Loogle!

Result

Found 15 declarations mentioning CliffordAlgebra.map.

• CliffordAlgebra.map 📋 Mathlib.LinearAlgebra.CliffordAlgebra.Basic
  {R : Type u_1} [CommRing R] {M₁ : Type u_4} {M₂ : Type u_5} [AddCommGroup M₁] [AddCommGroup M₂] [Module R M₁] [Module R M₂] {Q₁ : QuadraticForm R M₁} {Q₂ : QuadraticForm R M₂} (f : Q₁ →qᵢ Q₂) : CliffordAlgebra Q₁ →ₐ[R] CliffordAlgebra Q₂
• CliffordAlgebra.map_id 📋 Mathlib.LinearAlgebra.CliffordAlgebra.Basic
  {R : Type u_1} [CommRing R] {M₁ : Type u_4} [AddCommGroup M₁] [Module R M₁] (Q₁ : QuadraticForm R M₁) : CliffordAlgebra.map (QuadraticMap.Isometry.id Q₁) = AlgHom.id R (CliffordAlgebra Q₁)
• CliffordAlgebra.map_surjective 📋 Mathlib.LinearAlgebra.CliffordAlgebra.Basic
  {R : Type u_1} [CommRing R] {M₁ : Type u_4} {M₂ : Type u_5} [AddCommGroup M₁] [AddCommGroup M₂] [Module R M₁] [Module R M₂] {Q₁ : QuadraticForm R M₁} {Q₂ : QuadraticForm R M₂} (f : Q₁ →qᵢ Q₂) (hf : Function.Surjective ⇑f) : Function.Surjective ⇑(CliffordAlgebra.map f)
• CliffordAlgebra.map_comp_map 📋 Mathlib.LinearAlgebra.CliffordAlgebra.Basic
  {R : Type u_1} [CommRing R] {M₁ : Type u_4} {M₂ : Type u_5} {M₃ : Type u_6} [AddCommGroup M₁] [AddCommGroup M₂] [AddCommGroup M₃] [Module R M₁] [Module R M₂] [Module R M₃] {Q₁ : QuadraticForm R M₁} {Q₂ : QuadraticForm R M₂} {Q₃ : QuadraticForm R M₃} (f : Q₂ →qᵢ Q₃) (g : Q₁ →qᵢ Q₂) : (CliffordAlgebra.map f).comp (CliffordAlgebra.map g) = CliffordAlgebra.map (f.comp g)
• CliffordAlgebra.equivOfIsometry_apply 📋 Mathlib.LinearAlgebra.CliffordAlgebra.Basic
  {R : Type u_1} [CommRing R] {M₁ : Type u_4} {M₂ : Type u_5} [AddCommGroup M₁] [AddCommGroup M₂] [Module R M₁] [Module R M₂] {Q₁ : QuadraticForm R M₁} {Q₂ : QuadraticForm R M₂} (e : QuadraticMap.IsometryEquiv Q₁ Q₂) (a : CliffordAlgebra Q₁) : (CliffordAlgebra.equivOfIsometry e) a = (CliffordAlgebra.map e.toIsometry) a
• CliffordAlgebra.leftInverse_map_of_leftInverse 📋 Mathlib.LinearAlgebra.CliffordAlgebra.Basic
  {R : Type u_1} [CommRing R] {M₁ : Type u_4} {M₂ : Type u_5} [AddCommGroup M₁] [AddCommGroup M₂] [Module R M₁] [Module R M₂] {Q₁ : QuadraticForm R M₁} {Q₂ : QuadraticForm R M₂} (f : Q₁ →qᵢ Q₂) (g : Q₂ →qᵢ Q₁) (h : Function.LeftInverse ⇑g ⇑f) : Function.LeftInverse ⇑(CliffordAlgebra.map g) ⇑(CliffordAlgebra.map f)
• CliffordAlgebra.map_comp_ι 📋 Mathlib.LinearAlgebra.CliffordAlgebra.Basic
  {R : Type u_1} [CommRing R] {M₁ : Type u_4} {M₂ : Type u_5} [AddCommGroup M₁] [AddCommGroup M₂] [Module R M₁] [Module R M₂] {Q₁ : QuadraticForm R M₁} {Q₂ : QuadraticForm R M₂} (f : Q₁ →qᵢ Q₂) : (CliffordAlgebra.map f).toLinearMap ∘ₗ CliffordAlgebra.ι Q₁ = CliffordAlgebra.ι Q₂ ∘ₗ f.toLinearMap
• CliffordAlgebra.map_apply_ι 📋 Mathlib.LinearAlgebra.CliffordAlgebra.Basic
  {R : Type u_1} [CommRing R] {M₁ : Type u_4} {M₂ : Type u_5} [AddCommGroup M₁] [AddCommGroup M₂] [Module R M₁] [Module R M₂] {Q₁ : QuadraticForm R M₁} {Q₂ : QuadraticForm R M₂} (f : Q₁ →qᵢ Q₂) (m : M₁) : (CliffordAlgebra.map f) ((CliffordAlgebra.ι Q₁) m) = (CliffordAlgebra.ι Q₂) (f m)
• CliffordAlgebra.ι_range_map_map 📋 Mathlib.LinearAlgebra.CliffordAlgebra.Basic
  {R : Type u_1} [CommRing R] {M₁ : Type u_4} {M₂ : Type u_5} [AddCommGroup M₁] [AddCommGroup M₂] [Module R M₁] [Module R M₂] {Q₁ : QuadraticForm R M₁} {Q₂ : QuadraticForm R M₂} (f : Q₁ →qᵢ Q₂) : Submodule.map (CliffordAlgebra.map f).toLinearMap (LinearMap.range (CliffordAlgebra.ι Q₁)) = Submodule.map (CliffordAlgebra.ι Q₂) (LinearMap.range f)
• QuadraticModuleCat.cliffordAlgebra_map 📋 Mathlib.LinearAlgebra.CliffordAlgebra.CategoryTheory
  {R : Type u} [CommRing R] {_M _N : QuadraticModuleCat R} (f : _M ⟶ _N) : QuadraticModuleCat.cliffordAlgebra.map f = AlgebraCat.ofHom (CliffordAlgebra.map (QuadraticModuleCat.Hom.toIsometry f))
• CliffordAlgebra.toProd.eq_1 📋 Mathlib.LinearAlgebra.CliffordAlgebra.Prod
  {R : Type u_1} {M₁ : Type u_2} {M₂ : Type u_3} [CommRing R] [AddCommGroup M₁] [AddCommGroup M₂] [Module R M₁] [Module R M₂] (Q₁ : QuadraticForm R M₁) (Q₂ : QuadraticForm R M₂) : CliffordAlgebra.toProd Q₁ Q₂ = GradedTensorProduct.lift (CliffordAlgebra.evenOdd Q₁) (CliffordAlgebra.evenOdd Q₂) (CliffordAlgebra.map (QuadraticMap.Isometry.inl Q₁ Q₂)) (CliffordAlgebra.map (QuadraticMap.Isometry.inr Q₁ Q₂)) ⋯
• CliffordAlgebra.commute_map_mul_map_of_isOrtho_of_mem_evenOdd_zero_left 📋 Mathlib.LinearAlgebra.CliffordAlgebra.Prod
  {R : Type u_1} {M₁ : Type u_2} {M₂ : Type u_3} {N : Type u_4} [CommRing R] [AddCommGroup M₁] [AddCommGroup M₂] [AddCommGroup N] [Module R M₁] [Module R M₂] [Module R N] {Q₁ : QuadraticForm R M₁} {Q₂ : QuadraticForm R M₂} {Qₙ : QuadraticForm R N} (f₁ : Q₁ →qᵢ Qₙ) (f₂ : Q₂ →qᵢ Qₙ) (hf : ∀ (x : M₁) (y : M₂), QuadraticMap.IsOrtho Qₙ (f₁ x) (f₂ y)) (m₁ : CliffordAlgebra Q₁) (m₂ : CliffordAlgebra Q₂) {i₂ : ZMod 2} (hm₁ : m₁ ∈ CliffordAlgebra.evenOdd Q₁ 0) (hm₂ : m₂ ∈ CliffordAlgebra.evenOdd Q₂ i₂) : Commute ((CliffordAlgebra.map f₁) m₁) ((CliffordAlgebra.map f₂) m₂)
• CliffordAlgebra.commute_map_mul_map_of_isOrtho_of_mem_evenOdd_zero_right 📋 Mathlib.LinearAlgebra.CliffordAlgebra.Prod
  {R : Type u_1} {M₁ : Type u_2} {M₂ : Type u_3} {N : Type u_4} [CommRing R] [AddCommGroup M₁] [AddCommGroup M₂] [AddCommGroup N] [Module R M₁] [Module R M₂] [Module R N] {Q₁ : QuadraticForm R M₁} {Q₂ : QuadraticForm R M₂} {Qₙ : QuadraticForm R N} (f₁ : Q₁ →qᵢ Qₙ) (f₂ : Q₂ →qᵢ Qₙ) (hf : ∀ (x : M₁) (y : M₂), QuadraticMap.IsOrtho Qₙ (f₁ x) (f₂ y)) (m₁ : CliffordAlgebra Q₁) (m₂ : CliffordAlgebra Q₂) {i₁ : ZMod 2} (hm₁ : m₁ ∈ CliffordAlgebra.evenOdd Q₁ i₁) (hm₂ : m₂ ∈ CliffordAlgebra.evenOdd Q₂ 0) : Commute ((CliffordAlgebra.map f₁) m₁) ((CliffordAlgebra.map f₂) m₂)
• CliffordAlgebra.map_mul_map_eq_neg_of_isOrtho_of_mem_evenOdd_one 📋 Mathlib.LinearAlgebra.CliffordAlgebra.Prod
  {R : Type u_1} {M₁ : Type u_2} {M₂ : Type u_3} {N : Type u_4} [CommRing R] [AddCommGroup M₁] [AddCommGroup M₂] [AddCommGroup N] [Module R M₁] [Module R M₂] [Module R N] {Q₁ : QuadraticForm R M₁} {Q₂ : QuadraticForm R M₂} {Qₙ : QuadraticForm R N} (f₁ : Q₁ →qᵢ Qₙ) (f₂ : Q₂ →qᵢ Qₙ) (hf : ∀ (x : M₁) (y : M₂), QuadraticMap.IsOrtho Qₙ (f₁ x) (f₂ y)) (m₁ : CliffordAlgebra Q₁) (m₂ : CliffordAlgebra Q₂) (hm₁ : m₁ ∈ CliffordAlgebra.evenOdd Q₁ 1) (hm₂ : m₂ ∈ CliffordAlgebra.evenOdd Q₂ 1) : (CliffordAlgebra.map f₁) m₁ * (CliffordAlgebra.map f₂) m₂ = -(CliffordAlgebra.map f₂) m₂ * (CliffordAlgebra.map f₁) m₁
• CliffordAlgebra.map_mul_map_of_isOrtho_of_mem_evenOdd 📋 Mathlib.LinearAlgebra.CliffordAlgebra.Prod
  {R : Type u_1} {M₁ : Type u_2} {M₂ : Type u_3} {N : Type u_4} [CommRing R] [AddCommGroup M₁] [AddCommGroup M₂] [AddCommGroup N] [Module R M₁] [Module R M₂] [Module R N] {Q₁ : QuadraticForm R M₁} {Q₂ : QuadraticForm R M₂} {Qₙ : QuadraticForm R N} (f₁ : Q₁ →qᵢ Qₙ) (f₂ : Q₂ →qᵢ Qₙ) (hf : ∀ (x : M₁) (y : M₂), QuadraticMap.IsOrtho Qₙ (f₁ x) (f₂ y)) (m₁ : CliffordAlgebra Q₁) (m₂ : CliffordAlgebra Q₂) {i₁ i₂ : ZMod 2} (hm₁ : m₁ ∈ CliffordAlgebra.evenOdd Q₁ i₁) (hm₂ : m₂ ∈ CliffordAlgebra.evenOdd Q₂ i₂) : (CliffordAlgebra.map f₁) m₁ * (CliffordAlgebra.map f₂) m₂ = (-1) ^ (i₂ * i₁) • ((CliffordAlgebra.map f₂) m₂ * (CliffordAlgebra.map f₁) m₁)

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