l3lab/ntp-mathlib huggingface dataset:
theorem exp_one_near_10 : |exp 1 - 2244083 / 825552| ≤ 1 / 10 ^ 10 := by
  apply exp_approx_start
  iterate 13 refine' exp_1_approx_succ_eq (by norm_num1; rfl) (by norm_cast) _
  norm_num1
  refine' exp_approx_end' _ (by norm_num1; rfl) _ (by norm_cast) (by simp) _
  rw [_root_.abs_one, abs_of_pos] <;> 

Proof state:
case h
⊢ 1 - 5 / 7 ≤ 243243 / 390625 - 1 / 14 * ((14 + 1) / 14)

Next tactic:
norm_num1
theorem exp_one_near_10 : |exp 1 - 2244083 / 825552| ≤ 1 / 10 ^ 10 := by
  apply exp_approx_start
  iterate 13 refine' exp_1_approx_succ_eq (by norm_num1; rfl) (by norm_cast) _
  norm_num1
  refine' exp_approx_end' _ (by norm_num1; rfl) _ (by norm_cast) (by simp) _
  rw [_root_.abs_one, abs_of_pos] <;> 

Proof state:
case h
⊢ 0 < 1 - 5 / 7

Next tactic:
norm_num1
theorem exp_one_near_20 : |exp 1 - 363916618873 / 133877442384| ≤ 1 / 10 ^ 20 := by
  

Proof state:
⊢ |rexp 1 - 363916618873 / 133877442384| ≤ 1 / 10 ^ 20

Next tactic:
apply exp_approx_start
theorem exp_one_near_20 : |exp 1 - 363916618873 / 133877442384| ≤ 1 / 10 ^ 20 := by
  apply exp_approx_start
  

Proof state:
case h
⊢ |rexp 1 - expNear 0 1 (363916618873 / 133877442384)| ≤ |1| ^ 0 / ↑(Nat.factorial 0) * (1 / 10 ^ 20)

Next tactic:
iterate 21 refine' exp_1_approx_succ_eq (by norm_num1; rfl) (by norm_cast) _
theorem exp_one_near_20 : |exp 1 - 363916618873 / 133877442384| ≤ 1 / 10 ^ 20 := by
  apply exp_approx_start
  iterate 21 

Proof state:
case h
⊢ |rexp 1 - expNear 0 1 (363916618873 / 133877442384)| ≤ |1| ^ 0 / ↑(Nat.factorial 0) * (1 / 10 ^ 20)

Next tactic:
refine' exp_1_approx_succ_eq (by norm_num1; rfl) (by norm_cast) _
theorem exp_one_near_20 : |exp 1 - 363916618873 / 133877442384| ≤ 1 / 10 ^ 20 := by
  apply exp_approx_start
  iterate 21 refine' exp_1_approx_succ_eq (by 

Proof state:
⊢ 0 + 1 = ?m.3360

Next tactic:
norm_num1
theorem exp_one_near_20 : |exp 1 - 363916618873 / 133877442384| ≤ 1 / 10 ^ 20 := by
  apply exp_approx_start
  iterate 21 refine' exp_1_approx_succ_eq (by norm_num1; 

Proof state:
⊢ 1 = ?m.3360

Next tactic:
rfl
theorem exp_one_near_20 : |exp 1 - 363916618873 / 133877442384| ≤ 1 / 10 ^ 20 := by
  apply exp_approx_start
  iterate 21 refine' exp_1_approx_succ_eq (by norm_num1; rfl) (by 

Proof state:
⊢ ↑1 = ?m.3371

Next tactic:
norm_cast
theorem exp_one_near_20 : |exp 1 - 363916618873 / 133877442384| ≤ 1 / 10 ^ 20 := by
  apply exp_approx_start
  iterate 21 

Proof state:
case h
⊢ |rexp 1 - expNear 1 1 ((363916618873 / 133877442384 - 1) * 1)| ≤ |1| ^ 1 / ↑(Nat.factorial 1) * (1 / 10 ^ 20 * 1)

Next tactic:
refine' exp_1_approx_succ_eq (by norm_num1; rfl) (by norm_cast) _
theorem exp_one_near_20 : |exp 1 - 363916618873 / 133877442384| ≤ 1 / 10 ^ 20 := by
  apply exp_approx_start
  iterate 21 refine' exp_1_approx_succ_eq (by 

Proof state:
⊢ 1 + 1 = ?m.3703

Next tactic:
norm_num1
theorem exp_one_near_20 : |exp 1 - 363916618873 / 133877442384| ≤ 1 / 10 ^ 20 := by
  apply exp_approx_start
  iterate 21 refine' exp_1_approx_succ_eq (by norm_num1; 

Proof state:
⊢ 2 = ?m.3703

Next tactic:
rfl
theorem exp_one_near_20 : |exp 1 - 363916618873 / 133877442384| ≤ 1 / 10 ^ 20 := by
  apply exp_approx_start
  iterate 21 refine' exp_1_approx_succ_eq (by norm_num1; rfl) (by 

Proof state:
⊢ ↑2 = ?m.3705

Next tactic:
norm_cast
theorem exp_one_near_20 : |exp 1 - 363916618873 / 133877442384| ≤ 1 / 10 ^ 20 := by
  apply exp_approx_start
  iterate 21 

Proof state:
case h
⊢ |rexp 1 - expNear 2 1 (((363916618873 / 133877442384 - 1) * 1 - 1) * 2)| ≤
    |1| ^ 2 / ↑(Nat.factorial 2) * (1 / 10 ^ 20 * 1 * 2)

Next tactic:
refine' exp_1_approx_succ_eq (by norm_num1; rfl) (by norm_cast) _
theorem exp_one_near_20 : |exp 1 - 363916618873 / 133877442384| ≤ 1 / 10 ^ 20 := by
  apply exp_approx_start
  iterate 21 refine' exp_1_approx_succ_eq (by 

Proof state:
⊢ 2 + 1 = ?m.3773

Next tactic:
norm_num1
theorem exp_one_near_20 : |exp 1 - 363916618873 / 133877442384| ≤ 1 / 10 ^ 20 := by
  apply exp_approx_start
  iterate 21 refine' exp_1_approx_succ_eq (by norm_num1; 

Proof state:
⊢ 3 = ?m.3773

Next tactic:
rfl
theorem exp_one_near_20 : |exp 1 - 363916618873 / 133877442384| ≤ 1 / 10 ^ 20 := by
  apply exp_approx_start
  iterate 21 refine' exp_1_approx_succ_eq (by norm_num1; rfl) (by 

Proof state:
⊢ ↑3 = ?m.3775

Next tactic:
norm_cast
theorem exp_one_near_20 : |exp 1 - 363916618873 / 133877442384| ≤ 1 / 10 ^ 20 := by
  apply exp_approx_start
  iterate 21 

Proof state:
case h
⊢ |rexp 1 - expNear 3 1 ((((363916618873 / 133877442384 - 1) * 1 - 1) * 2 - 1) * 3)| ≤
    |1| ^ 3 / ↑(Nat.factorial 3) * (1 / 10 ^ 20 * 1 * 2 * 3)

Next tactic:
refine' exp_1_approx_succ_eq (by norm_num1; rfl) (by norm_cast) _
theorem exp_one_near_20 : |exp 1 - 363916618873 / 133877442384| ≤ 1 / 10 ^ 20 := by
  apply exp_approx_start
  iterate 21 refine' exp_1_approx_succ_eq (by 

Proof state:
⊢ 3 + 1 = ?m.3843

Next tactic:
norm_num1
theorem exp_one_near_20 : |exp 1 - 363916618873 / 133877442384| ≤ 1 / 10 ^ 20 := by
  apply exp_approx_start
  iterate 21 refine' exp_1_approx_succ_eq (by norm_num1; 

Proof state:
⊢ 4 = ?m.3843

Next tactic:
rfl
theorem exp_one_near_20 : |exp 1 - 363916618873 / 133877442384| ≤ 1 / 10 ^ 20 := by
  apply exp_approx_start
  iterate 21 refine' exp_1_approx_succ_eq (by norm_num1; rfl) (by 

Proof state:
⊢ ↑4 = ?m.3845

Next tactic:
norm_cast
theorem exp_one_near_20 : |exp 1 - 363916618873 / 133877442384| ≤ 1 / 10 ^ 20 := by
  apply exp_approx_start
  iterate 21 

Proof state:
case h
⊢ |rexp 1 - expNear 4 1 (((((363916618873 / 133877442384 - 1) * 1 - 1) * 2 - 1) * 3 - 1) * 4)| ≤
    |1| ^ 4 / ↑(Nat.factorial 4) * (1 / 10 ^ 20 * 1 * 2 * 3 * 4)

Next tactic:
refine' exp_1_approx_succ_eq (by norm_num1; rfl) (by norm_cast) _
theorem exp_one_near_20 : |exp 1 - 363916618873 / 133877442384| ≤ 1 / 10 ^ 20 := by
  apply exp_approx_start
  iterate 21 refine' exp_1_approx_succ_eq (by 

Proof state:
⊢ 4 + 1 = ?m.3913

Next tactic:
norm_num1
theorem exp_one_near_20 : |exp 1 - 363916618873 / 133877442384| ≤ 1 / 10 ^ 20 := by
  apply exp_approx_start
  iterate 21 refine' exp_1_approx_succ_eq (by norm_num1; 

Proof state:
⊢ 5 = ?m.3913

Next tactic:
rfl
theorem exp_one_near_20 : |exp 1 - 363916618873 / 133877442384| ≤ 1 / 10 ^ 20 := by
  apply exp_approx_start
  iterate 21 refine' exp_1_approx_succ_eq (by norm_num1; rfl) (by 

Proof state:
⊢ ↑5 = ?m.3915

Next tactic:
norm_cast
theorem exp_one_near_20 : |exp 1 - 363916618873 / 133877442384| ≤ 1 / 10 ^ 20 := by
  apply exp_approx_start
  iterate 21 

Proof state:
case h
⊢ |rexp 1 - expNear 5 1 ((((((363916618873 / 133877442384 - 1) * 1 - 1) * 2 - 1) * 3 - 1) * 4 - 1) * 5)| ≤
    |1| ^ 5 / ↑(Nat.factorial 5) * (1 / 10 ^ 20 * 1 * 2 * 3 * 4 * 5)

Next tactic:
refine' exp_1_approx_succ_eq (by norm_num1; rfl) (by norm_cast) _
theorem exp_one_near_20 : |exp 1 - 363916618873 / 133877442384| ≤ 1 / 10 ^ 20 := by
  apply exp_approx_start
  iterate 21 refine' exp_1_approx_succ_eq (by 

Proof state:
⊢ 5 + 1 = ?m.3983

Next tactic:
norm_num1
theorem exp_one_near_20 : |exp 1 - 363916618873 / 133877442384| ≤ 1 / 10 ^ 20 := by
  apply exp_approx_start
  iterate 21 refine' exp_1_approx_succ_eq (by norm_num1; 

Proof state:
⊢ 6 = ?m.3983

Next tactic:
rfl
theorem exp_one_near_20 : |exp 1 - 363916618873 / 133877442384| ≤ 1 / 10 ^ 20 := by
  apply exp_approx_start
  iterate 21 refine' exp_1_approx_succ_eq (by norm_num1; rfl) (by 

Proof state:
⊢ ↑6 = ?m.3985

Next tactic:
norm_cast
theorem exp_one_near_20 : |exp 1 - 363916618873 / 133877442384| ≤ 1 / 10 ^ 20 := by
  apply exp_approx_start
  iterate 21 

Proof state:
case h
⊢ |rexp 1 - expNear 6 1 (((((((363916618873 / 133877442384 - 1) * 1 - 1) * 2 - 1) * 3 - 1) * 4 - 1) * 5 - 1) * 6)| ≤
    |1| ^ 6 / ↑(Nat.factorial 6) * (1 / 10 ^ 20 * 1 * 2 * 3 * 4 * 5 * 6)

Next tactic:
refine' exp_1_approx_succ_eq (by norm_num1; rfl) (by norm_cast) _
theorem exp_one_near_20 : |exp 1 - 363916618873 / 133877442384| ≤ 1 / 10 ^ 20 := by
  apply exp_approx_start
  iterate 21 refine' exp_1_approx_succ_eq (by 

Proof state:
⊢ 6 + 1 = ?m.4053

Next tactic:
norm_num1
theorem exp_one_near_20 : |exp 1 - 363916618873 / 133877442384| ≤ 1 / 10 ^ 20 := by
  apply exp_approx_start
  iterate 21 refine' exp_1_approx_succ_eq (by norm_num1; 

Proof state:
⊢ 7 = ?m.4053

Next tactic:
rfl
theorem exp_one_near_20 : |exp 1 - 363916618873 / 133877442384| ≤ 1 / 10 ^ 20 := by
  apply exp_approx_start
  iterate 21 refine' exp_1_approx_succ_eq (by norm_num1; rfl) (by 

Proof state:
⊢ ↑7 = ?m.4055

Next tactic:
norm_cast
theorem exp_one_near_20 : |exp 1 - 363916618873 / 133877442384| ≤ 1 / 10 ^ 20 := by
  apply exp_approx_start
  iterate 21 

Proof state:
case h
⊢ |rexp 1 -
        expNear 7 1
          ((((((((363916618873 / 133877442384 - 1) * 1 - 1) * 2 - 1) * 3 - 1) * 4 - 1) * 5 - 1) * 6 - 1) * 7)| ≤
    |1| ^ 7 / ↑(Nat.factorial 7) * (1 / 10 ^ 20 * 1 * 2 * 3 * 4 * 5 * 6 * 7)

Next tactic:
refine' exp_1_approx_succ_eq (by norm_num1; rfl) (by norm_cast) _
theorem exp_one_near_20 : |exp 1 - 363916618873 / 133877442384| ≤ 1 / 10 ^ 20 := by
  apply exp_approx_start
  iterate 21 refine' exp_1_approx_succ_eq (by 

Proof state:
⊢ 7 + 1 = ?m.4123

Next tactic:
norm_num1
theorem exp_one_near_20 : |exp 1 - 363916618873 / 133877442384| ≤ 1 / 10 ^ 20 := by
  apply exp_approx_start
  iterate 21 refine' exp_1_approx_succ_eq (by norm_num1; 

Proof state:
⊢ 8 = ?m.4123

Next tactic:
rfl
theorem exp_one_near_20 : |exp 1 - 363916618873 / 133877442384| ≤ 1 / 10 ^ 20 := by
  apply exp_approx_start
  iterate 21 refine' exp_1_approx_succ_eq (by norm_num1; rfl) (by 

Proof state:
⊢ ↑8 = ?m.4125

Next tactic:
norm_cast
theorem exp_one_near_20 : |exp 1 - 363916618873 / 133877442384| ≤ 1 / 10 ^ 20 := by
  apply exp_approx_start
  iterate 21 

Proof state:
case h
⊢ |rexp 1 -
        expNear 8 1
          (((((((((363916618873 / 133877442384 - 1) * 1 - 1) * 2 - 1) * 3 - 1) * 4 - 1) * 5 - 1) * 6 - 1) * 7 - 1) *
            8)| ≤
    |1| ^ 8 / ↑(Nat.factorial 8) * (1 / 10 ^ 20 * 1 * 2 * 3 * 4 * 5 * 6 * 7 * 8)

Next tactic:
refine' exp_1_approx_succ_eq (by norm_num1; rfl) (by norm_cast) _
theorem exp_one_near_20 : |exp 1 - 363916618873 / 133877442384| ≤ 1 / 10 ^ 20 := by
  apply exp_approx_start
  iterate 21 refine' exp_1_approx_succ_eq (by 

Proof state:
⊢ 8 + 1 = ?m.4193

Next tactic:
norm_num1
theorem exp_one_near_20 : |exp 1 - 363916618873 / 133877442384| ≤ 1 / 10 ^ 20 := by
  apply exp_approx_start
  iterate 21 refine' exp_1_approx_succ_eq (by norm_num1; 

Proof state:
⊢ 9 = ?m.4193

Next tactic:
rfl
theorem exp_one_near_20 : |exp 1 - 363916618873 / 133877442384| ≤ 1 / 10 ^ 20 := by
  apply exp_approx_start
  iterate 21 refine' exp_1_approx_succ_eq (by norm_num1; rfl) (by 

Proof state:
⊢ ↑9 = ?m.4195

Next tactic:
norm_cast
theorem exp_one_near_20 : |exp 1 - 363916618873 / 133877442384| ≤ 1 / 10 ^ 20 := by
  apply exp_approx_start
  iterate 21 

Proof state:
case h
⊢ |rexp 1 -
        expNear 9 1
          ((((((((((363916618873 / 133877442384 - 1) * 1 - 1) * 2 - 1) * 3 - 1) * 4 - 1) * 5 - 1) * 6 - 1) * 7 - 1) *
                8 -
              1) *
            9)| ≤
    |1| ^ 9 / ↑(Nat.factorial 9) * (1 / 10 ^ 20 * 1 * 2 * 3 * 4 * 5 * 6 * 7 * 8 * 9)

Next tactic:
refine' exp_1_approx_succ_eq (by norm_num1; rfl) (by norm_cast) _
theorem exp_one_near_20 : |exp 1 - 363916618873 / 133877442384| ≤ 1 / 10 ^ 20 := by
  apply exp_approx_start
  iterate 21 refine' exp_1_approx_succ_eq (by 

Proof state:
⊢ 9 + 1 = ?m.4263

Next tactic:
norm_num1
theorem exp_one_near_20 : |exp 1 - 363916618873 / 133877442384| ≤ 1 / 10 ^ 20 := by
  apply exp_approx_start
  iterate 21 refine' exp_1_approx_succ_eq (by norm_num1; 

Proof state:
⊢ 10 = ?m.4263

Next tactic:
rfl
theorem exp_one_near_20 : |exp 1 - 363916618873 / 133877442384| ≤ 1 / 10 ^ 20 := by
  apply exp_approx_start
  iterate 21 refine' exp_1_approx_succ_eq (by norm_num1; rfl) (by 

Proof state:
⊢ ↑10 = ?m.4265

Next tactic:
norm_cast
theorem exp_one_near_20 : |exp 1 - 363916618873 / 133877442384| ≤ 1 / 10 ^ 20 := by
  apply exp_approx_start
  iterate 21 

Proof state:
case h
⊢ |rexp 1 -
        expNear 10 1
          (((((((((((363916618873 / 133877442384 - 1) * 1 - 1) * 2 - 1) * 3 - 1) * 4 - 1) * 5 - 1) * 6 - 1) * 7 - 1) *
                    8 -
                  1) *
                9 -
              1) *
            10)| ≤
    |1| ^ 10 / ↑(Nat.factorial 10) * (1 / 10 ^ 20 * 1 * 2 * 3 * 4 * 5 * 6 * 7 * 8 * 9 * 10)

Next tactic:
refine' exp_1_approx_succ_eq (by norm_num1; rfl) (by norm_cast) _
theorem exp_one_near_20 : |exp 1 - 363916618873 / 133877442384| ≤ 1 / 10 ^ 20 := by
  apply exp_approx_start
  iterate 21 refine' exp_1_approx_succ_eq (by 

Proof state:
⊢ 10 + 1 = ?m.4333

Next tactic:
norm_num1
theorem exp_one_near_20 : |exp 1 - 363916618873 / 133877442384| ≤ 1 / 10 ^ 20 := by
  apply exp_approx_start
  iterate 21 refine' exp_1_approx_succ_eq (by norm_num1; 

Proof state:
⊢ 11 = ?m.4333

Next tactic:
rfl
theorem exp_one_near_20 : |exp 1 - 363916618873 / 133877442384| ≤ 1 / 10 ^ 20 := by
  apply exp_approx_start
  iterate 21 refine' exp_1_approx_succ_eq (by norm_num1; rfl) (by 

Proof state:
⊢ ↑11 = ?m.4335

Next tactic:
norm_cast
theorem exp_one_near_20 : |exp 1 - 363916618873 / 133877442384| ≤ 1 / 10 ^ 20 := by
  apply exp_approx_start
  iterate 21 

Proof state:
case h
⊢ |rexp 1 -
        expNear 11 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| ^ 11 / ↑(Nat.factorial 11) * (1 / 10 ^ 20 * 1 * 2 * 3 * 4 * 5 * 6 * 7 * 8 * 9 * 10 * 11)

Next tactic:
refine' exp_1_approx_succ_eq (by norm_num1; rfl) (by norm_cast) _
theorem exp_one_near_20 : |exp 1 - 363916618873 / 133877442384| ≤ 1 / 10 ^ 20 := by
  apply exp_approx_start
  iterate 21 refine' exp_1_approx_succ_eq (by 

Proof state:
⊢ 11 + 1 = ?m.4403

Next tactic:
norm_num1
theorem exp_one_near_20 : |exp 1 - 363916618873 / 133877442384| ≤ 1 / 10 ^ 20 := by
  apply exp_approx_start
  iterate 21 refine' exp_1_approx_succ_eq (by norm_num1; 

Proof state:
⊢ 12 = ?m.4403

Next tactic:
rfl
theorem exp_one_near_20 : |exp 1 - 363916618873 / 133877442384| ≤ 1 / 10 ^ 20 := by
  apply exp_approx_start
  iterate 21 refine' exp_1_approx_succ_eq (by norm_num1; rfl) (by 

Proof state:
⊢ ↑12 = ?m.4405

Next tactic:
norm_cast
theorem exp_one_near_20 : |exp 1 - 363916618873 / 133877442384| ≤ 1 / 10 ^ 20 := by
  apply exp_approx_start
  iterate 21 

Proof state:
case h
⊢ |rexp 1 -
        expNear 12 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| ^ 12 / ↑(Nat.factorial 12) * (1 / 10 ^ 20 * 1 * 2 * 3 * 4 * 5 * 6 * 7 * 8 * 9 * 10 * 11 * 12)

Next tactic:
refine' exp_1_approx_succ_eq (by norm_num1; rfl) (by norm_cast) _
theorem exp_one_near_20 : |exp 1 - 363916618873 / 133877442384| ≤ 1 / 10 ^ 20 := by
  apply exp_approx_start
  iterate 21 refine' exp_1_approx_succ_eq (by 

Proof state:
⊢ 12 + 1 = ?m.4473

Next tactic:
norm_num1
theorem exp_one_near_20 : |exp 1 - 363916618873 / 133877442384| ≤ 1 / 10 ^ 20 := by
  apply exp_approx_start
  iterate 21 refine' exp_1_approx_succ_eq (by norm_num1; 

Proof state:
⊢ 13 = ?m.4473

Next tactic:
rfl
theorem exp_one_near_20 : |exp 1 - 363916618873 / 133877442384| ≤ 1 / 10 ^ 20 := by
  apply exp_approx_start
  iterate 21 refine' exp_1_approx_succ_eq (by norm_num1; rfl) (by 

Proof state:
⊢ ↑13 = ?m.4475

Next tactic:
norm_cast
theorem exp_one_near_20 : |exp 1 - 363916618873 / 133877442384| ≤ 1 / 10 ^ 20 := by
  apply exp_approx_start
  iterate 21 

Proof state:
case h
⊢ |rexp 1 -
        expNear 13 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| ^ 13 / ↑(Nat.factorial 13) * (1 / 10 ^ 20 * 1 * 2 * 3 * 4 * 5 * 6 * 7 * 8 * 9 * 10 * 11 * 12 * 13)

Next tactic:
refine' exp_1_approx_succ_eq (by norm_num1; rfl) (by norm_cast) _
theorem exp_one_near_20 : |exp 1 - 363916618873 / 133877442384| ≤ 1 / 10 ^ 20 := by
  apply exp_approx_start
  iterate 21 refine' exp_1_approx_succ_eq (by 

Proof state:
⊢ 13 + 1 = ?m.4543

Next tactic:
norm_num1
theorem exp_one_near_20 : |exp 1 - 363916618873 / 133877442384| ≤ 1 / 10 ^ 20 := by
  apply exp_approx_start
  iterate 21 refine' exp_1_approx_succ_eq (by norm_num1; 

Proof state:
⊢ 14 = ?m.4543

Next tactic:
rfl
theorem exp_one_near_20 : |exp 1 - 363916618873 / 133877442384| ≤ 1 / 10 ^ 20 := by
  apply exp_approx_start
  iterate 21 refine' exp_1_approx_succ_eq (by norm_num1; rfl) (by 

Proof state:
⊢ ↑14 = ?m.4545

Next tactic:
norm_cast
theorem exp_one_near_20 : |exp 1 - 363916618873 / 133877442384| ≤ 1 / 10 ^ 20 := by
  apply exp_approx_start
  iterate 21 

Proof state:
case h
⊢ |rexp 1 -
        expNear 14 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| ^ 14 / ↑(Nat.factorial 14) * (1 / 10 ^ 20 * 1 * 2 * 3 * 4 * 5 * 6 * 7 * 8 * 9 * 10 * 11 * 12 * 13 * 14)

Next tactic:
refine' exp_1_approx_succ_eq (by norm_num1; rfl) (by norm_cast) _
theorem exp_one_near_20 : |exp 1 - 363916618873 / 133877442384| ≤ 1 / 10 ^ 20 := by
  apply exp_approx_start
  iterate 21 refine' exp_1_approx_succ_eq (by 

Proof state:
⊢ 14 + 1 = ?m.4613

Next tactic:
norm_num1
theorem exp_one_near_20 : |exp 1 - 363916618873 / 133877442384| ≤ 1 / 10 ^ 20 := by
  apply exp_approx_start
  iterate 21 refine' exp_1_approx_succ_eq (by norm_num1; 

Proof state:
⊢ 15 = ?m.4613

Next tactic:
rfl
theorem exp_one_near_20 : |exp 1 - 363916618873 / 133877442384| ≤ 1 / 10 ^ 20 := by
  apply exp_approx_start
  iterate 21 refine' exp_1_approx_succ_eq (by norm_num1; rfl) (by 

Proof state:
⊢ ↑15 = ?m.4615

Next tactic:
norm_cast
theorem exp_one_near_20 : |exp 1 - 363916618873 / 133877442384| ≤ 1 / 10 ^ 20 := by
  apply exp_approx_start
  iterate 21 

Proof state:
case h
⊢ |rexp 1 -
        expNear 15 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| ^ 15 / ↑(Nat.factorial 15) * (1 / 10 ^ 20 * 1 * 2 * 3 * 4 * 5 * 6 * 7 * 8 * 9 * 10 * 11 * 12 * 13 * 14 * 15)

Next tactic:
refine' exp_1_approx_succ_eq (by norm_num1; rfl) (by norm_cast) _
theorem exp_one_near_20 : |exp 1 - 363916618873 / 133877442384| ≤ 1 / 10 ^ 20 := by
  apply exp_approx_start
  iterate 21 refine' exp_1_approx_succ_eq (by 

Proof state:
⊢ 15 + 1 = ?m.4683

Next tactic:
norm_num1
theorem exp_one_near_20 : |exp 1 - 363916618873 / 133877442384| ≤ 1 / 10 ^ 20 := by
  apply exp_approx_start
  iterate 21 refine' exp_1_approx_succ_eq (by norm_num1; 

Proof state:
⊢ 16 = ?m.4683

Next tactic:
rfl
theorem exp_one_near_20 : |exp 1 - 363916618873 / 133877442384| ≤ 1 / 10 ^ 20 := by
  apply exp_approx_start
  iterate 21 refine' exp_1_approx_succ_eq (by norm_num1; rfl) (by 

Proof state:
⊢ ↑16 = ?m.4685

Next tactic:
norm_cast
theorem exp_one_near_20 : |exp 1 - 363916618873 / 133877442384| ≤ 1 / 10 ^ 20 := by
  apply exp_approx_start
  iterate 21 

Proof state:
case h
⊢ |rexp 1 -
        expNear 16 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| ^ 16 / ↑(Nat.factorial 16) *
      (1 / 10 ^ 20 * 1 * 2 * 3 * 4 * 5 * 6 * 7 * 8 * 9 * 10 * 11 * 12 * 13 * 14 * 15 * 16)

Next tactic:
refine' exp_1_approx_succ_eq (by norm_num1; rfl) (by norm_cast) _
theorem exp_one_near_20 : |exp 1 - 363916618873 / 133877442384| ≤ 1 / 10 ^ 20 := by
  apply exp_approx_start
  iterate 21 refine' exp_1_approx_succ_eq (by 

Proof state:
⊢ 16 + 1 = ?m.4753

Next tactic:
norm_num1
theorem exp_one_near_20 : |exp 1 - 363916618873 / 133877442384| ≤ 1 / 10 ^ 20 := by
  apply exp_approx_start
  iterate 21 refine' exp_1_approx_succ_eq (by norm_num1; 

Proof state:
⊢ 17 = ?m.4753

Next tactic:
rfl
theorem exp_one_near_20 : |exp 1 - 363916618873 / 133877442384| ≤ 1 / 10 ^ 20 := by
  apply exp_approx_start
  iterate 21 refine' exp_1_approx_succ_eq (by norm_num1; rfl) (by 

Proof state:
⊢ ↑17 = ?m.4755

Next tactic:
norm_cast
theorem exp_one_near_20 : |exp 1 - 363916618873 / 133877442384| ≤ 1 / 10 ^ 20 := by
  apply exp_approx_start
  iterate 21 

Proof state:
case h
⊢ |rexp 1 -
        expNear 17 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| ^ 17 / ↑(Nat.factorial 17) *
      (1 / 10 ^ 20 * 1 * 2 * 3 * 4 * 5 * 6 * 7 * 8 * 9 * 10 * 11 * 12 * 13 * 14 * 15 * 16 * 17)

Next tactic:
refine' exp_1_approx_succ_eq (by norm_num1; rfl) (by norm_cast) _
theorem exp_one_near_20 : |exp 1 - 363916618873 / 133877442384| ≤ 1 / 10 ^ 20 := by
  apply exp_approx_start
  iterate 21 refine' exp_1_approx_succ_eq (by 

Proof state:
⊢ 17 + 1 = ?m.4823

Next tactic:
norm_num1
theorem exp_one_near_20 : |exp 1 - 363916618873 / 133877442384| ≤ 1 / 10 ^ 20 := by
  apply exp_approx_start
  iterate 21 refine' exp_1_approx_succ_eq (by norm_num1; 

Proof state:
⊢ 18 = ?m.4823

Next tactic:
rfl
theorem exp_one_near_20 : |exp 1 - 363916618873 / 133877442384| ≤ 1 / 10 ^ 20 := by
  apply exp_approx_start
  iterate 21 refine' exp_1_approx_succ_eq (by norm_num1; rfl) (by 

Proof state:
⊢ ↑18 = ?m.4825

Next tactic:
norm_cast