linear_algebra.clifford_algebra.basic | Mathlib porting status

Changes since the initial port

The following section lists changes to this file in mathlib3 and mathlib4 that occured after the initial port. Most recent changes are shown first. Hovering over a commit will show all commits associated with the same mathlib3 commit.

Changes in mathlib3

d46774d4

(last sync)

Changes in mathlib3port

mathlib3

mathlib3port

5d0c7689

bump to nightly-2024-04-06-08

mathlib commit https://github.com/leanprover-community/mathlib/commit/65a1391a0106c9204fe45bc73a039f056558cb83

e34029c9

Diff

@@ -5,7 +5,7 @@ Authors: Eric Wieser, Utensil Song
 -/
 import Algebra.RingQuot
 import LinearAlgebra.TensorAlgebra.Basic
-import LinearAlgebra.QuadraticForm.Isometry
+import LinearAlgebra.QuadraticForm.IsometryEquiv
 
 #align_import linear_algebra.clifford_algebra.basic from "leanprover-community/mathlib"@"5d0c76894ada7940957143163d7b921345474cbc"

5d0c7689

bump to nightly-2023-12-28-06

mathlib commit https://github.com/leanprover-community/mathlib/commit/65a1391a0106c9204fe45bc73a039f056558cb83

c30f5587

Diff

@@ -244,11 +244,11 @@ theorem ι_mul_ι_add_swap (a b : M) :
 #align clifford_algebra.ι_mul_ι_add_swap CliffordAlgebra.ι_mul_ι_add_swap
 -/
 
-#print CliffordAlgebra.ι_mul_comm /-
-theorem ι_mul_comm (a b : M) :
+#print CliffordAlgebra.ι_mul_ι_comm /-
+theorem ι_mul_ι_comm (a b : M) :
     ι Q a * ι Q b = algebraMap R _ (QuadraticForm.polar Q a b) - ι Q b * ι Q a :=
   eq_sub_of_add_eq (ι_mul_ι_add_swap a b)
-#align clifford_algebra.ι_mul_comm CliffordAlgebra.ι_mul_comm
+#align clifford_algebra.ι_mul_comm CliffordAlgebra.ι_mul_ι_comm
 -/
 
 #print CliffordAlgebra.ι_mul_ι_mul_ι /-

5d0c7689

bump to nightly-2023-09-24-06

mathlib commit https://github.com/leanprover-community/mathlib/commit/ce64cd319bb6b3e82f31c2d38e79080d377be451

5655435f

Diff

@@ -3,9 +3,9 @@ Copyright (c) 2020 Eric Wieser. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Eric Wieser, Utensil Song
 -/
-import Mathbin.Algebra.RingQuot
-import Mathbin.LinearAlgebra.TensorAlgebra.Basic
-import Mathbin.LinearAlgebra.QuadraticForm.Isometry
+import Algebra.RingQuot
+import LinearAlgebra.TensorAlgebra.Basic
+import LinearAlgebra.QuadraticForm.Isometry
 
 #align_import linear_algebra.clifford_algebra.basic from "leanprover-community/mathlib"@"5d0c76894ada7940957143163d7b921345474cbc"

5d0c7689

bump to nightly-2023-08-17-09

mathlib commit https://github.com/leanprover-community/mathlib/commit/32a7e535287f9c73f2e4d2aef306a39190f0b504

60285dcc

Diff

@@ -214,7 +214,7 @@ theorem induction {C : CliffordAlgebra Q → Prop}
   -- the arguments are enough to construct a subalgebra, and a mapping into it from M
   let s : Subalgebra R (CliffordAlgebra Q) :=
     { carrier := C
-      mul_mem' := h_mul
+      hMul_mem' := h_mul
       add_mem' := h_add
       algebraMap_mem' := h_grade0 }
   let of : { f : M →ₗ[R] s // ∀ m, f m * f m = algebraMap _ _ (Q m) } :=
@@ -385,9 +385,9 @@ variable (Q)
 def invertibleιOfInvertible (m : M) [Invertible (Q m)] : Invertible (ι Q m)
     where
   invOf := ι Q (⅟ (Q m) • m)
-  invOf_mul_self := by
+  invOf_hMul_self := by
     rw [map_smul, smul_mul_assoc, ι_sq_scalar, Algebra.smul_def, ← map_mul, invOf_mul_self, map_one]
-  mul_invOf_self := by
+  hMul_invOf_self := by
     rw [map_smul, mul_smul_comm, ι_sq_scalar, Algebra.smul_def, ← map_mul, invOf_mul_self, map_one]
 #align clifford_algebra.invertible_ι_of_invertible CliffordAlgebra.invertibleιOfInvertible
 -/

5d0c7689

bump to nightly-2023-08-03-09

mathlib commit https://github.com/leanprover-community/mathlib/commit/48a058d7e39a80ed56858505719a0b2197900999

b7c9a72c

Diff

@@ -337,7 +337,7 @@ variable {Q₁ Q₂ Q₃}
 /-- Two `clifford_algebra`s are equivalent as algebras if their quadratic forms are
 equivalent. -/
 @[simps apply]
-def equivOfIsometry (e : Q₁.Isometry Q₂) : CliffordAlgebra Q₁ ≃ₐ[R] CliffordAlgebra Q₂ :=
+def equivOfIsometry (e : Q₁.IsometryEquiv Q₂) : CliffordAlgebra Q₁ ≃ₐ[R] CliffordAlgebra Q₂ :=
   AlgEquiv.ofAlgHom (map Q₁ Q₂ e e.map_app) (map Q₂ Q₁ e.symm e.symm.map_app)
     ((map_comp_map _ _ _ _ _ _ _).trans <|
       by
@@ -354,7 +354,7 @@ def equivOfIsometry (e : Q₁.Isometry Q₂) : CliffordAlgebra Q₁ ≃ₐ[R] Cl
 
 #print CliffordAlgebra.equivOfIsometry_symm /-
 @[simp]
-theorem equivOfIsometry_symm (e : Q₁.Isometry Q₂) :
+theorem equivOfIsometry_symm (e : Q₁.IsometryEquiv Q₂) :
     (equivOfIsometry e).symm = equivOfIsometry e.symm :=
   rfl
 #align clifford_algebra.equiv_of_isometry_symm CliffordAlgebra.equivOfIsometry_symm
@@ -362,7 +362,7 @@ theorem equivOfIsometry_symm (e : Q₁.Isometry Q₂) :
 
 #print CliffordAlgebra.equivOfIsometry_trans /-
 @[simp]
-theorem equivOfIsometry_trans (e₁₂ : Q₁.Isometry Q₂) (e₂₃ : Q₂.Isometry Q₃) :
+theorem equivOfIsometry_trans (e₁₂ : Q₁.IsometryEquiv Q₂) (e₂₃ : Q₂.IsometryEquiv Q₃) :
     (equivOfIsometry e₁₂).trans (equivOfIsometry e₂₃) = equivOfIsometry (e₁₂.trans e₂₃) := by ext x;
   exact AlgHom.congr_fun (map_comp_map Q₁ Q₂ Q₃ _ _ _ _) x
 #align clifford_algebra.equiv_of_isometry_trans CliffordAlgebra.equivOfIsometry_trans
@@ -371,7 +371,7 @@ theorem equivOfIsometry_trans (e₁₂ : Q₁.Isometry Q₂) (e₂₃ : Q₂.Iso
 #print CliffordAlgebra.equivOfIsometry_refl /-
 @[simp]
 theorem equivOfIsometry_refl :
-    (equivOfIsometry <| QuadraticForm.Isometry.refl Q₁) = AlgEquiv.refl := by ext x;
+    (equivOfIsometry <| QuadraticForm.IsometryEquiv.refl Q₁) = AlgEquiv.refl := by ext x;
   exact AlgHom.congr_fun (map_id Q₁) x
 #align clifford_algebra.equiv_of_isometry_refl CliffordAlgebra.equivOfIsometry_refl
 -/

5d0c7689

bump to nightly-2023-07-20-09

mathlib commit https://github.com/leanprover-community/mathlib/commit/8ea5598db6caeddde6cb734aa179cc2408dbd345

9c56d0dd

Diff

@@ -2,16 +2,13 @@
 Copyright (c) 2020 Eric Wieser. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Eric Wieser, Utensil Song
-
-! This file was ported from Lean 3 source module linear_algebra.clifford_algebra.basic
-! leanprover-community/mathlib commit 5d0c76894ada7940957143163d7b921345474cbc
-! Please do not edit these lines, except to modify the commit id
-! if you have ported upstream changes.
 -/
 import Mathbin.Algebra.RingQuot
 import Mathbin.LinearAlgebra.TensorAlgebra.Basic
 import Mathbin.LinearAlgebra.QuadraticForm.Isometry
 
+#align_import linear_algebra.clifford_algebra.basic from "leanprover-community/mathlib"@"5d0c76894ada7940957143163d7b921345474cbc"
+
 /-!
 # Clifford Algebras

5d0c7689

bump to nightly-2023-06-24-03

mathlib commit https://github.com/leanprover-community/mathlib/commit/93f880918cb51905fd51b76add8273cbc27718ab

2e9fc2f2

Diff

@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Eric Wieser, Utensil Song
 
 ! This file was ported from Lean 3 source module linear_algebra.clifford_algebra.basic
-! leanprover-community/mathlib commit d46774d43797f5d1f507a63a6e904f7a533ae74a
+! leanprover-community/mathlib commit 5d0c76894ada7940957143163d7b921345474cbc
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/
@@ -15,6 +15,9 @@ import Mathbin.LinearAlgebra.QuadraticForm.Isometry
 /-!
 # Clifford Algebras
 
+> THIS FILE IS SYNCHRONIZED WITH MATHLIB4.
+> Any changes to this file require a corresponding PR to mathlib4.
+
 We construct the Clifford algebra of a module `M` over a commutative ring `R`, equipped with
 a quadratic_form `Q`.

d46774d4

bump to nightly-2023-06-22-09

mathlib commit https://github.com/leanprover-community/mathlib/commit/6285167a053ad0990fc88e56c48ccd9fae6550eb

c99553dc

Diff

@@ -61,6 +61,7 @@ namespace CliffordAlgebra
 
 open TensorAlgebra
 
+#print CliffordAlgebra.Rel /-
 /-- `rel` relates each `ι m * ι m`, for `m : M`, with `Q m`.
 
 The Clifford algebra of `M` is defined as the quotient modulo this relation.
@@ -68,10 +69,12 @@ The Clifford algebra of `M` is defined as the quotient modulo this relation.
 inductive Rel : TensorAlgebra R M → TensorAlgebra R M → Prop
   | of (m : M) : Rel (ι R m * ι R m) (algebraMap R _ (Q m))
 #align clifford_algebra.rel CliffordAlgebra.Rel
+-/
 
 end CliffordAlgebra
 
 /- ./././Mathport/Syntax/Translate/Command.lean:43:9: unsupported derive handler algebra[algebra] R -/
+#print CliffordAlgebra /-
 /-- The Clifford algebra of an `R`-module `M` equipped with a quadratic_form `Q`.
 -/
 def CliffordAlgebra :=
@@ -79,15 +82,19 @@ def CliffordAlgebra :=
 deriving Inhabited, Ring,
   «./././Mathport/Syntax/Translate/Command.lean:43:9: unsupported derive handler algebra[algebra] R»
 #align clifford_algebra CliffordAlgebra
+-/
 
 namespace CliffordAlgebra
 
+#print CliffordAlgebra.ι /-
 /-- The canonical linear map `M →ₗ[R] clifford_algebra Q`.
 -/
 def ι : M →ₗ[R] CliffordAlgebra Q :=
   (RingQuot.mkAlgHom R _).toLinearMap.comp (TensorAlgebra.ι R)
 #align clifford_algebra.ι CliffordAlgebra.ι
+-/
 
+#print CliffordAlgebra.ι_sq_scalar /-
 /-- As well as being linear, `ι Q` squares to the quadratic form -/
 @[simp]
 theorem ι_sq_scalar (m : M) : ι Q m * ι Q m = algebraMap R _ (Q m) :=
@@ -95,17 +102,21 @@ theorem ι_sq_scalar (m : M) : ι Q m * ι Q m = algebraMap R _ (Q m) :=
   erw [← AlgHom.map_mul, RingQuot.mkAlgHom_rel R (rel.of m), AlgHom.commutes]
   rfl
 #align clifford_algebra.ι_sq_scalar CliffordAlgebra.ι_sq_scalar
+-/
 
 variable {Q} {A : Type _} [Semiring A] [Algebra R A]
 
+#print CliffordAlgebra.comp_ι_sq_scalar /-
 @[simp]
 theorem comp_ι_sq_scalar (g : CliffordAlgebra Q →ₐ[R] A) (m : M) :
     g (ι Q m) * g (ι Q m) = algebraMap _ _ (Q m) := by
   rw [← AlgHom.map_mul, ι_sq_scalar, AlgHom.commutes]
 #align clifford_algebra.comp_ι_sq_scalar CliffordAlgebra.comp_ι_sq_scalar
+-/
 
 variable (Q)
 
+#print CliffordAlgebra.lift /-
 /-- Given a linear map `f : M →ₗ[R] A` into an `R`-algebra `A`, which satisfies the condition:
 `cond : ∀ m : M, f m * f m = Q(m)`, this is the canonical lift of `f` to a morphism of `R`-algebras
 from `clifford_algebra Q` to `A`.
@@ -132,21 +143,27 @@ def lift : { f : M →ₗ[R] A // ∀ m, f m * f m = algebraMap _ _ (Q m) } ≃
       LinearMap.coe_comp, Subtype.coe_mk, RingQuot.liftAlgHom_mkAlgHom_apply,
       TensorAlgebra.lift_ι_apply]
 #align clifford_algebra.lift CliffordAlgebra.lift
+-/
 
 variable {Q}
 
+#print CliffordAlgebra.ι_comp_lift /-
 @[simp]
 theorem ι_comp_lift (f : M →ₗ[R] A) (cond : ∀ m, f m * f m = algebraMap _ _ (Q m)) :
     (lift Q ⟨f, cond⟩).toLinearMap.comp (ι Q) = f :=
   Subtype.mk_eq_mk.mp <| (lift Q).symm_apply_apply ⟨f, cond⟩
 #align clifford_algebra.ι_comp_lift CliffordAlgebra.ι_comp_lift
+-/
 
+#print CliffordAlgebra.lift_ι_apply /-
 @[simp]
 theorem lift_ι_apply (f : M →ₗ[R] A) (cond : ∀ m, f m * f m = algebraMap _ _ (Q m)) (x) :
     lift Q ⟨f, cond⟩ (ι Q x) = f x :=
   (LinearMap.ext_iff.mp <| ι_comp_lift f cond) x
 #align clifford_algebra.lift_ι_apply CliffordAlgebra.lift_ι_apply
+-/
 
+#print CliffordAlgebra.lift_unique /-
 @[simp]
 theorem lift_unique (f : M →ₗ[R] A) (cond : ∀ m : M, f m * f m = algebraMap _ _ (Q m))
     (g : CliffordAlgebra Q →ₐ[R] A) : g.toLinearMap.comp (ι Q) = f ↔ g = lift Q ⟨f, cond⟩ :=
@@ -155,7 +172,9 @@ theorem lift_unique (f : M →ₗ[R] A) (cond : ∀ m : M, f m * f m = algebraMa
   rw [lift_symm_apply]
   simp only
 #align clifford_algebra.lift_unique CliffordAlgebra.lift_unique
+-/
 
+#print CliffordAlgebra.lift_comp_ι /-
 @[simp]
 theorem lift_comp_ι (g : CliffordAlgebra Q →ₐ[R] A) :
     lift Q ⟨g.toLinearMap.comp (ι Q), comp_ι_sq_scalar _⟩ = g :=
@@ -164,7 +183,9 @@ theorem lift_comp_ι (g : CliffordAlgebra Q →ₐ[R] A) :
   rw [lift_symm_apply]
   rfl
 #align clifford_algebra.lift_comp_ι CliffordAlgebra.lift_comp_ι
+-/
 
+#print CliffordAlgebra.hom_ext /-
 /-- See note [partially-applied ext lemmas]. -/
 @[ext]
 theorem hom_ext {A : Type _} [Semiring A] [Algebra R A] {f g : CliffordAlgebra Q →ₐ[R] A} :
@@ -175,7 +196,9 @@ theorem hom_ext {A : Type _} [Semiring A] [Algebra R A] {f g : CliffordAlgebra Q
   rw [lift_symm_apply, lift_symm_apply]
   simp only [h]
 #align clifford_algebra.hom_ext CliffordAlgebra.hom_ext
+-/
 
+#print CliffordAlgebra.induction /-
 -- This proof closely follows `tensor_algebra.induction`
 /-- If `C` holds for the `algebra_map` of `r : R` into `clifford_algebra Q`, the `ι` of `x : M`,
 and is preserved under addition and muliplication, then it holds for all of `clifford_algebra Q`.
@@ -205,7 +228,9 @@ theorem induction {C : CliffordAlgebra Q → Prop}
   convert Subtype.prop (lift Q of a)
   exact AlgHom.congr_fun of_id a
 #align clifford_algebra.induction CliffordAlgebra.induction
+-/
 
+#print CliffordAlgebra.ι_mul_ι_add_swap /-
 /-- The symmetric product of vectors is a scalar -/
 theorem ι_mul_ι_add_swap (a b : M) :
     ι Q a * ι Q b + ι Q b * ι Q a = algebraMap R _ (QuadraticForm.polar Q a b) :=
@@ -217,24 +242,31 @@ theorem ι_mul_ι_add_swap (a b : M) :
     _ = algebraMap R _ (Q (a + b) - Q a - Q b) := by rw [← RingHom.map_sub, ← RingHom.map_sub]
     _ = algebraMap R _ (QuadraticForm.polar Q a b) := rfl
 #align clifford_algebra.ι_mul_ι_add_swap CliffordAlgebra.ι_mul_ι_add_swap
+-/
 
+#print CliffordAlgebra.ι_mul_comm /-
 theorem ι_mul_comm (a b : M) :
     ι Q a * ι Q b = algebraMap R _ (QuadraticForm.polar Q a b) - ι Q b * ι Q a :=
   eq_sub_of_add_eq (ι_mul_ι_add_swap a b)
 #align clifford_algebra.ι_mul_comm CliffordAlgebra.ι_mul_comm
+-/
 
+#print CliffordAlgebra.ι_mul_ι_mul_ι /-
 /-- $aba$ is a vector. -/
 theorem ι_mul_ι_mul_ι (a b : M) :
     ι Q a * ι Q b * ι Q a = ι Q (QuadraticForm.polar Q a b • a - Q a • b) := by
   rw [ι_mul_comm, sub_mul, mul_assoc, ι_sq_scalar, ← Algebra.smul_def, ← Algebra.commutes, ←
     Algebra.smul_def, ← map_smul, ← map_smul, ← map_sub]
 #align clifford_algebra.ι_mul_ι_mul_ι CliffordAlgebra.ι_mul_ι_mul_ι
+-/
 
+#print CliffordAlgebra.ι_range_map_lift /-
 @[simp]
 theorem ι_range_map_lift (f : M →ₗ[R] A) (cond : ∀ m, f m * f m = algebraMap _ _ (Q m)) :
     (ι Q).range.map (lift Q ⟨f, cond⟩).toLinearMap = f.range := by
   rw [← LinearMap.range_comp, ι_comp_lift]
 #align clifford_algebra.ι_range_map_lift CliffordAlgebra.ι_range_map_lift
+-/
 
 section Map
 
@@ -246,6 +278,7 @@ variable [Module R M₁] [Module R M₂] [Module R M₃]
 
 variable (Q₁ : QuadraticForm R M₁) (Q₂ : QuadraticForm R M₂) (Q₃ : QuadraticForm R M₃)
 
+#print CliffordAlgebra.map /-
 /-- Any linear map that preserves the quadratic form lifts to an `alg_hom` between algebras.
 
 See `clifford_algebra.equiv_of_isometry` for the case when `f` is a `quadratic_form.isometry`. -/
@@ -254,24 +287,32 @@ def map (f : M₁ →ₗ[R] M₂) (hf : ∀ m, Q₂ (f m) = Q₁ m) :
   CliffordAlgebra.lift Q₁
     ⟨(CliffordAlgebra.ι Q₂).comp f, fun m => (ι_sq_scalar _ _).trans <| RingHom.congr_arg _ <| hf m⟩
 #align clifford_algebra.map CliffordAlgebra.map
+-/
 
+#print CliffordAlgebra.map_comp_ι /-
 @[simp]
 theorem map_comp_ι (f : M₁ →ₗ[R] M₂) (hf) :
     (map Q₁ Q₂ f hf).toLinearMap.comp (ι Q₁) = (ι Q₂).comp f :=
   ι_comp_lift _ _
 #align clifford_algebra.map_comp_ι CliffordAlgebra.map_comp_ι
+-/
 
+#print CliffordAlgebra.map_apply_ι /-
 @[simp]
 theorem map_apply_ι (f : M₁ →ₗ[R] M₂) (hf) (m : M₁) : map Q₁ Q₂ f hf (ι Q₁ m) = ι Q₂ (f m) :=
   lift_ι_apply _ _ m
 #align clifford_algebra.map_apply_ι CliffordAlgebra.map_apply_ι
+-/
 
+#print CliffordAlgebra.map_id /-
 @[simp]
 theorem map_id :
     (map Q₁ Q₁ (LinearMap.id : M₁ →ₗ[R] M₁) fun m => rfl) = AlgHom.id R (CliffordAlgebra Q₁) := by
   ext m; exact map_apply_ι _ _ _ _ m
 #align clifford_algebra.map_id CliffordAlgebra.map_id
+-/
 
+#print CliffordAlgebra.map_comp_map /-
 @[simp]
 theorem map_comp_map (f : M₂ →ₗ[R] M₃) (hf) (g : M₁ →ₗ[R] M₂) (hg) :
     (map Q₂ Q₃ f hf).comp (map Q₁ Q₂ g hg) = map Q₁ Q₃ (f.comp g) fun m => (hf _).trans <| hg m :=
@@ -280,15 +321,19 @@ theorem map_comp_map (f : M₂ →ₗ[R] M₃) (hf) (g : M₁ →ₗ[R] M₂) (h
   dsimp only [LinearMap.comp_apply, AlgHom.comp_apply, AlgHom.toLinearMap_apply, AlgHom.id_apply]
   rw [map_apply_ι, map_apply_ι, map_apply_ι, LinearMap.comp_apply]
 #align clifford_algebra.map_comp_map CliffordAlgebra.map_comp_map
+-/
 
+#print CliffordAlgebra.ι_range_map_map /-
 @[simp]
 theorem ι_range_map_map (f : M₁ →ₗ[R] M₂) (hf : ∀ m, Q₂ (f m) = Q₁ m) :
     (ι Q₁).range.map (map Q₁ Q₂ f hf).toLinearMap = f.range.map (ι Q₂) :=
   (ι_range_map_lift _ _).trans (LinearMap.range_comp _ _)
 #align clifford_algebra.ι_range_map_map CliffordAlgebra.ι_range_map_map
+-/
 
 variable {Q₁ Q₂ Q₃}
 
+#print CliffordAlgebra.equivOfIsometry /-
 /-- Two `clifford_algebra`s are equivalent as algebras if their quadratic forms are
 equivalent. -/
 @[simps apply]
@@ -305,29 +350,37 @@ def equivOfIsometry (e : Q₁.Isometry Q₂) : CliffordAlgebra Q₁ ≃ₐ[R] Cl
       ext m
       exact e.to_linear_equiv.symm_apply_apply m)
 #align clifford_algebra.equiv_of_isometry CliffordAlgebra.equivOfIsometry
+-/
 
+#print CliffordAlgebra.equivOfIsometry_symm /-
 @[simp]
 theorem equivOfIsometry_symm (e : Q₁.Isometry Q₂) :
     (equivOfIsometry e).symm = equivOfIsometry e.symm :=
   rfl
 #align clifford_algebra.equiv_of_isometry_symm CliffordAlgebra.equivOfIsometry_symm
+-/
 
+#print CliffordAlgebra.equivOfIsometry_trans /-
 @[simp]
 theorem equivOfIsometry_trans (e₁₂ : Q₁.Isometry Q₂) (e₂₃ : Q₂.Isometry Q₃) :
     (equivOfIsometry e₁₂).trans (equivOfIsometry e₂₃) = equivOfIsometry (e₁₂.trans e₂₃) := by ext x;
   exact AlgHom.congr_fun (map_comp_map Q₁ Q₂ Q₃ _ _ _ _) x
 #align clifford_algebra.equiv_of_isometry_trans CliffordAlgebra.equivOfIsometry_trans
+-/
 
+#print CliffordAlgebra.equivOfIsometry_refl /-
 @[simp]
 theorem equivOfIsometry_refl :
     (equivOfIsometry <| QuadraticForm.Isometry.refl Q₁) = AlgEquiv.refl := by ext x;
   exact AlgHom.congr_fun (map_id Q₁) x
 #align clifford_algebra.equiv_of_isometry_refl CliffordAlgebra.equivOfIsometry_refl
+-/
 
 end Map
 
 variable (Q)
 
+#print CliffordAlgebra.invertibleιOfInvertible /-
 /-- If the quadratic form of a vector is invertible, then so is that vector. -/
 def invertibleιOfInvertible (m : M) [Invertible (Q m)] : Invertible (ι Q m)
     where
@@ -337,34 +390,43 @@ def invertibleιOfInvertible (m : M) [Invertible (Q m)] : Invertible (ι Q m)
   mul_invOf_self := by
     rw [map_smul, mul_smul_comm, ι_sq_scalar, Algebra.smul_def, ← map_mul, invOf_mul_self, map_one]
 #align clifford_algebra.invertible_ι_of_invertible CliffordAlgebra.invertibleιOfInvertible
+-/
 
+#print CliffordAlgebra.invOf_ι /-
 /-- For a vector with invertible quadratic form, $v^{-1} = \frac{v}{Q(v)}$ -/
 theorem invOf_ι (m : M) [Invertible (Q m)] [Invertible (ι Q m)] : ⅟ (ι Q m) = ι Q (⅟ (Q m) • m) :=
   by
   letI := invertible_ι_of_invertible Q m
   convert (rfl : ⅟ (ι Q m) = _)
 #align clifford_algebra.inv_of_ι CliffordAlgebra.invOf_ι
+-/
 
+#print CliffordAlgebra.isUnit_ι_of_isUnit /-
 theorem isUnit_ι_of_isUnit {m : M} (h : IsUnit (Q m)) : IsUnit (ι Q m) :=
   by
   cases h.nonempty_invertible
   letI := invertible_ι_of_invertible Q m
   exact isUnit_of_invertible (ι Q m)
 #align clifford_algebra.is_unit_ι_of_is_unit CliffordAlgebra.isUnit_ι_of_isUnit
+-/
 
+#print CliffordAlgebra.ι_mul_ι_mul_invOf_ι /-
 /-- $aba^{-1}$ is a vector. -/
 theorem ι_mul_ι_mul_invOf_ι (a b : M) [Invertible (ι Q a)] [Invertible (Q a)] :
     ι Q a * ι Q b * ⅟ (ι Q a) = ι Q ((⅟ (Q a) * QuadraticForm.polar Q a b) • a - b) := by
   rw [inv_of_ι, map_smul, mul_smul_comm, ι_mul_ι_mul_ι, ← map_smul, smul_sub, smul_smul, smul_smul,
     invOf_mul_self, one_smul]
 #align clifford_algebra.ι_mul_ι_mul_inv_of_ι CliffordAlgebra.ι_mul_ι_mul_invOf_ι
+-/
 
+#print CliffordAlgebra.invOf_ι_mul_ι_mul_ι /-
 /-- $a^{-1}ba$ is a vector. -/
 theorem invOf_ι_mul_ι_mul_ι (a b : M) [Invertible (ι Q a)] [Invertible (Q a)] :
     ⅟ (ι Q a) * ι Q b * ι Q a = ι Q ((⅟ (Q a) * QuadraticForm.polar Q a b) • a - b) := by
   rw [inv_of_ι, map_smul, smul_mul_assoc, smul_mul_assoc, ι_mul_ι_mul_ι, ← map_smul, smul_sub,
     smul_smul, smul_smul, invOf_mul_self, one_smul]
 #align clifford_algebra.inv_of_ι_mul_ι_mul_ι CliffordAlgebra.invOf_ι_mul_ι_mul_ι
+-/
 
 end CliffordAlgebra
 
@@ -372,16 +434,20 @@ namespace TensorAlgebra
 
 variable {Q}
 
+#print TensorAlgebra.toClifford /-
 /-- The canonical image of the `tensor_algebra` in the `clifford_algebra`, which maps
 `tensor_algebra.ι R x` to `clifford_algebra.ι Q x`. -/
 def toClifford : TensorAlgebra R M →ₐ[R] CliffordAlgebra Q :=
   TensorAlgebra.lift R (CliffordAlgebra.ι Q)
 #align tensor_algebra.to_clifford TensorAlgebra.toClifford
+-/
 
+#print TensorAlgebra.toClifford_ι /-
 @[simp]
 theorem toClifford_ι (m : M) : (TensorAlgebra.ι R m).toClifford = CliffordAlgebra.ι Q m := by
   simp [to_clifford]
 #align tensor_algebra.to_clifford_ι TensorAlgebra.toClifford_ι
+-/
 
 end TensorAlgebra

d46774d4

bump to nightly-2023-06-09-07

mathlib commit https://github.com/leanprover-community/mathlib/commit/7e5137f579de09a059a5ce98f364a04e221aabf0

16afdc39

Diff

@@ -216,7 +216,6 @@ theorem ι_mul_ι_add_swap (a b : M) :
       rw [ι_sq_scalar, ι_sq_scalar, ι_sq_scalar]
     _ = algebraMap R _ (Q (a + b) - Q a - Q b) := by rw [← RingHom.map_sub, ← RingHom.map_sub]
     _ = algebraMap R _ (QuadraticForm.polar Q a b) := rfl
-    
 #align clifford_algebra.ι_mul_ι_add_swap CliffordAlgebra.ι_mul_ι_add_swap
 
 theorem ι_mul_comm (a b : M) :

d46774d4

bump to nightly-2023-06-04-18

mathlib commit https://github.com/leanprover-community/mathlib/commit/5f25c089cb34db4db112556f23c50d12da81b297

3fb4268b

Diff

@@ -71,13 +71,13 @@ inductive Rel : TensorAlgebra R M → TensorAlgebra R M → Prop
 
 end CliffordAlgebra
 
-/- ./././Mathport/Syntax/Translate/Command.lean:42:9: unsupported derive handler algebra[algebra] R -/
+/- ./././Mathport/Syntax/Translate/Command.lean:43:9: unsupported derive handler algebra[algebra] R -/
 /-- The Clifford algebra of an `R`-module `M` equipped with a quadratic_form `Q`.
 -/
 def CliffordAlgebra :=
   RingQuot (CliffordAlgebra.Rel Q)
 deriving Inhabited, Ring,
-  «./././Mathport/Syntax/Translate/Command.lean:42:9: unsupported derive handler algebra[algebra] R»
+  «./././Mathport/Syntax/Translate/Command.lean:43:9: unsupported derive handler algebra[algebra] R»
 #align clifford_algebra CliffordAlgebra
 
 namespace CliffordAlgebra
@@ -151,7 +151,7 @@ theorem lift_ι_apply (f : M →ₗ[R] A) (cond : ∀ m, f m * f m = algebraMap
 theorem lift_unique (f : M →ₗ[R] A) (cond : ∀ m : M, f m * f m = algebraMap _ _ (Q m))
     (g : CliffordAlgebra Q →ₐ[R] A) : g.toLinearMap.comp (ι Q) = f ↔ g = lift Q ⟨f, cond⟩ :=
   by
-  convert(lift Q).symm_apply_eq
+  convert (lift Q).symm_apply_eq
   rw [lift_symm_apply]
   simp only
 #align clifford_algebra.lift_unique CliffordAlgebra.lift_unique
@@ -160,7 +160,7 @@ theorem lift_unique (f : M →ₗ[R] A) (cond : ∀ m : M, f m * f m = algebraMa
 theorem lift_comp_ι (g : CliffordAlgebra Q →ₐ[R] A) :
     lift Q ⟨g.toLinearMap.comp (ι Q), comp_ι_sq_scalar _⟩ = g :=
   by
-  convert(lift Q).apply_symm_apply g
+  convert (lift Q).apply_symm_apply g
   rw [lift_symm_apply]
   rfl
 #align clifford_algebra.lift_comp_ι CliffordAlgebra.lift_comp_ι
@@ -343,7 +343,7 @@ def invertibleιOfInvertible (m : M) [Invertible (Q m)] : Invertible (ι Q m)
 theorem invOf_ι (m : M) [Invertible (Q m)] [Invertible (ι Q m)] : ⅟ (ι Q m) = ι Q (⅟ (Q m) • m) :=
   by
   letI := invertible_ι_of_invertible Q m
-  convert(rfl : ⅟ (ι Q m) = _)
+  convert (rfl : ⅟ (ι Q m) = _)
 #align clifford_algebra.inv_of_ι CliffordAlgebra.invOf_ι
 
 theorem isUnit_ι_of_isUnit {m : M} (h : IsUnit (Q m)) : IsUnit (ι Q m) :=

d46774d4

bump to nightly-2023-06-01-16

mathlib commit https://github.com/leanprover-community/mathlib/commit/cca40788df1b8755d5baf17ab2f27dacc2e17acb

b9c87c70

Diff

@@ -75,7 +75,8 @@ end CliffordAlgebra
 /-- The Clifford algebra of an `R`-module `M` equipped with a quadratic_form `Q`.
 -/
 def CliffordAlgebra :=
-  RingQuot (CliffordAlgebra.Rel Q)deriving Inhabited, Ring,
+  RingQuot (CliffordAlgebra.Rel Q)
+deriving Inhabited, Ring,
   «./././Mathport/Syntax/Translate/Command.lean:42:9: unsupported derive handler algebra[algebra] R»
 #align clifford_algebra CliffordAlgebra

d46774d4

bump to nightly-2023-05-28-04

mathlib commit https://github.com/leanprover-community/mathlib/commit/917c3c072e487b3cccdbfeff17e75b40e45f66cb

6797a637

Diff

@@ -209,10 +209,8 @@ theorem induction {C : CliffordAlgebra Q → Prop}
 theorem ι_mul_ι_add_swap (a b : M) :
     ι Q a * ι Q b + ι Q b * ι Q a = algebraMap R _ (QuadraticForm.polar Q a b) :=
   calc
-    ι Q a * ι Q b + ι Q b * ι Q a = ι Q (a + b) * ι Q (a + b) - ι Q a * ι Q a - ι Q b * ι Q b :=
-      by
-      rw [(ι Q).map_add, mul_add, add_mul, add_mul]
-      abel
+    ι Q a * ι Q b + ι Q b * ι Q a = ι Q (a + b) * ι Q (a + b) - ι Q a * ι Q a - ι Q b * ι Q b := by
+      rw [(ι Q).map_add, mul_add, add_mul, add_mul]; abel
     _ = algebraMap R _ (Q (a + b)) - algebraMap R _ (Q a) - algebraMap R _ (Q b) := by
       rw [ι_sq_scalar, ι_sq_scalar, ι_sq_scalar]
     _ = algebraMap R _ (Q (a + b) - Q a - Q b) := by rw [← RingHom.map_sub, ← RingHom.map_sub]
@@ -270,10 +268,8 @@ theorem map_apply_ι (f : M₁ →ₗ[R] M₂) (hf) (m : M₁) : map Q₁ Q₂ f
 
 @[simp]
 theorem map_id :
-    (map Q₁ Q₁ (LinearMap.id : M₁ →ₗ[R] M₁) fun m => rfl) = AlgHom.id R (CliffordAlgebra Q₁) :=
-  by
-  ext m
-  exact map_apply_ι _ _ _ _ m
+    (map Q₁ Q₁ (LinearMap.id : M₁ →ₗ[R] M₁) fun m => rfl) = AlgHom.id R (CliffordAlgebra Q₁) := by
+  ext m; exact map_apply_ι _ _ _ _ m
 #align clifford_algebra.map_id CliffordAlgebra.map_id
 
 @[simp]
@@ -318,17 +314,13 @@ theorem equivOfIsometry_symm (e : Q₁.Isometry Q₂) :
 
 @[simp]
 theorem equivOfIsometry_trans (e₁₂ : Q₁.Isometry Q₂) (e₂₃ : Q₂.Isometry Q₃) :
-    (equivOfIsometry e₁₂).trans (equivOfIsometry e₂₃) = equivOfIsometry (e₁₂.trans e₂₃) :=
-  by
-  ext x
+    (equivOfIsometry e₁₂).trans (equivOfIsometry e₂₃) = equivOfIsometry (e₁₂.trans e₂₃) := by ext x;
   exact AlgHom.congr_fun (map_comp_map Q₁ Q₂ Q₃ _ _ _ _) x
 #align clifford_algebra.equiv_of_isometry_trans CliffordAlgebra.equivOfIsometry_trans
 
 @[simp]
 theorem equivOfIsometry_refl :
-    (equivOfIsometry <| QuadraticForm.Isometry.refl Q₁) = AlgEquiv.refl :=
-  by
-  ext x
+    (equivOfIsometry <| QuadraticForm.Isometry.refl Q₁) = AlgEquiv.refl := by ext x;
   exact AlgHom.congr_fun (map_id Q₁) x
 #align clifford_algebra.equiv_of_isometry_refl CliffordAlgebra.equivOfIsometry_refl

d46774d4

bump to nightly-2023-03-17-00

mathlib commit https://github.com/leanprover-community/mathlib/commit/ce7e9d53d4bbc38065db3b595cd5bd73c323bc1d

c85864d4

Diff

@@ -150,7 +150,7 @@ theorem lift_ι_apply (f : M →ₗ[R] A) (cond : ∀ m, f m * f m = algebraMap
 theorem lift_unique (f : M →ₗ[R] A) (cond : ∀ m : M, f m * f m = algebraMap _ _ (Q m))
     (g : CliffordAlgebra Q →ₐ[R] A) : g.toLinearMap.comp (ι Q) = f ↔ g = lift Q ⟨f, cond⟩ :=
   by
-  convert (lift Q).symm_apply_eq
+  convert(lift Q).symm_apply_eq
   rw [lift_symm_apply]
   simp only
 #align clifford_algebra.lift_unique CliffordAlgebra.lift_unique
@@ -159,7 +159,7 @@ theorem lift_unique (f : M →ₗ[R] A) (cond : ∀ m : M, f m * f m = algebraMa
 theorem lift_comp_ι (g : CliffordAlgebra Q →ₐ[R] A) :
     lift Q ⟨g.toLinearMap.comp (ι Q), comp_ι_sq_scalar _⟩ = g :=
   by
-  convert (lift Q).apply_symm_apply g
+  convert(lift Q).apply_symm_apply g
   rw [lift_symm_apply]
   rfl
 #align clifford_algebra.lift_comp_ι CliffordAlgebra.lift_comp_ι
@@ -350,7 +350,7 @@ def invertibleιOfInvertible (m : M) [Invertible (Q m)] : Invertible (ι Q m)
 theorem invOf_ι (m : M) [Invertible (Q m)] [Invertible (ι Q m)] : ⅟ (ι Q m) = ι Q (⅟ (Q m) • m) :=
   by
   letI := invertible_ι_of_invertible Q m
-  convert (rfl : ⅟ (ι Q m) = _)
+  convert(rfl : ⅟ (ι Q m) = _)
 #align clifford_algebra.inv_of_ι CliffordAlgebra.invOf_ι
 
 theorem isUnit_ι_of_isUnit {m : M} (h : IsUnit (Q m)) : IsUnit (ι Q m) :=

d46774d4

bump to nightly-2023-02-21-16

mathlib commit https://github.com/leanprover-community/mathlib/commit/bd9851ca476957ea4549eb19b40e7b5ade9428cc

1107b979

Changes in mathlib4

mathlib3

mathlib4

d46774d4

style: replace '.-/' by '. -/' (#11938)

Purely automatic replacement. If this is in any way controversial; I'm happy to just close this PR.

3556ded2

Diff

@@ -359,7 +359,7 @@ lemma leftInverse_map_of_leftInverse {Q₁ : QuadraticForm R M₁} {Q₂ : Quadr
   rw [← AlgHom.comp_apply, map_comp_map, h, map_id, AlgHom.coe_id, id_eq]
 
 /-- If a linear map preserves the quadratic forms and is surjective, then the algebra
-maps it induces between Clifford algebras is also surjective.-/
+maps it induces between Clifford algebras is also surjective. -/
 lemma map_surjective {Q₁ : QuadraticForm R M₁} {Q₂ : QuadraticForm R M₂} (f : Q₁ →qᵢ Q₂)
     (hf : Function.Surjective f) : Function.Surjective (CliffordAlgebra.map f) :=
   CliffordAlgebra.induction

d46774d4

chore(*): remove empty lines between variable statements (#11418)

Empty lines were removed by executing the following Python script twice

import os
import re


# Loop through each file in the repository
for dir_path, dirs, files in os.walk('.'):
  for filename in files:
    if filename.endswith('.lean'):
      file_path = os.path.join(dir_path, filename)

      # Open the file and read its contents
      with open(file_path, 'r') as file:
        content = file.read()

      # Use a regular expression to replace sequences of "variable" lines separated by empty lines
      # with sequences without empty lines
      modified_content = re.sub(r'(variable.*\n)\n(variable(?! .* in))', r'\1\2', content)

      # Write the modified content back to the file
      with open(file_path, 'w') as file:
        file.write(modified_content)

75c86f04

Diff

@@ -46,11 +46,8 @@ This file is almost identical to `Mathlib/LinearAlgebra/ExteriorAlgebra/Basic.le
 
 
 variable {R : Type*} [CommRing R]
-
 variable {M : Type*} [AddCommGroup M] [Module R M]
-
 variable (Q : QuadraticForm R M)
-
 variable {n : ℕ}
 
 namespace CliffordAlgebra
@@ -307,11 +304,8 @@ theorem ι_range_map_lift (f : M →ₗ[R] A) (cond : ∀ m, f m * f m = algebra
 section Map
 
 variable {M₁ M₂ M₃ : Type*}
-
 variable [AddCommGroup M₁] [AddCommGroup M₂] [AddCommGroup M₃]
-
 variable [Module R M₁] [Module R M₂] [Module R M₃]
-
 variable {Q₁ : QuadraticForm R M₁} {Q₂ : QuadraticForm R M₂} {Q₃ : QuadraticForm R M₃}
 
 /-- Any linear map that preserves the quadratic form lifts to an `AlgHom` between algebras.

d46774d4

style: homogenise porting notes (#11145)

Homogenises porting notes via capitalisation and addition of whitespace.

It makes the following changes:

converts "--porting note" into "-- Porting note";
converts "porting note" into "Porting note".

8be0b69c

Diff

@@ -149,10 +149,10 @@ def lift : { f : M →ₗ[R] A // ∀ m, f m * f m = algebraMap _ _ (Q m) } ≃
       rw [LinearMap.comp_apply, AlgHom.toLinearMap_apply, comp_ι_sq_scalar]⟩
   left_inv f := by
     ext x
-    -- porting note: removed `simp only` proof which gets stuck simplifying `LinearMap.comp_apply`
+    -- Porting note: removed `simp only` proof which gets stuck simplifying `LinearMap.comp_apply`
     exact (RingQuot.liftAlgHom_mkAlgHom_apply _ _ _ _).trans (TensorAlgebra.lift_ι_apply _ x)
   right_inv F :=
-    -- porting note: replaced with proof derived from the one for `TensorAlgebra`
+    -- Porting note: replaced with proof derived from the one for `TensorAlgebra`
     RingQuot.ringQuot_ext' _ _ _ <|
       TensorAlgebra.hom_ext <|
         LinearMap.ext fun x => by
@@ -178,14 +178,14 @@ theorem lift_ι_apply (f : M →ₗ[R] A) (cond : ∀ m, f m * f m = algebraMap
 theorem lift_unique (f : M →ₗ[R] A) (cond : ∀ m : M, f m * f m = algebraMap _ _ (Q m))
     (g : CliffordAlgebra Q →ₐ[R] A) : g.toLinearMap.comp (ι Q) = f ↔ g = lift Q ⟨f, cond⟩ := by
   convert (lift Q : _ ≃ (CliffordAlgebra Q →ₐ[R] A)).symm_apply_eq
-  -- porting note: added `Subtype.mk_eq_mk`
+  -- Porting note: added `Subtype.mk_eq_mk`
   rw [lift_symm_apply, Subtype.mk_eq_mk]
 #align clifford_algebra.lift_unique CliffordAlgebra.lift_unique
 
 @[simp]
 theorem lift_comp_ι (g : CliffordAlgebra Q →ₐ[R] A) :
     lift Q ⟨g.toLinearMap.comp (ι Q), comp_ι_sq_scalar _⟩ = g := by
-  -- porting note: removed `rw [lift_symm_apply]; rfl`, changed `convert` to `exact`
+  -- Porting note: removed `rw [lift_symm_apply]; rfl`, changed `convert` to `exact`
   exact (lift Q : _ ≃ (CliffordAlgebra Q →ₐ[R] A)).apply_symm_apply g
 #align clifford_algebra.lift_comp_ι CliffordAlgebra.lift_comp_ι
 
@@ -216,7 +216,7 @@ theorem induction {C : CliffordAlgebra Q → Prop}
       mul_mem' := @mul
       add_mem' := @add
       algebraMap_mem' := algebraMap }
-  -- porting note: Added `h`. `h` is needed for `of`.
+  -- Porting note: Added `h`. `h` is needed for `of`.
   letI h : AddCommMonoid s := inferInstanceAs (AddCommMonoid (Subalgebra.toSubmodule s))
   let of : { f : M →ₗ[R] s // ∀ m, f m * f m = _root_.algebraMap _ _ (Q m) } :=
     ⟨(CliffordAlgebra.ι Q).codRestrict (Subalgebra.toSubmodule s) ι,
@@ -225,10 +225,10 @@ theorem induction {C : CliffordAlgebra Q → Prop}
   have of_id : AlgHom.id R (CliffordAlgebra Q) = s.val.comp (lift Q of) := by
     ext
     simp [of]
-    -- porting note: `simp` can't apply this
+    -- Porting note: `simp` can't apply this
     erw [LinearMap.codRestrict_apply]
   -- finding a proof is finding an element of the subalgebra
-  -- porting note: was `convert Subtype.prop (lift Q of a); exact AlgHom.congr_fun of_id a`
+  -- Porting note: was `convert Subtype.prop (lift Q of a); exact AlgHom.congr_fun of_id a`
   rw [← AlgHom.id_apply (R := R) a, of_id]
   exact Subtype.prop (lift Q of a)
 #align clifford_algebra.induction CliffordAlgebra.induction
@@ -380,11 +380,11 @@ equivalent. -/
 def equivOfIsometry (e : Q₁.IsometryEquiv Q₂) : CliffordAlgebra Q₁ ≃ₐ[R] CliffordAlgebra Q₂ :=
   AlgEquiv.ofAlgHom (map e.toIsometry) (map e.symm.toIsometry)
     ((map_comp_map _ _).trans <| by
-      convert map_id Q₂ using 2  -- porting note: replaced `_` with `Q₂`
+      convert map_id Q₂ using 2  -- Porting note: replaced `_` with `Q₂`
       ext m
       exact e.toLinearEquiv.apply_symm_apply m)
     ((map_comp_map _ _).trans <| by
-      convert map_id Q₁ using 2  -- porting note: replaced `_` with `Q₁`
+      convert map_id Q₁ using 2  -- Porting note: replaced `_` with `Q₁`
       ext m
       exact e.toLinearEquiv.symm_apply_apply m)
 #align clifford_algebra.equiv_of_isometry CliffordAlgebra.equivOfIsometry

d46774d4

chore: rename induction principle arguments around CliffordAlgebra (#10908)

In order to improve the ergonomics of the induction tactic, this renames the arguments of:

ExteriorAlgebra.induction
TensorAlgebra.induction
CliffordAlgebra.induction
CliffordAlgebra.left_induction
CliffordAlgebra.right_induction
CliffordAlgebra.even_induction
CliffordAlgebra.odd_induction
Submodule.iSup_induction'
Submodule.pow_induction_on_left'
Submodule.pow_induction_on_right'

This is slightly awkward for name-resolution within these induction principles, as the argument names end up clashing with the function they are about. Thankfully, this pain is not transferred to the caller using induction _ using _.

0257770c

Diff

@@ -207,19 +207,20 @@ See also the stronger `CliffordAlgebra.left_induction` and `CliffordAlgebra.righ
 -/
 @[elab_as_elim]
 theorem induction {C : CliffordAlgebra Q → Prop}
-    (h_grade0 : ∀ r, C (algebraMap R (CliffordAlgebra Q) r)) (h_grade1 : ∀ x, C (ι Q x))
-    (h_mul : ∀ a b, C a → C b → C (a * b)) (h_add : ∀ a b, C a → C b → C (a + b))
+    (algebraMap : ∀ r, C (algebraMap R (CliffordAlgebra Q) r)) (ι : ∀ x, C (ι Q x))
+    (mul : ∀ a b, C a → C b → C (a * b)) (add : ∀ a b, C a → C b → C (a + b))
     (a : CliffordAlgebra Q) : C a := by
   -- the arguments are enough to construct a subalgebra, and a mapping into it from M
   let s : Subalgebra R (CliffordAlgebra Q) :=
     { carrier := C
-      mul_mem' := @h_mul
-      add_mem' := @h_add
-      algebraMap_mem' := h_grade0 }
+      mul_mem' := @mul
+      add_mem' := @add
+      algebraMap_mem' := algebraMap }
   -- porting note: Added `h`. `h` is needed for `of`.
   letI h : AddCommMonoid s := inferInstanceAs (AddCommMonoid (Subalgebra.toSubmodule s))
-  let of : { f : M →ₗ[R] s // ∀ m, f m * f m = algebraMap _ _ (Q m) } :=
-    ⟨(ι Q).codRestrict (Subalgebra.toSubmodule s) h_grade1, fun m => Subtype.eq <| ι_sq_scalar Q m⟩
+  let of : { f : M →ₗ[R] s // ∀ m, f m * f m = _root_.algebraMap _ _ (Q m) } :=
+    ⟨(CliffordAlgebra.ι Q).codRestrict (Subalgebra.toSubmodule s) ι,
+      fun m => Subtype.eq <| ι_sq_scalar Q m⟩
   -- the mapping through the subalgebra is the identity
   have of_id : AlgHom.id R (CliffordAlgebra Q) = s.val.comp (lift Q of) := by
     ext

d46774d4

chore: prepare Lean version bump with explicit simp (#10999)

Co-authored-by: Scott Morrison <scott.morrison@gmail.com>

38bce757

Diff

@@ -223,7 +223,7 @@ theorem induction {C : CliffordAlgebra Q → Prop}
   -- the mapping through the subalgebra is the identity
   have of_id : AlgHom.id R (CliffordAlgebra Q) = s.val.comp (lift Q of) := by
     ext
-    simp
+    simp [of]
     -- porting note: `simp` can't apply this
     erw [LinearMap.codRestrict_apply]
   -- finding a proof is finding an element of the subalgebra

d46774d4

refactor(LinearAlgebra/QuadraticForm): Replace BilinForm with a scalar valued bi LinearMap (#10238)

Following on from #10097, which converted the companion of a quadratic form with a bilinear map, this PR replaces a number of results about quadratic forms and bilinear forms with results about quadratic forms and scalar valued bilinear maps. The long term aim is to be able to consider quadratic maps.

The main change is to LinearAlgebra/QuadraticForm/Basic, but this necessitates changes throughout LinearAlgebra/QuadraticForm/. Minor changes are also required elsewhere:

LinearAlgebra/CliffordAlgebra/
LinearAlgebra/Matrix/PosDef
LinearAlgebra/SesquilinearForm
A number of additional results about tensor products and linear maps are also required.

Co-authored-by: Eric Wieser <wieser.eric@gmail.com> Co-authored-by: Christopher Hoskin <christopher.hoskin@overleaf.com>

9f926b1a

Diff

@@ -251,7 +251,7 @@ theorem forall_mul_self_eq_iff {A : Type*} [Ring A] [Algebra R A] (h2 : IsUnit (
     (f : M →ₗ[R] A) :
     (∀ x, f x * f x = algebraMap _ _ (Q x)) ↔
       (LinearMap.mul R A).compl₂ f ∘ₗ f + (LinearMap.mul R A).flip.compl₂ f ∘ₗ f =
-        Q.polarBilin.toLin.compr₂ (Algebra.linearMap R A) := by
+        Q.polarBilin.compr₂ (Algebra.linearMap R A) := by
   simp_rw [DFunLike.ext_iff]
   refine ⟨mul_add_swap_eq_polar_of_forall_mul_self_eq _, fun h x => ?_⟩
   change ∀ x y : M, f x * f y + f y * f x = algebraMap R A (QuadraticForm.polar Q x y) at h

d46774d4

chore(diamonds): appropriate transparency levels for diamond checks (#10910)

Currently, we have multiple "no-diamond" tests of the form

example : x = y := rfl

where X and Y are instances of some class. The problem is that since simp and type class synthesis operate at reducible_and_instances transparency this check means little.

We went through all the mentions of diamonds and either added with_reducible_and_instancse or added a reference to the issue #10906.

Co-authored-by: Kevin Buzzard <k.buzzard@imperial.ac.uk>

29c854d2

Diff

@@ -88,7 +88,9 @@ instance (priority := 900) instAlgebra' {R A M} [CommSemiring R] [AddCommGroup M
   RingQuot.instAlgebra _
 
 -- verify there are no diamonds
+-- but doesn't work at `reducible_and_instances` #10906
 example : (algebraNat : Algebra ℕ (CliffordAlgebra Q)) = instAlgebra' _ := rfl
+-- but doesn't work at `reducible_and_instances` #10906
 example : (algebraInt _ : Algebra ℤ (CliffordAlgebra Q)) = instAlgebra' _ := rfl
 
 -- shortcut instance, as the other instance is slow

d46774d4

feat(LinearAlgebra/{ExteriorAlgebra,CliffordAlgebra}): Functoriality of the exterior algebra and some lemmas about generation (#9718)

This does a few things:

Define the algebra morphism TrivSqZeroExt.map f between trivial square-zero extensions induced by a linear mapf, and establish some of its basic properties (functoriality, composition with the basic maps to/from TrivSqZeroExt). Note that we only consider the case of a commutative base ring, because the case of a general base ring requires morphisms of bimodules, which we do not have.
Define the algebra morphism ExteriorAlgebra.map f between exterior algebras induced by a linear map f. This is a straightforward application of the similar construction for Clifford algebras, but I think it is still useful to have. Basic properties of this construction are proved: functoriality, composition with ExteriorAlgebra.ι, ExteriorAlgebra.ιInv (this part uses the first point) and ExteriorAlgebra.ιMulti. Then exactness properties of the construction is studied:
- If the linear map is surjective, then the map on exterior algebras is also surjective. This actually holds for Clifford algebras, so I added a lemma called CliffordAlgebra.map_surjective in LinearAlgebra/CliffordAlgebra/Basic.lean. For exterior algebras, the converse holds and is also proved.
- If the linear map has a retraction, then the map on exterior algebras is injective. So if the base ring is a field, the map on exterior algebras is injective if the linear map is injective.
Establish some properties of ExteriorAlgebra.ιMulti:
- ExteriorAlgebra.ιMulti_range: The range of ιMulti R n is contained in the nth exterior power (define here as LinearMap.range (ι R : M →ₗ[R] ExteriorAlgebra R M) ^ n).
- ExteriorAlgebra.ιMulti_span_fixedDegree: This range spans the nth exterior power.
- ExteriorAlgebra.ιMulti_span: The union over all n of the range of ιMulti R n spans the whole exterior algebra (this is in LinearAlgebra/ExteriorAlgebra/Grading.lean because the proof uses the graded module structure, but it might be possible to do something simpler).
Construct ExteriorAlgebra.ιMulti_family: This takes a natural number n and a family of vectors v indexed by a linearly ordered type I, and it returns the family of n-fold products of the v i in the exterior algebra, indexed by the set of finsets of I of cardinality n. (The point, to be proved in another PR, is that when v is a basis, then ExteriorAlgebra.ιMulti_family R n v is a basis of the nth exterior power.)

Co-authored-by: morel <smorel@math.princeton.edu> Co-authored-by: smorel394 <67864981+smorel394@users.noreply.github.com>

29c27c39

Diff

@@ -351,6 +351,26 @@ theorem ι_range_map_map (f : Q₁ →qᵢ Q₂) :
   (ι_range_map_lift _ _).trans (LinearMap.range_comp _ _)
 #align clifford_algebra.ι_range_map_map CliffordAlgebra.ι_range_map_map
 
+open Function in
+/-- If `f` is a linear map from `M₁` to `M₂` that preserves the quadratic forms, and if it has
+a linear retraction `g` that also preserves the quadratic forms, then `CliffordAlgebra.map g`
+is a retraction of `CliffordAlgebra.map f`. -/
+lemma leftInverse_map_of_leftInverse {Q₁ : QuadraticForm R M₁} {Q₂ : QuadraticForm R M₂}
+    (f : Q₁ →qᵢ Q₂) (g : Q₂ →qᵢ Q₁) (h : LeftInverse g f) : LeftInverse (map g) (map f) := by
+  refine fun x => ?_
+  replace h : g.comp f = QuadraticForm.Isometry.id Q₁ := DFunLike.ext _ _ h
+  rw [← AlgHom.comp_apply, map_comp_map, h, map_id, AlgHom.coe_id, id_eq]
+
+/-- If a linear map preserves the quadratic forms and is surjective, then the algebra
+maps it induces between Clifford algebras is also surjective.-/
+lemma map_surjective {Q₁ : QuadraticForm R M₁} {Q₂ : QuadraticForm R M₂} (f : Q₁ →qᵢ Q₂)
+    (hf : Function.Surjective f) : Function.Surjective (CliffordAlgebra.map f) :=
+  CliffordAlgebra.induction
+    (fun r ↦ ⟨algebraMap R (CliffordAlgebra Q₁) r, by simp only [AlgHom.commutes]⟩)
+    (fun y ↦ let ⟨x, hx⟩ := hf y; ⟨CliffordAlgebra.ι Q₁ x, by simp only [map_apply_ι, hx]⟩)
+    (fun _ _ ⟨x, hx⟩ ⟨y, hy⟩ ↦ ⟨x * y, by simp only [map_mul, hx, hy]⟩)
+    (fun _ _ ⟨x, hx⟩ ⟨y, hy⟩ ↦ ⟨x + y, by simp only [map_add, hx, hy]⟩)
+
 /-- Two `CliffordAlgebra`s are equivalent as algebras if their quadratic forms are
 equivalent. -/
 @[simps! apply]

d46774d4

chore(*): rename FunLike to DFunLike (#9785)

This prepares for the introduction of a non-dependent synonym of FunLike, which helps a lot with keeping #8386 readable.

This is entirely search-and-replace in 680197f combined with manual fixes in 4145626, e900597 and b8428f8. The commands that generated this change:

sed -i 's/\bFunLike\b/DFunLike/g' {Archive,Counterexamples,Mathlib,test}/**/*.lean
sed -i 's/\btoFunLike\b/toDFunLike/g' {Archive,Counterexamples,Mathlib,test}/**/*.lean
sed -i 's/import Mathlib.Data.DFunLike/import Mathlib.Data.FunLike/g' {Archive,Counterexamples,Mathlib,test}/**/*.lean
sed -i 's/\bHom_FunLike\b/Hom_DFunLike/g' {Archive,Counterexamples,Mathlib,test}/**/*.lean     
sed -i 's/\binstFunLike\b/instDFunLike/g' {Archive,Counterexamples,Mathlib,test}/**/*.lean
sed -i 's/\bfunLike\b/instDFunLike/g' {Archive,Counterexamples,Mathlib,test}/**/*.lean
sed -i 's/\btoo many metavariables to apply `fun_like.has_coe_to_fun`/too many metavariables to apply `DFunLike.hasCoeToFun`/g' {Archive,Counterexamples,Mathlib,test}/**/*.lean

Co-authored-by: Anne Baanen <Vierkantor@users.noreply.github.com>

82656743

Diff

@@ -250,7 +250,7 @@ theorem forall_mul_self_eq_iff {A : Type*} [Ring A] [Algebra R A] (h2 : IsUnit (
     (∀ x, f x * f x = algebraMap _ _ (Q x)) ↔
       (LinearMap.mul R A).compl₂ f ∘ₗ f + (LinearMap.mul R A).flip.compl₂ f ∘ₗ f =
         Q.polarBilin.toLin.compr₂ (Algebra.linearMap R A) := by
-  simp_rw [FunLike.ext_iff]
+  simp_rw [DFunLike.ext_iff]
   refine ⟨mul_add_swap_eq_polar_of_forall_mul_self_eq _, fun h x => ?_⟩
   change ∀ x y : M, f x * f y + f y * f x = algebraMap R A (QuadraticForm.polar Q x y) at h
   apply h2.mul_left_cancel

d46774d4

feat: define QuadraticForm.IsOrtho as Q (x + y) = Q x + Q y (#9141)

This includes some basic API, and the connection with BilinForm.IsOrtho.

The motivation for this definition are the results about vectors commuting in a clifford algebra.

499d5381

Diff

@@ -262,15 +262,36 @@ theorem ι_mul_ι_add_swap (a b : M) :
   mul_add_swap_eq_polar_of_forall_mul_self_eq _ (ι_sq_scalar _) _ _
 #align clifford_algebra.ι_mul_ι_add_swap CliffordAlgebra.ι_mul_ι_add_swap
 
-theorem ι_mul_comm (a b : M) :
+theorem ι_mul_ι_comm (a b : M) :
     ι Q a * ι Q b = algebraMap R _ (QuadraticForm.polar Q a b) - ι Q b * ι Q a :=
   eq_sub_of_add_eq (ι_mul_ι_add_swap a b)
-#align clifford_algebra.ι_mul_comm CliffordAlgebra.ι_mul_comm
+#align clifford_algebra.ι_mul_comm CliffordAlgebra.ι_mul_ι_comm
+
+section isOrtho
+
+@[simp] theorem ι_mul_ι_add_swap_of_isOrtho {a b : M} (h : Q.IsOrtho a b) :
+    ι Q a * ι Q b + ι Q b * ι Q a = 0 := by
+  rw [ι_mul_ι_add_swap, h.polar_eq_zero]
+  simp
+
+theorem ι_mul_ι_comm_of_isOrtho {a b : M} (h : Q.IsOrtho a b) :
+    ι Q a * ι Q b = -(ι Q b * ι Q a) :=
+  eq_neg_of_add_eq_zero_left <| ι_mul_ι_add_swap_of_isOrtho h
+
+theorem mul_ι_mul_ι_of_isOrtho (x : CliffordAlgebra Q) {a b : M} (h : Q.IsOrtho a b) :
+    x * ι Q a * ι Q b = -(x * ι Q b * ι Q a) := by
+  rw [mul_assoc, ι_mul_ι_comm_of_isOrtho h, mul_neg, mul_assoc]
+
+theorem ι_mul_ι_mul_of_isOrtho (x : CliffordAlgebra Q) {a b : M} (h : Q.IsOrtho a b) :
+    ι Q a * (ι Q b * x) = -(ι Q b * (ι Q a * x)) := by
+  rw [← mul_assoc, ι_mul_ι_comm_of_isOrtho h, neg_mul, mul_assoc]
+
+end isOrtho
 
 /-- $aba$ is a vector. -/
 theorem ι_mul_ι_mul_ι (a b : M) :
     ι Q a * ι Q b * ι Q a = ι Q (QuadraticForm.polar Q a b • a - Q a • b) := by
-  rw [ι_mul_comm, sub_mul, mul_assoc, ι_sq_scalar, ← Algebra.smul_def, ← Algebra.commutes, ←
+  rw [ι_mul_ι_comm, sub_mul, mul_assoc, ι_sq_scalar, ← Algebra.smul_def, ← Algebra.commutes, ←
     Algebra.smul_def, ← map_smul, ← map_smul, ← map_sub]
 #align clifford_algebra.ι_mul_ι_mul_ι CliffordAlgebra.ι_mul_ι_mul_ι

d46774d4

chore(LinearAlgebra/CliffordAlgebra): move results about inversion to a new file (#9075)

I plan to contribute a few more of these, and this provides a better place for them to live. No lemma statements or proofs have changed.

These were originally contributed in leanprover-community/mathlib#16077.

f1ba2113

Diff

@@ -367,44 +367,6 @@ theorem equivOfIsometry_refl :
 
 end Map
 
-variable (Q)
-
-/-- If the quadratic form of a vector is invertible, then so is that vector. -/
-def invertibleιOfInvertible (m : M) [Invertible (Q m)] : Invertible (ι Q m) where
-  invOf := ι Q (⅟ (Q m) • m)
-  invOf_mul_self := by
-    rw [map_smul, smul_mul_assoc, ι_sq_scalar, Algebra.smul_def, ← map_mul, invOf_mul_self, map_one]
-  mul_invOf_self := by
-    rw [map_smul, mul_smul_comm, ι_sq_scalar, Algebra.smul_def, ← map_mul, invOf_mul_self, map_one]
-#align clifford_algebra.invertible_ι_of_invertible CliffordAlgebra.invertibleιOfInvertible
-
-/-- For a vector with invertible quadratic form, $v^{-1} = \frac{v}{Q(v)}$ -/
-theorem invOf_ι (m : M) [Invertible (Q m)] [Invertible (ι Q m)] :
-    ⅟ (ι Q m) = ι Q (⅟ (Q m) • m) := by
-  letI := invertibleιOfInvertible Q m
-  convert (rfl : ⅟ (ι Q m) = _)
-#align clifford_algebra.inv_of_ι CliffordAlgebra.invOf_ι
-
-theorem isUnit_ι_of_isUnit {m : M} (h : IsUnit (Q m)) : IsUnit (ι Q m) := by
-  cases h.nonempty_invertible
-  letI := invertibleιOfInvertible Q m
-  exact isUnit_of_invertible (ι Q m)
-#align clifford_algebra.is_unit_ι_of_is_unit CliffordAlgebra.isUnit_ι_of_isUnit
-
-/-- $aba^{-1}$ is a vector. -/
-theorem ι_mul_ι_mul_invOf_ι (a b : M) [Invertible (ι Q a)] [Invertible (Q a)] :
-    ι Q a * ι Q b * ⅟ (ι Q a) = ι Q ((⅟ (Q a) * QuadraticForm.polar Q a b) • a - b) := by
-  rw [invOf_ι, map_smul, mul_smul_comm, ι_mul_ι_mul_ι, ← map_smul, smul_sub, smul_smul, smul_smul,
-    invOf_mul_self, one_smul]
-#align clifford_algebra.ι_mul_ι_mul_inv_of_ι CliffordAlgebra.ι_mul_ι_mul_invOf_ι
-
-/-- $a^{-1}ba$ is a vector. -/
-theorem invOf_ι_mul_ι_mul_ι (a b : M) [Invertible (ι Q a)] [Invertible (Q a)] :
-    ⅟ (ι Q a) * ι Q b * ι Q a = ι Q ((⅟ (Q a) * QuadraticForm.polar Q a b) • a - b) := by
-  rw [invOf_ι, map_smul, smul_mul_assoc, smul_mul_assoc, ι_mul_ι_mul_ι, ← map_smul, smul_sub,
-    smul_smul, smul_smul, invOf_mul_self, one_smul]
-#align clifford_algebra.inv_of_ι_mul_ι_mul_ι CliffordAlgebra.invOf_ι_mul_ι_mul_ι
-
 end CliffordAlgebra
 
 namespace TensorAlgebra

d46774d4

fix(LinearAlgebra/{Free,Tensor,Clifford}Algebra): remove an unused SMulCommClass argument (#8373)

SMulCommClass R S A is always true when Algebra R A and Algebra S A and A is commutative, since the two actions factor via multiplication in A. I don't think mathlib actually knows this fact yet, but in this particular case it's also true by definition.

30d39f9a

Diff

@@ -97,7 +97,7 @@ instance instAlgebra : Algebra R (CliffordAlgebra Q) := instAlgebra' _
 
 instance {R S A M} [CommSemiring R] [CommSemiring S] [AddCommGroup M] [CommRing A]
     [Algebra R A] [Algebra S A] [Module R M] [Module S M] [Module A M] (Q : QuadraticForm A M)
-    [IsScalarTower R A M] [IsScalarTower S A M] [SMulCommClass R S A] :
+    [IsScalarTower R A M] [IsScalarTower S A M] :
     SMulCommClass R S (CliffordAlgebra Q) :=
   RingQuot.instSMulCommClass _

d46774d4

feat: base change of Clifford algebras (#6778)

The main result here is CliffordAlgebra (Q.baseChange A) ≃ₐ[A] A ⊗[R] CliffordAlgebra Q; that is, the A ⊗[R] can be moved from _ : QuadraticForm A (A ⊗[R] _) to the outside and vice versa.

eec55af4

Diff

@@ -230,16 +230,36 @@ theorem induction {C : CliffordAlgebra Q → Prop}
   exact Subtype.prop (lift Q of a)
 #align clifford_algebra.induction CliffordAlgebra.induction
 
-/-- The symmetric product of vectors is a scalar -/
-theorem ι_mul_ι_add_swap (a b : M) :
-    ι Q a * ι Q b + ι Q b * ι Q a = algebraMap R _ (QuadraticForm.polar Q a b) :=
+theorem mul_add_swap_eq_polar_of_forall_mul_self_eq {A : Type*} [Ring A] [Algebra R A]
+    (f : M →ₗ[R] A) (hf : ∀ x, f x * f x = algebraMap _ _ (Q x)) (a b : M) :
+    f a * f b + f b * f a = algebraMap R _ (QuadraticForm.polar Q a b) :=
   calc
-    ι Q a * ι Q b + ι Q b * ι Q a = ι Q (a + b) * ι Q (a + b) - ι Q a * ι Q a - ι Q b * ι Q b := by
-      rw [(ι Q).map_add, mul_add, add_mul, add_mul]; abel
+    f a * f b + f b * f a = f (a + b) * f (a + b) - f a * f a - f b * f b := by
+      rw [f.map_add, mul_add, add_mul, add_mul]; abel
     _ = algebraMap R _ (Q (a + b)) - algebraMap R _ (Q a) - algebraMap R _ (Q b) := by
-      rw [ι_sq_scalar, ι_sq_scalar, ι_sq_scalar]
+      rw [hf, hf, hf]
     _ = algebraMap R _ (Q (a + b) - Q a - Q b) := by rw [← RingHom.map_sub, ← RingHom.map_sub]
     _ = algebraMap R _ (QuadraticForm.polar Q a b) := rfl
+
+/-- An alternative way to provide the argument to `CliffordAlgebra.lift` when `2` is invertible.
+
+To show a function squares to the quadratic form, it suffices to show that
+`f x * f y + f y * f x = algebraMap _ _ (polar Q x y)` -/
+theorem forall_mul_self_eq_iff {A : Type*} [Ring A] [Algebra R A] (h2 : IsUnit (2 : A))
+    (f : M →ₗ[R] A) :
+    (∀ x, f x * f x = algebraMap _ _ (Q x)) ↔
+      (LinearMap.mul R A).compl₂ f ∘ₗ f + (LinearMap.mul R A).flip.compl₂ f ∘ₗ f =
+        Q.polarBilin.toLin.compr₂ (Algebra.linearMap R A) := by
+  simp_rw [FunLike.ext_iff]
+  refine ⟨mul_add_swap_eq_polar_of_forall_mul_self_eq _, fun h x => ?_⟩
+  change ∀ x y : M, f x * f y + f y * f x = algebraMap R A (QuadraticForm.polar Q x y) at h
+  apply h2.mul_left_cancel
+  rw [two_mul, two_mul, h x x, QuadraticForm.polar_self, two_mul, map_add]
+
+/-- The symmetric product of vectors is a scalar -/
+theorem ι_mul_ι_add_swap (a b : M) :
+    ι Q a * ι Q b + ι Q b * ι Q a = algebraMap R _ (QuadraticForm.polar Q a b) :=
+  mul_add_swap_eq_polar_of_forall_mul_self_eq _ (ι_sq_scalar _) _ _
 #align clifford_algebra.ι_mul_ι_add_swap CliffordAlgebra.ι_mul_ι_add_swap
 
 theorem ι_mul_comm (a b : M) :

d46774d4

feat(LinearAlgebra/QuadraticForm): add QuadraticForm.Isometry (#6984)

Also use it to tidy the API of CliffordAlgebra.map.

351962aa

Diff

@@ -5,6 +5,7 @@ Authors: Eric Wieser, Utensil Song
 -/
 import Mathlib.Algebra.RingQuot
 import Mathlib.LinearAlgebra.TensorAlgebra.Basic
+import Mathlib.LinearAlgebra.QuadraticForm.Isometry
 import Mathlib.LinearAlgebra.QuadraticForm.IsometryEquiv
 
 #align_import linear_algebra.clifford_algebra.basic from "leanprover-community/mathlib"@"d46774d43797f5d1f507a63a6e904f7a533ae74a"
@@ -267,61 +268,58 @@ variable [AddCommGroup M₁] [AddCommGroup M₂] [AddCommGroup M₃]
 
 variable [Module R M₁] [Module R M₂] [Module R M₃]
 
-variable (Q₁ : QuadraticForm R M₁) (Q₂ : QuadraticForm R M₂) (Q₃ : QuadraticForm R M₃)
+variable {Q₁ : QuadraticForm R M₁} {Q₂ : QuadraticForm R M₂} {Q₃ : QuadraticForm R M₃}
 
 /-- Any linear map that preserves the quadratic form lifts to an `AlgHom` between algebras.
 
-See `CliffordAlgebra.equivOfIsometry` for the case when `f` is a `QuadraticForm.Isometry`. -/
-def map (f : M₁ →ₗ[R] M₂) (hf : ∀ m, Q₂ (f m) = Q₁ m) :
+See `CliffordAlgebra.equivOfIsometry` for the case when `f` is a `QuadraticForm.IsometryEquiv`. -/
+def map (f : Q₁ →qᵢ Q₂) :
     CliffordAlgebra Q₁ →ₐ[R] CliffordAlgebra Q₂ :=
   CliffordAlgebra.lift Q₁
-    ⟨(CliffordAlgebra.ι Q₂).comp f, fun m => (ι_sq_scalar _ _).trans <| RingHom.congr_arg _ <| hf m⟩
+    ⟨ι Q₂ ∘ₗ f.toLinearMap, fun m => (ι_sq_scalar _ _).trans <| RingHom.congr_arg _ <| f.map_app m⟩
 #align clifford_algebra.map CliffordAlgebra.map
 
 @[simp]
-theorem map_comp_ι (f : M₁ →ₗ[R] M₂) (hf) :
-    (map Q₁ Q₂ f hf).toLinearMap.comp (ι Q₁) = (ι Q₂).comp f :=
+theorem map_comp_ι (f : Q₁ →qᵢ Q₂) :
+    (map f).toLinearMap ∘ₗ ι Q₁ = ι Q₂ ∘ₗ f.toLinearMap :=
   ι_comp_lift _ _
 #align clifford_algebra.map_comp_ι CliffordAlgebra.map_comp_ι
 
 @[simp]
-theorem map_apply_ι (f : M₁ →ₗ[R] M₂) (hf) (m : M₁) : map Q₁ Q₂ f hf (ι Q₁ m) = ι Q₂ (f m) :=
+theorem map_apply_ι (f : Q₁ →qᵢ Q₂) (m : M₁) : map f (ι Q₁ m) = ι Q₂ (f m) :=
   lift_ι_apply _ _ m
 #align clifford_algebra.map_apply_ι CliffordAlgebra.map_apply_ι
 
+variable (Q₁) in
 @[simp]
-theorem map_id :
-    (map Q₁ Q₁ (LinearMap.id : M₁ →ₗ[R] M₁) fun m => rfl) = AlgHom.id R (CliffordAlgebra Q₁) := by
-  ext m; exact map_apply_ι _ _ _ _ m
+theorem map_id : map (QuadraticForm.Isometry.id Q₁) = AlgHom.id R (CliffordAlgebra Q₁) := by
+  ext m; exact map_apply_ι _ m
 #align clifford_algebra.map_id CliffordAlgebra.map_id
 
 @[simp]
-theorem map_comp_map (f : M₂ →ₗ[R] M₃) (hf) (g : M₁ →ₗ[R] M₂) (hg) :
-    (map Q₂ Q₃ f hf).comp (map Q₁ Q₂ g hg)
-      = map Q₁ Q₃ (f.comp g) fun m => (hf _).trans <| hg m := by
+theorem map_comp_map (f : Q₂ →qᵢ Q₃) (g : Q₁ →qᵢ Q₂) :
+    (map f).comp (map g) = map (f.comp g) := by
   ext m
   dsimp only [LinearMap.comp_apply, AlgHom.comp_apply, AlgHom.toLinearMap_apply, AlgHom.id_apply]
-  rw [map_apply_ι, map_apply_ι, map_apply_ι, LinearMap.comp_apply]
+  rw [map_apply_ι, map_apply_ι, map_apply_ι, QuadraticForm.Isometry.comp_apply]
 #align clifford_algebra.map_comp_map CliffordAlgebra.map_comp_map
 
 @[simp]
-theorem ι_range_map_map (f : M₁ →ₗ[R] M₂) (hf : ∀ m, Q₂ (f m) = Q₁ m) :
-    (ι Q₁).range.map (map Q₁ Q₂ f hf).toLinearMap = f.range.map (ι Q₂) :=
+theorem ι_range_map_map (f : Q₁ →qᵢ Q₂) :
+    (ι Q₁).range.map (map f).toLinearMap = f.range.map (ι Q₂) :=
   (ι_range_map_lift _ _).trans (LinearMap.range_comp _ _)
 #align clifford_algebra.ι_range_map_map CliffordAlgebra.ι_range_map_map
 
-variable {Q₁ Q₂ Q₃}
-
 /-- Two `CliffordAlgebra`s are equivalent as algebras if their quadratic forms are
 equivalent. -/
 @[simps! apply]
 def equivOfIsometry (e : Q₁.IsometryEquiv Q₂) : CliffordAlgebra Q₁ ≃ₐ[R] CliffordAlgebra Q₂ :=
-  AlgEquiv.ofAlgHom (map Q₁ Q₂ e e.map_app) (map Q₂ Q₁ e.symm e.symm.map_app)
-    ((map_comp_map _ _ _ _ _ _ _).trans <| by
+  AlgEquiv.ofAlgHom (map e.toIsometry) (map e.symm.toIsometry)
+    ((map_comp_map _ _).trans <| by
       convert map_id Q₂ using 2  -- porting note: replaced `_` with `Q₂`
       ext m
       exact e.toLinearEquiv.apply_symm_apply m)
-    ((map_comp_map _ _ _ _ _ _ _).trans <| by
+    ((map_comp_map _ _).trans <| by
       convert map_id Q₁ using 2  -- porting note: replaced `_` with `Q₁`
       ext m
       exact e.toLinearEquiv.symm_apply_apply m)
@@ -337,7 +335,7 @@ theorem equivOfIsometry_symm (e : Q₁.IsometryEquiv Q₂) :
 theorem equivOfIsometry_trans (e₁₂ : Q₁.IsometryEquiv Q₂) (e₂₃ : Q₂.IsometryEquiv Q₃) :
     (equivOfIsometry e₁₂).trans (equivOfIsometry e₂₃) = equivOfIsometry (e₁₂.trans e₂₃) := by
   ext x
-  exact AlgHom.congr_fun (map_comp_map Q₁ Q₂ Q₃ _ _ _ _) x
+  exact AlgHom.congr_fun (map_comp_map _ _) x
 #align clifford_algebra.equiv_of_isometry_trans CliffordAlgebra.equivOfIsometry_trans
 
 @[simp]

d46774d4

chore: tidy various files (#6838)

cda5caa7

Diff

@@ -75,7 +75,7 @@ def CliffordAlgebra :=
 namespace CliffordAlgebra
 
 -- Porting note: Expanded `deriving Inhabited, Semiring, Algebra`
-instance instInhabited : Inhabited (CliffordAlgebra Q) := RingQuot.instInhabitedRingQuot _
+instance instInhabited : Inhabited (CliffordAlgebra Q) := RingQuot.instInhabited _
 #align clifford_algebra.inhabited CliffordAlgebra.instInhabited
 instance instRing : Ring (CliffordAlgebra Q) := RingQuot.instRing _
 #align clifford_algebra.ring CliffordAlgebra.instRing
@@ -84,7 +84,7 @@ instance (priority := 900) instAlgebra' {R A M} [CommSemiring R] [AddCommGroup M
     [Algebra R A] [Module R M] [Module A M] (Q : QuadraticForm A M)
     [IsScalarTower R A M] :
     Algebra R (CliffordAlgebra Q) :=
-  RingQuot.instAlgebraRingQuot _
+  RingQuot.instAlgebra _
 
 -- verify there are no diamonds
 example : (algebraNat : Algebra ℕ (CliffordAlgebra Q)) = instAlgebra' _ := rfl
@@ -98,13 +98,13 @@ instance {R S A M} [CommSemiring R] [CommSemiring S] [AddCommGroup M] [CommRing
     [Algebra R A] [Algebra S A] [Module R M] [Module S M] [Module A M] (Q : QuadraticForm A M)
     [IsScalarTower R A M] [IsScalarTower S A M] [SMulCommClass R S A] :
     SMulCommClass R S (CliffordAlgebra Q) :=
-  RingQuot.instSMulCommClassRingQuot _
+  RingQuot.instSMulCommClass _
 
 instance {R S A M} [CommSemiring R] [CommSemiring S] [AddCommGroup M] [CommRing A]
     [SMul R S] [Algebra R A] [Algebra S A] [Module R M] [Module S M] [Module A M]
     [IsScalarTower R A M] [IsScalarTower S A M] [IsScalarTower R S A] (Q : QuadraticForm A M) :
     IsScalarTower R S (CliffordAlgebra Q) :=
-  RingQuot.instIsScalarTowerRingQuot _
+  RingQuot.instIsScalarTower _
 
 /-- The canonical linear map `M →ₗ[R] CliffordAlgebra Q`.
 -/

d46774d4

chore: banish Type _ and Sort _ (#6499)

We remove all possible occurences of Type _ and Sort _ in favor of Type* and Sort*.

This has nice performance benefits.

32de3f67

Diff

@@ -44,9 +44,9 @@ This file is almost identical to `Mathlib/LinearAlgebra/ExteriorAlgebra/Basic.le
 -/
 
 
-variable {R : Type _} [CommRing R]
+variable {R : Type*} [CommRing R]
 
-variable {M : Type _} [AddCommGroup M] [Module R M]
+variable {M : Type*} [AddCommGroup M] [Module R M]
 
 variable (Q : QuadraticForm R M)
 
@@ -119,7 +119,7 @@ theorem ι_sq_scalar (m : M) : ι Q m * ι Q m = algebraMap R _ (Q m) := by
   rfl
 #align clifford_algebra.ι_sq_scalar CliffordAlgebra.ι_sq_scalar
 
-variable {Q} {A : Type _} [Semiring A] [Algebra R A]
+variable {Q} {A : Type*} [Semiring A] [Algebra R A]
 
 @[simp]
 theorem comp_ι_sq_scalar (g : CliffordAlgebra Q →ₐ[R] A) (m : M) :
@@ -188,7 +188,7 @@ theorem lift_comp_ι (g : CliffordAlgebra Q →ₐ[R] A) :
 
 /-- See note [partially-applied ext lemmas]. -/
 @[ext high]
-theorem hom_ext {A : Type _} [Semiring A] [Algebra R A] {f g : CliffordAlgebra Q →ₐ[R] A} :
+theorem hom_ext {A : Type*} [Semiring A] [Algebra R A] {f g : CliffordAlgebra Q →ₐ[R] A} :
     f.toLinearMap.comp (ι Q) = g.toLinearMap.comp (ι Q) → f = g := by
   intro h
   apply (lift Q).symm.injective
@@ -261,7 +261,7 @@ theorem ι_range_map_lift (f : M →ₗ[R] A) (cond : ∀ m, f m * f m = algebra
 
 section Map
 
-variable {M₁ M₂ M₃ : Type _}
+variable {M₁ M₂ M₃ : Type*}
 
 variable [AddCommGroup M₁] [AddCommGroup M₂] [AddCommGroup M₃]

d46774d4

chore: ensure all instances referred to directly have explicit names (#6423)

Per https://github.com/leanprover/lean4/issues/2343, we are going to need to change the automatic generation of instance names, as they become too long.

This PR ensures that everywhere in Mathlib that refers to an instance by name, that name is given explicitly, rather than being automatically generated.

There are four exceptions, which are now commented, with links to https://github.com/leanprover/lean4/issues/2343.

This was implemented by running Mathlib against a modified Lean that appended _ᾰ to all automatically generated names, and fixing everything.

Co-authored-by: Scott Morrison <scott.morrison@gmail.com>

65a35f78

Diff

@@ -84,7 +84,7 @@ instance (priority := 900) instAlgebra' {R A M} [CommSemiring R] [AddCommGroup M
     [Algebra R A] [Module R M] [Module A M] (Q : QuadraticForm A M)
     [IsScalarTower R A M] :
     Algebra R (CliffordAlgebra Q) :=
-  RingQuot.instAlgebraRingQuotInstSemiring _
+  RingQuot.instAlgebraRingQuot _
 
 -- verify there are no diamonds
 example : (algebraNat : Algebra ℕ (CliffordAlgebra Q)) = instAlgebra' _ := rfl
@@ -98,13 +98,13 @@ instance {R S A M} [CommSemiring R] [CommSemiring S] [AddCommGroup M] [CommRing
     [Algebra R A] [Algebra S A] [Module R M] [Module S M] [Module A M] (Q : QuadraticForm A M)
     [IsScalarTower R A M] [IsScalarTower S A M] [SMulCommClass R S A] :
     SMulCommClass R S (CliffordAlgebra Q) :=
-  RingQuot.instSMulCommClassRingQuotInstSMulRingQuotInstSMulRingQuot _
+  RingQuot.instSMulCommClassRingQuot _
 
 instance {R S A M} [CommSemiring R] [CommSemiring S] [AddCommGroup M] [CommRing A]
     [SMul R S] [Algebra R A] [Algebra S A] [Module R M] [Module S M] [Module A M]
     [IsScalarTower R A M] [IsScalarTower S A M] [IsScalarTower R S A] (Q : QuadraticForm A M) :
     IsScalarTower R S (CliffordAlgebra Q) :=
-  RingQuot.instIsScalarTowerRingQuotInstSMulRingQuotInstSMulRingQuot _
+  RingQuot.instIsScalarTowerRingQuot _
 
 /-- The canonical linear map `M →ₗ[R] CliffordAlgebra Q`.
 -/

d46774d4

refactor(LinearAlgebra/QuadraticForm): rename Isometry to IsometryEquiv (#6305)

This is consistent with LinearIsometryEquiv vs LinearIsometry. The motivation is to make room for QuadraticForm.Isometry as the homomorphism.

b9cabefa

Diff

@@ -5,7 +5,7 @@ Authors: Eric Wieser, Utensil Song
 -/
 import Mathlib.Algebra.RingQuot
 import Mathlib.LinearAlgebra.TensorAlgebra.Basic
-import Mathlib.LinearAlgebra.QuadraticForm.Isometry
+import Mathlib.LinearAlgebra.QuadraticForm.IsometryEquiv
 
 #align_import linear_algebra.clifford_algebra.basic from "leanprover-community/mathlib"@"d46774d43797f5d1f507a63a6e904f7a533ae74a"
 
@@ -315,7 +315,7 @@ variable {Q₁ Q₂ Q₃}
 /-- Two `CliffordAlgebra`s are equivalent as algebras if their quadratic forms are
 equivalent. -/
 @[simps! apply]
-def equivOfIsometry (e : Q₁.Isometry Q₂) : CliffordAlgebra Q₁ ≃ₐ[R] CliffordAlgebra Q₂ :=
+def equivOfIsometry (e : Q₁.IsometryEquiv Q₂) : CliffordAlgebra Q₁ ≃ₐ[R] CliffordAlgebra Q₂ :=
   AlgEquiv.ofAlgHom (map Q₁ Q₂ e e.map_app) (map Q₂ Q₁ e.symm e.symm.map_app)
     ((map_comp_map _ _ _ _ _ _ _).trans <| by
       convert map_id Q₂ using 2  -- porting note: replaced `_` with `Q₂`
@@ -328,13 +328,13 @@ def equivOfIsometry (e : Q₁.Isometry Q₂) : CliffordAlgebra Q₁ ≃ₐ[R] Cl
 #align clifford_algebra.equiv_of_isometry CliffordAlgebra.equivOfIsometry
 
 @[simp]
-theorem equivOfIsometry_symm (e : Q₁.Isometry Q₂) :
+theorem equivOfIsometry_symm (e : Q₁.IsometryEquiv Q₂) :
     (equivOfIsometry e).symm = equivOfIsometry e.symm :=
   rfl
 #align clifford_algebra.equiv_of_isometry_symm CliffordAlgebra.equivOfIsometry_symm
 
 @[simp]
-theorem equivOfIsometry_trans (e₁₂ : Q₁.Isometry Q₂) (e₂₃ : Q₂.Isometry Q₃) :
+theorem equivOfIsometry_trans (e₁₂ : Q₁.IsometryEquiv Q₂) (e₂₃ : Q₂.IsometryEquiv Q₃) :
     (equivOfIsometry e₁₂).trans (equivOfIsometry e₂₃) = equivOfIsometry (e₁₂.trans e₂₃) := by
   ext x
   exact AlgHom.congr_fun (map_comp_map Q₁ Q₂ Q₃ _ _ _ _) x
@@ -342,7 +342,7 @@ theorem equivOfIsometry_trans (e₁₂ : Q₁.Isometry Q₂) (e₂₃ : Q₂.Iso
 
 @[simp]
 theorem equivOfIsometry_refl :
-    (equivOfIsometry <| QuadraticForm.Isometry.refl Q₁) = AlgEquiv.refl := by
+    (equivOfIsometry <| QuadraticForm.IsometryEquiv.refl Q₁) = AlgEquiv.refl := by
   ext x
   exact AlgHom.congr_fun (map_id Q₁) x
 #align clifford_algebra.equiv_of_isometry_refl CliffordAlgebra.equivOfIsometry_refl

d46774d4

feat(LinearAlgebra/CliffordAlgebra): support towers of algebras (#6074)

3cf73553

Diff

@@ -79,9 +79,33 @@ instance instInhabited : Inhabited (CliffordAlgebra Q) := RingQuot.instInhabited
 #align clifford_algebra.inhabited CliffordAlgebra.instInhabited
 instance instRing : Ring (CliffordAlgebra Q) := RingQuot.instRing _
 #align clifford_algebra.ring CliffordAlgebra.instRing
-instance instAlgebra: Algebra R (CliffordAlgebra Q) := RingQuot.instAlgebraRingQuotInstSemiring _
+
+instance (priority := 900) instAlgebra' {R A M} [CommSemiring R] [AddCommGroup M] [CommRing A]
+    [Algebra R A] [Module R M] [Module A M] (Q : QuadraticForm A M)
+    [IsScalarTower R A M] :
+    Algebra R (CliffordAlgebra Q) :=
+  RingQuot.instAlgebraRingQuotInstSemiring _
+
+-- verify there are no diamonds
+example : (algebraNat : Algebra ℕ (CliffordAlgebra Q)) = instAlgebra' _ := rfl
+example : (algebraInt _ : Algebra ℤ (CliffordAlgebra Q)) = instAlgebra' _ := rfl
+
+-- shortcut instance, as the other instance is slow
+instance instAlgebra : Algebra R (CliffordAlgebra Q) := instAlgebra' _
 #align clifford_algebra.algebra CliffordAlgebra.instAlgebra
 
+instance {R S A M} [CommSemiring R] [CommSemiring S] [AddCommGroup M] [CommRing A]
+    [Algebra R A] [Algebra S A] [Module R M] [Module S M] [Module A M] (Q : QuadraticForm A M)
+    [IsScalarTower R A M] [IsScalarTower S A M] [SMulCommClass R S A] :
+    SMulCommClass R S (CliffordAlgebra Q) :=
+  RingQuot.instSMulCommClassRingQuotInstSMulRingQuotInstSMulRingQuot _
+
+instance {R S A M} [CommSemiring R] [CommSemiring S] [AddCommGroup M] [CommRing A]
+    [SMul R S] [Algebra R A] [Algebra S A] [Module R M] [Module S M] [Module A M]
+    [IsScalarTower R A M] [IsScalarTower S A M] [IsScalarTower R S A] (Q : QuadraticForm A M) :
+    IsScalarTower R S (CliffordAlgebra Q) :=
+  RingQuot.instIsScalarTowerRingQuotInstSMulRingQuotInstSMulRingQuot _
+
 /-- The canonical linear map `M →ₗ[R] CliffordAlgebra Q`.
 -/
 def ι : M →ₗ[R] CliffordAlgebra Q :=

d46774d4

chore: script to replace headers with #align_import statements (#5979)

depends on: #5966

Co-authored-by: Eric Wieser <wieser.eric@gmail.com> Co-authored-by: Scott Morrison <scott.morrison@gmail.com>

89aa3a3b

Diff

@@ -2,16 +2,13 @@
 Copyright (c) 2020 Eric Wieser. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Eric Wieser, Utensil Song
-
-! This file was ported from Lean 3 source module linear_algebra.clifford_algebra.basic
-! leanprover-community/mathlib commit d46774d43797f5d1f507a63a6e904f7a533ae74a
-! Please do not edit these lines, except to modify the commit id
-! if you have ported upstream changes.
 -/
 import Mathlib.Algebra.RingQuot
 import Mathlib.LinearAlgebra.TensorAlgebra.Basic
 import Mathlib.LinearAlgebra.QuadraticForm.Isometry
 
+#align_import linear_algebra.clifford_algebra.basic from "leanprover-community/mathlib"@"d46774d43797f5d1f507a63a6e904f7a533ae74a"
+
 /-!
 # Clifford Algebras

d46774d4

chore: tidy various files (#5482)

0e36dc3b

Diff

@@ -16,11 +16,11 @@ import Mathlib.LinearAlgebra.QuadraticForm.Isometry
 # Clifford Algebras
 
 We construct the Clifford algebra of a module `M` over a commutative ring `R`, equipped with
-a quadratic_form `Q`.
+a quadratic form `Q`.
 
 ## Notation
 
-The Clifford algebra of the `R`-module `M` equipped with a quadratic_form `Q` is
+The Clifford algebra of the `R`-module `M` equipped with a quadratic form `Q` is
 an `R`-algebra denoted `CliffordAlgebra Q`.
 
 Given a linear morphism `f : M → A` from a module `M` to another `R`-algebra `A`, such that
@@ -36,8 +36,6 @@ of the Clifford algebra.
 1. `ι_comp_lift` is the fact that the composition of `ι Q` with `lift Q f cond` agrees with `f`.
 2. `lift_unique` ensures the uniqueness of `lift Q f cond` with respect to 1.
 
-Additionally, when `Q = 0` an `AlgEquiv` to the `exterior_algebra` is provided as `as_exterior`.
-
 ## Implementation details
 
 The Clifford algebra of `M` is constructed as a quotient of the tensor algebra, as follows.
@@ -45,7 +43,7 @@ The Clifford algebra of `M` is constructed as a quotient of the tensor algebra,
    This is the smallest relation which identifies squares of elements of `M` with `Q m`.
 2. The Clifford algebra is the quotient of the tensor algebra by this relation.
 
-This file is almost identical to `linear_algebra/exterior_algebra.lean`.
+This file is almost identical to `Mathlib/LinearAlgebra/ExteriorAlgebra/Basic.lean`.
 -/
 
 
@@ -181,7 +179,7 @@ theorem hom_ext {A : Type _} [Semiring A] [Algebra R A] {f g : CliffordAlgebra Q
 /-- If `C` holds for the `algebraMap` of `r : R` into `CliffordAlgebra Q`, the `ι` of `x : M`,
 and is preserved under addition and muliplication, then it holds for all of `CliffordAlgebra Q`.
 
-See also the stronger `clifford_algebra.left_induction` and `clifford_algebra.right_induction`.
+See also the stronger `CliffordAlgebra.left_induction` and `CliffordAlgebra.right_induction`.
 -/
 @[elab_as_elim]
 theorem induction {C : CliffordAlgebra Q → Prop}
@@ -220,7 +218,6 @@ theorem ι_mul_ι_add_swap (a b : M) :
       rw [ι_sq_scalar, ι_sq_scalar, ι_sq_scalar]
     _ = algebraMap R _ (Q (a + b) - Q a - Q b) := by rw [← RingHom.map_sub, ← RingHom.map_sub]
     _ = algebraMap R _ (QuadraticForm.polar Q a b) := rfl
-
 #align clifford_algebra.ι_mul_ι_add_swap CliffordAlgebra.ι_mul_ι_add_swap
 
 theorem ι_mul_comm (a b : M) :

d46774d4

feat: port LinearAlgebra.CliffordAlgebra.Conjugation (#5404)

Annoyingly, Lean 4 cannot work out that reverse x where x : clifford_algebra Q is referring to reverse (Q := Q) x, even though that's the only thing that type checks.

Otherwise the only difficulty when porting was the standard LinearMap.range dot notation failing thing.

Co-authored-by: Jeremy Tan Jie Rui <reddeloostw@gmail.com>

ef690afb

Diff

@@ -168,7 +168,7 @@ theorem lift_comp_ι (g : CliffordAlgebra Q →ₐ[R] A) :
 #align clifford_algebra.lift_comp_ι CliffordAlgebra.lift_comp_ι
 
 /-- See note [partially-applied ext lemmas]. -/
-@[ext]
+@[ext high]
 theorem hom_ext {A : Type _} [Semiring A] [Algebra R A] {f g : CliffordAlgebra Q →ₐ[R] A} :
     f.toLinearMap.comp (ι Q) = g.toLinearMap.comp (ι Q) → f = g := by
   intro h

d46774d4

feat: port LinearAlgebra.CliffordAlgebra.Basic (#4802)

96e68449

`linear_algebra.clifford_algebra.basic` ⟷ `Mathlib.LinearAlgebra.CliffordAlgebra.Basic`

Changes since the initial port

Changes in mathlib3

Changes in mathlib3port

Changes in mathlib4

Dependencies 10 + 630

linear_algebra.clifford_algebra.basic ⟷ Mathlib.LinearAlgebra.CliffordAlgebra.Basic

Changes since the initial port

Changes in mathlib3

Changes in mathlib3port

Changes in mathlib4

Dependencies 10 + 630

`linear_algebra.clifford_algebra.basic` ⟷ `Mathlib.LinearAlgebra.CliffordAlgebra.Basic`