D slash

lemma D_slash (k : ℤ) (F : ℍ → ℂ) (hF : MDiff F) (γ : SL(2, ℤ)) :    D (F ∣[k] γ) = (D F ∣[k + 2] γ) -        (fun z : ℍ => (k : ℂ) * (2 * π * I)⁻¹ * (γ 1 0 / denom γ z) * (F ∣[k] γ) z) := by  -- Strategy (all micro-lemmas proven above):  -- 1. SL_slash_apply: (F ∣[k] γ) z = F(γ•z) * denom(γ,z)^(-k)  -- 2. coe_smul_of_det_pos: (γ•z : ℂ) = num γ z / denom γ z (since det(SL₂) = 1 > 0)  -- 3. Product rule: deriv[f*g] = f*deriv[g] + deriv[f]*g  -- 4. Chain rule: deriv[F ∘ mobius] = deriv[F](mobius z) * deriv_moebius  -- 5. deriv_moebius: d/dz[num/denom] = 1/denom² (uses det = 1)  -- 6. deriv_denom_zpow: d/dz[denom^(-k)] = -k * c * denom^(-k-1)  --  -- Computation (product rule + chain rule):  -- D(F ∣[k] γ) = (2πi)⁻¹ * deriv[F(γ•·) * denom^(-k)]  --   = (2πi)⁻¹ * [F(γ•z)*(-k*c*denom^(-k-1)) + deriv[F](γ•z)*(1/denom²)*denom^(-k)]  --   = (D F ∣[k+2] γ) - k*(2πi)⁻¹*(c/denom)*(F ∣[k] γ)  ext z  unfold D  simp only [Pi.sub_apply]  -- Key facts about denom and determinant (used multiple times below)  have hz_denom_ne : denom γ z ≠ 0 := UpperHalfPlane.denom_ne_zero γ z  have hdet_pos : (0 : ℝ) < ((γ : GL (Fin 2) ℝ).det).val := by simp  -- The derivative computation on ℂ using Filter.EventuallyEq.deriv_eq  -- (F ∣[k] γ) ∘ ofComplex agrees with F(num/denom) * denom^(-k) on ℍ  have hcomp : deriv (((F ∣[k] γ)) ∘ ofComplex) z =      deriv (fun w => (F ∘ ofComplex) (num γ w / denom γ w) * (denom γ w) ^ (-k)) z := by    apply Filter.EventuallyEq.deriv_eq    filter_upwards [isOpen_upperHalfPlaneSet.mem_nhds z.im_pos] with w hw    simp only [Function.comp_apply, ofComplex_apply_of_im_pos hw]    rw [ModularForm.SL_slash_apply (f := F) (k := k) γ ⟨w, hw⟩]    -- Key: (γ • ⟨w, hw⟩ : ℂ) = num γ w / denom γ w    congr 1    · -- F (γ • ⟨w, hw⟩) = (F ∘ ofComplex) (num γ w / denom γ w)      -- Need: γ • ⟨w, hw⟩ = ofComplex (num γ w / denom γ w) as points in ℍ      -- The smul result as element of ℍ, then coerce to ℂ      let gz : ℍ := γ • ⟨w, hw⟩      -- The key: (gz : ℂ) = num/denom (using the lemma for GL coercion)      have hsmul_coe : (gz : ℂ) = num γ w / denom γ w := by        have h := UpperHalfPlane.coe_smul_of_det_pos hdet_pos ⟨w, hw⟩        simp only [gz] at h ⊢        exact h      -- im(num/denom) > 0 follows from gz ∈ ℍ      have hmob_im : (num γ w / denom γ w).im > 0 := by        rw [← hsmul_coe]; exact gz.im_pos      -- Now F(gz) = F(ofComplex(num/denom)) = (F ∘ ofComplex)(num/denom)      -- gz = γ • ⟨w, hw⟩, so F gz = F (γ • ⟨w, hw⟩)      congr 1      -- Show gz = ofComplex (num/denom) as points in ℍ      apply UpperHalfPlane.ext      rw [ofComplex_apply_of_im_pos hmob_im]      exact hsmul_coe  rw [hcomp]  -- Now apply product rule: deriv[f * g] = f * deriv[g] + deriv[f] * g  -- where f(w) = (F ∘ ofComplex)(num w / denom w) and g(w) = denom(w)^(-k)  --  -- Setup differentiability for product rule  have hdenom_ne : ∀ w : ℂ, w.im > 0 → denom γ w ≠ 0 := fun w hw =>    UpperHalfPlane.denom_ne_zero γ ⟨w, hw⟩  have hdiff_denom_zpow : DifferentiableAt ℂ (fun w => (denom γ w) ^ (-k)) z :=    DifferentiableAt.zpow (differentiableAt_denom γ z) (Or.inl (hdenom_ne z z.im_pos))  -- For the F ∘ (num/denom) term, we need differentiability of the Möbius and F  have hdiff_mobius : DifferentiableAt ℂ (fun w => num γ w / denom γ w) z :=    (differentiableAt_num γ z).div (differentiableAt_denom γ z) (hdenom_ne z z.im_pos)  -- The composition (F ∘ ofComplex) ∘ mobius is differentiable at z  -- because mobius(z) is in ℍ and F is MDifferentiable  have hmobius_in_H : (num γ z / denom γ z).im > 0 := by    rw [← UpperHalfPlane.coe_smul_of_det_pos hdet_pos z]    exact (γ • z).im_pos  have hdiff_F_comp : DifferentiableAt ℂ (F ∘ ofComplex) (num γ z / denom γ z) :=    MDifferentiableAt_DifferentiableAt (hF ⟨num γ z / denom γ z, hmobius_in_H⟩)  have hcomp_eq : (fun w => (F ∘ ofComplex) (num γ w / denom γ w)) =      (F ∘ ofComplex) ∘ (fun w => num γ w / denom γ w) := rfl  have hdiff_F_mobius : DifferentiableAt ℂ (fun w => (F ∘ ofComplex) (num γ w / denom γ w)) z := by    rw [hcomp_eq]    exact DifferentiableAt.comp (z : ℂ) hdiff_F_comp hdiff_mobius  -- Apply product rule  -- Note: need to show the functions are equal to use deriv_mul  have hfun_eq : (fun w => (F ∘ ofComplex) (num γ w / denom γ w) * (denom γ w) ^ (-k)) =      ((fun w => (F ∘ ofComplex) (num γ w / denom γ w)) * (fun w => (denom γ w) ^ (-k))) := rfl  rw [hfun_eq]  have hprod := deriv_mul hdiff_F_mobius hdiff_denom_zpow  rw [hprod]  -- Apply chain rule to (F ∘ ofComplex) ∘ mobius  have hchain : deriv (fun w => (F ∘ ofComplex) (num γ w / denom γ w)) z =      deriv (F ∘ ofComplex) (num γ z / denom γ z) * deriv (fun w => num γ w / denom γ w) z := by    rw [hcomp_eq, (hdiff_F_comp.hasDerivAt.comp (z : ℂ) hdiff_mobius.hasDerivAt).deriv]  -- Substitute the micro-lemmas  have hderiv_mob := deriv_moebius γ z  have hderiv_zpow := deriv_denom_zpow γ k z  rw [hchain, hderiv_mob, hderiv_zpow]  -- Now we have:  -- (2πi)⁻¹ * [deriv(F∘ofComplex)(mob z) * (1/denom²) * denom^(-k) +  --            (F∘ofComplex)(mob z) * (-k * c * denom^(-k-1))]  -- = (D F ∣[k+2] γ) z - k * (2πi)⁻¹ * (c/denom) * (F ∣[k] γ) z  --  -- Key observations:  -- - (2πi)⁻¹ * deriv(F∘ofComplex)(mob z) = D F (γ • z)  (by def of D)  -- - denom^(-k) / denom² = denom^(-k-2)  -- - (D F)(γ • z) * denom^(-k-2) = (D F ∣[k+2] γ) z  -- - (F∘ofComplex)(mob z) * denom^(-k) = F(γ • z) * denom^(-k) = (F ∣[k] γ) z  -- - -k * c * denom^(-k-1) * (2πi)⁻¹ = -k * (2πi)⁻¹ * c/denom * denom^(-k)  --  -- Relate mobius to γ • z: ↑(γ • z) = num/denom (explicit coercion from ℍ to ℂ)  have hmob_eq : ↑(γ • z) = num γ z / denom γ z :=    UpperHalfPlane.coe_smul_of_det_pos hdet_pos z  -- Relate (F ∘ ofComplex)(mob z) to F(γ • z)  have hF_mob : (F ∘ ofComplex) (num γ z / denom γ z) = F (γ • z) := by    simp only [Function.comp_apply, ← hmob_eq, ofComplex_apply]  -- Final algebraic manipulation  -- Goal: (2πi)⁻¹ * (deriv(F∘ofComplex)(mob z) * (1/denom²) * denom^(-k) +  --                   (F∘ofComplex)(mob z) * (-k * c * denom^(-k-1)))  --      = D F(γ•z) * denom^(-(k+2)) - k * (2πi)⁻¹ * (c/denom) * F(γ•z) * denom^(-k)  -- This follows from the above lemmas by algebraic manipulation  --  -- First expand the slash action on the RHS and normalize denom coercions  simp only [ModularForm.SL_slash_apply, hF_mob, hmob_eq]  -- Now both sides should have normalized denom (num/denom arguments and ℂ coercions)  -- Key identities for zpow:  -- (1/denom²) * denom^(-k) = denom^(-2) * denom^(-k) = denom^(-k-2) = denom^(-(k+2))  -- -k * c * denom^(-k-1) = -k * (c/denom) * denom^(-k)  --  -- Use zpow identities  have hpow_combine : 1 / (denom γ z) ^ 2 * (denom γ z) ^ (-k) = (denom γ z) ^ (-(k + 2)) := by    rw [one_div, ← zpow_natCast (denom γ z) 2, ← zpow_neg, ← zpow_add₀ hz_denom_ne]    congr 1    ring  have hpow_m1 : (denom γ z) ^ (-k - 1) = (denom γ z) ^ (-1 : ℤ) * (denom γ z) ^ (-k) := by    rw [← zpow_add₀ hz_denom_ne]    congr 1    ring  -- Rewrite powers on LHS  conv_lhs =>    rw [mul_assoc (deriv (F ∘ ofComplex) (num γ z / denom γ z)) (1 / denom γ z ^ 2) _]    rw [hpow_combine, hpow_m1]  -- Now the goal should be cleaner - distribute and simplify  simp only [zpow_neg_one]  ring