l3lab/ntp-mathlib huggingface dataset:
protected theorem nontrivial (R S : Type*) [CommRing R] [Nontrivial R] [CommRing S] [Algebra R S]
    [IsFractionRing R S] : Nontrivial S := by
  

Proof state:
R✝ : Type u_1
inst✝¹⁶ : CommRing R✝
M : Submonoid R✝
S✝ : Type u_2
inst✝¹⁵ : CommRing S✝
inst✝¹⁴ : Algebra R✝ S✝
P : Type u_3
inst✝¹³ : CommRing P
A : Type u_4
inst✝¹² : CommRing A
inst✝¹¹ : IsDomain A
K : Type u_5
B : Type u_6
inst✝¹⁰ : CommRing B
inst✝⁹ : IsDomain B
inst✝⁸ : Field K
L : Type u_7
inst✝⁷ : Field L
inst✝⁶ : Algebra A K
inst✝⁵ : IsFractionRing A K
g : A →+* L
R : Type u_8
S : Type u_9
inst✝⁴ : CommRing R
inst✝³ : Nontrivial R
inst✝² : CommRing S
inst✝¹ : Algebra R S
inst✝ : IsFractionRing R S
⊢ Nontrivial S

Next tactic:
apply nontrivial_of_ne
protected theorem nontrivial (R S : Type*) [CommRing R] [Nontrivial R] [CommRing S] [Algebra R S]
    [IsFractionRing R S] : Nontrivial S := by
  apply nontrivial_of_ne
  

Proof state:
case h
R✝ : Type u_1
inst✝¹⁶ : CommRing R✝
M : Submonoid R✝
S✝ : Type u_2
inst✝¹⁵ : CommRing S✝
inst✝¹⁴ : Algebra R✝ S✝
P : Type u_3
inst✝¹³ : CommRing P
A : Type u_4
inst✝¹² : CommRing A
inst✝¹¹ : IsDomain A
K : Type u_5
B : Type u_6
inst✝¹⁰ : CommRing B
inst✝⁹ : IsDomain B
inst✝⁸ : Field K
L : Type u_7
inst✝⁷ : Field L
inst✝⁶ : Algebra A K
inst✝⁵ : IsFractionRing A K
g : A →+* L
R : Type u_8
S : Type u_9
inst✝⁴ : CommRing R
inst✝³ : Nontrivial R
inst✝² : CommRing S
inst✝¹ : Algebra R S
inst✝ : IsFractionRing R S
⊢ ?x ≠ ?y
case x
R✝ : Type u_1
inst✝¹⁶ : CommRing R✝
M : Submonoid R✝
S✝ : Type u_2
inst✝¹⁵ : CommRing S✝
inst✝¹⁴ : Algebra R✝ S✝
P : Type u_3
inst✝¹³ : CommRing P
A : Type u_4
inst✝¹² : CommRing A
inst✝¹¹ : IsDomain A
K : Type u_5
B : Type u_6
inst✝¹⁰ : CommRing B
inst✝⁹ : IsDomain B
inst✝⁸ : Field K
L : Type u_7
inst✝⁷ : Field L
inst✝⁶ : Algebra A K
inst✝⁵ : IsFractionRing A K
g : A →+* L
R : Type u_8
S : Type u_9
inst✝⁴ : CommRing R
inst✝³ : Nontrivial R
inst✝² : CommRing S
inst✝¹ : Algebra R S
inst✝ : IsFractionRing R S
⊢ S
case y
R✝ : Type u_1
inst✝¹⁶ : CommRing R✝
M : Submonoid R✝
S✝ : Type u_2
inst✝¹⁵ : CommRing S✝
inst✝¹⁴ : Algebra R✝ S✝
P : Type u_3
inst✝¹³ : CommRing P
A : Type u_4
inst✝¹² : CommRing A
inst✝¹¹ : IsDomain A
K : Type u_5
B : Type u_6
inst✝¹⁰ : CommRing B
inst✝⁹ : IsDomain B
inst✝⁸ : Field K
L : Type u_7
inst✝⁷ : Field L
inst✝⁶ : Algebra A K
inst✝⁵ : IsFractionRing A K
g : A →+* L
R : Type u_8
S : Type u_9
inst✝⁴ : CommRing R
inst✝³ : Nontrivial R
inst✝² : CommRing S
inst✝¹ : Algebra R S
inst✝ : IsFractionRing R S
⊢ S

Next tactic:
intro h
protected theorem nontrivial (R S : Type*) [CommRing R] [Nontrivial R] [CommRing S] [Algebra R S]
    [IsFractionRing R S] : Nontrivial S := by
  apply nontrivial_of_ne
  intro h
  

Proof state:
case h
R✝ : Type u_1
inst✝¹⁶ : CommRing R✝
M : Submonoid R✝
S✝ : Type u_2
inst✝¹⁵ : CommRing S✝
inst✝¹⁴ : Algebra R✝ S✝
P : Type u_3
inst✝¹³ : CommRing P
A : Type u_4
inst✝¹² : CommRing A
inst✝¹¹ : IsDomain A
K : Type u_5
B : Type u_6
inst✝¹⁰ : CommRing B
inst✝⁹ : IsDomain B
inst✝⁸ : Field K
L : Type u_7
inst✝⁷ : Field L
inst✝⁶ : Algebra A K
inst✝⁵ : IsFractionRing A K
g : A →+* L
R : Type u_8
S : Type u_9
inst✝⁴ : CommRing R
inst✝³ : Nontrivial R
inst✝² : CommRing S
inst✝¹ : Algebra R S
inst✝ : IsFractionRing R S
h : ?x = ?y
⊢ False
case x
R✝ : Type u_1
inst✝¹⁶ : CommRing R✝
M : Submonoid R✝
S✝ : Type u_2
inst✝¹⁵ : CommRing S✝
inst✝¹⁴ : Algebra R✝ S✝
P : Type u_3
inst✝¹³ : CommRing P
A : Type u_4
inst✝¹² : CommRing A
inst✝¹¹ : IsDomain A
K : Type u_5
B : Type u_6
inst✝¹⁰ : CommRing B
inst✝⁹ : IsDomain B
inst✝⁸ : Field K
L : Type u_7
inst✝⁷ : Field L
inst✝⁶ : Algebra A K
inst✝⁵ : IsFractionRing A K
g : A →+* L
R : Type u_8
S : Type u_9
inst✝⁴ : CommRing R
inst✝³ : Nontrivial R
inst✝² : CommRing S
inst✝¹ : Algebra R S
inst✝ : IsFractionRing R S
⊢ S
case y
R✝ : Type u_1
inst✝¹⁶ : CommRing R✝
M : Submonoid R✝
S✝ : Type u_2
inst✝¹⁵ : CommRing S✝
inst✝¹⁴ : Algebra R✝ S✝
P : Type u_3
inst✝¹³ : CommRing P
A : Type u_4
inst✝¹² : CommRing A
inst✝¹¹ : IsDomain A
K : Type u_5
B : Type u_6
inst✝¹⁰ : CommRing B
inst✝⁹ : IsDomain B
inst✝⁸ : Field K
L : Type u_7
inst✝⁷ : Field L
inst✝⁶ : Algebra A K
inst✝⁵ : IsFractionRing A K
g : A →+* L
R : Type u_8
S : Type u_9
inst✝⁴ : CommRing R
inst✝³ : Nontrivial R
inst✝² : CommRing S
inst✝¹ : Algebra R S
inst✝ : IsFractionRing R S
⊢ S

Next tactic:
apply @zero_ne_one R
protected theorem nontrivial (R S : Type*) [CommRing R] [Nontrivial R] [CommRing S] [Algebra R S]
    [IsFractionRing R S] : Nontrivial S := by
  apply nontrivial_of_ne
  intro h
  apply @zero_ne_one R
  

Proof state:
case h.a
R✝ : Type u_1
inst✝¹⁶ : CommRing R✝
M : Submonoid R✝
S✝ : Type u_2
inst✝¹⁵ : CommRing S✝
inst✝¹⁴ : Algebra R✝ S✝
P : Type u_3
inst✝¹³ : CommRing P
A : Type u_4
inst✝¹² : CommRing A
inst✝¹¹ : IsDomain A
K : Type u_5
B : Type u_6
inst✝¹⁰ : CommRing B
inst✝⁹ : IsDomain B
inst✝⁸ : Field K
L : Type u_7
inst✝⁷ : Field L
inst✝⁶ : Algebra A K
inst✝⁵ : IsFractionRing A K
g : A →+* L
R : Type u_8
S : Type u_9
inst✝⁴ : CommRing R
inst✝³ : Nontrivial R
inst✝² : CommRing S
inst✝¹ : Algebra R S
inst✝ : IsFractionRing R S
h : ?x = ?y
⊢ 0 = 1
case x
R✝ : Type u_1
inst✝¹⁶ : CommRing R✝
M : Submonoid R✝
S✝ : Type u_2
inst✝¹⁵ : CommRing S✝
inst✝¹⁴ : Algebra R✝ S✝
P : Type u_3
inst✝¹³ : CommRing P
A : Type u_4
inst✝¹² : CommRing A
inst✝¹¹ : IsDomain A
K : Type u_5
B : Type u_6
inst✝¹⁰ : CommRing B
inst✝⁹ : IsDomain B
inst✝⁸ : Field K
L : Type u_7
inst✝⁷ : Field L
inst✝⁶ : Algebra A K
inst✝⁵ : IsFractionRing A K
g : A →+* L
R : Type u_8
S : Type u_9
inst✝⁴ : CommRing R
inst✝³ : Nontrivial R
inst✝² : CommRing S
inst✝¹ : Algebra R S
inst✝ : IsFractionRing R S
⊢ S
case y
R✝ : Type u_1
inst✝¹⁶ : CommRing R✝
M : Submonoid R✝
S✝ : Type u_2
inst✝¹⁵ : CommRing S✝
inst✝¹⁴ : Algebra R✝ S✝
P : Type u_3
inst✝¹³ : CommRing P
A : Type u_4
inst✝¹² : CommRing A
inst✝¹¹ : IsDomain A
K : Type u_5
B : Type u_6
inst✝¹⁰ : CommRing B
inst✝⁹ : IsDomain B
inst✝⁸ : Field K
L : Type u_7
inst✝⁷ : Field L
inst✝⁶ : Algebra A K
inst✝⁵ : IsFractionRing A K
g : A →+* L
R : Type u_8
S : Type u_9
inst✝⁴ : CommRing R
inst✝³ : Nontrivial R
inst✝² : CommRing S
inst✝¹ : Algebra R S
inst✝ : IsFractionRing R S
⊢ S

Next tactic:
exact
    IsLocalization.injective S (le_of_eq rfl)
      (((algebraMap R S).map_zero.trans h).trans (algebraMap R S).map_one.symm)
@[simp]
theorem mk_eq_div {r s} :
    (Localization.mk r s : FractionRing A) =
      (algebraMap _ _ r / algebraMap A _ s : FractionRing A) :=
  by 

Proof state:
R : Type u_1
inst✝⁵ : CommRing R
M : Submonoid R
S : Type u_2
inst✝⁴ : CommRing S
inst✝³ : Algebra R S
P : Type u_3
inst✝² : CommRing P
A : Type u_4
inst✝¹ : CommRing A
inst✝ : IsDomain A
K : Type u_5
r : A
s : ↥(nonZeroDivisors A)
⊢ Localization.mk r s = (algebraMap A (FractionRing A)) r / (algebraMap A (FractionRing A)) ↑s

Next tactic:
rw [Localization.mk_eq_mk', IsFractionRing.mk'_eq_div]
instance isScalarTower_liftAlgebra [IsDomain R] [Field K] [Algebra R K] [NoZeroSMulDivisors R K] :
    by 

Proof state:
R : Type u_1
inst✝⁹ : CommRing R
M : Submonoid R
S : Type u_2
inst✝⁸ : CommRing S
inst✝⁷ : Algebra R S
P : Type u_3
inst✝⁶ : CommRing P
A : Type u_4
inst✝⁵ : CommRing A
inst✝⁴ : IsDomain A
K : Type u_5
inst✝³ : IsDomain R
inst✝² : Field K
inst✝¹ : Algebra R K
inst✝ : NoZeroSMulDivisors R K
⊢ Sort ?u.531432

Next tactic:
letI := liftAlgebra R K
instance isScalarTower_liftAlgebra [IsDomain R] [Field K] [Algebra R K] [NoZeroSMulDivisors R K] :
    by letI := liftAlgebra R K; 

Proof state:
R : Type u_1
inst✝⁹ : CommRing R
M : Submonoid R
S : Type u_2
inst✝⁸ : CommRing S
inst✝⁷ : Algebra R S
P : Type u_3
inst✝⁶ : CommRing P
A : Type u_4
inst✝⁵ : CommRing A
inst✝⁴ : IsDomain A
K : Type u_5
inst✝³ : IsDomain R
inst✝² : Field K
inst✝¹ : Algebra R K
inst✝ : NoZeroSMulDivisors R K
this : Algebra (FractionRing R) K := liftAlgebra R K
⊢ Sort ?u.531432

Next tactic:
exact IsScalarTower R (FractionRing R) K
instance isScalarTower_liftAlgebra [IsDomain R] [Field K] [Algebra R K] [NoZeroSMulDivisors R K] :
    by letI := liftAlgebra R K; exact IsScalarTower R (FractionRing R) K := by
  

Proof state:
R : Type u_1
inst✝⁹ : CommRing R
M : Submonoid R
S : Type u_2
inst✝⁸ : CommRing S
inst✝⁷ : Algebra R S
P : Type u_3
inst✝⁶ : CommRing P
A : Type u_4
inst✝⁵ : CommRing A
inst✝⁴ : IsDomain A
K : Type u_5
inst✝³ : IsDomain R
inst✝² : Field K
inst✝¹ : Algebra R K
inst✝ : NoZeroSMulDivisors R K
⊢ IsScalarTower R (FractionRing R) K

Next tactic:
letI := liftAlgebra R K
instance isScalarTower_liftAlgebra [IsDomain R] [Field K] [Algebra R K] [NoZeroSMulDivisors R K] :
    by letI := liftAlgebra R K; exact IsScalarTower R (FractionRing R) K := by
  letI := liftAlgebra R K
  

Proof state:
R : Type u_1
inst✝⁹ : CommRing R
M : Submonoid R
S : Type u_2
inst✝⁸ : CommRing S
inst✝⁷ : Algebra R S
P : Type u_3
inst✝⁶ : CommRing P
A : Type u_4
inst✝⁵ : CommRing A
inst✝⁴ : IsDomain A
K : Type u_5
inst✝³ : IsDomain R
inst✝² : Field K
inst✝¹ : Algebra R K
inst✝ : NoZeroSMulDivisors R K
this : Algebra (FractionRing R) K := liftAlgebra R K
⊢ IsScalarTower R (FractionRing R) K

Next tactic:
exact IsScalarTower.of_algebraMap_eq fun x =>
    (IsFractionRing.lift_algebraMap (NoZeroSMulDivisors.algebraMap_injective R K) x).symm
instance [Algebra R A] [NoZeroSMulDivisors R A] : NoZeroSMulDivisors R (FractionRing A) := by
  

Proof state:
R : Type u_1
inst✝⁷ : CommRing R
M : Submonoid R
S : Type u_2
inst✝⁶ : CommRing S
inst✝⁵ : Algebra R S
P : Type u_3
inst✝⁴ : CommRing P
A : Type u_4
inst✝³ : CommRing A
inst✝² : IsDomain A
K : Type u_5
inst✝¹ : Algebra R A
inst✝ : NoZeroSMulDivisors R A
⊢ NoZeroSMulDivisors R (FractionRing A)

Next tactic:
apply NoZeroSMulDivisors.of_algebraMap_injective
instance [Algebra R A] [NoZeroSMulDivisors R A] : NoZeroSMulDivisors R (FractionRing A) := by
  apply NoZeroSMulDivisors.of_algebraMap_injective
  

Proof state:
case h
R : Type u_1
inst✝⁷ : CommRing R
M : Submonoid R
S : Type u_2
inst✝⁶ : CommRing S
inst✝⁵ : Algebra R S
P : Type u_3
inst✝⁴ : CommRing P
A : Type u_4
inst✝³ : CommRing A
inst✝² : IsDomain A
K : Type u_5
inst✝¹ : Algebra R A
inst✝ : NoZeroSMulDivisors R A
⊢ Function.Injective ⇑(algebraMap R (FractionRing A))

Next tactic:
rw [IsScalarTower.algebraMap_eq R A]
instance [Algebra R A] [NoZeroSMulDivisors R A] : NoZeroSMulDivisors R (FractionRing A) := by
  apply NoZeroSMulDivisors.of_algebraMap_injective
  rw [IsScalarTower.algebraMap_eq R A]
  

Proof state:
case h
R : Type u_1
inst✝⁷ : CommRing R
M : Submonoid R
S : Type u_2
inst✝⁶ : CommRing S
inst✝⁵ : Algebra R S
P : Type u_3
inst✝⁴ : CommRing P
A : Type u_4
inst✝³ : CommRing A
inst✝² : IsDomain A
K : Type u_5
inst✝¹ : Algebra R A
inst✝ : NoZeroSMulDivisors R A
⊢ Function.Injective ⇑(RingHom.comp (algebraMap A (FractionRing A)) (algebraMap R A))

Next tactic:
apply Function.Injective.comp (NoZeroSMulDivisors.algebraMap_injective A (FractionRing A))
    (NoZeroSMulDivisors.algebraMap_injective R A)
/-- **Ruzsa's triangle inequality**. Division version. -/
@[to_additive card_sub_mul_le_card_sub_mul_card_sub
      "**Ruzsa's triangle inequality**. Subtraction version."]
theorem card_div_mul_le_card_div_mul_card_div (A B C : Finset α) :
    (A / C).card * B.card ≤ (A / B).card * (B / C).card := by
  

Proof state:
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
A✝ B✝ C✝ A B C : Finset α
⊢ card (A / C) * card B ≤ card (A / B) * card (B / C)

Next tactic:
rw [← card_product (A / B), ← mul_one ((A / B) ×ˢ (B / C)).card]
/-- **Ruzsa's triangle inequality**. Division version. -/
@[to_additive card_sub_mul_le_card_sub_mul_card_sub
      "**Ruzsa's triangle inequality**. Subtraction version."]
theorem card_div_mul_le_card_div_mul_card_div (A B C : Finset α) :
    (A / C).card * B.card ≤ (A / B).card * (B / C).card := by
  rw [← card_product (A / B), ← mul_one ((A / B) ×ˢ (B / C)).card]
  

Proof state:
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
A✝ B✝ C✝ A B C : Finset α
⊢ card (A / C) * card B ≤ card ((A / B) ×ˢ (B / C)) * 1

Next tactic:
refine' card_mul_le_card_mul (fun b ac ↦ ac.1 * ac.2 = b) (fun x hx ↦ _)
    fun x _ ↦ card_le_one_iff.2 fun hu hv ↦
      ((mem_bipartiteBelow _).1 hu).2.symm.trans ((mem_bipartiteBelow _).1 hv).2
/-- **Ruzsa's triangle inequality**. Division version. -/
@[to_additive card_sub_mul_le_card_sub_mul_card_sub
      "**Ruzsa's triangle inequality**. Subtraction version."]
theorem card_div_mul_le_card_div_mul_card_div (A B C : Finset α) :
    (A / C).card * B.card ≤ (A / B).card * (B / C).card := by
  rw [← card_product (A / B), ← mul_one ((A / B) ×ˢ (B / C)).card]
  refine' card_mul_le_card_mul (fun b ac ↦ ac.1 * ac.2 = b) (fun x hx ↦ _)
    fun x _ ↦ card_le_one_iff.2 fun hu hv ↦
      ((mem_bipartiteBelow _).1 hu).2.symm.trans ((mem_bipartiteBelow _).1 hv).2
  

Proof state:
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
A✝ B✝ C✝ A B C : Finset α
x : α
hx : x ∈ A / C
⊢ card B ≤ card (bipartiteAbove (fun b ac => ac.1 * ac.2 = b) ((A / B) ×ˢ (B / C)) x)

Next tactic:
obtain ⟨a, c, ha, hc, rfl⟩ := mem_div.1 hx
/-- **Ruzsa's triangle inequality**. Division version. -/
@[to_additive card_sub_mul_le_card_sub_mul_card_sub
      "**Ruzsa's triangle inequality**. Subtraction version."]
theorem card_div_mul_le_card_div_mul_card_div (A B C : Finset α) :
    (A / C).card * B.card ≤ (A / B).card * (B / C).card := by
  rw [← card_product (A / B), ← mul_one ((A / B) ×ˢ (B / C)).card]
  refine' card_mul_le_card_mul (fun b ac ↦ ac.1 * ac.2 = b) (fun x hx ↦ _)
    fun x _ ↦ card_le_one_iff.2 fun hu hv ↦
      ((mem_bipartiteBelow _).1 hu).2.symm.trans ((mem_bipartiteBelow _).1 hv).2
  obtain ⟨a, c, ha, hc, rfl⟩ := mem_div.1 hx
  

Proof state:
case intro.intro.intro.intro
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
A✝ B✝ C✝ A B C : Finset α
a c : α
ha : a ∈ A
hc : c ∈ C
hx : a / c ∈ A / C
⊢ card B ≤ card (bipartiteAbove (fun b ac => ac.1 * ac.2 = b) ((A / B) ×ˢ (B / C)) (a / c))

Next tactic:
refine' card_le_card_of_inj_on (fun b ↦ (a / b, b / c)) (fun b hb ↦ _) fun b₁ _ b₂ _ h ↦ _
/-- **Ruzsa's triangle inequality**. Division version. -/
@[to_additive card_sub_mul_le_card_sub_mul_card_sub
      "**Ruzsa's triangle inequality**. Subtraction version."]
theorem card_div_mul_le_card_div_mul_card_div (A B C : Finset α) :
    (A / C).card * B.card ≤ (A / B).card * (B / C).card := by
  rw [← card_product (A / B), ← mul_one ((A / B) ×ˢ (B / C)).card]
  refine' card_mul_le_card_mul (fun b ac ↦ ac.1 * ac.2 = b) (fun x hx ↦ _)
    fun x _ ↦ card_le_one_iff.2 fun hu hv ↦
      ((mem_bipartiteBelow _).1 hu).2.symm.trans ((mem_bipartiteBelow _).1 hv).2
  obtain ⟨a, c, ha, hc, rfl⟩ := mem_div.1 hx
  refine' card_le_card_of_inj_on (fun b ↦ (a / b, b / c)) (fun b hb ↦ _) fun b₁ _ b₂ _ h ↦ _
  · 

Proof state:
case intro.intro.intro.intro.refine'_1
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
A✝ B✝ C✝ A B C : Finset α
a c : α
ha : a ∈ A
hc : c ∈ C
hx : a / c ∈ A / C
b : α
hb : b ∈ B
⊢ (fun b => (a / b, b / c)) b ∈ bipartiteAbove (fun b ac => ac.1 * ac.2 = b) ((A / B) ×ˢ (B / C)) (a / c)

Next tactic:
rw [mem_bipartiteAbove]
/-- **Ruzsa's triangle inequality**. Division version. -/
@[to_additive card_sub_mul_le_card_sub_mul_card_sub
      "**Ruzsa's triangle inequality**. Subtraction version."]
theorem card_div_mul_le_card_div_mul_card_div (A B C : Finset α) :
    (A / C).card * B.card ≤ (A / B).card * (B / C).card := by
  rw [← card_product (A / B), ← mul_one ((A / B) ×ˢ (B / C)).card]
  refine' card_mul_le_card_mul (fun b ac ↦ ac.1 * ac.2 = b) (fun x hx ↦ _)
    fun x _ ↦ card_le_one_iff.2 fun hu hv ↦
      ((mem_bipartiteBelow _).1 hu).2.symm.trans ((mem_bipartiteBelow _).1 hv).2
  obtain ⟨a, c, ha, hc, rfl⟩ := mem_div.1 hx
  refine' card_le_card_of_inj_on (fun b ↦ (a / b, b / c)) (fun b hb ↦ _) fun b₁ _ b₂ _ h ↦ _
  · rw [mem_bipartiteAbove]
    

Proof state:
case intro.intro.intro.intro.refine'_1
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
A✝ B✝ C✝ A B C : Finset α
a c : α
ha : a ∈ A
hc : c ∈ C
hx : a / c ∈ A / C
b : α
hb : b ∈ B
⊢ (fun b => (a / b, b / c)) b ∈ (A / B) ×ˢ (B / C) ∧
    ((fun b => (a / b, b / c)) b).1 * ((fun b => (a / b, b / c)) b).2 = a / c

Next tactic:
exact ⟨mk_mem_product (div_mem_div ha hb) (div_mem_div hb hc), div_mul_div_cancel' _ _ _⟩
/-- **Ruzsa's triangle inequality**. Division version. -/
@[to_additive card_sub_mul_le_card_sub_mul_card_sub
      "**Ruzsa's triangle inequality**. Subtraction version."]
theorem card_div_mul_le_card_div_mul_card_div (A B C : Finset α) :
    (A / C).card * B.card ≤ (A / B).card * (B / C).card := by
  rw [← card_product (A / B), ← mul_one ((A / B) ×ˢ (B / C)).card]
  refine' card_mul_le_card_mul (fun b ac ↦ ac.1 * ac.2 = b) (fun x hx ↦ _)
    fun x _ ↦ card_le_one_iff.2 fun hu hv ↦
      ((mem_bipartiteBelow _).1 hu).2.symm.trans ((mem_bipartiteBelow _).1 hv).2
  obtain ⟨a, c, ha, hc, rfl⟩ := mem_div.1 hx
  refine' card_le_card_of_inj_on (fun b ↦ (a / b, b / c)) (fun b hb ↦ _) fun b₁ _ b₂ _ h ↦ _
  · rw [mem_bipartiteAbove]
    exact ⟨mk_mem_product (div_mem_div ha hb) (div_mem_div hb hc), div_mul_div_cancel' _ _ _⟩
  · 

Proof state:
case intro.intro.intro.intro.refine'_2
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
A✝ B✝ C✝ A B C : Finset α
a c : α
ha : a ∈ A
hc : c ∈ C
hx : a / c ∈ A / C
b₁ : α
x✝¹ : b₁ ∈ B
b₂ : α
x✝ : b₂ ∈ B
h : (fun b => (a / b, b / c)) b₁ = (fun b => (a / b, b / c)) b₂
⊢ b₁ = b₂

Next tactic:
exact div_right_injective (Prod.ext_iff.1 h).1
/-- **Ruzsa's triangle inequality**. Div-mul-mul version. -/
@[to_additive card_sub_mul_le_card_add_mul_card_add
      "**Ruzsa's triangle inequality**. Sub-add-add version."]
theorem card_div_mul_le_card_mul_mul_card_mul (A B C : Finset α) :
    (A / C).card * B.card ≤ (A * B).card * (B * C).card := by
  

Proof state:
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
A✝ B✝ C✝ A B C : Finset α
⊢ card (A / C) * card B ≤ card (A * B) * card (B * C)

Next tactic:
rw [← div_inv_eq_mul, ← card_inv B, ← card_inv (B * C), mul_inv, ← div_eq_mul_inv]
/-- **Ruzsa's triangle inequality**. Div-mul-mul version. -/
@[to_additive card_sub_mul_le_card_add_mul_card_add
      "**Ruzsa's triangle inequality**. Sub-add-add version."]
theorem card_div_mul_le_card_mul_mul_card_mul (A B C : Finset α) :
    (A / C).card * B.card ≤ (A * B).card * (B * C).card := by
  rw [← div_inv_eq_mul, ← card_inv B, ← card_inv (B * C), mul_inv, ← div_eq_mul_inv]
  

Proof state:
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
A✝ B✝ C✝ A B C : Finset α
⊢ card (A / C) * card B⁻¹ ≤ card (A / B⁻¹) * card (B⁻¹ / C)

Next tactic:
exact card_div_mul_le_card_div_mul_card_div _ _ _
/-- **Ruzsa's triangle inequality**. Mul-div-div version. -/
@[to_additive card_add_mul_le_card_sub_mul_card_add
      "**Ruzsa's triangle inequality**. Add-sub-sub version."]
theorem card_mul_mul_le_card_div_mul_card_mul (A B C : Finset α) :
    (A * C).card * B.card ≤ (A / B).card * (B * C).card := by
  

Proof state:
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
A✝ B✝ C✝ A B C : Finset α
⊢ card (A * C) * card B ≤ card (A / B) * card (B * C)

Next tactic:
rw [← div_inv_eq_mul, ← div_inv_eq_mul B]
/-- **Ruzsa's triangle inequality**. Mul-div-div version. -/
@[to_additive card_add_mul_le_card_sub_mul_card_add
      "**Ruzsa's triangle inequality**. Add-sub-sub version."]
theorem card_mul_mul_le_card_div_mul_card_mul (A B C : Finset α) :
    (A * C).card * B.card ≤ (A / B).card * (B * C).card := by
  rw [← div_inv_eq_mul, ← div_inv_eq_mul B]
  

Proof state:
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
A✝ B✝ C✝ A B C : Finset α
⊢ card (A / C⁻¹) * card B ≤ card (A / B) * card (B / C⁻¹)

Next tactic:
exact card_div_mul_le_card_div_mul_card_div _ _ _
/-- **Ruzsa's triangle inequality**. Mul-mul-div version. -/
@[to_additive card_add_mul_le_card_add_mul_card_sub
      "**Ruzsa's triangle inequality**. Add-add-sub version."]
theorem card_mul_mul_le_card_mul_mul_card_div (A B C : Finset α) :
    (A * C).card * B.card ≤ (A * B).card * (B / C).card := by
  

Proof state:
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
A✝ B✝ C✝ A B C : Finset α
⊢ card (A * C) * card B ≤ card (A * B) * card (B / C)

Next tactic:
rw [← div_inv_eq_mul, div_eq_mul_inv B]
/-- **Ruzsa's triangle inequality**. Mul-mul-div version. -/
@[to_additive card_add_mul_le_card_add_mul_card_sub
      "**Ruzsa's triangle inequality**. Add-add-sub version."]
theorem card_mul_mul_le_card_mul_mul_card_div (A B C : Finset α) :
    (A * C).card * B.card ≤ (A * B).card * (B / C).card := by
  rw [← div_inv_eq_mul, div_eq_mul_inv B]
  

Proof state:
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
A✝ B✝ C✝ A B C : Finset α
⊢ card (A / C⁻¹) * card B ≤ card (A * B) * card (B * C⁻¹)

Next tactic:
exact card_div_mul_le_card_mul_mul_card_mul _ _ _
@[to_additive]
theorem mul_pluennecke_petridis (C : Finset α)
    (hA : ∀ (A') (_ : A' ⊆ A), (A * B).card * A'.card ≤ (A' * B).card * A.card) :
    (A * B * C).card * A.card ≤ (A * B).card * (A * C).card := by
  

Proof state:
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
A B C✝ C : Finset α
hA : ∀ A' ⊆ A, card (A * B) * card A' ≤ card (A' * B) * card A
⊢ card (A * B * C) * card A ≤ card (A * B) * card (A * C)

Next tactic:
induction' C using Finset.induction_on with x C _ ih
@[to_additive]
theorem mul_pluennecke_petridis (C : Finset α)
    (hA : ∀ (A') (_ : A' ⊆ A), (A * B).card * A'.card ≤ (A' * B).card * A.card) :
    (A * B * C).card * A.card ≤ (A * B).card * (A * C).card := by
  induction' C using Finset.induction_on with x C _ ih
  · 

Proof state:
case empty
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
A B C : Finset α
hA : ∀ A' ⊆ A, card (A * B) * card A' ≤ card (A' * B) * card A
⊢ card (A * B * ∅) * card A ≤ card (A * B) * card (A * ∅)

Next tactic:
simp
@[to_additive]
theorem mul_pluennecke_petridis (C : Finset α)
    (hA : ∀ (A') (_ : A' ⊆ A), (A * B).card * A'.card ≤ (A' * B).card * A.card) :
    (A * B * C).card * A.card ≤ (A * B).card * (A * C).card := by
  induction' C using Finset.induction_on with x C _ ih
  · simp
  

Proof state:
case insert
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
A B C✝ : Finset α
hA : ∀ A' ⊆ A, card (A * B) * card A' ≤ card (A' * B) * card A
x : α
C : Finset α
a✝ : x ∉ C
ih : card (A * B * C) * card A ≤ card (A * B) * card (A * C)
⊢ card (A * B * insert x C) * card A ≤ card (A * B) * card (A * insert x C)

Next tactic:
set A' := A ∩ (A * C / {x}) with hA'
@[to_additive]
theorem mul_pluennecke_petridis (C : Finset α)
    (hA : ∀ (A') (_ : A' ⊆ A), (A * B).card * A'.card ≤ (A' * B).card * A.card) :
    (A * B * C).card * A.card ≤ (A * B).card * (A * C).card := by
  induction' C using Finset.induction_on with x C _ ih
  · simp
  set A' := A ∩ (A * C / {x}) with hA'
  

Proof state:
case insert
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
A B C✝ : Finset α
hA : ∀ A' ⊆ A, card (A * B) * card A' ≤ card (A' * B) * card A
x : α
C : Finset α
a✝ : x ∉ C
ih : card (A * B * C) * card A ≤ card (A * B) * card (A * C)
A' : Finset α := A ∩ (A * C / {x})
hA' : A' = A ∩ (A * C / {x})
⊢ card (A * B * insert x C) * card A ≤ card (A * B) * card (A * insert x C)

Next tactic:
set C' := insert x C with hC'
@[to_additive]
theorem mul_pluennecke_petridis (C : Finset α)
    (hA : ∀ (A') (_ : A' ⊆ A), (A * B).card * A'.card ≤ (A' * B).card * A.card) :
    (A * B * C).card * A.card ≤ (A * B).card * (A * C).card := by
  induction' C using Finset.induction_on with x C _ ih
  · simp
  set A' := A ∩ (A * C / {x}) with hA'
  set C' := insert x C with hC'
  

Proof state:
case insert
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
A B C✝ : Finset α
hA : ∀ A' ⊆ A, card (A * B) * card A' ≤ card (A' * B) * card A
x : α
C : Finset α
a✝ : x ∉ C
ih : card (A * B * C) * card A ≤ card (A * B) * card (A * C)
A' : Finset α := A ∩ (A * C / {x})
hA' : A' = A ∩ (A * C / {x})
C' : Finset α := insert x C
hC' : C' = insert x C
⊢ card (A * B * C') * card A ≤ card (A * B) * card (A * C')

Next tactic:
have h₀ : A' * {x} = A * {x} ∩ (A * C) := by
    rw [hA', inter_mul_singleton, (isUnit_singleton x).div_mul_cancel]