-- Module: Mathlib.Combinatorics.Additive.VerySmallDoubling
Goal:
G : Type u_1
inst : Group G
inst_1 : DecidableEq G
K : ℝ
A_1 B S : Finset G
a_1 b_1 c_1 d_1 x y : G
A : Finset G
h : (↑(#(A * A)) : ℚ) < 3 / 2 * (↑(#A) : ℚ)
h₁ : ∀ (x : G) (hx : x ∈ A) (y : G) (hy : y ∈ A), 1 / 2 * (↑(#A) : ℚ) < (↑(#(x •> A ∩ y •> A)) : ℚ)
a : G
ha : a⁻¹ ∈ A
b : G
hb : b ∈ A
c : G
hc : c⁻¹ ∈ A
d : G
hd : d ∈ A
h₂ : 1 / 2 * (↑(#A) : ℚ) < (↑(#(A ∩ (a * b)⁻¹ •> A)) : ℚ)
h₃ : 1 / 2 * (↑(#A) : ℚ) < (↑(#(A ∩ (c * d) •> A)) : ℚ)
this_1 :
  HAdd.hAdd (α := ℚ) (↑(#(A ∩ (c * d) •> A ∩ (A ∩ (a * b)⁻¹ •> A))) : ℚ)
      (↑(#(A ∩ (c * d) •> A ∪ A ∩ (a * b)⁻¹ •> A)) : ℚ) =
    HAdd.hAdd (α := ℚ) (↑(#(A ∩ (c * d) •> A)) : ℚ) (↑(#(A ∩ (a * b)⁻¹ •> A)) : ℚ)
this : LE.le (α := ℚ) (↑(#(A ∩ (c * d) •> A ∪ A ∩ (a * b)⁻¹ •> A)) : ℚ) (↑(#A) : ℚ)
linarith : LE.le (α := ℚ) (↑(#(A ∩ (c * d) •> A ∩ (A ∩ (a * b)⁻¹ •> A))) : ℚ) 0 → 2 / 2 = 1
linarith₁ : LE.le (α := ℚ) (↑(#(A ∩ (c * d) •> A ∩ (A ∩ (a * b)⁻¹ •> A))) : ℚ) 0 → HMul.hMul (α := ℚ) 1 2 = 2
linarith₂ : LE.le (α := ℚ) (↑(#(A ∩ (c * d) •> A ∩ (A ∩ (a * b)⁻¹ •> A))) : ℚ) 0 → HMul.hMul (α := ℚ) 2 1 = 2
linarith₃ : LE.le (α := ℚ) (↑(#(A ∩ (c * d) •> A ∩ (A ∩ (a * b)⁻¹ •> A))) : ℚ) 0 → GT.gt (α := ℚ) 2 0
linarith₄ : LE.le (α := ℚ) (↑(#(A ∩ (c * d) •> A ∩ (A ∩ (a * b)⁻¹ •> A))) : ℚ) 0 → 2 / 2 = 1
linarith₅ : LE.le (α := ℚ) (↑(#(A ∩ (c * d) •> A ∩ (A ∩ (a * b)⁻¹ •> A))) : ℚ) 0 → HMul.hMul (α := ℚ) 1 2 = 2
linarith₆ : LE.le (α := ℚ) (↑(#(A ∩ (c * d) •> A ∩ (A ∩ (a * b)⁻¹ •> A))) : ℚ) 0 → HMul.hMul (α := ℚ) 2 1 = 2
linarith₇ : LE.le (α := ℚ) (↑(#(A ∩ (c * d) •> A ∩ (A ∩ (a * b)⁻¹ •> A))) : ℚ) 0 → GT.gt (α := ℚ) 2 0
linarith₈ : LE.le (α := ℚ) (↑(#(A ∩ (c * d) •> A ∩ (A ∩ (a * b)⁻¹ •> A))) : ℚ) 0 → 2 / 2 = 1
linarith₉ : LE.le (α := ℚ) (↑(#(A ∩ (c * d) •> A ∩ (A ∩ (a * b)⁻¹ •> A))) : ℚ) 0 → HMul.hMul (α := ℚ) 1 2 = 2
linarith₁₀ : LE.le (α := ℚ) (↑(#(A ∩ (c * d) •> A ∩ (A ∩ (a * b)⁻¹ •> A))) : ℚ) 0 → HMul.hMul (α := ℚ) 2 1 = 2
linarith₁₁ : LE.le (α := ℚ) (↑(#(A ∩ (c * d) •> A ∩ (A ∩ (a * b)⁻¹ •> A))) : ℚ) 0 → GT.gt (α := ℚ) 2 0
linarith₁₂ :
  LE.le (α := ℚ) (↑(#(A ∩ (c * d) •> A ∩ (A ∩ (a * b)⁻¹ •> A))) : ℚ) 0 →
    HMul.hMul (α := ℚ) 1 1 * HMul.hMul (α := ℚ) 1 (↑(#A) : ℚ) - HMul.hMul (α := ℚ) 2 (↑(#(A ∩ (a * b)⁻¹ •> A)) : ℚ) +
              (HMul.hMul (α := ℚ) 1 1 * HMul.hMul (α := ℚ) 1 (↑(#A) : ℚ) -
                HMul.hMul (α := ℚ) 2 (↑(#(A ∩ (c * d) •> A)) : ℚ)) +
            2 *
              -(HMul.hMul (α := ℚ) 1 (↑(#(A ∩ (c * d) •> A ∩ (A ∩ (a * b)⁻¹ •> A))) : ℚ) +
                    HMul.hMul (α := ℚ) 1 (↑(#(A ∩ (c * d) •> A ∪ A ∩ (a * b)⁻¹ •> A)) : ℚ) -
                  (HMul.hMul (α := ℚ) 1 (↑(#(A ∩ (c * d) •> A)) : ℚ) +
                    HMul.hMul (α := ℚ) 1 (↑(#(A ∩ (a * b)⁻¹ •> A)) : ℚ))) +
          2 *
            (HMul.hMul (α := ℚ) 1 (↑(#(A ∩ (c * d) •> A ∪ A ∩ (a * b)⁻¹ •> A)) : ℚ) -
              HMul.hMul (α := ℚ) 1 (↑(#A) : ℚ)) +
        2 * (HMul.hMul (α := ℚ) 1 (↑(#(A ∩ (c * d) •> A ∩ (A ∩ (a * b)⁻¹ •> A))) : ℚ) - HMul.hMul (α := ℚ) 1 0) =
      0
linarith₁₃ linarith₁₄ linarith₁₅ :
  LE.le (α := ℚ) (↑(#(A ∩ (c * d) •> A ∩ (A ∩ (a * b)⁻¹ •> A))) : ℚ) 0 → GT.gt (α := ℚ) 2 0
⊢ LT.lt (α := ℚ) 0 (↑(#(A ∩ (c * d) •> A ∩ (A ∩ (a * b)⁻¹ •> A))) : ℚ)

Tactic:
linarith