l3lab/ntp-mathlib huggingface dataset:
@[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'
  have h₀ : A' * {x} = A * {x} ∩ (A * C) := by
    

Proof state:
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
⊢ A' * {x} = A * {x} ∩ (A * C)

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

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
h₀ : A' * {x} = A * {x} ∩ (A * C)
⊢ card (A * B * C') * card A ≤ card (A * B) * card (A * C')

Next tactic:
have h₁ : A * B * C' = A * B * C ∪ (A * B * {x}) \ (A' * B * {x}) := by
    rw [hC', insert_eq, union_comm, mul_union]
    refine' (sup_sdiff_eq_sup _).symm
    rw [mul_right_comm, mul_right_comm A, h₀]
    exact mul_subset_mul_right (inter_subset_right _ _)
@[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'
  have h₀ : A' * {x} = A * {x} ∩ (A * C) := by
    rw [hA', inter_mul_singleton, (isUnit_singleton x).div_mul_cancel]
  have h₁ : A * B * C' = A * B * C ∪ (A * B * {x}) \ (A' * B * {x}) := by
    

Proof state:
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
h₀ : A' * {x} = A * {x} ∩ (A * C)
⊢ A * B * C' = A * B * C ∪ (A * B * {x}) \ (A' * B * {x})

Next tactic:
rw [hC', insert_eq, union_comm, mul_union]
@[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'
  have h₀ : A' * {x} = A * {x} ∩ (A * C) := by
    rw [hA', inter_mul_singleton, (isUnit_singleton x).div_mul_cancel]
  have h₁ : A * B * C' = A * B * C ∪ (A * B * {x}) \ (A' * B * {x}) := by
    rw [hC', insert_eq, union_comm, mul_union]
    

Proof state:
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
h₀ : A' * {x} = A * {x} ∩ (A * C)
⊢ A * B * C ∪ A * B * {x} = A * B * C ∪ (A * B * {x}) \ (A' * B * {x})

Next tactic:
refine' (sup_sdiff_eq_sup _).symm
@[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'
  have h₀ : A' * {x} = A * {x} ∩ (A * C) := by
    rw [hA', inter_mul_singleton, (isUnit_singleton x).div_mul_cancel]
  have h₁ : A * B * C' = A * B * C ∪ (A * B * {x}) \ (A' * B * {x}) := by
    rw [hC', insert_eq, union_comm, mul_union]
    refine' (sup_sdiff_eq_sup _).symm
    

Proof state:
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
h₀ : A' * {x} = A * {x} ∩ (A * C)
⊢ A' * B * {x} ≤ A * B * C

Next tactic:
rw [mul_right_comm, mul_right_comm A, h₀]
@[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'
  have h₀ : A' * {x} = A * {x} ∩ (A * C) := by
    rw [hA', inter_mul_singleton, (isUnit_singleton x).div_mul_cancel]
  have h₁ : A * B * C' = A * B * C ∪ (A * B * {x}) \ (A' * B * {x}) := by
    rw [hC', insert_eq, union_comm, mul_union]
    refine' (sup_sdiff_eq_sup _).symm
    rw [mul_right_comm, mul_right_comm A, h₀]
    

Proof state:
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
h₀ : A' * {x} = A * {x} ∩ (A * C)
⊢ A * {x} ∩ (A * C) * B ≤ A * C * B

Next tactic:
exact mul_subset_mul_right (inter_subset_right _ _)
@[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'
  have h₀ : A' * {x} = A * {x} ∩ (A * C) := by
    rw [hA', inter_mul_singleton, (isUnit_singleton x).div_mul_cancel]
  have h₁ : A * B * C' = A * B * C ∪ (A * B * {x}) \ (A' * B * {x}) := by
    rw [hC', insert_eq, union_comm, mul_union]
    refine' (sup_sdiff_eq_sup _).symm
    rw [mul_right_comm, mul_right_comm A, h₀]
    exact mul_subset_mul_right (inter_subset_right _ _)
  

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
h₀ : A' * {x} = A * {x} ∩ (A * C)
h₁ : A * B * C' = A * B * C ∪ (A * B * {x}) \ (A' * B * {x})
⊢ card (A * B * C') * card A ≤ card (A * B) * card (A * C')

Next tactic:
have h₂ : A' * B * {x} ⊆ A * B * {x} :=
    mul_subset_mul_right (mul_subset_mul_right <| inter_subset_left _ _)
@[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'
  have h₀ : A' * {x} = A * {x} ∩ (A * C) := by
    rw [hA', inter_mul_singleton, (isUnit_singleton x).div_mul_cancel]
  have h₁ : A * B * C' = A * B * C ∪ (A * B * {x}) \ (A' * B * {x}) := by
    rw [hC', insert_eq, union_comm, mul_union]
    refine' (sup_sdiff_eq_sup _).symm
    rw [mul_right_comm, mul_right_comm A, h₀]
    exact mul_subset_mul_right (inter_subset_right _ _)
  have h₂ : A' * B * {x} ⊆ A * B * {x} :=
    mul_subset_mul_right (mul_subset_mul_right <| inter_subset_left _ _)
  

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
h₀ : A' * {x} = A * {x} ∩ (A * C)
h₁ : A * B * C' = A * B * C ∪ (A * B * {x}) \ (A' * B * {x})
h₂ : A' * B * {x} ⊆ A * B * {x}
⊢ card (A * B * C') * card A ≤ card (A * B) * card (A * C')

Next tactic:
have h₃ : (A * B * C').card ≤ (A * B * C).card + (A * B).card - (A' * B).card := by
    rw [h₁]
    refine' (card_union_le _ _).trans_eq _
    rw [card_sdiff h₂, ← add_tsub_assoc_of_le (card_le_of_subset h₂), card_mul_singleton,
      card_mul_singleton]
@[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'
  have h₀ : A' * {x} = A * {x} ∩ (A * C) := by
    rw [hA', inter_mul_singleton, (isUnit_singleton x).div_mul_cancel]
  have h₁ : A * B * C' = A * B * C ∪ (A * B * {x}) \ (A' * B * {x}) := by
    rw [hC', insert_eq, union_comm, mul_union]
    refine' (sup_sdiff_eq_sup _).symm
    rw [mul_right_comm, mul_right_comm A, h₀]
    exact mul_subset_mul_right (inter_subset_right _ _)
  have h₂ : A' * B * {x} ⊆ A * B * {x} :=
    mul_subset_mul_right (mul_subset_mul_right <| inter_subset_left _ _)
  have h₃ : (A * B * C').card ≤ (A * B * C).card + (A * B).card - (A' * B).card := by
    

Proof state:
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
h₀ : A' * {x} = A * {x} ∩ (A * C)
h₁ : A * B * C' = A * B * C ∪ (A * B * {x}) \ (A' * B * {x})
h₂ : A' * B * {x} ⊆ A * B * {x}
⊢ card (A * B * C') ≤ card (A * B * C) + card (A * B) - card (A' * B)

Next tactic:
rw [h₁]
@[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'
  have h₀ : A' * {x} = A * {x} ∩ (A * C) := by
    rw [hA', inter_mul_singleton, (isUnit_singleton x).div_mul_cancel]
  have h₁ : A * B * C' = A * B * C ∪ (A * B * {x}) \ (A' * B * {x}) := by
    rw [hC', insert_eq, union_comm, mul_union]
    refine' (sup_sdiff_eq_sup _).symm
    rw [mul_right_comm, mul_right_comm A, h₀]
    exact mul_subset_mul_right (inter_subset_right _ _)
  have h₂ : A' * B * {x} ⊆ A * B * {x} :=
    mul_subset_mul_right (mul_subset_mul_right <| inter_subset_left _ _)
  have h₃ : (A * B * C').card ≤ (A * B * C).card + (A * B).card - (A' * B).card := by
    rw [h₁]
    

Proof state:
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
h₀ : A' * {x} = A * {x} ∩ (A * C)
h₁ : A * B * C' = A * B * C ∪ (A * B * {x}) \ (A' * B * {x})
h₂ : A' * B * {x} ⊆ A * B * {x}
⊢ card (A * B * C ∪ (A * B * {x}) \ (A' * B * {x})) ≤ card (A * B * C) + card (A * B) - card (A' * B)

Next tactic:
refine' (card_union_le _ _).trans_eq _
@[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'
  have h₀ : A' * {x} = A * {x} ∩ (A * C) := by
    rw [hA', inter_mul_singleton, (isUnit_singleton x).div_mul_cancel]
  have h₁ : A * B * C' = A * B * C ∪ (A * B * {x}) \ (A' * B * {x}) := by
    rw [hC', insert_eq, union_comm, mul_union]
    refine' (sup_sdiff_eq_sup _).symm
    rw [mul_right_comm, mul_right_comm A, h₀]
    exact mul_subset_mul_right (inter_subset_right _ _)
  have h₂ : A' * B * {x} ⊆ A * B * {x} :=
    mul_subset_mul_right (mul_subset_mul_right <| inter_subset_left _ _)
  have h₃ : (A * B * C').card ≤ (A * B * C).card + (A * B).card - (A' * B).card := by
    rw [h₁]
    refine' (card_union_le _ _).trans_eq _
    

Proof state:
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
h₀ : A' * {x} = A * {x} ∩ (A * C)
h₁ : A * B * C' = A * B * C ∪ (A * B * {x}) \ (A' * B * {x})
h₂ : A' * B * {x} ⊆ A * B * {x}
⊢ card (A * B * C) + card ((A * B * {x}) \ (A' * B * {x})) = card (A * B * C) + card (A * B) - card (A' * B)

Next tactic:
rw [card_sdiff h₂, ← add_tsub_assoc_of_le (card_le_of_subset h₂), card_mul_singleton,
      card_mul_singleton]
@[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'
  have h₀ : A' * {x} = A * {x} ∩ (A * C) := by
    rw [hA', inter_mul_singleton, (isUnit_singleton x).div_mul_cancel]
  have h₁ : A * B * C' = A * B * C ∪ (A * B * {x}) \ (A' * B * {x}) := by
    rw [hC', insert_eq, union_comm, mul_union]
    refine' (sup_sdiff_eq_sup _).symm
    rw [mul_right_comm, mul_right_comm A, h₀]
    exact mul_subset_mul_right (inter_subset_right _ _)
  have h₂ : A' * B * {x} ⊆ A * B * {x} :=
    mul_subset_mul_right (mul_subset_mul_right <| inter_subset_left _ _)
  have h₃ : (A * B * C').card ≤ (A * B * C).card + (A * B).card - (A' * B).card := by
    rw [h₁]
    refine' (card_union_le _ _).trans_eq _
    rw [card_sdiff h₂, ← add_tsub_assoc_of_le (card_le_of_subset h₂), card_mul_singleton,
      card_mul_singleton]
  

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
h₀ : A' * {x} = A * {x} ∩ (A * C)
h₁ : A * B * C' = A * B * C ∪ (A * B * {x}) \ (A' * B * {x})
h₂ : A' * B * {x} ⊆ A * B * {x}
h₃ : card (A * B * C') ≤ card (A * B * C) + card (A * B) - card (A' * B)
⊢ card (A * B * C') * card A ≤ card (A * B) * card (A * C')

Next tactic:
refine' (mul_le_mul_right' h₃ _).trans _
@[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'
  have h₀ : A' * {x} = A * {x} ∩ (A * C) := by
    rw [hA', inter_mul_singleton, (isUnit_singleton x).div_mul_cancel]
  have h₁ : A * B * C' = A * B * C ∪ (A * B * {x}) \ (A' * B * {x}) := by
    rw [hC', insert_eq, union_comm, mul_union]
    refine' (sup_sdiff_eq_sup _).symm
    rw [mul_right_comm, mul_right_comm A, h₀]
    exact mul_subset_mul_right (inter_subset_right _ _)
  have h₂ : A' * B * {x} ⊆ A * B * {x} :=
    mul_subset_mul_right (mul_subset_mul_right <| inter_subset_left _ _)
  have h₃ : (A * B * C').card ≤ (A * B * C).card + (A * B).card - (A' * B).card := by
    rw [h₁]
    refine' (card_union_le _ _).trans_eq _
    rw [card_sdiff h₂, ← add_tsub_assoc_of_le (card_le_of_subset h₂), card_mul_singleton,
      card_mul_singleton]
  refine' (mul_le_mul_right' h₃ _).trans _
  

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
h₀ : A' * {x} = A * {x} ∩ (A * C)
h₁ : A * B * C' = A * B * C ∪ (A * B * {x}) \ (A' * B * {x})
h₂ : A' * B * {x} ⊆ A * B * {x}
h₃ : card (A * B * C') ≤ card (A * B * C) + card (A * B) - card (A' * B)
⊢ (card (A * B * C) + card (A * B) - card (A' * B)) * card A ≤ card (A * B) * card (A * C')

Next tactic:
rw [tsub_mul, add_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
  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'
  have h₀ : A' * {x} = A * {x} ∩ (A * C) := by
    rw [hA', inter_mul_singleton, (isUnit_singleton x).div_mul_cancel]
  have h₁ : A * B * C' = A * B * C ∪ (A * B * {x}) \ (A' * B * {x}) := by
    rw [hC', insert_eq, union_comm, mul_union]
    refine' (sup_sdiff_eq_sup _).symm
    rw [mul_right_comm, mul_right_comm A, h₀]
    exact mul_subset_mul_right (inter_subset_right _ _)
  have h₂ : A' * B * {x} ⊆ A * B * {x} :=
    mul_subset_mul_right (mul_subset_mul_right <| inter_subset_left _ _)
  have h₃ : (A * B * C').card ≤ (A * B * C).card + (A * B).card - (A' * B).card := by
    rw [h₁]
    refine' (card_union_le _ _).trans_eq _
    rw [card_sdiff h₂, ← add_tsub_assoc_of_le (card_le_of_subset h₂), card_mul_singleton,
      card_mul_singleton]
  refine' (mul_le_mul_right' h₃ _).trans _
  rw [tsub_mul, add_mul]
  

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
h₀ : A' * {x} = A * {x} ∩ (A * C)
h₁ : A * B * C' = A * B * C ∪ (A * B * {x}) \ (A' * B * {x})
h₂ : A' * B * {x} ⊆ A * B * {x}
h₃ : card (A * B * C') ≤ card (A * B * C) + card (A * B) - card (A' * B)
⊢ card (A * B * C) * card A + card (A * B) * card A - card (A' * B) * card A ≤ card (A * B) * card (A * C')

Next tactic:
refine' (tsub_le_tsub (add_le_add_right ih _) <| hA _ <| inter_subset_left _ _).trans_eq _
@[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'
  have h₀ : A' * {x} = A * {x} ∩ (A * C) := by
    rw [hA', inter_mul_singleton, (isUnit_singleton x).div_mul_cancel]
  have h₁ : A * B * C' = A * B * C ∪ (A * B * {x}) \ (A' * B * {x}) := by
    rw [hC', insert_eq, union_comm, mul_union]
    refine' (sup_sdiff_eq_sup _).symm
    rw [mul_right_comm, mul_right_comm A, h₀]
    exact mul_subset_mul_right (inter_subset_right _ _)
  have h₂ : A' * B * {x} ⊆ A * B * {x} :=
    mul_subset_mul_right (mul_subset_mul_right <| inter_subset_left _ _)
  have h₃ : (A * B * C').card ≤ (A * B * C).card + (A * B).card - (A' * B).card := by
    rw [h₁]
    refine' (card_union_le _ _).trans_eq _
    rw [card_sdiff h₂, ← add_tsub_assoc_of_le (card_le_of_subset h₂), card_mul_singleton,
      card_mul_singleton]
  refine' (mul_le_mul_right' h₃ _).trans _
  rw [tsub_mul, add_mul]
  refine' (tsub_le_tsub (add_le_add_right ih _) <| hA _ <| inter_subset_left _ _).trans_eq _
  

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
h₀ : A' * {x} = A * {x} ∩ (A * C)
h₁ : A * B * C' = A * B * C ∪ (A * B * {x}) \ (A' * B * {x})
h₂ : A' * B * {x} ⊆ A * B * {x}
h₃ : card (A * B * C') ≤ card (A * B * C) + card (A * B) - card (A' * B)
⊢ card (A * B) * card (A * C) + card (A * B) * card A - card (A * B) * card (A ∩ (A * C / {x})) =
    card (A * B) * card (A * C')

Next tactic:
rw [← mul_add, ← mul_tsub, ← hA', hC', insert_eq, mul_union, ← card_mul_singleton A x, ←
    card_mul_singleton A' x, add_comm (card _), h₀,
    eq_tsub_of_add_eq (card_union_add_card_inter _ _)]
@[to_additive]
private theorem mul_aux (hA : A.Nonempty) (hAB : A ⊆ B)
    (h : ∀ A' ∈ B.powerset.erase ∅, ((A * C).card : ℚ≥0) / ↑A.card ≤ (A' * C).card / ↑A'.card) :
    ∀ (A') (_ : A' ⊆ A), (A * C).card * A'.card ≤ (A' * C).card * A.card := by
  

Proof state:
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
A B C : Finset α
hA : Finset.Nonempty A
hAB : A ⊆ B
h : ∀ A' ∈ erase (powerset B) ∅, ↑(card (A * C)) / ↑(card A) ≤ ↑(card (A' * C)) / ↑(card A')
⊢ ∀ A' ⊆ A, card (A * C) * card A' ≤ card (A' * C) * card A

Next tactic:
rintro A' hAA'
@[to_additive]
private theorem mul_aux (hA : A.Nonempty) (hAB : A ⊆ B)
    (h : ∀ A' ∈ B.powerset.erase ∅, ((A * C).card : ℚ≥0) / ↑A.card ≤ (A' * C).card / ↑A'.card) :
    ∀ (A') (_ : A' ⊆ A), (A * C).card * A'.card ≤ (A' * C).card * A.card := by
  rintro A' hAA'
  

Proof state:
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
A B C : Finset α
hA : Finset.Nonempty A
hAB : A ⊆ B
h : ∀ A' ∈ erase (powerset B) ∅, ↑(card (A * C)) / ↑(card A) ≤ ↑(card (A' * C)) / ↑(card A')
A' : Finset α
hAA' : A' ⊆ A
⊢ card (A * C) * card A' ≤ card (A' * C) * card A

Next tactic:
obtain rfl | hA' := A'.eq_empty_or_nonempty
@[to_additive]
private theorem mul_aux (hA : A.Nonempty) (hAB : A ⊆ B)
    (h : ∀ A' ∈ B.powerset.erase ∅, ((A * C).card : ℚ≥0) / ↑A.card ≤ (A' * C).card / ↑A'.card) :
    ∀ (A') (_ : A' ⊆ A), (A * C).card * A'.card ≤ (A' * C).card * A.card := by
  rintro A' hAA'
  obtain rfl | hA' := A'.eq_empty_or_nonempty
  · 

Proof state:
case inl
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
A B C : Finset α
hA : Finset.Nonempty A
hAB : A ⊆ B
h : ∀ A' ∈ erase (powerset B) ∅, ↑(card (A * C)) / ↑(card A) ≤ ↑(card (A' * C)) / ↑(card A')
hAA' : ∅ ⊆ A
⊢ card (A * C) * card ∅ ≤ card (∅ * C) * card A

Next tactic:
simp