Loogle!

Result

Found 14 declarations mentioning ArithmeticFunction.cardDistinctFactors.

• ArithmeticFunction.cardDistinctFactors 📋 Mathlib.NumberTheory.ArithmeticFunction.Misc
  : ArithmeticFunction ℕ
• ArithmeticFunction.cardDistinctFactors_apply 📋 Mathlib.NumberTheory.ArithmeticFunction.Misc
  {n : ℕ} : ArithmeticFunction.cardDistinctFactors n = n.primeFactorsList.dedup.length
• ArithmeticFunction.cardDistinctFactors_apply_prime 📋 Mathlib.NumberTheory.ArithmeticFunction.Misc
  {p : ℕ} (hp : Nat.Prime p) : ArithmeticFunction.cardDistinctFactors p = 1
• ArithmeticFunction.cardDistinctFactors_one 📋 Mathlib.NumberTheory.ArithmeticFunction.Misc
  : ArithmeticFunction.cardDistinctFactors 1 = 0
• ArithmeticFunction.cardDistinctFactors_zero 📋 Mathlib.NumberTheory.ArithmeticFunction.Misc
  : ArithmeticFunction.cardDistinctFactors 0 = 0
• ArithmeticFunction.cardDistinctFactors_eq_one_iff 📋 Mathlib.NumberTheory.ArithmeticFunction.Misc
  {n : ℕ} : ArithmeticFunction.cardDistinctFactors n = 1 ↔ IsPrimePow n
• ArithmeticFunction.cardDistinctFactors_eq_zero 📋 Mathlib.NumberTheory.ArithmeticFunction.Misc
  {n : ℕ} : ArithmeticFunction.cardDistinctFactors n = 0 ↔ n ≤ 1
• ArithmeticFunction.cardDistinctFactors_pos 📋 Mathlib.NumberTheory.ArithmeticFunction.Misc
  {n : ℕ} : 0 < ArithmeticFunction.cardDistinctFactors n ↔ 1 < n
• ArithmeticFunction.cardDistinctFactors_apply_prime_pow 📋 Mathlib.NumberTheory.ArithmeticFunction.Misc
  {p k : ℕ} (hp : Nat.Prime p) (hk : k ≠ 0) : ArithmeticFunction.cardDistinctFactors (p ^ k) = 1
• ArithmeticFunction.cardDistinctFactors_eq_cardFactors_iff_squarefree 📋 Mathlib.NumberTheory.ArithmeticFunction.Misc
  {n : ℕ} (h0 : n ≠ 0) : ArithmeticFunction.cardDistinctFactors n = ArithmeticFunction.cardFactors n ↔ Squarefree n
• ArithmeticFunction.cardDistinctFactors_prod 📋 Mathlib.NumberTheory.ArithmeticFunction.Misc
  {ι : Type u_2} {s : Finset ι} {f : ι → ℕ} (h : (↑s).Pairwise (Function.onFun Nat.Coprime f)) : ArithmeticFunction.cardDistinctFactors (∏ i ∈ s, f i) = ∑ i ∈ s, ArithmeticFunction.cardDistinctFactors (f i)
• ArithmeticFunction.cardDistinctFactors_mul 📋 Mathlib.NumberTheory.ArithmeticFunction.Misc
  {m n : ℕ} (h : m.Coprime n) : ArithmeticFunction.cardDistinctFactors (m * n) = ArithmeticFunction.cardDistinctFactors m + ArithmeticFunction.cardDistinctFactors n
• Nat.card_finMulAntidiag_of_squarefree 📋 Mathlib.Algebra.Order.Antidiag.Nat
  {d n : ℕ} (hn : Squarefree n) : (d.finMulAntidiag n).card = d ^ ArithmeticFunction.cardDistinctFactors n
• Nat.card_pair_lcm_eq 📋 Mathlib.Algebra.Order.Antidiag.Nat
  {n : ℕ} (hn : Squarefree n) : {p ∈ n.divisors ×ˢ n.divisors | p.1.lcm p.2 = n}.card = 3 ^ ArithmeticFunction.cardDistinctFactors n

About

Loogle searches Lean and Mathlib definitions and theorems.

You can use Loogle from within the Lean4 VSCode language extension using (by default) Ctrl-K Ctrl-S. You can also try the #loogle command from LeanSearchClient, the CLI version, the Loogle VS Code extension, the lean.nvim integration or the Zulip bot.

Usage

Loogle finds definitions and lemmas in various ways:

1. By constant:
   🔍 Real.sin
   finds all lemmas whose statement somehow mentions the sine function.

2. By lemma name substring:
   🔍 "differ"
   finds all lemmas that have "differ" somewhere in their lemma name.

3. By subexpression:
   🔍 _ * (_ ^ _)
   finds all lemmas whose statements somewhere include a product where the second argument is raised to some power.

   The pattern can also be non-linear, as in
   🔍 Real.sqrt ?a * Real.sqrt ?a

   If the pattern has parameters, they are matched in any order. Both of these will find List.map:
   🔍 (?a -> ?b) -> List ?a -> List ?b
   🔍 List ?a -> (?a -> ?b) -> List ?b

4. By main conclusion:
   🔍 |- tsum _ = _ * tsum _
   finds all lemmas where the conclusion (the subexpression to the right of all → and ∀) has the given shape.

   As before, if the pattern has parameters, they are matched against the hypotheses of the lemma in any order; for example,
   🔍 |- _ < _ → tsum _ < tsum _
   will find tsum_lt_tsum even though the hypothesis f i < g i is not the last.

5. You can filter for definitions vs theorems: Using ⊢ (_ : Type _) finds all definitions which provide data while ⊢ (_ : Prop) finds all theorems (and definitions of proofs).

If you pass more than one such search filter, separated by commas Loogle will return lemmas which match all of them. The search
🔍 Real.sin, "two", tsum, _ * _, _ ^ _, |- _ < _ → _
would find all lemmas which mention the constants Real.sin and tsum, have "two" as a substring of the lemma name, include a product and a power somewhere in the type, and have a hypothesis of the form _ < _ (if there were any such lemmas). Metavariables (?a) are assigned independently in each filter.

The #lucky button will directly send you to the documentation of the first hit.

Source code

You can find the source code for this service at https://github.com/nomeata/loogle. The https://loogle.lean-lang.org/ service is provided by the Lean FRO.

This is Loogle revision 128218b serving mathlib revision 4644b1d