-- Mathlib/Data/Complex/ExponentialBounds.lean
theorem exp_one_near_20 : |exp 1 - 363916618873 / 133877442384| ≤ 1 / 10 ^ 20

Proof state:
case h

⊢ |rexp 1 -
        expNear 20 1
          (((((((((((((((((((((363916618873 / 133877442384 - 1) * 1 - 1) * 2 - 1) * 3 - 1) * 4 - 1) * 5 - 1) * 6 - 1) *
                                                                7 -
                                                              1) *
                                                            8 -
                                                          1) *
                                                        9 -
                                                      1) *
                                                    10 -
                                                  1) *
                                                11 -
                                              1) *
                                            12 -
                                          1) *
                                        13 -
                                      1) *
                                    14 -
                                  1) *
                                15 -
                              1) *
                            16 -
                          1) *
                        17 -
                      1) *
                    18 -
                  1) *
                19 -
              1) *
            20)| ≤
    |1| ^ 20 / ↑(Nat.factorial 20) *
      (1 / 10 ^ 20 * 1 * 2 * 3 * 4 * 5 * 6 * 7 * 8 * 9 * 10 * 11 * 12 * 13 * 14 * 15 * 16 * 17 * 18 * 19 * 20)

Tactic:
refine exp_1_approx_succ_eq (by norm_num1; rfl) (by norm_cast) ?_

Result:
case h

⊢ |rexp 1 -
        expNear 21 1
          ((((((((((((((((((((((363916618873 / 133877442384 - 1) * 1 - 1) * 2 - 1) * 3 - 1) * 4 - 1) * 5 - 1) * 6 - 1) *
                                                                    7 -
                                                                  1) *
                                                                8 -
                                                              1) *
                                                            9 -
                                                          1) *
                                                        10 -
                                                      1) *
                                                    11 -
                                                  1) *
                                                12 -
                                              1) *
                                            13 -
                                          1) *
                                        14 -
                                      1) *
                                    15 -
                                  1) *
                                16 -
                              1) *
                            17 -
                          1) *
                        18 -
                      1) *
                    19 -
                  1) *
                20 -
              1) *
            21)| ≤
    |1| ^ 21 / ↑(Nat.factorial 21) *
      (1 / 10 ^ 20 * 1 * 2 * 3 * 4 * 5 * 6 * 7 * 8 * 9 * 10 * 11 * 12 * 13 * 14 * 15 * 16 * 17 * 18 * 19 * 20 * 21)