case h

⊢ Abs.abs (exp 1 - expNear 0 1 (2244083 / 825552)) ≤ Abs.abs 1 ^ 0 / ↑(Nat.factorial 0) * (1 / 10 ^ 10)
⊢ Abs.abs (exp 1 - 2244083 / 825552) ≤ 1 / 10 ^ 10
apply exp_approx_start
case h

⊢ Abs.abs
      (exp 1 -
        expNear 13 1
          ((((((((((((((2244083 / 825552 - 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)) ≤
    Abs.abs 1 ^ 13 / ↑(Nat.factorial 13) * (1 / 10 ^ 10 * 1 * 2 * 3 * 4 * 5 * 6 * 7 * 8 * 9 * 10 * 11 * 12 * 13)
case h

⊢ Abs.abs (exp 1 - expNear 0 1 (2244083 / 825552)) ≤ Abs.abs 1 ^ 0 / ↑(Nat.factorial 0) * (1 / 10 ^ 10)
iterate 13 refine' exp_1_approx_succ_eq (by norm_num1; rfl) (by norm_cast) _
case h

⊢ Abs.abs (exp 1 - expNear 13 1 (5 / 7)) ≤ Abs.abs 1 ^ 13 / ↑(Nat.factorial 13) * (243243 / 390625)
case h

⊢ Abs.abs
      (exp 1 -
        expNear 13 1
          ((((((((((((((2244083 / 825552 - 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)) ≤
    Abs.abs 1 ^ 13 / ↑(Nat.factorial 13) * (1 / 10 ^ 10 * 1 * 2 * 3 * 4 * 5 * 6 * 7 * 8 * 9 * 10 * 11 * 12 * 13)
norm_num1
case h

⊢ Abs.abs (1 - 5 / 7) ≤ 243243 / 390625 - Abs.abs 1 / 14 * ((14 + 1) / 14)
case h

⊢ Abs.abs (exp 1 - expNear 13 1 (5 / 7)) ≤ Abs.abs 1 ^ 13 / ↑(Nat.factorial 13) * (243243 / 390625)
refine' exp_approx_end' _ (by norm_num1; rfl) _ (by norm_cast) (by simp) _
no goals
case h

⊢ Abs.abs (1 - 5 / 7) ≤ 243243 / 390625 - Abs.abs 1 / 14 * ((14 + 1) / 14)
rw [_root_.abs_one, abs_of_pos] <;> norm_num1
case h

⊢ Abs.abs
      (exp 1 -
        expNear 13 1
          ((((((((((((((2244083 / 825552 - 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)) ≤
    Abs.abs 1 ^ 13 / ↑(Nat.factorial 13) * (1 / 10 ^ 10 * 1 * 2 * 3 * 4 * 5 * 6 * 7 * 8 * 9 * 10 * 11 * 12 * 13)
case h

⊢ Abs.abs
      (exp 1 -
        expNear 12 1
          (((((((((((((2244083 / 825552 - 1) * 1 - 1) * 2 - 1) * 3 - 1) * 4 - 1) * 5 - 1) * 6 - 1) * 7 - 1) * 8 - 1) *
                        9 -
                      1) *
                    10 -
                  1) *
                11 -
              1) *
            12)) ≤
    Abs.abs 1 ^ 12 / ↑(Nat.factorial 12) * (1 / 10 ^ 10 * 1 * 2 * 3 * 4 * 5 * 6 * 7 * 8 * 9 * 10 * 11 * 12)
refine' exp_1_approx_succ_eq (by norm_num1; rfl) (by norm_cast) _
⊢ 13 = ?m.1777
⊢ 12 + 1 = ?m.1777
norm_num1
no goals
⊢ 13 = ?m.1777
rfl
no goals
⊢ ↑13 = ?m.1779
norm_cast
⊢ 14 = ?m.2354
⊢ 13 + 1 = ?m.2354
norm_num1
no goals
⊢ 14 = ?m.2354
rfl
no goals
⊢ ↑14 = ?m.2356
norm_cast
no goals
⊢ Abs.abs 1 ≤ 1
simp