-- Mathlib/Combinatorics/Additive/PluenneckeRuzsa.lean
theorem pluennecke_ruzsa_inequality_pow_div_pow_mul (hA : A.Nonempty) (B : Finset G) (m n : ℕ) :
    #(B ^ m / B ^ n) ≤ (#(A * B) / #A : ℚ≥0) ^ (m + n) * #A

Proof state:
G : Type u_1
inst✝¹ : DecidableEq G
inst✝ : CommGroup G
A : Finset G
hA : A.Nonempty
B : Finset G
m n : ℕ
hA' : A ∈ A.powerset.erase ∅
C : Finset G
hCmin : ∀ x' ∈ A.powerset.erase ∅, ↑(#(C * B)) / ↑(#C) ≤ ↑(#(x' * B)) / ↑(#x')
hC : C.Nonempty
hCA : C ⊆ A
⊢ 0 < ↑(#C)

Tactic:
positivity

Result:
no goals

Proof state:
G : Type u_1
inst✝¹ : DecidableEq G
inst✝ : CommGroup G
A : Finset G
hA : A.Nonempty
B : Finset G
m n : ℕ
hA' : A ∈ A.powerset.erase ∅
C : Finset G
hCmin : ∀ x' ∈ A.powerset.erase ∅, ↑(#(C * B)) / ↑(#C) ≤ ↑(#(x' * B)) / ↑(#x')
hC : C.Nonempty
hCA : C ⊆ A
⊢ ↑(#(B ^ m * C)) * ↑(#(B ^ n * C)) = ↑(#(C * B ^ m)) * ↑(#(C * B ^ n))

Tactic:
simp_rw [mul_comm]

Result:
no goals

Proof state:
G : Type u_1
inst✝¹ : DecidableEq G
inst✝ : CommGroup G
A : Finset G
hA : A.Nonempty
B : Finset G
m n : ℕ
hA' : A ∈ A.powerset.erase ∅
C : Finset G
hCmin : ∀ x' ∈ A.powerset.erase ∅, ↑(#(C * B)) / ↑(#C) ≤ ↑(#(x' * B)) / ↑(#x')
hC : C.Nonempty
hCA : C ⊆ A
⊢ (↑(#(C * B)) / ↑(#C)) ^ (m + n) * ↑(#C) * ↑(#C) ≤ (↑(#(A * B)) / ↑(#A)) ^ (m + n) * ↑(#A) * ↑(#C)

Tactic:
gcongr (?_ ^ _) * #?_ * _

Result:
case bc.h₁.hab
G : Type u_1
inst✝¹ : DecidableEq G
inst✝ : CommGroup G
A : Finset G
hA : A.Nonempty
B : Finset G
m n : ℕ
hA' : A ∈ A.powerset.erase ∅
C : Finset G
hCmin : ∀ x' ∈ A.powerset.erase ∅, ↑(#(C * B)) / ↑(#C) ≤ ↑(#(x' * B)) / ↑(#x')
hC : C.Nonempty
hCA : C ⊆ A
⊢ ↑(#(C * B)) / ↑(#C) ≤ ↑(#(A * B)) / ↑(#A)

Proof state:
G : Type u_1
inst✝¹ : DecidableEq G
inst✝ : CommGroup G
A : Finset G
hA : A.Nonempty
B : Finset G
m n : ℕ
⊢ ↑(#(B ^ m / B ^ n)) ≤ (↑(#(A * B)) / ↑(#A)) ^ (m + n) * ↑(#A)

Tactic:
have hA' : A ∈ A.powerset.erase ∅ := mem_erase_of_ne_of_mem hA.ne_empty (mem_powerset_self _)

Result:
G : Type u_1
inst✝¹ : DecidableEq G
inst✝ : CommGroup G
A : Finset G
hA : A.Nonempty
B : Finset G
m n : ℕ
hA' : A ∈ A.powerset.erase ∅
⊢ ↑(#(B ^ m / B ^ n)) ≤ (↑(#(A * B)) / ↑(#A)) ^ (m + n) * ↑(#A)

Proof state:
case bc.h₁.hab
G : Type u_1
inst✝¹ : DecidableEq G
inst✝ : CommGroup G
A : Finset G
hA : A.Nonempty
B : Finset G
m n : ℕ
hA' : A ∈ A.powerset.erase ∅
C : Finset G
hCmin : ∀ x' ∈ A.powerset.erase ∅, ↑(#(C * B)) / ↑(#C) ≤ ↑(#(x' * B)) / ↑(#x')
hC : C.Nonempty
hCA : C ⊆ A
⊢ ↑(#(C * B)) / ↑(#C) ≤ ↑(#(A * B)) / ↑(#A)

Tactic:
exact hCmin _ hA'

Result:
no goals

Proof state:
G : Type u_1
inst✝¹ : DecidableEq G
inst✝ : CommGroup G
A : Finset G
hA : A.Nonempty
B : Finset G
m n : ℕ
hA' : A ∈ A.powerset.erase ∅
C : Finset G
hCmin : ∀ x' ∈ A.powerset.erase ∅, ↑(#(C * B)) / ↑(#C) ≤ ↑(#(x' * B)) / ↑(#x')
hC : C.Nonempty
hCA : C ⊆ A
⊢ (↑(#(C * B)) / ↑(#C)) ^ m * ↑(#C) * ((↑(#(C * B)) / ↑(#C)) ^ n * ↑(#C)) =
    (↑(#(C * B)) / ↑(#C)) ^ (m + n) * ↑(#C) * ↑(#C)

Tactic:
ring

Result:
no goals

Proof state:
G : Type u_1
inst✝¹ : DecidableEq G
inst✝ : CommGroup G
A : Finset G
hA : A.Nonempty
B : Finset G
m n : ℕ
hA' : A ∈ A.powerset.erase ∅
C : Finset G
hCmin : ∀ x' ∈ A.powerset.erase ∅, ↑(#(C * B)) / ↑(#C) ≤ ↑(#(x' * B)) / ↑(#x')
hC : C.Nonempty
hCA : C ⊆ A
⊢ ↑(#(C * B ^ m)) * ↑(#(C * B ^ n)) ≤ (↑(#(C * B)) / ↑(#C)) ^ m * ↑(#C) * ((↑(#(C * B)) / ↑(#C)) ^ n * ↑(#C))

Tactic:
gcongr <;> exact card_mul_pow_le (mul_aux hC hCA hCmin) _

Result:
no goals

Proof state:
case intro.intro.intro
G : Type u_1
inst✝¹ : DecidableEq G
inst✝ : CommGroup G
A : Finset G
hA : A.Nonempty
B : Finset G
m n : ℕ
hA' : A ∈ A.powerset.erase ∅
C : Finset G
hCmin : ∀ x' ∈ A.powerset.erase ∅, ↑(#(C * B)) / ↑(#C) ≤ ↑(#(x' * B)) / ↑(#x')
hC : C.Nonempty
hCA : C ⊆ A
⊢ ↑(#(B ^ m / B ^ n)) ≤ (↑(#(A * B)) / ↑(#A)) ^ (m + n) * ↑(#A)

Tactic:
refine le_of_mul_le_mul_right ?_ (by positivity : (0 : ℚ≥0) < #C)

Result:
case intro.intro.intro
G : Type u_1
inst✝¹ : DecidableEq G
inst✝ : CommGroup G
A : Finset G
hA : A.Nonempty
B : Finset G
m n : ℕ
hA' : A ∈ A.powerset.erase ∅
C : Finset G
hCmin : ∀ x' ∈ A.powerset.erase ∅, ↑(#(C * B)) / ↑(#C) ≤ ↑(#(x' * B)) / ↑(#x')
hC : C.Nonempty
hCA : C ⊆ A
⊢ ↑(#(B ^ m / B ^ n)) * ↑(#C) ≤ (↑(#(A * B)) / ↑(#A)) ^ (m + n) * ↑(#A) * ↑(#C)

Proof state:
case intro.intro
G : Type u_1
inst✝¹ : DecidableEq G
inst✝ : CommGroup G
A : Finset G
hA : A.Nonempty
B : Finset G
m n : ℕ
hA' : A ∈ A.powerset.erase ∅
C : Finset G
hC : C.Nonempty ∧ C ⊆ A
hCmin : ∀ x' ∈ A.powerset.erase ∅, ↑(#(C * B)) / ↑(#C) ≤ ↑(#(x' * B)) / ↑(#x')
⊢ ↑(#(B ^ m / B ^ n)) ≤ (↑(#(A * B)) / ↑(#A)) ^ (m + n) * ↑(#A)

Tactic:
obtain ⟨hC, hCA⟩ := hC

Result:
case intro.intro.intro
G : Type u_1
inst✝¹ : DecidableEq G
inst✝ : CommGroup G
A : Finset G
hA : A.Nonempty
B : Finset G
m n : ℕ
hA' : A ∈ A.powerset.erase ∅
C : Finset G
hCmin : ∀ x' ∈ A.powerset.erase ∅, ↑(#(C * B)) / ↑(#C) ≤ ↑(#(x' * B)) / ↑(#x')
hC : C.Nonempty
hCA : C ⊆ A
⊢ ↑(#(B ^ m / B ^ n)) ≤ (↑(#(A * B)) / ↑(#A)) ^ (m + n) * ↑(#A)

Proof state:
case intro.intro.intro
G : Type u_1
inst✝¹ : DecidableEq G
inst✝ : CommGroup G
A : Finset G
hA : A.Nonempty
B : Finset G
m n : ℕ
hA' : A ∈ A.powerset.erase ∅
C : Finset G
hCmin : ∀ x' ∈ A.powerset.erase ∅, ↑(#(C * B)) / ↑(#C) ≤ ↑(#(x' * B)) / ↑(#x')
hC : C.Nonempty
hCA : C ⊆ A
⊢ ↑(#(B ^ m / B ^ n)) * ↑(#C) ≤ (↑(#(A * B)) / ↑(#A)) ^ (m + n) * ↑(#A) * ↑(#C)

Tactic:
calc
  (#(B ^ m / B ^ n) * #C : ℚ≥0)
    ≤ #(B ^ m * C) * #(B ^ n * C) := mod_cast ruzsa_triangle_inequality_div_mul_mul ..
  _ = #(C * B ^ m) * #(C * B ^ n) := by simp_rw [mul_comm]
  _ ≤ ((#(C * B) / #C) ^ m * #C) * ((#(C * B) / #C : ℚ≥0) ^ n * #C) := by
    gcongr <;> exact card_mul_pow_le (mul_aux hC hCA hCmin) _
  _ = (#(C * B) / #C) ^ (m + n) * #C * #C := by ring
  _ ≤ (#(A * B) / #A) ^ (m + n) * #A * #C := by gcongr (?_ ^ _) * #?_ * _; exact hCmin _ hA'

Result:
no goals

Proof state:
G : Type u_1
inst✝¹ : DecidableEq G
inst✝ : CommGroup G
A : Finset G
hA : A.Nonempty
B : Finset G
m n : ℕ
hA' : A ∈ A.powerset.erase ∅
⊢ ↑(#(B ^ m / B ^ n)) ≤ (↑(#(A * B)) / ↑(#A)) ^ (m + n) * ↑(#A)

Tactic:
obtain ⟨C, hC, hCmin⟩ :=
  exists_min_image (A.powerset.erase ∅) (fun C ↦ #(C * B) / #C : _ → ℚ≥0) ⟨A, hA'⟩

Result:
case intro.intro
G : Type u_1
inst✝¹ : DecidableEq G
inst✝ : CommGroup G
A : Finset G
hA : A.Nonempty
B : Finset G
m n : ℕ
hA' : A ∈ A.powerset.erase ∅
C : Finset G
hC : C ∈ A.powerset.erase ∅
hCmin : ∀ x' ∈ A.powerset.erase ∅, ↑(#(C * B)) / ↑(#C) ≤ ↑(#(x' * B)) / ↑(#x')
⊢ ↑(#(B ^ m / B ^ n)) ≤ (↑(#(A * B)) / ↑(#A)) ^ (m + n) * ↑(#A)

Proof state:
case intro.intro
G : Type u_1
inst✝¹ : DecidableEq G
inst✝ : CommGroup G
A : Finset G
hA : A.Nonempty
B : Finset G
m n : ℕ
hA' : A ∈ A.powerset.erase ∅
C : Finset G
hC : C ∈ A.powerset.erase ∅
hCmin : ∀ x' ∈ A.powerset.erase ∅, ↑(#(C * B)) / ↑(#C) ≤ ↑(#(x' * B)) / ↑(#x')
⊢ ↑(#(B ^ m / B ^ n)) ≤ (↑(#(A * B)) / ↑(#A)) ^ (m + n) * ↑(#A)

Tactic:
rw [mem_erase, mem_powerset, ← nonempty_iff_ne_empty] at hC

Result:
case intro.intro
G : Type u_1
inst✝¹ : DecidableEq G
inst✝ : CommGroup G
A : Finset G
hA : A.Nonempty
B : Finset G
m n : ℕ
hA' : A ∈ A.powerset.erase ∅
C : Finset G
hC : C.Nonempty ∧ C ⊆ A
hCmin : ∀ x' ∈ A.powerset.erase ∅, ↑(#(C * B)) / ↑(#C) ≤ ↑(#(x' * B)) / ↑(#x')
⊢ ↑(#(B ^ m / B ^ n)) ≤ (↑(#(A * B)) / ↑(#A)) ^ (m + n) * ↑(#A)