306424.lean`

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L4DEL: Lean4 Domain Explanation Language

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

15 declarations, 1 contain sorry, 0 declared as axioms

Maximality of $k = 43$ with restricted digit counts in bases $3 \le b < k$

Numbers $k$ such that the base $b$ expansion of $k$ for each $b = 3..k-1$ never contains more than two distinct digits.

The sequence Numbers $k$ such that the base $b$ expansion of $k$ for each $b = 3..k-1$ never contains more than two distinct digits.

References:

A306424
arxiv/2605.22763 Advancing Mathematics Research with AI-Driven Formal Proof Search by George Tsoukalas et al.

Imports (1)

FormalConjecturesUtil

def OeisA306424.Condition Numbers $k$ such that the base $b$ expansion of $k$ for each $b = 3..k-1$ never contains more than two distinct digits line 45 source
Source documentation
Numbers $k$ such that the base $b$ expansion of $k$ for each $b = 3..k-1$ never contains more than two distinct digits.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat
Proof uses: And, LE.le, OfNat.ofNat, LT.lt, Finset.card, List.toFinset, Nat.digits
4 instances, coercions and auxiliary constants left out.
```
def Condition (k : ℕ) : Prop :=
…
```
def OeisA306424.a The $n$-th number $k$ such that the base $b$ expansion of $k$ for each $b = 3..k-1$ never contains more than two distinct digits line 53 source
Source documentation
The $n$-th number $k$ such that the base $b$ expansion of $k$ for each $b = 3..k-1$ never contains more than two distinct digits.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat
Proof uses: Nat.nth, OeisA306424.Condition
```
noncomputable def a (n : ℕ) : ℕ := n.nth Condition
…
```
Mentions: OeisA306424.Condition
lemma OeisA306424.Condition_lt_four line 56 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, LE.le, OfNat.ofNat, OeisA306424.Condition
Proof uses: And, LT.lt, Finset.card, List.toFinset, Nat.digits, Decidable.byContradiction, Not
9 instances, coercions and auxiliary constants left out.

Tactic trace: 2 steps
Open to load the checked before-and-after goal states.
lemma OeisA306424.Condition_zero line 61 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: OeisA306424.Condition, OfNat.ofNat, Nat
Proof uses: OeisA306424.Condition_lt_four, Decidable.byContradiction, LE.le, Not
5 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
lemma OeisA306424.Condition_one line 64 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: OeisA306424.Condition, OfNat.ofNat, Nat
Proof uses: OeisA306424.Condition_lt_four, Decidable.byContradiction, LE.le, Not
5 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
lemma OeisA306424.Condition_two line 67 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: OeisA306424.Condition, OfNat.ofNat, Nat
Proof uses: OeisA306424.Condition_lt_four, Decidable.byContradiction, LE.le, Not
5 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
lemma OeisA306424.Condition_three line 70 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: OeisA306424.Condition, OfNat.ofNat, Nat
Proof uses: OeisA306424.Condition_lt_four, Decidable.byContradiction, LE.le, Not
5 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
lemma OeisA306424.Condition_four line 73 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: OeisA306424.Condition, OfNat.ofNat, Nat
Proof uses: And, LE.le, LT.lt, Finset.card, List.toFinset, Nat.digits, Eq, Decidable.byContradiction, Not, of_decide_eq_true, id, Bool, Decidable.decide, Bool.true, Eq.refl, Eq.symm
9 instances, coercions and auxiliary constants left out.

Tactic trace: 5 steps
Open to load the checked before-and-after goal states.
lemma OeisA306424.Condition_five line 80 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: OeisA306424.Condition, OfNat.ofNat, Nat
Proof uses: And, LE.le, LT.lt, Finset.card, List.toFinset, Nat.digits, dite, of_decide_eq_true, id, Eq, Bool, Decidable.decide, Bool.true, Eq.refl, Eq.symm, le_antisymm, Nat.ge_of_not_lt, Mathlib.Tactic.IntervalCases.of_lt_right, Nat.rawCast, Mathlib.Meta.NormNum.IsNat.to_raw_eq and 3 more
11 instances, coercions and auxiliary constants left out.

Tactic trace: 6 steps
Open to load the checked before-and-after goal states.
lemma OeisA306424.a_1 line 88 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA306424.a, OfNat.ofNat
Proof uses: id, Eq.mpr, Nat.nth, OeisA306424.Condition, InfSet.sInf, Set.ofPred, And, LT.lt, congrArg, Nat.nth_eq_sInf, Nat.nth_zero_of_zero, OeisA306424.Condition_zero, Membership.mem, Set, And.intro, OeisA306424.Condition_one, zero_lt_one, Eq.symm, le_antisymm, Nat.ge_of_not_lt and 15 more
24 instances, coercions and auxiliary constants left out.

Tactic trace: 14 steps
Open to load the checked before-and-after goal states.
lemma OeisA306424.a_2 line 105 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA306424.a, OfNat.ofNat
Proof uses: id, Eq.mpr, Nat.nth, OeisA306424.Condition, InfSet.sInf, Set.ofPred, And, LT.lt, congrArg, Nat.nth_eq_sInf, Nat.nth_zero_of_zero, OeisA306424.Condition_zero, OeisA306424.a_1, Membership.mem, Set, And.intro, OeisA306424.Condition_two, dite, LE.le, one_lt_two and 18 more
32 instances, coercions and auxiliary constants left out.

Tactic trace: 19 steps
Open to load the checked before-and-after goal states.
lemma OeisA306424.a_3 line 123 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA306424.a, OfNat.ofNat
Proof uses: id, Eq.mpr, Nat.nth, OeisA306424.Condition, InfSet.sInf, Set.ofPred, And, LT.lt, congrArg, Nat.nth_eq_sInf, Nat.nth_zero_of_zero, OeisA306424.Condition_zero, OeisA306424.a_1, OeisA306424.a_2, Membership.mem, Set, And.intro, OeisA306424.Condition_three, dite, LE.le and 17 more
20 instances, coercions and auxiliary constants left out.

Tactic trace: 23 steps
Open to load the checked before-and-after goal states.
lemma OeisA306424.a_4 line 143 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA306424.a, OfNat.ofNat
Proof uses: id, Eq.mpr, Nat.nth, OeisA306424.Condition, InfSet.sInf, Set.ofPred, And, LT.lt, congrArg, Nat.nth_eq_sInf, Nat.nth_zero_of_zero, OeisA306424.Condition_zero, OeisA306424.a_1, OeisA306424.a_2, OeisA306424.a_3, Membership.mem, Set, And.intro, OeisA306424.Condition_four, dite and 18 more
20 instances, coercions and auxiliary constants left out.

Tactic trace: 27 steps
Open to load the checked before-and-after goal states.
lemma OeisA306424.a_5 line 165 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA306424.a, OfNat.ofNat
Proof uses: id, Eq.mpr, Nat.nth, OeisA306424.Condition, InfSet.sInf, Set.ofPred, And, LT.lt, congrArg, Nat.nth_eq_sInf, Nat.nth_zero_of_zero, OeisA306424.Condition_zero, OeisA306424.a_1, OeisA306424.a_2, OeisA306424.a_3, OeisA306424.a_4, Membership.mem, Set, And.intro, OeisA306424.Condition_five and 19 more
20 instances, coercions and auxiliary constants left out.

Tactic trace: 31 steps
Open to load the checked before-and-after goal states.
theorem OeisA306424.forty_three_is_max sorry line 197 source
depends on sorry what this means
Statement uses: And, OeisA306424.Condition, OfNat.ofNat, Nat, LT.lt, Not
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.