FKS2.floor_row2

theorem floor_row2 : ∀ x ∈ Set.Icc (Real.exp (8:ℝ)) (Real.exp (43:ℝ)),    Eπ x ≤ (Real.log x) ^ (3/2 : ℝ) * x ^ (-(1:ℝ) / 2) := by  have hcurve : ∀ y, Real.exp (8:ℝ) ≤ y →      Expr.eval (fun _ => Real.sqrt (Real.log y)) (xhalfCurveE 1 3)        ≤ (Real.log y) ^ (3/2 : ℝ) * y ^ (-(1:ℝ) / 2) := by    intro y hy    have hypos : (0:ℝ) < y := lt_of_lt_of_le (Real.exp_pos _) hy    have hyL : (0:ℝ) ≤ Real.log y := by      have h8 : (8:ℝ) ≤ Real.log y := by        rw [← Real.log_exp (8:ℝ)]; exact Real.log_le_log (Real.exp_pos _) hy      linarith    refine le_of_eq ?_    have hsq : (Real.sqrt (Real.log y)) ^ 3 = (Real.log y) ^ (3/2 : ℝ) := by      rw [Real.sqrt_eq_rpow, ← Real.rpow_natCast ((Real.log y) ^ (1/2 : ℝ)) 3,        ← Real.rpow_mul hyL]      congr 1      norm_num    rw [eval_xhalfCurveE 1 3 y hypos hyL, hsq]    push_cast; ring  exact floor_xhalf_of_check (xhalfCurveE 1 3)    (fun x => (Real.log x) ^ (3/2 : ℝ) * x ^ (-(1:ℝ) / 2)) 8 (282/100) 76 (by norm_num)    (by rw [show ((282/100:ℚ):ℝ) = 2.82 by norm_num,          show (2.82:ℝ) = Real.sqrt (2.82 ^ 2) from (Real.sqrt_sq (by norm_num)).symm]        exact Real.sqrt_le_sqrt (by norm_num))    (by have h656 : Real.sqrt 43 ≤ 6.56 := by          rw [show (6.56:ℝ) = Real.sqrt (6.56 ^ 2) from (Real.sqrt_sq (by norm_num)).symm]          exact Real.sqrt_le_sqrt (by norm_num)        push_cast; linarith [h656])    (xhalfCurve_sub_supported 1 3) floor_slab_check_row2 hcurve