Analysis I, Appendix B.2: The decimal representation of real numbers

An implementation of the decimal representation of Mathlib's real numbers ℝ : Typeℝ.

This is separate from the way decimal numerals are already represented in Mathlib. We also represent the integer part of the natural numbers just by ℕ : Typeℕ, avoiding using the decimal representation from the previous section, although we still retain the Unknown identifier `Digit`Digit class.

namespace AppendixBstructure NNRealDecimal where intPart : ℕ fracPart : ℕ → Digitopen NNReal NNRealDecimal

AppendixB.NNRealDecimal.mk (intPart : ℕ) (fracPart : ℕ → Digit) : NNRealDecimal

@[coe] noncomputable def NNRealDecimal.toNNReal (d:NNRealDecimal) : NNReal := d.intPart + ∑' i, (d.fracPart i) * (10:NNReal) ^ (-i-1:ℝ)noncomputable instance NNRealDecimal.instCoeNNReal : Coe NNRealDecimal NNReal where coe := toNNReal theorem NNRealDecimal.surj (x:NNReal) : ∃ d:NNRealDecimal, x = d := byx:ℝ≥0⊢ ∃ d, x = ↑d -- This proof is written to follow the structure of the original text. by_cases h : x = 0pos._@._hyg.162x:ℝ≥0h:x = 0⊢ ∃ d, x = ↑dneg._@._hyg.162x:ℝ≥0h:¬x = 0⊢ ∃ d, x = ↑d .pos._@._hyg.162x:ℝ≥0h:x = 0⊢ ∃ d, x = ↑d use mk 0 fun _ ↦ 0hx:ℝ≥0h:x = 0⊢ x = ↑{ intPart := 0, fracPart := fun x => 0 }; simp [h, toNNReal]All goals completed! 🐙 let s : ℕ → ℕ := fun n ↦ ⌊ x * 10^n ⌋₊neg._@._hyg.162x:ℝ≥0h:¬x = 0s:ℕ → ℕ := fun n => ⌊_fvar.8365 * 10 ^ n⌋₊⊢ ∃ d, x = ↑d have hs (n:ℕ) : s n ≤ x * 10^n := Nat.floor_le (byx:ℝ≥0h:¬x = 0s:ℕ → ℕ := fun n => ⌊_fvar.8365 * 10 ^ n⌋₊n:ℕ⊢ 0 ≤ x * 10 ^ n positivityAll goals completed! 🐙) have hs' (n:ℕ) : x * 10^n < s n + 1 := Nat.lt_floor_add_one _neg._@._hyg.162x:ℝ≥0h:¬x = 0s:ℕ → ℕ := fun n => ⌊_fvar.8365 * 10 ^ n⌋₊hs:∀ (n : ℕ), ↑(@_fvar.10801 n) ≤ _fvar.8365 * 10 ^ n := fun n => Nat.floor_le (@?_mvar.12729 n)hs':∀ (n : ℕ), _fvar.8365 * 10 ^ n < ↑(@_fvar.10801 n) + 1 := fun n => Nat.lt_floor_add_one (_fvar.8365 * 10 ^ n)⊢ ∃ d, x = ↑d have hdigit (n:ℕ) : ∃ a:Digit, s (n+1) = 10 * s n + (a:ℕ) := byx:ℝ≥0⊢ ∃ d, x = ↑d have hl : (10:NNReal) * s n < s (n+1) + 1 := calc _ ≤ 10 * (x * 10^n) := byx:ℝ≥0h:¬x = 0s:ℕ → ℕ := fun n => ⌊_fvar.8365 * 10 ^ n⌋₊hs:∀ (n : ℕ), ↑(@_fvar.10801 n) ≤ _fvar.8365 * 10 ^ n := fun n => Nat.floor_le (le_of_lt (mul_pos (lt_of_le_of_ne' (zero_le _fvar.8365) _fvar.8442) (pow_pos (Mathlib.Meta.Positivity.pos_of_isNat (Mathlib.Meta.NormNum.isNat_ofNat NNReal (Eq.refl 10)) (Eq.refl (Nat.ble 1 10))) n)))hs':∀ (n : ℕ), _fvar.8365 * 10 ^ n < ↑(@_fvar.10801 n) + 1 := fun n => Nat.lt_floor_add_one (_fvar.8365 * 10 ^ n)n:ℕ⊢ 10 * ↑(s n) ≤ 10 * (x * 10 ^ n) gcongrbcx:ℝ≥0h:¬x = 0s:ℕ → ℕ := fun n => ⌊_fvar.8365 * 10 ^ n⌋₊hs:∀ (n : ℕ), ↑(@_fvar.10801 n) ≤ _fvar.8365 * 10 ^ n := fun n => Nat.floor_le (le_of_lt (mul_pos (lt_of_le_of_ne' (zero_le _fvar.8365) _fvar.8442) (pow_pos (Mathlib.Meta.Positivity.pos_of_isNat (Mathlib.Meta.NormNum.isNat_ofNat NNReal (Eq.refl 10)) (Eq.refl (Nat.ble 1 10))) n)))hs':∀ (n : ℕ), _fvar.8365 * 10 ^ n < ↑(@_fvar.10801 n) + 1 := fun n => Nat.lt_floor_add_one (_fvar.8365 * 10 ^ n)n:ℕ⊢ ↑(s n) ≤ x * 10 ^ n; grindAll goals completed! 🐙 _ = x * 10^(n+1) := byx:ℝ≥0h:¬x = 0s:ℕ → ℕ := fun n => ⌊_fvar.8365 * 10 ^ n⌋₊hs:∀ (n : ℕ), ↑(@_fvar.10801 n) ≤ _fvar.8365 * 10 ^ n := fun n => Nat.floor_le (le_of_lt (mul_pos (lt_of_le_of_ne' (zero_le _fvar.8365) _fvar.8442) (pow_pos (Mathlib.Meta.Positivity.pos_of_isNat (Mathlib.Meta.NormNum.isNat_ofNat NNReal (Eq.refl 10)) (Eq.refl (Nat.ble 1 10))) n)))hs':∀ (n : ℕ), _fvar.8365 * 10 ^ n < ↑(@_fvar.10801 n) + 1 := fun n => Nat.lt_floor_add_one (_fvar.8365 * 10 ^ n)n:ℕ⊢ 10 * (x * 10 ^ n) = x * 10 ^ (n + 1) ring_nfAll goals completed! 🐙 _ < _ := hs' _ have hu : s (n+1) < (10:NNReal) * s n + 10 := calc _ ≤ x * 10^(n+1) := hs (n+1) _ = 10 * (x * 10^n) := byx:ℝ≥0h:¬x = 0s:ℕ → ℕ := fun n => ⌊_fvar.8365 * 10 ^ n⌋₊hs:∀ (n : ℕ), ↑(@_fvar.10801 n) ≤ _fvar.8365 * 10 ^ n := fun n => Nat.floor_le (le_of_lt (mul_pos (lt_of_le_of_ne' (zero_le _fvar.8365) _fvar.8442) (pow_pos (Mathlib.Meta.Positivity.pos_of_isNat (Mathlib.Meta.NormNum.isNat_ofNat NNReal (Eq.refl 10)) (Eq.refl (Nat.ble 1 10))) n)))hs':∀ (n : ℕ), _fvar.8365 * 10 ^ n < ↑(@_fvar.10801 n) + 1 := fun n => Nat.lt_floor_add_one (_fvar.8365 * 10 ^ n)n:ℕhl:10 * ↑(@_fvar.10801 _fvar.15937) < ↑(@_fvar.10801 (_fvar.15937 + 1)) + 1 := Trans.trans (Trans.trans (mul_le_mul_left' (AppendixB.NNRealDecimal.surj._proof_2 _fvar.8365 _fvar.12730 _fvar.15937) 10) (of_eq_true (Eq.trans (congr (congrArg Eq (Eq.trans (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat NNReal (Eq.refl 10))) (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.atom_pf _fvar.8365) (Mathlib.Tactic.Ring.pow_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat NNReal (Eq.refl 10))) (Mathlib.Tactic.Ring.atom_pf _fvar.15937) (Mathlib.Tactic.Ring.pow_add (Mathlib.Tactic.Ring.single_pow (Mathlib.Tactic.Ring.pow_prod_atom (Nat.rawCast 10) (_fvar.15937 ^ Nat.rawCast 1 * Nat.rawCast 1))) (Mathlib.Tactic.Ring.pow_zero (Nat.rawCast 10 + 0)) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_left (Nat.rawCast 10 + 0) (_fvar.15937 ^ Nat.rawCast 1 * Nat.rawCast 1) (Mathlib.Tactic.Ring.one_mul (Nat.rawCast 1))) (Mathlib.Tactic.Ring.mul_zero ((Nat.rawCast 10 + 0) ^ (_fvar.15937 ^ Nat.rawCast 1 * Nat.rawCast 1) * Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_add_zero ((Nat.rawCast 10 + 0) ^ (_fvar.15937 ^ Nat.rawCast 1 * Nat.rawCast 1) * Nat.rawCast 1 + 0))) (Mathlib.Tactic.Ring.zero_mul (Nat.rawCast 1 + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero ((Nat.rawCast 10 + 0) ^ (_fvar.15937 ^ Nat.rawCast 1 * Nat.rawCast 1) * Nat.rawCast 1 + 0))))) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_left _fvar.8365 (Nat.rawCast 1) (Mathlib.Tactic.Ring.mul_pf_right (Nat.rawCast 10 + 0) (_fvar.15937 ^ Nat.rawCast 1 * Nat.rawCast 1) (Mathlib.Tactic.Ring.one_mul (Nat.rawCast 1)))) (Mathlib.Tactic.Ring.mul_zero (_fvar.8365 ^ Nat.rawCast 1 * Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_add_zero (_fvar.8365 ^ Nat.rawCast 1 * ((Nat.rawCast 10 + 0) ^ (_fvar.15937 ^ Nat.rawCast 1 * Nat.rawCast 1) * Nat.rawCast 1) + 0))) (Mathlib.Tactic.Ring.zero_mul ((Nat.rawCast 10 + 0) ^ (_fvar.15937 ^ Nat.rawCast 1 * Nat.rawCast 1) * Nat.rawCast 1 + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (_fvar.8365 ^ Nat.rawCast 1 * ((Nat.rawCast 10 + 0) ^ (_fvar.15937 ^ Nat.rawCast 1 * Nat.rawCast 1) * Nat.rawCast 1) + 0)))) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_right _fvar.8365 (Nat.rawCast 1) (Mathlib.Tactic.Ring.mul_pf_right (Nat.rawCast 10 + 0) (_fvar.15937 ^ Nat.rawCast 1 * Nat.rawCast 1) (Mathlib.Tactic.Ring.mul_one (Nat.rawCast 10)))) (Mathlib.Tactic.Ring.mul_zero (Nat.rawCast 10)) (Mathlib.Tactic.Ring.add_pf_add_zero (_fvar.8365 ^ Nat.rawCast 1 * ((Nat.rawCast 10 + 0) ^ (_fvar.15937 ^ Nat.rawCast 1 * Nat.rawCast 1) * Nat.rawCast 10) + 0))) (Mathlib.Tactic.Ring.zero_mul (_fvar.8365 ^ Nat.rawCast 1 * ((Nat.rawCast 10 + 0) ^ (_fvar.15937 ^ Nat.rawCast 1 * Nat.rawCast 1) * Nat.rawCast 1) + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (_fvar.8365 ^ Nat.rawCast 1 * ((Nat.rawCast 10 + 0) ^ (_fvar.15937 ^ Nat.rawCast 1 * Nat.rawCast 1) * Nat.rawCast 10) + 0)))) (Eq.trans (congrArg (fun x => x + 0) (Eq.trans (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow _fvar.8365) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one _fvar.8365))) (congrArg (fun x => x * 10) (congr (congrArg HPow.hPow (add_zero 10)) (Eq.trans (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow _fvar.15937) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one _fvar.15937))) Mathlib.Tactic.RingNF.nat_rawCast_1) (mul_one _fvar.15937))))) (Mathlib.Tactic.RingNF.mul_assoc_rev _fvar.8365 (10 ^ _fvar.15937) 10))) (add_zero (_fvar.8365 * 10 ^ _fvar.15937 * 10))))) (Eq.trans (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.atom_pf _fvar.8365) (Mathlib.Tactic.Ring.pow_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat NNReal (Eq.refl 10))) (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.atom_pf _fvar.15937) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℕ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_pf_add_gt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_add_zero (_fvar.15937 ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))) (Mathlib.Tactic.Ring.pow_add (Mathlib.Tactic.Ring.pow_one_cast (Nat.rawCast 10 + 0)) (Mathlib.Tactic.Ring.pow_add (Mathlib.Tactic.Ring.single_pow (Mathlib.Tactic.Ring.pow_prod_atom (Nat.rawCast 10) (_fvar.15937 ^ Nat.rawCast 1 * Nat.rawCast 1))) (Mathlib.Tactic.Ring.pow_zero (Nat.rawCast 10 + 0)) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_left (Nat.rawCast 10 + 0) (_fvar.15937 ^ Nat.rawCast 1 * Nat.rawCast 1) (Mathlib.Tactic.Ring.one_mul (Nat.rawCast 1))) (Mathlib.Tactic.Ring.mul_zero ((Nat.rawCast 10 + 0) ^ (_fvar.15937 ^ Nat.rawCast 1 * Nat.rawCast 1) * Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_add_zero ((Nat.rawCast 10 + 0) ^ (_fvar.15937 ^ Nat.rawCast 1 * Nat.rawCast 1) * Nat.rawCast 1 + 0))) (Mathlib.Tactic.Ring.zero_mul (Nat.rawCast 1 + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero ((Nat.rawCast 10 + 0) ^ (_fvar.15937 ^ Nat.rawCast 1 * Nat.rawCast 1) * Nat.rawCast 1 + 0)))) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_right (Nat.rawCast 10 + 0) (_fvar.15937 ^ Nat.rawCast 1 * Nat.rawCast 1) (Mathlib.Tactic.Ring.mul_one (Nat.rawCast 10))) (Mathlib.Tactic.Ring.mul_zero (Nat.rawCast 10)) (Mathlib.Tactic.Ring.add_pf_add_zero ((Nat.rawCast 10 + 0) ^ (_fvar.15937 ^ Nat.rawCast 1 * Nat.rawCast 1) * Nat.rawCast 10 + 0))) (Mathlib.Tactic.Ring.zero_mul ((Nat.rawCast 10 + 0) ^ (_fvar.15937 ^ Nat.rawCast 1 * Nat.rawCast 1) * Nat.rawCast 1 + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero ((Nat.rawCast 10 + 0) ^ (_fvar.15937 ^ Nat.rawCast 1 * Nat.rawCast 1) * Nat.rawCast 10 + 0))))) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_left _fvar.8365 (Nat.rawCast 1) (Mathlib.Tactic.Ring.mul_pf_right (Nat.rawCast 10 + 0) (_fvar.15937 ^ Nat.rawCast 1 * Nat.rawCast 1) (Mathlib.Tactic.Ring.one_mul (Nat.rawCast 10)))) (Mathlib.Tactic.Ring.mul_zero (_fvar.8365 ^ Nat.rawCast 1 * Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_add_zero (_fvar.8365 ^ Nat.rawCast 1 * ((Nat.rawCast 10 + 0) ^ (_fvar.15937 ^ Nat.rawCast 1 * Nat.rawCast 1) * Nat.rawCast 10) + 0))) (Mathlib.Tactic.Ring.zero_mul ((Nat.rawCast 10 + 0) ^ (_fvar.15937 ^ Nat.rawCast 1 * Nat.rawCast 1) * Nat.rawCast 10 + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (_fvar.8365 ^ Nat.rawCast 1 * ((Nat.rawCast 10 + 0) ^ (_fvar.15937 ^ Nat.rawCast 1 * Nat.rawCast 1) * Nat.rawCast 10) + 0)))) (Eq.trans (congrArg (fun x => x + 0) (Eq.trans (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow _fvar.8365) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one _fvar.8365))) (congrArg (fun x => x * 10) (congr (congrArg HPow.hPow (add_zero 10)) (Eq.trans (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow _fvar.15937) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one _fvar.15937))) Mathlib.Tactic.RingNF.nat_rawCast_1) (mul_one _fvar.15937))))) (Mathlib.Tactic.RingNF.mul_assoc_rev _fvar.8365 (10 ^ _fvar.15937) 10))) (add_zero (_fvar.8365 * 10 ^ _fvar.15937 * 10))))) (eq_self (_fvar.8365 * 10 ^ _fvar.15937 * 10))))) (@_fvar.15930 (_fvar.15937 + 1))⊢ x * 10 ^ (n + 1) = 10 * (x * 10 ^ n) ring_nfAll goals completed! 🐙 _ < 10 * (s n + 1) := byx:ℝ≥0h:¬x = 0s:ℕ → ℕ := fun n => ⌊_fvar.8365 * 10 ^ n⌋₊hs:∀ (n : ℕ), ↑(@_fvar.10801 n) ≤ _fvar.8365 * 10 ^ n := fun n => Nat.floor_le (le_of_lt (mul_pos (lt_of_le_of_ne' (zero_le _fvar.8365) _fvar.8442) (pow_pos (Mathlib.Meta.Positivity.pos_of_isNat (Mathlib.Meta.NormNum.isNat_ofNat NNReal (Eq.refl 10)) (Eq.refl (Nat.ble 1 10))) n)))hs':∀ (n : ℕ), _fvar.8365 * 10 ^ n < ↑(@_fvar.10801 n) + 1 := fun n => Nat.lt_floor_add_one (_fvar.8365 * 10 ^ n)n:ℕhl:10 * ↑(@_fvar.10801 _fvar.15937) < ↑(@_fvar.10801 (_fvar.15937 + 1)) + 1 := Trans.trans (Trans.trans (mul_le_mul_left' (AppendixB.NNRealDecimal.surj._proof_2 _fvar.8365 _fvar.12730 _fvar.15937) 10) (of_eq_true (Eq.trans (congr (congrArg Eq (Eq.trans (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat NNReal (Eq.refl 10))) (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.atom_pf _fvar.8365) (Mathlib.Tactic.Ring.pow_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat NNReal (Eq.refl 10))) (Mathlib.Tactic.Ring.atom_pf _fvar.15937) (Mathlib.Tactic.Ring.pow_add (Mathlib.Tactic.Ring.single_pow (Mathlib.Tactic.Ring.pow_prod_atom (Nat.rawCast 10) (_fvar.15937 ^ Nat.rawCast 1 * Nat.rawCast 1))) (Mathlib.Tactic.Ring.pow_zero (Nat.rawCast 10 + 0)) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_left (Nat.rawCast 10 + 0) (_fvar.15937 ^ Nat.rawCast 1 * Nat.rawCast 1) (Mathlib.Tactic.Ring.one_mul (Nat.rawCast 1))) (Mathlib.Tactic.Ring.mul_zero ((Nat.rawCast 10 + 0) ^ (_fvar.15937 ^ Nat.rawCast 1 * Nat.rawCast 1) * Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_add_zero ((Nat.rawCast 10 + 0) ^ (_fvar.15937 ^ Nat.rawCast 1 * Nat.rawCast 1) * Nat.rawCast 1 + 0))) (Mathlib.Tactic.Ring.zero_mul (Nat.rawCast 1 + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero ((Nat.rawCast 10 + 0) ^ (_fvar.15937 ^ Nat.rawCast 1 * Nat.rawCast 1) * Nat.rawCast 1 + 0))))) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_left _fvar.8365 (Nat.rawCast 1) (Mathlib.Tactic.Ring.mul_pf_right (Nat.rawCast 10 + 0) (_fvar.15937 ^ Nat.rawCast 1 * Nat.rawCast 1) (Mathlib.Tactic.Ring.one_mul (Nat.rawCast 1)))) (Mathlib.Tactic.Ring.mul_zero (_fvar.8365 ^ Nat.rawCast 1 * Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_add_zero (_fvar.8365 ^ Nat.rawCast 1 * ((Nat.rawCast 10 + 0) ^ (_fvar.15937 ^ Nat.rawCast 1 * Nat.rawCast 1) * Nat.rawCast 1) + 0))) (Mathlib.Tactic.Ring.zero_mul ((Nat.rawCast 10 + 0) ^ (_fvar.15937 ^ Nat.rawCast 1 * Nat.rawCast 1) * Nat.rawCast 1 + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (_fvar.8365 ^ Nat.rawCast 1 * ((Nat.rawCast 10 + 0) ^ (_fvar.15937 ^ Nat.rawCast 1 * Nat.rawCast 1) * Nat.rawCast 1) + 0)))) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_right _fvar.8365 (Nat.rawCast 1) (Mathlib.Tactic.Ring.mul_pf_right (Nat.rawCast 10 + 0) (_fvar.15937 ^ Nat.rawCast 1 * Nat.rawCast 1) (Mathlib.Tactic.Ring.mul_one (Nat.rawCast 10)))) (Mathlib.Tactic.Ring.mul_zero (Nat.rawCast 10)) (Mathlib.Tactic.Ring.add_pf_add_zero (_fvar.8365 ^ Nat.rawCast 1 * ((Nat.rawCast 10 + 0) ^ (_fvar.15937 ^ Nat.rawCast 1 * Nat.rawCast 1) * Nat.rawCast 10) + 0))) (Mathlib.Tactic.Ring.zero_mul (_fvar.8365 ^ Nat.rawCast 1 * ((Nat.rawCast 10 + 0) ^ (_fvar.15937 ^ Nat.rawCast 1 * Nat.rawCast 1) * Nat.rawCast 1) + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (_fvar.8365 ^ Nat.rawCast 1 * ((Nat.rawCast 10 + 0) ^ (_fvar.15937 ^ Nat.rawCast 1 * Nat.rawCast 1) * Nat.rawCast 10) + 0)))) (Eq.trans (congrArg (fun x => x + 0) (Eq.trans (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow _fvar.8365) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one _fvar.8365))) (congrArg (fun x => x * 10) (congr (congrArg HPow.hPow (add_zero 10)) (Eq.trans (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow _fvar.15937) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one _fvar.15937))) Mathlib.Tactic.RingNF.nat_rawCast_1) (mul_one _fvar.15937))))) (Mathlib.Tactic.RingNF.mul_assoc_rev _fvar.8365 (10 ^ _fvar.15937) 10))) (add_zero (_fvar.8365 * 10 ^ _fvar.15937 * 10))))) (Eq.trans (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.atom_pf _fvar.8365) (Mathlib.Tactic.Ring.pow_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat NNReal (Eq.refl 10))) (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.atom_pf _fvar.15937) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℕ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_pf_add_gt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_add_zero (_fvar.15937 ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))) (Mathlib.Tactic.Ring.pow_add (Mathlib.Tactic.Ring.pow_one_cast (Nat.rawCast 10 + 0)) (Mathlib.Tactic.Ring.pow_add (Mathlib.Tactic.Ring.single_pow (Mathlib.Tactic.Ring.pow_prod_atom (Nat.rawCast 10) (_fvar.15937 ^ Nat.rawCast 1 * Nat.rawCast 1))) (Mathlib.Tactic.Ring.pow_zero (Nat.rawCast 10 + 0)) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_left (Nat.rawCast 10 + 0) (_fvar.15937 ^ Nat.rawCast 1 * Nat.rawCast 1) (Mathlib.Tactic.Ring.one_mul (Nat.rawCast 1))) (Mathlib.Tactic.Ring.mul_zero ((Nat.rawCast 10 + 0) ^ (_fvar.15937 ^ Nat.rawCast 1 * Nat.rawCast 1) * Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_add_zero ((Nat.rawCast 10 + 0) ^ (_fvar.15937 ^ Nat.rawCast 1 * Nat.rawCast 1) * Nat.rawCast 1 + 0))) (Mathlib.Tactic.Ring.zero_mul (Nat.rawCast 1 + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero ((Nat.rawCast 10 + 0) ^ (_fvar.15937 ^ Nat.rawCast 1 * Nat.rawCast 1) * Nat.rawCast 1 + 0)))) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_right (Nat.rawCast 10 + 0) (_fvar.15937 ^ Nat.rawCast 1 * Nat.rawCast 1) (Mathlib.Tactic.Ring.mul_one (Nat.rawCast 10))) (Mathlib.Tactic.Ring.mul_zero (Nat.rawCast 10)) (Mathlib.Tactic.Ring.add_pf_add_zero ((Nat.rawCast 10 + 0) ^ (_fvar.15937 ^ Nat.rawCast 1 * Nat.rawCast 1) * Nat.rawCast 10 + 0))) (Mathlib.Tactic.Ring.zero_mul ((Nat.rawCast 10 + 0) ^ (_fvar.15937 ^ Nat.rawCast 1 * Nat.rawCast 1) * Nat.rawCast 1 + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero ((Nat.rawCast 10 + 0) ^ (_fvar.15937 ^ Nat.rawCast 1 * Nat.rawCast 1) * Nat.rawCast 10 + 0))))) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_left _fvar.8365 (Nat.rawCast 1) (Mathlib.Tactic.Ring.mul_pf_right (Nat.rawCast 10 + 0) (_fvar.15937 ^ Nat.rawCast 1 * Nat.rawCast 1) (Mathlib.Tactic.Ring.one_mul (Nat.rawCast 10)))) (Mathlib.Tactic.Ring.mul_zero (_fvar.8365 ^ Nat.rawCast 1 * Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_add_zero (_fvar.8365 ^ Nat.rawCast 1 * ((Nat.rawCast 10 + 0) ^ (_fvar.15937 ^ Nat.rawCast 1 * Nat.rawCast 1) * Nat.rawCast 10) + 0))) (Mathlib.Tactic.Ring.zero_mul ((Nat.rawCast 10 + 0) ^ (_fvar.15937 ^ Nat.rawCast 1 * Nat.rawCast 1) * Nat.rawCast 10 + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (_fvar.8365 ^ Nat.rawCast 1 * ((Nat.rawCast 10 + 0) ^ (_fvar.15937 ^ Nat.rawCast 1 * Nat.rawCast 1) * Nat.rawCast 10) + 0)))) (Eq.trans (congrArg (fun x => x + 0) (Eq.trans (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow _fvar.8365) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one _fvar.8365))) (congrArg (fun x => x * 10) (congr (congrArg HPow.hPow (add_zero 10)) (Eq.trans (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow _fvar.15937) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one _fvar.15937))) Mathlib.Tactic.RingNF.nat_rawCast_1) (mul_one _fvar.15937))))) (Mathlib.Tactic.RingNF.mul_assoc_rev _fvar.8365 (10 ^ _fvar.15937) 10))) (add_zero (_fvar.8365 * 10 ^ _fvar.15937 * 10))))) (eq_self (_fvar.8365 * 10 ^ _fvar.15937 * 10))))) (@_fvar.15930 (_fvar.15937 + 1))⊢ 10 * (x * 10 ^ n) < 10 * (↑(s n) + 1) gcongrbcx:ℝ≥0h:¬x = 0s:ℕ → ℕ := fun n => ⌊_fvar.8365 * 10 ^ n⌋₊hs:∀ (n : ℕ), ↑(@_fvar.10801 n) ≤ _fvar.8365 * 10 ^ n := fun n => Nat.floor_le (le_of_lt (mul_pos (lt_of_le_of_ne' (zero_le _fvar.8365) _fvar.8442) (pow_pos (Mathlib.Meta.Positivity.pos_of_isNat (Mathlib.Meta.NormNum.isNat_ofNat NNReal (Eq.refl 10)) (Eq.refl (Nat.ble 1 10))) n)))hs':∀ (n : ℕ), _fvar.8365 * 10 ^ n < ↑(@_fvar.10801 n) + 1 := fun n => Nat.lt_floor_add_one (_fvar.8365 * 10 ^ n)n:ℕhl:10 * ↑(@_fvar.10801 _fvar.15937) < ↑(@_fvar.10801 (_fvar.15937 + 1)) + 1 := Trans.trans (Trans.trans (mul_le_mul_left' (AppendixB.NNRealDecimal.surj._proof_2 _fvar.8365 _fvar.12730 _fvar.15937) 10) (of_eq_true (Eq.trans (congr (congrArg Eq (Eq.trans (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat NNReal (Eq.refl 10))) (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.atom_pf _fvar.8365) (Mathlib.Tactic.Ring.pow_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat NNReal (Eq.refl 10))) (Mathlib.Tactic.Ring.atom_pf _fvar.15937) (Mathlib.Tactic.Ring.pow_add (Mathlib.Tactic.Ring.single_pow (Mathlib.Tactic.Ring.pow_prod_atom (Nat.rawCast 10) (_fvar.15937 ^ Nat.rawCast 1 * Nat.rawCast 1))) (Mathlib.Tactic.Ring.pow_zero (Nat.rawCast 10 + 0)) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_left (Nat.rawCast 10 + 0) (_fvar.15937 ^ Nat.rawCast 1 * Nat.rawCast 1) (Mathlib.Tactic.Ring.one_mul (Nat.rawCast 1))) (Mathlib.Tactic.Ring.mul_zero ((Nat.rawCast 10 + 0) ^ (_fvar.15937 ^ Nat.rawCast 1 * Nat.rawCast 1) * Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_add_zero ((Nat.rawCast 10 + 0) ^ (_fvar.15937 ^ Nat.rawCast 1 * Nat.rawCast 1) * Nat.rawCast 1 + 0))) (Mathlib.Tactic.Ring.zero_mul (Nat.rawCast 1 + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero ((Nat.rawCast 10 + 0) ^ (_fvar.15937 ^ Nat.rawCast 1 * Nat.rawCast 1) * Nat.rawCast 1 + 0))))) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_left _fvar.8365 (Nat.rawCast 1) (Mathlib.Tactic.Ring.mul_pf_right (Nat.rawCast 10 + 0) (_fvar.15937 ^ Nat.rawCast 1 * Nat.rawCast 1) (Mathlib.Tactic.Ring.one_mul (Nat.rawCast 1)))) (Mathlib.Tactic.Ring.mul_zero (_fvar.8365 ^ Nat.rawCast 1 * Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_add_zero (_fvar.8365 ^ Nat.rawCast 1 * ((Nat.rawCast 10 + 0) ^ (_fvar.15937 ^ Nat.rawCast 1 * Nat.rawCast 1) * Nat.rawCast 1) + 0))) (Mathlib.Tactic.Ring.zero_mul ((Nat.rawCast 10 + 0) ^ (_fvar.15937 ^ Nat.rawCast 1 * Nat.rawCast 1) * Nat.rawCast 1 + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (_fvar.8365 ^ Nat.rawCast 1 * ((Nat.rawCast 10 + 0) ^ (_fvar.15937 ^ Nat.rawCast 1 * Nat.rawCast 1) * Nat.rawCast 1) + 0)))) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_right _fvar.8365 (Nat.rawCast 1) (Mathlib.Tactic.Ring.mul_pf_right (Nat.rawCast 10 + 0) (_fvar.15937 ^ Nat.rawCast 1 * Nat.rawCast 1) (Mathlib.Tactic.Ring.mul_one (Nat.rawCast 10)))) (Mathlib.Tactic.Ring.mul_zero (Nat.rawCast 10)) (Mathlib.Tactic.Ring.add_pf_add_zero (_fvar.8365 ^ Nat.rawCast 1 * ((Nat.rawCast 10 + 0) ^ (_fvar.15937 ^ Nat.rawCast 1 * Nat.rawCast 1) * Nat.rawCast 10) + 0))) (Mathlib.Tactic.Ring.zero_mul (_fvar.8365 ^ Nat.rawCast 1 * ((Nat.rawCast 10 + 0) ^ (_fvar.15937 ^ Nat.rawCast 1 * Nat.rawCast 1) * Nat.rawCast 1) + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (_fvar.8365 ^ Nat.rawCast 1 * ((Nat.rawCast 10 + 0) ^ (_fvar.15937 ^ Nat.rawCast 1 * Nat.rawCast 1) * Nat.rawCast 10) + 0)))) (Eq.trans (congrArg (fun x => x + 0) (Eq.trans (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow _fvar.8365) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one _fvar.8365))) (congrArg (fun x => x * 10) (congr (congrArg HPow.hPow (add_zero 10)) (Eq.trans (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow _fvar.15937) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one _fvar.15937))) Mathlib.Tactic.RingNF.nat_rawCast_1) (mul_one _fvar.15937))))) (Mathlib.Tactic.RingNF.mul_assoc_rev _fvar.8365 (10 ^ _fvar.15937) 10))) (add_zero (_fvar.8365 * 10 ^ _fvar.15937 * 10))))) (Eq.trans (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.atom_pf _fvar.8365) (Mathlib.Tactic.Ring.pow_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat NNReal (Eq.refl 10))) (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.atom_pf _fvar.15937) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℕ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_pf_add_gt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_add_zero (_fvar.15937 ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))) (Mathlib.Tactic.Ring.pow_add (Mathlib.Tactic.Ring.pow_one_cast (Nat.rawCast 10 + 0)) (Mathlib.Tactic.Ring.pow_add (Mathlib.Tactic.Ring.single_pow (Mathlib.Tactic.Ring.pow_prod_atom (Nat.rawCast 10) (_fvar.15937 ^ Nat.rawCast 1 * Nat.rawCast 1))) (Mathlib.Tactic.Ring.pow_zero (Nat.rawCast 10 + 0)) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_left (Nat.rawCast 10 + 0) (_fvar.15937 ^ Nat.rawCast 1 * Nat.rawCast 1) (Mathlib.Tactic.Ring.one_mul (Nat.rawCast 1))) (Mathlib.Tactic.Ring.mul_zero ((Nat.rawCast 10 + 0) ^ (_fvar.15937 ^ Nat.rawCast 1 * Nat.rawCast 1) * Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_add_zero ((Nat.rawCast 10 + 0) ^ (_fvar.15937 ^ Nat.rawCast 1 * Nat.rawCast 1) * Nat.rawCast 1 + 0))) (Mathlib.Tactic.Ring.zero_mul (Nat.rawCast 1 + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero ((Nat.rawCast 10 + 0) ^ (_fvar.15937 ^ Nat.rawCast 1 * Nat.rawCast 1) * Nat.rawCast 1 + 0)))) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_right (Nat.rawCast 10 + 0) (_fvar.15937 ^ Nat.rawCast 1 * Nat.rawCast 1) (Mathlib.Tactic.Ring.mul_one (Nat.rawCast 10))) (Mathlib.Tactic.Ring.mul_zero (Nat.rawCast 10)) (Mathlib.Tactic.Ring.add_pf_add_zero ((Nat.rawCast 10 + 0) ^ (_fvar.15937 ^ Nat.rawCast 1 * Nat.rawCast 1) * Nat.rawCast 10 + 0))) (Mathlib.Tactic.Ring.zero_mul ((Nat.rawCast 10 + 0) ^ (_fvar.15937 ^ Nat.rawCast 1 * Nat.rawCast 1) * Nat.rawCast 1 + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero ((Nat.rawCast 10 + 0) ^ (_fvar.15937 ^ Nat.rawCast 1 * Nat.rawCast 1) * Nat.rawCast 10 + 0))))) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_left _fvar.8365 (Nat.rawCast 1) (Mathlib.Tactic.Ring.mul_pf_right (Nat.rawCast 10 + 0) (_fvar.15937 ^ Nat.rawCast 1 * Nat.rawCast 1) (Mathlib.Tactic.Ring.one_mul (Nat.rawCast 10)))) (Mathlib.Tactic.Ring.mul_zero (_fvar.8365 ^ Nat.rawCast 1 * Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_add_zero (_fvar.8365 ^ Nat.rawCast 1 * ((Nat.rawCast 10 + 0) ^ (_fvar.15937 ^ Nat.rawCast 1 * Nat.rawCast 1) * Nat.rawCast 10) + 0))) (Mathlib.Tactic.Ring.zero_mul ((Nat.rawCast 10 + 0) ^ (_fvar.15937 ^ Nat.rawCast 1 * Nat.rawCast 1) * Nat.rawCast 10 + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (_fvar.8365 ^ Nat.rawCast 1 * ((Nat.rawCast 10 + 0) ^ (_fvar.15937 ^ Nat.rawCast 1 * Nat.rawCast 1) * Nat.rawCast 10) + 0)))) (Eq.trans (congrArg (fun x => x + 0) (Eq.trans (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow _fvar.8365) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one _fvar.8365))) (congrArg (fun x => x * 10) (congr (congrArg HPow.hPow (add_zero 10)) (Eq.trans (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow _fvar.15937) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one _fvar.15937))) Mathlib.Tactic.RingNF.nat_rawCast_1) (mul_one _fvar.15937))))) (Mathlib.Tactic.RingNF.mul_assoc_rev _fvar.8365 (10 ^ _fvar.15937) 10))) (add_zero (_fvar.8365 * 10 ^ _fvar.15937 * 10))))) (eq_self (_fvar.8365 * 10 ^ _fvar.15937 * 10))))) (@_fvar.15930 (_fvar.15937 + 1))⊢ x * 10 ^ n < ↑(s n) + 1; grindAll goals completed! 🐙 _ = _ := byx:ℝ≥0h:¬x = 0s:ℕ → ℕ := fun n => ⌊_fvar.8365 * 10 ^ n⌋₊hs:∀ (n : ℕ), ↑(@_fvar.10801 n) ≤ _fvar.8365 * 10 ^ n := fun n => Nat.floor_le (le_of_lt (mul_pos (lt_of_le_of_ne' (zero_le _fvar.8365) _fvar.8442) (pow_pos (Mathlib.Meta.Positivity.pos_of_isNat (Mathlib.Meta.NormNum.isNat_ofNat NNReal (Eq.refl 10)) (Eq.refl (Nat.ble 1 10))) n)))hs':∀ (n : ℕ), _fvar.8365 * 10 ^ n < ↑(@_fvar.10801 n) + 1 := fun n => Nat.lt_floor_add_one (_fvar.8365 * 10 ^ n)n:ℕhl:10 * ↑(@_fvar.10801 _fvar.15937) < ↑(@_fvar.10801 (_fvar.15937 + 1)) + 1 := Trans.trans (Trans.trans (mul_le_mul_left' (AppendixB.NNRealDecimal.surj._proof_2 _fvar.8365 _fvar.12730 _fvar.15937) 10) (of_eq_true (Eq.trans (congr (congrArg Eq (Eq.trans (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat NNReal (Eq.refl 10))) (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.atom_pf _fvar.8365) (Mathlib.Tactic.Ring.pow_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat NNReal (Eq.refl 10))) (Mathlib.Tactic.Ring.atom_pf _fvar.15937) (Mathlib.Tactic.Ring.pow_add (Mathlib.Tactic.Ring.single_pow (Mathlib.Tactic.Ring.pow_prod_atom (Nat.rawCast 10) (_fvar.15937 ^ Nat.rawCast 1 * Nat.rawCast 1))) (Mathlib.Tactic.Ring.pow_zero (Nat.rawCast 10 + 0)) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_left (Nat.rawCast 10 + 0) (_fvar.15937 ^ Nat.rawCast 1 * Nat.rawCast 1) (Mathlib.Tactic.Ring.one_mul (Nat.rawCast 1))) (Mathlib.Tactic.Ring.mul_zero ((Nat.rawCast 10 + 0) ^ (_fvar.15937 ^ Nat.rawCast 1 * Nat.rawCast 1) * Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_add_zero ((Nat.rawCast 10 + 0) ^ (_fvar.15937 ^ Nat.rawCast 1 * Nat.rawCast 1) * Nat.rawCast 1 + 0))) (Mathlib.Tactic.Ring.zero_mul (Nat.rawCast 1 + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero ((Nat.rawCast 10 + 0) ^ (_fvar.15937 ^ Nat.rawCast 1 * Nat.rawCast 1) * Nat.rawCast 1 + 0))))) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_left _fvar.8365 (Nat.rawCast 1) (Mathlib.Tactic.Ring.mul_pf_right (Nat.rawCast 10 + 0) (_fvar.15937 ^ Nat.rawCast 1 * Nat.rawCast 1) (Mathlib.Tactic.Ring.one_mul (Nat.rawCast 1)))) (Mathlib.Tactic.Ring.mul_zero (_fvar.8365 ^ Nat.rawCast 1 * Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_add_zero (_fvar.8365 ^ Nat.rawCast 1 * ((Nat.rawCast 10 + 0) ^ (_fvar.15937 ^ Nat.rawCast 1 * Nat.rawCast 1) * Nat.rawCast 1) + 0))) (Mathlib.Tactic.Ring.zero_mul ((Nat.rawCast 10 + 0) ^ (_fvar.15937 ^ Nat.rawCast 1 * Nat.rawCast 1) * Nat.rawCast 1 + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (_fvar.8365 ^ Nat.rawCast 1 * ((Nat.rawCast 10 + 0) ^ (_fvar.15937 ^ Nat.rawCast 1 * Nat.rawCast 1) * Nat.rawCast 1) + 0)))) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_right _fvar.8365 (Nat.rawCast 1) (Mathlib.Tactic.Ring.mul_pf_right (Nat.rawCast 10 + 0) (_fvar.15937 ^ Nat.rawCast 1 * Nat.rawCast 1) (Mathlib.Tactic.Ring.mul_one (Nat.rawCast 10)))) (Mathlib.Tactic.Ring.mul_zero (Nat.rawCast 10)) (Mathlib.Tactic.Ring.add_pf_add_zero (_fvar.8365 ^ Nat.rawCast 1 * ((Nat.rawCast 10 + 0) ^ (_fvar.15937 ^ Nat.rawCast 1 * Nat.rawCast 1) * Nat.rawCast 10) + 0))) (Mathlib.Tactic.Ring.zero_mul (_fvar.8365 ^ Nat.rawCast 1 * ((Nat.rawCast 10 + 0) ^ (_fvar.15937 ^ Nat.rawCast 1 * Nat.rawCast 1) * Nat.rawCast 1) + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (_fvar.8365 ^ Nat.rawCast 1 * ((Nat.rawCast 10 + 0) ^ (_fvar.15937 ^ Nat.rawCast 1 * Nat.rawCast 1) * Nat.rawCast 10) + 0)))) (Eq.trans (congrArg (fun x => x + 0) (Eq.trans (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow _fvar.8365) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one _fvar.8365))) (congrArg (fun x => x * 10) (congr (congrArg HPow.hPow (add_zero 10)) (Eq.trans (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow _fvar.15937) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one _fvar.15937))) Mathlib.Tactic.RingNF.nat_rawCast_1) (mul_one _fvar.15937))))) (Mathlib.Tactic.RingNF.mul_assoc_rev _fvar.8365 (10 ^ _fvar.15937) 10))) (add_zero (_fvar.8365 * 10 ^ _fvar.15937 * 10))))) (Eq.trans (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.atom_pf _fvar.8365) (Mathlib.Tactic.Ring.pow_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat NNReal (Eq.refl 10))) (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.atom_pf _fvar.15937) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℕ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_pf_add_gt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_add_zero (_fvar.15937 ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))) (Mathlib.Tactic.Ring.pow_add (Mathlib.Tactic.Ring.pow_one_cast (Nat.rawCast 10 + 0)) (Mathlib.Tactic.Ring.pow_add (Mathlib.Tactic.Ring.single_pow (Mathlib.Tactic.Ring.pow_prod_atom (Nat.rawCast 10) (_fvar.15937 ^ Nat.rawCast 1 * Nat.rawCast 1))) (Mathlib.Tactic.Ring.pow_zero (Nat.rawCast 10 + 0)) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_left (Nat.rawCast 10 + 0) (_fvar.15937 ^ Nat.rawCast 1 * Nat.rawCast 1) (Mathlib.Tactic.Ring.one_mul (Nat.rawCast 1))) (Mathlib.Tactic.Ring.mul_zero ((Nat.rawCast 10 + 0) ^ (_fvar.15937 ^ Nat.rawCast 1 * Nat.rawCast 1) * Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_add_zero ((Nat.rawCast 10 + 0) ^ (_fvar.15937 ^ Nat.rawCast 1 * Nat.rawCast 1) * Nat.rawCast 1 + 0))) (Mathlib.Tactic.Ring.zero_mul (Nat.rawCast 1 + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero ((Nat.rawCast 10 + 0) ^ (_fvar.15937 ^ Nat.rawCast 1 * Nat.rawCast 1) * Nat.rawCast 1 + 0)))) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_right (Nat.rawCast 10 + 0) (_fvar.15937 ^ Nat.rawCast 1 * Nat.rawCast 1) (Mathlib.Tactic.Ring.mul_one (Nat.rawCast 10))) (Mathlib.Tactic.Ring.mul_zero (Nat.rawCast 10)) (Mathlib.Tactic.Ring.add_pf_add_zero ((Nat.rawCast 10 + 0) ^ (_fvar.15937 ^ Nat.rawCast 1 * Nat.rawCast 1) * Nat.rawCast 10 + 0))) (Mathlib.Tactic.Ring.zero_mul ((Nat.rawCast 10 + 0) ^ (_fvar.15937 ^ Nat.rawCast 1 * Nat.rawCast 1) * Nat.rawCast 1 + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero ((Nat.rawCast 10 + 0) ^ (_fvar.15937 ^ Nat.rawCast 1 * Nat.rawCast 1) * Nat.rawCast 10 + 0))))) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_left _fvar.8365 (Nat.rawCast 1) (Mathlib.Tactic.Ring.mul_pf_right (Nat.rawCast 10 + 0) (_fvar.15937 ^ Nat.rawCast 1 * Nat.rawCast 1) (Mathlib.Tactic.Ring.one_mul (Nat.rawCast 10)))) (Mathlib.Tactic.Ring.mul_zero (_fvar.8365 ^ Nat.rawCast 1 * Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_add_zero (_fvar.8365 ^ Nat.rawCast 1 * ((Nat.rawCast 10 + 0) ^ (_fvar.15937 ^ Nat.rawCast 1 * Nat.rawCast 1) * Nat.rawCast 10) + 0))) (Mathlib.Tactic.Ring.zero_mul ((Nat.rawCast 10 + 0) ^ (_fvar.15937 ^ Nat.rawCast 1 * Nat.rawCast 1) * Nat.rawCast 10 + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (_fvar.8365 ^ Nat.rawCast 1 * ((Nat.rawCast 10 + 0) ^ (_fvar.15937 ^ Nat.rawCast 1 * Nat.rawCast 1) * Nat.rawCast 10) + 0)))) (Eq.trans (congrArg (fun x => x + 0) (Eq.trans (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow _fvar.8365) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one _fvar.8365))) (congrArg (fun x => x * 10) (congr (congrArg HPow.hPow (add_zero 10)) (Eq.trans (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow _fvar.15937) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one _fvar.15937))) Mathlib.Tactic.RingNF.nat_rawCast_1) (mul_one _fvar.15937))))) (Mathlib.Tactic.RingNF.mul_assoc_rev _fvar.8365 (10 ^ _fvar.15937) 10))) (add_zero (_fvar.8365 * 10 ^ _fvar.15937 * 10))))) (eq_self (_fvar.8365 * 10 ^ _fvar.15937 * 10))))) (@_fvar.15930 (_fvar.15937 + 1))⊢ 10 * (↑(s n) + 1) = 10 * ↑(s n) + 10 ringAll goals completed! 🐙 norm_cast at hl hux:ℝ≥0h:¬x = 0s:ℕ → ℕ := fun n => ⌊_fvar.8365 * 10 ^ n⌋₊hs:∀ (n : ℕ), ↑(@_fvar.10801 n) ≤ _fvar.8365 * 10 ^ n := fun n => Nat.floor_le (@?_mvar.12729 n)hs':∀ (n : ℕ), _fvar.8365 * 10 ^ n < ↑(@_fvar.10801 n) + 1 := fun n => Nat.lt_floor_add_one (_fvar.8365 * 10 ^ n)n:ℕhl:10 * s n < s (n + 1) + 1hu:s (n + 1) < 10 * s n + 10⊢ ∃ a, s (n + 1) = 10 * s n + ↑a set d := s (n+1) - 10 * s nx:ℝ≥0h:¬x = 0s:ℕ → ℕ := fun n => ⌊_fvar.8365 * 10 ^ n⌋₊hs:∀ (n : ℕ), ↑(@_fvar.10801 n) ≤ _fvar.8365 * 10 ^ n := fun n => Nat.floor_le (@?_mvar.12729 n)hs':∀ (n : ℕ), _fvar.8365 * 10 ^ n < ↑(@_fvar.10801 n) + 1 := fun n => Nat.lt_floor_add_one (_fvar.8365 * 10 ^ n)n:ℕhl:10 * s n < s (n + 1) + 1hu:s (n + 1) < 10 * s n + 10d:ℕ := @_fvar.10801 (_fvar.15937 + 1) - 10 * @_fvar.10801 _fvar.15937⊢ ∃ a, s (n + 1) = 10 * s n + ↑a have hd : d < 10 := byx:ℝ≥0⊢ ∃ d, x = ↑d omegaAll goals completed! 🐙 have : s (n+1) = 10 * s n + d := byx:ℝ≥0⊢ ∃ d, x = ↑d omegaAll goals completed! 🐙 use Digit.mk hdAll goals completed! 🐙 choose a ha using hdigitneg._@._hyg.162x:ℝ≥0h:¬x = 0s:ℕ → ℕ := fun n => ⌊_fvar.8365 * 10 ^ n⌋₊hs:∀ (n : ℕ), ↑(@_fvar.10801 n) ≤ _fvar.8365 * 10 ^ n := fun n => Nat.floor_le (@?_mvar.12729 n)hs':∀ (n : ℕ), _fvar.8365 * 10 ^ n < ↑(@_fvar.10801 n) + 1 := fun n => Nat.lt_floor_add_one (_fvar.8365 * 10 ^ n)a:ℕ → Digitha:∀ (n : ℕ), s (n + 1) = 10 * s n + ↑(a n)⊢ ∃ d, x = ↑d set d := mk (s 0) aneg._@._hyg.162x:ℝ≥0h:¬x = 0s:ℕ → ℕ := fun n => ⌊_fvar.8365 * 10 ^ n⌋₊hs:∀ (n : ℕ), ↑(@_fvar.10801 n) ≤ _fvar.8365 * 10 ^ n := fun n => Nat.floor_le (@?_mvar.12729 n)hs':∀ (n : ℕ), _fvar.8365 * 10 ^ n < ↑(@_fvar.10801 n) + 1 := fun n => Nat.lt_floor_add_one (_fvar.8365 * 10 ^ n)a:ℕ → Digitha:∀ (n : ℕ), s (n + 1) = 10 * s n + ↑(a n)d:AppendixB.NNRealDecimal := { intPart := @_fvar.10801 0, fracPart := _fvar.51385 }⊢ ∃ d, x = ↑d; use dhx:ℝ≥0h:¬x = 0s:ℕ → ℕ := fun n => ⌊_fvar.8365 * 10 ^ n⌋₊hs:∀ (n : ℕ), ↑(@_fvar.10801 n) ≤ _fvar.8365 * 10 ^ n := fun n => Nat.floor_le (@?_mvar.12729 n)hs':∀ (n : ℕ), _fvar.8365 * 10 ^ n < ↑(@_fvar.10801 n) + 1 := fun n => Nat.lt_floor_add_one (_fvar.8365 * 10 ^ n)a:ℕ → Digitha:∀ (n : ℕ), s (n + 1) = 10 * s n + ↑(a n)d:AppendixB.NNRealDecimal := { intPart := @_fvar.10801 0, fracPart := _fvar.51385 }⊢ x = ↑d have hsum (n:ℕ) : s n * (10:NNReal)^(-n:ℝ) = s 0 + ∑ i ∈ .range n, a i * (10:NNReal)^(-i-1:ℝ) := byx:ℝ≥0⊢ ∃ d, x = ↑d induction' n with n hnzerox:ℝ≥0h:¬x = 0s:ℕ → ℕ := fun n => ⌊_fvar.8365 * 10 ^ n⌋₊hs:∀ (n : ℕ), ↑(@_fvar.10801 n) ≤ _fvar.8365 * 10 ^ n := fun n => Nat.floor_le (@?_mvar.12729 n)hs':∀ (n : ℕ), _fvar.8365 * 10 ^ n < ↑(@_fvar.10801 n) + 1 := fun n => Nat.lt_floor_add_one (_fvar.8365 * 10 ^ n)a:ℕ → Digitha:∀ (n : ℕ), s (n + 1) = 10 * s n + ↑(a n)d:AppendixB.NNRealDecimal := { intPart := @_fvar.10801 0, fracPart := _fvar.51385 }⊢ ↑(s 0) * 10 ^ (-↑0) = ↑(s 0) + ∑ i ∈ Finset.range 0, ↑↑(a i) * 10 ^ (-↑i - 1)succx:ℝ≥0h:¬x = 0s:ℕ → ℕ := fun n => ⌊_fvar.8365 * 10 ^ n⌋₊hs:∀ (n : ℕ), ↑(@_fvar.10801 n) ≤ _fvar.8365 * 10 ^ n := fun n => Nat.floor_le (@?_mvar.12729 n)hs':∀ (n : ℕ), _fvar.8365 * 10 ^ n < ↑(@_fvar.10801 n) + 1 := fun n => Nat.lt_floor_add_one (_fvar.8365 * 10 ^ n)a:ℕ → Digitha:∀ (n : ℕ), s (n + 1) = 10 * s n + ↑(a n)d:AppendixB.NNRealDecimal := { intPart := @_fvar.10801 0, fracPart := _fvar.51385 }n:ℕhn:↑(s n) * 10 ^ (-↑n) = ↑(s 0) + ∑ i ∈ Finset.range n, ↑↑(a i) * 10 ^ (-↑i - 1)⊢ ↑(s (n + 1)) * 10 ^ (-↑(n + 1)) = ↑(s 0) + ∑ i ∈ Finset.range (n + 1), ↑↑(a i) * 10 ^ (-↑i - 1); simpsuccx:ℝ≥0h:¬x = 0s:ℕ → ℕ := fun n => ⌊_fvar.8365 * 10 ^ n⌋₊hs:∀ (n : ℕ), ↑(@_fvar.10801 n) ≤ _fvar.8365 * 10 ^ n := fun n => Nat.floor_le (@?_mvar.12729 n)hs':∀ (n : ℕ), _fvar.8365 * 10 ^ n < ↑(@_fvar.10801 n) + 1 := fun n => Nat.lt_floor_add_one (_fvar.8365 * 10 ^ n)a:ℕ → Digitha:∀ (n : ℕ), s (n + 1) = 10 * s n + ↑(a n)d:AppendixB.NNRealDecimal := { intPart := @_fvar.10801 0, fracPart := _fvar.51385 }n:ℕhn:↑(s n) * 10 ^ (-↑n) = ↑(s 0) + ∑ i ∈ Finset.range n, ↑↑(a i) * 10 ^ (-↑i - 1)⊢ ↑(s (n + 1)) * 10 ^ (-↑(n + 1)) = ↑(s 0) + ∑ i ∈ Finset.range (n + 1), ↑↑(a i) * 10 ^ (-↑i - 1) rw [ha n]succx:ℝ≥0h:¬x = 0s:ℕ → ℕ := fun n => ⌊_fvar.8365 * 10 ^ n⌋₊hs:∀ (n : ℕ), ↑(@_fvar.10801 n) ≤ _fvar.8365 * 10 ^ n := fun n => Nat.floor_le (@?_mvar.12729 n)hs':∀ (n : ℕ), _fvar.8365 * 10 ^ n < ↑(@_fvar.10801 n) + 1 := fun n => Nat.lt_floor_add_one (_fvar.8365 * 10 ^ n)a:ℕ → Digitha:∀ (n : ℕ), s (n + 1) = 10 * s n + ↑(a n)d:AppendixB.NNRealDecimal := { intPart := @_fvar.10801 0, fracPart := _fvar.51385 }n:ℕhn:↑(s n) * 10 ^ (-↑n) = ↑(s 0) + ∑ i ∈ Finset.range n, ↑↑(a i) * 10 ^ (-↑i - 1)⊢ ↑(10 * s n + ↑(a n)) * 10 ^ (-↑(n + 1)) = ↑(s 0) + ∑ i ∈ Finset.range (n + 1), ↑↑(a i) * 10 ^ (-↑i - 1); calc _ = s n * (10:NNReal)^(-n:ℝ) + a n * 10^(-n-1:ℝ) := byx:ℝ≥0h:¬x = 0s:ℕ → ℕ := fun n => ⌊_fvar.8365 * 10 ^ n⌋₊hs:∀ (n : ℕ), ↑(@_fvar.10801 n) ≤ _fvar.8365 * 10 ^ n := fun n => Nat.floor_le (le_of_lt (mul_pos (lt_of_le_of_ne' (zero_le _fvar.8365) _fvar.8442) (pow_pos (Mathlib.Meta.Positivity.pos_of_isNat (Mathlib.Meta.NormNum.isNat_ofNat NNReal (Eq.refl 10)) (Eq.refl (Nat.ble 1 10))) n)))hs':∀ (n : ℕ), _fvar.8365 * 10 ^ n < ↑(@_fvar.10801 n) + 1 := fun n => Nat.lt_floor_add_one (_fvar.8365 * 10 ^ n)a:ℕ → Digitha:∀ (n : ℕ), s (n + 1) = 10 * s n + ↑(a n)d:AppendixB.NNRealDecimal := { intPart := @_fvar.10801 0, fracPart := _fvar.51385 }n:ℕhn:↑(s n) * 10 ^ (-↑n) = ↑(s 0) + ∑ i ∈ Finset.range n, ↑↑(a i) * 10 ^ (-↑i - 1)⊢ ↑(10 * s n + ↑(a n)) * 10 ^ (-↑(n + 1)) = ↑(s n) * 10 ^ (-↑n) + ↑↑(a n) * 10 ^ (-↑n - 1) simp [add_mul]x:ℝ≥0h:¬x = 0s:ℕ → ℕ := fun n => ⌊_fvar.8365 * 10 ^ n⌋₊hs:∀ (n : ℕ), ↑(@_fvar.10801 n) ≤ _fvar.8365 * 10 ^ n := fun n => Nat.floor_le (le_of_lt (mul_pos (lt_of_le_of_ne' (zero_le _fvar.8365) _fvar.8442) (pow_pos (Mathlib.Meta.Positivity.pos_of_isNat (Mathlib.Meta.NormNum.isNat_ofNat NNReal (Eq.refl 10)) (Eq.refl (Nat.ble 1 10))) n)))hs':∀ (n : ℕ), _fvar.8365 * 10 ^ n < ↑(@_fvar.10801 n) + 1 := fun n => Nat.lt_floor_add_one (_fvar.8365 * 10 ^ n)a:ℕ → Digitha:∀ (n : ℕ), s (n + 1) = 10 * s n + ↑(a n)d:AppendixB.NNRealDecimal := { intPart := @_fvar.10801 0, fracPart := _fvar.51385 }n:ℕhn:↑(s n) * 10 ^ (-↑n) = ↑(s 0) + ∑ i ∈ Finset.range n, ↑↑(a i) * 10 ^ (-↑i - 1)⊢ 10 * ↑(s n) * 10 ^ (-1 + -↑n) + ↑↑(a n) * 10 ^ (-1 + -↑n) = ↑(s n) * 10 ^ (-↑n) + ↑↑(a n) * 10 ^ (-↑n - 1); ring_nfx:ℝ≥0h:¬x = 0s:ℕ → ℕ := fun n => ⌊_fvar.8365 * 10 ^ n⌋₊hs:∀ (n : ℕ), ↑(@_fvar.10801 n) ≤ _fvar.8365 * 10 ^ n := fun n => Nat.floor_le (le_of_lt (mul_pos (lt_of_le_of_ne' (zero_le _fvar.8365) _fvar.8442) (pow_pos (Mathlib.Meta.Positivity.pos_of_isNat (Mathlib.Meta.NormNum.isNat_ofNat NNReal (Eq.refl 10)) (Eq.refl (Nat.ble 1 10))) n)))hs':∀ (n : ℕ), _fvar.8365 * 10 ^ n < ↑(@_fvar.10801 n) + 1 := fun n => Nat.lt_floor_add_one (_fvar.8365 * 10 ^ n)a:ℕ → Digitha:∀ (n : ℕ), s (n + 1) = 10 * s n + ↑(a n)d:AppendixB.NNRealDecimal := { intPart := @_fvar.10801 0, fracPart := _fvar.51385 }n:ℕhn:↑(s n) * 10 ^ (-↑n) = ↑(s 0) + ∑ i ∈ Finset.range n, ↑↑(a i) * 10 ^ (-↑i - 1)⊢ ↑(s n) * 10 ^ (-1 - ↑n) * 10 + 10 ^ (-1 - ↑n) * ↑↑(a n) = ↑(s n) * 10 ^ (-↑n) + 10 ^ (-1 - ↑n) * ↑↑(a n); congr 1e_a._@.Init.Prelude._hyg.2136x:ℝ≥0h:¬x = 0s:ℕ → ℕ := fun n => ⌊_fvar.8365 * 10 ^ n⌋₊hs:∀ (n : ℕ), ↑(@_fvar.10801 n) ≤ _fvar.8365 * 10 ^ n := fun n => Nat.floor_le (le_of_lt (mul_pos (lt_of_le_of_ne' (zero_le _fvar.8365) _fvar.8442) (pow_pos (Mathlib.Meta.Positivity.pos_of_isNat (Mathlib.Meta.NormNum.isNat_ofNat NNReal (Eq.refl 10)) (Eq.refl (Nat.ble 1 10))) n)))hs':∀ (n : ℕ), _fvar.8365 * 10 ^ n < ↑(@_fvar.10801 n) + 1 := fun n => Nat.lt_floor_add_one (_fvar.8365 * 10 ^ n)a:ℕ → Digitha:∀ (n : ℕ), s (n + 1) = 10 * s n + ↑(a n)d:AppendixB.NNRealDecimal := { intPart := @_fvar.10801 0, fracPart := _fvar.51385 }n:ℕhn:↑(s n) * 10 ^ (-↑n) = ↑(s 0) + ∑ i ∈ Finset.range n, ↑↑(a i) * 10 ^ (-↑i - 1)⊢ ↑(s n) * 10 ^ (-1 - ↑n) * 10 = ↑(s n) * 10 ^ (-↑n) rw [mul_assoc, ←rpow_add_one]e_a._@.Init.Prelude._hyg.2136x:ℝ≥0h:¬x = 0s:ℕ → ℕ := fun n => ⌊_fvar.8365 * 10 ^ n⌋₊hs:∀ (n : ℕ), ↑(@_fvar.10801 n) ≤ _fvar.8365 * 10 ^ n := fun n => Nat.floor_le (le_of_lt (mul_pos (lt_of_le_of_ne' (zero_le _fvar.8365) _fvar.8442) (pow_pos (Mathlib.Meta.Positivity.pos_of_isNat (Mathlib.Meta.NormNum.isNat_ofNat NNReal (Eq.refl 10)) (Eq.refl (Nat.ble 1 10))) n)))hs':∀ (n : ℕ), _fvar.8365 * 10 ^ n < ↑(@_fvar.10801 n) + 1 := fun n => Nat.lt_floor_add_one (_fvar.8365 * 10 ^ n)a:ℕ → Digitha:∀ (n : ℕ), s (n + 1) = 10 * s n + ↑(a n)d:AppendixB.NNRealDecimal := { intPart := @_fvar.10801 0, fracPart := _fvar.51385 }n:ℕhn:↑(s n) * 10 ^ (-↑n) = ↑(s 0) + ∑ i ∈ Finset.range n, ↑↑(a i) * 10 ^ (-↑i - 1)⊢ ↑(s n) * 10 ^ (-1 - ↑n + 1) = ↑(s n) * 10 ^ (-↑n)e_a.hx._@.Init.Prelude._hyg.2136x:ℝ≥0h:¬x = 0s:ℕ → ℕ := fun n => ⌊_fvar.8365 * 10 ^ n⌋₊hs:∀ (n : ℕ), ↑(@_fvar.10801 n) ≤ _fvar.8365 * 10 ^ n := fun n => Nat.floor_le (le_of_lt (mul_pos (lt_of_le_of_ne' (zero_le _fvar.8365) _fvar.8442) (pow_pos (Mathlib.Meta.Positivity.pos_of_isNat (Mathlib.Meta.NormNum.isNat_ofNat NNReal (Eq.refl 10)) (Eq.refl (Nat.ble 1 10))) n)))hs':∀ (n : ℕ), _fvar.8365 * 10 ^ n < ↑(@_fvar.10801 n) + 1 := fun n => Nat.lt_floor_add_one (_fvar.8365 * 10 ^ n)a:ℕ → Digitha:∀ (n : ℕ), s (n + 1) = 10 * s n + ↑(a n)d:AppendixB.NNRealDecimal := { intPart := @_fvar.10801 0, fracPart := _fvar.51385 }n:ℕhn:↑(s n) * 10 ^ (-↑n) = ↑(s 0) + ∑ i ∈ Finset.range n, ↑↑(a i) * 10 ^ (-↑i - 1)⊢ 10 ≠ 0; ring_nfe_a.hx._@.Init.Prelude._hyg.2136x:ℝ≥0h:¬x = 0s:ℕ → ℕ := fun n => ⌊_fvar.8365 * 10 ^ n⌋₊hs:∀ (n : ℕ), ↑(@_fvar.10801 n) ≤ _fvar.8365 * 10 ^ n := fun n => Nat.floor_le (le_of_lt (mul_pos (lt_of_le_of_ne' (zero_le _fvar.8365) _fvar.8442) (pow_pos (Mathlib.Meta.Positivity.pos_of_isNat (Mathlib.Meta.NormNum.isNat_ofNat NNReal (Eq.refl 10)) (Eq.refl (Nat.ble 1 10))) n)))hs':∀ (n : ℕ), _fvar.8365 * 10 ^ n < ↑(@_fvar.10801 n) + 1 := fun n => Nat.lt_floor_add_one (_fvar.8365 * 10 ^ n)a:ℕ → Digitha:∀ (n : ℕ), s (n + 1) = 10 * s n + ↑(a n)d:AppendixB.NNRealDecimal := { intPart := @_fvar.10801 0, fracPart := _fvar.51385 }n:ℕhn:↑(s n) * 10 ^ (-↑n) = ↑(s 0) + ∑ i ∈ Finset.range n, ↑↑(a i) * 10 ^ (-↑i - 1)⊢ 10 ≠ 0; norm_numAll goals completed! 🐙 _ = s 0 + (∑ i ∈ .range n, a i * (10:NNReal)^(-i-1:ℝ) + a n * 10^(-n-1:ℝ)) := byx:ℝ≥0h:¬x = 0s:ℕ → ℕ := fun n => ⌊_fvar.8365 * 10 ^ n⌋₊hs:∀ (n : ℕ), ↑(@_fvar.10801 n) ≤ _fvar.8365 * 10 ^ n := fun n => Nat.floor_le (le_of_lt (mul_pos (lt_of_le_of_ne' (zero_le _fvar.8365) _fvar.8442) (pow_pos (Mathlib.Meta.Positivity.pos_of_isNat (Mathlib.Meta.NormNum.isNat_ofNat NNReal (Eq.refl 10)) (Eq.refl (Nat.ble 1 10))) n)))hs':∀ (n : ℕ), _fvar.8365 * 10 ^ n < ↑(@_fvar.10801 n) + 1 := fun n => Nat.lt_floor_add_one (_fvar.8365 * 10 ^ n)a:ℕ → Digitha:∀ (n : ℕ), s (n + 1) = 10 * s n + ↑(a n)d:AppendixB.NNRealDecimal := { intPart := @_fvar.10801 0, fracPart := _fvar.51385 }n:ℕhn:↑(s n) * 10 ^ (-↑n) = ↑(s 0) + ∑ i ∈ Finset.range n, ↑↑(a i) * 10 ^ (-↑i - 1)⊢ ↑(s n) * 10 ^ (-↑n) + ↑↑(a n) * 10 ^ (-↑n - 1) = ↑(s 0) + (∑ i ∈ Finset.range n, ↑↑(a i) * 10 ^ (-↑i - 1) + ↑↑(a n) * 10 ^ (-↑n - 1)) grindAll goals completed! 🐙 _ = _ := byx:ℝ≥0h:¬x = 0s:ℕ → ℕ := fun n => ⌊_fvar.8365 * 10 ^ n⌋₊hs:∀ (n : ℕ), ↑(@_fvar.10801 n) ≤ _fvar.8365 * 10 ^ n := fun n => Nat.floor_le (le_of_lt (mul_pos (lt_of_le_of_ne' (zero_le _fvar.8365) _fvar.8442) (pow_pos (Mathlib.Meta.Positivity.pos_of_isNat (Mathlib.Meta.NormNum.isNat_ofNat NNReal (Eq.refl 10)) (Eq.refl (Nat.ble 1 10))) n)))hs':∀ (n : ℕ), _fvar.8365 * 10 ^ n < ↑(@_fvar.10801 n) + 1 := fun n => Nat.lt_floor_add_one (_fvar.8365 * 10 ^ n)a:ℕ → Digitha:∀ (n : ℕ), s (n + 1) = 10 * s n + ↑(a n)d:AppendixB.NNRealDecimal := { intPart := @_fvar.10801 0, fracPart := _fvar.51385 }n:ℕhn:↑(s n) * 10 ^ (-↑n) = ↑(s 0) + ∑ i ∈ Finset.range n, ↑↑(a i) * 10 ^ (-↑i - 1)⊢ ↑(s 0) + (∑ i ∈ Finset.range n, ↑↑(a i) * 10 ^ (-↑i - 1) + ↑↑(a n) * 10 ^ (-↑n - 1)) = ↑(s 0) + ∑ i ∈ Finset.range (n + 1), ↑↑(a i) * 10 ^ (-↑i - 1) congre_a._@.Init.Prelude._hyg.2138x:ℝ≥0h:¬x = 0s:ℕ → ℕ := fun n => ⌊_fvar.8365 * 10 ^ n⌋₊hs:∀ (n : ℕ), ↑(@_fvar.10801 n) ≤ _fvar.8365 * 10 ^ n := fun n => Nat.floor_le (le_of_lt (mul_pos (lt_of_le_of_ne' (zero_le _fvar.8365) _fvar.8442) (pow_pos (Mathlib.Meta.Positivity.pos_of_isNat (Mathlib.Meta.NormNum.isNat_ofNat NNReal (Eq.refl 10)) (Eq.refl (Nat.ble 1 10))) n)))hs':∀ (n : ℕ), _fvar.8365 * 10 ^ n < ↑(@_fvar.10801 n) + 1 := fun n => Nat.lt_floor_add_one (_fvar.8365 * 10 ^ n)a:ℕ → Digitha:∀ (n : ℕ), s (n + 1) = 10 * s n + ↑(a n)d:AppendixB.NNRealDecimal := { intPart := @_fvar.10801 0, fracPart := _fvar.51385 }n:ℕhn:↑(s n) * 10 ^ (-↑n) = ↑(s 0) + ∑ i ∈ Finset.range n, ↑↑(a i) * 10 ^ (-↑i - 1)⊢ ∑ i ∈ Finset.range n, ↑↑(a i) * 10 ^ (-↑i - 1) + ↑↑(a n) * 10 ^ (-↑n - 1) = ∑ i ∈ Finset.range (n + 1), ↑↑(a i) * 10 ^ (-↑i - 1); symme_a._@.Init.Prelude._hyg.2138x:ℝ≥0h:¬x = 0s:ℕ → ℕ := fun n => ⌊_fvar.8365 * 10 ^ n⌋₊hs:∀ (n : ℕ), ↑(@_fvar.10801 n) ≤ _fvar.8365 * 10 ^ n := fun n => Nat.floor_le (le_of_lt (mul_pos (lt_of_le_of_ne' (zero_le _fvar.8365) _fvar.8442) (pow_pos (Mathlib.Meta.Positivity.pos_of_isNat (Mathlib.Meta.NormNum.isNat_ofNat NNReal (Eq.refl 10)) (Eq.refl (Nat.ble 1 10))) n)))hs':∀ (n : ℕ), _fvar.8365 * 10 ^ n < ↑(@_fvar.10801 n) + 1 := fun n => Nat.lt_floor_add_one (_fvar.8365 * 10 ^ n)a:ℕ → Digitha:∀ (n : ℕ), s (n + 1) = 10 * s n + ↑(a n)d:AppendixB.NNRealDecimal := { intPart := @_fvar.10801 0, fracPart := _fvar.51385 }n:ℕhn:↑(s n) * 10 ^ (-↑n) = ↑(s 0) + ∑ i ∈ Finset.range n, ↑↑(a i) * 10 ^ (-↑i - 1)⊢ ∑ i ∈ Finset.range (n + 1), ↑↑(a i) * 10 ^ (-↑i - 1) = ∑ i ∈ Finset.range n, ↑↑(a i) * 10 ^ (-↑i - 1) + ↑↑(a n) * 10 ^ (-↑n - 1); apply Finset.sum_range_succAll goals completed! 🐙 have := (d.toNNReal_conv.tendsto_sum_tsum_nat).const_add (s 0:NNReal)hx:ℝ≥0h:¬x = 0s:ℕ → ℕ := fun n => ⌊_fvar.8365 * 10 ^ n⌋₊hs:∀ (n : ℕ), ↑(@_fvar.10801 n) ≤ _fvar.8365 * 10 ^ n := fun n => Nat.floor_le (@?_mvar.12729 n)hs':∀ (n : ℕ), _fvar.8365 * 10 ^ n < ↑(@_fvar.10801 n) + 1 := fun n => Nat.lt_floor_add_one (_fvar.8365 * 10 ^ n)a:ℕ → Digitha:∀ (n : ℕ), s (n + 1) = 10 * s n + ↑(a n)d:AppendixB.NNRealDecimal := { intPart := @_fvar.10801 0, fracPart := _fvar.51385 }hsum:∀ (n : ℕ), ↑(@_fvar.10801 n) * 10 ^ (-↑n) = ↑(@_fvar.10801 0) + ∑ i ∈ Finset.range n, ↑↑(@_fvar.51385 i) * 10 ^ (-↑i - 1) := fun n => @?_mvar.57011 nthis:?_mvar.101197 := Filter.Tendsto.const_add (↑(@_fvar.10801 0)) (Summable.tendsto_sum_tsum_nat (AppendixB.NNRealDecimal.toNNReal_conv _fvar.51407))⊢ x = ↑d convert_to Filter.atTop.Tendsto (fun n ↦ s n * (10:NNReal)^(-n:ℝ)) (nhds (d:NNReal)) at thish.e'_3x:ℝ≥0h:¬x = 0s:ℕ → ℕ := fun n => ⌊_fvar.8365 * 10 ^ n⌋₊hs:∀ (n : ℕ), ↑(@_fvar.10801 n) ≤ _fvar.8365 * 10 ^ n := fun n => Nat.floor_le (@?_mvar.12729 n)hs':∀ (n : ℕ), _fvar.8365 * 10 ^ n < ↑(@_fvar.10801 n) + 1 := fun n => Nat.lt_floor_add_one (_fvar.8365 * 10 ^ n)a:ℕ → Digitha:∀ (n : ℕ), s (n + 1) = 10 * s n + ↑(a n)d:AppendixB.NNRealDecimal := { intPart := @_fvar.10801 0, fracPart := _fvar.51385 }hsum:∀ (n : ℕ), ↑(@_fvar.10801 n) * 10 ^ (-↑n) = ↑(@_fvar.10801 0) + ∑ i ∈ Finset.range n, ↑↑(@_fvar.51385 i) * 10 ^ (-↑i - 1) := fun n => @?_mvar.57011 nthis:?_mvar.101197 := Filter.Tendsto.const_add (↑(@_fvar.10801 0)) (Summable.tendsto_sum_tsum_nat (AppendixB.NNRealDecimal.toNNReal_conv _fvar.51407))⊢ (fun k => ↑(s 0) + ∑ i ∈ Finset.range k, ↑↑(d.fracPart i) * 10 ^ (-↑i - 1)) = fun n => ↑(s n) * 10 ^ (-↑n)hx:ℝ≥0h:¬x = 0s:ℕ → ℕ := fun n => ⌊_fvar.8365 * 10 ^ n⌋₊hs:∀ (n : ℕ), ↑(@_fvar.10801 n) ≤ _fvar.8365 * 10 ^ n := fun n => Nat.floor_le (@?_mvar.12729 n)hs':∀ (n : ℕ), _fvar.8365 * 10 ^ n < ↑(@_fvar.10801 n) + 1 := fun n => Nat.lt_floor_add_one (_fvar.8365 * 10 ^ n)a:ℕ → Digitha:∀ (n : ℕ), s (n + 1) = 10 * s n + ↑(a n)d:AppendixB.NNRealDecimal := { intPart := @_fvar.10801 0, fracPart := _fvar.51385 }hsum:∀ (n : ℕ), ↑(@_fvar.10801 n) * 10 ^ (-↑n) = ↑(@_fvar.10801 0) + ∑ i ∈ Finset.range n, ↑↑(@_fvar.51385 i) * 10 ^ (-↑i - 1) := fun n => @?_mvar.57011 nthis:Filter.Tendsto (fun n => ↑(s n) * 10 ^ (-↑n)) Filter.atTop (nhds ↑d)⊢ x = ↑d .h.e'_3x:ℝ≥0h:¬x = 0s:ℕ → ℕ := fun n => ⌊_fvar.8365 * 10 ^ n⌋₊hs:∀ (n : ℕ), ↑(@_fvar.10801 n) ≤ _fvar.8365 * 10 ^ n := fun n => Nat.floor_le (@?_mvar.12729 n)hs':∀ (n : ℕ), _fvar.8365 * 10 ^ n < ↑(@_fvar.10801 n) + 1 := fun n => Nat.lt_floor_add_one (_fvar.8365 * 10 ^ n)a:ℕ → Digitha:∀ (n : ℕ), s (n + 1) = 10 * s n + ↑(a n)d:AppendixB.NNRealDecimal := { intPart := @_fvar.10801 0, fracPart := _fvar.51385 }hsum:∀ (n : ℕ), ↑(@_fvar.10801 n) * 10 ^ (-↑n) = ↑(@_fvar.10801 0) + ∑ i ∈ Finset.range n, ↑↑(@_fvar.51385 i) * 10 ^ (-↑i - 1) := fun n => @?_mvar.57011 nthis:?_mvar.101197 := Filter.Tendsto.const_add (↑(@_fvar.10801 0)) (Summable.tendsto_sum_tsum_nat (AppendixB.NNRealDecimal.toNNReal_conv _fvar.51407))⊢ (fun k => ↑(s 0) + ∑ i ∈ Finset.range k, ↑↑(d.fracPart i) * 10 ^ (-↑i - 1)) = fun n => ↑(s n) * 10 ^ (-↑n) ext nh.e'_3.h.ax:ℝ≥0h:¬x = 0s:ℕ → ℕ := fun n => ⌊_fvar.8365 * 10 ^ n⌋₊hs:∀ (n : ℕ), ↑(@_fvar.10801 n) ≤ _fvar.8365 * 10 ^ n := fun n => Nat.floor_le (@?_mvar.12729 n)hs':∀ (n : ℕ), _fvar.8365 * 10 ^ n < ↑(@_fvar.10801 n) + 1 := fun n => Nat.lt_floor_add_one (_fvar.8365 * 10 ^ n)a:ℕ → Digitha:∀ (n : ℕ), s (n + 1) = 10 * s n + ↑(a n)d:AppendixB.NNRealDecimal := { intPart := @_fvar.10801 0, fracPart := _fvar.51385 }hsum:∀ (n : ℕ), ↑(@_fvar.10801 n) * 10 ^ (-↑n) = ↑(@_fvar.10801 0) + ∑ i ∈ Finset.range n, ↑↑(@_fvar.51385 i) * 10 ^ (-↑i - 1) := fun n => @?_mvar.57011 nthis:?_mvar.101197 := Filter.Tendsto.const_add (↑(@_fvar.10801 0)) (Summable.tendsto_sum_tsum_nat (AppendixB.NNRealDecimal.toNNReal_conv _fvar.51407))n:ℕ⊢ ↑(↑(s 0) + ∑ i ∈ Finset.range n, ↑↑(d.fracPart i) * 10 ^ (-↑i - 1)) = ↑(↑(s n) * 10 ^ (-↑n)); rw [hsum n]All goals completed! 🐙 apply tendsto_nhds_unique _ thisx:ℝ≥0h:¬x = 0s:ℕ → ℕ := fun n => ⌊_fvar.8365 * 10 ^ n⌋₊hs:∀ (n : ℕ), ↑(@_fvar.10801 n) ≤ _fvar.8365 * 10 ^ n := fun n => Nat.floor_le (@?_mvar.12729 n)hs':∀ (n : ℕ), _fvar.8365 * 10 ^ n < ↑(@_fvar.10801 n) + 1 := fun n => Nat.lt_floor_add_one (_fvar.8365 * 10 ^ n)a:ℕ → Digitha:∀ (n : ℕ), s (n + 1) = 10 * s n + ↑(a n)d:AppendixB.NNRealDecimal := { intPart := @_fvar.10801 0, fracPart := _fvar.51385 }hsum:∀ (n : ℕ), ↑(@_fvar.10801 n) * 10 ^ (-↑n) = ↑(@_fvar.10801 0) + ∑ i ∈ Finset.range n, ↑↑(@_fvar.51385 i) * 10 ^ (-↑i - 1) := fun n => @?_mvar.57011 nthis:Filter.Tendsto (fun n => ↑(s n) * 10 ^ (-↑n)) Filter.atTop (nhds ↑d)⊢ Filter.Tendsto (fun n => ↑(s n) * 10 ^ (-↑n)) Filter.atTop (nhds x) apply Filter.Tendsto.squeeze (g := fun n:ℕ ↦ x - (10:NNReal)^(-n:ℝ)) (h := fun _ ↦ x)hgx:ℝ≥0h:¬x = 0s:ℕ → ℕ := fun n => ⌊_fvar.8365 * 10 ^ n⌋₊hs:∀ (n : ℕ), ↑(@_fvar.10801 n) ≤ _fvar.8365 * 10 ^ n := fun n => Nat.floor_le (@?_mvar.12729 n)hs':∀ (n : ℕ), _fvar.8365 * 10 ^ n < ↑(@_fvar.10801 n) + 1 := fun n => Nat.lt_floor_add_one (_fvar.8365 * 10 ^ n)a:ℕ → Digitha:∀ (n : ℕ), s (n + 1) = 10 * s n + ↑(a n)d:AppendixB.NNRealDecimal := { intPart := @_fvar.10801 0, fracPart := _fvar.51385 }hsum:∀ (n : ℕ), ↑(@_fvar.10801 n) * 10 ^ (-↑n) = ↑(@_fvar.10801 0) + ∑ i ∈ Finset.range n, ↑↑(@_fvar.51385 i) * 10 ^ (-↑i - 1) := fun n => @?_mvar.57011 nthis:Filter.Tendsto (fun n => ↑(s n) * 10 ^ (-↑n)) Filter.atTop (nhds ↑d)⊢ Filter.Tendsto (fun n => x - 10 ^ (-↑n)) Filter.atTop (nhds x)hhx:ℝ≥0h:¬x = 0s:ℕ → ℕ := fun n => ⌊_fvar.8365 * 10 ^ n⌋₊hs:∀ (n : ℕ), ↑(@_fvar.10801 n) ≤ _fvar.8365 * 10 ^ n := fun n => Nat.floor_le (@?_mvar.12729 n)hs':∀ (n : ℕ), _fvar.8365 * 10 ^ n < ↑(@_fvar.10801 n) + 1 := fun n => Nat.lt_floor_add_one (_fvar.8365 * 10 ^ n)a:ℕ → Digitha:∀ (n : ℕ), s (n + 1) = 10 * s n + ↑(a n)d:AppendixB.NNRealDecimal := { intPart := @_fvar.10801 0, fracPart := _fvar.51385 }hsum:∀ (n : ℕ), ↑(@_fvar.10801 n) * 10 ^ (-↑n) = ↑(@_fvar.10801 0) + ∑ i ∈ Finset.range n, ↑↑(@_fvar.51385 i) * 10 ^ (-↑i - 1) := fun n => @?_mvar.57011 nthis:Filter.Tendsto (fun n => ↑(s n) * 10 ^ (-↑n)) Filter.atTop (nhds ↑d)⊢ Filter.Tendsto (fun x_1 => x) Filter.atTop (nhds x)hgfx:ℝ≥0h:¬x = 0s:ℕ → ℕ := fun n => ⌊_fvar.8365 * 10 ^ n⌋₊hs:∀ (n : ℕ), ↑(@_fvar.10801 n) ≤ _fvar.8365 * 10 ^ n := fun n => Nat.floor_le (@?_mvar.12729 n)hs':∀ (n : ℕ), _fvar.8365 * 10 ^ n < ↑(@_fvar.10801 n) + 1 := fun n => Nat.lt_floor_add_one (_fvar.8365 * 10 ^ n)a:ℕ → Digitha:∀ (n : ℕ), s (n + 1) = 10 * s n + ↑(a n)d:AppendixB.NNRealDecimal := { intPart := @_fvar.10801 0, fracPart := _fvar.51385 }hsum:∀ (n : ℕ), ↑(@_fvar.10801 n) * 10 ^ (-↑n) = ↑(@_fvar.10801 0) + ∑ i ∈ Finset.range n, ↑↑(@_fvar.51385 i) * 10 ^ (-↑i - 1) := fun n => @?_mvar.57011 nthis:Filter.Tendsto (fun n => ↑(s n) * 10 ^ (-↑n)) Filter.atTop (nhds ↑d)⊢ (fun n => x - 10 ^ (-↑n)) ≤ fun n => ↑(s n) * 10 ^ (-↑n)hfhx:ℝ≥0h:¬x = 0s:ℕ → ℕ := fun n => ⌊_fvar.8365 * 10 ^ n⌋₊hs:∀ (n : ℕ), ↑(@_fvar.10801 n) ≤ _fvar.8365 * 10 ^ n := fun n => Nat.floor_le (@?_mvar.12729 n)hs':∀ (n : ℕ), _fvar.8365 * 10 ^ n < ↑(@_fvar.10801 n) + 1 := fun n => Nat.lt_floor_add_one (_fvar.8365 * 10 ^ n)a:ℕ → Digitha:∀ (n : ℕ), s (n + 1) = 10 * s n + ↑(a n)d:AppendixB.NNRealDecimal := { intPart := @_fvar.10801 0, fracPart := _fvar.51385 }hsum:∀ (n : ℕ), ↑(@_fvar.10801 n) * 10 ^ (-↑n) = ↑(@_fvar.10801 0) + ∑ i ∈ Finset.range n, ↑↑(@_fvar.51385 i) * 10 ^ (-↑i - 1) := fun n => @?_mvar.57011 nthis:Filter.Tendsto (fun n => ↑(s n) * 10 ^ (-↑n)) Filter.atTop (nhds ↑d)⊢ (fun n => ↑(s n) * 10 ^ (-↑n)) ≤ fun x_1 => x .hgx:ℝ≥0h:¬x = 0s:ℕ → ℕ := fun n => ⌊_fvar.8365 * 10 ^ n⌋₊hs:∀ (n : ℕ), ↑(@_fvar.10801 n) ≤ _fvar.8365 * 10 ^ n := fun n => Nat.floor_le (@?_mvar.12729 n)hs':∀ (n : ℕ), _fvar.8365 * 10 ^ n < ↑(@_fvar.10801 n) + 1 := fun n => Nat.lt_floor_add_one (_fvar.8365 * 10 ^ n)a:ℕ → Digitha:∀ (n : ℕ), s (n + 1) = 10 * s n + ↑(a n)d:AppendixB.NNRealDecimal := { intPart := @_fvar.10801 0, fracPart := _fvar.51385 }hsum:∀ (n : ℕ), ↑(@_fvar.10801 n) * 10 ^ (-↑n) = ↑(@_fvar.10801 0) + ∑ i ∈ Finset.range n, ↑↑(@_fvar.51385 i) * 10 ^ (-↑i - 1) := fun n => @?_mvar.57011 nthis:Filter.Tendsto (fun n => ↑(s n) * 10 ^ (-↑n)) Filter.atTop (nhds ↑d)⊢ Filter.Tendsto (fun n => x - 10 ^ (-↑n)) Filter.atTop (nhds x) convert Filter.Tendsto.const_sub (c := 0) x _h.e'_1.h.e'_3x:ℝ≥0h:¬x = 0s:ℕ → ℕ := fun n => ⌊_fvar.8365 * 10 ^ n⌋₊hs:∀ (n : ℕ), ↑(@_fvar.10801 n) ≤ _fvar.8365 * 10 ^ n := fun n => Nat.floor_le (@?_mvar.12729 n)hs':∀ (n : ℕ), _fvar.8365 * 10 ^ n < ↑(@_fvar.10801 n) + 1 := fun n => Nat.lt_floor_add_one (_fvar.8365 * 10 ^ n)a:ℕ → Digitha:∀ (n : ℕ), s (n + 1) = 10 * s n + ↑(a n)d:AppendixB.NNRealDecimal := { intPart := @_fvar.10801 0, fracPart := _fvar.51385 }hsum:∀ (n : ℕ), ↑(@_fvar.10801 n) * 10 ^ (-↑n) = ↑(@_fvar.10801 0) + ∑ i ∈ Finset.range n, ↑↑(@_fvar.51385 i) * 10 ^ (-↑i - 1) := fun n => @?_mvar.57011 nthis:Filter.Tendsto (fun n => ↑(s n) * 10 ^ (-↑n)) Filter.atTop (nhds ↑d)⊢ x = x - 0hg.convert_4x:ℝ≥0h:¬x = 0s:ℕ → ℕ := fun n => ⌊_fvar.8365 * 10 ^ n⌋₊hs:∀ (n : ℕ), ↑(@_fvar.10801 n) ≤ _fvar.8365 * 10 ^ n := fun n => Nat.floor_le (@?_mvar.12729 n)hs':∀ (n : ℕ), _fvar.8365 * 10 ^ n < ↑(@_fvar.10801 n) + 1 := fun n => Nat.lt_floor_add_one (_fvar.8365 * 10 ^ n)a:ℕ → Digitha:∀ (n : ℕ), s (n + 1) = 10 * s n + ↑(a n)d:AppendixB.NNRealDecimal := { intPart := @_fvar.10801 0, fracPart := _fvar.51385 }hsum:∀ (n : ℕ), ↑(@_fvar.10801 n) * 10 ^ (-↑n) = ↑(@_fvar.10801 0) + ∑ i ∈ Finset.range n, ↑↑(@_fvar.51385 i) * 10 ^ (-↑i - 1) := fun n => @?_mvar.57011 nthis:Filter.Tendsto (fun n => ↑(s n) * 10 ^ (-↑n)) Filter.atTop (nhds ↑d)⊢ Filter.Tendsto (fun n => 10 ^ (-↑n)) Filter.atTop (nhds 0) .h.e'_1.h.e'_3x:ℝ≥0h:¬x = 0s:ℕ → ℕ := fun n => ⌊_fvar.8365 * 10 ^ n⌋₊hs:∀ (n : ℕ), ↑(@_fvar.10801 n) ≤ _fvar.8365 * 10 ^ n := fun n => Nat.floor_le (@?_mvar.12729 n)hs':∀ (n : ℕ), _fvar.8365 * 10 ^ n < ↑(@_fvar.10801 n) + 1 := fun n => Nat.lt_floor_add_one (_fvar.8365 * 10 ^ n)a:ℕ → Digitha:∀ (n : ℕ), s (n + 1) = 10 * s n + ↑(a n)d:AppendixB.NNRealDecimal := { intPart := @_fvar.10801 0, fracPart := _fvar.51385 }hsum:∀ (n : ℕ), ↑(@_fvar.10801 n) * 10 ^ (-↑n) = ↑(@_fvar.10801 0) + ∑ i ∈ Finset.range n, ↑↑(@_fvar.51385 i) * 10 ^ (-↑i - 1) := fun n => @?_mvar.57011 nthis:Filter.Tendsto (fun n => ↑(s n) * 10 ^ (-↑n)) Filter.atTop (nhds ↑d)⊢ x = x - 0 simpAll goals completed! 🐙 convert tendsto_pow_atTop_nhds_zero_of_lt_one (?_:(1/10:NNReal) < 1) with nh.e'_3.hx:ℝ≥0h:¬x = 0s:ℕ → ℕ := fun n => ⌊_fvar.8365 * 10 ^ n⌋₊hs:∀ (n : ℕ), ↑(@_fvar.10801 n) ≤ _fvar.8365 * 10 ^ n := fun n => Nat.floor_le (@?_mvar.12729 n)hs':∀ (n : ℕ), _fvar.8365 * 10 ^ n < ↑(@_fvar.10801 n) + 1 := fun n => Nat.lt_floor_add_one (_fvar.8365 * 10 ^ n)a:ℕ → Digitha:∀ (n : ℕ), s (n + 1) = 10 * s n + ↑(a n)d:AppendixB.NNRealDecimal := { intPart := @_fvar.10801 0, fracPart := _fvar.51385 }hsum:∀ (n : ℕ), ↑(@_fvar.10801 n) * 10 ^ (-↑n) = ↑(@_fvar.10801 0) + ∑ i ∈ Finset.range n, ↑↑(@_fvar.51385 i) * 10 ^ (-↑i - 1) := fun n => @?_mvar.57011 nthis:Filter.Tendsto (fun n => ↑(s n) * 10 ^ (-↑n)) Filter.atTop (nhds ↑d)n:ℕ⊢ 10 ^ (-↑n) = (1 / 10) ^ nhg.convert_4x:ℝ≥0h:¬x = 0s:ℕ → ℕ := fun n => ⌊_fvar.8365 * 10 ^ n⌋₊hs:∀ (n : ℕ), ↑(@_fvar.10801 n) ≤ _fvar.8365 * 10 ^ n := fun n => Nat.floor_le (@?_mvar.12729 n)hs':∀ (n : ℕ), _fvar.8365 * 10 ^ n < ↑(@_fvar.10801 n) + 1 := fun n => Nat.lt_floor_add_one (_fvar.8365 * 10 ^ n)a:ℕ → Digitha:∀ (n : ℕ), s (n + 1) = 10 * s n + ↑(a n)d:AppendixB.NNRealDecimal := { intPart := @_fvar.10801 0, fracPart := _fvar.51385 }hsum:∀ (n : ℕ), ↑(@_fvar.10801 n) * 10 ^ (-↑n) = ↑(@_fvar.10801 0) + ∑ i ∈ Finset.range n, ↑↑(@_fvar.51385 i) * 10 ^ (-↑i - 1) := fun n => @?_mvar.57011 nthis:Filter.Tendsto (fun n => ↑(s n) * 10 ^ (-↑n)) Filter.atTop (nhds ↑d)⊢ 1 / 10 < 1 .h.e'_3.hx:ℝ≥0h:¬x = 0s:ℕ → ℕ := fun n => ⌊_fvar.8365 * 10 ^ n⌋₊hs:∀ (n : ℕ), ↑(@_fvar.10801 n) ≤ _fvar.8365 * 10 ^ n := fun n => Nat.floor_le (@?_mvar.12729 n)hs':∀ (n : ℕ), _fvar.8365 * 10 ^ n < ↑(@_fvar.10801 n) + 1 := fun n => Nat.lt_floor_add_one (_fvar.8365 * 10 ^ n)a:ℕ → Digitha:∀ (n : ℕ), s (n + 1) = 10 * s n + ↑(a n)d:AppendixB.NNRealDecimal := { intPart := @_fvar.10801 0, fracPart := _fvar.51385 }hsum:∀ (n : ℕ), ↑(@_fvar.10801 n) * 10 ^ (-↑n) = ↑(@_fvar.10801 0) + ∑ i ∈ Finset.range n, ↑↑(@_fvar.51385 i) * 10 ^ (-↑i - 1) := fun n => @?_mvar.57011 nthis:Filter.Tendsto (fun n => ↑(s n) * 10 ^ (-↑n)) Filter.atTop (nhds ↑d)n:ℕ⊢ 10 ^ (-↑n) = (1 / 10) ^ n rw [←rpow_natCast, one_div, inv_rpow, rpow_neg]All goals completed! 🐙 apply div_lt_one_of_lthg.convert_4.hx:ℝ≥0h:¬x = 0s:ℕ → ℕ := fun n => ⌊_fvar.8365 * 10 ^ n⌋₊hs:∀ (n : ℕ), ↑(@_fvar.10801 n) ≤ _fvar.8365 * 10 ^ n := fun n => Nat.floor_le (@?_mvar.12729 n)hs':∀ (n : ℕ), _fvar.8365 * 10 ^ n < ↑(@_fvar.10801 n) + 1 := fun n => Nat.lt_floor_add_one (_fvar.8365 * 10 ^ n)a:ℕ → Digitha:∀ (n : ℕ), s (n + 1) = 10 * s n + ↑(a n)d:AppendixB.NNRealDecimal := { intPart := @_fvar.10801 0, fracPart := _fvar.51385 }hsum:∀ (n : ℕ), ↑(@_fvar.10801 n) * 10 ^ (-↑n) = ↑(@_fvar.10801 0) + ∑ i ∈ Finset.range n, ↑↑(@_fvar.51385 i) * 10 ^ (-↑i - 1) := fun n => @?_mvar.57011 nthis:Filter.Tendsto (fun n => ↑(s n) * 10 ^ (-↑n)) Filter.atTop (nhds ↑d)⊢ 1 < 10; boundAll goals completed! 🐙 .hhx:ℝ≥0h:¬x = 0s:ℕ → ℕ := fun n => ⌊_fvar.8365 * 10 ^ n⌋₊hs:∀ (n : ℕ), ↑(@_fvar.10801 n) ≤ _fvar.8365 * 10 ^ n := fun n => Nat.floor_le (@?_mvar.12729 n)hs':∀ (n : ℕ), _fvar.8365 * 10 ^ n < ↑(@_fvar.10801 n) + 1 := fun n => Nat.lt_floor_add_one (_fvar.8365 * 10 ^ n)a:ℕ → Digitha:∀ (n : ℕ), s (n + 1) = 10 * s n + ↑(a n)d:AppendixB.NNRealDecimal := { intPart := @_fvar.10801 0, fracPart := _fvar.51385 }hsum:∀ (n : ℕ), ↑(@_fvar.10801 n) * 10 ^ (-↑n) = ↑(@_fvar.10801 0) + ∑ i ∈ Finset.range n, ↑↑(@_fvar.51385 i) * 10 ^ (-↑i - 1) := fun n => @?_mvar.57011 nthis:Filter.Tendsto (fun n => ↑(s n) * 10 ^ (-↑n)) Filter.atTop (nhds ↑d)⊢ Filter.Tendsto (fun x_1 => x) Filter.atTop (nhds x) exact tendsto_const_nhdsAll goals completed! 🐙 .hgfx:ℝ≥0h:¬x = 0s:ℕ → ℕ := fun n => ⌊_fvar.8365 * 10 ^ n⌋₊hs:∀ (n : ℕ), ↑(@_fvar.10801 n) ≤ _fvar.8365 * 10 ^ n := fun n => Nat.floor_le (@?_mvar.12729 n)hs':∀ (n : ℕ), _fvar.8365 * 10 ^ n < ↑(@_fvar.10801 n) + 1 := fun n => Nat.lt_floor_add_one (_fvar.8365 * 10 ^ n)a:ℕ → Digitha:∀ (n : ℕ), s (n + 1) = 10 * s n + ↑(a n)d:AppendixB.NNRealDecimal := { intPart := @_fvar.10801 0, fracPart := _fvar.51385 }hsum:∀ (n : ℕ), ↑(@_fvar.10801 n) * 10 ^ (-↑n) = ↑(@_fvar.10801 0) + ∑ i ∈ Finset.range n, ↑↑(@_fvar.51385 i) * 10 ^ (-↑i - 1) := fun n => @?_mvar.57011 nthis:Filter.Tendsto (fun n => ↑(s n) * 10 ^ (-↑n)) Filter.atTop (nhds ↑d)⊢ (fun n => x - 10 ^ (-↑n)) ≤ fun n => ↑(s n) * 10 ^ (-↑n) intro nhgfx:ℝ≥0h:¬x = 0s:ℕ → ℕ := fun n => ⌊_fvar.8365 * 10 ^ n⌋₊hs:∀ (n : ℕ), ↑(@_fvar.10801 n) ≤ _fvar.8365 * 10 ^ n := fun n => Nat.floor_le (@?_mvar.12729 n)hs':∀ (n : ℕ), _fvar.8365 * 10 ^ n < ↑(@_fvar.10801 n) + 1 := fun n => Nat.lt_floor_add_one (_fvar.8365 * 10 ^ n)a:ℕ → Digitha:∀ (n : ℕ), s (n + 1) = 10 * s n + ↑(a n)d:AppendixB.NNRealDecimal := { intPart := @_fvar.10801 0, fracPart := _fvar.51385 }hsum:∀ (n : ℕ), ↑(@_fvar.10801 n) * 10 ^ (-↑n) = ↑(@_fvar.10801 0) + ∑ i ∈ Finset.range n, ↑↑(@_fvar.51385 i) * 10 ^ (-↑i - 1) := fun n => @?_mvar.57011 nthis:Filter.Tendsto (fun n => ↑(s n) * 10 ^ (-↑n)) Filter.atTop (nhds ↑d)n:ℕ⊢ (fun n => x - 10 ^ (-↑n)) n ≤ (fun n => ↑(s n) * 10 ^ (-↑n)) n; simphgfx:ℝ≥0h:¬x = 0s:ℕ → ℕ := fun n => ⌊_fvar.8365 * 10 ^ n⌋₊hs:∀ (n : ℕ), ↑(@_fvar.10801 n) ≤ _fvar.8365 * 10 ^ n := fun n => Nat.floor_le (@?_mvar.12729 n)hs':∀ (n : ℕ), _fvar.8365 * 10 ^ n < ↑(@_fvar.10801 n) + 1 := fun n => Nat.lt_floor_add_one (_fvar.8365 * 10 ^ n)a:ℕ → Digitha:∀ (n : ℕ), s (n + 1) = 10 * s n + ↑(a n)d:AppendixB.NNRealDecimal := { intPart := @_fvar.10801 0, fracPart := _fvar.51385 }hsum:∀ (n : ℕ), ↑(@_fvar.10801 n) * 10 ^ (-↑n) = ↑(@_fvar.10801 0) + ∑ i ∈ Finset.range n, ↑↑(@_fvar.51385 i) * 10 ^ (-↑i - 1) := fun n => @?_mvar.57011 nthis:Filter.Tendsto (fun n => ↑(s n) * 10 ^ (-↑n)) Filter.atTop (nhds ↑d)n:ℕ⊢ x ≤ ↑(s n) * 10 ^ (-↑n) + 10 ^ (-↑n); calc _ = (x * 10^n) * (10:NNReal)^(-n:ℝ) := byx:ℝ≥0h:¬x = 0s:ℕ → ℕ := fun n => ⌊_fvar.8365 * 10 ^ n⌋₊hs:∀ (n : ℕ), ↑(@_fvar.10801 n) ≤ _fvar.8365 * 10 ^ n := fun n => Nat.floor_le (le_of_lt (mul_pos (lt_of_le_of_ne' (zero_le _fvar.8365) _fvar.8442) (pow_pos (Mathlib.Meta.Positivity.pos_of_isNat (Mathlib.Meta.NormNum.isNat_ofNat NNReal (Eq.refl 10)) (Eq.refl (Nat.ble 1 10))) n)))hs':∀ (n : ℕ), _fvar.8365 * 10 ^ n < ↑(@_fvar.10801 n) + 1 := fun n => Nat.lt_floor_add_one (_fvar.8365 * 10 ^ n)a:ℕ → Digitha:∀ (n : ℕ), s (n + 1) = 10 * s n + ↑(a n)d:AppendixB.NNRealDecimal := { intPart := @_fvar.10801 0, fracPart := _fvar.51385 }hsum:∀ (n : ℕ), ↑(@_fvar.10801 n) * 10 ^ (-↑n) = ↑(@_fvar.10801 0) + ∑ i ∈ Finset.range n, ↑↑(@_fvar.51385 i) * 10 ^ (-↑i - 1) := fun n => Nat.recAux (of_eq_true (Eq.trans (congr (congrArg Eq (Eq.trans (congrArg (HMul.hMul ↑(@_fvar.10801 0)) (Eq.trans (congrArg (HPow.hPow 10) (Eq.trans (congrArg Neg.neg (CharP.cast_eq_zero ℝ 0)) neg_zero)) (NNReal.rpow_zero 10))) (mul_one ↑(@_fvar.10801 0)))) (add_zero ↑(@_fvar.10801 0))) (eq_self ↑(@_fvar.10801 0)))) (fun n hn => Eq.mpr (id (congrArg (fun _a => ↑_a * 10 ^ (-↑(n + 1)) = ↑(@_fvar.10801 0) + ∑ i ∈ Finset.range (n + 1), ↑↑(@_fvar.51385 i) * 10 ^ (-↑i - 1)) (@_fvar.51390 n))) (Trans.trans (Trans.trans (Eq.mpr (id (congrArg (fun x => x = ↑(@_fvar.10801 n) * 10 ^ (-↑n) + ↑↑(@_fvar.51385 n) * 10 ^ (-↑n - 1)) (Eq.trans (congr (congrArg HMul.hMul (Eq.trans (Nat.cast_add (10 * @_fvar.10801 n) ↑(@_fvar.51385 n)) (congrArg (fun x => x + ↑↑(@_fvar.51385 n)) (Nat.cast_mul 10 (@_fvar.10801 n))))) (congrArg (HPow.hPow 10) (Eq.trans (congrArg Neg.neg (Eq.trans (Nat.cast_add n 1) (congrArg (HAdd.hAdd ↑n) Nat.cast_one))) (neg_add_rev (↑n) 1)))) (add_mul (10 * ↑(@_fvar.10801 n)) (↑↑(@_fvar.51385 n)) (10 ^ (-1 + -↑n)))))) (Eq.mpr (id (congr (congrArg Eq (Eq.trans (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat NNReal (Eq.refl 10))) (Mathlib.Tactic.Ring.atom_pf ↑(@_fvar.10801 n)) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_right (↑(@_fvar.10801 n)) (Nat.rawCast 1) (Mathlib.Tactic.Ring.mul_one (Nat.rawCast 10))) (Mathlib.Tactic.Ring.mul_zero (Nat.rawCast 10)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑(@_fvar.10801 n) ^ Nat.rawCast 1 * Nat.rawCast 10 + 0))) (Mathlib.Tactic.Ring.zero_mul (↑(@_fvar.10801 n) ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑(@_fvar.10801 n) ^ Nat.rawCast 1 * Nat.rawCast 10 + 0)))) (Mathlib.Tactic.Ring.atom_pf' (congrArg (HPow.hPow 10) (Eq.trans (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.atom_pf ↑n) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑n) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.add_pf_add_lt (Int.negOfNat 1).rawCast (Mathlib.Tactic.Ring.add_pf_zero_add (↑n ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))) (Eq.trans (congr (congrArg HAdd.hAdd (Eq.trans Mathlib.Tactic.RingNF.int_rawCast_neg (congrArg Neg.neg Mathlib.Tactic.RingNF.nat_rawCast_1))) (Eq.trans (congrArg (fun x => x + 0) (Eq.trans (Eq.trans (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow ↑n) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one ↑n))) (Eq.trans Mathlib.Tactic.RingNF.int_rawCast_neg (congrArg Neg.neg Mathlib.Tactic.RingNF.nat_rawCast_1))) (Mathlib.Tactic.RingNF.mul_neg (↑n) 1)) (congrArg Neg.neg (mul_one ↑n)))) (add_zero (-↑n)))) (Mathlib.Tactic.RingNF.add_neg (-1) ↑n))))) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_left (↑(@_fvar.10801 n)) (Nat.rawCast 1) (Mathlib.Tactic.Ring.mul_pf_right (10 ^ (-1 - ↑n)) (Nat.rawCast 1) (Mathlib.Tactic.Ring.mul_one (Nat.rawCast 10)))) (Mathlib.Tactic.Ring.mul_zero (↑(@_fvar.10801 n) ^ Nat.rawCast 1 * Nat.rawCast 10)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑(@_fvar.10801 n) ^ Nat.rawCast 1 * ((10 ^ (-1 - ↑n)) ^ Nat.rawCast 1 * Nat.rawCast 10) + 0))) (Mathlib.Tactic.Ring.zero_mul ((10 ^ (-1 - ↑n)) ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑(@_fvar.10801 n) ^ Nat.rawCast 1 * ((10 ^ (-1 - ↑n)) ^ Nat.rawCast 1 * Nat.rawCast 10) + 0)))) (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.atom_pf ↑↑(@_fvar.51385 n)) (Mathlib.Tactic.Ring.atom_pf' (congrArg (HPow.hPow 10) (Eq.trans (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.atom_pf ↑n) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑n) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.add_pf_add_lt (Int.negOfNat 1).rawCast (Mathlib.Tactic.Ring.add_pf_zero_add (↑n ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))) (Eq.trans (congr (congrArg HAdd.hAdd (Eq.trans Mathlib.Tactic.RingNF.int_rawCast_neg (congrArg Neg.neg Mathlib.Tactic.RingNF.nat_rawCast_1))) (Eq.trans (congrArg (fun x => x + 0) (Eq.trans (Eq.trans (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow ↑n) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one ↑n))) (Eq.trans Mathlib.Tactic.RingNF.int_rawCast_neg (congrArg Neg.neg Mathlib.Tactic.RingNF.nat_rawCast_1))) (Mathlib.Tactic.RingNF.mul_neg (↑n) 1)) (congrArg Neg.neg (mul_one ↑n)))) (add_zero (-↑n)))) (Mathlib.Tactic.RingNF.add_neg (-1) ↑n))))) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_right (10 ^ (-1 - ↑n)) (Nat.rawCast 1) (Mathlib.Tactic.Ring.mul_pf_left (↑↑(@_fvar.51385 n)) (Nat.rawCast 1) (Mathlib.Tactic.Ring.one_mul (Nat.rawCast 1)))) (Mathlib.Tactic.Ring.mul_zero (↑↑(@_fvar.51385 n) ^ Nat.rawCast 1 * Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_add_zero ((10 ^ (-1 - ↑n)) ^ Nat.rawCast 1 * (↑↑(@_fvar.51385 n) ^ Nat.rawCast 1 * Nat.rawCast 1) + 0))) (Mathlib.Tactic.Ring.zero_mul ((10 ^ (-1 - ↑n)) ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero ((10 ^ (-1 - ↑n)) ^ Nat.rawCast 1 * (↑↑(@_fvar.51385 n) ^ Nat.rawCast 1 * Nat.rawCast 1) + 0)))) (Mathlib.Tactic.Ring.add_pf_add_lt (↑(@_fvar.10801 n) ^ Nat.rawCast 1 * ((10 ^ (-1 - ↑n)) ^ Nat.rawCast 1 * Nat.rawCast 10)) (Mathlib.Tactic.Ring.add_pf_zero_add ((10 ^ (-1 - ↑n)) ^ Nat.rawCast 1 * (↑↑(@_fvar.51385 n) ^ Nat.rawCast 1 * Nat.rawCast 1) + 0)))) (congr (congrArg HAdd.hAdd (Eq.trans (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow ↑(@_fvar.10801 n)) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one ↑(@_fvar.10801 n)))) (congrArg (fun x => x * 10) (Eq.trans (congrArg (HPow.hPow (10 ^ (-1 - ↑n))) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one (10 ^ (-1 - ↑n)))))) (Mathlib.Tactic.RingNF.mul_assoc_rev (↑(@_fvar.10801 n)) (10 ^ (-1 - ↑n)) 10))) (Eq.trans (congrArg (fun x => x + 0) (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow (10 ^ (-1 - ↑n))) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one (10 ^ (-1 - ↑n))))) (Eq.trans (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow ↑↑(@_fvar.51385 n)) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one ↑↑(@_fvar.51385 n)))) Mathlib.Tactic.RingNF.nat_rawCast_1) (mul_one ↑↑(@_fvar.51385 n))))) (add_zero (10 ^ (-1 - ↑n) * ↑↑(@_fvar.51385 n))))))) (Eq.trans (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.atom_pf ↑(@_fvar.10801 n)) (Mathlib.Tactic.Ring.atom_pf (10 ^ (-↑n))) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_left (↑(@_fvar.10801 n)) (Nat.rawCast 1) (Mathlib.Tactic.Ring.mul_pf_right (10 ^ (-↑n)) (Nat.rawCast 1) (Mathlib.Tactic.Ring.one_mul (Nat.rawCast 1)))) (Mathlib.Tactic.Ring.mul_zero (↑(@_fvar.10801 n) ^ Nat.rawCast 1 * Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑(@_fvar.10801 n) ^ Nat.rawCast 1 * ((10 ^ (-↑n)) ^ Nat.rawCast 1 * Nat.rawCast 1) + 0))) (Mathlib.Tactic.Ring.zero_mul ((10 ^ (-↑n)) ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑(@_fvar.10801 n) ^ Nat.rawCast 1 * ((10 ^ (-↑n)) ^ Nat.rawCast 1 * Nat.rawCast 1) + 0)))) (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.atom_pf ↑↑(@_fvar.51385 n)) (Mathlib.Tactic.Ring.atom_pf' (congrArg (HPow.hPow 10) (Eq.trans (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.atom_pf ↑n) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑n) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_gt (Int.negOfNat 1).rawCast (Mathlib.Tactic.Ring.add_pf_add_zero (↑n ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))) (Eq.trans (congr (congrArg HAdd.hAdd (Eq.trans Mathlib.Tactic.RingNF.int_rawCast_neg (congrArg Neg.neg Mathlib.Tactic.RingNF.nat_rawCast_1))) (Eq.trans (congrArg (fun x => x + 0) (Eq.trans (Eq.trans (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow ↑n) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one ↑n))) (Eq.trans Mathlib.Tactic.RingNF.int_rawCast_neg (congrArg Neg.neg Mathlib.Tactic.RingNF.nat_rawCast_1))) (Mathlib.Tactic.RingNF.mul_neg (↑n) 1)) (congrArg Neg.neg (mul_one ↑n)))) (add_zero (-↑n)))) (Mathlib.Tactic.RingNF.add_neg (-1) ↑n))))) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_right (10 ^ (-1 - ↑n)) (Nat.rawCast 1) (Mathlib.Tactic.Ring.mul_pf_left (↑↑(@_fvar.51385 n)) (Nat.rawCast 1) (Mathlib.Tactic.Ring.one_mul (Nat.rawCast 1)))) (Mathlib.Tactic.Ring.mul_zero (↑↑(@_fvar.51385 n) ^ Nat.rawCast 1 * Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_add_zero ((10 ^ (-1 - ↑n)) ^ Nat.rawCast 1 * (↑↑(@_fvar.51385 n) ^ Nat.rawCast 1 * Nat.rawCast 1) + 0))) (Mathlib.Tactic.Ring.zero_mul ((10 ^ (-1 - ↑n)) ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero ((10 ^ (-1 - ↑n)) ^ Nat.rawCast 1 * (↑↑(@_fvar.51385 n) ^ Nat.rawCast 1 * Nat.rawCast 1) + 0)))) (Mathlib.Tactic.Ring.add_pf_add_lt (↑(@_fvar.10801 n) ^ Nat.rawCast 1 * ((10 ^ (-↑n)) ^ Nat.rawCast 1 * Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_zero_add ((10 ^ (-1 - ↑n)) ^ Nat.rawCast 1 * (↑↑(@_fvar.51385 n) ^ Nat.rawCast 1 * Nat.rawCast 1) + 0)))) (congr (congrArg HAdd.hAdd (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow ↑(@_fvar.10801 n)) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one ↑(@_fvar.10801 n)))) (Eq.trans (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow (10 ^ (-↑n))) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one (10 ^ (-↑n))))) Mathlib.Tactic.RingNF.nat_rawCast_1) (mul_one (10 ^ (-↑n)))))) (Eq.trans (congrArg (fun x => x + 0) (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow (10 ^ (-1 - ↑n))) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one (10 ^ (-1 - ↑n))))) (Eq.trans (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow ↑↑(@_fvar.51385 n)) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one ↑↑(@_fvar.51385 n)))) Mathlib.Tactic.RingNF.nat_rawCast_1) (mul_one ↑↑(@_fvar.51385 n))))) (add_zero (10 ^ (-1 - ↑n) * ↑↑(@_fvar.51385 n)))))))) ((fun {α β γ} [HAdd α β γ] a a_1 e_a => Eq.rec (motive := fun a_2 e_a => ∀ (a_3 a_4 : β), a_3 = a_4 → a + a_3 = a_2 + a_4) (fun a_2 a_3 e_a => e_a ▸ Eq.refl (a + a_2)) e_a) (↑(@_fvar.10801 n) * 10 ^ (-1 - ↑n) * 10) (↑(@_fvar.10801 n) * 10 ^ (-↑n)) (Eq.mpr (id (congrArg (fun _a => _a = ↑(@_fvar.10801 n) * 10 ^ (-↑n)) (mul_assoc (↑(@_fvar.10801 n)) (10 ^ (-1 - ↑n)) 10))) (Eq.mpr (id (congrArg (fun _a => ↑(@_fvar.10801 n) * _a = ↑(@_fvar.10801 n) * 10 ^ (-↑n)) (Eq.symm (NNReal.rpow_add_one (Mathlib.Meta.NormNum.isNat_eq_false (Mathlib.Meta.NormNum.isNat_ofNat NNReal (Eq.refl 10)) (Mathlib.Meta.NormNum.isNat_ofNat NNReal Nat.cast_zero) (Eq.refl false)) (-1 - ↑n))))) (of_eq_true (Eq.trans (congrArg (fun x => x = ↑(@_fvar.10801 n) * 10 ^ (-↑n)) (Eq.trans (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.atom_pf ↑(@_fvar.10801 n)) (Mathlib.Tactic.Ring.atom_pf' (congrArg (HPow.hPow 10) (Eq.trans (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.atom_pf ↑n) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑n) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_lt (Int.negOfNat 1).rawCast (Mathlib.Tactic.Ring.add_pf_zero_add (↑n ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1)) (Eq.refl (Int.ofNat 0)))) (Mathlib.Tactic.Ring.add_pf_add_zero (↑n ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))) (Eq.trans (congrArg (fun x => x + 0) (Eq.trans (Eq.trans (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow ↑n) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one ↑n))) (Eq.trans Mathlib.Tactic.RingNF.int_rawCast_neg (congrArg Neg.neg Mathlib.Tactic.RingNF.nat_rawCast_1))) (Mathlib.Tactic.RingNF.mul_neg (↑n) 1)) (congrArg Neg.neg (mul_one ↑n)))) (add_zero (-↑n)))))) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_left (↑(@_fvar.10801 n)) (Nat.rawCast 1) (Mathlib.Tactic.Ring.mul_pf_right (10 ^ (-↑n)) (Nat.rawCast 1) (Mathlib.Tactic.Ring.one_mul (Nat.rawCast 1)))) (Mathlib.Tactic.Ring.mul_zero (↑(@_fvar.10801 n) ^ Nat.rawCast 1 * Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑(@_fvar.10801 n) ^ Nat.rawCast 1 * ((10 ^ (-↑n)) ^ Nat.rawCast 1 * Nat.rawCast 1) + 0))) (Mathlib.Tactic.Ring.zero_mul ((10 ^ (-↑n)) ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑(@_fvar.10801 n) ^ Nat.rawCast 1 * ((10 ^ (-↑n)) ^ Nat.rawCast 1 * Nat.rawCast 1) + 0)))) (Eq.trans (congrArg (fun x => x + 0) (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow ↑(@_fvar.10801 n)) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one ↑(@_fvar.10801 n)))) (Eq.trans (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow (10 ^ (-↑n))) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one (10 ^ (-↑n))))) Mathlib.Tactic.RingNF.nat_rawCast_1) (mul_one (10 ^ (-↑n)))))) (add_zero (↑(@_fvar.10801 n) * 10 ^ (-↑n)))))) (eq_self (↑(@_fvar.10801 n) * 10 ^ (-↑n))))))) (10 ^ (-1 - ↑n) * ↑↑(@_fvar.51385 n)) (10 ^ (-1 - ↑n) * ↑↑(@_fvar.51385 n)) (Eq.refl (10 ^ (-1 - ↑n) * ↑↑(@_fvar.51385 n)))))) (AppendixB.NNRealDecimal.surj._proof_6 _fvar.8365 _fvar.51385 _fvar.51390 n hn)) ((fun {α β γ} [HAdd α β γ] a a_1 e_a => Eq.rec (motive := fun a_2 e_a => ∀ (a_3 a_4 : β), a_3 = a_4 → a + a_3 = a_2 + a_4) (fun a_2 a_3 e_a => e_a ▸ Eq.refl (a + a_2)) e_a) (↑(@_fvar.10801 0)) (↑(@_fvar.10801 0)) (Eq.refl ↑(@_fvar.10801 0)) (∑ i ∈ Finset.range n, ↑↑(@_fvar.51385 i) * 10 ^ (-↑i - 1) + ↑↑(@_fvar.51385 n) * 10 ^ (-↑n - 1)) (∑ i ∈ Finset.range (n + 1), ↑↑(@_fvar.51385 i) * 10 ^ (-↑i - 1)) (Eq.symm (Finset.sum_range_succ (fun x => ↑↑(@_fvar.51385 x) * 10 ^ (-↑x - 1)) n))))) nthis:Filter.Tendsto (fun n => ↑(s n) * 10 ^ (-↑n)) Filter.atTop (nhds ↑d)n:ℕ⊢ x = x * 10 ^ n * 10 ^ (-↑n) rw [mul_assoc, ←rpow_natCast, ←rpow_add]x:ℝ≥0h:¬x = 0s:ℕ → ℕ := fun n => ⌊_fvar.8365 * 10 ^ n⌋₊hs:∀ (n : ℕ), ↑(@_fvar.10801 n) ≤ _fvar.8365 * 10 ^ n := fun n => Nat.floor_le (le_of_lt (mul_pos (lt_of_le_of_ne' (zero_le _fvar.8365) _fvar.8442) (pow_pos (Mathlib.Meta.Positivity.pos_of_isNat (Mathlib.Meta.NormNum.isNat_ofNat NNReal (Eq.refl 10)) (Eq.refl (Nat.ble 1 10))) n)))hs':∀ (n : ℕ), _fvar.8365 * 10 ^ n < ↑(@_fvar.10801 n) + 1 := fun n => Nat.lt_floor_add_one (_fvar.8365 * 10 ^ n)a:ℕ → Digitha:∀ (n : ℕ), s (n + 1) = 10 * s n + ↑(a n)d:AppendixB.NNRealDecimal := { intPart := @_fvar.10801 0, fracPart := _fvar.51385 }hsum:∀ (n : ℕ), ↑(@_fvar.10801 n) * 10 ^ (-↑n) = ↑(@_fvar.10801 0) + ∑ i ∈ Finset.range n, ↑↑(@_fvar.51385 i) * 10 ^ (-↑i - 1) := fun n => Nat.recAux (of_eq_true (Eq.trans (congr (congrArg Eq (Eq.trans (congrArg (HMul.hMul ↑(@_fvar.10801 0)) (Eq.trans (congrArg (HPow.hPow 10) (Eq.trans (congrArg Neg.neg (CharP.cast_eq_zero ℝ 0)) neg_zero)) (NNReal.rpow_zero 10))) (mul_one ↑(@_fvar.10801 0)))) (add_zero ↑(@_fvar.10801 0))) (eq_self ↑(@_fvar.10801 0)))) (fun n hn => Eq.mpr (id (congrArg (fun _a => ↑_a * 10 ^ (-↑(n + 1)) = ↑(@_fvar.10801 0) + ∑ i ∈ Finset.range (n + 1), ↑↑(@_fvar.51385 i) * 10 ^ (-↑i - 1)) (@_fvar.51390 n))) (Trans.trans (Trans.trans (Eq.mpr (id (congrArg (fun x => x = ↑(@_fvar.10801 n) * 10 ^ (-↑n) + ↑↑(@_fvar.51385 n) * 10 ^ (-↑n - 1)) (Eq.trans (congr (congrArg HMul.hMul (Eq.trans (Nat.cast_add (10 * @_fvar.10801 n) ↑(@_fvar.51385 n)) (congrArg (fun x => x + ↑↑(@_fvar.51385 n)) (Nat.cast_mul 10 (@_fvar.10801 n))))) (congrArg (HPow.hPow 10) (Eq.trans (congrArg Neg.neg (Eq.trans (Nat.cast_add n 1) (congrArg (HAdd.hAdd ↑n) Nat.cast_one))) (neg_add_rev (↑n) 1)))) (add_mul (10 * ↑(@_fvar.10801 n)) (↑↑(@_fvar.51385 n)) (10 ^ (-1 + -↑n)))))) (Eq.mpr (id (congr (congrArg Eq (Eq.trans (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat NNReal (Eq.refl 10))) (Mathlib.Tactic.Ring.atom_pf ↑(@_fvar.10801 n)) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_right (↑(@_fvar.10801 n)) (Nat.rawCast 1) (Mathlib.Tactic.Ring.mul_one (Nat.rawCast 10))) (Mathlib.Tactic.Ring.mul_zero (Nat.rawCast 10)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑(@_fvar.10801 n) ^ Nat.rawCast 1 * Nat.rawCast 10 + 0))) (Mathlib.Tactic.Ring.zero_mul (↑(@_fvar.10801 n) ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑(@_fvar.10801 n) ^ Nat.rawCast 1 * Nat.rawCast 10 + 0)))) (Mathlib.Tactic.Ring.atom_pf' (congrArg (HPow.hPow 10) (Eq.trans (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.atom_pf ↑n) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑n) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.add_pf_add_lt (Int.negOfNat 1).rawCast (Mathlib.Tactic.Ring.add_pf_zero_add (↑n ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))) (Eq.trans (congr (congrArg HAdd.hAdd (Eq.trans Mathlib.Tactic.RingNF.int_rawCast_neg (congrArg Neg.neg Mathlib.Tactic.RingNF.nat_rawCast_1))) (Eq.trans (congrArg (fun x => x + 0) (Eq.trans (Eq.trans (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow ↑n) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one ↑n))) (Eq.trans Mathlib.Tactic.RingNF.int_rawCast_neg (congrArg Neg.neg Mathlib.Tactic.RingNF.nat_rawCast_1))) (Mathlib.Tactic.RingNF.mul_neg (↑n) 1)) (congrArg Neg.neg (mul_one ↑n)))) (add_zero (-↑n)))) (Mathlib.Tactic.RingNF.add_neg (-1) ↑n))))) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_left (↑(@_fvar.10801 n)) (Nat.rawCast 1) (Mathlib.Tactic.Ring.mul_pf_right (10 ^ (-1 - ↑n)) (Nat.rawCast 1) (Mathlib.Tactic.Ring.mul_one (Nat.rawCast 10)))) (Mathlib.Tactic.Ring.mul_zero (↑(@_fvar.10801 n) ^ Nat.rawCast 1 * Nat.rawCast 10)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑(@_fvar.10801 n) ^ Nat.rawCast 1 * ((10 ^ (-1 - ↑n)) ^ Nat.rawCast 1 * Nat.rawCast 10) + 0))) (Mathlib.Tactic.Ring.zero_mul ((10 ^ (-1 - ↑n)) ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑(@_fvar.10801 n) ^ Nat.rawCast 1 * ((10 ^ (-1 - ↑n)) ^ Nat.rawCast 1 * Nat.rawCast 10) + 0)))) (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.atom_pf ↑↑(@_fvar.51385 n)) (Mathlib.Tactic.Ring.atom_pf' (congrArg (HPow.hPow 10) (Eq.trans (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.atom_pf ↑n) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑n) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.add_pf_add_lt (Int.negOfNat 1).rawCast (Mathlib.Tactic.Ring.add_pf_zero_add (↑n ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))) (Eq.trans (congr (congrArg HAdd.hAdd (Eq.trans Mathlib.Tactic.RingNF.int_rawCast_neg (congrArg Neg.neg Mathlib.Tactic.RingNF.nat_rawCast_1))) (Eq.trans (congrArg (fun x => x + 0) (Eq.trans (Eq.trans (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow ↑n) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one ↑n))) (Eq.trans Mathlib.Tactic.RingNF.int_rawCast_neg (congrArg Neg.neg Mathlib.Tactic.RingNF.nat_rawCast_1))) (Mathlib.Tactic.RingNF.mul_neg (↑n) 1)) (congrArg Neg.neg (mul_one ↑n)))) (add_zero (-↑n)))) (Mathlib.Tactic.RingNF.add_neg (-1) ↑n))))) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_right (10 ^ (-1 - ↑n)) (Nat.rawCast 1) (Mathlib.Tactic.Ring.mul_pf_left (↑↑(@_fvar.51385 n)) (Nat.rawCast 1) (Mathlib.Tactic.Ring.one_mul (Nat.rawCast 1)))) (Mathlib.Tactic.Ring.mul_zero (↑↑(@_fvar.51385 n) ^ Nat.rawCast 1 * Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_add_zero ((10 ^ (-1 - ↑n)) ^ Nat.rawCast 1 * (↑↑(@_fvar.51385 n) ^ Nat.rawCast 1 * Nat.rawCast 1) + 0))) (Mathlib.Tactic.Ring.zero_mul ((10 ^ (-1 - ↑n)) ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero ((10 ^ (-1 - ↑n)) ^ Nat.rawCast 1 * (↑↑(@_fvar.51385 n) ^ Nat.rawCast 1 * Nat.rawCast 1) + 0)))) (Mathlib.Tactic.Ring.add_pf_add_lt (↑(@_fvar.10801 n) ^ Nat.rawCast 1 * ((10 ^ (-1 - ↑n)) ^ Nat.rawCast 1 * Nat.rawCast 10)) (Mathlib.Tactic.Ring.add_pf_zero_add ((10 ^ (-1 - ↑n)) ^ Nat.rawCast 1 * (↑↑(@_fvar.51385 n) ^ Nat.rawCast 1 * Nat.rawCast 1) + 0)))) (congr (congrArg HAdd.hAdd (Eq.trans (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow ↑(@_fvar.10801 n)) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one ↑(@_fvar.10801 n)))) (congrArg (fun x => x * 10) (Eq.trans (congrArg (HPow.hPow (10 ^ (-1 - ↑n))) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one (10 ^ (-1 - ↑n)))))) (Mathlib.Tactic.RingNF.mul_assoc_rev (↑(@_fvar.10801 n)) (10 ^ (-1 - ↑n)) 10))) (Eq.trans (congrArg (fun x => x + 0) (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow (10 ^ (-1 - ↑n))) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one (10 ^ (-1 - ↑n))))) (Eq.trans (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow ↑↑(@_fvar.51385 n)) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one ↑↑(@_fvar.51385 n)))) Mathlib.Tactic.RingNF.nat_rawCast_1) (mul_one ↑↑(@_fvar.51385 n))))) (add_zero (10 ^ (-1 - ↑n) * ↑↑(@_fvar.51385 n))))))) (Eq.trans (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.atom_pf ↑(@_fvar.10801 n)) (Mathlib.Tactic.Ring.atom_pf (10 ^ (-↑n))) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_left (↑(@_fvar.10801 n)) (Nat.rawCast 1) (Mathlib.Tactic.Ring.mul_pf_right (10 ^ (-↑n)) (Nat.rawCast 1) (Mathlib.Tactic.Ring.one_mul (Nat.rawCast 1)))) (Mathlib.Tactic.Ring.mul_zero (↑(@_fvar.10801 n) ^ Nat.rawCast 1 * Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑(@_fvar.10801 n) ^ Nat.rawCast 1 * ((10 ^ (-↑n)) ^ Nat.rawCast 1 * Nat.rawCast 1) + 0))) (Mathlib.Tactic.Ring.zero_mul ((10 ^ (-↑n)) ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑(@_fvar.10801 n) ^ Nat.rawCast 1 * ((10 ^ (-↑n)) ^ Nat.rawCast 1 * Nat.rawCast 1) + 0)))) (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.atom_pf ↑↑(@_fvar.51385 n)) (Mathlib.Tactic.Ring.atom_pf' (congrArg (HPow.hPow 10) (Eq.trans (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.atom_pf ↑n) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑n) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_gt (Int.negOfNat 1).rawCast (Mathlib.Tactic.Ring.add_pf_add_zero (↑n ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))) (Eq.trans (congr (congrArg HAdd.hAdd (Eq.trans Mathlib.Tactic.RingNF.int_rawCast_neg (congrArg Neg.neg Mathlib.Tactic.RingNF.nat_rawCast_1))) (Eq.trans (congrArg (fun x => x + 0) (Eq.trans (Eq.trans (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow ↑n) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one ↑n))) (Eq.trans Mathlib.Tactic.RingNF.int_rawCast_neg (congrArg Neg.neg Mathlib.Tactic.RingNF.nat_rawCast_1))) (Mathlib.Tactic.RingNF.mul_neg (↑n) 1)) (congrArg Neg.neg (mul_one ↑n)))) (add_zero (-↑n)))) (Mathlib.Tactic.RingNF.add_neg (-1) ↑n))))) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_right (10 ^ (-1 - ↑n)) (Nat.rawCast 1) (Mathlib.Tactic.Ring.mul_pf_left (↑↑(@_fvar.51385 n)) (Nat.rawCast 1) (Mathlib.Tactic.Ring.one_mul (Nat.rawCast 1)))) (Mathlib.Tactic.Ring.mul_zero (↑↑(@_fvar.51385 n) ^ Nat.rawCast 1 * Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_add_zero ((10 ^ (-1 - ↑n)) ^ Nat.rawCast 1 * (↑↑(@_fvar.51385 n) ^ Nat.rawCast 1 * Nat.rawCast 1) + 0))) (Mathlib.Tactic.Ring.zero_mul ((10 ^ (-1 - ↑n)) ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero ((10 ^ (-1 - ↑n)) ^ Nat.rawCast 1 * (↑↑(@_fvar.51385 n) ^ Nat.rawCast 1 * Nat.rawCast 1) + 0)))) (Mathlib.Tactic.Ring.add_pf_add_lt (↑(@_fvar.10801 n) ^ Nat.rawCast 1 * ((10 ^ (-↑n)) ^ Nat.rawCast 1 * Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_zero_add ((10 ^ (-1 - ↑n)) ^ Nat.rawCast 1 * (↑↑(@_fvar.51385 n) ^ Nat.rawCast 1 * Nat.rawCast 1) + 0)))) (congr (congrArg HAdd.hAdd (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow ↑(@_fvar.10801 n)) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one ↑(@_fvar.10801 n)))) (Eq.trans (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow (10 ^ (-↑n))) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one (10 ^ (-↑n))))) Mathlib.Tactic.RingNF.nat_rawCast_1) (mul_one (10 ^ (-↑n)))))) (Eq.trans (congrArg (fun x => x + 0) (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow (10 ^ (-1 - ↑n))) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one (10 ^ (-1 - ↑n))))) (Eq.trans (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow ↑↑(@_fvar.51385 n)) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one ↑↑(@_fvar.51385 n)))) Mathlib.Tactic.RingNF.nat_rawCast_1) (mul_one ↑↑(@_fvar.51385 n))))) (add_zero (10 ^ (-1 - ↑n) * ↑↑(@_fvar.51385 n)))))))) ((fun {α β γ} [HAdd α β γ] a a_1 e_a => Eq.rec (motive := fun a_2 e_a => ∀ (a_3 a_4 : β), a_3 = a_4 → a + a_3 = a_2 + a_4) (fun a_2 a_3 e_a => e_a ▸ Eq.refl (a + a_2)) e_a) (↑(@_fvar.10801 n) * 10 ^ (-1 - ↑n) * 10) (↑(@_fvar.10801 n) * 10 ^ (-↑n)) (Eq.mpr (id (congrArg (fun _a => _a = ↑(@_fvar.10801 n) * 10 ^ (-↑n)) (mul_assoc (↑(@_fvar.10801 n)) (10 ^ (-1 - ↑n)) 10))) (Eq.mpr (id (congrArg (fun _a => ↑(@_fvar.10801 n) * _a = ↑(@_fvar.10801 n) * 10 ^ (-↑n)) (Eq.symm (NNReal.rpow_add_one (Mathlib.Meta.NormNum.isNat_eq_false (Mathlib.Meta.NormNum.isNat_ofNat NNReal (Eq.refl 10)) (Mathlib.Meta.NormNum.isNat_ofNat NNReal Nat.cast_zero) (Eq.refl false)) (-1 - ↑n))))) (of_eq_true (Eq.trans (congrArg (fun x => x = ↑(@_fvar.10801 n) * 10 ^ (-↑n)) (Eq.trans (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.atom_pf ↑(@_fvar.10801 n)) (Mathlib.Tactic.Ring.atom_pf' (congrArg (HPow.hPow 10) (Eq.trans (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.atom_pf ↑n) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑n) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_lt (Int.negOfNat 1).rawCast (Mathlib.Tactic.Ring.add_pf_zero_add (↑n ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1)) (Eq.refl (Int.ofNat 0)))) (Mathlib.Tactic.Ring.add_pf_add_zero (↑n ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))) (Eq.trans (congrArg (fun x => x + 0) (Eq.trans (Eq.trans (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow ↑n) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one ↑n))) (Eq.trans Mathlib.Tactic.RingNF.int_rawCast_neg (congrArg Neg.neg Mathlib.Tactic.RingNF.nat_rawCast_1))) (Mathlib.Tactic.RingNF.mul_neg (↑n) 1)) (congrArg Neg.neg (mul_one ↑n)))) (add_zero (-↑n)))))) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_left (↑(@_fvar.10801 n)) (Nat.rawCast 1) (Mathlib.Tactic.Ring.mul_pf_right (10 ^ (-↑n)) (Nat.rawCast 1) (Mathlib.Tactic.Ring.one_mul (Nat.rawCast 1)))) (Mathlib.Tactic.Ring.mul_zero (↑(@_fvar.10801 n) ^ Nat.rawCast 1 * Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑(@_fvar.10801 n) ^ Nat.rawCast 1 * ((10 ^ (-↑n)) ^ Nat.rawCast 1 * Nat.rawCast 1) + 0))) (Mathlib.Tactic.Ring.zero_mul ((10 ^ (-↑n)) ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑(@_fvar.10801 n) ^ Nat.rawCast 1 * ((10 ^ (-↑n)) ^ Nat.rawCast 1 * Nat.rawCast 1) + 0)))) (Eq.trans (congrArg (fun x => x + 0) (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow ↑(@_fvar.10801 n)) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one ↑(@_fvar.10801 n)))) (Eq.trans (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow (10 ^ (-↑n))) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one (10 ^ (-↑n))))) Mathlib.Tactic.RingNF.nat_rawCast_1) (mul_one (10 ^ (-↑n)))))) (add_zero (↑(@_fvar.10801 n) * 10 ^ (-↑n)))))) (eq_self (↑(@_fvar.10801 n) * 10 ^ (-↑n))))))) (10 ^ (-1 - ↑n) * ↑↑(@_fvar.51385 n)) (10 ^ (-1 - ↑n) * ↑↑(@_fvar.51385 n)) (Eq.refl (10 ^ (-1 - ↑n) * ↑↑(@_fvar.51385 n)))))) (AppendixB.NNRealDecimal.surj._proof_6 _fvar.8365 _fvar.51385 _fvar.51390 n hn)) ((fun {α β γ} [HAdd α β γ] a a_1 e_a => Eq.rec (motive := fun a_2 e_a => ∀ (a_3 a_4 : β), a_3 = a_4 → a + a_3 = a_2 + a_4) (fun a_2 a_3 e_a => e_a ▸ Eq.refl (a + a_2)) e_a) (↑(@_fvar.10801 0)) (↑(@_fvar.10801 0)) (Eq.refl ↑(@_fvar.10801 0)) (∑ i ∈ Finset.range n, ↑↑(@_fvar.51385 i) * 10 ^ (-↑i - 1) + ↑↑(@_fvar.51385 n) * 10 ^ (-↑n - 1)) (∑ i ∈ Finset.range (n + 1), ↑↑(@_fvar.51385 i) * 10 ^ (-↑i - 1)) (Eq.symm (Finset.sum_range_succ (fun x => ↑↑(@_fvar.51385 x) * 10 ^ (-↑x - 1)) n))))) nthis:Filter.Tendsto (fun n => ↑(s n) * 10 ^ (-↑n)) Filter.atTop (nhds ↑d)n:ℕ⊢ x = x * 10 ^ (↑n + -↑n)hxx:ℝ≥0h:¬x = 0s:ℕ → ℕ := fun n => ⌊_fvar.8365 * 10 ^ n⌋₊hs:∀ (n : ℕ), ↑(@_fvar.10801 n) ≤ _fvar.8365 * 10 ^ n := fun n => Nat.floor_le (le_of_lt (mul_pos (lt_of_le_of_ne' (zero_le _fvar.8365) _fvar.8442) (pow_pos (Mathlib.Meta.Positivity.pos_of_isNat (Mathlib.Meta.NormNum.isNat_ofNat NNReal (Eq.refl 10)) (Eq.refl (Nat.ble 1 10))) n)))hs':∀ (n : ℕ), _fvar.8365 * 10 ^ n < ↑(@_fvar.10801 n) + 1 := fun n => Nat.lt_floor_add_one (_fvar.8365 * 10 ^ n)a:ℕ → Digitha:∀ (n : ℕ), s (n + 1) = 10 * s n + ↑(a n)d:AppendixB.NNRealDecimal := { intPart := @_fvar.10801 0, fracPart := _fvar.51385 }hsum:∀ (n : ℕ), ↑(@_fvar.10801 n) * 10 ^ (-↑n) = ↑(@_fvar.10801 0) + ∑ i ∈ Finset.range n, ↑↑(@_fvar.51385 i) * 10 ^ (-↑i - 1) := fun n => Nat.recAux (of_eq_true (Eq.trans (congr (congrArg Eq (Eq.trans (congrArg (HMul.hMul ↑(@_fvar.10801 0)) (Eq.trans (congrArg (HPow.hPow 10) (Eq.trans (congrArg Neg.neg (CharP.cast_eq_zero ℝ 0)) neg_zero)) (NNReal.rpow_zero 10))) (mul_one ↑(@_fvar.10801 0)))) (add_zero ↑(@_fvar.10801 0))) (eq_self ↑(@_fvar.10801 0)))) (fun n hn => Eq.mpr (id (congrArg (fun _a => ↑_a * 10 ^ (-↑(n + 1)) = ↑(@_fvar.10801 0) + ∑ i ∈ Finset.range (n + 1), ↑↑(@_fvar.51385 i) * 10 ^ (-↑i - 1)) (@_fvar.51390 n))) (Trans.trans (Trans.trans (Eq.mpr (id (congrArg (fun x => x = ↑(@_fvar.10801 n) * 10 ^ (-↑n) + ↑↑(@_fvar.51385 n) * 10 ^ (-↑n - 1)) (Eq.trans (congr (congrArg HMul.hMul (Eq.trans (Nat.cast_add (10 * @_fvar.10801 n) ↑(@_fvar.51385 n)) (congrArg (fun x => x + ↑↑(@_fvar.51385 n)) (Nat.cast_mul 10 (@_fvar.10801 n))))) (congrArg (HPow.hPow 10) (Eq.trans (congrArg Neg.neg (Eq.trans (Nat.cast_add n 1) (congrArg (HAdd.hAdd ↑n) Nat.cast_one))) (neg_add_rev (↑n) 1)))) (add_mul (10 * ↑(@_fvar.10801 n)) (↑↑(@_fvar.51385 n)) (10 ^ (-1 + -↑n)))))) (Eq.mpr (id (congr (congrArg Eq (Eq.trans (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat NNReal (Eq.refl 10))) (Mathlib.Tactic.Ring.atom_pf ↑(@_fvar.10801 n)) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_right (↑(@_fvar.10801 n)) (Nat.rawCast 1) (Mathlib.Tactic.Ring.mul_one (Nat.rawCast 10))) (Mathlib.Tactic.Ring.mul_zero (Nat.rawCast 10)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑(@_fvar.10801 n) ^ Nat.rawCast 1 * Nat.rawCast 10 + 0))) (Mathlib.Tactic.Ring.zero_mul (↑(@_fvar.10801 n) ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑(@_fvar.10801 n) ^ Nat.rawCast 1 * Nat.rawCast 10 + 0)))) (Mathlib.Tactic.Ring.atom_pf' (congrArg (HPow.hPow 10) (Eq.trans (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.atom_pf ↑n) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑n) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.add_pf_add_lt (Int.negOfNat 1).rawCast (Mathlib.Tactic.Ring.add_pf_zero_add (↑n ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))) (Eq.trans (congr (congrArg HAdd.hAdd (Eq.trans Mathlib.Tactic.RingNF.int_rawCast_neg (congrArg Neg.neg Mathlib.Tactic.RingNF.nat_rawCast_1))) (Eq.trans (congrArg (fun x => x + 0) (Eq.trans (Eq.trans (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow ↑n) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one ↑n))) (Eq.trans Mathlib.Tactic.RingNF.int_rawCast_neg (congrArg Neg.neg Mathlib.Tactic.RingNF.nat_rawCast_1))) (Mathlib.Tactic.RingNF.mul_neg (↑n) 1)) (congrArg Neg.neg (mul_one ↑n)))) (add_zero (-↑n)))) (Mathlib.Tactic.RingNF.add_neg (-1) ↑n))))) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_left (↑(@_fvar.10801 n)) (Nat.rawCast 1) (Mathlib.Tactic.Ring.mul_pf_right (10 ^ (-1 - ↑n)) (Nat.rawCast 1) (Mathlib.Tactic.Ring.mul_one (Nat.rawCast 10)))) (Mathlib.Tactic.Ring.mul_zero (↑(@_fvar.10801 n) ^ Nat.rawCast 1 * Nat.rawCast 10)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑(@_fvar.10801 n) ^ Nat.rawCast 1 * ((10 ^ (-1 - ↑n)) ^ Nat.rawCast 1 * Nat.rawCast 10) + 0))) (Mathlib.Tactic.Ring.zero_mul ((10 ^ (-1 - ↑n)) ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑(@_fvar.10801 n) ^ Nat.rawCast 1 * ((10 ^ (-1 - ↑n)) ^ Nat.rawCast 1 * Nat.rawCast 10) + 0)))) (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.atom_pf ↑↑(@_fvar.51385 n)) (Mathlib.Tactic.Ring.atom_pf' (congrArg (HPow.hPow 10) (Eq.trans (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.atom_pf ↑n) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑n) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.add_pf_add_lt (Int.negOfNat 1).rawCast (Mathlib.Tactic.Ring.add_pf_zero_add (↑n ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))) (Eq.trans (congr (congrArg HAdd.hAdd (Eq.trans Mathlib.Tactic.RingNF.int_rawCast_neg (congrArg Neg.neg Mathlib.Tactic.RingNF.nat_rawCast_1))) (Eq.trans (congrArg (fun x => x + 0) (Eq.trans (Eq.trans (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow ↑n) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one ↑n))) (Eq.trans Mathlib.Tactic.RingNF.int_rawCast_neg (congrArg Neg.neg Mathlib.Tactic.RingNF.nat_rawCast_1))) (Mathlib.Tactic.RingNF.mul_neg (↑n) 1)) (congrArg Neg.neg (mul_one ↑n)))) (add_zero (-↑n)))) (Mathlib.Tactic.RingNF.add_neg (-1) ↑n))))) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_right (10 ^ (-1 - ↑n)) (Nat.rawCast 1) (Mathlib.Tactic.Ring.mul_pf_left (↑↑(@_fvar.51385 n)) (Nat.rawCast 1) (Mathlib.Tactic.Ring.one_mul (Nat.rawCast 1)))) (Mathlib.Tactic.Ring.mul_zero (↑↑(@_fvar.51385 n) ^ Nat.rawCast 1 * Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_add_zero ((10 ^ (-1 - ↑n)) ^ Nat.rawCast 1 * (↑↑(@_fvar.51385 n) ^ Nat.rawCast 1 * Nat.rawCast 1) + 0))) (Mathlib.Tactic.Ring.zero_mul ((10 ^ (-1 - ↑n)) ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero ((10 ^ (-1 - ↑n)) ^ Nat.rawCast 1 * (↑↑(@_fvar.51385 n) ^ Nat.rawCast 1 * Nat.rawCast 1) + 0)))) (Mathlib.Tactic.Ring.add_pf_add_lt (↑(@_fvar.10801 n) ^ Nat.rawCast 1 * ((10 ^ (-1 - ↑n)) ^ Nat.rawCast 1 * Nat.rawCast 10)) (Mathlib.Tactic.Ring.add_pf_zero_add ((10 ^ (-1 - ↑n)) ^ Nat.rawCast 1 * (↑↑(@_fvar.51385 n) ^ Nat.rawCast 1 * Nat.rawCast 1) + 0)))) (congr (congrArg HAdd.hAdd (Eq.trans (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow ↑(@_fvar.10801 n)) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one ↑(@_fvar.10801 n)))) (congrArg (fun x => x * 10) (Eq.trans (congrArg (HPow.hPow (10 ^ (-1 - ↑n))) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one (10 ^ (-1 - ↑n)))))) (Mathlib.Tactic.RingNF.mul_assoc_rev (↑(@_fvar.10801 n)) (10 ^ (-1 - ↑n)) 10))) (Eq.trans (congrArg (fun x => x + 0) (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow (10 ^ (-1 - ↑n))) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one (10 ^ (-1 - ↑n))))) (Eq.trans (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow ↑↑(@_fvar.51385 n)) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one ↑↑(@_fvar.51385 n)))) Mathlib.Tactic.RingNF.nat_rawCast_1) (mul_one ↑↑(@_fvar.51385 n))))) (add_zero (10 ^ (-1 - ↑n) * ↑↑(@_fvar.51385 n))))))) (Eq.trans (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.atom_pf ↑(@_fvar.10801 n)) (Mathlib.Tactic.Ring.atom_pf (10 ^ (-↑n))) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_left (↑(@_fvar.10801 n)) (Nat.rawCast 1) (Mathlib.Tactic.Ring.mul_pf_right (10 ^ (-↑n)) (Nat.rawCast 1) (Mathlib.Tactic.Ring.one_mul (Nat.rawCast 1)))) (Mathlib.Tactic.Ring.mul_zero (↑(@_fvar.10801 n) ^ Nat.rawCast 1 * Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑(@_fvar.10801 n) ^ Nat.rawCast 1 * ((10 ^ (-↑n)) ^ Nat.rawCast 1 * Nat.rawCast 1) + 0))) (Mathlib.Tactic.Ring.zero_mul ((10 ^ (-↑n)) ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑(@_fvar.10801 n) ^ Nat.rawCast 1 * ((10 ^ (-↑n)) ^ Nat.rawCast 1 * Nat.rawCast 1) + 0)))) (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.atom_pf ↑↑(@_fvar.51385 n)) (Mathlib.Tactic.Ring.atom_pf' (congrArg (HPow.hPow 10) (Eq.trans (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.atom_pf ↑n) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑n) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_gt (Int.negOfNat 1).rawCast (Mathlib.Tactic.Ring.add_pf_add_zero (↑n ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))) (Eq.trans (congr (congrArg HAdd.hAdd (Eq.trans Mathlib.Tactic.RingNF.int_rawCast_neg (congrArg Neg.neg Mathlib.Tactic.RingNF.nat_rawCast_1))) (Eq.trans (congrArg (fun x => x + 0) (Eq.trans (Eq.trans (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow ↑n) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one ↑n))) (Eq.trans Mathlib.Tactic.RingNF.int_rawCast_neg (congrArg Neg.neg Mathlib.Tactic.RingNF.nat_rawCast_1))) (Mathlib.Tactic.RingNF.mul_neg (↑n) 1)) (congrArg Neg.neg (mul_one ↑n)))) (add_zero (-↑n)))) (Mathlib.Tactic.RingNF.add_neg (-1) ↑n))))) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_right (10 ^ (-1 - ↑n)) (Nat.rawCast 1) (Mathlib.Tactic.Ring.mul_pf_left (↑↑(@_fvar.51385 n)) (Nat.rawCast 1) (Mathlib.Tactic.Ring.one_mul (Nat.rawCast 1)))) (Mathlib.Tactic.Ring.mul_zero (↑↑(@_fvar.51385 n) ^ Nat.rawCast 1 * Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_add_zero ((10 ^ (-1 - ↑n)) ^ Nat.rawCast 1 * (↑↑(@_fvar.51385 n) ^ Nat.rawCast 1 * Nat.rawCast 1) + 0))) (Mathlib.Tactic.Ring.zero_mul ((10 ^ (-1 - ↑n)) ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero ((10 ^ (-1 - ↑n)) ^ Nat.rawCast 1 * (↑↑(@_fvar.51385 n) ^ Nat.rawCast 1 * Nat.rawCast 1) + 0)))) (Mathlib.Tactic.Ring.add_pf_add_lt (↑(@_fvar.10801 n) ^ Nat.rawCast 1 * ((10 ^ (-↑n)) ^ Nat.rawCast 1 * Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_zero_add ((10 ^ (-1 - ↑n)) ^ Nat.rawCast 1 * (↑↑(@_fvar.51385 n) ^ Nat.rawCast 1 * Nat.rawCast 1) + 0)))) (congr (congrArg HAdd.hAdd (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow ↑(@_fvar.10801 n)) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one ↑(@_fvar.10801 n)))) (Eq.trans (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow (10 ^ (-↑n))) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one (10 ^ (-↑n))))) Mathlib.Tactic.RingNF.nat_rawCast_1) (mul_one (10 ^ (-↑n)))))) (Eq.trans (congrArg (fun x => x + 0) (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow (10 ^ (-1 - ↑n))) 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false)) (-1 - ↑n))))) (of_eq_true (Eq.trans (congrArg (fun x => x = ↑(@_fvar.10801 n) * 10 ^ (-↑n)) (Eq.trans (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.atom_pf ↑(@_fvar.10801 n)) (Mathlib.Tactic.Ring.atom_pf' (congrArg (HPow.hPow 10) (Eq.trans (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.atom_pf ↑n) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑n) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_lt (Int.negOfNat 1).rawCast (Mathlib.Tactic.Ring.add_pf_zero_add (↑n ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1)) (Eq.refl (Int.ofNat 0)))) (Mathlib.Tactic.Ring.add_pf_add_zero (↑n ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))) (Eq.trans (congrArg (fun x => x + 0) (Eq.trans (Eq.trans (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow ↑n) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one ↑n))) (Eq.trans Mathlib.Tactic.RingNF.int_rawCast_neg (congrArg Neg.neg Mathlib.Tactic.RingNF.nat_rawCast_1))) (Mathlib.Tactic.RingNF.mul_neg (↑n) 1)) (congrArg Neg.neg (mul_one ↑n)))) (add_zero (-↑n)))))) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_left (↑(@_fvar.10801 n)) (Nat.rawCast 1) (Mathlib.Tactic.Ring.mul_pf_right (10 ^ (-↑n)) (Nat.rawCast 1) (Mathlib.Tactic.Ring.one_mul (Nat.rawCast 1)))) (Mathlib.Tactic.Ring.mul_zero (↑(@_fvar.10801 n) ^ Nat.rawCast 1 * Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑(@_fvar.10801 n) ^ Nat.rawCast 1 * ((10 ^ (-↑n)) ^ Nat.rawCast 1 * Nat.rawCast 1) + 0))) (Mathlib.Tactic.Ring.zero_mul ((10 ^ (-↑n)) ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑(@_fvar.10801 n) ^ Nat.rawCast 1 * ((10 ^ (-↑n)) ^ Nat.rawCast 1 * Nat.rawCast 1) + 0)))) (Eq.trans (congrArg (fun x => x + 0) (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow ↑(@_fvar.10801 n)) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one ↑(@_fvar.10801 n)))) (Eq.trans (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow (10 ^ (-↑n))) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one (10 ^ (-↑n))))) Mathlib.Tactic.RingNF.nat_rawCast_1) (mul_one (10 ^ (-↑n)))))) (add_zero (↑(@_fvar.10801 n) * 10 ^ (-↑n)))))) (eq_self (↑(@_fvar.10801 n) * 10 ^ (-↑n))))))) (10 ^ (-1 - ↑n) * ↑↑(@_fvar.51385 n)) (10 ^ (-1 - ↑n) * ↑↑(@_fvar.51385 n)) (Eq.refl (10 ^ (-1 - ↑n) * ↑↑(@_fvar.51385 n)))))) (AppendixB.NNRealDecimal.surj._proof_6 _fvar.8365 _fvar.51385 _fvar.51390 n hn)) ((fun {α β γ} [HAdd α β γ] a a_1 e_a => Eq.rec (motive := fun a_2 e_a => ∀ (a_3 a_4 : β), a_3 = a_4 → a + a_3 = a_2 + a_4) (fun a_2 a_3 e_a => e_a ▸ Eq.refl (a + a_2)) e_a) (↑(@_fvar.10801 0)) (↑(@_fvar.10801 0)) (Eq.refl ↑(@_fvar.10801 0)) (∑ i ∈ Finset.range n, ↑↑(@_fvar.51385 i) * 10 ^ (-↑i - 1) + ↑↑(@_fvar.51385 n) * 10 ^ (-↑n - 1)) (∑ i ∈ Finset.range (n + 1), ↑↑(@_fvar.51385 i) * 10 ^ (-↑i - 1)) (Eq.symm (Finset.sum_range_succ (fun x => ↑↑(@_fvar.51385 x) * 10 ^ (-↑x - 1)) n))))) nthis:Filter.Tendsto (fun n => ↑(s n) * 10 ^ (-↑n)) Filter.atTop (nhds ↑d)n:ℕ⊢ 10 ≠ 0; simphxx:ℝ≥0h:¬x = 0s:ℕ → ℕ := fun n => ⌊_fvar.8365 * 10 ^ n⌋₊hs:∀ (n : ℕ), ↑(@_fvar.10801 n) ≤ _fvar.8365 * 10 ^ n := fun n => Nat.floor_le (le_of_lt (mul_pos (lt_of_le_of_ne' (zero_le _fvar.8365) _fvar.8442) (pow_pos (Mathlib.Meta.Positivity.pos_of_isNat (Mathlib.Meta.NormNum.isNat_ofNat NNReal (Eq.refl 10)) (Eq.refl (Nat.ble 1 10))) n)))hs':∀ (n : ℕ), _fvar.8365 * 10 ^ n < ↑(@_fvar.10801 n) + 1 := fun n => Nat.lt_floor_add_one (_fvar.8365 * 10 ^ n)a:ℕ → Digitha:∀ (n : ℕ), s (n + 1) = 10 * s n + ↑(a n)d:AppendixB.NNRealDecimal := { intPart := @_fvar.10801 0, fracPart := _fvar.51385 }hsum:∀ (n : ℕ), ↑(@_fvar.10801 n) * 10 ^ (-↑n) = ↑(@_fvar.10801 0) + ∑ i ∈ Finset.range n, ↑↑(@_fvar.51385 i) * 10 ^ (-↑i - 1) := fun n => Nat.recAux (of_eq_true (Eq.trans (congr (congrArg Eq (Eq.trans (congrArg (HMul.hMul ↑(@_fvar.10801 0)) (Eq.trans (congrArg (HPow.hPow 10) (Eq.trans (congrArg Neg.neg (CharP.cast_eq_zero ℝ 0)) neg_zero)) (NNReal.rpow_zero 10))) (mul_one ↑(@_fvar.10801 0)))) (add_zero ↑(@_fvar.10801 0))) (eq_self ↑(@_fvar.10801 0)))) (fun n hn => Eq.mpr (id (congrArg (fun _a => ↑_a * 10 ^ (-↑(n + 1)) = ↑(@_fvar.10801 0) + ∑ i ∈ Finset.range (n + 1), ↑↑(@_fvar.51385 i) * 10 ^ (-↑i - 1)) (@_fvar.51390 n))) (Trans.trans (Trans.trans (Eq.mpr (id (congrArg (fun x => x = ↑(@_fvar.10801 n) * 10 ^ (-↑n) + ↑↑(@_fvar.51385 n) * 10 ^ (-↑n - 1)) (Eq.trans (congr (congrArg HMul.hMul (Eq.trans (Nat.cast_add (10 * @_fvar.10801 n) ↑(@_fvar.51385 n)) (congrArg (fun x => x + ↑↑(@_fvar.51385 n)) (Nat.cast_mul 10 (@_fvar.10801 n))))) (congrArg (HPow.hPow 10) (Eq.trans (congrArg Neg.neg (Eq.trans (Nat.cast_add n 1) (congrArg (HAdd.hAdd ↑n) Nat.cast_one))) (neg_add_rev (↑n) 1)))) (add_mul (10 * ↑(@_fvar.10801 n)) (↑↑(@_fvar.51385 n)) (10 ^ (-1 + -↑n)))))) (Eq.mpr (id (congr (congrArg Eq (Eq.trans (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat NNReal (Eq.refl 10))) (Mathlib.Tactic.Ring.atom_pf ↑(@_fvar.10801 n)) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_right (↑(@_fvar.10801 n)) (Nat.rawCast 1) (Mathlib.Tactic.Ring.mul_one (Nat.rawCast 10))) (Mathlib.Tactic.Ring.mul_zero (Nat.rawCast 10)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑(@_fvar.10801 n) ^ Nat.rawCast 1 * Nat.rawCast 10 + 0))) (Mathlib.Tactic.Ring.zero_mul (↑(@_fvar.10801 n) ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑(@_fvar.10801 n) ^ Nat.rawCast 1 * Nat.rawCast 10 + 0)))) (Mathlib.Tactic.Ring.atom_pf' (congrArg (HPow.hPow 10) (Eq.trans (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.atom_pf ↑n) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑n) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.add_pf_add_lt (Int.negOfNat 1).rawCast (Mathlib.Tactic.Ring.add_pf_zero_add (↑n ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))) (Eq.trans (congr (congrArg HAdd.hAdd (Eq.trans Mathlib.Tactic.RingNF.int_rawCast_neg (congrArg Neg.neg 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(Mathlib.Tactic.Ring.add_pf_add_zero (↑(@_fvar.10801 n) ^ Nat.rawCast 1 * ((10 ^ (-1 - ↑n)) ^ Nat.rawCast 1 * Nat.rawCast 10) + 0)))) (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.atom_pf ↑↑(@_fvar.51385 n)) (Mathlib.Tactic.Ring.atom_pf' (congrArg (HPow.hPow 10) (Eq.trans (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.atom_pf ↑n) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑n) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.add_pf_add_lt (Int.negOfNat 1).rawCast (Mathlib.Tactic.Ring.add_pf_zero_add (↑n ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))) (Eq.trans (congr (congrArg HAdd.hAdd (Eq.trans Mathlib.Tactic.RingNF.int_rawCast_neg (congrArg Neg.neg Mathlib.Tactic.RingNF.nat_rawCast_1))) (Eq.trans (congrArg (fun x => x + 0) (Eq.trans (Eq.trans (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow ↑n) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one ↑n))) (Eq.trans Mathlib.Tactic.RingNF.int_rawCast_neg (congrArg Neg.neg Mathlib.Tactic.RingNF.nat_rawCast_1))) (Mathlib.Tactic.RingNF.mul_neg (↑n) 1)) (congrArg Neg.neg (mul_one ↑n)))) (add_zero (-↑n)))) (Mathlib.Tactic.RingNF.add_neg (-1) ↑n))))) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_right (10 ^ (-1 - ↑n)) (Nat.rawCast 1) (Mathlib.Tactic.Ring.mul_pf_left (↑↑(@_fvar.51385 n)) (Nat.rawCast 1) (Mathlib.Tactic.Ring.one_mul (Nat.rawCast 1)))) (Mathlib.Tactic.Ring.mul_zero (↑↑(@_fvar.51385 n) ^ Nat.rawCast 1 * Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_add_zero ((10 ^ (-1 - ↑n)) ^ Nat.rawCast 1 * (↑↑(@_fvar.51385 n) ^ Nat.rawCast 1 * Nat.rawCast 1) + 0))) (Mathlib.Tactic.Ring.zero_mul ((10 ^ (-1 - ↑n)) ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero ((10 ^ (-1 - ↑n)) ^ Nat.rawCast 1 * (↑↑(@_fvar.51385 n) ^ Nat.rawCast 1 * Nat.rawCast 1) + 0)))) (Mathlib.Tactic.Ring.add_pf_add_lt (↑(@_fvar.10801 n) ^ Nat.rawCast 1 * ((10 ^ (-1 - ↑n)) ^ Nat.rawCast 1 * Nat.rawCast 10)) (Mathlib.Tactic.Ring.add_pf_zero_add ((10 ^ (-1 - ↑n)) ^ Nat.rawCast 1 * (↑↑(@_fvar.51385 n) ^ Nat.rawCast 1 * Nat.rawCast 1) + 0)))) (congr (congrArg HAdd.hAdd (Eq.trans (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow ↑(@_fvar.10801 n)) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one ↑(@_fvar.10801 n)))) (congrArg (fun x => x * 10) (Eq.trans (congrArg (HPow.hPow (10 ^ (-1 - ↑n))) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one (10 ^ (-1 - ↑n)))))) (Mathlib.Tactic.RingNF.mul_assoc_rev (↑(@_fvar.10801 n)) (10 ^ (-1 - ↑n)) 10))) (Eq.trans (congrArg (fun x => x + 0) (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow (10 ^ (-1 - ↑n))) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one (10 ^ (-1 - ↑n))))) (Eq.trans (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow ↑↑(@_fvar.51385 n)) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one ↑↑(@_fvar.51385 n)))) Mathlib.Tactic.RingNF.nat_rawCast_1) (mul_one ↑↑(@_fvar.51385 n))))) (add_zero (10 ^ (-1 - ↑n) * ↑↑(@_fvar.51385 n))))))) (Eq.trans (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.atom_pf ↑(@_fvar.10801 n)) (Mathlib.Tactic.Ring.atom_pf (10 ^ (-↑n))) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_left (↑(@_fvar.10801 n)) (Nat.rawCast 1) (Mathlib.Tactic.Ring.mul_pf_right (10 ^ (-↑n)) (Nat.rawCast 1) (Mathlib.Tactic.Ring.one_mul (Nat.rawCast 1)))) (Mathlib.Tactic.Ring.mul_zero (↑(@_fvar.10801 n) ^ Nat.rawCast 1 * Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑(@_fvar.10801 n) ^ Nat.rawCast 1 * ((10 ^ (-↑n)) ^ Nat.rawCast 1 * Nat.rawCast 1) + 0))) (Mathlib.Tactic.Ring.zero_mul ((10 ^ (-↑n)) ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑(@_fvar.10801 n) ^ Nat.rawCast 1 * ((10 ^ (-↑n)) ^ Nat.rawCast 1 * Nat.rawCast 1) + 0)))) (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.atom_pf ↑↑(@_fvar.51385 n)) (Mathlib.Tactic.Ring.atom_pf' (congrArg (HPow.hPow 10) (Eq.trans (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.atom_pf ↑n) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑n) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_gt (Int.negOfNat 1).rawCast (Mathlib.Tactic.Ring.add_pf_add_zero (↑n ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))) (Eq.trans (congr (congrArg HAdd.hAdd (Eq.trans Mathlib.Tactic.RingNF.int_rawCast_neg (congrArg Neg.neg Mathlib.Tactic.RingNF.nat_rawCast_1))) (Eq.trans (congrArg (fun x => x + 0) (Eq.trans (Eq.trans (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow ↑n) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one ↑n))) (Eq.trans Mathlib.Tactic.RingNF.int_rawCast_neg (congrArg Neg.neg Mathlib.Tactic.RingNF.nat_rawCast_1))) (Mathlib.Tactic.RingNF.mul_neg (↑n) 1)) (congrArg Neg.neg (mul_one ↑n)))) (add_zero (-↑n)))) (Mathlib.Tactic.RingNF.add_neg (-1) ↑n))))) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_right (10 ^ (-1 - ↑n)) (Nat.rawCast 1) (Mathlib.Tactic.Ring.mul_pf_left (↑↑(@_fvar.51385 n)) (Nat.rawCast 1) (Mathlib.Tactic.Ring.one_mul (Nat.rawCast 1)))) (Mathlib.Tactic.Ring.mul_zero (↑↑(@_fvar.51385 n) ^ Nat.rawCast 1 * Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_add_zero ((10 ^ (-1 - ↑n)) ^ Nat.rawCast 1 * (↑↑(@_fvar.51385 n) ^ Nat.rawCast 1 * Nat.rawCast 1) + 0))) (Mathlib.Tactic.Ring.zero_mul ((10 ^ (-1 - ↑n)) ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero ((10 ^ (-1 - ↑n)) ^ Nat.rawCast 1 * (↑↑(@_fvar.51385 n) ^ Nat.rawCast 1 * Nat.rawCast 1) + 0)))) (Mathlib.Tactic.Ring.add_pf_add_lt (↑(@_fvar.10801 n) ^ Nat.rawCast 1 * ((10 ^ (-↑n)) ^ Nat.rawCast 1 * Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_zero_add ((10 ^ (-1 - ↑n)) ^ Nat.rawCast 1 * (↑↑(@_fvar.51385 n) ^ Nat.rawCast 1 * Nat.rawCast 1) + 0)))) (congr (congrArg HAdd.hAdd (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow ↑(@_fvar.10801 n)) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one ↑(@_fvar.10801 n)))) (Eq.trans (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow (10 ^ (-↑n))) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one (10 ^ (-↑n))))) Mathlib.Tactic.RingNF.nat_rawCast_1) (mul_one (10 ^ (-↑n)))))) (Eq.trans (congrArg (fun x => x + 0) (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow (10 ^ (-1 - ↑n))) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one (10 ^ (-1 - ↑n))))) (Eq.trans (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow ↑↑(@_fvar.51385 n)) 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(Mathlib.Tactic.Ring.atom_pf ↑(@_fvar.10801 n)) (Mathlib.Tactic.Ring.atom_pf' (congrArg (HPow.hPow 10) (Eq.trans (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.atom_pf ↑n) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑n) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_lt (Int.negOfNat 1).rawCast (Mathlib.Tactic.Ring.add_pf_zero_add (↑n ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1)) (Eq.refl (Int.ofNat 0)))) (Mathlib.Tactic.Ring.add_pf_add_zero (↑n ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))) (Eq.trans (congrArg (fun x => x + 0) (Eq.trans (Eq.trans (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow ↑n) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one ↑n))) (Eq.trans Mathlib.Tactic.RingNF.int_rawCast_neg (congrArg Neg.neg Mathlib.Tactic.RingNF.nat_rawCast_1))) (Mathlib.Tactic.RingNF.mul_neg (↑n) 1)) (congrArg Neg.neg (mul_one ↑n)))) (add_zero (-↑n)))))) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_left (↑(@_fvar.10801 n)) (Nat.rawCast 1) (Mathlib.Tactic.Ring.mul_pf_right (10 ^ (-↑n)) (Nat.rawCast 1) (Mathlib.Tactic.Ring.one_mul (Nat.rawCast 1)))) (Mathlib.Tactic.Ring.mul_zero (↑(@_fvar.10801 n) ^ Nat.rawCast 1 * Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑(@_fvar.10801 n) ^ Nat.rawCast 1 * ((10 ^ (-↑n)) ^ Nat.rawCast 1 * Nat.rawCast 1) + 0))) (Mathlib.Tactic.Ring.zero_mul ((10 ^ (-↑n)) ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑(@_fvar.10801 n) ^ Nat.rawCast 1 * ((10 ^ (-↑n)) ^ Nat.rawCast 1 * Nat.rawCast 1) + 0)))) (Eq.trans (congrArg (fun x => x + 0) (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow ↑(@_fvar.10801 n)) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one ↑(@_fvar.10801 n)))) (Eq.trans (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow (10 ^ (-↑n))) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one (10 ^ (-↑n))))) Mathlib.Tactic.RingNF.nat_rawCast_1) (mul_one (10 ^ (-↑n)))))) (add_zero (↑(@_fvar.10801 n) * 10 ^ (-↑n)))))) (eq_self (↑(@_fvar.10801 n) * 10 ^ (-↑n))))))) (10 ^ (-1 - ↑n) * ↑↑(@_fvar.51385 n)) (10 ^ (-1 - ↑n) * ↑↑(@_fvar.51385 n)) (Eq.refl (10 ^ (-1 - ↑n) * ↑↑(@_fvar.51385 n)))))) (AppendixB.NNRealDecimal.surj._proof_6 _fvar.8365 _fvar.51385 _fvar.51390 n hn)) ((fun {α β γ} [HAdd α β γ] a a_1 e_a => Eq.rec (motive := fun a_2 e_a => ∀ (a_3 a_4 : β), a_3 = a_4 → a + a_3 = a_2 + a_4) (fun a_2 a_3 e_a => e_a ▸ Eq.refl (a + a_2)) e_a) (↑(@_fvar.10801 0)) (↑(@_fvar.10801 0)) (Eq.refl ↑(@_fvar.10801 0)) (∑ i ∈ Finset.range n, ↑↑(@_fvar.51385 i) * 10 ^ (-↑i - 1) + ↑↑(@_fvar.51385 n) * 10 ^ (-↑n - 1)) (∑ i ∈ Finset.range (n + 1), ↑↑(@_fvar.51385 i) * 10 ^ (-↑i - 1)) (Eq.symm (Finset.sum_range_succ (fun x => ↑↑(@_fvar.51385 x) * 10 ^ (-↑x - 1)) n))))) nthis:Filter.Tendsto (fun n => ↑(s n) * 10 ^ (-↑n)) Filter.atTop (nhds ↑d)n:ℕ⊢ 10 ≠ 0; norm_numAll goals completed! 🐙 _ ≤ ((s n:NNReal) + 1)*(10:NNReal)^(-n:ℝ) := byx:ℝ≥0h:¬x = 0s:ℕ → ℕ := fun n => ⌊_fvar.8365 * 10 ^ n⌋₊hs:∀ (n : ℕ), ↑(@_fvar.10801 n) ≤ _fvar.8365 * 10 ^ n := fun n => Nat.floor_le (le_of_lt (mul_pos (lt_of_le_of_ne' (zero_le _fvar.8365) _fvar.8442) (pow_pos (Mathlib.Meta.Positivity.pos_of_isNat (Mathlib.Meta.NormNum.isNat_ofNat NNReal (Eq.refl 10)) (Eq.refl (Nat.ble 1 10))) n)))hs':∀ (n : ℕ), _fvar.8365 * 10 ^ n < ↑(@_fvar.10801 n) + 1 := fun n => Nat.lt_floor_add_one (_fvar.8365 * 10 ^ n)a:ℕ → Digitha:∀ (n : ℕ), s (n + 1) = 10 * s n + ↑(a n)d:AppendixB.NNRealDecimal := { intPart := @_fvar.10801 0, fracPart := _fvar.51385 }hsum:∀ (n : ℕ), ↑(@_fvar.10801 n) * 10 ^ (-↑n) = ↑(@_fvar.10801 0) + ∑ i ∈ Finset.range n, ↑↑(@_fvar.51385 i) * 10 ^ (-↑i - 1) := fun n => Nat.recAux (of_eq_true (Eq.trans (congr (congrArg Eq (Eq.trans (congrArg (HMul.hMul ↑(@_fvar.10801 0)) (Eq.trans (congrArg (HPow.hPow 10) (Eq.trans (congrArg Neg.neg (CharP.cast_eq_zero ℝ 0)) neg_zero)) (NNReal.rpow_zero 10))) (mul_one ↑(@_fvar.10801 0)))) (add_zero ↑(@_fvar.10801 0))) (eq_self ↑(@_fvar.10801 0)))) (fun n hn => Eq.mpr (id (congrArg (fun _a => ↑_a * 10 ^ (-↑(n + 1)) = ↑(@_fvar.10801 0) + ∑ i ∈ Finset.range (n + 1), ↑↑(@_fvar.51385 i) * 10 ^ (-↑i - 1)) (@_fvar.51390 n))) (Trans.trans (Trans.trans (Eq.mpr (id (congrArg (fun x => x = ↑(@_fvar.10801 n) * 10 ^ (-↑n) + ↑↑(@_fvar.51385 n) * 10 ^ (-↑n - 1)) (Eq.trans (congr (congrArg HMul.hMul (Eq.trans (Nat.cast_add (10 * @_fvar.10801 n) ↑(@_fvar.51385 n)) (congrArg (fun x => x + ↑↑(@_fvar.51385 n)) (Nat.cast_mul 10 (@_fvar.10801 n))))) (congrArg (HPow.hPow 10) (Eq.trans (congrArg Neg.neg (Eq.trans (Nat.cast_add n 1) (congrArg (HAdd.hAdd ↑n) Nat.cast_one))) (neg_add_rev (↑n) 1)))) (add_mul (10 * ↑(@_fvar.10801 n)) (↑↑(@_fvar.51385 n)) (10 ^ (-1 + -↑n)))))) (Eq.mpr (id (congr (congrArg Eq (Eq.trans (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat NNReal (Eq.refl 10))) (Mathlib.Tactic.Ring.atom_pf ↑(@_fvar.10801 n)) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_right (↑(@_fvar.10801 n)) (Nat.rawCast 1) (Mathlib.Tactic.Ring.mul_one (Nat.rawCast 10))) (Mathlib.Tactic.Ring.mul_zero (Nat.rawCast 10)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑(@_fvar.10801 n) ^ Nat.rawCast 1 * Nat.rawCast 10 + 0))) (Mathlib.Tactic.Ring.zero_mul (↑(@_fvar.10801 n) ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑(@_fvar.10801 n) ^ Nat.rawCast 1 * Nat.rawCast 10 + 0)))) (Mathlib.Tactic.Ring.atom_pf' (congrArg (HPow.hPow 10) (Eq.trans (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.atom_pf ↑n) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑n) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.add_pf_add_lt (Int.negOfNat 1).rawCast (Mathlib.Tactic.Ring.add_pf_zero_add (↑n ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))) (Eq.trans (congr (congrArg HAdd.hAdd (Eq.trans Mathlib.Tactic.RingNF.int_rawCast_neg (congrArg Neg.neg Mathlib.Tactic.RingNF.nat_rawCast_1))) (Eq.trans (congrArg (fun x => x + 0) (Eq.trans (Eq.trans (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow ↑n) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one ↑n))) (Eq.trans Mathlib.Tactic.RingNF.int_rawCast_neg (congrArg Neg.neg Mathlib.Tactic.RingNF.nat_rawCast_1))) (Mathlib.Tactic.RingNF.mul_neg (↑n) 1)) (congrArg Neg.neg (mul_one ↑n)))) (add_zero (-↑n)))) (Mathlib.Tactic.RingNF.add_neg (-1) ↑n))))) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_left (↑(@_fvar.10801 n)) (Nat.rawCast 1) (Mathlib.Tactic.Ring.mul_pf_right (10 ^ (-1 - ↑n)) (Nat.rawCast 1) (Mathlib.Tactic.Ring.mul_one (Nat.rawCast 10)))) (Mathlib.Tactic.Ring.mul_zero (↑(@_fvar.10801 n) ^ Nat.rawCast 1 * Nat.rawCast 10)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑(@_fvar.10801 n) ^ Nat.rawCast 1 * ((10 ^ (-1 - ↑n)) ^ Nat.rawCast 1 * Nat.rawCast 10) + 0))) (Mathlib.Tactic.Ring.zero_mul ((10 ^ (-1 - ↑n)) ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑(@_fvar.10801 n) ^ Nat.rawCast 1 * ((10 ^ (-1 - ↑n)) ^ Nat.rawCast 1 * Nat.rawCast 10) + 0)))) (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.atom_pf ↑↑(@_fvar.51385 n)) (Mathlib.Tactic.Ring.atom_pf' (congrArg (HPow.hPow 10) (Eq.trans (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.atom_pf ↑n) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑n) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.add_pf_add_lt (Int.negOfNat 1).rawCast (Mathlib.Tactic.Ring.add_pf_zero_add (↑n ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))) (Eq.trans (congr (congrArg HAdd.hAdd (Eq.trans Mathlib.Tactic.RingNF.int_rawCast_neg (congrArg Neg.neg Mathlib.Tactic.RingNF.nat_rawCast_1))) (Eq.trans (congrArg (fun x => x + 0) (Eq.trans (Eq.trans (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow ↑n) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one ↑n))) (Eq.trans Mathlib.Tactic.RingNF.int_rawCast_neg (congrArg Neg.neg Mathlib.Tactic.RingNF.nat_rawCast_1))) (Mathlib.Tactic.RingNF.mul_neg (↑n) 1)) (congrArg Neg.neg (mul_one ↑n)))) (add_zero (-↑n)))) (Mathlib.Tactic.RingNF.add_neg (-1) ↑n))))) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_right (10 ^ (-1 - ↑n)) (Nat.rawCast 1) (Mathlib.Tactic.Ring.mul_pf_left (↑↑(@_fvar.51385 n)) (Nat.rawCast 1) (Mathlib.Tactic.Ring.one_mul (Nat.rawCast 1)))) (Mathlib.Tactic.Ring.mul_zero (↑↑(@_fvar.51385 n) ^ Nat.rawCast 1 * Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_add_zero ((10 ^ (-1 - ↑n)) ^ Nat.rawCast 1 * (↑↑(@_fvar.51385 n) ^ Nat.rawCast 1 * Nat.rawCast 1) + 0))) (Mathlib.Tactic.Ring.zero_mul ((10 ^ (-1 - ↑n)) ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero ((10 ^ (-1 - ↑n)) ^ Nat.rawCast 1 * (↑↑(@_fvar.51385 n) ^ Nat.rawCast 1 * Nat.rawCast 1) + 0)))) (Mathlib.Tactic.Ring.add_pf_add_lt (↑(@_fvar.10801 n) ^ Nat.rawCast 1 * ((10 ^ (-1 - ↑n)) ^ Nat.rawCast 1 * Nat.rawCast 10)) (Mathlib.Tactic.Ring.add_pf_zero_add ((10 ^ (-1 - ↑n)) ^ Nat.rawCast 1 * (↑↑(@_fvar.51385 n) ^ Nat.rawCast 1 * Nat.rawCast 1) + 0)))) (congr (congrArg HAdd.hAdd (Eq.trans (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow ↑(@_fvar.10801 n)) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one ↑(@_fvar.10801 n)))) (congrArg (fun x => x * 10) (Eq.trans (congrArg (HPow.hPow (10 ^ (-1 - ↑n))) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one (10 ^ (-1 - ↑n)))))) (Mathlib.Tactic.RingNF.mul_assoc_rev (↑(@_fvar.10801 n)) (10 ^ (-1 - ↑n)) 10))) (Eq.trans (congrArg (fun x => x + 0) (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow (10 ^ (-1 - ↑n))) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one (10 ^ (-1 - ↑n))))) (Eq.trans (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow ↑↑(@_fvar.51385 n)) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one ↑↑(@_fvar.51385 n)))) Mathlib.Tactic.RingNF.nat_rawCast_1) (mul_one ↑↑(@_fvar.51385 n))))) (add_zero (10 ^ (-1 - ↑n) * ↑↑(@_fvar.51385 n))))))) (Eq.trans (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.atom_pf ↑(@_fvar.10801 n)) (Mathlib.Tactic.Ring.atom_pf (10 ^ (-↑n))) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_left (↑(@_fvar.10801 n)) (Nat.rawCast 1) (Mathlib.Tactic.Ring.mul_pf_right (10 ^ (-↑n)) (Nat.rawCast 1) (Mathlib.Tactic.Ring.one_mul (Nat.rawCast 1)))) (Mathlib.Tactic.Ring.mul_zero (↑(@_fvar.10801 n) ^ Nat.rawCast 1 * Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑(@_fvar.10801 n) ^ Nat.rawCast 1 * ((10 ^ (-↑n)) ^ Nat.rawCast 1 * Nat.rawCast 1) + 0))) (Mathlib.Tactic.Ring.zero_mul ((10 ^ (-↑n)) ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑(@_fvar.10801 n) ^ Nat.rawCast 1 * ((10 ^ (-↑n)) ^ Nat.rawCast 1 * Nat.rawCast 1) + 0)))) (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.atom_pf ↑↑(@_fvar.51385 n)) (Mathlib.Tactic.Ring.atom_pf' (congrArg (HPow.hPow 10) (Eq.trans (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.atom_pf ↑n) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑n) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_gt (Int.negOfNat 1).rawCast (Mathlib.Tactic.Ring.add_pf_add_zero (↑n ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))) (Eq.trans (congr (congrArg HAdd.hAdd (Eq.trans Mathlib.Tactic.RingNF.int_rawCast_neg (congrArg Neg.neg Mathlib.Tactic.RingNF.nat_rawCast_1))) (Eq.trans (congrArg (fun x => x + 0) (Eq.trans (Eq.trans (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow ↑n) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one ↑n))) (Eq.trans Mathlib.Tactic.RingNF.int_rawCast_neg (congrArg Neg.neg Mathlib.Tactic.RingNF.nat_rawCast_1))) (Mathlib.Tactic.RingNF.mul_neg (↑n) 1)) (congrArg Neg.neg (mul_one ↑n)))) (add_zero (-↑n)))) (Mathlib.Tactic.RingNF.add_neg (-1) ↑n))))) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_right (10 ^ (-1 - ↑n)) (Nat.rawCast 1) (Mathlib.Tactic.Ring.mul_pf_left (↑↑(@_fvar.51385 n)) (Nat.rawCast 1) (Mathlib.Tactic.Ring.one_mul (Nat.rawCast 1)))) (Mathlib.Tactic.Ring.mul_zero (↑↑(@_fvar.51385 n) ^ Nat.rawCast 1 * Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_add_zero ((10 ^ (-1 - ↑n)) ^ Nat.rawCast 1 * (↑↑(@_fvar.51385 n) ^ Nat.rawCast 1 * Nat.rawCast 1) + 0))) (Mathlib.Tactic.Ring.zero_mul ((10 ^ (-1 - ↑n)) ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero ((10 ^ (-1 - ↑n)) ^ Nat.rawCast 1 * (↑↑(@_fvar.51385 n) ^ Nat.rawCast 1 * Nat.rawCast 1) + 0)))) (Mathlib.Tactic.Ring.add_pf_add_lt (↑(@_fvar.10801 n) ^ Nat.rawCast 1 * ((10 ^ (-↑n)) ^ Nat.rawCast 1 * Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_zero_add ((10 ^ (-1 - ↑n)) ^ Nat.rawCast 1 * (↑↑(@_fvar.51385 n) ^ Nat.rawCast 1 * Nat.rawCast 1) + 0)))) (congr (congrArg HAdd.hAdd (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow ↑(@_fvar.10801 n)) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one ↑(@_fvar.10801 n)))) (Eq.trans (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow (10 ^ (-↑n))) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one (10 ^ (-↑n))))) Mathlib.Tactic.RingNF.nat_rawCast_1) (mul_one (10 ^ (-↑n)))))) (Eq.trans (congrArg (fun x => x + 0) (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow (10 ^ (-1 - ↑n))) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one (10 ^ (-1 - ↑n))))) (Eq.trans (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow ↑↑(@_fvar.51385 n)) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one ↑↑(@_fvar.51385 n)))) Mathlib.Tactic.RingNF.nat_rawCast_1) (mul_one ↑↑(@_fvar.51385 n))))) (add_zero (10 ^ (-1 - ↑n) * ↑↑(@_fvar.51385 n)))))))) ((fun {α β γ} [HAdd α β γ] a a_1 e_a => Eq.rec (motive := fun a_2 e_a => ∀ (a_3 a_4 : β), a_3 = a_4 → a + a_3 = a_2 + a_4) (fun a_2 a_3 e_a => e_a ▸ Eq.refl (a + a_2)) e_a) (↑(@_fvar.10801 n) * 10 ^ (-1 - ↑n) * 10) (↑(@_fvar.10801 n) * 10 ^ (-↑n)) (Eq.mpr (id (congrArg (fun _a => _a = ↑(@_fvar.10801 n) * 10 ^ (-↑n)) (mul_assoc (↑(@_fvar.10801 n)) (10 ^ (-1 - ↑n)) 10))) (Eq.mpr (id (congrArg (fun _a => ↑(@_fvar.10801 n) * _a = ↑(@_fvar.10801 n) * 10 ^ (-↑n)) (Eq.symm (NNReal.rpow_add_one (Mathlib.Meta.NormNum.isNat_eq_false (Mathlib.Meta.NormNum.isNat_ofNat NNReal (Eq.refl 10)) (Mathlib.Meta.NormNum.isNat_ofNat NNReal Nat.cast_zero) (Eq.refl false)) (-1 - ↑n))))) (of_eq_true (Eq.trans (congrArg (fun x => x = ↑(@_fvar.10801 n) * 10 ^ (-↑n)) (Eq.trans (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.atom_pf ↑(@_fvar.10801 n)) (Mathlib.Tactic.Ring.atom_pf' (congrArg (HPow.hPow 10) (Eq.trans (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.atom_pf ↑n) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑n) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_lt (Int.negOfNat 1).rawCast (Mathlib.Tactic.Ring.add_pf_zero_add (↑n ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1)) (Eq.refl (Int.ofNat 0)))) (Mathlib.Tactic.Ring.add_pf_add_zero (↑n ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))) (Eq.trans (congrArg (fun x => x + 0) (Eq.trans (Eq.trans (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow ↑n) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one ↑n))) (Eq.trans Mathlib.Tactic.RingNF.int_rawCast_neg (congrArg Neg.neg Mathlib.Tactic.RingNF.nat_rawCast_1))) (Mathlib.Tactic.RingNF.mul_neg (↑n) 1)) (congrArg Neg.neg (mul_one ↑n)))) (add_zero (-↑n)))))) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_left (↑(@_fvar.10801 n)) (Nat.rawCast 1) (Mathlib.Tactic.Ring.mul_pf_right (10 ^ (-↑n)) (Nat.rawCast 1) (Mathlib.Tactic.Ring.one_mul (Nat.rawCast 1)))) (Mathlib.Tactic.Ring.mul_zero (↑(@_fvar.10801 n) ^ Nat.rawCast 1 * Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑(@_fvar.10801 n) ^ Nat.rawCast 1 * ((10 ^ (-↑n)) ^ Nat.rawCast 1 * Nat.rawCast 1) + 0))) (Mathlib.Tactic.Ring.zero_mul ((10 ^ (-↑n)) ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑(@_fvar.10801 n) ^ Nat.rawCast 1 * ((10 ^ (-↑n)) ^ Nat.rawCast 1 * Nat.rawCast 1) + 0)))) (Eq.trans (congrArg (fun x => x + 0) (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow ↑(@_fvar.10801 n)) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one ↑(@_fvar.10801 n)))) (Eq.trans (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow (10 ^ (-↑n))) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one (10 ^ (-↑n))))) Mathlib.Tactic.RingNF.nat_rawCast_1) (mul_one (10 ^ (-↑n)))))) (add_zero (↑(@_fvar.10801 n) * 10 ^ (-↑n)))))) (eq_self (↑(@_fvar.10801 n) * 10 ^ (-↑n))))))) (10 ^ (-1 - ↑n) * ↑↑(@_fvar.51385 n)) (10 ^ (-1 - ↑n) * ↑↑(@_fvar.51385 n)) (Eq.refl (10 ^ (-1 - ↑n) * ↑↑(@_fvar.51385 n)))))) (AppendixB.NNRealDecimal.surj._proof_6 _fvar.8365 _fvar.51385 _fvar.51390 n hn)) ((fun {α β γ} [HAdd α β γ] a a_1 e_a => Eq.rec (motive := fun a_2 e_a => ∀ (a_3 a_4 : β), a_3 = a_4 → a + a_3 = a_2 + a_4) (fun a_2 a_3 e_a => e_a ▸ Eq.refl (a + a_2)) e_a) (↑(@_fvar.10801 0)) (↑(@_fvar.10801 0)) (Eq.refl ↑(@_fvar.10801 0)) (∑ i ∈ Finset.range n, ↑↑(@_fvar.51385 i) * 10 ^ (-↑i - 1) + ↑↑(@_fvar.51385 n) * 10 ^ (-↑n - 1)) (∑ i ∈ Finset.range (n + 1), ↑↑(@_fvar.51385 i) * 10 ^ (-↑i - 1)) (Eq.symm (Finset.sum_range_succ (fun x => ↑↑(@_fvar.51385 x) * 10 ^ (-↑x - 1)) n))))) nthis:Filter.Tendsto (fun n => ↑(s n) * 10 ^ (-↑n)) Filter.atTop (nhds ↑d)n:ℕ⊢ x * 10 ^ n * 10 ^ (-↑n) ≤ (↑(s n) + 1) * 10 ^ (-↑n) gcongrbcx:ℝ≥0h:¬x = 0s:ℕ → ℕ := fun n => ⌊_fvar.8365 * 10 ^ n⌋₊hs:∀ (n : ℕ), ↑(@_fvar.10801 n) ≤ _fvar.8365 * 10 ^ n := fun n => Nat.floor_le (le_of_lt (mul_pos (lt_of_le_of_ne' (zero_le _fvar.8365) _fvar.8442) (pow_pos (Mathlib.Meta.Positivity.pos_of_isNat (Mathlib.Meta.NormNum.isNat_ofNat NNReal (Eq.refl 10)) (Eq.refl (Nat.ble 1 10))) n)))hs':∀ (n : ℕ), _fvar.8365 * 10 ^ n < ↑(@_fvar.10801 n) + 1 := fun n => Nat.lt_floor_add_one (_fvar.8365 * 10 ^ n)a:ℕ → Digitha:∀ (n : ℕ), s (n + 1) = 10 * s n + ↑(a n)d:AppendixB.NNRealDecimal := { intPart := @_fvar.10801 0, fracPart := _fvar.51385 }hsum:∀ (n : ℕ), ↑(@_fvar.10801 n) * 10 ^ (-↑n) = ↑(@_fvar.10801 0) + ∑ i ∈ Finset.range n, ↑↑(@_fvar.51385 i) * 10 ^ (-↑i - 1) := fun n => Nat.recAux (of_eq_true (Eq.trans (congr (congrArg Eq (Eq.trans (congrArg (HMul.hMul ↑(@_fvar.10801 0)) (Eq.trans (congrArg (HPow.hPow 10) (Eq.trans (congrArg Neg.neg (CharP.cast_eq_zero ℝ 0)) neg_zero)) (NNReal.rpow_zero 10))) (mul_one ↑(@_fvar.10801 0)))) (add_zero ↑(@_fvar.10801 0))) (eq_self ↑(@_fvar.10801 0)))) (fun n hn => Eq.mpr (id (congrArg (fun _a => ↑_a * 10 ^ (-↑(n + 1)) = ↑(@_fvar.10801 0) + ∑ i ∈ Finset.range (n + 1), ↑↑(@_fvar.51385 i) * 10 ^ (-↑i - 1)) (@_fvar.51390 n))) (Trans.trans (Trans.trans (Eq.mpr (id (congrArg (fun x => x = ↑(@_fvar.10801 n) * 10 ^ (-↑n) + ↑↑(@_fvar.51385 n) * 10 ^ (-↑n - 1)) (Eq.trans (congr (congrArg HMul.hMul (Eq.trans (Nat.cast_add (10 * @_fvar.10801 n) ↑(@_fvar.51385 n)) (congrArg (fun x => x + ↑↑(@_fvar.51385 n)) (Nat.cast_mul 10 (@_fvar.10801 n))))) (congrArg (HPow.hPow 10) (Eq.trans (congrArg Neg.neg (Eq.trans (Nat.cast_add n 1) (congrArg (HAdd.hAdd ↑n) Nat.cast_one))) (neg_add_rev (↑n) 1)))) (add_mul (10 * ↑(@_fvar.10801 n)) (↑↑(@_fvar.51385 n)) (10 ^ (-1 + -↑n)))))) (Eq.mpr (id (congr (congrArg Eq (Eq.trans (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat NNReal (Eq.refl 10))) (Mathlib.Tactic.Ring.atom_pf ↑(@_fvar.10801 n)) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_right (↑(@_fvar.10801 n)) (Nat.rawCast 1) (Mathlib.Tactic.Ring.mul_one (Nat.rawCast 10))) (Mathlib.Tactic.Ring.mul_zero (Nat.rawCast 10)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑(@_fvar.10801 n) ^ Nat.rawCast 1 * Nat.rawCast 10 + 0))) (Mathlib.Tactic.Ring.zero_mul (↑(@_fvar.10801 n) ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑(@_fvar.10801 n) ^ Nat.rawCast 1 * Nat.rawCast 10 + 0)))) (Mathlib.Tactic.Ring.atom_pf' (congrArg (HPow.hPow 10) (Eq.trans (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.atom_pf ↑n) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑n) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.add_pf_add_lt (Int.negOfNat 1).rawCast (Mathlib.Tactic.Ring.add_pf_zero_add (↑n ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))) (Eq.trans (congr (congrArg HAdd.hAdd (Eq.trans Mathlib.Tactic.RingNF.int_rawCast_neg (congrArg Neg.neg Mathlib.Tactic.RingNF.nat_rawCast_1))) (Eq.trans (congrArg (fun x => x + 0) (Eq.trans (Eq.trans (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow ↑n) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one ↑n))) (Eq.trans Mathlib.Tactic.RingNF.int_rawCast_neg (congrArg Neg.neg Mathlib.Tactic.RingNF.nat_rawCast_1))) (Mathlib.Tactic.RingNF.mul_neg (↑n) 1)) (congrArg Neg.neg (mul_one ↑n)))) (add_zero (-↑n)))) (Mathlib.Tactic.RingNF.add_neg (-1) ↑n))))) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_left (↑(@_fvar.10801 n)) (Nat.rawCast 1) (Mathlib.Tactic.Ring.mul_pf_right (10 ^ (-1 - ↑n)) (Nat.rawCast 1) (Mathlib.Tactic.Ring.mul_one (Nat.rawCast 10)))) (Mathlib.Tactic.Ring.mul_zero (↑(@_fvar.10801 n) ^ Nat.rawCast 1 * Nat.rawCast 10)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑(@_fvar.10801 n) ^ Nat.rawCast 1 * ((10 ^ (-1 - ↑n)) ^ Nat.rawCast 1 * Nat.rawCast 10) + 0))) (Mathlib.Tactic.Ring.zero_mul ((10 ^ (-1 - ↑n)) ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑(@_fvar.10801 n) ^ Nat.rawCast 1 * ((10 ^ (-1 - ↑n)) ^ Nat.rawCast 1 * Nat.rawCast 10) + 0)))) (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.atom_pf ↑↑(@_fvar.51385 n)) (Mathlib.Tactic.Ring.atom_pf' (congrArg (HPow.hPow 10) (Eq.trans (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.atom_pf ↑n) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑n) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.add_pf_add_lt (Int.negOfNat 1).rawCast (Mathlib.Tactic.Ring.add_pf_zero_add (↑n ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))) (Eq.trans (congr (congrArg HAdd.hAdd (Eq.trans Mathlib.Tactic.RingNF.int_rawCast_neg (congrArg Neg.neg Mathlib.Tactic.RingNF.nat_rawCast_1))) (Eq.trans (congrArg (fun x => x + 0) (Eq.trans (Eq.trans (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow ↑n) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one ↑n))) (Eq.trans Mathlib.Tactic.RingNF.int_rawCast_neg (congrArg Neg.neg Mathlib.Tactic.RingNF.nat_rawCast_1))) (Mathlib.Tactic.RingNF.mul_neg (↑n) 1)) (congrArg Neg.neg (mul_one ↑n)))) (add_zero (-↑n)))) (Mathlib.Tactic.RingNF.add_neg (-1) ↑n))))) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_right (10 ^ (-1 - ↑n)) (Nat.rawCast 1) (Mathlib.Tactic.Ring.mul_pf_left (↑↑(@_fvar.51385 n)) (Nat.rawCast 1) (Mathlib.Tactic.Ring.one_mul (Nat.rawCast 1)))) (Mathlib.Tactic.Ring.mul_zero (↑↑(@_fvar.51385 n) ^ Nat.rawCast 1 * Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_add_zero ((10 ^ (-1 - ↑n)) ^ Nat.rawCast 1 * (↑↑(@_fvar.51385 n) ^ Nat.rawCast 1 * Nat.rawCast 1) + 0))) (Mathlib.Tactic.Ring.zero_mul ((10 ^ (-1 - ↑n)) ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero ((10 ^ (-1 - ↑n)) ^ Nat.rawCast 1 * (↑↑(@_fvar.51385 n) ^ Nat.rawCast 1 * Nat.rawCast 1) + 0)))) (Mathlib.Tactic.Ring.add_pf_add_lt (↑(@_fvar.10801 n) ^ Nat.rawCast 1 * ((10 ^ (-1 - ↑n)) ^ Nat.rawCast 1 * Nat.rawCast 10)) (Mathlib.Tactic.Ring.add_pf_zero_add ((10 ^ (-1 - ↑n)) ^ Nat.rawCast 1 * (↑↑(@_fvar.51385 n) ^ Nat.rawCast 1 * Nat.rawCast 1) + 0)))) (congr (congrArg HAdd.hAdd (Eq.trans (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow ↑(@_fvar.10801 n)) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one ↑(@_fvar.10801 n)))) (congrArg (fun x => x * 10) (Eq.trans (congrArg (HPow.hPow (10 ^ (-1 - ↑n))) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one (10 ^ (-1 - ↑n)))))) (Mathlib.Tactic.RingNF.mul_assoc_rev (↑(@_fvar.10801 n)) (10 ^ (-1 - ↑n)) 10))) (Eq.trans (congrArg (fun x => x + 0) (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow (10 ^ (-1 - ↑n))) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one (10 ^ (-1 - ↑n))))) (Eq.trans (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow ↑↑(@_fvar.51385 n)) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one ↑↑(@_fvar.51385 n)))) Mathlib.Tactic.RingNF.nat_rawCast_1) (mul_one ↑↑(@_fvar.51385 n))))) (add_zero (10 ^ (-1 - ↑n) * ↑↑(@_fvar.51385 n))))))) (Eq.trans (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.atom_pf ↑(@_fvar.10801 n)) (Mathlib.Tactic.Ring.atom_pf (10 ^ (-↑n))) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_left (↑(@_fvar.10801 n)) (Nat.rawCast 1) (Mathlib.Tactic.Ring.mul_pf_right (10 ^ (-↑n)) (Nat.rawCast 1) (Mathlib.Tactic.Ring.one_mul (Nat.rawCast 1)))) (Mathlib.Tactic.Ring.mul_zero (↑(@_fvar.10801 n) ^ Nat.rawCast 1 * Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑(@_fvar.10801 n) ^ Nat.rawCast 1 * ((10 ^ (-↑n)) ^ Nat.rawCast 1 * Nat.rawCast 1) + 0))) (Mathlib.Tactic.Ring.zero_mul ((10 ^ (-↑n)) ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑(@_fvar.10801 n) ^ Nat.rawCast 1 * ((10 ^ (-↑n)) ^ Nat.rawCast 1 * Nat.rawCast 1) + 0)))) (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.atom_pf ↑↑(@_fvar.51385 n)) (Mathlib.Tactic.Ring.atom_pf' (congrArg (HPow.hPow 10) (Eq.trans (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.atom_pf ↑n) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑n) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_gt (Int.negOfNat 1).rawCast (Mathlib.Tactic.Ring.add_pf_add_zero (↑n ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))) (Eq.trans (congr (congrArg HAdd.hAdd (Eq.trans Mathlib.Tactic.RingNF.int_rawCast_neg (congrArg Neg.neg Mathlib.Tactic.RingNF.nat_rawCast_1))) (Eq.trans (congrArg (fun x => x + 0) (Eq.trans (Eq.trans (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow ↑n) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one ↑n))) (Eq.trans Mathlib.Tactic.RingNF.int_rawCast_neg (congrArg Neg.neg Mathlib.Tactic.RingNF.nat_rawCast_1))) (Mathlib.Tactic.RingNF.mul_neg (↑n) 1)) (congrArg Neg.neg (mul_one ↑n)))) (add_zero (-↑n)))) (Mathlib.Tactic.RingNF.add_neg (-1) ↑n))))) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_right (10 ^ (-1 - ↑n)) (Nat.rawCast 1) (Mathlib.Tactic.Ring.mul_pf_left (↑↑(@_fvar.51385 n)) (Nat.rawCast 1) (Mathlib.Tactic.Ring.one_mul (Nat.rawCast 1)))) (Mathlib.Tactic.Ring.mul_zero (↑↑(@_fvar.51385 n) ^ Nat.rawCast 1 * Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_add_zero ((10 ^ (-1 - ↑n)) ^ Nat.rawCast 1 * (↑↑(@_fvar.51385 n) ^ Nat.rawCast 1 * Nat.rawCast 1) + 0))) (Mathlib.Tactic.Ring.zero_mul ((10 ^ (-1 - ↑n)) ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero ((10 ^ (-1 - ↑n)) ^ Nat.rawCast 1 * (↑↑(@_fvar.51385 n) ^ Nat.rawCast 1 * Nat.rawCast 1) + 0)))) (Mathlib.Tactic.Ring.add_pf_add_lt (↑(@_fvar.10801 n) ^ Nat.rawCast 1 * ((10 ^ (-↑n)) ^ Nat.rawCast 1 * Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_zero_add ((10 ^ (-1 - ↑n)) ^ Nat.rawCast 1 * (↑↑(@_fvar.51385 n) ^ Nat.rawCast 1 * Nat.rawCast 1) + 0)))) (congr (congrArg HAdd.hAdd (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow ↑(@_fvar.10801 n)) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one ↑(@_fvar.10801 n)))) (Eq.trans (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow (10 ^ (-↑n))) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one (10 ^ (-↑n))))) Mathlib.Tactic.RingNF.nat_rawCast_1) (mul_one (10 ^ (-↑n)))))) (Eq.trans (congrArg (fun x => x + 0) (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow (10 ^ (-1 - ↑n))) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one (10 ^ (-1 - ↑n))))) (Eq.trans (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow ↑↑(@_fvar.51385 n)) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one ↑↑(@_fvar.51385 n)))) Mathlib.Tactic.RingNF.nat_rawCast_1) (mul_one ↑↑(@_fvar.51385 n))))) (add_zero (10 ^ (-1 - ↑n) * ↑↑(@_fvar.51385 n)))))))) ((fun {α β γ} [HAdd α β γ] a a_1 e_a => Eq.rec (motive := fun a_2 e_a => ∀ (a_3 a_4 : β), a_3 = a_4 → a + a_3 = a_2 + a_4) (fun a_2 a_3 e_a => e_a ▸ Eq.refl (a + a_2)) e_a) (↑(@_fvar.10801 n) * 10 ^ (-1 - ↑n) * 10) (↑(@_fvar.10801 n) * 10 ^ (-↑n)) (Eq.mpr (id (congrArg (fun _a => _a = ↑(@_fvar.10801 n) * 10 ^ (-↑n)) (mul_assoc (↑(@_fvar.10801 n)) (10 ^ (-1 - ↑n)) 10))) (Eq.mpr (id (congrArg (fun _a => ↑(@_fvar.10801 n) * _a = ↑(@_fvar.10801 n) * 10 ^ (-↑n)) (Eq.symm (NNReal.rpow_add_one (Mathlib.Meta.NormNum.isNat_eq_false (Mathlib.Meta.NormNum.isNat_ofNat NNReal (Eq.refl 10)) (Mathlib.Meta.NormNum.isNat_ofNat NNReal Nat.cast_zero) (Eq.refl false)) (-1 - ↑n))))) (of_eq_true (Eq.trans (congrArg (fun x => x = ↑(@_fvar.10801 n) * 10 ^ (-↑n)) (Eq.trans (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.atom_pf ↑(@_fvar.10801 n)) (Mathlib.Tactic.Ring.atom_pf' (congrArg (HPow.hPow 10) (Eq.trans (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.atom_pf ↑n) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑n) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_lt (Int.negOfNat 1).rawCast (Mathlib.Tactic.Ring.add_pf_zero_add (↑n ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1)) (Eq.refl (Int.ofNat 0)))) (Mathlib.Tactic.Ring.add_pf_add_zero (↑n ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))) (Eq.trans (congrArg (fun x => x + 0) (Eq.trans (Eq.trans (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow ↑n) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one ↑n))) (Eq.trans Mathlib.Tactic.RingNF.int_rawCast_neg (congrArg Neg.neg Mathlib.Tactic.RingNF.nat_rawCast_1))) (Mathlib.Tactic.RingNF.mul_neg (↑n) 1)) (congrArg Neg.neg (mul_one ↑n)))) (add_zero (-↑n)))))) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_left (↑(@_fvar.10801 n)) (Nat.rawCast 1) (Mathlib.Tactic.Ring.mul_pf_right (10 ^ (-↑n)) (Nat.rawCast 1) (Mathlib.Tactic.Ring.one_mul (Nat.rawCast 1)))) (Mathlib.Tactic.Ring.mul_zero (↑(@_fvar.10801 n) ^ Nat.rawCast 1 * Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑(@_fvar.10801 n) ^ Nat.rawCast 1 * ((10 ^ (-↑n)) ^ Nat.rawCast 1 * Nat.rawCast 1) + 0))) (Mathlib.Tactic.Ring.zero_mul ((10 ^ (-↑n)) ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑(@_fvar.10801 n) ^ Nat.rawCast 1 * ((10 ^ (-↑n)) ^ Nat.rawCast 1 * Nat.rawCast 1) + 0)))) (Eq.trans (congrArg (fun x => x + 0) (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow ↑(@_fvar.10801 n)) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one ↑(@_fvar.10801 n)))) (Eq.trans (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow (10 ^ (-↑n))) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one (10 ^ (-↑n))))) Mathlib.Tactic.RingNF.nat_rawCast_1) (mul_one (10 ^ (-↑n)))))) (add_zero (↑(@_fvar.10801 n) * 10 ^ (-↑n)))))) (eq_self (↑(@_fvar.10801 n) * 10 ^ (-↑n))))))) (10 ^ (-1 - ↑n) * ↑↑(@_fvar.51385 n)) (10 ^ (-1 - ↑n) * ↑↑(@_fvar.51385 n)) (Eq.refl (10 ^ (-1 - ↑n) * ↑↑(@_fvar.51385 n)))))) (AppendixB.NNRealDecimal.surj._proof_6 _fvar.8365 _fvar.51385 _fvar.51390 n hn)) ((fun {α β γ} [HAdd α β γ] a a_1 e_a => Eq.rec (motive := fun a_2 e_a => ∀ (a_3 a_4 : β), a_3 = a_4 → a + a_3 = a_2 + a_4) (fun a_2 a_3 e_a => e_a ▸ Eq.refl (a + a_2)) e_a) (↑(@_fvar.10801 0)) (↑(@_fvar.10801 0)) (Eq.refl ↑(@_fvar.10801 0)) (∑ i ∈ Finset.range n, ↑↑(@_fvar.51385 i) * 10 ^ (-↑i - 1) + ↑↑(@_fvar.51385 n) * 10 ^ (-↑n - 1)) (∑ i ∈ Finset.range (n + 1), ↑↑(@_fvar.51385 i) * 10 ^ (-↑i - 1)) (Eq.symm (Finset.sum_range_succ (fun x => ↑↑(@_fvar.51385 x) * 10 ^ (-↑x - 1)) n))))) nthis:Filter.Tendsto (fun n => ↑(s n) * 10 ^ (-↑n)) Filter.atTop (nhds ↑d)n:ℕ⊢ x * 10 ^ n ≤ ↑(s n) + 1; grind [le_of_lt]All goals completed! 🐙 _ = _ := byx:ℝ≥0h:¬x = 0s:ℕ → ℕ := fun n => ⌊_fvar.8365 * 10 ^ n⌋₊hs:∀ (n : ℕ), ↑(@_fvar.10801 n) ≤ _fvar.8365 * 10 ^ n := fun n => Nat.floor_le (le_of_lt (mul_pos (lt_of_le_of_ne' (zero_le _fvar.8365) _fvar.8442) (pow_pos (Mathlib.Meta.Positivity.pos_of_isNat (Mathlib.Meta.NormNum.isNat_ofNat NNReal (Eq.refl 10)) (Eq.refl (Nat.ble 1 10))) n)))hs':∀ (n : ℕ), _fvar.8365 * 10 ^ n < ↑(@_fvar.10801 n) + 1 := fun n => Nat.lt_floor_add_one (_fvar.8365 * 10 ^ n)a:ℕ → Digitha:∀ (n : ℕ), s (n + 1) = 10 * s n + ↑(a n)d:AppendixB.NNRealDecimal := { intPart := @_fvar.10801 0, fracPart := _fvar.51385 }hsum:∀ (n : ℕ), ↑(@_fvar.10801 n) * 10 ^ (-↑n) = ↑(@_fvar.10801 0) + ∑ i ∈ Finset.range n, ↑↑(@_fvar.51385 i) * 10 ^ (-↑i - 1) := fun n => Nat.recAux (of_eq_true (Eq.trans (congr (congrArg Eq (Eq.trans (congrArg (HMul.hMul ↑(@_fvar.10801 0)) (Eq.trans (congrArg (HPow.hPow 10) (Eq.trans (congrArg Neg.neg (CharP.cast_eq_zero ℝ 0)) neg_zero)) (NNReal.rpow_zero 10))) (mul_one ↑(@_fvar.10801 0)))) (add_zero ↑(@_fvar.10801 0))) (eq_self ↑(@_fvar.10801 0)))) (fun n hn => Eq.mpr (id (congrArg (fun _a => ↑_a * 10 ^ (-↑(n + 1)) = ↑(@_fvar.10801 0) + ∑ i ∈ Finset.range (n + 1), ↑↑(@_fvar.51385 i) * 10 ^ (-↑i - 1)) (@_fvar.51390 n))) (Trans.trans (Trans.trans (Eq.mpr (id (congrArg (fun x => x = ↑(@_fvar.10801 n) * 10 ^ (-↑n) + ↑↑(@_fvar.51385 n) * 10 ^ (-↑n - 1)) (Eq.trans (congr (congrArg HMul.hMul (Eq.trans (Nat.cast_add (10 * @_fvar.10801 n) ↑(@_fvar.51385 n)) (congrArg (fun x => x + ↑↑(@_fvar.51385 n)) (Nat.cast_mul 10 (@_fvar.10801 n))))) (congrArg (HPow.hPow 10) (Eq.trans (congrArg Neg.neg (Eq.trans (Nat.cast_add n 1) (congrArg (HAdd.hAdd ↑n) Nat.cast_one))) (neg_add_rev (↑n) 1)))) (add_mul (10 * ↑(@_fvar.10801 n)) (↑↑(@_fvar.51385 n)) (10 ^ (-1 + -↑n)))))) (Eq.mpr (id (congr (congrArg Eq (Eq.trans (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat NNReal (Eq.refl 10))) (Mathlib.Tactic.Ring.atom_pf ↑(@_fvar.10801 n)) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_right (↑(@_fvar.10801 n)) (Nat.rawCast 1) (Mathlib.Tactic.Ring.mul_one (Nat.rawCast 10))) (Mathlib.Tactic.Ring.mul_zero (Nat.rawCast 10)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑(@_fvar.10801 n) ^ Nat.rawCast 1 * Nat.rawCast 10 + 0))) (Mathlib.Tactic.Ring.zero_mul (↑(@_fvar.10801 n) ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑(@_fvar.10801 n) ^ Nat.rawCast 1 * Nat.rawCast 10 + 0)))) (Mathlib.Tactic.Ring.atom_pf' (congrArg (HPow.hPow 10) (Eq.trans (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.atom_pf ↑n) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑n) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.add_pf_add_lt (Int.negOfNat 1).rawCast (Mathlib.Tactic.Ring.add_pf_zero_add (↑n ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))) (Eq.trans (congr (congrArg HAdd.hAdd (Eq.trans Mathlib.Tactic.RingNF.int_rawCast_neg (congrArg Neg.neg Mathlib.Tactic.RingNF.nat_rawCast_1))) (Eq.trans (congrArg (fun x => x + 0) (Eq.trans (Eq.trans (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow ↑n) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one ↑n))) (Eq.trans Mathlib.Tactic.RingNF.int_rawCast_neg (congrArg Neg.neg Mathlib.Tactic.RingNF.nat_rawCast_1))) (Mathlib.Tactic.RingNF.mul_neg (↑n) 1)) (congrArg Neg.neg (mul_one ↑n)))) (add_zero (-↑n)))) (Mathlib.Tactic.RingNF.add_neg (-1) ↑n))))) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_left (↑(@_fvar.10801 n)) (Nat.rawCast 1) (Mathlib.Tactic.Ring.mul_pf_right (10 ^ (-1 - ↑n)) (Nat.rawCast 1) (Mathlib.Tactic.Ring.mul_one (Nat.rawCast 10)))) (Mathlib.Tactic.Ring.mul_zero (↑(@_fvar.10801 n) ^ Nat.rawCast 1 * Nat.rawCast 10)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑(@_fvar.10801 n) ^ Nat.rawCast 1 * ((10 ^ (-1 - ↑n)) ^ Nat.rawCast 1 * Nat.rawCast 10) + 0))) (Mathlib.Tactic.Ring.zero_mul ((10 ^ (-1 - ↑n)) ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑(@_fvar.10801 n) ^ Nat.rawCast 1 * ((10 ^ (-1 - ↑n)) ^ Nat.rawCast 1 * Nat.rawCast 10) + 0)))) (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.atom_pf ↑↑(@_fvar.51385 n)) (Mathlib.Tactic.Ring.atom_pf' (congrArg (HPow.hPow 10) (Eq.trans (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.atom_pf ↑n) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑n) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.add_pf_add_lt (Int.negOfNat 1).rawCast (Mathlib.Tactic.Ring.add_pf_zero_add (↑n ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))) (Eq.trans (congr (congrArg HAdd.hAdd (Eq.trans Mathlib.Tactic.RingNF.int_rawCast_neg (congrArg Neg.neg Mathlib.Tactic.RingNF.nat_rawCast_1))) (Eq.trans (congrArg (fun x => x + 0) (Eq.trans (Eq.trans (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow ↑n) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one ↑n))) (Eq.trans Mathlib.Tactic.RingNF.int_rawCast_neg (congrArg Neg.neg Mathlib.Tactic.RingNF.nat_rawCast_1))) (Mathlib.Tactic.RingNF.mul_neg (↑n) 1)) (congrArg Neg.neg (mul_one ↑n)))) (add_zero (-↑n)))) (Mathlib.Tactic.RingNF.add_neg (-1) ↑n))))) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_right (10 ^ (-1 - ↑n)) (Nat.rawCast 1) (Mathlib.Tactic.Ring.mul_pf_left (↑↑(@_fvar.51385 n)) (Nat.rawCast 1) (Mathlib.Tactic.Ring.one_mul (Nat.rawCast 1)))) (Mathlib.Tactic.Ring.mul_zero (↑↑(@_fvar.51385 n) ^ Nat.rawCast 1 * Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_add_zero ((10 ^ (-1 - ↑n)) ^ Nat.rawCast 1 * (↑↑(@_fvar.51385 n) ^ Nat.rawCast 1 * Nat.rawCast 1) + 0))) (Mathlib.Tactic.Ring.zero_mul ((10 ^ (-1 - ↑n)) ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero ((10 ^ (-1 - ↑n)) ^ Nat.rawCast 1 * (↑↑(@_fvar.51385 n) ^ Nat.rawCast 1 * Nat.rawCast 1) + 0)))) (Mathlib.Tactic.Ring.add_pf_add_lt (↑(@_fvar.10801 n) ^ Nat.rawCast 1 * ((10 ^ (-1 - ↑n)) ^ Nat.rawCast 1 * Nat.rawCast 10)) (Mathlib.Tactic.Ring.add_pf_zero_add ((10 ^ (-1 - ↑n)) ^ Nat.rawCast 1 * (↑↑(@_fvar.51385 n) ^ Nat.rawCast 1 * Nat.rawCast 1) + 0)))) (congr (congrArg HAdd.hAdd (Eq.trans (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow ↑(@_fvar.10801 n)) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one ↑(@_fvar.10801 n)))) (congrArg (fun x => x * 10) (Eq.trans (congrArg (HPow.hPow (10 ^ (-1 - ↑n))) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one (10 ^ (-1 - ↑n)))))) (Mathlib.Tactic.RingNF.mul_assoc_rev (↑(@_fvar.10801 n)) (10 ^ (-1 - ↑n)) 10))) (Eq.trans (congrArg (fun x => x + 0) (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow (10 ^ (-1 - ↑n))) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one (10 ^ (-1 - ↑n))))) (Eq.trans (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow ↑↑(@_fvar.51385 n)) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one ↑↑(@_fvar.51385 n)))) Mathlib.Tactic.RingNF.nat_rawCast_1) (mul_one ↑↑(@_fvar.51385 n))))) (add_zero (10 ^ (-1 - ↑n) * ↑↑(@_fvar.51385 n))))))) (Eq.trans (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.atom_pf ↑(@_fvar.10801 n)) (Mathlib.Tactic.Ring.atom_pf (10 ^ (-↑n))) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_left (↑(@_fvar.10801 n)) (Nat.rawCast 1) (Mathlib.Tactic.Ring.mul_pf_right (10 ^ (-↑n)) (Nat.rawCast 1) (Mathlib.Tactic.Ring.one_mul (Nat.rawCast 1)))) (Mathlib.Tactic.Ring.mul_zero (↑(@_fvar.10801 n) ^ Nat.rawCast 1 * Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑(@_fvar.10801 n) ^ Nat.rawCast 1 * ((10 ^ (-↑n)) ^ Nat.rawCast 1 * Nat.rawCast 1) + 0))) (Mathlib.Tactic.Ring.zero_mul ((10 ^ (-↑n)) ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑(@_fvar.10801 n) ^ Nat.rawCast 1 * ((10 ^ (-↑n)) ^ Nat.rawCast 1 * Nat.rawCast 1) + 0)))) (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.atom_pf ↑↑(@_fvar.51385 n)) (Mathlib.Tactic.Ring.atom_pf' (congrArg (HPow.hPow 10) (Eq.trans (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.atom_pf ↑n) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑n) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_gt (Int.negOfNat 1).rawCast (Mathlib.Tactic.Ring.add_pf_add_zero (↑n ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))) (Eq.trans (congr (congrArg HAdd.hAdd (Eq.trans Mathlib.Tactic.RingNF.int_rawCast_neg (congrArg Neg.neg Mathlib.Tactic.RingNF.nat_rawCast_1))) (Eq.trans (congrArg (fun x => x + 0) (Eq.trans (Eq.trans (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow ↑n) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one ↑n))) (Eq.trans Mathlib.Tactic.RingNF.int_rawCast_neg (congrArg Neg.neg Mathlib.Tactic.RingNF.nat_rawCast_1))) (Mathlib.Tactic.RingNF.mul_neg (↑n) 1)) (congrArg Neg.neg (mul_one ↑n)))) (add_zero (-↑n)))) (Mathlib.Tactic.RingNF.add_neg (-1) ↑n))))) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_right (10 ^ (-1 - ↑n)) (Nat.rawCast 1) (Mathlib.Tactic.Ring.mul_pf_left (↑↑(@_fvar.51385 n)) (Nat.rawCast 1) (Mathlib.Tactic.Ring.one_mul (Nat.rawCast 1)))) (Mathlib.Tactic.Ring.mul_zero (↑↑(@_fvar.51385 n) ^ Nat.rawCast 1 * Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_add_zero ((10 ^ (-1 - ↑n)) ^ Nat.rawCast 1 * (↑↑(@_fvar.51385 n) ^ Nat.rawCast 1 * Nat.rawCast 1) + 0))) (Mathlib.Tactic.Ring.zero_mul ((10 ^ (-1 - ↑n)) ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero ((10 ^ (-1 - ↑n)) ^ Nat.rawCast 1 * (↑↑(@_fvar.51385 n) ^ Nat.rawCast 1 * Nat.rawCast 1) + 0)))) (Mathlib.Tactic.Ring.add_pf_add_lt (↑(@_fvar.10801 n) ^ Nat.rawCast 1 * ((10 ^ (-↑n)) ^ Nat.rawCast 1 * Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_zero_add ((10 ^ (-1 - ↑n)) ^ Nat.rawCast 1 * (↑↑(@_fvar.51385 n) ^ Nat.rawCast 1 * Nat.rawCast 1) + 0)))) (congr (congrArg HAdd.hAdd (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow ↑(@_fvar.10801 n)) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one ↑(@_fvar.10801 n)))) (Eq.trans (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow (10 ^ (-↑n))) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one (10 ^ (-↑n))))) Mathlib.Tactic.RingNF.nat_rawCast_1) (mul_one (10 ^ (-↑n)))))) (Eq.trans (congrArg (fun x => x + 0) (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow (10 ^ (-1 - ↑n))) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one (10 ^ (-1 - ↑n))))) (Eq.trans (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow ↑↑(@_fvar.51385 n)) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one ↑↑(@_fvar.51385 n)))) Mathlib.Tactic.RingNF.nat_rawCast_1) (mul_one ↑↑(@_fvar.51385 n))))) (add_zero (10 ^ (-1 - ↑n) * ↑↑(@_fvar.51385 n)))))))) ((fun {α β γ} [HAdd α β γ] a a_1 e_a => Eq.rec (motive := fun a_2 e_a => ∀ (a_3 a_4 : β), a_3 = a_4 → a + a_3 = a_2 + a_4) (fun a_2 a_3 e_a => e_a ▸ Eq.refl (a + a_2)) e_a) (↑(@_fvar.10801 n) * 10 ^ (-1 - ↑n) * 10) (↑(@_fvar.10801 n) * 10 ^ (-↑n)) (Eq.mpr (id (congrArg (fun _a => _a = ↑(@_fvar.10801 n) * 10 ^ (-↑n)) (mul_assoc (↑(@_fvar.10801 n)) (10 ^ (-1 - ↑n)) 10))) (Eq.mpr (id (congrArg (fun _a => ↑(@_fvar.10801 n) * _a = ↑(@_fvar.10801 n) * 10 ^ (-↑n)) (Eq.symm (NNReal.rpow_add_one (Mathlib.Meta.NormNum.isNat_eq_false (Mathlib.Meta.NormNum.isNat_ofNat NNReal (Eq.refl 10)) (Mathlib.Meta.NormNum.isNat_ofNat NNReal Nat.cast_zero) (Eq.refl false)) (-1 - ↑n))))) (of_eq_true (Eq.trans (congrArg (fun x => x = ↑(@_fvar.10801 n) * 10 ^ (-↑n)) (Eq.trans (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.atom_pf ↑(@_fvar.10801 n)) (Mathlib.Tactic.Ring.atom_pf' (congrArg (HPow.hPow 10) (Eq.trans (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.atom_pf ↑n) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑n) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_lt (Int.negOfNat 1).rawCast (Mathlib.Tactic.Ring.add_pf_zero_add (↑n ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1)) (Eq.refl (Int.ofNat 0)))) (Mathlib.Tactic.Ring.add_pf_add_zero (↑n ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))) (Eq.trans (congrArg (fun x => x + 0) (Eq.trans (Eq.trans (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow ↑n) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one ↑n))) (Eq.trans Mathlib.Tactic.RingNF.int_rawCast_neg (congrArg Neg.neg Mathlib.Tactic.RingNF.nat_rawCast_1))) (Mathlib.Tactic.RingNF.mul_neg (↑n) 1)) (congrArg Neg.neg (mul_one ↑n)))) (add_zero (-↑n)))))) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_left (↑(@_fvar.10801 n)) (Nat.rawCast 1) (Mathlib.Tactic.Ring.mul_pf_right (10 ^ (-↑n)) (Nat.rawCast 1) (Mathlib.Tactic.Ring.one_mul (Nat.rawCast 1)))) (Mathlib.Tactic.Ring.mul_zero (↑(@_fvar.10801 n) ^ Nat.rawCast 1 * Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑(@_fvar.10801 n) ^ Nat.rawCast 1 * ((10 ^ (-↑n)) ^ Nat.rawCast 1 * Nat.rawCast 1) + 0))) (Mathlib.Tactic.Ring.zero_mul ((10 ^ (-↑n)) ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑(@_fvar.10801 n) ^ Nat.rawCast 1 * ((10 ^ (-↑n)) ^ Nat.rawCast 1 * Nat.rawCast 1) + 0)))) (Eq.trans (congrArg (fun x => x + 0) (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow ↑(@_fvar.10801 n)) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one ↑(@_fvar.10801 n)))) (Eq.trans (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow (10 ^ (-↑n))) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one (10 ^ (-↑n))))) Mathlib.Tactic.RingNF.nat_rawCast_1) (mul_one (10 ^ (-↑n)))))) (add_zero (↑(@_fvar.10801 n) * 10 ^ (-↑n)))))) (eq_self (↑(@_fvar.10801 n) * 10 ^ (-↑n))))))) (10 ^ (-1 - ↑n) * ↑↑(@_fvar.51385 n)) (10 ^ (-1 - ↑n) * ↑↑(@_fvar.51385 n)) (Eq.refl (10 ^ (-1 - ↑n) * ↑↑(@_fvar.51385 n)))))) (AppendixB.NNRealDecimal.surj._proof_6 _fvar.8365 _fvar.51385 _fvar.51390 n hn)) ((fun {α β γ} [HAdd α β γ] a a_1 e_a => Eq.rec (motive := fun a_2 e_a => ∀ (a_3 a_4 : β), a_3 = a_4 → a + a_3 = a_2 + a_4) (fun a_2 a_3 e_a => e_a ▸ Eq.refl (a + a_2)) e_a) (↑(@_fvar.10801 0)) (↑(@_fvar.10801 0)) (Eq.refl ↑(@_fvar.10801 0)) (∑ i ∈ Finset.range n, ↑↑(@_fvar.51385 i) * 10 ^ (-↑i - 1) + ↑↑(@_fvar.51385 n) * 10 ^ (-↑n - 1)) (∑ i ∈ Finset.range (n + 1), ↑↑(@_fvar.51385 i) * 10 ^ (-↑i - 1)) (Eq.symm (Finset.sum_range_succ (fun x => ↑↑(@_fvar.51385 x) * 10 ^ (-↑x - 1)) n))))) nthis:Filter.Tendsto (fun n => ↑(s n) * 10 ^ (-↑n)) Filter.atTop (nhds ↑d)n:ℕ⊢ (↑(s n) + 1) * 10 ^ (-↑n) = ↑(s n) * 10 ^ (-↑n) + 10 ^ (-↑n) ringAll goals completed! 🐙 intro nhfhx:ℝ≥0h:¬x = 0s:ℕ → ℕ := fun n => ⌊_fvar.8365 * 10 ^ n⌋₊hs:∀ (n : ℕ), ↑(@_fvar.10801 n) ≤ _fvar.8365 * 10 ^ n := fun n => Nat.floor_le (@?_mvar.12729 n)hs':∀ (n : ℕ), _fvar.8365 * 10 ^ n < ↑(@_fvar.10801 n) + 1 := fun n => Nat.lt_floor_add_one (_fvar.8365 * 10 ^ n)a:ℕ → Digitha:∀ (n : ℕ), s (n + 1) = 10 * s n + ↑(a n)d:AppendixB.NNRealDecimal := { intPart := @_fvar.10801 0, fracPart := _fvar.51385 }hsum:∀ (n : ℕ), ↑(@_fvar.10801 n) * 10 ^ (-↑n) = ↑(@_fvar.10801 0) + ∑ i ∈ Finset.range n, ↑↑(@_fvar.51385 i) * 10 ^ (-↑i - 1) := fun n => @?_mvar.57011 nthis:Filter.Tendsto (fun n => ↑(s n) * 10 ^ (-↑n)) Filter.atTop (nhds ↑d)n:ℕ⊢ (fun n => ↑(s n) * 10 ^ (-↑n)) n ≤ (fun x_1 => x) n; simphfhx:ℝ≥0h:¬x = 0s:ℕ → ℕ := fun n => ⌊_fvar.8365 * 10 ^ n⌋₊hs:∀ (n : ℕ), ↑(@_fvar.10801 n) ≤ _fvar.8365 * 10 ^ n := fun n => Nat.floor_le (@?_mvar.12729 n)hs':∀ (n : ℕ), _fvar.8365 * 10 ^ n < ↑(@_fvar.10801 n) + 1 := fun n => Nat.lt_floor_add_one (_fvar.8365 * 10 ^ n)a:ℕ → Digitha:∀ (n : ℕ), s (n + 1) = 10 * s n + ↑(a n)d:AppendixB.NNRealDecimal := { intPart := @_fvar.10801 0, fracPart := _fvar.51385 }hsum:∀ (n : ℕ), ↑(@_fvar.10801 n) * 10 ^ (-↑n) = ↑(@_fvar.10801 0) + ∑ i ∈ Finset.range n, ↑↑(@_fvar.51385 i) * 10 ^ (-↑i - 1) := fun n => @?_mvar.57011 nthis:Filter.Tendsto (fun n => ↑(s n) * 10 ^ (-↑n)) Filter.atTop (nhds ↑d)n:ℕ⊢ ↑(s n) * 10 ^ (-↑n) ≤ x; calc _ ≤ (x * 10^n) * (10:NNReal)^(-n:ℝ) := byx:ℝ≥0h:¬x = 0s:ℕ → ℕ := fun n => ⌊_fvar.8365 * 10 ^ n⌋₊hs:∀ (n : ℕ), ↑(@_fvar.10801 n) ≤ _fvar.8365 * 10 ^ n := fun n => Nat.floor_le (le_of_lt (mul_pos (lt_of_le_of_ne' (zero_le _fvar.8365) _fvar.8442) (pow_pos (Mathlib.Meta.Positivity.pos_of_isNat (Mathlib.Meta.NormNum.isNat_ofNat NNReal (Eq.refl 10)) (Eq.refl (Nat.ble 1 10))) n)))hs':∀ (n : ℕ), _fvar.8365 * 10 ^ n < ↑(@_fvar.10801 n) + 1 := fun n => Nat.lt_floor_add_one (_fvar.8365 * 10 ^ n)a:ℕ → Digitha:∀ (n : ℕ), s (n + 1) = 10 * s n + ↑(a n)d:AppendixB.NNRealDecimal := { intPart := @_fvar.10801 0, fracPart := _fvar.51385 }hsum:∀ (n : ℕ), ↑(@_fvar.10801 n) * 10 ^ (-↑n) = ↑(@_fvar.10801 0) + ∑ i ∈ Finset.range n, ↑↑(@_fvar.51385 i) * 10 ^ (-↑i - 1) := fun n => Nat.recAux (of_eq_true (Eq.trans (congr (congrArg Eq (Eq.trans (congrArg (HMul.hMul ↑(@_fvar.10801 0)) (Eq.trans (congrArg (HPow.hPow 10) (Eq.trans (congrArg Neg.neg (CharP.cast_eq_zero ℝ 0)) neg_zero)) (NNReal.rpow_zero 10))) (mul_one ↑(@_fvar.10801 0)))) (add_zero ↑(@_fvar.10801 0))) (eq_self ↑(@_fvar.10801 0)))) (fun n hn => Eq.mpr (id (congrArg (fun _a => ↑_a * 10 ^ (-↑(n + 1)) = ↑(@_fvar.10801 0) + ∑ i ∈ Finset.range (n + 1), ↑↑(@_fvar.51385 i) * 10 ^ (-↑i - 1)) (@_fvar.51390 n))) (Trans.trans (Trans.trans (Eq.mpr (id (congrArg (fun x => x = ↑(@_fvar.10801 n) * 10 ^ (-↑n) + ↑↑(@_fvar.51385 n) * 10 ^ (-↑n - 1)) (Eq.trans (congr (congrArg HMul.hMul (Eq.trans (Nat.cast_add (10 * @_fvar.10801 n) ↑(@_fvar.51385 n)) (congrArg (fun x => x + ↑↑(@_fvar.51385 n)) (Nat.cast_mul 10 (@_fvar.10801 n))))) (congrArg (HPow.hPow 10) (Eq.trans (congrArg Neg.neg (Eq.trans (Nat.cast_add n 1) (congrArg (HAdd.hAdd ↑n) Nat.cast_one))) (neg_add_rev (↑n) 1)))) (add_mul (10 * ↑(@_fvar.10801 n)) (↑↑(@_fvar.51385 n)) (10 ^ (-1 + -↑n)))))) (Eq.mpr (id (congr (congrArg Eq (Eq.trans (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat NNReal (Eq.refl 10))) (Mathlib.Tactic.Ring.atom_pf ↑(@_fvar.10801 n)) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_right (↑(@_fvar.10801 n)) (Nat.rawCast 1) (Mathlib.Tactic.Ring.mul_one (Nat.rawCast 10))) (Mathlib.Tactic.Ring.mul_zero (Nat.rawCast 10)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑(@_fvar.10801 n) ^ Nat.rawCast 1 * Nat.rawCast 10 + 0))) (Mathlib.Tactic.Ring.zero_mul (↑(@_fvar.10801 n) ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑(@_fvar.10801 n) ^ Nat.rawCast 1 * Nat.rawCast 10 + 0)))) (Mathlib.Tactic.Ring.atom_pf' (congrArg (HPow.hPow 10) (Eq.trans (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.atom_pf ↑n) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑n) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.add_pf_add_lt (Int.negOfNat 1).rawCast (Mathlib.Tactic.Ring.add_pf_zero_add (↑n ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))) (Eq.trans (congr (congrArg HAdd.hAdd (Eq.trans Mathlib.Tactic.RingNF.int_rawCast_neg (congrArg Neg.neg Mathlib.Tactic.RingNF.nat_rawCast_1))) (Eq.trans (congrArg (fun x => x + 0) (Eq.trans (Eq.trans (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow ↑n) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one ↑n))) (Eq.trans Mathlib.Tactic.RingNF.int_rawCast_neg (congrArg Neg.neg Mathlib.Tactic.RingNF.nat_rawCast_1))) (Mathlib.Tactic.RingNF.mul_neg (↑n) 1)) (congrArg Neg.neg (mul_one ↑n)))) (add_zero (-↑n)))) (Mathlib.Tactic.RingNF.add_neg (-1) ↑n))))) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_left (↑(@_fvar.10801 n)) (Nat.rawCast 1) (Mathlib.Tactic.Ring.mul_pf_right (10 ^ (-1 - ↑n)) (Nat.rawCast 1) (Mathlib.Tactic.Ring.mul_one (Nat.rawCast 10)))) (Mathlib.Tactic.Ring.mul_zero (↑(@_fvar.10801 n) ^ Nat.rawCast 1 * Nat.rawCast 10)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑(@_fvar.10801 n) ^ Nat.rawCast 1 * ((10 ^ (-1 - ↑n)) ^ Nat.rawCast 1 * Nat.rawCast 10) + 0))) (Mathlib.Tactic.Ring.zero_mul ((10 ^ (-1 - ↑n)) ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑(@_fvar.10801 n) ^ Nat.rawCast 1 * ((10 ^ (-1 - ↑n)) ^ Nat.rawCast 1 * Nat.rawCast 10) + 0)))) (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.atom_pf ↑↑(@_fvar.51385 n)) (Mathlib.Tactic.Ring.atom_pf' (congrArg (HPow.hPow 10) (Eq.trans (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.atom_pf ↑n) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑n) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.add_pf_add_lt (Int.negOfNat 1).rawCast (Mathlib.Tactic.Ring.add_pf_zero_add (↑n ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))) (Eq.trans (congr (congrArg HAdd.hAdd (Eq.trans Mathlib.Tactic.RingNF.int_rawCast_neg (congrArg Neg.neg Mathlib.Tactic.RingNF.nat_rawCast_1))) (Eq.trans (congrArg (fun x => x + 0) (Eq.trans (Eq.trans (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow ↑n) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one ↑n))) (Eq.trans Mathlib.Tactic.RingNF.int_rawCast_neg (congrArg Neg.neg Mathlib.Tactic.RingNF.nat_rawCast_1))) (Mathlib.Tactic.RingNF.mul_neg (↑n) 1)) (congrArg Neg.neg (mul_one ↑n)))) (add_zero (-↑n)))) (Mathlib.Tactic.RingNF.add_neg (-1) ↑n))))) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_right (10 ^ (-1 - ↑n)) (Nat.rawCast 1) (Mathlib.Tactic.Ring.mul_pf_left (↑↑(@_fvar.51385 n)) (Nat.rawCast 1) (Mathlib.Tactic.Ring.one_mul (Nat.rawCast 1)))) (Mathlib.Tactic.Ring.mul_zero (↑↑(@_fvar.51385 n) ^ Nat.rawCast 1 * Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_add_zero ((10 ^ (-1 - ↑n)) ^ Nat.rawCast 1 * (↑↑(@_fvar.51385 n) ^ Nat.rawCast 1 * Nat.rawCast 1) + 0))) (Mathlib.Tactic.Ring.zero_mul ((10 ^ (-1 - ↑n)) ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero ((10 ^ (-1 - ↑n)) ^ Nat.rawCast 1 * (↑↑(@_fvar.51385 n) ^ Nat.rawCast 1 * Nat.rawCast 1) + 0)))) (Mathlib.Tactic.Ring.add_pf_add_lt (↑(@_fvar.10801 n) ^ Nat.rawCast 1 * ((10 ^ (-1 - ↑n)) ^ Nat.rawCast 1 * Nat.rawCast 10)) (Mathlib.Tactic.Ring.add_pf_zero_add ((10 ^ (-1 - ↑n)) ^ Nat.rawCast 1 * (↑↑(@_fvar.51385 n) ^ Nat.rawCast 1 * Nat.rawCast 1) + 0)))) (congr (congrArg HAdd.hAdd (Eq.trans (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow ↑(@_fvar.10801 n)) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one ↑(@_fvar.10801 n)))) (congrArg (fun x => x * 10) (Eq.trans (congrArg (HPow.hPow (10 ^ (-1 - ↑n))) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one (10 ^ (-1 - ↑n)))))) (Mathlib.Tactic.RingNF.mul_assoc_rev (↑(@_fvar.10801 n)) (10 ^ (-1 - ↑n)) 10))) (Eq.trans (congrArg (fun x => x + 0) (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow (10 ^ (-1 - ↑n))) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one (10 ^ (-1 - ↑n))))) (Eq.trans (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow ↑↑(@_fvar.51385 n)) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one ↑↑(@_fvar.51385 n)))) Mathlib.Tactic.RingNF.nat_rawCast_1) (mul_one ↑↑(@_fvar.51385 n))))) (add_zero (10 ^ (-1 - ↑n) * ↑↑(@_fvar.51385 n))))))) (Eq.trans (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.atom_pf ↑(@_fvar.10801 n)) (Mathlib.Tactic.Ring.atom_pf (10 ^ (-↑n))) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_left (↑(@_fvar.10801 n)) (Nat.rawCast 1) (Mathlib.Tactic.Ring.mul_pf_right (10 ^ (-↑n)) (Nat.rawCast 1) (Mathlib.Tactic.Ring.one_mul (Nat.rawCast 1)))) (Mathlib.Tactic.Ring.mul_zero (↑(@_fvar.10801 n) ^ Nat.rawCast 1 * Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑(@_fvar.10801 n) ^ Nat.rawCast 1 * ((10 ^ (-↑n)) ^ Nat.rawCast 1 * Nat.rawCast 1) + 0))) (Mathlib.Tactic.Ring.zero_mul ((10 ^ (-↑n)) ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑(@_fvar.10801 n) ^ Nat.rawCast 1 * ((10 ^ (-↑n)) ^ Nat.rawCast 1 * Nat.rawCast 1) + 0)))) (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.atom_pf ↑↑(@_fvar.51385 n)) (Mathlib.Tactic.Ring.atom_pf' (congrArg (HPow.hPow 10) (Eq.trans (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.atom_pf ↑n) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑n) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_gt (Int.negOfNat 1).rawCast (Mathlib.Tactic.Ring.add_pf_add_zero (↑n ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))) (Eq.trans (congr (congrArg HAdd.hAdd (Eq.trans Mathlib.Tactic.RingNF.int_rawCast_neg (congrArg Neg.neg Mathlib.Tactic.RingNF.nat_rawCast_1))) (Eq.trans (congrArg (fun x => x + 0) (Eq.trans (Eq.trans (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow ↑n) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one ↑n))) (Eq.trans Mathlib.Tactic.RingNF.int_rawCast_neg (congrArg Neg.neg Mathlib.Tactic.RingNF.nat_rawCast_1))) (Mathlib.Tactic.RingNF.mul_neg (↑n) 1)) (congrArg Neg.neg (mul_one ↑n)))) (add_zero (-↑n)))) (Mathlib.Tactic.RingNF.add_neg (-1) ↑n))))) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_right (10 ^ (-1 - ↑n)) (Nat.rawCast 1) (Mathlib.Tactic.Ring.mul_pf_left (↑↑(@_fvar.51385 n)) (Nat.rawCast 1) (Mathlib.Tactic.Ring.one_mul (Nat.rawCast 1)))) (Mathlib.Tactic.Ring.mul_zero (↑↑(@_fvar.51385 n) ^ Nat.rawCast 1 * Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_add_zero ((10 ^ (-1 - ↑n)) ^ Nat.rawCast 1 * (↑↑(@_fvar.51385 n) ^ Nat.rawCast 1 * Nat.rawCast 1) + 0))) (Mathlib.Tactic.Ring.zero_mul ((10 ^ (-1 - ↑n)) ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero ((10 ^ (-1 - ↑n)) ^ Nat.rawCast 1 * (↑↑(@_fvar.51385 n) ^ Nat.rawCast 1 * Nat.rawCast 1) + 0)))) (Mathlib.Tactic.Ring.add_pf_add_lt (↑(@_fvar.10801 n) ^ Nat.rawCast 1 * ((10 ^ (-↑n)) ^ Nat.rawCast 1 * Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_zero_add ((10 ^ (-1 - ↑n)) ^ Nat.rawCast 1 * (↑↑(@_fvar.51385 n) ^ Nat.rawCast 1 * Nat.rawCast 1) + 0)))) (congr (congrArg HAdd.hAdd (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow ↑(@_fvar.10801 n)) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one ↑(@_fvar.10801 n)))) (Eq.trans (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow (10 ^ (-↑n))) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one (10 ^ (-↑n))))) Mathlib.Tactic.RingNF.nat_rawCast_1) (mul_one (10 ^ (-↑n)))))) (Eq.trans (congrArg (fun x => x + 0) (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow (10 ^ (-1 - ↑n))) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one (10 ^ (-1 - ↑n))))) (Eq.trans (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow ↑↑(@_fvar.51385 n)) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one ↑↑(@_fvar.51385 n)))) Mathlib.Tactic.RingNF.nat_rawCast_1) (mul_one ↑↑(@_fvar.51385 n))))) (add_zero (10 ^ (-1 - ↑n) * ↑↑(@_fvar.51385 n)))))))) ((fun {α β γ} [HAdd α β γ] a a_1 e_a => Eq.rec (motive := fun a_2 e_a => ∀ (a_3 a_4 : β), a_3 = a_4 → a + a_3 = a_2 + a_4) (fun a_2 a_3 e_a => e_a ▸ Eq.refl (a + a_2)) e_a) (↑(@_fvar.10801 n) * 10 ^ (-1 - ↑n) * 10) (↑(@_fvar.10801 n) * 10 ^ (-↑n)) (Eq.mpr (id (congrArg (fun _a => _a = ↑(@_fvar.10801 n) * 10 ^ (-↑n)) (mul_assoc (↑(@_fvar.10801 n)) (10 ^ (-1 - ↑n)) 10))) (Eq.mpr (id (congrArg (fun _a => ↑(@_fvar.10801 n) * _a = ↑(@_fvar.10801 n) * 10 ^ (-↑n)) (Eq.symm (NNReal.rpow_add_one (Mathlib.Meta.NormNum.isNat_eq_false (Mathlib.Meta.NormNum.isNat_ofNat NNReal (Eq.refl 10)) (Mathlib.Meta.NormNum.isNat_ofNat NNReal Nat.cast_zero) (Eq.refl false)) (-1 - ↑n))))) (of_eq_true (Eq.trans (congrArg (fun x => x = ↑(@_fvar.10801 n) * 10 ^ (-↑n)) (Eq.trans (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.atom_pf ↑(@_fvar.10801 n)) (Mathlib.Tactic.Ring.atom_pf' (congrArg (HPow.hPow 10) (Eq.trans (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.atom_pf ↑n) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑n) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_lt (Int.negOfNat 1).rawCast (Mathlib.Tactic.Ring.add_pf_zero_add (↑n ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1)) (Eq.refl (Int.ofNat 0)))) (Mathlib.Tactic.Ring.add_pf_add_zero (↑n ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))) (Eq.trans (congrArg (fun x => x + 0) (Eq.trans (Eq.trans (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow ↑n) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one ↑n))) (Eq.trans Mathlib.Tactic.RingNF.int_rawCast_neg (congrArg Neg.neg Mathlib.Tactic.RingNF.nat_rawCast_1))) (Mathlib.Tactic.RingNF.mul_neg (↑n) 1)) (congrArg Neg.neg (mul_one ↑n)))) (add_zero (-↑n)))))) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_left (↑(@_fvar.10801 n)) (Nat.rawCast 1) (Mathlib.Tactic.Ring.mul_pf_right (10 ^ (-↑n)) (Nat.rawCast 1) (Mathlib.Tactic.Ring.one_mul (Nat.rawCast 1)))) (Mathlib.Tactic.Ring.mul_zero (↑(@_fvar.10801 n) ^ Nat.rawCast 1 * Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑(@_fvar.10801 n) ^ Nat.rawCast 1 * ((10 ^ (-↑n)) ^ Nat.rawCast 1 * Nat.rawCast 1) + 0))) (Mathlib.Tactic.Ring.zero_mul ((10 ^ (-↑n)) ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑(@_fvar.10801 n) ^ Nat.rawCast 1 * ((10 ^ (-↑n)) ^ Nat.rawCast 1 * Nat.rawCast 1) + 0)))) (Eq.trans (congrArg (fun x => x + 0) (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow ↑(@_fvar.10801 n)) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one ↑(@_fvar.10801 n)))) (Eq.trans (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow (10 ^ (-↑n))) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one (10 ^ (-↑n))))) Mathlib.Tactic.RingNF.nat_rawCast_1) (mul_one (10 ^ (-↑n)))))) (add_zero (↑(@_fvar.10801 n) * 10 ^ (-↑n)))))) (eq_self (↑(@_fvar.10801 n) * 10 ^ (-↑n))))))) (10 ^ (-1 - ↑n) * ↑↑(@_fvar.51385 n)) (10 ^ (-1 - ↑n) * ↑↑(@_fvar.51385 n)) (Eq.refl (10 ^ (-1 - ↑n) * ↑↑(@_fvar.51385 n)))))) (AppendixB.NNRealDecimal.surj._proof_6 _fvar.8365 _fvar.51385 _fvar.51390 n hn)) ((fun {α β γ} [HAdd α β γ] a a_1 e_a => Eq.rec (motive := fun a_2 e_a => ∀ (a_3 a_4 : β), a_3 = a_4 → a + a_3 = a_2 + a_4) (fun a_2 a_3 e_a => e_a ▸ Eq.refl (a + a_2)) e_a) (↑(@_fvar.10801 0)) (↑(@_fvar.10801 0)) (Eq.refl ↑(@_fvar.10801 0)) (∑ i ∈ Finset.range n, ↑↑(@_fvar.51385 i) * 10 ^ (-↑i - 1) + ↑↑(@_fvar.51385 n) * 10 ^ (-↑n - 1)) (∑ i ∈ Finset.range (n + 1), ↑↑(@_fvar.51385 i) * 10 ^ (-↑i - 1)) (Eq.symm (Finset.sum_range_succ (fun x => ↑↑(@_fvar.51385 x) * 10 ^ (-↑x - 1)) n))))) nthis:Filter.Tendsto (fun n => ↑(s n) * 10 ^ (-↑n)) Filter.atTop (nhds ↑d)n:ℕ⊢ ↑(s n) * 10 ^ (-↑n) ≤ x * 10 ^ n * 10 ^ (-↑n) gcongrbcx:ℝ≥0h:¬x = 0s:ℕ → ℕ := fun n => ⌊_fvar.8365 * 10 ^ n⌋₊hs:∀ (n : ℕ), ↑(@_fvar.10801 n) ≤ _fvar.8365 * 10 ^ n := fun n => Nat.floor_le (le_of_lt (mul_pos (lt_of_le_of_ne' (zero_le _fvar.8365) _fvar.8442) (pow_pos (Mathlib.Meta.Positivity.pos_of_isNat (Mathlib.Meta.NormNum.isNat_ofNat NNReal (Eq.refl 10)) (Eq.refl (Nat.ble 1 10))) n)))hs':∀ (n : ℕ), _fvar.8365 * 10 ^ n < ↑(@_fvar.10801 n) + 1 := fun n => Nat.lt_floor_add_one (_fvar.8365 * 10 ^ n)a:ℕ → Digitha:∀ (n : ℕ), s (n + 1) = 10 * s n + ↑(a n)d:AppendixB.NNRealDecimal := { intPart := @_fvar.10801 0, fracPart := _fvar.51385 }hsum:∀ (n : ℕ), ↑(@_fvar.10801 n) * 10 ^ (-↑n) = ↑(@_fvar.10801 0) + ∑ i ∈ Finset.range n, ↑↑(@_fvar.51385 i) * 10 ^ (-↑i - 1) := fun n => Nat.recAux (of_eq_true (Eq.trans (congr (congrArg Eq (Eq.trans (congrArg (HMul.hMul ↑(@_fvar.10801 0)) (Eq.trans (congrArg (HPow.hPow 10) (Eq.trans (congrArg Neg.neg (CharP.cast_eq_zero ℝ 0)) neg_zero)) (NNReal.rpow_zero 10))) (mul_one ↑(@_fvar.10801 0)))) (add_zero ↑(@_fvar.10801 0))) (eq_self ↑(@_fvar.10801 0)))) (fun n hn => Eq.mpr (id (congrArg (fun _a => ↑_a * 10 ^ (-↑(n + 1)) = ↑(@_fvar.10801 0) + ∑ i ∈ Finset.range (n + 1), ↑↑(@_fvar.51385 i) * 10 ^ (-↑i - 1)) (@_fvar.51390 n))) (Trans.trans (Trans.trans (Eq.mpr (id (congrArg (fun x => x = ↑(@_fvar.10801 n) * 10 ^ (-↑n) + ↑↑(@_fvar.51385 n) * 10 ^ (-↑n - 1)) (Eq.trans (congr (congrArg HMul.hMul (Eq.trans (Nat.cast_add (10 * @_fvar.10801 n) ↑(@_fvar.51385 n)) (congrArg (fun x => x + ↑↑(@_fvar.51385 n)) (Nat.cast_mul 10 (@_fvar.10801 n))))) (congrArg (HPow.hPow 10) (Eq.trans (congrArg Neg.neg (Eq.trans (Nat.cast_add n 1) (congrArg (HAdd.hAdd ↑n) Nat.cast_one))) (neg_add_rev (↑n) 1)))) (add_mul (10 * ↑(@_fvar.10801 n)) (↑↑(@_fvar.51385 n)) (10 ^ (-1 + -↑n)))))) (Eq.mpr (id (congr (congrArg Eq (Eq.trans (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat NNReal (Eq.refl 10))) (Mathlib.Tactic.Ring.atom_pf ↑(@_fvar.10801 n)) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_right (↑(@_fvar.10801 n)) (Nat.rawCast 1) (Mathlib.Tactic.Ring.mul_one (Nat.rawCast 10))) (Mathlib.Tactic.Ring.mul_zero (Nat.rawCast 10)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑(@_fvar.10801 n) ^ Nat.rawCast 1 * Nat.rawCast 10 + 0))) (Mathlib.Tactic.Ring.zero_mul (↑(@_fvar.10801 n) ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑(@_fvar.10801 n) ^ Nat.rawCast 1 * Nat.rawCast 10 + 0)))) (Mathlib.Tactic.Ring.atom_pf' (congrArg (HPow.hPow 10) (Eq.trans (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.atom_pf ↑n) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑n) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.add_pf_add_lt (Int.negOfNat 1).rawCast (Mathlib.Tactic.Ring.add_pf_zero_add (↑n ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))) (Eq.trans (congr (congrArg HAdd.hAdd (Eq.trans Mathlib.Tactic.RingNF.int_rawCast_neg (congrArg Neg.neg Mathlib.Tactic.RingNF.nat_rawCast_1))) (Eq.trans (congrArg (fun x => x + 0) (Eq.trans (Eq.trans (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow ↑n) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one ↑n))) (Eq.trans Mathlib.Tactic.RingNF.int_rawCast_neg (congrArg Neg.neg Mathlib.Tactic.RingNF.nat_rawCast_1))) (Mathlib.Tactic.RingNF.mul_neg (↑n) 1)) (congrArg Neg.neg (mul_one ↑n)))) (add_zero (-↑n)))) (Mathlib.Tactic.RingNF.add_neg (-1) ↑n))))) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_left (↑(@_fvar.10801 n)) (Nat.rawCast 1) (Mathlib.Tactic.Ring.mul_pf_right (10 ^ (-1 - ↑n)) (Nat.rawCast 1) (Mathlib.Tactic.Ring.mul_one (Nat.rawCast 10)))) (Mathlib.Tactic.Ring.mul_zero (↑(@_fvar.10801 n) ^ Nat.rawCast 1 * Nat.rawCast 10)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑(@_fvar.10801 n) ^ Nat.rawCast 1 * ((10 ^ (-1 - ↑n)) ^ Nat.rawCast 1 * Nat.rawCast 10) + 0))) (Mathlib.Tactic.Ring.zero_mul ((10 ^ (-1 - ↑n)) ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑(@_fvar.10801 n) ^ Nat.rawCast 1 * ((10 ^ (-1 - ↑n)) ^ Nat.rawCast 1 * Nat.rawCast 10) + 0)))) (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.atom_pf ↑↑(@_fvar.51385 n)) (Mathlib.Tactic.Ring.atom_pf' (congrArg (HPow.hPow 10) (Eq.trans (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.atom_pf ↑n) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑n) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.add_pf_add_lt (Int.negOfNat 1).rawCast (Mathlib.Tactic.Ring.add_pf_zero_add (↑n ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))) (Eq.trans (congr (congrArg HAdd.hAdd (Eq.trans Mathlib.Tactic.RingNF.int_rawCast_neg (congrArg Neg.neg Mathlib.Tactic.RingNF.nat_rawCast_1))) (Eq.trans (congrArg (fun x => x + 0) (Eq.trans (Eq.trans (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow ↑n) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one ↑n))) (Eq.trans Mathlib.Tactic.RingNF.int_rawCast_neg (congrArg Neg.neg Mathlib.Tactic.RingNF.nat_rawCast_1))) (Mathlib.Tactic.RingNF.mul_neg (↑n) 1)) (congrArg Neg.neg (mul_one ↑n)))) (add_zero (-↑n)))) (Mathlib.Tactic.RingNF.add_neg (-1) ↑n))))) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_right (10 ^ (-1 - ↑n)) (Nat.rawCast 1) (Mathlib.Tactic.Ring.mul_pf_left (↑↑(@_fvar.51385 n)) (Nat.rawCast 1) (Mathlib.Tactic.Ring.one_mul (Nat.rawCast 1)))) (Mathlib.Tactic.Ring.mul_zero (↑↑(@_fvar.51385 n) ^ Nat.rawCast 1 * Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_add_zero ((10 ^ (-1 - ↑n)) ^ Nat.rawCast 1 * (↑↑(@_fvar.51385 n) ^ Nat.rawCast 1 * Nat.rawCast 1) + 0))) (Mathlib.Tactic.Ring.zero_mul ((10 ^ (-1 - ↑n)) ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero ((10 ^ (-1 - ↑n)) ^ Nat.rawCast 1 * (↑↑(@_fvar.51385 n) ^ Nat.rawCast 1 * Nat.rawCast 1) + 0)))) (Mathlib.Tactic.Ring.add_pf_add_lt (↑(@_fvar.10801 n) ^ Nat.rawCast 1 * ((10 ^ (-1 - ↑n)) ^ Nat.rawCast 1 * Nat.rawCast 10)) (Mathlib.Tactic.Ring.add_pf_zero_add ((10 ^ (-1 - ↑n)) ^ Nat.rawCast 1 * (↑↑(@_fvar.51385 n) ^ Nat.rawCast 1 * Nat.rawCast 1) + 0)))) (congr (congrArg HAdd.hAdd (Eq.trans (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow ↑(@_fvar.10801 n)) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one ↑(@_fvar.10801 n)))) (congrArg (fun x => x * 10) (Eq.trans (congrArg (HPow.hPow (10 ^ (-1 - ↑n))) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one (10 ^ (-1 - ↑n)))))) (Mathlib.Tactic.RingNF.mul_assoc_rev (↑(@_fvar.10801 n)) (10 ^ (-1 - ↑n)) 10))) (Eq.trans (congrArg (fun x => x + 0) (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow (10 ^ (-1 - ↑n))) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one (10 ^ (-1 - ↑n))))) (Eq.trans (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow ↑↑(@_fvar.51385 n)) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one ↑↑(@_fvar.51385 n)))) Mathlib.Tactic.RingNF.nat_rawCast_1) (mul_one ↑↑(@_fvar.51385 n))))) (add_zero (10 ^ (-1 - ↑n) * ↑↑(@_fvar.51385 n))))))) (Eq.trans (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.atom_pf ↑(@_fvar.10801 n)) (Mathlib.Tactic.Ring.atom_pf (10 ^ (-↑n))) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_left (↑(@_fvar.10801 n)) (Nat.rawCast 1) (Mathlib.Tactic.Ring.mul_pf_right (10 ^ (-↑n)) (Nat.rawCast 1) (Mathlib.Tactic.Ring.one_mul (Nat.rawCast 1)))) (Mathlib.Tactic.Ring.mul_zero (↑(@_fvar.10801 n) ^ Nat.rawCast 1 * Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑(@_fvar.10801 n) ^ Nat.rawCast 1 * ((10 ^ (-↑n)) ^ Nat.rawCast 1 * Nat.rawCast 1) + 0))) (Mathlib.Tactic.Ring.zero_mul ((10 ^ (-↑n)) ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑(@_fvar.10801 n) ^ Nat.rawCast 1 * ((10 ^ (-↑n)) ^ Nat.rawCast 1 * Nat.rawCast 1) + 0)))) (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.atom_pf ↑↑(@_fvar.51385 n)) (Mathlib.Tactic.Ring.atom_pf' (congrArg (HPow.hPow 10) (Eq.trans (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.atom_pf ↑n) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑n) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_gt (Int.negOfNat 1).rawCast (Mathlib.Tactic.Ring.add_pf_add_zero (↑n ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))) (Eq.trans (congr (congrArg HAdd.hAdd (Eq.trans Mathlib.Tactic.RingNF.int_rawCast_neg (congrArg Neg.neg Mathlib.Tactic.RingNF.nat_rawCast_1))) (Eq.trans (congrArg (fun x => x + 0) (Eq.trans (Eq.trans (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow ↑n) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one ↑n))) (Eq.trans Mathlib.Tactic.RingNF.int_rawCast_neg (congrArg Neg.neg Mathlib.Tactic.RingNF.nat_rawCast_1))) (Mathlib.Tactic.RingNF.mul_neg (↑n) 1)) (congrArg Neg.neg (mul_one ↑n)))) (add_zero (-↑n)))) (Mathlib.Tactic.RingNF.add_neg (-1) ↑n))))) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_right (10 ^ (-1 - ↑n)) (Nat.rawCast 1) (Mathlib.Tactic.Ring.mul_pf_left (↑↑(@_fvar.51385 n)) (Nat.rawCast 1) (Mathlib.Tactic.Ring.one_mul (Nat.rawCast 1)))) (Mathlib.Tactic.Ring.mul_zero (↑↑(@_fvar.51385 n) ^ Nat.rawCast 1 * Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_add_zero ((10 ^ (-1 - ↑n)) ^ Nat.rawCast 1 * (↑↑(@_fvar.51385 n) ^ Nat.rawCast 1 * Nat.rawCast 1) + 0))) (Mathlib.Tactic.Ring.zero_mul ((10 ^ (-1 - ↑n)) ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero ((10 ^ (-1 - ↑n)) ^ Nat.rawCast 1 * (↑↑(@_fvar.51385 n) ^ Nat.rawCast 1 * Nat.rawCast 1) + 0)))) (Mathlib.Tactic.Ring.add_pf_add_lt (↑(@_fvar.10801 n) ^ Nat.rawCast 1 * ((10 ^ (-↑n)) ^ Nat.rawCast 1 * Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_zero_add ((10 ^ (-1 - ↑n)) ^ Nat.rawCast 1 * (↑↑(@_fvar.51385 n) ^ Nat.rawCast 1 * Nat.rawCast 1) + 0)))) (congr (congrArg HAdd.hAdd (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow ↑(@_fvar.10801 n)) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one ↑(@_fvar.10801 n)))) (Eq.trans (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow (10 ^ (-↑n))) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one (10 ^ (-↑n))))) Mathlib.Tactic.RingNF.nat_rawCast_1) (mul_one (10 ^ (-↑n)))))) (Eq.trans (congrArg (fun x => x + 0) (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow (10 ^ (-1 - ↑n))) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one (10 ^ (-1 - ↑n))))) (Eq.trans (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow ↑↑(@_fvar.51385 n)) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one ↑↑(@_fvar.51385 n)))) Mathlib.Tactic.RingNF.nat_rawCast_1) (mul_one ↑↑(@_fvar.51385 n))))) (add_zero (10 ^ (-1 - ↑n) * ↑↑(@_fvar.51385 n)))))))) ((fun {α β γ} [HAdd α β γ] a a_1 e_a => Eq.rec (motive := fun a_2 e_a => ∀ (a_3 a_4 : β), a_3 = a_4 → a + a_3 = a_2 + a_4) (fun a_2 a_3 e_a => e_a ▸ Eq.refl (a + a_2)) e_a) (↑(@_fvar.10801 n) * 10 ^ (-1 - ↑n) * 10) (↑(@_fvar.10801 n) * 10 ^ (-↑n)) (Eq.mpr (id (congrArg (fun _a => _a = ↑(@_fvar.10801 n) * 10 ^ (-↑n)) (mul_assoc (↑(@_fvar.10801 n)) (10 ^ (-1 - ↑n)) 10))) (Eq.mpr (id (congrArg (fun _a => ↑(@_fvar.10801 n) * _a = ↑(@_fvar.10801 n) * 10 ^ (-↑n)) (Eq.symm (NNReal.rpow_add_one (Mathlib.Meta.NormNum.isNat_eq_false (Mathlib.Meta.NormNum.isNat_ofNat NNReal (Eq.refl 10)) (Mathlib.Meta.NormNum.isNat_ofNat NNReal Nat.cast_zero) (Eq.refl false)) (-1 - ↑n))))) (of_eq_true (Eq.trans (congrArg (fun x => x = ↑(@_fvar.10801 n) * 10 ^ (-↑n)) (Eq.trans (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.atom_pf ↑(@_fvar.10801 n)) (Mathlib.Tactic.Ring.atom_pf' (congrArg (HPow.hPow 10) (Eq.trans (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.atom_pf ↑n) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑n) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_lt (Int.negOfNat 1).rawCast (Mathlib.Tactic.Ring.add_pf_zero_add (↑n ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1)) (Eq.refl (Int.ofNat 0)))) (Mathlib.Tactic.Ring.add_pf_add_zero (↑n ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))) (Eq.trans (congrArg (fun x => x + 0) (Eq.trans (Eq.trans (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow ↑n) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one ↑n))) (Eq.trans Mathlib.Tactic.RingNF.int_rawCast_neg (congrArg Neg.neg Mathlib.Tactic.RingNF.nat_rawCast_1))) (Mathlib.Tactic.RingNF.mul_neg (↑n) 1)) (congrArg Neg.neg (mul_one ↑n)))) (add_zero (-↑n)))))) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_left (↑(@_fvar.10801 n)) (Nat.rawCast 1) (Mathlib.Tactic.Ring.mul_pf_right (10 ^ (-↑n)) (Nat.rawCast 1) (Mathlib.Tactic.Ring.one_mul (Nat.rawCast 1)))) (Mathlib.Tactic.Ring.mul_zero (↑(@_fvar.10801 n) ^ Nat.rawCast 1 * Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑(@_fvar.10801 n) ^ Nat.rawCast 1 * ((10 ^ (-↑n)) ^ Nat.rawCast 1 * Nat.rawCast 1) + 0))) (Mathlib.Tactic.Ring.zero_mul ((10 ^ (-↑n)) ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑(@_fvar.10801 n) ^ Nat.rawCast 1 * ((10 ^ (-↑n)) ^ Nat.rawCast 1 * Nat.rawCast 1) + 0)))) (Eq.trans (congrArg (fun x => x + 0) (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow ↑(@_fvar.10801 n)) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one ↑(@_fvar.10801 n)))) (Eq.trans (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow (10 ^ (-↑n))) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one (10 ^ (-↑n))))) Mathlib.Tactic.RingNF.nat_rawCast_1) (mul_one (10 ^ (-↑n)))))) (add_zero (↑(@_fvar.10801 n) * 10 ^ (-↑n)))))) (eq_self (↑(@_fvar.10801 n) * 10 ^ (-↑n))))))) (10 ^ (-1 - ↑n) * ↑↑(@_fvar.51385 n)) (10 ^ (-1 - ↑n) * ↑↑(@_fvar.51385 n)) (Eq.refl (10 ^ (-1 - ↑n) * ↑↑(@_fvar.51385 n)))))) (AppendixB.NNRealDecimal.surj._proof_6 _fvar.8365 _fvar.51385 _fvar.51390 n hn)) ((fun {α β γ} [HAdd α β γ] a a_1 e_a => Eq.rec (motive := fun a_2 e_a => ∀ (a_3 a_4 : β), a_3 = a_4 → a + a_3 = a_2 + a_4) (fun a_2 a_3 e_a => e_a ▸ Eq.refl (a + a_2)) e_a) (↑(@_fvar.10801 0)) (↑(@_fvar.10801 0)) (Eq.refl ↑(@_fvar.10801 0)) (∑ i ∈ Finset.range n, ↑↑(@_fvar.51385 i) * 10 ^ (-↑i - 1) + ↑↑(@_fvar.51385 n) * 10 ^ (-↑n - 1)) (∑ i ∈ Finset.range (n + 1), ↑↑(@_fvar.51385 i) * 10 ^ (-↑i - 1)) (Eq.symm (Finset.sum_range_succ (fun x => ↑↑(@_fvar.51385 x) * 10 ^ (-↑x - 1)) n))))) nthis:Filter.Tendsto (fun n => ↑(s n) * 10 ^ (-↑n)) Filter.atTop (nhds ↑d)n:ℕ⊢ ↑(s n) ≤ x * 10 ^ n; grindAll goals completed! 🐙 _ = x := byx:ℝ≥0h:¬x = 0s:ℕ → ℕ := fun n => ⌊_fvar.8365 * 10 ^ n⌋₊hs:∀ (n : ℕ), ↑(@_fvar.10801 n) ≤ _fvar.8365 * 10 ^ n := fun n => Nat.floor_le (le_of_lt (mul_pos (lt_of_le_of_ne' (zero_le _fvar.8365) _fvar.8442) (pow_pos (Mathlib.Meta.Positivity.pos_of_isNat (Mathlib.Meta.NormNum.isNat_ofNat NNReal (Eq.refl 10)) (Eq.refl (Nat.ble 1 10))) n)))hs':∀ (n : ℕ), _fvar.8365 * 10 ^ n < ↑(@_fvar.10801 n) + 1 := fun n => Nat.lt_floor_add_one (_fvar.8365 * 10 ^ n)a:ℕ → Digitha:∀ (n : ℕ), s (n + 1) = 10 * s n + ↑(a n)d:AppendixB.NNRealDecimal := { intPart := @_fvar.10801 0, fracPart := _fvar.51385 }hsum:∀ (n : ℕ), ↑(@_fvar.10801 n) * 10 ^ (-↑n) = ↑(@_fvar.10801 0) + ∑ i ∈ Finset.range n, ↑↑(@_fvar.51385 i) * 10 ^ (-↑i - 1) := fun n => Nat.recAux (of_eq_true (Eq.trans (congr (congrArg Eq (Eq.trans (congrArg (HMul.hMul ↑(@_fvar.10801 0)) (Eq.trans (congrArg (HPow.hPow 10) (Eq.trans (congrArg Neg.neg (CharP.cast_eq_zero ℝ 0)) neg_zero)) (NNReal.rpow_zero 10))) (mul_one ↑(@_fvar.10801 0)))) (add_zero ↑(@_fvar.10801 0))) (eq_self ↑(@_fvar.10801 0)))) (fun n hn => Eq.mpr (id (congrArg (fun _a => ↑_a * 10 ^ (-↑(n + 1)) = ↑(@_fvar.10801 0) + ∑ i ∈ Finset.range (n + 1), ↑↑(@_fvar.51385 i) * 10 ^ (-↑i - 1)) (@_fvar.51390 n))) (Trans.trans (Trans.trans (Eq.mpr (id (congrArg (fun x => x = ↑(@_fvar.10801 n) * 10 ^ (-↑n) + ↑↑(@_fvar.51385 n) * 10 ^ (-↑n - 1)) (Eq.trans (congr (congrArg HMul.hMul (Eq.trans (Nat.cast_add (10 * @_fvar.10801 n) ↑(@_fvar.51385 n)) (congrArg (fun x => x + ↑↑(@_fvar.51385 n)) (Nat.cast_mul 10 (@_fvar.10801 n))))) (congrArg (HPow.hPow 10) (Eq.trans (congrArg Neg.neg (Eq.trans (Nat.cast_add n 1) (congrArg (HAdd.hAdd ↑n) Nat.cast_one))) (neg_add_rev (↑n) 1)))) (add_mul (10 * ↑(@_fvar.10801 n)) (↑↑(@_fvar.51385 n)) (10 ^ (-1 + -↑n)))))) (Eq.mpr (id (congr (congrArg Eq (Eq.trans (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat NNReal (Eq.refl 10))) (Mathlib.Tactic.Ring.atom_pf ↑(@_fvar.10801 n)) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_right (↑(@_fvar.10801 n)) (Nat.rawCast 1) (Mathlib.Tactic.Ring.mul_one (Nat.rawCast 10))) (Mathlib.Tactic.Ring.mul_zero (Nat.rawCast 10)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑(@_fvar.10801 n) ^ Nat.rawCast 1 * Nat.rawCast 10 + 0))) (Mathlib.Tactic.Ring.zero_mul (↑(@_fvar.10801 n) ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑(@_fvar.10801 n) ^ Nat.rawCast 1 * Nat.rawCast 10 + 0)))) (Mathlib.Tactic.Ring.atom_pf' (congrArg (HPow.hPow 10) (Eq.trans (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.atom_pf ↑n) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑n) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.add_pf_add_lt (Int.negOfNat 1).rawCast (Mathlib.Tactic.Ring.add_pf_zero_add (↑n ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))) (Eq.trans (congr (congrArg HAdd.hAdd (Eq.trans Mathlib.Tactic.RingNF.int_rawCast_neg (congrArg Neg.neg Mathlib.Tactic.RingNF.nat_rawCast_1))) (Eq.trans (congrArg (fun x => x + 0) (Eq.trans (Eq.trans (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow ↑n) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one ↑n))) (Eq.trans Mathlib.Tactic.RingNF.int_rawCast_neg (congrArg Neg.neg Mathlib.Tactic.RingNF.nat_rawCast_1))) (Mathlib.Tactic.RingNF.mul_neg (↑n) 1)) (congrArg Neg.neg (mul_one ↑n)))) (add_zero (-↑n)))) (Mathlib.Tactic.RingNF.add_neg (-1) ↑n))))) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_left (↑(@_fvar.10801 n)) (Nat.rawCast 1) (Mathlib.Tactic.Ring.mul_pf_right (10 ^ (-1 - ↑n)) (Nat.rawCast 1) (Mathlib.Tactic.Ring.mul_one (Nat.rawCast 10)))) (Mathlib.Tactic.Ring.mul_zero (↑(@_fvar.10801 n) ^ Nat.rawCast 1 * Nat.rawCast 10)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑(@_fvar.10801 n) ^ Nat.rawCast 1 * ((10 ^ (-1 - ↑n)) ^ Nat.rawCast 1 * Nat.rawCast 10) + 0))) (Mathlib.Tactic.Ring.zero_mul ((10 ^ (-1 - ↑n)) ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑(@_fvar.10801 n) ^ Nat.rawCast 1 * ((10 ^ (-1 - ↑n)) ^ Nat.rawCast 1 * Nat.rawCast 10) + 0)))) (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.atom_pf ↑↑(@_fvar.51385 n)) (Mathlib.Tactic.Ring.atom_pf' (congrArg (HPow.hPow 10) (Eq.trans (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.atom_pf ↑n) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑n) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.add_pf_add_lt (Int.negOfNat 1).rawCast (Mathlib.Tactic.Ring.add_pf_zero_add (↑n ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))) (Eq.trans (congr (congrArg HAdd.hAdd (Eq.trans Mathlib.Tactic.RingNF.int_rawCast_neg (congrArg Neg.neg Mathlib.Tactic.RingNF.nat_rawCast_1))) (Eq.trans (congrArg (fun x => x + 0) (Eq.trans (Eq.trans (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow ↑n) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one ↑n))) (Eq.trans Mathlib.Tactic.RingNF.int_rawCast_neg (congrArg Neg.neg Mathlib.Tactic.RingNF.nat_rawCast_1))) (Mathlib.Tactic.RingNF.mul_neg (↑n) 1)) (congrArg Neg.neg (mul_one ↑n)))) (add_zero (-↑n)))) (Mathlib.Tactic.RingNF.add_neg (-1) ↑n))))) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_right (10 ^ (-1 - ↑n)) (Nat.rawCast 1) (Mathlib.Tactic.Ring.mul_pf_left (↑↑(@_fvar.51385 n)) (Nat.rawCast 1) (Mathlib.Tactic.Ring.one_mul (Nat.rawCast 1)))) (Mathlib.Tactic.Ring.mul_zero (↑↑(@_fvar.51385 n) ^ Nat.rawCast 1 * Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_add_zero ((10 ^ (-1 - ↑n)) ^ Nat.rawCast 1 * (↑↑(@_fvar.51385 n) ^ Nat.rawCast 1 * Nat.rawCast 1) + 0))) (Mathlib.Tactic.Ring.zero_mul ((10 ^ (-1 - ↑n)) ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero ((10 ^ (-1 - ↑n)) ^ Nat.rawCast 1 * (↑↑(@_fvar.51385 n) ^ Nat.rawCast 1 * Nat.rawCast 1) + 0)))) (Mathlib.Tactic.Ring.add_pf_add_lt (↑(@_fvar.10801 n) ^ Nat.rawCast 1 * ((10 ^ (-1 - ↑n)) ^ Nat.rawCast 1 * Nat.rawCast 10)) (Mathlib.Tactic.Ring.add_pf_zero_add ((10 ^ (-1 - ↑n)) ^ Nat.rawCast 1 * (↑↑(@_fvar.51385 n) ^ Nat.rawCast 1 * Nat.rawCast 1) + 0)))) (congr (congrArg HAdd.hAdd (Eq.trans (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow ↑(@_fvar.10801 n)) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one ↑(@_fvar.10801 n)))) (congrArg (fun x => x * 10) (Eq.trans (congrArg (HPow.hPow (10 ^ (-1 - ↑n))) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one (10 ^ (-1 - ↑n)))))) (Mathlib.Tactic.RingNF.mul_assoc_rev (↑(@_fvar.10801 n)) (10 ^ (-1 - ↑n)) 10))) (Eq.trans (congrArg (fun x => x + 0) (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow (10 ^ (-1 - ↑n))) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one (10 ^ (-1 - ↑n))))) (Eq.trans (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow ↑↑(@_fvar.51385 n)) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one ↑↑(@_fvar.51385 n)))) Mathlib.Tactic.RingNF.nat_rawCast_1) (mul_one ↑↑(@_fvar.51385 n))))) (add_zero (10 ^ (-1 - ↑n) * ↑↑(@_fvar.51385 n))))))) (Eq.trans (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.atom_pf ↑(@_fvar.10801 n)) (Mathlib.Tactic.Ring.atom_pf (10 ^ (-↑n))) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_left (↑(@_fvar.10801 n)) (Nat.rawCast 1) (Mathlib.Tactic.Ring.mul_pf_right (10 ^ (-↑n)) (Nat.rawCast 1) (Mathlib.Tactic.Ring.one_mul (Nat.rawCast 1)))) (Mathlib.Tactic.Ring.mul_zero (↑(@_fvar.10801 n) ^ Nat.rawCast 1 * Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑(@_fvar.10801 n) ^ Nat.rawCast 1 * ((10 ^ (-↑n)) ^ Nat.rawCast 1 * Nat.rawCast 1) + 0))) (Mathlib.Tactic.Ring.zero_mul ((10 ^ (-↑n)) ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑(@_fvar.10801 n) ^ Nat.rawCast 1 * ((10 ^ (-↑n)) ^ Nat.rawCast 1 * Nat.rawCast 1) + 0)))) (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.atom_pf ↑↑(@_fvar.51385 n)) (Mathlib.Tactic.Ring.atom_pf' (congrArg (HPow.hPow 10) (Eq.trans (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.atom_pf ↑n) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑n) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_gt (Int.negOfNat 1).rawCast (Mathlib.Tactic.Ring.add_pf_add_zero (↑n ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))) (Eq.trans (congr (congrArg HAdd.hAdd (Eq.trans Mathlib.Tactic.RingNF.int_rawCast_neg (congrArg Neg.neg Mathlib.Tactic.RingNF.nat_rawCast_1))) (Eq.trans (congrArg (fun x => x + 0) (Eq.trans (Eq.trans (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow ↑n) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one ↑n))) (Eq.trans Mathlib.Tactic.RingNF.int_rawCast_neg (congrArg Neg.neg Mathlib.Tactic.RingNF.nat_rawCast_1))) (Mathlib.Tactic.RingNF.mul_neg (↑n) 1)) (congrArg Neg.neg (mul_one ↑n)))) (add_zero (-↑n)))) (Mathlib.Tactic.RingNF.add_neg (-1) ↑n))))) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_right (10 ^ (-1 - ↑n)) (Nat.rawCast 1) (Mathlib.Tactic.Ring.mul_pf_left (↑↑(@_fvar.51385 n)) (Nat.rawCast 1) (Mathlib.Tactic.Ring.one_mul (Nat.rawCast 1)))) (Mathlib.Tactic.Ring.mul_zero (↑↑(@_fvar.51385 n) ^ Nat.rawCast 1 * Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_add_zero ((10 ^ (-1 - ↑n)) ^ Nat.rawCast 1 * (↑↑(@_fvar.51385 n) ^ Nat.rawCast 1 * Nat.rawCast 1) + 0))) (Mathlib.Tactic.Ring.zero_mul ((10 ^ (-1 - ↑n)) ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero ((10 ^ (-1 - ↑n)) ^ Nat.rawCast 1 * (↑↑(@_fvar.51385 n) ^ Nat.rawCast 1 * Nat.rawCast 1) + 0)))) (Mathlib.Tactic.Ring.add_pf_add_lt (↑(@_fvar.10801 n) ^ Nat.rawCast 1 * ((10 ^ (-↑n)) ^ Nat.rawCast 1 * Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_zero_add ((10 ^ (-1 - ↑n)) ^ Nat.rawCast 1 * (↑↑(@_fvar.51385 n) ^ Nat.rawCast 1 * Nat.rawCast 1) + 0)))) (congr (congrArg HAdd.hAdd (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow ↑(@_fvar.10801 n)) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one ↑(@_fvar.10801 n)))) (Eq.trans (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow (10 ^ (-↑n))) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one (10 ^ (-↑n))))) Mathlib.Tactic.RingNF.nat_rawCast_1) (mul_one (10 ^ (-↑n)))))) (Eq.trans (congrArg (fun x => x + 0) (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow (10 ^ (-1 - ↑n))) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one (10 ^ (-1 - ↑n))))) (Eq.trans (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow ↑↑(@_fvar.51385 n)) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one ↑↑(@_fvar.51385 n)))) Mathlib.Tactic.RingNF.nat_rawCast_1) (mul_one ↑↑(@_fvar.51385 n))))) (add_zero (10 ^ (-1 - ↑n) * ↑↑(@_fvar.51385 n)))))))) ((fun {α β γ} [HAdd α β γ] a a_1 e_a => Eq.rec (motive := fun a_2 e_a => ∀ (a_3 a_4 : β), a_3 = a_4 → a + a_3 = a_2 + a_4) (fun a_2 a_3 e_a => e_a ▸ Eq.refl (a + a_2)) e_a) (↑(@_fvar.10801 n) * 10 ^ (-1 - ↑n) * 10) (↑(@_fvar.10801 n) * 10 ^ (-↑n)) (Eq.mpr (id (congrArg (fun _a => _a = ↑(@_fvar.10801 n) * 10 ^ (-↑n)) (mul_assoc (↑(@_fvar.10801 n)) (10 ^ (-1 - ↑n)) 10))) (Eq.mpr (id (congrArg (fun _a => ↑(@_fvar.10801 n) * _a = ↑(@_fvar.10801 n) * 10 ^ (-↑n)) (Eq.symm (NNReal.rpow_add_one (Mathlib.Meta.NormNum.isNat_eq_false (Mathlib.Meta.NormNum.isNat_ofNat NNReal (Eq.refl 10)) (Mathlib.Meta.NormNum.isNat_ofNat NNReal Nat.cast_zero) (Eq.refl false)) (-1 - ↑n))))) (of_eq_true (Eq.trans (congrArg (fun x => x = ↑(@_fvar.10801 n) * 10 ^ (-↑n)) (Eq.trans (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.atom_pf ↑(@_fvar.10801 n)) (Mathlib.Tactic.Ring.atom_pf' (congrArg (HPow.hPow 10) (Eq.trans (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.atom_pf ↑n) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑n) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_lt (Int.negOfNat 1).rawCast (Mathlib.Tactic.Ring.add_pf_zero_add (↑n ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1)) (Eq.refl (Int.ofNat 0)))) (Mathlib.Tactic.Ring.add_pf_add_zero (↑n ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))) (Eq.trans (congrArg (fun x => x + 0) (Eq.trans (Eq.trans (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow ↑n) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one ↑n))) (Eq.trans Mathlib.Tactic.RingNF.int_rawCast_neg (congrArg Neg.neg Mathlib.Tactic.RingNF.nat_rawCast_1))) (Mathlib.Tactic.RingNF.mul_neg (↑n) 1)) (congrArg Neg.neg (mul_one ↑n)))) (add_zero (-↑n)))))) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_left (↑(@_fvar.10801 n)) (Nat.rawCast 1) (Mathlib.Tactic.Ring.mul_pf_right (10 ^ (-↑n)) (Nat.rawCast 1) (Mathlib.Tactic.Ring.one_mul (Nat.rawCast 1)))) (Mathlib.Tactic.Ring.mul_zero (↑(@_fvar.10801 n) ^ Nat.rawCast 1 * Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑(@_fvar.10801 n) ^ Nat.rawCast 1 * ((10 ^ (-↑n)) ^ Nat.rawCast 1 * Nat.rawCast 1) + 0))) (Mathlib.Tactic.Ring.zero_mul ((10 ^ (-↑n)) ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑(@_fvar.10801 n) ^ Nat.rawCast 1 * ((10 ^ (-↑n)) ^ Nat.rawCast 1 * Nat.rawCast 1) + 0)))) (Eq.trans (congrArg (fun x => x + 0) (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow ↑(@_fvar.10801 n)) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one ↑(@_fvar.10801 n)))) (Eq.trans (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow (10 ^ (-↑n))) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one (10 ^ (-↑n))))) Mathlib.Tactic.RingNF.nat_rawCast_1) (mul_one (10 ^ (-↑n)))))) (add_zero (↑(@_fvar.10801 n) * 10 ^ (-↑n)))))) (eq_self (↑(@_fvar.10801 n) * 10 ^ (-↑n))))))) (10 ^ (-1 - ↑n) * ↑↑(@_fvar.51385 n)) (10 ^ (-1 - ↑n) * ↑↑(@_fvar.51385 n)) (Eq.refl (10 ^ (-1 - ↑n) * ↑↑(@_fvar.51385 n)))))) (AppendixB.NNRealDecimal.surj._proof_6 _fvar.8365 _fvar.51385 _fvar.51390 n hn)) ((fun {α β γ} [HAdd α β γ] a a_1 e_a => Eq.rec (motive := fun a_2 e_a => ∀ (a_3 a_4 : β), a_3 = a_4 → a + a_3 = a_2 + a_4) (fun a_2 a_3 e_a => e_a ▸ Eq.refl (a + a_2)) e_a) (↑(@_fvar.10801 0)) (↑(@_fvar.10801 0)) (Eq.refl ↑(@_fvar.10801 0)) (∑ i ∈ Finset.range n, ↑↑(@_fvar.51385 i) * 10 ^ (-↑i - 1) + ↑↑(@_fvar.51385 n) * 10 ^ (-↑n - 1)) (∑ i ∈ Finset.range (n + 1), ↑↑(@_fvar.51385 i) * 10 ^ (-↑i - 1)) (Eq.symm (Finset.sum_range_succ (fun x => ↑↑(@_fvar.51385 x) * 10 ^ (-↑x - 1)) n))))) nthis:Filter.Tendsto (fun n => ↑(s n) * 10 ^ (-↑n)) Filter.atTop (nhds ↑d)n:ℕ⊢ x * 10 ^ n * 10 ^ (-↑n) = x rw [mul_assoc, ←rpow_natCast, ←rpow_add]x:ℝ≥0h:¬x = 0s:ℕ → ℕ := fun n => ⌊_fvar.8365 * 10 ^ n⌋₊hs:∀ (n : ℕ), ↑(@_fvar.10801 n) ≤ _fvar.8365 * 10 ^ n := fun n => Nat.floor_le (le_of_lt (mul_pos (lt_of_le_of_ne' (zero_le _fvar.8365) _fvar.8442) (pow_pos (Mathlib.Meta.Positivity.pos_of_isNat (Mathlib.Meta.NormNum.isNat_ofNat NNReal (Eq.refl 10)) (Eq.refl (Nat.ble 1 10))) n)))hs':∀ (n : ℕ), _fvar.8365 * 10 ^ n < ↑(@_fvar.10801 n) + 1 := fun n => Nat.lt_floor_add_one (_fvar.8365 * 10 ^ n)a:ℕ → Digitha:∀ (n : ℕ), s (n + 1) = 10 * s n + ↑(a n)d:AppendixB.NNRealDecimal := { intPart := @_fvar.10801 0, fracPart := _fvar.51385 }hsum:∀ (n : ℕ), ↑(@_fvar.10801 n) * 10 ^ (-↑n) = ↑(@_fvar.10801 0) + ∑ i ∈ Finset.range n, ↑↑(@_fvar.51385 i) * 10 ^ (-↑i - 1) := fun n => Nat.recAux (of_eq_true (Eq.trans (congr (congrArg Eq (Eq.trans (congrArg (HMul.hMul ↑(@_fvar.10801 0)) (Eq.trans (congrArg (HPow.hPow 10) (Eq.trans (congrArg Neg.neg (CharP.cast_eq_zero ℝ 0)) neg_zero)) (NNReal.rpow_zero 10))) (mul_one ↑(@_fvar.10801 0)))) (add_zero ↑(@_fvar.10801 0))) (eq_self ↑(@_fvar.10801 0)))) (fun n hn => Eq.mpr (id (congrArg (fun _a => ↑_a * 10 ^ (-↑(n + 1)) = ↑(@_fvar.10801 0) + ∑ i ∈ Finset.range (n + 1), ↑↑(@_fvar.51385 i) * 10 ^ (-↑i - 1)) (@_fvar.51390 n))) (Trans.trans (Trans.trans (Eq.mpr (id (congrArg (fun x => x = ↑(@_fvar.10801 n) * 10 ^ (-↑n) + ↑↑(@_fvar.51385 n) * 10 ^ (-↑n - 1)) (Eq.trans (congr (congrArg HMul.hMul (Eq.trans (Nat.cast_add (10 * @_fvar.10801 n) ↑(@_fvar.51385 n)) (congrArg (fun x => x + ↑↑(@_fvar.51385 n)) (Nat.cast_mul 10 (@_fvar.10801 n))))) (congrArg (HPow.hPow 10) (Eq.trans (congrArg Neg.neg (Eq.trans (Nat.cast_add n 1) (congrArg (HAdd.hAdd ↑n) Nat.cast_one))) (neg_add_rev (↑n) 1)))) (add_mul (10 * ↑(@_fvar.10801 n)) (↑↑(@_fvar.51385 n)) (10 ^ (-1 + -↑n)))))) (Eq.mpr (id (congr (congrArg Eq (Eq.trans (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat NNReal (Eq.refl 10))) (Mathlib.Tactic.Ring.atom_pf ↑(@_fvar.10801 n)) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_right (↑(@_fvar.10801 n)) (Nat.rawCast 1) (Mathlib.Tactic.Ring.mul_one (Nat.rawCast 10))) (Mathlib.Tactic.Ring.mul_zero (Nat.rawCast 10)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑(@_fvar.10801 n) ^ Nat.rawCast 1 * Nat.rawCast 10 + 0))) (Mathlib.Tactic.Ring.zero_mul (↑(@_fvar.10801 n) ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑(@_fvar.10801 n) ^ Nat.rawCast 1 * Nat.rawCast 10 + 0)))) (Mathlib.Tactic.Ring.atom_pf' (congrArg (HPow.hPow 10) (Eq.trans (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.atom_pf ↑n) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑n) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.add_pf_add_lt (Int.negOfNat 1).rawCast (Mathlib.Tactic.Ring.add_pf_zero_add (↑n ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))) (Eq.trans (congr (congrArg HAdd.hAdd (Eq.trans Mathlib.Tactic.RingNF.int_rawCast_neg (congrArg Neg.neg Mathlib.Tactic.RingNF.nat_rawCast_1))) (Eq.trans (congrArg (fun x => x + 0) (Eq.trans (Eq.trans (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow ↑n) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one ↑n))) (Eq.trans Mathlib.Tactic.RingNF.int_rawCast_neg (congrArg Neg.neg Mathlib.Tactic.RingNF.nat_rawCast_1))) (Mathlib.Tactic.RingNF.mul_neg (↑n) 1)) (congrArg Neg.neg (mul_one ↑n)))) (add_zero (-↑n)))) (Mathlib.Tactic.RingNF.add_neg (-1) ↑n))))) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_left (↑(@_fvar.10801 n)) (Nat.rawCast 1) (Mathlib.Tactic.Ring.mul_pf_right (10 ^ (-1 - ↑n)) (Nat.rawCast 1) (Mathlib.Tactic.Ring.mul_one (Nat.rawCast 10)))) (Mathlib.Tactic.Ring.mul_zero (↑(@_fvar.10801 n) ^ Nat.rawCast 1 * Nat.rawCast 10)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑(@_fvar.10801 n) ^ Nat.rawCast 1 * ((10 ^ (-1 - ↑n)) ^ Nat.rawCast 1 * Nat.rawCast 10) + 0))) (Mathlib.Tactic.Ring.zero_mul ((10 ^ (-1 - ↑n)) ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑(@_fvar.10801 n) ^ Nat.rawCast 1 * ((10 ^ (-1 - ↑n)) ^ Nat.rawCast 1 * Nat.rawCast 10) + 0)))) (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.atom_pf ↑↑(@_fvar.51385 n)) (Mathlib.Tactic.Ring.atom_pf' (congrArg (HPow.hPow 10) (Eq.trans (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.atom_pf ↑n) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑n) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.add_pf_add_lt (Int.negOfNat 1).rawCast (Mathlib.Tactic.Ring.add_pf_zero_add (↑n ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))) (Eq.trans (congr (congrArg HAdd.hAdd (Eq.trans Mathlib.Tactic.RingNF.int_rawCast_neg (congrArg Neg.neg Mathlib.Tactic.RingNF.nat_rawCast_1))) (Eq.trans (congrArg (fun x => x + 0) (Eq.trans (Eq.trans (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow ↑n) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one ↑n))) (Eq.trans Mathlib.Tactic.RingNF.int_rawCast_neg (congrArg Neg.neg Mathlib.Tactic.RingNF.nat_rawCast_1))) (Mathlib.Tactic.RingNF.mul_neg (↑n) 1)) (congrArg Neg.neg (mul_one ↑n)))) (add_zero (-↑n)))) (Mathlib.Tactic.RingNF.add_neg (-1) ↑n))))) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_right (10 ^ (-1 - ↑n)) (Nat.rawCast 1) (Mathlib.Tactic.Ring.mul_pf_left (↑↑(@_fvar.51385 n)) (Nat.rawCast 1) (Mathlib.Tactic.Ring.one_mul (Nat.rawCast 1)))) (Mathlib.Tactic.Ring.mul_zero (↑↑(@_fvar.51385 n) ^ Nat.rawCast 1 * Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_add_zero ((10 ^ (-1 - ↑n)) ^ Nat.rawCast 1 * (↑↑(@_fvar.51385 n) ^ Nat.rawCast 1 * Nat.rawCast 1) + 0))) (Mathlib.Tactic.Ring.zero_mul ((10 ^ (-1 - ↑n)) ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero ((10 ^ (-1 - ↑n)) ^ Nat.rawCast 1 * (↑↑(@_fvar.51385 n) ^ Nat.rawCast 1 * Nat.rawCast 1) + 0)))) (Mathlib.Tactic.Ring.add_pf_add_lt (↑(@_fvar.10801 n) ^ Nat.rawCast 1 * ((10 ^ (-1 - ↑n)) ^ Nat.rawCast 1 * Nat.rawCast 10)) (Mathlib.Tactic.Ring.add_pf_zero_add ((10 ^ (-1 - ↑n)) ^ Nat.rawCast 1 * (↑↑(@_fvar.51385 n) ^ Nat.rawCast 1 * Nat.rawCast 1) + 0)))) (congr (congrArg HAdd.hAdd (Eq.trans (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow ↑(@_fvar.10801 n)) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one ↑(@_fvar.10801 n)))) (congrArg (fun x => x * 10) (Eq.trans (congrArg (HPow.hPow (10 ^ (-1 - ↑n))) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one (10 ^ (-1 - ↑n)))))) (Mathlib.Tactic.RingNF.mul_assoc_rev (↑(@_fvar.10801 n)) (10 ^ (-1 - ↑n)) 10))) (Eq.trans (congrArg (fun x => x + 0) (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow (10 ^ (-1 - ↑n))) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one (10 ^ (-1 - ↑n))))) (Eq.trans (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow ↑↑(@_fvar.51385 n)) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one ↑↑(@_fvar.51385 n)))) Mathlib.Tactic.RingNF.nat_rawCast_1) (mul_one ↑↑(@_fvar.51385 n))))) (add_zero (10 ^ (-1 - ↑n) * ↑↑(@_fvar.51385 n))))))) (Eq.trans (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.atom_pf ↑(@_fvar.10801 n)) (Mathlib.Tactic.Ring.atom_pf (10 ^ (-↑n))) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_left (↑(@_fvar.10801 n)) (Nat.rawCast 1) (Mathlib.Tactic.Ring.mul_pf_right (10 ^ (-↑n)) (Nat.rawCast 1) (Mathlib.Tactic.Ring.one_mul (Nat.rawCast 1)))) (Mathlib.Tactic.Ring.mul_zero (↑(@_fvar.10801 n) ^ Nat.rawCast 1 * Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑(@_fvar.10801 n) ^ Nat.rawCast 1 * ((10 ^ (-↑n)) ^ Nat.rawCast 1 * Nat.rawCast 1) + 0))) (Mathlib.Tactic.Ring.zero_mul ((10 ^ (-↑n)) ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑(@_fvar.10801 n) ^ Nat.rawCast 1 * ((10 ^ (-↑n)) ^ Nat.rawCast 1 * Nat.rawCast 1) + 0)))) (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.atom_pf ↑↑(@_fvar.51385 n)) (Mathlib.Tactic.Ring.atom_pf' (congrArg (HPow.hPow 10) (Eq.trans (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.atom_pf ↑n) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑n) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_gt (Int.negOfNat 1).rawCast (Mathlib.Tactic.Ring.add_pf_add_zero (↑n ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))) (Eq.trans (congr (congrArg HAdd.hAdd (Eq.trans Mathlib.Tactic.RingNF.int_rawCast_neg (congrArg Neg.neg Mathlib.Tactic.RingNF.nat_rawCast_1))) (Eq.trans (congrArg (fun x => x + 0) (Eq.trans (Eq.trans (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow ↑n) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one ↑n))) (Eq.trans Mathlib.Tactic.RingNF.int_rawCast_neg (congrArg Neg.neg Mathlib.Tactic.RingNF.nat_rawCast_1))) (Mathlib.Tactic.RingNF.mul_neg (↑n) 1)) (congrArg Neg.neg (mul_one ↑n)))) (add_zero (-↑n)))) (Mathlib.Tactic.RingNF.add_neg (-1) ↑n))))) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_right (10 ^ (-1 - ↑n)) (Nat.rawCast 1) (Mathlib.Tactic.Ring.mul_pf_left (↑↑(@_fvar.51385 n)) (Nat.rawCast 1) (Mathlib.Tactic.Ring.one_mul (Nat.rawCast 1)))) (Mathlib.Tactic.Ring.mul_zero (↑↑(@_fvar.51385 n) ^ Nat.rawCast 1 * Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_add_zero ((10 ^ (-1 - ↑n)) ^ Nat.rawCast 1 * (↑↑(@_fvar.51385 n) ^ Nat.rawCast 1 * Nat.rawCast 1) + 0))) (Mathlib.Tactic.Ring.zero_mul ((10 ^ (-1 - ↑n)) ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero ((10 ^ (-1 - ↑n)) ^ Nat.rawCast 1 * (↑↑(@_fvar.51385 n) ^ Nat.rawCast 1 * Nat.rawCast 1) + 0)))) (Mathlib.Tactic.Ring.add_pf_add_lt (↑(@_fvar.10801 n) ^ Nat.rawCast 1 * ((10 ^ (-↑n)) ^ Nat.rawCast 1 * Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_zero_add ((10 ^ (-1 - ↑n)) ^ Nat.rawCast 1 * (↑↑(@_fvar.51385 n) ^ Nat.rawCast 1 * Nat.rawCast 1) + 0)))) (congr (congrArg HAdd.hAdd (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow ↑(@_fvar.10801 n)) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one ↑(@_fvar.10801 n)))) (Eq.trans (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow (10 ^ (-↑n))) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one (10 ^ (-↑n))))) Mathlib.Tactic.RingNF.nat_rawCast_1) (mul_one (10 ^ (-↑n)))))) (Eq.trans (congrArg (fun x => x + 0) (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow (10 ^ (-1 - ↑n))) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one (10 ^ (-1 - ↑n))))) (Eq.trans (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow ↑↑(@_fvar.51385 n)) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one ↑↑(@_fvar.51385 n)))) Mathlib.Tactic.RingNF.nat_rawCast_1) (mul_one ↑↑(@_fvar.51385 n))))) (add_zero (10 ^ (-1 - ↑n) * ↑↑(@_fvar.51385 n)))))))) ((fun {α β γ} [HAdd α β γ] a a_1 e_a => Eq.rec (motive := fun a_2 e_a => ∀ (a_3 a_4 : β), a_3 = a_4 → a + a_3 = a_2 + a_4) (fun a_2 a_3 e_a => e_a ▸ Eq.refl (a + a_2)) e_a) (↑(@_fvar.10801 n) * 10 ^ (-1 - ↑n) * 10) (↑(@_fvar.10801 n) * 10 ^ (-↑n)) (Eq.mpr (id (congrArg (fun _a => _a = ↑(@_fvar.10801 n) * 10 ^ (-↑n)) (mul_assoc (↑(@_fvar.10801 n)) (10 ^ (-1 - ↑n)) 10))) (Eq.mpr (id (congrArg (fun _a => ↑(@_fvar.10801 n) * _a = ↑(@_fvar.10801 n) * 10 ^ (-↑n)) (Eq.symm (NNReal.rpow_add_one (Mathlib.Meta.NormNum.isNat_eq_false (Mathlib.Meta.NormNum.isNat_ofNat NNReal (Eq.refl 10)) (Mathlib.Meta.NormNum.isNat_ofNat NNReal Nat.cast_zero) (Eq.refl false)) (-1 - ↑n))))) (of_eq_true (Eq.trans (congrArg (fun x => x = ↑(@_fvar.10801 n) * 10 ^ (-↑n)) (Eq.trans (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.atom_pf ↑(@_fvar.10801 n)) (Mathlib.Tactic.Ring.atom_pf' (congrArg (HPow.hPow 10) (Eq.trans (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.atom_pf ↑n) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑n) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_lt (Int.negOfNat 1).rawCast (Mathlib.Tactic.Ring.add_pf_zero_add (↑n ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1)) (Eq.refl (Int.ofNat 0)))) (Mathlib.Tactic.Ring.add_pf_add_zero (↑n ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))) (Eq.trans (congrArg (fun x => x + 0) (Eq.trans (Eq.trans (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow ↑n) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one ↑n))) (Eq.trans Mathlib.Tactic.RingNF.int_rawCast_neg (congrArg Neg.neg Mathlib.Tactic.RingNF.nat_rawCast_1))) (Mathlib.Tactic.RingNF.mul_neg (↑n) 1)) (congrArg Neg.neg (mul_one ↑n)))) (add_zero (-↑n)))))) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_left (↑(@_fvar.10801 n)) (Nat.rawCast 1) (Mathlib.Tactic.Ring.mul_pf_right (10 ^ (-↑n)) (Nat.rawCast 1) (Mathlib.Tactic.Ring.one_mul (Nat.rawCast 1)))) (Mathlib.Tactic.Ring.mul_zero (↑(@_fvar.10801 n) ^ Nat.rawCast 1 * Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑(@_fvar.10801 n) ^ Nat.rawCast 1 * ((10 ^ (-↑n)) ^ Nat.rawCast 1 * Nat.rawCast 1) + 0))) (Mathlib.Tactic.Ring.zero_mul ((10 ^ (-↑n)) ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑(@_fvar.10801 n) ^ Nat.rawCast 1 * ((10 ^ (-↑n)) ^ Nat.rawCast 1 * Nat.rawCast 1) + 0)))) (Eq.trans (congrArg (fun x => x + 0) (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow ↑(@_fvar.10801 n)) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one ↑(@_fvar.10801 n)))) (Eq.trans (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow (10 ^ (-↑n))) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one (10 ^ (-↑n))))) Mathlib.Tactic.RingNF.nat_rawCast_1) (mul_one (10 ^ (-↑n)))))) (add_zero (↑(@_fvar.10801 n) * 10 ^ (-↑n)))))) (eq_self (↑(@_fvar.10801 n) * 10 ^ (-↑n))))))) (10 ^ (-1 - ↑n) * ↑↑(@_fvar.51385 n)) (10 ^ (-1 - ↑n) * ↑↑(@_fvar.51385 n)) (Eq.refl (10 ^ (-1 - ↑n) * ↑↑(@_fvar.51385 n)))))) (AppendixB.NNRealDecimal.surj._proof_6 _fvar.8365 _fvar.51385 _fvar.51390 n hn)) ((fun {α β γ} [HAdd α β γ] a a_1 e_a => Eq.rec (motive := fun a_2 e_a => ∀ (a_3 a_4 : β), a_3 = a_4 → a + a_3 = a_2 + a_4) (fun a_2 a_3 e_a => e_a ▸ Eq.refl (a + a_2)) e_a) (↑(@_fvar.10801 0)) (↑(@_fvar.10801 0)) (Eq.refl ↑(@_fvar.10801 0)) (∑ i ∈ Finset.range n, ↑↑(@_fvar.51385 i) * 10 ^ (-↑i - 1) + ↑↑(@_fvar.51385 n) * 10 ^ (-↑n - 1)) (∑ i ∈ Finset.range (n + 1), ↑↑(@_fvar.51385 i) * 10 ^ (-↑i - 1)) (Eq.symm (Finset.sum_range_succ (fun x => ↑↑(@_fvar.51385 x) * 10 ^ (-↑x - 1)) n))))) nthis:Filter.Tendsto (fun n => ↑(s n) * 10 ^ (-↑n)) Filter.atTop (nhds ↑d)n:ℕ⊢ x * 10 ^ (↑n + -↑n) = xhxx:ℝ≥0h:¬x = 0s:ℕ → ℕ := fun n => ⌊_fvar.8365 * 10 ^ n⌋₊hs:∀ (n : ℕ), ↑(@_fvar.10801 n) ≤ _fvar.8365 * 10 ^ n := fun n => Nat.floor_le (le_of_lt (mul_pos (lt_of_le_of_ne' (zero_le _fvar.8365) _fvar.8442) (pow_pos (Mathlib.Meta.Positivity.pos_of_isNat (Mathlib.Meta.NormNum.isNat_ofNat NNReal (Eq.refl 10)) (Eq.refl (Nat.ble 1 10))) n)))hs':∀ (n : ℕ), _fvar.8365 * 10 ^ n < ↑(@_fvar.10801 n) + 1 := fun n => Nat.lt_floor_add_one (_fvar.8365 * 10 ^ n)a:ℕ → Digitha:∀ (n : ℕ), s (n + 1) = 10 * s n + ↑(a n)d:AppendixB.NNRealDecimal := { intPart := @_fvar.10801 0, fracPart := _fvar.51385 }hsum:∀ (n : ℕ), ↑(@_fvar.10801 n) * 10 ^ (-↑n) = ↑(@_fvar.10801 0) + ∑ i ∈ Finset.range n, ↑↑(@_fvar.51385 i) * 10 ^ (-↑i - 1) := fun n => Nat.recAux (of_eq_true (Eq.trans (congr (congrArg Eq (Eq.trans (congrArg (HMul.hMul ↑(@_fvar.10801 0)) (Eq.trans (congrArg (HPow.hPow 10) (Eq.trans (congrArg Neg.neg (CharP.cast_eq_zero ℝ 0)) neg_zero)) (NNReal.rpow_zero 10))) (mul_one ↑(@_fvar.10801 0)))) (add_zero ↑(@_fvar.10801 0))) (eq_self ↑(@_fvar.10801 0)))) (fun n hn => Eq.mpr (id (congrArg (fun _a => ↑_a * 10 ^ (-↑(n + 1)) = ↑(@_fvar.10801 0) + ∑ i ∈ Finset.range (n + 1), ↑↑(@_fvar.51385 i) * 10 ^ (-↑i - 1)) (@_fvar.51390 n))) (Trans.trans (Trans.trans (Eq.mpr (id (congrArg (fun x => x = ↑(@_fvar.10801 n) * 10 ^ (-↑n) + ↑↑(@_fvar.51385 n) * 10 ^ (-↑n - 1)) (Eq.trans (congr (congrArg HMul.hMul (Eq.trans (Nat.cast_add (10 * @_fvar.10801 n) ↑(@_fvar.51385 n)) (congrArg (fun x => x + ↑↑(@_fvar.51385 n)) (Nat.cast_mul 10 (@_fvar.10801 n))))) (congrArg (HPow.hPow 10) (Eq.trans (congrArg Neg.neg (Eq.trans (Nat.cast_add n 1) (congrArg (HAdd.hAdd ↑n) Nat.cast_one))) (neg_add_rev (↑n) 1)))) (add_mul (10 * ↑(@_fvar.10801 n)) (↑↑(@_fvar.51385 n)) (10 ^ (-1 + -↑n)))))) (Eq.mpr (id (congr (congrArg Eq (Eq.trans (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat NNReal (Eq.refl 10))) (Mathlib.Tactic.Ring.atom_pf ↑(@_fvar.10801 n)) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_right (↑(@_fvar.10801 n)) (Nat.rawCast 1) (Mathlib.Tactic.Ring.mul_one (Nat.rawCast 10))) (Mathlib.Tactic.Ring.mul_zero (Nat.rawCast 10)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑(@_fvar.10801 n) ^ Nat.rawCast 1 * Nat.rawCast 10 + 0))) (Mathlib.Tactic.Ring.zero_mul (↑(@_fvar.10801 n) ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑(@_fvar.10801 n) ^ Nat.rawCast 1 * Nat.rawCast 10 + 0)))) (Mathlib.Tactic.Ring.atom_pf' (congrArg (HPow.hPow 10) (Eq.trans (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.atom_pf ↑n) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑n) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.add_pf_add_lt (Int.negOfNat 1).rawCast (Mathlib.Tactic.Ring.add_pf_zero_add (↑n ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))) (Eq.trans (congr (congrArg HAdd.hAdd (Eq.trans Mathlib.Tactic.RingNF.int_rawCast_neg (congrArg Neg.neg Mathlib.Tactic.RingNF.nat_rawCast_1))) (Eq.trans (congrArg (fun x => x + 0) (Eq.trans (Eq.trans (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow ↑n) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one ↑n))) (Eq.trans Mathlib.Tactic.RingNF.int_rawCast_neg (congrArg Neg.neg Mathlib.Tactic.RingNF.nat_rawCast_1))) (Mathlib.Tactic.RingNF.mul_neg (↑n) 1)) (congrArg Neg.neg (mul_one ↑n)))) (add_zero (-↑n)))) (Mathlib.Tactic.RingNF.add_neg (-1) ↑n))))) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_left (↑(@_fvar.10801 n)) (Nat.rawCast 1) (Mathlib.Tactic.Ring.mul_pf_right (10 ^ (-1 - ↑n)) (Nat.rawCast 1) (Mathlib.Tactic.Ring.mul_one (Nat.rawCast 10)))) (Mathlib.Tactic.Ring.mul_zero (↑(@_fvar.10801 n) ^ Nat.rawCast 1 * Nat.rawCast 10)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑(@_fvar.10801 n) ^ Nat.rawCast 1 * ((10 ^ (-1 - ↑n)) ^ Nat.rawCast 1 * Nat.rawCast 10) + 0))) (Mathlib.Tactic.Ring.zero_mul ((10 ^ (-1 - ↑n)) ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑(@_fvar.10801 n) ^ Nat.rawCast 1 * ((10 ^ (-1 - ↑n)) ^ Nat.rawCast 1 * Nat.rawCast 10) + 0)))) (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.atom_pf ↑↑(@_fvar.51385 n)) (Mathlib.Tactic.Ring.atom_pf' (congrArg (HPow.hPow 10) (Eq.trans (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.atom_pf ↑n) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑n) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.add_pf_add_lt (Int.negOfNat 1).rawCast (Mathlib.Tactic.Ring.add_pf_zero_add (↑n ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))) (Eq.trans (congr (congrArg HAdd.hAdd (Eq.trans Mathlib.Tactic.RingNF.int_rawCast_neg (congrArg Neg.neg Mathlib.Tactic.RingNF.nat_rawCast_1))) (Eq.trans (congrArg (fun x => x + 0) (Eq.trans (Eq.trans (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow ↑n) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one ↑n))) (Eq.trans Mathlib.Tactic.RingNF.int_rawCast_neg (congrArg Neg.neg Mathlib.Tactic.RingNF.nat_rawCast_1))) (Mathlib.Tactic.RingNF.mul_neg (↑n) 1)) (congrArg Neg.neg (mul_one ↑n)))) (add_zero (-↑n)))) (Mathlib.Tactic.RingNF.add_neg (-1) ↑n))))) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_right (10 ^ (-1 - ↑n)) (Nat.rawCast 1) (Mathlib.Tactic.Ring.mul_pf_left (↑↑(@_fvar.51385 n)) (Nat.rawCast 1) (Mathlib.Tactic.Ring.one_mul (Nat.rawCast 1)))) (Mathlib.Tactic.Ring.mul_zero (↑↑(@_fvar.51385 n) ^ Nat.rawCast 1 * Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_add_zero ((10 ^ (-1 - ↑n)) ^ Nat.rawCast 1 * (↑↑(@_fvar.51385 n) ^ Nat.rawCast 1 * Nat.rawCast 1) + 0))) (Mathlib.Tactic.Ring.zero_mul ((10 ^ (-1 - ↑n)) ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero ((10 ^ (-1 - ↑n)) ^ Nat.rawCast 1 * (↑↑(@_fvar.51385 n) ^ Nat.rawCast 1 * Nat.rawCast 1) + 0)))) (Mathlib.Tactic.Ring.add_pf_add_lt (↑(@_fvar.10801 n) ^ Nat.rawCast 1 * ((10 ^ (-1 - ↑n)) ^ Nat.rawCast 1 * Nat.rawCast 10)) (Mathlib.Tactic.Ring.add_pf_zero_add ((10 ^ (-1 - ↑n)) ^ Nat.rawCast 1 * (↑↑(@_fvar.51385 n) ^ Nat.rawCast 1 * Nat.rawCast 1) + 0)))) (congr (congrArg HAdd.hAdd (Eq.trans (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow ↑(@_fvar.10801 n)) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one ↑(@_fvar.10801 n)))) (congrArg (fun x => x * 10) (Eq.trans (congrArg (HPow.hPow (10 ^ (-1 - ↑n))) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one (10 ^ (-1 - ↑n)))))) (Mathlib.Tactic.RingNF.mul_assoc_rev (↑(@_fvar.10801 n)) (10 ^ (-1 - ↑n)) 10))) (Eq.trans (congrArg (fun x => x + 0) (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow (10 ^ (-1 - ↑n))) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one (10 ^ (-1 - ↑n))))) (Eq.trans (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow ↑↑(@_fvar.51385 n)) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one ↑↑(@_fvar.51385 n)))) Mathlib.Tactic.RingNF.nat_rawCast_1) (mul_one ↑↑(@_fvar.51385 n))))) (add_zero (10 ^ (-1 - ↑n) * ↑↑(@_fvar.51385 n))))))) (Eq.trans (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.atom_pf ↑(@_fvar.10801 n)) (Mathlib.Tactic.Ring.atom_pf (10 ^ (-↑n))) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_left (↑(@_fvar.10801 n)) (Nat.rawCast 1) (Mathlib.Tactic.Ring.mul_pf_right (10 ^ (-↑n)) (Nat.rawCast 1) (Mathlib.Tactic.Ring.one_mul (Nat.rawCast 1)))) (Mathlib.Tactic.Ring.mul_zero (↑(@_fvar.10801 n) ^ Nat.rawCast 1 * Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑(@_fvar.10801 n) ^ Nat.rawCast 1 * ((10 ^ (-↑n)) ^ Nat.rawCast 1 * Nat.rawCast 1) + 0))) (Mathlib.Tactic.Ring.zero_mul ((10 ^ (-↑n)) ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑(@_fvar.10801 n) ^ Nat.rawCast 1 * ((10 ^ (-↑n)) ^ Nat.rawCast 1 * Nat.rawCast 1) + 0)))) (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.atom_pf ↑↑(@_fvar.51385 n)) (Mathlib.Tactic.Ring.atom_pf' (congrArg (HPow.hPow 10) (Eq.trans (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.atom_pf ↑n) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑n) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_gt (Int.negOfNat 1).rawCast (Mathlib.Tactic.Ring.add_pf_add_zero (↑n ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))) (Eq.trans (congr (congrArg HAdd.hAdd (Eq.trans Mathlib.Tactic.RingNF.int_rawCast_neg (congrArg Neg.neg Mathlib.Tactic.RingNF.nat_rawCast_1))) (Eq.trans (congrArg (fun x => x + 0) (Eq.trans (Eq.trans (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow ↑n) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one ↑n))) (Eq.trans Mathlib.Tactic.RingNF.int_rawCast_neg (congrArg Neg.neg Mathlib.Tactic.RingNF.nat_rawCast_1))) (Mathlib.Tactic.RingNF.mul_neg (↑n) 1)) (congrArg Neg.neg (mul_one ↑n)))) (add_zero (-↑n)))) (Mathlib.Tactic.RingNF.add_neg (-1) ↑n))))) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_right (10 ^ (-1 - ↑n)) (Nat.rawCast 1) (Mathlib.Tactic.Ring.mul_pf_left (↑↑(@_fvar.51385 n)) (Nat.rawCast 1) (Mathlib.Tactic.Ring.one_mul (Nat.rawCast 1)))) (Mathlib.Tactic.Ring.mul_zero (↑↑(@_fvar.51385 n) ^ Nat.rawCast 1 * Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_add_zero ((10 ^ (-1 - ↑n)) ^ Nat.rawCast 1 * (↑↑(@_fvar.51385 n) ^ Nat.rawCast 1 * Nat.rawCast 1) + 0))) (Mathlib.Tactic.Ring.zero_mul ((10 ^ (-1 - ↑n)) ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero ((10 ^ (-1 - ↑n)) ^ Nat.rawCast 1 * (↑↑(@_fvar.51385 n) ^ Nat.rawCast 1 * Nat.rawCast 1) + 0)))) (Mathlib.Tactic.Ring.add_pf_add_lt (↑(@_fvar.10801 n) ^ Nat.rawCast 1 * ((10 ^ (-↑n)) ^ Nat.rawCast 1 * Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_zero_add ((10 ^ (-1 - ↑n)) ^ Nat.rawCast 1 * (↑↑(@_fvar.51385 n) ^ Nat.rawCast 1 * Nat.rawCast 1) + 0)))) (congr (congrArg HAdd.hAdd (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow ↑(@_fvar.10801 n)) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one ↑(@_fvar.10801 n)))) (Eq.trans (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow (10 ^ (-↑n))) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one (10 ^ (-↑n))))) Mathlib.Tactic.RingNF.nat_rawCast_1) (mul_one (10 ^ (-↑n)))))) (Eq.trans (congrArg (fun x => x + 0) (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow (10 ^ (-1 - ↑n))) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one (10 ^ (-1 - ↑n))))) (Eq.trans (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow ↑↑(@_fvar.51385 n)) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one ↑↑(@_fvar.51385 n)))) Mathlib.Tactic.RingNF.nat_rawCast_1) (mul_one ↑↑(@_fvar.51385 n))))) (add_zero (10 ^ (-1 - ↑n) * ↑↑(@_fvar.51385 n)))))))) ((fun {α β γ} [HAdd α β γ] a a_1 e_a => Eq.rec (motive := fun a_2 e_a => ∀ (a_3 a_4 : β), a_3 = a_4 → a + a_3 = a_2 + a_4) (fun a_2 a_3 e_a => e_a ▸ Eq.refl (a + a_2)) e_a) (↑(@_fvar.10801 n) * 10 ^ (-1 - ↑n) * 10) (↑(@_fvar.10801 n) * 10 ^ (-↑n)) (Eq.mpr (id (congrArg (fun _a => _a = ↑(@_fvar.10801 n) * 10 ^ (-↑n)) (mul_assoc (↑(@_fvar.10801 n)) (10 ^ (-1 - ↑n)) 10))) (Eq.mpr (id (congrArg (fun _a => ↑(@_fvar.10801 n) * _a = ↑(@_fvar.10801 n) * 10 ^ (-↑n)) (Eq.symm (NNReal.rpow_add_one (Mathlib.Meta.NormNum.isNat_eq_false (Mathlib.Meta.NormNum.isNat_ofNat NNReal (Eq.refl 10)) (Mathlib.Meta.NormNum.isNat_ofNat NNReal Nat.cast_zero) (Eq.refl false)) (-1 - ↑n))))) (of_eq_true (Eq.trans (congrArg (fun x => x = ↑(@_fvar.10801 n) * 10 ^ (-↑n)) (Eq.trans (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.atom_pf ↑(@_fvar.10801 n)) (Mathlib.Tactic.Ring.atom_pf' (congrArg (HPow.hPow 10) (Eq.trans (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.atom_pf ↑n) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑n) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_lt (Int.negOfNat 1).rawCast (Mathlib.Tactic.Ring.add_pf_zero_add (↑n ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1)) (Eq.refl (Int.ofNat 0)))) (Mathlib.Tactic.Ring.add_pf_add_zero (↑n ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))) (Eq.trans (congrArg (fun x => x + 0) (Eq.trans (Eq.trans (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow ↑n) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one ↑n))) (Eq.trans Mathlib.Tactic.RingNF.int_rawCast_neg (congrArg Neg.neg Mathlib.Tactic.RingNF.nat_rawCast_1))) (Mathlib.Tactic.RingNF.mul_neg (↑n) 1)) (congrArg Neg.neg (mul_one ↑n)))) (add_zero (-↑n)))))) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_left (↑(@_fvar.10801 n)) (Nat.rawCast 1) (Mathlib.Tactic.Ring.mul_pf_right (10 ^ (-↑n)) (Nat.rawCast 1) (Mathlib.Tactic.Ring.one_mul (Nat.rawCast 1)))) (Mathlib.Tactic.Ring.mul_zero (↑(@_fvar.10801 n) ^ Nat.rawCast 1 * Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑(@_fvar.10801 n) ^ Nat.rawCast 1 * ((10 ^ (-↑n)) ^ Nat.rawCast 1 * Nat.rawCast 1) + 0))) (Mathlib.Tactic.Ring.zero_mul ((10 ^ (-↑n)) ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑(@_fvar.10801 n) ^ Nat.rawCast 1 * ((10 ^ (-↑n)) ^ Nat.rawCast 1 * Nat.rawCast 1) + 0)))) (Eq.trans (congrArg (fun x => x + 0) (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow ↑(@_fvar.10801 n)) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one ↑(@_fvar.10801 n)))) (Eq.trans (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow (10 ^ (-↑n))) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one (10 ^ (-↑n))))) Mathlib.Tactic.RingNF.nat_rawCast_1) (mul_one (10 ^ (-↑n)))))) (add_zero (↑(@_fvar.10801 n) * 10 ^ (-↑n)))))) (eq_self (↑(@_fvar.10801 n) * 10 ^ (-↑n))))))) (10 ^ (-1 - ↑n) * ↑↑(@_fvar.51385 n)) (10 ^ (-1 - ↑n) * ↑↑(@_fvar.51385 n)) (Eq.refl (10 ^ (-1 - ↑n) * ↑↑(@_fvar.51385 n)))))) (AppendixB.NNRealDecimal.surj._proof_6 _fvar.8365 _fvar.51385 _fvar.51390 n hn)) ((fun {α β γ} [HAdd α β γ] a a_1 e_a => Eq.rec (motive := fun a_2 e_a => ∀ (a_3 a_4 : β), a_3 = a_4 → a + a_3 = a_2 + a_4) (fun a_2 a_3 e_a => e_a ▸ Eq.refl (a + a_2)) e_a) (↑(@_fvar.10801 0)) (↑(@_fvar.10801 0)) (Eq.refl ↑(@_fvar.10801 0)) (∑ i ∈ Finset.range n, ↑↑(@_fvar.51385 i) * 10 ^ (-↑i - 1) + ↑↑(@_fvar.51385 n) * 10 ^ (-↑n - 1)) (∑ i ∈ Finset.range (n + 1), ↑↑(@_fvar.51385 i) * 10 ^ (-↑i - 1)) (Eq.symm (Finset.sum_range_succ (fun x => ↑↑(@_fvar.51385 x) * 10 ^ (-↑x - 1)) n))))) nthis:Filter.Tendsto (fun n => ↑(s n) * 10 ^ (-↑n)) Filter.atTop (nhds ↑d)n:ℕ⊢ 10 ≠ 0; simphxx:ℝ≥0h:¬x = 0s:ℕ → ℕ := fun n => ⌊_fvar.8365 * 10 ^ n⌋₊hs:∀ (n : ℕ), ↑(@_fvar.10801 n) ≤ _fvar.8365 * 10 ^ n := fun n => Nat.floor_le (le_of_lt (mul_pos (lt_of_le_of_ne' (zero_le _fvar.8365) _fvar.8442) (pow_pos (Mathlib.Meta.Positivity.pos_of_isNat (Mathlib.Meta.NormNum.isNat_ofNat NNReal (Eq.refl 10)) (Eq.refl (Nat.ble 1 10))) n)))hs':∀ (n : ℕ), _fvar.8365 * 10 ^ n < ↑(@_fvar.10801 n) + 1 := fun n => Nat.lt_floor_add_one (_fvar.8365 * 10 ^ n)a:ℕ → Digitha:∀ (n : ℕ), s (n + 1) = 10 * s n + ↑(a n)d:AppendixB.NNRealDecimal := { intPart := @_fvar.10801 0, fracPart := _fvar.51385 }hsum:∀ (n : ℕ), ↑(@_fvar.10801 n) * 10 ^ (-↑n) = ↑(@_fvar.10801 0) + ∑ i ∈ Finset.range n, ↑↑(@_fvar.51385 i) * 10 ^ (-↑i - 1) := fun n => Nat.recAux (of_eq_true (Eq.trans (congr (congrArg Eq (Eq.trans (congrArg (HMul.hMul ↑(@_fvar.10801 0)) (Eq.trans (congrArg (HPow.hPow 10) (Eq.trans (congrArg Neg.neg (CharP.cast_eq_zero ℝ 0)) neg_zero)) (NNReal.rpow_zero 10))) (mul_one ↑(@_fvar.10801 0)))) (add_zero ↑(@_fvar.10801 0))) (eq_self ↑(@_fvar.10801 0)))) (fun n hn => Eq.mpr (id (congrArg (fun _a => ↑_a * 10 ^ (-↑(n + 1)) = ↑(@_fvar.10801 0) + ∑ i ∈ Finset.range (n + 1), ↑↑(@_fvar.51385 i) * 10 ^ (-↑i - 1)) (@_fvar.51390 n))) (Trans.trans (Trans.trans (Eq.mpr (id (congrArg (fun x => x = ↑(@_fvar.10801 n) * 10 ^ (-↑n) + ↑↑(@_fvar.51385 n) * 10 ^ (-↑n - 1)) (Eq.trans (congr (congrArg HMul.hMul (Eq.trans (Nat.cast_add (10 * @_fvar.10801 n) ↑(@_fvar.51385 n)) (congrArg (fun x => x + ↑↑(@_fvar.51385 n)) (Nat.cast_mul 10 (@_fvar.10801 n))))) (congrArg (HPow.hPow 10) (Eq.trans (congrArg Neg.neg (Eq.trans (Nat.cast_add n 1) (congrArg (HAdd.hAdd ↑n) Nat.cast_one))) (neg_add_rev (↑n) 1)))) (add_mul (10 * ↑(@_fvar.10801 n)) (↑↑(@_fvar.51385 n)) (10 ^ (-1 + -↑n)))))) (Eq.mpr (id (congr (congrArg Eq (Eq.trans (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat NNReal (Eq.refl 10))) (Mathlib.Tactic.Ring.atom_pf ↑(@_fvar.10801 n)) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_right (↑(@_fvar.10801 n)) (Nat.rawCast 1) (Mathlib.Tactic.Ring.mul_one (Nat.rawCast 10))) (Mathlib.Tactic.Ring.mul_zero (Nat.rawCast 10)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑(@_fvar.10801 n) ^ Nat.rawCast 1 * Nat.rawCast 10 + 0))) (Mathlib.Tactic.Ring.zero_mul (↑(@_fvar.10801 n) ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑(@_fvar.10801 n) ^ Nat.rawCast 1 * Nat.rawCast 10 + 0)))) (Mathlib.Tactic.Ring.atom_pf' (congrArg (HPow.hPow 10) (Eq.trans (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.atom_pf ↑n) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑n) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.add_pf_add_lt (Int.negOfNat 1).rawCast (Mathlib.Tactic.Ring.add_pf_zero_add (↑n ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))) (Eq.trans (congr (congrArg HAdd.hAdd (Eq.trans Mathlib.Tactic.RingNF.int_rawCast_neg (congrArg Neg.neg Mathlib.Tactic.RingNF.nat_rawCast_1))) (Eq.trans (congrArg (fun x => x + 0) (Eq.trans (Eq.trans (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow ↑n) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one ↑n))) (Eq.trans Mathlib.Tactic.RingNF.int_rawCast_neg (congrArg Neg.neg Mathlib.Tactic.RingNF.nat_rawCast_1))) (Mathlib.Tactic.RingNF.mul_neg (↑n) 1)) (congrArg Neg.neg (mul_one ↑n)))) (add_zero (-↑n)))) (Mathlib.Tactic.RingNF.add_neg (-1) ↑n))))) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_left (↑(@_fvar.10801 n)) (Nat.rawCast 1) (Mathlib.Tactic.Ring.mul_pf_right (10 ^ (-1 - ↑n)) (Nat.rawCast 1) (Mathlib.Tactic.Ring.mul_one (Nat.rawCast 10)))) (Mathlib.Tactic.Ring.mul_zero (↑(@_fvar.10801 n) ^ Nat.rawCast 1 * Nat.rawCast 10)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑(@_fvar.10801 n) ^ Nat.rawCast 1 * ((10 ^ (-1 - ↑n)) ^ Nat.rawCast 1 * Nat.rawCast 10) + 0))) (Mathlib.Tactic.Ring.zero_mul ((10 ^ (-1 - ↑n)) ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑(@_fvar.10801 n) ^ Nat.rawCast 1 * ((10 ^ (-1 - ↑n)) ^ Nat.rawCast 1 * Nat.rawCast 10) + 0)))) (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.atom_pf ↑↑(@_fvar.51385 n)) (Mathlib.Tactic.Ring.atom_pf' (congrArg (HPow.hPow 10) (Eq.trans (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.atom_pf ↑n) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑n) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.add_pf_add_lt (Int.negOfNat 1).rawCast (Mathlib.Tactic.Ring.add_pf_zero_add (↑n ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))) (Eq.trans (congr (congrArg HAdd.hAdd (Eq.trans Mathlib.Tactic.RingNF.int_rawCast_neg (congrArg Neg.neg Mathlib.Tactic.RingNF.nat_rawCast_1))) (Eq.trans (congrArg (fun x => x + 0) (Eq.trans (Eq.trans (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow ↑n) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one ↑n))) (Eq.trans Mathlib.Tactic.RingNF.int_rawCast_neg (congrArg Neg.neg Mathlib.Tactic.RingNF.nat_rawCast_1))) (Mathlib.Tactic.RingNF.mul_neg (↑n) 1)) (congrArg Neg.neg (mul_one ↑n)))) (add_zero (-↑n)))) (Mathlib.Tactic.RingNF.add_neg (-1) ↑n))))) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_right (10 ^ (-1 - ↑n)) (Nat.rawCast 1) (Mathlib.Tactic.Ring.mul_pf_left (↑↑(@_fvar.51385 n)) (Nat.rawCast 1) (Mathlib.Tactic.Ring.one_mul (Nat.rawCast 1)))) (Mathlib.Tactic.Ring.mul_zero (↑↑(@_fvar.51385 n) ^ Nat.rawCast 1 * Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_add_zero ((10 ^ (-1 - ↑n)) ^ Nat.rawCast 1 * (↑↑(@_fvar.51385 n) ^ Nat.rawCast 1 * Nat.rawCast 1) + 0))) (Mathlib.Tactic.Ring.zero_mul ((10 ^ (-1 - ↑n)) ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero ((10 ^ (-1 - ↑n)) ^ Nat.rawCast 1 * (↑↑(@_fvar.51385 n) ^ Nat.rawCast 1 * Nat.rawCast 1) + 0)))) (Mathlib.Tactic.Ring.add_pf_add_lt (↑(@_fvar.10801 n) ^ Nat.rawCast 1 * ((10 ^ (-1 - ↑n)) ^ Nat.rawCast 1 * Nat.rawCast 10)) (Mathlib.Tactic.Ring.add_pf_zero_add ((10 ^ (-1 - ↑n)) ^ Nat.rawCast 1 * (↑↑(@_fvar.51385 n) ^ Nat.rawCast 1 * Nat.rawCast 1) + 0)))) (congr (congrArg HAdd.hAdd (Eq.trans (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow ↑(@_fvar.10801 n)) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one ↑(@_fvar.10801 n)))) (congrArg (fun x => x * 10) (Eq.trans (congrArg (HPow.hPow (10 ^ (-1 - ↑n))) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one (10 ^ (-1 - ↑n)))))) (Mathlib.Tactic.RingNF.mul_assoc_rev (↑(@_fvar.10801 n)) (10 ^ (-1 - ↑n)) 10))) (Eq.trans (congrArg (fun x => x + 0) (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow (10 ^ (-1 - ↑n))) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one (10 ^ (-1 - ↑n))))) (Eq.trans (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow ↑↑(@_fvar.51385 n)) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one ↑↑(@_fvar.51385 n)))) Mathlib.Tactic.RingNF.nat_rawCast_1) (mul_one ↑↑(@_fvar.51385 n))))) (add_zero (10 ^ (-1 - ↑n) * ↑↑(@_fvar.51385 n))))))) (Eq.trans (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.atom_pf ↑(@_fvar.10801 n)) (Mathlib.Tactic.Ring.atom_pf (10 ^ (-↑n))) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_left (↑(@_fvar.10801 n)) (Nat.rawCast 1) (Mathlib.Tactic.Ring.mul_pf_right (10 ^ (-↑n)) (Nat.rawCast 1) (Mathlib.Tactic.Ring.one_mul (Nat.rawCast 1)))) (Mathlib.Tactic.Ring.mul_zero (↑(@_fvar.10801 n) ^ Nat.rawCast 1 * Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑(@_fvar.10801 n) ^ Nat.rawCast 1 * ((10 ^ (-↑n)) ^ Nat.rawCast 1 * Nat.rawCast 1) + 0))) (Mathlib.Tactic.Ring.zero_mul ((10 ^ (-↑n)) ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑(@_fvar.10801 n) ^ Nat.rawCast 1 * ((10 ^ (-↑n)) ^ Nat.rawCast 1 * Nat.rawCast 1) + 0)))) (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.atom_pf ↑↑(@_fvar.51385 n)) (Mathlib.Tactic.Ring.atom_pf' (congrArg (HPow.hPow 10) (Eq.trans (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.atom_pf ↑n) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑n) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_gt (Int.negOfNat 1).rawCast (Mathlib.Tactic.Ring.add_pf_add_zero (↑n ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))) (Eq.trans (congr (congrArg HAdd.hAdd (Eq.trans Mathlib.Tactic.RingNF.int_rawCast_neg (congrArg Neg.neg Mathlib.Tactic.RingNF.nat_rawCast_1))) (Eq.trans (congrArg (fun x => x + 0) (Eq.trans (Eq.trans (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow ↑n) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one ↑n))) (Eq.trans Mathlib.Tactic.RingNF.int_rawCast_neg (congrArg Neg.neg Mathlib.Tactic.RingNF.nat_rawCast_1))) (Mathlib.Tactic.RingNF.mul_neg (↑n) 1)) (congrArg Neg.neg (mul_one ↑n)))) (add_zero (-↑n)))) (Mathlib.Tactic.RingNF.add_neg (-1) ↑n))))) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_right (10 ^ (-1 - ↑n)) (Nat.rawCast 1) (Mathlib.Tactic.Ring.mul_pf_left (↑↑(@_fvar.51385 n)) (Nat.rawCast 1) (Mathlib.Tactic.Ring.one_mul (Nat.rawCast 1)))) (Mathlib.Tactic.Ring.mul_zero (↑↑(@_fvar.51385 n) ^ Nat.rawCast 1 * Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_add_zero ((10 ^ (-1 - ↑n)) ^ Nat.rawCast 1 * (↑↑(@_fvar.51385 n) ^ Nat.rawCast 1 * Nat.rawCast 1) + 0))) (Mathlib.Tactic.Ring.zero_mul ((10 ^ (-1 - ↑n)) ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero ((10 ^ (-1 - ↑n)) ^ Nat.rawCast 1 * (↑↑(@_fvar.51385 n) ^ Nat.rawCast 1 * Nat.rawCast 1) + 0)))) (Mathlib.Tactic.Ring.add_pf_add_lt (↑(@_fvar.10801 n) ^ Nat.rawCast 1 * ((10 ^ (-↑n)) ^ Nat.rawCast 1 * Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_zero_add ((10 ^ (-1 - ↑n)) ^ Nat.rawCast 1 * (↑↑(@_fvar.51385 n) ^ Nat.rawCast 1 * Nat.rawCast 1) + 0)))) (congr (congrArg HAdd.hAdd (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow ↑(@_fvar.10801 n)) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one ↑(@_fvar.10801 n)))) (Eq.trans (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow (10 ^ (-↑n))) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one (10 ^ (-↑n))))) Mathlib.Tactic.RingNF.nat_rawCast_1) (mul_one (10 ^ (-↑n)))))) (Eq.trans (congrArg (fun x => x + 0) (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow (10 ^ (-1 - ↑n))) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one (10 ^ (-1 - ↑n))))) (Eq.trans (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow ↑↑(@_fvar.51385 n)) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one ↑↑(@_fvar.51385 n)))) Mathlib.Tactic.RingNF.nat_rawCast_1) (mul_one ↑↑(@_fvar.51385 n))))) (add_zero (10 ^ (-1 - ↑n) * ↑↑(@_fvar.51385 n)))))))) ((fun {α β γ} [HAdd α β γ] a a_1 e_a => Eq.rec (motive := fun a_2 e_a => ∀ (a_3 a_4 : β), a_3 = a_4 → a + a_3 = a_2 + a_4) (fun a_2 a_3 e_a => e_a ▸ Eq.refl (a + a_2)) e_a) (↑(@_fvar.10801 n) * 10 ^ (-1 - ↑n) * 10) (↑(@_fvar.10801 n) * 10 ^ (-↑n)) (Eq.mpr (id (congrArg (fun _a => _a = ↑(@_fvar.10801 n) * 10 ^ (-↑n)) (mul_assoc (↑(@_fvar.10801 n)) (10 ^ (-1 - ↑n)) 10))) (Eq.mpr (id (congrArg (fun _a => ↑(@_fvar.10801 n) * _a = ↑(@_fvar.10801 n) * 10 ^ (-↑n)) (Eq.symm (NNReal.rpow_add_one (Mathlib.Meta.NormNum.isNat_eq_false (Mathlib.Meta.NormNum.isNat_ofNat NNReal (Eq.refl 10)) (Mathlib.Meta.NormNum.isNat_ofNat NNReal Nat.cast_zero) (Eq.refl false)) (-1 - ↑n))))) (of_eq_true (Eq.trans (congrArg (fun x => x = ↑(@_fvar.10801 n) * 10 ^ (-↑n)) (Eq.trans (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.atom_pf ↑(@_fvar.10801 n)) (Mathlib.Tactic.Ring.atom_pf' (congrArg (HPow.hPow 10) (Eq.trans (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.atom_pf ↑n) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑n) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_lt (Int.negOfNat 1).rawCast (Mathlib.Tactic.Ring.add_pf_zero_add (↑n ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1)) (Eq.refl (Int.ofNat 0)))) (Mathlib.Tactic.Ring.add_pf_add_zero (↑n ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))) (Eq.trans (congrArg (fun x => x + 0) (Eq.trans (Eq.trans (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow ↑n) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one ↑n))) (Eq.trans Mathlib.Tactic.RingNF.int_rawCast_neg (congrArg Neg.neg Mathlib.Tactic.RingNF.nat_rawCast_1))) (Mathlib.Tactic.RingNF.mul_neg (↑n) 1)) (congrArg Neg.neg (mul_one ↑n)))) (add_zero (-↑n)))))) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_left (↑(@_fvar.10801 n)) (Nat.rawCast 1) (Mathlib.Tactic.Ring.mul_pf_right (10 ^ (-↑n)) (Nat.rawCast 1) (Mathlib.Tactic.Ring.one_mul (Nat.rawCast 1)))) (Mathlib.Tactic.Ring.mul_zero (↑(@_fvar.10801 n) ^ Nat.rawCast 1 * Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑(@_fvar.10801 n) ^ Nat.rawCast 1 * ((10 ^ (-↑n)) ^ Nat.rawCast 1 * Nat.rawCast 1) + 0))) (Mathlib.Tactic.Ring.zero_mul ((10 ^ (-↑n)) ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑(@_fvar.10801 n) ^ Nat.rawCast 1 * ((10 ^ (-↑n)) ^ Nat.rawCast 1 * Nat.rawCast 1) + 0)))) (Eq.trans (congrArg (fun x => x + 0) (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow ↑(@_fvar.10801 n)) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one ↑(@_fvar.10801 n)))) (Eq.trans (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow (10 ^ (-↑n))) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one (10 ^ (-↑n))))) Mathlib.Tactic.RingNF.nat_rawCast_1) (mul_one (10 ^ (-↑n)))))) (add_zero (↑(@_fvar.10801 n) * 10 ^ (-↑n)))))) (eq_self (↑(@_fvar.10801 n) * 10 ^ (-↑n))))))) (10 ^ (-1 - ↑n) * ↑↑(@_fvar.51385 n)) (10 ^ (-1 - ↑n) * ↑↑(@_fvar.51385 n)) (Eq.refl (10 ^ (-1 - ↑n) * ↑↑(@_fvar.51385 n)))))) (AppendixB.NNRealDecimal.surj._proof_6 _fvar.8365 _fvar.51385 _fvar.51390 n hn)) ((fun {α β γ} [HAdd α β γ] a a_1 e_a => Eq.rec (motive := fun a_2 e_a => ∀ (a_3 a_4 : β), a_3 = a_4 → a + a_3 = a_2 + a_4) (fun a_2 a_3 e_a => e_a ▸ Eq.refl (a + a_2)) e_a) (↑(@_fvar.10801 0)) (↑(@_fvar.10801 0)) (Eq.refl ↑(@_fvar.10801 0)) (∑ i ∈ Finset.range n, ↑↑(@_fvar.51385 i) * 10 ^ (-↑i - 1) + ↑↑(@_fvar.51385 n) * 10 ^ (-↑n - 1)) (∑ i ∈ Finset.range (n + 1), ↑↑(@_fvar.51385 i) * 10 ^ (-↑i - 1)) (Eq.symm (Finset.sum_range_succ (fun x => ↑↑(@_fvar.51385 x) * 10 ^ (-↑x - 1)) n))))) nthis:Filter.Tendsto (fun n => ↑(s n) * 10 ^ (-↑n)) Filter.atTop (nhds ↑d)n:ℕ⊢ 10 ≠ 0; norm_numAll goals completed! 🐙 inductive RealDecimal where | pos : NNRealDecimal → RealDecimal | neg : NNRealDecimal → RealDecimalnoncomputable instance RealDecimal.instCoeReal : Coe RealDecimal ℝ where coe := fun d ↦ match d with | RealDecimal.pos d => d.toNNReal | RealDecimal.neg d => -(d.toNNReal:ℝ)theorem RealDecimal.surj (x:ℝ) : ∃ d:RealDecimal, x = d := byx:ℝ⊢ ∃ d, x = match d with | pos d => ↑↑d | neg d => -↑↑d obtain h | h := le_or_gt 0 xinlx:ℝh:0 ≤ x⊢ ∃ d, x = match d with | pos d => ↑↑d | neg d => -↑↑dinrx:ℝh:x < 0⊢ ∃ d, x = match d with | pos d => ↑↑d | neg d => -↑↑d .inlx:ℝh:0 ≤ x⊢ ∃ d, x = match d with | pos d => ↑↑d | neg d => -↑↑d choose d hd using NNRealDecimal.surj (x.toNNReal)inlx:ℝh:0 ≤ xd:NNRealDecimalhd:x.toNNReal = ↑d⊢ ∃ d, x = match d with | pos d => ↑↑d | neg d => -↑↑d; use pos dhx:ℝh:0 ≤ xd:NNRealDecimalhd:x.toNNReal = ↑d⊢ x = match pos d with | pos d => ↑↑d | neg d => -↑↑d; simp [←hd, h]All goals completed! 🐙 .inrx:ℝh:x < 0⊢ ∃ d, x = match d with | pos d => ↑↑d | neg d => -↑↑d choose d hd using NNRealDecimal.surj ((-x).toNNReal)inrx:ℝh:x < 0d:NNRealDecimalhd:(-x).toNNReal = ↑d⊢ ∃ d, x = match d with | pos d => ↑↑d | neg d => -↑↑d; use neg dhx:ℝh:x < 0d:NNRealDecimalhd:(-x).toNNReal = ↑d⊢ x = match neg d with | pos d => ↑↑d | neg d => -↑↑d; simp [←hd, show 0 ≤ -x by linarith]All goals completed! 🐙 theorem declaration uses 'sorry'RealDecimal.not_inj_terminating {x:ℝ} (hx: TerminatingDecimal x) : ∃ d₁ d₂:RealDecimal, d₁ ≠ d₂ ∧ ∀ d: RealDecimal, d = x ↔ d = d₁ ∨ d = d₂ := byx:ℝhx:TerminatingDecimal x⊢ ∃ d₁ d₂, d₁ ≠ d₂ ∧ ∀ (d : RealDecimal), (match d with | pos d => ↑↑d | neg d => -↑↑d) = x ↔ d = d₁ ∨ d = d₂ sorryAll goals completed! 🐙theorem declaration uses 'sorry'RealDecimal.inj_nonterminating {x:ℝ} (hx: ¬TerminatingDecimal x) : ∃! d:RealDecimal, d = x := byx:ℝhx:¬TerminatingDecimal x⊢ ∃! d, (match d with | pos d => ↑↑d | neg d => -↑↑d) = x sorryAll goals completed! 🐙 end AppendixB