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refactor(Algebra/Quaternion): intermediate Module instance #30678

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refactor(Algebra/Quaternion): intermediate `Module` instance
YaelDillies Oct 19, 2025
4b87187
simp nf
YaelDillies Oct 19, 2025
3c84431
fast_instance%
YaelDillies Oct 20, 2025
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74 changes: 45 additions & 29 deletions Mathlib/Algebra/Quaternion.lean
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Original file line number Diff line number Diff line change
Expand Up @@ -175,8 +175,8 @@ theorem coe_injective : Function.Injective (coe : R → ℍ[R,c₁,c₂,c₃]) :
theorem coe_inj {x y : R} : (x : ℍ[R,c₁,c₂,c₃]) = y ↔ x = y :=
coe_injective.eq_iff

@[simps]
instance : Zero ℍ[R,c₁,c₂,c₃] := ⟨⟨0, 0, 0, 0⟩⟩
@[simps!]
instance : Zero ℍ[R,c₁,c₂,c₃] := fast_instance% (equivProd c₁ c₂ c₃).zero
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Doesn't this produce a worse term?

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Sure, but the simp lemmas are just the same as before, so that should be no difference

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In terms of unification I think the previous approach was better.

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But unification with what? With other instances? They are all like that now

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@eric-wieser eric-wieser Oct 20, 2025 •
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Are you sure? Can you show me the effect of #print on the AddCommGroup instance before and after this line being changed?

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This is not possible. fast_instance% complains about the data fields not matching up to some transparency

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Ah, that's useful information! I was hoping it would use a higher level of transparency here.

In that case, I think it would be best to define things via the Function.Injective constructors. One argument for why this new zero instance is bad is that at runtime it's now a bunch of if statements to index a vector, rather than a hard-coded zero.

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@YaelDillies YaelDillies Oct 20, 2025 •
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No actually it is not using vectors. It used to be, because I was using equivTuple : _ \equiv (Fin 4 \r R), but I am now using equivProd : _ \equiv R \times R \times R \times R... unless you're claiming this too is bad at runtime?

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Ah, I missed that! Indeed equivProd is less bad, but still not as good as the direct constructor.

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Bad enough that I should use Function.Injective.foo? I was quite happy that I finally got to use the TransferInstance files!


@[scoped simp] theorem im_zero : (0 : ℍ[R,c₁,c₂,c₃]).im = 0 := rfl

Expand Down Expand Up @@ -205,16 +205,26 @@ end Zero
section Add
variable [Add R]

@[simps]
instance : Add ℍ[R,c₁,c₂,c₃] :=
⟨fun a b => ⟨a.1 + b.1, a.2 + b.2, a.3 + b.3, a.4 + b.4⟩⟩
@[simps!]
instance : Add ℍ[R,c₁,c₂,c₃] := fast_instance% (equivProd c₁ c₂ c₃).add

@[simp]
theorem mk_add_mk (a₁ a₂ a₃ a₄ b₁ b₂ b₃ b₄ : R) :
(mk a₁ a₂ a₃ a₄ : ℍ[R,c₁,c₂,c₃]) + mk b₁ b₂ b₃ b₄ =
mk (a₁ + b₁) (a₂ + b₂) (a₃ + b₃) (a₄ + b₄) :=
rfl

/-- The additive equivalence between a quaternion algebra over `R` and `Fin 4 → R`. -/
def addEquivTuple (c₁ c₂ c₃ : R) : ℍ[R,c₁,c₂,c₃] ≃+ (Fin 4 → R) where
toEquiv := equivTuple ..
map_add' _ _ := by ext i; fin_cases i <;> rfl

@[simp]
lemma coe_addEquivTuple (c₁ c₂ c₃ : R) : ⇑(addEquivTuple c₁ c₂ c₃) = equivTuple c₁ c₂ c₃ := rfl

@[simp] lemma coe_symm_addEquivTuple (c₁ c₂ c₃ : R) :
⇑(addEquivTuple c₁ c₂ c₃).symm = (equivTuple c₁ c₂ c₃).symm := rfl

end Add

section AddZeroClass
Expand All @@ -233,8 +243,8 @@ end AddZeroClass
section Neg
variable [Neg R]

@[simps]
instance : Neg ℍ[R,c₁,c₂,c₃] := ⟨fun a => ⟨-a.1, -a.2, -a.3, -a.4⟩⟩
@[simps!]
instance : Neg ℍ[R,c₁,c₂,c₃] := fast_instance% (equivProd c₁ c₂ c₃).Neg

@[simp]
theorem neg_mk (a₁ a₂ a₃ a₄ : R) : -(mk a₁ a₂ a₃ a₄ : ℍ[R,c₁,c₂,c₃]) = ⟨-a₁, -a₂, -a₃, -a₄⟩ :=
Expand All @@ -253,9 +263,8 @@ variable [AddGroup R]
@[simp, norm_cast]
theorem coe_neg : ((-x : R) : ℍ[R,c₁,c₂,c₃]) = -x := by ext <;> simp

@[simps]
instance : Sub ℍ[R,c₁,c₂,c₃] :=
⟨fun a b => ⟨a.1 - b.1, a.2 - b.2, a.3 - b.3, a.4 - b.4⟩⟩
@[simps!]
instance : Sub ℍ[R,c₁,c₂,c₃] := fast_instance% (equivProd c₁ c₂ c₃).sub

@[simp] theorem im_sub : (a - b).im = a.im - b.im :=
QuaternionAlgebra.ext (sub_zero _).symm rfl rfl rfl
Expand Down Expand Up @@ -292,6 +301,9 @@ theorem sub_re_self : a - a.re = a.im :=

end AddGroup

instance [AddCommGroup R] : AddCommGroup ℍ[R,c₁,c₂,c₃] :=
fast_instance% (equivProd c₁ c₂ c₃).addCommGroup

section Ring
variable [Ring R]

Expand Down Expand Up @@ -327,14 +339,13 @@ section SMul

variable [SMul S R] [SMul T R] (s : S)

@[simps]
instance : SMul S ℍ[R,c₁,c₂,c₃] where smul s a := ⟨s • a.1, s • a.2, s • a.3, s • a.4⟩
@[simps!]
instance : SMul S ℍ[R,c₁,c₂,c₃] := fast_instance% (equivProd ..).smul _

instance [SMul S T] [IsScalarTower S T R] : IsScalarTower S T ℍ[R,c₁,c₂,c₃] where
smul_assoc s t x := by ext <;> exact smul_assoc _ _ _
instance [SMul S T] [IsScalarTower S T R] : IsScalarTower S T ℍ[R,c₁,c₂,c₃] :=
(equivTuple ..).isScalarTower ..

instance [SMulCommClass S T R] : SMulCommClass S T ℍ[R,c₁,c₂,c₃] where
smul_comm s t x := by ext <;> exact smul_comm _ _ _
instance [SMulCommClass S T R] : SMulCommClass S T ℍ[R,c₁,c₂,c₃] := (equivTuple ..).smulCommClass ..

@[simp] theorem im_smul {S} [CommRing R] [SMulZeroClass S R] (s : S) : (s • a).im = s • a.im :=
QuaternionAlgebra.ext (smul_zero s).symm rfl rfl rfl
Expand All @@ -348,14 +359,19 @@ theorem smul_mk (re im_i im_j im_k : R) :

end SMul

instance [Monoid S] [MulAction S R] : MulAction S ℍ[R,c₁,c₂,c₃] :=
fast_instance% (equivProd ..).mulAction _

@[simp, norm_cast]
theorem coe_smul [Zero R] [SMulZeroClass S R] (s : S) (r : R) :
(↑(s • r) : ℍ[R,c₁,c₂,c₃]) = s • (r : ℍ[R,c₁,c₂,c₃]) :=
QuaternionAlgebra.ext rfl (smul_zero _).symm (smul_zero _).symm (smul_zero _).symm

instance [AddCommGroup R] : AddCommGroup ℍ[R,c₁,c₂,c₃] :=
(equivProd c₁ c₂ c₃).injective.addCommGroup _ rfl (fun _ _ ↦ rfl) (fun _ ↦ rfl) (fun _ _ ↦ rfl)
(fun _ _ ↦ rfl) (fun _ _ ↦ rfl)
instance [Semiring S] [AddCommGroup R] [DistribMulAction S R] : DistribMulAction S ℍ[R,c₁,c₂,c₃] :=
fast_instance% (equivProd ..).distribMulAction _

instance [Semiring S] [AddCommGroup R] [Module S R] : Module S ℍ[R,c₁,c₂,c₃] :=
fast_instance% (equivProd ..).module _

section AddCommGroupWithOne
variable [AddCommGroupWithOne R]
Expand Down Expand Up @@ -486,10 +502,7 @@ lemma coe_ofNat {n : ℕ} [n.AtLeastTwo] :
((ofNat(n) : R) : ℍ[R,c₁,c₂,c₃]) = (ofNat(n) : ℍ[R,c₁,c₂,c₃]) :=
rfl

-- TODO: add weaker `MulAction`, `DistribMulAction`, and `Module` instances (and repeat them
-- for `ℍ[R]`)
instance [CommSemiring S] [Algebra S R] : Algebra S ℍ[R,c₁,c₂,c₃] where
smul := (· • ·)
algebraMap :=
{ toFun s := coe (algebraMap S R s)
map_one' := by simp only [map_one, coe_one]
Expand Down Expand Up @@ -544,13 +557,7 @@ def imKₗ : ℍ[R,c₁,c₂,c₃] →ₗ[R] R where
map_smul' _ _ := rfl

/-- `QuaternionAlgebra.equivTuple` as a linear equivalence. -/
def linearEquivTuple : ℍ[R,c₁,c₂,c₃] ≃ₗ[R] Fin 4 → R :=
LinearEquiv.symm -- proofs are not `rfl` in the forward direction
{ (equivTuple c₁ c₂ c₃).symm with
toFun := (equivTuple c₁ c₂ c₃).symm
invFun := equivTuple c₁ c₂ c₃
map_add' := fun _ _ => rfl
map_smul' := fun _ _ => rfl }
def linearEquivTuple : ℍ[R,c₁,c₂,c₃] ≃ₗ[R] Fin 4 → R := (equivTuple ..).linearEquiv _

@[simp]
theorem coe_linearEquivTuple :
Expand Down Expand Up @@ -779,6 +786,15 @@ instance [SMul S T] [SMul S R] [SMul T R] [IsScalarTower S T R] : IsScalarTower
instance [SMul S R] [SMul T R] [SMulCommClass S T R] : SMulCommClass S T ℍ[R] :=
inferInstanceAs <| SMulCommClass S T ℍ[R,-1,0,-1]

instance [Monoid S] [MulAction S R] : MulAction S ℍ[R] :=
inferInstanceAs <| MulAction S ℍ[R,-1,0,-1]

instance [Semiring S] [DistribMulAction S R] : DistribMulAction S ℍ[R] :=
inferInstanceAs <| DistribMulAction S ℍ[R,-1,0,-1]

instance [Semiring S] [Module S R] : Module S ℍ[R] :=
inferInstanceAs <| Module S ℍ[R,-1,0,-1]

protected instance algebra [CommSemiring S] [Algebra S R] : Algebra S ℍ[R] :=
inferInstanceAs <| Algebra S ℍ[R,-1,0,-1]

Expand Down