361.lean`

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

6 declarations, 1 contain sorry, 0 declared as axioms

Erdős Problem 361

Reference: erdosproblems.com/361

Imports (1)

FormalConjecturesUtil

def Erdos361.AvoidsSubsetSum AvoidsSubsetSum n A means that no subset $B \subseteq A$ has sum equal to $n$ line 33 source
Source documentation
AvoidsSubsetSum n A means that no subset $B \subseteq A$ has sum equal to $n$.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Finset
Proof uses: Membership.mem, Finset.powerset, Ne, Finset.sum
3 instances, coercions and auxiliary constants left out.

Tactic trace: 2 steps
Open to load the checked before-and-after goal states.
```
def AvoidsSubsetSum (n : ℕ) (A : Finset ℕ) : Prop :=
…
```
instance Erdos361.(instance at line 36) line 36 source
def Erdos361.maxSubsetSumAvoidingCard The largest cardinality of a set $A \subseteq \{1, \ldots, N\}$ such that no subset of $A$ sums to $n$ line 44 source
Source documentation
The largest cardinality of a set $A \subseteq \{1, \ldots, N\}$ such that no subset of $A$ sums to $n$.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat
Proof uses: Finset.sup, Finset, Finset.filter, Erdos361.AvoidsSubsetSum, Finset.powerset, Finset.Icc, OfNat.ofNat, Finset.card
7 instances, coercions and auxiliary constants left out.
```
def maxSubsetSumAvoidingCard (N n : ℕ) : ℕ :=
…
```
def Erdos361.subsetSumAvoidanceNumber For fixed $c$ and $n$, this is the size of the largest set $A \subseteq \{1, \ldots, \lfloor cn \rfloor\}$ such that no subset of $A$ sums to $n$ line 51 source
Source documentation
For fixed $c$ and $n$, this is the size of the largest set $A \subseteq \{1, \ldots, \lfloor cn \rfloor\}$ such that no subset of $A$ sums to $n$.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Real, Nat
Proof uses: Erdos361.maxSubsetSumAvoidingCard, Nat.floor, HMul.hMul
10 instances, coercions and auxiliary constants left out.
```
noncomputable def subsetSumAvoidanceNumber (c : ℝ) (n : ℕ) : ℕ :=
…
```
Mentions: c
theorem Erdos361.maxSubsetSumAvoidingCard_three_four For target $4$ and universe $\{1, 2, 3\}$, the maximum is $2$: the full set is invalid because its subset $\{1, 3\}$ sums to $4$ line 59 source
Source documentation
For target $4$ and universe $\{1, 2, 3\}$, the maximum is $2$: the full set is invalid because its subset $\{1, 3\}$ sums to $4$.

extra axioms: _private.0.Erdos361.maxSubsetSumAvoidingCard_three_four._native.native_decide.ax_1 what this means
Statement uses: Eq, Nat, Erdos361.maxSubsetSumAvoidingCard, OfNat.ofNat
Proof uses: of_decide_eq_true
4 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem maxSubsetSumAvoidingCard_three_four : maxSubsetSumAvoidingCard 3 4 = 2 := by
…
```
Mentions: Erdos361.maxSubsetSumAvoidingCard
theorem Erdos361.erdos_361 Let $c > 0$ and $n$ be some large integer sorry line 68 source
Source documentation
Let $c > 0$ and $n$ be some large integer. What is the size of the largest set $A \subseteq \{1, \ldots, \lfloor c n \rfloor\}$ such that $n$ is not a sum of a subset of $A$? Does this depend on $n$ in an irregular way?

depends on sorry what this means
Statement uses: Real, LT.lt, OfNat.ofNat, Eq, Nat, Erdos361.subsetSumAvoidanceNumber, 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.
```
theorem erdos_361 (c : ℝ) (hc : 0 < c) :
    subsetSumAvoidanceNumber c = answer(sorry) := by
…
```
Mentions: c, Erdos361.subsetSumAvoidanceNumber