87571.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

Smallest prime formed by concatenation $n, n-1, \dots, n-k$

$a(n)$ is the smallest prime which has the form of the concatenation $n, n-1, n-2, \dots, n-k$ for some $k < n$, or $0$ if no such prime exists.

References:

A087571

Imports (1)

FormalConjecturesUtil

def OeisA87571.getAllDigitsMsf Concatenate a list of numbers into a list of decimal digits in most-significant-first order line 35 source
Source documentation
Concatenate a list of numbers into a list of decimal digits in most-significant-first order.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: List, Nat
Proof uses: List.flatten, List.map, List.reverse, Nat.digits, OfNat.ofNat
1 instances, coercions and auxiliary constants left out.
```
def getAllDigitsMsf (L : List ℕ) : List ℕ :=
…
```
def OeisA87571.ofDigitsMsf Convert a list of decimal digits (most significant first) to a natural number line 39 source
Source documentation
Convert a list of decimal digits (most significant first) to a natural number.

no axioms what this means
Statement uses: List, Nat
Proof uses: List.foldl, HAdd.hAdd, HMul.hMul, OfNat.ofNat
5 instances, coercions and auxiliary constants left out.
```
def ofDigitsMsf (D : List ℕ) : ℕ :=
…
```
def OeisA87571.concatNum Concatenation of numbers $n, n-1, \dots, n-k$ line 43 source
Source documentation
Concatenation of numbers $n, n-1, \dots, n-k$.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat
Proof uses: OeisA87571.ofDigitsMsf, OeisA87571.getAllDigitsMsf, List.map, HSub.hSub, List.range, HAdd.hAdd, OfNat.ofNat
5 instances, coercions and auxiliary constants left out.
```
def concatNum (n : ℕ) (k : ℕ) : ℕ :=
…
```
def OeisA87571.a Smallest prime of the form $n, n-1, \dots, n-k$ for $k < n$, or $0$ if none exists line 47 source
Source documentation
Smallest prime of the form $n, n-1, \dots, n-k$ for $k < n$, or $0$ if none exists.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat
Proof uses: OeisA87571.a.match_1, Option, List.find?, Decidable.decide, Nat.Prime, List.map, OeisA87571.concatNum, List.range, Unit, OfNat.ofNat
2 instances, coercions and auxiliary constants left out.
```
def a (n : ℕ) : ℕ :=
…
```
theorem OeisA87571.a_0 Value of the sequence a at 0 line 54 source
Source documentation
Value of the sequence a at 0.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA87571.a, OfNat.ofNat
Proof uses: of_decide_eq_true, id, Bool, Decidable.decide, Bool.true, Eq.refl
3 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem a_0 : a 0 = 0 := by decide
…
```
Mentions: OeisA87571.a
theorem OeisA87571.a_1 Value of the sequence a at 1 line 58 source
Source documentation
Value of the sequence a at 1.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA87571.a, OfNat.ofNat
Proof uses: of_decide_eq_true, id, Bool, Decidable.decide, Bool.true, Eq.refl
3 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem a_1 : a 1 = 0 := by decide
…
```
Mentions: OeisA87571.a
theorem OeisA87571.a_2 Value of the sequence a at 2 line 62 source
Source documentation
Value of the sequence a at 2.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA87571.a, OfNat.ofNat
Proof uses: of_decide_eq_true, id, Bool, Decidable.decide, Bool.true, Eq.refl
3 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem a_2 : a 2 = 2 := by decide
…
```
Mentions: OeisA87571.a
theorem OeisA87571.a_3 Value of the sequence a at 3 line 66 source
Source documentation
Value of the sequence a at 3.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA87571.a, OfNat.ofNat
Proof uses: of_decide_eq_true, id, Bool, Decidable.decide, Bool.true, Eq.refl
3 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem a_3 : a 3 = 3 := by decide
…
```
Mentions: OeisA87571.a
theorem OeisA87571.a_4 Value of the sequence a at 4 line 70 source
Source documentation
Value of the sequence a at 4.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA87571.a, OfNat.ofNat
Proof uses: of_decide_eq_true, id, Bool, Decidable.decide, Bool.true, Eq.refl
3 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem a_4 : a 4 = 43 := by decide
…
```
Mentions: OeisA87571.a
theorem OeisA87571.conjecture Conjecture: There are infinitely many composite numbers $n$ such that $a(n)$ is nonzero sorry line 75 source
Source documentation
Conjecture: There are infinitely many composite numbers $n$ such that $a(n)$ is nonzero.

depends on sorry what this means
Statement uses: Nat, Exists, And, GT.gt, LT.lt, OfNat.ofNat, Not, Nat.Prime, Ne, OeisA87571.a
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
4 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem conjecture (M : ℕ) : ∃ n > M, 1 < n ∧ ¬ n.Prime ∧ a n ≠ 0 := by
…
```
Mentions: OeisA87571.a