52709.lean`

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L4DEL: Lean4 Domain Explanation Language

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Credit: The upstream authors and contributors retain credit for the project and mathematics. lean4-explainer only renders their pinned source and adds the separately disclosed explanation layer.

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

13 declarations, 1 contain sorry, 0 declared as axioms

Expansion of g.f. $(1-\sqrt{1-4x-4x^2})/(2(1+x))$

The $n$-th term $a(n)$ is given by $$a(n) = \sum_{k=0}^{n-1} \frac{1}{k+1} \binom{2k}{k} \binom{k}{n-1-k}$$

References:

A052709

Imports (1)

FormalConjecturesUtil

def OeisA52709.a $a(n) = \sum_{k=0}^{n-1} \frac{1}{k+1} \binom{2k}{k} \binom{k}{n-1-k}$ line 34 source
Source documentation
$a(n) = \sum_{k=0}^{n-1} \frac{1}{k+1} \binom{2k}{k} \binom{k}{n-1-k}$.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat
Proof uses: Finset.sum, Finset.range, HMul.hMul, HDiv.hDiv, Nat.choose, OfNat.ofNat, HAdd.hAdd, HSub.hSub
10 instances, coercions and auxiliary constants left out.
```
def a (n : ℕ) : ℕ :=
…
```
theorem OeisA52709.a_0 line 38 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA52709.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 OeisA52709.a_1 line 42 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA52709.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 OeisA52709.a_2 line 46 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA52709.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 OeisA52709.a_3 line 50 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA52709.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 OeisA52709.a_4 line 54 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA52709.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 OeisA52709.a_5 line 58 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA52709.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.
def OeisA52709.isPositiveList A list of natural numbers is composed of positive integers line 62 source
Source documentation
A list of natural numbers is composed of positive integers.

no axioms what this means
Statement uses: List, Nat
Proof uses: Membership.mem, LT.lt, OfNat.ofNat
3 instances, coercions and auxiliary constants left out.
```
def isPositiveList (l : List ℕ) : Prop :=
…
```
def OeisA52709.coversInitialInterval A list covers an initial interval of positive integers if the set of its elements is $\{1, 2, \dots, \max(l)\}$ line 67 source
Source documentation
A list covers an initial interval of positive integers if the set of its elements is $\{1, 2, \dots, \max(l)\}$.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: List, Nat
Proof uses: And, OeisA52709.isPositiveList, Finset, List.toFinset, OeisA52709.coversInitialInterval.match_1, WithBot, Finset.max, LT.lt, OfNat.ofNat, Iff, Membership.mem, LE.le, Unit, Eq, Bool, List.isEmpty, Bool.true
7 instances, coercions and auxiliary constants left out.
```
def coversInitialInterval (l : List ℕ) : Prop :=
…
```
def OeisA52709.hasNondecreasingPattern3 A list has a non-decreasing subsequence of length 3 (the pattern $x \le y \le z$) line 75 source
Source documentation
A list has a non-decreasing subsequence of length 3 (the pattern $x \le y \le z$).

no axioms what this means
Statement uses: List, Nat
Proof uses: Exists, Fin, List.length, And, LT.lt, LE.le, List.get
2 instances, coercions and auxiliary constants left out.
```
def hasNondecreasingPattern3 (l : List ℕ) : Prop :=
…
```
def OeisA52709.avoidsPatternXYZ A list avoids the pattern $x \le y \le z$ if it does not have a non-decreasing subsequence of length 3 line 81 source
Source documentation
A list avoids the pattern $x \le y \le z$ if it does not have a non-decreasing subsequence of length 3.

no axioms what this means
Statement uses: List, Nat
Proof uses: Not, OeisA52709.hasNondecreasingPattern3
```
def avoidsPatternXYZ (l : List ℕ) : Prop :=
…
```
def OeisA52709.sequencesCountedByA052709 The set of sequences of length $n-1$ satisfying the conditions of the conjecture line 85 source
Source documentation
The set of sequences of length $n-1$ satisfying the conditions of the conjecture.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Set, List
Proof uses: Set.ofPred, And, Eq, List.length, HSub.hSub, OfNat.ofNat, OeisA52709.coversInitialInterval, OeisA52709.avoidsPatternXYZ
3 instances, coercions and auxiliary constants left out.
```
def sequencesCountedByA052709 (n : ℕ) : Set (List ℕ) :=
…
```
theorem OeisA52709.conjecture sorry line 96 source
depends on sorry what this means
Statement uses: Nat, LT.lt, OfNat.ofNat, Fintype, List, OeisA52709.sequencesCountedByA052709, Eq, OeisA52709.a, Fintype.card
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
6 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.