url stringclasses 147 values | commit stringclasses 147 values | file_path stringlengths 7 101 | full_name stringlengths 1 94 | start stringlengths 6 10 | end stringlengths 6 11 | tactic stringlengths 1 11.2k | state_before stringlengths 3 2.09M | state_after stringlengths 6 2.09M | input stringlengths 73 2.09M |
|---|---|---|---|---|---|---|---|---|---|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/Multipartite.lean | SimpleGraph.mp_cliqueFree | [452, 1] | [476, 53] | have uni := biUnion_parts M | case intro.mk
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
t : ℕ
M : MultiPart α
ht : M.t = t
hA : M.A = univ
S : Finset α
hs1 : Set.Pairwise (↑S) (mp M).Adj
hs2 : card S = t + 2
h : ∀ (i : ℕ), i ∈ range (M.t + 1) → card (S ∩ MultiPart.P M i) ≤ 1
ub : ∑ i in range (M.t + 1), card (S ∩ MultiPart.P M i) ≤ t + 1
⊢ False | case intro.mk
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
t : ℕ
M : MultiPart α
ht : M.t = t
hA : M.A = univ
S : Finset α
hs1 : Set.Pairwise (↑S) (mp M).Adj
hs2 : card S = t + 2
h : ∀ (i : ℕ), i ∈ range (M.t + 1) → card (S ∩ MultiPart.P M i) ≤ 1
ub : ∑ i in range (M.t + 1), card (S ∩ MultiPart.P M i) ≤ t + 1
uni : M.A = Finset.biUnion (range (M.t + 1)) fun i => MultiPart.P M i
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.mk
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
t : ℕ
M : MultiPart α
ht : M.t = t
hA : M.A = univ
S : Finset α
hs1 : Set.Pairwise (↑S) (mp M).Adj
hs2 : card S = t + 2
h : ∀ (i : ℕ), i ∈ range (M.t + 1) → card (S ∩ MultiPart.P M i) ≤ 1
ub : ∑ i in range (M.t + 1), card (S ∩ MultiPart.P M i) ≤ t + 1
⊢ False
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/Multipartite.lean | SimpleGraph.mp_cliqueFree | [452, 1] | [476, 53] | rw [hA] at uni | case intro.mk
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
t : ℕ
M : MultiPart α
ht : M.t = t
hA : M.A = univ
S : Finset α
hs1 : Set.Pairwise (↑S) (mp M).Adj
hs2 : card S = t + 2
h : ∀ (i : ℕ), i ∈ range (M.t + 1) → card (S ∩ MultiPart.P M i) ≤ 1
ub : ∑ i in range (M.t + 1), card (S ∩ MultiPart.P M i) ≤ t + 1
uni : M.A = Finset.biUnion (range (M.t + 1)) fun i => MultiPart.P M i
⊢ False | case intro.mk
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
t : ℕ
M : MultiPart α
ht : M.t = t
hA : M.A = univ
S : Finset α
hs1 : Set.Pairwise (↑S) (mp M).Adj
hs2 : card S = t + 2
h : ∀ (i : ℕ), i ∈ range (M.t + 1) → card (S ∩ MultiPart.P M i) ≤ 1
ub : ∑ i in range (M.t + 1), card (S ∩ MultiPart.P M i) ≤ t + 1
uni : univ = Finset.biUnion (range (M.t + 1)) fun i => MultiPart.P M i
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.mk
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
t : ℕ
M : MultiPart α
ht : M.t = t
hA : M.A = univ
S : Finset α
hs1 : Set.Pairwise (↑S) (mp M).Adj
hs2 : card S = t + 2
h : ∀ (i : ℕ), i ∈ range (M.t + 1) → card (S ∩ MultiPart.P M i) ≤ 1
ub : ∑ i in range (M.t + 1), card (S ∩ MultiPart.P M i) ≤ t + 1
uni : M.A = Finset.biUnion (range (M.t + 1)) fun i => MultiPart.P M i
⊢ False
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/Multipartite.lean | SimpleGraph.mp_cliqueFree | [452, 1] | [476, 53] | have sin := inter_univ S | case intro.mk
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
t : ℕ
M : MultiPart α
ht : M.t = t
hA : M.A = univ
S : Finset α
hs1 : Set.Pairwise (↑S) (mp M).Adj
hs2 : card S = t + 2
h : ∀ (i : ℕ), i ∈ range (M.t + 1) → card (S ∩ MultiPart.P M i) ≤ 1
ub : ∑ i in range (M.t + 1), card (S ∩ MultiPart.P M i) ≤ t + 1
uni : univ = Finset.biUnion (range (M.t + 1)) fun i => MultiPart.P M i
⊢ False | case intro.mk
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
t : ℕ
M : MultiPart α
ht : M.t = t
hA : M.A = univ
S : Finset α
hs1 : Set.Pairwise (↑S) (mp M).Adj
hs2 : card S = t + 2
h : ∀ (i : ℕ), i ∈ range (M.t + 1) → card (S ∩ MultiPart.P M i) ≤ 1
ub : ∑ i in range (M.t + 1), card (S ∩ MultiPart.P M i) ≤ t + 1
uni : univ = Finset.biUnion (range (M.t + 1)) fun i => MultiPart.P M i
sin : S ∩ univ = S
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.mk
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
t : ℕ
M : MultiPart α
ht : M.t = t
hA : M.A = univ
S : Finset α
hs1 : Set.Pairwise (↑S) (mp M).Adj
hs2 : card S = t + 2
h : ∀ (i : ℕ), i ∈ range (M.t + 1) → card (S ∩ MultiPart.P M i) ≤ 1
ub : ∑ i in range (M.t + 1), card (S ∩ MultiPart.P M i) ≤ t + 1
uni : univ = Finset.biUnion (range (M.t + 1)) fun i => MultiPart.P M i
⊢ False
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/Multipartite.lean | SimpleGraph.mp_cliqueFree | [452, 1] | [476, 53] | rw [uni, inter_biUnion] at sin | case intro.mk
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
t : ℕ
M : MultiPart α
ht : M.t = t
hA : M.A = univ
S : Finset α
hs1 : Set.Pairwise (↑S) (mp M).Adj
hs2 : card S = t + 2
h : ∀ (i : ℕ), i ∈ range (M.t + 1) → card (S ∩ MultiPart.P M i) ≤ 1
ub : ∑ i in range (M.t + 1), card (S ∩ MultiPart.P M i) ≤ t + 1
uni : univ = Finset.biUnion (range (M.t + 1)) fun i => MultiPart.P M i
sin : S ∩ univ = S
⊢ False | case intro.mk
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
t : ℕ
M : MultiPart α
ht : M.t = t
hA : M.A = univ
S : Finset α
hs1 : Set.Pairwise (↑S) (mp M).Adj
hs2 : card S = t + 2
h : ∀ (i : ℕ), i ∈ range (M.t + 1) → card (S ∩ MultiPart.P M i) ≤ 1
ub : ∑ i in range (M.t + 1), card (S ∩ MultiPart.P M i) ≤ t + 1
uni : univ = Finset.biUnion (range (M.t + 1)) fun i => MultiPart.P M i
sin : (Finset.biUnion (range (M.t + 1)) fun x => S ∩ MultiPart.P M x) = S
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.mk
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
t : ℕ
M : MultiPart α
ht : M.t = t
hA : M.A = univ
S : Finset α
hs1 : Set.Pairwise (↑S) (mp M).Adj
hs2 : card S = t + 2
h : ∀ (i : ℕ), i ∈ range (M.t + 1) → card (S ∩ MultiPart.P M i) ≤ 1
ub : ∑ i in range (M.t + 1), card (S ∩ MultiPart.P M i) ≤ t + 1
uni : univ = Finset.biUnion (range (M.t + 1)) fun i => MultiPart.P M i
sin : S ∩ univ = S
⊢ False
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/Multipartite.lean | SimpleGraph.mp_cliqueFree | [452, 1] | [476, 53] | rw [← sin, card_biUnion] at hs2 | case intro.mk
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
t : ℕ
M : MultiPart α
ht : M.t = t
hA : M.A = univ
S : Finset α
hs1 : Set.Pairwise (↑S) (mp M).Adj
hs2 : card S = t + 2
h : ∀ (i : ℕ), i ∈ range (M.t + 1) → card (S ∩ MultiPart.P M i) ≤ 1
ub : ∑ i in range (M.t + 1), card (S ∩ MultiPart.P M i) ≤ t + 1
uni : univ = Finset.biUnion (range (M.t + 1)) fun i => MultiPart.P M i
sin : (Finset.biUnion (range (M.t + 1)) fun x => S ∩ MultiPart.P M x) = S
⊢ False | case intro.mk
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
t : ℕ
M : MultiPart α
ht : M.t = t
hA : M.A = univ
S : Finset α
hs1 : Set.Pairwise (↑S) (mp M).Adj
hs2 : ∑ u in range (M.t + 1), card (S ∩ MultiPart.P M u) = t + 2
h : ∀ (i : ℕ), i ∈ range (M.t + 1) → card (S ∩ MultiPart.P M i) ≤ 1
ub : ∑ i in range (M.t + 1), card (S ∩ MultiPart.P M i) ≤ t + 1
uni : univ = Finset.biUnion (range (M.t + 1)) fun i => MultiPart.P M i
sin : (Finset.biUnion (range (M.t + 1)) fun x => S ∩ MultiPart.P M x) = S
⊢ False
case intro.mk
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
t : ℕ
M : MultiPart α
ht : M.t = t
hA : M.A = univ
S : Finset α
hs1 : Set.Pairwise (↑S) (mp M).Adj
hs2 : card (Finset.biUnion (range (M.t + 1)) fun x => S ∩ MultiPart.P M x) = t + 2
h : ∀ (i : ℕ), i ∈ range (M.t + 1) → card (S ∩ MultiPart.P M i) ≤ 1
ub : ∑ i in range (M.t + 1), card (S ∩ MultiPart.P M i) ≤ t + 1
uni : univ = Finset.biUnion (range (M.t + 1)) fun i => MultiPart.P M i
sin : (Finset.biUnion (range (M.t + 1)) fun x => S ∩ MultiPart.P M x) = S
⊢ ∀ (x : ℕ),
x ∈ range (M.t + 1) → ∀ (y : ℕ), y ∈ range (M.t + 1) → x ≠ y → Disjoint (S ∩ MultiPart.P M x) (S ∩ MultiPart.P M y) | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.mk
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
t : ℕ
M : MultiPart α
ht : M.t = t
hA : M.A = univ
S : Finset α
hs1 : Set.Pairwise (↑S) (mp M).Adj
hs2 : card S = t + 2
h : ∀ (i : ℕ), i ∈ range (M.t + 1) → card (S ∩ MultiPart.P M i) ≤ 1
ub : ∑ i in range (M.t + 1), card (S ∩ MultiPart.P M i) ≤ t + 1
uni : univ = Finset.biUnion (range (M.t + 1)) fun i => MultiPart.P M i
sin : (Finset.biUnion (range (M.t + 1)) fun x => S ∩ MultiPart.P M x) = S
⊢ False
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/Multipartite.lean | SimpleGraph.mp_cliqueFree | [452, 1] | [476, 53] | apply not_succ_le_self _ (hs2 ▸ ub) | case intro.mk
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
t : ℕ
M : MultiPart α
ht : M.t = t
hA : M.A = univ
S : Finset α
hs1 : Set.Pairwise (↑S) (mp M).Adj
hs2 : ∑ u in range (M.t + 1), card (S ∩ MultiPart.P M u) = t + 2
h : ∀ (i : ℕ), i ∈ range (M.t + 1) → card (S ∩ MultiPart.P M i) ≤ 1
ub : ∑ i in range (M.t + 1), card (S ∩ MultiPart.P M i) ≤ t + 1
uni : univ = Finset.biUnion (range (M.t + 1)) fun i => MultiPart.P M i
sin : (Finset.biUnion (range (M.t + 1)) fun x => S ∩ MultiPart.P M x) = S
⊢ False
case intro.mk
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
t : ℕ
M : MultiPart α
ht : M.t = t
hA : M.A = univ
S : Finset α
hs1 : Set.Pairwise (↑S) (mp M).Adj
hs2 : card (Finset.biUnion (range (M.t + 1)) fun x => S ∩ MultiPart.P M x) = t + 2
h : ∀ (i : ℕ), i ∈ range (M.t + 1) → card (S ∩ MultiPart.P M i) ≤ 1
ub : ∑ i in range (M.t + 1), card (S ∩ MultiPart.P M i) ≤ t + 1
uni : univ = Finset.biUnion (range (M.t + 1)) fun i => MultiPart.P M i
sin : (Finset.biUnion (range (M.t + 1)) fun x => S ∩ MultiPart.P M x) = S
⊢ ∀ (x : ℕ),
x ∈ range (M.t + 1) → ∀ (y : ℕ), y ∈ range (M.t + 1) → x ≠ y → Disjoint (S ∩ MultiPart.P M x) (S ∩ MultiPart.P M y) | case intro.mk
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
t : ℕ
M : MultiPart α
ht : M.t = t
hA : M.A = univ
S : Finset α
hs1 : Set.Pairwise (↑S) (mp M).Adj
hs2 : card (Finset.biUnion (range (M.t + 1)) fun x => S ∩ MultiPart.P M x) = t + 2
h : ∀ (i : ℕ), i ∈ range (M.t + 1) → card (S ∩ MultiPart.P M i) ≤ 1
ub : ∑ i in range (M.t + 1), card (S ∩ MultiPart.P M i) ≤ t + 1
uni : univ = Finset.biUnion (range (M.t + 1)) fun i => MultiPart.P M i
sin : (Finset.biUnion (range (M.t + 1)) fun x => S ∩ MultiPart.P M x) = S
⊢ ∀ (x : ℕ),
x ∈ range (M.t + 1) → ∀ (y : ℕ), y ∈ range (M.t + 1) → x ≠ y → Disjoint (S ∩ MultiPart.P M x) (S ∩ MultiPart.P M y) | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.mk
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
t : ℕ
M : MultiPart α
ht : M.t = t
hA : M.A = univ
S : Finset α
hs1 : Set.Pairwise (↑S) (mp M).Adj
hs2 : ∑ u in range (M.t + 1), card (S ∩ MultiPart.P M u) = t + 2
h : ∀ (i : ℕ), i ∈ range (M.t + 1) → card (S ∩ MultiPart.P M i) ≤ 1
ub : ∑ i in range (M.t + 1), card (S ∩ MultiPart.P M i) ≤ t + 1
uni : univ = Finset.biUnion (range (M.t + 1)) fun i => MultiPart.P M i
sin : (Finset.biUnion (range (M.t + 1)) fun x => S ∩ MultiPart.P M x) = S
⊢ False
case intro.mk
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
t : ℕ
M : MultiPart α
ht : M.t = t
hA : M.A = univ
S : Finset α
hs1 : Set.Pairwise (↑S) (mp M).Adj
hs2 : card (Finset.biUnion (range (M.t + 1)) fun x => S ∩ MultiPart.P M x) = t + 2
h : ∀ (i : ℕ), i ∈ range (M.t + 1) → card (S ∩ MultiPart.P M i) ≤ 1
ub : ∑ i in range (M.t + 1), card (S ∩ MultiPart.P M i) ≤ t + 1
uni : univ = Finset.biUnion (range (M.t + 1)) fun i => MultiPart.P M i
sin : (Finset.biUnion (range (M.t + 1)) fun x => S ∩ MultiPart.P M x) = S
⊢ ∀ (x : ℕ),
x ∈ range (M.t + 1) → ∀ (y : ℕ), y ∈ range (M.t + 1) → x ≠ y → Disjoint (S ∩ MultiPart.P M x) (S ∩ MultiPart.P M y)
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/Multipartite.lean | SimpleGraph.mp_cliqueFree | [452, 1] | [476, 53] | intro x hx y hy ne | case intro.mk
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
t : ℕ
M : MultiPart α
ht : M.t = t
hA : M.A = univ
S : Finset α
hs1 : Set.Pairwise (↑S) (mp M).Adj
hs2 : card (Finset.biUnion (range (M.t + 1)) fun x => S ∩ MultiPart.P M x) = t + 2
h : ∀ (i : ℕ), i ∈ range (M.t + 1) → card (S ∩ MultiPart.P M i) ≤ 1
ub : ∑ i in range (M.t + 1), card (S ∩ MultiPart.P M i) ≤ t + 1
uni : univ = Finset.biUnion (range (M.t + 1)) fun i => MultiPart.P M i
sin : (Finset.biUnion (range (M.t + 1)) fun x => S ∩ MultiPart.P M x) = S
⊢ ∀ (x : ℕ),
x ∈ range (M.t + 1) → ∀ (y : ℕ), y ∈ range (M.t + 1) → x ≠ y → Disjoint (S ∩ MultiPart.P M x) (S ∩ MultiPart.P M y) | case intro.mk
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
t : ℕ
M : MultiPart α
ht : M.t = t
hA : M.A = univ
S : Finset α
hs1 : Set.Pairwise (↑S) (mp M).Adj
hs2 : card (Finset.biUnion (range (M.t + 1)) fun x => S ∩ MultiPart.P M x) = t + 2
h : ∀ (i : ℕ), i ∈ range (M.t + 1) → card (S ∩ MultiPart.P M i) ≤ 1
ub : ∑ i in range (M.t + 1), card (S ∩ MultiPart.P M i) ≤ t + 1
uni : univ = Finset.biUnion (range (M.t + 1)) fun i => MultiPart.P M i
sin : (Finset.biUnion (range (M.t + 1)) fun x => S ∩ MultiPart.P M x) = S
x : ℕ
hx : x ∈ range (M.t + 1)
y : ℕ
hy : y ∈ range (M.t + 1)
ne : x ≠ y
⊢ Disjoint (S ∩ MultiPart.P M x) (S ∩ MultiPart.P M y) | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.mk
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
t : ℕ
M : MultiPart α
ht : M.t = t
hA : M.A = univ
S : Finset α
hs1 : Set.Pairwise (↑S) (mp M).Adj
hs2 : card (Finset.biUnion (range (M.t + 1)) fun x => S ∩ MultiPart.P M x) = t + 2
h : ∀ (i : ℕ), i ∈ range (M.t + 1) → card (S ∩ MultiPart.P M i) ≤ 1
ub : ∑ i in range (M.t + 1), card (S ∩ MultiPart.P M i) ≤ t + 1
uni : univ = Finset.biUnion (range (M.t + 1)) fun i => MultiPart.P M i
sin : (Finset.biUnion (range (M.t + 1)) fun x => S ∩ MultiPart.P M x) = S
⊢ ∀ (x : ℕ),
x ∈ range (M.t + 1) → ∀ (y : ℕ), y ∈ range (M.t + 1) → x ≠ y → Disjoint (S ∩ MultiPart.P M x) (S ∩ MultiPart.P M y)
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/Multipartite.lean | SimpleGraph.mp_cliqueFree | [452, 1] | [476, 53] | apply disj_of_disj_inter S S (M.disj x hx y hy ne) | case intro.mk
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
t : ℕ
M : MultiPart α
ht : M.t = t
hA : M.A = univ
S : Finset α
hs1 : Set.Pairwise (↑S) (mp M).Adj
hs2 : card (Finset.biUnion (range (M.t + 1)) fun x => S ∩ MultiPart.P M x) = t + 2
h : ∀ (i : ℕ), i ∈ range (M.t + 1) → card (S ∩ MultiPart.P M i) ≤ 1
ub : ∑ i in range (M.t + 1), card (S ∩ MultiPart.P M i) ≤ t + 1
uni : univ = Finset.biUnion (range (M.t + 1)) fun i => MultiPart.P M i
sin : (Finset.biUnion (range (M.t + 1)) fun x => S ∩ MultiPart.P M x) = S
x : ℕ
hx : x ∈ range (M.t + 1)
y : ℕ
hy : y ∈ range (M.t + 1)
ne : x ≠ y
⊢ Disjoint (S ∩ MultiPart.P M x) (S ∩ MultiPart.P M y) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.mk
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
t : ℕ
M : MultiPart α
ht : M.t = t
hA : M.A = univ
S : Finset α
hs1 : Set.Pairwise (↑S) (mp M).Adj
hs2 : card (Finset.biUnion (range (M.t + 1)) fun x => S ∩ MultiPart.P M x) = t + 2
h : ∀ (i : ℕ), i ∈ range (M.t + 1) → card (S ∩ MultiPart.P M i) ≤ 1
ub : ∑ i in range (M.t + 1), card (S ∩ MultiPart.P M i) ≤ t + 1
uni : univ = Finset.biUnion (range (M.t + 1)) fun i => MultiPart.P M i
sin : (Finset.biUnion (range (M.t + 1)) fun x => S ∩ MultiPart.P M x) = S
x : ℕ
hx : x ∈ range (M.t + 1)
y : ℕ
hy : y ∈ range (M.t + 1)
ne : x ≠ y
⊢ Disjoint (S ∩ MultiPart.P M x) (S ∩ MultiPart.P M y)
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/Multipartite.lean | SimpleGraph.mp_cliqueFree | [452, 1] | [476, 53] | obtain ⟨i, hi, hc⟩ := this | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
t : ℕ
M : MultiPart α
ht : M.t = t
hA : M.A = univ
S : Finset α
hs1 : Set.Pairwise (↑S) (mp M).Adj
hs2 : card S = t + 2
this : ∃ i, i ∈ range (M.t + 1) ∧ 1 < card (S ∩ MultiPart.P M i)
⊢ False | case intro.intro
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
t : ℕ
M : MultiPart α
ht : M.t = t
hA : M.A = univ
S : Finset α
hs1 : Set.Pairwise (↑S) (mp M).Adj
hs2 : card S = t + 2
i : ℕ
hi : i ∈ range (M.t + 1)
hc : 1 < card (S ∩ MultiPart.P M i)
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
t : ℕ
M : MultiPart α
ht : M.t = t
hA : M.A = univ
S : Finset α
hs1 : Set.Pairwise (↑S) (mp M).Adj
hs2 : card S = t + 2
this : ∃ i, i ∈ range (M.t + 1) ∧ 1 < card (S ∩ MultiPart.P M i)
⊢ False
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/Multipartite.lean | SimpleGraph.mp_cliqueFree | [452, 1] | [476, 53] | rw [one_lt_card_iff] at hc | case intro.intro
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
t : ℕ
M : MultiPart α
ht : M.t = t
hA : M.A = univ
S : Finset α
hs1 : Set.Pairwise (↑S) (mp M).Adj
hs2 : card S = t + 2
i : ℕ
hi : i ∈ range (M.t + 1)
hc : 1 < card (S ∩ MultiPart.P M i)
⊢ False | case intro.intro
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
t : ℕ
M : MultiPart α
ht : M.t = t
hA : M.A = univ
S : Finset α
hs1 : Set.Pairwise (↑S) (mp M).Adj
hs2 : card S = t + 2
i : ℕ
hi : i ∈ range (M.t + 1)
hc : ∃ a b, a ∈ S ∩ MultiPart.P M i ∧ b ∈ S ∩ MultiPart.P M i ∧ a ≠ b
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
t : ℕ
M : MultiPart α
ht : M.t = t
hA : M.A = univ
S : Finset α
hs1 : Set.Pairwise (↑S) (mp M).Adj
hs2 : card S = t + 2
i : ℕ
hi : i ∈ range (M.t + 1)
hc : 1 < card (S ∩ MultiPart.P M i)
⊢ False
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/Multipartite.lean | SimpleGraph.mp_cliqueFree | [452, 1] | [476, 53] | obtain ⟨a, b, ha, hb, ne⟩ := hc | case intro.intro
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
t : ℕ
M : MultiPart α
ht : M.t = t
hA : M.A = univ
S : Finset α
hs1 : Set.Pairwise (↑S) (mp M).Adj
hs2 : card S = t + 2
i : ℕ
hi : i ∈ range (M.t + 1)
hc : ∃ a b, a ∈ S ∩ MultiPart.P M i ∧ b ∈ S ∩ MultiPart.P M i ∧ a ≠ b
⊢ False | case intro.intro.intro.intro.intro.intro
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
t : ℕ
M : MultiPart α
ht : M.t = t
hA : M.A = univ
S : Finset α
hs1 : Set.Pairwise (↑S) (mp M).Adj
hs2 : card S = t + 2
i : ℕ
hi : i ∈ range (M.t + 1)
a b : α
ha : a ∈ S ∩ MultiPart.P M i
hb : b ∈ S ∩ MultiPart.P M i
ne : a ≠ b
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
t : ℕ
M : MultiPart α
ht : M.t = t
hA : M.A = univ
S : Finset α
hs1 : Set.Pairwise (↑S) (mp M).Adj
hs2 : card S = t + 2
i : ℕ
hi : i ∈ range (M.t + 1)
hc : ∃ a b, a ∈ S ∩ MultiPart.P M i ∧ b ∈ S ∩ MultiPart.P M i ∧ a ≠ b
⊢ False
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/Multipartite.lean | SimpleGraph.mp_cliqueFree | [452, 1] | [476, 53] | rw [mem_inter] at * | case intro.intro.intro.intro.intro.intro
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
t : ℕ
M : MultiPart α
ht : M.t = t
hA : M.A = univ
S : Finset α
hs1 : Set.Pairwise (↑S) (mp M).Adj
hs2 : card S = t + 2
i : ℕ
hi : i ∈ range (M.t + 1)
a b : α
ha : a ∈ S ∩ MultiPart.P M i
hb : b ∈ S ∩ MultiPart.P M i
ne : a ≠ b
⊢ False | case intro.intro.intro.intro.intro.intro
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
t : ℕ
M : MultiPart α
ht : M.t = t
hA : M.A = univ
S : Finset α
hs1 : Set.Pairwise (↑S) (mp M).Adj
hs2 : card S = t + 2
i : ℕ
hi : i ∈ range (M.t + 1)
a b : α
ha : a ∈ S ∧ a ∈ MultiPart.P M i
hb : b ∈ S ∧ b ∈ MultiPart.P M i
ne : a ≠ b
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro.intro
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
t : ℕ
M : MultiPart α
ht : M.t = t
hA : M.A = univ
S : Finset α
hs1 : Set.Pairwise (↑S) (mp M).Adj
hs2 : card S = t + 2
i : ℕ
hi : i ∈ range (M.t + 1)
a b : α
ha : a ∈ S ∩ MultiPart.P M i
hb : b ∈ S ∩ MultiPart.P M i
ne : a ≠ b
⊢ False
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/Multipartite.lean | SimpleGraph.mp_cliqueFree | [452, 1] | [476, 53] | have nadj := not_nbhr_same_part' hi ha.2 hb.2 | case intro.intro.intro.intro.intro.intro
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
t : ℕ
M : MultiPart α
ht : M.t = t
hA : M.A = univ
S : Finset α
hs1 : Set.Pairwise (↑S) (mp M).Adj
hs2 : card S = t + 2
i : ℕ
hi : i ∈ range (M.t + 1)
a b : α
ha : a ∈ S ∧ a ∈ MultiPart.P M i
hb : b ∈ S ∧ b ∈ MultiPart.P M i
ne : a ≠ b
⊢ False | case intro.intro.intro.intro.intro.intro
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
t : ℕ
M : MultiPart α
ht : M.t = t
hA : M.A = univ
S : Finset α
hs1 : Set.Pairwise (↑S) (mp M).Adj
hs2 : card S = t + 2
i : ℕ
hi : i ∈ range (M.t + 1)
a b : α
ha : a ∈ S ∧ a ∈ MultiPart.P M i
hb : b ∈ S ∧ b ∈ MultiPart.P M i
ne : a ≠ b
nadj : ¬Adj (mp M) a b
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro.intro
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
t : ℕ
M : MultiPart α
ht : M.t = t
hA : M.A = univ
S : Finset α
hs1 : Set.Pairwise (↑S) (mp M).Adj
hs2 : card S = t + 2
i : ℕ
hi : i ∈ range (M.t + 1)
a b : α
ha : a ∈ S ∧ a ∈ MultiPart.P M i
hb : b ∈ S ∧ b ∈ MultiPart.P M i
ne : a ≠ b
⊢ False
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/Multipartite.lean | SimpleGraph.mp_cliqueFree | [452, 1] | [476, 53] | exact nadj (hs1 ha.1 hb.1 ne) | case intro.intro.intro.intro.intro.intro
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
t : ℕ
M : MultiPart α
ht : M.t = t
hA : M.A = univ
S : Finset α
hs1 : Set.Pairwise (↑S) (mp M).Adj
hs2 : card S = t + 2
i : ℕ
hi : i ∈ range (M.t + 1)
a b : α
ha : a ∈ S ∧ a ∈ MultiPart.P M i
hb : b ∈ S ∧ b ∈ MultiPart.P M i
ne : a ≠ b
nadj : ¬Adj (mp M) a b
⊢ False | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro.intro
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
t : ℕ
M : MultiPart α
ht : M.t = t
hA : M.A = univ
S : Finset α
hs1 : Set.Pairwise (↑S) (mp M).Adj
hs2 : card S = t + 2
i : ℕ
hi : i ∈ range (M.t + 1)
a b : α
ha : a ∈ S ∧ a ∈ MultiPart.P M i
hb : b ∈ S ∧ b ∈ MultiPart.P M i
ne : a ≠ b
nadj : ¬Adj (mp M) a b
⊢ False
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/Multipartite.lean | SimpleGraph.mp_cliqueFree | [452, 1] | [476, 53] | nth_rw 2 [← card_range (M.t + 1)] | case ub
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
t : ℕ
M : MultiPart α
ht : M.t = t
hA : M.A = univ
S : Finset α
hs1 : Set.Pairwise (↑S) (mp M).Adj
hs2 : card S = t + 2
h : ∀ (i : ℕ), i ∈ range (M.t + 1) → card (S ∩ MultiPart.P M i) ≤ 1
⊢ ∑ i in range (M.t + 1), card (S ∩ MultiPart.P M i) ≤ M.t + 1 | case ub
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
t : ℕ
M : MultiPart α
ht : M.t = t
hA : M.A = univ
S : Finset α
hs1 : Set.Pairwise (↑S) (mp M).Adj
hs2 : card S = t + 2
h : ∀ (i : ℕ), i ∈ range (M.t + 1) → card (S ∩ MultiPart.P M i) ≤ 1
⊢ ∑ i in range (M.t + 1), card (S ∩ MultiPart.P M i) ≤ card (range (M.t + 1)) | Please generate a tactic in lean4 to solve the state.
STATE:
case ub
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
t : ℕ
M : MultiPart α
ht : M.t = t
hA : M.A = univ
S : Finset α
hs1 : Set.Pairwise (↑S) (mp M).Adj
hs2 : card S = t + 2
h : ∀ (i : ℕ), i ∈ range (M.t + 1) → card (S ∩ MultiPart.P M i) ≤ 1
⊢ ∑ i in range (M.t + 1), card (S ∩ MultiPart.P M i) ≤ M.t + 1
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/Multipartite.lean | SimpleGraph.mp_cliqueFree | [452, 1] | [476, 53] | nth_rw 1 [card_eq_sum_ones] | case ub
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
t : ℕ
M : MultiPart α
ht : M.t = t
hA : M.A = univ
S : Finset α
hs1 : Set.Pairwise (↑S) (mp M).Adj
hs2 : card S = t + 2
h : ∀ (i : ℕ), i ∈ range (M.t + 1) → card (S ∩ MultiPart.P M i) ≤ 1
⊢ ∑ i in range (M.t + 1), card (S ∩ MultiPart.P M i) ≤ card (range (M.t + 1)) | case ub
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
t : ℕ
M : MultiPart α
ht : M.t = t
hA : M.A = univ
S : Finset α
hs1 : Set.Pairwise (↑S) (mp M).Adj
hs2 : card S = t + 2
h : ∀ (i : ℕ), i ∈ range (M.t + 1) → card (S ∩ MultiPart.P M i) ≤ 1
⊢ ∑ i in range (M.t + 1), card (S ∩ MultiPart.P M i) ≤ ∑ x in range (M.t + 1), 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case ub
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
t : ℕ
M : MultiPart α
ht : M.t = t
hA : M.A = univ
S : Finset α
hs1 : Set.Pairwise (↑S) (mp M).Adj
hs2 : card S = t + 2
h : ∀ (i : ℕ), i ∈ range (M.t + 1) → card (S ∩ MultiPart.P M i) ≤ 1
⊢ ∑ i in range (M.t + 1), card (S ∩ MultiPart.P M i) ≤ card (range (M.t + 1))
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/Multipartite.lean | SimpleGraph.mp_cliqueFree | [452, 1] | [476, 53] | apply sum_le_sum h | case ub
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
t : ℕ
M : MultiPart α
ht : M.t = t
hA : M.A = univ
S : Finset α
hs1 : Set.Pairwise (↑S) (mp M).Adj
hs2 : card S = t + 2
h : ∀ (i : ℕ), i ∈ range (M.t + 1) → card (S ∩ MultiPart.P M i) ≤ 1
⊢ ∑ i in range (M.t + 1), card (S ∩ MultiPart.P M i) ≤ ∑ x in range (M.t + 1), 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case ub
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
t : ℕ
M : MultiPart α
ht : M.t = t
hA : M.A = univ
S : Finset α
hs1 : Set.Pairwise (↑S) (mp M).Adj
hs2 : card S = t + 2
h : ∀ (i : ℕ), i ∈ range (M.t + 1) → card (S ∩ MultiPart.P M i) ≤ 1
⊢ ∑ i in range (M.t + 1), card (S ∩ MultiPart.P M i) ≤ ∑ x in range (M.t + 1), 1
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/NbhdRes.lean | SimpleGraph.nbhdRes_filter | [29, 1] | [31, 62] | ext x | α : Type u_1
G : SimpleGraph α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
v : α
A : Finset α
⊢ nbhdRes G v A = filter (fun x => Adj G v x) A | case a
α : Type u_1
G : SimpleGraph α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
v : α
A : Finset α
x : α
⊢ x ∈ nbhdRes G v A ↔ x ∈ filter (fun x => Adj G v 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 : α
A : Finset α
⊢ nbhdRes G v A = filter (fun x => Adj G v x) A
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/NbhdRes.lean | SimpleGraph.nbhdRes_filter | [29, 1] | [31, 62] | rw [nbhdRes,mem_inter,mem_filter,mem_neighborFinset] | case a
α : Type u_1
G : SimpleGraph α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
v : α
A : Finset α
x : α
⊢ x ∈ nbhdRes G v A ↔ x ∈ filter (fun x => Adj G v x) A | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case a
α : Type u_1
G : SimpleGraph α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
v : α
A : Finset α
x : α
⊢ x ∈ nbhdRes G v A ↔ x ∈ filter (fun x => Adj G v x) A
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/NbhdRes.lean | SimpleGraph.degRes_univ | [39, 1] | [42, 27] | rw [degRes, degree] | α : Type u_1
G : SimpleGraph α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
v : α
⊢ degRes G v univ = degree G v | α : Type u_1
G : SimpleGraph α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
v : α
⊢ card (nbhdRes G v univ) = card (neighborFinset G v) | 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
⊢ degRes G v univ = degree G v
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/NbhdRes.lean | SimpleGraph.degRes_univ | [39, 1] | [42, 27] | congr | α : Type u_1
G : SimpleGraph α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
v : α
⊢ card (nbhdRes G v univ) = card (neighborFinset G v) | case e_s
α : Type u_1
G : SimpleGraph α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
v : α
⊢ nbhdRes G v univ = neighborFinset G v | 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
⊢ card (nbhdRes G v univ) = card (neighborFinset G v)
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/NbhdRes.lean | SimpleGraph.degRes_univ | [39, 1] | [42, 27] | rw [nbhdRes, univ_inter] | case e_s
α : Type u_1
G : SimpleGraph α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
v : α
⊢ nbhdRes G v univ = neighborFinset G v | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case e_s
α : Type u_1
G : SimpleGraph α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
v : α
⊢ nbhdRes G v univ = neighborFinset G v
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/NbhdRes.lean | SimpleGraph.exists_max_res_deg_vertex | [51, 1] | [58, 6] | obtain ⟨t, ht⟩ := max_of_nonempty (Nonempty.image hA (fun v => G.degRes v A)) | α : Type u_1
G : SimpleGraph α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
A : Finset α
hA : Finset.Nonempty A
⊢ ∃ v, v ∈ A ∧ maxDegRes G A = degRes G v A | case intro
α : Type u_1
G : SimpleGraph α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
A : Finset α
hA : Finset.Nonempty A
t : ℕ
ht : Finset.max (image (fun v => degRes G v A) A) = ↑t
⊢ ∃ v, v ∈ A ∧ maxDegRes G A = degRes 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 : Finset.Nonempty A
⊢ ∃ v, v ∈ A ∧ maxDegRes G A = degRes G v A
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/NbhdRes.lean | SimpleGraph.exists_max_res_deg_vertex | [51, 1] | [58, 6] | obtain ⟨a, ha1, ha2⟩:=mem_image.1 (mem_of_max ht) | case intro
α : Type u_1
G : SimpleGraph α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
A : Finset α
hA : Finset.Nonempty A
t : ℕ
ht : Finset.max (image (fun v => degRes G v A) A) = ↑t
⊢ ∃ v, v ∈ A ∧ maxDegRes G A = degRes G v A | case intro.intro.intro
α : Type u_1
G : SimpleGraph α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
A : Finset α
hA : Finset.Nonempty A
t : ℕ
ht : Finset.max (image (fun v => degRes G v A) A) = ↑t
a : α
ha1 : a ∈ A
ha2 : degRes G a A = t
⊢ ∃ v, v ∈ A ∧ maxDegRes G A = degRes G v A | 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
hA : Finset.Nonempty A
t : ℕ
ht : Finset.max (image (fun v => degRes G v A) A) = ↑t
⊢ ∃ v, v ∈ A ∧ maxDegRes G A = degRes G v A
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/NbhdRes.lean | SimpleGraph.exists_max_res_deg_vertex | [51, 1] | [58, 6] | refine ⟨a, ha1,?_⟩ | case intro.intro.intro
α : Type u_1
G : SimpleGraph α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
A : Finset α
hA : Finset.Nonempty A
t : ℕ
ht : Finset.max (image (fun v => degRes G v A) A) = ↑t
a : α
ha1 : a ∈ A
ha2 : degRes G a A = t
⊢ ∃ v, v ∈ A ∧ maxDegRes G A = degRes G v A | case intro.intro.intro
α : Type u_1
G : SimpleGraph α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
A : Finset α
hA : Finset.Nonempty A
t : ℕ
ht : Finset.max (image (fun v => degRes G v A) A) = ↑t
a : α
ha1 : a ∈ A
ha2 : degRes G a A = t
⊢ maxDegRes G A = degRes G a A | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro
α : Type u_1
G : SimpleGraph α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
A : Finset α
hA : Finset.Nonempty A
t : ℕ
ht : Finset.max (image (fun v => degRes G v A) A) = ↑t
a : α
ha1 : a ∈ A
ha2 : degRes G a A = t
⊢ ∃ v, v ∈ A ∧ maxDegRes G A = degRes G v A
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/NbhdRes.lean | SimpleGraph.exists_max_res_deg_vertex | [51, 1] | [58, 6] | rw [maxDegRes,ht,ha2] | case intro.intro.intro
α : Type u_1
G : SimpleGraph α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
A : Finset α
hA : Finset.Nonempty A
t : ℕ
ht : Finset.max (image (fun v => degRes G v A) A) = ↑t
a : α
ha1 : a ∈ A
ha2 : degRes G a A = t
⊢ maxDegRes G A = degRes G a A | case intro.intro.intro
α : Type u_1
G : SimpleGraph α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
A : Finset α
hA : Finset.Nonempty A
t : ℕ
ht : Finset.max (image (fun v => degRes G v A) A) = ↑t
a : α
ha1 : a ∈ A
ha2 : degRes G a A = t
⊢ Option.getD (↑t) 0 = t | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro
α : Type u_1
G : SimpleGraph α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
A : Finset α
hA : Finset.Nonempty A
t : ℕ
ht : Finset.max (image (fun v => degRes G v A) A) = ↑t
a : α
ha1 : a ∈ A
ha2 : degRes G a A = t
⊢ maxDegRes G A = degRes G a A
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/NbhdRes.lean | SimpleGraph.exists_max_res_deg_vertex | [51, 1] | [58, 6] | rfl | case intro.intro.intro
α : Type u_1
G : SimpleGraph α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
A : Finset α
hA : Finset.Nonempty A
t : ℕ
ht : Finset.max (image (fun v => degRes G v A) A) = ↑t
a : α
ha1 : a ∈ A
ha2 : degRes G a A = t
⊢ Option.getD (↑t) 0 = t | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro
α : Type u_1
G : SimpleGraph α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
A : Finset α
hA : Finset.Nonempty A
t : ℕ
ht : Finset.max (image (fun v => degRes G v A) A) = ↑t
a : α
ha1 : a ∈ A
ha2 : degRes G a A = t
⊢ Option.getD (↑t) 0 = t
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/NbhdRes.lean | SimpleGraph.degRes_le_maxDegRes | [61, 1] | [66, 22] | obtain ⟨t, ht⟩ := Finset.max_of_mem (mem_image_of_mem (fun v => G.degRes v A) hvA) | α : Type u_1
G : SimpleGraph α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
v : α
A : Finset α
hvA : v ∈ A
⊢ degRes G v A ≤ maxDegRes G A | case intro
α : Type u_1
G : SimpleGraph α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
v : α
A : Finset α
hvA : v ∈ A
t : ℕ
ht : Finset.max (image (fun v => degRes G v A) A) = ↑t
⊢ degRes G v A ≤ maxDegRes G 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 : α
A : Finset α
hvA : v ∈ A
⊢ degRes G v A ≤ maxDegRes G A
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/NbhdRes.lean | SimpleGraph.degRes_le_maxDegRes | [61, 1] | [66, 22] | have := mem_image_of_mem (fun v => G.degRes v A) hvA | case intro
α : Type u_1
G : SimpleGraph α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
v : α
A : Finset α
hvA : v ∈ A
t : ℕ
ht : Finset.max (image (fun v => degRes G v A) A) = ↑t
⊢ degRes G v A ≤ maxDegRes G A | case intro
α : Type u_1
G : SimpleGraph α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
v : α
A : Finset α
hvA : v ∈ A
t : ℕ
ht : Finset.max (image (fun v => degRes G v A) A) = ↑t
this : (fun v => degRes G v A) v ∈ image (fun v => degRes G v A) A
⊢ degRes G v A ≤ maxDegRes G A | 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
v : α
A : Finset α
hvA : v ∈ A
t : ℕ
ht : Finset.max (image (fun v => degRes G v A) A) = ↑t
⊢ degRes G v A ≤ maxDegRes G A
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/NbhdRes.lean | SimpleGraph.degRes_le_maxDegRes | [61, 1] | [66, 22] | have := Finset.le_max_of_eq this ht | case intro
α : Type u_1
G : SimpleGraph α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
v : α
A : Finset α
hvA : v ∈ A
t : ℕ
ht : Finset.max (image (fun v => degRes G v A) A) = ↑t
this : (fun v => degRes G v A) v ∈ image (fun v => degRes G v A) A
⊢ degRes G v A ≤ maxDegRes G A | case intro
α : Type u_1
G : SimpleGraph α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
v : α
A : Finset α
hvA : v ∈ A
t : ℕ
ht : Finset.max (image (fun v => degRes G v A) A) = ↑t
this✝ : (fun v => degRes G v A) v ∈ image (fun v => degRes G v A) A
this : (fun v => degRes G v A) v ≤ t
⊢ degRes G v A ≤ maxDegRes G A | 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
v : α
A : Finset α
hvA : v ∈ A
t : ℕ
ht : Finset.max (image (fun v => degRes G v A) A) = ↑t
this : (fun v => degRes G v A) v ∈ image (fun v => degRes G v A) A
⊢ degRes G v A ≤ maxDegRes G A
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/NbhdRes.lean | SimpleGraph.degRes_le_maxDegRes | [61, 1] | [66, 22] | rwa [maxDegRes, ht] | case intro
α : Type u_1
G : SimpleGraph α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
v : α
A : Finset α
hvA : v ∈ A
t : ℕ
ht : Finset.max (image (fun v => degRes G v A) A) = ↑t
this✝ : (fun v => degRes G v A) v ∈ image (fun v => degRes G v A) A
this : (fun v => degRes G v A) v ≤ t
⊢ degRes G v A ≤ maxDegRes G A | 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
v : α
A : Finset α
hvA : v ∈ A
t : ℕ
ht : Finset.max (image (fun v => degRes G v A) A) = ↑t
this✝ : (fun v => degRes G v A) v ∈ image (fun v => degRes G v A) A
this : (fun v => degRes G v A) v ≤ t
⊢ degRes G v A ≤ maxDegRes G A
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/NbhdRes.lean | SimpleGraph.maxDegRes_sum_le | [70, 1] | [74, 70] | rw [card_eq_sum_ones, mul_sum, mul_one] | α : Type u_1
G : SimpleGraph α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
C A : Finset α
hC : C ⊆ A
⊢ ∑ v in C, degRes G v A ≤ maxDegRes G A * card C | α : Type u_1
G : SimpleGraph α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
C A : Finset α
hC : C ⊆ A
⊢ ∑ v in C, degRes G v A ≤ ∑ x in C, maxDegRes G 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
C A : Finset α
hC : C ⊆ A
⊢ ∑ v in C, degRes G v A ≤ maxDegRes G A * card C
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/NbhdRes.lean | SimpleGraph.maxDegRes_sum_le | [70, 1] | [74, 70] | apply sum_le_sum _ | α : Type u_1
G : SimpleGraph α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
C A : Finset α
hC : C ⊆ A
⊢ ∑ v in C, degRes G v A ≤ ∑ x in C, maxDegRes G A | α : Type u_1
G : SimpleGraph α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
C A : Finset α
hC : C ⊆ A
⊢ ∀ (i : α), i ∈ C → degRes G i A ≤ maxDegRes G 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
C A : Finset α
hC : C ⊆ A
⊢ ∑ v in C, degRes G v A ≤ ∑ x in C, maxDegRes G A
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/NbhdRes.lean | SimpleGraph.maxDegRes_sum_le | [70, 1] | [74, 70] | intro i hi | α : Type u_1
G : SimpleGraph α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
C A : Finset α
hC : C ⊆ A
⊢ ∀ (i : α), i ∈ C → degRes G i A ≤ maxDegRes G A | α : Type u_1
G : SimpleGraph α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
C A : Finset α
hC : C ⊆ A
i : α
hi : i ∈ C
⊢ degRes G i A ≤ maxDegRes G 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
C A : Finset α
hC : C ⊆ A
⊢ ∀ (i : α), i ∈ C → degRes G i A ≤ maxDegRes G A
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/NbhdRes.lean | SimpleGraph.maxDegRes_sum_le | [70, 1] | [74, 70] | exact G.degRes_le_maxDegRes (hC hi) | α : Type u_1
G : SimpleGraph α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
C A : Finset α
hC : C ⊆ A
i : α
hi : i ∈ C
⊢ degRes G i A ≤ maxDegRes G 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
C A : Finset α
hC : C ⊆ A
i : α
hi : i ∈ C
⊢ degRes G i A ≤ maxDegRes G A
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/NbhdRes.lean | SimpleGraph.exists_mem_nempty | [81, 1] | [86, 23] | rw [nbhdRes] at hA | α : Type u_1
G : SimpleGraph α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
v : α
A : Finset α
hA : ¬nbhdRes G v A = ∅
⊢ ∃ w, w ∈ A ∧ Adj G v w | α : Type u_1
G : SimpleGraph α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
v : α
A : Finset α
hA : ¬A ∩ neighborFinset G v = ∅
⊢ ∃ w, w ∈ A ∧ Adj G v w | 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 : Finset α
hA : ¬nbhdRes G v A = ∅
⊢ ∃ w, w ∈ A ∧ Adj G v w
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/NbhdRes.lean | SimpleGraph.exists_mem_nempty | [81, 1] | [86, 23] | contrapose! hA | α : Type u_1
G : SimpleGraph α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
v : α
A : Finset α
hA : ¬A ∩ neighborFinset G v = ∅
⊢ ∃ w, w ∈ A ∧ Adj G v w | α : Type u_1
G : SimpleGraph α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
v : α
A : Finset α
hA : ∀ (w : α), w ∈ A → ¬Adj G v w
⊢ A ∩ neighborFinset G v = ∅ | 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 : Finset α
hA : ¬A ∩ neighborFinset G v = ∅
⊢ ∃ w, w ∈ A ∧ Adj G v w
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/NbhdRes.lean | SimpleGraph.exists_mem_nempty | [81, 1] | [86, 23] | rw [eq_empty_iff_forall_not_mem] | α : Type u_1
G : SimpleGraph α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
v : α
A : Finset α
hA : ∀ (w : α), w ∈ A → ¬Adj G v w
⊢ A ∩ neighborFinset G v = ∅ | α : Type u_1
G : SimpleGraph α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
v : α
A : Finset α
hA : ∀ (w : α), w ∈ A → ¬Adj G v w
⊢ ∀ (x : α), ¬x ∈ A ∩ neighborFinset G v | 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 : Finset α
hA : ∀ (w : α), w ∈ A → ¬Adj G v w
⊢ A ∩ neighborFinset G v = ∅
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/NbhdRes.lean | SimpleGraph.exists_mem_nempty | [81, 1] | [86, 23] | intro x hx | α : Type u_1
G : SimpleGraph α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
v : α
A : Finset α
hA : ∀ (w : α), w ∈ A → ¬Adj G v w
⊢ ∀ (x : α), ¬x ∈ A ∩ neighborFinset G v | α : Type u_1
G : SimpleGraph α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
v : α
A : Finset α
hA : ∀ (w : α), w ∈ A → ¬Adj G v w
x : α
hx : x ∈ A ∩ neighborFinset G v
⊢ False | 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 : Finset α
hA : ∀ (w : α), w ∈ A → ¬Adj G v w
⊢ ∀ (x : α), ¬x ∈ A ∩ neighborFinset G v
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/NbhdRes.lean | SimpleGraph.exists_mem_nempty | [81, 1] | [86, 23] | rw [mem_inter, mem_neighborFinset] at hx | α : Type u_1
G : SimpleGraph α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
v : α
A : Finset α
hA : ∀ (w : α), w ∈ A → ¬Adj G v w
x : α
hx : x ∈ A ∩ neighborFinset G v
⊢ False | α : Type u_1
G : SimpleGraph α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
v : α
A : Finset α
hA : ∀ (w : α), w ∈ A → ¬Adj G v w
x : α
hx : x ∈ A ∧ Adj G v x
⊢ False | 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 : Finset α
hA : ∀ (w : α), w ∈ A → ¬Adj G v w
x : α
hx : x ∈ A ∩ neighborFinset G v
⊢ False
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/NbhdRes.lean | SimpleGraph.exists_mem_nempty | [81, 1] | [86, 23] | exact hA x hx.1 hx.2 | α : Type u_1
G : SimpleGraph α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
v : α
A : Finset α
hA : ∀ (w : α), w ∈ A → ¬Adj G v w
x : α
hx : x ∈ A ∧ Adj G v x
⊢ False | 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 : Finset α
hA : ∀ (w : α), w ∈ A → ¬Adj G v w
x : α
hx : x ∈ A ∧ Adj G v x
⊢ False
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/NbhdRes.lean | SimpleGraph.mem_res_nbhd_iff | [89, 1] | [92, 26] | rw [nbhdRes, mem_inter] | 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
⊢ w ∈ nbhdRes G v A ↔ w ∈ A ∧ w ∈ neighborFinset G v | 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
⊢ w ∈ nbhdRes G v A ↔ w ∈ A ∧ w ∈ neighborFinset G v
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/NbhdRes.lean | SimpleGraph.subset_res_nbhd | [94, 1] | [96, 77] | exact (mem_neighborFinset _ _ _).1 ((G.mem_res_nbhd_iff _ _ _).1 (h hb)).2 | α : Type u_1
G : SimpleGraph α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
B : Finset α
v : α
A : Finset α
b : α
h : B ⊆ nbhdRes G v A
hb : b ∈ B
⊢ Adj G v b | 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 : Finset α
v : α
A : Finset α
b : α
h : B ⊆ nbhdRes G v A
hb : b ∈ B
⊢ Adj G v b
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/NbhdRes.lean | SimpleGraph.not_mem_nbhd | [99, 1] | [102, 21] | rw [mem_neighborFinset] | α : Type u_1
G : SimpleGraph α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
v : α
⊢ ¬v ∈ neighborFinset G v | α : Type u_1
G : SimpleGraph α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
v : α
⊢ ¬Adj G v v | 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
⊢ ¬v ∈ neighborFinset G v
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/NbhdRes.lean | SimpleGraph.not_mem_nbhd | [99, 1] | [102, 21] | exact G.loopless v | α : Type u_1
G : SimpleGraph α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
v : α
⊢ ¬Adj G v v | 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
⊢ ¬Adj G v v
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/NbhdRes.lean | SimpleGraph.not_mem_res_nbhd | [105, 1] | [108, 44] | rw [mem_res_nbhd_iff] | α : Type u_1
G : SimpleGraph α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
v : α
A : Finset α
⊢ ¬v ∈ nbhdRes G v A | α : Type u_1
G : SimpleGraph α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
v : α
A : Finset α
⊢ ¬(v ∈ A ∧ v ∈ neighborFinset G v) | 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 : Finset α
⊢ ¬v ∈ nbhdRes G v A
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/NbhdRes.lean | SimpleGraph.not_mem_res_nbhd | [105, 1] | [108, 44] | push_neg | α : Type u_1
G : SimpleGraph α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
v : α
A : Finset α
⊢ ¬(v ∈ A ∧ v ∈ neighborFinset G v) | α : Type u_1
G : SimpleGraph α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
v : α
A : Finset α
⊢ v ∈ A → ¬v ∈ neighborFinset G v | 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 : Finset α
⊢ ¬(v ∈ A ∧ v ∈ neighborFinset G v)
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/NbhdRes.lean | SimpleGraph.not_mem_res_nbhd | [105, 1] | [108, 44] | intro _ | α : Type u_1
G : SimpleGraph α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
v : α
A : Finset α
⊢ v ∈ A → ¬v ∈ neighborFinset G v | α : Type u_1
G : SimpleGraph α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
v : α
A : Finset α
_✝ : v ∈ A
⊢ ¬v ∈ neighborFinset G v | 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 : Finset α
⊢ v ∈ A → ¬v ∈ neighborFinset G v
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/NbhdRes.lean | SimpleGraph.not_mem_res_nbhd | [105, 1] | [108, 44] | exact G.not_mem_nbhd v | α : Type u_1
G : SimpleGraph α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
v : α
A : Finset α
_✝ : v ∈ A
⊢ ¬v ∈ neighborFinset G v | 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 : Finset α
_✝ : v ∈ A
⊢ ¬v ∈ neighborFinset G v
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/NbhdRes.lean | SimpleGraph.sub_res_nbhd_A | [111, 1] | [114, 21] | intro x | α : Type u_1
G : SimpleGraph α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
v : α
A : Finset α
⊢ nbhdRes G v A ⊆ A | α : Type u_1
G : SimpleGraph α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
v : α
A : Finset α
x : α
⊢ x ∈ nbhdRes G v A → 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 : α
A : Finset α
⊢ nbhdRes G v A ⊆ A
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/NbhdRes.lean | SimpleGraph.sub_res_nbhd_A | [111, 1] | [114, 21] | rw [mem_res_nbhd_iff] | α : Type u_1
G : SimpleGraph α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
v : α
A : Finset α
x : α
⊢ x ∈ nbhdRes G v A → x ∈ A | α : Type u_1
G : SimpleGraph α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
v : α
A : Finset α
x : α
⊢ x ∈ A ∧ x ∈ neighborFinset G v → 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 : α
A : Finset α
x : α
⊢ x ∈ nbhdRes G v A → x ∈ A
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/NbhdRes.lean | SimpleGraph.sub_res_nbhd_A | [111, 1] | [114, 21] | intro h | α : Type u_1
G : SimpleGraph α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
v : α
A : Finset α
x : α
⊢ x ∈ A ∧ x ∈ neighborFinset G v → x ∈ A | α : Type u_1
G : SimpleGraph α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
v : α
A : Finset α
x : α
h : x ∈ A ∧ x ∈ neighborFinset G v
⊢ 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 : α
A : Finset α
x : α
⊢ x ∈ A ∧ x ∈ neighborFinset G v → x ∈ A
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/NbhdRes.lean | SimpleGraph.sub_res_nbhd_A | [111, 1] | [114, 21] | exact h.1 | α : Type u_1
G : SimpleGraph α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
v : α
A : Finset α
x : α
h : x ∈ A ∧ x ∈ neighborFinset G v
⊢ 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 : α
A : Finset α
x : α
h : x ∈ A ∧ x ∈ neighborFinset G v
⊢ x ∈ A
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/NbhdRes.lean | SimpleGraph.sum_sdf | [128, 1] | [131, 72] | nth_rw 1 [← union_sdiff_of_subset hB] | α : Type u_1
G : SimpleGraph α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
B A : Finset α
hB : B ⊆ A
C : Finset α
⊢ ∑ v in A, degRes G v C = ∑ v in B, degRes G v C + ∑ v in A \ B, degRes G v C | α : Type u_1
G : SimpleGraph α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
B A : Finset α
hB : B ⊆ A
C : Finset α
⊢ ∑ v in B ∪ A \ B, degRes G v C = ∑ v in B, degRes G v C + ∑ v in A \ B, degRes G v C | 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 : Finset α
hB : B ⊆ A
C : Finset α
⊢ ∑ v in A, degRes G v C = ∑ v in B, degRes G v C + ∑ v in A \ B, degRes G v C
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/NbhdRes.lean | SimpleGraph.sum_sdf | [128, 1] | [131, 72] | exact sum_union disjoint_sdiff | α : Type u_1
G : SimpleGraph α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
B A : Finset α
hB : B ⊆ A
C : Finset α
⊢ ∑ v in B ∪ A \ B, degRes G v C = ∑ v in B, degRes G v C + ∑ v in A \ B, degRes G v C | 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 : Finset α
hB : B ⊆ A
C : Finset α
⊢ ∑ v in B ∪ A \ B, degRes G v C = ∑ v in B, degRes G v C + ∑ v in A \ B, degRes G v C
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/NbhdRes.lean | SimpleGraph.degRes_add_sub | [134, 1] | [139, 53] | simp_rw [degRes, nbhdRes] | α : Type u_1
G : SimpleGraph α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
B A : Finset α
v : α
hB : B ⊆ A
⊢ degRes G v A = degRes G v B + degRes G v (A \ B) | α : Type u_1
G : SimpleGraph α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
B A : Finset α
v : α
hB : B ⊆ A
⊢ card (A ∩ neighborFinset G v) = card (B ∩ neighborFinset G v) + card (A \ B ∩ neighborFinset G v) | 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 : Finset α
v : α
hB : B ⊆ A
⊢ degRes G v A = degRes G v B + degRes G v (A \ B)
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/NbhdRes.lean | SimpleGraph.degRes_add_sub | [134, 1] | [139, 53] | nth_rw 1 [← union_sdiff_of_subset hB] | α : Type u_1
G : SimpleGraph α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
B A : Finset α
v : α
hB : B ⊆ A
⊢ card (A ∩ neighborFinset G v) = card (B ∩ neighborFinset G v) + card (A \ B ∩ neighborFinset G v) | α : Type u_1
G : SimpleGraph α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
B A : Finset α
v : α
hB : B ⊆ A
⊢ card ((B ∪ A \ B) ∩ neighborFinset G v) = card (B ∩ neighborFinset G v) + card (A \ B ∩ neighborFinset G v) | 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 : Finset α
v : α
hB : B ⊆ A
⊢ card (A ∩ neighborFinset G v) = card (B ∩ neighborFinset G v) + card (A \ B ∩ neighborFinset G v)
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/NbhdRes.lean | SimpleGraph.degRes_add_sub | [134, 1] | [139, 53] | rw [inter_distrib_right B (A \ B) _] | α : Type u_1
G : SimpleGraph α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
B A : Finset α
v : α
hB : B ⊆ A
⊢ card ((B ∪ A \ B) ∩ neighborFinset G v) = card (B ∩ neighborFinset G v) + card (A \ B ∩ neighborFinset G v) | α : Type u_1
G : SimpleGraph α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
B A : Finset α
v : α
hB : B ⊆ A
⊢ card (B ∩ neighborFinset G v ∪ A \ B ∩ neighborFinset G v) =
card (B ∩ neighborFinset G v) + card (A \ B ∩ neighborFinset G v) | 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 : Finset α
v : α
hB : B ⊆ A
⊢ card ((B ∪ A \ B) ∩ neighborFinset G v) = card (B ∩ neighborFinset G v) + card (A \ B ∩ neighborFinset G v)
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/NbhdRes.lean | SimpleGraph.degRes_add_sub | [134, 1] | [139, 53] | exact card_disjoint_union (sdiff_inter_disj A B _) | α : Type u_1
G : SimpleGraph α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
B A : Finset α
v : α
hB : B ⊆ A
⊢ card (B ∩ neighborFinset G v ∪ A \ B ∩ neighborFinset G v) =
card (B ∩ neighborFinset G v) + card (A \ B ∩ neighborFinset G v) | 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 : Finset α
v : α
hB : B ⊆ A
⊢ card (B ∩ neighborFinset G v ∪ A \ B ∩ neighborFinset G v) =
card (B ∩ neighborFinset G v) + card (A \ B ∩ neighborFinset G v)
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/NbhdRes.lean | SimpleGraph.degRes_add_sum | [142, 1] | [144, 77] | rw [← sum_add_distrib] | α : Type u_1
G : SimpleGraph α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
A B C : Finset α
hB : B ⊆ A
⊢ ∑ v in C, degRes G v A = ∑ v in C, degRes G v B + ∑ v in C, degRes G v (A \ B) | α : Type u_1
G : SimpleGraph α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
A B C : Finset α
hB : B ⊆ A
⊢ ∑ v in C, degRes G v A = ∑ x in C, (degRes G x B + degRes G x (A \ 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 B C : Finset α
hB : B ⊆ A
⊢ ∑ v in C, degRes G v A = ∑ v in C, degRes G v B + ∑ v in C, degRes G v (A \ B)
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/NbhdRes.lean | SimpleGraph.degRes_add_sum | [142, 1] | [144, 77] | exact sum_congr rfl fun _ _ => G.degRes_add_sub hB | α : Type u_1
G : SimpleGraph α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
A B C : Finset α
hB : B ⊆ A
⊢ ∑ v in C, degRes G v A = ∑ x in C, (degRes G x B + degRes G x (A \ B)) | 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 C : Finset α
hB : B ⊆ A
⊢ ∑ v in C, degRes G v A = ∑ x in C, (degRes G x B + degRes G x (A \ B))
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/NbhdRes.lean | SimpleGraph.degRes_add' | [147, 1] | [151, 55] | rw [degRes, nbhdRes,inter_distrib_right] | α : Type u_1
G : SimpleGraph α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
A B : Finset α
v : α
h : Disjoint A B
⊢ degRes G v (A ∪ B) = degRes G v A + degRes G v B | α : Type u_1
G : SimpleGraph α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
A B : Finset α
v : α
h : Disjoint A B
⊢ card (A ∩ neighborFinset G v ∪ B ∩ neighborFinset G v) = degRes G v A + degRes G v 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 B : Finset α
v : α
h : Disjoint A B
⊢ degRes G v (A ∪ B) = degRes G v A + degRes G v B
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/NbhdRes.lean | SimpleGraph.degRes_add' | [147, 1] | [151, 55] | exact card_disjoint_union (disj_of_inter_disj _ _ h) | α : Type u_1
G : SimpleGraph α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
A B : Finset α
v : α
h : Disjoint A B
⊢ card (A ∩ neighborFinset G v ∪ B ∩ neighborFinset G v) = degRes G v A + degRes G v B | 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 : Finset α
v : α
h : Disjoint A B
⊢ card (A ∩ neighborFinset G v ∪ B ∩ neighborFinset G v) = degRes G v A + degRes G v B
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/NbhdRes.lean | SimpleGraph.degRes_add_sum' | [154, 1] | [157, 73] | rw [← sum_add_distrib] | α : Type u_1
G : SimpleGraph α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
A B C : Finset α
h : Disjoint A B
⊢ ∑ v in C, degRes G v (A ∪ B) = ∑ v in C, degRes G v A + ∑ v in C, degRes G v B | α : Type u_1
G : SimpleGraph α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
A B C : Finset α
h : Disjoint A B
⊢ ∑ v in C, degRes G v (A ∪ B) = ∑ x in C, (degRes G x A + degRes G x 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 B C : Finset α
h : Disjoint A B
⊢ ∑ v in C, degRes G v (A ∪ B) = ∑ v in C, degRes G v A + ∑ v in C, degRes G v B
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/NbhdRes.lean | SimpleGraph.degRes_add_sum' | [154, 1] | [157, 73] | exact sum_congr rfl fun _ _ => G.degRes_add' h | α : Type u_1
G : SimpleGraph α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
A B C : Finset α
h : Disjoint A B
⊢ ∑ v in C, degRes G v (A ∪ B) = ∑ x in C, (degRes G x A + degRes G x B) | 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 C : Finset α
h : Disjoint A B
⊢ ∑ v in C, degRes G v (A ∪ B) = ∑ x in C, (degRes G x A + degRes G x B)
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/NbhdRes.lean | SimpleGraph.bip_count_help | [160, 1] | [165, 68] | simp only [degRes_ones] | α : Type u_1
G : SimpleGraph α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
A B : Finset α
⊢ ∑ v in B, degRes G v (A \ B) = ∑ v in B, ∑ w in A \ B, if Adj G v w then 1 else 0 | α : Type u_1
G : SimpleGraph α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
A B : Finset α
⊢ ∑ x in B, ∑ x in nbhdRes G x (A \ B), 1 = ∑ v in B, ∑ w in A \ B, if Adj G v w then 1 else 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 B : Finset α
⊢ ∑ v in B, degRes G v (A \ B) = ∑ v in B, ∑ w in A \ B, if Adj G v w then 1 else 0
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/NbhdRes.lean | SimpleGraph.bip_count_help | [160, 1] | [165, 68] | congr | α : Type u_1
G : SimpleGraph α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
A B : Finset α
⊢ ∑ x in B, ∑ x in nbhdRes G x (A \ B), 1 = ∑ v in B, ∑ w in A \ B, if Adj G v w then 1 else 0 | case e_f
α : Type u_1
G : SimpleGraph α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
A B : Finset α
⊢ (fun x => ∑ x in nbhdRes G x (A \ B), 1) = fun v => ∑ w in A \ B, if Adj G v w then 1 else 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 B : Finset α
⊢ ∑ x in B, ∑ x in nbhdRes G x (A \ B), 1 = ∑ v in B, ∑ w in A \ B, if Adj G v w then 1 else 0
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/NbhdRes.lean | SimpleGraph.bip_count_help | [160, 1] | [165, 68] | ext x | case e_f
α : Type u_1
G : SimpleGraph α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
A B : Finset α
⊢ (fun x => ∑ x in nbhdRes G x (A \ B), 1) = fun v => ∑ w in A \ B, if Adj G v w then 1 else 0 | case e_f.h
α : Type u_1
G : SimpleGraph α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
A B : Finset α
x : α
⊢ ∑ x in nbhdRes G x (A \ B), 1 = ∑ w in A \ B, if Adj G x w then 1 else 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case e_f
α : Type u_1
G : SimpleGraph α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
A B : Finset α
⊢ (fun x => ∑ x in nbhdRes G x (A \ B), 1) = fun v => ∑ w in A \ B, if Adj G v w then 1 else 0
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/NbhdRes.lean | SimpleGraph.bip_count_help | [160, 1] | [165, 68] | simp only [sum_const, Algebra.id.smul_eq_mul, mul_one, sum_boole, cast_id] | case e_f.h
α : Type u_1
G : SimpleGraph α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
A B : Finset α
x : α
⊢ ∑ x in nbhdRes G x (A \ B), 1 = ∑ w in A \ B, if Adj G x w then 1 else 0 | case e_f.h
α : Type u_1
G : SimpleGraph α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
A B : Finset α
x : α
⊢ card (nbhdRes G x (A \ B)) = card (filter (fun x_1 => Adj G x x_1) (A \ B)) | Please generate a tactic in lean4 to solve the state.
STATE:
case e_f.h
α : Type u_1
G : SimpleGraph α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
A B : Finset α
x : α
⊢ ∑ x in nbhdRes G x (A \ B), 1 = ∑ w in A \ B, if Adj G x w then 1 else 0
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/NbhdRes.lean | SimpleGraph.bip_count_help | [160, 1] | [165, 68] | congr | case e_f.h
α : Type u_1
G : SimpleGraph α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
A B : Finset α
x : α
⊢ card (nbhdRes G x (A \ B)) = card (filter (fun x_1 => Adj G x x_1) (A \ B)) | case e_f.h.e_s
α : Type u_1
G : SimpleGraph α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
A B : Finset α
x : α
⊢ nbhdRes G x (A \ B) = filter (fun x_1 => Adj G x x_1) (A \ B) | Please generate a tactic in lean4 to solve the state.
STATE:
case e_f.h
α : Type u_1
G : SimpleGraph α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
A B : Finset α
x : α
⊢ card (nbhdRes G x (A \ B)) = card (filter (fun x_1 => Adj G x x_1) (A \ B))
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/NbhdRes.lean | SimpleGraph.bip_count_help | [160, 1] | [165, 68] | ext | case e_f.h.e_s
α : Type u_1
G : SimpleGraph α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
A B : Finset α
x : α
⊢ nbhdRes G x (A \ B) = filter (fun x_1 => Adj G x x_1) (A \ B) | case e_f.h.e_s.a
α : Type u_1
G : SimpleGraph α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
A B : Finset α
x a✝ : α
⊢ a✝ ∈ nbhdRes G x (A \ B) ↔ a✝ ∈ filter (fun x_1 => Adj G x x_1) (A \ B) | Please generate a tactic in lean4 to solve the state.
STATE:
case e_f.h.e_s
α : Type u_1
G : SimpleGraph α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
A B : Finset α
x : α
⊢ nbhdRes G x (A \ B) = filter (fun x_1 => Adj G x x_1) (A \ B)
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/NbhdRes.lean | SimpleGraph.bip_count_help | [160, 1] | [165, 68] | rw [mem_res_nbhd_iff, mem_filter, mem_neighborFinset] | case e_f.h.e_s.a
α : Type u_1
G : SimpleGraph α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
A B : Finset α
x a✝ : α
⊢ a✝ ∈ nbhdRes G x (A \ B) ↔ a✝ ∈ filter (fun x_1 => Adj G x x_1) (A \ B) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case e_f.h.e_s.a
α : Type u_1
G : SimpleGraph α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
A B : Finset α
x a✝ : α
⊢ a✝ ∈ nbhdRes G x (A \ B) ↔ a✝ ∈ filter (fun x_1 => Adj G x x_1) (A \ B)
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/NbhdRes.lean | SimpleGraph.bip_count | [168, 1] | [182, 8] | rw [G.bip_count_help] | α : Type u_1
G : SimpleGraph α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
B A : Finset α
hB : B ⊆ A
⊢ ∑ v in B, degRes G v (A \ B) = ∑ v in A \ B, degRes G v B | α : Type u_1
G : SimpleGraph α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
B A : Finset α
hB : B ⊆ A
⊢ (∑ v in B, ∑ w in A \ B, if Adj G v w then 1 else 0) = ∑ v in A \ B, degRes G v 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
B A : Finset α
hB : B ⊆ A
⊢ ∑ v in B, degRes G v (A \ B) = ∑ v in A \ B, degRes G v B
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/NbhdRes.lean | SimpleGraph.bip_count | [168, 1] | [182, 8] | have := _root_.sdiff_sdiff_eq_self hB | α : Type u_1
G : SimpleGraph α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
B A : Finset α
hB : B ⊆ A
⊢ (∑ v in B, ∑ w in A \ B, if Adj G v w then 1 else 0) = ∑ v in A \ B, degRes G v B | α : Type u_1
G : SimpleGraph α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
B A : Finset α
hB : B ⊆ A
this : A \ (A \ B) = B
⊢ (∑ v in B, ∑ w in A \ B, if Adj G v w then 1 else 0) = ∑ v in A \ B, degRes G v 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
B A : Finset α
hB : B ⊆ A
⊢ (∑ v in B, ∑ w in A \ B, if Adj G v w then 1 else 0) = ∑ v in A \ B, degRes G v B
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/NbhdRes.lean | SimpleGraph.bip_count | [168, 1] | [182, 8] | nth_rw 4 [←this ] | α : Type u_1
G : SimpleGraph α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
B A : Finset α
hB : B ⊆ A
this : A \ (A \ B) = B
⊢ (∑ v in B, ∑ w in A \ B, if Adj G v w then 1 else 0) = ∑ v in A \ B, degRes G v B | α : Type u_1
G : SimpleGraph α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
B A : Finset α
hB : B ⊆ A
this : A \ (A \ B) = B
⊢ (∑ v in B, ∑ w in A \ B, if Adj G v w then 1 else 0) = ∑ v in A \ B, degRes G v (A \ (A \ 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
B A : Finset α
hB : B ⊆ A
this : A \ (A \ B) = B
⊢ (∑ v in B, ∑ w in A \ B, if Adj G v w then 1 else 0) = ∑ v in A \ B, degRes G v B
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/NbhdRes.lean | SimpleGraph.bip_count | [168, 1] | [182, 8] | rw [G.bip_count_help, this, sum_comm] | α : Type u_1
G : SimpleGraph α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
B A : Finset α
hB : B ⊆ A
this : A \ (A \ B) = B
⊢ (∑ v in B, ∑ w in A \ B, if Adj G v w then 1 else 0) = ∑ v in A \ B, degRes G v (A \ (A \ B)) | α : Type u_1
G : SimpleGraph α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
B A : Finset α
hB : B ⊆ A
this : A \ (A \ B) = B
⊢ (∑ y in A \ B, ∑ x in B, if Adj G x y then 1 else 0) = ∑ v in A \ B, ∑ w in B, if Adj G v w then 1 else 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
B A : Finset α
hB : B ⊆ A
this : A \ (A \ B) = B
⊢ (∑ v in B, ∑ w in A \ B, if Adj G v w then 1 else 0) = ∑ v in A \ B, degRes G v (A \ (A \ B))
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/NbhdRes.lean | SimpleGraph.bip_count | [168, 1] | [182, 8] | congr | α : Type u_1
G : SimpleGraph α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
B A : Finset α
hB : B ⊆ A
this : A \ (A \ B) = B
⊢ (∑ y in A \ B, ∑ x in B, if Adj G x y then 1 else 0) = ∑ v in A \ B, ∑ w in B, if Adj G v w then 1 else 0 | case e_f
α : Type u_1
G : SimpleGraph α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
B A : Finset α
hB : B ⊆ A
this : A \ (A \ B) = B
⊢ (fun y => ∑ x in B, if Adj G x y then 1 else 0) = fun v => ∑ w in B, if Adj G v w then 1 else 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
B A : Finset α
hB : B ⊆ A
this : A \ (A \ B) = B
⊢ (∑ y in A \ B, ∑ x in B, if Adj G x y then 1 else 0) = ∑ v in A \ B, ∑ w in B, if Adj G v w then 1 else 0
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/NbhdRes.lean | SimpleGraph.bip_count | [168, 1] | [182, 8] | ext y | case e_f
α : Type u_1
G : SimpleGraph α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
B A : Finset α
hB : B ⊆ A
this : A \ (A \ B) = B
⊢ (fun y => ∑ x in B, if Adj G x y then 1 else 0) = fun v => ∑ w in B, if Adj G v w then 1 else 0 | case e_f.h
α : Type u_1
G : SimpleGraph α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
B A : Finset α
hB : B ⊆ A
this : A \ (A \ B) = B
y : α
⊢ (∑ x in B, if Adj G x y then 1 else 0) = ∑ w in B, if Adj G y w then 1 else 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case e_f
α : Type u_1
G : SimpleGraph α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
B A : Finset α
hB : B ⊆ A
this : A \ (A \ B) = B
⊢ (fun y => ∑ x in B, if Adj G x y then 1 else 0) = fun v => ∑ w in B, if Adj G v w then 1 else 0
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/NbhdRes.lean | SimpleGraph.bip_count | [168, 1] | [182, 8] | congr | case e_f.h
α : Type u_1
G : SimpleGraph α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
B A : Finset α
hB : B ⊆ A
this : A \ (A \ B) = B
y : α
⊢ (∑ x in B, if Adj G x y then 1 else 0) = ∑ w in B, if Adj G y w then 1 else 0 | case e_f.h.e_f
α : Type u_1
G : SimpleGraph α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
B A : Finset α
hB : B ⊆ A
this : A \ (A \ B) = B
y : α
⊢ (fun x => if Adj G x y then 1 else 0) = fun w => if Adj G y w then 1 else 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case e_f.h
α : Type u_1
G : SimpleGraph α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
B A : Finset α
hB : B ⊆ A
this : A \ (A \ B) = B
y : α
⊢ (∑ x in B, if Adj G x y then 1 else 0) = ∑ w in B, if Adj G y w then 1 else 0
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/NbhdRes.lean | SimpleGraph.bip_count | [168, 1] | [182, 8] | ext x | case e_f.h.e_f
α : Type u_1
G : SimpleGraph α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
B A : Finset α
hB : B ⊆ A
this : A \ (A \ B) = B
y : α
⊢ (fun x => if Adj G x y then 1 else 0) = fun w => if Adj G y w then 1 else 0 | case e_f.h.e_f.h
α : Type u_1
G : SimpleGraph α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
B A : Finset α
hB : B ⊆ A
this : A \ (A \ B) = B
y x : α
⊢ (if Adj G x y then 1 else 0) = if Adj G y x then 1 else 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case e_f.h.e_f
α : Type u_1
G : SimpleGraph α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
B A : Finset α
hB : B ⊆ A
this : A \ (A \ B) = B
y : α
⊢ (fun x => if Adj G x y then 1 else 0) = fun w => if Adj G y w then 1 else 0
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/NbhdRes.lean | SimpleGraph.bip_count | [168, 1] | [182, 8] | split_ifs with h1 h2 h3 | case e_f.h.e_f.h
α : Type u_1
G : SimpleGraph α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
B A : Finset α
hB : B ⊆ A
this : A \ (A \ B) = B
y x : α
⊢ (if Adj G x y then 1 else 0) = if Adj G y x then 1 else 0 | 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
hB : B ⊆ A
this : A \ (A \ B) = B
y x : α
h1 : Adj G x y
h2 : Adj G y x
⊢ 1 = 1
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
hB : B ⊆ A
this : A \ (A \ B) = B
y x : α
h1 : Adj G x y
h2 : ¬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
hB : B ⊆ A
this : A \ (A \ B) = B
y x : α
h1 : ¬Adj G x y
h3 : 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
hB : B ⊆ A
this : A \ (A \ B) = B
y x : α
h1 : ¬Adj G x y
h3 : ¬Adj G y x
⊢ 0 = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case e_f.h.e_f.h
α : Type u_1
G : SimpleGraph α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
B A : Finset α
hB : B ⊆ A
this : A \ (A \ B) = B
y x : α
⊢ (if Adj G x y then 1 else 0) = if Adj G y x then 1 else 0
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/NbhdRes.lean | SimpleGraph.bip_count | [168, 1] | [182, 8] | rfl | 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
hB : B ⊆ A
this : A \ (A \ B) = B
y x : α
h1 : Adj G x y
h2 : Adj G y x
⊢ 1 = 1 | 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
hB : B ⊆ A
this : A \ (A \ B) = B
y x : α
h1 : Adj G x y
h2 : Adj G y x
⊢ 1 = 1
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/NbhdRes.lean | SimpleGraph.bip_count | [168, 1] | [182, 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
hB : B ⊆ A
this : A \ (A \ B) = B
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
hB : B ⊆ A
this : A \ (A \ B) = B
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
hB : B ⊆ A
this : A \ (A \ B) = B
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 | [168, 1] | [182, 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
hB : B ⊆ A
this : A \ (A \ B) = B
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
hB : B ⊆ A
this : A \ (A \ B) = B
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 | [168, 1] | [182, 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
hB : B ⊆ A
this : A \ (A \ B) = B
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
hB : B ⊆ A
this : A \ (A \ B) = B
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
hB : B ⊆ A
this : A \ (A \ B) = B
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 | [168, 1] | [182, 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
hB : B ⊆ A
this : A \ (A \ B) = B
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
hB : B ⊆ A
this : A \ (A \ B) = B
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 | [168, 1] | [182, 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
hB : B ⊆ A
this : A \ (A \ B) = B
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
hB : B ⊆ A
this : A \ (A \ B) = B
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.bip_count_help' | [185, 1] | [189, 33] | simp_rw [degRes_ones] | α : Type u_1
G : SimpleGraph α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
B A : Finset α
⊢ ∑ v in B, degRes G v A = ∑ v in B, ∑ w in A, if Adj G v w then 1 else 0 | α : Type u_1
G : SimpleGraph α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
B A : Finset α
⊢ ∑ x in B, ∑ x in nbhdRes G x A, 1 = ∑ v in B, ∑ w in A, if Adj G v w then 1 else 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
B A : Finset α
⊢ ∑ v in B, degRes G v A = ∑ v in B, ∑ w in A, if Adj G v w then 1 else 0
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/NbhdRes.lean | SimpleGraph.bip_count_help' | [185, 1] | [189, 33] | congr | α : Type u_1
G : SimpleGraph α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
B A : Finset α
⊢ ∑ x in B, ∑ x in nbhdRes G x A, 1 = ∑ v in B, ∑ w in A, if Adj G v w then 1 else 0 | case e_f
α : Type u_1
G : SimpleGraph α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
B A : Finset α
⊢ (fun x => ∑ x in nbhdRes G x A, 1) = fun v => ∑ w in A, if Adj G v w then 1 else 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
B A : Finset α
⊢ ∑ x in B, ∑ x in nbhdRes G x A, 1 = ∑ v in B, ∑ w in A, if Adj G v w then 1 else 0
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/NbhdRes.lean | SimpleGraph.bip_count_help' | [185, 1] | [189, 33] | ext x | case e_f
α : Type u_1
G : SimpleGraph α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
B A : Finset α
⊢ (fun x => ∑ x in nbhdRes G x A, 1) = fun v => ∑ w in A, if Adj G v w then 1 else 0 | case e_f.h
α : Type u_1
G : SimpleGraph α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
B A : Finset α
x : α
⊢ ∑ x in nbhdRes G x A, 1 = ∑ w in A, if Adj G x w then 1 else 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case e_f
α : Type u_1
G : SimpleGraph α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
B A : Finset α
⊢ (fun x => ∑ x in nbhdRes G x A, 1) = fun v => ∑ w in A, if Adj G v w then 1 else 0
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/NbhdRes.lean | SimpleGraph.bip_count_help' | [185, 1] | [189, 33] | rw [nbhdRes_filter,sum_filter] | case e_f.h
α : Type u_1
G : SimpleGraph α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
B A : Finset α
x : α
⊢ ∑ x in nbhdRes G x A, 1 = ∑ w in A, if Adj G x w then 1 else 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case e_f.h
α : Type u_1
G : SimpleGraph α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
B A : Finset α
x : α
⊢ ∑ x in nbhdRes G x A, 1 = ∑ w in A, if Adj G x w then 1 else 0
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/NbhdRes.lean | SimpleGraph.bip_count' | [192, 1] | [203, 8] | rw [G.bip_count_help' ] | α : Type u_1
G : SimpleGraph α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
B A : Finset α
⊢ ∑ v in B, degRes G v A = ∑ v in A, degRes G v B | α : Type u_1
G : SimpleGraph α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
B A : Finset α
⊢ (∑ v in B, ∑ w in A, if Adj G v w then 1 else 0) = ∑ v in A, degRes G v 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
B A : Finset α
⊢ ∑ v in B, degRes G v A = ∑ v in A, degRes G v B
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/NbhdRes.lean | SimpleGraph.bip_count' | [192, 1] | [203, 8] | rw [G.bip_count_help'] | α : Type u_1
G : SimpleGraph α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
B A : Finset α
⊢ (∑ v in B, ∑ w in A, if Adj G v w then 1 else 0) = ∑ v in A, degRes G v B | α : Type u_1
G : SimpleGraph α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
B A : Finset α
⊢ (∑ v in B, ∑ w in A, if Adj G v w then 1 else 0) = ∑ v in A, ∑ w in B, if Adj G v w then 1 else 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
B A : Finset α
⊢ (∑ v in B, ∑ w in A, if Adj G v w then 1 else 0) = ∑ v in A, degRes G v B
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/NbhdRes.lean | SimpleGraph.bip_count' | [192, 1] | [203, 8] | rw [sum_comm] | α : Type u_1
G : SimpleGraph α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
B A : Finset α
⊢ (∑ v in B, ∑ w in A, if Adj G v w then 1 else 0) = ∑ v in A, ∑ w in B, if Adj G v w then 1 else 0 | α : Type u_1
G : SimpleGraph α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
B A : Finset α
⊢ (∑ y in A, ∑ x in B, if Adj G x y then 1 else 0) = ∑ v in A, ∑ w in B, if Adj G v w then 1 else 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
B A : Finset α
⊢ (∑ v in B, ∑ w in A, if Adj G v w then 1 else 0) = ∑ v in A, ∑ w in B, if Adj G v w then 1 else 0
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/NbhdRes.lean | SimpleGraph.bip_count' | [192, 1] | [203, 8] | congr | α : Type u_1
G : SimpleGraph α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
B A : Finset α
⊢ (∑ y in A, ∑ x in B, if Adj G x y then 1 else 0) = ∑ v in A, ∑ w in B, if Adj G v w then 1 else 0 | case e_f
α : Type u_1
G : SimpleGraph α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
B A : Finset α
⊢ (fun y => ∑ x in B, if Adj G x y then 1 else 0) = fun v => ∑ w in B, if Adj G v w then 1 else 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
B A : Finset α
⊢ (∑ y in A, ∑ x in B, if Adj G x y then 1 else 0) = ∑ v in A, ∑ w in B, if Adj G v w then 1 else 0
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/NbhdRes.lean | SimpleGraph.bip_count' | [192, 1] | [203, 8] | ext y | case e_f
α : Type u_1
G : SimpleGraph α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
B A : Finset α
⊢ (fun y => ∑ x in B, if Adj G x y then 1 else 0) = fun v => ∑ w in B, if Adj G v w then 1 else 0 | case e_f.h
α : Type u_1
G : SimpleGraph α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
B A : Finset α
y : α
⊢ (∑ x in B, if Adj G x y then 1 else 0) = ∑ w in B, if Adj G y w then 1 else 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case e_f
α : Type u_1
G : SimpleGraph α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
B A : Finset α
⊢ (fun y => ∑ x in B, if Adj G x y then 1 else 0) = fun v => ∑ w in B, if Adj G v w then 1 else 0
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/NbhdRes.lean | SimpleGraph.bip_count' | [192, 1] | [203, 8] | congr | case e_f.h
α : Type u_1
G : SimpleGraph α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
B A : Finset α
y : α
⊢ (∑ x in B, if Adj G x y then 1 else 0) = ∑ w in B, if Adj G y w then 1 else 0 | case e_f.h.e_f
α : Type u_1
G : SimpleGraph α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
B A : Finset α
y : α
⊢ (fun x => if Adj G x y then 1 else 0) = fun w => if Adj G y w then 1 else 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case e_f.h
α : Type u_1
G : SimpleGraph α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
B A : Finset α
y : α
⊢ (∑ x in B, if Adj G x y then 1 else 0) = ∑ w in B, if Adj G y w then 1 else 0
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/NbhdRes.lean | SimpleGraph.bip_count' | [192, 1] | [203, 8] | ext x | case e_f.h.e_f
α : Type u_1
G : SimpleGraph α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
B A : Finset α
y : α
⊢ (fun x => if Adj G x y then 1 else 0) = fun w => if Adj G y w then 1 else 0 | case e_f.h.e_f.h
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
⊢ (if Adj G x y then 1 else 0) = if Adj G y x then 1 else 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case e_f.h.e_f
α : Type u_1
G : SimpleGraph α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
B A : Finset α
y : α
⊢ (fun x => if Adj G x y then 1 else 0) = fun w => if Adj G y w then 1 else 0
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/NbhdRes.lean | SimpleGraph.bip_count' | [192, 1] | [203, 8] | split_ifs with h1 h2 h3 | case e_f.h.e_f.h
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
⊢ (if Adj G x y then 1 else 0) = if Adj G y x then 1 else 0 | 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
h2 : Adj G y x
⊢ 1 = 1
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 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 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 | Please generate a tactic in lean4 to solve the state.
STATE:
case e_f.h.e_f.h
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
⊢ (if Adj G x y then 1 else 0) = if Adj G y x then 1 else 0
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/NbhdRes.lean | SimpleGraph.bip_count' | [192, 1] | [203, 8] | rfl | 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
h2 : Adj G y x
⊢ 1 = 1 | 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 x y
h2 : Adj G y x
⊢ 1 = 1
TACTIC:
|
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.