350.lean`

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

7 declarations, 2 contain sorry, 0 declared as axioms

Erdős Problem 350

References:

erdosproblems.com/350
[BeEr74] Benkoski, S. J. and Erdős, P., On weird and pseudoperfect numbers. Math. Comp. (1974), 617-623.
[HSS77] Hanson, F. and Steele, J. M. and Stenger, F., Distinct sums over subsets. Proc. Amer. Math. Soc. (1977), 179-180.

Imports (1)

FormalConjecturesUtil

Imported by 1 file

FormalConjectures/Subsets/FC100SolvedSet1.lean

def Erdos350.DistinctSubsetSums The predicate that all (finite) subsets of A have distinct sums line 36 source
Source documentation
The predicate that all (finite) subsets of A have distinct sums.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: AddCommMonoid, Set
Proof uses: Set.Pairwise, Finset, Set.ofPred, LE.le, Ne, Finset.sum, id
3 instances, coercions and auxiliary constants left out.
```
def DistinctSubsetSums {M : Type*} [AddCommMonoid M] (A : Set M) : Prop :=
…
```
def Erdos350.DecidableDistinctSubsetSums The predicate that all (finite) subsets of A have distinct sums, decidable version line 40 source
Source documentation
The predicate that all (finite) subsets of A have distinct sums, decidable version

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: AddCommMonoid, DecidableEq, Finset
Proof uses: LE.le, Ne, Finset.sum, id
3 instances, coercions and auxiliary constants left out.
```
def DecidableDistinctSubsetSums {M : Type*} [AddCommMonoid M] [DecidableEq M] (A : Finset M) : Prop :=
…
```
theorem Erdos350.decidableDistinctSubsetSums_1_2 line 44 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Erdos350.DecidableDistinctSubsetSums, Nat, Insert.insert, Finset, OfNat.ofNat, Singleton.singleton
Proof uses: Eq.mpr, LE.le, Ne, Finset.sum, id, Eq, congrArg, Erdos350.DecidableDistinctSubsetSums.eq_1, of_decide_eq_true, Bool, Decidable.decide, Bool.true, Eq.refl
17 instances, coercions and auxiliary constants left out.

Tactic trace: 2 steps
Open to load the checked before-and-after goal states.
theorem Erdos350.distinctSubsetSums_1_2 line 48 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Erdos350.DistinctSubsetSums, Nat, Insert.insert, Set, OfNat.ofNat, Singleton.singleton
Proof uses: Eq.mpr, Finset, LE.le, Not, Eq, Finset.sum, id, forall_congr, Ne, implies_congr, Eq.refl, congr, congrArg, Finset.sum_congr, Membership.mem, List, List.Mem, List.pmap, Multiset, Multiset.Nodup and 38 more
52 instances, coercions and auxiliary constants left out.

Tactic trace: 6 steps
Open to load the checked before-and-after goal states.
theorem Erdos350.DistinctSubsetSums_iff_DecidableDistinctSubsetSums Small sanity check: the two predicates are saying the same thing line 57 source
Source documentation
Small sanity check: the two predicates are saying the same thing.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: AddCommMonoid, DecidableEq, Finset, Iff, Erdos350.DistinctSubsetSums, Erdos350.DecidableDistinctSubsetSums
Proof uses: Eq.mpr, Set.Pairwise, Set.ofPred, LE.le, Set, Ne, Finset.sum, id, Eq, congrArg, Erdos350.DistinctSubsetSums.eq_1, Erdos350.DecidableDistinctSubsetSums.eq_1, Membership.mem, Set.Pairwise.eq_1, of_eq_true, Eq.trans, Not, True, congr, forall_congr and 6 more
12 instances, coercions and auxiliary constants left out.

Tactic trace: 2 steps
Open to load the checked before-and-after goal states.
```
theorem DistinctSubsetSums_iff_DecidableDistinctSubsetSums
    {M : Type*} [AddCommMonoid M] [DecidableEq M] (A : Finset M) :
    DistinctSubsetSums (A : Set M) ↔ DecidableDistinctSubsetSums A := by
…
```
Mentions: Erdos350.DistinctSubsetSums, Erdos350.DecidableDistinctSubsetSums
theorem Erdos350.erdos_350 sorry line 76 source
depends on sorry what this means
Statement uses: Finset, Nat, Erdos350.DecidableDistinctSubsetSums, LT.lt, Real, Finset.sum, HDiv.hDiv, OfNat.ofNat
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
30 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
theorem Erdos350.erdos_350.variants.strengthening If A ⊂ ℕ is a finite set of integers all of whose subset sums are distinct then ∑ n ∈ A, 1/n^s < 1/(1 - 2^(-s)), for any s > 0 sorry line 87 source
Source documentation
If A ⊂ ℕ is a finite set of integers all of whose subset sums are distinct then ∑ n ∈ A, 1/n^s < 1/(1 - 2^(-s)), for any s > 0. Proved by Hanson, Steele, and Stenger [HSS77].
We exclude here the case s = 0, because in the informal formulation then the right hand side is to be interpreted as ∞, while the left hand side counts the elements in A.

depends on sorry what this means
Statement uses: Finset, Nat, Erdos350.DecidableDistinctSubsetSums, Real, LT.lt, OfNat.ofNat, Finset.sum, HPow.hPow, HDiv.hDiv, HSub.hSub, Neg.neg
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
44 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_350.variants.strengthening (A : Finset ℕ) (hA : DecidableDistinctSubsetSums A)
    (s : ℝ) (hs : 0 < s) : ∑ n ∈ A, (1 / n : ℝ)^s < 1 / (1 - 2^(-s)) := by
…
```
Mentions: Erdos350.DecidableDistinctSubsetSums