Skip to content

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

New issue

Have a question about this project? Sign up for a free GitHub account to open an issue and contact its maintainers and the community.

Sign up for GitHub

By clicking “Sign up for GitHub”, you agree to our terms of service and privacy statement. We’ll occasionally send you account related emails.

Already on GitHub? Sign in to your account

Open
wants to merge 3 commits into
base: master
Choose a base branch
from
Open

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

Changes from 2 commits
Commits
Show all changes
3 commits
Select commit Hold shift + click to select a range
4b04f33
refactor(Algebra/Quaternion): intermediate `Module` instance
YaelDillies Oct 19, 2025
4b87187
simp nf
YaelDillies Oct 19, 2025
3c84431
fast_instance%
YaelDillies Oct 20, 2025
File filter

Filter by extension

Filter by extension
Conversations
Failed to load comments.
Loading
Jump to
Jump to file
Failed to load files.
Loading
Diff view
Diff view
30 changes: 20 additions & 10 deletions Mathlib/Algebra/Quaternion.lean
View file
Open in desktop
Original file line number Diff line number Diff line change
Expand Up @@ -215,6 +215,17 @@ theorem mk_add_mk (a₁ a₂ a₃ a₄ b₁ b₂ b₃ b₄ : R) :
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 Down Expand Up @@ -486,10 +497,15 @@ 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 [Monoid S] [MulAction S R] : MulAction S ℍ[R,c₁,c₂,c₃] := (equivTuple ..).mulAction _

instance [Semiring S] [DistribMulAction S R] : DistribMulAction S ℍ[R,c₁,c₂,c₃] :=
(equivTuple ..).distribMulAction _

instance [Semiring S] [Module S R] : Module S ℍ[R,c₁,c₂,c₃] := (equivTuple ..).module _

-- TODO: repeat weaker `MulAction`, `DistribMulAction`, and `Module` instances 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 +560,7 @@ 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