1059.lean`

Explanation legend Lean4 Domain Explanation Language (L4DEL)

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L4DEL: Lean4 Domain Explanation Language

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Credit: The upstream authors and contributors retain credit for the project and mathematics. lean4-explainer only renders their pinned source and adds the separately disclosed explanation layer.

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

14 declarations, 1 contain sorry, 0 declared as axioms

Erdős Problem 1059

Reference: erdosproblems.com/1059

Imports (1)

FormalConjecturesUtil

def Erdos1059.IsFactorial line 30 source
no axioms what this means
Statement uses: Nat
Proof uses: Membership.mem, Set, Set.range, Nat.factorial
1 instances, coercions and auxiliary constants left out.
def Erdos1059.factorialsLessThanN line 33 source
no axioms what this means
Statement uses: Nat, Set
Proof uses: Set.ofPred, And, LT.lt, Erdos1059.IsFactorial
1 instances, coercions and auxiliary constants left out.
def Erdos1059.AllFactorialSubtractionsComposite line 36 source
standard axioms: propext what this means
Statement uses: Nat
Proof uses: Membership.mem, Set, Erdos1059.factorialsLessThanN, Nat.Composite, HSub.hSub
3 instances, coercions and auxiliary constants left out.
theorem Erdos1059.erdos_1059 Are there infinitely many primes $p$ such that $p - k!$ is composite for each $k$ such that $1 ≤ k! < p$? sorry line 41 source
Source documentation
Are there infinitely many primes $p$ such that $p - k!$ is composite for each $k$ such that $1 ≤ k! < p$?

depends on sorry what this means
Statement uses: Iff, True, Set.Infinite, Nat, Set.ofPred, And, Nat.Prime, Erdos1059.AllFactorialSubtractionsComposite
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, OfNat.ofNat
1 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_1059 :
    answer(sorry) ↔ Set.Infinite {p | p.Prime ∧ AllFactorialSubtractionsComposite p} := by
…
```
Mentions: Erdos1059.AllFactorialSubtractionsComposite
abbrev Erdos1059.DecidableIsFactorial line 45 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat
Proof uses: Finset.Nonempty, Finset.filter, Eq, Nat.factorial, Finset.Icc, OfNat.ofNat
4 instances, coercions and auxiliary constants left out.
def Erdos1059.decidableFactorialsLessThanN line 48 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Finset
Proof uses: Finset.filter, Erdos1059.DecidableIsFactorial, Eq, Nat.factorial, Finset.Icc, OfNat.ofNat, Finset.range
5 instances, coercions and auxiliary constants left out.
def Erdos1059.DecidableAllFactorialSubtractionsComposite line 51 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat
Proof uses: Membership.mem, Finset, Erdos1059.decidableFactorialsLessThanN, Nat.Composite, HSub.hSub
4 instances, coercions and auxiliary constants left out.
lemma Erdos1059.isFactorial_equivalent line 55 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Iff, Erdos1059.IsFactorial, Erdos1059.DecidableIsFactorial
Proof uses: id, Membership.mem, Set, Set.range, Nat.factorial, Eq.mpr, Finset.Nonempty, Finset.filter, Eq, Finset.Icc, OfNat.ofNat, Exists, congrFun', congrArg, Iff.intro, Exists.intro, Finset, And, propext, Finset.mem_filter and 8 more
11 instances, coercions and auxiliary constants left out.

Tactic trace: 21 steps
Open to load the checked before-and-after goal states.
lemma Erdos1059.factorialsLessThanN_equivalent line 76 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Eq, Set, Erdos1059.factorialsLessThanN, Finset, Erdos1059.decidableFactorialsLessThanN
Proof uses: Set.ext, id, Iff, Membership.mem, Set.ofPred, And, LT.lt, Erdos1059.IsFactorial, Eq.mpr, Finset.filter, Erdos1059.DecidableIsFactorial, Nat.factorial, Finset.Icc, OfNat.ofNat, Finset.range, Eq.trans, congrArg, congrFun', Finset.coe_filter, funext and 1 more
14 instances, coercions and auxiliary constants left out.

Tactic trace: 4 steps
Open to load the checked before-and-after goal states.
lemma Erdos1059.allFactorialSubtractionsComposite_equivalent line 84 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Iff, Erdos1059.DecidableAllFactorialSubtractionsComposite, Erdos1059.AllFactorialSubtractionsComposite
Proof uses: id, Membership.mem, Set, Erdos1059.factorialsLessThanN, Nat.Composite, HSub.hSub, Eq.mpr, Finset, Erdos1059.decidableFactorialsLessThanN, Eq, congrArg, Erdos1059.factorialsLessThanN_equivalent, of_eq_true, Eq.trans, True, forall_congr, implies_congr, Eq.refl, iff_self
7 instances, coercions and auxiliary constants left out.

Tactic trace: 3 steps
Open to load the checked before-and-after goal states.
theorem Erdos1059.allFactorialSubtractionsComposite_101 line 91 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Erdos1059.AllFactorialSubtractionsComposite, OfNat.ofNat, Nat
Proof uses: Erdos1059.DecidableAllFactorialSubtractionsComposite, Eq.mpr, LT.lt, Erdos1059.DecidableIsFactorial, Nat.Composite, HSub.hSub, id, Eq, forall_congr, Membership.mem, Finset, Finset.filter, Nat.factorial, Finset.Icc, Finset.range, Eq.trans, And, implies_congr, congrFun', congrArg and 3 more
15 instances, coercions and auxiliary constants left out.

Tactic trace: 4 steps
Open to load the checked before-and-after goal states.
theorem Erdos1059.allFactorialSubtractionsComposite_211 line 98 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Erdos1059.AllFactorialSubtractionsComposite, OfNat.ofNat, Nat
Proof uses: Erdos1059.DecidableAllFactorialSubtractionsComposite, Eq.mpr, LT.lt, Erdos1059.DecidableIsFactorial, Nat.Composite, HSub.hSub, id, Eq, forall_congr, Membership.mem, Finset, Finset.filter, Nat.factorial, Finset.Icc, Finset.range, Eq.trans, And, implies_congr, congrFun', congrArg and 3 more
15 instances, coercions and auxiliary constants left out.

Tactic trace: 4 steps
Open to load the checked before-and-after goal states.
theorem Erdos1059.notAllFactorialSubtractionsComposite_89 line 105 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Not, Erdos1059.AllFactorialSubtractionsComposite, OfNat.ofNat, Nat
Proof uses: Erdos1059.DecidableAllFactorialSubtractionsComposite, id, Membership.mem, Finset, Erdos1059.decidableFactorialsLessThanN, Nat.Composite, HSub.hSub, Finset.filter, Erdos1059.DecidableIsFactorial, Eq, Nat.factorial, Finset.Icc, Finset.range, Nat.Prime, Mathlib.Meta.NormNum.isNat_prime_2, Mathlib.Meta.NormNum.isNat_natSub, Mathlib.Meta.NormNum.isNat_ofNat, Eq.refl, Bool, Bool.true and 14 more
19 instances, coercions and auxiliary constants left out.

Tactic trace: 9 steps
Open to load the checked before-and-after goal states.
theorem Erdos1059.testFactorialsLessThanN line 116 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Set, Nat, Erdos1059.factorialsLessThanN, OfNat.ofNat, Insert.insert, Singleton.singleton
Proof uses: Finset, Erdos1059.decidableFactorialsLessThanN, id, Eq.mpr, congrArg, Erdos1059.factorialsLessThanN_equivalent, of_eq_true, Eq.trans, True, congrFun', Finset.coe_insert, Finset.coe_singleton, eq_self
12 instances, coercions and auxiliary constants left out.

Tactic trace: 5 steps
Open to load the checked before-and-after goal states.