108081.lean`

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

12 declarations, 1 contain sorry, 0 declared as axioms

$a(n) = \sum_{i=0}^n \binom{2n-i}{n+i}$

Alternatively the sequence a can be defined as $a(n) = \sum_{k=0}^n \binom{n+k-1}{k} F(n-k+1)$, where $F(m)$ is the $m$-th Fibonacci number. We formalize a conjecture about the number of words of length $n$ in a set $X$ being related to this sequence.

References:

A108081

Imports (1)

FormalConjecturesUtil

def OeisA108081.a The primary defining sequence a line 40 source
Source documentation
The primary defining sequence a. $a(n) = \sum_{k=0}^n \binom{n+k-1}{k} F(n-k+1)$, where $F(m)$ is the $m$-th Fibonacci number.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat
Proof uses: Finset.sum, Finset.range, HAdd.hAdd, OfNat.ofNat, HMul.hMul, Nat.choose, HSub.hSub, Nat.fib
8 instances, coercions and auxiliary constants left out.
```
def a (n : ℕ) : ℕ :=
…
```
theorem OeisA108081.a_0 Term theorems verifying the first few values of the sequence against the official OEIS b-file line 45 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, OeisA108081.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 = 1 := by decide
…
```
Mentions: OeisA108081.a
theorem OeisA108081.a_1 line 48 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA108081.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 OeisA108081.a_2 line 51 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA108081.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 OeisA108081.a_3 line 54 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA108081.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 OeisA108081.a_4 line 57 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA108081.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.
abbrev OeisA108081.Word Words are finite sequences of integers line 60 source
Source documentation
Words are finite sequences of integers.

no axioms what this means
Proof uses: List, Int
```
abbrev Word := List ℤ
```
def OeisA108081.l l w is the word obtained by reversing w and subtracting 1 from every term line 63 source
Source documentation
l w is the word obtained by reversing w and subtracting 1 from every term.

no axioms what this means
Statement uses: OeisA108081.Word
Proof uses: List.map, Int, HSub.hSub, OfNat.ofNat, List.reverse
3 instances, coercions and auxiliary constants left out.
```
def l (w : Word) : Word :=
…
```
Mentions: OeisA108081.Word
def OeisA108081.r r w is the word obtained by reversing w and adding 1 to every term line 67 source
Source documentation
r w is the word obtained by reversing w and adding 1 to every term.

no axioms what this means
Statement uses: OeisA108081.Word
Proof uses: List.map, Int, HAdd.hAdd, OfNat.ofNat, List.reverse
3 instances, coercions and auxiliary constants left out.
```
def r (w : Word) : Word :=
…
```
Mentions: OeisA108081.Word
inductive OeisA108081.XWord XWord is the smallest set of words satisfying the inductive properties described in A108081 line 73 source
Source documentation
XWord is the smallest set of words satisfying the inductive properties described in A108081.

no axioms what this means
Statement uses: OeisA108081.Word
```
inductive XWord : Word → Prop
  | base : XWord [0]
  | step_left {u v : Word} (hu : XWord u) (hv : XWord v) : XWord (l u ++ v)
  | step_right {u v : Word} (hu : XWord u) (hv : XWord v) : XWord (u ++ r v)

/-- `xN n` is the set of words in `XWord` of length `n`. -/
```
Mentions: OeisA108081.Word, OeisA108081.l, OeisA108081.r, OeisA108081.xN
def OeisA108081.xN xN n is the set of words in XWord of length n line 79 source
Source documentation
xN n is the set of words in XWord of length n.

no axioms what this means
Statement uses: Nat, Set, OeisA108081.Word
Proof uses: Set.ofPred, And, OeisA108081.XWord, Eq, List.length, Int
```
def xN (n : ℕ) : Set Word :=
…
```
Mentions: OeisA108081.Word
theorem OeisA108081.count_words_in_x_is_a_shifted sorry line 95 source
depends on sorry what this means
Statement uses: Nat, GE.ge, OfNat.ofNat, Eq, Set.ncard, OeisA108081.Word, OeisA108081.xN, OeisA108081.a, HSub.hSub
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
8 instances, coercions and auxiliary constants left out.

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