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/NbhdRes.lean | SimpleGraph.bip_count' | [192, 1] | [203, 8] | rw [adj_comm] at h1 | case neg
α : Type u_1
G : SimpleGraph α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
B A : Finset α
y x : α
h1 : Adj G x y
h2 : ¬Adj G y x
⊢ False | case neg
α : Type u_1
G : SimpleGraph α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
B A : Finset α
y x : α
h1 : Adj G y x
h2 : ¬Adj G y x
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
α : Type u_1
G : SimpleGraph α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
B A : Finset α
y x : α
h1 : Adj G x y
h2 : ¬Adj G y x
⊢ False
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/NbhdRes.lean | SimpleGraph.bip_count' | [192, 1] | [203, 8] | exact h2 h1 | case neg
α : Type u_1
G : SimpleGraph α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
B A : Finset α
y x : α
h1 : Adj G y x
h2 : ¬Adj G y x
⊢ False | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
α : Type u_1
G : SimpleGraph α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
B A : Finset α
y x : α
h1 : Adj G y x
h2 : ¬Adj G y x
⊢ False
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/NbhdRes.lean | SimpleGraph.bip_count' | [192, 1] | [203, 8] | rw [adj_comm] at h1 | case pos
α : Type u_1
G : SimpleGraph α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
B A : Finset α
y x : α
h1 : ¬Adj G x y
h3 : Adj G y x
⊢ False | case pos
α : Type u_1
G : SimpleGraph α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
B A : Finset α
y x : α
h1 : ¬Adj G y x
h3 : Adj G y x
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
α : Type u_1
G : SimpleGraph α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
B A : Finset α
y x : α
h1 : ¬Adj G x y
h3 : Adj G y x
⊢ False
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/NbhdRes.lean | SimpleGraph.bip_count' | [192, 1] | [203, 8] | exact h1 h3 | case pos
α : Type u_1
G : SimpleGraph α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
B A : Finset α
y x : α
h1 : ¬Adj G y x
h3 : Adj G y x
⊢ False | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
α : Type u_1
G : SimpleGraph α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
B A : Finset α
y x : α
h1 : ¬Adj G y x
h3 : Adj G y x
⊢ False
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/NbhdRes.lean | SimpleGraph.bip_count' | [192, 1] | [203, 8] | rfl | case neg
α : Type u_1
G : SimpleGraph α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
B A : Finset α
y x : α
h1 : ¬Adj G x y
h3 : ¬Adj G y x
⊢ 0 = 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
α : Type u_1
G : SimpleGraph α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
B A : Finset α
y x : α
h1 : ¬Adj G x y
h3 : ¬Adj G y x
⊢ 0 = 0
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/NbhdRes.lean | SimpleGraph.sum_res_le | [207, 1] | [214, 24] | apply sum_le_sum _ | α : Type u_1
G : SimpleGraph α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
B A C : Finset α
hB : B ⊆ A
⊢ ∑ v in C, degRes G v B ≤ ∑ v in C, degRes G v A | α : Type u_1
G : SimpleGraph α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
B A C : Finset α
hB : B ⊆ A
⊢ ∀ (i : α), i ∈ C → degRes G i B ≤ degRes G i A | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
G : SimpleGraph α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
B A C : Finset α
hB : B ⊆ A
⊢ ∑ v in C, degRes G v B ≤ ∑ v in C, degRes G v A
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/NbhdRes.lean | SimpleGraph.sum_res_le | [207, 1] | [214, 24] | intro i _ | α : Type u_1
G : SimpleGraph α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
B A C : Finset α
hB : B ⊆ A
⊢ ∀ (i : α), i ∈ C → degRes G i B ≤ degRes G i A | α : Type u_1
G : SimpleGraph α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
B A C : Finset α
hB : B ⊆ A
i : α
a✝ : i ∈ C
⊢ degRes G i B ≤ degRes G i A | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
G : SimpleGraph α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
B A C : Finset α
hB : B ⊆ A
⊢ ∀ (i : α), i ∈ C → degRes G i B ≤ degRes G i A
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/NbhdRes.lean | SimpleGraph.sum_res_le | [207, 1] | [214, 24] | rw [degRes, degRes] | α : Type u_1
G : SimpleGraph α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
B A C : Finset α
hB : B ⊆ A
i : α
a✝ : i ∈ C
⊢ degRes G i B ≤ degRes G i A | α : Type u_1
G : SimpleGraph α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
B A C : Finset α
hB : B ⊆ A
i : α
a✝ : i ∈ C
⊢ card (nbhdRes G i B) ≤ card (nbhdRes G i A) | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
G : SimpleGraph α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
B A C : Finset α
hB : B ⊆ A
i : α
a✝ : i ∈ C
⊢ degRes G i B ≤ degRes G i A
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/NbhdRes.lean | SimpleGraph.sum_res_le | [207, 1] | [214, 24] | apply card_le_of_subset _ | α : Type u_1
G : SimpleGraph α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
B A C : Finset α
hB : B ⊆ A
i : α
a✝ : i ∈ C
⊢ card (nbhdRes G i B) ≤ card (nbhdRes G i A) | α : Type u_1
G : SimpleGraph α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
B A C : Finset α
hB : B ⊆ A
i : α
a✝ : i ∈ C
⊢ nbhdRes G i B ⊆ nbhdRes G i A | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
G : SimpleGraph α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
B A C : Finset α
hB : B ⊆ A
i : α
a✝ : i ∈ C
⊢ card (nbhdRes G i B) ≤ card (nbhdRes G i A)
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/NbhdRes.lean | SimpleGraph.sum_res_le | [207, 1] | [214, 24] | intro x hx | α : Type u_1
G : SimpleGraph α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
B A C : Finset α
hB : B ⊆ A
i : α
a✝ : i ∈ C
⊢ nbhdRes G i B ⊆ nbhdRes G i A | α : Type u_1
G : SimpleGraph α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
B A C : Finset α
hB : B ⊆ A
i : α
a✝ : i ∈ C
x : α
hx : x ∈ nbhdRes G i B
⊢ x ∈ nbhdRes G i A | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
G : SimpleGraph α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
B A C : Finset α
hB : B ⊆ A
i : α
a✝ : i ∈ C
⊢ nbhdRes G i B ⊆ nbhdRes G i A
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/NbhdRes.lean | SimpleGraph.sum_res_le | [207, 1] | [214, 24] | rw [mem_res_nbhd_iff] at * | α : Type u_1
G : SimpleGraph α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
B A C : Finset α
hB : B ⊆ A
i : α
a✝ : i ∈ C
x : α
hx : x ∈ nbhdRes G i B
⊢ x ∈ nbhdRes G i A | α : Type u_1
G : SimpleGraph α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
B A C : Finset α
hB : B ⊆ A
i : α
a✝ : i ∈ C
x : α
hx : x ∈ B ∧ x ∈ neighborFinset G i
⊢ x ∈ A ∧ x ∈ neighborFinset G i | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
G : SimpleGraph α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
B A C : Finset α
hB : B ⊆ A
i : α
a✝ : i ∈ C
x : α
hx : x ∈ nbhdRes G i B
⊢ x ∈ nbhdRes G i A
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/NbhdRes.lean | SimpleGraph.sum_res_le | [207, 1] | [214, 24] | exact ⟨hB hx.1, hx.2⟩ | α : Type u_1
G : SimpleGraph α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
B A C : Finset α
hB : B ⊆ A
i : α
a✝ : i ∈ C
x : α
hx : x ∈ B ∧ x ∈ neighborFinset G i
⊢ x ∈ A ∧ x ∈ neighborFinset G i | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
G : SimpleGraph α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
B A C : Finset α
hB : B ⊆ A
i : α
a✝ : i ∈ C
x : α
hx : x ∈ B ∧ x ∈ neighborFinset G i
⊢ x ∈ A ∧ x ∈ neighborFinset G i
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/CliqueFreeSets.lean | SimpleGraph.pair_subset | [25, 1] | [31, 26] | intro x | α : Type u_1
G : SimpleGraph α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
v w : α
A : Finset α
hv : v ∈ A
hw : w ∈ A
⊢ {v, w} ⊆ A | α : Type u_1
G : SimpleGraph α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
v w : α
A : Finset α
hv : v ∈ A
hw : w ∈ A
x : α
⊢ x ∈ {v, w} → x ∈ A | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
G : SimpleGraph α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
v w : α
A : Finset α
hv : v ∈ A
hw : w ∈ A
⊢ {v, w} ⊆ A
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/CliqueFreeSets.lean | SimpleGraph.pair_subset | [25, 1] | [31, 26] | rw [mem_insert,mem_singleton] | α : Type u_1
G : SimpleGraph α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
v w : α
A : Finset α
hv : v ∈ A
hw : w ∈ A
x : α
⊢ x ∈ {v, w} → x ∈ A | α : Type u_1
G : SimpleGraph α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
v w : α
A : Finset α
hv : v ∈ A
hw : w ∈ A
x : α
⊢ x = v ∨ x = w → x ∈ A | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
G : SimpleGraph α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
v w : α
A : Finset α
hv : v ∈ A
hw : w ∈ A
x : α
⊢ x ∈ {v, w} → x ∈ A
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/CliqueFreeSets.lean | SimpleGraph.pair_subset | [25, 1] | [31, 26] | intro h | α : Type u_1
G : SimpleGraph α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
v w : α
A : Finset α
hv : v ∈ A
hw : w ∈ A
x : α
⊢ x = v ∨ x = w → x ∈ A | α : Type u_1
G : SimpleGraph α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
v w : α
A : Finset α
hv : v ∈ A
hw : w ∈ A
x : α
h : x = v ∨ x = w
⊢ x ∈ A | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
G : SimpleGraph α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
v w : α
A : Finset α
hv : v ∈ A
hw : w ∈ A
x : α
⊢ x = v ∨ x = w → x ∈ A
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/CliqueFreeSets.lean | SimpleGraph.pair_subset | [25, 1] | [31, 26] | cases h with
| inl h => exact h ▸ hv
| inr h => exact h ▸ hw | α : Type u_1
G : SimpleGraph α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
v w : α
A : Finset α
hv : v ∈ A
hw : w ∈ A
x : α
h : x = v ∨ x = w
⊢ x ∈ A | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
G : SimpleGraph α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
v w : α
A : Finset α
hv : v ∈ A
hw : w ∈ A
x : α
h : x = v ∨ x = w
⊢ x ∈ A
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/CliqueFreeSets.lean | SimpleGraph.pair_subset | [25, 1] | [31, 26] | exact h ▸ hv | case inl
α : Type u_1
G : SimpleGraph α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
v w : α
A : Finset α
hv : v ∈ A
hw : w ∈ A
x : α
h : x = v
⊢ x ∈ A | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case inl
α : Type u_1
G : SimpleGraph α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
v w : α
A : Finset α
hv : v ∈ A
hw : w ∈ A
x : α
h : x = v
⊢ x ∈ A
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/CliqueFreeSets.lean | SimpleGraph.pair_subset | [25, 1] | [31, 26] | exact h ▸ hw | case inr
α : Type u_1
G : SimpleGraph α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
v w : α
A : Finset α
hv : v ∈ A
hw : w ∈ A
x : α
h : x = w
⊢ x ∈ A | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case inr
α : Type u_1
G : SimpleGraph α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
v w : α
A : Finset α
hv : v ∈ A
hw : w ∈ A
x : α
h : x = w
⊢ x ∈ A
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/CliqueFreeSets.lean | SimpleGraph.adj_is2Clique | [33, 1] | [36, 60] | rw [isNClique_iff, coe_insert, coe_singleton, isClique_iff, Set.pairwise_pair_of_symmetric G.symm] | α : Type u_1
G : SimpleGraph α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
a b : α
h : Adj G a b
⊢ IsNClique G 2 {a, b} | α : Type u_1
G : SimpleGraph α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
a b : α
h : Adj G a b
⊢ (a ≠ b → Adj G a b) ∧ card {a, b} = 2 | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
G : SimpleGraph α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
a b : α
h : Adj G a b
⊢ IsNClique G 2 {a, b}
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/CliqueFreeSets.lean | SimpleGraph.adj_is2Clique | [33, 1] | [36, 60] | exact ⟨fun _ => h, card_eq_two.2 ⟨a,b,G.ne_of_adj h,rfl⟩⟩ | α : Type u_1
G : SimpleGraph α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
a b : α
h : Adj G a b
⊢ (a ≠ b → Adj G a b) ∧ card {a, b} = 2 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
G : SimpleGraph α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
a b : α
h : Adj G a b
⊢ (a ≠ b → Adj G a b) ∧ card {a, b} = 2
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/CliqueFreeSets.lean | SimpleGraph.clique_free_card_lt | [42, 1] | [46, 66] | rw [CliqueFreeSet] | α : Type u_1
G : SimpleGraph α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
s : ℕ
A : Finset α
h : card A < s
⊢ CliqueFreeSet A s | α : Type u_1
G : SimpleGraph α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
s : ℕ
A : Finset α
h : card A < s
⊢ ∀ (B : Finset α), B ⊆ A → ¬IsNClique G s B | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
G : SimpleGraph α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
s : ℕ
A : Finset α
h : card A < s
⊢ CliqueFreeSet A s
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/CliqueFreeSets.lean | SimpleGraph.clique_free_card_lt | [42, 1] | [46, 66] | intro B hB | α : Type u_1
G : SimpleGraph α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
s : ℕ
A : Finset α
h : card A < s
⊢ ∀ (B : Finset α), B ⊆ A → ¬IsNClique G s B | α : Type u_1
G : SimpleGraph α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
s : ℕ
A : Finset α
h : card A < s
B : Finset α
hB : B ⊆ A
⊢ ¬IsNClique G s B | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
G : SimpleGraph α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
s : ℕ
A : Finset α
h : card A < s
⊢ ∀ (B : Finset α), B ⊆ A → ¬IsNClique G s B
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/CliqueFreeSets.lean | SimpleGraph.clique_free_card_lt | [42, 1] | [46, 66] | rw [isNClique_iff] | α : Type u_1
G : SimpleGraph α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
s : ℕ
A : Finset α
h : card A < s
B : Finset α
hB : B ⊆ A
⊢ ¬IsNClique G s B | α : Type u_1
G : SimpleGraph α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
s : ℕ
A : Finset α
h : card A < s
B : Finset α
hB : B ⊆ A
⊢ ¬(IsClique G ↑B ∧ card B = s) | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
G : SimpleGraph α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
s : ℕ
A : Finset α
h : card A < s
B : Finset α
hB : B ⊆ A
⊢ ¬IsNClique G s B
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/CliqueFreeSets.lean | SimpleGraph.clique_free_card_lt | [42, 1] | [46, 66] | push_neg | α : Type u_1
G : SimpleGraph α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
s : ℕ
A : Finset α
h : card A < s
B : Finset α
hB : B ⊆ A
⊢ ¬(IsClique G ↑B ∧ card B = s) | α : Type u_1
G : SimpleGraph α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
s : ℕ
A : Finset α
h : card A < s
B : Finset α
hB : B ⊆ A
⊢ IsClique G ↑B → card B ≠ s | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
G : SimpleGraph α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
s : ℕ
A : Finset α
h : card A < s
B : Finset α
hB : B ⊆ A
⊢ ¬(IsClique G ↑B ∧ card B = s)
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/CliqueFreeSets.lean | SimpleGraph.clique_free_card_lt | [42, 1] | [46, 66] | intro | α : Type u_1
G : SimpleGraph α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
s : ℕ
A : Finset α
h : card A < s
B : Finset α
hB : B ⊆ A
⊢ IsClique G ↑B → card B ≠ s | α : Type u_1
G : SimpleGraph α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
s : ℕ
A : Finset α
h : card A < s
B : Finset α
hB : B ⊆ A
_✝ : IsClique G ↑B
⊢ card B ≠ s | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
G : SimpleGraph α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
s : ℕ
A : Finset α
h : card A < s
B : Finset α
hB : B ⊆ A
⊢ IsClique G ↑B → card B ≠ s
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/CliqueFreeSets.lean | SimpleGraph.clique_free_card_lt | [42, 1] | [46, 66] | exact _root_.ne_of_lt (lt_of_le_of_lt (card_le_of_subset hB) h) | α : Type u_1
G : SimpleGraph α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
s : ℕ
A : Finset α
h : card A < s
B : Finset α
hB : B ⊆ A
_✝ : IsClique G ↑B
⊢ card B ≠ s | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
G : SimpleGraph α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
s : ℕ
A : Finset α
h : card A < s
B : Finset α
hB : B ⊆ A
_✝ : IsClique G ↑B
⊢ card B ≠ s
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/CliqueFreeSets.lean | SimpleGraph.clique_free_empty | [49, 1] | [51, 64] | rw [← Finset.card_empty] at h | α : Type u_1
G : SimpleGraph α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
s : ℕ
h : 0 < s
⊢ CliqueFreeSet ∅ s | α : Type u_1
G : SimpleGraph α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
s : ℕ
h✝ : 0 < s
h : card ∅ < s
⊢ CliqueFreeSet ∅ s
α : Type u_1
G : SimpleGraph α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
s : ℕ
h : 0 < s
⊢ Type ?u.2299 | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
G : SimpleGraph α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
s : ℕ
h : 0 < s
⊢ CliqueFreeSet ∅ s
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/CliqueFreeSets.lean | SimpleGraph.clique_free_empty | [49, 1] | [51, 64] | exact G.clique_free_card_lt h | α : Type u_1
G : SimpleGraph α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
s : ℕ
h✝ : 0 < s
h : card ∅ < s
⊢ CliqueFreeSet ∅ s
α : Type u_1
G : SimpleGraph α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
s : ℕ
h : 0 < s
⊢ Type ?u.2299 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
G : SimpleGraph α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
s : ℕ
h✝ : 0 < s
h : card ∅ < s
⊢ CliqueFreeSet ∅ s
α : Type u_1
G : SimpleGraph α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
s : ℕ
h : 0 < s
⊢ Type ?u.2299
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/CliqueFreeSets.lean | SimpleGraph.cliqueFree_graph_imp_set | [54, 1] | [58, 40] | revert h | α : Type u_1
G : SimpleGraph α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
s : ℕ
h : CliqueFree G s
U : Finset α
⊢ CliqueFreeSet U s | α : Type u_1
G : SimpleGraph α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
s : ℕ
U : Finset α
⊢ CliqueFree G s → CliqueFreeSet U s | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
G : SimpleGraph α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
s : ℕ
h : CliqueFree G s
U : Finset α
⊢ CliqueFreeSet U s
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/CliqueFreeSets.lean | SimpleGraph.cliqueFree_graph_imp_set | [54, 1] | [58, 40] | contrapose | α : Type u_1
G : SimpleGraph α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
s : ℕ
U : Finset α
⊢ CliqueFree G s → CliqueFreeSet U s | α : Type u_1
G : SimpleGraph α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
s : ℕ
U : Finset α
⊢ ¬CliqueFreeSet U s → ¬CliqueFree G s | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
G : SimpleGraph α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
s : ℕ
U : Finset α
⊢ CliqueFree G s → CliqueFreeSet U s
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/CliqueFreeSets.lean | SimpleGraph.cliqueFree_graph_imp_set | [54, 1] | [58, 40] | rw [CliqueFreeSet] | α : Type u_1
G : SimpleGraph α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
s : ℕ
U : Finset α
⊢ ¬CliqueFreeSet U s → ¬CliqueFree G s | α : Type u_1
G : SimpleGraph α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
s : ℕ
U : Finset α
⊢ (¬∀ (B : Finset α), B ⊆ U → ¬IsNClique G s B) → ¬CliqueFree G s | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
G : SimpleGraph α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
s : ℕ
U : Finset α
⊢ ¬CliqueFreeSet U s → ¬CliqueFree G s
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/CliqueFreeSets.lean | SimpleGraph.cliqueFree_graph_imp_set | [54, 1] | [58, 40] | push_neg | α : Type u_1
G : SimpleGraph α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
s : ℕ
U : Finset α
⊢ (¬∀ (B : Finset α), B ⊆ U → ¬IsNClique G s B) → ¬CliqueFree G s | α : Type u_1
G : SimpleGraph α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
s : ℕ
U : Finset α
⊢ (∃ B, B ⊆ U ∧ IsNClique G s B) → ¬CliqueFree G s | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
G : SimpleGraph α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
s : ℕ
U : Finset α
⊢ (¬∀ (B : Finset α), B ⊆ U → ¬IsNClique G s B) → ¬CliqueFree G s
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/CliqueFreeSets.lean | SimpleGraph.cliqueFree_graph_imp_set | [54, 1] | [58, 40] | intro h | α : Type u_1
G : SimpleGraph α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
s : ℕ
U : Finset α
⊢ (∃ B, B ⊆ U ∧ IsNClique G s B) → ¬CliqueFree G s | α : Type u_1
G : SimpleGraph α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
s : ℕ
U : Finset α
h : ∃ B, B ⊆ U ∧ IsNClique G s B
⊢ ¬CliqueFree G s | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
G : SimpleGraph α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
s : ℕ
U : Finset α
⊢ (∃ B, B ⊆ U ∧ IsNClique G s B) → ¬CliqueFree G s
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/CliqueFreeSets.lean | SimpleGraph.cliqueFree_graph_imp_set | [54, 1] | [58, 40] | rw [CliqueFree] | α : Type u_1
G : SimpleGraph α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
s : ℕ
U : Finset α
h : ∃ B, B ⊆ U ∧ IsNClique G s B
⊢ ¬CliqueFree G s | α : Type u_1
G : SimpleGraph α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
s : ℕ
U : Finset α
h : ∃ B, B ⊆ U ∧ IsNClique G s B
⊢ ¬∀ (t : Finset α), ¬IsNClique G s t | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
G : SimpleGraph α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
s : ℕ
U : Finset α
h : ∃ B, B ⊆ U ∧ IsNClique G s B
⊢ ¬CliqueFree G s
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/CliqueFreeSets.lean | SimpleGraph.cliqueFree_graph_imp_set | [54, 1] | [58, 40] | push_neg | α : Type u_1
G : SimpleGraph α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
s : ℕ
U : Finset α
h : ∃ B, B ⊆ U ∧ IsNClique G s B
⊢ ¬∀ (t : Finset α), ¬IsNClique G s t | α : Type u_1
G : SimpleGraph α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
s : ℕ
U : Finset α
h : ∃ B, B ⊆ U ∧ IsNClique G s B
⊢ ∃ t, IsNClique G s t | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
G : SimpleGraph α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
s : ℕ
U : Finset α
h : ∃ B, B ⊆ U ∧ IsNClique G s B
⊢ ¬∀ (t : Finset α), ¬IsNClique G s t
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/CliqueFreeSets.lean | SimpleGraph.cliqueFree_graph_imp_set | [54, 1] | [58, 40] | obtain ⟨B, _, h2⟩ := h | α : Type u_1
G : SimpleGraph α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
s : ℕ
U : Finset α
h : ∃ B, B ⊆ U ∧ IsNClique G s B
⊢ ∃ t, IsNClique G s t | case intro.intro
α : Type u_1
G : SimpleGraph α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
s : ℕ
U B : Finset α
left✝ : B ⊆ U
h2 : IsNClique G s B
⊢ ∃ t, IsNClique G s t | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
G : SimpleGraph α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
s : ℕ
U : Finset α
h : ∃ B, B ⊆ U ∧ IsNClique G s B
⊢ ∃ t, IsNClique G s t
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/CliqueFreeSets.lean | SimpleGraph.cliqueFree_graph_imp_set | [54, 1] | [58, 40] | exact ⟨B, h2⟩ | case intro.intro
α : Type u_1
G : SimpleGraph α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
s : ℕ
U B : Finset α
left✝ : B ⊆ U
h2 : IsNClique G s B
⊢ ∃ t, IsNClique G s t | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
α : Type u_1
G : SimpleGraph α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
s : ℕ
U B : Finset α
left✝ : B ⊆ U
h2 : IsNClique G s B
⊢ ∃ t, IsNClique G s t
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/CliqueFreeSets.lean | SimpleGraph.two_clique_free | [63, 1] | [69, 59] | intro v hv | α : Type u_1
G : SimpleGraph α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
A : Finset α
hA : CliqueFreeSet A 2
⊢ ∀ (v : α), v ∈ A → degRes G v A = 0 | α : Type u_1
G : SimpleGraph α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
A : Finset α
hA : CliqueFreeSet A 2
v : α
hv : v ∈ A
⊢ degRes G v A = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
G : SimpleGraph α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
A : Finset α
hA : CliqueFreeSet A 2
⊢ ∀ (v : α), v ∈ A → degRes G v A = 0
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/CliqueFreeSets.lean | SimpleGraph.two_clique_free | [63, 1] | [69, 59] | rw [degRes, card_eq_zero] | α : Type u_1
G : SimpleGraph α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
A : Finset α
hA : CliqueFreeSet A 2
v : α
hv : v ∈ A
⊢ degRes G v A = 0 | α : Type u_1
G : SimpleGraph α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
A : Finset α
hA : CliqueFreeSet A 2
v : α
hv : v ∈ A
⊢ nbhdRes G v A = ∅ | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
G : SimpleGraph α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
A : Finset α
hA : CliqueFreeSet A 2
v : α
hv : v ∈ A
⊢ degRes G v A = 0
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/CliqueFreeSets.lean | SimpleGraph.two_clique_free | [63, 1] | [69, 59] | contrapose hA | α : Type u_1
G : SimpleGraph α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
A : Finset α
hA : CliqueFreeSet A 2
v : α
hv : v ∈ A
⊢ nbhdRes G v A = ∅ | α : Type u_1
G : SimpleGraph α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
A : Finset α
v : α
hv : v ∈ A
hA : ¬nbhdRes G v A = ∅
⊢ ¬CliqueFreeSet A 2 | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
G : SimpleGraph α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
A : Finset α
hA : CliqueFreeSet A 2
v : α
hv : v ∈ A
⊢ nbhdRes G v A = ∅
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/CliqueFreeSets.lean | SimpleGraph.two_clique_free | [63, 1] | [69, 59] | obtain ⟨w, hw⟩ := exists_mem_nempty G hA | α : Type u_1
G : SimpleGraph α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
A : Finset α
v : α
hv : v ∈ A
hA : ¬nbhdRes G v A = ∅
⊢ ¬CliqueFreeSet A 2 | case intro
α : Type u_1
G : SimpleGraph α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
A : Finset α
v : α
hv : v ∈ A
hA : ¬nbhdRes G v A = ∅
w : α
hw : w ∈ A ∧ Adj G v w
⊢ ¬CliqueFreeSet A 2 | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
G : SimpleGraph α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
A : Finset α
v : α
hv : v ∈ A
hA : ¬nbhdRes G v A = ∅
⊢ ¬CliqueFreeSet A 2
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/CliqueFreeSets.lean | SimpleGraph.two_clique_free | [63, 1] | [69, 59] | rw [CliqueFreeSet] | case intro
α : Type u_1
G : SimpleGraph α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
A : Finset α
v : α
hv : v ∈ A
hA : ¬nbhdRes G v A = ∅
w : α
hw : w ∈ A ∧ Adj G v w
⊢ ¬CliqueFreeSet A 2 | case intro
α : Type u_1
G : SimpleGraph α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
A : Finset α
v : α
hv : v ∈ A
hA : ¬nbhdRes G v A = ∅
w : α
hw : w ∈ A ∧ Adj G v w
⊢ ¬∀ (B : Finset α), B ⊆ A → ¬IsNClique G 2 B | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
α : Type u_1
G : SimpleGraph α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
A : Finset α
v : α
hv : v ∈ A
hA : ¬nbhdRes G v A = ∅
w : α
hw : w ∈ A ∧ Adj G v w
⊢ ¬CliqueFreeSet A 2
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/CliqueFreeSets.lean | SimpleGraph.two_clique_free | [63, 1] | [69, 59] | push_neg | case intro
α : Type u_1
G : SimpleGraph α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
A : Finset α
v : α
hv : v ∈ A
hA : ¬nbhdRes G v A = ∅
w : α
hw : w ∈ A ∧ Adj G v w
⊢ ¬∀ (B : Finset α), B ⊆ A → ¬IsNClique G 2 B | case intro
α : Type u_1
G : SimpleGraph α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
A : Finset α
v : α
hv : v ∈ A
hA : ¬nbhdRes G v A = ∅
w : α
hw : w ∈ A ∧ Adj G v w
⊢ ∃ B, B ⊆ A ∧ IsNClique G 2 B | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
α : Type u_1
G : SimpleGraph α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
A : Finset α
v : α
hv : v ∈ A
hA : ¬nbhdRes G v A = ∅
w : α
hw : w ∈ A ∧ Adj G v w
⊢ ¬∀ (B : Finset α), B ⊆ A → ¬IsNClique G 2 B
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/CliqueFreeSets.lean | SimpleGraph.two_clique_free | [63, 1] | [69, 59] | exact ⟨{v, w}, pair_subset hv hw.1, adj_is2Clique hw.2⟩ | case intro
α : Type u_1
G : SimpleGraph α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
A : Finset α
v : α
hv : v ∈ A
hA : ¬nbhdRes G v A = ∅
w : α
hw : w ∈ A ∧ Adj G v w
⊢ ∃ B, B ⊆ A ∧ IsNClique G 2 B | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
α : Type u_1
G : SimpleGraph α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
A : Finset α
v : α
hv : v ∈ A
hA : ¬nbhdRes G v A = ∅
w : α
hw : w ∈ A ∧ Adj G v w
⊢ ∃ B, B ⊆ A ∧ IsNClique G 2 B
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/CliqueFreeSets.lean | SimpleGraph.isClique_insert | [79, 1] | [94, 34] | intro x hx y hy hne | α : Type u_1
G : SimpleGraph α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
v : α
A B : Finset α
hv : v ∈ A
hB : B ⊆ nbhdRes G v A
h : IsClique G ↑B
⊢ IsClique G (insert v ↑B) | α : Type u_1
G : SimpleGraph α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
v : α
A B : Finset α
hv : v ∈ A
hB : B ⊆ nbhdRes G v A
h : IsClique G ↑B
x : α
hx : x ∈ insert v ↑B
y : α
hy : y ∈ insert v ↑B
hne : x ≠ y
⊢ Adj G x y | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
G : SimpleGraph α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
v : α
A B : Finset α
hv : v ∈ A
hB : B ⊆ nbhdRes G v A
h : IsClique G ↑B
⊢ IsClique G (insert v ↑B)
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/CliqueFreeSets.lean | SimpleGraph.isClique_insert | [79, 1] | [94, 34] | cases hx with
| inl hx =>
cases hy with
| inl hy => rw [hx,hy] at hne; contradiction
| inr hy =>
rw [hx];
apply G.subset_res_nbhd hB hy
| inr hx =>
cases hy with
| inl hy =>
rw [hy, G.adj_comm]
apply G.subset_res_nbhd hB hx
| inr hy => exact h hx hy hne | α : Type u_1
G : SimpleGraph α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
v : α
A B : Finset α
hv : v ∈ A
hB : B ⊆ nbhdRes G v A
h : IsClique G ↑B
x : α
hx : x ∈ insert v ↑B
y : α
hy : y ∈ insert v ↑B
hne : x ≠ y
⊢ Adj G x y | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
G : SimpleGraph α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
v : α
A B : Finset α
hv : v ∈ A
hB : B ⊆ nbhdRes G v A
h : IsClique G ↑B
x : α
hx : x ∈ insert v ↑B
y : α
hy : y ∈ insert v ↑B
hne : x ≠ y
⊢ Adj G x y
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/CliqueFreeSets.lean | SimpleGraph.isClique_insert | [79, 1] | [94, 34] | cases hy with
| inl hy => rw [hx,hy] at hne; contradiction
| inr hy =>
rw [hx];
apply G.subset_res_nbhd hB hy | case inl
α : Type u_1
G : SimpleGraph α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
v : α
A B : Finset α
hv : v ∈ A
hB : B ⊆ nbhdRes G v A
h : IsClique G ↑B
x y : α
hy : y ∈ insert v ↑B
hne : x ≠ y
hx : x = v
⊢ Adj G x y | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case inl
α : Type u_1
G : SimpleGraph α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
v : α
A B : Finset α
hv : v ∈ A
hB : B ⊆ nbhdRes G v A
h : IsClique G ↑B
x y : α
hy : y ∈ insert v ↑B
hne : x ≠ y
hx : x = v
⊢ Adj G x y
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/CliqueFreeSets.lean | SimpleGraph.isClique_insert | [79, 1] | [94, 34] | rw [hx,hy] at hne | case inl.inl
α : Type u_1
G : SimpleGraph α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
v : α
A B : Finset α
hv : v ∈ A
hB : B ⊆ nbhdRes G v A
h : IsClique G ↑B
x y : α
hne : x ≠ y
hx : x = v
hy : y = v
⊢ Adj G x y | case inl.inl
α : Type u_1
G : SimpleGraph α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
v : α
A B : Finset α
hv : v ∈ A
hB : B ⊆ nbhdRes G v A
h : IsClique G ↑B
x y : α
hne : v ≠ v
hx : x = v
hy : y = v
⊢ Adj G x y | Please generate a tactic in lean4 to solve the state.
STATE:
case inl.inl
α : Type u_1
G : SimpleGraph α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
v : α
A B : Finset α
hv : v ∈ A
hB : B ⊆ nbhdRes G v A
h : IsClique G ↑B
x y : α
hne : x ≠ y
hx : x = v
hy : y = v
⊢ Adj G x y
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/CliqueFreeSets.lean | SimpleGraph.isClique_insert | [79, 1] | [94, 34] | contradiction | case inl.inl
α : Type u_1
G : SimpleGraph α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
v : α
A B : Finset α
hv : v ∈ A
hB : B ⊆ nbhdRes G v A
h : IsClique G ↑B
x y : α
hne : v ≠ v
hx : x = v
hy : y = v
⊢ Adj G x y | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case inl.inl
α : Type u_1
G : SimpleGraph α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
v : α
A B : Finset α
hv : v ∈ A
hB : B ⊆ nbhdRes G v A
h : IsClique G ↑B
x y : α
hne : v ≠ v
hx : x = v
hy : y = v
⊢ Adj G x y
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/CliqueFreeSets.lean | SimpleGraph.isClique_insert | [79, 1] | [94, 34] | rw [hx] | case inl.inr
α : Type u_1
G : SimpleGraph α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
v : α
A B : Finset α
hv : v ∈ A
hB : B ⊆ nbhdRes G v A
h : IsClique G ↑B
x y : α
hne : x ≠ y
hx : x = v
hy : y ∈ ↑B
⊢ Adj G x y | case inl.inr
α : Type u_1
G : SimpleGraph α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
v : α
A B : Finset α
hv : v ∈ A
hB : B ⊆ nbhdRes G v A
h : IsClique G ↑B
x y : α
hne : x ≠ y
hx : x = v
hy : y ∈ ↑B
⊢ Adj G v y | Please generate a tactic in lean4 to solve the state.
STATE:
case inl.inr
α : Type u_1
G : SimpleGraph α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
v : α
A B : Finset α
hv : v ∈ A
hB : B ⊆ nbhdRes G v A
h : IsClique G ↑B
x y : α
hne : x ≠ y
hx : x = v
hy : y ∈ ↑B
⊢ Adj G x y
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/CliqueFreeSets.lean | SimpleGraph.isClique_insert | [79, 1] | [94, 34] | apply G.subset_res_nbhd hB hy | case inl.inr
α : Type u_1
G : SimpleGraph α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
v : α
A B : Finset α
hv : v ∈ A
hB : B ⊆ nbhdRes G v A
h : IsClique G ↑B
x y : α
hne : x ≠ y
hx : x = v
hy : y ∈ ↑B
⊢ Adj G v y | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case inl.inr
α : Type u_1
G : SimpleGraph α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
v : α
A B : Finset α
hv : v ∈ A
hB : B ⊆ nbhdRes G v A
h : IsClique G ↑B
x y : α
hne : x ≠ y
hx : x = v
hy : y ∈ ↑B
⊢ Adj G v y
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/CliqueFreeSets.lean | SimpleGraph.isClique_insert | [79, 1] | [94, 34] | cases hy with
| inl hy =>
rw [hy, G.adj_comm]
apply G.subset_res_nbhd hB hx
| inr hy => exact h hx hy hne | case inr
α : Type u_1
G : SimpleGraph α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
v : α
A B : Finset α
hv : v ∈ A
hB : B ⊆ nbhdRes G v A
h : IsClique G ↑B
x y : α
hy : y ∈ insert v ↑B
hne : x ≠ y
hx : x ∈ ↑B
⊢ Adj G x y | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case inr
α : Type u_1
G : SimpleGraph α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
v : α
A B : Finset α
hv : v ∈ A
hB : B ⊆ nbhdRes G v A
h : IsClique G ↑B
x y : α
hy : y ∈ insert v ↑B
hne : x ≠ y
hx : x ∈ ↑B
⊢ Adj G x y
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/CliqueFreeSets.lean | SimpleGraph.isClique_insert | [79, 1] | [94, 34] | rw [hy, G.adj_comm] | case inr.inl
α : Type u_1
G : SimpleGraph α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
v : α
A B : Finset α
hv : v ∈ A
hB : B ⊆ nbhdRes G v A
h : IsClique G ↑B
x y : α
hne : x ≠ y
hx : x ∈ ↑B
hy : y = v
⊢ Adj G x y | case inr.inl
α : Type u_1
G : SimpleGraph α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
v : α
A B : Finset α
hv : v ∈ A
hB : B ⊆ nbhdRes G v A
h : IsClique G ↑B
x y : α
hne : x ≠ y
hx : x ∈ ↑B
hy : y = v
⊢ Adj G v x | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.inl
α : Type u_1
G : SimpleGraph α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
v : α
A B : Finset α
hv : v ∈ A
hB : B ⊆ nbhdRes G v A
h : IsClique G ↑B
x y : α
hne : x ≠ y
hx : x ∈ ↑B
hy : y = v
⊢ Adj G x y
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/CliqueFreeSets.lean | SimpleGraph.isClique_insert | [79, 1] | [94, 34] | apply G.subset_res_nbhd hB hx | case inr.inl
α : Type u_1
G : SimpleGraph α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
v : α
A B : Finset α
hv : v ∈ A
hB : B ⊆ nbhdRes G v A
h : IsClique G ↑B
x y : α
hne : x ≠ y
hx : x ∈ ↑B
hy : y = v
⊢ Adj G v x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.inl
α : Type u_1
G : SimpleGraph α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
v : α
A B : Finset α
hv : v ∈ A
hB : B ⊆ nbhdRes G v A
h : IsClique G ↑B
x y : α
hne : x ≠ y
hx : x ∈ ↑B
hy : y = v
⊢ Adj G v x
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/CliqueFreeSets.lean | SimpleGraph.isClique_insert | [79, 1] | [94, 34] | exact h hx hy hne | case inr.inr
α : Type u_1
G : SimpleGraph α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
v : α
A B : Finset α
hv : v ∈ A
hB : B ⊆ nbhdRes G v A
h : IsClique G ↑B
x y : α
hne : x ≠ y
hx : x ∈ ↑B
hy : y ∈ ↑B
⊢ Adj G x y | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.inr
α : Type u_1
G : SimpleGraph α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
v : α
A B : Finset α
hv : v ∈ A
hB : B ⊆ nbhdRes G v A
h : IsClique G ↑B
x y : α
hne : x ≠ y
hx : x ∈ ↑B
hy : y ∈ ↑B
⊢ Adj G x y
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/CliqueFreeSets.lean | SimpleGraph.isSuccClique | [97, 1] | [103, 51] | constructor | α : Type u_1
G : SimpleGraph α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
v : α
A B : Finset α
t : ℕ
hv : v ∈ A
hB : B ⊆ nbhdRes G v A
h : IsNClique G t B
⊢ IsNClique G (t + 1) (insert v B) | case clique
α : Type u_1
G : SimpleGraph α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
v : α
A B : Finset α
t : ℕ
hv : v ∈ A
hB : B ⊆ nbhdRes G v A
h : IsNClique G t B
⊢ IsClique G ↑(insert v B)
case card_eq
α : Type u_1
G : SimpleGraph α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
v : α
A B : Finset α
t : ℕ
hv : v ∈ A
hB : B ⊆ nbhdRes G v A
h : IsNClique G t B
⊢ card (insert v B) = t + 1 | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
G : SimpleGraph α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
v : α
A B : Finset α
t : ℕ
hv : v ∈ A
hB : B ⊆ nbhdRes G v A
h : IsNClique G t B
⊢ IsNClique G (t + 1) (insert v B)
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/CliqueFreeSets.lean | SimpleGraph.isSuccClique | [97, 1] | [103, 51] | rw [coe_insert] | case clique
α : Type u_1
G : SimpleGraph α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
v : α
A B : Finset α
t : ℕ
hv : v ∈ A
hB : B ⊆ nbhdRes G v A
h : IsNClique G t B
⊢ IsClique G ↑(insert v B) | case clique
α : Type u_1
G : SimpleGraph α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
v : α
A B : Finset α
t : ℕ
hv : v ∈ A
hB : B ⊆ nbhdRes G v A
h : IsNClique G t B
⊢ IsClique G (insert v ↑B) | Please generate a tactic in lean4 to solve the state.
STATE:
case clique
α : Type u_1
G : SimpleGraph α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
v : α
A B : Finset α
t : ℕ
hv : v ∈ A
hB : B ⊆ nbhdRes G v A
h : IsNClique G t B
⊢ IsClique G ↑(insert v B)
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/CliqueFreeSets.lean | SimpleGraph.isSuccClique | [97, 1] | [103, 51] | exact isClique_insert hv hB h.1 | case clique
α : Type u_1
G : SimpleGraph α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
v : α
A B : Finset α
t : ℕ
hv : v ∈ A
hB : B ⊆ nbhdRes G v A
h : IsNClique G t B
⊢ IsClique G (insert v ↑B) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case clique
α : Type u_1
G : SimpleGraph α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
v : α
A B : Finset α
t : ℕ
hv : v ∈ A
hB : B ⊆ nbhdRes G v A
h : IsNClique G t B
⊢ IsClique G (insert v ↑B)
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/CliqueFreeSets.lean | SimpleGraph.isSuccClique | [97, 1] | [103, 51] | rw [card_insert_of_not_mem,add_left_inj,h.2] | case card_eq
α : Type u_1
G : SimpleGraph α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
v : α
A B : Finset α
t : ℕ
hv : v ∈ A
hB : B ⊆ nbhdRes G v A
h : IsNClique G t B
⊢ card (insert v B) = t + 1 | case card_eq
α : Type u_1
G : SimpleGraph α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
v : α
A B : Finset α
t : ℕ
hv : v ∈ A
hB : B ⊆ nbhdRes G v A
h : IsNClique G t B
⊢ ¬v ∈ B | Please generate a tactic in lean4 to solve the state.
STATE:
case card_eq
α : Type u_1
G : SimpleGraph α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
v : α
A B : Finset α
t : ℕ
hv : v ∈ A
hB : B ⊆ nbhdRes G v A
h : IsNClique G t B
⊢ card (insert v B) = t + 1
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/CliqueFreeSets.lean | SimpleGraph.isSuccClique | [97, 1] | [103, 51] | apply not_mem_mono hB (G.not_mem_res_nbhd v A) | case card_eq
α : Type u_1
G : SimpleGraph α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
v : α
A B : Finset α
t : ℕ
hv : v ∈ A
hB : B ⊆ nbhdRes G v A
h : IsNClique G t B
⊢ ¬v ∈ B | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case card_eq
α : Type u_1
G : SimpleGraph α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
v : α
A B : Finset α
t : ℕ
hv : v ∈ A
hB : B ⊆ nbhdRes G v A
h : IsNClique G t B
⊢ ¬v ∈ B
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/CliqueFreeSets.lean | SimpleGraph.t_clique_free | [109, 1] | [115, 66] | rw [CliqueFreeSet] at * | α : Type u_1
G : SimpleGraph α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
A : Finset α
t : ℕ
v : α
hA : CliqueFreeSet A (t + 2)
hv : v ∈ A
⊢ CliqueFreeSet (nbhdRes G v A) (t + 1) | α : Type u_1
G : SimpleGraph α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
A : Finset α
t : ℕ
v : α
hA : ∀ (B : Finset α), B ⊆ A → ¬IsNClique G (t + 2) B
hv : v ∈ A
⊢ ∀ (B : Finset α), B ⊆ nbhdRes G v A → ¬IsNClique G (t + 1) B | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
G : SimpleGraph α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
A : Finset α
t : ℕ
v : α
hA : CliqueFreeSet A (t + 2)
hv : v ∈ A
⊢ CliqueFreeSet (nbhdRes G v A) (t + 1)
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/CliqueFreeSets.lean | SimpleGraph.t_clique_free | [109, 1] | [115, 66] | intro B hB | α : Type u_1
G : SimpleGraph α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
A : Finset α
t : ℕ
v : α
hA : ∀ (B : Finset α), B ⊆ A → ¬IsNClique G (t + 2) B
hv : v ∈ A
⊢ ∀ (B : Finset α), B ⊆ nbhdRes G v A → ¬IsNClique G (t + 1) B | α : Type u_1
G : SimpleGraph α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
A : Finset α
t : ℕ
v : α
hA : ∀ (B : Finset α), B ⊆ A → ¬IsNClique G (t + 2) B
hv : v ∈ A
B : Finset α
hB : B ⊆ nbhdRes G v A
⊢ ¬IsNClique G (t + 1) B | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
G : SimpleGraph α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
A : Finset α
t : ℕ
v : α
hA : ∀ (B : Finset α), B ⊆ A → ¬IsNClique G (t + 2) B
hv : v ∈ A
⊢ ∀ (B : Finset α), B ⊆ nbhdRes G v A → ¬IsNClique G (t + 1) B
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/CliqueFreeSets.lean | SimpleGraph.t_clique_free | [109, 1] | [115, 66] | contrapose! hA | α : Type u_1
G : SimpleGraph α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
A : Finset α
t : ℕ
v : α
hA : ∀ (B : Finset α), B ⊆ A → ¬IsNClique G (t + 2) B
hv : v ∈ A
B : Finset α
hB : B ⊆ nbhdRes G v A
⊢ ¬IsNClique G (t + 1) B | α : Type u_1
G : SimpleGraph α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
A : Finset α
t : ℕ
v : α
hv : v ∈ A
B : Finset α
hB : B ⊆ nbhdRes G v A
hA : IsNClique G (t + 1) B
⊢ ∃ B, B ⊆ A ∧ IsNClique G (t + 2) B | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
G : SimpleGraph α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
A : Finset α
t : ℕ
v : α
hA : ∀ (B : Finset α), B ⊆ A → ¬IsNClique G (t + 2) B
hv : v ∈ A
B : Finset α
hB : B ⊆ nbhdRes G v A
⊢ ¬IsNClique G (t + 1) B
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/CliqueFreeSets.lean | SimpleGraph.t_clique_free | [109, 1] | [115, 66] | refine ⟨insert v B,?_, isSuccClique hv hB hA⟩ | α : Type u_1
G : SimpleGraph α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
A : Finset α
t : ℕ
v : α
hv : v ∈ A
B : Finset α
hB : B ⊆ nbhdRes G v A
hA : IsNClique G (t + 1) B
⊢ ∃ B, B ⊆ A ∧ IsNClique G (t + 2) B | α : Type u_1
G : SimpleGraph α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
A : Finset α
t : ℕ
v : α
hv : v ∈ A
B : Finset α
hB : B ⊆ nbhdRes G v A
hA : IsNClique G (t + 1) B
⊢ insert v B ⊆ A | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
G : SimpleGraph α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
A : Finset α
t : ℕ
v : α
hv : v ∈ A
B : Finset α
hB : B ⊆ nbhdRes G v A
hA : IsNClique G (t + 1) B
⊢ ∃ B, B ⊆ A ∧ IsNClique G (t + 2) B
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/CliqueFreeSets.lean | SimpleGraph.t_clique_free | [109, 1] | [115, 66] | exact insert_subset hv (subset_trans hB (G.sub_res_nbhd_A v A)) | α : Type u_1
G : SimpleGraph α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
A : Finset α
t : ℕ
v : α
hv : v ∈ A
B : Finset α
hB : B ⊆ nbhdRes G v A
hA : IsNClique G (t + 1) B
⊢ insert v B ⊆ A | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
G : SimpleGraph α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
A : Finset α
t : ℕ
v : α
hv : v ∈ A
B : Finset α
hB : B ⊆ nbhdRes G v A
hA : IsNClique G (t + 1) B
⊢ insert v B ⊆ A
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/Turanpartition.lean | Turanpartition.two | [55, 1] | [65, 27] | induction a with
| zero => rfl
| succ n ih =>
cases n with
| zero => rfl
| succ n =>
rw [Nat.choose,Nat.choose_one_right,mul_add,ih]
rw [succ_sub_succ_eq_sub,tsub_zero,mul_comm _ n,← add_mul];
congr; rw [add_comm] | a : ℕ
⊢ 2 * Nat.choose a 2 = a * (a - 1) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
a : ℕ
⊢ 2 * Nat.choose a 2 = a * (a - 1)
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/Turanpartition.lean | Turanpartition.two | [55, 1] | [65, 27] | rfl | case zero
⊢ 2 * Nat.choose zero 2 = zero * (zero - 1) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case zero
⊢ 2 * Nat.choose zero 2 = zero * (zero - 1)
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/Turanpartition.lean | Turanpartition.two | [55, 1] | [65, 27] | cases n with
| zero => rfl
| succ n =>
rw [Nat.choose,Nat.choose_one_right,mul_add,ih]
rw [succ_sub_succ_eq_sub,tsub_zero,mul_comm _ n,← add_mul];
congr; rw [add_comm] | case succ
n : ℕ
ih : 2 * Nat.choose n 2 = n * (n - 1)
⊢ 2 * Nat.choose (succ n) 2 = succ n * (succ n - 1) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case succ
n : ℕ
ih : 2 * Nat.choose n 2 = n * (n - 1)
⊢ 2 * Nat.choose (succ n) 2 = succ n * (succ n - 1)
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/Turanpartition.lean | Turanpartition.two | [55, 1] | [65, 27] | rfl | case succ.zero
ih : 2 * Nat.choose zero 2 = zero * (zero - 1)
⊢ 2 * Nat.choose (succ zero) 2 = succ zero * (succ zero - 1) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case succ.zero
ih : 2 * Nat.choose zero 2 = zero * (zero - 1)
⊢ 2 * Nat.choose (succ zero) 2 = succ zero * (succ zero - 1)
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/Turanpartition.lean | Turanpartition.two | [55, 1] | [65, 27] | rw [Nat.choose,Nat.choose_one_right,mul_add,ih] | case succ.succ
n : ℕ
ih : 2 * Nat.choose (succ n) 2 = succ n * (succ n - 1)
⊢ 2 * Nat.choose (succ (succ n)) 2 = succ (succ n) * (succ (succ n) - 1) | case succ.succ
n : ℕ
ih : 2 * Nat.choose (succ n) 2 = succ n * (succ n - 1)
⊢ 2 * (n + 1) + succ n * (succ n - 1) = succ (succ n) * (succ (succ n) - 1) | Please generate a tactic in lean4 to solve the state.
STATE:
case succ.succ
n : ℕ
ih : 2 * Nat.choose (succ n) 2 = succ n * (succ n - 1)
⊢ 2 * Nat.choose (succ (succ n)) 2 = succ (succ n) * (succ (succ n) - 1)
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/Turanpartition.lean | Turanpartition.two | [55, 1] | [65, 27] | rw [succ_sub_succ_eq_sub,tsub_zero,mul_comm _ n,← add_mul] | case succ.succ
n : ℕ
ih : 2 * Nat.choose (succ n) 2 = succ n * (succ n - 1)
⊢ 2 * (n + 1) + succ n * (succ n - 1) = succ (succ n) * (succ (succ n) - 1) | case succ.succ
n : ℕ
ih : 2 * Nat.choose (succ n) 2 = succ n * (succ n - 1)
⊢ (2 + n) * (n + 1) = succ (succ n) * (succ (succ n) - 1) | Please generate a tactic in lean4 to solve the state.
STATE:
case succ.succ
n : ℕ
ih : 2 * Nat.choose (succ n) 2 = succ n * (succ n - 1)
⊢ 2 * (n + 1) + succ n * (succ n - 1) = succ (succ n) * (succ (succ n) - 1)
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/Turanpartition.lean | Turanpartition.two | [55, 1] | [65, 27] | congr | case succ.succ
n : ℕ
ih : 2 * Nat.choose (succ n) 2 = succ n * (succ n - 1)
⊢ (2 + n) * (n + 1) = succ (succ n) * (succ (succ n) - 1) | case succ.succ.e_a
n : ℕ
ih : 2 * Nat.choose (succ n) 2 = succ n * (succ n - 1)
⊢ 2 + n = succ (succ n) | Please generate a tactic in lean4 to solve the state.
STATE:
case succ.succ
n : ℕ
ih : 2 * Nat.choose (succ n) 2 = succ n * (succ n - 1)
⊢ (2 + n) * (n + 1) = succ (succ n) * (succ (succ n) - 1)
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/Turanpartition.lean | Turanpartition.two | [55, 1] | [65, 27] | rw [add_comm] | case succ.succ.e_a
n : ℕ
ih : 2 * Nat.choose (succ n) 2 = succ n * (succ n - 1)
⊢ 2 + n = succ (succ n) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case succ.succ.e_a
n : ℕ
ih : 2 * Nat.choose (succ n) 2 = succ n * (succ n - 1)
⊢ 2 + n = succ (succ n)
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/Turanpartition.lean | Turanpartition.square | [69, 1] | [71, 113] | rw [pow_two,add_mul, mul_add,mul_add,pow_two,pow_two,mul_comm c,two_mul,add_mul,add_assoc,add_assoc,add_assoc] | b c : ℕ
⊢ (b + c) ^ 2 = b ^ 2 + 2 * b * c + c ^ 2 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
b c : ℕ
⊢ (b + c) ^ 2 = b ^ 2 + 2 * b * c + c ^ 2
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/Turanpartition.lean | Turanpartition.tn_turanNumb' | [76, 1] | [119, 111] | rw [tN,tN'] | t n : ℕ
⊢ turanNumb' t n + 2 * turanNumb t n = n ^ 2 | t n : ℕ
⊢ (t + 1 - n % (t + 1)) * (n / (t + 1)) ^ 2 + n % (t + 1) * (n / (t + 1) + 1) ^ 2 +
2 *
((n / (t + 1)) ^ 2 * Nat.choose (t + 1 - n % (t + 1)) 2 +
n / (t + 1) * (n / (t + 1) + 1) * (n % (t + 1)) * (t + 1 - n % (t + 1)) +
(n / (t + 1) + 1) ^ 2 * Nat.choose (n % (t + 1)) 2) =
n ^ 2 | Please generate a tactic in lean4 to solve the state.
STATE:
t n : ℕ
⊢ turanNumb' t n + 2 * turanNumb t n = n ^ 2
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/Turanpartition.lean | Turanpartition.tn_turanNumb' | [76, 1] | [119, 111] | have t1 : t + 1 - 1 = t := tsub_eq_of_eq_add rfl | t n : ℕ
⊢ (t + 1 - n % (t + 1)) * (n / (t + 1)) ^ 2 + n % (t + 1) * (n / (t + 1) + 1) ^ 2 +
2 *
((n / (t + 1)) ^ 2 * Nat.choose (t + 1 - n % (t + 1)) 2 +
n / (t + 1) * (n / (t + 1) + 1) * (n % (t + 1)) * (t + 1 - n % (t + 1)) +
(n / (t + 1) + 1) ^ 2 * Nat.choose (n % (t + 1)) 2) =
n ^ 2 | t n : ℕ
t1 : t + 1 - 1 = t
⊢ (t + 1 - n % (t + 1)) * (n / (t + 1)) ^ 2 + n % (t + 1) * (n / (t + 1) + 1) ^ 2 +
2 *
((n / (t + 1)) ^ 2 * Nat.choose (t + 1 - n % (t + 1)) 2 +
n / (t + 1) * (n / (t + 1) + 1) * (n % (t + 1)) * (t + 1 - n % (t + 1)) +
(n / (t + 1) + 1) ^ 2 * Nat.choose (n % (t + 1)) 2) =
n ^ 2 | Please generate a tactic in lean4 to solve the state.
STATE:
t n : ℕ
⊢ (t + 1 - n % (t + 1)) * (n / (t + 1)) ^ 2 + n % (t + 1) * (n / (t + 1) + 1) ^ 2 +
2 *
((n / (t + 1)) ^ 2 * Nat.choose (t + 1 - n % (t + 1)) 2 +
n / (t + 1) * (n / (t + 1) + 1) * (n % (t + 1)) * (t + 1 - n % (t + 1)) +
(n / (t + 1) + 1) ^ 2 * Nat.choose (n % (t + 1)) 2) =
n ^ 2
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/Turanpartition.lean | Turanpartition.tn_turanNumb' | [76, 1] | [119, 111] | have n1 := div_add_mod n (t + 1) | t n : ℕ
t1 : t + 1 - 1 = t
⊢ (t + 1 - n % (t + 1)) * (n / (t + 1)) ^ 2 + n % (t + 1) * (n / (t + 1) + 1) ^ 2 +
2 *
((n / (t + 1)) ^ 2 * Nat.choose (t + 1 - n % (t + 1)) 2 +
n / (t + 1) * (n / (t + 1) + 1) * (n % (t + 1)) * (t + 1 - n % (t + 1)) +
(n / (t + 1) + 1) ^ 2 * Nat.choose (n % (t + 1)) 2) =
n ^ 2 | t n : ℕ
t1 : t + 1 - 1 = t
n1 : (t + 1) * (n / (t + 1)) + n % (t + 1) = n
⊢ (t + 1 - n % (t + 1)) * (n / (t + 1)) ^ 2 + n % (t + 1) * (n / (t + 1) + 1) ^ 2 +
2 *
((n / (t + 1)) ^ 2 * Nat.choose (t + 1 - n % (t + 1)) 2 +
n / (t + 1) * (n / (t + 1) + 1) * (n % (t + 1)) * (t + 1 - n % (t + 1)) +
(n / (t + 1) + 1) ^ 2 * Nat.choose (n % (t + 1)) 2) =
n ^ 2 | Please generate a tactic in lean4 to solve the state.
STATE:
t n : ℕ
t1 : t + 1 - 1 = t
⊢ (t + 1 - n % (t + 1)) * (n / (t + 1)) ^ 2 + n % (t + 1) * (n / (t + 1) + 1) ^ 2 +
2 *
((n / (t + 1)) ^ 2 * Nat.choose (t + 1 - n % (t + 1)) 2 +
n / (t + 1) * (n / (t + 1) + 1) * (n % (t + 1)) * (t + 1 - n % (t + 1)) +
(n / (t + 1) + 1) ^ 2 * Nat.choose (n % (t + 1)) 2) =
n ^ 2
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/Turanpartition.lean | Turanpartition.tn_turanNumb' | [76, 1] | [119, 111] | have n2 := mod_lt n (succ_pos t) | t n : ℕ
t1 : t + 1 - 1 = t
n1 : (t + 1) * (n / (t + 1)) + n % (t + 1) = n
⊢ (t + 1 - n % (t + 1)) * (n / (t + 1)) ^ 2 + n % (t + 1) * (n / (t + 1) + 1) ^ 2 +
2 *
((n / (t + 1)) ^ 2 * Nat.choose (t + 1 - n % (t + 1)) 2 +
n / (t + 1) * (n / (t + 1) + 1) * (n % (t + 1)) * (t + 1 - n % (t + 1)) +
(n / (t + 1) + 1) ^ 2 * Nat.choose (n % (t + 1)) 2) =
n ^ 2 | t n : ℕ
t1 : t + 1 - 1 = t
n1 : (t + 1) * (n / (t + 1)) + n % (t + 1) = n
n2 : n % succ t < succ t
⊢ (t + 1 - n % (t + 1)) * (n / (t + 1)) ^ 2 + n % (t + 1) * (n / (t + 1) + 1) ^ 2 +
2 *
((n / (t + 1)) ^ 2 * Nat.choose (t + 1 - n % (t + 1)) 2 +
n / (t + 1) * (n / (t + 1) + 1) * (n % (t + 1)) * (t + 1 - n % (t + 1)) +
(n / (t + 1) + 1) ^ 2 * Nat.choose (n % (t + 1)) 2) =
n ^ 2 | Please generate a tactic in lean4 to solve the state.
STATE:
t n : ℕ
t1 : t + 1 - 1 = t
n1 : (t + 1) * (n / (t + 1)) + n % (t + 1) = n
⊢ (t + 1 - n % (t + 1)) * (n / (t + 1)) ^ 2 + n % (t + 1) * (n / (t + 1) + 1) ^ 2 +
2 *
((n / (t + 1)) ^ 2 * Nat.choose (t + 1 - n % (t + 1)) 2 +
n / (t + 1) * (n / (t + 1) + 1) * (n % (t + 1)) * (t + 1 - n % (t + 1)) +
(n / (t + 1) + 1) ^ 2 * Nat.choose (n % (t + 1)) 2) =
n ^ 2
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/Turanpartition.lean | Turanpartition.tn_turanNumb' | [76, 1] | [119, 111] | rw [succ_eq_add_one] at n2 | t n : ℕ
t1 : t + 1 - 1 = t
n1 : (t + 1) * (n / (t + 1)) + n % (t + 1) = n
n2 : n % succ t < succ t
⊢ (t + 1 - n % (t + 1)) * (n / (t + 1)) ^ 2 + n % (t + 1) * (n / (t + 1) + 1) ^ 2 +
2 *
((n / (t + 1)) ^ 2 * Nat.choose (t + 1 - n % (t + 1)) 2 +
n / (t + 1) * (n / (t + 1) + 1) * (n % (t + 1)) * (t + 1 - n % (t + 1)) +
(n / (t + 1) + 1) ^ 2 * Nat.choose (n % (t + 1)) 2) =
n ^ 2 | t n : ℕ
t1 : t + 1 - 1 = t
n1 : (t + 1) * (n / (t + 1)) + n % (t + 1) = n
n2 : n % (t + 1) < t + 1
⊢ (t + 1 - n % (t + 1)) * (n / (t + 1)) ^ 2 + n % (t + 1) * (n / (t + 1) + 1) ^ 2 +
2 *
((n / (t + 1)) ^ 2 * Nat.choose (t + 1 - n % (t + 1)) 2 +
n / (t + 1) * (n / (t + 1) + 1) * (n % (t + 1)) * (t + 1 - n % (t + 1)) +
(n / (t + 1) + 1) ^ 2 * Nat.choose (n % (t + 1)) 2) =
n ^ 2 | Please generate a tactic in lean4 to solve the state.
STATE:
t n : ℕ
t1 : t + 1 - 1 = t
n1 : (t + 1) * (n / (t + 1)) + n % (t + 1) = n
n2 : n % succ t < succ t
⊢ (t + 1 - n % (t + 1)) * (n / (t + 1)) ^ 2 + n % (t + 1) * (n / (t + 1) + 1) ^ 2 +
2 *
((n / (t + 1)) ^ 2 * Nat.choose (t + 1 - n % (t + 1)) 2 +
n / (t + 1) * (n / (t + 1) + 1) * (n % (t + 1)) * (t + 1 - n % (t + 1)) +
(n / (t + 1) + 1) ^ 2 * Nat.choose (n % (t + 1)) 2) =
n ^ 2
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/Turanpartition.lean | Turanpartition.tn_turanNumb' | [76, 1] | [119, 111] | have n3 : t + 1 - n % (t + 1) + n % (t + 1) = t + 1 := tsub_add_cancel_of_le (le_of_lt n2) | t n : ℕ
t1 : t + 1 - 1 = t
n1 : (t + 1) * (n / (t + 1)) + n % (t + 1) = n
n2 : n % (t + 1) < t + 1
⊢ (t + 1 - n % (t + 1)) * (n / (t + 1)) ^ 2 + n % (t + 1) * (n / (t + 1) + 1) ^ 2 +
2 *
((n / (t + 1)) ^ 2 * Nat.choose (t + 1 - n % (t + 1)) 2 +
n / (t + 1) * (n / (t + 1) + 1) * (n % (t + 1)) * (t + 1 - n % (t + 1)) +
(n / (t + 1) + 1) ^ 2 * Nat.choose (n % (t + 1)) 2) =
n ^ 2 | t n : ℕ
t1 : t + 1 - 1 = t
n1 : (t + 1) * (n / (t + 1)) + n % (t + 1) = n
n2 : n % (t + 1) < t + 1
n3 : t + 1 - n % (t + 1) + n % (t + 1) = t + 1
⊢ (t + 1 - n % (t + 1)) * (n / (t + 1)) ^ 2 + n % (t + 1) * (n / (t + 1) + 1) ^ 2 +
2 *
((n / (t + 1)) ^ 2 * Nat.choose (t + 1 - n % (t + 1)) 2 +
n / (t + 1) * (n / (t + 1) + 1) * (n % (t + 1)) * (t + 1 - n % (t + 1)) +
(n / (t + 1) + 1) ^ 2 * Nat.choose (n % (t + 1)) 2) =
n ^ 2 | Please generate a tactic in lean4 to solve the state.
STATE:
t n : ℕ
t1 : t + 1 - 1 = t
n1 : (t + 1) * (n / (t + 1)) + n % (t + 1) = n
n2 : n % (t + 1) < t + 1
⊢ (t + 1 - n % (t + 1)) * (n / (t + 1)) ^ 2 + n % (t + 1) * (n / (t + 1) + 1) ^ 2 +
2 *
((n / (t + 1)) ^ 2 * Nat.choose (t + 1 - n % (t + 1)) 2 +
n / (t + 1) * (n / (t + 1) + 1) * (n % (t + 1)) * (t + 1 - n % (t + 1)) +
(n / (t + 1) + 1) ^ 2 * Nat.choose (n % (t + 1)) 2) =
n ^ 2
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/Turanpartition.lean | Turanpartition.tn_turanNumb' | [76, 1] | [119, 111] | set a := n / (t + 1) | t n : ℕ
t1 : t + 1 - 1 = t
n1 : (t + 1) * (n / (t + 1)) + n % (t + 1) = n
n2 : n % (t + 1) < t + 1
n3 : t + 1 - n % (t + 1) + n % (t + 1) = t + 1
⊢ (t + 1 - n % (t + 1)) * (n / (t + 1)) ^ 2 + n % (t + 1) * (n / (t + 1) + 1) ^ 2 +
2 *
((n / (t + 1)) ^ 2 * Nat.choose (t + 1 - n % (t + 1)) 2 +
n / (t + 1) * (n / (t + 1) + 1) * (n % (t + 1)) * (t + 1 - n % (t + 1)) +
(n / (t + 1) + 1) ^ 2 * Nat.choose (n % (t + 1)) 2) =
n ^ 2 | t n : ℕ
t1 : t + 1 - 1 = t
n2 : n % (t + 1) < t + 1
n3 : t + 1 - n % (t + 1) + n % (t + 1) = t + 1
a : ℕ := n / (t + 1)
n1 : (t + 1) * a + n % (t + 1) = n
⊢ (t + 1 - n % (t + 1)) * a ^ 2 + n % (t + 1) * (a + 1) ^ 2 +
2 *
(a ^ 2 * Nat.choose (t + 1 - n % (t + 1)) 2 + a * (a + 1) * (n % (t + 1)) * (t + 1 - n % (t + 1)) +
(a + 1) ^ 2 * Nat.choose (n % (t + 1)) 2) =
n ^ 2 | Please generate a tactic in lean4 to solve the state.
STATE:
t n : ℕ
t1 : t + 1 - 1 = t
n1 : (t + 1) * (n / (t + 1)) + n % (t + 1) = n
n2 : n % (t + 1) < t + 1
n3 : t + 1 - n % (t + 1) + n % (t + 1) = t + 1
⊢ (t + 1 - n % (t + 1)) * (n / (t + 1)) ^ 2 + n % (t + 1) * (n / (t + 1) + 1) ^ 2 +
2 *
((n / (t + 1)) ^ 2 * Nat.choose (t + 1 - n % (t + 1)) 2 +
n / (t + 1) * (n / (t + 1) + 1) * (n % (t + 1)) * (t + 1 - n % (t + 1)) +
(n / (t + 1) + 1) ^ 2 * Nat.choose (n % (t + 1)) 2) =
n ^ 2
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/Turanpartition.lean | Turanpartition.tn_turanNumb' | [76, 1] | [119, 111] | set b := n % (t + 1) with hb | t n : ℕ
t1 : t + 1 - 1 = t
n2 : n % (t + 1) < t + 1
n3 : t + 1 - n % (t + 1) + n % (t + 1) = t + 1
a : ℕ := n / (t + 1)
n1 : (t + 1) * a + n % (t + 1) = n
⊢ (t + 1 - n % (t + 1)) * a ^ 2 + n % (t + 1) * (a + 1) ^ 2 +
2 *
(a ^ 2 * Nat.choose (t + 1 - n % (t + 1)) 2 + a * (a + 1) * (n % (t + 1)) * (t + 1 - n % (t + 1)) +
(a + 1) ^ 2 * Nat.choose (n % (t + 1)) 2) =
n ^ 2 | t n : ℕ
t1 : t + 1 - 1 = t
a : ℕ := n / (t + 1)
b : ℕ := n % (t + 1)
n2 : b < t + 1
n3 : t + 1 - b + b = t + 1
n1 : (t + 1) * a + b = n
hb : b = n % (t + 1)
⊢ (t + 1 - b) * a ^ 2 + b * (a + 1) ^ 2 +
2 * (a ^ 2 * Nat.choose (t + 1 - b) 2 + a * (a + 1) * b * (t + 1 - b) + (a + 1) ^ 2 * Nat.choose b 2) =
n ^ 2 | Please generate a tactic in lean4 to solve the state.
STATE:
t n : ℕ
t1 : t + 1 - 1 = t
n2 : n % (t + 1) < t + 1
n3 : t + 1 - n % (t + 1) + n % (t + 1) = t + 1
a : ℕ := n / (t + 1)
n1 : (t + 1) * a + n % (t + 1) = n
⊢ (t + 1 - n % (t + 1)) * a ^ 2 + n % (t + 1) * (a + 1) ^ 2 +
2 *
(a ^ 2 * Nat.choose (t + 1 - n % (t + 1)) 2 + a * (a + 1) * (n % (t + 1)) * (t + 1 - n % (t + 1)) +
(a + 1) ^ 2 * Nat.choose (n % (t + 1)) 2) =
n ^ 2
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/Turanpartition.lean | Turanpartition.tn_turanNumb' | [76, 1] | [119, 111] | cases Nat.eq_zero_or_pos n with
| inl hn =>
rw [hn] at hb n1 ⊢
rw [Nat.zero_mod] at hb
rw [hb] at n3 n1 ⊢
simp_rw [tsub_zero,hn,Nat.choose_zero_succ,Nat.zero_div,zero_mul,mul_zero,add_zero,zero_pow' ,zero_mul,mul_zero]
| inr hn =>
cases Nat.eq_zero_or_pos b with
| inl hb' =>
rw [hb', tsub_zero,mul_add,mul_add,zero_mul,←mul_assoc 2,mul_comm 2 (a^2),
mul_assoc,two,t1,mul_zero, zero_mul,mul_zero,add_zero,add_zero,Nat.choose_zero_succ,mul_zero,mul_zero,add_zero];
rw [hb',add_zero] at n1
rw [←n1,mul_pow,mul_comm (a^2),←add_mul]
nth_rw 1 [←mul_one (t+1)]; rw [←mul_add,add_comm 1,←pow_two]
|inr hb' =>
rw [← n3, add_mul] at n1 ; nth_rw 3 [← mul_one b] at n1 ; rw [add_assoc, ← mul_add b] at n1
rw [← n1]; rw [mul_add, mul_add, ← mul_assoc 2, ← mul_assoc 2 ((a + 1) ^ 2)]
nth_rw 1 [mul_comm 2 _];
rw [mul_assoc, two,mul_comm 2 ((a+1)^2),mul_assoc ((a+1)^2) 2,two, square ((t + 1 - b) * a) (b * (a + 1)),
mul_comm _ (a ^ 2)]
set c := t + 1 - b;
have hc2 : c - 1 + 1 = c := tsub_add_cancel_of_le (tsub_pos_of_lt n2)
have hb2 : b - 1 + 1 = b := tsub_add_cancel_of_le hb'
rw [add_comm (a^2*c),add_assoc,←add_assoc (a^2*c),←add_assoc (a^2*c),← mul_add (a^2)]
nth_rw 1 [←mul_one c,←mul_add c,add_comm _ (c-1),hc2]
rw [add_comm (b*(a+1)^2),add_assoc,mul_comm b ((a+1)^2),←mul_add ((a+1)^2)]
nth_rw 4 [←mul_one b];
rw[←mul_add b,hb2,mul_comm c a,mul_pow,←pow_two c,add_assoc,add_assoc,
add_right_inj,mul_pow,mul_add,pow_two,mul_add,mul_one,add_mul,mul_one,pow_two,mul_add,add_mul,
add_mul,add_mul,add_mul,one_mul,mul_one,add_mul,one_mul,mul_add,mul_add,mul_add,mul_add,mul_add,mul_one,
← add_assoc,← add_assoc,← add_assoc,← add_assoc,←add_assoc,add_left_inj,mul_comm a (b*b),add_left_inj,
←add_assoc,add_left_inj,mul_comm (b*b), add_left_inj,mul_assoc 2 _ b,mul_assoc a c b,mul_comm c b,
←mul_assoc a,add_left_inj,mul_assoc 2,mul_assoc,mul_assoc,mul_assoc,mul_comm b a,mul_comm c,mul_assoc] | t n : ℕ
t1 : t + 1 - 1 = t
a : ℕ := n / (t + 1)
b : ℕ := n % (t + 1)
n2 : b < t + 1
n3 : t + 1 - b + b = t + 1
n1 : (t + 1) * a + b = n
hb : b = n % (t + 1)
⊢ (t + 1 - b) * a ^ 2 + b * (a + 1) ^ 2 +
2 * (a ^ 2 * Nat.choose (t + 1 - b) 2 + a * (a + 1) * b * (t + 1 - b) + (a + 1) ^ 2 * Nat.choose b 2) =
n ^ 2 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
t n : ℕ
t1 : t + 1 - 1 = t
a : ℕ := n / (t + 1)
b : ℕ := n % (t + 1)
n2 : b < t + 1
n3 : t + 1 - b + b = t + 1
n1 : (t + 1) * a + b = n
hb : b = n % (t + 1)
⊢ (t + 1 - b) * a ^ 2 + b * (a + 1) ^ 2 +
2 * (a ^ 2 * Nat.choose (t + 1 - b) 2 + a * (a + 1) * b * (t + 1 - b) + (a + 1) ^ 2 * Nat.choose b 2) =
n ^ 2
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/Turanpartition.lean | Turanpartition.tn_turanNumb' | [76, 1] | [119, 111] | rw [hn] at hb n1 ⊢ | case inl
t n : ℕ
t1 : t + 1 - 1 = t
a : ℕ := n / (t + 1)
b : ℕ := n % (t + 1)
n2 : b < t + 1
n3 : t + 1 - b + b = t + 1
n1 : (t + 1) * a + b = n
hb : b = n % (t + 1)
hn : n = 0
⊢ (t + 1 - b) * a ^ 2 + b * (a + 1) ^ 2 +
2 * (a ^ 2 * Nat.choose (t + 1 - b) 2 + a * (a + 1) * b * (t + 1 - b) + (a + 1) ^ 2 * Nat.choose b 2) =
n ^ 2 | case inl
t n : ℕ
t1 : t + 1 - 1 = t
a : ℕ := n / (t + 1)
b : ℕ := n % (t + 1)
n2 : b < t + 1
n3 : t + 1 - b + b = t + 1
n1 : (t + 1) * a + b = 0
hb : b = 0 % (t + 1)
hn : n = 0
⊢ (t + 1 - b) * a ^ 2 + b * (a + 1) ^ 2 +
2 * (a ^ 2 * Nat.choose (t + 1 - b) 2 + a * (a + 1) * b * (t + 1 - b) + (a + 1) ^ 2 * Nat.choose b 2) =
0 ^ 2 | Please generate a tactic in lean4 to solve the state.
STATE:
case inl
t n : ℕ
t1 : t + 1 - 1 = t
a : ℕ := n / (t + 1)
b : ℕ := n % (t + 1)
n2 : b < t + 1
n3 : t + 1 - b + b = t + 1
n1 : (t + 1) * a + b = n
hb : b = n % (t + 1)
hn : n = 0
⊢ (t + 1 - b) * a ^ 2 + b * (a + 1) ^ 2 +
2 * (a ^ 2 * Nat.choose (t + 1 - b) 2 + a * (a + 1) * b * (t + 1 - b) + (a + 1) ^ 2 * Nat.choose b 2) =
n ^ 2
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/Turanpartition.lean | Turanpartition.tn_turanNumb' | [76, 1] | [119, 111] | rw [Nat.zero_mod] at hb | case inl
t n : ℕ
t1 : t + 1 - 1 = t
a : ℕ := n / (t + 1)
b : ℕ := n % (t + 1)
n2 : b < t + 1
n3 : t + 1 - b + b = t + 1
n1 : (t + 1) * a + b = 0
hb : b = 0 % (t + 1)
hn : n = 0
⊢ (t + 1 - b) * a ^ 2 + b * (a + 1) ^ 2 +
2 * (a ^ 2 * Nat.choose (t + 1 - b) 2 + a * (a + 1) * b * (t + 1 - b) + (a + 1) ^ 2 * Nat.choose b 2) =
0 ^ 2 | case inl
t n : ℕ
t1 : t + 1 - 1 = t
a : ℕ := n / (t + 1)
b : ℕ := n % (t + 1)
n2 : b < t + 1
n3 : t + 1 - b + b = t + 1
n1 : (t + 1) * a + b = 0
hb : b = 0
hn : n = 0
⊢ (t + 1 - b) * a ^ 2 + b * (a + 1) ^ 2 +
2 * (a ^ 2 * Nat.choose (t + 1 - b) 2 + a * (a + 1) * b * (t + 1 - b) + (a + 1) ^ 2 * Nat.choose b 2) =
0 ^ 2 | Please generate a tactic in lean4 to solve the state.
STATE:
case inl
t n : ℕ
t1 : t + 1 - 1 = t
a : ℕ := n / (t + 1)
b : ℕ := n % (t + 1)
n2 : b < t + 1
n3 : t + 1 - b + b = t + 1
n1 : (t + 1) * a + b = 0
hb : b = 0 % (t + 1)
hn : n = 0
⊢ (t + 1 - b) * a ^ 2 + b * (a + 1) ^ 2 +
2 * (a ^ 2 * Nat.choose (t + 1 - b) 2 + a * (a + 1) * b * (t + 1 - b) + (a + 1) ^ 2 * Nat.choose b 2) =
0 ^ 2
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/Turanpartition.lean | Turanpartition.tn_turanNumb' | [76, 1] | [119, 111] | rw [hb] at n3 n1 ⊢ | case inl
t n : ℕ
t1 : t + 1 - 1 = t
a : ℕ := n / (t + 1)
b : ℕ := n % (t + 1)
n2 : b < t + 1
n3 : t + 1 - b + b = t + 1
n1 : (t + 1) * a + b = 0
hb : b = 0
hn : n = 0
⊢ (t + 1 - b) * a ^ 2 + b * (a + 1) ^ 2 +
2 * (a ^ 2 * Nat.choose (t + 1 - b) 2 + a * (a + 1) * b * (t + 1 - b) + (a + 1) ^ 2 * Nat.choose b 2) =
0 ^ 2 | case inl
t n : ℕ
t1 : t + 1 - 1 = t
a : ℕ := n / (t + 1)
b : ℕ := n % (t + 1)
n2 : b < t + 1
n3 : t + 1 - 0 + 0 = t + 1
n1 : (t + 1) * a + 0 = 0
hb : b = 0
hn : n = 0
⊢ (t + 1 - 0) * a ^ 2 + 0 * (a + 1) ^ 2 +
2 * (a ^ 2 * Nat.choose (t + 1 - 0) 2 + a * (a + 1) * 0 * (t + 1 - 0) + (a + 1) ^ 2 * Nat.choose 0 2) =
0 ^ 2 | Please generate a tactic in lean4 to solve the state.
STATE:
case inl
t n : ℕ
t1 : t + 1 - 1 = t
a : ℕ := n / (t + 1)
b : ℕ := n % (t + 1)
n2 : b < t + 1
n3 : t + 1 - b + b = t + 1
n1 : (t + 1) * a + b = 0
hb : b = 0
hn : n = 0
⊢ (t + 1 - b) * a ^ 2 + b * (a + 1) ^ 2 +
2 * (a ^ 2 * Nat.choose (t + 1 - b) 2 + a * (a + 1) * b * (t + 1 - b) + (a + 1) ^ 2 * Nat.choose b 2) =
0 ^ 2
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/Turanpartition.lean | Turanpartition.tn_turanNumb' | [76, 1] | [119, 111] | simp_rw [tsub_zero,hn,Nat.choose_zero_succ,Nat.zero_div,zero_mul,mul_zero,add_zero,zero_pow' ,zero_mul,mul_zero] | case inl
t n : ℕ
t1 : t + 1 - 1 = t
a : ℕ := n / (t + 1)
b : ℕ := n % (t + 1)
n2 : b < t + 1
n3 : t + 1 - 0 + 0 = t + 1
n1 : (t + 1) * a + 0 = 0
hb : b = 0
hn : n = 0
⊢ (t + 1 - 0) * a ^ 2 + 0 * (a + 1) ^ 2 +
2 * (a ^ 2 * Nat.choose (t + 1 - 0) 2 + a * (a + 1) * 0 * (t + 1 - 0) + (a + 1) ^ 2 * Nat.choose 0 2) =
0 ^ 2 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case inl
t n : ℕ
t1 : t + 1 - 1 = t
a : ℕ := n / (t + 1)
b : ℕ := n % (t + 1)
n2 : b < t + 1
n3 : t + 1 - 0 + 0 = t + 1
n1 : (t + 1) * a + 0 = 0
hb : b = 0
hn : n = 0
⊢ (t + 1 - 0) * a ^ 2 + 0 * (a + 1) ^ 2 +
2 * (a ^ 2 * Nat.choose (t + 1 - 0) 2 + a * (a + 1) * 0 * (t + 1 - 0) + (a + 1) ^ 2 * Nat.choose 0 2) =
0 ^ 2
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/Turanpartition.lean | Turanpartition.tn_turanNumb' | [76, 1] | [119, 111] | cases Nat.eq_zero_or_pos b with
| inl hb' =>
rw [hb', tsub_zero,mul_add,mul_add,zero_mul,←mul_assoc 2,mul_comm 2 (a^2),
mul_assoc,two,t1,mul_zero, zero_mul,mul_zero,add_zero,add_zero,Nat.choose_zero_succ,mul_zero,mul_zero,add_zero];
rw [hb',add_zero] at n1
rw [←n1,mul_pow,mul_comm (a^2),←add_mul]
nth_rw 1 [←mul_one (t+1)]; rw [←mul_add,add_comm 1,←pow_two]
|inr hb' =>
rw [← n3, add_mul] at n1 ; nth_rw 3 [← mul_one b] at n1 ; rw [add_assoc, ← mul_add b] at n1
rw [← n1]; rw [mul_add, mul_add, ← mul_assoc 2, ← mul_assoc 2 ((a + 1) ^ 2)]
nth_rw 1 [mul_comm 2 _];
rw [mul_assoc, two,mul_comm 2 ((a+1)^2),mul_assoc ((a+1)^2) 2,two, square ((t + 1 - b) * a) (b * (a + 1)),
mul_comm _ (a ^ 2)]
set c := t + 1 - b;
have hc2 : c - 1 + 1 = c := tsub_add_cancel_of_le (tsub_pos_of_lt n2)
have hb2 : b - 1 + 1 = b := tsub_add_cancel_of_le hb'
rw [add_comm (a^2*c),add_assoc,←add_assoc (a^2*c),←add_assoc (a^2*c),← mul_add (a^2)]
nth_rw 1 [←mul_one c,←mul_add c,add_comm _ (c-1),hc2]
rw [add_comm (b*(a+1)^2),add_assoc,mul_comm b ((a+1)^2),←mul_add ((a+1)^2)]
nth_rw 4 [←mul_one b];
rw[←mul_add b,hb2,mul_comm c a,mul_pow,←pow_two c,add_assoc,add_assoc,
add_right_inj,mul_pow,mul_add,pow_two,mul_add,mul_one,add_mul,mul_one,pow_two,mul_add,add_mul,
add_mul,add_mul,add_mul,one_mul,mul_one,add_mul,one_mul,mul_add,mul_add,mul_add,mul_add,mul_add,mul_one,
← add_assoc,← add_assoc,← add_assoc,← add_assoc,←add_assoc,add_left_inj,mul_comm a (b*b),add_left_inj,
←add_assoc,add_left_inj,mul_comm (b*b), add_left_inj,mul_assoc 2 _ b,mul_assoc a c b,mul_comm c b,
←mul_assoc a,add_left_inj,mul_assoc 2,mul_assoc,mul_assoc,mul_assoc,mul_comm b a,mul_comm c,mul_assoc] | case inr
t n : ℕ
t1 : t + 1 - 1 = t
a : ℕ := n / (t + 1)
b : ℕ := n % (t + 1)
n2 : b < t + 1
n3 : t + 1 - b + b = t + 1
n1 : (t + 1) * a + b = n
hb : b = n % (t + 1)
hn : n > 0
⊢ (t + 1 - b) * a ^ 2 + b * (a + 1) ^ 2 +
2 * (a ^ 2 * Nat.choose (t + 1 - b) 2 + a * (a + 1) * b * (t + 1 - b) + (a + 1) ^ 2 * Nat.choose b 2) =
n ^ 2 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case inr
t n : ℕ
t1 : t + 1 - 1 = t
a : ℕ := n / (t + 1)
b : ℕ := n % (t + 1)
n2 : b < t + 1
n3 : t + 1 - b + b = t + 1
n1 : (t + 1) * a + b = n
hb : b = n % (t + 1)
hn : n > 0
⊢ (t + 1 - b) * a ^ 2 + b * (a + 1) ^ 2 +
2 * (a ^ 2 * Nat.choose (t + 1 - b) 2 + a * (a + 1) * b * (t + 1 - b) + (a + 1) ^ 2 * Nat.choose b 2) =
n ^ 2
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/Turanpartition.lean | Turanpartition.tn_turanNumb' | [76, 1] | [119, 111] | rw [hb', tsub_zero,mul_add,mul_add,zero_mul,←mul_assoc 2,mul_comm 2 (a^2),
mul_assoc,two,t1,mul_zero, zero_mul,mul_zero,add_zero,add_zero,Nat.choose_zero_succ,mul_zero,mul_zero,add_zero] | case inr.inl
t n : ℕ
t1 : t + 1 - 1 = t
a : ℕ := n / (t + 1)
b : ℕ := n % (t + 1)
n2 : b < t + 1
n3 : t + 1 - b + b = t + 1
n1 : (t + 1) * a + b = n
hb : b = n % (t + 1)
hn : n > 0
hb' : b = 0
⊢ (t + 1 - b) * a ^ 2 + b * (a + 1) ^ 2 +
2 * (a ^ 2 * Nat.choose (t + 1 - b) 2 + a * (a + 1) * b * (t + 1 - b) + (a + 1) ^ 2 * Nat.choose b 2) =
n ^ 2 | case inr.inl
t n : ℕ
t1 : t + 1 - 1 = t
a : ℕ := n / (t + 1)
b : ℕ := n % (t + 1)
n2 : b < t + 1
n3 : t + 1 - b + b = t + 1
n1 : (t + 1) * a + b = n
hb : b = n % (t + 1)
hn : n > 0
hb' : b = 0
⊢ (t + 1) * a ^ 2 + a ^ 2 * ((t + 1) * t) = n ^ 2 | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.inl
t n : ℕ
t1 : t + 1 - 1 = t
a : ℕ := n / (t + 1)
b : ℕ := n % (t + 1)
n2 : b < t + 1
n3 : t + 1 - b + b = t + 1
n1 : (t + 1) * a + b = n
hb : b = n % (t + 1)
hn : n > 0
hb' : b = 0
⊢ (t + 1 - b) * a ^ 2 + b * (a + 1) ^ 2 +
2 * (a ^ 2 * Nat.choose (t + 1 - b) 2 + a * (a + 1) * b * (t + 1 - b) + (a + 1) ^ 2 * Nat.choose b 2) =
n ^ 2
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/Turanpartition.lean | Turanpartition.tn_turanNumb' | [76, 1] | [119, 111] | rw [hb',add_zero] at n1 | case inr.inl
t n : ℕ
t1 : t + 1 - 1 = t
a : ℕ := n / (t + 1)
b : ℕ := n % (t + 1)
n2 : b < t + 1
n3 : t + 1 - b + b = t + 1
n1 : (t + 1) * a + b = n
hb : b = n % (t + 1)
hn : n > 0
hb' : b = 0
⊢ (t + 1) * a ^ 2 + a ^ 2 * ((t + 1) * t) = n ^ 2 | case inr.inl
t n : ℕ
t1 : t + 1 - 1 = t
a : ℕ := n / (t + 1)
b : ℕ := n % (t + 1)
n2 : b < t + 1
n3 : t + 1 - b + b = t + 1
n1 : (t + 1) * a = n
hb : b = n % (t + 1)
hn : n > 0
hb' : b = 0
⊢ (t + 1) * a ^ 2 + a ^ 2 * ((t + 1) * t) = n ^ 2 | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.inl
t n : ℕ
t1 : t + 1 - 1 = t
a : ℕ := n / (t + 1)
b : ℕ := n % (t + 1)
n2 : b < t + 1
n3 : t + 1 - b + b = t + 1
n1 : (t + 1) * a + b = n
hb : b = n % (t + 1)
hn : n > 0
hb' : b = 0
⊢ (t + 1) * a ^ 2 + a ^ 2 * ((t + 1) * t) = n ^ 2
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/Turanpartition.lean | Turanpartition.tn_turanNumb' | [76, 1] | [119, 111] | rw [←n1,mul_pow,mul_comm (a^2),←add_mul] | case inr.inl
t n : ℕ
t1 : t + 1 - 1 = t
a : ℕ := n / (t + 1)
b : ℕ := n % (t + 1)
n2 : b < t + 1
n3 : t + 1 - b + b = t + 1
n1 : (t + 1) * a = n
hb : b = n % (t + 1)
hn : n > 0
hb' : b = 0
⊢ (t + 1) * a ^ 2 + a ^ 2 * ((t + 1) * t) = n ^ 2 | case inr.inl
t n : ℕ
t1 : t + 1 - 1 = t
a : ℕ := n / (t + 1)
b : ℕ := n % (t + 1)
n2 : b < t + 1
n3 : t + 1 - b + b = t + 1
n1 : (t + 1) * a = n
hb : b = n % (t + 1)
hn : n > 0
hb' : b = 0
⊢ (t + 1 + (t + 1) * t) * a ^ 2 = (t + 1) ^ 2 * a ^ 2 | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.inl
t n : ℕ
t1 : t + 1 - 1 = t
a : ℕ := n / (t + 1)
b : ℕ := n % (t + 1)
n2 : b < t + 1
n3 : t + 1 - b + b = t + 1
n1 : (t + 1) * a = n
hb : b = n % (t + 1)
hn : n > 0
hb' : b = 0
⊢ (t + 1) * a ^ 2 + a ^ 2 * ((t + 1) * t) = n ^ 2
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/Turanpartition.lean | Turanpartition.tn_turanNumb' | [76, 1] | [119, 111] | nth_rw 1 [←mul_one (t+1)] | case inr.inl
t n : ℕ
t1 : t + 1 - 1 = t
a : ℕ := n / (t + 1)
b : ℕ := n % (t + 1)
n2 : b < t + 1
n3 : t + 1 - b + b = t + 1
n1 : (t + 1) * a = n
hb : b = n % (t + 1)
hn : n > 0
hb' : b = 0
⊢ (t + 1 + (t + 1) * t) * a ^ 2 = (t + 1) ^ 2 * a ^ 2 | case inr.inl
t n : ℕ
t1 : t + 1 - 1 = t
a : ℕ := n / (t + 1)
b : ℕ := n % (t + 1)
n2 : b < t + 1
n3 : t + 1 - b + b = t + 1
n1 : (t + 1) * a = n
hb : b = n % (t + 1)
hn : n > 0
hb' : b = 0
⊢ ((t + 1) * 1 + (t + 1) * t) * a ^ 2 = (t + 1) ^ 2 * a ^ 2 | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.inl
t n : ℕ
t1 : t + 1 - 1 = t
a : ℕ := n / (t + 1)
b : ℕ := n % (t + 1)
n2 : b < t + 1
n3 : t + 1 - b + b = t + 1
n1 : (t + 1) * a = n
hb : b = n % (t + 1)
hn : n > 0
hb' : b = 0
⊢ (t + 1 + (t + 1) * t) * a ^ 2 = (t + 1) ^ 2 * a ^ 2
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/Turanpartition.lean | Turanpartition.tn_turanNumb' | [76, 1] | [119, 111] | rw [←mul_add,add_comm 1,←pow_two] | case inr.inl
t n : ℕ
t1 : t + 1 - 1 = t
a : ℕ := n / (t + 1)
b : ℕ := n % (t + 1)
n2 : b < t + 1
n3 : t + 1 - b + b = t + 1
n1 : (t + 1) * a = n
hb : b = n % (t + 1)
hn : n > 0
hb' : b = 0
⊢ ((t + 1) * 1 + (t + 1) * t) * a ^ 2 = (t + 1) ^ 2 * a ^ 2 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.inl
t n : ℕ
t1 : t + 1 - 1 = t
a : ℕ := n / (t + 1)
b : ℕ := n % (t + 1)
n2 : b < t + 1
n3 : t + 1 - b + b = t + 1
n1 : (t + 1) * a = n
hb : b = n % (t + 1)
hn : n > 0
hb' : b = 0
⊢ ((t + 1) * 1 + (t + 1) * t) * a ^ 2 = (t + 1) ^ 2 * a ^ 2
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/Turanpartition.lean | Turanpartition.tn_turanNumb' | [76, 1] | [119, 111] | rw [← n3, add_mul] at n1 | case inr.inr
t n : ℕ
t1 : t + 1 - 1 = t
a : ℕ := n / (t + 1)
b : ℕ := n % (t + 1)
n2 : b < t + 1
n3 : t + 1 - b + b = t + 1
n1 : (t + 1) * a + b = n
hb : b = n % (t + 1)
hn : n > 0
hb' : b > 0
⊢ (t + 1 - b) * a ^ 2 + b * (a + 1) ^ 2 +
2 * (a ^ 2 * Nat.choose (t + 1 - b) 2 + a * (a + 1) * b * (t + 1 - b) + (a + 1) ^ 2 * Nat.choose b 2) =
n ^ 2 | case inr.inr
t n : ℕ
t1 : t + 1 - 1 = t
a : ℕ := n / (t + 1)
b : ℕ := n % (t + 1)
n2 : b < t + 1
n3 : t + 1 - b + b = t + 1
n1 : (t + 1 - b) * a + b * a + b = n
hb : b = n % (t + 1)
hn : n > 0
hb' : b > 0
⊢ (t + 1 - b) * a ^ 2 + b * (a + 1) ^ 2 +
2 * (a ^ 2 * Nat.choose (t + 1 - b) 2 + a * (a + 1) * b * (t + 1 - b) + (a + 1) ^ 2 * Nat.choose b 2) =
n ^ 2 | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.inr
t n : ℕ
t1 : t + 1 - 1 = t
a : ℕ := n / (t + 1)
b : ℕ := n % (t + 1)
n2 : b < t + 1
n3 : t + 1 - b + b = t + 1
n1 : (t + 1) * a + b = n
hb : b = n % (t + 1)
hn : n > 0
hb' : b > 0
⊢ (t + 1 - b) * a ^ 2 + b * (a + 1) ^ 2 +
2 * (a ^ 2 * Nat.choose (t + 1 - b) 2 + a * (a + 1) * b * (t + 1 - b) + (a + 1) ^ 2 * Nat.choose b 2) =
n ^ 2
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/Turanpartition.lean | Turanpartition.tn_turanNumb' | [76, 1] | [119, 111] | nth_rw 3 [← mul_one b] at n1 | case inr.inr
t n : ℕ
t1 : t + 1 - 1 = t
a : ℕ := n / (t + 1)
b : ℕ := n % (t + 1)
n2 : b < t + 1
n3 : t + 1 - b + b = t + 1
n1 : (t + 1 - b) * a + b * a + b = n
hb : b = n % (t + 1)
hn : n > 0
hb' : b > 0
⊢ (t + 1 - b) * a ^ 2 + b * (a + 1) ^ 2 +
2 * (a ^ 2 * Nat.choose (t + 1 - b) 2 + a * (a + 1) * b * (t + 1 - b) + (a + 1) ^ 2 * Nat.choose b 2) =
n ^ 2 | case inr.inr
t n : ℕ
t1 : t + 1 - 1 = t
a : ℕ := n / (t + 1)
b : ℕ := n % (t + 1)
n2 : b < t + 1
n3 : t + 1 - b + b = t + 1
n1 : (t + 1 - b) * a + b * a + b * 1 = n
hb : b = n % (t + 1)
hn : n > 0
hb' : b > 0
⊢ (t + 1 - b) * a ^ 2 + b * (a + 1) ^ 2 +
2 * (a ^ 2 * Nat.choose (t + 1 - b) 2 + a * (a + 1) * b * (t + 1 - b) + (a + 1) ^ 2 * Nat.choose b 2) =
n ^ 2 | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.inr
t n : ℕ
t1 : t + 1 - 1 = t
a : ℕ := n / (t + 1)
b : ℕ := n % (t + 1)
n2 : b < t + 1
n3 : t + 1 - b + b = t + 1
n1 : (t + 1 - b) * a + b * a + b = n
hb : b = n % (t + 1)
hn : n > 0
hb' : b > 0
⊢ (t + 1 - b) * a ^ 2 + b * (a + 1) ^ 2 +
2 * (a ^ 2 * Nat.choose (t + 1 - b) 2 + a * (a + 1) * b * (t + 1 - b) + (a + 1) ^ 2 * Nat.choose b 2) =
n ^ 2
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/Turanpartition.lean | Turanpartition.tn_turanNumb' | [76, 1] | [119, 111] | rw [add_assoc, ← mul_add b] at n1 | case inr.inr
t n : ℕ
t1 : t + 1 - 1 = t
a : ℕ := n / (t + 1)
b : ℕ := n % (t + 1)
n2 : b < t + 1
n3 : t + 1 - b + b = t + 1
n1 : (t + 1 - b) * a + b * a + b * 1 = n
hb : b = n % (t + 1)
hn : n > 0
hb' : b > 0
⊢ (t + 1 - b) * a ^ 2 + b * (a + 1) ^ 2 +
2 * (a ^ 2 * Nat.choose (t + 1 - b) 2 + a * (a + 1) * b * (t + 1 - b) + (a + 1) ^ 2 * Nat.choose b 2) =
n ^ 2 | case inr.inr
t n : ℕ
t1 : t + 1 - 1 = t
a : ℕ := n / (t + 1)
b : ℕ := n % (t + 1)
n2 : b < t + 1
n3 : t + 1 - b + b = t + 1
n1 : (t + 1 - b) * a + b * (a + 1) = n
hb : b = n % (t + 1)
hn : n > 0
hb' : b > 0
⊢ (t + 1 - b) * a ^ 2 + b * (a + 1) ^ 2 +
2 * (a ^ 2 * Nat.choose (t + 1 - b) 2 + a * (a + 1) * b * (t + 1 - b) + (a + 1) ^ 2 * Nat.choose b 2) =
n ^ 2 | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.inr
t n : ℕ
t1 : t + 1 - 1 = t
a : ℕ := n / (t + 1)
b : ℕ := n % (t + 1)
n2 : b < t + 1
n3 : t + 1 - b + b = t + 1
n1 : (t + 1 - b) * a + b * a + b * 1 = n
hb : b = n % (t + 1)
hn : n > 0
hb' : b > 0
⊢ (t + 1 - b) * a ^ 2 + b * (a + 1) ^ 2 +
2 * (a ^ 2 * Nat.choose (t + 1 - b) 2 + a * (a + 1) * b * (t + 1 - b) + (a + 1) ^ 2 * Nat.choose b 2) =
n ^ 2
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/Turanpartition.lean | Turanpartition.tn_turanNumb' | [76, 1] | [119, 111] | rw [← n1] | case inr.inr
t n : ℕ
t1 : t + 1 - 1 = t
a : ℕ := n / (t + 1)
b : ℕ := n % (t + 1)
n2 : b < t + 1
n3 : t + 1 - b + b = t + 1
n1 : (t + 1 - b) * a + b * (a + 1) = n
hb : b = n % (t + 1)
hn : n > 0
hb' : b > 0
⊢ (t + 1 - b) * a ^ 2 + b * (a + 1) ^ 2 +
2 * (a ^ 2 * Nat.choose (t + 1 - b) 2 + a * (a + 1) * b * (t + 1 - b) + (a + 1) ^ 2 * Nat.choose b 2) =
n ^ 2 | case inr.inr
t n : ℕ
t1 : t + 1 - 1 = t
a : ℕ := n / (t + 1)
b : ℕ := n % (t + 1)
n2 : b < t + 1
n3 : t + 1 - b + b = t + 1
n1 : (t + 1 - b) * a + b * (a + 1) = n
hb : b = n % (t + 1)
hn : n > 0
hb' : b > 0
⊢ (t + 1 - b) * a ^ 2 + b * (a + 1) ^ 2 +
2 * (a ^ 2 * Nat.choose (t + 1 - b) 2 + a * (a + 1) * b * (t + 1 - b) + (a + 1) ^ 2 * Nat.choose b 2) =
((t + 1 - b) * a + b * (a + 1)) ^ 2 | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.inr
t n : ℕ
t1 : t + 1 - 1 = t
a : ℕ := n / (t + 1)
b : ℕ := n % (t + 1)
n2 : b < t + 1
n3 : t + 1 - b + b = t + 1
n1 : (t + 1 - b) * a + b * (a + 1) = n
hb : b = n % (t + 1)
hn : n > 0
hb' : b > 0
⊢ (t + 1 - b) * a ^ 2 + b * (a + 1) ^ 2 +
2 * (a ^ 2 * Nat.choose (t + 1 - b) 2 + a * (a + 1) * b * (t + 1 - b) + (a + 1) ^ 2 * Nat.choose b 2) =
n ^ 2
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/Turanpartition.lean | Turanpartition.tn_turanNumb' | [76, 1] | [119, 111] | rw [mul_add, mul_add, ← mul_assoc 2, ← mul_assoc 2 ((a + 1) ^ 2)] | case inr.inr
t n : ℕ
t1 : t + 1 - 1 = t
a : ℕ := n / (t + 1)
b : ℕ := n % (t + 1)
n2 : b < t + 1
n3 : t + 1 - b + b = t + 1
n1 : (t + 1 - b) * a + b * (a + 1) = n
hb : b = n % (t + 1)
hn : n > 0
hb' : b > 0
⊢ (t + 1 - b) * a ^ 2 + b * (a + 1) ^ 2 +
2 * (a ^ 2 * Nat.choose (t + 1 - b) 2 + a * (a + 1) * b * (t + 1 - b) + (a + 1) ^ 2 * Nat.choose b 2) =
((t + 1 - b) * a + b * (a + 1)) ^ 2 | case inr.inr
t n : ℕ
t1 : t + 1 - 1 = t
a : ℕ := n / (t + 1)
b : ℕ := n % (t + 1)
n2 : b < t + 1
n3 : t + 1 - b + b = t + 1
n1 : (t + 1 - b) * a + b * (a + 1) = n
hb : b = n % (t + 1)
hn : n > 0
hb' : b > 0
⊢ (t + 1 - b) * a ^ 2 + b * (a + 1) ^ 2 +
(2 * a ^ 2 * Nat.choose (t + 1 - b) 2 + 2 * (a * (a + 1) * b * (t + 1 - b)) + 2 * (a + 1) ^ 2 * Nat.choose b 2) =
((t + 1 - b) * a + b * (a + 1)) ^ 2 | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.inr
t n : ℕ
t1 : t + 1 - 1 = t
a : ℕ := n / (t + 1)
b : ℕ := n % (t + 1)
n2 : b < t + 1
n3 : t + 1 - b + b = t + 1
n1 : (t + 1 - b) * a + b * (a + 1) = n
hb : b = n % (t + 1)
hn : n > 0
hb' : b > 0
⊢ (t + 1 - b) * a ^ 2 + b * (a + 1) ^ 2 +
2 * (a ^ 2 * Nat.choose (t + 1 - b) 2 + a * (a + 1) * b * (t + 1 - b) + (a + 1) ^ 2 * Nat.choose b 2) =
((t + 1 - b) * a + b * (a + 1)) ^ 2
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/Turanpartition.lean | Turanpartition.tn_turanNumb' | [76, 1] | [119, 111] | nth_rw 1 [mul_comm 2 _] | case inr.inr
t n : ℕ
t1 : t + 1 - 1 = t
a : ℕ := n / (t + 1)
b : ℕ := n % (t + 1)
n2 : b < t + 1
n3 : t + 1 - b + b = t + 1
n1 : (t + 1 - b) * a + b * (a + 1) = n
hb : b = n % (t + 1)
hn : n > 0
hb' : b > 0
⊢ (t + 1 - b) * a ^ 2 + b * (a + 1) ^ 2 +
(2 * a ^ 2 * Nat.choose (t + 1 - b) 2 + 2 * (a * (a + 1) * b * (t + 1 - b)) + 2 * (a + 1) ^ 2 * Nat.choose b 2) =
((t + 1 - b) * a + b * (a + 1)) ^ 2 | case inr.inr
t n : ℕ
t1 : t + 1 - 1 = t
a : ℕ := n / (t + 1)
b : ℕ := n % (t + 1)
n2 : b < t + 1
n3 : t + 1 - b + b = t + 1
n1 : (t + 1 - b) * a + b * (a + 1) = n
hb : b = n % (t + 1)
hn : n > 0
hb' : b > 0
⊢ (t + 1 - b) * a ^ 2 + b * (a + 1) ^ 2 +
(a ^ 2 * 2 * Nat.choose (t + 1 - b) 2 + 2 * (a * (a + 1) * b * (t + 1 - b)) + 2 * (a + 1) ^ 2 * Nat.choose b 2) =
((t + 1 - b) * a + b * (a + 1)) ^ 2 | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.inr
t n : ℕ
t1 : t + 1 - 1 = t
a : ℕ := n / (t + 1)
b : ℕ := n % (t + 1)
n2 : b < t + 1
n3 : t + 1 - b + b = t + 1
n1 : (t + 1 - b) * a + b * (a + 1) = n
hb : b = n % (t + 1)
hn : n > 0
hb' : b > 0
⊢ (t + 1 - b) * a ^ 2 + b * (a + 1) ^ 2 +
(2 * a ^ 2 * Nat.choose (t + 1 - b) 2 + 2 * (a * (a + 1) * b * (t + 1 - b)) + 2 * (a + 1) ^ 2 * Nat.choose b 2) =
((t + 1 - b) * a + b * (a + 1)) ^ 2
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/Turanpartition.lean | Turanpartition.tn_turanNumb' | [76, 1] | [119, 111] | rw [mul_assoc, two,mul_comm 2 ((a+1)^2),mul_assoc ((a+1)^2) 2,two, square ((t + 1 - b) * a) (b * (a + 1)),
mul_comm _ (a ^ 2)] | case inr.inr
t n : ℕ
t1 : t + 1 - 1 = t
a : ℕ := n / (t + 1)
b : ℕ := n % (t + 1)
n2 : b < t + 1
n3 : t + 1 - b + b = t + 1
n1 : (t + 1 - b) * a + b * (a + 1) = n
hb : b = n % (t + 1)
hn : n > 0
hb' : b > 0
⊢ (t + 1 - b) * a ^ 2 + b * (a + 1) ^ 2 +
(a ^ 2 * 2 * Nat.choose (t + 1 - b) 2 + 2 * (a * (a + 1) * b * (t + 1 - b)) + 2 * (a + 1) ^ 2 * Nat.choose b 2) =
((t + 1 - b) * a + b * (a + 1)) ^ 2 | case inr.inr
t n : ℕ
t1 : t + 1 - 1 = t
a : ℕ := n / (t + 1)
b : ℕ := n % (t + 1)
n2 : b < t + 1
n3 : t + 1 - b + b = t + 1
n1 : (t + 1 - b) * a + b * (a + 1) = n
hb : b = n % (t + 1)
hn : n > 0
hb' : b > 0
⊢ a ^ 2 * (t + 1 - b) + b * (a + 1) ^ 2 +
(a ^ 2 * ((t + 1 - b) * (t + 1 - b - 1)) + 2 * (a * (a + 1) * b * (t + 1 - b)) + (a + 1) ^ 2 * (b * (b - 1))) =
((t + 1 - b) * a) ^ 2 + 2 * ((t + 1 - b) * a) * (b * (a + 1)) + (b * (a + 1)) ^ 2 | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.inr
t n : ℕ
t1 : t + 1 - 1 = t
a : ℕ := n / (t + 1)
b : ℕ := n % (t + 1)
n2 : b < t + 1
n3 : t + 1 - b + b = t + 1
n1 : (t + 1 - b) * a + b * (a + 1) = n
hb : b = n % (t + 1)
hn : n > 0
hb' : b > 0
⊢ (t + 1 - b) * a ^ 2 + b * (a + 1) ^ 2 +
(a ^ 2 * 2 * Nat.choose (t + 1 - b) 2 + 2 * (a * (a + 1) * b * (t + 1 - b)) + 2 * (a + 1) ^ 2 * Nat.choose b 2) =
((t + 1 - b) * a + b * (a + 1)) ^ 2
TACTIC:
|
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