Primitive.lean`

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

12 declarations, 0 contain sorry, 0 declared as axioms

Primitive elements of a set with respect to divisibility

An element n is primitive in a set S ⊆ ℕ if n ∈ S and no proper divisor of n is in S.

This concept appears naturally when studying sequences closed under "divisor inheritance" - if every member n of a sequence has the property that m * n is also in the sequence for coprime m, then the primitive elements generate the entire sequence.

Main definitions

Set.IsPrimitive: A natural number n is primitive in S if n ∈ S and n.properDivisors is disjoint from S.
Set.primitives: The set of primitive elements of S.

Main results

Set.IsPrimitive.mem: A primitive element is in the set.
Set.IsPrimitive.not_mem_of_properDivisor: Proper divisors of a primitive element are not in the set.
Set.one_isPrimitive_iff: 1 is primitive in S iff 1 ∈ S.
Set.prime_isPrimitive_iff: A prime p is primitive in S iff p ∈ S and 1 ∉ S.

Imports (1)

Mathlib.Data.Nat.Squarefree

Imported by 1 file

FormalConjecturesForMathlib.lean

def Set.IsPrimitive An element n is primitive in a set S ⊆ ℕ if n ∈ S and no proper divisor of n is in S line 51 source
Source documentation
An element n is primitive in a set S ⊆ ℕ if n ∈ S and no proper divisor of n is in S.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Set, Nat
Proof uses: And, Membership.mem, Disjoint, Finset, Nat.properDivisors
14 instances, coercions and auxiliary constants left out.
```
def IsPrimitive (S : Set ℕ) (n : ℕ) : Prop :=
…
```
def Set.primitives The set of primitive elements of S line 55 source
Source documentation
The set of primitive elements of S.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Set, Nat
Proof uses: Set.ofPred, Set.IsPrimitive
```
def primitives (S : Set ℕ) : Set ℕ :=
…
```
theorem Set.IsPrimitive.mem line 58 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Set, Nat, Set.IsPrimitive, Membership.mem
Proof uses: And.left, Disjoint, Finset, Nat.properDivisors
15 instances, coercions and auxiliary constants left out.
theorem Set.IsPrimitive.disjoint_properDivisors line 60 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Set, Nat, Set.IsPrimitive, Disjoint, Finset, Nat.properDivisors
Proof uses: And.right, Membership.mem
27 instances, coercions and auxiliary constants left out.
theorem Set.IsPrimitive.not_mem_of_properDivisor line 63 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Set, Nat, Set.IsPrimitive, Membership.mem, Finset, Nat.properDivisors, Not
Proof uses: Iff.mp, Disjoint, Set.disjoint_left, And.right
26 instances, coercions and auxiliary constants left out.
theorem Set.IsPrimitive.not_mem_of_dvd_of_lt line 67 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Set, Nat, Set.IsPrimitive, Dvd.dvd, LT.lt, Not, Membership.mem
Proof uses: Set.IsPrimitive.not_mem_of_properDivisor, Iff.mpr, Finset, Nat.properDivisors, And, Nat.mem_properDivisors, And.intro
7 instances, coercions and auxiliary constants left out.
theorem Set.isPrimitive_iff line 71 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Set, Nat, Iff, Set.IsPrimitive, And, Membership.mem, Finset, Nat.properDivisors, Not
Proof uses: of_eq_true, Eq.trans, True, congrFun', congrArg, Disjoint, forall_congr, implies_congr, Eq.refl, iff_self
28 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
theorem Set.isPrimitive_iff' line 75 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Set, Nat, LT.lt, OfNat.ofNat, Iff, Set.IsPrimitive, And, Membership.mem, Dvd.dvd, Not
Proof uses: Eq.mpr, Finset, Nat.properDivisors, id, Eq, congrArg, propext, Set.isPrimitive_iff, Iff.intro, And.intro, Iff.mpr, Nat.mem_properDivisors, And.left, Iff.mp, And.right
12 instances, coercions and auxiliary constants left out.

Tactic trace: 7 steps
Open to load the checked before-and-after goal states.
theorem Set.mem_primitives_iff line 82 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Set, Nat, Iff, Membership.mem, Set.primitives, Set.IsPrimitive
Proof uses: Iff.rfl
2 instances, coercions and auxiliary constants left out.
theorem Set.primitives_subset line 85 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Set, Nat, LE.le, Set.primitives
Proof uses: Membership.mem, Set.IsPrimitive.mem
2 instances, coercions and auxiliary constants left out.
theorem Set.one_isPrimitive_iff line 87 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Set, Nat, Iff, Set.IsPrimitive, OfNat.ofNat, Membership.mem
Proof uses: of_eq_true, Eq.trans, True, congrFun', congrArg, And, Finset, Nat.properDivisors, Not, forall_congr, False, implies_congr, EmptyCollection.emptyCollection, Nat.properDivisors_one, Eq.refl, implies_true, and_true, iff_self
11 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
theorem Set.prime_isPrimitive_iff line 90 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Set, Nat, Nat.Prime, Iff, Set.IsPrimitive, And, Membership.mem, Not, OfNat.ofNat
Proof uses: of_eq_true, Eq.trans, True, congrFun', congrArg, Finset, Nat.properDivisors, Eq, forall_congr, implies_congr, Singleton.singleton, Nat.Prime.properDivisors, Eq.refl, iff_self
10 instances, coercions and auxiliary constants left out.

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