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

Goal state before:
pp: α : Type u_1
G H : SimpleGraph α
inst✝³ : Fintype α
inst✝² : DecidableRel G.Adj
inst✝¹ : DecidableRel H.Adj
inst✝ : DecidableEq α
tris : Finset (Finset α)
htris : tris ⊆ G.cliqueFinset 3
pd : (↑tris).Pairwise fun x y ↦ (↑x ∩ ↑y).Subsingleton
hHG : H ≤ G
hH : H.CliqueFree 3
hG : #(G.edgeFinset \ H.edgeFinset) < #tris.attach
t : Finset α
ht : t ∈ tris
⊢ ∃ x y, x ∈ t ∧ y ∈ t ∧ x ≠ y ∧ s(x, y) ∈ G.edgeFinset \ H.edgeFinset

Tactic:
by

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

Goal state before:
pp: V : Type u
V' : Type v
G : SimpleGraph V
G' : SimpleGraph V'
f : G →g G'
u : V
p : G.Walk u u
hinj : Injective ⇑f
⊢ (Walk.map f p).IsCircuit ↔ p.IsCircuit

Tactic:
rw [isCircuit_def, isCircuit_def, isTrail_map_iff_of_injective hinj, ne_eq, ne_eq, eq_nil_iff_nil,
    eq_nil_iff_nil, nil_map_iff]

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

Goal state before:
pp: V : Type u
V' : Type v
G : SimpleGraph V
G' : SimpleGraph V'
f : G →g G'
u : V
p : G.Walk u u
hinj : Injective ⇑f
⊢ (Walk.map f p).IsCircuit ↔ p.IsCircuit

Tactic:
by

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

Goal state before:
pp: α : Type u_1
G : SimpleGraph α
inst✝ : DecidableRel G.Adj
ε : ℝ
s t : Finset α
⊢ ↑(#(Rel.interedges G.Adj (G.badVertices ε s t) t)) ≤ ↑(#(G.badVertices ε s t)) * ↑(#t) * (↑(G.edgeDensity s t) - ε)

Tactic:
by

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

Goal state before:
pp: α : Type u_1
G : SimpleGraph α
inst✝ : DecidableRel G.Adj
ε : ℝ
s t : Finset α
hε : 0 ≤ ε
dst : 2 * ε ≤ ↑(G.edgeDensity s t)
⊢ 0 ≤ ↑(#(G.badVertices ε s t)) * ↑(#t)

Tactic:
by

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

Goal state before:
pp: V : Type u
V' : Type v
G : SimpleGraph V
G' : SimpleGraph V'
f : G →g G'
u : V
p : G.Walk u u
hp : (Walk.map f p).IsCycle
⊢ p.IsCycle

Tactic:
by

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

Goal state before:
pp: α : Type u_1
G : SimpleGraph α
inst✝ : DecidableRel G.Adj
ε : ℝ
s t : Finset α
hε : 0 ≤ ε
dst : 2 * ε ≤ ↑(G.edgeDensity s t)
⊢ ε ≤ ↑(G.edgeDensity s t)

Tactic:
by

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

Goal state before:
pp: 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
⊢ (∑ ab ∈ A.product B, ↑(G.edgeDensity ab.1 ab.2)) / (↑(#A) * ↑(#B)) ≤
    ↑(G.edgeDensity (A.biUnion id) (B.biUnion id)) + ε ^ 5 / 49

Tactic:
by

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

Goal state before:
pp: V : Type u
V' : Type v
G : SimpleGraph V
G' : SimpleGraph V'
f : G →g G'
u : V
p : G.Walk u u
hinj : Injective ⇑f
⊢ (Walk.map f p).IsCycle ↔ p.IsCycle

Tactic:
by

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

Goal state before:
pp: α : Type u_1
G : SimpleGraph α
inst✝ : DecidableRel G.Adj
ε : ℝ
s t : Finset α
hε : 0 ≤ ε
dst : 2 * ε ≤ ↑(G.edgeDensity s t)
⊢ ↑(G.edgeDensity (G.badVertices ε s t) t) ≤ ↑(G.edgeDensity s t) - ε

Tactic:
by

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

Goal state before:
pp: V : Type u
V' : Type v
G : SimpleGraph V
G' : SimpleGraph V'
f : G →g G'
hinj : Injective ⇑f
u v : V
⊢ Injective (Path.map f hinj)

Tactic:
by

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

Goal state before:
pp: α : Type u_1
G : SimpleGraph α
inst✝ : DecidableRel G.Adj
ε : ℝ
s t : Finset α
dst : 2 * ε ≤ ↑(G.edgeDensity s t)
hst : G.IsUniform ε s t
⊢ 1 ≤ 2

Tactic:
by

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

Goal state before:
pp: V : Type u
G : SimpleGraph V
u v : V
H : SimpleGraph V
p : G.Walk u v
hp : ∀ e ∈ p.edges, e ∈ H.edgeSet
pp : p.IsPath
⊢ (p.transfer H hp).IsPath

Tactic:
by

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

Goal state before:
pp: V : Type u
G : SimpleGraph V
u : V
H : SimpleGraph V
q : G.Walk u u
qc : q.IsCycle
hq : ∀ e ∈ q.edges, e ∈ H.edgeSet
⊢ (q.transfer H hq).IsCycle

Tactic:
by

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

Goal state before:
pp: α : Type u_1
G : SimpleGraph α
inst✝ : DecidableRel G.Adj
ε : ℝ
s t : Finset α
dst : 2 * ε ≤ ↑(G.edgeDensity s t)
hst : G.IsUniform ε s t
⊢ ↑(#(G.badVertices ε s t)) ≤ ↑(#s) * ε

Tactic:
by

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

Goal state before:
pp: V : Type u
V' : Type v
G : SimpleGraph V
G' : SimpleGraph V'
v✝ w v u u' v' : V
s : Set (Sym2 V)
p : G.Walk u v
hu : u = u'
hv : v = v'
h : ∀ e ∈ (p.copy hu hv).edges, e ∉ s
⊢ ∀ e ∈ p.edges, e ∉ s

Tactic:
by

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

Goal state before:
pp: V : Type u
G : SimpleGraph V
v u u' v' : V
s : Set (Sym2 V)
p : G.Walk u v
hu : u = u'
hv : v = v'
h : ∀ e ∈ (p.copy hu hv).edges, e ∉ s
⊢ toDeleteEdges s (p.copy hu hv) h = (toDeleteEdges s p ⋯).copy hu hv

Tactic:
by

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

Goal state before:
pp: α : Type u_1
G : SimpleGraph α
inst✝¹ : DecidableRel G.Adj
ε : ℝ
s t u : Finset α
inst✝ : DecidableEq α
⊢ ((s \ (G.badVertices ε s t ∪ G.badVertices ε s u)).biUnion fun x ↦
      image (fun x_1 ↦ (x, x_1)) (G.interedges ({y ∈ t | G.Adj x y}) ({y ∈ u | G.Adj x y}))) ⊆
    {x ∈ s ×ˢ t ×ˢ u |
      match x with
      | (x, y, z) => G.Adj x y ∧ G.Adj x z ∧ G.Adj y z}

Tactic:
by

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

Goal state before:
pp: α : Type u_1
G H : SimpleGraph α
inst✝³ : Fintype α
inst✝² : DecidableRel G.Adj
inst✝¹ : DecidableRel H.Adj
inst✝ : DecidableEq α
tris : Finset (Finset α)
htris : tris ⊆ G.cliqueFinset 3
pd : (↑tris).Pairwise fun x y ↦ (↑x ∩ ↑y).Subsingleton
hHG : H ≤ G
hH : H.CliqueFree 3
⊢ #tris ≤ #G.edgeFinset - #H.edgeFinset

Tactic:
by

Goal state after:
Module: Mathlib.Data.Finset.CastCard

Goal state before:
pp: α : Type u_1
R : Type u_2
s : Finset α
a : α
inst✝¹ : DecidableEq α
inst✝ : AddGroupWithOne R
hs : a ∈ s
⊢ ↑(#(s.erase a)) = ↑(#s) - 1

Tactic:
by

Goal state after:
Module: Mathlib.Data.Finset.CastCard

Goal state before:
pp: α : Type u_1
R : Type u_2
s t : Finset α
inst✝¹ : DecidableEq α
inst✝ : AddGroupWithOne R
⊢ ↑(#(s ∩ t)) = ↑(#s) + ↑(#t) - ↑(#(s ∪ t))

Tactic:
by

Goal state after:
Module: Mathlib.Data.Finset.CastCard

Goal state before:
pp: α : Type u_1
R : Type u_2
s t : Finset α
inst✝¹ : DecidableEq α
inst✝ : AddGroupWithOne R
⊢ ↑(#(s ∪ t)) = ↑(#s) + ↑(#t) - ↑(#(s ∩ t))

Tactic:
by

Goal state after:
Module: Mathlib.Data.Finset.CastCard

Goal state before:
pp: α : Type u_1
R : Type u_2
s t : Finset α
inst✝¹ : DecidableEq α
inst✝ : AddGroupWithOne R
h : s ⊆ t
⊢ ↑(#(t \ s)) = ↑(#t) - ↑(#s)

Tactic:
by

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

Goal state before:
pp: ε : ℝ
hε : 0 < ε
⊢ 0 < 1 - min 1 ε / 4

Tactic:
by

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

Goal state before:
pp: ε : ℝ
hε : 0 < ε
⊢ 0 < triangleRemovalBound ε

Tactic:
by

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

Goal state before:
pp: ε : ℝ
hε : ε ≤ 0
⊢ triangleRemovalBound ε ≤ 0

Tactic:
by

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

Goal state before:
pp: ε : ℝ
hε : 0 < ε
⊢ triangleRemovalBound ε * ↑⌈4 / ε⌉₊ ^ 3 ≤ (2 * ↑⌈4 / ε⌉₊ ^ 3)⁻¹ * ↑⌈4 / ε⌉₊ ^ 3

Tactic:
by

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

Goal state before:
pp: ε : ℝ
hε : 0 < ε
⊢ (2 * ↑⌈4 / ε⌉₊ ^ 3)⁻¹ * ↑⌈4 / ε⌉₊ ^ 3 = 2⁻¹

Tactic:
by

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

Goal state before:
pp: ε : ℝ
hε : 0 < ε
⊢ 2⁻¹ < 1

Tactic:
by

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

Goal state before:
pp: ε : ℝ
hε : 0 < ε
⊢ triangleRemovalBound ε * ↑⌈4 / ε⌉₊ ^ 3 < 1

Tactic:
by

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

Goal state before:
pp: ε : ℝ
hε₁ : ε ≤ 1
⊢ triangleRemovalBound ε ≤ (1 - ε / 4) * (ε / (16 * ↑(bound (ε / 8) ⌈4 / ε⌉₊))) ^ 3

Tactic:
by

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

Goal state before:
pp: α : Type u_1
G : SimpleGraph α
inst✝¹ : DecidableRel G.Adj
ε : ℝ
s t u : Finset α
inst✝ : DecidableEq α
dst : 2 * ε ≤ ↑(G.edgeDensity s t)
dsu : 2 * ε ≤ ↑(G.edgeDensity s u)
dtu : 2 * ε ≤ ↑(G.edgeDensity t u)
utu : G.IsUniform ε t u
x : α
hxY : ↑(#t) * (↑(G.edgeDensity s t) - ε) ≤ ↑(#({y ∈ t | G.Adj x y}))
hsu : ↑(#u) * (↑(G.edgeDensity s u) - ε) ≤ ↑(#({y ∈ u | G.Adj x y}))
⊢ ↑(#t) * ε ≤ ↑(#({y ∈ t | G.Adj x y}))

Tactic:
by

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

Goal state before:
pp: α : Type u_1
𝕜 : Type u_3
inst✝⁵ : Field 𝕜
inst✝⁴ : LinearOrder 𝕜
inst✝³ : IsStrictOrderedRing 𝕜
G : SimpleGraph α
ε : 𝕜
inst✝² : Fintype α
inst✝¹ : DecidableRel G.Adj
inst✝ : DecidableEq α
tris : Finset (Finset α)
htris : tris ⊆ G.cliqueFinset 3
pd : (↑tris).Pairwise fun x y ↦ (↑x ∩ ↑y).Subsingleton
tris_big : ε * ↑(Fintype.card α ^ 2) ≤ ↑(#tris)
⊢ G.FarFromTriangleFree ε

Tactic:
by

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

Goal state before:
pp: 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 :
  ↑(G.edgeDensity (A.biUnion id) (B.biUnion id)) - (∑ ab ∈ A.product B, ↑(G.edgeDensity ab.1 ab.2)) / (↑(#A) * ↑(#B)) ≤
    ε ^ 5 / 50
⊢ ↑(G.edgeDensity (A.biUnion id) (B.biUnion id)) - (∑ ab ∈ A.product B, ↑(G.edgeDensity ab.1 ab.2)) / (↑(#A) * ↑(#B)) ≤
    ε ^ 5 / 49

Tactic:
by

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

Goal state before:
pp: α : Type u_1
𝕜 : Type u_3
inst✝⁵ : Field 𝕜
inst✝⁴ : LinearOrder 𝕜
inst✝³ : IsStrictOrderedRing 𝕜
G : SimpleGraph α
ε : 𝕜
inst✝² : Fintype α
inst✝¹ : DecidableRel G.Adj
inst✝ : DecidableEq α
hG : G.EdgeDisjointTriangles
tris_big : ε * ↑(Fintype.card α ^ 2) ≤ ↑(#(G.cliqueFinset 3))
⊢ (↑(G.cliqueFinset 3)).Pairwise fun x y ↦ (↑x ∩ ↑y).Subsingleton

Tactic:
by

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

Goal state before:
pp: α : Type u_1
𝕜 : Type u_3
inst✝⁵ : Field 𝕜
inst✝⁴ : LinearOrder 𝕜
inst✝³ : IsStrictOrderedRing 𝕜
G : SimpleGraph α
ε : 𝕜
inst✝² : Fintype α
inst✝¹ : DecidableRel G.Adj
inst✝ : Nonempty α
hε : G.FarFromTriangleFree ε
⊢ 0 ≤ ↑(Fintype.card α) ^ 2

Tactic:
by

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

Goal state before:
pp: n k : ℕ
hk : 0 < k
hn : k ≤ n
⊢ n < 2 * k * (n / k)

Tactic:
by

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

Goal state before:
pp: α : Type u_1
𝕜 : Type u_3
inst✝⁵ : Field 𝕜
inst✝⁴ : LinearOrder 𝕜
inst✝³ : IsStrictOrderedRing 𝕜
G : SimpleGraph α
ε : 𝕜
inst✝² : Fintype α
inst✝¹ : DecidableRel G.Adj
inst✝ : Nonempty α
hε : G.FarFromTriangleFree ε
⊢ ⊥.CliqueFree 3

Tactic:
by

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

Goal state before:
pp: α : Type u_1
𝕜 : Type u_3
inst✝⁵ : Field 𝕜
inst✝⁴ : LinearOrder 𝕜
inst✝³ : IsStrictOrderedRing 𝕜
G : SimpleGraph α
ε : 𝕜
inst✝² : Fintype α
inst✝¹ : DecidableRel G.Adj
inst✝ : Nonempty α
hε : G.FarFromTriangleFree ε
⊢ ε * ↑(Fintype.card α) ^ 2 ≤ ↑(#G.edgeFinset)

Tactic:
by

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

Goal state before:
pp: α : Type u_1
𝕜 : Type u_3
inst✝⁵ : Field 𝕜
inst✝⁴ : LinearOrder 𝕜
inst✝³ : IsStrictOrderedRing 𝕜
G : SimpleGraph α
ε : 𝕜
inst✝² : Fintype α
inst✝¹ : DecidableRel G.Adj
inst✝ : Nonempty α
hε : G.FarFromTriangleFree ε
⊢ ↑(#G.edgeFinset) ≤ ↑((Fintype.card α).choose 2)

Tactic:
by

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

Goal state before:
pp: α : Type u_1
G : SimpleGraph α
inst✝¹ : DecidableRel G.Adj
ε : ℝ
s t u : Finset α
inst✝ : DecidableEq α
dst : 2 * ε ≤ ↑(G.edgeDensity s t)
dsu : 2 * ε ≤ ↑(G.edgeDensity s u)
dtu : 2 * ε ≤ ↑(G.edgeDensity t u)
utu : G.IsUniform ε t u
x : α
hxY : ↑(#t) * (↑(G.edgeDensity s t) - ε) ≤ ↑(#({y ∈ t | G.Adj x y}))
hsu : ↑(#u) * (↑(G.edgeDensity s u) - ε) ≤ ↑(#({y ∈ u | G.Adj x y}))
hY : ↑(#t) * ε ≤ ↑(#({y ∈ t | G.Adj x y}))
⊢ ↑(#u) * ε ≤ ↑(#({y ∈ u | G.Adj x y}))

Tactic:
by

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

Goal state before:
pp: α : Type u_1
𝕜 : Type u_3
inst✝⁵ : Field 𝕜
inst✝⁴ : LinearOrder 𝕜
inst✝³ : IsStrictOrderedRing 𝕜
G : SimpleGraph α
ε : 𝕜
inst✝² : Fintype α
inst✝¹ : DecidableRel G.Adj
inst✝ : Nonempty α
hε : G.FarFromTriangleFree ε
⊢ Fintype.card α ≠ 0

Tactic:
by

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

Goal state before:
pp: α : Type u_1
𝕜 : Type u_3
inst✝⁵ : Field 𝕜
inst✝⁴ : LinearOrder 𝕜
inst✝³ : IsStrictOrderedRing 𝕜
G : SimpleGraph α
ε : 𝕜
inst✝² : Fintype α
inst✝¹ : DecidableRel G.Adj
inst✝ : Nonempty α
hε : G.FarFromTriangleFree ε
⊢ ↑((Fintype.card α).choose 2) < 2⁻¹ * ↑(Fintype.card α) ^ 2

Tactic:
by

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

Goal state before:
pp: α : Type u_1
𝕜 : Type u_3
inst✝⁵ : Field 𝕜
inst✝⁴ : LinearOrder 𝕜
inst✝³ : IsStrictOrderedRing 𝕜
G : SimpleGraph α
ε : 𝕜
inst✝² : Fintype α
inst✝¹ : DecidableRel G.Adj
inst✝ : Nonempty α
hε : G.FarFromTriangleFree ε
⊢ ε < 2⁻¹

Tactic:
by

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

Goal state before:
pp: α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
s : Finset α
P : Finpartition univ
ε : ℝ
hP₁ : P.IsEquipartition
hP₃ : #P.parts ≤ bound (ε / 8) ⌈4 / ε⌉₊
hX : s ∈ P.parts
h✝ : Nonempty α
this : 0 < #P.parts
⊢ ↑(Fintype.card α) / (2 * ↑(bound (ε / 8) ⌈4 / ε⌉₊)) ≤ ↑(Fintype.card α) / (2 * ↑(#P.parts))

Tactic:
by

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

Goal state before:
pp: α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
s : Finset α
P : Finpartition univ
ε : ℝ
hP₁ : P.IsEquipartition
hP₃ : #P.parts ≤ bound (ε / 8) ⌈4 / ε⌉₊
hX : s ∈ P.parts
h✝ : Nonempty α
this : 0 < #P.parts
⊢ 0 < 2 * ↑(#P.parts)

Tactic:
by

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

Goal state before:
pp: α : Type u_1
𝕜 : Type u_3
inst✝⁵ : Field 𝕜
inst✝⁴ : LinearOrder 𝕜
inst✝³ : IsStrictOrderedRing 𝕜
G : SimpleGraph α
ε : 𝕜
inst✝² : Fintype α
inst✝¹ : DecidableRel G.Adj
inst✝ : Nonempty α
h₀ : G.FarFromTriangleFree ε
h₁ : G.CliqueFree 3
⊢ ε ≤ 0

Tactic:
by

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

Goal state before:
pp: α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
s : Finset α
P : Finpartition univ
ε : ℝ
hP₁ : P.IsEquipartition
hP₃ : #P.parts ≤ bound (ε / 8) ⌈4 / ε⌉₊
hX : s ∈ P.parts
⊢ ↑(Fintype.card α) / (2 * ↑(bound (ε / 8) ⌈4 / ε⌉₊)) ≤ ↑(#s)

Tactic:
by

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

Goal state before:
pp: 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
⊢ |(∑ ab ∈ A.product B, ↑(G.edgeDensity ab.1 ab.2)) / (↑(#A) * ↑(#B)) -
        ↑(G.edgeDensity (A.biUnion id) (B.biUnion id))| ≤
    ε ^ 5 / 49

Tactic:
by

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

Goal state before:
pp: α : Type u_1
β : Type u_2
γ : Type u_3
𝕜 : Type u_4
inst✝² : Field 𝕜
inst✝¹ : LinearOrder 𝕜
inst✝ : IsStrictOrderedRing 𝕜
t : Finset (α × β × γ)
x y : α ⊕ β ⊕ γ
h : Rel t x y
⊢ Rel t y x

Tactic:
by

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

Goal state before:
pp: α : Type u_1
G : SimpleGraph α
inst✝¹ : DecidableRel G.Adj
ε : ℝ
s t u : Finset α
inst✝ : DecidableEq α
dst : 2 * ε ≤ ↑(G.edgeDensity s t)
dsu : 2 * ε ≤ ↑(G.edgeDensity s u)
dtu : 2 * ε ≤ ↑(G.edgeDensity t u)
utu : G.IsUniform ε t u
x : α
hxY : ↑(#t) * (↑(G.edgeDensity s t) - ε) ≤ ↑(#({y ∈ t | G.Adj x y}))
hsu : ↑(#u) * (↑(G.edgeDensity s u) - ε) ≤ ↑(#({y ∈ u | G.Adj x y}))
hY : ↑(#t) * ε ≤ ↑(#({y ∈ t | G.Adj x y}))
hZ : ↑(#u) * ε ≤ ↑(#({y ∈ u | G.Adj x y}))
this : |↑(G.edgeDensity (filter (G.Adj x) t) (filter (G.Adj x) u)) - ↑(G.edgeDensity t u)| < ε
⊢ ε ≤ ↑(G.edgeDensity ({y ∈ t | G.Adj x y}) ({y ∈ u | G.Adj x y}))

Tactic:
by

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

Goal state before:
pp: α : Type u_1
β : Type u_2
γ : Type u_3
t : Finset (α × β × γ)
a : α
b : β
⊢ (graph t).Adj (in₀ a) (in₁ b) → ∃ c, (a, b, c) ∈ t

Tactic:
by

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

Goal state before:
pp: α : Type u_1
β : Type u_2
γ : Type u_3
t : Finset (α × β × γ)
a : α
b : β
⊢ (graph t).Adj (in₁ b) (in₀ a) → ∃ c, (a, b, c) ∈ t

Tactic:
by

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

Goal state before:
pp: α : Type u_1
β : Type u_2
γ : Type u_3
t : Finset (α × β × γ)
a : α
c : γ
⊢ (graph t).Adj (in₀ a) (in₂ c) → ∃ b, (a, b, c) ∈ t

Tactic:
by

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

Goal state before:
pp: α : Type u_1
β : Type u_2
γ : Type u_3
t : Finset (α × β × γ)
a : α
c : γ
⊢ (graph t).Adj (in₂ c) (in₀ a) → ∃ b, (a, b, c) ∈ t

Tactic:
by

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

Goal state before:
pp: α : Type u_1
β : Type u_2
γ : Type u_3
t : Finset (α × β × γ)
b : β
c : γ
⊢ (graph t).Adj (in₁ b) (in₂ c) → ∃ a, (a, b, c) ∈ t

Tactic:
by

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

Goal state before:
pp: α : Type u_1
β : Type u_2
γ : Type u_3
t : Finset (α × β × γ)
b : β
c : γ
⊢ (graph t).Adj (in₂ c) (in₁ b) → ∃ a, (a, b, c) ∈ t

Tactic:
by

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

Goal state before:
pp: α : Type u_1
β : Type u_2
γ : Type u_3
t : Finset (α × β × γ)
a : α
b : β
c✝ : γ
a✝ : (a, b, c✝) ∈ t
⊢ (a, b, c✝).1 = a ∧ (a, b, c✝).2.1 = b

Tactic:
by

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

Goal state before:
pp: α : Type u_1
β : Type u_2
γ : Type u_3
t : Finset (α × β × γ)
a : α
b : β
⊢ (graph t).Adj (in₀ a) (in₁ b) → ∃ x ∈ t, x.1 = a ∧ x.2.1 = b

Tactic:
by

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

Goal state before:
pp: α : Type u_1
β : Type u_2
γ : Type u_3
t : Finset (α × β × γ)
a : α
b : β
⊢ (∃ x ∈ t, x.1 = a ∧ x.2.1 = b) → (graph t).Adj (in₀ a) (in₁ b)

Tactic:
by

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

Goal state before:
pp: α : Type u_1
β : Type u_2
γ : Type u_3
t : Finset (α × β × γ)
a : α
b : β
c✝ : γ
a✝ : (a, b, c✝) ∈ t
⊢ (a, b, c✝).2.1 = b ∧ (a, b, c✝).1 = a

Tactic:
by

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

Goal state before:
pp: α : Type u_1
β : Type u_2
γ : Type u_3
t : Finset (α × β × γ)
a : α
b : β
⊢ (graph t).Adj (in₁ b) (in₀ a) → ∃ x ∈ t, x.2.1 = b ∧ x.1 = a

Tactic:
by

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

Goal state before:
pp: α : Type u_1
β : Type u_2
γ : Type u_3
t : Finset (α × β × γ)
a : α
b : β
⊢ (∃ x ∈ t, x.2.1 = b ∧ x.1 = a) → (graph t).Adj (in₁ b) (in₀ a)

Tactic:
by

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

Goal state before:
pp: α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : Fintype α
G : SimpleGraph α
inst✝ : DecidableRel G.Adj
P : Finpartition univ
ε : ℝ
hε : 0 < ε
hε₁ : ε ≤ 1
hP₁ : P.IsEquipartition
hP₃ : #P.parts ≤ bound (ε / 8) ⌈4 / ε⌉₊
x y z : α
s : Finset α
hX : s ∈ P.parts
Y : Finset α
hY : Y ∈ P.parts
xX : x ∈ s
yY : y ∈ Y
nXY : s ≠ Y
uXY : G.IsUniform (ε / 8) s Y
dXY✝ : ε / 4 ≤ ↑(G.edgeDensity s Y)
Z : Finset α
hZ : Z ∈ P.parts
zZ : z ∈ Z
hX' : s ∈ P.parts
xX' : x ∈ s
nXZ : s ≠ Z
uXZ : G.IsUniform (ε / 8) s Z
dXZ✝ : ε / 4 ≤ ↑(G.edgeDensity s Z)
hY' : Y ∈ P.parts
yY' : y ∈ Y
hZ' : Z ∈ P.parts
zZ' : z ∈ Z
nYZ : Y ≠ Z
uYZ : G.IsUniform (ε / 8) Y Z
dYZ✝ : ε / 4 ≤ ↑(G.edgeDensity Y Z)
dXY : Function.onFun Disjoint id s Y
dXZ : Function.onFun Disjoint id s Z
dYZ : Function.onFun Disjoint id Y Z
⊢ 2 * (ε / 8) = ε / 4

Tactic:
by

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

Goal state before:
pp: α : Type u_1
β : Type u_2
γ : Type u_3
t : Finset (α × β × γ)
a : α
c : γ
b✝ : β
a✝ : (a, b✝, c) ∈ t
⊢ (a, b✝, c).1 = a ∧ (a, b✝, c).2.2 = c

Tactic:
by

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

Goal state before:
pp: α : Type u_1
β : Type u_2
γ : Type u_3
t : Finset (α × β × γ)
a : α
c : γ
⊢ (graph t).Adj (in₀ a) (in₂ c) → ∃ x ∈ t, x.1 = a ∧ x.2.2 = c

Tactic:
by

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

Goal state before:
pp: α : Type u_1
inst✝³ : Fintype α
inst✝² : DecidableEq α
P : Finpartition univ
G : SimpleGraph α
inst✝¹ : DecidableRel G.Adj
ε : ℝ
U V : Finset α
inst✝ : Nonempty α
hP : P.IsEquipartition
hPα : #P.parts * 16 ^ #P.parts ≤ Fintype.card α
hPε : 100 ≤ 4 ^ #P.parts * ε ^ 5
hU : U ∈ P.parts
hV : V ∈ P.parts
hGε : ↑(G.edgeDensity U V) ≤ ε ^ 5 / 50
⊢ 0 ≤ ε ^ 5

Tactic:
by

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

Goal state before:
pp: α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : Fintype α
G : SimpleGraph α
inst✝ : DecidableRel G.Adj
P : Finpartition univ
ε : ℝ
hε : 0 < ε
hε₁ : ε ≤ 1
hP₁ : P.IsEquipartition
hP₃ : #P.parts ≤ bound (ε / 8) ⌈4 / ε⌉₊
x y z : α
s : Finset α
hX : s ∈ P.parts
Y : Finset α
hY : Y ∈ P.parts
xX : x ∈ s
yY : y ∈ Y
nXY : s ≠ Y
uXY : G.IsUniform (ε / 8) s Y
dXY✝ : ε / 4 ≤ ↑(G.edgeDensity s Y)
Z : Finset α
hZ : Z ∈ P.parts
zZ : z ∈ Z
hX' : s ∈ P.parts
xX' : x ∈ s
nXZ : s ≠ Z
uXZ : G.IsUniform (ε / 8) s Z
dXZ✝ : ε / 4 ≤ ↑(G.edgeDensity s Z)
hY' : Y ∈ P.parts
yY' : y ∈ Y
hZ' : Z ∈ P.parts
zZ' : z ∈ Z
nYZ : Y ≠ Z
uYZ : G.IsUniform (ε / 8) Y Z
dYZ✝ : ε / 4 ≤ ↑(G.edgeDensity Y Z)
dXY : Function.onFun Disjoint id s Y
dXZ : Function.onFun Disjoint id s Z
dYZ : Function.onFun Disjoint id Y Z
that : 2 * (ε / 8) = ε / 4
⊢ ↑(G.edgeDensity Y Z) ≤ 1

Tactic:
by

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

Goal state before:
pp: α : Type u_1
inst✝³ : Fintype α
inst✝² : DecidableEq α
P : Finpartition univ
G : SimpleGraph α
inst✝¹ : DecidableRel G.Adj
ε : ℝ
U V : Finset α
inst✝ : Nonempty α
hP : P.IsEquipartition
hPα : #P.parts * 16 ^ #P.parts ≤ Fintype.card α
hPε : 100 ≤ 4 ^ #P.parts * ε ^ 5
hU : U ∈ P.parts
hV : V ∈ P.parts
hGε : ↑(G.edgeDensity U V) ≤ ε ^ 5 / 50
⊢ 0 < 25

Tactic:
by

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

Goal state before:
pp: α : Type u_1
β : Type u_2
γ : Type u_3
t : Finset (α × β × γ)
a : α
c : γ
⊢ (∃ x ∈ t, x.1 = a ∧ x.2.2 = c) → (graph t).Adj (in₀ a) (in₂ c)

Tactic:
by

Goal state after: