BeurlingPrimes.lean`

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

8 declarations, 0 contain sorry, 0 declared as axioms

Beurling primes

References:

Imports (3)

Imported by 1 file

FormalConjecturesForMathlib.lean

def IsBeurlingPrimes A sequence of real numbers 1 < a 0 < a 1 < ... is called a set of Beurling prime numbers if it tends to infinity line 36 source
Source documentation
A sequence of real numbers 1 < a 0 < a 1 < ... is called a set of Beurling prime numbers if it tends to infinity.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Real
Proof uses: And, LT.lt, OfNat.ofNat, StrictMono, Filter.Tendsto, Filter.atTop
6 instances, coercions and auxiliary constants left out.
```
noncomputable def IsBeurlingPrimes (a : ℕ → ℝ) : Prop :=
…
```
def beurlingInteger A Beurling integer is a number of the form ∏ i, (a i) ^ (k i) for a given sequence a and a finitely-supported sequence of naturals k line 41 source
Source documentation
A Beurling integer is a number of the form ∏ i, (a i) ^ (k i) for a given sequence a and a finitely-supported sequence of naturals k.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Real, Finsupp
Proof uses: Finsupp.prod, HPow.hPow
9 instances, coercions and auxiliary constants left out.
```
def beurlingInteger (a : ℕ → ℝ) (k : ℕ →₀ ℕ) : ℝ := k.prod fun x y ↦ (a x) ^ y
```
theorem beurlingInteger_def line 43 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Real, Finsupp, Eq, beurlingInteger, Finsupp.prod, HPow.hPow
Proof uses: rfl
9 instances, coercions and auxiliary constants left out.
def BeurlingIntegers The set of Beurling integers are numbers of the form ∏ i, (a i) ^ (k i), where k has finite support line 47 source
Source documentation
The set of Beurling integers are numbers of the form ∏ i, (a i) ^ (k i), where k has finite support.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Real, Set
Proof uses: Set.range, Finsupp, beurlingInteger
2 instances, coercions and auxiliary constants left out.
```
def BeurlingIntegers (a : ℕ → ℝ) : Set ℝ := .range (beurlingInteger a)
```
Mentions: beurlingInteger
theorem beurlingInteger_mem line 49 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Real, Finsupp, Membership.mem, Set, BeurlingIntegers, beurlingInteger
Proof uses: id, Exists.intro, Eq, Finsupp.prod, HPow.hPow, rfl
11 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
lemma generator_mem_beurling Every element of the sequence a is a Beurling integer line 53 source
Source documentation
Every element of the sequence a is a Beurling integer.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Real, Membership.mem, Set, BeurlingIntegers
Proof uses: Exists.intro, Finsupp, Eq, beurlingInteger, Finsupp.single, OfNat.ofNat, of_eq_true, Eq.trans, True, congrFun', congrArg, Finsupp.prod, HPow.hPow, Finsupp.prod_single_index, pow_zero, eq_self, pow_one
15 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
lemma generator_mem_beurling (a : ℕ → ℝ) (i : ℕ) : a i ∈ BeurlingIntegers a :=
…
```
Mentions: BeurlingIntegers
lemma mul_mem_beurling The set of Beurling integers is closed under multiplication line 57 source
Source documentation
The set of Beurling integers is closed under multiplication.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Real, Membership.mem, Set, BeurlingIntegers, HMul.hMul
Proof uses: Finsupp, Eq, beurlingInteger, Exists.intro, HAdd.hAdd, of_eq_true, Finsupp.prod, HPow.hPow, Eq.trans, True, congrFun', congrArg, Finsupp.prod_add_index', OfNat.ofNat, forall_congr, pow_zero, eq_self, implies_true, pow_add
29 instances, coercions and auxiliary constants left out.

Tactic trace: 4 steps
Open to load the checked before-and-after goal states.
```
lemma mul_mem_beurling {a : ℕ → ℝ} {x y : ℝ} (hx : x ∈ BeurlingIntegers a)
    (hy : y ∈ BeurlingIntegers a) : x * y ∈ BeurlingIntegers a := by
…
```
Mentions: BeurlingIntegers
lemma pow_mem_beurling The set of Beurling integers is closed under taking powers line 64 source
Source documentation
The set of Beurling integers is closed under taking powers.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Real, Membership.mem, Set, BeurlingIntegers, HPow.hPow
Proof uses: Nat.recAux, Exists.intro, Finsupp, Eq, beurlingInteger, OfNat.ofNat, of_eq_true, Eq.trans, True, congrArg, pow_zero, eq_self, Eq.mpr, HAdd.hAdd, HMul.hMul, id, pow_succ', mul_mem_beurling
25 instances, coercions and auxiliary constants left out.

Tactic trace: 4 steps
Open to load the checked before-and-after goal states.
```
lemma pow_mem_beurling {a : ℕ → ℝ} {x : ℝ} (k : ℕ) (hx : x ∈ BeurlingIntegers a) :
    x ^ k ∈ BeurlingIntegers a := by
…
```
Mentions: BeurlingIntegers