analysis.inner_product_space.orientation ⟷ Mathlib.Analysis.InnerProductSpace.Orientation

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

@@ -357,7 +357,7 @@ theorem abs_volumeForm_apply_of_pairwise_orthogonal {v : Fin n → E}
   congr
   ext i
   have hb : b i = ‖v i‖⁻¹ • v i := gramSchmidtOrthonormalBasis_apply_of_orthogonal hdim hv (h i)
-  simp only [hb, inner_smul_left, real_inner_self_eq_norm_mul_norm, IsROrC.conj_to_real]
+  simp only [hb, inner_smul_left, real_inner_self_eq_norm_mul_norm, RCLike.conj_to_real]
   rw [abs_of_nonneg]
   · have : ‖v i‖ ≠ 0 := by simpa using h i
     field_simp

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

@@ -61,7 +61,7 @@ theorem det_to_matrix_orthonormalBasis_of_same_orientation
   apply (e.det_to_matrix_orthonormal_basis_real f).resolve_right
   have : 0 < e.to_basis.det f :=
     by
-    rw [e.to_basis.orientation_eq_iff_det_pos] at h 
+    rw [e.to_basis.orientation_eq_iff_det_pos] at h
     simpa using h
   linarith
 #align orthonormal_basis.det_to_matrix_orthonormal_basis_of_same_orientation OrthonormalBasis.det_to_matrix_orthonormalBasis_of_same_orientation
@@ -270,7 +270,7 @@ theorem volumeForm_robust_neg (b : OrthonormalBasis (Fin n) ℝ E) (hb : b.toBas
   cases n
   ·
     classical
-    have : positive_orientation ≠ o := by rwa [b.to_basis.orientation_is_empty] at hb 
+    have : positive_orientation ≠ o := by rwa [b.to_basis.orientation_is_empty] at hb
     simp [volume_form, Or.by_cases, dif_neg this.symm]
   let e : OrthonormalBasis (Fin n.succ) ℝ E := o.fin_orthonormal_basis n.succ_pos (Fact.out _)
   dsimp [volume_form]
@@ -353,7 +353,7 @@ theorem abs_volumeForm_apply_of_pairwise_orthogonal {v : Fin n → E}
   ·
     rw [Finset.prod_eq_zero (Finset.mem_univ i), Finset.prod_eq_zero (Finset.mem_univ i)] <;>
       simp [hi]
-  push_neg at h 
+  push_neg at h
   congr
   ext i
   have hb : b i = ‖v i‖⁻¹ • v i := gramSchmidtOrthonormalBasis_apply_of_orthogonal hdim hv (h i)
@@ -402,7 +402,7 @@ theorem volumeForm_comp_linearIsometryEquiv (φ : E ≃ₗᵢ[ℝ] E)
   by
   convert o.volume_form_map φ (φ ∘ x)
   · symm
-    rwa [← o.map_eq_iff_det_pos φ.to_linear_equiv] at hφ 
+    rwa [← o.map_eq_iff_det_pos φ.to_linear_equiv] at hφ
     rw [_i.out, Fintype.card_fin]
   · ext
     simp

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

@@ -214,7 +214,12 @@ variable [_i : Fact (finrank ℝ E = n)] (o : Orientation ℝ E (Fin n))
 /-- The volume form on an oriented real inner product space, a nonvanishing top-dimensional
 alternating form uniquely defined by compatibility with the orientation and inner product structure.
 -/
-irreducible_def volumeForm : AlternatingMap ℝ E ℝ (Fin n) := by classical
+irreducible_def volumeForm : AlternatingMap ℝ E ℝ (Fin n) := by
+  classical
+  cases n
+  · let opos : AlternatingMap ℝ E ℝ (Fin 0) := AlternatingMap.constOfIsEmpty ℝ E (Fin 0) (1 : ℝ)
+    exact o.eq_or_eq_neg_of_is_empty.by_cases (fun _ => opos) fun _ => -opos
+  · exact (o.fin_orthonormal_basis n.succ_pos _i.out).toBasis.det
 #align orientation.volume_form Orientation.volumeForm
 -/
 
@@ -247,7 +252,10 @@ respect to any orthonormal basis of the space compatible with the orientation. -
 theorem volumeForm_robust (b : OrthonormalBasis (Fin n) ℝ E) (hb : b.toBasis.Orientation = o) :
     o.volumeForm = b.toBasis.det := by
   cases n
-  · classical
+  ·
+    classical
+    have : o = positive_orientation := hb.symm.trans b.to_basis.orientation_is_empty
+    simp [volume_form, Or.by_cases, dif_pos this]
   · dsimp [volume_form]
     rw [same_orientation_iff_det_eq_det, hb]
     exact o.fin_orthonormal_basis_orientation _ _
@@ -260,7 +268,10 @@ respect to any orthonormal basis of the space compatible with the orientation. -
 theorem volumeForm_robust_neg (b : OrthonormalBasis (Fin n) ℝ E) (hb : b.toBasis.Orientation ≠ o) :
     o.volumeForm = -b.toBasis.det := by
   cases n
-  · classical
+  ·
+    classical
+    have : positive_orientation ≠ o := by rwa [b.to_basis.orientation_is_empty] at hb 
+    simp [volume_form, Or.by_cases, dif_neg this.symm]
   let e : OrthonormalBasis (Fin n.succ) ℝ E := o.fin_orthonormal_basis n.succ_pos (Fact.out _)
   dsimp [volume_form]
   apply e.det_eq_neg_det_of_opposite_orientation b

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

@@ -214,12 +214,7 @@ variable [_i : Fact (finrank ℝ E = n)] (o : Orientation ℝ E (Fin n))
 /-- The volume form on an oriented real inner product space, a nonvanishing top-dimensional
 alternating form uniquely defined by compatibility with the orientation and inner product structure.
 -/
-irreducible_def volumeForm : AlternatingMap ℝ E ℝ (Fin n) := by
-  classical
-  cases n
-  · let opos : AlternatingMap ℝ E ℝ (Fin 0) := AlternatingMap.constOfIsEmpty ℝ E (Fin 0) (1 : ℝ)
-    exact o.eq_or_eq_neg_of_is_empty.by_cases (fun _ => opos) fun _ => -opos
-  · exact (o.fin_orthonormal_basis n.succ_pos _i.out).toBasis.det
+irreducible_def volumeForm : AlternatingMap ℝ E ℝ (Fin n) := by classical
 #align orientation.volume_form Orientation.volumeForm
 -/
 
@@ -252,10 +247,7 @@ respect to any orthonormal basis of the space compatible with the orientation. -
 theorem volumeForm_robust (b : OrthonormalBasis (Fin n) ℝ E) (hb : b.toBasis.Orientation = o) :
     o.volumeForm = b.toBasis.det := by
   cases n
-  ·
-    classical
-    have : o = positive_orientation := hb.symm.trans b.to_basis.orientation_is_empty
-    simp [volume_form, Or.by_cases, dif_pos this]
+  · classical
   · dsimp [volume_form]
     rw [same_orientation_iff_det_eq_det, hb]
     exact o.fin_orthonormal_basis_orientation _ _
@@ -268,10 +260,7 @@ respect to any orthonormal basis of the space compatible with the orientation. -
 theorem volumeForm_robust_neg (b : OrthonormalBasis (Fin n) ℝ E) (hb : b.toBasis.Orientation ≠ o) :
     o.volumeForm = -b.toBasis.det := by
   cases n
-  ·
-    classical
-    have : positive_orientation ≠ o := by rwa [b.to_basis.orientation_is_empty] at hb 
-    simp [volume_form, Or.by_cases, dif_neg this.symm]
+  · classical
   let e : OrthonormalBasis (Fin n.succ) ℝ E := o.fin_orthonormal_basis n.succ_pos (Fact.out _)
   dsimp [volume_form]
   apply e.det_eq_neg_det_of_opposite_orientation b

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

@@ -3,8 +3,8 @@ Copyright (c) 2022 Joseph Myers. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Joseph Myers, Heather Macbeth
 -/
-import Mathbin.Analysis.InnerProductSpace.GramSchmidtOrtho
-import Mathbin.LinearAlgebra.Orientation
+import Analysis.InnerProductSpace.GramSchmidtOrtho
+import LinearAlgebra.Orientation
 
 #align_import analysis.inner_product_space.orientation from "leanprover-community/mathlib"@"af471b9e3ce868f296626d33189b4ce730fa4c00"
 

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

@@ -2,15 +2,12 @@
 Copyright (c) 2022 Joseph Myers. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Joseph Myers, Heather Macbeth
-
-! This file was ported from Lean 3 source module analysis.inner_product_space.orientation
-! leanprover-community/mathlib commit af471b9e3ce868f296626d33189b4ce730fa4c00
-! Please do not edit these lines, except to modify the commit id
-! if you have ported upstream changes.
 -/
 import Mathbin.Analysis.InnerProductSpace.GramSchmidtOrtho
 import Mathbin.LinearAlgebra.Orientation
 
+#align_import analysis.inner_product_space.orientation from "leanprover-community/mathlib"@"af471b9e3ce868f296626d33189b4ce730fa4c00"
+
 /-!
 # Orientations of real inner product spaces.
 

mathlib commit https://github.com/leanprover-community/mathlib/commit/9fb8964792b4237dac6200193a0d533f1b3f7423

@@ -55,6 +55,7 @@ namespace OrthonormalBasis
 variable {ι : Type _} [Fintype ι] [DecidableEq ι] [ne : Nonempty ι] (e f : OrthonormalBasis ι ℝ E)
   (x : Orientation ℝ E ι)
 
+#print OrthonormalBasis.det_to_matrix_orthonormalBasis_of_same_orientation /-
 /-- The change-of-basis matrix between two orthonormal bases with the same orientation has
 determinant 1. -/
 theorem det_to_matrix_orthonormalBasis_of_same_orientation
@@ -67,7 +68,9 @@ theorem det_to_matrix_orthonormalBasis_of_same_orientation
     simpa using h
   linarith
 #align orthonormal_basis.det_to_matrix_orthonormal_basis_of_same_orientation OrthonormalBasis.det_to_matrix_orthonormalBasis_of_same_orientation
+-/
 
+#print OrthonormalBasis.det_to_matrix_orthonormalBasis_of_opposite_orientation /-
 /-- The change-of-basis matrix between two orthonormal bases with the opposite orientations has
 determinant -1. -/
 theorem det_to_matrix_orthonormalBasis_of_opposite_orientation
@@ -77,9 +80,11 @@ theorem det_to_matrix_orthonormalBasis_of_opposite_orientation
   simp [e.to_basis.orientation_eq_iff_det_pos,
     (e.det_to_matrix_orthonormal_basis_real f).resolve_right h]
 #align orthonormal_basis.det_to_matrix_orthonormal_basis_of_opposite_orientation OrthonormalBasis.det_to_matrix_orthonormalBasis_of_opposite_orientation
+-/
 
 variable {e f}
 
+#print OrthonormalBasis.same_orientation_iff_det_eq_det /-
 /-- Two orthonormal bases with the same orientation determine the same "determinant" top-dimensional
 form on `E`, and conversely. -/
 theorem same_orientation_iff_det_eq_det :
@@ -93,9 +98,11 @@ theorem same_orientation_iff_det_eq_det :
     rw [e.to_basis.det.eq_smul_basis_det f.to_basis]
     simp [e.det_to_matrix_orthonormal_basis_of_same_orientation f h]
 #align orthonormal_basis.same_orientation_iff_det_eq_det OrthonormalBasis.same_orientation_iff_det_eq_det
+-/
 
 variable (e f)
 
+#print OrthonormalBasis.det_eq_neg_det_of_opposite_orientation /-
 /-- Two orthonormal bases with opposite orientations determine opposite "determinant"
 top-dimensional forms on `E`. -/
 theorem det_eq_neg_det_of_opposite_orientation (h : e.toBasis.Orientation ≠ f.toBasis.Orientation) :
@@ -104,11 +111,11 @@ theorem det_eq_neg_det_of_opposite_orientation (h : e.toBasis.Orientation ≠ f.
   rw [e.to_basis.det.eq_smul_basis_det f.to_basis]
   simp [e.det_to_matrix_orthonormal_basis_of_opposite_orientation f h]
 #align orthonormal_basis.det_eq_neg_det_of_opposite_orientation OrthonormalBasis.det_eq_neg_det_of_opposite_orientation
+-/
 
 section AdjustToOrientation
 
-include ne
-
+#print OrthonormalBasis.orthonormal_adjustToOrientation /-
 /-- `orthonormal_basis.adjust_to_orientation`, applied to an orthonormal basis, preserves the
 property of orthonormality. -/
 theorem orthonormal_adjustToOrientation : Orthonormal ℝ (e.toBasis.adjustToOrientation x) :=
@@ -116,6 +123,7 @@ theorem orthonormal_adjustToOrientation : Orthonormal ℝ (e.toBasis.adjustToOri
   apply e.orthonormal.orthonormal_of_forall_eq_or_eq_neg
   simpa using e.to_basis.adjust_to_orientation_apply_eq_or_eq_neg x
 #align orthonormal_basis.orthonormal_adjust_to_orientation OrthonormalBasis.orthonormal_adjustToOrientation
+-/
 
 #print OrthonormalBasis.adjustToOrientation /-
 /-- Given an orthonormal basis and an orientation, return an orthonormal basis giving that
@@ -126,11 +134,14 @@ def adjustToOrientation : OrthonormalBasis ι ℝ E :=
 #align orthonormal_basis.adjust_to_orientation OrthonormalBasis.adjustToOrientation
 -/
 
+#print OrthonormalBasis.toBasis_adjustToOrientation /-
 theorem toBasis_adjustToOrientation :
     (e.adjustToOrientation x).toBasis = e.toBasis.adjustToOrientation x :=
   (e.toBasis.adjustToOrientation x).toBasis_toOrthonormalBasis _
 #align orthonormal_basis.to_basis_adjust_to_orientation OrthonormalBasis.toBasis_adjustToOrientation
+-/
 
+#print OrthonormalBasis.orientation_adjustToOrientation /-
 /-- `adjust_to_orientation` gives an orthonormal basis with the required orientation. -/
 @[simp]
 theorem orientation_adjustToOrientation : (e.adjustToOrientation x).toBasis.Orientation = x :=
@@ -138,7 +149,9 @@ theorem orientation_adjustToOrientation : (e.adjustToOrientation x).toBasis.Orie
   rw [e.to_basis_adjust_to_orientation]
   exact e.to_basis.orientation_adjust_to_orientation x
 #align orthonormal_basis.orientation_adjust_to_orientation OrthonormalBasis.orientation_adjustToOrientation
+-/
 
+#print OrthonormalBasis.adjustToOrientation_apply_eq_or_eq_neg /-
 /-- Every basis vector from `adjust_to_orientation` is either that from the original basis or its
 negation. -/
 theorem adjustToOrientation_apply_eq_or_eq_neg (i : ι) :
@@ -146,17 +159,22 @@ theorem adjustToOrientation_apply_eq_or_eq_neg (i : ι) :
   simpa [← e.to_basis_adjust_to_orientation] using
     e.to_basis.adjust_to_orientation_apply_eq_or_eq_neg x i
 #align orthonormal_basis.adjust_to_orientation_apply_eq_or_eq_neg OrthonormalBasis.adjustToOrientation_apply_eq_or_eq_neg
+-/
 
+#print OrthonormalBasis.det_adjustToOrientation /-
 theorem det_adjustToOrientation :
     (e.adjustToOrientation x).toBasis.det = e.toBasis.det ∨
       (e.adjustToOrientation x).toBasis.det = -e.toBasis.det :=
   by simpa using e.to_basis.det_adjust_to_orientation x
 #align orthonormal_basis.det_adjust_to_orientation OrthonormalBasis.det_adjustToOrientation
+-/
 
+#print OrthonormalBasis.abs_det_adjustToOrientation /-
 theorem abs_det_adjustToOrientation (v : ι → E) :
     |(e.adjustToOrientation x).toBasis.det v| = |e.toBasis.det v| := by
   simp [to_basis_adjust_to_orientation]
 #align orthonormal_basis.abs_det_adjust_to_orientation OrthonormalBasis.abs_det_adjustToOrientation
+-/
 
 end AdjustToOrientation
 
@@ -179,6 +197,7 @@ protected def finOrthonormalBasis (hn : 0 < n) (h : finrank ℝ E = n) (x : Orie
 #align orientation.fin_orthonormal_basis Orientation.finOrthonormalBasis
 -/
 
+#print Orientation.finOrthonormalBasis_orientation /-
 /-- `orientation.fin_orthonormal_basis` gives a basis with the required orientation. -/
 @[simp]
 theorem finOrthonormalBasis_orientation (hn : 0 < n) (h : finrank ℝ E = n)
@@ -188,13 +207,12 @@ theorem finOrthonormalBasis_orientation (hn : 0 < n) (h : finrank ℝ E = n)
   haveI := finite_dimensional_of_finrank (h.symm ▸ hn : 0 < finrank ℝ E)
   exact ((stdOrthonormalBasis _ _).reindex <| finCongr h).orientation_adjustToOrientation x
 #align orientation.fin_orthonormal_basis_orientation Orientation.finOrthonormalBasis_orientation
+-/
 
 section VolumeForm
 
 variable [_i : Fact (finrank ℝ E = n)] (o : Orientation ℝ E (Fin n))
 
-include _i o
-
 #print Orientation.volumeForm /-
 /-- The volume form on an oriented real inner product space, a nonvanishing top-dimensional
 alternating form uniquely defined by compatibility with the orientation and inner product structure.
@@ -208,15 +226,16 @@ irreducible_def volumeForm : AlternatingMap ℝ E ℝ (Fin n) := by
 #align orientation.volume_form Orientation.volumeForm
 -/
 
-omit _i o
-
+#print Orientation.volumeForm_zero_pos /-
 @[simp]
 theorem volumeForm_zero_pos [_i : Fact (finrank ℝ E = 0)] :
     Orientation.volumeForm (positiveOrientation : Orientation ℝ E (Fin 0)) =
       AlternatingMap.constLinearEquivOfIsEmpty 1 :=
   by simp [volume_form, Or.by_cases, if_pos]
 #align orientation.volume_form_zero_pos Orientation.volumeForm_zero_pos
+-/
 
+#print Orientation.volumeForm_zero_neg /-
 theorem volumeForm_zero_neg [_i : Fact (finrank ℝ E = 0)] :
     Orientation.volumeForm (-positiveOrientation : Orientation ℝ E (Fin 0)) =
       -AlternatingMap.constLinearEquivOfIsEmpty 1 :=
@@ -228,9 +247,9 @@ theorem volumeForm_zero_neg [_i : Fact (finrank ℝ E = 0)] :
   simpa using
     congr_arg alternating_map.const_linear_equiv_of_is_empty.symm (eq_zero_of_sameRay_self_neg h)
 #align orientation.volume_form_zero_neg Orientation.volumeForm_zero_neg
+-/
 
-include _i o
-
+#print Orientation.volumeForm_robust /-
 /-- The volume form on an oriented real inner product space can be evaluated as the determinant with
 respect to any orthonormal basis of the space compatible with the orientation. -/
 theorem volumeForm_robust (b : OrthonormalBasis (Fin n) ℝ E) (hb : b.toBasis.Orientation = o) :
@@ -244,7 +263,9 @@ theorem volumeForm_robust (b : OrthonormalBasis (Fin n) ℝ E) (hb : b.toBasis.O
     rw [same_orientation_iff_det_eq_det, hb]
     exact o.fin_orthonormal_basis_orientation _ _
 #align orientation.volume_form_robust Orientation.volumeForm_robust
+-/
 
+#print Orientation.volumeForm_robust_neg /-
 /-- The volume form on an oriented real inner product space can be evaluated as the determinant with
 respect to any orthonormal basis of the space compatible with the orientation. -/
 theorem volumeForm_robust_neg (b : OrthonormalBasis (Fin n) ℝ E) (hb : b.toBasis.Orientation ≠ o) :
@@ -260,7 +281,9 @@ theorem volumeForm_robust_neg (b : OrthonormalBasis (Fin n) ℝ E) (hb : b.toBas
   convert hb.symm
   exact o.fin_orthonormal_basis_orientation _ _
 #align orientation.volume_form_robust_neg Orientation.volumeForm_robust_neg
+-/
 
+#print Orientation.volumeForm_neg_orientation /-
 @[simp]
 theorem volumeForm_neg_orientation : (-o).volumeForm = -o.volumeForm :=
   by
@@ -273,7 +296,9 @@ theorem volumeForm_neg_orientation : (-o).volumeForm = -o.volumeForm :=
     rw [e.to_basis.orientation_ne_iff_eq_neg, h₁]
   rw [o.volume_form_robust e h₁, (-o).volumeForm_robust_neg e h₂]
 #align orientation.volume_form_neg_orientation Orientation.volumeForm_neg_orientation
+-/
 
+#print Orientation.volumeForm_robust' /-
 theorem volumeForm_robust' (b : OrthonormalBasis (Fin n) ℝ E) (v : Fin n → E) :
     |o.volumeForm v| = |b.toBasis.det v| := by
   cases n
@@ -282,7 +307,9 @@ theorem volumeForm_robust' (b : OrthonormalBasis (Fin n) ℝ E) (v : Fin n → E
     rw [o.volume_form_robust (b.adjust_to_orientation o) (b.orientation_adjust_to_orientation o),
       b.abs_det_adjust_to_orientation]
 #align orientation.volume_form_robust' Orientation.volumeForm_robust'
+-/
 
+#print Orientation.abs_volumeForm_apply_le /-
 /-- Let `v` be an indexed family of `n` vectors in an oriented `n`-dimensional real inner
 product space `E`. The output of the volume form of `E` when evaluated on `v` is bounded in absolute
 value by the product of the norms of the vectors `v i`. -/
@@ -302,11 +329,15 @@ theorem abs_volumeForm_apply_le (v : Fin n → E) : |o.volumeForm v| ≤ ∏ i :
   convert abs_real_inner_le_norm (b i) (v i)
   simp [b.orthonormal.1 i]
 #align orientation.abs_volume_form_apply_le Orientation.abs_volumeForm_apply_le
+-/
 
+#print Orientation.volumeForm_apply_le /-
 theorem volumeForm_apply_le (v : Fin n → E) : o.volumeForm v ≤ ∏ i : Fin n, ‖v i‖ :=
   (le_abs_self _).trans (o.abs_volumeForm_apply_le v)
 #align orientation.volume_form_apply_le Orientation.volumeForm_apply_le
+-/
 
+#print Orientation.abs_volumeForm_apply_of_pairwise_orthogonal /-
 /-- Let `v` be an indexed family of `n` orthogonal vectors in an oriented `n`-dimensional
 real inner product space `E`. The output of the volume form of `E` when evaluated on `v` is, up to
 sign, the product of the norms of the vectors `v i`. -/
@@ -335,13 +366,16 @@ theorem abs_volumeForm_apply_of_pairwise_orthogonal {v : Fin n → E}
     field_simp
   · positivity
 #align orientation.abs_volume_form_apply_of_pairwise_orthogonal Orientation.abs_volumeForm_apply_of_pairwise_orthogonal
+-/
 
+#print Orientation.abs_volumeForm_apply_of_orthonormal /-
 /-- The output of the volume form of an oriented real inner product space `E` when evaluated on an
 orthonormal basis is ±1. -/
 theorem abs_volumeForm_apply_of_orthonormal (v : OrthonormalBasis (Fin n) ℝ E) :
     |o.volumeForm v| = 1 := by
   simpa [o.volume_form_robust' v v] using congr_arg abs v.to_basis.det_self
 #align orientation.abs_volume_form_apply_of_orthonormal Orientation.abs_volumeForm_apply_of_orthonormal
+-/
 
 #print Orientation.volumeForm_map /-
 theorem volumeForm_map {F : Type _} [NormedAddCommGroup F] [InnerProductSpace ℝ F]
@@ -363,6 +397,7 @@ theorem volumeForm_map {F : Type _} [NormedAddCommGroup F] [InnerProductSpace 
 #align orientation.volume_form_map Orientation.volumeForm_map
 -/
 
+#print Orientation.volumeForm_comp_linearIsometryEquiv /-
 /-- The volume form is invariant under pullback by a positively-oriented isometric automorphism. -/
 theorem volumeForm_comp_linearIsometryEquiv (φ : E ≃ₗᵢ[ℝ] E)
     (hφ : 0 < (φ.toLinearEquiv : E →ₗ[ℝ] E).det) (x : Fin n → E) :
@@ -375,6 +410,7 @@ theorem volumeForm_comp_linearIsometryEquiv (φ : E ≃ₗᵢ[ℝ] E)
   · ext
     simp
 #align orientation.volume_form_comp_linear_isometry_equiv Orientation.volumeForm_comp_linearIsometryEquiv
+-/
 
 end VolumeForm
 

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

@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Joseph Myers, Heather Macbeth
 
 ! This file was ported from Lean 3 source module analysis.inner_product_space.orientation
-! leanprover-community/mathlib commit bd65478311e4dfd41f48bf38c7e3b02fb75d0163
+! leanprover-community/mathlib commit af471b9e3ce868f296626d33189b4ce730fa4c00
 ! 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.Orientation
 /-!
 # Orientations of real inner product spaces.
 
+> THIS FILE IS SYNCHRONIZED WITH MATHLIB4.
+> Any changes to this file require a corresponding PR to mathlib4.
+
 This file provides definitions and proves lemmas about orientations of real inner product spaces.
 
 ## Main definitions
@@ -114,12 +117,14 @@ theorem orthonormal_adjustToOrientation : Orthonormal ℝ (e.toBasis.adjustToOri
   simpa using e.to_basis.adjust_to_orientation_apply_eq_or_eq_neg x
 #align orthonormal_basis.orthonormal_adjust_to_orientation OrthonormalBasis.orthonormal_adjustToOrientation
 
+#print OrthonormalBasis.adjustToOrientation /-
 /-- Given an orthonormal basis and an orientation, return an orthonormal basis giving that
 orientation: either the original basis, or one constructed by negating a single (arbitrary) basis
 vector. -/
 def adjustToOrientation : OrthonormalBasis ι ℝ E :=
   (e.toBasis.adjustToOrientation x).toOrthonormalBasis (e.orthonormal_adjustToOrientation x)
 #align orthonormal_basis.adjust_to_orientation OrthonormalBasis.adjustToOrientation
+-/
 
 theorem toBasis_adjustToOrientation :
     (e.adjustToOrientation x).toBasis = e.toBasis.adjustToOrientation x :=
@@ -163,6 +168,7 @@ variable {n : ℕ}
 
 open OrthonormalBasis
 
+#print Orientation.finOrthonormalBasis /-
 /-- An orthonormal basis, indexed by `fin n`, with the given orientation. -/
 protected def finOrthonormalBasis (hn : 0 < n) (h : finrank ℝ E = n) (x : Orientation ℝ E (Fin n)) :
     OrthonormalBasis (Fin n) ℝ E :=
@@ -171,6 +177,7 @@ protected def finOrthonormalBasis (hn : 0 < n) (h : finrank ℝ E = n) (x : Orie
   haveI := finite_dimensional_of_finrank (h.symm ▸ hn : 0 < finrank ℝ E)
   exact ((stdOrthonormalBasis _ _).reindex <| finCongr h).adjustToOrientation x
 #align orientation.fin_orthonormal_basis Orientation.finOrthonormalBasis
+-/
 
 /-- `orientation.fin_orthonormal_basis` gives a basis with the required orientation. -/
 @[simp]
@@ -188,16 +195,18 @@ variable [_i : Fact (finrank ℝ E = n)] (o : Orientation ℝ E (Fin n))
 
 include _i o
 
+#print Orientation.volumeForm /-
 /-- The volume form on an oriented real inner product space, a nonvanishing top-dimensional
 alternating form uniquely defined by compatibility with the orientation and inner product structure.
 -/
 irreducible_def volumeForm : AlternatingMap ℝ E ℝ (Fin n) := by
   classical
-    cases n
-    · let opos : AlternatingMap ℝ E ℝ (Fin 0) := AlternatingMap.constOfIsEmpty ℝ E (Fin 0) (1 : ℝ)
-      exact o.eq_or_eq_neg_of_is_empty.by_cases (fun _ => opos) fun _ => -opos
-    · exact (o.fin_orthonormal_basis n.succ_pos _i.out).toBasis.det
+  cases n
+  · let opos : AlternatingMap ℝ E ℝ (Fin 0) := AlternatingMap.constOfIsEmpty ℝ E (Fin 0) (1 : ℝ)
+    exact o.eq_or_eq_neg_of_is_empty.by_cases (fun _ => opos) fun _ => -opos
+  · exact (o.fin_orthonormal_basis n.succ_pos _i.out).toBasis.det
 #align orientation.volume_form Orientation.volumeForm
+-/
 
 omit _i o
 
@@ -229,8 +238,8 @@ theorem volumeForm_robust (b : OrthonormalBasis (Fin n) ℝ E) (hb : b.toBasis.O
   cases n
   ·
     classical
-      have : o = positive_orientation := hb.symm.trans b.to_basis.orientation_is_empty
-      simp [volume_form, Or.by_cases, dif_pos this]
+    have : o = positive_orientation := hb.symm.trans b.to_basis.orientation_is_empty
+    simp [volume_form, Or.by_cases, dif_pos this]
   · dsimp [volume_form]
     rw [same_orientation_iff_det_eq_det, hb]
     exact o.fin_orthonormal_basis_orientation _ _
@@ -243,8 +252,8 @@ theorem volumeForm_robust_neg (b : OrthonormalBasis (Fin n) ℝ E) (hb : b.toBas
   cases n
   ·
     classical
-      have : positive_orientation ≠ o := by rwa [b.to_basis.orientation_is_empty] at hb 
-      simp [volume_form, Or.by_cases, dif_neg this.symm]
+    have : positive_orientation ≠ o := by rwa [b.to_basis.orientation_is_empty] at hb 
+    simp [volume_form, Or.by_cases, dif_neg this.symm]
   let e : OrthonormalBasis (Fin n.succ) ℝ E := o.fin_orthonormal_basis n.succ_pos (Fact.out _)
   dsimp [volume_form]
   apply e.det_eq_neg_det_of_opposite_orientation b
@@ -316,7 +325,7 @@ theorem abs_volumeForm_apply_of_pairwise_orthogonal {v : Fin n → E}
   ·
     rw [Finset.prod_eq_zero (Finset.mem_univ i), Finset.prod_eq_zero (Finset.mem_univ i)] <;>
       simp [hi]
-  push_neg  at h 
+  push_neg at h 
   congr
   ext i
   have hb : b i = ‖v i‖⁻¹ • v i := gramSchmidtOrthonormalBasis_apply_of_orthogonal hdim hv (h i)
@@ -334,6 +343,7 @@ theorem abs_volumeForm_apply_of_orthonormal (v : OrthonormalBasis (Fin n) ℝ E)
   simpa [o.volume_form_robust' v v] using congr_arg abs v.to_basis.det_self
 #align orientation.abs_volume_form_apply_of_orthonormal Orientation.abs_volumeForm_apply_of_orthonormal
 
+#print Orientation.volumeForm_map /-
 theorem volumeForm_map {F : Type _} [NormedAddCommGroup F] [InnerProductSpace ℝ F]
     [Fact (finrank ℝ F = n)] (φ : E ≃ₗᵢ[ℝ] F) (x : Fin n → F) :
     (Orientation.map (Fin n) φ.toLinearEquiv o).volumeForm x = o.volumeForm (φ.symm ∘ x) :=
@@ -351,6 +361,7 @@ theorem volumeForm_map {F : Type _} [NormedAddCommGroup F] [InnerProductSpace 
   rw [o.volume_form_robust e he]
   simp
 #align orientation.volume_form_map Orientation.volumeForm_map
+-/
 
 /-- The volume form is invariant under pullback by a positively-oriented isometric automorphism. -/
 theorem volumeForm_comp_linearIsometryEquiv (φ : E ≃ₗᵢ[ℝ] E)

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

@@ -60,7 +60,7 @@ theorem det_to_matrix_orthonormalBasis_of_same_orientation
   apply (e.det_to_matrix_orthonormal_basis_real f).resolve_right
   have : 0 < e.to_basis.det f :=
     by
-    rw [e.to_basis.orientation_eq_iff_det_pos] at h
+    rw [e.to_basis.orientation_eq_iff_det_pos] at h 
     simpa using h
   linarith
 #align orthonormal_basis.det_to_matrix_orthonormal_basis_of_same_orientation OrthonormalBasis.det_to_matrix_orthonormalBasis_of_same_orientation
@@ -243,7 +243,7 @@ theorem volumeForm_robust_neg (b : OrthonormalBasis (Fin n) ℝ E) (hb : b.toBas
   cases n
   ·
     classical
-      have : positive_orientation ≠ o := by rwa [b.to_basis.orientation_is_empty] at hb
+      have : positive_orientation ≠ o := by rwa [b.to_basis.orientation_is_empty] at hb 
       simp [volume_form, Or.by_cases, dif_neg this.symm]
   let e : OrthonormalBasis (Fin n.succ) ℝ E := o.fin_orthonormal_basis n.succ_pos (Fact.out _)
   dsimp [volume_form]
@@ -316,7 +316,7 @@ theorem abs_volumeForm_apply_of_pairwise_orthogonal {v : Fin n → E}
   ·
     rw [Finset.prod_eq_zero (Finset.mem_univ i), Finset.prod_eq_zero (Finset.mem_univ i)] <;>
       simp [hi]
-  push_neg  at h
+  push_neg  at h 
   congr
   ext i
   have hb : b i = ‖v i‖⁻¹ • v i := gramSchmidtOrthonormalBasis_apply_of_orthogonal hdim hv (h i)
@@ -359,7 +359,7 @@ theorem volumeForm_comp_linearIsometryEquiv (φ : E ≃ₗᵢ[ℝ] E)
   by
   convert o.volume_form_map φ (φ ∘ x)
   · symm
-    rwa [← o.map_eq_iff_det_pos φ.to_linear_equiv] at hφ
+    rwa [← o.map_eq_iff_det_pos φ.to_linear_equiv] at hφ 
     rw [_i.out, Fintype.card_fin]
   · ext
     simp

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

@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Joseph Myers, Heather Macbeth
 
 ! This file was ported from Lean 3 source module analysis.inner_product_space.orientation
-! leanprover-community/mathlib commit 46b633fd842bef9469441c0209906f6dddd2b4f5
+! leanprover-community/mathlib commit bd65478311e4dfd41f48bf38c7e3b02fb75d0163
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/
@@ -194,7 +194,7 @@ alternating form uniquely defined by compatibility with the orientation and inne
 irreducible_def volumeForm : AlternatingMap ℝ E ℝ (Fin n) := by
   classical
     cases n
-    · let opos : AlternatingMap ℝ E ℝ (Fin 0) := AlternatingMap.constOfIsEmpty ℝ E (1 : ℝ)
+    · let opos : AlternatingMap ℝ E ℝ (Fin 0) := AlternatingMap.constOfIsEmpty ℝ E (Fin 0) (1 : ℝ)
       exact o.eq_or_eq_neg_of_is_empty.by_cases (fun _ => opos) fun _ => -opos
     · exact (o.fin_orthonormal_basis n.succ_pos _i.out).toBasis.det
 #align orientation.volume_form Orientation.volumeForm

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

@@ -45,7 +45,7 @@ variable {E : Type _} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
 
 open FiniteDimensional
 
-open BigOperators RealInnerProductSpace
+open scoped BigOperators RealInnerProductSpace
 
 namespace OrthonormalBasis
 

mathlib commit https://github.com/leanprover-community/mathlib/commit/ef95945cd48c932c9e034872bd25c3c220d9c946

@@ -108,17 +108,17 @@ include ne
 
 /-- `orthonormal_basis.adjust_to_orientation`, applied to an orthonormal basis, preserves the
 property of orthonormality. -/
-theorem orthonormalAdjustToOrientation : Orthonormal ℝ (e.toBasis.adjustToOrientation x) :=
+theorem orthonormal_adjustToOrientation : Orthonormal ℝ (e.toBasis.adjustToOrientation x) :=
   by
   apply e.orthonormal.orthonormal_of_forall_eq_or_eq_neg
   simpa using e.to_basis.adjust_to_orientation_apply_eq_or_eq_neg x
-#align orthonormal_basis.orthonormal_adjust_to_orientation OrthonormalBasis.orthonormalAdjustToOrientation
+#align orthonormal_basis.orthonormal_adjust_to_orientation OrthonormalBasis.orthonormal_adjustToOrientation
 
 /-- Given an orthonormal basis and an orientation, return an orthonormal basis giving that
 orientation: either the original basis, or one constructed by negating a single (arbitrary) basis
 vector. -/
 def adjustToOrientation : OrthonormalBasis ι ℝ E :=
-  (e.toBasis.adjustToOrientation x).toOrthonormalBasis (e.orthonormalAdjustToOrientation x)
+  (e.toBasis.adjustToOrientation x).toOrthonormalBasis (e.orthonormal_adjustToOrientation x)
 #align orthonormal_basis.adjust_to_orientation OrthonormalBasis.adjustToOrientation
 
 theorem toBasis_adjustToOrientation :

mathlib commit https://github.com/leanprover-community/mathlib/commit/2f8347015b12b0864dfaf366ec4909eb70c78740

@@ -108,17 +108,17 @@ include ne
 
 /-- `orthonormal_basis.adjust_to_orientation`, applied to an orthonormal basis, preserves the
 property of orthonormality. -/
-theorem orthonormal_adjustToOrientation : Orthonormal ℝ (e.toBasis.adjustToOrientation x) :=
+theorem orthonormalAdjustToOrientation : Orthonormal ℝ (e.toBasis.adjustToOrientation x) :=
   by
   apply e.orthonormal.orthonormal_of_forall_eq_or_eq_neg
   simpa using e.to_basis.adjust_to_orientation_apply_eq_or_eq_neg x
-#align orthonormal_basis.orthonormal_adjust_to_orientation OrthonormalBasis.orthonormal_adjustToOrientation
+#align orthonormal_basis.orthonormal_adjust_to_orientation OrthonormalBasis.orthonormalAdjustToOrientation
 
 /-- Given an orthonormal basis and an orientation, return an orthonormal basis giving that
 orientation: either the original basis, or one constructed by negating a single (arbitrary) basis
 vector. -/
 def adjustToOrientation : OrthonormalBasis ι ℝ E :=
-  (e.toBasis.adjustToOrientation x).toOrthonormalBasis (e.orthonormal_adjustToOrientation x)
+  (e.toBasis.adjustToOrientation x).toOrthonormalBasis (e.orthonormalAdjustToOrientation x)
 #align orthonormal_basis.adjust_to_orientation OrthonormalBasis.adjustToOrientation
 
 theorem toBasis_adjustToOrientation :

mathlib commit https://github.com/leanprover-community/mathlib/commit/55d771df074d0dd020139ee1cd4b95521422df9f

@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Joseph Myers, Heather Macbeth
 
 ! This file was ported from Lean 3 source module analysis.inner_product_space.orientation
-! leanprover-community/mathlib commit 6623e6af705e97002a9054c1c05a980180276fc1
+! leanprover-community/mathlib commit 46b633fd842bef9469441c0209906f6dddd2b4f5
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/
@@ -41,7 +41,7 @@ This file provides definitions and proves lemmas about orientations of real inne
 
 noncomputable section
 
-variable {E : Type _} [InnerProductSpace ℝ E]
+variable {E : Type _} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
 
 open FiniteDimensional
 
@@ -334,8 +334,8 @@ theorem abs_volumeForm_apply_of_orthonormal (v : OrthonormalBasis (Fin n) ℝ E)
   simpa [o.volume_form_robust' v v] using congr_arg abs v.to_basis.det_self
 #align orientation.abs_volume_form_apply_of_orthonormal Orientation.abs_volumeForm_apply_of_orthonormal
 
-theorem volumeForm_map {F : Type _} [InnerProductSpace ℝ F] [Fact (finrank ℝ F = n)]
-    (φ : E ≃ₗᵢ[ℝ] F) (x : Fin n → F) :
+theorem volumeForm_map {F : Type _} [NormedAddCommGroup F] [InnerProductSpace ℝ F]
+    [Fact (finrank ℝ F = n)] (φ : E ≃ₗᵢ[ℝ] F) (x : Fin n → F) :
     (Orientation.map (Fin n) φ.toLinearEquiv o).volumeForm x = o.volumeForm (φ.symm ∘ x) :=
   by
   cases n

mathlib commit https://github.com/leanprover-community/mathlib/commit/2af0836443b4cfb5feda0df0051acdb398304931

@@ -108,17 +108,17 @@ include ne
 
 /-- `orthonormal_basis.adjust_to_orientation`, applied to an orthonormal basis, preserves the
 property of orthonormality. -/
-theorem orthonormalAdjustToOrientation : Orthonormal ℝ (e.toBasis.adjustToOrientation x) :=
+theorem orthonormal_adjustToOrientation : Orthonormal ℝ (e.toBasis.adjustToOrientation x) :=
   by
   apply e.orthonormal.orthonormal_of_forall_eq_or_eq_neg
   simpa using e.to_basis.adjust_to_orientation_apply_eq_or_eq_neg x
-#align orthonormal_basis.orthonormal_adjust_to_orientation OrthonormalBasis.orthonormalAdjustToOrientation
+#align orthonormal_basis.orthonormal_adjust_to_orientation OrthonormalBasis.orthonormal_adjustToOrientation
 
 /-- Given an orthonormal basis and an orientation, return an orthonormal basis giving that
 orientation: either the original basis, or one constructed by negating a single (arbitrary) basis
 vector. -/
 def adjustToOrientation : OrthonormalBasis ι ℝ E :=
-  (e.toBasis.adjustToOrientation x).toOrthonormalBasis (e.orthonormalAdjustToOrientation x)
+  (e.toBasis.adjustToOrientation x).toOrthonormalBasis (e.orthonormal_adjustToOrientation x)
 #align orthonormal_basis.adjust_to_orientation OrthonormalBasis.adjustToOrientation
 
 theorem toBasis_adjustToOrientation :

mathlib commit https://github.com/leanprover-community/mathlib/commit/eb0cb4511aaef0da2462207b67358a0e1fe1e2ee

@@ -214,7 +214,7 @@ theorem volumeForm_zero_neg [_i : Fact (finrank ℝ E = 0)] :
   by
   dsimp [volume_form, Or.by_cases, positive_orientation]
   apply if_neg
-  rw [ray_eq_iff, sameRay_comm]
+  rw [ray_eq_iff, SameRay.sameRay_comm]
   intro h
   simpa using
     congr_arg alternating_map.const_linear_equiv_of_is_empty.symm (eq_zero_of_sameRay_self_neg h)

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

A PR analogous to #12338: reformatting proofs following the multiple goals linter of #12339.

@@ -292,8 +292,8 @@ theorem abs_volumeForm_apply_of_pairwise_orthogonal {v : Fin n → E}
   have hb : b.toBasis.det v = ∏ i, ⟪b i, v i⟫ := gramSchmidtOrthonormalBasis_det hdim v
   rw [o.volumeForm_robust' b, hb, Finset.abs_prod]
   by_cases h : ∃ i, v i = 0
-  obtain ⟨i, hi⟩ := h
-  · rw [Finset.prod_eq_zero (Finset.mem_univ i), Finset.prod_eq_zero (Finset.mem_univ i)] <;>
+  · obtain ⟨i, hi⟩ := h
+    rw [Finset.prod_eq_zero (Finset.mem_univ i), Finset.prod_eq_zero (Finset.mem_univ i)] <;>
       simp [hi]
   push_neg at h
   congr

IsROrC contains data, which goes against the expectation that classes prefixed with Is are prop-valued. People have been complaining about this on and off, so this PR renames IsROrC to RCLike.

@@ -299,7 +299,7 @@ theorem abs_volumeForm_apply_of_pairwise_orthogonal {v : Fin n → E}
   congr
   ext i
   have hb : b i = ‖v i‖⁻¹ • v i := gramSchmidtOrthonormalBasis_apply_of_orthogonal hdim hv (h i)
-  simp only [hb, inner_smul_left, real_inner_self_eq_norm_mul_norm, IsROrC.conj_to_real]
+  simp only [hb, inner_smul_left, real_inner_self_eq_norm_mul_norm, RCLike.conj_to_real]
   rw [abs_of_nonneg]
   · field_simp
   · positivity

That is, \bigwedge, not \Lambda

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

@@ -93,7 +93,7 @@ theorem det_eq_neg_det_of_opposite_orientation (h : e.toBasis.orientation ≠ f.
   rw [e.toBasis.det.eq_smul_basis_det f.toBasis]
   -- Porting note: added `neg_one_smul` with explicit type
   simp [e.det_to_matrix_orthonormalBasis_of_opposite_orientation f h,
-    neg_one_smul ℝ (M := E [Λ^ι]→ₗ[ℝ] ℝ)]
+    neg_one_smul ℝ (M := E [⋀^ι]→ₗ[ℝ] ℝ)]
 #align orthonormal_basis.det_eq_neg_det_of_opposite_orientation OrthonormalBasis.det_eq_neg_det_of_opposite_orientation
 
 section AdjustToOrientation
@@ -178,10 +178,10 @@ variable [_i : Fact (finrank ℝ E = n)] (o : Orientation ℝ E (Fin n))
 /-- The volume form on an oriented real inner product space, a nonvanishing top-dimensional
 alternating form uniquely defined by compatibility with the orientation and inner product structure.
 -/
-irreducible_def volumeForm : E [Λ^Fin n]→ₗ[ℝ] ℝ := by
+irreducible_def volumeForm : E [⋀^Fin n]→ₗ[ℝ] ℝ := by
   classical
     cases' n with n
-    · let opos : E [Λ^Fin 0]→ₗ[ℝ] ℝ := .constOfIsEmpty ℝ E (Fin 0) (1 : ℝ)
+    · let opos : E [⋀^Fin 0]→ₗ[ℝ] ℝ := .constOfIsEmpty ℝ E (Fin 0) (1 : ℝ)
       exact o.eq_or_eq_neg_of_isEmpty.by_cases (fun _ => opos) fun _ => -opos
     · exact (o.finOrthonormalBasis n.succ_pos _i.out).toBasis.det
 #align orientation.volume_form Orientation.volumeForm

Rename lemmas to enable new-style dot notation or drop repeating FiniteDimensional.finiteDimensional_*. Restore old names as deprecated aliases.

@@ -157,7 +157,7 @@ open OrthonormalBasis
 protected def finOrthonormalBasis (hn : 0 < n) (h : finrank ℝ E = n) (x : Orientation ℝ E (Fin n)) :
     OrthonormalBasis (Fin n) ℝ E := by
   haveI := Fin.pos_iff_nonempty.1 hn
-  haveI := finiteDimensional_of_finrank (h.symm ▸ hn : 0 < finrank ℝ E)
+  haveI : FiniteDimensional ℝ E := .of_finrank_pos <| h.symm ▸ hn
   exact ((@stdOrthonormalBasis _ _ _ _ _ this).reindex <| finCongr h).adjustToOrientation x
 #align orientation.fin_orthonormal_basis Orientation.finOrthonormalBasis
 
@@ -166,7 +166,7 @@ protected def finOrthonormalBasis (hn : 0 < n) (h : finrank ℝ E = n) (x : Orie
 theorem finOrthonormalBasis_orientation (hn : 0 < n) (h : finrank ℝ E = n)
     (x : Orientation ℝ E (Fin n)) : (x.finOrthonormalBasis hn h).toBasis.orientation = x := by
   haveI := Fin.pos_iff_nonempty.1 hn
-  haveI := finiteDimensional_of_finrank (h.symm ▸ hn : 0 < finrank ℝ E)
+  haveI : FiniteDimensional ℝ E := .of_finrank_pos <| h.symm ▸ hn
   exact ((@stdOrthonormalBasis _ _ _ _ _ this).reindex <|
     finCongr h).orientation_adjustToOrientation x
 #align orientation.fin_orthonormal_basis_orientation Orientation.finOrthonormalBasis_orientation
@@ -262,7 +262,7 @@ value by the product of the norms of the vectors `v i`. -/
 theorem abs_volumeForm_apply_le (v : Fin n → E) : |o.volumeForm v| ≤ ∏ i : Fin n, ‖v i‖ := by
   cases' n with n
   · refine' o.eq_or_eq_neg_of_isEmpty.elim _ _ <;> rintro rfl <;> simp
-  haveI : FiniteDimensional ℝ E := fact_finiteDimensional_of_finrank_eq_succ n
+  haveI : FiniteDimensional ℝ E := .of_fact_finrank_eq_succ n
   have : finrank ℝ E = Fintype.card (Fin n.succ) := by simpa using _i.out
   let b : OrthonormalBasis (Fin n.succ) ℝ E := gramSchmidtOrthonormalBasis this v
   have hb : b.toBasis.det v = ∏ i, ⟪b i, v i⟫ := gramSchmidtOrthonormalBasis_det this v
@@ -286,7 +286,7 @@ theorem abs_volumeForm_apply_of_pairwise_orthogonal {v : Fin n → E}
     (hv : Pairwise fun i j => ⟪v i, v j⟫ = 0) : |o.volumeForm v| = ∏ i : Fin n, ‖v i‖ := by
   cases' n with n
   · refine' o.eq_or_eq_neg_of_isEmpty.elim _ _ <;> rintro rfl <;> simp
-  haveI : FiniteDimensional ℝ E := fact_finiteDimensional_of_finrank_eq_succ n
+  haveI : FiniteDimensional ℝ E := .of_fact_finrank_eq_succ n
   have hdim : finrank ℝ E = Fintype.card (Fin n.succ) := by simpa using _i.out
   let b : OrthonormalBasis (Fin n.succ) ℝ E := gramSchmidtOrthonormalBasis hdim v
   have hb : b.toBasis.det v = ∏ i, ⟪b i, v i⟫ := gramSchmidtOrthonormalBasis_det hdim v
@@ -334,7 +334,7 @@ theorem volumeForm_comp_linearIsometryEquiv (φ : E ≃ₗᵢ[ℝ] E)
     o.volumeForm (φ ∘ x) = o.volumeForm x := by
   cases' n with n -- Porting note: need to explicitly prove `FiniteDimensional ℝ E`
   · refine' o.eq_or_eq_neg_of_isEmpty.elim _ _ <;> rintro rfl <;> simp
-  haveI : FiniteDimensional ℝ E := fact_finiteDimensional_of_finrank_eq_succ n
+  haveI : FiniteDimensional ℝ E := .of_fact_finrank_eq_succ n
   convert o.volumeForm_map φ (φ ∘ x)
   · symm
     rwa [← o.map_eq_iff_det_pos φ.toLinearEquiv] at hφ

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.

@@ -93,7 +93,7 @@ theorem det_eq_neg_det_of_opposite_orientation (h : e.toBasis.orientation ≠ f.
   rw [e.toBasis.det.eq_smul_basis_det f.toBasis]
   -- Porting note: added `neg_one_smul` with explicit type
   simp [e.det_to_matrix_orthonormalBasis_of_opposite_orientation f h,
-    neg_one_smul ℝ (M := AlternatingMap ℝ E ℝ ι)]
+    neg_one_smul ℝ (M := E [Λ^ι]→ₗ[ℝ] ℝ)]
 #align orthonormal_basis.det_eq_neg_det_of_opposite_orientation OrthonormalBasis.det_eq_neg_det_of_opposite_orientation
 
 section AdjustToOrientation
@@ -178,10 +178,10 @@ variable [_i : Fact (finrank ℝ E = n)] (o : Orientation ℝ E (Fin n))
 /-- The volume form on an oriented real inner product space, a nonvanishing top-dimensional
 alternating form uniquely defined by compatibility with the orientation and inner product structure.
 -/
-irreducible_def volumeForm : AlternatingMap ℝ E ℝ (Fin n) := by
+irreducible_def volumeForm : E [Λ^Fin n]→ₗ[ℝ] ℝ := by
   classical
     cases' n with n
-    · let opos : AlternatingMap ℝ E ℝ (Fin 0) := AlternatingMap.constOfIsEmpty ℝ E (Fin 0) (1 : ℝ)
+    · let opos : E [Λ^Fin 0]→ₗ[ℝ] ℝ := .constOfIsEmpty ℝ E (Fin 0) (1 : ℝ)
       exact o.eq_or_eq_neg_of_isEmpty.by_cases (fun _ => opos) fun _ => -opos
     · exact (o.finOrthonormalBasis n.succ_pos _i.out).toBasis.det
 #align orientation.volume_form Orientation.volumeForm

@@ -175,9 +175,6 @@ section VolumeForm
 
 variable [_i : Fact (finrank ℝ E = n)] (o : Orientation ℝ E (Fin n))
 
--- Porting note: added instance
-instance : IsEmpty (Fin Nat.zero) := by simp only [Nat.zero_eq]; infer_instance
-
 /-- The volume form on an oriented real inner product space, a nonvanishing top-dimensional
 alternating form uniquely defined by compatibility with the orientation and inner product structure.
 -/

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

This has nice performance benefits.

@@ -38,7 +38,7 @@ This file provides definitions and proves lemmas about orientations of real inne
 
 noncomputable section
 
-variable {E : Type _} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
+variable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
 
 open FiniteDimensional
 
@@ -46,7 +46,7 @@ open scoped BigOperators RealInnerProductSpace
 
 namespace OrthonormalBasis
 
-variable {ι : Type _} [Fintype ι] [DecidableEq ι] [ne : Nonempty ι] (e f : OrthonormalBasis ι ℝ E)
+variable {ι : Type*} [Fintype ι] [DecidableEq ι] [ne : Nonempty ι] (e f : OrthonormalBasis ι ℝ E)
   (x : Orientation ℝ E ι)
 
 /-- The change-of-basis matrix between two orthonormal bases with the same orientation has
@@ -315,7 +315,7 @@ theorem abs_volumeForm_apply_of_orthonormal (v : OrthonormalBasis (Fin n) ℝ E)
   simpa [o.volumeForm_robust' v v] using congr_arg abs v.toBasis.det_self
 #align orientation.abs_volume_form_apply_of_orthonormal Orientation.abs_volumeForm_apply_of_orthonormal
 
-theorem volumeForm_map {F : Type _} [NormedAddCommGroup F] [InnerProductSpace ℝ F]
+theorem volumeForm_map {F : Type*} [NormedAddCommGroup F] [InnerProductSpace ℝ F]
     [Fact (finrank ℝ F = n)] (φ : E ≃ₗᵢ[ℝ] F) (x : Fin n → F) :
     (Orientation.map (Fin n) φ.toLinearEquiv o).volumeForm x = o.volumeForm (φ.symm ∘ x) := by
   cases' n with n

• 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) 2022 Joseph Myers. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Joseph Myers, Heather Macbeth
-
-! This file was ported from Lean 3 source module analysis.inner_product_space.orientation
-! leanprover-community/mathlib commit bd65478311e4dfd41f48bf38c7e3b02fb75d0163
-! Please do not edit these lines, except to modify the commit id
-! if you have ported upstream changes.
 -/
 import Mathlib.Analysis.InnerProductSpace.GramSchmidtOrtho
 import Mathlib.LinearAlgebra.Orientation
 
+#align_import analysis.inner_product_space.orientation from "leanprover-community/mathlib"@"bd65478311e4dfd41f48bf38c7e3b02fb75d0163"
+
 /-!
 # Orientations of real inner product spaces.
 

Co-authored-by: Moritz Firsching <firsching@google.com> Co-authored-by: Ruben Van de Velde <65514131+Ruben-VandeVelde@users.noreply.github.com> Co-authored-by: Jeremy Tan Jie Rui <reddeloostw@gmail.com> Co-authored-by: Parcly Taxel <reddeloostw@gmail.com>