Loogle!

Result

Found 42 declarations mentioning HSub.hSub, instHMod, Int, HMod.hMod, instHSub, Int.instMod, and Int.instSub. Of these, 17 match your pattern(s).

• Int.sub_emod 📋 Init.Data.Int.DivMod.Bootstrap
  (a b n : ℤ) : (a - b) % n = (a % n - b % n) % n
• Int.sub_emod_right 📋 Init.Data.Int.DivMod.Lemmas
  (a b : ℤ) : (a - b) % b = a % b
• Int.neg_emod_eq_sub_emod 📋 Init.Data.Int.DivMod.Lemmas
  {a b : ℤ} : -a % b = (b - a) % b
• Int.sub_emod_left 📋 Init.Data.Int.DivMod.Lemmas
  (a b : ℤ) : (a - b) % a = -b % a
• Int.sub_mul_emod_self_left 📋 Init.Data.Int.DivMod.Lemmas
  (a b c : ℤ) : (a - b * c) % b = a % b
• Int.sub_mul_emod_self_right 📋 Init.Data.Int.DivMod.Lemmas
  (a b c : ℤ) : (a - b * c) % c = a % c
• Int.mul_sub_emod_self_left 📋 Init.Data.Int.DivMod.Lemmas
  (a b c : ℤ) : (a * b - c) % a = -c % a
• Int.mul_sub_emod_self_right 📋 Init.Data.Int.DivMod.Lemmas
  (a b c : ℤ) : (a * b - c) % b = -c % b
• Int.emod_sub_emod 📋 Init.Data.Int.DivMod.Lemmas
  (m n k : ℤ) : (m % n - k) % n = (m - k) % n
• Int.sub_emod_emod 📋 Init.Data.Int.DivMod.Lemmas
  (m n k : ℤ) : (m - n % k) % k = (m - n) % k
• Int.emod_eq_emod_iff_emod_sub_eq_zero 📋 Init.Data.Int.DivMod.Lemmas
  {m n k : ℤ} : m % n = k % n ↔ (m - k) % n = 0
• Int.emod_sub_cancel_left 📋 Init.Data.Int.DivMod.Lemmas
  {m n k : ℤ} (i : ℤ) : (i - m) % n = (i - k) % n ↔ m % n = k % n
• Int.emod_sub_cancel_right 📋 Init.Data.Int.DivMod.Lemmas
  {m n k : ℤ} (i : ℤ) : (m - i) % n = (k - i) % n ↔ m % n = k % n
• Lean.Grind.IntInterval.wrap_eq_wrap_iff 📋 Init.Grind.ToInt
  {lo hi x y : ℤ} : (Lean.Grind.IntInterval.co lo hi).wrap x = (Lean.Grind.IntInterval.co lo hi).wrap y ↔ (x - y) % (hi - lo) = 0
• Int.emod_eq_sub_self_emod 📋 Mathlib.Data.Int.DivMod
  {a b : ℤ} : a % b = (a - b) % b
• Fin.coe_int_sub_eq_mod 📋 Mathlib.Data.Fin.Basic
  {n : ℕ} (u v : Fin n) : ↑↑(u - v) = (↑↑u - ↑↑v) % ↑n
• ZMod.intCast_cast_sub 📋 Mathlib.Data.ZMod.Basic
  {n : ℕ} (x y : ZMod n) : (x - y).cast = (x.cast - y.cast) % ↑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