analysis.inner_product_space.gram_schmidt_ortho ⟷ Mathlib.Analysis.InnerProductSpace.GramSchmidtOrtho

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

@@ -44,7 +44,7 @@ open scoped BigOperators
 
 open Finset Submodule FiniteDimensional
 
-variable (𝕜 : Type _) {E : Type _} [IsROrC 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E]
+variable (𝕜 : Type _) {E : Type _} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E]
 
 variable {ι : Type _} [LinearOrder ι] [LocallyFiniteOrderBot ι] [IsWellOrder ι (· < ·)]
 
@@ -118,7 +118,7 @@ theorem gramSchmidt_orthogonal (f : ι → E) {a b : ι} (h₀ : a ≠ b) :
   rw [Finset.sum_eq_single_of_mem a (finset.mem_Iio.mpr h₀)]
   · by_cases h : gramSchmidt 𝕜 f a = 0
     · simp only [h, inner_zero_left, zero_div, MulZeroClass.zero_mul, sub_zero]
-    · rw [← inner_self_eq_norm_sq_to_K, div_mul_cancel, sub_self]
+    · rw [← inner_self_eq_norm_sq_to_K, div_mul_cancel₀, sub_self]
       rwa [inner_self_ne_zero]
   simp_intro i hi hia only [Finset.mem_range]
   simp only [mul_eq_zero, div_eq_zero_iff, inner_self_eq_zero]
@@ -361,8 +361,8 @@ theorem gramSchmidt_orthonormal {f : ι → E} (h₀ : LinearIndependent 𝕜 f)
   constructor
   · simp only [gramSchmidtNormed_unit_length, h₀, eq_self_iff_true, imp_true_iff]
   · intro i j hij
-    simp only [gramSchmidtNormed, inner_smul_left, inner_smul_right, IsROrC.conj_inv,
-      IsROrC.conj_ofReal, mul_eq_zero, inv_eq_zero, IsROrC.ofReal_eq_zero, norm_eq_zero]
+    simp only [gramSchmidtNormed, inner_smul_left, inner_smul_right, RCLike.conj_inv,
+      RCLike.conj_ofReal, mul_eq_zero, inv_eq_zero, RCLike.ofReal_eq_zero, norm_eq_zero]
     repeat' right
     exact gramSchmidt_orthogonal 𝕜 f hij
 #align gram_schmidt_orthonormal gramSchmidt_orthonormal

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

@@ -252,7 +252,7 @@ theorem gramSchmidt_ne_zero_coe {f : ι → E} (n : ι)
     simp_intro a ha only [Finset.mem_Ico]
     simp only [Set.mem_image, Set.mem_Iio, orthogonalProjection_singleton]
     apply Submodule.smul_mem _ _ _
-    rw [Finset.mem_Iio] at ha 
+    rw [Finset.mem_Iio] at ha
     refine' subset_span ⟨a, ha, by rfl⟩
   have h₂ :
     (f ∘ (coe : Set.Iic n → ι)) ⟨n, le_refl n⟩ ∈
@@ -377,7 +377,7 @@ theorem gramSchmidt_orthonormal' (f : ι → E) :
   by
   refine' ⟨fun i => gramSchmidtNormed_unit_length' i.Prop, _⟩
   rintro i j (hij : ¬_)
-  rw [Subtype.ext_iff] at hij 
+  rw [Subtype.ext_iff] at hij
   simp [gramSchmidtNormed, inner_smul_left, inner_smul_right, gramSchmidt_orthogonal 𝕜 f hij]
 #align gram_schmidt_orthonormal' gramSchmidt_orthonormal'
 -/

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

@@ -223,7 +223,7 @@ theorem gramSchmidt_of_orthogonal {f : ι → E} (hf : Pairwise fun i j => ⟪f
   · congr
     apply Finset.sum_eq_zero
     intro j hj
-    rw [coe_eq_zero]
+    rw [NNReal.coe_eq_zero]
     suffices span 𝕜 (f '' Set.Iic j) ⟂ 𝕜 ∙ f i
       by
       apply orthogonalProjection_mem_subspace_orthogonalComplement_eq_zero

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

@@ -3,8 +3,8 @@ Copyright (c) 2022 Jiale Miao. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Jiale Miao, Kevin Buzzard, Alexander Bentkamp
 -/
-import Mathbin.Analysis.InnerProductSpace.PiL2
-import Mathbin.LinearAlgebra.Matrix.Block
+import Analysis.InnerProductSpace.PiL2
+import LinearAlgebra.Matrix.Block
 
 #align_import analysis.inner_product_space.gram_schmidt_ortho from "leanprover-community/mathlib"@"61b5e2755ccb464b68d05a9acf891ae04992d09d"
 

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

@@ -2,15 +2,12 @@
 Copyright (c) 2022 Jiale Miao. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Jiale Miao, Kevin Buzzard, Alexander Bentkamp
-
-! This file was ported from Lean 3 source module analysis.inner_product_space.gram_schmidt_ortho
-! leanprover-community/mathlib commit 61b5e2755ccb464b68d05a9acf891ae04992d09d
-! Please do not edit these lines, except to modify the commit id
-! if you have ported upstream changes.
 -/
 import Mathbin.Analysis.InnerProductSpace.PiL2
 import Mathbin.LinearAlgebra.Matrix.Block
 
+#align_import analysis.inner_product_space.gram_schmidt_ortho from "leanprover-community/mathlib"@"61b5e2755ccb464b68d05a9acf891ae04992d09d"
+
 /-!
 # Gram-Schmidt Orthogonalization and Orthonormalization
 

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

@@ -53,7 +53,6 @@ variable {ι : Type _} [LinearOrder ι] [LocallyFiniteOrderBot ι] [IsWellOrder
 
 attribute [local instance] IsWellOrder.toHasWellFounded
 
--- mathport name: «expr⟪ , ⟫»
 local notation "⟪" x ", " y "⟫" => @inner 𝕜 _ _ x y
 
 #print gramSchmidt /-
@@ -65,17 +64,22 @@ decreasing_by exact mem_Iio.1 i.2
 #align gram_schmidt gramSchmidt
 -/
 
+#print gramSchmidt_def /-
 /-- This lemma uses `∑ i in` instead of `∑ i :`.-/
 theorem gramSchmidt_def (f : ι → E) (n : ι) :
     gramSchmidt 𝕜 f n = f n - ∑ i in Iio n, orthogonalProjection (𝕜 ∙ gramSchmidt 𝕜 f i) (f n) := by
   rw [← sum_attach, attach_eq_univ, gramSchmidt]; rfl
 #align gram_schmidt_def gramSchmidt_def
+-/
 
+#print gramSchmidt_def' /-
 theorem gramSchmidt_def' (f : ι → E) (n : ι) :
     f n = gramSchmidt 𝕜 f n + ∑ i in Iio n, orthogonalProjection (𝕜 ∙ gramSchmidt 𝕜 f i) (f n) := by
   rw [gramSchmidt_def, sub_add_cancel]
 #align gram_schmidt_def' gramSchmidt_def'
+-/
 
+#print gramSchmidt_def'' /-
 theorem gramSchmidt_def'' (f : ι → E) (n : ι) :
     f n =
       gramSchmidt 𝕜 f n +
@@ -85,13 +89,17 @@ theorem gramSchmidt_def'' (f : ι → E) (n : ι) :
   ext i
   rw [orthogonalProjection_singleton]
 #align gram_schmidt_def'' gramSchmidt_def''
+-/
 
+#print gramSchmidt_zero /-
 @[simp]
 theorem gramSchmidt_zero {ι : Type _} [LinearOrder ι] [LocallyFiniteOrder ι] [OrderBot ι]
     [IsWellOrder ι (· < ·)] (f : ι → E) : gramSchmidt 𝕜 f ⊥ = f ⊥ := by
   rw [gramSchmidt_def, Iio_eq_Ico, Finset.Ico_self, Finset.sum_empty, sub_zero]
 #align gram_schmidt_zero gramSchmidt_zero
+-/
 
+#print gramSchmidt_orthogonal /-
 /-- **Gram-Schmidt Orthogonalisation**:
 `gram_schmidt` produces an orthogonal system of vectors. -/
 theorem gramSchmidt_orthogonal (f : ι → E) {a b : ι} (h₀ : a ≠ b) :
@@ -123,13 +131,17 @@ theorem gramSchmidt_orthogonal (f : ι → E) {a b : ι} (h₀ : a ≠ b) :
     exact ih a h₀ i hia₁
   · exact ih i (mem_Iio.1 hi) a hia₂
 #align gram_schmidt_orthogonal gramSchmidt_orthogonal
+-/
 
+#print gramSchmidt_pairwise_orthogonal /-
 /-- This is another version of `gram_schmidt_orthogonal` using `pairwise` instead. -/
 theorem gramSchmidt_pairwise_orthogonal (f : ι → E) :
     Pairwise fun a b => ⟪gramSchmidt 𝕜 f a, gramSchmidt 𝕜 f b⟫ = 0 := fun a b =>
   gramSchmidt_orthogonal 𝕜 f
 #align gram_schmidt_pairwise_orthogonal gramSchmidt_pairwise_orthogonal
+-/
 
+#print gramSchmidt_inv_triangular /-
 theorem gramSchmidt_inv_triangular (v : ι → E) {i j : ι} (hij : i < j) :
     ⟪gramSchmidt 𝕜 v j, v i⟫ = 0 := by
   rw [gramSchmidt_def'' 𝕜 v]
@@ -143,6 +155,7 @@ theorem gramSchmidt_inv_triangular (v : ι → E) {i j : ι} (hij : i < j) :
   have : ⟪b j, b k⟫ = 0 := gramSchmidt_orthogonal 𝕜 v (hki.trans hij).ne'
   simp [this]
 #align gram_schmidt_inv_triangular gramSchmidt_inv_triangular
+-/
 
 open Submodule Set Order
 
@@ -174,12 +187,15 @@ theorem gramSchmidt_mem_span (f : ι → E) : ∀ {j i}, i ≤ j → gramSchmidt
 #align gram_schmidt_mem_span gramSchmidt_mem_span
 -/
 
+#print span_gramSchmidt_Iic /-
 theorem span_gramSchmidt_Iic (f : ι → E) (c : ι) :
     span 𝕜 (gramSchmidt 𝕜 f '' Iic c) = span 𝕜 (f '' Iic c) :=
   span_eq_span (Set.image_subset_iff.2 fun i => gramSchmidt_mem_span _ _) <|
     Set.image_subset_iff.2 fun i => mem_span_gramSchmidt _ _
 #align span_gram_schmidt_Iic span_gramSchmidt_Iic
+-/
 
+#print span_gramSchmidt_Iio /-
 theorem span_gramSchmidt_Iio (f : ι → E) (c : ι) :
     span 𝕜 (gramSchmidt 𝕜 f '' Iio c) = span 𝕜 (f '' Iio c) :=
   span_eq_span
@@ -188,7 +204,9 @@ theorem span_gramSchmidt_Iio (f : ι → E) (c : ι) :
     Set.image_subset_iff.2 fun i hi =>
       span_mono (image_subset _ <| Iic_subset_Iio.2 hi) <| mem_span_gramSchmidt _ _ le_rfl
 #align span_gram_schmidt_Iio span_gramSchmidt_Iio
+-/
 
+#print span_gramSchmidt /-
 /-- `gram_schmidt` preserves span of vectors. -/
 theorem span_gramSchmidt (f : ι → E) : span 𝕜 (range (gramSchmidt 𝕜 f)) = span 𝕜 (range f) :=
   span_eq_span
@@ -197,7 +215,9 @@ theorem span_gramSchmidt (f : ι → E) : span 𝕜 (range (gramSchmidt 𝕜 f))
     range_subset_iff.2 fun i =>
       span_mono (image_subset_range _ _) <| mem_span_gramSchmidt _ _ le_rfl
 #align span_gram_schmidt span_gramSchmidt
+-/
 
+#print gramSchmidt_of_orthogonal /-
 theorem gramSchmidt_of_orthogonal {f : ι → E} (hf : Pairwise fun i j => ⟪f i, f j⟫ = 0) :
     gramSchmidt 𝕜 f = f := by
   ext i
@@ -219,9 +239,11 @@ theorem gramSchmidt_of_orthogonal {f : ι → E} (hf : Pairwise fun i j => ⟪f
     exact (lt_of_le_of_lt hk (finset.mem_Iio.mp hj)).Ne
   · simp
 #align gram_schmidt_of_orthogonal gramSchmidt_of_orthogonal
+-/
 
 variable {𝕜}
 
+#print gramSchmidt_ne_zero_coe /-
 theorem gramSchmidt_ne_zero_coe {f : ι → E} (n : ι)
     (h₀ : LinearIndependent 𝕜 (f ∘ (coe : Set.Iic n → ι))) : gramSchmidt 𝕜 f n ≠ 0 :=
   by
@@ -246,14 +268,18 @@ theorem gramSchmidt_ne_zero_coe {f : ι → E} (n : ι)
   apply LinearIndependent.not_mem_span_image h₀ _ h₂
   simp only [Set.mem_Iio, lt_self_iff_false, not_false_iff]
 #align gram_schmidt_ne_zero_coe gramSchmidt_ne_zero_coe
+-/
 
+#print gramSchmidt_ne_zero /-
 /-- If the input vectors of `gram_schmidt` are linearly independent,
 then the output vectors are non-zero. -/
 theorem gramSchmidt_ne_zero {f : ι → E} (n : ι) (h₀ : LinearIndependent 𝕜 f) :
     gramSchmidt 𝕜 f n ≠ 0 :=
   gramSchmidt_ne_zero_coe _ (LinearIndependent.comp h₀ _ Subtype.coe_injective)
 #align gram_schmidt_ne_zero gramSchmidt_ne_zero
+-/
 
+#print gramSchmidt_triangular /-
 /-- `gram_schmidt` produces a triangular matrix of vectors when given a basis. -/
 theorem gramSchmidt_triangular {i j : ι} (hij : i < j) (b : Basis ι 𝕜 E) :
     b.repr (gramSchmidt 𝕜 b i) j = 0 :=
@@ -265,13 +291,16 @@ theorem gramSchmidt_triangular {i j : ι} (hij : i < j) (b : Basis ι 𝕜 E) :
     Basis.repr_support_subset_of_mem_span b (Set.Iio j) this
   exact (Finsupp.mem_supported' _ _).1 ((Finsupp.mem_supported 𝕜 _).2 this) j Set.not_mem_Iio_self
 #align gram_schmidt_triangular gramSchmidt_triangular
+-/
 
+#print gramSchmidt_linearIndependent /-
 /-- `gram_schmidt` produces linearly independent vectors when given linearly independent vectors. -/
 theorem gramSchmidt_linearIndependent {f : ι → E} (h₀ : LinearIndependent 𝕜 f) :
     LinearIndependent 𝕜 (gramSchmidt 𝕜 f) :=
   linearIndependent_of_ne_zero_of_inner_eq_zero (fun i => gramSchmidt_ne_zero _ h₀) fun i j =>
     gramSchmidt_orthogonal 𝕜 f
 #align gram_schmidt_linear_independent gramSchmidt_linearIndependent
+-/
 
 #print gramSchmidtBasis /-
 /-- When given a basis, `gram_schmidt` produces a basis. -/
@@ -281,9 +310,11 @@ noncomputable def gramSchmidtBasis (b : Basis ι 𝕜 E) : Basis ι 𝕜 E :=
 #align gram_schmidt_basis gramSchmidtBasis
 -/
 
+#print coe_gramSchmidtBasis /-
 theorem coe_gramSchmidtBasis (b : Basis ι 𝕜 E) : (gramSchmidtBasis b : ι → E) = gramSchmidt 𝕜 b :=
   Basis.coe_mk _ _
 #align coe_gram_schmidt_basis coe_gramSchmidtBasis
+-/
 
 variable (𝕜)
 
@@ -297,17 +328,22 @@ noncomputable def gramSchmidtNormed (f : ι → E) (n : ι) : E :=
 
 variable {𝕜}
 
+#print gramSchmidtNormed_unit_length_coe /-
 theorem gramSchmidtNormed_unit_length_coe {f : ι → E} (n : ι)
     (h₀ : LinearIndependent 𝕜 (f ∘ (coe : Set.Iic n → ι))) : ‖gramSchmidtNormed 𝕜 f n‖ = 1 := by
   simp only [gramSchmidt_ne_zero_coe n h₀, gramSchmidtNormed, norm_smul_inv_norm, Ne.def,
     not_false_iff]
 #align gram_schmidt_normed_unit_length_coe gramSchmidtNormed_unit_length_coe
+-/
 
+#print gramSchmidtNormed_unit_length /-
 theorem gramSchmidtNormed_unit_length {f : ι → E} (n : ι) (h₀ : LinearIndependent 𝕜 f) :
     ‖gramSchmidtNormed 𝕜 f n‖ = 1 :=
   gramSchmidtNormed_unit_length_coe _ (LinearIndependent.comp h₀ _ Subtype.coe_injective)
 #align gram_schmidt_normed_unit_length gramSchmidtNormed_unit_length
+-/
 
+#print gramSchmidtNormed_unit_length' /-
 theorem gramSchmidtNormed_unit_length' {f : ι → E} {n : ι} (hn : gramSchmidtNormed 𝕜 f n ≠ 0) :
     ‖gramSchmidtNormed 𝕜 f n‖ = 1 :=
   by
@@ -315,7 +351,9 @@ theorem gramSchmidtNormed_unit_length' {f : ι → E} {n : ι} (hn : gramSchmidt
   rw [norm_smul_inv_norm]
   simpa using hn
 #align gram_schmidt_normed_unit_length' gramSchmidtNormed_unit_length'
+-/
 
+#print gramSchmidt_orthonormal /-
 /-- **Gram-Schmidt Orthonormalization**:
 `gram_schmidt_normed` applied to a linearly independent set of vectors produces an orthornormal
 system of vectors. -/
@@ -331,7 +369,9 @@ theorem gramSchmidt_orthonormal {f : ι → E} (h₀ : LinearIndependent 𝕜 f)
     repeat' right
     exact gramSchmidt_orthogonal 𝕜 f hij
 #align gram_schmidt_orthonormal gramSchmidt_orthonormal
+-/
 
+#print gramSchmidt_orthonormal' /-
 /-- **Gram-Schmidt Orthonormalization**:
 `gram_schmidt_normed` produces an orthornormal system of vectors after removing the vectors which
 become zero in the process. -/
@@ -343,6 +383,7 @@ theorem gramSchmidt_orthonormal' (f : ι → E) :
   rw [Subtype.ext_iff] at hij 
   simp [gramSchmidtNormed, inner_smul_left, inner_smul_right, gramSchmidt_orthogonal 𝕜 f hij]
 #align gram_schmidt_orthonormal' gramSchmidt_orthonormal'
+-/
 
 #print span_gramSchmidtNormed /-
 theorem span_gramSchmidtNormed (f : ι → E) (s : Set ι) :
@@ -361,17 +402,17 @@ theorem span_gramSchmidtNormed (f : ι → E) (s : Set ι) :
 #align span_gram_schmidt_normed span_gramSchmidtNormed
 -/
 
+#print span_gramSchmidtNormed_range /-
 theorem span_gramSchmidtNormed_range (f : ι → E) :
     span 𝕜 (range (gramSchmidtNormed 𝕜 f)) = span 𝕜 (range (gramSchmidt 𝕜 f)) := by
   simpa only [image_univ.symm] using span_gramSchmidtNormed f univ
 #align span_gram_schmidt_normed_range span_gramSchmidtNormed_range
+-/
 
 section OrthonormalBasis
 
 variable [Fintype ι] [FiniteDimensional 𝕜 E] (h : finrank 𝕜 E = Fintype.card ι) (f : ι → E)
 
-include h
-
 #print gramSchmidtOrthonormalBasis /-
 /-- Given an indexed family `f : ι → E` of vectors in an inner product space `E`, for which the
 size of the index set is the dimension of `E`, produce an orthonormal basis for `E` which agrees
@@ -382,11 +423,14 @@ noncomputable def gramSchmidtOrthonormalBasis : OrthonormalBasis ι 𝕜 E :=
 #align gram_schmidt_orthonormal_basis gramSchmidtOrthonormalBasis
 -/
 
+#print gramSchmidtOrthonormalBasis_apply /-
 theorem gramSchmidtOrthonormalBasis_apply {f : ι → E} {i : ι} (hi : gramSchmidtNormed 𝕜 f i ≠ 0) :
     gramSchmidtOrthonormalBasis h f i = gramSchmidtNormed 𝕜 f i :=
   ((gramSchmidt_orthonormal' f).exists_orthonormalBasis_extension_of_card_eq h).choose_spec i hi
 #align gram_schmidt_orthonormal_basis_apply gramSchmidtOrthonormalBasis_apply
+-/
 
+#print gramSchmidtOrthonormalBasis_apply_of_orthogonal /-
 theorem gramSchmidtOrthonormalBasis_apply_of_orthogonal {f : ι → E}
     (hf : Pairwise fun i j => ⟪f i, f j⟫ = 0) {i : ι} (hi : f i ≠ 0) :
     gramSchmidtOrthonormalBasis h f i = (‖f i‖⁻¹ : 𝕜) • f i :=
@@ -396,7 +440,9 @@ theorem gramSchmidtOrthonormalBasis_apply_of_orthogonal {f : ι → E}
   rw [gramSchmidtOrthonormalBasis_apply h, H]
   simpa [H] using hi
 #align gram_schmidt_orthonormal_basis_apply_of_orthogonal gramSchmidtOrthonormalBasis_apply_of_orthogonal
+-/
 
+#print inner_gramSchmidtOrthonormalBasis_eq_zero /-
 theorem inner_gramSchmidtOrthonormalBasis_eq_zero {f : ι → E} {i : ι}
     (hi : gramSchmidtNormed 𝕜 f i = 0) (j : ι) : ⟪gramSchmidtOrthonormalBasis h f i, f j⟫ = 0 :=
   by
@@ -416,7 +462,9 @@ theorem inner_gramSchmidtOrthonormalBasis_eq_zero {f : ι → E} {i : ι}
     exact hk hi
   exact (gramSchmidtOrthonormalBasis h f).Orthonormal.2 this
 #align inner_gram_schmidt_orthonormal_basis_eq_zero inner_gramSchmidtOrthonormalBasis_eq_zero
+-/
 
+#print gramSchmidtOrthonormalBasis_inv_triangular /-
 theorem gramSchmidtOrthonormalBasis_inv_triangular {i j : ι} (hij : i < j) :
     ⟪gramSchmidtOrthonormalBasis h f j, f i⟫ = 0 :=
   by
@@ -426,12 +474,16 @@ theorem gramSchmidtOrthonormalBasis_inv_triangular {i j : ι} (hij : i < j) :
     simp [gramSchmidtOrthonormalBasis_apply h hi, gramSchmidtNormed, inner_smul_left,
       gramSchmidt_inv_triangular 𝕜 f hij]
 #align gram_schmidt_orthonormal_basis_inv_triangular gramSchmidtOrthonormalBasis_inv_triangular
+-/
 
+#print gramSchmidtOrthonormalBasis_inv_triangular' /-
 theorem gramSchmidtOrthonormalBasis_inv_triangular' {i j : ι} (hij : i < j) :
     (gramSchmidtOrthonormalBasis h f).repr (f i) j = 0 := by
   simpa [OrthonormalBasis.repr_apply_apply] using gramSchmidtOrthonormalBasis_inv_triangular h f hij
 #align gram_schmidt_orthonormal_basis_inv_triangular' gramSchmidtOrthonormalBasis_inv_triangular'
+-/
 
+#print gramSchmidtOrthonormalBasis_inv_blockTriangular /-
 /-- Given an indexed family `f : ι → E` of vectors in an inner product space `E`, for which the
 size of the index set is the dimension of `E`, the matrix of coefficients of `f` with respect to the
 orthonormal basis `gram_schmidt_orthonormal_basis` constructed from `f` is upper-triangular. -/
@@ -439,7 +491,9 @@ theorem gramSchmidtOrthonormalBasis_inv_blockTriangular :
     ((gramSchmidtOrthonormalBasis h f).toBasis.toMatrix f).BlockTriangular id := fun i j =>
   gramSchmidtOrthonormalBasis_inv_triangular' h f
 #align gram_schmidt_orthonormal_basis_inv_block_triangular gramSchmidtOrthonormalBasis_inv_blockTriangular
+-/
 
+#print gramSchmidtOrthonormalBasis_det /-
 theorem gramSchmidtOrthonormalBasis_det :
     (gramSchmidtOrthonormalBasis h f).toBasis.det f =
       ∏ i, ⟪gramSchmidtOrthonormalBasis h f i, f i⟫ :=
@@ -448,6 +502,7 @@ theorem gramSchmidtOrthonormalBasis_det :
   ext i
   exact ((gramSchmidtOrthonormalBasis h f).repr_apply_apply (f i) i).symm
 #align gram_schmidt_orthonormal_basis_det gramSchmidtOrthonormalBasis_det
+-/
 
 end OrthonormalBasis
 

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

@@ -336,7 +336,7 @@ theorem gramSchmidt_orthonormal {f : ι → E} (h₀ : LinearIndependent 𝕜 f)
 `gram_schmidt_normed` produces an orthornormal system of vectors after removing the vectors which
 become zero in the process. -/
 theorem gramSchmidt_orthonormal' (f : ι → E) :
-    Orthonormal 𝕜 fun i : { i | gramSchmidtNormed 𝕜 f i ≠ 0 } => gramSchmidtNormed 𝕜 f i :=
+    Orthonormal 𝕜 fun i : {i | gramSchmidtNormed 𝕜 f i ≠ 0} => gramSchmidtNormed 𝕜 f i :=
   by
   refine' ⟨fun i => gramSchmidtNormed_unit_length' i.Prop, _⟩
   rintro i j (hij : ¬_)

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

@@ -60,8 +60,8 @@ local notation "⟪" x ", " y "⟫" => @inner 𝕜 _ _ x y
 /-- The Gram-Schmidt process takes a set of vectors as input
 and outputs a set of orthogonal vectors which have the same span. -/
 noncomputable def gramSchmidt (f : ι → E) : ι → E
-  | n => f n - ∑ i : Iio n, orthogonalProjection (𝕜 ∙ gramSchmidt i) (f n)decreasing_by
-  exact mem_Iio.1 i.2
+  | n => f n - ∑ i : Iio n, orthogonalProjection (𝕜 ∙ gramSchmidt i) (f n)
+decreasing_by exact mem_Iio.1 i.2
 #align gram_schmidt gramSchmidt
 -/
 
@@ -233,7 +233,7 @@ theorem gramSchmidt_ne_zero_coe {f : ι → E} (n : ι)
     simp_intro a ha only [Finset.mem_Ico]
     simp only [Set.mem_image, Set.mem_Iio, orthogonalProjection_singleton]
     apply Submodule.smul_mem _ _ _
-    rw [Finset.mem_Iio] at ha
+    rw [Finset.mem_Iio] at ha 
     refine' subset_span ⟨a, ha, by rfl⟩
   have h₂ :
     (f ∘ (coe : Set.Iic n → ι)) ⟨n, le_refl n⟩ ∈
@@ -340,7 +340,7 @@ theorem gramSchmidt_orthonormal' (f : ι → E) :
   by
   refine' ⟨fun i => gramSchmidtNormed_unit_length' i.Prop, _⟩
   rintro i j (hij : ¬_)
-  rw [Subtype.ext_iff] at hij
+  rw [Subtype.ext_iff] at hij 
   simp [gramSchmidtNormed, inner_smul_left, inner_smul_right, gramSchmidt_orthogonal 𝕜 f hij]
 #align gram_schmidt_orthonormal' gramSchmidt_orthonormal'
 

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: Jiale Miao, Kevin Buzzard, Alexander Bentkamp
 
 ! This file was ported from Lean 3 source module analysis.inner_product_space.gram_schmidt_ortho
-! leanprover-community/mathlib commit 1a4df69ca1a9a0e5e26bfe12e2b92814216016d0
+! leanprover-community/mathlib commit 61b5e2755ccb464b68d05a9acf891ae04992d09d
 ! 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.Matrix.Block
 /-!
 # Gram-Schmidt Orthogonalization and Orthonormalization
 
+> THIS FILE IS SYNCHRONIZED WITH MATHLIB4.
+> Any changes to this file require a corresponding PR to mathlib4.
+
 In this file we introduce Gram-Schmidt Orthogonalization and Orthonormalization.
 
 The Gram-Schmidt process takes a set of vectors as input
@@ -53,12 +56,14 @@ attribute [local instance] IsWellOrder.toHasWellFounded
 -- mathport name: «expr⟪ , ⟫»
 local notation "⟪" x ", " y "⟫" => @inner 𝕜 _ _ x y
 
+#print gramSchmidt /-
 /-- The Gram-Schmidt process takes a set of vectors as input
 and outputs a set of orthogonal vectors which have the same span. -/
 noncomputable def gramSchmidt (f : ι → E) : ι → E
   | n => f n - ∑ i : Iio n, orthogonalProjection (𝕜 ∙ gramSchmidt i) (f n)decreasing_by
   exact mem_Iio.1 i.2
 #align gram_schmidt gramSchmidt
+-/
 
 /-- This lemma uses `∑ i in` instead of `∑ i :`.-/
 theorem gramSchmidt_def (f : ι → E) (n : ι) :
@@ -141,6 +146,7 @@ theorem gramSchmidt_inv_triangular (v : ι → E) {i j : ι} (hij : i < j) :
 
 open Submodule Set Order
 
+#print mem_span_gramSchmidt /-
 theorem mem_span_gramSchmidt (f : ι → E) {i j : ι} (hij : i ≤ j) :
     f i ∈ span 𝕜 (gramSchmidt 𝕜 f '' Iic j) :=
   by
@@ -152,7 +158,9 @@ theorem mem_span_gramSchmidt (f : ι → E) {i j : ι} (hij : i ≤ j) :
         smul_mem (span 𝕜 (gramSchmidt 𝕜 f '' Iic j)) _ <|
           subset_span <| mem_image_of_mem (gramSchmidt 𝕜 f) <| (Finset.mem_Iio.1 hk).le.trans hij)
 #align mem_span_gram_schmidt mem_span_gramSchmidt
+-/
 
+#print gramSchmidt_mem_span /-
 theorem gramSchmidt_mem_span (f : ι → E) : ∀ {j i}, i ≤ j → gramSchmidt 𝕜 f i ∈ span 𝕜 (f '' Iic j)
   | j => fun i hij => by
     rw [gramSchmidt_def 𝕜 f i]
@@ -164,6 +172,7 @@ theorem gramSchmidt_mem_span (f : ι → E) : ∀ {j i}, i ≤ j → gramSchmidt
       smul_mem _ _
         (span_mono (image_subset f <| Iic_subset_Iic.2 hkj.le) <| gramSchmidt_mem_span le_rfl)
 #align gram_schmidt_mem_span gramSchmidt_mem_span
+-/
 
 theorem span_gramSchmidt_Iic (f : ι → E) (c : ι) :
     span 𝕜 (gramSchmidt 𝕜 f '' Iic c) = span 𝕜 (f '' Iic c) :=
@@ -264,11 +273,13 @@ theorem gramSchmidt_linearIndependent {f : ι → E} (h₀ : LinearIndependent 
     gramSchmidt_orthogonal 𝕜 f
 #align gram_schmidt_linear_independent gramSchmidt_linearIndependent
 
+#print gramSchmidtBasis /-
 /-- When given a basis, `gram_schmidt` produces a basis. -/
 noncomputable def gramSchmidtBasis (b : Basis ι 𝕜 E) : Basis ι 𝕜 E :=
   Basis.mk (gramSchmidt_linearIndependent b.LinearIndependent)
     ((span_gramSchmidt 𝕜 b).trans b.span_eq).ge
 #align gram_schmidt_basis gramSchmidtBasis
+-/
 
 theorem coe_gramSchmidtBasis (b : Basis ι 𝕜 E) : (gramSchmidtBasis b : ι → E) = gramSchmidt 𝕜 b :=
   Basis.coe_mk _ _
@@ -276,11 +287,13 @@ theorem coe_gramSchmidtBasis (b : Basis ι 𝕜 E) : (gramSchmidtBasis b : ι 
 
 variable (𝕜)
 
+#print gramSchmidtNormed /-
 /-- the normalized `gram_schmidt`
 (i.e each vector in `gram_schmidt_normed` has unit length.) -/
 noncomputable def gramSchmidtNormed (f : ι → E) (n : ι) : E :=
   (‖gramSchmidt 𝕜 f n‖ : 𝕜)⁻¹ • gramSchmidt 𝕜 f n
 #align gram_schmidt_normed gramSchmidtNormed
+-/
 
 variable {𝕜}
 
@@ -306,7 +319,7 @@ theorem gramSchmidtNormed_unit_length' {f : ι → E} {n : ι} (hn : gramSchmidt
 /-- **Gram-Schmidt Orthonormalization**:
 `gram_schmidt_normed` applied to a linearly independent set of vectors produces an orthornormal
 system of vectors. -/
-theorem gram_schmidt_orthonormal {f : ι → E} (h₀ : LinearIndependent 𝕜 f) :
+theorem gramSchmidt_orthonormal {f : ι → E} (h₀ : LinearIndependent 𝕜 f) :
     Orthonormal 𝕜 (gramSchmidtNormed 𝕜 f) :=
   by
   unfold Orthonormal
@@ -317,20 +330,21 @@ theorem gram_schmidt_orthonormal {f : ι → E} (h₀ : LinearIndependent 𝕜 f
       IsROrC.conj_ofReal, mul_eq_zero, inv_eq_zero, IsROrC.ofReal_eq_zero, norm_eq_zero]
     repeat' right
     exact gramSchmidt_orthogonal 𝕜 f hij
-#align gram_schmidt_orthonormal gram_schmidt_orthonormal
+#align gram_schmidt_orthonormal gramSchmidt_orthonormal
 
 /-- **Gram-Schmidt Orthonormalization**:
 `gram_schmidt_normed` produces an orthornormal system of vectors after removing the vectors which
 become zero in the process. -/
-theorem gram_schmidt_orthonormal' (f : ι → E) :
+theorem gramSchmidt_orthonormal' (f : ι → E) :
     Orthonormal 𝕜 fun i : { i | gramSchmidtNormed 𝕜 f i ≠ 0 } => gramSchmidtNormed 𝕜 f i :=
   by
   refine' ⟨fun i => gramSchmidtNormed_unit_length' i.Prop, _⟩
   rintro i j (hij : ¬_)
   rw [Subtype.ext_iff] at hij
   simp [gramSchmidtNormed, inner_smul_left, inner_smul_right, gramSchmidt_orthogonal 𝕜 f hij]
-#align gram_schmidt_orthonormal' gram_schmidt_orthonormal'
+#align gram_schmidt_orthonormal' gramSchmidt_orthonormal'
 
+#print span_gramSchmidtNormed /-
 theorem span_gramSchmidtNormed (f : ι → E) (s : Set ι) :
     span 𝕜 (gramSchmidtNormed 𝕜 f '' s) = span 𝕜 (gramSchmidt 𝕜 f '' s) :=
   by
@@ -345,6 +359,7 @@ theorem span_gramSchmidtNormed (f : ι → E) (s : Set ι) :
   · refine' mem_span_singleton.2 ⟨‖gramSchmidt 𝕜 f i‖, smul_inv_smul₀ _ _⟩
     exact_mod_cast norm_ne_zero_iff.2 h
 #align span_gram_schmidt_normed span_gramSchmidtNormed
+-/
 
 theorem span_gramSchmidtNormed_range (f : ι → E) :
     span 𝕜 (range (gramSchmidtNormed 𝕜 f)) = span 𝕜 (range (gramSchmidt 𝕜 f)) := by
@@ -357,17 +372,19 @@ variable [Fintype ι] [FiniteDimensional 𝕜 E] (h : finrank 𝕜 E = Fintype.c
 
 include h
 
+#print gramSchmidtOrthonormalBasis /-
 /-- Given an indexed family `f : ι → E` of vectors in an inner product space `E`, for which the
 size of the index set is the dimension of `E`, produce an orthonormal basis for `E` which agrees
 with the orthonormal set produced by the Gram-Schmidt orthonormalization process on the elements of
 `ι` for which this process gives a nonzero number. -/
 noncomputable def gramSchmidtOrthonormalBasis : OrthonormalBasis ι 𝕜 E :=
-  ((gram_schmidt_orthonormal' f).exists_orthonormalBasis_extension_of_card_eq h).some
+  ((gramSchmidt_orthonormal' f).exists_orthonormalBasis_extension_of_card_eq h).some
 #align gram_schmidt_orthonormal_basis gramSchmidtOrthonormalBasis
+-/
 
 theorem gramSchmidtOrthonormalBasis_apply {f : ι → E} {i : ι} (hi : gramSchmidtNormed 𝕜 f i ≠ 0) :
     gramSchmidtOrthonormalBasis h f i = gramSchmidtNormed 𝕜 f i :=
-  ((gram_schmidt_orthonormal' f).exists_orthonormalBasis_extension_of_card_eq h).choose_spec i hi
+  ((gramSchmidt_orthonormal' f).exists_orthonormalBasis_extension_of_card_eq h).choose_spec i hi
 #align gram_schmidt_orthonormal_basis_apply gramSchmidtOrthonormalBasis_apply
 
 theorem gramSchmidtOrthonormalBasis_apply_of_orthogonal {f : ι → E}

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

@@ -40,7 +40,7 @@ and outputs a set of orthogonal vectors which have the same span.
 -/
 
 
-open BigOperators
+open scoped BigOperators
 
 open Finset Submodule FiniteDimensional
 

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

@@ -62,10 +62,8 @@ noncomputable def gramSchmidt (f : ι → E) : ι → E
 
 /-- This lemma uses `∑ i in` instead of `∑ i :`.-/
 theorem gramSchmidt_def (f : ι → E) (n : ι) :
-    gramSchmidt 𝕜 f n = f n - ∑ i in Iio n, orthogonalProjection (𝕜 ∙ gramSchmidt 𝕜 f i) (f n) :=
-  by
-  rw [← sum_attach, attach_eq_univ, gramSchmidt]
-  rfl
+    gramSchmidt 𝕜 f n = f n - ∑ i in Iio n, orthogonalProjection (𝕜 ∙ gramSchmidt 𝕜 f i) (f n) := by
+  rw [← sum_attach, attach_eq_univ, gramSchmidt]; rfl
 #align gram_schmidt_def gramSchmidt_def
 
 theorem gramSchmidt_def' (f : ι → E) (n : ι) :

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

@@ -202,7 +202,7 @@ theorem gramSchmidt_of_orthogonal {f : ι → E} (hf : Pairwise fun i j => ⟪f
     rw [coe_eq_zero]
     suffices span 𝕜 (f '' Set.Iic j) ⟂ 𝕜 ∙ f i
       by
-      apply orthogonalProjection_mem_subspace_orthogonal_complement_eq_zero
+      apply orthogonalProjection_mem_subspace_orthogonalComplement_eq_zero
       rw [mem_orthogonal_singleton_iff_inner_left]
       rw [← mem_orthogonal_singleton_iff_inner_right]
       exact this (gramSchmidt_mem_span 𝕜 f (le_refl j))

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

@@ -308,7 +308,7 @@ theorem gramSchmidtNormed_unit_length' {f : ι → E} {n : ι} (hn : gramSchmidt
 /-- **Gram-Schmidt Orthonormalization**:
 `gram_schmidt_normed` applied to a linearly independent set of vectors produces an orthornormal
 system of vectors. -/
-theorem gramSchmidtOrthonormal {f : ι → E} (h₀ : LinearIndependent 𝕜 f) :
+theorem gram_schmidt_orthonormal {f : ι → E} (h₀ : LinearIndependent 𝕜 f) :
     Orthonormal 𝕜 (gramSchmidtNormed 𝕜 f) :=
   by
   unfold Orthonormal
@@ -319,19 +319,19 @@ theorem gramSchmidtOrthonormal {f : ι → E} (h₀ : LinearIndependent 𝕜 f)
       IsROrC.conj_ofReal, mul_eq_zero, inv_eq_zero, IsROrC.ofReal_eq_zero, norm_eq_zero]
     repeat' right
     exact gramSchmidt_orthogonal 𝕜 f hij
-#align gram_schmidt_orthonormal gramSchmidtOrthonormal
+#align gram_schmidt_orthonormal gram_schmidt_orthonormal
 
 /-- **Gram-Schmidt Orthonormalization**:
 `gram_schmidt_normed` produces an orthornormal system of vectors after removing the vectors which
 become zero in the process. -/
-theorem gramSchmidtOrthonormal' (f : ι → E) :
+theorem gram_schmidt_orthonormal' (f : ι → E) :
     Orthonormal 𝕜 fun i : { i | gramSchmidtNormed 𝕜 f i ≠ 0 } => gramSchmidtNormed 𝕜 f i :=
   by
   refine' ⟨fun i => gramSchmidtNormed_unit_length' i.Prop, _⟩
   rintro i j (hij : ¬_)
   rw [Subtype.ext_iff] at hij
   simp [gramSchmidtNormed, inner_smul_left, inner_smul_right, gramSchmidt_orthogonal 𝕜 f hij]
-#align gram_schmidt_orthonormal' gramSchmidtOrthonormal'
+#align gram_schmidt_orthonormal' gram_schmidt_orthonormal'
 
 theorem span_gramSchmidtNormed (f : ι → E) (s : Set ι) :
     span 𝕜 (gramSchmidtNormed 𝕜 f '' s) = span 𝕜 (gramSchmidt 𝕜 f '' s) :=
@@ -364,12 +364,12 @@ size of the index set is the dimension of `E`, produce an orthonormal basis for
 with the orthonormal set produced by the Gram-Schmidt orthonormalization process on the elements of
 `ι` for which this process gives a nonzero number. -/
 noncomputable def gramSchmidtOrthonormalBasis : OrthonormalBasis ι 𝕜 E :=
-  ((gramSchmidtOrthonormal' f).exists_orthonormalBasis_extension_of_card_eq h).some
+  ((gram_schmidt_orthonormal' f).exists_orthonormalBasis_extension_of_card_eq h).some
 #align gram_schmidt_orthonormal_basis gramSchmidtOrthonormalBasis
 
 theorem gramSchmidtOrthonormalBasis_apply {f : ι → E} {i : ι} (hi : gramSchmidtNormed 𝕜 f i ≠ 0) :
     gramSchmidtOrthonormalBasis h f i = gramSchmidtNormed 𝕜 f i :=
-  ((gramSchmidtOrthonormal' f).exists_orthonormalBasis_extension_of_card_eq h).choose_spec i hi
+  ((gram_schmidt_orthonormal' f).exists_orthonormalBasis_extension_of_card_eq h).choose_spec i hi
 #align gram_schmidt_orthonormal_basis_apply gramSchmidtOrthonormalBasis_apply
 
 theorem gramSchmidtOrthonormalBasis_apply_of_orthogonal {f : ι → E}

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

@@ -308,7 +308,7 @@ theorem gramSchmidtNormed_unit_length' {f : ι → E} {n : ι} (hn : gramSchmidt
 /-- **Gram-Schmidt Orthonormalization**:
 `gram_schmidt_normed` applied to a linearly independent set of vectors produces an orthornormal
 system of vectors. -/
-theorem gram_schmidt_orthonormal {f : ι → E} (h₀ : LinearIndependent 𝕜 f) :
+theorem gramSchmidtOrthonormal {f : ι → E} (h₀ : LinearIndependent 𝕜 f) :
     Orthonormal 𝕜 (gramSchmidtNormed 𝕜 f) :=
   by
   unfold Orthonormal
@@ -316,22 +316,22 @@ theorem gram_schmidt_orthonormal {f : ι → E} (h₀ : LinearIndependent 𝕜 f
   · simp only [gramSchmidtNormed_unit_length, h₀, eq_self_iff_true, imp_true_iff]
   · intro i j hij
     simp only [gramSchmidtNormed, inner_smul_left, inner_smul_right, IsROrC.conj_inv,
-      IsROrC.conj_of_real, mul_eq_zero, inv_eq_zero, IsROrC.of_real_eq_zero, norm_eq_zero]
+      IsROrC.conj_ofReal, mul_eq_zero, inv_eq_zero, IsROrC.ofReal_eq_zero, norm_eq_zero]
     repeat' right
     exact gramSchmidt_orthogonal 𝕜 f hij
-#align gram_schmidt_orthonormal gram_schmidt_orthonormal
+#align gram_schmidt_orthonormal gramSchmidtOrthonormal
 
 /-- **Gram-Schmidt Orthonormalization**:
 `gram_schmidt_normed` produces an orthornormal system of vectors after removing the vectors which
 become zero in the process. -/
-theorem gram_schmidt_orthonormal' (f : ι → E) :
+theorem gramSchmidtOrthonormal' (f : ι → E) :
     Orthonormal 𝕜 fun i : { i | gramSchmidtNormed 𝕜 f i ≠ 0 } => gramSchmidtNormed 𝕜 f i :=
   by
   refine' ⟨fun i => gramSchmidtNormed_unit_length' i.Prop, _⟩
   rintro i j (hij : ¬_)
   rw [Subtype.ext_iff] at hij
   simp [gramSchmidtNormed, inner_smul_left, inner_smul_right, gramSchmidt_orthogonal 𝕜 f hij]
-#align gram_schmidt_orthonormal' gram_schmidt_orthonormal'
+#align gram_schmidt_orthonormal' gramSchmidtOrthonormal'
 
 theorem span_gramSchmidtNormed (f : ι → E) (s : Set ι) :
     span 𝕜 (gramSchmidtNormed 𝕜 f '' s) = span 𝕜 (gramSchmidt 𝕜 f '' s) :=
@@ -364,12 +364,12 @@ size of the index set is the dimension of `E`, produce an orthonormal basis for
 with the orthonormal set produced by the Gram-Schmidt orthonormalization process on the elements of
 `ι` for which this process gives a nonzero number. -/
 noncomputable def gramSchmidtOrthonormalBasis : OrthonormalBasis ι 𝕜 E :=
-  ((gram_schmidt_orthonormal' f).exists_orthonormalBasis_extension_of_card_eq h).some
+  ((gramSchmidtOrthonormal' f).exists_orthonormalBasis_extension_of_card_eq h).some
 #align gram_schmidt_orthonormal_basis gramSchmidtOrthonormalBasis
 
 theorem gramSchmidtOrthonormalBasis_apply {f : ι → E} {i : ι} (hi : gramSchmidtNormed 𝕜 f i ≠ 0) :
     gramSchmidtOrthonormalBasis h f i = gramSchmidtNormed 𝕜 f i :=
-  ((gram_schmidt_orthonormal' f).exists_orthonormalBasis_extension_of_card_eq h).choose_spec i hi
+  ((gramSchmidtOrthonormal' f).exists_orthonormalBasis_extension_of_card_eq h).choose_spec i hi
 #align gram_schmidt_orthonormal_basis_apply gramSchmidtOrthonormalBasis_apply
 
 theorem gramSchmidtOrthonormalBasis_apply_of_orthogonal {f : ι → E}

mathlib commit https://github.com/leanprover-community/mathlib/commit/9b2b58d6b14b895b2f375108e765cb47de71aebd

@@ -429,7 +429,7 @@ theorem gramSchmidtOrthonormalBasis_det :
     (gramSchmidtOrthonormalBasis h f).toBasis.det f =
       ∏ i, ⟪gramSchmidtOrthonormalBasis h f i, f i⟫ :=
   by
-  convert Matrix.det_of_upper_triangular (gramSchmidtOrthonormalBasis_inv_blockTriangular h f)
+  convert Matrix.det_of_upperTriangular (gramSchmidtOrthonormalBasis_inv_blockTriangular h f)
   ext i
   exact ((gramSchmidtOrthonormalBasis h f).repr_apply_apply (f i) i).symm
 #align gram_schmidt_orthonormal_basis_det gramSchmidtOrthonormalBasis_det

mathlib commit https://github.com/leanprover-community/mathlib/commit/1a4df69ca1a9a0e5e26bfe12e2b92814216016d0

@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Jiale Miao, Kevin Buzzard, Alexander Bentkamp
 
 ! This file was ported from Lean 3 source module analysis.inner_product_space.gram_schmidt_ortho
-! leanprover-community/mathlib commit 46b633fd842bef9469441c0209906f6dddd2b4f5
+! leanprover-community/mathlib commit 1a4df69ca1a9a0e5e26bfe12e2b92814216016d0
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/
@@ -200,18 +200,16 @@ theorem gramSchmidt_of_orthogonal {f : ι → E} (hf : Pairwise fun i j => ⟪f
     apply Finset.sum_eq_zero
     intro j hj
     rw [coe_eq_zero]
-    suffices span 𝕜 (f '' Set.Iic j) ≤ (𝕜 ∙ f i)ᗮ
+    suffices span 𝕜 (f '' Set.Iic j) ⟂ 𝕜 ∙ f i
       by
       apply orthogonalProjection_mem_subspace_orthogonal_complement_eq_zero
       rw [mem_orthogonal_singleton_iff_inner_left]
       rw [← mem_orthogonal_singleton_iff_inner_right]
       exact this (gramSchmidt_mem_span 𝕜 f (le_refl j))
-    rw [span_le]
-    rintro - ⟨k, hk, rfl⟩
-    rw [SetLike.mem_coe, mem_orthogonal_singleton_iff_inner_left]
+    rw [is_ortho_span]
+    rintro u ⟨k, hk, rfl⟩ v (rfl : v = f i)
     apply hf
-    refine' (lt_of_le_of_lt hk _).Ne
-    simpa using hj
+    exact (lt_of_le_of_lt hk (finset.mem_Iio.mp hj)).Ne
   · simp
 #align gram_schmidt_of_orthogonal gramSchmidt_of_orthogonal
 
@@ -388,16 +386,15 @@ theorem inner_gramSchmidtOrthonormalBasis_eq_zero {f : ι → E} {i : ι}
     (hi : gramSchmidtNormed 𝕜 f i = 0) (j : ι) : ⟪gramSchmidtOrthonormalBasis h f i, f j⟫ = 0 :=
   by
   rw [← mem_orthogonal_singleton_iff_inner_right]
-  suffices span 𝕜 (gramSchmidtNormed 𝕜 f '' Iic j) ≤ (𝕜 ∙ gramSchmidtOrthonormalBasis h f i)ᗮ
+  suffices span 𝕜 (gramSchmidtNormed 𝕜 f '' Iic j) ⟂ 𝕜 ∙ gramSchmidtOrthonormalBasis h f i
     by
     apply this
     rw [span_gramSchmidtNormed]
-    simpa using mem_span_gramSchmidt 𝕜 f (le_refl j)
-  rw [span_le]
-  rintro - ⟨k, -, rfl⟩
-  rw [SetLike.mem_coe, mem_orthogonal_singleton_iff_inner_left]
+    exact mem_span_gramSchmidt 𝕜 f le_rfl
+  rw [is_ortho_span]
+  rintro u ⟨k, hk, rfl⟩ v (rfl : v = _)
   by_cases hk : gramSchmidtNormed 𝕜 f k = 0
-  · simp [hk]
+  · rw [hk, inner_zero_left]
   rw [← gramSchmidtOrthonormalBasis_apply h hk]
   have : k ≠ i := by
     rintro rfl

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: Jiale Miao, Kevin Buzzard, Alexander Bentkamp
 
 ! This file was ported from Lean 3 source module analysis.inner_product_space.gram_schmidt_ortho
-! leanprover-community/mathlib commit a37865088599172dc923253bb7b31998297d9c8a
+! leanprover-community/mathlib commit 46b633fd842bef9469441c0209906f6dddd2b4f5
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/
@@ -44,7 +44,7 @@ open BigOperators
 
 open Finset Submodule FiniteDimensional
 
-variable (𝕜 : Type _) {E : Type _} [IsROrC 𝕜] [InnerProductSpace 𝕜 E]
+variable (𝕜 : Type _) {E : Type _} [IsROrC 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E]
 
 variable {ι : Type _} [LinearOrder ι] [LocallyFiniteOrderBot ι] [IsWellOrder ι (· < ·)]
 

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

@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Jiale Miao, Kevin Buzzard, Alexander Bentkamp
 
 ! This file was ported from Lean 3 source module analysis.inner_product_space.gram_schmidt_ortho
-! leanprover-community/mathlib commit 3fc0b254310908f70a1a75f01147d52e53e9f8a2
+! leanprover-community/mathlib commit a37865088599172dc923253bb7b31998297d9c8a
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/
@@ -111,7 +111,7 @@ theorem gramSchmidt_orthogonal (f : ι → E) {a b : ι} (h₀ : a ≠ b) :
   · by_cases h : gramSchmidt 𝕜 f a = 0
     · simp only [h, inner_zero_left, zero_div, MulZeroClass.zero_mul, sub_zero]
     · rw [← inner_self_eq_norm_sq_to_K, div_mul_cancel, sub_self]
-      rwa [Ne.def, inner_self_eq_zero]
+      rwa [inner_self_ne_zero]
   simp_intro i hi hia only [Finset.mem_range]
   simp only [mul_eq_zero, div_eq_zero_iff, inner_self_eq_zero]
   right

mathlib commit https://github.com/leanprover-community/mathlib/commit/1a313d8bba1bad05faba71a4a4e9742ab5bd9efd

@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Jiale Miao, Kevin Buzzard, Alexander Bentkamp
 
 ! This file was ported from Lean 3 source module analysis.inner_product_space.gram_schmidt_ortho
-! leanprover-community/mathlib commit e99e8c8934985b3bed8c6badbd333043a18c49c9
+! leanprover-community/mathlib commit 3fc0b254310908f70a1a75f01147d52e53e9f8a2
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/
@@ -98,7 +98,7 @@ theorem gramSchmidt_orthogonal (f : ι → E) {a b : ι} (h₀ : a ≠ b) :
     by
     cases' h₀.lt_or_lt with ha hb
     · exact this _ _ ha
-    · rw [inner_eq_zero_sym]
+    · rw [inner_eq_zero_symm]
       exact this _ _ hb
   clear h₀ a b
   intro a b h₀
@@ -116,7 +116,7 @@ theorem gramSchmidt_orthogonal (f : ι → E) {a b : ι} (h₀ : a ≠ b) :
   simp only [mul_eq_zero, div_eq_zero_iff, inner_self_eq_zero]
   right
   cases' hia.lt_or_lt with hia₁ hia₂
-  · rw [inner_eq_zero_sym]
+  · rw [inner_eq_zero_symm]
     exact ih a h₀ i hia₁
   · exact ih i (mem_Iio.1 hi) a hia₂
 #align gram_schmidt_orthogonal gramSchmidt_orthogonal

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

@@ -310,7 +310,7 @@ theorem gramSchmidtNormed_unit_length' {f : ι → E} {n : ι} (hn : gramSchmidt
 /-- **Gram-Schmidt Orthonormalization**:
 `gram_schmidt_normed` applied to a linearly independent set of vectors produces an orthornormal
 system of vectors. -/
-theorem gramSchmidtOrthonormal {f : ι → E} (h₀ : LinearIndependent 𝕜 f) :
+theorem gram_schmidt_orthonormal {f : ι → E} (h₀ : LinearIndependent 𝕜 f) :
     Orthonormal 𝕜 (gramSchmidtNormed 𝕜 f) :=
   by
   unfold Orthonormal
@@ -321,19 +321,19 @@ theorem gramSchmidtOrthonormal {f : ι → E} (h₀ : LinearIndependent 𝕜 f)
       IsROrC.conj_of_real, mul_eq_zero, inv_eq_zero, IsROrC.of_real_eq_zero, norm_eq_zero]
     repeat' right
     exact gramSchmidt_orthogonal 𝕜 f hij
-#align gram_schmidt_orthonormal gramSchmidtOrthonormal
+#align gram_schmidt_orthonormal gram_schmidt_orthonormal
 
 /-- **Gram-Schmidt Orthonormalization**:
 `gram_schmidt_normed` produces an orthornormal system of vectors after removing the vectors which
 become zero in the process. -/
-theorem gramSchmidtOrthonormal' (f : ι → E) :
+theorem gram_schmidt_orthonormal' (f : ι → E) :
     Orthonormal 𝕜 fun i : { i | gramSchmidtNormed 𝕜 f i ≠ 0 } => gramSchmidtNormed 𝕜 f i :=
   by
   refine' ⟨fun i => gramSchmidtNormed_unit_length' i.Prop, _⟩
   rintro i j (hij : ¬_)
   rw [Subtype.ext_iff] at hij
   simp [gramSchmidtNormed, inner_smul_left, inner_smul_right, gramSchmidt_orthogonal 𝕜 f hij]
-#align gram_schmidt_orthonormal' gramSchmidtOrthonormal'
+#align gram_schmidt_orthonormal' gram_schmidt_orthonormal'
 
 theorem span_gramSchmidtNormed (f : ι → E) (s : Set ι) :
     span 𝕜 (gramSchmidtNormed 𝕜 f '' s) = span 𝕜 (gramSchmidt 𝕜 f '' s) :=
@@ -366,12 +366,12 @@ size of the index set is the dimension of `E`, produce an orthonormal basis for
 with the orthonormal set produced by the Gram-Schmidt orthonormalization process on the elements of
 `ι` for which this process gives a nonzero number. -/
 noncomputable def gramSchmidtOrthonormalBasis : OrthonormalBasis ι 𝕜 E :=
-  ((gramSchmidtOrthonormal' f).exists_orthonormalBasis_extension_of_card_eq h).some
+  ((gram_schmidt_orthonormal' f).exists_orthonormalBasis_extension_of_card_eq h).some
 #align gram_schmidt_orthonormal_basis gramSchmidtOrthonormalBasis
 
 theorem gramSchmidtOrthonormalBasis_apply {f : ι → E} {i : ι} (hi : gramSchmidtNormed 𝕜 f i ≠ 0) :
     gramSchmidtOrthonormalBasis h f i = gramSchmidtNormed 𝕜 f i :=
-  ((gramSchmidtOrthonormal' f).exists_orthonormalBasis_extension_of_card_eq h).choose_spec i hi
+  ((gram_schmidt_orthonormal' f).exists_orthonormalBasis_extension_of_card_eq h).choose_spec i hi
 #align gram_schmidt_orthonormal_basis_apply gramSchmidtOrthonormalBasis_apply
 
 theorem gramSchmidtOrthonormalBasis_apply_of_orthogonal {f : ι → E}

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

@@ -109,7 +109,7 @@ theorem gramSchmidt_orthogonal (f : ι → E) {a b : ι} (h₀ : a ≠ b) :
     inner_smul_right]
   rw [Finset.sum_eq_single_of_mem a (finset.mem_Iio.mpr h₀)]
   · by_cases h : gramSchmidt 𝕜 f a = 0
-    · simp only [h, inner_zero_left, zero_div, zero_mul, sub_zero]
+    · simp only [h, inner_zero_left, zero_div, MulZeroClass.zero_mul, sub_zero]
     · rw [← inner_self_eq_norm_sq_to_K, div_mul_cancel, sub_self]
       rwa [Ne.def, inner_self_eq_zero]
   simp_intro i hi hia only [Finset.mem_range]

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

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

@@ -56,7 +56,7 @@ termination_by n
 decreasing_by exact mem_Iio.1 i.2
 #align gram_schmidt gramSchmidt
 
-/-- This lemma uses `∑ i in` instead of `∑ i :`.-/
+/-- This lemma uses `∑ i in` instead of `∑ i :`. -/
 theorem gramSchmidt_def (f : ι → E) (n : ι) :
     gramSchmidt 𝕜 f n = f n - ∑ i in Iio n, orthogonalProjection (𝕜 ∙ gramSchmidt 𝕜 f i) (f n) := by
   rw [← sum_attach, attach_eq_univ, gramSchmidt]

@@ -110,7 +110,7 @@ theorem gramSchmidt_orthogonal (f : ι → E) {a b : ι} (h₀ : a ≠ b) :
   · exact ih i (mem_Iio.1 hi) a hia₂
 #align gram_schmidt_orthogonal gramSchmidt_orthogonal
 
-/-- This is another version of `gramSchmidt_orthogonal` using `pairwise` instead. -/
+/-- This is another version of `gramSchmidt_orthogonal` using `Pairwise` instead. -/
 theorem gramSchmidt_pairwise_orthogonal (f : ι → E) :
     Pairwise fun a b => ⟪gramSchmidt 𝕜 f a, gramSchmidt 𝕜 f b⟫ = 0 := fun _ _ =>
   gramSchmidt_orthogonal 𝕜 f

@@ -265,7 +265,7 @@ variable {𝕜}
 
 theorem gramSchmidtNormed_unit_length_coe {f : ι → E} (n : ι)
     (h₀ : LinearIndependent 𝕜 (f ∘ ((↑) : Set.Iic n → ι))) : ‖gramSchmidtNormed 𝕜 f n‖ = 1 := by
-  simp only [gramSchmidt_ne_zero_coe n h₀, gramSchmidtNormed, norm_smul_inv_norm, Ne.def,
+  simp only [gramSchmidt_ne_zero_coe n h₀, gramSchmidtNormed, norm_smul_inv_norm, Ne,
     not_false_iff]
 #align gram_schmidt_normed_unit_length_coe gramSchmidtNormed_unit_length_coe
 

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.

@@ -41,7 +41,7 @@ open scoped BigOperators
 
 open Finset Submodule FiniteDimensional
 
-variable (𝕜 : Type*) {E : Type*} [IsROrC 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E]
+variable (𝕜 : Type*) {E : Type*} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E]
 variable {ι : Type*} [LinearOrder ι] [LocallyFiniteOrderBot ι] [IsWellOrder ι (· < ·)]
 
 attribute [local instance] IsWellOrder.toHasWellFounded
@@ -71,7 +71,7 @@ theorem gramSchmidt_def'' (f : ι → E) (n : ι) :
     f n = gramSchmidt 𝕜 f n + ∑ i in Iio n,
       (⟪gramSchmidt 𝕜 f i, f n⟫ / (‖gramSchmidt 𝕜 f i‖ : 𝕜) ^ 2) • gramSchmidt 𝕜 f i := by
   convert gramSchmidt_def' 𝕜 f n
-  rw [orthogonalProjection_singleton, IsROrC.ofReal_pow]
+  rw [orthogonalProjection_singleton, RCLike.ofReal_pow]
 #align gram_schmidt_def'' gramSchmidt_def''
 
 @[simp]
@@ -99,7 +99,7 @@ theorem gramSchmidt_orthogonal (f : ι → E) {a b : ι} (h₀ : a ≠ b) :
   rw [Finset.sum_eq_single_of_mem a (Finset.mem_Iio.mpr h₀)]
   · by_cases h : gramSchmidt 𝕜 f a = 0
     · simp only [h, inner_zero_left, zero_div, zero_mul, sub_zero]
-    · rw [IsROrC.ofReal_pow, ← inner_self_eq_norm_sq_to_K, div_mul_cancel₀, sub_self]
+    · rw [RCLike.ofReal_pow, ← inner_self_eq_norm_sq_to_K, div_mul_cancel₀, sub_self]
       rwa [inner_self_ne_zero]
   intro i hi hia
   simp only [mul_eq_zero, div_eq_zero_iff, inner_self_eq_zero]
@@ -290,8 +290,8 @@ theorem gramSchmidt_orthonormal {f : ι → E} (h₀ : LinearIndependent 𝕜 f)
   constructor
   · simp only [gramSchmidtNormed_unit_length, h₀, eq_self_iff_true, imp_true_iff]
   · intro i j hij
-    simp only [gramSchmidtNormed, inner_smul_left, inner_smul_right, IsROrC.conj_inv,
-      IsROrC.conj_ofReal, mul_eq_zero, inv_eq_zero, IsROrC.ofReal_eq_zero, norm_eq_zero]
+    simp only [gramSchmidtNormed, inner_smul_left, inner_smul_right, RCLike.conj_inv,
+      RCLike.conj_ofReal, mul_eq_zero, inv_eq_zero, RCLike.ofReal_eq_zero, norm_eq_zero]
     repeat' right
     exact gramSchmidt_orthogonal 𝕜 f hij
 #align gram_schmidt_orthonormal gramSchmidt_orthonormal

Lemma names around cancellation of multiplication and division are a mess.

This PR renames a handful of them according to the following table (each big row contains the multiplicative statement, then the three rows contain the GroupWithZero lemma name, the Group lemma, the AddGroup lemma name).

| Statement | New name | Old name | |

@@ -99,7 +99,7 @@ theorem gramSchmidt_orthogonal (f : ι → E) {a b : ι} (h₀ : a ≠ b) :
   rw [Finset.sum_eq_single_of_mem a (Finset.mem_Iio.mpr h₀)]
   · by_cases h : gramSchmidt 𝕜 f a = 0
     · simp only [h, inner_zero_left, zero_div, zero_mul, sub_zero]
-    · rw [IsROrC.ofReal_pow, ← inner_self_eq_norm_sq_to_K, div_mul_cancel, sub_self]
+    · rw [IsROrC.ofReal_pow, ← inner_self_eq_norm_sq_to_K, div_mul_cancel₀, sub_self]
       rwa [inner_self_ne_zero]
   intro i hi hia
   simp only [mul_eq_zero, div_eq_zero_iff, inner_self_eq_zero]

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)

@@ -42,7 +42,6 @@ open scoped BigOperators
 open Finset Submodule FiniteDimensional
 
 variable (𝕜 : Type*) {E : Type*} [IsROrC 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E]
-
 variable {ι : Type*} [LinearOrder ι] [LocallyFiniteOrderBot ι] [IsWellOrder ι (· < ·)]
 
 attribute [local instance] IsWellOrder.toHasWellFounded

The termination checker has been getting more capable, and many of the termination_by or decreasing_by clauses in Mathlib are no longer needed.

(Note that termination_by? will show the automatically derived termination expression, so no information is being lost by removing these.)

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

@@ -153,7 +153,6 @@ theorem gramSchmidt_mem_span (f : ι → E) :
   exact smul_mem _ _
     (span_mono (image_subset f <| Iic_subset_Iic.2 hkj.le) <| gramSchmidt_mem_span _ le_rfl)
 termination_by j => j
-decreasing_by exact hkj
 #align gram_schmidt_mem_span gramSchmidt_mem_span
 
 theorem span_gramSchmidt_Iic (f : ι → E) (c : ι) :

I replaced a few "terminal" refine/refine's with exact.

The strategy was very simple-minded: essentially any refine whose following line had smaller indentation got replaced by exact and then I cleaned up the mess.

This PR certainly leaves some further terminal refines, but maybe the current change is beneficial.

@@ -211,7 +211,7 @@ theorem gramSchmidt_ne_zero_coe {f : ι → E} (n : ι)
     simp only [Set.mem_image, Set.mem_Iio, orthogonalProjection_singleton]
     apply Submodule.smul_mem _ _ _
     rw [Finset.mem_Iio] at ha
-    refine' subset_span ⟨a, ha, by rfl⟩
+    exact subset_span ⟨a, ha, by rfl⟩
   have h₂ : (f ∘ ((↑) : Set.Iic n → ι)) ⟨n, le_refl n⟩ ∈
       span 𝕜 (f ∘ ((↑) : Set.Iic n → ι) '' Set.Iio ⟨n, le_refl n⟩) := by
     rw [image_comp]

Co-authored-by: Scott Morrison <scott.morrison@gmail.com> Co-authored-by: Eric Wieser <wieser.eric@gmail.com> Co-authored-by: Joachim Breitner <mail@joachim-breitner.de>

@@ -53,7 +53,7 @@ local notation "⟪" x ", " y "⟫" => @inner 𝕜 _ _ x y
 and outputs a set of orthogonal vectors which have the same span. -/
 noncomputable def gramSchmidt [IsWellOrder ι (· < ·)] (f : ι → E) (n : ι) : E :=
   f n - ∑ i : Iio n, orthogonalProjection (𝕜 ∙ gramSchmidt f i) (f n)
-termination_by _ n => n
+termination_by n
 decreasing_by exact mem_Iio.1 i.2
 #align gram_schmidt gramSchmidt
 
@@ -152,7 +152,7 @@ theorem gramSchmidt_mem_span (f : ι → E) :
   let hkj : k < j := (Finset.mem_Iio.1 hk).trans_le hij
   exact smul_mem _ _
     (span_mono (image_subset f <| Iic_subset_Iic.2 hkj.le) <| gramSchmidt_mem_span _ le_rfl)
-termination_by _ => j
+termination_by j => j
 decreasing_by exact hkj
 #align gram_schmidt_mem_span gramSchmidt_mem_span
 

and other simple pointwise lemmas for Set and Finset. Also add supporting Fintype.piFinset lemmas and fix the names of two lemmas.

From LeanAPAP and LeanCamCombi

@@ -186,7 +186,7 @@ theorem gramSchmidt_of_orthogonal {f : ι → E} (hf : Pairwise fun i j => ⟪f
   · congr
     apply Finset.sum_eq_zero
     intro j hj
-    rw [coe_eq_zero]
+    rw [Submodule.coe_eq_zero]
     suffices span 𝕜 (f '' Set.Iic j) ⟂ 𝕜 ∙ f i by
       apply orthogonalProjection_mem_subspace_orthogonalComplement_eq_zero
       rw [mem_orthogonal_singleton_iff_inner_left]

We still have the exact_mod_cast tactic, used in a few places, which somehow (?) works a little bit harder to prevent the expected type influencing the elaboration of the term. I would like to get to the bottom of this, and it will be easier once the only usages of exact_mod_cast are the ones that don't work using the term elaborator by itself.

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

@@ -319,7 +319,7 @@ theorem span_gramSchmidtNormed (f : ι → E) (s : Set ι) :
   by_cases h : gramSchmidt 𝕜 f i = 0
   · simp [h]
   · refine' mem_span_singleton.2 ⟨‖gramSchmidt 𝕜 f i‖, smul_inv_smul₀ _ _⟩
-    exact_mod_cast norm_ne_zero_iff.2 h
+    exact mod_cast norm_ne_zero_iff.2 h
 #align span_gram_schmidt_normed span_gramSchmidtNormed
 
 theorem span_gramSchmidtNormed_range (f : ι → E) :

We already have WellFoundedLT/WellFoundedGT as wrappers around IsWellFounded, but we didn't have the corresponding wrapper lemmas.

@@ -93,7 +93,7 @@ theorem gramSchmidt_orthogonal (f : ι → E) {a b : ι} (h₀ : a ≠ b) :
   clear h₀ a b
   intro a b h₀
   revert a
-  apply WellFounded.induction (@IsWellFounded.wf ι (· < ·) _) b
+  apply wellFounded_lt.induction b
   intro b ih a h₀
   simp only [gramSchmidt_def 𝕜 f b, inner_sub_right, inner_sum, orthogonalProjection_singleton,
     inner_smul_right]

Add simp lemmas for images of intervals in other intervals under Subtype.val.

@@ -215,11 +215,7 @@ theorem gramSchmidt_ne_zero_coe {f : ι → E} (n : ι)
   have h₂ : (f ∘ ((↑) : Set.Iic n → ι)) ⟨n, le_refl n⟩ ∈
       span 𝕜 (f ∘ ((↑) : Set.Iic n → ι) '' Set.Iio ⟨n, le_refl n⟩) := by
     rw [image_comp]
-    convert h₁ using 3
-    ext i
-    apply Iff.intro <;> simp -- Porting note: was `simpa using @le_of_lt _ _ i n`
-    · intros; simp_all only
-    · intros q; use i; exact ⟨q, le_of_lt q, rfl⟩
+    simpa using h₁
   apply LinearIndependent.not_mem_span_image h₀ _ h₂
   simp only [Set.mem_Iio, lt_self_iff_false, not_false_iff]
 #align gram_schmidt_ne_zero_coe gramSchmidt_ne_zero_coe

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).

@@ -99,7 +99,7 @@ theorem gramSchmidt_orthogonal (f : ι → E) {a b : ι} (h₀ : a ≠ b) :
     inner_smul_right]
   rw [Finset.sum_eq_single_of_mem a (Finset.mem_Iio.mpr h₀)]
   · by_cases h : gramSchmidt 𝕜 f a = 0
-    · simp only [h, inner_zero_left, zero_div, MulZeroClass.zero_mul, sub_zero]
+    · simp only [h, inner_zero_left, zero_div, zero_mul, sub_zero]
     · rw [IsROrC.ofReal_pow, ← inner_self_eq_norm_sq_to_K, div_mul_cancel, sub_self]
       rwa [inner_self_ne_zero]
   intro i hi hia

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

This has nice performance benefits.

@@ -41,9 +41,9 @@ open scoped BigOperators
 
 open Finset Submodule FiniteDimensional
 
-variable (𝕜 : Type _) {E : Type _} [IsROrC 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E]
+variable (𝕜 : Type*) {E : Type*} [IsROrC 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E]
 
-variable {ι : Type _} [LinearOrder ι] [LocallyFiniteOrderBot ι] [IsWellOrder ι (· < ·)]
+variable {ι : Type*} [LinearOrder ι] [LocallyFiniteOrderBot ι] [IsWellOrder ι (· < ·)]
 
 attribute [local instance] IsWellOrder.toHasWellFounded
 
@@ -76,7 +76,7 @@ theorem gramSchmidt_def'' (f : ι → E) (n : ι) :
 #align gram_schmidt_def'' gramSchmidt_def''
 
 @[simp]
-theorem gramSchmidt_zero {ι : Type _} [LinearOrder ι] [LocallyFiniteOrder ι] [OrderBot ι]
+theorem gramSchmidt_zero {ι : Type*} [LinearOrder ι] [LocallyFiniteOrder ι] [OrderBot ι]
     [IsWellOrder ι (· < ·)] (f : ι → E) : gramSchmidt 𝕜 f ⊥ = f ⊥ := by
   rw [gramSchmidt_def, Iio_eq_Ico, Finset.Ico_self, Finset.sum_empty, sub_zero]
 #align gram_schmidt_zero gramSchmidt_zero

• 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 Jiale Miao. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Jiale Miao, Kevin Buzzard, Alexander Bentkamp
-
-! This file was ported from Lean 3 source module analysis.inner_product_space.gram_schmidt_ortho
-! leanprover-community/mathlib commit 1a4df69ca1a9a0e5e26bfe12e2b92814216016d0
-! Please do not edit these lines, except to modify the commit id
-! if you have ported upstream changes.
 -/
 import Mathlib.Analysis.InnerProductSpace.PiL2
 import Mathlib.LinearAlgebra.Matrix.Block
 
+#align_import analysis.inner_product_space.gram_schmidt_ortho from "leanprover-community/mathlib"@"1a4df69ca1a9a0e5e26bfe12e2b92814216016d0"
+
 /-!
 # Gram-Schmidt Orthogonalization and Orthonormalization
 

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>

@@ -401,8 +401,11 @@ theorem gramSchmidtOrthonormalBasis_inv_blockTriangular :
   gramSchmidtOrthonormalBasis_inv_triangular' h f
 #align gram_schmidt_orthonormal_basis_inv_block_triangular gramSchmidtOrthonormalBasis_inv_blockTriangular
 
-theorem gramSchmidtOrthonormalBasis_det : (gramSchmidtOrthonormalBasis h f).toBasis.det f =
-    ∏ i, ⟪gramSchmidtOrthonormalBasis h f i, f i⟫ := by
+-- Porting note: added a `DecidableEq` argument to help with timeouts in
+-- `Mathlib/Analysis/InnerProductSpace/Orientation.lean`
+theorem gramSchmidtOrthonormalBasis_det [DecidableEq ι] :
+    (gramSchmidtOrthonormalBasis h f).toBasis.det f =
+      ∏ i, ⟪gramSchmidtOrthonormalBasis h f i, f i⟫ := by
   convert Matrix.det_of_upperTriangular (gramSchmidtOrthonormalBasis_inv_blockTriangular h f)
   exact ((gramSchmidtOrthonormalBasis h f).repr_apply_apply (f _) _).symm
 #align gram_schmidt_orthonormal_basis_det gramSchmidtOrthonormalBasis_det