Loogle!

Result

Found 1045 declarations mentioning HAdd.hAdd and Prod. Of these, 14 match your pattern(s).

• Prod.fst_add 📋 Mathlib.Algebra.Notation.Prod
  {M : Type u_8} {N : Type u_9} [Add M] [Add N] (p q : M × N) : (p + q).1 = p.1 + q.1
• Prod.snd_add 📋 Mathlib.Algebra.Notation.Prod
  {M : Type u_8} {N : Type u_9} [Add M] [Add N] (p q : M × N) : (p + q).2 = p.2 + q.2
• Prod.mk_add_mk 📋 Mathlib.Algebra.Notation.Prod
  {M : Type u_8} {N : Type u_9} [Add M] [Add N] (a₁ a₂ : M) (b₁ b₂ : N) : (a₁, b₁) + (a₂, b₂) = (a₁ + a₂, b₁ + b₂)
• Prod.swap_add 📋 Mathlib.Algebra.Notation.Prod
  {M : Type u_8} {N : Type u_9} [Add M] [Add N] (p q : M × N) : (p + q).swap = p.swap + q.swap
• Prod.add_def 📋 Mathlib.Algebra.Notation.Prod
  {M : Type u_8} {N : Type u_9} [Add M] [Add N] (p q : M × N) : p + q = (p.1 + q.1, p.2 + q.2)
• Prod.fst_add_snd 📋 Mathlib.Algebra.Group.Prod
  {M : Type u_3} {N : Type u_4} [AddZeroClass M] [AddZeroClass N] (p : M × N) : (p.1, 0) + (0, p.2) = p
• Prod.mk_zero_add_mk_zero 📋 Mathlib.Algebra.Group.Prod
  {M : Type u_3} {N : Type u_4} [Add M] [AddZeroClass N] (a₁ a₂ : M) : (a₁, 0) + (a₂, 0) = (a₁ + a₂, 0)
• Prod.zero_mk_add_zero_mk 📋 Mathlib.Algebra.Group.Prod
  {M : Type u_3} {N : Type u_4} [AddZeroClass M] [Add N] (b₁ b₂ : N) : (0, b₁) + (0, b₂) = (0, b₁ + b₂)
• DFinsupp.equivProdDFinsupp_add 📋 Mathlib.Data.DFinsupp.Defs
  {ι : Type u} {α : Option ι → Type v} [(i : Option ι) → AddZeroClass (α i)] (f g : Π₀ (i : Option ι), α i) : DFinsupp.equivProdDFinsupp (f + g) = DFinsupp.equivProdDFinsupp f + DFinsupp.equivProdDFinsupp g
• LieAlgebra.of_add 📋 Mathlib.Algebra.Lie.Extension
  {R : Type u_1} {L : Type u_3} {M : Type u_4} [CommRing R] [LieRing L] [LieAlgebra R L] [AddCommGroup M] [Module R M] [LieRingModule L M] [LieModule R L M] (c : ↥(LieModule.Cohomology.twoCocycle R L M)) (x y : L × M) : (LieAlgebra.ofProd c) (x + y) = (LieAlgebra.ofProd c) x + (LieAlgebra.ofProd c) y
• LieAlgebra.of_symm_add 📋 Mathlib.Algebra.Lie.Extension
  {R : Type u_1} {L : Type u_3} {M : Type u_4} [CommRing R] [LieRing L] [LieAlgebra R L] [AddCommGroup M] [Module R M] [LieRingModule L M] [LieModule R L M] (c : ↥(LieModule.Cohomology.twoCocycle R L M)) (x y : LieAlgebra.ofTwoCocycle c) : (LieAlgebra.ofProd c).symm (x + y) = (LieAlgebra.ofProd c).symm x + (LieAlgebra.ofProd c).symm y
• OrderAddMonoidHom.inl_add_inr_eq_mk 📋 Mathlib.Algebra.Order.Monoid.Lex
  (α : Type u_1) (β : Type u_2) [AddMonoid α] [PartialOrder α] [AddMonoid β] [Preorder β] (m : α) (n : β) : (OrderAddMonoidHom.inl α β) m + (OrderAddMonoidHom.inr α β) n = (m, n)
• NonemptyInterval.toProd_add 📋 Mathlib.Algebra.Order.Interval.Basic
  {α : Type u_2} [Preorder α] [Add α] [AddLeftMono α] [AddRightMono α] (s t : NonemptyInterval α) : (s + t).toProd = s.toProd + t.toProd
• DoubleCentralizer.add_toProd 📋 Mathlib.Analysis.CStarAlgebra.Multiplier
  {𝕜 : Type u_1} {A : Type u_2} [NontriviallyNormedField 𝕜] [NonUnitalNormedRing A] [NormedSpace 𝕜 A] [SMulCommClass 𝕜 A A] [IsScalarTower 𝕜 A A] (a b : DoubleCentralizer 𝕜 A) : (a + b).toProd = a.toProd + b.toProd

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