mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Combinatorics.SimpleGraph.Operations

Goal state before:
pp: V : Type u_1
G : SimpleGraph V
s t : V
H : (G ⊔ edge s t).Subgraph
h : ¬H.Adj s t
⊢ H.spanningCoe ≤ G

Tactic:
intro v w hvw

Goal state after:
V : Type u_1
G : SimpleGraph V
s t : V
H : (G ⊔ edge s t).Subgraph
h : ¬H.Adj s t
v w : V
hvw : H.spanningCoe.Adj v w
⊢ G.Adj v w
Module: Mathlib.Combinatorics.SimpleGraph.Subgraph

Goal state before:
pp: V : Type u
G G' : SimpleGraph V
u : Set V
h : G ≤ G'
⊢ (⊤.deleteVerts u).coe ≤ (⊤.deleteVerts u).coe

Tactic:
intro v w hvw

Goal state after:
V : Type u
G G' : SimpleGraph V
u : Set V
h : G ≤ G'
v w : ↑(⊤.deleteVerts u).verts
hvw : (⊤.deleteVerts u).coe.Adj v w
⊢ (⊤.deleteVerts u).coe.Adj v w
Module: Mathlib.Combinatorics.SimpleGraph.Walk.Basic

Goal state before:
pp: V : Type u
G : SimpleGraph V
u v : V
h : u ≠ v
p : G.Walk u v
⊢ v ∈ p.support.tail

Tactic:
by
  obtain ⟨_, _, _, rfl⟩ := exists_eq_cons_of_ne h p
  simp

Goal state after:
Module: Mathlib.Combinatorics.SimpleGraph.Walk.Operations

Goal state before:
pp: case nil
V : Type u
G : SimpleGraph V
u v w u✝ : V
p' : G.Walk u✝ w
⊢ (nil.append p').support = nil.support.dropLast ++ p'.support

Tactic:
simp_all [List.dropLast_cons_of_ne_nil]

Goal state after:
Module: Mathlib.Combinatorics.SimpleGraph.Walk.Operations

Goal state before:
pp: case cons
V : Type u
G : SimpleGraph V
u v w u✝ v✝ w✝ : V
h✝ : G.Adj u✝ v✝
p✝ : G.Walk v✝ w✝
p_ih✝ : ∀ (p' : G.Walk w✝ w), (p✝.append p').support = p✝.support.dropLast ++ p'.support
p' : G.Walk w✝ w
⊢ ((cons h✝ p✝).append p').support = (cons h✝ p✝).support.dropLast ++ p'.support

Tactic:
simp_all [List.dropLast_cons_of_ne_nil]

Goal state after:
Module: Mathlib.Combinatorics.SimpleGraph.Walk.Decomp

Goal state before:
pp: case neg
V : Type u
G : SimpleGraph V
inst✝ : DecidableEq V
u v w x u✝ v✝ w✝ : V
ha : G.Adj u✝ v✝
p' : G.Walk v✝ w✝
ih : ∀ (h : u ∈ p'.support), List.count s(u, x) (p'.takeUntil u h).edges ≤ 1
a✝ : List.Mem u p'.support
h' : ¬u✝ = u
⊢ List.count s(u, x) (cons ha (p'.takeUntil u ⋯)).edges ≤ 1

Tactic:
rw [edges_cons, List.count_cons]

Goal state after:
case neg
V : Type u
G : SimpleGraph V
inst✝ : DecidableEq V
u v w x u✝ v✝ w✝ : V
ha : G.Adj u✝ v✝
p' : G.Walk v✝ w✝
ih : ∀ (h : u ∈ p'.support), List.count s(u, x) (p'.takeUntil u h).edges ≤ 1
a✝ : List.Mem u p'.support
h' : ¬u✝ = u
⊢ (List.count s(u, x) (p'.takeUntil u ⋯).edges + if (s(u✝, v✝) == s(u, x)) = true then 1 else 0) ≤ 1
Module: Mathlib.Combinatorics.SimpleGraph.Walk.Subwalks

Goal state before:
pp: V : Type u_1
G : SimpleGraph V
u v u' v' : V
p : G.Walk u v
h : G.Adj u' v'
⊢ { fst := u', snd := v', adj := h } ∈ p.darts ∨ { fst := v', snd := u', adj := ⋯ } ∈ p.darts ↔
    s(u', v') ∈ List.map Dart.edge p.darts

Tactic:
List.mem_map

Goal state after:
V : Type u_1
G : SimpleGraph V
u v u' v' : V
p : G.Walk u v
h : G.Adj u' v'
⊢ { fst := u', snd := v', adj := h } ∈ p.darts ∨ { fst := v', snd := u', adj := ⋯ } ∈ p.darts ↔
    ∃ a ∈ p.darts, a.edge = s(u', v')
Module: Mathlib.Combinatorics.SimpleGraph.Walk.Decomp

Goal state before:
pp: case neg
V : Type u
G : SimpleGraph V
inst✝ : DecidableEq V
u v w x u✝ v✝ w✝ : V
ha : G.Adj u✝ v✝
p' : G.Walk v✝ w✝
ih : ∀ (h : u ∈ p'.support), List.count s(u, x) (p'.takeUntil u h).edges ≤ 1
a✝ : List.Mem u p'.support
h' : ¬u✝ = u
h'' : ¬(s(u✝, v✝) == s(u, x)) = true
⊢ List.count s(u, x) (p'.takeUntil u ⋯).edges + 0 ≤ 1

Tactic:
· apply ih

Goal state after:
Module: Mathlib.Combinatorics.SimpleGraph.Walk.Decomp

Goal state before:
pp: V : Type u
G : SimpleGraph V
w : V
inst✝ : DecidableEq V
u v : V
n : ℕ
p : G.Walk u v
hw : w ∈ p.support
hn : n ≤ (p.takeUntil w hw).length
⊢ (p.takeUntil w hw).getVert n =
    if n < (p.takeUntil w hw).length then (p.takeUntil w hw).getVert n
    else (p.dropUntil w hw).getVert (n - (p.takeUntil w hw).length)

Tactic:
cases hn.lt_or_eq

Goal state after:
case inl
V : Type u
G : SimpleGraph V
w : V
inst✝ : DecidableEq V
u v : V
n : ℕ
p : G.Walk u v
hw : w ∈ p.support
hn : n ≤ (p.takeUntil w hw).length
h✝ : n < (p.takeUntil w hw).length
⊢ (p.takeUntil w hw).getVert n =
    if n < (p.takeUntil w hw).length then (p.takeUntil w hw).getVert n
    else (p.dropUntil w hw).getVert (n - (p.takeUntil w hw).length)

case inr
V : Type u
G : SimpleGraph V
w : V
inst✝ : DecidableEq V
u v : V
n : ℕ
p : G.Walk u v
hw : w ∈ p.support
hn : n ≤ (p.takeUntil w hw).length
h✝ : n = (p.takeUntil w hw).length
⊢ (p.takeUntil w hw).getVert n =
    if n < (p.takeUntil w hw).length then (p.takeUntil w hw).getVert n
    else (p.dropUntil w hw).getVert (n - (p.takeUntil w hw).length)
Module: Mathlib.Combinatorics.SimpleGraph.Walk.Decomp

Goal state before:
pp: V : Type u
G : SimpleGraph V
v : V
inst✝ : DecidableEq V
c : G.Walk v v
u : V
h : u ∈ c.support
⊢ (c.takeUntil u h).support.tail ++ (c.dropUntil u h).support.tail ~r c.support.tail

Tactic:
rw [← tail_support_append, take_spec]

Goal state after:
Module: Mathlib.Combinatorics.SimpleGraph.Walk.Operations

Goal state before:
pp: V : Type u
G : SimpleGraph V
u v : V
p : G.Walk u v
n : ℕ
⊢ (p.take n).length = min n p.length

Tactic:
induction p generalizing n <;> cases n <;> simp [*, take]

Goal state after:
Module: Mathlib.Combinatorics.SimpleGraph.Paths

Goal state before:
pp: V : Type u
G : SimpleGraph V
u v w : V
p : G.Walk u v
q : G.Walk v w
h : (q.reverse.append p.reverse).IsPath
⊢ q.reverse.IsPath

Tactic:
apply h.of_append_left

Goal state after:
Module: Mathlib.Combinatorics.SimpleGraph.Walk.Operations

Goal state before:
pp: V : Type u
G : SimpleGraph V
u v : V
p : G.Walk u v
n : ℕ
⊢ (p.take n).support = List.take (n + 1) p.support

Tactic:
induction p generalizing n <;> cases n <;> simp [*, take]

Goal state after:
Module: Mathlib.Combinatorics.SimpleGraph.Walk.Operations

Goal state before:
pp: V : Type u
G : SimpleGraph V
u v : V
p : G.Walk u v
n : ℕ
⊢ (p.take n).darts = List.take n p.darts

Tactic:
induction p generalizing n <;> cases n <;> simp [*, take]

Goal state after:
Module: Mathlib.Combinatorics.SimpleGraph.Walk.Operations

Goal state before:
pp: V : Type u
G : SimpleGraph V
u v : V
p : G.Walk u v
n : ℕ
⊢ (p.take n).edges = List.take n p.edges

Tactic:
induction p generalizing n <;> cases n <;> simp [*, take]

Goal state after:
Module: Mathlib.Combinatorics.SimpleGraph.Regularity.Chunk

Goal state before:
pp: case refine_1
α : Type u_1
inst✝³ : Fintype α
inst✝² : DecidableEq α
P : Finpartition univ
hP : P.IsEquipartition
G : SimpleGraph α
inst✝¹ : DecidableRel G.Adj
ε : ℝ
U V : Finset α
inst✝ : Nonempty α
hPα : #P.parts * 16 ^ #P.parts ≤ Fintype.card α
hPε : 100 ≤ 4 ^ #P.parts * ε ^ 5
hU : U ∈ P.parts
hV : V ∈ P.parts
A B : Finset (Finset α)
hA : A ⊆ (chunk hP G ε hU).parts
hB : B ⊆ (chunk hP G ε hV).parts
this :
  ↑(G.edgeDensity (A.biUnion id) (B.biUnion id)) - ε ^ 5 / 50 ≤
    (1 - ε ^ 5 / 50) * ↑(G.edgeDensity (A.biUnion id) (B.biUnion id))
x y : Finset α
hx : x ∈ A
hy : y ∈ B
⊢ 0 < ↑(#A)

Tactic:
rw [cast_pos, Finset.card_pos]

Goal state after:
case refine_1
α : Type u_1
inst✝³ : Fintype α
inst✝² : DecidableEq α
P : Finpartition univ
hP : P.IsEquipartition
G : SimpleGraph α
inst✝¹ : DecidableRel G.Adj
ε : ℝ
U V : Finset α
inst✝ : Nonempty α
hPα : #P.parts * 16 ^ #P.parts ≤ Fintype.card α
hPε : 100 ≤ 4 ^ #P.parts * ε ^ 5
hU : U ∈ P.parts
hV : V ∈ P.parts
A B : Finset (Finset α)
hA : A ⊆ (chunk hP G ε hU).parts
hB : B ⊆ (chunk hP G ε hV).parts
this :
  ↑(G.edgeDensity (A.biUnion id) (B.biUnion id)) - ε ^ 5 / 50 ≤
    (1 - ε ^ 5 / 50) * ↑(G.edgeDensity (A.biUnion id) (B.biUnion id))
x y : Finset α
hx : x ∈ A
hy : y ∈ B
⊢ A.Nonempty
Module: Mathlib.Combinatorics.SimpleGraph.Regularity.Chunk

Goal state before:
pp: case refine_2
α : Type u_1
inst✝³ : Fintype α
inst✝² : DecidableEq α
P : Finpartition univ
hP : P.IsEquipartition
G : SimpleGraph α
inst✝¹ : DecidableRel G.Adj
ε : ℝ
U V : Finset α
inst✝ : Nonempty α
hPα : #P.parts * 16 ^ #P.parts ≤ Fintype.card α
hPε : 100 ≤ 4 ^ #P.parts * ε ^ 5
hU : U ∈ P.parts
hV : V ∈ P.parts
A B : Finset (Finset α)
hA : A ⊆ (chunk hP G ε hU).parts
hB : B ⊆ (chunk hP G ε hV).parts
this :
  ↑(G.edgeDensity (A.biUnion id) (B.biUnion id)) - ε ^ 5 / 50 ≤
    (1 - ε ^ 5 / 50) * ↑(G.edgeDensity (A.biUnion id) (B.biUnion id))
x y : Finset α
hx : x ∈ A
hy : y ∈ B
⊢ 0 < ↑(#x)

Tactic:
rw [cast_pos, Finset.card_pos]

Goal state after:
case refine_2
α : Type u_1
inst✝³ : Fintype α
inst✝² : DecidableEq α
P : Finpartition univ
hP : P.IsEquipartition
G : SimpleGraph α
inst✝¹ : DecidableRel G.Adj
ε : ℝ
U V : Finset α
inst✝ : Nonempty α
hPα : #P.parts * 16 ^ #P.parts ≤ Fintype.card α
hPε : 100 ≤ 4 ^ #P.parts * ε ^ 5
hU : U ∈ P.parts
hV : V ∈ P.parts
A B : Finset (Finset α)
hA : A ⊆ (chunk hP G ε hU).parts
hB : B ⊆ (chunk hP G ε hV).parts
this :
  ↑(G.edgeDensity (A.biUnion id) (B.biUnion id)) - ε ^ 5 / 50 ≤
    (1 - ε ^ 5 / 50) * ↑(G.edgeDensity (A.biUnion id) (B.biUnion id))
x y : Finset α
hx : x ∈ A
hy : y ∈ B
⊢ x.Nonempty
Module: Mathlib.Combinatorics.SimpleGraph.Regularity.Chunk

Goal state before:
pp: case refine_3
α : Type u_1
inst✝³ : Fintype α
inst✝² : DecidableEq α
P : Finpartition univ
hP : P.IsEquipartition
G : SimpleGraph α
inst✝¹ : DecidableRel G.Adj
ε : ℝ
U V : Finset α
inst✝ : Nonempty α
hPα : #P.parts * 16 ^ #P.parts ≤ Fintype.card α
hPε : 100 ≤ 4 ^ #P.parts * ε ^ 5
hU : U ∈ P.parts
hV : V ∈ P.parts
A B : Finset (Finset α)
hA : A ⊆ (chunk hP G ε hU).parts
hB : B ⊆ (chunk hP G ε hV).parts
this :
  ↑(G.edgeDensity (A.biUnion id) (B.biUnion id)) - ε ^ 5 / 50 ≤
    (1 - ε ^ 5 / 50) * ↑(G.edgeDensity (A.biUnion id) (B.biUnion id))
x y : Finset α
hx : x ∈ A
hy : y ∈ B
⊢ 0 < ↑(#B)

Tactic:
rw [cast_pos, Finset.card_pos]

Goal state after:
case refine_3
α : Type u_1
inst✝³ : Fintype α
inst✝² : DecidableEq α
P : Finpartition univ
hP : P.IsEquipartition
G : SimpleGraph α
inst✝¹ : DecidableRel G.Adj
ε : ℝ
U V : Finset α
inst✝ : Nonempty α
hPα : #P.parts * 16 ^ #P.parts ≤ Fintype.card α
hPε : 100 ≤ 4 ^ #P.parts * ε ^ 5
hU : U ∈ P.parts
hV : V ∈ P.parts
A B : Finset (Finset α)
hA : A ⊆ (chunk hP G ε hU).parts
hB : B ⊆ (chunk hP G ε hV).parts
this :
  ↑(G.edgeDensity (A.biUnion id) (B.biUnion id)) - ε ^ 5 / 50 ≤
    (1 - ε ^ 5 / 50) * ↑(G.edgeDensity (A.biUnion id) (B.biUnion id))
x y : Finset α
hx : x ∈ A
hy : y ∈ B
⊢ B.Nonempty
Module: Mathlib.Combinatorics.SimpleGraph.Regularity.Chunk

Goal state before:
pp: case refine_4
α : Type u_1
inst✝³ : Fintype α
inst✝² : DecidableEq α
P : Finpartition univ
hP : P.IsEquipartition
G : SimpleGraph α
inst✝¹ : DecidableRel G.Adj
ε : ℝ
U V : Finset α
inst✝ : Nonempty α
hPα : #P.parts * 16 ^ #P.parts ≤ Fintype.card α
hPε : 100 ≤ 4 ^ #P.parts * ε ^ 5
hU : U ∈ P.parts
hV : V ∈ P.parts
A B : Finset (Finset α)
hA : A ⊆ (chunk hP G ε hU).parts
hB : B ⊆ (chunk hP G ε hV).parts
this :
  ↑(G.edgeDensity (A.biUnion id) (B.biUnion id)) - ε ^ 5 / 50 ≤
    (1 - ε ^ 5 / 50) * ↑(G.edgeDensity (A.biUnion id) (B.biUnion id))
x y : Finset α
hx : x ∈ A
hy : y ∈ B
⊢ 0 < ↑(#y)

Tactic:
rw [cast_pos, Finset.card_pos]

Goal state after:
case refine_4
α : Type u_1
inst✝³ : Fintype α
inst✝² : DecidableEq α
P : Finpartition univ
hP : P.IsEquipartition
G : SimpleGraph α
inst✝¹ : DecidableRel G.Adj
ε : ℝ
U V : Finset α
inst✝ : Nonempty α
hPα : #P.parts * 16 ^ #P.parts ≤ Fintype.card α
hPε : 100 ≤ 4 ^ #P.parts * ε ^ 5
hU : U ∈ P.parts
hV : V ∈ P.parts
A B : Finset (Finset α)
hA : A ⊆ (chunk hP G ε hU).parts
hB : B ⊆ (chunk hP G ε hV).parts
this :
  ↑(G.edgeDensity (A.biUnion id) (B.biUnion id)) - ε ^ 5 / 50 ≤
    (1 - ε ^ 5 / 50) * ↑(G.edgeDensity (A.biUnion id) (B.biUnion id))
x y : Finset α
hx : x ∈ A
hy : y ∈ B
⊢ y.Nonempty
Module: Mathlib.Combinatorics.SimpleGraph.Clique

Goal state before:
pp: α : Type u_1
β : Type u_2
G : SimpleGraph α
s : Set α
f : α ↪ β
⊢ (SimpleGraph.map (⇑f) G).IsClique (⇑f '' s) ↔ G.IsClique s

Tactic:
rw [isClique_map_iff, f.injective.subsingleton_image_iff]
  obtain (hs | hs) := s.subsingleton_or_nontrivial
  · simp [hs, IsClique.of_subsingleton]
  simp [or_iff_right hs.not_subsingleton, Set.image_eq_image f.injective]

Goal state after:
Module: Mathlib.Combinatorics.SimpleGraph.Triangle.Basic

Goal state before:
pp: case mp.inl.inr.inl
α : Type u_1
G : SimpleGraph α
a b : α
hab : ¬a = b
e : α
hce : G.Adj a e
hde : G.Adj b e
hcd : G.Adj a b
⊢ ∃ c, G.Adj a c ∧ G.Adj b c ∧ {a, b, e} = {a, b, c}

Tactic:
exact ⟨d, by aesop⟩

Goal state after:
case mp.inl.inr.inl
α : Type u_1
G : SimpleGraph α
a b : α
hab : ¬a = b
e : α
hce : G.Adj a e
hde : G.Adj b e
hcd : G.Adj a b
⊢ ∃ c, G.Adj a c ∧ G.Adj b c ∧ {a, b, e} = {a, b, c}
Module: Mathlib.Combinatorics.SimpleGraph.Triangle.Basic

Goal state before:
pp: case mp.inl.inr.inr
α : Type u_1
G : SimpleGraph α
a b : α
hab : ¬a = b
d : α
hcd : G.Adj a d
hce : G.Adj a b
hde : G.Adj b d
⊢ ∃ c, G.Adj a c ∧ G.Adj b c ∧ {a, d, b} = {a, b, c}

Tactic:
exact ⟨d, by aesop⟩

Goal state after:
Module: Mathlib.Combinatorics.SimpleGraph.Triangle.Basic

Goal state before:
pp: case mp.inr.inl.inl
α : Type u_1
G : SimpleGraph α
a b : α
hab : ¬a = b
e : α
hde : G.Adj a e
hce : G.Adj b e
hcd : G.Adj a b
⊢ G.Adj a b ∧ ∃ c, G.Adj a c ∧ G.Adj b c ∧ {b, a, e} = {a, b, c}

Tactic:
exact ⟨d, by aesop⟩

Goal state after:
case mp.inr.inl.inl
α : Type u_1
G : SimpleGraph α
a b : α
hab : ¬a = b
e : α
hde : G.Adj a e
hce : G.Adj b e
hcd : G.Adj a b
⊢ G.Adj a b ∧ ∃ c, G.Adj a c ∧ G.Adj b c ∧ {b, a, e} = {a, b, c}
Module: Mathlib.Combinatorics.SimpleGraph.Triangle.Basic

Goal state before:
pp: case mp.inr.inl.inl
α : Type u_1
G : SimpleGraph α
a b : α
hab : ¬a = b
e : α
hde : G.Adj a e
hce : G.Adj b e
hcd : G.Adj a b
⊢ ∃ c, G.Adj a c ∧ G.Adj b c ∧ {b, a, e} = {a, b, c}

Tactic:
exact ⟨d, by aesop⟩

Goal state after:
case mp.inr.inl.inl
α : Type u_1
G : SimpleGraph α
a b : α
hab : ¬a = b
e : α
hde : G.Adj a e
hce : G.Adj b e
hcd : G.Adj a b
⊢ ∃ c, G.Adj a c ∧ G.Adj b c ∧ {b, a, e} = {a, b, c}
Module: Mathlib.Combinatorics.SimpleGraph.Triangle.Basic

Goal state before:
pp: case mp.inr.inr.inl
α : Type u_1
G : SimpleGraph α
a b : α
hab : ¬a = b
d : α
hcd : G.Adj b d
hde : G.Adj a d
hce : G.Adj a b
⊢ G.Adj a b ∧ ∃ c, G.Adj a c ∧ G.Adj b c ∧ {b, d, a} = {a, b, c}

Tactic:
exact ⟨d, by aesop⟩

Goal state after:
case mp.inr.inr.inl
α : Type u_1
G : SimpleGraph α
a b : α
hab : ¬a = b
d : α
hcd : G.Adj b d
hde : G.Adj a d
hce : G.Adj a b
⊢ G.Adj a b ∧ ∃ c, G.Adj a c ∧ G.Adj b c ∧ {b, d, a} = {a, b, c}
Module: Mathlib.Combinatorics.SimpleGraph.Triangle.Basic

Goal state before:
pp: case mp.inr.inr.inl
α : Type u_1
G : SimpleGraph α
a b : α
hab : ¬a = b
d : α
hcd : G.Adj b d
hde : G.Adj a d
hce : G.Adj a b
⊢ ∃ c, G.Adj a c ∧ G.Adj b c ∧ {b, d, a} = {a, b, c}

Tactic:
exact ⟨d, by aesop⟩

Goal state after:
Module: Mathlib.Combinatorics.SimpleGraph.Triangle.Basic

Goal state before:
pp: case mp.inr.inr.inr.inl
α : Type u_1
G : SimpleGraph α
a b : α
hab : ¬a = b
c : α
hce : G.Adj a c
hcd : G.Adj b c
hde : G.Adj a b
⊢ G.Adj a b ∧ ∃ c_1, G.Adj a c_1 ∧ G.Adj b c_1 ∧ {c, b, a} = {a, b, c_1}

Tactic:
exact ⟨d, by aesop⟩

Goal state after:
case mp.inr.inr.inr.inl
α : Type u_1
G : SimpleGraph α
a b : α
hab : ¬a = b
c : α
hce : G.Adj a c
hcd : G.Adj b c
hde : G.Adj a b
⊢ G.Adj a b ∧ ∃ c_1, G.Adj a c_1 ∧ G.Adj b c_1 ∧ {c, b, a} = {a, b, c_1}
Module: Mathlib.Combinatorics.SimpleGraph.Clique

Goal state before:
pp: case mp
α : Type u_1
G : SimpleGraph α
⊢ G.CliqueFree 2 → G.edgeSet = ∅

Tactic:
Set.eq_empty_iff_forall_notMem,

Goal state after:
case mp
α : Type u_1
G : SimpleGraph α
⊢ G.CliqueFree 2 → ∀ (x : Sym2 α), x ∉ G.edgeSet
Module: Mathlib.Combinatorics.SimpleGraph.Clique

Goal state before:
pp: α : Type u_1
G : SimpleGraph α
n : ℕ
⊢ G.cliqueSet n = ∅ ↔ ∀ (t : Finset α), ¬G.IsNClique n t

Tactic:
Set.eq_empty_iff_forall_notMem,

Goal state after:
α : Type u_1
G : SimpleGraph α
n : ℕ
⊢ (∀ (x : Finset α), x ∉ G.cliqueSet n) ↔ ∀ (t : Finset α), ¬G.IsNClique n t
Module: Mathlib.Combinatorics.SimpleGraph.Regularity.Chunk

Goal state before:
pp: case refine_1
α : Type u_1
inst✝³ : Fintype α
inst✝² : DecidableEq α
P : Finpartition univ
hP : P.IsEquipartition
G : SimpleGraph α
inst✝¹ : DecidableRel G.Adj
ε : ℝ
U V : Finset α
inst✝ : Nonempty α
hPα : #P.parts * 16 ^ #P.parts ≤ Fintype.card α
hPε : 100 ≤ 4 ^ #P.parts * ε ^ 5
hε₁ : ε ≤ 1
hU : U ∈ P.parts
hV : V ∈ P.parts
A B : Finset (Finset α)
hA : A ⊆ (chunk hP G ε hU).parts
hB : B ⊆ (chunk hP G ε hV).parts
this :
  (1 + ε ^ 5 / 49) * ↑(G.edgeDensity (A.biUnion id) (B.biUnion id)) ≤
    ↑(G.edgeDensity (A.biUnion id) (B.biUnion id)) + ε ^ 5 / 49
x y : Finset α
hx : x ∈ A
hy : y ∈ B
⊢ 0 < ↑(#A)

Tactic:
rw [cast_pos, Finset.card_pos]

Goal state after:
case refine_1
α : Type u_1
inst✝³ : Fintype α
inst✝² : DecidableEq α
P : Finpartition univ
hP : P.IsEquipartition
G : SimpleGraph α
inst✝¹ : DecidableRel G.Adj
ε : ℝ
U V : Finset α
inst✝ : Nonempty α
hPα : #P.parts * 16 ^ #P.parts ≤ Fintype.card α
hPε : 100 ≤ 4 ^ #P.parts * ε ^ 5
hε₁ : ε ≤ 1
hU : U ∈ P.parts
hV : V ∈ P.parts
A B : Finset (Finset α)
hA : A ⊆ (chunk hP G ε hU).parts
hB : B ⊆ (chunk hP G ε hV).parts
this :
  (1 + ε ^ 5 / 49) * ↑(G.edgeDensity (A.biUnion id) (B.biUnion id)) ≤
    ↑(G.edgeDensity (A.biUnion id) (B.biUnion id)) + ε ^ 5 / 49
x y : Finset α
hx : x ∈ A
hy : y ∈ B
⊢ A.Nonempty
Module: Mathlib.Combinatorics.SimpleGraph.Regularity.Chunk

Goal state before:
pp: case refine_2
α : Type u_1
inst✝³ : Fintype α
inst✝² : DecidableEq α
P : Finpartition univ
hP : P.IsEquipartition
G : SimpleGraph α
inst✝¹ : DecidableRel G.Adj
ε : ℝ
U V : Finset α
inst✝ : Nonempty α
hPα : #P.parts * 16 ^ #P.parts ≤ Fintype.card α
hPε : 100 ≤ 4 ^ #P.parts * ε ^ 5
hε₁ : ε ≤ 1
hU : U ∈ P.parts
hV : V ∈ P.parts
A B : Finset (Finset α)
hA : A ⊆ (chunk hP G ε hU).parts
hB : B ⊆ (chunk hP G ε hV).parts
this :
  (1 + ε ^ 5 / 49) * ↑(G.edgeDensity (A.biUnion id) (B.biUnion id)) ≤
    ↑(G.edgeDensity (A.biUnion id) (B.biUnion id)) + ε ^ 5 / 49
x y : Finset α
hx : x ∈ A
hy : y ∈ B
⊢ 0 < ↑(#x)

Tactic:
rw [cast_pos, Finset.card_pos]

Goal state after:
case refine_2
α : Type u_1
inst✝³ : Fintype α
inst✝² : DecidableEq α
P : Finpartition univ
hP : P.IsEquipartition
G : SimpleGraph α
inst✝¹ : DecidableRel G.Adj
ε : ℝ
U V : Finset α
inst✝ : Nonempty α
hPα : #P.parts * 16 ^ #P.parts ≤ Fintype.card α
hPε : 100 ≤ 4 ^ #P.parts * ε ^ 5
hε₁ : ε ≤ 1
hU : U ∈ P.parts
hV : V ∈ P.parts
A B : Finset (Finset α)
hA : A ⊆ (chunk hP G ε hU).parts
hB : B ⊆ (chunk hP G ε hV).parts
this :
  (1 + ε ^ 5 / 49) * ↑(G.edgeDensity (A.biUnion id) (B.biUnion id)) ≤
    ↑(G.edgeDensity (A.biUnion id) (B.biUnion id)) + ε ^ 5 / 49
x y : Finset α
hx : x ∈ A
hy : y ∈ B
⊢ x.Nonempty
Module: Mathlib.Combinatorics.SimpleGraph.Regularity.Chunk

Goal state before:
pp: case refine_3
α : Type u_1
inst✝³ : Fintype α
inst✝² : DecidableEq α
P : Finpartition univ
hP : P.IsEquipartition
G : SimpleGraph α
inst✝¹ : DecidableRel G.Adj
ε : ℝ
U V : Finset α
inst✝ : Nonempty α
hPα : #P.parts * 16 ^ #P.parts ≤ Fintype.card α
hPε : 100 ≤ 4 ^ #P.parts * ε ^ 5
hε₁ : ε ≤ 1
hU : U ∈ P.parts
hV : V ∈ P.parts
A B : Finset (Finset α)
hA : A ⊆ (chunk hP G ε hU).parts
hB : B ⊆ (chunk hP G ε hV).parts
this :
  (1 + ε ^ 5 / 49) * ↑(G.edgeDensity (A.biUnion id) (B.biUnion id)) ≤
    ↑(G.edgeDensity (A.biUnion id) (B.biUnion id)) + ε ^ 5 / 49
x y : Finset α
hx : x ∈ A
hy : y ∈ B
⊢ 0 < ↑(#B)

Tactic:
rw [cast_pos, Finset.card_pos]

Goal state after:
case refine_3
α : Type u_1
inst✝³ : Fintype α
inst✝² : DecidableEq α
P : Finpartition univ
hP : P.IsEquipartition
G : SimpleGraph α
inst✝¹ : DecidableRel G.Adj
ε : ℝ
U V : Finset α
inst✝ : Nonempty α
hPα : #P.parts * 16 ^ #P.parts ≤ Fintype.card α
hPε : 100 ≤ 4 ^ #P.parts * ε ^ 5
hε₁ : ε ≤ 1
hU : U ∈ P.parts
hV : V ∈ P.parts
A B : Finset (Finset α)
hA : A ⊆ (chunk hP G ε hU).parts
hB : B ⊆ (chunk hP G ε hV).parts
this :
  (1 + ε ^ 5 / 49) * ↑(G.edgeDensity (A.biUnion id) (B.biUnion id)) ≤
    ↑(G.edgeDensity (A.biUnion id) (B.biUnion id)) + ε ^ 5 / 49
x y : Finset α
hx : x ∈ A
hy : y ∈ B
⊢ B.Nonempty
Module: Mathlib.Combinatorics.SimpleGraph.Regularity.Chunk

Goal state before:
pp: case refine_4
α : Type u_1
inst✝³ : Fintype α
inst✝² : DecidableEq α
P : Finpartition univ
hP : P.IsEquipartition
G : SimpleGraph α
inst✝¹ : DecidableRel G.Adj
ε : ℝ
U V : Finset α
inst✝ : Nonempty α
hPα : #P.parts * 16 ^ #P.parts ≤ Fintype.card α
hPε : 100 ≤ 4 ^ #P.parts * ε ^ 5
hε₁ : ε ≤ 1
hU : U ∈ P.parts
hV : V ∈ P.parts
A B : Finset (Finset α)
hA : A ⊆ (chunk hP G ε hU).parts
hB : B ⊆ (chunk hP G ε hV).parts
this :
  (1 + ε ^ 5 / 49) * ↑(G.edgeDensity (A.biUnion id) (B.biUnion id)) ≤
    ↑(G.edgeDensity (A.biUnion id) (B.biUnion id)) + ε ^ 5 / 49
x y : Finset α
hx : x ∈ A
hy : y ∈ B
⊢ 0 < ↑(#y)

Tactic:
rw [cast_pos, Finset.card_pos]

Goal state after:
case refine_4
α : Type u_1
inst✝³ : Fintype α
inst✝² : DecidableEq α
P : Finpartition univ
hP : P.IsEquipartition
G : SimpleGraph α
inst✝¹ : DecidableRel G.Adj
ε : ℝ
U V : Finset α
inst✝ : Nonempty α
hPα : #P.parts * 16 ^ #P.parts ≤ Fintype.card α
hPε : 100 ≤ 4 ^ #P.parts * ε ^ 5
hε₁ : ε ≤ 1
hU : U ∈ P.parts
hV : V ∈ P.parts
A B : Finset (Finset α)
hA : A ⊆ (chunk hP G ε hU).parts
hB : B ⊆ (chunk hP G ε hV).parts
this :
  (1 + ε ^ 5 / 49) * ↑(G.edgeDensity (A.biUnion id) (B.biUnion id)) ≤
    ↑(G.edgeDensity (A.biUnion id) (B.biUnion id)) + ε ^ 5 / 49
x y : Finset α
hx : x ∈ A
hy : y ∈ B
⊢ y.Nonempty
Module: Mathlib.Combinatorics.SimpleGraph.Triangle.Removal

Goal state before:
pp: α : Type u_1
inst✝³ : DecidableEq α
inst✝² : Fintype α
G : SimpleGraph α
inst✝¹ : DecidableRel G.Adj
P : Finpartition univ
ε : ℝ
inst✝ : Nonempty α
hε : 0 < ε
hP : P.IsEquipartition
hPε : P.IsUniform G (ε / 8)
hP' : 4 / ε ≤ ↑(#P.parts)
A : Finset (α × α) :=
  (P.nonUniforms G (ε / 8)).biUnion fun x ↦
    match x with
    | (U, V) => U ×ˢ V
B : Finset (α × α) := P.parts.biUnion offDiag
C : Finset (α × α) :=
  (P.sparsePairs G (ε / 4)).biUnion fun x ↦
    match x with
    | (U, V) => G.interedges U V
⊢ 2 * ↑(#G.edgeFinset) - 2 * ↑(#(regularityReduced P G (ε / 8) (ε / 4)).edgeFinset) =
    ↑(#{x |
          match x with
          | (x, y) => G.Adj x y ∧ ¬(regularityReduced P G (ε / 8) (ε / 4)).Adj x y})

Tactic:
filter_and_not,

Goal state after:
α : Type u_1
inst✝³ : DecidableEq α
inst✝² : Fintype α
G : SimpleGraph α
inst✝¹ : DecidableRel G.Adj
P : Finpartition univ
ε : ℝ
inst✝ : Nonempty α
hε : 0 < ε
hP : P.IsEquipartition
hPε : P.IsUniform G (ε / 8)
hP' : 4 / ε ≤ ↑(#P.parts)
A : Finset (α × α) :=
  (P.nonUniforms G (ε / 8)).biUnion fun x ↦
    match x with
    | (U, V) => U ×ˢ V
B : Finset (α × α) := P.parts.biUnion offDiag
C : Finset (α × α) :=
  (P.sparsePairs G (ε / 4)).biUnion fun x ↦
    match x with
    | (U, V) => G.interedges U V
⊢ 2 * ↑(#G.edgeFinset) - 2 * ↑(#(regularityReduced P G (ε / 8) (ε / 4)).edgeFinset) =
    ↑(#({x | G.Adj x.1 x.2} \ {x | (regularityReduced P G (ε / 8) (ε / 4)).Adj x.1 x.2}))
Module: Mathlib.Combinatorics.Additive.ErdosGinzburgZiv

Goal state before:
pp: m : ℕ
hm : 2 ≤ m
ihm : ∀ {ι : Type u_1} {s : Finset ι} (a : ι → ℤ), 2 * m - 1 ≤ #s → ∃ t ⊆ s, #t = m ∧ ↑m ∣ ∑ i ∈ t, a i
n : ℕ
hn : 2 ≤ n
ihn : ∀ {ι : Type u_1} {s : Finset ι} (a : ι → ℤ), 2 * n - 1 ≤ #s → ∃ t ⊆ s, #t = n ∧ ↑n ∣ ∑ i ∈ t, a i
ι : Type u_1
s : Finset ι
a : ι → ℤ
hs : 2 * (m * n) - 1 ≤ #s
k : ℕ
ih : k ≤ 2 * m - 1 → ∃ 𝒜, #𝒜 = k ∧ (↑𝒜).Pairwise Disjoint ∧ ∀ ⦃t : Finset ι⦄, t ∈ 𝒜 → t ⊆ s ∧ #t = n ∧ ↑n ∣ ∑ i ∈ t, a i
hk : k + 1 ≤ 2 * m - 1
𝒜 : Finset (Finset ι)
h𝒜card : #𝒜 = k
h𝒜disj : (↑𝒜).Pairwise Disjoint
h𝒜 : ∀ ⦃t : Finset ι⦄, t ∈ 𝒜 → t ⊆ s ∧ #t = n ∧ ↑n ∣ ∑ i ∈ t, a i
⊢ 2 * (m * n) - 1 - k * n = 2 * (m * n) - 1 - #𝒜 * n

Tactic:
h𝒜card

Goal state after:
m : ℕ
hm : 2 ≤ m
ihm : ∀ {ι : Type u_1} {s : Finset ι} (a : ι → ℤ), 2 * m - 1 ≤ #s → ∃ t ⊆ s, #t = m ∧ ↑m ∣ ∑ i ∈ t, a i
n : ℕ
hn : 2 ≤ n
ihn : ∀ {ι : Type u_1} {s : Finset ι} (a : ι → ℤ), 2 * n - 1 ≤ #s → ∃ t ⊆ s, #t = n ∧ ↑n ∣ ∑ i ∈ t, a i
ι : Type u_1
s : Finset ι
a : ι → ℤ
hs : 2 * (m * n) - 1 ≤ #s
k : ℕ
ih : k ≤ 2 * m - 1 → ∃ 𝒜, #𝒜 = k ∧ (↑𝒜).Pairwise Disjoint ∧ ∀ ⦃t : Finset ι⦄, t ∈ 𝒜 → t ⊆ s ∧ #t = n ∧ ↑n ∣ ∑ i ∈ t, a i
hk : k + 1 ≤ 2 * m - 1
𝒜 : Finset (Finset ι)
h𝒜card : #𝒜 = k
h𝒜disj : (↑𝒜).Pairwise Disjoint
h𝒜 : ∀ ⦃t : Finset ι⦄, t ∈ 𝒜 → t ⊆ s ∧ #t = n ∧ ↑n ∣ ∑ i ∈ t, a i
⊢ 2 * (m * n) - 1 - k * n = 2 * (m * n) - 1 - k * n