linear_algebra.exterior_algebra.basic ⟷ Mathlib.LinearAlgebra.ExteriorAlgebra.Basic

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

@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Zhangir Azerbayev, Adam Topaz, Eric Wieser
 -/
 import LinearAlgebra.CliffordAlgebra.Basic
-import LinearAlgebra.Alternating
+import LinearAlgebra.Alternating.Basic
 
 #align_import linear_algebra.exterior_algebra.basic from "leanprover-community/mathlib"@"5d0c76894ada7940957143163d7b921345474cbc"
 

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

@@ -277,7 +277,7 @@ theorem ι_eq_algebraMap_iff (x : M) (r : R) : ι R x = algebraMap R _ r ↔ x =
   · letI : Module Rᵐᵒᵖ M := Module.compHom _ ((RingHom.id R).fromOpposite mul_comm)
     haveI : IsCentralScalar R M := ⟨fun r m => rfl⟩
     have hf0 : to_triv_sq_zero_ext (ι R x) = (0, x) := to_triv_sq_zero_ext_ι _
-    rw [h, AlgHom.commutes] at hf0 
+    rw [h, AlgHom.commutes] at hf0
     have : r = 0 ∧ 0 = x := Prod.ext_iff.1 hf0
     exact this.symm.imp_left Eq.symm
   · rintro ⟨rfl, rfl⟩
@@ -302,7 +302,7 @@ theorem ι_range_disjoint_one :
   by
   rw [Submodule.disjoint_def]
   rintro _ ⟨x, hx⟩ ⟨r, rfl : algebraMap _ _ _ = _⟩
-  rw [ι_eq_algebra_map_iff x] at hx 
+  rw [ι_eq_algebra_map_iff x] at hx
   rw [hx.2, RingHom.map_zero]
 #align exterior_algebra.ι_range_disjoint_one ExteriorAlgebra.ι_range_disjoint_one
 -/
@@ -326,8 +326,8 @@ theorem ι_mul_prod_list {n : ℕ} (f : Fin n → M) (i : Fin n) :
     by_cases h : i = 0
     · rw [h, ι_sq_zero, MulZeroClass.zero_mul]
     · replace hn := congr_arg ((· * ·) <| ι R <| f 0) (hn (fun i => f <| Fin.succ i) (i.pred h))
-      simp only at hn 
-      rw [Fin.succ_pred, ← mul_assoc, MulZeroClass.mul_zero] at hn 
+      simp only at hn
+      rw [Fin.succ_pred, ← mul_assoc, MulZeroClass.mul_zero] at hn
       refine' (eq_zero_iff_eq_zero_of_add_eq_zero _).mp hn
       rw [← add_mul, ι_add_mul_swap, MulZeroClass.zero_mul]
 #align exterior_algebra.ι_mul_prod_list ExteriorAlgebra.ι_mul_prod_list
@@ -356,7 +356,7 @@ def ιMulti (n : ℕ) : AlternatingMap R M (ExteriorAlgebra R M) (Fin n) :=
       · exact x.elim0
       · rw [List.ofFn_succ, List.prod_cons]
         by_cases hx : x = 0
-        · rw [hx] at hfxy h 
+        · rw [hx] at hfxy h
           rw [hfxy, ← Fin.succ_pred y (ne_of_lt h).symm]
           exact ι_mul_prod_list (f ∘ Fin.succ) _
         · convert MulZeroClass.mul_zero _

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

@@ -3,8 +3,8 @@ Copyright (c) 2020 Adam Topaz. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Zhangir Azerbayev, Adam Topaz, Eric Wieser
 -/
-import Mathbin.LinearAlgebra.CliffordAlgebra.Basic
-import Mathbin.LinearAlgebra.Alternating
+import LinearAlgebra.CliffordAlgebra.Basic
+import LinearAlgebra.Alternating
 
 #align_import linear_algebra.exterior_algebra.basic from "leanprover-community/mathlib"@"5d0c76894ada7940957143163d7b921345474cbc"
 

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

@@ -2,15 +2,12 @@
 Copyright (c) 2020 Adam Topaz. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Zhangir Azerbayev, Adam Topaz, Eric Wieser
-
-! This file was ported from Lean 3 source module linear_algebra.exterior_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.LinearAlgebra.CliffordAlgebra.Basic
 import Mathbin.LinearAlgebra.Alternating
 
+#align_import linear_algebra.exterior_algebra.basic from "leanprover-community/mathlib"@"5d0c76894ada7940957143163d7b921345474cbc"
+
 /-!
 # Exterior Algebras
 

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

@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Zhangir Azerbayev, Adam Topaz, Eric Wieser
 
 ! This file was ported from Lean 3 source module linear_algebra.exterior_algebra.basic
-! leanprover-community/mathlib commit b8d2eaa69d69ce8f03179a5cda774fc0cde984e4
+! leanprover-community/mathlib commit 5d0c76894ada7940957143163d7b921345474cbc
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/
@@ -14,6 +14,9 @@ import Mathbin.LinearAlgebra.Alternating
 /-!
 # Exterior Algebras
 
+> THIS FILE IS SYNCHRONIZED WITH MATHLIB4.
+> Any changes to this file require a corresponding PR to mathlib4.
+
 We construct the exterior algebra of a module `M` over a commutative semiring `R`.
 
 ## Notation

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

@@ -51,41 +51,50 @@ variable (R : Type u1) [CommRing R]
 
 variable (M : Type u2) [AddCommGroup M] [Module R M]
 
+#print ExteriorAlgebra /-
 /-- The exterior algebra of an `R`-module `M`.
 -/
 @[reducible]
 def ExteriorAlgebra :=
   CliffordAlgebra (0 : QuadraticForm R M)
 #align exterior_algebra ExteriorAlgebra
+-/
 
 namespace ExteriorAlgebra
 
 variable {M}
 
+#print ExteriorAlgebra.ι /-
 /-- The canonical linear map `M →ₗ[R] exterior_algebra R M`.
 -/
 @[reducible]
 def ι : M →ₗ[R] ExteriorAlgebra R M :=
   CliffordAlgebra.ι _
 #align exterior_algebra.ι ExteriorAlgebra.ι
+-/
 
 variable {R}
 
+#print ExteriorAlgebra.ι_sq_zero /-
 /-- As well as being linear, `ι m` squares to zero -/
 @[simp]
 theorem ι_sq_zero (m : M) : ι R m * ι R m = 0 :=
   (CliffordAlgebra.ι_sq_scalar _ m).trans <| map_zero _
 #align exterior_algebra.ι_sq_zero ExteriorAlgebra.ι_sq_zero
+-/
 
 variable {A : Type _} [Semiring A] [Algebra R A]
 
+#print ExteriorAlgebra.comp_ι_sq_zero /-
 @[simp]
 theorem comp_ι_sq_zero (g : ExteriorAlgebra R M →ₐ[R] A) (m : M) : g (ι R m) * g (ι R m) = 0 := by
   rw [← AlgHom.map_mul, ι_sq_zero, AlgHom.map_zero]
 #align exterior_algebra.comp_ι_sq_zero ExteriorAlgebra.comp_ι_sq_zero
+-/
 
 variable (R)
 
+#print ExteriorAlgebra.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 = 0`, this is the canonical lift of `f` to a morphism of `R`-algebras
 from `exterior_algebra R M` to `A`.
@@ -94,40 +103,52 @@ from `exterior_algebra R M` to `A`.
 def lift : { f : M →ₗ[R] A // ∀ m, f m * f m = 0 } ≃ (ExteriorAlgebra R M →ₐ[R] A) :=
   Equiv.trans (Equiv.subtypeEquiv (Equiv.refl _) <| by simp) <| CliffordAlgebra.lift _
 #align exterior_algebra.lift ExteriorAlgebra.lift
+-/
 
+#print ExteriorAlgebra.ι_comp_lift /-
 @[simp]
 theorem ι_comp_lift (f : M →ₗ[R] A) (cond : ∀ m, f m * f m = 0) :
     (lift R ⟨f, cond⟩).toLinearMap.comp (ι R) = f :=
   CliffordAlgebra.ι_comp_lift f _
 #align exterior_algebra.ι_comp_lift ExteriorAlgebra.ι_comp_lift
+-/
 
+#print ExteriorAlgebra.lift_ι_apply /-
 @[simp]
 theorem lift_ι_apply (f : M →ₗ[R] A) (cond : ∀ m, f m * f m = 0) (x) :
     lift R ⟨f, cond⟩ (ι R x) = f x :=
   CliffordAlgebra.lift_ι_apply f _ x
 #align exterior_algebra.lift_ι_apply ExteriorAlgebra.lift_ι_apply
+-/
 
+#print ExteriorAlgebra.lift_unique /-
 @[simp]
 theorem lift_unique (f : M →ₗ[R] A) (cond : ∀ m, f m * f m = 0) (g : ExteriorAlgebra R M →ₐ[R] A) :
     g.toLinearMap.comp (ι R) = f ↔ g = lift R ⟨f, cond⟩ :=
   CliffordAlgebra.lift_unique f _ _
 #align exterior_algebra.lift_unique ExteriorAlgebra.lift_unique
+-/
 
 variable {R M}
 
+#print ExteriorAlgebra.lift_comp_ι /-
 @[simp]
 theorem lift_comp_ι (g : ExteriorAlgebra R M →ₐ[R] A) :
     lift R ⟨g.toLinearMap.comp (ι R), comp_ι_sq_zero _⟩ = g :=
   CliffordAlgebra.lift_comp_ι g
 #align exterior_algebra.lift_comp_ι ExteriorAlgebra.lift_comp_ι
+-/
 
+#print ExteriorAlgebra.hom_ext /-
 /-- See note [partially-applied ext lemmas]. -/
 @[ext]
 theorem hom_ext {f g : ExteriorAlgebra R M →ₐ[R] A}
     (h : f.toLinearMap.comp (ι R) = g.toLinearMap.comp (ι R)) : f = g :=
   CliffordAlgebra.hom_ext h
 #align exterior_algebra.hom_ext ExteriorAlgebra.hom_ext
+-/
 
+#print ExteriorAlgebra.induction /-
 /-- If `C` holds for the `algebra_map` of `r : R` into `exterior_algebra R M`, the `ι` of `x : M`,
 and is preserved under addition and muliplication, then it holds for all of `exterior_algebra R M`.
 -/
@@ -138,61 +159,81 @@ theorem induction {C : ExteriorAlgebra R M → Prop}
     (a : ExteriorAlgebra R M) : C a :=
   CliffordAlgebra.induction h_grade0 h_grade1 h_mul h_add a
 #align exterior_algebra.induction ExteriorAlgebra.induction
+-/
 
+#print ExteriorAlgebra.algebraMapInv /-
 /-- The left-inverse of `algebra_map`. -/
 def algebraMapInv : ExteriorAlgebra R M →ₐ[R] R :=
   ExteriorAlgebra.lift R ⟨(0 : M →ₗ[R] R), fun m => by simp⟩
 #align exterior_algebra.algebra_map_inv ExteriorAlgebra.algebraMapInv
+-/
 
 variable (M)
 
+#print ExteriorAlgebra.algebraMap_leftInverse /-
 theorem algebraMap_leftInverse :
     Function.LeftInverse algebraMapInv (algebraMap R <| ExteriorAlgebra R M) := fun x => by
   simp [algebra_map_inv]
 #align exterior_algebra.algebra_map_left_inverse ExteriorAlgebra.algebraMap_leftInverse
+-/
 
+#print ExteriorAlgebra.algebraMap_inj /-
 @[simp]
 theorem algebraMap_inj (x y : R) :
     algebraMap R (ExteriorAlgebra R M) x = algebraMap R (ExteriorAlgebra R M) y ↔ x = y :=
   (algebraMap_leftInverse M).Injective.eq_iff
 #align exterior_algebra.algebra_map_inj ExteriorAlgebra.algebraMap_inj
+-/
 
+#print ExteriorAlgebra.algebraMap_eq_zero_iff /-
 @[simp]
 theorem algebraMap_eq_zero_iff (x : R) : algebraMap R (ExteriorAlgebra R M) x = 0 ↔ x = 0 :=
   map_eq_zero_iff (algebraMap _ _) (algebraMap_leftInverse _).Injective
 #align exterior_algebra.algebra_map_eq_zero_iff ExteriorAlgebra.algebraMap_eq_zero_iff
+-/
 
+#print ExteriorAlgebra.algebraMap_eq_one_iff /-
 @[simp]
 theorem algebraMap_eq_one_iff (x : R) : algebraMap R (ExteriorAlgebra R M) x = 1 ↔ x = 1 :=
   map_eq_one_iff (algebraMap _ _) (algebraMap_leftInverse _).Injective
 #align exterior_algebra.algebra_map_eq_one_iff ExteriorAlgebra.algebraMap_eq_one_iff
+-/
 
+#print ExteriorAlgebra.isUnit_algebraMap /-
 theorem isUnit_algebraMap (r : R) : IsUnit (algebraMap R (ExteriorAlgebra R M) r) ↔ IsUnit r :=
   isUnit_map_of_leftInverse _ (algebraMap_leftInverse M)
 #align exterior_algebra.is_unit_algebra_map ExteriorAlgebra.isUnit_algebraMap
+-/
 
+#print ExteriorAlgebra.invertibleAlgebraMapEquiv /-
 /-- Invertibility in the exterior algebra is the same as invertibility of the base ring. -/
 @[simps]
 def invertibleAlgebraMapEquiv (r : R) :
     Invertible (algebraMap R (ExteriorAlgebra R M) r) ≃ Invertible r :=
   invertibleEquivOfLeftInverse _ _ _ (algebraMap_leftInverse M)
 #align exterior_algebra.invertible_algebra_map_equiv ExteriorAlgebra.invertibleAlgebraMapEquiv
+-/
 
 variable {M}
 
+#print ExteriorAlgebra.toTrivSqZeroExt /-
 /-- The canonical map from `exterior_algebra R M` into `triv_sq_zero_ext R M` that sends
 `exterior_algebra.ι` to `triv_sq_zero_ext.inr`. -/
 def toTrivSqZeroExt [Module Rᵐᵒᵖ M] [IsCentralScalar R M] :
     ExteriorAlgebra R M →ₐ[R] TrivSqZeroExt R M :=
   lift R ⟨TrivSqZeroExt.inrHom R M, fun m => TrivSqZeroExt.inr_mul_inr R m m⟩
 #align exterior_algebra.to_triv_sq_zero_ext ExteriorAlgebra.toTrivSqZeroExt
+-/
 
+#print ExteriorAlgebra.toTrivSqZeroExt_ι /-
 @[simp]
 theorem toTrivSqZeroExt_ι [Module Rᵐᵒᵖ M] [IsCentralScalar R M] (x : M) :
     toTrivSqZeroExt (ι R x) = TrivSqZeroExt.inr x :=
   lift_ι_apply _ _ _ _
 #align exterior_algebra.to_triv_sq_zero_ext_ι ExteriorAlgebra.toTrivSqZeroExt_ι
+-/
 
+#print ExteriorAlgebra.ιInv /-
 /-- The left-inverse of `ι`.
 
 As an implementation detail, we implement this using `triv_sq_zero_ext` which has a suitable
@@ -203,24 +244,32 @@ def ιInv : ExteriorAlgebra R M →ₗ[R] M :=
   haveI : IsCentralScalar R M := ⟨fun r m => rfl⟩
   exact (TrivSqZeroExt.sndHom R M).comp to_triv_sq_zero_ext.to_linear_map
 #align exterior_algebra.ι_inv ExteriorAlgebra.ιInv
+-/
 
+#print ExteriorAlgebra.ι_leftInverse /-
 theorem ι_leftInverse : Function.LeftInverse ιInv (ι R : M → ExteriorAlgebra R M) := fun x => by
   simp [ι_inv]
 #align exterior_algebra.ι_left_inverse ExteriorAlgebra.ι_leftInverse
+-/
 
 variable (R)
 
+#print ExteriorAlgebra.ι_inj /-
 @[simp]
 theorem ι_inj (x y : M) : ι R x = ι R y ↔ x = y :=
   ι_leftInverse.Injective.eq_iff
 #align exterior_algebra.ι_inj ExteriorAlgebra.ι_inj
+-/
 
 variable {R}
 
+#print ExteriorAlgebra.ι_eq_zero_iff /-
 @[simp]
 theorem ι_eq_zero_iff (x : M) : ι R x = 0 ↔ x = 0 := by rw [← ι_inj R x 0, LinearMap.map_zero]
 #align exterior_algebra.ι_eq_zero_iff ExteriorAlgebra.ι_eq_zero_iff
+-/
 
+#print ExteriorAlgebra.ι_eq_algebraMap_iff /-
 @[simp]
 theorem ι_eq_algebraMap_iff (x : M) (r : R) : ι R x = algebraMap R _ r ↔ x = 0 ∧ r = 0 :=
   by
@@ -234,14 +283,18 @@ theorem ι_eq_algebraMap_iff (x : M) (r : R) : ι R x = algebraMap R _ r ↔ x =
   · rintro ⟨rfl, rfl⟩
     rw [LinearMap.map_zero, RingHom.map_zero]
 #align exterior_algebra.ι_eq_algebra_map_iff ExteriorAlgebra.ι_eq_algebraMap_iff
+-/
 
+#print ExteriorAlgebra.ι_ne_one /-
 @[simp]
 theorem ι_ne_one [Nontrivial R] (x : M) : ι R x ≠ 1 :=
   by
   rw [← (algebraMap R (ExteriorAlgebra R M)).map_one, Ne.def, ι_eq_algebra_map_iff]
   exact one_ne_zero ∘ And.right
 #align exterior_algebra.ι_ne_one ExteriorAlgebra.ι_ne_one
+-/
 
+#print ExteriorAlgebra.ι_range_disjoint_one /-
 /-- The generators of the exterior algebra are disjoint from its scalars. -/
 theorem ι_range_disjoint_one :
     Disjoint (LinearMap.range (ι R : M →ₗ[R] ExteriorAlgebra R M))
@@ -252,14 +305,18 @@ theorem ι_range_disjoint_one :
   rw [ι_eq_algebra_map_iff x] at hx 
   rw [hx.2, RingHom.map_zero]
 #align exterior_algebra.ι_range_disjoint_one ExteriorAlgebra.ι_range_disjoint_one
+-/
 
+#print ExteriorAlgebra.ι_add_mul_swap /-
 @[simp]
 theorem ι_add_mul_swap (x y : M) : ι R x * ι R y + ι R y * ι R x = 0 :=
   calc
     _ = ι R (x + y) * ι R (x + y) := by simp [mul_add, add_mul]
     _ = _ := ι_sq_zero _
 #align exterior_algebra.ι_add_mul_swap ExteriorAlgebra.ι_add_mul_swap
+-/
 
+#print ExteriorAlgebra.ι_mul_prod_list /-
 theorem ι_mul_prod_list {n : ℕ} (f : Fin n → M) (i : Fin n) :
     (ι R <| f i) * (List.ofFn fun i => ι R <| f i).Prod = 0 :=
   by
@@ -274,9 +331,11 @@ theorem ι_mul_prod_list {n : ℕ} (f : Fin n → M) (i : Fin n) :
       refine' (eq_zero_iff_eq_zero_of_add_eq_zero _).mp hn
       rw [← add_mul, ι_add_mul_swap, MulZeroClass.zero_mul]
 #align exterior_algebra.ι_mul_prod_list ExteriorAlgebra.ι_mul_prod_list
+-/
 
 variable (R)
 
+#print ExteriorAlgebra.ιMulti /-
 /-- The product of `n` terms of the form `ι R m` is an alternating map.
 
 This is a special case of `multilinear_map.mk_pi_algebra_fin`, and the exterior algebra version of
@@ -308,24 +367,32 @@ def ιMulti (n : ℕ) : AlternatingMap R M (ExteriorAlgebra R M) (Fin n) :=
           exact hfxy
     toFun := F }
 #align exterior_algebra.ι_multi ExteriorAlgebra.ιMulti
+-/
 
 variable {R}
 
+#print ExteriorAlgebra.ιMulti_apply /-
 theorem ιMulti_apply {n : ℕ} (v : Fin n → M) : ιMulti R n v = (List.ofFn fun i => ι R (v i)).Prod :=
   rfl
 #align exterior_algebra.ι_multi_apply ExteriorAlgebra.ιMulti_apply
+-/
 
+#print ExteriorAlgebra.ιMulti_zero_apply /-
 @[simp]
 theorem ιMulti_zero_apply (v : Fin 0 → M) : ιMulti R 0 v = 1 :=
   rfl
 #align exterior_algebra.ι_multi_zero_apply ExteriorAlgebra.ιMulti_zero_apply
+-/
 
+#print ExteriorAlgebra.ιMulti_succ_apply /-
 @[simp]
 theorem ιMulti_succ_apply {n : ℕ} (v : Fin n.succ → M) :
     ιMulti R _ v = ι R (v 0) * ιMulti R _ (Matrix.vecTail v) :=
   (congr_arg List.prod (List.ofFn_succ _)).trans List.prod_cons
 #align exterior_algebra.ι_multi_succ_apply ExteriorAlgebra.ιMulti_succ_apply
+-/
 
+#print ExteriorAlgebra.ιMulti_succ_curryLeft /-
 theorem ιMulti_succ_curryLeft {n : ℕ} (m : M) :
     (ιMulti R n.succ).curryLeft m = (LinearMap.mulLeft R (ι R m)).compAlternatingMap (ιMulti R n) :=
   AlternatingMap.ext fun v =>
@@ -333,6 +400,7 @@ theorem ιMulti_succ_curryLeft {n : ℕ} (m : M) :
       simp_rw [Matrix.tail_cons]
       rfl
 #align exterior_algebra.ι_multi_succ_curry_left ExteriorAlgebra.ιMulti_succ_curryLeft
+-/
 
 end ExteriorAlgebra
 
@@ -340,16 +408,20 @@ namespace TensorAlgebra
 
 variable {R M}
 
+#print TensorAlgebra.toExterior /-
 /-- The canonical image of the `tensor_algebra` in the `exterior_algebra`, which maps
 `tensor_algebra.ι R x` to `exterior_algebra.ι R x`. -/
 def toExterior : TensorAlgebra R M →ₐ[R] ExteriorAlgebra R M :=
   TensorAlgebra.lift R (ExteriorAlgebra.ι R : M →ₗ[R] ExteriorAlgebra R M)
 #align tensor_algebra.to_exterior TensorAlgebra.toExterior
+-/
 
+#print TensorAlgebra.toExterior_ι /-
 @[simp]
 theorem toExterior_ι (m : M) : (TensorAlgebra.ι R m).toExterior = ExteriorAlgebra.ι R m := by
   simp [to_exterior]
 #align tensor_algebra.to_exterior_ι TensorAlgebra.toExterior_ι
+-/
 
 end TensorAlgebra
 

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

@@ -258,7 +258,6 @@ theorem ι_add_mul_swap (x y : M) : ι R x * ι R y + ι R y * ι R x = 0 :=
   calc
     _ = ι R (x + y) * ι R (x + y) := by simp [mul_add, add_mul]
     _ = _ := ι_sq_zero _
-    
 #align exterior_algebra.ι_add_mul_swap ExteriorAlgebra.ι_add_mul_swap
 
 theorem ι_mul_prod_list {n : ℕ} (f : Fin n → M) (i : Fin n) :

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

@@ -228,7 +228,7 @@ theorem ι_eq_algebraMap_iff (x : M) (r : R) : ι R x = algebraMap R _ r ↔ x =
   · letI : Module Rᵐᵒᵖ M := Module.compHom _ ((RingHom.id R).fromOpposite mul_comm)
     haveI : IsCentralScalar R M := ⟨fun r m => rfl⟩
     have hf0 : to_triv_sq_zero_ext (ι R x) = (0, x) := to_triv_sq_zero_ext_ι _
-    rw [h, AlgHom.commutes] at hf0
+    rw [h, AlgHom.commutes] at hf0 
     have : r = 0 ∧ 0 = x := Prod.ext_iff.1 hf0
     exact this.symm.imp_left Eq.symm
   · rintro ⟨rfl, rfl⟩
@@ -249,7 +249,7 @@ theorem ι_range_disjoint_one :
   by
   rw [Submodule.disjoint_def]
   rintro _ ⟨x, hx⟩ ⟨r, rfl : algebraMap _ _ _ = _⟩
-  rw [ι_eq_algebra_map_iff x] at hx
+  rw [ι_eq_algebra_map_iff x] at hx 
   rw [hx.2, RingHom.map_zero]
 #align exterior_algebra.ι_range_disjoint_one ExteriorAlgebra.ι_range_disjoint_one
 
@@ -270,8 +270,8 @@ theorem ι_mul_prod_list {n : ℕ} (f : Fin n → M) (i : Fin n) :
     by_cases h : i = 0
     · rw [h, ι_sq_zero, MulZeroClass.zero_mul]
     · replace hn := congr_arg ((· * ·) <| ι R <| f 0) (hn (fun i => f <| Fin.succ i) (i.pred h))
-      simp only at hn
-      rw [Fin.succ_pred, ← mul_assoc, MulZeroClass.mul_zero] at hn
+      simp only at hn 
+      rw [Fin.succ_pred, ← mul_assoc, MulZeroClass.mul_zero] at hn 
       refine' (eq_zero_iff_eq_zero_of_add_eq_zero _).mp hn
       rw [← add_mul, ι_add_mul_swap, MulZeroClass.zero_mul]
 #align exterior_algebra.ι_mul_prod_list ExteriorAlgebra.ι_mul_prod_list
@@ -298,7 +298,7 @@ def ιMulti (n : ℕ) : AlternatingMap R M (ExteriorAlgebra R M) (Fin n) :=
       · exact x.elim0
       · rw [List.ofFn_succ, List.prod_cons]
         by_cases hx : x = 0
-        · rw [hx] at hfxy h
+        · rw [hx] at hfxy h 
           rw [hfxy, ← Fin.succ_pred y (ne_of_lt h).symm]
           exact ι_mul_prod_list (f ∘ Fin.succ) _
         · convert MulZeroClass.mul_zero _

mathlib commit https://github.com/leanprover-community/mathlib/commit/3180fab693e2cee3bff62675571264cb8778b212

@@ -268,12 +268,12 @@ theorem ι_mul_prod_list {n : ℕ} (f : Fin n → M) (i : Fin n) :
   · exact i.elim0
   · rw [List.ofFn_succ, List.prod_cons, ← mul_assoc]
     by_cases h : i = 0
-    · rw [h, ι_sq_zero, zero_mul]
+    · rw [h, ι_sq_zero, MulZeroClass.zero_mul]
     · replace hn := congr_arg ((· * ·) <| ι R <| f 0) (hn (fun i => f <| Fin.succ i) (i.pred h))
       simp only at hn
-      rw [Fin.succ_pred, ← mul_assoc, mul_zero] at hn
+      rw [Fin.succ_pred, ← mul_assoc, MulZeroClass.mul_zero] at hn
       refine' (eq_zero_iff_eq_zero_of_add_eq_zero _).mp hn
-      rw [← add_mul, ι_add_mul_swap, zero_mul]
+      rw [← add_mul, ι_add_mul_swap, MulZeroClass.zero_mul]
 #align exterior_algebra.ι_mul_prod_list ExteriorAlgebra.ι_mul_prod_list
 
 variable (R)
@@ -301,7 +301,7 @@ def ιMulti (n : ℕ) : AlternatingMap R M (ExteriorAlgebra R M) (Fin n) :=
         · rw [hx] at hfxy h
           rw [hfxy, ← Fin.succ_pred y (ne_of_lt h).symm]
           exact ι_mul_prod_list (f ∘ Fin.succ) _
-        · convert mul_zero _
+        · convert MulZeroClass.mul_zero _
           refine'
             hn (fun i => f <| Fin.succ i) (x.pred hx)
               (y.pred (ne_of_lt <| lt_of_le_of_lt x.zero_le h).symm) _ (fin.pred_lt_pred_iff.mpr h)

mathlib commit https://github.com/leanprover-community/mathlib/commit/22131150f88a2d125713ffa0f4693e3355b1eb49

@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Zhangir Azerbayev, Adam Topaz, Eric Wieser
 
 ! This file was ported from Lean 3 source module linear_algebra.exterior_algebra.basic
-! leanprover-community/mathlib commit 92ca63f0fb391a9ca5f22d2409a6080e786d99f7
+! leanprover-community/mathlib commit b8d2eaa69d69ce8f03179a5cda774fc0cde984e4
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/
@@ -182,12 +182,14 @@ variable {M}
 
 /-- The canonical map from `exterior_algebra R M` into `triv_sq_zero_ext R M` that sends
 `exterior_algebra.ι` to `triv_sq_zero_ext.inr`. -/
-def toTrivSqZeroExt : ExteriorAlgebra R M →ₐ[R] TrivSqZeroExt R M :=
+def toTrivSqZeroExt [Module Rᵐᵒᵖ M] [IsCentralScalar R M] :
+    ExteriorAlgebra R M →ₐ[R] TrivSqZeroExt R M :=
   lift R ⟨TrivSqZeroExt.inrHom R M, fun m => TrivSqZeroExt.inr_mul_inr R m m⟩
 #align exterior_algebra.to_triv_sq_zero_ext ExteriorAlgebra.toTrivSqZeroExt
 
 @[simp]
-theorem toTrivSqZeroExt_ι (x : M) : toTrivSqZeroExt (ι R x) = TrivSqZeroExt.inr x :=
+theorem toTrivSqZeroExt_ι [Module Rᵐᵒᵖ M] [IsCentralScalar R M] (x : M) :
+    toTrivSqZeroExt (ι R x) = TrivSqZeroExt.inr x :=
   lift_ι_apply _ _ _ _
 #align exterior_algebra.to_triv_sq_zero_ext_ι ExteriorAlgebra.toTrivSqZeroExt_ι
 
@@ -196,7 +198,10 @@ theorem toTrivSqZeroExt_ι (x : M) : toTrivSqZeroExt (ι R x) = TrivSqZeroExt.in
 As an implementation detail, we implement this using `triv_sq_zero_ext` which has a suitable
 algebra structure. -/
 def ιInv : ExteriorAlgebra R M →ₗ[R] M :=
-  (TrivSqZeroExt.sndHom R M).comp toTrivSqZeroExt.toLinearMap
+  by
+  letI : Module Rᵐᵒᵖ M := Module.compHom _ ((RingHom.id R).fromOpposite mul_comm)
+  haveI : IsCentralScalar R M := ⟨fun r m => rfl⟩
+  exact (TrivSqZeroExt.sndHom R M).comp to_triv_sq_zero_ext.to_linear_map
 #align exterior_algebra.ι_inv ExteriorAlgebra.ιInv
 
 theorem ι_leftInverse : Function.LeftInverse ιInv (ι R : M → ExteriorAlgebra R M) := fun x => by
@@ -220,7 +225,9 @@ theorem ι_eq_zero_iff (x : M) : ι R x = 0 ↔ x = 0 := by rw [← ι_inj R x 0
 theorem ι_eq_algebraMap_iff (x : M) (r : R) : ι R x = algebraMap R _ r ↔ x = 0 ∧ r = 0 :=
   by
   refine' ⟨fun h => _, _⟩
-  · have hf0 : to_triv_sq_zero_ext (ι R x) = (0, x) := to_triv_sq_zero_ext_ι _
+  · letI : Module Rᵐᵒᵖ M := Module.compHom _ ((RingHom.id R).fromOpposite mul_comm)
+    haveI : IsCentralScalar R M := ⟨fun r m => rfl⟩
+    have hf0 : to_triv_sq_zero_ext (ι R x) = (0, x) := to_triv_sq_zero_ext_ι _
     rw [h, AlgHom.commutes] at hf0
     have : r = 0 ∧ 0 = x := Prod.ext_iff.1 hf0
     exact this.symm.imp_left Eq.symm

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

@@ -254,7 +254,7 @@ theorem ι_eq_algebraMap_iff (x : M) (r : R) : ι R x = algebraMap R _ r ↔ x =
 
 @[simp]
 theorem ι_ne_one [Nontrivial R] (x : M) : ι R x ≠ 1 := by
-  rw [← (algebraMap R (ExteriorAlgebra R M)).map_one, Ne.def, ι_eq_algebraMap_iff]
+  rw [← (algebraMap R (ExteriorAlgebra R M)).map_one, Ne, ι_eq_algebraMap_iff]
   exact one_ne_zero ∘ And.right
 #align exterior_algebra.ι_ne_one ExteriorAlgebra.ι_ne_one
 

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)

@@ -50,7 +50,6 @@ as this avoids us having to duplicate API.
 universe u1 u2 u3 u4 u5
 
 variable (R : Type u1) [CommRing R]
-
 variable (M : Type u2) [AddCommGroup M] [Module R M]
 
 /-- The exterior algebra of an `R`-module `M`.

That is, \bigwedge, not \Lambda

Co-authored-by: Richard Copley <rcopley@gmail.com> Co-authored-by: Patrick Massot <patrickmassot@free.fr>

@@ -295,7 +295,7 @@ variable (R)
 
 This is a special case of `MultilinearMap.mkPiAlgebraFin`, and the exterior algebra version of
 `TensorAlgebra.tprod`. -/
-def ιMulti (n : ℕ) : M [Λ^Fin n]→ₗ[R] ExteriorAlgebra R M :=
+def ιMulti (n : ℕ) : M [⋀^Fin n]→ₗ[R] ExteriorAlgebra R M :=
   let F := (MultilinearMap.mkPiAlgebraFin R n (ExteriorAlgebra R M)).compLinearMap fun _ => ι R
   { F with
     map_eq_zero_of_eq' := fun f x y hfxy hxy => by

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 _.

@@ -150,10 +150,10 @@ and is preserved under addition and muliplication, then it holds for all of `Ext
 -/
 @[elab_as_elim]
 theorem induction {C : ExteriorAlgebra R M → Prop}
-    (h_grade0 : ∀ r, C (algebraMap R (ExteriorAlgebra R M) r)) (h_grade1 : ∀ x, C (ι R 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 (ExteriorAlgebra R M) r)) (ι : ∀ x, C (ι R x))
+    (mul : ∀ a b, C a → C b → C (a * b)) (add : ∀ a b, C a → C b → C (a + b))
     (a : ExteriorAlgebra R M) : C a :=
-  CliffordAlgebra.induction h_grade0 h_grade1 h_mul h_add a
+  CliffordAlgebra.induction algebraMap ι mul add a
 #align exterior_algebra.induction ExteriorAlgebra.induction
 
 /-- The left-inverse of `algebraMap`. -/

This is split off from PR #10654 (that actually proves some properties of the exterior powers). It introduces the reducible definition exteriorPower R n M for the nth exterior power of the R-module M; this is of type Submodule R (ExteriorAlgebra R M) and defined as LinearMap.range (ExteriorAlgebra.ι R : M →ₗ[R] ExteriorAlgebra R M) ^ n. It also introduces the notation Λ[R]^n M for exteriorPower R n M.

Note: for a reason that I don't understand, Lean becomes unable to synthesize the SetLike.GradedMonoid instance on fun (i : ℕ) ↦ (Λ[R]^i) M, so I added it manually in ExteriorAlgebra/Graded.lean. I am far from sure that this is the correct solution.

Co-authored-by: Eric Wieser <wieser.eric@gmail.com> Co-authored-by: Richard Copley <buster@buster.me.uk>

@@ -18,6 +18,11 @@ We construct the exterior algebra of a module `M` over a commutative semiring `R
 The exterior algebra of the `R`-module `M` is denoted as `ExteriorAlgebra R M`.
 It is endowed with the structure of an `R`-algebra.
 
+The `n`th exterior power of the `R`-module `M` is denoted by `exteriorPower R n M`;
+it is of type `Submodule R (ExteriorAlgebra R M)` and defined as
+`LinearMap.range (ExteriorAlgebra.ι R : M →ₗ[R] ExteriorAlgebra R M) ^ n`.
+We also introduce the notation `⋀[R]^n M` for `exteriorPower R n M`.
+
 Given a linear morphism `f : M → A` from a module `M` to another `R`-algebra `A`, such that
 `cond : ∀ m : M, f m * f m = 0`, there is a (unique) lift of `f` to an `R`-algebra morphism,
 which is denoted `ExteriorAlgebra.lift R f cond`.
@@ -66,6 +71,21 @@ def ι : M →ₗ[R] ExteriorAlgebra R M :=
   CliffordAlgebra.ι _
 #align exterior_algebra.ι ExteriorAlgebra.ι
 
+section exteriorPower
+
+-- New variables `n` and `M`, to get the correct order of variables in the notation.
+variable (n : ℕ) (M : Type u2) [AddCommGroup M] [Module R M]
+
+/-- Definition of the `n`th exterior power of a `R`-module `N`. We introduce the notation
+`⋀[R]^n M` for `exteriorPower R n M`. -/
+abbrev exteriorPower : Submodule R (ExteriorAlgebra R M) :=
+  LinearMap.range (ι R : M →ₗ[R] ExteriorAlgebra R M) ^ n
+
+@[inherit_doc exteriorPower]
+notation:max "⋀[" R "]^" n:arg => exteriorPower R n
+
+end exteriorPower
+
 variable {R}
 
 /-- As well as being linear, `ι m` squares to zero. -/
@@ -333,7 +353,7 @@ variable (R)
 
 /-- The image of `ExteriorAlgebra.ιMulti R n` is contained in the `n`th exterior power. -/
 lemma ιMulti_range (n : ℕ) :
-    Set.range (ιMulti R n (M := M)) ⊆ ↑(LinearMap.range (ι R (M := M)) ^ n) := by
+    Set.range (ιMulti R n (M := M)) ⊆ ↑(⋀[R]^n M) := by
   rw [Set.range_subset_iff]
   intro v
   rw [ιMulti_apply]
@@ -344,10 +364,9 @@ lemma ιMulti_range (n : ℕ) :
 /-- The image of `ExteriorAlgebra.ιMulti R n` spans the `n`th exterior power, as a submodule
 of the exterior algebra. -/
 lemma ιMulti_span_fixedDegree (n : ℕ) :
-    Submodule.span R (Set.range (ιMulti R n)) =
-    LinearMap.range (ι R : M →ₗ[R] ExteriorAlgebra R M) ^ n := by
+    Submodule.span R (Set.range (ιMulti R n)) = ⋀[R]^n M := by
   refine le_antisymm (Submodule.span_le.2 (ιMulti_range R n)) ?_
-  rw [Submodule.pow_eq_span_pow_set, Submodule.span_le]
+  rw [exteriorPower, Submodule.pow_eq_span_pow_set, Submodule.span_le]
   refine fun u hu ↦ Submodule.subset_span ?_
   obtain ⟨f, rfl⟩ := Set.mem_pow.mp hu
   refine ⟨fun i => ιInv (f i).1, ?_⟩

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

@@ -279,7 +279,7 @@ def ιMulti (n : ℕ) : M [Λ^Fin n]→ₗ[R] ExteriorAlgebra R M :=
   let F := (MultilinearMap.mkPiAlgebraFin R n (ExteriorAlgebra R M)).compLinearMap fun _ => ι R
   { F with
     map_eq_zero_of_eq' := fun f x y hfxy hxy => by
-      dsimp
+      dsimp [F]
       clear F
       wlog h : x < y
       · exact this R (A := A) n f y x hfxy.symm hxy.symm (hxy.lt_or_lt.resolve_left h)

Classify by adding issue number (#10618) to porting notes claiming anything semantically equivalent to simp can prove this or simp can simplify this.

@@ -69,14 +69,14 @@ def ι : M →ₗ[R] ExteriorAlgebra R M :=
 variable {R}
 
 /-- As well as being linear, `ι m` squares to zero. -/
--- @[simp] -- Porting note: simp can prove this
+-- @[simp] -- Porting note (#10618): simp can prove this
 theorem ι_sq_zero (m : M) : ι R m * ι R m = 0 :=
   (CliffordAlgebra.ι_sq_scalar _ m).trans <| map_zero _
 #align exterior_algebra.ι_sq_zero ExteriorAlgebra.ι_sq_zero
 
 variable {A : Type*} [Semiring A] [Algebra R A]
 
--- @[simp] -- Porting note: simp can prove this
+-- @[simp] -- Porting note (#10618): simp can prove this
 theorem comp_ι_sq_zero (g : ExteriorAlgebra R M →ₐ[R] A) (m : M) : g (ι R m) * g (ι R m) = 0 := by
   rw [← AlgHom.map_mul, ι_sq_zero, AlgHom.map_zero]
 #align exterior_algebra.comp_ι_sq_zero ExteriorAlgebra.comp_ι_sq_zero

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>

@@ -42,7 +42,7 @@ as this avoids us having to duplicate API.
 -/
 
 
-universe u1 u2 u3
+universe u1 u2 u3 u4 u5
 
 variable (R : Type u1) [CommRing R]
 
@@ -200,8 +200,7 @@ def ιInv : ExteriorAlgebra R M →ₗ[R] M := by
   exact (TrivSqZeroExt.sndHom R M).comp toTrivSqZeroExt.toLinearMap
 #align exterior_algebra.ι_inv ExteriorAlgebra.ιInv
 
--- Porting note: In the type, changed `ιInv` to `ιInv.1`
-theorem ι_leftInverse : Function.LeftInverse ιInv.1 (ι R : M → ExteriorAlgebra R M) := fun x => by
+theorem ι_leftInverse : Function.LeftInverse ιInv (ι R : M → ExteriorAlgebra R M) := fun x => by
   -- Porting note: Original proof didn't have `letI` and `haveI`
   letI : Module Rᵐᵒᵖ M := Module.compHom _ ((RingHom.id R).fromOpposite mul_comm)
   haveI : IsCentralScalar R M := ⟨fun r m => rfl⟩
@@ -330,10 +329,143 @@ theorem ιMulti_succ_curryLeft {n : ℕ} (m : M) :
       rfl
 #align exterior_algebra.ι_multi_succ_curry_left ExteriorAlgebra.ιMulti_succ_curryLeft
 
+variable (R)
+
+/-- The image of `ExteriorAlgebra.ιMulti R n` is contained in the `n`th exterior power. -/
+lemma ιMulti_range (n : ℕ) :
+    Set.range (ιMulti R n (M := M)) ⊆ ↑(LinearMap.range (ι R (M := M)) ^ n) := by
+  rw [Set.range_subset_iff]
+  intro v
+  rw [ιMulti_apply]
+  apply Submodule.pow_subset_pow
+  rw [Set.mem_pow]
+  exact ⟨fun i => ⟨ι R (v i), LinearMap.mem_range_self _ _⟩, rfl⟩
+
+/-- The image of `ExteriorAlgebra.ιMulti R n` spans the `n`th exterior power, as a submodule
+of the exterior algebra. -/
+lemma ιMulti_span_fixedDegree (n : ℕ) :
+    Submodule.span R (Set.range (ιMulti R n)) =
+    LinearMap.range (ι R : M →ₗ[R] ExteriorAlgebra R M) ^ n := by
+  refine le_antisymm (Submodule.span_le.2 (ιMulti_range R n)) ?_
+  rw [Submodule.pow_eq_span_pow_set, Submodule.span_le]
+  refine fun u hu ↦ Submodule.subset_span ?_
+  obtain ⟨f, rfl⟩ := Set.mem_pow.mp hu
+  refine ⟨fun i => ιInv (f i).1, ?_⟩
+  rw [ιMulti_apply]
+  congr with i
+  obtain ⟨v, hv⟩ := (f i).prop
+  rw [← hv, ι_leftInverse]
+
+/-- Given a linearly ordered family `v` of vectors of `M` and a natural number `n`, produce the
+family of `n`fold exterior products of elements of `v`, seen as members of the exterior algebra. -/
+abbrev ιMulti_family (n : ℕ) {I : Type*} [LinearOrder I] (v : I → M)
+    (s : {s : Finset I // Finset.card s = n}) : ExteriorAlgebra R M :=
+  ιMulti R n fun i => v (Finset.orderIsoOfFin _ s.prop i)
+
+variable {R}
+
 /-- An `ExteriorAlgebra` over a nontrivial ring is nontrivial. -/
 instance [Nontrivial R] : Nontrivial (ExteriorAlgebra R M) :=
   (algebraMap_leftInverse M).injective.nontrivial
 
+/-! Functoriality of the exterior algebra. -/
+
+variable {N : Type u4} {N' : Type u5} [AddCommGroup N] [Module R N] [AddCommGroup N'] [Module R N']
+
+/-- The morphism of exterior algebras induced by a linear map. -/
+def map (f : M →ₗ[R] N) : ExteriorAlgebra R M →ₐ[R] ExteriorAlgebra R N :=
+  CliffordAlgebra.map { f with map_app' := fun _ => rfl }
+
+@[simp]
+theorem map_comp_ι (f : M →ₗ[R] N) : (map f).toLinearMap ∘ₗ ι R = ι R ∘ₗ f :=
+  CliffordAlgebra.map_comp_ι _
+
+@[simp]
+theorem map_apply_ι (f : M →ₗ[R] N) (m : M) : map f (ι R m) = ι R (f m) :=
+  CliffordAlgebra.map_apply_ι _ m
+
+@[simp]
+theorem map_apply_ιMulti {n : ℕ} (f : M →ₗ[R] N) (m : Fin n → M) :
+    map f (ιMulti R n m) = ιMulti R n (f ∘ m) := by
+  rw [ιMulti_apply, ιMulti_apply, map_list_prod]
+  simp only [List.map_ofFn, Function.comp, map_apply_ι]
+
+@[simp]
+theorem map_comp_ιMulti {n : ℕ} (f : M →ₗ[R] N) :
+    (map f).toLinearMap.compAlternatingMap (ιMulti R n (M := M)) =
+    (ιMulti R n (M := N)).compLinearMap f := by
+  ext m
+  exact map_apply_ιMulti _ _
+
+@[simp]
+theorem map_id :
+    map LinearMap.id = AlgHom.id R (ExteriorAlgebra R M) :=
+  CliffordAlgebra.map_id 0
+
+@[simp]
+theorem map_comp_map (f : M →ₗ[R] N) (g : N →ₗ[R] N') :
+    AlgHom.comp (map g) (map f) = map (LinearMap.comp g f) :=
+  CliffordAlgebra.map_comp_map _ _
+
+@[simp]
+theorem ι_range_map_map (f : M →ₗ[R] N) :
+    Submodule.map (AlgHom.toLinearMap (map f)) (LinearMap.range (ι R (M := M))) =
+    Submodule.map (ι R) (LinearMap.range f) :=
+  CliffordAlgebra.ι_range_map_map _
+
+theorem toTrivSqZeroExt_comp_map [Module Rᵐᵒᵖ M] [IsCentralScalar R M] [Module Rᵐᵒᵖ N]
+    [IsCentralScalar R N] (f : M →ₗ[R] N) :
+    toTrivSqZeroExt.comp (map f) = (TrivSqZeroExt.map f).comp toTrivSqZeroExt := by
+  apply hom_ext
+  apply LinearMap.ext
+  simp only [AlgHom.comp_toLinearMap, LinearMap.coe_comp, Function.comp_apply,
+    AlgHom.toLinearMap_apply, map_apply_ι, toTrivSqZeroExt_ι, TrivSqZeroExt.map_inr, forall_const]
+
+theorem ιInv_comp_map (f : M →ₗ[R] N) :
+    ιInv.comp (map f).toLinearMap = f.comp ιInv := by
+  letI : Module Rᵐᵒᵖ M := Module.compHom _ ((RingHom.id R).fromOpposite mul_comm)
+  haveI : IsCentralScalar R M := ⟨fun r m => rfl⟩
+  letI : Module Rᵐᵒᵖ N := Module.compHom _ ((RingHom.id R).fromOpposite mul_comm)
+  haveI : IsCentralScalar R N := ⟨fun r m => rfl⟩
+  unfold ιInv
+  conv_lhs => rw [LinearMap.comp_assoc, ← AlgHom.comp_toLinearMap, toTrivSqZeroExt_comp_map,
+                AlgHom.comp_toLinearMap, ← LinearMap.comp_assoc, TrivSqZeroExt.sndHom_comp_map]
+
+open Function in
+/-- For a linear map `f` from `M` to `N`,
+`ExteriorAlgebra.map g` is a retraction of `ExteriorAlgebra.map f` iff
+`g` is a retraction of `f`. -/
+@[simp]
+lemma leftInverse_map_iff {f : M →ₗ[R] N} {g : N →ₗ[R] M} :
+    LeftInverse (map g) (map f) ↔ LeftInverse g f := by
+  refine ⟨fun h x => ?_, fun h => CliffordAlgebra.leftInverse_map_of_leftInverse _ _ h⟩
+  simpa using h (ι _ x)
+
+/-- A morphism of modules that admits a linear retraction induces an injective morphism of
+exterior algebras. -/
+lemma map_injective {f : M →ₗ[R] N} (hf : ∃ (g : N →ₗ[R] M), g.comp f = LinearMap.id) :
+    Function.Injective (map f) :=
+  let ⟨_, hgf⟩ := hf; (leftInverse_map_iff.mpr (DFunLike.congr_fun hgf)).injective
+
+/-- A morphism of modules is surjective if and only the morphism of exterior algebras that it
+induces is surjective. -/
+@[simp]
+lemma map_surjective_iff {f : M →ₗ[R] N} :
+    Function.Surjective (map f) ↔ Function.Surjective f := by
+  refine ⟨fun h y ↦ ?_, fun h ↦ CliffordAlgebra.map_surjective _ h⟩
+  obtain ⟨x, hx⟩ := h (ι R y)
+  existsi ιInv x
+  rw [← LinearMap.comp_apply, ← ιInv_comp_map, LinearMap.comp_apply]
+  erw [hx, ExteriorAlgebra.ι_leftInverse]
+
+variable {K E F : Type*} [Field K] [AddCommGroup E]
+  [Module K E] [AddCommGroup F] [Module K F]
+
+/-- An injective morphism of vector spaces induces an injective morphism of exterior algebras. -/
+lemma map_injective_field {f : E →ₗ[K] F} (hf : LinearMap.ker f = ⊥) :
+    Function.Injective (map f) :=
+  map_injective (LinearMap.exists_leftInverse_of_injective f hf)
+
 end ExteriorAlgebra
 
 namespace TensorAlgebra

See Zulip thread for the discussion.

@@ -263,7 +263,7 @@ theorem ι_mul_prod_list {n : ℕ} (f : Fin n → M) (i : Fin n) :
     by_cases h : i = 0
     · rw [h, ι_sq_zero, zero_mul]
     · replace hn :=
-        congr_arg (ι R (f 0) * ·) $ hn (fun i => f <| Fin.succ i) (i.pred h)
+        congr_arg (ι R (f 0) * ·) <| hn (fun i => f <| Fin.succ i) (i.pred h)
       simp only at hn
       rw [Fin.succ_pred, ← mul_assoc, mul_zero] at hn
       refine' (eq_zero_iff_eq_zero_of_add_eq_zero _).mp hn

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.

@@ -252,9 +252,7 @@ theorem ι_range_disjoint_one :
 
 @[simp]
 theorem ι_add_mul_swap (x y : M) : ι R x * ι R y + ι R y * ι R x = 0 :=
-  calc
-    _ = ι R (y + x) * ι R (y + x) := by simp [mul_add, add_mul]
-    _ = _ := ι_sq_zero _
+  CliffordAlgebra.ι_mul_ι_add_swap_of_isOrtho <| .all _ _
 #align exterior_algebra.ι_add_mul_swap ExteriorAlgebra.ι_add_mul_swap
 
 theorem ι_mul_prod_list {n : ℕ} (f : Fin n → M) (i : Fin n) :

I used the regex \(\(· (.) ·\) (.)\), replacing with ($2 $1 ·).

@@ -265,8 +265,7 @@ theorem ι_mul_prod_list {n : ℕ} (f : Fin n → M) (i : Fin n) :
     by_cases h : i = 0
     · rw [h, ι_sq_zero, zero_mul]
     · replace hn :=
-        congr_arg
-          ((· * ·) <| ι R <| f 0) (hn (fun i => f <| Fin.succ i) (i.pred h))
+        congr_arg (ι R (f 0) * ·) $ hn (fun i => f <| Fin.succ i) (i.pred h)
       simp only at hn
       rw [Fin.succ_pred, ← mul_assoc, mul_zero] at hn
       refine' (eq_zero_iff_eq_zero_of_add_eq_zero _).mp hn

Use M [Λ^ι]→ₗ[R] N for AlternatingMap R M N ι, similarly to the existing notation M [Λ^ι]→L[R] N for ContinuousAlternatingMap R M N ι.

@@ -279,7 +279,7 @@ variable (R)
 
 This is a special case of `MultilinearMap.mkPiAlgebraFin`, and the exterior algebra version of
 `TensorAlgebra.tprod`. -/
-def ιMulti (n : ℕ) : AlternatingMap R M (ExteriorAlgebra R M) (Fin n) :=
+def ιMulti (n : ℕ) : M [Λ^Fin n]→ₗ[R] ExteriorAlgebra R M :=
   let F := (MultilinearMap.mkPiAlgebraFin R n (ExteriorAlgebra R M)).compLinearMap fun _ => ι R
   { F with
     map_eq_zero_of_eq' := fun f x y hfxy hxy => by

This eliminates (fun a ↦ β) α in the type when applying a FunLike.

Co-authored-by: Matthew Ballard <matt@mrb.email> Co-authored-by: Eric Wieser <wieser.eric@gmail.com>

@@ -138,7 +138,7 @@ theorem induction {C : ExteriorAlgebra R M → Prop}
 
 /-- The left-inverse of `algebraMap`. -/
 def algebraMapInv : ExteriorAlgebra R M →ₐ[R] R :=
-  ExteriorAlgebra.lift R ⟨(0 : M →ₗ[R] R), fun m => by simp⟩
+  ExteriorAlgebra.lift R ⟨(0 : M →ₗ[R] R), fun _ => by simp⟩
 #align exterior_algebra.algebra_map_inv ExteriorAlgebra.algebraMapInv
 
 variable (M)

Using methods suggested by @eric-wieser Zulip discussion

@@ -333,6 +333,10 @@ theorem ιMulti_succ_curryLeft {n : ℕ} (m : M) :
       rfl
 #align exterior_algebra.ι_multi_succ_curry_left ExteriorAlgebra.ιMulti_succ_curryLeft
 
+/-- An `ExteriorAlgebra` over a nontrivial ring is nontrivial. -/
+instance [Nontrivial R] : Nontrivial (ExteriorAlgebra R M) :=
+  (algebraMap_leftInverse M).injective.nontrivial
+
 end ExteriorAlgebra
 
 namespace TensorAlgebra

I'm going to refactor/expand API about domCoprod, and I don't want to rebuild the whole library when I do it.

@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Zhangir Azerbayev, Adam Topaz, Eric Wieser
 -/
 import Mathlib.LinearAlgebra.CliffordAlgebra.Basic
-import Mathlib.LinearAlgebra.Alternating
+import Mathlib.LinearAlgebra.Alternating.Basic
 
 #align_import linear_algebra.exterior_algebra.basic from "leanprover-community/mathlib"@"b8d2eaa69d69ce8f03179a5cda774fc0cde984e4"
 

Search&replace MulZeroClass.mul_zero -> mul_zero, MulZeroClass.zero_mul -> zero_mul.

These were introduced by Mathport, as the full name of mul_zero is actually MulZeroClass.mul_zero (it's exported with the short name).

@@ -263,14 +263,14 @@ theorem ι_mul_prod_list {n : ℕ} (f : Fin n → M) (i : Fin n) :
   · exact i.elim0
   · rw [List.ofFn_succ, List.prod_cons, ← mul_assoc]
     by_cases h : i = 0
-    · rw [h, ι_sq_zero, MulZeroClass.zero_mul]
+    · rw [h, ι_sq_zero, zero_mul]
     · replace hn :=
         congr_arg
           ((· * ·) <| ι R <| f 0) (hn (fun i => f <| Fin.succ i) (i.pred h))
       simp only at hn
-      rw [Fin.succ_pred, ← mul_assoc, MulZeroClass.mul_zero] at hn
+      rw [Fin.succ_pred, ← mul_assoc, mul_zero] at hn
       refine' (eq_zero_iff_eq_zero_of_add_eq_zero _).mp hn
-      rw [← add_mul, ι_add_mul_swap, MulZeroClass.zero_mul]
+      rw [← add_mul, ι_add_mul_swap, zero_mul]
 #align exterior_algebra.ι_mul_prod_list ExteriorAlgebra.ι_mul_prod_list
 
 variable (R)
@@ -297,7 +297,7 @@ def ιMulti (n : ℕ) : AlternatingMap R M (ExteriorAlgebra R M) (Fin n) :=
           rw [hfxy, ← Fin.succ_pred y (ne_of_lt h).symm]
           exact ι_mul_prod_list (f ∘ Fin.succ) _
         -- ignore the left-most term and induct on the remaining ones, decrementing indices
-        · convert MulZeroClass.mul_zero (ι R (f 0))
+        · convert mul_zero (ι R (f 0))
           refine'
             hn
               (fun i => f <| Fin.succ i) (x.pred hx)

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

This has nice performance benefits.

@@ -74,7 +74,7 @@ theorem ι_sq_zero (m : M) : ι R m * ι R m = 0 :=
   (CliffordAlgebra.ι_sq_scalar _ m).trans <| map_zero _
 #align exterior_algebra.ι_sq_zero ExteriorAlgebra.ι_sq_zero
 
-variable {A : Type _} [Semiring A] [Algebra R A]
+variable {A : Type*} [Semiring A] [Algebra R A]
 
 -- @[simp] -- Porting note: simp can prove this
 theorem comp_ι_sq_zero (g : ExteriorAlgebra R M →ₐ[R] A) (m : M) : g (ι R m) * g (ι R m) = 0 := by

Various adaptations to changes when Fin API was moved to Std. One notable change is that many lemmas are now stated in terms of i ≠ 0 (for i : Fin n) rather then i.1 ≠ 0, and as a consequence many Fin.vne_of_ne applications have been added or removed, mostly removed.

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

@@ -266,7 +266,7 @@ theorem ι_mul_prod_list {n : ℕ} (f : Fin n → M) (i : Fin n) :
     · rw [h, ι_sq_zero, MulZeroClass.zero_mul]
     · replace hn :=
         congr_arg
-          ((· * ·) <| ι R <| f 0) (hn (fun i => f <| Fin.succ i) (i.pred <| Fin.vne_of_ne h))
+          ((· * ·) <| ι R <| f 0) (hn (fun i => f <| Fin.succ i) (i.pred h))
       simp only at hn
       rw [Fin.succ_pred, ← mul_assoc, MulZeroClass.mul_zero] at hn
       refine' (eq_zero_iff_eq_zero_of_add_eq_zero _).mp hn
@@ -294,14 +294,14 @@ def ιMulti (n : ℕ) : AlternatingMap R M (ExteriorAlgebra R M) (Fin n) :=
         by_cases hx : x = 0
         -- one of the repeated terms is on the left
         · rw [hx] at hfxy h
-          rw [hfxy, ← Fin.succ_pred y (Fin.vne_of_ne <| (ne_of_lt h).symm)]
+          rw [hfxy, ← Fin.succ_pred y (ne_of_lt h).symm]
           exact ι_mul_prod_list (f ∘ Fin.succ) _
         -- ignore the left-most term and induct on the remaining ones, decrementing indices
         · convert MulZeroClass.mul_zero (ι R (f 0))
           refine'
             hn
-              (fun i => f <| Fin.succ i) (x.pred <| Fin.vne_of_ne hx)
-              (y.pred (Fin.vne_of_ne <| ne_of_lt <| lt_of_le_of_lt x.zero_le h).symm) _
+              (fun i => f <| Fin.succ i) (x.pred hx)
+              (y.pred (ne_of_lt <| lt_of_le_of_lt x.zero_le h).symm) _
               (Fin.pred_lt_pred_iff.mpr h)
           simp only [Fin.succ_pred]
           exact hfxy

• depends on: #5966

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

@@ -2,15 +2,12 @@
 Copyright (c) 2020 Adam Topaz. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Zhangir Azerbayev, Adam Topaz, Eric Wieser
-
-! This file was ported from Lean 3 source module linear_algebra.exterior_algebra.basic
-! leanprover-community/mathlib commit b8d2eaa69d69ce8f03179a5cda774fc0cde984e4
-! Please do not edit these lines, except to modify the commit id
-! if you have ported upstream changes.
 -/
 import Mathlib.LinearAlgebra.CliffordAlgebra.Basic
 import Mathlib.LinearAlgebra.Alternating
 
+#align_import linear_algebra.exterior_algebra.basic from "leanprover-community/mathlib"@"b8d2eaa69d69ce8f03179a5cda774fc0cde984e4"
+
 /-!
 # Exterior Algebras
 

• depends on: https://github.com/leanprover/std4/pull/163
• depends on: #5584
• depends on: #5715
• depends on: #5729
• depends on: https://github.com/leanprover/std4/pull/173

Co-authored-by: Komyyy <pol_tta@outlook.jp> Co-authored-by: Scott Morrison <scott.morrison@gmail.com> Co-authored-by: Scott Morrison <scott.morrison@anu.edu.au> Co-authored-by: Ruben Van de Velde <65514131+Ruben-VandeVelde@users.noreply.github.com> Co-authored-by: Mario Carneiro <di.gama@gmail.com>

@@ -267,7 +267,9 @@ theorem ι_mul_prod_list {n : ℕ} (f : Fin n → M) (i : Fin n) :
   · rw [List.ofFn_succ, List.prod_cons, ← mul_assoc]
     by_cases h : i = 0
     · rw [h, ι_sq_zero, MulZeroClass.zero_mul]
-    · replace hn := congr_arg ((· * ·) <| ι R <| f 0) (hn (fun i => f <| Fin.succ i) (i.pred h))
+    · replace hn :=
+        congr_arg
+          ((· * ·) <| ι R <| f 0) (hn (fun i => f <| Fin.succ i) (i.pred <| Fin.vne_of_ne h))
       simp only at hn
       rw [Fin.succ_pred, ← mul_assoc, MulZeroClass.mul_zero] at hn
       refine' (eq_zero_iff_eq_zero_of_add_eq_zero _).mp hn
@@ -295,12 +297,15 @@ def ιMulti (n : ℕ) : AlternatingMap R M (ExteriorAlgebra R M) (Fin n) :=
         by_cases hx : x = 0
         -- one of the repeated terms is on the left
         · rw [hx] at hfxy h
-          rw [hfxy, ← Fin.succ_pred y (ne_of_lt h).symm]
+          rw [hfxy, ← Fin.succ_pred y (Fin.vne_of_ne <| (ne_of_lt h).symm)]
           exact ι_mul_prod_list (f ∘ Fin.succ) _
         -- ignore the left-most term and induct on the remaining ones, decrementing indices
         · convert MulZeroClass.mul_zero (ι R (f 0))
-          refine' hn (fun i => f <| Fin.succ i) (x.pred hx)
-            (y.pred (ne_of_lt <| lt_of_le_of_lt x.zero_le h).symm) _ (Fin.pred_lt_pred_iff.mpr h)
+          refine'
+            hn
+              (fun i => f <| Fin.succ i) (x.pred <| Fin.vne_of_ne hx)
+              (y.pred (Fin.vne_of_ne <| ne_of_lt <| lt_of_le_of_lt x.zero_le h).symm) _
+              (Fin.pred_lt_pred_iff.mpr h)
           simp only [Fin.succ_pred]
           exact hfxy
     toFun := F }