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

@@ -40,7 +40,7 @@ A slightly more elaborate corollary is that if `E` is complete and `T` is a comp
 -/
 
 
-variable {𝕜 : Type _} [IsROrC 𝕜]
+variable {𝕜 : Type _} [RCLike 𝕜]
 
 variable {E : Type _} [NormedAddCommGroup E] [InnerProductSpace 𝕜 E]
 
@@ -195,7 +195,7 @@ theorem eq_smul_self_of_isLocalExtrOn (hT : IsSelfAdjoint T) {x₀ : E}
     (hextr : IsLocalExtrOn T.reApplyInnerSelf (sphere (0 : E) ‖x₀‖) x₀) :
     T x₀ = (↑(rayleigh_quotient x₀) : 𝕜) • x₀ :=
   by
-  letI := InnerProductSpace.isROrCToReal 𝕜 E
+  letI := InnerProductSpace.rclikeToReal 𝕜 E
   let hSA := hT.is_symmetric.restrict_scalars.to_self_adjoint.prop
   exact hSA.eq_smul_self_of_is_local_extr_on_real hextr
 #align is_self_adjoint.eq_smul_self_of_is_local_extr_on IsSelfAdjoint.eq_smul_self_of_isLocalExtrOn
@@ -272,9 +272,9 @@ namespace IsSymmetric
 /-- The supremum of the Rayleigh quotient of a symmetric operator `T` on a nontrivial
 finite-dimensional vector space is an eigenvalue for that operator. -/
 theorem hasEigenvalue_iSup_of_finiteDimensional (hT : T.IsSymmetric) :
-    HasEigenvalue T ↑(⨆ x : { x : E // x ≠ 0 }, IsROrC.re ⟪T x, x⟫ / ‖(x : E)‖ ^ 2) :=
+    HasEigenvalue T ↑(⨆ x : { x : E // x ≠ 0 }, RCLike.re ⟪T x, x⟫ / ‖(x : E)‖ ^ 2) :=
   by
-  haveI := FiniteDimensional.proper_isROrC 𝕜 E
+  haveI := FiniteDimensional.proper_rclike 𝕜 E
   let T' := hT.to_self_adjoint
   obtain ⟨x, hx⟩ : ∃ x : E, x ≠ 0 := exists_ne 0
   have H₁ : IsCompact (sphere (0 : E) ‖x‖) := isCompact_sphere _ _
@@ -296,9 +296,9 @@ theorem hasEigenvalue_iSup_of_finiteDimensional (hT : T.IsSymmetric) :
 /-- The infimum of the Rayleigh quotient of a symmetric operator `T` on a nontrivial
 finite-dimensional vector space is an eigenvalue for that operator. -/
 theorem hasEigenvalue_iInf_of_finiteDimensional (hT : T.IsSymmetric) :
-    HasEigenvalue T ↑(⨅ x : { x : E // x ≠ 0 }, IsROrC.re ⟪T x, x⟫ / ‖(x : E)‖ ^ 2) :=
+    HasEigenvalue T ↑(⨅ x : { x : E // x ≠ 0 }, RCLike.re ⟪T x, x⟫ / ‖(x : E)‖ ^ 2) :=
   by
-  haveI := FiniteDimensional.proper_isROrC 𝕜 E
+  haveI := FiniteDimensional.proper_rclike 𝕜 E
   let T' := hT.to_self_adjoint
   obtain ⟨x, hx⟩ : ∃ x : E, x ≠ 0 := exists_ne 0
   have H₁ : IsCompact (sphere (0 : E) ‖x‖) := isCompact_sphere _ _

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

@@ -122,7 +122,7 @@ theorem LinearMap.IsSymmetric.hasStrictFDerivAt_reApplyInnerSelf {T : F →L[ℝ
   convert T.has_strict_fderiv_at.inner _ (hasStrictFDerivAt_id x₀)
   ext y
   simp_rw [_root_.bit0, ContinuousLinearMap.comp_apply, ContinuousLinearMap.add_apply,
-    innerSL_apply, fderivInnerClm_apply, id.def, ContinuousLinearMap.prod_apply,
+    innerSL_apply, fderivInnerCLM_apply, id.def, ContinuousLinearMap.prod_apply,
     ContinuousLinearMap.id_apply, hT.apply_clm x₀ y, real_inner_comm _ x₀]
 #align linear_map.is_symmetric.has_strict_fderiv_at_re_apply_inner_self LinearMap.IsSymmetric.hasStrictFDerivAt_reApplyInnerSelf
 -/

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

@@ -3,11 +3,11 @@ Copyright (c) 2021 Heather Macbeth. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Heather Macbeth, Frédéric Dupuis
 -/
-import Mathbin.Analysis.InnerProductSpace.Calculus
-import Mathbin.Analysis.InnerProductSpace.Dual
-import Mathbin.Analysis.InnerProductSpace.Adjoint
-import Mathbin.Analysis.Calculus.LagrangeMultipliers
-import Mathbin.LinearAlgebra.Eigenspace.Basic
+import Analysis.InnerProductSpace.Calculus
+import Analysis.InnerProductSpace.Dual
+import Analysis.InnerProductSpace.Adjoint
+import Analysis.Calculus.LagrangeMultipliers
+import LinearAlgebra.Eigenspace.Basic
 
 #align_import analysis.inner_product_space.rayleigh from "leanprover-community/mathlib"@"cff8231f04dfa33fd8f2f45792eebd862ef30cad"
 

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

@@ -2,11 +2,6 @@
 Copyright (c) 2021 Heather Macbeth. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Heather Macbeth, Frédéric Dupuis
-
-! This file was ported from Lean 3 source module analysis.inner_product_space.rayleigh
-! leanprover-community/mathlib commit cff8231f04dfa33fd8f2f45792eebd862ef30cad
-! Please do not edit these lines, except to modify the commit id
-! if you have ported upstream changes.
 -/
 import Mathbin.Analysis.InnerProductSpace.Calculus
 import Mathbin.Analysis.InnerProductSpace.Dual
@@ -14,6 +9,8 @@ import Mathbin.Analysis.InnerProductSpace.Adjoint
 import Mathbin.Analysis.Calculus.LagrangeMultipliers
 import Mathbin.LinearAlgebra.Eigenspace.Basic
 
+#align_import analysis.inner_product_space.rayleigh from "leanprover-community/mathlib"@"cff8231f04dfa33fd8f2f45792eebd862ef30cad"
+
 /-!
 # The Rayleigh quotient
 

mathlib commit https://github.com/leanprover-community/mathlib/commit/2a0ce625dbb0ffbc7d1316597de0b25c1ec75303

@@ -59,6 +59,7 @@ variable (T : E →L[𝕜] E)
 
 local notation "rayleigh_quotient" => fun x : E => T.reApplyInnerSelf x / ‖(x : E)‖ ^ 2
 
+#print ContinuousLinearMap.rayleigh_smul /-
 theorem rayleigh_smul (x : E) {c : 𝕜} (hc : c ≠ 0) :
     rayleigh_quotient (c • x) = rayleigh_quotient x :=
   by
@@ -69,7 +70,9 @@ theorem rayleigh_smul (x : E) {c : 𝕜} (hc : c ≠ 0) :
   field_simp [norm_smul, T.re_apply_inner_self_smul]
   ring
 #align continuous_linear_map.rayleigh_smul ContinuousLinearMap.rayleigh_smul
+-/
 
+#print ContinuousLinearMap.image_rayleigh_eq_image_rayleigh_sphere /-
 theorem image_rayleigh_eq_image_rayleigh_sphere {r : ℝ} (hr : 0 < r) :
     rayleigh_quotient '' {0}ᶜ = rayleigh_quotient '' sphere 0 r :=
   by
@@ -86,20 +89,25 @@ theorem image_rayleigh_eq_image_rayleigh_sphere {r : ℝ} (hr : 0 < r) :
   · rintro ⟨x, hx, hxT⟩
     exact ⟨x, ne_zero_of_mem_sphere hr.ne' ⟨x, hx⟩, hxT⟩
 #align continuous_linear_map.image_rayleigh_eq_image_rayleigh_sphere ContinuousLinearMap.image_rayleigh_eq_image_rayleigh_sphere
+-/
 
+#print ContinuousLinearMap.iSup_rayleigh_eq_iSup_rayleigh_sphere /-
 theorem iSup_rayleigh_eq_iSup_rayleigh_sphere {r : ℝ} (hr : 0 < r) :
     (⨆ x : { x : E // x ≠ 0 }, rayleigh_quotient x) = ⨆ x : sphere (0 : E) r, rayleigh_quotient x :=
   show (⨆ x : ({0} : Set E)ᶜ, rayleigh_quotient x) = _ by
     simp only [← @sSup_image' _ _ _ _ rayleigh_quotient,
       T.image_rayleigh_eq_image_rayleigh_sphere hr]
 #align continuous_linear_map.supr_rayleigh_eq_supr_rayleigh_sphere ContinuousLinearMap.iSup_rayleigh_eq_iSup_rayleigh_sphere
+-/
 
+#print ContinuousLinearMap.iInf_rayleigh_eq_iInf_rayleigh_sphere /-
 theorem iInf_rayleigh_eq_iInf_rayleigh_sphere {r : ℝ} (hr : 0 < r) :
     (⨅ x : { x : E // x ≠ 0 }, rayleigh_quotient x) = ⨅ x : sphere (0 : E) r, rayleigh_quotient x :=
   show (⨅ x : ({0} : Set E)ᶜ, rayleigh_quotient x) = _ by
     simp only [← @sInf_image' _ _ _ _ rayleigh_quotient,
       T.image_rayleigh_eq_image_rayleigh_sphere hr]
 #align continuous_linear_map.infi_rayleigh_eq_infi_rayleigh_sphere ContinuousLinearMap.iInf_rayleigh_eq_iInf_rayleigh_sphere
+-/
 
 end ContinuousLinearMap
 
@@ -109,6 +117,7 @@ section Real
 
 variable {F : Type _} [NormedAddCommGroup F] [InnerProductSpace ℝ F]
 
+#print LinearMap.IsSymmetric.hasStrictFDerivAt_reApplyInnerSelf /-
 theorem LinearMap.IsSymmetric.hasStrictFDerivAt_reApplyInnerSelf {T : F →L[ℝ] F}
     (hT : (T : F →ₗ[ℝ] F).IsSymmetric) (x₀ : F) :
     HasStrictFDerivAt T.reApplyInnerSelf (bit0 (innerSL ℝ (T x₀))) x₀ :=
@@ -119,11 +128,13 @@ theorem LinearMap.IsSymmetric.hasStrictFDerivAt_reApplyInnerSelf {T : F →L[ℝ
     innerSL_apply, fderivInnerClm_apply, id.def, ContinuousLinearMap.prod_apply,
     ContinuousLinearMap.id_apply, hT.apply_clm x₀ y, real_inner_comm _ x₀]
 #align linear_map.is_symmetric.has_strict_fderiv_at_re_apply_inner_self LinearMap.IsSymmetric.hasStrictFDerivAt_reApplyInnerSelf
+-/
 
 variable [CompleteSpace F] {T : F →L[ℝ] F}
 
 local notation "rayleigh_quotient" => fun x : F => T.reApplyInnerSelf x / ‖(x : F)‖ ^ 2
 
+#print IsSelfAdjoint.linearly_dependent_of_isLocalExtrOn /-
 theorem linearly_dependent_of_isLocalExtrOn (hT : IsSelfAdjoint T) {x₀ : F}
     (hextr : IsLocalExtrOn T.reApplyInnerSelf (sphere (0 : F) ‖x₀‖) x₀) :
     ∃ a b : ℝ, (a, b) ≠ 0 ∧ a • x₀ + b • T x₀ = 0 :=
@@ -145,7 +156,9 @@ theorem linearly_dependent_of_isLocalExtrOn (hT : IsSelfAdjoint T) {x₀ : F}
   apply smul_right_injective (F →L[ℝ] ℝ) (two_ne_zero : (2 : ℝ) ≠ 0)
   simpa only [_root_.bit0, add_smul, smul_add, one_smul, add_zero] using h₂
 #align is_self_adjoint.linearly_dependent_of_is_local_extr_on IsSelfAdjoint.linearly_dependent_of_isLocalExtrOn
+-/
 
+#print IsSelfAdjoint.eq_smul_self_of_isLocalExtrOn_real /-
 theorem eq_smul_self_of_isLocalExtrOn_real (hT : IsSelfAdjoint T) {x₀ : F}
     (hextr : IsLocalExtrOn T.reApplyInnerSelf (sphere (0 : F) ‖x₀‖) x₀) :
     T x₀ = rayleigh_quotient x₀ • x₀ :=
@@ -170,6 +183,7 @@ theorem eq_smul_self_of_isLocalExtrOn_real (hT : IsSelfAdjoint T) {x₀ : F}
   simpa [inner_smul_left, real_inner_self_eq_norm_mul_norm, sq] using
     congr_arg (fun x => ⟪x, x₀⟫_ℝ) hc
 #align is_self_adjoint.eq_smul_self_of_is_local_extr_on_real IsSelfAdjoint.eq_smul_self_of_isLocalExtrOn_real
+-/
 
 end Real
 
@@ -179,6 +193,7 @@ variable [CompleteSpace E] {T : E →L[𝕜] E}
 
 local notation "rayleigh_quotient" => fun x : E => T.reApplyInnerSelf x / ‖(x : E)‖ ^ 2
 
+#print IsSelfAdjoint.eq_smul_self_of_isLocalExtrOn /-
 theorem eq_smul_self_of_isLocalExtrOn (hT : IsSelfAdjoint T) {x₀ : E}
     (hextr : IsLocalExtrOn T.reApplyInnerSelf (sphere (0 : E) ‖x₀‖) x₀) :
     T x₀ = (↑(rayleigh_quotient x₀) : 𝕜) • x₀ :=
@@ -187,7 +202,9 @@ theorem eq_smul_self_of_isLocalExtrOn (hT : IsSelfAdjoint T) {x₀ : E}
   let hSA := hT.is_symmetric.restrict_scalars.to_self_adjoint.prop
   exact hSA.eq_smul_self_of_is_local_extr_on_real hextr
 #align is_self_adjoint.eq_smul_self_of_is_local_extr_on IsSelfAdjoint.eq_smul_self_of_isLocalExtrOn
+-/
 
+#print IsSelfAdjoint.hasEigenvector_of_isLocalExtrOn /-
 /-- For a self-adjoint operator `T`, a local extremum of the Rayleigh quotient of `T` on a sphere
 centred at the origin is an eigenvector of `T`. -/
 theorem hasEigenvector_of_isLocalExtrOn (hT : IsSelfAdjoint T) {x₀ : E} (hx₀ : x₀ ≠ 0)
@@ -198,7 +215,9 @@ theorem hasEigenvector_of_isLocalExtrOn (hT : IsSelfAdjoint T) {x₀ : E} (hx₀
   rw [Module.End.mem_eigenspace_iff]
   exact hT.eq_smul_self_of_is_local_extr_on hextr
 #align is_self_adjoint.has_eigenvector_of_is_local_extr_on IsSelfAdjoint.hasEigenvector_of_isLocalExtrOn
+-/
 
+#print IsSelfAdjoint.hasEigenvector_of_isMaxOn /-
 /-- For a self-adjoint operator `T`, a maximum of the Rayleigh quotient of `T` on a sphere centred
 at the origin is an eigenvector of `T`, with eigenvalue the global supremum of the Rayleigh
 quotient. -/
@@ -217,7 +236,9 @@ theorem hasEigenvector_of_isMaxOn (hT : IsSelfAdjoint T) {x₀ : E} (hx₀ : x
   rw [this]
   exact div_le_div_of_le (sq_nonneg ‖x₀‖) (hextr hx)
 #align is_self_adjoint.has_eigenvector_of_is_max_on IsSelfAdjoint.hasEigenvector_of_isMaxOn
+-/
 
+#print IsSelfAdjoint.hasEigenvector_of_isMinOn /-
 /-- For a self-adjoint operator `T`, a minimum of the Rayleigh quotient of `T` on a sphere centred
 at the origin is an eigenvector of `T`, with eigenvalue the global infimum of the Rayleigh
 quotient. -/
@@ -236,6 +257,7 @@ theorem hasEigenvector_of_isMinOn (hT : IsSelfAdjoint T) {x₀ : E} (hx₀ : x
   rw [this]
   exact div_le_div_of_le (sq_nonneg ‖x₀‖) (hextr hx)
 #align is_self_adjoint.has_eigenvector_of_is_min_on IsSelfAdjoint.hasEigenvector_of_isMinOn
+-/
 
 end CompleteSpace
 
@@ -249,6 +271,7 @@ namespace LinearMap
 
 namespace IsSymmetric
 
+#print LinearMap.IsSymmetric.hasEigenvalue_iSup_of_finiteDimensional /-
 /-- The supremum of the Rayleigh quotient of a symmetric operator `T` on a nontrivial
 finite-dimensional vector space is an eigenvalue for that operator. -/
 theorem hasEigenvalue_iSup_of_finiteDimensional (hT : T.IsSymmetric) :
@@ -270,7 +293,9 @@ theorem hasEigenvalue_iSup_of_finiteDimensional (hT : T.IsSymmetric) :
     simpa [← norm_eq_zero, Ne.def]
   exact has_eigenvalue_of_has_eigenvector (T'.prop.has_eigenvector_of_is_max_on hx₀_ne this)
 #align linear_map.is_symmetric.has_eigenvalue_supr_of_finite_dimensional LinearMap.IsSymmetric.hasEigenvalue_iSup_of_finiteDimensional
+-/
 
+#print LinearMap.IsSymmetric.hasEigenvalue_iInf_of_finiteDimensional /-
 /-- The infimum of the Rayleigh quotient of a symmetric operator `T` on a nontrivial
 finite-dimensional vector space is an eigenvalue for that operator. -/
 theorem hasEigenvalue_iInf_of_finiteDimensional (hT : T.IsSymmetric) :
@@ -292,12 +317,15 @@ theorem hasEigenvalue_iInf_of_finiteDimensional (hT : T.IsSymmetric) :
     simpa [← norm_eq_zero, Ne.def]
   exact has_eigenvalue_of_has_eigenvector (T'.prop.has_eigenvector_of_is_min_on hx₀_ne this)
 #align linear_map.is_symmetric.has_eigenvalue_infi_of_finite_dimensional LinearMap.IsSymmetric.hasEigenvalue_iInf_of_finiteDimensional
+-/
 
+#print LinearMap.IsSymmetric.subsingleton_of_no_eigenvalue_finiteDimensional /-
 theorem subsingleton_of_no_eigenvalue_finiteDimensional (hT : T.IsSymmetric)
     (hT' : ∀ μ : 𝕜, Module.End.eigenspace (T : E →ₗ[𝕜] E) μ = ⊥) : Subsingleton E :=
   (subsingleton_or_nontrivial E).resolve_right fun h =>
     absurd (hT' _) hT.has_eigenvalue_supr_of_finite_dimensional
 #align linear_map.is_symmetric.subsingleton_of_no_eigenvalue_finite_dimensional LinearMap.IsSymmetric.subsingleton_of_no_eigenvalue_finiteDimensional
+-/
 
 end IsSymmetric
 

mathlib commit https://github.com/leanprover-community/mathlib/commit/893964fc28cefbcffc7cb784ed00a2895b4e65cf

@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Heather Macbeth, Frédéric Dupuis
 
 ! This file was ported from Lean 3 source module analysis.inner_product_space.rayleigh
-! leanprover-community/mathlib commit 6b0169218d01f2837d79ea2784882009a0da1aa1
+! leanprover-community/mathlib commit cff8231f04dfa33fd8f2f45792eebd862ef30cad
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/
@@ -17,6 +17,9 @@ import Mathbin.LinearAlgebra.Eigenspace.Basic
 /-!
 # The Rayleigh quotient
 
+> THIS FILE IS SYNCHRONIZED WITH MATHLIB4.
+> Any changes to this file require a corresponding PR to mathlib4.
+
 The Rayleigh quotient of a self-adjoint operator `T` on an inner product space `E` is the function
 `λ x, ⟪T x, x⟫ / ‖x‖ ^ 2`.
 

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

@@ -44,7 +44,6 @@ variable {𝕜 : Type _} [IsROrC 𝕜]
 
 variable {E : Type _} [NormedAddCommGroup E] [InnerProductSpace 𝕜 E]
 
--- mathport name: «expr⟪ , ⟫»
 local notation "⟪" x ", " y "⟫" => @inner 𝕜 _ _ x y
 
 open scoped NNReal
@@ -55,7 +54,6 @@ namespace ContinuousLinearMap
 
 variable (T : E →L[𝕜] E)
 
--- mathport name: exprrayleigh_quotient
 local notation "rayleigh_quotient" => fun x : E => T.reApplyInnerSelf x / ‖(x : E)‖ ^ 2
 
 theorem rayleigh_smul (x : E) {c : 𝕜} (hc : c ≠ 0) :
@@ -121,7 +119,6 @@ theorem LinearMap.IsSymmetric.hasStrictFDerivAt_reApplyInnerSelf {T : F →L[ℝ
 
 variable [CompleteSpace F] {T : F →L[ℝ] F}
 
--- mathport name: exprrayleigh_quotient
 local notation "rayleigh_quotient" => fun x : F => T.reApplyInnerSelf x / ‖(x : F)‖ ^ 2
 
 theorem linearly_dependent_of_isLocalExtrOn (hT : IsSelfAdjoint T) {x₀ : F}
@@ -177,7 +174,6 @@ section CompleteSpace
 
 variable [CompleteSpace E] {T : E →L[𝕜] E}
 
--- mathport name: exprrayleigh_quotient
 local notation "rayleigh_quotient" => fun x : E => T.reApplyInnerSelf x / ‖(x : E)‖ ^ 2
 
 theorem eq_smul_self_of_isLocalExtrOn (hT : IsSelfAdjoint T) {x₀ : E}
@@ -250,8 +246,6 @@ namespace LinearMap
 
 namespace IsSymmetric
 
-include _i
-
 /-- The supremum of the Rayleigh quotient of a symmetric operator `T` on a nontrivial
 finite-dimensional vector space is an eigenvalue for that operator. -/
 theorem hasEigenvalue_iSup_of_finiteDimensional (hT : T.IsSymmetric) :
@@ -296,8 +290,6 @@ theorem hasEigenvalue_iInf_of_finiteDimensional (hT : T.IsSymmetric) :
   exact has_eigenvalue_of_has_eigenvector (T'.prop.has_eigenvector_of_is_min_on hx₀_ne this)
 #align linear_map.is_symmetric.has_eigenvalue_infi_of_finite_dimensional LinearMap.IsSymmetric.hasEigenvalue_iInf_of_finiteDimensional
 
-omit _i
-
 theorem subsingleton_of_no_eigenvalue_finiteDimensional (hT : T.IsSymmetric)
     (hT' : ∀ μ : 𝕜, Module.End.eigenspace (T : E →ₗ[𝕜] E) μ = ⊥) : Subsingleton E :=
   (subsingleton_or_nontrivial E).resolve_right fun h =>

mathlib commit https://github.com/leanprover-community/mathlib/commit/31c24aa72e7b3e5ed97a8412470e904f82b81004

@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Heather Macbeth, Frédéric Dupuis
 
 ! This file was ported from Lean 3 source module analysis.inner_product_space.rayleigh
-! leanprover-community/mathlib commit 3f655f5297b030a87d641ad4e825af8d9679eb0b
+! leanprover-community/mathlib commit 6b0169218d01f2837d79ea2784882009a0da1aa1
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/
@@ -12,7 +12,7 @@ import Mathbin.Analysis.InnerProductSpace.Calculus
 import Mathbin.Analysis.InnerProductSpace.Dual
 import Mathbin.Analysis.InnerProductSpace.Adjoint
 import Mathbin.Analysis.Calculus.LagrangeMultipliers
-import Mathbin.LinearAlgebra.Eigenspace
+import Mathbin.LinearAlgebra.Eigenspace.Basic
 
 /-!
 # The Rayleigh quotient

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

@@ -128,7 +128,7 @@ theorem linearly_dependent_of_isLocalExtrOn (hT : IsSelfAdjoint T) {x₀ : F}
     (hextr : IsLocalExtrOn T.reApplyInnerSelf (sphere (0 : F) ‖x₀‖) x₀) :
     ∃ a b : ℝ, (a, b) ≠ 0 ∧ a • x₀ + b • T x₀ = 0 :=
   by
-  have H : IsLocalExtrOn T.re_apply_inner_self { x : F | ‖x‖ ^ 2 = ‖x₀‖ ^ 2 } x₀ :=
+  have H : IsLocalExtrOn T.re_apply_inner_self {x : F | ‖x‖ ^ 2 = ‖x₀‖ ^ 2} x₀ :=
     by
     convert hextr
     ext x

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

@@ -47,7 +47,7 @@ variable {E : Type _} [NormedAddCommGroup E] [InnerProductSpace 𝕜 E]
 -- mathport name: «expr⟪ , ⟫»
 local notation "⟪" x ", " y "⟫" => @inner 𝕜 _ _ x y
 
-open NNReal
+open scoped NNReal
 
 open Module.End Metric
 

mathlib commit https://github.com/leanprover-community/mathlib/commit/33c67ae661dd8988516ff7f247b0be3018cdd952

@@ -108,16 +108,16 @@ section Real
 
 variable {F : Type _} [NormedAddCommGroup F] [InnerProductSpace ℝ F]
 
-theorem LinearMap.IsSymmetric.hasStrictFderivAt_reApplyInnerSelf {T : F →L[ℝ] F}
+theorem LinearMap.IsSymmetric.hasStrictFDerivAt_reApplyInnerSelf {T : F →L[ℝ] F}
     (hT : (T : F →ₗ[ℝ] F).IsSymmetric) (x₀ : F) :
-    HasStrictFderivAt T.reApplyInnerSelf (bit0 (innerSL ℝ (T x₀))) x₀ :=
+    HasStrictFDerivAt T.reApplyInnerSelf (bit0 (innerSL ℝ (T x₀))) x₀ :=
   by
-  convert T.has_strict_fderiv_at.inner _ (hasStrictFderivAt_id x₀)
+  convert T.has_strict_fderiv_at.inner _ (hasStrictFDerivAt_id x₀)
   ext y
   simp_rw [_root_.bit0, ContinuousLinearMap.comp_apply, ContinuousLinearMap.add_apply,
     innerSL_apply, fderivInnerClm_apply, id.def, ContinuousLinearMap.prod_apply,
     ContinuousLinearMap.id_apply, hT.apply_clm x₀ y, real_inner_comm _ x₀]
-#align linear_map.is_symmetric.has_strict_fderiv_at_re_apply_inner_self LinearMap.IsSymmetric.hasStrictFderivAt_reApplyInnerSelf
+#align linear_map.is_symmetric.has_strict_fderiv_at_re_apply_inner_self LinearMap.IsSymmetric.hasStrictFDerivAt_reApplyInnerSelf
 
 variable [CompleteSpace F] {T : F →L[ℝ] F}
 
@@ -136,7 +136,7 @@ theorem linearly_dependent_of_isLocalExtrOn (hT : IsSelfAdjoint T) {x₀ : F}
   -- find Lagrange multipliers for the function `T.re_apply_inner_self` and the
   -- hypersurface-defining function `λ x, ‖x‖ ^ 2`
   obtain ⟨a, b, h₁, h₂⟩ :=
-    IsLocalExtrOn.exists_multipliers_of_hasStrictFderivAt_1d H (hasStrictFderivAt_norm_sq x₀)
+    IsLocalExtrOn.exists_multipliers_of_hasStrictFDerivAt_1d H (hasStrictFDerivAt_norm_sq x₀)
       (hT.is_symmetric.has_strict_fderiv_at_re_apply_inner_self x₀)
   refine' ⟨a, b, h₁, _⟩
   apply (InnerProductSpace.toDualMap ℝ F).Injective

mathlib commit https://github.com/leanprover-community/mathlib/commit/e3fb84046afd187b710170887195d50bada934ee

@@ -86,19 +86,19 @@ theorem image_rayleigh_eq_image_rayleigh_sphere {r : ℝ} (hr : 0 < r) :
     exact ⟨x, ne_zero_of_mem_sphere hr.ne' ⟨x, hx⟩, hxT⟩
 #align continuous_linear_map.image_rayleigh_eq_image_rayleigh_sphere ContinuousLinearMap.image_rayleigh_eq_image_rayleigh_sphere
 
-theorem supᵢ_rayleigh_eq_supᵢ_rayleigh_sphere {r : ℝ} (hr : 0 < r) :
+theorem iSup_rayleigh_eq_iSup_rayleigh_sphere {r : ℝ} (hr : 0 < r) :
     (⨆ x : { x : E // x ≠ 0 }, rayleigh_quotient x) = ⨆ x : sphere (0 : E) r, rayleigh_quotient x :=
   show (⨆ x : ({0} : Set E)ᶜ, rayleigh_quotient x) = _ by
-    simp only [← @supₛ_image' _ _ _ _ rayleigh_quotient,
+    simp only [← @sSup_image' _ _ _ _ rayleigh_quotient,
       T.image_rayleigh_eq_image_rayleigh_sphere hr]
-#align continuous_linear_map.supr_rayleigh_eq_supr_rayleigh_sphere ContinuousLinearMap.supᵢ_rayleigh_eq_supᵢ_rayleigh_sphere
+#align continuous_linear_map.supr_rayleigh_eq_supr_rayleigh_sphere ContinuousLinearMap.iSup_rayleigh_eq_iSup_rayleigh_sphere
 
-theorem infᵢ_rayleigh_eq_infᵢ_rayleigh_sphere {r : ℝ} (hr : 0 < r) :
+theorem iInf_rayleigh_eq_iInf_rayleigh_sphere {r : ℝ} (hr : 0 < r) :
     (⨅ x : { x : E // x ≠ 0 }, rayleigh_quotient x) = ⨅ x : sphere (0 : E) r, rayleigh_quotient x :=
   show (⨅ x : ({0} : Set E)ᶜ, rayleigh_quotient x) = _ by
-    simp only [← @infₛ_image' _ _ _ _ rayleigh_quotient,
+    simp only [← @sInf_image' _ _ _ _ rayleigh_quotient,
       T.image_rayleigh_eq_image_rayleigh_sphere hr]
-#align continuous_linear_map.infi_rayleigh_eq_infi_rayleigh_sphere ContinuousLinearMap.infᵢ_rayleigh_eq_infᵢ_rayleigh_sphere
+#align continuous_linear_map.infi_rayleigh_eq_infi_rayleigh_sphere ContinuousLinearMap.iInf_rayleigh_eq_iInf_rayleigh_sphere
 
 end ContinuousLinearMap
 
@@ -211,7 +211,7 @@ theorem hasEigenvector_of_isMaxOn (hT : IsSelfAdjoint T) {x₀ : E} (hx₀ : x
   have hx₀' : 0 < ‖x₀‖ := by simp [hx₀]
   have hx₀'' : x₀ ∈ sphere (0 : E) ‖x₀‖ := by simp
   rw [T.supr_rayleigh_eq_supr_rayleigh_sphere hx₀']
-  refine' IsMaxOn.supᵢ_eq hx₀'' _
+  refine' IsMaxOn.iSup_eq hx₀'' _
   intro x hx
   dsimp
   have : ‖x‖ = ‖x₀‖ := by simpa using hx
@@ -230,7 +230,7 @@ theorem hasEigenvector_of_isMinOn (hT : IsSelfAdjoint T) {x₀ : E} (hx₀ : x
   have hx₀' : 0 < ‖x₀‖ := by simp [hx₀]
   have hx₀'' : x₀ ∈ sphere (0 : E) ‖x₀‖ := by simp
   rw [T.infi_rayleigh_eq_infi_rayleigh_sphere hx₀']
-  refine' IsMinOn.infᵢ_eq hx₀'' _
+  refine' IsMinOn.iInf_eq hx₀'' _
   intro x hx
   dsimp
   have : ‖x‖ = ‖x₀‖ := by simpa using hx
@@ -254,7 +254,7 @@ include _i
 
 /-- The supremum of the Rayleigh quotient of a symmetric operator `T` on a nontrivial
 finite-dimensional vector space is an eigenvalue for that operator. -/
-theorem hasEigenvalue_supᵢ_of_finiteDimensional (hT : T.IsSymmetric) :
+theorem hasEigenvalue_iSup_of_finiteDimensional (hT : T.IsSymmetric) :
     HasEigenvalue T ↑(⨆ x : { x : E // x ≠ 0 }, IsROrC.re ⟪T x, x⟫ / ‖(x : E)‖ ^ 2) :=
   by
   haveI := FiniteDimensional.proper_isROrC 𝕜 E
@@ -272,11 +272,11 @@ theorem hasEigenvalue_supᵢ_of_finiteDimensional (hT : T.IsSymmetric) :
     have : ‖x₀‖ ≠ 0 := by simp only [hx₀, norm_eq_zero, hx, Ne.def, not_false_iff]
     simpa [← norm_eq_zero, Ne.def]
   exact has_eigenvalue_of_has_eigenvector (T'.prop.has_eigenvector_of_is_max_on hx₀_ne this)
-#align linear_map.is_symmetric.has_eigenvalue_supr_of_finite_dimensional LinearMap.IsSymmetric.hasEigenvalue_supᵢ_of_finiteDimensional
+#align linear_map.is_symmetric.has_eigenvalue_supr_of_finite_dimensional LinearMap.IsSymmetric.hasEigenvalue_iSup_of_finiteDimensional
 
 /-- The infimum of the Rayleigh quotient of a symmetric operator `T` on a nontrivial
 finite-dimensional vector space is an eigenvalue for that operator. -/
-theorem hasEigenvalue_infᵢ_of_finiteDimensional (hT : T.IsSymmetric) :
+theorem hasEigenvalue_iInf_of_finiteDimensional (hT : T.IsSymmetric) :
     HasEigenvalue T ↑(⨅ x : { x : E // x ≠ 0 }, IsROrC.re ⟪T x, x⟫ / ‖(x : E)‖ ^ 2) :=
   by
   haveI := FiniteDimensional.proper_isROrC 𝕜 E
@@ -294,7 +294,7 @@ theorem hasEigenvalue_infᵢ_of_finiteDimensional (hT : T.IsSymmetric) :
     have : ‖x₀‖ ≠ 0 := by simp only [hx₀, norm_eq_zero, hx, Ne.def, not_false_iff]
     simpa [← norm_eq_zero, Ne.def]
   exact has_eigenvalue_of_has_eigenvector (T'.prop.has_eigenvector_of_is_min_on hx₀_ne this)
-#align linear_map.is_symmetric.has_eigenvalue_infi_of_finite_dimensional LinearMap.IsSymmetric.hasEigenvalue_infᵢ_of_finiteDimensional
+#align linear_map.is_symmetric.has_eigenvalue_infi_of_finite_dimensional LinearMap.IsSymmetric.hasEigenvalue_iInf_of_finiteDimensional
 
 omit _i
 

mathlib commit https://github.com/leanprover-community/mathlib/commit/d4437c68c8d350fc9d4e95e1e174409db35e30d7

@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Heather Macbeth, Frédéric Dupuis
 
 ! This file was ported from Lean 3 source module analysis.inner_product_space.rayleigh
-! leanprover-community/mathlib commit 46b633fd842bef9469441c0209906f6dddd2b4f5
+! leanprover-community/mathlib commit 3f655f5297b030a87d641ad4e825af8d9679eb0b
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/
@@ -79,7 +79,7 @@ theorem image_rayleigh_eq_image_rayleigh_sphere {r : ℝ} (hr : 0 < r) :
     let c : 𝕜 := ↑‖x‖⁻¹ * r
     have : c ≠ 0 := by simp [c, hx, hr.ne']
     refine' ⟨c • x, _, _⟩
-    · field_simp [norm_smul, IsROrC.norm_eq_abs, abs_of_nonneg hr.le]
+    · field_simp [norm_smul, abs_of_pos hr]
     · rw [T.rayleigh_smul x this]
       exact hxT
   · rintro ⟨x, hx, hxT⟩

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: Heather Macbeth, Frédéric Dupuis
 
 ! This file was ported from Lean 3 source module analysis.inner_product_space.rayleigh
-! leanprover-community/mathlib commit c78cad350eb321c81e1eacf68d14e3d3ba1e17f7
+! leanprover-community/mathlib commit 46b633fd842bef9469441c0209906f6dddd2b4f5
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/
@@ -42,7 +42,7 @@ A slightly more elaborate corollary is that if `E` is complete and `T` is a comp
 
 variable {𝕜 : Type _} [IsROrC 𝕜]
 
-variable {E : Type _} [InnerProductSpace 𝕜 E]
+variable {E : Type _} [NormedAddCommGroup E] [InnerProductSpace 𝕜 E]
 
 -- mathport name: «expr⟪ , ⟫»
 local notation "⟪" x ", " y "⟫" => @inner 𝕜 _ _ x y
@@ -106,13 +106,13 @@ namespace IsSelfAdjoint
 
 section Real
 
-variable {F : Type _} [InnerProductSpace ℝ F]
+variable {F : Type _} [NormedAddCommGroup F] [InnerProductSpace ℝ F]
 
 theorem LinearMap.IsSymmetric.hasStrictFderivAt_reApplyInnerSelf {T : F →L[ℝ] F}
     (hT : (T : F →ₗ[ℝ] F).IsSymmetric) (x₀ : F) :
     HasStrictFderivAt T.reApplyInnerSelf (bit0 (innerSL ℝ (T x₀))) x₀ :=
   by
-  convert T.has_strict_fderiv_at.inner (hasStrictFderivAt_id x₀)
+  convert T.has_strict_fderiv_at.inner _ (hasStrictFderivAt_id x₀)
   ext y
   simp_rw [_root_.bit0, ContinuousLinearMap.comp_apply, ContinuousLinearMap.add_apply,
     innerSL_apply, fderivInnerClm_apply, id.def, ContinuousLinearMap.prod_apply,

mathlib commit https://github.com/leanprover-community/mathlib/commit/7ec294687917cbc5c73620b4414ae9b5dd9ae1b4

@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Heather Macbeth, Frédéric Dupuis
 
 ! This file was ported from Lean 3 source module analysis.inner_product_space.rayleigh
-! leanprover-community/mathlib commit ce613251beb65cb746070b76d9be5765e5968003
+! leanprover-community/mathlib commit c78cad350eb321c81e1eacf68d14e3d3ba1e17f7
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/
@@ -110,7 +110,7 @@ variable {F : Type _} [InnerProductSpace ℝ F]
 
 theorem LinearMap.IsSymmetric.hasStrictFderivAt_reApplyInnerSelf {T : F →L[ℝ] F}
     (hT : (T : F →ₗ[ℝ] F).IsSymmetric) (x₀ : F) :
-    HasStrictFderivAt T.reApplyInnerSelf (bit0 (innerSL (T x₀) : F →L[ℝ] ℝ)) x₀ :=
+    HasStrictFderivAt T.reApplyInnerSelf (bit0 (innerSL ℝ (T x₀))) x₀ :=
   by
   convert T.has_strict_fderiv_at.inner (hasStrictFderivAt_id x₀)
   ext y
@@ -141,7 +141,7 @@ theorem linearly_dependent_of_isLocalExtrOn (hT : IsSelfAdjoint T) {x₀ : F}
   refine' ⟨a, b, h₁, _⟩
   apply (InnerProductSpace.toDualMap ℝ F).Injective
   simp only [LinearIsometry.map_add, LinearIsometry.map_smul, LinearIsometry.map_zero]
-  change a • innerSL x₀ + b • innerSL (T x₀) = 0
+  change a • innerSL _ x₀ + b • innerSL _ (T x₀) = 0
   apply smul_right_injective (F →L[ℝ] ℝ) (two_ne_zero : (2 : ℝ) ≠ 0)
   simpa only [_root_.bit0, add_smul, smul_add, one_smul, add_zero] using h₂
 #align is_self_adjoint.linearly_dependent_of_is_local_extr_on IsSelfAdjoint.linearly_dependent_of_isLocalExtrOn

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

Co-authored-by: Ruben Van de Velde <65514131+Ruben-VandeVelde@users.noreply.github.com>

@@ -110,7 +110,7 @@ theorem _root_.LinearMap.IsSymmetric.hasStrictFDerivAt_reApplyInnerSelf {T : F 
   convert T.hasStrictFDerivAt.inner ℝ (hasStrictFDerivAt_id x₀) using 1
   ext y
   rw [ContinuousLinearMap.smul_apply, ContinuousLinearMap.comp_apply, fderivInnerCLM_apply,
-    ContinuousLinearMap.prod_apply, innerSL_apply, id.def, ContinuousLinearMap.id_apply,
+    ContinuousLinearMap.prod_apply, innerSL_apply, id, ContinuousLinearMap.id_apply,
     hT.apply_clm x₀ y, real_inner_comm _ x₀, two_smul]
 #align linear_map.is_symmetric.has_strict_fderiv_at_re_apply_inner_self LinearMap.IsSymmetric.hasStrictFDerivAt_reApplyInnerSelf
 

@@ -252,8 +252,8 @@ theorem hasEigenvalue_iSup_of_finiteDimensional (hT : T.IsSymmetric) :
   have hx₀ : ‖x₀‖ = ‖x‖ := by simpa using hx₀'
   have : IsMaxOn T'.val.reApplyInnerSelf (sphere 0 ‖x₀‖) x₀ := by simpa only [← hx₀] using hTx₀
   have hx₀_ne : x₀ ≠ 0 := by
-    have : ‖x₀‖ ≠ 0 := by simp only [hx₀, norm_eq_zero, hx, Ne.def, not_false_iff]
-    simpa [← norm_eq_zero, Ne.def]
+    have : ‖x₀‖ ≠ 0 := by simp only [hx₀, norm_eq_zero, hx, Ne, not_false_iff]
+    simpa [← norm_eq_zero, Ne]
   exact hasEigenvalue_of_hasEigenvector (T'.prop.hasEigenvector_of_isMaxOn hx₀_ne this)
 #align linear_map.is_symmetric.has_eigenvalue_supr_of_finite_dimensional LinearMap.IsSymmetric.hasEigenvalue_iSup_of_finiteDimensional
 
@@ -272,8 +272,8 @@ theorem hasEigenvalue_iInf_of_finiteDimensional (hT : T.IsSymmetric) :
   have hx₀ : ‖x₀‖ = ‖x‖ := by simpa using hx₀'
   have : IsMinOn T'.val.reApplyInnerSelf (sphere 0 ‖x₀‖) x₀ := by simpa only [← hx₀] using hTx₀
   have hx₀_ne : x₀ ≠ 0 := by
-    have : ‖x₀‖ ≠ 0 := by simp only [hx₀, norm_eq_zero, hx, Ne.def, not_false_iff]
-    simpa [← norm_eq_zero, Ne.def]
+    have : ‖x₀‖ ≠ 0 := by simp only [hx₀, norm_eq_zero, hx, Ne, not_false_iff]
+    simpa [← norm_eq_zero, Ne]
   exact hasEigenvalue_of_hasEigenvector (T'.prop.hasEigenvector_of_isMinOn hx₀_ne this)
 #align linear_map.is_symmetric.has_eigenvalue_infi_of_finite_dimensional LinearMap.IsSymmetric.hasEigenvalue_iInf_of_finiteDimensional
 

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.

@@ -37,7 +37,7 @@ A slightly more elaborate corollary is that if `E` is complete and `T` is a comp
 -/
 
 
-variable {𝕜 : Type*} [IsROrC 𝕜]
+variable {𝕜 : Type*} [RCLike 𝕜]
 variable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace 𝕜 E]
 
 local notation "⟪" x ", " y "⟫" => @inner 𝕜 _ _ x y
@@ -132,7 +132,7 @@ theorem linearly_dependent_of_isLocalExtrOn (hT : IsSelfAdjoint T) {x₀ : F}
   apply (InnerProductSpace.toDualMap ℝ F).injective
   simp only [LinearIsometry.map_add, LinearIsometry.map_smul, LinearIsometry.map_zero]
   -- Note: #8386 changed `map_smulₛₗ` into `map_smulₛₗ _`
-  simp only [map_smulₛₗ _, IsROrC.conj_to_real]
+  simp only [map_smulₛₗ _, RCLike.conj_to_real]
   change a • innerSL ℝ x₀ + b • innerSL ℝ (T x₀) = 0
   apply smul_right_injective (F →L[ℝ] ℝ) (two_ne_zero : (2 : ℝ) ≠ 0)
   simpa only [two_smul, smul_add, add_smul, add_zero] using h₂
@@ -170,7 +170,7 @@ variable [CompleteSpace E] {T : E →L[𝕜] E}
 theorem eq_smul_self_of_isLocalExtrOn (hT : IsSelfAdjoint T) {x₀ : E}
     (hextr : IsLocalExtrOn T.reApplyInnerSelf (sphere (0 : E) ‖x₀‖) x₀) :
     T x₀ = (↑(T.rayleighQuotient x₀) : 𝕜) • x₀ := by
-  letI := InnerProductSpace.isROrCToReal 𝕜 E
+  letI := InnerProductSpace.rclikeToReal 𝕜 E
   let hSA := hT.isSymmetric.restrictScalars.toSelfAdjoint.prop
   exact hSA.eq_smul_self_of_isLocalExtrOn_real hextr
 #align is_self_adjoint.eq_smul_self_of_is_local_extr_on IsSelfAdjoint.eq_smul_self_of_isLocalExtrOn
@@ -240,8 +240,8 @@ namespace IsSymmetric
 /-- The supremum of the Rayleigh quotient of a symmetric operator `T` on a nontrivial
 finite-dimensional vector space is an eigenvalue for that operator. -/
 theorem hasEigenvalue_iSup_of_finiteDimensional (hT : T.IsSymmetric) :
-    HasEigenvalue T ↑(⨆ x : { x : E // x ≠ 0 }, IsROrC.re ⟪T x, x⟫ / ‖(x : E)‖ ^ 2 : ℝ) := by
-  haveI := FiniteDimensional.proper_isROrC 𝕜 E
+    HasEigenvalue T ↑(⨆ x : { x : E // x ≠ 0 }, RCLike.re ⟪T x, x⟫ / ‖(x : E)‖ ^ 2 : ℝ) := by
+  haveI := FiniteDimensional.proper_rclike 𝕜 E
   let T' := hT.toSelfAdjoint
   obtain ⟨x, hx⟩ : ∃ x : E, x ≠ 0 := exists_ne 0
   have H₁ : IsCompact (sphere (0 : E) ‖x‖) := isCompact_sphere _ _
@@ -260,8 +260,8 @@ theorem hasEigenvalue_iSup_of_finiteDimensional (hT : T.IsSymmetric) :
 /-- The infimum of the Rayleigh quotient of a symmetric operator `T` on a nontrivial
 finite-dimensional vector space is an eigenvalue for that operator. -/
 theorem hasEigenvalue_iInf_of_finiteDimensional (hT : T.IsSymmetric) :
-    HasEigenvalue T ↑(⨅ x : { x : E // x ≠ 0 }, IsROrC.re ⟪T x, x⟫ / ‖(x : E)‖ ^ 2 : ℝ) := by
-  haveI := FiniteDimensional.proper_isROrC 𝕜 E
+    HasEigenvalue T ↑(⨅ x : { x : E // x ≠ 0 }, RCLike.re ⟪T x, x⟫ / ‖(x : E)‖ ^ 2 : ℝ) := by
+  haveI := FiniteDimensional.proper_rclike 𝕜 E
   let T' := hT.toSelfAdjoint
   obtain ⟨x, hx⟩ : ∃ x : E, x ≠ 0 := exists_ne 0
   have H₁ : IsCompact (sphere (0 : E) ‖x‖) := isCompact_sphere _ _

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)

@@ -38,7 +38,6 @@ A slightly more elaborate corollary is that if `E` is complete and `T` is a comp
 
 
 variable {𝕜 : Type*} [IsROrC 𝕜]
-
 variable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace 𝕜 E]
 
 local notation "⟪" x ", " y "⟫" => @inner 𝕜 _ _ x y

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

@@ -72,9 +72,9 @@ theorem image_rayleigh_eq_image_rayleigh_sphere {r : ℝ} (hr : 0 < r) :
   · rintro ⟨x, hx : x ≠ 0, hxT⟩
     have : ‖x‖ ≠ 0 := by simp [hx]
     let c : 𝕜 := ↑‖x‖⁻¹ * r
-    have : c ≠ 0 := by simp [hx, hr.ne']
+    have : c ≠ 0 := by simp [c, hx, hr.ne']
     refine' ⟨c • x, _, _⟩
-    · field_simp [norm_smul, abs_of_pos hr]
+    · field_simp [c, norm_smul, abs_of_pos hr]
     · rw [T.rayleigh_smul x this]
       exact hxT
   · rintro ⟨x, hx, hxT⟩
@@ -154,7 +154,7 @@ theorem eq_smul_self_of_isLocalExtrOn_real (hT : IsSelfAdjoint T) {x₀ : F}
   have hc : T x₀ = c • x₀ := by
     have : b * (b⁻¹ * a) = a := by field_simp [mul_comm]
     apply smul_right_injective F hb
-    simp [eq_neg_of_add_eq_zero_left h₂, ← mul_smul, this]
+    simp [c, eq_neg_of_add_eq_zero_left h₂, ← mul_smul, this]
   convert hc
   have : ‖x₀‖ ≠ 0 := by simp [hx₀]
   have := congr_arg (fun x => ⟪x, x₀⟫_ℝ) hc

Use Lean 4 syntax fun x ↦ instead, matching the style guide. This is close to exhaustive for doc comments; mathlib has about 460 remaining uses of λ (not all in Lean 3 syntax).

@@ -15,18 +15,18 @@ import Mathlib.LinearAlgebra.Eigenspace.Basic
 # The Rayleigh quotient
 
 The Rayleigh quotient of a self-adjoint operator `T` on an inner product space `E` is the function
-`λ x, ⟪T x, x⟫ / ‖x‖ ^ 2`.
+`fun x ↦ ⟪T x, x⟫ / ‖x‖ ^ 2`.
 
 The main results of this file are `IsSelfAdjoint.hasEigenvector_of_isMaxOn` and
 `IsSelfAdjoint.hasEigenvector_of_isMinOn`, which state that if `E` is complete, and if the
 Rayleigh quotient attains its global maximum/minimum over some sphere at the point `x₀`, then `x₀`
-is an eigenvector of `T`, and the `iSup`/`iInf` of `λ x, ⟪T x, x⟫ / ‖x‖ ^ 2` is the corresponding
+is an eigenvector of `T`, and the `iSup`/`iInf` of `fun x ↦ ⟪T x, x⟫ / ‖x‖ ^ 2` is the corresponding
 eigenvalue.
 
 The corollaries `LinearMap.IsSymmetric.hasEigenvalue_iSup_of_finiteDimensional` and
 `LinearMap.IsSymmetric.hasEigenvalue_iSup_of_finiteDimensional` state that if `E` is
 finite-dimensional and nontrivial, then `T` has some (nonzero) eigenvectors with eigenvalue the
-`iSup`/`iInf` of `λ x, ⟪T x, x⟫ / ‖x‖ ^ 2`.
+`iSup`/`iInf` of `fun x ↦ ⟪T x, x⟫ / ‖x‖ ^ 2`.
 
 ## TODO
 
@@ -125,7 +125,7 @@ theorem linearly_dependent_of_isLocalExtrOn (hT : IsSelfAdjoint T) {x₀ : F}
     ext x
     simp [dist_eq_norm]
   -- find Lagrange multipliers for the function `T.re_apply_inner_self` and the
-  -- hypersurface-defining function `λ x, ‖x‖ ^ 2`
+  -- hypersurface-defining function `fun x ↦ ‖x‖ ^ 2`
   obtain ⟨a, b, h₁, h₂⟩ :=
     IsLocalExtrOn.exists_multipliers_of_hasStrictFDerivAt_1d H (hasStrictFDerivAt_norm_sq x₀)
       (hT.isSymmetric.hasStrictFDerivAt_reApplyInnerSelf x₀)

The FunLike hierarchy is very big and gets scanned through each time we need a coercion (via the CoeFun instance). It looks like unbundled inheritance suits Lean 4 better here. The only class that still extends FunLike is EquivLike, since that has a custom coe_injective' field that is easier to implement. All other classes should take FunLike or EquivLike as a parameter.

Zulip thread

Important changes

Previously, morphism classes would be Type-valued and extend FunLike:

/-- `MyHomClass F A B` states that `F` is a type of `MyClass.op`-preserving morphisms.
You should extend this class when you extend `MyHom`. -/
class MyHomClass (F : Type*) (A B : outParam <| Type*) [MyClass A] [MyClass B]
  extends FunLike F A B :=
(map_op : ∀ (f : F) (x y : A), f (MyClass.op x y) = MyClass.op (f x) (f y))

After this PR, they should be Prop-valued and take FunLike as a parameter:

/-- `MyHomClass F A B` states that `F` is a type of `MyClass.op`-preserving morphisms.
You should extend this class when you extend `MyHom`. -/
class MyHomClass (F : Type*) (A B : outParam <| Type*) [MyClass A] [MyClass B]
  [FunLike F A B] : Prop :=
(map_op : ∀ (f : F) (x y : A), f (MyClass.op x y) = MyClass.op (f x) (f y))

(Note that A B stay marked as outParam even though they are not purely required to be so due to the FunLike parameter already filling them in. This is required to see through type synonyms, which is important in the category theory library. Also, I think keeping them as outParam is slightly faster.)

Similarly, MyEquivClass should take EquivLike as a parameter.

As a result, every mention of [MyHomClass F A B] should become [FunLike F A B] [MyHomClass F A B].

Remaining issues

Slower (failing) search

While overall this gives some great speedups, there are some cases that are noticeably slower. In particular, a failing application of a lemma such as map_mul is more expensive. This is due to suboptimal processing of arguments. For example:

variable [FunLike F M N] [Mul M] [Mul N] (f : F) (x : M) (y : M)

theorem map_mul [MulHomClass F M N] : f (x * y) = f x * f y

example [AddHomClass F A B] : f (x * y) = f x * f y := map_mul f _ _

Before this PR, applying map_mul f gives the goals [Mul ?M] [Mul ?N] [MulHomClass F ?M ?N]. Since M and N are out_params, [MulHomClass F ?M ?N] is synthesized first, supplies values for ?M and ?N and then the Mul M and Mul N instances can be found.

After this PR, the goals become [FunLike F ?M ?N] [Mul ?M] [Mul ?N] [MulHomClass F ?M ?N]. Now [FunLike F ?M ?N] is synthesized first, supplies values for ?M and ?N and then the Mul M and Mul N instances can be found, before trying MulHomClass F M N which fails. Since the Mul hierarchy is very big, this can be slow to fail, especially when there is no such Mul instance.

A long-term but harder to achieve solution would be to specify the order in which instance goals get solved. For example, we'd like to change the arguments to map_mul to look like [FunLike F M N] [Mul M] [Mul N] [highPriority <| MulHomClass F M N] because MulHomClass fails or succeeds much faster than the others.

As a consequence, the simpNF linter is much slower since by design it tries and fails to apply many map_ lemmas. The same issue occurs a few times in existing calls to simp [map_mul], where map_mul is tried "too soon" and fails. Thanks to the speedup of leanprover/lean4#2478 the impact is very limited, only in files that already were close to the timeout.

simp not firing sometimes

This affects map_smulₛₗ and related definitions. For simp lemmas Lean apparently uses a slightly different mechanism to find instances, so that rw can find every argument to map_smulₛₗ successfully but simp can't: leanprover/lean4#3701.

Missing instances due to unification failing

Especially in the category theory library, we might sometimes have a type A which is also accessible as a synonym (Bundled A hA).1. Instance synthesis doesn't always work if we have f : A →* B but x * y : (Bundled A hA).1 or vice versa. This seems to be mostly fixed by keeping A B as outParams in MulHomClass F A B. (Presumably because Lean will do a definitional check A =?= (Bundled A hA).1 instead of using the syntax in the discrimination tree.)

Workaround for issues

The timeouts can be worked around for now by specifying which map_mul we mean, either as map_mul f for some explicit f, or as e.g. MonoidHomClass.map_mul.

map_smulₛₗ not firing as simp lemma can be worked around by going back to the pre-FunLike situation and making LinearMap.map_smulₛₗ a simp lemma instead of the generic map_smulₛₗ. Writing simp [map_smulₛₗ _] also works.

Co-authored-by: Matthew Ballard <matt@mrb.email> Co-authored-by: Scott Morrison <scott.morrison@gmail.com> Co-authored-by: Scott Morrison <scott@tqft.net> Co-authored-by: Anne Baanen <Vierkantor@users.noreply.github.com>

@@ -132,7 +132,8 @@ theorem linearly_dependent_of_isLocalExtrOn (hT : IsSelfAdjoint T) {x₀ : F}
   refine' ⟨a, b, h₁, _⟩
   apply (InnerProductSpace.toDualMap ℝ F).injective
   simp only [LinearIsometry.map_add, LinearIsometry.map_smul, LinearIsometry.map_zero]
-  simp only [map_smulₛₗ, IsROrC.conj_to_real]
+  -- Note: #8386 changed `map_smulₛₗ` into `map_smulₛₗ _`
+  simp only [map_smulₛₗ _, IsROrC.conj_to_real]
   change a • innerSL ℝ x₀ + b • innerSL ℝ (T x₀) = 0
   apply smul_right_injective (F →L[ℝ] ℝ) (two_ne_zero : (2 : ℝ) ≠ 0)
   simpa only [two_smul, smul_add, add_smul, add_zero] using h₂

Rename

• Complex.equivRealProdClm → Complex.equivRealProdCLM;
  • TODO: should this one use CLE?
• Complex.reClm → Complex.reCLM;
• Complex.imClm → Complex.imCLM;
• Complex.conjLie → Complex.conjLIE;
• Complex.conjCle → Complex.conjCLE;
• Complex.ofRealLi → Complex.ofRealLI;
• Complex.ofRealClm → Complex.ofRealCLM;
• fderivInnerClm → fderivInnerCLM;
• LinearPMap.adjointDomainMkClm → LinearPMap.adjointDomainMkCLM;
• LinearPMap.adjointDomainMkClmExtend → LinearPMap.adjointDomainMkCLMExtend;
• IsROrC.reClm → IsROrC.reCLM;
• IsROrC.imClm → IsROrC.imCLM;
• IsROrC.conjLie → IsROrC.conjLIE;
• IsROrC.conjCle → IsROrC.conjCLE;
• IsROrC.ofRealLi → IsROrC.ofRealLI;
• IsROrC.ofRealClm → IsROrC.ofRealCLM;
• MeasureTheory.condexpL1Clm → MeasureTheory.condexpL1CLM;
• algebraMapClm → algebraMapCLM;
• WeakDual.CharacterSpace.toClm → WeakDual.CharacterSpace.toCLM;
• BoundedContinuousFunction.evalClm → BoundedContinuousFunction.evalCLM;
• ContinuousMap.evalClm → ContinuousMap.evalCLM;
• TrivSqZeroExt.fstClm → TrivSqZeroExt.fstClm;
• TrivSqZeroExt.sndClm → TrivSqZeroExt.sndCLM;
• TrivSqZeroExt.inlClm → TrivSqZeroExt.inlCLM;
• TrivSqZeroExt.inrClm → TrivSqZeroExt.inrCLM

and related theorems.

@@ -110,7 +110,7 @@ theorem _root_.LinearMap.IsSymmetric.hasStrictFDerivAt_reApplyInnerSelf {T : F 
     HasStrictFDerivAt T.reApplyInnerSelf (2 • (innerSL ℝ (T x₀))) x₀ := by
   convert T.hasStrictFDerivAt.inner ℝ (hasStrictFDerivAt_id x₀) using 1
   ext y
-  rw [ContinuousLinearMap.smul_apply, ContinuousLinearMap.comp_apply, fderivInnerClm_apply,
+  rw [ContinuousLinearMap.smul_apply, ContinuousLinearMap.comp_apply, fderivInnerCLM_apply,
     ContinuousLinearMap.prod_apply, innerSL_apply, id.def, ContinuousLinearMap.id_apply,
     hT.apply_clm x₀ y, real_inner_comm _ x₀, two_smul]
 #align linear_map.is_symmetric.has_strict_fderiv_at_re_apply_inner_self LinearMap.IsSymmetric.hasStrictFDerivAt_reApplyInnerSelf

PR contents

This is the supremum of

• #8284
• #8056
• #8023
• #8332
• #8226 (already approved)
• #7834 (already approved)

along with some minor fixes from failures on nightly-testing as Mathlib master is merged into it.

Note that some PRs for changes that are already compatible with the current toolchain and will be necessary have already been split out: #8380.

I am hopeful that in future we will be able to progressively merge adaptation PRs into a bump/v4.X.0 branch, so we never end up with a "big merge" like this. However one of these adaptation PRs (#8056) predates my new scheme for combined CI, and it wasn't possible to keep that PR viable in the meantime.

Lean PRs involved in this bump

In particular this includes adjustments for the Lean PRs

• leanprover/lean4#2778
• leanprover/lean4#2790
• leanprover/lean4#2783
• leanprover/lean4#2825
• leanprover/lean4#2722

leanprover/lean4#2778

We can get rid of all the

local macro_rules | `($x ^ $y) => `(HPow.hPow $x $y) -- Porting note: See issue [lean4#2220](https://github.com/leanprover/lean4/pull/2220)

macros across Mathlib (and in any projects that want to write natural number powers of reals).

leanprover/lean4#2722

Changes the default behaviour of simp to (config := {decide := false}). This makes simp (and consequentially norm_num) less powerful, but also more consistent, and less likely to blow up in long failures. This requires a variety of changes: changing some previously by simp or norm_num to decide or rfl, or adding (config := {decide := true}).

leanprover/lean4#2783

This changed the behaviour of simp so that simp [f] will only unfold "fully applied" occurrences of f. The old behaviour can be recovered with simp (config := { unfoldPartialApp := true }). We may in future add a syntax for this, e.g. simp [!f]; please provide feedback! In the meantime, we have made the following changes:

• switching to using explicit lemmas that have the intended level of application
• (config := { unfoldPartialApp := true }) in some places, to recover the old behaviour
• Using @[eqns] to manually adjust the equation lemmas for a particular definition, recovering the old behaviour just for that definition. See #8371, where we do this for Function.comp and Function.flip.

This change in Lean may require further changes down the line (e.g. adding the !f syntax, and/or upstreaming the special treatment for Function.comp and Function.flip, and/or removing this special treatment). Please keep an open and skeptical mind about these changes!

Co-authored-by: leanprover-community-mathlib4-bot <leanprover-community-mathlib4-bot@users.noreply.github.com> Co-authored-by: Scott Morrison <scott.morrison@gmail.com> Co-authored-by: Eric Wieser <wieser.eric@gmail.com> Co-authored-by: Mauricio Collares <mauricio@collares.org>

@@ -117,8 +117,6 @@ theorem _root_.LinearMap.IsSymmetric.hasStrictFDerivAt_reApplyInnerSelf {T : F 
 
 variable [CompleteSpace F] {T : F →L[ℝ] F}
 
-local macro_rules | `($x ^ $y) => `(HPow.hPow $x $y) -- Porting note: See issue lean4#2220
-
 theorem linearly_dependent_of_isLocalExtrOn (hT : IsSelfAdjoint T) {x₀ : F}
     (hextr : IsLocalExtrOn T.reApplyInnerSelf (sphere (0 : F) ‖x₀‖) x₀) :
     ∃ a b : ℝ, (a, b) ≠ 0 ∧ a • x₀ + b • T x₀ = 0 := by
@@ -242,7 +240,7 @@ namespace IsSymmetric
 /-- The supremum of the Rayleigh quotient of a symmetric operator `T` on a nontrivial
 finite-dimensional vector space is an eigenvalue for that operator. -/
 theorem hasEigenvalue_iSup_of_finiteDimensional (hT : T.IsSymmetric) :
-    HasEigenvalue T ↑(⨆ x : { x : E // x ≠ 0 }, IsROrC.re ⟪T x, x⟫ / ‖(x : E)‖ ^ 2) := by
+    HasEigenvalue T ↑(⨆ x : { x : E // x ≠ 0 }, IsROrC.re ⟪T x, x⟫ / ‖(x : E)‖ ^ 2 : ℝ) := by
   haveI := FiniteDimensional.proper_isROrC 𝕜 E
   let T' := hT.toSelfAdjoint
   obtain ⟨x, hx⟩ : ∃ x : E, x ≠ 0 := exists_ne 0
@@ -262,7 +260,7 @@ theorem hasEigenvalue_iSup_of_finiteDimensional (hT : T.IsSymmetric) :
 /-- The infimum of the Rayleigh quotient of a symmetric operator `T` on a nontrivial
 finite-dimensional vector space is an eigenvalue for that operator. -/
 theorem hasEigenvalue_iInf_of_finiteDimensional (hT : T.IsSymmetric) :
-    HasEigenvalue T ↑(⨅ x : { x : E // x ≠ 0 }, IsROrC.re ⟪T x, x⟫ / ‖(x : E)‖ ^ 2) := by
+    HasEigenvalue T ↑(⨅ x : { x : E // x ≠ 0 }, IsROrC.re ⟪T x, x⟫ / ‖(x : E)‖ ^ 2 : ℝ) := by
   haveI := FiniteDimensional.proper_isROrC 𝕜 E
   let T' := hT.toSelfAdjoint
   obtain ⟨x, hx⟩ : ∃ x : E, x ≠ 0 := exists_ne 0

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

This has nice performance benefits.

@@ -37,9 +37,9 @@ A slightly more elaborate corollary is that if `E` is complete and `T` is a comp
 -/
 
 
-variable {𝕜 : Type _} [IsROrC 𝕜]
+variable {𝕜 : Type*} [IsROrC 𝕜]
 
-variable {E : Type _} [NormedAddCommGroup E] [InnerProductSpace 𝕜 E]
+variable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace 𝕜 E]
 
 local notation "⟪" x ", " y "⟫" => @inner 𝕜 _ _ x y
 
@@ -103,7 +103,7 @@ namespace IsSelfAdjoint
 
 section Real
 
-variable {F : Type _} [NormedAddCommGroup F] [InnerProductSpace ℝ F]
+variable {F : Type*} [NormedAddCommGroup F] [InnerProductSpace ℝ F]
 
 theorem _root_.LinearMap.IsSymmetric.hasStrictFDerivAt_reApplyInnerSelf {T : F →L[ℝ] F}
     (hT : (T : F →ₗ[ℝ] F).IsSymmetric) (x₀ : F) :

@@ -117,7 +117,7 @@ theorem _root_.LinearMap.IsSymmetric.hasStrictFDerivAt_reApplyInnerSelf {T : F 
 
 variable [CompleteSpace F] {T : F →L[ℝ] F}
 
-local macro_rules | `($x ^ $y)   => `(HPow.hPow $x $y)
+local macro_rules | `($x ^ $y) => `(HPow.hPow $x $y) -- Porting note: See issue lean4#2220
 
 theorem linearly_dependent_of_isLocalExtrOn (hT : IsSelfAdjoint T) {x₀ : F}
     (hextr : IsLocalExtrOn T.reApplyInnerSelf (sphere (0 : F) ‖x₀‖) x₀) :

• depends on: #5966

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

@@ -2,11 +2,6 @@
 Copyright (c) 2021 Heather Macbeth. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Heather Macbeth, Frédéric Dupuis
-
-! This file was ported from Lean 3 source module analysis.inner_product_space.rayleigh
-! leanprover-community/mathlib commit 6b0169218d01f2837d79ea2784882009a0da1aa1
-! Please do not edit these lines, except to modify the commit id
-! if you have ported upstream changes.
 -/
 import Mathlib.Analysis.InnerProductSpace.Calculus
 import Mathlib.Analysis.InnerProductSpace.Dual
@@ -14,6 +9,8 @@ import Mathlib.Analysis.InnerProductSpace.Adjoint
 import Mathlib.Analysis.Calculus.LagrangeMultipliers
 import Mathlib.LinearAlgebra.Eigenspace.Basic
 
+#align_import analysis.inner_product_space.rayleigh from "leanprover-community/mathlib"@"6b0169218d01f2837d79ea2784882009a0da1aa1"
+
 /-!
 # The Rayleigh quotient
 

@@ -85,17 +85,17 @@ theorem image_rayleigh_eq_image_rayleigh_sphere {r : ℝ} (hr : 0 < r) :
 #align continuous_linear_map.image_rayleigh_eq_image_rayleigh_sphere ContinuousLinearMap.image_rayleigh_eq_image_rayleigh_sphere
 
 theorem iSup_rayleigh_eq_iSup_rayleigh_sphere {r : ℝ} (hr : 0 < r) :
-    (⨆ x : { x : E // x ≠ 0 }, rayleighQuotient T x) =
+    ⨆ x : { x : E // x ≠ 0 }, rayleighQuotient T x =
       ⨆ x : sphere (0 : E) r, rayleighQuotient T x :=
-  show (⨆ x : ({0}ᶜ : Set E), rayleighQuotient T x) = _ by
+  show ⨆ x : ({0}ᶜ : Set E), rayleighQuotient T x = _ by
     simp only [← @sSup_image' _ _ _ _ (rayleighQuotient T),
       T.image_rayleigh_eq_image_rayleigh_sphere hr]
 #align continuous_linear_map.supr_rayleigh_eq_supr_rayleigh_sphere ContinuousLinearMap.iSup_rayleigh_eq_iSup_rayleigh_sphere
 
 theorem iInf_rayleigh_eq_iInf_rayleigh_sphere {r : ℝ} (hr : 0 < r) :
-    (⨅ x : { x : E // x ≠ 0 }, rayleighQuotient T x) =
+    ⨅ x : { x : E // x ≠ 0 }, rayleighQuotient T x =
       ⨅ x : sphere (0 : E) r, rayleighQuotient T x :=
-  show (⨅ x : ({0}ᶜ : Set E), rayleighQuotient T x) = _ by
+  show ⨅ x : ({0}ᶜ : Set E), rayleighQuotient T x = _ by
     simp only [← @sInf_image' _ _ _ _ (rayleighQuotient T),
       T.image_rayleigh_eq_image_rayleigh_sphere hr]
 #align continuous_linear_map.infi_rayleigh_eq_infi_rayleigh_sphere ContinuousLinearMap.iInf_rayleigh_eq_iInf_rayleigh_sphere

Co-authored-by: Jireh Loreaux <loreaujy@gmail.com>