Divisors.lean`

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2 declarations, 0 contain sorry, 0 declared as axioms

The increasing enumeration of the divisors of a natural number

Basic facts about Nat.nth (· ∈ n.divisors), the increasing enumeration of the divisors of n.

Imports (2)

Imported by 1 file

FormalConjecturesForMathlib.lean

lemma Nat.nth_divisors_zero The smallest divisor of a positive number is 1, i.e. the 0th entry of the increasing enumeration of its divisors line 33 source
Source documentation
The smallest divisor of a positive number is 1, i.e. the 0th entry of the increasing enumeration of its divisors.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Ne, OfNat.ofNat, Eq, Nat.nth, Membership.mem, Finset, Nat.divisors
Proof uses: Eq.mpr, InfSet.sInf, Set.ofPred, id, congrArg, Nat.nth_zero, IsLeast.csInf_eq, And.intro, Set, lowerBounds, Iff.mpr, Nat.one_mem_divisors, Nat.pos_of_mem_divisors
16 instances, coercions and auxiliary constants left out.

Tactic trace: 2 steps
Open to load the checked before-and-after goal states.
```
lemma nth_divisors_zero {n : ℕ} (hn : n ≠ 0) : Nat.nth (· ∈ n.divisors) 0 = 1 := by
…
```
lemma Nat.two_le_nth_divisors Every divisor enumerated after index 0 is at least 2 line 38 source
Source documentation
Every divisor enumerated after index 0 is at least 2.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Ne, OfNat.ofNat, Nat.nth, Membership.mem, Finset, Nat.divisors, LE.le
Proof uses: Set.Finite, Set.ofPred, Finset.finite_toSet, Nat.pos_of_mem_divisors, Nat.nth_mem_of_ne_zero, LT.lt, Eq, Or, LE.le.lt_or_eq, Decidable.byContradiction, Not, absurd, Nat.nth_injOn, Iff.mpr, Set, Set.Iio, Finset.card, Set.Finite.toFinset, Set.mem_Iio, Nat.lt_card_toFinset_of_nth_ne_zero and 4 more
19 instances, coercions and auxiliary constants left out.

Tactic trace: 11 steps
Open to load the checked before-and-after goal states.
```
lemma two_le_nth_divisors {n : ℕ} (hn : n ≠ 0) {i : ℕ} (hi : i ≠ 0)
    (h : Nat.nth (· ∈ n.divisors) i ≠ 0) : 2 ≤ Nat.nth (· ∈ n.divisors) i := by
…
```