Module: Mathlib.Combinatorics.SimpleGraph.Triangle.Counting

Goal state before:
pp: case refine_2
α : Type u_1
G : SimpleGraph α
inst✝² : DecidableRel G.Adj
ε : ℝ
s t✝ u : Finset α
inst✝¹ : DecidableEq α
inst✝ : Fintype α
dst : 2 * ε ≤ ↑(G.edgeDensity s t✝)
ust : G.IsUniform ε s t✝
hst : Disjoint s t✝
dsu : 2 * ε ≤ ↑(G.edgeDensity s u)
usu : G.IsUniform ε s u
hsu : Disjoint s u
dtu : 2 * ε ≤ ↑(G.edgeDensity t✝ u)
utu : G.IsUniform ε t✝ u
htu : Disjoint t✝ u
x₁ y₁ z₁ x₂ y₂ z₂ : α
t :
  (fun x ↦
        match x with
        | (x, y, z) => {x, y, z})
      (x₁, y₁, z₁) =
    (fun x ↦
        match x with
        | (x, y, z) => {x, y, z})
      (x₂, y₂, z₂)
h₁ : (x₁ ∈ s ∧ y₁ ∈ t✝ ∧ z₁ ∈ u) ∧ G.Adj x₁ y₁ ∧ G.Adj x₁ z₁ ∧ G.Adj y₁ z₁
h₂ : (x₂ ∈ s ∧ y₂ ∈ t✝ ∧ z₂ ∈ u) ∧ G.Adj x₂ y₂ ∧ G.Adj x₂ z₂ ∧ G.Adj y₂ z₂
⊢ (x₁, y₁, z₁) = (x₂, y₂, z₂)

Tactic:
apply triple_eq_triple_of_mem hst hsu htu t

Goal state after:
case refine_2.hx₁
α : Type u_1
G : SimpleGraph α
inst✝² : DecidableRel G.Adj
ε : ℝ
s t✝ u : Finset α
inst✝¹ : DecidableEq α
inst✝ : Fintype α
dst : 2 * ε ≤ ↑(G.edgeDensity s t✝)
ust : G.IsUniform ε s t✝
hst : Disjoint s t✝
dsu : 2 * ε ≤ ↑(G.edgeDensity s u)
usu : G.IsUniform ε s u
hsu : Disjoint s u
dtu : 2 * ε ≤ ↑(G.edgeDensity t✝ u)
utu : G.IsUniform ε t✝ u
htu : Disjoint t✝ u
x₁ y₁ z₁ x₂ y₂ z₂ : α
t :
  (fun x ↦
        match x with
        | (x, y, z) => {x, y, z})
      (x₁, y₁, z₁) =
    (fun x ↦
        match x with
        | (x, y, z) => {x, y, z})
      (x₂, y₂, z₂)
h₁ : (x₁ ∈ s ∧ y₁ ∈ t✝ ∧ z₁ ∈ u) ∧ G.Adj x₁ y₁ ∧ G.Adj x₁ z₁ ∧ G.Adj y₁ z₁
h₂ : (x₂ ∈ s ∧ y₂ ∈ t✝ ∧ z₂ ∈ u) ∧ G.Adj x₂ y₂ ∧ G.Adj x₂ z₂ ∧ G.Adj y₂ z₂
⊢ x₁ ∈ s

case refine_2.hx₂
α : Type u_1
G : SimpleGraph α
inst✝² : DecidableRel G.Adj
ε : ℝ
s t✝ u : Finset α
inst✝¹ : DecidableEq α
inst✝ : Fintype α
dst : 2 * ε ≤ ↑(G.edgeDensity s t✝)
ust : G.IsUniform ε s t✝
hst : Disjoint s t✝
dsu : 2 * ε ≤ ↑(G.edgeDensity s u)
usu : G.IsUniform ε s u
hsu : Disjoint s u
dtu : 2 * ε ≤ ↑(G.edgeDensity t✝ u)
utu : G.IsUniform ε t✝ u
htu : Disjoint t✝ u
x₁ y₁ z₁ x₂ y₂ z₂ : α
t :
  (fun x ↦
        match x with
        | (x, y, z) => {x, y, z})
      (x₁, y₁, z₁) =
    (fun x ↦
        match x with
        | (x, y, z) => {x, y, z})
      (x₂, y₂, z₂)
h₁ : (x₁ ∈ s ∧ y₁ ∈ t✝ ∧ z₁ ∈ u) ∧ G.Adj x₁ y₁ ∧ G.Adj x₁ z₁ ∧ G.Adj y₁ z₁
h₂ : (x₂ ∈ s ∧ y₂ ∈ t✝ ∧ z₂ ∈ u) ∧ G.Adj x₂ y₂ ∧ G.Adj x₂ z₂ ∧ G.Adj y₂ z₂
⊢ x₂ ∈ s

case refine_2.hy₁
α : Type u_1
G : SimpleGraph α
inst✝² : DecidableRel G.Adj
ε : ℝ
s t✝ u : Finset α
inst✝¹ : DecidableEq α
inst✝ : Fintype α
dst : 2 * ε ≤ ↑(G.edgeDensity s t✝)
ust : G.IsUniform ε s t✝
hst : Disjoint s t✝
dsu : 2 * ε ≤ ↑(G.edgeDensity s u)
usu : G.IsUniform ε s u
hsu : Disjoint s u
dtu : 2 * ε ≤ ↑(G.edgeDensity t✝ u)
utu : G.IsUniform ε t✝ u
htu : Disjoint t✝ u
x₁ y₁ z₁ x₂ y₂ z₂ : α
t :
  (fun x ↦
        match x with
        | (x, y, z) => {x, y, z})
      (x₁, y₁, z₁) =
    (fun x ↦
        match x with
        | (x, y, z) => {x, y, z})
      (x₂, y₂, z₂)
h₁ : (x₁ ∈ s ∧ y₁ ∈ t✝ ∧ z₁ ∈ u) ∧ G.Adj x₁ y₁ ∧ G.Adj x₁ z₁ ∧ G.Adj y₁ z₁
h₂ : (x₂ ∈ s ∧ y₂ ∈ t✝ ∧ z₂ ∈ u) ∧ G.Adj x₂ y₂ ∧ G.Adj x₂ z₂ ∧ G.Adj y₂ z₂
⊢ y₁ ∈ t✝

case refine_2.hy₂
α : Type u_1
G : SimpleGraph α
inst✝² : DecidableRel G.Adj
ε : ℝ
s t✝ u : Finset α
inst✝¹ : DecidableEq α
inst✝ : Fintype α
dst : 2 * ε ≤ ↑(G.edgeDensity s t✝)
ust : G.IsUniform ε s t✝
hst : Disjoint s t✝
dsu : 2 * ε ≤ ↑(G.edgeDensity s u)
usu : G.IsUniform ε s u
hsu : Disjoint s u
dtu : 2 * ε ≤ ↑(G.edgeDensity t✝ u)
utu : G.IsUniform ε t✝ u
htu : Disjoint t✝ u
x₁ y₁ z₁ x₂ y₂ z₂ : α
t :
  (fun x ↦
        match x with
        | (x, y, z) => {x, y, z})
      (x₁, y₁, z₁) =
    (fun x ↦
        match x with
        | (x, y, z) => {x, y, z})
      (x₂, y₂, z₂)
h₁ : (x₁ ∈ s ∧ y₁ ∈ t✝ ∧ z₁ ∈ u) ∧ G.Adj x₁ y₁ ∧ G.Adj x₁ z₁ ∧ G.Adj y₁ z₁
h₂ : (x₂ ∈ s ∧ y₂ ∈ t✝ ∧ z₂ ∈ u) ∧ G.Adj x₂ y₂ ∧ G.Adj x₂ z₂ ∧ G.Adj y₂ z₂
⊢ y₂ ∈ t✝

case refine_2.hz₁
α : Type u_1
G : SimpleGraph α
inst✝² : DecidableRel G.Adj
ε : ℝ
s t✝ u : Finset α
inst✝¹ : DecidableEq α
inst✝ : Fintype α
dst : 2 * ε ≤ ↑(G.edgeDensity s t✝)
ust : G.IsUniform ε s t✝
hst : Disjoint s t✝
dsu : 2 * ε ≤ ↑(G.edgeDensity s u)
usu : G.IsUniform ε s u
hsu : Disjoint s u
dtu : 2 * ε ≤ ↑(G.edgeDensity t✝ u)
utu : G.IsUniform ε t✝ u
htu : Disjoint t✝ u
x₁ y₁ z₁ x₂ y₂ z₂ : α
t :
  (fun x ↦
        match x with
        | (x, y, z) => {x, y, z})
      (x₁, y₁, z₁) =
    (fun x ↦
        match x with
        | (x, y, z) => {x, y, z})
      (x₂, y₂, z₂)
h₁ : (x₁ ∈ s ∧ y₁ ∈ t✝ ∧ z₁ ∈ u) ∧ G.Adj x₁ y₁ ∧ G.Adj x₁ z₁ ∧ G.Adj y₁ z₁
h₂ : (x₂ ∈ s ∧ y₂ ∈ t✝ ∧ z₂ ∈ u) ∧ G.Adj x₂ y₂ ∧ G.Adj x₂ z₂ ∧ G.Adj y₂ z₂
⊢ z₁ ∈ u

case refine_2.hz₂
α : Type u_1
G : SimpleGraph α
inst✝² : DecidableRel G.Adj
ε : ℝ
s t✝ u : Finset α
inst✝¹ : DecidableEq α
inst✝ : Fintype α
dst : 2 * ε ≤ ↑(G.edgeDensity s t✝)
ust : G.IsUniform ε s t✝
hst : Disjoint s t✝
dsu : 2 * ε ≤ ↑(G.edgeDensity s u)
usu : G.IsUniform ε s u
hsu : Disjoint s u
dtu : 2 * ε ≤ ↑(G.edgeDensity t✝ u)
utu : G.IsUniform ε t✝ u
htu : Disjoint t✝ u
x₁ y₁ z₁ x₂ y₂ z₂ : α
t :
  (fun x ↦
        match x with
        | (x, y, z) => {x, y, z})
      (x₁, y₁, z₁) =
    (fun x ↦
        match x with
        | (x, y, z) => {x, y, z})
      (x₂, y₂, z₂)
h₁ : (x₁ ∈ s ∧ y₁ ∈ t✝ ∧ z₁ ∈ u) ∧ G.Adj x₁ y₁ ∧ G.Adj x₁ z₁ ∧ G.Adj y₁ z₁
h₂ : (x₂ ∈ s ∧ y₂ ∈ t✝ ∧ z₂ ∈ u) ∧ G.Adj x₂ y₂ ∧ G.Adj x₂ z₂ ∧ G.Adj y₂ z₂
⊢ z₂ ∈ u