analysis.normed_space.star.spectrum ⟷ Mathlib.Analysis.NormedSpace.Star.Spectrum

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

@@ -212,7 +212,7 @@ noncomputable instance (priority := 100) : StarHomClass F A ℂ
       have := AlgHom.apply_mem_spectrum φ (s : A)
       rw [selfAdjoint.val_re_map_spectrum s] at this
       rcases this with ⟨⟨_, _⟩, _, heq⟩
-      rw [← HEq, IsROrC.star_def, IsROrC.conj_ofReal]
+      rw [← HEq, RCLike.star_def, RCLike.conj_ofReal]
 
 #print AlgHomClass.instStarAlgHomClass /-
 /-- This is not an instance to avoid type class inference loops. See

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

@@ -37,9 +37,9 @@ theorem unitary.spectrum_subset_circle (u : unitary E) : spectrum 𝕜 (u : E) 
   nontriviality E
   refine' fun k hk => mem_sphere_zero_iff_norm.mpr (le_antisymm _ _)
   · simpa only [CstarRing.norm_coe_unitary u] using norm_le_norm_of_mem hk
-  · rw [← unitary.coe_toUnits_apply u] at hk 
+  · rw [← unitary.coe_toUnits_apply u] at hk
     have hnk := ne_zero_of_mem_of_unit hk
-    rw [← inv_inv (unitary.toUnits u), ← spectrum.map_inv, Set.mem_inv] at hk 
+    rw [← inv_inv (unitary.toUnits u), ← spectrum.map_inv, Set.mem_inv] at hk
     have : ‖k‖⁻¹ ≤ ‖↑(unitary.toUnits u)⁻¹‖; simpa only [norm_inv] using norm_le_norm_of_mem hk
     simpa using inv_le_of_inv_le (norm_pos_iff.mpr hnk) this
 #align unitary.spectrum_subset_circle unitary.spectrum_subset_circle
@@ -93,7 +93,7 @@ theorem IsStarNormal.spectralRadius_eq_nnnorm (a : A) [IsStarNormal a] :
   have h₂ :=
     ((ENNReal.continuous_pow 2).Tendsto (spectralRadius ℂ a)).comp
       (spectrum.pow_nnnorm_pow_one_div_tendsto_nhds_spectralRadius a)
-  rw [← HEq] at h₂ 
+  rw [← HEq] at h₂
   convert tendsto_nhds_unique h₂ (pow_nnnorm_pow_one_div_tendsto_nhds_spectral_radius (a⋆ * a))
   rw [(IsSelfAdjoint.star_mul_self a).spectralRadius_eq_nnnorm, sq, nnnorm_star_mul_self, coe_mul]
 #align is_star_normal.spectral_radius_eq_nnnorm IsStarNormal.spectralRadius_eq_nnnorm
@@ -210,7 +210,7 @@ noncomputable instance (priority := 100) : StarHomClass F A ℂ
       simp only [map_add, map_smul, star_add, star_smul, hsa, selfAdjoint.star_val_eq]
     · intro s
       have := AlgHom.apply_mem_spectrum φ (s : A)
-      rw [selfAdjoint.val_re_map_spectrum s] at this 
+      rw [selfAdjoint.val_re_map_spectrum s] at this
       rcases this with ⟨⟨_, _⟩, _, heq⟩
       rw [← HEq, IsROrC.star_def, IsROrC.conj_ofReal]
 

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

@@ -203,7 +203,7 @@ See note [lower instance priority] -/
 noncomputable instance (priority := 100) : StarHomClass F A ℂ
     where
   coe φ := φ
-  coe_injective' := FunLike.coe_injective'
+  coe_injective' := DFunLike.coe_injective'
   map_star φ a := by
     suffices hsa : ∀ s : selfAdjoint A, (φ s)⋆ = φ s
     · rw [← realPart_add_I_smul_imaginaryPart a]

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

@@ -104,9 +104,9 @@ theorem IsStarNormal.spectralRadius_eq_nnnorm (a : A) [IsStarNormal a] :
 theorem IsSelfAdjoint.mem_spectrum_eq_re [StarModule ℂ A] {a : A} (ha : IsSelfAdjoint a) {z : ℂ}
     (hz : z ∈ spectrum ℂ a) : z = z.re :=
   by
-  have hu := exp_mem_unitary_of_mem_skewAdjoint ℂ (ha.smul_mem_skew_adjoint conj_I)
+  have hu := NormedSpace.exp_mem_unitary_of_mem_skewAdjoint ℂ (ha.smul_mem_skew_adjoint conj_I)
   let Iu := Units.mk0 I I_ne_zero
-  have : exp ℂ (I • z) ∈ spectrum ℂ (exp ℂ (I • a)) := by
+  have : NormedSpace.exp ℂ (I • z) ∈ spectrum ℂ (NormedSpace.exp ℂ (I • a)) := by
     simpa only [Units.smul_def, Units.val_mk0] using
       spectrum.exp_mem_exp (Iu • a) (smul_mem_smul_iff.mpr hz)
   exact

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

@@ -3,10 +3,10 @@ Copyright (c) 2022 Jireh Loreaux. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Jireh Loreaux
 -/
-import Mathbin.Analysis.NormedSpace.Star.Basic
-import Mathbin.Analysis.NormedSpace.Spectrum
-import Mathbin.Analysis.SpecialFunctions.Exponential
-import Mathbin.Algebra.Star.StarAlgHom
+import Analysis.NormedSpace.Star.Basic
+import Analysis.NormedSpace.Spectrum
+import Analysis.SpecialFunctions.Exponential
+import Algebra.Star.StarAlgHom
 
 #align_import analysis.normed_space.star.spectrum from "leanprover-community/mathlib"@"087c325ae0ab42dbdd5dee55bc37d3d5a0bf2197"
 

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

@@ -2,17 +2,14 @@
 Copyright (c) 2022 Jireh Loreaux. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Jireh Loreaux
-
-! This file was ported from Lean 3 source module analysis.normed_space.star.spectrum
-! leanprover-community/mathlib commit 087c325ae0ab42dbdd5dee55bc37d3d5a0bf2197
-! Please do not edit these lines, except to modify the commit id
-! if you have ported upstream changes.
 -/
 import Mathbin.Analysis.NormedSpace.Star.Basic
 import Mathbin.Analysis.NormedSpace.Spectrum
 import Mathbin.Analysis.SpecialFunctions.Exponential
 import Mathbin.Algebra.Star.StarAlgHom
 
+#align_import analysis.normed_space.star.spectrum from "leanprover-community/mathlib"@"087c325ae0ab42dbdd5dee55bc37d3d5a0bf2197"
+
 /-! # Spectral properties in C⋆-algebras
 
 > THIS FILE IS SYNCHRONIZED WITH MATHLIB4.

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

@@ -21,7 +21,6 @@ In this file, we establish various properties related to the spectrum of element
 -/
 
 
--- mathport name: «expr ⋆»
 local postfix:max "⋆" => star
 
 section
@@ -35,6 +34,7 @@ section UnitarySpectrum
 variable {𝕜 : Type _} [NormedField 𝕜] {E : Type _} [NormedRing E] [StarRing E] [CstarRing E]
   [NormedAlgebra 𝕜 E] [CompleteSpace E]
 
+#print unitary.spectrum_subset_circle /-
 theorem unitary.spectrum_subset_circle (u : unitary E) : spectrum 𝕜 (u : E) ⊆ Metric.sphere 0 1 :=
   by
   nontriviality E
@@ -46,11 +46,14 @@ theorem unitary.spectrum_subset_circle (u : unitary E) : spectrum 𝕜 (u : E) 
     have : ‖k‖⁻¹ ≤ ‖↑(unitary.toUnits u)⁻¹‖; simpa only [norm_inv] using norm_le_norm_of_mem hk
     simpa using inv_le_of_inv_le (norm_pos_iff.mpr hnk) this
 #align unitary.spectrum_subset_circle unitary.spectrum_subset_circle
+-/
 
+#print spectrum.subset_circle_of_unitary /-
 theorem spectrum.subset_circle_of_unitary {u : E} (h : u ∈ unitary E) :
     spectrum 𝕜 u ⊆ Metric.sphere 0 1 :=
   unitary.spectrum_subset_circle ⟨u, h⟩
 #align spectrum.subset_circle_of_unitary spectrum.subset_circle_of_unitary
+-/
 
 end UnitarySpectrum
 
@@ -61,9 +64,9 @@ open Complex
 variable {A : Type _} [NormedRing A] [NormedAlgebra ℂ A] [CompleteSpace A] [StarRing A]
   [CstarRing A]
 
--- mathport name: «expr↑ₐ»
 local notation "↑ₐ" => algebraMap ℂ A
 
+#print IsSelfAdjoint.spectralRadius_eq_nnnorm /-
 theorem IsSelfAdjoint.spectralRadius_eq_nnnorm {a : A} (ha : IsSelfAdjoint a) :
     spectralRadius ℂ a = ‖a‖₊ :=
   by
@@ -76,7 +79,9 @@ theorem IsSelfAdjoint.spectralRadius_eq_nnnorm {a : A} (ha : IsSelfAdjoint a) :
   rw [Function.comp_apply, ha.nnnorm_pow_two_pow, ENNReal.coe_pow, ← rpow_nat_cast, ← rpow_mul]
   simp
 #align is_self_adjoint.spectral_radius_eq_nnnorm IsSelfAdjoint.spectralRadius_eq_nnnorm
+-/
 
+#print IsStarNormal.spectralRadius_eq_nnnorm /-
 theorem IsStarNormal.spectralRadius_eq_nnnorm (a : A) [IsStarNormal a] :
     spectralRadius ℂ a = ‖a‖₊ :=
   by
@@ -95,7 +100,9 @@ theorem IsStarNormal.spectralRadius_eq_nnnorm (a : A) [IsStarNormal a] :
   convert tendsto_nhds_unique h₂ (pow_nnnorm_pow_one_div_tendsto_nhds_spectral_radius (a⋆ * a))
   rw [(IsSelfAdjoint.star_mul_self a).spectralRadius_eq_nnnorm, sq, nnnorm_star_mul_self, coe_mul]
 #align is_star_normal.spectral_radius_eq_nnnorm IsStarNormal.spectralRadius_eq_nnnorm
+-/
 
+#print IsSelfAdjoint.mem_spectrum_eq_re /-
 /-- Any element of the spectrum of a selfadjoint is real. -/
 theorem IsSelfAdjoint.mem_spectrum_eq_re [StarModule ℂ A] {a : A} (ha : IsSelfAdjoint a) {z : ℂ}
     (hz : z ∈ spectrum ℂ a) : z = z.re :=
@@ -112,13 +119,17 @@ theorem IsSelfAdjoint.mem_spectrum_eq_re [StarModule ℂ A] {a : A} (ha : IsSelf
           Real.exp_eq_one_iff, smul_eq_mul, I_mul, neg_eq_zero] using
           spectrum.subset_circle_of_unitary hu this)
 #align is_self_adjoint.mem_spectrum_eq_re IsSelfAdjoint.mem_spectrum_eq_re
+-/
 
+#print selfAdjoint.mem_spectrum_eq_re /-
 /-- Any element of the spectrum of a selfadjoint is real. -/
 theorem selfAdjoint.mem_spectrum_eq_re [StarModule ℂ A] (a : selfAdjoint A) {z : ℂ}
     (hz : z ∈ spectrum ℂ (a : A)) : z = z.re :=
   a.Prop.mem_spectrum_eq_re hz
 #align self_adjoint.mem_spectrum_eq_re selfAdjoint.mem_spectrum_eq_re
+-/
 
+#print IsSelfAdjoint.val_re_map_spectrum /-
 /-- The spectrum of a selfadjoint is real -/
 theorem IsSelfAdjoint.val_re_map_spectrum [StarModule ℂ A] {a : A} (ha : IsSelfAdjoint a) :
     spectrum ℂ a = (coe ∘ re '' spectrum ℂ a : Set ℂ) :=
@@ -127,12 +138,15 @@ theorem IsSelfAdjoint.val_re_map_spectrum [StarModule ℂ A] {a : A} (ha : IsSel
     rintro ⟨z, hz, rfl⟩
     simpa only [(ha.mem_spectrum_eq_re hz).symm, Function.comp_apply] using hz
 #align is_self_adjoint.coe_re_map_spectrum IsSelfAdjoint.val_re_map_spectrum
+-/
 
+#print selfAdjoint.val_re_map_spectrum /-
 /-- The spectrum of a selfadjoint is real -/
 theorem selfAdjoint.val_re_map_spectrum [StarModule ℂ A] (a : selfAdjoint A) :
     spectrum ℂ (a : A) = (coe ∘ re '' spectrum ℂ (a : A) : Set ℂ) :=
   a.property.val_re_map_spectrum
 #align self_adjoint.coe_re_map_spectrum selfAdjoint.val_re_map_spectrum
+-/
 
 end ComplexScalars
 
@@ -142,8 +156,7 @@ variable {F A B : Type _} [NormedRing A] [NormedAlgebra ℂ A] [CompleteSpace A]
   [CstarRing A] [NormedRing B] [NormedAlgebra ℂ B] [CompleteSpace B] [StarRing B] [CstarRing B]
   [hF : StarAlgHomClass F ℂ A B] (φ : F)
 
-include hF
-
+#print StarAlgHom.nnnorm_apply_le /-
 /-- A star algebra homomorphism of complex C⋆-algebras is norm contractive. -/
 theorem nnnorm_apply_le (a : A) : ‖(φ a : B)‖₊ ≤ ‖a‖₊ :=
   by
@@ -159,11 +172,14 @@ theorem nnnorm_apply_le (a : A) : ‖(φ a : B)‖₊ ≤ ‖a‖₊ :=
       show spectralRadius ℂ (φ s) ≤ spectralRadius ℂ s from
         iSup_le_iSup_of_subset (AlgHom.spectrum_apply_subset φ s)
 #align star_alg_hom.nnnorm_apply_le StarAlgHom.nnnorm_apply_le
+-/
 
+#print StarAlgHom.norm_apply_le /-
 /-- A star algebra homomorphism of complex C⋆-algebras is norm contractive. -/
 theorem norm_apply_le (a : A) : ‖(φ a : B)‖ ≤ ‖a‖ :=
   nnnorm_apply_le φ a
 #align star_alg_hom.norm_apply_le StarAlgHom.norm_apply_le
+-/
 
 /-- Star algebra homomorphisms between C⋆-algebras are continuous linear maps.
 See note [lower instance priority] -/
@@ -185,8 +201,6 @@ open scoped ComplexStarModule
 variable {F A : Type _} [NormedRing A] [NormedAlgebra ℂ A] [CompleteSpace A] [StarRing A]
   [CstarRing A] [StarModule ℂ A] [hF : AlgHomClass F ℂ A ℂ]
 
-include hF
-
 /-- This instance is provided instead of `star_alg_hom_class` to avoid type class inference loops.
 See note [lower instance priority] -/
 noncomputable instance (priority := 100) : StarHomClass F A ℂ
@@ -203,13 +217,13 @@ noncomputable instance (priority := 100) : StarHomClass F A ℂ
       rcases this with ⟨⟨_, _⟩, _, heq⟩
       rw [← HEq, IsROrC.star_def, IsROrC.conj_ofReal]
 
+#print AlgHomClass.instStarAlgHomClass /-
 /-- This is not an instance to avoid type class inference loops. See
 `weak_dual.complex.star_hom_class`. -/
 noncomputable def AlgHomClass.instStarAlgHomClass : StarAlgHomClass F ℂ A ℂ :=
   { WeakDual.Complex.starHomClass, hF with coe := fun f => f }
 #align alg_hom_class.star_alg_hom_class AlgHomClass.instStarAlgHomClass
-
-omit hF
+-/
 
 namespace CharacterSpace
 

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

@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Jireh Loreaux
 
 ! This file was ported from Lean 3 source module analysis.normed_space.star.spectrum
-! leanprover-community/mathlib commit f0c8bf9245297a541f468be517f1bde6195105e9
+! leanprover-community/mathlib commit 087c325ae0ab42dbdd5dee55bc37d3d5a0bf2197
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/
@@ -14,6 +14,9 @@ import Mathbin.Analysis.SpecialFunctions.Exponential
 import Mathbin.Algebra.Star.StarAlgHom
 
 /-! # Spectral properties in C⋆-algebras
+
+> THIS FILE IS SYNCHRONIZED WITH MATHLIB4.
+> Any changes to this file require a corresponding PR to mathlib4.
 In this file, we establish various properties related to the spectrum of elements in C⋆-algebras.
 -/
 

mathlib commit https://github.com/leanprover-community/mathlib/commit/c471da714c044131b90c133701e51b877c246677

@@ -117,19 +117,19 @@ theorem selfAdjoint.mem_spectrum_eq_re [StarModule ℂ A] (a : selfAdjoint A) {z
 #align self_adjoint.mem_spectrum_eq_re selfAdjoint.mem_spectrum_eq_re
 
 /-- The spectrum of a selfadjoint is real -/
-theorem IsSelfAdjoint.coe_re_map_spectrum [StarModule ℂ A] {a : A} (ha : IsSelfAdjoint a) :
+theorem IsSelfAdjoint.val_re_map_spectrum [StarModule ℂ A] {a : A} (ha : IsSelfAdjoint a) :
     spectrum ℂ a = (coe ∘ re '' spectrum ℂ a : Set ℂ) :=
   le_antisymm (fun z hz => ⟨z, hz, (ha.mem_spectrum_eq_re hz).symm⟩) fun z =>
     by
     rintro ⟨z, hz, rfl⟩
     simpa only [(ha.mem_spectrum_eq_re hz).symm, Function.comp_apply] using hz
-#align is_self_adjoint.coe_re_map_spectrum IsSelfAdjoint.coe_re_map_spectrum
+#align is_self_adjoint.coe_re_map_spectrum IsSelfAdjoint.val_re_map_spectrum
 
 /-- The spectrum of a selfadjoint is real -/
-theorem selfAdjoint.coe_re_map_spectrum [StarModule ℂ A] (a : selfAdjoint A) :
+theorem selfAdjoint.val_re_map_spectrum [StarModule ℂ A] (a : selfAdjoint A) :
     spectrum ℂ (a : A) = (coe ∘ re '' spectrum ℂ (a : A) : Set ℂ) :=
-  a.property.coe_re_map_spectrum
-#align self_adjoint.coe_re_map_spectrum selfAdjoint.coe_re_map_spectrum
+  a.property.val_re_map_spectrum
+#align self_adjoint.coe_re_map_spectrum selfAdjoint.val_re_map_spectrum
 
 end ComplexScalars
 
@@ -196,22 +196,22 @@ noncomputable instance (priority := 100) : StarHomClass F A ℂ
       simp only [map_add, map_smul, star_add, star_smul, hsa, selfAdjoint.star_val_eq]
     · intro s
       have := AlgHom.apply_mem_spectrum φ (s : A)
-      rw [selfAdjoint.coe_re_map_spectrum s] at this 
+      rw [selfAdjoint.val_re_map_spectrum s] at this 
       rcases this with ⟨⟨_, _⟩, _, heq⟩
       rw [← HEq, IsROrC.star_def, IsROrC.conj_ofReal]
 
 /-- This is not an instance to avoid type class inference loops. See
 `weak_dual.complex.star_hom_class`. -/
-noncomputable def AlgHomClass.starAlgHomClass : StarAlgHomClass F ℂ A ℂ :=
+noncomputable def AlgHomClass.instStarAlgHomClass : StarAlgHomClass F ℂ A ℂ :=
   { WeakDual.Complex.starHomClass, hF with coe := fun f => f }
-#align alg_hom_class.star_alg_hom_class AlgHomClass.starAlgHomClass
+#align alg_hom_class.star_alg_hom_class AlgHomClass.instStarAlgHomClass
 
 omit hF
 
 namespace CharacterSpace
 
 noncomputable instance : StarAlgHomClass (characterSpace ℂ A) ℂ A ℂ :=
-  { AlgHomClass.starAlgHomClass with coe := fun f => f }
+  { AlgHomClass.instStarAlgHomClass with coe := fun f => f }
 
 end CharacterSpace
 

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

@@ -66,7 +66,8 @@ theorem IsSelfAdjoint.spectralRadius_eq_nnnorm {a : A} (ha : IsSelfAdjoint a) :
   by
   have hconst : tendsto (fun n : ℕ => (‖a‖₊ : ℝ≥0∞)) at_top _ := tendsto_const_nhds
   refine' tendsto_nhds_unique _ hconst
-  convert(spectrum.pow_nnnorm_pow_one_div_tendsto_nhds_spectralRadius (a : A)).comp
+  convert
+    (spectrum.pow_nnnorm_pow_one_div_tendsto_nhds_spectralRadius (a : A)).comp
       (Nat.tendsto_pow_atTop_atTop_of_one_lt one_lt_two)
   refine' funext fun n => _
   rw [Function.comp_apply, ha.nnnorm_pow_two_pow, ENNReal.coe_pow, ← rpow_nat_cast, ← rpow_mul]

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

@@ -37,9 +37,9 @@ theorem unitary.spectrum_subset_circle (u : unitary E) : spectrum 𝕜 (u : E) 
   nontriviality E
   refine' fun k hk => mem_sphere_zero_iff_norm.mpr (le_antisymm _ _)
   · simpa only [CstarRing.norm_coe_unitary u] using norm_le_norm_of_mem hk
-  · rw [← unitary.coe_toUnits_apply u] at hk
+  · rw [← unitary.coe_toUnits_apply u] at hk 
     have hnk := ne_zero_of_mem_of_unit hk
-    rw [← inv_inv (unitary.toUnits u), ← spectrum.map_inv, Set.mem_inv] at hk
+    rw [← inv_inv (unitary.toUnits u), ← spectrum.map_inv, Set.mem_inv] at hk 
     have : ‖k‖⁻¹ ≤ ‖↑(unitary.toUnits u)⁻¹‖; simpa only [norm_inv] using norm_le_norm_of_mem hk
     simpa using inv_le_of_inv_le (norm_pos_iff.mpr hnk) this
 #align unitary.spectrum_subset_circle unitary.spectrum_subset_circle
@@ -87,7 +87,7 @@ theorem IsStarNormal.spectralRadius_eq_nnnorm (a : A) [IsStarNormal a] :
   have h₂ :=
     ((ENNReal.continuous_pow 2).Tendsto (spectralRadius ℂ a)).comp
       (spectrum.pow_nnnorm_pow_one_div_tendsto_nhds_spectralRadius a)
-  rw [← HEq] at h₂
+  rw [← HEq] at h₂ 
   convert tendsto_nhds_unique h₂ (pow_nnnorm_pow_one_div_tendsto_nhds_spectral_radius (a⋆ * a))
   rw [(IsSelfAdjoint.star_mul_self a).spectralRadius_eq_nnnorm, sq, nnnorm_star_mul_self, coe_mul]
 #align is_star_normal.spectral_radius_eq_nnnorm IsStarNormal.spectralRadius_eq_nnnorm
@@ -195,7 +195,7 @@ noncomputable instance (priority := 100) : StarHomClass F A ℂ
       simp only [map_add, map_smul, star_add, star_smul, hsa, selfAdjoint.star_val_eq]
     · intro s
       have := AlgHom.apply_mem_spectrum φ (s : A)
-      rw [selfAdjoint.coe_re_map_spectrum s] at this
+      rw [selfAdjoint.coe_re_map_spectrum s] at this 
       rcases this with ⟨⟨_, _⟩, _, heq⟩
       rw [← HEq, IsROrC.star_def, IsROrC.conj_ofReal]
 

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

@@ -23,7 +23,7 @@ local postfix:max "⋆" => star
 
 section
 
-open Topology ENNReal
+open scoped Topology ENNReal
 
 open Filter ENNReal spectrum CstarRing
 
@@ -176,7 +176,7 @@ namespace WeakDual
 
 open ContinuousMap Complex
 
-open ComplexStarModule
+open scoped ComplexStarModule
 
 variable {F A : Type _} [NormedRing A] [NormedAlgebra ℂ A] [CompleteSpace A] [StarRing A]
   [CstarRing A] [StarModule ℂ A] [hF : AlgHomClass F ℂ A ℂ]

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

@@ -40,8 +40,7 @@ theorem unitary.spectrum_subset_circle (u : unitary E) : spectrum 𝕜 (u : E) 
   · rw [← unitary.coe_toUnits_apply u] at hk
     have hnk := ne_zero_of_mem_of_unit hk
     rw [← inv_inv (unitary.toUnits u), ← spectrum.map_inv, Set.mem_inv] at hk
-    have : ‖k‖⁻¹ ≤ ‖↑(unitary.toUnits u)⁻¹‖
-    simpa only [norm_inv] using norm_le_norm_of_mem hk
+    have : ‖k‖⁻¹ ≤ ‖↑(unitary.toUnits u)⁻¹‖; simpa only [norm_inv] using norm_le_norm_of_mem hk
     simpa using inv_le_of_inv_le (norm_pos_iff.mpr hnk) this
 #align unitary.spectrum_subset_circle unitary.spectrum_subset_circle
 

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

@@ -154,7 +154,7 @@ theorem nnnorm_apply_le (a : A) : ‖(φ a : B)‖₊ ≤ ‖a‖₊ :=
     simpa only [hs.spectral_radius_eq_nnnorm, (hs.star_hom_apply φ).spectralRadius_eq_nnnorm,
       coe_le_coe] using
       show spectralRadius ℂ (φ s) ≤ spectralRadius ℂ s from
-        supᵢ_le_supᵢ_of_subset (AlgHom.spectrum_apply_subset φ s)
+        iSup_le_iSup_of_subset (AlgHom.spectrum_apply_subset φ s)
 #align star_alg_hom.nnnorm_apply_le StarAlgHom.nnnorm_apply_le
 
 /-- A star algebra homomorphism of complex C⋆-algebras is norm contractive. -/

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

@@ -198,7 +198,7 @@ noncomputable instance (priority := 100) : StarHomClass F A ℂ
       have := AlgHom.apply_mem_spectrum φ (s : A)
       rw [selfAdjoint.coe_re_map_spectrum s] at this
       rcases this with ⟨⟨_, _⟩, _, heq⟩
-      rw [← HEq, IsROrC.star_def, IsROrC.conj_of_real]
+      rw [← HEq, IsROrC.star_def, IsROrC.conj_ofReal]
 
 /-- This is not an instance to avoid type class inference loops. See
 `weak_dual.complex.star_hom_class`. -/

mathlib commit https://github.com/leanprover-community/mathlib/commit/738054fa93d43512da144ec45ce799d18fd44248

@@ -4,13 +4,12 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Jireh Loreaux
 
 ! This file was ported from Lean 3 source module analysis.normed_space.star.spectrum
-! leanprover-community/mathlib commit 1e3201306d4d9eb1fd54c60d7c4510ad5126f6f9
+! leanprover-community/mathlib commit f0c8bf9245297a541f468be517f1bde6195105e9
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/
 import Mathbin.Analysis.NormedSpace.Star.Basic
 import Mathbin.Analysis.NormedSpace.Spectrum
-import Mathbin.Analysis.NormedSpace.Star.Exponential
 import Mathbin.Analysis.SpecialFunctions.Exponential
 import Mathbin.Algebra.Star.StarAlgHom
 

mathlib commit https://github.com/leanprover-community/mathlib/commit/39478763114722f0ec7613cb2f3f7701f9b86c8d

@@ -193,7 +193,7 @@ noncomputable instance (priority := 100) : StarHomClass F A ℂ
   coe_injective' := FunLike.coe_injective'
   map_star φ a := by
     suffices hsa : ∀ s : selfAdjoint A, (φ s)⋆ = φ s
-    · rw [← realPart_add_i_smul_imaginaryPart a]
+    · rw [← realPart_add_I_smul_imaginaryPart a]
       simp only [map_add, map_smul, star_add, star_smul, hsa, selfAdjoint.star_val_eq]
     · intro s
       have := AlgHom.apply_mem_spectrum φ (s : A)

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

@@ -68,8 +68,7 @@ theorem IsSelfAdjoint.spectralRadius_eq_nnnorm {a : A} (ha : IsSelfAdjoint a) :
   by
   have hconst : tendsto (fun n : ℕ => (‖a‖₊ : ℝ≥0∞)) at_top _ := tendsto_const_nhds
   refine' tendsto_nhds_unique _ hconst
-  convert
-    (spectrum.pow_nnnorm_pow_one_div_tendsto_nhds_spectralRadius (a : A)).comp
+  convert(spectrum.pow_nnnorm_pow_one_div_tendsto_nhds_spectralRadius (a : A)).comp
       (Nat.tendsto_pow_atTop_atTop_of_one_lt one_lt_two)
   refine' funext fun n => _
   rw [Function.comp_apply, ha.nnnorm_pow_two_pow, ENNReal.coe_pow, ← rpow_nat_cast, ← rpow_mul]

mathlib commit https://github.com/leanprover-community/mathlib/commit/2196ab363eb097c008d4497125e0dde23fb36db2

@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Jireh Loreaux
 
 ! This file was ported from Lean 3 source module analysis.normed_space.star.spectrum
-! leanprover-community/mathlib commit f2ce6086713c78a7f880485f7917ea547a215982
+! leanprover-community/mathlib commit 1e3201306d4d9eb1fd54c60d7c4510ad5126f6f9
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/
@@ -15,7 +15,7 @@ import Mathbin.Analysis.SpecialFunctions.Exponential
 import Mathbin.Algebra.Star.StarAlgHom
 
 /-! # Spectral properties in C⋆-algebras
-In this file, we establish various propreties related to the spectrum of elements in C⋆-algebras.
+In this file, we establish various properties related to the spectrum of elements in C⋆-algebras.
 -/
 
 
@@ -99,6 +99,7 @@ theorem IsStarNormal.spectralRadius_eq_nnnorm (a : A) [IsStarNormal a] :
 theorem IsSelfAdjoint.mem_spectrum_eq_re [StarModule ℂ A] {a : A} (ha : IsSelfAdjoint a) {z : ℂ}
     (hz : z ∈ spectrum ℂ a) : z = z.re :=
   by
+  have hu := exp_mem_unitary_of_mem_skewAdjoint ℂ (ha.smul_mem_skew_adjoint conj_I)
   let Iu := Units.mk0 I I_ne_zero
   have : exp ℂ (I • z) ∈ spectrum ℂ (exp ℂ (I • a)) := by
     simpa only [Units.smul_def, Units.val_mk0] using
@@ -108,7 +109,7 @@ theorem IsSelfAdjoint.mem_spectrum_eq_re [StarModule ℂ A] {a : A} (ha : IsSelf
       (by
         simpa only [← Complex.exp_eq_exp_ℂ, mem_sphere_zero_iff_norm, norm_eq_abs, abs_exp,
           Real.exp_eq_one_iff, smul_eq_mul, I_mul, neg_eq_zero] using
-          spectrum.subset_circle_of_unitary ha.exp_i_smul_unitary this)
+          spectrum.subset_circle_of_unitary hu this)
 #align is_self_adjoint.mem_spectrum_eq_re IsSelfAdjoint.mem_spectrum_eq_re
 
 /-- Any element of the spectrum of a selfadjoint is real. -/

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

@@ -24,9 +24,9 @@ local postfix:max "⋆" => star
 
 section
 
-open Topology Ennreal
+open Topology ENNReal
 
-open Filter Ennreal spectrum CstarRing
+open Filter ENNReal spectrum CstarRing
 
 section UnitarySpectrum
 
@@ -72,14 +72,14 @@ theorem IsSelfAdjoint.spectralRadius_eq_nnnorm {a : A} (ha : IsSelfAdjoint a) :
     (spectrum.pow_nnnorm_pow_one_div_tendsto_nhds_spectralRadius (a : A)).comp
       (Nat.tendsto_pow_atTop_atTop_of_one_lt one_lt_two)
   refine' funext fun n => _
-  rw [Function.comp_apply, ha.nnnorm_pow_two_pow, Ennreal.coe_pow, ← rpow_nat_cast, ← rpow_mul]
+  rw [Function.comp_apply, ha.nnnorm_pow_two_pow, ENNReal.coe_pow, ← rpow_nat_cast, ← rpow_mul]
   simp
 #align is_self_adjoint.spectral_radius_eq_nnnorm IsSelfAdjoint.spectralRadius_eq_nnnorm
 
 theorem IsStarNormal.spectralRadius_eq_nnnorm (a : A) [IsStarNormal a] :
     spectralRadius ℂ a = ‖a‖₊ :=
   by
-  refine' (Ennreal.pow_strictMono two_ne_zero).Injective _
+  refine' (ENNReal.pow_strictMono two_ne_zero).Injective _
   have heq :
     (fun n : ℕ => (‖(a⋆ * a) ^ n‖₊ ^ (1 / n : ℝ) : ℝ≥0∞)) =
       (fun x => x ^ 2) ∘ fun n : ℕ => (‖a ^ n‖₊ ^ (1 / n : ℝ) : ℝ≥0∞) :=
@@ -88,7 +88,7 @@ theorem IsStarNormal.spectralRadius_eq_nnnorm (a : A) [IsStarNormal a] :
     rw [Function.comp_apply, ← rpow_nat_cast, ← rpow_mul, mul_comm, rpow_mul, rpow_nat_cast, ←
       coe_pow, sq, ← nnnorm_star_mul_self, Commute.mul_pow (star_comm_self' a), star_pow]
   have h₂ :=
-    ((Ennreal.continuous_pow 2).Tendsto (spectralRadius ℂ a)).comp
+    ((ENNReal.continuous_pow 2).Tendsto (spectralRadius ℂ a)).comp
       (spectrum.pow_nnnorm_pow_one_div_tendsto_nhds_spectralRadius a)
   rw [← HEq] at h₂
   convert tendsto_nhds_unique h₂ (pow_nnnorm_pow_one_div_tendsto_nhds_spectral_radius (a⋆ * a))

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

Now that I am defining NNRat.cast, I want a definitive answer to this naming issue. Plenty of lemmas in mathlib already use natCast/intCast/ratCast over nat_cast/int_cast/rat_cast, and this matches with the general expectation that underscore-separated name parts correspond to a single declaration.

@@ -65,7 +65,7 @@ theorem IsSelfAdjoint.spectralRadius_eq_nnnorm {a : A} (ha : IsSelfAdjoint a) :
     (spectrum.pow_nnnorm_pow_one_div_tendsto_nhds_spectralRadius (a : A)).comp
       (Nat.tendsto_pow_atTop_atTop_of_one_lt one_lt_two) using 1
   refine' funext fun n => _
-  rw [Function.comp_apply, ha.nnnorm_pow_two_pow, ENNReal.coe_pow, ← rpow_nat_cast, ← rpow_mul]
+  rw [Function.comp_apply, ha.nnnorm_pow_two_pow, ENNReal.coe_pow, ← rpow_natCast, ← rpow_mul]
   simp
 #align is_self_adjoint.spectral_radius_eq_nnnorm IsSelfAdjoint.spectralRadius_eq_nnnorm
 
@@ -76,7 +76,7 @@ theorem IsStarNormal.spectralRadius_eq_nnnorm (a : A) [IsStarNormal a] :
     (fun n : ℕ => (‖(a⋆ * a) ^ n‖₊ : ℝ≥0∞) ^ (1 / n : ℝ)) =
       (fun x => x ^ 2) ∘ fun n : ℕ => (‖a ^ n‖₊ : ℝ≥0∞) ^ (1 / n : ℝ) := by
     funext n
-    rw [Function.comp_apply, ← rpow_nat_cast, ← rpow_mul, mul_comm, rpow_mul, rpow_nat_cast, ←
+    rw [Function.comp_apply, ← rpow_natCast, ← rpow_mul, mul_comm, rpow_mul, rpow_natCast, ←
       coe_pow, sq, ← nnnorm_star_mul_self, Commute.mul_pow (star_comm_self' a), star_pow]
   have h₂ :=
     ((ENNReal.continuous_pow 2).tendsto (spectralRadius ℂ a)).comp

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.

@@ -196,8 +196,8 @@ noncomputable instance (priority := 100) Complex.instStarHomClass : StarHomClass
     rw [selfAdjoint.val_re_map_spectrum s] at this
     rcases this with ⟨⟨_, _⟩, _, heq⟩
     simp only [Function.comp_apply] at heq
-    rw [← heq, IsROrC.star_def]
-    exact IsROrC.conj_ofReal _
+    rw [← heq, RCLike.star_def]
+    exact RCLike.conj_ofReal _
 
 /-- This is not an instance to avoid type class inference loops. See
 `WeakDual.Complex.instStarHomClass`. -/

@@ -155,6 +155,24 @@ noncomputable instance (priority := 100) : ContinuousLinearMapClass F ℂ A B :=
 
 end StarAlgHom
 
+namespace StarAlgEquiv
+
+variable {F A B : Type*} [NormedRing A] [NormedAlgebra ℂ A] [CompleteSpace A] [StarRing A]
+  [CstarRing A] [NormedRing B] [NormedAlgebra ℂ B] [CompleteSpace B] [StarRing B] [CstarRing B]
+  [EquivLike F A B] [NonUnitalAlgEquivClass F ℂ A B] [StarAlgEquivClass F ℂ A B]
+
+lemma nnnorm_map (φ : F) (a : A) : ‖φ a‖₊ = ‖a‖₊ :=
+  le_antisymm (StarAlgHom.nnnorm_apply_le φ a) <| by
+    simpa using StarAlgHom.nnnorm_apply_le (symm (φ : A ≃⋆ₐ[ℂ] B)) ((φ : A ≃⋆ₐ[ℂ] B) a)
+
+lemma norm_map (φ : F) (a : A) : ‖φ a‖ = ‖a‖ :=
+  congr_arg NNReal.toReal (nnnorm_map φ a)
+
+lemma isometry (φ : F) : Isometry φ :=
+  AddMonoidHomClass.isometry_of_norm φ (norm_map φ)
+
+end StarAlgEquiv
+
 end
 
 namespace WeakDual

No changes to tactic file, it's just boring fixes throughout the library.

This follows on from #6964.

Co-authored-by: sgouezel <sebastien.gouezel@univ-rennes1.fr> Co-authored-by: Eric Wieser <wieser.eric@gmail.com>

@@ -170,16 +170,16 @@ variable {F A : Type*} [NormedRing A] [NormedAlgebra ℂ A] [CompleteSpace A] [S
 See note [lower instance priority] -/
 noncomputable instance (priority := 100) Complex.instStarHomClass : StarHomClass F A ℂ where
   map_star φ a := by
-    suffices hsa : ∀ s : selfAdjoint A, (φ s)⋆ = φ s
-    · rw [← realPart_add_I_smul_imaginaryPart a]
+    suffices hsa : ∀ s : selfAdjoint A, (φ s)⋆ = φ s by
+      rw [← realPart_add_I_smul_imaginaryPart a]
       simp only [map_add, map_smul, star_add, star_smul, hsa, selfAdjoint.star_val_eq]
-    · intro s
-      have := AlgHom.apply_mem_spectrum φ (s : A)
-      rw [selfAdjoint.val_re_map_spectrum s] at this
-      rcases this with ⟨⟨_, _⟩, _, heq⟩
-      simp only [Function.comp_apply] at heq
-      rw [← heq, IsROrC.star_def]
-      exact IsROrC.conj_ofReal _
+    intro s
+    have := AlgHom.apply_mem_spectrum φ (s : A)
+    rw [selfAdjoint.val_re_map_spectrum s] at this
+    rcases this with ⟨⟨_, _⟩, _, heq⟩
+    simp only [Function.comp_apply] at heq
+    rw [← heq, IsROrC.star_def]
+    exact IsROrC.conj_ofReal _
 
 /-- This is not an instance to avoid type class inference loops. See
 `WeakDual.Complex.instStarHomClass`. -/

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>

@@ -126,7 +126,7 @@ namespace StarAlgHom
 
 variable {F A B : Type*} [NormedRing A] [NormedAlgebra ℂ A] [CompleteSpace A] [StarRing A]
   [CstarRing A] [NormedRing B] [NormedAlgebra ℂ B] [CompleteSpace B] [StarRing B] [CstarRing B]
-  [hF : StarAlgHomClass F ℂ A B] (φ : F)
+  [FunLike F A B] [AlgHomClass F ℂ A B] [StarAlgHomClass F ℂ A B] (φ : F)
 
 /-- A star algebra homomorphism of complex C⋆-algebras is norm contractive. -/
 theorem nnnorm_apply_le (a : A) : ‖(φ a : B)‖₊ ≤ ‖a‖₊ := by
@@ -164,13 +164,11 @@ open ContinuousMap Complex
 open scoped ComplexStarModule
 
 variable {F A : Type*} [NormedRing A] [NormedAlgebra ℂ A] [CompleteSpace A] [StarRing A]
-  [CstarRing A] [StarModule ℂ A] [hF : AlgHomClass F ℂ A ℂ]
+  [CstarRing A] [StarModule ℂ A] [FunLike F A ℂ] [hF : AlgHomClass F ℂ A ℂ]
 
 /-- This instance is provided instead of `StarAlgHomClass` to avoid type class inference loops.
 See note [lower instance priority] -/
 noncomputable instance (priority := 100) Complex.instStarHomClass : StarHomClass F A ℂ where
-  coe φ := φ
-  coe_injective' := DFunLike.coe_injective'
   map_star φ a := by
     suffices hsa : ∀ s : selfAdjoint A, (φ s)⋆ = φ s
     · rw [← realPart_add_I_smul_imaginaryPart a]
@@ -185,14 +183,14 @@ noncomputable instance (priority := 100) Complex.instStarHomClass : StarHomClass
 
 /-- This is not an instance to avoid type class inference loops. See
 `WeakDual.Complex.instStarHomClass`. -/
-noncomputable def _root_.AlgHomClass.instStarAlgHomClass : StarAlgHomClass F ℂ A ℂ :=
-  { WeakDual.Complex.instStarHomClass, hF with coe := fun f => f }
+lemma _root_.AlgHomClass.instStarAlgHomClass : StarAlgHomClass F ℂ A ℂ :=
+  { WeakDual.Complex.instStarHomClass, hF with }
 #align alg_hom_class.star_alg_hom_class AlgHomClass.instStarAlgHomClass
 
 namespace CharacterSpace
 
 noncomputable instance instStarAlgHomClass : StarAlgHomClass (characterSpace ℂ A) ℂ A ℂ :=
-  { AlgHomClass.instStarAlgHomClass with coe := fun f => f }
+  { AlgHomClass.instStarAlgHomClass with }
 
 end CharacterSpace
 

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

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

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

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

@@ -170,7 +170,7 @@ variable {F A : Type*} [NormedRing A] [NormedAlgebra ℂ A] [CompleteSpace A] [S
 See note [lower instance priority] -/
 noncomputable instance (priority := 100) Complex.instStarHomClass : StarHomClass F A ℂ where
   coe φ := φ
-  coe_injective' := FunLike.coe_injective'
+  coe_injective' := DFunLike.coe_injective'
   map_star φ a := by
     suffices hsa : ∀ s : selfAdjoint A, (φ s)⋆ = φ s
     · rw [← realPart_add_I_smul_imaginaryPart a]

Per discussion at zulip

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

@@ -21,7 +21,7 @@ section
 
 open scoped Topology ENNReal
 
-open Filter ENNReal spectrum CstarRing
+open Filter ENNReal spectrum CstarRing NormedSpace
 
 section UnitarySpectrum
 
@@ -91,7 +91,7 @@ theorem IsSelfAdjoint.mem_spectrum_eq_re [StarModule ℂ A] {a : A} (ha : IsSelf
     (hz : z ∈ spectrum ℂ a) : z = z.re := by
   have hu := exp_mem_unitary_of_mem_skewAdjoint ℂ (ha.smul_mem_skewAdjoint conj_I)
   let Iu := Units.mk0 I I_ne_zero
-  have : _root_.exp ℂ (I • z) ∈ spectrum ℂ (_root_.exp ℂ (I • a)) := by
+  have : NormedSpace.exp ℂ (I • z) ∈ spectrum ℂ (NormedSpace.exp ℂ (I • a)) := by
     simpa only [Units.smul_def, Units.val_mk0] using
       spectrum.exp_mem_exp (Iu • a) (smul_mem_smul_iff.mpr hz)
   exact Complex.ext (ofReal_re _) <| by

@@ -33,12 +33,12 @@ theorem unitary.spectrum_subset_circle (u : unitary E) :
   nontriviality E
   refine' fun k hk => mem_sphere_zero_iff_norm.mpr (le_antisymm _ _)
   · simpa only [CstarRing.norm_coe_unitary u] using norm_le_norm_of_mem hk
-  · rw [← unitary.toUnits_apply_val u] at hk
+  · rw [← unitary.val_toUnits_apply u] at hk
     have hnk := ne_zero_of_mem_of_unit hk
     rw [← inv_inv (unitary.toUnits u), ← spectrum.map_inv, Set.mem_inv] at hk
     have : ‖k‖⁻¹ ≤ ‖(↑(unitary.toUnits u)⁻¹ : E)‖ :=
       by simpa only [norm_inv] using norm_le_norm_of_mem hk
-    simpa [←Units.inv_eq_val_inv] using inv_le_of_inv_le (norm_pos_iff.mpr hnk) this
+    simpa using inv_le_of_inv_le (norm_pos_iff.mpr hnk) this
 #align unitary.spectrum_subset_circle unitary.spectrum_subset_circle
 
 theorem spectrum.subset_circle_of_unitary {u : E} (h : u ∈ unitary E) :

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

This has nice performance benefits.

@@ -25,7 +25,7 @@ open Filter ENNReal spectrum CstarRing
 
 section UnitarySpectrum
 
-variable {𝕜 : Type _} [NormedField 𝕜] {E : Type _} [NormedRing E] [StarRing E] [CstarRing E]
+variable {𝕜 : Type*} [NormedField 𝕜] {E : Type*} [NormedRing E] [StarRing E] [CstarRing E]
   [NormedAlgebra 𝕜 E] [CompleteSpace E]
 
 theorem unitary.spectrum_subset_circle (u : unitary E) :
@@ -52,7 +52,7 @@ section ComplexScalars
 
 open Complex
 
-variable {A : Type _} [NormedRing A] [NormedAlgebra ℂ A] [CompleteSpace A] [StarRing A]
+variable {A : Type*} [NormedRing A] [NormedAlgebra ℂ A] [CompleteSpace A] [StarRing A]
   [CstarRing A]
 
 local notation "↑ₐ" => algebraMap ℂ A
@@ -124,7 +124,7 @@ end ComplexScalars
 
 namespace StarAlgHom
 
-variable {F A B : Type _} [NormedRing A] [NormedAlgebra ℂ A] [CompleteSpace A] [StarRing A]
+variable {F A B : Type*} [NormedRing A] [NormedAlgebra ℂ A] [CompleteSpace A] [StarRing A]
   [CstarRing A] [NormedRing B] [NormedAlgebra ℂ B] [CompleteSpace B] [StarRing B] [CstarRing B]
   [hF : StarAlgHomClass F ℂ A B] (φ : F)
 
@@ -163,7 +163,7 @@ open ContinuousMap Complex
 
 open scoped ComplexStarModule
 
-variable {F A : Type _} [NormedRing A] [NormedAlgebra ℂ A] [CompleteSpace A] [StarRing A]
+variable {F A : Type*} [NormedRing A] [NormedAlgebra ℂ A] [CompleteSpace A] [StarRing A]
   [CstarRing A] [StarModule ℂ A] [hF : AlgHomClass F ℂ A ℂ]
 
 /-- This instance is provided instead of `StarAlgHomClass` to avoid type class inference loops.

• depends on: #5966

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

@@ -2,17 +2,14 @@
 Copyright (c) 2022 Jireh Loreaux. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Jireh Loreaux
-
-! This file was ported from Lean 3 source module analysis.normed_space.star.spectrum
-! leanprover-community/mathlib commit f0c8bf9245297a541f468be517f1bde6195105e9
-! Please do not edit these lines, except to modify the commit id
-! if you have ported upstream changes.
 -/
 import Mathlib.Analysis.NormedSpace.Star.Basic
 import Mathlib.Analysis.NormedSpace.Spectrum
 import Mathlib.Analysis.SpecialFunctions.Exponential
 import Mathlib.Algebra.Star.StarAlgHom
 
+#align_import analysis.normed_space.star.spectrum from "leanprover-community/mathlib"@"f0c8bf9245297a541f468be517f1bde6195105e9"
+
 /-! # Spectral properties in C⋆-algebras
 In this file, we establish various properties related to the spectrum of elements in C⋆-algebras.
 -/

Grepping for [^ .:{-] [^ :] and reviewing the results. Once I started I couldn't stop. :-)

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

@@ -77,7 +77,7 @@ theorem IsStarNormal.spectralRadius_eq_nnnorm (a : A) [IsStarNormal a] :
   refine' (ENNReal.pow_strictMono two_ne_zero).injective _
   have heq :
     (fun n : ℕ => (‖(a⋆ * a) ^ n‖₊ : ℝ≥0∞) ^ (1 / n : ℝ)) =
-      (fun x => x ^ 2) ∘ fun n : ℕ => (‖a ^ n‖₊ : ℝ≥0∞)  ^ (1 / n : ℝ) := by
+      (fun x => x ^ 2) ∘ fun n : ℕ => (‖a ^ n‖₊ : ℝ≥0∞) ^ (1 / n : ℝ) := by
     funext n
     rw [Function.comp_apply, ← rpow_nat_cast, ← rpow_mul, mul_comm, rpow_mul, rpow_nat_cast, ←
       coe_pow, sq, ← nnnorm_star_mul_self, Commute.mul_pow (star_comm_self' a), star_pow]

• depends on: #4946

Co-authored-by: Chris Hughes <chrishughes24@gmail.com>