/-- Effective **Szemerédi Regularity Lemma**: For any sufficiently large graph, there is an
`ε`-uniform equipartition of bounded size (where the bound does not depend on the graph). -/
theorem szemeredi_regularity (hε : 0 < ε) (hl : l ≤ card α) :
    ∃ P : Finpartition univ,
      P.IsEquipartition ∧ l ≤ P.parts.card ∧ P.parts.card ≤ bound ε l ∧ P.IsUniform G ε := by
  obtain hα | hα := le_total (card α) (bound ε l)
  -- If `card α ≤ bound ε l`, then the partition into singletons is acceptable.
  · refine' ⟨⊥, bot_isEquipartition _, _⟩
    rw [card_bot, card_univ]
    exact ⟨hl, hα, botIsUniform _ hε⟩
  -- Else, let's start from a dummy equipartition of size `initialBound ε l`.
  let t := initialBound ε l
  have htα : t ≤ (univ : Finset α).card :=
    (initialBound_le_bound _ _).trans (by rwa [Finset.card_univ])
  obtain ⟨dum, hdum₁, hdum₂⟩ :=
    exists_equipartition_card_eq (univ : Finset α) (initialBound_pos _ _).ne' htα
  obtain hε₁ | hε₁ := le_total 1 ε
  -- If `ε ≥ 1`, then this dummy equipartition is `ε`-uniform, so we're done.
  · exact ⟨dum, hdum₁, (le_initialBound ε l).trans hdum₂.ge,
      hdum₂.le.trans (initialBound_le_bound ε l), (dum.isUniformOne G).mono hε₁⟩
  -- Else, set up the induction on energy. We phrase it through the existence for each `i` of an
  -- equipartition of size bounded by `stepBound^[i] (initialBound ε l)` and which is either
  -- `ε`-uniform or has energy at least `ε ^ 5 / 4 * i`.
  have : Nonempty α := by
    rw [← Fintype.card_pos_iff]
    exact (bound_pos _ _).trans_le hα
  suffices h : ∀ i, ∃ P : Finpartition (univ : Finset α), P.IsEquipartition ∧ t ≤ P.parts.card ∧
    P.parts.card ≤ stepBound^[i] t ∧ (P.IsUniform G ε ∨ ε ^ 5 / 4 * i ≤ P.energy G)
  -- For `i > 4 / ε ^ 5` we know that the partition we get can't have energy `≥ ε ^ 5 / 4 * i > 1`,
  -- so it must instead be `ε`-uniform and we won.
  · obtain ⟨P, hP₁, hP₂, hP₃, hP₄⟩ := h (⌊4 / ε ^ 5⌋₊ + 1)
    refine' ⟨P, hP₁, (le_initialBound _ _).trans hP₂, hP₃.trans _,
      hP₄.resolve_right fun hPenergy => lt_irrefl (1 : ℝ) _⟩
    · rw [iterate_succ_apply']
      exact mul_le_mul_left' (pow_le_pow_left (by norm_num) (by norm_num) _) _
    calc
      (1 : ℝ) = ε ^ 5 / ↑4 * (↑4 / ε ^ 5) := by
        rw [mul_comm, div_mul_div_cancel 4 (pow_pos hε 5).ne']; norm_num
      _ < ε ^ 5 / 4 * (⌊4 / ε ^ 5⌋₊ + 1) :=
        ((mul_lt_mul_left <| by positivity).2 (Nat.lt_floor_add_one _))
      _ ≤ (P.energy G : ℝ) := by rwa [← Nat.cast_add_one]
      _ ≤ 1 := mod_cast P.energy_le_one G
  -- Let's do the actual induction.
  intro i
  induction' i with i ih
  -- For `i = 0`, the dummy equipartition is enough.
  · refine' ⟨dum, hdum₁, hdum₂.ge, hdum₂.le, Or.inr _⟩
    rw [Nat.cast_zero, mul_zero]
    exact mod_cast dum.energy_nonneg G
  -- For the induction step at `i + 1`, find `P` the equipartition at `i`.
  obtain ⟨P, hP₁, hP₂, hP₃, hP₄⟩ := ih
  by_cases huniform : P.IsUniform G ε
  -- If `P` is already uniform, then no need to break it up further. We can just return `P` again.
  · refine' ⟨P, hP₁, hP₂, _, Or.inl huniform⟩
    rw [iterate_succ_apply']
    exact hP₃.trans (le_stepBound _)
  -- Else, `P` must instead have energy at least `ε ^ 5 / 4 * i`.
  replace hP₄ := hP₄.resolve_left huniform
  -- We gather a few numerical facts.
  have hεl' : ↑100 < ↑4 ^ P.parts.card * ε ^ 5 :=
    (hundred_lt_pow_initialBound_mul hε l).trans_le
      (mul_le_mul_of_nonneg_right (pow_le_pow_right (by norm_num) hP₂) <| by 

Proof state:
α : Type u_1
inst✝ : Fintype α
G : SimpleGraph α
ε : ℝ
l : ℕ
hε : 0 < ε
hl : l ≤ Fintype.card α
hα : bound ε l ≤ Fintype.card α
t : ℕ := initialBound ε l
htα : t ≤ Finset.card univ
dum : Finpartition univ
hdum₁ : IsEquipartition dum
hdum₂ : Finset.card dum.parts = initialBound ε l
hε₁ : ε ≤ 1
this : Nonempty α
i : ℕ
P : Finpartition univ
hP₁ : IsEquipartition P
hP₂ : t ≤ Finset.card P.parts
hP₃ : Finset.card P.parts ≤ stepBound^[i] t
huniform : ¬IsUniform P G ε
hP₄ : ε ^ 5 / 4 * ↑i ≤ ↑(energy P G)
⊢ 0 ≤ ε ^ 5

Next tactic:
positivity