100800.lean`

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L4DEL: Lean4 Domain Explanation Language

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Project sources: Pinned project documentation, Formal Conjectures paper, arXiv v1

10 declarations, 1 contain sorry, 0 declared as axioms

Conjectures associated with A100800

Let $f(n) = n + \text{sum of the digits of } n$. If $f(n)$ is multiple of $n$ then $a(n)= f(n)$ else $a(n) = f(f(f(n)))\dots$ until one gets a multiple of $n$; $a(n) = 0$ if no such number exists.

References:

A100800

Imports (1)

FormalConjecturesUtil

def OeisA100800.sumDigits The sum of the decimal digits of a natural number line 38 source
Source documentation
The sum of the decimal digits of a natural number.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat
Proof uses: List.sum, Nat.digits, OfNat.ofNat
4 instances, coercions and auxiliary constants left out.
```
def sumDigits (n : ℕ) : ℕ := ((10).digits n).sum
```
def OeisA100800.f The function $f(n) = n + \text{sum of the digits of } n$ line 41 source
Source documentation
The function $f(n) = n + \text{sum of the digits of } n$.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat
Proof uses: HAdd.hAdd, OeisA100800.sumDigits
2 instances, coercions and auxiliary constants left out.
```
def f (n : ℕ) : ℕ := n + sumDigits n
…
```
Mentions: OeisA100800.sumDigits
def OeisA100800.a a n is the first iteration of $f(n) = n + \text{sum of the digits of } n$ that is a multiple of $n$ line 48 source
Source documentation
a n is the first iteration of $f(n) = n + \text{sum of the digits of } n$ that is a multiple of $n$. $a(n) = 0$ if no such number exists.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat
Proof uses: Dvd.dvd, Nat.iterate, OeisA100800.f, HAdd.hAdd, OfNat.ofNat, dite, Exists, Classical.propDecidable, Nat.find, Not
5 instances, coercions and auxiliary constants left out.
```
noncomputable def a (n : ℕ) : ℕ :=
…
```
lemma OeisA100800.a_of_dvd If n already divides f n, the search stops immediately and a n = f n line 63 source
Source documentation
If n already divides f n, the search stops immediately and a n = f n.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Dvd.dvd, OeisA100800.f, Eq, OeisA100800.a
Proof uses: Exists, Nat.iterate, HAdd.hAdd, OfNat.ofNat, Exists.intro, Eq.mpr, id, congrArg, Eq.trans, congrFun', zero_add, congrFun, Function.iterate_one, Nat.find, Iff.mpr, Nat.find_eq_zero, dite, Classical.propDecidable, Not, OeisA100800.a.eq_1 and 4 more
8 instances, coercions and auxiliary constants left out.

Tactic trace: 6 steps
Open to load the checked before-and-after goal states.
```
lemma a_of_dvd {n : ℕ} (h : n ∣ f n) : a n = f n := by
…
```
Mentions: OeisA100800.f, OeisA100800.a
theorem OeisA100800.a_1 Term theorems verifying the first few values of the sequence against the official OEIS b-file line 71 source
Source documentation
Term theorems verifying the first few values of the sequence against the official OEIS b-file

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA100800.a, OfNat.ofNat
Proof uses: Eq.mpr, OeisA100800.f, id, congrArg, OeisA100800.a_of_dvd, of_eq_true, Dvd.dvd, Eq.trans, True, IsUnit, Mathlib.Meta.NormNum.IsNat.to_eq, Mathlib.Meta.NormNum.isNat_ofNat, Nat.cast_one, Units, eq_true, Mathlib.Meta.NormNum.isNat_eq_true, Eq.refl, True.intro, congrFun', HAdd.hAdd and 21 more
35 instances, coercions and auxiliary constants left out.

Tactic trace: 3 steps
Open to load the checked before-and-after goal states.
```
theorem a_1 : a 1 = 2 := by rw [a_of_dvd (by norm_num)]; norm_num [f, sumDigits]
…
```
Mentions: OeisA100800.a, OeisA100800.a_of_dvd, OeisA100800.f, OeisA100800.sumDigits
theorem OeisA100800.a_2 line 74 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA100800.a, OfNat.ofNat
Proof uses: Eq.mpr, OeisA100800.f, id, congrArg, OeisA100800.a_of_dvd, of_eq_true, Dvd.dvd, Eq.trans, True, HAdd.hAdd, OeisA100800.sumDigits, List.sum, Nat.digits, List.cons, List.nil, List, HMod.hMod, HDiv.hDiv, Nat.digits_of_two_le_of_pos, Mathlib.Meta.NormNum.isNat_le_true and 18 more
24 instances, coercions and auxiliary constants left out.

Tactic trace: 3 steps
Open to load the checked before-and-after goal states.
theorem OeisA100800.a_3 line 77 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA100800.a, OfNat.ofNat
Proof uses: Eq.mpr, OeisA100800.f, id, congrArg, OeisA100800.a_of_dvd, of_eq_true, Dvd.dvd, Eq.trans, True, HAdd.hAdd, OeisA100800.sumDigits, List.sum, Nat.digits, List.cons, List.nil, List, HMod.hMod, HDiv.hDiv, Nat.digits_of_two_le_of_pos, Mathlib.Meta.NormNum.isNat_le_true and 18 more
24 instances, coercions and auxiliary constants left out.

Tactic trace: 3 steps
Open to load the checked before-and-after goal states.
theorem OeisA100800.a_4 line 80 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA100800.a, OfNat.ofNat
Proof uses: Eq.mpr, OeisA100800.f, id, congrArg, OeisA100800.a_of_dvd, of_eq_true, Dvd.dvd, Eq.trans, True, HAdd.hAdd, OeisA100800.sumDigits, List.sum, Nat.digits, List.cons, List.nil, List, HMod.hMod, HDiv.hDiv, Nat.digits_of_two_le_of_pos, Mathlib.Meta.NormNum.isNat_le_true and 18 more
24 instances, coercions and auxiliary constants left out.

Tactic trace: 3 steps
Open to load the checked before-and-after goal states.
theorem OeisA100800.a_5 line 83 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA100800.a, OfNat.ofNat
Proof uses: Eq.mpr, OeisA100800.f, id, congrArg, OeisA100800.a_of_dvd, of_eq_true, Dvd.dvd, Eq.trans, True, HAdd.hAdd, OeisA100800.sumDigits, List.sum, Nat.digits, List.cons, List.nil, List, HMod.hMod, HDiv.hDiv, Nat.digits_of_two_le_of_pos, Mathlib.Meta.NormNum.isNat_le_true and 18 more
24 instances, coercions and auxiliary constants left out.

Tactic trace: 3 steps
Open to load the checked before-and-after goal states.
theorem OeisA100800.conjecture A100800 Conjecture: No term is zero sorry line 87 source
Source documentation
A100800 Conjecture: No term is zero.

depends on sorry what this means
Statement uses: Nat, Ne, OfNat.ofNat, OeisA100800.a
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
2 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem conjecture : ∀ (n : ℕ), n ≠ 0 → a n ≠ 0 := by
…
```
Mentions: OeisA100800.a