url stringclasses 147 values | commit stringclasses 147 values | file_path stringlengths 7 101 | full_name stringlengths 1 94 | start stringlengths 6 10 | end stringlengths 6 11 | tactic stringlengths 1 11.2k | state_before stringlengths 3 2.09M | state_after stringlengths 6 2.09M | input stringlengths 73 2.09M |
|---|---|---|---|---|---|---|---|---|---|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/Turanpartition.lean | Turanpartition.sdiff_part | [567, 1] | [571, 60] | have := sub_part hi | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i : ℕ
M : MultiPart α
hi : i ∈ range (M.t + 1)
⊢ M.A = M.A \ MultiPart.P M i ∪ MultiPart.P M i | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i : ℕ
M : MultiPart α
hi : i ∈ range (M.t + 1)
this : MultiPart.P M i ⊆ M.A
⊢ M.A = M.A \ MultiPart.P M i ∪ MultiPart.P M i | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i : ℕ
M : MultiPart α
hi : i ∈ range (M.t + 1)
⊢ M.A = M.A \ MultiPart.P M i ∪ MultiPart.P M i
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/Turanpartition.lean | Turanpartition.sdiff_part | [567, 1] | [571, 60] | rwa [sdiff_union_self_eq_union, left_eq_union_iff_subset] | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i : ℕ
M : MultiPart α
hi : i ∈ range (M.t + 1)
this : MultiPart.P M i ⊆ M.A
⊢ M.A = M.A \ MultiPart.P M i ∪ MultiPart.P M i | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i : ℕ
M : MultiPart α
hi : i ∈ range (M.t + 1)
this : MultiPart.P M i ⊆ M.A
⊢ M.A = M.A \ MultiPart.P M i ∪ MultiPart.P M i
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/Turanpartition.lean | Turanpartition.card_part_uni | [578, 1] | [582, 43] | nth_rw 1 [sdiff_part hi] | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i : ℕ
M : MultiPart α
hi : i ∈ range (M.t + 1)
⊢ card M.A = card (M.A \ MultiPart.P M i) + card (MultiPart.P M i) | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i : ℕ
M : MultiPart α
hi : i ∈ range (M.t + 1)
⊢ card (M.A \ MultiPart.P M i ∪ MultiPart.P M i) = card (M.A \ MultiPart.P M i) + card (MultiPart.P M i) | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i : ℕ
M : MultiPart α
hi : i ∈ range (M.t + 1)
⊢ card M.A = card (M.A \ MultiPart.P M i) + card (MultiPart.P M i)
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/Turanpartition.lean | Turanpartition.card_part_uni | [578, 1] | [582, 43] | apply card_disjoint_union sdiff_disjoint | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i : ℕ
M : MultiPart α
hi : i ∈ range (M.t + 1)
⊢ card (M.A \ MultiPart.P M i ∪ MultiPart.P M i) = card (M.A \ MultiPart.P M i) + card (MultiPart.P M i) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i : ℕ
M : MultiPart α
hi : i ∈ range (M.t + 1)
⊢ card (M.A \ MultiPart.P M i ∪ MultiPart.P M i) = card (M.A \ MultiPart.P M i) + card (MultiPart.P M i)
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/Turanpartition.lean | Turanpartition.move_p | [663, 1] | [668, 18] | intro _ | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i : ℕ
v : α
j k : ℕ
M : MultiPart α
hvi : i ∈ range (M.t + 1) ∧ v ∈ MultiPart.P M i
hj : j ∈ range (M.t + 1) ∧ j ≠ i
⊢ k ∈ range (M.t + 1) →
MultiPart.P (move M hvi hj) k =
if k ≠ i ∧ k ≠ j then MultiPart.P M k else if k = i then erase (MultiPart.P M i) v else insert v (MultiPart.P M j) | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i : ℕ
v : α
j k : ℕ
M : MultiPart α
hvi : i ∈ range (M.t + 1) ∧ v ∈ MultiPart.P M i
hj : j ∈ range (M.t + 1) ∧ j ≠ i
a✝ : k ∈ range (M.t + 1)
⊢ MultiPart.P (move M hvi hj) k =
if k ≠ i ∧ k ≠ j then MultiPart.P M k else if k = i then erase (MultiPart.P M i) v else insert v (MultiPart.P M j) | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i : ℕ
v : α
j k : ℕ
M : MultiPart α
hvi : i ∈ range (M.t + 1) ∧ v ∈ MultiPart.P M i
hj : j ∈ range (M.t + 1) ∧ j ≠ i
⊢ k ∈ range (M.t + 1) →
MultiPart.P (move M hvi hj) k =
if k ≠ i ∧ k ≠ j then MultiPart.P M k else if k = i then erase (MultiPart.P M i) v else insert v (MultiPart.P M j)
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/Turanpartition.lean | Turanpartition.move_p | [663, 1] | [668, 18] | rfl | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i : ℕ
v : α
j k : ℕ
M : MultiPart α
hvi : i ∈ range (M.t + 1) ∧ v ∈ MultiPart.P M i
hj : j ∈ range (M.t + 1) ∧ j ≠ i
a✝ : k ∈ range (M.t + 1)
⊢ MultiPart.P (move M hvi hj) k =
if k ≠ i ∧ k ≠ j then MultiPart.P M k else if k = i then erase (MultiPart.P M i) v else insert v (MultiPart.P M j) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i : ℕ
v : α
j k : ℕ
M : MultiPart α
hvi : i ∈ range (M.t + 1) ∧ v ∈ MultiPart.P M i
hj : j ∈ range (M.t + 1) ∧ j ≠ i
a✝ : k ∈ range (M.t + 1)
⊢ MultiPart.P (move M hvi hj) k =
if k ≠ i ∧ k ≠ j then MultiPart.P M k else if k = i then erase (MultiPart.P M i) v else insert v (MultiPart.P M j)
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/Turanpartition.lean | Turanpartition.move_Pcard | [671, 1] | [681, 73] | intro hk | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i : ℕ
v : α
j k : ℕ
M : MultiPart α
hvi : i ∈ range (M.t + 1) ∧ v ∈ MultiPart.P M i
hj : j ∈ range (M.t + 1) ∧ j ≠ i
⊢ k ∈ range (M.t + 1) →
card (MultiPart.P (move M hvi hj) k) =
if k ≠ i ∧ k ≠ j then card (MultiPart.P M k)
else if k = i then card (MultiPart.P M i) - 1 else card (MultiPart.P M j) + 1 | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i : ℕ
v : α
j k : ℕ
M : MultiPart α
hvi : i ∈ range (M.t + 1) ∧ v ∈ MultiPart.P M i
hj : j ∈ range (M.t + 1) ∧ j ≠ i
hk : k ∈ range (M.t + 1)
⊢ card (MultiPart.P (move M hvi hj) k) =
if k ≠ i ∧ k ≠ j then card (MultiPart.P M k)
else if k = i then card (MultiPart.P M i) - 1 else card (MultiPart.P M j) + 1 | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i : ℕ
v : α
j k : ℕ
M : MultiPart α
hvi : i ∈ range (M.t + 1) ∧ v ∈ MultiPart.P M i
hj : j ∈ range (M.t + 1) ∧ j ≠ i
⊢ k ∈ range (M.t + 1) →
card (MultiPart.P (move M hvi hj) k) =
if k ≠ i ∧ k ≠ j then card (MultiPart.P M k)
else if k = i then card (MultiPart.P M i) - 1 else card (MultiPart.P M j) + 1
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/Turanpartition.lean | Turanpartition.move_Pcard | [671, 1] | [681, 73] | rw [move_p hvi hj hk] | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i : ℕ
v : α
j k : ℕ
M : MultiPart α
hvi : i ∈ range (M.t + 1) ∧ v ∈ MultiPart.P M i
hj : j ∈ range (M.t + 1) ∧ j ≠ i
hk : k ∈ range (M.t + 1)
⊢ card (MultiPart.P (move M hvi hj) k) =
if k ≠ i ∧ k ≠ j then card (MultiPart.P M k)
else if k = i then card (MultiPart.P M i) - 1 else card (MultiPart.P M j) + 1 | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i : ℕ
v : α
j k : ℕ
M : MultiPart α
hvi : i ∈ range (M.t + 1) ∧ v ∈ MultiPart.P M i
hj : j ∈ range (M.t + 1) ∧ j ≠ i
hk : k ∈ range (M.t + 1)
⊢ card
(if k ≠ i ∧ k ≠ j then MultiPart.P M k
else if k = i then erase (MultiPart.P M i) v else insert v (MultiPart.P M j)) =
if k ≠ i ∧ k ≠ j then card (MultiPart.P M k)
else if k = i then card (MultiPart.P M i) - 1 else card (MultiPart.P M j) + 1 | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i : ℕ
v : α
j k : ℕ
M : MultiPart α
hvi : i ∈ range (M.t + 1) ∧ v ∈ MultiPart.P M i
hj : j ∈ range (M.t + 1) ∧ j ≠ i
hk : k ∈ range (M.t + 1)
⊢ card (MultiPart.P (move M hvi hj) k) =
if k ≠ i ∧ k ≠ j then card (MultiPart.P M k)
else if k = i then card (MultiPart.P M i) - 1 else card (MultiPart.P M j) + 1
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/Turanpartition.lean | Turanpartition.move_Pcard | [671, 1] | [681, 73] | split_ifs with h h_1 | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i : ℕ
v : α
j k : ℕ
M : MultiPart α
hvi : i ∈ range (M.t + 1) ∧ v ∈ MultiPart.P M i
hj : j ∈ range (M.t + 1) ∧ j ≠ i
hk : k ∈ range (M.t + 1)
⊢ card
(if k ≠ i ∧ k ≠ j then MultiPart.P M k
else if k = i then erase (MultiPart.P M i) v else insert v (MultiPart.P M j)) =
if k ≠ i ∧ k ≠ j then card (MultiPart.P M k)
else if k = i then card (MultiPart.P M i) - 1 else card (MultiPart.P M j) + 1 | case pos
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i : ℕ
v : α
j k : ℕ
M : MultiPart α
hvi : i ∈ range (M.t + 1) ∧ v ∈ MultiPart.P M i
hj : j ∈ range (M.t + 1) ∧ j ≠ i
hk : k ∈ range (M.t + 1)
h : k ≠ i ∧ k ≠ j
⊢ card (MultiPart.P M k) = card (MultiPart.P M k)
case pos
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i : ℕ
v : α
j k : ℕ
M : MultiPart α
hvi : i ∈ range (M.t + 1) ∧ v ∈ MultiPart.P M i
hj : j ∈ range (M.t + 1) ∧ j ≠ i
hk : k ∈ range (M.t + 1)
h : ¬(k ≠ i ∧ k ≠ j)
h_1 : k = i
⊢ card (erase (MultiPart.P M i) v) = card (MultiPart.P M i) - 1
case neg
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i : ℕ
v : α
j k : ℕ
M : MultiPart α
hvi : i ∈ range (M.t + 1) ∧ v ∈ MultiPart.P M i
hj : j ∈ range (M.t + 1) ∧ j ≠ i
hk : k ∈ range (M.t + 1)
h : ¬(k ≠ i ∧ k ≠ j)
h_1 : ¬k = i
⊢ card (insert v (MultiPart.P M j)) = card (MultiPart.P M j) + 1 | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i : ℕ
v : α
j k : ℕ
M : MultiPart α
hvi : i ∈ range (M.t + 1) ∧ v ∈ MultiPart.P M i
hj : j ∈ range (M.t + 1) ∧ j ≠ i
hk : k ∈ range (M.t + 1)
⊢ card
(if k ≠ i ∧ k ≠ j then MultiPart.P M k
else if k = i then erase (MultiPart.P M i) v else insert v (MultiPart.P M j)) =
if k ≠ i ∧ k ≠ j then card (MultiPart.P M k)
else if k = i then card (MultiPart.P M i) - 1 else card (MultiPart.P M j) + 1
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/Turanpartition.lean | Turanpartition.move_Pcard | [671, 1] | [681, 73] | rfl | case pos
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i : ℕ
v : α
j k : ℕ
M : MultiPart α
hvi : i ∈ range (M.t + 1) ∧ v ∈ MultiPart.P M i
hj : j ∈ range (M.t + 1) ∧ j ≠ i
hk : k ∈ range (M.t + 1)
h : k ≠ i ∧ k ≠ j
⊢ card (MultiPart.P M k) = card (MultiPart.P M k) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i : ℕ
v : α
j k : ℕ
M : MultiPart α
hvi : i ∈ range (M.t + 1) ∧ v ∈ MultiPart.P M i
hj : j ∈ range (M.t + 1) ∧ j ≠ i
hk : k ∈ range (M.t + 1)
h : k ≠ i ∧ k ≠ j
⊢ card (MultiPart.P M k) = card (MultiPart.P M k)
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/Turanpartition.lean | Turanpartition.move_Pcard | [671, 1] | [681, 73] | exact card_erase_of_mem hvi.2 | case pos
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i : ℕ
v : α
j k : ℕ
M : MultiPart α
hvi : i ∈ range (M.t + 1) ∧ v ∈ MultiPart.P M i
hj : j ∈ range (M.t + 1) ∧ j ≠ i
hk : k ∈ range (M.t + 1)
h : ¬(k ≠ i ∧ k ≠ j)
h_1 : k = i
⊢ card (erase (MultiPart.P M i) v) = card (MultiPart.P M i) - 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i : ℕ
v : α
j k : ℕ
M : MultiPart α
hvi : i ∈ range (M.t + 1) ∧ v ∈ MultiPart.P M i
hj : j ∈ range (M.t + 1) ∧ j ≠ i
hk : k ∈ range (M.t + 1)
h : ¬(k ≠ i ∧ k ≠ j)
h_1 : k = i
⊢ card (erase (MultiPart.P M i) v) = card (MultiPart.P M i) - 1
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/Turanpartition.lean | Turanpartition.move_Pcard | [671, 1] | [681, 73] | apply card_insert_of_not_mem (uniq_part' hvi.1 hj.1 hj.2.symm hvi.2) | case neg
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i : ℕ
v : α
j k : ℕ
M : MultiPart α
hvi : i ∈ range (M.t + 1) ∧ v ∈ MultiPart.P M i
hj : j ∈ range (M.t + 1) ∧ j ≠ i
hk : k ∈ range (M.t + 1)
h : ¬(k ≠ i ∧ k ≠ j)
h_1 : ¬k = i
⊢ card (insert v (MultiPart.P M j)) = card (MultiPart.P M j) + 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i : ℕ
v : α
j k : ℕ
M : MultiPart α
hvi : i ∈ range (M.t + 1) ∧ v ∈ MultiPart.P M i
hj : j ∈ range (M.t + 1) ∧ j ≠ i
hk : k ∈ range (M.t + 1)
h : ¬(k ≠ i ∧ k ≠ j)
h_1 : ¬k = i
⊢ card (insert v (MultiPart.P M j)) = card (MultiPart.P M j) + 1
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/Turanpartition.lean | Turanpartition.sdiff_erase | [684, 1] | [699, 27] | ext a | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Finset α
v : α
A : Finset α
hB : B ⊆ A
hv : v ∈ B
⊢ A \ erase B v = A \ B ∪ {v} | case a
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Finset α
v : α
A : Finset α
hB : B ⊆ A
hv : v ∈ B
a : α
⊢ a ∈ A \ erase B v ↔ a ∈ A \ B ∪ {v} | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Finset α
v : α
A : Finset α
hB : B ⊆ A
hv : v ∈ B
⊢ A \ erase B v = A \ B ∪ {v}
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/Turanpartition.lean | Turanpartition.sdiff_erase | [684, 1] | [699, 27] | constructor | case a
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Finset α
v : α
A : Finset α
hB : B ⊆ A
hv : v ∈ B
a : α
⊢ a ∈ A \ erase B v ↔ a ∈ A \ B ∪ {v} | case a.mp
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Finset α
v : α
A : Finset α
hB : B ⊆ A
hv : v ∈ B
a : α
⊢ a ∈ A \ erase B v → a ∈ A \ B ∪ {v}
case a.mpr
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Finset α
v : α
A : Finset α
hB : B ⊆ A
hv : v ∈ B
a : α
⊢ a ∈ A \ B ∪ {v} → a ∈ A \ erase B v | Please generate a tactic in lean4 to solve the state.
STATE:
case a
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Finset α
v : α
A : Finset α
hB : B ⊆ A
hv : v ∈ B
a : α
⊢ a ∈ A \ erase B v ↔ a ∈ A \ B ∪ {v}
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/Turanpartition.lean | Turanpartition.sdiff_erase | [684, 1] | [699, 27] | intro h | case a.mp
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Finset α
v : α
A : Finset α
hB : B ⊆ A
hv : v ∈ B
a : α
⊢ a ∈ A \ erase B v → a ∈ A \ B ∪ {v} | case a.mp
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Finset α
v : α
A : Finset α
hB : B ⊆ A
hv : v ∈ B
a : α
h : a ∈ A \ erase B v
⊢ a ∈ A \ B ∪ {v} | Please generate a tactic in lean4 to solve the state.
STATE:
case a.mp
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Finset α
v : α
A : Finset α
hB : B ⊆ A
hv : v ∈ B
a : α
⊢ a ∈ A \ erase B v → a ∈ A \ B ∪ {v}
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/Turanpartition.lean | Turanpartition.sdiff_erase | [684, 1] | [699, 27] | rw [mem_union, mem_sdiff] at * | case a.mp
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Finset α
v : α
A : Finset α
hB : B ⊆ A
hv : v ∈ B
a : α
h : a ∈ A \ erase B v
⊢ a ∈ A \ B ∪ {v} | case a.mp
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Finset α
v : α
A : Finset α
hB : B ⊆ A
hv : v ∈ B
a : α
h : a ∈ A ∧ ¬a ∈ erase B v
⊢ a ∈ A ∧ ¬a ∈ B ∨ a ∈ {v} | Please generate a tactic in lean4 to solve the state.
STATE:
case a.mp
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Finset α
v : α
A : Finset α
hB : B ⊆ A
hv : v ∈ B
a : α
h : a ∈ A \ erase B v
⊢ a ∈ A \ B ∪ {v}
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/Turanpartition.lean | Turanpartition.sdiff_erase | [684, 1] | [699, 27] | rw [mem_erase] at h | case a.mp
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Finset α
v : α
A : Finset α
hB : B ⊆ A
hv : v ∈ B
a : α
h : a ∈ A ∧ ¬a ∈ erase B v
⊢ a ∈ A ∧ ¬a ∈ B ∨ a ∈ {v} | case a.mp
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Finset α
v : α
A : Finset α
hB : B ⊆ A
hv : v ∈ B
a : α
h : a ∈ A ∧ ¬(a ≠ v ∧ a ∈ B)
⊢ a ∈ A ∧ ¬a ∈ B ∨ a ∈ {v} | Please generate a tactic in lean4 to solve the state.
STATE:
case a.mp
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Finset α
v : α
A : Finset α
hB : B ⊆ A
hv : v ∈ B
a : α
h : a ∈ A ∧ ¬a ∈ erase B v
⊢ a ∈ A ∧ ¬a ∈ B ∨ a ∈ {v}
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/Turanpartition.lean | Turanpartition.sdiff_erase | [684, 1] | [699, 27] | push_neg at h | case a.mp
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Finset α
v : α
A : Finset α
hB : B ⊆ A
hv : v ∈ B
a : α
h : a ∈ A ∧ ¬(a ≠ v ∧ a ∈ B)
⊢ a ∈ A ∧ ¬a ∈ B ∨ a ∈ {v} | case a.mp
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Finset α
v : α
A : Finset α
hB : B ⊆ A
hv : v ∈ B
a : α
h : a ∈ A ∧ (a ≠ v → ¬a ∈ B)
⊢ a ∈ A ∧ ¬a ∈ B ∨ a ∈ {v} | Please generate a tactic in lean4 to solve the state.
STATE:
case a.mp
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Finset α
v : α
A : Finset α
hB : B ⊆ A
hv : v ∈ B
a : α
h : a ∈ A ∧ ¬(a ≠ v ∧ a ∈ B)
⊢ a ∈ A ∧ ¬a ∈ B ∨ a ∈ {v}
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/Turanpartition.lean | Turanpartition.sdiff_erase | [684, 1] | [699, 27] | by_cases h' : a = v | case a.mp
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Finset α
v : α
A : Finset α
hB : B ⊆ A
hv : v ∈ B
a : α
h : a ∈ A ∧ (a ≠ v → ¬a ∈ B)
⊢ a ∈ A ∧ ¬a ∈ B ∨ a ∈ {v} | case pos
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Finset α
v : α
A : Finset α
hB : B ⊆ A
hv : v ∈ B
a : α
h : a ∈ A ∧ (a ≠ v → ¬a ∈ B)
h' : a = v
⊢ a ∈ A ∧ ¬a ∈ B ∨ a ∈ {v}
case neg
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Finset α
v : α
A : Finset α
hB : B ⊆ A
hv : v ∈ B
a : α
h : a ∈ A ∧ (a ≠ v → ¬a ∈ B)
h' : ¬a = v
⊢ a ∈ A ∧ ¬a ∈ B ∨ a ∈ {v} | Please generate a tactic in lean4 to solve the state.
STATE:
case a.mp
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Finset α
v : α
A : Finset α
hB : B ⊆ A
hv : v ∈ B
a : α
h : a ∈ A ∧ (a ≠ v → ¬a ∈ B)
⊢ a ∈ A ∧ ¬a ∈ B ∨ a ∈ {v}
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/Turanpartition.lean | Turanpartition.sdiff_erase | [684, 1] | [699, 27] | right | case pos
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Finset α
v : α
A : Finset α
hB : B ⊆ A
hv : v ∈ B
a : α
h : a ∈ A ∧ (a ≠ v → ¬a ∈ B)
h' : a = v
⊢ a ∈ A ∧ ¬a ∈ B ∨ a ∈ {v} | case pos.h
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Finset α
v : α
A : Finset α
hB : B ⊆ A
hv : v ∈ B
a : α
h : a ∈ A ∧ (a ≠ v → ¬a ∈ B)
h' : a = v
⊢ a ∈ {v} | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Finset α
v : α
A : Finset α
hB : B ⊆ A
hv : v ∈ B
a : α
h : a ∈ A ∧ (a ≠ v → ¬a ∈ B)
h' : a = v
⊢ a ∈ A ∧ ¬a ∈ B ∨ a ∈ {v}
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/Turanpartition.lean | Turanpartition.sdiff_erase | [684, 1] | [699, 27] | exact mem_singleton.mpr h' | case pos.h
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Finset α
v : α
A : Finset α
hB : B ⊆ A
hv : v ∈ B
a : α
h : a ∈ A ∧ (a ≠ v → ¬a ∈ B)
h' : a = v
⊢ a ∈ {v} | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case pos.h
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Finset α
v : α
A : Finset α
hB : B ⊆ A
hv : v ∈ B
a : α
h : a ∈ A ∧ (a ≠ v → ¬a ∈ B)
h' : a = v
⊢ a ∈ {v}
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/Turanpartition.lean | Turanpartition.sdiff_erase | [684, 1] | [699, 27] | left | case neg
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Finset α
v : α
A : Finset α
hB : B ⊆ A
hv : v ∈ B
a : α
h : a ∈ A ∧ (a ≠ v → ¬a ∈ B)
h' : ¬a = v
⊢ a ∈ A ∧ ¬a ∈ B ∨ a ∈ {v} | case neg.h
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Finset α
v : α
A : Finset α
hB : B ⊆ A
hv : v ∈ B
a : α
h : a ∈ A ∧ (a ≠ v → ¬a ∈ B)
h' : ¬a = v
⊢ a ∈ A ∧ ¬a ∈ B | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Finset α
v : α
A : Finset α
hB : B ⊆ A
hv : v ∈ B
a : α
h : a ∈ A ∧ (a ≠ v → ¬a ∈ B)
h' : ¬a = v
⊢ a ∈ A ∧ ¬a ∈ B ∨ a ∈ {v}
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/Turanpartition.lean | Turanpartition.sdiff_erase | [684, 1] | [699, 27] | exact ⟨h.1, h.2 h'⟩ | case neg.h
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Finset α
v : α
A : Finset α
hB : B ⊆ A
hv : v ∈ B
a : α
h : a ∈ A ∧ (a ≠ v → ¬a ∈ B)
h' : ¬a = v
⊢ a ∈ A ∧ ¬a ∈ B | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.h
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Finset α
v : α
A : Finset α
hB : B ⊆ A
hv : v ∈ B
a : α
h : a ∈ A ∧ (a ≠ v → ¬a ∈ B)
h' : ¬a = v
⊢ a ∈ A ∧ ¬a ∈ B
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/Turanpartition.lean | Turanpartition.sdiff_erase | [684, 1] | [699, 27] | intro h | case a.mpr
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Finset α
v : α
A : Finset α
hB : B ⊆ A
hv : v ∈ B
a : α
⊢ a ∈ A \ B ∪ {v} → a ∈ A \ erase B v | case a.mpr
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Finset α
v : α
A : Finset α
hB : B ⊆ A
hv : v ∈ B
a : α
h : a ∈ A \ B ∪ {v}
⊢ a ∈ A \ erase B v | Please generate a tactic in lean4 to solve the state.
STATE:
case a.mpr
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Finset α
v : α
A : Finset α
hB : B ⊆ A
hv : v ∈ B
a : α
⊢ a ∈ A \ B ∪ {v} → a ∈ A \ erase B v
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/Turanpartition.lean | Turanpartition.sdiff_erase | [684, 1] | [699, 27] | rw [mem_sdiff] | case a.mpr
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Finset α
v : α
A : Finset α
hB : B ⊆ A
hv : v ∈ B
a : α
h : a ∈ A \ B ∪ {v}
⊢ a ∈ A \ erase B v | case a.mpr
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Finset α
v : α
A : Finset α
hB : B ⊆ A
hv : v ∈ B
a : α
h : a ∈ A \ B ∪ {v}
⊢ a ∈ A ∧ ¬a ∈ erase B v | Please generate a tactic in lean4 to solve the state.
STATE:
case a.mpr
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Finset α
v : α
A : Finset α
hB : B ⊆ A
hv : v ∈ B
a : α
h : a ∈ A \ B ∪ {v}
⊢ a ∈ A \ erase B v
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/Turanpartition.lean | Turanpartition.sdiff_erase | [684, 1] | [699, 27] | rw [mem_erase] | case a.mpr
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Finset α
v : α
A : Finset α
hB : B ⊆ A
hv : v ∈ B
a : α
h : a ∈ A \ B ∪ {v}
⊢ a ∈ A ∧ ¬a ∈ erase B v | case a.mpr
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Finset α
v : α
A : Finset α
hB : B ⊆ A
hv : v ∈ B
a : α
h : a ∈ A \ B ∪ {v}
⊢ a ∈ A ∧ ¬(a ≠ v ∧ a ∈ B) | Please generate a tactic in lean4 to solve the state.
STATE:
case a.mpr
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Finset α
v : α
A : Finset α
hB : B ⊆ A
hv : v ∈ B
a : α
h : a ∈ A \ B ∪ {v}
⊢ a ∈ A ∧ ¬a ∈ erase B v
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/Turanpartition.lean | Turanpartition.sdiff_erase | [684, 1] | [699, 27] | rw [mem_union, mem_sdiff] at h | case a.mpr
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Finset α
v : α
A : Finset α
hB : B ⊆ A
hv : v ∈ B
a : α
h : a ∈ A \ B ∪ {v}
⊢ a ∈ A ∧ ¬(a ≠ v ∧ a ∈ B) | case a.mpr
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Finset α
v : α
A : Finset α
hB : B ⊆ A
hv : v ∈ B
a : α
h : a ∈ A ∧ ¬a ∈ B ∨ a ∈ {v}
⊢ a ∈ A ∧ ¬(a ≠ v ∧ a ∈ B) | Please generate a tactic in lean4 to solve the state.
STATE:
case a.mpr
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Finset α
v : α
A : Finset α
hB : B ⊆ A
hv : v ∈ B
a : α
h : a ∈ A \ B ∪ {v}
⊢ a ∈ A ∧ ¬(a ≠ v ∧ a ∈ B)
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/Turanpartition.lean | Turanpartition.sdiff_erase | [684, 1] | [699, 27] | push_neg | case a.mpr
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Finset α
v : α
A : Finset α
hB : B ⊆ A
hv : v ∈ B
a : α
h : a ∈ A ∧ ¬a ∈ B ∨ a ∈ {v}
⊢ a ∈ A ∧ ¬(a ≠ v ∧ a ∈ B) | case a.mpr
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Finset α
v : α
A : Finset α
hB : B ⊆ A
hv : v ∈ B
a : α
h : a ∈ A ∧ ¬a ∈ B ∨ a ∈ {v}
⊢ a ∈ A ∧ (a ≠ v → ¬a ∈ B) | Please generate a tactic in lean4 to solve the state.
STATE:
case a.mpr
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Finset α
v : α
A : Finset α
hB : B ⊆ A
hv : v ∈ B
a : α
h : a ∈ A ∧ ¬a ∈ B ∨ a ∈ {v}
⊢ a ∈ A ∧ ¬(a ≠ v ∧ a ∈ B)
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/Turanpartition.lean | Turanpartition.sdiff_erase | [684, 1] | [699, 27] | cases h with
| inl h => exact ⟨h.1, fun _ => h.2⟩;
| inr h =>
by_contra h'; push_neg at h'
have ha := hB hv
have := mem_singleton.mp h; rw [← this] at ha
exact (h' ha).1 this | case a.mpr
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Finset α
v : α
A : Finset α
hB : B ⊆ A
hv : v ∈ B
a : α
h : a ∈ A ∧ ¬a ∈ B ∨ a ∈ {v}
⊢ a ∈ A ∧ (a ≠ v → ¬a ∈ B) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case a.mpr
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Finset α
v : α
A : Finset α
hB : B ⊆ A
hv : v ∈ B
a : α
h : a ∈ A ∧ ¬a ∈ B ∨ a ∈ {v}
⊢ a ∈ A ∧ (a ≠ v → ¬a ∈ B)
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/Turanpartition.lean | Turanpartition.sdiff_erase | [684, 1] | [699, 27] | exact ⟨h.1, fun _ => h.2⟩ | case a.mpr.inl
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Finset α
v : α
A : Finset α
hB : B ⊆ A
hv : v ∈ B
a : α
h : a ∈ A ∧ ¬a ∈ B
⊢ a ∈ A ∧ (a ≠ v → ¬a ∈ B) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case a.mpr.inl
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Finset α
v : α
A : Finset α
hB : B ⊆ A
hv : v ∈ B
a : α
h : a ∈ A ∧ ¬a ∈ B
⊢ a ∈ A ∧ (a ≠ v → ¬a ∈ B)
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/Turanpartition.lean | Turanpartition.sdiff_erase | [684, 1] | [699, 27] | by_contra h' | case a.mpr.inr
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Finset α
v : α
A : Finset α
hB : B ⊆ A
hv : v ∈ B
a : α
h : a ∈ {v}
⊢ a ∈ A ∧ (a ≠ v → ¬a ∈ B) | case a.mpr.inr
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Finset α
v : α
A : Finset α
hB : B ⊆ A
hv : v ∈ B
a : α
h : a ∈ {v}
h' : ¬(a ∈ A ∧ (a ≠ v → ¬a ∈ B))
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case a.mpr.inr
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Finset α
v : α
A : Finset α
hB : B ⊆ A
hv : v ∈ B
a : α
h : a ∈ {v}
⊢ a ∈ A ∧ (a ≠ v → ¬a ∈ B)
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/Turanpartition.lean | Turanpartition.sdiff_erase | [684, 1] | [699, 27] | push_neg at h' | case a.mpr.inr
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Finset α
v : α
A : Finset α
hB : B ⊆ A
hv : v ∈ B
a : α
h : a ∈ {v}
h' : ¬(a ∈ A ∧ (a ≠ v → ¬a ∈ B))
⊢ False | case a.mpr.inr
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Finset α
v : α
A : Finset α
hB : B ⊆ A
hv : v ∈ B
a : α
h : a ∈ {v}
h' : a ∈ A → a ≠ v ∧ a ∈ B
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case a.mpr.inr
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Finset α
v : α
A : Finset α
hB : B ⊆ A
hv : v ∈ B
a : α
h : a ∈ {v}
h' : ¬(a ∈ A ∧ (a ≠ v → ¬a ∈ B))
⊢ False
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/Turanpartition.lean | Turanpartition.sdiff_erase | [684, 1] | [699, 27] | have ha := hB hv | case a.mpr.inr
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Finset α
v : α
A : Finset α
hB : B ⊆ A
hv : v ∈ B
a : α
h : a ∈ {v}
h' : a ∈ A → a ≠ v ∧ a ∈ B
⊢ False | case a.mpr.inr
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Finset α
v : α
A : Finset α
hB : B ⊆ A
hv : v ∈ B
a : α
h : a ∈ {v}
h' : a ∈ A → a ≠ v ∧ a ∈ B
ha : v ∈ A
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case a.mpr.inr
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Finset α
v : α
A : Finset α
hB : B ⊆ A
hv : v ∈ B
a : α
h : a ∈ {v}
h' : a ∈ A → a ≠ v ∧ a ∈ B
⊢ False
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/Turanpartition.lean | Turanpartition.sdiff_erase | [684, 1] | [699, 27] | have := mem_singleton.mp h | case a.mpr.inr
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Finset α
v : α
A : Finset α
hB : B ⊆ A
hv : v ∈ B
a : α
h : a ∈ {v}
h' : a ∈ A → a ≠ v ∧ a ∈ B
ha : v ∈ A
⊢ False | case a.mpr.inr
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Finset α
v : α
A : Finset α
hB : B ⊆ A
hv : v ∈ B
a : α
h : a ∈ {v}
h' : a ∈ A → a ≠ v ∧ a ∈ B
ha : v ∈ A
this : a = v
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case a.mpr.inr
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Finset α
v : α
A : Finset α
hB : B ⊆ A
hv : v ∈ B
a : α
h : a ∈ {v}
h' : a ∈ A → a ≠ v ∧ a ∈ B
ha : v ∈ A
⊢ False
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/Turanpartition.lean | Turanpartition.sdiff_erase | [684, 1] | [699, 27] | rw [← this] at ha | case a.mpr.inr
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Finset α
v : α
A : Finset α
hB : B ⊆ A
hv : v ∈ B
a : α
h : a ∈ {v}
h' : a ∈ A → a ≠ v ∧ a ∈ B
ha : v ∈ A
this : a = v
⊢ False | case a.mpr.inr
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Finset α
v : α
A : Finset α
hB : B ⊆ A
hv : v ∈ B
a : α
h : a ∈ {v}
h' : a ∈ A → a ≠ v ∧ a ∈ B
ha : a ∈ A
this : a = v
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case a.mpr.inr
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Finset α
v : α
A : Finset α
hB : B ⊆ A
hv : v ∈ B
a : α
h : a ∈ {v}
h' : a ∈ A → a ≠ v ∧ a ∈ B
ha : v ∈ A
this : a = v
⊢ False
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/Turanpartition.lean | Turanpartition.sdiff_erase | [684, 1] | [699, 27] | exact (h' ha).1 this | case a.mpr.inr
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Finset α
v : α
A : Finset α
hB : B ⊆ A
hv : v ∈ B
a : α
h : a ∈ {v}
h' : a ∈ A → a ≠ v ∧ a ∈ B
ha : a ∈ A
this : a = v
⊢ False | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case a.mpr.inr
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Finset α
v : α
A : Finset α
hB : B ⊆ A
hv : v ∈ B
a : α
h : a ∈ {v}
h' : a ∈ A → a ≠ v ∧ a ∈ B
ha : a ∈ A
this : a = v
⊢ False
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/Turanpartition.lean | Turanpartition.card_sdiff_erase | [702, 1] | [706, 87] | have hv2 : v ∉ A \ B := by rw [mem_sdiff]; push_neg; intro _; exact hv | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Finset α
v : α
A : Finset α
hB : B ⊆ A
hv : v ∈ B
⊢ card (A \ erase B v) = card (A \ B) + 1 | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Finset α
v : α
A : Finset α
hB : B ⊆ A
hv : v ∈ B
hv2 : ¬v ∈ A \ B
⊢ card (A \ erase B v) = card (A \ B) + 1 | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Finset α
v : α
A : Finset α
hB : B ⊆ A
hv : v ∈ B
⊢ card (A \ erase B v) = card (A \ B) + 1
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/Turanpartition.lean | Turanpartition.card_sdiff_erase | [702, 1] | [706, 87] | rw [sdiff_erase hB hv] | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Finset α
v : α
A : Finset α
hB : B ⊆ A
hv : v ∈ B
hv2 : ¬v ∈ A \ B
⊢ card (A \ erase B v) = card (A \ B) + 1 | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Finset α
v : α
A : Finset α
hB : B ⊆ A
hv : v ∈ B
hv2 : ¬v ∈ A \ B
⊢ card (A \ B ∪ {v}) = card (A \ B) + 1 | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Finset α
v : α
A : Finset α
hB : B ⊆ A
hv : v ∈ B
hv2 : ¬v ∈ A \ B
⊢ card (A \ erase B v) = card (A \ B) + 1
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/Turanpartition.lean | Turanpartition.card_sdiff_erase | [702, 1] | [706, 87] | exact card_disjoint_union (disjoint_singleton_right.mpr hv2) | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Finset α
v : α
A : Finset α
hB : B ⊆ A
hv : v ∈ B
hv2 : ¬v ∈ A \ B
⊢ card (A \ B ∪ {v}) = card (A \ B) + 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Finset α
v : α
A : Finset α
hB : B ⊆ A
hv : v ∈ B
hv2 : ¬v ∈ A \ B
⊢ card (A \ B ∪ {v}) = card (A \ B) + 1
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/Turanpartition.lean | Turanpartition.card_sdiff_erase | [702, 1] | [706, 87] | rw [mem_sdiff] | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Finset α
v : α
A : Finset α
hB : B ⊆ A
hv : v ∈ B
⊢ ¬v ∈ A \ B | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Finset α
v : α
A : Finset α
hB : B ⊆ A
hv : v ∈ B
⊢ ¬(v ∈ A ∧ ¬v ∈ B) | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Finset α
v : α
A : Finset α
hB : B ⊆ A
hv : v ∈ B
⊢ ¬v ∈ A \ B
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/Turanpartition.lean | Turanpartition.card_sdiff_erase | [702, 1] | [706, 87] | push_neg | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Finset α
v : α
A : Finset α
hB : B ⊆ A
hv : v ∈ B
⊢ ¬(v ∈ A ∧ ¬v ∈ B) | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Finset α
v : α
A : Finset α
hB : B ⊆ A
hv : v ∈ B
⊢ v ∈ A → v ∈ B | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Finset α
v : α
A : Finset α
hB : B ⊆ A
hv : v ∈ B
⊢ ¬(v ∈ A ∧ ¬v ∈ B)
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/Turanpartition.lean | Turanpartition.card_sdiff_erase | [702, 1] | [706, 87] | intro _ | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Finset α
v : α
A : Finset α
hB : B ⊆ A
hv : v ∈ B
⊢ v ∈ A → v ∈ B | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Finset α
v : α
A : Finset α
hB : B ⊆ A
hv : v ∈ B
_✝ : v ∈ A
⊢ v ∈ B | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Finset α
v : α
A : Finset α
hB : B ⊆ A
hv : v ∈ B
⊢ v ∈ A → v ∈ B
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/Turanpartition.lean | Turanpartition.card_sdiff_erase | [702, 1] | [706, 87] | exact hv | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Finset α
v : α
A : Finset α
hB : B ⊆ A
hv : v ∈ B
_✝ : v ∈ A
⊢ v ∈ B | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Finset α
v : α
A : Finset α
hB : B ⊆ A
hv : v ∈ B
_✝ : v ∈ A
⊢ v ∈ B
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/Turanpartition.lean | Turanpartition.sdiff_insert | [709, 1] | [716, 52] | ext | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
v : α
B A : Finset α
⊢ A \ insert v B = erase (A \ B) v | case a
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
v : α
B A : Finset α
a✝ : α
⊢ a✝ ∈ A \ insert v B ↔ a✝ ∈ erase (A \ B) v | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
v : α
B A : Finset α
⊢ A \ insert v B = erase (A \ B) v
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/Turanpartition.lean | Turanpartition.sdiff_insert | [709, 1] | [716, 52] | constructor | case a
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
v : α
B A : Finset α
a✝ : α
⊢ a✝ ∈ A \ insert v B ↔ a✝ ∈ erase (A \ B) v | case a.mp
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
v : α
B A : Finset α
a✝ : α
⊢ a✝ ∈ A \ insert v B → a✝ ∈ erase (A \ B) v
case a.mpr
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
v : α
B A : Finset α
a✝ : α
⊢ a✝ ∈ erase (A \ B) v → a✝ ∈ A \ insert v B | Please generate a tactic in lean4 to solve the state.
STATE:
case a
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
v : α
B A : Finset α
a✝ : α
⊢ a✝ ∈ A \ insert v B ↔ a✝ ∈ erase (A \ B) v
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/Turanpartition.lean | Turanpartition.sdiff_insert | [709, 1] | [716, 52] | intro h | case a.mp
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
v : α
B A : Finset α
a✝ : α
⊢ a✝ ∈ A \ insert v B → a✝ ∈ erase (A \ B) v | case a.mp
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
v : α
B A : Finset α
a✝ : α
h : a✝ ∈ A \ insert v B
⊢ a✝ ∈ erase (A \ B) v | Please generate a tactic in lean4 to solve the state.
STATE:
case a.mp
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
v : α
B A : Finset α
a✝ : α
⊢ a✝ ∈ A \ insert v B → a✝ ∈ erase (A \ B) v
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/Turanpartition.lean | Turanpartition.sdiff_insert | [709, 1] | [716, 52] | rw [mem_erase] | case a.mp
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
v : α
B A : Finset α
a✝ : α
h : a✝ ∈ A \ insert v B
⊢ a✝ ∈ erase (A \ B) v | case a.mp
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
v : α
B A : Finset α
a✝ : α
h : a✝ ∈ A \ insert v B
⊢ a✝ ≠ v ∧ a✝ ∈ A \ B | Please generate a tactic in lean4 to solve the state.
STATE:
case a.mp
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
v : α
B A : Finset α
a✝ : α
h : a✝ ∈ A \ insert v B
⊢ a✝ ∈ erase (A \ B) v
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/Turanpartition.lean | Turanpartition.sdiff_insert | [709, 1] | [716, 52] | rw [mem_sdiff] at * | case a.mp
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
v : α
B A : Finset α
a✝ : α
h : a✝ ∈ A \ insert v B
⊢ a✝ ≠ v ∧ a✝ ∈ A \ B | case a.mp
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
v : α
B A : Finset α
a✝ : α
h : a✝ ∈ A ∧ ¬a✝ ∈ insert v B
⊢ a✝ ≠ v ∧ a✝ ∈ A ∧ ¬a✝ ∈ B | Please generate a tactic in lean4 to solve the state.
STATE:
case a.mp
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
v : α
B A : Finset α
a✝ : α
h : a✝ ∈ A \ insert v B
⊢ a✝ ≠ v ∧ a✝ ∈ A \ B
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/Turanpartition.lean | Turanpartition.sdiff_insert | [709, 1] | [716, 52] | rw [mem_insert] at h | case a.mp
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
v : α
B A : Finset α
a✝ : α
h : a✝ ∈ A ∧ ¬a✝ ∈ insert v B
⊢ a✝ ≠ v ∧ a✝ ∈ A ∧ ¬a✝ ∈ B | case a.mp
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
v : α
B A : Finset α
a✝ : α
h : a✝ ∈ A ∧ ¬(a✝ = v ∨ a✝ ∈ B)
⊢ a✝ ≠ v ∧ a✝ ∈ A ∧ ¬a✝ ∈ B | Please generate a tactic in lean4 to solve the state.
STATE:
case a.mp
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
v : α
B A : Finset α
a✝ : α
h : a✝ ∈ A ∧ ¬a✝ ∈ insert v B
⊢ a✝ ≠ v ∧ a✝ ∈ A ∧ ¬a✝ ∈ B
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/Turanpartition.lean | Turanpartition.sdiff_insert | [709, 1] | [716, 52] | push_neg at h | case a.mp
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
v : α
B A : Finset α
a✝ : α
h : a✝ ∈ A ∧ ¬(a✝ = v ∨ a✝ ∈ B)
⊢ a✝ ≠ v ∧ a✝ ∈ A ∧ ¬a✝ ∈ B | case a.mp
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
v : α
B A : Finset α
a✝ : α
h : a✝ ∈ A ∧ a✝ ≠ v ∧ ¬a✝ ∈ B
⊢ a✝ ≠ v ∧ a✝ ∈ A ∧ ¬a✝ ∈ B | Please generate a tactic in lean4 to solve the state.
STATE:
case a.mp
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
v : α
B A : Finset α
a✝ : α
h : a✝ ∈ A ∧ ¬(a✝ = v ∨ a✝ ∈ B)
⊢ a✝ ≠ v ∧ a✝ ∈ A ∧ ¬a✝ ∈ B
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/Turanpartition.lean | Turanpartition.sdiff_insert | [709, 1] | [716, 52] | exact ⟨h.2.1, h.1, h.2.2⟩ | case a.mp
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
v : α
B A : Finset α
a✝ : α
h : a✝ ∈ A ∧ a✝ ≠ v ∧ ¬a✝ ∈ B
⊢ a✝ ≠ v ∧ a✝ ∈ A ∧ ¬a✝ ∈ B | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case a.mp
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
v : α
B A : Finset α
a✝ : α
h : a✝ ∈ A ∧ a✝ ≠ v ∧ ¬a✝ ∈ B
⊢ a✝ ≠ v ∧ a✝ ∈ A ∧ ¬a✝ ∈ B
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/Turanpartition.lean | Turanpartition.sdiff_insert | [709, 1] | [716, 52] | intro h | case a.mpr
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
v : α
B A : Finset α
a✝ : α
⊢ a✝ ∈ erase (A \ B) v → a✝ ∈ A \ insert v B | case a.mpr
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
v : α
B A : Finset α
a✝ : α
h : a✝ ∈ erase (A \ B) v
⊢ a✝ ∈ A \ insert v B | Please generate a tactic in lean4 to solve the state.
STATE:
case a.mpr
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
v : α
B A : Finset α
a✝ : α
⊢ a✝ ∈ erase (A \ B) v → a✝ ∈ A \ insert v B
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/Turanpartition.lean | Turanpartition.sdiff_insert | [709, 1] | [716, 52] | rw [mem_erase] at h | case a.mpr
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
v : α
B A : Finset α
a✝ : α
h : a✝ ∈ erase (A \ B) v
⊢ a✝ ∈ A \ insert v B | case a.mpr
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
v : α
B A : Finset α
a✝ : α
h : a✝ ≠ v ∧ a✝ ∈ A \ B
⊢ a✝ ∈ A \ insert v B | Please generate a tactic in lean4 to solve the state.
STATE:
case a.mpr
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
v : α
B A : Finset α
a✝ : α
h : a✝ ∈ erase (A \ B) v
⊢ a✝ ∈ A \ insert v B
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/Turanpartition.lean | Turanpartition.sdiff_insert | [709, 1] | [716, 52] | rw [mem_sdiff] | case a.mpr
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
v : α
B A : Finset α
a✝ : α
h : a✝ ≠ v ∧ a✝ ∈ A \ B
⊢ a✝ ∈ A \ insert v B | case a.mpr
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
v : α
B A : Finset α
a✝ : α
h : a✝ ≠ v ∧ a✝ ∈ A \ B
⊢ a✝ ∈ A ∧ ¬a✝ ∈ insert v B | Please generate a tactic in lean4 to solve the state.
STATE:
case a.mpr
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
v : α
B A : Finset α
a✝ : α
h : a✝ ≠ v ∧ a✝ ∈ A \ B
⊢ a✝ ∈ A \ insert v B
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/Turanpartition.lean | Turanpartition.sdiff_insert | [709, 1] | [716, 52] | rw [mem_insert] | case a.mpr
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
v : α
B A : Finset α
a✝ : α
h : a✝ ≠ v ∧ a✝ ∈ A \ B
⊢ a✝ ∈ A ∧ ¬a✝ ∈ insert v B | case a.mpr
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
v : α
B A : Finset α
a✝ : α
h : a✝ ≠ v ∧ a✝ ∈ A \ B
⊢ a✝ ∈ A ∧ ¬(a✝ = v ∨ a✝ ∈ B) | Please generate a tactic in lean4 to solve the state.
STATE:
case a.mpr
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
v : α
B A : Finset α
a✝ : α
h : a✝ ≠ v ∧ a✝ ∈ A \ B
⊢ a✝ ∈ A ∧ ¬a✝ ∈ insert v B
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/Turanpartition.lean | Turanpartition.sdiff_insert | [709, 1] | [716, 52] | push_neg | case a.mpr
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
v : α
B A : Finset α
a✝ : α
h : a✝ ≠ v ∧ a✝ ∈ A \ B
⊢ a✝ ∈ A ∧ ¬(a✝ = v ∨ a✝ ∈ B) | case a.mpr
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
v : α
B A : Finset α
a✝ : α
h : a✝ ≠ v ∧ a✝ ∈ A \ B
⊢ a✝ ∈ A ∧ a✝ ≠ v ∧ ¬a✝ ∈ B | Please generate a tactic in lean4 to solve the state.
STATE:
case a.mpr
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
v : α
B A : Finset α
a✝ : α
h : a✝ ≠ v ∧ a✝ ∈ A \ B
⊢ a✝ ∈ A ∧ ¬(a✝ = v ∨ a✝ ∈ B)
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/Turanpartition.lean | Turanpartition.sdiff_insert | [709, 1] | [716, 52] | rw [mem_sdiff] at h | case a.mpr
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
v : α
B A : Finset α
a✝ : α
h : a✝ ≠ v ∧ a✝ ∈ A \ B
⊢ a✝ ∈ A ∧ a✝ ≠ v ∧ ¬a✝ ∈ B | case a.mpr
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
v : α
B A : Finset α
a✝ : α
h : a✝ ≠ v ∧ a✝ ∈ A ∧ ¬a✝ ∈ B
⊢ a✝ ∈ A ∧ a✝ ≠ v ∧ ¬a✝ ∈ B | Please generate a tactic in lean4 to solve the state.
STATE:
case a.mpr
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
v : α
B A : Finset α
a✝ : α
h : a✝ ≠ v ∧ a✝ ∈ A \ B
⊢ a✝ ∈ A ∧ a✝ ≠ v ∧ ¬a✝ ∈ B
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/Turanpartition.lean | Turanpartition.sdiff_insert | [709, 1] | [716, 52] | exact ⟨h.2.1, h.1, h.2.2⟩ | case a.mpr
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
v : α
B A : Finset α
a✝ : α
h : a✝ ≠ v ∧ a✝ ∈ A ∧ ¬a✝ ∈ B
⊢ a✝ ∈ A ∧ a✝ ≠ v ∧ ¬a✝ ∈ B | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case a.mpr
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
v : α
B A : Finset α
a✝ : α
h : a✝ ≠ v ∧ a✝ ∈ A ∧ ¬a✝ ∈ B
⊢ a✝ ∈ A ∧ a✝ ≠ v ∧ ¬a✝ ∈ B
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/Turanpartition.lean | Turanpartition.card_sdiff_insert | [719, 1] | [722, 72] | rw [sdiff_insert] | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
v : α
B A : Finset α
hvB : ¬v ∈ B
hvA : v ∈ A
⊢ card (A \ insert v B) = card (A \ B) - 1 | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
v : α
B A : Finset α
hvB : ¬v ∈ B
hvA : v ∈ A
⊢ card (erase (A \ B) v) = card (A \ B) - 1 | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
v : α
B A : Finset α
hvB : ¬v ∈ B
hvA : v ∈ A
⊢ card (A \ insert v B) = card (A \ B) - 1
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/Turanpartition.lean | Turanpartition.card_sdiff_insert | [719, 1] | [722, 72] | exact card_erase_of_mem (mem_sdiff.mpr ⟨hvA, hvB⟩) | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
v : α
B A : Finset α
hvB : ¬v ∈ B
hvA : v ∈ A
⊢ card (erase (A \ B) v) = card (A \ B) - 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
v : α
B A : Finset α
hvB : ¬v ∈ B
hvA : v ∈ A
⊢ card (erase (A \ B) v) = card (A \ B) - 1
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/Turanpartition.lean | Turanpartition.move_Pcard_sdiff | [726, 1] | [739, 31] | intro hk | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i : ℕ
v : α
j k : ℕ
M : MultiPart α
hvi : i ∈ range (M.t + 1) ∧ v ∈ MultiPart.P M i
hj : j ∈ range (M.t + 1) ∧ j ≠ i
⊢ k ∈ range (M.t + 1) →
card ((move M hvi hj).A \ MultiPart.P (move M hvi hj) k) =
if k ≠ i ∧ k ≠ j then card (M.A \ MultiPart.P M k)
else if k = i then card (M.A \ MultiPart.P M i) + 1 else card (M.A \ MultiPart.P M j) - 1 | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i : ℕ
v : α
j k : ℕ
M : MultiPart α
hvi : i ∈ range (M.t + 1) ∧ v ∈ MultiPart.P M i
hj : j ∈ range (M.t + 1) ∧ j ≠ i
hk : k ∈ range (M.t + 1)
⊢ card ((move M hvi hj).A \ MultiPart.P (move M hvi hj) k) =
if k ≠ i ∧ k ≠ j then card (M.A \ MultiPart.P M k)
else if k = i then card (M.A \ MultiPart.P M i) + 1 else card (M.A \ MultiPart.P M j) - 1 | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i : ℕ
v : α
j k : ℕ
M : MultiPart α
hvi : i ∈ range (M.t + 1) ∧ v ∈ MultiPart.P M i
hj : j ∈ range (M.t + 1) ∧ j ≠ i
⊢ k ∈ range (M.t + 1) →
card ((move M hvi hj).A \ MultiPart.P (move M hvi hj) k) =
if k ≠ i ∧ k ≠ j then card (M.A \ MultiPart.P M k)
else if k = i then card (M.A \ MultiPart.P M i) + 1 else card (M.A \ MultiPart.P M j) - 1
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/Turanpartition.lean | Turanpartition.move_Pcard_sdiff | [726, 1] | [739, 31] | rw [move_p hvi hj hk] | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i : ℕ
v : α
j k : ℕ
M : MultiPart α
hvi : i ∈ range (M.t + 1) ∧ v ∈ MultiPart.P M i
hj : j ∈ range (M.t + 1) ∧ j ≠ i
hk : k ∈ range (M.t + 1)
⊢ card ((move M hvi hj).A \ MultiPart.P (move M hvi hj) k) =
if k ≠ i ∧ k ≠ j then card (M.A \ MultiPart.P M k)
else if k = i then card (M.A \ MultiPart.P M i) + 1 else card (M.A \ MultiPart.P M j) - 1 | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i : ℕ
v : α
j k : ℕ
M : MultiPart α
hvi : i ∈ range (M.t + 1) ∧ v ∈ MultiPart.P M i
hj : j ∈ range (M.t + 1) ∧ j ≠ i
hk : k ∈ range (M.t + 1)
⊢ card
((move M hvi hj).A \
if k ≠ i ∧ k ≠ j then MultiPart.P M k
else if k = i then erase (MultiPart.P M i) v else insert v (MultiPart.P M j)) =
if k ≠ i ∧ k ≠ j then card (M.A \ MultiPart.P M k)
else if k = i then card (M.A \ MultiPart.P M i) + 1 else card (M.A \ MultiPart.P M j) - 1 | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i : ℕ
v : α
j k : ℕ
M : MultiPart α
hvi : i ∈ range (M.t + 1) ∧ v ∈ MultiPart.P M i
hj : j ∈ range (M.t + 1) ∧ j ≠ i
hk : k ∈ range (M.t + 1)
⊢ card ((move M hvi hj).A \ MultiPart.P (move M hvi hj) k) =
if k ≠ i ∧ k ≠ j then card (M.A \ MultiPart.P M k)
else if k = i then card (M.A \ MultiPart.P M i) + 1 else card (M.A \ MultiPart.P M j) - 1
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/Turanpartition.lean | Turanpartition.move_Pcard_sdiff | [726, 1] | [739, 31] | rw [move_a hvi hj] | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i : ℕ
v : α
j k : ℕ
M : MultiPart α
hvi : i ∈ range (M.t + 1) ∧ v ∈ MultiPart.P M i
hj : j ∈ range (M.t + 1) ∧ j ≠ i
hk : k ∈ range (M.t + 1)
⊢ card
((move M hvi hj).A \
if k ≠ i ∧ k ≠ j then MultiPart.P M k
else if k = i then erase (MultiPart.P M i) v else insert v (MultiPart.P M j)) =
if k ≠ i ∧ k ≠ j then card (M.A \ MultiPart.P M k)
else if k = i then card (M.A \ MultiPart.P M i) + 1 else card (M.A \ MultiPart.P M j) - 1 | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i : ℕ
v : α
j k : ℕ
M : MultiPart α
hvi : i ∈ range (M.t + 1) ∧ v ∈ MultiPart.P M i
hj : j ∈ range (M.t + 1) ∧ j ≠ i
hk : k ∈ range (M.t + 1)
⊢ card
(M.A \
if k ≠ i ∧ k ≠ j then MultiPart.P M k
else if k = i then erase (MultiPart.P M i) v else insert v (MultiPart.P M j)) =
if k ≠ i ∧ k ≠ j then card (M.A \ MultiPart.P M k)
else if k = i then card (M.A \ MultiPart.P M i) + 1 else card (M.A \ MultiPart.P M j) - 1 | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i : ℕ
v : α
j k : ℕ
M : MultiPart α
hvi : i ∈ range (M.t + 1) ∧ v ∈ MultiPart.P M i
hj : j ∈ range (M.t + 1) ∧ j ≠ i
hk : k ∈ range (M.t + 1)
⊢ card
((move M hvi hj).A \
if k ≠ i ∧ k ≠ j then MultiPart.P M k
else if k = i then erase (MultiPart.P M i) v else insert v (MultiPart.P M j)) =
if k ≠ i ∧ k ≠ j then card (M.A \ MultiPart.P M k)
else if k = i then card (M.A \ MultiPart.P M i) + 1 else card (M.A \ MultiPart.P M j) - 1
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/Turanpartition.lean | Turanpartition.move_Pcard_sdiff | [726, 1] | [739, 31] | split_ifs | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i : ℕ
v : α
j k : ℕ
M : MultiPart α
hvi : i ∈ range (M.t + 1) ∧ v ∈ MultiPart.P M i
hj : j ∈ range (M.t + 1) ∧ j ≠ i
hk : k ∈ range (M.t + 1)
⊢ card
(M.A \
if k ≠ i ∧ k ≠ j then MultiPart.P M k
else if k = i then erase (MultiPart.P M i) v else insert v (MultiPart.P M j)) =
if k ≠ i ∧ k ≠ j then card (M.A \ MultiPart.P M k)
else if k = i then card (M.A \ MultiPart.P M i) + 1 else card (M.A \ MultiPart.P M j) - 1 | case pos
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i : ℕ
v : α
j k : ℕ
M : MultiPart α
hvi : i ∈ range (M.t + 1) ∧ v ∈ MultiPart.P M i
hj : j ∈ range (M.t + 1) ∧ j ≠ i
hk : k ∈ range (M.t + 1)
h✝ : k ≠ i ∧ k ≠ j
⊢ card (M.A \ MultiPart.P M k) = card (M.A \ MultiPart.P M k)
case pos
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i : ℕ
v : α
j k : ℕ
M : MultiPart α
hvi : i ∈ range (M.t + 1) ∧ v ∈ MultiPart.P M i
hj : j ∈ range (M.t + 1) ∧ j ≠ i
hk : k ∈ range (M.t + 1)
h✝¹ : ¬(k ≠ i ∧ k ≠ j)
h✝ : k = i
⊢ card (M.A \ erase (MultiPart.P M i) v) = card (M.A \ MultiPart.P M i) + 1
case neg
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i : ℕ
v : α
j k : ℕ
M : MultiPart α
hvi : i ∈ range (M.t + 1) ∧ v ∈ MultiPart.P M i
hj : j ∈ range (M.t + 1) ∧ j ≠ i
hk : k ∈ range (M.t + 1)
h✝¹ : ¬(k ≠ i ∧ k ≠ j)
h✝ : ¬k = i
⊢ card (M.A \ insert v (MultiPart.P M j)) = card (M.A \ MultiPart.P M j) - 1 | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i : ℕ
v : α
j k : ℕ
M : MultiPart α
hvi : i ∈ range (M.t + 1) ∧ v ∈ MultiPart.P M i
hj : j ∈ range (M.t + 1) ∧ j ≠ i
hk : k ∈ range (M.t + 1)
⊢ card
(M.A \
if k ≠ i ∧ k ≠ j then MultiPart.P M k
else if k = i then erase (MultiPart.P M i) v else insert v (MultiPart.P M j)) =
if k ≠ i ∧ k ≠ j then card (M.A \ MultiPart.P M k)
else if k = i then card (M.A \ MultiPart.P M i) + 1 else card (M.A \ MultiPart.P M j) - 1
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/Turanpartition.lean | Turanpartition.move_Pcard_sdiff | [726, 1] | [739, 31] | rfl | case pos
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i : ℕ
v : α
j k : ℕ
M : MultiPart α
hvi : i ∈ range (M.t + 1) ∧ v ∈ MultiPart.P M i
hj : j ∈ range (M.t + 1) ∧ j ≠ i
hk : k ∈ range (M.t + 1)
h✝ : k ≠ i ∧ k ≠ j
⊢ card (M.A \ MultiPart.P M k) = card (M.A \ MultiPart.P M k) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i : ℕ
v : α
j k : ℕ
M : MultiPart α
hvi : i ∈ range (M.t + 1) ∧ v ∈ MultiPart.P M i
hj : j ∈ range (M.t + 1) ∧ j ≠ i
hk : k ∈ range (M.t + 1)
h✝ : k ≠ i ∧ k ≠ j
⊢ card (M.A \ MultiPart.P M k) = card (M.A \ MultiPart.P M k)
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/Turanpartition.lean | Turanpartition.move_Pcard_sdiff | [726, 1] | [739, 31] | exact card_sdiff_erase (sub_part hvi.1) hvi.2 | case pos
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i : ℕ
v : α
j k : ℕ
M : MultiPart α
hvi : i ∈ range (M.t + 1) ∧ v ∈ MultiPart.P M i
hj : j ∈ range (M.t + 1) ∧ j ≠ i
hk : k ∈ range (M.t + 1)
h✝¹ : ¬(k ≠ i ∧ k ≠ j)
h✝ : k = i
⊢ card (M.A \ erase (MultiPart.P M i) v) = card (M.A \ MultiPart.P M i) + 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i : ℕ
v : α
j k : ℕ
M : MultiPart α
hvi : i ∈ range (M.t + 1) ∧ v ∈ MultiPart.P M i
hj : j ∈ range (M.t + 1) ∧ j ≠ i
hk : k ∈ range (M.t + 1)
h✝¹ : ¬(k ≠ i ∧ k ≠ j)
h✝ : k = i
⊢ card (M.A \ erase (MultiPart.P M i) v) = card (M.A \ MultiPart.P M i) + 1
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/Turanpartition.lean | Turanpartition.move_Pcard_sdiff | [726, 1] | [739, 31] | apply card_sdiff_insert
(uniq_part' hvi.1 hj.1 hj.2.symm hvi.2)
(mem_part hvi.1 hvi.2) | case neg
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i : ℕ
v : α
j k : ℕ
M : MultiPart α
hvi : i ∈ range (M.t + 1) ∧ v ∈ MultiPart.P M i
hj : j ∈ range (M.t + 1) ∧ j ≠ i
hk : k ∈ range (M.t + 1)
h✝¹ : ¬(k ≠ i ∧ k ≠ j)
h✝ : ¬k = i
⊢ card (M.A \ insert v (MultiPart.P M j)) = card (M.A \ MultiPart.P M j) - 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i : ℕ
v : α
j k : ℕ
M : MultiPart α
hvi : i ∈ range (M.t + 1) ∧ v ∈ MultiPart.P M i
hj : j ∈ range (M.t + 1) ∧ j ≠ i
hk : k ∈ range (M.t + 1)
h✝¹ : ¬(k ≠ i ∧ k ≠ j)
h✝ : ¬k = i
⊢ card (M.A \ insert v (MultiPart.P M j)) = card (M.A \ MultiPart.P M j) - 1
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/Turanpartition.lean | Turanpartition.move_change | [742, 1] | [760, 30] | rw [mul_add,add_mul,mul_one,one_mul] | α : Type ?u.138093
inst✝¹ : Fintype α
inst✝ : DecidableEq α
b a n : ℕ
hb : b + 1 < a
hn : a + b ≤ n
⊢ a * (n - a) + b * (n - b) < (a - 1) * (n - a + 1) + (b + 1) * (n - b - 1) | α : Type ?u.138093
inst✝¹ : Fintype α
inst✝ : DecidableEq α
b a n : ℕ
hb : b + 1 < a
hn : a + b ≤ n
⊢ a * (n - a) + b * (n - b) < (a - 1) * (n - a) + (a - 1) + (b * (n - b - 1) + (n - b - 1)) | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type ?u.138093
inst✝¹ : Fintype α
inst✝ : DecidableEq α
b a n : ℕ
hb : b + 1 < a
hn : a + b ≤ n
⊢ a * (n - a) + b * (n - b) < (a - 1) * (n - a + 1) + (b + 1) * (n - b - 1)
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/Turanpartition.lean | Turanpartition.move_change | [742, 1] | [760, 30] | have ha := tsub_add_cancel_of_le ((le_tsub_of_add_le_left hb.le).trans tsub_le_self : 1 ≤ a) | α : Type ?u.138093
inst✝¹ : Fintype α
inst✝ : DecidableEq α
b a n : ℕ
hb : b + 1 < a
hn : a + b ≤ n
⊢ a * (n - a) + b * (n - b) < (a - 1) * (n - a) + (a - 1) + (b * (n - b - 1) + (n - b - 1)) | α : Type ?u.138093
inst✝¹ : Fintype α
inst✝ : DecidableEq α
b a n : ℕ
hb : b + 1 < a
hn : a + b ≤ n
ha : a - 1 + 1 = a
⊢ a * (n - a) + b * (n - b) < (a - 1) * (n - a) + (a - 1) + (b * (n - b - 1) + (n - b - 1)) | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type ?u.138093
inst✝¹ : Fintype α
inst✝ : DecidableEq α
b a n : ℕ
hb : b + 1 < a
hn : a + b ≤ n
⊢ a * (n - a) + b * (n - b) < (a - 1) * (n - a) + (a - 1) + (b * (n - b - 1) + (n - b - 1))
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/Turanpartition.lean | Turanpartition.move_change | [742, 1] | [760, 30] | have h2 : a ≤ n - b := le_tsub_of_add_le_right hn | α : Type ?u.138093
inst✝¹ : Fintype α
inst✝ : DecidableEq α
b a n : ℕ
hb : b + 1 < a
hn : a + b ≤ n
ha : a - 1 + 1 = a
⊢ a * (n - a) + b * (n - b) < (a - 1) * (n - a) + (a - 1) + (b * (n - b - 1) + (n - b - 1)) | α : Type ?u.138093
inst✝¹ : Fintype α
inst✝ : DecidableEq α
b a n : ℕ
hb : b + 1 < a
hn : a + b ≤ n
ha : a - 1 + 1 = a
h2 : a ≤ n - b
⊢ a * (n - a) + b * (n - b) < (a - 1) * (n - a) + (a - 1) + (b * (n - b - 1) + (n - b - 1)) | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type ?u.138093
inst✝¹ : Fintype α
inst✝ : DecidableEq α
b a n : ℕ
hb : b + 1 < a
hn : a + b ≤ n
ha : a - 1 + 1 = a
⊢ a * (n - a) + b * (n - b) < (a - 1) * (n - a) + (a - 1) + (b * (n - b - 1) + (n - b - 1))
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/Turanpartition.lean | Turanpartition.move_change | [742, 1] | [760, 30] | have hnb := tsub_add_cancel_of_le (le_trans ((le_tsub_of_add_le_left hb.le).trans tsub_le_self : 1 ≤ a) h2) | α : Type ?u.138093
inst✝¹ : Fintype α
inst✝ : DecidableEq α
b a n : ℕ
hb : b + 1 < a
hn : a + b ≤ n
ha : a - 1 + 1 = a
h2 : a ≤ n - b
⊢ a * (n - a) + b * (n - b) < (a - 1) * (n - a) + (a - 1) + (b * (n - b - 1) + (n - b - 1)) | α : Type ?u.138093
inst✝¹ : Fintype α
inst✝ : DecidableEq α
b a n : ℕ
hb : b + 1 < a
hn : a + b ≤ n
ha : a - 1 + 1 = a
h2 : a ≤ n - b
hnb : n - b - 1 + 1 = n - b
⊢ a * (n - a) + b * (n - b) < (a - 1) * (n - a) + (a - 1) + (b * (n - b - 1) + (n - b - 1)) | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type ?u.138093
inst✝¹ : Fintype α
inst✝ : DecidableEq α
b a n : ℕ
hb : b + 1 < a
hn : a + b ≤ n
ha : a - 1 + 1 = a
h2 : a ≤ n - b
⊢ a * (n - a) + b * (n - b) < (a - 1) * (n - a) + (a - 1) + (b * (n - b - 1) + (n - b - 1))
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/Turanpartition.lean | Turanpartition.move_change | [742, 1] | [760, 30] | nth_rw 1 [← ha,← hnb] | α : Type ?u.138093
inst✝¹ : Fintype α
inst✝ : DecidableEq α
b a n : ℕ
hb : b + 1 < a
hn : a + b ≤ n
ha : a - 1 + 1 = a
h2 : a ≤ n - b
hnb : n - b - 1 + 1 = n - b
⊢ a * (n - a) + b * (n - b) < (a - 1) * (n - a) + (a - 1) + (b * (n - b - 1) + (n - b - 1)) | α : Type ?u.138093
inst✝¹ : Fintype α
inst✝ : DecidableEq α
b a n : ℕ
hb : b + 1 < a
hn : a + b ≤ n
ha : a - 1 + 1 = a
h2 : a ≤ n - b
hnb : n - b - 1 + 1 = n - b
⊢ (a - 1 + 1) * (n - a) + b * (n - b - 1 + 1) < (a - 1) * (n - a) + (a - 1) + (b * (n - b - 1) + (n - b - 1)) | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type ?u.138093
inst✝¹ : Fintype α
inst✝ : DecidableEq α
b a n : ℕ
hb : b + 1 < a
hn : a + b ≤ n
ha : a - 1 + 1 = a
h2 : a ≤ n - b
hnb : n - b - 1 + 1 = n - b
⊢ a * (n - a) + b * (n - b) < (a - 1) * (n - a) + (a - 1) + (b * (n - b - 1) + (n - b - 1))
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/Turanpartition.lean | Turanpartition.move_change | [742, 1] | [760, 30] | rw [add_mul, mul_add, one_mul, mul_one, add_assoc, add_assoc] | α : Type ?u.138093
inst✝¹ : Fintype α
inst✝ : DecidableEq α
b a n : ℕ
hb : b + 1 < a
hn : a + b ≤ n
ha : a - 1 + 1 = a
h2 : a ≤ n - b
hnb : n - b - 1 + 1 = n - b
⊢ (a - 1 + 1) * (n - a) + b * (n - b - 1 + 1) < (a - 1) * (n - a) + (a - 1) + (b * (n - b - 1) + (n - b - 1)) | α : Type ?u.138093
inst✝¹ : Fintype α
inst✝ : DecidableEq α
b a n : ℕ
hb : b + 1 < a
hn : a + b ≤ n
ha : a - 1 + 1 = a
h2 : a ≤ n - b
hnb : n - b - 1 + 1 = n - b
⊢ (a - 1) * (n - a) + (n - a + (b * (n - b - 1) + b)) < (a - 1) * (n - a) + (a - 1 + (b * (n - b - 1) + (n - b - 1))) | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type ?u.138093
inst✝¹ : Fintype α
inst✝ : DecidableEq α
b a n : ℕ
hb : b + 1 < a
hn : a + b ≤ n
ha : a - 1 + 1 = a
h2 : a ≤ n - b
hnb : n - b - 1 + 1 = n - b
⊢ (a - 1 + 1) * (n - a) + b * (n - b - 1 + 1) < (a - 1) * (n - a) + (a - 1) + (b * (n - b - 1) + (n - b - 1))
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/Turanpartition.lean | Turanpartition.move_change | [742, 1] | [760, 30] | apply Nat.add_lt_add_left | α : Type ?u.138093
inst✝¹ : Fintype α
inst✝ : DecidableEq α
b a n : ℕ
hb : b + 1 < a
hn : a + b ≤ n
ha : a - 1 + 1 = a
h2 : a ≤ n - b
hnb : n - b - 1 + 1 = n - b
⊢ (a - 1) * (n - a) + (n - a + (b * (n - b - 1) + b)) < (a - 1) * (n - a) + (a - 1 + (b * (n - b - 1) + (n - b - 1))) | case h
α : Type ?u.138093
inst✝¹ : Fintype α
inst✝ : DecidableEq α
b a n : ℕ
hb : b + 1 < a
hn : a + b ≤ n
ha : a - 1 + 1 = a
h2 : a ≤ n - b
hnb : n - b - 1 + 1 = n - b
⊢ n - a + (b * (n - b - 1) + b) < a - 1 + (b * (n - b - 1) + (n - b - 1)) | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type ?u.138093
inst✝¹ : Fintype α
inst✝ : DecidableEq α
b a n : ℕ
hb : b + 1 < a
hn : a + b ≤ n
ha : a - 1 + 1 = a
h2 : a ≤ n - b
hnb : n - b - 1 + 1 = n - b
⊢ (a - 1) * (n - a) + (n - a + (b * (n - b - 1) + b)) < (a - 1) * (n - a) + (a - 1 + (b * (n - b - 1) + (n - b - 1)))
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/Turanpartition.lean | Turanpartition.move_change | [742, 1] | [760, 30] | rw [add_comm, ← add_assoc, add_comm (a - 1), add_assoc, add_assoc] | case h
α : Type ?u.138093
inst✝¹ : Fintype α
inst✝ : DecidableEq α
b a n : ℕ
hb : b + 1 < a
hn : a + b ≤ n
ha : a - 1 + 1 = a
h2 : a ≤ n - b
hnb : n - b - 1 + 1 = n - b
⊢ n - a + (b * (n - b - 1) + b) < a - 1 + (b * (n - b - 1) + (n - b - 1)) | case h
α : Type ?u.138093
inst✝¹ : Fintype α
inst✝ : DecidableEq α
b a n : ℕ
hb : b + 1 < a
hn : a + b ≤ n
ha : a - 1 + 1 = a
h2 : a ≤ n - b
hnb : n - b - 1 + 1 = n - b
⊢ b * (n - b - 1) + (b + (n - a)) < b * (n - b - 1) + (a - 1 + (n - b - 1)) | Please generate a tactic in lean4 to solve the state.
STATE:
case h
α : Type ?u.138093
inst✝¹ : Fintype α
inst✝ : DecidableEq α
b a n : ℕ
hb : b + 1 < a
hn : a + b ≤ n
ha : a - 1 + 1 = a
h2 : a ≤ n - b
hnb : n - b - 1 + 1 = n - b
⊢ n - a + (b * (n - b - 1) + b) < a - 1 + (b * (n - b - 1) + (n - b - 1))
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/Turanpartition.lean | Turanpartition.move_change | [742, 1] | [760, 30] | apply Nat.add_lt_add_left | case h
α : Type ?u.138093
inst✝¹ : Fintype α
inst✝ : DecidableEq α
b a n : ℕ
hb : b + 1 < a
hn : a + b ≤ n
ha : a - 1 + 1 = a
h2 : a ≤ n - b
hnb : n - b - 1 + 1 = n - b
⊢ b * (n - b - 1) + (b + (n - a)) < b * (n - b - 1) + (a - 1 + (n - b - 1)) | case h.h
α : Type ?u.138093
inst✝¹ : Fintype α
inst✝ : DecidableEq α
b a n : ℕ
hb : b + 1 < a
hn : a + b ≤ n
ha : a - 1 + 1 = a
h2 : a ≤ n - b
hnb : n - b - 1 + 1 = n - b
⊢ b + (n - a) < a - 1 + (n - b - 1) | Please generate a tactic in lean4 to solve the state.
STATE:
case h
α : Type ?u.138093
inst✝¹ : Fintype α
inst✝ : DecidableEq α
b a n : ℕ
hb : b + 1 < a
hn : a + b ≤ n
ha : a - 1 + 1 = a
h2 : a ≤ n - b
hnb : n - b - 1 + 1 = n - b
⊢ b * (n - b - 1) + (b + (n - a)) < b * (n - b - 1) + (a - 1 + (n - b - 1))
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/Turanpartition.lean | Turanpartition.move_change | [742, 1] | [760, 30] | have ab : b < a - 1 := lt_tsub_of_add_lt_right hb | case h.h
α : Type ?u.138093
inst✝¹ : Fintype α
inst✝ : DecidableEq α
b a n : ℕ
hb : b + 1 < a
hn : a + b ≤ n
ha : a - 1 + 1 = a
h2 : a ≤ n - b
hnb : n - b - 1 + 1 = n - b
⊢ b + (n - a) < a - 1 + (n - b - 1) | case h.h
α : Type ?u.138093
inst✝¹ : Fintype α
inst✝ : DecidableEq α
b a n : ℕ
hb : b + 1 < a
hn : a + b ≤ n
ha : a - 1 + 1 = a
h2 : a ≤ n - b
hnb : n - b - 1 + 1 = n - b
ab : b < a - 1
⊢ b + (n - a) < a - 1 + (n - b - 1) | Please generate a tactic in lean4 to solve the state.
STATE:
case h.h
α : Type ?u.138093
inst✝¹ : Fintype α
inst✝ : DecidableEq α
b a n : ℕ
hb : b + 1 < a
hn : a + b ≤ n
ha : a - 1 + 1 = a
h2 : a ≤ n - b
hnb : n - b - 1 + 1 = n - b
⊢ b + (n - a) < a - 1 + (n - b - 1)
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/Turanpartition.lean | Turanpartition.move_change | [742, 1] | [760, 30] | have nba : n - a < n - b - 1 | case h.h
α : Type ?u.138093
inst✝¹ : Fintype α
inst✝ : DecidableEq α
b a n : ℕ
hb : b + 1 < a
hn : a + b ≤ n
ha : a - 1 + 1 = a
h2 : a ≤ n - b
hnb : n - b - 1 + 1 = n - b
ab : b < a - 1
⊢ b + (n - a) < a - 1 + (n - b - 1) | case nba
α : Type ?u.138093
inst✝¹ : Fintype α
inst✝ : DecidableEq α
b a n : ℕ
hb : b + 1 < a
hn : a + b ≤ n
ha : a - 1 + 1 = a
h2 : a ≤ n - b
hnb : n - b - 1 + 1 = n - b
ab : b < a - 1
⊢ n - a < n - b - 1
case h.h
α : Type ?u.138093
inst✝¹ : Fintype α
inst✝ : DecidableEq α
b a n : ℕ
hb : b + 1 < a
hn : a + b ≤ n
ha : a - 1 + 1 = a
h2 : a ≤ n - b
hnb : n - b - 1 + 1 = n - b
ab : b < a - 1
nba : n - a < n - b - 1
⊢ b + (n - a) < a - 1 + (n - b - 1) | Please generate a tactic in lean4 to solve the state.
STATE:
case h.h
α : Type ?u.138093
inst✝¹ : Fintype α
inst✝ : DecidableEq α
b a n : ℕ
hb : b + 1 < a
hn : a + b ≤ n
ha : a - 1 + 1 = a
h2 : a ≤ n - b
hnb : n - b - 1 + 1 = n - b
ab : b < a - 1
⊢ b + (n - a) < a - 1 + (n - b - 1)
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/Turanpartition.lean | Turanpartition.move_change | [742, 1] | [760, 30] | exact Nat.add_lt_add ab nba | case h.h
α : Type ?u.138093
inst✝¹ : Fintype α
inst✝ : DecidableEq α
b a n : ℕ
hb : b + 1 < a
hn : a + b ≤ n
ha : a - 1 + 1 = a
h2 : a ≤ n - b
hnb : n - b - 1 + 1 = n - b
ab : b < a - 1
nba : n - a < n - b - 1
⊢ b + (n - a) < a - 1 + (n - b - 1) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.h
α : Type ?u.138093
inst✝¹ : Fintype α
inst✝ : DecidableEq α
b a n : ℕ
hb : b + 1 < a
hn : a + b ≤ n
ha : a - 1 + 1 = a
h2 : a ≤ n - b
hnb : n - b - 1 + 1 = n - b
ab : b < a - 1
nba : n - a < n - b - 1
⊢ b + (n - a) < a - 1 + (n - b - 1)
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/Turanpartition.lean | Turanpartition.move_change | [742, 1] | [760, 30] | have nba' : n - a < n - (b + 1) | case nba
α : Type ?u.138093
inst✝¹ : Fintype α
inst✝ : DecidableEq α
b a n : ℕ
hb : b + 1 < a
hn : a + b ≤ n
ha : a - 1 + 1 = a
h2 : a ≤ n - b
hnb : n - b - 1 + 1 = n - b
ab : b < a - 1
⊢ n - a < n - b - 1 | case nba'
α : Type ?u.138093
inst✝¹ : Fintype α
inst✝ : DecidableEq α
b a n : ℕ
hb : b + 1 < a
hn : a + b ≤ n
ha : a - 1 + 1 = a
h2 : a ≤ n - b
hnb : n - b - 1 + 1 = n - b
ab : b < a - 1
⊢ n - a < n - (b + 1)
case nba
α : Type ?u.138093
inst✝¹ : Fintype α
inst✝ : DecidableEq α
b a n : ℕ
hb : b + 1 < a
hn : a + b ≤ n
ha : a - 1 + 1 = a
h2 : a ≤ n - b
hnb : n - b - 1 + 1 = n - b
ab : b < a - 1
nba' : n - a < n - (b + 1)
⊢ n - a < n - b - 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case nba
α : Type ?u.138093
inst✝¹ : Fintype α
inst✝ : DecidableEq α
b a n : ℕ
hb : b + 1 < a
hn : a + b ≤ n
ha : a - 1 + 1 = a
h2 : a ≤ n - b
hnb : n - b - 1 + 1 = n - b
ab : b < a - 1
⊢ n - a < n - b - 1
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/Turanpartition.lean | Turanpartition.move_change | [742, 1] | [760, 30] | rwa [tsub_add_eq_tsub_tsub] at nba' | case nba
α : Type ?u.138093
inst✝¹ : Fintype α
inst✝ : DecidableEq α
b a n : ℕ
hb : b + 1 < a
hn : a + b ≤ n
ha : a - 1 + 1 = a
h2 : a ≤ n - b
hnb : n - b - 1 + 1 = n - b
ab : b < a - 1
nba' : n - a < n - (b + 1)
⊢ n - a < n - b - 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case nba
α : Type ?u.138093
inst✝¹ : Fintype α
inst✝ : DecidableEq α
b a n : ℕ
hb : b + 1 < a
hn : a + b ≤ n
ha : a - 1 + 1 = a
h2 : a ≤ n - b
hnb : n - b - 1 + 1 = n - b
ab : b < a - 1
nba' : n - a < n - (b + 1)
⊢ n - a < n - b - 1
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/Turanpartition.lean | Turanpartition.move_change | [742, 1] | [760, 30] | rw [tsub_lt_tsub_iff_left_of_le <| h2.trans tsub_le_self ] | case nba'
α : Type ?u.138093
inst✝¹ : Fintype α
inst✝ : DecidableEq α
b a n : ℕ
hb : b + 1 < a
hn : a + b ≤ n
ha : a - 1 + 1 = a
h2 : a ≤ n - b
hnb : n - b - 1 + 1 = n - b
ab : b < a - 1
⊢ n - a < n - (b + 1) | case nba'
α : Type ?u.138093
inst✝¹ : Fintype α
inst✝ : DecidableEq α
b a n : ℕ
hb : b + 1 < a
hn : a + b ≤ n
ha : a - 1 + 1 = a
h2 : a ≤ n - b
hnb : n - b - 1 + 1 = n - b
ab : b < a - 1
⊢ b + 1 < a | Please generate a tactic in lean4 to solve the state.
STATE:
case nba'
α : Type ?u.138093
inst✝¹ : Fintype α
inst✝ : DecidableEq α
b a n : ℕ
hb : b + 1 < a
hn : a + b ≤ n
ha : a - 1 + 1 = a
h2 : a ≤ n - b
hnb : n - b - 1 + 1 = n - b
ab : b < a - 1
⊢ n - a < n - (b + 1)
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/Turanpartition.lean | Turanpartition.move_change | [742, 1] | [760, 30] | exact tsub_pos_iff_lt.1 <|tsub_pos_of_lt hb | case nba'
α : Type ?u.138093
inst✝¹ : Fintype α
inst✝ : DecidableEq α
b a n : ℕ
hb : b + 1 < a
hn : a + b ≤ n
ha : a - 1 + 1 = a
h2 : a ≤ n - b
hnb : n - b - 1 + 1 = n - b
ab : b < a - 1
⊢ b + 1 < a | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case nba'
α : Type ?u.138093
inst✝¹ : Fintype α
inst✝ : DecidableEq α
b a n : ℕ
hb : b + 1 < a
hn : a + b ≤ n
ha : a - 1 + 1 = a
h2 : a ≤ n - b
hnb : n - b - 1 + 1 = n - b
ab : b < a - 1
⊢ b + 1 < a
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/Fedges.lean | SimpleGraph.subgraph_edge_subset | [21, 1] | [23, 46] | exact Iff.symm edgeFinset_subset_edgeFinset | α : Type u_1
inst✝³ : Fintype α
inst✝² : DecidableEq α
G H : SimpleGraph α
inst✝¹ : DecidableRel G.Adj
inst✝ : DecidableRel H.Adj
⊢ G ≤ H ↔ edgeFinset G ⊆ edgeFinset H | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝³ : Fintype α
inst✝² : DecidableEq α
G H : SimpleGraph α
inst✝¹ : DecidableRel G.Adj
inst✝ : DecidableRel H.Adj
⊢ G ≤ H ↔ edgeFinset G ⊆ edgeFinset H
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/Fedges.lean | SimpleGraph.eq_imp_edges_card_eq | [30, 1] | [33, 28] | intro h | α : Type u_1
inst✝³ : Fintype α
inst✝² : DecidableEq α
G H : SimpleGraph α
inst✝¹ : DecidableRel G.Adj
inst✝ : DecidableRel H.Adj
⊢ G = H → card (edgeFinset G) = card (edgeFinset H) | α : Type u_1
inst✝³ : Fintype α
inst✝² : DecidableEq α
G H : SimpleGraph α
inst✝¹ : DecidableRel G.Adj
inst✝ : DecidableRel H.Adj
h : G = H
⊢ card (edgeFinset G) = card (edgeFinset H) | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝³ : Fintype α
inst✝² : DecidableEq α
G H : SimpleGraph α
inst✝¹ : DecidableRel G.Adj
inst✝ : DecidableRel H.Adj
⊢ G = H → card (edgeFinset G) = card (edgeFinset H)
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/Fedges.lean | SimpleGraph.eq_imp_edges_card_eq | [30, 1] | [33, 28] | rw [eq_iff_edges_eq.mp h] | α : Type u_1
inst✝³ : Fintype α
inst✝² : DecidableEq α
G H : SimpleGraph α
inst✝¹ : DecidableRel G.Adj
inst✝ : DecidableRel H.Adj
h : G = H
⊢ card (edgeFinset G) = card (edgeFinset H) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝³ : Fintype α
inst✝² : DecidableEq α
G H : SimpleGraph α
inst✝¹ : DecidableRel G.Adj
inst✝ : DecidableRel H.Adj
h : G = H
⊢ card (edgeFinset G) = card (edgeFinset H)
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/Fedges.lean | SimpleGraph.empty_iff_edge_empty | [43, 1] | [45, 39] | rw [← edgeFinset_inj,edgeFinset_bot] | α : Type u_1
inst✝³ : Fintype α
inst✝² : DecidableEq α
G H : SimpleGraph α
inst✝¹ : DecidableRel G.Adj
inst✝ : DecidableRel H.Adj
⊢ G = ⊥ ↔ edgeFinset G = ∅ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝³ : Fintype α
inst✝² : DecidableEq α
G H : SimpleGraph α
inst✝¹ : DecidableRel G.Adj
inst✝ : DecidableRel H.Adj
⊢ G = ⊥ ↔ edgeFinset G = ∅
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/Fedges.lean | SimpleGraph.edge_of_not_empty | [48, 1] | [51, 73] | contrapose | α : Type u_1
inst✝³ : Fintype α
inst✝² : DecidableEq α
G H : SimpleGraph α
inst✝¹ : DecidableRel G.Adj
inst✝ : DecidableRel H.Adj
⊢ G ≠ ⊥ → ∃ v w, v ≠ w ∧ Adj G v w | α : Type u_1
inst✝³ : Fintype α
inst✝² : DecidableEq α
G H : SimpleGraph α
inst✝¹ : DecidableRel G.Adj
inst✝ : DecidableRel H.Adj
⊢ (¬∃ v w, v ≠ w ∧ Adj G v w) → ¬G ≠ ⊥ | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝³ : Fintype α
inst✝² : DecidableEq α
G H : SimpleGraph α
inst✝¹ : DecidableRel G.Adj
inst✝ : DecidableRel H.Adj
⊢ G ≠ ⊥ → ∃ v w, v ≠ w ∧ Adj G v w
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/Fedges.lean | SimpleGraph.edge_of_not_empty | [48, 1] | [51, 73] | intro h | α : Type u_1
inst✝³ : Fintype α
inst✝² : DecidableEq α
G H : SimpleGraph α
inst✝¹ : DecidableRel G.Adj
inst✝ : DecidableRel H.Adj
⊢ (¬∃ v w, v ≠ w ∧ Adj G v w) → ¬G ≠ ⊥ | α : Type u_1
inst✝³ : Fintype α
inst✝² : DecidableEq α
G H : SimpleGraph α
inst✝¹ : DecidableRel G.Adj
inst✝ : DecidableRel H.Adj
h : ¬∃ v w, v ≠ w ∧ Adj G v w
⊢ ¬G ≠ ⊥ | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝³ : Fintype α
inst✝² : DecidableEq α
G H : SimpleGraph α
inst✝¹ : DecidableRel G.Adj
inst✝ : DecidableRel H.Adj
⊢ (¬∃ v w, v ≠ w ∧ Adj G v w) → ¬G ≠ ⊥
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/Fedges.lean | SimpleGraph.edge_of_not_empty | [48, 1] | [51, 73] | push_neg at h | α : Type u_1
inst✝³ : Fintype α
inst✝² : DecidableEq α
G H : SimpleGraph α
inst✝¹ : DecidableRel G.Adj
inst✝ : DecidableRel H.Adj
h : ¬∃ v w, v ≠ w ∧ Adj G v w
⊢ ¬G ≠ ⊥ | α : Type u_1
inst✝³ : Fintype α
inst✝² : DecidableEq α
G H : SimpleGraph α
inst✝¹ : DecidableRel G.Adj
inst✝ : DecidableRel H.Adj
h : ∀ (v w : α), v ≠ w → ¬Adj G v w
⊢ ¬G ≠ ⊥ | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝³ : Fintype α
inst✝² : DecidableEq α
G H : SimpleGraph α
inst✝¹ : DecidableRel G.Adj
inst✝ : DecidableRel H.Adj
h : ¬∃ v w, v ≠ w ∧ Adj G v w
⊢ ¬G ≠ ⊥
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/Fedges.lean | SimpleGraph.edge_of_not_empty | [48, 1] | [51, 73] | push_neg | α : Type u_1
inst✝³ : Fintype α
inst✝² : DecidableEq α
G H : SimpleGraph α
inst✝¹ : DecidableRel G.Adj
inst✝ : DecidableRel H.Adj
h : ∀ (v w : α), v ≠ w → ¬Adj G v w
⊢ ¬G ≠ ⊥ | α : Type u_1
inst✝³ : Fintype α
inst✝² : DecidableEq α
G H : SimpleGraph α
inst✝¹ : DecidableRel G.Adj
inst✝ : DecidableRel H.Adj
h : ∀ (v w : α), v ≠ w → ¬Adj G v w
⊢ G = ⊥ | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝³ : Fintype α
inst✝² : DecidableEq α
G H : SimpleGraph α
inst✝¹ : DecidableRel G.Adj
inst✝ : DecidableRel H.Adj
h : ∀ (v w : α), v ≠ w → ¬Adj G v w
⊢ ¬G ≠ ⊥
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/Fedges.lean | SimpleGraph.edge_of_not_empty | [48, 1] | [51, 73] | ext x x_1 | α : Type u_1
inst✝³ : Fintype α
inst✝² : DecidableEq α
G H : SimpleGraph α
inst✝¹ : DecidableRel G.Adj
inst✝ : DecidableRel H.Adj
h : ∀ (v w : α), v ≠ w → ¬Adj G v w
⊢ G = ⊥ | case Adj.h.h.a
α : Type u_1
inst✝³ : Fintype α
inst✝² : DecidableEq α
G H : SimpleGraph α
inst✝¹ : DecidableRel G.Adj
inst✝ : DecidableRel H.Adj
h : ∀ (v w : α), v ≠ w → ¬Adj G v w
x x_1 : α
⊢ Adj G x x_1 ↔ Adj ⊥ x x_1 | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝³ : Fintype α
inst✝² : DecidableEq α
G H : SimpleGraph α
inst✝¹ : DecidableRel G.Adj
inst✝ : DecidableRel H.Adj
h : ∀ (v w : α), v ≠ w → ¬Adj G v w
⊢ G = ⊥
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/Fedges.lean | SimpleGraph.edge_of_not_empty | [48, 1] | [51, 73] | rw [bot_adj] | case Adj.h.h.a
α : Type u_1
inst✝³ : Fintype α
inst✝² : DecidableEq α
G H : SimpleGraph α
inst✝¹ : DecidableRel G.Adj
inst✝ : DecidableRel H.Adj
h : ∀ (v w : α), v ≠ w → ¬Adj G v w
x x_1 : α
⊢ Adj G x x_1 ↔ Adj ⊥ x x_1 | case Adj.h.h.a
α : Type u_1
inst✝³ : Fintype α
inst✝² : DecidableEq α
G H : SimpleGraph α
inst✝¹ : DecidableRel G.Adj
inst✝ : DecidableRel H.Adj
h : ∀ (v w : α), v ≠ w → ¬Adj G v w
x x_1 : α
⊢ Adj G x x_1 ↔ False | Please generate a tactic in lean4 to solve the state.
STATE:
case Adj.h.h.a
α : Type u_1
inst✝³ : Fintype α
inst✝² : DecidableEq α
G H : SimpleGraph α
inst✝¹ : DecidableRel G.Adj
inst✝ : DecidableRel H.Adj
h : ∀ (v w : α), v ≠ w → ¬Adj G v w
x x_1 : α
⊢ Adj G x x_1 ↔ Adj ⊥ x x_1
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/Fedges.lean | SimpleGraph.edge_of_not_empty | [48, 1] | [51, 73] | specialize h x x_1 | case Adj.h.h.a
α : Type u_1
inst✝³ : Fintype α
inst✝² : DecidableEq α
G H : SimpleGraph α
inst✝¹ : DecidableRel G.Adj
inst✝ : DecidableRel H.Adj
h : ∀ (v w : α), v ≠ w → ¬Adj G v w
x x_1 : α
⊢ Adj G x x_1 ↔ False | case Adj.h.h.a
α : Type u_1
inst✝³ : Fintype α
inst✝² : DecidableEq α
G H : SimpleGraph α
inst✝¹ : DecidableRel G.Adj
inst✝ : DecidableRel H.Adj
x x_1 : α
h : x ≠ x_1 → ¬Adj G x x_1
⊢ Adj G x x_1 ↔ False | Please generate a tactic in lean4 to solve the state.
STATE:
case Adj.h.h.a
α : Type u_1
inst✝³ : Fintype α
inst✝² : DecidableEq α
G H : SimpleGraph α
inst✝¹ : DecidableRel G.Adj
inst✝ : DecidableRel H.Adj
h : ∀ (v w : α), v ≠ w → ¬Adj G v w
x x_1 : α
⊢ Adj G x x_1 ↔ False
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/Fedges.lean | SimpleGraph.edge_of_not_empty | [48, 1] | [51, 73] | by_cases h' : x = x_1 | case Adj.h.h.a
α : Type u_1
inst✝³ : Fintype α
inst✝² : DecidableEq α
G H : SimpleGraph α
inst✝¹ : DecidableRel G.Adj
inst✝ : DecidableRel H.Adj
x x_1 : α
h : x ≠ x_1 → ¬Adj G x x_1
⊢ Adj G x x_1 ↔ False | case pos
α : Type u_1
inst✝³ : Fintype α
inst✝² : DecidableEq α
G H : SimpleGraph α
inst✝¹ : DecidableRel G.Adj
inst✝ : DecidableRel H.Adj
x x_1 : α
h : x ≠ x_1 → ¬Adj G x x_1
h' : x = x_1
⊢ Adj G x x_1 ↔ False
case neg
α : Type u_1
inst✝³ : Fintype α
inst✝² : DecidableEq α
G H : SimpleGraph α
inst✝¹ : DecidableRel G.Adj
inst✝ : DecidableRel H.Adj
x x_1 : α
h : x ≠ x_1 → ¬Adj G x x_1
h' : ¬x = x_1
⊢ Adj G x x_1 ↔ False | Please generate a tactic in lean4 to solve the state.
STATE:
case Adj.h.h.a
α : Type u_1
inst✝³ : Fintype α
inst✝² : DecidableEq α
G H : SimpleGraph α
inst✝¹ : DecidableRel G.Adj
inst✝ : DecidableRel H.Adj
x x_1 : α
h : x ≠ x_1 → ¬Adj G x x_1
⊢ Adj G x x_1 ↔ False
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/Fedges.lean | SimpleGraph.edge_of_not_empty | [48, 1] | [51, 73] | simp only [h,h', G.irrefl] | case pos
α : Type u_1
inst✝³ : Fintype α
inst✝² : DecidableEq α
G H : SimpleGraph α
inst✝¹ : DecidableRel G.Adj
inst✝ : DecidableRel H.Adj
x x_1 : α
h : x ≠ x_1 → ¬Adj G x x_1
h' : x = x_1
⊢ Adj G x x_1 ↔ False
case neg
α : Type u_1
inst✝³ : Fintype α
inst✝² : DecidableEq α
G H : SimpleGraph α
inst✝¹ : DecidableRel G.Adj
inst✝ : DecidableRel H.Adj
x x_1 : α
h : x ≠ x_1 → ¬Adj G x x_1
h' : ¬x = x_1
⊢ Adj G x x_1 ↔ False | case neg
α : Type u_1
inst✝³ : Fintype α
inst✝² : DecidableEq α
G H : SimpleGraph α
inst✝¹ : DecidableRel G.Adj
inst✝ : DecidableRel H.Adj
x x_1 : α
h : x ≠ x_1 → ¬Adj G x x_1
h' : ¬x = x_1
⊢ Adj G x x_1 ↔ False | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
α : Type u_1
inst✝³ : Fintype α
inst✝² : DecidableEq α
G H : SimpleGraph α
inst✝¹ : DecidableRel G.Adj
inst✝ : DecidableRel H.Adj
x x_1 : α
h : x ≠ x_1 → ¬Adj G x x_1
h' : x = x_1
⊢ Adj G x x_1 ↔ False
case neg
α : Type u_1
inst✝³ : Fintype α
inst✝² : DecidableEq α
G H : SimpleGraph α
inst✝¹ : DecidableRel G.Adj
inst✝ : DecidableRel H.Adj
x x_1 : α
h : x ≠ x_1 → ¬Adj G x x_1
h' : ¬x = x_1
⊢ Adj G x x_1 ↔ False
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/Fedges.lean | SimpleGraph.edge_of_not_empty | [48, 1] | [51, 73] | have := h h' | case neg
α : Type u_1
inst✝³ : Fintype α
inst✝² : DecidableEq α
G H : SimpleGraph α
inst✝¹ : DecidableRel G.Adj
inst✝ : DecidableRel H.Adj
x x_1 : α
h : x ≠ x_1 → ¬Adj G x x_1
h' : ¬x = x_1
⊢ Adj G x x_1 ↔ False | case neg
α : Type u_1
inst✝³ : Fintype α
inst✝² : DecidableEq α
G H : SimpleGraph α
inst✝¹ : DecidableRel G.Adj
inst✝ : DecidableRel H.Adj
x x_1 : α
h : x ≠ x_1 → ¬Adj G x x_1
h' : ¬x = x_1
this : ¬Adj G x x_1
⊢ Adj G x x_1 ↔ False | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
α : Type u_1
inst✝³ : Fintype α
inst✝² : DecidableEq α
G H : SimpleGraph α
inst✝¹ : DecidableRel G.Adj
inst✝ : DecidableRel H.Adj
x x_1 : α
h : x ≠ x_1 → ¬Adj G x x_1
h' : ¬x = x_1
⊢ Adj G x x_1 ↔ False
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/Fedges.lean | SimpleGraph.edge_of_not_empty | [48, 1] | [51, 73] | tauto | case neg
α : Type u_1
inst✝³ : Fintype α
inst✝² : DecidableEq α
G H : SimpleGraph α
inst✝¹ : DecidableRel G.Adj
inst✝ : DecidableRel H.Adj
x x_1 : α
h : x ≠ x_1 → ¬Adj G x x_1
h' : ¬x = x_1
this : ¬Adj G x x_1
⊢ Adj G x x_1 ↔ False | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
α : Type u_1
inst✝³ : Fintype α
inst✝² : DecidableEq α
G H : SimpleGraph α
inst✝¹ : DecidableRel G.Adj
inst✝ : DecidableRel H.Adj
x x_1 : α
h : x ≠ x_1 → ¬Adj G x x_1
h' : ¬x = x_1
this : ¬Adj G x x_1
⊢ Adj G x x_1 ↔ False
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/Fedges.lean | SimpleGraph.two_cliqueFree_imp_empty | [54, 1] | [58, 45] | intro h | α : Type u_1
inst✝³ : Fintype α
inst✝² : DecidableEq α
G H : SimpleGraph α
inst✝¹ : DecidableRel G.Adj
inst✝ : DecidableRel H.Adj
⊢ CliqueFree G 2 → G = ⊥ | α : Type u_1
inst✝³ : Fintype α
inst✝² : DecidableEq α
G H : SimpleGraph α
inst✝¹ : DecidableRel G.Adj
inst✝ : DecidableRel H.Adj
h : CliqueFree G 2
⊢ G = ⊥ | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝³ : Fintype α
inst✝² : DecidableEq α
G H : SimpleGraph α
inst✝¹ : DecidableRel G.Adj
inst✝ : DecidableRel H.Adj
⊢ CliqueFree G 2 → G = ⊥
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/Fedges.lean | SimpleGraph.two_cliqueFree_imp_empty | [54, 1] | [58, 45] | contrapose h | α : Type u_1
inst✝³ : Fintype α
inst✝² : DecidableEq α
G H : SimpleGraph α
inst✝¹ : DecidableRel G.Adj
inst✝ : DecidableRel H.Adj
h : CliqueFree G 2
⊢ G = ⊥ | α : Type u_1
inst✝³ : Fintype α
inst✝² : DecidableEq α
G H : SimpleGraph α
inst✝¹ : DecidableRel G.Adj
inst✝ : DecidableRel H.Adj
h : ¬G = ⊥
⊢ ¬CliqueFree G 2 | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝³ : Fintype α
inst✝² : DecidableEq α
G H : SimpleGraph α
inst✝¹ : DecidableRel G.Adj
inst✝ : DecidableRel H.Adj
h : CliqueFree G 2
⊢ G = ⊥
TACTIC:
|
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