219.lean`

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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

7 declarations, 1 contain sorry, 0 declared as axioms

Erdős Problem 219

Reference: erdosproblems.com/219

Imports (1)

FormalConjecturesUtil

def Erdos219.primeArithmeticProgressions The set of arithmetic progressions of primes line 33 source
Source documentation
The set of arithmetic progressions of primes

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Set, Nat
Proof uses: Set.ofPred, And, Membership.mem, Nat.Prime, Exists, ENat, GT.gt, OfNat.ofNat, Set.IsAPOfLength
8 instances, coercions and auxiliary constants left out.
```
def primeArithmeticProgressions : Set (Set ℕ) :=
…
```
theorem Erdos219.primeArithmeticProgression_3_5_7 line 37 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Membership.mem, Set, Nat, Erdos219.primeArithmeticProgressions, Insert.insert, OfNat.ofNat, Singleton.singleton
Proof uses: Eq.mpr, And, Nat.Prime, Exists, Eq, Set.ofPred, LT.lt, HAdd.hAdd, HMul.hMul, id, Eq.trans, ENat, ENat.card, congrFun', congrArg, GT.gt, Set.IsAPOfLength, funext, congr, Set.IsAPOfLengthWith and 63 more
103 instances, coercions and auxiliary constants left out.

Tactic trace: 15 steps
Open to load the checked before-and-after goal states.
theorem Erdos219.not_primeArithmeticProgression_1_2 line 53 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Not, Membership.mem, Set, Nat, Erdos219.primeArithmeticProgressions, Insert.insert, OfNat.ofNat, Singleton.singleton
Proof uses: of_eq_true, Eq.trans, False, True, congrArg, Set.ofPred, And, Nat.Prime, Exists, ENat, LT.lt, Set.IsAPOfLength, congrFun', GT.gt, funext, Eq, Or, forall_congr, implies_congr, Eq.refl and 3 more
21 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
theorem Erdos219.empty_not_primeArithmeticProgression line 57 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Not, Membership.mem, Set, Nat, Erdos219.primeArithmeticProgressions, EmptyCollection.emptyCollection
Proof uses: Eq.mpr, ENat, LT.lt, OfNat.ofNat, Set.IsAPOfLength, id, Eq, Eq.trans, And, Exists, congrArg, Set.ofPred, Nat.Prime, congrFun', GT.gt, funext, True, forall_congr, False, implies_congr and 4 more
17 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
lemma Erdos219.singleton_mem_primeArithmeticProgressions line 61 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Nat.Prime, Membership.mem, Set, Erdos219.primeArithmeticProgressions, Singleton.singleton
Proof uses: Eq.mpr, Exists, ENat, And, LT.lt, OfNat.ofNat, Set.IsAPOfLength, id, Eq, Eq.trans, Set.ofPred, congrFun', congrArg, GT.gt, funext, True, forall_congr, implies_congr, Eq.refl, eq_true and 5 more
28 instances, coercions and auxiliary constants left out.

Tactic trace: 2 steps
Open to load the checked before-and-after goal states.
lemma Erdos219.pair_mem_primeArithmeticProgressions line 66 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Nat.Prime, LT.lt, Membership.mem, Set, Erdos219.primeArithmeticProgressions, Insert.insert, Singleton.singleton
Proof uses: Exists, Eq, HAdd.hAdd, OfNat.ofNat, Nat.exists_eq_add_of_lt, Eq.mpr, ENat, And, Set.IsAPOfLength, id, Eq.trans, Set.ofPred, congrFun', congrArg, GT.gt, funext, True, Or, forall_congr, implies_congr and 14 more
37 instances, coercions and auxiliary constants left out.

Tactic trace: 3 steps
Open to load the checked before-and-after goal states.
theorem Erdos219.erdos_219 sorry line 80 source
depends on sorry what this means
Statement uses: Iff, True, Nat, Exists, Set, And, Membership.mem, Erdos219.primeArithmeticProgressions, LE.le, ENat, ENat.card
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, OfNat.ofNat
11 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.