Loogle!

Result

Found 8 declarations mentioning TensorProduct and PiTensorProduct.

• PiTensorProduct.tmulEquiv 📋 Mathlib.LinearAlgebra.PiTensorProduct
  {ι : Type u_1} {ι₂ : Type u_2} (R : Type u_4) [CommSemiring R] (M : Type u_8) [AddCommMonoid M] [Module R M] : TensorProduct R (PiTensorProduct R fun x => M) (PiTensorProduct R fun x => M) ≃ₗ[R] PiTensorProduct R fun x => M
• PiTensorProduct.tmulEquiv.eq_1 📋 Mathlib.LinearAlgebra.PiTensorProduct
  {ι : Type u_1} {ι₂ : Type u_2} (R : Type u_4) [CommSemiring R] (M : Type u_8) [AddCommMonoid M] [Module R M] : PiTensorProduct.tmulEquiv R M = PiTensorProduct.tmulEquivDep R fun x => M
• PiTensorProduct.tmulEquivDep 📋 Mathlib.LinearAlgebra.PiTensorProduct
  {ι : Type u_1} {ι₂ : Type u_2} (R : Type u_4) [CommSemiring R] (N : ι ⊕ ι₂ → Type u_12) [(i : ι ⊕ ι₂) → AddCommMonoid (N i)] [(i : ι ⊕ ι₂) → Module R (N i)] : TensorProduct R (PiTensorProduct R fun i₁ => N (Sum.inl i₁)) (PiTensorProduct R fun i₂ => N (Sum.inr i₂)) ≃ₗ[R] PiTensorProduct R fun i => N i
• PiTensorProduct.tmulEquiv_apply 📋 Mathlib.LinearAlgebra.PiTensorProduct
  {ι : Type u_1} {ι₂ : Type u_2} (R : Type u_4) [CommSemiring R] (M : Type u_8) [AddCommMonoid M] [Module R M] (a : ι → M) (b : ι₂ → M) : (PiTensorProduct.tmulEquiv R M) (((PiTensorProduct.tprod R) fun i => a i) ⊗ₜ[R] (PiTensorProduct.tprod R) fun i => b i) = (PiTensorProduct.tprod R) fun i => Sum.elim a b i
• PiTensorProduct.tmulEquiv_symm_apply 📋 Mathlib.LinearAlgebra.PiTensorProduct
  {ι : Type u_1} {ι₂ : Type u_2} (R : Type u_4) [CommSemiring R] (M : Type u_8) [AddCommMonoid M] [Module R M] (a : ι ⊕ ι₂ → M) : (PiTensorProduct.tmulEquiv R M).symm ((PiTensorProduct.tprod R) fun i => a i) = ((PiTensorProduct.tprod R) fun i => a (Sum.inl i)) ⊗ₜ[R] (PiTensorProduct.tprod R) fun i => a (Sum.inr i)
• PiTensorProduct.tmulEquivDep_apply 📋 Mathlib.LinearAlgebra.PiTensorProduct
  {ι : Type u_1} {ι₂ : Type u_2} (R : Type u_4) [CommSemiring R] (N : ι ⊕ ι₂ → Type u_12) [(i : ι ⊕ ι₂) → AddCommMonoid (N i)] [(i : ι ⊕ ι₂) → Module R (N i)] (a : (i₁ : ι) → N (Sum.inl i₁)) (b : (i₂ : ι₂) → N (Sum.inr i₂)) : (PiTensorProduct.tmulEquivDep R N) (((PiTensorProduct.tprod R) fun i₁ => a i₁) ⊗ₜ[R] (PiTensorProduct.tprod R) fun i₂ => b i₂) = ⨂ₜ[R] (i : ι ⊕ ι₂), Sum.rec a b i
• PiTensorProduct.tmulEquivDep_symm_apply 📋 Mathlib.LinearAlgebra.PiTensorProduct
  {ι : Type u_1} {ι₂ : Type u_2} (R : Type u_4) [CommSemiring R] (N : ι ⊕ ι₂ → Type u_12) [(i : ι ⊕ ι₂) → AddCommMonoid (N i)] [(i : ι ⊕ ι₂) → Module R (N i)] (f : (i : ι ⊕ ι₂) → N i) : (PiTensorProduct.tmulEquivDep R N).symm (⨂ₜ[R] (i : ι ⊕ ι₂), f i) = ((PiTensorProduct.tprod R) fun i₁ => f (Sum.inl i₁)) ⊗ₜ[R] (PiTensorProduct.tprod R) fun i₂ => f (Sum.inr i₂)
• PiTensorProduct.tmulEquivDep.eq_1 📋 Mathlib.LinearAlgebra.PiTensorProduct
  {ι : Type u_1} {ι₂ : Type u_2} (R : Type u_4) [CommSemiring R] (N : ι ⊕ ι₂ → Type u_12) [(i : ι ⊕ ι₂) → AddCommMonoid (N i)] [(i : ι ⊕ ι₂) → Module R (N i)] : PiTensorProduct.tmulEquivDep R N = LinearEquiv.ofLinear (TensorProduct.lift { toFun := fun a => PiTensorProduct.lift ((PiTensorProduct.lift (MultilinearMap.currySumEquiv (PiTensorProduct.tprod R))) a), map_add' := ⋯, map_smul' := ⋯ }) (PiTensorProduct.lift ((PiTensorProduct.tprod R).domCoprodDep (PiTensorProduct.tprod R))) ⋯ ⋯

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 e0654b0