Loogle!

Result

Found 16 declarations mentioning HMod.hMod and Int.bmod. Of these, 16 match your pattern(s).

• Int.bmod_emod 📋 Init.Data.Int.DivMod.Bootstrap
  {x : ℤ} {m : ℕ} : x.bmod m % ↑m = x % ↑m
• Int.bmod_def 📋 Init.Data.Int.DivMod.Bootstrap
  (x : ℤ) (m : ℕ) : x.bmod m = if x % ↑m < (↑m + 1) / 2 then x % ↑m else x % ↑m - ↑m
• Int.emod_bmod 📋 Init.Data.Int.DivMod.Lemmas
  (x : ℤ) (n : ℕ) : (x % ↑n).bmod n = x.bmod n
• Int.add_emod_bmod 📋 Init.Data.Int.DivMod.Lemmas
  {y : ℤ} (x : ℤ) (n : ℕ) : (y + x % ↑n).bmod n = (y + x).bmod n
• Int.emod_add_bmod 📋 Init.Data.Int.DivMod.Lemmas
  {y : ℤ} (x : ℤ) (n : ℕ) : (x % ↑n + y).bmod n = (x + y).bmod n
• Int.emod_mul_bmod 📋 Init.Data.Int.DivMod.Lemmas
  {y : ℤ} (x : ℤ) (n : ℕ) : (x % ↑n * y).bmod n = (x * y).bmod n
• Int.emod_sub_bmod 📋 Init.Data.Int.DivMod.Lemmas
  {y : ℤ} (x : ℤ) (n : ℕ) : (x % ↑n - y).bmod n = (x - y).bmod n
• Int.mul_emod_bmod 📋 Init.Data.Int.DivMod.Lemmas
  {y : ℤ} (x : ℤ) (n : ℕ) : (y * (x % ↑n)).bmod n = (y * x).bmod n
• Int.sub_emod_bmod 📋 Init.Data.Int.DivMod.Lemmas
  {y : ℤ} (x : ℤ) (n : ℕ) : (x - y % ↑n).bmod n = (x - y).bmod n
• Int.bmod_eq_emod_of_lt 📋 Init.Data.Int.DivMod.Lemmas
  {x : ℤ} {m : ℕ} (hx : x % ↑m < (↑m + 1) / 2) : x.bmod m = x % ↑m
• Int.bmod_pos 📋 Init.Data.Int.DivMod.Lemmas
  (x : ℤ) (m : ℕ) (p : x % ↑m < (↑m + 1) / 2) : x.bmod m = x % ↑m
• Int.bmod_neg 📋 Init.Data.Int.DivMod.Lemmas
  (x : ℤ) (m : ℕ) (p : x % ↑m ≥ (↑m + 1) / 2) : x.bmod m = x % ↑m - ↑m
• Int.bmod_eq_emod 📋 Init.Data.Int.DivMod.Lemmas
  (x : ℤ) (m : ℕ) : x.bmod m = x % ↑m - ↑(if x % ↑m ≥ (↑m + 1) / 2 then m else 0)
• BitVec.toInt_umod 📋 Init.Data.BitVec.Lemmas
  {w : ℕ} {x y : BitVec w} : (x % y).toInt = (↑x.toNat % ↑y.toNat).bmod (2 ^ w)
• BitVec.toInt_rotateLeft 📋 Init.Data.BitVec.Lemmas
  {w : ℕ} {x : BitVec w} {r : ℕ} : (x.rotateLeft r).toInt = (↑((x <<< (r % w)).toNat ||| (x >>> (w - r % w)).toNat)).bmod (2 ^ w)
• BitVec.toInt_rotateRight 📋 Init.Data.BitVec.Lemmas
  {w : ℕ} {x : BitVec w} {r : ℕ} : (x.rotateRight r).toInt = (↑((x >>> (r % w)).toNat ||| (x <<< (w - r % w)).toNat)).bmod (2 ^ w)

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 401c76f serving mathlib revision d9804c6