H₄ imag axis pos

@[fun_prop]theorem H₄_imag_axis_pos : ResToImagAxis.Pos H₄ := by  constructor  · exact H₄_imag_axis_real  · intro t ht    -- Strategy: Use H₄_S_action and ResToImagAxis.SlashActionS to relate    -- H₄ positivity to H₂ positivity via the modular S-transformation    have h1t_pos : 0 < 1 / t := one_div_pos.mpr ht    -- Apply SlashActionS at 1/t    have hSlash := ResToImagAxis.SlashActionS H₄ 2 h1t_pos    -- Use H₄_S_action: (H₄ ∣[2] S) = -H₂    rw [H₄_S_action] at hSlash    -- Now hSlash : (-H₂).resToImagAxis (1/t) = I^(-2) * (1/t)^(-2) * H₄.resToImagAxis t    -- Simplify: I^(-2) = -1    have hI_neg2 : (I : ℂ) ^ (-2 : ℤ) = -1 := by      change (I ^ 2)⁻¹ = -1      rw [I_sq]      norm_num    -- Simplify: (1/t)^(-2) = t^2    have h1t_neg2 : ((1 / t : ℝ) : ℂ) ^ (-2 : ℤ) = (t : ℂ) ^ 2 := by      have ht_ne : (t : ℂ) ≠ 0 := ofReal_ne_zero.mpr (ne_of_gt ht)      simp only [one_div, ofReal_inv, _root_.zpow_neg]      -- Goal: ((↑t)⁻¹ ^ 2)⁻¹ = ↑t ^ 2      field_simp    -- Simplify 1/(1/t) = t    have h1_div_1t : 1 / (1 / t) = t := by field_simp    -- The negation of resToImagAxis    have hNeg : (-H₂).resToImagAxis (1 / t) = -(H₂.resToImagAxis (1 / t)) := by      simp only [Function.resToImagAxis_apply, ResToImagAxis, h1t_pos, ↓reduceDIte, Pi.neg_apply]    -- Substitute into hSlash    rw [hNeg, hI_neg2, h1t_neg2, h1_div_1t] at hSlash    -- hSlash : -(H₂.resToImagAxis (1/t)) = -1 * t^2 * H₄.resToImagAxis t    -- Simplify: H₂.resToImagAxis (1/t) = t^2 * H₄.resToImagAxis t    have hEq : H₂.resToImagAxis (1 / t) = (t : ℂ) ^ 2 * H₄.resToImagAxis t := by      have h : -H₂.resToImagAxis (1 / t) = -(↑t ^ 2 * H₄.resToImagAxis t) := by        simp only [neg_mul, one_mul] at hSlash ⊢        exact hSlash      exact neg_inj.mp h    -- H₂.resToImagAxis (1/t).re > 0 from H₂_imag_axis_pos    have hH₂_pos := H₂_imag_axis_pos.2 (1 / t) h1t_pos    -- H₄.resToImagAxis t is real (im = 0)    have hH₄_real := H₄_imag_axis_real t ht    -- From hEq, extract real parts    have hRe : (H₂.resToImagAxis (1 / t)).re = ((t : ℂ) ^ 2 * H₄.resToImagAxis t).re := by      rw [hEq]    -- Since t^2 is real positive and H₄.resToImagAxis t is real:    -- (t^2 * H₄.resToImagAxis t).re = t^2 * (H₄.resToImagAxis t).re    have hProd_re : ((t : ℂ) ^ 2 * H₄.resToImagAxis t).re =        (t : ℝ) ^ 2 * (H₄.resToImagAxis t).re := by      simp only [Function.resToImagAxis_apply, ResToImagAxis, ht, ↓reduceDIte] at hH₄_real ⊢      simp only [sq, Complex.mul_re, ofReal_re, ofReal_im, zero_mul, sub_zero]      ring_nf      simp only [hH₄_real, mul_zero, sub_zero]    -- Combine: t^2 * (H₄.resToImagAxis t).re > 0 and t^2 > 0 imply (H₄.resToImagAxis t).re > 0    rw [hRe, hProd_re] at hH₂_pos    have ht2_pos : 0 < (t : ℝ) ^ 2 := sq_pos_of_pos ht    rw [mul_comm] at hH₂_pos    exact pos_of_mul_pos_left hH₂_pos (le_of_lt ht2_pos)