60957.lean`

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

9 declarations, 1 contain sorry, 0 declared as axioms

Number of different products of subsets of $\{1, 2, \dots, n\}$

The number of distinct products (including the empty product 1) of any subset of $\{1, 2, \dots, n\}$.

References:

A060957

Imports (1)

FormalConjecturesUtil

def OeisA60957.a Number of different products of any subset of $\{1, 2, \dots, n\}$ line 34 source
Source documentation
Number of different products of any subset of $\{1, 2, \dots, n\}$.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat
Proof uses: Finset.card, Finset.image, Finset, Finset.prod, id, Finset.powerset, Finset.Icc, OfNat.ofNat
5 instances, coercions and auxiliary constants left out.
```
def a (n : ℕ) : ℕ :=
…
```
theorem OeisA60957.a_0 line 38 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA60957.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 OeisA60957.a_1 line 42 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA60957.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 OeisA60957.a_2 line 46 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA60957.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 OeisA60957.a_3 line 50 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA60957.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 OeisA60957.a_4 line 54 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA60957.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 OeisA60957.a_5 line 58 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA60957.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 OeisA60957.productsOfSubsets The set of products of subsets of $\{1, \dots, n\}$ line 62 source
Source documentation
The set of products of subsets of $\{1, \dots, n\}$.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Set
Proof uses: Set.ofPred, Exists, Finset, And, LE.le, Finset.Icc, OfNat.ofNat, Eq, Finset.prod, id
7 instances, coercions and auxiliary constants left out.
```
def productsOfSubsets (n : ℕ) : Set ℕ :=
…
```
theorem OeisA60957.conjecture Conjecture: let $p \le n$ be prime sorry line 71 source
Source documentation
Conjecture: let $p \le n$ be prime. If $m$ and $p^a m$ are two such products, then so is $p^k m$ for all $0 < k < a$.
- Yan Sheng Ang, Feb 13 2020
depends on sorry what this means
Statement uses: Nat, Nat.Prime, LE.le, Membership.mem, Set, OeisA60957.productsOfSubsets, HMul.hMul, HPow.hPow, LT.lt, OfNat.ofNat
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
20 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) (p : ℕ) (hp : p.Prime) (hpn : p ≤ n)
    (m a_exp : ℕ) (h1 : m ∈ productsOfSubsets n) (h2 : p ^ a_exp * m ∈ productsOfSubsets n)
    (k : ℕ) (hk1 : 0 < k) (hk2 : k < a_exp) :
    p ^ k * m ∈ productsOfSubsets n := by
…
```
Mentions: OeisA60957.productsOfSubsets