l3lab/ntp-mathlib huggingface dataset:
/-- 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
  

Proof state:
α : Type u_1
inst✝ : Fintype α
G : SimpleGraph α
ε : ℝ
l : ℕ
hε : 0 < ε
hl : l ≤ Fintype.card α
⊢ ∃ P, IsEquipartition P ∧ l ≤ Finset.card P.parts ∧ Finset.card P.parts ≤ bound ε l ∧ IsUniform P G ε

Next tactic:
obtain hα | hα := le_total (card α) (bound ε l)
/-- 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.
  · 

Proof state:
case inl
α : Type u_1
inst✝ : Fintype α
G : SimpleGraph α
ε : ℝ
l : ℕ
hε : 0 < ε
hl : l ≤ Fintype.card α
hα : Fintype.card α ≤ bound ε l
⊢ ∃ P, IsEquipartition P ∧ l ≤ Finset.card P.parts ∧ Finset.card P.parts ≤ bound ε l ∧ IsUniform P G ε

Next tactic:
refine' ⟨⊥, bot_isEquipartition _, _⟩
/-- 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 _, _⟩
    

Proof state:
case inl
α : Type u_1
inst✝ : Fintype α
G : SimpleGraph α
ε : ℝ
l : ℕ
hε : 0 < ε
hl : l ≤ Fintype.card α
hα : Fintype.card α ≤ bound ε l
⊢ l ≤ Finset.card ⊥.parts ∧ Finset.card ⊥.parts ≤ bound ε l ∧ IsUniform ⊥ G ε

Next tactic:
rw [card_bot, card_univ]
/-- 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]
    

Proof state:
case inl
α : Type u_1
inst✝ : Fintype α
G : SimpleGraph α
ε : ℝ
l : ℕ
hε : 0 < ε
hl : l ≤ Fintype.card α
hα : Fintype.card α ≤ bound ε l
⊢ l ≤ Fintype.card α ∧ Fintype.card α ≤ bound ε l ∧ IsUniform ⊥ G ε

Next tactic:
exact ⟨hl, hα, botIsUniform _ hε⟩
/-- 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`.
  

Proof state:
case inr
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
⊢ ∃ P, IsEquipartition P ∧ l ≤ Finset.card P.parts ∧ Finset.card P.parts ≤ bound ε l ∧ IsUniform P G ε

Next tactic:
let t := initialBound ε l
/-- 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
  

Proof state:
case inr
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
⊢ ∃ P, IsEquipartition P ∧ l ≤ Finset.card P.parts ∧ Finset.card P.parts ≤ bound ε l ∧ IsUniform P G ε

Next tactic:
have htα : t ≤ (univ : Finset α).card :=
    (initialBound_le_bound _ _).trans (by rwa [Finset.card_univ])
/-- 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 

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
⊢ bound ε l ≤ Finset.card univ

Next tactic:
rwa [Finset.card_univ]
/-- 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])
  

Proof state:
case inr
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
⊢ ∃ P, IsEquipartition P ∧ l ≤ Finset.card P.parts ∧ Finset.card P.parts ≤ bound ε l ∧ IsUniform P G ε

Next tactic:
obtain ⟨dum, hdum₁, hdum₂⟩ :=
    exists_equipartition_card_eq (univ : Finset α) (initialBound_pos _ _).ne' htα
/-- 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
  

Proof state:
case inr.intro.intro
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
⊢ ∃ P, IsEquipartition P ∧ l ≤ Finset.card P.parts ∧ Finset.card P.parts ≤ bound ε l ∧ IsUniform P G ε

Next tactic:
obtain hε₁ | hε₁ := le_total 1 ε
/-- 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.
  · 

Proof state:
case inr.intro.intro.inl
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 : 1 ≤ ε
⊢ ∃ P, IsEquipartition P ∧ l ≤ Finset.card P.parts ∧ Finset.card P.parts ≤ bound ε l ∧ IsUniform P G ε

Next tactic:
exact ⟨dum, hdum₁, (le_initialBound ε l).trans hdum₂.ge,
      hdum₂.le.trans (initialBound_le_bound ε l), (dum.isUniformOne G).mono hε₁⟩
/-- 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`.
  

Proof state:
case inr.intro.intro.inr
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
⊢ ∃ P, IsEquipartition P ∧ l ≤ Finset.card P.parts ∧ Finset.card P.parts ≤ bound ε l ∧ IsUniform P G ε

Next tactic:
have : Nonempty α := by
    rw [← Fintype.card_pos_iff]
    exact (bound_pos _ _).trans_le hα
/-- 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
    

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
⊢ Nonempty α

Next tactic:
rw [← Fintype.card_pos_iff]
/-- 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]
    

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
⊢ 0 < Fintype.card α

Next tactic:
exact (bound_pos _ _).trans_le hα
/-- 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
  

Proof state:
case inr.intro.intro.inr
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
⊢ ∃ P, IsEquipartition P ∧ l ≤ Finset.card P.parts ∧ Finset.card P.parts ≤ bound ε l ∧ IsUniform P G ε

Next tactic:
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)
/-- 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.
  · 

Proof state:
case inr.intro.intro.inr
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
h :
  ∀ (i : ℕ),
    ∃ P,
      IsEquipartition P ∧
        t ≤ Finset.card P.parts ∧
          Finset.card P.parts ≤ stepBound^[i] t ∧ (IsUniform P G ε ∨ ε ^ 5 / 4 * ↑i ≤ ↑(energy P G))
⊢ ∃ P, IsEquipartition P ∧ l ≤ Finset.card P.parts ∧ Finset.card P.parts ≤ bound ε l ∧ IsUniform P G ε

Next tactic:
obtain ⟨P, hP₁, hP₂, hP₃, hP₄⟩ := h (⌊4 / ε ^ 5⌋₊ + 1)
/-- 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)
    

Proof state:
case inr.intro.intro.inr.intro.intro.intro.intro
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
h :
  ∀ (i : ℕ),
    ∃ P,
      IsEquipartition P ∧
        t ≤ Finset.card P.parts ∧
          Finset.card P.parts ≤ stepBound^[i] t ∧ (IsUniform P G ε ∨ ε ^ 5 / 4 * ↑i ≤ ↑(energy P G))
P : Finpartition univ
hP₁ : IsEquipartition P
hP₂ : t ≤ Finset.card P.parts
hP₃ : Finset.card P.parts ≤ stepBound^[⌊4 / ε ^ 5⌋₊ + 1] t
hP₄ : IsUniform P G ε ∨ ε ^ 5 / 4 * ↑(⌊4 / ε ^ 5⌋₊ + 1) ≤ ↑(energy P G)
⊢ ∃ P, IsEquipartition P ∧ l ≤ Finset.card P.parts ∧ Finset.card P.parts ≤ bound ε l ∧ IsUniform P G ε

Next tactic:
refine' ⟨P, hP₁, (le_initialBound _ _).trans hP₂, hP₃.trans _,
      hP₄.resolve_right fun hPenergy => lt_irrefl (1 : ℝ) _⟩