l3lab/ntp-mathlib huggingface dataset:
/-- **Ruzsa's covering lemma**. -/
@[to_additive "**Ruzsa's covering lemma**"]
theorem exists_subset_mul_div (ht : t.Nonempty) :
    ∃ u : Finset α, u.card * t.card ≤ (s * t).card ∧ s ⊆ u * t / t := by
  haveI : ∀ u, Decidable ((u : Set α).PairwiseDisjoint (· • t)) := fun u ↦ Classical.dec _
  set C := s.powerset.filter fun u ↦ u.toSet.PairwiseDisjoint (· • t)
  obtain ⟨u, hu, hCmax⟩ := C.exists_maximal (filter_nonempty_iff.2
    ⟨∅, empty_mem_powerset _, by rw [coe_empty]; exact Set.pairwiseDisjoint_empty⟩)
  rw [mem_filter, mem_powerset] at hu
  refine' ⟨u,
    (card_mul_iff.2 <| pairwiseDisjoint_smul_iff.1 hu.2).ge.trans
      (card_le_of_subset <| mul_subset_mul_right hu.1),
    fun a ha ↦ _⟩
  rw [mul_div_assoc]
  by_cases hau : a ∈ u
  · exact subset_mul_left _ ht.one_mem_div hau
  by_cases H : ∀ b ∈ u, Disjoint (a • t) (b • t)
  · refine' (hCmax _ _ <| ssubset_insert hau).elim
    

Proof state:
case pos
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : CommGroup α
s t : Finset α
ht : Finset.Nonempty t
this : (u : Set α) → Decidable (Set.PairwiseDisjoint u fun x => x • t)
C : Finset (Finset α) := filter (fun u => Set.PairwiseDisjoint ↑u fun x => x • t) (powerset s)
u : Finset α
hu : u ⊆ s ∧ Set.PairwiseDisjoint ↑u fun x => x • t
hCmax : ∀ x ∈ C, ¬u < x
a : α
ha : a ∈ s
hau : a ∉ u
H : ∀ b ∈ u, Disjoint (a • t) (b • t)
⊢ insert a u ∈ C

Next tactic:
rw [mem_filter, mem_powerset, insert_subset_iff, coe_insert]
/-- **Ruzsa's covering lemma**. -/
@[to_additive "**Ruzsa's covering lemma**"]
theorem exists_subset_mul_div (ht : t.Nonempty) :
    ∃ u : Finset α, u.card * t.card ≤ (s * t).card ∧ s ⊆ u * t / t := by
  haveI : ∀ u, Decidable ((u : Set α).PairwiseDisjoint (· • t)) := fun u ↦ Classical.dec _
  set C := s.powerset.filter fun u ↦ u.toSet.PairwiseDisjoint (· • t)
  obtain ⟨u, hu, hCmax⟩ := C.exists_maximal (filter_nonempty_iff.2
    ⟨∅, empty_mem_powerset _, by rw [coe_empty]; exact Set.pairwiseDisjoint_empty⟩)
  rw [mem_filter, mem_powerset] at hu
  refine' ⟨u,
    (card_mul_iff.2 <| pairwiseDisjoint_smul_iff.1 hu.2).ge.trans
      (card_le_of_subset <| mul_subset_mul_right hu.1),
    fun a ha ↦ _⟩
  rw [mul_div_assoc]
  by_cases hau : a ∈ u
  · exact subset_mul_left _ ht.one_mem_div hau
  by_cases H : ∀ b ∈ u, Disjoint (a • t) (b • t)
  · refine' (hCmax _ _ <| ssubset_insert hau).elim
    rw [mem_filter, mem_powerset, insert_subset_iff, coe_insert]
    

Proof state:
case pos
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : CommGroup α
s t : Finset α
ht : Finset.Nonempty t
this : (u : Set α) → Decidable (Set.PairwiseDisjoint u fun x => x • t)
C : Finset (Finset α) := filter (fun u => Set.PairwiseDisjoint ↑u fun x => x • t) (powerset s)
u : Finset α
hu : u ⊆ s ∧ Set.PairwiseDisjoint ↑u fun x => x • t
hCmax : ∀ x ∈ C, ¬u < x
a : α
ha : a ∈ s
hau : a ∉ u
H : ∀ b ∈ u, Disjoint (a • t) (b • t)
⊢ (a ∈ s ∧ u ⊆ s) ∧ Set.PairwiseDisjoint (insert a ↑u) fun x => x • t

Next tactic:
exact ⟨⟨ha, hu.1⟩, hu.2.insert fun _ hb _ ↦ H _ hb⟩
/-- **Ruzsa's covering lemma**. -/
@[to_additive "**Ruzsa's covering lemma**"]
theorem exists_subset_mul_div (ht : t.Nonempty) :
    ∃ u : Finset α, u.card * t.card ≤ (s * t).card ∧ s ⊆ u * t / t := by
  haveI : ∀ u, Decidable ((u : Set α).PairwiseDisjoint (· • t)) := fun u ↦ Classical.dec _
  set C := s.powerset.filter fun u ↦ u.toSet.PairwiseDisjoint (· • t)
  obtain ⟨u, hu, hCmax⟩ := C.exists_maximal (filter_nonempty_iff.2
    ⟨∅, empty_mem_powerset _, by rw [coe_empty]; exact Set.pairwiseDisjoint_empty⟩)
  rw [mem_filter, mem_powerset] at hu
  refine' ⟨u,
    (card_mul_iff.2 <| pairwiseDisjoint_smul_iff.1 hu.2).ge.trans
      (card_le_of_subset <| mul_subset_mul_right hu.1),
    fun a ha ↦ _⟩
  rw [mul_div_assoc]
  by_cases hau : a ∈ u
  · exact subset_mul_left _ ht.one_mem_div hau
  by_cases H : ∀ b ∈ u, Disjoint (a • t) (b • t)
  · refine' (hCmax _ _ <| ssubset_insert hau).elim
    rw [mem_filter, mem_powerset, insert_subset_iff, coe_insert]
    exact ⟨⟨ha, hu.1⟩, hu.2.insert fun _ hb _ ↦ H _ hb⟩
  

Proof state:
case neg
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : CommGroup α
s t : Finset α
ht : Finset.Nonempty t
this : (u : Set α) → Decidable (Set.PairwiseDisjoint u fun x => x • t)
C : Finset (Finset α) := filter (fun u => Set.PairwiseDisjoint ↑u fun x => x • t) (powerset s)
u : Finset α
hu : u ⊆ s ∧ Set.PairwiseDisjoint ↑u fun x => x • t
hCmax : ∀ x ∈ C, ¬u < x
a : α
ha : a ∈ s
hau : a ∉ u
H : ¬∀ b ∈ u, Disjoint (a • t) (b • t)
⊢ a ∈ u * (t / t)

Next tactic:
push_neg at H
/-- **Ruzsa's covering lemma**. -/
@[to_additive "**Ruzsa's covering lemma**"]
theorem exists_subset_mul_div (ht : t.Nonempty) :
    ∃ u : Finset α, u.card * t.card ≤ (s * t).card ∧ s ⊆ u * t / t := by
  haveI : ∀ u, Decidable ((u : Set α).PairwiseDisjoint (· • t)) := fun u ↦ Classical.dec _
  set C := s.powerset.filter fun u ↦ u.toSet.PairwiseDisjoint (· • t)
  obtain ⟨u, hu, hCmax⟩ := C.exists_maximal (filter_nonempty_iff.2
    ⟨∅, empty_mem_powerset _, by rw [coe_empty]; exact Set.pairwiseDisjoint_empty⟩)
  rw [mem_filter, mem_powerset] at hu
  refine' ⟨u,
    (card_mul_iff.2 <| pairwiseDisjoint_smul_iff.1 hu.2).ge.trans
      (card_le_of_subset <| mul_subset_mul_right hu.1),
    fun a ha ↦ _⟩
  rw [mul_div_assoc]
  by_cases hau : a ∈ u
  · exact subset_mul_left _ ht.one_mem_div hau
  by_cases H : ∀ b ∈ u, Disjoint (a • t) (b • t)
  · refine' (hCmax _ _ <| ssubset_insert hau).elim
    rw [mem_filter, mem_powerset, insert_subset_iff, coe_insert]
    exact ⟨⟨ha, hu.1⟩, hu.2.insert fun _ hb _ ↦ H _ hb⟩
  push_neg at H
  

Proof state:
case neg
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : CommGroup α
s t : Finset α
ht : Finset.Nonempty t
this : (u : Set α) → Decidable (Set.PairwiseDisjoint u fun x => x • t)
C : Finset (Finset α) := filter (fun u => Set.PairwiseDisjoint ↑u fun x => x • t) (powerset s)
u : Finset α
hu : u ⊆ s ∧ Set.PairwiseDisjoint ↑u fun x => x • t
hCmax : ∀ x ∈ C, ¬u < x
a : α
ha : a ∈ s
hau : a ∉ u
H : ∃ b ∈ u, ¬Disjoint (a • t) (b • t)
⊢ a ∈ u * (t / t)

Next tactic:
simp_rw [not_disjoint_iff, ← inv_smul_mem_iff] at H
/-- **Ruzsa's covering lemma**. -/
@[to_additive "**Ruzsa's covering lemma**"]
theorem exists_subset_mul_div (ht : t.Nonempty) :
    ∃ u : Finset α, u.card * t.card ≤ (s * t).card ∧ s ⊆ u * t / t := by
  haveI : ∀ u, Decidable ((u : Set α).PairwiseDisjoint (· • t)) := fun u ↦ Classical.dec _
  set C := s.powerset.filter fun u ↦ u.toSet.PairwiseDisjoint (· • t)
  obtain ⟨u, hu, hCmax⟩ := C.exists_maximal (filter_nonempty_iff.2
    ⟨∅, empty_mem_powerset _, by rw [coe_empty]; exact Set.pairwiseDisjoint_empty⟩)
  rw [mem_filter, mem_powerset] at hu
  refine' ⟨u,
    (card_mul_iff.2 <| pairwiseDisjoint_smul_iff.1 hu.2).ge.trans
      (card_le_of_subset <| mul_subset_mul_right hu.1),
    fun a ha ↦ _⟩
  rw [mul_div_assoc]
  by_cases hau : a ∈ u
  · exact subset_mul_left _ ht.one_mem_div hau
  by_cases H : ∀ b ∈ u, Disjoint (a • t) (b • t)
  · refine' (hCmax _ _ <| ssubset_insert hau).elim
    rw [mem_filter, mem_powerset, insert_subset_iff, coe_insert]
    exact ⟨⟨ha, hu.1⟩, hu.2.insert fun _ hb _ ↦ H _ hb⟩
  push_neg at H
  simp_rw [not_disjoint_iff, ← inv_smul_mem_iff] at H
  

Proof state:
case neg
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : CommGroup α
s t : Finset α
ht : Finset.Nonempty t
this : (u : Set α) → Decidable (Set.PairwiseDisjoint u fun x => x • t)
C : Finset (Finset α) := filter (fun u => Set.PairwiseDisjoint ↑u fun x => x • t) (powerset s)
u : Finset α
hu : u ⊆ s ∧ Set.PairwiseDisjoint ↑u fun x => x • t
hCmax : ∀ x ∈ C, ¬u < x
a : α
ha : a ∈ s
hau : a ∉ u
H : ∃ b ∈ u, ∃ a_1, a⁻¹ • a_1 ∈ t ∧ b⁻¹ • a_1 ∈ t
⊢ a ∈ u * (t / t)

Next tactic:
obtain ⟨b, hb, c, hc₁, hc₂⟩ := H
/-- **Ruzsa's covering lemma**. -/
@[to_additive "**Ruzsa's covering lemma**"]
theorem exists_subset_mul_div (ht : t.Nonempty) :
    ∃ u : Finset α, u.card * t.card ≤ (s * t).card ∧ s ⊆ u * t / t := by
  haveI : ∀ u, Decidable ((u : Set α).PairwiseDisjoint (· • t)) := fun u ↦ Classical.dec _
  set C := s.powerset.filter fun u ↦ u.toSet.PairwiseDisjoint (· • t)
  obtain ⟨u, hu, hCmax⟩ := C.exists_maximal (filter_nonempty_iff.2
    ⟨∅, empty_mem_powerset _, by rw [coe_empty]; exact Set.pairwiseDisjoint_empty⟩)
  rw [mem_filter, mem_powerset] at hu
  refine' ⟨u,
    (card_mul_iff.2 <| pairwiseDisjoint_smul_iff.1 hu.2).ge.trans
      (card_le_of_subset <| mul_subset_mul_right hu.1),
    fun a ha ↦ _⟩
  rw [mul_div_assoc]
  by_cases hau : a ∈ u
  · exact subset_mul_left _ ht.one_mem_div hau
  by_cases H : ∀ b ∈ u, Disjoint (a • t) (b • t)
  · refine' (hCmax _ _ <| ssubset_insert hau).elim
    rw [mem_filter, mem_powerset, insert_subset_iff, coe_insert]
    exact ⟨⟨ha, hu.1⟩, hu.2.insert fun _ hb _ ↦ H _ hb⟩
  push_neg at H
  simp_rw [not_disjoint_iff, ← inv_smul_mem_iff] at H
  obtain ⟨b, hb, c, hc₁, hc₂⟩ := H
  

Proof state:
case neg.intro.intro.intro.intro
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : CommGroup α
s t : Finset α
ht : Finset.Nonempty t
this : (u : Set α) → Decidable (Set.PairwiseDisjoint u fun x => x • t)
C : Finset (Finset α) := filter (fun u => Set.PairwiseDisjoint ↑u fun x => x • t) (powerset s)
u : Finset α
hu : u ⊆ s ∧ Set.PairwiseDisjoint ↑u fun x => x • t
hCmax : ∀ x ∈ C, ¬u < x
a : α
ha : a ∈ s
hau : a ∉ u
b : α
hb : b ∈ u
c : α
hc₁ : a⁻¹ • c ∈ t
hc₂ : b⁻¹ • c ∈ t
⊢ a ∈ u * (t / t)

Next tactic:
refine' mem_mul.2 ⟨b, a / b, hb, _, by simp⟩
/-- **Ruzsa's covering lemma**. -/
@[to_additive "**Ruzsa's covering lemma**"]
theorem exists_subset_mul_div (ht : t.Nonempty) :
    ∃ u : Finset α, u.card * t.card ≤ (s * t).card ∧ s ⊆ u * t / t := by
  haveI : ∀ u, Decidable ((u : Set α).PairwiseDisjoint (· • t)) := fun u ↦ Classical.dec _
  set C := s.powerset.filter fun u ↦ u.toSet.PairwiseDisjoint (· • t)
  obtain ⟨u, hu, hCmax⟩ := C.exists_maximal (filter_nonempty_iff.2
    ⟨∅, empty_mem_powerset _, by rw [coe_empty]; exact Set.pairwiseDisjoint_empty⟩)
  rw [mem_filter, mem_powerset] at hu
  refine' ⟨u,
    (card_mul_iff.2 <| pairwiseDisjoint_smul_iff.1 hu.2).ge.trans
      (card_le_of_subset <| mul_subset_mul_right hu.1),
    fun a ha ↦ _⟩
  rw [mul_div_assoc]
  by_cases hau : a ∈ u
  · exact subset_mul_left _ ht.one_mem_div hau
  by_cases H : ∀ b ∈ u, Disjoint (a • t) (b • t)
  · refine' (hCmax _ _ <| ssubset_insert hau).elim
    rw [mem_filter, mem_powerset, insert_subset_iff, coe_insert]
    exact ⟨⟨ha, hu.1⟩, hu.2.insert fun _ hb _ ↦ H _ hb⟩
  push_neg at H
  simp_rw [not_disjoint_iff, ← inv_smul_mem_iff] at H
  obtain ⟨b, hb, c, hc₁, hc₂⟩ := H
  refine' mem_mul.2 ⟨b, a / b, hb, _, by 

Proof state:
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : CommGroup α
s t : Finset α
ht : Finset.Nonempty t
this : (u : Set α) → Decidable (Set.PairwiseDisjoint u fun x => x • t)
C : Finset (Finset α) := filter (fun u => Set.PairwiseDisjoint ↑u fun x => x • t) (powerset s)
u : Finset α
hu : u ⊆ s ∧ Set.PairwiseDisjoint ↑u fun x => x • t
hCmax : ∀ x ∈ C, ¬u < x
a : α
ha : a ∈ s
hau : a ∉ u
b : α
hb : b ∈ u
c : α
hc₁ : a⁻¹ • c ∈ t
hc₂ : b⁻¹ • c ∈ t
⊢ b * (a / b) = a

Next tactic:
simp
/-- **Ruzsa's covering lemma**. -/
@[to_additive "**Ruzsa's covering lemma**"]
theorem exists_subset_mul_div (ht : t.Nonempty) :
    ∃ u : Finset α, u.card * t.card ≤ (s * t).card ∧ s ⊆ u * t / t := by
  haveI : ∀ u, Decidable ((u : Set α).PairwiseDisjoint (· • t)) := fun u ↦ Classical.dec _
  set C := s.powerset.filter fun u ↦ u.toSet.PairwiseDisjoint (· • t)
  obtain ⟨u, hu, hCmax⟩ := C.exists_maximal (filter_nonempty_iff.2
    ⟨∅, empty_mem_powerset _, by rw [coe_empty]; exact Set.pairwiseDisjoint_empty⟩)
  rw [mem_filter, mem_powerset] at hu
  refine' ⟨u,
    (card_mul_iff.2 <| pairwiseDisjoint_smul_iff.1 hu.2).ge.trans
      (card_le_of_subset <| mul_subset_mul_right hu.1),
    fun a ha ↦ _⟩
  rw [mul_div_assoc]
  by_cases hau : a ∈ u
  · exact subset_mul_left _ ht.one_mem_div hau
  by_cases H : ∀ b ∈ u, Disjoint (a • t) (b • t)
  · refine' (hCmax _ _ <| ssubset_insert hau).elim
    rw [mem_filter, mem_powerset, insert_subset_iff, coe_insert]
    exact ⟨⟨ha, hu.1⟩, hu.2.insert fun _ hb _ ↦ H _ hb⟩
  push_neg at H
  simp_rw [not_disjoint_iff, ← inv_smul_mem_iff] at H
  obtain ⟨b, hb, c, hc₁, hc₂⟩ := H
  refine' mem_mul.2 ⟨b, a / b, hb, _, by simp⟩
  

Proof state:
case neg.intro.intro.intro.intro
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : CommGroup α
s t : Finset α
ht : Finset.Nonempty t
this : (u : Set α) → Decidable (Set.PairwiseDisjoint u fun x => x • t)
C : Finset (Finset α) := filter (fun u => Set.PairwiseDisjoint ↑u fun x => x • t) (powerset s)
u : Finset α
hu : u ⊆ s ∧ Set.PairwiseDisjoint ↑u fun x => x • t
hCmax : ∀ x ∈ C, ¬u < x
a : α
ha : a ∈ s
hau : a ∉ u
b : α
hb : b ∈ u
c : α
hc₁ : a⁻¹ • c ∈ t
hc₂ : b⁻¹ • c ∈ t
⊢ a / b ∈ t / t

Next tactic:
exact mem_div.2 ⟨_, _, hc₂, hc₁, by simp [div_eq_mul_inv a b, mul_comm]⟩
/-- **Ruzsa's covering lemma**. -/
@[to_additive "**Ruzsa's covering lemma**"]
theorem exists_subset_mul_div (ht : t.Nonempty) :
    ∃ u : Finset α, u.card * t.card ≤ (s * t).card ∧ s ⊆ u * t / t := by
  haveI : ∀ u, Decidable ((u : Set α).PairwiseDisjoint (· • t)) := fun u ↦ Classical.dec _
  set C := s.powerset.filter fun u ↦ u.toSet.PairwiseDisjoint (· • t)
  obtain ⟨u, hu, hCmax⟩ := C.exists_maximal (filter_nonempty_iff.2
    ⟨∅, empty_mem_powerset _, by rw [coe_empty]; exact Set.pairwiseDisjoint_empty⟩)
  rw [mem_filter, mem_powerset] at hu
  refine' ⟨u,
    (card_mul_iff.2 <| pairwiseDisjoint_smul_iff.1 hu.2).ge.trans
      (card_le_of_subset <| mul_subset_mul_right hu.1),
    fun a ha ↦ _⟩
  rw [mul_div_assoc]
  by_cases hau : a ∈ u
  · exact subset_mul_left _ ht.one_mem_div hau
  by_cases H : ∀ b ∈ u, Disjoint (a • t) (b • t)
  · refine' (hCmax _ _ <| ssubset_insert hau).elim
    rw [mem_filter, mem_powerset, insert_subset_iff, coe_insert]
    exact ⟨⟨ha, hu.1⟩, hu.2.insert fun _ hb _ ↦ H _ hb⟩
  push_neg at H
  simp_rw [not_disjoint_iff, ← inv_smul_mem_iff] at H
  obtain ⟨b, hb, c, hc₁, hc₂⟩ := H
  refine' mem_mul.2 ⟨b, a / b, hb, _, by simp⟩
  exact mem_div.2 ⟨_, _, hc₂, hc₁, by 

Proof state:
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : CommGroup α
s t : Finset α
ht : Finset.Nonempty t
this : (u : Set α) → Decidable (Set.PairwiseDisjoint u fun x => x • t)
C : Finset (Finset α) := filter (fun u => Set.PairwiseDisjoint ↑u fun x => x • t) (powerset s)
u : Finset α
hu : u ⊆ s ∧ Set.PairwiseDisjoint ↑u fun x => x • t
hCmax : ∀ x ∈ C, ¬u < x
a : α
ha : a ∈ s
hau : a ∉ u
b : α
hb : b ∈ u
c : α
hc₁ : a⁻¹ • c ∈ t
hc₂ : b⁻¹ • c ∈ t
⊢ b⁻¹ • c / a⁻¹ • c = a / b

Next tactic:
simp [div_eq_mul_inv a b, mul_comm]
/-- **Ruzsa's covering lemma** for sets. See also `Finset.exists_subset_mul_div`. -/
@[to_additive "**Ruzsa's covering lemma**. Version for sets. For finsets,
see `Finset.exists_subset_add_sub`."]
lemma exists_subset_mul_div (hs : s.Finite) (ht' : t.Finite) (ht : t.Nonempty) :
    ∃ u : Set α, Nat.card u * Nat.card t ≤ Nat.card (s * t) ∧ s ⊆ u * t / t ∧ u.Finite := by
  

Proof state:
α : Type u_1
inst✝ : CommGroup α
s t : Set α
hs : Set.Finite s
ht' : Set.Finite t
ht : Set.Nonempty t
⊢ ∃ u, Nat.card ↑u * Nat.card ↑t ≤ Nat.card ↑(s * t) ∧ s ⊆ u * t / t ∧ Set.Finite u

Next tactic:
lift s to Finset α using hs
/-- **Ruzsa's covering lemma** for sets. See also `Finset.exists_subset_mul_div`. -/
@[to_additive "**Ruzsa's covering lemma**. Version for sets. For finsets,
see `Finset.exists_subset_add_sub`."]
lemma exists_subset_mul_div (hs : s.Finite) (ht' : t.Finite) (ht : t.Nonempty) :
    ∃ u : Set α, Nat.card u * Nat.card t ≤ Nat.card (s * t) ∧ s ⊆ u * t / t ∧ u.Finite := by
  lift s to Finset α using hs
  

Proof state:
case intro
α : Type u_1
inst✝ : CommGroup α
t : Set α
ht' : Set.Finite t
ht : Set.Nonempty t
s : Finset α
⊢ ∃ u, Nat.card ↑u * Nat.card ↑t ≤ Nat.card ↑(↑s * t) ∧ ↑s ⊆ u * t / t ∧ Set.Finite u

Next tactic:
lift t to Finset α using ht'
/-- **Ruzsa's covering lemma** for sets. See also `Finset.exists_subset_mul_div`. -/
@[to_additive "**Ruzsa's covering lemma**. Version for sets. For finsets,
see `Finset.exists_subset_add_sub`."]
lemma exists_subset_mul_div (hs : s.Finite) (ht' : t.Finite) (ht : t.Nonempty) :
    ∃ u : Set α, Nat.card u * Nat.card t ≤ Nat.card (s * t) ∧ s ⊆ u * t / t ∧ u.Finite := by
  lift s to Finset α using hs
  lift t to Finset α using ht'
  

Proof state:
case intro.intro
α : Type u_1
inst✝ : CommGroup α
s t : Finset α
ht : Set.Nonempty ↑t
⊢ ∃ u, Nat.card ↑u * Nat.card ↑↑t ≤ Nat.card ↑(↑s * ↑t) ∧ ↑s ⊆ u * ↑t / ↑t ∧ Set.Finite u

Next tactic:
classical
  obtain ⟨u, hu, hsut⟩ := Finset.exists_subset_mul_div s ht
  refine ⟨u, ?_⟩
  -- `norm_cast` would find these automatically, but breaks `to_additive` when it does so
  rw [← Finset.coe_mul, ← Finset.coe_mul, ← Finset.coe_div]
  norm_cast
  simp [*]
/-- **Ruzsa's covering lemma** for sets. See also `Finset.exists_subset_mul_div`. -/
@[to_additive "**Ruzsa's covering lemma**. Version for sets. For finsets,
see `Finset.exists_subset_add_sub`."]
lemma exists_subset_mul_div (hs : s.Finite) (ht' : t.Finite) (ht : t.Nonempty) :
    ∃ u : Set α, Nat.card u * Nat.card t ≤ Nat.card (s * t) ∧ s ⊆ u * t / t ∧ u.Finite := by
  lift s to Finset α using hs
  lift t to Finset α using ht'
  classical
  

Proof state:
case intro.intro
α : Type u_1
inst✝ : CommGroup α
s t : Finset α
ht : Set.Nonempty ↑t
⊢ ∃ u, Nat.card ↑u * Nat.card ↑↑t ≤ Nat.card ↑(↑s * ↑t) ∧ ↑s ⊆ u * ↑t / ↑t ∧ Set.Finite u

Next tactic:
obtain ⟨u, hu, hsut⟩ := Finset.exists_subset_mul_div s ht
/-- **Ruzsa's covering lemma** for sets. See also `Finset.exists_subset_mul_div`. -/
@[to_additive "**Ruzsa's covering lemma**. Version for sets. For finsets,
see `Finset.exists_subset_add_sub`."]
lemma exists_subset_mul_div (hs : s.Finite) (ht' : t.Finite) (ht : t.Nonempty) :
    ∃ u : Set α, Nat.card u * Nat.card t ≤ Nat.card (s * t) ∧ s ⊆ u * t / t ∧ u.Finite := by
  lift s to Finset α using hs
  lift t to Finset α using ht'
  classical
  obtain ⟨u, hu, hsut⟩ := Finset.exists_subset_mul_div s ht
  

Proof state:
case intro.intro.intro.intro
α : Type u_1
inst✝ : CommGroup α
s t : Finset α
ht : Set.Nonempty ↑t
u : Finset α
hu : Finset.card u * Finset.card t ≤ Finset.card (s * t)
hsut : s ⊆ u * t / t
⊢ ∃ u, Nat.card ↑u * Nat.card ↑↑t ≤ Nat.card ↑(↑s * ↑t) ∧ ↑s ⊆ u * ↑t / ↑t ∧ Set.Finite u

Next tactic:
refine ⟨u, ?_⟩
/-- **Ruzsa's covering lemma** for sets. See also `Finset.exists_subset_mul_div`. -/
@[to_additive "**Ruzsa's covering lemma**. Version for sets. For finsets,
see `Finset.exists_subset_add_sub`."]
lemma exists_subset_mul_div (hs : s.Finite) (ht' : t.Finite) (ht : t.Nonempty) :
    ∃ u : Set α, Nat.card u * Nat.card t ≤ Nat.card (s * t) ∧ s ⊆ u * t / t ∧ u.Finite := by
  lift s to Finset α using hs
  lift t to Finset α using ht'
  classical
  obtain ⟨u, hu, hsut⟩ := Finset.exists_subset_mul_div s ht
  refine ⟨u, ?_⟩
  -- `norm_cast` would find these automatically, but breaks `to_additive` when it does so
  

Proof state:
case intro.intro.intro.intro
α : Type u_1
inst✝ : CommGroup α
s t : Finset α
ht : Set.Nonempty ↑t
u : Finset α
hu : Finset.card u * Finset.card t ≤ Finset.card (s * t)
hsut : s ⊆ u * t / t
⊢ Nat.card ↑↑u * Nat.card ↑↑t ≤ Nat.card ↑(↑s * ↑t) ∧ ↑s ⊆ ↑u * ↑t / ↑t ∧ Set.Finite ↑u

Next tactic:
rw [← Finset.coe_mul, ← Finset.coe_mul, ← Finset.coe_div]
/-- **Ruzsa's covering lemma** for sets. See also `Finset.exists_subset_mul_div`. -/
@[to_additive "**Ruzsa's covering lemma**. Version for sets. For finsets,
see `Finset.exists_subset_add_sub`."]
lemma exists_subset_mul_div (hs : s.Finite) (ht' : t.Finite) (ht : t.Nonempty) :
    ∃ u : Set α, Nat.card u * Nat.card t ≤ Nat.card (s * t) ∧ s ⊆ u * t / t ∧ u.Finite := by
  lift s to Finset α using hs
  lift t to Finset α using ht'
  classical
  obtain ⟨u, hu, hsut⟩ := Finset.exists_subset_mul_div s ht
  refine ⟨u, ?_⟩
  -- `norm_cast` would find these automatically, but breaks `to_additive` when it does so
  rw [← Finset.coe_mul, ← Finset.coe_mul, ← Finset.coe_div]
  

Proof state:
case intro.intro.intro.intro
α : Type u_1
inst✝ : CommGroup α
s t : Finset α
ht : Set.Nonempty ↑t
u : Finset α
hu : Finset.card u * Finset.card t ≤ Finset.card (s * t)
hsut : s ⊆ u * t / t
⊢ Nat.card ↑↑u * Nat.card ↑↑t ≤ Nat.card ↑↑(s * t) ∧ ↑s ⊆ ↑(u * t / t) ∧ Set.Finite ↑u

Next tactic:
norm_cast
/-- **Ruzsa's covering lemma** for sets. See also `Finset.exists_subset_mul_div`. -/
@[to_additive "**Ruzsa's covering lemma**. Version for sets. For finsets,
see `Finset.exists_subset_add_sub`."]
lemma exists_subset_mul_div (hs : s.Finite) (ht' : t.Finite) (ht : t.Nonempty) :
    ∃ u : Set α, Nat.card u * Nat.card t ≤ Nat.card (s * t) ∧ s ⊆ u * t / t ∧ u.Finite := by
  lift s to Finset α using hs
  lift t to Finset α using ht'
  classical
  obtain ⟨u, hu, hsut⟩ := Finset.exists_subset_mul_div s ht
  refine ⟨u, ?_⟩
  -- `norm_cast` would find these automatically, but breaks `to_additive` when it does so
  rw [← Finset.coe_mul, ← Finset.coe_mul, ← Finset.coe_div]
  norm_cast
  

Proof state:
case intro.intro.intro.intro
α : Type u_1
inst✝ : CommGroup α
s t : Finset α
ht : Set.Nonempty ↑t
u : Finset α
hu : Finset.card u * Finset.card t ≤ Finset.card (s * t)
hsut : s ⊆ u * t / t
⊢ Nat.card { x // x ∈ u } * Nat.card { x // x ∈ t } ≤ Nat.card { x // x ∈ s * t } ∧ s ⊆ u * t / t ∧ Set.Finite ↑u

Next tactic:
simp [*]
instance : Top (Subpresheaf F) :=
  ⟨⟨fun U => ⊤, @fun U V _ x _ => by 

Proof state:
C : Type u
inst✝ : Category.{v, u} C
J : GrothendieckTopology C
F F' F'' : Cᵒᵖ ⥤ Type w
G G' : Subpresheaf F
U V : Cᵒᵖ
x✝¹ : U ⟶ V
x : F.obj U
x✝ : x ∈ (fun U => ⊤) U
⊢ x ∈ F.map x✝¹ ⁻¹' (fun U => ⊤) V

Next tactic:
aesop_cat
/-- The subpresheaf as a presheaf. -/
@[simps!]
def Subpresheaf.toPresheaf : Cᵒᵖ ⥤ Type w where
  obj U := G.obj U
  map := @fun U V i x => ⟨F.map i x, G.map i x.prop⟩
  map_id X := by
    

Proof state:
C : Type u
inst✝ : Category.{v, u} C
J : GrothendieckTopology C
F F' F'' : Cᵒᵖ ⥤ Type w
G G' : Subpresheaf F
X : Cᵒᵖ
⊢ { obj := fun U => ↑(obj G U),
          map := fun U V i x => { val := F.map i ↑x, property := (_ : ↑x ∈ F.map i ⁻¹' obj G V) } }.map
      (𝟙 X) =
    𝟙
      ({ obj := fun U => ↑(obj G U),
            map := fun U V i x => { val := F.map i ↑x, property := (_ : ↑x ∈ F.map i ⁻¹' obj G V) } }.obj
        X)

Next tactic:
ext ⟨x, _⟩
/-- The subpresheaf as a presheaf. -/
@[simps!]
def Subpresheaf.toPresheaf : Cᵒᵖ ⥤ Type w where
  obj U := G.obj U
  map := @fun U V i x => ⟨F.map i x, G.map i x.prop⟩
  map_id X := by
    ext ⟨x, _⟩
    

Proof state:
case h.mk.a
C : Type u
inst✝ : Category.{v, u} C
J : GrothendieckTopology C
F F' F'' : Cᵒᵖ ⥤ Type w
G G' : Subpresheaf F
X : Cᵒᵖ
x : F.obj X
property✝ : x ∈ obj G X
⊢ ↑({ obj := fun U => ↑(obj G U),
            map := fun U V i x => { val := F.map i ↑x, property := (_ : ↑x ∈ F.map i ⁻¹' obj G V) } }.map
        (𝟙 X) { val := x, property := property✝ }) =
    ↑(𝟙
        ({ obj := fun U => ↑(obj G U),
              map := fun U V i x => { val := F.map i ↑x, property := (_ : ↑x ∈ F.map i ⁻¹' obj G V) } }.obj
          X)
        { val := x, property := property✝ })

Next tactic:
dsimp
/-- The subpresheaf as a presheaf. -/
@[simps!]
def Subpresheaf.toPresheaf : Cᵒᵖ ⥤ Type w where
  obj U := G.obj U
  map := @fun U V i x => ⟨F.map i x, G.map i x.prop⟩
  map_id X := by
    ext ⟨x, _⟩
    dsimp
    

Proof state:
case h.mk.a
C : Type u
inst✝ : Category.{v, u} C
J : GrothendieckTopology C
F F' F'' : Cᵒᵖ ⥤ Type w
G G' : Subpresheaf F
X : Cᵒᵖ
x : F.obj X
property✝ : x ∈ obj G X
⊢ F.map (𝟙 X) x = x

Next tactic:
simp only [FunctorToTypes.map_id_apply]
/-- The subpresheaf as a presheaf. -/
@[simps!]
def Subpresheaf.toPresheaf : Cᵒᵖ ⥤ Type w where
  obj U := G.obj U
  map := @fun U V i x => ⟨F.map i x, G.map i x.prop⟩
  map_id X := by
    ext ⟨x, _⟩
    dsimp
    simp only [FunctorToTypes.map_id_apply]
  map_comp := @fun X Y Z i j => by
    

Proof state:
C : Type u
inst✝ : Category.{v, u} C
J : GrothendieckTopology C
F F' F'' : Cᵒᵖ ⥤ Type w
G G' : Subpresheaf F
X Y Z : Cᵒᵖ
i : X ⟶ Y
j : Y ⟶ Z
⊢ { obj := fun U => ↑(obj G U),
          map := fun U V i x => { val := F.map i ↑x, property := (_ : ↑x ∈ F.map i ⁻¹' obj G V) } }.map
      (i ≫ j) =
    { obj := fun U => ↑(obj G U),
            map := fun U V i x => { val := F.map i ↑x, property := (_ : ↑x ∈ F.map i ⁻¹' obj G V) } }.map
        i ≫
      { obj := fun U => ↑(obj G U),
            map := fun U V i x => { val := F.map i ↑x, property := (_ : ↑x ∈ F.map i ⁻¹' obj G V) } }.map
        j

Next tactic:
ext ⟨x, _⟩