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/Multipartite.lean | SimpleGraph.mp_deg_sum | [129, 1] | [135, 36] | exact mp_deg ⟨hx, hv⟩ | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
M : MultiPart α
x : ℕ
hx : x ∈ range (M.t + 1)
v : α
hv : v ∈ MultiPart.P M x
⊢ degree (mp M) v = card (M.A \ MultiPart.P M x) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
M : MultiPart α
x : ℕ
hx : x ∈ range (M.t + 1)
v : α
hv : v ∈ MultiPart.P M x
⊢ degree (mp M) v = card (M.A \ MultiPart.P M x)
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/Multipartite.lean | SimpleGraph.mp_deg_sum_sq' | [138, 1] | [143, 66] | rw [mp_deg_sum M] | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
M : MultiPart α
⊢ ∑ v in M.A, degree (mp M) v + ∑ i in range (M.t + 1), card (MultiPart.P M i) ^ 2 = card M.A ^ 2 | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
M : MultiPart α
⊢ ∑ i in range (M.t + 1), card (MultiPart.P M i) * card (M.A \ MultiPart.P M i) +
∑ i in range (M.t + 1), card (MultiPart.P M i) ^ 2 =
card M.A ^ 2 | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
M : MultiPart α
⊢ ∑ v in M.A, degree (mp M) v + ∑ i in range (M.t + 1), card (MultiPart.P M i) ^ 2 = card M.A ^ 2
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/Multipartite.lean | SimpleGraph.mp_deg_sum_sq' | [138, 1] | [143, 66] | rw [pow_two] | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
M : MultiPart α
⊢ ∑ i in range (M.t + 1), card (MultiPart.P M i) * card (M.A \ MultiPart.P M i) +
∑ i in range (M.t + 1), card (MultiPart.P M i) ^ 2 =
card M.A ^ 2 | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
M : MultiPart α
⊢ ∑ i in range (M.t + 1), card (MultiPart.P M i) * card (M.A \ MultiPart.P M i) +
∑ i in range (M.t + 1), card (MultiPart.P M i) ^ 2 =
card M.A * card M.A | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
M : MultiPart α
⊢ ∑ i in range (M.t + 1), card (MultiPart.P M i) * card (M.A \ MultiPart.P M i) +
∑ i in range (M.t + 1), card (MultiPart.P M i) ^ 2 =
card M.A ^ 2
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/Multipartite.lean | SimpleGraph.mp_deg_sum_sq' | [138, 1] | [143, 66] | nth_rw 1 [card_uni] | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
M : MultiPart α
⊢ ∑ i in range (M.t + 1), card (MultiPart.P M i) * card (M.A \ MultiPart.P M i) +
∑ i in range (M.t + 1), card (MultiPart.P M i) ^ 2 =
card M.A * card M.A | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
M : MultiPart α
⊢ ∑ i in range (M.t + 1), card (MultiPart.P M i) * card (M.A \ MultiPart.P M i) +
∑ i in range (M.t + 1), card (MultiPart.P M i) ^ 2 =
(∑ i in range (M.t + 1), card (MultiPart.P M i)) * card M.A | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
M : MultiPart α
⊢ ∑ i in range (M.t + 1), card (MultiPart.P M i) * card (M.A \ MultiPart.P M i) +
∑ i in range (M.t + 1), card (MultiPart.P M i) ^ 2 =
card M.A * card M.A
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/Multipartite.lean | SimpleGraph.mp_deg_sum_sq' | [138, 1] | [143, 66] | rw [← sum_add_distrib] | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
M : MultiPart α
⊢ ∑ i in range (M.t + 1), card (MultiPart.P M i) * card (M.A \ MultiPart.P M i) +
∑ i in range (M.t + 1), card (MultiPart.P M i) ^ 2 =
(∑ i in range (M.t + 1), card (MultiPart.P M i)) * card M.A | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
M : MultiPart α
⊢ ∑ x in range (M.t + 1), (card (MultiPart.P M x) * card (M.A \ MultiPart.P M x) + card (MultiPart.P M x) ^ 2) =
(∑ i in range (M.t + 1), card (MultiPart.P M i)) * card M.A | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
M : MultiPart α
⊢ ∑ i in range (M.t + 1), card (MultiPart.P M i) * card (M.A \ MultiPart.P M i) +
∑ i in range (M.t + 1), card (MultiPart.P M i) ^ 2 =
(∑ i in range (M.t + 1), card (MultiPart.P M i)) * card M.A
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/Multipartite.lean | SimpleGraph.mp_deg_sum_sq' | [138, 1] | [143, 66] | rw [sum_mul] | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
M : MultiPart α
⊢ ∑ x in range (M.t + 1), (card (MultiPart.P M x) * card (M.A \ MultiPart.P M x) + card (MultiPart.P M x) ^ 2) =
(∑ i in range (M.t + 1), card (MultiPart.P M i)) * card M.A | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
M : MultiPart α
⊢ ∑ x in range (M.t + 1), (card (MultiPart.P M x) * card (M.A \ MultiPart.P M x) + card (MultiPart.P M x) ^ 2) =
∑ x in range (M.t + 1), card (MultiPart.P M x) * card M.A | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
M : MultiPart α
⊢ ∑ x in range (M.t + 1), (card (MultiPart.P M x) * card (M.A \ MultiPart.P M x) + card (MultiPart.P M x) ^ 2) =
(∑ i in range (M.t + 1), card (MultiPart.P M i)) * card M.A
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/Multipartite.lean | SimpleGraph.mp_deg_sum_sq' | [138, 1] | [143, 66] | refine' Finset.sum_congr rfl _ | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
M : MultiPart α
⊢ ∑ x in range (M.t + 1), (card (MultiPart.P M x) * card (M.A \ MultiPart.P M x) + card (MultiPart.P M x) ^ 2) =
∑ x in range (M.t + 1), card (MultiPart.P M x) * card M.A | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
M : MultiPart α
⊢ ∀ (x : ℕ),
x ∈ range (M.t + 1) →
card (MultiPart.P M x) * card (M.A \ MultiPart.P M x) + card (MultiPart.P M x) ^ 2 =
card (MultiPart.P M x) * card M.A | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
M : MultiPart α
⊢ ∑ x in range (M.t + 1), (card (MultiPart.P M x) * card (M.A \ MultiPart.P M x) + card (MultiPart.P M x) ^ 2) =
∑ x in range (M.t + 1), card (MultiPart.P M x) * card M.A
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/Multipartite.lean | SimpleGraph.mp_deg_sum_sq' | [138, 1] | [143, 66] | intro x hx | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
M : MultiPart α
⊢ ∀ (x : ℕ),
x ∈ range (M.t + 1) →
card (MultiPart.P M x) * card (M.A \ MultiPart.P M x) + card (MultiPart.P M x) ^ 2 =
card (MultiPart.P M x) * card M.A | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
M : MultiPart α
x : ℕ
hx : x ∈ range (M.t + 1)
⊢ card (MultiPart.P M x) * card (M.A \ MultiPart.P M x) + card (MultiPart.P M x) ^ 2 = card (MultiPart.P M x) * card M.A | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
M : MultiPart α
⊢ ∀ (x : ℕ),
x ∈ range (M.t + 1) →
card (MultiPart.P M x) * card (M.A \ MultiPart.P M x) + card (MultiPart.P M x) ^ 2 =
card (MultiPart.P M x) * card M.A
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/Multipartite.lean | SimpleGraph.mp_deg_sum_sq' | [138, 1] | [143, 66] | rw [pow_two] | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
M : MultiPart α
x : ℕ
hx : x ∈ range (M.t + 1)
⊢ card (MultiPart.P M x) * card (M.A \ MultiPart.P M x) + card (MultiPart.P M x) ^ 2 = card (MultiPart.P M x) * card M.A | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
M : MultiPart α
x : ℕ
hx : x ∈ range (M.t + 1)
⊢ card (MultiPart.P M x) * card (M.A \ MultiPart.P M x) + card (MultiPart.P M x) * card (MultiPart.P M x) =
card (MultiPart.P M x) * card M.A | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
M : MultiPart α
x : ℕ
hx : x ∈ range (M.t + 1)
⊢ card (MultiPart.P M x) * card (M.A \ MultiPart.P M x) + card (MultiPart.P M x) ^ 2 = card (MultiPart.P M x) * card M.A
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/Multipartite.lean | SimpleGraph.mp_deg_sum_sq' | [138, 1] | [143, 66] | rw [← mul_add] | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
M : MultiPart α
x : ℕ
hx : x ∈ range (M.t + 1)
⊢ card (MultiPart.P M x) * card (M.A \ MultiPart.P M x) + card (MultiPart.P M x) * card (MultiPart.P M x) =
card (MultiPart.P M x) * card M.A | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
M : MultiPart α
x : ℕ
hx : x ∈ range (M.t + 1)
⊢ card (MultiPart.P M x) * (card (M.A \ MultiPart.P M x) + card (MultiPart.P M x)) = card (MultiPart.P M x) * card M.A | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
M : MultiPart α
x : ℕ
hx : x ∈ range (M.t + 1)
⊢ card (MultiPart.P M x) * card (M.A \ MultiPart.P M x) + card (MultiPart.P M x) * card (MultiPart.P M x) =
card (MultiPart.P M x) * card M.A
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/Multipartite.lean | SimpleGraph.mp_deg_sum_sq' | [138, 1] | [143, 66] | rw [card_part_uni hx] | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
M : MultiPart α
x : ℕ
hx : x ∈ range (M.t + 1)
⊢ card (MultiPart.P M x) * (card (M.A \ MultiPart.P M x) + card (MultiPart.P M x)) = card (MultiPart.P M x) * card M.A | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
M : MultiPart α
x : ℕ
hx : x ∈ range (M.t + 1)
⊢ card (MultiPart.P M x) * (card (M.A \ MultiPart.P M x) + card (MultiPart.P M x)) = card (MultiPart.P M x) * card M.A
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/Multipartite.lean | SimpleGraph.turanPartition_deg_sum_eq | [152, 1] | [159, 56] | intro hA ht iM iN | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
M N : MultiPart α
⊢ M.A = N.A → M.t = N.t → TuranPartition M → TuranPartition N → mpDsum M = mpDsum N | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
M N : MultiPart α
hA : M.A = N.A
ht : M.t = N.t
iM : TuranPartition M
iN : TuranPartition N
⊢ mpDsum M = mpDsum N | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
M N : MultiPart α
⊢ M.A = N.A → M.t = N.t → TuranPartition M → TuranPartition N → mpDsum M = mpDsum N
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/Multipartite.lean | SimpleGraph.turanPartition_deg_sum_eq | [152, 1] | [159, 56] | unfold mpDsum | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
M N : MultiPart α
hA : M.A = N.A
ht : M.t = N.t
iM : TuranPartition M
iN : TuranPartition N
⊢ mpDsum M = mpDsum N | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
M N : MultiPart α
hA : M.A = N.A
ht : M.t = N.t
iM : TuranPartition M
iN : TuranPartition N
⊢ ∑ v in M.A, degree (mp M) v = ∑ v in N.A, degree (mp N) v | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
M N : MultiPart α
hA : M.A = N.A
ht : M.t = N.t
iM : TuranPartition M
iN : TuranPartition N
⊢ mpDsum M = mpDsum N
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/Multipartite.lean | SimpleGraph.turanPartition_deg_sum_eq | [152, 1] | [159, 56] | rw [mp_deg_sum_sq, mp_deg_sum_sq, hA] | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
M N : MultiPart α
hA : M.A = N.A
ht : M.t = N.t
iM : TuranPartition M
iN : TuranPartition N
⊢ ∑ v in M.A, degree (mp M) v = ∑ v in N.A, degree (mp N) v | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
M N : MultiPart α
hA : M.A = N.A
ht : M.t = N.t
iM : TuranPartition M
iN : TuranPartition N
⊢ card N.A ^ 2 - ∑ i in range (M.t + 1), card (MultiPart.P M i) ^ 2 =
card N.A ^ 2 - ∑ i in range (N.t + 1), card (MultiPart.P N i) ^ 2 | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
M N : MultiPart α
hA : M.A = N.A
ht : M.t = N.t
iM : TuranPartition M
iN : TuranPartition N
⊢ ∑ v in M.A, degree (mp M) v = ∑ v in N.A, degree (mp N) v
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/Multipartite.lean | SimpleGraph.turanPartition_deg_sum_eq | [152, 1] | [159, 56] | rw [turanPartition_iff_not_moveable, Moveable, Classical.not_not] at * | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
M N : MultiPart α
hA : M.A = N.A
ht : M.t = N.t
iM : TuranPartition M
iN : TuranPartition N
⊢ card N.A ^ 2 - ∑ i in range (M.t + 1), card (MultiPart.P M i) ^ 2 =
card N.A ^ 2 - ∑ i in range (N.t + 1), card (MultiPart.P N i) ^ 2 | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
M N : MultiPart α
hA : M.A = N.A
ht : M.t = N.t
iM : Balanced M.t fun i => card (MultiPart.P M i)
iN : Balanced N.t fun i => card (MultiPart.P N i)
⊢ card N.A ^ 2 - ∑ i in range (M.t + 1), card (MultiPart.P M i) ^ 2 =
card N.A ^ 2 - ∑ i in range (N.t + 1), card (MultiPart.P N i) ^ 2 | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
M N : MultiPart α
hA : M.A = N.A
ht : M.t = N.t
iM : TuranPartition M
iN : TuranPartition N
⊢ card N.A ^ 2 - ∑ i in range (M.t + 1), card (MultiPart.P M i) ^ 2 =
card N.A ^ 2 - ∑ i in range (N.t + 1), card (MultiPart.P N i) ^ 2
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/Multipartite.lean | SimpleGraph.turanPartition_deg_sum_eq | [152, 1] | [159, 56] | apply congr_arg _ | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
M N : MultiPart α
hA : M.A = N.A
ht : M.t = N.t
iM : Balanced M.t fun i => card (MultiPart.P M i)
iN : Balanced N.t fun i => card (MultiPart.P N i)
⊢ card N.A ^ 2 - ∑ i in range (M.t + 1), card (MultiPart.P M i) ^ 2 =
card N.A ^ 2 - ∑ i in range (N.t + 1), card (MultiPart.P N i) ^ 2 | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
M N : MultiPart α
hA : M.A = N.A
ht : M.t = N.t
iM : Balanced M.t fun i => card (MultiPart.P M i)
iN : Balanced N.t fun i => card (MultiPart.P N i)
⊢ ∑ i in range (M.t + 1), card (MultiPart.P M i) ^ 2 = ∑ i in range (N.t + 1), card (MultiPart.P N i) ^ 2 | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
M N : MultiPart α
hA : M.A = N.A
ht : M.t = N.t
iM : Balanced M.t fun i => card (MultiPart.P M i)
iN : Balanced N.t fun i => card (MultiPart.P N i)
⊢ card N.A ^ 2 - ∑ i in range (M.t + 1), card (MultiPart.P M i) ^ 2 =
card N.A ^ 2 - ∑ i in range (N.t + 1), card (MultiPart.P N i) ^ 2
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/Multipartite.lean | SimpleGraph.turanPartition_deg_sum_eq | [152, 1] | [159, 56] | have hN := turan_bal iN | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
M N : MultiPart α
hA : M.A = N.A
ht : M.t = N.t
iM : Balanced M.t fun i => card (MultiPart.P M i)
iN : Balanced N.t fun i => card (MultiPart.P N i)
⊢ ∑ i in range (M.t + 1), card (MultiPart.P M i) ^ 2 = ∑ i in range (N.t + 1), card (MultiPart.P N i) ^ 2 | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
M N : MultiPart α
hA : M.A = N.A
ht : M.t = N.t
iM : Balanced M.t fun i => card (MultiPart.P M i)
iN : Balanced N.t fun i => card (MultiPart.P N i)
hN : Bal N.t (card N.A) fun i => card (MultiPart.P N i)
⊢ ∑ i in range (M.t + 1), card (MultiPart.P M i) ^ 2 = ∑ i in range (N.t + 1), card (MultiPart.P N i) ^ 2 | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
M N : MultiPart α
hA : M.A = N.A
ht : M.t = N.t
iM : Balanced M.t fun i => card (MultiPart.P M i)
iN : Balanced N.t fun i => card (MultiPart.P N i)
⊢ ∑ i in range (M.t + 1), card (MultiPart.P M i) ^ 2 = ∑ i in range (N.t + 1), card (MultiPart.P N i) ^ 2
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/Multipartite.lean | SimpleGraph.turanPartition_deg_sum_eq | [152, 1] | [159, 56] | rw [← ht, ← hA] at hN | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
M N : MultiPart α
hA : M.A = N.A
ht : M.t = N.t
iM : Balanced M.t fun i => card (MultiPart.P M i)
iN : Balanced N.t fun i => card (MultiPart.P N i)
hN : Bal N.t (card N.A) fun i => card (MultiPart.P N i)
⊢ ∑ i in range (M.t + 1), card (MultiPart.P M i) ^ 2 = ∑ i in range (N.t + 1), card (MultiPart.P N i) ^ 2 | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
M N : MultiPart α
hA : M.A = N.A
ht : M.t = N.t
iM : Balanced M.t fun i => card (MultiPart.P M i)
iN : Balanced N.t fun i => card (MultiPart.P N i)
hN : Bal M.t (card M.A) fun i => card (MultiPart.P N i)
⊢ ∑ i in range (M.t + 1), card (MultiPart.P M i) ^ 2 = ∑ i in range (N.t + 1), card (MultiPart.P N i) ^ 2 | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
M N : MultiPart α
hA : M.A = N.A
ht : M.t = N.t
iM : Balanced M.t fun i => card (MultiPart.P M i)
iN : Balanced N.t fun i => card (MultiPart.P N i)
hN : Bal N.t (card N.A) fun i => card (MultiPart.P N i)
⊢ ∑ i in range (M.t + 1), card (MultiPart.P M i) ^ 2 = ∑ i in range (N.t + 1), card (MultiPart.P N i) ^ 2
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/Multipartite.lean | SimpleGraph.turanPartition_deg_sum_eq | [152, 1] | [159, 56] | have := bal_turan_help' (turan_bal iM) hN | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
M N : MultiPart α
hA : M.A = N.A
ht : M.t = N.t
iM : Balanced M.t fun i => card (MultiPart.P M i)
iN : Balanced N.t fun i => card (MultiPart.P N i)
hN : Bal M.t (card M.A) fun i => card (MultiPart.P N i)
⊢ ∑ i in range (M.t + 1), card (MultiPart.P M i) ^ 2 = ∑ i in range (N.t + 1), card (MultiPart.P N i) ^ 2 | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
M N : MultiPart α
hA : M.A = N.A
ht : M.t = N.t
iM : Balanced M.t fun i => card (MultiPart.P M i)
iN : Balanced N.t fun i => card (MultiPart.P N i)
hN : Bal M.t (card M.A) fun i => card (MultiPart.P N i)
this : (sumSq M.t fun i => card (MultiPart.P M i)) = sumSq M.t fun i => card (MultiPart.P N i)
⊢ ∑ i in range (M.t + 1), card (MultiPart.P M i) ^ 2 = ∑ i in range (N.t + 1), card (MultiPart.P N i) ^ 2 | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
M N : MultiPart α
hA : M.A = N.A
ht : M.t = N.t
iM : Balanced M.t fun i => card (MultiPart.P M i)
iN : Balanced N.t fun i => card (MultiPart.P N i)
hN : Bal M.t (card M.A) fun i => card (MultiPart.P N i)
⊢ ∑ i in range (M.t + 1), card (MultiPart.P M i) ^ 2 = ∑ i in range (N.t + 1), card (MultiPart.P N i) ^ 2
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/Multipartite.lean | SimpleGraph.turanPartition_deg_sum_eq | [152, 1] | [159, 56] | rwa [← ht] | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
M N : MultiPart α
hA : M.A = N.A
ht : M.t = N.t
iM : Balanced M.t fun i => card (MultiPart.P M i)
iN : Balanced N.t fun i => card (MultiPart.P N i)
hN : Bal M.t (card M.A) fun i => card (MultiPart.P N i)
this : (sumSq M.t fun i => card (MultiPart.P M i)) = sumSq M.t fun i => card (MultiPart.P N i)
⊢ ∑ i in range (M.t + 1), card (MultiPart.P M i) ^ 2 = ∑ i in range (N.t + 1), card (MultiPart.P N i) ^ 2 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
M N : MultiPart α
hA : M.A = N.A
ht : M.t = N.t
iM : Balanced M.t fun i => card (MultiPart.P M i)
iN : Balanced N.t fun i => card (MultiPart.P N i)
hN : Bal M.t (card M.A) fun i => card (MultiPart.P N i)
this : (sumSq M.t fun i => card (MultiPart.P M i)) = sumSq M.t fun i => card (MultiPart.P N i)
⊢ ∑ i in range (M.t + 1), card (MultiPart.P M i) ^ 2 = ∑ i in range (N.t + 1), card (MultiPart.P N i) ^ 2
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/Multipartite.lean | SimpleGraph.mp_deg_sum_move_help | [162, 1] | [180, 18] | rw [move_Pcard hvi hj hvi.1, move_Pcard hvi hj hj.1, move_Pcard_sdiff hvi hj hvi.1,
move_Pcard_sdiff hvi hj hj.1] | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i : ℕ
v : α
j : ℕ
M : MultiPart α
hvi : i ∈ range (M.t + 1) ∧ v ∈ MultiPart.P M i
hj : j ∈ range (M.t + 1) ∧ j ≠ i
hc : card (MultiPart.P M j) + 1 < card (MultiPart.P M i)
⊢ card (MultiPart.P M i) * card (M.A \ MultiPart.P M i) + card (MultiPart.P M j) * card (M.A \ MultiPart.P M j) <
card (MultiPart.P (move M hvi hj) i) * card ((move M hvi hj).A \ MultiPart.P (move M hvi hj) i) +
card (MultiPart.P (move M hvi hj) j) * card ((move M hvi hj).A \ MultiPart.P (move M hvi hj) j) | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i : ℕ
v : α
j : ℕ
M : MultiPart α
hvi : i ∈ range (M.t + 1) ∧ v ∈ MultiPart.P M i
hj : j ∈ range (M.t + 1) ∧ j ≠ i
hc : card (MultiPart.P M j) + 1 < card (MultiPart.P M i)
⊢ card (MultiPart.P M i) * card (M.A \ MultiPart.P M i) + card (MultiPart.P M j) * card (M.A \ MultiPart.P M j) <
((if i ≠ i ∧ i ≠ j then card (MultiPart.P M i)
else if i = i then card (MultiPart.P M i) - 1 else card (MultiPart.P M j) + 1) *
if i ≠ i ∧ i ≠ j then card (M.A \ MultiPart.P M i)
else if i = i then card (M.A \ MultiPart.P M i) + 1 else card (M.A \ MultiPart.P M j) - 1) +
(if j ≠ i ∧ j ≠ j then card (MultiPart.P M j)
else if j = i then card (MultiPart.P M i) - 1 else card (MultiPart.P M j) + 1) *
if j ≠ i ∧ j ≠ j then card (M.A \ MultiPart.P M j)
else if j = 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 : ℕ
M : MultiPart α
hvi : i ∈ range (M.t + 1) ∧ v ∈ MultiPart.P M i
hj : j ∈ range (M.t + 1) ∧ j ≠ i
hc : card (MultiPart.P M j) + 1 < card (MultiPart.P M i)
⊢ card (MultiPart.P M i) * card (M.A \ MultiPart.P M i) + card (MultiPart.P M j) * card (M.A \ MultiPart.P M j) <
card (MultiPart.P (move M hvi hj) i) * card ((move M hvi hj).A \ MultiPart.P (move M hvi hj) i) +
card (MultiPart.P (move M hvi hj) j) * card ((move M hvi hj).A \ MultiPart.P (move M hvi hj) j)
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/Multipartite.lean | SimpleGraph.mp_deg_sum_move_help | [162, 1] | [180, 18] | split_ifs with h h_1 h_2 h_3 h_4 h_5 h_6 h_7 | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i : ℕ
v : α
j : ℕ
M : MultiPart α
hvi : i ∈ range (M.t + 1) ∧ v ∈ MultiPart.P M i
hj : j ∈ range (M.t + 1) ∧ j ≠ i
hc : card (MultiPart.P M j) + 1 < card (MultiPart.P M i)
⊢ card (MultiPart.P M i) * card (M.A \ MultiPart.P M i) + card (MultiPart.P M j) * card (M.A \ MultiPart.P M j) <
((if i ≠ i ∧ i ≠ j then card (MultiPart.P M i)
else if i = i then card (MultiPart.P M i) - 1 else card (MultiPart.P M j) + 1) *
if i ≠ i ∧ i ≠ j then card (M.A \ MultiPart.P M i)
else if i = i then card (M.A \ MultiPart.P M i) + 1 else card (M.A \ MultiPart.P M j) - 1) +
(if j ≠ i ∧ j ≠ j then card (MultiPart.P M j)
else if j = i then card (MultiPart.P M i) - 1 else card (MultiPart.P M j) + 1) *
if j ≠ i ∧ j ≠ j then card (M.A \ MultiPart.P M j)
else if j = 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 : ℕ
M : MultiPart α
hvi : i ∈ range (M.t + 1) ∧ v ∈ MultiPart.P M i
hj : j ∈ range (M.t + 1) ∧ j ≠ i
hc : card (MultiPart.P M j) + 1 < card (MultiPart.P M i)
h : i ≠ i ∧ i ≠ j
h_1 : j ≠ i ∧ j ≠ j
⊢ card (MultiPart.P M i) * card (M.A \ MultiPart.P M i) + card (MultiPart.P M j) * card (M.A \ MultiPart.P M j) <
card (MultiPart.P M i) * card (M.A \ MultiPart.P M i) + card (MultiPart.P M j) * card (M.A \ MultiPart.P M j)
case pos
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i : ℕ
v : α
j : ℕ
M : MultiPart α
hvi : i ∈ range (M.t + 1) ∧ v ∈ MultiPart.P M i
hj : j ∈ range (M.t + 1) ∧ j ≠ i
hc : card (MultiPart.P M j) + 1 < card (MultiPart.P M i)
h : i ≠ i ∧ i ≠ j
h_1 : ¬(j ≠ i ∧ j ≠ j)
h_2 : j = i
⊢ card (MultiPart.P M i) * card (M.A \ MultiPart.P M i) + card (MultiPart.P M j) * card (M.A \ MultiPart.P M j) <
card (MultiPart.P M i) * card (M.A \ MultiPart.P M i) +
(card (MultiPart.P M i) - 1) * (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 : ℕ
M : MultiPart α
hvi : i ∈ range (M.t + 1) ∧ v ∈ MultiPart.P M i
hj : j ∈ range (M.t + 1) ∧ j ≠ i
hc : card (MultiPart.P M j) + 1 < card (MultiPart.P M i)
h : i ≠ i ∧ i ≠ j
h_1 : ¬(j ≠ i ∧ j ≠ j)
h_2 : ¬j = i
⊢ card (MultiPart.P M i) * card (M.A \ MultiPart.P M i) + card (MultiPart.P M j) * card (M.A \ MultiPart.P M j) <
card (MultiPart.P M i) * card (M.A \ MultiPart.P M i) +
(card (MultiPart.P M j) + 1) * (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 : ℕ
M : MultiPart α
hvi : i ∈ range (M.t + 1) ∧ v ∈ MultiPart.P M i
hj : j ∈ range (M.t + 1) ∧ j ≠ i
hc : card (MultiPart.P M j) + 1 < card (MultiPart.P M i)
h : ¬(i ≠ i ∧ i ≠ j)
h_3 : i = i
h_4 : j ≠ i ∧ j ≠ j
⊢ card (MultiPart.P M i) * card (M.A \ MultiPart.P M i) + card (MultiPart.P M j) * card (M.A \ MultiPart.P M j) <
(card (MultiPart.P M i) - 1) * (card (M.A \ MultiPart.P M i) + 1) +
card (MultiPart.P M j) * card (M.A \ MultiPart.P M j)
case pos
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i : ℕ
v : α
j : ℕ
M : MultiPart α
hvi : i ∈ range (M.t + 1) ∧ v ∈ MultiPart.P M i
hj : j ∈ range (M.t + 1) ∧ j ≠ i
hc : card (MultiPart.P M j) + 1 < card (MultiPart.P M i)
h : ¬(i ≠ i ∧ i ≠ j)
h_3 : i = i
h_4 : ¬(j ≠ i ∧ j ≠ j)
h_5 : j = i
⊢ card (MultiPart.P M i) * card (M.A \ MultiPart.P M i) + card (MultiPart.P M j) * card (M.A \ MultiPart.P M j) <
(card (MultiPart.P M i) - 1) * (card (M.A \ MultiPart.P M i) + 1) +
(card (MultiPart.P M i) - 1) * (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 : ℕ
M : MultiPart α
hvi : i ∈ range (M.t + 1) ∧ v ∈ MultiPart.P M i
hj : j ∈ range (M.t + 1) ∧ j ≠ i
hc : card (MultiPart.P M j) + 1 < card (MultiPart.P M i)
h : ¬(i ≠ i ∧ i ≠ j)
h_3 : i = i
h_4 : ¬(j ≠ i ∧ j ≠ j)
h_5 : ¬j = i
⊢ card (MultiPart.P M i) * card (M.A \ MultiPart.P M i) + card (MultiPart.P M j) * card (M.A \ MultiPart.P M j) <
(card (MultiPart.P M i) - 1) * (card (M.A \ MultiPart.P M i) + 1) +
(card (MultiPart.P M j) + 1) * (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 : ℕ
M : MultiPart α
hvi : i ∈ range (M.t + 1) ∧ v ∈ MultiPart.P M i
hj : j ∈ range (M.t + 1) ∧ j ≠ i
hc : card (MultiPart.P M j) + 1 < card (MultiPart.P M i)
h : ¬(i ≠ i ∧ i ≠ j)
h_3 : ¬i = i
h_6 : j ≠ i ∧ j ≠ j
⊢ card (MultiPart.P M i) * card (M.A \ MultiPart.P M i) + card (MultiPart.P M j) * card (M.A \ MultiPart.P M j) <
(card (MultiPart.P M j) + 1) * (card (M.A \ MultiPart.P M j) - 1) +
card (MultiPart.P M j) * card (M.A \ MultiPart.P M j)
case pos
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i : ℕ
v : α
j : ℕ
M : MultiPart α
hvi : i ∈ range (M.t + 1) ∧ v ∈ MultiPart.P M i
hj : j ∈ range (M.t + 1) ∧ j ≠ i
hc : card (MultiPart.P M j) + 1 < card (MultiPart.P M i)
h : ¬(i ≠ i ∧ i ≠ j)
h_3 : ¬i = i
h_6 : ¬(j ≠ i ∧ j ≠ j)
h_7 : j = i
⊢ card (MultiPart.P M i) * card (M.A \ MultiPart.P M i) + card (MultiPart.P M j) * card (M.A \ MultiPart.P M j) <
(card (MultiPart.P M j) + 1) * (card (M.A \ MultiPart.P M j) - 1) +
(card (MultiPart.P M i) - 1) * (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 : ℕ
M : MultiPart α
hvi : i ∈ range (M.t + 1) ∧ v ∈ MultiPart.P M i
hj : j ∈ range (M.t + 1) ∧ j ≠ i
hc : card (MultiPart.P M j) + 1 < card (MultiPart.P M i)
h : ¬(i ≠ i ∧ i ≠ j)
h_3 : ¬i = i
h_6 : ¬(j ≠ i ∧ j ≠ j)
h_7 : ¬j = i
⊢ card (MultiPart.P M i) * card (M.A \ MultiPart.P M i) + card (MultiPart.P M j) * card (M.A \ MultiPart.P M j) <
(card (MultiPart.P M j) + 1) * (card (M.A \ MultiPart.P M j) - 1) +
(card (MultiPart.P M j) + 1) * (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 : ℕ
M : MultiPart α
hvi : i ∈ range (M.t + 1) ∧ v ∈ MultiPart.P M i
hj : j ∈ range (M.t + 1) ∧ j ≠ i
hc : card (MultiPart.P M j) + 1 < card (MultiPart.P M i)
⊢ card (MultiPart.P M i) * card (M.A \ MultiPart.P M i) + card (MultiPart.P M j) * card (M.A \ MultiPart.P M j) <
((if i ≠ i ∧ i ≠ j then card (MultiPart.P M i)
else if i = i then card (MultiPart.P M i) - 1 else card (MultiPart.P M j) + 1) *
if i ≠ i ∧ i ≠ j then card (M.A \ MultiPart.P M i)
else if i = i then card (M.A \ MultiPart.P M i) + 1 else card (M.A \ MultiPart.P M j) - 1) +
(if j ≠ i ∧ j ≠ j then card (MultiPart.P M j)
else if j = i then card (MultiPart.P M i) - 1 else card (MultiPart.P M j) + 1) *
if j ≠ i ∧ j ≠ j then card (M.A \ MultiPart.P M j)
else if j = 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/Multipartite.lean | SimpleGraph.mp_deg_sum_move_help | [162, 1] | [180, 18] | exfalso | case pos
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i : ℕ
v : α
j : ℕ
M : MultiPart α
hvi : i ∈ range (M.t + 1) ∧ v ∈ MultiPart.P M i
hj : j ∈ range (M.t + 1) ∧ j ≠ i
hc : card (MultiPart.P M j) + 1 < card (MultiPart.P M i)
h : i ≠ i ∧ i ≠ j
h_1 : j ≠ i ∧ j ≠ j
⊢ card (MultiPart.P M i) * card (M.A \ MultiPart.P M i) + card (MultiPart.P M j) * card (M.A \ MultiPart.P M j) <
card (MultiPart.P M i) * card (M.A \ MultiPart.P M i) + card (MultiPart.P M j) * card (M.A \ MultiPart.P M j) | case pos.h
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i : ℕ
v : α
j : ℕ
M : MultiPart α
hvi : i ∈ range (M.t + 1) ∧ v ∈ MultiPart.P M i
hj : j ∈ range (M.t + 1) ∧ j ≠ i
hc : card (MultiPart.P M j) + 1 < card (MultiPart.P M i)
h : i ≠ i ∧ i ≠ j
h_1 : j ≠ i ∧ j ≠ j
⊢ False | 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 : ℕ
M : MultiPart α
hvi : i ∈ range (M.t + 1) ∧ v ∈ MultiPart.P M i
hj : j ∈ range (M.t + 1) ∧ j ≠ i
hc : card (MultiPart.P M j) + 1 < card (MultiPart.P M i)
h : i ≠ i ∧ i ≠ j
h_1 : j ≠ i ∧ j ≠ j
⊢ card (MultiPart.P M i) * card (M.A \ MultiPart.P M i) + card (MultiPart.P M j) * card (M.A \ MultiPart.P M j) <
card (MultiPart.P M i) * card (M.A \ MultiPart.P M i) + card (MultiPart.P M j) * card (M.A \ MultiPart.P M j)
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/Multipartite.lean | SimpleGraph.mp_deg_sum_move_help | [162, 1] | [180, 18] | exact h.1 rfl | case pos.h
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i : ℕ
v : α
j : ℕ
M : MultiPart α
hvi : i ∈ range (M.t + 1) ∧ v ∈ MultiPart.P M i
hj : j ∈ range (M.t + 1) ∧ j ≠ i
hc : card (MultiPart.P M j) + 1 < card (MultiPart.P M i)
h : i ≠ i ∧ i ≠ j
h_1 : j ≠ i ∧ j ≠ j
⊢ False | 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
i : ℕ
v : α
j : ℕ
M : MultiPart α
hvi : i ∈ range (M.t + 1) ∧ v ∈ MultiPart.P M i
hj : j ∈ range (M.t + 1) ∧ j ≠ i
hc : card (MultiPart.P M j) + 1 < card (MultiPart.P M i)
h : i ≠ i ∧ i ≠ j
h_1 : j ≠ i ∧ j ≠ j
⊢ False
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/Multipartite.lean | SimpleGraph.mp_deg_sum_move_help | [162, 1] | [180, 18] | exfalso | case pos
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i : ℕ
v : α
j : ℕ
M : MultiPart α
hvi : i ∈ range (M.t + 1) ∧ v ∈ MultiPart.P M i
hj : j ∈ range (M.t + 1) ∧ j ≠ i
hc : card (MultiPart.P M j) + 1 < card (MultiPart.P M i)
h : i ≠ i ∧ i ≠ j
h_1 : ¬(j ≠ i ∧ j ≠ j)
h_2 : j = i
⊢ card (MultiPart.P M i) * card (M.A \ MultiPart.P M i) + card (MultiPart.P M j) * card (M.A \ MultiPart.P M j) <
card (MultiPart.P M i) * card (M.A \ MultiPart.P M i) +
(card (MultiPart.P M i) - 1) * (card (M.A \ MultiPart.P M i) + 1) | case pos.h
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i : ℕ
v : α
j : ℕ
M : MultiPart α
hvi : i ∈ range (M.t + 1) ∧ v ∈ MultiPart.P M i
hj : j ∈ range (M.t + 1) ∧ j ≠ i
hc : card (MultiPart.P M j) + 1 < card (MultiPart.P M i)
h : i ≠ i ∧ i ≠ j
h_1 : ¬(j ≠ i ∧ j ≠ j)
h_2 : j = i
⊢ False | 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 : ℕ
M : MultiPart α
hvi : i ∈ range (M.t + 1) ∧ v ∈ MultiPart.P M i
hj : j ∈ range (M.t + 1) ∧ j ≠ i
hc : card (MultiPart.P M j) + 1 < card (MultiPart.P M i)
h : i ≠ i ∧ i ≠ j
h_1 : ¬(j ≠ i ∧ j ≠ j)
h_2 : j = i
⊢ card (MultiPart.P M i) * card (M.A \ MultiPart.P M i) + card (MultiPart.P M j) * card (M.A \ MultiPart.P M j) <
card (MultiPart.P M i) * card (M.A \ MultiPart.P M i) +
(card (MultiPart.P M i) - 1) * (card (M.A \ MultiPart.P M i) + 1)
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/Multipartite.lean | SimpleGraph.mp_deg_sum_move_help | [162, 1] | [180, 18] | exact h.1 rfl | case pos.h
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i : ℕ
v : α
j : ℕ
M : MultiPart α
hvi : i ∈ range (M.t + 1) ∧ v ∈ MultiPart.P M i
hj : j ∈ range (M.t + 1) ∧ j ≠ i
hc : card (MultiPart.P M j) + 1 < card (MultiPart.P M i)
h : i ≠ i ∧ i ≠ j
h_1 : ¬(j ≠ i ∧ j ≠ j)
h_2 : j = i
⊢ False | 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
i : ℕ
v : α
j : ℕ
M : MultiPart α
hvi : i ∈ range (M.t + 1) ∧ v ∈ MultiPart.P M i
hj : j ∈ range (M.t + 1) ∧ j ≠ i
hc : card (MultiPart.P M j) + 1 < card (MultiPart.P M i)
h : i ≠ i ∧ i ≠ j
h_1 : ¬(j ≠ i ∧ j ≠ j)
h_2 : j = i
⊢ False
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/Multipartite.lean | SimpleGraph.mp_deg_sum_move_help | [162, 1] | [180, 18] | exfalso | case neg
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i : ℕ
v : α
j : ℕ
M : MultiPart α
hvi : i ∈ range (M.t + 1) ∧ v ∈ MultiPart.P M i
hj : j ∈ range (M.t + 1) ∧ j ≠ i
hc : card (MultiPart.P M j) + 1 < card (MultiPart.P M i)
h : i ≠ i ∧ i ≠ j
h_1 : ¬(j ≠ i ∧ j ≠ j)
h_2 : ¬j = i
⊢ card (MultiPart.P M i) * card (M.A \ MultiPart.P M i) + card (MultiPart.P M j) * card (M.A \ MultiPart.P M j) <
card (MultiPart.P M i) * card (M.A \ MultiPart.P M i) +
(card (MultiPart.P M j) + 1) * (card (M.A \ MultiPart.P M j) - 1) | case neg.h
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i : ℕ
v : α
j : ℕ
M : MultiPart α
hvi : i ∈ range (M.t + 1) ∧ v ∈ MultiPart.P M i
hj : j ∈ range (M.t + 1) ∧ j ≠ i
hc : card (MultiPart.P M j) + 1 < card (MultiPart.P M i)
h : i ≠ i ∧ i ≠ j
h_1 : ¬(j ≠ i ∧ j ≠ j)
h_2 : ¬j = i
⊢ False | 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 : ℕ
M : MultiPart α
hvi : i ∈ range (M.t + 1) ∧ v ∈ MultiPart.P M i
hj : j ∈ range (M.t + 1) ∧ j ≠ i
hc : card (MultiPart.P M j) + 1 < card (MultiPart.P M i)
h : i ≠ i ∧ i ≠ j
h_1 : ¬(j ≠ i ∧ j ≠ j)
h_2 : ¬j = i
⊢ card (MultiPart.P M i) * card (M.A \ MultiPart.P M i) + card (MultiPart.P M j) * card (M.A \ MultiPart.P M j) <
card (MultiPart.P M i) * card (M.A \ MultiPart.P M i) +
(card (MultiPart.P M j) + 1) * (card (M.A \ MultiPart.P M j) - 1)
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/Multipartite.lean | SimpleGraph.mp_deg_sum_move_help | [162, 1] | [180, 18] | exact h.1 rfl | case neg.h
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i : ℕ
v : α
j : ℕ
M : MultiPart α
hvi : i ∈ range (M.t + 1) ∧ v ∈ MultiPart.P M i
hj : j ∈ range (M.t + 1) ∧ j ≠ i
hc : card (MultiPart.P M j) + 1 < card (MultiPart.P M i)
h : i ≠ i ∧ i ≠ j
h_1 : ¬(j ≠ i ∧ j ≠ j)
h_2 : ¬j = i
⊢ False | 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
i : ℕ
v : α
j : ℕ
M : MultiPart α
hvi : i ∈ range (M.t + 1) ∧ v ∈ MultiPart.P M i
hj : j ∈ range (M.t + 1) ∧ j ≠ i
hc : card (MultiPart.P M j) + 1 < card (MultiPart.P M i)
h : i ≠ i ∧ i ≠ j
h_1 : ¬(j ≠ i ∧ j ≠ j)
h_2 : ¬j = i
⊢ False
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/Multipartite.lean | SimpleGraph.mp_deg_sum_move_help | [162, 1] | [180, 18] | exfalso | case pos
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i : ℕ
v : α
j : ℕ
M : MultiPart α
hvi : i ∈ range (M.t + 1) ∧ v ∈ MultiPart.P M i
hj : j ∈ range (M.t + 1) ∧ j ≠ i
hc : card (MultiPart.P M j) + 1 < card (MultiPart.P M i)
h : ¬(i ≠ i ∧ i ≠ j)
h_3 : i = i
h_4 : j ≠ i ∧ j ≠ j
⊢ card (MultiPart.P M i) * card (M.A \ MultiPart.P M i) + card (MultiPart.P M j) * card (M.A \ MultiPart.P M j) <
(card (MultiPart.P M i) - 1) * (card (M.A \ MultiPart.P M i) + 1) +
card (MultiPart.P M j) * card (M.A \ MultiPart.P M j) | case pos.h
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i : ℕ
v : α
j : ℕ
M : MultiPart α
hvi : i ∈ range (M.t + 1) ∧ v ∈ MultiPart.P M i
hj : j ∈ range (M.t + 1) ∧ j ≠ i
hc : card (MultiPart.P M j) + 1 < card (MultiPart.P M i)
h : ¬(i ≠ i ∧ i ≠ j)
h_3 : i = i
h_4 : j ≠ i ∧ j ≠ j
⊢ False | 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 : ℕ
M : MultiPart α
hvi : i ∈ range (M.t + 1) ∧ v ∈ MultiPart.P M i
hj : j ∈ range (M.t + 1) ∧ j ≠ i
hc : card (MultiPart.P M j) + 1 < card (MultiPart.P M i)
h : ¬(i ≠ i ∧ i ≠ j)
h_3 : i = i
h_4 : j ≠ i ∧ j ≠ j
⊢ card (MultiPart.P M i) * card (M.A \ MultiPart.P M i) + card (MultiPart.P M j) * card (M.A \ MultiPart.P M j) <
(card (MultiPart.P M i) - 1) * (card (M.A \ MultiPart.P M i) + 1) +
card (MultiPart.P M j) * card (M.A \ MultiPart.P M j)
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/Multipartite.lean | SimpleGraph.mp_deg_sum_move_help | [162, 1] | [180, 18] | apply h_4.2 rfl | case pos.h
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i : ℕ
v : α
j : ℕ
M : MultiPart α
hvi : i ∈ range (M.t + 1) ∧ v ∈ MultiPart.P M i
hj : j ∈ range (M.t + 1) ∧ j ≠ i
hc : card (MultiPart.P M j) + 1 < card (MultiPart.P M i)
h : ¬(i ≠ i ∧ i ≠ j)
h_3 : i = i
h_4 : j ≠ i ∧ j ≠ j
⊢ False | 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
i : ℕ
v : α
j : ℕ
M : MultiPart α
hvi : i ∈ range (M.t + 1) ∧ v ∈ MultiPart.P M i
hj : j ∈ range (M.t + 1) ∧ j ≠ i
hc : card (MultiPart.P M j) + 1 < card (MultiPart.P M i)
h : ¬(i ≠ i ∧ i ≠ j)
h_3 : i = i
h_4 : j ≠ i ∧ j ≠ j
⊢ False
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/Multipartite.lean | SimpleGraph.mp_deg_sum_move_help | [162, 1] | [180, 18] | exfalso | case pos
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i : ℕ
v : α
j : ℕ
M : MultiPart α
hvi : i ∈ range (M.t + 1) ∧ v ∈ MultiPart.P M i
hj : j ∈ range (M.t + 1) ∧ j ≠ i
hc : card (MultiPart.P M j) + 1 < card (MultiPart.P M i)
h : ¬(i ≠ i ∧ i ≠ j)
h_3 : i = i
h_4 : ¬(j ≠ i ∧ j ≠ j)
h_5 : j = i
⊢ card (MultiPart.P M i) * card (M.A \ MultiPart.P M i) + card (MultiPart.P M j) * card (M.A \ MultiPart.P M j) <
(card (MultiPart.P M i) - 1) * (card (M.A \ MultiPart.P M i) + 1) +
(card (MultiPart.P M i) - 1) * (card (M.A \ MultiPart.P M i) + 1) | case pos.h
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i : ℕ
v : α
j : ℕ
M : MultiPart α
hvi : i ∈ range (M.t + 1) ∧ v ∈ MultiPart.P M i
hj : j ∈ range (M.t + 1) ∧ j ≠ i
hc : card (MultiPart.P M j) + 1 < card (MultiPart.P M i)
h : ¬(i ≠ i ∧ i ≠ j)
h_3 : i = i
h_4 : ¬(j ≠ i ∧ j ≠ j)
h_5 : j = i
⊢ False | 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 : ℕ
M : MultiPart α
hvi : i ∈ range (M.t + 1) ∧ v ∈ MultiPart.P M i
hj : j ∈ range (M.t + 1) ∧ j ≠ i
hc : card (MultiPart.P M j) + 1 < card (MultiPart.P M i)
h : ¬(i ≠ i ∧ i ≠ j)
h_3 : i = i
h_4 : ¬(j ≠ i ∧ j ≠ j)
h_5 : j = i
⊢ card (MultiPart.P M i) * card (M.A \ MultiPart.P M i) + card (MultiPart.P M j) * card (M.A \ MultiPart.P M j) <
(card (MultiPart.P M i) - 1) * (card (M.A \ MultiPart.P M i) + 1) +
(card (MultiPart.P M i) - 1) * (card (M.A \ MultiPart.P M i) + 1)
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/Multipartite.lean | SimpleGraph.mp_deg_sum_move_help | [162, 1] | [180, 18] | exact hj.2 h_5 | case pos.h
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i : ℕ
v : α
j : ℕ
M : MultiPart α
hvi : i ∈ range (M.t + 1) ∧ v ∈ MultiPart.P M i
hj : j ∈ range (M.t + 1) ∧ j ≠ i
hc : card (MultiPart.P M j) + 1 < card (MultiPart.P M i)
h : ¬(i ≠ i ∧ i ≠ j)
h_3 : i = i
h_4 : ¬(j ≠ i ∧ j ≠ j)
h_5 : j = i
⊢ False | 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
i : ℕ
v : α
j : ℕ
M : MultiPart α
hvi : i ∈ range (M.t + 1) ∧ v ∈ MultiPart.P M i
hj : j ∈ range (M.t + 1) ∧ j ≠ i
hc : card (MultiPart.P M j) + 1 < card (MultiPart.P M i)
h : ¬(i ≠ i ∧ i ≠ j)
h_3 : i = i
h_4 : ¬(j ≠ i ∧ j ≠ j)
h_5 : j = i
⊢ False
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/Multipartite.lean | SimpleGraph.mp_deg_sum_move_help | [162, 1] | [180, 18] | rw [card_sdiff (sub_part hvi.1)] | case neg
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i : ℕ
v : α
j : ℕ
M : MultiPart α
hvi : i ∈ range (M.t + 1) ∧ v ∈ MultiPart.P M i
hj : j ∈ range (M.t + 1) ∧ j ≠ i
hc : card (MultiPart.P M j) + 1 < card (MultiPart.P M i)
h : ¬(i ≠ i ∧ i ≠ j)
h_3 : i = i
h_4 : ¬(j ≠ i ∧ j ≠ j)
h_5 : ¬j = i
⊢ card (MultiPart.P M i) * card (M.A \ MultiPart.P M i) + card (MultiPart.P M j) * card (M.A \ MultiPart.P M j) <
(card (MultiPart.P M i) - 1) * (card (M.A \ MultiPart.P M i) + 1) +
(card (MultiPart.P M j) + 1) * (card (M.A \ MultiPart.P M j) - 1) | case neg
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i : ℕ
v : α
j : ℕ
M : MultiPart α
hvi : i ∈ range (M.t + 1) ∧ v ∈ MultiPart.P M i
hj : j ∈ range (M.t + 1) ∧ j ≠ i
hc : card (MultiPart.P M j) + 1 < card (MultiPart.P M i)
h : ¬(i ≠ i ∧ i ≠ j)
h_3 : i = i
h_4 : ¬(j ≠ i ∧ j ≠ j)
h_5 : ¬j = i
⊢ card (MultiPart.P M i) * (card M.A - card (MultiPart.P M i)) + card (MultiPart.P M j) * card (M.A \ MultiPart.P M j) <
(card (MultiPart.P M i) - 1) * (card M.A - card (MultiPart.P M i) + 1) +
(card (MultiPart.P M j) + 1) * (card (M.A \ MultiPart.P M j) - 1) | 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 : ℕ
M : MultiPart α
hvi : i ∈ range (M.t + 1) ∧ v ∈ MultiPart.P M i
hj : j ∈ range (M.t + 1) ∧ j ≠ i
hc : card (MultiPart.P M j) + 1 < card (MultiPart.P M i)
h : ¬(i ≠ i ∧ i ≠ j)
h_3 : i = i
h_4 : ¬(j ≠ i ∧ j ≠ j)
h_5 : ¬j = i
⊢ card (MultiPart.P M i) * card (M.A \ MultiPart.P M i) + card (MultiPart.P M j) * card (M.A \ MultiPart.P M j) <
(card (MultiPart.P M i) - 1) * (card (M.A \ MultiPart.P M i) + 1) +
(card (MultiPart.P M j) + 1) * (card (M.A \ MultiPart.P M j) - 1)
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/Multipartite.lean | SimpleGraph.mp_deg_sum_move_help | [162, 1] | [180, 18] | rw [card_sdiff (sub_part hj.1)] | case neg
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i : ℕ
v : α
j : ℕ
M : MultiPart α
hvi : i ∈ range (M.t + 1) ∧ v ∈ MultiPart.P M i
hj : j ∈ range (M.t + 1) ∧ j ≠ i
hc : card (MultiPart.P M j) + 1 < card (MultiPart.P M i)
h : ¬(i ≠ i ∧ i ≠ j)
h_3 : i = i
h_4 : ¬(j ≠ i ∧ j ≠ j)
h_5 : ¬j = i
⊢ card (MultiPart.P M i) * (card M.A - card (MultiPart.P M i)) + card (MultiPart.P M j) * card (M.A \ MultiPart.P M j) <
(card (MultiPart.P M i) - 1) * (card M.A - card (MultiPart.P M i) + 1) +
(card (MultiPart.P M j) + 1) * (card (M.A \ MultiPart.P M j) - 1) | case neg
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i : ℕ
v : α
j : ℕ
M : MultiPart α
hvi : i ∈ range (M.t + 1) ∧ v ∈ MultiPart.P M i
hj : j ∈ range (M.t + 1) ∧ j ≠ i
hc : card (MultiPart.P M j) + 1 < card (MultiPart.P M i)
h : ¬(i ≠ i ∧ i ≠ j)
h_3 : i = i
h_4 : ¬(j ≠ i ∧ j ≠ j)
h_5 : ¬j = i
⊢ card (MultiPart.P M i) * (card M.A - card (MultiPart.P M i)) +
card (MultiPart.P M j) * (card M.A - card (MultiPart.P M j)) <
(card (MultiPart.P M i) - 1) * (card M.A - card (MultiPart.P M i) + 1) +
(card (MultiPart.P M j) + 1) * (card M.A - card (MultiPart.P M j) - 1) | 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 : ℕ
M : MultiPart α
hvi : i ∈ range (M.t + 1) ∧ v ∈ MultiPart.P M i
hj : j ∈ range (M.t + 1) ∧ j ≠ i
hc : card (MultiPart.P M j) + 1 < card (MultiPart.P M i)
h : ¬(i ≠ i ∧ i ≠ j)
h_3 : i = i
h_4 : ¬(j ≠ i ∧ j ≠ j)
h_5 : ¬j = i
⊢ card (MultiPart.P M i) * (card M.A - card (MultiPart.P M i)) + card (MultiPart.P M j) * card (M.A \ MultiPart.P M j) <
(card (MultiPart.P M i) - 1) * (card M.A - card (MultiPart.P M i) + 1) +
(card (MultiPart.P M j) + 1) * (card (M.A \ MultiPart.P M j) - 1)
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/Multipartite.lean | SimpleGraph.mp_deg_sum_move_help | [162, 1] | [180, 18] | exact move_change hc (two_parts hvi.1 hj.1 hj.2.symm) | case neg
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i : ℕ
v : α
j : ℕ
M : MultiPart α
hvi : i ∈ range (M.t + 1) ∧ v ∈ MultiPart.P M i
hj : j ∈ range (M.t + 1) ∧ j ≠ i
hc : card (MultiPart.P M j) + 1 < card (MultiPart.P M i)
h : ¬(i ≠ i ∧ i ≠ j)
h_3 : i = i
h_4 : ¬(j ≠ i ∧ j ≠ j)
h_5 : ¬j = i
⊢ card (MultiPart.P M i) * (card M.A - card (MultiPart.P M i)) +
card (MultiPart.P M j) * (card M.A - card (MultiPart.P M j)) <
(card (MultiPart.P M i) - 1) * (card M.A - card (MultiPart.P M i) + 1) +
(card (MultiPart.P M j) + 1) * (card M.A - 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 : ℕ
M : MultiPart α
hvi : i ∈ range (M.t + 1) ∧ v ∈ MultiPart.P M i
hj : j ∈ range (M.t + 1) ∧ j ≠ i
hc : card (MultiPart.P M j) + 1 < card (MultiPart.P M i)
h : ¬(i ≠ i ∧ i ≠ j)
h_3 : i = i
h_4 : ¬(j ≠ i ∧ j ≠ j)
h_5 : ¬j = i
⊢ card (MultiPart.P M i) * (card M.A - card (MultiPart.P M i)) +
card (MultiPart.P M j) * (card M.A - card (MultiPart.P M j)) <
(card (MultiPart.P M i) - 1) * (card M.A - card (MultiPart.P M i) + 1) +
(card (MultiPart.P M j) + 1) * (card M.A - card (MultiPart.P M j) - 1)
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/Multipartite.lean | SimpleGraph.mp_deg_sum_move_help | [162, 1] | [180, 18] | exfalso | case pos
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i : ℕ
v : α
j : ℕ
M : MultiPart α
hvi : i ∈ range (M.t + 1) ∧ v ∈ MultiPart.P M i
hj : j ∈ range (M.t + 1) ∧ j ≠ i
hc : card (MultiPart.P M j) + 1 < card (MultiPart.P M i)
h : ¬(i ≠ i ∧ i ≠ j)
h_3 : ¬i = i
h_6 : j ≠ i ∧ j ≠ j
⊢ card (MultiPart.P M i) * card (M.A \ MultiPart.P M i) + card (MultiPart.P M j) * card (M.A \ MultiPart.P M j) <
(card (MultiPart.P M j) + 1) * (card (M.A \ MultiPart.P M j) - 1) +
card (MultiPart.P M j) * card (M.A \ MultiPart.P M j) | case pos.h
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i : ℕ
v : α
j : ℕ
M : MultiPart α
hvi : i ∈ range (M.t + 1) ∧ v ∈ MultiPart.P M i
hj : j ∈ range (M.t + 1) ∧ j ≠ i
hc : card (MultiPart.P M j) + 1 < card (MultiPart.P M i)
h : ¬(i ≠ i ∧ i ≠ j)
h_3 : ¬i = i
h_6 : j ≠ i ∧ j ≠ j
⊢ False | 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 : ℕ
M : MultiPart α
hvi : i ∈ range (M.t + 1) ∧ v ∈ MultiPart.P M i
hj : j ∈ range (M.t + 1) ∧ j ≠ i
hc : card (MultiPart.P M j) + 1 < card (MultiPart.P M i)
h : ¬(i ≠ i ∧ i ≠ j)
h_3 : ¬i = i
h_6 : j ≠ i ∧ j ≠ j
⊢ card (MultiPart.P M i) * card (M.A \ MultiPart.P M i) + card (MultiPart.P M j) * card (M.A \ MultiPart.P M j) <
(card (MultiPart.P M j) + 1) * (card (M.A \ MultiPart.P M j) - 1) +
card (MultiPart.P M j) * card (M.A \ MultiPart.P M j)
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/Multipartite.lean | SimpleGraph.mp_deg_sum_move_help | [162, 1] | [180, 18] | exact h_6.2 rfl | case pos.h
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i : ℕ
v : α
j : ℕ
M : MultiPart α
hvi : i ∈ range (M.t + 1) ∧ v ∈ MultiPart.P M i
hj : j ∈ range (M.t + 1) ∧ j ≠ i
hc : card (MultiPart.P M j) + 1 < card (MultiPart.P M i)
h : ¬(i ≠ i ∧ i ≠ j)
h_3 : ¬i = i
h_6 : j ≠ i ∧ j ≠ j
⊢ False | 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
i : ℕ
v : α
j : ℕ
M : MultiPart α
hvi : i ∈ range (M.t + 1) ∧ v ∈ MultiPart.P M i
hj : j ∈ range (M.t + 1) ∧ j ≠ i
hc : card (MultiPart.P M j) + 1 < card (MultiPart.P M i)
h : ¬(i ≠ i ∧ i ≠ j)
h_3 : ¬i = i
h_6 : j ≠ i ∧ j ≠ j
⊢ False
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/Multipartite.lean | SimpleGraph.mp_deg_sum_move_help | [162, 1] | [180, 18] | contradiction | case pos
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i : ℕ
v : α
j : ℕ
M : MultiPart α
hvi : i ∈ range (M.t + 1) ∧ v ∈ MultiPart.P M i
hj : j ∈ range (M.t + 1) ∧ j ≠ i
hc : card (MultiPart.P M j) + 1 < card (MultiPart.P M i)
h : ¬(i ≠ i ∧ i ≠ j)
h_3 : ¬i = i
h_6 : ¬(j ≠ i ∧ j ≠ j)
h_7 : j = i
⊢ card (MultiPart.P M i) * card (M.A \ MultiPart.P M i) + card (MultiPart.P M j) * card (M.A \ MultiPart.P M j) <
(card (MultiPart.P M j) + 1) * (card (M.A \ MultiPart.P M j) - 1) +
(card (MultiPart.P M i) - 1) * (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 : ℕ
M : MultiPart α
hvi : i ∈ range (M.t + 1) ∧ v ∈ MultiPart.P M i
hj : j ∈ range (M.t + 1) ∧ j ≠ i
hc : card (MultiPart.P M j) + 1 < card (MultiPart.P M i)
h : ¬(i ≠ i ∧ i ≠ j)
h_3 : ¬i = i
h_6 : ¬(j ≠ i ∧ j ≠ j)
h_7 : j = i
⊢ card (MultiPart.P M i) * card (M.A \ MultiPart.P M i) + card (MultiPart.P M j) * card (M.A \ MultiPart.P M j) <
(card (MultiPart.P M j) + 1) * (card (M.A \ MultiPart.P M j) - 1) +
(card (MultiPart.P M i) - 1) * (card (M.A \ MultiPart.P M i) + 1)
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/Multipartite.lean | SimpleGraph.mp_deg_sum_move_help | [162, 1] | [180, 18] | contradiction | case neg
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i : ℕ
v : α
j : ℕ
M : MultiPart α
hvi : i ∈ range (M.t + 1) ∧ v ∈ MultiPart.P M i
hj : j ∈ range (M.t + 1) ∧ j ≠ i
hc : card (MultiPart.P M j) + 1 < card (MultiPart.P M i)
h : ¬(i ≠ i ∧ i ≠ j)
h_3 : ¬i = i
h_6 : ¬(j ≠ i ∧ j ≠ j)
h_7 : ¬j = i
⊢ card (MultiPart.P M i) * card (M.A \ MultiPart.P M i) + card (MultiPart.P M j) * card (M.A \ MultiPart.P M j) <
(card (MultiPart.P M j) + 1) * (card (M.A \ MultiPart.P M j) - 1) +
(card (MultiPart.P M j) + 1) * (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 : ℕ
M : MultiPart α
hvi : i ∈ range (M.t + 1) ∧ v ∈ MultiPart.P M i
hj : j ∈ range (M.t + 1) ∧ j ≠ i
hc : card (MultiPart.P M j) + 1 < card (MultiPart.P M i)
h : ¬(i ≠ i ∧ i ≠ j)
h_3 : ¬i = i
h_6 : ¬(j ≠ i ∧ j ≠ j)
h_7 : ¬j = i
⊢ card (MultiPart.P M i) * card (M.A \ MultiPart.P M i) + card (MultiPart.P M j) * card (M.A \ MultiPart.P M j) <
(card (MultiPart.P M j) + 1) * (card (M.A \ MultiPart.P M j) - 1) +
(card (MultiPart.P M j) + 1) * (card (M.A \ MultiPart.P M j) - 1)
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/Multipartite.lean | SimpleGraph.mp_deg_sum_move_help2 | [183, 1] | [195, 49] | apply sum_congr rfl _ | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i : ℕ
v : α
j : ℕ
M : MultiPart α
hvi : i ∈ range (M.t + 1) ∧ v ∈ MultiPart.P M i
hj : j ∈ range (M.t + 1) ∧ j ≠ i
⊢ ∑ x in erase (erase (range (M.t + 1)) j) i,
card (MultiPart.P (move M hvi hj) x) * card ((move M hvi hj).A \ MultiPart.P (move M hvi hj) x) =
∑ y in erase (erase (range (M.t + 1)) j) i, card (MultiPart.P M y) * card (M.A \ MultiPart.P M y) | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i : ℕ
v : α
j : ℕ
M : MultiPart α
hvi : i ∈ range (M.t + 1) ∧ v ∈ MultiPart.P M i
hj : j ∈ range (M.t + 1) ∧ j ≠ i
⊢ ∀ (x : ℕ),
x ∈ erase (erase (range (M.t + 1)) j) i →
card (MultiPart.P (move M hvi hj) x) * card ((move M hvi hj).A \ MultiPart.P (move M hvi hj) x) =
card (MultiPart.P M x) * card (M.A \ MultiPart.P M x) | 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 : ℕ
M : MultiPart α
hvi : i ∈ range (M.t + 1) ∧ v ∈ MultiPart.P M i
hj : j ∈ range (M.t + 1) ∧ j ≠ i
⊢ ∑ x in erase (erase (range (M.t + 1)) j) i,
card (MultiPart.P (move M hvi hj) x) * card ((move M hvi hj).A \ MultiPart.P (move M hvi hj) x) =
∑ y in erase (erase (range (M.t + 1)) j) i, card (MultiPart.P M y) * card (M.A \ MultiPart.P M y)
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/Multipartite.lean | SimpleGraph.mp_deg_sum_move_help2 | [183, 1] | [195, 49] | intro k hk | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i : ℕ
v : α
j : ℕ
M : MultiPart α
hvi : i ∈ range (M.t + 1) ∧ v ∈ MultiPart.P M i
hj : j ∈ range (M.t + 1) ∧ j ≠ i
⊢ ∀ (x : ℕ),
x ∈ erase (erase (range (M.t + 1)) j) i →
card (MultiPart.P (move M hvi hj) x) * card ((move M hvi hj).A \ MultiPart.P (move M hvi hj) x) =
card (MultiPart.P M x) * card (M.A \ MultiPart.P M x) | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i : ℕ
v : α
j : ℕ
M : MultiPart α
hvi : i ∈ range (M.t + 1) ∧ v ∈ MultiPart.P M i
hj : j ∈ range (M.t + 1) ∧ j ≠ i
k : ℕ
hk : k ∈ erase (erase (range (M.t + 1)) j) i
⊢ card (MultiPart.P (move M hvi hj) k) * card ((move M hvi hj).A \ MultiPart.P (move M hvi hj) k) =
card (MultiPart.P M k) * card (M.A \ MultiPart.P M k) | 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 : ℕ
M : MultiPart α
hvi : i ∈ range (M.t + 1) ∧ v ∈ MultiPart.P M i
hj : j ∈ range (M.t + 1) ∧ j ≠ i
⊢ ∀ (x : ℕ),
x ∈ erase (erase (range (M.t + 1)) j) i →
card (MultiPart.P (move M hvi hj) x) * card ((move M hvi hj).A \ MultiPart.P (move M hvi hj) x) =
card (MultiPart.P M x) * card (M.A \ MultiPart.P M x)
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/Multipartite.lean | SimpleGraph.mp_deg_sum_move_help2 | [183, 1] | [195, 49] | rw [move_Pcard hvi hj] | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i : ℕ
v : α
j : ℕ
M : MultiPart α
hvi : i ∈ range (M.t + 1) ∧ v ∈ MultiPart.P M i
hj : j ∈ range (M.t + 1) ∧ j ≠ i
k : ℕ
hk : k ∈ erase (erase (range (M.t + 1)) j) i
⊢ card (MultiPart.P (move M hvi hj) k) * card ((move M hvi hj).A \ MultiPart.P (move M hvi hj) k) =
card (MultiPart.P M k) * card (M.A \ MultiPart.P M k) | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i : ℕ
v : α
j : ℕ
M : MultiPart α
hvi : i ∈ range (M.t + 1) ∧ v ∈ MultiPart.P M i
hj : j ∈ range (M.t + 1) ∧ j ≠ i
k : ℕ
hk : k ∈ erase (erase (range (M.t + 1)) j) i
⊢ (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) *
card ((move M hvi hj).A \ MultiPart.P (move M hvi hj) k) =
card (MultiPart.P M k) * card (M.A \ MultiPart.P M k)
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i : ℕ
v : α
j : ℕ
M : MultiPart α
hvi : i ∈ range (M.t + 1) ∧ v ∈ MultiPart.P M i
hj : j ∈ range (M.t + 1) ∧ j ≠ i
k : ℕ
hk : k ∈ erase (erase (range (M.t + 1)) j) i
⊢ k ∈ range (M.t + 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 : ℕ
M : MultiPart α
hvi : i ∈ range (M.t + 1) ∧ v ∈ MultiPart.P M i
hj : j ∈ range (M.t + 1) ∧ j ≠ i
k : ℕ
hk : k ∈ erase (erase (range (M.t + 1)) j) i
⊢ card (MultiPart.P (move M hvi hj) k) * card ((move M hvi hj).A \ MultiPart.P (move M hvi hj) k) =
card (MultiPart.P M k) * card (M.A \ MultiPart.P M k)
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/Multipartite.lean | SimpleGraph.mp_deg_sum_move_help2 | [183, 1] | [195, 49] | rw [move_Pcard_sdiff hvi hj] | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i : ℕ
v : α
j : ℕ
M : MultiPart α
hvi : i ∈ range (M.t + 1) ∧ v ∈ MultiPart.P M i
hj : j ∈ range (M.t + 1) ∧ j ≠ i
k : ℕ
hk : k ∈ erase (erase (range (M.t + 1)) j) i
⊢ (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) *
card ((move M hvi hj).A \ MultiPart.P (move M hvi hj) k) =
card (MultiPart.P M k) * card (M.A \ MultiPart.P M k)
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i : ℕ
v : α
j : ℕ
M : MultiPart α
hvi : i ∈ range (M.t + 1) ∧ v ∈ MultiPart.P M i
hj : j ∈ range (M.t + 1) ∧ j ≠ i
k : ℕ
hk : k ∈ erase (erase (range (M.t + 1)) j) i
⊢ k ∈ range (M.t + 1) | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i : ℕ
v : α
j : ℕ
M : MultiPart α
hvi : i ∈ range (M.t + 1) ∧ v ∈ MultiPart.P M i
hj : j ∈ range (M.t + 1) ∧ j ≠ i
k : ℕ
hk : k ∈ erase (erase (range (M.t + 1)) j) i
⊢ ((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) *
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) =
card (MultiPart.P M k) * card (M.A \ MultiPart.P M k)
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i : ℕ
v : α
j : ℕ
M : MultiPart α
hvi : i ∈ range (M.t + 1) ∧ v ∈ MultiPart.P M i
hj : j ∈ range (M.t + 1) ∧ j ≠ i
k : ℕ
hk : k ∈ erase (erase (range (M.t + 1)) j) i
⊢ k ∈ range (M.t + 1)
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i : ℕ
v : α
j : ℕ
M : MultiPart α
hvi : i ∈ range (M.t + 1) ∧ v ∈ MultiPart.P M i
hj : j ∈ range (M.t + 1) ∧ j ≠ i
k : ℕ
hk : k ∈ erase (erase (range (M.t + 1)) j) i
⊢ k ∈ range (M.t + 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 : ℕ
M : MultiPart α
hvi : i ∈ range (M.t + 1) ∧ v ∈ MultiPart.P M i
hj : j ∈ range (M.t + 1) ∧ j ≠ i
k : ℕ
hk : k ∈ erase (erase (range (M.t + 1)) j) i
⊢ (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) *
card ((move M hvi hj).A \ MultiPart.P (move M hvi hj) k) =
card (MultiPart.P M k) * card (M.A \ MultiPart.P M k)
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i : ℕ
v : α
j : ℕ
M : MultiPart α
hvi : i ∈ range (M.t + 1) ∧ v ∈ MultiPart.P M i
hj : j ∈ range (M.t + 1) ∧ j ≠ i
k : ℕ
hk : k ∈ erase (erase (range (M.t + 1)) j) i
⊢ k ∈ range (M.t + 1)
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/Multipartite.lean | SimpleGraph.mp_deg_sum_move_help2 | [183, 1] | [195, 49] | split_ifs with h h_1 | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i : ℕ
v : α
j : ℕ
M : MultiPart α
hvi : i ∈ range (M.t + 1) ∧ v ∈ MultiPart.P M i
hj : j ∈ range (M.t + 1) ∧ j ≠ i
k : ℕ
hk : k ∈ erase (erase (range (M.t + 1)) j) i
⊢ ((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) *
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) =
card (MultiPart.P M k) * card (M.A \ MultiPart.P M k)
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i : ℕ
v : α
j : ℕ
M : MultiPart α
hvi : i ∈ range (M.t + 1) ∧ v ∈ MultiPart.P M i
hj : j ∈ range (M.t + 1) ∧ j ≠ i
k : ℕ
hk : k ∈ erase (erase (range (M.t + 1)) j) i
⊢ k ∈ range (M.t + 1)
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i : ℕ
v : α
j : ℕ
M : MultiPart α
hvi : i ∈ range (M.t + 1) ∧ v ∈ MultiPart.P M i
hj : j ∈ range (M.t + 1) ∧ j ≠ i
k : ℕ
hk : k ∈ erase (erase (range (M.t + 1)) j) i
⊢ k ∈ range (M.t + 1) | case pos
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i : ℕ
v : α
j : ℕ
M : MultiPart α
hvi : i ∈ range (M.t + 1) ∧ v ∈ MultiPart.P M i
hj : j ∈ range (M.t + 1) ∧ j ≠ i
k : ℕ
hk : k ∈ erase (erase (range (M.t + 1)) j) i
h : k ≠ i ∧ k ≠ j
⊢ card (MultiPart.P M k) * card (M.A \ MultiPart.P M k) = card (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 : ℕ
M : MultiPart α
hvi : i ∈ range (M.t + 1) ∧ v ∈ MultiPart.P M i
hj : j ∈ range (M.t + 1) ∧ j ≠ i
k : ℕ
hk : k ∈ erase (erase (range (M.t + 1)) j) i
h : ¬(k ≠ i ∧ k ≠ j)
h_1 : k = i
⊢ (card (MultiPart.P M i) - 1) * (card (M.A \ MultiPart.P M i) + 1) =
card (MultiPart.P M k) * card (M.A \ MultiPart.P M k)
case neg
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i : ℕ
v : α
j : ℕ
M : MultiPart α
hvi : i ∈ range (M.t + 1) ∧ v ∈ MultiPart.P M i
hj : j ∈ range (M.t + 1) ∧ j ≠ i
k : ℕ
hk : k ∈ erase (erase (range (M.t + 1)) j) i
h : ¬(k ≠ i ∧ k ≠ j)
h_1 : ¬k = i
⊢ (card (MultiPart.P M j) + 1) * (card (M.A \ MultiPart.P M j) - 1) =
card (MultiPart.P M k) * card (M.A \ MultiPart.P M k)
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i : ℕ
v : α
j : ℕ
M : MultiPart α
hvi : i ∈ range (M.t + 1) ∧ v ∈ MultiPart.P M i
hj : j ∈ range (M.t + 1) ∧ j ≠ i
k : ℕ
hk : k ∈ erase (erase (range (M.t + 1)) j) i
⊢ k ∈ range (M.t + 1)
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i : ℕ
v : α
j : ℕ
M : MultiPart α
hvi : i ∈ range (M.t + 1) ∧ v ∈ MultiPart.P M i
hj : j ∈ range (M.t + 1) ∧ j ≠ i
k : ℕ
hk : k ∈ erase (erase (range (M.t + 1)) j) i
⊢ k ∈ range (M.t + 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 : ℕ
M : MultiPart α
hvi : i ∈ range (M.t + 1) ∧ v ∈ MultiPart.P M i
hj : j ∈ range (M.t + 1) ∧ j ≠ i
k : ℕ
hk : k ∈ erase (erase (range (M.t + 1)) j) i
⊢ ((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) *
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) =
card (MultiPart.P M k) * card (M.A \ MultiPart.P M k)
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i : ℕ
v : α
j : ℕ
M : MultiPart α
hvi : i ∈ range (M.t + 1) ∧ v ∈ MultiPart.P M i
hj : j ∈ range (M.t + 1) ∧ j ≠ i
k : ℕ
hk : k ∈ erase (erase (range (M.t + 1)) j) i
⊢ k ∈ range (M.t + 1)
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i : ℕ
v : α
j : ℕ
M : MultiPart α
hvi : i ∈ range (M.t + 1) ∧ v ∈ MultiPart.P M i
hj : j ∈ range (M.t + 1) ∧ j ≠ i
k : ℕ
hk : k ∈ erase (erase (range (M.t + 1)) j) i
⊢ k ∈ range (M.t + 1)
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/Multipartite.lean | SimpleGraph.mp_deg_sum_move_help2 | [183, 1] | [195, 49] | rfl | case pos
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i : ℕ
v : α
j : ℕ
M : MultiPart α
hvi : i ∈ range (M.t + 1) ∧ v ∈ MultiPart.P M i
hj : j ∈ range (M.t + 1) ∧ j ≠ i
k : ℕ
hk : k ∈ erase (erase (range (M.t + 1)) j) i
h : k ≠ i ∧ k ≠ j
⊢ card (MultiPart.P M k) * card (M.A \ MultiPart.P M k) = card (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 : ℕ
M : MultiPart α
hvi : i ∈ range (M.t + 1) ∧ v ∈ MultiPart.P M i
hj : j ∈ range (M.t + 1) ∧ j ≠ i
k : ℕ
hk : k ∈ erase (erase (range (M.t + 1)) j) i
h : k ≠ i ∧ k ≠ j
⊢ card (MultiPart.P M k) * card (M.A \ MultiPart.P M k) = card (MultiPart.P M k) * card (M.A \ MultiPart.P M k)
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/Multipartite.lean | SimpleGraph.mp_deg_sum_move_help2 | [183, 1] | [195, 49] | exfalso | case pos
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i : ℕ
v : α
j : ℕ
M : MultiPart α
hvi : i ∈ range (M.t + 1) ∧ v ∈ MultiPart.P M i
hj : j ∈ range (M.t + 1) ∧ j ≠ i
k : ℕ
hk : k ∈ erase (erase (range (M.t + 1)) j) i
h : ¬(k ≠ i ∧ k ≠ j)
h_1 : k = i
⊢ (card (MultiPart.P M i) - 1) * (card (M.A \ MultiPart.P M i) + 1) =
card (MultiPart.P M k) * card (M.A \ MultiPart.P M k) | case pos.h
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i : ℕ
v : α
j : ℕ
M : MultiPart α
hvi : i ∈ range (M.t + 1) ∧ v ∈ MultiPart.P M i
hj : j ∈ range (M.t + 1) ∧ j ≠ i
k : ℕ
hk : k ∈ erase (erase (range (M.t + 1)) j) i
h : ¬(k ≠ i ∧ k ≠ j)
h_1 : k = i
⊢ False | 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 : ℕ
M : MultiPart α
hvi : i ∈ range (M.t + 1) ∧ v ∈ MultiPart.P M i
hj : j ∈ range (M.t + 1) ∧ j ≠ i
k : ℕ
hk : k ∈ erase (erase (range (M.t + 1)) j) i
h : ¬(k ≠ i ∧ k ≠ j)
h_1 : k = i
⊢ (card (MultiPart.P M i) - 1) * (card (M.A \ MultiPart.P M i) + 1) =
card (MultiPart.P M k) * card (M.A \ MultiPart.P M k)
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/Multipartite.lean | SimpleGraph.mp_deg_sum_move_help2 | [183, 1] | [195, 49] | rw [h_1] at hk | case pos.h
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i : ℕ
v : α
j : ℕ
M : MultiPart α
hvi : i ∈ range (M.t + 1) ∧ v ∈ MultiPart.P M i
hj : j ∈ range (M.t + 1) ∧ j ≠ i
k : ℕ
hk : k ∈ erase (erase (range (M.t + 1)) j) i
h : ¬(k ≠ i ∧ k ≠ j)
h_1 : k = i
⊢ False | case pos.h
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i : ℕ
v : α
j : ℕ
M : MultiPart α
hvi : i ∈ range (M.t + 1) ∧ v ∈ MultiPart.P M i
hj : j ∈ range (M.t + 1) ∧ j ≠ i
k : ℕ
hk : i ∈ erase (erase (range (M.t + 1)) j) i
h : ¬(k ≠ i ∧ k ≠ j)
h_1 : k = i
⊢ False | 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
i : ℕ
v : α
j : ℕ
M : MultiPart α
hvi : i ∈ range (M.t + 1) ∧ v ∈ MultiPart.P M i
hj : j ∈ range (M.t + 1) ∧ j ≠ i
k : ℕ
hk : k ∈ erase (erase (range (M.t + 1)) j) i
h : ¬(k ≠ i ∧ k ≠ j)
h_1 : k = i
⊢ False
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/Multipartite.lean | SimpleGraph.mp_deg_sum_move_help2 | [183, 1] | [195, 49] | exact not_mem_erase i _ hk | case pos.h
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i : ℕ
v : α
j : ℕ
M : MultiPart α
hvi : i ∈ range (M.t + 1) ∧ v ∈ MultiPart.P M i
hj : j ∈ range (M.t + 1) ∧ j ≠ i
k : ℕ
hk : i ∈ erase (erase (range (M.t + 1)) j) i
h : ¬(k ≠ i ∧ k ≠ j)
h_1 : k = i
⊢ False | 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
i : ℕ
v : α
j : ℕ
M : MultiPart α
hvi : i ∈ range (M.t + 1) ∧ v ∈ MultiPart.P M i
hj : j ∈ range (M.t + 1) ∧ j ≠ i
k : ℕ
hk : i ∈ erase (erase (range (M.t + 1)) j) i
h : ¬(k ≠ i ∧ k ≠ j)
h_1 : k = i
⊢ False
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/Multipartite.lean | SimpleGraph.mp_deg_sum_move_help2 | [183, 1] | [195, 49] | exfalso | case neg
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i : ℕ
v : α
j : ℕ
M : MultiPart α
hvi : i ∈ range (M.t + 1) ∧ v ∈ MultiPart.P M i
hj : j ∈ range (M.t + 1) ∧ j ≠ i
k : ℕ
hk : k ∈ erase (erase (range (M.t + 1)) j) i
h : ¬(k ≠ i ∧ k ≠ j)
h_1 : ¬k = i
⊢ (card (MultiPart.P M j) + 1) * (card (M.A \ MultiPart.P M j) - 1) =
card (MultiPart.P M k) * card (M.A \ MultiPart.P M k) | case neg.h
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i : ℕ
v : α
j : ℕ
M : MultiPart α
hvi : i ∈ range (M.t + 1) ∧ v ∈ MultiPart.P M i
hj : j ∈ range (M.t + 1) ∧ j ≠ i
k : ℕ
hk : k ∈ erase (erase (range (M.t + 1)) j) i
h : ¬(k ≠ i ∧ k ≠ j)
h_1 : ¬k = i
⊢ False | 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 : ℕ
M : MultiPart α
hvi : i ∈ range (M.t + 1) ∧ v ∈ MultiPart.P M i
hj : j ∈ range (M.t + 1) ∧ j ≠ i
k : ℕ
hk : k ∈ erase (erase (range (M.t + 1)) j) i
h : ¬(k ≠ i ∧ k ≠ j)
h_1 : ¬k = i
⊢ (card (MultiPart.P M j) + 1) * (card (M.A \ MultiPart.P M j) - 1) =
card (MultiPart.P M k) * card (M.A \ MultiPart.P M k)
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/Multipartite.lean | SimpleGraph.mp_deg_sum_move_help2 | [183, 1] | [195, 49] | push_neg at h | case neg.h
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i : ℕ
v : α
j : ℕ
M : MultiPart α
hvi : i ∈ range (M.t + 1) ∧ v ∈ MultiPart.P M i
hj : j ∈ range (M.t + 1) ∧ j ≠ i
k : ℕ
hk : k ∈ erase (erase (range (M.t + 1)) j) i
h : ¬(k ≠ i ∧ k ≠ j)
h_1 : ¬k = i
⊢ False | case neg.h
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i : ℕ
v : α
j : ℕ
M : MultiPart α
hvi : i ∈ range (M.t + 1) ∧ v ∈ MultiPart.P M i
hj : j ∈ range (M.t + 1) ∧ j ≠ i
k : ℕ
hk : k ∈ erase (erase (range (M.t + 1)) j) i
h_1 : ¬k = i
h : k ≠ i → k = j
⊢ False | 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
i : ℕ
v : α
j : ℕ
M : MultiPart α
hvi : i ∈ range (M.t + 1) ∧ v ∈ MultiPart.P M i
hj : j ∈ range (M.t + 1) ∧ j ≠ i
k : ℕ
hk : k ∈ erase (erase (range (M.t + 1)) j) i
h : ¬(k ≠ i ∧ k ≠ j)
h_1 : ¬k = i
⊢ False
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/Multipartite.lean | SimpleGraph.mp_deg_sum_move_help2 | [183, 1] | [195, 49] | simp_all only [Ne.def, not_false_iff, mem_erase, eq_self_iff_true] | case neg.h
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i : ℕ
v : α
j : ℕ
M : MultiPart α
hvi : i ∈ range (M.t + 1) ∧ v ∈ MultiPart.P M i
hj : j ∈ range (M.t + 1) ∧ j ≠ i
k : ℕ
hk : k ∈ erase (erase (range (M.t + 1)) j) i
h_1 : ¬k = i
h : k ≠ i → k = j
⊢ False | 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
i : ℕ
v : α
j : ℕ
M : MultiPart α
hvi : i ∈ range (M.t + 1) ∧ v ∈ MultiPart.P M i
hj : j ∈ range (M.t + 1) ∧ j ≠ i
k : ℕ
hk : k ∈ erase (erase (range (M.t + 1)) j) i
h_1 : ¬k = i
h : k ≠ i → k = j
⊢ False
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/Multipartite.lean | SimpleGraph.mp_deg_sum_move_help2 | [183, 1] | [195, 49] | exact mem_of_mem_erase (mem_of_mem_erase hk) | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i : ℕ
v : α
j : ℕ
M : MultiPart α
hvi : i ∈ range (M.t + 1) ∧ v ∈ MultiPart.P M i
hj : j ∈ range (M.t + 1) ∧ j ≠ i
k : ℕ
hk : k ∈ erase (erase (range (M.t + 1)) j) i
⊢ k ∈ range (M.t + 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
i : ℕ
v : α
j : ℕ
M : MultiPart α
hvi : i ∈ range (M.t + 1) ∧ v ∈ MultiPart.P M i
hj : j ∈ range (M.t + 1) ∧ j ≠ i
k : ℕ
hk : k ∈ erase (erase (range (M.t + 1)) j) i
⊢ k ∈ range (M.t + 1)
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/Multipartite.lean | SimpleGraph.mp_deg_sum_move | [199, 1] | [209, 39] | rw [mp_deg_sum M, mp_deg_sum (move M hvi hj), move_t hvi hj] | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i : ℕ
v : α
j : ℕ
M : MultiPart α
hvi : i ∈ range (M.t + 1) ∧ v ∈ MultiPart.P M i
hj : j ∈ range (M.t + 1) ∧ j ≠ i
hc : card (MultiPart.P M j) + 1 < card (MultiPart.P M i)
⊢ ∑ w in M.A, degree (mp M) w < ∑ w in (move M hvi hj).A, degree (mp (move M hvi hj)) w | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i : ℕ
v : α
j : ℕ
M : MultiPart α
hvi : i ∈ range (M.t + 1) ∧ v ∈ MultiPart.P M i
hj : j ∈ range (M.t + 1) ∧ j ≠ i
hc : card (MultiPart.P M j) + 1 < card (MultiPart.P M i)
⊢ ∑ i in range (M.t + 1), card (MultiPart.P M i) * card (M.A \ MultiPart.P M i) <
∑ i_1 in range (M.t + 1),
card (MultiPart.P (move M hvi hj) i_1) * card ((move M hvi hj).A \ MultiPart.P (move M hvi hj) i_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 : ℕ
M : MultiPart α
hvi : i ∈ range (M.t + 1) ∧ v ∈ MultiPart.P M i
hj : j ∈ range (M.t + 1) ∧ j ≠ i
hc : card (MultiPart.P M j) + 1 < card (MultiPart.P M i)
⊢ ∑ w in M.A, degree (mp M) w < ∑ w in (move M hvi hj).A, degree (mp (move M hvi hj)) w
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/Multipartite.lean | SimpleGraph.mp_deg_sum_move | [199, 1] | [209, 39] | rw [← sum_erase_add (range (M.t + 1)) _ hj.1, ← sum_erase_add (range (M.t + 1)) _ hj.1] | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i : ℕ
v : α
j : ℕ
M : MultiPart α
hvi : i ∈ range (M.t + 1) ∧ v ∈ MultiPart.P M i
hj : j ∈ range (M.t + 1) ∧ j ≠ i
hc : card (MultiPart.P M j) + 1 < card (MultiPart.P M i)
⊢ ∑ i in range (M.t + 1), card (MultiPart.P M i) * card (M.A \ MultiPart.P M i) <
∑ i_1 in range (M.t + 1),
card (MultiPart.P (move M hvi hj) i_1) * card ((move M hvi hj).A \ MultiPart.P (move M hvi hj) i_1) | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i : ℕ
v : α
j : ℕ
M : MultiPart α
hvi : i ∈ range (M.t + 1) ∧ v ∈ MultiPart.P M i
hj : j ∈ range (M.t + 1) ∧ j ≠ i
hc : card (MultiPart.P M j) + 1 < card (MultiPart.P M i)
⊢ ∑ x in erase (range (M.t + 1)) j, card (MultiPart.P M x) * card (M.A \ MultiPart.P M x) +
card (MultiPart.P M j) * card (M.A \ MultiPart.P M j) <
∑ x in erase (range (M.t + 1)) j,
card (MultiPart.P (move M hvi hj) x) * card ((move M hvi hj).A \ MultiPart.P (move M hvi hj) x) +
card (MultiPart.P (move M hvi hj) j) * card ((move M hvi hj).A \ MultiPart.P (move M hvi hj) 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 : ℕ
M : MultiPart α
hvi : i ∈ range (M.t + 1) ∧ v ∈ MultiPart.P M i
hj : j ∈ range (M.t + 1) ∧ j ≠ i
hc : card (MultiPart.P M j) + 1 < card (MultiPart.P M i)
⊢ ∑ i in range (M.t + 1), card (MultiPart.P M i) * card (M.A \ MultiPart.P M i) <
∑ i_1 in range (M.t + 1),
card (MultiPart.P (move M hvi hj) i_1) * card ((move M hvi hj).A \ MultiPart.P (move M hvi hj) i_1)
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/Multipartite.lean | SimpleGraph.mp_deg_sum_move | [199, 1] | [209, 39] | rw [← sum_erase_add ((range (M.t + 1)).erase j) _ (mem_erase_of_ne_of_mem hj.2.symm hvi.1)] | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i : ℕ
v : α
j : ℕ
M : MultiPart α
hvi : i ∈ range (M.t + 1) ∧ v ∈ MultiPart.P M i
hj : j ∈ range (M.t + 1) ∧ j ≠ i
hc : card (MultiPart.P M j) + 1 < card (MultiPart.P M i)
⊢ ∑ x in erase (range (M.t + 1)) j, card (MultiPart.P M x) * card (M.A \ MultiPart.P M x) +
card (MultiPart.P M j) * card (M.A \ MultiPart.P M j) <
∑ x in erase (range (M.t + 1)) j,
card (MultiPart.P (move M hvi hj) x) * card ((move M hvi hj).A \ MultiPart.P (move M hvi hj) x) +
card (MultiPart.P (move M hvi hj) j) * card ((move M hvi hj).A \ MultiPart.P (move M hvi hj) j) | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i : ℕ
v : α
j : ℕ
M : MultiPart α
hvi : i ∈ range (M.t + 1) ∧ v ∈ MultiPart.P M i
hj : j ∈ range (M.t + 1) ∧ j ≠ i
hc : card (MultiPart.P M j) + 1 < card (MultiPart.P M i)
⊢ ∑ x in erase (erase (range (M.t + 1)) j) i, card (MultiPart.P M x) * card (M.A \ MultiPart.P M x) +
card (MultiPart.P M i) * card (M.A \ MultiPart.P M i) +
card (MultiPart.P M j) * card (M.A \ MultiPart.P M j) <
∑ x in erase (range (M.t + 1)) j,
card (MultiPart.P (move M hvi hj) x) * card ((move M hvi hj).A \ MultiPart.P (move M hvi hj) x) +
card (MultiPart.P (move M hvi hj) j) * card ((move M hvi hj).A \ MultiPart.P (move M hvi hj) 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 : ℕ
M : MultiPart α
hvi : i ∈ range (M.t + 1) ∧ v ∈ MultiPart.P M i
hj : j ∈ range (M.t + 1) ∧ j ≠ i
hc : card (MultiPart.P M j) + 1 < card (MultiPart.P M i)
⊢ ∑ x in erase (range (M.t + 1)) j, card (MultiPart.P M x) * card (M.A \ MultiPart.P M x) +
card (MultiPart.P M j) * card (M.A \ MultiPart.P M j) <
∑ x in erase (range (M.t + 1)) j,
card (MultiPart.P (move M hvi hj) x) * card ((move M hvi hj).A \ MultiPart.P (move M hvi hj) x) +
card (MultiPart.P (move M hvi hj) j) * card ((move M hvi hj).A \ MultiPart.P (move M hvi hj) j)
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/Multipartite.lean | SimpleGraph.mp_deg_sum_move | [199, 1] | [209, 39] | rw [← sum_erase_add ((range (M.t + 1)).erase j) _ (mem_erase_of_ne_of_mem hj.2.symm hvi.1)] | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i : ℕ
v : α
j : ℕ
M : MultiPart α
hvi : i ∈ range (M.t + 1) ∧ v ∈ MultiPart.P M i
hj : j ∈ range (M.t + 1) ∧ j ≠ i
hc : card (MultiPart.P M j) + 1 < card (MultiPart.P M i)
⊢ ∑ x in erase (erase (range (M.t + 1)) j) i, card (MultiPart.P M x) * card (M.A \ MultiPart.P M x) +
card (MultiPart.P M i) * card (M.A \ MultiPart.P M i) +
card (MultiPart.P M j) * card (M.A \ MultiPart.P M j) <
∑ x in erase (range (M.t + 1)) j,
card (MultiPart.P (move M hvi hj) x) * card ((move M hvi hj).A \ MultiPart.P (move M hvi hj) x) +
card (MultiPart.P (move M hvi hj) j) * card ((move M hvi hj).A \ MultiPart.P (move M hvi hj) j) | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i : ℕ
v : α
j : ℕ
M : MultiPart α
hvi : i ∈ range (M.t + 1) ∧ v ∈ MultiPart.P M i
hj : j ∈ range (M.t + 1) ∧ j ≠ i
hc : card (MultiPart.P M j) + 1 < card (MultiPart.P M i)
⊢ ∑ x in erase (erase (range (M.t + 1)) j) i, card (MultiPart.P M x) * card (M.A \ MultiPart.P M x) +
card (MultiPart.P M i) * card (M.A \ MultiPart.P M i) +
card (MultiPart.P M j) * card (M.A \ MultiPart.P M j) <
∑ x in erase (erase (range (M.t + 1)) j) i,
card (MultiPart.P (move M hvi hj) x) * card ((move M hvi hj).A \ MultiPart.P (move M hvi hj) x) +
card (MultiPart.P (move M hvi hj) i) * card ((move M hvi hj).A \ MultiPart.P (move M hvi hj) i) +
card (MultiPart.P (move M hvi hj) j) * card ((move M hvi hj).A \ MultiPart.P (move M hvi hj) 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 : ℕ
M : MultiPart α
hvi : i ∈ range (M.t + 1) ∧ v ∈ MultiPart.P M i
hj : j ∈ range (M.t + 1) ∧ j ≠ i
hc : card (MultiPart.P M j) + 1 < card (MultiPart.P M i)
⊢ ∑ x in erase (erase (range (M.t + 1)) j) i, card (MultiPart.P M x) * card (M.A \ MultiPart.P M x) +
card (MultiPart.P M i) * card (M.A \ MultiPart.P M i) +
card (MultiPart.P M j) * card (M.A \ MultiPart.P M j) <
∑ x in erase (range (M.t + 1)) j,
card (MultiPart.P (move M hvi hj) x) * card ((move M hvi hj).A \ MultiPart.P (move M hvi hj) x) +
card (MultiPart.P (move M hvi hj) j) * card ((move M hvi hj).A \ MultiPart.P (move M hvi hj) j)
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/Multipartite.lean | SimpleGraph.mp_deg_sum_move | [199, 1] | [209, 39] | rw [mp_deg_sum_move_help2] | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i : ℕ
v : α
j : ℕ
M : MultiPart α
hvi : i ∈ range (M.t + 1) ∧ v ∈ MultiPart.P M i
hj : j ∈ range (M.t + 1) ∧ j ≠ i
hc : card (MultiPart.P M j) + 1 < card (MultiPart.P M i)
⊢ ∑ x in erase (erase (range (M.t + 1)) j) i, card (MultiPart.P M x) * card (M.A \ MultiPart.P M x) +
card (MultiPart.P M i) * card (M.A \ MultiPart.P M i) +
card (MultiPart.P M j) * card (M.A \ MultiPart.P M j) <
∑ x in erase (erase (range (M.t + 1)) j) i,
card (MultiPart.P (move M hvi hj) x) * card ((move M hvi hj).A \ MultiPart.P (move M hvi hj) x) +
card (MultiPart.P (move M hvi hj) i) * card ((move M hvi hj).A \ MultiPart.P (move M hvi hj) i) +
card (MultiPart.P (move M hvi hj) j) * card ((move M hvi hj).A \ MultiPart.P (move M hvi hj) j) | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i : ℕ
v : α
j : ℕ
M : MultiPart α
hvi : i ∈ range (M.t + 1) ∧ v ∈ MultiPart.P M i
hj : j ∈ range (M.t + 1) ∧ j ≠ i
hc : card (MultiPart.P M j) + 1 < card (MultiPart.P M i)
⊢ ∑ x in erase (erase (range (M.t + 1)) j) i, card (MultiPart.P M x) * card (M.A \ MultiPart.P M x) +
card (MultiPart.P M i) * card (M.A \ MultiPart.P M i) +
card (MultiPart.P M j) * card (M.A \ MultiPart.P M j) <
∑ y in erase (erase (range (M.t + 1)) j) i, card (MultiPart.P M y) * card (M.A \ MultiPart.P M y) +
card (MultiPart.P (move M hvi hj) i) * card ((move M hvi hj).A \ MultiPart.P (move M hvi hj) i) +
card (MultiPart.P (move M hvi hj) j) * card ((move M hvi hj).A \ MultiPart.P (move M hvi hj) 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 : ℕ
M : MultiPart α
hvi : i ∈ range (M.t + 1) ∧ v ∈ MultiPart.P M i
hj : j ∈ range (M.t + 1) ∧ j ≠ i
hc : card (MultiPart.P M j) + 1 < card (MultiPart.P M i)
⊢ ∑ x in erase (erase (range (M.t + 1)) j) i, card (MultiPart.P M x) * card (M.A \ MultiPart.P M x) +
card (MultiPart.P M i) * card (M.A \ MultiPart.P M i) +
card (MultiPart.P M j) * card (M.A \ MultiPart.P M j) <
∑ x in erase (erase (range (M.t + 1)) j) i,
card (MultiPart.P (move M hvi hj) x) * card ((move M hvi hj).A \ MultiPart.P (move M hvi hj) x) +
card (MultiPart.P (move M hvi hj) i) * card ((move M hvi hj).A \ MultiPart.P (move M hvi hj) i) +
card (MultiPart.P (move M hvi hj) j) * card ((move M hvi hj).A \ MultiPart.P (move M hvi hj) j)
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/Multipartite.lean | SimpleGraph.mp_deg_sum_move | [199, 1] | [209, 39] | rw [add_assoc, add_assoc] | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i : ℕ
v : α
j : ℕ
M : MultiPart α
hvi : i ∈ range (M.t + 1) ∧ v ∈ MultiPart.P M i
hj : j ∈ range (M.t + 1) ∧ j ≠ i
hc : card (MultiPart.P M j) + 1 < card (MultiPart.P M i)
⊢ ∑ x in erase (erase (range (M.t + 1)) j) i, card (MultiPart.P M x) * card (M.A \ MultiPart.P M x) +
card (MultiPart.P M i) * card (M.A \ MultiPart.P M i) +
card (MultiPart.P M j) * card (M.A \ MultiPart.P M j) <
∑ y in erase (erase (range (M.t + 1)) j) i, card (MultiPart.P M y) * card (M.A \ MultiPart.P M y) +
card (MultiPart.P (move M hvi hj) i) * card ((move M hvi hj).A \ MultiPart.P (move M hvi hj) i) +
card (MultiPart.P (move M hvi hj) j) * card ((move M hvi hj).A \ MultiPart.P (move M hvi hj) j) | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i : ℕ
v : α
j : ℕ
M : MultiPart α
hvi : i ∈ range (M.t + 1) ∧ v ∈ MultiPart.P M i
hj : j ∈ range (M.t + 1) ∧ j ≠ i
hc : card (MultiPart.P M j) + 1 < card (MultiPart.P M i)
⊢ ∑ x in erase (erase (range (M.t + 1)) j) i, card (MultiPart.P M x) * card (M.A \ MultiPart.P M x) +
(card (MultiPart.P M i) * card (M.A \ MultiPart.P M i) + card (MultiPart.P M j) * card (M.A \ MultiPart.P M j)) <
∑ y in erase (erase (range (M.t + 1)) j) i, card (MultiPart.P M y) * card (M.A \ MultiPart.P M y) +
(card (MultiPart.P (move M hvi hj) i) * card ((move M hvi hj).A \ MultiPart.P (move M hvi hj) i) +
card (MultiPart.P (move M hvi hj) j) * card ((move M hvi hj).A \ MultiPart.P (move M hvi hj) 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 : ℕ
M : MultiPart α
hvi : i ∈ range (M.t + 1) ∧ v ∈ MultiPart.P M i
hj : j ∈ range (M.t + 1) ∧ j ≠ i
hc : card (MultiPart.P M j) + 1 < card (MultiPart.P M i)
⊢ ∑ x in erase (erase (range (M.t + 1)) j) i, card (MultiPart.P M x) * card (M.A \ MultiPart.P M x) +
card (MultiPart.P M i) * card (M.A \ MultiPart.P M i) +
card (MultiPart.P M j) * card (M.A \ MultiPart.P M j) <
∑ y in erase (erase (range (M.t + 1)) j) i, card (MultiPart.P M y) * card (M.A \ MultiPart.P M y) +
card (MultiPart.P (move M hvi hj) i) * card ((move M hvi hj).A \ MultiPart.P (move M hvi hj) i) +
card (MultiPart.P (move M hvi hj) j) * card ((move M hvi hj).A \ MultiPart.P (move M hvi hj) j)
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/Multipartite.lean | SimpleGraph.mp_deg_sum_move | [199, 1] | [209, 39] | refine' (add_lt_add_iff_left _).mpr _ | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i : ℕ
v : α
j : ℕ
M : MultiPart α
hvi : i ∈ range (M.t + 1) ∧ v ∈ MultiPart.P M i
hj : j ∈ range (M.t + 1) ∧ j ≠ i
hc : card (MultiPart.P M j) + 1 < card (MultiPart.P M i)
⊢ ∑ x in erase (erase (range (M.t + 1)) j) i, card (MultiPart.P M x) * card (M.A \ MultiPart.P M x) +
(card (MultiPart.P M i) * card (M.A \ MultiPart.P M i) + card (MultiPart.P M j) * card (M.A \ MultiPart.P M j)) <
∑ y in erase (erase (range (M.t + 1)) j) i, card (MultiPart.P M y) * card (M.A \ MultiPart.P M y) +
(card (MultiPart.P (move M hvi hj) i) * card ((move M hvi hj).A \ MultiPart.P (move M hvi hj) i) +
card (MultiPart.P (move M hvi hj) j) * card ((move M hvi hj).A \ MultiPart.P (move M hvi hj) j)) | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i : ℕ
v : α
j : ℕ
M : MultiPart α
hvi : i ∈ range (M.t + 1) ∧ v ∈ MultiPart.P M i
hj : j ∈ range (M.t + 1) ∧ j ≠ i
hc : card (MultiPart.P M j) + 1 < card (MultiPart.P M i)
⊢ card (MultiPart.P M i) * card (M.A \ MultiPart.P M i) + card (MultiPart.P M j) * card (M.A \ MultiPart.P M j) <
card (MultiPart.P (move M hvi hj) i) * card ((move M hvi hj).A \ MultiPart.P (move M hvi hj) i) +
card (MultiPart.P (move M hvi hj) j) * card ((move M hvi hj).A \ MultiPart.P (move M hvi hj) 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 : ℕ
M : MultiPart α
hvi : i ∈ range (M.t + 1) ∧ v ∈ MultiPart.P M i
hj : j ∈ range (M.t + 1) ∧ j ≠ i
hc : card (MultiPart.P M j) + 1 < card (MultiPart.P M i)
⊢ ∑ x in erase (erase (range (M.t + 1)) j) i, card (MultiPart.P M x) * card (M.A \ MultiPart.P M x) +
(card (MultiPart.P M i) * card (M.A \ MultiPart.P M i) + card (MultiPart.P M j) * card (M.A \ MultiPart.P M j)) <
∑ y in erase (erase (range (M.t + 1)) j) i, card (MultiPart.P M y) * card (M.A \ MultiPart.P M y) +
(card (MultiPart.P (move M hvi hj) i) * card ((move M hvi hj).A \ MultiPart.P (move M hvi hj) i) +
card (MultiPart.P (move M hvi hj) j) * card ((move M hvi hj).A \ MultiPart.P (move M hvi hj) j))
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/Multipartite.lean | SimpleGraph.mp_deg_sum_move | [199, 1] | [209, 39] | exact mp_deg_sum_move_help hvi hj hc | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i : ℕ
v : α
j : ℕ
M : MultiPart α
hvi : i ∈ range (M.t + 1) ∧ v ∈ MultiPart.P M i
hj : j ∈ range (M.t + 1) ∧ j ≠ i
hc : card (MultiPart.P M j) + 1 < card (MultiPart.P M i)
⊢ card (MultiPart.P M i) * card (M.A \ MultiPart.P M i) + card (MultiPart.P M j) * card (M.A \ MultiPart.P M j) <
card (MultiPart.P (move M hvi hj) i) * card ((move M hvi hj).A \ MultiPart.P (move M hvi hj) i) +
card (MultiPart.P (move M hvi hj) j) * card ((move M hvi hj).A \ MultiPart.P (move M hvi hj) 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 : ℕ
M : MultiPart α
hvi : i ∈ range (M.t + 1) ∧ v ∈ MultiPart.P M i
hj : j ∈ range (M.t + 1) ∧ j ≠ i
hc : card (MultiPart.P M j) + 1 < card (MultiPart.P M i)
⊢ card (MultiPart.P M i) * card (M.A \ MultiPart.P M i) + card (MultiPart.P M j) * card (M.A \ MultiPart.P M j) <
card (MultiPart.P (move M hvi hj) i) * card ((move M hvi hj).A \ MultiPart.P (move M hvi hj) i) +
card (MultiPart.P (move M hvi hj) j) * card ((move M hvi hj).A \ MultiPart.P (move M hvi hj) j)
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/Multipartite.lean | SimpleGraph.sumSqC_dec | [212, 1] | [220, 30] | unfold sumSqC | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i : ℕ
v : α
j : ℕ
M : MultiPart α
hvi : i ∈ range (M.t + 1) ∧ v ∈ MultiPart.P M i
hj : j ∈ range (M.t + 1) ∧ j ≠ i
hc : card (MultiPart.P M j) + 1 < card (MultiPart.P M i)
⊢ sumSqC (move M hvi hj) < sumSqC M | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i : ℕ
v : α
j : ℕ
M : MultiPart α
hvi : i ∈ range (M.t + 1) ∧ v ∈ MultiPart.P M i
hj : j ∈ range (M.t + 1) ∧ j ≠ i
hc : card (MultiPart.P M j) + 1 < card (MultiPart.P M i)
⊢ ∑ i_1 in range ((move M hvi hj).t + 1), card (MultiPart.P (move M hvi hj) i_1) ^ 2 <
∑ i in range (M.t + 1), card (MultiPart.P M i) ^ 2 | 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 : ℕ
M : MultiPart α
hvi : i ∈ range (M.t + 1) ∧ v ∈ MultiPart.P M i
hj : j ∈ range (M.t + 1) ∧ j ≠ i
hc : card (MultiPart.P M j) + 1 < card (MultiPart.P M i)
⊢ sumSqC (move M hvi hj) < sumSqC M
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/Multipartite.lean | SimpleGraph.sumSqC_dec | [212, 1] | [220, 30] | have h3 := mp_deg_sum_move hvi hj hc | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i : ℕ
v : α
j : ℕ
M : MultiPart α
hvi : i ∈ range (M.t + 1) ∧ v ∈ MultiPart.P M i
hj : j ∈ range (M.t + 1) ∧ j ≠ i
hc : card (MultiPart.P M j) + 1 < card (MultiPart.P M i)
⊢ ∑ i_1 in range ((move M hvi hj).t + 1), card (MultiPart.P (move M hvi hj) i_1) ^ 2 <
∑ i in range (M.t + 1), card (MultiPart.P M i) ^ 2 | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i : ℕ
v : α
j : ℕ
M : MultiPart α
hvi : i ∈ range (M.t + 1) ∧ v ∈ MultiPart.P M i
hj : j ∈ range (M.t + 1) ∧ j ≠ i
hc : card (MultiPart.P M j) + 1 < card (MultiPart.P M i)
h3 : ∑ w in M.A, degree (mp M) w < ∑ w in (move M hvi hj).A, degree (mp (move M hvi hj)) w
⊢ ∑ i_1 in range ((move M hvi hj).t + 1), card (MultiPart.P (move M hvi hj) i_1) ^ 2 <
∑ i in range (M.t + 1), card (MultiPart.P M i) ^ 2 | 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 : ℕ
M : MultiPart α
hvi : i ∈ range (M.t + 1) ∧ v ∈ MultiPart.P M i
hj : j ∈ range (M.t + 1) ∧ j ≠ i
hc : card (MultiPart.P M j) + 1 < card (MultiPart.P M i)
⊢ ∑ i_1 in range ((move M hvi hj).t + 1), card (MultiPart.P (move M hvi hj) i_1) ^ 2 <
∑ i in range (M.t + 1), card (MultiPart.P M i) ^ 2
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/Multipartite.lean | SimpleGraph.sumSqC_dec | [212, 1] | [220, 30] | have h1 := mp_deg_sum_sq' M | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i : ℕ
v : α
j : ℕ
M : MultiPart α
hvi : i ∈ range (M.t + 1) ∧ v ∈ MultiPart.P M i
hj : j ∈ range (M.t + 1) ∧ j ≠ i
hc : card (MultiPart.P M j) + 1 < card (MultiPart.P M i)
h3 : ∑ w in M.A, degree (mp M) w < ∑ w in (move M hvi hj).A, degree (mp (move M hvi hj)) w
⊢ ∑ i_1 in range ((move M hvi hj).t + 1), card (MultiPart.P (move M hvi hj) i_1) ^ 2 <
∑ i in range (M.t + 1), card (MultiPart.P M i) ^ 2 | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i : ℕ
v : α
j : ℕ
M : MultiPart α
hvi : i ∈ range (M.t + 1) ∧ v ∈ MultiPart.P M i
hj : j ∈ range (M.t + 1) ∧ j ≠ i
hc : card (MultiPart.P M j) + 1 < card (MultiPart.P M i)
h3 : ∑ w in M.A, degree (mp M) w < ∑ w in (move M hvi hj).A, degree (mp (move M hvi hj)) w
h1 : ∑ v in M.A, degree (mp M) v + ∑ i in range (M.t + 1), card (MultiPart.P M i) ^ 2 = card M.A ^ 2
⊢ ∑ i_1 in range ((move M hvi hj).t + 1), card (MultiPart.P (move M hvi hj) i_1) ^ 2 <
∑ i in range (M.t + 1), card (MultiPart.P M i) ^ 2 | 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 : ℕ
M : MultiPart α
hvi : i ∈ range (M.t + 1) ∧ v ∈ MultiPart.P M i
hj : j ∈ range (M.t + 1) ∧ j ≠ i
hc : card (MultiPart.P M j) + 1 < card (MultiPart.P M i)
h3 : ∑ w in M.A, degree (mp M) w < ∑ w in (move M hvi hj).A, degree (mp (move M hvi hj)) w
⊢ ∑ i_1 in range ((move M hvi hj).t + 1), card (MultiPart.P (move M hvi hj) i_1) ^ 2 <
∑ i in range (M.t + 1), card (MultiPart.P M i) ^ 2
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/Multipartite.lean | SimpleGraph.sumSqC_dec | [212, 1] | [220, 30] | have h2 := mp_deg_sum_sq' (move M hvi hj) | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i : ℕ
v : α
j : ℕ
M : MultiPart α
hvi : i ∈ range (M.t + 1) ∧ v ∈ MultiPart.P M i
hj : j ∈ range (M.t + 1) ∧ j ≠ i
hc : card (MultiPart.P M j) + 1 < card (MultiPart.P M i)
h3 : ∑ w in M.A, degree (mp M) w < ∑ w in (move M hvi hj).A, degree (mp (move M hvi hj)) w
h1 : ∑ v in M.A, degree (mp M) v + ∑ i in range (M.t + 1), card (MultiPart.P M i) ^ 2 = card M.A ^ 2
⊢ ∑ i_1 in range ((move M hvi hj).t + 1), card (MultiPart.P (move M hvi hj) i_1) ^ 2 <
∑ i in range (M.t + 1), card (MultiPart.P M i) ^ 2 | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i : ℕ
v : α
j : ℕ
M : MultiPart α
hvi : i ∈ range (M.t + 1) ∧ v ∈ MultiPart.P M i
hj : j ∈ range (M.t + 1) ∧ j ≠ i
hc : card (MultiPart.P M j) + 1 < card (MultiPart.P M i)
h3 : ∑ w in M.A, degree (mp M) w < ∑ w in (move M hvi hj).A, degree (mp (move M hvi hj)) w
h1 : ∑ v in M.A, degree (mp M) v + ∑ i in range (M.t + 1), card (MultiPart.P M i) ^ 2 = card M.A ^ 2
h2 :
∑ v_1 in (move M hvi hj).A, degree (mp (move M hvi hj)) v_1 +
∑ i_1 in range ((move M hvi hj).t + 1), card (MultiPart.P (move M hvi hj) i_1) ^ 2 =
card (move M hvi hj).A ^ 2
⊢ ∑ i_1 in range ((move M hvi hj).t + 1), card (MultiPart.P (move M hvi hj) i_1) ^ 2 <
∑ i in range (M.t + 1), card (MultiPart.P M i) ^ 2 | 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 : ℕ
M : MultiPart α
hvi : i ∈ range (M.t + 1) ∧ v ∈ MultiPart.P M i
hj : j ∈ range (M.t + 1) ∧ j ≠ i
hc : card (MultiPart.P M j) + 1 < card (MultiPart.P M i)
h3 : ∑ w in M.A, degree (mp M) w < ∑ w in (move M hvi hj).A, degree (mp (move M hvi hj)) w
h1 : ∑ v in M.A, degree (mp M) v + ∑ i in range (M.t + 1), card (MultiPart.P M i) ^ 2 = card M.A ^ 2
⊢ ∑ i_1 in range ((move M hvi hj).t + 1), card (MultiPart.P (move M hvi hj) i_1) ^ 2 <
∑ i in range (M.t + 1), card (MultiPart.P M i) ^ 2
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/Multipartite.lean | SimpleGraph.sumSqC_dec | [212, 1] | [220, 30] | rw [move_a, move_t] at * | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i : ℕ
v : α
j : ℕ
M : MultiPart α
hvi : i ∈ range (M.t + 1) ∧ v ∈ MultiPart.P M i
hj : j ∈ range (M.t + 1) ∧ j ≠ i
hc : card (MultiPart.P M j) + 1 < card (MultiPart.P M i)
h3 : ∑ w in M.A, degree (mp M) w < ∑ w in (move M hvi hj).A, degree (mp (move M hvi hj)) w
h1 : ∑ v in M.A, degree (mp M) v + ∑ i in range (M.t + 1), card (MultiPart.P M i) ^ 2 = card M.A ^ 2
h2 :
∑ v_1 in (move M hvi hj).A, degree (mp (move M hvi hj)) v_1 +
∑ i_1 in range ((move M hvi hj).t + 1), card (MultiPart.P (move M hvi hj) i_1) ^ 2 =
card (move M hvi hj).A ^ 2
⊢ ∑ i_1 in range ((move M hvi hj).t + 1), card (MultiPart.P (move M hvi hj) i_1) ^ 2 <
∑ i in range (M.t + 1), card (MultiPart.P M i) ^ 2 | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i : ℕ
v : α
j : ℕ
M : MultiPart α
hvi : i ∈ range (M.t + 1) ∧ v ∈ MultiPart.P M i
hj : j ∈ range (M.t + 1) ∧ j ≠ i
hc : card (MultiPart.P M j) + 1 < card (MultiPart.P M i)
h3 : ∑ w in M.A, degree (mp M) w < ∑ w in M.A, degree (mp (move M hvi hj)) w
h1 : ∑ v in M.A, degree (mp M) v + ∑ i in range (M.t + 1), card (MultiPart.P M i) ^ 2 = card M.A ^ 2
h2 :
∑ v_1 in M.A, degree (mp (move M hvi hj)) v_1 + ∑ i_1 in range (M.t + 1), card (MultiPart.P (move M hvi hj) i_1) ^ 2 =
card M.A ^ 2
⊢ ∑ i_1 in range (M.t + 1), card (MultiPart.P (move M hvi hj) i_1) ^ 2 <
∑ i in range (M.t + 1), card (MultiPart.P M i) ^ 2 | 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 : ℕ
M : MultiPart α
hvi : i ∈ range (M.t + 1) ∧ v ∈ MultiPart.P M i
hj : j ∈ range (M.t + 1) ∧ j ≠ i
hc : card (MultiPart.P M j) + 1 < card (MultiPart.P M i)
h3 : ∑ w in M.A, degree (mp M) w < ∑ w in (move M hvi hj).A, degree (mp (move M hvi hj)) w
h1 : ∑ v in M.A, degree (mp M) v + ∑ i in range (M.t + 1), card (MultiPart.P M i) ^ 2 = card M.A ^ 2
h2 :
∑ v_1 in (move M hvi hj).A, degree (mp (move M hvi hj)) v_1 +
∑ i_1 in range ((move M hvi hj).t + 1), card (MultiPart.P (move M hvi hj) i_1) ^ 2 =
card (move M hvi hj).A ^ 2
⊢ ∑ i_1 in range ((move M hvi hj).t + 1), card (MultiPart.P (move M hvi hj) i_1) ^ 2 <
∑ i in range (M.t + 1), card (MultiPart.P M i) ^ 2
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/Multipartite.lean | SimpleGraph.sumSqC_dec | [212, 1] | [220, 30] | rw [← h2] at h1 | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i : ℕ
v : α
j : ℕ
M : MultiPart α
hvi : i ∈ range (M.t + 1) ∧ v ∈ MultiPart.P M i
hj : j ∈ range (M.t + 1) ∧ j ≠ i
hc : card (MultiPart.P M j) + 1 < card (MultiPart.P M i)
h3 : ∑ w in M.A, degree (mp M) w < ∑ w in M.A, degree (mp (move M hvi hj)) w
h1 : ∑ v in M.A, degree (mp M) v + ∑ i in range (M.t + 1), card (MultiPart.P M i) ^ 2 = card M.A ^ 2
h2 :
∑ v_1 in M.A, degree (mp (move M hvi hj)) v_1 + ∑ i_1 in range (M.t + 1), card (MultiPart.P (move M hvi hj) i_1) ^ 2 =
card M.A ^ 2
⊢ ∑ i_1 in range (M.t + 1), card (MultiPart.P (move M hvi hj) i_1) ^ 2 <
∑ i in range (M.t + 1), card (MultiPart.P M i) ^ 2 | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i : ℕ
v : α
j : ℕ
M : MultiPart α
hvi : i ∈ range (M.t + 1) ∧ v ∈ MultiPart.P M i
hj : j ∈ range (M.t + 1) ∧ j ≠ i
hc : card (MultiPart.P M j) + 1 < card (MultiPart.P M i)
h3 : ∑ w in M.A, degree (mp M) w < ∑ w in M.A, degree (mp (move M hvi hj)) w
h1 :
∑ v in M.A, degree (mp M) v + ∑ i in range (M.t + 1), card (MultiPart.P M i) ^ 2 =
∑ v_1 in M.A, degree (mp (move M hvi hj)) v_1 + ∑ i_1 in range (M.t + 1), card (MultiPart.P (move M hvi hj) i_1) ^ 2
h2 :
∑ v_1 in M.A, degree (mp (move M hvi hj)) v_1 + ∑ i_1 in range (M.t + 1), card (MultiPart.P (move M hvi hj) i_1) ^ 2 =
card M.A ^ 2
⊢ ∑ i_1 in range (M.t + 1), card (MultiPart.P (move M hvi hj) i_1) ^ 2 <
∑ i in range (M.t + 1), card (MultiPart.P M i) ^ 2 | 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 : ℕ
M : MultiPart α
hvi : i ∈ range (M.t + 1) ∧ v ∈ MultiPart.P M i
hj : j ∈ range (M.t + 1) ∧ j ≠ i
hc : card (MultiPart.P M j) + 1 < card (MultiPart.P M i)
h3 : ∑ w in M.A, degree (mp M) w < ∑ w in M.A, degree (mp (move M hvi hj)) w
h1 : ∑ v in M.A, degree (mp M) v + ∑ i in range (M.t + 1), card (MultiPart.P M i) ^ 2 = card M.A ^ 2
h2 :
∑ v_1 in M.A, degree (mp (move M hvi hj)) v_1 + ∑ i_1 in range (M.t + 1), card (MultiPart.P (move M hvi hj) i_1) ^ 2 =
card M.A ^ 2
⊢ ∑ i_1 in range (M.t + 1), card (MultiPart.P (move M hvi hj) i_1) ^ 2 <
∑ i in range (M.t + 1), card (MultiPart.P M i) ^ 2
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/Multipartite.lean | SimpleGraph.sumSqC_dec | [212, 1] | [220, 30] | linarith | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i : ℕ
v : α
j : ℕ
M : MultiPart α
hvi : i ∈ range (M.t + 1) ∧ v ∈ MultiPart.P M i
hj : j ∈ range (M.t + 1) ∧ j ≠ i
hc : card (MultiPart.P M j) + 1 < card (MultiPart.P M i)
h3 : ∑ w in M.A, degree (mp M) w < ∑ w in M.A, degree (mp (move M hvi hj)) w
h1 :
∑ v in M.A, degree (mp M) v + ∑ i in range (M.t + 1), card (MultiPart.P M i) ^ 2 =
∑ v_1 in M.A, degree (mp (move M hvi hj)) v_1 + ∑ i_1 in range (M.t + 1), card (MultiPart.P (move M hvi hj) i_1) ^ 2
h2 :
∑ v_1 in M.A, degree (mp (move M hvi hj)) v_1 + ∑ i_1 in range (M.t + 1), card (MultiPart.P (move M hvi hj) i_1) ^ 2 =
card M.A ^ 2
⊢ ∑ i_1 in range (M.t + 1), card (MultiPart.P (move M hvi hj) i_1) ^ 2 <
∑ i in range (M.t + 1), card (MultiPart.P M i) ^ 2 | 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 : ℕ
M : MultiPart α
hvi : i ∈ range (M.t + 1) ∧ v ∈ MultiPart.P M i
hj : j ∈ range (M.t + 1) ∧ j ≠ i
hc : card (MultiPart.P M j) + 1 < card (MultiPart.P M i)
h3 : ∑ w in M.A, degree (mp M) w < ∑ w in M.A, degree (mp (move M hvi hj)) w
h1 :
∑ v in M.A, degree (mp M) v + ∑ i in range (M.t + 1), card (MultiPart.P M i) ^ 2 =
∑ v_1 in M.A, degree (mp (move M hvi hj)) v_1 + ∑ i_1 in range (M.t + 1), card (MultiPart.P (move M hvi hj) i_1) ^ 2
h2 :
∑ v_1 in M.A, degree (mp (move M hvi hj)) v_1 + ∑ i_1 in range (M.t + 1), card (MultiPart.P (move M hvi hj) i_1) ^ 2 =
card M.A ^ 2
⊢ ∑ i_1 in range (M.t + 1), card (MultiPart.P (move M hvi hj) i_1) ^ 2 <
∑ i in range (M.t + 1), card (MultiPart.P M i) ^ 2
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/Multipartite.lean | SimpleGraph.moved | [224, 1] | [240, 27] | apply WellFounded.recursion (InvImage.wf sumSqC Nat.lt_wfRel.wf) M | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
M : MultiPart α
⊢ ∃ N, N.A = M.A ∧ N.t = M.t ∧ TuranPartition N ∧ mpDsum M ≤ mpDsum N | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
M : MultiPart α
⊢ ∀ (x : MultiPart α),
(∀ (y : MultiPart α),
InvImage WellFoundedRelation.rel sumSqC y x →
∃ N, N.A = y.A ∧ N.t = y.t ∧ TuranPartition N ∧ mpDsum y ≤ mpDsum N) →
∃ N, N.A = x.A ∧ N.t = x.t ∧ TuranPartition N ∧ mpDsum x ≤ mpDsum N | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
M : MultiPart α
⊢ ∃ N, N.A = M.A ∧ N.t = M.t ∧ TuranPartition N ∧ mpDsum M ≤ mpDsum N
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/Multipartite.lean | SimpleGraph.moved | [224, 1] | [240, 27] | intro X h | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
M : MultiPart α
⊢ ∀ (x : MultiPart α),
(∀ (y : MultiPart α),
InvImage WellFoundedRelation.rel sumSqC y x →
∃ N, N.A = y.A ∧ N.t = y.t ∧ TuranPartition N ∧ mpDsum y ≤ mpDsum N) →
∃ N, N.A = x.A ∧ N.t = x.t ∧ TuranPartition N ∧ mpDsum x ≤ mpDsum N | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
M X : MultiPart α
h :
∀ (y : MultiPart α),
InvImage WellFoundedRelation.rel sumSqC y X → ∃ N, N.A = y.A ∧ N.t = y.t ∧ TuranPartition N ∧ mpDsum y ≤ mpDsum N
⊢ ∃ N, N.A = X.A ∧ N.t = X.t ∧ TuranPartition N ∧ mpDsum X ≤ mpDsum N | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
M : MultiPart α
⊢ ∀ (x : MultiPart α),
(∀ (y : MultiPart α),
InvImage WellFoundedRelation.rel sumSqC y x →
∃ N, N.A = y.A ∧ N.t = y.t ∧ TuranPartition N ∧ mpDsum y ≤ mpDsum N) →
∃ N, N.A = x.A ∧ N.t = x.t ∧ TuranPartition N ∧ mpDsum x ≤ mpDsum N
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/Multipartite.lean | SimpleGraph.moved | [224, 1] | [240, 27] | by_cases h' : TuranPartition X | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
M X : MultiPart α
h :
∀ (y : MultiPart α),
InvImage WellFoundedRelation.rel sumSqC y X → ∃ N, N.A = y.A ∧ N.t = y.t ∧ TuranPartition N ∧ mpDsum y ≤ mpDsum N
⊢ ∃ N, N.A = X.A ∧ N.t = X.t ∧ TuranPartition N ∧ mpDsum X ≤ mpDsum N | case pos
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
M X : MultiPart α
h :
∀ (y : MultiPart α),
InvImage WellFoundedRelation.rel sumSqC y X → ∃ N, N.A = y.A ∧ N.t = y.t ∧ TuranPartition N ∧ mpDsum y ≤ mpDsum N
h' : TuranPartition X
⊢ ∃ N, N.A = X.A ∧ N.t = X.t ∧ TuranPartition N ∧ mpDsum X ≤ mpDsum N
case neg
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
M X : MultiPart α
h :
∀ (y : MultiPart α),
InvImage WellFoundedRelation.rel sumSqC y X → ∃ N, N.A = y.A ∧ N.t = y.t ∧ TuranPartition N ∧ mpDsum y ≤ mpDsum N
h' : ¬TuranPartition X
⊢ ∃ N, N.A = X.A ∧ N.t = X.t ∧ TuranPartition N ∧ mpDsum X ≤ mpDsum N | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
M X : MultiPart α
h :
∀ (y : MultiPart α),
InvImage WellFoundedRelation.rel sumSqC y X → ∃ N, N.A = y.A ∧ N.t = y.t ∧ TuranPartition N ∧ mpDsum y ≤ mpDsum N
⊢ ∃ N, N.A = X.A ∧ N.t = X.t ∧ TuranPartition N ∧ mpDsum X ≤ mpDsum N
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/Multipartite.lean | SimpleGraph.moved | [224, 1] | [240, 27] | exact ⟨X, rfl, rfl, h', le_rfl⟩ | case pos
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
M X : MultiPart α
h :
∀ (y : MultiPart α),
InvImage WellFoundedRelation.rel sumSqC y X → ∃ N, N.A = y.A ∧ N.t = y.t ∧ TuranPartition N ∧ mpDsum y ≤ mpDsum N
h' : TuranPartition X
⊢ ∃ N, N.A = X.A ∧ N.t = X.t ∧ TuranPartition N ∧ mpDsum X ≤ mpDsum N | 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
M X : MultiPart α
h :
∀ (y : MultiPart α),
InvImage WellFoundedRelation.rel sumSqC y X → ∃ N, N.A = y.A ∧ N.t = y.t ∧ TuranPartition N ∧ mpDsum y ≤ mpDsum N
h' : TuranPartition X
⊢ ∃ N, N.A = X.A ∧ N.t = X.t ∧ TuranPartition N ∧ mpDsum X ≤ mpDsum N
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/Multipartite.lean | SimpleGraph.moved | [224, 1] | [240, 27] | obtain ⟨i, hi, j, hj, v, hv, ne, hc⟩ := not_turanPartition_imp h' | case neg
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
M X : MultiPart α
h :
∀ (y : MultiPart α),
InvImage WellFoundedRelation.rel sumSqC y X → ∃ N, N.A = y.A ∧ N.t = y.t ∧ TuranPartition N ∧ mpDsum y ≤ mpDsum N
h' : ¬TuranPartition X
⊢ ∃ N, N.A = X.A ∧ N.t = X.t ∧ TuranPartition N ∧ mpDsum X ≤ mpDsum N | case neg.intro.intro.intro.intro.intro.intro.intro
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
M X : MultiPart α
h :
∀ (y : MultiPart α),
InvImage WellFoundedRelation.rel sumSqC y X → ∃ N, N.A = y.A ∧ N.t = y.t ∧ TuranPartition N ∧ mpDsum y ≤ mpDsum N
h' : ¬TuranPartition X
i : ℕ
hi : i ∈ range (X.t + 1)
j : ℕ
hj : j ∈ range (X.t + 1)
v : α
hv : v ∈ MultiPart.P X i
ne : j ≠ i
hc : card (MultiPart.P X j) + 1 < card (MultiPart.P X i)
⊢ ∃ N, N.A = X.A ∧ N.t = X.t ∧ TuranPartition N ∧ mpDsum X ≤ mpDsum N | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
M X : MultiPart α
h :
∀ (y : MultiPart α),
InvImage WellFoundedRelation.rel sumSqC y X → ∃ N, N.A = y.A ∧ N.t = y.t ∧ TuranPartition N ∧ mpDsum y ≤ mpDsum N
h' : ¬TuranPartition X
⊢ ∃ N, N.A = X.A ∧ N.t = X.t ∧ TuranPartition N ∧ mpDsum X ≤ mpDsum N
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/Multipartite.lean | SimpleGraph.moved | [224, 1] | [240, 27] | set Y := move X ⟨hi, hv⟩ ⟨hj, ne⟩ with hY | case neg.intro.intro.intro.intro.intro.intro.intro
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
M X : MultiPart α
h :
∀ (y : MultiPart α),
InvImage WellFoundedRelation.rel sumSqC y X → ∃ N, N.A = y.A ∧ N.t = y.t ∧ TuranPartition N ∧ mpDsum y ≤ mpDsum N
h' : ¬TuranPartition X
i : ℕ
hi : i ∈ range (X.t + 1)
j : ℕ
hj : j ∈ range (X.t + 1)
v : α
hv : v ∈ MultiPart.P X i
ne : j ≠ i
hc : card (MultiPart.P X j) + 1 < card (MultiPart.P X i)
⊢ ∃ N, N.A = X.A ∧ N.t = X.t ∧ TuranPartition N ∧ mpDsum X ≤ mpDsum N | case neg.intro.intro.intro.intro.intro.intro.intro
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
M X : MultiPart α
h :
∀ (y : MultiPart α),
InvImage WellFoundedRelation.rel sumSqC y X → ∃ N, N.A = y.A ∧ N.t = y.t ∧ TuranPartition N ∧ mpDsum y ≤ mpDsum N
h' : ¬TuranPartition X
i : ℕ
hi : i ∈ range (X.t + 1)
j : ℕ
hj : j ∈ range (X.t + 1)
v : α
hv : v ∈ MultiPart.P X i
ne : j ≠ i
hc : card (MultiPart.P X j) + 1 < card (MultiPart.P X i)
Y : MultiPart α := move X (_ : i ∈ range (X.t + 1) ∧ v ∈ MultiPart.P X i) (_ : j ∈ range (X.t + 1) ∧ j ≠ i)
hY : Y = move X (_ : i ∈ range (X.t + 1) ∧ v ∈ MultiPart.P X i) (_ : j ∈ range (X.t + 1) ∧ j ≠ i)
⊢ ∃ N, N.A = X.A ∧ N.t = X.t ∧ TuranPartition N ∧ mpDsum X ≤ mpDsum N | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.intro.intro.intro.intro.intro.intro.intro
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
M X : MultiPart α
h :
∀ (y : MultiPart α),
InvImage WellFoundedRelation.rel sumSqC y X → ∃ N, N.A = y.A ∧ N.t = y.t ∧ TuranPartition N ∧ mpDsum y ≤ mpDsum N
h' : ¬TuranPartition X
i : ℕ
hi : i ∈ range (X.t + 1)
j : ℕ
hj : j ∈ range (X.t + 1)
v : α
hv : v ∈ MultiPart.P X i
ne : j ≠ i
hc : card (MultiPart.P X j) + 1 < card (MultiPart.P X i)
⊢ ∃ N, N.A = X.A ∧ N.t = X.t ∧ TuranPartition N ∧ mpDsum X ≤ mpDsum N
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/Multipartite.lean | SimpleGraph.moved | [224, 1] | [240, 27] | specialize h Y (sumSqC_dec X ⟨hi, hv⟩ ⟨hj, ne⟩ hc) | case neg.intro.intro.intro.intro.intro.intro.intro
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
M X : MultiPart α
h :
∀ (y : MultiPart α),
InvImage WellFoundedRelation.rel sumSqC y X → ∃ N, N.A = y.A ∧ N.t = y.t ∧ TuranPartition N ∧ mpDsum y ≤ mpDsum N
h' : ¬TuranPartition X
i : ℕ
hi : i ∈ range (X.t + 1)
j : ℕ
hj : j ∈ range (X.t + 1)
v : α
hv : v ∈ MultiPart.P X i
ne : j ≠ i
hc : card (MultiPart.P X j) + 1 < card (MultiPart.P X i)
Y : MultiPart α := move X (_ : i ∈ range (X.t + 1) ∧ v ∈ MultiPart.P X i) (_ : j ∈ range (X.t + 1) ∧ j ≠ i)
hY : Y = move X (_ : i ∈ range (X.t + 1) ∧ v ∈ MultiPart.P X i) (_ : j ∈ range (X.t + 1) ∧ j ≠ i)
⊢ ∃ N, N.A = X.A ∧ N.t = X.t ∧ TuranPartition N ∧ mpDsum X ≤ mpDsum N | case neg.intro.intro.intro.intro.intro.intro.intro
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
M X : MultiPart α
h' : ¬TuranPartition X
i : ℕ
hi : i ∈ range (X.t + 1)
j : ℕ
hj : j ∈ range (X.t + 1)
v : α
hv : v ∈ MultiPart.P X i
ne : j ≠ i
hc : card (MultiPart.P X j) + 1 < card (MultiPart.P X i)
Y : MultiPart α := move X (_ : i ∈ range (X.t + 1) ∧ v ∈ MultiPart.P X i) (_ : j ∈ range (X.t + 1) ∧ j ≠ i)
hY : Y = move X (_ : i ∈ range (X.t + 1) ∧ v ∈ MultiPart.P X i) (_ : j ∈ range (X.t + 1) ∧ j ≠ i)
h : ∃ N, N.A = Y.A ∧ N.t = Y.t ∧ TuranPartition N ∧ mpDsum Y ≤ mpDsum N
⊢ ∃ N, N.A = X.A ∧ N.t = X.t ∧ TuranPartition N ∧ mpDsum X ≤ mpDsum N | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.intro.intro.intro.intro.intro.intro.intro
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
M X : MultiPart α
h :
∀ (y : MultiPart α),
InvImage WellFoundedRelation.rel sumSqC y X → ∃ N, N.A = y.A ∧ N.t = y.t ∧ TuranPartition N ∧ mpDsum y ≤ mpDsum N
h' : ¬TuranPartition X
i : ℕ
hi : i ∈ range (X.t + 1)
j : ℕ
hj : j ∈ range (X.t + 1)
v : α
hv : v ∈ MultiPart.P X i
ne : j ≠ i
hc : card (MultiPart.P X j) + 1 < card (MultiPart.P X i)
Y : MultiPart α := move X (_ : i ∈ range (X.t + 1) ∧ v ∈ MultiPart.P X i) (_ : j ∈ range (X.t + 1) ∧ j ≠ i)
hY : Y = move X (_ : i ∈ range (X.t + 1) ∧ v ∈ MultiPart.P X i) (_ : j ∈ range (X.t + 1) ∧ j ≠ i)
⊢ ∃ N, N.A = X.A ∧ N.t = X.t ∧ TuranPartition N ∧ mpDsum X ≤ mpDsum N
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/Multipartite.lean | SimpleGraph.moved | [224, 1] | [240, 27] | rw [move_t, move_a] at h | case neg.intro.intro.intro.intro.intro.intro.intro
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
M X : MultiPart α
h' : ¬TuranPartition X
i : ℕ
hi : i ∈ range (X.t + 1)
j : ℕ
hj : j ∈ range (X.t + 1)
v : α
hv : v ∈ MultiPart.P X i
ne : j ≠ i
hc : card (MultiPart.P X j) + 1 < card (MultiPart.P X i)
Y : MultiPart α := move X (_ : i ∈ range (X.t + 1) ∧ v ∈ MultiPart.P X i) (_ : j ∈ range (X.t + 1) ∧ j ≠ i)
hY : Y = move X (_ : i ∈ range (X.t + 1) ∧ v ∈ MultiPart.P X i) (_ : j ∈ range (X.t + 1) ∧ j ≠ i)
h : ∃ N, N.A = Y.A ∧ N.t = Y.t ∧ TuranPartition N ∧ mpDsum Y ≤ mpDsum N
⊢ ∃ N, N.A = X.A ∧ N.t = X.t ∧ TuranPartition N ∧ mpDsum X ≤ mpDsum N | case neg.intro.intro.intro.intro.intro.intro.intro
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
M X : MultiPart α
h' : ¬TuranPartition X
i : ℕ
hi : i ∈ range (X.t + 1)
j : ℕ
hj : j ∈ range (X.t + 1)
v : α
hv : v ∈ MultiPart.P X i
ne : j ≠ i
hc : card (MultiPart.P X j) + 1 < card (MultiPart.P X i)
Y : MultiPart α := move X (_ : i ∈ range (X.t + 1) ∧ v ∈ MultiPart.P X i) (_ : j ∈ range (X.t + 1) ∧ j ≠ i)
hY : Y = move X (_ : i ∈ range (X.t + 1) ∧ v ∈ MultiPart.P X i) (_ : j ∈ range (X.t + 1) ∧ j ≠ i)
h : ∃ N, N.A = X.A ∧ N.t = X.t ∧ TuranPartition N ∧ mpDsum Y ≤ mpDsum N
⊢ ∃ N, N.A = X.A ∧ N.t = X.t ∧ TuranPartition N ∧ mpDsum X ≤ mpDsum N | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.intro.intro.intro.intro.intro.intro.intro
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
M X : MultiPart α
h' : ¬TuranPartition X
i : ℕ
hi : i ∈ range (X.t + 1)
j : ℕ
hj : j ∈ range (X.t + 1)
v : α
hv : v ∈ MultiPart.P X i
ne : j ≠ i
hc : card (MultiPart.P X j) + 1 < card (MultiPart.P X i)
Y : MultiPart α := move X (_ : i ∈ range (X.t + 1) ∧ v ∈ MultiPart.P X i) (_ : j ∈ range (X.t + 1) ∧ j ≠ i)
hY : Y = move X (_ : i ∈ range (X.t + 1) ∧ v ∈ MultiPart.P X i) (_ : j ∈ range (X.t + 1) ∧ j ≠ i)
h : ∃ N, N.A = Y.A ∧ N.t = Y.t ∧ TuranPartition N ∧ mpDsum Y ≤ mpDsum N
⊢ ∃ N, N.A = X.A ∧ N.t = X.t ∧ TuranPartition N ∧ mpDsum X ≤ mpDsum N
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/Multipartite.lean | SimpleGraph.moved | [224, 1] | [240, 27] | have := mp_deg_sum_move ⟨hi, hv⟩ ⟨hj, ne⟩ hc | case neg.intro.intro.intro.intro.intro.intro.intro
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
M X : MultiPart α
h' : ¬TuranPartition X
i : ℕ
hi : i ∈ range (X.t + 1)
j : ℕ
hj : j ∈ range (X.t + 1)
v : α
hv : v ∈ MultiPart.P X i
ne : j ≠ i
hc : card (MultiPart.P X j) + 1 < card (MultiPart.P X i)
Y : MultiPart α := move X (_ : i ∈ range (X.t + 1) ∧ v ∈ MultiPart.P X i) (_ : j ∈ range (X.t + 1) ∧ j ≠ i)
hY : Y = move X (_ : i ∈ range (X.t + 1) ∧ v ∈ MultiPart.P X i) (_ : j ∈ range (X.t + 1) ∧ j ≠ i)
h : ∃ N, N.A = X.A ∧ N.t = X.t ∧ TuranPartition N ∧ mpDsum Y ≤ mpDsum N
⊢ ∃ N, N.A = X.A ∧ N.t = X.t ∧ TuranPartition N ∧ mpDsum X ≤ mpDsum N | case neg.intro.intro.intro.intro.intro.intro.intro
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
M X : MultiPart α
h' : ¬TuranPartition X
i : ℕ
hi : i ∈ range (X.t + 1)
j : ℕ
hj : j ∈ range (X.t + 1)
v : α
hv : v ∈ MultiPart.P X i
ne : j ≠ i
hc : card (MultiPart.P X j) + 1 < card (MultiPart.P X i)
Y : MultiPart α := move X (_ : i ∈ range (X.t + 1) ∧ v ∈ MultiPart.P X i) (_ : j ∈ range (X.t + 1) ∧ j ≠ i)
hY : Y = move X (_ : i ∈ range (X.t + 1) ∧ v ∈ MultiPart.P X i) (_ : j ∈ range (X.t + 1) ∧ j ≠ i)
h : ∃ N, N.A = X.A ∧ N.t = X.t ∧ TuranPartition N ∧ mpDsum Y ≤ mpDsum N
this :
∑ w in X.A, degree (mp X) w <
∑ w in (move X (_ : i ∈ range (X.t + 1) ∧ v ∈ MultiPart.P X i) (_ : j ∈ range (X.t + 1) ∧ j ≠ i)).A,
degree (mp (move X (_ : i ∈ range (X.t + 1) ∧ v ∈ MultiPart.P X i) (_ : j ∈ range (X.t + 1) ∧ j ≠ i))) w
⊢ ∃ N, N.A = X.A ∧ N.t = X.t ∧ TuranPartition N ∧ mpDsum X ≤ mpDsum N | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.intro.intro.intro.intro.intro.intro.intro
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
M X : MultiPart α
h' : ¬TuranPartition X
i : ℕ
hi : i ∈ range (X.t + 1)
j : ℕ
hj : j ∈ range (X.t + 1)
v : α
hv : v ∈ MultiPart.P X i
ne : j ≠ i
hc : card (MultiPart.P X j) + 1 < card (MultiPart.P X i)
Y : MultiPart α := move X (_ : i ∈ range (X.t + 1) ∧ v ∈ MultiPart.P X i) (_ : j ∈ range (X.t + 1) ∧ j ≠ i)
hY : Y = move X (_ : i ∈ range (X.t + 1) ∧ v ∈ MultiPart.P X i) (_ : j ∈ range (X.t + 1) ∧ j ≠ i)
h : ∃ N, N.A = X.A ∧ N.t = X.t ∧ TuranPartition N ∧ mpDsum Y ≤ mpDsum N
⊢ ∃ N, N.A = X.A ∧ N.t = X.t ∧ TuranPartition N ∧ mpDsum X ≤ mpDsum N
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/Multipartite.lean | SimpleGraph.moved | [224, 1] | [240, 27] | rw [← mpDsum, ← mpDsum, ← hY] at this | case neg.intro.intro.intro.intro.intro.intro.intro
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
M X : MultiPart α
h' : ¬TuranPartition X
i : ℕ
hi : i ∈ range (X.t + 1)
j : ℕ
hj : j ∈ range (X.t + 1)
v : α
hv : v ∈ MultiPart.P X i
ne : j ≠ i
hc : card (MultiPart.P X j) + 1 < card (MultiPart.P X i)
Y : MultiPart α := move X (_ : i ∈ range (X.t + 1) ∧ v ∈ MultiPart.P X i) (_ : j ∈ range (X.t + 1) ∧ j ≠ i)
hY : Y = move X (_ : i ∈ range (X.t + 1) ∧ v ∈ MultiPart.P X i) (_ : j ∈ range (X.t + 1) ∧ j ≠ i)
h : ∃ N, N.A = X.A ∧ N.t = X.t ∧ TuranPartition N ∧ mpDsum Y ≤ mpDsum N
this :
∑ w in X.A, degree (mp X) w <
∑ w in (move X (_ : i ∈ range (X.t + 1) ∧ v ∈ MultiPart.P X i) (_ : j ∈ range (X.t + 1) ∧ j ≠ i)).A,
degree (mp (move X (_ : i ∈ range (X.t + 1) ∧ v ∈ MultiPart.P X i) (_ : j ∈ range (X.t + 1) ∧ j ≠ i))) w
⊢ ∃ N, N.A = X.A ∧ N.t = X.t ∧ TuranPartition N ∧ mpDsum X ≤ mpDsum N | case neg.intro.intro.intro.intro.intro.intro.intro
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
M X : MultiPart α
h' : ¬TuranPartition X
i : ℕ
hi : i ∈ range (X.t + 1)
j : ℕ
hj : j ∈ range (X.t + 1)
v : α
hv : v ∈ MultiPart.P X i
ne : j ≠ i
hc : card (MultiPart.P X j) + 1 < card (MultiPart.P X i)
Y : MultiPart α := move X (_ : i ∈ range (X.t + 1) ∧ v ∈ MultiPart.P X i) (_ : j ∈ range (X.t + 1) ∧ j ≠ i)
hY : Y = move X (_ : i ∈ range (X.t + 1) ∧ v ∈ MultiPart.P X i) (_ : j ∈ range (X.t + 1) ∧ j ≠ i)
h : ∃ N, N.A = X.A ∧ N.t = X.t ∧ TuranPartition N ∧ mpDsum Y ≤ mpDsum N
this : mpDsum X < mpDsum Y
⊢ ∃ N, N.A = X.A ∧ N.t = X.t ∧ TuranPartition N ∧ mpDsum X ≤ mpDsum N | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.intro.intro.intro.intro.intro.intro.intro
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
M X : MultiPart α
h' : ¬TuranPartition X
i : ℕ
hi : i ∈ range (X.t + 1)
j : ℕ
hj : j ∈ range (X.t + 1)
v : α
hv : v ∈ MultiPart.P X i
ne : j ≠ i
hc : card (MultiPart.P X j) + 1 < card (MultiPart.P X i)
Y : MultiPart α := move X (_ : i ∈ range (X.t + 1) ∧ v ∈ MultiPart.P X i) (_ : j ∈ range (X.t + 1) ∧ j ≠ i)
hY : Y = move X (_ : i ∈ range (X.t + 1) ∧ v ∈ MultiPart.P X i) (_ : j ∈ range (X.t + 1) ∧ j ≠ i)
h : ∃ N, N.A = X.A ∧ N.t = X.t ∧ TuranPartition N ∧ mpDsum Y ≤ mpDsum N
this :
∑ w in X.A, degree (mp X) w <
∑ w in (move X (_ : i ∈ range (X.t + 1) ∧ v ∈ MultiPart.P X i) (_ : j ∈ range (X.t + 1) ∧ j ≠ i)).A,
degree (mp (move X (_ : i ∈ range (X.t + 1) ∧ v ∈ MultiPart.P X i) (_ : j ∈ range (X.t + 1) ∧ j ≠ i))) w
⊢ ∃ N, N.A = X.A ∧ N.t = X.t ∧ TuranPartition N ∧ mpDsum X ≤ mpDsum N
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/Multipartite.lean | SimpleGraph.moved | [224, 1] | [240, 27] | obtain ⟨N, h1, h2, h3, h4⟩ := h | case neg.intro.intro.intro.intro.intro.intro.intro
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
M X : MultiPart α
h' : ¬TuranPartition X
i : ℕ
hi : i ∈ range (X.t + 1)
j : ℕ
hj : j ∈ range (X.t + 1)
v : α
hv : v ∈ MultiPart.P X i
ne : j ≠ i
hc : card (MultiPart.P X j) + 1 < card (MultiPart.P X i)
Y : MultiPart α := move X (_ : i ∈ range (X.t + 1) ∧ v ∈ MultiPart.P X i) (_ : j ∈ range (X.t + 1) ∧ j ≠ i)
hY : Y = move X (_ : i ∈ range (X.t + 1) ∧ v ∈ MultiPart.P X i) (_ : j ∈ range (X.t + 1) ∧ j ≠ i)
h : ∃ N, N.A = X.A ∧ N.t = X.t ∧ TuranPartition N ∧ mpDsum Y ≤ mpDsum N
this : mpDsum X < mpDsum Y
⊢ ∃ N, N.A = X.A ∧ N.t = X.t ∧ TuranPartition N ∧ mpDsum X ≤ mpDsum N | case neg.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
M X : MultiPart α
h' : ¬TuranPartition X
i : ℕ
hi : i ∈ range (X.t + 1)
j : ℕ
hj : j ∈ range (X.t + 1)
v : α
hv : v ∈ MultiPart.P X i
ne : j ≠ i
hc : card (MultiPart.P X j) + 1 < card (MultiPart.P X i)
Y : MultiPart α := move X (_ : i ∈ range (X.t + 1) ∧ v ∈ MultiPart.P X i) (_ : j ∈ range (X.t + 1) ∧ j ≠ i)
hY : Y = move X (_ : i ∈ range (X.t + 1) ∧ v ∈ MultiPart.P X i) (_ : j ∈ range (X.t + 1) ∧ j ≠ i)
this : mpDsum X < mpDsum Y
N : MultiPart α
h1 : N.A = X.A
h2 : N.t = X.t
h3 : TuranPartition N
h4 : mpDsum Y ≤ mpDsum N
⊢ ∃ N, N.A = X.A ∧ N.t = X.t ∧ TuranPartition N ∧ mpDsum X ≤ mpDsum N | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.intro.intro.intro.intro.intro.intro.intro
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
M X : MultiPart α
h' : ¬TuranPartition X
i : ℕ
hi : i ∈ range (X.t + 1)
j : ℕ
hj : j ∈ range (X.t + 1)
v : α
hv : v ∈ MultiPart.P X i
ne : j ≠ i
hc : card (MultiPart.P X j) + 1 < card (MultiPart.P X i)
Y : MultiPart α := move X (_ : i ∈ range (X.t + 1) ∧ v ∈ MultiPart.P X i) (_ : j ∈ range (X.t + 1) ∧ j ≠ i)
hY : Y = move X (_ : i ∈ range (X.t + 1) ∧ v ∈ MultiPart.P X i) (_ : j ∈ range (X.t + 1) ∧ j ≠ i)
h : ∃ N, N.A = X.A ∧ N.t = X.t ∧ TuranPartition N ∧ mpDsum Y ≤ mpDsum N
this : mpDsum X < mpDsum Y
⊢ ∃ N, N.A = X.A ∧ N.t = X.t ∧ TuranPartition N ∧ mpDsum X ≤ mpDsum N
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/Multipartite.lean | SimpleGraph.moved | [224, 1] | [240, 27] | refine ⟨N, h1, h2, h3, ?_⟩ | case neg.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
M X : MultiPart α
h' : ¬TuranPartition X
i : ℕ
hi : i ∈ range (X.t + 1)
j : ℕ
hj : j ∈ range (X.t + 1)
v : α
hv : v ∈ MultiPart.P X i
ne : j ≠ i
hc : card (MultiPart.P X j) + 1 < card (MultiPart.P X i)
Y : MultiPart α := move X (_ : i ∈ range (X.t + 1) ∧ v ∈ MultiPart.P X i) (_ : j ∈ range (X.t + 1) ∧ j ≠ i)
hY : Y = move X (_ : i ∈ range (X.t + 1) ∧ v ∈ MultiPart.P X i) (_ : j ∈ range (X.t + 1) ∧ j ≠ i)
this : mpDsum X < mpDsum Y
N : MultiPart α
h1 : N.A = X.A
h2 : N.t = X.t
h3 : TuranPartition N
h4 : mpDsum Y ≤ mpDsum N
⊢ ∃ N, N.A = X.A ∧ N.t = X.t ∧ TuranPartition N ∧ mpDsum X ≤ mpDsum N | case neg.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
M X : MultiPart α
h' : ¬TuranPartition X
i : ℕ
hi : i ∈ range (X.t + 1)
j : ℕ
hj : j ∈ range (X.t + 1)
v : α
hv : v ∈ MultiPart.P X i
ne : j ≠ i
hc : card (MultiPart.P X j) + 1 < card (MultiPart.P X i)
Y : MultiPart α := move X (_ : i ∈ range (X.t + 1) ∧ v ∈ MultiPart.P X i) (_ : j ∈ range (X.t + 1) ∧ j ≠ i)
hY : Y = move X (_ : i ∈ range (X.t + 1) ∧ v ∈ MultiPart.P X i) (_ : j ∈ range (X.t + 1) ∧ j ≠ i)
this : mpDsum X < mpDsum Y
N : MultiPart α
h1 : N.A = X.A
h2 : N.t = X.t
h3 : TuranPartition N
h4 : mpDsum Y ≤ mpDsum N
⊢ mpDsum X ≤ mpDsum N | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
M X : MultiPart α
h' : ¬TuranPartition X
i : ℕ
hi : i ∈ range (X.t + 1)
j : ℕ
hj : j ∈ range (X.t + 1)
v : α
hv : v ∈ MultiPart.P X i
ne : j ≠ i
hc : card (MultiPart.P X j) + 1 < card (MultiPart.P X i)
Y : MultiPart α := move X (_ : i ∈ range (X.t + 1) ∧ v ∈ MultiPart.P X i) (_ : j ∈ range (X.t + 1) ∧ j ≠ i)
hY : Y = move X (_ : i ∈ range (X.t + 1) ∧ v ∈ MultiPart.P X i) (_ : j ∈ range (X.t + 1) ∧ j ≠ i)
this : mpDsum X < mpDsum Y
N : MultiPart α
h1 : N.A = X.A
h2 : N.t = X.t
h3 : TuranPartition N
h4 : mpDsum Y ≤ mpDsum N
⊢ ∃ N, N.A = X.A ∧ N.t = X.t ∧ TuranPartition N ∧ mpDsum X ≤ mpDsum N
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/Multipartite.lean | SimpleGraph.moved | [224, 1] | [240, 27] | exact this.le.trans h4 | case neg.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
M X : MultiPart α
h' : ¬TuranPartition X
i : ℕ
hi : i ∈ range (X.t + 1)
j : ℕ
hj : j ∈ range (X.t + 1)
v : α
hv : v ∈ MultiPart.P X i
ne : j ≠ i
hc : card (MultiPart.P X j) + 1 < card (MultiPart.P X i)
Y : MultiPart α := move X (_ : i ∈ range (X.t + 1) ∧ v ∈ MultiPart.P X i) (_ : j ∈ range (X.t + 1) ∧ j ≠ i)
hY : Y = move X (_ : i ∈ range (X.t + 1) ∧ v ∈ MultiPart.P X i) (_ : j ∈ range (X.t + 1) ∧ j ≠ i)
this : mpDsum X < mpDsum Y
N : MultiPart α
h1 : N.A = X.A
h2 : N.t = X.t
h3 : TuranPartition N
h4 : mpDsum Y ≤ mpDsum N
⊢ mpDsum X ≤ mpDsum N | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
M X : MultiPart α
h' : ¬TuranPartition X
i : ℕ
hi : i ∈ range (X.t + 1)
j : ℕ
hj : j ∈ range (X.t + 1)
v : α
hv : v ∈ MultiPart.P X i
ne : j ≠ i
hc : card (MultiPart.P X j) + 1 < card (MultiPart.P X i)
Y : MultiPart α := move X (_ : i ∈ range (X.t + 1) ∧ v ∈ MultiPart.P X i) (_ : j ∈ range (X.t + 1) ∧ j ≠ i)
hY : Y = move X (_ : i ∈ range (X.t + 1) ∧ v ∈ MultiPart.P X i) (_ : j ∈ range (X.t + 1) ∧ j ≠ i)
this : mpDsum X < mpDsum Y
N : MultiPart α
h1 : N.A = X.A
h2 : N.t = X.t
h3 : TuranPartition N
h4 : mpDsum Y ≤ mpDsum N
⊢ mpDsum X ≤ mpDsum N
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/Multipartite.lean | SimpleGraph.moved_max | [245, 1] | [259, 14] | intro hA ht him h1 | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
M N : MultiPart α
⊢ M.A = N.A → M.t = N.t → TuranPartition M → ¬TuranPartition N → mpDsum N < mpDsum M | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
M N : MultiPart α
hA : M.A = N.A
ht : M.t = N.t
him : TuranPartition M
h1 : ¬TuranPartition N
⊢ mpDsum N < mpDsum M | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
M N : MultiPart α
⊢ M.A = N.A → M.t = N.t → TuranPartition M → ¬TuranPartition N → mpDsum N < mpDsum M
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/Multipartite.lean | SimpleGraph.moved_max | [245, 1] | [259, 14] | obtain ⟨i, hi, j, hj, v, hv, ne, hc⟩ := not_turanPartition_imp h1 | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
M N : MultiPart α
hA : M.A = N.A
ht : M.t = N.t
him : TuranPartition M
h1 : ¬TuranPartition N
⊢ mpDsum N < mpDsum M | case intro.intro.intro.intro.intro.intro.intro
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
M N : MultiPart α
hA : M.A = N.A
ht : M.t = N.t
him : TuranPartition M
h1 : ¬TuranPartition N
i : ℕ
hi : i ∈ range (N.t + 1)
j : ℕ
hj : j ∈ range (N.t + 1)
v : α
hv : v ∈ MultiPart.P N i
ne : j ≠ i
hc : card (MultiPart.P N j) + 1 < card (MultiPart.P N i)
⊢ mpDsum N < mpDsum M | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
M N : MultiPart α
hA : M.A = N.A
ht : M.t = N.t
him : TuranPartition M
h1 : ¬TuranPartition N
⊢ mpDsum N < mpDsum M
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/Multipartite.lean | SimpleGraph.moved_max | [245, 1] | [259, 14] | set O := move N ⟨hi, hv⟩ ⟨hj, ne⟩ | case intro.intro.intro.intro.intro.intro.intro
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
M N : MultiPart α
hA : M.A = N.A
ht : M.t = N.t
him : TuranPartition M
h1 : ¬TuranPartition N
i : ℕ
hi : i ∈ range (N.t + 1)
j : ℕ
hj : j ∈ range (N.t + 1)
v : α
hv : v ∈ MultiPart.P N i
ne : j ≠ i
hc : card (MultiPart.P N j) + 1 < card (MultiPart.P N i)
⊢ mpDsum N < mpDsum M | case intro.intro.intro.intro.intro.intro.intro
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
M N : MultiPart α
hA : M.A = N.A
ht : M.t = N.t
him : TuranPartition M
h1 : ¬TuranPartition N
i : ℕ
hi : i ∈ range (N.t + 1)
j : ℕ
hj : j ∈ range (N.t + 1)
v : α
hv : v ∈ MultiPart.P N i
ne : j ≠ i
hc : card (MultiPart.P N j) + 1 < card (MultiPart.P N i)
O : MultiPart α := move N (_ : i ∈ range (N.t + 1) ∧ v ∈ MultiPart.P N i) (_ : j ∈ range (N.t + 1) ∧ j ≠ i)
⊢ mpDsum N < mpDsum M | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro.intro.intro
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
M N : MultiPart α
hA : M.A = N.A
ht : M.t = N.t
him : TuranPartition M
h1 : ¬TuranPartition N
i : ℕ
hi : i ∈ range (N.t + 1)
j : ℕ
hj : j ∈ range (N.t + 1)
v : α
hv : v ∈ MultiPart.P N i
ne : j ≠ i
hc : card (MultiPart.P N j) + 1 < card (MultiPart.P N i)
⊢ mpDsum N < mpDsum M
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/Multipartite.lean | SimpleGraph.moved_max | [245, 1] | [259, 14] | have Ns : mpDsum N < mpDsum O := mp_deg_sum_move ⟨hi, hv⟩ ⟨hj, ne⟩ hc | case intro.intro.intro.intro.intro.intro.intro
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
M N : MultiPart α
hA : M.A = N.A
ht : M.t = N.t
him : TuranPartition M
h1 : ¬TuranPartition N
i : ℕ
hi : i ∈ range (N.t + 1)
j : ℕ
hj : j ∈ range (N.t + 1)
v : α
hv : v ∈ MultiPart.P N i
ne : j ≠ i
hc : card (MultiPart.P N j) + 1 < card (MultiPart.P N i)
O : MultiPart α := move N (_ : i ∈ range (N.t + 1) ∧ v ∈ MultiPart.P N i) (_ : j ∈ range (N.t + 1) ∧ j ≠ i)
⊢ mpDsum N < mpDsum M | case intro.intro.intro.intro.intro.intro.intro
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
M N : MultiPart α
hA : M.A = N.A
ht : M.t = N.t
him : TuranPartition M
h1 : ¬TuranPartition N
i : ℕ
hi : i ∈ range (N.t + 1)
j : ℕ
hj : j ∈ range (N.t + 1)
v : α
hv : v ∈ MultiPart.P N i
ne : j ≠ i
hc : card (MultiPart.P N j) + 1 < card (MultiPart.P N i)
O : MultiPart α := move N (_ : i ∈ range (N.t + 1) ∧ v ∈ MultiPart.P N i) (_ : j ∈ range (N.t + 1) ∧ j ≠ i)
Ns : mpDsum N < mpDsum O
⊢ mpDsum N < mpDsum M | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro.intro.intro
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
M N : MultiPart α
hA : M.A = N.A
ht : M.t = N.t
him : TuranPartition M
h1 : ¬TuranPartition N
i : ℕ
hi : i ∈ range (N.t + 1)
j : ℕ
hj : j ∈ range (N.t + 1)
v : α
hv : v ∈ MultiPart.P N i
ne : j ≠ i
hc : card (MultiPart.P N j) + 1 < card (MultiPart.P N i)
O : MultiPart α := move N (_ : i ∈ range (N.t + 1) ∧ v ∈ MultiPart.P N i) (_ : j ∈ range (N.t + 1) ∧ j ≠ i)
⊢ mpDsum N < mpDsum M
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/Multipartite.lean | SimpleGraph.moved_max | [245, 1] | [259, 14] | obtain ⟨Q, QA, Qt, Qim, Qs⟩ := moved O | case intro.intro.intro.intro.intro.intro.intro
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
M N : MultiPart α
hA : M.A = N.A
ht : M.t = N.t
him : TuranPartition M
h1 : ¬TuranPartition N
i : ℕ
hi : i ∈ range (N.t + 1)
j : ℕ
hj : j ∈ range (N.t + 1)
v : α
hv : v ∈ MultiPart.P N i
ne : j ≠ i
hc : card (MultiPart.P N j) + 1 < card (MultiPart.P N i)
O : MultiPart α := move N (_ : i ∈ range (N.t + 1) ∧ v ∈ MultiPart.P N i) (_ : j ∈ range (N.t + 1) ∧ j ≠ i)
Ns : mpDsum N < mpDsum O
⊢ mpDsum N < mpDsum M | case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
M N : MultiPart α
hA : M.A = N.A
ht : M.t = N.t
him : TuranPartition M
h1 : ¬TuranPartition N
i : ℕ
hi : i ∈ range (N.t + 1)
j : ℕ
hj : j ∈ range (N.t + 1)
v : α
hv : v ∈ MultiPart.P N i
ne : j ≠ i
hc : card (MultiPart.P N j) + 1 < card (MultiPart.P N i)
O : MultiPart α := move N (_ : i ∈ range (N.t + 1) ∧ v ∈ MultiPart.P N i) (_ : j ∈ range (N.t + 1) ∧ j ≠ i)
Ns : mpDsum N < mpDsum O
Q : MultiPart α
QA : Q.A = O.A
Qt : Q.t = O.t
Qim : TuranPartition Q
Qs : mpDsum O ≤ mpDsum Q
⊢ mpDsum N < mpDsum M | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro.intro.intro
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
M N : MultiPart α
hA : M.A = N.A
ht : M.t = N.t
him : TuranPartition M
h1 : ¬TuranPartition N
i : ℕ
hi : i ∈ range (N.t + 1)
j : ℕ
hj : j ∈ range (N.t + 1)
v : α
hv : v ∈ MultiPart.P N i
ne : j ≠ i
hc : card (MultiPart.P N j) + 1 < card (MultiPart.P N i)
O : MultiPart α := move N (_ : i ∈ range (N.t + 1) ∧ v ∈ MultiPart.P N i) (_ : j ∈ range (N.t + 1) ∧ j ≠ i)
Ns : mpDsum N < mpDsum O
⊢ mpDsum N < mpDsum M
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/Multipartite.lean | SimpleGraph.moved_max | [245, 1] | [259, 14] | have := turanPartition_deg_sum_eq M Q | case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
M N : MultiPart α
hA : M.A = N.A
ht : M.t = N.t
him : TuranPartition M
h1 : ¬TuranPartition N
i : ℕ
hi : i ∈ range (N.t + 1)
j : ℕ
hj : j ∈ range (N.t + 1)
v : α
hv : v ∈ MultiPart.P N i
ne : j ≠ i
hc : card (MultiPart.P N j) + 1 < card (MultiPart.P N i)
O : MultiPart α := move N (_ : i ∈ range (N.t + 1) ∧ v ∈ MultiPart.P N i) (_ : j ∈ range (N.t + 1) ∧ j ≠ i)
Ns : mpDsum N < mpDsum O
Q : MultiPart α
QA : Q.A = O.A
Qt : Q.t = O.t
Qim : TuranPartition Q
Qs : mpDsum O ≤ mpDsum Q
⊢ mpDsum N < mpDsum M | case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
M N : MultiPart α
hA : M.A = N.A
ht : M.t = N.t
him : TuranPartition M
h1 : ¬TuranPartition N
i : ℕ
hi : i ∈ range (N.t + 1)
j : ℕ
hj : j ∈ range (N.t + 1)
v : α
hv : v ∈ MultiPart.P N i
ne : j ≠ i
hc : card (MultiPart.P N j) + 1 < card (MultiPart.P N i)
O : MultiPart α := move N (_ : i ∈ range (N.t + 1) ∧ v ∈ MultiPart.P N i) (_ : j ∈ range (N.t + 1) ∧ j ≠ i)
Ns : mpDsum N < mpDsum O
Q : MultiPart α
QA : Q.A = O.A
Qt : Q.t = O.t
Qim : TuranPartition Q
Qs : mpDsum O ≤ mpDsum Q
this : M.A = Q.A → M.t = Q.t → TuranPartition M → TuranPartition Q → mpDsum M = mpDsum Q
⊢ mpDsum N < mpDsum M | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
M N : MultiPart α
hA : M.A = N.A
ht : M.t = N.t
him : TuranPartition M
h1 : ¬TuranPartition N
i : ℕ
hi : i ∈ range (N.t + 1)
j : ℕ
hj : j ∈ range (N.t + 1)
v : α
hv : v ∈ MultiPart.P N i
ne : j ≠ i
hc : card (MultiPart.P N j) + 1 < card (MultiPart.P N i)
O : MultiPart α := move N (_ : i ∈ range (N.t + 1) ∧ v ∈ MultiPart.P N i) (_ : j ∈ range (N.t + 1) ∧ j ≠ i)
Ns : mpDsum N < mpDsum O
Q : MultiPart α
QA : Q.A = O.A
Qt : Q.t = O.t
Qim : TuranPartition Q
Qs : mpDsum O ≤ mpDsum Q
⊢ mpDsum N < mpDsum M
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/Multipartite.lean | SimpleGraph.moved_max | [245, 1] | [259, 14] | rw [this] | case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
M N : MultiPart α
hA : M.A = N.A
ht : M.t = N.t
him : TuranPartition M
h1 : ¬TuranPartition N
i : ℕ
hi : i ∈ range (N.t + 1)
j : ℕ
hj : j ∈ range (N.t + 1)
v : α
hv : v ∈ MultiPart.P N i
ne : j ≠ i
hc : card (MultiPart.P N j) + 1 < card (MultiPart.P N i)
O : MultiPart α := move N (_ : i ∈ range (N.t + 1) ∧ v ∈ MultiPart.P N i) (_ : j ∈ range (N.t + 1) ∧ j ≠ i)
Ns : mpDsum N < mpDsum O
Q : MultiPart α
QA : Q.A = O.A
Qt : Q.t = O.t
Qim : TuranPartition Q
Qs : mpDsum O ≤ mpDsum Q
this : M.A = Q.A → M.t = Q.t → TuranPartition M → TuranPartition Q → mpDsum M = mpDsum Q
⊢ mpDsum N < mpDsum M | case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
M N : MultiPart α
hA : M.A = N.A
ht : M.t = N.t
him : TuranPartition M
h1 : ¬TuranPartition N
i : ℕ
hi : i ∈ range (N.t + 1)
j : ℕ
hj : j ∈ range (N.t + 1)
v : α
hv : v ∈ MultiPart.P N i
ne : j ≠ i
hc : card (MultiPart.P N j) + 1 < card (MultiPart.P N i)
O : MultiPart α := move N (_ : i ∈ range (N.t + 1) ∧ v ∈ MultiPart.P N i) (_ : j ∈ range (N.t + 1) ∧ j ≠ i)
Ns : mpDsum N < mpDsum O
Q : MultiPart α
QA : Q.A = O.A
Qt : Q.t = O.t
Qim : TuranPartition Q
Qs : mpDsum O ≤ mpDsum Q
this : M.A = Q.A → M.t = Q.t → TuranPartition M → TuranPartition Q → mpDsum M = mpDsum Q
⊢ mpDsum N < mpDsum Q
case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.a
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
M N : MultiPart α
hA : M.A = N.A
ht : M.t = N.t
him : TuranPartition M
h1 : ¬TuranPartition N
i : ℕ
hi : i ∈ range (N.t + 1)
j : ℕ
hj : j ∈ range (N.t + 1)
v : α
hv : v ∈ MultiPart.P N i
ne : j ≠ i
hc : card (MultiPart.P N j) + 1 < card (MultiPart.P N i)
O : MultiPart α := move N (_ : i ∈ range (N.t + 1) ∧ v ∈ MultiPart.P N i) (_ : j ∈ range (N.t + 1) ∧ j ≠ i)
Ns : mpDsum N < mpDsum O
Q : MultiPart α
QA : Q.A = O.A
Qt : Q.t = O.t
Qim : TuranPartition Q
Qs : mpDsum O ≤ mpDsum Q
this : M.A = Q.A → M.t = Q.t → TuranPartition M → TuranPartition Q → mpDsum M = mpDsum Q
⊢ M.A = Q.A
case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.a
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
M N : MultiPart α
hA : M.A = N.A
ht : M.t = N.t
him : TuranPartition M
h1 : ¬TuranPartition N
i : ℕ
hi : i ∈ range (N.t + 1)
j : ℕ
hj : j ∈ range (N.t + 1)
v : α
hv : v ∈ MultiPart.P N i
ne : j ≠ i
hc : card (MultiPart.P N j) + 1 < card (MultiPart.P N i)
O : MultiPart α := move N (_ : i ∈ range (N.t + 1) ∧ v ∈ MultiPart.P N i) (_ : j ∈ range (N.t + 1) ∧ j ≠ i)
Ns : mpDsum N < mpDsum O
Q : MultiPart α
QA : Q.A = O.A
Qt : Q.t = O.t
Qim : TuranPartition Q
Qs : mpDsum O ≤ mpDsum Q
this : M.A = Q.A → M.t = Q.t → TuranPartition M → TuranPartition Q → mpDsum M = mpDsum Q
⊢ M.t = Q.t
case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.a
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
M N : MultiPart α
hA : M.A = N.A
ht : M.t = N.t
him : TuranPartition M
h1 : ¬TuranPartition N
i : ℕ
hi : i ∈ range (N.t + 1)
j : ℕ
hj : j ∈ range (N.t + 1)
v : α
hv : v ∈ MultiPart.P N i
ne : j ≠ i
hc : card (MultiPart.P N j) + 1 < card (MultiPart.P N i)
O : MultiPart α := move N (_ : i ∈ range (N.t + 1) ∧ v ∈ MultiPart.P N i) (_ : j ∈ range (N.t + 1) ∧ j ≠ i)
Ns : mpDsum N < mpDsum O
Q : MultiPart α
QA : Q.A = O.A
Qt : Q.t = O.t
Qim : TuranPartition Q
Qs : mpDsum O ≤ mpDsum Q
this : M.A = Q.A → M.t = Q.t → TuranPartition M → TuranPartition Q → mpDsum M = mpDsum Q
⊢ TuranPartition M
case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.a
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
M N : MultiPart α
hA : M.A = N.A
ht : M.t = N.t
him : TuranPartition M
h1 : ¬TuranPartition N
i : ℕ
hi : i ∈ range (N.t + 1)
j : ℕ
hj : j ∈ range (N.t + 1)
v : α
hv : v ∈ MultiPart.P N i
ne : j ≠ i
hc : card (MultiPart.P N j) + 1 < card (MultiPart.P N i)
O : MultiPart α := move N (_ : i ∈ range (N.t + 1) ∧ v ∈ MultiPart.P N i) (_ : j ∈ range (N.t + 1) ∧ j ≠ i)
Ns : mpDsum N < mpDsum O
Q : MultiPart α
QA : Q.A = O.A
Qt : Q.t = O.t
Qim : TuranPartition Q
Qs : mpDsum O ≤ mpDsum Q
this : M.A = Q.A → M.t = Q.t → TuranPartition M → TuranPartition Q → mpDsum M = mpDsum Q
⊢ TuranPartition Q | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
M N : MultiPart α
hA : M.A = N.A
ht : M.t = N.t
him : TuranPartition M
h1 : ¬TuranPartition N
i : ℕ
hi : i ∈ range (N.t + 1)
j : ℕ
hj : j ∈ range (N.t + 1)
v : α
hv : v ∈ MultiPart.P N i
ne : j ≠ i
hc : card (MultiPart.P N j) + 1 < card (MultiPart.P N i)
O : MultiPart α := move N (_ : i ∈ range (N.t + 1) ∧ v ∈ MultiPart.P N i) (_ : j ∈ range (N.t + 1) ∧ j ≠ i)
Ns : mpDsum N < mpDsum O
Q : MultiPart α
QA : Q.A = O.A
Qt : Q.t = O.t
Qim : TuranPartition Q
Qs : mpDsum O ≤ mpDsum Q
this : M.A = Q.A → M.t = Q.t → TuranPartition M → TuranPartition Q → mpDsum M = mpDsum Q
⊢ mpDsum N < mpDsum M
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/Multipartite.lean | SimpleGraph.moved_max | [245, 1] | [259, 14] | exact lt_of_lt_of_le Ns Qs | case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
M N : MultiPart α
hA : M.A = N.A
ht : M.t = N.t
him : TuranPartition M
h1 : ¬TuranPartition N
i : ℕ
hi : i ∈ range (N.t + 1)
j : ℕ
hj : j ∈ range (N.t + 1)
v : α
hv : v ∈ MultiPart.P N i
ne : j ≠ i
hc : card (MultiPart.P N j) + 1 < card (MultiPart.P N i)
O : MultiPart α := move N (_ : i ∈ range (N.t + 1) ∧ v ∈ MultiPart.P N i) (_ : j ∈ range (N.t + 1) ∧ j ≠ i)
Ns : mpDsum N < mpDsum O
Q : MultiPart α
QA : Q.A = O.A
Qt : Q.t = O.t
Qim : TuranPartition Q
Qs : mpDsum O ≤ mpDsum Q
this : M.A = Q.A → M.t = Q.t → TuranPartition M → TuranPartition Q → mpDsum M = mpDsum Q
⊢ mpDsum N < mpDsum Q | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
M N : MultiPart α
hA : M.A = N.A
ht : M.t = N.t
him : TuranPartition M
h1 : ¬TuranPartition N
i : ℕ
hi : i ∈ range (N.t + 1)
j : ℕ
hj : j ∈ range (N.t + 1)
v : α
hv : v ∈ MultiPart.P N i
ne : j ≠ i
hc : card (MultiPart.P N j) + 1 < card (MultiPart.P N i)
O : MultiPart α := move N (_ : i ∈ range (N.t + 1) ∧ v ∈ MultiPart.P N i) (_ : j ∈ range (N.t + 1) ∧ j ≠ i)
Ns : mpDsum N < mpDsum O
Q : MultiPart α
QA : Q.A = O.A
Qt : Q.t = O.t
Qim : TuranPartition Q
Qs : mpDsum O ≤ mpDsum Q
this : M.A = Q.A → M.t = Q.t → TuranPartition M → TuranPartition Q → mpDsum M = mpDsum Q
⊢ mpDsum N < mpDsum Q
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/Multipartite.lean | SimpleGraph.moved_max | [245, 1] | [259, 14] | rw [hA, QA] | case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.a
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
M N : MultiPart α
hA : M.A = N.A
ht : M.t = N.t
him : TuranPartition M
h1 : ¬TuranPartition N
i : ℕ
hi : i ∈ range (N.t + 1)
j : ℕ
hj : j ∈ range (N.t + 1)
v : α
hv : v ∈ MultiPart.P N i
ne : j ≠ i
hc : card (MultiPart.P N j) + 1 < card (MultiPart.P N i)
O : MultiPart α := move N (_ : i ∈ range (N.t + 1) ∧ v ∈ MultiPart.P N i) (_ : j ∈ range (N.t + 1) ∧ j ≠ i)
Ns : mpDsum N < mpDsum O
Q : MultiPart α
QA : Q.A = O.A
Qt : Q.t = O.t
Qim : TuranPartition Q
Qs : mpDsum O ≤ mpDsum Q
this : M.A = Q.A → M.t = Q.t → TuranPartition M → TuranPartition Q → mpDsum M = mpDsum Q
⊢ M.A = Q.A | case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.a
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
M N : MultiPart α
hA : M.A = N.A
ht : M.t = N.t
him : TuranPartition M
h1 : ¬TuranPartition N
i : ℕ
hi : i ∈ range (N.t + 1)
j : ℕ
hj : j ∈ range (N.t + 1)
v : α
hv : v ∈ MultiPart.P N i
ne : j ≠ i
hc : card (MultiPart.P N j) + 1 < card (MultiPart.P N i)
O : MultiPart α := move N (_ : i ∈ range (N.t + 1) ∧ v ∈ MultiPart.P N i) (_ : j ∈ range (N.t + 1) ∧ j ≠ i)
Ns : mpDsum N < mpDsum O
Q : MultiPart α
QA : Q.A = O.A
Qt : Q.t = O.t
Qim : TuranPartition Q
Qs : mpDsum O ≤ mpDsum Q
this : M.A = Q.A → M.t = Q.t → TuranPartition M → TuranPartition Q → mpDsum M = mpDsum Q
⊢ N.A = O.A | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.a
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
M N : MultiPart α
hA : M.A = N.A
ht : M.t = N.t
him : TuranPartition M
h1 : ¬TuranPartition N
i : ℕ
hi : i ∈ range (N.t + 1)
j : ℕ
hj : j ∈ range (N.t + 1)
v : α
hv : v ∈ MultiPart.P N i
ne : j ≠ i
hc : card (MultiPart.P N j) + 1 < card (MultiPart.P N i)
O : MultiPart α := move N (_ : i ∈ range (N.t + 1) ∧ v ∈ MultiPart.P N i) (_ : j ∈ range (N.t + 1) ∧ j ≠ i)
Ns : mpDsum N < mpDsum O
Q : MultiPart α
QA : Q.A = O.A
Qt : Q.t = O.t
Qim : TuranPartition Q
Qs : mpDsum O ≤ mpDsum Q
this : M.A = Q.A → M.t = Q.t → TuranPartition M → TuranPartition Q → mpDsum M = mpDsum Q
⊢ M.A = Q.A
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/Multipartite.lean | SimpleGraph.moved_max | [245, 1] | [259, 14] | have NOA : N.A = O.A := move_a ⟨hi, hv⟩ ⟨hj, ne⟩ | case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.a
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
M N : MultiPart α
hA : M.A = N.A
ht : M.t = N.t
him : TuranPartition M
h1 : ¬TuranPartition N
i : ℕ
hi : i ∈ range (N.t + 1)
j : ℕ
hj : j ∈ range (N.t + 1)
v : α
hv : v ∈ MultiPart.P N i
ne : j ≠ i
hc : card (MultiPart.P N j) + 1 < card (MultiPart.P N i)
O : MultiPart α := move N (_ : i ∈ range (N.t + 1) ∧ v ∈ MultiPart.P N i) (_ : j ∈ range (N.t + 1) ∧ j ≠ i)
Ns : mpDsum N < mpDsum O
Q : MultiPart α
QA : Q.A = O.A
Qt : Q.t = O.t
Qim : TuranPartition Q
Qs : mpDsum O ≤ mpDsum Q
this : M.A = Q.A → M.t = Q.t → TuranPartition M → TuranPartition Q → mpDsum M = mpDsum Q
⊢ N.A = O.A | case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.a
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
M N : MultiPart α
hA : M.A = N.A
ht : M.t = N.t
him : TuranPartition M
h1 : ¬TuranPartition N
i : ℕ
hi : i ∈ range (N.t + 1)
j : ℕ
hj : j ∈ range (N.t + 1)
v : α
hv : v ∈ MultiPart.P N i
ne : j ≠ i
hc : card (MultiPart.P N j) + 1 < card (MultiPart.P N i)
O : MultiPart α := move N (_ : i ∈ range (N.t + 1) ∧ v ∈ MultiPart.P N i) (_ : j ∈ range (N.t + 1) ∧ j ≠ i)
Ns : mpDsum N < mpDsum O
Q : MultiPart α
QA : Q.A = O.A
Qt : Q.t = O.t
Qim : TuranPartition Q
Qs : mpDsum O ≤ mpDsum Q
this : M.A = Q.A → M.t = Q.t → TuranPartition M → TuranPartition Q → mpDsum M = mpDsum Q
NOA : N.A = O.A
⊢ N.A = O.A | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.a
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
M N : MultiPart α
hA : M.A = N.A
ht : M.t = N.t
him : TuranPartition M
h1 : ¬TuranPartition N
i : ℕ
hi : i ∈ range (N.t + 1)
j : ℕ
hj : j ∈ range (N.t + 1)
v : α
hv : v ∈ MultiPart.P N i
ne : j ≠ i
hc : card (MultiPart.P N j) + 1 < card (MultiPart.P N i)
O : MultiPart α := move N (_ : i ∈ range (N.t + 1) ∧ v ∈ MultiPart.P N i) (_ : j ∈ range (N.t + 1) ∧ j ≠ i)
Ns : mpDsum N < mpDsum O
Q : MultiPart α
QA : Q.A = O.A
Qt : Q.t = O.t
Qim : TuranPartition Q
Qs : mpDsum O ≤ mpDsum Q
this : M.A = Q.A → M.t = Q.t → TuranPartition M → TuranPartition Q → mpDsum M = mpDsum Q
⊢ N.A = O.A
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/Multipartite.lean | SimpleGraph.moved_max | [245, 1] | [259, 14] | exact NOA | case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.a
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
M N : MultiPart α
hA : M.A = N.A
ht : M.t = N.t
him : TuranPartition M
h1 : ¬TuranPartition N
i : ℕ
hi : i ∈ range (N.t + 1)
j : ℕ
hj : j ∈ range (N.t + 1)
v : α
hv : v ∈ MultiPart.P N i
ne : j ≠ i
hc : card (MultiPart.P N j) + 1 < card (MultiPart.P N i)
O : MultiPart α := move N (_ : i ∈ range (N.t + 1) ∧ v ∈ MultiPart.P N i) (_ : j ∈ range (N.t + 1) ∧ j ≠ i)
Ns : mpDsum N < mpDsum O
Q : MultiPart α
QA : Q.A = O.A
Qt : Q.t = O.t
Qim : TuranPartition Q
Qs : mpDsum O ≤ mpDsum Q
this : M.A = Q.A → M.t = Q.t → TuranPartition M → TuranPartition Q → mpDsum M = mpDsum Q
NOA : N.A = O.A
⊢ N.A = O.A | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.a
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
M N : MultiPart α
hA : M.A = N.A
ht : M.t = N.t
him : TuranPartition M
h1 : ¬TuranPartition N
i : ℕ
hi : i ∈ range (N.t + 1)
j : ℕ
hj : j ∈ range (N.t + 1)
v : α
hv : v ∈ MultiPart.P N i
ne : j ≠ i
hc : card (MultiPart.P N j) + 1 < card (MultiPart.P N i)
O : MultiPart α := move N (_ : i ∈ range (N.t + 1) ∧ v ∈ MultiPart.P N i) (_ : j ∈ range (N.t + 1) ∧ j ≠ i)
Ns : mpDsum N < mpDsum O
Q : MultiPart α
QA : Q.A = O.A
Qt : Q.t = O.t
Qim : TuranPartition Q
Qs : mpDsum O ≤ mpDsum Q
this : M.A = Q.A → M.t = Q.t → TuranPartition M → TuranPartition Q → mpDsum M = mpDsum Q
NOA : N.A = O.A
⊢ N.A = O.A
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/Multipartite.lean | SimpleGraph.moved_max | [245, 1] | [259, 14] | rw [ht, Qt] | case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.a
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
M N : MultiPart α
hA : M.A = N.A
ht : M.t = N.t
him : TuranPartition M
h1 : ¬TuranPartition N
i : ℕ
hi : i ∈ range (N.t + 1)
j : ℕ
hj : j ∈ range (N.t + 1)
v : α
hv : v ∈ MultiPart.P N i
ne : j ≠ i
hc : card (MultiPart.P N j) + 1 < card (MultiPart.P N i)
O : MultiPart α := move N (_ : i ∈ range (N.t + 1) ∧ v ∈ MultiPart.P N i) (_ : j ∈ range (N.t + 1) ∧ j ≠ i)
Ns : mpDsum N < mpDsum O
Q : MultiPart α
QA : Q.A = O.A
Qt : Q.t = O.t
Qim : TuranPartition Q
Qs : mpDsum O ≤ mpDsum Q
this : M.A = Q.A → M.t = Q.t → TuranPartition M → TuranPartition Q → mpDsum M = mpDsum Q
⊢ M.t = Q.t | case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.a
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
M N : MultiPart α
hA : M.A = N.A
ht : M.t = N.t
him : TuranPartition M
h1 : ¬TuranPartition N
i : ℕ
hi : i ∈ range (N.t + 1)
j : ℕ
hj : j ∈ range (N.t + 1)
v : α
hv : v ∈ MultiPart.P N i
ne : j ≠ i
hc : card (MultiPart.P N j) + 1 < card (MultiPart.P N i)
O : MultiPart α := move N (_ : i ∈ range (N.t + 1) ∧ v ∈ MultiPart.P N i) (_ : j ∈ range (N.t + 1) ∧ j ≠ i)
Ns : mpDsum N < mpDsum O
Q : MultiPart α
QA : Q.A = O.A
Qt : Q.t = O.t
Qim : TuranPartition Q
Qs : mpDsum O ≤ mpDsum Q
this : M.A = Q.A → M.t = Q.t → TuranPartition M → TuranPartition Q → mpDsum M = mpDsum Q
⊢ N.t = O.t | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.a
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
M N : MultiPart α
hA : M.A = N.A
ht : M.t = N.t
him : TuranPartition M
h1 : ¬TuranPartition N
i : ℕ
hi : i ∈ range (N.t + 1)
j : ℕ
hj : j ∈ range (N.t + 1)
v : α
hv : v ∈ MultiPart.P N i
ne : j ≠ i
hc : card (MultiPart.P N j) + 1 < card (MultiPart.P N i)
O : MultiPart α := move N (_ : i ∈ range (N.t + 1) ∧ v ∈ MultiPart.P N i) (_ : j ∈ range (N.t + 1) ∧ j ≠ i)
Ns : mpDsum N < mpDsum O
Q : MultiPart α
QA : Q.A = O.A
Qt : Q.t = O.t
Qim : TuranPartition Q
Qs : mpDsum O ≤ mpDsum Q
this : M.A = Q.A → M.t = Q.t → TuranPartition M → TuranPartition Q → mpDsum M = mpDsum Q
⊢ M.t = Q.t
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/Multipartite.lean | SimpleGraph.moved_max | [245, 1] | [259, 14] | have NOt : N.t = O.t := move_t ⟨hi, hv⟩ ⟨hj, ne⟩ | case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.a
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
M N : MultiPart α
hA : M.A = N.A
ht : M.t = N.t
him : TuranPartition M
h1 : ¬TuranPartition N
i : ℕ
hi : i ∈ range (N.t + 1)
j : ℕ
hj : j ∈ range (N.t + 1)
v : α
hv : v ∈ MultiPart.P N i
ne : j ≠ i
hc : card (MultiPart.P N j) + 1 < card (MultiPart.P N i)
O : MultiPart α := move N (_ : i ∈ range (N.t + 1) ∧ v ∈ MultiPart.P N i) (_ : j ∈ range (N.t + 1) ∧ j ≠ i)
Ns : mpDsum N < mpDsum O
Q : MultiPart α
QA : Q.A = O.A
Qt : Q.t = O.t
Qim : TuranPartition Q
Qs : mpDsum O ≤ mpDsum Q
this : M.A = Q.A → M.t = Q.t → TuranPartition M → TuranPartition Q → mpDsum M = mpDsum Q
⊢ N.t = O.t | case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.a
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
M N : MultiPart α
hA : M.A = N.A
ht : M.t = N.t
him : TuranPartition M
h1 : ¬TuranPartition N
i : ℕ
hi : i ∈ range (N.t + 1)
j : ℕ
hj : j ∈ range (N.t + 1)
v : α
hv : v ∈ MultiPart.P N i
ne : j ≠ i
hc : card (MultiPart.P N j) + 1 < card (MultiPart.P N i)
O : MultiPart α := move N (_ : i ∈ range (N.t + 1) ∧ v ∈ MultiPart.P N i) (_ : j ∈ range (N.t + 1) ∧ j ≠ i)
Ns : mpDsum N < mpDsum O
Q : MultiPart α
QA : Q.A = O.A
Qt : Q.t = O.t
Qim : TuranPartition Q
Qs : mpDsum O ≤ mpDsum Q
this : M.A = Q.A → M.t = Q.t → TuranPartition M → TuranPartition Q → mpDsum M = mpDsum Q
NOt : N.t = O.t
⊢ N.t = O.t | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.a
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
M N : MultiPart α
hA : M.A = N.A
ht : M.t = N.t
him : TuranPartition M
h1 : ¬TuranPartition N
i : ℕ
hi : i ∈ range (N.t + 1)
j : ℕ
hj : j ∈ range (N.t + 1)
v : α
hv : v ∈ MultiPart.P N i
ne : j ≠ i
hc : card (MultiPart.P N j) + 1 < card (MultiPart.P N i)
O : MultiPart α := move N (_ : i ∈ range (N.t + 1) ∧ v ∈ MultiPart.P N i) (_ : j ∈ range (N.t + 1) ∧ j ≠ i)
Ns : mpDsum N < mpDsum O
Q : MultiPart α
QA : Q.A = O.A
Qt : Q.t = O.t
Qim : TuranPartition Q
Qs : mpDsum O ≤ mpDsum Q
this : M.A = Q.A → M.t = Q.t → TuranPartition M → TuranPartition Q → mpDsum M = mpDsum Q
⊢ N.t = O.t
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/Multipartite.lean | SimpleGraph.moved_max | [245, 1] | [259, 14] | exact NOt | case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.a
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
M N : MultiPart α
hA : M.A = N.A
ht : M.t = N.t
him : TuranPartition M
h1 : ¬TuranPartition N
i : ℕ
hi : i ∈ range (N.t + 1)
j : ℕ
hj : j ∈ range (N.t + 1)
v : α
hv : v ∈ MultiPart.P N i
ne : j ≠ i
hc : card (MultiPart.P N j) + 1 < card (MultiPart.P N i)
O : MultiPart α := move N (_ : i ∈ range (N.t + 1) ∧ v ∈ MultiPart.P N i) (_ : j ∈ range (N.t + 1) ∧ j ≠ i)
Ns : mpDsum N < mpDsum O
Q : MultiPart α
QA : Q.A = O.A
Qt : Q.t = O.t
Qim : TuranPartition Q
Qs : mpDsum O ≤ mpDsum Q
this : M.A = Q.A → M.t = Q.t → TuranPartition M → TuranPartition Q → mpDsum M = mpDsum Q
NOt : N.t = O.t
⊢ N.t = O.t | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.a
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
M N : MultiPart α
hA : M.A = N.A
ht : M.t = N.t
him : TuranPartition M
h1 : ¬TuranPartition N
i : ℕ
hi : i ∈ range (N.t + 1)
j : ℕ
hj : j ∈ range (N.t + 1)
v : α
hv : v ∈ MultiPart.P N i
ne : j ≠ i
hc : card (MultiPart.P N j) + 1 < card (MultiPart.P N i)
O : MultiPart α := move N (_ : i ∈ range (N.t + 1) ∧ v ∈ MultiPart.P N i) (_ : j ∈ range (N.t + 1) ∧ j ≠ i)
Ns : mpDsum N < mpDsum O
Q : MultiPart α
QA : Q.A = O.A
Qt : Q.t = O.t
Qim : TuranPartition Q
Qs : mpDsum O ≤ mpDsum Q
this : M.A = Q.A → M.t = Q.t → TuranPartition M → TuranPartition Q → mpDsum M = mpDsum Q
NOt : N.t = O.t
⊢ N.t = O.t
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/Multipartite.lean | SimpleGraph.moved_max | [245, 1] | [259, 14] | exact him | case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.a
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
M N : MultiPart α
hA : M.A = N.A
ht : M.t = N.t
him : TuranPartition M
h1 : ¬TuranPartition N
i : ℕ
hi : i ∈ range (N.t + 1)
j : ℕ
hj : j ∈ range (N.t + 1)
v : α
hv : v ∈ MultiPart.P N i
ne : j ≠ i
hc : card (MultiPart.P N j) + 1 < card (MultiPart.P N i)
O : MultiPart α := move N (_ : i ∈ range (N.t + 1) ∧ v ∈ MultiPart.P N i) (_ : j ∈ range (N.t + 1) ∧ j ≠ i)
Ns : mpDsum N < mpDsum O
Q : MultiPart α
QA : Q.A = O.A
Qt : Q.t = O.t
Qim : TuranPartition Q
Qs : mpDsum O ≤ mpDsum Q
this : M.A = Q.A → M.t = Q.t → TuranPartition M → TuranPartition Q → mpDsum M = mpDsum Q
⊢ TuranPartition M | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.a
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
M N : MultiPart α
hA : M.A = N.A
ht : M.t = N.t
him : TuranPartition M
h1 : ¬TuranPartition N
i : ℕ
hi : i ∈ range (N.t + 1)
j : ℕ
hj : j ∈ range (N.t + 1)
v : α
hv : v ∈ MultiPart.P N i
ne : j ≠ i
hc : card (MultiPart.P N j) + 1 < card (MultiPart.P N i)
O : MultiPart α := move N (_ : i ∈ range (N.t + 1) ∧ v ∈ MultiPart.P N i) (_ : j ∈ range (N.t + 1) ∧ j ≠ i)
Ns : mpDsum N < mpDsum O
Q : MultiPart α
QA : Q.A = O.A
Qt : Q.t = O.t
Qim : TuranPartition Q
Qs : mpDsum O ≤ mpDsum Q
this : M.A = Q.A → M.t = Q.t → TuranPartition M → TuranPartition Q → mpDsum M = mpDsum Q
⊢ TuranPartition M
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/Multipartite.lean | SimpleGraph.moved_max | [245, 1] | [259, 14] | exact Qim | case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.a
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
M N : MultiPart α
hA : M.A = N.A
ht : M.t = N.t
him : TuranPartition M
h1 : ¬TuranPartition N
i : ℕ
hi : i ∈ range (N.t + 1)
j : ℕ
hj : j ∈ range (N.t + 1)
v : α
hv : v ∈ MultiPart.P N i
ne : j ≠ i
hc : card (MultiPart.P N j) + 1 < card (MultiPart.P N i)
O : MultiPart α := move N (_ : i ∈ range (N.t + 1) ∧ v ∈ MultiPart.P N i) (_ : j ∈ range (N.t + 1) ∧ j ≠ i)
Ns : mpDsum N < mpDsum O
Q : MultiPart α
QA : Q.A = O.A
Qt : Q.t = O.t
Qim : TuranPartition Q
Qs : mpDsum O ≤ mpDsum Q
this : M.A = Q.A → M.t = Q.t → TuranPartition M → TuranPartition Q → mpDsum M = mpDsum Q
⊢ TuranPartition Q | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.a
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
M N : MultiPart α
hA : M.A = N.A
ht : M.t = N.t
him : TuranPartition M
h1 : ¬TuranPartition N
i : ℕ
hi : i ∈ range (N.t + 1)
j : ℕ
hj : j ∈ range (N.t + 1)
v : α
hv : v ∈ MultiPart.P N i
ne : j ≠ i
hc : card (MultiPart.P N j) + 1 < card (MultiPart.P N i)
O : MultiPart α := move N (_ : i ∈ range (N.t + 1) ∧ v ∈ MultiPart.P N i) (_ : j ∈ range (N.t + 1) ∧ j ≠ i)
Ns : mpDsum N < mpDsum O
Q : MultiPart α
QA : Q.A = O.A
Qt : Q.t = O.t
Qim : TuranPartition Q
Qs : mpDsum O ≤ mpDsum Q
this : M.A = Q.A → M.t = Q.t → TuranPartition M → TuranPartition Q → mpDsum M = mpDsum Q
⊢ TuranPartition Q
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/Multipartite.lean | SimpleGraph.turan_bound_M | [262, 1] | [270, 59] | obtain ⟨N, hA, ht, iN, sN⟩ := moved M | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
M : MultiPart α
⊢ mpDsum M ≤ 2 * turanNumb M.t (card M.A) | case intro.intro.intro.intro
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
M N : MultiPart α
hA : N.A = M.A
ht : N.t = M.t
iN : TuranPartition N
sN : mpDsum M ≤ mpDsum N
⊢ mpDsum M ≤ 2 * turanNumb M.t (card M.A) | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
M : MultiPart α
⊢ mpDsum M ≤ 2 * turanNumb M.t (card M.A)
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/Multipartite.lean | SimpleGraph.turan_bound_M | [262, 1] | [270, 59] | apply le_trans sN _ | case intro.intro.intro.intro
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
M N : MultiPart α
hA : N.A = M.A
ht : N.t = M.t
iN : TuranPartition N
sN : mpDsum M ≤ mpDsum N
⊢ mpDsum M ≤ 2 * turanNumb M.t (card M.A) | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
M N : MultiPart α
hA : N.A = M.A
ht : N.t = M.t
iN : TuranPartition N
sN : mpDsum M ≤ mpDsum N
⊢ mpDsum N ≤ 2 * turanNumb M.t (card M.A) | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
M N : MultiPart α
hA : N.A = M.A
ht : N.t = M.t
iN : TuranPartition N
sN : mpDsum M ≤ mpDsum N
⊢ mpDsum M ≤ 2 * turanNumb M.t (card M.A)
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/Multipartite.lean | SimpleGraph.turan_bound_M | [262, 1] | [270, 59] | apply le_of_eq | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
M N : MultiPart α
hA : N.A = M.A
ht : N.t = M.t
iN : TuranPartition N
sN : mpDsum M ≤ mpDsum N
⊢ mpDsum N ≤ 2 * turanNumb M.t (card M.A) | case a
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
M N : MultiPart α
hA : N.A = M.A
ht : N.t = M.t
iN : TuranPartition N
sN : mpDsum M ≤ mpDsum N
⊢ mpDsum N = 2 * turanNumb M.t (card M.A) | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
M N : MultiPart α
hA : N.A = M.A
ht : N.t = M.t
iN : TuranPartition N
sN : mpDsum M ≤ mpDsum N
⊢ mpDsum N ≤ 2 * turanNumb M.t (card M.A)
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/Multipartite.lean | SimpleGraph.turan_bound_M | [262, 1] | [270, 59] | rw [turanPartition_iff_not_moveable] at iN | case a
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
M N : MultiPart α
hA : N.A = M.A
ht : N.t = M.t
iN : TuranPartition N
sN : mpDsum M ≤ mpDsum N
⊢ mpDsum N = 2 * turanNumb M.t (card M.A) | case a
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
M N : MultiPart α
hA : N.A = M.A
ht : N.t = M.t
iN : ¬Moveable N
sN : mpDsum M ≤ mpDsum N
⊢ mpDsum N = 2 * turanNumb M.t (card M.A) | Please generate a tactic in lean4 to solve the state.
STATE:
case a
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
M N : MultiPart α
hA : N.A = M.A
ht : N.t = M.t
iN : TuranPartition N
sN : mpDsum M ≤ mpDsum N
⊢ mpDsum N = 2 * turanNumb M.t (card M.A)
TACTIC:
|
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.