＼ D).E
⊢ e ∈ M.closure (C ∩ M.E) \ (I ∪ D) ↔ e ∈ M.closure (C ∩ M.E) \ C | [Goal(assumptions=[Declaration(ident='α', lean_type='Type u_1'), Declaration(ident='M', lean_type='Matroid α'), Declaration(ident="M'", lean_type='Matroid α'), Declaration(ident='N', lean_type='Matroid α'), Declaration(ident='e', lean_type='α'), Declaration(ident='f', lean_type='α'), Declaration(ident='I✝', lean_type='Set α'), Declaration(ident='J', lean_type='Set α'), Declaration(ident='R', lean_type='Set α'), Declaration(ident='B', lean_type='Set α'), Declaration(ident='X', lean_type='Set α'), Declaration(ident='Y', lean_type='Set α'), Declaration(ident='Z', lean_type='Set α'), Declaration(ident='K', lean_type='Set α'), Declaration(ident='C', lean_type='Set α'), Declaration(ident='C₁', lean_type='Set α'), Declaration(ident='C₂', lean_type='Set α'), Declaration(ident='I', lean_type='Set α'), Declaration(ident='D', lean_type='Set α'), Declaration(ident='hI', lean_type='M.Basis I (C ∩ M.E)'), Declaration(ident='hM', lean_type='(M ／ C).E = (M ／ I ＼ D).E')], conclusion='e ∈ M.closure (C ∩ M.E) \\ (I ∪ D) ↔ e ∈ M.closure (C ∩ M.E) \\ C')] | rw [delete_ground, contract_ground, contract_ground, diff_diff, diff_eq_diff_iff_inter_eq_inter,
inter_comm, inter_comm M.E] at hM | <class 'lean_dojo.interaction.dojo.TacticState'> | TacticState(pp="case intro.intro.intro.intro.intro\nα : Type u_1\nM M' N : Matroid α\ne f : α\nI✝ J R B X Y Z K C C₁ C₂ I D : Set α\nhI : M.Basis I (C ∩ M.E)\nhM : C ∩ M.E = (I ∪ D) ∩ M.E\n⊢ e ∈ M.closure (C ∩ M.E) \\ (I ∪ D) ↔ e ∈ M.closure (C ∩ M.E) \\ C", id=4, message='') | case intro.intro.intro.intro.intro
α : Type u_1
M M' N : Matroid α
e f : α
I✝ J R B X Y Z K C C₁ C₂ I D : Set α
hI : M.Basis I (C ∩ M.E)
hM : C ∩ M.E = (I ∪ D) ∩ M.E
⊢ e ∈ M.closure (C ∩ M.E) \ (I ∪ D) ↔ e ∈ M.closure (C ∩ M.E) \ C |
|
https://github.com/apnelson1/Matroid | 510262f56c0025be19502ab3457a788e2f6d2d1c | Matroid/Minor/Basic.lean | Matroid.contract_loop_iff_mem_closure | Status.PROVED | case intro.intro.intro.intro.intro
α : Type u_1
M M' N : Matroid α
e f : α
I✝ J R B X Y Z K C C₁ C₂ I D : Set α
hI : M.Basis I (C ∩ M.E)
hM : C ∩ M.E = (I ∪ D) ∩ M.E
⊢ e ∈ M.closure (C ∩ M.E) \ (I ∪ D) ↔ e ∈ M.closure (C ∩ M.E) \ C | [Goal(assumptions=[Declaration(ident='α', lean_type='Type u_1'), Declaration(ident='M', lean_type='Matroid α'), Declaration(ident="M'", lean_type='Matroid α'), Declaration(ident='N', lean_type='Matroid α'), Declaration(ident='e', lean_type='α'), Declaration(ident='f', lean_type='α'), Declaration(ident='I✝', lean_type='Set α'), Declaration(ident='J', lean_type='Set α'), Declaration(ident='R', lean_type='Set α'), Declaration(ident='B', lean_type='Set α'), Declaration(ident='X', lean_type='Set α'), Declaration(ident='Y', lean_type='Set α'), Declaration(ident='Z', lean_type='Set α'), Declaration(ident='K', lean_type='Set α'), Declaration(ident='C', lean_type='Set α'), Declaration(ident='C₁', lean_type='Set α'), Declaration(ident='C₂', lean_type='Set α'), Declaration(ident='I', lean_type='Set α'), Declaration(ident='D', lean_type='Set α'), Declaration(ident='hI', lean_type='M.Basis I (C ∩ M.E)'), Declaration(ident='hM', lean_type='C ∩ M.E = (I ∪ D) ∩ M.E')], conclusion='e ∈ M.closure (C ∩ M.E) \\ (I ∪ D) ↔ e ∈ M.closure (C ∩ M.E) \\ C')] | exact
⟨fun h ↦ ⟨h.1, fun heC ↦ h.2 (hM.subset ⟨heC, M.closure_subset_ground _ h.1⟩).1⟩, fun h ↦
⟨h.1, fun h' ↦ h.2 (hM.symm.subset ⟨h', M.closure_subset_ground _ h.1⟩).1⟩⟩ | <class 'lean_dojo.interaction.dojo.ProofFinished'> | ProofFinished(tactic_state_id=5, message='') | ||
https://github.com/apnelson1/Matroid | 510262f56c0025be19502ab3457a788e2f6d2d1c | Matroid/Extension.lean | Matroid.ModularCut.inter_mem | Status.PROVED | α : Type u_1
M : Matroid α
I J B F₀ F F' X Y : Set α
e f : α
U✝ U : M.ModularCut
hF : F ∈ U
hF' : F' ∈ U
h : M.ModularPair F F'
⊢ F ∩ F' ∈ U | None | [Goal(assumptions=[Declaration(ident='α', lean_type='Type u_1'), Declaration(ident='M', lean_type='Matroid α'), Declaration(ident='I', lean_type='Set α'), Declaration(ident='J', lean_type='Set α'), Declaration(ident='B', lean_type='Set α'), Declaration(ident='F₀', lean_type='Set α'), Declaration(ident='F', lean_type='Set α'), Declaration(ident="F'", lean_type='Set α'), Declaration(ident='X', lean_type='Set α'), Declaration(ident='Y', lean_type='Set α'), Declaration(ident='e', lean_type='α'), Declaration(ident='f', lean_type='α'), Declaration(ident='U✝', lean_type='M.ModularCut'), Declaration(ident='U', lean_type='M.ModularCut'), Declaration(ident='hF', lean_type='F ∈ U'), Declaration(ident="hF'", lean_type="F' ∈ U"), Declaration(ident='h', lean_type="M.ModularPair F F'")], conclusion="F ∩ F' ∈ U")] | rw [inter_eq_iInter] | <class 'lean_dojo.interaction.dojo.TacticState'> | TacticState(pp="α : Type u_1\nM : Matroid α\nI J B F₀ F F' X Y : Set α\ne f : α\nU✝ U : M.ModularCut\nhF : F ∈ U\nhF' : F' ∈ U\nh : M.ModularPair F F'\n⊢ (⋂ b, bif b then F else F') ∈ U", id=1, message='') | α : Type u_1
M : Matroid α
I J B F₀ F F' X Y : Set α
e f : α
U✝ U : M.ModularCut
hF : F ∈ U
hF' : F' ∈ U
h : M.ModularPair F F'
⊢ (⋂ b, bif b then F else F') ∈ U |
https://github.com/apnelson1/Matroid | 510262f56c0025be19502ab3457a788e2f6d2d1c | Matroid/Extension.lean | Matroid.ModularCut.inter_mem | Status.PROVED | α : Type u_1
M : Matroid α
I J B F₀ F F' X Y : Set α
e f : α
U✝ U : M.ModularCut
hF : F ∈ U
hF' : F' ∈ U
h : M.ModularPair F F'
⊢ (⋂ b, bif b then F else F') ∈ U | [Goal(assumptions=[Declaration(ident='α', lean_type='Type u_1'), Declaration(ident='M', lean_type='Matroid α'), Declaration(ident='I', lean_type='Set α'), Declaration(ident='J', lean_type='Set α'), Declaration(ident='B', lean_type='Set α'), Declaration(ident='F₀', lean_type='Set α'), Declaration(ident='F', lean_type='Set α'), Declaration(ident="F'", lean_type='Set α'), Declaration(ident='X', lean_type='Set α'), Declaration(ident='Y', lean_type='Set α'), Declaration(ident='e', lean_type='α'), Declaration(ident='f', lean_type='α'), Declaration(ident='U✝', lean_type='M.ModularCut'), Declaration(ident='U', lean_type='M.ModularCut'), Declaration(ident='hF', lean_type='F ∈ U'), Declaration(ident="hF'", lean_type="F' ∈ U"), Declaration(ident='h', lean_type="M.ModularPair F F'")], conclusion="(⋂ b, bif b then F else F') ∈ U")] | apply U.iInter_mem _ _ h | <class 'lean_dojo.interaction.dojo.TacticState'> | TacticState(pp="α : Type u_1\nM : Matroid α\nI J B F₀ F F' X Y : Set α\ne f : α\nU✝ U : M.ModularCut\nhF : F ∈ U\nhF' : F' ∈ U\nh : M.ModularPair F F'\n⊢ ∀ (i : Bool), (bif i then F else F') ∈ U", id=2, message='') | α : Type u_1
M : Matroid α
I J B F₀ F F' X Y : Set α
e f : α
U✝ U : M.ModularCut
hF : F ∈ U
hF' : F' ∈ U
h : M.ModularPair F F'
⊢ ∀ (i : Bool), (bif i then F else F') ∈ U |
|
https://github.com/apnelson1/Matroid | 510262f56c0025be19502ab3457a788e2f6d2d1c | Matroid/Extension.lean | Matroid.ModularCut.inter_mem | Status.PROVED | α : Type u_1
M : Matroid α
I J B F₀ F F' X Y : Set α
e f : α
U✝ U : M.ModularCut
hF : F ∈ U
hF' : F' ∈ U
h : M.ModularPair F F'
⊢ ∀ (i : Bool), (bif i then F else F') ∈ U | [Goal(assumptions=[Declaration(ident='α', lean_type='Type u_1'), Declaration(ident='M', lean_type='Matroid α'), Declaration(ident='I', lean_type='Set α'), Declaration(ident='J', lean_type='Set α'), Declaration(ident='B', lean_type='Set α'), Declaration(ident='F₀', lean_type='Set α'), Declaration(ident='F', lean_type='Set α'), Declaration(ident="F'", lean_type='Set α'), Declaration(ident='X', lean_type='Set α'), Declaration(ident='Y', lean_type='Set α'), Declaration(ident='e', lean_type='α'), Declaration(ident='f', lean_type='α'), Declaration(ident='U✝', lean_type='M.ModularCut'), Declaration(ident='U', lean_type='M.ModularCut'), Declaration(ident='hF', lean_type='F ∈ U'), Declaration(ident="hF'", lean_type="F' ∈ U"), Declaration(ident='h', lean_type="M.ModularPair F F'")], conclusion="∀ (i : Bool), (bif i then F else F') ∈ U")] | simp [hF, hF'] | <class 'lean_dojo.interaction.dojo.ProofFinished'> | ProofFinished(tactic_state_id=3, message='') | ||
https://github.com/apnelson1/Matroid | 510262f56c0025be19502ab3457a788e2f6d2d1c | Matroid/Circuit.lean | Matroid.loopyOn_dep_iff | Status.PROVED | α : Type u_1
M : Matroid α
C C' I X K C₁ C₂ R : Set α
e f x y : α
E D : Set α
⊢ (loopyOn E).Dep D ↔ D.Nonempty ∧ D ⊆ E | None | [Goal(assumptions=[Declaration(ident='α', lean_type='Type u_1'), Declaration(ident='M', lean_type='Matroid α'), Declaration(ident='C', lean_type='Set α'), Declaration(ident="C'", lean_type='Set α'), Declaration(ident='I', lean_type='Set α'), Declaration(ident='X', lean_type='Set α'), Declaration(ident='K', lean_type='Set α'), Declaration(ident='C₁', lean_type='Set α'), Declaration(ident='C₂', lean_type='Set α'), Declaration(ident='R', lean_type='Set α'), Declaration(ident='e', lean_type='α'), Declaration(ident='f', lean_type='α'), Declaration(ident='x', lean_type='α'), Declaration(ident='y', lean_type='α'), Declaration(ident='E', lean_type='Set α'), Declaration(ident='D', lean_type='Set α')], conclusion='(loopyOn E).Dep D ↔ D.Nonempty ∧ D ⊆ E')] | simp [Dep, nonempty_iff_ne_empty] | <class 'lean_dojo.interaction.dojo.ProofFinished'> | ProofFinished(tactic_state_id=1, message='') | |
https://github.com/apnelson1/Matroid | 510262f56c0025be19502ab3457a788e2f6d2d1c | Matroid/Rank.lean | Matroid.rFin.r_le_r_of_er_le_er | Status.PROVED | 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
hY : M.rFin Y
hle : M.er X ≤ M.er Y
⊢ M.r X ≤ M.r Y | None | [Goal(assumptions=[Declaration(ident='α', lean_type='Type u_1'), Declaration(ident='ι', lean_type='Type u_2'), Declaration(ident='M', lean_type='Matroid α'), Declaration(ident='N', lean_type='Matroid α'), Declaration(ident='I', lean_type='Set α'), Declaration(ident='B', lean_type='Set α'), Declaration(ident='X', lean_type='Set α'), Declaration(ident="X'", lean_type='Set α'), Declaration(ident='Y', lean_type='Set α'), Declaration(ident="Y'", lean_type='Set α'), Declaration(ident='Z', lean_type='Set α'), Declaration(ident='R', lean_type='Set α'), Declaration(ident='n', lean_type='ℕ∞'), Declaration(ident='e', lean_type='α'), Declaration(ident='f', lean_type='α'), Declaration(ident='hY', lean_type='M.rFin Y'), Declaration(ident='hle', lean_type='M.er X ≤ M.er Y')], conclusion='M.r X ≤ M.r Y')] | rwa [← rFin.er_le_er_iff _ hY] | <class 'lean_dojo.interaction.dojo.TacticState'> | TacticState(pp="α : Type u_1\nι : Type u_2\nM N : Matroid α\nI B X X' Y Y' Z R : Set α\nn : ℕ∞\ne f : α\nhY : M.rFin Y\nhle : M.er X ≤ M.er Y\n⊢ M.rFin X", id=1, message='') | 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
hY : M.rFin Y
hle : M.er X ≤ M.er Y
⊢ M.rFin X |
https://github.com/apnelson1/Matroid | 510262f56c0025be19502ab3457a788e2f6d2d1c | Matroid/Rank.lean | Matroid.rFin.r_le_r_of_er_le_er | Status.PROVED | 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
hY : M.rFin Y
hle : M.er X ≤ M.er Y
⊢ M.rFin X | [Goal(assumptions=[Declaration(ident='α', lean_type='Type u_1'), Declaration(ident='ι', lean_type='Type u_2'), Declaration(ident='M', lean_type='Matroid α'), Declaration(ident='N', lean_type='Matroid α'), Declaration(ident='I', lean_type='Set α'), Declaration(ident='B', lean_type='Set α'), Declaration(ident='X', lean_type='Set α'), Declaration(ident="X'", lean_type='Set α'), Declaration(ident='Y', lean_type='Set α'), Declaration(ident="Y'", lean_type='Set α'), Declaration(ident='Z', lean_type='Set α'), Declaration(ident='R', lean_type='Set α'), Declaration(ident='n', lean_type='ℕ∞'), Declaration(ident='e', lean_type='α'), Declaration(ident='f', lean_type='α'), Declaration(ident='hY', lean_type='M.rFin Y'), Declaration(ident='hle', lean_type='M.er X ≤ M.er Y')], conclusion='M.rFin X')] | exact hle.trans_lt hY.lt | <class 'lean_dojo.interaction.dojo.ProofFinished'> | ProofFinished(tactic_state_id=2, message='') |
End of preview. Expand
in Dataset Viewer.
YAML Metadata
Warning:
empty or missing yaml metadata in repo card
(https://huggingface.co/docs/hub/datasets-cards)
Repo: Matroid
Total Theorems: 1119. Proved: 860
Repo: lean-math-workshop
Total Theorems: 78. Proved: 47
Repo: mathematics_in_lean_source
Total Theorems: 139. Proved: 68
Repo: NazgandLean4
Total Theorems: 0. Proved: 0
Repo: std4
Total Theorems: 1018. Proved: 680
Repo: extra4
Total Theorems: 289. Proved: 256
Repo: lean-workbook
Total Theorems: 13469. Proved: 10286
Repo: flt-regular
Total Theorems: 351. Proved: 255
Repo: lean4lean
Total Theorems: 248. Proved: 166
Repo: Weights
Total Theorems: 113. Proved: 81
Repo: EulerProducts
Total Theorems: 65. Proved: 47
Repo: lean4-ascii
Total Theorems: 31. Proved: 28
Repo: compfiles
Total Theorems: 346. Proved: 246
Repo: mathlib4
Total Theorems: 60971. Proved: 45487
- Downloads last month
- 106