chore(LinearAlgebra/QuadraticForm): fixups to #10097 (#10167)

I didn't get a chance to review #10097 before it was merged; this contains some minor fixups.

This removes `LinearMap.linMulLin`, as it can be recovered easily via `LinearMap.mul`.

Mathlib/Algebra/Algebra/Bilinear.lean

@@ -22,6 +22,26 @@ open TensorProduct Module
 
 namespace LinearMap
 
+section RestrictScalars
+
+variable
+  (R : Type*) {A M N P : Type*}
+  [CommSemiring R] [CommSemiring A]
+  [AddCommMonoid M] [AddCommMonoid N] [AddCommMonoid P]
+  [Algebra R A]
+  [Module R M] [Module R N] [Module R P]
+  [Module A M] [Module A N] [Module A P]
+  [IsScalarTower R A M] [IsScalarTower R A N] [IsScalarTower R A P]
+
+/-- A version of `LinearMap.restrictScalars` for bilinear maps.
+
+The double subscript in the name is to match `LinearMap.compl₁₂`. -/
+@[simps!]
+def restrictScalars₁₂ (f : M →ₗ[A] N →ₗ[A] P) : M →ₗ[R] N →ₗ[R] P :=
+  (f.flip.restrictScalars _).flip.restrictScalars _
+
+end RestrictScalars
+
 section NonUnitalNonAssoc
 
 variable (R A : Type*) [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]

Mathlib/LinearAlgebra/BilinearMap.lean

@@ -404,13 +404,6 @@ abbrev _root_.Submodule.restrictBilinear (p : Submodule R M) (f : M →ₗ[R] M
     p →ₗ[R] p →ₗ[R] R :=
   f.compl₁₂ p.subtype p.subtype
 
-/-- `linMulLin f g` is the bilinear form mapping `x` and `y` to `f x * g y` -/
-def linMulLin (f g : M →ₗ[R] R) : M →ₗ[R] M →ₗ[R] R :=
-  LinearMap.mk₂ R (fun x y => f x * g y) (fun x y z => by simp only [map_add, add_mul])
-  (fun _ _ => by simp only [SMulHomClass.map_smul, smul_eq_mul, mul_assoc, forall_const])
-  (fun _ _ _ => by simp only [map_add, mul_add])
-  (fun _ _ => by simp only [SMulHomClass.map_smul, smul_eq_mul, mul_left_comm, forall_const])
-
 end CommSemiring
 
 section CommRing

Mathlib/LinearAlgebra/QuadraticForm/Basic.lean

@@ -382,7 +382,7 @@ variable [CommSemiring R] [AddCommMonoid M] [Module R M]
 section SMul
 
 variable [Monoid S] [Monoid T] [DistribMulAction S R] [DistribMulAction T R]
-variable [SMulCommClass R S R] [SMulCommClass S R R] [SMulCommClass T R R] [SMulCommClass R T R]
+variable [SMulCommClass S R R] [SMulCommClass T R R]
 
 /-- `QuadraticForm R M` inherits the scalar action from any algebra over `R`.
 
@@ -393,6 +393,7 @@ instance : SMul S (QuadraticForm R M) :=
       toFun_smul := fun b x => by rw [Pi.smul_apply, map_smul, Pi.smul_apply, mul_smul_comm]
       exists_companion' :=
         let ⟨B, h⟩ := Q.exists_companion
+        letI := SMulCommClass.symm S R R
         ⟨a • B, by simp [h]⟩ }⟩
 
 @[simp]
@@ -486,7 +487,7 @@ theorem sum_apply {ι : Type*} (Q : ι → QuadraticForm R M) (s : Finset ι) (x
 
 end Sum
 
-instance [Monoid S] [DistribMulAction S R] [SMulCommClass S R R] [SMulCommClass R S R] :
+instance [Monoid S] [DistribMulAction S R] [SMulCommClass S R R] :
     DistribMulAction S (QuadraticForm R M) where
   mul_smul a b Q := ext fun x => by simp only [smul_apply, mul_smul]
   one_smul Q := ext fun x => by simp only [QuadraticForm.smul_apply, one_smul]
@@ -497,7 +498,7 @@ instance [Monoid S] [DistribMulAction S R] [SMulCommClass S R R] [SMulCommClass
     ext
     simp only [zero_apply, smul_apply, smul_zero]
 
-instance [Semiring S] [Module S R] [SMulCommClass S R R] [SMulCommClass R S R] :
+instance [Semiring S] [Module S R] [SMulCommClass S R R] :
     Module S (QuadraticForm R M) where
   zero_smul Q := by
     ext
@@ -577,9 +578,8 @@ def _root_.LinearMap.compQuadraticForm [CommSemiring S] [Algebra S R] [Module S
   toFun_smul b x := by simp only [Q.map_smul_of_tower b x, f.map_smul, smul_eq_mul]
   exists_companion' :=
     let ⟨B, h⟩ := Q.exists_companion
-    ⟨(LinearMap.restrictScalars S (LinearMap.restrictScalars S B.flip).flip).compr₂ f, fun x y => by
-      simp_rw [h, f.map_add]
-      rfl⟩
+    ⟨(B.restrictScalars₁₂ S).compr₂ f, fun x y => by
+      simp_rw [h, f.map_add, LinearMap.compr₂_apply, LinearMap.restrictScalars₁₂_apply_apply]⟩
 #align linear_map.comp_quadratic_form LinearMap.compQuadraticForm
 
 end Comp
@@ -595,9 +595,9 @@ def linMulLin (f g : M →ₗ[R] R) : QuadraticForm R M where
     simp only [smul_eq_mul, RingHom.id_apply, Pi.mul_apply, LinearMap.map_smulₛₗ]
     ring
   exists_companion' :=
-    ⟨LinearMap.linMulLin f g + LinearMap.linMulLin g f, fun x y => by
-      simp only [Pi.mul_apply, map_add, LinearMap.linMulLin, LinearMap.add_apply,
-        LinearMap.mk₂_apply]
+    ⟨(LinearMap.mul R R).compl₁₂ f g + (LinearMap.mul R R).compl₁₂ g f, fun x y => by
+      simp only [Pi.mul_apply, map_add, LinearMap.compl₁₂_apply, LinearMap.mul_apply',
+        LinearMap.add_apply]
       ring_nf⟩
 #align quadratic_form.lin_mul_lin QuadraticForm.linMulLin
 
@@ -664,7 +664,7 @@ section Semiring
 variable [CommSemiring R] [AddCommMonoid M] [Module R M]
 
 /-- A bilinear map into `R` gives a quadratic form by applying the argument twice. -/
-def _root_.LinearMap.toQuadraticForm (B: M →ₗ[R] M →ₗ[R] R) : QuadraticForm R M where
+def _root_.LinearMap.toQuadraticForm (B : M →ₗ[R] M →ₗ[R] R) : QuadraticForm R M where
   toFun x := B x x
   toFun_smul a x := by
     simp only [SMulHomClass.map_smul, LinearMap.smul_apply, smul_eq_mul, mul_assoc]
@@ -702,7 +702,7 @@ theorem toQuadraticForm_add (B₁ B₂ : BilinForm R M) :
 
 @[simp]
 theorem toQuadraticForm_smul [Monoid S] [DistribMulAction S R] [SMulCommClass S R R]
-    [SMulCommClass R S R] (a : S) (B : BilinForm R M) :
+    (a : S) (B : BilinForm R M) :
     (a • B).toQuadraticForm = a • B.toQuadraticForm :=
   rfl
 #align bilin_form.to_quadratic_form_smul BilinForm.toQuadraticForm_smul
@@ -721,7 +721,7 @@ def toQuadraticFormAddMonoidHom : BilinForm R M →+ QuadraticForm R M where
 
 /-- `BilinForm.toQuadraticForm` as a linear map -/
 @[simps!]
-def toQuadraticFormLinearMap [Semiring S] [Module S R] [SMulCommClass S R R] [SMulCommClass R S R] :
+def toQuadraticFormLinearMap [Semiring S] [Module S R] [SMulCommClass S R R] :
     BilinForm R M →ₗ[S] QuadraticForm R M where
   toFun := toQuadraticForm
   map_smul' := toQuadraticForm_smul
@@ -1292,16 +1292,16 @@ variable (R)
 
 The weights are applied using `•`; typically this definition is used either with `S = R` or
 `[Algebra S R]`, although this is stated more generally. -/
-def weightedSumSquares [Monoid S] [DistribMulAction S R] [SMulCommClass S R R] [SMulCommClass R S R]
-    (w : ι → S) : QuadraticForm R (ι → R) :=
+def weightedSumSquares [Monoid S] [DistribMulAction S R] [SMulCommClass S R R] (w : ι → S) :
+    QuadraticForm R (ι → R) :=
   ∑ i : ι, w i • proj i i
 #align quadratic_form.weighted_sum_squares QuadraticForm.weightedSumSquares
 
 end
 
 @[simp]
 theorem weightedSumSquares_apply [Monoid S] [DistribMulAction S R] [SMulCommClass S R R]
-    [SMulCommClass R S R] (w : ι → S) (v : ι → R) :
+    (w : ι → S) (v : ι → R) :
     weightedSumSquares R w v = ∑ i : ι, w i • (v i * v i) :=
   QuadraticForm.sum_apply _ _ _
 #align quadratic_form.weighted_sum_squares_apply QuadraticForm.weightedSumSquares_apply