-- Module: Mathlib.NumberTheory.ModularForms.LevelOne.DimensionFormula
Goal:
k : ℕ
ihn :
  ∀ m < k,
    ∀ (hk2 : Even m),
      Module.rank ℂ (ModularForm (MonoidHom.range (G := SL(2, ℤ)) (Matrix.SpecialLinearGroup.mapGL ℝ)) (↑m : ℤ)) =
        (↑(if m ≡ 2 [MOD 12] then m / 12 else m / 12 + 1) : Cardinal.{0})
hk2 : Even k
hk : 3 ≤ k ∧ k < 12
this : Membership.mem (γ := Finset ℕ) (Finset.filter Even (Finset.Icc 3 11)) k
h :
  ∀
    (ihn :
      ∀ m < HAdd.hAdd (α := ℕ) 3 1,
        ∀ (hk2 : Even m),
          Module.rank ℂ (ModularForm (MonoidHom.range (G := SL(2, ℤ)) (Matrix.SpecialLinearGroup.mapGL ℝ)) (↑m : ℤ)) =
            (↑(if m ≡ 2 [MOD 12] then m / 12 else m / 12 + 1) : Cardinal.{0}))
    (hk2 : Even (HAdd.hAdd (α := ℕ) 3 1)) (hk : 3 ≤ HAdd.hAdd (α := ℕ) 3 1 ∧ HAdd.hAdd (α := ℕ) 3 1 < 12),
    HAdd.hAdd (α := Cardinal.{0}) 1 0 =
      (↑(if HAdd.hAdd (α := ℕ) 3 1 ≡ 2 [MOD 12] then HAdd.hAdd (α := ℕ) 3 1 / 12 else HAdd.hAdd (α := ℕ) 3 1 / 12 + 1) :
        Cardinal.{0})
h₁ :
  ∀
    (ihn :
      ∀ m < HAdd.hAdd (α := ℕ) 3 1 + 1 + 1,
        ∀ (hk2 : Even m),
          Module.rank ℂ (ModularForm (MonoidHom.range (G := SL(2, ℤ)) (Matrix.SpecialLinearGroup.mapGL ℝ)) (↑m : ℤ)) =
            (↑(if m ≡ 2 [MOD 12] then m / 12 else m / 12 + 1) : Cardinal.{0}))
    (hk2 : Even (HAdd.hAdd (α := ℕ) 3 1 + 1 + 1))
    (hk : 3 ≤ HAdd.hAdd (α := ℕ) 3 1 + 1 + 1 ∧ HAdd.hAdd (α := ℕ) 3 1 + 1 + 1 < 12),
    HAdd.hAdd (α := Cardinal.{0}) 1 0 =
      (↑(if HAdd.hAdd (α := ℕ) 3 1 + 1 + 1 ≡ 2 [MOD 12] then (HAdd.hAdd (α := ℕ) 3 1 + 1 + 1) / 12
          else (HAdd.hAdd (α := ℕ) 3 1 + 1 + 1) / 12 + 1) :
        Cardinal.{0})
h₂ :
  ∀
    (ihn :
      ∀ m < HAdd.hAdd (α := ℕ) 3 1 + 1 + 1 + 1 + 1,
        ∀ (hk2 : Even m),
          Module.rank ℂ (ModularForm (MonoidHom.range (G := SL(2, ℤ)) (Matrix.SpecialLinearGroup.mapGL ℝ)) (↑m : ℤ)) =
            (↑(if m ≡ 2 [MOD 12] then m / 12 else m / 12 + 1) : Cardinal.{0}))
    (hk2 : Even (HAdd.hAdd (α := ℕ) 3 1 + 1 + 1 + 1 + 1))
    (hk : 3 ≤ HAdd.hAdd (α := ℕ) 3 1 + 1 + 1 + 1 + 1 ∧ HAdd.hAdd (α := ℕ) 3 1 + 1 + 1 + 1 + 1 < 12),
    HAdd.hAdd (α := Cardinal.{0}) 1 0 =
      (↑(if HAdd.hAdd (α := ℕ) 3 1 + 1 + 1 + 1 + 1 ≡ 2 [MOD 12] then (HAdd.hAdd (α := ℕ) 3 1 + 1 + 1 + 1 + 1) / 12
          else (HAdd.hAdd (α := ℕ) 3 1 + 1 + 1 + 1 + 1) / 12 + 1) :
        Cardinal.{0})
h₃ :
  ∀
    (ihn :
      ∀ m < HAdd.hAdd (α := ℕ) 3 1 + 1 + 1 + 1 + 1 + 1 + 1,
        ∀ (hk2 : Even m),
          Module.rank ℂ (ModularForm (MonoidHom.range (G := SL(2, ℤ)) (Matrix.SpecialLinearGroup.mapGL ℝ)) (↑m : ℤ)) =
            (↑(if m ≡ 2 [MOD 12] then m / 12 else m / 12 + 1) : Cardinal.{0}))
    (hk2 : Even (HAdd.hAdd (α := ℕ) 3 1 + 1 + 1 + 1 + 1 + 1 + 1))
    (hk : 3 ≤ HAdd.hAdd (α := ℕ) 3 1 + 1 + 1 + 1 + 1 + 1 + 1 ∧ HAdd.hAdd (α := ℕ) 3 1 + 1 + 1 + 1 + 1 + 1 + 1 < 12),
    HAdd.hAdd (α := Cardinal.{0}) 1 0 =
      (↑(if HAdd.hAdd (α := ℕ) 3 1 + 1 + 1 + 1 + 1 + 1 + 1 ≡ 2 [MOD 12] then
            (HAdd.hAdd (α := ℕ) 3 1 + 1 + 1 + 1 + 1 + 1 + 1) / 12
          else (HAdd.hAdd (α := ℕ) 3 1 + 1 + 1 + 1 + 1 + 1 + 1) / 12 + 1) :
        Cardinal.{0})
⊢ HAdd.hAdd (α := Cardinal.{0}) 1 0 = (↑(if k ≡ 2 [MOD 12] then k / 12 else k / 12 + 1) : Cardinal.{0})

Tactic:
fin_cases this