Theorem: Real.exp_one_near_10
File: Mathlib/Data/Complex/ExponentialBounds.lean

tactic: apply exp_approx_start
state_before: ⊢ |rexp 1 - 2244083 / 825552| ≤ 1 / 10 ^ 10
state_after: case h

⊢ |rexp 1 - expNear 0 1 (2244083 / 825552)| ≤ |1| ^ 0 / ↑(Nat.factorial 0) * (1 / 10 ^ 10)

tactic: iterate 13 refine exp_1_approx_succ_eq (by norm_num1; rfl) (by norm_cast) ?_
state_before: case h

⊢ |rexp 1 - expNear 0 1 (2244083 / 825552)| ≤ |1| ^ 0 / ↑(Nat.factorial 0) * (1 / 10 ^ 10)
state_after: case h

⊢ |rexp 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)| ≤
    |1| ^ 13 / ↑(Nat.factorial 13) * (1 / 10 ^ 10 * 1 * 2 * 3 * 4 * 5 * 6 * 7 * 8 * 9 * 10 * 11 * 12 * 13)

tactic: norm_num1
state_before: case h

⊢ |rexp 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)| ≤
    |1| ^ 13 / ↑(Nat.factorial 13) * (1 / 10 ^ 10 * 1 * 2 * 3 * 4 * 5 * 6 * 7 * 8 * 9 * 10 * 11 * 12 * 13)
state_after: case h

⊢ |rexp 1 - expNear 13 1 (5 / 7)| ≤ |1| ^ 13 / ↑(Nat.factorial 13) * (243243 / 390625)

tactic: refine exp_approx_end' _ (by norm_num1; rfl) _ (by norm_cast) (by simp) ?_
state_before: case h

⊢ |rexp 1 - expNear 13 1 (5 / 7)| ≤ |1| ^ 13 / ↑(Nat.factorial 13) * (243243 / 390625)
state_after: case h

⊢ |1 - 5 / 7| ≤ 243243 / 390625 - |1| / 14 * ((14 + 1) / 14)

tactic: rw [_root_.abs_one, abs_of_pos] <;> norm_num1
state_before: case h

⊢ |1 - 5 / 7| ≤ 243243 / 390625 - |1| / 14 * ((14 + 1) / 14)
state_after: no goals

tactic: refine exp_1_approx_succ_eq (by norm_num1; rfl) (by norm_cast) ?_
state_before: case h

⊢ |rexp 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)| ≤
    |1| ^ 12 / ↑(Nat.factorial 12) * (1 / 10 ^ 10 * 1 * 2 * 3 * 4 * 5 * 6 * 7 * 8 * 9 * 10 * 11 * 12)
state_after: case h

⊢ |rexp 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)| ≤
    |1| ^ 13 / ↑(Nat.factorial 13) * (1 / 10 ^ 10 * 1 * 2 * 3 * 4 * 5 * 6 * 7 * 8 * 9 * 10 * 11 * 12 * 13)

tactic: norm_num1
state_before: ⊢ 12 + 1 = ?m.1527
state_after: ⊢ 13 = ?m.1527

tactic: rfl
state_before: ⊢ 13 = ?m.1527
state_after: no goals

tactic: norm_cast
state_before: ⊢ ↑13 = ?m.1529
state_after: no goals

tactic: norm_num1
state_before: ⊢ 13 + 1 = ?m.2233
state_after: ⊢ 14 = ?m.2233

tactic: rfl
state_before: ⊢ 14 = ?m.2233
state_after: no goals

tactic: norm_cast
state_before: ⊢ ↑14 = ?m.2235
state_after: no goals

tactic: simp
state_before: ⊢ |1| ≤ 1
state_after: no goals