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[Merged by Bors] - fix(linear_algebra/basic): generalize linear_map.add_comm_group to semilinear maps #9402

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[Merged by Bors] - fix(linear_algebra/basic): generalize linear_map.add_comm_group to semilinear maps #9402

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65b0c33
fix(linear_algebra/basic): generalize `linear_map.add_comm_group` to …
eric-wieser Sep 27, 2021
58e5d1f
Update src/linear_algebra/basic.lean
eric-wieser Sep 27, 2021
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5 changes: 2 additions & 3 deletions src/algebra/module/linear_map.lean
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Expand Up @@ -103,9 +103,8 @@ instance {σ : R →+* S} : has_coe_to_fun (M →ₛₗ[σ] M₃) := ⟨_, to_fu

initialize_simps_projections linear_map (to_fun → apply)

@[simp] lemma coe_mk (f : M → M₂) (h₁ h₂) :
((linear_map.mk f h₁ h₂ : M →ₗ[R] M₂) : M → M₂) = f := rfl

@[simp] lemma coe_mk {σ : R →+* S} (f : M → M₃) (h₁ h₂) :
⇑(linear_map.mk f h₁ h₂ : M →ₛₗ[σ] M₃) = f := rfl

/-- Identity map as a `linear_map` -/
def id : M →ₗ[R] M :=
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37 changes: 15 additions & 22 deletions src/linear_algebra/basic.lean
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Expand Up @@ -222,10 +222,10 @@ instance : has_add (M →ₛₗ[σ₁₂] M₂) :=
instance : add_comm_monoid (M →ₛₗ[σ₁₂] M₂) :=
{ zero := 0,
add := (+),
add_assoc := by intros; ext; simp [add_comm, add_left_comm],
zero_add := by intros; ext; simp [add_comm, add_left_comm],
add_zero := by intros; ext; simp [add_comm, add_left_comm],
add_comm := by intros; ext; simp [add_comm, add_left_comm],
add_assoc := λ f g h, linear_map.ext $ λ x, add_assoc _ _ _,
zero_add := λ f, linear_map.ext $ λ x, zero_add _,
add_zero := λ f, linear_map.ext $ λ x, add_zero _,
add_comm := λ f g, linear_map.ext $ λ x, add_comm _ _,
nsmul := λ n f,
{ to_fun := λ x, n • (f x),
map_add' := λ x y, by rw [f.map_add, smul_add],
Expand All @@ -234,12 +234,8 @@ instance : add_comm_monoid (M →ₛₗ[σ₁₂] M₂) :=
rw [f.map_smulₛₗ],
simp [smul_comm n (σ₁₂ c) (f x)],
end },
nsmul_zero' := λ f, by { ext x, change 0 • f x = 0, simp only [zero_smul] },
nsmul_succ' := λ n f, linear_map.ext $ λ x,
begin
change n.succ • (f x) = f x + n • (f x),
simp [nat.succ_eq_one_add, add_nsmul],
end }
nsmul_zero' := λ f, linear_map.ext $ λ x, add_comm_monoid.nsmul_zero' _,
nsmul_succ' := λ n f, linear_map.ext $ λ x, add_comm_monoid.nsmul_succ' _ _ }

/-- Evaluation of a `σ₁₂`-linear map at a fixed `a`, as an `add_monoid_hom`. -/
def eval_add_monoid_hom (a : M) : (M →ₛₗ[σ₁₂] M₂) →+ M₂ :=
Expand Down Expand Up @@ -447,28 +443,25 @@ lemma comp_sub (f g : M →ₛₗ[σ₁₃] M₃) (h : M₃ →ₛₗ[σ₃₄]
omit σ₁₄

/-- The type of linear maps is an additive group. -/
instance [module R M₃] : add_comm_group (M →ₗ[R] M₃) :=
by refine
instance : add_comm_group (M →ₛₗ[σ₁₃] M₃) :=
{ zero := 0,
add := (+),
neg := has_neg.neg,
sub := has_sub.sub,
sub_eq_add_neg := _,
add_left_neg := _,
sub_eq_add_neg := λ f g, linear_map.ext $ λ m, sub_eq_add_neg _ _,
add_left_neg := λ f, linear_map.ext $ λ m, add_left_neg _,
nsmul := λ n f, {
to_fun := λ x, n • (f x),
map_add' := λ x y, by rw [f.map_add, smul_add],
map_smul' := λ c x, by dsimp; rw [f.map_smul, smul_comm n c (f x)] },
map_smul' := λ c x, by rw [f.map_smulₛₗ, smul_comm n (σ₁₃ c) (f x)] },
gsmul := λ n f, {
to_fun := λ x, n • (f x),
map_add' := λ x y, by rw [f.map_add, smul_add],
map_smul' := λ c x, by dsimp; rw [f.map_smul, smul_comm n c (f x)] },
gsmul_zero' := _,
gsmul_succ' := _,
gsmul_neg' := _,
.. linear_map.add_comm_monoid };
intros; apply linear_map.ext;
simp [add_comm, add_left_comm, sub_eq_add_neg, add_smul, nat.succ_eq_add_one]
map_smul' := λ c x, by rw [f.map_smulₛₗ, smul_comm n (σ₁₃ c) (f x)] },
gsmul_zero' := λ a, linear_map.ext $ λ m, zero_smul _ _,
gsmul_succ' := λ n a, linear_map.ext $ λ m, add_comm_group.gsmul_succ' n _,
gsmul_neg' := λ n a, linear_map.ext $ λ m, add_comm_group.gsmul_neg' n _,
.. linear_map.add_comm_monoid }

end add_comm_group

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