analysis.inner_product_space.positive ⟷ Mathlib.Analysis.InnerProductSpace.Positive

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

@@ -39,13 +39,13 @@ Positive operator
 -/
 
 
-open InnerProductSpace IsROrC ContinuousLinearMap
+open InnerProductSpace RCLike ContinuousLinearMap
 
 open scoped InnerProduct ComplexConjugate
 
 namespace ContinuousLinearMap
 
-variable {𝕜 E F : Type _} [IsROrC 𝕜]
+variable {𝕜 E F : Type _} [RCLike 𝕜]
 
 variable [NormedAddCommGroup E] [NormedAddCommGroup F]
 

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

@@ -132,7 +132,7 @@ theorem IsPositive.conj_orthogonalProjection (U : Submodule 𝕜 E) {T : E →L[
         orthogonalProjection U ∘L T ∘L U.subtypeL ∘L orthogonalProjection U).IsPositive :=
   by
   have := hT.conj_adjoint (U.subtypeL ∘L orthogonalProjection U)
-  rwa [(orthogonalProjection_isSelfAdjoint U).adjoint_eq] at this 
+  rwa [(orthogonalProjection_isSelfAdjoint U).adjoint_eq] at this
 #align continuous_linear_map.is_positive.conj_orthogonal_projection ContinuousLinearMap.IsPositive.conj_orthogonalProjection
 -/
 
@@ -141,7 +141,7 @@ theorem IsPositive.orthogonalProjection_comp {T : E →L[𝕜] E} (hT : T.IsPosi
     [CompleteSpace U] : (orthogonalProjection U ∘L T ∘L U.subtypeL).IsPositive :=
   by
   have := hT.conj_adjoint (orthogonalProjection U : E →L[𝕜] U)
-  rwa [U.adjoint_orthogonal_projection] at this 
+  rwa [U.adjoint_orthogonal_projection] at this
 #align continuous_linear_map.is_positive.orthogonal_projection_comp ContinuousLinearMap.IsPositive.orthogonalProjection_comp
 -/
 

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

@@ -3,7 +3,7 @@ Copyright (c) 2022 Anatole Dedecker. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Anatole Dedecker
 -/
-import Mathbin.Analysis.InnerProductSpace.Adjoint
+import Analysis.InnerProductSpace.Adjoint
 
 #align_import analysis.inner_product_space.positive from "leanprover-community/mathlib"@"2ebc1d6c2fed9f54c95bbc3998eaa5570527129a"
 

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

@@ -2,14 +2,11 @@
 Copyright (c) 2022 Anatole Dedecker. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Anatole Dedecker
-
-! This file was ported from Lean 3 source module analysis.inner_product_space.positive
-! leanprover-community/mathlib commit 2ebc1d6c2fed9f54c95bbc3998eaa5570527129a
-! Please do not edit these lines, except to modify the commit id
-! if you have ported upstream changes.
 -/
 import Mathbin.Analysis.InnerProductSpace.Adjoint
 
+#align_import analysis.inner_product_space.positive from "leanprover-community/mathlib"@"2ebc1d6c2fed9f54c95bbc3998eaa5570527129a"
+
 /-!
 # Positive operators
 

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

@@ -56,7 +56,6 @@ variable [InnerProductSpace 𝕜 E] [InnerProductSpace 𝕜 F]
 
 variable [CompleteSpace E] [CompleteSpace F]
 
--- mathport name: «expr⟪ , ⟫»
 local notation "⟪" x ", " y "⟫" => @inner 𝕜 _ _ x y
 
 #print ContinuousLinearMap.IsPositive /-
@@ -67,30 +66,41 @@ def IsPositive (T : E →L[𝕜] E) : Prop :=
 #align continuous_linear_map.is_positive ContinuousLinearMap.IsPositive
 -/
 
+#print ContinuousLinearMap.IsPositive.isSelfAdjoint /-
 theorem IsPositive.isSelfAdjoint {T : E →L[𝕜] E} (hT : IsPositive T) : IsSelfAdjoint T :=
   hT.1
 #align continuous_linear_map.is_positive.is_self_adjoint ContinuousLinearMap.IsPositive.isSelfAdjoint
+-/
 
+#print ContinuousLinearMap.IsPositive.inner_nonneg_left /-
 theorem IsPositive.inner_nonneg_left {T : E →L[𝕜] E} (hT : IsPositive T) (x : E) :
     0 ≤ re ⟪T x, x⟫ :=
   hT.2 x
 #align continuous_linear_map.is_positive.inner_nonneg_left ContinuousLinearMap.IsPositive.inner_nonneg_left
+-/
 
+#print ContinuousLinearMap.IsPositive.inner_nonneg_right /-
 theorem IsPositive.inner_nonneg_right {T : E →L[𝕜] E} (hT : IsPositive T) (x : E) :
     0 ≤ re ⟪x, T x⟫ := by rw [inner_re_symm] <;> exact hT.inner_nonneg_left x
 #align continuous_linear_map.is_positive.inner_nonneg_right ContinuousLinearMap.IsPositive.inner_nonneg_right
+-/
 
+#print ContinuousLinearMap.isPositive_zero /-
 theorem isPositive_zero : IsPositive (0 : E →L[𝕜] E) :=
   by
   refine' ⟨isSelfAdjoint_zero _, fun x => _⟩
   change 0 ≤ re ⟪_, _⟫
   rw [zero_apply, inner_zero_left, ZeroHomClass.map_zero]
 #align continuous_linear_map.is_positive_zero ContinuousLinearMap.isPositive_zero
+-/
 
+#print ContinuousLinearMap.isPositive_one /-
 theorem isPositive_one : IsPositive (1 : E →L[𝕜] E) :=
   ⟨isSelfAdjoint_one _, fun x => inner_self_nonneg⟩
 #align continuous_linear_map.is_positive_one ContinuousLinearMap.isPositive_one
+-/
 
+#print ContinuousLinearMap.IsPositive.add /-
 theorem IsPositive.add {T S : E →L[𝕜] E} (hT : T.IsPositive) (hS : S.IsPositive) :
     (T + S).IsPositive :=
   by
@@ -98,7 +108,9 @@ theorem IsPositive.add {T S : E →L[𝕜] E} (hT : T.IsPositive) (hS : S.IsPosi
   rw [re_apply_inner_self, add_apply, inner_add_left, map_add]
   exact add_nonneg (hT.inner_nonneg_left x) (hS.inner_nonneg_left x)
 #align continuous_linear_map.is_positive.add ContinuousLinearMap.IsPositive.add
+-/
 
+#print ContinuousLinearMap.IsPositive.conj_adjoint /-
 theorem IsPositive.conj_adjoint {T : E →L[𝕜] E} (hT : T.IsPositive) (S : E →L[𝕜] F) :
     (S ∘L T ∘L S†).IsPositive :=
   by
@@ -106,13 +118,17 @@ theorem IsPositive.conj_adjoint {T : E →L[𝕜] E} (hT : T.IsPositive) (S : E
   rw [re_apply_inner_self, comp_apply, ← adjoint_inner_right]
   exact hT.inner_nonneg_left _
 #align continuous_linear_map.is_positive.conj_adjoint ContinuousLinearMap.IsPositive.conj_adjoint
+-/
 
+#print ContinuousLinearMap.IsPositive.adjoint_conj /-
 theorem IsPositive.adjoint_conj {T : E →L[𝕜] E} (hT : T.IsPositive) (S : F →L[𝕜] E) :
     (S† ∘L T ∘L S).IsPositive := by
   convert hT.conj_adjoint (S†)
   rw [adjoint_adjoint]
 #align continuous_linear_map.is_positive.adjoint_conj ContinuousLinearMap.IsPositive.adjoint_conj
+-/
 
+#print ContinuousLinearMap.IsPositive.conj_orthogonalProjection /-
 theorem IsPositive.conj_orthogonalProjection (U : Submodule 𝕜 E) {T : E →L[𝕜] E} (hT : T.IsPositive)
     [CompleteSpace U] :
     (U.subtypeL ∘L
@@ -121,18 +137,22 @@ theorem IsPositive.conj_orthogonalProjection (U : Submodule 𝕜 E) {T : E →L[
   have := hT.conj_adjoint (U.subtypeL ∘L orthogonalProjection U)
   rwa [(orthogonalProjection_isSelfAdjoint U).adjoint_eq] at this 
 #align continuous_linear_map.is_positive.conj_orthogonal_projection ContinuousLinearMap.IsPositive.conj_orthogonalProjection
+-/
 
+#print ContinuousLinearMap.IsPositive.orthogonalProjection_comp /-
 theorem IsPositive.orthogonalProjection_comp {T : E →L[𝕜] E} (hT : T.IsPositive) (U : Submodule 𝕜 E)
     [CompleteSpace U] : (orthogonalProjection U ∘L T ∘L U.subtypeL).IsPositive :=
   by
   have := hT.conj_adjoint (orthogonalProjection U : E →L[𝕜] U)
   rwa [U.adjoint_orthogonal_projection] at this 
 #align continuous_linear_map.is_positive.orthogonal_projection_comp ContinuousLinearMap.IsPositive.orthogonalProjection_comp
+-/
 
 section Complex
 
 variable {E' : Type _} [NormedAddCommGroup E'] [InnerProductSpace ℂ E'] [CompleteSpace E']
 
+#print ContinuousLinearMap.isPositive_iff_complex /-
 theorem isPositive_iff_complex (T : E' →L[ℂ] E') :
     IsPositive T ↔ ∀ x, (re ⟪T x, x⟫_ℂ : ℂ) = ⟪T x, x⟫_ℂ ∧ 0 ≤ re ⟪T x, x⟫_ℂ :=
   by
@@ -140,6 +160,7 @@ theorem isPositive_iff_complex (T : E' →L[ℂ] E') :
     LinearMap.isSymmetric_iff_inner_map_self_real, conj_eq_iff_re]
   rfl
 #align continuous_linear_map.is_positive_iff_complex ContinuousLinearMap.isPositive_iff_complex
+-/
 
 end Complex
 

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

@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Anatole Dedecker
 
 ! This file was ported from Lean 3 source module analysis.inner_product_space.positive
-! leanprover-community/mathlib commit caa58cbf5bfb7f81ccbaca4e8b8ac4bc2b39cc1c
+! leanprover-community/mathlib commit 2ebc1d6c2fed9f54c95bbc3998eaa5570527129a
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/
@@ -13,6 +13,9 @@ import Mathbin.Analysis.InnerProductSpace.Adjoint
 /-!
 # Positive operators
 
+> THIS FILE IS SYNCHRONIZED WITH MATHLIB4.
+> Any changes to this file require a corresponding PR to mathlib4.
+
 In this file we define positive operators in a Hilbert space. We follow Bourbaki's choice
 of requiring self adjointness in the definition.
 

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

@@ -56,11 +56,13 @@ variable [CompleteSpace E] [CompleteSpace F]
 -- mathport name: «expr⟪ , ⟫»
 local notation "⟪" x ", " y "⟫" => @inner 𝕜 _ _ x y
 
+#print ContinuousLinearMap.IsPositive /-
 /-- A continuous linear endomorphism `T` of a Hilbert space is **positive** if it is self adjoint
   and `∀ x, 0 ≤ re ⟪T x, x⟫`. -/
 def IsPositive (T : E →L[𝕜] E) : Prop :=
   IsSelfAdjoint T ∧ ∀ x, 0 ≤ T.reApplyInnerSelf x
 #align continuous_linear_map.is_positive ContinuousLinearMap.IsPositive
+-/
 
 theorem IsPositive.isSelfAdjoint {T : E →L[𝕜] E} (hT : IsPositive T) : IsSelfAdjoint T :=
   hT.1
@@ -114,14 +116,14 @@ theorem IsPositive.conj_orthogonalProjection (U : Submodule 𝕜 E) {T : E →L[
         orthogonalProjection U ∘L T ∘L U.subtypeL ∘L orthogonalProjection U).IsPositive :=
   by
   have := hT.conj_adjoint (U.subtypeL ∘L orthogonalProjection U)
-  rwa [(orthogonalProjection_isSelfAdjoint U).adjoint_eq] at this
+  rwa [(orthogonalProjection_isSelfAdjoint U).adjoint_eq] at this 
 #align continuous_linear_map.is_positive.conj_orthogonal_projection ContinuousLinearMap.IsPositive.conj_orthogonalProjection
 
 theorem IsPositive.orthogonalProjection_comp {T : E →L[𝕜] E} (hT : T.IsPositive) (U : Submodule 𝕜 E)
     [CompleteSpace U] : (orthogonalProjection U ∘L T ∘L U.subtypeL).IsPositive :=
   by
   have := hT.conj_adjoint (orthogonalProjection U : E →L[𝕜] U)
-  rwa [U.adjoint_orthogonal_projection] at this
+  rwa [U.adjoint_orthogonal_projection] at this 
 #align continuous_linear_map.is_positive.orthogonal_projection_comp ContinuousLinearMap.IsPositive.orthogonalProjection_comp
 
 section Complex

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

@@ -41,7 +41,7 @@ Positive operator
 
 open InnerProductSpace IsROrC ContinuousLinearMap
 
-open InnerProduct ComplexConjugate
+open scoped InnerProduct ComplexConjugate
 
 namespace ContinuousLinearMap
 

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

@@ -75,16 +75,16 @@ theorem IsPositive.inner_nonneg_right {T : E →L[𝕜] E} (hT : IsPositive T) (
     0 ≤ re ⟪x, T x⟫ := by rw [inner_re_symm] <;> exact hT.inner_nonneg_left x
 #align continuous_linear_map.is_positive.inner_nonneg_right ContinuousLinearMap.IsPositive.inner_nonneg_right
 
-theorem isPositiveZero : IsPositive (0 : E →L[𝕜] E) :=
+theorem isPositive_zero : IsPositive (0 : E →L[𝕜] E) :=
   by
   refine' ⟨isSelfAdjoint_zero _, fun x => _⟩
   change 0 ≤ re ⟪_, _⟫
   rw [zero_apply, inner_zero_left, ZeroHomClass.map_zero]
-#align continuous_linear_map.is_positive_zero ContinuousLinearMap.isPositiveZero
+#align continuous_linear_map.is_positive_zero ContinuousLinearMap.isPositive_zero
 
-theorem isPositiveOne : IsPositive (1 : E →L[𝕜] E) :=
+theorem isPositive_one : IsPositive (1 : E →L[𝕜] E) :=
   ⟨isSelfAdjoint_one _, fun x => inner_self_nonneg⟩
-#align continuous_linear_map.is_positive_one ContinuousLinearMap.isPositiveOne
+#align continuous_linear_map.is_positive_one ContinuousLinearMap.isPositive_one
 
 theorem IsPositive.add {T S : E →L[𝕜] E} (hT : T.IsPositive) (hS : S.IsPositive) :
     (T + S).IsPositive :=
@@ -94,35 +94,35 @@ theorem IsPositive.add {T S : E →L[𝕜] E} (hT : T.IsPositive) (hS : S.IsPosi
   exact add_nonneg (hT.inner_nonneg_left x) (hS.inner_nonneg_left x)
 #align continuous_linear_map.is_positive.add ContinuousLinearMap.IsPositive.add
 
-theorem IsPositive.conjAdjoint {T : E →L[𝕜] E} (hT : T.IsPositive) (S : E →L[𝕜] F) :
+theorem IsPositive.conj_adjoint {T : E →L[𝕜] E} (hT : T.IsPositive) (S : E →L[𝕜] F) :
     (S ∘L T ∘L S†).IsPositive :=
   by
   refine' ⟨hT.is_self_adjoint.conj_adjoint S, fun x => _⟩
   rw [re_apply_inner_self, comp_apply, ← adjoint_inner_right]
   exact hT.inner_nonneg_left _
-#align continuous_linear_map.is_positive.conj_adjoint ContinuousLinearMap.IsPositive.conjAdjoint
+#align continuous_linear_map.is_positive.conj_adjoint ContinuousLinearMap.IsPositive.conj_adjoint
 
-theorem IsPositive.adjointConj {T : E →L[𝕜] E} (hT : T.IsPositive) (S : F →L[𝕜] E) :
+theorem IsPositive.adjoint_conj {T : E →L[𝕜] E} (hT : T.IsPositive) (S : F →L[𝕜] E) :
     (S† ∘L T ∘L S).IsPositive := by
   convert hT.conj_adjoint (S†)
   rw [adjoint_adjoint]
-#align continuous_linear_map.is_positive.adjoint_conj ContinuousLinearMap.IsPositive.adjointConj
+#align continuous_linear_map.is_positive.adjoint_conj ContinuousLinearMap.IsPositive.adjoint_conj
 
-theorem IsPositive.conjOrthogonalProjection (U : Submodule 𝕜 E) {T : E →L[𝕜] E} (hT : T.IsPositive)
+theorem IsPositive.conj_orthogonalProjection (U : Submodule 𝕜 E) {T : E →L[𝕜] E} (hT : T.IsPositive)
     [CompleteSpace U] :
     (U.subtypeL ∘L
         orthogonalProjection U ∘L T ∘L U.subtypeL ∘L orthogonalProjection U).IsPositive :=
   by
   have := hT.conj_adjoint (U.subtypeL ∘L orthogonalProjection U)
   rwa [(orthogonalProjection_isSelfAdjoint U).adjoint_eq] at this
-#align continuous_linear_map.is_positive.conj_orthogonal_projection ContinuousLinearMap.IsPositive.conjOrthogonalProjection
+#align continuous_linear_map.is_positive.conj_orthogonal_projection ContinuousLinearMap.IsPositive.conj_orthogonalProjection
 
-theorem IsPositive.orthogonalProjectionComp {T : E →L[𝕜] E} (hT : T.IsPositive) (U : Submodule 𝕜 E)
+theorem IsPositive.orthogonalProjection_comp {T : E →L[𝕜] E} (hT : T.IsPositive) (U : Submodule 𝕜 E)
     [CompleteSpace U] : (orthogonalProjection U ∘L T ∘L U.subtypeL).IsPositive :=
   by
   have := hT.conj_adjoint (orthogonalProjection U : E →L[𝕜] U)
   rwa [U.adjoint_orthogonal_projection] at this
-#align continuous_linear_map.is_positive.orthogonal_projection_comp ContinuousLinearMap.IsPositive.orthogonalProjectionComp
+#align continuous_linear_map.is_positive.orthogonal_projection_comp ContinuousLinearMap.IsPositive.orthogonalProjection_comp
 
 section Complex
 

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

@@ -75,16 +75,16 @@ theorem IsPositive.inner_nonneg_right {T : E →L[𝕜] E} (hT : IsPositive T) (
     0 ≤ re ⟪x, T x⟫ := by rw [inner_re_symm] <;> exact hT.inner_nonneg_left x
 #align continuous_linear_map.is_positive.inner_nonneg_right ContinuousLinearMap.IsPositive.inner_nonneg_right
 
-theorem isPositive_zero : IsPositive (0 : E →L[𝕜] E) :=
+theorem isPositiveZero : IsPositive (0 : E →L[𝕜] E) :=
   by
   refine' ⟨isSelfAdjoint_zero _, fun x => _⟩
   change 0 ≤ re ⟪_, _⟫
   rw [zero_apply, inner_zero_left, ZeroHomClass.map_zero]
-#align continuous_linear_map.is_positive_zero ContinuousLinearMap.isPositive_zero
+#align continuous_linear_map.is_positive_zero ContinuousLinearMap.isPositiveZero
 
-theorem isPositive_one : IsPositive (1 : E →L[𝕜] E) :=
+theorem isPositiveOne : IsPositive (1 : E →L[𝕜] E) :=
   ⟨isSelfAdjoint_one _, fun x => inner_self_nonneg⟩
-#align continuous_linear_map.is_positive_one ContinuousLinearMap.isPositive_one
+#align continuous_linear_map.is_positive_one ContinuousLinearMap.isPositiveOne
 
 theorem IsPositive.add {T S : E →L[𝕜] E} (hT : T.IsPositive) (hS : S.IsPositive) :
     (T + S).IsPositive :=
@@ -94,35 +94,35 @@ theorem IsPositive.add {T S : E →L[𝕜] E} (hT : T.IsPositive) (hS : S.IsPosi
   exact add_nonneg (hT.inner_nonneg_left x) (hS.inner_nonneg_left x)
 #align continuous_linear_map.is_positive.add ContinuousLinearMap.IsPositive.add
 
-theorem IsPositive.conj_adjoint {T : E →L[𝕜] E} (hT : T.IsPositive) (S : E →L[𝕜] F) :
+theorem IsPositive.conjAdjoint {T : E →L[𝕜] E} (hT : T.IsPositive) (S : E →L[𝕜] F) :
     (S ∘L T ∘L S†).IsPositive :=
   by
   refine' ⟨hT.is_self_adjoint.conj_adjoint S, fun x => _⟩
   rw [re_apply_inner_self, comp_apply, ← adjoint_inner_right]
   exact hT.inner_nonneg_left _
-#align continuous_linear_map.is_positive.conj_adjoint ContinuousLinearMap.IsPositive.conj_adjoint
+#align continuous_linear_map.is_positive.conj_adjoint ContinuousLinearMap.IsPositive.conjAdjoint
 
-theorem IsPositive.adjoint_conj {T : E →L[𝕜] E} (hT : T.IsPositive) (S : F →L[𝕜] E) :
+theorem IsPositive.adjointConj {T : E →L[𝕜] E} (hT : T.IsPositive) (S : F →L[𝕜] E) :
     (S† ∘L T ∘L S).IsPositive := by
   convert hT.conj_adjoint (S†)
   rw [adjoint_adjoint]
-#align continuous_linear_map.is_positive.adjoint_conj ContinuousLinearMap.IsPositive.adjoint_conj
+#align continuous_linear_map.is_positive.adjoint_conj ContinuousLinearMap.IsPositive.adjointConj
 
-theorem IsPositive.conj_orthogonalProjection (U : Submodule 𝕜 E) {T : E →L[𝕜] E} (hT : T.IsPositive)
+theorem IsPositive.conjOrthogonalProjection (U : Submodule 𝕜 E) {T : E →L[𝕜] E} (hT : T.IsPositive)
     [CompleteSpace U] :
     (U.subtypeL ∘L
         orthogonalProjection U ∘L T ∘L U.subtypeL ∘L orthogonalProjection U).IsPositive :=
   by
   have := hT.conj_adjoint (U.subtypeL ∘L orthogonalProjection U)
   rwa [(orthogonalProjection_isSelfAdjoint U).adjoint_eq] at this
-#align continuous_linear_map.is_positive.conj_orthogonal_projection ContinuousLinearMap.IsPositive.conj_orthogonalProjection
+#align continuous_linear_map.is_positive.conj_orthogonal_projection ContinuousLinearMap.IsPositive.conjOrthogonalProjection
 
-theorem IsPositive.orthogonalProjection_comp {T : E →L[𝕜] E} (hT : T.IsPositive) (U : Submodule 𝕜 E)
+theorem IsPositive.orthogonalProjectionComp {T : E →L[𝕜] E} (hT : T.IsPositive) (U : Submodule 𝕜 E)
     [CompleteSpace U] : (orthogonalProjection U ∘L T ∘L U.subtypeL).IsPositive :=
   by
   have := hT.conj_adjoint (orthogonalProjection U : E →L[𝕜] U)
   rwa [U.adjoint_orthogonal_projection] at this
-#align continuous_linear_map.is_positive.orthogonal_projection_comp ContinuousLinearMap.IsPositive.orthogonalProjection_comp
+#align continuous_linear_map.is_positive.orthogonal_projection_comp ContinuousLinearMap.IsPositive.orthogonalProjectionComp
 
 section Complex
 

mathlib commit https://github.com/leanprover-community/mathlib/commit/92c69b77c5a7dc0f7eeddb552508633305157caa

@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Anatole Dedecker
 
 ! This file was ported from Lean 3 source module analysis.inner_product_space.positive
-! leanprover-community/mathlib commit 46b633fd842bef9469441c0209906f6dddd2b4f5
+! leanprover-community/mathlib commit caa58cbf5bfb7f81ccbaca4e8b8ac4bc2b39cc1c
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/
@@ -132,7 +132,7 @@ theorem isPositive_iff_complex (T : E' →L[ℂ] E') :
     IsPositive T ↔ ∀ x, (re ⟪T x, x⟫_ℂ : ℂ) = ⟪T x, x⟫_ℂ ∧ 0 ≤ re ⟪T x, x⟫_ℂ :=
   by
   simp_rw [is_positive, forall_and, is_self_adjoint_iff_is_symmetric,
-    LinearMap.isSymmetric_iff_inner_map_self_real, eq_conj_iff_re]
+    LinearMap.isSymmetric_iff_inner_map_self_real, conj_eq_iff_re]
   rfl
 #align continuous_linear_map.is_positive_iff_complex ContinuousLinearMap.isPositive_iff_complex
 

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: Anatole Dedecker
 
 ! This file was ported from Lean 3 source module analysis.inner_product_space.positive
-! leanprover-community/mathlib commit 70fd9563a21e7b963887c9360bd29b2393e6225a
+! leanprover-community/mathlib commit 46b633fd842bef9469441c0209906f6dddd2b4f5
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/
@@ -45,8 +45,13 @@ open InnerProduct ComplexConjugate
 
 namespace ContinuousLinearMap
 
-variable {𝕜 E F : Type _} [IsROrC 𝕜] [InnerProductSpace 𝕜 E] [InnerProductSpace 𝕜 F]
-  [CompleteSpace E] [CompleteSpace F]
+variable {𝕜 E F : Type _} [IsROrC 𝕜]
+
+variable [NormedAddCommGroup E] [NormedAddCommGroup F]
+
+variable [InnerProductSpace 𝕜 E] [InnerProductSpace 𝕜 F]
+
+variable [CompleteSpace E] [CompleteSpace F]
 
 -- mathport name: «expr⟪ , ⟫»
 local notation "⟪" x ", " y "⟫" => @inner 𝕜 _ _ x y
@@ -121,7 +126,7 @@ theorem IsPositive.orthogonalProjection_comp {T : E →L[𝕜] E} (hT : T.IsPosi
 
 section Complex
 
-variable {E' : Type _} [InnerProductSpace ℂ E'] [CompleteSpace E']
+variable {E' : Type _} [NormedAddCommGroup E'] [InnerProductSpace ℂ E'] [CompleteSpace E']
 
 theorem isPositive_iff_complex (T : E' →L[ℂ] E') :
     IsPositive T ↔ ∀ x, (re ⟪T x, x⟫_ℂ : ℂ) = ⟪T x, x⟫_ℂ ∧ 0 ≤ re ⟪T x, x⟫_ℂ :=

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

@@ -70,16 +70,16 @@ theorem IsPositive.inner_nonneg_right {T : E →L[𝕜] E} (hT : IsPositive T) (
     0 ≤ re ⟪x, T x⟫ := by rw [inner_re_symm] <;> exact hT.inner_nonneg_left x
 #align continuous_linear_map.is_positive.inner_nonneg_right ContinuousLinearMap.IsPositive.inner_nonneg_right
 
-theorem isPositiveZero : IsPositive (0 : E →L[𝕜] E) :=
+theorem isPositive_zero : IsPositive (0 : E →L[𝕜] E) :=
   by
   refine' ⟨isSelfAdjoint_zero _, fun x => _⟩
   change 0 ≤ re ⟪_, _⟫
   rw [zero_apply, inner_zero_left, ZeroHomClass.map_zero]
-#align continuous_linear_map.is_positive_zero ContinuousLinearMap.isPositiveZero
+#align continuous_linear_map.is_positive_zero ContinuousLinearMap.isPositive_zero
 
-theorem isPositiveOne : IsPositive (1 : E →L[𝕜] E) :=
+theorem isPositive_one : IsPositive (1 : E →L[𝕜] E) :=
   ⟨isSelfAdjoint_one _, fun x => inner_self_nonneg⟩
-#align continuous_linear_map.is_positive_one ContinuousLinearMap.isPositiveOne
+#align continuous_linear_map.is_positive_one ContinuousLinearMap.isPositive_one
 
 theorem IsPositive.add {T S : E →L[𝕜] E} (hT : T.IsPositive) (hS : S.IsPositive) :
     (T + S).IsPositive :=
@@ -89,35 +89,35 @@ theorem IsPositive.add {T S : E →L[𝕜] E} (hT : T.IsPositive) (hS : S.IsPosi
   exact add_nonneg (hT.inner_nonneg_left x) (hS.inner_nonneg_left x)
 #align continuous_linear_map.is_positive.add ContinuousLinearMap.IsPositive.add
 
-theorem IsPositive.conjAdjoint {T : E →L[𝕜] E} (hT : T.IsPositive) (S : E →L[𝕜] F) :
+theorem IsPositive.conj_adjoint {T : E →L[𝕜] E} (hT : T.IsPositive) (S : E →L[𝕜] F) :
     (S ∘L T ∘L S†).IsPositive :=
   by
   refine' ⟨hT.is_self_adjoint.conj_adjoint S, fun x => _⟩
   rw [re_apply_inner_self, comp_apply, ← adjoint_inner_right]
   exact hT.inner_nonneg_left _
-#align continuous_linear_map.is_positive.conj_adjoint ContinuousLinearMap.IsPositive.conjAdjoint
+#align continuous_linear_map.is_positive.conj_adjoint ContinuousLinearMap.IsPositive.conj_adjoint
 
-theorem IsPositive.adjointConj {T : E →L[𝕜] E} (hT : T.IsPositive) (S : F →L[𝕜] E) :
+theorem IsPositive.adjoint_conj {T : E →L[𝕜] E} (hT : T.IsPositive) (S : F →L[𝕜] E) :
     (S† ∘L T ∘L S).IsPositive := by
   convert hT.conj_adjoint (S†)
   rw [adjoint_adjoint]
-#align continuous_linear_map.is_positive.adjoint_conj ContinuousLinearMap.IsPositive.adjointConj
+#align continuous_linear_map.is_positive.adjoint_conj ContinuousLinearMap.IsPositive.adjoint_conj
 
-theorem IsPositive.conjOrthogonalProjection (U : Submodule 𝕜 E) {T : E →L[𝕜] E} (hT : T.IsPositive)
+theorem IsPositive.conj_orthogonalProjection (U : Submodule 𝕜 E) {T : E →L[𝕜] E} (hT : T.IsPositive)
     [CompleteSpace U] :
     (U.subtypeL ∘L
         orthogonalProjection U ∘L T ∘L U.subtypeL ∘L orthogonalProjection U).IsPositive :=
   by
   have := hT.conj_adjoint (U.subtypeL ∘L orthogonalProjection U)
   rwa [(orthogonalProjection_isSelfAdjoint U).adjoint_eq] at this
-#align continuous_linear_map.is_positive.conj_orthogonal_projection ContinuousLinearMap.IsPositive.conjOrthogonalProjection
+#align continuous_linear_map.is_positive.conj_orthogonal_projection ContinuousLinearMap.IsPositive.conj_orthogonalProjection
 
-theorem IsPositive.orthogonalProjectionComp {T : E →L[𝕜] E} (hT : T.IsPositive) (U : Submodule 𝕜 E)
+theorem IsPositive.orthogonalProjection_comp {T : E →L[𝕜] E} (hT : T.IsPositive) (U : Submodule 𝕜 E)
     [CompleteSpace U] : (orthogonalProjection U ∘L T ∘L U.subtypeL).IsPositive :=
   by
   have := hT.conj_adjoint (orthogonalProjection U : E →L[𝕜] U)
   rwa [U.adjoint_orthogonal_projection] at this
-#align continuous_linear_map.is_positive.orthogonal_projection_comp ContinuousLinearMap.IsPositive.orthogonalProjectionComp
+#align continuous_linear_map.is_positive.orthogonal_projection_comp ContinuousLinearMap.IsPositive.orthogonalProjection_comp
 
 section Complex
 

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

The (Loewner) partial order on continuous linear maps on a Hilbert space determined by f ≤ g if and only if g - f is a positive linear map (in the sense of ContinuousLinearMap.IsPositive). With this partial order, the continuous linear maps form a StarOrderedRing.

@@ -125,4 +125,34 @@ theorem isPositive_iff_complex (T : E' →L[ℂ] E') :
 
 end Complex
 
+section PartialOrder
+
+/-- The (Loewner) partial order on continuous linear maps on a Hilbert space determined by
+`f ≤ g` if and only if `g - f` is a positive linear map (in the sense of
+`ContinuousLinearMap.IsPositive`). With this partial order, the continuous linear maps form a
+`StarOrderedRing`. -/
+instance instLoewnerPartialOrder : PartialOrder (E →L[𝕜] E) where
+  le f g := (g - f).IsPositive
+  le_refl _ := by simpa using isPositive_zero
+  le_trans _ _ _ h₁ h₂ := by simpa using h₁.add h₂
+  le_antisymm f₁ f₂ h₁ h₂ := by
+    rw [← sub_eq_zero]
+    have h_isSymm := isSelfAdjoint_iff_isSymmetric.mp h₂.isSelfAdjoint
+    exact_mod_cast h_isSymm.inner_map_self_eq_zero.mp fun x ↦ by
+      apply RCLike.ext
+      · rw [map_zero]
+        apply le_antisymm
+        · rw [← neg_nonneg, ← map_neg, ← inner_neg_left]
+          simpa using h₁.inner_nonneg_left _
+        · exact h₂.inner_nonneg_left _
+      · rw [coe_sub, LinearMap.sub_apply, coe_coe, coe_coe, map_zero, ← sub_apply,
+          ← h_isSymm.coe_reApplyInnerSelf_apply (T := f₁ - f₂) x, RCLike.ofReal_im]
+
+lemma le_def (f g : E →L[𝕜] E) : f ≤ g ↔ (g - f).IsPositive := Iff.rfl
+
+lemma nonneg_iff_isPositive (f : E →L[𝕜] E) : 0 ≤ f ↔ f.IsPositive := by
+  simpa using le_def 0 f
+
+end PartialOrder
+
 end ContinuousLinearMap

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.

@@ -36,13 +36,13 @@ Positive operator
 -/
 
 
-open InnerProductSpace IsROrC ContinuousLinearMap
+open InnerProductSpace RCLike ContinuousLinearMap
 
 open scoped InnerProduct ComplexConjugate
 
 namespace ContinuousLinearMap
 
-variable {𝕜 E F : Type*} [IsROrC 𝕜]
+variable {𝕜 E F : Type*} [RCLike 𝕜]
 variable [NormedAddCommGroup E] [NormedAddCommGroup F]
 variable [InnerProductSpace 𝕜 E] [InnerProductSpace 𝕜 F]
 variable [CompleteSpace E] [CompleteSpace F]

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)

@@ -43,11 +43,8 @@ open scoped InnerProduct ComplexConjugate
 namespace ContinuousLinearMap
 
 variable {𝕜 E F : Type*} [IsROrC 𝕜]
-
 variable [NormedAddCommGroup E] [NormedAddCommGroup F]
-
 variable [InnerProductSpace 𝕜 E] [InnerProductSpace 𝕜 F]
-
 variable [CompleteSpace E] [CompleteSpace F]
 
 local notation "⟪" x ", " y "⟫" => @inner 𝕜 _ _ x y

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

This has nice performance benefits.

@@ -42,7 +42,7 @@ open scoped InnerProduct ComplexConjugate
 
 namespace ContinuousLinearMap
 
-variable {𝕜 E F : Type _} [IsROrC 𝕜]
+variable {𝕜 E F : Type*} [IsROrC 𝕜]
 
 variable [NormedAddCommGroup E] [NormedAddCommGroup F]
 
@@ -117,7 +117,7 @@ theorem IsPositive.orthogonalProjection_comp {T : E →L[𝕜] E} (hT : T.IsPosi
 
 section Complex
 
-variable {E' : Type _} [NormedAddCommGroup E'] [InnerProductSpace ℂ E'] [CompleteSpace E']
+variable {E' : Type*} [NormedAddCommGroup E'] [InnerProductSpace ℂ E'] [CompleteSpace E']
 
 theorem isPositive_iff_complex (T : E' →L[ℂ] E') :
     IsPositive T ↔ ∀ x, (re ⟪T x, x⟫_ℂ : ℂ) = ⟪T x, x⟫_ℂ ∧ 0 ≤ re ⟪T x, x⟫_ℂ := by

• depends on: #5966

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

@@ -2,14 +2,11 @@
 Copyright (c) 2022 Anatole Dedecker. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Anatole Dedecker
-
-! This file was ported from Lean 3 source module analysis.inner_product_space.positive
-! leanprover-community/mathlib commit caa58cbf5bfb7f81ccbaca4e8b8ac4bc2b39cc1c
-! Please do not edit these lines, except to modify the commit id
-! if you have ported upstream changes.
 -/
 import Mathlib.Analysis.InnerProductSpace.Adjoint
 
+#align_import analysis.inner_product_space.positive from "leanprover-community/mathlib"@"caa58cbf5bfb7f81ccbaca4e8b8ac4bc2b39cc1c"
+
 /-!
 # Positive operators