Loogle!

Result

Found 189 declarations mentioning Nat.Prime and Iff. Of these, 12 match your pattern(s).

• Nat.prime_iff 📋 Mathlib.Data.Nat.Prime.Defs
  {p : ℕ} : Nat.Prime p ↔ Prime p
• Nat.prime_def_minFac 📋 Mathlib.Data.Nat.Prime.Defs
  {p : ℕ} : Nat.Prime p ↔ 2 ≤ p ∧ p.minFac = p
• Nat.prime_def 📋 Mathlib.Data.Nat.Prime.Defs
  {p : ℕ} : Nat.Prime p ↔ 2 ≤ p ∧ ∀ (m : ℕ), m ∣ p → m = 1 ∨ m = p
• Nat.prime_def_lt 📋 Mathlib.Data.Nat.Prime.Defs
  {p : ℕ} : Nat.Prime p ↔ 2 ≤ p ∧ ∀ m < p, m ∣ p → m = 1
• Nat.prime_def_lt'' 📋 Mathlib.Data.Nat.Prime.Defs
  {p : ℕ} : Nat.Prime p ↔ 2 ≤ p ∧ ∀ (m : ℕ), m ∣ p → m = 1 ∨ m = p
• Nat.prime_def_lt' 📋 Mathlib.Data.Nat.Prime.Defs
  {p : ℕ} : Nat.Prime p ↔ 2 ≤ p ∧ ∀ (m : ℕ), 2 ≤ m → m < p → ¬m ∣ p
• Nat.prime_def_le_sqrt 📋 Mathlib.Data.Nat.Prime.Defs
  {p : ℕ} : Nat.Prime p ↔ 2 ≤ p ∧ ∀ (m : ℕ), 2 ≤ m → m ≤ p.sqrt → ¬m ∣ p
• Nat.prime_mul_iff 📋 Mathlib.Data.Nat.Prime.Basic
  {a b : ℕ} : Nat.Prime (a * b) ↔ Nat.Prime a ∧ b = 1 ∨ Nat.Prime b ∧ a = 1
• Nat.prime_iff_card_units 📋 Mathlib.Data.Nat.Totient
  (p : ℕ) [Fintype (ZMod p)ˣ] : Nat.Prime p ↔ Fintype.card (ZMod p)ˣ = p - 1
• Nat.prime_iff_prime_int 📋 Mathlib.Data.Nat.Prime.Int
  {p : ℕ} : Nat.Prime p ↔ Prime ↑p
• lucas_primality_iff 📋 Mathlib.NumberTheory.LucasPrimality
  (p : ℕ) : Nat.Prime p ↔ ∃ a, a ^ (p - 1) = 1 ∧ ∀ (q : ℕ), Nat.Prime q → q ∣ p - 1 → a ^ ((p - 1) / q) ≠ 1
• Nat.prime_iff_fac_equiv_neg_one 📋 Mathlib.NumberTheory.Wilson
  {n : ℕ} (h : n ≠ 1) : Nat.Prime n ↔ ↑(n - 1).factorial = -1

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.

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 19971e9 serving mathlib revision bce1d65