diff --git a/thys/Complex_Bounded_Operators/Cblinfun_Matrix.thy b/thys/Complex_Bounded_Operators/Cblinfun_Matrix.thy
--- a/thys/Complex_Bounded_Operators/Cblinfun_Matrix.thy
+++ b/thys/Complex_Bounded_Operators/Cblinfun_Matrix.thy
@@ -1,1575 +1,1573 @@
 section \<open>\<open>Cblinfun_Matrix\<close> -- Matrix representation of bounded operators\<close>
 
 theory Cblinfun_Matrix
   imports
     Complex_L2
 
     "Jordan_Normal_Form.Gram_Schmidt"
     "HOL-Analysis.Starlike"
     "Complex_Bounded_Operators.Extra_Jordan_Normal_Form"
 begin
 
 hide_const (open) Order.bottom Order.top
 hide_type (open) Finite_Cartesian_Product.vec
 hide_const (open) Finite_Cartesian_Product.mat
 hide_fact (open) Finite_Cartesian_Product.mat_def
 hide_const (open) Finite_Cartesian_Product.vec
 hide_fact (open) Finite_Cartesian_Product.vec_def
 hide_const (open) Finite_Cartesian_Product.row
 hide_fact (open) Finite_Cartesian_Product.row_def
 no_notation Finite_Cartesian_Product.vec_nth (infixl "$" 90)
 
 unbundle jnf_notation
 unbundle cblinfun_notation
 
 subsection \<open>Isomorphism between vectors\<close>
 
 text \<open>We define the canonical isomorphism between vectors in some complex vector space \<^typ>\<open>'a::basis_enum\<close> and the
   complex \<^term>\<open>n\<close>-dimensional vectors (where \<^term>\<open>n\<close> is the dimension of \<^typ>\<open>'a\<close>).
   This is possible if \<^typ>\<open>'a\<close>, \<^typ>\<open>'b\<close> are of class \<^class>\<open>basis_enum\<close>
   since that class fixes a finite canonical basis. Vector are represented using
   the \<^typ>\<open>complex vec\<close> type from \<^session>\<open>Jordan_Normal_Form\<close>.
   (The isomorphism will be called \<^term>\<open>vec_of_onb_enum\<close> below.)\<close>
 
 definition vec_of_basis_enum :: \<open>'a::basis_enum \<Rightarrow> complex vec\<close> where
   \<comment> \<open>Maps \<^term>\<open>v\<close> to a \<^typ>\<open>'a vec\<close> represented in basis \<^const>\<open>canonical_basis\<close>\<close>
   \<open>vec_of_basis_enum v = vec_of_list (map (crepresentation (set canonical_basis) v) canonical_basis)\<close>
 
 lemma dim_vec_of_basis_enum'[simp]:
   \<open>dim_vec (vec_of_basis_enum (v::'a)) = length (canonical_basis::'a::basis_enum list)\<close>
   unfolding vec_of_basis_enum_def
   by simp
 
 
 definition basis_enum_of_vec :: \<open>complex vec \<Rightarrow> 'a::basis_enum\<close> where
   \<open>basis_enum_of_vec v =
     (if dim_vec v = length (canonical_basis :: 'a list)
      then sum_list (map2 (*\<^sub>C) (list_of_vec v) (canonical_basis::'a list))
      else 0)\<close>
 
 lemma vec_of_basis_enum_inverse[simp]:
   fixes w::"'a::basis_enum"
   shows  "basis_enum_of_vec (vec_of_basis_enum w) = w"
   unfolding vec_of_basis_enum_def basis_enum_of_vec_def
   unfolding list_vec zip_map1 zip_same_conv_map map_map
   apply (simp add: o_def)
   apply (subst sum.distinct_set_conv_list[symmetric], simp)
   apply (rule complex_vector.sum_representation_eq)
   using  is_generator_set by auto
 
 lemma basis_enum_of_vec_inverse[simp]:
   fixes v::"complex vec"
   defines "n \<equiv> length (canonical_basis :: 'a::basis_enum list)"
   assumes f1: "dim_vec v = n"
   shows "vec_of_basis_enum ((basis_enum_of_vec v)::'a) = v"
 proof (rule eq_vecI)
   show \<open>dim_vec (vec_of_basis_enum (basis_enum_of_vec v :: 'a)) = dim_vec v\<close>
     by (auto simp: vec_of_basis_enum_def f1 n_def)
 next
   fix j assume j_v: \<open>j < dim_vec v\<close>
   define w where "w = list_of_vec v"
   define basis where "basis = (canonical_basis::'a list)"
   have [simp]: "length w = n" "length basis = n" \<open>dim_vec v = n\<close> \<open>length (canonical_basis::'a list) = n\<close>
     \<open>j < n\<close>
     using j_v by (auto simp: f1 basis_def w_def n_def)
   have [simp]: \<open>cindependent (set basis)\<close> \<open>cspan (set basis) = UNIV\<close>
     by (auto simp: basis_def is_cindependent_set is_generator_set)
 
   have \<open>vec_of_basis_enum ((basis_enum_of_vec v)::'a) $ j
        = map (crepresentation (set basis) (sum_list (map2 (*\<^sub>C) w basis))) basis ! j\<close>
     by (auto simp: vec_of_list_index vec_of_basis_enum_def basis_enum_of_vec_def simp flip: w_def basis_def)
   also have \<open>\<dots> = crepresentation (set basis) (sum_list (map2 (*\<^sub>C) w basis)) (basis!j)\<close>
     by simp
   also have \<open>\<dots> = crepresentation (set basis) (\<Sum>i<n. (w!i) *\<^sub>C (basis!i)) (basis!j)\<close>
     by (auto simp: sum_list_sum_nth atLeast0LessThan)
   also have \<open>\<dots> = (\<Sum>i<n. (w!i) *\<^sub>C crepresentation (set basis) (basis!i) (basis!j))\<close>
     by (auto simp: complex_vector.representation_sum complex_vector.representation_scale)
   also have \<open>\<dots> = w!j\<close>
     apply (subst sum_single[where i=j])
       apply (auto simp: complex_vector.representation_basis)
     using \<open>j < n\<close> \<open>length basis = n\<close> basis_def distinct_canonical_basis nth_eq_iff_index_eq by blast
   also have \<open>\<dots> = v $ j\<close>
     by (simp add: w_def)
   finally show \<open>vec_of_basis_enum (basis_enum_of_vec v :: 'a) $ j = v $ j\<close>
     by -
 qed
 
 lemma basis_enum_eq_vec_of_basis_enumI:
   fixes a b :: "_::basis_enum"
   assumes "vec_of_basis_enum a = vec_of_basis_enum b"
   shows "a = b"
   by (metis assms vec_of_basis_enum_inverse)
 
 subsection \<open>Operations on vectors\<close>
 
 
 lemma basis_enum_of_vec_add:
   assumes [simp]: \<open>dim_vec v1 = length (canonical_basis :: 'a::basis_enum list)\<close>
     \<open>dim_vec v2 = length (canonical_basis :: 'a list)\<close>
   shows \<open>((basis_enum_of_vec (v1 + v2)) :: 'a) = basis_enum_of_vec v1 + basis_enum_of_vec v2\<close>
 proof -
   have \<open>length (list_of_vec v1) = length (list_of_vec v2)\<close> and \<open>length (list_of_vec v2) = length (canonical_basis :: 'a list)\<close>
     by simp_all
   then have \<open>sum_list (map2 (*\<^sub>C) (map2 (+) (list_of_vec v1) (list_of_vec v2)) (canonical_basis::'a list))
     = sum_list (map2 (*\<^sub>C) (list_of_vec v1) canonical_basis) + sum_list (map2 (*\<^sub>C) (list_of_vec v2) canonical_basis)\<close>
     apply (induction rule: list_induct3)
     by (auto simp: scaleC_add_left)
   then show ?thesis
     using assms by (auto simp: basis_enum_of_vec_def list_of_vec_plus)
 qed
 
 lemma basis_enum_of_vec_mult:
   assumes [simp]: \<open>dim_vec v = length (canonical_basis :: 'a::basis_enum list)\<close>
   shows \<open>((basis_enum_of_vec (c \<cdot>\<^sub>v v)) :: 'a) =  c *\<^sub>C basis_enum_of_vec v\<close>
 proof -
   have *: \<open>monoid_add_hom ((*\<^sub>C) c :: 'a \<Rightarrow> _)\<close>
     by (simp add: monoid_add_hom_def plus_hom.intro scaleC_add_right semigroup_add_hom.intro zero_hom.intro)
   show ?thesis
     apply (auto simp: basis_enum_of_vec_def list_of_vec_mult map_zip_map
         monoid_add_hom.hom_sum_list[OF *])
     by (metis case_prod_unfold comp_apply scaleC_scaleC)
 qed
 
 
 lemma vec_of_basis_enum_add:
   "vec_of_basis_enum (b1 + b2) = vec_of_basis_enum b1 + vec_of_basis_enum b2"
   by (auto simp: vec_of_basis_enum_def complex_vector.representation_add)
 
 lemma vec_of_basis_enum_scaleC:
   "vec_of_basis_enum (c *\<^sub>C b) = c \<cdot>\<^sub>v (vec_of_basis_enum b)"
   by (auto simp: vec_of_basis_enum_def complex_vector.representation_scale)
 
 lemma vec_of_basis_enum_scaleR:
   "vec_of_basis_enum (r *\<^sub>R b) = complex_of_real r \<cdot>\<^sub>v (vec_of_basis_enum b)"
   by (simp add: scaleR_scaleC vec_of_basis_enum_scaleC)
 
 lemma vec_of_basis_enum_uminus:
   "vec_of_basis_enum (- b2) = - vec_of_basis_enum b2"
   unfolding scaleC_minus1_left[symmetric, of b2]
   unfolding scaleC_minus1_left_vec[symmetric]
   by (rule vec_of_basis_enum_scaleC)
 
 
 lemma vec_of_basis_enum_minus:
   "vec_of_basis_enum (b1 - b2) = vec_of_basis_enum b1 - vec_of_basis_enum b2"
   by (metis (mono_tags, opaque_lifting) carrier_vec_dim_vec diff_conv_add_uminus diff_zero index_add_vec(2) minus_add_uminus_vec vec_of_basis_enum_add vec_of_basis_enum_uminus)
 
 lemma cinner_basis_enum_of_vec:
   defines "n \<equiv> length (canonical_basis :: 'a::onb_enum list)"
   assumes [simp]: "dim_vec x = n" "dim_vec y = n"
-  shows  "\<langle>basis_enum_of_vec x :: 'a, basis_enum_of_vec y\<rangle> = y \<bullet>c x"
+  shows  "(basis_enum_of_vec x :: 'a) \<bullet>\<^sub>C basis_enum_of_vec y = y \<bullet>c x"
 proof -
-  have \<open>\<langle>basis_enum_of_vec x :: 'a, basis_enum_of_vec y\<rangle>
+  have \<open>(basis_enum_of_vec x :: 'a) \<bullet>\<^sub>C basis_enum_of_vec y
     = (\<Sum>i<n. x$i *\<^sub>C canonical_basis ! i :: 'a) \<bullet>\<^sub>C (\<Sum>i<n. y$i *\<^sub>C canonical_basis ! i)\<close>
     by (auto simp: basis_enum_of_vec_def sum_list_sum_nth atLeast0LessThan simp flip: n_def)
   also have \<open>\<dots> = (\<Sum>i<n. \<Sum>j<n. cnj (x$i) *\<^sub>C y$j *\<^sub>C ((canonical_basis ! i :: 'a) \<bullet>\<^sub>C (canonical_basis ! j)))\<close>
     apply (subst cinner_sum_left)
     apply (subst cinner_sum_right)
     by (auto simp: mult_ac)
   also have \<open>\<dots> = (\<Sum>i<n. \<Sum>j<n. cnj (x$i) *\<^sub>C y$j *\<^sub>C (if i=j then 1 else 0))\<close>
     apply (rule sum.cong[OF refl])
     apply (rule sum.cong[OF refl])
     by (auto simp: cinner_canonical_basis n_def)
   also have \<open>\<dots> = (\<Sum>i<n. cnj (x$i) *\<^sub>C y$i)\<close>
     apply (rule sum.cong[OF refl])
     apply (subst sum_single)
     by auto
   also have \<open>\<dots> = y \<bullet>c x\<close>
     by (smt (z3) assms(2) complex_scaleC_def conjugate_complex_def dim_vec_conjugate lessThan_atLeast0 lessThan_iff mult.commute scalar_prod_def sum.cong vec_index_conjugate)
   finally show ?thesis
     by -
 qed
 
 lemma cscalar_prod_vec_of_basis_enum: "cscalar_prod (vec_of_basis_enum \<phi>) (vec_of_basis_enum \<psi>) = cinner \<psi> \<phi>"
   for \<psi> :: "'a::onb_enum"
   apply (subst cinner_basis_enum_of_vec[symmetric, where 'a='a])
   by simp_all
 
 lemma norm_ell2_vec_of_basis_enum: "norm \<psi> =
   (let \<psi>' = vec_of_basis_enum \<psi> in
     sqrt (\<Sum> i \<in> {0 ..< dim_vec \<psi>'}. let z = vec_index \<psi>' i in (Re z)\<^sup>2 + (Im z)\<^sup>2))"
   (is "_ = ?rhs") for \<psi> :: "'a::onb_enum"
 proof -
   have "norm \<psi> = sqrt (cmod (\<Sum>i = 0..<dim_vec (vec_of_basis_enum \<psi>).
             vec_of_basis_enum \<psi> $ i * conjugate (vec_of_basis_enum \<psi>) $ i))"
     unfolding norm_eq_sqrt_cinner[where 'a='a] cscalar_prod_vec_of_basis_enum[symmetric] scalar_prod_def dim_vec_conjugate
     by rule
   also have "\<dots> = sqrt (cmod (\<Sum>x = 0..<dim_vec (vec_of_basis_enum \<psi>).
                     let z = vec_of_basis_enum \<psi> $ x in (Re z)\<^sup>2 + (Im z)\<^sup>2))"
     apply (subst sum.cong, rule refl)
      apply (subst vec_index_conjugate)
     by (auto simp: Let_def complex_mult_cnj)
   also have "\<dots> = ?rhs"
     unfolding Let_def norm_of_real
     apply (subst abs_of_nonneg)
      apply (rule sum_nonneg)
     by auto
   finally show ?thesis
     by -
 qed
 
 lemma basis_enum_of_vec_unit_vec:
   defines "basis \<equiv> (canonical_basis::'a::basis_enum list)"
     and "n \<equiv> length (canonical_basis :: 'a list)"
   assumes a3: "i < n"
   shows "basis_enum_of_vec (unit_vec n i) = basis!i"
 proof-
   define L::"complex list" where "L = list_of_vec (unit_vec n i)"
   define I::"nat list" where "I = [0..<n]"
   have "length L = n"
     by (simp add: L_def)
   moreover have "length basis = n"
     by (simp add: basis_def n_def)
   ultimately have "map2 (*\<^sub>C) L basis = map (\<lambda>j. L!j *\<^sub>C basis!j) I"
     by (simp add: I_def list_eq_iff_nth_eq)
   hence "sum_list (map2 (*\<^sub>C) L basis) = sum_list (map (\<lambda>j. L!j *\<^sub>C basis!j) I)"
     by simp
   also have "\<dots> = sum (\<lambda>j. L!j *\<^sub>C basis!j) {0..n-1}"
   proof-
     have "set I = {0..n-1}"
       using I_def a3 by auto
     thus ?thesis
       using Groups_List.sum_code[where xs = I and g = "(\<lambda>j. L!j *\<^sub>C basis!j)"]
       by (simp add: I_def)
   qed
   also have "\<dots> = sum (\<lambda>j. (list_of_vec (unit_vec n i))!j *\<^sub>C basis!j) {0..n-1}"
     unfolding L_def by blast
   finally have "sum_list (map2 (*\<^sub>C) (list_of_vec (unit_vec n i)) basis)
        = sum (\<lambda>j. (list_of_vec (unit_vec n i))!j *\<^sub>C basis!j) {0..n-1}"
     using L_def by blast
   also have "\<dots> = basis ! i"
   proof-
     have "(\<Sum>j = 0..n - 1. list_of_vec (unit_vec n i) ! j *\<^sub>C basis ! j) =
           (\<Sum>j \<in> {0..n - 1}. list_of_vec (unit_vec n i) ! j *\<^sub>C basis ! j)"
       by simp
     also have "\<dots> = list_of_vec (unit_vec n i) ! i *\<^sub>C basis ! i
                + (\<Sum>j \<in> {0..n - 1}-{i}. list_of_vec (unit_vec n i) ! j *\<^sub>C basis ! j)"
     proof-
       define a where "a j = list_of_vec (unit_vec n i) ! j *\<^sub>C basis ! j" for j
       define S where "S = {0..n - 1}"
       have "finite S"
         by (simp add: S_def)
       hence "(\<Sum>j \<in> insert i S. a j) = a i + (\<Sum>j\<in>S-{i}. a j)"
         using Groups_Big.comm_monoid_add_class.sum.insert_remove
         by auto
       moreover have "S-{i} = {0..n-1}-{i}"
         unfolding S_def
         by blast
       moreover have "insert i S = {0..n-1}"
         using S_def Suc_diff_1 a3 atLeastAtMost_iff diff_is_0_eq' le_SucE le_numeral_extra(4)
           less_imp_le not_gr_zero
         by fastforce
       ultimately show ?thesis
         using \<open>a \<equiv> \<lambda>j. list_of_vec (unit_vec n i) ! j *\<^sub>C basis ! j\<close>
         by simp
     qed
     also have "\<dots> = list_of_vec (unit_vec n i) ! i *\<^sub>C basis ! i"
     proof-
       have "j \<in> {0..n - 1}-{i} \<Longrightarrow> list_of_vec (unit_vec n i) ! j = 0"
         for j
         using a3 atMost_atLeast0 atMost_iff diff_Suc_less index_unit_vec(1) le_less_trans
           list_of_vec_index member_remove zero_le by fastforce
       hence "j \<in> {0..n - 1}-{i} \<Longrightarrow> list_of_vec (unit_vec n i) ! j *\<^sub>C basis ! j = 0"
         for j
         by auto
       hence "(\<Sum>j \<in> {0..n - 1}-{i}. list_of_vec (unit_vec n i) ! j *\<^sub>C basis ! j) = 0"
         by (simp add: \<open>\<And>j. j \<in> {0..n - 1} - {i} \<Longrightarrow> list_of_vec (unit_vec n i) ! j *\<^sub>C basis ! j = 0\<close>)
       thus ?thesis by simp
     qed
     also have "\<dots> = basis ! i"
       by (simp add: a3)
     finally show ?thesis
       using \<open>(\<Sum>j = 0..n - 1. list_of_vec (unit_vec n i) ! j *\<^sub>C basis ! j)
              = list_of_vec (unit_vec n i) ! i *\<^sub>C basis ! i + (\<Sum>j\<in>{0..n - 1} - {i}. list_of_vec (unit_vec n i) ! j *\<^sub>C basis ! j)\<close>
         \<open>list_of_vec (unit_vec n i) ! i *\<^sub>C basis ! i + (\<Sum>j\<in>{0..n - 1} - {i}. list_of_vec (unit_vec n i) ! j *\<^sub>C basis ! j)
            = list_of_vec (unit_vec n i) ! i *\<^sub>C basis ! i\<close>
         \<open>list_of_vec (unit_vec n i) ! i *\<^sub>C basis ! i = basis ! i\<close>
       by auto
   qed
   finally have "sum_list (map2 (*\<^sub>C) (list_of_vec (unit_vec n i)) basis)
       = basis ! i"
     by (simp add: assms)
   hence "sum_list (map2 scaleC (list_of_vec (unit_vec n i)) (canonical_basis::'a list))
       = (canonical_basis::'a list) ! i"
     by (simp add: assms)
   thus ?thesis
     unfolding basis_enum_of_vec_def
     by (simp add: assms)
 qed
 
 
 lemma vec_of_basis_enum_ket:
   "vec_of_basis_enum (ket i) = unit_vec (CARD('a)) (enum_idx i)"
   for i::"'a::enum"
 proof-
   have "dim_vec (vec_of_basis_enum (ket i))
       = dim_vec (unit_vec (CARD('a)) (enum_idx i))"
   proof-
     have "dim_vec (unit_vec (CARD('a)) (enum_idx i))
       = CARD('a)"
       by simp
     moreover have "dim_vec (vec_of_basis_enum (ket i)) = CARD('a)"
       unfolding vec_of_basis_enum_def vec_of_basis_enum_def by auto
     ultimately show ?thesis by simp
   qed
   moreover have "vec_of_basis_enum (ket i) $ j =
     (unit_vec (CARD('a)) (enum_idx i)) $ j"
     if "j < dim_vec (vec_of_basis_enum (ket i))"
     for j
   proof-
     have j_bound: "j < length (canonical_basis::'a ell2 list)"
       by (metis dim_vec_of_basis_enum' that)
     have y1: "cindependent (set (canonical_basis::'a ell2 list))"
       using is_cindependent_set by blast
     have y2: "canonical_basis ! j \<in> set (canonical_basis::'a ell2 list)"
       using j_bound by auto
     have p1: "enum_class.enum ! enum_idx i = i"
       using enum_idx_correct by blast
     moreover have p2: "(canonical_basis::'a ell2 list) ! t  = ket ((enum_class.enum::'a list) ! t)"
       if "t < length (enum_class.enum::'a list)"
       for t
       unfolding canonical_basis_ell2_def
       using that by auto
     moreover have p3: "enum_idx i < length (enum_class.enum::'a list)"
     proof-
       have "set (enum_class.enum::'a list) = UNIV"
         using UNIV_enum by blast
       hence "i \<in> set (enum_class.enum::'a list)"
         by blast
       thus ?thesis
         unfolding enum_idx_def
         by (metis index_of_bound length_greater_0_conv length_pos_if_in_set)
     qed
     ultimately have p4: "(canonical_basis::'a ell2 list) ! (enum_idx i)  = ket i"
       by auto
     have "enum_idx i < length (enum_class.enum::'a list)"
       using p3
       by auto
     moreover have "length (enum_class.enum::'a list) = dim_vec (vec_of_basis_enum (ket i))"
       unfolding vec_of_basis_enum_def canonical_basis_ell2_def
       using dim_vec_of_basis_enum'[where v = "ket i"]
       unfolding canonical_basis_ell2_def by simp
     ultimately have enum_i_dim_vec: "enum_idx i < dim_vec (unit_vec (CARD('a)) (enum_idx i))"
       using \<open>dim_vec (vec_of_basis_enum (ket i)) = dim_vec (unit_vec (CARD('a)) (enum_idx i))\<close> by auto
     hence r1: "(unit_vec (CARD('a)) (enum_idx i)) $ j
         = (if enum_idx i = j then 1 else 0)"
       using \<open>dim_vec (vec_of_basis_enum (ket i)) = dim_vec (unit_vec (CARD('a)) (enum_idx i))\<close> that by auto
     moreover have "vec_of_basis_enum (ket i) $ j = (if enum_idx i = j then 1 else 0)"
     proof(cases "enum_idx i = j")
       case True
       have "crepresentation (set (canonical_basis::'a ell2 list))
           ((canonical_basis::'a ell2 list) ! j) ((canonical_basis::'a ell2 list) ! j) = 1"
         using y1 y2 complex_vector.representation_basis[where
             basis = "set (canonical_basis::'a ell2 list)"
             and b = "(canonical_basis::'a ell2 list) ! j"]
         by smt
 
       hence "vec_of_basis_enum ((canonical_basis::'a ell2 list) ! j) $ j = 1"
         unfolding vec_of_basis_enum_def
         by (metis j_bound nth_map vec_of_list_index)
       hence "vec_of_basis_enum ((canonical_basis::'a ell2 list) ! (enum_idx i))
             $ enum_idx i = 1"
         using True by simp
       hence "vec_of_basis_enum (ket i) $ enum_idx i = 1"
         using p4
         by simp
       thus ?thesis using True unfolding vec_of_basis_enum_def by auto
     next
       case False
       have "crepresentation (set (canonical_basis::'a ell2 list))
           ((canonical_basis::'a ell2 list) ! (enum_idx i)) ((canonical_basis::'a ell2 list) ! j) = 0"
         using y1 y2 complex_vector.representation_basis[where
             basis = "set (canonical_basis::'a ell2 list)"
             and b = "(canonical_basis::'a ell2 list) ! j"]
         by (metis (mono_tags, opaque_lifting) False enum_i_dim_vec basis_enum_of_vec_inverse
             basis_enum_of_vec_unit_vec canonical_basis_length_ell2 index_unit_vec(3) j_bound
             list_of_vec_index list_vec nth_map r1 vec_of_basis_enum_def)
       hence "vec_of_basis_enum ((canonical_basis::'a ell2 list) ! (enum_idx i)) $ j = 0"
         unfolding vec_of_basis_enum_def by (smt j_bound nth_map vec_of_list_index)
       hence "vec_of_basis_enum ((canonical_basis::'a ell2 list) ! (enum_idx i)) $ j = 0"
         by auto
       hence "vec_of_basis_enum (ket i) $ j = 0"
         using p4
         by simp
       thus ?thesis using False unfolding vec_of_basis_enum_def by simp
     qed
     ultimately show ?thesis by auto
   qed
   ultimately show ?thesis
     using Matrix.eq_vecI
     by auto
 qed
 
 lemma vec_of_basis_enum_zero:
   defines "nA \<equiv> length (canonical_basis :: 'a::basis_enum list)"
   shows "vec_of_basis_enum (0::'a) = 0\<^sub>v nA"
   by (metis assms carrier_vecI dim_vec_of_basis_enum' minus_cancel_vec right_minus_eq vec_of_basis_enum_minus)
 
 lemma (in complex_vec_space) vec_of_basis_enum_cspan:
   fixes X :: "'a::basis_enum set"
   assumes "length (canonical_basis :: 'a list) = n"
   shows "vec_of_basis_enum ` cspan X = span (vec_of_basis_enum ` X)"
 proof -
   have carrier: "vec_of_basis_enum ` X \<subseteq> carrier_vec n"
     by (metis assms carrier_vecI dim_vec_of_basis_enum' image_subsetI)
   have lincomb_sum: "lincomb c A = vec_of_basis_enum (\<Sum>b\<in>B. c' b *\<^sub>C b)"
     if B_def: "B = basis_enum_of_vec ` A" and \<open>finite A\<close>
       and AX: "A \<subseteq> vec_of_basis_enum ` X" and c'_def: "\<And>b. c' b = c (vec_of_basis_enum b)"
     for c c' A and B::"'a set"
     unfolding B_def using \<open>finite A\<close> AX
   proof induction
     case empty
     then show ?case
       by (simp add: assms vec_of_basis_enum_zero)
   next
     case (insert x F)
     then have IH: "lincomb c F = vec_of_basis_enum (\<Sum>b\<in>basis_enum_of_vec ` F. c' b *\<^sub>C b)"
       (is "_ = ?sum")
       by simp
     have xx: "vec_of_basis_enum (basis_enum_of_vec x :: 'a) = x"
       apply (rule basis_enum_of_vec_inverse)
       using assms carrier carrier_vecD insert.prems by auto
     have "lincomb c (insert x F) = c x \<cdot>\<^sub>v x + lincomb c F"
       apply (rule lincomb_insert2)
       using insert.hyps insert.prems carrier by auto
     also have "c x \<cdot>\<^sub>v x = vec_of_basis_enum (c' (basis_enum_of_vec x) *\<^sub>C (basis_enum_of_vec x :: 'a))"
       by (simp add: xx vec_of_basis_enum_scaleC c'_def)
     also note IH
     also have
       "vec_of_basis_enum (c' (basis_enum_of_vec x) *\<^sub>C (basis_enum_of_vec x :: 'a)) + ?sum
           = vec_of_basis_enum (\<Sum>b\<in>basis_enum_of_vec ` insert x F. c' b *\<^sub>C b)"
       apply simp apply (rule sym)
       apply (subst sum.insert)
       using \<open>finite F\<close> \<open>x \<notin> F\<close> dim_vec_of_basis_enum' insert.prems
         vec_of_basis_enum_add c'_def by auto
     finally show ?case
       by -
   qed
 
   show ?thesis
   proof auto
     fix x assume "x \<in> local.span (vec_of_basis_enum ` X)"
     then obtain c A where xA: "x = lincomb c A" and "finite A" and AX: "A \<subseteq> vec_of_basis_enum ` X"
       unfolding span_def by auto
     define B::"'a set" and c' where "B = basis_enum_of_vec ` A"
       and "c' b = c (vec_of_basis_enum b)" for b::'a
     note xA
     also have "lincomb c A = vec_of_basis_enum (\<Sum>b\<in>B. c' b *\<^sub>C b)"
       using B_def \<open>finite A\<close> AX c'_def by (rule lincomb_sum)
     also have "\<dots> \<in> vec_of_basis_enum ` cspan X"
       unfolding complex_vector.span_explicit
       apply (rule imageI) apply (rule CollectI)
       apply (rule exI) apply (rule exI)
       using \<open>finite A\<close> AX by (auto simp: B_def)
     finally show "x \<in> vec_of_basis_enum ` cspan X"
       by -
   next
     fix x assume "x \<in> cspan X"
     then obtain c' B where x: "x = (\<Sum>b\<in>B. c' b *\<^sub>C b)" and "finite B" and BX: "B \<subseteq> X"
       unfolding complex_vector.span_explicit by auto
     define A and c where "A = vec_of_basis_enum ` B"
       and "c b = c' (basis_enum_of_vec b)" for b
     have "vec_of_basis_enum x = vec_of_basis_enum (\<Sum>b\<in>B. c' b *\<^sub>C b)"
       using x by simp
     also have "\<dots> = lincomb c A"
       apply (rule lincomb_sum[symmetric])
       unfolding A_def c_def using BX \<open>finite B\<close>
       by (auto simp: image_image)
     also have "\<dots> \<in> span (vec_of_basis_enum ` X)"
       unfolding span_def apply (rule CollectI)
       apply (rule exI, rule exI)
       unfolding A_def using \<open>finite B\<close> BX by auto
     finally show "vec_of_basis_enum x \<in> local.span (vec_of_basis_enum ` X)"
       by -
   qed
 qed
 
 lemma (in complex_vec_space) basis_enum_of_vec_span:
   assumes "length (canonical_basis :: 'a list) = n"
   assumes "Y \<subseteq> carrier_vec n"
   shows "basis_enum_of_vec ` local.span Y = cspan (basis_enum_of_vec ` Y :: 'a::basis_enum set)"
 proof -
   define X::"'a set" where "X = basis_enum_of_vec ` Y"
   then have "cspan (basis_enum_of_vec ` Y :: 'a set) = basis_enum_of_vec ` vec_of_basis_enum ` cspan X"
     by (simp add: image_image)
   also have "\<dots> = basis_enum_of_vec ` local.span (vec_of_basis_enum ` X)"
     apply (subst vec_of_basis_enum_cspan)
     using assms by simp_all
   also have "vec_of_basis_enum ` X = Y"
     unfolding X_def image_image
     apply (rule image_cong[where g=id and M=Y and N=Y, simplified])
     using assms(1) assms(2) by auto
   finally show ?thesis
     by simp
 qed
 
 lemma vec_of_basis_enum_canonical_basis:
   assumes "i < length (canonical_basis :: 'a list)"
   shows "vec_of_basis_enum (canonical_basis!i :: 'a)
        = unit_vec (length (canonical_basis :: 'a::basis_enum list)) i"
   by (metis assms basis_enum_of_vec_inverse basis_enum_of_vec_unit_vec index_unit_vec(3))
 
 lemma vec_of_basis_enum_times:
   fixes \<psi> \<phi> :: "'a::one_dim"
   shows "vec_of_basis_enum (\<psi> * \<phi>)
    = vec_of_list [vec_index (vec_of_basis_enum \<psi>) 0 * vec_index (vec_of_basis_enum \<phi>) 0]"
 proof -
   have [simp]: \<open>crepresentation {1} x 1 = one_dim_iso x\<close> for x :: 'a
     apply (subst one_dim_scaleC_1[where x=x, symmetric])
     apply (subst complex_vector.representation_scale)
     by (auto simp add: complex_vector.span_base complex_vector.representation_basis)
   show ?thesis
     apply (rule eq_vecI)
     by (auto simp: vec_of_basis_enum_def)
 qed
 
 lemma vec_of_basis_enum_to_inverse:
   fixes \<psi> :: "'a::one_dim"
   shows "vec_of_basis_enum (inverse \<psi>) = vec_of_list [inverse (vec_index (vec_of_basis_enum \<psi>) 0)]"
 proof -
   have [simp]: \<open>crepresentation {1} x 1 = one_dim_iso x\<close> for x :: 'a
     apply (subst one_dim_scaleC_1[where x=x, symmetric])
     apply (subst complex_vector.representation_scale)
     by (auto simp add: complex_vector.span_base complex_vector.representation_basis)
   show ?thesis
     apply (rule eq_vecI)
      apply (auto simp: vec_of_basis_enum_def)
     by (metis complex_vector.scale_cancel_right one_dim_inverse one_dim_scaleC_1 zero_neq_one)
 qed
 
 lemma vec_of_basis_enum_divide:
   fixes \<psi> \<phi> :: "'a::one_dim"
   shows "vec_of_basis_enum (\<psi> / \<phi>)
    = vec_of_list [vec_index (vec_of_basis_enum \<psi>) 0 / vec_index (vec_of_basis_enum \<phi>) 0]"
   by (simp add: divide_inverse vec_of_basis_enum_to_inverse vec_of_basis_enum_times)
 
 lemma vec_of_basis_enum_1: "vec_of_basis_enum (1 :: 'a::one_dim) = vec_of_list [1]"
 proof -
   have [simp]: \<open>crepresentation {1} x 1 = one_dim_iso x\<close> for x :: 'a
     apply (subst one_dim_scaleC_1[where x=x, symmetric])
     apply (subst complex_vector.representation_scale)
     by (auto simp add: complex_vector.span_base complex_vector.representation_basis)
   show ?thesis
     apply (rule eq_vecI)
     by (auto simp: vec_of_basis_enum_def)
 qed
 
 lemma vec_of_basis_enum_ell2_component:
   fixes \<psi> :: \<open>'a::enum ell2\<close>
   assumes [simp]: \<open>i < CARD('a)\<close>
   shows \<open>vec_of_basis_enum \<psi> $ i = Rep_ell2 \<psi> (Enum.enum ! i)\<close>
 proof -
   let ?i = \<open>Enum.enum ! i\<close>
-  have \<open>Rep_ell2 \<psi> (Enum.enum ! i) = \<langle>ket ?i, \<psi>\<rangle>\<close>
+  have \<open>Rep_ell2 \<psi> (Enum.enum ! i) = ket ?i \<bullet>\<^sub>C \<psi>\<close>
     by (simp add: cinner_ket_left)
   also have \<open>\<dots> = vec_of_basis_enum \<psi> \<bullet>c vec_of_basis_enum (ket ?i :: 'a ell2)\<close>
     by (rule cscalar_prod_vec_of_basis_enum[symmetric])
   also have \<open>\<dots> = vec_of_basis_enum \<psi> \<bullet>c unit_vec (CARD('a)) i\<close>
     by (simp add: vec_of_basis_enum_ket enum_idx_enum)
   also have \<open>\<dots> = vec_of_basis_enum \<psi> \<bullet> unit_vec (CARD('a)) i\<close>
     by (smt (verit, best) assms carrier_vecI conjugate_conjugate_sprod conjugate_id conjugate_vec_sprod_comm dim_vec_conjugate eq_vecI index_unit_vec(3) scalar_prod_left_unit vec_index_conjugate)
   also have \<open>\<dots> = vec_of_basis_enum \<psi> $ i\<close>
     by simp
   finally show ?thesis
     by simp
 qed
 
 
 lemma corthogonal_vec_of_basis_enum:
   fixes S :: "'a::onb_enum list"
   shows "corthogonal (map vec_of_basis_enum S) \<longleftrightarrow> is_ortho_set (set S) \<and> distinct S"
 proof auto
   assume assm: \<open>corthogonal (map vec_of_basis_enum S)\<close>
   then show \<open>is_ortho_set (set S)\<close>
     by (smt (verit, ccfv_SIG) cinner_eq_zero_iff corthogonal_def cscalar_prod_vec_of_basis_enum in_set_conv_nth is_ortho_set_def length_map nth_map)
   show \<open>distinct S\<close>
     using assm corthogonal_distinct distinct_map by blast
 next
   assume \<open>is_ortho_set (set S)\<close> and \<open>distinct S\<close>
   then show \<open>corthogonal (map vec_of_basis_enum S)\<close>
     by (smt (verit, ccfv_threshold) cinner_eq_zero_iff corthogonalI cscalar_prod_vec_of_basis_enum is_ortho_set_def length_map length_map nth_eq_iff_index_eq nth_map nth_map nth_mem nth_mem)
 qed
 
 subsection \<open>Isomorphism between bounded linear functions and matrices\<close>
 
 
 text \<open>We define the canonical isomorphism between \<^typ>\<open>'a::basis_enum \<Rightarrow>\<^sub>C\<^sub>L'b::basis_enum\<close>
   and the complex \<^term>\<open>n*m\<close>-matrices (where n,m are the dimensions of \<^typ>\<open>'a\<close>, \<^typ>\<open>'b\<close>,
   respectively). This is possible if \<^typ>\<open>'a\<close>, \<^typ>\<open>'b\<close> are of class \<^class>\<open>basis_enum\<close>
   since that class fixes a finite canonical basis. Matrices are represented using
   the \<^typ>\<open>complex mat\<close> type from \<^session>\<open>Jordan_Normal_Form\<close>.
   (The isomorphism will be called \<^term>\<open>mat_of_cblinfun\<close> below.)\<close>
 
 definition mat_of_cblinfun :: \<open>'a::{basis_enum,complex_normed_vector} \<Rightarrow>\<^sub>C\<^sub>L'b::{basis_enum,complex_normed_vector} \<Rightarrow> complex mat\<close> where
   \<open>mat_of_cblinfun f =
     mat (length (canonical_basis :: 'b list)) (length (canonical_basis :: 'a list)) (
     \<lambda> (i, j). crepresentation (set (canonical_basis::'b list)) (f *\<^sub>V ((canonical_basis::'a list)!j)) ((canonical_basis::'b list)!i))\<close>
   for f
 
 lift_definition cblinfun_of_mat :: \<open>complex mat \<Rightarrow> 'a::{basis_enum,complex_normed_vector} \<Rightarrow>\<^sub>C\<^sub>L'b::{basis_enum,complex_normed_vector}\<close> is
   \<open>\<lambda>M. \<lambda>v. (if M\<in>carrier_mat (length (canonical_basis :: 'b list)) (length (canonical_basis :: 'a list))
            then basis_enum_of_vec (M *\<^sub>v vec_of_basis_enum v)
            else 0)\<close>
 proof
   fix M :: "complex mat"
   define m where "m = length (canonical_basis :: 'b list)"
   define n where "n = length (canonical_basis :: 'a list)"
   define f::"complex mat \<Rightarrow> 'a \<Rightarrow> 'b"
     where "f M v = (if M\<in>carrier_mat m n
         then basis_enum_of_vec (M *\<^sub>v vec_of_basis_enum (v::'a))
         else (0::'b))"
     for M::"complex mat" and v::'a
 
   show add: \<open>f M (b1 + b2) = f M b1 + f M b2\<close> for b1 b2
     apply (auto simp: f_def)
     by (metis (mono_tags, lifting) carrier_matD(1) carrier_vec_dim_vec dim_mult_mat_vec dim_vec_of_basis_enum' m_def mult_add_distrib_mat_vec n_def basis_enum_of_vec_add vec_of_basis_enum_add)
 
   show scale: \<open>f M (c *\<^sub>C b) = c *\<^sub>C f M b\<close> for c b
     apply (auto simp: f_def)
     by (metis carrier_matD(1) carrier_vec_dim_vec dim_mult_mat_vec dim_vec_of_basis_enum' m_def mult_mat_vec n_def basis_enum_of_vec_mult vec_of_basis_enum_scaleC)
 
   from add scale have \<open>clinear (f M)\<close>
     by (simp add: clinear_iff)
 
   show \<open>\<exists>K. \<forall>b. norm (f M b) \<le> norm b * K\<close>
   proof (cases "M\<in>carrier_mat m n")
     case True
     have f_def': "f M v = basis_enum_of_vec (M *\<^sub>v (vec_of_basis_enum v))" for v
       using True f_def
         m_def n_def by auto
     have M_carrier_mat:
       "M \<in> carrier_mat m n"
       by (simp add: True)
     have "bounded_clinear (f M)"
       apply (rule bounded_clinear_finite_dim) using \<open>clinear (f M)\<close> by auto
     thus ?thesis
       by (simp add: bounded_clinear.bounded)
   next
     case False
     thus ?thesis
       unfolding f_def m_def n_def
       by (metis (full_types) order_refl mult_eq_0_iff norm_eq_zero)
   qed
 qed
 
 lemma mat_of_cblinfun_ell2_carrier[simp]: \<open>mat_of_cblinfun (a::'a::enum ell2 \<Rightarrow>\<^sub>C\<^sub>L 'b::enum ell2) \<in> carrier_mat CARD('b) CARD('a)\<close>
   by (simp add: mat_of_cblinfun_def)
 
 lemma dim_row_mat_of_cblinfun[simp]: \<open>dim_row (mat_of_cblinfun (a::'a::enum ell2 \<Rightarrow>\<^sub>C\<^sub>L 'b::enum ell2)) = CARD('b)\<close>
   by (simp add: mat_of_cblinfun_def)
 
 lemma dim_col_mat_of_cblinfun[simp]: \<open>dim_col (mat_of_cblinfun (a::'a::enum ell2 \<Rightarrow>\<^sub>C\<^sub>L 'b::enum ell2)) = CARD('a)\<close>
   by (simp add: mat_of_cblinfun_def)
 
 lemma mat_of_cblinfun_cblinfun_apply:
   "vec_of_basis_enum (F *\<^sub>V u) = mat_of_cblinfun F *\<^sub>v vec_of_basis_enum u"
   for F::"'a::{basis_enum,complex_normed_vector}  \<Rightarrow>\<^sub>C\<^sub>L 'b::{basis_enum,complex_normed_vector}" and u::'a
 proof (rule eq_vecI)
   show \<open>dim_vec (vec_of_basis_enum (F *\<^sub>V u)) = dim_vec (mat_of_cblinfun F *\<^sub>v vec_of_basis_enum u)\<close>
     by (simp add: dim_vec_of_basis_enum' mat_of_cblinfun_def)
 next
   fix i
   define BasisA where "BasisA = (canonical_basis::'a list)"
   define BasisB where "BasisB = (canonical_basis::'b list)"
   define nA where "nA = length (canonical_basis :: 'a list)"
   define nB where "nB = length (canonical_basis :: 'b list)"
   assume \<open>i < dim_vec (mat_of_cblinfun F *\<^sub>v vec_of_basis_enum u)\<close>
   then have [simp]: \<open>i < nB\<close>
     by (simp add: mat_of_cblinfun_def nB_def)
   define v where \<open>v = BasisB ! i\<close>
 
   have [simp]: \<open>dim_row (mat_of_cblinfun F) = nB\<close>
     by (simp add: mat_of_cblinfun_def nB_def)
   have [simp]: \<open>length BasisB = nB\<close>
     by (simp add: nB_def BasisB_def)
   have [simp]: \<open>length BasisA = nA\<close>
     using BasisA_def nA_def by auto
   have [simp]: \<open>cindependent (set BasisA)\<close>
     using BasisA_def is_cindependent_set by auto
   have [simp]: \<open>cindependent (set BasisB)\<close>
     using BasisB_def is_cindependent_set by blast
   have [simp]: \<open>cspan (set BasisB) = UNIV\<close>
     by (simp add: BasisB_def is_generator_set)
   have [simp]: \<open>cspan (set BasisA) = UNIV\<close>
     by (simp add: BasisA_def is_generator_set)
 
   have \<open>(mat_of_cblinfun F *\<^sub>v vec_of_basis_enum u) $ i =
           (\<Sum>j = 0..<nA. row (mat_of_cblinfun F) i $ j * crepresentation (set BasisA) u (vec_of_list BasisA $ j))\<close>
     by (auto simp: vec_of_basis_enum_def scalar_prod_def simp flip: BasisA_def)
   also have \<open>\<dots> = (\<Sum>j = 0..<nA. crepresentation (set BasisB) (F *\<^sub>V BasisA ! j) v
                                  * crepresentation (set BasisA) u (BasisA ! j))\<close>
     apply (rule sum.cong[OF refl])
     by (auto simp: vec_of_list_index mat_of_cblinfun_def scalar_prod_def v_def simp flip: BasisA_def BasisB_def)
   also have \<open>\<dots> = crepresentation (set BasisB) (F *\<^sub>V u) v\<close> (is \<open>(\<Sum>j=_..<_. ?lhs v j) = _\<close>)
   proof (rule complex_vector.representation_eqI[symmetric, THEN fun_cong])
     show \<open>cindependent (set BasisB)\<close> \<open>F *\<^sub>V u \<in> cspan (set BasisB)\<close>
       by simp_all
     show only_basis: \<open>(\<Sum>j = 0..<nA. ?lhs b j) \<noteq> 0 \<Longrightarrow> b \<in> set BasisB\<close> for b
       by (metis (mono_tags, lifting) complex_vector.representation_ne_zero mult_not_zero sum.not_neutral_contains_not_neutral)
     then show \<open>finite {b. (\<Sum>j = 0..<nA. ?lhs b j) \<noteq> 0}\<close>
       by (smt (z3) List.finite_set finite_subset mem_Collect_eq subsetI)
     have \<open>(\<Sum>b | (\<Sum>j = 0..<nA. ?lhs b j) \<noteq> 0. (\<Sum>j = 0..<nA. ?lhs b j) *\<^sub>C b) =
             (\<Sum>b\<in>set BasisB. (\<Sum>j = 0..<nA. ?lhs b j) *\<^sub>C b)\<close>
       apply (rule sum.mono_neutral_left)
       using only_basis by auto
     also have \<open>\<dots> = (\<Sum>b\<in>set BasisB. (\<Sum>a\<in>set BasisA. crepresentation (set BasisB) (F *\<^sub>V a) b * crepresentation (set BasisA) u a) *\<^sub>C b)\<close>
       apply (subst sum.reindex_bij_betw[where h=\<open>nth BasisA\<close> and T=\<open>set BasisA\<close>])
        apply (metis BasisA_def \<open>length BasisA = nA\<close> atLeast0LessThan bij_betw_nth distinct_canonical_basis)
       by simp
     also have \<open>\<dots> = (\<Sum>a\<in>set BasisA. crepresentation (set BasisA) u a *\<^sub>C (\<Sum>b\<in>set BasisB. crepresentation (set BasisB) (F *\<^sub>V a) b *\<^sub>C b))\<close>
       apply (simp add: scaleC_sum_left scaleC_sum_right)
       apply (subst sum.swap)
       by (metis (no_types, lifting) mult.commute sum.cong)
     also have \<open>\<dots> = (\<Sum>a\<in>set BasisA. crepresentation (set BasisA) u a *\<^sub>C (F *\<^sub>V a))\<close>
       apply (subst complex_vector.sum_representation_eq)
       by auto
     also have \<open>\<dots> = F *\<^sub>V (\<Sum>a\<in>set BasisA. crepresentation (set BasisA) u a *\<^sub>C a)\<close>
       by (simp flip: cblinfun.scaleC_right cblinfun.sum_right)
     also have \<open>\<dots> = F *\<^sub>V u\<close>
       apply (subst complex_vector.sum_representation_eq)
       by auto
     finally show \<open>(\<Sum>b | (\<Sum>j = 0..<nA. ?lhs b j) \<noteq> 0. (\<Sum>j = 0..<nA. ?lhs b j) *\<^sub>C b) = F *\<^sub>V u\<close>
       by auto
   qed
   also have \<open>crepresentation (set BasisB) (F *\<^sub>V u) v = vec_of_basis_enum (F *\<^sub>V u) $ i\<close>
     by (auto simp: vec_of_list_index vec_of_basis_enum_def v_def simp flip: BasisB_def)
   finally show \<open>vec_of_basis_enum (F *\<^sub>V u) $ i = (mat_of_cblinfun F *\<^sub>v vec_of_basis_enum u) $ i\<close>
     by simp
 qed
 
 lemma basis_enum_of_vec_cblinfun_apply:
   fixes M :: "complex mat"
   defines "nA \<equiv> length (canonical_basis :: 'a::{basis_enum,complex_normed_vector} list)"
     and "nB \<equiv> length (canonical_basis :: 'b::{basis_enum,complex_normed_vector} list)"
   assumes "M \<in> carrier_mat nB nA" and "dim_vec x = nA"
   shows "basis_enum_of_vec (M *\<^sub>v x) = (cblinfun_of_mat M :: 'a \<Rightarrow>\<^sub>C\<^sub>L 'b) *\<^sub>V basis_enum_of_vec x"
   by (metis assms basis_enum_of_vec_inverse cblinfun_of_mat.rep_eq)
 
 
 lemma mat_of_cblinfun_inverse:
   "cblinfun_of_mat (mat_of_cblinfun B) = B"
   for B::"'a::{basis_enum,complex_normed_vector}  \<Rightarrow>\<^sub>C\<^sub>L 'b::{basis_enum,complex_normed_vector}"
 proof (rule cblinfun_eqI)
   fix x :: 'a define y where \<open>y = vec_of_basis_enum x\<close>
   then have \<open>cblinfun_of_mat (mat_of_cblinfun B) *\<^sub>V x = ((cblinfun_of_mat (mat_of_cblinfun B) :: 'a\<Rightarrow>\<^sub>C\<^sub>L'b) *\<^sub>V basis_enum_of_vec y)\<close>
     by simp
   also have \<open>\<dots> = basis_enum_of_vec (mat_of_cblinfun B *\<^sub>v vec_of_basis_enum (basis_enum_of_vec y :: 'a))\<close>
     apply (transfer fixing: B)
     by (simp add: mat_of_cblinfun_def)
   also have \<open>\<dots> = basis_enum_of_vec (vec_of_basis_enum (B *\<^sub>V x))\<close>
     by (simp add: mat_of_cblinfun_cblinfun_apply y_def)
   also have \<open>\<dots> = B *\<^sub>V x\<close>
     by simp
   finally show \<open>cblinfun_of_mat (mat_of_cblinfun B) *\<^sub>V x = B *\<^sub>V x\<close>
     by -
 qed
 
 lemma mat_of_cblinfun_inj: "inj mat_of_cblinfun"
   by (metis inj_on_def mat_of_cblinfun_inverse)
 
 lemma cblinfun_of_mat_inverse:
   fixes M::"complex mat"
   defines "nA \<equiv> length (canonical_basis :: 'a::{basis_enum,complex_normed_vector} list)"
     and "nB \<equiv> length (canonical_basis :: 'b::{basis_enum,complex_normed_vector} list)"
   assumes "M \<in> carrier_mat nB nA"
   shows "mat_of_cblinfun (cblinfun_of_mat M :: 'a \<Rightarrow>\<^sub>C\<^sub>L 'b) = M"
   by (smt (verit) assms(3) basis_enum_of_vec_inverse carrier_matD(1) carrier_vecD cblinfun_of_mat.rep_eq dim_mult_mat_vec eq_mat_on_vecI mat_carrier mat_of_cblinfun_def mat_of_cblinfun_cblinfun_apply nA_def nB_def)
 
 lemma cblinfun_of_mat_inj: "inj_on (cblinfun_of_mat::complex mat \<Rightarrow> 'a \<Rightarrow>\<^sub>C\<^sub>L 'b)
       (carrier_mat (length (canonical_basis :: 'b::{basis_enum,complex_normed_vector} list))
                    (length (canonical_basis :: 'a::{basis_enum,complex_normed_vector} list)))"
   using cblinfun_of_mat_inverse
   by (metis inj_onI)
 
 
 lemma cblinfun_eq_mat_of_cblinfunI:
   assumes "mat_of_cblinfun a = mat_of_cblinfun b"
   shows "a = b"
   by (metis assms mat_of_cblinfun_inverse)
 
 
 subsection \<open>Matrix operations\<close>
 
 lemma cblinfun_of_mat_plus:
   defines "nA \<equiv> length (canonical_basis :: 'a::{basis_enum,complex_normed_vector} list)"
     and "nB \<equiv> length (canonical_basis :: 'b::{basis_enum,complex_normed_vector} list)"
   assumes [simp,intro]: "M \<in> carrier_mat nB nA" and [simp,intro]: "N \<in> carrier_mat nB nA"
   shows "(cblinfun_of_mat (M + N) :: 'a \<Rightarrow>\<^sub>C\<^sub>L 'b) = ((cblinfun_of_mat M + cblinfun_of_mat N))"
 proof -
   have [simp]: \<open>vec_of_basis_enum (v :: 'a) \<in> carrier_vec nA\<close> for v
     by (auto simp add: carrier_dim_vec dim_vec_of_basis_enum' nA_def)
   have [simp]: \<open>dim_row M = nB\<close> \<open>dim_row N = nB\<close>
     using carrier_matD(1) by auto
   show ?thesis
     apply (transfer fixing: M N)
     by (auto intro!: ext simp add: add_mult_distrib_mat_vec nA_def[symmetric] nB_def[symmetric]
         add_mult_distrib_mat_vec[where nr=nB and nc=nA] basis_enum_of_vec_add)
 qed
 
 lemma mat_of_cblinfun_zero:
   "mat_of_cblinfun (0 :: ('a::{basis_enum,complex_normed_vector}  \<Rightarrow>\<^sub>C\<^sub>L 'b::{basis_enum,complex_normed_vector}))
   = 0\<^sub>m (length (canonical_basis :: 'b list)) (length (canonical_basis :: 'a list))"
   unfolding mat_of_cblinfun_def
   by (auto simp: complex_vector.representation_zero)
 
 lemma mat_of_cblinfun_plus:
   "mat_of_cblinfun (F + G) = mat_of_cblinfun F + mat_of_cblinfun G"
   for F G::"'a::{basis_enum,complex_normed_vector} \<Rightarrow>\<^sub>C\<^sub>L'b::{basis_enum,complex_normed_vector}"
   by (auto simp add: mat_of_cblinfun_def cblinfun.add_left complex_vector.representation_add)
 
 lemma mat_of_cblinfun_id:
   "mat_of_cblinfun (id_cblinfun :: ('a::{basis_enum,complex_normed_vector} \<Rightarrow>\<^sub>C\<^sub>L'a)) = 1\<^sub>m (length (canonical_basis :: 'a list))"
   apply (rule eq_matI)
   by (auto simp: mat_of_cblinfun_def complex_vector.representation_basis is_cindependent_set nth_eq_iff_index_eq)
 
 lemma mat_of_cblinfun_1:
   "mat_of_cblinfun (1 :: ('a::one_dim \<Rightarrow>\<^sub>C\<^sub>L'b::one_dim)) = 1\<^sub>m 1"
   apply (rule eq_matI)
   by (auto simp: mat_of_cblinfun_def complex_vector.representation_basis nth_eq_iff_index_eq)
 
 lemma mat_of_cblinfun_uminus:
   "mat_of_cblinfun (- M) = - mat_of_cblinfun M"
   for M::"'a::{basis_enum,complex_normed_vector} \<Rightarrow>\<^sub>C\<^sub>L'b::{basis_enum,complex_normed_vector}"
 proof-
   define nA where "nA = length (canonical_basis :: 'a list)"
   define nB where "nB = length (canonical_basis :: 'b list)"
   have M1: "mat_of_cblinfun M \<in> carrier_mat nB nA"
     unfolding nB_def nA_def
     by (metis add.right_neutral add_carrier_mat mat_of_cblinfun_plus mat_of_cblinfun_zero nA_def
         nB_def zero_carrier_mat)
   have M2: "mat_of_cblinfun (-M) \<in> carrier_mat nB nA"
     by (metis add_carrier_mat mat_of_cblinfun_plus mat_of_cblinfun_zero diff_0 nA_def nB_def
         uminus_add_conv_diff zero_carrier_mat)
   have "mat_of_cblinfun (M - M) =  0\<^sub>m nB nA"
     unfolding nA_def nB_def
     by (simp add: mat_of_cblinfun_zero)
   moreover have "mat_of_cblinfun (M - M) = mat_of_cblinfun M + mat_of_cblinfun (- M)"
     by (metis mat_of_cblinfun_plus pth_2)
   ultimately have "mat_of_cblinfun M + mat_of_cblinfun (- M) = 0\<^sub>m nB nA"
     by simp
   thus ?thesis
     using M1 M2
     by (smt add_uminus_minus_mat assoc_add_mat comm_add_mat left_add_zero_mat minus_r_inv_mat
         uminus_carrier_mat)
 qed
 
 lemma mat_of_cblinfun_minus:
   "mat_of_cblinfun (M - N) = mat_of_cblinfun M - mat_of_cblinfun N"
   for M::"'a::{basis_enum,complex_normed_vector} \<Rightarrow>\<^sub>C\<^sub>L 'b::{basis_enum,complex_normed_vector}" and N::"'a \<Rightarrow>\<^sub>C\<^sub>L'b"
   by (smt (z3) add_uminus_minus_mat mat_of_cblinfun_uminus mat_carrier mat_of_cblinfun_def mat_of_cblinfun_plus pth_2)
 
 lemma cblinfun_of_mat_uminus:
   defines "nA \<equiv> length (canonical_basis :: 'a::{basis_enum,complex_normed_vector} list)"
     and "nB \<equiv> length (canonical_basis :: 'b::{basis_enum,complex_normed_vector} list)"
   assumes "M \<in> carrier_mat nB nA"
   shows "(cblinfun_of_mat (-M) :: 'a \<Rightarrow>\<^sub>C\<^sub>L 'b) = - cblinfun_of_mat M"
   by (smt assms add.group_axioms add.right_neutral add_minus_cancel add_uminus_minus_mat
       cblinfun_of_mat_plus group.inverse_inverse mat_of_cblinfun_inverse mat_of_cblinfun_zero
       minus_r_inv_mat uminus_carrier_mat)
 
 lemma cblinfun_of_mat_minus:
   fixes M::"complex mat"
   defines "nA \<equiv> length (canonical_basis :: 'a::{basis_enum,complex_normed_vector} list)"
     and "nB \<equiv> length (canonical_basis :: 'b::{basis_enum,complex_normed_vector} list)"
   assumes "M \<in> carrier_mat nB nA" and "N \<in> carrier_mat nB nA"
   shows "(cblinfun_of_mat (M - N) :: 'a \<Rightarrow>\<^sub>C\<^sub>L 'b) = cblinfun_of_mat M - cblinfun_of_mat N"
   by (metis assms add_uminus_minus_mat cblinfun_of_mat_plus cblinfun_of_mat_uminus pth_2 uminus_carrier_mat)
 
 lemma cblinfun_of_mat_times:
   fixes M N ::"complex mat"
   defines "nA \<equiv> length (canonical_basis :: 'a::{basis_enum,complex_normed_vector} list)"
     and "nB \<equiv> length (canonical_basis :: 'b::{basis_enum,complex_normed_vector} list)"
     and "nC \<equiv> length (canonical_basis :: 'c::{basis_enum,complex_normed_vector} list)"
   assumes a1: "M \<in> carrier_mat nC nB" and a2: "N \<in> carrier_mat nB nA"
   shows  "cblinfun_of_mat (M * N) = ((cblinfun_of_mat M)::'b \<Rightarrow>\<^sub>C\<^sub>L'c) o\<^sub>C\<^sub>L ((cblinfun_of_mat N)::'a \<Rightarrow>\<^sub>C\<^sub>L'b)"
 proof -
   have b1: "((cblinfun_of_mat M)::'b \<Rightarrow>\<^sub>C\<^sub>L'c) v = basis_enum_of_vec (M *\<^sub>v vec_of_basis_enum v)"
     for v
     by (metis assms(4) cblinfun_of_mat.rep_eq nB_def nC_def)
   have b2: "((cblinfun_of_mat N)::'a \<Rightarrow>\<^sub>C\<^sub>L'b) v = basis_enum_of_vec (N *\<^sub>v vec_of_basis_enum v)"
     for v
     by (metis assms(5) cblinfun_of_mat.rep_eq nA_def nB_def)
   have b3: "((cblinfun_of_mat (M * N))::'a \<Rightarrow>\<^sub>C\<^sub>L'c) v
        = basis_enum_of_vec ((M * N) *\<^sub>v vec_of_basis_enum v)"
     for v
     by (metis assms(4) assms(5) cblinfun_of_mat.rep_eq mult_carrier_mat nA_def nC_def)
   have "(basis_enum_of_vec ((M * N) *\<^sub>v vec_of_basis_enum v)::'c)
       = (basis_enum_of_vec (M *\<^sub>v ( vec_of_basis_enum ( (basis_enum_of_vec (N *\<^sub>v vec_of_basis_enum v))::'b ))))"
     for v::'a
   proof-
     have c1: "vec_of_basis_enum (basis_enum_of_vec x :: 'b) = x"
       if "dim_vec x = nB"
       for x::"complex vec"
       using that unfolding nB_def
       by simp
     have c2: "vec_of_basis_enum v \<in> carrier_vec nA"
       by (metis (mono_tags, opaque_lifting) add.commute carrier_vec_dim_vec index_add_vec(2)
           index_zero_vec(2) nA_def vec_of_basis_enum_add basis_enum_of_vec_inverse)
     have "(M * N) *\<^sub>v vec_of_basis_enum v = M *\<^sub>v (N *\<^sub>v vec_of_basis_enum v)"
       using Matrix.assoc_mult_mat_vec a1 a2 c2 by blast
     hence "(basis_enum_of_vec ((M * N) *\<^sub>v vec_of_basis_enum v)::'c)
         = (basis_enum_of_vec (M *\<^sub>v (N *\<^sub>v vec_of_basis_enum v))::'c)"
       by simp
     also have "\<dots> =
       (basis_enum_of_vec (M *\<^sub>v ( vec_of_basis_enum ( (basis_enum_of_vec (N *\<^sub>v vec_of_basis_enum v))::'b ))))"
       using c1 a2 by auto
     finally show ?thesis by simp
   qed
   thus ?thesis using b1 b2 b3
     by (simp add: cblinfun_eqI scaleC_cblinfun.rep_eq)
 qed
 
 lemma cblinfun_of_mat_adjoint:
   defines "nA \<equiv> length (canonical_basis :: 'a::onb_enum list)"
     and "nB \<equiv> length (canonical_basis :: 'b::onb_enum list)"
   fixes M:: "complex mat"
   assumes "M \<in> carrier_mat nB nA"
   shows "((cblinfun_of_mat (mat_adjoint M)) :: 'b \<Rightarrow>\<^sub>C\<^sub>L 'a) = (cblinfun_of_mat M)*"
 proof (rule adjoint_eqI)
-  show "\<langle>cblinfun_of_mat (mat_adjoint M) *\<^sub>V x, y\<rangle> =
-           \<langle>x, cblinfun_of_mat M *\<^sub>V y\<rangle>"
+  show "(cblinfun_of_mat (mat_adjoint M) *\<^sub>V x) \<bullet>\<^sub>C y = x \<bullet>\<^sub>C (cblinfun_of_mat M *\<^sub>V y)"
     for x::'b and y::'a
   proof-
     define u where "u = vec_of_basis_enum x"
     define v where "v = vec_of_basis_enum y"
     have c1: "vec_of_basis_enum ((cblinfun_of_mat (mat_adjoint M) *\<^sub>V x)::'a) = (mat_adjoint M) *\<^sub>v u"
       unfolding u_def
       by (metis (mono_tags, lifting) assms(3) cblinfun_of_mat_inverse map_carrier_mat mat_adjoint_def' mat_of_cblinfun_cblinfun_apply nA_def nB_def transpose_carrier_mat)
     have c2: "(vec_of_basis_enum ((cblinfun_of_mat M *\<^sub>V y)::'b))
         = M *\<^sub>v v"
       by (metis assms(3) cblinfun_of_mat_inverse mat_of_cblinfun_cblinfun_apply nA_def nB_def v_def)
     have c3: "dim_vec v = nA"
       unfolding v_def nA_def vec_of_basis_enum_def
       by (simp add:)
     have c4: "dim_vec u = nB"
       unfolding u_def nB_def vec_of_basis_enum_def
       by (simp add:)
     have "v \<bullet>c ((mat_adjoint M) *\<^sub>v u) = (M *\<^sub>v v) \<bullet>c u"
       using c3 c4 cscalar_prod_adjoint assms(3) by blast
     hence "v \<bullet>c (vec_of_basis_enum ((cblinfun_of_mat (mat_adjoint M) *\<^sub>V x)::'a))
         = (vec_of_basis_enum ((cblinfun_of_mat M *\<^sub>V y)::'b)) \<bullet>c u"
       using c1 c2 by simp
-    thus "\<langle>cblinfun_of_mat (mat_adjoint M) *\<^sub>V x, y\<rangle> =
-          \<langle>x, cblinfun_of_mat M *\<^sub>V y\<rangle>"
+    thus "(cblinfun_of_mat (mat_adjoint M) *\<^sub>V x) \<bullet>\<^sub>C y = x \<bullet>\<^sub>C (cblinfun_of_mat M *\<^sub>V y)"
       unfolding u_def v_def
       by (simp add: cscalar_prod_vec_of_basis_enum)
   qed
 qed
 
 lemma mat_of_cblinfun_classical_operator:
   fixes f::"'a::enum \<Rightarrow> 'b::enum option"
   shows "mat_of_cblinfun (classical_operator f) = mat (CARD('b)) (CARD('a))
            (\<lambda>(r,c). if f (Enum.enum!c) = Some (Enum.enum!r) then 1 else 0)"
 proof -
   define nA where "nA = CARD('a)"
   define nB where "nB = CARD('b)"
   define BasisA where "BasisA = (canonical_basis::'a ell2 list)"
   define BasisB where "BasisB = (canonical_basis::'b ell2 list)"
   have "mat_of_cblinfun (classical_operator f) \<in> carrier_mat nB nA"
     unfolding nA_def nB_def by simp
   moreover have "nA = CARD ('a)"
     unfolding nA_def
     by (simp add:)
   moreover have "nB = CARD ('b)"
     unfolding nB_def
     by (simp add:)
   ultimately have "mat_of_cblinfun (classical_operator f) \<in> carrier_mat (CARD('b)) (CARD('a))"
     unfolding nA_def nB_def
     by simp
   moreover have "(mat_of_cblinfun (classical_operator f))$$(r,c)
   = (mat (CARD('b)) (CARD('a))
     (\<lambda>(r,c). if f (Enum.enum!c) = Some (Enum.enum!r) then 1 else 0))$$(r,c)"
     if a1: "r < CARD('b)" and a2: "c < CARD('a)"
     for r c
   proof-
     have "CARD('a) = length (enum_class.enum::'a list)"
       using card_UNIV_length_enum[where 'a = 'a] .
     hence x1: "BasisA!c = ket ((Enum.enum::'a list)!c)"
       unfolding BasisA_def using a2 canonical_basis_ell2_def
         nth_map[where n = c and xs = "Enum.enum::'a list" and f = ket] by metis
     have cardb: "CARD('b) = length (enum_class.enum::'b list)"
       using card_UNIV_length_enum[where 'a = 'b] .
     hence x2: "BasisB!r = ket ((Enum.enum::'b list)!r)"
       unfolding BasisB_def using a1 canonical_basis_ell2_def
         nth_map[where n = r and xs = "Enum.enum::'b list" and f = ket] by metis
     have "inj (map (ket::'b \<Rightarrow>_))"
       by (meson injI ket_injective list.inj_map)
     hence "length (Enum.enum::'b list) = length (map (ket::'b \<Rightarrow>_) (Enum.enum::'b list))"
       by simp
     hence lengthBasisB: "CARD('b) = length BasisB"
       unfolding BasisB_def canonical_basis_ell2_def using cardb
       by smt
     have v1: "(mat_of_cblinfun (classical_operator f))$$(r,c) = 0"
       if c1: "f (Enum.enum!c) = None"
     proof-
       have "(classical_operator f) *\<^sub>V ket (Enum.enum!c)
           = (case f (Enum.enum!c) of None \<Rightarrow> 0 | Some i \<Rightarrow> ket i)"
         using classical_operator_ket_finite .
       also have "\<dots> = 0"
         using c1 by simp
       finally have "(classical_operator f) *\<^sub>V ket (Enum.enum!c) = 0" .
       hence *: "(classical_operator f) *\<^sub>V BasisA!c = 0"
         using x1 by simp
-      hence "\<langle>BasisB!r, (classical_operator f) *\<^sub>V BasisA!c\<rangle> = 0"
+      hence "is_orthogonal (BasisB!r) (classical_operator f *\<^sub>V BasisA!c)"
         by simp
       thus ?thesis
         unfolding mat_of_cblinfun_def BasisA_def BasisB_def
         by (smt (verit, del_insts) BasisA_def * a1 a2 canonical_basis_length_ell2 complex_vector.representation_zero index_mat(1) old.prod.case)
     qed
     have v2: "(mat_of_cblinfun (classical_operator f))$$(r,c) = 0"
       if c1: "f (Enum.enum!c) = Some (Enum.enum!r')" and c2: "r\<noteq>r'"
         and c3: "r' < CARD('b)"
       for r'
     proof-
       have x3: "BasisB!r' = ket ((Enum.enum::'b list)!r')"
         unfolding BasisB_def using cardb c3 canonical_basis_ell2_def
           nth_map[where n = r' and xs = "Enum.enum::'b list" and f = ket]
         by smt
       have "distinct BasisB"
         unfolding BasisB_def
         by simp
       moreover have "r < length BasisB"
         using a1 lengthBasisB by simp
       moreover have "r' < length BasisB"
         using c3 lengthBasisB by simp
       ultimately have h1: "BasisB!r \<noteq> BasisB!r'"
         using nth_eq_iff_index_eq[where xs = BasisB and i = r and j = r'] c2
         by blast
       have "(classical_operator f) *\<^sub>V ket (Enum.enum!c)
           = (case f (Enum.enum!c) of None \<Rightarrow> 0 | Some i \<Rightarrow> ket i)"
         using classical_operator_ket_finite .
       also have "\<dots> = ket (Enum.enum!r')"
         using c1 by simp
       finally have "(classical_operator f) *\<^sub>V ket (Enum.enum!c) = ket (Enum.enum!r')" .
       hence *: "(classical_operator f) *\<^sub>V BasisA!c = BasisB!r'"
         using x1 x3 by simp
-      moreover have "\<langle>BasisB!r, BasisB!r'\<rangle> = 0"
+      moreover have "is_orthogonal (BasisB!r) (BasisB!r')"
         using h1
         using BasisB_def \<open>r < length BasisB\<close> \<open>r' < length BasisB\<close> is_ortho_set_def is_orthonormal nth_mem
         by metis
-      ultimately have "\<langle>BasisB!r, (classical_operator f) *\<^sub>V BasisA!c\<rangle> = 0"
+      ultimately have "is_orthogonal (BasisB!r) (classical_operator f *\<^sub>V BasisA!c)"
         by simp
       thus ?thesis
         unfolding mat_of_cblinfun_def BasisA_def BasisB_def
         by (smt (z3) BasisA_def BasisB_def * \<open>r < length BasisB\<close> \<open>r' < length BasisB\<close> a2 canonical_basis_length_ell2 case_prod_conv complex_vector.representation_basis h1 index_mat(1) is_cindependent_set nth_mem)
     qed
     have "(mat_of_cblinfun (classical_operator f))$$(r,c) = 0"
       if b1: "f (Enum.enum!c) \<noteq> Some (Enum.enum!r)"
     proof (cases "f (Enum.enum!c) = None")
       case True thus ?thesis using v1 by blast
     next
       case False
       hence "\<exists>R. f (Enum.enum!c) = Some R"
         apply (induction "f (Enum.enum!c)")
          apply simp
         by simp
       then obtain R where R0: "f (Enum.enum!c) = Some R"
         by blast
       have "R \<in> set (Enum.enum::'b list)"
         using UNIV_enum by blast
       hence "\<exists>r'. R = (Enum.enum::'b list)!r' \<and> r' < length (Enum.enum::'b list)"
         by (metis in_set_conv_nth)
       then obtain r' where u1: "R = (Enum.enum::'b list)!r'"
         and u2: "r' < length (Enum.enum::'b list)"
         by blast
       have R1: "f (Enum.enum!c) = Some (Enum.enum!r')"
         using R0 u1 by blast
       have "Some ((Enum.enum::'b list)!r') \<noteq> Some ((Enum.enum::'b list)!r)"
       proof(rule classical)
         assume "\<not>(Some ((Enum.enum::'b list)!r') \<noteq> Some ((Enum.enum::'b list)!r))"
         hence "Some ((Enum.enum::'b list)!r') = Some ((Enum.enum::'b list)!r)"
           by blast
         hence "f (Enum.enum!c) = Some ((Enum.enum::'b list)!r)"
           using R1 by auto
         thus ?thesis
           using b1 by blast
       qed
       hence "((Enum.enum::'b list)!r') \<noteq> ((Enum.enum::'b list)!r)"
         by simp
       hence "r' \<noteq> r"
         by blast
       moreover have "r' < CARD('b)"
         using u2 cardb by simp
       ultimately show ?thesis using R1 v2[where r' = r'] by blast
     qed
     moreover have "(mat_of_cblinfun (classical_operator f))$$(r,c) = 1"
       if b1: "f (Enum.enum!c) = Some (Enum.enum!r)"
     proof-
       have "CARD('b) = length (enum_class.enum::'b list)"
         using card_UNIV_length_enum[where 'a = 'b].
       hence "length (map (ket::'b\<Rightarrow>_) enum_class.enum) = CARD('b)"
         by simp
       hence w0: "map (ket::'b\<Rightarrow>_) enum_class.enum ! r = ket (enum_class.enum ! r)"
         by (simp add: a1)
       have "CARD('a) = length (enum_class.enum::'a list)"
         using card_UNIV_length_enum[where 'a = 'a].
       hence "length (map (ket::'a\<Rightarrow>_) enum_class.enum) = CARD('a)"
         by simp
       hence "map (ket::'a\<Rightarrow>_) enum_class.enum ! c = ket (enum_class.enum ! c)"
         by (simp add: a2)
       hence "(classical_operator f) *\<^sub>V (BasisA!c) = (classical_operator f) *\<^sub>V (ket (Enum.enum!c))"
         unfolding BasisA_def canonical_basis_ell2_def by simp
       also have "... = (case f (enum_class.enum ! c) of None \<Rightarrow> 0 | Some x \<Rightarrow> ket x)"
         by (rule classical_operator_ket_finite)
       also have "\<dots> = BasisB!r"
         using w0 b1 by (simp add: BasisB_def canonical_basis_ell2_def)
       finally have w1: "(classical_operator f) *\<^sub>V (BasisA!c) = BasisB!r"
         by simp
       have "(mat_of_cblinfun (classical_operator f))$$(r,c)
-        = \<langle>BasisB!r, (classical_operator f) *\<^sub>V (BasisA!c)\<rangle>"
+        = (BasisB!r) \<bullet>\<^sub>C (classical_operator f *\<^sub>V (BasisA!c))"
         unfolding BasisB_def BasisA_def mat_of_cblinfun_def
         using \<open>nA = CARD('a)\<close> \<open>nB = CARD('b)\<close> a1 a2 nA_def nB_def apply auto
         by (metis BasisA_def BasisB_def canonical_basis_length_ell2 cinner_canonical_basis complex_vector.representation_basis is_cindependent_set nth_mem w1)
-      also have "\<dots> = \<langle>BasisB!r, BasisB!r\<rangle>"
+      also have "\<dots> = (BasisB!r) \<bullet>\<^sub>C (BasisB!r)"
         using w1 by simp
       also have "\<dots> = 1"
         unfolding BasisB_def
         using \<open>nB = CARD('b)\<close> a1 nB_def
         by (simp add: cinner_canonical_basis)
       finally show ?thesis by blast
     qed
     ultimately show ?thesis
       by (simp add: a1 a2)
   qed
   ultimately show ?thesis
     apply (rule_tac eq_matI) by auto
 qed
 
 lemma mat_of_cblinfun_compose:
   "mat_of_cblinfun (F o\<^sub>C\<^sub>L G) = mat_of_cblinfun F * mat_of_cblinfun G"
   for F::"'b::{basis_enum,complex_normed_vector} \<Rightarrow>\<^sub>C\<^sub>L 'c::{basis_enum,complex_normed_vector}"
     and G::"'a::{basis_enum,complex_normed_vector}  \<Rightarrow>\<^sub>C\<^sub>L 'b"
   by (smt (verit, del_insts) cblinfun_of_mat_inverse mat_carrier mat_of_cblinfun_def mat_of_cblinfun_inverse cblinfun_of_mat_times mult_carrier_mat)
 
 lemma mat_of_cblinfun_scaleC:
   "mat_of_cblinfun ((a::complex) *\<^sub>C F) = a \<cdot>\<^sub>m (mat_of_cblinfun F)"
   for F :: "'a::{basis_enum,complex_normed_vector} \<Rightarrow>\<^sub>C\<^sub>L 'b::{basis_enum,complex_normed_vector}"
   by (auto simp add: complex_vector.representation_scale mat_of_cblinfun_def)
 
 lemma mat_of_cblinfun_scaleR:
   "mat_of_cblinfun ((a::real) *\<^sub>R F) = (complex_of_real a) \<cdot>\<^sub>m (mat_of_cblinfun F)"
   unfolding scaleR_scaleC by (rule mat_of_cblinfun_scaleC)
 
 lemma mat_of_cblinfun_adj:
   "mat_of_cblinfun (F*) = mat_adjoint (mat_of_cblinfun F)"
   for F :: "'a::onb_enum \<Rightarrow>\<^sub>C\<^sub>L 'b::onb_enum"
   by (metis (no_types, lifting) cblinfun_of_mat_inverse map_carrier_mat mat_adjoint_def' mat_carrier cblinfun_of_mat_adjoint mat_of_cblinfun_def mat_of_cblinfun_inverse transpose_carrier_mat)
 
 lemma mat_of_cblinfun_vector_to_cblinfun:
   "mat_of_cblinfun (vector_to_cblinfun \<psi>)
        = mat_of_cols (length (canonical_basis :: 'a list)) [vec_of_basis_enum \<psi>]"
   for \<psi>::"'a::{basis_enum,complex_normed_vector}"
   by (auto simp: mat_of_cols_Cons_index_0 mat_of_cblinfun_def vec_of_basis_enum_def vec_of_list_index)
 
 lemma mat_of_cblinfun_proj:
   fixes a::"'a::onb_enum"
   defines   "d \<equiv> length (canonical_basis :: 'a list)"
     and "norm2 \<equiv> (vec_of_basis_enum a) \<bullet>c (vec_of_basis_enum a)"
   shows  "mat_of_cblinfun (proj a) =
       1 / norm2 \<cdot>\<^sub>m (mat_of_cols d [vec_of_basis_enum a]
                  * mat_of_rows d [conjugate (vec_of_basis_enum a)])" (is \<open>_ = ?rhs\<close>)
 proof (cases "a = 0")
   case False
   have norm2: \<open>norm2 = (norm a)\<^sup>2\<close>
     by (simp add: cscalar_prod_vec_of_basis_enum norm2_def cdot_square_norm[symmetric, simplified])
   have [simp]: \<open>vec_of_basis_enum a \<in> carrier_vec d\<close>
     by (simp add: carrier_vecI d_def)
 
   have \<open>mat_of_cblinfun (proj a) = mat_of_cblinfun (proj (a /\<^sub>R norm a))\<close>
     by (metis (mono_tags, opaque_lifting) ccspan_singleton_scaleC complex_vector.scale_eq_0_iff
         nonzero_imp_inverse_nonzero norm_eq_zero scaleR_scaleC scale_left_imp_eq)
   also have \<open>\<dots> = mat_of_cblinfun (selfbutter (a /\<^sub>R norm a))\<close>
     apply (subst butterfly_eq_proj)
     by (auto simp add: False)
   also have \<open>\<dots> = 1/norm2 \<cdot>\<^sub>m mat_of_cblinfun (selfbutter a)\<close>
     apply (simp add: mat_of_cblinfun_scaleC norm2)
     by (simp add: inverse_eq_divide power2_eq_square)
   also have \<open>\<dots> = 1 / norm2 \<cdot>\<^sub>m (mat_of_cblinfun (vector_to_cblinfun a :: complex \<Rightarrow>\<^sub>C\<^sub>L 'a) * mat_adjoint (mat_of_cblinfun (vector_to_cblinfun a :: complex \<Rightarrow>\<^sub>C\<^sub>L 'a)))\<close>
     by (simp add: butterfly_def mat_of_cblinfun_compose mat_of_cblinfun_adj)
   also have \<open>\<dots> = ?rhs\<close>
     by (simp add: mat_of_cblinfun_vector_to_cblinfun mat_adjoint_def flip: d_def)
   finally show ?thesis
     by -
 next
   case True
   show ?thesis
     apply (rule eq_matI)
     by (auto simp: True mat_of_cblinfun_zero vec_of_basis_enum_zero scalar_prod_def  mat_of_rows_index
         simp flip: d_def)
 qed
 
 
 lemma mat_of_cblinfun_ell2_component:
   fixes a :: \<open>'a::enum ell2 \<Rightarrow>\<^sub>C\<^sub>L 'b::enum ell2\<close>
   assumes [simp]: \<open>i < CARD('b)\<close> \<open>j < CARD('a)\<close>
   shows \<open>mat_of_cblinfun a $$ (i,j) = Rep_ell2 (a *\<^sub>V ket (Enum.enum ! j)) (Enum.enum ! i)\<close>
 proof -
   let ?i = \<open>Enum.enum ! i\<close> and ?j = \<open>Enum.enum ! j\<close> and ?aj = \<open>a *\<^sub>V ket (Enum.enum ! j)\<close>
   have \<open>Rep_ell2 ?aj (Enum.enum ! i) = vec_of_basis_enum ?aj $ i\<close>
     by (rule vec_of_basis_enum_ell2_component[symmetric], simp)
   also have \<open>\<dots> = (mat_of_cblinfun a *\<^sub>v vec_of_basis_enum (ket (enum_class.enum ! j) :: 'a ell2)) $ i\<close>
     by (simp add: mat_of_cblinfun_cblinfun_apply)
   also have \<open>\<dots> = (mat_of_cblinfun a *\<^sub>v unit_vec CARD('a) j) $ i\<close>
     by (simp add: vec_of_basis_enum_ket enum_idx_enum)
   also have \<open>\<dots> = mat_of_cblinfun a $$ (i, j)\<close>
     apply (subst mat_entry_explicit[where m=\<open>CARD('b)\<close>])
     by auto
   finally show ?thesis
     by auto
 qed
 
 
 lemma mat_of_cblinfun_sandwich:
   fixes a :: "(_::onb_enum, _::onb_enum) cblinfun"
   shows \<open>mat_of_cblinfun (sandwich a *\<^sub>V b) = (let a' = mat_of_cblinfun a in a' * mat_of_cblinfun b * mat_adjoint a')\<close>
   by (simp add: mat_of_cblinfun_compose sandwich_apply Let_def mat_of_cblinfun_adj)
 
 
 subsection \<open>Operations on subspaces\<close>
 
 lemma ccspan_gram_schmidt0_invariant:
   defines "basis_enum \<equiv> (basis_enum_of_vec :: _ \<Rightarrow> 'a::{basis_enum,complex_normed_vector})"
   defines "n \<equiv> length (canonical_basis :: 'a list)"
   assumes "set ws \<subseteq> carrier_vec n"
   shows "ccspan (set (map basis_enum (gram_schmidt0 n ws))) = ccspan (set (map basis_enum ws))"
 proof (transfer fixing: n ws basis_enum)
   interpret complex_vec_space.
   define gs where "gs = gram_schmidt0 n ws"
   have "closure (cspan (set (map basis_enum gs)))
      = cspan (set (map basis_enum gs))"
     apply (rule closure_finite_cspan)
     by simp
   also have "\<dots> = cspan (basis_enum ` set gs)"
     by simp
   also have "\<dots> = basis_enum ` span (set gs)"
     unfolding basis_enum_def
     apply (rule basis_enum_of_vec_span[symmetric])
     using n_def apply simp
     by (simp add: assms(3) cof_vec_space.gram_schmidt0_result(1) gs_def)
   also have "\<dots> = basis_enum ` span (set ws)"
     unfolding gs_def
     apply (subst gram_schmidt0_result(4)[where ws=ws, symmetric])
     using assms by auto
   also have "\<dots> = cspan (basis_enum ` set ws)"
     unfolding basis_enum_def
     apply (rule basis_enum_of_vec_span)
     using n_def apply simp
     by (simp add: assms(3))
   also have "\<dots> = cspan (set (map basis_enum ws))"
     by simp
   also have "\<dots> = closure (cspan (set (map basis_enum ws)))"
     apply (rule closure_finite_cspan[symmetric])
     by simp
   finally show "closure (cspan (set (map basis_enum gs)))
               = closure (cspan (set (map basis_enum ws)))".
 qed
 
 definition "is_subspace_of_vec_list n vs ws =
   (let ws' = gram_schmidt0 n ws in
      \<forall>v\<in>set vs. adjuster n v ws' = - v)"
 
 lemma ccspan_leq_using_vec:
   fixes A B :: "'a::{basis_enum,complex_normed_vector} list"
   shows "(ccspan (set A) \<le> ccspan (set B)) \<longleftrightarrow>
     is_subspace_of_vec_list (length (canonical_basis :: 'a list))
       (map vec_of_basis_enum A) (map vec_of_basis_enum B)"
 proof -
   define d Av Bv Bo
     where "d = length (canonical_basis :: 'a list)"
       and "Av = map vec_of_basis_enum A"
       and "Bv = map vec_of_basis_enum B"
       and "Bo = gram_schmidt0 d Bv"
   interpret complex_vec_space d.
 
   have Av_carrier: "set Av \<subseteq> carrier_vec d"
     unfolding Av_def apply auto
     by (simp add: carrier_vecI d_def dim_vec_of_basis_enum')
   have Bv_carrier: "set Bv \<subseteq> carrier_vec d"
     unfolding Bv_def apply auto
     by (simp add: carrier_vecI d_def dim_vec_of_basis_enum')
   have Bo_carrier: "set Bo \<subseteq> carrier_vec d"
     apply (simp add: Bo_def)
     using Bv_carrier by (rule gram_schmidt0_result(1))
   have orth_Bo: "corthogonal Bo"
     apply (simp add: Bo_def)
     using Bv_carrier by (rule gram_schmidt0_result(3))
   have distinct_Bo: "distinct Bo"
     apply (simp add: Bo_def)
     using Bv_carrier by (rule gram_schmidt0_result(2))
 
   have "ccspan (set A) \<le> ccspan (set B) \<longleftrightarrow> cspan (set A) \<subseteq> cspan (set B)"
     apply (transfer fixing: A B)
     apply (subst closure_finite_cspan, simp)
     by (subst closure_finite_cspan, simp_all)
   also have "\<dots> \<longleftrightarrow> span (set Av) \<subseteq> span (set Bv)"
     apply (simp add: Av_def Bv_def)
     apply (subst vec_of_basis_enum_cspan[symmetric], simp add: d_def)
     apply (subst vec_of_basis_enum_cspan[symmetric], simp add: d_def)
     by (metis inj_image_subset_iff inj_on_def vec_of_basis_enum_inverse)
   also have "\<dots> \<longleftrightarrow> span (set Av) \<subseteq> span (set Bo)"
     unfolding Bo_def Av_def Bv_def
     apply (subst gram_schmidt0_result(4)[symmetric])
     by (simp_all add: carrier_dim_vec d_def dim_vec_of_basis_enum' image_subset_iff)
   also have "\<dots> \<longleftrightarrow> (\<forall>v\<in>set Av. adjuster d v Bo = - v)"
   proof (intro iffI ballI)
     fix v assume "v \<in> set Av" and "span (set Av) \<subseteq> span (set Bo)"
     then have "v \<in> span (set Bo)"
       using Av_carrier span_mem by auto
     have "adjuster d v Bo + v = 0\<^sub>v d"
       apply (rule adjuster_already_in_span)
       using Av_carrier \<open>v \<in> set Av\<close> Bo_carrier orth_Bo
         \<open>v \<in> span (set Bo)\<close> by simp_all
     then show "adjuster d v Bo = - v"
       using Av_carrier Bo_carrier \<open>v \<in> set Av\<close>
       by (metis (no_types, lifting) add.inv_equality adjuster_carrier' local.vec_neg subsetD)
   next
     fix v
     assume adj_minusv: "\<forall>v\<in>set Av. adjuster d v Bo = - v"
     have *: "adjuster d v Bo \<in> span (set Bo)" if "v \<in> set Av" for v
       apply (rule adjuster_in_span)
       using Bo_carrier that Av_carrier distinct_Bo by auto
     have "v \<in> span (set Bo)" if "v \<in> set Av" for v
       using *[OF that] adj_minusv[rule_format, OF that]
       apply auto
       by (metis (no_types, lifting) Av_carrier Bo_carrier adjust_nonzero distinct_Bo subsetD that uminus_l_inv_vec)
     then show "span (set Av) \<subseteq> span (set Bo)"
       apply auto
       by (meson Bo_carrier in_mono span_subsetI subsetI)
   qed
   also have "\<dots> = is_subspace_of_vec_list d Av Bv"
     by (simp add: is_subspace_of_vec_list_def d_def Bo_def)
   finally show "ccspan (set A) \<le> ccspan (set B) \<longleftrightarrow> is_subspace_of_vec_list d Av Bv"
     by simp
 qed
 
 lemma cblinfun_apply_ccspan_using_vec:
   "A *\<^sub>S ccspan (set S) = ccspan (basis_enum_of_vec ` set (map ((*\<^sub>v) (mat_of_cblinfun A)) (map vec_of_basis_enum S)))"
   apply (auto simp: cblinfun_image_ccspan image_image)
   by (metis mat_of_cblinfun_cblinfun_apply vec_of_basis_enum_inverse)
 
 text \<open>\<^term>\<open>mk_projector_orthog d L\<close> takes a list L of d-dimensional vectors
 and returns the projector onto the span of L. (Assuming that all vectors in L are
 orthogonal and nonzero.)\<close>
 fun mk_projector_orthog :: "nat \<Rightarrow> complex vec list \<Rightarrow> complex mat" where
   "mk_projector_orthog d [] = zero_mat d d"
 | "mk_projector_orthog d [v] = (let norm2 = cscalar_prod v v in
                                 smult_mat (1/norm2) (mat_of_cols d [v] * mat_of_rows d [conjugate v]))"
 | "mk_projector_orthog d (v#vs) = (let norm2 = cscalar_prod v v in
                                    smult_mat (1/norm2) (mat_of_cols d [v] * mat_of_rows d [conjugate v])
                                         + mk_projector_orthog d vs)"
 
 lemma mk_projector_orthog_correct:
   fixes S :: "'a::onb_enum list"
   defines "d \<equiv> length (canonical_basis :: 'a list)"
   assumes ortho: "is_ortho_set (set S)" and distinct: "distinct S"
   shows "mk_projector_orthog d (map vec_of_basis_enum S)
        = mat_of_cblinfun (Proj (ccspan (set S)))"
 proof -
   define Snorm where "Snorm = map (\<lambda>s. s /\<^sub>R norm s) S"
 
   have "distinct Snorm"
   proof (insert ortho distinct, unfold Snorm_def, induction S)
     case Nil
     show ?case by simp
   next
     case (Cons s S)
     then have "is_ortho_set (set S)" and "distinct S"
       unfolding is_ortho_set_def by auto
     note IH = Cons.IH[OF this]
     have "s /\<^sub>R norm s \<notin> (\<lambda>s. s /\<^sub>R norm s) ` set S"
     proof auto
       fix s' assume "s' \<in> set S" and same: "s /\<^sub>R norm s = s' /\<^sub>R norm s'"
       with Cons.prems have "s \<noteq> s'" by auto
       have "s \<noteq> 0"
         by (metis Cons.prems(1) is_ortho_set_def list.set_intros(1))
-      then have "0 \<noteq> \<langle>s /\<^sub>R norm s, s /\<^sub>R norm s\<rangle>"
+      then have "0 \<noteq> (s /\<^sub>R norm s) \<bullet>\<^sub>C (s /\<^sub>R norm s)"
         by simp
-      also have \<open>\<langle>s /\<^sub>R norm s, s /\<^sub>R norm s\<rangle> = \<langle>s /\<^sub>R norm s, s' /\<^sub>R norm s'\<rangle>\<close>
+      also have \<open>(s /\<^sub>R norm s) \<bullet>\<^sub>C (s /\<^sub>R norm s) = (s /\<^sub>R norm s) \<bullet>\<^sub>C (s' /\<^sub>R norm s')\<close>
         by (simp add: same)
-      also have \<open>\<langle>s /\<^sub>R norm s, s' /\<^sub>R norm s'\<rangle> = \<langle>s, s'\<rangle> / (norm s * norm s')\<close>
+      also have \<open>(s /\<^sub>R norm s) \<bullet>\<^sub>C (s' /\<^sub>R norm s') = (s \<bullet>\<^sub>C s') / (norm s * norm s')\<close>
         by (simp add: scaleR_scaleC divide_inverse_commute)
       also from \<open>s' \<in> set S\<close> \<open>s \<noteq> s'\<close> have "\<dots> = 0"
         using Cons.prems unfolding is_ortho_set_def by simp
       finally show False
         by simp
     qed
     then show ?case
       using IH by simp
   qed
 
   have norm_Snorm: "norm s = 1" if "s \<in> set Snorm" for s
     using that ortho unfolding Snorm_def is_ortho_set_def apply auto
     by (metis left_inverse norm_eq_zero)
 
   have ortho_Snorm: "is_ortho_set (set Snorm)"
     unfolding is_ortho_set_def
   proof (intro conjI ballI impI)
     fix x y
     show "0 \<notin> set Snorm"
       using norm_Snorm[of 0] by auto
     assume "x \<in> set Snorm" and "y \<in> set Snorm" and "x \<noteq> y"
     from \<open>x \<in> set Snorm\<close>
     obtain x' where x: "x = x' /\<^sub>R norm x'" and x': "x' \<in> set S"
       unfolding Snorm_def by auto
     from \<open>y \<in> set Snorm\<close>
     obtain y' where y: "y = y' /\<^sub>R norm y'" and y': "y' \<in> set S"
       unfolding Snorm_def by auto
     from \<open>x \<noteq> y\<close> x y have \<open>x' \<noteq> y'\<close> by auto
     with x' y' ortho have "cinner x' y' = 0"
       unfolding is_ortho_set_def by auto
     then show "cinner x y = 0"
       unfolding x y scaleR_scaleC by auto
   qed
 
   have inj_butter: "inj_on selfbutter (set Snorm)"
   proof (rule inj_onI)
     fix x y
     assume "x \<in> set Snorm" and "y \<in> set Snorm"
     assume "selfbutter x = selfbutter y"
     then obtain c where xcy: "x = c *\<^sub>C y" and "cmod c = 1"
       using inj_selfbutter_upto_phase by auto
     have "0 \<noteq> cmod (cinner x x)"
       using \<open>x \<in> set Snorm\<close> norm_Snorm
       by force
-    also have "cmod (cinner x x) = cmod (c * \<langle>x, y\<rangle>)"
+    also have "cmod (cinner x x) = cmod (c * (x \<bullet>\<^sub>C y))"
       apply (subst (2) xcy) by simp
-    also have "\<dots> = cmod \<langle>x, y\<rangle>"
+    also have "\<dots> = cmod (x \<bullet>\<^sub>C y)"
       using \<open>cmod c = 1\<close> by (simp add: norm_mult)
-    finally have "\<langle>x, y\<rangle> \<noteq> 0"
+    finally have "(x \<bullet>\<^sub>C y) \<noteq> 0"
       by simp
     then show "x = y"
       using ortho_Snorm \<open>x \<in> set Snorm\<close> \<open>y \<in> set Snorm\<close>
       unfolding is_ortho_set_def by auto
   qed
 
   from \<open>distinct Snorm\<close> inj_butter
   have distinct': "distinct (map selfbutter Snorm)"
     unfolding distinct_map by simp
 
   have Span_Snorm: "ccspan (set Snorm) = ccspan (set S)"
     apply (transfer fixing: Snorm S)
     apply (simp add: scaleR_scaleC Snorm_def)
     apply (subst complex_vector.span_image_scale)
     using is_ortho_set_def ortho by fastforce+
 
   have "mk_projector_orthog d (map vec_of_basis_enum S)
       = mat_of_cblinfun (sum_list (map selfbutter Snorm))"
     unfolding Snorm_def
   proof (induction S)
     case Nil
     show ?case
       by (simp add: d_def mat_of_cblinfun_zero)
   next
     case (Cons a S)
     define sumS where "sumS = sum_list (map selfbutter (map (\<lambda>s. s /\<^sub>R norm s) S))"
     with Cons have IH: "mk_projector_orthog d (map vec_of_basis_enum S)
                   = mat_of_cblinfun sumS"
       by simp
 
     define factor where "factor = inverse ((complex_of_real (norm a))\<^sup>2)"
     have factor': "factor = 1 / (vec_of_basis_enum a \<bullet>c vec_of_basis_enum a)"
       unfolding factor_def cscalar_prod_vec_of_basis_enum
       by (simp add: inverse_eq_divide power2_norm_eq_cinner)
 
     have "mk_projector_orthog d (map vec_of_basis_enum (a # S))
           = factor \<cdot>\<^sub>m (mat_of_cols d [vec_of_basis_enum a]
                     * mat_of_rows d [conjugate (vec_of_basis_enum a)])
             + mat_of_cblinfun sumS"
       apply (cases S)
        apply (auto simp add: factor' sumS_def d_def mat_of_cblinfun_zero)[1]
       by (auto simp add: IH[symmetric] factor' d_def)
 
     also have "\<dots> = factor \<cdot>\<^sub>m (mat_of_cols d [vec_of_basis_enum a] *
          mat_adjoint (mat_of_cols d [vec_of_basis_enum a])) + mat_of_cblinfun sumS"
       apply (rule arg_cong[where f="\<lambda>x. _ \<cdot>\<^sub>m (_ * x) + _"])
       apply (rule mat_eq_iff[THEN iffD2])
       apply (auto simp add: mat_adjoint_def)
       apply (subst mat_of_rows_index) apply auto
       apply (subst mat_of_rows_index) apply auto
       apply (subst mat_of_cols_index) apply auto
       by (simp add: assms(1) dim_vec_of_basis_enum')
     also have "\<dots> = mat_of_cblinfun (selfbutter (a /\<^sub>R norm a)) + mat_of_cblinfun sumS"
       apply (simp add: butterfly_scaleR_left butterfly_scaleR_right power_inverse mat_of_cblinfun_scaleR factor_def)
       apply (simp add: butterfly_def mat_of_cblinfun_compose
           mat_of_cblinfun_adj mat_of_cblinfun_vector_to_cblinfun d_def)
       by (simp add: mat_of_cblinfun_compose mat_of_cblinfun_adj mat_of_cblinfun_vector_to_cblinfun mat_of_cblinfun_scaleC power2_eq_square)
     finally show ?case
       by (simp add: mat_of_cblinfun_plus sumS_def)
   qed
   also have "\<dots> = mat_of_cblinfun (\<Sum>s\<in>set Snorm. selfbutter s)"
     by (metis distinct' distinct_map sum.distinct_set_conv_list)
   also have "\<dots> = mat_of_cblinfun (\<Sum>s\<in>set Snorm. proj s)"
     apply (rule arg_cong[where f="mat_of_cblinfun"])
     apply (rule sum.cong[OF refl])
     apply (rule butterfly_eq_proj)
     using norm_Snorm by simp
   also have "\<dots> = mat_of_cblinfun (Proj (ccspan (set Snorm)))"
     apply (rule arg_cong[of _ _ mat_of_cblinfun])
   proof (insert ortho_Snorm, insert \<open>distinct Snorm\<close>, induction Snorm)
     case Nil
     show ?case
       by simp
   next
     case (Cons a Snorm)
     from Cons.prems have [simp]: "a \<notin> set Snorm"
       by simp
 
     have "sum proj (set (a # Snorm))
         = proj a + sum proj (set Snorm)"
       by auto
     also have "\<dots> = proj a + Proj (ccspan (set Snorm))"
       apply (subst Cons.IH)
       using Cons.prems apply auto
       by (meson Cons.prems(1) is_ortho_set_antimono set_subset_Cons)
     also have "\<dots> = Proj (ccspan ({a} \<union> set Snorm))"
       apply (rule Proj_orthog_ccspan_union[symmetric])
       by (metis Cons.prems(1) \<open>a \<notin> set Snorm\<close> is_ortho_set_def list.set_intros(1) list.set_intros(2) singleton_iff)
     finally show ?case
       by simp
   qed
   also have "\<dots> = mat_of_cblinfun (Proj (ccspan (set S)))"
     unfolding Span_Snorm by simp
   finally show ?thesis
     by -
 qed
 
 lemma mat_of_cblinfun_Proj_ccspan:
   fixes S :: "'a::onb_enum list"
   shows "mat_of_cblinfun (Proj (ccspan (set S))) =
     (let d = length (canonical_basis :: 'a list) in
       mk_projector_orthog d (gram_schmidt0 d (map vec_of_basis_enum S)))"
 proof-
   define d gs
     where "d = length (canonical_basis :: 'a list)"
       and "gs = gram_schmidt0 d (map vec_of_basis_enum S)"
   interpret complex_vec_space d.
   have gs_dim: "x \<in> set gs \<Longrightarrow> dim_vec x = d" for x
     by (smt carrier_vecD carrier_vec_dim_vec d_def dim_vec_of_basis_enum' ex_map_conv gram_schmidt0_result(1) gs_def subset_code(1))
   have ortho_gs: "is_ortho_set (set (map basis_enum_of_vec gs :: 'a list))"
     apply (subst corthogonal_vec_of_basis_enum[THEN iffD1], auto)
     by (smt carrier_dim_vec cof_vec_space.gram_schmidt0_result(1) d_def dim_vec_of_basis_enum' gram_schmidt0_result(3) gs_def imageE map_idI map_map o_apply set_map subset_code(1) basis_enum_of_vec_inverse)
   have distinct_gs: "distinct (map basis_enum_of_vec gs :: 'a list)"
     by (metis (mono_tags, opaque_lifting) carrier_vec_dim_vec cof_vec_space.gram_schmidt0_result(2) d_def dim_vec_of_basis_enum' distinct_map gs_def gs_dim image_iff inj_on_inverseI set_map subsetI basis_enum_of_vec_inverse)
 
   have "mk_projector_orthog d gs
       = mk_projector_orthog d (map vec_of_basis_enum (map basis_enum_of_vec gs :: 'a list))"
     apply simp
     apply (subst map_cong[where ys=gs and g=id], simp)
     using gs_dim by (auto intro!: vec_of_basis_enum_inverse simp: d_def)
   also have "\<dots> = mat_of_cblinfun (Proj (ccspan (set (map basis_enum_of_vec gs :: 'a list))))"
     unfolding d_def
     apply (subst mk_projector_orthog_correct)
     using ortho_gs distinct_gs by auto
   also have "\<dots> = mat_of_cblinfun (Proj (ccspan (set S)))"
     apply (rule arg_cong[where f="\<lambda>x. mat_of_cblinfun (Proj x)"])
     unfolding gs_def d_def
     apply (subst ccspan_gram_schmidt0_invariant)
     by (auto simp add: carrier_vecI dim_vec_of_basis_enum')
   finally show ?thesis
     unfolding d_def gs_def by auto
 qed
 
 unbundle no_jnf_notation
 unbundle no_cblinfun_notation
 
 end
diff --git a/thys/Complex_Bounded_Operators/Complex_Bounded_Linear_Function.thy b/thys/Complex_Bounded_Operators/Complex_Bounded_Linear_Function.thy
--- a/thys/Complex_Bounded_Operators/Complex_Bounded_Linear_Function.thy
+++ b/thys/Complex_Bounded_Operators/Complex_Bounded_Linear_Function.thy
@@ -1,4530 +1,4576 @@
 section \<open>\<open>Complex_Bounded_Linear_Function\<close> -- Complex bounded linear functions (bounded operators)\<close>
 
 (*
 Authors:
 
   Dominique Unruh, University of Tartu, unruh@ut.ee
   Jose Manuel Rodriguez Caballero, University of Tartu, jose.manuel.rodriguez.caballero@ut.ee
 
 *)
 
 theory Complex_Bounded_Linear_Function
   imports
     "HOL-Types_To_Sets.Types_To_Sets"
     Banach_Steinhaus.Banach_Steinhaus
     Complex_Inner_Product
     One_Dimensional_Spaces
     Complex_Bounded_Linear_Function0
     "HOL-Library.Function_Algebras"
 begin
 
 unbundle lattice_syntax
 
 subsection \<open>Misc basic facts and declarations\<close>
 
 notation cblinfun_apply (infixr "*\<^sub>V" 70)
 
 lemma id_cblinfun_apply[simp]: "id_cblinfun *\<^sub>V \<psi> = \<psi>"
   apply transfer by simp
 
 lemma apply_id_cblinfun[simp]: \<open>(*\<^sub>V) id_cblinfun = id\<close>
   by auto
 
 lemma isCont_cblinfun_apply[simp]: "isCont ((*\<^sub>V) A) \<psi>"
   apply transfer
   by (simp add: clinear_continuous_at)
 
 declare cblinfun.scaleC_left[simp]
 
 lemma cblinfun_apply_clinear[simp]: \<open>clinear (cblinfun_apply A)\<close>
   using bounded_clinear.axioms(1) cblinfun_apply by blast
 
 lemma cblinfun_cinner_eqI:
   fixes A B :: \<open>'a::chilbert_space \<Rightarrow>\<^sub>C\<^sub>L 'a\<close>
   assumes \<open>\<And>\<psi>. cinner \<psi> (A *\<^sub>V \<psi>) = cinner \<psi> (B *\<^sub>V \<psi>)\<close>
   shows \<open>A = B\<close>
 proof -
   define C where \<open>C = A - B\<close>
   have C0[simp]: \<open>cinner \<psi> (C \<psi>) = 0\<close> for \<psi>
     by (simp add: C_def assms cblinfun.diff_left cinner_diff_right)
   { fix f g \<alpha>
     have \<open>0 = cinner (f + \<alpha> *\<^sub>C g) (C *\<^sub>V (f + \<alpha> *\<^sub>C g))\<close>
       by (simp add: cinner_diff_right minus_cblinfun.rep_eq)
     also have \<open>\<dots> = \<alpha> *\<^sub>C cinner f (C g) + cnj \<alpha> *\<^sub>C cinner g (C f)\<close>
       by (smt (z3) C0 add.commute add.right_neutral cblinfun.add_right cblinfun.scaleC_right cblinfun_cinner_right.rep_eq cinner_add_left cinner_scaleC_left complex_scaleC_def)
     finally have \<open>\<alpha> *\<^sub>C cinner f (C g) = - cnj \<alpha> *\<^sub>C cinner g (C f)\<close>
       by (simp add: eq_neg_iff_add_eq_0)
   }
   then have \<open>cinner f (C g) = 0\<close> for f g
     by (metis complex_cnj_i complex_cnj_one complex_vector.scale_cancel_right complex_vector.scale_left_imp_eq equation_minus_iff i_squared mult_eq_0_iff one_neq_neg_one)
   then have \<open>C g = 0\<close> for g
     using cinner_eq_zero_iff by blast
   then have \<open>C = 0\<close>
     by (simp add: cblinfun_eqI)
   then show \<open>A = B\<close>
     using C_def by auto
 qed
 
 lemma id_cblinfun_not_0[simp]: \<open>(id_cblinfun :: 'a::{complex_normed_vector, not_singleton} \<Rightarrow>\<^sub>C\<^sub>L _) \<noteq> 0\<close>
   by (metis (full_types) Extra_General.UNIV_not_singleton cblinfun.zero_left cblinfun_id_cblinfun_apply ex_norm1 norm_zero one_neq_zero)
 
 lemma cblinfun_norm_geqI:
   assumes \<open>norm (f *\<^sub>V x) / norm x \<ge> K\<close>
   shows \<open>norm f \<ge> K\<close>
   using assms apply transfer
   by (smt (z3) bounded_clinear.bounded_linear le_onorm)
 
 (* This lemma is proven in Complex_Bounded_Linear_Function0 but we add the [simp]
    only here because we try to keep Complex_Bounded_Linear_Function0 as close to
    Bounded_Linear_Function as possible. *)
 declare scaleC_conv_of_complex[simp]
 
 lemma cblinfun_eq_0_on_span:
   fixes S::\<open>'a::complex_normed_vector set\<close>
   assumes "x \<in> cspan S"
     and "\<And>s. s\<in>S \<Longrightarrow> F *\<^sub>V s = 0"
   shows \<open>F *\<^sub>V x = 0\<close>
   apply (rule complex_vector.linear_eq_0_on_span[where f=F])
   using bounded_clinear.axioms(1) cblinfun_apply assms by auto
 
 lemma cblinfun_eq_on_span:
   fixes S::\<open>'a::complex_normed_vector set\<close>
   assumes "x \<in> cspan S"
     and "\<And>s. s\<in>S \<Longrightarrow> F *\<^sub>V s = G *\<^sub>V s"
   shows \<open>F *\<^sub>V x = G *\<^sub>V x\<close>
   apply (rule complex_vector.linear_eq_on_span[where f=F])
   using bounded_clinear.axioms(1) cblinfun_apply assms by auto
 
 lemma cblinfun_eq_0_on_UNIV_span:
   fixes basis::\<open>'a::complex_normed_vector set\<close>
   assumes "cspan basis = UNIV"
     and "\<And>s. s\<in>basis \<Longrightarrow> F *\<^sub>V s = 0"
   shows \<open>F = 0\<close>
   by (metis cblinfun_eq_0_on_span UNIV_I assms cblinfun.zero_left cblinfun_eqI)
 
 lemma cblinfun_eq_on_UNIV_span:
   fixes basis::"'a::complex_normed_vector set" and \<phi>::"'a \<Rightarrow> 'b::complex_normed_vector"
   assumes "cspan basis = UNIV"
     and "\<And>s. s\<in>basis \<Longrightarrow> F *\<^sub>V s = G *\<^sub>V s"
   shows \<open>F = G\<close>
 proof-
   have "F - G = 0"
     apply (rule cblinfun_eq_0_on_UNIV_span[where basis=basis])
     using assms by (auto simp add: cblinfun.diff_left)
   thus ?thesis by simp
 qed
 
 lemma cblinfun_eq_on_canonical_basis:
   fixes f g::"'a::{basis_enum,complex_normed_vector} \<Rightarrow>\<^sub>C\<^sub>L 'b::complex_normed_vector"
   defines "basis == set (canonical_basis::'a list)"
   assumes "\<And>u. u \<in> basis \<Longrightarrow> f *\<^sub>V u = g *\<^sub>V u"
   shows  "f = g"
   apply (rule cblinfun_eq_on_UNIV_span[where basis=basis])
   using assms is_generator_set is_cindependent_set by auto
 
 lemma cblinfun_eq_0_on_canonical_basis:
   fixes f ::"'a::{basis_enum,complex_normed_vector} \<Rightarrow>\<^sub>C\<^sub>L 'b::complex_normed_vector"
   defines "basis == set (canonical_basis::'a list)"
   assumes "\<And>u. u \<in> basis \<Longrightarrow> f *\<^sub>V u = 0"
   shows  "f = 0"
   by (simp add: assms cblinfun_eq_on_canonical_basis)
 
 lemma cinner_canonical_basis_eq_0:
   defines "basisA == set (canonical_basis::'a::onb_enum list)"
     and   "basisB == set (canonical_basis::'b::onb_enum list)"
-  assumes "\<And>u v. u\<in>basisA \<Longrightarrow> v\<in>basisB \<Longrightarrow> \<langle>v, F *\<^sub>V u\<rangle> = 0"
+  assumes "\<And>u v. u\<in>basisA \<Longrightarrow> v\<in>basisB \<Longrightarrow> is_orthogonal v (F *\<^sub>V u)"
   shows "F = 0"
 proof-
   have "F *\<^sub>V u = 0"
     if "u\<in>basisA" for u
   proof-
-    have "\<And>v. v\<in>basisB \<Longrightarrow> \<langle>v, F *\<^sub>V u\<rangle> = 0"
+    have "\<And>v. v\<in>basisB \<Longrightarrow> is_orthogonal v (F *\<^sub>V u)"
       by (simp add: assms(3) that)
-    moreover have "(\<And>v. v\<in>basisB \<Longrightarrow> \<langle>v, x\<rangle> = 0) \<Longrightarrow> x = 0"
+    moreover have "(\<And>v. v\<in>basisB \<Longrightarrow> is_orthogonal v x) \<Longrightarrow> x = 0"
       for x
     proof-
-      assume r1: "\<And>v. v\<in>basisB \<Longrightarrow> \<langle>v, x\<rangle> = 0"
-      have "\<langle>v, x\<rangle> = 0" for v
+      assume r1: "\<And>v. v\<in>basisB \<Longrightarrow> is_orthogonal v x"
+      have "is_orthogonal v x" for v
       proof-
         have "cspan basisB = UNIV"
           using basisB_def is_generator_set  by auto
         hence "v \<in> cspan basisB"
           by (smt iso_tuple_UNIV_I)
         hence "\<exists>t s. v = (\<Sum>a\<in>t. s a *\<^sub>C a) \<and> finite t \<and> t \<subseteq> basisB"
           using complex_vector.span_explicit
           by (smt mem_Collect_eq)
         then obtain t s where b1: "v = (\<Sum>a\<in>t. s a *\<^sub>C a)" and b2: "finite t" and b3: "t \<subseteq> basisB"
           by blast
-        have "\<langle>v, x\<rangle> = \<langle>(\<Sum>a\<in>t. s a *\<^sub>C a), x\<rangle>"
+        have "v \<bullet>\<^sub>C x = (\<Sum>a\<in>t. s a *\<^sub>C a) \<bullet>\<^sub>C x"
           by (simp add: b1)
-        also have "\<dots> = (\<Sum>a\<in>t. \<langle>s a *\<^sub>C a, x\<rangle>)"
+        also have "\<dots> = (\<Sum>a\<in>t. (s a *\<^sub>C a) \<bullet>\<^sub>C x)"
           using cinner_sum_left by blast
-        also have "\<dots> = (\<Sum>a\<in>t. cnj (s a) * \<langle>a, x\<rangle>)"
+        also have "\<dots> = (\<Sum>a\<in>t. cnj (s a) * (a \<bullet>\<^sub>C x))"
           by auto
         also have "\<dots> = 0"
           using b3 r1 subsetD by force
         finally show ?thesis by simp
       qed
       thus ?thesis
-        by (simp add: \<open>\<And>v. \<langle>v, x\<rangle> = 0\<close> cinner_extensionality)
+        by (simp add: \<open>\<And>v. (v \<bullet>\<^sub>C x) = 0\<close> cinner_extensionality)
     qed
     ultimately show ?thesis by simp
   qed
   thus ?thesis
     using basisA_def cblinfun_eq_0_on_canonical_basis by auto
 qed
 
 lemma cinner_canonical_basis_eq:
   defines "basisA == set (canonical_basis::'a::onb_enum list)"
     and   "basisB == set (canonical_basis::'b::onb_enum list)"
-  assumes "\<And>u v. u\<in>basisA \<Longrightarrow> v\<in>basisB \<Longrightarrow> \<langle>v, F *\<^sub>V u\<rangle> = \<langle>v, G *\<^sub>V u\<rangle>"
+  assumes "\<And>u v. u\<in>basisA \<Longrightarrow> v\<in>basisB \<Longrightarrow> v \<bullet>\<^sub>C (F *\<^sub>V u) = v \<bullet>\<^sub>C (G *\<^sub>V u)"
   shows "F = G"
 proof-
   define H where "H = F - G"
-  have "\<And>u v. u\<in>basisA \<Longrightarrow> v\<in>basisB \<Longrightarrow> \<langle>v, H *\<^sub>V u\<rangle> = 0"
+  have "\<And>u v. u\<in>basisA \<Longrightarrow> v\<in>basisB \<Longrightarrow> is_orthogonal v (H *\<^sub>V u)"
     unfolding H_def
     by (simp add: assms(3) cinner_diff_right minus_cblinfun.rep_eq)
   hence "H = 0"
     by (simp add: basisA_def basisB_def cinner_canonical_basis_eq_0)
   thus ?thesis unfolding H_def by simp
 qed
 
 lemma cinner_canonical_basis_eq':
   defines "basisA == set (canonical_basis::'a::onb_enum list)"
     and   "basisB == set (canonical_basis::'b::onb_enum list)"
-  assumes "\<And>u v. u\<in>basisA \<Longrightarrow> v\<in>basisB \<Longrightarrow> \<langle>F *\<^sub>V u, v\<rangle> = \<langle>G *\<^sub>V u, v\<rangle>"
+  assumes "\<And>u v. u\<in>basisA \<Longrightarrow> v\<in>basisB \<Longrightarrow> (F *\<^sub>V u) \<bullet>\<^sub>C v = (G *\<^sub>V u) \<bullet>\<^sub>C v"
   shows "F = G"
   using cinner_canonical_basis_eq assms
   by (metis cinner_commute')
 
 lemma cblinfun_norm_approx_witness:
   fixes A :: \<open>'a::{not_singleton,complex_normed_vector} \<Rightarrow>\<^sub>C\<^sub>L 'b::complex_normed_vector\<close>
   assumes \<open>\<epsilon> > 0\<close>
   shows \<open>\<exists>\<psi>. norm (A *\<^sub>V \<psi>) \<ge> norm A - \<epsilon> \<and> norm \<psi> = 1\<close>
 proof (transfer fixing: \<epsilon>)
   fix A :: \<open>'a \<Rightarrow> 'b\<close> assume [simp]: \<open>bounded_clinear A\<close>
   have \<open>\<exists>y\<in>{norm (A x) |x. norm x = 1}. y > \<Squnion> {norm (A x) |x. norm x = 1} - \<epsilon>\<close>
     apply (rule Sup_real_close)
     using assms by (auto simp: ex_norm1 bounded_clinear.bounded_linear bdd_above_norm_f)
   also have \<open>\<Squnion> {norm (A x) |x. norm x = 1} = onorm A\<close>
     by (simp add: bounded_clinear.bounded_linear onorm_sphere)
   finally
   show \<open>\<exists>\<psi>. onorm A - \<epsilon> \<le> norm (A \<psi>) \<and> norm \<psi> = 1\<close>
     by force
 qed
 
 lemma cblinfun_norm_approx_witness_mult:
   fixes A :: \<open>'a::{not_singleton,complex_normed_vector} \<Rightarrow>\<^sub>C\<^sub>L 'b::complex_normed_vector\<close>
   assumes \<open>\<epsilon> < 1\<close>
   shows \<open>\<exists>\<psi>. norm (A *\<^sub>V \<psi>) \<ge> norm A * \<epsilon> \<and> norm \<psi> = 1\<close>
 proof (cases \<open>norm A = 0\<close>)
   case True
   then show ?thesis
     apply auto
     by (simp add: ex_norm1)
 next
   case False
   then have \<open>(1 - \<epsilon>) * norm A > 0\<close>
     using assms by fastforce
   then obtain \<psi> where geq: \<open>norm (A *\<^sub>V \<psi>) \<ge> norm A - ((1 - \<epsilon>) * norm A)\<close> and \<open>norm \<psi> = 1\<close>
     using cblinfun_norm_approx_witness by blast
   have \<open>norm A * \<epsilon> = norm A - (1 - \<epsilon>) * norm A\<close>
     by (simp add: mult.commute right_diff_distrib')
   also have \<open>\<dots> \<le> norm (A *\<^sub>V \<psi>)\<close>
     by (rule geq)
   finally show ?thesis
     using \<open>norm \<psi> = 1\<close> by auto
 qed
 
 
 lemma cblinfun_to_CARD_1_0[simp]: \<open>(A :: _ \<Rightarrow>\<^sub>C\<^sub>L _::CARD_1) = 0\<close>
   apply (rule cblinfun_eqI)
   by auto
 
 lemma cblinfun_from_CARD_1_0[simp]: \<open>(A :: _::CARD_1 \<Rightarrow>\<^sub>C\<^sub>L _) = 0\<close>
   apply (rule cblinfun_eqI)
   apply (subst CARD_1_vec_0)
   by auto
 
 
 lemma cblinfun_cspan_UNIV:
   fixes basis :: \<open>('a::{complex_normed_vector,cfinite_dim} \<Rightarrow>\<^sub>C\<^sub>L 'b::complex_normed_vector) set\<close>
     and basisA :: \<open>'a set\<close> and basisB :: \<open>'b set\<close>
   assumes \<open>cspan basisA = UNIV\<close> and \<open>cspan basisB = UNIV\<close>
   assumes basis: \<open>\<And>a b. a\<in>basisA \<Longrightarrow> b\<in>basisB \<Longrightarrow> \<exists>F\<in>basis. \<forall>a'\<in>basisA. F *\<^sub>V a' = (if a'=a then b else 0)\<close>
   shows \<open>cspan basis = UNIV\<close>
 proof -
   obtain basisA' where \<open>basisA' \<subseteq> basisA\<close> and \<open>cindependent basisA'\<close> and \<open>cspan basisA' = UNIV\<close>
     by (metis assms(1) complex_vector.maximal_independent_subset complex_vector.span_eq top_greatest)
   then have [simp]: \<open>finite basisA'\<close>
     by (simp add: cindependent_cfinite_dim_finite)
   have basis': \<open>\<And>a b. a\<in>basisA' \<Longrightarrow> b\<in>basisB \<Longrightarrow> \<exists>F\<in>basis. \<forall>a'\<in>basisA'. F *\<^sub>V a' = (if a'=a then b else 0)\<close>
     using basis \<open>basisA' \<subseteq> basisA\<close> by fastforce
 
   obtain F where F: \<open>F a b \<in> basis \<and> F a b *\<^sub>V a' = (if a'=a then b else 0)\<close>
     if \<open>a\<in>basisA'\<close> \<open>b\<in>basisB\<close> \<open>a'\<in>basisA'\<close> for a b a'
     apply atomize_elim apply (intro choice allI)
     using basis' by metis
   then have F_apply: \<open>F a b *\<^sub>V a' = (if a'=a then b else 0)\<close>
     if \<open>a\<in>basisA'\<close> \<open>b\<in>basisB\<close> \<open>a'\<in>basisA'\<close> for a b a'
     using that by auto
   have F_basis: \<open>F a b \<in> basis\<close>
     if \<open>a\<in>basisA'\<close> \<open>b\<in>basisB\<close> for a b
     using that F by auto
   have b_span: \<open>\<exists>G\<in>cspan {F a b|b. b\<in>basisB}. \<forall>a'\<in>basisA'. G *\<^sub>V a' = (if a'=a then b else 0)\<close> if \<open>a\<in>basisA'\<close> for a b
   proof -
     from \<open>cspan basisB = UNIV\<close>
     obtain r t where \<open>finite t\<close> and \<open>t \<subseteq> basisB\<close> and b_lincom: \<open>b = (\<Sum>a\<in>t. r a *\<^sub>C a)\<close>
       unfolding complex_vector.span_alt apply atomize_elim by blast
     define G where \<open>G = (\<Sum>i\<in>t. r i *\<^sub>C F a i)\<close>
     have \<open>G \<in> cspan {F a b|b. b\<in>basisB}\<close>
       using \<open>finite t\<close> \<open>t \<subseteq> basisB\<close> unfolding G_def
       by (smt (verit, ccfv_threshold) complex_vector.span_base complex_vector.span_scale complex_vector.span_sum mem_Collect_eq subset_eq)
     moreover have \<open>G *\<^sub>V a' = (if a'=a then b else 0)\<close> if \<open>a'\<in>basisA'\<close> for a'
       apply (cases \<open>a'=a\<close>)
       using \<open>t \<subseteq> basisB\<close> \<open>a\<in>basisA'\<close> \<open>a'\<in>basisA'\<close>
       by (auto simp: b_lincom G_def cblinfun.sum_left F_apply intro!: sum.neutral sum.cong)
     ultimately show ?thesis
       by blast
   qed
 
   have a_span: \<open>cspan (\<Union>a\<in>basisA'. cspan {F a b|b. b\<in>basisB}) = UNIV\<close>
   proof (intro equalityI subset_UNIV subsetI, rename_tac H)
     fix H
     obtain G where G: \<open>G a b \<in> cspan {F a b|b. b\<in>basisB} \<and> G a b *\<^sub>V a' = (if a'=a then b else 0)\<close> if \<open>a\<in>basisA'\<close> and \<open>a'\<in>basisA'\<close> for a b a'
       apply atomize_elim apply (intro choice allI)
       using b_span by blast
     then have G_cspan: \<open>G a b \<in> cspan {F a b|b. b\<in>basisB}\<close> if \<open>a\<in>basisA'\<close> for a b
       using that by auto
     from G have G: \<open>G a b *\<^sub>V a' = (if a'=a then b else 0)\<close> if \<open>a\<in>basisA'\<close> and \<open>a'\<in>basisA'\<close> for a b a'
       using that by auto
     define H' where \<open>H' = (\<Sum>a\<in>basisA'. G a (H *\<^sub>V a))\<close>
     have \<open>H' \<in> cspan (\<Union>a\<in>basisA'. cspan {F a b|b. b\<in>basisB})\<close>
       unfolding H'_def using G_cspan
       by (smt (verit, del_insts) UN_iff complex_vector.span_clauses(1) complex_vector.span_sum)
     moreover have \<open>H' = H\<close>
       using \<open>cspan basisA' = UNIV\<close> apply (rule cblinfun_eq_on_UNIV_span)
       apply (auto simp: H'_def cblinfun.sum_left)
       apply (subst sum_single)
       by (auto simp: G)
     ultimately show \<open>H \<in> cspan (\<Union>a\<in>basisA'. cspan {F a b |b. b \<in> basisB})\<close>
       by simp
   qed
 
   moreover have \<open>cspan basis \<supseteq> cspan (\<Union>a\<in>basisA'. cspan {F a b|b. b\<in>basisB})\<close>
     using F_basis
     by (smt (z3) UN_subset_iff complex_vector.span_alt complex_vector.span_minimal complex_vector.subspace_span mem_Collect_eq subset_iff)
 
   ultimately show \<open>cspan basis = UNIV\<close>
     by auto
 qed
 
 
 instance cblinfun :: (\<open>{cfinite_dim,complex_normed_vector}\<close>, \<open>{cfinite_dim,complex_normed_vector}\<close>) cfinite_dim
 proof intro_classes
   obtain basisA :: \<open>'a set\<close> where [simp]: \<open>cspan basisA = UNIV\<close> \<open>cindependent basisA\<close> \<open>finite basisA\<close>
     using finite_basis by blast
   obtain basisB :: \<open>'b set\<close> where [simp]: \<open>cspan basisB = UNIV\<close> \<open>cindependent basisB\<close> \<open>finite basisB\<close>
     using finite_basis by blast
   define f where \<open>f a b = cconstruct basisA (\<lambda>x. if x=a then b else 0)\<close> for a :: 'a and b :: 'b
   have f_a: \<open>f a b a = b\<close> if \<open>a : basisA\<close> for a b
     by (simp add: complex_vector.construct_basis f_def that)
   have f_not_a: \<open>f a b c = 0\<close> if \<open>a : basisA\<close> and \<open>c : basisA\<close> and \<open>a \<noteq> c\<close>for a b c
     using that by (simp add: complex_vector.construct_basis f_def)
   define F where \<open>F a b = CBlinfun (f a b)\<close> for a b
   have \<open>clinear (f a b)\<close> for a b
     by (auto intro: complex_vector.linear_construct simp: f_def)
   then have \<open>bounded_clinear (f a b)\<close> for a b
     by auto
   then have F_apply: \<open>cblinfun_apply (F a b) = f a b\<close> for a b
     by (simp add: F_def bounded_clinear_CBlinfun_apply)
   define basis where \<open>basis = {F a b| a b. a\<in>basisA \<and> b\<in>basisB}\<close>
   have \<open>cspan basis = UNIV\<close>
     apply (rule cblinfun_cspan_UNIV[where basisA=basisA and basisB=basisB])
       apply (auto simp: basis_def)
     by (metis F_apply f_a f_not_a)
 
   moreover have \<open>finite basis\<close>
     unfolding basis_def apply (rule finite_image_set2) by auto
 
   ultimately show \<open>\<exists>S :: ('a \<Rightarrow>\<^sub>C\<^sub>L 'b) set. finite S \<and> cspan S = UNIV\<close>
     by auto
 qed
 
 lemma norm_cblinfun_bound_dense:
   assumes \<open>0 \<le> b\<close>
   assumes S: \<open>closure S = UNIV\<close>
   assumes bound: \<open>\<And>x. x\<in>S \<Longrightarrow> norm (cblinfun_apply f x) \<le> b * norm x\<close>
   shows \<open>norm f \<le> b\<close>
 proof -
   have 1: \<open>continuous_on UNIV (\<lambda>a. norm (f *\<^sub>V a))\<close>
     apply (intro continuous_on_norm linear_continuous_on)
     by (simp add: Complex_Vector_Spaces.bounded_clinear.bounded_linear cblinfun.bounded_clinear_right)
   have 2: \<open>continuous_on UNIV (\<lambda>a. b * norm a)\<close>
     using continuous_on_mult_left continuous_on_norm_id by blast
   have \<open>norm (cblinfun_apply f x) \<le> b * norm x\<close> for x
     apply (rule on_closure_leI[where x=x and S=S])
     using S bound 1 2 by auto
   then show \<open>norm f \<le> b\<close>
     apply (rule_tac norm_cblinfun_bound)
     using \<open>0 \<le> b\<close> by auto
 qed
 
 lemma dense_span_separating: \<open>closure (cspan S) = UNIV \<Longrightarrow> bounded_clinear F \<Longrightarrow> bounded_clinear G \<Longrightarrow> (\<forall>x\<in>S. F x = G x) \<Longrightarrow> F = G\<close>
 proof -
   fix F G :: \<open>'a \<Rightarrow> 'b\<close>
   assume dense: \<open>closure (cspan S) = UNIV\<close>
   assume [simp]: \<open>bounded_clinear F\<close> \<open>bounded_clinear G\<close>
   assume \<open>\<forall>x\<in>S. F x = G x\<close>
   then have \<open>F x = G x\<close> if \<open>x \<in> cspan S\<close> for x
     apply (rule_tac complex_vector.linear_eq_on[of F G _ S])
     using that by (auto simp: bounded_clinear.clinear)
   then show \<open>F = G\<close>
     apply (rule_tac ext)
     apply (rule on_closure_eqI[of \<open>cspan S\<close> F G])
     using dense by (auto intro!: continuous_at_imp_continuous_on clinear_continuous_at)
 qed
 
 
 lemma infsum_cblinfun_apply:
   assumes \<open>g summable_on S\<close>
   shows \<open>infsum (\<lambda>x. A *\<^sub>V g x) S = A *\<^sub>V (infsum g S)\<close>
   apply (rule infsum_bounded_linear[unfolded o_def, of \<open>cblinfun_apply A\<close>])
   using assms 
   by (auto simp add: bounded_clinear.bounded_linear bounded_cbilinear.bounded_clinear_right
       bounded_cbilinear_cblinfun_apply)
 
 lemma has_sum_cblinfun_apply:
   assumes \<open>has_sum g S x\<close>
   shows \<open>has_sum (\<lambda>x. A *\<^sub>V g x) S (A *\<^sub>V x)\<close>
   apply (rule has_sum_bounded_linear[unfolded o_def, of \<open>cblinfun_apply A\<close>])
   using assms by (auto simp add: bounded_clinear.bounded_linear cblinfun.bounded_clinear_right)
 
 lemma abs_summable_on_cblinfun_apply:
   assumes \<open>g abs_summable_on S\<close>
   shows \<open>(\<lambda>x. A *\<^sub>V g x) abs_summable_on S\<close>
   using bounded_clinear.bounded_linear[OF cblinfun.bounded_clinear_right] assms
   by (rule abs_summable_on_bounded_linear[unfolded o_def])
 
 text \<open>The next eight lemmas logically belong in \<^theory>\<open>Complex_Bounded_Operators.Complex_Inner_Product\<close> 
   but the proofs use facts from this theory.\<close>
 lemma has_sum_cinner_left:
   assumes \<open>has_sum f I x\<close>
   shows \<open>has_sum (\<lambda>i. cinner a (f i)) I (cinner a x)\<close>
   by (metis assms cblinfun_cinner_right.rep_eq has_sum_cblinfun_apply)
 
 lemma summable_on_cinner_left:
   assumes \<open>f summable_on I\<close>
   shows \<open>(\<lambda>i. cinner a (f i)) summable_on I\<close>
   by (metis assms has_sum_cinner_left summable_on_def)
 
 lemma infsum_cinner_left:
   assumes \<open>\<phi> summable_on I\<close>
   shows \<open>cinner \<psi> (\<Sum>\<^sub>\<infinity>i\<in>I. \<phi> i) = (\<Sum>\<^sub>\<infinity>i\<in>I. cinner \<psi> (\<phi> i))\<close>
   by (metis assms has_sum_cinner_left has_sum_infsum infsumI)
 
 lemma has_sum_cinner_right:
   assumes \<open>has_sum f I x\<close>
   shows \<open>has_sum (\<lambda>i. f i \<bullet>\<^sub>C a) I (x \<bullet>\<^sub>C a)\<close>
   apply (rule has_sum_bounded_linear[where f=\<open>\<lambda>x. x \<bullet>\<^sub>C a\<close>, unfolded o_def])
   using assms by (simp_all add: bounded_antilinear.bounded_linear bounded_antilinear_cinner_left)
 
 lemma summable_on_cinner_right:
   assumes \<open>f summable_on I\<close>
   shows \<open>(\<lambda>i. f i \<bullet>\<^sub>C a) summable_on I\<close>
   by (metis assms has_sum_cinner_right summable_on_def)
 
 lemma infsum_cinner_right:
   assumes \<open>\<phi> summable_on I\<close>
   shows \<open>(\<Sum>\<^sub>\<infinity>i\<in>I. \<phi> i) \<bullet>\<^sub>C \<psi> = (\<Sum>\<^sub>\<infinity>i\<in>I. \<phi> i \<bullet>\<^sub>C \<psi>)\<close>
   by (metis assms has_sum_cinner_right has_sum_infsum infsumI)
 
 lemma Cauchy_cinner_product_summable:
   assumes asum: \<open>a summable_on UNIV\<close>
   assumes bsum: \<open>b summable_on UNIV\<close>
   assumes \<open>finite X\<close> \<open>finite Y\<close>
   assumes pos: \<open>\<And>x y. x \<notin> X \<Longrightarrow> y \<notin> Y \<Longrightarrow> cinner (a x) (b y) \<ge> 0\<close>
   shows absum: \<open>(\<lambda>(x, y). cinner (a x) (b y)) summable_on UNIV\<close>
 proof -
   have \<open>(\<Sum>(x,y)\<in>F. norm (cinner (a x) (b y))) \<le> norm (cinner (infsum a (-X)) (infsum b (-Y))) + norm (infsum a (-X)) + norm (infsum b (-Y)) + 1\<close> 
     if \<open>finite F\<close> and \<open>F \<subseteq> (-X)\<times>(-Y)\<close> for F
   proof -
     from asum \<open>finite X\<close>
     have \<open>a summable_on (-X)\<close>
       by (simp add: Compl_eq_Diff_UNIV summable_on_cofin_subset)
     then obtain MA where \<open>finite MA\<close> and \<open>MA \<subseteq> -X\<close>
       and MA: \<open>G \<supseteq> MA \<Longrightarrow> G \<subseteq> -X \<Longrightarrow> finite G \<Longrightarrow> norm (sum a G - infsum a (-X)) \<le> 1\<close> for G
       apply (simp add: summable_iff_has_sum_infsum has_sum_metric dist_norm)
       by (meson less_eq_real_def zero_less_one)
     
     from bsum \<open>finite Y\<close>
     have \<open>b summable_on (-Y)\<close>
       by (simp add: Compl_eq_Diff_UNIV summable_on_cofin_subset)
     then obtain MB where \<open>finite MB\<close> and \<open>MB \<subseteq> -Y\<close>
       and MB: \<open>G \<supseteq> MB \<Longrightarrow> G \<subseteq> -Y \<Longrightarrow> finite G \<Longrightarrow> norm (sum b G - infsum b (-Y)) \<le> 1\<close> for G
       apply (simp add: summable_iff_has_sum_infsum has_sum_metric dist_norm)
       by (meson less_eq_real_def zero_less_one)
 
     define F1 F2 where \<open>F1 = fst ` F \<union> MA\<close> and \<open>F2 = snd ` F \<union> MB\<close>
     define t1 t2 where \<open>t1 = sum a F1 - infsum a (-X)\<close> and \<open>t2 = sum b F2 - infsum b (-Y)\<close>
   
     have [simp]: \<open>finite F1\<close> \<open>finite F2\<close>
       using F1_def F2_def \<open>finite MA\<close> \<open>finite MB\<close> that by auto
     have [simp]: \<open>F1 \<subseteq> -X\<close> \<open>F2 \<subseteq> -Y\<close>
       using \<open>F \<subseteq> (-X)\<times>(-Y)\<close> \<open>MA \<subseteq> -X\<close> \<open>MB \<subseteq> -Y\<close>
       by (auto simp: F1_def F2_def)
     from MA[OF _ \<open>F1 \<subseteq> -X\<close> \<open>finite F1\<close>] have \<open>norm t1 \<le> 1\<close> 
       by (auto simp: t1_def F1_def)
     from MB[OF _ \<open>F2 \<subseteq> -Y\<close> \<open>finite F2\<close>] have \<open>norm t2 \<le> 1\<close> 
       by (auto simp: t2_def F2_def)
     have [simp]: \<open>F \<subseteq> F1 \<times> F2\<close>
       apply (auto simp: F1_def F2_def image_def)
       by force+
     have \<open>(\<Sum>(x,y)\<in>F. norm (cinner (a x) (b y))) \<le> (\<Sum>(x,y)\<in>F1\<times>F2. norm (cinner (a x) (b y)))\<close>
       apply (rule sum_mono2)
       by auto
     also from pos have \<open>\<dots> = norm (\<Sum>(x,y)\<in>F1\<times>F2. cinner (a x) (b y))\<close>
       apply (auto intro!: of_real_eq_iff[THEN iffD1] simp: case_prod_beta)
       apply (subst abs_complex_def[unfolded o_def, symmetric, THEN fun_cong])+
       apply (subst (2) abs_pos)
        apply (rule sum_nonneg)
        apply (metis Compl_eq_Diff_UNIV Diff_iff SigmaE \<open>F1 \<subseteq> - X\<close> \<open>F2 \<subseteq> - Y\<close> fst_conv prod.sel(2) subsetD)
       apply (rule sum.cong)
       apply simp
       by (metis Compl_iff SigmaE \<open>F1 \<subseteq> - X\<close> \<open>F2 \<subseteq> - Y\<close> abs_pos fst_conv prod.sel(2) subset_eq)
     also have \<open>\<dots> = norm (cinner (sum a F1) (sum b F2))\<close>
       by (simp add: sum.cartesian_product sum_cinner)
     also have \<open>\<dots> = norm (cinner (infsum a (-X) + t1) (infsum b (-Y) + t2))\<close>
       by (simp add: t1_def t2_def)
     also have \<open>\<dots> \<le> norm (cinner (infsum a (-X)) (infsum b (-Y))) + norm (infsum a (-X)) * norm t2 + norm t1 * norm (infsum b (-Y)) + norm t1 * norm t2\<close>
       apply (simp add: cinner_add_right cinner_add_left)
       by (smt (verit, del_insts) complex_inner_class.Cauchy_Schwarz_ineq2 norm_triangle_ineq)
     also from \<open>norm t1 \<le> 1\<close> \<open>norm t2 \<le> 1\<close>
     have \<open>\<dots> \<le> norm (cinner (infsum a (-X)) (infsum b (-Y))) + norm (infsum a (-X)) + norm (infsum b (-Y)) + 1\<close>
       by (auto intro!: add_mono mult_left_le mult_left_le_one_le mult_le_one)
     finally show ?thesis
       by -
   qed
 
   then have \<open>(\<lambda>(x, y). cinner (a x) (b y)) abs_summable_on (-X)\<times>(-Y)\<close>
     apply (rule_tac nonneg_bdd_above_summable_on)
     by (auto intro!: bdd_aboveI2 simp: case_prod_unfold)
   then have 1: \<open>(\<lambda>(x, y). cinner (a x) (b y)) summable_on (-X)\<times>(-Y)\<close>
     using abs_summable_summable by blast
 
   from bsum
   have \<open>(\<lambda>y. b y) summable_on (-Y)\<close>
     by (simp add: Compl_eq_Diff_UNIV assms(4) summable_on_cofin_subset)
   then have \<open>(\<lambda>y. cinner (a x) (b y)) summable_on (-Y)\<close> for x
     using summable_on_cinner_left by blast
   with \<open>finite X\<close> have 2: \<open>(\<lambda>(x, y). cinner (a x) (b y)) summable_on X\<times>(-Y)\<close>
     apply (rule_tac summable_on_product_finite_left)
     by auto
 
   from asum
   have \<open>(\<lambda>x. a x) summable_on (-X)\<close>
     by (simp add: Compl_eq_Diff_UNIV assms(3) summable_on_cofin_subset)
   then have \<open>(\<lambda>x. cinner (a x) (b y)) summable_on (-X)\<close> for y
     using summable_on_cinner_right by blast
   with \<open>finite Y\<close> have 3: \<open>(\<lambda>(x, y). cinner (a x) (b y)) summable_on (-X)\<times>Y\<close>
     apply (rule_tac summable_on_product_finite_right)
     by auto
 
   have 4: \<open>(\<lambda>(x, y). cinner (a x) (b y)) summable_on X\<times>Y\<close>
     by (simp add: \<open>finite X\<close> \<open>finite Y\<close>)
 
   show ?thesis
     apply (subst asm_rl[of \<open>UNIV = (-X)\<times>(-Y) \<union> X\<times>(-Y) \<union> (-X)\<times>Y \<union> X\<times>Y\<close>])
     using 1 2 3 4 by (auto intro!: summable_on_Un_disjoint)
 qed
 
 
 text \<open>A variant of @{thm [source] Series.Cauchy_product_sums} with \<^term>\<open>(*)\<close> replaced by \<^term>\<open>cinner\<close>.
    Differently from @{thm [source] Series.Cauchy_product_sums}, we do not require absolute summability
    of \<^term>\<open>a\<close> and \<^term>\<open>b\<close> individually but only unconditional summability of \<^term>\<open>a\<close>, \<^term>\<open>b\<close>, and their product.
    While on, e.g., reals, unconditional summability is equivalent to absolute summability, in
    general unconditional summability is a weaker requirement.\<close>
 lemma 
   fixes a b :: "nat \<Rightarrow> 'a::complex_inner"
   assumes asum: \<open>a summable_on UNIV\<close>
   assumes bsum: \<open>b summable_on UNIV\<close>
   assumes absum: \<open>(\<lambda>(x, y). cinner (a x) (b y)) summable_on UNIV\<close>
     (* \<comment> \<open>See @{thm [source] Cauchy_cinner_product_summable} or @{thm [source] Cauchy_cinner_product_summable'} for a way to rewrite this premise.\<close> *)
   shows Cauchy_cinner_product_infsum: \<open>(\<Sum>\<^sub>\<infinity>k. \<Sum>i\<le>k. cinner (a i) (b (k - i))) = cinner (\<Sum>\<^sub>\<infinity>k. a k) (\<Sum>\<^sub>\<infinity>k. b k)\<close>
     and Cauchy_cinner_product_infsum_exists: \<open>(\<lambda>k. \<Sum>i\<le>k. cinner (a i) (b (k - i))) summable_on UNIV\<close>
 proof -
   have img: \<open>(\<lambda>(k::nat, i). (i, k - i)) ` {(k, i). i \<le> k} = UNIV\<close>
     apply (auto simp: image_def)
     by (metis add.commute add_diff_cancel_right' diff_le_self)
   have inj: \<open>inj_on (\<lambda>(k::nat, i). (i, k - i)) {(k, i). i \<le> k}\<close>
     by (smt (verit, del_insts) Pair_inject case_prodE case_prod_conv eq_diff_iff inj_onI mem_Collect_eq)
   have sigma: \<open>(SIGMA k:UNIV. {i. i \<le> k}) = {(k, i). i \<le> k}\<close>
     by auto
 
   from absum
   have \<open>(\<lambda>(x, y). cinner (a y) (b (x - y))) summable_on {(k, i). i \<le> k}\<close>
     by (rule Cauchy_cinner_product_summable'[THEN iffD1])
   then have \<open>(\<lambda>k. \<Sum>\<^sub>\<infinity>i|i\<le>k. cinner (a i) (b (k-i))) summable_on UNIV\<close>
     by (metis (mono_tags, lifting) sigma summable_on_Sigma_banach summable_on_cong)
   then show \<open>(\<lambda>k. \<Sum>i\<le>k. cinner (a i) (b (k - i))) summable_on UNIV\<close>
     by (metis (mono_tags, lifting) atMost_def finite_Collect_le_nat infsum_finite summable_on_cong)
 
   have \<open>cinner (\<Sum>\<^sub>\<infinity>k. a k) (\<Sum>\<^sub>\<infinity>k. b k) = (\<Sum>\<^sub>\<infinity>k. \<Sum>\<^sub>\<infinity>l. cinner (a k) (b l))\<close>
     apply (subst infsum_cinner_right)
      apply (rule asum)
     apply (subst infsum_cinner_left)
      apply (rule bsum)
     by simp
   also have \<open>\<dots> = (\<Sum>\<^sub>\<infinity>(k,l). cinner (a k) (b l))\<close>
     apply (subst infsum_Sigma'_banach)
     using absum by auto
   also have \<open>\<dots> = (\<Sum>\<^sub>\<infinity>(k, l)\<in>(\<lambda>(k, i). (i, k - i)) ` {(k, i). i \<le> k}. cinner (a k) (b l))\<close>
     by (simp only: img)
   also have \<open>\<dots> = infsum ((\<lambda>(k, l). a k \<bullet>\<^sub>C b l) \<circ> (\<lambda>(k, i). (i, k - i))) {(k, i). i \<le> k}\<close>
     using inj by (rule infsum_reindex)
   also have \<open>\<dots> = (\<Sum>\<^sub>\<infinity>(k,i)|i\<le>k. a i \<bullet>\<^sub>C b (k-i))\<close>
     by (simp add: o_def case_prod_unfold)
   also have \<open>\<dots> = (\<Sum>\<^sub>\<infinity>k. \<Sum>\<^sub>\<infinity>i|i\<le>k. a i \<bullet>\<^sub>C b (k-i))\<close>
     apply (subst infsum_Sigma'_banach)
     using absum by (auto simp: sigma Cauchy_cinner_product_summable')
   also have \<open>\<dots> = (\<Sum>\<^sub>\<infinity>k. \<Sum>i\<le>k. a i \<bullet>\<^sub>C b (k-i))\<close>
     apply (subst infsum_finite[symmetric])
     by (auto simp add: atMost_def)
   finally show \<open>(\<Sum>\<^sub>\<infinity>k. \<Sum>i\<le>k. a i \<bullet>\<^sub>C b (k - i)) = (\<Sum>\<^sub>\<infinity>k. a k) \<bullet>\<^sub>C (\<Sum>\<^sub>\<infinity>k. b k)\<close>
     by simp
 qed
 
 
 lemma CBlinfun_plus: 
   assumes [simp]: \<open>bounded_clinear f\<close> \<open>bounded_clinear g\<close>
   shows \<open>CBlinfun (f + g) = CBlinfun f + CBlinfun g\<close>
   by (auto intro!: cblinfun_eqI simp: plus_fun_def bounded_clinear_add CBlinfun_inverse cblinfun.add_left)
 
 lemma CBlinfun_scaleC:
   assumes \<open>bounded_clinear f\<close>
   shows \<open>CBlinfun (\<lambda>y. c *\<^sub>C f y) = c *\<^sub>C CBlinfun f\<close>
   by (simp add: assms eq_onp_same_args scaleC_cblinfun.abs_eq)
 
 
-lemma bounded_clinear_inv:
-  assumes [simp]: \<open>bounded_clinear f\<close>
-  assumes b: \<open>b > 0\<close>
-  assumes bound: \<open>\<And>x. norm (f x) \<ge> b * norm x\<close>
-  assumes \<open>surj f\<close>
-  shows \<open>bounded_clinear (inv f)\<close>
-proof (rule bounded_clinear_intro)
-  fix x y :: 'b and r :: complex
-  define x' y' where \<open>x' = inv f x\<close> and \<open>y' = inv f y\<close>
-  have [simp]: \<open>clinear f\<close>
-    by (simp add: bounded_clinear.clinear)
-  have [simp]: \<open>inj f\<close>
-  proof (rule injI)
-    fix x y assume \<open>f x = f y\<close>
-    then have \<open>norm (f (x - y)) = 0\<close>
-      by (simp add: complex_vector.linear_diff)
-    with bound b have \<open>norm (x - y) = 0\<close>
-      by (metis linorder_not_le mult_le_0_iff nle_le norm_ge_zero)
-    then show \<open>x = y\<close>
-      by simp
-  qed
-
-  from \<open>surj f\<close>
-  have [simp]: \<open>x = f x'\<close> \<open>y = f y'\<close>
-    by (simp_all add: surj_f_inv_f x'_def y'_def)
-  show "inv f (x + y) = inv f x + inv f y"
-    by (simp flip: complex_vector.linear_add)
-  show "inv f (r *\<^sub>C x) = r *\<^sub>C inv f x"
-    by (simp flip: clinear.scaleC)
-  from bound have "b * norm (inv f x) \<le> norm x" 
-    by (simp flip: clinear.scaleC)
-  with b show "norm (inv f x) \<le> norm x * inverse b" 
-    by (smt (verit, ccfv_threshold) left_inverse mult.commute mult_cancel_right1 mult_le_cancel_left_pos vector_space_over_itself.scale_scale)
-qed
-
 lemma cinner_sup_norm_cblinfun:
   fixes A :: \<open>'a::{complex_normed_vector,not_singleton} \<Rightarrow>\<^sub>C\<^sub>L 'b::complex_inner\<close>
   shows \<open>norm A = (SUP (\<psi>,\<phi>). cmod (cinner \<psi> (A *\<^sub>V \<phi>)) / (norm \<psi> * norm \<phi>))\<close>
   apply transfer
   apply (rule cinner_sup_onorm)
   by (simp add: bounded_clinear.bounded_linear)
 
 
 lemma norm_cblinfun_Sup: \<open>norm A = (SUP \<psi>. norm (A *\<^sub>V \<psi>) / norm \<psi>)\<close>
   by (simp add: norm_cblinfun.rep_eq onorm_def)
 
+lemma cblinfun_eq_on:
+  fixes A B :: "'a::cbanach \<Rightarrow>\<^sub>C\<^sub>L'b::complex_normed_vector"
+  assumes "\<And>x. x \<in> G \<Longrightarrow> A *\<^sub>V x = B *\<^sub>V x" and \<open>t \<in> closure (cspan G)\<close>
+  shows "A *\<^sub>V t = B *\<^sub>V t"
+  using assms
+  apply transfer
+  using bounded_clinear_eq_on by blast
+
+lemma cblinfun_eq_gen_eqI:
+  fixes A B :: "'a::cbanach \<Rightarrow>\<^sub>C\<^sub>L'b::complex_normed_vector"
+  assumes "\<And>x. x \<in> G \<Longrightarrow> A *\<^sub>V x = B *\<^sub>V x" and \<open>ccspan G = \<top>\<close>
+  shows "A = B"
+  apply (rule cblinfun_eqI)
+  apply (rule cblinfun_eq_on[where G=G])
+  using assms apply auto
+  by (metis ccspan.rep_eq iso_tuple_UNIV_I top_ccsubspace.rep_eq)
+
+
 subsection \<open>Relationship to real bounded operators (\<^typ>\<open>_ \<Rightarrow>\<^sub>L _\<close>)\<close>
 
 instantiation blinfun :: (real_normed_vector, complex_normed_vector) "complex_normed_vector"
 begin
 lift_definition scaleC_blinfun :: \<open>complex \<Rightarrow>
  ('a::real_normed_vector, 'b::complex_normed_vector) blinfun \<Rightarrow>
  ('a, 'b) blinfun\<close>
   is \<open>\<lambda> c::complex. \<lambda> f::'a\<Rightarrow>'b. (\<lambda> x. c *\<^sub>C (f x) )\<close>
 proof
   fix c::complex and f :: \<open>'a\<Rightarrow>'b\<close> and b1::'a and b2::'a
   assume \<open>bounded_linear f\<close>
   show \<open>c *\<^sub>C f (b1 + b2) = c *\<^sub>C f b1 + c *\<^sub>C f b2\<close>
     by (simp add: \<open>bounded_linear f\<close> linear_simps scaleC_add_right)
 
   fix c::complex and f :: \<open>'a\<Rightarrow>'b\<close> and b::'a and r::real
   assume \<open>bounded_linear f\<close>
   show \<open>c *\<^sub>C f (r *\<^sub>R b) = r *\<^sub>R (c *\<^sub>C f b)\<close>
     by (simp add: \<open>bounded_linear f\<close> linear_simps(5) scaleR_scaleC)
 
   fix c::complex and f :: \<open>'a\<Rightarrow>'b\<close>
   assume \<open>bounded_linear f\<close>
 
   have \<open>\<exists> K. \<forall> x. norm (f x) \<le> norm x * K\<close>
     using \<open>bounded_linear f\<close>
     by (simp add: bounded_linear.bounded)
   then obtain K where \<open>\<forall> x. norm (f x) \<le> norm x * K\<close>
     by blast
   have \<open>cmod c \<ge> 0\<close>
     by simp
   hence \<open>\<forall> x. (cmod c) * norm (f x) \<le> (cmod c) * norm x * K\<close>
     using  \<open>\<forall> x. norm (f x) \<le> norm x * K\<close>
     by (metis ordered_comm_semiring_class.comm_mult_left_mono vector_space_over_itself.scale_scale)
   moreover have \<open>norm (c *\<^sub>C f x) = (cmod c) * norm (f x)\<close>
     for x
     by simp
   ultimately show \<open>\<exists>K. \<forall>x. norm (c *\<^sub>C f x) \<le> norm x * K\<close>
     by (metis ab_semigroup_mult_class.mult_ac(1) mult.commute)
 qed
 
 instance
 proof
   have "r *\<^sub>R x = complex_of_real r *\<^sub>C x"
     for x :: "('a, 'b) blinfun" and r
     apply transfer
     by (simp add: scaleR_scaleC)
   thus "((*\<^sub>R) r::'a \<Rightarrow>\<^sub>L 'b \<Rightarrow> _) = (*\<^sub>C) (complex_of_real r)" for r
     by auto
   show "a *\<^sub>C (x + y) = a *\<^sub>C x + a *\<^sub>C y"
     for a :: complex and x y :: "'a \<Rightarrow>\<^sub>L 'b"
     apply transfer
     by (simp add: scaleC_add_right)
 
   show "(a + b) *\<^sub>C x = a *\<^sub>C x + b *\<^sub>C x"
     for a b :: complex and x :: "'a \<Rightarrow>\<^sub>L 'b"
     apply transfer
     by (simp add: scaleC_add_left)
 
   show "a *\<^sub>C b *\<^sub>C x = (a * b) *\<^sub>C x"
     for a b :: complex and x :: "'a \<Rightarrow>\<^sub>L 'b"
     apply transfer
     by simp
 
   have \<open>1 *\<^sub>C f x = f x\<close>
     for f :: \<open>'a\<Rightarrow>'b\<close> and x
     by auto
   thus "1 *\<^sub>C x = x"
     for x :: "'a \<Rightarrow>\<^sub>L 'b"
     by (simp add: scaleC_blinfun.rep_eq blinfun_eqI)
 
   have \<open>onorm (\<lambda>x. a *\<^sub>C f x) = cmod a * onorm f\<close>
     if \<open>bounded_linear f\<close>
     for f :: \<open>'a \<Rightarrow> 'b\<close> and a :: complex
   proof-
     have \<open>cmod a \<ge> 0\<close>
       by simp
     have \<open>\<exists> K::real. \<forall> x. (\<bar> ereal ((norm (f x)) / (norm x)) \<bar>) \<le> K\<close>
       using \<open>bounded_linear f\<close> le_onorm by fastforce
     then obtain K::real where \<open>\<forall> x. (\<bar> ereal ((norm (f x)) / (norm x)) \<bar>) \<le> K\<close>
       by blast
     hence  \<open>\<forall> x. (cmod a) *(\<bar> ereal ((norm (f x)) / (norm x)) \<bar>) \<le> (cmod a) * K\<close>
       using \<open>cmod a \<ge> 0\<close>
       by (metis abs_ereal.simps(1) abs_ereal_pos   abs_pos ereal_mult_left_mono  times_ereal.simps(1))
     hence  \<open>\<forall> x.  (\<bar> ereal ((cmod a) * (norm (f x)) / (norm x)) \<bar>) \<le> (cmod a) * K\<close>
       by simp
     hence \<open>bdd_above {ereal (cmod a * (norm (f x)) / (norm x)) | x. True}\<close>
       by simp
     moreover have \<open>{ereal (cmod a * (norm (f x)) / (norm x)) | x. True} \<noteq> {}\<close>
       by auto
     ultimately have p1: \<open>(SUP x. \<bar>ereal (cmod a * (norm (f x)) / (norm x))\<bar>) \<le> cmod a * K\<close>
       using \<open>\<forall> x. \<bar> ereal (cmod a * (norm (f x)) / (norm x)) \<bar> \<le> cmod a * K\<close>
         Sup_least mem_Collect_eq
       by (simp add: SUP_le_iff)
     have  p2: \<open>\<And>i. i \<in> UNIV \<Longrightarrow> 0 \<le> ereal (cmod a * norm (f i) / norm i)\<close>
       by simp
     hence \<open>\<bar>SUP x. ereal (cmod a * (norm (f x)) / (norm x))\<bar>
               \<le> (SUP x. \<bar>ereal (cmod a * (norm (f x)) / (norm x))\<bar>)\<close>
       using  \<open>bdd_above {ereal (cmod a * (norm (f x)) / (norm x)) | x. True}\<close>
         \<open>{ereal (cmod a * (norm (f x)) / (norm x)) | x. True} \<noteq> {}\<close>
       by (metis (mono_tags, lifting) SUP_upper2 Sup.SUP_cong UNIV_I
           p2 abs_ereal_ge0 ereal_le_real)
     hence \<open>\<bar>SUP x. ereal (cmod a * (norm (f x)) / (norm x))\<bar> \<le> cmod a * K\<close>
       using  \<open>(SUP x. \<bar>ereal (cmod a * (norm (f x)) / (norm x))\<bar>) \<le> cmod a * K\<close>
       by simp
     hence \<open>\<bar> ( SUP i\<in>UNIV::'a set. ereal ((\<lambda> x. (cmod a) * (norm (f x)) / norm x) i)) \<bar> \<noteq> \<infinity>\<close>
       by auto
     hence w2: \<open>( SUP i\<in>UNIV::'a set. ereal ((\<lambda> x. cmod a * (norm (f x)) / norm x) i))
              = ereal ( Sup ((\<lambda> x. cmod a * (norm (f x)) / norm x) ` (UNIV::'a set) ))\<close>
       by (simp add: ereal_SUP)
     have \<open>(UNIV::('a set)) \<noteq> {}\<close>
       by simp
     moreover have \<open>\<And> i. i \<in> (UNIV::('a set)) \<Longrightarrow> (\<lambda> x. (norm (f x)) / norm x :: ereal) i \<ge> 0\<close>
       by simp
     moreover have \<open>cmod a \<ge> 0\<close>
       by simp
     ultimately have \<open>(SUP i\<in>(UNIV::('a set)). ((cmod a)::ereal) * (\<lambda> x. (norm (f x)) / norm x :: ereal) i )
         = ((cmod a)::ereal) * ( SUP i\<in>(UNIV::('a set)). (\<lambda> x. (norm (f x)) / norm x :: ereal) i )\<close>
       by (simp add: Sup_ereal_mult_left')
     hence \<open>(SUP x. ((cmod a)::ereal) * ( (norm (f x)) / norm x :: ereal) )
         = ((cmod a)::ereal) * ( SUP x. ( (norm (f x)) / norm x :: ereal) )\<close>
       by simp
     hence z1: \<open>real_of_ereal ( (SUP x. ((cmod a)::ereal) * ( (norm (f x)) / norm x :: ereal) ) )
         = real_of_ereal ( ((cmod a)::ereal) * ( SUP x. ( (norm (f x)) / norm x :: ereal) ) )\<close>
       by simp
     have z2: \<open>real_of_ereal (SUP x. ((cmod a)::ereal) * ( (norm (f x)) / norm x :: ereal) )
                   = (SUP x. cmod a * (norm (f x) / norm x))\<close>
       using w2
       by auto
     have \<open>real_of_ereal ( ((cmod a)::ereal) * ( SUP x. ( (norm (f x)) / norm x :: ereal) ) )
                 =  (cmod a) * real_of_ereal ( SUP x. ( (norm (f x)) / norm x :: ereal) )\<close>
       by simp
     moreover have \<open>real_of_ereal ( SUP x. ( (norm (f x)) / norm x :: ereal) )
                   = ( SUP x. ((norm (f x)) / norm x) )\<close>
     proof-
       have \<open>\<bar> ( SUP i\<in>UNIV::'a set. ereal ((\<lambda> x. (norm (f x)) / norm x) i)) \<bar> \<noteq> \<infinity>\<close>
       proof-
         have \<open>\<exists> K::real. \<forall> x. (\<bar> ereal ((norm (f x)) / (norm x)) \<bar>) \<le> K\<close>
           using \<open>bounded_linear f\<close> le_onorm by fastforce
         then obtain K::real where \<open>\<forall> x. (\<bar> ereal ((norm (f x)) / (norm x)) \<bar>) \<le> K\<close>
           by blast
         hence \<open>bdd_above {ereal ((norm (f x)) / (norm x)) | x. True}\<close>
           by simp
         moreover have \<open>{ereal ((norm (f x)) / (norm x)) | x. True} \<noteq> {}\<close>
           by auto
         ultimately have \<open>(SUP x. \<bar>ereal ((norm (f x)) / (norm x))\<bar>) \<le> K\<close>
           using \<open>\<forall> x. \<bar> ereal ((norm (f x)) / (norm x)) \<bar> \<le> K\<close>
             Sup_least mem_Collect_eq
           by (simp add: SUP_le_iff)
         hence \<open>\<bar>SUP x. ereal ((norm (f x)) / (norm x))\<bar>
               \<le> (SUP x. \<bar>ereal ((norm (f x)) / (norm x))\<bar>)\<close>
           using  \<open>bdd_above {ereal ((norm (f x)) / (norm x)) | x. True}\<close>
             \<open>{ereal ((norm (f x)) / (norm x)) | x. True} \<noteq> {}\<close>
           by (metis (mono_tags, lifting) SUP_upper2 Sup.SUP_cong UNIV_I \<open>\<And>i. i \<in> UNIV \<Longrightarrow> 0 \<le> ereal (norm (f i) / norm i)\<close> abs_ereal_ge0 ereal_le_real)
         hence \<open>\<bar>SUP x. ereal ((norm (f x)) / (norm x))\<bar> \<le> K\<close>
           using  \<open>(SUP x. \<bar>ereal ((norm (f x)) / (norm x))\<bar>) \<le> K\<close>
           by simp
         thus ?thesis
           by auto
       qed
       hence \<open> ( SUP i\<in>UNIV::'a set. ereal ((\<lambda> x. (norm (f x)) / norm x) i))
              = ereal ( Sup ((\<lambda> x. (norm (f x)) / norm x) ` (UNIV::'a set) ))\<close>
         by (simp add: ereal_SUP)
       thus ?thesis
         by simp
     qed
     have z3: \<open>real_of_ereal ( ((cmod a)::ereal) * ( SUP x. ( (norm (f x)) / norm x :: ereal) ) )
                 = cmod a * (SUP x. norm (f x) / norm x)\<close>
       by (simp add: \<open>real_of_ereal (SUP x. ereal (norm (f x) / norm x)) = (SUP x. norm (f x) / norm x)\<close>)
     hence w1: \<open>(SUP x. cmod a * (norm (f x) / norm x)) =
           cmod a * (SUP x. norm (f x) / norm x)\<close>
       using z1 z2 by linarith
     have v1: \<open>onorm (\<lambda>x. a *\<^sub>C f x) = (SUP x. norm (a *\<^sub>C f x) / norm x)\<close>
       by (simp add: onorm_def)
     have v2: \<open>(SUP x. norm (a *\<^sub>C f x) / norm x) = (SUP x. ((cmod a) * norm (f x)) / norm x)\<close>
       by simp
     have v3: \<open>(SUP x. ((cmod a) * norm (f x)) / norm x) =  (SUP x. (cmod a) * ((norm (f x)) / norm x))\<close>
       by simp
     have v4: \<open>(SUP x. (cmod a) * ((norm (f x)) / norm x)) = (cmod a) *  (SUP x. ((norm (f x)) / norm x))\<close>
       using w1
       by blast
     show \<open>onorm (\<lambda>x. a *\<^sub>C f x) = cmod a * onorm f\<close>
       using v1 v2 v3 v4
       by (metis (mono_tags, lifting) onorm_def)
   qed
   thus \<open>norm (a *\<^sub>C x) = cmod a * norm x\<close>
     for a::complex and x::\<open>('a, 'b) blinfun\<close>
     apply transfer
     by blast
 qed
 end
 
 (* We do not have clinear_blinfun_compose_right *)
 lemma clinear_blinfun_compose_left: \<open>clinear (\<lambda>x. blinfun_compose x y)\<close>
   by (auto intro!: clinearI simp: blinfun_eqI scaleC_blinfun.rep_eq bounded_bilinear.add_left
                                   bounded_bilinear_blinfun_compose)
 
 instantiation blinfun :: (real_normed_vector, cbanach) "cbanach"
 begin
 instance..
 end
 
 lemma blinfun_compose_assoc: "(A o\<^sub>L B) o\<^sub>L C = A o\<^sub>L (B  o\<^sub>L C)"
   by (simp add: blinfun_eqI)
 
 lift_definition blinfun_of_cblinfun::\<open>'a::complex_normed_vector \<Rightarrow>\<^sub>C\<^sub>L 'b::complex_normed_vector
   \<Rightarrow> 'a \<Rightarrow>\<^sub>L 'b\<close> is "id"
   apply transfer by (simp add: bounded_clinear.bounded_linear)
 
 lift_definition blinfun_cblinfun_eq ::
   \<open>'a \<Rightarrow>\<^sub>L 'b \<Rightarrow> 'a::complex_normed_vector \<Rightarrow>\<^sub>C\<^sub>L 'b::complex_normed_vector \<Rightarrow> bool\<close> is "(=)" .
 
 lemma blinfun_cblinfun_eq_bi_unique[transfer_rule]: \<open>bi_unique blinfun_cblinfun_eq\<close>
   unfolding bi_unique_def apply transfer by auto
 
 lemma blinfun_cblinfun_eq_right_total[transfer_rule]: \<open>right_total blinfun_cblinfun_eq\<close>
   unfolding right_total_def apply transfer
   by (simp add: bounded_clinear.bounded_linear)
 
 named_theorems cblinfun_blinfun_transfer
 
 lemma cblinfun_blinfun_transfer_0[cblinfun_blinfun_transfer]:
   "blinfun_cblinfun_eq (0::(_,_) blinfun) (0::(_,_) cblinfun)"
   apply transfer by simp
 
 lemma cblinfun_blinfun_transfer_plus[cblinfun_blinfun_transfer]:
   includes lifting_syntax
   shows "(blinfun_cblinfun_eq ===> blinfun_cblinfun_eq ===> blinfun_cblinfun_eq) (+) (+)"
   unfolding rel_fun_def apply transfer by auto
 
 lemma cblinfun_blinfun_transfer_minus[cblinfun_blinfun_transfer]:
   includes lifting_syntax
   shows "(blinfun_cblinfun_eq ===> blinfun_cblinfun_eq ===> blinfun_cblinfun_eq) (-) (-)"
   unfolding rel_fun_def apply transfer by auto
 
 lemma cblinfun_blinfun_transfer_uminus[cblinfun_blinfun_transfer]:
   includes lifting_syntax
   shows "(blinfun_cblinfun_eq ===> blinfun_cblinfun_eq) (uminus) (uminus)"
   unfolding rel_fun_def apply transfer by auto
 
 definition "real_complex_eq r c \<longleftrightarrow> complex_of_real r = c"
 
 lemma bi_unique_real_complex_eq[transfer_rule]: \<open>bi_unique real_complex_eq\<close>
   unfolding real_complex_eq_def bi_unique_def by auto
 
 lemma left_total_real_complex_eq[transfer_rule]: \<open>left_total real_complex_eq\<close>
   unfolding real_complex_eq_def left_total_def by auto
 
 lemma cblinfun_blinfun_transfer_scaleC[cblinfun_blinfun_transfer]:
   includes lifting_syntax
   shows "(real_complex_eq ===> blinfun_cblinfun_eq ===> blinfun_cblinfun_eq) (scaleR) (scaleC)"
   unfolding rel_fun_def apply transfer
   by (simp add: real_complex_eq_def scaleR_scaleC)
 
 lemma cblinfun_blinfun_transfer_CBlinfun[cblinfun_blinfun_transfer]:
   includes lifting_syntax
   shows "(eq_onp bounded_clinear ===> blinfun_cblinfun_eq) Blinfun CBlinfun"
   unfolding rel_fun_def blinfun_cblinfun_eq.rep_eq eq_onp_def
   by (auto simp: CBlinfun_inverse Blinfun_inverse bounded_clinear.bounded_linear)
 
 lemma cblinfun_blinfun_transfer_norm[cblinfun_blinfun_transfer]:
   includes lifting_syntax
   shows "(blinfun_cblinfun_eq ===> (=)) norm norm"
   unfolding rel_fun_def apply transfer by auto
 
 lemma cblinfun_blinfun_transfer_dist[cblinfun_blinfun_transfer]:
   includes lifting_syntax
   shows "(blinfun_cblinfun_eq ===> blinfun_cblinfun_eq ===> (=)) dist dist"
   unfolding rel_fun_def dist_norm apply transfer by auto
 
 lemma cblinfun_blinfun_transfer_sgn[cblinfun_blinfun_transfer]:
   includes lifting_syntax
   shows "(blinfun_cblinfun_eq ===> blinfun_cblinfun_eq) sgn sgn"
   unfolding rel_fun_def sgn_blinfun_def sgn_cblinfun_def apply transfer
   by (auto simp: scaleR_scaleC)
 
 lemma cblinfun_blinfun_transfer_Cauchy[cblinfun_blinfun_transfer]:
   includes lifting_syntax
   shows "(((=) ===> blinfun_cblinfun_eq) ===> (=)) Cauchy Cauchy"
 proof -
   note cblinfun_blinfun_transfer[transfer_rule]
   show ?thesis
     unfolding Cauchy_def
     by transfer_prover
 qed
 
 lemma cblinfun_blinfun_transfer_tendsto[cblinfun_blinfun_transfer]:
   includes lifting_syntax
   shows "(((=) ===> blinfun_cblinfun_eq) ===> blinfun_cblinfun_eq ===> (=) ===> (=)) tendsto tendsto"
 proof -
   note cblinfun_blinfun_transfer[transfer_rule]
   show ?thesis
     unfolding tendsto_iff
     by transfer_prover
 qed
 
 lemma cblinfun_blinfun_transfer_compose[cblinfun_blinfun_transfer]:
   includes lifting_syntax
   shows "(blinfun_cblinfun_eq ===> blinfun_cblinfun_eq ===> blinfun_cblinfun_eq) (o\<^sub>L) (o\<^sub>C\<^sub>L)"
   unfolding rel_fun_def apply transfer by auto
 
 lemma cblinfun_blinfun_transfer_apply[cblinfun_blinfun_transfer]:
   includes lifting_syntax
   shows "(blinfun_cblinfun_eq ===> (=) ===> (=)) blinfun_apply cblinfun_apply"
   unfolding rel_fun_def apply transfer by auto
 
 lemma blinfun_of_cblinfun_inj:
   \<open>blinfun_of_cblinfun f = blinfun_of_cblinfun g \<Longrightarrow> f = g\<close>
   by (metis cblinfun_apply_inject blinfun_of_cblinfun.rep_eq)
 
 lemma blinfun_of_cblinfun_inv:
   assumes "\<And>c. \<And>x. f *\<^sub>v (c *\<^sub>C x) = c *\<^sub>C (f *\<^sub>v x)"
   shows "\<exists>g. blinfun_of_cblinfun g = f"
   using assms
 proof transfer
   show "\<exists>g\<in>Collect bounded_clinear. id g = f"
     if "bounded_linear f"
       and "\<And>c x. f (c *\<^sub>C x) = c *\<^sub>C f x"
     for f :: "'a \<Rightarrow> 'b"
     using that bounded_linear_bounded_clinear by auto
 qed
 
 lemma blinfun_of_cblinfun_zero:
   \<open>blinfun_of_cblinfun 0 = 0\<close>
   apply transfer by simp
 
 lemma blinfun_of_cblinfun_uminus:
   \<open>blinfun_of_cblinfun (- f) = - (blinfun_of_cblinfun f)\<close>
   apply transfer
   by auto
 
 lemma blinfun_of_cblinfun_minus:
   \<open>blinfun_of_cblinfun (f - g) = blinfun_of_cblinfun f - blinfun_of_cblinfun g\<close>
   apply transfer
   by auto
 
 lemma blinfun_of_cblinfun_scaleC:
   \<open>blinfun_of_cblinfun (c *\<^sub>C f) = c *\<^sub>C (blinfun_of_cblinfun f)\<close>
   apply transfer
   by auto
 
 lemma blinfun_of_cblinfun_scaleR:
   \<open>blinfun_of_cblinfun (c *\<^sub>R f) = c *\<^sub>R (blinfun_of_cblinfun f)\<close>
   apply transfer by auto
 
 lemma blinfun_of_cblinfun_norm:
   fixes f::\<open>'a::complex_normed_vector \<Rightarrow>\<^sub>C\<^sub>L 'b::complex_normed_vector\<close>
   shows \<open>norm f = norm (blinfun_of_cblinfun f)\<close>
   apply transfer by auto
 
-subsection \<open>Composition\<close>
-
 lemma blinfun_of_cblinfun_cblinfun_compose:
   fixes f::\<open>'b::complex_normed_vector \<Rightarrow>\<^sub>C\<^sub>L 'c::complex_normed_vector\<close>
     and g::\<open>'a::complex_normed_vector \<Rightarrow>\<^sub>C\<^sub>L 'b\<close>
   shows \<open>blinfun_of_cblinfun (f  o\<^sub>C\<^sub>L g) = (blinfun_of_cblinfun f) o\<^sub>L (blinfun_of_cblinfun g)\<close>
   apply transfer by auto
 
+subsection \<open>Composition\<close>
+
 lemma cblinfun_compose_assoc:
   shows "(A o\<^sub>C\<^sub>L B) o\<^sub>C\<^sub>L C = A o\<^sub>C\<^sub>L (B o\<^sub>C\<^sub>L C)"
   by (metis (no_types, lifting) cblinfun_apply_inject fun.map_comp cblinfun_compose.rep_eq)
 
 lemma cblinfun_compose_zero_right[simp]: "U o\<^sub>C\<^sub>L 0 = 0"
   using bounded_cbilinear.zero_right bounded_cbilinear_cblinfun_compose by blast
 
 lemma cblinfun_compose_zero_left[simp]: "0 o\<^sub>C\<^sub>L U = 0"
   using bounded_cbilinear.zero_left bounded_cbilinear_cblinfun_compose by blast
 
 lemma cblinfun_compose_scaleC_left[simp]:
   fixes A::"'b::complex_normed_vector \<Rightarrow>\<^sub>C\<^sub>L 'c::complex_normed_vector"
     and B::"'a::complex_normed_vector \<Rightarrow>\<^sub>C\<^sub>L 'b"
   shows \<open>(a *\<^sub>C A) o\<^sub>C\<^sub>L B = a *\<^sub>C (A o\<^sub>C\<^sub>L B)\<close>
   by (simp add: bounded_cbilinear.scaleC_left bounded_cbilinear_cblinfun_compose)
 
 lemma cblinfun_compose_scaleR_left[simp]:
   fixes A::"'b::complex_normed_vector \<Rightarrow>\<^sub>C\<^sub>L 'c::complex_normed_vector"
     and B::"'a::complex_normed_vector \<Rightarrow>\<^sub>C\<^sub>L 'b"
   shows \<open>(a *\<^sub>R A) o\<^sub>C\<^sub>L B = a *\<^sub>R (A o\<^sub>C\<^sub>L B)\<close>
   by (simp add: scaleR_scaleC)
 
 lemma cblinfun_compose_scaleC_right[simp]:
   fixes A::"'b::complex_normed_vector \<Rightarrow>\<^sub>C\<^sub>L 'c::complex_normed_vector"
     and B::"'a::complex_normed_vector \<Rightarrow>\<^sub>C\<^sub>L 'b"
   shows \<open>A o\<^sub>C\<^sub>L (a *\<^sub>C B) = a *\<^sub>C (A o\<^sub>C\<^sub>L B)\<close>
   apply transfer by (auto intro!: ext bounded_clinear.clinear complex_vector.linear_scale)
 
 lemma cblinfun_compose_scaleR_right[simp]:
   fixes A::"'b::complex_normed_vector \<Rightarrow>\<^sub>C\<^sub>L 'c::complex_normed_vector"
     and B::"'a::complex_normed_vector \<Rightarrow>\<^sub>C\<^sub>L 'b"
   shows \<open>A o\<^sub>C\<^sub>L (a *\<^sub>R B) = a *\<^sub>R (A o\<^sub>C\<^sub>L B)\<close>
   by (simp add: scaleR_scaleC)
 
 lemma cblinfun_compose_id_right[simp]:
   shows "U o\<^sub>C\<^sub>L id_cblinfun = U"
   apply transfer by auto
 
 lemma cblinfun_compose_id_left[simp]:
   shows "id_cblinfun o\<^sub>C\<^sub>L U  = U"
   apply transfer by auto
 
-lemma cblinfun_eq_on:
-  fixes A B :: "'a::cbanach \<Rightarrow>\<^sub>C\<^sub>L'b::complex_normed_vector"
-  assumes "\<And>x. x \<in> G \<Longrightarrow> A *\<^sub>V x = B *\<^sub>V x" and \<open>t \<in> closure (cspan G)\<close>
-  shows "A *\<^sub>V t = B *\<^sub>V t"
-  using assms
-  apply transfer
-  using bounded_clinear_eq_on by blast
-
-lemma cblinfun_eq_gen_eqI:
-  fixes A B :: "'a::cbanach \<Rightarrow>\<^sub>C\<^sub>L'b::complex_normed_vector"
-  assumes "\<And>x. x \<in> G \<Longrightarrow> A *\<^sub>V x = B *\<^sub>V x" and \<open>ccspan G = \<top>\<close>
-  shows "A = B"
-  apply (rule cblinfun_eqI)
-  apply (rule cblinfun_eq_on[where G=G])
-  using assms apply auto
-  by (metis ccspan.rep_eq iso_tuple_UNIV_I top_ccsubspace.rep_eq)
-
 lemma cblinfun_compose_add_left: \<open>(a + b) o\<^sub>C\<^sub>L c = (a o\<^sub>C\<^sub>L c) + (b o\<^sub>C\<^sub>L c)\<close>
   by (simp add: bounded_cbilinear.add_left bounded_cbilinear_cblinfun_compose)
 
 lemma cblinfun_compose_add_right: \<open>a o\<^sub>C\<^sub>L (b + c) = (a o\<^sub>C\<^sub>L b) + (a o\<^sub>C\<^sub>L c)\<close>
   by (simp add: bounded_cbilinear.add_right bounded_cbilinear_cblinfun_compose)
 
 lemma cbilinear_cblinfun_compose[simp]: "cbilinear cblinfun_compose"
   by (auto intro!: clinearI simp add: cbilinear_def bounded_cbilinear.add_left bounded_cbilinear.add_right bounded_cbilinear_cblinfun_compose)
 
 lemma cblinfun_compose_sum_left: \<open>(\<Sum>i\<in>S. g i) o\<^sub>C\<^sub>L x = (\<Sum>i\<in>S. g i o\<^sub>C\<^sub>L x)\<close>
   apply (induction S rule:infinite_finite_induct)
   by (auto simp: cblinfun_compose_add_left)
 
 lemma cblinfun_compose_sum_right: \<open>x o\<^sub>C\<^sub>L (\<Sum>i\<in>S. g i) = (\<Sum>i\<in>S. x o\<^sub>C\<^sub>L g i)\<close>
   apply (induction S rule:infinite_finite_induct)
   by (auto simp: cblinfun_compose_add_right)
 
 lemma cblinfun_compose_minus_right: \<open>a o\<^sub>C\<^sub>L (b - c) = (a o\<^sub>C\<^sub>L b) - (a o\<^sub>C\<^sub>L c)\<close>
   by (simp add: bounded_cbilinear.diff_right bounded_cbilinear_cblinfun_compose)
 lemma cblinfun_compose_minus_left: \<open>(a - b) o\<^sub>C\<^sub>L c = (a o\<^sub>C\<^sub>L c) - (b o\<^sub>C\<^sub>L c)\<close>
   by (simp add: bounded_cbilinear.diff_left bounded_cbilinear_cblinfun_compose)
 
 
 lemma simp_a_oCL_b: \<open>a o\<^sub>C\<^sub>L b = c \<Longrightarrow> a o\<^sub>C\<^sub>L (b o\<^sub>C\<^sub>L d) = c o\<^sub>C\<^sub>L d\<close>
   \<comment> \<open>A convenience lemma to transform simplification rules of the form \<^term>\<open>a o\<^sub>C\<^sub>L b = c\<close>.
      E.g., \<open>simp_a_oCL_b[OF isometryD, simp]\<close> declares a simp-rule for simplifying \<^term>\<open>adj x o\<^sub>C\<^sub>L (x o\<^sub>C\<^sub>L y) = id_cblinfun o\<^sub>C\<^sub>L y\<close>.\<close>
   by (auto simp: cblinfun_compose_assoc)
 
 lemma simp_a_oCL_b': \<open>a o\<^sub>C\<^sub>L b = c \<Longrightarrow> a *\<^sub>V (b *\<^sub>V d) = c *\<^sub>V d\<close>
   \<comment> \<open>A convenience lemma to transform simplification rules of the form \<^term>\<open>a o\<^sub>C\<^sub>L b = c\<close>.
      E.g., \<open>simp_a_oCL_b'[OF isometryD, simp]\<close> declares a simp-rule for simplifying \<^term>\<open>adj x *\<^sub>V x *\<^sub>V y = id_cblinfun *\<^sub>V y\<close>.\<close>
   by auto
 
 lemma cblinfun_compose_uminus_left: \<open>(- a) o\<^sub>C\<^sub>L b = - (a o\<^sub>C\<^sub>L b)\<close>
   by (simp add: bounded_cbilinear.minus_left bounded_cbilinear_cblinfun_compose)
 
 lemma cblinfun_compose_uminus_right: \<open>a o\<^sub>C\<^sub>L (- b) = - (a o\<^sub>C\<^sub>L b)\<close>
   by (simp add: bounded_cbilinear.minus_right bounded_cbilinear_cblinfun_compose)
 
 subsection \<open>Adjoint\<close>
 
 lift_definition
   adj :: "'a::chilbert_space \<Rightarrow>\<^sub>C\<^sub>L 'b::complex_inner \<Rightarrow> 'b \<Rightarrow>\<^sub>C\<^sub>L 'a" ("_*" [99] 100)
   is cadjoint by (fact cadjoint_bounded_clinear)
 
 lemma id_cblinfun_adjoint[simp]: "id_cblinfun* = id_cblinfun"
   apply transfer using cadjoint_id
   by (metis eq_id_iff)
 
 lemma double_adj[simp]: "(A*)* = A"
   apply transfer using double_cadjoint by blast
 
 lemma adj_cblinfun_compose[simp]:
   fixes B::\<open>'a::chilbert_space \<Rightarrow>\<^sub>C\<^sub>L 'b::chilbert_space\<close>
     and A::\<open>'b \<Rightarrow>\<^sub>C\<^sub>L 'c::complex_inner\<close>
   shows "(A o\<^sub>C\<^sub>L B)* =  (B*) o\<^sub>C\<^sub>L (A*)"
 proof transfer
   fix  A :: \<open>'b \<Rightarrow> 'c\<close> and B :: \<open>'a \<Rightarrow> 'b\<close>
   assume \<open>bounded_clinear A\<close> and \<open>bounded_clinear B\<close>
   hence \<open>bounded_clinear (A \<circ> B)\<close>
     by (simp add: comp_bounded_clinear)
-  have \<open>\<langle> (A \<circ> B) u, v \<rangle> = \<langle> u, (B\<^sup>\<dagger> \<circ> A\<^sup>\<dagger>) v \<rangle>\<close>
+  have \<open>((A \<circ> B) u \<bullet>\<^sub>C v) = (u \<bullet>\<^sub>C (B\<^sup>\<dagger> \<circ> A\<^sup>\<dagger>) v)\<close>
     for u v
     by (metis (no_types, lifting) cadjoint_univ_prop \<open>bounded_clinear A\<close> \<open>bounded_clinear B\<close> cinner_commute' comp_def)
   thus \<open>(A \<circ> B)\<^sup>\<dagger> = B\<^sup>\<dagger> \<circ> A\<^sup>\<dagger>\<close>
     using \<open>bounded_clinear (A \<circ> B)\<close>
     by (metis cadjoint_eqI cinner_commute')
 qed
 
 lemma scaleC_adj[simp]: "(a *\<^sub>C A)* = (cnj a) *\<^sub>C (A*)"
   apply transfer
   by (simp add: bounded_clinear.bounded_linear bounded_clinear_def complex_vector.linear_scale scaleC_cadjoint)
 
 lemma scaleR_adj[simp]: "(a *\<^sub>R A)* = a *\<^sub>R (A*)"
   by (simp add: scaleR_scaleC)
 
 lemma adj_plus: \<open>(A + B)* = (A*) + (B*)\<close>
 proof transfer
   fix A B::\<open>'b \<Rightarrow> 'a\<close>
   assume a1: \<open>bounded_clinear A\<close> and a2: \<open>bounded_clinear B\<close>
   define F where \<open>F = (\<lambda>x. (A\<^sup>\<dagger>) x + (B\<^sup>\<dagger>) x)\<close>
   define G where \<open>G = (\<lambda>x. A x + B x)\<close>
   have \<open>bounded_clinear G\<close>
     unfolding G_def
     by (simp add: a1 a2 bounded_clinear_add)
-  moreover have \<open>\<langle>F u,  v\<rangle> = \<langle>u, G v\<rangle>\<close> for u v
+  moreover have \<open>(F u \<bullet>\<^sub>C v) = (u \<bullet>\<^sub>C G v)\<close> for u v
     unfolding F_def G_def
     using cadjoint_univ_prop a1 a2 cinner_add_left
     by (simp add: cadjoint_univ_prop cinner_add_left cinner_add_right)
   ultimately have \<open>F = G\<^sup>\<dagger> \<close>
     using cadjoint_eqI by blast
   thus \<open>(\<lambda>x. A x + B x)\<^sup>\<dagger> = (\<lambda>x. (A\<^sup>\<dagger>) x + (B\<^sup>\<dagger>) x)\<close>
     unfolding F_def G_def
     by auto
 qed
 
 lemma cinner_adj_left:
   fixes G :: "'b::chilbert_space \<Rightarrow>\<^sub>C\<^sub>L 'a::complex_inner"
-  shows \<open>\<langle>G* *\<^sub>V x, y\<rangle> = \<langle>x, G *\<^sub>V y\<rangle>\<close>
+  shows \<open>(G* *\<^sub>V x) \<bullet>\<^sub>C y = x \<bullet>\<^sub>C (G *\<^sub>V y)\<close>
   apply transfer using cadjoint_univ_prop by blast
 
 lemma cinner_adj_right:
   fixes G :: "'b::chilbert_space \<Rightarrow>\<^sub>C\<^sub>L 'a::complex_inner"
-  shows \<open>\<langle>x, G* *\<^sub>V y\<rangle> = \<langle>G *\<^sub>V x, y\<rangle>\<close>
+  shows \<open>x \<bullet>\<^sub>C (G* *\<^sub>V y) = (G *\<^sub>V x) \<bullet>\<^sub>C y\<close>
   apply transfer using cadjoint_univ_prop' by blast
 
 lemma adj_0[simp]: \<open>0* = 0\<close>
   by (metis add_cancel_right_left adj_plus)
 
 lemma norm_adj[simp]: \<open>norm (A*) = norm A\<close>
   for A :: \<open>'b::chilbert_space \<Rightarrow>\<^sub>C\<^sub>L 'c::complex_inner\<close>
 proof (cases \<open>(\<exists>x y :: 'b. x \<noteq> y) \<and> (\<exists>x y :: 'c. x \<noteq> y)\<close>)
   case True
   then have c1: \<open>class.not_singleton TYPE('b)\<close>
     apply intro_classes by simp
   from True have c2: \<open>class.not_singleton TYPE('c)\<close>
     apply intro_classes by simp
   have normA: \<open>norm A = (SUP (\<psi>, \<phi>). cmod (\<psi> \<bullet>\<^sub>C (A *\<^sub>V \<phi>)) / (norm \<psi> * norm \<phi>))\<close>
     apply (rule cinner_sup_norm_cblinfun[internalize_sort \<open>'a::{complex_normed_vector,not_singleton}\<close>])
      apply (rule complex_normed_vector_axioms)
     by (rule c1)
   have normAadj: \<open>norm (A*) = (SUP (\<psi>, \<phi>). cmod (\<psi> \<bullet>\<^sub>C (A* *\<^sub>V \<phi>)) / (norm \<psi> * norm \<phi>))\<close>
     apply (rule cinner_sup_norm_cblinfun[internalize_sort \<open>'a::{complex_normed_vector,not_singleton}\<close>])
      apply (rule complex_normed_vector_axioms)
     by (rule c2)
 
   have \<open>norm (A*) = (SUP (\<psi>, \<phi>). cmod (\<phi> \<bullet>\<^sub>C (A *\<^sub>V \<psi>)) / (norm \<psi> * norm \<phi>))\<close>
     unfolding normAadj
     apply (subst cinner_adj_right)
     apply (subst cinner_commute)
     apply (subst complex_mod_cnj)
     by rule
   also have \<open>\<dots> = Sup ((\<lambda>(\<psi>, \<phi>). cmod (\<phi> \<bullet>\<^sub>C (A *\<^sub>V \<psi>)) / (norm \<psi> * norm \<phi>)) ` prod.swap ` UNIV)\<close>
     by auto
   also have \<open>\<dots> = (SUP (\<phi>, \<psi>). cmod (\<phi> \<bullet>\<^sub>C (A *\<^sub>V \<psi>)) / (norm \<psi> * norm \<phi>))\<close>
     apply (subst image_image)
     by auto
   also have \<open>\<dots> = norm A\<close>
     unfolding normA
     by (simp add: mult.commute)
   finally show ?thesis
     by -
 next
   case False
   then consider (b) \<open>\<And>x::'b. x = 0\<close> | (c) \<open>\<And>x::'c. x = 0\<close>
     by auto
   then have \<open>A = 0\<close>
     apply (cases; transfer)
      apply (metis (full_types) bounded_clinear_def complex_vector.linear_0)
     by auto
   then show \<open>norm (A*) = norm A\<close>
     by simp
 qed
 
 
 lemma antilinear_adj[simp]: \<open>antilinear adj\<close>
   apply (rule antilinearI) by (auto simp add: adj_plus)
 
 lemma bounded_antilinear_adj[bounded_antilinear, simp]: \<open>bounded_antilinear adj\<close>
   by (auto intro!: antilinearI exI[of _ 1] simp: bounded_antilinear_def bounded_antilinear_axioms_def adj_plus)
 
 lemma adjoint_eqI:
   fixes G:: \<open>'b::chilbert_space \<Rightarrow>\<^sub>C\<^sub>L 'a::complex_inner\<close>
     and F:: \<open>'a \<Rightarrow>\<^sub>C\<^sub>L 'b\<close>
-  assumes \<open>\<And>x y. \<langle>(cblinfun_apply F) x, y\<rangle> = \<langle>x, (cblinfun_apply G) y\<rangle>\<close>
+  assumes \<open>\<And>x y. ((cblinfun_apply F) x \<bullet>\<^sub>C y) = (x \<bullet>\<^sub>C (cblinfun_apply G) y)\<close>
   shows \<open>F = G*\<close>
   using assms apply transfer using cadjoint_eqI by auto
 
 lemma adj_uminus: \<open>(-A)* = - (A*)\<close>
   apply (rule adjoint_eqI[symmetric])
   by (simp add: cblinfun.minus_left cinner_adj_left)
 
 lemma cinner_real_hermiteanI:
   \<comment> \<open>Prop. II.2.12 in @{cite conway2013course}\<close>
   assumes \<open>\<And>\<psi>. cinner \<psi> (A *\<^sub>V \<psi>) \<in> \<real>\<close>
   shows \<open>A = A*\<close>
 proof -
   { fix g h :: 'a
     {
       fix \<alpha> :: complex
       have \<open>cinner h (A h) + cnj \<alpha> *\<^sub>C cinner g (A h) + \<alpha> *\<^sub>C cinner h (A g) + (abs \<alpha>)\<^sup>2 * cinner g (A g)
         = cinner (h + \<alpha> *\<^sub>C g) (A *\<^sub>V (h + \<alpha> *\<^sub>C g))\<close> (is \<open>?sum4 = _\<close>)
         apply (auto simp: cinner_add_right cinner_add_left cblinfun.add_right cblinfun.scaleC_right ring_class.ring_distribs)
         by (metis cnj_x_x mult.commute)
       also have \<open>\<dots> \<in> \<real>\<close>
         using assms by auto
       finally have \<open>?sum4 = cnj ?sum4\<close>
         using Reals_cnj_iff by fastforce
       then have \<open>cnj \<alpha> *\<^sub>C cinner g (A h) + \<alpha> *\<^sub>C cinner h (A g)
             = \<alpha> *\<^sub>C cinner (A h) g + cnj \<alpha> *\<^sub>C cinner (A g) h\<close>
         using Reals_cnj_iff abs_complex_real assms by force
       also have \<open>\<dots> = \<alpha> *\<^sub>C cinner h (A* *\<^sub>V g) + cnj \<alpha> *\<^sub>C cinner g (A* *\<^sub>V h)\<close>
         by (simp add: cinner_adj_right)
       finally have \<open>cnj \<alpha> *\<^sub>C cinner g (A h) + \<alpha> *\<^sub>C cinner h (A g) = \<alpha> *\<^sub>C cinner h (A* *\<^sub>V g) + cnj \<alpha> *\<^sub>C cinner g (A* *\<^sub>V h)\<close>
         by -
     }
     from this[where \<alpha>2=1] this[where \<alpha>2=\<i>]
     have 1: \<open>cinner g (A h) + cinner h (A g) = cinner h (A* *\<^sub>V g) + cinner g (A* *\<^sub>V h)\<close>
       and i: \<open>- \<i> * cinner g (A h) + \<i> *\<^sub>C cinner h (A g) =  \<i> *\<^sub>C cinner h (A* *\<^sub>V g) - \<i> *\<^sub>C cinner g (A* *\<^sub>V h)\<close>
       by auto
     from arg_cong2[OF 1 arg_cong[OF i, where f=\<open>(*) (-\<i>)\<close>], where f=plus]
     have \<open>cinner h (A g) = cinner h (A* *\<^sub>V g)\<close>
       by (auto simp: ring_class.ring_distribs)
   }
   then show "A = A*"
     by (simp add: adjoint_eqI cinner_adj_right)
 qed
 
 
 lemma norm_AAadj[simp]: \<open>norm (A o\<^sub>C\<^sub>L A*) = (norm A)\<^sup>2\<close> for A :: \<open>'a::chilbert_space \<Rightarrow>\<^sub>C\<^sub>L 'b::{complex_inner}\<close>
 proof (cases \<open>class.not_singleton TYPE('b)\<close>)
   case True
   then have [simp]: \<open>class.not_singleton TYPE('b)\<close>
     by -
   have 1: \<open>(norm A)\<^sup>2 * \<epsilon> \<le> norm (A o\<^sub>C\<^sub>L A*)\<close> if \<open>\<epsilon> < 1\<close> and \<open>\<epsilon> \<ge> 0\<close> for \<epsilon>
   proof -
     obtain \<psi> where \<psi>: \<open>norm ((A*) *\<^sub>V \<psi>) \<ge> norm (A*) * sqrt \<epsilon>\<close> and [simp]: \<open>norm \<psi> = 1\<close>
       apply atomize_elim
       apply (rule cblinfun_norm_approx_witness_mult[internalize_sort' 'a])
       using \<open>\<epsilon> < 1\<close> by (auto intro: complex_normed_vector_class.complex_normed_vector_axioms)
     have \<open>complex_of_real ((norm A)\<^sup>2 * \<epsilon>) = (norm (A*) * sqrt \<epsilon>)\<^sup>2\<close>
       by (simp add: ordered_field_class.sign_simps(23) that(2))
     also have \<open>\<dots> \<le> (norm ((A* *\<^sub>V \<psi>)))\<^sup>2\<close>
       apply (rule complex_of_real_mono)
       using \<psi> apply (rule power_mono)
       using \<open>\<epsilon> \<ge> 0\<close> by auto
     also have \<open>\<dots> \<le> cinner (A* *\<^sub>V \<psi>) (A* *\<^sub>V \<psi>)\<close>
       by (auto simp flip: power2_norm_eq_cinner)
     also have \<open>\<dots> = cinner \<psi> (A *\<^sub>V A* *\<^sub>V \<psi>)\<close>
       by (simp add: cinner_adj_left)
     also have \<open>\<dots> = cinner \<psi> ((A o\<^sub>C\<^sub>L A*) *\<^sub>V \<psi>)\<close>
       by auto
     also have \<open>\<dots> \<le> norm (A o\<^sub>C\<^sub>L A*)\<close>
       using \<open>norm \<psi> = 1\<close>
       by (smt (verit, best) Im_complex_of_real Re_complex_of_real \<open>(A* *\<^sub>V \<psi>) \<bullet>\<^sub>C (A* *\<^sub>V \<psi>) = \<psi> \<bullet>\<^sub>C (A *\<^sub>V A* *\<^sub>V \<psi>)\<close> \<open>\<psi> \<bullet>\<^sub>C (A *\<^sub>V A* *\<^sub>V \<psi>) = \<psi> \<bullet>\<^sub>C ((A o\<^sub>C\<^sub>L A*) *\<^sub>V \<psi>)\<close> cdot_square_norm cinner_ge_zero cmod_Re complex_inner_class.Cauchy_Schwarz_ineq2 less_eq_complex_def mult_cancel_left1 mult_cancel_right1 norm_cblinfun)
     finally show ?thesis
       by (auto simp: less_eq_complex_def)
   qed
   then have 1: \<open>(norm A)\<^sup>2 \<le> norm (A o\<^sub>C\<^sub>L A*)\<close>
     by (metis field_le_mult_one_interval less_eq_real_def ordered_field_class.sign_simps(5))
 
   have 2: \<open>norm (A o\<^sub>C\<^sub>L A*) \<le> (norm A)\<^sup>2\<close>
   proof (rule norm_cblinfun_bound)
     show \<open>0 \<le> (norm A)\<^sup>2\<close> by simp
     fix \<psi>
     have \<open>norm ((A o\<^sub>C\<^sub>L A*) *\<^sub>V \<psi>) = norm (A *\<^sub>V A* *\<^sub>V \<psi>)\<close>
       by auto
     also have \<open>\<dots> \<le> norm A * norm (A* *\<^sub>V \<psi>)\<close>
       by (simp add: norm_cblinfun)
     also have \<open>\<dots> \<le> norm A * norm (A*) * norm \<psi>\<close>
       by (metis mult.assoc norm_cblinfun norm_imp_pos_and_ge ordered_comm_semiring_class.comm_mult_left_mono)
     also have \<open>\<dots> = (norm A)\<^sup>2 * norm \<psi>\<close>
       by (simp add: power2_eq_square)
     finally show \<open>norm ((A o\<^sub>C\<^sub>L A*) *\<^sub>V \<psi>) \<le> (norm A)\<^sup>2 * norm \<psi>\<close>
       by -
   qed
 
   from 1 2 show ?thesis by simp
 next
   case False
   then have [simp]: \<open>class.CARD_1 TYPE('b)\<close>
     by (rule not_singleton_vs_CARD_1)
   have \<open>A = 0\<close>
     apply (rule cblinfun_to_CARD_1_0[internalize_sort' 'b])
     by (auto intro: complex_normed_vector_class.complex_normed_vector_axioms)
   then show ?thesis
     by auto
 qed
 
 lemma sum_adj: \<open>(sum a F)* = sum (\<lambda>i. (a i)*) F\<close>
   apply (induction rule:infinite_finite_induct)
   by (auto simp add: adj_plus)
 
 lemma has_sum_adj:
   assumes \<open>has_sum f I x\<close>
   shows \<open>has_sum (\<lambda>x. adj (f x)) I (adj x)\<close>
   apply (rule has_sum_comm_additive[where f=adj, unfolded o_def])
   apply (simp add: antilinear.axioms(1))
   apply (metis (no_types, lifting) LIM_eq adj_plus adj_uminus norm_adj uminus_add_conv_diff)
   by (simp add: assms)
 
 lemma adj_minus: \<open>(A - B)* = (A*) - (B*)\<close>
   by (metis add_implies_diff adj_plus diff_add_cancel)
 
 lemma cinner_hermitian_real: \<open>x \<bullet>\<^sub>C (A *\<^sub>V x) \<in> \<real>\<close> if \<open>A* = A\<close>
   by (metis Reals_cnj_iff cinner_adj_right cinner_commute' that)
 
 lemma adj_inject: \<open>adj a = adj b \<longleftrightarrow> a = b\<close>
   by (metis (no_types, opaque_lifting) adj_minus eq_iff_diff_eq_0 norm_adj norm_eq_zero)
 
 lemma norm_AadjA[simp]: \<open>norm (A* o\<^sub>C\<^sub>L A) = (norm A)\<^sup>2\<close> for A :: \<open>'a::chilbert_space \<Rightarrow>\<^sub>C\<^sub>L 'b::chilbert_space\<close>
   by (metis double_adj norm_AAadj norm_adj)
 
 subsection \<open>Powers of operators\<close>
 
 lift_definition cblinfun_power :: \<open>'a::complex_normed_vector \<Rightarrow>\<^sub>C\<^sub>L 'a \<Rightarrow> nat \<Rightarrow> 'a \<Rightarrow>\<^sub>C\<^sub>L 'a\<close> is
   \<open>\<lambda>(a::'a\<Rightarrow>'a) n. a ^^ n\<close>
   apply (rename_tac f n, induct_tac n, auto simp: Nat.funpow_code_def)
   by (simp add: bounded_clinear_compose)
 
 lemma cblinfun_power_0[simp]: \<open>cblinfun_power A 0 = id_cblinfun\<close> 
   apply transfer by auto
 
 lemma cblinfun_power_Suc': \<open>cblinfun_power A (Suc n) = A o\<^sub>C\<^sub>L cblinfun_power A n\<close> 
   apply transfer by auto
 
 lemma cblinfun_power_Suc: \<open>cblinfun_power A (Suc n) = cblinfun_power A n o\<^sub>C\<^sub>L A\<close>
   apply (induction n)
   by (auto simp: cblinfun_power_Suc' simp flip:  cblinfun_compose_assoc)
 
 lemma cblinfun_power_compose[simp]: \<open>cblinfun_power A n o\<^sub>C\<^sub>L cblinfun_power A m = cblinfun_power A (n+m)\<close>
   apply (induction n)
   by (auto simp: cblinfun_power_Suc' cblinfun_compose_assoc)
 
 lemma cblinfun_power_scaleC: \<open>cblinfun_power (c *\<^sub>C a) n = c^n *\<^sub>C cblinfun_power a n\<close>
   apply (induction n)
   by (auto simp: cblinfun_power_Suc)
 
 lemma cblinfun_power_scaleR: \<open>cblinfun_power (c *\<^sub>R a) n = c^n *\<^sub>R cblinfun_power a n\<close>
   apply (induction n)
   by (auto simp: cblinfun_power_Suc)
 
 lemma cblinfun_power_uminus: \<open>cblinfun_power (-a) n = (-1)^n *\<^sub>R cblinfun_power a n\<close>
   apply (subst asm_rl[of \<open>-a = (-1) *\<^sub>R a\<close>])
    apply simp
   by (rule cblinfun_power_scaleR)
 
 lemma cblinfun_power_adj: \<open>(cblinfun_power S n)* = cblinfun_power (S*) n\<close>
   apply (induction n)
    apply simp
   apply (subst cblinfun_power_Suc)
   apply (subst cblinfun_power_Suc')
   by auto
 
 subsection \<open>Unitaries / isometries\<close>
 
 
 definition isometry::\<open>'a::chilbert_space \<Rightarrow>\<^sub>C\<^sub>L 'b::complex_inner \<Rightarrow> bool\<close> where
   \<open>isometry U \<longleftrightarrow> U* o\<^sub>C\<^sub>L U = id_cblinfun\<close>
 
 definition unitary::\<open>'a::chilbert_space \<Rightarrow>\<^sub>C\<^sub>L 'b::complex_inner \<Rightarrow> bool\<close> where
   \<open>unitary U \<longleftrightarrow> (U* o\<^sub>C\<^sub>L U  = id_cblinfun) \<and> (U o\<^sub>C\<^sub>L U* = id_cblinfun)\<close>
 
 lemma unitary_twosided_isometry: "unitary U \<longleftrightarrow> isometry U \<and> isometry (U*)"
   unfolding unitary_def isometry_def by simp
 
 lemma isometryD[simp]: "isometry U \<Longrightarrow> U* o\<^sub>C\<^sub>L U = id_cblinfun"
   unfolding isometry_def by simp
 
 (* Not [simp] because isometryD[simp] + unitary_isometry[simp] already have the same effect *)
 lemma unitaryD1: "unitary U \<Longrightarrow> U* o\<^sub>C\<^sub>L U = id_cblinfun"
   unfolding unitary_def by simp
 
 lemma unitaryD2[simp]: "unitary U \<Longrightarrow> U o\<^sub>C\<^sub>L U* = id_cblinfun"
   unfolding unitary_def by simp
 
 lemma unitary_isometry[simp]: "unitary U \<Longrightarrow> isometry U"
   unfolding unitary_def isometry_def by simp
 
 lemma unitary_adj[simp]: "unitary (U*) = unitary U"
   unfolding unitary_def by auto
 
 lemma isometry_cblinfun_compose[simp]:
   assumes "isometry A" and "isometry B"
   shows "isometry (A o\<^sub>C\<^sub>L B)"
 proof-
   have "B* o\<^sub>C\<^sub>L A* o\<^sub>C\<^sub>L (A o\<^sub>C\<^sub>L B) = id_cblinfun" if "A* o\<^sub>C\<^sub>L A = id_cblinfun" and "B* o\<^sub>C\<^sub>L B = id_cblinfun"
     using that
     by (smt (verit, del_insts) adjoint_eqI cblinfun_apply_cblinfun_compose cblinfun_id_cblinfun_apply)
   thus ?thesis
     using assms unfolding isometry_def by simp
 qed
 
 lemma unitary_cblinfun_compose[simp]: "unitary (A o\<^sub>C\<^sub>L B)"
   if "unitary A" and "unitary B"
   using that
   by (smt (z3) adj_cblinfun_compose cblinfun_compose_assoc cblinfun_compose_id_right double_adj isometryD isometry_cblinfun_compose unitary_def unitary_isometry)
 
 lemma unitary_surj:
   assumes "unitary U"
   shows "surj (cblinfun_apply U)"
   apply (rule surjI[where f=\<open>cblinfun_apply (U*)\<close>])
   using assms unfolding unitary_def apply transfer
   using comp_eq_dest_lhs by force
 
 lemma unitary_id[simp]: "unitary id_cblinfun"
   by (simp add: unitary_def)
 
 lemma orthogonal_on_basis_is_isometry:
   assumes spanB: \<open>ccspan B = \<top>\<close>
   assumes orthoU: \<open>\<And>b c. b\<in>B \<Longrightarrow> c\<in>B \<Longrightarrow> cinner (U *\<^sub>V b) (U *\<^sub>V c) = cinner b c\<close>
   shows \<open>isometry U\<close>
 proof -
   have [simp]: \<open>b \<in> closure (cspan B)\<close> for b
     using spanB apply transfer by simp
   have *: \<open>cinner (U* *\<^sub>V U *\<^sub>V \<psi>) \<phi> = cinner \<psi> \<phi>\<close> if \<open>\<psi>\<in>B\<close> and \<open>\<phi>\<in>B\<close> for \<psi> \<phi>
     by (simp add: cinner_adj_left orthoU that(1) that(2))
   have *: \<open>cinner (U* *\<^sub>V U *\<^sub>V \<psi>) \<phi> = cinner \<psi> \<phi>\<close> if \<open>\<psi>\<in>B\<close> for \<psi> \<phi>
     apply (rule bounded_clinear_eq_on[where t=\<phi> and G=B])
     using bounded_clinear_cinner_right *[OF that]
     by auto
   have \<open>U* *\<^sub>V U *\<^sub>V \<phi> = \<phi>\<close> if \<open>\<phi>\<in>B\<close> for \<phi>
     apply (rule cinner_extensionality)
     apply (subst cinner_eq_flip)
     by (simp add: * that)
   then have \<open>U* o\<^sub>C\<^sub>L U = id_cblinfun\<close>
     by (metis cblinfun_apply_cblinfun_compose cblinfun_eq_gen_eqI cblinfun_id_cblinfun_apply spanB)
   then show \<open>isometry U\<close>
     using isometry_def by blast
 qed
 
 lemma isometry_preserves_norm: \<open>isometry U \<Longrightarrow> norm (U *\<^sub>V \<psi>) = norm \<psi>\<close>
   by (metis (no_types, lifting) cblinfun_apply_cblinfun_compose cblinfun_id_cblinfun_apply cinner_adj_right cnorm_eq isometryD)
 
 lemma norm_isometry_compose: 
   assumes \<open>isometry U\<close>
   shows \<open>norm (U o\<^sub>C\<^sub>L A) = norm A\<close>
 proof -
   have *: \<open>norm (U *\<^sub>V A *\<^sub>V \<psi>) = norm (A *\<^sub>V \<psi>)\<close> for \<psi>
     by (smt (verit, ccfv_threshold) assms cblinfun_apply_cblinfun_compose cinner_adj_right cnorm_eq id_cblinfun_apply isometryD)
 
   have \<open>norm (U o\<^sub>C\<^sub>L A) = (SUP \<psi>. norm (U *\<^sub>V A *\<^sub>V \<psi>) / norm \<psi>)\<close>
     unfolding norm_cblinfun_Sup by auto
   also have \<open>\<dots> = (SUP \<psi>. norm (A *\<^sub>V \<psi>) / norm \<psi>)\<close>
     using * by auto
   also have \<open>\<dots> = norm A\<close>
     unfolding norm_cblinfun_Sup by auto
   finally show ?thesis
     by simp
 qed
 
 lemma norm_isometry:
   fixes U :: \<open>'a::{chilbert_space,not_singleton} \<Rightarrow>\<^sub>C\<^sub>L 'b::complex_inner\<close>
   assumes \<open>isometry U\<close>
   shows \<open>norm U = 1\<close>
   apply (subst asm_rl[of \<open>U = U o\<^sub>C\<^sub>L id_cblinfun\<close>], simp)
   apply (subst norm_isometry_compose, simp add: assms)
   by simp
 
 lemma norm_preserving_isometry: \<open>isometry U\<close> if \<open>\<And>\<psi>. norm (U *\<^sub>V \<psi>) = norm \<psi>\<close>
   by (smt (verit, ccfv_SIG) cblinfun_cinner_eqI cblinfun_id_cblinfun_apply cinner_adj_right cnorm_eq isometry_def simp_a_oCL_b' that)
 
 subsection \<open>Product spaces\<close>
 
 lift_definition cblinfun_left :: \<open>'a::complex_normed_vector \<Rightarrow>\<^sub>C\<^sub>L ('a\<times>'b::complex_normed_vector)\<close> is \<open>(\<lambda>x. (x,0))\<close>
   by (auto intro!: bounded_clinearI[where K=1])
 lift_definition cblinfun_right :: \<open>'b::complex_normed_vector \<Rightarrow>\<^sub>C\<^sub>L ('a::complex_normed_vector\<times>'b)\<close> is \<open>(\<lambda>x. (0,x))\<close>
   by (auto intro!: bounded_clinearI[where K=1])
 
 lemma isometry_cblinfun_left[simp]: \<open>isometry cblinfun_left\<close>
   apply (rule orthogonal_on_basis_is_isometry[of some_chilbert_basis])
    apply simp
   apply transfer
   by simp
 
 lemma isometry_cblinfun_right[simp]: \<open>isometry cblinfun_right\<close>
   apply (rule orthogonal_on_basis_is_isometry[of some_chilbert_basis])
    apply simp
   apply transfer
   by simp
 
 lemma cblinfun_left_right_ortho[simp]: \<open>cblinfun_left* o\<^sub>C\<^sub>L cblinfun_right = 0\<close>
 proof -
   have \<open>x \<bullet>\<^sub>C ((cblinfun_left* o\<^sub>C\<^sub>L cblinfun_right) *\<^sub>V y) = 0\<close> for x :: 'b and y :: 'a
     apply (simp add: cinner_adj_right)
     apply transfer
     by auto
   then show ?thesis
     by (metis cblinfun.zero_left cblinfun_eqI cinner_eq_zero_iff)
 qed
 
 lemma cblinfun_right_left_ortho[simp]: \<open>cblinfun_right* o\<^sub>C\<^sub>L cblinfun_left = 0\<close>
 proof -
   have \<open>x \<bullet>\<^sub>C ((cblinfun_right* o\<^sub>C\<^sub>L cblinfun_left) *\<^sub>V y) = 0\<close> for x :: 'b and y :: 'a
     apply (simp add: cinner_adj_right)
     apply transfer
     by auto
   then show ?thesis
     by (metis cblinfun.zero_left cblinfun_eqI cinner_eq_zero_iff)
 qed
 
 lemma cblinfun_left_apply[simp]: \<open>cblinfun_left *\<^sub>V \<psi> = (\<psi>,0)\<close>
   apply transfer by simp
 
 lemma cblinfun_left_adj_apply[simp]: \<open>cblinfun_left* *\<^sub>V \<psi> = fst \<psi>\<close>
   apply (cases \<psi>)
   by (auto intro!: cinner_extensionality[of \<open>_ *\<^sub>V _\<close>] simp: cinner_adj_right)
 
 lemma cblinfun_right_apply[simp]: \<open>cblinfun_right *\<^sub>V \<psi> = (0,\<psi>)\<close>
   apply transfer by simp
 
 lemma cblinfun_right_adj_apply[simp]: \<open>cblinfun_right* *\<^sub>V \<psi> = snd \<psi>\<close>
   apply (cases \<psi>)
   by (auto intro!: cinner_extensionality[of \<open>_ *\<^sub>V _\<close>] simp: cinner_adj_right)
 
 lift_definition ccsubspace_Times :: \<open>'a::complex_normed_vector ccsubspace \<Rightarrow> 'b::complex_normed_vector ccsubspace \<Rightarrow> ('a\<times>'b) ccsubspace\<close> is
   Product_Type.Times
 proof -
   fix S :: \<open>'a set\<close> and T :: \<open>'b set\<close>
   assume [simp]: \<open>closed_csubspace S\<close> \<open>closed_csubspace T\<close>
   have \<open>csubspace (S \<times> T)\<close>
     by (simp add: complex_vector.subspace_Times)
   moreover have \<open>closed (S \<times> T)\<close>
     by (simp add: closed_Times closed_csubspace.closed)
   ultimately show \<open>closed_csubspace (S \<times> T)\<close>
     by (rule closed_csubspace.intro)
 qed
 
 
 lemma ccspan_Times: \<open>ccspan (S \<times> T) = ccsubspace_Times (ccspan S) (ccspan T)\<close> if \<open>0 \<in> S\<close> and \<open>0 \<in> T\<close>
 proof (transfer fixing: S T)
   from that have \<open>closure (cspan (S \<times> T)) = closure (cspan S \<times> cspan T)\<close>
     by (simp add: cspan_Times)
   also have \<open>\<dots> = closure (cspan S) \<times> closure (cspan T)\<close>
     using closure_Times by blast
   finally   show \<open>closure (cspan (S \<times> T)) = closure (cspan S) \<times> closure (cspan T)\<close>
     by -
 qed
 
 lemma ccspan_Times_sing1: \<open>ccspan ({0::'a::complex_normed_vector} \<times> B) = ccsubspace_Times 0 (ccspan B)\<close>
 proof (transfer fixing: B)
   have \<open>closure (cspan ({0::'a} \<times> B)) = closure ({0} \<times> cspan B)\<close>
     by (simp add: complex_vector.span_Times_sing1)
   also have \<open>\<dots> = closure {0} \<times> closure (cspan B)\<close>
     using closure_Times by blast
   also have \<open>\<dots> = {0} \<times> closure (cspan B)\<close>
     by simp
   finally show \<open>closure (cspan ({0::'a} \<times> B)) = {0} \<times> closure (cspan B)\<close>
     by -
 qed
 
 lemma ccspan_Times_sing2: \<open>ccspan (B \<times> {0::'a::complex_normed_vector}) = ccsubspace_Times (ccspan B) 0\<close>
 proof (transfer fixing: B)
   have \<open>closure (cspan (B \<times> {0::'a})) = closure (cspan B \<times> {0})\<close>
     by (simp add: complex_vector.span_Times_sing2)
   also have \<open>\<dots> = closure (cspan B) \<times> closure {0}\<close>
     using closure_Times by blast
   also have \<open>\<dots> = closure (cspan B) \<times> {0}\<close>
     by simp
   finally show \<open>closure (cspan (B \<times> {0::'a})) = closure (cspan B) \<times> {0}\<close>
     by -
 qed
 
 lemma ccsubspace_Times_sup: \<open>sup (ccsubspace_Times A B) (ccsubspace_Times C D) = ccsubspace_Times (sup A C) (sup B D)\<close>
 proof transfer
   fix A C :: \<open>'a set\<close> and B D :: \<open>'b set\<close>
   have \<open>A \<times> B +\<^sub>M C \<times> D = closure ((A \<times> B) + (C \<times> D))\<close>
     using closed_sum_def by blast
   also have \<open>\<dots> = closure ((A + C) \<times> (B + D))\<close>
     by (simp add: set_Times_plus_distrib)
   also have \<open>\<dots> = closure (A + C) \<times> closure (B + D)\<close>
     by (simp add: closure_Times)
   also have \<open>\<dots> = (A +\<^sub>M C) \<times> (B +\<^sub>M D)\<close>
     by (simp add: closed_sum_def)
   finally show \<open>A \<times> B +\<^sub>M C \<times> D = (A +\<^sub>M C) \<times> (B +\<^sub>M D)\<close>
     by -
 qed
 
 lemma ccsubspace_Times_top_top[simp]: \<open>ccsubspace_Times top top = top\<close>
   apply transfer
   by simp
 
 lemma is_onb_prod:
   assumes \<open>is_onb B\<close> \<open>is_onb B'\<close>
   shows \<open>is_onb ((B \<times> {0}) \<union> ({0} \<times> B'))\<close>
 proof -
   from assms
   have 1: \<open>is_ortho_set ((B \<times> {0}) \<union> ({0} \<times> B'))\<close>
     unfolding is_ortho_set_def
     apply (auto simp: is_onb_def is_ortho_set_def zero_prod_def)
     by (meson is_onb_def is_ortho_set_def)+
 
   have 2: \<open>(l, r) \<in> B \<times> {0} \<Longrightarrow> norm (l, r) = 1\<close> for l :: 'a and r :: 'b
     using \<open>is_onb B\<close> is_onb_def by auto
 
   have 3: \<open>(l, r) \<in> {0} \<times> B' \<Longrightarrow> norm (l, r) = 1\<close> for l :: 'a and r :: 'b
     using \<open>is_onb B'\<close> is_onb_def by auto
 
   have [simp]: \<open>ccspan B = top\<close> \<open>ccspan B' = top\<close>
     using assms is_onb_def by auto
 
   have 4: \<open>ccspan ((B \<times> {0}) \<union> ({0} \<times> B')) = top\<close>
     by (auto simp: ccspan_Times_sing1 ccspan_Times_sing2 ccsubspace_Times_sup simp flip: ccspan_union)
 
   from 1 2 3 4
   show \<open>is_onb ((B \<times> {0}) \<union> ({0} \<times> B'))\<close>
     by (auto simp add: is_onb_def)
 qed
 
 subsection \<open>Images\<close>
 
-
-(* Closure is necessary. See email 47a3bb3d-3cc3-0934-36eb-3ef0f7b70a85@ut.ee *)
+text \<open>The following definition defines the image of a closed subspace \<^term>\<open>S\<close> under a bounded operator \<^term>\<open>A\<close>.
+We do not define that image as the image of \<^term>\<open>A\<close> seen as a function (\<^term>\<open>A ` S\<close>) but as the topological closure of that image.
+This is because \<^term>\<open>A ` S\<close> might in general not be closed.
+
+For example, if $e_i$ ($i\in\mathbb N$) form an orthonormal basis, and $A$ maps $e_i$ to $e_i/i$, 
+then all $e_i$ are in \<^term>\<open>A ` S\<close>, so the closure of \<^term>\<open>A ` S\<close> is the whole space.
+However, $\sum_i e_i/i$ is not in \<^term>\<open>A ` S\<close> because its preimage would have to be $\sum_i e_i$ which does not converge.
+So \<^term>\<open>A ` S\<close> does not contain the whole space, hence it is not closed.\<close>
+
 lift_definition cblinfun_image :: \<open>'a::complex_normed_vector \<Rightarrow>\<^sub>C\<^sub>L 'b::complex_normed_vector
 \<Rightarrow> 'a ccsubspace \<Rightarrow> 'b ccsubspace\<close>  (infixr "*\<^sub>S" 70)
   is "\<lambda>A S. closure (A ` S)"
   using  bounded_clinear_def closed_closure  closed_csubspace.intro
   by (simp add: bounded_clinear_def complex_vector.linear_subspace_image closure_is_closed_csubspace)
 
 lemma cblinfun_image_mono:
   assumes a1: "S \<le> T"
   shows "A *\<^sub>S S \<le> A *\<^sub>S T"
   using a1
   by (simp add: cblinfun_image.rep_eq closure_mono image_mono less_eq_ccsubspace.rep_eq)
 
 lemma cblinfun_image_0[simp]:
   shows "U *\<^sub>S 0 = 0"
   thm zero_ccsubspace_def
   apply transfer
   by (simp add: bounded_clinear_def complex_vector.linear_0)
 
 lemma cblinfun_image_bot[simp]: "U *\<^sub>S bot = bot"
   using cblinfun_image_0 by auto
 
 lemma cblinfun_image_sup[simp]:
   fixes A B :: \<open>'a::chilbert_space ccsubspace\<close> and U :: "'a \<Rightarrow>\<^sub>C\<^sub>L'b::chilbert_space"
   shows \<open>U *\<^sub>S (sup A B) = sup (U *\<^sub>S A) (U *\<^sub>S B)\<close>
   apply transfer using bounded_clinear.bounded_linear closure_image_closed_sum by blast
 
 lemma scaleC_cblinfun_image[simp]:
   fixes A :: \<open>'a::chilbert_space \<Rightarrow>\<^sub>C\<^sub>L 'b :: chilbert_space\<close>
     and S :: \<open>'a ccsubspace\<close> and \<alpha> :: complex
   shows \<open>(\<alpha> *\<^sub>C A) *\<^sub>S S  = \<alpha> *\<^sub>C (A *\<^sub>S S)\<close>
 proof-
   have \<open>closure ( ( ((*\<^sub>C) \<alpha>) \<circ> (cblinfun_apply A) ) ` space_as_set S) =
    ((*\<^sub>C) \<alpha>) ` (closure (cblinfun_apply A ` space_as_set S))\<close>
     by (metis closure_scaleC image_comp)
   hence \<open>(closure (cblinfun_apply (\<alpha> *\<^sub>C A) ` space_as_set S)) =
    ((*\<^sub>C) \<alpha>) ` (closure (cblinfun_apply A ` space_as_set S))\<close>
     by (metis (mono_tags, lifting) comp_apply image_cong scaleC_cblinfun.rep_eq)
   hence \<open>Abs_clinear_space (closure (cblinfun_apply (\<alpha> *\<^sub>C A) ` space_as_set S)) =
             \<alpha> *\<^sub>C Abs_clinear_space (closure (cblinfun_apply A ` space_as_set S))\<close>
     by (metis space_as_set_inverse cblinfun_image.rep_eq scaleC_ccsubspace.rep_eq)
   have x1: "Abs_clinear_space (closure ((*\<^sub>V) (\<alpha> *\<^sub>C A) ` space_as_set S)) =
             \<alpha> *\<^sub>C Abs_clinear_space (closure ((*\<^sub>V) A ` space_as_set S))"
     using \<open>Abs_clinear_space (closure (cblinfun_apply (\<alpha> *\<^sub>C A) ` space_as_set S)) =
             \<alpha> *\<^sub>C Abs_clinear_space (closure (cblinfun_apply A ` space_as_set S))\<close>
     by blast
   show ?thesis
     unfolding cblinfun_image_def using x1 by force
 qed
 
 lemma cblinfun_image_id[simp]:
   "id_cblinfun *\<^sub>S \<psi> = \<psi>"
   apply transfer
   by (simp add: closed_csubspace.closed)
 
 lemma cblinfun_compose_image:
   \<open>(A o\<^sub>C\<^sub>L B) *\<^sub>S S =  A *\<^sub>S (B *\<^sub>S S)\<close>
   apply transfer unfolding image_comp[symmetric]
   apply (rule closure_bounded_linear_image_subset_eq[symmetric])
   by (simp add: bounded_clinear.bounded_linear)
 
 lemmas cblinfun_assoc_left = cblinfun_compose_assoc[symmetric] cblinfun_compose_image[symmetric]
   add.assoc[where ?'a="'a::chilbert_space \<Rightarrow>\<^sub>C\<^sub>L 'b::chilbert_space", symmetric]
 lemmas cblinfun_assoc_right = cblinfun_compose_assoc cblinfun_compose_image
   add.assoc[where ?'a="'a::chilbert_space \<Rightarrow>\<^sub>C\<^sub>L 'b::chilbert_space"]
 
 lemma cblinfun_image_INF_leq[simp]:
   fixes U :: "'b::complex_normed_vector \<Rightarrow>\<^sub>C\<^sub>L 'c::cbanach"
     and V :: "'a \<Rightarrow> 'b ccsubspace"
   shows \<open>U *\<^sub>S (INF i. V i) \<le> (INF i. U *\<^sub>S (V i))\<close>
   apply transfer
   by (simp add: INT_greatest Inter_lower closure_mono image_mono)
 
 lemma isometry_cblinfun_image_inf_distrib':
   fixes U::\<open>'a::complex_normed_vector \<Rightarrow>\<^sub>C\<^sub>L 'b::cbanach\<close> and B C::"'a ccsubspace"
   shows "U *\<^sub>S (inf B C) \<le> inf (U *\<^sub>S B) (U *\<^sub>S C)"
 proof -
   define V where \<open>V b = (if b then B else C)\<close> for b
   have \<open>U *\<^sub>S (INF i. V i) \<le> (INF i. U *\<^sub>S (V i))\<close>
     by auto
   then show ?thesis
     unfolding V_def
     by (metis (mono_tags, lifting) INF_UNIV_bool_expand)
 qed
 
 lemma cblinfun_image_eq:
   fixes S :: "'a::cbanach ccsubspace"
     and A B :: "'a::cbanach \<Rightarrow>\<^sub>C\<^sub>L'b::cbanach"
   assumes "\<And>x. x \<in> G \<Longrightarrow> A *\<^sub>V x = B *\<^sub>V x" and "ccspan G \<ge> S"
   shows "A *\<^sub>S S = B *\<^sub>S S"
 proof (use assms in transfer)
   fix G :: "'a set" and A :: "'a \<Rightarrow> 'b" and B :: "'a \<Rightarrow> 'b" and S :: "'a set"
   assume a1: "bounded_clinear A"
   assume a2: "bounded_clinear B"
   assume a3: "\<And>x. x \<in> G \<Longrightarrow> A x = B x"
   assume a4: "S \<subseteq> closure (cspan G)"
 
   have "A ` closure S = B ` closure S"
     by (smt (verit, best) UnCI a1 a2 a3 a4 bounded_clinear_eq_on closure_Un closure_closure image_cong sup.absorb_iff1)
   then show "closure (A ` S) = closure (B ` S)"
     by (metis bounded_clinear.bounded_linear a1 a2 closure_bounded_linear_image_subset_eq)
 qed
 
 lemma cblinfun_fixes_range:
   assumes "A o\<^sub>C\<^sub>L B = B" and "\<psi> \<in> space_as_set (B *\<^sub>S top)"
   shows "A *\<^sub>V \<psi> = \<psi>"
 proof-
   define rangeB rangeB' where "rangeB = space_as_set (B *\<^sub>S top)"
     and "rangeB' = range (cblinfun_apply B)"
   from assms have "\<psi> \<in> closure rangeB'"
     by (simp add: cblinfun_image.rep_eq rangeB'_def top_ccsubspace.rep_eq)
 
   then obtain \<psi>i where \<psi>i_lim: "\<psi>i \<longlonglongrightarrow> \<psi>" and \<psi>i_B: "\<psi>i i \<in> rangeB'" for i
     using closure_sequential by blast
   have A_invariant: "A *\<^sub>V \<psi>i i = \<psi>i i"
     for i
   proof-
     from \<psi>i_B obtain \<phi> where \<phi>: "\<psi>i i = B *\<^sub>V \<phi>"
       using rangeB'_def by blast
     hence "A *\<^sub>V \<psi>i i = (A o\<^sub>C\<^sub>L B) *\<^sub>V \<phi>"
       by (simp add: cblinfun_compose.rep_eq)
     also have "\<dots> = B *\<^sub>V \<phi>"
       by (simp add: assms)
     also have "\<dots> = \<psi>i i"
       by (simp add: \<phi>)
     finally show ?thesis.
   qed
   from \<psi>i_lim have "(\<lambda>i. A *\<^sub>V (\<psi>i i)) \<longlonglongrightarrow> A *\<^sub>V \<psi>"
     by (rule isCont_tendsto_compose[rotated], simp)
   with A_invariant have "(\<lambda>i. \<psi>i i) \<longlonglongrightarrow> A *\<^sub>V \<psi>"
     by auto
   with \<psi>i_lim show "A *\<^sub>V \<psi> = \<psi>"
     using LIMSEQ_unique by blast
 qed
 
 lemma zero_cblinfun_image[simp]: "0 *\<^sub>S S = (0::_ ccsubspace)"
   apply transfer by (simp add: complex_vector.subspace_0 image_constant[where x=0])
 
 lemma cblinfun_image_INF_eq_general:
   fixes V :: "'a \<Rightarrow> 'b::chilbert_space ccsubspace"
     and U :: "'b \<Rightarrow>\<^sub>C\<^sub>L'c::chilbert_space"
     and Uinv :: "'c \<Rightarrow>\<^sub>C\<^sub>L'b"
   assumes UinvUUinv: "Uinv o\<^sub>C\<^sub>L U o\<^sub>C\<^sub>L Uinv = Uinv" and UUinvU: "U o\<^sub>C\<^sub>L Uinv o\<^sub>C\<^sub>L U = U"
     \<comment> \<open>Meaning: \<^term>\<open>Uinv\<close> is a Pseudoinverse of \<^term>\<open>U\<close>\<close>
     and V: "\<And>i. V i \<le> Uinv *\<^sub>S top"
   shows "U *\<^sub>S (INF i. V i) = (INF i. U *\<^sub>S V i)"
 proof (rule antisym)
   show "U *\<^sub>S (INF i. V i) \<le> (INF i. U *\<^sub>S V i)"
     by (rule cblinfun_image_INF_leq)
 next
   define rangeU rangeUinv where "rangeU = U *\<^sub>S top" and "rangeUinv = Uinv *\<^sub>S top"
   define INFUV INFV where INFUV_def: "INFUV = (INF i. U *\<^sub>S V i)" and INFV_def: "INFV = (INF i. V i)"
   from assms have "V i \<le> rangeUinv"
     for i
     unfolding rangeUinv_def by simp
   moreover have "(Uinv o\<^sub>C\<^sub>L U) *\<^sub>V \<psi> = \<psi>" if "\<psi> \<in> space_as_set rangeUinv"
     for \<psi>
     using UinvUUinv cblinfun_fixes_range rangeUinv_def that by fastforce
   ultimately have "(Uinv o\<^sub>C\<^sub>L U) *\<^sub>V \<psi> = \<psi>" if "\<psi> \<in> space_as_set (V i)"
     for \<psi> i
     using less_eq_ccsubspace.rep_eq that by blast
   hence d1: "(Uinv o\<^sub>C\<^sub>L U) *\<^sub>S (V i) = (V i)" for i
   proof transfer
     show "closure ((Uinv \<circ> U) ` V i) = V i"
       if "pred_fun \<top> closed_csubspace V"
         and "bounded_clinear Uinv"
         and "bounded_clinear U"
         and "\<And>\<psi> i. \<psi> \<in> V i \<Longrightarrow> (Uinv \<circ> U) \<psi> = \<psi>"
       for V :: "'a \<Rightarrow> 'b set"
         and Uinv :: "'c \<Rightarrow> 'b"
         and U :: "'b \<Rightarrow> 'c"
         and i :: 'a
       using that proof auto
       show "x \<in> V i"
         if "\<forall>x. closed_csubspace (V x)"
           and "bounded_clinear Uinv"
           and "bounded_clinear U"
           and "\<And>\<psi> i. \<psi> \<in> V i \<Longrightarrow> Uinv (U \<psi>) = \<psi>"
           and "x \<in> closure (V i)"
         for x :: 'b
         using that
         by (metis orthogonal_complement_of_closure closed_csubspace.subspace double_orthogonal_complement_id closure_is_closed_csubspace)
       show "x \<in> closure (V i)"
         if "\<forall>x. closed_csubspace (V x)"
           and "bounded_clinear Uinv"
           and "bounded_clinear U"
           and "\<And>\<psi> i. \<psi> \<in> V i \<Longrightarrow> Uinv (U \<psi>) = \<psi>"
           and "x \<in> V i"
         for x :: 'b
         using that
         using setdist_eq_0_sing_1 setdist_sing_in_set
         by blast
     qed
   qed
   have "U *\<^sub>S V i \<le> rangeU" for i
     by (simp add: cblinfun_image_mono rangeU_def)
   hence "INFUV \<le> rangeU"
     unfolding INFUV_def by (meson INF_lower UNIV_I order_trans)
   moreover have "(U o\<^sub>C\<^sub>L Uinv) *\<^sub>V \<psi> = \<psi>" if "\<psi> \<in> space_as_set rangeU" for \<psi>
     using UUinvU cblinfun_fixes_range rangeU_def that by fastforce
   ultimately have x: "(U o\<^sub>C\<^sub>L Uinv) *\<^sub>V \<psi> = \<psi>" if "\<psi> \<in> space_as_set INFUV" for \<psi>
     by (simp add: in_mono less_eq_ccsubspace.rep_eq that)
 
   have "closure ((U \<circ> Uinv) ` INFUV) = INFUV"
     if "closed_csubspace INFUV"
       and "bounded_clinear U"
       and "bounded_clinear Uinv"
       and "\<And>\<psi>. \<psi> \<in> INFUV \<Longrightarrow> (U \<circ> Uinv) \<psi> = \<psi>"
     for INFUV :: "'c set"
       and U :: "'b \<Rightarrow> 'c"
       and Uinv :: "'c \<Rightarrow> 'b"
     using that proof auto
     show "x \<in> INFUV"
       if "closed_csubspace INFUV"
         and "bounded_clinear U"
         and "bounded_clinear Uinv"
         and "\<And>\<psi>. \<psi> \<in> INFUV \<Longrightarrow> U (Uinv \<psi>) = \<psi>"
         and "x \<in> closure INFUV"
       for x :: 'c
       using that
       by (metis orthogonal_complement_of_closure closed_csubspace.subspace double_orthogonal_complement_id closure_is_closed_csubspace)
     show "x \<in> closure INFUV"
       if "closed_csubspace INFUV"
         and "bounded_clinear U"
         and "bounded_clinear Uinv"
         and "\<And>\<psi>. \<psi> \<in> INFUV \<Longrightarrow> U (Uinv \<psi>) = \<psi>"
         and "x \<in> INFUV"
       for x :: 'c
       using that
       using setdist_eq_0_sing_1 setdist_sing_in_set
       by (simp add: closed_csubspace.closed)
   qed
   hence "(U o\<^sub>C\<^sub>L Uinv) *\<^sub>S INFUV = INFUV"
     by (metis (mono_tags, opaque_lifting) x cblinfun_image.rep_eq cblinfun_image_id id_cblinfun_apply image_cong
         space_as_set_inject)
   hence "INFUV = U *\<^sub>S Uinv *\<^sub>S INFUV"
     by (simp add: cblinfun_compose_image)
   also have "\<dots> \<le> U *\<^sub>S (INF i. Uinv *\<^sub>S U *\<^sub>S V i)"
     unfolding INFUV_def
     by (metis cblinfun_image_mono cblinfun_image_INF_leq)
   also have "\<dots> = U *\<^sub>S INFV"
     using d1
     by (metis (no_types, lifting) INFV_def cblinfun_assoc_left(2) image_cong)
   finally show "INFUV \<le> U *\<^sub>S INFV".
 qed
 
 lemma unitary_range[simp]:
   assumes "unitary U"
   shows "U *\<^sub>S top = top"
   using assms unfolding unitary_def apply transfer
   by (metis closure_UNIV comp_apply surj_def)
 
 lemma range_adjoint_isometry:
   assumes "isometry U"
   shows "U* *\<^sub>S top = top"
 proof-
   from assms have "top = U* *\<^sub>S U *\<^sub>S top"
     by (simp add: cblinfun_assoc_left(2))
   also have "\<dots> \<le> U* *\<^sub>S top"
     by (simp add: cblinfun_image_mono)
   finally show ?thesis
     using top.extremum_unique by blast
 qed
 
 lemma cblinfun_image_INF_eq[simp]:
   fixes V :: "'a \<Rightarrow> 'b::chilbert_space ccsubspace"
     and U :: "'b \<Rightarrow>\<^sub>C\<^sub>L 'c::chilbert_space"
   assumes \<open>isometry U\<close>
   shows "U *\<^sub>S (INF i. V i) = (INF i. U *\<^sub>S V i)"
 proof -
   from \<open>isometry U\<close> have "U* o\<^sub>C\<^sub>L U o\<^sub>C\<^sub>L U* = U*"
     unfolding isometry_def by simp
   moreover from \<open>isometry U\<close> have "U o\<^sub>C\<^sub>L U* o\<^sub>C\<^sub>L U = U"
     unfolding isometry_def
     by (simp add: cblinfun_compose_assoc)
   moreover have "V i \<le> U* *\<^sub>S top" for i
     by (simp add: range_adjoint_isometry assms)
   ultimately show ?thesis
     by (rule cblinfun_image_INF_eq_general)
 qed
 
 lemma isometry_cblinfun_image_inf_distrib[simp]:
   fixes U::\<open>'a::chilbert_space \<Rightarrow>\<^sub>C\<^sub>L 'b::chilbert_space\<close>
     and X Y::"'a ccsubspace"
   assumes "isometry U"
   shows "U *\<^sub>S (inf X Y) = inf (U *\<^sub>S X) (U *\<^sub>S Y)"
   using cblinfun_image_INF_eq[where V="\<lambda>b. if b then X else Y" and U=U]
   unfolding INF_UNIV_bool_expand
   using assms by auto
 
 lemma cblinfun_image_ccspan:
   shows "A *\<^sub>S ccspan G = ccspan ((*\<^sub>V) A ` G)"
   apply transfer
   by (simp add: bounded_clinear.bounded_linear bounded_clinear_def closure_bounded_linear_image_subset_eq complex_vector.linear_span_image)
 
 
 lemma cblinfun_apply_in_image[simp]: "A *\<^sub>V \<psi> \<in> space_as_set (A *\<^sub>S \<top>)"
   by (metis cblinfun_image.rep_eq closure_subset in_mono range_eqI top_ccsubspace.rep_eq)
 
 
 lemma cblinfun_plus_image_distr:
   \<open>(A + B) *\<^sub>S S \<le> A *\<^sub>S S \<squnion> B *\<^sub>S S\<close>
   apply transfer
   by (smt (verit, ccfv_threshold) closed_closure closed_sum_def closure_minimal closure_subset image_subset_iff set_plus_intro subset_eq)
 
 lemma cblinfun_sum_image_distr:
   \<open>(\<Sum>i\<in>I. A i) *\<^sub>S S \<le> (SUP i\<in>I. A i *\<^sub>S S)\<close>
 proof (cases \<open>finite I\<close>)
   case True
   then show ?thesis
   proof induction
     case empty
     then show ?case
       by auto
   next
     case (insert x F)
     then show ?case
       apply auto by (smt (z3) cblinfun_plus_image_distr inf_sup_aci(6) le_iff_sup)
   qed
 next
   case False
   then show ?thesis
     by auto
 qed
 
 lemma space_as_set_image_commute:
   assumes UV: \<open>U o\<^sub>C\<^sub>L V = id_cblinfun\<close> and VU: \<open>V o\<^sub>C\<^sub>L U = id_cblinfun\<close>
     (* I think only one of them is needed, can the lemma be strengthened? *)
   shows \<open>(*\<^sub>V) U ` space_as_set T = space_as_set (U *\<^sub>S T)\<close>
 proof -
   have \<open>space_as_set (U *\<^sub>S T) = U ` V ` space_as_set (U *\<^sub>S T)\<close>
     by (simp add: image_image UV flip: cblinfun_apply_cblinfun_compose)
   also have \<open>\<dots> \<subseteq> U ` space_as_set (V *\<^sub>S U *\<^sub>S T)\<close>
     by (metis cblinfun_image.rep_eq closure_subset image_mono)
   also have \<open>\<dots> = U ` space_as_set T\<close>
     by (simp add: VU cblinfun_assoc_left(2))
   finally have 1: \<open>space_as_set (U *\<^sub>S T) \<subseteq> U ` space_as_set T\<close>
     by -
   have 2: \<open>U ` space_as_set T \<subseteq> space_as_set (U *\<^sub>S T)\<close>
     by (simp add: cblinfun_image.rep_eq closure_subset)
   from 1 2 show ?thesis
     by simp
 qed
 
 lemma right_total_rel_ccsubspace:
   fixes R :: \<open>'a::complex_normed_vector \<Rightarrow> 'b::complex_normed_vector \<Rightarrow> bool\<close>
   assumes UV: \<open>U o\<^sub>C\<^sub>L V = id_cblinfun\<close>
   assumes VU: \<open>V o\<^sub>C\<^sub>L U = id_cblinfun\<close>
   assumes R_def: \<open>\<And>x y. R x y \<longleftrightarrow> x = U *\<^sub>V y\<close>
   shows \<open>right_total (rel_ccsubspace R)\<close>
 proof (rule right_totalI)
   fix T :: \<open>'b ccsubspace\<close>
   show \<open>\<exists>S. rel_ccsubspace R S T\<close>
     apply (rule exI[of _ \<open>U *\<^sub>S T\<close>])
     using UV VU by (auto simp add: rel_ccsubspace_def R_def rel_set_def simp flip: space_as_set_image_commute)
 qed
 
 lemma left_total_rel_ccsubspace:
   fixes R :: \<open>'a::complex_normed_vector \<Rightarrow> 'b::complex_normed_vector \<Rightarrow> bool\<close>
   assumes UV: \<open>U o\<^sub>C\<^sub>L V = id_cblinfun\<close>
   assumes VU: \<open>V o\<^sub>C\<^sub>L U = id_cblinfun\<close>
   assumes R_def: \<open>\<And>x y. R x y \<longleftrightarrow> y = U *\<^sub>V x\<close>
   shows \<open>left_total (rel_ccsubspace R)\<close>
 proof -
   have \<open>right_total (rel_ccsubspace (conversep R))\<close>
     apply (rule right_total_rel_ccsubspace)
     using assms by auto
   then show ?thesis
     by (auto intro!: right_total_conversep[THEN iffD1] simp: converse_rel_ccsubspace)
 qed
 
 lemma cblinfun_image_bot_zero[simp]: \<open>A *\<^sub>S top = bot \<longleftrightarrow> A = 0\<close>
   by (metis Complex_Bounded_Linear_Function.zero_cblinfun_image bot_ccsubspace.rep_eq cblinfun_apply_in_image cblinfun_eqI empty_iff insert_iff zero_ccsubspace_def)
 
+lemma surj_isometry_is_unitary:
+  fixes U :: \<open>'a::chilbert_space \<Rightarrow>\<^sub>C\<^sub>L 'b::chilbert_space\<close>
+  assumes \<open>isometry U\<close>
+  assumes \<open>U *\<^sub>S \<top> = \<top>\<close>
+  shows \<open>unitary U\<close>
+  by (metis UNIV_I assms(1) assms(2) cblinfun_assoc_left(1) cblinfun_compose_id_right cblinfun_eqI cblinfun_fixes_range id_cblinfun_apply isometry_def space_as_set_top unitary_def)
+
 subsection \<open>Sandwiches\<close>
 
-
 lift_definition sandwich :: \<open>('a::chilbert_space \<Rightarrow>\<^sub>C\<^sub>L 'b::complex_inner) \<Rightarrow> (('a \<Rightarrow>\<^sub>C\<^sub>L 'a) \<Rightarrow>\<^sub>C\<^sub>L ('b \<Rightarrow>\<^sub>C\<^sub>L 'b))\<close> is
   \<open>\<lambda>(A::'a\<Rightarrow>\<^sub>C\<^sub>L'b) B. A o\<^sub>C\<^sub>L B o\<^sub>C\<^sub>L A*\<close>
 proof
   fix A :: \<open>'a \<Rightarrow>\<^sub>C\<^sub>L 'b\<close> and B B1 B2 :: \<open>'a \<Rightarrow>\<^sub>C\<^sub>L 'a\<close> and c :: complex
   show \<open>A o\<^sub>C\<^sub>L (B1 + B2) o\<^sub>C\<^sub>L A* = (A o\<^sub>C\<^sub>L B1 o\<^sub>C\<^sub>L A*) + (A o\<^sub>C\<^sub>L B2 o\<^sub>C\<^sub>L A*)\<close>
     by (simp add: cblinfun_compose_add_left cblinfun_compose_add_right)
   show \<open>A o\<^sub>C\<^sub>L (c *\<^sub>C B) o\<^sub>C\<^sub>L A* = c *\<^sub>C (A o\<^sub>C\<^sub>L B o\<^sub>C\<^sub>L A*)\<close>
     by auto
   show \<open>\<exists>K. \<forall>B. norm (A o\<^sub>C\<^sub>L B o\<^sub>C\<^sub>L A*) \<le> norm B * K\<close>
   proof (rule exI[of _ \<open>norm A * norm (A*)\<close>], rule allI)
     fix B
     have \<open>norm (A o\<^sub>C\<^sub>L B o\<^sub>C\<^sub>L A*) \<le> norm (A o\<^sub>C\<^sub>L B) * norm (A*)\<close>
       using norm_cblinfun_compose by blast
     also have \<open>\<dots> \<le> (norm A * norm B) * norm (A*)\<close>
       by (simp add: mult_right_mono norm_cblinfun_compose)
     finally show \<open>norm (A o\<^sub>C\<^sub>L B o\<^sub>C\<^sub>L A*) \<le> norm B * (norm A * norm (A*))\<close>
       by (simp add: mult.assoc vector_space_over_itself.scale_left_commute)
   qed
 qed
 
 lemma sandwich_0[simp]: \<open>sandwich 0 = 0\<close>
   by (simp add: cblinfun_eqI sandwich.rep_eq)
 
 lemma sandwich_apply: \<open>sandwich A *\<^sub>V B = A o\<^sub>C\<^sub>L B o\<^sub>C\<^sub>L A*\<close>
   apply (transfer fixing: A B) by auto
 
 lemma norm_sandwich: \<open>norm (sandwich A) = (norm A)\<^sup>2\<close> for A :: \<open>'a::{chilbert_space} \<Rightarrow>\<^sub>C\<^sub>L 'b::{complex_inner}\<close>
 proof -
   have main: \<open>norm (sandwich A) = (norm A)\<^sup>2\<close> for A :: \<open>'c::{chilbert_space,not_singleton} \<Rightarrow>\<^sub>C\<^sub>L 'd::{complex_inner}\<close>
   proof (rule norm_cblinfun_eqI)
     show \<open>(norm A)\<^sup>2 \<le> norm (sandwich A *\<^sub>V id_cblinfun) / norm (id_cblinfun :: 'c \<Rightarrow>\<^sub>C\<^sub>L _)\<close>
       apply (auto simp: sandwich_apply)
       by -
     fix B
     have \<open>norm (sandwich A *\<^sub>V B) \<le> norm (A o\<^sub>C\<^sub>L B) * norm (A*)\<close>
       using norm_cblinfun_compose by (auto simp: sandwich_apply simp del: norm_adj)
     also have \<open>\<dots> \<le> (norm A * norm B) * norm (A*)\<close>
       by (simp add: mult_right_mono norm_cblinfun_compose)
     also have \<open>\<dots> \<le> (norm A)\<^sup>2 * norm B\<close>
       by (simp add: power2_eq_square mult.assoc vector_space_over_itself.scale_left_commute)
     finally show \<open>norm (sandwich A *\<^sub>V B) \<le> (norm A)\<^sup>2 * norm B\<close>
       by -
     show \<open>0 \<le> (norm A)\<^sup>2\<close>
       by auto
   qed
 
   show ?thesis
   proof (cases \<open>class.not_singleton TYPE('a)\<close>)
     case True
     show ?thesis
       apply (rule main[internalize_sort' 'c2])
        apply standard[1]
       using True by simp
   next
     case False
     have \<open>A = 0\<close>
       apply (rule cblinfun_from_CARD_1_0[internalize_sort' 'a])
        apply (rule not_singleton_vs_CARD_1)
        apply (rule False)
       by standard
     then show ?thesis
       by simp
   qed
 qed
 
 lemma sandwich_apply_adj: \<open>sandwich A (B*) = (sandwich A B)*\<close>
   by (simp add: cblinfun_assoc_left(1) sandwich_apply)
 
 lemma sandwich_id[simp]: "sandwich id_cblinfun = id_cblinfun"
   apply (rule cblinfun_eqI)
   by (auto simp: sandwich_apply)
 
 
 subsection \<open>Projectors\<close>
 
 lift_definition Proj :: "('a::chilbert_space) ccsubspace \<Rightarrow> 'a \<Rightarrow>\<^sub>C\<^sub>L'a"
   is \<open>projection\<close>
   by (rule projection_bounded_clinear)
 
 lemma Proj_range[simp]: "Proj S *\<^sub>S top = S"
 proof transfer
   fix S :: \<open>'a set\<close> assume \<open>closed_csubspace S\<close>
   then have "closure (range (projection S)) \<subseteq> S"
     by (metis closed_csubspace.closed closed_csubspace.subspace closure_closed complex_vector.subspace_0 csubspace_is_convex dual_order.eq_iff insert_absorb insert_not_empty projection_image)
   moreover have "S \<subseteq> closure (range (projection S))"
     using \<open>closed_csubspace S\<close>
     by (metis closed_csubspace_def closure_subset csubspace_is_convex equals0D projection_image subset_iff)
   ultimately show \<open>closure (range (projection S)) = S\<close>
     by auto
 qed
 
 lemma adj_Proj: \<open>(Proj M)* = Proj M\<close>
   apply transfer by (simp add: projection_cadjoint)
 
 lemma Proj_idempotent[simp]: \<open>Proj M o\<^sub>C\<^sub>L Proj M = Proj M\<close>
 proof -
   have u1: \<open>(cblinfun_apply (Proj M)) = projection (space_as_set M)\<close>
     apply transfer
     by blast
   have \<open>closed_csubspace (space_as_set M)\<close>
     using space_as_set by auto
   hence u2: \<open>(projection (space_as_set M))\<circ>(projection (space_as_set M))
                 = (projection (space_as_set M))\<close>
     using projection_idem by fastforce
   have \<open>(cblinfun_apply (Proj M)) \<circ> (cblinfun_apply (Proj M)) = cblinfun_apply (Proj M)\<close>
     using u1 u2
     by simp
   hence \<open>cblinfun_apply ((Proj M) o\<^sub>C\<^sub>L (Proj M)) = cblinfun_apply (Proj M)\<close>
     by (simp add: cblinfun_compose.rep_eq)
   thus ?thesis using cblinfun_apply_inject
     by auto
 qed
 
-(* Widen the type class *)
-lift_definition is_Proj :: \<open>'a::chilbert_space \<Rightarrow>\<^sub>C\<^sub>L 'a \<Rightarrow> bool\<close> is
-  \<open>\<lambda>P. \<exists>M. closed_csubspace M \<and> is_projection_on P M\<close> .
+lift_definition is_Proj :: \<open>'a::complex_normed_vector \<Rightarrow>\<^sub>C\<^sub>L 'a \<Rightarrow> bool\<close> is
+  \<open>\<lambda>P. \<exists>M. is_projection_on P M\<close> .
 
 lemma Proj_top[simp]: \<open>Proj \<top> = id_cblinfun\<close>
   by (metis Proj_idempotent Proj_range cblinfun_eqI cblinfun_fixes_range id_cblinfun_apply iso_tuple_UNIV_I space_as_set_top)
 
 lemma Proj_on_own_range':
   fixes P :: \<open>'a::chilbert_space \<Rightarrow>\<^sub>C\<^sub>L'a\<close>
   assumes \<open>P o\<^sub>C\<^sub>L P = P\<close> and \<open>P = P*\<close>
   shows \<open>Proj (P *\<^sub>S top) = P\<close>
-proof-
+proof -
   define M where "M = P *\<^sub>S top"
   have v3: "x \<in> (\<lambda>x. x - P *\<^sub>V x) -` {0}"
     if "x \<in> range (cblinfun_apply P)"
     for x :: 'a
   proof-
     have v3_1: \<open>cblinfun_apply P \<circ> cblinfun_apply P = cblinfun_apply P\<close>
       by (metis \<open>P o\<^sub>C\<^sub>L P = P\<close> cblinfun_compose.rep_eq)
     have \<open>\<exists>t. P *\<^sub>V t = x\<close>
       using that by blast
     then obtain t where t_def: \<open>P *\<^sub>V t = x\<close>
       by blast
     hence \<open>x - P *\<^sub>V x = x - P *\<^sub>V (P *\<^sub>V t)\<close>
       by simp
     also have \<open>\<dots> = x - (P *\<^sub>V t)\<close>
       using v3_1
       by (metis comp_apply)
     also have \<open>\<dots> = 0\<close>
       by (simp add: t_def)
     finally have \<open>x - P *\<^sub>V x = 0\<close>
       by blast
     thus ?thesis
       by simp
   qed
 
   have v1: "range (cblinfun_apply P) \<subseteq> (\<lambda>x. x - cblinfun_apply P x) -` {0}"
     using v3
     by blast
 
   have "x \<in> range (cblinfun_apply P)"
     if "x \<in> (\<lambda>x. x - P *\<^sub>V x) -` {0}"
     for x :: 'a
   proof-
     have x1:\<open>x - P *\<^sub>V x = 0\<close>
       using that by blast
     have \<open>x = P *\<^sub>V x\<close>
       by (simp add: x1 eq_iff_diff_eq_0)
     thus ?thesis
       by blast
   qed
   hence v2: "(\<lambda>x. x - cblinfun_apply P x) -` {0} \<subseteq> range (cblinfun_apply P)"
     by blast
   have i1: \<open>range (cblinfun_apply P) = (\<lambda> x. x - cblinfun_apply P x) -` {0}\<close>
     using v1 v2
     by (simp add: v1 dual_order.antisym)
   have p1: \<open>closed {(0::'a)}\<close>
     by simp
   have p2: \<open>continuous (at x) (\<lambda> x. x - P *\<^sub>V x)\<close>
     for x
   proof-
     have \<open>cblinfun_apply (id_cblinfun - P) = (\<lambda> x. x - P *\<^sub>V x)\<close>
       by (simp add: id_cblinfun.rep_eq minus_cblinfun.rep_eq)
     hence \<open>bounded_clinear (cblinfun_apply (id_cblinfun - P))\<close>
       using cblinfun_apply
       by blast
     hence \<open>continuous (at x) (cblinfun_apply (id_cblinfun - P))\<close>
       by (simp add: clinear_continuous_at)
     thus ?thesis
       using \<open>cblinfun_apply (id_cblinfun - P) = (\<lambda> x. x - P *\<^sub>V x)\<close>
       by simp
   qed
 
   have i2: \<open>closed ( (\<lambda> x. x - P *\<^sub>V x) -` {0} )\<close>
     using p1 p2
     by (rule Abstract_Topology.continuous_closed_vimage)
 
   have \<open>closed (range (cblinfun_apply P))\<close>
     using i1 i2
     by simp
   have u2: \<open>cblinfun_apply P x \<in> space_as_set M\<close>
     for x
     by (simp add: M_def \<open>closed (range ((*\<^sub>V) P))\<close> cblinfun_image.rep_eq top_ccsubspace.rep_eq)
 
-  have xy: \<open>\<langle> x - P *\<^sub>V x, y \<rangle> = 0\<close>
+  have xy: \<open>is_orthogonal (x - P *\<^sub>V x) y\<close>
     if y1: \<open>y \<in> space_as_set M\<close>
     for x y
   proof-
     have \<open>\<exists>t. y = P *\<^sub>V t\<close>
       using y1
       by (simp add:  M_def \<open>closed (range ((*\<^sub>V) P))\<close> cblinfun_image.rep_eq image_iff
           top_ccsubspace.rep_eq)
     then obtain t where t_def: \<open>y = P *\<^sub>V t\<close>
       by blast
-    have \<open>\<langle> x - P *\<^sub>V x, y \<rangle> = \<langle> x - P *\<^sub>V x, P *\<^sub>V t \<rangle>\<close>
+    have \<open>(x - P *\<^sub>V x) \<bullet>\<^sub>C y = (x - P *\<^sub>V x) \<bullet>\<^sub>C (P *\<^sub>V t)\<close>
       by (simp add: t_def)
-    also have \<open>\<dots> = \<langle> P *\<^sub>V (x - P *\<^sub>V x), t \<rangle>\<close>
+    also have \<open>\<dots> = (P *\<^sub>V (x - P *\<^sub>V x)) \<bullet>\<^sub>C t\<close>
       by (metis \<open>P = P*\<close> cinner_adj_left)
-    also have \<open>\<dots> = \<langle> P *\<^sub>V x - P *\<^sub>V (P *\<^sub>V x), t \<rangle>\<close>
+    also have \<open>\<dots> = (P *\<^sub>V x - P *\<^sub>V (P *\<^sub>V x)) \<bullet>\<^sub>C t\<close>
       by (simp add: cblinfun.diff_right)
-    also have \<open>\<dots> = \<langle> P *\<^sub>V x - P *\<^sub>V x, t \<rangle>\<close>
+    also have \<open>\<dots> = (P *\<^sub>V x - P *\<^sub>V x) \<bullet>\<^sub>C t\<close>
       by (metis assms(1) comp_apply cblinfun_compose.rep_eq)
-    also have \<open>\<dots> = \<langle> 0, t \<rangle>\<close>
+    also have \<open>\<dots> = (0 \<bullet>\<^sub>C t)\<close>
       by simp
     also have \<open>\<dots> = 0\<close>
       by simp
     finally show ?thesis by blast
   qed
   hence u1: \<open>x - P *\<^sub>V x \<in> orthogonal_complement (space_as_set M)\<close>
     for x
     by (simp add: orthogonal_complementI)
   have "closed_csubspace (space_as_set M)"
     using space_as_set by auto
   hence f1: "(Proj M) *\<^sub>V a = P *\<^sub>V a" for a
     by (simp add: Proj.rep_eq projection_eqI u1 u2)
   have "(+) ((P - Proj M) *\<^sub>V a) = id" for a
     using f1
     by (auto intro!: ext simp add: minus_cblinfun.rep_eq)
   hence "b - b = cblinfun_apply (P - Proj M) a"
     for a b
     by (metis (no_types) add_diff_cancel_right' id_apply)
   hence "cblinfun_apply (id_cblinfun - (P - Proj M)) a = a"
     for a
     by (simp add: minus_cblinfun.rep_eq)
   thus ?thesis
     using u1 u2 cblinfun_apply_inject diff_diff_eq2 diff_eq_diff_eq eq_id_iff id_cblinfun.rep_eq
     by (metis (no_types, opaque_lifting) M_def)
 qed
 
 lemma Proj_range_closed:
   assumes "is_Proj P"
   shows "closed (range (cblinfun_apply P))"
-  using assms apply transfer
-  using closed_csubspace.closed is_projection_on_image by blast
+  apply (rule is_projection_on_closed[where f=\<open>cblinfun_apply P\<close>])
+  using assms is_Proj.rep_eq is_projection_on_image by auto
 
 lemma Proj_is_Proj[simp]:
   fixes M::\<open>'a::chilbert_space ccsubspace\<close>
   shows \<open>is_Proj (Proj M)\<close>
 proof-
   have u1: "closed_csubspace (space_as_set M)"
     using space_as_set by blast
   have v1: "h - Proj M *\<^sub>V h
          \<in> orthogonal_complement (space_as_set M)" for h
     by (simp add: Proj.rep_eq orthogonal_complementI projection_orthogonal u1)
   have v2: "Proj M *\<^sub>V h \<in> space_as_set M" for h
     by (metis Proj.rep_eq mem_Collect_eq orthog_proj_exists projection_eqI space_as_set)
   have u2: "is_projection_on ((*\<^sub>V) (Proj M)) (space_as_set M)"
     unfolding is_projection_on_def
     by (simp add: smallest_dist_is_ortho u1 v1 v2)
   show ?thesis
     using u1 u2 is_Proj.rep_eq by blast
 qed
 
 lemma is_Proj_algebraic:
   fixes P::\<open>'a::chilbert_space \<Rightarrow>\<^sub>C\<^sub>L 'a\<close>
   shows \<open>is_Proj P \<longleftrightarrow> P o\<^sub>C\<^sub>L P = P \<and> P = P*\<close>
 proof
   have "P o\<^sub>C\<^sub>L P = P"
     if "is_Proj P"
     using that apply transfer
     using is_projection_on_idem
     by fastforce
   moreover have "P = P*"
     if "is_Proj P"
-    using that apply transfer
-    by (metis is_projection_on_cadjoint)
+    using that Proj_range_closed[OF that] is_projection_on_cadjoint[where \<pi>=P and M=\<open>range P\<close>]
+    apply transfer
+    by (metis bounded_clinear.axioms(1) closed_csubspace_UNIV closed_csubspace_def complex_vector.linear_subspace_image is_projection_on_image)
   ultimately show "P o\<^sub>C\<^sub>L P = P \<and> P = P*"
     if "is_Proj P"
     using that
     by blast
   show "is_Proj P"
     if "P o\<^sub>C\<^sub>L P = P \<and> P = P*"
     using that Proj_on_own_range' Proj_is_Proj by metis
 qed
 
 lemma Proj_on_own_range:
   fixes P :: \<open>'a::chilbert_space \<Rightarrow>\<^sub>C\<^sub>L'a\<close>
   assumes \<open>is_Proj P\<close>
   shows \<open>Proj (P *\<^sub>S top) = P\<close>
   using Proj_on_own_range' assms is_Proj_algebraic by blast
 
 lemma Proj_image_leq: "(Proj S) *\<^sub>S A \<le> S"
   by (metis Proj_range inf_top_left le_inf_iff isometry_cblinfun_image_inf_distrib')
 
 lemma Proj_sandwich:
   fixes A::"'a::chilbert_space \<Rightarrow>\<^sub>C\<^sub>L 'b::chilbert_space"
   assumes "isometry A"
   shows "sandwich A *\<^sub>V Proj S = Proj (A *\<^sub>S S)"
-proof-
+proof -
   define P where \<open>P = A o\<^sub>C\<^sub>L Proj S o\<^sub>C\<^sub>L (A*)\<close>
   have \<open>P o\<^sub>C\<^sub>L P = P\<close>
     using assms
     unfolding P_def isometry_def
     by (metis (no_types, lifting) Proj_idempotent cblinfun_assoc_left(1) cblinfun_compose_id_left)
   moreover have \<open>P = P*\<close>
     unfolding P_def
     by (metis adj_Proj adj_cblinfun_compose cblinfun_assoc_left(1) double_adj)
   ultimately have
     \<open>\<exists>M. P = Proj M \<and> space_as_set M = range (cblinfun_apply (A o\<^sub>C\<^sub>L (Proj S) o\<^sub>C\<^sub>L (A*)))\<close>
     using P_def Proj_on_own_range'
     by (metis Proj_is_Proj Proj_range_closed cblinfun_image.rep_eq closure_closed top_ccsubspace.rep_eq)
   then obtain M where \<open>P = Proj M\<close>
     and \<open>space_as_set M = range (cblinfun_apply (A o\<^sub>C\<^sub>L (Proj S) o\<^sub>C\<^sub>L (A*)))\<close>
     by blast
 
   have f1: "A o\<^sub>C\<^sub>L Proj S = P o\<^sub>C\<^sub>L A"
     by (simp add: P_def assms cblinfun_compose_assoc)
   hence "P o\<^sub>C\<^sub>L A o\<^sub>C\<^sub>L A* = P"
     using P_def by presburger
   hence "(P o\<^sub>C\<^sub>L A) *\<^sub>S (c \<squnion> A* *\<^sub>S d) = P *\<^sub>S (A *\<^sub>S c \<squnion> d)"
     for c d
 
     by (simp add: cblinfun_assoc_left(2))
   hence "P *\<^sub>S (A *\<^sub>S \<top> \<squnion> c) = (P o\<^sub>C\<^sub>L A) *\<^sub>S \<top>"
     for c
     by (metis sup_top_left)
   hence \<open>M = A *\<^sub>S S\<close>
     using f1
     by (metis \<open>P = Proj M\<close> cblinfun_assoc_left(2) Proj_range sup_top_right)
   thus ?thesis
     using \<open>P = Proj M\<close>
     unfolding P_def sandwich_apply by blast
 qed
 
 lemma Proj_orthog_ccspan_union:
   assumes "\<And>x y. x \<in> X \<Longrightarrow> y \<in> Y \<Longrightarrow> is_orthogonal x y"
   shows \<open>Proj (ccspan (X \<union> Y)) = Proj (ccspan X) + Proj (ccspan Y)\<close>
 proof -
   have \<open>x \<in> cspan X \<Longrightarrow> y \<in> cspan Y \<Longrightarrow> is_orthogonal x y\<close> for x y
     apply (rule is_orthogonal_closure_cspan[where X=X and Y=Y])
     using closure_subset assms by auto
   then have \<open>x \<in> closure (cspan X) \<Longrightarrow> y \<in> closure (cspan Y) \<Longrightarrow> is_orthogonal x y\<close> for x y
     by (metis orthogonal_complementI orthogonal_complement_of_closure orthogonal_complement_orthoI')
   then show ?thesis
     apply (transfer fixing: X Y)
     apply (subst projection_plus[symmetric])
     by auto
 qed
 
 abbreviation proj :: "'a::chilbert_space \<Rightarrow> 'a \<Rightarrow>\<^sub>C\<^sub>L 'a" where "proj \<psi> \<equiv> Proj (ccspan {\<psi>})"
 
 lemma proj_0[simp]: \<open>proj 0 = 0\<close>
   apply transfer by auto
 
-lemma surj_isometry_is_unitary:
-  fixes U :: \<open>'a::chilbert_space \<Rightarrow>\<^sub>C\<^sub>L 'b::chilbert_space\<close>
-  assumes \<open>isometry U\<close>
-  assumes \<open>U *\<^sub>S \<top> = \<top>\<close>
-  shows \<open>unitary U\<close>
-  by (metis Proj_sandwich sandwich_apply Proj_on_own_range' assms(1) assms(2) cblinfun_compose_id_right isometry_def unitary_def unitary_id unitary_range)
-
 lemma ccsubspace_supI_via_Proj:
   fixes A B C::"'a::chilbert_space ccsubspace"
   assumes a1: \<open>Proj (- C) *\<^sub>S A \<le> B\<close>
-  shows  "A \<le> sup B C"
+  shows  "A \<le> B \<squnion> C"
 proof-
   have x2: \<open>x \<in> space_as_set B\<close>
     if "x \<in>  closure ( (projection (orthogonal_complement (space_as_set C))) ` space_as_set A)" for x
     using that
     by (metis Proj.rep_eq cblinfun_image.rep_eq assms less_eq_ccsubspace.rep_eq subsetD
         uminus_ccsubspace.rep_eq)
   have q1: \<open>x \<in> closure {\<psi> + \<phi> |\<psi> \<phi>. \<psi> \<in> space_as_set B \<and> \<phi> \<in> space_as_set C}\<close>
     if \<open>x \<in> space_as_set A\<close>
     for x
   proof-
     have p1: \<open>closed_csubspace (space_as_set C)\<close>
       using space_as_set by auto
     hence \<open>x = (projection (space_as_set C)) x
        + (projection (orthogonal_complement (space_as_set C))) x\<close>
       by simp
     hence \<open>x = (projection (orthogonal_complement (space_as_set C))) x
               + (projection (space_as_set C)) x\<close>
       by (metis ordered_field_class.sign_simps(2))
     moreover have \<open>(projection (orthogonal_complement (space_as_set C))) x \<in> space_as_set B\<close>
       using x2
       by (meson closure_subset image_subset_iff that)
     moreover have \<open>(projection (space_as_set C)) x \<in> space_as_set C\<close>
       by (metis mem_Collect_eq orthog_proj_exists projection_eqI space_as_set)
     ultimately show ?thesis
       using closure_subset by fastforce
   qed
   have x1: \<open>x \<in> (space_as_set B +\<^sub>M space_as_set C)\<close>
     if "x \<in> space_as_set A" for x
   proof -
     have f1: "x \<in> closure {a + b |a b. a \<in> space_as_set B \<and> b \<in> space_as_set C}"
       by (simp add: q1 that)
     have "{a + b |a b. a \<in> space_as_set B \<and> b \<in> space_as_set C} = {a. \<exists>p. p \<in> space_as_set B
       \<and> (\<exists>q. q \<in> space_as_set C \<and> a = p + q)}"
       by blast
     hence "x \<in> closure {a. \<exists>b\<in>space_as_set B. \<exists>c\<in>space_as_set C. a = b + c}"
       using f1 by (simp add: Bex_def_raw)
     thus ?thesis
       using that
       unfolding closed_sum_def set_plus_def
       by blast
   qed
 
   hence \<open>x \<in> space_as_set (Abs_clinear_space (space_as_set B +\<^sub>M space_as_set C))\<close>
     if "x \<in> space_as_set A" for x
     using that
     by (metis space_as_set_inverse sup_ccsubspace.rep_eq)
   thus ?thesis
     by (simp add: x1 less_eq_ccsubspace.rep_eq subset_eq sup_ccsubspace.rep_eq)
 qed
 
 lemma is_Proj_idempotent:
   assumes "is_Proj P"
   shows "P o\<^sub>C\<^sub>L P = P"
-  using assms
-  unfolding is_Proj_def
-  using assms is_Proj_algebraic by auto
+  using assms apply transfer
+  using is_projection_on_fixes_image is_projection_on_in_image by fastforce
 
 lemma is_proj_selfadj:
   assumes "is_Proj P"
   shows "P* = P"
   using assms
   unfolding is_Proj_def
   by (metis is_Proj_algebraic is_Proj_def)
 
 lemma is_Proj_I:
   assumes "P o\<^sub>C\<^sub>L P = P" and "P* = P"
   shows "is_Proj P"
   using assms is_Proj_algebraic by metis
 
 lemma is_Proj_0[simp]: "is_Proj 0"
-  by (metis add_left_cancel adj_plus bounded_cbilinear.zero_left bounded_cbilinear_cblinfun_compose group_cancel.rule0 is_Proj_I)
+  apply transfer apply (rule exI[of _ 0])
+  by (simp add: is_projection_on_zero)
 
 lemma is_Proj_complement[simp]:
+  fixes P :: \<open>'a::chilbert_space \<Rightarrow>\<^sub>C\<^sub>L 'a\<close>
   assumes a1: "is_Proj P"
-  shows "is_Proj (id_cblinfun-P)"
+  shows "is_Proj (id_cblinfun - P)"
   by (smt (z3) add_diff_cancel_left add_diff_cancel_left' adj_cblinfun_compose adj_plus assms bounded_cbilinear.add_left bounded_cbilinear_cblinfun_compose diff_add_cancel id_cblinfun_adjoint is_Proj_algebraic cblinfun_compose_id_left)
 
 lemma Proj_bot[simp]: "Proj bot = 0"
   by (metis zero_cblinfun_image Proj_on_own_range' is_Proj_0 is_Proj_algebraic
       zero_ccsubspace_def)
 
 lemma Proj_ortho_compl:
   "Proj (- X) = id_cblinfun - Proj X"
-  by (transfer , auto)
+  by (transfer, auto)
 
 lemma Proj_inj:
   assumes "Proj X = Proj Y"
   shows "X = Y"
   by (metis assms Proj_range)
 
-lemma norm_Proj_leq1: \<open>norm (Proj M) \<le> 1\<close>
+lemma norm_Proj_leq1: \<open>norm (Proj M) \<le> 1\<close> for M :: \<open>'a :: chilbert_space ccsubspace\<close>
   apply transfer
   by (metis (no_types, opaque_lifting) mult.left_neutral onorm_bound projection_reduces_norm zero_less_one_class.zero_le_one)
 
 lemma Proj_orthog_ccspan_insert:
   assumes "\<And>y. y \<in> Y \<Longrightarrow> is_orthogonal x y"
   shows \<open>Proj (ccspan (insert x Y)) = proj x + Proj (ccspan Y)\<close>
   apply (subst asm_rl[of \<open>insert x Y = {x} \<union> Y\<close>], simp)
   apply (rule Proj_orthog_ccspan_union)
   using assms by auto
 
-lemma cancel_apply_Proj:
-  assumes \<open>\<psi> \<in> space_as_set S\<close>
-  shows \<open>Proj S *\<^sub>V \<psi> = \<psi>\<close>
-  by (metis Proj_idempotent Proj_range assms cblinfun_fixes_range)
-
 lemma Proj_fixes_image: \<open>Proj S *\<^sub>V \<psi> = \<psi>\<close> if \<open>\<psi> \<in> space_as_set S\<close>
-  by (simp add: Proj.rep_eq closed_csubspace_def projection_fixes_image that)
-
-lemma norm_is_Proj: \<open>norm P \<le> 1\<close> if \<open>is_Proj P\<close>
+  by (metis Proj_idempotent Proj_range that cblinfun_fixes_range)
+
+lemma norm_is_Proj: \<open>norm P \<le> 1\<close> if \<open>is_Proj P\<close> for P :: \<open>'a :: chilbert_space \<Rightarrow>\<^sub>C\<^sub>L 'a\<close>
   by (metis Proj_on_own_range norm_Proj_leq1 that)
 
 lemma Proj_sup: \<open>orthogonal_spaces S T \<Longrightarrow> Proj (sup S T) = Proj S + Proj T\<close>
   unfolding orthogonal_spaces_def
   apply transfer
   by (simp add: projection_plus)
 
 lemma Proj_sum_spaces:
   assumes \<open>finite X\<close>
   assumes \<open>\<And>x y. x\<in>X \<Longrightarrow> y\<in>X \<Longrightarrow> x\<noteq>y \<Longrightarrow> orthogonal_spaces (J x) (J y)\<close>
   shows \<open>Proj (\<Sum>x\<in>X. J x) = (\<Sum>x\<in>X. Proj (J x))\<close>
   using assms
 proof induction
   case empty
   show ?case 
     by auto
 next
   case (insert x F)
   have \<open>Proj (sum J (insert x F)) = Proj (J x \<squnion> sum J F)\<close>
     by (simp add: insert)
   also have \<open>\<dots> = Proj (J x) + Proj (sum J F)\<close>
     apply (rule Proj_sup)
     apply (rule orthogonal_sum)
     using insert by auto
   also have \<open>\<dots> = (\<Sum>x\<in>insert x F. Proj (J x))\<close>
     by (simp add: insert.IH insert.hyps(1) insert.hyps(2) insert.prems)
   finally show ?case
     by -
 qed
 
 lemma is_Proj_reduces_norm:
+  fixes P :: \<open>'a::complex_inner \<Rightarrow>\<^sub>C\<^sub>L 'a\<close>
   assumes \<open>is_Proj P\<close>
   shows \<open>norm (P *\<^sub>V \<psi>) \<le> norm \<psi>\<close>
-  using assms apply transfer
-  using is_projection_on_reduces_norm by blast
+  apply (rule is_projection_on_reduces_norm[where M=\<open>range P\<close>])
+  using assms is_Proj.rep_eq is_projection_on_image apply blast
+  by (simp add: Proj_range_closed assms closed_csubspace.intro)
 
 lemma norm_Proj_apply: \<open>norm (Proj T *\<^sub>V \<psi>) = norm \<psi> \<longleftrightarrow> \<psi> \<in> space_as_set T\<close>
 proof (rule iffI)
   show \<open>norm (Proj T *\<^sub>V \<psi>) = norm \<psi>\<close> if \<open>\<psi> \<in> space_as_set T\<close>
-    by (simp add: cancel_apply_Proj that)
+    by (simp add: Proj_fixes_image that)
   assume assm: \<open>norm (Proj T *\<^sub>V \<psi>) = norm \<psi>\<close>
   have \<psi>_decomp: \<open>\<psi> = Proj T \<psi> + Proj (-T) \<psi>\<close>
     by (simp add: Proj_ortho_compl cblinfun.real.diff_left)
   have \<open>(norm (Proj (-T) \<psi>))\<^sup>2 = (norm (Proj T \<psi>))\<^sup>2 + (norm (Proj (-T) \<psi>))\<^sup>2 - (norm (Proj T \<psi>))\<^sup>2\<close>
     by auto
   also have \<open>\<dots> = (norm (Proj T \<psi> + Proj (-T) \<psi>))\<^sup>2 - (norm (Proj T \<psi>))\<^sup>2\<close>
     apply (subst pythagorean_theorem)
      apply (metis (no_types, lifting) Proj_idempotent \<psi>_decomp add_cancel_right_left adj_Proj cblinfun.real.add_right cblinfun_apply_cblinfun_compose cinner_adj_left cinner_zero_left)
     by simp
   also with \<psi>_decomp have \<open>\<dots> = (norm \<psi>)\<^sup>2 - (norm (Proj T \<psi>))\<^sup>2\<close>
     by metis
   also with assm have \<open>\<dots> = 0\<close>
     by simp
   finally have \<open>norm (Proj (-T) \<psi>) = 0\<close>
     by auto
   with \<psi>_decomp have \<open>\<psi> = Proj T \<psi>\<close>
     by auto
   then show \<open>\<psi> \<in> space_as_set T\<close>
     by (metis Proj_range cblinfun_apply_in_image)
 qed
 
 lemma norm_Proj_apply_1: \<open>norm \<psi> = 1 \<Longrightarrow> norm (Proj T *\<^sub>V \<psi>) = 1 \<longleftrightarrow> \<psi> \<in> space_as_set T\<close>
   using norm_Proj_apply by metis
 
-subsection \<open>Kernel\<close>
+subsection \<open>Kernel / eigenspaces\<close>
 
 lift_definition kernel :: "'a::complex_normed_vector \<Rightarrow>\<^sub>C\<^sub>L'b::complex_normed_vector
    \<Rightarrow> 'a ccsubspace"
   is "\<lambda> f. f -` {0}"
   by (metis kernel_is_closed_csubspace)
 
 definition eigenspace :: "complex \<Rightarrow> 'a::complex_normed_vector \<Rightarrow>\<^sub>C\<^sub>L'a \<Rightarrow> 'a ccsubspace" where
   "eigenspace a A = kernel (A - a *\<^sub>C id_cblinfun)"
 
 lemma kernel_scaleC[simp]: "a\<noteq>0 \<Longrightarrow> kernel (a *\<^sub>C A) = kernel A"
   for a :: complex and A :: "(_,_) cblinfun"
   apply transfer
   using complex_vector.scale_eq_0_iff by blast
 
 lemma kernel_0[simp]: "kernel 0 = top"
   apply transfer by auto
 
 lemma kernel_id[simp]: "kernel id_cblinfun = 0"
   apply transfer by simp
 
 lemma eigenspace_scaleC[simp]:
   assumes a1: "a \<noteq> 0"
   shows "eigenspace b (a *\<^sub>C A) = eigenspace (b/a) A"
 proof -
   have "b *\<^sub>C (id_cblinfun::('a, _) cblinfun) = a *\<^sub>C (b / a) *\<^sub>C id_cblinfun"
     using a1
     by (metis ceq_vector_fraction_iff)
   hence "kernel (a *\<^sub>C A - b *\<^sub>C id_cblinfun) = kernel (A - (b / a) *\<^sub>C id_cblinfun)"
     using a1 by (metis (no_types) complex_vector.scale_right_diff_distrib kernel_scaleC)
   thus ?thesis
     unfolding eigenspace_def
     by blast
 qed
 
 lemma eigenspace_memberD:
   assumes "x \<in> space_as_set (eigenspace e A)"
   shows "A *\<^sub>V x = e *\<^sub>C x"
   using assms unfolding eigenspace_def apply transfer by auto
 
 lemma kernel_memberD:
   assumes "x \<in> space_as_set (kernel A)"
   shows "A *\<^sub>V x = 0"
   using assms apply transfer by auto
 
 lemma eigenspace_memberI:
   assumes "A *\<^sub>V x = e *\<^sub>C x"
   shows "x \<in> space_as_set (eigenspace e A)"
   using assms unfolding eigenspace_def apply transfer by auto
 
 lemma kernel_memberI:
   assumes "A *\<^sub>V x = 0"
   shows "x \<in> space_as_set (kernel A)"
   using assms apply transfer by auto
 
 lemma kernel_Proj[simp]: \<open>kernel (Proj S) = - S\<close>
   apply transfer
   apply auto
   apply (metis diff_0_right is_projection_on_iff_orthog projection_is_projection_on')
   by (simp add: complex_vector.subspace_0 projection_eqI)
 
 lemma orthogonal_projectors_orthogonal_spaces:
   \<comment> \<open>Logically belongs in section "Projectors".\<close>
-  fixes S T :: \<open>'a::chilbert_space set\<close>
-  shows \<open>(\<forall>x\<in>S. \<forall>y\<in>T. is_orthogonal x y) \<longleftrightarrow> Proj (ccspan S) o\<^sub>C\<^sub>L Proj (ccspan T) = 0\<close>
+  fixes S T :: \<open>'a::chilbert_space ccsubspace\<close>
+  shows \<open>orthogonal_spaces S T \<longleftrightarrow> Proj S o\<^sub>C\<^sub>L Proj T = 0\<close>
 proof (intro ballI iffI)
-  fix x y assume \<open>Proj (ccspan S) o\<^sub>C\<^sub>L Proj (ccspan T) = 0\<close> \<open>x \<in> S\<close> \<open>y \<in> T\<close>
-  then show \<open>is_orthogonal x y\<close>
-    by (smt (verit, del_insts) Proj_idempotent Proj_range adj_Proj cblinfun.zero_left cblinfun_apply_cblinfun_compose cblinfun_fixes_range ccspan_superset cinner_adj_right cinner_zero_right in_mono)
+  assume \<open>Proj S o\<^sub>C\<^sub>L Proj T = 0\<close> 
+  then have \<open>is_orthogonal x y\<close> if \<open>x \<in> space_as_set S\<close> \<open>y \<in> space_as_set T\<close> for x y
+    by (metis (no_types, opaque_lifting) Proj_fixes_image adj_Proj cblinfun.zero_left cblinfun_apply_cblinfun_compose cinner_adj_right cinner_zero_right that(1) that(2))
+  then show \<open>orthogonal_spaces S T\<close>
+    by (simp add: orthogonal_spaces_def)
 next 
-  assume \<open>\<forall>x\<in>S. \<forall>y\<in>T. is_orthogonal x y\<close>
-  then have \<open>ccspan S \<le> - ccspan T\<close>
-    by (simp add: ccspan_leq_ortho_ccspan)
-  then show \<open>Proj (ccspan S) o\<^sub>C\<^sub>L Proj (ccspan T) = 0\<close>
+  assume \<open>orthogonal_spaces S T\<close>
+  then have \<open>S \<le> - T\<close>
+    by (simp add: orthogonal_spaces_leq_compl)
+  then show \<open>Proj S o\<^sub>C\<^sub>L Proj T = 0\<close>
     by (metis (no_types, opaque_lifting) Proj_range adj_Proj adj_cblinfun_compose basic_trans_rules(31) cblinfun.zero_left cblinfun_apply_cblinfun_compose cblinfun_apply_in_image cblinfun_eqI kernel_Proj kernel_memberD less_eq_ccsubspace.rep_eq)
 qed
 
+
 lemma cblinfun_compose_Proj_kernel[simp]: \<open>a o\<^sub>C\<^sub>L Proj (kernel a) = 0\<close>
   apply (rule cblinfun_eqI)
   apply simp
   by (metis Proj_range cblinfun_apply_in_image kernel_memberD)
 
 lemma kernel_compl_adj_range:
   shows \<open>kernel a = - (a* *\<^sub>S top)\<close>
 proof (rule ccsubspace_eqI)
   fix x
   have \<open>x \<in> space_as_set (kernel a) \<longleftrightarrow> a x = 0\<close>
     apply transfer by simp
   also have \<open>a x = 0 \<longleftrightarrow> (\<forall>y. is_orthogonal y (a x))\<close>
     by (metis cinner_gt_zero_iff cinner_zero_right)
   also have \<open>\<dots> \<longleftrightarrow> (\<forall>y. is_orthogonal (a* *\<^sub>V y) x)\<close>
     by (simp add: cinner_adj_left)
   also have \<open>\<dots> \<longleftrightarrow> x \<in> space_as_set (- (a* *\<^sub>S top))\<close>
     apply transfer
     by (metis (mono_tags, opaque_lifting) UNIV_I image_iff is_orthogonal_sym orthogonal_complementI orthogonal_complement_of_closure orthogonal_complement_orthoI')
   finally show \<open>x \<in> space_as_set (kernel a) \<longleftrightarrow> x \<in> space_as_set (- (a* *\<^sub>S top))\<close>
     by -
 qed
 
 
 subsection \<open>Partial isometries\<close>
 
 definition partial_isometry where
   \<open>partial_isometry A \<longleftrightarrow> (\<forall>h \<in> space_as_set (- kernel A). norm (A h) = norm h)\<close>
 
 lemma partial_isometryI: 
   assumes \<open>\<And>h. h \<in> space_as_set (- kernel A) \<Longrightarrow> norm (A h) = norm h\<close>
   shows \<open>partial_isometry A\<close>
   using assms partial_isometry_def by blast
 
-
 lemma
   fixes A :: \<open>'a :: chilbert_space \<Rightarrow>\<^sub>C\<^sub>L 'b :: complex_normed_vector\<close>
   assumes iso: \<open>\<And>\<psi>. \<psi> \<in> space_as_set V \<Longrightarrow> norm (A *\<^sub>V \<psi>) = norm \<psi>\<close>
   assumes zero: \<open>\<And>\<psi>. \<psi> \<in> space_as_set (- V) \<Longrightarrow> A *\<^sub>V \<psi> = 0\<close>
   shows partial_isometryI': \<open>partial_isometry A\<close>
     and partial_isometry_initial: \<open>kernel A = - V\<close>
 proof -
   from zero
   have \<open>- V \<le> kernel A\<close>
     by (simp add: kernel_memberI less_eq_ccsubspace.rep_eq subsetI)
   moreover have \<open>kernel A \<le> -V\<close>
     by (smt (verit, ccfv_threshold) Proj_ortho_compl Proj_range assms(1) cblinfun.diff_left cblinfun.diff_right cblinfun_apply_in_image cblinfun_id_cblinfun_apply ccsubspace_leI kernel_Proj kernel_memberD kernel_memberI norm_eq_zero ortho_involution subsetI zero)
   ultimately show kerA: \<open>kernel A = -V\<close>
     by simp
 
   show \<open>partial_isometry A\<close>
     apply (rule partial_isometryI)
     by (simp add: kerA iso)
 qed
 
-lemma Proj_partial_isometry: \<open>partial_isometry (Proj S)\<close>
+lemma Proj_partial_isometry[simp]: \<open>partial_isometry (Proj S)\<close>
   apply (rule partial_isometryI)
-  by (simp add: cancel_apply_Proj)
-
-lemma is_Proj_partial_isometry: \<open>is_Proj P \<Longrightarrow> partial_isometry P\<close>
+  by (simp add: Proj_fixes_image)
+
+lemma is_Proj_partial_isometry: \<open>is_Proj P \<Longrightarrow> partial_isometry P\<close> for P :: \<open>_ :: chilbert_space \<Rightarrow>\<^sub>C\<^sub>L _\<close>
   by (metis Proj_on_own_range Proj_partial_isometry)
 
 lemma isometry_partial_isometry: \<open>isometry P \<Longrightarrow> partial_isometry P\<close>
   by (simp add: isometry_preserves_norm partial_isometry_def)
 
 lemma unitary_partial_isometry: \<open>unitary P \<Longrightarrow> partial_isometry P\<close>
   using isometry_partial_isometry unitary_isometry by blast
 
 lemma norm_partial_isometry:
   fixes A :: \<open>'a :: chilbert_space \<Rightarrow>\<^sub>C\<^sub>L 'b::complex_normed_vector\<close>
   assumes \<open>partial_isometry A\<close> and \<open>A \<noteq> 0\<close>
   shows \<open>norm A = 1\<close> 
 proof -
   from \<open>A \<noteq> 0\<close>
   have \<open>- (kernel A) \<noteq> 0\<close>
     by (metis cblinfun_eqI diff_zero id_cblinfun_apply kernel_id kernel_memberD ortho_involution orthog_proj_exists orthogonal_complement_closed_subspace uminus_ccsubspace.rep_eq zero_cblinfun.rep_eq)
   then obtain h where \<open>h \<in> space_as_set (- kernel A)\<close> and \<open>h \<noteq> 0\<close>
     by (metis cblinfun_id_cblinfun_apply ccsubspace_eqI closed_csubspace.subspace complex_vector.subspace_0 kernel_id kernel_memberD kernel_memberI orthogonal_complement_closed_subspace uminus_ccsubspace.rep_eq)
   with \<open>partial_isometry A\<close>
   have \<open>norm (A h) = norm h\<close>
     using partial_isometry_def by blast
   then have \<open>norm A \<ge> 1\<close>
     by (smt (verit) \<open>h \<noteq> 0\<close> mult_cancel_right1 mult_left_le_one_le norm_cblinfun norm_eq_zero norm_ge_zero)
 
   have \<open>norm A \<le> 1\<close>
   proof (rule norm_cblinfun_bound)
     show \<open>0 \<le> (1::real)\<close>
       by simp
     fix \<psi> :: 'a
     define g h where \<open>g = Proj (kernel A) \<psi>\<close> and \<open>h = Proj (- kernel A) \<psi>\<close>
     have \<open>A g = 0\<close>
       by (metis Proj_range cblinfun_apply_in_image g_def kernel_memberD)
     moreover from \<open>partial_isometry A\<close>
     have \<open>norm (A h) = norm h\<close>
       by (metis Proj_range cblinfun_apply_in_image h_def partial_isometry_def)
     ultimately have \<open>norm (A \<psi>) = norm h\<close>
       by (simp add: Proj_ortho_compl cblinfun.diff_left cblinfun.diff_right g_def h_def)
     also have \<open>norm h \<le> norm \<psi>\<close>
       by (smt (verit, del_insts) h_def mult_left_le_one_le norm_Proj_leq1 norm_cblinfun norm_ge_zero)
     finally show \<open>norm (A *\<^sub>V \<psi>) \<le> 1 * norm \<psi>\<close>
       by simp
   qed
 
   from \<open>norm A \<le> 1\<close> and \<open>norm A \<ge> 1\<close>
   show \<open>norm A = 1\<close>
     by auto
 qed
 
 lemma partial_isometry_adj_a_o_a:
   assumes \<open>partial_isometry a\<close>
   shows \<open>a* o\<^sub>C\<^sub>L a = Proj (- kernel a)\<close>
 proof (rule cblinfun_cinner_eqI)
   define P where \<open>P = Proj (- kernel a)\<close>
   have aP: \<open>a o\<^sub>C\<^sub>L P = a\<close>
     by (auto intro!: simp: cblinfun_compose_minus_right P_def Proj_ortho_compl)
   have is_Proj_P[simp]: \<open>is_Proj P\<close>
     by (simp add: P_def)
 
   fix \<psi> :: 'a
   have \<open>\<psi> \<bullet>\<^sub>C ((a* o\<^sub>C\<^sub>L a) *\<^sub>V \<psi>) = a \<psi> \<bullet>\<^sub>C a \<psi>\<close>
     by (simp add: cinner_adj_right)
   also have \<open>\<dots> = a (P \<psi>) \<bullet>\<^sub>C a (P \<psi>)\<close>
     by (metis aP cblinfun_apply_cblinfun_compose)
   also have \<open>\<dots> = P \<psi> \<bullet>\<^sub>C P \<psi>\<close>
     by (metis P_def Proj_range assms cblinfun_apply_in_image cdot_square_norm partial_isometry_def)
   also have \<open>\<dots> = \<psi> \<bullet>\<^sub>C P \<psi>\<close>
     by (simp flip: cinner_adj_right add: is_proj_selfadj is_Proj_idempotent[THEN simp_a_oCL_b'])
   finally show \<open>\<psi> \<bullet>\<^sub>C ((a* o\<^sub>C\<^sub>L a) *\<^sub>V \<psi>) = \<psi> \<bullet>\<^sub>C P \<psi>\<close>
     by -
 qed
 
 lemma partial_isometry_square_proj: \<open>is_Proj (a* o\<^sub>C\<^sub>L a)\<close> if \<open>partial_isometry a\<close>
   by (simp add: partial_isometry_adj_a_o_a that)
 
 lemma partial_isometry_adj[simp]: \<open>partial_isometry (a*)\<close> if \<open>partial_isometry a\<close>
   for a :: \<open>'a::chilbert_space \<Rightarrow>\<^sub>C\<^sub>L 'b::chilbert_space\<close>
 proof -
   have ran_ker: \<open>a *\<^sub>S top = - kernel (a*)\<close>
     by (simp add: kernel_compl_adj_range)
 
   have \<open>norm (a* *\<^sub>V h) = norm h\<close> if \<open>h \<in> range a\<close> for h
   proof -
     from that obtain x where h: \<open>h = a x\<close>
       by auto
     have \<open>norm (a* *\<^sub>V h) = norm (a* *\<^sub>V a *\<^sub>V x)\<close>
       by (simp add: h)
     also have \<open>\<dots> = norm (Proj (- kernel a) *\<^sub>V x)\<close>
       by (simp add: \<open>partial_isometry a\<close> partial_isometry_adj_a_o_a simp_a_oCL_b')
     also have \<open>\<dots> = norm (a *\<^sub>V Proj (- kernel a) *\<^sub>V x)\<close>
       by (metis Proj_range \<open>partial_isometry a\<close> cblinfun_apply_in_image partial_isometry_def)
     also have \<open>\<dots> = norm (a *\<^sub>V x)\<close>
       by (smt (verit, best) Proj_idempotent \<open>partial_isometry a\<close> adj_Proj cblinfun_apply_cblinfun_compose cinner_adj_right cnorm_eq partial_isometry_adj_a_o_a)
     also have \<open>\<dots> = norm h\<close>
       using h by auto
     finally show ?thesis
       by -
   qed
 
   then have norm_pres: \<open>norm (a* *\<^sub>V h) = norm h\<close> if \<open>h \<in> closure (range a)\<close> for h
     using that apply (rule on_closure_eqI)
       apply assumption
     by (intro continuous_intros)+
 
   show ?thesis
     apply (rule partial_isometryI)
     by (auto simp: cblinfun_image.rep_eq norm_pres simp flip: ran_ker)
 qed
 
 subsection \<open>Isomorphisms and inverses\<close>
 
 definition iso_cblinfun :: \<open>('a::complex_normed_vector, 'b::complex_normed_vector) cblinfun \<Rightarrow> bool\<close> where
   \<open>iso_cblinfun A = (\<exists> B. A o\<^sub>C\<^sub>L B = id_cblinfun \<and> B o\<^sub>C\<^sub>L A = id_cblinfun)\<close>
 
 definition cblinfun_inv :: \<open>('a::complex_normed_vector, 'b::complex_normed_vector) cblinfun \<Rightarrow> ('b,'a) cblinfun\<close> where
   \<open>cblinfun_inv A = (SOME B. B o\<^sub>C\<^sub>L A = id_cblinfun)\<close>
 
 lemma
   assumes \<open>iso_cblinfun A\<close>
   shows cblinfun_inv_left: \<open>cblinfun_inv A o\<^sub>C\<^sub>L A = id_cblinfun\<close>
     and cblinfun_inv_right: \<open>A o\<^sub>C\<^sub>L cblinfun_inv A = id_cblinfun\<close>
 proof -
   from assms
   obtain B where AB: \<open>A o\<^sub>C\<^sub>L B = id_cblinfun\<close> and BA: \<open>B o\<^sub>C\<^sub>L A = id_cblinfun\<close>
     using iso_cblinfun_def by blast
   from BA have \<open>cblinfun_inv A o\<^sub>C\<^sub>L A = id_cblinfun\<close>
     by (metis (mono_tags, lifting) cblinfun_inv_def someI_ex)
   with AB BA have \<open>cblinfun_inv A = B\<close>
     by (metis cblinfun_assoc_left(1) cblinfun_compose_id_right)
   with AB BA show \<open>cblinfun_inv A o\<^sub>C\<^sub>L A = id_cblinfun\<close>
     and \<open>A o\<^sub>C\<^sub>L cblinfun_inv A = id_cblinfun\<close>
     by auto
 qed
 
 
 lemma cblinfun_inv_uniq:
   assumes "A o\<^sub>C\<^sub>L B = id_cblinfun" and "B o\<^sub>C\<^sub>L A = id_cblinfun"
   shows "cblinfun_inv A = B"
   using assms by (metis cblinfun_compose_assoc cblinfun_compose_id_right cblinfun_inv_left iso_cblinfun_def)
 
 subsection \<open>One-dimensional spaces\<close>
 
 
 instantiation cblinfun :: (one_dim, one_dim) complex_inner begin
 text \<open>Once we have a theory for the trace, we could instead define the Hilbert-Schmidt inner product
   and relax the \<^class>\<open>one_dim\<close>-sort constraint to (\<^class>\<open>cfinite_dim\<close>,\<^class>\<open>complex_normed_vector\<close>) or similar\<close>
 definition "cinner_cblinfun (A::'a \<Rightarrow>\<^sub>C\<^sub>L 'b) (B::'a \<Rightarrow>\<^sub>C\<^sub>L 'b)
              = cnj (one_dim_iso (A *\<^sub>V 1)) * one_dim_iso (B *\<^sub>V 1)"
 instance
 proof intro_classes
   fix A B C :: "'a \<Rightarrow>\<^sub>C\<^sub>L 'b"
     and c c' :: complex
-  show "\<langle>A, B\<rangle> = cnj \<langle>B, A\<rangle>"
+  show "(A \<bullet>\<^sub>C B) = cnj (B \<bullet>\<^sub>C A)"
     unfolding cinner_cblinfun_def by auto
-  show "\<langle>A + B, C\<rangle> = \<langle>A, C\<rangle> + \<langle>B, C\<rangle>"
+  show "(A + B) \<bullet>\<^sub>C C = (A \<bullet>\<^sub>C C) + (B \<bullet>\<^sub>C C)"
     by (simp add: cinner_cblinfun_def algebra_simps plus_cblinfun.rep_eq)
-  show "\<langle>c *\<^sub>C A, B\<rangle> = cnj c * \<langle>A, B\<rangle>"
+  show "(c *\<^sub>C A \<bullet>\<^sub>C B) = cnj c * (A \<bullet>\<^sub>C B)"
     by (simp add: cblinfun.scaleC_left cinner_cblinfun_def)
-  show "0 \<le> \<langle>A, A\<rangle>"
+  show "0 \<le> (A \<bullet>\<^sub>C A)"
     unfolding cinner_cblinfun_def by auto
   have "bounded_clinear A \<Longrightarrow> A 1 = 0 \<Longrightarrow> A = (\<lambda>_. 0)"
     for A::"'a \<Rightarrow> 'b"
   proof (rule one_dim_clinear_eqI [where x = 1] , auto)
     show "clinear A"
       if "bounded_clinear A"
         and "A 1 = 0"
       for A :: "'a \<Rightarrow> 'b"
       using that
       by (simp add: bounded_clinear.clinear)
     show "clinear ((\<lambda>_. 0)::'a \<Rightarrow> 'b)"
       if "bounded_clinear A"
         and "A 1 = 0"
       for A :: "'a \<Rightarrow> 'b"
       using that
       by (simp add: complex_vector.module_hom_zero)
   qed
   hence "A *\<^sub>V 1 = 0 \<Longrightarrow> A = 0"
     by transfer
   hence "one_dim_iso (A *\<^sub>V 1) = 0 \<Longrightarrow> A = 0"
     by (metis one_dim_iso_of_zero one_dim_iso_inj)
-  thus "(\<langle>A, A\<rangle> = 0) = (A = 0)"
+  thus "((A \<bullet>\<^sub>C A) = 0) = (A = 0)"
     by (auto simp: cinner_cblinfun_def)
 
-  show "norm A = sqrt (cmod \<langle>A, A\<rangle>)"
+  show "norm A = sqrt (cmod (A \<bullet>\<^sub>C A))"
     unfolding cinner_cblinfun_def
     apply transfer
     by (simp add: norm_mult abs_complex_def one_dim_onorm' cnj_x_x power2_eq_square bounded_clinear.clinear)
 qed
 end
 
 instantiation cblinfun :: (one_dim, one_dim) one_dim begin
 lift_definition one_cblinfun :: "'a \<Rightarrow>\<^sub>C\<^sub>L 'b" is "one_dim_iso"
   by (rule bounded_clinear_one_dim_iso)
 lift_definition times_cblinfun :: "'a \<Rightarrow>\<^sub>C\<^sub>L 'b \<Rightarrow> 'a \<Rightarrow>\<^sub>C\<^sub>L 'b \<Rightarrow> 'a \<Rightarrow>\<^sub>C\<^sub>L 'b"
   is "\<lambda>f g. f o one_dim_iso o g"
   by (simp add: comp_bounded_clinear)
 lift_definition inverse_cblinfun :: "'a \<Rightarrow>\<^sub>C\<^sub>L 'b \<Rightarrow> 'a \<Rightarrow>\<^sub>C\<^sub>L 'b" is
   "\<lambda>f. ((*) (one_dim_iso (inverse (f 1)))) o one_dim_iso"
   by (auto intro!: comp_bounded_clinear bounded_clinear_mult_right)
 definition divide_cblinfun :: "'a \<Rightarrow>\<^sub>C\<^sub>L 'b \<Rightarrow> 'a \<Rightarrow>\<^sub>C\<^sub>L 'b \<Rightarrow> 'a \<Rightarrow>\<^sub>C\<^sub>L 'b" where
   "divide_cblinfun A B = A * inverse B"
 definition "canonical_basis_cblinfun = [1 :: 'a \<Rightarrow>\<^sub>C\<^sub>L 'b]"
 instance
 proof intro_classes
   let ?basis = "canonical_basis :: ('a \<Rightarrow>\<^sub>C\<^sub>L 'b) list"
   fix A B C :: "'a \<Rightarrow>\<^sub>C\<^sub>L 'b"
     and c c' :: complex
   show "distinct ?basis"
     unfolding canonical_basis_cblinfun_def by simp
   have "(1::'a \<Rightarrow>\<^sub>C\<^sub>L 'b) \<noteq> (0::'a \<Rightarrow>\<^sub>C\<^sub>L 'b)"
     by (metis cblinfun.zero_left one_cblinfun.rep_eq one_dim_iso_of_one zero_neq_one)
   thus "cindependent (set ?basis)"
     unfolding canonical_basis_cblinfun_def by simp
 
   have "A \<in> cspan (set ?basis)" for A
   proof -
     define c :: complex where "c = one_dim_iso (A *\<^sub>V 1)"
     have "A x = one_dim_iso (A 1) *\<^sub>C one_dim_iso x" for x
       by (smt (z3) cblinfun.scaleC_right complex_vector.scale_left_commute one_dim_iso_idem one_dim_scaleC_1)
     hence "A = one_dim_iso (A *\<^sub>V 1) *\<^sub>C 1"
       apply transfer by metis
     thus "A \<in> cspan (set ?basis)"
       unfolding canonical_basis_cblinfun_def
       by (smt complex_vector.span_base complex_vector.span_scale list.set_intros(1))
   qed
   thus "cspan (set ?basis) = UNIV" by auto
 
   have "A = (1::'a \<Rightarrow>\<^sub>C\<^sub>L 'b) \<Longrightarrow>
     norm (1::'a \<Rightarrow>\<^sub>C\<^sub>L 'b) = (1::real)"
     apply transfer by simp
   thus "A \<in> set ?basis \<Longrightarrow> norm A = 1"
     unfolding canonical_basis_cblinfun_def
     by simp
 
   show "?basis = [1]"
     unfolding canonical_basis_cblinfun_def by simp
   show "c *\<^sub>C 1 * c' *\<^sub>C 1 = (c * c') *\<^sub>C (1::'a\<Rightarrow>\<^sub>C\<^sub>L'b)"
     apply transfer by auto
   have "(1::'a \<Rightarrow>\<^sub>C\<^sub>L 'b) = (0::'a \<Rightarrow>\<^sub>C\<^sub>L 'b) \<Longrightarrow> False"
     by (metis cblinfun.zero_left one_cblinfun.rep_eq one_dim_iso_of_zero' zero_neq_neg_one)
   thus "is_ortho_set (set ?basis)"
     unfolding is_ortho_set_def canonical_basis_cblinfun_def by auto
   show "A div B = A * inverse B"
     by (simp add: divide_cblinfun_def)
   show "inverse (c *\<^sub>C 1) = (1::'a\<Rightarrow>\<^sub>C\<^sub>L'b) /\<^sub>C c"
     apply transfer by (simp add: o_def one_dim_inverse)
 qed
 end
 
 lemma id_cblinfun_eq_1[simp]: \<open>id_cblinfun = 1\<close>
   apply transfer by auto
 
 lemma one_dim_apply_is_times[simp]:
   fixes A :: "'a::one_dim \<Rightarrow>\<^sub>C\<^sub>L 'a" and B :: "'a \<Rightarrow>\<^sub>C\<^sub>L 'a"
   shows "A o\<^sub>C\<^sub>L B = A * B"
   apply transfer by simp
 
 lemma one_comp_one_cblinfun[simp]: "1 o\<^sub>C\<^sub>L 1 = 1"
   apply transfer unfolding o_def by simp
 
 lemma one_cblinfun_adj[simp]: "1* = 1"
   apply transfer by simp
 
-
-lemma scaleC_1_right[simp]: \<open>scaleC x (1::'a::one_dim) = of_complex x\<close>
-  unfolding of_complex_def by simp
-
-lemma scaleC_of_complex[simp]: \<open>scaleC x (of_complex y) = of_complex (x * y)\<close>
-  unfolding of_complex_def using scaleC_scaleC by blast
-
 lemma scaleC_1_apply[simp]: \<open>(x *\<^sub>C 1) *\<^sub>V y = x *\<^sub>C y\<close>
   by (metis cblinfun.scaleC_left cblinfun_id_cblinfun_apply id_cblinfun_eq_1)
 
 lemma cblinfun_apply_1_left[simp]: \<open>1 *\<^sub>V y = y\<close>
   by (metis cblinfun_id_cblinfun_apply id_cblinfun_eq_1)
 
-lemma of_complex_cblinfun_apply[simp]: \<open>of_complex x *\<^sub>V y = x *\<^sub>C y\<close>
-  unfolding of_complex_def
-  by (metis cblinfun.scaleC_left cblinfun_id_cblinfun_apply id_cblinfun_eq_1)
+lemma of_complex_cblinfun_apply[simp]: \<open>of_complex x *\<^sub>V y = one_dim_iso (x *\<^sub>C y)\<close>
+  by (metis of_complex_def cblinfun.scaleC_right one_cblinfun.rep_eq scaleC_cblinfun.rep_eq)
 
 lemma cblinfun_compose_1_left[simp]: \<open>1 o\<^sub>C\<^sub>L x = x\<close>
   apply transfer by auto
 
 lemma cblinfun_compose_1_right[simp]: \<open>x o\<^sub>C\<^sub>L 1 = x\<close>
   apply transfer by auto
 
 lemma one_dim_iso_id_cblinfun: \<open>one_dim_iso id_cblinfun = id_cblinfun\<close>
   by simp
 
 lemma one_dim_iso_id_cblinfun_eq_1: \<open>one_dim_iso id_cblinfun = 1\<close>
   by simp
 
 lemma one_dim_iso_comp_distr[simp]: \<open>one_dim_iso (a o\<^sub>C\<^sub>L b) = one_dim_iso a o\<^sub>C\<^sub>L one_dim_iso b\<close>
   by (smt (z3) cblinfun_compose_scaleC_left cblinfun_compose_scaleC_right one_cinner_a_scaleC_one one_comp_one_cblinfun one_dim_iso_of_one one_dim_iso_scaleC)
 
 lemma one_dim_iso_comp_distr_times[simp]: \<open>one_dim_iso (a o\<^sub>C\<^sub>L b) = one_dim_iso a * one_dim_iso b\<close>
   by (smt (verit, del_insts) mult.left_neutral mult_scaleC_left one_cinner_a_scaleC_one one_comp_one_cblinfun one_dim_iso_of_one one_dim_iso_scaleC cblinfun_compose_scaleC_right cblinfun_compose_scaleC_left)
 
 lemma one_dim_iso_adjoint[simp]: \<open>one_dim_iso (A*) = (one_dim_iso A)*\<close>
   by (smt (z3) one_cblinfun_adj one_cinner_a_scaleC_one one_dim_iso_of_one one_dim_iso_scaleC scaleC_adj)
 
 lemma one_dim_iso_adjoint_complex[simp]: \<open>one_dim_iso (A*) = cnj (one_dim_iso A)\<close>
   by (metis (mono_tags, lifting) one_cblinfun_adj one_dim_iso_idem one_dim_scaleC_1 scaleC_adj)
 
 lemma one_dim_cblinfun_compose_commute: \<open>a o\<^sub>C\<^sub>L b = b o\<^sub>C\<^sub>L a\<close> for a b :: \<open>('a::one_dim,'a) cblinfun\<close>
   by (simp add: one_dim_iso_inj)
 
 lemma one_cblinfun_apply_one[simp]: \<open>1 *\<^sub>V 1 = 1\<close>
   by (simp add: one_cblinfun.rep_eq)
 
-lemma ccspan_one_dim[simp]: \<open>ccspan {x} = top\<close> if \<open>x \<noteq> 0\<close> for x :: \<open>_ :: one_dim\<close>
-proof -
-  have \<open>y \<in> cspan {x}\<close> for y
-    using that by (metis complex_vector.span_base complex_vector.span_zero cspan_singleton_scaleC insertI1 one_dim_scaleC_1 scaleC_zero_left)
-  then show ?thesis
-    by (auto intro!: order.antisym ccsubspace_leI
-      simp: top_ccsubspace.rep_eq ccspan.rep_eq)
-qed
-
 lemma is_onb_one_dim[simp]: \<open>norm x = 1 \<Longrightarrow> is_onb {x}\<close> for x :: \<open>_ :: one_dim\<close>
   by (auto simp: is_onb_def intro!: ccspan_one_dim)
 
 lemma one_dim_iso_cblinfun_comp: \<open>one_dim_iso (a o\<^sub>C\<^sub>L b) = of_complex (cinner (a* *\<^sub>V 1) (b *\<^sub>V 1))\<close>
   for a :: \<open>'a::chilbert_space \<Rightarrow>\<^sub>C\<^sub>L 'b::one_dim\<close> and b :: \<open>'c::one_dim \<Rightarrow>\<^sub>C\<^sub>L 'a\<close>
   by (simp add: cinner_adj_left cinner_cblinfun_def one_dim_iso_def)
 
+lemma one_dim_iso_cblinfun_apply[simp]: \<open>one_dim_iso \<psi> *\<^sub>V \<phi> = one_dim_iso (one_dim_iso \<psi> *\<^sub>C \<phi>)\<close>
+  by (smt (verit) cblinfun.scaleC_left one_cblinfun.rep_eq one_dim_iso_of_one one_dim_iso_scaleC one_dim_scaleC_1)
+
 subsection \<open>Loewner order\<close>
 
 lift_definition heterogenous_cblinfun_id :: \<open>'a::complex_normed_vector \<Rightarrow>\<^sub>C\<^sub>L 'b::complex_normed_vector\<close>
   is \<open>if bounded_clinear (heterogenous_identity :: 'a::complex_normed_vector \<Rightarrow> 'b::complex_normed_vector) then heterogenous_identity else (\<lambda>_. 0)\<close>
   by auto
 
 lemma heterogenous_cblinfun_id_def'[simp]: "heterogenous_cblinfun_id = id_cblinfun"
   apply transfer by auto
 
 definition "heterogenous_same_type_cblinfun (x::'a::chilbert_space itself) (y::'b::chilbert_space itself) \<longleftrightarrow>
   unitary (heterogenous_cblinfun_id :: 'a \<Rightarrow>\<^sub>C\<^sub>L 'b) \<and> unitary (heterogenous_cblinfun_id :: 'b \<Rightarrow>\<^sub>C\<^sub>L 'a)"
 
 lemma heterogenous_same_type_cblinfun[simp]: \<open>heterogenous_same_type_cblinfun (x::'a::chilbert_space itself) (y::'a::chilbert_space itself)\<close>
   unfolding heterogenous_same_type_cblinfun_def by auto
 
 instantiation cblinfun :: (chilbert_space, chilbert_space) ord begin
 definition less_eq_cblinfun :: \<open>('a \<Rightarrow>\<^sub>C\<^sub>L 'b) \<Rightarrow> ('a \<Rightarrow>\<^sub>C\<^sub>L 'b) \<Rightarrow> bool\<close>
   where less_eq_cblinfun_def_heterogenous: \<open>less_eq_cblinfun A B =
   (if heterogenous_same_type_cblinfun TYPE('a) TYPE('b) then
     \<forall>\<psi>::'b. cinner \<psi> ((B-A) *\<^sub>V heterogenous_cblinfun_id *\<^sub>V \<psi>) \<ge> 0 else (A=B))\<close>
 definition \<open>less_cblinfun (A :: 'a \<Rightarrow>\<^sub>C\<^sub>L 'b) B \<longleftrightarrow> A \<le> B \<and> \<not> B \<le> A\<close>
 instance..
 end
 
 lemma less_eq_cblinfun_def: \<open>A \<le> B \<longleftrightarrow>
     (\<forall>\<psi>. cinner \<psi> (A *\<^sub>V \<psi>) \<le> cinner \<psi> (B *\<^sub>V \<psi>))\<close>
   unfolding less_eq_cblinfun_def_heterogenous
   by (auto simp del: less_eq_complex_def simp: cblinfun.diff_left cinner_diff_right)
 
 instantiation cblinfun :: (chilbert_space, chilbert_space) ordered_complex_vector begin
 instance
 proof intro_classes
-  note less_eq_complex_def[simp del]
-
   fix x y z :: \<open>'a \<Rightarrow>\<^sub>C\<^sub>L 'b\<close>
   fix a b :: complex
 
   define pos where \<open>pos X \<longleftrightarrow> (\<forall>\<psi>. cinner \<psi> (X *\<^sub>V \<psi>) \<ge> 0)\<close> for X :: \<open>'b \<Rightarrow>\<^sub>C\<^sub>L 'b\<close>
   consider (unitary) \<open>heterogenous_same_type_cblinfun TYPE('a) TYPE('b)\<close>
       \<open>\<And>A B :: 'a \<Rightarrow>\<^sub>C\<^sub>L 'b. A \<le> B = pos ((B-A) o\<^sub>C\<^sub>L (heterogenous_cblinfun_id :: 'b\<Rightarrow>\<^sub>C\<^sub>L'a))\<close>
     | (trivial) \<open>\<And>A B :: 'a \<Rightarrow>\<^sub>C\<^sub>L 'b. A \<le> B \<longleftrightarrow> A = B\<close>
     apply atomize_elim by (auto simp: pos_def less_eq_cblinfun_def_heterogenous)
   note cases = this
 
   have [simp]: \<open>pos 0\<close>
     unfolding pos_def by auto
 
   have pos_nondeg: \<open>X = 0\<close> if \<open>pos X\<close> and \<open>pos (-X)\<close> for X
     apply (rule cblinfun_cinner_eqI, simp)
     using that by (metis (no_types, lifting) cblinfun.minus_left cinner_minus_right dual_order.antisym equation_minus_iff neg_le_0_iff_le pos_def)
 
   have pos_add: \<open>pos (X+Y)\<close> if \<open>pos X\<close> and \<open>pos Y\<close> for X Y
     by (smt (z3) pos_def cblinfun.diff_left cinner_minus_right cinner_simps(3) diff_ge_0_iff_ge diff_minus_eq_add neg_le_0_iff_le order_trans that(1) that(2) uminus_cblinfun.rep_eq)
 
   have pos_scaleC: \<open>pos (a *\<^sub>C X)\<close> if \<open>a\<ge>0\<close> and \<open>pos X\<close> for X a
     using that unfolding pos_def by (auto simp: cblinfun.scaleC_left)
 
   let ?id = \<open>heterogenous_cblinfun_id :: 'b \<Rightarrow>\<^sub>C\<^sub>L 'a\<close>
 
   show \<open>x \<le> x\<close>
     apply (cases rule:cases) by auto
   show \<open>(x < y) \<longleftrightarrow> (x \<le> y \<and> \<not> y \<le> x)\<close>
     unfolding less_cblinfun_def by simp
   show \<open>x \<le> z\<close> if \<open>x \<le> y\<close> and \<open>y \<le> z\<close>
   proof (cases rule:cases)
     case unitary
     define a b :: \<open>'b \<Rightarrow>\<^sub>C\<^sub>L 'b\<close> where \<open>a = (y-x) o\<^sub>C\<^sub>L heterogenous_cblinfun_id\<close>
       and \<open>b = (z-y) o\<^sub>C\<^sub>L heterogenous_cblinfun_id\<close>
     with unitary that have \<open>pos a\<close> and \<open>pos b\<close>
       by auto
     then have \<open>pos (a + b)\<close>
       by (rule pos_add)
     moreover have \<open>a + b = (z - x) o\<^sub>C\<^sub>L heterogenous_cblinfun_id\<close>
       unfolding a_def b_def
       by (metis (no_types, lifting) bounded_cbilinear.add_left bounded_cbilinear_cblinfun_compose diff_add_cancel ordered_field_class.sign_simps(2) ordered_field_class.sign_simps(8))
     ultimately show ?thesis
       using unitary by auto
   next
     case trivial
     with that show ?thesis by auto
   qed
   show \<open>x = y\<close> if \<open>x \<le> y\<close> and \<open>y \<le> x\<close>
   proof (cases rule:cases)
     case unitary
     then have \<open>unitary ?id\<close>
       by (auto simp: heterogenous_same_type_cblinfun_def)
     define a b :: \<open>'b \<Rightarrow>\<^sub>C\<^sub>L 'b\<close> where \<open>a = (y-x) o\<^sub>C\<^sub>L ?id\<close>
       and \<open>b = (x-y) o\<^sub>C\<^sub>L ?id\<close>
     with unitary that have \<open>pos a\<close> and \<open>pos b\<close>
       by auto
     then have \<open>a = 0\<close>
       apply (rule_tac pos_nondeg)
        apply (auto simp: a_def b_def)
       by (smt (verit, best) add.commute bounded_cbilinear.add_left bounded_cbilinear_cblinfun_compose cblinfun_compose_zero_left diff_0 diff_add_cancel group_cancel.rule0 group_cancel.sub1)
     then show ?thesis
       unfolding a_def using \<open>unitary ?id\<close>
       by (metis cblinfun_compose_assoc cblinfun_compose_id_right cblinfun_compose_zero_left eq_iff_diff_eq_0 unitaryD2)
   next
     case trivial
     with that show ?thesis by simp
   qed
   show \<open>x + y \<le> x + z\<close> if \<open>y \<le> z\<close>
   proof (cases rule:cases)
     case unitary
     with that show ?thesis
       by auto
   next
     case trivial
     with that show ?thesis
       by auto
   qed
 
   show \<open>a *\<^sub>C x \<le> a *\<^sub>C y\<close> if \<open>x \<le> y\<close> and \<open>0 \<le> a\<close>
   proof (cases rule:cases)
     case unitary
     with that pos_scaleC show ?thesis
       by (metis cblinfun_compose_scaleC_left complex_vector.scale_right_diff_distrib)
   next
     case trivial
     with that show ?thesis
       by auto
   qed
 
   show \<open>a *\<^sub>C x \<le> b *\<^sub>C x\<close> if \<open>a \<le> b\<close> and \<open>0 \<le> x\<close>
   proof (cases rule:cases)
     case unitary
     with that show ?thesis
       by (auto intro!: pos_scaleC simp flip: scaleC_diff_left)
   next
     case trivial
     with that show ?thesis
       by auto
   qed
 qed
 end
 
 lemma positive_id_cblinfun[simp]: "id_cblinfun \<ge> 0"
   unfolding less_eq_cblinfun_def using cinner_ge_zero by auto
 
 lemma positive_hermitianI: \<open>A = A*\<close> if \<open>A \<ge> 0\<close>
   apply (rule cinner_real_hermiteanI)
   using that by (auto simp: complex_is_real_iff_compare0 less_eq_cblinfun_def)
 
 lemma cblinfun_leI:
   assumes \<open>\<And>x. norm x = 1 \<Longrightarrow> x \<bullet>\<^sub>C (A *\<^sub>V x) \<le> x \<bullet>\<^sub>C (B *\<^sub>V x)\<close>
   shows \<open>A \<le> B\<close>
 proof (unfold less_eq_cblinfun_def, intro allI, case_tac \<open>\<psi> = 0\<close>)
   fix \<psi> :: 'a assume \<open>\<psi> = 0\<close>
   then show \<open>\<psi> \<bullet>\<^sub>C (A *\<^sub>V \<psi>) \<le> \<psi> \<bullet>\<^sub>C (B *\<^sub>V \<psi>)\<close>
     by simp
 next
   fix \<psi> :: 'a assume \<open>\<psi> \<noteq> 0\<close>
   define \<phi> where \<open>\<phi> = \<psi> /\<^sub>R norm \<psi>\<close>
   have \<open>\<phi> \<bullet>\<^sub>C (A *\<^sub>V \<phi>) \<le> \<phi> \<bullet>\<^sub>C (B *\<^sub>V \<phi>)\<close>
     apply (rule assms)
     unfolding \<phi>_def
     by (simp add: \<open>\<psi> \<noteq> 0\<close>)
   with \<open>\<psi> \<noteq> 0\<close> show \<open>\<psi> \<bullet>\<^sub>C (A *\<^sub>V \<psi>) \<le> \<psi> \<bullet>\<^sub>C (B *\<^sub>V \<psi>)\<close>
     unfolding \<phi>_def
     by (smt (verit) cinner_adj_left cinner_scaleR_left cinner_simps(6) complex_of_real_nn_iff mult_cancel_right1 mult_left_mono norm_eq_zero norm_ge_zero of_real_1 right_inverse scaleR_scaleC scaleR_scaleR)
 qed
 
 lemma positive_cblinfunI: \<open>A \<ge> 0\<close> if \<open>\<And>x. norm x = 1 \<Longrightarrow> cinner x (A *\<^sub>V x) \<ge> 0\<close>
   apply (rule cblinfun_leI)
   using that by simp
 
 (* Note: this does not require B to be a square operator *)
 lemma positive_cblinfun_squareI: \<open>A = B* o\<^sub>C\<^sub>L B \<Longrightarrow> A \<ge> 0\<close>
   apply (rule positive_cblinfunI)
   by (metis cblinfun_apply_cblinfun_compose cinner_adj_right cinner_ge_zero)
 
 lemma one_dim_loewner_order: \<open>A \<ge> B \<longleftrightarrow> one_dim_iso A \<ge> (one_dim_iso B :: complex)\<close> for A B :: \<open>'a \<Rightarrow>\<^sub>C\<^sub>L 'a::{chilbert_space, one_dim}\<close>
 proof -
-  note less_eq_complex_def[simp del]
   have A: \<open>A = one_dim_iso A *\<^sub>C id_cblinfun\<close>
     by simp
   have B: \<open>B = one_dim_iso B *\<^sub>C id_cblinfun\<close>
     by simp
   have \<open>A \<ge> B \<longleftrightarrow> (\<forall>\<psi>. cinner \<psi> (A \<psi>) \<ge> cinner \<psi> (B \<psi>))\<close>
     by (simp add: less_eq_cblinfun_def)
   also have \<open>\<dots> \<longleftrightarrow> (\<forall>\<psi>::'a. one_dim_iso B * (\<psi> \<bullet>\<^sub>C \<psi>) \<le> one_dim_iso A * (\<psi> \<bullet>\<^sub>C \<psi>))\<close>
     apply (subst A, subst B)
     by (metis (no_types, opaque_lifting) cinner_scaleC_right id_cblinfun_apply scaleC_cblinfun.rep_eq)
   also have \<open>\<dots> \<longleftrightarrow> one_dim_iso A \<ge> (one_dim_iso B :: complex)\<close>
     by (auto intro!: mult_right_mono elim!: allE[where x=1])
   finally show ?thesis
     by -
 qed
 
 lemma one_dim_positive: \<open>A \<ge> 0 \<longleftrightarrow> one_dim_iso A \<ge> (0::complex)\<close> for A :: \<open>'a \<Rightarrow>\<^sub>C\<^sub>L 'a::{chilbert_space, one_dim}\<close>
   using one_dim_loewner_order[where B=0] by auto
 
 lemma op_square_nondegenerate: \<open>a = 0\<close> if \<open>a* o\<^sub>C\<^sub>L a = 0\<close>
 proof (rule cblinfun_eq_0_on_UNIV_span[where basis=UNIV]; simp)
   fix s
   from that have \<open>s \<bullet>\<^sub>C ((a* o\<^sub>C\<^sub>L a) *\<^sub>V s) = 0\<close>
     by simp
   then have \<open>(a *\<^sub>V s) \<bullet>\<^sub>C (a *\<^sub>V s) = 0\<close>
     by (simp add: cinner_adj_right)
   then show \<open>a *\<^sub>V s = 0\<close>
     by simp
 qed
 
 lemma comparable_hermitean:
   assumes \<open>a \<le> b\<close>
   assumes \<open>a* = a\<close>
   shows \<open>b* = b\<close>
   by (smt (verit, best) assms(1) assms(2) cinner_hermitian_real cinner_real_hermiteanI comparable complex_is_real_iff_compare0 less_eq_cblinfun_def)
 
 lemma comparable_hermitean':
   assumes \<open>a \<le> b\<close>
   assumes \<open>b* = b\<close>
   shows \<open>a* = a\<close>
   by (smt (verit, best) assms(1) assms(2) cinner_hermitian_real cinner_real_hermiteanI comparable complex_is_real_iff_compare0 less_eq_cblinfun_def)
 
 lemma Proj_mono: \<open>Proj S \<le> Proj T \<longleftrightarrow> S \<le> T\<close>
 proof (rule iffI)
   assume \<open>S \<le> T\<close>
   define D where \<open>D = Proj T - Proj S\<close>
   from \<open>S \<le> T\<close> have TS_S[simp]: \<open>Proj T o\<^sub>C\<^sub>L Proj S = Proj S\<close>
     by (smt (verit, ccfv_threshold) Proj_idempotent Proj_range cblinfun_apply_cblinfun_compose cblinfun_apply_in_image cblinfun_eqI cblinfun_fixes_range less_eq_ccsubspace.rep_eq subset_iff)
   then have ST_S[simp]: \<open>Proj S o\<^sub>C\<^sub>L Proj T = Proj S\<close>
     by (metis adj_Proj adj_cblinfun_compose)
   have \<open>D* o\<^sub>C\<^sub>L D = D\<close>
     by (simp add: D_def cblinfun_compose_minus_left cblinfun_compose_minus_right adj_minus adj_Proj)
   then have \<open>D \<ge> 0\<close>
     by (metis positive_cblinfun_squareI)
   then show \<open>Proj S \<le> Proj T\<close>
     by (simp add: D_def)
 next
   assume PS_PT: \<open>Proj S \<le> Proj T\<close>
   show \<open>S \<le> T\<close>
   proof (rule ccsubspace_leI_unit)
     fix \<psi> assume \<open>\<psi> \<in> space_as_set S\<close> and [simp]: \<open>norm \<psi> = 1\<close>
     then have \<open>1 = norm (Proj S *\<^sub>V \<psi>)\<close>
-      by (simp add: cancel_apply_Proj)
+      by (simp add: Proj_fixes_image)
     also from PS_PT have \<open>\<dots> \<le> norm (Proj T *\<^sub>V \<psi>)\<close>
       by (metis (no_types, lifting) Proj_idempotent adj_Proj cblinfun_apply_cblinfun_compose cinner_adj_left cnorm_le less_eq_cblinfun_def)
     also have \<open>\<dots> \<le> 1\<close>
       by (metis Proj_is_Proj \<open>norm \<psi> = 1\<close> is_Proj_reduces_norm)
     ultimately have \<open>norm (Proj T *\<^sub>V \<psi>) = 1\<close>
       by auto
     then show \<open>\<psi> \<in> space_as_set T\<close>
       by (simp add: norm_Proj_apply_1)
   qed
 qed
 
 subsection \<open>Embedding vectors to operators\<close>
 
 lift_definition vector_to_cblinfun :: \<open>'a::complex_normed_vector \<Rightarrow> 'b::one_dim \<Rightarrow>\<^sub>C\<^sub>L 'a\<close> is
   \<open>\<lambda>\<psi> \<phi>. one_dim_iso \<phi> *\<^sub>C \<psi>\<close>
   by (simp add: bounded_clinear_scaleC_const)
 
-lemma vector_to_cblinfun_cblinfun_apply:
-  "vector_to_cblinfun (A *\<^sub>V \<psi>) = A  o\<^sub>C\<^sub>L (vector_to_cblinfun \<psi>)"
+lemma vector_to_cblinfun_cblinfun_compose[simp]:
+  "A  o\<^sub>C\<^sub>L (vector_to_cblinfun \<psi>) = vector_to_cblinfun (A *\<^sub>V \<psi>)"
   apply transfer
   unfolding comp_def bounded_clinear_def clinear_def Vector_Spaces.linear_def
     module_hom_def module_hom_axioms_def
   by simp
 
 lemma vector_to_cblinfun_add: \<open>vector_to_cblinfun (x + y) = vector_to_cblinfun x + vector_to_cblinfun y\<close>
   apply transfer
   by (simp add: scaleC_add_right)
 
 lemma norm_vector_to_cblinfun[simp]: "norm (vector_to_cblinfun x) = norm x"
 proof transfer
   have "bounded_clinear (one_dim_iso::'a \<Rightarrow> complex)"
     by simp
   moreover have "onorm (one_dim_iso::'a \<Rightarrow> complex) * norm x = norm x"
     for x :: 'b
     by simp
   ultimately show "onorm (\<lambda>\<phi>. one_dim_iso (\<phi>::'a) *\<^sub>C x) = norm x"
     for x :: 'b
     by (subst onorm_scaleC_left)
 qed
 
 lemma bounded_clinear_vector_to_cblinfun[bounded_clinear]: "bounded_clinear vector_to_cblinfun"
   apply (rule bounded_clinearI[where K=1])
     apply (transfer, simp add: scaleC_add_right)
    apply (transfer, simp add: mult.commute)
   by simp
 
 lemma vector_to_cblinfun_scaleC[simp]:
   "vector_to_cblinfun (a *\<^sub>C \<psi>) = a *\<^sub>C vector_to_cblinfun \<psi>" for a::complex
-proof (subst asm_rl [of "a *\<^sub>C \<psi> = (a *\<^sub>C id_cblinfun) *\<^sub>V \<psi>"])
-  show "a *\<^sub>C \<psi> = a *\<^sub>C id_cblinfun *\<^sub>V \<psi>"
-    by (simp add: scaleC_cblinfun.rep_eq)
-  show "vector_to_cblinfun (a *\<^sub>C id_cblinfun *\<^sub>V \<psi>) = a *\<^sub>C (vector_to_cblinfun \<psi>::'a \<Rightarrow>\<^sub>C\<^sub>L 'b)"
-    by (metis cblinfun_id_cblinfun_apply cblinfun_compose_scaleC_left vector_to_cblinfun_cblinfun_apply)
-qed
+  by (intro clinear.scaleC bounded_clinear.clinear bounded_clinear_vector_to_cblinfun)
 
 lemma vector_to_cblinfun_apply_one_dim[simp]:
   shows "vector_to_cblinfun \<phi> *\<^sub>V \<gamma> = one_dim_iso \<gamma> *\<^sub>C \<phi>"
   apply transfer by (rule refl)
 
+lemma vector_to_cblinfun_one_dim_iso[simp]: \<open>vector_to_cblinfun = one_dim_iso\<close>
+  by (auto intro!: ext cblinfun_eqI)
+
 lemma vector_to_cblinfun_adj_apply[simp]:
   shows "vector_to_cblinfun \<psi>* *\<^sub>V \<phi> = of_complex (cinner \<psi> \<phi>)"
   by (simp add: cinner_adj_right one_dim_iso_def one_dim_iso_inj)
 
 lemma vector_to_cblinfun_comp_one[simp]:
   "(vector_to_cblinfun s :: 'a::one_dim \<Rightarrow>\<^sub>C\<^sub>L _) o\<^sub>C\<^sub>L 1
      = (vector_to_cblinfun s :: 'b::one_dim \<Rightarrow>\<^sub>C\<^sub>L _)"
   apply (transfer fixing: s)
   by fastforce
 
 lemma vector_to_cblinfun_0[simp]: "vector_to_cblinfun 0 = 0"
-  by (metis cblinfun.zero_left cblinfun_compose_zero_left vector_to_cblinfun_cblinfun_apply)
-
-lemma image_vector_to_cblinfun[simp]: "vector_to_cblinfun x *\<^sub>S top = ccspan {x}"
+  by (metis cblinfun.zero_left cblinfun_compose_zero_left vector_to_cblinfun_cblinfun_compose)
+
+lemma image_vector_to_cblinfun[simp]: "vector_to_cblinfun x *\<^sub>S \<top> = ccspan {x}"
+  \<comment> \<open>Not that the general case \<^term>\<open>vector_to_cblinfun x *\<^sub>S S\<close> can be handled by using
+      that \<open>S = \<top>\<close> or \<open>S = \<bottom>\<close> by @{thm [source] one_dim_ccsubspace_all_or_nothing}\<close>
 proof transfer
   show "closure (range (\<lambda>\<phi>::'b. one_dim_iso \<phi> *\<^sub>C x)) = closure (cspan {x})"
     for x :: 'a
   proof (rule arg_cong [where f = closure])
     have "k *\<^sub>C x \<in> range (\<lambda>\<phi>. one_dim_iso \<phi> *\<^sub>C x)" for k
       by (smt (z3) id_apply one_dim_iso_id one_dim_iso_idem range_eqI)
     thus "range (\<lambda>\<phi>. one_dim_iso (\<phi>::'b) *\<^sub>C x) = cspan {x}"
       unfolding complex_vector.span_singleton
       by auto
   qed
 qed
 
 lemma vector_to_cblinfun_adj_comp_vector_to_cblinfun[simp]:
   shows "vector_to_cblinfun \<psi>* o\<^sub>C\<^sub>L vector_to_cblinfun \<phi> = cinner \<psi> \<phi> *\<^sub>C id_cblinfun"
 proof -
-  have "one_dim_iso \<gamma> *\<^sub>C one_dim_iso (of_complex \<langle>\<psi>, \<phi>\<rangle>) =
-    \<langle>\<psi>, \<phi>\<rangle> *\<^sub>C one_dim_iso \<gamma>"
+  have "one_dim_iso \<gamma> *\<^sub>C one_dim_iso (of_complex (\<psi> \<bullet>\<^sub>C \<phi>)) =
+    (\<psi> \<bullet>\<^sub>C \<phi>) *\<^sub>C one_dim_iso \<gamma>"
     for \<gamma> :: "'c::one_dim"
     by (metis complex_vector.scale_left_commute of_complex_def one_dim_iso_of_one one_dim_iso_scaleC one_dim_scaleC_1)
   hence "one_dim_iso ((vector_to_cblinfun \<psi>* o\<^sub>C\<^sub>L vector_to_cblinfun \<phi>) *\<^sub>V \<gamma>)
       = one_dim_iso ((cinner \<psi> \<phi> *\<^sub>C id_cblinfun) *\<^sub>V \<gamma>)"
     for \<gamma> :: "'c::one_dim"
     by simp
   hence "((vector_to_cblinfun \<psi>* o\<^sub>C\<^sub>L vector_to_cblinfun \<phi>) *\<^sub>V \<gamma>) = ((cinner \<psi> \<phi> *\<^sub>C id_cblinfun) *\<^sub>V \<gamma>)"
     for \<gamma> :: "'c::one_dim"
     by (rule one_dim_iso_inj)
   thus ?thesis
     using cblinfun_eqI[where x = "vector_to_cblinfun \<psi>* o\<^sub>C\<^sub>L vector_to_cblinfun \<phi>"
-        and y = "\<langle>\<psi>, \<phi>\<rangle> *\<^sub>C id_cblinfun"]
+        and y = "(\<psi> \<bullet>\<^sub>C \<phi>) *\<^sub>C id_cblinfun"]
     by auto
 qed
 
 lemma isometry_vector_to_cblinfun[simp]:
   assumes "norm x = 1"
   shows "isometry (vector_to_cblinfun x)"
   using assms cnorm_eq_1 isometry_def by force
 
-subsection \<open>Butterflies (rank-1 projectors)\<close>
-
-definition butterfly_def: "butterfly (s::'a::complex_normed_vector) (t::'b::chilbert_space)
+lemma image_vector_to_cblinfun_adj: 
+  assumes \<open>\<psi> \<notin> space_as_set (- S)\<close>
+  shows \<open>(vector_to_cblinfun \<psi>)* *\<^sub>S S = \<top>\<close>
+proof -
+  from assms obtain \<phi> where \<open>\<phi> \<in> space_as_set S\<close> and \<open>\<not> is_orthogonal \<psi> \<phi>\<close>
+    by (metis orthogonal_complementI uminus_ccsubspace.rep_eq)
+  have \<open>((vector_to_cblinfun \<psi>)* *\<^sub>S S :: 'b ccsubspace) \<ge> (vector_to_cblinfun \<psi>)* *\<^sub>S ccspan {\<phi>}\<close> (is \<open>_ \<ge> \<dots>\<close>)
+    by (simp add: \<open>\<phi> \<in> space_as_set S\<close> cblinfun_image_mono ccspan_leqI)
+  also have \<open>\<dots> = ccspan {(vector_to_cblinfun \<psi>)* *\<^sub>V \<phi>}\<close>
+    by (auto simp: cblinfun_image_ccspan)
+  also have \<open>\<dots> = ccspan {of_complex (\<psi> \<bullet>\<^sub>C \<phi>)}\<close>
+    by auto
+  also have \<open>\<dots> > \<bottom>\<close>
+    by (simp add: \<open>\<psi> \<bullet>\<^sub>C \<phi> \<noteq> 0\<close> flip: bot.not_eq_extremum )
+  finally(dual_order.strict_trans1) show ?thesis
+    using one_dim_ccsubspace_all_or_nothing bot.not_eq_extremum by auto
+qed
+
+
+lemma image_vector_to_cblinfun_adj': 
+  assumes \<open>\<psi> \<noteq> 0\<close>
+  shows \<open>(vector_to_cblinfun \<psi>)* *\<^sub>S \<top> = \<top>\<close>
+  apply (rule image_vector_to_cblinfun_adj)
+  using assms by simp
+
+subsection \<open>Rank-1 operators / butterflies\<close>
+
+definition rank1 where \<open>rank1 A \<longleftrightarrow> (\<exists>\<psi>\<noteq>0. A *\<^sub>S \<top> = ccspan {\<psi>})\<close>
+
+definition "butterfly (s::'a::complex_normed_vector) (t::'b::chilbert_space)
    = vector_to_cblinfun s o\<^sub>C\<^sub>L (vector_to_cblinfun t :: complex \<Rightarrow>\<^sub>C\<^sub>L _)*"
 
 abbreviation "selfbutter s \<equiv> butterfly s s"
 
 lemma butterfly_add_left: \<open>butterfly (a + a') b = butterfly a b + butterfly a' b\<close>
   by (simp add: butterfly_def vector_to_cblinfun_add cbilinear_add_left bounded_cbilinear.add_left bounded_cbilinear_cblinfun_compose)
 
 lemma butterfly_add_right: \<open>butterfly a (b + b') = butterfly a b + butterfly a b'\<close>
   by (simp add: butterfly_def adj_plus vector_to_cblinfun_add cblinfun_compose_add_right)
 
 lemma butterfly_def_one_dim: "butterfly s t = (vector_to_cblinfun s :: 'c::one_dim \<Rightarrow>\<^sub>C\<^sub>L _)
                                           o\<^sub>C\<^sub>L (vector_to_cblinfun t :: 'c \<Rightarrow>\<^sub>C\<^sub>L _)*"
   (is "_ = ?rhs") for s :: "'a::complex_normed_vector" and t :: "'b::chilbert_space"
 proof -
   let ?isoAC = "1 :: 'c \<Rightarrow>\<^sub>C\<^sub>L complex"
   let ?isoCA = "1 :: complex \<Rightarrow>\<^sub>C\<^sub>L 'c"
   let ?vector = "vector_to_cblinfun :: _ \<Rightarrow> ('c \<Rightarrow>\<^sub>C\<^sub>L _)"
 
   have "butterfly s t =
     (?vector s o\<^sub>C\<^sub>L ?isoCA) o\<^sub>C\<^sub>L (?vector t o\<^sub>C\<^sub>L ?isoCA)*"
     unfolding butterfly_def vector_to_cblinfun_comp_one by simp
   also have "\<dots> = ?vector s o\<^sub>C\<^sub>L (?isoCA o\<^sub>C\<^sub>L ?isoCA*) o\<^sub>C\<^sub>L (?vector t)*"
     by (metis (no_types, lifting) cblinfun_compose_assoc adj_cblinfun_compose)
   also have "\<dots> = ?rhs"
     by simp
   finally show ?thesis
     by simp
 qed
 
 lemma butterfly_comp_cblinfun: "butterfly \<psi> \<phi> o\<^sub>C\<^sub>L a = butterfly \<psi> (a* *\<^sub>V \<phi>)"
   unfolding butterfly_def
-  by (simp add: cblinfun_compose_assoc vector_to_cblinfun_cblinfun_apply)
+  by (simp add: cblinfun_compose_assoc flip: vector_to_cblinfun_cblinfun_compose)
 
 lemma cblinfun_comp_butterfly: "a o\<^sub>C\<^sub>L butterfly \<psi> \<phi> = butterfly (a *\<^sub>V \<psi>) \<phi>"
   unfolding butterfly_def
-  by (simp add: cblinfun_compose_assoc vector_to_cblinfun_cblinfun_apply)
-
-lemma butterfly_apply[simp]: "butterfly \<psi> \<psi>' *\<^sub>V \<phi> = \<langle>\<psi>', \<phi>\<rangle> *\<^sub>C \<psi>"
+  by (simp add: cblinfun_compose_assoc flip: vector_to_cblinfun_cblinfun_compose)
+
+lemma butterfly_apply[simp]: "butterfly \<psi> \<psi>' *\<^sub>V \<phi> = (\<psi>' \<bullet>\<^sub>C \<phi>) *\<^sub>C \<psi>"
   by (simp add: butterfly_def scaleC_cblinfun.rep_eq)
 
 lemma butterfly_scaleC_left[simp]: "butterfly (c *\<^sub>C \<psi>) \<phi> = c *\<^sub>C butterfly \<psi> \<phi>"
   unfolding butterfly_def vector_to_cblinfun_scaleC scaleC_adj
   by (simp add: cnj_x_x)
 
 lemma butterfly_scaleC_right[simp]: "butterfly \<psi> (c *\<^sub>C \<phi>) = cnj c *\<^sub>C butterfly \<psi> \<phi>"
   unfolding butterfly_def vector_to_cblinfun_scaleC scaleC_adj
   by (simp add: cnj_x_x)
 
 lemma butterfly_scaleR_left[simp]: "butterfly (r *\<^sub>R \<psi>) \<phi> = r *\<^sub>C butterfly \<psi> \<phi>"
   by (simp add: scaleR_scaleC)
 
 lemma butterfly_scaleR_right[simp]: "butterfly \<psi> (r *\<^sub>R \<phi>) = r *\<^sub>C butterfly \<psi> \<phi>"
   by (simp add: butterfly_scaleC_right scaleR_scaleC)
 
 lemma butterfly_adjoint[simp]: "(butterfly \<psi> \<phi>)* = butterfly \<phi> \<psi>"
   unfolding butterfly_def by auto
 
-lemma butterfly_comp_butterfly[simp]: "butterfly \<psi>1 \<psi>2 o\<^sub>C\<^sub>L butterfly \<psi>3 \<psi>4 = \<langle>\<psi>2, \<psi>3\<rangle> *\<^sub>C butterfly \<psi>1 \<psi>4"
+lemma butterfly_comp_butterfly[simp]: "butterfly \<psi>1 \<psi>2 o\<^sub>C\<^sub>L butterfly \<psi>3 \<psi>4 = (\<psi>2 \<bullet>\<^sub>C \<psi>3) *\<^sub>C butterfly \<psi>1 \<psi>4"
   by (simp add: butterfly_comp_cblinfun)
 
 lemma butterfly_0_left[simp]: "butterfly 0 a = 0"
   by (simp add: butterfly_def)
 
 lemma butterfly_0_right[simp]: "butterfly a 0 = 0"
   by (simp add: butterfly_def)
 
+lemma butterfly_is_rank1:
+  assumes \<open>\<phi> \<noteq> 0\<close>
+  shows \<open>butterfly \<psi> \<phi> *\<^sub>S \<top> = ccspan {\<psi>}\<close>
+  using assms by (simp add: butterfly_def cblinfun_compose_image image_vector_to_cblinfun_adj')
+
+
+lemma rank1_is_butterfly:
+  assumes \<open>A *\<^sub>S \<top> = ccspan {\<psi>::_::chilbert_space}\<close>
+  shows \<open>\<exists>\<phi>. A = butterfly \<psi> \<phi>\<close>
+proof (rule exI[of _ \<open>A* *\<^sub>V (\<psi> /\<^sub>R (norm \<psi>)\<^sup>2)\<close>], rule cblinfun_eqI)
+  fix \<gamma> :: 'b
+  from assms have \<open>A *\<^sub>V \<gamma> \<in> space_as_set (ccspan {\<psi>})\<close>
+    by (simp flip: assms)
+  then obtain c where c: \<open>A *\<^sub>V \<gamma> = c *\<^sub>C \<psi>\<close>
+    apply atomize_elim
+    apply (auto simp: ccspan.rep_eq)
+    by (metis complex_vector.span_breakdown_eq complex_vector.span_empty eq_iff_diff_eq_0 singletonD)
+  have \<open>A *\<^sub>V \<gamma> = butterfly \<psi> (\<psi> /\<^sub>R (norm \<psi>)\<^sup>2) *\<^sub>V (A *\<^sub>V \<gamma>)\<close>
+    apply (auto simp: c simp flip: scaleC_scaleC)
+    by (metis cinner_eq_zero_iff divideC_field_simps(1) power2_norm_eq_cinner scaleC_left_commute scaleC_zero_right)
+  also have \<open>\<dots> = (butterfly \<psi> (\<psi> /\<^sub>R (norm \<psi>)\<^sup>2) o\<^sub>C\<^sub>L A) *\<^sub>V \<gamma>\<close>
+    by simp
+  also have \<open>\<dots> = butterfly \<psi> (A* *\<^sub>V (\<psi> /\<^sub>R (norm \<psi>)\<^sup>2)) *\<^sub>V \<gamma>\<close>
+    by (simp add: cinner_adj_left)
+  finally show \<open>A *\<^sub>V \<gamma> = \<dots>\<close>
+    by -
+qed
+
+lemma zero_not_rank1[simp]: \<open>\<not> rank1 0\<close>
+  unfolding rank1_def
+  apply auto
+  by (metis ccspan_superset insert_not_empty singleton_insert_inj_eq space_as_set_bot subset_singletonD)
+
+lemma rank1_iff_butterfly: \<open>rank1 A \<longleftrightarrow> (\<exists>\<psi> \<phi>. A = butterfly \<psi> \<phi>) \<and> A \<noteq> 0\<close>
+  for A :: \<open>_::complex_inner \<Rightarrow>\<^sub>C\<^sub>L _::chilbert_space\<close>
+proof (rule iffI)
+  assume \<open>rank1 A\<close>
+  then obtain \<psi> where \<open>A *\<^sub>S \<top> = ccspan {\<psi>}\<close>
+    using rank1_def by auto
+  then have \<open>\<exists>\<phi>. A = butterfly \<psi> \<phi>\<close>
+    by (rule rank1_is_butterfly)
+  moreover from \<open>rank1 A\<close> have \<open>A \<noteq> 0\<close>
+    by auto
+  ultimately show \<open>(\<exists>\<psi> \<phi>. A = butterfly \<psi> \<phi>) \<and> A \<noteq> 0\<close>
+    by auto
+next
+  assume asm: \<open>(\<exists>\<psi> \<phi>. A = butterfly \<psi> \<phi>) \<and> A \<noteq> 0\<close>
+  then obtain \<psi> \<phi> where A: \<open>A = butterfly \<psi> \<phi>\<close>
+    by auto
+  from asm have \<open>A \<noteq> 0\<close>
+    by simp
+  with A have \<open>\<psi> \<noteq> 0\<close> and \<open>\<phi> \<noteq> 0\<close>
+    by auto
+  then have \<open>butterfly \<psi> \<phi> *\<^sub>S \<top> = ccspan {\<psi>}\<close>
+    by (rule_tac butterfly_is_rank1)
+  with A \<open>\<psi> \<noteq> 0\<close> show \<open>rank1 A\<close>
+    by (auto intro!: exI[of _ \<psi>] simp: rank1_def)
+qed
+
+lemma butterfly_if_rank1: \<open>(\<exists>\<psi> \<phi>. A = butterfly \<psi> \<phi>) \<longleftrightarrow> rank1 A \<or> A = 0\<close>
+  for A :: \<open>_::complex_inner \<Rightarrow>\<^sub>C\<^sub>L _::chilbert_space\<close>
+  by (metis butterfly_0_left rank1_iff_butterfly)
+
 lemma norm_butterfly: "norm (butterfly \<psi> \<phi>) = norm \<psi> * norm \<phi>"
 proof (cases "\<phi>=0")
   case True
   then show ?thesis by simp
 next
   case False
   show ?thesis
     unfolding norm_cblinfun.rep_eq
     thm onormI[OF _ False]
   proof (rule onormI[OF _ False])
     fix x
 
-    have "cmod \<langle>\<phi>, x\<rangle> * norm \<psi> \<le> norm \<psi> * norm \<phi> * norm x"
+    have "cmod (\<phi> \<bullet>\<^sub>C x) * norm \<psi> \<le> norm \<psi> * norm \<phi> * norm x"
       by (metis ab_semigroup_mult_class.mult_ac(1) complex_inner_class.Cauchy_Schwarz_ineq2 mult.commute mult_left_mono norm_ge_zero)
     thus "norm (butterfly \<psi> \<phi> *\<^sub>V x) \<le> norm \<psi> * norm \<phi> * norm x"
       by (simp add: power2_eq_square)
 
     show "norm (butterfly \<psi> \<phi> *\<^sub>V \<phi>) = norm \<psi> * norm \<phi> * norm \<phi>"
       by (smt (z3) ab_semigroup_mult_class.mult_ac(1) butterfly_apply mult.commute norm_eq_sqrt_cinner norm_ge_zero norm_scaleC power2_eq_square real_sqrt_abs real_sqrt_eq_iff)
   qed
 qed
 
 lemma bounded_sesquilinear_butterfly[bounded_sesquilinear]: \<open>bounded_sesquilinear (\<lambda>(b::'b::chilbert_space) (a::'a::chilbert_space). butterfly a b)\<close>
 proof standard
   fix a a' :: 'a and b b' :: 'b and r :: complex
   show \<open>butterfly (a + a') b = butterfly a b + butterfly a' b\<close>
     by (rule butterfly_add_left)
   show \<open>butterfly a (b + b') = butterfly a b + butterfly a b'\<close>
     by (rule butterfly_add_right)
   show \<open>butterfly (r *\<^sub>C a) b = r *\<^sub>C butterfly a b\<close>
     by simp
   show \<open>butterfly a (r *\<^sub>C b) = cnj r *\<^sub>C butterfly a b\<close>
     by simp
   show \<open>\<exists>K. \<forall>b a. norm (butterfly a b) \<le> norm b * norm a * K \<close>
     apply (rule exI[of _ 1])
     by (simp add: norm_butterfly)
 qed
 
 lemma inj_selfbutter_upto_phase:
   assumes "selfbutter x = selfbutter y"
   shows "\<exists>c. cmod c = 1 \<and> x = c *\<^sub>C y"
 proof (cases "x = 0")
   case True
   from assms have "y = 0"
     using norm_butterfly
     by (metis True butterfly_0_left divisors_zero norm_eq_zero)
   with True show ?thesis
     using norm_one by fastforce
 next
   case False
-  define c where "c = \<langle>y, x\<rangle> / \<langle>x, x\<rangle>"
-  have "\<langle>x, x\<rangle> *\<^sub>C x = selfbutter x *\<^sub>V x"
+  define c where "c = (y \<bullet>\<^sub>C x) / (x \<bullet>\<^sub>C x)"
+  have "(x \<bullet>\<^sub>C x) *\<^sub>C x = selfbutter x *\<^sub>V x"
     by (simp add: butterfly_apply)
   also have "\<dots> = selfbutter y *\<^sub>V x"
     using assms by simp
-  also have "\<dots> = \<langle>y, x\<rangle> *\<^sub>C y"
+  also have "\<dots> = (y \<bullet>\<^sub>C x) *\<^sub>C y"
     by (simp add: butterfly_apply)
   finally have xcy: "x = c *\<^sub>C y"
     by (simp add: c_def ceq_vector_fraction_iff)
   have "cmod c * norm x = cmod c * norm y"
     using assms norm_butterfly
-    by (smt (verit, ccfv_SIG) \<open>\<langle>x, x\<rangle> *\<^sub>C x = selfbutter x *\<^sub>V x\<close> \<open>selfbutter y *\<^sub>V x = \<langle>y, x\<rangle> *\<^sub>C y\<close> cinner_scaleC_right complex_vector.scale_left_commute complex_vector.scale_right_imp_eq mult_cancel_left norm_eq_sqrt_cinner norm_eq_zero scaleC_scaleC xcy)
+    by (smt (verit, ccfv_SIG) \<open>(x \<bullet>\<^sub>C x) *\<^sub>C x = selfbutter x *\<^sub>V x\<close> \<open>selfbutter y *\<^sub>V x = (y \<bullet>\<^sub>C x) *\<^sub>C y\<close> cinner_scaleC_right complex_vector.scale_left_commute complex_vector.scale_right_imp_eq mult_cancel_left norm_eq_sqrt_cinner norm_eq_zero scaleC_scaleC xcy)
   also have "cmod c * norm y = norm (c *\<^sub>C y)"
     by simp
   also have "\<dots> = norm x"
     unfolding xcy[symmetric] by simp
   finally have c: "cmod c = 1"
     by (simp add: False)
   from c xcy show ?thesis
     by auto
 qed
 
 lemma butterfly_eq_proj:
   assumes "norm x = 1"
   shows "selfbutter x = proj x"
 proof -
   define B and \<phi> :: "complex \<Rightarrow>\<^sub>C\<^sub>L 'a"
     where "B = selfbutter x" and "\<phi> = vector_to_cblinfun x"
   then have B: "B = \<phi> o\<^sub>C\<^sub>L \<phi>*"
     unfolding butterfly_def by simp
   have \<phi>adj\<phi>: "\<phi>* o\<^sub>C\<^sub>L \<phi> = id_cblinfun"
     using \<phi>_def assms isometry_def isometry_vector_to_cblinfun by blast
   have "B o\<^sub>C\<^sub>L B = \<phi> o\<^sub>C\<^sub>L (\<phi>* o\<^sub>C\<^sub>L \<phi>) o\<^sub>C\<^sub>L \<phi>*"
     by (simp add: B cblinfun_assoc_left(1))
   also have "\<dots> = B"
     unfolding \<phi>adj\<phi> by (simp add: B)
   finally have idem: "B o\<^sub>C\<^sub>L B = B".
   have herm: "B = B*"
     unfolding B by simp
   from idem herm have BProj: "B = Proj (B *\<^sub>S top)"
     by (rule Proj_on_own_range'[symmetric])
   have "B *\<^sub>S top = ccspan {x}"
     by (simp add: B \<phi>_def assms cblinfun_compose_image range_adjoint_isometry)
   with BProj show "B = proj x"
     by simp
 qed
 
 lemma butterfly_sgn_eq_proj:
   shows "selfbutter (sgn x) = proj x"
 proof (cases \<open>x = 0\<close>)
   case True
   then show ?thesis
     by simp
 next
   case False
   then have \<open>selfbutter (sgn x) = proj (sgn x)\<close>
     by (simp add: butterfly_eq_proj norm_sgn)
   also have \<open>ccspan {sgn x} = ccspan {x}\<close>
     by (metis ccspan_singleton_scaleC scaleC_eq_0_iff scaleR_scaleC sgn_div_norm sgn_zero_iff)
   finally show ?thesis
     by -
 qed
 
 lemma butterfly_is_Proj:
   \<open>norm x = 1 \<Longrightarrow> is_Proj (selfbutter x)\<close>
   by (subst butterfly_eq_proj, simp_all)
 
 lemma cspan_butterfly_UNIV:
   assumes \<open>cspan basisA = UNIV\<close>
   assumes \<open>cspan basisB = UNIV\<close>
   assumes \<open>is_ortho_set basisB\<close>
   assumes \<open>\<And>b. b \<in> basisB \<Longrightarrow> norm b = 1\<close>
   shows \<open>cspan {butterfly a b| (a::'a::{complex_normed_vector}) (b::'b::{chilbert_space,cfinite_dim}). a \<in> basisA \<and> b \<in> basisB} = UNIV\<close>
 proof -
   have F: \<open>\<exists>F\<in>{butterfly a b |a b. a \<in> basisA \<and> b \<in> basisB}. \<forall>b'\<in>basisB. F *\<^sub>V b' = (if b' = b then a else 0)\<close>
     if \<open>a \<in> basisA\<close> and \<open>b \<in> basisB\<close> for a b
     apply (rule bexI[where x=\<open>butterfly a b\<close>])
     using assms that by (auto simp: is_ortho_set_def cnorm_eq_1)
   show ?thesis
     apply (rule cblinfun_cspan_UNIV[where basisA=basisB and basisB=basisA])
     using assms apply auto[2]
     using F by (smt (verit, ccfv_SIG) image_iff)
 qed
 
 lemma cindependent_butterfly:
   fixes basisA :: \<open>'a::chilbert_space set\<close> and basisB :: \<open>'b::chilbert_space set\<close>
   assumes \<open>is_ortho_set basisA\<close> \<open>is_ortho_set basisB\<close>
   assumes normA: \<open>\<And>a. a\<in>basisA \<Longrightarrow> norm a = 1\<close> and normB: \<open>\<And>b. b\<in>basisB \<Longrightarrow> norm b = 1\<close>
   shows \<open>cindependent {butterfly a b| a b. a\<in>basisA \<and> b\<in>basisB}\<close>
 proof (unfold complex_vector.independent_explicit_module, intro allI impI, rename_tac T f g)
   fix T :: \<open>('b \<Rightarrow>\<^sub>C\<^sub>L 'a) set\<close> and f :: \<open>'b \<Rightarrow>\<^sub>C\<^sub>L 'a \<Rightarrow> complex\<close> and g :: \<open>'b \<Rightarrow>\<^sub>C\<^sub>L 'a\<close>
   assume \<open>finite T\<close>
   assume T_subset: \<open>T \<subseteq> {butterfly a b |a b. a \<in> basisA \<and> b \<in> basisB}\<close>
   define lin where \<open>lin = (\<Sum>g\<in>T. f g *\<^sub>C g)\<close>
   assume \<open>lin = 0\<close>
   assume \<open>g \<in> T\<close>
   (* To show: f g = 0 *)
   then obtain a b where g: \<open>g = butterfly a b\<close> and [simp]: \<open>a \<in> basisA\<close> \<open>b \<in> basisB\<close>
     using T_subset by auto
 
   have *: "(vector_to_cblinfun a)* *\<^sub>V f g *\<^sub>C g *\<^sub>V b = 0"
     if \<open>g \<in> T - {butterfly a b}\<close> for g
   proof -
     from that
     obtain a' b' where g: \<open>g = butterfly a' b'\<close> and [simp]: \<open>a' \<in> basisA\<close> \<open>b' \<in> basisB\<close>
       using T_subset by auto
     from that have \<open>g \<noteq> butterfly a b\<close> by auto
     with g consider (a) \<open>a\<noteq>a'\<close> | (b) \<open>b\<noteq>b'\<close>
       by auto
     then show \<open>(vector_to_cblinfun a)* *\<^sub>V f g *\<^sub>C g *\<^sub>V b = 0\<close>
     proof cases
       case a
       then show ?thesis
         using  \<open>is_ortho_set basisA\<close> unfolding g
         by (auto simp: is_ortho_set_def butterfly_def scaleC_cblinfun.rep_eq)
     next
       case b
       then show ?thesis
         using  \<open>is_ortho_set basisB\<close> unfolding g
         by (auto simp: is_ortho_set_def butterfly_def scaleC_cblinfun.rep_eq)
     qed
   qed
 
   have \<open>0 = (vector_to_cblinfun a)* *\<^sub>V lin *\<^sub>V b\<close>
     using \<open>lin = 0\<close> by auto
   also have \<open>\<dots> = (\<Sum>g\<in>T. (vector_to_cblinfun a)* *\<^sub>V (f g *\<^sub>C g) *\<^sub>V b)\<close>
     unfolding lin_def
     apply (rule complex_vector.linear_sum)
     by (smt (z3) cblinfun.scaleC_left cblinfun.scaleC_right cblinfun.add_right clinearI plus_cblinfun.rep_eq)
   also have \<open>\<dots> = (\<Sum>g\<in>{butterfly a b}. (vector_to_cblinfun a)* *\<^sub>V (f g *\<^sub>C g) *\<^sub>V b)\<close>
     apply (rule sum.mono_neutral_right)
     using \<open>finite T\<close> * \<open>g \<in> T\<close> g by auto
   also have \<open>\<dots> = (vector_to_cblinfun a)* *\<^sub>V (f g *\<^sub>C g) *\<^sub>V b\<close>
     by (simp add: g)
   also have \<open>\<dots> = f g\<close>
     unfolding g
     using normA normB by (auto simp: butterfly_def scaleC_cblinfun.rep_eq cnorm_eq_1)
   finally show \<open>f g = 0\<close>
     by simp
 qed
 
 lemma clinear_eq_butterflyI:
   fixes F G :: \<open>('a::{chilbert_space,cfinite_dim} \<Rightarrow>\<^sub>C\<^sub>L 'b::complex_inner) \<Rightarrow> 'c::complex_vector\<close>
   assumes "clinear F" and "clinear G"
   assumes \<open>cspan basisA = UNIV\<close> \<open>cspan basisB = UNIV\<close>
   assumes \<open>is_ortho_set basisA\<close> \<open>is_ortho_set basisB\<close>
   assumes "\<And>a b. a\<in>basisA \<Longrightarrow> b\<in>basisB \<Longrightarrow> F (butterfly a b) = G (butterfly a b)"
   assumes \<open>\<And>b. b\<in>basisB \<Longrightarrow> norm b = 1\<close>
   shows "F = G"
   apply (rule complex_vector.linear_eq_on_span[where f=F, THEN ext, rotated 3])
      apply (subst cspan_butterfly_UNIV)
   using assms by auto
 
 lemma sum_butterfly_is_Proj:
   assumes \<open>is_ortho_set E\<close>
   assumes \<open>\<And>e. e\<in>E \<Longrightarrow> norm e = 1\<close>
   shows \<open>is_Proj (\<Sum>e\<in>E. butterfly e e)\<close>
 proof (cases \<open>finite E\<close>)
   case True
   show ?thesis
   proof (rule is_Proj_I)
     show \<open>(\<Sum>e\<in>E. butterfly e e)* = (\<Sum>e\<in>E. butterfly e e)\<close>
       by (simp add: sum_adj)
     have ortho: \<open>f \<noteq> e \<Longrightarrow> e \<in> E \<Longrightarrow> f \<in> E \<Longrightarrow> is_orthogonal f e\<close> for f e
       by (meson assms(1) is_ortho_set_def)
     have unit: \<open>e \<bullet>\<^sub>C e = 1\<close> if \<open>e \<in> E\<close> for e
       using assms(2) cnorm_eq_1 that by blast
     have *: \<open>(\<Sum>f\<in>E. (f \<bullet>\<^sub>C e) *\<^sub>C butterfly f e) = butterfly e e\<close> if \<open>e \<in> E\<close> for e
       apply (subst sum_single[where i=e])
       by (auto intro!: simp: that ortho unit True)
     show \<open>(\<Sum>e\<in>E. butterfly e e) o\<^sub>C\<^sub>L (\<Sum>e\<in>E. butterfly e e) = (\<Sum>e\<in>E. butterfly e e)\<close>
       by (auto simp: * cblinfun_compose_sum_right cblinfun_compose_sum_left)
   qed
 next
   case False
   then show ?thesis
     by simp
 qed
 
+
 subsection \<open>Bifunctionals\<close>
 
 lift_definition bifunctional :: \<open>'a::complex_normed_vector \<Rightarrow>\<^sub>C\<^sub>L (('a \<Rightarrow>\<^sub>C\<^sub>L complex) \<Rightarrow>\<^sub>C\<^sub>L complex)\<close>
   is \<open>\<lambda>x f. f *\<^sub>V x\<close>
   by (simp add: cblinfun.flip)
 
 lemma bifunctional_apply[simp]: \<open>(bifunctional *\<^sub>V x) *\<^sub>V f = f *\<^sub>V x\<close>
   by (transfer fixing: x f, simp)
 
 lemma bifunctional_isometric[simp]: \<open>norm (bifunctional *\<^sub>V x) = norm x\<close> for x :: \<open>'a::complex_inner\<close>
 proof -
   define f :: \<open>'a \<Rightarrow>\<^sub>C\<^sub>L complex\<close> where \<open>f = CBlinfun (\<lambda>y. cinner x y)\<close>
   then have [simp]: \<open>f *\<^sub>V y = cinner x y\<close> for y
     by (simp add: bounded_clinear_CBlinfun_apply bounded_clinear_cinner_right)
   then have [simp]: \<open>norm f = norm x\<close>
     apply (auto intro!: norm_cblinfun_eqI[where x=x] simp: power2_norm_eq_cinner[symmetric])
      apply (smt (verit, best) norm_eq_sqrt_cinner norm_ge_zero power2_norm_eq_cinner real_div_sqrt)
     using Cauchy_Schwarz_ineq2 by blast
   show ?thesis
     apply (auto intro!: norm_cblinfun_eqI[where x=f])
      apply (metis norm_eq_sqrt_cinner norm_imp_pos_and_ge real_div_sqrt)
     by (metis norm_cblinfun ordered_field_class.sign_simps(33))
 qed
 
 lemma norm_bifunctional[simp]: \<open>norm (bifunctional :: 'a::{complex_inner, not_singleton} \<Rightarrow>\<^sub>C\<^sub>L _) = 1\<close>
 proof -
   obtain x :: 'a where [simp]: \<open>norm x = 1\<close>
     by (meson UNIV_not_singleton ex_norm1)
   show ?thesis
     by (auto intro!: norm_cblinfun_eqI[where x=x])
 qed
 
 subsection \<open>Banach-Steinhaus\<close>
 
 theorem cbanach_steinhaus:
   fixes F :: \<open>'c \<Rightarrow> 'a::cbanach \<Rightarrow>\<^sub>C\<^sub>L 'b::complex_normed_vector\<close>
   assumes \<open>\<And>x. \<exists>M. \<forall>n.  norm ((F n) *\<^sub>V x) \<le> M\<close>
   shows  \<open>\<exists>M. \<forall> n. norm (F n) \<le> M\<close>
   using cblinfun_blinfun_transfer[transfer_rule] apply (rule TrueI)? (* Deletes current facts *)
 proof (use assms in transfer)
   fix F :: \<open>'c \<Rightarrow> 'a \<Rightarrow>\<^sub>L 'b\<close> assume \<open>(\<And>x. \<exists>M. \<forall>n. norm (F n *\<^sub>v x) \<le> M)\<close>
   hence \<open>\<And>x. bounded (range (\<lambda>n. blinfun_apply (F n) x))\<close>
     by (metis (no_types, lifting) boundedI rangeE)
   hence \<open>bounded (range F)\<close>
     by (simp add: banach_steinhaus)
   thus  \<open>\<exists>M. \<forall>n. norm (F n) \<le> M\<close>
     by (simp add: bounded_iff)
 qed
 
 subsection \<open>Riesz-representation theorem\<close>
 
 theorem riesz_frechet_representation_cblinfun_existence:
   \<comment> \<open>Theorem 3.4 in @{cite conway2013course}\<close>
   fixes f::\<open>'a::chilbert_space \<Rightarrow>\<^sub>C\<^sub>L complex\<close>
-  shows \<open>\<exists>t. \<forall>x.  f *\<^sub>V x = \<langle>t, x\<rangle>\<close>
+  shows \<open>\<exists>t. \<forall>x.  f *\<^sub>V x = (t \<bullet>\<^sub>C x)\<close>
   apply transfer by (rule riesz_frechet_representation_existence)
 
 lemma riesz_frechet_representation_cblinfun_unique:
   \<comment> \<open>Theorem 3.4 in @{cite conway2013course}\<close>
   fixes f::\<open>'a::complex_inner \<Rightarrow>\<^sub>C\<^sub>L complex\<close>
-  assumes \<open>\<And>x. f *\<^sub>V x = \<langle>t, x\<rangle>\<close>
-  assumes \<open>\<And>x. f *\<^sub>V x = \<langle>u, x\<rangle>\<close>
+  assumes \<open>\<And>x. f *\<^sub>V x = (t \<bullet>\<^sub>C x)\<close>
+  assumes \<open>\<And>x. f *\<^sub>V x = (u \<bullet>\<^sub>C x)\<close>
   shows \<open>t = u\<close>
   using assms by (rule riesz_frechet_representation_unique)
 
 theorem riesz_frechet_representation_cblinfun_norm:
   includes notation_norm
   fixes f::\<open>'a::chilbert_space \<Rightarrow>\<^sub>C\<^sub>L complex\<close>
-  assumes \<open>\<And>x.  f *\<^sub>V x = \<langle>t, x\<rangle>\<close>
+  assumes \<open>\<And>x.  f *\<^sub>V x = (t \<bullet>\<^sub>C x)\<close>
   shows \<open>\<parallel>f\<parallel> = \<parallel>t\<parallel>\<close>
   using assms
 proof transfer
   fix f::\<open>'a \<Rightarrow> complex\<close> and t
-  assume \<open>bounded_clinear f\<close> and \<open>\<And>x. f x = \<langle>t, x\<rangle>\<close>
-  from  \<open>\<And>x. f x = \<langle>t, x\<rangle>\<close>
+  assume \<open>bounded_clinear f\<close> and \<open>\<And>x. f x = (t \<bullet>\<^sub>C x)\<close>
+  from  \<open>\<And>x. f x = (t \<bullet>\<^sub>C x)\<close>
   have \<open>(norm (f x)) / (norm x) \<le> norm t\<close>
     for x
   proof(cases \<open>norm x = 0\<close>)
     case True
     thus ?thesis by simp
   next
     case False
-    have \<open>norm (f x) = norm (\<langle>t, x\<rangle>)\<close>
-      using \<open>\<And>x. f x = \<langle>t, x\<rangle>\<close> by simp
-    also have \<open>norm \<langle>t, x\<rangle> \<le> norm t * norm x\<close>
+    have \<open>norm (f x) = norm ((t \<bullet>\<^sub>C x))\<close>
+      using \<open>\<And>x. f x = (t \<bullet>\<^sub>C x)\<close> by simp
+    also have \<open>norm (t \<bullet>\<^sub>C x) \<le> norm t * norm x\<close>
       by (simp add: complex_inner_class.Cauchy_Schwarz_ineq2)
     finally have \<open>norm (f x) \<le> norm t * norm x\<close>
       by blast
     thus ?thesis
       by (metis False linordered_field_class.divide_right_mono nonzero_mult_div_cancel_right norm_ge_zero)
   qed
   moreover have \<open>(norm (f t)) / (norm t) = norm t\<close>
   proof(cases \<open>norm t = 0\<close>)
     case True
     thus ?thesis
       by simp
   next
     case False
-    have \<open>f t = \<langle>t, t\<rangle>\<close>
-      using \<open>\<And>x. f x = \<langle>t, x\<rangle>\<close> by blast
+    have \<open>f t = (t \<bullet>\<^sub>C t)\<close>
+      using \<open>\<And>x. f x = (t \<bullet>\<^sub>C x)\<close> by blast
     also have \<open>\<dots> = (norm t)^2\<close>
       by (meson cnorm_eq_square)
     also have \<open>\<dots> = (norm t)*(norm t)\<close>
       by (simp add: power2_eq_square)
     finally have \<open>f t = (norm t)*(norm t)\<close>
       by blast
     thus ?thesis
       by (metis False Re_complex_of_real \<open>\<And>x. f x = cinner t x\<close> cinner_ge_zero complex_of_real_cmod nonzero_divide_eq_eq)
   qed
   ultimately have \<open>Sup {(norm (f x)) / (norm x)| x. True} = norm t\<close>
     by (smt cSup_eq_maximum mem_Collect_eq)
   moreover have \<open>Sup {(norm (f x)) / (norm x)| x. True} = (SUP x. (norm (f x)) / (norm x))\<close>
     by (simp add: full_SetCompr_eq)
   ultimately show \<open>onorm f = norm t\<close>
     by (simp add: onorm_def)
 qed
 
 definition the_riesz_rep :: \<open>('a::chilbert_space \<Rightarrow>\<^sub>C\<^sub>L complex) \<Rightarrow> 'a\<close> where
-  \<open>the_riesz_rep f = (SOME t. \<forall>x. f x = \<langle>t, x\<rangle>)\<close>
+  \<open>the_riesz_rep f = (SOME t. \<forall>x. f x = (t \<bullet>\<^sub>C x))\<close>
 
 lemma the_riesz_rep[simp]: \<open>the_riesz_rep f \<bullet>\<^sub>C x = f *\<^sub>V x\<close>
   unfolding the_riesz_rep_def
   apply (rule someI2_ex)
   by (simp_all add: riesz_frechet_representation_cblinfun_existence)
 
 lemma the_riesz_rep_unique:
   assumes \<open>\<And>x. f *\<^sub>V x = t \<bullet>\<^sub>C x\<close>
   shows \<open>t = the_riesz_rep f\<close>
   using assms riesz_frechet_representation_cblinfun_unique the_riesz_rep by metis
 
 lemma the_riesz_rep_scaleC: \<open>the_riesz_rep (c *\<^sub>C f) = cnj c *\<^sub>C the_riesz_rep f\<close>
   apply (rule the_riesz_rep_unique[symmetric])
   by auto
 
 lemma the_riesz_rep_add: \<open>the_riesz_rep (f + g) = the_riesz_rep f + the_riesz_rep g\<close>
   apply (rule the_riesz_rep_unique[symmetric])
   by (auto simp: cinner_add_left cblinfun.add_left)
 
 lemma the_riesz_rep_norm[simp]: \<open>norm (the_riesz_rep f) = norm f\<close>
   apply (rule riesz_frechet_representation_cblinfun_norm[symmetric])
   by simp
 
 lemma bounded_antilinear_the_riesz_rep[bounded_antilinear]: \<open>bounded_antilinear the_riesz_rep\<close>
   by (metis (no_types, opaque_lifting) bounded_antilinear_intro dual_order.refl mult.commute mult_1 the_riesz_rep_add the_riesz_rep_norm the_riesz_rep_scaleC)
 
 definition the_riesz_rep_bilinear' :: \<open>('a::complex_normed_vector \<Rightarrow> 'b::chilbert_space \<Rightarrow> complex) \<Rightarrow> ('a \<Rightarrow> 'b)\<close> where
   \<open>the_riesz_rep_bilinear' p x = the_riesz_rep (CBlinfun (p x))\<close>
 
 lemma the_riesz_rep_bilinear'_correct:
   assumes \<open>bounded_clinear (p x)\<close>
   shows \<open>(the_riesz_rep_bilinear' p x) \<bullet>\<^sub>C y = p x y\<close>
   by (auto simp add: the_riesz_rep_bilinear'_def assms bounded_clinear_CBlinfun_apply)
 
 
 lemma the_riesz_rep_bilinear'_plus1:
   assumes \<open>\<And>x. bounded_clinear (p x)\<close> and \<open>\<And>x. bounded_clinear (q x)\<close>
   shows \<open>the_riesz_rep_bilinear' (p + q) = the_riesz_rep_bilinear' p + the_riesz_rep_bilinear' q\<close>
   by (auto intro!: ext simp add: the_riesz_rep_add CBlinfun_plus the_riesz_rep_bilinear'_def 
       assms bounded_clinear_CBlinfun_apply)
 
 lemma the_riesz_rep_bilinear'_plus2:
   assumes \<open>bounded_sesquilinear p\<close>
   shows \<open>the_riesz_rep_bilinear' p (x + y) = the_riesz_rep_bilinear' p x + the_riesz_rep_bilinear' p y\<close>
 proof -
   have [simp]: \<open>p (x + y) = p x + p y\<close>
     using assms bounded_sesquilinear.add_left by fastforce
   have [simp]: \<open>bounded_clinear (p x)\<close> for x
     by (simp add: assms bounded_sesquilinear.bounded_clinear_right)
   show ?thesis
     by (auto intro!: ext simp add: the_riesz_rep_add CBlinfun_plus the_riesz_rep_bilinear'_def
       assms bounded_clinear_CBlinfun_apply)
 qed
 
 lemma the_riesz_rep_bilinear'_scaleC1:
   assumes \<open>\<And>x. bounded_clinear (p x)\<close>
   shows \<open>the_riesz_rep_bilinear' (\<lambda>x y. c *\<^sub>C p x y) = (\<lambda>x. cnj c *\<^sub>C the_riesz_rep_bilinear' p x)\<close>
   by (auto intro!: ext simp add: the_riesz_rep_scaleC CBlinfun_scaleC the_riesz_rep_bilinear'_def 
       assms bounded_clinear_CBlinfun_apply
       simp del: complex_scaleC_def scaleC_conv_of_complex)
 
 lemma the_riesz_rep_bilinear'_scaleC2:
   assumes \<open>bounded_sesquilinear p\<close>
   shows \<open>the_riesz_rep_bilinear' p (c *\<^sub>C x) = c *\<^sub>C the_riesz_rep_bilinear' p x\<close>
 proof -
   have [simp]: \<open>p (c *\<^sub>C x) = (\<lambda>y. cnj c *\<^sub>C p x y)\<close>
     using assms bounded_sesquilinear.scaleC_left by blast
   have [simp]: \<open>bounded_clinear (p x)\<close> for x
     by (simp add: assms bounded_sesquilinear.bounded_clinear_right)
   show ?thesis
     by (auto intro!: ext simp add: the_riesz_rep_scaleC CBlinfun_scaleC the_riesz_rep_bilinear'_def
       assms bounded_clinear_CBlinfun_apply
       simp del: complex_scaleC_def scaleC_conv_of_complex)
 qed
 
 lemma bounded_clinear_the_riesz_rep_bilinear'2:
   assumes \<open>bounded_sesquilinear p\<close>
   shows \<open>bounded_clinear (the_riesz_rep_bilinear' p)\<close>
 proof -
   obtain K0 where K0: \<open>cmod (p x y) \<le> norm x * norm y * K0\<close> for x y
     using assms bounded_sesquilinear.bounded  by blast
   define K where \<open>K = max K0 0\<close>
   with K0 have K: \<open>cmod (p x y) \<le> norm x * norm y * K\<close> for x y
     by (smt (verit, del_insts) mult_nonneg_nonneg mult_nonneg_nonpos norm_ge_zero)
   have [simp]: \<open>K \<ge> 0\<close>
     by (simp add: K_def)
   have [simp]: \<open>bounded_clinear (p x)\<close> for x
     by (simp add: assms bounded_sesquilinear.bounded_clinear_right)
   have "norm (the_riesz_rep_bilinear' p x) \<le> norm x * K" for x
     unfolding the_riesz_rep_bilinear'_def
     using K by (auto intro!: norm_cblinfun_bound simp: bounded_clinear_CBlinfun_apply mult.commute mult.left_commute)
   then show ?thesis
     apply (rule_tac bounded_clinearI)
     by (auto simp: assms the_riesz_rep_bilinear'_plus2 the_riesz_rep_bilinear'_scaleC2)
 qed
 
 lift_definition the_riesz_rep_bilinear:: \<open>('a::complex_normed_vector \<Rightarrow> 'b::chilbert_space \<Rightarrow> complex) \<Rightarrow> ('a \<Rightarrow>\<^sub>C\<^sub>L 'b)\<close> is
   \<open>\<lambda>p. if bounded_sesquilinear p then the_riesz_rep_bilinear' p else 0\<close>
   by (auto simp: bounded_clinear_the_riesz_rep_bilinear'2 zero_fun_def)
 
 lemma the_riesz_rep_bilinear_correct:
   assumes \<open>bounded_sesquilinear p\<close>
   shows \<open>(the_riesz_rep_bilinear p x) \<bullet>\<^sub>C y = p x y\<close>
   apply (transfer fixing: p)
   by (simp add: assms bounded_sesquilinear.bounded_clinear_right the_riesz_rep_bilinear'_correct)
 
 
 subsection \<open>Extension of complex bounded operators\<close>
 
 definition cblinfun_extension where
   "cblinfun_extension S \<phi> = (SOME B. \<forall>x\<in>S. B *\<^sub>V x = \<phi> x)"
 
 definition cblinfun_extension_exists where
   "cblinfun_extension_exists S \<phi> = (\<exists>B. \<forall>x\<in>S. B *\<^sub>V x = \<phi> x)"
 
 lemma cblinfun_extension_existsI:
   assumes "\<And>x. x\<in>S \<Longrightarrow> B *\<^sub>V x = \<phi> x"
   shows "cblinfun_extension_exists S \<phi>"
   using assms cblinfun_extension_exists_def by blast
 
 lemma cblinfun_extension_exists_finite_dim:
   fixes \<phi>::"'a::{complex_normed_vector,cfinite_dim} \<Rightarrow> 'b::complex_normed_vector"
   assumes "cindependent S"
     and "cspan S = UNIV"
   shows "cblinfun_extension_exists S \<phi>"
 proof-
   define f::"'a \<Rightarrow> 'b"
     where "f = complex_vector.construct S \<phi>"
   have "clinear f"
     by (simp add: complex_vector.linear_construct assms linear_construct f_def)
   have "bounded_clinear f"
     using \<open>clinear f\<close> assms by auto
   then obtain B::"'a \<Rightarrow>\<^sub>C\<^sub>L 'b"
     where "B *\<^sub>V x = f x" for x
     using cblinfun_apply_cases by blast
   have "B *\<^sub>V x = \<phi> x"
     if c1: "x\<in>S"
     for x
   proof-
     have "B *\<^sub>V x = f x"
       by (simp add: \<open>\<And>x. B *\<^sub>V x = f x\<close>)
     also have "\<dots> = \<phi> x"
       using assms complex_vector.construct_basis f_def that
       by (simp add: complex_vector.construct_basis)
     finally show?thesis by blast
   qed
   thus ?thesis
     unfolding cblinfun_extension_exists_def
     by blast
 qed
 
 lemma cblinfun_extension_apply:
   assumes "cblinfun_extension_exists S f"
     and "v \<in> S"
   shows "(cblinfun_extension S f) *\<^sub>V v = f v"
   by (smt assms cblinfun_extension_def cblinfun_extension_exists_def tfl_some)
 
 lemma
   fixes f :: \<open>'a::complex_normed_vector \<Rightarrow> 'b::cbanach\<close>
   assumes \<open>csubspace S\<close>
   assumes \<open>closure S = UNIV\<close>
   assumes f_add: \<open>\<And>x y. x \<in> S \<Longrightarrow> y \<in> S \<Longrightarrow> f (x + y) = f x + f y\<close>
   assumes f_scale: \<open>\<And>c x y. x \<in> S \<Longrightarrow> f (c *\<^sub>C x) = c *\<^sub>C f x\<close>
   assumes bounded: \<open>\<And>x. x \<in> S \<Longrightarrow> norm (f x) \<le> B * norm x\<close>
   shows cblinfun_extension_exists_bounded_dense: \<open>cblinfun_extension_exists S f\<close>
     and cblinfun_extension_exists_norm: \<open>B \<ge> 0 \<Longrightarrow> norm (cblinfun_extension S f) \<le> B\<close>
 proof -
   define B' where \<open>B' = (if B\<le>0 then 1 else B)\<close>
   then have bounded': \<open>\<And>x. x \<in> S \<Longrightarrow> norm (f x) \<le> B' * norm x\<close>
     using bounded by (metis mult_1 mult_le_0_iff norm_ge_zero order_trans)
   have \<open>B' > 0\<close>
     by (simp add: B'_def)
 
   have \<open>\<exists>xi. (xi \<longlonglongrightarrow> x) \<and> (\<forall>i. xi i \<in> S)\<close> for x
     using assms(2) closure_sequential by blast
   then obtain seq :: \<open>'a \<Rightarrow> nat \<Rightarrow> 'a\<close> where seq_lim: \<open>seq x \<longlonglongrightarrow> x\<close> and seq_S: \<open>seq x i \<in> S\<close> for x i
     apply (atomize_elim, subst all_conj_distrib[symmetric])
     apply (rule choice)
     by auto
   define g where \<open>g x = lim (\<lambda>i. f (seq x i))\<close> for x
   have \<open>Cauchy (\<lambda>i. f (seq x i))\<close> for x
   proof (rule CauchyI)
     fix e :: real assume \<open>e > 0\<close>
     have \<open>Cauchy (seq x)\<close>
       using LIMSEQ_imp_Cauchy seq_lim by blast
     then obtain M where less_eB: \<open>norm (seq x m - seq x n) < e/B'\<close> if \<open>n \<ge> M\<close> and \<open>m \<ge> M\<close> for n m
       apply atomize_elim by (meson CauchyD \<open>0 < B'\<close> \<open>0 < e\<close> linordered_field_class.divide_pos_pos)
     have \<open>norm (f (seq x m) - f (seq x n)) < e\<close> if \<open>n \<ge> M\<close> and \<open>m \<ge> M\<close> for n m
     proof -
       have \<open>norm (f (seq x m) - f (seq x n)) = norm (f (seq x m - seq x n))\<close>
         using f_add f_scale seq_S
         by (metis add_diff_cancel assms(1) complex_vector.subspace_diff diff_add_cancel)
       also have \<open>\<dots> \<le> B' * norm (seq x m - seq x n)\<close>
         apply (rule bounded')
         by (simp add: assms(1) complex_vector.subspace_diff seq_S)
       also from less_eB have \<open>\<dots> < B' * (e/B')\<close>
         by (meson \<open>0 < B'\<close> linordered_semiring_strict_class.mult_strict_left_mono that)
       also have \<open>\<dots> \<le> e\<close>
         using \<open>0 < B'\<close> by auto
       finally show ?thesis
         by -
     qed
     then show \<open>\<exists>M. \<forall>m\<ge>M. \<forall>n\<ge>M. norm (f (seq x m) - f (seq x n)) < e\<close>
       by auto
   qed
   then have f_seq_lim: \<open>(\<lambda>i. f (seq x i)) \<longlonglongrightarrow> g x\<close> for x
     by (simp add: Cauchy_convergent_iff convergent_LIMSEQ_iff g_def)
   have f_xi_lim: \<open>(\<lambda>i. f (xi i)) \<longlonglongrightarrow> g x\<close> if \<open>xi \<longlonglongrightarrow> x\<close> and \<open>\<And>i. xi i \<in> S\<close> for xi x
   proof -
     from seq_lim that
     have \<open>(\<lambda>i. B' * norm (xi i - seq x i)) \<longlonglongrightarrow> 0\<close>
       by (metis (no_types) \<open>0 < B'\<close> cancel_comm_monoid_add_class.diff_cancel norm_not_less_zero norm_zero tendsto_diff tendsto_norm_zero_iff tendsto_zero_mult_left_iff)
     then have \<open>(\<lambda>i. f (xi i + (-1) *\<^sub>C seq x i)) \<longlonglongrightarrow> 0\<close>
       apply (rule Lim_null_comparison[rotated])
       using bounded' by (simp add: assms(1) complex_vector.subspace_diff seq_S that(2))
     then have \<open>(\<lambda>i. f (xi i) - f (seq x i)) \<longlonglongrightarrow> 0\<close>
       apply (subst (asm) f_add)
         apply (auto simp: that \<open>csubspace S\<close> complex_vector.subspace_neg seq_S)[2]
       apply (subst (asm) f_scale)
       by (auto simp: that \<open>csubspace S\<close> complex_vector.subspace_neg seq_S)
     then show \<open>(\<lambda>i. f (xi i)) \<longlonglongrightarrow> g x\<close>
       using Lim_transform f_seq_lim by fastforce
   qed
   have g_add: \<open>g (x + y) = g x + g y\<close> for x y
   proof -
     obtain xi :: \<open>nat \<Rightarrow> 'a\<close> where \<open>xi \<longlonglongrightarrow> x\<close> and \<open>xi i \<in> S\<close> for i
       using seq_S seq_lim by auto
     obtain yi :: \<open>nat \<Rightarrow> 'a\<close> where \<open>yi \<longlonglongrightarrow> y\<close> and \<open>yi i \<in> S\<close> for i
       using seq_S seq_lim by auto
     have \<open>(\<lambda>i. xi i + yi i) \<longlonglongrightarrow> x + y\<close>
       using \<open>xi \<longlonglongrightarrow> x\<close> \<open>yi \<longlonglongrightarrow> y\<close> tendsto_add by blast
     then have lim1: \<open>(\<lambda>i. f (xi i + yi i)) \<longlonglongrightarrow> g (x + y)\<close>
       by (simp add: \<open>\<And>i. xi i \<in> S\<close> \<open>\<And>i. yi i \<in> S\<close> assms(1) complex_vector.subspace_add f_xi_lim)
     have \<open>(\<lambda>i. f (xi i + yi i)) = (\<lambda>i. f (xi i) + f (yi i))\<close>
       by (simp add: \<open>\<And>i. xi i \<in> S\<close> \<open>\<And>i. yi i \<in> S\<close> f_add)
     also have \<open>\<dots> \<longlonglongrightarrow> g x + g y\<close>
       by (simp add: \<open>\<And>i. xi i \<in> S\<close> \<open>\<And>i. yi i \<in> S\<close> \<open>xi \<longlonglongrightarrow> x\<close> \<open>yi \<longlonglongrightarrow> y\<close> f_xi_lim tendsto_add)
     finally show ?thesis
       using lim1 LIMSEQ_unique by blast
   qed
   have g_scale: \<open>g (c *\<^sub>C x) = c *\<^sub>C g x\<close> for c x
   proof -
     obtain xi :: \<open>nat \<Rightarrow> 'a\<close> where \<open>xi \<longlonglongrightarrow> x\<close> and \<open>xi i \<in> S\<close> for i
       using seq_S seq_lim by auto
     have \<open>(\<lambda>i. c *\<^sub>C xi i) \<longlonglongrightarrow> c *\<^sub>C x\<close>
       using \<open>xi \<longlonglongrightarrow> x\<close> bounded_clinear_scaleC_right clinear_continuous_at isCont_tendsto_compose by blast
     then have lim1: \<open>(\<lambda>i. f (c *\<^sub>C xi i)) \<longlonglongrightarrow> g (c *\<^sub>C x)\<close>
       by (simp add: \<open>\<And>i. xi i \<in> S\<close> assms(1) complex_vector.subspace_scale f_xi_lim)
     have \<open>(\<lambda>i. f (c *\<^sub>C xi i)) = (\<lambda>i. c *\<^sub>C f (xi i))\<close>
       by (simp add: \<open>\<And>i. xi i \<in> S\<close> f_scale)
     also have \<open>\<dots> \<longlonglongrightarrow> c *\<^sub>C g x\<close>
       using \<open>\<And>i. xi i \<in> S\<close> \<open>xi \<longlonglongrightarrow> x\<close> bounded_clinear_scaleC_right clinear_continuous_at f_xi_lim isCont_tendsto_compose by blast
     finally show ?thesis
       using lim1 LIMSEQ_unique by blast
   qed
 
   have [simp]: \<open>f x = g x\<close> if \<open>x \<in> S\<close> for x
   proof -
     have \<open>(\<lambda>_. x) \<longlonglongrightarrow> x\<close>
       by auto
     then have \<open>(\<lambda>_. f x) \<longlonglongrightarrow> g x\<close>
       using that by (rule f_xi_lim)
     then show \<open>f x = g x\<close>
       by (simp add: LIMSEQ_const_iff)
   qed
 
   have g_bounded: \<open>norm (g x) \<le> B' * norm x\<close> for x
   proof -
     obtain xi :: \<open>nat \<Rightarrow> 'a\<close> where \<open>xi \<longlonglongrightarrow> x\<close> and \<open>xi i \<in> S\<close> for i
       using seq_S seq_lim by auto
     then have \<open>(\<lambda>i. f (xi i)) \<longlonglongrightarrow> g x\<close>
       using f_xi_lim by presburger
     then have \<open>(\<lambda>i. norm (f (xi i))) \<longlonglongrightarrow> norm (g x)\<close>
       by (metis tendsto_norm)
     moreover have \<open>(\<lambda>i. B' * norm (xi i)) \<longlonglongrightarrow> B' * norm x\<close>
       by (simp add: \<open>xi \<longlonglongrightarrow> x\<close> tendsto_mult_left tendsto_norm)
     ultimately show \<open>norm (g x) \<le> B' * norm x\<close>
       apply (rule lim_mono[rotated])
       using bounded' using \<open>xi _ \<in> S\<close> by blast
   qed
 
   have \<open>bounded_clinear g\<close>
     using g_add g_scale apply (rule bounded_clinearI[where K=B'])
     using g_bounded by (simp add: ordered_field_class.sign_simps(5))
   then have [simp]: \<open>CBlinfun g *\<^sub>V x = g x\<close> for x
     by (subst CBlinfun_inverse, auto)
 
   show \<open>cblinfun_extension_exists S f\<close>
     apply (rule cblinfun_extension_existsI[where B=\<open>CBlinfun g\<close>])
     by auto
 
   then have \<open>cblinfun_extension S f *\<^sub>V \<psi> = CBlinfun g *\<^sub>V \<psi>\<close> if \<open>\<psi> \<in> S\<close> for \<psi>
     by (simp add: cblinfun_extension_apply that)
 
   then have ext_is_g: \<open>(*\<^sub>V) (cblinfun_extension S f) = g\<close>
     apply (rule_tac ext)
     apply (rule on_closure_eqI[where S=S])
     using  \<open>closure S = UNIV\<close> \<open>bounded_clinear g\<close>
     by (auto simp add: continuous_at_imp_continuous_on clinear_continuous_within)
 
   show \<open>norm (cblinfun_extension S f) \<le> B\<close> if \<open>B \<ge> 0\<close>
   proof (cases \<open>B > 0\<close>)
     case True
     then have \<open>B = B'\<close>
       unfolding B'_def
       by auto
     moreover have *: \<open>norm (cblinfun_extension S f) \<le> B'\<close>
       by (metis ext_is_g \<open>0 < B'\<close> g_bounded norm_cblinfun_bound order_le_less)
     ultimately show ?thesis
       by simp
   next
     case False
     with bounded have \<open>f x = 0\<close> if \<open>x \<in> S\<close> for x
       by (smt (verit) mult_nonpos_nonneg norm_ge_zero norm_le_zero_iff that)
     then have \<open>g x = (\<lambda>_. 0) x\<close> if \<open>x \<in> S\<close> for x
       using that by simp
     then have \<open>g x = 0\<close> for x
       apply (rule on_closure_eqI[where S=S])
       using \<open>closure S = UNIV\<close> \<open>bounded_clinear g\<close>
       by (auto simp add: continuous_at_imp_continuous_on clinear_continuous_within)
     with ext_is_g have \<open>cblinfun_extension S f = 0\<close>
       by (simp add: cblinfun_eqI)
     then show ?thesis
       using that by simp
   qed
 qed
 
 lemma cblinfun_extension_cong:
   assumes \<open>cspan A = cspan B\<close>
   assumes \<open>B \<subseteq> A\<close>
   assumes fg: \<open>\<And>x. x\<in>B \<Longrightarrow> f x = g x\<close>
   assumes \<open>cblinfun_extension_exists A f\<close>
   shows \<open>cblinfun_extension A f = cblinfun_extension B g\<close>
 proof -
   from \<open>cblinfun_extension_exists A f\<close> fg \<open>B \<subseteq> A\<close>
   have \<open>cblinfun_extension_exists B g\<close>
     by (metis assms(2) cblinfun_extension_exists_def subset_eq)
 
   have \<open>(\<forall>x\<in>A. C *\<^sub>V x = f x) \<longleftrightarrow> (\<forall>x\<in>B. C *\<^sub>V x = f x)\<close> for C
     by (smt (verit, ccfv_SIG) assms(1) assms(2) assms(4) cblinfun_eq_on_span cblinfun_extension_exists_def complex_vector.span_eq subset_iff)
   also from fg have \<open>\<dots> C \<longleftrightarrow> (\<forall>x\<in>B. C *\<^sub>V x = g x)\<close> for C
     by auto
   finally show \<open>cblinfun_extension A f = cblinfun_extension B g\<close>
     unfolding cblinfun_extension_def
     by auto
 qed
 
 lemma
   fixes f :: \<open>'a::complex_inner \<Rightarrow> 'b::chilbert_space\<close> and S 
   assumes \<open>is_ortho_set S\<close> and \<open>closure (cspan S) = UNIV\<close>
   assumes ortho_f: \<open>\<And>x y. x\<in>S \<Longrightarrow> y\<in>S \<Longrightarrow> x\<noteq>y \<Longrightarrow> is_orthogonal (f x) (f y)\<close>
   assumes bounded: \<open>\<And>x. x \<in> S \<Longrightarrow> norm (f x) \<le> B * norm x\<close>
   shows cblinfun_extension_exists_ortho: \<open>cblinfun_extension_exists S f\<close>
     and cblinfun_extension_exists_ortho_norm: \<open>B \<ge> 0 \<Longrightarrow> norm (cblinfun_extension S f) \<le> B\<close>
 proof -
   define g where \<open>g = cconstruct S f\<close>
   have \<open>cindependent S\<close>
     using assms(1) is_ortho_set_cindependent by blast
   have g_f: \<open>g x = f x\<close> if \<open>x\<in>S\<close> for x
     unfolding g_def using \<open>cindependent S\<close> that by (rule complex_vector.construct_basis)
   have [simp]: \<open>clinear g\<close>
     unfolding g_def using \<open>cindependent S\<close> by (rule complex_vector.linear_construct)
   then have g_add: \<open>g (x + y) = g x + g y\<close> if \<open>x \<in> cspan S\<close> and \<open>y \<in> cspan S\<close> for x y
     using clinear_iff by blast
   from \<open>clinear g\<close> have g_scale: \<open>g (c *\<^sub>C x) = c *\<^sub>C g x\<close> if \<open>x \<in> cspan S\<close> for x c
     by (simp add: complex_vector.linear_scale)
   moreover have g_bounded: \<open>norm (g x) \<le> abs B * norm x\<close> if \<open>x \<in> cspan S\<close> for x
   proof -
     from that obtain t r where x_sum: \<open>x = (\<Sum>a\<in>t. r a *\<^sub>C a)\<close> and \<open>finite t\<close> and \<open>t \<subseteq> S\<close>
       unfolding complex_vector.span_explicit by auto
     have \<open>(norm (g x))\<^sup>2 = (norm (\<Sum>a\<in>t. r a *\<^sub>C g a))\<^sup>2\<close>
       by (simp add: x_sum complex_vector.linear_sum clinear.scaleC)
     also have \<open>\<dots> = (norm (\<Sum>a\<in>t. r a *\<^sub>C f a))\<^sup>2\<close>
       by (smt (verit) \<open>t \<subseteq> S\<close> g_f in_mono sum.cong)
     also have \<open>\<dots> = (\<Sum>a\<in>t. (norm (r a *\<^sub>C f a))\<^sup>2)\<close>
       using _ \<open>finite t\<close> apply (rule pythagorean_theorem_sum)
       using \<open>t \<subseteq> S\<close> ortho_f in_mono by fastforce
     also have \<open>\<dots> = (\<Sum>a\<in>t. (cmod (r a) * norm (f a))\<^sup>2)\<close>
       by simp
     also have \<open>\<dots> \<le> (\<Sum>a\<in>t. (cmod (r a) * B * norm a)\<^sup>2)\<close>
       apply (rule sum_mono)
       by (metis \<open>t \<subseteq> S\<close> assms(4) in_mono mult_left_mono mult_nonneg_nonneg norm_ge_zero power_mono vector_space_over_itself.scale_scale)
     also have \<open>\<dots> = B\<^sup>2 * (\<Sum>a\<in>t. (norm (r a *\<^sub>C a))\<^sup>2)\<close>
       by (simp add: sum_distrib_left mult.commute vector_space_over_itself.scale_left_commute flip: power_mult_distrib)
     also have \<open>\<dots> = B\<^sup>2 * (norm (\<Sum>a\<in>t. (r a *\<^sub>C a)))\<^sup>2\<close>
       apply (subst pythagorean_theorem_sum)
       using \<open>finite t\<close> apply auto
       by (meson \<open>t \<subseteq> S\<close> assms(1) is_ortho_set_def subsetD)
     also have \<open>\<dots> = (abs B * norm x)\<^sup>2\<close>
       by (simp add: power_mult_distrib x_sum)
     finally show \<open>norm (g x) \<le> abs B * norm x\<close>
       by auto
   qed
   
   from g_add g_scale g_bounded
   have extg_exists: \<open>cblinfun_extension_exists (cspan S) g\<close>
     apply (rule_tac cblinfun_extension_exists_bounded_dense[where B=\<open>abs B\<close>])
     using \<open>closure (cspan S) = UNIV\<close> by auto
 
   then show \<open>cblinfun_extension_exists S f\<close>
     by (metis (mono_tags, opaque_lifting) g_f cblinfun_extension_apply cblinfun_extension_existsI complex_vector.span_base)
 
   have norm_extg: \<open>norm (cblinfun_extension (cspan S) g) \<le> B\<close> if \<open>B \<ge> 0\<close>
     apply (rule cblinfun_extension_exists_norm)
     using g_add g_scale g_bounded \<open>closure (cspan S) = UNIV\<close> that by auto
 
   have extg_extf: \<open>cblinfun_extension (cspan S) g = cblinfun_extension S f\<close>
     apply (rule cblinfun_extension_cong)
     by (auto simp add: complex_vector.span_base g_f extg_exists)
 
   from norm_extg extg_extf
   show \<open>norm (cblinfun_extension S f) \<le> B\<close> if \<open>B \<ge> 0\<close>
     using that by simp
 qed
 
 
 lemma cblinfun_extension_exists_proj:
   fixes f :: \<open>'a::complex_normed_vector \<Rightarrow> 'b::cbanach\<close>
   assumes \<open>csubspace S\<close>
   assumes \<open>\<exists>P. is_projection_on P (closure S) \<and> bounded_clinear P\<close> (* Maybe can be replaced by is_Proj if the latter's type class is widened *)
   assumes f_add: \<open>\<And>x y. x \<in> S \<Longrightarrow> y \<in> S \<Longrightarrow> f (x + y) = f x + f y\<close>
   assumes f_scale: \<open>\<And>c x y. x \<in> S \<Longrightarrow> f (c *\<^sub>C x) = c *\<^sub>C f x\<close>
   assumes bounded: \<open>\<And>x. x \<in> S \<Longrightarrow> norm (f x) \<le> B * norm x\<close>
   shows \<open>cblinfun_extension_exists S f\<close>
 proof (cases \<open>B \<ge> 0\<close>)
   case True
   note True[simp]
   obtain P where P_proj: \<open>is_projection_on P (closure S)\<close> and P_blin[simp]: \<open>bounded_clinear P\<close>
     using assms(2) by blast 
   have P_lin[simp]: \<open>clinear P\<close>
     by (simp add: bounded_clinear.clinear)
   define f' S' where \<open>f' \<psi> = f (P \<psi>)\<close> and \<open>S' = S + (P -` {0})\<close> for \<psi>
   have \<open>csubspace S'\<close>
     by (simp add: S'_def assms(1) csubspace_set_plus)
   moreover have \<open>closure S' = UNIV\<close>
   proof auto
     fix \<psi>
     have \<open>\<psi> = P \<psi> + (id - P) \<psi>\<close>
       by simp
     also have \<open>\<dots> \<in> closure S + (P -` {0})\<close>
       apply (rule set_plus_intro) 
       using P_proj is_projection_on_in_image 
       by (auto simp: complex_vector.linear_diff is_projection_on_fixes_image is_projection_on_in_image)
     also have \<open>\<dots> \<subseteq> closure (closure S + (P -` {0}))\<close>
       using closure_subset by blast
     also have \<open>\<dots> = closure (S + (P -` {0}))\<close>
       using closed_sum_closure_left closed_sum_def by blast
     also have \<open>\<dots> = closure S'\<close>
       using S'_def by fastforce
     finally show \<open>\<psi> \<in> closure S'\<close>
       by -
   qed
 
   moreover have \<open>f' (x + y) = f' x + f' y\<close> if \<open>x \<in> S'\<close> and \<open>y \<in> S'\<close> for x y
     by (smt (z3) P_blin P_proj S'_def f'_def add.right_neutral bounded_clinear_CBlinfun_apply cblinfun.add_right closure_subset f_add is_projection_on_fixes_image set_plus_elim singletonD subset_eq that(1) that(2) vimageE)
   moreover have \<open>f' (c *\<^sub>C x) = c *\<^sub>C f' x\<close> if \<open>x \<in> S'\<close> for c x
     by (smt (verit, ccfv_SIG) P_blin P_proj S'_def f'_def add.right_neutral bounded_clinear_CBlinfun_apply cblinfun.add_right cblinfun.scaleC_right closure_subset f_scale is_projection_on_fixes_image set_plus_elim singletonD subset_eq that vimageE)
 
   moreover 
   from P_blin obtain B' where B': \<open>norm (P x) \<le> B' * norm x\<close> for x
     by (metis bounded_clinear.bounded mult.commute)
   have \<open>norm (f' x) \<le> (B * B') * norm x\<close> if \<open>x \<in> S'\<close> for x
   proof -
     have \<open>norm (f' x) \<le> B* norm (P x)\<close>
       apply (auto simp: f'_def)
       by (smt (verit) P_blin P_proj S'_def add.right_neutral bounded bounded_clinear_CBlinfun_apply cblinfun.add_right closure_subset is_projection_on_fixes_image set_plus_elim singletonD subset_eq that vimageE)
     also have \<open>\<dots> \<le> B * B' * norm x\<close>
       by (simp add: B' mult.assoc mult_mono)
     finally show ?thesis
       by auto
   qed
 
   ultimately have F_ex: \<open>cblinfun_extension_exists S' f'\<close>
     by (rule cblinfun_extension_exists_bounded_dense)
   define F where \<open>F = cblinfun_extension S' f'\<close>
   from F_ex have *: \<open>F \<psi> = f' \<psi>\<close> if \<open>\<psi> \<in> S'\<close> for \<psi>
     by (simp add: F_def cblinfun_extension_apply that)
   then have \<open>F \<psi> = f \<psi>\<close> if \<open>\<psi> \<in> S\<close> for \<psi>
     apply (auto simp: S'_def f'_def)
     by (metis (no_types, lifting) P_lin P_proj add.right_neutral closure_subset complex_vector.linear_subspace_vimage complex_vector.subspace_0 complex_vector.subspace_single_0 is_projection_on_fixes_image set_plus_intro subset_eq that)
   then show \<open>cblinfun_extension_exists S f\<close>
     using cblinfun_extension_exists_def by blast
 next
   case False
   then have \<open>S \<subseteq> {0}\<close>
     using bounded apply auto
     by (meson norm_ge_zero norm_le_zero_iff order_trans zero_le_mult_iff)
   then show \<open>cblinfun_extension_exists S f\<close>
     apply (rule_tac cblinfun_extension_existsI[where B=0])
     apply auto
     using bounded by fastforce
 qed
 
 subsection \<open>Bijections between different ONBs\<close>
 
 text \<open>Some of the theorems here logically belong into \<^theory>\<open>Complex_Bounded_Operators.Complex_Inner_Product\<close>
   but the proof uses some concepts from the present theory.\<close>
 
-(* TODO mention: follows conway functional, Prop 4.14 *)
 lemma all_ortho_bases_same_card:
+  \<comment> \<open>Follows @{cite conway2013course}, Proposition 4.14\<close>
   fixes E F :: \<open>'a::chilbert_space set\<close>
   assumes \<open>is_ortho_set E\<close> \<open>is_ortho_set F\<close> \<open>ccspan E = top\<close> \<open>ccspan F = top\<close>
   shows \<open>\<exists>f. bij_betw f E F\<close>
 proof -
   have \<open>|F| \<le>o |E|\<close> if \<open>infinite E\<close> and
     \<open>is_ortho_set E\<close> \<open>is_ortho_set F\<close> \<open>ccspan E = top\<close> \<open>ccspan F = top\<close> for E F :: \<open>'a set\<close>
   proof -
     define F' where \<open>F' e = {f\<in>F. \<not> is_orthogonal f e}\<close> for e
     have \<open>\<exists>e\<in>E. cinner f e \<noteq> 0\<close> if \<open>f \<in> F\<close> for f
     proof (rule ccontr, simp)
       assume \<open>\<forall>e\<in>E. is_orthogonal f e\<close>
       then have \<open>f \<in> orthogonal_complement E\<close>
         by (simp add: orthogonal_complementI)
       also have \<open>\<dots> = orthogonal_complement (closure (cspan E))\<close>
         using orthogonal_complement_of_closure orthogonal_complement_of_cspan by blast
       also have \<open>\<dots> = space_as_set (- ccspan E)\<close>
         apply transfer by simp
       also have \<open>\<dots> = space_as_set (- top)\<close>
         by (simp add: \<open>ccspan E = top\<close>)
       also have \<open>\<dots> = {0}\<close>
         by (auto simp add: top_ccsubspace.rep_eq uminus_ccsubspace.rep_eq)
       finally have \<open>f = 0\<close>
         by simp
       with \<open>f \<in> F\<close> \<open>is_ortho_set F\<close> show False
         by (simp add: is_onb_def is_ortho_set_def)
     qed
     then have F'_union: \<open>F = (\<Union>e\<in>E. F' e)\<close>
       unfolding F'_def by auto
     have \<open>countable (F' e)\<close> for e
     proof -
       have \<open>(\<Sum>f\<in>M. (cmod (cinner (sgn f) e))\<^sup>2) \<le> (norm e)\<^sup>2\<close> if \<open>finite M\<close> and \<open>M \<subseteq> F\<close> for M
       proof -
         have [simp]: \<open>is_ortho_set M\<close>
           using \<open>is_ortho_set F\<close> is_ortho_set_antimono that(2) by blast
         have [simp]: \<open>norm (sgn f) = 1\<close> if \<open>f \<in> M\<close> for f
           by (metis \<open>is_ortho_set M\<close> is_ortho_set_def norm_sgn that)
         have \<open>(\<Sum>f\<in>M. (cmod (cinner (sgn f) e))\<^sup>2) = (\<Sum>f\<in>M. (norm ((cinner (sgn f) e) *\<^sub>C sgn f))\<^sup>2)\<close>
           apply (rule sum.cong[OF refl])
           by simp
         also have \<open>\<dots> = (norm (\<Sum>f\<in>M. ((cinner (sgn f) e) *\<^sub>C sgn f)))\<^sup>2\<close>
           apply (rule pythagorean_theorem_sum[symmetric])
           using that apply auto
           by (meson \<open>is_ortho_set M\<close> is_ortho_set_def)
         also have \<open>\<dots> = (norm (\<Sum>f\<in>M. proj f e))\<^sup>2\<close>
           by (metis butterfly_apply butterfly_sgn_eq_proj)
         also have \<open>\<dots> = (norm (Proj (ccspan M) e))\<^sup>2\<close>
           apply (rule arg_cong[where f=\<open>\<lambda>x. (norm x)\<^sup>2\<close>])
           using \<open>finite M\<close> \<open>is_ortho_set M\<close> apply induction
            apply simp
           by (smt (verit, ccfv_threshold) Proj_orthog_ccspan_insert insertCI is_ortho_set_def plus_cblinfun.rep_eq sum.insert)
         also have \<open>\<dots> \<le> (norm (Proj (ccspan M)) * norm e)\<^sup>2\<close>
           by (auto simp: norm_cblinfun intro!: power_mono)
         also have \<open>\<dots> \<le> (norm e)\<^sup>2\<close>
           apply (rule power_mono)
            apply (metis norm_Proj_leq1 mult_left_le_one_le norm_ge_zero)
           by simp
         finally show ?thesis
           by -
       qed
       then have \<open>(\<lambda>f. (cmod (cinner (sgn f) e))\<^sup>2) abs_summable_on F\<close>
         apply (intro nonneg_bdd_above_summable_on bdd_aboveI)
         by auto
       then have \<open>countable {f \<in> F. (cmod (sgn f \<bullet>\<^sub>C e))\<^sup>2 \<noteq> 0}\<close>
         by (rule abs_summable_countable)
       then have \<open>countable {f \<in> F. \<not> is_orthogonal (sgn f) e}\<close>
         by force
       then have \<open>countable {f \<in> F. \<not> is_orthogonal f e}\<close>
         by force
       then show ?thesis
         unfolding F'_def by simp
     qed
     then have F'_leq: \<open>|F' e| \<le>o natLeq\<close> for e
       using countable_leq_natLeq by auto
 
     from F'_union have \<open>|F| \<le>o |Sigma E F'|\<close> (is \<open>_ \<le>o \<dots>\<close>)
       using card_of_UNION_Sigma by blast
     also have \<open>\<dots> \<le>o |E \<times> (UNIV::nat set)|\<close> (is \<open>_ \<le>o \<dots>\<close>)
       apply (rule card_of_Sigma_mono1)
       using F'_leq
       using card_of_nat ordIso_symmetric ordLeq_ordIso_trans by blast
     also have \<open>\<dots> =o |E| *c natLeq\<close> (is \<open>ordIso2 _ \<dots>\<close>)
       by (metis Field_card_of Field_natLeq card_of_ordIso_subst cprod_def)
     also have \<open>\<dots> =o |E|\<close>
       apply (rule card_prod_omega)
       using that by (simp add: cinfinite_def)
     finally show \<open>|F| \<le>o |E|\<close>
       by -
   qed
   then have infinite: \<open>|E| =o |F|\<close> if \<open>infinite E\<close> and \<open>infinite F\<close>
     by (simp add: assms ordIso_iff_ordLeq that(1) that(2))
 
   have \<open>|E| =o |F|\<close> if \<open>finite E\<close> and \<open>is_ortho_set E\<close> \<open>is_ortho_set F\<close> \<open>ccspan E = top\<close> \<open>ccspan F = top\<close>
     for E F :: \<open>'a set\<close>
   proof -
     have \<open>card E = card F\<close>
       using that 
       by (metis bij_betw_same_card ccspan.rep_eq closure_finite_cspan complex_vector.bij_if_span_eq_span_bases 
           complex_vector.independent_span_bound is_ortho_set_cindependent top_ccsubspace.rep_eq top_greatest)
     with \<open>finite E\<close> have \<open>finite F\<close>
       by (metis ccspan.rep_eq closure_finite_cspan complex_vector.independent_span_bound is_ortho_set_cindependent that(3) that(4) top_ccsubspace.rep_eq top_greatest)
     with \<open>card E = card F\<close> that show ?thesis
       apply (rule_tac finite_card_of_iff_card[THEN iffD2])
       by auto
   qed
 
   with infinite assms have \<open>|E| =o |F|\<close>
     by (meson ordIso_symmetric)
 
   then show ?thesis
     by (simp add: card_of_ordIso)
 qed
 
 lemma all_onbs_same_card:
   fixes E F :: \<open>'a::chilbert_space set\<close>
   assumes \<open>is_onb E\<close> \<open>is_onb F\<close>
   shows \<open>\<exists>f. bij_betw f E F\<close>
   apply (rule all_ortho_bases_same_card)
   using assms by (auto simp: is_onb_def)
 
 definition bij_between_bases where \<open>bij_between_bases E F = (SOME f. bij_betw f E F)\<close> for E F :: \<open>'a::chilbert_space set\<close>
 
 lemma
   fixes E F :: \<open>'a::chilbert_space set\<close>
   assumes \<open>is_onb E\<close> \<open>is_onb F\<close>
   shows bij_between_bases_bij: \<open>bij_betw (bij_between_bases E F) E F\<close>
   using all_onbs_same_card
   by (metis assms(1) assms(2) bij_between_bases_def someI)
 
 definition unitary_between where \<open>unitary_between E F = cblinfun_extension E (bij_between_bases E F)\<close>
 
 lemma unitary_between_apply:
   fixes E F :: \<open>'a::chilbert_space set\<close>
   assumes \<open>is_onb E\<close> \<open>is_onb F\<close> \<open>e \<in> E\<close>
   shows \<open>unitary_between E F *\<^sub>V e = bij_between_bases E F e\<close>
 proof -
   from \<open>is_onb E\<close> \<open>is_onb F\<close>
   have EF: \<open>bij_between_bases E F e \<in> F\<close> if \<open>e \<in> E\<close> for e
     by (meson bij_betwE bij_between_bases_bij that)
   have ortho: \<open>is_orthogonal (bij_between_bases E F x) (bij_between_bases E F y)\<close> if \<open>x \<noteq> y\<close> and \<open>x \<in> E\<close> and \<open>y \<in> E\<close> for x y
     by (smt (verit, del_insts) assms(1) assms(2) bij_betw_iff_bijections bij_between_bases_bij is_onb_def is_ortho_set_def that(1) that(2) that(3))
   have spanE: \<open>closure (cspan E) = UNIV\<close>
     by (metis assms(1) ccspan.rep_eq is_onb_def top_ccsubspace.rep_eq)
   show ?thesis
     unfolding unitary_between_def
     apply (rule cblinfun_extension_apply)
      apply (rule cblinfun_extension_exists_ortho[where B=1])
     using assms EF ortho spanE
     by (auto simp: is_onb_def)
 qed
 
 lemma unitary_between_unitary:
   fixes E F :: \<open>'a::chilbert_space set\<close>
   assumes \<open>is_onb E\<close> \<open>is_onb F\<close>
   shows \<open>unitary (unitary_between E F)\<close>
 proof -
   have \<open>(unitary_between E F *\<^sub>V b) \<bullet>\<^sub>C (unitary_between E F *\<^sub>V c) = b \<bullet>\<^sub>C c\<close> if \<open>b \<in> E\<close> and \<open>c \<in> E\<close> for b c
   proof (cases \<open>b = c\<close>)
     case True
     from \<open>is_onb E\<close> that have 1: \<open>b \<bullet>\<^sub>C b = 1\<close>
       using cnorm_eq_1 is_onb_def by blast
     from that have \<open>unitary_between E F *\<^sub>V b \<in> F\<close>
       by (metis assms(1) assms(2) bij_betw_apply bij_between_bases_bij unitary_between_apply)
     with \<open>is_onb F\<close> have 2: \<open>(unitary_between E F *\<^sub>V b) \<bullet>\<^sub>C (unitary_between E F *\<^sub>V b) = 1\<close>
       by (simp add: cnorm_eq_1 is_onb_def)
     from 1 2 True show ?thesis
       by simp
   next
     case False
     from \<open>is_onb E\<close> that have 1: \<open>b \<bullet>\<^sub>C c = 0\<close>
       by (simp add: False is_onb_def is_ortho_set_def)
     from that have inF: \<open>unitary_between E F *\<^sub>V b \<in> F\<close> \<open>unitary_between E F *\<^sub>V c \<in> F\<close>
       by (metis assms(1) assms(2) bij_betw_apply bij_between_bases_bij unitary_between_apply)+
     have neq: \<open>unitary_between E F *\<^sub>V b \<noteq> unitary_between E F *\<^sub>V c\<close>
       by (metis (no_types, lifting) False assms(1) assms(2) bij_betw_iff_bijections bij_between_bases_bij that(1) that(2) unitary_between_apply)
     from inF neq \<open>is_onb F\<close> have 2: \<open>(unitary_between E F *\<^sub>V b) \<bullet>\<^sub>C (unitary_between E F *\<^sub>V c) = 0\<close>
       by (simp add: is_onb_def is_ortho_set_def)
     from 1 2 show ?thesis
       by simp
   qed
   then have iso: \<open>isometry (unitary_between E F)\<close>
     apply (rule_tac orthogonal_on_basis_is_isometry[where B=E])
     using assms(1) is_onb_def by auto
   have \<open>unitary_between E F *\<^sub>S top = unitary_between E F *\<^sub>S ccspan E\<close>
     by (metis assms(1) is_onb_def)
   also have \<open>\<dots> \<ge> ccspan (unitary_between E F ` E)\<close> (is \<open>_ \<ge> \<dots>\<close>)
     by (simp add: cblinfun_image_ccspan)
   also have \<open>\<dots> = ccspan (bij_between_bases E F ` E)\<close>
     by (metis assms(1) assms(2) image_cong unitary_between_apply)
   also have \<open>\<dots> = ccspan F\<close>
     by (metis assms(1) assms(2) bij_betw_imp_surj_on bij_between_bases_bij)
   also have \<open>\<dots> = top\<close>
     using assms(2) is_onb_def by blast
   finally have surj: \<open>unitary_between E F *\<^sub>S top = top\<close>
     by (simp add: top.extremum_unique)
   from iso surj show ?thesis
     by (rule surj_isometry_is_unitary)
 qed
 
 subsection \<open>Notation\<close>
 
 bundle cblinfun_notation begin
 notation cblinfun_compose (infixl "o\<^sub>C\<^sub>L" 67)
 notation cblinfun_apply (infixr "*\<^sub>V" 70)
 notation cblinfun_image (infixr "*\<^sub>S" 70)
 notation adj ("_*" [99] 100)
 end
 
 bundle no_cblinfun_notation begin
 no_notation cblinfun_compose (infixl "o\<^sub>C\<^sub>L" 67)
 no_notation cblinfun_apply (infixr "*\<^sub>V" 70)
 no_notation cblinfun_image (infixr "*\<^sub>S" 70)
 no_notation adj ("_*" [99] 100)
 end
 
 bundle blinfun_notation begin
 notation blinfun_apply (infixr "*\<^sub>V" 70)
 end
 
 bundle no_blinfun_notation begin
 no_notation blinfun_apply (infixr "*\<^sub>V" 70)
 end
 
 unbundle no_cblinfun_notation
 unbundle no_lattice_syntax
 
 end
diff --git a/thys/Complex_Bounded_Operators/Complex_Bounded_Linear_Function0.thy b/thys/Complex_Bounded_Operators/Complex_Bounded_Linear_Function0.thy
--- a/thys/Complex_Bounded_Operators/Complex_Bounded_Linear_Function0.thy
+++ b/thys/Complex_Bounded_Operators/Complex_Bounded_Linear_Function0.thy
@@ -1,924 +1,924 @@
 (*  Title:      HOL/Analysis/Bounded_Linear_Function.thy
     Author:     Fabian Immler, TU München
 *)
 
 section \<open>\<open>Complex_Bounded_Linear_Function0\<close> -- Bounded Linear Function\<close>
 
 theory Complex_Bounded_Linear_Function0
   imports
     "HOL-Analysis.Bounded_Linear_Function"
     Complex_Inner_Product
     Complex_Euclidean_Space0
 begin
 
 unbundle cinner_syntax
 
 lemma conorm_componentwise:
   assumes "bounded_clinear f"
   shows "onorm f \<le> (\<Sum>i\<in>CBasis. norm (f i))"
 proof -
   {
     fix i::'a
     assume "i \<in> CBasis"
     hence "onorm (\<lambda>x. (i \<bullet>\<^sub>C x) *\<^sub>C f i) \<le> onorm (\<lambda>x. (i \<bullet>\<^sub>C x)) * norm (f i)"
       by (auto intro!: onorm_scaleC_left_lemma bounded_clinear_cinner_right)
     also have "\<dots> \<le>  norm i * norm (f i)"
       apply (rule mult_right_mono)
        apply (simp add: complex_inner_class.Cauchy_Schwarz_ineq2 onorm_bound)
       by simp
     finally have "onorm (\<lambda>x. (i \<bullet>\<^sub>C x) *\<^sub>C f i) \<le> norm (f i)" using \<open>i \<in> CBasis\<close>
       by simp
   } hence "onorm (\<lambda>x. \<Sum>i\<in>CBasis. (i \<bullet>\<^sub>C x) *\<^sub>C f i) \<le> (\<Sum>i\<in>CBasis. norm (f i))"
     by (auto intro!: order_trans[OF onorm_sum_le] bounded_clinear_scaleC_const
         sum_mono bounded_clinear_cinner_right bounded_clinear.bounded_linear)
   also have "(\<lambda>x. \<Sum>i\<in>CBasis. (i \<bullet>\<^sub>C x) *\<^sub>C f i) = (\<lambda>x. f (\<Sum>i\<in>CBasis. (i \<bullet>\<^sub>C x) *\<^sub>C i))"
     by (simp add: clinear.scaleC linear_sum bounded_clinear.clinear clinear.linear assms)
   also have "\<dots> = f"
     by (simp add: ceuclidean_representation)
   finally show ?thesis .
 qed
 
 lemmas conorm_componentwise_le = order_trans[OF conorm_componentwise]
 
 subsection\<^marker>\<open>tag unimportant\<close> \<open>Intro rules for \<^term>\<open>bounded_linear\<close>\<close>
 
 (* We share the same attribute [bounded_linear_intros] with Bounded_Linear_Function *)
 (* named_theorems bounded_linear_intros *)
 
 lemma onorm_cinner_left:
   assumes "bounded_linear r"
   shows "onorm (\<lambda>x. r x \<bullet>\<^sub>C f) \<le> onorm r * norm f"
 proof (rule onorm_bound)
   fix x
   have "norm (r x \<bullet>\<^sub>C f) \<le> norm (r x) * norm f"
     by (simp add: Cauchy_Schwarz_ineq2)
   also have "\<dots> \<le> onorm r * norm x * norm f"
     by (simp add: assms mult.commute mult_left_mono onorm)
   finally show "norm (r x \<bullet>\<^sub>C f) \<le> onorm r * norm f * norm x"
     by (simp add: ac_simps)
 qed (intro mult_nonneg_nonneg norm_ge_zero onorm_pos_le assms)
 
 lemma onorm_cinner_right:
   assumes "bounded_linear r"
   shows "onorm (\<lambda>x. f \<bullet>\<^sub>C r x) \<le> norm f * onorm r"
 proof (rule onorm_bound)
   fix x
   have "norm (f \<bullet>\<^sub>C r x) \<le> norm f * norm (r x)"
     by (simp add: Cauchy_Schwarz_ineq2)
   also have "\<dots> \<le> onorm r * norm x * norm f"
     by (simp add: assms mult.commute mult_left_mono onorm)
   finally show "norm (f \<bullet>\<^sub>C r x) \<le> norm f * onorm r * norm x"
     by (simp add: ac_simps)
 qed (intro mult_nonneg_nonneg norm_ge_zero onorm_pos_le assms)
 
 lemmas [bounded_linear_intros] =
   bounded_clinear_zero
   bounded_clinear_add
   bounded_clinear_const_mult
   bounded_clinear_mult_const
   bounded_clinear_scaleC_const
   bounded_clinear_const_scaleC
   bounded_clinear_const_scaleR
   bounded_clinear_ident
   bounded_clinear_sum
   (* bounded_clinear_Pair *) (* The Product_Vector theory does not instantiate Pair for complex vector spaces *)
   bounded_clinear_sub
   (* bounded_clinear_fst_comp *) (* The Product_Vector theory does not instantiate Pair for complex vector spaces *)
   (* bounded_clinear_snd_comp *) (* The Product_Vector theory does not instantiate Pair for complex vector spaces *)
   bounded_antilinear_cinner_left_comp
   bounded_clinear_cinner_right_comp
 
 
 subsection\<^marker>\<open>tag unimportant\<close> \<open>declaration of derivative/continuous/tendsto introduction rules for bounded linear functions\<close>
 
 attribute_setup bounded_clinear =
   \<open>let val bounded_linear = Attrib.attribute \<^context> (the_single @{attributes [bounded_linear]}) in
    Scan.succeed (Thm.declaration_attribute (fn thm =>
     Thm.attribute_declaration bounded_linear (thm RS @{thm bounded_clinear.bounded_linear}) o
     fold (fn (r, s) => Named_Theorems.add_thm s (thm RS r))
       [
         (* Not present in Bounded_Linear_Function *)
         (@{thm bounded_clinear_compose}, \<^named_theorems>\<open>bounded_linear_intros\<close>),
         (@{thm bounded_clinear_o_bounded_antilinear[unfolded o_def]}, \<^named_theorems>\<open>bounded_linear_intros\<close>)
       ]))
   end\<close>
 
 (* Analogue to [bounded_clinear], not present in Bounded_Linear_Function *)
 attribute_setup bounded_antilinear =
   \<open>let val bounded_linear = Attrib.attribute \<^context> (the_single @{attributes [bounded_linear]}) in
    Scan.succeed (Thm.declaration_attribute (fn thm =>
     Thm.attribute_declaration bounded_linear (thm RS @{thm bounded_antilinear.bounded_linear}) o
     fold (fn (r, s) => Named_Theorems.add_thm s (thm RS r))
       [
         (* Not present in Bounded_Linear_Function *)
         (@{thm bounded_antilinear_o_bounded_clinear[unfolded o_def]}, \<^named_theorems>\<open>bounded_linear_intros\<close>),
         (@{thm bounded_antilinear_o_bounded_antilinear[unfolded o_def]}, \<^named_theorems>\<open>bounded_linear_intros\<close>)
       ]))
   end\<close>
 
 attribute_setup bounded_cbilinear =
   \<open>let val bounded_bilinear = Attrib.attribute \<^context> (the_single @{attributes [bounded_bilinear]}) in
    Scan.succeed (Thm.declaration_attribute (fn thm =>
     Thm.attribute_declaration bounded_bilinear (thm RS @{thm bounded_cbilinear.bounded_bilinear}) o
     fold (fn (r, s) => Named_Theorems.add_thm s (thm RS r))
       [
         (@{thm bounded_clinear_compose[OF bounded_cbilinear.bounded_clinear_left]},
           \<^named_theorems>\<open>bounded_linear_intros\<close>),
         (@{thm bounded_clinear_compose[OF bounded_cbilinear.bounded_clinear_right]},
           \<^named_theorems>\<open>bounded_linear_intros\<close>),
         (@{thm bounded_clinear_o_bounded_antilinear[unfolded o_def, OF bounded_cbilinear.bounded_clinear_left]},
           \<^named_theorems>\<open>bounded_linear_intros\<close>),
         (@{thm bounded_clinear_o_bounded_antilinear[unfolded o_def, OF bounded_cbilinear.bounded_clinear_right]},
           \<^named_theorems>\<open>bounded_linear_intros\<close>)
       ]))
   end\<close>
 
 (* Analogue to [bounded_sesquilinear], not present in Bounded_Linear_Function *)
 attribute_setup bounded_sesquilinear =
   \<open>let val bounded_bilinear = Attrib.attribute \<^context> (the_single @{attributes [bounded_bilinear]}) in
    Scan.succeed (Thm.declaration_attribute (fn thm =>
     Thm.attribute_declaration bounded_bilinear (thm RS @{thm bounded_sesquilinear.bounded_bilinear}) o
     fold (fn (r, s) => Named_Theorems.add_thm s (thm RS r))
       [
         (@{thm bounded_antilinear_o_bounded_clinear[unfolded o_def, OF bounded_sesquilinear.bounded_antilinear_left]},
           \<^named_theorems>\<open>bounded_linear_intros\<close>),
         (@{thm bounded_clinear_compose[OF bounded_sesquilinear.bounded_clinear_right]},
           \<^named_theorems>\<open>bounded_linear_intros\<close>),
         (@{thm bounded_antilinear_o_bounded_antilinear[unfolded o_def, OF bounded_sesquilinear.bounded_antilinear_left]},
           \<^named_theorems>\<open>bounded_linear_intros\<close>),
         (@{thm bounded_clinear_o_bounded_antilinear[unfolded o_def, OF bounded_sesquilinear.bounded_clinear_right]},
           \<^named_theorems>\<open>bounded_linear_intros\<close>)
       ]))
   end\<close>
 
 
 subsection \<open>Type of complex bounded linear functions\<close>
 
 typedef\<^marker>\<open>tag important\<close> (overloaded) ('a, 'b) cblinfun ("(_ \<Rightarrow>\<^sub>C\<^sub>L /_)" [22, 21] 21) =
   "{f::'a::complex_normed_vector\<Rightarrow>'b::complex_normed_vector. bounded_clinear f}"
   morphisms cblinfun_apply CBlinfun
   by (blast intro: bounded_linear_intros)
 
 declare [[coercion
       "cblinfun_apply :: ('a::complex_normed_vector \<Rightarrow>\<^sub>C\<^sub>L'b::complex_normed_vector) \<Rightarrow> 'a \<Rightarrow> 'b"]]
 
 lemma bounded_clinear_cblinfun_apply[bounded_linear_intros]:
   "bounded_clinear g \<Longrightarrow> bounded_clinear (\<lambda>x. cblinfun_apply f (g x))"
   by (metis cblinfun_apply mem_Collect_eq bounded_clinear_compose)
 
 setup_lifting type_definition_cblinfun
 
 lemma cblinfun_eqI: "(\<And>i. cblinfun_apply x i = cblinfun_apply y i) \<Longrightarrow> x = y"
   by transfer auto
 
 lemma bounded_clinear_CBlinfun_apply: "bounded_clinear f \<Longrightarrow> cblinfun_apply (CBlinfun f) = f"
   by (auto simp: CBlinfun_inverse)
 
 
 subsection \<open>Type class instantiations\<close>
 
 instantiation cblinfun :: (complex_normed_vector, complex_normed_vector) complex_normed_vector
 begin
 
 lift_definition\<^marker>\<open>tag important\<close> norm_cblinfun :: "'a \<Rightarrow>\<^sub>C\<^sub>L 'b \<Rightarrow> real" is onorm .
 
 lift_definition minus_cblinfun :: "'a \<Rightarrow>\<^sub>C\<^sub>L 'b \<Rightarrow> 'a \<Rightarrow>\<^sub>C\<^sub>L 'b \<Rightarrow> 'a \<Rightarrow>\<^sub>C\<^sub>L 'b"
   is "\<lambda>f g x. f x - g x"
   by (rule bounded_clinear_sub)
 
 definition dist_cblinfun :: "'a \<Rightarrow>\<^sub>C\<^sub>L 'b \<Rightarrow> 'a \<Rightarrow>\<^sub>C\<^sub>L 'b \<Rightarrow> real"
   where "dist_cblinfun a b = norm (a - b)"
 
 definition [code del]:
   "(uniformity :: (('a \<Rightarrow>\<^sub>C\<^sub>L 'b) \<times> ('a \<Rightarrow>\<^sub>C\<^sub>L 'b)) filter) = (INF e\<in>{0 <..}. principal {(x, y). dist x y < e})"
 
 definition open_cblinfun :: "('a \<Rightarrow>\<^sub>C\<^sub>L 'b) set \<Rightarrow> bool"
   where [code del]: "open_cblinfun S = (\<forall>x\<in>S. \<forall>\<^sub>F (x', y) in uniformity. x' = x \<longrightarrow> y \<in> S)"
 
 lift_definition uminus_cblinfun :: "'a \<Rightarrow>\<^sub>C\<^sub>L 'b \<Rightarrow> 'a \<Rightarrow>\<^sub>C\<^sub>L 'b" is "\<lambda>f x. - f x"
   by (rule bounded_clinear_minus)
 
 lift_definition\<^marker>\<open>tag important\<close> zero_cblinfun :: "'a \<Rightarrow>\<^sub>C\<^sub>L 'b" is "\<lambda>x. 0"
   by (rule bounded_clinear_zero)
 
 lift_definition\<^marker>\<open>tag important\<close> plus_cblinfun :: "'a \<Rightarrow>\<^sub>C\<^sub>L 'b \<Rightarrow> 'a \<Rightarrow>\<^sub>C\<^sub>L 'b \<Rightarrow> 'a \<Rightarrow>\<^sub>C\<^sub>L 'b"
   is "\<lambda>f g x. f x + g x"
   by (metis bounded_clinear_add)
 
 lift_definition\<^marker>\<open>tag important\<close> scaleC_cblinfun::"complex \<Rightarrow> 'a \<Rightarrow>\<^sub>C\<^sub>L 'b \<Rightarrow> 'a \<Rightarrow>\<^sub>C\<^sub>L 'b" is "\<lambda>r f x. r *\<^sub>C f x"
   by (metis bounded_clinear_compose bounded_clinear_scaleC_right)
 lift_definition\<^marker>\<open>tag important\<close> scaleR_cblinfun::"real \<Rightarrow> 'a \<Rightarrow>\<^sub>C\<^sub>L 'b \<Rightarrow> 'a \<Rightarrow>\<^sub>C\<^sub>L 'b" is "\<lambda>r f x. r *\<^sub>R f x"
   by (rule bounded_clinear_const_scaleR)
 
 definition sgn_cblinfun :: "'a \<Rightarrow>\<^sub>C\<^sub>L 'b \<Rightarrow> 'a \<Rightarrow>\<^sub>C\<^sub>L 'b"
   where "sgn_cblinfun x = scaleC (inverse (norm x)) x"
 
 instance
 proof
   fix a b c :: "'a \<Rightarrow>\<^sub>C\<^sub>L'b" and r q :: real and s t :: complex
 
   show \<open>a + b + c = a + (b + c)\<close>
     apply transfer by auto
   show \<open>0 + a = a\<close>
     apply transfer by auto
   show \<open>a + b = b + a\<close>
     apply transfer by auto
   show \<open>- a + a = 0\<close>
     apply transfer by auto
   show \<open>a - b = a + - b\<close>
     apply transfer by auto
   show scaleR_scaleC: \<open>((*\<^sub>R) r::('a \<Rightarrow>\<^sub>C\<^sub>L 'b) \<Rightarrow> _) = (*\<^sub>C) (complex_of_real r)\<close> for r
     apply (rule ext, transfer fixing: r) by (simp add: scaleR_scaleC)
   show \<open>s *\<^sub>C (b + c) = s *\<^sub>C b + s *\<^sub>C c\<close>
     apply transfer by (simp add: scaleC_add_right) 
   show \<open>(s + t) *\<^sub>C a = s *\<^sub>C a + t *\<^sub>C a\<close>
     apply transfer by (simp add: scaleC_left.add) 
   show \<open>s *\<^sub>C t *\<^sub>C a = (s * t) *\<^sub>C a\<close>
     apply transfer by auto
   show \<open>1 *\<^sub>C a = a\<close>
     apply transfer by auto
   show \<open>dist a b = norm (a - b)\<close>
     unfolding dist_cblinfun_def by simp
   show \<open>sgn a = (inverse (norm a)) *\<^sub>R a\<close>
     unfolding sgn_cblinfun_def unfolding scaleR_scaleC by auto
   show \<open>uniformity = (INF e\<in>{0<..}. principal {(x, y). dist (x::('a \<Rightarrow>\<^sub>C\<^sub>L 'b)) y < e})\<close>
     by (simp add: uniformity_cblinfun_def)
   show \<open>open U = (\<forall>x\<in>U. \<forall>\<^sub>F (x', y) in uniformity. (x'::('a \<Rightarrow>\<^sub>C\<^sub>L 'b)) = x \<longrightarrow> y \<in> U)\<close> for U
     by (simp add: open_cblinfun_def)
   show \<open>(norm a = 0) = (a = 0)\<close>
     apply transfer using bounded_clinear.bounded_linear onorm_eq_0 by blast
   show \<open>norm (a + b) \<le> norm a + norm b\<close>
     apply transfer by (simp add: bounded_clinear.bounded_linear onorm_triangle)
   show \<open>norm (s *\<^sub>C a) = cmod s * norm a\<close>
     apply transfer using onorm_scalarC by blast
   show \<open>norm (r *\<^sub>R a) = \<bar>r\<bar> * norm a\<close>
     apply transfer using bounded_clinear.bounded_linear onorm_scaleR by blast
   show \<open>r *\<^sub>R (a + b) = r *\<^sub>R a +  r *\<^sub>R b\<close>
     apply transfer by (simp add: scaleR_add_right) 
   show \<open>(r + q) *\<^sub>R a = r *\<^sub>R a +  q *\<^sub>R a\<close>
     apply transfer by (simp add: scaleR_add_left)
   show \<open>r *\<^sub>R q *\<^sub>R a = (r * q) *\<^sub>R a\<close>
     apply transfer by auto
   show \<open>1 *\<^sub>R a = a\<close>
     apply transfer by auto
 qed
 
 end
 
 declare uniformity_Abort[where 'a="('a :: complex_normed_vector) \<Rightarrow>\<^sub>C\<^sub>L ('b :: complex_normed_vector)", code]
 
 lemma norm_cblinfun_eqI:
   assumes "n \<le> norm (cblinfun_apply f x) / norm x"
   assumes "\<And>x. norm (cblinfun_apply f x) \<le> n * norm x"
   assumes "0 \<le> n"
   shows "norm f = n"
   by (auto simp: norm_cblinfun_def
       intro!: antisym onorm_bound assms order_trans[OF _ le_onorm] bounded_clinear.bounded_linear
       bounded_linear_intros)
 
 lemma norm_cblinfun: "norm (cblinfun_apply f x) \<le> norm f * norm x"
   apply transfer by (simp add: bounded_clinear.bounded_linear onorm)
 
 lemma norm_cblinfun_bound: "0 \<le> b \<Longrightarrow> (\<And>x. norm (cblinfun_apply f x) \<le> b * norm x) \<Longrightarrow> norm f \<le> b"
   by transfer (rule onorm_bound)
 
 lemma bounded_cbilinear_cblinfun_apply[bounded_cbilinear]: "bounded_cbilinear cblinfun_apply"
 proof
   fix f g::"'a \<Rightarrow>\<^sub>C\<^sub>L 'b" and a b::'a and r::complex
   show "(f + g) a = f a + g a" "(r *\<^sub>C f) a = r *\<^sub>C f a"
     by (transfer, simp)+
   interpret bounded_clinear f for f::"'a \<Rightarrow>\<^sub>C\<^sub>L 'b"
     by (auto intro!: bounded_linear_intros)
   show "f (a + b) = f a + f b" "f (r *\<^sub>C a) = r *\<^sub>C f a"
     by (simp_all add: add scaleC)
   show "\<exists>K. \<forall>a b. norm (cblinfun_apply a b) \<le> norm a * norm b * K"
     by (auto intro!: exI[where x=1] norm_cblinfun)
 qed
 
 interpretation cblinfun: bounded_cbilinear cblinfun_apply
   by (rule bounded_cbilinear_cblinfun_apply)
 
 lemmas bounded_clinear_apply_cblinfun[intro, simp] = cblinfun.bounded_clinear_left
 
 declare cblinfun.zero_left [simp] cblinfun.zero_right [simp]
 
 
 context bounded_cbilinear
 begin
 
 named_theorems cbilinear_simps
 
 lemmas [cbilinear_simps] =
   add_left
   add_right
   diff_left
   diff_right
   minus_left
   minus_right
   scaleC_left
   scaleC_right
   zero_left
   zero_right
   sum_left
   sum_right
 
 end
 
 
 instance cblinfun :: (complex_normed_vector, cbanach) cbanach
 (* The proof is almost the same as for \<open>instance blinfun :: (real_normed_vector, banach) banach\<close> *)
 proof
   fix X::"nat \<Rightarrow> 'a \<Rightarrow>\<^sub>C\<^sub>L 'b"
   assume "Cauchy X"
   {
     fix x::'a
     {
       fix x::'a
       assume "norm x \<le> 1"
       have "Cauchy (\<lambda>n. X n x)"
       proof (rule CauchyI)
         fix e::real
         assume "0 < e"
         from CauchyD[OF \<open>Cauchy X\<close> \<open>0 < e\<close>] obtain M
           where M: "\<And>m n. m \<ge> M \<Longrightarrow> n \<ge> M \<Longrightarrow> norm (X m - X n) < e"
           by auto
         show "\<exists>M. \<forall>m\<ge>M. \<forall>n\<ge>M. norm (X m x - X n x) < e"
         proof (safe intro!: exI[where x=M])
           fix m n
           assume le: "M \<le> m" "M \<le> n"
           have "norm (X m x - X n x) = norm ((X m - X n) x)"
             by (simp add: cblinfun.cbilinear_simps)
           also have "\<dots> \<le> norm (X m - X n) * norm x"
             by (rule norm_cblinfun)
           also have "\<dots> \<le> norm (X m - X n) * 1"
             using \<open>norm x \<le> 1\<close> norm_ge_zero by (rule mult_left_mono)
           also have "\<dots> = norm (X m - X n)" by simp
           also have "\<dots> < e" using le by fact
           finally show "norm (X m x - X n x) < e" .
         qed
       qed
       hence "convergent (\<lambda>n. X n x)"
         by (metis Cauchy_convergent_iff)
     } note convergent_norm1 = this
     define y where "y = x /\<^sub>R norm x"
     have y: "norm y \<le> 1" and xy: "x = norm x *\<^sub>R y"
       by (simp_all add: y_def inverse_eq_divide)
     have "convergent (\<lambda>n. norm x *\<^sub>R X n y)"
       by (intro bounded_bilinear.convergent[OF bounded_bilinear_scaleR] convergent_const
           convergent_norm1 y)
     also have "(\<lambda>n. norm x *\<^sub>R X n y) = (\<lambda>n. X n x)"
       by (metis cblinfun.scaleC_right scaleR_scaleC xy)
     finally have "convergent (\<lambda>n. X n x)" .
   }
   then obtain v where v: "\<And>x. (\<lambda>n. X n x) \<longlonglongrightarrow> v x"
     unfolding convergent_def
     by metis
 
   have "Cauchy (\<lambda>n. norm (X n))"
   proof (rule CauchyI)
     fix e::real
     assume "e > 0"
     from CauchyD[OF \<open>Cauchy X\<close> \<open>0 < e\<close>] obtain M
       where M: "\<And>m n. m \<ge> M \<Longrightarrow> n \<ge> M \<Longrightarrow> norm (X m - X n) < e"
       by auto
     show "\<exists>M. \<forall>m\<ge>M. \<forall>n\<ge>M. norm (norm (X m) - norm (X n)) < e"
     proof (safe intro!: exI[where x=M])
       fix m n assume mn: "m \<ge> M" "n \<ge> M"
       have "norm (norm (X m) - norm (X n)) \<le> norm (X m - X n)"
         by (metis norm_triangle_ineq3 real_norm_def)
       also have "\<dots> < e" using mn by fact
       finally show "norm (norm (X m) - norm (X n)) < e" .
     qed
   qed
   then obtain K where K: "(\<lambda>n. norm (X n)) \<longlonglongrightarrow> K"
     unfolding Cauchy_convergent_iff convergent_def
     by metis
 
   have "bounded_clinear v"
   proof
     fix x y and r::complex
     from tendsto_add[OF v[of x] v [of y]] v[of "x + y", unfolded cblinfun.cbilinear_simps]
       tendsto_scaleC[OF tendsto_const[of r] v[of x]] v[of "r *\<^sub>C x", unfolded cblinfun.cbilinear_simps]
     show "v (x + y) = v x + v y" "v (r *\<^sub>C x) = r *\<^sub>C v x"
       by (metis (poly_guards_query) LIMSEQ_unique)+
     show "\<exists>K. \<forall>x. norm (v x) \<le> norm x * K"
     proof (safe intro!: exI[where x=K])
       fix x
       have "norm (v x) \<le> K * norm x"
         apply (rule tendsto_le[OF _ tendsto_mult[OF K tendsto_const] tendsto_norm[OF v]])
         by (auto simp: norm_cblinfun)
       thus "norm (v x) \<le> norm x * K"
         by (simp add: ac_simps)
     qed
   qed
   hence Bv: "\<And>x. (\<lambda>n. X n x) \<longlonglongrightarrow> CBlinfun v x"
     by (auto simp: bounded_clinear_CBlinfun_apply v)
 
   have "X \<longlonglongrightarrow> CBlinfun v"
   proof (rule LIMSEQ_I)
     fix r::real assume "r > 0"
     define r' where "r' = r / 2"
     have "0 < r'" "r' < r" using \<open>r > 0\<close> by (simp_all add: r'_def)
     from CauchyD[OF \<open>Cauchy X\<close> \<open>r' > 0\<close>]
     obtain M where M: "\<And>m n. m \<ge> M \<Longrightarrow> n \<ge> M \<Longrightarrow> norm (X m - X n) < r'"
       by metis
     show "\<exists>no. \<forall>n\<ge>no. norm (X n - CBlinfun v) < r"
     proof (safe intro!: exI[where x=M])
       fix n assume n: "M \<le> n"
       have "norm (X n - CBlinfun v) \<le> r'"
       proof (rule norm_cblinfun_bound)
         fix x
         have "eventually (\<lambda>m. m \<ge> M) sequentially"
           by (metis eventually_ge_at_top)
         hence ev_le: "eventually (\<lambda>m. norm (X n x - X m x) \<le> r' * norm x) sequentially"
         proof eventually_elim
           case (elim m)
           have "norm (X n x - X m x) = norm ((X n - X m) x)"
             by (simp add: cblinfun.cbilinear_simps)
           also have "\<dots> \<le> norm ((X n - X m)) * norm x"
             by (rule norm_cblinfun)
           also have "\<dots> \<le> r' * norm x"
             using M[OF n elim] by (simp add: mult_right_mono)
           finally show ?case .
         qed
         have tendsto_v: "(\<lambda>m. norm (X n x - X m x)) \<longlonglongrightarrow> norm (X n x - CBlinfun v x)"
           by (auto intro!: tendsto_intros Bv)
         show "norm ((X n - CBlinfun v) x) \<le> r' * norm x"
           by (auto intro!: tendsto_upperbound tendsto_v ev_le simp: cblinfun.cbilinear_simps)
       qed (simp add: \<open>0 < r'\<close> less_imp_le)
       thus "norm (X n - CBlinfun v) < r"
         by (metis \<open>r' < r\<close> le_less_trans)
     qed
   qed
   thus "convergent X"
     by (rule convergentI)
 qed
 
 subsection\<^marker>\<open>tag unimportant\<close> \<open>On Euclidean Space\<close>
 
 (* No different in complex case *)
 (* lemma Zfun_sum:
   assumes "finite s"
   assumes f: "\<And>i. i \<in> s \<Longrightarrow> Zfun (f i) F"
   shows "Zfun (\<lambda>x. sum (\<lambda>i. f i x) s) F" *)
 
 lemma norm_cblinfun_ceuclidean_le:
   fixes a::"'a::ceuclidean_space \<Rightarrow>\<^sub>C\<^sub>L 'b::complex_normed_vector"
   shows "norm a \<le> sum (\<lambda>x. norm (a x)) CBasis"
   apply (rule norm_cblinfun_bound)
    apply (simp add: sum_nonneg)
   apply (subst ceuclidean_representation[symmetric, where 'a='a])
   apply (simp only: cblinfun.cbilinear_simps sum_distrib_right)
   apply (rule order.trans[OF norm_sum sum_mono])
   apply (simp add: abs_mult mult_right_mono ac_simps CBasis_le_norm)
   by (metis complex_inner_class.Cauchy_Schwarz_ineq2 mult.commute mult.left_neutral mult_right_mono norm_CBasis norm_ge_zero)
 
 lemma ctendsto_componentwise1:
   fixes a::"'a::ceuclidean_space \<Rightarrow>\<^sub>C\<^sub>L 'b::complex_normed_vector"
     and b::"'c \<Rightarrow> 'a \<Rightarrow>\<^sub>C\<^sub>L 'b"
   assumes "(\<And>j. j \<in> CBasis \<Longrightarrow> ((\<lambda>n. b n j) \<longlongrightarrow> a j) F)"
   shows "(b \<longlongrightarrow> a) F"
 proof -
   have "\<And>j. j \<in> CBasis \<Longrightarrow> Zfun (\<lambda>x. norm (b x j - a j)) F"
     using assms unfolding tendsto_Zfun_iff Zfun_norm_iff .
   hence "Zfun (\<lambda>x. \<Sum>j\<in>CBasis. norm (b x j - a j)) F"
     by (auto intro!: Zfun_sum)
   thus ?thesis
     unfolding tendsto_Zfun_iff
     by (rule Zfun_le)
       (auto intro!: order_trans[OF norm_cblinfun_ceuclidean_le] simp: cblinfun.cbilinear_simps)
 qed
 
 lift_definition
   cblinfun_of_matrix::"('b::ceuclidean_space \<Rightarrow> 'a::ceuclidean_space \<Rightarrow> complex) \<Rightarrow> 'a \<Rightarrow>\<^sub>C\<^sub>L 'b"
   is "\<lambda>a x. \<Sum>i\<in>CBasis. \<Sum>j\<in>CBasis. ((j \<bullet>\<^sub>C x) * a i j) *\<^sub>C i"
   by (intro bounded_linear_intros)
 
 lemma cblinfun_of_matrix_works:
   fixes f::"'a::ceuclidean_space \<Rightarrow>\<^sub>C\<^sub>L 'b::ceuclidean_space"
   shows "cblinfun_of_matrix (\<lambda>i j. i \<bullet>\<^sub>C (f j)) = f"
 proof (transfer, rule,  rule ceuclidean_eqI)
   fix f::"'a \<Rightarrow> 'b" and x::'a and b::'b assume "bounded_clinear f" and b: "b \<in> CBasis"
   then interpret bounded_clinear f by simp
   have "(\<Sum>j\<in>CBasis. \<Sum>i\<in>CBasis. (i \<bullet>\<^sub>C x * (j \<bullet>\<^sub>C f i)) *\<^sub>C j) \<bullet>\<^sub>C b
     = (\<Sum>j\<in>CBasis. if j = b then (\<Sum>i\<in>CBasis. (x \<bullet>\<^sub>C i * (f i \<bullet>\<^sub>C j))) else 0)"
     using b
     apply (simp add: cinner_sum_left cinner_CBasis if_distrib cong: if_cong) 
     by (simp add: sum.swap)
   also have "\<dots> = (\<Sum>i\<in>CBasis. ((x \<bullet>\<^sub>C i) * (f i \<bullet>\<^sub>C b)))"
     using b by (simp)
   also have "\<dots> = f x \<bullet>\<^sub>C b"
   proof -
     have \<open>(\<Sum>i\<in>CBasis. (x \<bullet>\<^sub>C i) * (f i \<bullet>\<^sub>C b)) = (\<Sum>i\<in>CBasis. (i \<bullet>\<^sub>C x) *\<^sub>C f i) \<bullet>\<^sub>C b\<close>
       by (auto simp: cinner_sum_left)
     also have \<open>\<dots> = f x \<bullet>\<^sub>C b\<close>
       by (simp add: ceuclidean_representation sum[symmetric] scale[symmetric])
     finally show ?thesis by -
   qed
   finally show "(\<Sum>j\<in>CBasis. \<Sum>i\<in>CBasis. (i \<bullet>\<^sub>C x * (j \<bullet>\<^sub>C f i)) *\<^sub>C j) \<bullet>\<^sub>C b = f x \<bullet>\<^sub>C b" .
 qed
 
 
 lemma cblinfun_of_matrix_apply:
   "cblinfun_of_matrix a x = (\<Sum>i\<in>CBasis. \<Sum>j\<in>CBasis. ((j \<bullet>\<^sub>C x) * a i j) *\<^sub>C i)"
   apply transfer by simp
 
 lemma cblinfun_of_matrix_minus: "cblinfun_of_matrix x - cblinfun_of_matrix y = cblinfun_of_matrix (x - y)"
   by transfer (auto simp: algebra_simps sum_subtractf)
 
 lemma norm_cblinfun_of_matrix:
   "norm (cblinfun_of_matrix a) \<le> (\<Sum>i\<in>CBasis. \<Sum>j\<in>CBasis. cmod (a i j))"
   apply (rule norm_cblinfun_bound)
    apply (simp add: sum_nonneg)
   apply (simp only: cblinfun_of_matrix_apply sum_distrib_right)
   apply (rule order_trans[OF norm_sum sum_mono])
   apply (rule order_trans[OF norm_sum sum_mono])
   apply (simp add: abs_mult mult_right_mono ac_simps Basis_le_norm)
   by (metis complex_inner_class.Cauchy_Schwarz_ineq2 complex_scaleC_def mult.left_neutral mult_right_mono norm_CBasis norm_ge_zero norm_scaleC)
 
 lemma tendsto_cblinfun_of_matrix:
   assumes "\<And>i j. i \<in> CBasis \<Longrightarrow> j \<in> CBasis \<Longrightarrow> ((\<lambda>n. b n i j) \<longlongrightarrow> a i j) F"
   shows "((\<lambda>n. cblinfun_of_matrix (b n)) \<longlongrightarrow> cblinfun_of_matrix a) F"
 proof -
   have "\<And>i j. i \<in> CBasis \<Longrightarrow> j \<in> CBasis \<Longrightarrow> Zfun (\<lambda>x. norm (b x i j - a i j)) F"
     using assms unfolding tendsto_Zfun_iff Zfun_norm_iff .
   hence "Zfun (\<lambda>x. (\<Sum>i\<in>CBasis. \<Sum>j\<in>CBasis. cmod (b x i j - a i j))) F"
     by (auto intro!: Zfun_sum)
   thus ?thesis
     unfolding tendsto_Zfun_iff cblinfun_of_matrix_minus
     by (rule Zfun_le) (auto intro!: order_trans[OF norm_cblinfun_of_matrix])
 qed
 
 
 lemma ctendsto_componentwise:
   fixes a::"'a::ceuclidean_space \<Rightarrow>\<^sub>C\<^sub>L 'b::ceuclidean_space"
     and b::"'c \<Rightarrow> 'a \<Rightarrow>\<^sub>C\<^sub>L 'b"
   shows "(\<And>i j. i \<in> CBasis \<Longrightarrow> j \<in> CBasis \<Longrightarrow> ((\<lambda>n. b n j \<bullet>\<^sub>C i) \<longlongrightarrow> a j \<bullet>\<^sub>C i) F) \<Longrightarrow> (b \<longlongrightarrow> a) F"
   apply (subst cblinfun_of_matrix_works[of a, symmetric])
   apply (subst cblinfun_of_matrix_works[of "b x" for x, symmetric, abs_def])
   apply (rule tendsto_cblinfun_of_matrix)
   apply (subst (1) cinner_commute, subst (2) cinner_commute)
   by (metis lim_cnj)
 
 lemma
   continuous_cblinfun_componentwiseI:
   fixes f:: "'b::t2_space \<Rightarrow> 'a::ceuclidean_space \<Rightarrow>\<^sub>C\<^sub>L 'c::ceuclidean_space"
   assumes "\<And>i j. i \<in> CBasis \<Longrightarrow> j \<in> CBasis \<Longrightarrow> continuous F (\<lambda>x. (f x) j \<bullet>\<^sub>C i)"
   shows "continuous F f"
   using assms by (auto simp: continuous_def intro!: ctendsto_componentwise)
 
 lemma
   continuous_cblinfun_componentwiseI1:
   fixes f:: "'b::t2_space \<Rightarrow> 'a::ceuclidean_space \<Rightarrow>\<^sub>C\<^sub>L 'c::complex_normed_vector"
   assumes "\<And>i. i \<in> CBasis \<Longrightarrow> continuous F (\<lambda>x. f x i)"
   shows "continuous F f"
   using assms by (auto simp: continuous_def intro!: ctendsto_componentwise1)
 
 lemma
   continuous_on_cblinfun_componentwise:
   fixes f:: "'d::t2_space \<Rightarrow> 'e::ceuclidean_space \<Rightarrow>\<^sub>C\<^sub>L 'f::complex_normed_vector"
   assumes "\<And>i. i \<in> CBasis \<Longrightarrow> continuous_on s (\<lambda>x. f x i)"
   shows "continuous_on s f"
   using assms
   by (auto intro!: continuous_at_imp_continuous_on intro!: ctendsto_componentwise1
       simp: continuous_on_eq_continuous_within continuous_def)
 
 lemma bounded_antilinear_cblinfun_matrix: "bounded_antilinear (\<lambda>x. (x::_\<Rightarrow>\<^sub>C\<^sub>L _) j \<bullet>\<^sub>C i)"
   by (auto intro!: bounded_linear_intros)
 
 lemma continuous_cblinfun_matrix:
   fixes f:: "'b::t2_space \<Rightarrow> 'a::complex_normed_vector \<Rightarrow>\<^sub>C\<^sub>L 'c::complex_inner"
   assumes "continuous F f"
   shows "continuous F (\<lambda>x. (f x) j \<bullet>\<^sub>C i)"
   by (rule bounded_antilinear.continuous[OF bounded_antilinear_cblinfun_matrix assms])
 
 lemma continuous_on_cblinfun_matrix:
   fixes f::"'a::t2_space \<Rightarrow> 'b::complex_normed_vector \<Rightarrow>\<^sub>C\<^sub>L 'c::complex_inner"
   assumes "continuous_on S f"
   shows "continuous_on S (\<lambda>x. (f x) j \<bullet>\<^sub>C i)"
   using assms
   by (auto simp: continuous_on_eq_continuous_within continuous_cblinfun_matrix)
 
 lemma continuous_on_cblinfun_of_matrix[continuous_intros]:
   assumes "\<And>i j. i \<in> CBasis \<Longrightarrow> j \<in> CBasis \<Longrightarrow> continuous_on S (\<lambda>s. g s i j)"
   shows "continuous_on S (\<lambda>s. cblinfun_of_matrix (g s))"
   using assms
   by (auto simp: continuous_on intro!: tendsto_cblinfun_of_matrix)
 
 (* Not specific to complex/real *)
 (* lemma mult_if_delta:
   "(if P then (1::'a::comm_semiring_1) else 0) * q = (if P then q else 0)" *)
 
 (* Needs that ceuclidean_space is heine_borel. This is shown for euclidean_space in Toplogy_Euclidean_Space
    which has not been ported to complex *)
 (* lemma compact_cblinfun_lemma:
   fixes f :: "nat \<Rightarrow> 'a::ceuclidean_space \<Rightarrow>\<^sub>C\<^sub>L 'b::ceuclidean_space"
   assumes "bounded (range f)"
   shows "\<forall>d\<subseteq>CBasis. \<exists>l::'a \<Rightarrow>\<^sub>C\<^sub>L 'b. \<exists> r::nat\<Rightarrow>nat.
     strict_mono r \<and> (\<forall>e>0. eventually (\<lambda>n. \<forall>i\<in>d. dist (f (r n) i) (l i) < e) sequentially)"
   apply (rule compact_lemma_general[where unproj = "\<lambda>e. cblinfun_of_matrix (\<lambda>i j. e j \<bullet>\<^sub>C i)"])
   by (auto intro!: euclidean_eqI[where 'a='b] bounded_linear_image assms
     simp: blinfun_of_matrix_works blinfun_of_matrix_apply inner_Basis mult_if_delta sum.delta'
       scaleR_sum_left[symmetric]) *)
 
 
 lemma cblinfun_euclidean_eqI: "(\<And>i. i \<in> CBasis \<Longrightarrow> cblinfun_apply x i = cblinfun_apply y i) \<Longrightarrow> x = y"
   apply (auto intro!: cblinfun_eqI)
   apply (subst (2) ceuclidean_representation[symmetric, where 'a='a])
   apply (subst (1) ceuclidean_representation[symmetric, where 'a='a])
   by (simp add: cblinfun.cbilinear_simps)
 
 lemma CBlinfun_eq_matrix: "bounded_clinear f \<Longrightarrow> CBlinfun f = cblinfun_of_matrix (\<lambda>i j. i \<bullet>\<^sub>C f j)"
   apply (intro cblinfun_euclidean_eqI)
   by (auto simp: cblinfun_of_matrix_apply bounded_clinear_CBlinfun_apply cinner_CBasis if_distrib
       if_distribR sum.delta' ceuclidean_representation
       cong: if_cong)
 
 (* Conflicts with: cblinfun :: (complex_normed_vector, cbanach) complete_space *)
 (* instance cblinfun :: (ceuclidean_space, ceuclidean_space) heine_borel *)
 
 
 subsection\<^marker>\<open>tag unimportant\<close> \<open>concrete bounded linear functions\<close>
 
 lemma transfer_bounded_cbilinear_bounded_clinearI:
   assumes "g = (\<lambda>i x. (cblinfun_apply (f i) x))"
   shows "bounded_cbilinear g = bounded_clinear f"
 proof
   assume "bounded_cbilinear g"
   then interpret bounded_cbilinear f by (simp add: assms)
   show "bounded_clinear f"
   proof (unfold_locales, safe intro!: cblinfun_eqI)
     fix i
     show "f (x + y) i = (f x + f y) i" "f (r *\<^sub>C x) i = (r *\<^sub>C f x) i" for r x y
       by (auto intro!: cblinfun_eqI simp: cblinfun.cbilinear_simps)
     from _ nonneg_bounded show "\<exists>K. \<forall>x. norm (f x) \<le> norm x * K"
       by (rule ex_reg) (auto intro!: onorm_bound simp: norm_cblinfun.rep_eq ac_simps)
   qed
 qed (auto simp: assms intro!: cblinfun.comp)
 
 lemma transfer_bounded_cbilinear_bounded_clinear[transfer_rule]:
   "(rel_fun (rel_fun (=) (pcr_cblinfun (=) (=))) (=)) bounded_cbilinear bounded_clinear"
   by (auto simp: pcr_cblinfun_def cr_cblinfun_def rel_fun_def OO_def
       intro!: transfer_bounded_cbilinear_bounded_clinearI)
 
 (* Not present in Bounded_Linear_Function *)
 lemma transfer_bounded_sesquilinear_bounded_antilinearI:
   assumes "g = (\<lambda>i x. (cblinfun_apply (f i) x))"
   shows "bounded_sesquilinear g = bounded_antilinear f"
 proof
   assume "bounded_sesquilinear g"
   then interpret bounded_sesquilinear f by (simp add: assms)
   show "bounded_antilinear f"
   proof (unfold_locales, safe intro!: cblinfun_eqI)
     fix i
     show "f (x + y) i = (f x + f y) i" "f (r *\<^sub>C x) i = (cnj r *\<^sub>C f x) i" for r x y
       by (auto intro!: cblinfun_eqI simp: cblinfun.scaleC_left scaleC_left add_left cblinfun.add_left)
     from _ real.nonneg_bounded show "\<exists>K. \<forall>x. norm (f x) \<le> norm x * K"
       by (rule ex_reg) (auto intro!: onorm_bound simp: norm_cblinfun.rep_eq ac_simps)
   qed
 next
   assume "bounded_antilinear f"
   then obtain K where K: \<open>norm (f x) \<le> norm x * K\<close> for x
     using bounded_antilinear.bounded by blast
   have \<open>norm (cblinfun_apply (f a) b) \<le> norm (f a) * norm b\<close> for a b
     by (simp add: norm_cblinfun)
   also have \<open>\<dots> a b \<le> norm a * norm b * K\<close> for a b
     by (smt (verit, best) K mult.assoc mult.commute mult_mono' norm_ge_zero)
   finally have *: \<open>norm (cblinfun_apply (f a) b) \<le> norm a * norm b * K\<close> for a b
     by simp
   show "bounded_sesquilinear g"
     using \<open>bounded_antilinear f\<close>
     apply (auto intro!: bounded_sesquilinear.intro simp: assms cblinfun.add_left cblinfun.add_right 
         linear_simps bounded_antilinear.bounded_linear antilinear.scaleC bounded_antilinear.antilinear
         cblinfun.scaleC_left cblinfun.scaleC_right)
     using * by blast
 qed
 
 lemma transfer_bounded_sesquilinear_bounded_antilinear[transfer_rule]:
   "(rel_fun (rel_fun (=) (pcr_cblinfun (=) (=))) (=)) bounded_sesquilinear bounded_antilinear"
   by (auto simp: pcr_cblinfun_def cr_cblinfun_def rel_fun_def OO_def
       intro!: transfer_bounded_sesquilinear_bounded_antilinearI)
 
 context bounded_cbilinear
 begin
 
 lift_definition prod_left::"'b \<Rightarrow> 'a \<Rightarrow>\<^sub>C\<^sub>L 'c" is "(\<lambda>b a. prod a b)"
   by (rule bounded_clinear_left)
 declare prod_left.rep_eq[simp]
 
 lemma bounded_clinear_prod_left[bounded_clinear]: "bounded_clinear prod_left"
   by transfer (rule flip)
 
 lift_definition prod_right::"'a \<Rightarrow> 'b \<Rightarrow>\<^sub>C\<^sub>L 'c" is "(\<lambda>a b. prod a b)"
   by (rule bounded_clinear_right)
 declare prod_right.rep_eq[simp]
 
 lemma bounded_clinear_prod_right[bounded_clinear]: "bounded_clinear prod_right"
   by transfer (rule bounded_cbilinear_axioms)
 
 end
 
 lift_definition id_cblinfun::"'a::complex_normed_vector \<Rightarrow>\<^sub>C\<^sub>L 'a" is "\<lambda>x. x"
   by (rule bounded_clinear_ident)
 
 lemmas cblinfun_id_cblinfun_apply[simp] = id_cblinfun.rep_eq
 
-(* Strong than norm_blinfun_id because we replaced the perfect_space typeclass by not_singleton *)
+(* Stronger than norm_blinfun_id because we replaced the perfect_space typeclass by not_singleton *)
 lemma norm_cblinfun_id[simp]:
   "norm (id_cblinfun::'a::{complex_normed_vector, not_singleton} \<Rightarrow>\<^sub>C\<^sub>L 'a) = 1"
   apply transfer
   apply (rule onorm_id[internalize_sort' 'a])
    apply standard[1]
   by simp
 
 lemma norm_cblinfun_id_le:
   "norm (id_cblinfun::'a::complex_normed_vector \<Rightarrow>\<^sub>C\<^sub>L 'a) \<le> 1"
   by transfer (auto simp: onorm_id_le)
 
 (* Skipped because we do not have "prod :: (cbanach, cbanach) cbanach" (Product_Vector not ported to complex)*)
 (* lift_definition fst_cblinfun::"('a::complex_normed_vector \<times> 'b::complex_normed_vector) \<Rightarrow>\<^sub>C\<^sub>L 'a" is fst *)
 
 (* lemma cblinfun_apply_fst_cblinfun[simp]: "cblinfun_apply fst_cblinfun = fst" *)
 
 (* lift_definition snd_cblinfun::"('a::complex_normed_vector \<times> 'b::complex_normed_vector) \<Rightarrow>\<^sub>C\<^sub>L 'b" is snd *)
 
 (* lemma blinfun_apply_snd_blinfun[simp]: "blinfun_apply snd_blinfun = snd" *)
 
 lift_definition cblinfun_compose::
   "'a::complex_normed_vector \<Rightarrow>\<^sub>C\<^sub>L 'b::complex_normed_vector \<Rightarrow>
     'c::complex_normed_vector \<Rightarrow>\<^sub>C\<^sub>L 'a \<Rightarrow>
     'c \<Rightarrow>\<^sub>C\<^sub>L 'b" (infixl "o\<^sub>C\<^sub>L" 67) is "(o)"
   (* Difference from Real_Vector_Spaces: Priority of o\<^sub>C\<^sub>L is 55 there.
      But we want "a - b o\<^sub>C\<^sub>L c" to parse as "a - (b o\<^sub>C\<^sub>L c)". *)
   parametric comp_transfer
   unfolding o_def
   by (rule bounded_clinear_compose)
 
 lemma cblinfun_apply_cblinfun_compose[simp]: "(a o\<^sub>C\<^sub>L b) c = a (b c)"
   by (simp add: cblinfun_compose.rep_eq)
 
 lemma norm_cblinfun_compose:
   "norm (f o\<^sub>C\<^sub>L g) \<le> norm f * norm g"
   apply transfer
   by (auto intro!: onorm_compose simp: bounded_clinear.bounded_linear)
 
 lemma bounded_cbilinear_cblinfun_compose[bounded_cbilinear]: "bounded_cbilinear (o\<^sub>C\<^sub>L)"
   by unfold_locales
     (auto intro!: cblinfun_eqI exI[where x=1] simp: cblinfun.cbilinear_simps norm_cblinfun_compose)
 
 lemma cblinfun_compose_zero[simp]:
   "blinfun_compose 0 = (\<lambda>_. 0)"
   "blinfun_compose x 0 = 0"
   by (auto simp: blinfun.bilinear_simps intro!: blinfun_eqI)
 
 lemma cblinfun_bij2:
   fixes f::"'a \<Rightarrow>\<^sub>C\<^sub>L 'a::ceuclidean_space"
   assumes "f o\<^sub>C\<^sub>L g = id_cblinfun"
   shows "bij (cblinfun_apply g)"
 proof (rule bijI)
   show "inj g"
     using assms
     by (metis cblinfun_id_cblinfun_apply cblinfun_compose.rep_eq injI inj_on_imageI2)
   then show "surj g"
     using bounded_clinear_def cblinfun.bounded_clinear_right ceucl.linear_inj_imp_surj by blast
 qed
 
 lemma cblinfun_bij1:
   fixes f::"'a \<Rightarrow>\<^sub>C\<^sub>L 'a::ceuclidean_space"
   assumes "f o\<^sub>C\<^sub>L g = id_cblinfun"
   shows "bij (cblinfun_apply f)"
 proof (rule bijI)
   show "surj (cblinfun_apply f)"
     by (metis assms cblinfun_apply_cblinfun_compose cblinfun_id_cblinfun_apply surjI)
   then show "inj (cblinfun_apply f)"
     using bounded_clinear_def cblinfun.bounded_clinear_right ceucl.linear_surjective_imp_injective by blast
 qed
 
 lift_definition cblinfun_cinner_right::"'a::complex_inner \<Rightarrow> 'a \<Rightarrow>\<^sub>C\<^sub>L complex" is "(\<bullet>\<^sub>C)"
   by (rule bounded_clinear_cinner_right)
 declare cblinfun_cinner_right.rep_eq[simp]
 
 lemma bounded_antilinear_cblinfun_cinner_right[bounded_antilinear]: "bounded_antilinear cblinfun_cinner_right"
   apply transfer by (simp add: bounded_sesquilinear_cinner)
 
 (* Cannot be defined. cinner is antilinear in first argument. *)
 (* lift_definition cblinfun_cinner_left::"'a::complex_inner \<Rightarrow> 'a \<Rightarrow>\<^sub>C\<^sub>L complex" is "\<lambda>x y. y \<bullet>\<^sub>C x" *)
 (* declare cblinfun_cinner_left.rep_eq[simp] *)
 
 (* lemma bounded_clinear_cblinfun_cinner_left[bounded_clinear]: "bounded_clinear cblinfun_cinner_left" *)
 
 lift_definition cblinfun_scaleC_right::"complex \<Rightarrow> 'a \<Rightarrow>\<^sub>C\<^sub>L 'a::complex_normed_vector" is "(*\<^sub>C)"
   by (rule bounded_clinear_scaleC_right)
 declare cblinfun_scaleC_right.rep_eq[simp]
 
 lemma bounded_clinear_cblinfun_scaleC_right[bounded_clinear]: "bounded_clinear cblinfun_scaleC_right"
   by transfer (rule bounded_cbilinear_scaleC)
 
 lift_definition cblinfun_scaleC_left::"'a::complex_normed_vector \<Rightarrow> complex \<Rightarrow>\<^sub>C\<^sub>L 'a" is "\<lambda>x y. y *\<^sub>C x"
   by (rule bounded_clinear_scaleC_left)
 lemmas [simp] = cblinfun_scaleC_left.rep_eq
 
 lemma bounded_clinear_cblinfun_scaleC_left[bounded_clinear]: "bounded_clinear cblinfun_scaleC_left"
   by transfer (rule bounded_cbilinear.flip[OF bounded_cbilinear_scaleC])
 
 lift_definition cblinfun_mult_right::"'a \<Rightarrow> 'a \<Rightarrow>\<^sub>C\<^sub>L 'a::complex_normed_algebra" is "(*)"
   by (rule bounded_clinear_mult_right)
 declare cblinfun_mult_right.rep_eq[simp]
 
 lemma bounded_clinear_cblinfun_mult_right[bounded_clinear]: "bounded_clinear cblinfun_mult_right"
   by transfer (rule bounded_cbilinear_mult)
 
 lift_definition cblinfun_mult_left::"'a::complex_normed_algebra \<Rightarrow> 'a \<Rightarrow>\<^sub>C\<^sub>L 'a" is "\<lambda>x y. y * x"
   by (rule bounded_clinear_mult_left)
 lemmas [simp] = cblinfun_mult_left.rep_eq
 
 lemma bounded_clinear_cblinfun_mult_left[bounded_clinear]: "bounded_clinear cblinfun_mult_left"
   by transfer (rule bounded_cbilinear.flip[OF bounded_cbilinear_mult])
 
 lemmas bounded_clinear_function_uniform_limit_intros[uniform_limit_intros] =
   bounded_clinear.uniform_limit[OF bounded_clinear_apply_cblinfun]
   bounded_clinear.uniform_limit[OF bounded_clinear_cblinfun_apply]
   bounded_antilinear.uniform_limit[OF bounded_antilinear_cblinfun_matrix]
 
 
 subsection \<open>The strong operator topology on continuous linear operators\<close>
 
 text \<open>Let \<open>'a\<close> and \<open>'b\<close> be two normed real vector spaces. Then the space of linear continuous
 operators from \<open>'a\<close> to \<open>'b\<close> has a canonical norm, and therefore a canonical corresponding topology
 (the type classes instantiation are given in \<^file>\<open>Complex_Bounded_Linear_Function0.thy\<close>).
 
 However, there is another topology on this space, the strong operator topology, where \<open>T\<^sub>n\<close> tends to
 \<open>T\<close> iff, for all \<open>x\<close> in \<open>'a\<close>, then \<open>T\<^sub>n x\<close> tends to \<open>T x\<close>. This is precisely the product topology
 where the target space is endowed with the norm topology. It is especially useful when \<open>'b\<close> is the set
 of real numbers, since then this topology is compact.
 
 We can not implement it using type classes as there is already a topology, but at least we
 can define it as a topology.
 
 Note that there is yet another (common and useful) topology on operator spaces, the weak operator
 topology, defined analogously using the product topology, but where the target space is given the
 weak-* topology, i.e., the pullback of the weak topology on the bidual of the space under the
 canonical embedding of a space into its bidual. We do not define it there, although it could also be
 defined analogously.
 \<close>
 
 definition\<^marker>\<open>tag important\<close> cstrong_operator_topology::"('a::complex_normed_vector \<Rightarrow>\<^sub>C\<^sub>L'b::complex_normed_vector) topology"
   where "cstrong_operator_topology = pullback_topology UNIV cblinfun_apply euclidean"
 
 lemma cstrong_operator_topology_topspace:
   "topspace cstrong_operator_topology = UNIV"
   unfolding cstrong_operator_topology_def topspace_pullback_topology topspace_euclidean by auto
 
 lemma cstrong_operator_topology_basis:
   fixes f::"('a::complex_normed_vector \<Rightarrow>\<^sub>C\<^sub>L'b::complex_normed_vector)" and U::"'i \<Rightarrow> 'b set" and x::"'i \<Rightarrow> 'a"
   assumes "finite I" "\<And>i. i \<in> I \<Longrightarrow> open (U i)"
   shows "openin cstrong_operator_topology {f. \<forall>i\<in>I. cblinfun_apply f (x i) \<in> U i}"
 proof -
   have "open {g::('a\<Rightarrow>'b). \<forall>i\<in>I. g (x i) \<in> U i}"
     by (rule product_topology_basis'[OF assms])
   moreover have "{f. \<forall>i\<in>I. cblinfun_apply f (x i) \<in> U i}
                 = cblinfun_apply-`{g::('a\<Rightarrow>'b). \<forall>i\<in>I. g (x i) \<in> U i} \<inter> UNIV"
     by auto
   ultimately show ?thesis
     unfolding cstrong_operator_topology_def by (subst openin_pullback_topology) auto
 qed
 
 lemma cstrong_operator_topology_continuous_evaluation:
   "continuous_map cstrong_operator_topology euclidean (\<lambda>f. cblinfun_apply f x)"
 proof -
   have "continuous_map cstrong_operator_topology euclidean ((\<lambda>f. f x) o cblinfun_apply)"
     unfolding cstrong_operator_topology_def apply (rule continuous_map_pullback)
     using continuous_on_product_coordinates by fastforce
   then show ?thesis unfolding comp_def by simp
 qed
 
 lemma continuous_on_cstrong_operator_topo_iff_coordinatewise:
   "continuous_map T cstrong_operator_topology f
     \<longleftrightarrow> (\<forall>x. continuous_map T euclidean (\<lambda>y. cblinfun_apply (f y) x))"
 proof (auto)
   fix x::"'b"
   assume "continuous_map T cstrong_operator_topology f"
   with continuous_map_compose[OF this cstrong_operator_topology_continuous_evaluation]
   have "continuous_map T euclidean ((\<lambda>z. cblinfun_apply z x) o f)"
     by simp
   then show "continuous_map T euclidean (\<lambda>y. cblinfun_apply (f y) x)"
     unfolding comp_def by auto
 next
   assume *: "\<forall>x. continuous_map T euclidean (\<lambda>y. cblinfun_apply (f y) x)"
   have "\<And>i. continuous_map T euclidean (\<lambda>x. cblinfun_apply (f x) i)"
     using * unfolding comp_def by auto
   then have "continuous_map T euclidean (cblinfun_apply o f)"
     unfolding o_def
     by (metis (no_types) continuous_map_componentwise_UNIV euclidean_product_topology)
   show "continuous_map T cstrong_operator_topology f"
     unfolding cstrong_operator_topology_def
     apply (rule continuous_map_pullback')
     by (auto simp add: \<open>continuous_map T euclidean (cblinfun_apply o f)\<close>)
 qed
 
 lemma cstrong_operator_topology_weaker_than_euclidean:
   "continuous_map euclidean cstrong_operator_topology (\<lambda>f. f)"
   apply (subst continuous_on_cstrong_operator_topo_iff_coordinatewise)
   by (auto simp add: linear_continuous_on continuous_at_imp_continuous_on linear_continuous_at 
       bounded_clinear.bounded_linear)
 end
diff --git a/thys/Complex_Bounded_Operators/Complex_Inner_Product.thy b/thys/Complex_Bounded_Operators/Complex_Inner_Product.thy
--- a/thys/Complex_Bounded_Operators/Complex_Inner_Product.thy
+++ b/thys/Complex_Bounded_Operators/Complex_Inner_Product.thy
@@ -1,2523 +1,2446 @@
 (*
 Authors:
 
   Dominique Unruh, University of Tartu, unruh@ut.ee
   Jose Manuel Rodriguez Caballero, University of Tartu, jose.manuel.rodriguez.caballero@ut.ee
 
 *)
 
 section \<open>\<open>Complex_Inner_Product\<close> -- Complex Inner Product Spaces\<close>
 
 theory Complex_Inner_Product
   imports
     Complex_Inner_Product0
 begin
 
 subsection \<open>Complex inner product spaces\<close>
 
+unbundle cinner_syntax
+
+(* TODO: Remove this eventually. Do not use this syntax. *)
 bundle cinner_bracket_notation begin
 notation cinner ("\<langle>_, _\<rangle>")
 end
-unbundle cinner_bracket_notation
 
+(* TODO: Remove this eventually. Do not use this syntax. *)
 bundle no_cinner_bracket_notation begin
 no_notation cinner ("\<langle>_, _\<rangle>")
 end
 
 lemma cinner_real: "cinner x x \<in> \<real>"
   by (simp add: cdot_square_norm)
 
 lemmas cinner_commute' [simp] = cinner_commute[symmetric]
 
 lemma (in complex_inner) cinner_eq_flip: \<open>(cinner x y = cinner z w) \<longleftrightarrow> (cinner y x = cinner w z)\<close>
   by (metis cinner_commute)
 
-lemma Im_cinner_x_x[simp]: "Im \<langle>x , x\<rangle> = 0"
+lemma Im_cinner_x_x[simp]: "Im (x \<bullet>\<^sub>C x) = 0"
   using comp_Im_same[OF cinner_ge_zero] by simp
 
 
 lemma of_complex_inner_1' [simp]:
   "cinner (1 :: 'a :: {complex_inner, complex_normed_algebra_1}) (of_complex x) = x"
   by (metis cinner_commute complex_cnj_cnj of_complex_inner_1)
 
 
 class chilbert_space =  complex_inner + complete_space
 begin
 subclass cbanach by standard
 end
 
 instantiation complex :: "chilbert_space" begin
 instance ..
 end
 
 subsection \<open>Misc facts\<close>
 
 lemma cinner_scaleR_left [simp]: "cinner (scaleR r x) y = of_real r * (cinner x y)"
   by (simp add: scaleR_scaleC)
 
 lemma cinner_scaleR_right [simp]: "cinner x (scaleR r y) = of_real r * (cinner x y)"
   by (simp add: scaleR_scaleC)
 
-
 text \<open>This is a useful rule for establishing the equality of vectors\<close>
 lemma cinner_extensionality:
-  assumes \<open>\<And>\<gamma>. \<langle>\<gamma>, \<psi>\<rangle> = \<langle>\<gamma>, \<phi>\<rangle>\<close>
+  assumes \<open>\<And>\<gamma>. \<gamma> \<bullet>\<^sub>C \<psi> = \<gamma> \<bullet>\<^sub>C \<phi>\<close>
   shows \<open>\<psi> = \<phi>\<close>
   by (metis assms cinner_eq_zero_iff cinner_simps(3) right_minus_eq)
 
 lemma polar_identity:
   includes notation_norm
-  shows \<open>\<parallel>x + y\<parallel>^2 = \<parallel>x\<parallel>^2 + \<parallel>y\<parallel>^2 + 2*Re \<langle>x, y\<rangle>\<close>
+  shows \<open>\<parallel>x + y\<parallel>^2 = \<parallel>x\<parallel>^2 + \<parallel>y\<parallel>^2 + 2 * Re (x \<bullet>\<^sub>C y)\<close>
     \<comment> \<open>Shown in the proof of Corollary 1.5 in @{cite conway2013course}\<close>
 proof -
-  have \<open>\<langle>x , y\<rangle> + \<langle>y , x\<rangle> = \<langle>x , y\<rangle> + cnj \<langle>x , y\<rangle>\<close>
+  have \<open>(x \<bullet>\<^sub>C y) + (y \<bullet>\<^sub>C x) = (x \<bullet>\<^sub>C y) + cnj (x \<bullet>\<^sub>C y)\<close>
     by simp
-  hence \<open>\<langle>x , y\<rangle> + \<langle>y , x\<rangle> = 2 * Re \<langle>x , y\<rangle> \<close>
+  hence \<open>(x \<bullet>\<^sub>C y) + (y \<bullet>\<^sub>C x) = 2 * Re (x \<bullet>\<^sub>C y) \<close>
     using complex_add_cnj by presburger
-  have \<open>\<parallel>x + y\<parallel>^2 = \<langle>x+y, x+y\<rangle>\<close>
+  have \<open>\<parallel>x + y\<parallel>^2 = (x+y) \<bullet>\<^sub>C (x+y)\<close>
     by (simp add: cdot_square_norm)
-  hence \<open>\<parallel>x + y\<parallel>^2 = \<langle>x , x\<rangle> + \<langle>x , y\<rangle> + \<langle>y , x\<rangle> + \<langle>y , y\<rangle>\<close>
+  hence \<open>\<parallel>x + y\<parallel>^2 = (x \<bullet>\<^sub>C x) + (x \<bullet>\<^sub>C y) + (y \<bullet>\<^sub>C x) + (y \<bullet>\<^sub>C y)\<close>
     by (simp add: cinner_add_left cinner_add_right)
-  thus ?thesis using  \<open>\<langle>x , y\<rangle> + \<langle>y , x\<rangle> = 2 * Re \<langle>x , y\<rangle>\<close>
+  thus ?thesis using  \<open>(x \<bullet>\<^sub>C y) + (y \<bullet>\<^sub>C x) = 2 * Re (x \<bullet>\<^sub>C y)\<close>
     by (smt (verit, ccfv_SIG) Re_complex_of_real plus_complex.simps(1) power2_norm_eq_cinner')
 qed
 
 lemma polar_identity_minus:
   includes notation_norm
-  shows \<open>\<parallel>x - y\<parallel>^2 = \<parallel>x\<parallel>^2 + \<parallel>y\<parallel>^2 - 2 * Re \<langle>x, y\<rangle>\<close>
+  shows \<open>\<parallel>x - y\<parallel>^2 = \<parallel>x\<parallel>^2 + \<parallel>y\<parallel>^2 - 2 * Re (x \<bullet>\<^sub>C y)\<close>
 proof-
-  have \<open>\<parallel>x + (-y)\<parallel>^2 = \<parallel>x\<parallel>^2 + \<parallel>-y\<parallel>^2 + 2 * Re \<langle>x , (-y)\<rangle>\<close>
+  have \<open>\<parallel>x + (-y)\<parallel>^2 = \<parallel>x\<parallel>^2 + \<parallel>-y\<parallel>^2 + 2 * Re (x \<bullet>\<^sub>C -y)\<close>
     using polar_identity by blast
-  hence \<open>\<parallel>x - y\<parallel>^2 = \<parallel>x\<parallel>^2 + \<parallel>y\<parallel>^2 - 2*Re \<langle>x , y\<rangle>\<close>
+  hence \<open>\<parallel>x - y\<parallel>^2 = \<parallel>x\<parallel>^2 + \<parallel>y\<parallel>^2 - 2*Re (x \<bullet>\<^sub>C y)\<close>
     by simp
   thus ?thesis
     by blast
 qed
 
 proposition parallelogram_law:
   includes notation_norm
   fixes x y :: "'a::complex_inner"
   shows \<open>\<parallel>x+y\<parallel>^2 + \<parallel>x-y\<parallel>^2 = 2*( \<parallel>x\<parallel>^2 + \<parallel>y\<parallel>^2 )\<close>
     \<comment> \<open>Shown in the proof of Theorem 2.3 in @{cite conway2013course}\<close>
   by (simp add: polar_identity_minus polar_identity)
 
 
 theorem pythagorean_theorem:
   includes notation_norm
-  shows \<open>\<langle>x , y\<rangle> = 0 \<Longrightarrow> \<parallel> x + y \<parallel>^2 = \<parallel> x \<parallel>^2 + \<parallel> y \<parallel>^2\<close>
+  shows \<open>(x \<bullet>\<^sub>C y) = 0 \<Longrightarrow> \<parallel> x + y \<parallel>^2 = \<parallel> x \<parallel>^2 + \<parallel> y \<parallel>^2\<close>
     \<comment> \<open>Shown in the proof of Theorem 2.2 in @{cite conway2013course}\<close>
   by (simp add: polar_identity)
 
 lemma pythagorean_theorem_sum:
-  assumes q1: "\<And>a a'. a \<in> t \<Longrightarrow> a' \<in> t \<Longrightarrow> a \<noteq> a' \<Longrightarrow> \<langle>f a, f a'\<rangle> = 0"
+  assumes q1: "\<And>a a'. a \<in> t \<Longrightarrow> a' \<in> t \<Longrightarrow> a \<noteq> a' \<Longrightarrow> f a \<bullet>\<^sub>C f a' = 0"
     and q2: "finite t"
   shows "(norm  (\<Sum>a\<in>t. f a))^2 = (\<Sum>a\<in>t.(norm (f a))^2)"
 proof (insert q1, use q2 in induction)
   case empty
   show ?case
     by auto
 next
   case (insert x F)
-  have r1: "\<langle>f x, f a\<rangle> = 0"
+  have r1: "f x \<bullet>\<^sub>C f a = 0"
     if "a \<in> F"
     for a
     using that insert.hyps(2) insert.prems by auto
   have "sum f F = (\<Sum>a\<in>F. f a)"
     by simp
-  hence s4: "\<langle>f x, sum f F\<rangle> = \<langle>f x, (\<Sum>a\<in>F. f a)\<rangle>"
+  hence s4: "f x \<bullet>\<^sub>C sum f F = f x \<bullet>\<^sub>C (\<Sum>a\<in>F. f a)"
     by simp
-  also have s3: "\<dots> = (\<Sum>a\<in>F. \<langle>f x, f a\<rangle>)"
+  also have s3: "\<dots> = (\<Sum>a\<in>F. f x \<bullet>\<^sub>C f a)"
     using cinner_sum_right by auto
   also have s2: "\<dots> = (\<Sum>a\<in>F. 0)"
     using r1
     by simp
   also have s1: "\<dots> = 0"
     by simp
-  finally have xF_ortho: "\<langle>f x, sum f F\<rangle> = 0"
+  finally have xF_ortho: "f x \<bullet>\<^sub>C sum f F = 0"
     using s2 s3 by auto
   have "(norm (sum f (insert x F)))\<^sup>2 = (norm (f x + sum f F))\<^sup>2"
     by (simp add: insert.hyps(1) insert.hyps(2))
   also have "\<dots> = (norm (f x))\<^sup>2 + (norm (sum f F))\<^sup>2"
     using xF_ortho by (rule pythagorean_theorem)
   also have "\<dots> = (norm (f x))\<^sup>2 + (\<Sum>a\<in>F.(norm (f a))^2)"
     apply (subst insert.IH) using insert.prems by auto
   also have "\<dots> = (\<Sum>a\<in>insert x F.(norm (f a))^2)"
     by (simp add: insert.hyps(1) insert.hyps(2))
   finally show ?case
     by simp
 qed
 
 
 lemma Cauchy_cinner_Cauchy:
   fixes x y :: \<open>nat \<Rightarrow> 'a::complex_inner\<close>
   assumes a1: \<open>Cauchy x\<close> and a2: \<open>Cauchy y\<close>
-  shows \<open>Cauchy (\<lambda> n. \<langle> x n, y n \<rangle>)\<close>
+  shows \<open>Cauchy (\<lambda> n. x n \<bullet>\<^sub>C y n)\<close>
 proof-
   have \<open>bounded (range x)\<close>
     using a1
     by (simp add: Elementary_Metric_Spaces.cauchy_imp_bounded)
   hence b1: \<open>\<exists>M. \<forall>n. norm (x n) < M\<close>
     by (meson bounded_pos_less rangeI)
   have \<open>bounded (range y)\<close>
     using a2
     by (simp add: Elementary_Metric_Spaces.cauchy_imp_bounded)
   hence b2: \<open>\<exists> M. \<forall> n. norm (y n) < M\<close>
     by (meson bounded_pos_less rangeI)
   have \<open>\<exists>M. \<forall>n. norm (x n) < M \<and> norm (y n) < M\<close>
     using b1 b2
     by (metis dual_order.strict_trans linorder_neqE_linordered_idom)
   then obtain M where M1: \<open>\<And>n. norm (x n) < M\<close> and M2: \<open>\<And>n. norm (y n) < M\<close>
     by blast
   have M3: \<open>M > 0\<close>
     by (smt M2 norm_not_less_zero)
-  have \<open>\<exists>N. \<forall>n \<ge> N. \<forall>m \<ge> N. norm ( (\<lambda> i. \<langle> x i, y i \<rangle>) n -  (\<lambda> i. \<langle> x i, y i \<rangle>) m ) < e\<close>
+  have \<open>\<exists>N. \<forall>n \<ge> N. \<forall>m \<ge> N. norm ( (\<lambda> i. x i \<bullet>\<^sub>C y i) n -  (\<lambda> i. x i \<bullet>\<^sub>C y i) m ) < e\<close>
     if "e > 0" for e
   proof-
     have \<open>e / (2*M) > 0\<close>
       using M3
       by (simp add: that)
     hence \<open>\<exists>N. \<forall>n\<ge>N. \<forall>m\<ge>N. norm (x n - x m) < e / (2*M)\<close>
       using a1
       by (simp add: Cauchy_iff)
     then obtain N1 where N1_def: \<open>\<And>n m. n\<ge>N1 \<Longrightarrow> m\<ge>N1 \<Longrightarrow> norm (x n - x m) < e / (2*M)\<close>
       by blast
     have x1: \<open>\<exists>N. \<forall> n\<ge>N. \<forall> m\<ge>N. norm (y n - y m) < e / (2*M)\<close>
       using a2 \<open>e / (2*M) > 0\<close>
       by (simp add: Cauchy_iff)
     obtain N2 where N2_def: \<open>\<And>n m.  n\<ge>N2 \<Longrightarrow> m\<ge>N2 \<Longrightarrow> norm (y n - y m) < e / (2*M)\<close>
       using x1
       by blast
     define N where N_def: \<open>N = N1 + N2\<close>
     hence \<open>N \<ge> N1\<close>
       by auto
     have \<open>N \<ge> N2\<close>
       using N_def
       by auto
-    have \<open>norm ( \<langle> x n, y n \<rangle> - \<langle> x m, y m \<rangle> ) < e\<close>
+    have \<open>norm (x n \<bullet>\<^sub>C y n - x m \<bullet>\<^sub>C y m) < e\<close>
       if \<open>n \<ge> N\<close> and \<open>m \<ge> N\<close>
       for n m
     proof -
-      have \<open>\<langle> x n, y n \<rangle> - \<langle> x m, y m \<rangle> = (\<langle> x n, y n \<rangle> - \<langle> x m, y n \<rangle>) + (\<langle> x m, y n \<rangle> - \<langle> x m, y m \<rangle>)\<close>
+      have \<open>x n \<bullet>\<^sub>C y n - x m \<bullet>\<^sub>C y m = (x n \<bullet>\<^sub>C y n - x m \<bullet>\<^sub>C y n) + (x m \<bullet>\<^sub>C y n - x m \<bullet>\<^sub>C y m)\<close>
         by simp
-      hence y1: \<open>norm (\<langle> x n, y n \<rangle> - \<langle> x m, y m \<rangle>) \<le> norm (\<langle> x n, y n \<rangle> - \<langle> x m, y n \<rangle>)
-           + norm (\<langle> x m, y n \<rangle> - \<langle> x m, y m \<rangle>)\<close>
+      hence y1: \<open>norm (x n \<bullet>\<^sub>C y n - x m \<bullet>\<^sub>C y m) \<le> norm (x n \<bullet>\<^sub>C y n - x m \<bullet>\<^sub>C y n)
+           + norm (x m \<bullet>\<^sub>C y n - x m \<bullet>\<^sub>C y m)\<close>
         by (metis norm_triangle_ineq)
 
-      have \<open>\<langle> x n, y n \<rangle> - \<langle> x m, y n \<rangle> = \<langle> x n - x m, y n \<rangle>\<close>
+      have \<open>x n \<bullet>\<^sub>C y n - x m \<bullet>\<^sub>C y n = (x n - x m) \<bullet>\<^sub>C y n\<close>
         by (simp add: cinner_diff_left)
-      hence \<open>norm (\<langle> x n, y n \<rangle> - \<langle> x m, y n \<rangle>) = norm \<langle> x n - x m, y n \<rangle>\<close>
+      hence \<open>norm (x n \<bullet>\<^sub>C y n - x m \<bullet>\<^sub>C y n) = norm ((x n - x m) \<bullet>\<^sub>C y n)\<close>
         by simp
-      moreover have \<open>norm \<langle> x n - x m, y n \<rangle> \<le> norm (x n - x m) * norm (y n)\<close>
+      moreover have \<open>norm ((x n - x m) \<bullet>\<^sub>C y n) \<le> norm (x n - x m) * norm (y n)\<close>
         using complex_inner_class.Cauchy_Schwarz_ineq2 by blast
       moreover have \<open>norm (y n) < M\<close>
         by (simp add: M2)
       moreover have \<open>norm (x n - x m) < e/(2*M)\<close>
         using \<open>N \<le> m\<close> \<open>N \<le> n\<close> \<open>N1 \<le> N\<close> N1_def by auto
-      ultimately have \<open>norm (\<langle> x n, y n \<rangle> - \<langle> x m, y n \<rangle>) < (e/(2*M)) * M\<close>
+      ultimately have \<open>norm ((x n \<bullet>\<^sub>C y n) - (x m \<bullet>\<^sub>C y n)) < (e/(2*M)) * M\<close>
         by (smt linordered_semiring_strict_class.mult_strict_mono norm_ge_zero)
       moreover have \<open> (e/(2*M)) * M = e/2\<close>
         using \<open>M > 0\<close> by simp
-      ultimately have  \<open>norm (\<langle> x n, y n \<rangle> - \<langle> x m, y n \<rangle>) < e/2\<close>
+      ultimately have  \<open>norm ((x n \<bullet>\<^sub>C y n) - (x m \<bullet>\<^sub>C y n)) < e/2\<close>
         by simp
-      hence y2: \<open>norm (\<langle> x n, y n \<rangle> - \<langle> x m, y n \<rangle>) < e/2\<close>
+      hence y2: \<open>norm (x n \<bullet>\<^sub>C y n - x m \<bullet>\<^sub>C y n) < e/2\<close>
         by blast
-      have \<open>\<langle> x m, y n \<rangle> - \<langle> x m, y m \<rangle> = \<langle> x m, y n - y m \<rangle>\<close>
+      have \<open>x m \<bullet>\<^sub>C y n - x m \<bullet>\<^sub>C y m = x m \<bullet>\<^sub>C (y n - y m)\<close>
         by (simp add: cinner_diff_right)
-      hence \<open>norm (\<langle> x m, y n \<rangle> - \<langle> x m, y m \<rangle>) = norm \<langle> x m, y n - y m \<rangle>\<close>
+      hence \<open>norm ((x m \<bullet>\<^sub>C y n) - (x m \<bullet>\<^sub>C y m)) = norm (x m \<bullet>\<^sub>C (y n - y m))\<close>
         by simp
-      moreover have \<open>norm \<langle> x m, y n - y m \<rangle> \<le> norm (x m) * norm (y n - y m)\<close>
+      moreover have \<open>norm (x m \<bullet>\<^sub>C (y n - y m)) \<le> norm (x m) * norm (y n - y m)\<close>
         by (meson complex_inner_class.Cauchy_Schwarz_ineq2)
       moreover have \<open>norm (x m) < M\<close>
         by (simp add: M1)
       moreover have \<open>norm (y n - y m) < e/(2*M)\<close>
         using \<open>N \<le> m\<close> \<open>N \<le> n\<close> \<open>N2 \<le> N\<close> N2_def by auto
-      ultimately have \<open>norm (\<langle> x m, y n \<rangle> - \<langle> x m, y m \<rangle>) < M * (e/(2*M))\<close>
+      ultimately have \<open>norm ((x m \<bullet>\<^sub>C y n) - (x m \<bullet>\<^sub>C y m)) < M * (e/(2*M))\<close>
         by (smt linordered_semiring_strict_class.mult_strict_mono norm_ge_zero)
       moreover have \<open>M * (e/(2*M)) = e/2\<close>
         using \<open>M > 0\<close> by simp
-      ultimately have  \<open>norm (\<langle> x m, y n \<rangle> - \<langle> x m, y m \<rangle>) < e/2\<close>
+      ultimately have  \<open>norm ((x m \<bullet>\<^sub>C y n) - (x m \<bullet>\<^sub>C y m)) < e/2\<close>
         by simp
-      hence y3: \<open>norm (\<langle> x m, y n \<rangle> - \<langle> x m, y m \<rangle>) < e/2\<close>
+      hence y3: \<open>norm ((x m \<bullet>\<^sub>C y n) - (x m \<bullet>\<^sub>C y m)) < e/2\<close>
         by blast
-      show \<open>norm ( \<langle> x n, y n \<rangle> - \<langle> x m, y m \<rangle> ) < e\<close>
+      show \<open>norm ( (x n \<bullet>\<^sub>C y n) - (x m \<bullet>\<^sub>C y m) ) < e\<close>
         using y1 y2 y3 by simp
     qed
     thus ?thesis by blast
   qed
   thus ?thesis
     by (simp add: CauchyI)
 qed
 
 
 lemma cinner_sup_norm: \<open>norm \<psi> = (SUP \<phi>. cmod (cinner \<phi> \<psi>) / norm \<phi>)\<close>
 proof (rule sym, rule cSup_eq_maximum)
   have \<open>norm \<psi> = cmod (cinner \<psi> \<psi>) / norm \<psi>\<close>
     by (metis norm_eq_sqrt_cinner norm_ge_zero real_div_sqrt)
   then show \<open>norm \<psi> \<in> range (\<lambda>\<phi>. cmod (cinner \<phi> \<psi>) / norm \<phi>)\<close>
     by blast
 next
   fix n assume \<open>n \<in> range (\<lambda>\<phi>. cmod (cinner \<phi> \<psi>) / norm \<phi>)\<close>
   then obtain \<phi> where n\<phi>: \<open>n = cmod (cinner \<phi> \<psi>) / norm \<phi>\<close>
     by auto
   show \<open>n \<le> norm \<psi>\<close>
     unfolding n\<phi>
     by (simp add: complex_inner_class.Cauchy_Schwarz_ineq2 divide_le_eq ordered_field_class.sign_simps(33))
 qed
 
 lemma cinner_sup_onorm:
   fixes A :: \<open>'a::{real_normed_vector,not_singleton} \<Rightarrow> 'b::complex_inner\<close>
   assumes \<open>bounded_linear A\<close>
   shows \<open>onorm A = (SUP (\<psi>,\<phi>). cmod (cinner \<psi> (A \<phi>)) / (norm \<psi> * norm \<phi>))\<close>
 proof (unfold onorm_def, rule cSup_eq_cSup)
   show \<open>bdd_above (range (\<lambda>x. norm (A x) / norm x))\<close>
     by (meson assms bdd_aboveI2 le_onorm)
 next
   fix a
   assume \<open>a \<in> range (\<lambda>\<phi>. norm (A \<phi>) / norm \<phi>)\<close>
   then obtain \<phi> where \<open>a = norm (A \<phi>) / norm \<phi>\<close>
     by auto
   then have \<open>a \<le> cmod (cinner (A \<phi>) (A \<phi>)) / (norm (A \<phi>) * norm \<phi>)\<close>
     apply auto
     by (smt (verit) divide_divide_eq_left norm_eq_sqrt_cinner norm_imp_pos_and_ge real_div_sqrt)
   then show \<open>\<exists>b\<in>range (\<lambda>(\<psi>, \<phi>). cmod (cinner \<psi> (A \<phi>)) / (norm \<psi> * norm \<phi>)). a \<le> b\<close>
     by force
 next
   fix b
   assume \<open>b \<in> range (\<lambda>(\<psi>, \<phi>). cmod (cinner \<psi> (A \<phi>)) / (norm \<psi> * norm \<phi>))\<close>
   then obtain \<psi> \<phi> where b: \<open>b = cmod (cinner \<psi> (A \<phi>)) / (norm \<psi> * norm \<phi>)\<close>
     by auto
   then have \<open>b \<le> norm (A \<phi>) / norm \<phi>\<close>
     apply auto
     by (smt (verit, ccfv_threshold) complex_inner_class.Cauchy_Schwarz_ineq2 division_ring_divide_zero linordered_field_class.divide_right_mono mult_cancel_left1 nonzero_mult_divide_mult_cancel_left2 norm_imp_pos_and_ge ordered_field_class.sign_simps(33) zero_le_divide_iff)
   then show \<open>\<exists>a\<in>range (\<lambda>x. norm (A x) / norm x). b \<le> a\<close>
     by auto
 qed
 
 
 lemma sum_cinner:
   fixes f :: "'a \<Rightarrow> 'b::complex_inner"
   shows "cinner (sum f A) (sum g B) = (\<Sum>i\<in>A. \<Sum>j\<in>B. cinner (f i) (g j))"
   by (simp add: cinner_sum_right cinner_sum_left) (rule sum.swap)
 
 lemma Cauchy_cinner_product_summable':
   fixes a b :: "nat \<Rightarrow> 'a::complex_inner"
   shows \<open>(\<lambda>(x, y). cinner (a x) (b y)) summable_on UNIV \<longleftrightarrow> (\<lambda>(x, y). cinner (a y) (b (x - y))) summable_on {(k, i). i \<le> k}\<close>
 proof -
   have img: \<open>(\<lambda>(k::nat, i). (i, k - i)) ` {(k, i). i \<le> k} = UNIV\<close>
     apply (auto simp: image_def)
     by (metis add.commute add_diff_cancel_right' diff_le_self)
   have inj: \<open>inj_on (\<lambda>(k::nat, i). (i, k - i)) {(k, i). i \<le> k}\<close>
     by (smt (verit, del_insts) Pair_inject case_prodE case_prod_conv eq_diff_iff inj_onI mem_Collect_eq)
 
   have \<open>(\<lambda>(x, y). cinner (a x) (b y)) summable_on UNIV \<longleftrightarrow> (\<lambda>(k, l). cinner (a k) (b l)) summable_on (\<lambda>(k, i). (i, k - i)) ` {(k, i). i \<le> k}\<close>
     by (simp only: img)
   also have \<open>\<dots> \<longleftrightarrow> ((\<lambda>(k, l). cinner (a k) (b l)) \<circ> (\<lambda>(k, i). (i, k - i))) summable_on {(k, i). i \<le> k}\<close>
     using inj by (rule summable_on_reindex)
   also have \<open>\<dots> \<longleftrightarrow> (\<lambda>(x, y). cinner (a y) (b (x - y))) summable_on {(k, i). i \<le> k}\<close>
     by (simp add: o_def case_prod_unfold)
   finally show ?thesis
     by -
 qed
 
 instantiation prod :: (complex_inner, complex_inner) complex_inner
 begin
 
 definition cinner_prod_def:
   "cinner x y = cinner (fst x) (fst y) + cinner (snd x) (snd y)"
 
 instance
 proof
   fix r :: complex
   fix x y z :: "'a::complex_inner \<times> 'b::complex_inner"
   show "cinner x y = cnj (cinner y x)"
     unfolding cinner_prod_def
     by simp
   show "cinner (x + y) z = cinner x z + cinner y z"
     unfolding cinner_prod_def
     by (simp add: cinner_add_left)
   show "cinner (scaleC r x) y = cnj r * cinner x y"
     unfolding cinner_prod_def
     by (simp add: distrib_left)
   show "0 \<le> cinner x x"
     unfolding cinner_prod_def
     by (intro add_nonneg_nonneg cinner_ge_zero)
   show "cinner x x = 0 \<longleftrightarrow> x = 0"
     unfolding cinner_prod_def prod_eq_iff
     by (metis antisym cinner_eq_zero_iff cinner_ge_zero fst_zero le_add_same_cancel2 snd_zero verit_sum_simplify)
   show "norm x = sqrt (cmod (cinner x x))"
     unfolding norm_prod_def cinner_prod_def
     by (metis (no_types, lifting) Re_complex_of_real add_nonneg_nonneg cinner_ge_zero complex_of_real_cmod plus_complex.simps(1) power2_norm_eq_cinner')
 qed
 
 end
 
+lemma sgn_cinner[simp]: \<open>sgn \<psi> \<bullet>\<^sub>C \<psi> = norm \<psi>\<close>
+  apply (cases \<open>\<psi> = 0\<close>)
+   apply (auto simp: sgn_div_norm)
+  by (smt (verit, ccfv_SIG) cinner_scaleR_left cinner_scaleR_right cnorm_eq cnorm_eq_1 complex_of_real_cmod complex_of_real_nn_iff left_inverse mult.right_neutral mult_scaleR_right norm_eq_zero norm_not_less_zero norm_one of_real_def of_real_eq_iff)
+
 instance prod :: (chilbert_space, chilbert_space) chilbert_space..
 
 subsection \<open>Orthogonality\<close>
 
 definition "orthogonal_complement S = {x| x. \<forall>y\<in>S. cinner x y = 0}"
 
 lemma orthogonal_complement_orthoI:
-  \<open>x \<in> orthogonal_complement M \<Longrightarrow> y \<in> M \<Longrightarrow> \<langle> x, y \<rangle> = 0\<close>
+  \<open>x \<in> orthogonal_complement M \<Longrightarrow> y \<in> M \<Longrightarrow> x \<bullet>\<^sub>C y = 0\<close>
   unfolding orthogonal_complement_def by auto
 
 lemma orthogonal_complement_orthoI':
-  \<open>x \<in> M \<Longrightarrow> y \<in> orthogonal_complement M \<Longrightarrow> \<langle> x, y \<rangle> = 0\<close>
+  \<open>x \<in> M \<Longrightarrow> y \<in> orthogonal_complement M \<Longrightarrow> x \<bullet>\<^sub>C y = 0\<close>
   by (metis cinner_commute' complex_cnj_zero orthogonal_complement_orthoI)
 
 lemma orthogonal_complementI:
-  \<open>(\<And>x. x \<in> M \<Longrightarrow> \<langle> y, x \<rangle> = 0) \<Longrightarrow> y \<in> orthogonal_complement M\<close>
+  \<open>(\<And>x. x \<in> M \<Longrightarrow> y \<bullet>\<^sub>C x = 0) \<Longrightarrow> y \<in> orthogonal_complement M\<close>
   unfolding orthogonal_complement_def
   by simp
 
 abbreviation is_orthogonal::\<open>'a::complex_inner \<Rightarrow> 'a \<Rightarrow> bool\<close>  where
-  \<open>is_orthogonal x y \<equiv> \<langle> x, y \<rangle> = 0\<close>
+  \<open>is_orthogonal x y \<equiv> x \<bullet>\<^sub>C y = 0\<close>
 
 bundle orthogonal_notation begin
 notation is_orthogonal (infixl "\<bottom>" 69)
 end
 
 bundle no_orthogonal_notation begin
 no_notation is_orthogonal (infixl "\<bottom>" 69)
 end
 
 
 lemma is_orthogonal_sym: "is_orthogonal \<psi> \<phi> = is_orthogonal \<phi> \<psi>"
   by (metis cinner_commute' complex_cnj_zero)
 
 lemma is_orthogonal_sgn_right[simp]: \<open>is_orthogonal e (sgn f) \<longleftrightarrow> is_orthogonal e f\<close>
 proof (cases \<open>f = 0\<close>)
   case True
   then show ?thesis
     by simp
 next
   case False
   have \<open>cinner e (sgn f) = cinner e f / norm f\<close>
     by (simp add: sgn_div_norm divide_inverse scaleR_scaleC)
   moreover have \<open>norm f \<noteq> 0\<close>
     by (simp add: False)
   ultimately show ?thesis
     by force
 qed
 
 lemma is_orthogonal_sgn_left[simp]: \<open>is_orthogonal (sgn e) f \<longleftrightarrow> is_orthogonal e f\<close>
   by (simp add: is_orthogonal_sym)
 
 lemma orthogonal_complement_closed_subspace[simp]:
   "closed_csubspace (orthogonal_complement A)"
   for A :: \<open>('a::complex_inner) set\<close>
 proof (intro closed_csubspace.intro complex_vector.subspaceI)
   fix x y and c
   show \<open>0 \<in> orthogonal_complement A\<close>
     by (rule orthogonal_complementI, simp)
   show \<open>x + y \<in> orthogonal_complement A\<close>
     if \<open>x \<in> orthogonal_complement A\<close> and \<open>y \<in> orthogonal_complement A\<close>
     using that by (auto intro!: orthogonal_complementI dest!: orthogonal_complement_orthoI
         simp add: cinner_add_left)
   show \<open>c *\<^sub>C x \<in> orthogonal_complement A\<close> if \<open>x \<in> orthogonal_complement A\<close>
     using that by (auto intro!: orthogonal_complementI dest!: orthogonal_complement_orthoI)
 
   show "closed (orthogonal_complement A)"
   proof (auto simp add: closed_sequential_limits, rename_tac an a)
     fix an a
     assume ortho: \<open>\<forall>n::nat. an n \<in> orthogonal_complement A\<close>
     assume lim: \<open>an \<longlonglongrightarrow> a\<close>
 
-    have \<open>\<forall> y \<in> A. \<forall> n. \<langle> y , an n \<rangle> = 0\<close>
+    have \<open>\<forall> y \<in> A. \<forall> n. is_orthogonal y (an n)\<close>
       using orthogonal_complement_orthoI'
       by (simp add: orthogonal_complement_orthoI' ortho)
-    moreover have \<open>isCont (\<lambda> x. \<langle> y , x \<rangle>) a\<close> for y
+    moreover have \<open>isCont (\<lambda> x. y \<bullet>\<^sub>C x) a\<close> for y
       using bounded_clinear_cinner_right clinear_continuous_at
       by (simp add: clinear_continuous_at bounded_clinear_cinner_right)
-    ultimately have \<open>(\<lambda> n. (\<lambda> v. \<langle> y , v \<rangle>) (an n)) \<longlonglongrightarrow> (\<lambda> v. \<langle> y , v \<rangle>) a\<close> for y
+    ultimately have \<open>(\<lambda> n. (\<lambda> v. y \<bullet>\<^sub>C v) (an n)) \<longlonglongrightarrow> (\<lambda> v. y \<bullet>\<^sub>C v) a\<close> for y
       using isCont_tendsto_compose
       by (simp add: isCont_tendsto_compose lim)
-    hence  \<open>\<forall> y\<in>A. (\<lambda> n. \<langle> y , an n \<rangle>  ) \<longlonglongrightarrow>  \<langle> y , a \<rangle>\<close>
+    hence  \<open>\<forall> y\<in>A. (\<lambda> n. y \<bullet>\<^sub>C an n) \<longlonglongrightarrow>  y \<bullet>\<^sub>C a\<close>
       by simp
-    hence  \<open>\<forall> y\<in>A. (\<lambda> n. 0  ) \<longlonglongrightarrow>  \<langle> y , a \<rangle>\<close>
-      using \<open>\<forall> y \<in> A. \<forall> n. \<langle> y , an n \<rangle> = 0\<close>
+    hence  \<open>\<forall> y\<in>A. (\<lambda> n. 0) \<longlonglongrightarrow>  y \<bullet>\<^sub>C a\<close>
+      using \<open>\<forall> y \<in> A. \<forall> n. is_orthogonal y (an n)\<close>
       by fastforce
-    hence  \<open>\<forall> y \<in> A. \<langle> y , a \<rangle> = 0\<close>
+    hence  \<open>\<forall> y \<in> A. is_orthogonal y a\<close>
       using limI by fastforce
     then show \<open>a \<in> orthogonal_complement A\<close>
       by (simp add: orthogonal_complementI is_orthogonal_sym)
   qed
 qed
 
 lemma orthogonal_complement_zero_intersection:
   assumes "0\<in>M"
   shows \<open>M \<inter> (orthogonal_complement M) = {0}\<close>
 proof -
   have "x=0" if "x\<in>M" and "x\<in>orthogonal_complement M" for x
   proof -
-    from that have "\<langle> x, x \<rangle> = 0"
+    from that have "is_orthogonal x x"
       unfolding orthogonal_complement_def by auto
     thus "x=0"
       by auto
   qed
   with assms show ?thesis
     unfolding orthogonal_complement_def by auto
 qed
 
 lemma is_orthogonal_closure_cspan:
   assumes "\<And>x y. x \<in> X \<Longrightarrow> y \<in> Y \<Longrightarrow> is_orthogonal x y"
   assumes \<open>x \<in> closure (cspan X)\<close> \<open>y \<in> closure (cspan Y)\<close>
   shows "is_orthogonal x y"
 proof -
   have *: \<open>cinner x y = 0\<close> if \<open>y \<in> Y\<close> for y
     using bounded_antilinear_cinner_left apply (rule bounded_antilinear_eq_on[where G=X])
     using assms that by auto
   show \<open>cinner x y = 0\<close>
     using bounded_clinear_cinner_right apply (rule bounded_clinear_eq_on[where G=Y])
     using * assms by auto
 qed
 
 
 instantiation ccsubspace :: (complex_inner) "uminus"
 begin
 lift_definition uminus_ccsubspace::\<open>'a ccsubspace  \<Rightarrow> 'a ccsubspace\<close>
   is \<open>orthogonal_complement\<close>
   by simp
 
 instance ..
 end
 
+lemma orthocomplement_top[simp]: \<open>- top = (bot :: 'a::complex_inner ccsubspace)\<close>
+  \<comment> \<open>For \<^typ>\<open>'a\<close> of sort \<^class>\<open>chilbert_space\<close>, this is covered by @{thm [source] orthocomplemented_lattice_class.compl_top_eq} already.
+      But here we give it a wider sort.\<close>
+  apply transfer
+  by (metis Int_UNIV_left UNIV_I orthogonal_complement_zero_intersection)
 
 instantiation ccsubspace :: (complex_inner) minus begin
 lift_definition minus_ccsubspace :: "'a ccsubspace \<Rightarrow> 'a ccsubspace \<Rightarrow> 'a ccsubspace"
   is "\<lambda>A B. A \<inter> (orthogonal_complement B)"
   by simp
 instance..
 end
 
 definition is_ortho_set :: "'a::complex_inner set \<Rightarrow> bool" where
   \<comment> \<open>Orthogonal set\<close>
-  \<open>is_ortho_set S = ((\<forall>x\<in>S. \<forall>y\<in>S. x \<noteq> y \<longrightarrow> \<langle>x, y\<rangle> = 0) \<and> 0 \<notin> S)\<close>
+  \<open>is_ortho_set S = ((\<forall>x\<in>S. \<forall>y\<in>S. x \<noteq> y \<longrightarrow> (x \<bullet>\<^sub>C y) = 0) \<and> 0 \<notin> S)\<close>
 
 definition is_onb where \<open>is_onb E \<longleftrightarrow> is_ortho_set E \<and> (\<forall>b\<in>E. norm b = 1) \<and> ccspan E = top\<close>
 
 lemma is_ortho_set_empty[simp]: "is_ortho_set {}"
   unfolding is_ortho_set_def by auto
 
 lemma is_ortho_set_antimono: \<open>A \<subseteq> B \<Longrightarrow> is_ortho_set B \<Longrightarrow> is_ortho_set A\<close>
   unfolding is_ortho_set_def by auto
 
 lemma orthogonal_complement_of_closure:
   fixes A ::"('a::complex_inner) set"
   shows "orthogonal_complement A = orthogonal_complement (closure A)"
 proof-
-  have s1: \<open>\<langle> y, x \<rangle> = 0\<close>
+  have s1: \<open>is_orthogonal y x\<close>
     if a1: "x \<in> (orthogonal_complement A)"
       and a2: \<open>y \<in> closure A\<close>
     for x y
   proof-
-    have \<open>\<forall> y \<in> A. \<langle> y , x \<rangle> = 0\<close>
+    have \<open>\<forall> y \<in> A. is_orthogonal y x\<close>
       by (simp add: a1 orthogonal_complement_orthoI')
     then obtain yy where \<open>\<forall> n. yy n \<in> A\<close> and \<open>yy \<longlonglongrightarrow> y\<close>
       using a2 closure_sequential by blast
-    have \<open>isCont (\<lambda> t. \<langle> t , x \<rangle>) y\<close>
+    have \<open>isCont (\<lambda> t. t \<bullet>\<^sub>C x) y\<close>
       by simp
-    hence \<open>(\<lambda> n. \<langle> yy n , x \<rangle>) \<longlonglongrightarrow>  \<langle> y , x \<rangle>\<close>
+    hence \<open>(\<lambda> n. yy n \<bullet>\<^sub>C x) \<longlonglongrightarrow> y \<bullet>\<^sub>C x\<close>
       using \<open>yy \<longlonglongrightarrow> y\<close> isCont_tendsto_compose
       by fastforce
-    hence \<open>(\<lambda> n. 0) \<longlonglongrightarrow>  \<langle> y , x \<rangle>\<close>
-      using \<open>\<forall> y \<in> A. \<langle> y , x \<rangle> = 0\<close>  \<open>\<forall> n. yy n \<in> A\<close> by simp
+    hence \<open>(\<lambda> n. 0) \<longlonglongrightarrow> y \<bullet>\<^sub>C x\<close>
+      using \<open>\<forall> y \<in> A. is_orthogonal y x\<close>  \<open>\<forall> n. yy n \<in> A\<close> by simp
     thus ?thesis
       using limI by force
   qed
   hence "x \<in> orthogonal_complement (closure A)"
     if a1: "x \<in> (orthogonal_complement A)"
     for x
     using that
     by (meson orthogonal_complementI is_orthogonal_sym)
   moreover have \<open>x \<in> (orthogonal_complement A)\<close>
     if "x \<in> (orthogonal_complement (closure A))"
     for x
     using that
     by (meson closure_subset orthogonal_complement_orthoI orthogonal_complementI subset_eq)
   ultimately show ?thesis by blast
 qed
 
 
 lemma is_orthogonal_closure:
   assumes \<open>\<And>s. s \<in> S \<Longrightarrow> is_orthogonal a  s\<close>
   assumes \<open>x \<in> closure S\<close>
   shows \<open>is_orthogonal a x\<close>
   by (metis assms(1) assms(2) orthogonal_complementI orthogonal_complement_of_closure orthogonal_complement_orthoI)
 
 
 lemma is_orthogonal_cspan:
   assumes a1: "\<And>s. s \<in> S \<Longrightarrow> is_orthogonal a s" and a3: "x \<in> cspan S"
-  shows "\<langle>a, x\<rangle> = 0"
+  shows "is_orthogonal a x"
 proof-
   have "\<exists>t r. finite t \<and> t \<subseteq> S \<and> (\<Sum>a\<in>t. r a *\<^sub>C a) = x"
     using complex_vector.span_explicit
     by (smt a3 mem_Collect_eq)
   then obtain t r where b1: "finite t" and b2: "t \<subseteq> S" and b3: "(\<Sum>a\<in>t. r a *\<^sub>C a) = x"
     by blast
-  have x1: "\<langle>a, i\<rangle> = 0"
+  have x1: "is_orthogonal a i"
     if "i\<in>t" for i
     using b2 a1 that by blast
-  have  "\<langle>a, x\<rangle> = \<langle>a, (\<Sum>i\<in>t. r i *\<^sub>C i)\<rangle>"
+  have  "a \<bullet>\<^sub>C x = a \<bullet>\<^sub>C (\<Sum>i\<in>t. r i *\<^sub>C i)"
     by (simp add: b3)
-  also have  "\<dots> = (\<Sum>i\<in>t. r i *\<^sub>C \<langle>a, i\<rangle>)"
+  also have  "\<dots> = (\<Sum>i\<in>t. r i *\<^sub>C (a \<bullet>\<^sub>C i))"
     by (simp add: cinner_sum_right)
   also have  "\<dots> = 0"
     using x1 by simp
   finally show ?thesis.
 qed
 
 lemma ccspan_leq_ortho_ccspan:
   assumes "\<And>s t. s\<in>S \<Longrightarrow> t\<in>T \<Longrightarrow> is_orthogonal s t"
   shows "ccspan S \<le> - (ccspan T)"
   using assms apply transfer
   by (smt (verit, ccfv_threshold) is_orthogonal_closure is_orthogonal_cspan is_orthogonal_sym orthogonal_complementI subsetI)
 
 lemma double_orthogonal_complement_increasing[simp]:
   shows "M \<subseteq> orthogonal_complement (orthogonal_complement M)"
 proof (rule subsetI)
   fix x assume s1: "x \<in> M"
-  have \<open>\<forall> y \<in> (orthogonal_complement M). \<langle> x, y \<rangle> = 0\<close>
+  have \<open>\<forall> y \<in> (orthogonal_complement M). is_orthogonal x y\<close>
     using s1 orthogonal_complement_orthoI' by auto
   hence \<open>x \<in> orthogonal_complement (orthogonal_complement M)\<close>
     by (simp add: orthogonal_complement_def)
   then show "x \<in> orthogonal_complement (orthogonal_complement M)"
     by blast
 qed
 
 
 lemma orthonormal_basis_of_cspan:
   fixes S::"'a::complex_inner set"
   assumes "finite S"
   shows "\<exists>A. is_ortho_set A \<and> (\<forall>x\<in>A. norm x = 1) \<and> cspan A = cspan S \<and> finite A"
 proof (use assms in induction)
   case empty
   show ?case
     apply (rule exI[of _ "{}"])
     by auto
 next
   case (insert s S)
   from insert.IH
   obtain A where orthoA: "is_ortho_set A" and normA: "\<And>x. x\<in>A \<Longrightarrow> norm x = 1" and spanA: "cspan A = cspan S" and finiteA: "finite A"
     by auto
   show ?case
   proof (cases \<open>s \<in> cspan S\<close>)
     case True
     then have \<open>cspan (insert s S) = cspan S\<close>
       by (simp add: complex_vector.span_redundant)
     with orthoA normA spanA finiteA
     show ?thesis
       by auto
   next
     case False
     obtain a where a_ortho: \<open>\<And>x. x\<in>A \<Longrightarrow> is_orthogonal x a\<close> and sa_span: \<open>s - a \<in> cspan A\<close>
     proof (atomize_elim, use \<open>finite A\<close> \<open>is_ortho_set A\<close> in induction)
       case empty
       then show ?case
         by auto
     next
       case (insert x A)
       then obtain a where orthoA: \<open>\<And>x. x \<in> A \<Longrightarrow> is_orthogonal x a\<close> and sa: \<open>s - a \<in> cspan A\<close>
         by (meson is_ortho_set_antimono subset_insertI)
       define a' where \<open>a' = a - cinner x a *\<^sub>C inverse (cinner x x) *\<^sub>C x\<close>
       have \<open>is_orthogonal x a'\<close>
         unfolding a'_def cinner_diff_right cinner_scaleC_right
         apply (cases \<open>cinner x x = 0\<close>)
         by auto
       have orthoA: \<open>is_orthogonal y a'\<close> if \<open>y \<in> A\<close> for y
         unfolding a'_def cinner_diff_right cinner_scaleC_right
         apply auto by (metis insert.prems insertCI is_ortho_set_def mult_not_zero orthoA that)
       have \<open>s - a' \<in> cspan (insert x A)\<close>
         unfolding a'_def apply auto
         by (metis (no_types, lifting) complex_vector.span_breakdown_eq diff_add_cancel diff_diff_add sa)
       with \<open>is_orthogonal x a'\<close> orthoA
       show ?case
         apply (rule_tac exI[of _ a'])
         by auto
     qed
 
     from False sa_span
     have \<open>a \<noteq> 0\<close>
       unfolding spanA by auto
     define a' where \<open>a' = inverse (norm a) *\<^sub>C a\<close>
     with \<open>a \<noteq> 0\<close> have \<open>norm a' = 1\<close>
       by (simp add: norm_inverse)
     have a: \<open>a = norm a *\<^sub>C a'\<close>
       by (simp add: \<open>a \<noteq> 0\<close> a'_def)
 
     from sa_span spanA
     have a'_span: \<open>a' \<in> cspan (insert s S)\<close>
       unfolding a'_def
       by (metis complex_vector.eq_span_insert_eq complex_vector.span_scale complex_vector.span_superset in_mono insertI1)
     from sa_span
     have s_span: \<open>s \<in> cspan (insert a' A)\<close>
       apply (subst (asm) a)
       using complex_vector.span_breakdown_eq by blast
 
     from \<open>a \<noteq> 0\<close> a_ortho orthoA
     have ortho: "is_ortho_set (insert a' A)"
       unfolding is_ortho_set_def a'_def
       apply auto
       by (meson is_orthogonal_sym)
 
     have span: \<open>cspan (insert a' A) = cspan (insert s S)\<close>
       using a'_span s_span spanA apply auto
        apply (metis (full_types) complex_vector.span_breakdown_eq complex_vector.span_redundant insert_commute s_span)
       by (metis (full_types) complex_vector.span_breakdown_eq complex_vector.span_redundant insert_commute s_span)
 
     show ?thesis
       apply (rule exI[of _ \<open>insert a' A\<close>])
       by (simp add: ortho \<open>norm a' = 1\<close> normA finiteA span)
   qed
 qed
 
 lemma is_ortho_set_cindependent:
   assumes "is_ortho_set A"
   shows "cindependent A"
 proof -
   have "u v = 0"
     if b1: "finite t" and b2: "t \<subseteq> A" and b3: "(\<Sum>v\<in>t. u v *\<^sub>C v) = 0" and b4: "v \<in> t"
     for t u v
   proof -
-    have "\<langle>v, v'\<rangle> = 0" if c1: "v'\<in>t-{v}" for v'
+    have "is_orthogonal v v'" if c1: "v'\<in>t-{v}" for v'
       by (metis DiffE assms b2 b4 insertI1 is_ortho_set_antimono is_ortho_set_def that)
-    hence sum0: "(\<Sum>v'\<in>t-{v}. u v' * \<langle>v, v'\<rangle>) = 0"
+    hence sum0: "(\<Sum>v'\<in>t-{v}. u v' * (v \<bullet>\<^sub>C v')) = 0"
       by simp
-    have "\<langle>v, (\<Sum>v'\<in>t. u v' *\<^sub>C v')\<rangle> = (\<Sum>v'\<in>t. u v' * \<langle>v, v'\<rangle>)"
+    have "v \<bullet>\<^sub>C (\<Sum>v'\<in>t. u v' *\<^sub>C v') = (\<Sum>v'\<in>t. u v' * (v \<bullet>\<^sub>C v'))"
       using b1
       by (metis (mono_tags, lifting) cinner_scaleC_right cinner_sum_right sum.cong)
-    also have "\<dots> = u v * \<langle>v, v\<rangle> + (\<Sum>v'\<in>t-{v}. u v' * \<langle>v, v'\<rangle>)"
+    also have "\<dots> = u v * (v \<bullet>\<^sub>C v) + (\<Sum>v'\<in>t-{v}. u v' * (v \<bullet>\<^sub>C v'))"
       by (meson b1 b4 sum.remove)
-    also have "\<dots> = u v * \<langle>v, v\<rangle>"
+    also have "\<dots> = u v * (v \<bullet>\<^sub>C v)"
       using sum0 by simp
-    finally have "\<langle>v, (\<Sum>v'\<in>t. u v' *\<^sub>C v')\<rangle> =  u v * \<langle>v, v\<rangle>"
+    finally have "v \<bullet>\<^sub>C (\<Sum>v'\<in>t. u v' *\<^sub>C v') =  u v * (v \<bullet>\<^sub>C v)"
       by blast
-    hence "u v * \<langle>v, v\<rangle> = 0" using b3 by simp
-    moreover have "\<langle>v, v\<rangle> \<noteq> 0"
+    hence "u v * (v \<bullet>\<^sub>C v) = 0" using b3 by simp
+    moreover have "(v \<bullet>\<^sub>C v) \<noteq> 0"
       using assms is_ortho_set_def b2 b4 by auto
     ultimately show "u v = 0" by simp
   qed
   thus ?thesis using complex_vector.independent_explicit_module
     by (smt cdependent_raw_def)
 qed
 
 
 lemma onb_expansion_finite:
   includes notation_norm
   fixes T::\<open>'a::{complex_inner,cfinite_dim} set\<close>
   assumes a1: \<open>cspan T = UNIV\<close> and a3: \<open>is_ortho_set T\<close>
     and a4: \<open>\<And>t. t\<in>T \<Longrightarrow> \<parallel>t\<parallel> = 1\<close>
-  shows \<open>x = (\<Sum>t\<in>T. \<langle> t, x \<rangle> *\<^sub>C t)\<close>
+  shows \<open>x = (\<Sum>t\<in>T. (t \<bullet>\<^sub>C x) *\<^sub>C t)\<close>
 proof -
   have \<open>finite T\<close>
     apply (rule cindependent_cfinite_dim_finite)
     by (simp add: a3 is_ortho_set_cindependent)
   have \<open>closure (complex_vector.span T)  = complex_vector.span T\<close>
     by (simp add: a1)
   have \<open>{\<Sum>a\<in>t. r a *\<^sub>C a |t r. finite t \<and> t \<subseteq> T} = {\<Sum>a\<in>T. r a *\<^sub>C a |r. True}\<close>
     apply auto
      apply (rule_tac x=\<open>\<lambda>a. if a \<in> t then r a else 0\<close> in exI)
      apply (simp add: \<open>finite T\<close> sum.mono_neutral_cong_right)
     using \<open>finite T\<close> by blast
 
   have f1: "\<forall>A. {a. \<exists>Aa f. (a::'a) = (\<Sum>a\<in>Aa. f a *\<^sub>C a) \<and> finite Aa \<and> Aa \<subseteq> A} = cspan A"
     by (simp add: complex_vector.span_explicit)
   have f2: "\<forall>a. (\<exists>f. a = (\<Sum>a\<in>T. f a *\<^sub>C a)) \<or> (\<forall>A. (\<forall>f. a \<noteq> (\<Sum>a\<in>A. f a *\<^sub>C a)) \<or> infinite A \<or> \<not> A \<subseteq> T)"
     using \<open>{\<Sum>a\<in>t. r a *\<^sub>C a |t r. finite t \<and> t \<subseteq> T} = {\<Sum>a\<in>T. r a *\<^sub>C a |r. True}\<close> by auto
   have f3: "\<forall>A a. (\<exists>Aa f. (a::'a) = (\<Sum>a\<in>Aa. f a *\<^sub>C a) \<and> finite Aa \<and> Aa \<subseteq> A) \<or> a \<notin> cspan A"
     using f1 by blast
   have "cspan T = UNIV"
     by (metis (full_types, lifting)  \<open>complex_vector.span T = UNIV\<close>)
   hence \<open>\<exists> r. x = (\<Sum> a\<in>T. r a *\<^sub>C a)\<close>
     using f3 f2 by blast
   then obtain r where \<open>x = (\<Sum> a\<in>T. r a *\<^sub>C a)\<close>
     by blast
 
-  have \<open>r a = \<langle>a, x\<rangle>\<close> if \<open>a \<in> T\<close> for a
+  have \<open>r a = a \<bullet>\<^sub>C x\<close> if \<open>a \<in> T\<close> for a
   proof-
     have \<open>norm a = 1\<close>
       using a4
       by (simp add: \<open>a \<in> T\<close>)
-    moreover have \<open>norm a = sqrt (norm \<langle>a, a\<rangle>)\<close>
+    moreover have \<open>norm a = sqrt (norm (a \<bullet>\<^sub>C a))\<close>
       using norm_eq_sqrt_cinner by auto
-    ultimately have \<open>sqrt (norm \<langle>a, a\<rangle>) = 1\<close>
+    ultimately have \<open>sqrt (norm (a \<bullet>\<^sub>C a)) = 1\<close>
       by simp
-    hence \<open>norm \<langle>a, a\<rangle> = 1\<close>
+    hence \<open>norm (a \<bullet>\<^sub>C a) = 1\<close>
       using real_sqrt_eq_1_iff by blast
-    moreover have \<open>\<langle>a, a\<rangle> \<in> \<real>\<close>
+    moreover have \<open>(a \<bullet>\<^sub>C a) \<in> \<real>\<close>
       by (simp add: cinner_real)
-    moreover have \<open>\<langle>a, a\<rangle> \<ge> 0\<close>
+    moreover have \<open>(a \<bullet>\<^sub>C a) \<ge> 0\<close>
       using cinner_ge_zero by blast
-    ultimately have w1: \<open>\<langle>a, a\<rangle> = 1\<close>
-      by (metis \<open>0 \<le> \<langle>a, a\<rangle>\<close> \<open>cmod \<langle>a, a\<rangle> = 1\<close> complex_of_real_cmod of_real_1)
+    ultimately have w1: \<open>(a \<bullet>\<^sub>C a) = 1\<close>
+      by (metis \<open>0 \<le> (a \<bullet>\<^sub>C a)\<close> \<open>cmod (a \<bullet>\<^sub>C a) = 1\<close> complex_of_real_cmod of_real_1)
 
-    have \<open>r t * \<langle>a, t\<rangle> = 0\<close> if \<open>t \<in> T-{a}\<close> for t
+    have \<open>r t * (a \<bullet>\<^sub>C t) = 0\<close> if \<open>t \<in> T-{a}\<close> for t
       by (metis DiffD1 DiffD2 \<open>a \<in> T\<close> a3 is_ortho_set_def mult_eq_0_iff singletonI that)
-    hence s1: \<open>(\<Sum> t\<in>T-{a}. r t * \<langle>a, t\<rangle>) = 0\<close>
-      by (simp add: \<open>\<And>t. t \<in> T - {a} \<Longrightarrow> r t * \<langle>a, t\<rangle> = 0\<close>)
-    have \<open>\<langle>a, x\<rangle> = \<langle>a, (\<Sum> t\<in>T. r t *\<^sub>C t)\<rangle>\<close>
+    hence s1: \<open>(\<Sum> t\<in>T-{a}. r t * (a \<bullet>\<^sub>C t)) = 0\<close>
+      by (simp add: \<open>\<And>t. t \<in> T - {a} \<Longrightarrow> r t * (a \<bullet>\<^sub>C t) = 0\<close>)
+    have \<open>(a \<bullet>\<^sub>C x) = a \<bullet>\<^sub>C (\<Sum> t\<in>T. r t *\<^sub>C t)\<close>
       using \<open>x = (\<Sum> a\<in>T. r a *\<^sub>C a)\<close>
       by simp
-    also have \<open>\<dots> = (\<Sum> t\<in>T. \<langle>a, r t *\<^sub>C t\<rangle>)\<close>
+    also have \<open>\<dots> = (\<Sum> t\<in>T. a \<bullet>\<^sub>C (r t *\<^sub>C t))\<close>
       using cinner_sum_right by blast
-    also have \<open>\<dots> = (\<Sum> t\<in>T. r t * \<langle>a, t\<rangle>)\<close>
+    also have \<open>\<dots> = (\<Sum> t\<in>T. r t * (a \<bullet>\<^sub>C t))\<close>
       by simp
-    also have \<open>\<dots> = r a * \<langle>a, a\<rangle> + (\<Sum> t\<in>T-{a}. r t * \<langle>a, t\<rangle>)\<close>
+    also have \<open>\<dots> = r a * (a \<bullet>\<^sub>C a) + (\<Sum> t\<in>T-{a}. r t * (a \<bullet>\<^sub>C t))\<close>
       using \<open>a \<in> T\<close>
       by (meson \<open>finite T\<close> sum.remove)
-    also have \<open>\<dots> = r a * \<langle>a, a\<rangle>\<close>
+    also have \<open>\<dots> = r a * (a \<bullet>\<^sub>C a)\<close>
       using s1
       by simp
     also have \<open>\<dots> = r a\<close>
       by (simp add: w1)
     finally show ?thesis by auto
   qed
   thus ?thesis
     using \<open>x = (\<Sum> a\<in>T. r a *\<^sub>C a)\<close>
     by fastforce
 qed
 
 lemma is_ortho_set_singleton[simp]: \<open>is_ortho_set {x} \<longleftrightarrow> x \<noteq> 0\<close>
   by (simp add: is_ortho_set_def)
 
+lemma orthogonal_complement_antimono[simp]:
+  fixes  A B :: \<open>('a::complex_inner) set\<close>
+  assumes "A \<supseteq> B"
+  shows \<open>orthogonal_complement A \<subseteq> orthogonal_complement B\<close>
+  by (meson assms orthogonal_complementI orthogonal_complement_orthoI' subsetD subsetI)
+
+lemma orthogonal_complement_UNIV[simp]:
+  "orthogonal_complement UNIV = {0}"
+  by (metis Int_UNIV_left complex_vector.subspace_UNIV complex_vector.subspace_def orthogonal_complement_zero_intersection)
+
+lemma orthogonal_complement_zero[simp]:
+  "orthogonal_complement {0} = UNIV"
+  unfolding orthogonal_complement_def by auto
+
 subsection \<open>Projections\<close>
 
 lemma smallest_norm_exists:
   \<comment> \<open>Theorem 2.5 in @{cite conway2013course} (inside the proof)\<close>
   includes notation_norm
   fixes M :: \<open>'a::chilbert_space set\<close>
   assumes q1: \<open>convex M\<close> and q2: \<open>closed M\<close> and q3: \<open>M \<noteq> {}\<close>
   shows  \<open>\<exists>k. is_arg_min (\<lambda> x. \<parallel>x\<parallel>) (\<lambda> t. t \<in> M) k\<close>
-proof-
+proof -
   define d where \<open>d = Inf { \<parallel>x\<parallel>^2 | x. x \<in> M }\<close>
   have w4: \<open>{ \<parallel>x\<parallel>^2 | x. x \<in> M } \<noteq> {}\<close>
     by (simp add: assms(3))
   have \<open>\<forall> x. \<parallel>x\<parallel>^2 \<ge> 0\<close>
     by simp
   hence bdd_below1: \<open>bdd_below { \<parallel>x\<parallel>^2 | x. x \<in> M }\<close>
     by fastforce
-  have \<open>d \<le> \<parallel>x\<parallel>^2\<close>
-    if a1: "x \<in> M"
-    for x
+  have \<open>d \<le> \<parallel>x\<parallel>^2\<close> if a1: "x \<in> M" for x
   proof-
-    have "\<forall>v. (\<exists>w. Re (\<langle>v , v\<rangle> ) = \<parallel>w\<parallel>\<^sup>2 \<and> w \<in> M) \<or> v \<notin> M"
+    have "\<forall>v. (\<exists>w. Re (v \<bullet>\<^sub>C v) = \<parallel>w\<parallel>\<^sup>2 \<and> w \<in> M) \<or> v \<notin> M"
       by (metis (no_types) power2_norm_eq_cinner')
-    hence "Re (\<langle>x , x\<rangle> ) \<in> {\<parallel>v\<parallel>\<^sup>2 |v. v \<in> M}"
+    hence "Re (x \<bullet>\<^sub>C x) \<in> {\<parallel>v\<parallel>\<^sup>2 |v. v \<in> M}"
       using a1 by blast
     thus ?thesis
       unfolding d_def
       by (metis (lifting) bdd_below1 cInf_lower power2_norm_eq_cinner')
   qed
 
   have \<open>\<forall> \<epsilon> > 0. \<exists> t \<in> { \<parallel>x\<parallel>^2 | x. x \<in> M }.  t < d + \<epsilon>\<close>
     unfolding d_def
     using w4  bdd_below1
     by (meson cInf_lessD less_add_same_cancel1)
   hence \<open>\<forall> \<epsilon> > 0. \<exists> x \<in> M.  \<parallel>x\<parallel>^2 < d + \<epsilon>\<close>
     by auto
   hence \<open>\<forall> \<epsilon> > 0. \<exists> x \<in> M.  \<parallel>x\<parallel>^2 < d + \<epsilon>\<close>
     by (simp add: \<open>\<And>x. x \<in> M \<Longrightarrow> d \<le> \<parallel>x\<parallel>\<^sup>2\<close>)
   hence w1: \<open>\<forall> n::nat. \<exists> x \<in> M.  \<parallel>x\<parallel>^2 < d + 1/(n+1)\<close> by auto
 
   then obtain r::\<open>nat \<Rightarrow> 'a\<close> where w2: \<open>\<forall> n. r n \<in> M \<and>  \<parallel> r n \<parallel>^2 < d + 1/(n+1)\<close>
     by metis
   have w3: \<open>\<forall> n. r n \<in> M\<close>
     by (simp add: w2)
   have \<open>\<forall> n. \<parallel> r n \<parallel>^2 < d + 1/(n+1)\<close>
     by (simp add: w2)
   have w5: \<open>\<parallel> (r n) - (r m) \<parallel>^2 < 2*(1/(n+1) + 1/(m+1))\<close>
     for m n
   proof-
     have w6: \<open>\<parallel> r n \<parallel>^2 < d + 1/(n+1)\<close>
       by (metis w2  of_nat_1 of_nat_add)
     have \<open> \<parallel> r m \<parallel>^2 < d + 1/(m+1)\<close>
       by (metis w2 of_nat_1 of_nat_add)
     have \<open>(r n) \<in> M\<close>
       by (simp add: \<open>\<forall>n. r n \<in> M\<close>)
     moreover have \<open>(r m) \<in> M\<close>
       by (simp add: \<open>\<forall>n. r n \<in> M\<close>)
     ultimately have \<open>(1/2) *\<^sub>R (r n) + (1/2) *\<^sub>R (r m) \<in> M\<close>
       using \<open>convex M\<close>
       by (simp add: convexD)
     hence \<open>\<parallel> (1/2) *\<^sub>R (r n) + (1/2) *\<^sub>R (r m) \<parallel>^2 \<ge> d\<close>
       by (simp add: \<open>\<And>x. x \<in> M \<Longrightarrow> d \<le> \<parallel>x\<parallel>\<^sup>2\<close>)
     have \<open>\<parallel> (1/2) *\<^sub>R (r n) - (1/2) *\<^sub>R (r m) \<parallel>^2
               = (1/2)*( \<parallel> r n \<parallel>^2 + \<parallel> r m \<parallel>^2 ) - \<parallel> (1/2) *\<^sub>R (r n) + (1/2) *\<^sub>R (r m) \<parallel>^2\<close>
       by (smt (z3) div_by_1 field_sum_of_halves nonzero_mult_div_cancel_left parallelogram_law polar_identity power2_norm_eq_cinner' scaleR_collapse times_divide_eq_left)
     also have  \<open>...
               < (1/2)*( d + 1/(n+1) + \<parallel> r m \<parallel>^2 ) - \<parallel> (1/2) *\<^sub>R (r n) + (1/2) *\<^sub>R (r m) \<parallel>^2\<close>
       using \<open>\<parallel>r n\<parallel>\<^sup>2 < d + 1 / real (n + 1)\<close> by auto
     also have  \<open>...
               < (1/2)*( d + 1/(n+1) + d + 1/(m+1) ) - \<parallel> (1/2) *\<^sub>R (r n) + (1/2) *\<^sub>R (r m) \<parallel>^2\<close>
       using \<open>\<parallel>r m\<parallel>\<^sup>2 < d + 1 / real (m + 1)\<close> by auto
     also have  \<open>...
               \<le> (1/2)*( d + 1/(n+1) + d + 1/(m+1) ) - d\<close>
       by (simp add: \<open>d \<le> \<parallel>(1 / 2) *\<^sub>R r n + (1 / 2) *\<^sub>R r m\<parallel>\<^sup>2\<close>)
     also have  \<open>...
               \<le> (1/2)*( 1/(n+1) + 1/(m+1) + 2*d ) - d\<close>
       by simp
     also have  \<open>...
               \<le> (1/2)*( 1/(n+1) + 1/(m+1) ) + (1/2)*(2*d) - d\<close>
       by (simp add: distrib_left)
     also have  \<open>...
               \<le> (1/2)*( 1/(n+1) + 1/(m+1) ) + d - d\<close>
       by simp
     also have  \<open>...
               \<le> (1/2)*( 1/(n+1) + 1/(m+1) )\<close>
       by simp
     finally have \<open> \<parallel>(1 / 2) *\<^sub>R r n - (1 / 2) *\<^sub>R r m\<parallel>\<^sup>2 < 1 / 2 * (1 / real (n + 1) + 1 / real (m + 1)) \<close>
       by blast
     hence \<open> \<parallel>(1 / 2) *\<^sub>R (r n - r m) \<parallel>\<^sup>2 < (1 / 2) * (1 / real (n + 1) + 1 / real (m + 1)) \<close>
       by (simp add: real_vector.scale_right_diff_distrib)
     hence \<open> ((1 / 2)*\<parallel> (r n - r m) \<parallel>)\<^sup>2 < (1 / 2) * (1 / real (n + 1) + 1 / real (m + 1)) \<close>
       by simp
     hence \<open> (1 / 2)^2*(\<parallel> (r n - r m) \<parallel>)\<^sup>2 < (1 / 2) * (1 / real (n + 1) + 1 / real (m + 1)) \<close>
       by (metis power_mult_distrib)
     hence \<open> (1 / 4) *(\<parallel> (r n - r m) \<parallel>)\<^sup>2 < (1 / 2) * (1 / real (n + 1) + 1 / real (m + 1)) \<close>
       by (simp add: power_divide)
     hence \<open> \<parallel> (r n - r m) \<parallel>\<^sup>2 < 2 * (1 / real (n + 1) + 1 / real (m + 1)) \<close>
       by simp
     thus ?thesis
       by (metis of_nat_1 of_nat_add)
   qed
   hence "\<exists> N. \<forall> n m. n \<ge> N \<and> m \<ge> N \<longrightarrow> \<parallel> (r n) - (r m) \<parallel>^2 < \<epsilon>^2"
     if "\<epsilon> > 0"
     for \<epsilon>
   proof-
     obtain N::nat where \<open>1/(N + 1) < \<epsilon>^2/4\<close>
       using LIMSEQ_ignore_initial_segment[OF lim_inverse_n', where k=1]
       by (metis Suc_eq_plus1 \<open>0 < \<epsilon>\<close> nat_approx_posE zero_less_divide_iff zero_less_numeral
           zero_less_power )
     hence \<open>4/(N + 1) < \<epsilon>^2\<close>
       by simp
     have "2*(1/(n+1) + 1/(m+1)) < \<epsilon>^2"
       if f1: "n \<ge> N" and f2: "m \<ge> N"
       for m n::nat
     proof-
       have \<open>1/(n+1) \<le> 1/(N+1)\<close>
         by (simp add: f1 linordered_field_class.frac_le)
       moreover have \<open>1/(m+1) \<le> 1/(N+1)\<close>
         by (simp add: f2 linordered_field_class.frac_le)
       ultimately have  \<open>2*(1/(n+1) + 1/(m+1)) \<le> 4/(N+1)\<close>
         by simp
       thus ?thesis using \<open>4/(N + 1) < \<epsilon>^2\<close>
         by linarith
     qed
     hence "\<parallel> (r n) - (r m) \<parallel>^2 < \<epsilon>^2"
       if y1: "n \<ge> N" and y2: "m \<ge> N"
       for m n::nat
       using that
       by (smt \<open>\<And>n m. \<parallel>r n - r m\<parallel>\<^sup>2 < 2 * (1 / (real n + 1) + 1 / (real m + 1))\<close> of_nat_1 of_nat_add)
     thus ?thesis
       by blast
   qed
   hence  \<open>\<forall> \<epsilon> > 0. \<exists> N::nat. \<forall> n m::nat. n \<ge> N \<and> m \<ge> N \<longrightarrow> \<parallel> (r n) - (r m) \<parallel>^2 < \<epsilon>^2\<close>
     by blast
   hence  \<open>\<forall> \<epsilon> > 0. \<exists> N::nat. \<forall> n m::nat. n \<ge> N \<and> m \<ge> N \<longrightarrow> \<parallel> (r n) - (r m) \<parallel> < \<epsilon>\<close>
     by (meson less_eq_real_def power_less_imp_less_base)
   hence \<open>Cauchy r\<close>
     using CauchyI by fastforce
   then obtain k where \<open>r \<longlonglongrightarrow> k\<close>
     using  convergent_eq_Cauchy by auto
   have \<open>k \<in> M\<close> using \<open>closed M\<close>
     using \<open>\<forall>n. r n \<in> M\<close> \<open>r \<longlonglongrightarrow> k\<close> closed_sequentially by auto
   have  \<open>(\<lambda> n.  \<parallel> r n \<parallel>^2) \<longlonglongrightarrow>  \<parallel> k \<parallel>^2\<close>
     by (simp add: \<open>r \<longlonglongrightarrow> k\<close> tendsto_norm tendsto_power)
   moreover  have  \<open>(\<lambda> n.  \<parallel> r n \<parallel>^2) \<longlonglongrightarrow>  d\<close>
   proof-
     have \<open>\<bar>\<parallel> r n \<parallel>^2 - d\<bar> < 1/(n+1)\<close> for n :: nat
       using \<open>\<And>x. x \<in> M \<Longrightarrow> d \<le> \<parallel>x\<parallel>\<^sup>2\<close> \<open>\<forall>n. r n \<in> M \<and> \<parallel>r n\<parallel>\<^sup>2 < d + 1 / (real n + 1)\<close> of_nat_1 of_nat_add
       by smt
     moreover have \<open>(\<lambda>n. 1 / real (n + 1)) \<longlonglongrightarrow> 0\<close>
       using  LIMSEQ_ignore_initial_segment[OF lim_inverse_n', where k=1] by blast
     ultimately have \<open>(\<lambda> n. \<bar>\<parallel> r n \<parallel>^2 - d\<bar> ) \<longlonglongrightarrow> 0\<close>
       by (simp add: LIMSEQ_norm_0)
     hence \<open>(\<lambda> n. \<parallel> r n \<parallel>^2 - d ) \<longlonglongrightarrow> 0\<close>
       by (simp add: tendsto_rabs_zero_iff)
     moreover have \<open>(\<lambda> n. d ) \<longlonglongrightarrow> d\<close>
       by simp
     ultimately have \<open>(\<lambda> n. (\<parallel> r n \<parallel>^2 - d)+d ) \<longlonglongrightarrow> 0+d\<close>
       using tendsto_add by fastforce
     thus ?thesis by simp
   qed
   ultimately have \<open>d = \<parallel> k \<parallel>^2\<close>
     using LIMSEQ_unique by auto
   hence \<open>t \<in> M \<Longrightarrow> \<parallel> k \<parallel>^2 \<le> \<parallel> t \<parallel>^2\<close> for t
     using \<open>\<And>x. x \<in> M \<Longrightarrow> d \<le> \<parallel>x\<parallel>\<^sup>2\<close> by auto
   hence q1: \<open>\<exists> k. is_arg_min (\<lambda> x. \<parallel>x\<parallel>^2) (\<lambda> t. t \<in> M) k\<close>
     using \<open>k \<in> M\<close>
       is_arg_min_def \<open>d = \<parallel>k\<parallel>\<^sup>2\<close>
     by smt
   thus \<open>\<exists> k. is_arg_min (\<lambda> x. \<parallel>x\<parallel>) (\<lambda> t. t \<in> M) k\<close>
     by (smt is_arg_min_def norm_ge_zero power2_eq_square power2_le_imp_le)
 qed
 
 
 lemma smallest_norm_unique:
   \<comment> \<open>Theorem 2.5 in @{cite conway2013course} (inside the proof)\<close>
   includes notation_norm
   fixes M :: \<open>'a::complex_inner set\<close>
   assumes q1: \<open>convex M\<close>
   assumes r: \<open>is_arg_min (\<lambda> x. \<parallel>x\<parallel>) (\<lambda> t. t \<in> M) r\<close>
   assumes s: \<open>is_arg_min (\<lambda> x. \<parallel>x\<parallel>) (\<lambda> t. t \<in> M) s\<close>
   shows \<open>r = s\<close>
 proof -
   have \<open>r \<in> M\<close>
     using \<open>is_arg_min (\<lambda>x. \<parallel>x\<parallel>) (\<lambda> t. t \<in> M) r\<close>
     by (simp add: is_arg_min_def)
   moreover have \<open>s \<in> M\<close>
     using \<open>is_arg_min (\<lambda>x. \<parallel>x\<parallel>) (\<lambda> t. t \<in> M) s\<close>
     by (simp add: is_arg_min_def)
   ultimately have \<open>((1/2) *\<^sub>R r + (1/2) *\<^sub>R s) \<in> M\<close> using \<open>convex M\<close>
     by (simp add: convexD)
   hence \<open>\<parallel>r\<parallel> \<le> \<parallel> (1/2) *\<^sub>R r + (1/2) *\<^sub>R s \<parallel>\<close>
     by (metis is_arg_min_linorder r)
   hence u2: \<open>\<parallel>r\<parallel>^2 \<le> \<parallel> (1/2) *\<^sub>R r + (1/2) *\<^sub>R s \<parallel>^2\<close>
     using norm_ge_zero power_mono by blast
 
   have \<open>\<parallel>r\<parallel> \<le> \<parallel>s\<parallel>\<close>
     using r s is_arg_min_def
     by (metis is_arg_min_linorder)
   moreover have \<open>\<parallel>s\<parallel> \<le> \<parallel>r\<parallel>\<close>
     using r s is_arg_min_def
     by (metis is_arg_min_linorder)
   ultimately have u3: \<open>\<parallel>r\<parallel> = \<parallel>s\<parallel>\<close> by simp
 
   have \<open>\<parallel> (1/2) *\<^sub>R r - (1/2) *\<^sub>R s \<parallel>^2 \<le> 0\<close>
     using u2 u3 parallelogram_law
     by (smt (verit, ccfv_SIG) polar_identity_minus power2_norm_eq_cinner' scaleR_add_right scaleR_half_double)
   hence \<open>\<parallel> (1/2) *\<^sub>R r - (1/2) *\<^sub>R s \<parallel>^2 = 0\<close>
     by simp
   hence \<open>\<parallel> (1/2) *\<^sub>R r - (1/2) *\<^sub>R s \<parallel> = 0\<close>
     by auto
   hence \<open>(1/2) *\<^sub>R r - (1/2) *\<^sub>R s = 0\<close>
     using norm_eq_zero by blast
   thus ?thesis by simp
 qed
 
 theorem smallest_dist_exists:
   \<comment> \<open>Theorem 2.5 in @{cite conway2013course}\<close>
   fixes M::\<open>'a::chilbert_space set\<close> and h
   assumes a1: \<open>convex M\<close> and a2: \<open>closed M\<close> and a3: \<open>M \<noteq> {}\<close>
   shows  \<open>\<exists>k. is_arg_min (\<lambda> x. dist x h) (\<lambda> x. x \<in> M) k\<close>
-proof-
+proof -
   have *: "is_arg_min (\<lambda>x. dist x h) (\<lambda>x. x\<in>M) (k+h) \<longleftrightarrow> is_arg_min (\<lambda>x. norm x) (\<lambda>x. x\<in>(\<lambda>x. x-h) ` M) k" for k
     unfolding dist_norm is_arg_min_def apply auto using add_implies_diff by blast
   have \<open>\<exists>k. is_arg_min (\<lambda>x. dist x h) (\<lambda>x. x\<in>M) (k+h)\<close>
     apply (subst *)
     apply (rule smallest_norm_exists)
     using assms by (auto simp: closed_translation_subtract)
   then show \<open>\<exists>k. is_arg_min (\<lambda> x. dist x h) (\<lambda> x. x \<in> M) k\<close>
     by metis
 qed
 
 theorem smallest_dist_unique:
   \<comment> \<open>Theorem 2.5 in @{cite conway2013course}\<close>
   fixes M::\<open>'a::complex_inner set\<close> and h
   assumes a1: \<open>convex M\<close>
   assumes \<open>is_arg_min (\<lambda> x. dist x h) (\<lambda> x. x \<in> M) r\<close>
   assumes \<open>is_arg_min (\<lambda> x. dist x h) (\<lambda> x. x \<in> M) s\<close>
   shows  \<open>r = s\<close>
 proof-
   have *: "is_arg_min (\<lambda>x. dist x h) (\<lambda>x. x\<in>M) k \<longleftrightarrow> is_arg_min (\<lambda>x. norm x) (\<lambda>x. x\<in>(\<lambda>x. x-h) ` M) (k-h)" for k
     unfolding dist_norm is_arg_min_def by auto
   have \<open>r - h = s - h\<close>
     using _ assms(2,3)[unfolded *] apply (rule smallest_norm_unique)
     by (simp add: a1)
   thus \<open>r = s\<close>
     by auto
 qed
 
 
 \<comment> \<open>Theorem 2.6 in @{cite conway2013course}\<close>
 theorem smallest_dist_is_ortho:
   fixes M::\<open>'a::complex_inner set\<close> and h k::'a
   assumes b1: \<open>closed_csubspace M\<close>
   shows  \<open>(is_arg_min (\<lambda> x. dist x h) (\<lambda> x. x \<in> M) k) \<longleftrightarrow>
-          h - k \<in> (orthogonal_complement M) \<and> k \<in> M\<close>
-proof-
+          h - k \<in> orthogonal_complement M \<and> k \<in> M\<close>
+proof -
   include notation_norm
   have  \<open>csubspace M\<close>
     using \<open>closed_csubspace M\<close> unfolding closed_csubspace_def by blast
-  have r1: \<open>2 * Re (\<langle> h - k , f \<rangle>) \<le> \<parallel> f \<parallel>^2\<close>
+  have r1: \<open>2 * Re ((h - k) \<bullet>\<^sub>C f) \<le> \<parallel> f \<parallel>^2\<close>
     if "f \<in> M" and \<open>k \<in> M\<close> and \<open>is_arg_min (\<lambda>x. dist x h) (\<lambda> x. x \<in> M) k\<close>
     for f
   proof-
     have \<open>k + f \<in>  M\<close>
       using \<open>csubspace M\<close>
       by (simp add:complex_vector.subspace_add that)
     have "\<forall>f A a b. \<not> is_arg_min f (\<lambda> x. x \<in> A) (a::'a) \<or> (f a::real) \<le> f b \<or> b \<notin> A"
       by (metis (no_types) is_arg_min_linorder)
     hence "dist k h \<le> dist (f + k) h"
       by (metis \<open>is_arg_min (\<lambda>x. dist x h) (\<lambda> x. x \<in> M) k\<close> \<open>k + f \<in> M\<close> add.commute)
     hence \<open>dist h k \<le> dist  h (k + f)\<close>
       by (simp add: add.commute dist_commute)
     hence \<open>\<parallel> h - k \<parallel> \<le> \<parallel> h - (k + f) \<parallel>\<close>
       by (simp add: dist_norm)
     hence \<open>\<parallel> h - k \<parallel>^2 \<le> \<parallel> h - (k + f) \<parallel>^2\<close>
       by (simp add: power_mono)
     also have \<open>... \<le> \<parallel> (h - k) - f \<parallel>^2\<close>
       by (simp add: diff_diff_add)
-    also have \<open>... \<le> \<parallel> (h - k) \<parallel>^2 + \<parallel> f \<parallel>^2 -  2 * Re (\<langle> h - k , f \<rangle>)\<close>
+    also have \<open>... \<le> \<parallel> (h - k) \<parallel>^2 + \<parallel> f \<parallel>^2 -  2 * Re ((h - k) \<bullet>\<^sub>C f)\<close>
       by (simp add: polar_identity_minus)
-    finally have \<open>\<parallel> (h - k) \<parallel>^2 \<le> \<parallel> (h - k) \<parallel>^2 + \<parallel> f \<parallel>^2 -  2 * Re (\<langle> h - k , f \<rangle>)\<close>
+    finally have \<open>\<parallel> (h - k) \<parallel>^2 \<le> \<parallel> (h - k) \<parallel>^2 + \<parallel> f \<parallel>^2 -  2 * Re ((h - k) \<bullet>\<^sub>C f)\<close>
       by simp
     thus ?thesis by simp
   qed
 
-  have q4: \<open>\<forall> c > 0.  2 * Re (\<langle> h - k , f \<rangle>) \<le> c\<close>
-    if  \<open>\<forall>c>0. 2 * Re (\<langle>h - k , f\<rangle> ) \<le> c * \<parallel>f\<parallel>\<^sup>2\<close>
+  have q4: \<open>\<forall> c > 0.  2 * Re ((h - k) \<bullet>\<^sub>C f) \<le> c\<close>
+    if  \<open>\<forall>c>0. 2 * Re ((h - k) \<bullet>\<^sub>C f) \<le> c * \<parallel>f\<parallel>\<^sup>2\<close>
     for f
   proof (cases \<open>\<parallel> f \<parallel>^2 > 0\<close>)
     case True
-    hence \<open>\<forall> c > 0.  2 * Re (\<langle> h - k , f \<rangle>) \<le> (c/\<parallel> f \<parallel>^2)*\<parallel> f \<parallel>^2\<close>
+    hence \<open>\<forall> c > 0.  2 * Re (((h - k) \<bullet>\<^sub>C f)) \<le> (c/\<parallel> f \<parallel>^2)*\<parallel> f \<parallel>^2\<close>
       using that linordered_field_class.divide_pos_pos by blast
     thus ?thesis
       using True by auto
   next
     case False
     hence \<open>\<parallel> f \<parallel>^2 = 0\<close>
       by simp
     thus ?thesis
       by auto
   qed
-  have q3: \<open>\<forall> c::real. c > 0 \<longrightarrow> 2 * Re (\<langle> h - k , f \<rangle>) \<le> 0\<close>
-    if a3: \<open>\<forall>f. f \<in> M \<longrightarrow> (\<forall>c>0. 2 * Re \<langle>h - k , f\<rangle> \<le> c * \<parallel>f\<parallel>\<^sup>2)\<close>
+  have q3: \<open>\<forall> c::real. c > 0 \<longrightarrow> 2 * Re (((h - k) \<bullet>\<^sub>C f)) \<le> 0\<close>
+    if a3: \<open>\<forall>f. f \<in> M \<longrightarrow> (\<forall>c>0. 2 * Re ((h - k) \<bullet>\<^sub>C f) \<le> c * \<parallel>f\<parallel>\<^sup>2)\<close>
       and a2: "f \<in>  M"
       and a1: "is_arg_min (\<lambda> x. dist x h) (\<lambda> x. x \<in> M) k"
     for f
   proof-
-    have \<open>\<forall> c > 0.  2 * Re (\<langle> h - k , f \<rangle>) \<le> c*\<parallel> f \<parallel>^2\<close>
+    have \<open>\<forall> c > 0.  2 * Re (((h - k) \<bullet>\<^sub>C f)) \<le> c*\<parallel> f \<parallel>^2\<close>
       by (simp add: that )
     thus ?thesis
       using q4 by smt
   qed
   have w2: "h - k \<in> orthogonal_complement M \<and> k \<in> M"
     if a1: "is_arg_min (\<lambda> x. dist x h) (\<lambda> x. x \<in> M) k"
   proof-
     have  \<open>k \<in> M\<close>
       using is_arg_min_def that by fastforce
-    hence \<open>\<forall> f. f \<in>  M \<longrightarrow> 2 * Re (\<langle> h - k , f \<rangle>) \<le> \<parallel> f \<parallel>^2\<close>
+    hence \<open>\<forall> f. f \<in>  M \<longrightarrow> 2 * Re (((h - k) \<bullet>\<^sub>C f)) \<le> \<parallel> f \<parallel>^2\<close>
       using r1
       by (simp add: that)
     have \<open>\<forall> f. f \<in>  M \<longrightarrow>
-                (\<forall> c::real.  2 * Re (\<langle> h - k , c *\<^sub>R f \<rangle>) \<le> \<parallel> c *\<^sub>R f \<parallel>^2)\<close>
+                (\<forall> c::real.  2 * Re ((h - k) \<bullet>\<^sub>C (c *\<^sub>R f)) \<le> \<parallel> c *\<^sub>R f \<parallel>^2)\<close>
       using  assms scaleR_scaleC complex_vector.subspace_def \<open>csubspace M\<close>
-      by (metis \<open>\<forall>f. f \<in> M \<longrightarrow> 2 * Re \<langle>h - k, f\<rangle> \<le> \<parallel>f\<parallel>\<^sup>2\<close>)
+      by (metis \<open>\<forall>f. f \<in> M \<longrightarrow> 2 * Re ((h - k) \<bullet>\<^sub>C f) \<le> \<parallel>f\<parallel>\<^sup>2\<close>)
     hence  \<open>\<forall> f. f \<in>  M \<longrightarrow>
-                (\<forall> c::real. c * (2 * Re (\<langle> h - k , f \<rangle>)) \<le> \<parallel> c *\<^sub>R f \<parallel>^2)\<close>
+                (\<forall> c::real. c * (2 * Re (((h - k) \<bullet>\<^sub>C f))) \<le> \<parallel> c *\<^sub>R f \<parallel>^2)\<close>
       by (metis Re_complex_of_real cinner_scaleC_right complex_add_cnj complex_cnj_complex_of_real
           complex_cnj_mult of_real_mult scaleR_scaleC semiring_normalization_rules(34))
     hence  \<open>\<forall> f. f \<in>  M \<longrightarrow>
-                (\<forall> c::real. c * (2 * Re (\<langle> h - k , f \<rangle>)) \<le> \<bar>c\<bar>^2*\<parallel> f \<parallel>^2)\<close>
+                (\<forall> c::real. c * (2 * Re (((h - k) \<bullet>\<^sub>C f))) \<le> \<bar>c\<bar>^2*\<parallel> f \<parallel>^2)\<close>
       by (simp add: power_mult_distrib)
     hence  \<open>\<forall> f. f \<in>  M \<longrightarrow>
-                (\<forall> c::real. c * (2 * Re (\<langle> h - k , f \<rangle>)) \<le> c^2*\<parallel> f \<parallel>^2)\<close>
+                (\<forall> c::real. c * (2 * Re (((h - k) \<bullet>\<^sub>C f))) \<le> c^2*\<parallel> f \<parallel>^2)\<close>
       by auto
     hence  \<open>\<forall> f. f \<in>  M \<longrightarrow>
-                (\<forall> c::real. c > 0 \<longrightarrow> c * (2 * Re (\<langle> h - k , f \<rangle>)) \<le> c^2*\<parallel> f \<parallel>^2)\<close>
+                (\<forall> c::real. c > 0 \<longrightarrow> c * (2 * Re (((h - k) \<bullet>\<^sub>C f))) \<le> c^2*\<parallel> f \<parallel>^2)\<close>
       by simp
     hence  \<open>\<forall> f. f \<in>  M \<longrightarrow>
-                (\<forall> c::real. c > 0 \<longrightarrow> c*(2 * Re (\<langle> h - k , f \<rangle>)) \<le> c*(c*\<parallel> f \<parallel>^2))\<close>
+                (\<forall> c::real. c > 0 \<longrightarrow> c*(2 * Re (((h - k) \<bullet>\<^sub>C f))) \<le> c*(c*\<parallel> f \<parallel>^2))\<close>
       by (simp add: power2_eq_square)
     hence  q4: \<open>\<forall> f. f \<in>  M \<longrightarrow>
-                (\<forall> c::real. c > 0 \<longrightarrow> 2 * Re (\<langle> h - k , f \<rangle>) \<le> c*\<parallel> f \<parallel>^2)\<close>
+                (\<forall> c::real. c > 0 \<longrightarrow> 2 * Re (((h - k) \<bullet>\<^sub>C f)) \<le> c*\<parallel> f \<parallel>^2)\<close>
       by simp
     have  \<open>\<forall> f. f \<in>  M \<longrightarrow>
-                (\<forall> c::real. c > 0 \<longrightarrow> 2 * Re (\<langle> h - k , f \<rangle>) \<le> 0)\<close>
+                (\<forall> c::real. c > 0 \<longrightarrow> 2 * Re (((h - k) \<bullet>\<^sub>C f)) \<le> 0)\<close>
       using q3
       by (simp add: q4 that)
     hence  \<open>\<forall> f. f \<in>  M \<longrightarrow>
-                (\<forall> c::real. c > 0 \<longrightarrow> (2 * Re (\<langle> h - k , (-1) *\<^sub>R f \<rangle>)) \<le> 0)\<close>
+                (\<forall> c::real. c > 0 \<longrightarrow> (2 * Re ((h - k) \<bullet>\<^sub>C (-1 *\<^sub>R f))) \<le> 0)\<close>
       using assms scaleR_scaleC complex_vector.subspace_def
       by (metis \<open>csubspace M\<close>)
     hence  \<open>\<forall> f. f \<in>  M \<longrightarrow>
-                (\<forall> c::real. c > 0 \<longrightarrow> -(2 * Re (\<langle> h - k , f \<rangle>)) \<le> 0)\<close>
+                (\<forall> c::real. c > 0 \<longrightarrow> -(2 * Re (((h - k) \<bullet>\<^sub>C f))) \<le> 0)\<close>
       by simp
     hence  \<open>\<forall> f. f \<in>  M \<longrightarrow>
-                (\<forall> c::real. c > 0 \<longrightarrow> 2 * Re (\<langle> h - k , f \<rangle>) \<ge> 0)\<close>
+                (\<forall> c::real. c > 0 \<longrightarrow> 2 * Re (((h - k) \<bullet>\<^sub>C f)) \<ge> 0)\<close>
       by simp
     hence \<open>\<forall> f. f \<in>  M \<longrightarrow>
-                (\<forall> c::real. c > 0 \<longrightarrow> 2 * Re (\<langle> h - k , f \<rangle>) = 0)\<close>
+                (\<forall> c::real. c > 0 \<longrightarrow> 2 * Re (((h - k) \<bullet>\<^sub>C f)) = 0)\<close>
       using  \<open>\<forall> f. f \<in>  M \<longrightarrow>
-                (\<forall> c::real. c > 0 \<longrightarrow> (2 * Re (\<langle> h - k , f \<rangle>)) \<le> 0)\<close>
+                (\<forall> c::real. c > 0 \<longrightarrow> (2 * Re (((h - k) \<bullet>\<^sub>C f))) \<le> 0)\<close>
       by fastforce
 
     have \<open>\<forall> f. f \<in>  M \<longrightarrow>
-                 ((1::real) > 0 \<longrightarrow> 2 * Re (\<langle> h - k , f \<rangle>) = 0)\<close>
-      using \<open>\<forall>f. f \<in>  M \<longrightarrow> (\<forall>c>0. 2 * Re (\<langle>h - k , f\<rangle> ) = 0)\<close> by blast
-    hence \<open>\<forall> f. f \<in>  M \<longrightarrow> 2 * Re (\<langle> h - k , f \<rangle>) = 0\<close>
-      by simp
-    hence \<open>\<forall> f. f \<in>  M \<longrightarrow> Re (\<langle> h - k , f \<rangle>) = 0\<close>
+                 ((1::real) > 0 \<longrightarrow> 2 * Re (((h - k) \<bullet>\<^sub>C f)) = 0)\<close>
+      using \<open>\<forall>f. f \<in>  M \<longrightarrow> (\<forall>c>0. 2 * Re (((h - k) \<bullet>\<^sub>C f) ) = 0)\<close> by blast
+    hence \<open>\<forall> f. f \<in>  M \<longrightarrow> 2 * Re (((h - k) \<bullet>\<^sub>C f)) = 0\<close>
       by simp
-    have  \<open>\<forall> f. f \<in>  M \<longrightarrow> Re (\<langle> h - k , (Complex 0 (-1)) *\<^sub>C f \<rangle>) = 0\<close>
+    hence \<open>\<forall> f. f \<in>  M \<longrightarrow> Re (((h - k) \<bullet>\<^sub>C f)) = 0\<close>
+      by simp
+    have  \<open>\<forall> f. f \<in>  M \<longrightarrow> Re ((h - k) \<bullet>\<^sub>C ((Complex 0 (-1)) *\<^sub>C f)) = 0\<close>
       using assms  complex_vector.subspace_def \<open>csubspace M\<close>
-      by (metis \<open>\<forall>f. f \<in> M \<longrightarrow> Re \<langle>h - k, f\<rangle> = 0\<close>)
-    hence  \<open>\<forall> f. f \<in>  M \<longrightarrow> Re ( (Complex 0 (-1))*(\<langle> h - k , f \<rangle>) ) = 0\<close>
+      by (metis \<open>\<forall>f. f \<in> M \<longrightarrow> Re ((h - k) \<bullet>\<^sub>C f) = 0\<close>)
+    hence  \<open>\<forall> f. f \<in>  M \<longrightarrow> Re ( (Complex 0 (-1))*(((h - k) \<bullet>\<^sub>C f)) ) = 0\<close>
       by simp
-    hence \<open>\<forall> f. f \<in>  M \<longrightarrow> Im (\<langle> h - k , f \<rangle>) = 0\<close>
+    hence \<open>\<forall> f. f \<in>  M \<longrightarrow> Im (((h - k) \<bullet>\<^sub>C f)) = 0\<close>
       using Complex_eq_neg_1 Re_i_times cinner_scaleC_right complex_of_real_def by auto
 
-    have \<open>\<forall> f. f \<in>  M \<longrightarrow> (\<langle> h - k , f \<rangle>) = 0\<close>
+    have \<open>\<forall> f. f \<in>  M \<longrightarrow> (((h - k) \<bullet>\<^sub>C f)) = 0\<close>
       using complex_eq_iff
-      by (simp add: \<open>\<forall>f. f \<in> M \<longrightarrow> Im \<langle>h - k, f\<rangle> = 0\<close> \<open>\<forall>f. f \<in> M \<longrightarrow> Re \<langle>h - k, f\<rangle> = 0\<close>)
+      by (simp add: \<open>\<forall>f. f \<in> M \<longrightarrow> Im ((h - k) \<bullet>\<^sub>C f) = 0\<close> \<open>\<forall>f. f \<in> M \<longrightarrow> Re ((h - k) \<bullet>\<^sub>C f) = 0\<close>)
     hence \<open>h - k \<in> orthogonal_complement M \<and> k \<in> M\<close>
       by (simp add: \<open>k \<in> M\<close> orthogonal_complementI)
     have  \<open>\<forall> c. c *\<^sub>R f \<in> M\<close>
       if "f \<in> M"
       for f
       using that scaleR_scaleC  \<open>csubspace M\<close> complex_vector.subspace_def
       by (simp add: complex_vector.subspace_def scaleR_scaleC)
-    have \<open>\<langle> h - k , f \<rangle> = 0\<close>
+    have \<open>((h - k) \<bullet>\<^sub>C f) = 0\<close>
       if "f \<in> M"
       for f
       using \<open>h - k \<in> orthogonal_complement M \<and> k \<in> M\<close> orthogonal_complement_orthoI that by auto
     hence \<open>h - k \<in> orthogonal_complement M\<close>
       by (simp add: orthogonal_complement_def)
     thus ?thesis
       using \<open>k \<in> M\<close> by auto
   qed
 
   have q1: \<open>dist h k \<le> dist h f \<close>
     if "f \<in> M" and  \<open>h - k \<in> orthogonal_complement M \<and> k \<in> M\<close>
     for f
   proof-
-    have \<open>\<langle> h - k,  k - f \<rangle> = 0\<close>
+    have \<open>(h - k) \<bullet>\<^sub>C (k - f) = 0\<close>
       by (metis (no_types, lifting) that
           cinner_diff_right diff_0_right orthogonal_complement_orthoI that)
     have \<open>\<parallel> h - f \<parallel>^2 = \<parallel> (h - k) + (k - f) \<parallel>^2\<close>
       by simp
     also have \<open>... = \<parallel> h - k \<parallel>^2 + \<parallel> k - f \<parallel>^2\<close>
-      using  \<open>\<langle> h - k, k - f \<rangle> = 0\<close> pythagorean_theorem by blast
+      using  \<open>((h - k) \<bullet>\<^sub>C (k - f)) = 0\<close> pythagorean_theorem by blast
     also have \<open>... \<ge> \<parallel> h - k \<parallel>^2\<close>
       by simp
     finally have \<open>\<parallel>h - k\<parallel>\<^sup>2 \<le> \<parallel>h - f\<parallel>\<^sup>2 \<close>
       by blast
     hence \<open>\<parallel>h - k\<parallel> \<le> \<parallel>h - f\<parallel>\<close>
       using norm_ge_zero power2_le_imp_le by blast
     thus ?thesis
       by (simp add: dist_norm)
   qed
 
   have  w1: "is_arg_min (\<lambda> x. dist x h) (\<lambda> x. x \<in> M) k"
     if "h - k \<in> orthogonal_complement M \<and> k \<in>  M"
   proof-
     have \<open>h - k \<in> orthogonal_complement M\<close>
       using that by blast
     have \<open>k \<in> M\<close> using \<open>h - k \<in> orthogonal_complement M \<and> k \<in>  M\<close>
       by blast
     thus ?thesis
       by (metis (no_types, lifting) dist_commute is_arg_min_linorder q1 that)
   qed
   show ?thesis
     using w1 w2 by blast
 qed
 
 corollary orthog_proj_exists:
   fixes M :: \<open>'a::chilbert_space set\<close>
   assumes \<open>closed_csubspace M\<close>
   shows  \<open>\<exists>k. h - k \<in> orthogonal_complement M \<and> k \<in> M\<close>
-proof-
+proof -
   from  \<open>closed_csubspace M\<close>
   have \<open>M \<noteq> {}\<close>
     using closed_csubspace.subspace complex_vector.subspace_0 by blast
   have \<open>closed  M\<close>
     using  \<open>closed_csubspace M\<close>
     by (simp add: closed_csubspace.closed)
   have \<open>convex  M\<close>
     using  \<open>closed_csubspace M\<close>
     by (simp)
   have \<open>\<exists>k.  is_arg_min (\<lambda> x. dist x h) (\<lambda> x. x \<in> M) k\<close>
     by (simp add: smallest_dist_exists \<open>closed M\<close> \<open>convex M\<close> \<open>M \<noteq> {}\<close>)
   thus ?thesis
     by (simp add: assms smallest_dist_is_ortho)
 qed
 
 corollary orthog_proj_unique:
   fixes M :: \<open>'a::complex_inner set\<close>
   assumes \<open>closed_csubspace M\<close>
   assumes \<open>h - r \<in> orthogonal_complement M \<and> r \<in> M\<close>
   assumes \<open>h - s \<in> orthogonal_complement M \<and> s \<in> M\<close>
   shows  \<open>r = s\<close>
   using _ assms(2,3) unfolding smallest_dist_is_ortho[OF assms(1), symmetric]
   apply (rule smallest_dist_unique)
   using assms(1) by (simp)
 
 definition is_projection_on::\<open>('a \<Rightarrow> 'a) \<Rightarrow> ('a::metric_space) set \<Rightarrow> bool\<close> where
   \<open>is_projection_on \<pi> M \<longleftrightarrow> (\<forall>h. is_arg_min (\<lambda> x. dist x h) (\<lambda> x. x \<in> M) (\<pi> h))\<close>
 
 lemma is_projection_on_iff_orthog:
   \<open>closed_csubspace M \<Longrightarrow> is_projection_on \<pi> M \<longleftrightarrow> (\<forall>h. h - \<pi> h \<in> orthogonal_complement M \<and> \<pi> h \<in> M)\<close>
   by (simp add: is_projection_on_def smallest_dist_is_ortho)
 
 lemma is_projection_on_exists:
   fixes M :: \<open>'a::chilbert_space set\<close>
   assumes \<open>convex M\<close> and \<open>closed M\<close> and \<open>M \<noteq> {}\<close>
   shows "\<exists>\<pi>. is_projection_on \<pi> M"
   unfolding is_projection_on_def apply (rule choice)
   using smallest_dist_exists[OF assms] by auto
 
 lemma is_projection_on_unique:
   fixes M :: \<open>'a::complex_inner set\<close>
   assumes \<open>convex M\<close>
   assumes "is_projection_on \<pi>\<^sub>1 M"
   assumes "is_projection_on \<pi>\<^sub>2 M"
   shows "\<pi>\<^sub>1 = \<pi>\<^sub>2"
   using smallest_dist_unique[OF assms(1)] using assms(2,3)
   unfolding is_projection_on_def by blast
 
 definition projection :: \<open>'a::metric_space set \<Rightarrow> ('a \<Rightarrow> 'a)\<close> where
   \<open>projection M \<equiv> SOME \<pi>. is_projection_on \<pi> M\<close>
 
 lemma projection_is_projection_on:
   fixes M :: \<open>'a::chilbert_space set\<close>
   assumes \<open>convex M\<close> and \<open>closed M\<close> and \<open>M \<noteq> {}\<close>
   shows "is_projection_on (projection M) M"
   by (metis assms(1) assms(2) assms(3) is_projection_on_exists projection_def someI)
 
 lemma projection_is_projection_on'[simp]:
   \<comment> \<open>Common special case of @{thm projection_is_projection_on}\<close>
   fixes M :: \<open>'a::chilbert_space set\<close>
   assumes \<open>closed_csubspace M\<close>
   shows "is_projection_on (projection M) M"
   apply (rule projection_is_projection_on)
     apply (auto simp add: assms closed_csubspace.closed)
   using assms closed_csubspace.subspace complex_vector.subspace_0 by blast
 
 lemma projection_orthogonal:
   fixes M :: \<open>'a::chilbert_space set\<close>
   assumes "closed_csubspace M" and \<open>m \<in> M\<close>
   shows \<open>is_orthogonal (h - projection M h) m\<close>
   by (metis assms(1) assms(2) closed_csubspace.closed closed_csubspace.subspace csubspace_is_convex empty_iff is_projection_on_iff_orthog orthogonal_complement_orthoI projection_is_projection_on)
 
 lemma is_projection_on_in_image:
   assumes "is_projection_on \<pi> M"
   shows "\<pi> h \<in> M"
   using assms
   by (simp add: is_arg_min_def is_projection_on_def)
 
 lemma is_projection_on_image:
   assumes "is_projection_on \<pi> M"
   shows "range \<pi> = M"
   using assms
   apply (auto simp: is_projection_on_in_image)
   by (smt (verit, ccfv_threshold) dist_pos_lt dist_self is_arg_min_def is_projection_on_def rangeI)
 
 lemma projection_in_image[simp]:
   fixes M :: \<open>'a::chilbert_space set\<close>
   assumes \<open>convex M\<close> and \<open>closed M\<close> and \<open>M \<noteq> {}\<close>
   shows \<open>projection M h \<in> M\<close>
   by (simp add: assms(1) assms(2) assms(3) is_projection_on_in_image projection_is_projection_on)
 
 lemma projection_image[simp]:
   fixes M :: \<open>'a::chilbert_space set\<close>
   assumes \<open>convex M\<close> and \<open>closed M\<close> and \<open>M \<noteq> {}\<close>
   shows \<open>range (projection M) = M\<close>
   by (simp add: assms(1) assms(2) assms(3) is_projection_on_image projection_is_projection_on)
 
 lemma projection_eqI':
   fixes M :: \<open>'a::complex_inner set\<close>
   assumes \<open>convex M\<close>
   assumes \<open>is_projection_on f M\<close>
   shows \<open>projection M = f\<close>
   by (metis assms(1) assms(2) is_projection_on_unique projection_def someI_ex)
 
 lemma is_projection_on_eqI:
   fixes  M :: \<open>'a::complex_inner set\<close>
   assumes a1: \<open>closed_csubspace M\<close> and a2: \<open>h - x \<in> orthogonal_complement M\<close> and a3: \<open>x \<in> M\<close>
     and a4: \<open>is_projection_on \<pi> M\<close>
   shows \<open>\<pi> h = x\<close>
   by (meson a1 a2 a3 a4 closed_csubspace.subspace csubspace_is_convex is_projection_on_def smallest_dist_is_ortho smallest_dist_unique)
 
 lemma projection_eqI:
   fixes  M :: \<open>('a::chilbert_space) set\<close>
   assumes  \<open>closed_csubspace M\<close> and \<open>h - x \<in> orthogonal_complement M\<close> and \<open>x \<in> M\<close>
   shows \<open>projection M h = x\<close>
   by (metis assms(1) assms(2) assms(3) is_projection_on_iff_orthog orthog_proj_exists projection_def is_projection_on_eqI tfl_some)
 
 lemma is_projection_on_fixes_image:
   fixes M :: \<open>'a::metric_space set\<close>
   assumes a1: "is_projection_on \<pi> M" and a3: "x \<in> M"
   shows "\<pi> x = x"
   by (metis a1 a3 dist_pos_lt dist_self is_arg_min_def is_projection_on_def)
 
 lemma projection_fixes_image:
   fixes M :: \<open>('a::chilbert_space) set\<close>
-  assumes a1: "closed_csubspace M" and a2: "x \<in> M"
-  shows "(projection M) x = x"
+  assumes "closed_csubspace M" and "x \<in> M"
+  shows "projection M x = x"
   using is_projection_on_fixes_image
     \<comment> \<open>Theorem 2.7 in @{cite conway2013course}\<close>
-  by (simp add: a1 a2 complex_vector.subspace_0 projection_eqI)
+  by (simp add: assms complex_vector.subspace_0 projection_eqI)
+
+lemma is_projection_on_closed:
+  assumes cont_f: \<open>\<And>x. x \<in> closure M \<Longrightarrow> isCont f x\<close>
+  assumes \<open>is_projection_on f M\<close>
+  shows \<open>closed M\<close>
+proof -
+  have \<open>x \<in> M\<close> if \<open>s \<longlonglongrightarrow> x\<close> and \<open>range s \<subseteq> M\<close> for s x
+  proof -
+    from \<open>is_projection_on f M\<close> \<open>range s \<subseteq> M\<close>
+    have \<open>s = (f o s)\<close>
+      by (simp add: comp_def is_projection_on_fixes_image range_subsetD)
+    also from cont_f \<open>s \<longlonglongrightarrow> x\<close> 
+    have \<open>(f o s) \<longlonglongrightarrow> f x\<close>
+      apply (rule continuous_imp_tendsto)
+      using \<open>s \<longlonglongrightarrow> x\<close> \<open>range s \<subseteq> M\<close>
+      by (meson closure_sequential range_subsetD)
+    finally have \<open>x = f x\<close>
+      using \<open>s \<longlonglongrightarrow> x\<close>
+      by (simp add: LIMSEQ_unique)
+    then have \<open>x \<in> range f\<close>
+      by simp
+    with \<open>is_projection_on f M\<close> show \<open>x \<in> M\<close>
+      by (simp add: is_projection_on_image)
+  qed
+  then show ?thesis
+    by (metis closed_sequential_limits image_subset_iff)
+qed
 
 proposition is_projection_on_reduces_norm:
   includes notation_norm
   fixes M :: \<open>('a::complex_inner) set\<close>
   assumes \<open>is_projection_on \<pi> M\<close> and \<open>closed_csubspace M\<close>
   shows \<open>\<parallel> \<pi>  h \<parallel> \<le> \<parallel> h \<parallel>\<close>
 proof-
   have \<open>h - \<pi> h \<in> orthogonal_complement M\<close>
     using assms is_projection_on_iff_orthog by blast
-  hence \<open>\<forall> k \<in> M. \<langle>  h - \<pi> h , k \<rangle> = 0\<close>
+  hence \<open>\<forall> k \<in> M. is_orthogonal (h - \<pi> h) k\<close>
     using orthogonal_complement_orthoI by blast
   also have \<open>\<pi> h \<in>  M\<close>
     using \<open>is_projection_on \<pi> M\<close>
     by (simp add: is_projection_on_in_image)
-  ultimately have \<open>\<langle>  h - \<pi> h , \<pi> h \<rangle> = 0\<close>
+  ultimately have \<open>is_orthogonal (h - \<pi> h) (\<pi> h)\<close>
     by auto
   hence \<open>\<parallel> \<pi> h \<parallel>^2 + \<parallel> h - \<pi> h \<parallel>^2 = \<parallel> h \<parallel>^2\<close>
     using pythagorean_theorem by fastforce
   hence \<open>\<parallel>\<pi> h \<parallel>^2 \<le> \<parallel> h \<parallel>^2\<close>
     by (smt zero_le_power2)
   thus ?thesis
     using norm_ge_zero power2_le_imp_le by blast
 qed
 
 proposition projection_reduces_norm:
   includes notation_norm
   fixes M :: \<open>'a::chilbert_space set\<close>
   assumes a1: "closed_csubspace M"
   shows \<open>\<parallel> projection M h \<parallel> \<le> \<parallel> h \<parallel>\<close>
   using assms is_projection_on_iff_orthog orthog_proj_exists is_projection_on_reduces_norm projection_eqI by blast
 
 \<comment> \<open>Theorem 2.7 (version) in @{cite conway2013course}\<close>
 theorem is_projection_on_bounded_clinear:
   fixes M :: \<open>'a::complex_inner set\<close>
   assumes a1: "is_projection_on \<pi> M" and a2: "closed_csubspace M"
   shows "bounded_clinear \<pi>"
 proof
   have b1:  \<open>csubspace (orthogonal_complement M)\<close>
     by (simp add: a2)
   have f1: "\<forall>a. a - \<pi> a \<in> orthogonal_complement M \<and> \<pi> a \<in> M"
     using a1 a2 is_projection_on_iff_orthog by blast
   hence "c *\<^sub>C x - c *\<^sub>C \<pi> x \<in> orthogonal_complement M"
     for c x
     by (metis (no_types) b1
         add_diff_cancel_right' complex_vector.subspace_def diff_add_cancel scaleC_add_right)
   thus r1: \<open>\<pi> (c *\<^sub>C x) = c *\<^sub>C (\<pi> x)\<close> for x c
     using f1 by (meson a2 a1 closed_csubspace.subspace
         complex_vector.subspace_def is_projection_on_eqI)
   show r2: \<open>\<pi> (x + y) =  (\<pi> x) + (\<pi> y)\<close>
     for x y
   proof-
     have "\<forall>A. \<not> closed_csubspace (A::'a set) \<or> csubspace A"
       by (metis closed_csubspace.subspace)
     hence "csubspace M"
       using a2 by auto
     hence \<open>\<pi> (x + y) - ( (\<pi> x) + (\<pi> y) ) \<in> M\<close>
       by (simp add: complex_vector.subspace_add complex_vector.subspace_diff f1)
     have \<open>closed_csubspace (orthogonal_complement M)\<close>
       using a2
       by simp
     have f1: "\<forall>a b. (b::'a) + (a - b) = a"
       by (metis add.commute diff_add_cancel)
     have f2: "\<forall>a b. (b::'a) - b = a - a"
       by auto
     hence f3: "\<forall>a. a - a \<in> orthogonal_complement M"
       by (simp add: complex_vector.subspace_0)
     have "\<forall>a b. (a \<in> orthogonal_complement M \<or> a + b \<notin> orthogonal_complement M)
              \<or> b \<notin> orthogonal_complement M"
       using add_diff_cancel_right' b1 complex_vector.subspace_diff
       by metis
     hence "\<forall>a b c. (a \<in> orthogonal_complement M \<or> c - (b + a) \<notin> orthogonal_complement M)
               \<or> c - b \<notin> orthogonal_complement M"
       using f1 by (metis diff_diff_add)
     hence f4: "\<forall>a b f. (f a - b \<in> orthogonal_complement M \<or> a - b \<notin> orthogonal_complement M)
               \<or> \<not> is_projection_on f M"
       using f1
       by (metis a2 is_projection_on_iff_orthog)
     have f5: "\<forall>a b c d. (d::'a) - (c + (b - a)) = d + (a - (b + c))"
       by auto
     have "x - \<pi> x \<in> orthogonal_complement M"
       using a1 a2 is_projection_on_iff_orthog by blast
     hence q1: \<open>\<pi> (x + y) - ( (\<pi> x) + (\<pi> y) ) \<in> orthogonal_complement M\<close>
       using f5 f4 f3 by (metis \<open>csubspace (orthogonal_complement M)\<close>
           \<open>is_projection_on \<pi> M\<close> add_diff_eq complex_vector.subspace_diff diff_diff_add
           diff_diff_eq2)
     hence \<open>\<pi> (x + y) - ( (\<pi> x) + (\<pi> y) ) \<in> M \<inter> (orthogonal_complement M)\<close>
       by (simp add: \<open>\<pi> (x + y) - (\<pi> x + \<pi> y) \<in> M\<close>)
     moreover have \<open>M \<inter> (orthogonal_complement M) = {0}\<close>
       by (simp add: \<open>closed_csubspace M\<close> complex_vector.subspace_0 orthogonal_complement_zero_intersection)
     ultimately have \<open>\<pi> (x + y) - ( (\<pi> x) + (\<pi> y) ) = 0\<close>
       by auto
     thus ?thesis by simp
   qed
   from is_projection_on_reduces_norm
   show t1: \<open>\<exists> K. \<forall> x. norm (\<pi> x) \<le> norm x * K\<close>
     by (metis a1 a2 mult.left_neutral ordered_field_class.sign_simps(5))
 qed
 
 theorem projection_bounded_clinear:
   fixes M :: \<open>('a::chilbert_space) set\<close>
   assumes a1: "closed_csubspace M"
   shows \<open>bounded_clinear (projection M)\<close>
     \<comment> \<open>Theorem 2.7 in @{cite conway2013course}\<close>
   using assms is_projection_on_iff_orthog orthog_proj_exists is_projection_on_bounded_clinear projection_eqI by blast
 
 proposition is_projection_on_idem:
   fixes M :: \<open>('a::complex_inner) set\<close>
   assumes "is_projection_on \<pi> M"
   shows "\<pi> (\<pi> x) = \<pi> x"
   using is_projection_on_fixes_image is_projection_on_in_image assms by blast
 
 proposition projection_idem:
   fixes M :: "'a::chilbert_space set"
   assumes a1: "closed_csubspace M"
   shows "projection M (projection M x) = projection M x"
   by (metis assms closed_csubspace.closed closed_csubspace.subspace complex_vector.subspace_0 csubspace_is_convex equals0D projection_fixes_image projection_in_image)
 
 
 proposition is_projection_on_kernel_is_orthogonal_complement:
   fixes M :: \<open>'a::complex_inner set\<close>
   assumes a1: "is_projection_on \<pi> M" and a2: "closed_csubspace M"
   shows "\<pi> -` {0} = orthogonal_complement M"
 proof-
   have "x \<in> (\<pi> -` {0})"
     if "x \<in> orthogonal_complement M"
     for x
     by (smt (verit, ccfv_SIG) a1 a2 closed_csubspace_def complex_vector.subspace_def complex_vector.subspace_diff is_projection_on_eqI orthogonal_complement_closed_subspace that vimage_singleton_eq)
   moreover have "x \<in> orthogonal_complement M"
     if s1: "x \<in> \<pi> -` {0}" for x
     by (metis a1 a2 diff_zero is_projection_on_iff_orthog that vimage_singleton_eq)
   ultimately show ?thesis
     by blast
 qed
 
 \<comment> \<open>Theorem 2.7 in @{cite conway2013course}\<close>
 proposition projection_kernel_is_orthogonal_complement:
   fixes M :: \<open>'a::chilbert_space set\<close>
   assumes "closed_csubspace M"
   shows "(projection M) -` {0} = (orthogonal_complement M)"
   by (metis assms closed_csubspace_def complex_vector.subspace_def csubspace_is_convex insert_absorb insert_not_empty is_projection_on_kernel_is_orthogonal_complement projection_is_projection_on)
 
 lemma is_projection_on_id_minus:
   fixes M :: \<open>'a::complex_inner set\<close>
   assumes is_proj: "is_projection_on \<pi> M"
     and cc: "closed_csubspace M"
   shows "is_projection_on (id - \<pi>) (orthogonal_complement M)"
   using is_proj apply (simp add: cc is_projection_on_iff_orthog)
   using double_orthogonal_complement_increasing by blast
 
 
 text \<open>Exercise 2 (section 2, chapter I) in  @{cite conway2013course}\<close>
 lemma projection_on_orthogonal_complement[simp]:
   fixes M :: "'a::chilbert_space set"
   assumes a1: "closed_csubspace M"
   shows "projection (orthogonal_complement M) = id - projection M"
   apply (auto intro!: ext)
   by (smt (verit, ccfv_SIG) add_diff_cancel_left' assms closed_csubspace.closed closed_csubspace.subspace complex_vector.subspace_0 csubspace_is_convex diff_add_cancel double_orthogonal_complement_increasing insert_absorb insert_not_empty is_projection_on_iff_orthog orthogonal_complement_closed_subspace projection_eqI projection_is_projection_on subset_eq)
 
 lemma is_projection_on_zero:
   "is_projection_on (\<lambda>_. 0) {0}"
   by (simp add: is_projection_on_def is_arg_min_def)
 
 lemma projection_zero[simp]:
   "projection {0} = (\<lambda>_. 0)"
   using is_projection_on_zero
   by (metis (full_types) is_projection_on_in_image projection_def singletonD someI_ex)
 
 lemma is_projection_on_rank1:
   fixes t :: \<open>'a::complex_inner\<close>
-  shows \<open>is_projection_on (\<lambda>x. (\<langle>t , x\<rangle> / \<langle>t , t\<rangle>) *\<^sub>C t) (cspan {t})\<close>
+  shows \<open>is_projection_on (\<lambda>x. ((t \<bullet>\<^sub>C x) / (t \<bullet>\<^sub>C t)) *\<^sub>C t) (cspan {t})\<close>
 proof (cases \<open>t = 0\<close>)
   case True
   then show ?thesis
     by (simp add: is_projection_on_zero)
 next
   case False
-  define P where \<open>P x = (\<langle>t , x\<rangle> / \<langle>t , t\<rangle>) *\<^sub>C t\<close> for x
+  define P where \<open>P x = ((t \<bullet>\<^sub>C x) / (t \<bullet>\<^sub>C t)) *\<^sub>C t\<close> for x
   define t' where \<open>t' = t /\<^sub>C norm t\<close>
   with False have \<open>norm t' = 1\<close>
     by (simp add: norm_inverse)
   have P_def': \<open>P x = cinner t' x *\<^sub>C t'\<close> for x
     unfolding P_def t'_def apply auto
     by (metis divide_divide_eq_left divide_inverse mult.commute power2_eq_square power2_norm_eq_cinner)
   have spant': \<open>cspan {t} = cspan {t'}\<close>
     by (simp add: False t'_def)
   have cc: \<open>closed_csubspace (cspan {t})\<close>
     by (auto intro!: finite_cspan_closed closed_csubspace.intro)
   have ortho: \<open>h - P h \<in> orthogonal_complement (cspan {t})\<close> for h
     unfolding orthogonal_complement_def P_def' spant' apply auto
     by (smt (verit, ccfv_threshold) \<open>norm t' = 1\<close> add_cancel_right_left cinner_add_right cinner_commute' cinner_scaleC_right cnorm_eq_1 complex_vector.span_breakdown_eq complex_vector.span_empty diff_add_cancel mult_cancel_left1 singletonD)
   have inspan: \<open>P h \<in> cspan {t}\<close> for h
     unfolding P_def' spant'
     by (simp add: complex_vector.span_base complex_vector.span_scale)
   show \<open>is_projection_on P (cspan {t})\<close>
     apply (subst is_projection_on_iff_orthog)
     using cc ortho inspan by auto
 qed
 
 lemma projection_rank1:
   fixes t x :: \<open>'a::complex_inner\<close>
-  shows \<open>projection (cspan {t}) x = (\<langle>t , x\<rangle> / \<langle>t , t\<rangle>) *\<^sub>C t\<close>
+  shows \<open>projection (cspan {t}) x = ((t \<bullet>\<^sub>C x) / (t \<bullet>\<^sub>C t)) *\<^sub>C t\<close>
   apply (rule fun_cong, rule projection_eqI', simp)
   by (rule is_projection_on_rank1)
 
 subsection \<open>More orthogonal complement\<close>
 
 text \<open>The following lemmas logically fit into the "orthogonality" section but depend on projections for their proofs.\<close>
 
-text \<open>Corollary 2.8 in  @{cite conway2013course}\<close>
+text \<open>Corollary 2.8 in @{cite conway2013course}\<close>
 theorem double_orthogonal_complement_id[simp]:
   fixes M :: \<open>'a::chilbert_space set\<close>
   assumes a1: "closed_csubspace M"
   shows "orthogonal_complement (orthogonal_complement M) = M"
 proof-
   have b2: "x \<in> (id - projection M) -` {0}"
     if c1: "x \<in> M" for x
     by (simp add: assms projection_fixes_image that)
 
   have b3: \<open>x \<in> M\<close>
     if c1: \<open>x \<in> (id - projection M) -` {0}\<close> for x
     by (metis assms closed_csubspace.closed closed_csubspace.subspace complex_vector.subspace_0 csubspace_is_convex eq_id_iff equals0D fun_diff_def projection_in_image right_minus_eq that vimage_singleton_eq)
   have \<open>x \<in>  M \<longleftrightarrow> x \<in> (id - projection M) -` {0}\<close> for x
     using b2 b3 by blast
   hence b4: \<open>( id - (projection M) ) -` {0} =  M\<close>
     by blast
   have b1: "orthogonal_complement (orthogonal_complement M)
           = (projection (orthogonal_complement M)) -` {0}"
     by (simp add: a1 projection_kernel_is_orthogonal_complement del: projection_on_orthogonal_complement)
   also have \<open>... = ( id - (projection M) ) -` {0}\<close>
     by (simp add: a1)
   also have \<open>... = M\<close>
     by (simp add: b4)
   finally show ?thesis by blast
 qed
 
-lemma orthogonal_complement_antimono[simp]:
-  fixes  A B :: \<open>('a::complex_inner) set\<close>
-  assumes "A \<supseteq> B"
-  shows \<open>orthogonal_complement A \<subseteq> orthogonal_complement B\<close>
-  by (meson assms orthogonal_complementI orthogonal_complement_orthoI' subsetD subsetI)
-
 lemma orthogonal_complement_antimono_iff[simp]:
   fixes  A B :: \<open>('a::chilbert_space) set\<close>
   assumes \<open>closed_csubspace A\<close> and  \<open>closed_csubspace B\<close>
   shows \<open>orthogonal_complement A \<subseteq> orthogonal_complement B \<longleftrightarrow> A \<supseteq> B\<close>
-proof
+proof (rule iffI)
   show \<open>orthogonal_complement A \<subseteq> orthogonal_complement B\<close> if \<open>A \<supseteq> B\<close>
     using that by auto
 
   assume \<open>orthogonal_complement A \<subseteq> orthogonal_complement B\<close>
   then have \<open>orthogonal_complement (orthogonal_complement A) \<supseteq> orthogonal_complement (orthogonal_complement B)\<close>
     by simp
   then show \<open>A \<supseteq> B\<close>
     using assms by auto
 qed
 
-lemma orthogonal_complement_UNIV[simp]:
-  "orthogonal_complement UNIV = {0}"
-  by (metis Int_UNIV_left complex_vector.subspace_UNIV complex_vector.subspace_def orthogonal_complement_zero_intersection)
-
-lemma orthogonal_complement_zero[simp]:
-  "orthogonal_complement {0} = UNIV"
-  unfolding orthogonal_complement_def by auto
-
-
 lemma de_morgan_orthogonal_complement_plus:
   fixes A B::"('a::complex_inner) set"
   assumes \<open>0 \<in> A\<close> and \<open>0 \<in> B\<close>
-  shows \<open>orthogonal_complement (A +\<^sub>M B) = (orthogonal_complement A) \<inter> (orthogonal_complement B)\<close>
-proof-
+  shows \<open>orthogonal_complement (A +\<^sub>M B) = orthogonal_complement A \<inter> orthogonal_complement B\<close>
+proof -
   have "x \<in> (orthogonal_complement A) \<inter> (orthogonal_complement B)"
-    if "x \<in> orthogonal_complement (A +\<^sub>M B)"
-    for x
-  proof-
+    if "x \<in> orthogonal_complement (A +\<^sub>M B)" for x
+  proof -
     have \<open>orthogonal_complement (A +\<^sub>M B) = orthogonal_complement (A + B)\<close>
       unfolding closed_sum_def by (subst orthogonal_complement_of_closure[symmetric], simp)
     hence \<open>x \<in> orthogonal_complement (A + B)\<close>
       using that by blast
-    hence t1: \<open>\<forall>z \<in> (A + B). \<langle> z , x \<rangle> = 0\<close>
+    hence t1: \<open>\<forall>z \<in> (A + B). (z \<bullet>\<^sub>C x) = 0\<close>
       by (simp add: orthogonal_complement_orthoI')
     have \<open>A \<subseteq> A + B\<close>
       using subset_iff add.commute set_zero_plus2 \<open>0 \<in> B\<close>
       by fastforce
-    hence \<open>\<forall>z \<in> A. \<langle> z , x \<rangle> = 0\<close>
+    hence \<open>\<forall>z \<in> A. (z \<bullet>\<^sub>C x) = 0\<close>
       using t1 by auto
     hence w1: \<open>x \<in> (orthogonal_complement A)\<close>
       by (smt mem_Collect_eq is_orthogonal_sym orthogonal_complement_def)
     have \<open>B \<subseteq> A + B\<close>
       using \<open>0 \<in> A\<close> subset_iff set_zero_plus2 by blast
-    hence \<open>\<forall> z \<in> B. \<langle> z , x \<rangle> = 0\<close>
+    hence \<open>\<forall> z \<in> B. (z \<bullet>\<^sub>C x) = 0\<close>
       using t1 by auto
     hence \<open>x \<in> (orthogonal_complement B)\<close>
       by (smt mem_Collect_eq is_orthogonal_sym orthogonal_complement_def)
     thus ?thesis
       using w1 by auto
   qed
   moreover have "x \<in> (orthogonal_complement (A +\<^sub>M B))"
     if v1: "x \<in> (orthogonal_complement A) \<inter> (orthogonal_complement B)"
     for x
   proof-
     have \<open>x \<in> (orthogonal_complement A)\<close>
       using v1
       by blast
-    hence \<open>\<forall>y\<in> A. \<langle> y , x \<rangle> = 0\<close>
+    hence \<open>\<forall>y\<in> A. (y \<bullet>\<^sub>C x) = 0\<close>
       by (simp add: orthogonal_complement_orthoI')
     have \<open>x \<in> (orthogonal_complement B)\<close>
       using v1
       by blast
-    hence \<open>\<forall> y\<in> B. \<langle> y , x \<rangle> = 0\<close>
+    hence \<open>\<forall> y\<in> B. (y \<bullet>\<^sub>C x) = 0\<close>
       by (simp add: orthogonal_complement_orthoI')
-    have \<open>\<forall> a\<in>A. \<forall> b\<in>B. \<langle> a+b , x \<rangle> = 0\<close>
-      by (simp add: \<open>\<forall>y\<in>A. \<langle>y , x\<rangle> = 0\<close> \<open>\<forall>y\<in>B. \<langle>y , x\<rangle> = 0\<close> cinner_add_left)
-    hence \<open>\<forall> y \<in> (A + B). \<langle> y , x \<rangle> = 0\<close>
+    have \<open>\<forall> a\<in>A. \<forall> b\<in>B. (a+b) \<bullet>\<^sub>C x = 0\<close>
+      by (simp add: \<open>\<forall>y\<in>A. y \<bullet>\<^sub>C x = 0\<close> \<open>\<forall>y\<in>B. (y \<bullet>\<^sub>C x) = 0\<close> cinner_add_left)
+    hence \<open>\<forall> y \<in> (A + B). y \<bullet>\<^sub>C x = 0\<close>
       using set_plus_elim by force
     hence \<open>x \<in> (orthogonal_complement (A + B))\<close>
       by (smt mem_Collect_eq is_orthogonal_sym orthogonal_complement_def)
     moreover have \<open>(orthogonal_complement (A + B)) = (orthogonal_complement (A +\<^sub>M B))\<close>
       unfolding closed_sum_def by (subst orthogonal_complement_of_closure[symmetric], simp)
     ultimately have \<open>x \<in> (orthogonal_complement (A +\<^sub>M B))\<close>
       by blast
     thus ?thesis
       by blast
   qed
   ultimately show ?thesis by blast
 qed
 
 lemma de_morgan_orthogonal_complement_inter:
   fixes A B::"'a::chilbert_space set"
   assumes a1: \<open>closed_csubspace A\<close> and a2: \<open>closed_csubspace B\<close>
   shows  \<open>orthogonal_complement (A \<inter> B) = orthogonal_complement A +\<^sub>M orthogonal_complement B\<close>
 proof-
   have \<open>orthogonal_complement A +\<^sub>M orthogonal_complement B
     = orthogonal_complement (orthogonal_complement (orthogonal_complement A +\<^sub>M orthogonal_complement B))\<close>
     by (simp add: closed_subspace_closed_sum)
   also have \<open>\<dots> = orthogonal_complement (orthogonal_complement (orthogonal_complement A) \<inter> orthogonal_complement (orthogonal_complement B))\<close>
     by (simp add: de_morgan_orthogonal_complement_plus orthogonal_complementI)
   also have \<open>\<dots> = orthogonal_complement (A \<inter> B)\<close>
     by (simp add: a1 a2)
   finally show ?thesis
     by simp
 qed
 
 lemma orthogonal_complement_of_cspan: \<open>orthogonal_complement A = orthogonal_complement (cspan A)\<close>
   by (metis (no_types, opaque_lifting) closed_csubspace.subspace complex_vector.span_minimal complex_vector.span_superset double_orthogonal_complement_increasing orthogonal_complement_antimono orthogonal_complement_closed_subspace subset_antisym)
 
 lemma orthogonal_complement_orthogonal_complement_closure_cspan:
   \<open>orthogonal_complement (orthogonal_complement S) = closure (cspan S)\<close> for S :: \<open>'a::chilbert_space set\<close>
 proof -
   have \<open>orthogonal_complement (orthogonal_complement S) = orthogonal_complement (orthogonal_complement (closure (cspan S)))\<close>
     by (simp flip: orthogonal_complement_of_closure orthogonal_complement_of_cspan)
   also have \<open>\<dots> = closure (cspan S)\<close>
     by simp
   finally show \<open>orthogonal_complement (orthogonal_complement S) = closure (cspan S)\<close>
     by -
 qed
 
+instance ccsubspace :: (chilbert_space) complete_orthomodular_lattice
+proof
+  fix X Y :: \<open>'a ccsubspace\<close>
+
+  show "inf X (- X) = bot"
+    apply transfer
+    by (simp add: closed_csubspace_def complex_vector.subspace_0 orthogonal_complement_zero_intersection)
+
+  have \<open>t \<in> M +\<^sub>M orthogonal_complement M\<close>
+    if \<open>closed_csubspace M\<close> for t::'a and M
+    by (metis (no_types, lifting) UNIV_I closed_csubspace.subspace complex_vector.subspace_def de_morgan_orthogonal_complement_inter double_orthogonal_complement_id orthogonal_complement_closed_subspace orthogonal_complement_zero orthogonal_complement_zero_intersection that)
+  hence b1: \<open>M +\<^sub>M orthogonal_complement M = UNIV\<close>
+    if \<open>closed_csubspace M\<close> for M :: \<open>'a set\<close>
+    using that by blast
+  show "sup X (- X) = top"
+    apply transfer
+    using b1 by auto
+  show "- (- X) = X"
+    apply transfer by simp
+
+  show "- Y \<le> - X"
+    if "X \<le> Y"
+    using that apply transfer by simp
+
+  have c1: "M +\<^sub>M orthogonal_complement M \<inter> N \<subseteq> N"
+    if "closed_csubspace M" and "closed_csubspace N" and "M \<subseteq> N"
+    for M N :: "'a set"
+    using that
+    by (simp add: closed_sum_is_sup)
+
+  have c2: \<open>u \<in> M +\<^sub>M (orthogonal_complement M \<inter> N)\<close>
+    if a1: "closed_csubspace M" and a2: "closed_csubspace N" and a3: "M \<subseteq> N" and x1: \<open>u \<in> N\<close>
+    for M :: "'a set" and N :: "'a set"  and u
+  proof -
+    have d4: \<open>(projection M) u \<in> M\<close>
+      by (metis a1 closed_csubspace_def csubspace_is_convex equals0D orthog_proj_exists projection_in_image)
+    hence d2: \<open>(projection M) u \<in> N\<close>
+      using a3 by auto
+    have d1: \<open>csubspace N\<close>
+      by (simp add: a2)
+    have \<open>u - (projection M) u \<in> orthogonal_complement M\<close>
+      by (simp add: a1 orthogonal_complementI projection_orthogonal)
+    moreover have  \<open>u - (projection M) u \<in> N\<close>
+      by (simp add: d1 d2 complex_vector.subspace_diff x1)
+    ultimately have d3: \<open>u - (projection M) u \<in> ((orthogonal_complement M) \<inter> N)\<close>
+      by simp
+    hence \<open>\<exists> v \<in> ((orthogonal_complement M) \<inter> N). u = (projection M) u + v\<close>
+      by (metis d3 diff_add_cancel ordered_field_class.sign_simps(2))
+    then obtain v where \<open>v \<in> ((orthogonal_complement M) \<inter> N)\<close> and \<open>u = (projection M) u + v\<close>
+      by blast
+    hence \<open>u \<in> M + ((orthogonal_complement M) \<inter> N)\<close>
+      by (metis d4 set_plus_intro)
+    thus ?thesis
+      unfolding closed_sum_def
+      using closure_subset by blast
+  qed
+
+  have c3: "N \<subseteq> M +\<^sub>M ((orthogonal_complement M) \<inter> N)"
+    if "closed_csubspace M" and "closed_csubspace N" and "M \<subseteq> N"
+    for M N :: "'a set"
+    using c2 that by auto
+
+  show "sup X (inf (- X) Y) = Y"
+    if "X \<le> Y"
+    using that apply transfer
+    using c1 c3
+    by (simp add: subset_antisym)
+
+  show "X - Y = inf X (- Y)"
+    apply transfer by simp
+qed
+
 subsection \<open>Orthogonal spaces\<close>
-(* TODO: Add to report overview *)
 
 definition \<open>orthogonal_spaces S T \<longleftrightarrow> (\<forall>x\<in>space_as_set S. \<forall>y\<in>space_as_set T. is_orthogonal x y)\<close>
 
 lemma orthogonal_spaces_leq_compl: \<open>orthogonal_spaces S T \<longleftrightarrow> S \<le> -T\<close>
   unfolding orthogonal_spaces_def apply transfer
   by (auto simp: orthogonal_complement_def)
 
 lemma orthogonal_bot[simp]: \<open>orthogonal_spaces S bot\<close>
   by (simp add: orthogonal_spaces_def)
 
 lemma orthogonal_spaces_sym: \<open>orthogonal_spaces S T \<Longrightarrow> orthogonal_spaces T S\<close>
   unfolding orthogonal_spaces_def
   using is_orthogonal_sym by blast
 
 lemma orthogonal_sup: \<open>orthogonal_spaces S T1 \<Longrightarrow> orthogonal_spaces S T2 \<Longrightarrow> orthogonal_spaces S (sup T1 T2)\<close>
   apply (rule orthogonal_spaces_sym)
   apply (simp add: orthogonal_spaces_leq_compl)
   using orthogonal_spaces_leq_compl orthogonal_spaces_sym by blast
 
 lemma orthogonal_sum:
   assumes \<open>finite F\<close> and \<open>\<And>x. x\<in>F \<Longrightarrow> orthogonal_spaces S (T x)\<close> 
   shows \<open>orthogonal_spaces S (sum T F)\<close>
   using assms
   apply induction
   by (auto intro!: orthogonal_sup)
 
+lemma orthogonal_spaces_ccspan: \<open>(\<forall>x\<in>S. \<forall>y\<in>T. is_orthogonal x y) \<longleftrightarrow> orthogonal_spaces (ccspan S) (ccspan T)\<close>
+  by (meson ccspan_leq_ortho_ccspan ccspan_superset orthogonal_spaces_def orthogonal_spaces_leq_compl subset_iff)
 
 subsection \<open>Orthonormal bases\<close>
 
 lemma ortho_basis_exists: 
   fixes S :: \<open>'a::chilbert_space set\<close>
   assumes \<open>is_ortho_set S\<close>
   shows \<open>\<exists>B. B \<supseteq> S \<and> is_ortho_set B \<and> closure (cspan B) = UNIV\<close>
 proof -
   define on where \<open>on B \<longleftrightarrow> B \<supseteq> S \<and> is_ortho_set B\<close> for B :: \<open>'a set\<close>
   have \<open>\<exists>B\<in>Collect on. \<forall>B'\<in>Collect on. B \<subseteq> B' \<longrightarrow> B' = B\<close>
   proof (rule subset_Zorn_nonempty; simp)
     show \<open>\<exists>S. on S\<close>
       apply (rule exI[of _ S])
       using assms on_def by fastforce
   next
     fix C :: \<open>'a set set\<close>
     assume \<open>C \<noteq> {}\<close>
     assume \<open>subset.chain (Collect on) C\<close>
     then have C_on: \<open>B \<in> C \<Longrightarrow> on B\<close> and C_order: \<open>B \<in> C \<Longrightarrow> B' \<in> C \<Longrightarrow> B \<subseteq> B' \<or> B' \<subseteq> B\<close> for B B'
       by (auto simp: subset.chain_def)
     have \<open>is_orthogonal x y\<close> if \<open>x\<in>\<Union>C\<close> \<open>y\<in>\<Union>C\<close> \<open>x \<noteq> y\<close> for x y
       by (smt (verit) UnionE C_order C_on on_def is_ortho_set_def subsetD that(1) that(2) that(3))
     moreover have \<open>0 \<notin> \<Union> C\<close>
       by (meson UnionE C_on is_ortho_set_def on_def)
     moreover have \<open>\<Union>C \<supseteq> S\<close>
       using C_on \<open>C \<noteq> {}\<close> on_def by blast
     ultimately show \<open>on (\<Union> C)\<close>
       unfolding on_def is_ortho_set_def by simp
   qed
   then obtain B where \<open>on B\<close> and B_max: \<open>B' \<supseteq> B \<Longrightarrow> on B' \<Longrightarrow> B=B'\<close> for B'
     by auto
   have \<open>\<psi> = 0\<close> if \<psi>ortho: \<open>\<forall>b\<in>B. is_orthogonal \<psi> b\<close> for \<psi> :: 'a
   proof (rule ccontr)
     assume \<open>\<psi> \<noteq> 0\<close>
     define \<phi> B' where \<open>\<phi> = \<psi> /\<^sub>R norm \<psi>\<close> and \<open>B' = B \<union> {\<phi>}\<close>
     have [simp]: \<open>norm \<phi> = 1\<close>
       using \<open>\<psi> \<noteq> 0\<close> by (auto simp: \<phi>_def)
     have \<phi>ortho: \<open>is_orthogonal \<phi> b\<close> if \<open>b \<in> B\<close> for b
       using \<psi>ortho that \<phi>_def  by auto
     have orthoB': \<open>is_orthogonal x y\<close> if \<open>x\<in>B'\<close> \<open>y\<in>B'\<close> \<open>x \<noteq> y\<close> for x y
       using that \<open>on B\<close> \<phi>ortho \<phi>ortho[THEN is_orthogonal_sym[THEN iffD1]]
       by (auto simp: B'_def on_def is_ortho_set_def)
     have B'0: \<open>0 \<notin> B'\<close>
       using B'_def \<open>norm \<phi> = 1\<close> \<open>on B\<close> is_ortho_set_def on_def by fastforce
     have \<open>S \<subseteq> B'\<close>
       using B'_def \<open>on B\<close> on_def by auto
     from orthoB' B'0 \<open>S \<subseteq> B'\<close> have \<open>on B'\<close>
       by (simp add: on_def is_ortho_set_def)
     with B_max have \<open>B = B'\<close>
       by (metis B'_def Un_upper1)
     then have \<open>\<phi> \<in> B\<close>
       using B'_def by blast
     then have \<open>is_orthogonal \<phi> \<phi>\<close>
       using \<phi>ortho by blast
     then show False
       using B'0 \<open>B = B'\<close> \<open>\<phi> \<in> B\<close> by fastforce
   qed 
   then have \<open>orthogonal_complement B = {0}\<close>
     by (auto simp: orthogonal_complement_def)
   then have \<open>UNIV = orthogonal_complement (orthogonal_complement B)\<close>
     by simp
   also have \<open>\<dots> = orthogonal_complement (orthogonal_complement (closure (cspan B)))\<close>
     by (metis (mono_tags, opaque_lifting) \<open>orthogonal_complement B = {0}\<close> cinner_zero_left complex_vector.span_superset empty_iff insert_iff orthogonal_complementI orthogonal_complement_antimono orthogonal_complement_of_closure subsetI subset_antisym)
   also have \<open>\<dots> = closure (cspan B)\<close>
     apply (rule double_orthogonal_complement_id)
     by simp
   finally have \<open>closure (cspan B) = UNIV\<close>
     by simp
   with \<open>on B\<close> show ?thesis
     by (auto simp: on_def)
 qed
 
 lemma orthonormal_basis_exists: 
   fixes S :: \<open>'a::chilbert_space set\<close>
   assumes \<open>is_ortho_set S\<close> and \<open>\<And>x. x\<in>S \<Longrightarrow> norm x = 1\<close>
   shows \<open>\<exists>B. B \<supseteq> S \<and> is_onb B\<close>
 proof -
   from \<open>is_ortho_set S\<close>
   obtain B where \<open>is_ortho_set B\<close> and \<open>B \<supseteq> S\<close> and \<open>closure (cspan B) = UNIV\<close>
     using ortho_basis_exists by blast
   define B' where \<open>B' = (\<lambda>x. x /\<^sub>R norm x) ` B\<close>
   have \<open>S = (\<lambda>x. x /\<^sub>R norm x) ` S\<close>
     by (simp add: assms(2))
   then have \<open>B' \<supseteq> S\<close>
     using B'_def \<open>S \<subseteq> B\<close> by blast
   moreover 
   have \<open>ccspan B' = top\<close>
     apply (transfer fixing: B')
     apply (simp add: B'_def scaleR_scaleC)
     apply (subst complex_vector.span_image_scale')
     using \<open>is_ortho_set B\<close> \<open>closure (cspan B) = UNIV\<close> is_ortho_set_def 
     by auto
   moreover have \<open>is_ortho_set B'\<close>
     using \<open>is_ortho_set B\<close> by (auto simp: B'_def is_ortho_set_def)
   moreover have \<open>\<forall>b\<in>B'. norm b = 1\<close>
     using \<open>is_ortho_set B\<close> apply (auto simp: B'_def is_ortho_set_def)
     by (metis field_class.field_inverse norm_eq_zero)
   ultimately show ?thesis
     by (auto simp: is_onb_def)
 qed
 
 
 definition some_chilbert_basis :: \<open>'a::chilbert_space set\<close> where
   \<open>some_chilbert_basis = (SOME B::'a set. is_onb B)\<close>
 
 lemma is_onb_some_chilbert_basis[simp]: \<open>is_onb (some_chilbert_basis :: 'a::chilbert_space set)\<close>
   using orthonormal_basis_exists[OF is_ortho_set_empty]
   by (auto simp add: some_chilbert_basis_def intro: someI2)
 
 lemma is_ortho_set_some_chilbert_basis[simp]: \<open>is_ortho_set some_chilbert_basis\<close>
   using is_onb_def is_onb_some_chilbert_basis by blast
+
 lemma is_normal_some_chilbert_basis: \<open>\<And>x. x \<in> some_chilbert_basis \<Longrightarrow> norm x = 1\<close>
   using is_onb_def is_onb_some_chilbert_basis by blast
+
 lemma ccspan_some_chilbert_basis[simp]: \<open>ccspan some_chilbert_basis = top\<close>
   using is_onb_def is_onb_some_chilbert_basis by blast
+
 lemma span_some_chilbert_basis[simp]: \<open>closure (cspan some_chilbert_basis) = UNIV\<close>
   by (metis ccspan.rep_eq ccspan_some_chilbert_basis top_ccsubspace.rep_eq)
 
 lemma cindependent_some_chilbert_basis[simp]: \<open>cindependent some_chilbert_basis\<close>
   using is_ortho_set_cindependent is_ortho_set_some_chilbert_basis by blast
 
 lemma finite_some_chilbert_basis[simp]: \<open>finite (some_chilbert_basis :: 'a :: {chilbert_space, cfinite_dim} set)\<close>
   apply (rule cindependent_cfinite_dim_finite)
   by simp
 
 lemma some_chilbert_basis_nonempty: \<open>(some_chilbert_basis :: 'a::{chilbert_space, not_singleton} set) \<noteq> {}\<close>
 proof (rule ccontr, simp)
   define B :: \<open>'a set\<close> where \<open>B = some_chilbert_basis\<close>
   assume [simp]: \<open>B = {}\<close>
   have \<open>UNIV = closure (cspan B)\<close>
     using B_def span_some_chilbert_basis by blast
   also have \<open>\<dots> = {0}\<close>
     by simp
   also have \<open>\<dots> \<noteq> UNIV\<close>
     using Extra_General.UNIV_not_singleton by blast
   finally show False
     by simp
 qed
 
 subsection \<open>Riesz-representation theorem\<close>
 
 lemma orthogonal_complement_kernel_functional:
   fixes f :: \<open>'a::complex_inner \<Rightarrow> complex\<close>
   assumes \<open>bounded_clinear f\<close>
   shows \<open>\<exists>x. orthogonal_complement (f -` {0}) = cspan {x}\<close>
 proof (cases \<open>orthogonal_complement (f -` {0}) = {0}\<close>)
   case True
   then show ?thesis
     apply (rule_tac x=0 in exI) by auto
 next
   case False
   then obtain x where xortho: \<open>x \<in> orthogonal_complement (f -` {0})\<close> and xnon0: \<open>x \<noteq> 0\<close>
     using complex_vector.subspace_def by fastforce
 
   from xnon0 xortho
   have r1: \<open>f x \<noteq> 0\<close>
     by (metis cinner_eq_zero_iff orthogonal_complement_orthoI vimage_singleton_eq)
 
   have \<open>\<exists> k. y = k *\<^sub>C x\<close> if \<open>y \<in> orthogonal_complement (f -` {0})\<close> for y
   proof (cases \<open>y = 0\<close>)
     case True
     then show ?thesis by auto
   next
     case False
     with that
     have \<open>f y \<noteq> 0\<close>
       by (metis cinner_eq_zero_iff orthogonal_complement_orthoI vimage_singleton_eq)
     then obtain k where k_def: \<open>f x = k * f y\<close>
       by (metis add.inverse_inverse minus_divide_eq_eq)
     with assms have \<open>f x = f (k *\<^sub>C y)\<close>
       by (simp add: bounded_clinear.axioms(1) clinear.scaleC)
     hence \<open>f x - f (k *\<^sub>C y) = 0\<close>
       by simp
     with assms have s1: \<open>f (x - k *\<^sub>C y) = 0\<close>
       by (simp add: bounded_clinear.axioms(1) complex_vector.linear_diff)
     from that have \<open>k *\<^sub>C y \<in> orthogonal_complement (f -` {0})\<close>
       by (simp add: complex_vector.subspace_scale)
     with xortho have s2: \<open>x - (k *\<^sub>C y) \<in> orthogonal_complement (f -` {0})\<close>
       by (simp add: complex_vector.subspace_diff)
     have s3: \<open>(x - (k *\<^sub>C y)) \<in> f -` {0}\<close>
       using s1 by simp
     moreover have \<open>(f -` {0}) \<inter> (orthogonal_complement (f -` {0})) = {0}\<close>
       by (meson assms closed_csubspace_def complex_vector.subspace_def kernel_is_closed_csubspace
           orthogonal_complement_zero_intersection)
     ultimately have \<open>x - (k *\<^sub>C y) = 0\<close>
       using s2 by blast
     thus ?thesis
       by (metis ceq_vector_fraction_iff eq_iff_diff_eq_0 k_def r1 scaleC_scaleC)
   qed
   then have \<open>orthogonal_complement (f -` {0}) \<subseteq> cspan {x}\<close>
     using complex_vector.span_superset complex_vector.subspace_scale by blast
 
   moreover from xortho have \<open>orthogonal_complement (f -` {0}) \<supseteq> cspan {x}\<close>
     by (simp add: complex_vector.span_minimal)
 
   ultimately show ?thesis
     by auto
 qed
 
 lemma riesz_frechet_representation_existence:
   \<comment> \<open>Theorem 3.4 in @{cite conway2013course}\<close>
   fixes f::\<open>'a::chilbert_space \<Rightarrow> complex\<close>
   assumes a1: \<open>bounded_clinear f\<close>
-  shows \<open>\<exists>t. \<forall>x.  f x = \<langle>t, x\<rangle>\<close>
+  shows \<open>\<exists>t. \<forall>x.  f x = t \<bullet>\<^sub>C x\<close>
 proof(cases \<open>\<forall> x. f x = 0\<close>)
   case True
   thus ?thesis
     by (metis cinner_zero_left)
 next
   case False
   obtain t where spant: \<open>orthogonal_complement (f -` {0}) = cspan {t}\<close>
     using orthogonal_complement_kernel_functional
     using assms by blast
-  have \<open>projection (orthogonal_complement (f -` {0})) x = (\<langle>t , x\<rangle>/\<langle>t , t\<rangle>) *\<^sub>C t\<close> for x
+  have \<open>projection (orthogonal_complement (f -` {0})) x = ((t \<bullet>\<^sub>C x)/(t \<bullet>\<^sub>C t)) *\<^sub>C t\<close> for x
     apply (subst spant) by (rule projection_rank1)
-  hence \<open>f (projection (orthogonal_complement (f -` {0})) x) = ((\<langle>t , x\<rangle>)/(\<langle>t , t\<rangle>)) * (f t)\<close> for x
+  hence \<open>f (projection (orthogonal_complement (f -` {0})) x) = (((t \<bullet>\<^sub>C x))/(t \<bullet>\<^sub>C t)) * (f t)\<close> for x
     using a1 unfolding bounded_clinear_def
     by (simp add: complex_vector.linear_scale)
-  hence l2: \<open>f (projection (orthogonal_complement (f -` {0})) x) = \<langle>((cnj (f t)/\<langle>t , t\<rangle>) *\<^sub>C t) , x\<rangle>\<close> for x
+  hence l2: \<open>f (projection (orthogonal_complement (f -` {0})) x) = ((cnj (f t)/(t \<bullet>\<^sub>C t)) *\<^sub>C t) \<bullet>\<^sub>C x\<close> for x
     using complex_cnj_divide by force
   have \<open>f (projection (f -` {0}) x) = 0\<close> for x
     by (metis (no_types, lifting) assms bounded_clinear_def closed_csubspace.closed
         complex_vector.linear_subspace_vimage complex_vector.subspace_0 complex_vector.subspace_single_0
         csubspace_is_convex insert_absorb insert_not_empty kernel_is_closed_csubspace projection_in_image vimage_singleton_eq)
   hence "\<And>a b. f (projection (f -` {0}) a + b) = 0 + f b"
     using additive.add assms
     by (simp add: bounded_clinear_def complex_vector.linear_add)
   hence "\<And>a. 0 + f (projection (orthogonal_complement (f -` {0})) a) = f a"
     apply (simp add: assms)
     by (metis add.commute diff_add_cancel)
-  hence \<open>f x = \<langle>(cnj (f t)/\<langle>t , t\<rangle>) *\<^sub>C t, x\<rangle>\<close> for x
+  hence \<open>f x = ((cnj (f t)/(t \<bullet>\<^sub>C t)) *\<^sub>C t) \<bullet>\<^sub>C x\<close> for x
     by (simp add: l2)
   thus ?thesis
     by blast
 qed
 
 lemma riesz_frechet_representation_unique:
   \<comment> \<open>Theorem 3.4 in @{cite conway2013course}\<close>
   fixes f::\<open>'a::complex_inner \<Rightarrow> complex\<close>
-  assumes \<open>\<And>x. f x = \<langle>t, x\<rangle>\<close>
-  assumes \<open>\<And>x. f x = \<langle>u, x\<rangle>\<close>
+  assumes \<open>\<And>x. f x = (t \<bullet>\<^sub>C x)\<close>
+  assumes \<open>\<And>x. f x = (u \<bullet>\<^sub>C x)\<close>
   shows \<open>t = u\<close>
   by (metis add_diff_cancel_left' assms(1) assms(2) cinner_diff_left cinner_gt_zero_iff diff_add_cancel diff_zero)
 
-
 subsection \<open>Adjoints\<close>
 
-definition "is_cadjoint F G \<longleftrightarrow> (\<forall>x. \<forall>y. \<langle>F x, y\<rangle> = \<langle>x, G y\<rangle>)"
+definition "is_cadjoint F G \<longleftrightarrow> (\<forall>x. \<forall>y. (F x \<bullet>\<^sub>C y) = (x \<bullet>\<^sub>C G y))"
 
 lemma is_adjoint_sym:
   \<open>is_cadjoint F G \<Longrightarrow> is_cadjoint G F\<close>
   unfolding is_cadjoint_def apply auto
   by (metis cinner_commute')
 
 definition \<open>cadjoint G = (SOME F. is_cadjoint F G)\<close>
   for G :: "'b::complex_inner \<Rightarrow> 'a::complex_inner"
 
 lemma cadjoint_exists:
   fixes G :: "'b::chilbert_space \<Rightarrow> 'a::complex_inner"
   assumes [simp]: \<open>bounded_clinear G\<close>
   shows \<open>\<exists>F. is_cadjoint F G\<close>
 proof -
   include notation_norm
   have [simp]: \<open>clinear G\<close>
     using assms unfolding bounded_clinear_def by blast
   define g :: \<open>'a \<Rightarrow> 'b \<Rightarrow> complex\<close>
-    where \<open>g x y = \<langle>x , G y\<rangle>\<close> for x y
+    where \<open>g x y = (x \<bullet>\<^sub>C G y)\<close> for x y
   have \<open>bounded_clinear (g x)\<close> for x
   proof -
     have \<open>g x (a + b) = g x a + g x b\<close> for a b
       unfolding g_def
       using additive.add cinner_add_right clinear_def
       by (simp add: cinner_add_right complex_vector.linear_add)
     moreover have  \<open>g x (k *\<^sub>C a) = k *\<^sub>C (g x a)\<close>
       for a k
       unfolding g_def
       by (simp add: complex_vector.linear_scale)
     ultimately have \<open>clinear (g x)\<close>
       by (simp add: clinearI)
     moreover
     have \<open>\<exists> M. \<forall> y. \<parallel> G y \<parallel> \<le> \<parallel> y \<parallel> * M\<close>
       using \<open>bounded_clinear G\<close>
       unfolding bounded_clinear_def bounded_clinear_axioms_def by blast
     then have \<open>\<exists>M. \<forall>y. \<parallel> g x y \<parallel> \<le> \<parallel> y \<parallel> * M\<close>
       using g_def
       by (simp add: bounded_clinear.bounded bounded_clinear_cinner_right_comp)
     ultimately show ?thesis unfolding bounded_linear_def
       using bounded_clinear.intro
       using bounded_clinear_axioms_def by blast
   qed
-  hence \<open>\<forall>x. \<exists>t. \<forall>y.  g x y = \<langle>t, y\<rangle>\<close>
+  hence \<open>\<forall>x. \<exists>t. \<forall>y.  g x y = (t \<bullet>\<^sub>C y)\<close>
     using riesz_frechet_representation_existence by blast
-  then obtain F where \<open>\<forall>x. \<forall>y. g x y = \<langle>F x, y\<rangle>\<close>
+  then obtain F where \<open>\<forall>x. \<forall>y. g x y = (F x \<bullet>\<^sub>C y)\<close>
     by metis
   then have \<open>is_cadjoint F G\<close>
     unfolding is_cadjoint_def g_def by simp
   thus ?thesis
     by auto
 qed
 
 lemma cadjoint_is_cadjoint[simp]:
   fixes G :: "'b::chilbert_space \<Rightarrow> 'a::complex_inner"
   assumes [simp]: \<open>bounded_clinear G\<close>
   shows \<open>is_cadjoint (cadjoint G) G\<close>
   by (metis assms cadjoint_def cadjoint_exists someI_ex)
 
 lemma is_cadjoint_unique:
   assumes \<open>is_cadjoint F1 G\<close>
   assumes \<open>is_cadjoint F2 G\<close>
   shows \<open>F1 = F2\<close>
-proof (rule ext)
-  fix x
-  {
-    fix y
-    have \<open>cinner (F1 x - F2 x) y = cinner (F1 x) y - cinner (F2 x) y\<close>
-      by (simp add: cinner_diff_left)
-    also have \<open>\<dots> = cinner x (G y) - cinner x (G y)\<close>
-      by (metis assms(1) assms(2) is_cadjoint_def)
-    also have \<open>\<dots> = 0\<close>
-      by simp
-    finally have \<open>cinner (F1 x - F2 x) y = 0\<close>
-      by -
-  }
-  then show \<open>F1 x = F2 x\<close>
-    by fastforce
-qed
+  by (metis (full_types) assms(1) assms(2) ext is_cadjoint_def riesz_frechet_representation_unique)
 
 lemma cadjoint_univ_prop:
   fixes G :: "'b::chilbert_space \<Rightarrow> 'a::complex_inner"
   assumes a1: \<open>bounded_clinear G\<close>
-  shows \<open>\<forall>x. \<forall>y. \<langle>cadjoint G x, y\<rangle> = \<langle>x, G y\<rangle>\<close>
+  shows \<open>cadjoint G x \<bullet>\<^sub>C y = x \<bullet>\<^sub>C G y\<close>
   using assms cadjoint_is_cadjoint is_cadjoint_def by blast
 
 lemma cadjoint_univ_prop':
   fixes G :: "'b::chilbert_space \<Rightarrow> 'a::complex_inner"
   assumes a1: \<open>bounded_clinear G\<close>
-  shows \<open>\<forall>x. \<forall>y. \<langle>x, cadjoint G y\<rangle> = \<langle>G x, y\<rangle>\<close>
+  shows \<open>x \<bullet>\<^sub>C cadjoint G y = G x \<bullet>\<^sub>C y\<close>
   by (metis cadjoint_univ_prop assms cinner_commute')
 
 notation cadjoint ("_\<^sup>\<dagger>" [99] 100)
 
 lemma cadjoint_eqI:
   fixes G:: \<open>'b::complex_inner \<Rightarrow> 'a::complex_inner\<close>
     and F:: \<open>'a \<Rightarrow> 'b\<close>
-  assumes \<open>\<And>x y. \<langle>F x, y\<rangle> = \<langle>x, G y\<rangle>\<close>
+  assumes \<open>\<And>x y. (F x \<bullet>\<^sub>C y) = (x \<bullet>\<^sub>C G y)\<close>
   shows \<open>G\<^sup>\<dagger> = F\<close>
   by (metis assms cadjoint_def is_cadjoint_def is_cadjoint_unique someI_ex)
 
 lemma cadjoint_bounded_clinear:
   fixes A :: "'a::chilbert_space \<Rightarrow> 'b::complex_inner"
   assumes a1: "bounded_clinear A"
   shows \<open>bounded_clinear (A\<^sup>\<dagger>)\<close>
 proof
   include notation_norm
-  have b1: \<open>\<langle>(A\<^sup>\<dagger>) x, y\<rangle> = \<langle>x , A y\<rangle>\<close> for x y
+  have b1: \<open>((A\<^sup>\<dagger>) x \<bullet>\<^sub>C y) = (x \<bullet>\<^sub>C A y)\<close> for x y
     using cadjoint_univ_prop a1 by auto
-  have \<open>\<langle>(A\<^sup>\<dagger>) (x1 + x2) - ((A\<^sup>\<dagger>) x1 + (A\<^sup>\<dagger>) x2) , y\<rangle> = 0\<close> for x1 x2 y
+  have \<open>is_orthogonal ((A\<^sup>\<dagger>) (x1 + x2) - ((A\<^sup>\<dagger>) x1 + (A\<^sup>\<dagger>) x2)) y\<close> for x1 x2 y
     by (simp add: b1 cinner_diff_left cinner_add_left)
   hence b2: \<open>(A\<^sup>\<dagger>) (x1 + x2) - ((A\<^sup>\<dagger>) x1 + (A\<^sup>\<dagger>) x2) = 0\<close> for x1 x2
     using cinner_eq_zero_iff by blast
   thus z1: \<open>(A\<^sup>\<dagger>) (x1 + x2) = (A\<^sup>\<dagger>) x1 + (A\<^sup>\<dagger>) x2\<close> for x1 x2
     by (simp add: b2 eq_iff_diff_eq_0)
 
-  have f1: \<open>\<langle>(A\<^sup>\<dagger>) (r *\<^sub>C x) - (r *\<^sub>C (A\<^sup>\<dagger>) x ), y\<rangle> = 0\<close> for r x y
+  have f1: \<open>is_orthogonal ((A\<^sup>\<dagger>) (r *\<^sub>C x) - (r *\<^sub>C (A\<^sup>\<dagger>) x )) y\<close> for r x y
     by (simp add: b1 cinner_diff_left)
   thus z2: \<open>(A\<^sup>\<dagger>) (r *\<^sub>C x) = r *\<^sub>C (A\<^sup>\<dagger>) x\<close> for r x
     using cinner_eq_zero_iff eq_iff_diff_eq_0 by blast
-  have \<open>\<parallel> (A\<^sup>\<dagger>) x \<parallel>^2 = \<langle>(A\<^sup>\<dagger>) x, (A\<^sup>\<dagger>) x\<rangle>\<close> for x
+  have \<open>\<parallel> (A\<^sup>\<dagger>) x \<parallel>^2 = ((A\<^sup>\<dagger>) x \<bullet>\<^sub>C (A\<^sup>\<dagger>) x)\<close> for x
     by (metis cnorm_eq_square)
   moreover have \<open>\<parallel> (A\<^sup>\<dagger>) x \<parallel>^2 \<ge> 0\<close> for x
     by simp
-  ultimately have \<open>\<parallel> (A\<^sup>\<dagger>) x \<parallel>^2 = \<bar> \<langle>(A\<^sup>\<dagger>) x, (A\<^sup>\<dagger>) x\<rangle> \<bar>\<close> for x
+  ultimately have \<open>\<parallel> (A\<^sup>\<dagger>) x \<parallel>^2 = \<bar> ((A\<^sup>\<dagger>) x \<bullet>\<^sub>C (A\<^sup>\<dagger>) x) \<bar>\<close> for x
     by (metis abs_pos cinner_ge_zero)
-  hence \<open>\<parallel> (A\<^sup>\<dagger>) x \<parallel>^2 = \<bar> \<langle>x, A ((A\<^sup>\<dagger>) x)\<rangle> \<bar>\<close> for x
+  hence \<open>\<parallel> (A\<^sup>\<dagger>) x \<parallel>^2 = \<bar> (x \<bullet>\<^sub>C A ((A\<^sup>\<dagger>) x)) \<bar>\<close> for x
     by (simp add: b1)
-  moreover have  \<open>\<bar>\<langle>x , A ((A\<^sup>\<dagger>) x)\<rangle>\<bar> \<le> \<parallel>x\<parallel> *  \<parallel>A ((A\<^sup>\<dagger>) x)\<parallel>\<close> for x
+  moreover have  \<open>\<bar>(x \<bullet>\<^sub>C A ((A\<^sup>\<dagger>) x))\<bar> \<le> \<parallel>x\<parallel> *  \<parallel>A ((A\<^sup>\<dagger>) x)\<parallel>\<close> for x
     by (simp add: abs_complex_def complex_inner_class.Cauchy_Schwarz_ineq2 less_eq_complex_def)
   ultimately have b5: \<open>\<parallel> (A\<^sup>\<dagger>) x \<parallel>^2  \<le> \<parallel>x\<parallel> * \<parallel>A ((A\<^sup>\<dagger>) x)\<parallel>\<close> for x
     by (metis complex_of_real_mono_iff)
   have \<open>\<exists>M. M \<ge> 0 \<and> (\<forall> x. \<parallel>A ((A\<^sup>\<dagger>) x)\<parallel> \<le> M *  \<parallel>(A\<^sup>\<dagger>) x\<parallel>)\<close>
     using a1
     by (metis (mono_tags, opaque_lifting) bounded_clinear.bounded linear mult_nonneg_nonpos
         mult_zero_right norm_ge_zero order.trans semiring_normalization_rules(7))
   then obtain M where q1: \<open>M \<ge> 0\<close> and q2: \<open>\<forall> x. \<parallel>A ((A\<^sup>\<dagger>) x)\<parallel> \<le> M * \<parallel>(A\<^sup>\<dagger>) x\<parallel>\<close>
     by blast
   have \<open>\<forall> x::'b. \<parallel>x\<parallel> \<ge> 0\<close>
     by simp
   hence b6: \<open>\<parallel>x\<parallel> * \<parallel>A ((A\<^sup>\<dagger>) x)\<parallel> \<le>  \<parallel>x\<parallel> * M * \<parallel>(A\<^sup>\<dagger>) x\<parallel>\<close> for x
     using q2
     by (smt ordered_comm_semiring_class.comm_mult_left_mono vector_space_over_itself.scale_scale)
   have z3: \<open>\<parallel> (A\<^sup>\<dagger>) x \<parallel> \<le> \<parallel>x\<parallel> * M\<close> for x
   proof(cases \<open>\<parallel>(A\<^sup>\<dagger>) x\<parallel> = 0\<close>)
     case True
     thus ?thesis
       by (simp add: \<open>0 \<le> M\<close>)
   next
     case False
     have \<open>\<parallel> (A\<^sup>\<dagger>) x \<parallel>^2 \<le> \<parallel>x\<parallel> *  M *  \<parallel>(A\<^sup>\<dagger>) x\<parallel>\<close>
       by (smt b5 b6)
     thus ?thesis
       by (smt False mult_right_cancel mult_right_mono norm_ge_zero semiring_normalization_rules(29))
   qed
   thus \<open>\<exists>K. \<forall>x. \<parallel>(A\<^sup>\<dagger>) x\<parallel> \<le> \<parallel>x\<parallel> * K\<close>
     by auto
 qed
 
 proposition double_cadjoint:
   fixes U :: \<open>'a::chilbert_space \<Rightarrow> 'b::complex_inner\<close>
   assumes a1: "bounded_clinear U"
   shows "U\<^sup>\<dagger>\<^sup>\<dagger> = U"
   by (metis assms cadjoint_def cadjoint_is_cadjoint is_adjoint_sym is_cadjoint_unique someI_ex)
 
-lemma cadjoint_id: \<open>(id::'a::complex_inner\<Rightarrow>'a)\<^sup>\<dagger> = id\<close>
+lemma cadjoint_id[simp]: \<open>id\<^sup>\<dagger> = id\<close>
   by (simp add: cadjoint_eqI id_def)
 
 lemma scaleC_cadjoint:
   fixes A::"'a::chilbert_space \<Rightarrow> 'b::complex_inner"
   assumes "bounded_clinear A"
-  shows \<open>(\<lambda>t. a *\<^sub>C (A t))\<^sup>\<dagger> = (\<lambda>s. (cnj a) *\<^sub>C ((A\<^sup>\<dagger>) s))\<close>
-proof-
-  have b3: \<open>\<langle>(\<lambda> s. (cnj a) *\<^sub>C ((A\<^sup>\<dagger>) s)) x, y \<rangle> = \<langle>x, (\<lambda> t. a *\<^sub>C (A t)) y \<rangle>\<close>
+  shows \<open>(\<lambda>t. a *\<^sub>C A t)\<^sup>\<dagger> = (\<lambda>s. cnj a *\<^sub>C (A\<^sup>\<dagger>) s)\<close>
+proof -
+  have b3: \<open>((\<lambda> s. (cnj a) *\<^sub>C ((A\<^sup>\<dagger>) s)) x \<bullet>\<^sub>C y) = (x \<bullet>\<^sub>C (\<lambda> t. a *\<^sub>C (A t)) y)\<close>
     for x y
     by (simp add: assms cadjoint_univ_prop)
 
   have "((\<lambda>t. a *\<^sub>C A t)\<^sup>\<dagger>) b = cnj a *\<^sub>C (A\<^sup>\<dagger>) b"
     for b::'b
   proof-
     have "bounded_clinear (\<lambda>t. a *\<^sub>C A t)"
       by (simp add: assms bounded_clinear_const_scaleC)
     thus ?thesis
       by (metis (no_types) cadjoint_eqI b3)
   qed
   thus ?thesis
     by blast
 qed
 
 
 lemma is_projection_on_is_cadjoint:
   fixes M :: \<open>'a::complex_inner set\<close>
   assumes a1: \<open>is_projection_on \<pi> M\<close> and a2: \<open>closed_csubspace M\<close>
   shows \<open>is_cadjoint \<pi> \<pi>\<close>
-proof -
-  have \<open>cinner (x - \<pi> x) y = 0\<close> if \<open>y\<in>M\<close> for x y
-    using a1 a2 is_projection_on_iff_orthog orthogonal_complement_orthoI that by blast
-  then have \<open>cinner x y = cinner (\<pi> x) y\<close> if \<open>y\<in>M\<close> for x y
-    by (metis cinner_diff_left eq_iff_diff_eq_0 that)
-  moreover have \<open>cinner x y = cinner x (\<pi> y)\<close> if \<open>y\<in>M\<close> for x y
-    using a1 is_projection_on_fixes_image that by fastforce
-  ultimately have 1: \<open>cinner (\<pi> x) y = cinner x (\<pi> y)\<close> if \<open>y\<in>M\<close> for x y
-    using that by metis
-
-  have \<open>cinner (\<pi> x) y = 0\<close> if \<open>y \<in> orthogonal_complement M\<close> for x y
-    by (meson a1 is_projection_on_in_image orthogonal_complement_orthoI' that)
-  also have \<open>0 = cinner x (\<pi> y)\<close> if \<open>y \<in> orthogonal_complement M\<close> for x y
-    by (metis a1 a2 cinner_zero_right closed_csubspace.subspace complex_vector.subspace_0 diff_zero is_projection_on_eqI that)
-  finally have 2: \<open>cinner (\<pi> x) y = cinner x (\<pi> y)\<close> if \<open>y \<in> orthogonal_complement M\<close> for x y
-    using that by simp
-
-  from 1 2
-  have \<open>cinner (\<pi> x) y = cinner x (\<pi> y)\<close> for x y
-    by (smt (verit, ccfv_threshold) a1 a2 cinner_commute cinner_diff_left eq_iff_diff_eq_0 is_projection_on_iff_orthog orthogonal_complement_orthoI)
-  then show ?thesis
-    by (simp add: is_cadjoint_def)
-qed
+  by (smt (verit, ccfv_threshold) a1 a2 cinner_diff_left cinner_eq_flip is_cadjoint_def is_projection_on_iff_orthog orthogonal_complement_orthoI right_minus_eq)
 
 lemma is_projection_on_cadjoint:
   fixes M :: \<open>'a::complex_inner set\<close>
   assumes \<open>is_projection_on \<pi> M\<close> and \<open>closed_csubspace M\<close>
   shows \<open>\<pi>\<^sup>\<dagger> = \<pi>\<close>
   using assms is_projection_on_is_cadjoint cadjoint_eqI is_cadjoint_def by blast
 
 lemma projection_cadjoint:
   fixes M :: \<open>'a::chilbert_space set\<close>
   assumes \<open>closed_csubspace M\<close>
   shows \<open>(projection M)\<^sup>\<dagger> = projection M\<close>
   using is_projection_on_cadjoint assms
   by (metis closed_csubspace.closed closed_csubspace.subspace csubspace_is_convex empty_iff orthog_proj_exists projection_is_projection_on)
 
-instance ccsubspace :: (chilbert_space) complete_orthomodular_lattice
-proof
-  show "inf x (- x) = bot"
-    for x :: "'a ccsubspace"
-    apply transfer
-    by (simp add: closed_csubspace_def complex_vector.subspace_0 orthogonal_complement_zero_intersection)
-
-  have \<open>t \<in> x +\<^sub>M orthogonal_complement x\<close>
-    if a1: \<open>closed_csubspace x\<close>
-    for t::'a and x
-  proof-
-    have e1: \<open>t = (projection x) t + (projection (orthogonal_complement x)) t\<close>
-      by (simp add: that)
-    have e2: \<open>(projection x) t \<in> x\<close>
-      by (metis closed_csubspace.closed closed_csubspace.subspace csubspace_is_convex empty_iff orthog_proj_exists projection_in_image that)
-    have e3: \<open>(projection (orthogonal_complement x)) t \<in> orthogonal_complement x\<close>
-      by (metis add_diff_cancel_left' e1 orthogonal_complementI projection_orthogonal that)
-    have "orthogonal_complement x \<subseteq> x +\<^sub>M orthogonal_complement x"
-      by (simp add: closed_sum_right_subset complex_vector.subspace_0 that)
-    thus ?thesis
-      using \<open>closed_csubspace x\<close>
-        \<open>projection (orthogonal_complement x) t \<in> orthogonal_complement x\<close> \<open>projection x t \<in> x\<close>
-        \<open>t = projection x t + projection (orthogonal_complement x) t\<close> in_mono
-        closed_sum_left_subset complex_vector.subspace_def
-      by (metis closed_csubspace.subspace closed_subspace_closed_sum orthogonal_complement_closed_subspace)
-  qed
-  hence b1: \<open>x +\<^sub>M orthogonal_complement x = UNIV\<close>
-    if a1: \<open>closed_csubspace x\<close>
-    for x::\<open>'a set\<close>
-    using that by blast
-  show "sup x (- x) = top"
-    for x :: "'a ccsubspace"
-    apply transfer
-    using b1 by auto
-  show "- (- x) = x"
-    for x :: "'a ccsubspace"
-    apply transfer
-    by (simp)
-
-  show "- y \<le> - x"
-    if "x \<le> y"
-    for x :: "'a ccsubspace"
-      and y :: "'a ccsubspace"
-    using that apply transfer
-    by simp
-
-  have c1: "x +\<^sub>M orthogonal_complement x \<inter> y \<subseteq> y"
-    if "closed_csubspace x"
-      and "closed_csubspace y"
-      and "x \<subseteq> y"
-    for x :: "'a set"
-      and y :: "'a set"
-    using that
-    by (simp add: closed_sum_is_sup)
-
-  have c2: \<open>u \<in> x +\<^sub>M ((orthogonal_complement x) \<inter> y)\<close>
-    if a1: "closed_csubspace x" and a2: "closed_csubspace y" and a3: "x \<subseteq> y" and x1: \<open>u \<in> y\<close>
-    for x :: "'a set" and y :: "'a set"  and u
-  proof-
-    have d4: \<open>(projection x) u \<in> x\<close>
-      by (metis a1 closed_csubspace_def csubspace_is_convex equals0D orthog_proj_exists projection_in_image)
-    hence d2: \<open>(projection x) u \<in> y\<close>
-      using a3 by auto
-    have d1: \<open>csubspace y\<close>
-      by (simp add: a2)
-    have \<open>u - (projection x) u \<in> orthogonal_complement x\<close>
-      by (simp add: a1 orthogonal_complementI projection_orthogonal)
-    moreover have  \<open>u - (projection x) u \<in> y\<close>
-      by (simp add: d1 d2 complex_vector.subspace_diff x1)
-    ultimately have d3: \<open>u - (projection x) u \<in> ((orthogonal_complement x) \<inter> y)\<close>
-      by simp
-    hence \<open>\<exists> v \<in> ((orthogonal_complement x) \<inter> y). u = (projection x) u + v\<close>
-      by (metis d3 diff_add_cancel ordered_field_class.sign_simps(2))
-    then obtain v where \<open>v \<in> ((orthogonal_complement x) \<inter> y)\<close> and \<open>u = (projection x) u + v\<close>
-      by blast
-    hence \<open>u \<in> x + ((orthogonal_complement x) \<inter> y)\<close>
-      by (metis d4 set_plus_intro)
-    thus ?thesis
-      unfolding closed_sum_def
-      using closure_subset by blast
-  qed
-
-  have c3: "y \<subseteq> x +\<^sub>M ((orthogonal_complement x) \<inter> y)"
-    if a1: "closed_csubspace x" and a2: "closed_csubspace y" and a3: "x \<subseteq> y"
-    for x y :: "'a set"
-    using c2 a1 a2 a3 by auto
-
-  show "sup x (inf (- x) y) = y"
-    if "x \<le> y"
-    for x y :: "'a ccsubspace"
-    using that apply transfer
-    using c1 c3
-    by (simp add: subset_antisym)
-
-  show "x - y = inf x (- y)"
-    for x y :: "'a ccsubspace"
-    apply transfer
-    by simp
-qed
 
 subsection \<open>More projections\<close>
 
 text \<open>These lemmas logically belong in the "projections" section above but depend on lemmas developed later.\<close>
 
 lemma is_projection_on_plus:
-  assumes "\<And>x y. x:A \<Longrightarrow> y:B \<Longrightarrow> is_orthogonal x y"
+  assumes "\<And>x y. x \<in> A \<Longrightarrow> y \<in> B \<Longrightarrow> is_orthogonal x y"
   assumes \<open>closed_csubspace A\<close>
   assumes \<open>closed_csubspace B\<close>
   assumes \<open>is_projection_on \<pi>A A\<close>
   assumes \<open>is_projection_on \<pi>B B\<close>
   shows \<open>is_projection_on (\<lambda>x. \<pi>A x + \<pi>B x) (A +\<^sub>M B)\<close>
 proof (rule is_projection_on_iff_orthog[THEN iffD2, rule_format])
   show clAB: \<open>closed_csubspace (A +\<^sub>M B)\<close>
     by (simp add: assms(2) assms(3) closed_subspace_closed_sum)
   fix h
   have 1: \<open>\<pi>A h + \<pi>B h \<in> A +\<^sub>M B\<close>
     by (meson clAB assms(2) assms(3) assms(4) assms(5) closed_csubspace_def closed_sum_left_subset closed_sum_right_subset complex_vector.subspace_def in_mono is_projection_on_in_image)
 
   have \<open>\<pi>A (\<pi>B h) = 0\<close>
     by (smt (verit, del_insts) assms(1) assms(2) assms(4) assms(5) cinner_eq_zero_iff is_cadjoint_def is_projection_on_in_image is_projection_on_is_cadjoint)
   then have \<open>h - (\<pi>A h + \<pi>B h) = (h - \<pi>B h) - \<pi>A (h - \<pi>B h)\<close>
     by (smt (verit) add.right_neutral add_diff_cancel_left' assms(2) assms(4) closed_csubspace.subspace complex_vector.subspace_diff diff_add_eq_diff_diff_swap diff_diff_add is_projection_on_iff_orthog orthog_proj_unique orthogonal_complement_closed_subspace)
   also have \<open>\<dots> \<in> orthogonal_complement A\<close>
     using assms(2) assms(4) is_projection_on_iff_orthog by blast
   finally have orthoA: \<open>h - (\<pi>A h + \<pi>B h) \<in> orthogonal_complement A\<close>
     by -
 
   have \<open>\<pi>B (\<pi>A h) = 0\<close>
     by (smt (verit, del_insts) assms(1) assms(3) assms(4) assms(5) cinner_eq_zero_iff is_cadjoint_def is_projection_on_in_image is_projection_on_is_cadjoint)
   then have \<open>h - (\<pi>A h + \<pi>B h) = (h - \<pi>A h) - \<pi>B (h - \<pi>A h)\<close>
     by (smt (verit) add.right_neutral add_diff_cancel assms(3) assms(5) closed_csubspace.subspace complex_vector.subspace_diff diff_add_eq_diff_diff_swap diff_diff_add is_projection_on_iff_orthog orthog_proj_unique orthogonal_complement_closed_subspace)
   also have \<open>\<dots> \<in> orthogonal_complement B\<close>
     using assms(3) assms(5) is_projection_on_iff_orthog by blast
   finally have orthoB: \<open>h - (\<pi>A h + \<pi>B h) \<in> orthogonal_complement B\<close>
     by -
 
   from orthoA orthoB
   have 2: \<open>h - (\<pi>A h + \<pi>B h) \<in> orthogonal_complement (A +\<^sub>M B)\<close>
     by (metis IntI assms(2) assms(3) closed_csubspace_def complex_vector.subspace_def de_morgan_orthogonal_complement_plus)
 
   from 1 2 show \<open>h - (\<pi>A h + \<pi>B h) \<in> orthogonal_complement (A +\<^sub>M B) \<and> \<pi>A h + \<pi>B h \<in> A +\<^sub>M B\<close>
     by simp
 qed
 
 lemma projection_plus:
   fixes A B :: "'a::chilbert_space set"
   assumes "\<And>x y. x:A \<Longrightarrow> y:B \<Longrightarrow> is_orthogonal x y"
   assumes \<open>closed_csubspace A\<close>
   assumes \<open>closed_csubspace B\<close>
   shows \<open>projection (A +\<^sub>M B) = (\<lambda>x. projection A x + projection B x)\<close>
 proof -
   have \<open>is_projection_on (\<lambda>x. projection A x + projection B x) (A +\<^sub>M B)\<close>
     apply (rule is_projection_on_plus)
     using assms by auto
   then show ?thesis
     by (meson assms(2) assms(3) closed_csubspace.subspace closed_subspace_closed_sum csubspace_is_convex projection_eqI')
 qed
 
 lemma is_projection_on_insert:
-  assumes ortho: "\<And>s. s \<in> S \<Longrightarrow> \<langle>a, s\<rangle> = 0"
+  assumes ortho: "\<And>s. s \<in> S \<Longrightarrow> is_orthogonal a s"
   assumes \<open>is_projection_on \<pi> (closure (cspan S))\<close>
   assumes \<open>is_projection_on \<pi>a (cspan {a})\<close>
   shows "is_projection_on (\<lambda>x. \<pi>a x + \<pi> x) (closure (cspan (insert a S)))"
 proof -
   from ortho
   have \<open>x \<in> cspan {a} \<Longrightarrow> y \<in> closure (cspan S) \<Longrightarrow> is_orthogonal x y\<close> for x y
     using is_orthogonal_cspan is_orthogonal_closure is_orthogonal_sym
     by (smt (verit, ccfv_threshold) empty_iff insert_iff)
   then have \<open>is_projection_on (\<lambda>x. \<pi>a x + \<pi> x) (cspan {a} +\<^sub>M closure (cspan S))\<close>
     apply (rule is_projection_on_plus)
     using assms by (auto simp add: closed_csubspace.intro)
   also have \<open>\<dots> = closure (cspan (insert a S))\<close>
     using closed_sum_cspan[where X=\<open>{a}\<close>] by simp
   finally show ?thesis
     by -
 qed
 
 lemma projection_insert:
   fixes a :: \<open>'a::chilbert_space\<close>
-  assumes a1: "\<And>s. s \<in> S \<Longrightarrow> \<langle>a, s\<rangle> = 0"
+  assumes a1: "\<And>s. s \<in> S \<Longrightarrow> is_orthogonal a s"
   shows "projection (closure (cspan (insert a S))) u
         = projection (cspan {a}) u + projection (closure (cspan S)) u"
   using is_projection_on_insert[where S=S, OF a1]
   by (metis (no_types, lifting) closed_closure closed_csubspace.intro closure_is_csubspace complex_vector.subspace_span csubspace_is_convex finite.intros(1) finite.intros(2) finite_cspan_closed_csubspace projection_eqI' projection_is_projection_on')
 
 lemma projection_insert_finite:
-  assumes a1: "\<And>s. s \<in> S \<Longrightarrow> \<langle>a, s\<rangle> = 0" and a2: "finite (S::'a::chilbert_space set)"
+  fixes S :: \<open>'a::chilbert_space set\<close>
+  assumes a1: "\<And>s. s \<in> S \<Longrightarrow> is_orthogonal a s" and a2: "finite S"
   shows "projection (cspan (insert a S)) u
         = projection (cspan {a}) u + projection (cspan S) u"
   using projection_insert
   by (metis a1 a2 closure_finite_cspan finite.insertI)
 
 subsection \<open>Canonical basis (\<open>onb_enum\<close>)\<close>
 
 setup \<open>Sign.add_const_constraint (\<^const_name>\<open>is_ortho_set\<close>, SOME \<^typ>\<open>'a set \<Rightarrow> bool\<close>)\<close>
 
 class onb_enum = basis_enum + complex_inner +
   assumes is_orthonormal: "is_ortho_set (set canonical_basis)"
     and is_normal: "\<And>x. x \<in> (set canonical_basis) \<Longrightarrow> norm x = 1"
 
 setup \<open>Sign.add_const_constraint (\<^const_name>\<open>is_ortho_set\<close>, SOME \<^typ>\<open>'a::complex_inner set \<Rightarrow> bool\<close>)\<close>
 
 lemma cinner_canonical_basis:
   assumes \<open>i < length (canonical_basis :: 'a::onb_enum list)\<close>
   assumes \<open>j < length (canonical_basis :: 'a::onb_enum list)\<close>
   shows \<open>cinner (canonical_basis!i :: 'a) (canonical_basis!j) = (if i=j then 1 else 0)\<close>
   by (metis assms(1) assms(2) distinct_canonical_basis is_normal is_ortho_set_def is_orthonormal nth_eq_iff_index_eq nth_mem of_real_1 power2_norm_eq_cinner power_one)
 
 instance onb_enum \<subseteq> chilbert_space
 proof
-  show "convergent X"
-    if "Cauchy X"
-    for X :: "nat \<Rightarrow> 'a"
-  proof-
-    have \<open>finite (set canonical_basis)\<close>
-      by simp
-    have \<open>Cauchy (\<lambda> n. \<langle> t, X n \<rangle>)\<close> for t
-      by (simp add: bounded_clinear.Cauchy bounded_clinear_cinner_right that)
-    hence \<open>convergent (\<lambda> n. \<langle> t, X n \<rangle>)\<close>
-      for t
-      by (simp add: Cauchy_convergent_iff)
-    hence \<open>\<forall> t\<in>set canonical_basis. \<exists> L. (\<lambda> n. \<langle> t, X n \<rangle>) \<longlonglongrightarrow> L\<close>
-      by (simp add: convergentD)
-    hence \<open>\<exists> L. \<forall> t\<in>set canonical_basis. (\<lambda> n. \<langle> t, X n \<rangle>) \<longlonglongrightarrow> L t\<close>
-      by metis
-    then obtain L where \<open>\<And> t. t\<in>set canonical_basis \<Longrightarrow> (\<lambda> n. \<langle> t, X n \<rangle>) \<longlonglongrightarrow> L t\<close>
-      by blast
-    define l where \<open>l = (\<Sum>t\<in>set canonical_basis. L t *\<^sub>C t)\<close>
-    have x1: \<open>X n = (\<Sum>t\<in>set canonical_basis. \<langle> t, X n \<rangle> *\<^sub>C t)\<close>
-      for n
-      using onb_expansion_finite[where T = "set canonical_basis" and x = "X n"]
-        \<open>finite (set canonical_basis)\<close>
-      by (smt is_generator_set is_normal is_orthonormal)
-
-    have \<open>(\<lambda> n. \<langle> t, X n \<rangle> *\<^sub>C t) \<longlonglongrightarrow> L t *\<^sub>C t\<close>
-      if r1: \<open>t\<in>set canonical_basis\<close>
-      for t
-    proof-
-      have \<open>(\<lambda> n. \<langle> t, X n \<rangle>) \<longlonglongrightarrow> L t\<close>
-        using r1  \<open>\<And> t. t\<in>set canonical_basis \<Longrightarrow> (\<lambda> n. \<langle> t, X n \<rangle>) \<longlonglongrightarrow> L t\<close>
-        by blast
-      define f where \<open>f x = x *\<^sub>C t\<close> for x
-      have \<open>isCont f r\<close>
-        for r
-        unfolding f_def
-        by (simp add: bounded_clinear_scaleC_left clinear_continuous_at)
-      hence \<open>(\<lambda> n. f \<langle> t, X n \<rangle>) \<longlonglongrightarrow> f (L t)\<close>
-        using \<open>(\<lambda>n. \<langle>t, X n\<rangle>) \<longlonglongrightarrow> L t\<close> isCont_tendsto_compose by blast
-      hence \<open>(\<lambda> n. \<langle> t, X n \<rangle> *\<^sub>C t) \<longlonglongrightarrow> L t *\<^sub>C t\<close>
-        unfolding f_def
-        by simp
-      thus ?thesis by blast
-    qed
-    hence \<open>(\<lambda> n. (\<Sum>t\<in>set canonical_basis. \<langle> t, X n \<rangle> *\<^sub>C t))
-    \<longlonglongrightarrow>  (\<Sum>t\<in>set canonical_basis. L t *\<^sub>C t)\<close>
-      using \<open>finite (set canonical_basis)\<close>
-        tendsto_sum[where I = "set canonical_basis" and f = "\<lambda> t. \<lambda> n. \<langle>t, X n\<rangle> *\<^sub>C t"
-          and a = "\<lambda> t. L t *\<^sub>C t"]
-      by auto
-    hence x2: \<open>(\<lambda> n. (\<Sum>t\<in>set canonical_basis. \<langle> t, X n \<rangle> *\<^sub>C t)) \<longlonglongrightarrow> l\<close>
-      using l_def by blast
-    have \<open>X \<longlonglongrightarrow> l\<close>
-      using x1 x2 by simp
-    thus ?thesis
-      unfolding convergent_def by blast
-  qed
+  have \<open>complete (UNIV :: 'a set)\<close>
+    using finite_cspan_complete[where B=\<open>set canonical_basis\<close>]
+    by simp
+  then show "convergent X" if "Cauchy X" for X :: "nat \<Rightarrow> 'a"
+    by (simp add: complete_def convergent_def that)
 qed
 
 subsection \<open>Conjugate space\<close>
 
 instantiation conjugate_space :: (complex_inner) complex_inner begin
 lift_definition cinner_conjugate_space :: "'a conjugate_space \<Rightarrow> 'a conjugate_space \<Rightarrow> complex" is
   \<open>\<lambda>x y. cinner y x\<close>.
 instance
   apply (intro_classes; transfer)
        apply (simp_all add: )
     apply (simp add: cinner_add_right)
   using cinner_ge_zero norm_eq_sqrt_cinner by auto
 end
 
+instance conjugate_space :: (chilbert_space) chilbert_space..
+
 subsection \<open>Misc (ctd.)\<close>
 
 
 lemma separating_dense_span: 
   assumes \<open>\<And>F G :: 'a::chilbert_space \<Rightarrow> 'b::{complex_normed_vector,not_singleton}. 
            bounded_clinear F \<Longrightarrow> bounded_clinear G \<Longrightarrow> (\<forall>x\<in>S. F x = G x) \<Longrightarrow> F = G\<close>
   shows \<open>closure (cspan S) = UNIV\<close>
 proof -
   have \<open>\<psi> = 0\<close> if \<open>\<psi> \<in> orthogonal_complement S\<close> for \<psi>
   proof -
     obtain \<phi> :: 'b where \<open>\<phi> \<noteq> 0\<close>
       by fastforce
     have \<open>(\<lambda>x. cinner \<psi> x *\<^sub>C \<phi>) = (\<lambda>_. 0)\<close> 
       apply (rule assms[rule_format])
       using orthogonal_complement_orthoI that
       by (auto simp add: bounded_clinear_cinner_right bounded_clinear_scaleC_const)
     then have \<open>cinner \<psi> \<psi> = 0\<close>
       by (meson \<open>\<phi> \<noteq> 0\<close> scaleC_eq_0_iff)
     then show \<open>\<psi> = 0\<close>
       by auto
   qed
   then have \<open>orthogonal_complement (orthogonal_complement S) = UNIV\<close>
     by (metis UNIV_eq_I cinner_zero_right orthogonal_complementI)
   then show \<open>closure (cspan S) = UNIV\<close>
     by (simp add: orthogonal_complement_orthogonal_complement_closure_cspan)
 qed
 
-
-instance conjugate_space :: (chilbert_space) chilbert_space..
-
 end
diff --git a/thys/Complex_Bounded_Operators/Complex_Inner_Product0.thy b/thys/Complex_Bounded_Operators/Complex_Inner_Product0.thy
--- a/thys/Complex_Bounded_Operators/Complex_Inner_Product0.thy
+++ b/thys/Complex_Bounded_Operators/Complex_Inner_Product0.thy
@@ -1,548 +1,548 @@
 (*  Based on HOL/Real_Vector_Spaces.thy by Brian Huffman
     Adapted to the complex case by Dominique Unruh *)
 
 section \<open>\<open>Complex_Inner_Product0\<close> -- Inner Product Spaces and Gradient Derivative\<close>
 
 theory Complex_Inner_Product0
   imports
     Complex_Main Complex_Vector_Spaces
     "HOL-Analysis.Inner_Product"
     "Complex_Bounded_Operators.Extra_Ordered_Fields"
 begin
 
 subsection \<open>Complex inner product spaces\<close>
 
 text \<open>
   Temporarily relax type constraints for \<^term>\<open>open\<close>, \<^term>\<open>uniformity\<close>,
   \<^term>\<open>dist\<close>, and \<^term>\<open>norm\<close>.
 \<close>
 
 setup \<open>Sign.add_const_constraint
   (\<^const_name>\<open>open\<close>, SOME \<^typ>\<open>'a::open set \<Rightarrow> bool\<close>)\<close>
 
 setup \<open>Sign.add_const_constraint
   (\<^const_name>\<open>dist\<close>, SOME \<^typ>\<open>'a::dist \<Rightarrow> 'a \<Rightarrow> real\<close>)\<close>
 
 setup \<open>Sign.add_const_constraint
   (\<^const_name>\<open>uniformity\<close>, SOME \<^typ>\<open>('a::uniformity \<times> 'a) filter\<close>)\<close>
 
 setup \<open>Sign.add_const_constraint
   (\<^const_name>\<open>norm\<close>, SOME \<^typ>\<open>'a::norm \<Rightarrow> real\<close>)\<close>
 
 class complex_inner = complex_vector + sgn_div_norm + dist_norm + uniformity_dist + open_uniformity +
   fixes cinner :: "'a \<Rightarrow> 'a \<Rightarrow> complex"
   assumes cinner_commute: "cinner x y = cnj (cinner y x)"
     and cinner_add_left: "cinner (x + y) z = cinner x z + cinner y z"
     and cinner_scaleC_left [simp]: "cinner (scaleC r x) y = (cnj r) * (cinner x y)"
     and cinner_ge_zero [simp]: "0 \<le> cinner x x"
     and cinner_eq_zero_iff [simp]: "cinner x x = 0 \<longleftrightarrow> x = 0"
     and norm_eq_sqrt_cinner: "norm x = sqrt (cmod (cinner x x))"
 begin
 
 lemma cinner_zero_left [simp]: "cinner 0 x = 0"
   using cinner_add_left [of 0 0 x] by simp
 
 lemma cinner_minus_left [simp]: "cinner (- x) y = - cinner x y"
   using cinner_add_left [of x "- x" y]
   by (simp add: group_add_class.add_eq_0_iff)
 
 lemma cinner_diff_left: "cinner (x - y) z = cinner x z - cinner y z"
   using cinner_add_left [of x "- y" z] by simp
 
 lemma cinner_sum_left: "cinner (\<Sum>x\<in>A. f x) y = (\<Sum>x\<in>A. cinner (f x) y)"
   by (cases "finite A", induct set: finite, simp_all add: cinner_add_left)
 
 lemma call_zero_iff [simp]: "(\<forall>u. cinner x u = 0) \<longleftrightarrow> (x = 0)"
   by auto (use cinner_eq_zero_iff in blast)
 
 text \<open>Transfer distributivity rules to right argument.\<close>
 
 lemma cinner_add_right: "cinner x (y + z) = cinner x y + cinner x z"
   using cinner_add_left [of y z x]
   by (metis complex_cnj_add local.cinner_commute)
 
 lemma cinner_scaleC_right [simp]: "cinner x (scaleC r y) = r * (cinner x y)"
   using cinner_scaleC_left [of r y x]
   by (metis complex_cnj_cnj complex_cnj_mult local.cinner_commute)
 
 lemma cinner_zero_right [simp]: "cinner x 0 = 0"
   using cinner_zero_left [of x]
   by (metis (mono_tags, opaque_lifting) complex_cnj_zero local.cinner_commute)
 
 lemma cinner_minus_right [simp]: "cinner x (- y) = - cinner x y"
   using cinner_minus_left [of y x]
   by (metis complex_cnj_minus local.cinner_commute)
 
 lemma cinner_diff_right: "cinner x (y - z) = cinner x y - cinner x z"
   using cinner_diff_left [of y z x]
   by (metis complex_cnj_diff local.cinner_commute)
 
 lemma cinner_sum_right: "cinner x (\<Sum>y\<in>A. f y) = (\<Sum>y\<in>A. cinner x (f y))"
 proof (subst cinner_commute)
   have "(\<Sum>y\<in>A. cinner (f y) x) = (\<Sum>y\<in>A. cinner (f y) x)"
     by blast
   hence "cnj (\<Sum>y\<in>A. cinner (f y) x) = cnj (\<Sum>y\<in>A. (cinner (f y) x))"
     by simp
   hence "cnj (cinner (sum f A) x) = (\<Sum>y\<in>A. cnj (cinner (f y) x))"
     by (simp add: cinner_sum_left)
   thus "cnj (cinner (sum f A) x) = (\<Sum>y\<in>A. (cinner x (f y)))"
     by (subst (2) cinner_commute)
 qed
 
 lemmas cinner_add [algebra_simps] = cinner_add_left cinner_add_right
 lemmas cinner_diff [algebra_simps]  = cinner_diff_left cinner_diff_right
 lemmas cinner_scaleC = cinner_scaleC_left cinner_scaleC_right
 
 (* text \<open>Legacy theorem names\<close>
 lemmas cinner_left_distrib = cinner_add_left
 lemmas cinner_right_distrib = cinner_add_right
 lemmas cinner_distrib = cinner_left_distrib cinner_right_distrib *)
 
 lemma cinner_gt_zero_iff [simp]: "0 < cinner x x \<longleftrightarrow> x \<noteq> 0"
   by (smt (verit) less_irrefl local.cinner_eq_zero_iff local.cinner_ge_zero order.not_eq_order_implies_strict)
 
 (* In Inner_Product, we have
   lemma power2_norm_eq_cinner: "(norm x)\<^sup>2 = cinner x x"
 The following are two ways of inserting the conversions between real and complex into this:
 *)
 
 lemma power2_norm_eq_cinner:
   shows "(complex_of_real (norm x))\<^sup>2 = (cinner x x)"
   by (smt (verit, del_insts) Im_complex_of_real Re_complex_of_real cinner_gt_zero_iff cinner_zero_right cmod_def complex_eq_0 complex_eq_iff less_complex_def local.norm_eq_sqrt_cinner of_real_power real_sqrt_abs real_sqrt_pow2_iff zero_complex.sel(1))
 
 lemma power2_norm_eq_cinner':
   shows "(norm x)\<^sup>2 = Re (cinner x x)"
   by (metis Re_complex_of_real of_real_power power2_norm_eq_cinner)
 
 text \<open>Identities involving real multiplication and division.\<close>
 
 lemma cinner_mult_left: "cinner (of_complex m * a) b = cnj m * (cinner a b)"
   by (simp add: of_complex_def)
 
 lemma cinner_mult_right: "cinner a (of_complex m * b) = m * (cinner a b)"
   by (metis complex_inner_class.cinner_scaleC_right scaleC_conv_of_complex)
 
 lemma cinner_mult_left': "cinner (a * of_complex m) b = cnj m * (cinner a b)"
   by (metis cinner_mult_left mult.right_neutral mult_scaleC_right scaleC_conv_of_complex)
 
 lemma cinner_mult_right': "cinner a (b * of_complex m) = (cinner a b) * m"
   by (simp add: complex_inner_class.cinner_scaleC_right of_complex_def)
 
 (* In Inner_Product, we have
 lemma Cauchy_Schwarz_ineq:
   "(cinner x y)\<^sup>2 \<le> cinner x x * cinner y y"
 The following are two ways of inserting the conversions between real and complex into this:
 *)
 
 lemma Cauchy_Schwarz_ineq:
   "(cinner x y) * (cinner y x) \<le> cinner x x * cinner y y"
 proof (cases)
   assume "y = 0"
   thus ?thesis by simp
 next
   assume y: "y \<noteq> 0"
   have [simp]: "cnj (cinner y y) = cinner y y" for y
     by (metis cinner_commute)
   define r where "r = cnj (cinner x y) / cinner y y"
   have "0 \<le> cinner (x - scaleC r y) (x - scaleC r y)"
     by (rule cinner_ge_zero)
   also have "\<dots> = cinner x x - r * cinner x y - cnj r * cinner y x + r * cnj r * cinner y y"
     unfolding cinner_diff_left cinner_diff_right cinner_scaleC_left cinner_scaleC_right
     by (smt (z3) cancel_comm_monoid_add_class.diff_cancel cancel_comm_monoid_add_class.diff_zero complex_cnj_divide group_add_class.diff_add_cancel local.cinner_commute local.cinner_eq_zero_iff local.cinner_scaleC_left mult.assoc mult.commute mult_eq_0_iff nonzero_eq_divide_eq r_def y)
   also have "\<dots> = cinner x x - cinner y x * cnj r"
     unfolding r_def by auto
   also have "\<dots> = cinner x x - cinner x y * cnj (cinner x y) / cinner y y"
     unfolding r_def
     by (metis complex_cnj_divide local.cinner_commute mult.commute times_divide_eq_left)
   finally have "0 \<le> cinner x x - cinner x y * cnj (cinner x y) / cinner y y" .
   hence "cinner x y * cnj (cinner x y) / cinner y y \<le> cinner x x"
     by (simp add: le_diff_eq)
   thus "cinner x y * cinner y x \<le> cinner x x * cinner y y"
     by (metis cinner_gt_zero_iff local.cinner_commute nice_ordered_field_class.pos_divide_le_eq y)
 qed
 
 
 lemma Cauchy_Schwarz_ineq2:
   shows "norm (cinner x y) \<le> norm x * norm y"
 proof (rule power2_le_imp_le)
   have "(norm (cinner x y))^2 = Re (cinner x y * cinner y x)"
     by (metis (full_types) Re_complex_of_real complex_norm_square local.cinner_commute)
   also have "\<dots> \<le> Re (cinner x x * cinner y y)"
     using Cauchy_Schwarz_ineq by (rule Re_mono)
   also have "\<dots> = Re (complex_of_real ((norm x)^2) * complex_of_real ((norm y)^2))"
     by (simp add: power2_norm_eq_cinner)
   also have "\<dots> = (norm x * norm y)\<^sup>2"
     by (simp add: power_mult_distrib)
   finally show "(cmod (cinner x y))^2 \<le> (norm x * norm y)\<^sup>2" .
   show "0 \<le> norm x * norm y"
     by (simp add: local.norm_eq_sqrt_cinner)
 qed
 
 (* The following variant does not hold in the complex case: *)
 (* lemma norm_cauchy_schwarz: "cinner x y \<le> norm x * norm y"
   using Cauchy_Schwarz_ineq2 [of x y] by auto *)
 
 subclass complex_normed_vector
 proof
   fix a :: complex and r :: real and x y :: 'a
   show "norm x = 0 \<longleftrightarrow> x = 0"
     unfolding norm_eq_sqrt_cinner by simp
   show "norm (x + y) \<le> norm x + norm y"
   proof (rule power2_le_imp_le)
     have "Re (cinner x y) \<le> cmod (cinner x y)"
       if "\<And>x. Re x \<le> cmod x" and
         "\<And>x y. x \<le> y \<Longrightarrow> complex_of_real x \<le> complex_of_real y"
       using that by simp
     hence a1: "2 * Re (cinner x y) \<le> 2 * cmod (cinner x y)"
       if "\<And>x. Re x \<le> cmod x" and
         "\<And>x y. x \<le> y \<Longrightarrow> complex_of_real x \<le> complex_of_real y"
       using that by simp
     have "cinner x y + cinner y x = complex_of_real (2 * Re (cinner x y))"
       by (metis complex_add_cnj local.cinner_commute)
     also have "\<dots> \<le> complex_of_real (2 * cmod (cinner x y))"
       using complex_Re_le_cmod complex_of_real_mono a1
       by blast
     also have "\<dots> = 2 * abs (cinner x y)"
       unfolding abs_complex_def by simp
     also have "\<dots> \<le> 2 * complex_of_real (norm x) * complex_of_real (norm y)"
       using Cauchy_Schwarz_ineq2 unfolding abs_complex_def less_eq_complex_def by auto
     finally have xyyx: "cinner x y + cinner y x \<le> complex_of_real (2 * norm x * norm y)"
       by auto
     have "complex_of_real ((norm (x + y))\<^sup>2) = cinner (x+y) (x+y)"
       by (simp add: power2_norm_eq_cinner)
     also have "\<dots> = cinner x x + cinner x y + cinner y x + cinner y y"
       by (simp add: cinner_add)
     also have "\<dots> = complex_of_real ((norm x)\<^sup>2) + complex_of_real ((norm y)\<^sup>2) + cinner x y + cinner y x"
       by (simp add: power2_norm_eq_cinner)
     also have "\<dots> \<le> complex_of_real ((norm x)\<^sup>2) + complex_of_real ((norm y)\<^sup>2) + complex_of_real (2 * norm x * norm y)"
       using xyyx by auto
     also have "\<dots> = complex_of_real ((norm x + norm y)\<^sup>2)"
       unfolding power2_sum by auto
     finally show "(norm (x + y))\<^sup>2 \<le> (norm x + norm y)\<^sup>2"
       using complex_of_real_mono_iff by blast
     show "0 \<le> norm x + norm y"
       unfolding norm_eq_sqrt_cinner by simp
   qed
   show norm_scaleC: "norm (a *\<^sub>C x) = cmod a * norm x" for a
   proof (rule power2_eq_imp_eq)
     show "(norm (a *\<^sub>C x))\<^sup>2 = (cmod a * norm x)\<^sup>2"
       by (simp_all add: norm_eq_sqrt_cinner norm_mult power2_eq_square)
     show "0 \<le> norm (a *\<^sub>C x)"
       by (simp_all add: norm_eq_sqrt_cinner)
     show "0 \<le> cmod a * norm x"
       by (simp_all add: norm_eq_sqrt_cinner)
   qed
   show "norm (r *\<^sub>R x) = \<bar>r\<bar> * norm x"
     unfolding scaleR_scaleC norm_scaleC by auto
 qed
 
 end
 
 (* Does not hold in the complex case *)
 (* lemma csquare_bound_lemma:
   fixes x :: complex
   shows "x < (1 + x) * (1 + x)" *)
 
 lemma csquare_continuous:
   fixes e :: real
   shows "e > 0 \<Longrightarrow> \<exists>d. 0 < d \<and> (\<forall>y. cmod (y - x) < d \<longrightarrow> cmod (y * y - x * x) < e)"
   using isCont_power[OF continuous_ident, of x, unfolded isCont_def LIM_eq, rule_format, of e 2]
   by (force simp add: power2_eq_square)
 
 lemma cnorm_le: "norm x \<le> norm y \<longleftrightarrow> cinner x x \<le> cinner y y"
   by (smt (verit) complex_of_real_mono_iff norm_eq_sqrt_cinner norm_ge_zero of_real_power power2_norm_eq_cinner real_sqrt_le_mono real_sqrt_pow2)
 
 lemma cnorm_lt: "norm x < norm y \<longleftrightarrow> cinner x x < cinner y y"
   by (meson cnorm_le less_le_not_le)
 
 lemma cnorm_eq: "norm x = norm y \<longleftrightarrow> cinner x x = cinner y y"
   by (metis norm_eq_sqrt_cinner power2_norm_eq_cinner)
 
 lemma cnorm_eq_1: "norm x = 1 \<longleftrightarrow> cinner x x = 1"
   by (metis cinner_ge_zero complex_of_real_cmod norm_eq_sqrt_cinner norm_one of_real_1 real_sqrt_eq_iff real_sqrt_one)
 
 lemma cinner_divide_left:
   fixes a :: "'a :: {complex_inner,complex_div_algebra}"
   shows "cinner (a / of_complex m) b = (cinner a b) / cnj m"
   by (metis cinner_mult_left' complex_cnj_inverse divide_inverse mult.commute of_complex_inverse)
 
 lemma cinner_divide_right:
   fixes a :: "'a :: {complex_inner,complex_div_algebra}"
   shows "cinner a (b / of_complex m) = (cinner a b) / m"
   by (metis cinner_mult_right' divide_inverse of_complex_inverse)
 
 text \<open>
   Re-enable constraints for \<^term>\<open>open\<close>, \<^term>\<open>uniformity\<close>,
   \<^term>\<open>dist\<close>, and \<^term>\<open>norm\<close>.
 \<close>
 
 setup \<open>Sign.add_const_constraint
   (\<^const_name>\<open>open\<close>, SOME \<^typ>\<open>'a::topological_space set \<Rightarrow> bool\<close>)\<close>
 
 setup \<open>Sign.add_const_constraint
   (\<^const_name>\<open>uniformity\<close>, SOME \<^typ>\<open>('a::uniform_space \<times> 'a) filter\<close>)\<close>
 
 setup \<open>Sign.add_const_constraint
   (\<^const_name>\<open>dist\<close>, SOME \<^typ>\<open>'a::metric_space \<Rightarrow> 'a \<Rightarrow> real\<close>)\<close>
 
 setup \<open>Sign.add_const_constraint
   (\<^const_name>\<open>norm\<close>, SOME \<^typ>\<open>'a::real_normed_vector \<Rightarrow> real\<close>)\<close>
 
 
 lemma bounded_sesquilinear_cinner:
   "bounded_sesquilinear (cinner::'a::complex_inner \<Rightarrow> 'a \<Rightarrow> complex)"
 proof
   fix x y z :: 'a and r :: complex
   show "cinner (x + y) z = cinner x z + cinner y z"
     by (rule cinner_add_left)
   show "cinner x (y + z) = cinner x y + cinner x z"
     by (rule cinner_add_right)
   show "cinner (scaleC r x) y = scaleC (cnj r) (cinner x y)"
     unfolding complex_scaleC_def by (rule cinner_scaleC_left)
   show "cinner x (scaleC r y) = scaleC r (cinner x y)"
     unfolding complex_scaleC_def by (rule cinner_scaleC_right)
   have "\<forall>x y::'a. norm (cinner x y) \<le> norm x * norm y * 1"
     by (simp add: complex_inner_class.Cauchy_Schwarz_ineq2)
   thus "\<exists>K. \<forall>x y::'a. norm (cinner x y) \<le> norm x * norm y * K"
     by metis
 qed
 
 lemmas tendsto_cinner [tendsto_intros] =
   bounded_bilinear.tendsto [OF bounded_sesquilinear_cinner[THEN bounded_sesquilinear.bounded_bilinear]]
 
 lemmas isCont_cinner [simp] =
   bounded_bilinear.isCont [OF bounded_sesquilinear_cinner[THEN bounded_sesquilinear.bounded_bilinear]]
 
 lemmas has_derivative_cinner [derivative_intros] =
   bounded_bilinear.FDERIV [OF bounded_sesquilinear_cinner[THEN bounded_sesquilinear.bounded_bilinear]]
 
 lemmas bounded_antilinear_cinner_left =
   bounded_sesquilinear.bounded_antilinear_left [OF bounded_sesquilinear_cinner]
 
 lemmas bounded_clinear_cinner_right =
   bounded_sesquilinear.bounded_clinear_right [OF bounded_sesquilinear_cinner]
 
 lemmas bounded_antilinear_cinner_left_comp = bounded_antilinear_cinner_left[THEN bounded_antilinear_o_bounded_clinear]
 
 lemmas bounded_clinear_cinner_right_comp = bounded_clinear_cinner_right[THEN bounded_clinear_compose]
 
 lemmas has_derivative_cinner_right [derivative_intros] =
   bounded_linear.has_derivative [OF bounded_clinear_cinner_right[THEN bounded_clinear.bounded_linear]]
 
 lemmas has_derivative_cinner_left [derivative_intros] =
   bounded_linear.has_derivative [OF bounded_antilinear_cinner_left[THEN bounded_antilinear.bounded_linear]]
 
 lemma differentiable_cinner [simp]:
   "f differentiable (at x within s) \<Longrightarrow> g differentiable at x within s \<Longrightarrow> (\<lambda>x. cinner (f x) (g x)) differentiable at x within s"
   unfolding differentiable_def by (blast intro: has_derivative_cinner)
 
 
 subsection \<open>Class instances\<close>
 
 instantiation complex :: complex_inner
 begin
 
 definition cinner_complex_def [simp]: "cinner x y = cnj x * y"
 
 instance
 proof
   fix x y z r :: complex
   show "cinner x y = cnj (cinner y x)"
     unfolding cinner_complex_def by auto
   show "cinner (x + y) z = cinner x z + cinner y z"
     unfolding cinner_complex_def
     by (simp add: ring_class.ring_distribs(2))
   show "cinner (scaleC r x) y = cnj r * cinner x y"
     unfolding cinner_complex_def complex_scaleC_def by simp
   show "0 \<le> cinner x x"
     by simp
   show "cinner x x = 0 \<longleftrightarrow> x = 0"
     unfolding cinner_complex_def by simp
   have "cmod (Complex x1 x2) = sqrt (cmod (cinner (Complex x1 x2) (Complex x1 x2)))"
     for x1 x2
     unfolding cinner_complex_def complex_cnj complex_mult complex_norm
     by (simp add: power2_eq_square)
   thus "norm x = sqrt (cmod (cinner x x))"
     by (cases x, hypsubst_thin)
 qed
 
 end
 
 lemma
   shows complex_inner_1_left[simp]: "cinner 1 x = x"
     and complex_inner_1_right[simp]: "cinner x 1 = cnj x"
   by simp_all
 
-(* No analogous to \<open>instantiation complex :: real_inner\<close> or to
+(* No analogue to \<open>instantiation complex :: real_inner\<close> or to
 lemma complex_inner_1 [simp]: "inner 1 x = Re x"
 lemma complex_inner_1_right [simp]: "inner x 1 = Re x"
 lemma complex_inner_i_left [simp]: "inner \<i> x = Im x"
 lemma complex_inner_i_right [simp]: "inner x \<i> = Im x"
  *)
 
 lemma cdot_square_norm: "cinner x x = complex_of_real ((norm x)\<^sup>2)"
   by (metis Im_complex_of_real Re_complex_of_real cinner_ge_zero complex_eq_iff less_eq_complex_def power2_norm_eq_cinner' zero_complex.simps(2))
 
 lemma cnorm_eq_square: "norm x = a \<longleftrightarrow> 0 \<le> a \<and> cinner x x = complex_of_real (a\<^sup>2)"
   by (metis cdot_square_norm norm_ge_zero of_real_eq_iff power2_eq_iff_nonneg)
 
 lemma cnorm_le_square: "norm x \<le> a \<longleftrightarrow> 0 \<le> a \<and> cinner x x \<le> complex_of_real (a\<^sup>2)"
   by (smt (verit) cdot_square_norm complex_of_real_mono_iff norm_ge_zero power2_le_imp_le)
 
 lemma cnorm_ge_square: "norm x \<ge> a \<longleftrightarrow> a \<le> 0 \<or> cinner x x \<ge> complex_of_real (a\<^sup>2)"
   by (smt (verit, best) antisym_conv cnorm_eq_square cnorm_le_square complex_of_real_nn_iff nn_comparable zero_le_power2)
 
 lemma norm_lt_square: "norm x < a \<longleftrightarrow> 0 < a \<and> cinner x x < complex_of_real (a\<^sup>2)"
   by (meson cnorm_ge_square cnorm_le_square less_le_not_le)
 
 lemma norm_gt_square: "norm x > a \<longleftrightarrow> a < 0 \<or> cinner x x > complex_of_real (a\<^sup>2)"
   by (smt (verit, ccfv_SIG) cdot_square_norm complex_of_real_strict_mono_iff norm_ge_zero power2_eq_imp_eq power_mono)
 
 text\<open>Dot product in terms of the norm rather than conversely.\<close>
 
 lemmas cinner_simps = cinner_add_left cinner_add_right cinner_diff_right cinner_diff_left
   cinner_scaleC_left cinner_scaleC_right
 
 (* Analogue to both dot_norm and dot_norm_neg *)
 lemma cdot_norm: "cinner x y = ((norm (x+y))\<^sup>2 - (norm (x-y))\<^sup>2 - \<i> * (norm (x + \<i> *\<^sub>C y))\<^sup>2 + \<i> * (norm (x - \<i> *\<^sub>C y))\<^sup>2) / 4"
   unfolding power2_norm_eq_cinner
   by (simp add: power2_norm_eq_cinner cinner_add_left cinner_add_right
       cinner_diff_left cinner_diff_right ring_distribs)
 
 lemma of_complex_inner_1 [simp]:
   "cinner (of_complex x) (1 :: 'a :: {complex_inner, complex_normed_algebra_1}) = cnj x"
   by (metis Complex_Inner_Product0.complex_inner_1_right cinner_complex_def cinner_mult_left complex_cnj_one norm_one of_complex_def power2_norm_eq_cinner scaleC_conv_of_complex)
 
 lemma summable_of_complex_iff:
   "summable (\<lambda>x. of_complex (f x) :: 'a :: {complex_normed_algebra_1,complex_inner}) \<longleftrightarrow> summable f"
 proof
   assume *: "summable (\<lambda>x. of_complex (f x) :: 'a)"
   have "bounded_clinear (cinner (1::'a))"
     by (rule bounded_clinear_cinner_right)
   then interpret bounded_linear "\<lambda>x::'a. cinner 1 x"
     by (rule bounded_clinear.bounded_linear)
   from summable [OF *] show "summable f"
     apply (subst (asm) cinner_commute) by simp
 next
   assume sum: "summable f"
   thus "summable (\<lambda>x. of_complex (f x) :: 'a)"
     by (rule summable_of_complex)
 qed
 
 subsection \<open>Gradient derivative\<close>
 
 definition\<^marker>\<open>tag important\<close>
   cgderiv :: "['a::complex_inner \<Rightarrow> complex, 'a, 'a] \<Rightarrow> bool"
   ("(cGDERIV (_)/ (_)/ :> (_))" [1000, 1000, 60] 60)
   where
     (* Must be "cinner D" not "\<lambda>h. cinner h D", otherwise not even "cGDERIV id x :> 1" holds *)
     "cGDERIV f x :> D \<longleftrightarrow> FDERIV f x :> cinner D"
 
 lemma cgderiv_deriv [simp]: "cGDERIV f x :> D \<longleftrightarrow> DERIV f x :> cnj D"
   by (simp only: cgderiv_def has_field_derivative_def cinner_complex_def[THEN ext])
 
 lemma cGDERIV_DERIV_compose:
   assumes "cGDERIV f x :> df" and "DERIV g (f x) :> cnj dg"
   shows "cGDERIV (\<lambda>x. g (f x)) x :> scaleC dg df"
 proof (insert assms)
   show "cGDERIV (\<lambda>x. g (f x)) x :> dg *\<^sub>C df"
     if "cGDERIV f x :> df"
       and "(g has_field_derivative cnj dg) (at (f x))"
     unfolding cgderiv_def has_field_derivative_def cinner_scaleC_left complex_cnj_cnj
     using that
     by (simp add: cgderiv_def has_derivative_compose has_field_derivative_imp_has_derivative)
 
 qed
 
 (* Not specific to complex/real *)
 (* lemma has_derivative_subst: "\<lbrakk>FDERIV f x :> df; df = d\<rbrakk> \<Longrightarrow> FDERIV f x :> d" *)
 
 lemma cGDERIV_subst: "\<lbrakk>cGDERIV f x :> df; df = d\<rbrakk> \<Longrightarrow> cGDERIV f x :> d"
   by simp
 
 lemma cGDERIV_const: "cGDERIV (\<lambda>x. k) x :> 0"
   unfolding cgderiv_def cinner_zero_left[THEN ext] by (rule has_derivative_const)
 
 lemma cGDERIV_add:
   "\<lbrakk>cGDERIV f x :> df; cGDERIV g x :> dg\<rbrakk>
      \<Longrightarrow> cGDERIV (\<lambda>x. f x + g x) x :> df + dg"
   unfolding cgderiv_def cinner_add_left[THEN ext] by (rule has_derivative_add)
 
 lemma cGDERIV_minus:
   "cGDERIV f x :> df \<Longrightarrow> cGDERIV (\<lambda>x. - f x) x :> - df"
   unfolding cgderiv_def cinner_minus_left[THEN ext] by (rule has_derivative_minus)
 
 lemma cGDERIV_diff:
   "\<lbrakk>cGDERIV f x :> df; cGDERIV g x :> dg\<rbrakk>
      \<Longrightarrow> cGDERIV (\<lambda>x. f x - g x) x :> df - dg"
   unfolding cgderiv_def cinner_diff_left by (rule has_derivative_diff)
 
 lemma cGDERIV_scaleC:
   "\<lbrakk>DERIV f x :> df; cGDERIV g x :> dg\<rbrakk>
      \<Longrightarrow> cGDERIV (\<lambda>x. scaleC (f x) (g x)) x
       :> (scaleC (cnj (f x)) dg + scaleC (cnj df) (cnj (g x)))"
   unfolding cgderiv_def has_field_derivative_def cinner_add_left cinner_scaleC_left
   apply (rule has_derivative_subst)
    apply (erule (1) has_derivative_scaleC)
   by (simp add: ac_simps)
 
 lemma GDERIV_mult:
   "\<lbrakk>cGDERIV f x :> df; cGDERIV g x :> dg\<rbrakk>
      \<Longrightarrow> cGDERIV (\<lambda>x. f x * g x) x :> cnj (f x) *\<^sub>C dg + cnj (g x) *\<^sub>C df"
   unfolding cgderiv_def
   apply (rule has_derivative_subst)
    apply (erule (1) has_derivative_mult)
   apply (rule ext)
   by (simp add: cinner_add ac_simps)
 
 lemma cGDERIV_inverse:
   "\<lbrakk>cGDERIV f x :> df; f x \<noteq> 0\<rbrakk>
      \<Longrightarrow> cGDERIV (\<lambda>x. inverse (f x)) x :> - cnj ((inverse (f x))\<^sup>2) *\<^sub>C df"
   by (metis DERIV_inverse cGDERIV_DERIV_compose complex_cnj_cnj complex_cnj_minus numerals(2))
 
 (* Don't know if this holds: *)
 (* lemma cGDERIV_norm:
   assumes "x \<noteq> 0" shows "cGDERIV (\<lambda>x. norm x) x :> sgn x"
 *)
 
 
 lemma has_derivative_norm[derivative_intros]:
   fixes x :: "'a::complex_inner"
   assumes "x \<noteq> 0"
   shows "(norm has_derivative (\<lambda>h. Re (cinner (sgn x) h))) (at x)"
   thm has_derivative_norm
 proof -
   have Re_pos: "0 < Re (cinner x x)"
     using assms
     by (metis Re_strict_mono cinner_gt_zero_iff zero_complex.simps(1))
   have Re_plus_Re: "Re (cinner x y) + Re (cinner y x) = 2 * Re (cinner x y)"
     for x y :: 'a
     by (metis cinner_commute cnj.simps(1) mult_2_right semiring_normalization_rules(7))
   have norm: "norm x = sqrt (Re (cinner x x))" for x :: 'a
     apply (subst norm_eq_sqrt_cinner, subst cmod_Re)
     using cinner_ge_zero by auto
   have v2:"((\<lambda>x. sqrt (Re (cinner x x))) has_derivative
           (\<lambda>xa. (Re (cinner x xa) + Re (cinner xa x)) * (inverse (sqrt (Re (cinner x x))) / 2))) (at x)"
     by (rule derivative_eq_intros | simp add: Re_pos)+
   have v1: "((\<lambda>x. sqrt (Re (cinner x x))) has_derivative (\<lambda>y. Re (cinner x y) / sqrt (Re (cinner x x)))) (at x)"
     if "((\<lambda>x. sqrt (Re (cinner x x))) has_derivative (\<lambda>xa. Re (cinner x xa) * inverse (sqrt (Re (cinner x x))))) (at x)"
     using that apply (subst divide_real_def)
     by simp
   have \<open>(norm has_derivative (\<lambda>y. Re (cinner x y) / norm x)) (at x)\<close>
     using v2
     apply (auto simp: Re_plus_Re norm [abs_def])
     using v1 by blast
   then show ?thesis
     by (auto simp: power2_eq_square sgn_div_norm scaleR_scaleC)
 qed
 
 
 bundle cinner_syntax begin
 notation cinner (infix "\<bullet>\<^sub>C" 70)
 end
 
 bundle no_cinner_syntax begin
 no_notation cinner (infix "\<bullet>\<^sub>C" 70)
 end
 
 end
diff --git a/thys/Complex_Bounded_Operators/Complex_L2.thy b/thys/Complex_Bounded_Operators/Complex_L2.thy
--- a/thys/Complex_Bounded_Operators/Complex_L2.thy
+++ b/thys/Complex_Bounded_Operators/Complex_L2.thy
@@ -1,1563 +1,1557 @@
 section \<open>\<open>Complex_L2\<close> -- Hilbert space of square-summable functions\<close>
 
 (*
 Authors:
 
   Dominique Unruh, University of Tartu, unruh@ut.ee
   Jose Manuel Rodriguez Caballero, University of Tartu, jose.manuel.rodriguez.caballero@ut.ee
 
 *)
 
 theory Complex_L2
   imports 
     Complex_Bounded_Linear_Function
 
     "HOL-Analysis.L2_Norm"
     "HOL-Library.Rewrite"
     "HOL-Analysis.Infinite_Sum"
 begin
 
 unbundle lattice_syntax
 unbundle cblinfun_notation
 unbundle no_notation_blinfun_apply
 
 subsection \<open>l2 norm of functions\<close>
 
 definition "has_ell2_norm (x::_\<Rightarrow>complex) \<longleftrightarrow> (\<lambda>i. (x i)\<^sup>2) abs_summable_on UNIV"
 
 lemma has_ell2_norm_bdd_above: \<open>has_ell2_norm x \<longleftrightarrow> bdd_above (sum (\<lambda>xa. norm ((x xa)\<^sup>2)) ` Collect finite)\<close>
   by (simp add: has_ell2_norm_def abs_summable_iff_bdd_above)
 
 lemma has_ell2_norm_L2_set: "has_ell2_norm x = bdd_above (L2_set (norm o x) ` Collect finite)"
 proof (rule iffI)
   have \<open>mono sqrt\<close>
     using monoI real_sqrt_le_mono by blast
   assume \<open>has_ell2_norm x\<close>
   then have *: \<open>bdd_above (sum (\<lambda>xa. norm ((x xa)\<^sup>2)) ` Collect finite)\<close>
     by (subst (asm) has_ell2_norm_bdd_above)
   have \<open>bdd_above ((\<lambda>F. sqrt (sum (\<lambda>xa. norm ((x xa)\<^sup>2)) F)) ` Collect finite)\<close>
     using bdd_above_image_mono[OF \<open>mono sqrt\<close> *]
     by (auto simp: image_image)
   then show \<open>bdd_above (L2_set (norm o x) ` Collect finite)\<close>
     by (auto simp: L2_set_def norm_power)
 next
   define p2 where \<open>p2 x = (if x < 0 then 0 else x^2)\<close> for x :: real
   have \<open>mono p2\<close>
     by (simp add: monoI p2_def)
   have [simp]: \<open>p2 (L2_set f F) = (\<Sum>i\<in>F. (f i)\<^sup>2)\<close> for f and F :: \<open>'a set\<close>
     by (smt (verit) L2_set_def L2_set_nonneg p2_def power2_less_0 real_sqrt_pow2 sum.cong sum_nonneg)
   assume *: \<open>bdd_above (L2_set (norm o x) ` Collect finite)\<close>
   have \<open>bdd_above (p2 ` L2_set (norm o x) ` Collect finite)\<close>
     using bdd_above_image_mono[OF \<open>mono p2\<close> *]
     by auto
   then show \<open>has_ell2_norm x\<close>
     apply (simp add: image_image has_ell2_norm_def abs_summable_iff_bdd_above)
     by (simp add: norm_power)
 qed
 
 definition ell2_norm :: \<open>('a \<Rightarrow> complex) \<Rightarrow> real\<close> where \<open>ell2_norm x = sqrt (\<Sum>\<^sub>\<infinity>i. norm (x i)^2)\<close>
 
 lemma ell2_norm_SUP:
   assumes \<open>has_ell2_norm x\<close>
   shows "ell2_norm x = sqrt (SUP F\<in>{F. finite F}. sum (\<lambda>i. norm (x i)^2) F)"
   using assms apply (auto simp add: ell2_norm_def has_ell2_norm_def)
   apply (subst infsum_nonneg_is_SUPREMUM_real)
   by (auto simp: norm_power)
 
 lemma ell2_norm_L2_set: 
   assumes "has_ell2_norm x"
   shows "ell2_norm x = (SUP F\<in>{F. finite F}. L2_set (norm o x) F)"
 proof-
   have "sqrt (\<Squnion> (sum (\<lambda>i. (cmod (x i))\<^sup>2) ` Collect finite)) =
       (SUP F\<in>{F. finite F}. sqrt (\<Sum>i\<in>F. (cmod (x i))\<^sup>2))"
   proof (subst continuous_at_Sup_mono)
     show "mono sqrt"
       by (simp add: mono_def)      
     show "continuous (at_left (\<Squnion> (sum (\<lambda>i. (cmod (x i))\<^sup>2) ` Collect finite))) sqrt"
       using continuous_at_split isCont_real_sqrt by blast    
     show "sum (\<lambda>i. (cmod (x i))\<^sup>2) ` Collect finite \<noteq> {}"
       by auto      
     show "bdd_above (sum (\<lambda>i. (cmod (x i))\<^sup>2) ` Collect finite)"
       using has_ell2_norm_bdd_above[THEN iffD1, OF assms] by (auto simp: norm_power)
     show "\<Squnion> (sqrt ` sum (\<lambda>i. (cmod (x i))\<^sup>2) ` Collect finite) = (SUP F\<in>Collect finite. sqrt (\<Sum>i\<in>F. (cmod (x i))\<^sup>2))"
       by (metis image_image)      
   qed  
   thus ?thesis 
     using assms by (auto simp: ell2_norm_SUP L2_set_def)
 qed
 
 lemma has_ell2_norm_finite[simp]: "has_ell2_norm (x::'a::finite\<Rightarrow>_)"
   unfolding has_ell2_norm_def by simp
 
 lemma ell2_norm_finite: 
   "ell2_norm (x::'a::finite\<Rightarrow>complex) = sqrt (sum (\<lambda>i. (norm(x i))^2) UNIV)"
   by (simp add: ell2_norm_def)
 
 lemma ell2_norm_finite_L2_set: "ell2_norm (x::'a::finite\<Rightarrow>complex) = L2_set (norm o x) UNIV"
   by (simp add: ell2_norm_finite L2_set_def)
 
 lemma ell2_norm_square: \<open>(ell2_norm x)\<^sup>2 = (\<Sum>\<^sub>\<infinity>i. (cmod (x i))\<^sup>2)\<close>
   unfolding ell2_norm_def
   apply (subst real_sqrt_pow2)
   by (simp_all add: infsum_nonneg)
 
 lemma ell2_ket:
   fixes a
   defines \<open>f \<equiv> (\<lambda>i. if a = i then 1 else 0)\<close>
   shows has_ell2_norm_ket: \<open>has_ell2_norm f\<close>
     and ell2_norm_ket: \<open>ell2_norm f = 1\<close>
 proof -
   have \<open>(\<lambda>x. (f x)\<^sup>2) abs_summable_on {a}\<close>
     apply (rule summable_on_finite) by simp
   then show \<open>has_ell2_norm f\<close>
     unfolding has_ell2_norm_def
     apply (rule summable_on_cong_neutral[THEN iffD1, rotated -1])
     unfolding f_def by auto
 
   have \<open>(\<Sum>\<^sub>\<infinity>x\<in>{a}. (f x)\<^sup>2) = 1\<close>
     apply (subst infsum_finite)
     by (auto simp: f_def)
   then show \<open>ell2_norm f = 1\<close>
     unfolding ell2_norm_def
     apply (subst infsum_cong_neutral[where T=\<open>{a}\<close> and g=\<open>\<lambda>x. (cmod (f x))\<^sup>2\<close>])
     by (auto simp: f_def)
 qed
 
 lemma ell2_norm_geq0: \<open>ell2_norm x \<ge> 0\<close>
   by (auto simp: ell2_norm_def intro!: infsum_nonneg)
 
 lemma ell2_norm_point_bound:
   assumes \<open>has_ell2_norm x\<close>
   shows \<open>ell2_norm x \<ge> cmod (x i)\<close>
 proof -
   have \<open>(cmod (x i))\<^sup>2 = norm ((x i)\<^sup>2)\<close>
     by (simp add: norm_power)
   also have \<open>norm ((x i)\<^sup>2) = sum (\<lambda>i. (norm ((x i)\<^sup>2))) {i}\<close>
     by auto
   also have \<open>\<dots> = infsum (\<lambda>i. (norm ((x i)\<^sup>2))) {i}\<close>
     by (rule infsum_finite[symmetric], simp)
   also have \<open>\<dots> \<le> infsum (\<lambda>i. (norm ((x i)\<^sup>2))) UNIV\<close>
     apply (rule infsum_mono_neutral)
     using assms by (auto simp: has_ell2_norm_def)
   also have \<open>\<dots> = (ell2_norm x)\<^sup>2\<close>
     by (metis (no_types, lifting) ell2_norm_def ell2_norm_geq0 infsum_cong norm_power real_sqrt_eq_iff real_sqrt_unique)
   finally show ?thesis
     using ell2_norm_geq0 power2_le_imp_le by blast
 qed
 
 lemma ell2_norm_0:
   assumes "has_ell2_norm x"
   shows "(ell2_norm x = 0) = (x = (\<lambda>_. 0))"
 proof
   assume u1: "x = (\<lambda>_. 0)"
   have u2: "(SUP x::'a set\<in>Collect finite. (0::real)) = 0"
     if "x = (\<lambda>_. 0)"
     by (metis cSUP_const empty_Collect_eq finite.emptyI)
   show "ell2_norm x = 0"
     unfolding ell2_norm_def
     using u1 u2 by auto 
 next
   assume norm0: "ell2_norm x = 0"
   show "x = (\<lambda>_. 0)"
   proof
     fix i
     have \<open>cmod (x i) \<le> ell2_norm x\<close>
       using assms by (rule ell2_norm_point_bound)
     also have \<open>\<dots> = 0\<close>
       by (fact norm0)
     finally show "x i = 0" by auto
   qed
 qed
 
 
 lemma ell2_norm_smult:
   assumes "has_ell2_norm x"
   shows "has_ell2_norm (\<lambda>i. c * x i)" and "ell2_norm (\<lambda>i. c * x i) = cmod c * ell2_norm x"
 proof -
   have L2_set_mul: "L2_set (cmod \<circ> (\<lambda>i. c * x i)) F = cmod c * L2_set (cmod \<circ> x) F" for F
   proof-
     have "L2_set (cmod \<circ> (\<lambda>i. c * x i)) F = L2_set (\<lambda>i. (cmod c * (cmod o x) i)) F"
       by (metis comp_def norm_mult)
     also have "\<dots> = cmod c * L2_set (cmod o x) F"
       by (metis norm_ge_zero L2_set_right_distrib)
     finally show ?thesis .
   qed
 
   from assms obtain M where M: "M \<ge> L2_set (cmod o x) F" if "finite F" for F
     unfolding has_ell2_norm_L2_set bdd_above_def by auto
   hence "cmod c * M \<ge> L2_set (cmod o (\<lambda>i. c * x i)) F" if "finite F" for F
     unfolding L2_set_mul
     by (simp add: ordered_comm_semiring_class.comm_mult_left_mono that) 
   thus has: "has_ell2_norm (\<lambda>i. c * x i)"
     unfolding has_ell2_norm_L2_set bdd_above_def using L2_set_mul[symmetric] by auto
   have "ell2_norm (\<lambda>i. c * x i) = (SUP F \<in> Collect finite. (L2_set (cmod \<circ> (\<lambda>i. c * x i)) F))"
     by (simp add: ell2_norm_L2_set has)
   also have "\<dots> = (SUP F \<in> Collect finite. (cmod c * L2_set (cmod \<circ> x) F))"
     using L2_set_mul by auto   
   also have "\<dots> = cmod c * ell2_norm x" 
   proof (subst ell2_norm_L2_set)
     show "has_ell2_norm x"
       by (simp add: assms)      
     show "(SUP F\<in>Collect finite. cmod c * L2_set (cmod \<circ> x) F) = cmod c * \<Squnion> (L2_set (cmod \<circ> x) ` Collect finite)"
     proof (subst continuous_at_Sup_mono [where f = "\<lambda>x. cmod c * x"])
       show "mono ((*) (cmod c))"
         by (simp add: mono_def ordered_comm_semiring_class.comm_mult_left_mono)
       show "continuous (at_left (\<Squnion> (L2_set (cmod \<circ> x) ` Collect finite))) ((*) (cmod c))"
       proof (rule continuous_mult)
         show "continuous (at_left (\<Squnion> (L2_set (cmod \<circ> x) ` Collect finite))) (\<lambda>x. cmod c)"
           by simp
         show "continuous (at_left (\<Squnion> (L2_set (cmod \<circ> x) ` Collect finite))) (\<lambda>x. x)"
           by simp
       qed    
       show "L2_set (cmod \<circ> x) ` Collect finite \<noteq> {}"
         by auto        
       show "bdd_above (L2_set (cmod \<circ> x) ` Collect finite)"
         by (meson assms has_ell2_norm_L2_set)        
       show "(SUP F\<in>Collect finite. cmod c * L2_set (cmod \<circ> x) F) = \<Squnion> ((*) (cmod c) ` L2_set (cmod \<circ> x) ` Collect finite)"
         by (metis image_image)        
     qed   
   qed     
   finally show "ell2_norm (\<lambda>i. c * x i) = cmod c * ell2_norm x".
 qed
 
 
 lemma ell2_norm_triangle:
   assumes "has_ell2_norm x" and "has_ell2_norm y"
   shows "has_ell2_norm (\<lambda>i. x i + y i)" and "ell2_norm (\<lambda>i. x i + y i) \<le> ell2_norm x + ell2_norm y"
 proof -
   have triangle: "L2_set (cmod \<circ> (\<lambda>i. x i + y i)) F \<le> L2_set (cmod \<circ> x) F + L2_set (cmod \<circ> y) F" 
     (is "?lhs\<le>?rhs") 
     if "finite F" for F
   proof -
     have "?lhs \<le> L2_set (\<lambda>i. (cmod o x) i + (cmod o y) i) F"
     proof (rule L2_set_mono)
       show "(cmod \<circ> (\<lambda>i. x i + y i)) i \<le> (cmod \<circ> x) i + (cmod \<circ> y) i"
         if "i \<in> F"
         for i :: 'a
         using that norm_triangle_ineq by auto 
       show "0 \<le> (cmod \<circ> (\<lambda>i. x i + y i)) i"
         if "i \<in> F"
         for i :: 'a
         using that
         by simp 
     qed
     also have "\<dots> \<le> ?rhs"
       by (rule L2_set_triangle_ineq)
     finally show ?thesis .
   qed
   obtain Mx My where Mx: "Mx \<ge> L2_set (cmod o x) F" and My: "My \<ge> L2_set (cmod o y) F" 
     if "finite F" for F
     using assms unfolding has_ell2_norm_L2_set bdd_above_def by auto
   hence MxMy: "Mx + My \<ge> L2_set (cmod \<circ> x) F + L2_set (cmod \<circ> y) F" if "finite F" for F
     using that by fastforce
   hence bdd_plus: "bdd_above ((\<lambda>xa. L2_set (cmod \<circ> x) xa + L2_set (cmod \<circ> y) xa) ` Collect finite)"
     unfolding bdd_above_def by auto
   from MxMy have MxMy': "Mx + My \<ge> L2_set (cmod \<circ> (\<lambda>i. x i + y i)) F" if "finite F" for F 
     using triangle that by fastforce
   thus has: "has_ell2_norm (\<lambda>i. x i + y i)"
     unfolding has_ell2_norm_L2_set bdd_above_def by auto
   have SUP_plus: "(SUP x\<in>A. f x + g x) \<le> (SUP x\<in>A. f x) + (SUP x\<in>A. g x)" 
     if notempty: "A\<noteq>{}" and bddf: "bdd_above (f`A)"and bddg: "bdd_above (g`A)"
     for f g :: "'a set \<Rightarrow> real" and A
   proof-
     have xleq: "x \<le> (SUP x\<in>A. f x) + (SUP x\<in>A. g x)" if x: "x \<in> (\<lambda>x. f x + g x) ` A" for x
     proof -
       obtain a where aA: "a:A" and ax: "x = f a + g a"
         using x by blast
       have fa: "f a \<le> (SUP x\<in>A. f x)"
         by (simp add: bddf aA cSUP_upper)
       moreover have "g a \<le> (SUP x\<in>A. g x)"
         by (simp add: bddg aA cSUP_upper)
       ultimately have "f a + g a \<le> (SUP x\<in>A. f x) + (SUP x\<in>A. g x)" by simp
       with ax show ?thesis by simp
     qed
     have "(\<lambda>x. f x + g x) ` A \<noteq> {}"
       using notempty by auto        
     moreover have "x \<le> \<Squnion> (f ` A) + \<Squnion> (g ` A)"
       if "x \<in> (\<lambda>x. f x + g x) ` A"
       for x :: real
       using that
       by (simp add: xleq) 
     ultimately show ?thesis
       by (meson bdd_above_def cSup_le_iff)      
   qed
   have a2: "bdd_above (L2_set (cmod \<circ> x) ` Collect finite)"
     by (meson assms(1) has_ell2_norm_L2_set)    
   have a3: "bdd_above (L2_set (cmod \<circ> y) ` Collect finite)"
     by (meson assms(2) has_ell2_norm_L2_set)    
   have a1: "Collect finite \<noteq> {}"
     by auto    
   have a4: "\<Squnion> (L2_set (cmod \<circ> (\<lambda>i. x i + y i)) ` Collect finite)
     \<le> (SUP xa\<in>Collect finite.
            L2_set (cmod \<circ> x) xa + L2_set (cmod \<circ> y) xa)"
     by (metis (mono_tags, lifting) a1 bdd_plus cSUP_mono mem_Collect_eq triangle)    
   have "\<forall>r. \<Squnion> (L2_set (cmod \<circ> (\<lambda>a. x a + y a)) ` Collect finite) \<le> r \<or> \<not> (SUP A\<in>Collect finite. L2_set (cmod \<circ> x) A + L2_set (cmod \<circ> y) A) \<le> r"
     using a4 by linarith
   hence "\<Squnion> (L2_set (cmod \<circ> (\<lambda>i. x i + y i)) ` Collect finite)
     \<le> \<Squnion> (L2_set (cmod \<circ> x) ` Collect finite) +
        \<Squnion> (L2_set (cmod \<circ> y) ` Collect finite)"
     by (metis (no_types) SUP_plus a1 a2 a3)
   hence "\<Squnion> (L2_set (cmod \<circ> (\<lambda>i. x i + y i)) ` Collect finite) \<le> ell2_norm x + ell2_norm y"
     by (simp add: assms(1) assms(2) ell2_norm_L2_set)
   thus "ell2_norm (\<lambda>i. x i + y i) \<le> ell2_norm x + ell2_norm y"
     by (simp add: ell2_norm_L2_set has)  
 qed
 
 lemma ell2_norm_uminus:
   assumes "has_ell2_norm x"
   shows \<open>has_ell2_norm (\<lambda>i. - x i)\<close> and \<open>ell2_norm (\<lambda>i. - x i) = ell2_norm x\<close>
   using assms by (auto simp: has_ell2_norm_def ell2_norm_def)
 
 subsection \<open>The type \<open>ell2\<close> of square-summable functions\<close>
 
 typedef 'a ell2 = "{x::'a\<Rightarrow>complex. has_ell2_norm x}"
   unfolding has_ell2_norm_def by (rule exI[of _ "\<lambda>_.0"], auto)
 setup_lifting type_definition_ell2
 
 instantiation ell2 :: (type)complex_vector begin
 lift_definition zero_ell2 :: "'a ell2" is "\<lambda>_. 0" by (auto simp: has_ell2_norm_def)
 lift_definition uminus_ell2 :: "'a ell2 \<Rightarrow> 'a ell2" is uminus by (simp add: has_ell2_norm_def)
 lift_definition plus_ell2 :: "'a ell2 \<Rightarrow> 'a ell2 \<Rightarrow> 'a ell2" is "\<lambda>f g x. f x + g x"
   by (rule ell2_norm_triangle) 
 lift_definition minus_ell2 :: "'a ell2 \<Rightarrow> 'a ell2 \<Rightarrow> 'a ell2" is "\<lambda>f g x. f x - g x"
   apply (subst add_uminus_conv_diff[symmetric])
   apply (rule ell2_norm_triangle)
   by (auto simp add: ell2_norm_uminus)
 lift_definition scaleR_ell2 :: "real \<Rightarrow> 'a ell2 \<Rightarrow> 'a ell2" is "\<lambda>r f x. complex_of_real r * f x"
   by (rule ell2_norm_smult)
 lift_definition scaleC_ell2 :: "complex \<Rightarrow> 'a ell2 \<Rightarrow> 'a ell2" is "\<lambda>c f x. c * f x"
   by (rule ell2_norm_smult)
 
 instance
 proof
   fix a b c :: "'a ell2"
 
   show "((*\<^sub>R) r::'a ell2 \<Rightarrow> _) = (*\<^sub>C) (complex_of_real r)" for r
     apply (rule ext) apply transfer by auto
   show "a + b + c = a + (b + c)"
     by (transfer; rule ext; simp)
   show "a + b = b + a"
     by (transfer; rule ext; simp)
   show "0 + a = a"
     by (transfer; rule ext; simp)
   show "- a + a = 0"
     by (transfer; rule ext; simp)
   show "a - b = a + - b"
     by (transfer; rule ext; simp)
   show "r *\<^sub>C (a + b) = r *\<^sub>C a + r *\<^sub>C b" for r
     apply (transfer; rule ext)
     by (simp add: vector_space_over_itself.scale_right_distrib)
   show "(r + r') *\<^sub>C a = r *\<^sub>C a + r' *\<^sub>C a" for r r'
     apply (transfer; rule ext)
     by (simp add: ring_class.ring_distribs(2)) 
   show "r *\<^sub>C r' *\<^sub>C a = (r * r') *\<^sub>C a" for r r'
     by (transfer; rule ext; simp)
   show "1 *\<^sub>C a = a"
     by (transfer; rule ext; simp)
 qed
 end
 
 instantiation ell2 :: (type)complex_normed_vector begin
 lift_definition norm_ell2 :: "'a ell2 \<Rightarrow> real" is ell2_norm .
 declare norm_ell2_def[code del]
 definition "dist x y = norm (x - y)" for x y::"'a ell2"
 definition "sgn x = x /\<^sub>R norm x" for x::"'a ell2"
 definition [code del]: "uniformity = (INF e\<in>{0<..}. principal {(x::'a ell2, y). norm (x - y) < e})"
 definition [code del]: "open U = (\<forall>x\<in>U. \<forall>\<^sub>F (x', y) in INF e\<in>{0<..}. principal {(x, y). norm (x - y) < e}. x' = x \<longrightarrow> y \<in> U)" for U :: "'a ell2 set"
 instance
 proof
   fix a b :: "'a ell2"
   show "dist a b = norm (a - b)"
     by (simp add: dist_ell2_def)    
   show "sgn a = a /\<^sub>R norm a"
     by (simp add: sgn_ell2_def)    
   show "uniformity = (INF e\<in>{0<..}. principal {(x, y). dist (x::'a ell2) y < e})"
     unfolding dist_ell2_def  uniformity_ell2_def by simp
   show "open U = (\<forall>x\<in>U. \<forall>\<^sub>F (x', y) in uniformity. (x'::'a ell2) = x \<longrightarrow> y \<in> U)" for U :: "'a ell2 set"
     unfolding uniformity_ell2_def open_ell2_def by simp_all        
   show "(norm a = 0) = (a = 0)"
     apply transfer by (fact ell2_norm_0)    
   show "norm (a + b) \<le> norm a + norm b"
     apply transfer by (fact ell2_norm_triangle)
   show "norm (r *\<^sub>R (a::'a ell2)) = \<bar>r\<bar> * norm a" for r
     and a :: "'a ell2"
     apply transfer
     by (simp add: ell2_norm_smult(2)) 
   show "norm (r *\<^sub>C a) = cmod r * norm a" for r
     apply transfer
     by (simp add: ell2_norm_smult(2)) 
 qed  
 end
 
 lemma norm_point_bound_ell2: "norm (Rep_ell2 x i) \<le> norm x"
   apply transfer
   by (simp add: ell2_norm_point_bound)
 
 lemma ell2_norm_finite_support:
   assumes \<open>finite S\<close> \<open>\<And> i. i \<notin> S \<Longrightarrow> Rep_ell2 x i = 0\<close>
   shows \<open>norm x = sqrt ((sum (\<lambda>i. (cmod (Rep_ell2 x i))\<^sup>2)) S)\<close>
 proof (insert assms(2), transfer fixing: S)
   fix x :: \<open>'a \<Rightarrow> complex\<close>
   assume zero: \<open>\<And>i. i \<notin> S \<Longrightarrow> x i = 0\<close>
   have \<open>ell2_norm x = sqrt (\<Sum>\<^sub>\<infinity>i. (cmod (x i))\<^sup>2)\<close>
     by (auto simp: ell2_norm_def)
   also have \<open>\<dots> = sqrt (\<Sum>\<^sub>\<infinity>i\<in>S. (cmod (x i))\<^sup>2)\<close>
     apply (subst infsum_cong_neutral[where g=\<open>\<lambda>i. (cmod (x i))\<^sup>2\<close> and S=UNIV and T=S])
     using zero by auto
   also have \<open>\<dots> = sqrt (\<Sum>i\<in>S. (cmod (x i))\<^sup>2)\<close>
     using \<open>finite S\<close> by simp
   finally show \<open>ell2_norm x = sqrt (\<Sum>i\<in>S. (cmod (x i))\<^sup>2)\<close>
     by -
 qed
 
 instantiation ell2 :: (type) complex_inner begin
 lift_definition cinner_ell2 :: "'a ell2 \<Rightarrow> 'a ell2 \<Rightarrow> complex" is 
   "\<lambda>x y. infsum (\<lambda>i. (cnj (x i) * y i)) UNIV" .
 declare cinner_ell2_def[code del]
 
 instance
 proof standard
   fix x y z :: "'a ell2" fix c :: complex
   show "cinner x y = cnj (cinner y x)"
   proof transfer
     fix x y :: "'a\<Rightarrow>complex" assume "has_ell2_norm x" and "has_ell2_norm y"
     have "(\<Sum>\<^sub>\<infinity>i. cnj (x i) * y i) = (\<Sum>\<^sub>\<infinity>i. cnj (cnj (y i) * x i))"
       by (metis complex_cnj_cnj complex_cnj_mult mult.commute)
     also have "\<dots> = cnj (\<Sum>\<^sub>\<infinity>i. cnj (y i) * x i)"
       by (metis infsum_cnj) 
     finally show "(\<Sum>\<^sub>\<infinity>i. cnj (x i) * y i) = cnj (\<Sum>\<^sub>\<infinity>i. cnj (y i) * x i)" .
   qed
 
   show "cinner (x + y) z = cinner x z + cinner y z"
   proof transfer
     fix x y z :: "'a \<Rightarrow> complex"
     assume "has_ell2_norm x"
     hence cnj_x: "(\<lambda>i. cnj (x i) * cnj (x i)) abs_summable_on UNIV"
       by (simp del: complex_cnj_mult add: norm_mult[symmetric] complex_cnj_mult[symmetric] has_ell2_norm_def power2_eq_square)
     assume "has_ell2_norm y"
     hence cnj_y: "(\<lambda>i. cnj (y i) * cnj (y i)) abs_summable_on UNIV"
       by (simp del: complex_cnj_mult add: norm_mult[symmetric] complex_cnj_mult[symmetric] has_ell2_norm_def power2_eq_square)
     assume "has_ell2_norm z"
     hence z: "(\<lambda>i. z i * z i) abs_summable_on UNIV" 
       by (simp add: norm_mult[symmetric] has_ell2_norm_def power2_eq_square)
     have cnj_x_z:"(\<lambda>i. cnj (x i) * z i) abs_summable_on UNIV"
       using cnj_x z by (rule abs_summable_product) 
     have cnj_y_z:"(\<lambda>i. cnj (y i) * z i) abs_summable_on UNIV"
       using cnj_y z by (rule abs_summable_product) 
     show "(\<Sum>\<^sub>\<infinity>i. cnj (x i + y i) * z i) = (\<Sum>\<^sub>\<infinity>i. cnj (x i) * z i) + (\<Sum>\<^sub>\<infinity>i. cnj (y i) * z i)"
       apply (subst infsum_add [symmetric])
       using cnj_x_z cnj_y_z 
       by (auto simp add: summable_on_iff_abs_summable_on_complex distrib_left mult.commute)
   qed
 
   show "cinner (c *\<^sub>C x) y = cnj c * cinner x y"
   proof transfer
     fix x y :: "'a \<Rightarrow> complex" and c :: complex
     assume "has_ell2_norm x"
     hence cnj_x: "(\<lambda>i. cnj (x i) * cnj (x i)) abs_summable_on UNIV"
       by (simp del: complex_cnj_mult add: norm_mult[symmetric] complex_cnj_mult[symmetric] has_ell2_norm_def power2_eq_square)
     assume "has_ell2_norm y"
     hence y: "(\<lambda>i. y i * y i) abs_summable_on UNIV" 
       by (simp add: norm_mult[symmetric] has_ell2_norm_def power2_eq_square)
     have cnj_x_y:"(\<lambda>i. cnj (x i) * y i) abs_summable_on UNIV"
       using cnj_x y by (rule abs_summable_product) 
     thus "(\<Sum>\<^sub>\<infinity>i. cnj (c * x i) * y i) = cnj c * (\<Sum>\<^sub>\<infinity>i. cnj (x i) * y i)"
       by (auto simp flip: infsum_cmult_right simp add: abs_summable_summable mult.commute vector_space_over_itself.scale_left_commute)
   qed
 
   show "0 \<le> cinner x x"
   proof transfer
     fix x :: "'a \<Rightarrow> complex"
     assume "has_ell2_norm x"
     hence "(\<lambda>i. cmod (cnj (x i) * x i)) abs_summable_on UNIV"
       by (simp add: norm_mult has_ell2_norm_def power2_eq_square)
     hence "(\<lambda>i. cnj (x i) * x i) abs_summable_on UNIV"
       by auto
     hence sum: "(\<lambda>i. cnj (x i) * x i) abs_summable_on UNIV"
       unfolding has_ell2_norm_def power2_eq_square.
     have "0 = (\<Sum>\<^sub>\<infinity>i::'a. 0)" by auto
     also have "\<dots> \<le> (\<Sum>\<^sub>\<infinity>i. cnj (x i) * x i)"
       apply (rule infsum_mono_complex)
       by (auto simp add: abs_summable_summable sum)
     finally show "0 \<le> (\<Sum>\<^sub>\<infinity>i. cnj (x i) * x i)" by assumption
   qed
 
   show "(cinner x x = 0) = (x = 0)"
   proof (transfer, auto)
     fix x :: "'a \<Rightarrow> complex"
     assume "has_ell2_norm x"
     hence "(\<lambda>i::'a. cmod (cnj (x i) * x i)) abs_summable_on UNIV"
       by (smt (verit, del_insts) complex_mod_mult_cnj has_ell2_norm_def mult.commute norm_ge_zero norm_power real_norm_def summable_on_cong)
     hence cmod_x2: "(\<lambda>i. cnj (x i) * x i) abs_summable_on UNIV"
       unfolding has_ell2_norm_def power2_eq_square
       by simp
     assume eq0: "(\<Sum>\<^sub>\<infinity>i. cnj (x i) * x i) = 0"
     show "x = (\<lambda>_. 0)"
     proof (rule ccontr)
       assume "x \<noteq> (\<lambda>_. 0)"
       then obtain i where "x i \<noteq> 0" by auto
       hence "0 < cnj (x i) * x i"
         by (metis le_less cnj_x_x_geq0 complex_cnj_zero_iff vector_space_over_itself.scale_eq_0_iff)
       also have "\<dots> = (\<Sum>\<^sub>\<infinity>i\<in>{i}. cnj (x i) * x i)" by auto
       also have "\<dots> \<le> (\<Sum>\<^sub>\<infinity>i. cnj (x i) * x i)"
         apply (rule infsum_mono_neutral_complex)
         by (auto simp add: abs_summable_summable cmod_x2)
       also from eq0 have "\<dots> = 0" by assumption
       finally show False by simp
     qed
   qed
 
   show "norm x = sqrt (cmod (cinner x x))"
   proof transfer 
     fix x :: "'a \<Rightarrow> complex" 
     assume x: "has_ell2_norm x"
     have "(\<lambda>i::'a. cmod (x i) * cmod (x i)) abs_summable_on UNIV \<Longrightarrow>
     (\<lambda>i::'a. cmod (cnj (x i) * x i)) abs_summable_on UNIV"
       by (simp add: norm_mult has_ell2_norm_def power2_eq_square)
     hence sum: "(\<lambda>i. cnj (x i) * x i) abs_summable_on UNIV"
       by (metis (no_types, lifting) complex_mod_mult_cnj has_ell2_norm_def mult.commute norm_power summable_on_cong x)
     from x have "ell2_norm x = sqrt (\<Sum>\<^sub>\<infinity>i. (cmod (x i))\<^sup>2)"
       unfolding ell2_norm_def by simp
     also have "\<dots> = sqrt (\<Sum>\<^sub>\<infinity>i. cmod (cnj (x i) * x i))"
       unfolding norm_complex_def power2_eq_square by auto
     also have "\<dots> = sqrt (cmod (\<Sum>\<^sub>\<infinity>i. cnj (x i) * x i))"
       by (auto simp: infsum_cmod abs_summable_summable sum)
     finally show "ell2_norm x = sqrt (cmod (\<Sum>\<^sub>\<infinity>i. cnj (x i) * x i))" by assumption
   qed
 qed
 end
 
 instance ell2 :: (type) chilbert_space
 proof
   fix X :: \<open>nat \<Rightarrow> 'a ell2\<close>
   define x where \<open>x n a = Rep_ell2 (X n) a\<close> for n a
   have [simp]: \<open>has_ell2_norm (x n)\<close> for n
     using Rep_ell2 x_def[abs_def] by simp
 
   assume \<open>Cauchy X\<close>
   moreover have "dist (x n a) (x m a) \<le> dist (X n) (X m)" for n m a
     by (metis Rep_ell2 x_def dist_norm ell2_norm_point_bound mem_Collect_eq minus_ell2.rep_eq norm_ell2.rep_eq)
   ultimately have \<open>Cauchy (\<lambda>n. x n a)\<close> for a
     by (meson Cauchy_def le_less_trans)
   then obtain l where x_lim: \<open>(\<lambda>n. x n a) \<longlonglongrightarrow> l a\<close> for a
     apply atomize_elim apply (rule choice)
     by (simp add: convergent_eq_Cauchy)
   define L where \<open>L = Abs_ell2 l\<close>
   define normF where \<open>normF F x = L2_set (cmod \<circ> x) F\<close> for F :: \<open>'a set\<close> and x
   have normF_triangle: \<open>normF F (\<lambda>a. x a + y a) \<le> normF F x + normF F y\<close> if \<open>finite F\<close> for F x y
   proof -
     have \<open>normF F (\<lambda>a. x a + y a) = L2_set (\<lambda>a. cmod (x a + y a)) F\<close>
       by (metis (mono_tags, lifting) L2_set_cong comp_apply normF_def)
     also have \<open>\<dots> \<le> L2_set (\<lambda>a. cmod (x a) + cmod (y a)) F\<close>
       by (meson L2_set_mono norm_ge_zero norm_triangle_ineq)
     also have \<open>\<dots> \<le> L2_set (\<lambda>a. cmod (x a)) F + L2_set (\<lambda>a. cmod (y a)) F\<close>
       by (simp add: L2_set_triangle_ineq)
     also have \<open>\<dots> \<le> normF F x + normF F y\<close>
       by (smt (verit, best) L2_set_cong normF_def comp_apply)
     finally show ?thesis
       by -
   qed
   have normF_negate: \<open>normF F (\<lambda>a. - x a) = normF F x\<close> if \<open>finite F\<close> for F x
     unfolding normF_def o_def by simp
   have normF_ell2norm: \<open>normF F x \<le> ell2_norm x\<close> if \<open>finite F\<close> and \<open>has_ell2_norm x\<close> for F x
     apply (auto intro!: cSUP_upper2[where x=F] simp: that normF_def ell2_norm_L2_set)
     by (meson has_ell2_norm_L2_set that(2))
 
   note Lim_bounded2[rotated, rule_format, trans]
 
   from \<open>Cauchy X\<close>
   obtain I where cauchyX: \<open>norm (X n - X m) \<le> \<epsilon>\<close> if \<open>\<epsilon>>0\<close> \<open>n\<ge>I \<epsilon>\<close> \<open>m\<ge>I \<epsilon>\<close> for \<epsilon> n m
     by (metis Cauchy_def dist_norm less_eq_real_def)
   have normF_xx: \<open>normF F (\<lambda>a. x n a - x m a) \<le> \<epsilon>\<close> if \<open>finite F\<close> \<open>\<epsilon>>0\<close> \<open>n\<ge>I \<epsilon>\<close> \<open>m\<ge>I \<epsilon>\<close> for \<epsilon> n m F
     apply (subst asm_rl[of \<open>(\<lambda>a. x n a - x m a) = Rep_ell2 (X n - X m)\<close>])
      apply (simp add: x_def minus_ell2.rep_eq)
     using that cauchyX by (metis Rep_ell2 mem_Collect_eq normF_ell2norm norm_ell2.rep_eq order_trans)
   have normF_xl_lim: \<open>(\<lambda>m. normF F (\<lambda>a. x m a - l a)) \<longlonglongrightarrow> 0\<close> if \<open>finite F\<close> for F
   proof -
     have \<open>(\<lambda>xa. cmod (x xa m - l m)) \<longlonglongrightarrow> 0\<close> for m
       using x_lim by (simp add: LIM_zero_iff tendsto_norm_zero)
     then have \<open>(\<lambda>m. \<Sum>i\<in>F. ((cmod \<circ> (\<lambda>a. x m a - l a)) i)\<^sup>2) \<longlonglongrightarrow> 0\<close>
       by (auto intro: tendsto_null_sum)
     then show ?thesis
       unfolding normF_def L2_set_def
       using tendsto_real_sqrt by force
   qed
   have normF_xl: \<open>normF F (\<lambda>a. x n a - l a) \<le> \<epsilon>\<close>
     if \<open>n \<ge> I \<epsilon>\<close> and \<open>\<epsilon> > 0\<close> and \<open>finite F\<close> for n \<epsilon> F
   proof -
     have \<open>normF F (\<lambda>a. x n a - l a) - \<epsilon> \<le> normF F (\<lambda>a. x n a - x m a) + normF F (\<lambda>a. x m a - l a) - \<epsilon>\<close> for m
       using normF_triangle[OF \<open>finite F\<close>, where x=\<open>(\<lambda>a. x n a - x m a)\<close> and y=\<open>(\<lambda>a. x m a - l a)\<close>]
       by auto
     also have \<open>\<dots> m \<le> normF F (\<lambda>a. x m a - l a)\<close> if \<open>m \<ge> I \<epsilon>\<close> for m
       using normF_xx[OF \<open>finite F\<close> \<open>\<epsilon>>0\<close> \<open>n \<ge> I \<epsilon>\<close> \<open>m \<ge> I \<epsilon>\<close>]
       by auto
     also have \<open>(\<lambda>m. \<dots> m) \<longlonglongrightarrow> 0\<close>
       using \<open>finite F\<close> by (rule normF_xl_lim)
     finally show ?thesis
       by auto
   qed
   have \<open>normF F l \<le> 1 + normF F (x (I 1))\<close> if [simp]: \<open>finite F\<close> for F
     using normF_xl[where F=F and \<epsilon>=1 and n=\<open>I 1\<close>]
     using normF_triangle[where F=F and x=\<open>x (I 1)\<close> and y=\<open>\<lambda>a. l a - x (I 1) a\<close>]
     using normF_negate[where F=F and x=\<open>(\<lambda>a. x (I 1) a - l a)\<close>]
     by auto
   also have \<open>\<dots> F \<le> 1 + ell2_norm (x (I 1))\<close> if \<open>finite F\<close> for F
     using normF_ell2norm that by simp
   finally have [simp]: \<open>has_ell2_norm l\<close>
     unfolding has_ell2_norm_L2_set
     by (auto intro!: bdd_aboveI simp flip: normF_def)
   then have \<open>l = Rep_ell2 L\<close>
     by (simp add: Abs_ell2_inverse L_def)
   have [simp]: \<open>has_ell2_norm (\<lambda>a. x n a - l a)\<close> for n
     apply (subst diff_conv_add_uminus)
     apply (rule ell2_norm_triangle)
     by (auto intro!: ell2_norm_uminus)
   from normF_xl have ell2norm_xl: \<open>ell2_norm (\<lambda>a. x n a - l a) \<le> \<epsilon>\<close>
     if \<open>n \<ge> I \<epsilon>\<close> and \<open>\<epsilon> > 0\<close> for n \<epsilon>
     apply (subst ell2_norm_L2_set)
     using that by (auto intro!: cSUP_least simp: normF_def)
   have \<open>norm (X n - L) \<le> \<epsilon>\<close> if \<open>n \<ge> I \<epsilon>\<close> and \<open>\<epsilon> > 0\<close> for n \<epsilon>
     using ell2norm_xl[OF that]
     by (simp add: x_def norm_ell2.rep_eq \<open>l = Rep_ell2 L\<close> minus_ell2.rep_eq)
   then have \<open>X \<longlonglongrightarrow> L\<close>
     unfolding tendsto_iff
     apply (auto simp: dist_norm eventually_sequentially)
     by (meson field_lbound_gt_zero le_less_trans)
   then show \<open>convergent X\<close>
     by (rule convergentI)
 qed
 
 instantiation ell2 :: (CARD_1) complex_algebra_1 
 begin
 lift_definition one_ell2 :: "'a ell2" is "\<lambda>_. 1" by simp
 lift_definition times_ell2 :: "'a ell2 \<Rightarrow> 'a ell2 \<Rightarrow> 'a ell2" is "\<lambda>a b x. a x * b x"
   by simp   
 instance 
 proof
   fix a b c :: "'a ell2" and r :: complex
   show "a * b * c = a * (b * c)"
     by (transfer, auto)
   show "(a + b) * c = a * c + b * c"
     apply (transfer, rule ext)
     by (simp add: distrib_left mult.commute)
   show "a * (b + c) = a * b + a * c"
     apply transfer
     by (simp add: ring_class.ring_distribs(1))
   show "r *\<^sub>C a * b = r *\<^sub>C (a * b)"
     by (transfer, auto)
   show "(a::'a ell2) * r *\<^sub>C b = r *\<^sub>C (a * b)"
     by (transfer, auto)
   show "1 * a = a"
     by (transfer, rule ext, auto)
   show "a * 1 = a"
     by (transfer, rule ext, auto)
   show "(0::'a ell2) \<noteq> 1"
     apply transfer
     by (meson zero_neq_one)
 qed
 end
 
 instantiation ell2 :: (CARD_1) field begin
 lift_definition divide_ell2 :: "'a ell2 \<Rightarrow> 'a ell2 \<Rightarrow> 'a ell2" is "\<lambda>a b x. a x / b x"
   by simp   
 lift_definition inverse_ell2 :: "'a ell2 \<Rightarrow> 'a ell2" is "\<lambda>a x. inverse (a x)"
   by simp
 instance
 proof (intro_classes; transfer)
   fix a :: "'a \<Rightarrow> complex"
   assume "a \<noteq> (\<lambda>_. 0)"
   then obtain y where ay: "a y \<noteq> 0"
     by auto
   show "(\<lambda>x. inverse (a x) * a x) = (\<lambda>_. 1)"
   proof (rule ext)
     fix x
     have "x = y"
       by auto
     with ay have "a x \<noteq> 0"
       by metis
     then show "inverse (a x) * a x = 1"
       by auto
   qed
 qed (auto simp add: divide_complex_def mult.commute ring_class.ring_distribs)
 end
 
 lemma sum_ell2_transfer[transfer_rule]:
   includes lifting_syntax
   shows \<open>(((=) ===> pcr_ell2 (=)) ===> rel_set (=) ===> pcr_ell2 (=)) 
           (\<lambda>f X x. sum (\<lambda>y. f y x) X) sum\<close>
 proof (intro rel_funI, rename_tac f f' X X')
   fix f and f' :: \<open>'a \<Rightarrow> 'b ell2\<close> 
   assume [transfer_rule]: \<open>((=) ===> pcr_ell2 (=)) f f'\<close>
   fix X X' :: \<open>'a set\<close>
   assume \<open>rel_set (=) X X'\<close>
   then have [simp]: \<open>X' = X\<close>
     by (simp add: rel_set_eq)
   show \<open>pcr_ell2 (=) (\<lambda>x. \<Sum>y\<in>X. f y x) (sum f' X')\<close>
     unfolding \<open>X' = X\<close>
   proof (induction X rule: infinite_finite_induct)
     case (infinite X)
     show ?case
       apply (simp add: infinite)
       by transfer_prover
   next
     case empty
     show ?case
       apply (simp add: empty)
       by transfer_prover
   next
     case (insert x F)
     note [transfer_rule] = insert.IH
     show ?case
       apply (simp add: insert)
       by transfer_prover
   qed
 qed
 
 subsection \<open>Orthogonality\<close>
 
 lemma ell2_pointwise_ortho:
   assumes \<open>\<And> i. Rep_ell2 x i = 0 \<or> Rep_ell2 y i = 0\<close>
   shows \<open>is_orthogonal x y\<close>
   using assms apply transfer
   by (simp add: infsum_0)
 
 subsection \<open>Truncated vectors\<close>
 
 lift_definition trunc_ell2:: \<open>'a set \<Rightarrow> 'a ell2 \<Rightarrow> 'a ell2\<close>
   is \<open>\<lambda> S x. (\<lambda> i. (if i \<in> S then x i else 0))\<close>
 proof (rename_tac S x)
   fix x :: \<open>'a \<Rightarrow> complex\<close> and S :: \<open>'a set\<close>
   assume \<open>has_ell2_norm x\<close>
   then have \<open>(\<lambda>i. (x i)\<^sup>2) abs_summable_on UNIV\<close>
     unfolding has_ell2_norm_def by -
   then have \<open>(\<lambda>i. (x i)\<^sup>2) abs_summable_on S\<close>
     using summable_on_subset_banach by blast
   then have \<open>(\<lambda>xa. (if xa \<in> S then x xa else 0)\<^sup>2) abs_summable_on UNIV\<close>
     apply (rule summable_on_cong_neutral[THEN iffD1, rotated -1])
     by auto
   then show \<open>has_ell2_norm (\<lambda>i. if i \<in> S then x i else 0)\<close>
     unfolding has_ell2_norm_def by -
 qed
 
 lemma trunc_ell2_empty[simp]: \<open>trunc_ell2 {} x = 0\<close>
   apply transfer by simp
 
 lemma trunc_ell2_UNIV[simp]: \<open>trunc_ell2 UNIV \<psi> = \<psi>\<close>
   apply transfer by simp
 
 lemma norm_id_minus_trunc_ell2:
   \<open>(norm (x - trunc_ell2 S x))^2 = (norm x)^2 - (norm (trunc_ell2 S x))^2\<close>
 proof-
   have \<open>Rep_ell2 (trunc_ell2 S x) i = 0 \<or> Rep_ell2 (x - trunc_ell2 S x) i = 0\<close> for i
     apply transfer
     by auto
-  hence \<open>\<langle> (trunc_ell2 S x), (x - trunc_ell2 S x) \<rangle> = 0\<close>
+  hence \<open>((trunc_ell2 S x) \<bullet>\<^sub>C (x - trunc_ell2 S x)) = 0\<close>
     using ell2_pointwise_ortho by blast
   hence \<open>(norm x)^2 = (norm (trunc_ell2 S x))^2 + (norm (x - trunc_ell2 S x))^2\<close>
     using pythagorean_theorem by fastforce    
   thus ?thesis by simp
 qed
 
 lemma norm_trunc_ell2_finite:
   \<open>finite S \<Longrightarrow> (norm (trunc_ell2 S x)) = sqrt ((sum (\<lambda>i. (cmod (Rep_ell2 x i))\<^sup>2)) S)\<close>
 proof-
   assume \<open>finite S\<close>
   moreover have \<open>\<And> i. i \<notin> S \<Longrightarrow> Rep_ell2 ((trunc_ell2 S x)) i = 0\<close>
     by (simp add: trunc_ell2.rep_eq)    
   ultimately have \<open>(norm (trunc_ell2 S x)) = sqrt ((sum (\<lambda>i. (cmod (Rep_ell2 ((trunc_ell2 S x)) i))\<^sup>2)) S)\<close>
     using ell2_norm_finite_support
     by blast 
   moreover have \<open>\<And> i. i \<in> S \<Longrightarrow> Rep_ell2 ((trunc_ell2 S x)) i = Rep_ell2 x i\<close>
     by (simp add: trunc_ell2.rep_eq)
   ultimately show ?thesis by simp
 qed
 
 lemma trunc_ell2_lim_at_UNIV:
   \<open>((\<lambda>S. trunc_ell2 S \<psi>) \<longlongrightarrow> \<psi>) (finite_subsets_at_top UNIV)\<close>
 proof -
   define f where \<open>f i = (cmod (Rep_ell2 \<psi> i))\<^sup>2\<close> for i
 
   have has: \<open>has_ell2_norm (Rep_ell2 \<psi>)\<close>
     using Rep_ell2 by blast
   then have summable: "f abs_summable_on UNIV"
     by (smt (verit, del_insts) f_def has_ell2_norm_def norm_ge_zero norm_power real_norm_def summable_on_cong)
 
   have \<open>norm \<psi> = (ell2_norm (Rep_ell2 \<psi>))\<close>
     apply transfer by simp
   also have \<open>\<dots> = sqrt (infsum f UNIV)\<close>
     by (simp add: ell2_norm_def f_def[symmetric])
   finally have norm\<psi>: \<open>norm \<psi> = sqrt (infsum f UNIV)\<close>
     by -
 
   have norm_trunc: \<open>norm (trunc_ell2 S \<psi>) = sqrt (sum f S)\<close> if \<open>finite S\<close> for S
     using f_def that norm_trunc_ell2_finite by fastforce
 
   have \<open>(sum f \<longlongrightarrow> infsum f UNIV) (finite_subsets_at_top UNIV)\<close>
     using f_def[abs_def] infsum_tendsto local.summable by fastforce
   then have \<open>((\<lambda>S. sqrt (sum f S)) \<longlongrightarrow> sqrt (infsum f UNIV)) (finite_subsets_at_top UNIV)\<close>
     using tendsto_real_sqrt by blast
   then have \<open>((\<lambda>S. norm (trunc_ell2 S \<psi>)) \<longlongrightarrow> norm \<psi>) (finite_subsets_at_top UNIV)\<close>
     apply (subst tendsto_cong[where g=\<open>\<lambda>S. sqrt (sum f S)\<close>])
     by (auto simp add: eventually_finite_subsets_at_top_weakI norm_trunc norm\<psi>)
   then have \<open>((\<lambda>S. (norm (trunc_ell2 S \<psi>))\<^sup>2) \<longlongrightarrow> (norm \<psi>)\<^sup>2) (finite_subsets_at_top UNIV)\<close>
     by (simp add: tendsto_power)
   then have \<open>((\<lambda>S. (norm \<psi>)\<^sup>2 - (norm (trunc_ell2 S \<psi>))\<^sup>2) \<longlongrightarrow> 0) (finite_subsets_at_top UNIV)\<close>
     apply (rule tendsto_diff[where a=\<open>(norm \<psi>)^2\<close> and b=\<open>(norm \<psi>)^2\<close>, simplified, rotated])
     by auto
   then have \<open>((\<lambda>S. (norm (\<psi> - trunc_ell2 S \<psi>))\<^sup>2) \<longlongrightarrow> 0) (finite_subsets_at_top UNIV)\<close>
     unfolding norm_id_minus_trunc_ell2 by simp
   then have \<open>((\<lambda>S. norm (\<psi> - trunc_ell2 S \<psi>)) \<longlongrightarrow> 0) (finite_subsets_at_top UNIV)\<close>
     by auto
   then have \<open>((\<lambda>S. \<psi> - trunc_ell2 S \<psi>) \<longlongrightarrow> 0) (finite_subsets_at_top UNIV)\<close>
     by (rule tendsto_norm_zero_cancel)
   then show ?thesis
     apply (rule Lim_transform2[where f=\<open>\<lambda>_. \<psi>\<close>, rotated])
     by simp
 qed
 
 lemma trunc_ell2_norm_mono: \<open>M \<subseteq> N \<Longrightarrow> norm (trunc_ell2 M \<psi>) \<le> norm (trunc_ell2 N \<psi>)\<close>
 proof (rule power2_le_imp_le[rotated], force, transfer)
   fix M N :: \<open>'a set\<close> and \<psi> :: \<open>'a \<Rightarrow> complex\<close>
   assume \<open>M \<subseteq> N\<close> and \<open>has_ell2_norm \<psi>\<close>
   have \<open>(ell2_norm (\<lambda>i. if i \<in> M then \<psi> i else 0))\<^sup>2 = (\<Sum>\<^sub>\<infinity>i\<in>M. (cmod (\<psi> i))\<^sup>2)\<close>
     unfolding ell2_norm_square
     apply (rule infsum_cong_neutral)
     by auto
   also have \<open>\<dots> \<le> (\<Sum>\<^sub>\<infinity>i\<in>N. (cmod (\<psi> i))\<^sup>2)\<close>
     apply (rule infsum_mono2)
     using \<open>has_ell2_norm \<psi>\<close> \<open>M \<subseteq> N\<close>
     by (auto simp add: ell2_norm_square has_ell2_norm_def simp flip: norm_power intro: summable_on_subset_banach)
   also have \<open>\<dots> = (ell2_norm (\<lambda>i. if i \<in> N then \<psi> i else 0))\<^sup>2\<close>
     unfolding ell2_norm_square
     apply (rule infsum_cong_neutral)
     by auto
   finally show \<open>(ell2_norm (\<lambda>i. if i \<in> M then \<psi> i else 0))\<^sup>2 \<le> (ell2_norm (\<lambda>i. if i \<in> N then \<psi> i else 0))\<^sup>2\<close>
     by -
 qed
 
 lemma trunc_ell2_twice[simp]: \<open>trunc_ell2 M (trunc_ell2 N \<psi>) = trunc_ell2 (M\<inter>N) \<psi>\<close>
   apply transfer by auto
 
 lemma trunc_ell2_union: \<open>trunc_ell2 (M \<union> N) \<psi> = trunc_ell2 M \<psi> + trunc_ell2 N \<psi> - trunc_ell2 (M\<inter>N) \<psi>\<close>
   apply transfer by auto
 
 lemma trunc_ell2_union_disjoint: \<open>M\<inter>N = {} \<Longrightarrow> trunc_ell2 (M \<union> N) \<psi> = trunc_ell2 M \<psi> + trunc_ell2 N \<psi>\<close>
   by (simp add: trunc_ell2_union)
 
 lemma trunc_ell2_union_Diff: \<open>M \<subseteq> N \<Longrightarrow> trunc_ell2 (N-M) \<psi> = trunc_ell2 N \<psi> - trunc_ell2 M \<psi>\<close>
   using trunc_ell2_union_disjoint[where M=\<open>N-M\<close> and N=M and \<psi>=\<psi>]
   by (simp add: Un_commute inf.commute le_iff_sup)
 
 lemma trunc_ell2_lim: \<open>((\<lambda>S. trunc_ell2 S \<psi>) \<longlongrightarrow> trunc_ell2 M \<psi>) (finite_subsets_at_top M)\<close>
 proof -
   have \<open>((\<lambda>S. trunc_ell2 S (trunc_ell2 M \<psi>)) \<longlongrightarrow> trunc_ell2 M \<psi>) (finite_subsets_at_top UNIV)\<close>
     using trunc_ell2_lim_at_UNIV by blast
   then have \<open>((\<lambda>S. trunc_ell2 (S\<inter>M) \<psi>) \<longlongrightarrow> trunc_ell2 M \<psi>) (finite_subsets_at_top UNIV)\<close>
     by simp
   then show \<open>((\<lambda>S. trunc_ell2 S \<psi>) \<longlongrightarrow> trunc_ell2 M \<psi>) (finite_subsets_at_top M)\<close>
     unfolding filterlim_def
     apply (subst (asm) filtermap_filtermap[where g=\<open>\<lambda>S. S\<inter>M\<close>, symmetric])
     apply (subst (asm) finite_subsets_at_top_inter[where A=M and B=UNIV])
     by auto
 qed
 
 lemma trunc_ell2_lim_general:
   assumes big: \<open>\<And>G. finite G \<Longrightarrow> G \<subseteq> M \<Longrightarrow> (\<forall>\<^sub>F H in F. H \<supseteq> G)\<close>
   assumes small: \<open>\<forall>\<^sub>F H in F. H \<subseteq> M\<close>
   shows \<open>((\<lambda>S. trunc_ell2 S \<psi>) \<longlongrightarrow> trunc_ell2 M \<psi>) F\<close>
 proof (rule tendstoI)
   fix e :: real assume \<open>e > 0\<close>
   from trunc_ell2_lim[THEN tendsto_iff[THEN iffD1], rule_format, OF \<open>e > 0\<close>, where M=M and \<psi>=\<psi>]
   obtain G where \<open>finite G\<close> and \<open>G \<subseteq> M\<close> and 
     close: \<open>dist (trunc_ell2 G \<psi>) (trunc_ell2 M \<psi>) < e\<close>
     apply atomize_elim
     unfolding eventually_finite_subsets_at_top
     by blast
   from \<open>finite G\<close> \<open>G \<subseteq> M\<close> and big
   have \<open>\<forall>\<^sub>F H in F. H \<supseteq> G\<close>
     by -
   with small have \<open>\<forall>\<^sub>F H in F. H \<subseteq> M \<and> H \<supseteq> G\<close>
     by (simp add: eventually_conj_iff)
   then show \<open>\<forall>\<^sub>F H in F. dist (trunc_ell2 H \<psi>) (trunc_ell2 M \<psi>) < e\<close>
   proof (rule eventually_mono)
     fix H assume GHM: \<open>H \<subseteq> M \<and> H \<supseteq> G\<close>
     have \<open>dist (trunc_ell2 H \<psi>) (trunc_ell2 M \<psi>) = norm (trunc_ell2 (M-H) \<psi>)\<close>
       by (simp add: GHM dist_ell2_def norm_minus_commute trunc_ell2_union_Diff)
     also have \<open>\<dots> \<le> norm (trunc_ell2 (M-G) \<psi>)\<close>
       by (simp add: Diff_mono GHM trunc_ell2_norm_mono)
     also have \<open>\<dots>  = dist (trunc_ell2 G \<psi>) (trunc_ell2 M \<psi>)\<close>
       by (simp add: \<open>G \<subseteq> M\<close> dist_ell2_def norm_minus_commute trunc_ell2_union_Diff)
     also have \<open>\<dots> < e\<close>
       using close by simp
     finally show \<open>dist (trunc_ell2 H \<psi>) (trunc_ell2 M \<psi>) < e\<close>
       by -
   qed
 qed
 
 subsection \<open>Kets and bras\<close>
 
 lift_definition ket :: "'a \<Rightarrow> 'a ell2" is "\<lambda>x y. if x=y then 1 else 0"
   by (rule has_ell2_norm_ket)
 
 abbreviation bra :: "'a \<Rightarrow> (_,complex) cblinfun" where "bra i \<equiv> vector_to_cblinfun (ket i)*" for i
 
 instance ell2 :: (type) not_singleton
 proof standard
   have "ket undefined \<noteq> (0::'a ell2)"
   proof transfer
     show "(\<lambda>y. if (undefined::'a) = y then 1::complex else 0) \<noteq> (\<lambda>_. 0)"
       by (meson one_neq_zero)
   qed   
   thus \<open>\<exists>x y::'a ell2. x \<noteq> y\<close>
     by blast    
 qed
 
-lemma cinner_ket_left: \<open>\<langle>ket i, \<psi>\<rangle> = Rep_ell2 \<psi> i\<close>
+lemma cinner_ket_left: \<open>(ket i \<bullet>\<^sub>C \<psi>) = Rep_ell2 \<psi> i\<close>
   apply (transfer fixing: i)
   apply (subst infsum_cong_neutral[where T=\<open>{i}\<close>])
   by auto
 
-lemma cinner_ket_right: \<open>\<langle>\<psi>, ket i\<rangle> = cnj (Rep_ell2 \<psi> i)\<close>
+lemma cinner_ket_right: \<open>(\<psi> \<bullet>\<^sub>C ket i) = cnj (Rep_ell2 \<psi> i)\<close>
   apply (transfer fixing: i)
   apply (subst infsum_cong_neutral[where T=\<open>{i}\<close>])
   by auto
 
 lemma cinner_ket_eqI:
   assumes \<open>\<And>i. cinner (ket i) \<psi> = cinner (ket i) \<phi>\<close>
   shows \<open>\<psi> = \<phi>\<close>
   by (metis Rep_ell2_inject assms cinner_ket_left ext)
 
 lemma norm_ket[simp]: "norm (ket i) = 1"
   apply transfer by (rule ell2_norm_ket)
 
 lemma cinner_ket_same[simp]:
-  \<open>\<langle>ket i, ket i\<rangle> = 1\<close>
+  \<open>(ket i \<bullet>\<^sub>C ket i) = 1\<close>
 proof-
   have \<open>norm (ket i) = 1\<close>
     by simp
-  hence \<open>sqrt (cmod \<langle>ket i, ket i\<rangle>) = 1\<close>
+  hence \<open>sqrt (cmod (ket i \<bullet>\<^sub>C ket i)) = 1\<close>
     by (metis norm_eq_sqrt_cinner)
-  hence \<open>cmod \<langle>ket i, ket i\<rangle> = 1\<close>
+  hence \<open>cmod (ket i \<bullet>\<^sub>C ket i) = 1\<close>
     using real_sqrt_eq_1_iff by blast
-  moreover have \<open>\<langle>ket i, ket i\<rangle> = cmod \<langle>ket i, ket i\<rangle>\<close>
+  moreover have \<open>(ket i \<bullet>\<^sub>C ket i) = cmod (ket i \<bullet>\<^sub>C ket i)\<close>
   proof-
-    have \<open>\<langle>ket i, ket i\<rangle> \<in> \<real>\<close>
+    have \<open>(ket i \<bullet>\<^sub>C ket i) \<in> \<real>\<close>
       by (simp add: cinner_real)      
     thus ?thesis 
       by (metis cinner_ge_zero complex_of_real_cmod) 
   qed
   ultimately show ?thesis by simp
 qed
 
 lemma orthogonal_ket[simp]:
   \<open>is_orthogonal (ket i) (ket j) \<longleftrightarrow> i \<noteq> j\<close>
   by (simp add: cinner_ket_left ket.rep_eq)
 
-lemma cinner_ket: \<open>\<langle>ket i, ket j\<rangle> = (if i=j then 1 else 0)\<close>
+lemma cinner_ket: \<open>(ket i \<bullet>\<^sub>C ket j) = (if i=j then 1 else 0)\<close>
   by (simp add: cinner_ket_left ket.rep_eq)
 
 lemma ket_injective[simp]: \<open>ket i = ket j \<longleftrightarrow> i = j\<close>
   by (metis cinner_ket one_neq_zero)
 
 lemma inj_ket[simp]: \<open>inj ket\<close>
   by (simp add: inj_on_def)
 
 
 lemma trunc_ell2_ket_cspan:
   \<open>trunc_ell2 S x \<in> (cspan (range ket))\<close> if \<open>finite S\<close>
 proof (use that in induction)
   case empty
   then show ?case 
     by (auto intro: complex_vector.span_zero)
 next
   case (insert a F)
   from insert.hyps have \<open>trunc_ell2 (insert a F) x = trunc_ell2 F x + Rep_ell2 x a *\<^sub>C ket a\<close>
     apply (transfer fixing: F a)
     by auto
   with insert.IH
   show ?case
     by (simp add: complex_vector.span_add_eq complex_vector.span_base complex_vector.span_scale)
 qed
 
 lemma closed_cspan_range_ket[simp]:
   \<open>closure (cspan (range ket)) = UNIV\<close>
 proof (intro set_eqI iffI UNIV_I closure_approachable[THEN iffD2] allI impI)
   fix \<psi> :: \<open>'a ell2\<close>
   fix e :: real assume \<open>e > 0\<close>
   have \<open>((\<lambda>S. trunc_ell2 S \<psi>) \<longlongrightarrow> \<psi>) (finite_subsets_at_top UNIV)\<close>
     by (rule trunc_ell2_lim_at_UNIV)
   then obtain F where \<open>finite F\<close> and \<open>dist (trunc_ell2 F \<psi>) \<psi> < e\<close>
     apply (drule_tac tendstoD[OF _ \<open>e > 0\<close>])
     by (auto dest: simp: eventually_finite_subsets_at_top)
   moreover have \<open>trunc_ell2 F \<psi> \<in> cspan (range ket)\<close>
     using \<open>finite F\<close> trunc_ell2_ket_cspan by blast
   ultimately show \<open>\<exists>\<phi>\<in>cspan (range ket). dist \<phi> \<psi> < e\<close>
     by auto
 qed
 
 lemma ccspan_range_ket[simp]: "ccspan (range ket) = (top::('a ell2 ccsubspace))"
 proof-
   have \<open>closure (complex_vector.span (range ket)) = (UNIV::'a ell2 set)\<close>
     using Complex_L2.closed_cspan_range_ket by blast
   thus ?thesis
     by (simp add: ccspan.abs_eq top_ccsubspace.abs_eq)
 qed
 
 lemma cspan_range_ket_finite[simp]: "cspan (range ket :: 'a::finite ell2 set) = UNIV"
   by (metis closed_cspan_range_ket closure_finite_cspan finite_class.finite_UNIV finite_imageI)
 
 instance ell2 :: (finite) cfinite_dim
 proof
   define basis :: \<open>'a ell2 set\<close> where \<open>basis = range ket\<close>
   have \<open>finite basis\<close>
     unfolding basis_def by simp
   moreover have \<open>cspan basis = UNIV\<close>
     by (simp add: basis_def)
   ultimately show \<open>\<exists>basis::'a ell2 set. finite basis \<and> cspan basis = UNIV\<close>
     by auto
 qed
 
 instantiation ell2 :: (enum) onb_enum begin
 definition "canonical_basis_ell2 = map ket Enum.enum"
 instance
 proof
   show "distinct (canonical_basis::'a ell2 list)"
   proof-
     have \<open>finite (UNIV::'a set)\<close>
       by simp
     have \<open>distinct (enum_class.enum::'a list)\<close>
       using enum_distinct by blast
     moreover have \<open>inj_on ket (set enum_class.enum)\<close>
       by (meson inj_onI ket_injective)         
     ultimately show ?thesis
       unfolding canonical_basis_ell2_def
       using distinct_map
       by blast
   qed    
 
   show "is_ortho_set (set (canonical_basis::'a ell2 list))"
     apply (auto simp: canonical_basis_ell2_def enum_UNIV)
     by (smt (z3) norm_ket f_inv_into_f is_ortho_set_def orthogonal_ket norm_zero)
 
   show "cindependent (set (canonical_basis::'a ell2 list))"
     apply (auto simp: canonical_basis_ell2_def enum_UNIV)
     by (smt (verit, best) norm_ket f_inv_into_f is_ortho_set_def is_ortho_set_cindependent orthogonal_ket norm_zero)
 
   show "cspan (set (canonical_basis::'a ell2 list)) = UNIV"
     by (auto simp: canonical_basis_ell2_def enum_UNIV)
 
   show "norm (x::'a ell2) = 1"
     if "(x::'a ell2) \<in> set canonical_basis"
     for x :: "'a ell2"
     using that unfolding canonical_basis_ell2_def 
     by auto
 qed
 
 end
 
 lemma canonical_basis_length_ell2[code_unfold, simp]:
   "length (canonical_basis ::'a::enum ell2 list) = CARD('a)"
   unfolding canonical_basis_ell2_def apply simp
   using card_UNIV_length_enum by metis
 
 lemma ket_canonical_basis: "ket x = canonical_basis ! enum_idx x"
 proof-
   have "x = (enum_class.enum::'a list) ! enum_idx x"
     using enum_idx_correct[where i = x] by simp
   hence p1: "ket x = ket ((enum_class.enum::'a list) ! enum_idx x)"
     by simp
   have "enum_idx x < length (enum_class.enum::'a list)"
     using enum_idx_bound[where x = x].
   hence "(map ket (enum_class.enum::'a list)) ! enum_idx x 
         = ket ((enum_class.enum::'a list) ! enum_idx x)"
     by auto      
   thus ?thesis
     unfolding canonical_basis_ell2_def using p1 by auto    
 qed
 
 lemma clinear_equal_ket:
   fixes f g :: \<open>'a::finite ell2 \<Rightarrow> _\<close>
   assumes \<open>clinear f\<close>
   assumes \<open>clinear g\<close>
   assumes \<open>\<And>i. f (ket i) = g (ket i)\<close>
   shows \<open>f = g\<close>
   apply (rule ext)
   apply (rule complex_vector.linear_eq_on_span[where f=f and g=g and B=\<open>range ket\<close>])
   using assms by auto
 
 lemma equal_ket:
   fixes A B :: \<open>('a ell2, 'b::complex_normed_vector) cblinfun\<close>
   assumes \<open>\<And> x. cblinfun_apply A (ket x) = cblinfun_apply B (ket x)\<close>
   shows \<open>A = B\<close>
   apply (rule cblinfun_eq_gen_eqI[where G=\<open>range ket\<close>])
   using assms by auto
 
 lemma antilinear_equal_ket:
   fixes f g :: \<open>'a::finite ell2 \<Rightarrow> _\<close>
   assumes \<open>antilinear f\<close>
   assumes \<open>antilinear g\<close>
   assumes \<open>\<And>i. f (ket i) = g (ket i)\<close>
   shows \<open>f = g\<close>
 proof -
   have [simp]: \<open>clinear (f \<circ> from_conjugate_space)\<close>
     apply (rule antilinear_o_antilinear)
     using assms by (simp_all add: antilinear_from_conjugate_space)
   have [simp]: \<open>clinear (g \<circ> from_conjugate_space)\<close>
     apply (rule antilinear_o_antilinear)
     using assms by (simp_all add: antilinear_from_conjugate_space)
   have [simp]: \<open>cspan (to_conjugate_space ` (range ket :: 'a ell2 set)) = UNIV\<close>
     by simp
   have "f o from_conjugate_space = g o from_conjugate_space"
     apply (rule ext)
     apply (rule complex_vector.linear_eq_on_span[where f="f o from_conjugate_space" and g="g o from_conjugate_space" and B=\<open>to_conjugate_space ` range ket\<close>])
        apply (simp, simp)
     using assms(3) by (auto simp: to_conjugate_space_inverse)
   then show "f = g"
     by (smt (verit) UNIV_I from_conjugate_space_inverse surj_def surj_fun_eq to_conjugate_space_inject) 
 qed
 
 lemma cinner_ket_adjointI:
   fixes F::"'a ell2 \<Rightarrow>\<^sub>C\<^sub>L _" and G::"'b ell2 \<Rightarrow>\<^sub>C\<^sub>L_"
-  assumes "\<And> i j. \<langle>F *\<^sub>V ket i, ket j\<rangle> = \<langle>ket i, G *\<^sub>V ket j\<rangle>"
+  assumes "\<And> i j. (F *\<^sub>V ket i) \<bullet>\<^sub>C ket j = ket i \<bullet>\<^sub>C (G *\<^sub>V ket j)"
   shows "F = G*"
 proof -
   from assms
   have \<open>(F *\<^sub>V x) \<bullet>\<^sub>C y = x \<bullet>\<^sub>C (G *\<^sub>V y)\<close> if \<open>x \<in> range ket\<close> and \<open>y \<in> range ket\<close> for x y
     using that by auto
   then have \<open>(F *\<^sub>V x) \<bullet>\<^sub>C y = x \<bullet>\<^sub>C (G *\<^sub>V y)\<close> if \<open>x \<in> range ket\<close> for x y
     apply (rule bounded_clinear_eq_on[where G=\<open>range ket\<close> and t=y, rotated 2])
     using that by (auto intro!: bounded_linear_intros)
   then have \<open>(F *\<^sub>V x) \<bullet>\<^sub>C y = x \<bullet>\<^sub>C (G *\<^sub>V y)\<close> for x y
     apply (rule bounded_antilinear_eq_on[where G=\<open>range ket\<close> and t=x, rotated 2])
     by (auto intro!: bounded_linear_intros)
   then show ?thesis
     by (rule adjoint_eqI)
 qed
 
 lemma ket_nonzero[simp]: "ket i \<noteq> 0"
   using norm_ket[of i] by force
 
 
 lemma cindependent_ket:
   "cindependent (range (ket::'a\<Rightarrow>_))"
 proof-
   define S where "S = range (ket::'a\<Rightarrow>_)"
   have "is_ortho_set S"
     unfolding S_def is_ortho_set_def by auto
   moreover have "0 \<notin> S"
     unfolding S_def
     using ket_nonzero
     by (simp add: image_iff)
   ultimately show ?thesis
     using is_ortho_set_cindependent[where A = S] unfolding S_def 
     by blast
 qed
 
 lemma cdim_UNIV_ell2[simp]: \<open>cdim (UNIV::'a::finite ell2 set) = CARD('a)\<close>
   apply (subst cspan_range_ket_finite[symmetric])
   by (metis card_image cindependent_ket complex_vector.dim_span_eq_card_independent inj_ket)
 
 lemma is_ortho_set_ket[simp]: \<open>is_ortho_set (range ket)\<close>
   using is_ortho_set_def by fastforce
 
 lemma bounded_clinear_equal_ket:
   fixes f g :: \<open>'a ell2 \<Rightarrow> _\<close>
   assumes \<open>bounded_clinear f\<close>
   assumes \<open>bounded_clinear g\<close>
   assumes \<open>\<And>i. f (ket i) = g (ket i)\<close>
   shows \<open>f = g\<close>
   apply (rule ext)
   apply (rule bounded_clinear_eq_on[of f g \<open>range ket\<close>])
   using assms by auto
 
 lemma bounded_antilinear_equal_ket:
   fixes f g :: \<open>'a ell2 \<Rightarrow> _\<close>
   assumes \<open>bounded_antilinear f\<close>
   assumes \<open>bounded_antilinear g\<close>
   assumes \<open>\<And>i. f (ket i) = g (ket i)\<close>
   shows \<open>f = g\<close>
   apply (rule ext)
   apply (rule bounded_antilinear_eq_on[of f g \<open>range ket\<close>])
   using assms by auto
 
 lemma ket_CARD_1_is_1: \<open>ket x = 1\<close> for x :: \<open>'a::CARD_1\<close>
   apply transfer by simp
 
 lemma is_onb_ket[simp]: \<open>is_onb (range ket)\<close>
   by (auto simp: is_onb_def)
 
 lemma ell2_sum_ket: \<open>\<psi> = (\<Sum>i\<in>UNIV. Rep_ell2 \<psi> i *\<^sub>C ket i)\<close> for \<psi> :: \<open>_::finite ell2\<close>
   apply transfer apply (rule ext)
   apply (subst sum_single)
   by auto
 
 subsection \<open>Butterflies\<close>
 
 lemma cspan_butterfly_ket: \<open>cspan {butterfly (ket i) (ket j)| (i::'b::finite) (j::'a::finite). True} = UNIV\<close>
 proof -
   have *: \<open>{butterfly (ket i) (ket j)| (i::'b::finite) (j::'a::finite). True} = {butterfly a b |a b. a \<in> range ket \<and> b \<in> range ket}\<close>
     by auto
   show ?thesis
     apply (subst *)
     apply (rule cspan_butterfly_UNIV)
     by auto
 qed
 
 lemma cindependent_butterfly_ket: \<open>cindependent {butterfly (ket i) (ket j)| (i::'b) (j::'a). True}\<close>
 proof -
   have *: \<open>{butterfly (ket i) (ket j)| (i::'b) (j::'a). True} = {butterfly a b |a b. a \<in> range ket \<and> b \<in> range ket}\<close>
     by auto
   show ?thesis
     apply (subst *)
     apply (rule cindependent_butterfly)
     by auto
 qed
 
 lemma clinear_eq_butterfly_ketI:
   fixes F G :: \<open>('a::finite ell2 \<Rightarrow>\<^sub>C\<^sub>L 'b::finite ell2) \<Rightarrow> 'c::complex_vector\<close>
   assumes "clinear F" and "clinear G"
   assumes "\<And>i j. F (butterfly (ket i) (ket j)) = G (butterfly (ket i) (ket j))"
   shows "F = G"
   apply (rule complex_vector.linear_eq_on_span[where f=F, THEN ext, rotated 3])
      apply (subst cspan_butterfly_ket)
   using assms by auto
 
 lemma sum_butterfly_ket[simp]: \<open>(\<Sum>(i::'a::finite)\<in>UNIV. butterfly (ket i) (ket i)) = id_cblinfun\<close>
   apply (rule equal_ket)
   apply (subst complex_vector.linear_sum[where f=\<open>\<lambda>y. y *\<^sub>V ket _\<close>])
    apply (auto simp add: scaleC_cblinfun.rep_eq cblinfun.add_left clinearI butterfly_def cblinfun_compose_image cinner_ket)
   apply (subst sum.mono_neutral_cong_right[where S=\<open>{_}\<close>])
   by auto
 
 subsection \<open>One-dimensional spaces\<close>
 
 instantiation ell2 :: ("{enum,CARD_1}") one_dim begin
 text \<open>Note: enum is not needed logically, but without it this instantiation
             clashes with \<open>instantiation ell2 :: (enum) onb_enum\<close>\<close>
 instance
 proof
   show "canonical_basis = [1::'a ell2]"
     unfolding canonical_basis_ell2_def
     apply transfer
     by (simp add: enum_CARD_1[of undefined])
   show "a *\<^sub>C 1 * b *\<^sub>C 1 = (a * b) *\<^sub>C (1::'a ell2)" for a b
     apply (transfer fixing: a b) by simp
   show "x / y = x * inverse y" for x y :: "'a ell2"
     by (simp add: divide_inverse)
   show "inverse (c *\<^sub>C 1) = inverse c *\<^sub>C (1::'a ell2)" for c :: complex
     apply transfer by auto
 qed
 end
 
 
 subsection \<open>Classical operators\<close>
 
 text \<open>We call an operator mapping \<^term>\<open>ket x\<close> to \<^term>\<open>ket (\<pi> x)\<close> or \<^term>\<open>0\<close> "classical".
 (The meaning is inspired by the fact that in quantum mechanics, such operators usually correspond
 to operations with classical interpretation (such as Pauli-X, CNOT, measurement in the computational
 basis, etc.))\<close>
 
 definition classical_operator :: "('a\<Rightarrow>'b option) \<Rightarrow> 'a ell2 \<Rightarrow>\<^sub>C\<^sub>L'b ell2" where
   "classical_operator \<pi> = 
     (let f = (\<lambda>t. (case \<pi> (inv (ket::'a\<Rightarrow>_) t) 
                            of None \<Rightarrow> (0::'b ell2) 
                           | Some i \<Rightarrow> ket i))
      in
       cblinfun_extension (range (ket::'a\<Rightarrow>_)) f)"
 
 
 definition "classical_operator_exists \<pi> \<longleftrightarrow>
   cblinfun_extension_exists (range ket)
     (\<lambda>t. case \<pi> (inv ket t) of None \<Rightarrow> 0 | Some i \<Rightarrow> ket i)"
 
 lemma classical_operator_existsI:
   assumes "\<And>x. B *\<^sub>V (ket x) = (case \<pi> x of Some i \<Rightarrow> ket i | None \<Rightarrow> 0)"
   shows "classical_operator_exists \<pi>"
   unfolding classical_operator_exists_def
   apply (rule cblinfun_extension_existsI[of _ B])
   using assms 
   by (auto simp: inv_f_f[OF inj_ket])
 
 lemma 
   assumes "inj_map \<pi>"
   shows classical_operator_exists_inj: "classical_operator_exists \<pi>"
     and classical_operator_norm_inj: \<open>norm (classical_operator \<pi>) \<le> 1\<close>
 proof -
   have \<open>is_orthogonal (case \<pi> x of None \<Rightarrow> 0 | Some x' \<Rightarrow> ket x')
                       (case \<pi> y of None \<Rightarrow> 0 | Some y' \<Rightarrow> ket y')\<close>
     if \<open>x \<noteq> y\<close> for x y
     apply (cases \<open>\<pi> x\<close>; cases \<open>\<pi> y\<close>)
     using that assms
     by (auto simp add: inj_map_def)
   then have 1: \<open>is_orthogonal (case \<pi> (inv ket x) of None \<Rightarrow> 0 | Some x' \<Rightarrow> ket x')
                       (case \<pi> (inv ket y) of None \<Rightarrow> 0 | Some y' \<Rightarrow> ket y')\<close>
     if \<open>x \<in> range ket\<close> and \<open>y \<in> range ket\<close> and \<open>x \<noteq> y\<close> for x y
     using that by auto
 
   have \<open>norm (case \<pi> x of None \<Rightarrow> 0 | Some x \<Rightarrow> ket x) \<le> 1 * norm (ket x)\<close> for x
     apply (cases \<open>\<pi> x\<close>) by auto
   then have 2: \<open>norm (case \<pi> (inv ket x) of None \<Rightarrow> 0 | Some x \<Rightarrow> ket x) \<le> 1 * norm x\<close>
     if \<open>x \<in> range ket\<close> for x
     using that by auto
 
   show \<open>classical_operator_exists \<pi>\<close>
     unfolding classical_operator_exists_def
     using _ _ 1 2 apply (rule cblinfun_extension_exists_ortho)
     by simp_all
 
   show \<open>norm (classical_operator \<pi>) \<le> 1\<close>
     unfolding classical_operator_def Let_def
     using _ _ 1 2 apply (rule cblinfun_extension_exists_ortho_norm)
     by simp_all
 qed
 
 lemma classical_operator_exists_finite[simp]: "classical_operator_exists (\<pi> :: _::finite \<Rightarrow> _)"
   unfolding classical_operator_exists_def
   apply (rule cblinfun_extension_exists_finite_dim)
   using cindependent_ket apply blast
   using finite_class.finite_UNIV finite_imageI closed_cspan_range_ket closure_finite_cspan by blast
 
 lemma classical_operator_ket:
   assumes "classical_operator_exists \<pi>"
   shows "(classical_operator \<pi>) *\<^sub>V (ket x) = (case \<pi> x of Some i \<Rightarrow> ket i | None \<Rightarrow> 0)"
   unfolding classical_operator_def 
   using f_inv_into_f ket_injective rangeI
   by (metis assms cblinfun_extension_apply classical_operator_exists_def)
 
 lemma classical_operator_ket_finite:
   "(classical_operator \<pi>) *\<^sub>V (ket (x::'a::finite)) = (case \<pi> x of Some i \<Rightarrow> ket i | None \<Rightarrow> 0)"
   by (rule classical_operator_ket, simp)
 
 lemma classical_operator_adjoint[simp]:
   fixes \<pi> :: "'a \<Rightarrow> 'b option"
   assumes a1: "inj_map \<pi>"
   shows  "(classical_operator \<pi>)* = classical_operator (inv_map \<pi>)"
 proof-
   define F where "F = classical_operator (inv_map \<pi>)"
   define G where "G = classical_operator \<pi>"
-  have "\<langle>F *\<^sub>V ket i, ket j\<rangle> = \<langle>ket i, G *\<^sub>V ket j\<rangle>" for i j
+  have "(F *\<^sub>V ket i) \<bullet>\<^sub>C ket j = ket i \<bullet>\<^sub>C (G *\<^sub>V ket j)" for i j
   proof-
     have w1: "(classical_operator (inv_map \<pi>)) *\<^sub>V (ket i)
      = (case inv_map \<pi> i of Some k \<Rightarrow> ket k | None \<Rightarrow> 0)"
       by (simp add: classical_operator_ket classical_operator_exists_inj)
     have w2: "(classical_operator \<pi>) *\<^sub>V (ket j)
      = (case \<pi> j of Some k \<Rightarrow> ket k | None \<Rightarrow> 0)"
       by (simp add: assms classical_operator_ket classical_operator_exists_inj)
-    have "\<langle>F *\<^sub>V ket i, ket j\<rangle> = \<langle>classical_operator (inv_map \<pi>) *\<^sub>V ket i, ket j\<rangle>"
+    have "(F *\<^sub>V ket i) \<bullet>\<^sub>C ket j = (classical_operator (inv_map \<pi>) *\<^sub>V ket i) \<bullet>\<^sub>C ket j"
       unfolding F_def by blast
-    also have "\<dots> = \<langle>(case inv_map \<pi> i of Some k \<Rightarrow> ket k | None \<Rightarrow> 0), ket j\<rangle>"
+    also have "\<dots> = ((case inv_map \<pi> i of Some k \<Rightarrow> ket k | None \<Rightarrow> 0) \<bullet>\<^sub>C ket j)"
       using w1 by simp
-    also have "\<dots> = \<langle>ket i, (case \<pi> j of Some k \<Rightarrow> ket k | None \<Rightarrow> 0)\<rangle>"
+    also have "\<dots> = (ket i \<bullet>\<^sub>C (case \<pi> j of Some k \<Rightarrow> ket k | None \<Rightarrow> 0))"
     proof(induction "inv_map \<pi> i")
       case None
       hence pi1: "None = inv_map \<pi> i".
       show ?case 
       proof (induction "\<pi> j")
         case None
         thus ?case
           using pi1 by auto
       next
         case (Some c)
         have "c \<noteq> i"
         proof(rule classical)
           assume "\<not>(c \<noteq> i)"
           hence "c = i"
             by blast
           hence "inv_map \<pi> c = inv_map \<pi> i"
             by simp
           hence "inv_map \<pi> c = None"
             by (simp add: pi1)
           moreover have "inv_map \<pi> c = Some j"
             using Some.hyps unfolding inv_map_def
             apply auto
             by (metis a1 f_inv_into_f inj_map_def option.distinct(1) rangeI)
           ultimately show ?thesis by simp
         qed
         thus ?thesis
           by (metis None.hyps Some.hyps cinner_zero_left orthogonal_ket option.simps(4) 
               option.simps(5)) 
       qed       
     next
       case (Some d)
       hence s1: "Some d = inv_map \<pi> i".
-      show "\<langle>case inv_map \<pi> i of 
-            None \<Rightarrow> 0
-        | Some a \<Rightarrow> ket a, ket j\<rangle> =
-       \<langle>ket i, case \<pi> j of 
-            None \<Rightarrow> 0 
-        | Some a \<Rightarrow> ket a\<rangle>" 
+      show "(case inv_map \<pi> i of None \<Rightarrow> 0| Some a \<Rightarrow> ket a) \<bullet>\<^sub>C ket j
+           = ket i \<bullet>\<^sub>C (case \<pi> j of None \<Rightarrow> 0 | Some a \<Rightarrow> ket a)" 
       proof(induction "\<pi> j")
         case None
         have "d \<noteq> j"
         proof(rule classical)
           assume "\<not>(d \<noteq> j)"
           hence "d = j"
             by blast
           hence "\<pi> d = \<pi> j"
             by simp
           hence "\<pi> d = None"
             by (simp add: None.hyps)
           moreover have "\<pi> d = Some i"
             using Some.hyps unfolding inv_map_def
             apply auto
             by (metis f_inv_into_f option.distinct(1) option.inject)
           ultimately show ?thesis 
             by simp
         qed
         thus ?case
           by (metis None.hyps Some.hyps cinner_zero_right orthogonal_ket option.case_eq_if 
               option.simps(5)) 
       next
         case (Some c)
         hence s2: "\<pi> j = Some c" by simp
-        have "\<langle>ket d, ket j\<rangle> = \<langle>ket i, ket c\<rangle>"
+        have "(ket d \<bullet>\<^sub>C ket j) = (ket i \<bullet>\<^sub>C ket c)"
         proof(cases "\<pi> j = Some i")
           case True
           hence ij: "Some j = inv_map \<pi> i"
             unfolding inv_map_def apply auto
              apply (metis a1 f_inv_into_f inj_map_def option.discI range_eqI)
             by (metis range_eqI)
           have "i = c"
             using True s2 by auto
           moreover have "j = d"
             by (metis option.inject s1 ij)
           ultimately show ?thesis
             by (simp add: cinner_ket_same) 
         next
           case False
           moreover have "\<pi> d = Some i"
             using s1 unfolding inv_map_def
             by (metis f_inv_into_f option.distinct(1) option.inject)            
           ultimately have "j \<noteq> d"
             by auto            
           moreover have "i \<noteq> c"
             using False s2 by auto            
           ultimately show ?thesis
             by (metis orthogonal_ket) 
         qed
-        hence "\<langle>case Some d of None \<Rightarrow> 0
-        | Some a \<Rightarrow> ket a, ket j\<rangle> =
-       \<langle>ket i, case Some c of None \<Rightarrow> 0 | Some a \<Rightarrow> ket a\<rangle>"
+        hence "(case Some d of None \<Rightarrow> 0 | Some a \<Rightarrow> ket a) \<bullet>\<^sub>C ket j
+             = ket i \<bullet>\<^sub>C (case Some c of None \<Rightarrow> 0 | Some a \<Rightarrow> ket a)"
           by simp          
-        thus "\<langle>case inv_map \<pi> i of None \<Rightarrow> 0
-        | Some a \<Rightarrow> ket a, ket j\<rangle> =
-       \<langle>ket i, case \<pi> j of None \<Rightarrow> 0 | Some a \<Rightarrow> ket a\<rangle>"
+        thus "(case inv_map \<pi> i of None \<Rightarrow> 0 | Some a \<Rightarrow> ket a) \<bullet>\<^sub>C ket j
+             = ket i \<bullet>\<^sub>C (case \<pi> j of None \<Rightarrow> 0 | Some a \<Rightarrow> ket a)"
           by (simp add: Some.hyps s1)          
       qed
     qed
-    also have "\<dots> = \<langle>ket i, classical_operator \<pi> *\<^sub>V ket j\<rangle>"
+    also have "\<dots> = ket i \<bullet>\<^sub>C (classical_operator \<pi> *\<^sub>V ket j)"
       by (simp add: w2)
-    also have "\<dots> = \<langle>ket i, G *\<^sub>V ket j\<rangle>"
+    also have "\<dots> = ket i \<bullet>\<^sub>C (G *\<^sub>V ket j)"
       unfolding G_def by blast
     finally show ?thesis .
   qed
   hence "G* = F"
     using cinner_ket_adjointI
     by auto
   thus ?thesis unfolding G_def F_def .
 qed
 
 lemma
   fixes \<pi>::"'b \<Rightarrow> 'c option" and \<rho>::"'a \<Rightarrow> 'b option"
   assumes "classical_operator_exists \<pi>"
   assumes "classical_operator_exists \<rho>"
   shows classical_operator_exists_comp[simp]: "classical_operator_exists (\<pi> \<circ>\<^sub>m \<rho>)"
     and classical_operator_mult[simp]: "classical_operator \<pi> o\<^sub>C\<^sub>L classical_operator \<rho> = classical_operator (\<pi> \<circ>\<^sub>m \<rho>)"
 proof -
   define C\<pi> C\<rho> C\<pi>\<rho> where "C\<pi> = classical_operator \<pi>" and "C\<rho> = classical_operator \<rho>" 
     and "C\<pi>\<rho> = classical_operator (\<pi> \<circ>\<^sub>m \<rho>)"
   have C\<pi>x: "C\<pi> *\<^sub>V (ket x) = (case \<pi> x of Some i \<Rightarrow> ket i | None \<Rightarrow> 0)" for x
     unfolding C\<pi>_def using \<open>classical_operator_exists \<pi>\<close> by (rule classical_operator_ket)
   have C\<rho>x: "C\<rho> *\<^sub>V (ket x) = (case \<rho> x of Some i \<Rightarrow> ket i | None \<Rightarrow> 0)" for x
     unfolding C\<rho>_def using \<open>classical_operator_exists \<rho>\<close> by (rule classical_operator_ket)
   have C\<pi>\<rho>x': "(C\<pi> o\<^sub>C\<^sub>L C\<rho>) *\<^sub>V (ket x) = (case (\<pi> \<circ>\<^sub>m \<rho>) x of Some i \<Rightarrow> ket i | None \<Rightarrow> 0)" for x
     apply (simp add: scaleC_cblinfun.rep_eq C\<rho>x)
     apply (cases "\<rho> x")
     by (auto simp: C\<pi>x)
   thus \<open>classical_operator_exists (\<pi> \<circ>\<^sub>m \<rho>)\<close>
     by (rule classical_operator_existsI)
   hence "C\<pi>\<rho> *\<^sub>V (ket x) = (case (\<pi> \<circ>\<^sub>m \<rho>) x of Some i \<Rightarrow> ket i | None \<Rightarrow> 0)" for x
     unfolding C\<pi>\<rho>_def
     by (rule classical_operator_ket)
   with C\<pi>\<rho>x' have "(C\<pi> o\<^sub>C\<^sub>L C\<rho>) *\<^sub>V (ket x) = C\<pi>\<rho> *\<^sub>V (ket x)" for x
     by simp
   thus "C\<pi> o\<^sub>C\<^sub>L C\<rho> = C\<pi>\<rho>"
     by (simp add: equal_ket)
 qed
 
 lemma classical_operator_Some[simp]: "classical_operator (Some::'a\<Rightarrow>_) = id_cblinfun"
 proof-
   have "(classical_operator Some) *\<^sub>V (ket i)  = id_cblinfun *\<^sub>V (ket i)"
     for i::'a
     apply (subst classical_operator_ket)
      apply (rule classical_operator_exists_inj)
     by auto
   thus ?thesis
     using equal_ket[where A = "classical_operator (Some::'a \<Rightarrow> _ option)"
         and B = "id_cblinfun::'a ell2 \<Rightarrow>\<^sub>C\<^sub>L _"]
     by blast
 qed
 
 lemma isometry_classical_operator[simp]:
   fixes \<pi>::"'a \<Rightarrow> 'b"
   assumes a1: "inj \<pi>"
   shows "isometry (classical_operator (Some o \<pi>))"
 proof -
   have b0: "inj_map (Some \<circ> \<pi>)"
     by (simp add: a1)
   have b0': "inj_map (inv_map (Some \<circ> \<pi>))"
     by simp
   have b1: "inv_map (Some \<circ> \<pi>) \<circ>\<^sub>m (Some \<circ> \<pi>) = Some" 
     apply (rule ext) unfolding inv_map_def o_def 
     using assms unfolding inj_def inv_def by auto
   have b3: "classical_operator (inv_map (Some \<circ> \<pi>)) o\<^sub>C\<^sub>L
             classical_operator (Some \<circ> \<pi>) = classical_operator (inv_map (Some \<circ> \<pi>) \<circ>\<^sub>m (Some \<circ> \<pi>))"
     by (metis b0 b0' b1 classical_operator_Some classical_operator_exists_inj 
         classical_operator_mult)
   show ?thesis
     unfolding isometry_def
     apply (subst classical_operator_adjoint)
     using b0 by (auto simp add: b1 b3)
 qed
 
 lemma unitary_classical_operator[simp]:
   fixes \<pi>::"'a \<Rightarrow> 'b"
   assumes a1: "bij \<pi>"
   shows "unitary (classical_operator (Some o \<pi>))"
 proof (unfold unitary_def, rule conjI)
   have "inj \<pi>"
     using a1 bij_betw_imp_inj_on by auto
   hence "isometry (classical_operator (Some o \<pi>))"
     by simp
   hence "classical_operator (Some \<circ> \<pi>)* o\<^sub>C\<^sub>L classical_operator (Some \<circ> \<pi>) = id_cblinfun"
     unfolding isometry_def by simp
   thus \<open>classical_operator (Some \<circ> \<pi>)* o\<^sub>C\<^sub>L classical_operator (Some \<circ> \<pi>) = id_cblinfun\<close>
     by simp 
 next
   have "inj \<pi>"
     by (simp add: assms bij_is_inj)
   have comp: "Some \<circ> \<pi> \<circ>\<^sub>m inv_map (Some \<circ> \<pi>) = Some"
     apply (rule ext)
     unfolding inv_map_def o_def map_comp_def
     unfolding inv_def apply auto
      apply (metis \<open>inj \<pi>\<close> inv_def inv_f_f)
     using bij_def image_iff range_eqI
     by (metis a1)
   have "classical_operator (Some \<circ> \<pi>) o\<^sub>C\<^sub>L classical_operator (Some \<circ> \<pi>)*
       = classical_operator (Some \<circ> \<pi>) o\<^sub>C\<^sub>L classical_operator (inv_map (Some \<circ> \<pi>))"
     by (simp add: \<open>inj \<pi>\<close>)
   also have "\<dots> = classical_operator ((Some \<circ> \<pi>) \<circ>\<^sub>m (inv_map (Some \<circ> \<pi>)))"
     by (simp add: \<open>inj \<pi>\<close> classical_operator_exists_inj)
   also have "\<dots> = classical_operator (Some::'b\<Rightarrow>_)"
     using comp
     by simp 
   also have "\<dots> = (id_cblinfun:: 'b ell2 \<Rightarrow>\<^sub>C\<^sub>L _)"
     by simp
   finally show "classical_operator (Some \<circ> \<pi>) o\<^sub>C\<^sub>L classical_operator (Some \<circ> \<pi>)* = id_cblinfun".
 qed
 
 
 unbundle no_lattice_syntax
 unbundle no_cblinfun_notation
 
 end
diff --git a/thys/Complex_Bounded_Operators/Complex_Vector_Spaces.thy b/thys/Complex_Bounded_Operators/Complex_Vector_Spaces.thy
--- a/thys/Complex_Bounded_Operators/Complex_Vector_Spaces.thy
+++ b/thys/Complex_Bounded_Operators/Complex_Vector_Spaces.thy
@@ -1,3158 +1,3121 @@
 section \<open>\<open>Complex_Vector_Spaces\<close> -- Complex Vector Spaces\<close>
 
 (*
 Authors:
 
   Dominique Unruh, University of Tartu, unruh@ut.ee
   Jose Manuel Rodriguez Caballero, University of Tartu, jose.manuel.rodriguez.caballero@ut.ee
 *)
 
 theory Complex_Vector_Spaces
   imports
     "HOL-Analysis.Elementary_Topology"
     "HOL-Analysis.Operator_Norm"
     "HOL-Analysis.Elementary_Normed_Spaces"
     "HOL-Library.Set_Algebras"
     "HOL-Analysis.Starlike"
     "HOL-Types_To_Sets.Types_To_Sets"
     "HOL-Library.Complemented_Lattices"
 
     Extra_Vector_Spaces
     Extra_Ordered_Fields
     Extra_Operator_Norm
     Extra_General
 
     Complex_Vector_Spaces0
 begin
 
 bundle notation_norm begin
 notation norm ("\<parallel>_\<parallel>")
 end
 
 unbundle lattice_syntax
 
 subsection \<open>Misc\<close>
 
 
 (* Should rather be in Extra_Vector_Spaces but then complex_vector.span_image_scale' does not exist for some reason.
    Ideally this would replace Vector_Spaces.vector_space.span_image_scale. *)
 lemma (in vector_space) span_image_scale':
   \<comment> \<open>Strengthening of @{thm [source] vector_space.span_image_scale} without the condition \<open>finite S\<close>\<close>
   assumes nz: "\<And>x. x \<in> S \<Longrightarrow> c x \<noteq> 0"
   shows "span ((\<lambda>x. c x *s x) ` S) = span S"
 proof
   have \<open>((\<lambda>x. c x *s x) ` S) \<subseteq> span S\<close>
     by (metis (mono_tags, lifting) image_subsetI in_mono local.span_superset local.subspace_scale local.subspace_span)
   then show \<open>span ((\<lambda>x. c x *s x) ` S) \<subseteq> span S\<close>
     by (simp add: local.span_minimal)
 next
   have \<open>x \<in> span ((\<lambda>x. c x *s x) ` S)\<close> if \<open>x \<in> S\<close> for x
   proof -
     have \<open>x = inverse (c x) *s c x *s x\<close>
       by (simp add: nz that)
     moreover have \<open>c x *s x \<in> (\<lambda>x. c x *s x) ` S\<close>
       using that by blast
     ultimately show ?thesis
       by (metis local.span_base local.span_scale)
   qed
   then show \<open>span S \<subseteq> span ((\<lambda>x. c x *s x) ` S)\<close>
     by (simp add: local.span_minimal subsetI)
 qed
 
 lemma (in scaleC) scaleC_real: assumes "r\<in>\<real>" shows "r *\<^sub>C x = Re r *\<^sub>R x"
   unfolding scaleR_scaleC using assms by simp
 
 lemma of_complex_of_real_eq [simp]: "of_complex (of_real n) = of_real n"
   unfolding of_complex_def of_real_def unfolding scaleR_scaleC by simp
 
 lemma Complexs_of_real [simp]: "of_real r \<in> \<complex>"
   unfolding Complexs_def of_real_def of_complex_def
   apply (subst scaleR_scaleC) by simp
 
 lemma Reals_in_Complexs: "\<real> \<subseteq> \<complex>"
   unfolding Reals_def by auto
 
 lemma (in bounded_clinear) bounded_linear: "bounded_linear f"
   by standard
 
 lemma clinear_times: "clinear (\<lambda>x. c * x)"
   for c :: "'a::complex_algebra"
   by (auto simp: clinearI distrib_left)
 
 lemma (in clinear) linear: \<open>linear f\<close>
   by standard
 
 lemma bounded_clinearI:
   assumes \<open>\<And>b1 b2. f (b1 + b2) = f b1 + f b2\<close>
   assumes \<open>\<And>r b. f (r *\<^sub>C b) = r *\<^sub>C f b\<close>
   assumes \<open>\<And>x. norm (f x) \<le> norm x * K\<close>
   shows "bounded_clinear f"
   using assms by (auto intro!: exI bounded_clinear.intro clinearI simp: bounded_clinear_axioms_def)
 
 lemma bounded_clinear_id[simp]: \<open>bounded_clinear id\<close>
   by (simp add: id_def)
 
 definition cbilinear :: \<open>('a::complex_vector \<Rightarrow> 'b::complex_vector \<Rightarrow> 'c::complex_vector) \<Rightarrow> bool\<close>
   where \<open>cbilinear = (\<lambda> f. (\<forall> y. clinear (\<lambda> x. f x y)) \<and> (\<forall> x. clinear (\<lambda> y. f x y)) )\<close>
 
 lemma cbilinear_add_left:
   assumes \<open>cbilinear f\<close>
   shows \<open>f (a + b) c = f a c + f b c\<close>
   by (smt (verit, del_insts) assms cbilinear_def complex_vector.linear_add)
 
 lemma cbilinear_add_right:
   assumes \<open>cbilinear f\<close>
   shows \<open>f a (b + c) = f a b + f a c\<close>
   by (smt (verit, del_insts) assms cbilinear_def complex_vector.linear_add)
 
 lemma cbilinear_times:
   fixes g' :: \<open>'a::complex_vector \<Rightarrow> complex\<close> and g :: \<open>'b::complex_vector \<Rightarrow> complex\<close>
   assumes \<open>\<And> x y. h x y = (g' x)*(g y)\<close> and \<open>clinear g\<close> and \<open>clinear g'\<close>
   shows \<open>cbilinear h\<close>
 proof -
   have w1: "h (b1 + b2) y = h b1 y + h b2 y"
     for b1 :: 'a
       and b2 :: 'a
       and y
   proof-
     have \<open>h (b1 + b2) y = g' (b1 + b2) * g y\<close>
       using \<open>\<And> x y. h x y = (g' x)*(g y)\<close>
       by auto
     also have \<open>\<dots> = (g' b1 + g' b2) * g y\<close>
       using \<open>clinear g'\<close>
       unfolding clinear_def
       by (simp add: assms(3) complex_vector.linear_add)
     also have \<open>\<dots> = g' b1 * g y + g' b2 * g y\<close>
       by (simp add: ring_class.ring_distribs(2))
     also have \<open>\<dots> = h b1 y + h b2 y\<close>
       using assms(1) by auto
     finally show ?thesis by blast
   qed
   have w2: "h (r *\<^sub>C b) y = r *\<^sub>C h b y"
     for r :: complex
       and b :: 'a
       and y
   proof-
     have \<open>h (r *\<^sub>C b) y = g' (r *\<^sub>C b) * g y\<close>
       by (simp add: assms(1))
     also have \<open>\<dots> = r *\<^sub>C (g' b * g y)\<close>
       by (simp add: assms(3) complex_vector.linear_scale)
     also have \<open>\<dots> = r *\<^sub>C (h b y)\<close>
       by (simp add: assms(1))
     finally show ?thesis by blast
   qed
   have "clinear (\<lambda>x. h x y)"
     for y :: 'b
     unfolding clinear_def
     by (meson clinearI clinear_def w1 w2)
   hence t2: "\<forall>y. clinear (\<lambda>x. h x y)"
     by simp
   have v1: "h x (b1 + b2) = h x b1 + h x b2"
     for b1 :: 'b
       and b2 :: 'b
       and x
   proof-
     have \<open>h x (b1 + b2)  = g' x * g (b1 + b2)\<close>
       using \<open>\<And> x y. h x y = (g' x)*(g y)\<close>
       by auto
     also have \<open>\<dots> = g' x * (g b1 + g b2)\<close>
       using \<open>clinear g'\<close>
       unfolding clinear_def
       by (simp add: assms(2) complex_vector.linear_add)
     also have \<open>\<dots> = g' x * g b1 + g' x * g b2\<close>
       by (simp add: ring_class.ring_distribs(1))
     also have \<open>\<dots> = h x b1 + h x b2\<close>
       using assms(1) by auto
     finally show ?thesis by blast
   qed
 
   have v2:  "h x (r *\<^sub>C b) = r *\<^sub>C h x b"
     for r :: complex
       and b :: 'b
       and x
   proof-
     have \<open>h x (r *\<^sub>C b) =  g' x * g (r *\<^sub>C b)\<close>
       by (simp add: assms(1))
     also have \<open>\<dots> = r *\<^sub>C (g' x * g b)\<close>
       by (simp add: assms(2) complex_vector.linear_scale)
     also have \<open>\<dots> = r *\<^sub>C (h x b)\<close>
       by (simp add: assms(1))
     finally show ?thesis by blast
   qed
   have "Vector_Spaces.linear (*\<^sub>C) (*\<^sub>C) (h x)"
     for x :: 'a
     using v1 v2
     by (meson clinearI clinear_def)
   hence t1: "\<forall>x. clinear (h x)"
     unfolding clinear_def
     by simp
   show ?thesis
     unfolding cbilinear_def
     by (simp add: t1 t2)
 qed
 
 lemma csubspace_is_subspace: "csubspace A \<Longrightarrow> subspace A"
   apply (rule subspaceI)
   by (auto simp: complex_vector.subspace_def scaleR_scaleC)
 
 lemma span_subset_cspan: "span A \<subseteq> cspan A"
   unfolding span_def complex_vector.span_def
   by (simp add: csubspace_is_subspace hull_antimono)
 
 
 lemma cindependent_implies_independent:
   assumes "cindependent (S::'a::complex_vector set)"
   shows "independent S"
   using assms unfolding dependent_def complex_vector.dependent_def
   using span_subset_cspan by blast
 
 lemma cspan_singleton: "cspan {x} = {\<alpha> *\<^sub>C x| \<alpha>. True}"
 proof -
   have \<open>cspan {x} = {y. y\<in>cspan {x}}\<close>
     by auto
   also have \<open>\<dots> = {\<alpha> *\<^sub>C x| \<alpha>. True}\<close>
     apply (subst complex_vector.span_breakdown_eq)
     by auto
   finally show ?thesis
     by -
 qed
 
 
 lemma cspan_as_span:
   "cspan (B::'a::complex_vector set) = span (B \<union> scaleC \<i> ` B)"
-proof auto
+proof (intro set_eqI iffI)
   let ?cspan = complex_vector.span
   let ?rspan = real_vector.span
   fix \<psi>
   assume cspan: "\<psi> \<in> ?cspan B"
   have "\<exists>B' r. finite B' \<and> B' \<subseteq> B \<and> \<psi> = (\<Sum>b\<in>B'. r b *\<^sub>C b)"
     using complex_vector.span_explicit[of B] cspan
     by auto
   then obtain B' r where "finite B'" and "B' \<subseteq> B" and \<psi>_explicit: "\<psi> = (\<Sum>b\<in>B'. r b *\<^sub>C b)"
     by atomize_elim
   define R where "R = B \<union> scaleC \<i> ` B"
 
   have x2: "(case x of (b, i) \<Rightarrow> if i
             then Im (r b) *\<^sub>R \<i> *\<^sub>C b
             else Re (r b) *\<^sub>R b) \<in> span (B \<union> (*\<^sub>C) \<i> ` B)"
     if "x \<in> B' \<times> (UNIV::bool set)"
     for x :: "'a \<times> bool"
     using that \<open>B' \<subseteq> B\<close> by (auto simp add: real_vector.span_base real_vector.span_scale subset_iff)
   have x1: "\<psi> = (\<Sum>x\<in>B'. \<Sum>i\<in>UNIV. if i then Im (r x) *\<^sub>R \<i> *\<^sub>C x else Re (r x) *\<^sub>R x)"
     if "\<And>b. r b *\<^sub>C b = Re (r b) *\<^sub>R b + Im (r b) *\<^sub>R \<i> *\<^sub>C b"
     using that by (simp add: UNIV_bool \<psi>_explicit)
   moreover have "r b *\<^sub>C b = Re (r b) *\<^sub>R b + Im (r b) *\<^sub>R \<i> *\<^sub>C b" for b
     using complex_eq scaleC_add_left scaleC_scaleC scaleR_scaleC
     by (metis (no_types, lifting) complex_of_real_i i_complex_of_real)
   ultimately have "\<psi> = (\<Sum>(b,i)\<in>(B'\<times>UNIV). if i then Im (r b) *\<^sub>R (\<i> *\<^sub>C b) else Re (r b) *\<^sub>R b)"
     by (simp add: sum.cartesian_product)
   also have "\<dots> \<in> ?rspan R"
     unfolding R_def
     using x2
     by (rule real_vector.span_sum)
   finally show "\<psi> \<in> ?rspan R" by -
 next
   let ?cspan = complex_vector.span
   let ?rspan = real_vector.span
   define R where "R = B \<union> scaleC \<i> ` B"
   fix \<psi>
   assume rspan: "\<psi> \<in> ?rspan R"
   have "subspace {a. a \<in> cspan B}"
     by (rule real_vector.subspaceI, auto simp add: complex_vector.span_zero
         complex_vector.span_add_eq2 complex_vector.span_scale scaleR_scaleC)
   moreover have "x \<in> cspan B"
     if "x \<in> R"
     for x :: 'a
     using that R_def complex_vector.span_base complex_vector.span_scale by fastforce
   ultimately show "\<psi> \<in> ?cspan B"
     using real_vector.span_induct rspan by blast
 qed
 
 
 lemma isomorphic_equal_cdim:
   assumes lin_f: \<open>clinear f\<close>
   assumes inj_f: \<open>inj_on f (cspan S)\<close>
   assumes im_S: \<open>f ` S = T\<close>
   shows \<open>cdim S = cdim T\<close>
 proof -
   obtain SB where SB_span: "cspan SB = cspan S" and indep_SB: \<open>cindependent SB\<close>
     by (metis complex_vector.basis_exists complex_vector.span_mono complex_vector.span_span subset_antisym)
   with lin_f inj_f have indep_fSB: \<open>cindependent (f ` SB)\<close>
     apply (rule_tac complex_vector.linear_independent_injective_image)
     by auto
   from lin_f have \<open>cspan (f ` SB) = f ` cspan SB\<close>
     by (meson complex_vector.linear_span_image)
   also from SB_span lin_f have \<open>\<dots> = cspan T\<close>
     by (metis complex_vector.linear_span_image im_S)
   finally have \<open>cdim T = card (f ` SB)\<close>
     using indep_fSB complex_vector.dim_eq_card by blast
   also have \<open>\<dots> = card SB\<close>
     apply (rule card_image) using inj_f
     by (metis SB_span complex_vector.linear_inj_on_span_iff_independent_image indep_fSB lin_f)
   also have \<open>\<dots> = cdim S\<close>
     using indep_SB SB_span
     by (metis complex_vector.dim_eq_card)
   finally show ?thesis by simp
 qed
 
 
 lemma cindependent_inter_scaleC_cindependent:
   assumes a1: "cindependent (B::'a::complex_vector set)" and a3: "c \<noteq> 1"
   shows "B \<inter> (*\<^sub>C) c ` B = {}"
 proof (rule classical, cases \<open>c = 0\<close>)
   case True
   then show ?thesis
     using a1 by (auto simp add: complex_vector.dependent_zero)
 next
   case False
   assume "\<not>(B \<inter> (*\<^sub>C) c ` B = {})"
   hence "B \<inter> (*\<^sub>C) c ` B \<noteq> {}"
     by blast
   then obtain x where u1: "x \<in> B \<inter> (*\<^sub>C) c ` B"
     by blast
   then obtain b where u2: "x = b" and u3: "b\<in>B"
     by blast
   then  obtain b' where u2': "x = c *\<^sub>C b'" and u3': "b'\<in>B"
     using u1
     by blast
   have g1: "b = c *\<^sub>C b'"
     using u2 and u2' by simp
   hence "b \<in> complex_vector.span {b'}"
     using False
     by (simp add: complex_vector.span_base complex_vector.span_scale)
   hence "b = b'"
     by (metis  u3' a1 complex_vector.dependent_def complex_vector.span_base
         complex_vector.span_scale insertE insert_Diff u2 u2' u3)
   hence "b' = c *\<^sub>C b'"
     using g1 by blast
   thus ?thesis
     by (metis a1 a3 complex_vector.dependent_zero complex_vector.scale_right_imp_eq
         mult_cancel_right2 scaleC_scaleC u3')
 qed
 
 lemma real_independent_from_complex_independent:
   assumes "cindependent (B::'a::complex_vector set)"
   defines "B' == ((*\<^sub>C) \<i> ` B)"
   shows "independent (B \<union> B')"
 proof (rule notI)
   assume \<open>dependent (B \<union> B')\<close>
   then obtain T f0 x where [simp]: \<open>finite T\<close> and \<open>T \<subseteq> B \<union> B'\<close> and f0_sum: \<open>(\<Sum>v\<in>T. f0 v *\<^sub>R v) = 0\<close>
     and x: \<open>x \<in> T\<close> and f0_x: \<open>f0 x \<noteq> 0\<close>
     by (auto simp: real_vector.dependent_explicit)
   define f T1 T2 T' f' x' where \<open>f v = (if v \<in> T then f0 v else 0)\<close>
     and \<open>T1 = T \<inter> B\<close> and \<open>T2 = scaleC (-\<i>) ` (T \<inter> B')\<close>
     and \<open>T' = T1 \<union> T2\<close> and \<open>f' v = f v + \<i> * f (\<i> *\<^sub>C v)\<close>
     and \<open>x' = (if x \<in> T1 then x else -\<i> *\<^sub>C x)\<close> for v
   have \<open>B \<inter> B' = {}\<close>
     by (simp add: assms cindependent_inter_scaleC_cindependent)
   have \<open>T' \<subseteq> B\<close>
     by (auto simp: T'_def T1_def T2_def B'_def)
   have [simp]: \<open>finite T'\<close> \<open>finite T1\<close> \<open>finite T2\<close>
     by (auto simp add: T'_def T1_def T2_def)
   have f_sum: \<open>(\<Sum>v\<in>T. f v *\<^sub>R v) = 0\<close>
     unfolding f_def using f0_sum by auto
   have f_x: \<open>f x \<noteq> 0\<close>
     using f0_x x by (auto simp: f_def)
   have f'_sum: \<open>(\<Sum>v\<in>T'. f' v *\<^sub>C v) = 0\<close>
   proof -
     have \<open>(\<Sum>v\<in>T'. f' v *\<^sub>C v) = (\<Sum>v\<in>T'. complex_of_real (f v) *\<^sub>C v) + (\<Sum>v\<in>T'. (\<i> * complex_of_real (f (\<i> *\<^sub>C v))) *\<^sub>C v)\<close>
       by (auto simp: f'_def sum.distrib scaleC_add_left)
     also have \<open>(\<Sum>v\<in>T'. complex_of_real (f v) *\<^sub>C v) = (\<Sum>v\<in>T1. f v *\<^sub>R v)\<close> (is \<open>_ = ?left\<close>)
       apply (auto simp: T'_def scaleR_scaleC intro!: sum.mono_neutral_cong_right)
       using T'_def T1_def \<open>T' \<subseteq> B\<close> f_def by auto
     also have \<open>(\<Sum>v\<in>T'. (\<i> * complex_of_real (f (\<i> *\<^sub>C v))) *\<^sub>C v) = (\<Sum>v\<in>T2. (\<i> * complex_of_real (f (\<i> *\<^sub>C v))) *\<^sub>C v)\<close> (is \<open>_ = ?right\<close>)
       apply (auto simp: T'_def intro!: sum.mono_neutral_cong_right)
       by (smt (z3) B'_def IntE IntI T1_def T2_def \<open>f \<equiv> \<lambda>v. if v \<in> T then f0 v else 0\<close> add.inverse_inverse complex_vector.vector_space_axioms i_squared imageI mult_minus_left vector_space.vector_space_assms(3) vector_space.vector_space_assms(4))
     also have \<open>?right = (\<Sum>v\<in>T\<inter>B'. f v *\<^sub>R v)\<close> (is \<open>_ = ?right\<close>)
       apply (rule sum.reindex_cong[symmetric, where l=\<open>scaleC \<i>\<close>])
         apply (auto simp: T2_def image_image scaleR_scaleC)
       using inj_on_def by fastforce
     also have \<open>?left + ?right = (\<Sum>v\<in>T. f v *\<^sub>R v)\<close>
       apply (subst sum.union_disjoint[symmetric])
       using \<open>B \<inter> B' = {}\<close> \<open>T \<subseteq> B \<union> B'\<close> apply (auto simp: T1_def)
       by (metis Int_Un_distrib Un_Int_eq(4) sup.absorb_iff1)
     also have \<open>\<dots> = 0\<close>
       by (rule f_sum)
     finally show ?thesis
       by -
   qed
 
   have x': \<open>x' \<in> T'\<close>
     using \<open>T \<subseteq> B \<union> B'\<close> x by (auto simp: x'_def T'_def T1_def T2_def)
 
   have f'_x': \<open>f' x' \<noteq> 0\<close>
     using Complex_eq Complex_eq_0 f'_def f_x x'_def by auto
 
   from \<open>finite T'\<close> \<open>T' \<subseteq> B\<close> f'_sum x' f'_x'
   have \<open>cdependent B\<close>
     using complex_vector.independent_explicit_module by blast
   with assms show False
     by auto
 qed
 
 lemma crepresentation_from_representation:
   assumes a1: "cindependent B" and a2: "b \<in> B" and a3: "finite B"
   shows "crepresentation B \<psi> b = (representation (B \<union> (*\<^sub>C) \<i> ` B) \<psi> b)
                            + \<i> *\<^sub>C (representation (B \<union> (*\<^sub>C) \<i> ` B) \<psi> (\<i> *\<^sub>C b))"
 proof (cases "\<psi> \<in> cspan B")
   define B' where "B' = B \<union> (*\<^sub>C) \<i> ` B"
   case True
   define r  where "r v = real_vector.representation B' \<psi> v" for v
   define r' where "r' v = real_vector.representation B' \<psi> (\<i> *\<^sub>C v)" for v
   define f  where "f v = r v + \<i> *\<^sub>C r' v" for v
   define g  where "g v = crepresentation B \<psi> v" for v
   have "(\<Sum>v | g v \<noteq> 0. g v *\<^sub>C v) = \<psi>"
     unfolding g_def
     using Collect_cong Collect_mono_iff DiffD1 DiffD2 True a1
       complex_vector.finite_representation
       complex_vector.sum_nonzero_representation_eq sum.mono_neutral_cong_left
     by fastforce
   moreover have "finite {v. g v \<noteq> 0}"
     unfolding g_def
     by (simp add: complex_vector.finite_representation)
   moreover have "v \<in> B"
     if "g v \<noteq> 0" for v
     using that unfolding g_def
     by (simp add: complex_vector.representation_ne_zero)
   ultimately have rep1: "(\<Sum>v\<in>B. g v *\<^sub>C v) = \<psi>"
     unfolding g_def
     using a3 True a1 complex_vector.sum_representation_eq by blast
   have l0': "inj ((*\<^sub>C) \<i>::'a \<Rightarrow>'a)"
     unfolding inj_def
     by simp
   have l0: "inj ((*\<^sub>C) (- \<i>)::'a \<Rightarrow>'a)"
     unfolding inj_def
     by simp
   have l1: "(*\<^sub>C) (- \<i>) ` B \<inter> B = {}"
     using cindependent_inter_scaleC_cindependent[where B=B and c = "- \<i>"]
     by (metis Int_commute a1 add.inverse_inverse complex_i_not_one i_squared mult_cancel_left1
         neg_equal_0_iff_equal)
   have l2: "B \<inter> (*\<^sub>C) \<i> ` B = {}"
     by (simp add: a1 cindependent_inter_scaleC_cindependent)
   have rr1: "r (\<i> *\<^sub>C v) = r' v" for v
     unfolding r_def r'_def
     by simp
   have k1: "independent B'"
     unfolding B'_def using a1 real_independent_from_complex_independent by simp
   have "\<psi> \<in> span B'"
     using B'_def True cspan_as_span by blast
   have "v \<in> B'"
     if "r v \<noteq> 0"
     for v
     unfolding r_def
     using r_def real_vector.representation_ne_zero that by auto
   have "finite B'"
     unfolding B'_def using a3
     by simp
   have "(\<Sum>v\<in>B'. r v *\<^sub>R v) = \<psi>"
     unfolding r_def
     using True  Real_Vector_Spaces.real_vector.sum_representation_eq[where B = B' and basis = B'
         and v = \<psi>]
     by (smt Real_Vector_Spaces.dependent_raw_def \<open>\<psi> \<in> Real_Vector_Spaces.span B'\<close> \<open>finite B'\<close>
         equalityD2 k1)
   have d1: "(\<Sum>v\<in>B. r (\<i> *\<^sub>C v) *\<^sub>R (\<i> *\<^sub>C v)) = (\<Sum>v\<in>(*\<^sub>C) \<i> ` B. r v *\<^sub>R v)"
     using l0'
     by (metis (mono_tags, lifting) inj_eq inj_on_def sum.reindex_cong)
   have "(\<Sum>v\<in>B. (r v + \<i> * (r' v)) *\<^sub>C v) = (\<Sum>v\<in>B. r v *\<^sub>C v + (\<i> * r' v) *\<^sub>C v)"
     by (meson scaleC_left.add)
   also have "\<dots> = (\<Sum>v\<in>B. r v *\<^sub>C v) + (\<Sum>v\<in>B. (\<i> * r' v) *\<^sub>C v)"
     using sum.distrib by fastforce
   also have "\<dots> = (\<Sum>v\<in>B. r v *\<^sub>C v) + (\<Sum>v\<in>B. \<i> *\<^sub>C (r' v *\<^sub>C v))"
     by auto
   also have "\<dots> = (\<Sum>v\<in>B. r v *\<^sub>R v) + (\<Sum>v\<in>B. \<i> *\<^sub>C (r (\<i> *\<^sub>C v) *\<^sub>R v))"
     unfolding r'_def r_def
     by (metis (mono_tags, lifting) scaleR_scaleC sum.cong)
   also have "\<dots> = (\<Sum>v\<in>B. r v *\<^sub>R v) + (\<Sum>v\<in>B. r (\<i> *\<^sub>C v) *\<^sub>R (\<i> *\<^sub>C v))"
     by (metis (no_types, lifting) complex_vector.scale_left_commute scaleR_scaleC)
   also have "\<dots> = (\<Sum>v\<in>B. r v *\<^sub>R v) + (\<Sum>v\<in>(*\<^sub>C) \<i> ` B. r v *\<^sub>R v)"
     using d1
     by simp
   also have "\<dots> = \<psi>"
     using l2 \<open>(\<Sum>v\<in>B'. r v *\<^sub>R v) = \<psi>\<close>
     unfolding B'_def
     by (simp add: a3 sum.union_disjoint)
   finally have "(\<Sum>v\<in>B. f v *\<^sub>C v) = \<psi>" unfolding r'_def r_def f_def by simp
   hence "0 = (\<Sum>v\<in>B. f v *\<^sub>C v) - (\<Sum>v\<in>B. crepresentation B \<psi> v *\<^sub>C v)"
     using rep1
     unfolding g_def
     by simp
   also have "\<dots> = (\<Sum>v\<in>B. f v *\<^sub>C v - crepresentation B \<psi> v *\<^sub>C v)"
     by (simp add: sum_subtractf)
   also have "\<dots> = (\<Sum>v\<in>B. (f v - crepresentation B \<psi> v) *\<^sub>C v)"
     by (metis scaleC_left.diff)
   finally have "0 = (\<Sum>v\<in>B. (f v - crepresentation B \<psi> v) *\<^sub>C v)".
   hence "(\<Sum>v\<in>B. (f v - crepresentation B \<psi> v) *\<^sub>C v) = 0"
     by simp
   hence "f b - crepresentation B \<psi> b = 0"
     using a1 a2 a3 complex_vector.independentD[where s = B and t = B
         and u = "\<lambda>v. f v - crepresentation B \<psi> v" and v = b]
       order_refl  by smt
   hence "crepresentation B \<psi> b = f b"
     by simp
   thus ?thesis unfolding f_def r_def r'_def B'_def by auto
 next
   define B' where "B' = B \<union> (*\<^sub>C) \<i> ` B"
   case False
   have b2: "\<psi> \<notin> real_vector.span B'"
     unfolding B'_def
     using False cspan_as_span by auto
   have "\<psi> \<notin> complex_vector.span B"
     using False by blast
   have "crepresentation B \<psi> b = 0"
     unfolding complex_vector.representation_def
     by (simp add: False)
   moreover have "real_vector.representation B' \<psi> b = 0"
     unfolding real_vector.representation_def
     by (simp add: b2)
   moreover have "real_vector.representation B' \<psi> ((*\<^sub>C) \<i> b) = 0"
     unfolding real_vector.representation_def
     by (simp add: b2)
   ultimately show ?thesis unfolding B'_def by simp
 qed
 
 
 lemma CARD_1_vec_0[simp]: \<open>(\<psi> :: _ ::{complex_vector,CARD_1}) = 0\<close>
   by auto
 
 
 lemma scaleC_cindependent:
   assumes a1: "cindependent (B::'a::complex_vector set)" and a3: "c \<noteq> 0"
   shows "cindependent ((*\<^sub>C) c ` B)"
 proof-
   have "u y = 0"
     if g1: "y\<in>S" and g2: "(\<Sum>x\<in>S. u x *\<^sub>C x) = 0" and g3: "finite S" and g4: "S\<subseteq>(*\<^sub>C) c ` B"
     for u y S
   proof-
     define v where "v x = u (c *\<^sub>C x)" for x
     obtain S' where "S'\<subseteq>B" and S_S': "S = (*\<^sub>C) c ` S'"
       by (meson g4 subset_imageE)
     have "inj ((*\<^sub>C) c::'a\<Rightarrow>_)"
       unfolding inj_def
       using a3 by auto
     hence "finite S'"
       using S_S' finite_imageD g3 subset_inj_on by blast
     have "t \<in> (*\<^sub>C) (inverse c) ` S"
       if "t \<in> S'" for t
     proof-
       have "c *\<^sub>C t \<in> S"
         using \<open>S = (*\<^sub>C) c ` S'\<close> that by blast
       hence "(inverse c) *\<^sub>C (c *\<^sub>C t) \<in> (*\<^sub>C) (inverse c) ` S"
         by blast
       moreover have "(inverse c) *\<^sub>C (c *\<^sub>C t) = t"
         by (simp add: a3)
       ultimately show ?thesis by simp
     qed
     moreover have "t \<in> S'"
       if "t \<in> (*\<^sub>C) (inverse c) ` S" for t
     proof-
       obtain t' where "t = (inverse c) *\<^sub>C t'" and "t' \<in> S"
         using \<open>t \<in> (*\<^sub>C) (inverse c) ` S\<close> by auto
       have "c *\<^sub>C t = c *\<^sub>C ((inverse c) *\<^sub>C t')"
         using \<open>t = (inverse c) *\<^sub>C t'\<close> by simp
       also have "\<dots> = (c * (inverse c)) *\<^sub>C t'"
         by simp
       also have "\<dots> = t'"
         by (simp add: a3)
       finally have "c *\<^sub>C t = t'".
       thus ?thesis using \<open>t' \<in> S\<close>
         using \<open>S = (*\<^sub>C) c ` S'\<close> a3 complex_vector.scale_left_imp_eq by blast
     qed
     ultimately have "S' = (*\<^sub>C) (inverse c) ` S"
       by blast
     hence "inverse c *\<^sub>C y \<in> S'"
       using that(1) by blast
     have t: "inj (((*\<^sub>C) c)::'a \<Rightarrow> _)"
       using a3 complex_vector.injective_scale[where c = c]
       by blast
     have "0 = (\<Sum>x\<in>(*\<^sub>C) c ` S'. u x *\<^sub>C x)"
       using \<open>S = (*\<^sub>C) c ` S'\<close> that(2) by auto
     also have "\<dots> = (\<Sum>x\<in>S'. v x *\<^sub>C (c *\<^sub>C x))"
       unfolding v_def
       using t Groups_Big.comm_monoid_add_class.sum.reindex[where h = "((*\<^sub>C) c)" and A = S'
           and g = "\<lambda>x. u x *\<^sub>C x"] subset_inj_on by auto
     also have "\<dots> = c *\<^sub>C (\<Sum>x\<in>S'. v x *\<^sub>C x)"
       by (metis (mono_tags, lifting) complex_vector.scale_left_commute scaleC_right.sum sum.cong)
     finally have "0 = c *\<^sub>C (\<Sum>x\<in>S'. v x *\<^sub>C x)".
     hence "(\<Sum>x\<in>S'. v x *\<^sub>C x) = 0"
       using a3 by auto
     hence "v (inverse c *\<^sub>C y) = 0"
       using \<open>inverse c *\<^sub>C y \<in> S'\<close> \<open>finite S'\<close> \<open>S' \<subseteq> B\<close> a1
         complex_vector.independentD
       by blast
     thus "u y = 0"
       unfolding v_def
       by (simp add: a3)
   qed
   thus ?thesis
     using complex_vector.dependent_explicit
     by (simp add: complex_vector.dependent_explicit )
 qed
 
 lemma cspan_eqI:
   assumes \<open>\<And>a. a\<in>A \<Longrightarrow> a\<in>cspan B\<close>
   assumes \<open>\<And>b. b\<in>B \<Longrightarrow> b\<in>cspan A\<close>
   shows \<open>cspan A = cspan B\<close>
   apply (rule complex_vector.span_subspace[rotated])
     apply (rule complex_vector.span_minimal)
   using assms by auto
 
 lemma (in bounded_cbilinear) bounded_bilinear[simp]: "bounded_bilinear prod"
   by standard
 
+lemma norm_scaleC_sgn[simp]: \<open>complex_of_real (norm \<psi>) *\<^sub>C sgn \<psi> = \<psi>\<close> for \<psi> :: "'a::complex_normed_vector"
+  apply (cases \<open>\<psi> = 0\<close>)
+  by (auto simp: sgn_div_norm scaleR_scaleC)
+
+lemma scaleC_of_complex[simp]: \<open>scaleC x (of_complex y) = of_complex (x * y)\<close>
+  unfolding of_complex_def using scaleC_scaleC by blast
+
+lemma bounded_clinear_inv:
+  assumes [simp]: \<open>bounded_clinear f\<close>
+  assumes b: \<open>b > 0\<close>
+  assumes bound: \<open>\<And>x. norm (f x) \<ge> b * norm x\<close>
+  assumes \<open>surj f\<close>
+  shows \<open>bounded_clinear (inv f)\<close>
+proof (rule bounded_clinear_intro)
+  fix x y :: 'b and r :: complex
+  define x' y' where \<open>x' = inv f x\<close> and \<open>y' = inv f y\<close>
+  have [simp]: \<open>clinear f\<close>
+    by (simp add: bounded_clinear.clinear)
+  have [simp]: \<open>inj f\<close>
+  proof (rule injI)
+    fix x y assume \<open>f x = f y\<close>
+    then have \<open>norm (f (x - y)) = 0\<close>
+      by (simp add: complex_vector.linear_diff)
+    with bound b have \<open>norm (x - y) = 0\<close>
+      by (metis linorder_not_le mult_le_0_iff nle_le norm_ge_zero)
+    then show \<open>x = y\<close>
+      by simp
+  qed
+
+  from \<open>surj f\<close>
+  have [simp]: \<open>x = f x'\<close> \<open>y = f y'\<close>
+    by (simp_all add: surj_f_inv_f x'_def y'_def)
+  show "inv f (x + y) = inv f x + inv f y"
+    by (simp flip: complex_vector.linear_add)
+  show "inv f (r *\<^sub>C x) = r *\<^sub>C inv f x"
+    by (simp flip: clinear.scaleC)
+  from bound have "b * norm (inv f x) \<le> norm x" 
+    by (simp flip: clinear.scaleC)
+  with b show "norm (inv f x) \<le> norm x * inverse b" 
+    by (smt (verit, ccfv_threshold) left_inverse mult.commute mult_cancel_right1 mult_le_cancel_left_pos vector_space_over_itself.scale_scale)
+qed
+
+lemma range_is_csubspace[simp]:
+  assumes a1: "clinear f"
+  shows "csubspace (range f)"
+  using assms complex_vector.linear_subspace_image complex_vector.subspace_UNIV by blast
+
+lemma csubspace_is_convex[simp]:
+  assumes a1: "csubspace M"
+  shows "convex M"
+proof-
+  have \<open>\<forall>x\<in>M. \<forall>y\<in> M. \<forall>u. \<forall>v. u *\<^sub>C x + v *\<^sub>C y \<in>  M\<close>
+    using a1
+    by (simp add:  complex_vector.subspace_def)
+  hence \<open>\<forall>x\<in>M. \<forall>y\<in>M. \<forall>u::real. \<forall>v::real. u *\<^sub>R x + v *\<^sub>R y \<in> M\<close>
+    by (simp add: scaleR_scaleC)
+  hence \<open>\<forall>x\<in>M. \<forall>y\<in>M. \<forall>u\<ge>0. \<forall>v\<ge>0. u + v = 1 \<longrightarrow> u *\<^sub>R x + v *\<^sub>R y \<in>M\<close>
+    by blast
+  thus ?thesis using convex_def by blast
+qed
+
+lemma kernel_is_csubspace[simp]:
+  assumes a1: "clinear f"
+  shows "csubspace  (f -` {0})"
+  by (simp add: assms complex_vector.linear_subspace_vimage)
+
+
 subsection \<open>Antilinear maps and friends\<close>
 
 locale antilinear = additive f for f :: "'a::complex_vector \<Rightarrow> 'b::complex_vector" +
   assumes scaleC: "f (scaleC r x) = cnj r *\<^sub>C f x"
 
 sublocale antilinear \<subseteq> linear
 proof (rule linearI)
   show "f (b1 + b2) = f b1 + f b2"
     for b1 :: 'a
       and b2 :: 'a
     by (simp add: add)
   show "f (r *\<^sub>R b) = r *\<^sub>R f b"
     for r :: real
       and b :: 'a
     unfolding scaleR_scaleC by (subst scaleC, simp)
 qed
 
 lemma antilinear_imp_scaleC:
   fixes D :: "complex \<Rightarrow> 'a::complex_vector"
   assumes "antilinear D"
   obtains d where "D = (\<lambda>x. cnj x *\<^sub>C d)"
 proof -
   interpret clinear "D o cnj"
     apply standard apply auto
      apply (simp add: additive.add assms antilinear.axioms(1))
     using assms antilinear.scaleC by fastforce
   obtain d where "D o cnj = (\<lambda>x. x *\<^sub>C d)"
     using clinear_axioms complex_vector.linear_imp_scale by blast
   then have \<open>D = (\<lambda>x. cnj x *\<^sub>C d)\<close>
     by (metis comp_apply complex_cnj_cnj)
   then show ?thesis
     by (rule that)
 qed
 
 corollary complex_antilinearD:
   fixes f :: "complex \<Rightarrow> complex"
   assumes "antilinear f" obtains c where "f = (\<lambda>x. c * cnj x)"
   by (rule antilinear_imp_scaleC [OF assms]) (force simp: scaleC_conv_of_complex)
 
 lemma antilinearI:
   assumes "\<And>x y. f (x + y) = f x + f y"
     and "\<And>c x. f (c *\<^sub>C x) = cnj c *\<^sub>C f x"
   shows "antilinear f"
   by standard (rule assms)+
 
 lemma antilinear_o_antilinear: "antilinear f \<Longrightarrow> antilinear g \<Longrightarrow> clinear (g o f)"
   apply (rule clinearI)
    apply (simp add: additive.add antilinear_def)
   by (simp add: antilinear.scaleC)
 
 lemma clinear_o_antilinear: "antilinear f \<Longrightarrow> clinear g \<Longrightarrow> antilinear (g o f)"
   apply (rule antilinearI)
    apply (simp add: additive.add complex_vector.linear_add antilinear_def)
   by (simp add: complex_vector.linear_scale antilinear.scaleC)
 
 lemma antilinear_o_clinear: "clinear f \<Longrightarrow> antilinear g \<Longrightarrow> antilinear (g o f)"
   apply (rule antilinearI)
    apply (simp add: additive.add complex_vector.linear_add antilinear_def)
   by (simp add: complex_vector.linear_scale antilinear.scaleC)
 
 locale bounded_antilinear = antilinear f for f :: "'a::complex_normed_vector \<Rightarrow> 'b::complex_normed_vector" +
   assumes bounded: "\<exists>K. \<forall>x. norm (f x) \<le> norm x * K"
 
 lemma bounded_antilinearI:
   assumes \<open>\<And>b1 b2. f (b1 + b2) = f b1 + f b2\<close>
   assumes \<open>\<And>r b. f (r *\<^sub>C b) = cnj r *\<^sub>C f b\<close>
   assumes \<open>\<forall>x. norm (f x) \<le> norm x * K\<close>
   shows "bounded_antilinear f"
   using assms by (auto intro!: exI bounded_antilinear.intro antilinearI simp: bounded_antilinear_axioms_def)
 
 sublocale bounded_antilinear \<subseteq> real: bounded_linear
   \<comment> \<open>Gives access to all lemmas from \<^locale>\<open>linear\<close> using prefix \<open>real.\<close>\<close>
   apply standard by (fact bounded)
 
 lemma (in bounded_antilinear) bounded_linear: "bounded_linear f"
   by (fact real.bounded_linear)
 
 lemma (in bounded_antilinear) antilinear: "antilinear f"
   by (fact antilinear_axioms)
 
 lemma bounded_antilinear_intro:
   assumes "\<And>x y. f (x + y) = f x + f y"
     and "\<And>r x. f (scaleC r x) = scaleC (cnj r) (f x)"
     and "\<And>x. norm (f x) \<le> norm x * K"
   shows "bounded_antilinear f"
   by standard (blast intro: assms)+
 
 lemma bounded_antilinear_0[simp]: \<open>bounded_antilinear (\<lambda>_. 0)\<close>
   by (rule bounded_antilinear_intro[where K=0], auto)
 
 lemma cnj_bounded_antilinear[simp]: "bounded_antilinear cnj"
   apply (rule bounded_antilinear_intro [where K = 1])
   by auto
 
 lemma bounded_antilinear_o_bounded_antilinear:
   assumes "bounded_antilinear f"
     and "bounded_antilinear g"
   shows "bounded_clinear (\<lambda>x. f (g x))"
 proof
   interpret f: bounded_antilinear f by fact
   interpret g: bounded_antilinear g by fact
   fix b1 b2 b r
   show "f (g (b1 + b2)) = f (g b1) + f (g b2)"
     by (simp add: f.add g.add)
   show "f (g (r *\<^sub>C b)) = r *\<^sub>C f (g b)"
     by (simp add: f.scaleC g.scaleC)
   have "bounded_linear (\<lambda>x. f (g x))"
     using f.bounded_linear g.bounded_linear by (rule bounded_linear_compose)
   then show "\<exists>K. \<forall>x. norm (f (g x)) \<le> norm x * K"
     by (rule bounded_linear.bounded)
 qed
 
 lemma bounded_antilinear_o_bounded_clinear:
   assumes "bounded_antilinear f"
     and "bounded_clinear g"
   shows "bounded_antilinear (\<lambda>x. f (g x))"
 proof
   interpret f: bounded_antilinear f by fact
   interpret g: bounded_clinear g by fact
   show "f (g (x + y)) = f (g x) + f (g y)" for x y
     by (simp only: f.add g.add)
   show "f (g (scaleC r x)) = scaleC (cnj r) (f (g x))" for r x
     by (simp add: f.scaleC g.scaleC)
   have "bounded_linear (\<lambda>x. f (g x))"
     using f.bounded_linear g.bounded_linear by (rule bounded_linear_compose)
   then show "\<exists>K. \<forall>x. norm (f (g x)) \<le> norm x * K"
     by (rule bounded_linear.bounded)
 qed
 
 lemma bounded_clinear_o_bounded_antilinear:
   assumes "bounded_clinear f"
     and "bounded_antilinear g"
   shows "bounded_antilinear (\<lambda>x. f (g x))"
 proof
   interpret f: bounded_clinear f by fact
   interpret g: bounded_antilinear g by fact
   show "f (g (x + y)) = f (g x) + f (g y)" for x y
     by (simp only: f.add g.add)
   show "f (g (scaleC r x)) = scaleC (cnj r) (f (g x))" for r x
     using f.scaleC g.scaleC by fastforce
   have "bounded_linear (\<lambda>x. f (g x))"
     using f.bounded_linear g.bounded_linear by (rule bounded_linear_compose)
   then show "\<exists>K. \<forall>x. norm (f (g x)) \<le> norm x * K"
     by (rule bounded_linear.bounded)
 qed
 
 lemma bij_clinear_imp_inv_clinear: "clinear (inv f)"
   if a1: "clinear f" and a2: "bij f"
 proof
   fix b1 b2 r b
   show "inv f (b1 + b2) = inv f b1 + inv f b2"
     by (simp add: a1 a2 bij_is_inj bij_is_surj complex_vector.linear_add inv_f_eq surj_f_inv_f)
   show "inv f (r *\<^sub>C b) = r *\<^sub>C inv f b"
     using that
     by (smt bij_inv_eq_iff clinear_def complex_vector.linear_scale)
 qed
 
 
 locale bounded_sesquilinear =
   fixes
     prod :: "'a::complex_normed_vector \<Rightarrow> 'b::complex_normed_vector \<Rightarrow> 'c::complex_normed_vector"
       (infixl "**" 70)
   assumes add_left: "prod (a + a') b = prod a b + prod a' b"
     and add_right: "prod a (b + b') = prod a b + prod a b'"
     and scaleC_left: "prod (r *\<^sub>C a) b = (cnj r) *\<^sub>C (prod a b)"
     and scaleC_right: "prod a (r *\<^sub>C b) = r *\<^sub>C (prod a b)"
     and bounded: "\<exists>K. \<forall>a b. norm (prod a b) \<le> norm a * norm b * K"
 
 sublocale bounded_sesquilinear \<subseteq> real: bounded_bilinear
   \<comment> \<open>Gives access to all lemmas from \<^locale>\<open>linear\<close> using prefix \<open>real.\<close>\<close>
   apply standard
   by (auto simp: add_left add_right scaleC_left scaleC_right bounded scaleR_scaleC)
 
 lemma (in bounded_sesquilinear) bounded_bilinear[simp]: "bounded_bilinear prod"
   by intro_locales
 
 lemma (in bounded_sesquilinear) bounded_antilinear_left: "bounded_antilinear (\<lambda>a. prod a b)"
   apply standard
     apply (auto simp add: scaleC_left add_left)
   by (metis ab_semigroup_mult_class.mult_ac(1) bounded)
 
 lemma (in bounded_sesquilinear) bounded_clinear_right: "bounded_clinear (\<lambda>b. prod a b)"
   apply standard
     apply (auto simp add: scaleC_right add_right)
   by (metis ab_semigroup_mult_class.mult_ac(1) ordered_field_class.sign_simps(34) real.pos_bounded)
 
 lemma (in bounded_sesquilinear) comp1:
   assumes \<open>bounded_clinear g\<close>
   shows \<open>bounded_sesquilinear (\<lambda>x. prod (g x))\<close>
 proof
   interpret bounded_clinear g by fact
   fix a a' b b' r
   show "prod (g (a + a')) b = prod (g a) b + prod (g a') b"
     by (simp add: add add_left)
   show "prod (g a) (b + b') = prod (g a) b + prod (g a) b'"
     by (simp add: add add_right)
   show "prod (g (r *\<^sub>C a)) b = cnj r *\<^sub>C prod (g a) b"
     by (simp add: scaleC scaleC_left)
   show "prod (g a) (r *\<^sub>C b) = r *\<^sub>C prod (g a) b"
     by (simp add: scaleC_right)
   interpret bi: bounded_bilinear \<open>(\<lambda>x. prod (g x))\<close>
     by (simp add: bounded_linear real.comp1)
   show "\<exists>K. \<forall>a b. norm (prod (g a) b) \<le> norm a * norm b * K"
     using bi.bounded by blast
 qed
 
 lemma (in bounded_sesquilinear) comp2:
   assumes \<open>bounded_clinear g\<close>
   shows \<open>bounded_sesquilinear (\<lambda>x y. prod x (g y))\<close>
 proof
   interpret bounded_clinear g by fact
   fix a a' b b' r
   show "prod (a + a') (g b) = prod a (g b) + prod a' (g b)"
     by (simp add: add add_left)
   show "prod a (g (b + b')) = prod a (g b) + prod a (g b')"
     by (simp add: add add_right)
   show "prod (r *\<^sub>C a) (g b) = cnj r *\<^sub>C prod a (g b)"
     by (simp add: scaleC scaleC_left)
   show "prod a (g (r *\<^sub>C b)) = r *\<^sub>C prod a (g b)"
     by (simp add: scaleC scaleC_right)
   interpret bi: bounded_bilinear \<open>(\<lambda>x y. prod x (g y))\<close>
     apply (rule bounded_bilinear.flip)
     using _ bounded_linear apply (rule bounded_bilinear.comp1)
     using bounded_bilinear by (rule bounded_bilinear.flip)
   show "\<exists>K. \<forall>a b. norm (prod a (g b)) \<le> norm a * norm b * K"
     using bi.bounded by blast
 qed
 
 lemma (in bounded_sesquilinear) comp: "bounded_clinear f \<Longrightarrow> bounded_clinear g \<Longrightarrow> bounded_sesquilinear (\<lambda>x y. prod (f x) (g y))"
   using comp1 bounded_sesquilinear.comp2 by auto
 
 lemma bounded_clinear_const_scaleR:
   fixes c :: real
   assumes \<open>bounded_clinear f\<close>
   shows \<open>bounded_clinear (\<lambda> x. c *\<^sub>R f x )\<close>
 proof-
   have  \<open>bounded_clinear (\<lambda> x. (complex_of_real c) *\<^sub>C f x )\<close>
     by (simp add: assms bounded_clinear_const_scaleC)
   thus ?thesis
     by (simp add: scaleR_scaleC)
 qed
 
 lemma bounded_linear_bounded_clinear:
   \<open>bounded_linear A \<Longrightarrow> \<forall>c x. A (c *\<^sub>C x) = c *\<^sub>C A x \<Longrightarrow> bounded_clinear A\<close>
   apply standard
   by (simp_all add: linear_simps bounded_linear.bounded)
 
 lemma comp_bounded_clinear:
   fixes  A :: \<open>'b::complex_normed_vector \<Rightarrow> 'c::complex_normed_vector\<close>
     and B :: \<open>'a::complex_normed_vector \<Rightarrow> 'b\<close>
   assumes \<open>bounded_clinear A\<close> and \<open>bounded_clinear B\<close>
   shows \<open>bounded_clinear (A \<circ> B)\<close>
   by (metis clinear_compose assms(1) assms(2) bounded_clinear_axioms_def bounded_clinear_compose bounded_clinear_def o_def)
 
 
+lemma bounded_sesquilinear_add:
+  \<open>bounded_sesquilinear (\<lambda> x y. A x y + B x y)\<close> if \<open>bounded_sesquilinear A\<close> and \<open>bounded_sesquilinear B\<close>
+proof
+  fix a a' :: 'a and b b' :: 'b and r :: complex
+  show "A (a + a') b + B (a + a') b = (A a b + B a b) + (A a' b + B a' b)"
+    by (simp add: bounded_sesquilinear.add_left that(1) that(2))
+  show \<open>A a (b + b') + B a (b + b') = (A a b + B a b) + (A a b' + B a b')\<close>
+    by (simp add: bounded_sesquilinear.add_right that(1) that(2))
+  show \<open>A (r *\<^sub>C a) b + B (r *\<^sub>C a) b = cnj r *\<^sub>C (A a b + B a b)\<close>
+    by (simp add: bounded_sesquilinear.scaleC_left scaleC_add_right that(1) that(2))
+  show \<open>A a (r *\<^sub>C b) + B a (r *\<^sub>C b) = r *\<^sub>C (A a b + B a b)\<close>
+    by (simp add: bounded_sesquilinear.scaleC_right scaleC_add_right that(1) that(2))
+  show \<open>\<exists>K. \<forall>a b. norm (A a b + B a b) \<le> norm a * norm b * K\<close>
+  proof-
+    have \<open>\<exists> KA. \<forall> a b. norm (A a b) \<le> norm a * norm b * KA\<close>
+      by (simp add: bounded_sesquilinear.bounded that(1))
+    then obtain KA where \<open>\<forall> a b. norm (A a b) \<le> norm a * norm b * KA\<close>
+      by blast
+    have \<open>\<exists> KB. \<forall> a b. norm (B a b) \<le> norm a * norm b * KB\<close>
+      by (simp add: bounded_sesquilinear.bounded that(2))
+    then obtain KB where \<open>\<forall> a b. norm (B a b) \<le> norm a * norm b * KB\<close>
+      by blast
+    have \<open>norm (A a b + B a b) \<le> norm a * norm b * (KA + KB)\<close>
+      for a b
+    proof-
+      have \<open>norm (A a b + B a b) \<le> norm (A a b) +  norm (B a b)\<close>
+        using norm_triangle_ineq by blast
+      also have \<open>\<dots> \<le> norm a * norm b * KA + norm a * norm b * KB\<close>
+        using  \<open>\<forall> a b. norm (A a b) \<le> norm a * norm b * KA\<close>
+          \<open>\<forall> a b. norm (B a b) \<le> norm a * norm b * KB\<close>
+        using add_mono by blast
+      also have \<open>\<dots>=  norm a * norm b * (KA + KB)\<close>
+        by (simp add: mult.commute ring_class.ring_distribs(2))
+      finally show ?thesis
+        by blast
+    qed
+    thus ?thesis by blast
+  qed
+qed
+
+lemma bounded_sesquilinear_uminus:
+  \<open>bounded_sesquilinear (\<lambda> x y. - A x y)\<close> if \<open>bounded_sesquilinear A\<close>
+proof
+  fix a a' :: 'a and b b' :: 'b and r :: complex
+  show "- A (a + a') b = (- A a b) + (- A a' b)"
+    by (simp add: bounded_sesquilinear.add_left that)
+  show \<open>- A a (b + b') = (- A a b) + (- A a b')\<close>
+    by (simp add: bounded_sesquilinear.add_right that)
+  show \<open>- A (r *\<^sub>C a) b = cnj r *\<^sub>C (- A a b)\<close>
+    by (simp add: bounded_sesquilinear.scaleC_left that)
+  show \<open>- A a (r *\<^sub>C b) = r *\<^sub>C (- A a b)\<close>
+    by (simp add: bounded_sesquilinear.scaleC_right that)
+  show \<open>\<exists>K. \<forall>a b. norm (- A a b) \<le> norm a * norm b * K\<close>
+  proof-
+    have \<open>\<exists> KA. \<forall> a b. norm (A a b) \<le> norm a * norm b * KA\<close>
+      by (simp add: bounded_sesquilinear.bounded that(1))
+    then obtain KA where \<open>\<forall> a b. norm (A a b) \<le> norm a * norm b * KA\<close>
+      by blast
+    have \<open>norm (- A a b) \<le> norm a * norm b * KA\<close>
+      for a b
+      by (simp add: \<open>\<forall>a b. norm (A a b) \<le> norm a * norm b * KA\<close>)
+    thus ?thesis by blast
+  qed
+qed
+
+lemma bounded_sesquilinear_diff:
+  \<open>bounded_sesquilinear (\<lambda> x y. A x y - B x y)\<close> if \<open>bounded_sesquilinear A\<close> and \<open>bounded_sesquilinear B\<close>
+proof -
+  have \<open>bounded_sesquilinear (\<lambda> x y. - B x y)\<close>
+    using that(2) by (rule bounded_sesquilinear_uminus)
+  then have \<open>bounded_sesquilinear (\<lambda> x y. A x y + (- B x y))\<close>
+    using that(1) by (rule bounded_sesquilinear_add[rotated])
+  then show ?thesis
+    by auto
+qed
+
 lemmas isCont_scaleC [simp] =
   bounded_bilinear.isCont [OF bounded_cbilinear_scaleC[THEN bounded_cbilinear.bounded_bilinear]]
 
 subsection \<open>Misc 2\<close>
 
 lemma summable_on_scaleC_left [intro]:
   fixes c :: \<open>'a :: complex_normed_vector\<close>
   assumes "c \<noteq> 0 \<Longrightarrow> f summable_on A"
   shows   "(\<lambda>x. f x *\<^sub>C c) summable_on A"
   apply (cases \<open>c \<noteq> 0\<close>)
    apply (subst asm_rl[of \<open>(\<lambda>x. f x *\<^sub>C c) = (\<lambda>y. y *\<^sub>C c) o f\<close>], simp add: o_def)
    apply (rule summable_on_comm_additive)
   using assms by (auto simp add: scaleC_left.additive_axioms)
 
 lemma summable_on_scaleC_right [intro]:
   fixes f :: \<open>'a \<Rightarrow> 'b :: complex_normed_vector\<close>
   assumes "c \<noteq> 0 \<Longrightarrow> f summable_on A"
   shows   "(\<lambda>x. c *\<^sub>C f x) summable_on A"
   apply (cases \<open>c \<noteq> 0\<close>)
    apply (subst asm_rl[of \<open>(\<lambda>x. c *\<^sub>C f x) = (\<lambda>y. c *\<^sub>C y) o f\<close>], simp add: o_def)
    apply (rule summable_on_comm_additive)
   using assms by (auto simp add: scaleC_right.additive_axioms)
 
 lemma infsum_scaleC_left:
   fixes c :: \<open>'a :: complex_normed_vector\<close>
   assumes "c \<noteq> 0 \<Longrightarrow> f summable_on A"
   shows   "infsum (\<lambda>x. f x *\<^sub>C c) A = infsum f A *\<^sub>C c"
   apply (cases \<open>c \<noteq> 0\<close>)
    apply (subst asm_rl[of \<open>(\<lambda>x. f x *\<^sub>C c) = (\<lambda>y. y *\<^sub>C c) o f\<close>], simp add: o_def)
    apply (rule infsum_comm_additive)
   using assms by (auto simp add: scaleC_left.additive_axioms)
 
 lemma infsum_scaleC_right:
   fixes f :: \<open>'a \<Rightarrow> 'b :: complex_normed_vector\<close>
   shows   "infsum (\<lambda>x. c *\<^sub>C f x) A = c *\<^sub>C infsum f A"
 proof -
   consider (summable) \<open>f summable_on A\<close> | (c0) \<open>c = 0\<close> | (not_summable) \<open>\<not> f summable_on A\<close> \<open>c \<noteq> 0\<close>
     by auto
   then show ?thesis
   proof cases
     case summable
     then show ?thesis
       apply (subst asm_rl[of \<open>(\<lambda>x. c *\<^sub>C f x) = (\<lambda>y. c *\<^sub>C y) o f\<close>], simp add: o_def)
       apply (rule infsum_comm_additive)
       using summable by (auto simp add: scaleC_right.additive_axioms)
   next
     case c0
     then show ?thesis by auto
   next
     case not_summable
     have \<open>\<not> (\<lambda>x. c *\<^sub>C f x) summable_on A\<close>
     proof (rule notI)
       assume \<open>(\<lambda>x. c *\<^sub>C f x) summable_on A\<close>
       then have \<open>(\<lambda>x. inverse c *\<^sub>C c *\<^sub>C f x) summable_on A\<close>
         using summable_on_scaleC_right by blast
       then have \<open>f summable_on A\<close>
         using not_summable by auto
       with not_summable show False
         by simp
     qed
     then show ?thesis
       by (simp add: infsum_not_exists not_summable(1)) 
   qed
 qed
 
 
 
 lemmas sums_of_complex = bounded_linear.sums [OF bounded_clinear_of_complex[THEN bounded_clinear.bounded_linear]]
 lemmas summable_of_complex = bounded_linear.summable [OF bounded_clinear_of_complex[THEN bounded_clinear.bounded_linear]]
 lemmas suminf_of_complex = bounded_linear.suminf [OF bounded_clinear_of_complex[THEN bounded_clinear.bounded_linear]]
 
 lemmas sums_scaleC_left = bounded_linear.sums[OF bounded_clinear_scaleC_left[THEN bounded_clinear.bounded_linear]]
 lemmas summable_scaleC_left = bounded_linear.summable[OF bounded_clinear_scaleC_left[THEN bounded_clinear.bounded_linear]]
 lemmas suminf_scaleC_left = bounded_linear.suminf[OF bounded_clinear_scaleC_left[THEN bounded_clinear.bounded_linear]]
 
 lemmas sums_scaleC_right = bounded_linear.sums[OF bounded_clinear_scaleC_right[THEN bounded_clinear.bounded_linear]]
 lemmas summable_scaleC_right = bounded_linear.summable[OF bounded_clinear_scaleC_right[THEN bounded_clinear.bounded_linear]]
 lemmas suminf_scaleC_right = bounded_linear.suminf[OF bounded_clinear_scaleC_right[THEN bounded_clinear.bounded_linear]]
 
 lemma closed_scaleC:
   fixes S::\<open>'a::complex_normed_vector set\<close> and a :: complex
   assumes \<open>closed S\<close>
   shows \<open>closed ((*\<^sub>C) a ` S)\<close>
 proof (cases \<open>a = 0\<close>)
   case True
   then show ?thesis
     apply (cases \<open>S = {}\<close>)
     by (auto simp: image_constant)
 next
   case False
   then have \<open>(*\<^sub>C) a ` S = (*\<^sub>C) (inverse a) -` S\<close>
     by (auto simp add: rev_image_eqI)
   moreover have \<open>closed ((*\<^sub>C) (inverse a) -` S)\<close>
     by (simp add: assms continuous_closed_vimage)
   ultimately show ?thesis
     by simp
 qed
 
 lemma closure_scaleC:
   fixes S::\<open>'a::complex_normed_vector set\<close>
   shows \<open>closure ((*\<^sub>C) a ` S) = (*\<^sub>C) a ` closure S\<close>
 proof
   have \<open>closed (closure S)\<close>
     by simp
   show "closure ((*\<^sub>C) a ` S) \<subseteq> (*\<^sub>C) a ` closure S"
     by (simp add: closed_scaleC closure_minimal closure_subset image_mono)
 
   have "x \<in> closure ((*\<^sub>C) a ` S)"
     if "x \<in> (*\<^sub>C) a ` closure S"
     for x :: 'a
   proof-
     obtain t where \<open>x = ((*\<^sub>C) a) t\<close> and \<open>t \<in> closure S\<close>
       using \<open>x \<in> (*\<^sub>C) a ` closure S\<close> by auto
     have \<open>\<exists>s. (\<forall>n. s n \<in> S) \<and> s \<longlonglongrightarrow> t\<close>
       using \<open>t \<in> closure S\<close> Elementary_Topology.closure_sequential
       by blast
     then obtain s where \<open>\<forall>n. s n \<in> S\<close> and \<open>s \<longlonglongrightarrow> t\<close>
       by blast
     have \<open>(\<forall> n. scaleC a (s n) \<in> ((*\<^sub>C) a ` S))\<close>
       using \<open>\<forall>n. s n \<in> S\<close> by blast
     moreover have \<open>(\<lambda> n. scaleC a (s n)) \<longlonglongrightarrow> x\<close>
     proof-
       have \<open>isCont (scaleC a) t\<close>
         by simp
       thus ?thesis
         using  \<open>s \<longlonglongrightarrow> t\<close>  \<open>x = ((*\<^sub>C) a) t\<close>
         by (simp add: isCont_tendsto_compose)
     qed
     ultimately show ?thesis using Elementary_Topology.closure_sequential
       by metis
   qed
   thus "(*\<^sub>C) a ` closure S \<subseteq> closure ((*\<^sub>C) a ` S)" by blast
 qed
 
 lemma onorm_scalarC:
   fixes f :: \<open>'a::complex_normed_vector \<Rightarrow> 'b::complex_normed_vector\<close>
   assumes a1: \<open>bounded_clinear f\<close>
   shows  \<open>onorm (\<lambda> x. r *\<^sub>C (f x)) = (cmod r) * onorm f\<close>
 proof-
   have \<open>(norm (f x)) / norm x \<le> onorm f\<close>
     for x
     using a1
     by (simp add: bounded_clinear.bounded_linear le_onorm)
   hence t2: \<open>bdd_above {(norm (f x)) / norm x | x. True}\<close>
     by fastforce
   have \<open>continuous_on UNIV ( (*) w ) \<close>
     for w::real
     by simp
   hence \<open>isCont ( ((*) (cmod r)) ) x\<close>
     for x
     by simp
   hence t3: \<open>continuous (at_left (Sup {(norm (f x)) / norm x | x. True})) ((*) (cmod r))\<close>
     using Elementary_Topology.continuous_at_imp_continuous_within
     by blast
   have \<open>{(norm (f x)) / norm x | x. True} \<noteq> {}\<close>
     by blast
   moreover have \<open>mono ((*) (cmod r))\<close>
     by (simp add: monoI ordered_comm_semiring_class.comm_mult_left_mono)
   ultimately have \<open>Sup {((*) (cmod r)) ((norm (f x)) / norm x) | x. True}
          = ((*) (cmod r)) (Sup {(norm (f x)) / norm x | x. True})\<close>
     using t2 t3
     by (simp add:  continuous_at_Sup_mono full_SetCompr_eq image_image)
   hence  \<open>Sup {(cmod r) * ((norm (f x)) / norm x) | x. True}
          = (cmod r) * (Sup {(norm (f x)) / norm x | x. True})\<close>
     by blast
   moreover have \<open>Sup {(cmod r) * ((norm (f x)) / norm x) | x. True}
                 = (SUP x. cmod r * norm (f x) / norm x)\<close>
     by (simp add: full_SetCompr_eq)
   moreover have \<open>(Sup {(norm (f x)) / norm x | x. True})
                 = (SUP x. norm (f x) / norm x)\<close>
     by (simp add: full_SetCompr_eq)
   ultimately have t1: "(SUP x. cmod r * norm (f x) / norm x)
       = cmod r * (SUP x. norm (f x) / norm x)"
     by simp
   have \<open>onorm (\<lambda> x. r *\<^sub>C (f x)) = (SUP x. norm ( (\<lambda> t. r *\<^sub>C (f t)) x) / norm x)\<close>
     by (simp add: onorm_def)
   hence \<open>onorm (\<lambda> x. r *\<^sub>C (f x)) = (SUP x. (cmod r) * (norm (f x)) / norm x)\<close>
     by simp
   also have \<open>... = (cmod r) * (SUP x. (norm (f x)) / norm x)\<close>
     using t1.
   finally show ?thesis
     by (simp add: onorm_def)
 qed
 
 lemma onorm_scaleC_left_lemma:
   fixes f :: "'a::complex_normed_vector"
   assumes r: "bounded_clinear r"
   shows "onorm (\<lambda>x. r x *\<^sub>C f) \<le> onorm r * norm f"
 proof (rule onorm_bound)
   fix x
   have "norm (r x *\<^sub>C f) = norm (r x) * norm f"
     by simp
   also have "\<dots> \<le> onorm r * norm x * norm f"
     by (simp add: bounded_clinear.bounded_linear mult.commute mult_left_mono onorm r)
   finally show "norm (r x *\<^sub>C f) \<le> onorm r * norm f * norm x"
     by (simp add: ac_simps)
   show "0 \<le> onorm r * norm f"
     by (simp add: bounded_clinear.bounded_linear onorm_pos_le r)
 qed
 
 lemma onorm_scaleC_left:
   fixes f :: "'a::complex_normed_vector"
   assumes f: "bounded_clinear r"
   shows "onorm (\<lambda>x. r x *\<^sub>C f) = onorm r * norm f"
 proof (cases "f = 0")
   assume "f \<noteq> 0"
   show ?thesis
   proof (rule order_antisym)
     show "onorm (\<lambda>x. r x *\<^sub>C f) \<le> onorm r * norm f"
       using f by (rule onorm_scaleC_left_lemma)
   next
     have bl1: "bounded_clinear (\<lambda>x. r x *\<^sub>C f)"
       by (metis bounded_clinear_scaleC_const f)
     have x1:"bounded_clinear (\<lambda>x. r x * norm f)"
       by (metis bounded_clinear_mult_const f)
 
     have "onorm r \<le> onorm (\<lambda>x. r x * complex_of_real (norm f)) / norm f"
       if "onorm r \<le> onorm (\<lambda>x. r x * complex_of_real (norm f)) * cmod (1 / complex_of_real (norm f))"
         and "f \<noteq> 0"
       using that
       by (metis complex_of_real_cmod complex_of_real_nn_iff field_class.field_divide_inverse
           inverse_eq_divide nice_ordered_field_class.zero_le_divide_1_iff norm_ge_zero of_real_1
           of_real_divide of_real_eq_iff)
     hence "onorm r \<le> onorm (\<lambda>x. r x * norm f) * inverse (norm f)"
       using \<open>f \<noteq> 0\<close> onorm_scaleC_left_lemma[OF x1, of "inverse (norm f)"]
       by (simp add: inverse_eq_divide)
     also have "onorm (\<lambda>x. r x * norm f) \<le> onorm (\<lambda>x. r x *\<^sub>C f)"
     proof (rule onorm_bound)
       have "bounded_linear (\<lambda>x. r x *\<^sub>C f)"
         using bl1 bounded_clinear.bounded_linear by auto
       thus "0 \<le> onorm (\<lambda>x. r x *\<^sub>C f)"
         by (rule Operator_Norm.onorm_pos_le)
       show "cmod (r x * complex_of_real (norm f)) \<le> onorm (\<lambda>x. r x *\<^sub>C f) * norm x"
         for x :: 'b
         by (smt \<open>bounded_linear (\<lambda>x. r x *\<^sub>C f)\<close> complex_of_real_cmod complex_of_real_nn_iff
             complex_scaleC_def norm_ge_zero norm_scaleC of_real_eq_iff onorm)
     qed
     finally show "onorm r * norm f \<le> onorm (\<lambda>x. r x *\<^sub>C f)"
       using \<open>f \<noteq> 0\<close>
       by (simp add: inverse_eq_divide pos_le_divide_eq mult.commute)
   qed
 qed (simp add: onorm_zero)
 
 subsection \<open>Finite dimension and canonical basis\<close>
 
 lemma vector_finitely_spanned:
   assumes \<open>z \<in> cspan T\<close>
   shows \<open>\<exists> S. finite S \<and> S \<subseteq> T \<and> z \<in> cspan S\<close>
 proof-
   have \<open>\<exists> S r. finite S \<and> S \<subseteq> T \<and> z = (\<Sum>a\<in>S. r a *\<^sub>C a)\<close>
     using complex_vector.span_explicit[where b = "T"]
       assms by auto
   then obtain S r where \<open>finite S\<close> and \<open>S \<subseteq> T\<close> and \<open>z = (\<Sum>a\<in>S. r a *\<^sub>C a)\<close>
     by blast
   thus ?thesis
     by (meson complex_vector.span_scale complex_vector.span_sum complex_vector.span_superset subset_iff)
 qed
 
 setup \<open>Sign.add_const_constraint ("Complex_Vector_Spaces0.cindependent", SOME \<^typ>\<open>'a set \<Rightarrow> bool\<close>)\<close>
 setup \<open>Sign.add_const_constraint (\<^const_name>\<open>cdependent\<close>, SOME \<^typ>\<open>'a set \<Rightarrow> bool\<close>)\<close>
 setup \<open>Sign.add_const_constraint (\<^const_name>\<open>cspan\<close>, SOME \<^typ>\<open>'a set \<Rightarrow> 'a set\<close>)\<close>
 
 class cfinite_dim = complex_vector +
   assumes cfinitely_spanned: "\<exists>S::'a set. finite S \<and> cspan S = UNIV"
 
 class basis_enum = complex_vector +
   fixes canonical_basis :: "'a list"
   assumes distinct_canonical_basis[simp]:
     "distinct canonical_basis"
     and is_cindependent_set[simp]:
     "cindependent (set canonical_basis)"
     and is_generator_set[simp]:
     "cspan (set canonical_basis) = UNIV"
 
 setup \<open>Sign.add_const_constraint ("Complex_Vector_Spaces0.cindependent", SOME \<^typ>\<open>'a::complex_vector set \<Rightarrow> bool\<close>)\<close>
 setup \<open>Sign.add_const_constraint (\<^const_name>\<open>cdependent\<close>, SOME \<^typ>\<open>'a::complex_vector set \<Rightarrow> bool\<close>)\<close>
 setup \<open>Sign.add_const_constraint (\<^const_name>\<open>cspan\<close>, SOME \<^typ>\<open>'a::complex_vector set \<Rightarrow> 'a set\<close>)\<close>
 
+
+instantiation complex :: basis_enum begin
+definition "canonical_basis = [1::complex]"
+instance
+proof
+  show "distinct (canonical_basis::complex list)"
+    by (simp add: canonical_basis_complex_def)
+  show "cindependent (set (canonical_basis::complex list))"
+    unfolding canonical_basis_complex_def
+    by auto
+  show "cspan (set (canonical_basis::complex list)) = UNIV"
+    unfolding canonical_basis_complex_def
+    apply (auto simp add: cspan_raw_def vector_space_over_itself.span_Basis)
+    by (metis complex_scaleC_def complex_vector.span_base complex_vector.span_scale cspan_raw_def insertI1 mult.right_neutral)
+qed
+end
+
 lemma cdim_UNIV_basis_enum[simp]: \<open>cdim (UNIV::'a::basis_enum set) = length (canonical_basis::'a list)\<close>
   apply (subst is_generator_set[symmetric])
   apply (subst complex_vector.dim_span_eq_card_independent)
    apply (rule is_cindependent_set)
   using distinct_canonical_basis distinct_card by blast
 
 lemma finite_basis: "\<exists>basis::'a::cfinite_dim set. finite basis \<and> cindependent basis \<and> cspan basis = UNIV"
 proof -
   from cfinitely_spanned
   obtain S :: \<open>'a set\<close> where \<open>finite S\<close> and \<open>cspan S = UNIV\<close>
     by auto
   from complex_vector.maximal_independent_subset
   obtain B :: \<open>'a set\<close> where \<open>B \<subseteq> S\<close> and \<open>cindependent B\<close> and \<open>S \<subseteq> cspan B\<close>
     by metis
   moreover have \<open>finite B\<close>
     using \<open>B \<subseteq> S\<close> \<open>finite S\<close>
     by (meson finite_subset)
   moreover have \<open>cspan B = UNIV\<close>
     using \<open>cspan S = UNIV\<close> \<open>S \<subseteq> cspan B\<close>
     by (metis complex_vector.span_eq top_greatest)
   ultimately show ?thesis
     by auto
 qed
 
 instance basis_enum \<subseteq> cfinite_dim
   apply intro_classes
   apply (rule exI[of _ \<open>set canonical_basis\<close>])
   using is_cindependent_set is_generator_set by auto
 
 lemma cindependent_cfinite_dim_finite:
   assumes \<open>cindependent (S::'a::cfinite_dim set)\<close>
   shows \<open>finite S\<close>
   by (metis assms cfinitely_spanned complex_vector.independent_span_bound top_greatest)
 
 lemma cfinite_dim_finite_subspace_basis:
   assumes \<open>csubspace X\<close>
   shows "\<exists>basis::'a::cfinite_dim set. finite basis \<and> cindependent basis \<and> cspan basis = X"
   by (meson assms cindependent_cfinite_dim_finite complex_vector.basis_exists complex_vector.span_subspace)
 
 
 text \<open>The following auxiliary lemma (\<open>finite_span_complete_aux\<close>) shows more or less the same as \<open>finite_span_representation_bounded\<close>,
    \<open>finite_span_complete\<close> below (see there for an intuition about the mathematical
    content of the lemmas). However, there is one difference: Here we additionally assume here
    that there is a bijection rep/abs between a finite type \<^typ>\<open>'basis\<close> and the set $B$.
    This is needed to be able to use results about euclidean spaces that are formulated w.r.t.
    the type class \<^class>\<open>finite\<close>
 
    Since we anyway assume that $B$ is finite, this added assumption does not make the lemma
    weaker. However, we cannot derive the existence of \<^typ>\<open>'basis\<close> inside the proof
    (HOL does not support such reasoning). Therefore we have the type \<^typ>\<open>'basis\<close> as
    an explicit assumption and remove it using @{attribute internalize_sort} after the proof.\<close>
 
+(* TODO: Maybe this should be in Extra_Vector_Spaces *)
 lemma finite_span_complete_aux:
   fixes b :: "'b::real_normed_vector" and B :: "'b set"
     and  rep :: "'basis::finite \<Rightarrow> 'b" and abs :: "'b \<Rightarrow> 'basis"
   assumes t: "type_definition rep abs B"
     and t1: "finite B" and t2: "b\<in>B" and t3: "independent B"
   shows "\<exists>D>0. \<forall>\<psi>. norm (representation B \<psi> b) \<le> norm \<psi> * D"
     and "complete (span B)"
 proof -
   define repr  where "repr = real_vector.representation B"
   define repr' where "repr' \<psi> = Abs_euclidean_space (repr \<psi> o rep)" for \<psi>
   define comb  where "comb l = (\<Sum>b\<in>B. l b *\<^sub>R b)" for l
   define comb' where "comb' l = comb (Rep_euclidean_space l o abs)" for l
 
   have comb_cong: "comb x = comb y" if "\<And>z. z\<in>B \<Longrightarrow> x z = y z" for x y
     unfolding comb_def using that by auto
   have comb_repr[simp]: "comb (repr \<psi>) = \<psi>" if "\<psi> \<in> real_vector.span B" for \<psi>
     using \<open>comb \<equiv> \<lambda>l. \<Sum>b\<in>B. l b *\<^sub>R b\<close> local.repr_def real_vector.sum_representation_eq t1 t3 that
     by fastforce
 
   have w5:"(\<Sum>b | (b \<in> B \<longrightarrow> x b \<noteq> 0) \<and> b \<in> B. x b *\<^sub>R b) =
     (\<Sum>b\<in>B. x b *\<^sub>R b)" for x
     using \<open>finite B\<close>
     by (smt DiffD1 DiffD2 mem_Collect_eq real_vector.scale_eq_0_iff subset_eq sum.mono_neutral_left)
   have "representation B (\<Sum>b\<in>B. x b *\<^sub>R b) =  (\<lambda>b. if b \<in> B then x b else 0)"
     for x
   proof (rule real_vector.representation_eqI)
     show "independent B"
       by (simp add: t3)
     show "(\<Sum>b\<in>B. x b *\<^sub>R b) \<in> span B"
       by (meson real_vector.span_scale real_vector.span_sum real_vector.span_superset subset_iff)
     show "b \<in> B"
       if "(if b \<in> B then x b else 0) \<noteq> 0"
       for b :: 'b
       using that
       by meson
     show "finite {b. (if b \<in> B then x b else 0) \<noteq> 0}"
       using t1 by auto
     show "(\<Sum>b | (if b \<in> B then x b else 0) \<noteq> 0. (if b \<in> B then x b else 0) *\<^sub>R b) = (\<Sum>b\<in>B. x b *\<^sub>R b)"
       using w5
       by simp
   qed
   hence repr_comb[simp]: "repr (comb x) = (\<lambda>b. if b\<in>B then x b else 0)" for x
     unfolding repr_def comb_def.
   have repr_bad[simp]: "repr \<psi> = (\<lambda>_. 0)" if "\<psi> \<notin> real_vector.span B" for \<psi>
     unfolding repr_def using that
     by (simp add: real_vector.representation_def)
   have [simp]: "repr' \<psi> = 0" if "\<psi> \<notin> real_vector.span B" for \<psi>
     unfolding repr'_def repr_bad[OF that]
     apply transfer
     by auto
   have comb'_repr'[simp]: "comb' (repr' \<psi>) = \<psi>"
     if "\<psi> \<in> real_vector.span B" for \<psi>
   proof -
     have x1: "(repr \<psi> \<circ> rep \<circ> abs) z = repr \<psi> z"
       if "z \<in> B"
       for z
       unfolding o_def
       using t that type_definition.Abs_inverse by fastforce
     have "comb' (repr' \<psi>) = comb ((repr \<psi> \<circ> rep) \<circ> abs)"
       unfolding comb'_def repr'_def
       by (subst Abs_euclidean_space_inverse; simp)
     also have "\<dots> = comb (repr \<psi>)"
       using x1 comb_cong by blast
     also have "\<dots> = \<psi>"
       using that by simp
     finally show ?thesis by -
   qed
 
   have t1: "Abs_euclidean_space (Rep_euclidean_space t) = t"
     if "\<And>x. rep x \<in> B"
     for t::"'a euclidean_space"
     apply (subst Rep_euclidean_space_inverse)
     by simp
   have "Abs_euclidean_space
      (\<lambda>y. if rep y \<in> B
            then Rep_euclidean_space x y
            else 0) = x"
     for x
     using type_definition.Rep[OF t] apply simp
     using t1 by blast
   hence "Abs_euclidean_space
      (\<lambda>y. if rep y \<in> B
            then Rep_euclidean_space x (abs (rep y))
            else 0) = x"
     for x
     apply (subst type_definition.Rep_inverse[OF t])
     by simp
   hence repr'_comb'[simp]: "repr' (comb' x) = x" for x
     unfolding comb'_def repr'_def o_def
     by simp
   have sphere: "compact (sphere 0 d :: 'basis euclidean_space set)" for d
     using compact_sphere by blast
   have "complete (UNIV :: 'basis euclidean_space set)"
     by (simp add: complete_UNIV)
 
 
   have "(\<Sum>b\<in>B. (Rep_euclidean_space (x + y) \<circ> abs) b *\<^sub>R b) = (\<Sum>b\<in>B. (Rep_euclidean_space x \<circ> abs) b *\<^sub>R b) + (\<Sum>b\<in>B. (Rep_euclidean_space y \<circ> abs) b *\<^sub>R b)"
     for x :: "'basis euclidean_space"
       and y :: "'basis euclidean_space"
     apply (transfer fixing: abs)
     by (simp add: scaleR_add_left sum.distrib)
   moreover have "(\<Sum>b\<in>B. (Rep_euclidean_space (c *\<^sub>R x) \<circ> abs) b *\<^sub>R b) = c *\<^sub>R (\<Sum>b\<in>B. (Rep_euclidean_space x \<circ> abs) b *\<^sub>R b)"
     for c :: real
       and x :: "'basis euclidean_space"
     apply (transfer fixing: abs)
     by (simp add: real_vector.scale_sum_right)
   ultimately have blin_comb': "bounded_linear comb'"
     unfolding comb_def comb'_def
     by (rule bounded_linearI')
   hence "continuous_on X comb'" for X
     by (simp add: linear_continuous_on)
   hence "compact (comb' ` sphere 0 d)" for d
     using sphere
     by (rule compact_continuous_image)
   hence compact_norm_comb': "compact (norm ` comb' ` sphere 0 1)"
     using compact_continuous_image continuous_on_norm_id by blast
   have not0: "0 \<notin> norm ` comb' ` sphere 0 1"
   proof (rule ccontr, simp)
     assume "0 \<in> norm ` comb' ` sphere 0 1"
     then obtain x where nc0: "norm (comb' x) = 0" and x: "x \<in> sphere 0 1"
       by auto
     hence "comb' x = 0"
       by simp
     hence "repr' (comb' x) = 0"
       unfolding repr'_def o_def repr_def apply simp
       by (smt repr'_comb' blin_comb' dist_0_norm linear_simps(3) mem_sphere norm_zero x)
     hence "x = 0"
       by auto
     with x show False
       by simp
   qed
 
   have "closed (norm ` comb' ` sphere 0 1)"
     using compact_imp_closed compact_norm_comb' by blast
   moreover have "0 \<notin> norm ` comb' ` sphere 0 1"
     by (simp add: not0)
   ultimately have "\<exists>d>0. \<forall>x\<in>norm ` comb' ` sphere 0 1. d \<le> dist 0 x"
     by (meson separate_point_closed)
 
   then obtain d where d: "x\<in>norm ` comb' ` sphere 0 1 \<Longrightarrow> d \<le> dist 0 x"
     and "d > 0" for x
     by metis
   define D where "D = 1/d"
   hence "D > 0"
     using \<open>d>0\<close> unfolding D_def by auto
   have "x \<ge> d"
     if "x\<in>norm ` comb' ` sphere 0 1"
     for x
     using d that
     apply auto
     by fastforce
   hence *: "norm (comb' x) \<ge> d" if "norm x = 1" for x
     using that by auto
   have norm_comb': "norm (comb' x) \<ge> d * norm x" for x
   proof (cases "x=0")
     show "d * norm x \<le> norm (comb' x)"
       if "x = 0"
       using that
       by simp
     show "d * norm x \<le> norm (comb' x)"
       if "x \<noteq> 0"
       using that
       using *[of "(1/norm x) *\<^sub>R x"]
       unfolding linear_simps(5)[OF blin_comb']
       apply auto
       by (simp add: le_divide_eq)
   qed
 
   have *:  "norm (repr' \<psi>) \<le> norm \<psi> * D" for \<psi>
   proof (cases "\<psi> \<in> real_vector.span B")
     show "norm (repr' \<psi>) \<le> norm \<psi> * D"
       if "\<psi> \<in> span B"
       using that     unfolding D_def
       using norm_comb'[of "repr' \<psi>"] \<open>d>0\<close>
       by (simp_all add: linordered_field_class.mult_imp_le_div_pos mult.commute)
 
     show "norm (repr' \<psi>) \<le> norm \<psi> * D"
       if "\<psi> \<notin> span B"
       using that \<open>0 < D\<close> by auto
   qed
 
   hence "norm (Rep_euclidean_space (repr' \<psi>) (abs b)) \<le> norm \<psi> * D" for \<psi>
   proof -
     have "(Rep_euclidean_space (repr' \<psi>) (abs b)) = repr' \<psi> \<bullet> euclidean_space_basis_vector (abs b)"
       apply (transfer fixing: abs b)
       by auto
     also have "\<bar>\<dots>\<bar> \<le> norm (repr' \<psi>)"
       apply (rule Basis_le_norm)
       unfolding Basis_euclidean_space_def by simp
     also have "\<dots> \<le> norm \<psi> * D"
       using * by auto
     finally show ?thesis by simp
   qed
   hence "norm (repr \<psi> b) \<le> norm \<psi> * D" for \<psi>
     unfolding repr'_def
     by (smt \<open>comb' \<equiv> \<lambda>l. comb (Rep_euclidean_space l \<circ> abs)\<close>
         \<open>repr' \<equiv> \<lambda>\<psi>. Abs_euclidean_space (repr \<psi> \<circ> rep)\<close> comb'_repr' comp_apply norm_le_zero_iff
         repr_bad repr_comb)
   thus "\<exists>D>0. \<forall>\<psi>. norm (repr \<psi> b) \<le> norm \<psi> * D"
     using \<open>D>0\<close> by auto
   from \<open>d>0\<close>
   have complete_comb': "complete (comb' ` UNIV)"
   proof (rule complete_isometric_image)
     show "subspace (UNIV::'basis euclidean_space set)"
       by simp
     show "bounded_linear comb'"
       by (simp add: blin_comb')
     show "\<forall>x\<in>UNIV. d * norm x \<le> norm (comb' x)"
       by (simp add: norm_comb')
     show "complete (UNIV::'basis euclidean_space set)"
       by (simp add: \<open>complete UNIV\<close>)
   qed
 
   have range_comb': "comb' ` UNIV = real_vector.span B"
   proof (auto simp: image_def)
     show "comb' x \<in> real_vector.span B" for x
       by (metis comb'_def comb_cong comb_repr local.repr_def repr_bad repr_comb real_vector.representation_zero real_vector.span_zero)
   next
     fix \<psi> assume "\<psi> \<in> real_vector.span B"
     then obtain f where f: "comb f = \<psi>"
       apply atomize_elim
       unfolding span_finite[OF \<open>finite B\<close>] comb_def
       by auto
     define f' where "f' b = (if b\<in>B then f b else 0)" for b :: 'b
     have f': "comb f' = \<psi>"
       unfolding f[symmetric]
       apply (rule comb_cong)
       unfolding f'_def by simp
     define x :: "'basis euclidean_space" where "x = Abs_euclidean_space (f' o rep)"
     have "\<psi> = comb' x"
       by (metis (no_types, lifting) \<open>\<psi> \<in> span B\<close> \<open>repr' \<equiv> \<lambda>\<psi>. Abs_euclidean_space (repr \<psi> \<circ> rep)\<close>
           comb'_repr' f' fun.map_cong repr_comb t type_definition.Rep_range x_def)
     thus "\<exists>x. \<psi> = comb' x"
       by auto
   qed
 
   from range_comb' complete_comb'
   show "complete (real_vector.span B)"
     by simp
 qed
 
+(* TODO: Maybe this should be in Extra_Vector_Spaces *)
 lemma finite_span_complete[simp]:
   fixes A :: "'a::real_normed_vector set"
   assumes "finite A"
   shows "complete (span A)"
   text \<open>The span of a finite set is complete.\<close>
 proof (cases "A \<noteq> {} \<and> A \<noteq> {0}")
   case True
   obtain B where
     BT: "real_vector.span B = real_vector.span A"
     and "independent B"
     and "finite B"
     by (meson True assms finite_subset real_vector.maximal_independent_subset real_vector.span_eq
         real_vector.span_superset subset_trans)
 
   have "B\<noteq>{}"
     apply (rule ccontr, simp)
     using BT True
     by (metis real_vector.span_superset real_vector.span_empty subset_singletonD)
 
 (* The following generalizes finite_span_complete_aux to hold without the assumption
      that 'basis has type class finite *)
   {
     (* The type variable 'basisT must not be the same as the one used in finite_span_complete_aux,
        otherwise "internalize_sort" below fails *)
     assume "\<exists>(Rep :: 'basisT\<Rightarrow>'a) Abs. type_definition Rep Abs B"
     then obtain rep :: "'basisT \<Rightarrow> 'a" and abs :: "'a \<Rightarrow> 'basisT" where t: "type_definition rep abs B"
       by auto
     have basisT_finite: "class.finite TYPE('basisT)"
       apply intro_classes
       using \<open>finite B\<close> t
       by (metis (mono_tags, opaque_lifting) ex_new_if_finite finite_imageI image_eqI type_definition_def)
     note finite_span_complete_aux(2)[internalize_sort "'basis::finite"]
     note this[OF basisT_finite t]
   }
   note this[cancel_type_definition, OF \<open>B\<noteq>{}\<close> \<open>finite B\<close> _ \<open>independent B\<close>]
   hence "complete (real_vector.span B)"
     using \<open>B\<noteq>{}\<close> by auto
   thus "complete (real_vector.span A)"
     unfolding BT by simp
 next
   case False
   thus ?thesis
     using complete_singleton by auto
 qed
 
-
+(* TODO: Maybe this should be in Extra_Vector_Spaces *)
 lemma finite_span_representation_bounded:
   fixes B :: "'a::real_normed_vector set"
   assumes "finite B" and "independent B"
   shows "\<exists>D>0. \<forall>\<psi> b. abs (representation B \<psi> b) \<le> norm \<psi> * D"
 
   text \<open>
   Assume $B$ is a finite linear independent set of vectors (in a real normed vector space).
   Let $\alpha^\psi_b$ be the coefficients of $\psi$ expressed as a linear combination over $B$.
   Then $\alpha$ is is uniformly cblinfun (i.e., $\lvert\alpha^\psi_b \leq D \lVert\psi\rVert\psi$
   for some $D$ independent of $\psi,b$).
 
   (This also holds when $b$ is not in the span of $B$ because of the way \<open>real_vector.representation\<close>
   is defined in this corner case.)\<close>
 
 proof (cases "B\<noteq>{}")
   case True
 
 (* The following generalizes finite_span_complete_aux to hold without the assumption
      that 'basis has type class finite *)
   define repr  where "repr = real_vector.representation B"
   {
     (* Step 1: Create a fake type definition by introducing a new type variable 'basis
                and then assuming the existence of the morphisms Rep/Abs to B
                This is then roughly equivalent to "typedef 'basis = B" *)
     (* The type variable 'basisT must not be the same as the one used in finite_span_complete_aux
        (I.e., we cannot call it 'basis) *)
     assume "\<exists>(Rep :: 'basisT\<Rightarrow>'a) Abs. type_definition Rep Abs B"
     then obtain rep :: "'basisT \<Rightarrow> 'a" and abs :: "'a \<Rightarrow> 'basisT" where t: "type_definition rep abs B"
       by auto
         (* Step 2: We show that our fake typedef 'basisT could be instantiated as type class finite *)
     have basisT_finite: "class.finite TYPE('basisT)"
       apply intro_classes
       using \<open>finite B\<close> t
       by (metis (mono_tags, opaque_lifting) ex_new_if_finite finite_imageI image_eqI type_definition_def)
         (* Step 3: We take the finite_span_complete_aux and remove the requirement that 'basis::finite
                (instead, a precondition "class.finite TYPE('basisT)" is introduced) *)
     note finite_span_complete_aux(1)[internalize_sort "'basis::finite"]
       (* Step 4: We instantiate the premises *)
     note this[OF basisT_finite t]
   }
     (* Now we have the desired fact, except that it still assumes that B is isomorphic to some type 'basis
      together with the assumption that there are morphisms between 'basis and B. 'basis and that premise
      are removed using cancel_type_definition
   *)
   note this[cancel_type_definition, OF True \<open>finite B\<close> _ \<open>independent B\<close>]
 
   hence d2:"\<exists>D. \<forall>\<psi>. D>0 \<and> norm (repr \<psi> b) \<le> norm \<psi> * D" if \<open>b\<in>B\<close> for b
     by (simp add: repr_def that True)
   have d1: " (\<And>b. b \<in> B \<Longrightarrow>
           \<exists>D. \<forall>\<psi>. 0 < D \<and> norm (repr \<psi> b) \<le> norm \<psi> * D) \<Longrightarrow>
     \<exists>D. \<forall>b \<psi>. b \<in> B \<longrightarrow>
                0 < D b \<and> norm (repr \<psi> b) \<le> norm \<psi> * D b"
     apply (rule choice) by auto
   then obtain D where D: "D b > 0 \<and> norm (repr \<psi> b) \<le> norm \<psi> * D b" if "b\<in>B" for b \<psi>
     apply atomize_elim
     using d2 by blast
 
   hence Dpos: "D b > 0" and Dbound: "norm (repr \<psi> b) \<le> norm \<psi> * D b"
     if "b\<in>B" for b \<psi>
     using that by auto
   define Dall where "Dall = Max (D`B)"
   have "Dall > 0"
     unfolding Dall_def using \<open>finite B\<close> \<open>B\<noteq>{}\<close> Dpos
     by (metis (mono_tags, lifting) Max_in finite_imageI image_iff image_is_empty)
   have "Dall \<ge> D b" if "b\<in>B" for b
     unfolding Dall_def using \<open>finite B\<close> that by auto
   with Dbound
   have "norm (repr \<psi> b) \<le> norm \<psi> * Dall" if "b\<in>B" for b \<psi>
     using that
     by (smt mult_left_mono norm_not_less_zero)
   moreover have "norm (repr \<psi> b) \<le> norm \<psi> * Dall" if "b\<notin>B" for b \<psi>
     unfolding repr_def using real_vector.representation_ne_zero True
     by (metis calculation empty_subsetI less_le_trans local.repr_def norm_ge_zero norm_zero not_less
         subsetI subset_antisym)
   ultimately show "\<exists>D>0. \<forall>\<psi> b. abs (repr \<psi> b) \<le> norm \<psi> * D"
     using \<open>Dall > 0\<close> real_norm_def by metis
 next
   case False
   thus ?thesis
     unfolding repr_def using real_vector.representation_ne_zero[of B]
     using nice_ordered_field_class.linordered_field_no_ub by fastforce
 qed
 
 hide_fact finite_span_complete_aux
 
 
 lemma finite_cspan_complete[simp]:
   fixes B :: "'a::complex_normed_vector set"
   assumes "finite B"
   shows "complete (cspan B)"
   by (simp add: assms cspan_as_span)
 
-
+(* TODO: Maybe this should be in Extra_Vector_Spaces *)
 lemma finite_span_closed[simp]:
   fixes B :: "'a::real_normed_vector set"
   assumes "finite B"
   shows "closed (real_vector.span B)"
   by (simp add: assms complete_imp_closed)
 
 
 lemma finite_cspan_closed[simp]:
   fixes S::\<open>'a::complex_normed_vector set\<close>
   assumes a1: \<open>finite S\<close>
   shows \<open>closed (cspan S)\<close>
   by (simp add: assms complete_imp_closed)
 
 lemma closure_finite_cspan:
   fixes T::\<open>'a::complex_normed_vector set\<close>
   assumes \<open>finite T\<close>
-  shows \<open>closure (cspan T)  = cspan T\<close>
+  shows \<open>closure (cspan T) = cspan T\<close>
   by (simp add: assms)
 
 
 lemma finite_cspan_crepresentation_bounded:
   fixes B :: "'a::complex_normed_vector set"
   assumes a1: "finite B" and a2: "cindependent B"
-  shows "\<exists>D>0. \<forall>\<psi> b. norm (crepresentation B \<psi> b) \<le> norm \<psi> * D"
+  shows "\<exists>D>0. \<forall>\<psi> b. cmod (crepresentation B \<psi> b) \<le> norm \<psi> * D"
 proof -
   define B' where "B' = (B \<union> scaleC \<i> ` B)"
   have independent_B': "independent B'"
     using B'_def \<open>cindependent B\<close>
     by (simp add: real_independent_from_complex_independent a1)
   have "finite B'"
     unfolding B'_def using \<open>finite B\<close> by simp
   obtain D' where "D' > 0" and D': "norm (real_vector.representation B' \<psi> b) \<le> norm \<psi> * D'"
     for \<psi> b
     apply atomize_elim
     using independent_B' \<open>finite B'\<close>
     by (simp add: finite_span_representation_bounded)
 
   define D where "D = 2*D'"
   from \<open>D' > 0\<close> have \<open>D > 0\<close>
     unfolding D_def by simp
   have "norm (crepresentation B \<psi> b) \<le> norm \<psi> * D" for \<psi> b
   proof (cases "b\<in>B")
     case True
     have d3: "norm \<i> = 1"
       by simp
 
     have "norm (\<i> *\<^sub>C complex_of_real (real_vector.representation B' \<psi> (\<i> *\<^sub>C b)))
           = norm \<i> * norm (complex_of_real (real_vector.representation B' \<psi> (\<i> *\<^sub>C b)))"
       using norm_scaleC by blast
     also have "\<dots> = norm (complex_of_real (real_vector.representation B' \<psi> (\<i> *\<^sub>C b)))"
       using d3 by simp
     finally have d2:"norm (\<i> *\<^sub>C complex_of_real (real_vector.representation B' \<psi> (\<i> *\<^sub>C b)))
           = norm (complex_of_real (real_vector.representation B' \<psi> (\<i> *\<^sub>C b)))".
     have "norm (crepresentation B \<psi> b)
         = norm (complex_of_real (real_vector.representation B' \<psi> b)
             + \<i> *\<^sub>C complex_of_real (real_vector.representation B' \<psi> (\<i> *\<^sub>C b)))"
       by (simp add: B'_def True a1 a2 crepresentation_from_representation)
     also have "\<dots> \<le> norm (complex_of_real (real_vector.representation B' \<psi> b))
              + norm (\<i> *\<^sub>C complex_of_real (real_vector.representation B' \<psi> (\<i> *\<^sub>C b)))"
       using norm_triangle_ineq by blast
     also have "\<dots> = norm (complex_of_real (real_vector.representation B' \<psi> b))
                   + norm (complex_of_real (real_vector.representation B' \<psi> (\<i> *\<^sub>C b)))"
       using d2 by simp
     also have "\<dots> = norm (real_vector.representation B' \<psi> b)
                   + norm (real_vector.representation B' \<psi> (\<i> *\<^sub>C b))"
       by simp
     also have "\<dots> \<le> norm \<psi> * D' + norm \<psi> * D'"
       by (rule add_mono; rule D')
     also have "\<dots> \<le> norm \<psi> * D"
       unfolding D_def by linarith
     finally show ?thesis
       by auto
   next
     case False
     hence "crepresentation B \<psi> b = 0"
       using complex_vector.representation_ne_zero by blast
     thus ?thesis
       by (smt \<open>0 < D\<close> norm_ge_zero norm_zero split_mult_pos_le)
   qed
   with \<open>D > 0\<close>
   show ?thesis
     by auto
 qed
 
 lemma bounded_clinear_finite_dim[simp]:
   fixes f :: \<open>'a::{cfinite_dim,complex_normed_vector} \<Rightarrow> 'b::complex_normed_vector\<close>
   assumes \<open>clinear f\<close>
   shows \<open>bounded_clinear f\<close>
 proof -
   include notation_norm
   obtain basis :: \<open>'a set\<close> where b1: "complex_vector.span basis = UNIV"
     and b2: "cindependent basis"
     and b3:"finite basis"
     using finite_basis by auto
   have "\<exists>C>0. \<forall>\<psi> b. cmod (crepresentation basis \<psi> b) \<le> \<parallel>\<psi>\<parallel> * C"
     using finite_cspan_crepresentation_bounded[where B = basis] b2 b3 by blast
   then obtain C where s1: "cmod (crepresentation basis \<psi> b) \<le> \<parallel>\<psi>\<parallel> * C"
     and s2: "C > 0"
   for \<psi> b by blast
   define M where "M = C * (\<Sum>a\<in>basis. \<parallel>f a\<parallel>)"
   have "\<parallel>f x\<parallel> \<le> \<parallel>x\<parallel> * M"
     for x
   proof-
     define r where "r b = crepresentation basis x b" for b
     have x_span: "x \<in> complex_vector.span basis"
       by (simp add: b1)
     have f0: "v \<in> basis"
       if "r v \<noteq> 0" for v
       using complex_vector.representation_ne_zero r_def that by auto
     have w:"{a|a. r a \<noteq> 0} \<subseteq> basis"
       using f0 by blast
     hence f1: "finite {a|a. r a \<noteq> 0}"
       using b3 rev_finite_subset by auto
     have f2: "(\<Sum>a| r a \<noteq> 0. r a *\<^sub>C a) = x"
       unfolding r_def
       using b2 complex_vector.sum_nonzero_representation_eq x_span
         Collect_cong  by fastforce
     have g1: "(\<Sum>a\<in>basis. crepresentation basis x a *\<^sub>C a) = x"
       by (simp add: b2 b3 complex_vector.sum_representation_eq x_span)
     have f3: "(\<Sum>a\<in>basis. r a *\<^sub>C a) = x"
       unfolding r_def
       by (simp add: g1)
     hence "f x = f (\<Sum>a\<in>basis. r a *\<^sub>C a)"
       by simp
     also have "\<dots> = (\<Sum>a\<in>basis. r a *\<^sub>C f a)"
       by (smt (verit, ccfv_SIG) assms complex_vector.linear_scale complex_vector.linear_sum sum.cong)
     finally have "f x = (\<Sum>a\<in>basis. r a *\<^sub>C f a)".
     hence "\<parallel>f x\<parallel> = \<parallel>(\<Sum>a\<in>basis. r a *\<^sub>C f a)\<parallel>"
       by simp
     also have "\<dots> \<le> (\<Sum>a\<in>basis. \<parallel>r a *\<^sub>C f a\<parallel>)"
       by (simp add: sum_norm_le)
     also have "\<dots> \<le> (\<Sum>a\<in>basis. \<parallel>r a\<parallel> * \<parallel>f a\<parallel>)"
       by simp
     also have "\<dots> \<le> (\<Sum>a\<in>basis. \<parallel>x\<parallel> * C * \<parallel>f a\<parallel>)"
       using sum_mono s1 unfolding r_def
       by (simp add: sum_mono mult_right_mono)
     also have "\<dots> \<le> \<parallel>x\<parallel> * C * (\<Sum>a\<in>basis. \<parallel>f a\<parallel>)"
       using sum_distrib_left
       by (smt sum.cong)
     also have "\<dots> = \<parallel>x\<parallel> * M"
       unfolding M_def
       by linarith
     finally show ?thesis .
   qed
   thus ?thesis
     using assms bounded_clinear_def bounded_clinear_axioms_def by blast
 qed
 
-
+(* TODO: Maybe this should be in Extra_Vector_Spaces if we move finite_span_closed, too. *)
 lemma summable_on_scaleR_left_converse:
   \<comment> \<open>This result has nothing to do with the bounded operator library but it
       uses @{thm [source] finite_span_closed} so it is proven here.\<close>
   fixes f :: \<open>'b \<Rightarrow> real\<close>
     and c :: \<open>'a :: real_normed_vector\<close>
   assumes \<open>c \<noteq> 0\<close>
   assumes \<open>(\<lambda>x. f x *\<^sub>R c) summable_on A\<close>
   shows \<open>f summable_on A\<close>
 proof -
   define fromR toR T where \<open>fromR x = x *\<^sub>R c\<close> and \<open>toR = inv fromR\<close> and \<open>T = range fromR\<close> for x :: real
   have \<open>additive fromR\<close>
     by (simp add: fromR_def additive.intro scaleR_left_distrib)
   have \<open>inj fromR\<close>
     by (simp add: fromR_def \<open>c \<noteq> 0\<close> inj_def)
   have toR_fromR: \<open>toR (fromR x) = x\<close> for x
     by (simp add: \<open>inj fromR\<close> toR_def)
   have fromR_toR: \<open>fromR (toR x) = x\<close> if \<open>x \<in> T\<close> for x
     by (metis T_def f_inv_into_f that toR_def)
 
   have 1: \<open>sum (toR \<circ> (fromR \<circ> f)) F = toR (sum (fromR \<circ> f) F)\<close> if \<open>finite F\<close> for F
     by (simp add: o_def additive.sum[OF \<open>additive fromR\<close>, symmetric] toR_fromR)
   have 2: \<open>sum (fromR \<circ> f) F \<in> T\<close> if \<open>finite F\<close> for F
     by (simp add: o_def additive.sum[OF \<open>additive fromR\<close>, symmetric] T_def)
   have 3: \<open>(toR \<longlongrightarrow> toR x) (at x within T)\<close> for x
   proof (cases \<open>x \<in> T\<close>)
     case True
     have \<open>dist (toR y) (toR x) < e\<close> if \<open>y\<in>T\<close> \<open>e>0\<close> \<open>dist y x < e * norm c\<close> for e y
     proof -
       obtain x' y' where x: \<open>x = fromR x'\<close> and y: \<open>y = fromR y'\<close>
         using T_def True \<open>y \<in> T\<close> by blast
       have \<open>dist (toR y) (toR x) = dist (fromR (toR y)) (fromR (toR x)) / norm c\<close>
         by (auto simp: dist_real_def fromR_def \<open>c \<noteq> 0\<close>)
       also have \<open>\<dots> = dist y x / norm c\<close>
         using \<open>x\<in>T\<close> \<open>y\<in>T\<close> by (simp add: fromR_toR)
       also have \<open>\<dots> < e\<close>
         using \<open>dist y x < e * norm c\<close>
         by (simp add: divide_less_eq that(2))
       finally show ?thesis
         by (simp add: x y toR_fromR)
     qed
     then show ?thesis
       apply (auto simp: tendsto_iff at_within_def eventually_inf_principal eventually_nhds_metric)
       by (metis assms(1) div_0 divide_less_eq zero_less_norm_iff)
   next
     case False
     have \<open>T = span {c}\<close>
       by (simp add: T_def fromR_def span_singleton)
     then have \<open>closed T\<close>
       by simp
     with False have \<open>x \<notin> closure T\<close>
       by simp
     then have \<open>(at x within T) = bot\<close>
       by (rule not_in_closure_trivial_limitI)
     then show ?thesis
       by simp
   qed
   have 4: \<open>(fromR \<circ> f) summable_on A\<close>
     by (simp add: assms(2) fromR_def summable_on_cong)
 
   have \<open>(toR o (fromR o f)) summable_on A\<close>
     using 1 2 3 4 
     by (rule summable_on_comm_additive_general[where T=T])
   with toR_fromR
   show ?thesis
     by (auto simp: o_def)
 qed
 
+(* TODO: Maybe this should be in Extra_Vector_Spaces if we move finite_span_closed, too. *)
 lemma infsum_scaleR_left:
   \<comment> \<open>This result has nothing to do with the bounded operator library but it
       uses @{thm [source] finite_span_closed} so it is proven here.
 
       It is a strengthening of @{thm [source] infsum_scaleR_left}.\<close>
   fixes c :: \<open>'a :: real_normed_vector\<close>
   shows "infsum (\<lambda>x. f x *\<^sub>R c) A = infsum f A *\<^sub>R c"
 proof (cases \<open>f summable_on A\<close>)
   case True
   then show ?thesis 
    apply (subst asm_rl[of \<open>(\<lambda>x. f x *\<^sub>R c) = (\<lambda>y. y *\<^sub>R c) o f\<close>], simp add: o_def)
    apply (rule infsum_comm_additive)
   using True by (auto simp add: scaleR_left.additive_axioms)
 next
   case False
   then have \<open>\<not> (\<lambda>x. f x *\<^sub>R c) summable_on A\<close> if \<open>c \<noteq> 0\<close>
     using summable_on_scaleR_left_converse[where A=A and f=f and c=c]
     using that by auto
   with False show ?thesis
     apply (cases \<open>c = 0\<close>)
     by (auto simp add: infsum_not_exists)
 qed
 
+(* TODO: Maybe this should be in Extra_Vector_Spaces if we move finite_span_closed, too. *)
 lemma infsum_of_real: 
   shows \<open>(\<Sum>\<^sub>\<infinity>x\<in>A. of_real (f x) :: 'b::{real_normed_vector, real_algebra_1}) = of_real (\<Sum>\<^sub>\<infinity>x\<in>A. f x)\<close>
   \<comment> \<open>This result has nothing to do with the bounded operator library but it
       uses @{thm [source] finite_span_closed} so it is proven here.\<close>
   unfolding of_real_def
   by (rule infsum_scaleR_left)
 
 
 
 subsection \<open>Closed subspaces\<close>
 
 lemma csubspace_INF[simp]: "(\<And>x. x \<in> A \<Longrightarrow> csubspace x) \<Longrightarrow> csubspace (\<Inter>A)"
   by (simp add: complex_vector.subspace_Inter)
 
 locale closed_csubspace =
   fixes A::"('a::{complex_vector,topological_space}) set"
   assumes subspace: "csubspace A"
   assumes closed: "closed A"
 
 declare closed_csubspace.subspace[simp]
 
 lemma closure_is_csubspace[simp]:
   fixes A::"('a::complex_normed_vector) set"
   assumes \<open>csubspace A\<close>
   shows \<open>csubspace (closure A)\<close>
 proof-
   have "x \<in> closure A \<Longrightarrow> y \<in> closure A \<Longrightarrow> x+y \<in> closure A" for x y
   proof-
     assume \<open>x\<in>(closure A)\<close>
     then obtain xx where \<open>\<forall> n::nat. xx n \<in> A\<close> and \<open>xx \<longlonglongrightarrow> x\<close>
       using closure_sequential by blast
     assume \<open>y\<in>(closure A)\<close>
     then obtain yy where \<open>\<forall> n::nat. yy n \<in> A\<close> and \<open>yy \<longlonglongrightarrow> y\<close>
       using closure_sequential by blast
     have \<open>\<forall> n::nat. (xx n) + (yy n) \<in> A\<close>
       using \<open>\<forall>n. xx n \<in> A\<close> \<open>\<forall>n. yy n \<in> A\<close> assms complex_vector.subspace_def
       by (simp add: complex_vector.subspace_def)
     hence  \<open>(\<lambda> n. (xx n) + (yy n)) \<longlonglongrightarrow> x + y\<close> using  \<open>xx \<longlonglongrightarrow> x\<close> \<open>yy \<longlonglongrightarrow> y\<close>
       by (simp add: tendsto_add)
     thus ?thesis using  \<open>\<forall> n::nat. (xx n) + (yy n) \<in> A\<close>
       by (meson closure_sequential)
   qed
   moreover have "x\<in>(closure A) \<Longrightarrow> c *\<^sub>C x \<in> (closure A)" for x c
   proof-
     assume \<open>x\<in>(closure A)\<close>
     then obtain xx where \<open>\<forall> n::nat. xx n \<in> A\<close> and \<open>xx \<longlonglongrightarrow> x\<close>
       using closure_sequential by blast
     have \<open>\<forall> n::nat. c *\<^sub>C (xx n) \<in> A\<close>
       using \<open>\<forall>n. xx n \<in> A\<close> assms complex_vector.subspace_def
       by (simp add: complex_vector.subspace_def)
     have \<open>isCont (\<lambda> t. c *\<^sub>C t) x\<close>
       using bounded_clinear.bounded_linear bounded_clinear_scaleC_right linear_continuous_at by auto
     hence  \<open>(\<lambda> n. c *\<^sub>C (xx n)) \<longlonglongrightarrow> c *\<^sub>C x\<close> using  \<open>xx \<longlonglongrightarrow> x\<close>
       by (simp add: isCont_tendsto_compose)
     thus ?thesis using  \<open>\<forall> n::nat. c *\<^sub>C (xx n) \<in> A\<close>
       by (meson closure_sequential)
   qed
   moreover have "0 \<in> (closure A)"
     using assms closure_subset complex_vector.subspace_def
     by (metis in_mono)
   ultimately show ?thesis
     by (simp add: complex_vector.subspaceI)
 qed
 
 lemma csubspace_set_plus:
   assumes \<open>csubspace A\<close> and \<open>csubspace B\<close>
   shows \<open>csubspace (A + B)\<close>
 proof -
   define C where \<open>C = {\<psi>+\<phi>| \<psi> \<phi>. \<psi>\<in>A \<and> \<phi>\<in>B}\<close>
   have  "x\<in>C \<Longrightarrow> y\<in>C \<Longrightarrow> x+y\<in>C" for x y
     using C_def assms(1) assms(2) complex_vector.subspace_add complex_vector.subspace_sums by blast
   moreover have "c *\<^sub>C x \<in> C" if \<open>x\<in>C\<close> for x c
   proof -
     have "csubspace C"
       by (simp add: C_def assms(1) assms(2) complex_vector.subspace_sums)
     then show ?thesis
       using that by (simp add: complex_vector.subspace_def)
   qed
   moreover have  "0 \<in> C"
     using  \<open>C = {\<psi> + \<phi> |\<psi> \<phi>. \<psi> \<in> A \<and> \<phi> \<in> B}\<close> add.inverse_neutral add_uminus_conv_diff assms(1) assms(2) diff_0  mem_Collect_eq
       add.right_inverse
     by (metis (mono_tags, lifting) complex_vector.subspace_0)
   ultimately show ?thesis
     unfolding C_def complex_vector.subspace_def
     by (smt mem_Collect_eq set_plus_elim set_plus_intro)
 qed
 
 lemma closed_csubspace_0[simp]:
   "closed_csubspace ({0} :: ('a::{complex_vector,t1_space}) set)"
 proof-
   have \<open>csubspace {0}\<close>
     using add.right_neutral complex_vector.subspace_def scaleC_right.zero
     by blast
   moreover have "closed ({0} :: 'a set)"
     by simp
   ultimately show ?thesis
     by (simp add: closed_csubspace_def)
 qed
 
 lemma closed_csubspace_UNIV[simp]: "closed_csubspace (UNIV::('a::{complex_vector,topological_space}) set)"
 proof-
   have \<open>csubspace UNIV\<close>
     by simp
   moreover have \<open>closed UNIV\<close>
     by simp
   ultimately show ?thesis
     unfolding closed_csubspace_def by auto
 qed
 
 lemma closed_csubspace_inter[simp]:
   assumes "closed_csubspace A" and "closed_csubspace B"
   shows "closed_csubspace (A\<inter>B)"
 proof-
   obtain C where \<open>C = A \<inter> B\<close> by blast
   have \<open>csubspace C\<close>
   proof-
     have "x\<in>C \<Longrightarrow> y\<in>C \<Longrightarrow> x+y\<in>C" for x y
       by (metis IntD1 IntD2 IntI \<open>C = A \<inter> B\<close> assms(1) assms(2) complex_vector.subspace_def closed_csubspace_def)
     moreover have "x\<in>C \<Longrightarrow> c *\<^sub>C x \<in> C" for x c
       by (metis IntD1 IntD2 IntI \<open>C = A \<inter> B\<close> assms(1) assms(2) complex_vector.subspace_def closed_csubspace_def)
     moreover have "0 \<in> C"
       using  \<open>C = A \<inter> B\<close> assms(1) assms(2) complex_vector.subspace_def closed_csubspace_def by fastforce
     ultimately show ?thesis
       by (simp add: complex_vector.subspace_def)
   qed
   moreover have \<open>closed C\<close>
     using  \<open>C = A \<inter> B\<close>
     by (simp add: assms(1) assms(2) closed_Int closed_csubspace.closed)
   ultimately show ?thesis
     using  \<open>C = A \<inter> B\<close>
     by (simp add: closed_csubspace_def)
 qed
 
 
 lemma closed_csubspace_INF[simp]:
   assumes a1: "\<forall>A\<in>\<A>. closed_csubspace A"
   shows "closed_csubspace (\<Inter>\<A>)"
 proof-
   have \<open>csubspace (\<Inter>\<A>)\<close>
     by (simp add: assms closed_csubspace.subspace complex_vector.subspace_Inter)
   moreover have \<open>closed (\<Inter>\<A>)\<close>
     by (simp add: assms closed_Inter closed_csubspace.closed)
   ultimately show ?thesis
     by (simp add: closed_csubspace.intro)
 qed
 
 
 typedef (overloaded) ('a::"{complex_vector,topological_space}")
   ccsubspace = \<open>{S::'a set. closed_csubspace S}\<close>
   morphisms space_as_set Abs_clinear_space
   using Complex_Vector_Spaces.closed_csubspace_UNIV by blast
 
 setup_lifting type_definition_ccsubspace
 
 lemma csubspace_space_as_set[simp]: \<open>csubspace (space_as_set S)\<close>
   by (metis closed_csubspace_def mem_Collect_eq space_as_set)
 
 lemma closed_space_as_set[simp]: \<open>closed (space_as_set S)\<close>
   apply transfer by (simp add: closed_csubspace.closed)
 
 lemma zero_space_as_set[simp]: \<open>0 \<in> space_as_set A\<close>
   by (simp add: complex_vector.subspace_0)
 
 instantiation ccsubspace :: (complex_normed_vector) scaleC begin
 lift_definition scaleC_ccsubspace :: "complex \<Rightarrow> 'a ccsubspace \<Rightarrow> 'a ccsubspace" is
   "\<lambda>c S. (*\<^sub>C) c ` S"
 proof
-  show "csubspace ((*\<^sub>C) c ` S)"
-    if "closed_csubspace S"
-    for c :: complex
-      and S :: "'a set"
+  show "csubspace ((*\<^sub>C) c ` S)" if "closed_csubspace S" for c :: complex and S :: "'a set"
     using that
-    by (simp add: closed_csubspace.subspace complex_vector.linear_subspace_image)
-  show "closed ((*\<^sub>C) c ` S)"
-    if "closed_csubspace S"
-    for c :: complex
-      and S :: "'a set"
+    by (simp add: complex_vector.linear_subspace_image)
+  show "closed ((*\<^sub>C) c ` S)" if "closed_csubspace S" for c :: complex and S :: "'a set"
     using that
     by (simp add: closed_scaleC closed_csubspace.closed)
 qed
 
 lift_definition scaleR_ccsubspace :: "real \<Rightarrow> 'a ccsubspace \<Rightarrow> 'a ccsubspace" is
   "\<lambda>c S. (*\<^sub>R) c ` S"
 proof
   show "csubspace ((*\<^sub>R) r ` S)"
     if "closed_csubspace S"
     for r :: real
       and S :: "'a set"
     using that   using bounded_clinear_def bounded_clinear_scaleC_right scaleR_scaleC
-    by (simp add: scaleR_scaleC closed_csubspace.subspace complex_vector.linear_subspace_image)
+    by (simp add: scaleR_scaleC complex_vector.linear_subspace_image)
   show "closed ((*\<^sub>R) r ` S)"
     if "closed_csubspace S"
     for r :: real
       and S :: "'a set"
     using that
     by (simp add: closed_scaling closed_csubspace.closed)
 qed
 
 instance
 proof
   show "((*\<^sub>R) r::'a ccsubspace \<Rightarrow> _) = (*\<^sub>C) (complex_of_real r)" for r :: real
     by (simp add: scaleR_scaleC scaleC_ccsubspace_def scaleR_ccsubspace_def)
 qed
 end
 
 instantiation ccsubspace :: ("{complex_vector,t1_space}") bot begin
 lift_definition bot_ccsubspace :: \<open>'a ccsubspace\<close> is \<open>{0}\<close>
   by simp
 instance..
 end
 
 lemma zero_cblinfun_image[simp]: "0 *\<^sub>C S = bot" for S :: "_ ccsubspace"
 proof transfer
   have "(0::'b) \<in> (\<lambda>x. 0) ` S"
     if "closed_csubspace S"
     for S::"'b set"
     using that unfolding closed_csubspace_def
     by (simp add: complex_vector.linear_subspace_image complex_vector.module_hom_zero
         complex_vector.subspace_0)
   thus "(*\<^sub>C) 0 ` S = {0::'b}"
     if "closed_csubspace (S::'b set)"
     for S :: "'b set"
     using that
     by (auto intro !: exI [of _ 0])
 qed
 
 lemma csubspace_scaleC_invariant:
   fixes a S
   assumes \<open>a \<noteq> 0\<close> and \<open>csubspace S\<close>
   shows \<open>(*\<^sub>C) a ` S = S\<close>
 proof-
   have  \<open>x \<in> (*\<^sub>C) a ` S \<Longrightarrow> x \<in> S\<close>
     for x
     using assms(2) complex_vector.subspace_scale by blast
   moreover have  \<open>x \<in> S \<Longrightarrow> x \<in> (*\<^sub>C) a ` S\<close>
     for x
   proof -
     assume "x \<in> S"
     hence "\<exists>c aa. (c / a) *\<^sub>C aa \<in> S \<and> c *\<^sub>C aa = x"
       using assms(2) complex_vector.subspace_def scaleC_one by metis
     hence "\<exists>aa. aa \<in> S \<and> a *\<^sub>C aa = x"
       using assms(1) by auto
     thus ?thesis
       by (meson image_iff)
   qed
   ultimately show ?thesis by blast
 qed
 
 
 lemma ccsubspace_scaleC_invariant[simp]: "a \<noteq> 0 \<Longrightarrow> a *\<^sub>C S = S" for S :: "_ ccsubspace"
   apply transfer
   by (simp add: closed_csubspace.subspace csubspace_scaleC_invariant)
 
 
 instantiation ccsubspace :: ("{complex_vector,topological_space}") "top"
 begin
 lift_definition top_ccsubspace :: \<open>'a ccsubspace\<close> is \<open>UNIV\<close>
   by simp
 
 instance ..
 end
 
 lemma space_as_set_bot[simp]: \<open>space_as_set bot = {0}\<close>
   by (rule bot_ccsubspace.rep_eq)
 
 lemma ccsubspace_top_not_bot[simp]:
   "(top::'a::{complex_vector,t1_space,not_singleton} ccsubspace) \<noteq> bot"
   (* The type class t1_space is needed because the definition of bot in ccsubspace needs it *)
   by (metis UNIV_not_singleton bot_ccsubspace.rep_eq top_ccsubspace.rep_eq)
 
 lemma ccsubspace_bot_not_top[simp]:
   "(bot::'a::{complex_vector,t1_space,not_singleton} ccsubspace) \<noteq> top"
   using ccsubspace_top_not_bot by metis
 
 instantiation ccsubspace :: ("{complex_vector,topological_space}") "Inf"
 begin
 lift_definition Inf_ccsubspace::\<open>'a ccsubspace set \<Rightarrow> 'a ccsubspace\<close>
   is \<open>\<lambda> S. \<Inter> S\<close>
 proof
   fix S :: "'a set set"
   assume closed: "closed_csubspace x" if \<open>x \<in> S\<close> for x
   show "csubspace (\<Inter> S::'a set)"
     by (simp add: closed closed_csubspace.subspace)
   show "closed (\<Inter> S::'a set)"
     by (simp add: closed closed_csubspace.closed)
 qed
 
 instance ..
 end
 
 lift_definition ccspan :: "'a::complex_normed_vector set \<Rightarrow> 'a ccsubspace"
   is "\<lambda>G. closure (cspan G)"
 proof (rule closed_csubspace.intro)
   fix S :: "'a set"
   show "csubspace (closure (cspan S))"
     by (simp add: closure_is_csubspace)
   show "closed (closure (cspan S))"
     by simp
 qed
 
+lemma ccspan_superset:
+  \<open>A \<subseteq> space_as_set (ccspan A)\<close>
+  for A :: \<open>'a::complex_normed_vector set\<close>
+  apply transfer
+  by (meson closure_subset complex_vector.span_superset subset_trans)
+
 lemma ccspan_canonical_basis[simp]: "ccspan (set canonical_basis) = top"
   using ccspan.rep_eq space_as_set_inject top_ccsubspace.rep_eq
     closure_UNIV is_generator_set
   by metis
 
 lemma ccspan_Inf_def: \<open>ccspan A = Inf {S. A \<subseteq> space_as_set S}\<close>
   for A::\<open>('a::cbanach) set\<close>
-proof-
+proof -
   have \<open>x \<in> space_as_set (ccspan A)
     \<Longrightarrow> x \<in> space_as_set (Inf {S. A \<subseteq> space_as_set S})\<close>
     for x::'a
   proof-
     assume \<open>x \<in> space_as_set (ccspan A)\<close>
     hence "x \<in> closure (cspan A)"
       by (simp add: ccspan.rep_eq)
     hence \<open>x \<in> closure (complex_vector.span A)\<close>
       unfolding ccspan_def
       by simp
     hence \<open>\<exists> y::nat \<Rightarrow> 'a. (\<forall> n. y n \<in> (complex_vector.span A)) \<and> y \<longlonglongrightarrow> x\<close>
       by (simp add: closure_sequential)
     then obtain y where \<open>\<forall> n. y n \<in> (complex_vector.span A)\<close> and \<open>y \<longlonglongrightarrow> x\<close>
       by blast
     have \<open>y n \<in> \<Inter> {S. (complex_vector.span A) \<subseteq> S \<and> closed_csubspace S}\<close>
       for n
       using  \<open>\<forall> n. y n \<in> (complex_vector.span A)\<close>
       by auto
 
     have \<open>closed_csubspace S \<Longrightarrow> closed S\<close>
       for S::\<open>'a set\<close>
       by (simp add: closed_csubspace.closed)
     hence \<open>closed ( \<Inter> {S. (complex_vector.span A) \<subseteq> S \<and> closed_csubspace S})\<close>
       by simp
     hence \<open>x \<in> \<Inter> {S. (complex_vector.span A) \<subseteq> S \<and> closed_csubspace S}\<close> using \<open>y \<longlonglongrightarrow> x\<close>
       using \<open>\<And>n. y n \<in> \<Inter> {S. complex_vector.span A \<subseteq> S \<and> closed_csubspace S}\<close> closed_sequentially
       by blast
     moreover have \<open>{S. A \<subseteq> S \<and> closed_csubspace S} \<subseteq> {S. (complex_vector.span A) \<subseteq> S \<and> closed_csubspace S}\<close>
       using Collect_mono_iff
       by (simp add: Collect_mono_iff closed_csubspace.subspace complex_vector.span_minimal)
     ultimately have \<open>x \<in> \<Inter> {S. A \<subseteq> S \<and> closed_csubspace S}\<close>
       by blast
     moreover have "(x::'a) \<in> \<Inter> {x. A \<subseteq> x \<and> closed_csubspace x}"
       if "(x::'a) \<in> \<Inter> {S. A \<subseteq> S \<and> closed_csubspace S}"
       for x :: 'a
         and A :: "'a set"
       using that
       by simp
     ultimately show \<open>x \<in> space_as_set (Inf {S. A \<subseteq> space_as_set S})\<close>
       apply transfer.
   qed
   moreover have \<open>x \<in> space_as_set (Inf {S. A \<subseteq> space_as_set S})
              \<Longrightarrow> x \<in> space_as_set (ccspan A)\<close>
     for x::'a
   proof-
     assume \<open>x \<in> space_as_set (Inf {S. A \<subseteq> space_as_set S})\<close>
     hence \<open>x \<in> \<Inter> {S. A \<subseteq> S \<and> closed_csubspace S}\<close>
       apply transfer
       by blast
     moreover have \<open>{S. (complex_vector.span A) \<subseteq> S \<and> closed_csubspace S} \<subseteq> {S. A \<subseteq> S \<and> closed_csubspace S}\<close>
       using Collect_mono_iff complex_vector.span_superset by fastforce
     ultimately have \<open>x \<in> \<Inter> {S. (complex_vector.span A) \<subseteq> S \<and> closed_csubspace S}\<close>
       by blast
     thus \<open>x \<in> space_as_set (ccspan A)\<close>
       by (metis (no_types, lifting) Inter_iff space_as_set closure_subset mem_Collect_eq ccspan.rep_eq)
   qed
   ultimately have \<open>space_as_set (ccspan A) = space_as_set (Inf {S. A \<subseteq> space_as_set S})\<close>
     by blast
   thus ?thesis
     using space_as_set_inject by auto
 qed
 
 lemma cspan_singleton_scaleC[simp]: "(a::complex)\<noteq>0 \<Longrightarrow> cspan { a *\<^sub>C \<psi> } = cspan {\<psi>}"
   for \<psi>::"'a::complex_vector"
   by (smt complex_vector.dependent_single complex_vector.independent_insert
       complex_vector.scale_eq_0_iff complex_vector.span_base complex_vector.span_redundant
       complex_vector.span_scale doubleton_eq_iff insert_absorb insert_absorb2 insert_commute
       singletonI)
 
 lemma closure_is_closed_csubspace[simp]:
   fixes S::\<open>'a::complex_normed_vector set\<close>
   assumes \<open>csubspace S\<close>
   shows \<open>closed_csubspace (closure S)\<close>
-proof-
-  fix x y :: 'a and c :: complex
-  have "x + y \<in> closure S"
-    if "x \<in> closure S"
-      and "y \<in> closure S"
-  proof-
-    have \<open>\<exists> r. (\<forall> n::nat. r n \<in> S) \<and> r \<longlonglongrightarrow> x\<close>
-      using closure_sequential that(1) by auto
-    then obtain r where \<open>\<forall> n::nat. r n \<in> S\<close> and \<open>r \<longlonglongrightarrow> x\<close>
-      by blast
-    have \<open>\<exists> s. (\<forall> n::nat. s n \<in> S) \<and> s \<longlonglongrightarrow> y\<close>
-      using closure_sequential that(2) by auto
-    then obtain s where \<open>\<forall> n::nat. s n \<in> S\<close> and \<open>s \<longlonglongrightarrow> y\<close>
-      by blast
-    have \<open>\<forall> n::nat. r n + s n \<in> S\<close>
-      using \<open>\<forall>n. r n \<in> S\<close> \<open>\<forall>n. s n \<in> S\<close> assms complex_vector.subspace_add by blast
-    moreover have \<open>(\<lambda> n. r n + s n) \<longlonglongrightarrow> x + y\<close>
-      by (simp add: \<open>r \<longlonglongrightarrow> x\<close> \<open>s \<longlonglongrightarrow> y\<close> tendsto_add)
-    ultimately show ?thesis
-      using assms that(1) that(2)
-      by (simp add: complex_vector.subspace_add)
-  qed
-  moreover have "c *\<^sub>C x \<in> closure S"
-    if "x \<in> closure S"
-  proof-
-    have \<open>\<exists> y. (\<forall> n::nat. y n \<in> S) \<and> y \<longlonglongrightarrow> x\<close>
-      using Elementary_Topology.closure_sequential that by auto
-    then obtain y where \<open>\<forall> n::nat. y n \<in> S\<close> and \<open>y \<longlonglongrightarrow> x\<close>
-      by blast
-    have \<open>isCont (scaleC c) x\<close>
-      by simp
-    hence \<open>(\<lambda> n. scaleC c (y n)) \<longlonglongrightarrow> scaleC c x\<close>
-      using  \<open>y \<longlonglongrightarrow> x\<close>
-      by (simp add: isCont_tendsto_compose)
-    from  \<open>\<forall> n::nat. y n \<in> S\<close>
-    have  \<open>\<forall> n::nat. scaleC c (y n) \<in> S\<close>
-      using assms complex_vector.subspace_scale by auto
-    thus ?thesis
-      using assms that
-      by (simp add: complex_vector.subspace_scale)
-  qed
-  moreover have "0 \<in> closure S"
-    by (simp add: assms complex_vector.subspace_0)
-  moreover have "closed (closure S)"
-    by auto
-  ultimately show ?thesis
-    by (simp add: assms closed_csubspace_def)
-qed
+  using assms closed_csubspace.intro closure_is_csubspace by blast
 
 lemma ccspan_singleton_scaleC[simp]: "(a::complex)\<noteq>0 \<Longrightarrow> ccspan {a *\<^sub>C \<psi>} = ccspan {\<psi>}"
   apply transfer by simp
 
 lemma clinear_continuous_at:
   assumes \<open>bounded_clinear f\<close>
   shows \<open>isCont f x\<close>
   by (simp add: assms bounded_clinear.bounded_linear linear_continuous_at)
 
 lemma clinear_continuous_within:
   assumes \<open>bounded_clinear f\<close>
   shows \<open>continuous (at x within s) f\<close>
   by (simp add: assms bounded_clinear.bounded_linear linear_continuous_within)
 
 lemma antilinear_continuous_at:
   assumes \<open>bounded_antilinear f\<close>
   shows \<open>isCont f x\<close>
   by (simp add: assms bounded_antilinear.bounded_linear linear_continuous_at)
 
 lemma antilinear_continuous_within:
   assumes \<open>bounded_antilinear f\<close>
   shows \<open>continuous (at x within s) f\<close>
   by (simp add: assms bounded_antilinear.bounded_linear linear_continuous_within)
 
 lemma bounded_clinear_eq_on:
   fixes A B :: "'a::complex_normed_vector \<Rightarrow> 'b::complex_normed_vector"
   assumes \<open>bounded_clinear A\<close> and \<open>bounded_clinear B\<close> and
     eq: \<open>\<And>x. x \<in> G \<Longrightarrow> A x = B x\<close> and t: \<open>t \<in> closure (cspan G)\<close>
   shows \<open>A t = B t\<close>
 proof -
   have eq': \<open>A t = B t\<close> if \<open>t \<in> cspan G\<close> for t
     using _ _ that eq apply (rule complex_vector.linear_eq_on)
     by (auto simp: assms bounded_clinear.clinear)
   have \<open>A t - B t = 0\<close>
     using _ _ t apply (rule continuous_constant_on_closure)
     by (auto simp add: eq' assms(1) assms(2) clinear_continuous_at continuous_at_imp_continuous_on)
   then show ?thesis
     by auto
 qed
 
 instantiation ccsubspace :: ("{complex_vector,topological_space}") "order"
 begin
 lift_definition less_eq_ccsubspace :: \<open>'a ccsubspace \<Rightarrow> 'a ccsubspace \<Rightarrow> bool\<close>
   is  \<open>(\<subseteq>)\<close>.
 declare less_eq_ccsubspace_def[code del]
 lift_definition less_ccsubspace :: \<open>'a ccsubspace \<Rightarrow> 'a ccsubspace \<Rightarrow> bool\<close>
   is \<open>(\<subset>)\<close>.
 declare less_ccsubspace_def[code del]
 instance
 proof
   fix x y z :: "'a ccsubspace"
   show "(x < y) = (x \<le> y \<and> \<not> y \<le> x)"
     by (simp add: less_eq_ccsubspace.rep_eq less_le_not_le less_ccsubspace.rep_eq)
   show "x \<le> x"
     by (simp add: less_eq_ccsubspace.rep_eq)
   show "x \<le> z" if "x \<le> y" and "y \<le> z"
     using that less_eq_ccsubspace.rep_eq by auto
   show "x = y" if "x \<le> y" and "y \<le> x"
     using that by (simp add: space_as_set_inject less_eq_ccsubspace.rep_eq)
 qed
 end
 
 lemma ccspan_leqI:
   assumes \<open>M \<subseteq> space_as_set S\<close>
   shows \<open>ccspan M \<le> S\<close>
   using assms apply transfer
   by (simp add: closed_csubspace.closed closure_minimal complex_vector.span_minimal)
 
 lemma ccspan_mono:
   assumes \<open>A \<subseteq> B\<close>
   shows \<open>ccspan A \<le> ccspan B\<close>
   apply (transfer fixing: A B)
   by (simp add: assms closure_mono complex_vector.span_mono)
 
-lemma bounded_sesquilinear_add:
-  \<open>bounded_sesquilinear (\<lambda> x y. A x y + B x y)\<close> if \<open>bounded_sesquilinear A\<close> and \<open>bounded_sesquilinear B\<close>
-proof
-  fix a a' :: 'a and b b' :: 'b and r :: complex
-  show "A (a + a') b + B (a + a') b = (A a b + B a b) + (A a' b + B a' b)"
-    by (simp add: bounded_sesquilinear.add_left that(1) that(2))
-  show \<open>A a (b + b') + B a (b + b') = (A a b + B a b) + (A a b' + B a b')\<close>
-    by (simp add: bounded_sesquilinear.add_right that(1) that(2))
-  show \<open>A (r *\<^sub>C a) b + B (r *\<^sub>C a) b = cnj r *\<^sub>C (A a b + B a b)\<close>
-    by (simp add: bounded_sesquilinear.scaleC_left scaleC_add_right that(1) that(2))
-  show \<open>A a (r *\<^sub>C b) + B a (r *\<^sub>C b) = r *\<^sub>C (A a b + B a b)\<close>
-    by (simp add: bounded_sesquilinear.scaleC_right scaleC_add_right that(1) that(2))
-  show \<open>\<exists>K. \<forall>a b. norm (A a b + B a b) \<le> norm a * norm b * K\<close>
-  proof-
-    have \<open>\<exists> KA. \<forall> a b. norm (A a b) \<le> norm a * norm b * KA\<close>
-      by (simp add: bounded_sesquilinear.bounded that(1))
-    then obtain KA where \<open>\<forall> a b. norm (A a b) \<le> norm a * norm b * KA\<close>
-      by blast
-    have \<open>\<exists> KB. \<forall> a b. norm (B a b) \<le> norm a * norm b * KB\<close>
-      by (simp add: bounded_sesquilinear.bounded that(2))
-    then obtain KB where \<open>\<forall> a b. norm (B a b) \<le> norm a * norm b * KB\<close>
-      by blast
-    have \<open>norm (A a b + B a b) \<le> norm a * norm b * (KA + KB)\<close>
-      for a b
-    proof-
-      have \<open>norm (A a b + B a b) \<le> norm (A a b) +  norm (B a b)\<close>
-        using norm_triangle_ineq by blast
-      also have \<open>\<dots> \<le> norm a * norm b * KA + norm a * norm b * KB\<close>
-        using  \<open>\<forall> a b. norm (A a b) \<le> norm a * norm b * KA\<close>
-          \<open>\<forall> a b. norm (B a b) \<le> norm a * norm b * KB\<close>
-        using add_mono by blast
-      also have \<open>\<dots>=  norm a * norm b * (KA + KB)\<close>
-        by (simp add: mult.commute ring_class.ring_distribs(2))
-      finally show ?thesis
-        by blast
-    qed
-    thus ?thesis by blast
-  qed
-qed
-
-lemma bounded_sesquilinear_uminus:
-  \<open>bounded_sesquilinear (\<lambda> x y. - A x y)\<close> if \<open>bounded_sesquilinear A\<close>
-proof
-  fix a a' :: 'a and b b' :: 'b and r :: complex
-  show "- A (a + a') b = (- A a b) + (- A a' b)"
-    by (simp add: bounded_sesquilinear.add_left that)
-  show \<open>- A a (b + b') = (- A a b) + (- A a b')\<close>
-    by (simp add: bounded_sesquilinear.add_right that)
-  show \<open>- A (r *\<^sub>C a) b = cnj r *\<^sub>C (- A a b)\<close>
-    by (simp add: bounded_sesquilinear.scaleC_left that)
-  show \<open>- A a (r *\<^sub>C b) = r *\<^sub>C (- A a b)\<close>
-    by (simp add: bounded_sesquilinear.scaleC_right that)
-  show \<open>\<exists>K. \<forall>a b. norm (- A a b) \<le> norm a * norm b * K\<close>
-  proof-
-    have \<open>\<exists> KA. \<forall> a b. norm (A a b) \<le> norm a * norm b * KA\<close>
-      by (simp add: bounded_sesquilinear.bounded that(1))
-    then obtain KA where \<open>\<forall> a b. norm (A a b) \<le> norm a * norm b * KA\<close>
-      by blast
-    have \<open>norm (- A a b) \<le> norm a * norm b * KA\<close>
-      for a b
-      by (simp add: \<open>\<forall>a b. norm (A a b) \<le> norm a * norm b * KA\<close>)
-    thus ?thesis by blast
-  qed
-qed
-
-lemma bounded_sesquilinear_diff:
-  \<open>bounded_sesquilinear (\<lambda> x y. A x y - B x y)\<close> if \<open>bounded_sesquilinear A\<close> and \<open>bounded_sesquilinear B\<close>
-proof -
-  have \<open>bounded_sesquilinear (\<lambda> x y. - B x y)\<close>
-    using that(2) by (rule bounded_sesquilinear_uminus)
-  then have \<open>bounded_sesquilinear (\<lambda> x y. A x y + (- B x y))\<close>
-    using that(1) by (rule bounded_sesquilinear_add[rotated])
-  then show ?thesis
-    by auto
-qed
-
 lemma ccsubspace_leI:
   assumes t1: "space_as_set A \<subseteq> space_as_set B"
   shows "A \<le> B"
   using t1 apply transfer by -
 
 lemma ccspan_of_empty[simp]: "ccspan {} = bot"
 proof transfer
   show "closure (cspan {}) = {0::'a}"
     by simp
 qed
 
 
 instantiation ccsubspace :: ("{complex_vector,topological_space}") inf begin
 lift_definition inf_ccsubspace :: "'a ccsubspace \<Rightarrow> 'a ccsubspace \<Rightarrow> 'a ccsubspace"
   is "(\<inter>)" by simp
 instance .. end
 
 lemma space_as_set_inf[simp]: "space_as_set (A \<sqinter> B) = space_as_set A \<inter> space_as_set B"
   by (rule inf_ccsubspace.rep_eq)
 
 instantiation ccsubspace :: ("{complex_vector,topological_space}") order_top begin
 instance
 proof
   show "a \<le> \<top>"
     for a :: "'a ccsubspace"
     apply transfer
     by simp
 qed
 end
 
 
 instantiation ccsubspace :: ("{complex_vector,t1_space}") order_bot begin
 instance
 proof
   show "(\<bottom>::'a ccsubspace) \<le> a"
     for a :: "'a ccsubspace"
     apply transfer
     apply auto
     using closed_csubspace.subspace complex_vector.subspace_0 by blast
 qed
 end
 
 
 instantiation ccsubspace :: ("{complex_vector,topological_space}") semilattice_inf begin
 instance
 proof
   fix x y z :: \<open>'a ccsubspace\<close>
   show "x \<sqinter> y \<le> x"
     apply transfer by simp
   show "x \<sqinter> y \<le> y"
     apply transfer by simp
   show "x \<le> y \<sqinter> z" if "x \<le> y" and "x \<le> z"
     using that apply transfer by simp
 qed
 end
 
 
 instantiation ccsubspace :: ("{complex_vector,t1_space}") zero begin
 definition zero_ccsubspace :: "'a ccsubspace" where [simp]: "zero_ccsubspace = bot"
 lemma zero_ccsubspace_transfer[transfer_rule]: \<open>pcr_ccsubspace (=) {0} 0\<close>
   unfolding zero_ccsubspace_def by transfer_prover
 instance ..
 end
 
 lemma ccspan_0[simp]: \<open>ccspan {0} = 0\<close>
   apply transfer
   by simp
 
 definition \<open>rel_ccsubspace R x y = rel_set R (space_as_set x) (space_as_set y)\<close>
 
 
 lemma left_unique_rel_ccsubspace[transfer_rule]: \<open>left_unique (rel_ccsubspace R)\<close> if \<open>left_unique R\<close>
 proof (rule left_uniqueI)
   fix S T :: \<open>'a ccsubspace\<close> and U
   assume assms: \<open>rel_ccsubspace R S U\<close> \<open>rel_ccsubspace R T U\<close>
   have \<open>space_as_set S = space_as_set T\<close>
     apply (rule left_uniqueD)
       using that apply (rule left_unique_rel_set)
     using assms unfolding rel_ccsubspace_def by auto
   then show \<open>S = T\<close>
     by (simp add: space_as_set_inject)
 qed
 
 lemma right_unique_rel_ccsubspace[transfer_rule]: \<open>right_unique (rel_ccsubspace R)\<close> if \<open>right_unique R\<close>
   by (metis rel_ccsubspace_def right_unique_def right_unique_rel_set space_as_set_inject that)
 
 lemma bi_unique_rel_ccsubspace[transfer_rule]: \<open>bi_unique (rel_ccsubspace R)\<close> if \<open>bi_unique R\<close>
   by (metis (no_types, lifting) bi_unique_def bi_unique_rel_set rel_ccsubspace_def space_as_set_inject that)
 
 lemma converse_rel_ccsubspace: \<open>conversep (rel_ccsubspace R) = rel_ccsubspace (conversep R)\<close>
   by (auto simp: rel_ccsubspace_def[abs_def])
 
 lemma space_as_set_top[simp]: \<open>space_as_set top = UNIV\<close>
   by (rule top_ccsubspace.rep_eq)
 
 lemma ccsubspace_eqI:
   assumes \<open>\<And>x. x \<in> space_as_set S \<longleftrightarrow> x \<in> space_as_set T\<close>
   shows \<open>S = T\<close>
   by (metis Abs_clinear_space_cases Abs_clinear_space_inverse antisym assms subsetI)
 
 lemma ccspan_remove_0: \<open>ccspan (A - {0}) = ccspan A\<close>
   apply transfer
   by auto
 
 lemma sgn_in_spaceD: \<open>\<psi> \<in> space_as_set A\<close> if \<open>sgn \<psi> \<in> space_as_set A\<close> and \<open>\<psi> \<noteq> 0\<close>
   for \<psi> :: \<open>_ :: complex_normed_vector\<close>
   using that
   apply (transfer fixing: \<psi>)
   by (metis closed_csubspace.subspace complex_vector.subspace_scale divideC_field_simps(1) scaleR_eq_0_iff scaleR_scaleC sgn_div_norm sgn_zero_iff)
 
 lemma sgn_in_spaceI: \<open>sgn \<psi> \<in> space_as_set A\<close> if \<open>\<psi> \<in> space_as_set A\<close> 
   for \<psi> :: \<open>_ :: complex_normed_vector\<close>
   using that by (auto simp: sgn_div_norm scaleR_scaleC complex_vector.subspace_scale)
 
 lemma ccsubspace_leI_unit:
   fixes A B :: \<open>_ :: complex_normed_vector ccsubspace\<close>
   assumes "\<And>\<psi>. norm \<psi> = 1 \<Longrightarrow> \<psi> \<in> space_as_set A \<Longrightarrow> \<psi> \<in> space_as_set B"
   shows "A \<le> B"
 proof (rule ccsubspace_leI, rule subsetI)
   fix \<psi> assume \<psi>A: \<open>\<psi> \<in> space_as_set A\<close>
   show \<open>\<psi> \<in> space_as_set B\<close>
     apply (cases \<open>\<psi> = 0\<close>)
     apply simp
     using assms[of \<open>sgn \<psi>\<close>] \<psi>A sgn_in_spaceD sgn_in_spaceI 
     by (auto simp: norm_sgn)
 qed
 
+lemma kernel_is_closed_csubspace[simp]:
+  assumes a1: "bounded_clinear f"
+  shows "closed_csubspace (f -` {0})"
+proof-
+  have \<open>csubspace (f -` {0})\<close>
+    using assms bounded_clinear.clinear complex_vector.linear_subspace_vimage complex_vector.subspace_single_0 by blast
+  have "L \<in> {x. f x = 0}"
+    if "r \<longlonglongrightarrow> L" and "\<forall> n. r n \<in> {x. f x = 0}"
+    for r and  L
+  proof-
+    have d1: \<open>\<forall> n. f (r n) = 0\<close>
+      using that(2) by auto
+    have \<open>(\<lambda> n. f (r n)) \<longlonglongrightarrow> f L\<close>
+      using assms clinear_continuous_at continuous_within_tendsto_compose' that(1)
+      by fastforce
+    hence \<open>(\<lambda> n. 0) \<longlonglongrightarrow> f L\<close>
+      using d1 by simp
+    hence \<open>f L = 0\<close>
+      using limI by fastforce
+    thus ?thesis by blast
+  qed
+  then have s3: \<open>closed (f -` {0})\<close>
+    using closed_sequential_limits by force
+  with \<open>csubspace (f -` {0})\<close>
+  show ?thesis
+    using closed_csubspace.intro by blast
+qed
+
 subsection \<open>Closed sums\<close>
 
 definition closed_sum:: \<open>'a::{semigroup_add,topological_space} set \<Rightarrow> 'a set \<Rightarrow> 'a set\<close> where
   \<open>closed_sum A B = closure (A + B)\<close>
 
 notation closed_sum (infixl "+\<^sub>M" 65)
 
 lemma closed_sum_comm: \<open>A +\<^sub>M B = B +\<^sub>M A\<close> for A B :: "_::ab_semigroup_add"
   by (simp add: add.commute closed_sum_def)
 
 lemma closed_sum_left_subset: \<open>0 \<in> B \<Longrightarrow> A \<subseteq> A +\<^sub>M B\<close> for A B :: "_::monoid_add"
   by (metis add.right_neutral closed_sum_def closure_subset in_mono set_plus_intro subsetI)
 
 lemma closed_sum_right_subset: \<open>0 \<in> A \<Longrightarrow> B \<subseteq> A +\<^sub>M B\<close> for A B :: "_::monoid_add"
   by (metis add.left_neutral closed_sum_def closure_subset set_plus_intro subset_iff)
 
 lemma finite_cspan_closed_csubspace:
   assumes "finite (S::'a::complex_normed_vector set)"
   shows "closed_csubspace (cspan S)"
   by (simp add: assms closed_csubspace.intro)
 
 lemma closed_sum_is_sup:
   fixes A B C:: \<open>('a::{complex_vector,topological_space}) set\<close>
   assumes \<open>closed_csubspace C\<close>
   assumes \<open>A \<subseteq> C\<close> and \<open>B \<subseteq> C\<close>
   shows \<open>(A +\<^sub>M B) \<subseteq> C\<close>
 proof -
   have \<open>A + B \<subseteq> C\<close>
     using assms unfolding set_plus_def
     using closed_csubspace.subspace complex_vector.subspace_add by blast
   then show \<open>(A +\<^sub>M B) \<subseteq> C\<close>
     unfolding closed_sum_def
     using \<open>closed_csubspace C\<close>
     by (simp add: closed_csubspace.closed closure_minimal)
 qed
 
 lemma closed_subspace_closed_sum:
   fixes A B::"('a::complex_normed_vector) set"
   assumes a1: \<open>csubspace A\<close> and a2: \<open>csubspace B\<close>
   shows \<open>closed_csubspace (A +\<^sub>M B)\<close>
   using a1 a2 closed_sum_def
   by (metis closure_is_closed_csubspace csubspace_set_plus)
 
 
 lemma closed_sum_assoc:
   fixes A B C::"'a::real_normed_vector set"
   shows \<open>A +\<^sub>M (B +\<^sub>M C) = (A +\<^sub>M B) +\<^sub>M C\<close>
 proof -
   have \<open>A + closure B \<subseteq> closure (A + B)\<close> for A B :: "'a set"
     by (meson closure_subset closure_sum dual_order.trans order_refl set_plus_mono2)
   then have \<open>A +\<^sub>M (B +\<^sub>M C) = closure (A + (B + C))\<close>
     unfolding closed_sum_def
     by (meson antisym_conv closed_closure closure_minimal closure_mono closure_subset equalityD1 set_plus_mono2)
   moreover
   have \<open>closure A + B \<subseteq> closure (A + B)\<close> for A B :: "'a set"
     by (meson closure_subset closure_sum dual_order.trans order_refl set_plus_mono2)
   then have \<open>(A +\<^sub>M B) +\<^sub>M C = closure ((A + B) + C)\<close>
     unfolding closed_sum_def
     by (meson closed_closure closure_minimal closure_mono closure_subset eq_iff set_plus_mono2)
   ultimately show ?thesis
     by (simp add: ab_semigroup_add_class.add_ac(1))
 qed
 
 
 lemma closed_sum_zero_left[simp]:
   fixes A :: \<open>('a::{monoid_add, topological_space}) set\<close>
   shows \<open>{0} +\<^sub>M A = closure A\<close>
   unfolding closed_sum_def
   by (metis add.left_neutral set_zero)
 
 lemma closed_sum_zero_right[simp]:
   fixes A :: \<open>('a::{monoid_add, topological_space}) set\<close>
   shows \<open>A +\<^sub>M {0} = closure A\<close>
   unfolding closed_sum_def
   by (metis add.right_neutral set_zero)
 
 lemma closed_sum_closure_right[simp]:
   fixes A B :: \<open>'a::real_normed_vector set\<close>
   shows \<open>A +\<^sub>M closure B = A +\<^sub>M B\<close>
   by (metis closed_sum_assoc closed_sum_def closed_sum_zero_right closure_closure)
 
 lemma closed_sum_closure_left[simp]:
   fixes A B :: \<open>'a::real_normed_vector set\<close>
   shows \<open>closure A +\<^sub>M B = A +\<^sub>M B\<close>
   by (simp add: closed_sum_comm)
 
 lemma closed_sum_mono_left:
   assumes \<open>A \<subseteq> B\<close>
   shows \<open>A +\<^sub>M C \<subseteq> B +\<^sub>M C\<close>
   by (simp add: assms closed_sum_def closure_mono set_plus_mono2)
 
 lemma closed_sum_mono_right:
   assumes \<open>A \<subseteq> B\<close>
   shows \<open>C +\<^sub>M A \<subseteq> C +\<^sub>M B\<close>
   by (simp add: assms closed_sum_def closure_mono set_plus_mono2)
 
 instantiation ccsubspace :: (complex_normed_vector) sup begin
 lift_definition sup_ccsubspace :: "'a ccsubspace \<Rightarrow> 'a ccsubspace \<Rightarrow> 'a ccsubspace"
   \<comment> \<open>Note that \<^term>\<open>A+B\<close> would not be a closed subspace, we need the closure. See, e.g., \<^url>\<open>https://math.stackexchange.com/a/1786792/403528\<close>.\<close>
   is "\<lambda>A B::'a set. A +\<^sub>M B"
   by (simp add: closed_subspace_closed_sum)
 instance ..
 end
 
 lemma closed_sum_cspan[simp]:
   shows \<open>cspan X +\<^sub>M cspan Y = closure (cspan (X \<union> Y))\<close>
   by (smt (verit, best) Collect_cong closed_sum_def complex_vector.span_Un set_plus_def)
 
 lemma closure_image_closed_sum:
   assumes \<open>bounded_linear U\<close>
   shows \<open>closure (U ` (A +\<^sub>M B)) = closure (U ` A) +\<^sub>M closure (U ` B)\<close>
 proof -
   have \<open>closure (U ` (A +\<^sub>M B)) = closure (U ` closure (closure A + closure B))\<close>
     unfolding closed_sum_def
     by (smt (verit, best) closed_closure closure_minimal closure_mono closure_subset closure_sum set_plus_mono2 subset_antisym)
   also have \<open>\<dots> = closure (U ` (closure A + closure B))\<close>
     using assms closure_bounded_linear_image_subset_eq by blast
   also have \<open>\<dots> = closure (U ` closure A + U ` closure B)\<close>
     apply (subst image_set_plus)
     by (simp_all add: assms bounded_linear.linear)
   also have \<open>\<dots> = closure (closure (U ` A) + closure (U ` B))\<close>
     by (smt (verit, ccfv_SIG) assms closed_closure closure_bounded_linear_image_subset closure_bounded_linear_image_subset_eq closure_minimal closure_mono closure_sum dual_order.eq_iff set_plus_mono2)
   also have \<open>\<dots> = closure (U ` A) +\<^sub>M closure (U ` B)\<close>
     using closed_sum_def by blast
   finally show ?thesis
     by -
 qed
 
 
 
 lemma ccspan_union: "ccspan A \<squnion> ccspan B = ccspan (A \<union> B)"
   apply transfer by simp
 
 instantiation ccsubspace :: (complex_normed_vector) "Sup"
 begin
 lift_definition Sup_ccsubspace::\<open>'a ccsubspace set \<Rightarrow> 'a ccsubspace\<close>
   is \<open>\<lambda>S. closure (complex_vector.span (Union S))\<close>
 proof
   show "csubspace (closure (complex_vector.span (\<Union> S::'a set)))"
     if "\<And>x::'a set. x \<in> S \<Longrightarrow> closed_csubspace x"
     for S :: "'a set set"
     using that
     by (simp add: closure_is_closed_csubspace)
   show "closed (closure (complex_vector.span (\<Union> S::'a set)))"
     if "\<And>x. (x::'a set) \<in> S \<Longrightarrow> closed_csubspace x"
     for S :: "'a set set"
     using that
     by simp
 qed
 
 instance..
 end
 
 
 instance ccsubspace :: ("{complex_normed_vector}") semilattice_sup
 proof
   fix x y z :: \<open>'a ccsubspace\<close>
   show \<open>x \<le> sup x y\<close>
     apply transfer
     by (simp add: closed_csubspace_def closed_sum_left_subset complex_vector.subspace_0)
 
   show "y \<le> sup x y"
     apply transfer
     by (simp add: closed_csubspace_def closed_sum_right_subset complex_vector.subspace_0)
 
   show "sup x y \<le> z" if "x \<le> z" and "y \<le> z"
     using that apply transfer
     apply (rule closed_sum_is_sup) by auto
 qed
 
-instance ccsubspace :: ("{complex_normed_vector}") complete_lattice
+instance ccsubspace :: (complex_normed_vector) complete_lattice
 proof
-  show "Inf A \<le> x"
-    if "x \<in> A"
-    for x :: "'a ccsubspace"
-      and A :: "'a ccsubspace set"
+  show "Inf A \<le> x" if "x \<in> A"
+    for x :: "'a ccsubspace" and A :: "'a ccsubspace set"
     using that
     apply transfer
     by auto
 
   have b1: "z \<subseteq> \<Inter> A"
     if "Ball A closed_csubspace" and
       "closed_csubspace z" and
       "(\<And>x. closed_csubspace x \<Longrightarrow> x \<in> A \<Longrightarrow> z \<subseteq> x)"
     for z::"'a set" and A
     using that
     by auto
   show "z \<le> Inf A"
     if "\<And>x::'a ccsubspace. x \<in> A \<Longrightarrow> z \<le> x"
     for A :: "'a ccsubspace set"
       and z :: "'a ccsubspace"
     using that
     apply transfer
     using b1 by blast
 
   show "x \<le> Sup A"
     if "x \<in> A"
     for x :: "'a ccsubspace"
       and A :: "'a ccsubspace set"
     using that
     apply transfer
     by (meson Union_upper closure_subset complex_vector.span_superset dual_order.trans)
 
   show "Sup A \<le> z"
     if "\<And>x::'a ccsubspace. x \<in> A \<Longrightarrow> x \<le> z"
     for A :: "'a ccsubspace set"
       and z :: "'a ccsubspace"
     using that apply transfer
   proof -
     fix A :: "'a set set" and z :: "'a set"
     assume A_closed: "Ball A closed_csubspace"
     assume "closed_csubspace z"
     assume in_z: "\<And>x. closed_csubspace x \<Longrightarrow> x \<in> A \<Longrightarrow> x \<subseteq> z"
     from A_closed in_z
     have \<open>V \<subseteq> z\<close> if \<open>V \<in> A\<close> for V
       by (simp add: that)
     then have \<open>\<Union> A \<subseteq> z\<close>
       by (simp add: Sup_le_iff)
     with \<open>closed_csubspace z\<close>
     show "closure (cspan (\<Union> A)) \<subseteq> z"
       by (simp add: closed_csubspace_def closure_minimal complex_vector.span_def subset_hull)
   qed
 
   show "Inf {} = (top::'a ccsubspace)"
     using \<open>\<And>z A. (\<And>x. x \<in> A \<Longrightarrow> z \<le> x) \<Longrightarrow> z \<le> Inf A\<close> top.extremum_uniqueI by auto
 
   show "Sup {} = (bot::'a ccsubspace)"
     using \<open>\<And>z A. (\<And>x. x \<in> A \<Longrightarrow> x \<le> z) \<Longrightarrow> Sup A \<le> z\<close> bot.extremum_uniqueI by auto
 qed
 
 instantiation ccsubspace :: (complex_normed_vector) comm_monoid_add begin
 definition plus_ccsubspace :: "'a ccsubspace \<Rightarrow> _ \<Rightarrow> _"
   where [simp]: "plus_ccsubspace = sup"
 instance
 proof
   fix a b c :: \<open>'a ccsubspace\<close>
   show "a + b + c = a + (b + c)"
     using sup.assoc by auto
   show "a + b = b + a"
     by (simp add: sup.commute)
   show "0 + a = a"
     by (simp add: zero_ccsubspace_def)
 qed
 end
 
 lemma ccsubspace_plus_sup: "y \<le> x \<Longrightarrow> z \<le> x \<Longrightarrow> y + z \<le> x"
   for x y z :: "'a::complex_normed_vector ccsubspace"
   unfolding plus_ccsubspace_def by auto
 
 lemma ccsubspace_Sup_empty: "Sup {} = (0::_ ccsubspace)"
   unfolding zero_ccsubspace_def by auto
 
 lemma ccsubspace_add_right_incr[simp]: "a \<le> a + c" for a::"_ ccsubspace"
   by (simp add: add_increasing2)
 
 lemma ccsubspace_add_left_incr[simp]: "a \<le> c + a" for a::"_ ccsubspace"
   by (simp add: add_increasing)
 
 lemma sum_bot_ccsubspace[simp]: \<open>(\<Sum>x\<in>X. \<bottom>) = (\<bottom> :: _ ccsubspace)\<close>
   by (simp flip: zero_ccsubspace_def)
 
 subsection \<open>Conjugate space\<close>
 
 typedef 'a conjugate_space = "UNIV :: 'a set"
   morphisms from_conjugate_space to_conjugate_space ..
 setup_lifting type_definition_conjugate_space
 
 instantiation conjugate_space :: (complex_vector) complex_vector begin
 lift_definition scaleC_conjugate_space :: \<open>complex \<Rightarrow> 'a conjugate_space \<Rightarrow> 'a conjugate_space\<close> is \<open>\<lambda>c x. cnj c *\<^sub>C x\<close>.
 lift_definition scaleR_conjugate_space :: \<open>real \<Rightarrow> 'a conjugate_space \<Rightarrow> 'a conjugate_space\<close> is \<open>\<lambda>r x. r *\<^sub>R x\<close>.
 lift_definition plus_conjugate_space :: "'a conjugate_space \<Rightarrow> 'a conjugate_space \<Rightarrow> 'a conjugate_space" is "(+)".
 lift_definition uminus_conjugate_space :: "'a conjugate_space \<Rightarrow> 'a conjugate_space" is \<open>\<lambda>x. -x\<close>.
 lift_definition zero_conjugate_space :: "'a conjugate_space" is 0.
 lift_definition minus_conjugate_space :: "'a conjugate_space \<Rightarrow> 'a conjugate_space \<Rightarrow> 'a conjugate_space" is "(-)".
 instance
   apply (intro_classes; transfer)
   by (simp_all add: scaleR_scaleC scaleC_add_right scaleC_left.add)
 end
 
 instantiation conjugate_space :: (complex_normed_vector) complex_normed_vector begin
 lift_definition sgn_conjugate_space :: "'a conjugate_space \<Rightarrow> 'a conjugate_space" is "sgn".
 lift_definition norm_conjugate_space :: "'a conjugate_space \<Rightarrow> real" is norm.
 lift_definition dist_conjugate_space :: "'a conjugate_space \<Rightarrow> 'a conjugate_space \<Rightarrow> real" is dist.
 lift_definition uniformity_conjugate_space :: "('a conjugate_space \<times> 'a conjugate_space) filter" is uniformity.
 lift_definition  open_conjugate_space :: "'a conjugate_space set \<Rightarrow> bool" is "open".
 instance
   apply (intro_classes; transfer)
   by (simp_all add: dist_norm sgn_div_norm open_uniformity uniformity_dist norm_triangle_ineq)
 end
 
 instantiation conjugate_space :: (cbanach) cbanach begin
 instance
   apply intro_classes
   unfolding Cauchy_def convergent_def LIMSEQ_def apply transfer
   using Cauchy_convergent unfolding Cauchy_def convergent_def LIMSEQ_def by metis
 end
 
 lemma bounded_antilinear_to_conjugate_space[simp]: \<open>bounded_antilinear to_conjugate_space\<close>
   by (rule bounded_antilinear_intro[where K=1]; transfer; auto)
 
 lemma bounded_antilinear_from_conjugate_space[simp]: \<open>bounded_antilinear from_conjugate_space\<close>
   by (rule bounded_antilinear_intro[where K=1]; transfer; auto)
 
 lemma antilinear_to_conjugate_space[simp]: \<open>antilinear to_conjugate_space\<close>
   by (rule antilinearI; transfer, auto)
 
 lemma antilinear_from_conjugate_space[simp]: \<open>antilinear from_conjugate_space\<close>
   by (rule antilinearI; transfer, auto)
 
 lemma cspan_to_conjugate_space[simp]: "cspan (to_conjugate_space ` X) = to_conjugate_space ` cspan X"
   unfolding complex_vector.span_def complex_vector.subspace_def hull_def
   apply transfer
   apply simp
   by (metis (no_types, opaque_lifting) complex_cnj_cnj)
 
 lemma surj_to_conjugate_space[simp]: "surj to_conjugate_space"
   by (meson surj_def to_conjugate_space_cases)
 
 lemmas has_derivative_scaleC[simp, derivative_intros] =
   bounded_bilinear.FDERIV[OF bounded_cbilinear_scaleC[THEN bounded_cbilinear.bounded_bilinear]]
 
 lemma norm_to_conjugate_space[simp]: \<open>norm (to_conjugate_space x) = norm x\<close>
   by (fact norm_conjugate_space.abs_eq)
 
 lemma norm_from_conjugate_space[simp]: \<open>norm (from_conjugate_space x) = norm x\<close>
   by (simp add: norm_conjugate_space.rep_eq)
 
 lemma closure_to_conjugate_space: \<open>closure (to_conjugate_space ` X) = to_conjugate_space ` closure X\<close>
 proof -
   have 1: \<open>to_conjugate_space ` closure X \<subseteq> closure (to_conjugate_space ` X)\<close>
     apply (rule closure_bounded_linear_image_subset)
     by (simp add: bounded_antilinear.bounded_linear)
   have \<open>\<dots> = to_conjugate_space ` from_conjugate_space ` closure (to_conjugate_space ` X)\<close>
     by (simp add: from_conjugate_space_inverse image_image)
   also have \<open>\<dots> \<subseteq> to_conjugate_space ` closure (from_conjugate_space ` to_conjugate_space ` X)\<close>
     apply (rule image_mono)
     apply (rule closure_bounded_linear_image_subset)
     by (simp add: bounded_antilinear.bounded_linear)
   also have \<open>\<dots> = to_conjugate_space ` closure X\<close>
     by (simp add: to_conjugate_space_inverse image_image)
   finally show ?thesis
     using 1 by simp
 qed
 
 lemma closure_from_conjugate_space: \<open>closure (from_conjugate_space ` X) = from_conjugate_space ` closure X\<close>
 proof -
   have 1: \<open>from_conjugate_space ` closure X \<subseteq> closure (from_conjugate_space ` X)\<close>
     apply (rule closure_bounded_linear_image_subset)
     by (simp add: bounded_antilinear.bounded_linear)
   have \<open>\<dots> = from_conjugate_space ` to_conjugate_space ` closure (from_conjugate_space ` X)\<close>
     by (simp add: to_conjugate_space_inverse image_image)
   also have \<open>\<dots> \<subseteq> from_conjugate_space ` closure (to_conjugate_space ` from_conjugate_space ` X)\<close>
     apply (rule image_mono)
     apply (rule closure_bounded_linear_image_subset)
     by (simp add: bounded_antilinear.bounded_linear)
   also have \<open>\<dots> = from_conjugate_space ` closure X\<close>
     by (simp add: from_conjugate_space_inverse image_image)
   finally show ?thesis
     using 1 by simp
 qed
 
 lemma bounded_antilinear_eq_on:
   fixes A B :: "'a::complex_normed_vector \<Rightarrow> 'b::complex_normed_vector"
   assumes \<open>bounded_antilinear A\<close> and \<open>bounded_antilinear B\<close> and
     eq: \<open>\<And>x. x \<in> G \<Longrightarrow> A x = B x\<close> and t: \<open>t \<in> closure (cspan G)\<close>
   shows \<open>A t = B t\<close>
 proof -
   let ?A = \<open>\<lambda>x. A (from_conjugate_space x)\<close> and ?B = \<open>\<lambda>x. B (from_conjugate_space x)\<close>
     and ?G = \<open>to_conjugate_space ` G\<close> and ?t = \<open>to_conjugate_space t\<close>
   have \<open>bounded_clinear ?A\<close> and \<open>bounded_clinear ?B\<close>
     by (auto intro!: bounded_antilinear_o_bounded_antilinear[OF \<open>bounded_antilinear A\<close>]
         bounded_antilinear_o_bounded_antilinear[OF \<open>bounded_antilinear B\<close>])
   moreover from eq have \<open>\<And>x. x \<in> ?G \<Longrightarrow> ?A x = ?B x\<close>
     by (metis image_iff iso_tuple_UNIV_I to_conjugate_space_inverse)
   moreover from t have \<open>?t \<in> closure (cspan ?G)\<close>
     by (metis bounded_antilinear.bounded_linear bounded_antilinear_to_conjugate_space closure_bounded_linear_image_subset cspan_to_conjugate_space imageI subsetD)
   ultimately have \<open>?A ?t = ?B ?t\<close>
     by (rule bounded_clinear_eq_on)
   then show \<open>A t = B t\<close>
     by (simp add: to_conjugate_space_inverse)
 qed
 
-instantiation complex :: basis_enum begin
-definition "canonical_basis = [1::complex]"
-instance
-proof
-  show "distinct (canonical_basis::complex list)"
-    by (simp add: canonical_basis_complex_def)
-  show "cindependent (set (canonical_basis::complex list))"
-    unfolding canonical_basis_complex_def
-    by auto
-  show "cspan (set (canonical_basis::complex list)) = UNIV"
-    unfolding canonical_basis_complex_def
-    apply (auto simp add: cspan_raw_def vector_space_over_itself.span_Basis)
-    by (metis complex_scaleC_def complex_vector.span_base complex_vector.span_scale cspan_raw_def insertI1 mult.right_neutral)
-qed
-end
-
-lemma csubspace_is_convex[simp]:
-  assumes a1: "csubspace M"
-  shows "convex M"
-proof-
-  have \<open>\<forall>x\<in>M. \<forall>y\<in> M. \<forall>u. \<forall>v. u *\<^sub>C x + v *\<^sub>C y \<in>  M\<close>
-    using a1
-    by (simp add:  complex_vector.subspace_def)
-  hence \<open>\<forall>x\<in>M. \<forall>y\<in>M. \<forall>u::real. \<forall>v::real. u *\<^sub>R x + v *\<^sub>R y \<in> M\<close>
-    by (simp add: scaleR_scaleC)
-  hence \<open>\<forall>x\<in>M. \<forall>y\<in>M. \<forall>u\<ge>0. \<forall>v\<ge>0. u + v = 1 \<longrightarrow> u *\<^sub>R x + v *\<^sub>R y \<in>M\<close>
-    by blast
-  thus ?thesis using convex_def by blast
-qed
-
-lemma kernel_is_csubspace[simp]:
-  assumes a1: "clinear f"
-  shows "csubspace  (f -` {0})"
-proof-
-  have w3: \<open>t *\<^sub>C x \<in> {x. f x = 0}\<close>
-    if b1: "x \<in> {x. f x = 0}"
-    for x t
-    by (metis assms complex_vector.linear_subspace_kernel complex_vector.subspace_def that)
-  have \<open>f 0 = 0\<close>
-    by (simp add: assms complex_vector.linear_0)
-  hence s2: \<open>0 \<in> {x. f x = 0}\<close>
-    by blast
-
-  have w4: "x + y \<in> {x. f x = 0}"
-    if c1: "x \<in> {x. f x = 0}" and c2: "y \<in> {x. f x = 0}"
-    for x y
-    using assms c1 c2 complex_vector.linear_add by fastforce
-  have s4: \<open>c *\<^sub>C t \<in> {x. f x = 0}\<close>
-    if "t \<in> {x. f x = 0}"
-    for t c
-    using that w3 by auto
-  have s5: "u + v \<in> {x. f x = 0}"
-    if "u \<in> {x. f x = 0}" and "v \<in> {x. f x = 0}"
-    for u v
-    using w4 that(1) that(2) by auto
-  have f3: "f -` {b. b = 0 \<or> b \<in> {}} = {a. f a = 0}"
-    by blast
-  have "csubspace {a. f a = 0}"
-    by (metis complex_vector.subspace_def s2 s4 s5)
-  thus ?thesis
-    using f3 by auto
-qed
-
-
-lemma kernel_is_closed_csubspace[simp]:
-  assumes a1: "bounded_clinear f"
-  shows "closed_csubspace (f -` {0})"
-proof-
-  have \<open>csubspace (f -` {0})\<close>
-    using assms bounded_clinear.clinear complex_vector.linear_subspace_vimage complex_vector.subspace_single_0 by blast
-  have "L \<in> {x. f x = 0}"
-    if "r \<longlonglongrightarrow> L" and "\<forall> n. r n \<in> {x. f x = 0}"
-    for r and  L
-  proof-
-    have d1: \<open>\<forall> n. f (r n) = 0\<close>
-      using that(2) by auto
-    have \<open>(\<lambda> n. f (r n)) \<longlonglongrightarrow> f L\<close>
-      using assms clinear_continuous_at continuous_within_tendsto_compose' that(1)
-      by fastforce
-    hence \<open>(\<lambda> n. 0) \<longlonglongrightarrow> f L\<close>
-      using d1 by simp
-    hence \<open>f L = 0\<close>
-      using limI by fastforce
-    thus ?thesis by blast
-  qed
-  then have s3: \<open>closed (f -` {0})\<close>
-    using closed_sequential_limits by force
-  with \<open>csubspace (f -` {0})\<close>
-  show ?thesis
-    using closed_csubspace.intro by blast
-qed
-
-lemma range_is_clinear[simp]:
-  assumes a1: "clinear f"
-  shows "csubspace (range f)"
-  using assms complex_vector.linear_subspace_image complex_vector.subspace_UNIV by blast
-
-lemma ccspan_superset:
-  \<open>A \<subseteq> space_as_set (ccspan A)\<close>
-  for A :: \<open>'a::complex_normed_vector set\<close>
-  apply transfer
-  by (meson closure_subset complex_vector.span_superset subset_trans)
-
-
 subsection \<open>Product is a Complex Vector Space\<close>
 
 (* Follows closely Product_Vector.thy *)
 
 instantiation prod :: (complex_vector, complex_vector) complex_vector
 begin
 
 definition scaleC_prod_def:
   "scaleC r A = (scaleC r (fst A), scaleC r (snd A))"
 
 lemma fst_scaleC [simp]: "fst (scaleC r A) = scaleC r (fst A)"
   unfolding scaleC_prod_def by simp
 
 lemma snd_scaleC [simp]: "snd (scaleC r A) = scaleC r (snd A)"
   unfolding scaleC_prod_def by simp
 
 proposition scaleC_Pair [simp]: "scaleC r (a, b) = (scaleC r a, scaleC r b)"
   unfolding scaleC_prod_def by simp
 
 instance
 proof
   fix a b :: complex and x y :: "'a \<times> 'b"
   show "scaleC a (x + y) = scaleC a x + scaleC a y"
     by (simp add: scaleC_add_right scaleC_prod_def)
   show "scaleC (a + b) x = scaleC a x + scaleC b x"
     by (simp add: Complex_Vector_Spaces.scaleC_prod_def scaleC_left.add)
   show "scaleC a (scaleC b x) = scaleC (a * b) x"
     by (simp add: prod_eq_iff)
   show "scaleC 1 x = x"
     by (simp add: prod_eq_iff)
   show \<open>(scaleR :: _ \<Rightarrow> _ \<Rightarrow> 'a*'b) r = (*\<^sub>C) (complex_of_real r)\<close> for r
     by (auto intro!: ext simp: scaleR_scaleC scaleC_prod_def scaleR_prod_def)
 qed
 
 end
 
 lemma module_prod_scale_eq_scaleC: "module_prod.scale (*\<^sub>C) (*\<^sub>C) = scaleC"
   apply (rule ext) apply (rule ext)
   apply (subst module_prod.scale_def)
   subgoal by unfold_locales
   by (simp add: scaleC_prod_def)
 
 interpretation complex_vector?: vector_space_prod "scaleC::_\<Rightarrow>_\<Rightarrow>'a::complex_vector" "scaleC::_\<Rightarrow>_\<Rightarrow>'b::complex_vector"
   rewrites "scale = ((*\<^sub>C)::_\<Rightarrow>_\<Rightarrow>('a \<times> 'b))"
     and "module.dependent (*\<^sub>C) = cdependent"
     and "module.representation (*\<^sub>C) = crepresentation"
     and "module.subspace (*\<^sub>C) = csubspace"
     and "module.span (*\<^sub>C) = cspan"
     and "vector_space.extend_basis (*\<^sub>C) = cextend_basis"
     and "vector_space.dim (*\<^sub>C) = cdim"
     and "Vector_Spaces.linear (*\<^sub>C) (*\<^sub>C) = clinear"
   subgoal by unfold_locales
   subgoal by (fact module_prod_scale_eq_scaleC)
   unfolding cdependent_raw_def crepresentation_raw_def csubspace_raw_def cspan_raw_def
     cextend_basis_raw_def cdim_raw_def clinear_def
   by (rule refl)+
 
 instance prod :: (complex_normed_vector, complex_normed_vector) complex_normed_vector 
 proof
   fix c :: complex and x y :: "'a \<times> 'b"
   show "norm (c *\<^sub>C x) = cmod c * norm x"
     unfolding norm_prod_def
     apply (simp add: power_mult_distrib)
     apply (simp add: distrib_left [symmetric])
     by (simp add: real_sqrt_mult)
 qed
 
 
 lemma cspan_Times: \<open>cspan (S \<times> T) = cspan S \<times> cspan T\<close> if \<open>0 \<in> S\<close> and \<open>0 \<in> T\<close>
 proof 
   have \<open>fst ` cspan (S \<times> T) \<subseteq> cspan S\<close>
     apply (subst complex_vector.linear_span_image[symmetric])
     using that complex_vector.module_hom_fst by auto
   moreover have \<open>snd ` cspan (S \<times> T) \<subseteq> cspan T\<close>
     apply (subst complex_vector.linear_span_image[symmetric])
     using that complex_vector.module_hom_snd by auto
   ultimately show \<open>cspan (S \<times> T) \<subseteq> cspan S \<times> cspan T\<close>
     by auto
 
   show \<open>cspan S \<times> cspan T \<subseteq> cspan (S \<times> T)\<close>
   proof
     fix x assume assm: \<open>x \<in> cspan S \<times> cspan T\<close>
     then have \<open>fst x \<in> cspan S\<close>
       by auto
     then obtain t1 r1 where fst_x: \<open>fst x = (\<Sum>a\<in>t1. r1 a *\<^sub>C a)\<close> and [simp]: \<open>finite t1\<close> and \<open>t1 \<subseteq> S\<close>
       by (auto simp add: complex_vector.span_explicit)
     from assm
     have \<open>snd x \<in> cspan T\<close>
       by auto
     then obtain t2 r2 where snd_x: \<open>snd x = (\<Sum>a\<in>t2. r2 a *\<^sub>C a)\<close> and [simp]: \<open>finite t2\<close> and \<open>t2 \<subseteq> T\<close>
       by (auto simp add: complex_vector.span_explicit)
     define t :: \<open>('a+'b) set\<close> and r :: \<open>('a+'b) \<Rightarrow> complex\<close> and f :: \<open>('a+'b) \<Rightarrow> ('a\<times>'b)\<close>
       where \<open>t = t1 <+> t2\<close>
       and \<open>r a = (case a of Inl a1 \<Rightarrow> r1 a1 | Inr a2 \<Rightarrow> r2 a2)\<close>
       and \<open>f a = (case a of Inl a1 \<Rightarrow> (a1,0) | Inr a2 \<Rightarrow> (0,a2))\<close>
     for a
     have \<open>finite t\<close>
       by (simp add: t_def)
     moreover have \<open>f ` t \<subseteq> S \<times> T\<close>
       using  \<open>t1 \<subseteq> S\<close> \<open>t2 \<subseteq> T\<close> that
       by (auto simp: f_def t_def)
     moreover have \<open>(fst x, snd x) = (\<Sum>a\<in>t. r a *\<^sub>C f a)\<close>
       apply (simp only: fst_x snd_x)
       by (auto simp: t_def sum.Plus r_def f_def sum_prod)
     ultimately show \<open>x \<in> cspan (S \<times> T)\<close>
       apply auto
       by (smt (verit, best) complex_vector.span_scale complex_vector.span_sum complex_vector.span_superset image_subset_iff subset_iff)
   qed
 qed
 
 lemma onorm_case_prod_plus: \<open>onorm (case_prod plus :: _ \<Rightarrow> 'a::{real_normed_vector, not_singleton}) = sqrt 2\<close>
 proof -
   obtain x :: 'a where \<open>x \<noteq> 0\<close>
     apply atomize_elim by auto
   show ?thesis
     apply (rule onormI[where x=\<open>(x,x)\<close>])
     using norm_plus_leq_norm_prod apply force
     using  \<open>x \<noteq> 0\<close>
     by (auto simp add: zero_prod_def norm_prod_def real_sqrt_mult
         simp flip: scaleR_2)
 qed
 
 
 subsection \<open>Copying existing theorems into sublocales\<close>
 
 context bounded_clinear begin
 interpretation bounded_linear f by (rule bounded_linear)
 lemmas continuous = real.continuous
 lemmas uniform_limit = real.uniform_limit
 lemmas Cauchy = real.Cauchy
 end
 
 context bounded_antilinear begin
 interpretation bounded_linear f by (rule bounded_linear)
 lemmas continuous = real.continuous
 lemmas uniform_limit = real.uniform_limit
 end
 
 
 context bounded_cbilinear begin
 interpretation bounded_bilinear prod by simp
 lemmas tendsto = real.tendsto
 lemmas isCont = real.isCont
 lemmas scaleR_right = real.scaleR_right
 lemmas scaleR_left = real.scaleR_left
 end
 
 context bounded_sesquilinear begin
 interpretation bounded_bilinear prod by simp
 lemmas tendsto = real.tendsto
 lemmas isCont = real.isCont
 lemmas scaleR_right = real.scaleR_right
 lemmas scaleR_left = real.scaleR_left
 end
 
 lemmas tendsto_scaleC [tendsto_intros] =
   bounded_cbilinear.tendsto [OF bounded_cbilinear_scaleC]
 
 unbundle no_lattice_syntax
 
 end
diff --git a/thys/Complex_Bounded_Operators/Complex_Vector_Spaces0.thy b/thys/Complex_Bounded_Operators/Complex_Vector_Spaces0.thy
--- a/thys/Complex_Bounded_Operators/Complex_Vector_Spaces0.thy
+++ b/thys/Complex_Bounded_Operators/Complex_Vector_Spaces0.thy
@@ -1,1559 +1,1559 @@
 (*  Based on HOL/Real_Vector_Spaces.thy by Brian Huffman, Johannes Hölzl
     Adapted to the complex case by Dominique Unruh *)
 
 section \<open>\<open>Complex_Vector_Spaces0\<close> -- Vector Spaces and Algebras over the Complex Numbers\<close>
 
 theory Complex_Vector_Spaces0
   imports HOL.Real_Vector_Spaces HOL.Topological_Spaces HOL.Vector_Spaces
     Complex_Main 
     "HOL-Library.Complex_Order"
     "HOL-Analysis.Product_Vector"
 begin                              
 
 subsection \<open>Complex vector spaces\<close>
 
 class scaleC = scaleR +
   fixes scaleC :: "complex \<Rightarrow> 'a \<Rightarrow> 'a" (infixr "*\<^sub>C" 75)
   assumes scaleR_scaleC: "scaleR r = scaleC (complex_of_real r)"
 begin
 
 abbreviation divideC :: "'a \<Rightarrow> complex \<Rightarrow> 'a"  (infixl "'/\<^sub>C" 70)
   where "x /\<^sub>C c \<equiv> inverse c *\<^sub>C x"
 
 end
 
 class complex_vector = scaleC + ab_group_add +
   assumes scaleC_add_right: "a *\<^sub>C (x + y) = (a *\<^sub>C x) + (a *\<^sub>C y)"
     and scaleC_add_left: "(a + b) *\<^sub>C x = (a *\<^sub>C x) + (b *\<^sub>C x)"
     and scaleC_scaleC[simp]: "a *\<^sub>C (b *\<^sub>C x) =  (a * b) *\<^sub>C x"
     and scaleC_one[simp]: "1 *\<^sub>C x = x"
 
 (* Not present in Real_Vector_Spaces *)
 subclass (in complex_vector) real_vector
   by (standard, simp_all add: scaleR_scaleC scaleC_add_right scaleC_add_left)
 
 class complex_algebra = complex_vector + ring +
   assumes mult_scaleC_left [simp]: "a *\<^sub>C x * y = a *\<^sub>C (x * y)"
     and mult_scaleC_right [simp]: "x * a *\<^sub>C y = a *\<^sub>C (x * y)"
 
 (* Not present in Real_Vector_Spaces *)
 subclass (in complex_algebra) real_algebra
   by (standard, simp_all add: scaleR_scaleC)
 
 class complex_algebra_1 = complex_algebra + ring_1
 
 (* Not present in Real_Vector_Spaces *)
 subclass (in complex_algebra_1) real_algebra_1 ..
 
 class complex_div_algebra = complex_algebra_1 + division_ring
 
 (* Not present in Real_Vector_Spaces *)
 subclass (in complex_div_algebra) real_div_algebra ..
 
 class complex_field = complex_div_algebra + field
 
 (* Not present in Real_Vector_Spaces *)
 subclass (in complex_field) real_field ..
 
 instantiation complex :: complex_field
 begin
 
 definition complex_scaleC_def [simp]: "scaleC a x = a * x"
 
 instance
 proof intro_classes
   fix r :: real and a b x y :: complex
   show "((*\<^sub>R) r::complex \<Rightarrow> _) = (*\<^sub>C) (complex_of_real r)"
     by (auto simp add: scaleR_conv_of_real)
   show "a *\<^sub>C (x + y) = a *\<^sub>C x + a *\<^sub>C y"
     by (simp add: ring_class.ring_distribs(1))
   show "(a + b) *\<^sub>C x = a *\<^sub>C x + b *\<^sub>C x"
     by (simp add: algebra_simps)
   show "a *\<^sub>C b *\<^sub>C x = (a * b) *\<^sub>C x"
     by simp
   show "1 *\<^sub>C x = x"
     by simp
   show "a *\<^sub>C (x::complex) * y = a *\<^sub>C (x * y)"
     by simp
   show "(x::complex) * a *\<^sub>C y = a *\<^sub>C (x * y)"
     by simp
 qed
 
 end
 
 locale clinear = Vector_Spaces.linear "scaleC::_\<Rightarrow>_\<Rightarrow>'a::complex_vector" "scaleC::_\<Rightarrow>_\<Rightarrow>'b::complex_vector"
 begin
 
 (* Not present in Real_Vector_Spaces. *)
 sublocale real: linear
   \<comment> \<open>Gives access to all lemmas from \<^locale>\<open>linear\<close> using prefix \<open>real.\<close>\<close>
   apply standard
   by (auto simp add: add scale scaleR_scaleC)
 
 lemmas scaleC = scale
 
 end
 
 global_interpretation complex_vector: vector_space "scaleC :: complex \<Rightarrow> 'a \<Rightarrow> 'a :: complex_vector"
   rewrites "Vector_Spaces.linear (*\<^sub>C) (*\<^sub>C) = clinear"
     and "Vector_Spaces.linear (*) (*\<^sub>C) = clinear"
   defines cdependent_raw_def: cdependent = complex_vector.dependent
     and crepresentation_raw_def: crepresentation = complex_vector.representation
     and csubspace_raw_def: csubspace = complex_vector.subspace
     and cspan_raw_def: cspan = complex_vector.span
     and cextend_basis_raw_def: cextend_basis = complex_vector.extend_basis
     and cdim_raw_def: cdim = complex_vector.dim
 proof unfold_locales
   show "Vector_Spaces.linear (*\<^sub>C) (*\<^sub>C) = clinear" "Vector_Spaces.linear (*) (*\<^sub>C) = clinear"
     by (force simp: clinear_def complex_scaleC_def[abs_def])+
 qed (use scaleC_add_right scaleC_add_left in auto)
 
 
 (* Not needed since we did the global_interpretation with mandatory complex_vector-prefix:
 hide_const (open)\<comment> \<open>locale constants\<close>
   complex_vector.dependent
   complex_vector.independent
   complex_vector.representation
   complex_vector.subspace
   complex_vector.span
   complex_vector.extend_basis
   complex_vector.dim *)
 
 abbreviation "cindependent x \<equiv> \<not> cdependent x"
 
 global_interpretation complex_vector: vector_space_pair "scaleC::_\<Rightarrow>_\<Rightarrow>'a::complex_vector" "scaleC::_\<Rightarrow>_\<Rightarrow>'b::complex_vector"
   rewrites  "Vector_Spaces.linear (*\<^sub>C) (*\<^sub>C) = clinear"
     and "Vector_Spaces.linear (*) (*\<^sub>C) = clinear"
   defines cconstruct_raw_def: cconstruct = complex_vector.construct
 proof unfold_locales
   show "Vector_Spaces.linear (*) (*\<^sub>C) = clinear"
     unfolding clinear_def complex_scaleC_def by auto
 qed (auto simp: clinear_def)
 
 (* Not needed since we did the global_interpretation with mandatory complex_vector-prefix:
 hide_const (open)\<comment> \<open>locale constants\<close>
   complex_vector.construct *)
 
 lemma clinear_compose: "clinear f \<Longrightarrow> clinear g \<Longrightarrow> clinear (g \<circ> f)"
   unfolding clinear_def by (rule Vector_Spaces.linear_compose)
 
 text \<open>Recover original theorem names\<close>
 
 lemmas scaleC_left_commute = complex_vector.scale_left_commute
 lemmas scaleC_zero_left = complex_vector.scale_zero_left
 lemmas scaleC_minus_left = complex_vector.scale_minus_left
 lemmas scaleC_diff_left = complex_vector.scale_left_diff_distrib
 lemmas scaleC_sum_left = complex_vector.scale_sum_left
 lemmas scaleC_zero_right = complex_vector.scale_zero_right
 lemmas scaleC_minus_right = complex_vector.scale_minus_right
 lemmas scaleC_diff_right = complex_vector.scale_right_diff_distrib
 lemmas scaleC_sum_right = complex_vector.scale_sum_right
 lemmas scaleC_eq_0_iff = complex_vector.scale_eq_0_iff
 lemmas scaleC_left_imp_eq = complex_vector.scale_left_imp_eq
 lemmas scaleC_right_imp_eq = complex_vector.scale_right_imp_eq
 lemmas scaleC_cancel_left = complex_vector.scale_cancel_left
 lemmas scaleC_cancel_right = complex_vector.scale_cancel_right
 
 lemma divideC_field_simps[field_simps]: (* In Real_Vector_Spaces, these lemmas are unnamed *)
   "c \<noteq> 0 \<Longrightarrow> a = b /\<^sub>C c \<longleftrightarrow> c *\<^sub>C a = b"
   "c \<noteq> 0 \<Longrightarrow> b /\<^sub>C c = a \<longleftrightarrow> b = c *\<^sub>C a"
   "c \<noteq> 0 \<Longrightarrow> a + b /\<^sub>C c = (c *\<^sub>C a + b) /\<^sub>C c"
   "c \<noteq> 0 \<Longrightarrow> a /\<^sub>C c + b = (a + c *\<^sub>C b) /\<^sub>C c"
   "c \<noteq> 0 \<Longrightarrow> a - b /\<^sub>C c = (c *\<^sub>C a - b) /\<^sub>C c"
   "c \<noteq> 0 \<Longrightarrow> a /\<^sub>C c - b = (a - c *\<^sub>C b) /\<^sub>C c"
   "c \<noteq> 0 \<Longrightarrow> - (a /\<^sub>C c) + b = (- a + c *\<^sub>C b) /\<^sub>C c"
   "c \<noteq> 0 \<Longrightarrow> - (a /\<^sub>C c) - b = (- a - c *\<^sub>C b) /\<^sub>C c"
   for a b :: "'a :: complex_vector"
   by (auto simp add: scaleC_add_right scaleC_add_left scaleC_diff_right scaleC_diff_left)
 
 
 text \<open>Legacy names -- omitted\<close>
 
 (* lemmas scaleC_left_distrib = scaleC_add_left
 lemmas scaleC_right_distrib = scaleC_add_right
 lemmas scaleC_left_diff_distrib = scaleC_diff_left
 lemmas scaleC_right_diff_distrib = scaleC_diff_right *)
 
 lemmas clinear_injective_0 = linear_inj_iff_eq_0
   and clinear_injective_on_subspace_0 = linear_inj_on_iff_eq_0
   and clinear_cmul = linear_scale
   and clinear_scaleC = linear_scale_self
   and csubspace_mul = subspace_scale
   and cspan_linear_image = linear_span_image
   and cspan_0 = span_zero
   and cspan_mul = span_scale
   and injective_scaleC = injective_scale
 
 lemma scaleC_minus1_left [simp]: "scaleC (-1) x = - x"
   for x :: "'a::complex_vector"
   using scaleC_minus_left [of 1 x] by simp
 
 lemma scaleC_2:
   fixes x :: "'a::complex_vector"
   shows "scaleC 2 x = x + x"
   unfolding one_add_one [symmetric] scaleC_add_left by simp
 
 lemma scaleC_half_double [simp]:
   fixes a :: "'a::complex_vector"
   shows "(1 / 2) *\<^sub>C (a + a) = a"
 proof -
   have "\<And>r. r *\<^sub>C (a + a) = (r * 2) *\<^sub>C a"
     by (metis scaleC_2 scaleC_scaleC)
   thus ?thesis
     by simp
 qed
 
 lemma clinear_scale_complex:
   fixes c::complex shows "clinear f \<Longrightarrow> f (c * b) = c * f b"
   using complex_vector.linear_scale by fastforce
 
 
 interpretation scaleC_left: additive "(\<lambda>a. scaleC a x :: 'a::complex_vector)"
   by standard (rule scaleC_add_left)
 
 interpretation scaleC_right: additive "(\<lambda>x. scaleC a x :: 'a::complex_vector)"
   by standard (rule scaleC_add_right)
 
 lemma nonzero_inverse_scaleC_distrib:
   "a \<noteq> 0 \<Longrightarrow> x \<noteq> 0 \<Longrightarrow> inverse (scaleC a x) = scaleC (inverse a) (inverse x)"
   for x :: "'a::complex_div_algebra"
   by (rule inverse_unique) simp
 
 lemma inverse_scaleC_distrib: "inverse (scaleC a x) = scaleC (inverse a) (inverse x)"
   for x :: "'a::{complex_div_algebra,division_ring}"
   by (metis inverse_zero nonzero_inverse_scaleC_distrib complex_vector.scale_eq_0_iff)
 
 (* lemmas sum_constant_scaleC = real_vector.sum_constant_scale\<comment> \<open>legacy name\<close> *)
 
 (* Defined in Real_Vector_Spaces:
 named_theorems vector_add_divide_simps "to simplify sums of scaled vectors" *)
 
 lemma complex_add_divide_simps[vector_add_divide_simps]:  (* In Real_Vector_Spaces, these lemmas are unnamed *)
   "v + (b / z) *\<^sub>C w = (if z = 0 then v else (z *\<^sub>C v + b *\<^sub>C w) /\<^sub>C z)"
   "a *\<^sub>C v + (b / z) *\<^sub>C w = (if z = 0 then a *\<^sub>C v else ((a * z) *\<^sub>C v + b *\<^sub>C w) /\<^sub>C z)"
   "(a / z) *\<^sub>C v + w = (if z = 0 then w else (a *\<^sub>C v + z *\<^sub>C w) /\<^sub>C z)"
   "(a / z) *\<^sub>C v + b *\<^sub>C w = (if z = 0 then b *\<^sub>C w else (a *\<^sub>C v + (b * z) *\<^sub>C w) /\<^sub>C z)"
   "v - (b / z) *\<^sub>C w = (if z = 0 then v else (z *\<^sub>C v - b *\<^sub>C w) /\<^sub>C z)"
   "a *\<^sub>C v - (b / z) *\<^sub>C w = (if z = 0 then a *\<^sub>C v else ((a * z) *\<^sub>C v - b *\<^sub>C w) /\<^sub>C z)"
   "(a / z) *\<^sub>C v - w = (if z = 0 then -w else (a *\<^sub>C v - z *\<^sub>C w) /\<^sub>C z)"
   "(a / z) *\<^sub>C v - b *\<^sub>C w = (if z = 0 then -b *\<^sub>C w else (a *\<^sub>C v - (b * z) *\<^sub>C w) /\<^sub>C z)"
   for v :: "'a :: complex_vector"
   by (simp_all add: divide_inverse_commute scaleC_add_right scaleC_diff_right)
 
 lemma ceq_vector_fraction_iff [vector_add_divide_simps]:
   fixes x :: "'a :: complex_vector"
   shows "(x = (u / v) *\<^sub>C a) \<longleftrightarrow> (if v=0 then x = 0 else v *\<^sub>C x = u *\<^sub>C a)"
   by auto (metis (no_types) divide_eq_1_iff divide_inverse_commute scaleC_one scaleC_scaleC)
 
 lemma cvector_fraction_eq_iff [vector_add_divide_simps]:
   fixes x :: "'a :: complex_vector"
   shows "((u / v) *\<^sub>C a = x) \<longleftrightarrow> (if v=0 then x = 0 else u *\<^sub>C a = v *\<^sub>C x)"
   by (metis ceq_vector_fraction_iff)
 
 lemma complex_vector_affinity_eq:
   fixes x :: "'a :: complex_vector"
   assumes m0: "m \<noteq> 0"
   shows "m *\<^sub>C x + c = y \<longleftrightarrow> x = inverse m *\<^sub>C y - (inverse m *\<^sub>C c)"
     (is "?lhs \<longleftrightarrow> ?rhs")
 proof
   assume ?lhs
   hence "m *\<^sub>C x = y - c" by (simp add: field_simps)
   hence "inverse m *\<^sub>C (m *\<^sub>C x) = inverse m *\<^sub>C (y - c)" by simp
   thus "x = inverse m *\<^sub>C y - (inverse m *\<^sub>C c)"
     using m0
     by (simp add: complex_vector.scale_right_diff_distrib)
 next
   assume ?rhs
   with m0 show "m *\<^sub>C x + c = y"
     by (simp add: complex_vector.scale_right_diff_distrib)
 qed
 
 lemma complex_vector_eq_affinity: "m \<noteq> 0 \<Longrightarrow> y = m *\<^sub>C x + c \<longleftrightarrow> inverse m *\<^sub>C y - (inverse m *\<^sub>C c) = x"
   for x :: "'a::complex_vector"
   using complex_vector_affinity_eq[where m=m and x=x and y=y and c=c]
   by metis
 
 lemma scaleC_eq_iff [simp]: "b + u *\<^sub>C a = a + u *\<^sub>C b \<longleftrightarrow> a = b \<or> u = 1"
   for a :: "'a::complex_vector"
 proof (cases "u = 1")
   case True
   thus ?thesis by auto
 next
   case False
   have "a = b" if "b + u *\<^sub>C a = a + u *\<^sub>C b"
   proof -
     from that have "(u - 1) *\<^sub>C a = (u - 1) *\<^sub>C b"
       by (simp add: algebra_simps)
     with False show ?thesis
       by auto
   qed
   thus ?thesis by auto
 qed
 
 lemma scaleC_collapse [simp]: "(1 - u) *\<^sub>C a + u *\<^sub>C a = a"
   for a :: "'a::complex_vector"
   by (simp add: algebra_simps)
 
 subsection \<open>Embedding of the Complex Numbers into any \<open>complex_algebra_1\<close>: \<open>of_complex\<close>\<close>
 
 
 definition of_complex :: "complex \<Rightarrow> 'a::complex_algebra_1"
   where "of_complex c = scaleC c 1"
 
 
 lemma scaleC_conv_of_complex: "scaleC r x = of_complex r * x"
   by (simp add: of_complex_def)
 
 lemma of_complex_0 [simp]: "of_complex 0 = 0"
   by (simp add: of_complex_def)
 
 lemma of_complex_1 [simp]: "of_complex 1 = 1"
   by (simp add: of_complex_def)
 
 lemma of_complex_add [simp]: "of_complex (x + y) = of_complex x + of_complex y"
   by (simp add: of_complex_def scaleC_add_left)
 
 lemma of_complex_minus [simp]: "of_complex (- x) = - of_complex x"
   by (simp add: of_complex_def)
 
 lemma of_complex_diff [simp]: "of_complex (x - y) = of_complex x - of_complex y"
   by (simp add: of_complex_def scaleC_diff_left)
 
 lemma of_complex_mult [simp]: "of_complex (x * y) = of_complex x * of_complex y"
   by (simp add: of_complex_def mult.commute)
 
 lemma of_complex_sum[simp]: "of_complex (sum f s) = (\<Sum>x\<in>s. of_complex (f x))"
   by (induct s rule: infinite_finite_induct) auto
 
 lemma of_complex_prod[simp]: "of_complex (prod f s) = (\<Prod>x\<in>s. of_complex (f x))"
   by (induct s rule: infinite_finite_induct) auto
 
 lemma nonzero_of_complex_inverse:
   "x \<noteq> 0 \<Longrightarrow> of_complex (inverse x) = inverse (of_complex x :: 'a::complex_div_algebra)"
   by (simp add: of_complex_def nonzero_inverse_scaleC_distrib)
 
 lemma of_complex_inverse [simp]:
   "of_complex (inverse x) = inverse (of_complex x :: 'a::{complex_div_algebra,division_ring})"
   by (simp add: of_complex_def inverse_scaleC_distrib)
 
 lemma nonzero_of_complex_divide:
   "y \<noteq> 0 \<Longrightarrow> of_complex (x / y) = (of_complex x / of_complex y :: 'a::complex_field)"
   by (simp add: divide_inverse nonzero_of_complex_inverse)
 
 lemma of_complex_divide [simp]:
   "of_complex (x / y) = (of_complex x / of_complex y :: 'a::complex_div_algebra)"
   by (simp add: divide_inverse)
 
 lemma of_complex_power [simp]:
   "of_complex (x ^ n) = (of_complex x :: 'a::{complex_algebra_1}) ^ n"
   by (induct n) simp_all
 
 lemma of_complex_power_int [simp]:
   "of_complex (power_int x n) = power_int (of_complex x :: 'a :: {complex_div_algebra,division_ring}) n"
   by (auto simp: power_int_def)
 
 lemma of_complex_eq_iff [simp]: "of_complex x = of_complex y \<longleftrightarrow> x = y"
   by (simp add: of_complex_def)
 
 lemma inj_of_complex: "inj of_complex"
   by (auto intro: injI)
 
 lemmas of_complex_eq_0_iff [simp] = of_complex_eq_iff [of _ 0, simplified]
 lemmas of_complex_eq_1_iff [simp] = of_complex_eq_iff [of _ 1, simplified]
 
 lemma minus_of_complex_eq_of_complex_iff [simp]: "-of_complex x = of_complex y \<longleftrightarrow> -x = y"
   using of_complex_eq_iff[of "-x" y] by (simp only: of_complex_minus)
 
 lemma of_complex_eq_minus_of_complex_iff [simp]: "of_complex x = -of_complex y \<longleftrightarrow> x = -y"
   using of_complex_eq_iff[of x "-y"] by (simp only: of_complex_minus)
 
 lemma of_complex_eq_id [simp]: "of_complex = (id :: complex \<Rightarrow> complex)"
   by (rule ext) (simp add: of_complex_def)
 
 text \<open>Collapse nested embeddings.\<close>
 lemma of_complex_of_nat_eq [simp]: "of_complex (of_nat n) = of_nat n"
   by (induct n) auto
 
 lemma of_complex_of_int_eq [simp]: "of_complex (of_int z) = of_int z"
   by (cases z rule: int_diff_cases) simp
 
 lemma of_complex_numeral [simp]: "of_complex (numeral w) = numeral w"
   using of_complex_of_int_eq [of "numeral w"] by simp
 
 lemma of_complex_neg_numeral [simp]: "of_complex (- numeral w) = - numeral w"
   using of_complex_of_int_eq [of "- numeral w"] by simp
 
 lemma numeral_power_int_eq_of_complex_cancel_iff [simp]:
   "power_int (numeral x) n = (of_complex y :: 'a :: {complex_div_algebra, division_ring}) \<longleftrightarrow>
      power_int (numeral x) n = y"
 proof -
   have "power_int (numeral x) n = (of_complex (power_int (numeral x) n) :: 'a)"
     by simp
   also have "\<dots> = of_complex y \<longleftrightarrow> power_int (numeral x) n = y"
     by (subst of_complex_eq_iff) auto
   finally show ?thesis .
 qed
 
 lemma of_complex_eq_numeral_power_int_cancel_iff [simp]:
   "(of_complex y :: 'a :: {complex_div_algebra, division_ring}) = power_int (numeral x) n \<longleftrightarrow>
      y = power_int (numeral x) n"
   by (subst (1 2) eq_commute) simp
 
 lemma of_complex_eq_of_complex_power_int_cancel_iff [simp]:
   "power_int (of_complex b :: 'a :: {complex_div_algebra, division_ring}) w = of_complex x \<longleftrightarrow>
      power_int b w = x"
   by (metis of_complex_power_int of_complex_eq_iff)
 
 lemma of_complex_in_Ints_iff [simp]: "of_complex x \<in> \<int> \<longleftrightarrow> x \<in> \<int>"
 proof safe
   fix x assume "(of_complex x :: 'a) \<in> \<int>"
   then obtain n where "(of_complex x :: 'a) = of_int n"
     by (auto simp: Ints_def)
   also have "of_int n = of_complex (of_int n)"
     by simp
   finally have "x = of_int n"
     by (subst (asm) of_complex_eq_iff)
   thus "x \<in> \<int>"
     by auto
 qed (auto simp: Ints_def)
 
 lemma Ints_of_complex [intro]: "x \<in> \<int> \<Longrightarrow> of_complex x \<in> \<int>"
   by simp
 
 
 text \<open>Every complex algebra has characteristic zero.\<close>
 
 (* Inherited from real_algebra_1 *)
 (* instance complex_algebra_1 < ring_char_0 .. *)
 
 lemma fraction_scaleC_times [simp]:
   fixes a :: "'a::complex_algebra_1"
   shows "(numeral u / numeral v) *\<^sub>C (numeral w * a) = (numeral u * numeral w / numeral v) *\<^sub>C a"
   by (metis (no_types, lifting) of_complex_numeral scaleC_conv_of_complex scaleC_scaleC times_divide_eq_left)
 
 lemma inverse_scaleC_times [simp]:
   fixes a :: "'a::complex_algebra_1"
   shows "(1 / numeral v) *\<^sub>C (numeral w * a) = (numeral w / numeral v) *\<^sub>C a"
   by (metis divide_inverse_commute inverse_eq_divide of_complex_numeral scaleC_conv_of_complex scaleC_scaleC)
 
 lemma scaleC_times [simp]:
   fixes a :: "'a::complex_algebra_1"
   shows "(numeral u) *\<^sub>C (numeral w * a) = (numeral u * numeral w) *\<^sub>C a"
   by (simp add: scaleC_conv_of_complex)
 
 (* Inherited from real_field *)
 (* instance complex_field < field_char_0 .. *)
 
 
 subsection \<open>The Set of Real Numbers\<close>
 
 definition Complexs :: "'a::complex_algebra_1 set"  ("\<complex>")
   where "\<complex> = range of_complex"
 
 lemma Complexs_of_complex [simp]: "of_complex r \<in> \<complex>"
   by (simp add: Complexs_def)
 
 lemma Complexs_of_int [simp]: "of_int z \<in> \<complex>"
   by (subst of_complex_of_int_eq [symmetric], rule Complexs_of_complex)
 
 lemma Complexs_of_nat [simp]: "of_nat n \<in> \<complex>"
   by (subst of_complex_of_nat_eq [symmetric], rule Complexs_of_complex)
 
 lemma Complexs_numeral [simp]: "numeral w \<in> \<complex>"
   by (subst of_complex_numeral [symmetric], rule Complexs_of_complex)
 
 lemma Complexs_0 [simp]: "0 \<in> \<complex>" and Complexs_1 [simp]: "1 \<in> \<complex>"
   by (simp_all add: Complexs_def)
 
 lemma Complexs_add [simp]: "a \<in> \<complex> \<Longrightarrow> b \<in> \<complex> \<Longrightarrow> a + b \<in> \<complex>"
   apply (auto simp add: Complexs_def)
   by (metis of_complex_add range_eqI)
 
 lemma Complexs_minus [simp]: "a \<in> \<complex> \<Longrightarrow> - a \<in> \<complex>"
   by (auto simp: Complexs_def)
 
 lemma Complexs_minus_iff [simp]: "- a \<in> \<complex> \<longleftrightarrow> a \<in> \<complex>"
   using Complexs_minus by fastforce
 
 lemma Complexs_diff [simp]: "a \<in> \<complex> \<Longrightarrow> b \<in> \<complex> \<Longrightarrow> a - b \<in> \<complex>"
   by (metis Complexs_add Complexs_minus_iff add_uminus_conv_diff)
 
 lemma Complexs_mult [simp]: "a \<in> \<complex> \<Longrightarrow> b \<in> \<complex> \<Longrightarrow> a * b \<in> \<complex>"
   apply (auto simp add: Complexs_def)
   by (metis of_complex_mult rangeI)
 
 lemma nonzero_Complexs_inverse: "a \<in> \<complex> \<Longrightarrow> a \<noteq> 0 \<Longrightarrow> inverse a \<in> \<complex>"
   for a :: "'a::complex_div_algebra"
   apply (auto simp add: Complexs_def)
   by (metis of_complex_inverse range_eqI)
 
 lemma Complexs_inverse: "a \<in> \<complex> \<Longrightarrow> inverse a \<in> \<complex>"
   for a :: "'a::{complex_div_algebra,division_ring}"
   using nonzero_Complexs_inverse by fastforce
 
 lemma Complexs_inverse_iff [simp]: "inverse x \<in> \<complex> \<longleftrightarrow> x \<in> \<complex>"
   for x :: "'a::{complex_div_algebra,division_ring}"
   by (metis Complexs_inverse inverse_inverse_eq)
 
 lemma nonzero_Complexs_divide: "a \<in> \<complex> \<Longrightarrow> b \<in> \<complex> \<Longrightarrow> b \<noteq> 0 \<Longrightarrow> a / b \<in> \<complex>"
   for a b :: "'a::complex_field"
   by (simp add: divide_inverse)
 
 lemma Complexs_divide [simp]: "a \<in> \<complex> \<Longrightarrow> b \<in> \<complex> \<Longrightarrow> a / b \<in> \<complex>"
   for a b :: "'a::{complex_field,field}"
   using nonzero_Complexs_divide by fastforce
 
 lemma Complexs_power [simp]: "a \<in> \<complex> \<Longrightarrow> a ^ n \<in> \<complex>"
   for a :: "'a::complex_algebra_1"
   apply (auto simp add: Complexs_def)
   by (metis range_eqI of_complex_power[symmetric])
 
 lemma Complexs_cases [cases set: Complexs]:
   assumes "q \<in> \<complex>"
   obtains (of_complex) c where "q = of_complex c"
   unfolding Complexs_def
 proof -
   from \<open>q \<in> \<complex>\<close> have "q \<in> range of_complex" unfolding Complexs_def .
   then obtain c where "q = of_complex c" ..
   then show thesis ..
 qed
 
 lemma sum_in_Complexs [intro,simp]: "(\<And>i. i \<in> s \<Longrightarrow> f i \<in> \<complex>) \<Longrightarrow> sum f s \<in> \<complex>"
 proof (induct s rule: infinite_finite_induct)
   case infinite
   then show ?case by (metis Complexs_0 sum.infinite)
 qed simp_all
 
 lemma prod_in_Complexs [intro,simp]: "(\<And>i. i \<in> s \<Longrightarrow> f i \<in> \<complex>) \<Longrightarrow> prod f s \<in> \<complex>"
 proof (induct s rule: infinite_finite_induct)
   case infinite
   then show ?case by (metis Complexs_1 prod.infinite)
 qed simp_all
 
 lemma Complexs_induct [case_names of_complex, induct set: Complexs]:
   "q \<in> \<complex> \<Longrightarrow> (\<And>r. P (of_complex r)) \<Longrightarrow> P q"
   by (rule Complexs_cases) auto
 
 
 
 subsection \<open>Ordered complex vector spaces\<close>
 
 class ordered_complex_vector = complex_vector + ordered_ab_group_add +
   assumes scaleC_left_mono: "x \<le> y \<Longrightarrow> 0 \<le> a \<Longrightarrow> a *\<^sub>C x \<le> a *\<^sub>C y"
     and scaleC_right_mono: "a \<le> b \<Longrightarrow> 0 \<le> x \<Longrightarrow> a *\<^sub>C x \<le> b *\<^sub>C x"
 begin
 
 subclass (in ordered_complex_vector) ordered_real_vector
   apply standard
   by (auto simp add: less_eq_complex_def scaleC_left_mono scaleC_right_mono scaleR_scaleC)
 
 lemma scaleC_mono:
   "a \<le> b \<Longrightarrow> x \<le> y \<Longrightarrow> 0 \<le> b \<Longrightarrow> 0 \<le> x \<Longrightarrow> a *\<^sub>C x \<le> b *\<^sub>C y"
   by (meson order_trans scaleC_left_mono scaleC_right_mono)
 
 lemma scaleC_mono':
   "a \<le> b \<Longrightarrow> c \<le> d \<Longrightarrow> 0 \<le> a \<Longrightarrow> 0 \<le> c \<Longrightarrow> a *\<^sub>C c \<le> b *\<^sub>C d"
   by (rule scaleC_mono) (auto intro: order.trans)
 
 lemma pos_le_divideC_eq [field_simps]:
   "a \<le> b /\<^sub>C c \<longleftrightarrow> c *\<^sub>C a \<le> b" (is "?P \<longleftrightarrow> ?Q") if "0 < c"
 proof
   assume ?P
   with scaleC_left_mono that have "c *\<^sub>C a \<le> c *\<^sub>C (b /\<^sub>C c)"
     using preorder_class.less_imp_le by blast
   with that show ?Q
     by auto
 next
   assume ?Q
   with scaleC_left_mono that have "c *\<^sub>C a /\<^sub>C c \<le> b /\<^sub>C c"
     using less_complex_def less_eq_complex_def by fastforce
   with that show ?P
     by auto
 qed
 
 lemma pos_less_divideC_eq [field_simps]:
   "a < b /\<^sub>C c \<longleftrightarrow> c *\<^sub>C a < b" if "c > 0"
   using that pos_le_divideC_eq [of c a b]
   by (auto simp add: le_less)
 
 lemma pos_divideC_le_eq [field_simps]:
   "b /\<^sub>C c \<le> a \<longleftrightarrow> b \<le> c *\<^sub>C a" if "c > 0"
   using that pos_le_divideC_eq [of "inverse c" b a]
     less_complex_def by auto
 
 lemma pos_divideC_less_eq [field_simps]:
   "b /\<^sub>C c < a \<longleftrightarrow> b < c *\<^sub>C a" if "c > 0"
   using that pos_less_divideC_eq [of "inverse c" b a]
   by (simp add: local.less_le_not_le local.pos_divideC_le_eq local.pos_le_divideC_eq)
 
 lemma pos_le_minus_divideC_eq [field_simps]:
   "a \<le> - (b /\<^sub>C c) \<longleftrightarrow> c *\<^sub>C a \<le> - b" if "c > 0"
   using that
   by (metis local.ab_left_minus local.add.inverse_unique local.add.right_inverse local.add_minus_cancel local.le_minus_iff local.pos_divideC_le_eq local.scaleC_add_right local.scaleC_one local.scaleC_scaleC)
 
 lemma pos_less_minus_divideC_eq [field_simps]:
   "a < - (b /\<^sub>C c) \<longleftrightarrow> c *\<^sub>C a < - b" if "c > 0"
   using that
   by (metis le_less less_le_not_le pos_divideC_le_eq pos_divideC_less_eq pos_le_minus_divideC_eq)
 
 lemma pos_minus_divideC_le_eq [field_simps]:
   "- (b /\<^sub>C c) \<le> a \<longleftrightarrow> - b \<le> c *\<^sub>C a" if "c > 0"
   using that
   by (metis local.add_minus_cancel local.left_minus local.pos_divideC_le_eq local.scaleC_add_right)
 
 lemma pos_minus_divideC_less_eq [field_simps]:
   "- (b /\<^sub>C c) < a \<longleftrightarrow> - b < c *\<^sub>C a" if "c > 0"
   using that by (simp add: less_le_not_le pos_le_minus_divideC_eq pos_minus_divideC_le_eq)
 
 lemma scaleC_image_atLeastAtMost: "c > 0 \<Longrightarrow> scaleC c ` {x..y} = {c *\<^sub>C x..c *\<^sub>C y}"
   apply (auto intro!: scaleC_left_mono simp: image_iff Bex_def)
   by (meson order.eq_iff local.order.refl pos_divideC_le_eq pos_le_divideC_eq)
 
 end (* class ordered_complex_vector *)
 
 lemma neg_le_divideC_eq [field_simps]:
   "a \<le> b /\<^sub>C c \<longleftrightarrow> b \<le> c *\<^sub>C a" (is "?P \<longleftrightarrow> ?Q") if "c < 0"
     for a b :: "'a :: ordered_complex_vector"
   using that pos_le_divideC_eq [of "- c" a "- b"]
   by (simp add: less_complex_def)
 
 lemma neg_less_divideC_eq [field_simps]:
   "a < b /\<^sub>C c \<longleftrightarrow> b < c *\<^sub>C a" if "c < 0"
     for a b :: "'a :: ordered_complex_vector"
   using that neg_le_divideC_eq [of c a b]
   by (smt (verit, ccfv_SIG) neg_le_divideC_eq antisym_conv2 complex_vector.scale_minus_right dual_order.strict_implies_order le_less_trans neg_le_iff_le scaleC_scaleC)
 
 lemma neg_divideC_le_eq [field_simps]:
   "b /\<^sub>C c \<le> a \<longleftrightarrow> c *\<^sub>C a \<le> b" if "c < 0"
     for a b :: "'a :: ordered_complex_vector"
   using that pos_divideC_le_eq [of "- c" "- b" a]
   by (simp add: less_complex_def)
 
 lemma neg_divideC_less_eq [field_simps]:
   "b /\<^sub>C c < a \<longleftrightarrow> c *\<^sub>C a < b" if "c < 0"
     for a b :: "'a :: ordered_complex_vector"
   using that neg_divideC_le_eq [of c b a]
   by (meson neg_le_divideC_eq less_le_not_le)
 
 lemma neg_le_minus_divideC_eq [field_simps]:
   "a \<le> - (b /\<^sub>C c) \<longleftrightarrow> - b \<le> c *\<^sub>C a" if "c < 0"
     for a b :: "'a :: ordered_complex_vector"
   using that pos_le_minus_divideC_eq [of "- c" a "- b"]
   by (metis neg_le_divideC_eq complex_vector.scale_minus_right)
 
 lemma neg_less_minus_divideC_eq [field_simps]:
   "a < - (b /\<^sub>C c) \<longleftrightarrow> - b < c *\<^sub>C a" if "c < 0"
     for a b :: "'a :: ordered_complex_vector"
 proof -
   have *: "- b = c *\<^sub>C a \<longleftrightarrow> b = - (c *\<^sub>C a)"
     by (metis add.inverse_inverse)
   from that neg_le_minus_divideC_eq [of c a b]
   show ?thesis by (auto simp add: le_less *)
 qed
 
 lemma neg_minus_divideC_le_eq [field_simps]:
   "- (b /\<^sub>C c) \<le> a \<longleftrightarrow> c *\<^sub>C a \<le> - b" if "c < 0"
 for a b :: "'a :: ordered_complex_vector"
   using that pos_minus_divideC_le_eq [of "- c" "- b" a]
   by (metis Complex_Vector_Spaces0.neg_divideC_le_eq complex_vector.scale_minus_right)
 
 lemma neg_minus_divideC_less_eq [field_simps]:
   "- (b /\<^sub>C c) < a \<longleftrightarrow> c *\<^sub>C a < - b" if "c < 0"
 for a b :: "'a :: ordered_complex_vector"
   using that by (simp add: less_le_not_le neg_le_minus_divideC_eq neg_minus_divideC_le_eq)
 
 lemma divideC_field_splits_simps_1 [field_split_simps]: (* In Real_Vector_Spaces, these lemmas are unnamed *)
   "a = b /\<^sub>C c \<longleftrightarrow> (if c = 0 then a = 0 else c *\<^sub>C a = b)"
   "b /\<^sub>C c = a \<longleftrightarrow> (if c = 0 then a = 0 else b = c *\<^sub>C a)"
   "a + b /\<^sub>C c = (if c = 0 then a else (c *\<^sub>C a + b) /\<^sub>C c)"
   "a /\<^sub>C c + b = (if c = 0 then b else (a + c *\<^sub>C b) /\<^sub>C c)"
   "a - b /\<^sub>C c = (if c = 0 then a else (c *\<^sub>C a - b) /\<^sub>C c)"
   "a /\<^sub>C c - b = (if c = 0 then - b else (a - c *\<^sub>C b) /\<^sub>C c)"
   "- (a /\<^sub>C c) + b = (if c = 0 then b else (- a + c *\<^sub>C b) /\<^sub>C c)"
   "- (a /\<^sub>C c) - b = (if c = 0 then - b else (- a - c *\<^sub>C b) /\<^sub>C c)"
   for a b :: "'a :: complex_vector"
   by (auto simp add: field_simps)
 
 lemma divideC_field_splits_simps_2 [field_split_simps]: (* In Real_Vector_Spaces, these lemmas are unnamed *)
   "0 < c \<Longrightarrow> a \<le> b /\<^sub>C c \<longleftrightarrow> (if c > 0 then c *\<^sub>C a \<le> b else if c < 0 then b \<le> c *\<^sub>C a else a \<le> 0)"
   "0 < c \<Longrightarrow> a < b /\<^sub>C c \<longleftrightarrow> (if c > 0 then c *\<^sub>C a < b else if c < 0 then b < c *\<^sub>C a else a < 0)"
   "0 < c \<Longrightarrow> b /\<^sub>C c \<le> a \<longleftrightarrow> (if c > 0 then b \<le> c *\<^sub>C a else if c < 0 then c *\<^sub>C a \<le> b else a \<ge> 0)"
   "0 < c \<Longrightarrow> b /\<^sub>C c < a \<longleftrightarrow> (if c > 0 then b < c *\<^sub>C a else if c < 0 then c *\<^sub>C a < b else a > 0)"
   "0 < c \<Longrightarrow> a \<le> - (b /\<^sub>C c) \<longleftrightarrow> (if c > 0 then c *\<^sub>C a \<le> - b else if c < 0 then - b \<le> c *\<^sub>C a else a \<le> 0)"
   "0 < c \<Longrightarrow> a < - (b /\<^sub>C c) \<longleftrightarrow> (if c > 0 then c *\<^sub>C a < - b else if c < 0 then - b < c *\<^sub>C a else a < 0)"
   "0 < c \<Longrightarrow> - (b /\<^sub>C c) \<le> a \<longleftrightarrow> (if c > 0 then - b \<le> c *\<^sub>C a else if c < 0 then c *\<^sub>C a \<le> - b else a \<ge> 0)"
   "0 < c \<Longrightarrow> - (b /\<^sub>C c) < a \<longleftrightarrow> (if c > 0 then - b < c *\<^sub>C a else if c < 0 then c *\<^sub>C a < - b else a > 0)"
   for a b :: "'a :: ordered_complex_vector"
   by (clarsimp intro!: field_simps)+
 
 lemma scaleC_nonneg_nonneg: "0 \<le> a \<Longrightarrow> 0 \<le> x \<Longrightarrow> 0 \<le> a *\<^sub>C x"
   for x :: "'a::ordered_complex_vector"
   using scaleC_left_mono [of 0 x a] by simp
 
 lemma scaleC_nonneg_nonpos: "0 \<le> a \<Longrightarrow> x \<le> 0 \<Longrightarrow> a *\<^sub>C x \<le> 0"
   for x :: "'a::ordered_complex_vector"
   using scaleC_left_mono [of x 0 a] by simp
 
 lemma scaleC_nonpos_nonneg: "a \<le> 0 \<Longrightarrow> 0 \<le> x \<Longrightarrow> a *\<^sub>C x \<le> 0"
   for x :: "'a::ordered_complex_vector"
   using scaleC_right_mono [of a 0 x] by simp
 
 lemma split_scaleC_neg_le: "(0 \<le> a \<and> x \<le> 0) \<or> (a \<le> 0 \<and> 0 \<le> x) \<Longrightarrow> a *\<^sub>C x \<le> 0"
   for x :: "'a::ordered_complex_vector"
   by (auto simp: scaleC_nonneg_nonpos scaleC_nonpos_nonneg)
 
 lemma cle_add_iff1: "a *\<^sub>C e + c \<le> b *\<^sub>C e + d \<longleftrightarrow> (a - b) *\<^sub>C e + c \<le> d"
   for c d e :: "'a::ordered_complex_vector"
   by (simp add: algebra_simps)
 
 lemma cle_add_iff2: "a *\<^sub>C e + c \<le> b *\<^sub>C e + d \<longleftrightarrow> c \<le> (b - a) *\<^sub>C e + d"
   for c d e :: "'a::ordered_complex_vector"
   by (simp add: algebra_simps)
 
 lemma scaleC_left_mono_neg: "b \<le> a \<Longrightarrow> c \<le> 0 \<Longrightarrow> c *\<^sub>C a \<le> c *\<^sub>C b"
   for a b :: "'a::ordered_complex_vector"
   by (drule scaleC_left_mono [of _ _ "- c"], simp_all add: less_eq_complex_def)
 
 lemma scaleC_right_mono_neg: "b \<le> a \<Longrightarrow> c \<le> 0 \<Longrightarrow> a *\<^sub>C c \<le> b *\<^sub>C c"
   for c :: "'a::ordered_complex_vector"
   by (drule scaleC_right_mono [of _ _ "- c"], simp_all)
 
 lemma scaleC_nonpos_nonpos: "a \<le> 0 \<Longrightarrow> b \<le> 0 \<Longrightarrow> 0 \<le> a *\<^sub>C b"
   for b :: "'a::ordered_complex_vector"
   using scaleC_right_mono_neg [of a 0 b] by simp
 
 lemma split_scaleC_pos_le: "(0 \<le> a \<and> 0 \<le> b) \<or> (a \<le> 0 \<and> b \<le> 0) \<Longrightarrow> 0 \<le> a *\<^sub>C b"
   for b :: "'a::ordered_complex_vector"
   by (auto simp: scaleC_nonneg_nonneg scaleC_nonpos_nonpos)
 
 lemma zero_le_scaleC_iff:
   fixes b :: "'a::ordered_complex_vector"
   assumes "a \<in> \<real>" (* Not present in Real_Vector_Spaces.thy *)
   shows "0 \<le> a *\<^sub>C b \<longleftrightarrow> 0 < a \<and> 0 \<le> b \<or> a < 0 \<and> b \<le> 0 \<or> a = 0"
     (is "?lhs = ?rhs")
 proof (cases "a = 0")
   case True
   then show ?thesis by simp
 next
   case False
   show ?thesis
   proof
     assume ?lhs
     from \<open>a \<noteq> 0\<close> consider "a > 0" | "a < 0"
       by (metis assms complex_is_Real_iff less_complex_def less_eq_complex_def not_le order.not_eq_order_implies_strict that(1) zero_complex.sel(2))
     then show ?rhs
     proof cases
       case 1
       with \<open>?lhs\<close> have "inverse a *\<^sub>C 0 \<le> inverse a *\<^sub>C (a *\<^sub>C b)"
         by (metis complex_vector.scale_zero_right ordered_complex_vector_class.pos_le_divideC_eq)
       with 1 show ?thesis
         by simp
     next
       case 2
       with \<open>?lhs\<close> have "- inverse a *\<^sub>C 0 \<le> - inverse a *\<^sub>C (a *\<^sub>C b)"
         by (metis Complex_Vector_Spaces0.neg_le_minus_divideC_eq complex_vector.scale_zero_right neg_le_0_iff_le scaleC_left.minus)
       with 2 show ?thesis
         by simp
     qed
   next
     assume ?rhs
     then show ?lhs
       using less_imp_le split_scaleC_pos_le by auto
   qed
 qed
 
 lemma scaleC_le_0_iff:
   "a *\<^sub>C b \<le> 0 \<longleftrightarrow> 0 < a \<and> b \<le> 0 \<or> a < 0 \<and> 0 \<le> b \<or> a = 0"
   if "a \<in> \<real>" (* Not present in Real_Vector_Spaces *)
   for b::"'a::ordered_complex_vector"
   apply (insert zero_le_scaleC_iff [of "-a" b])
   using less_complex_def that by force
 
 
 lemma scaleC_le_cancel_left: "c *\<^sub>C a \<le> c *\<^sub>C b \<longleftrightarrow> (0 < c \<longrightarrow> a \<le> b) \<and> (c < 0 \<longrightarrow> b \<le> a)"
   if "c \<in> \<real>" (* Not present in Real_Vector_Spaces *)
   for b :: "'a::ordered_complex_vector"
   by (smt (verit, ccfv_threshold) Complex_Vector_Spaces0.neg_divideC_le_eq complex_vector.scale_cancel_left complex_vector.scale_zero_right dual_order.eq_iff dual_order.trans ordered_complex_vector_class.pos_le_divideC_eq that zero_le_scaleC_iff)
 
 lemma scaleC_le_cancel_left_pos: "0 < c \<Longrightarrow> c *\<^sub>C a \<le> c *\<^sub>C b \<longleftrightarrow> a \<le> b"
   for b :: "'a::ordered_complex_vector"
   by (simp add: complex_is_Real_iff less_complex_def scaleC_le_cancel_left)
 
 lemma scaleC_le_cancel_left_neg: "c < 0 \<Longrightarrow> c *\<^sub>C a \<le> c *\<^sub>C b \<longleftrightarrow> b \<le> a"
   for b :: "'a::ordered_complex_vector"
   by (simp add: complex_is_Real_iff less_complex_def scaleC_le_cancel_left)
 
 lemma scaleC_left_le_one_le: "0 \<le> x \<Longrightarrow> a \<le> 1 \<Longrightarrow> a *\<^sub>C x \<le> x"
   for x :: "'a::ordered_complex_vector" and a :: complex
   using scaleC_right_mono[of a 1 x] by simp
 
 subsection \<open>Complex normed vector spaces\<close>
 
 (* Classes dist, norm, sgn_div_norm, dist_norm, uniformity_dist
    defined in Real_Vector_Spaces are unchanged in the complex setting.
    No need to define them here. *)
 
 class complex_normed_vector = complex_vector + sgn_div_norm + dist_norm + uniformity_dist + open_uniformity +
   real_normed_vector + (* Not present in Real_Normed_Vector *)
   assumes norm_scaleC [simp]: "norm (scaleC a x) = cmod a * norm x"
 begin
 (* lemma norm_ge_zero [simp]: "0 \<le> norm x" *) (* Not needed, included from real_normed_vector *)
 end
 
 class complex_normed_algebra = complex_algebra + complex_normed_vector +
   real_normed_algebra (* Not present in Real_Normed_Vector *)
   (* assumes norm_mult_ineq: "norm (x * y) \<le> norm x * norm y" *) (* Not needed, included from real_normed_algebra *)
 
 class complex_normed_algebra_1 = complex_algebra_1 + complex_normed_algebra +
   real_normed_algebra_1 (* Not present in Real_Normed_Vector *)
   (* assumes norm_one [simp]: "norm 1 = 1" *) (* Not needed, included from real_normed_algebra_1 *)
 
 lemma (in complex_normed_algebra_1) scaleC_power [simp]: "(scaleC x y) ^ n = scaleC (x^n) (y^n)"
   by (induct n) (simp_all add: mult_ac)
 
 class complex_normed_div_algebra = complex_div_algebra + complex_normed_vector +
   real_normed_div_algebra (* Not present in Real_Normed_Vector *)
   (* assumes norm_mult: "norm (x * y) = norm x * norm y" *) (* Not needed, included from real_normed_div_algebra *)
 
 class complex_normed_field = complex_field + complex_normed_div_algebra
 
 subclass (in complex_normed_field) real_normed_field ..
 
 instance complex_normed_div_algebra < complex_normed_algebra_1 ..
 
 context complex_normed_vector begin
 (* Inherited from real_normed_vector:
 lemma norm_zero [simp]: "norm (0::'a) = 0"
 lemma zero_less_norm_iff [simp]: "norm x > 0 \<longleftrightarrow> x \<noteq> 0"
 lemma norm_not_less_zero [simp]: "\<not> norm x < 0"
 lemma norm_le_zero_iff [simp]: "norm x \<le> 0 \<longleftrightarrow> x = 0"
 lemma norm_minus_cancel [simp]: "norm (- x) = norm x"
 lemma norm_minus_commute: "norm (a - b) = norm (b - a)"
 lemma dist_add_cancel [simp]: "dist (a + b) (a + c) = dist b c"
 lemma dist_add_cancel2 [simp]: "dist (b + a) (c + a) = dist b c"
 lemma norm_uminus_minus: "norm (- x - y) = norm (x + y)"
 lemma norm_triangle_ineq2: "norm a - norm b \<le> norm (a - b)"
 lemma norm_triangle_ineq3: "\<bar>norm a - norm b\<bar> \<le> norm (a - b)"
 lemma norm_triangle_ineq4: "norm (a - b) \<le> norm a + norm b"
 lemma norm_triangle_le_diff: "norm x + norm y \<le> e \<Longrightarrow> norm (x - y) \<le> e"
 lemma norm_diff_ineq: "norm a - norm b \<le> norm (a + b)"
 lemma norm_triangle_sub: "norm x \<le> norm y + norm (x - y)"
 lemma norm_triangle_le: "norm x + norm y \<le> e \<Longrightarrow> norm (x + y) \<le> e"
 lemma norm_triangle_lt: "norm x + norm y < e \<Longrightarrow> norm (x + y) < e"
 lemma norm_add_leD: "norm (a + b) \<le> c \<Longrightarrow> norm b \<le> norm a + c"
 lemma norm_diff_triangle_ineq: "norm ((a + b) - (c + d)) \<le> norm (a - c) + norm (b - d)"
 lemma norm_diff_triangle_le: "norm (x - z) \<le> e1 + e2"
   if "norm (x - y) \<le> e1"  "norm (y - z) \<le> e2"
 lemma norm_diff_triangle_less: "norm (x - z) < e1 + e2"
   if "norm (x - y) < e1"  "norm (y - z) < e2"
 lemma norm_triangle_mono:
   "norm a \<le> r \<Longrightarrow> norm b \<le> s \<Longrightarrow> norm (a + b) \<le> r + s"
 lemma norm_sum: "norm (sum f A) \<le> (\<Sum>i\<in>A. norm (f i))"
   for f::"'b \<Rightarrow> 'a"
 lemma sum_norm_le: "norm (sum f S) \<le> sum g S"
   if "\<And>x. x \<in> S \<Longrightarrow> norm (f x) \<le> g x"
   for f::"'b \<Rightarrow> 'a"
 lemma abs_norm_cancel [simp]: "\<bar>norm a\<bar> = norm a"
 lemma sum_norm_bound:
   "norm (sum f S) \<le> of_nat (card S)*K"
   if "\<And>x. x \<in> S \<Longrightarrow> norm (f x) \<le> K"
   for f :: "'b \<Rightarrow> 'a"
 lemma norm_add_less: "norm x < r \<Longrightarrow> norm y < s \<Longrightarrow> norm (x + y) < r + s"
 *)
 end
 
 lemma dist_scaleC [simp]: "dist (x *\<^sub>C a) (y *\<^sub>C a) = \<bar>x - y\<bar> * norm a"
   for a :: "'a::complex_normed_vector"
   by (metis dist_scaleR scaleR_scaleC)
 
 (* Inherited from real_normed_vector *)
 (* lemma norm_mult_less: "norm x < r \<Longrightarrow> norm y < s \<Longrightarrow> norm (x * y) < r * s"
   for x y :: "'a::complex_normed_algebra" *)
 
 lemma norm_of_complex [simp]: "norm (of_complex c :: 'a::complex_normed_algebra_1) = cmod c"
   by (simp add: of_complex_def)
 
 (* Inherited from real_normed_vector:
 lemma norm_numeral [simp]: "norm (numeral w::'a::complex_normed_algebra_1) = numeral w"
 lemma norm_neg_numeral [simp]: "norm (- numeral w::'a::complex_normed_algebra_1) = numeral w"
 lemma norm_of_complex_add1 [simp]: "norm (of_real x + 1 :: 'a :: complex_normed_div_algebra) = \<bar>x + 1\<bar>"
 lemma norm_of_complex_addn [simp]:
   "norm (of_real x + numeral b :: 'a :: complex_normed_div_algebra) = \<bar>x + numeral b\<bar>"
 lemma norm_of_int [simp]: "norm (of_int z::'a::complex_normed_algebra_1) = \<bar>of_int z\<bar>"
 lemma norm_of_nat [simp]: "norm (of_nat n::'a::complex_normed_algebra_1) = of_nat n"
 lemma nonzero_norm_inverse: "a \<noteq> 0 \<Longrightarrow> norm (inverse a) = inverse (norm a)"
   for a :: "'a::complex_normed_div_algebra"
 lemma norm_inverse: "norm (inverse a) = inverse (norm a)"
   for a :: "'a::{complex_normed_div_algebra,division_ring}"
 lemma nonzero_norm_divide: "b \<noteq> 0 \<Longrightarrow> norm (a / b) = norm a / norm b"
   for a b :: "'a::complex_normed_field"
 lemma norm_divide: "norm (a / b) = norm a / norm b"
   for a b :: "'a::{complex_normed_field,field}"
 lemma norm_inverse_le_norm:
   fixes x :: "'a::complex_normed_div_algebra"
   shows "r \<le> norm x \<Longrightarrow> 0 < r \<Longrightarrow> norm (inverse x) \<le> inverse r"
 lemma norm_power_ineq: "norm (x ^ n) \<le> norm x ^ n"
   for x :: "'a::complex_normed_algebra_1"
 lemma norm_power: "norm (x ^ n) = norm x ^ n"
   for x :: "'a::complex_normed_div_algebra"
 lemma norm_power_int: "norm (power_int x n) = power_int (norm x) n"
   for x :: "'a::complex_normed_div_algebra"
 lemma power_eq_imp_eq_norm:
   fixes w :: "'a::complex_normed_div_algebra"
   assumes eq: "w ^ n = z ^ n" and "n > 0"
     shows "norm w = norm z"
 lemma power_eq_1_iff:
   fixes w :: "'a::complex_normed_div_algebra"
   shows "w ^ n = 1 \<Longrightarrow> norm w = 1 \<or> n = 0"
 lemma norm_mult_numeral1 [simp]: "norm (numeral w * a) = numeral w * norm a"
   for a b :: "'a::{complex_normed_field,field}"
 lemma norm_mult_numeral2 [simp]: "norm (a * numeral w) = norm a * numeral w"
   for a b :: "'a::{complex_normed_field,field}"
 lemma norm_divide_numeral [simp]: "norm (a / numeral w) = norm a / numeral w"
   for a b :: "'a::{complex_normed_field,field}"
 lemma square_norm_one:
   fixes x :: "'a::complex_normed_div_algebra"
   assumes "x\<^sup>2 = 1"
   shows "norm x = 1"
 lemma norm_less_p1: "norm x < norm (of_real (norm x) + 1 :: 'a)"
   for x :: "'a::complex_normed_algebra_1"
 lemma prod_norm: "prod (\<lambda>x. norm (f x)) A = norm (prod f A)"
   for f :: "'a \<Rightarrow> 'b::{comm_semiring_1,complex_normed_div_algebra}"
 lemma norm_prod_le:
   "norm (prod f A) \<le> (\<Prod>a\<in>A. norm (f a :: 'a :: {complex_normed_algebra_1,comm_monoid_mult}))"
 lemma norm_prod_diff:
   fixes z w :: "'i \<Rightarrow> 'a::{complex_normed_algebra_1, comm_monoid_mult}"
   shows "(\<And>i. i \<in> I \<Longrightarrow> norm (z i) \<le> 1) \<Longrightarrow> (\<And>i. i \<in> I \<Longrightarrow> norm (w i) \<le> 1) \<Longrightarrow>
     norm ((\<Prod>i\<in>I. z i) - (\<Prod>i\<in>I. w i)) \<le> (\<Sum>i\<in>I. norm (z i - w i))"
 lemma norm_power_diff:
   fixes z w :: "'a::{complex_normed_algebra_1, comm_monoid_mult}"
   assumes "norm z \<le> 1" "norm w \<le> 1"
   shows "norm (z^m - w^m) \<le> m * norm (z - w)"
 *)
 
 lemma norm_of_complex_add1 [simp]: "norm (of_complex x + 1 :: 'a :: complex_normed_div_algebra) = cmod (x + 1)"
   by (metis norm_of_complex of_complex_1 of_complex_add)
 
 lemma norm_of_complex_addn [simp]:
   "norm (of_complex x + numeral b :: 'a :: complex_normed_div_algebra) = cmod (x + numeral b)"
   by (metis norm_of_complex of_complex_add of_complex_numeral)
 
 lemma norm_of_complex_diff [simp]:
   "norm (of_complex b - of_complex a :: 'a::complex_normed_algebra_1) \<le> cmod (b - a)"
   by (metis norm_of_complex of_complex_diff order_refl)
 
 subsection \<open>Metric spaces\<close>
 
 (* Class metric_space is already defined in Real_Vector_Spaces and does not need changing here *)
 
 text \<open>Every normed vector space is a metric space.\<close>
 (* Already follows from complex_normed_vector < real_normed_vector < metric_space *)
 (* instance complex_normed_vector < metric_space *)
 
 subsection \<open>Class instances for complex numbers\<close>
 
 instantiation complex :: complex_normed_field
 begin
 
 instance
   apply intro_classes
   by (simp add: norm_mult)
 
 end
 
 declare uniformity_Abort[where 'a=complex, code]
 
 lemma dist_of_complex [simp]: "dist (of_complex x :: 'a) (of_complex y) = dist x y"
   for a :: "'a::complex_normed_div_algebra"
   by (metis dist_norm norm_of_complex of_complex_diff)
 
 declare [[code abort: "open :: complex set \<Rightarrow> bool"]]
 
 (* As far as I can tell, there is no analogue to this for complex:
 instance real :: order_topology
 instance real :: linear_continuum_topology ..
 
 lemmas open_complex_greaterThan = open_greaterThan[where 'a=complex]
 lemmas open_complex_lessThan = open_lessThan[where 'a=complex]
 lemmas open_complex_greaterThanLessThan = open_greaterThanLessThan[where 'a=complex]
 *)
 
 lemma closed_complex_atMost: \<open>closed {..a::complex}\<close>
 proof -
   have \<open>{..a} = Im -` {Im a} \<inter> Re -` {..Re a}\<close>
     by (auto simp: less_eq_complex_def)
   also have \<open>closed \<dots>\<close>
     by (auto intro!: closed_Int closed_vimage continuous_on_Im continuous_on_Re)
   finally show ?thesis
     by -
 qed
 
 lemma closed_complex_atLeast: \<open>closed {a::complex..}\<close>
 proof -
   have \<open>{a..} = Im -` {Im a} \<inter> Re -` {Re a..}\<close>
     by (auto simp: less_eq_complex_def)
   also have \<open>closed \<dots>\<close>
     by (auto intro!: closed_Int closed_vimage continuous_on_Im continuous_on_Re)
   finally show ?thesis
     by -
 qed
 
 lemma closed_complex_atLeastAtMost: \<open>closed {a::complex .. b}\<close>
 proof (cases \<open>Im a = Im b\<close>)
   case True
   have \<open>{a..b} = Im -` {Im a} \<inter> Re -` {Re a..Re b}\<close>
     by (auto simp add: less_eq_complex_def intro!: True)
   also have \<open>closed \<dots>\<close>
     by (auto intro!: closed_Int closed_vimage continuous_on_Im continuous_on_Re)
   finally show ?thesis
     by -
 next
   case False
   then have *: \<open>{a..b} = {}\<close>
     using less_eq_complex_def by auto
   show ?thesis
     by (simp add: *)
 qed
 
 (* As far as I can tell, there is no analogue to this for complex:
 instance real :: ordered_real_vector
   by standard (auto intro: mult_left_mono mult_right_mono)
 *)
 
 (* subsection \<open>Extra type constraints\<close> *)
 (* Everything is commented out, so we comment out the heading, too. *)
 
 (* These are already configured in Real_Vector_Spaces:
 
 text \<open>Only allow \<^term>\<open>open\<close> in class \<open>topological_space\<close>.\<close>
 setup \<open>Sign.add_const_constraint
   (\<^const_name>\<open>open\<close>, SOME \<^typ>\<open>'a::topological_space set \<Rightarrow> bool\<close>)\<close>
 
 text \<open>Only allow \<^term>\<open>uniformity\<close> in class \<open>uniform_space\<close>.\<close>
 setup \<open>Sign.add_const_constraint
   (\<^const_name>\<open>uniformity\<close>, SOME \<^typ>\<open>('a::uniformity \<times> 'a) filter\<close>)\<close>
 
 text \<open>Only allow \<^term>\<open>dist\<close> in class \<open>metric_space\<close>.\<close>
 setup \<open>Sign.add_const_constraint
   (\<^const_name>\<open>dist\<close>, SOME \<^typ>\<open>'a::metric_space \<Rightarrow> 'a \<Rightarrow> real\<close>)\<close>
 
 text \<open>Only allow \<^term>\<open>norm\<close> in class \<open>complex_normed_vector\<close>.\<close>
 setup \<open>Sign.add_const_constraint
   (\<^const_name>\<open>norm\<close>, SOME \<^typ>\<open>'a::complex_normed_vector \<Rightarrow> real\<close>)\<close>
 *)
 
 subsection \<open>Sign function\<close>
 
 (* Inherited from real_normed_vector:
 lemma norm_sgn: "norm (sgn x) = (if x = 0 then 0 else 1)"
   for x :: "'a::complex_normed_vector"
 lemma sgn_zero [simp]: "sgn (0::'a::complex_normed_vector) = 0"
 lemma sgn_zero_iff: "sgn x = 0 \<longleftrightarrow> x = 0"
   for x :: "'a::complex_normed_vector"
 lemma sgn_minus: "sgn (- x) = - sgn x"
   for x :: "'a::complex_normed_vector"
 lemma sgn_one [simp]: "sgn (1::'a::complex_normed_algebra_1) = 1"
 lemma sgn_mult: "sgn (x * y) = sgn x * sgn y"
   for x y :: "'a::complex_normed_div_algebra"
 hide_fact (open) sgn_mult
 lemma norm_conv_dist: "norm x = dist x 0"
 declare norm_conv_dist [symmetric, simp]
 lemma dist_0_norm [simp]: "dist 0 x = norm x"
   for x :: "'a::complex_normed_vector"
 lemma dist_diff [simp]: "dist a (a - b) = norm b"  "dist (a - b) a = norm b"
 lemma dist_of_int: "dist (of_int m) (of_int n :: 'a :: complex_normed_algebra_1) = of_int \<bar>m - n\<bar>"
 lemma dist_of_nat:
   "dist (of_nat m) (of_nat n :: 'a :: complex_normed_algebra_1) = of_int \<bar>int m - int n\<bar>"
 *)
 
 lemma sgn_scaleC: "sgn (scaleC r x) = scaleC (sgn r) (sgn x)"
   for x :: "'a::complex_normed_vector"
   by (simp add: scaleR_scaleC sgn_div_norm ac_simps)
 
 lemma sgn_of_complex: "sgn (of_complex r :: 'a::complex_normed_algebra_1) = of_complex (sgn r)"
   unfolding of_complex_def by (simp only: sgn_scaleC sgn_one)
 
 lemma complex_sgn_eq: "sgn x = x / \<bar>x\<bar>"
   for x :: complex
   by (simp add: abs_complex_def scaleR_scaleC sgn_div_norm divide_inverse)
 
 lemma czero_le_sgn_iff [simp]: "0 \<le> sgn x \<longleftrightarrow> 0 \<le> x"
   for x :: complex
   using cmod_eq_Re divide_eq_0_iff less_eq_complex_def by auto
 
 lemma csgn_le_0_iff [simp]: "sgn x \<le> 0 \<longleftrightarrow> x \<le> 0"
   for x :: complex
   by (smt (verit, best) czero_le_sgn_iff Im_sgn Re_sgn divide_eq_0_iff dual_order.eq_iff less_eq_complex_def sgn_zero_iff zero_complex.sel(1) zero_complex.sel(2))
 
 
 subsection \<open>Bounded Linear and Bilinear Operators\<close>
 
 lemma clinearI: "clinear f"
   if "\<And>b1 b2. f (b1 + b2) = f b1 + f b2"
     "\<And>r b. f (r *\<^sub>C b) = r *\<^sub>C f b"
   using that
   by unfold_locales (auto simp: algebra_simps)
 
 lemma clinear_iff:
   "clinear f \<longleftrightarrow> (\<forall>x y. f (x + y) = f x + f y) \<and> (\<forall>c x. f (c *\<^sub>C x) = c *\<^sub>C f x)"
   (is "clinear f \<longleftrightarrow> ?rhs")
 proof
   assume "clinear f"
   then interpret f: clinear f .
   show "?rhs"
     by (simp add: f.add f.scale complex_vector.linear_scale f.clinear_axioms)
 next
   assume "?rhs"
   then show "clinear f" by (intro clinearI) auto
 qed
 
 lemmas clinear_scaleC_left = complex_vector.linear_scale_left
 lemmas clinear_imp_scaleC = complex_vector.linear_imp_scale
 
 corollary complex_clinearD:
   fixes f :: "complex \<Rightarrow> complex"
   assumes "clinear f" obtains c where "f = (*) c"
   by (rule clinear_imp_scaleC [OF assms]) (force simp: scaleC_conv_of_complex)
 
 lemma clinear_times_of_complex: "clinear (\<lambda>x. a * of_complex x)"
   by (auto intro!: clinearI simp: distrib_left)
     (metis mult_scaleC_right scaleC_conv_of_complex)
 
 locale bounded_clinear = clinear f for f :: "'a::complex_normed_vector \<Rightarrow> 'b::complex_normed_vector" +
   assumes bounded: "\<exists>K. \<forall>x. norm (f x) \<le> norm x * K"
 begin
 
 (* Not present in Real_Vector_Spaces. *)
 sublocale real: bounded_linear
   \<comment> \<open>Gives access to all lemmas from \<^locale>\<open>bounded_linear\<close> using prefix \<open>real.\<close>\<close>
   apply standard
   by (auto simp add: add scaleR_scaleC scale bounded)
 
 lemmas pos_bounded = real.pos_bounded (* "\<exists>K>0. \<forall>x. norm (f x) \<le> norm x * K" *)
 
 (* Inherited from bounded_linear *)
 lemmas nonneg_bounded = real.nonneg_bounded (* "\<exists>K\<ge>0. \<forall>x. norm (f x) \<le> norm x * K" *)
 
 lemma clinear: "clinear f"
   by (fact local.clinear_axioms)
 
 end
 
 lemma bounded_clinear_intro:
   assumes "\<And>x y. f (x + y) = f x + f y"
     and "\<And>r x. f (scaleC r x) = scaleC r (f x)"
     and "\<And>x. norm (f x) \<le> norm x * K"
   shows "bounded_clinear f"
   by standard (blast intro: assms)+
 
 locale bounded_cbilinear =
   fixes prod :: "'a::complex_normed_vector \<Rightarrow> 'b::complex_normed_vector \<Rightarrow> 'c::complex_normed_vector"
     (infixl "**" 70)
   assumes add_left: "prod (a + a') b = prod a b + prod a' b"
     and add_right: "prod a (b + b') = prod a b + prod a b'"
     and scaleC_left: "prod (scaleC r a) b = scaleC r (prod a b)"
     and scaleC_right: "prod a (scaleC r b) = scaleC r (prod a b)"
     and bounded: "\<exists>K. \<forall>a b. norm (prod a b) \<le> norm a * norm b * K"
 begin
 
 (* Not present in Real_Vector_Spaces. *)
 sublocale real: bounded_bilinear
   \<comment> \<open>Gives access to all lemmas from \<^locale>\<open>bounded_bilinear\<close> using prefix \<open>real.\<close>\<close>
   apply standard
   by (auto simp add: add_left add_right scaleR_scaleC scaleC_left scaleC_right bounded)
 
 lemmas pos_bounded = real.pos_bounded (* "\<exists>K>0. \<forall>a b. norm (a ** b) \<le> norm a * norm b * K" *)
 lemmas nonneg_bounded = real.nonneg_bounded (* "\<exists>K\<ge>0. \<forall>a b. norm (a ** b) \<le> norm a * norm b * K" *)
 lemmas additive_right = real.additive_right (* "additive (\<lambda>b. prod a b)" *)
 lemmas additive_left = real.additive_left (* "additive (\<lambda>a. prod a b)" *)
 lemmas zero_left = real.zero_left (* "prod 0 b = 0" *)
 lemmas zero_right = real.zero_right (* "prod a 0 = 0" *)
 lemmas minus_left = real.minus_left (* "prod (- a) b = - prod a b" *)
 lemmas minus_right = real.minus_right (* "prod a (- b) = - prod a b" *)
 lemmas diff_left = real.diff_left (* "prod (a - a') b = prod a b - prod a' b" *)
 lemmas diff_right = real.diff_right (* "prod a (b - b') = prod a b - prod a b'" *)
 lemmas sum_left = real.sum_left (* "prod (sum g S) x = sum ((\<lambda>i. prod (g i) x)) S" *)
 lemmas sum_right = real.sum_right (* "prod x (sum g S) = sum ((\<lambda>i. (prod x (g i)))) S" *)
 lemmas prod_diff_prod = real.prod_diff_prod (* "(x ** y - a ** b) = (x - a) ** (y - b) + (x - a) ** b + a ** (y - b)" *)
 
 lemma bounded_clinear_left: "bounded_clinear (\<lambda>a. a ** b)"
 proof -
   obtain K where "\<And>a b. norm (a ** b) \<le> norm a * norm b * K"
     using pos_bounded by blast
   then show ?thesis
     by (rule_tac K="norm b * K" in bounded_clinear_intro) (auto simp: algebra_simps scaleC_left add_left)
 qed
 
 lemma bounded_clinear_right: "bounded_clinear (\<lambda>b. a ** b)"
 proof -
   obtain K where "\<And>a b. norm (a ** b) \<le> norm a * norm b * K"
     using pos_bounded by blast
   then show ?thesis
     by (rule_tac K="norm a * K" in bounded_clinear_intro) (auto simp: algebra_simps scaleC_right add_right)
 qed
 
 lemma flip: "bounded_cbilinear (\<lambda>x y. y ** x)"
 proof
   show "\<exists>K. \<forall>a b. norm (b ** a) \<le> norm a * norm b * K"
     by (metis bounded mult.commute)
 qed (simp_all add: add_right add_left scaleC_right scaleC_left)
 
 lemma comp1:
   assumes "bounded_clinear g"
   shows "bounded_cbilinear (\<lambda>x. (**) (g x))"
 proof
   interpret g: bounded_clinear g by fact
   show "\<And>a a' b. g (a + a') ** b = g a ** b + g a' ** b"
     "\<And>a b b'. g a ** (b + b') = g a ** b + g a ** b'"
     "\<And>r a b. g (r *\<^sub>C a) ** b = r *\<^sub>C (g a ** b)"
     "\<And>a r b. g a ** (r *\<^sub>C b) = r *\<^sub>C (g a ** b)"
     by (auto simp: g.add add_left add_right g.scaleC scaleC_left scaleC_right)
   have "bounded_bilinear (\<lambda>a b. g a ** b)"
     using g.real.bounded_linear by (rule real.comp1)
   then show "\<exists>K. \<forall>a b. norm (g a ** b) \<le> norm a * norm b * K"
     by (rule bounded_bilinear.bounded)
 qed
 
 lemma comp: "bounded_clinear f \<Longrightarrow> bounded_clinear g \<Longrightarrow> bounded_cbilinear (\<lambda>x y. f x ** g y)"
   by (rule bounded_cbilinear.flip[OF bounded_cbilinear.comp1[OF bounded_cbilinear.flip[OF comp1]]])
 
 end (* locale bounded_cbilinear *)
 
 lemma bounded_clinear_ident[simp]: "bounded_clinear (\<lambda>x. x)"
   by standard (auto intro!: exI[of _ 1])
 
 lemma bounded_clinear_zero[simp]: "bounded_clinear (\<lambda>x. 0)"
   by standard (auto intro!: exI[of _ 1])
 
 lemma bounded_clinear_add:
   assumes "bounded_clinear f"
     and "bounded_clinear g"
   shows "bounded_clinear (\<lambda>x. f x + g x)"
 proof -
   interpret f: bounded_clinear f by fact
   interpret g: bounded_clinear g by fact
   show ?thesis
   proof
     from f.bounded obtain Kf where Kf: "norm (f x) \<le> norm x * Kf" for x
       by blast
     from g.bounded obtain Kg where Kg: "norm (g x) \<le> norm x * Kg" for x
       by blast
     show "\<exists>K. \<forall>x. norm (f x + g x) \<le> norm x * K"
       using add_mono[OF Kf Kg]
       by (intro exI[of _ "Kf + Kg"]) (auto simp: field_simps intro: norm_triangle_ineq order_trans)
   qed (simp_all add: f.add g.add f.scaleC g.scaleC scaleC_add_right)
 qed
 
 lemma bounded_clinear_minus:
   assumes "bounded_clinear f"
   shows "bounded_clinear (\<lambda>x. - f x)"
 proof -
   interpret f: bounded_clinear f by fact
   show ?thesis
     by unfold_locales (simp_all add: f.add f.scaleC f.bounded)
 qed
 
 lemma bounded_clinear_sub: "bounded_clinear f \<Longrightarrow> bounded_clinear g \<Longrightarrow> bounded_clinear (\<lambda>x. f x - g x)"
   using bounded_clinear_add[of f "\<lambda>x. - g x"] bounded_clinear_minus[of g]
   by (auto simp: algebra_simps)
 
 lemma bounded_clinear_sum:
   fixes f :: "'i \<Rightarrow> 'a::complex_normed_vector \<Rightarrow> 'b::complex_normed_vector"
   shows "(\<And>i. i \<in> I \<Longrightarrow> bounded_clinear (f i)) \<Longrightarrow> bounded_clinear (\<lambda>x. \<Sum>i\<in>I. f i x)"
   by (induct I rule: infinite_finite_induct) (auto intro!: bounded_clinear_add)
 
 lemma bounded_clinear_compose:
   assumes "bounded_clinear f"
     and "bounded_clinear g"
   shows "bounded_clinear (\<lambda>x. f (g x))"
 proof
   interpret f: bounded_clinear f by fact
   interpret g: bounded_clinear g by fact
   show "f (g (x + y)) = f (g x) + f (g y)" for x y
     by (simp only: f.add g.add)
   show "f (g (scaleC r x)) = scaleC r (f (g x))" for r x
     by (simp only: f.scaleC g.scaleC)
   from f.pos_bounded obtain Kf where f: "\<And>x. norm (f x) \<le> norm x * Kf" and Kf: "0 < Kf"
     by blast
   from g.pos_bounded obtain Kg where g: "\<And>x. norm (g x) \<le> norm x * Kg"
     by blast
   show "\<exists>K. \<forall>x. norm (f (g x)) \<le> norm x * K"
   proof (intro exI allI)
     fix x
     have "norm (f (g x)) \<le> norm (g x) * Kf"
       using f .
     also have "\<dots> \<le> (norm x * Kg) * Kf"
       using g Kf [THEN order_less_imp_le] by (rule mult_right_mono)
     also have "(norm x * Kg) * Kf = norm x * (Kg * Kf)"
       by (rule mult.assoc)
     finally show "norm (f (g x)) \<le> norm x * (Kg * Kf)" .
   qed
 qed
 
 lemma bounded_cbilinear_mult: "bounded_cbilinear ((*) :: 'a \<Rightarrow> 'a \<Rightarrow> 'a::complex_normed_algebra)"
 proof (rule bounded_cbilinear.intro)
   show "\<exists>K. \<forall>a b::'a. norm (a * b) \<le> norm a * norm b * K"
     by (rule_tac x=1 in exI) (simp add: norm_mult_ineq)
 qed (auto simp: algebra_simps)
 
 lemma bounded_clinear_mult_left: "bounded_clinear (\<lambda>x::'a::complex_normed_algebra. x * y)"
   using bounded_cbilinear_mult
   by (rule bounded_cbilinear.bounded_clinear_left)
 
 lemma bounded_clinear_mult_right: "bounded_clinear (\<lambda>y::'a::complex_normed_algebra. x * y)"
   using bounded_cbilinear_mult
   by (rule bounded_cbilinear.bounded_clinear_right)
 
 lemmas bounded_clinear_mult_const =
   bounded_clinear_mult_left [THEN bounded_clinear_compose]
 
 lemmas bounded_clinear_const_mult =
   bounded_clinear_mult_right [THEN bounded_clinear_compose]
 
 lemma bounded_clinear_divide: "bounded_clinear (\<lambda>x. x / y)"
   for y :: "'a::complex_normed_field"
   unfolding divide_inverse by (rule bounded_clinear_mult_left)
 
 lemma bounded_cbilinear_scaleC: "bounded_cbilinear scaleC"
 proof (rule bounded_cbilinear.intro)
   obtain K where K: \<open>\<forall>a (b::'a). norm b \<le> norm b * K\<close>
     using less_eq_real_def by auto
   show "\<exists>K. \<forall>a (b::'a). norm (a *\<^sub>C b) \<le> norm a * norm b * K"
     apply (rule exI[where x=K]) using K
     by (metis norm_scaleC)
 qed (auto simp: algebra_simps)
 
 lemma bounded_clinear_scaleC_left: "bounded_clinear (\<lambda>c. scaleC c x)"
   using bounded_cbilinear_scaleC
   by (rule bounded_cbilinear.bounded_clinear_left)
 
 lemma bounded_clinear_scaleC_right: "bounded_clinear (\<lambda>x. scaleC c x)"
   using bounded_cbilinear_scaleC
   by (rule bounded_cbilinear.bounded_clinear_right)
 
 lemmas bounded_clinear_scaleC_const =
   bounded_clinear_scaleC_left[THEN bounded_clinear_compose]
 
 lemmas bounded_clinear_const_scaleC =
   bounded_clinear_scaleC_right[THEN bounded_clinear_compose]
 
 lemma bounded_clinear_of_complex: "bounded_clinear (\<lambda>r. of_complex r)"
   unfolding of_complex_def by (rule bounded_clinear_scaleC_left)
 
 lemma complex_bounded_clinear: "bounded_clinear f \<longleftrightarrow> (\<exists>c::complex. f = (\<lambda>x. x * c))"
   for f :: "complex \<Rightarrow> complex"
 proof -
   {
     fix x
     assume "bounded_clinear f"
     then interpret bounded_clinear f .
     from scaleC[of x 1] have "f x = x * f 1"
       by simp
   }
   then show ?thesis
     by (auto intro: exI[of _ "f 1"] bounded_clinear_mult_left)
 qed
 
 (* Inherited from real_normed_algebra_1 *)
 (* instance complex_normed_algebra_1 \<subseteq> perfect_space *)
 
 (* subsection \<open>Filters and Limits on Metric Space\<close> *)
 (* Everything is commented out, so we comment out the heading, too. *)
 
 (* Not specific to real/complex *)
 (* lemma (in metric_space) nhds_metric: "nhds x = (INF e\<in>{0 <..}. principal {y. dist y x < e})" *)
 (* lemma (in metric_space) tendsto_iff: "(f \<longlongrightarrow> l) F \<longleftrightarrow> (\<forall>e>0. eventually (\<lambda>x. dist (f x) l < e) F)" *)
 (* lemma tendsto_dist_iff: "((f \<longlongrightarrow> l) F) \<longleftrightarrow> (((\<lambda>x. dist (f x) l) \<longlongrightarrow> 0) F)" *)
 (* lemma (in metric_space) tendstoI [intro?]:
   "(\<And>e. 0 < e \<Longrightarrow> eventually (\<lambda>x. dist (f x) l < e) F) \<Longrightarrow> (f \<longlongrightarrow> l) F" *)
 (* lemma (in metric_space) tendstoD: "(f \<longlongrightarrow> l) F \<Longrightarrow> 0 < e \<Longrightarrow> eventually (\<lambda>x. dist (f x) l < e) F" *)
 (* lemma (in metric_space) eventually_nhds_metric:
   "eventually P (nhds a) \<longleftrightarrow> (\<exists>d>0. \<forall>x. dist x a < d \<longrightarrow> P x)" *)
 (* lemma eventually_at: "eventually P (at a within S) \<longleftrightarrow> (\<exists>d>0. \<forall>x\<in>S. x \<noteq> a \<and> dist x a < d \<longrightarrow> P x)"
   for a :: "'a :: metric_space" *)
 (* lemma frequently_at: "frequently P (at a within S) \<longleftrightarrow> (\<forall>d>0. \<exists>x\<in>S. x \<noteq> a \<and> dist x a < d \<and> P x)"
   for a :: "'a :: metric_space" *)
 (* lemma eventually_at_le: "eventually P (at a within S) \<longleftrightarrow> (\<exists>d>0. \<forall>x\<in>S. x \<noteq> a \<and> dist x a \<le> d \<longrightarrow> P x)"
   for a :: "'a::metric_space" *)
 
 (* Does not work in complex case because it needs complex :: order_toplogy *)
 (* lemma eventually_at_left_real: "a > (b :: real) \<Longrightarrow> eventually (\<lambda>x. x \<in> {b<..<a}) (at_left a)" *)
 (* lemma eventually_at_right_real: "a < (b :: real) \<Longrightarrow> eventually (\<lambda>x. x \<in> {a<..<b}) (at_right a)" *)
 
 (* Not specific to real/complex *)
 (* lemma metric_tendsto_imp_tendsto:
   fixes a :: "'a :: metric_space"
     and b :: "'b :: metric_space"
   assumes f: "(f \<longlongrightarrow> a) F"
     and le: "eventually (\<lambda>x. dist (g x) b \<le> dist (f x) a) F"
   shows "(g \<longlongrightarrow> b) F" *)
 
 (* Not sure if this makes sense in the complex case *)
 (* lemma filterlim_complex_sequentially: "LIM x sequentially. (of_nat x :: complex) :> at_top" *)
 
 (* Not specific to real/complex *)
 (* lemma filterlim_nat_sequentially: "filterlim nat sequentially at_top" *)
 (* lemma filterlim_floor_sequentially: "filterlim floor at_top at_top" *)
 
 (* Not sure if this makes sense in the complex case *)
 (* lemma filterlim_sequentially_iff_filterlim_real:
   "filterlim f sequentially F \<longleftrightarrow> filterlim (\<lambda>x. real (f x)) at_top F" (is "?lhs = ?rhs") *)
 
 
 subsubsection \<open>Limits of Sequences\<close>
 
 (* Not specific to real/complex *)
 (* lemma lim_sequentially: "X \<longlonglongrightarrow> L \<longleftrightarrow> (\<forall>r>0. \<exists>no. \<forall>n\<ge>no. dist (X n) L < r)"
   for L :: "'a::metric_space" *)
 (* lemmas LIMSEQ_def = lim_sequentially  (*legacy binding*) *)
 (* lemma LIMSEQ_iff_nz: "X \<longlonglongrightarrow> L \<longleftrightarrow> (\<forall>r>0. \<exists>no>0. \<forall>n\<ge>no. dist (X n) L < r)"
   for L :: "'a::metric_space" *)
 (* lemma metric_LIMSEQ_I: "(\<And>r. 0 < r \<Longrightarrow> \<exists>no. \<forall>n\<ge>no. dist (X n) L < r) \<Longrightarrow> X \<longlonglongrightarrow> L"
   for L :: "'a::metric_space" *)
 (* lemma metric_LIMSEQ_D: "X \<longlonglongrightarrow> L \<Longrightarrow> 0 < r \<Longrightarrow> \<exists>no. \<forall>n\<ge>no. dist (X n) L < r"
   for L :: "'a::metric_space" *)
 (* lemma LIMSEQ_norm_0:
   assumes  "\<And>n::nat. norm (f n) < 1 / real (Suc n)"
   shows "f \<longlonglongrightarrow> 0" *)
 
 (* subsubsection \<open>Limits of Functions\<close> *)
 (* Everything is commented out, so we comment out the heading, too. *)
 
 (* Not specific to real/complex *)
 (* lemma LIM_def: "f \<midarrow>a\<rightarrow> L \<longleftrightarrow> (\<forall>r > 0. \<exists>s > 0. \<forall>x. x \<noteq> a \<and> dist x a < s \<longrightarrow> dist (f x) L < r)"
   for a :: "'a::metric_space" and L :: "'b::metric_space" *)
 (* lemma metric_LIM_I:
   "(\<And>r. 0 < r \<Longrightarrow> \<exists>s>0. \<forall>x. x \<noteq> a \<and> dist x a < s \<longrightarrow> dist (f x) L < r) \<Longrightarrow> f \<midarrow>a\<rightarrow> L"
   for a :: "'a::metric_space" and L :: "'b::metric_space" *)
 (* lemma metric_LIM_D: "f \<midarrow>a\<rightarrow> L \<Longrightarrow> 0 < r \<Longrightarrow> \<exists>s>0. \<forall>x. x \<noteq> a \<and> dist x a < s \<longrightarrow> dist (f x) L < r"
   for a :: "'a::metric_space" and L :: "'b::metric_space" *)
 (* lemma metric_LIM_imp_LIM:
   fixes l :: "'a::metric_space"
     and m :: "'b::metric_space"
   assumes f: "f \<midarrow>a\<rightarrow> l"
     and le: "\<And>x. x \<noteq> a \<Longrightarrow> dist (g x) m \<le> dist (f x) l"
   shows "g \<midarrow>a\<rightarrow> m" *)
 (* lemma metric_LIM_equal2:
   fixes a :: "'a::metric_space"
   assumes "g \<midarrow>a\<rightarrow> l" "0 < R"
     and "\<And>x. x \<noteq> a \<Longrightarrow> dist x a < R \<Longrightarrow> f x = g x"
   shows "f \<midarrow>a\<rightarrow> l" *)
 (* lemma metric_LIM_compose2:
   fixes a :: "'a::metric_space"
   assumes f: "f \<midarrow>a\<rightarrow> b"
     and g: "g \<midarrow>b\<rightarrow> c"
     and inj: "\<exists>d>0. \<forall>x. x \<noteq> a \<and> dist x a < d \<longrightarrow> f x \<noteq> b"
   shows "(\<lambda>x. g (f x)) \<midarrow>a\<rightarrow> c" *)
 (* lemma metric_isCont_LIM_compose2:
   fixes f :: "'a :: metric_space \<Rightarrow> _"
   assumes f [unfolded isCont_def]: "isCont f a"
     and g: "g \<midarrow>f a\<rightarrow> l"
     and inj: "\<exists>d>0. \<forall>x. x \<noteq> a \<and> dist x a < d \<longrightarrow> f x \<noteq> f a"
   shows "(\<lambda>x. g (f x)) \<midarrow>a\<rightarrow> l" *)
 
 
 (* subsection \<open>Complete metric spaces\<close> *)
 (* Everything is commented out, so we comment out the heading, too. *)
 
 subsection \<open>Cauchy sequences\<close>
 
 (* Not specific to real/complex *)
 (* lemma (in metric_space) Cauchy_def: "Cauchy X = (\<forall>e>0. \<exists>M. \<forall>m\<ge>M. \<forall>n\<ge>M. dist (X m) (X n) < e)" *)
 (* lemma (in metric_space) Cauchy_altdef: "Cauchy f \<longleftrightarrow> (\<forall>e>0. \<exists>M. \<forall>m\<ge>M. \<forall>n>m. dist (f m) (f n) < e)" *)
 (* lemma (in metric_space) Cauchy_altdef2: "Cauchy s \<longleftrightarrow> (\<forall>e>0. \<exists>N::nat. \<forall>n\<ge>N. dist(s n)(s N) < e)" (is "?lhs = ?rhs") *)
 (* lemma (in metric_space) metric_CauchyI:
   "(\<And>e. 0 < e \<Longrightarrow> \<exists>M. \<forall>m\<ge>M. \<forall>n\<ge>M. dist (X m) (X n) < e) \<Longrightarrow> Cauchy X" *)
 (* lemma (in metric_space) CauchyI':
   "(\<And>e. 0 < e \<Longrightarrow> \<exists>M. \<forall>m\<ge>M. \<forall>n>m. dist (X m) (X n) < e) \<Longrightarrow> Cauchy X" *)
 (* lemma (in metric_space) metric_CauchyD:
   "Cauchy X \<Longrightarrow> 0 < e \<Longrightarrow> \<exists>M. \<forall>m\<ge>M. \<forall>n\<ge>M. dist (X m) (X n) < e" *)
 (* lemma (in metric_space) metric_Cauchy_iff2:
   "Cauchy X = (\<forall>j. (\<exists>M. \<forall>m \<ge> M. \<forall>n \<ge> M. dist (X m) (X n) < inverse(real (Suc j))))" *)
 
 lemma cCauchy_iff2: "Cauchy X \<longleftrightarrow> (\<forall>j. (\<exists>M. \<forall>m \<ge> M. \<forall>n \<ge> M. cmod (X m - X n) < inverse (real (Suc j))))"
   by (simp only: metric_Cauchy_iff2 dist_complex_def)
 
 (* Not specific to real/complex *)
 (* lemma lim_1_over_n [tendsto_intros]: "((\<lambda>n. 1 / of_nat n) \<longlongrightarrow> (0::'a::complex_normed_field)) sequentially" *)
 (* lemma (in metric_space) complete_def:
   shows "complete S = (\<forall>f. (\<forall>n. f n \<in> S) \<and> Cauchy f \<longrightarrow> (\<exists>l\<in>S. f \<longlonglongrightarrow> l))" *)
 (* lemma (in metric_space) totally_bounded_metric:
   "totally_bounded S \<longleftrightarrow> (\<forall>e>0. \<exists>k. finite k \<and> S \<subseteq> (\<Union>x\<in>k. {y. dist x y < e}))" *)
 
 (* subsubsection \<open>Cauchy Sequences are Convergent\<close> *)
 (* Everything is commented out, so we comment out the heading, too. *)
 
 (* Not specific to real/complex *)
 (* class complete_space *)
 (* lemma Cauchy_convergent_iff: "Cauchy X \<longleftrightarrow> convergent X"
   for X :: "nat \<Rightarrow> 'a::complete_space" *)
 
 (* text \<open>To prove that a Cauchy sequence converges, it suffices to show that a subsequence converges.\<close> *)
 
 (* Not specific to real/complex *)
 (* lemma Cauchy_converges_subseq:
   fixes u::"nat \<Rightarrow> 'a::metric_space"
   assumes "Cauchy u"
     "strict_mono r"
     "(u \<circ> r) \<longlonglongrightarrow> l"
   shows "u \<longlonglongrightarrow> l" *)
 
-subsection \<open>The set of real numbers is a complete metric space\<close>
+subsection \<open>The set of complex numbers is a complete metric space\<close>
 
 text \<open>
   Proof that Cauchy sequences converge based on the one from
   \<^url>\<open>http://pirate.shu.edu/~wachsmut/ira/numseq/proofs/cauconv.html\<close>
 \<close>
 
 text \<open>
   If sequence \<^term>\<open>X\<close> is Cauchy, then its limit is the lub of
   \<^term>\<open>{r::real. \<exists>N. \<forall>n\<ge>N. r < X n}\<close>
 \<close>
 
 lemma complex_increasing_LIMSEQ:
   fixes f :: "nat \<Rightarrow> complex"
   assumes inc: "\<And>n. f n \<le> f (Suc n)"
     and bdd: "\<And>n. f n \<le> l"
     and en: "\<And>e. 0 < e \<Longrightarrow> \<exists>n. l \<le> f n + e"
   shows "f \<longlonglongrightarrow> l"
 proof -
   have \<open>(\<lambda>n. Re (f n)) \<longlonglongrightarrow> Re l\<close>
     apply (rule increasing_LIMSEQ)
     using assms apply (auto simp: less_eq_complex_def less_complex_def)
     by (metis Im_complex_of_real Re_complex_of_real)
   moreover have \<open>Im (f n) = Im l\<close> for n
     using bdd by (auto simp: less_eq_complex_def)
   then have \<open>(\<lambda>n. Im (f n)) \<longlonglongrightarrow> Im l\<close>
     by auto
   ultimately show \<open>f \<longlonglongrightarrow> l\<close>
     by (simp add: tendsto_complex_iff)
 qed
 
 lemma complex_Cauchy_convergent:
   fixes X :: "nat \<Rightarrow> complex"
   assumes X: "Cauchy X"
   shows "convergent X"
   using assms by (rule Cauchy_convergent)
 
 instance complex :: complete_space
   by intro_classes (rule complex_Cauchy_convergent)
 
 class cbanach = complex_normed_vector + complete_space
 
 (* Not present in Real_Vector_Spaces *)
 subclass (in cbanach) banach ..
 
 instance complex :: banach ..
 
 (* Don't know if this holds in the complex case *)
 (* lemma tendsto_at_topI_sequentially:
   fixes f :: "complex \<Rightarrow> 'b::first_countable_topology"
   assumes *: "\<And>X. filterlim X at_top sequentially \<Longrightarrow> (\<lambda>n. f (X n)) \<longlonglongrightarrow> y"
   shows "(f \<longlongrightarrow> y) at_top" *)
 (* lemma tendsto_at_topI_sequentially_real:
   fixes f :: "real \<Rightarrow> real"
   assumes mono: "mono f"
     and limseq: "(\<lambda>n. f (real n)) \<longlonglongrightarrow> y"
   shows "(f \<longlongrightarrow> y) at_top" *)
 
 end
diff --git a/thys/Complex_Bounded_Operators/One_Dimensional_Spaces.thy b/thys/Complex_Bounded_Operators/One_Dimensional_Spaces.thy
--- a/thys/Complex_Bounded_Operators/One_Dimensional_Spaces.thy
+++ b/thys/Complex_Bounded_Operators/One_Dimensional_Spaces.thy
@@ -1,287 +1,318 @@
 section \<open>\<open>One_Dimensional_Spaces\<close> -- One dimensional complex vector spaces\<close>
 
 theory One_Dimensional_Spaces
   imports
     Complex_Inner_Product
     "Complex_Bounded_Operators.Extra_Operator_Norm"
 begin
 
 text \<open>The class \<open>one_dim\<close> applies to one-dimensional vector spaces.
      Those are additionally interpreted as \<^class>\<open>complex_algebra_1\<close>s 
      via the canonical isomorphism between a one-dimensional vector space and 
      \<^typ>\<open>complex\<close>.\<close>
-class one_dim = onb_enum + one + times + complex_inner + inverse +
+class one_dim = onb_enum + one + times + inverse +
   assumes one_dim_canonical_basis[simp]: "canonical_basis = [1]"
-  assumes one_dim_prod_scale1: "(a *\<^sub>C 1) * (b *\<^sub>C 1) = (a*b) *\<^sub>C 1"
+  assumes one_dim_prod_scale1: "(a *\<^sub>C 1) * (b *\<^sub>C 1) = (a * b) *\<^sub>C 1"
   assumes divide_inverse: "x / y = x * inverse y"
   assumes one_dim_inverse: "inverse (a *\<^sub>C 1) = inverse a *\<^sub>C 1"
 
-hide_fact (open) divide_inverse (* divide_inverse from field_class, instantiated below, subsumed this one *)
+hide_fact (open) divide_inverse
+  \<comment> \<open>@{thm [source] divide_inverse} from class \<^class>\<open>field\<close>, instantiated below, subsumes this fact.\<close>
 
 instance complex :: one_dim
   apply intro_classes
   unfolding canonical_basis_complex_def is_ortho_set_def
   by (auto simp: divide_complex_def)
 
-lemma one_cinner_one[simp]: \<open>\<langle>(1::('a::one_dim)), 1\<rangle> = 1\<close>
+lemma one_cinner_one[simp]: \<open>(1::('a::one_dim)) \<bullet>\<^sub>C 1 = 1\<close>
 proof-
   include notation_norm
   have \<open>(canonical_basis::'a list) = [1::('a::one_dim)]\<close>
-    by (simp add: one_dim_canonical_basis)    
+    by simp
   hence \<open>\<parallel>1::'a::one_dim\<parallel> = 1\<close>
     by (metis is_normal list.set_intros(1))
   hence \<open>\<parallel>1::'a::one_dim\<parallel>^2 = 1\<close>
     by simp
-  moreover have  \<open>\<parallel>(1::('a::one_dim))\<parallel>^2 = \<langle>(1::('a::one_dim)), 1\<rangle>\<close>
+  moreover have  \<open>\<parallel>(1::('a::one_dim))\<parallel>^2 = (1::('a::one_dim)) \<bullet>\<^sub>C 1\<close>
     by (metis cnorm_eq_square)
   ultimately show ?thesis by simp
 qed
 
-lemma one_cinner_a_scaleC_one[simp]: \<open>\<langle>1::('a::one_dim), a\<rangle> *\<^sub>C 1 = a\<close>
+lemma one_cinner_a_scaleC_one[simp]: \<open>((1::'a::one_dim) \<bullet>\<^sub>C a) *\<^sub>C 1 = a\<close>
 proof-
   have \<open>(canonical_basis::'a list) = [1]\<close>
-    by (simp add: one_dim_canonical_basis)
+    by simp
   hence r2: \<open>a \<in> complex_vector.span ({1::'a})\<close>        
     using  iso_tuple_UNIV_I empty_set is_generator_set list.simps(15)
     by metis
   have "(1::'a) \<notin> {}"
     by (metis equals0D)
   hence r1: \<open>\<exists> s. a = s *\<^sub>C 1\<close>
     by (metis Diff_insert_absorb r2 complex_vector.span_breakdown 
         complex_vector.span_empty eq_iff_diff_eq_0 singleton_iff)
   then obtain s where s_def: \<open>a = s *\<^sub>C 1\<close>
     by blast
-  have  \<open>\<langle>(1::'a), a\<rangle> = \<langle>(1::'a), s *\<^sub>C 1\<rangle>\<close>
+  have  \<open>(1::'a) \<bullet>\<^sub>C a = (1::'a) \<bullet>\<^sub>C (s *\<^sub>C 1)\<close>
     using \<open>a = s *\<^sub>C 1\<close>
     by simp 
-  also have \<open>\<dots> = s * \<langle>(1::'a), 1\<rangle>\<close>
+  also have \<open>\<dots> = s * ((1::'a) \<bullet>\<^sub>C 1)\<close>
     by simp
   also have \<open>\<dots> = s\<close>
     using one_cinner_one by auto
   finally show ?thesis
     by (simp add: s_def) 
 qed
 
 lemma one_dim_apply_is_times_def:
-  "\<psi> * \<phi> = (\<langle>1, \<psi>\<rangle> * \<langle>1, \<phi>\<rangle>) *\<^sub>C 1" for \<psi> :: \<open>'a::one_dim\<close>
+  "\<psi> * \<phi> = ((1 \<bullet>\<^sub>C \<psi>) * (1 \<bullet>\<^sub>C \<phi>)) *\<^sub>C 1" for \<psi> :: \<open>'a::one_dim\<close>
   by (metis one_cinner_a_scaleC_one one_dim_prod_scale1)
 
 instance one_dim \<subseteq> complex_algebra_1
 proof
   fix x y z :: \<open>'a::one_dim\<close> and c :: complex
   show "(x * y) * z = x * (y * z)"
     by (simp add: one_dim_apply_is_times_def[where ?'a='a])
   show "(x + y) * z = x * z + y * z"
     by (metis (no_types, lifting) cinner_simps(2) complex_vector.vector_space_assms(2) complex_vector.vector_space_assms(3) one_dim_apply_is_times_def)
   show "x * (y + z) = x * y + x * z"
     by (metis (mono_tags, lifting) cinner_simps(2) complex_vector.vector_space_assms(2) distrib_left one_dim_apply_is_times_def)
   show "(c *\<^sub>C x) * y = c *\<^sub>C (x * y)"
     by (simp add: one_dim_apply_is_times_def[where ?'a='a])
   show "x * (c *\<^sub>C y) = c *\<^sub>C (x * y)"
     by (simp add: one_dim_apply_is_times_def[where ?'a='a])
   show "1 * x = x"
     by (metis mult.left_neutral one_cinner_a_scaleC_one one_cinner_one one_dim_apply_is_times_def)
   show "x * 1 = x"
     by (simp add: one_dim_apply_is_times_def [where ?'a = 'a])
   show "(0::'a) \<noteq> 1"
     by (metis cinner_eq_zero_iff one_cinner_one zero_neq_one)
 qed
 
 instance one_dim \<subseteq> complex_normed_algebra
 proof
   fix x y :: \<open>'a::one_dim\<close>
   show "norm (x * y) \<le> norm x * norm y"
   proof-
-    have r1:  "cmod (\<langle>1::'a, x\<rangle>) \<le> norm (1::'a) * norm x"
+    have r1:  "cmod ((1::'a) \<bullet>\<^sub>C x) \<le> norm (1::'a) * norm x"
       by (simp add: complex_inner_class.Cauchy_Schwarz_ineq2)
-    have r2: "cmod (\<langle>1::'a, y\<rangle>) \<le> norm (1::'a) * norm y"
+    have r2: "cmod ((1::'a) \<bullet>\<^sub>C y) \<le> norm (1::'a) * norm y"
       by (simp add: complex_inner_class.Cauchy_Schwarz_ineq2)
 
-    have q1: "\<langle>(1::'a), 1\<rangle> = 1"
+    have q1: "(1::'a) \<bullet>\<^sub>C 1 = 1"
       by simp
     hence "(norm (1::'a))^2 = 1"
       by (simp add: cnorm_eq_1 power2_eq_1_iff)
     hence "norm (1::'a) = 1"
       by (smt abs_norm_cancel power2_eq_1_iff)
-    hence "cmod (\<langle>1::'a, x\<rangle> * \<langle>1::'a, y\<rangle>) * norm (1::'a) = cmod (\<langle>1::'a, x\<rangle> * \<langle>1::'a, y\<rangle>)"
+    hence "cmod (((1::'a) \<bullet>\<^sub>C x) * ((1::'a) \<bullet>\<^sub>C y)) * norm (1::'a) = cmod (((1::'a) \<bullet>\<^sub>C x) * ((1::'a) \<bullet>\<^sub>C y))"
       by simp
-    also have "\<dots> = cmod (\<langle>1::'a, x\<rangle>) * cmod (\<langle>1::'a, y\<rangle>)"
+    also have "\<dots> = cmod (((1::'a) \<bullet>\<^sub>C x)) * cmod (((1::'a) \<bullet>\<^sub>C y))"
       by (simp add: norm_mult)
     also have "\<dots> \<le> norm (1::'a) * norm x * norm (1::'a) * norm y"
       by (smt \<open>norm 1 = 1\<close> mult.commute mult_cancel_right1 norm_scaleC one_cinner_a_scaleC_one)
     also have "\<dots> = norm x * norm y"
       by (simp add: \<open>norm 1 = 1\<close>)
     finally show ?thesis
       by (simp add: one_dim_apply_is_times_def[where ?'a='a])
   qed
 qed
 
 instance one_dim \<subseteq> complex_normed_algebra_1
 proof intro_classes
   show "norm (1::'a) = 1"
     by (metis complex_inner_1_left norm_eq_sqrt_cinner norm_one one_cinner_one)
 qed
 
 
 text \<open>This is the canonical isomorphism between any two one dimensional spaces. Specifically,
   if 1 denotes the element of the canonical basis (which is specified by type class \<^class>\<open>basis_enum\<close>),
   then \<^term>\<open>one_dim_iso\<close> is the unique isomorphism that maps 1 to 1.\<close>
 definition one_dim_iso :: "'a::one_dim \<Rightarrow> 'b::one_dim"
-  where "one_dim_iso a = of_complex (\<langle>1, a\<rangle>)"
+  where "one_dim_iso a = of_complex (1 \<bullet>\<^sub>C a)"
 
 lemma one_dim_iso_idem[simp]: "one_dim_iso (one_dim_iso x) = one_dim_iso x"
   by (simp add: one_dim_iso_def)
 
 lemma one_dim_iso_id[simp]: "one_dim_iso = id"
   unfolding one_dim_iso_def
   by (auto simp add: of_complex_def)
 
 lemma one_dim_iso_adjoint[simp]: \<open>cadjoint one_dim_iso = one_dim_iso\<close>
   apply (rule cadjoint_eqI)
   by (simp add: one_dim_iso_def of_complex_def)
 
 lemma one_dim_iso_is_of_complex[simp]: "one_dim_iso = of_complex"
   unfolding one_dim_iso_def by auto
 
 lemma of_complex_one_dim_iso[simp]: "of_complex (one_dim_iso \<psi>) = one_dim_iso \<psi>"
   by (metis one_dim_iso_is_of_complex one_dim_iso_idem)
 
 lemma one_dim_iso_of_complex[simp]: "one_dim_iso (of_complex c) = of_complex c"
   by (metis one_dim_iso_is_of_complex one_dim_iso_idem)
 
 lemma one_dim_iso_add[simp]:
   \<open>one_dim_iso (a + b) = one_dim_iso a + one_dim_iso b\<close>
   by (simp add: cinner_simps(2) one_dim_iso_def)
 
 lemma one_dim_iso_minus[simp]:
   \<open>one_dim_iso (a - b) = one_dim_iso a - one_dim_iso b\<close>
   by (simp add: cinner_simps(3) one_dim_iso_def)
 
 lemma one_dim_iso_scaleC[simp]: "one_dim_iso (c *\<^sub>C \<psi>) = c *\<^sub>C one_dim_iso \<psi>"
   by (metis cinner_scaleC_right of_complex_mult one_dim_iso_def scaleC_conv_of_complex)
 
 lemma clinear_one_dim_iso[simp]: "clinear one_dim_iso"
   by (rule clinearI, auto)
 
 lemma bounded_clinear_one_dim_iso[simp]: "bounded_clinear one_dim_iso"
 proof (rule bounded_clinear_intro [where K = 1] , auto)
   fix x :: \<open>'a::one_dim\<close>
   show "norm (one_dim_iso x) \<le> norm x"
     by (metis (full_types) norm_of_complex of_complex_def one_cinner_a_scaleC_one one_dim_iso_def 
         order_refl)
 qed
 
 lemma one_dim_iso_of_one[simp]: "one_dim_iso 1 = 1"
   by (simp add: one_dim_iso_def)
 
 lemma onorm_one_dim_iso[simp]: "onorm one_dim_iso = 1"
 proof (rule onormI [where b = 1 and x = 1])
   fix x :: \<open>'a::one_dim\<close>
   have "norm (one_dim_iso x ::'b) \<le> norm x"
     by (metis eq_iff norm_of_complex of_complex_def one_cinner_a_scaleC_one one_dim_iso_def)
   thus "norm (one_dim_iso (x::'a)::'b) \<le> 1 * norm x"
     by auto
   show "(1::'a) \<noteq> 0"
     by simp
   show "norm (one_dim_iso (1::'a)::'b) = 1 * norm (1::'a)"
     by auto
 qed
 
 lemma one_dim_iso_times[simp]: "one_dim_iso (\<psi> * \<phi>) = one_dim_iso \<psi> * one_dim_iso \<phi>"
   by (metis mult.left_neutral mult_scaleC_left of_complex_def one_cinner_a_scaleC_one one_dim_iso_def one_dim_iso_scaleC)
 
 lemma one_dim_iso_of_zero[simp]: "one_dim_iso 0 = 0"
   by (simp add: complex_vector.linear_0)
 
 lemma one_dim_iso_of_zero': "one_dim_iso x = 0 \<Longrightarrow> x = 0"
   by (metis of_complex_def of_complex_eq_0_iff one_cinner_a_scaleC_one one_dim_iso_def)
 
 lemma one_dim_scaleC_1[simp]: "one_dim_iso x *\<^sub>C 1 = x"
   by (simp add: one_dim_iso_def)
 
 lemma one_dim_clinear_eqI: 
   assumes "(x::'a::one_dim) \<noteq> 0" and "clinear f" and "clinear g" and "f x = g x"
   shows "f = g"
 proof (rule ext)
   fix y :: 'a
   from \<open>f x = g x\<close>
   have \<open>one_dim_iso x *\<^sub>C f 1 = one_dim_iso x *\<^sub>C g 1\<close>
     by (metis assms(2) assms(3) complex_vector.linear_scale one_dim_scaleC_1)
   hence "f 1 = g 1"
     using assms(1) one_dim_iso_of_zero' by auto
   then show "f y = g y"
     by (metis assms(2) assms(3) complex_vector.linear_scale one_dim_scaleC_1)
 qed
 
 lemma one_dim_norm: "norm x = cmod (one_dim_iso x)"
 proof (subst one_dim_scaleC_1 [symmetric])
   show "norm (one_dim_iso x *\<^sub>C (1::'a)) = cmod (one_dim_iso x)"
     by (metis norm_of_complex of_complex_def)    
 qed
 
 lemma one_dim_onorm:
   fixes f :: "'a::one_dim \<Rightarrow> 'b::complex_normed_vector"
   assumes "clinear f"
   shows "onorm f = norm (f 1)"
 proof (rule onormI[where x=1])
   fix x :: 'a
   have "norm x * norm (f 1) \<le> norm (f 1) * norm x"
     by simp    
   hence "norm (f (one_dim_iso x *\<^sub>C 1)) \<le> norm (f 1) * norm x"
     by (metis (mono_tags, lifting) assms complex_vector.linear_scale norm_scaleC one_dim_norm)
   thus "norm (f x) \<le> norm (f 1) * norm x"
     by (subst one_dim_scaleC_1 [symmetric]) 
 qed auto
 
 lemma one_dim_onorm':
   fixes f :: "'a::one_dim \<Rightarrow> 'b::one_dim"
   assumes "clinear f"
   shows "onorm f = cmod (one_dim_iso (f 1))"
   using assms one_dim_norm one_dim_onorm by fastforce
 
 instance one_dim \<subseteq> zero_neq_one ..
 
 lemma one_dim_iso_inj: "one_dim_iso x = one_dim_iso y \<Longrightarrow> x = y"
   by (metis one_dim_iso_idem one_dim_scaleC_1)
 
 instance one_dim \<subseteq> comm_ring
 proof intro_classes
   fix x y z :: 'a
   show "x * y = y * x"
     by (metis one_dim_apply_is_times_def ordered_field_class.sign_simps(5))
   show "(x + y) * z = x * z + y * z"
     by (simp add: ring_class.ring_distribs(2))
 qed
 
 instance one_dim \<subseteq> field
 proof intro_classes
   fix x y z :: \<open>'a::one_dim\<close>
   show "1 * x = x"
     by simp
 
-  have "(inverse \<langle>1, x\<rangle> * \<langle>1, x\<rangle>) *\<^sub>C (1::'a) = 1" if "x \<noteq> 0"
+  have "(inverse ((1::'a) \<bullet>\<^sub>C x) * ((1::'a) \<bullet>\<^sub>C x)) *\<^sub>C (1::'a) = 1" if "x \<noteq> 0"
     by (metis left_inverse of_complex_def one_cinner_a_scaleC_one one_dim_iso_of_zero 
         one_dim_iso_is_of_complex one_dim_iso_of_one that)
-  hence "inverse (\<langle>1, x\<rangle> *\<^sub>C 1) * \<langle>1, x\<rangle> *\<^sub>C 1 = (1::'a)" if "x \<noteq> 0"
+  hence "inverse (((1::'a) \<bullet>\<^sub>C x) *\<^sub>C 1) * ((1::'a) \<bullet>\<^sub>C x) *\<^sub>C 1 = (1::'a)" if "x \<noteq> 0"
     by (metis one_dim_inverse one_dim_prod_scale1 that)    
-  hence "inverse (\<langle>1, x\<rangle> *\<^sub>C 1) * x = 1" if "x \<noteq> 0"
+  hence "inverse (((1::'a) \<bullet>\<^sub>C x) *\<^sub>C 1) * x = 1" if "x \<noteq> 0"
     using one_cinner_a_scaleC_one[of x, symmetric] that by auto
   thus "inverse x * x = 1" if "x \<noteq> 0"
     by (simp add: that)    
   show "x / y = x * inverse y"
     by (simp add: one_dim_class.divide_inverse)
   show "inverse (0::'a) = 0"
     by (subst complex_vector.scale_zero_left[symmetric], subst one_dim_inverse, simp)
 qed
 
 
 instance one_dim \<subseteq> complex_normed_field
 proof intro_classes
   fix x y :: 'a
   show "norm (x * y) = norm x * norm y"
     by (metis norm_mult one_dim_iso_times one_dim_norm)
 qed
 
 instance one_dim \<subseteq> chilbert_space..
 
+lemma ccspan_one_dim[simp]: \<open>ccspan {x} = top\<close> if \<open>x \<noteq> 0\<close> for x :: \<open>_ :: one_dim\<close>
+proof -
+  have \<open>y \<in> cspan {x}\<close> for y
+    using that by (metis complex_vector.span_base complex_vector.span_zero cspan_singleton_scaleC insertI1 one_dim_scaleC_1 scaleC_zero_left)
+  then show ?thesis
+    by (auto intro!: order.antisym ccsubspace_leI
+      simp: top_ccsubspace.rep_eq ccspan.rep_eq)
+qed
+
+lemma one_dim_ccsubspace_all_or_nothing: \<open>A = bot \<or> A = top\<close> for A :: \<open>_::one_dim ccsubspace\<close>
+proof (rule Meson.disj_comm, rule disjCI)
+  assume \<open>A \<noteq> bot\<close>
+  then obtain \<psi> where \<open>\<psi> \<in> space_as_set A\<close> and \<open>\<psi> \<noteq> 0\<close>
+    by (metis ccsubspace_eqI singleton_iff space_as_set_bot zero_space_as_set)
+  then have \<open>A \<ge> ccspan {\<psi>}\<close> (is \<open>_ \<ge> \<dots>\<close>)
+    by (metis bot.extremum ccspan_leqI insert_absorb insert_mono)
+  also have \<open>\<dots> = ccspan {one_dim_iso \<psi> *\<^sub>C 1}\<close>
+    by auto
+  also have \<open>\<dots> = ccspan {1}\<close>
+    apply (rule ccspan_singleton_scaleC)
+    using \<open>\<psi> \<noteq> 0\<close> one_dim_iso_of_zero' by auto
+  also have \<open>\<dots> = top\<close>
+    by auto
+  finally show \<open>A = top\<close>
+    by (simp add: top.extremum_uniqueI)
+qed
+
+lemma scaleC_1_right[simp]: \<open>scaleC x (1::'a::one_dim) = of_complex x\<close>
+  unfolding of_complex_def by simp
+
 end
diff --git a/thys/Complex_Bounded_Operators/extra/Extra_General.thy b/thys/Complex_Bounded_Operators/extra/Extra_General.thy
--- a/thys/Complex_Bounded_Operators/extra/Extra_General.thy
+++ b/thys/Complex_Bounded_Operators/extra/Extra_General.thy
@@ -1,720 +1,711 @@
 section \<open>\<open>Extra_General\<close> -- General missing things\<close>
 
 theory Extra_General
   imports
     "HOL-Library.Cardinality"
     "HOL-Analysis.Elementary_Topology"
     "HOL-Analysis.Uniform_Limit"
     "HOL-Library.Set_Algebras"
     "HOL-Types_To_Sets.Types_To_Sets"
     "HOL-Library.Complex_Order"
     "HOL-Analysis.Infinite_Sum"
     "HOL-Cardinals.Cardinals"
     "HOL-Library.Complemented_Lattices"
 begin
 
 subsection \<open>Misc\<close>
 
 lemma reals_zero_comparable:
   fixes x::complex
   assumes "x\<in>\<real>"
   shows "x \<le> 0 \<or> x \<ge> 0"
   using assms unfolding complex_is_real_iff_compare0 by assumption
 
 lemma unique_choice: "\<forall>x. \<exists>!y. Q x y \<Longrightarrow> \<exists>!f. \<forall>x. Q x (f x)"
   apply (auto intro!: choice ext) by metis
 
 lemma image_set_plus: 
   assumes \<open>linear U\<close>
   shows \<open>U ` (A + B) = U ` A + U ` B\<close>
   unfolding image_def set_plus_def
   using assms by (force simp: linear_add)
 
 consts heterogenous_identity :: \<open>'a \<Rightarrow> 'b\<close>
 overloading heterogenous_identity_id \<equiv> "heterogenous_identity :: 'a \<Rightarrow> 'a" begin
 definition heterogenous_identity_def[simp]: \<open>heterogenous_identity_id = id\<close>
 end
 
 lemma L2_set_mono2:
   assumes a1: "finite L" and a2: "K \<le> L"
   shows "L2_set f K \<le> L2_set f L"
 proof-
   have "(\<Sum>i\<in>K. (f i)\<^sup>2) \<le> (\<Sum>i\<in>L. (f i)\<^sup>2)"
-  proof (rule sum_mono2)
-    show "finite L"
-      using a1.
-    show "K \<subseteq> L"
-      using a2.
-    show "0 \<le> (f b)\<^sup>2"
-      if "b \<in> L - K"
-      for b :: 'a
-      using that
-      by simp 
-  qed
+    apply (rule sum_mono2) 
+    using assms by auto
   hence "sqrt (\<Sum>i\<in>K. (f i)\<^sup>2) \<le> sqrt (\<Sum>i\<in>L. (f i)\<^sup>2)"
     by (rule real_sqrt_le_mono)
   thus ?thesis
     unfolding L2_set_def.
 qed
 
 lemma Sup_real_close:
   fixes e :: real
   assumes "0 < e"
     and S: "bdd_above S" "S \<noteq> {}"
   shows "\<exists>x\<in>S. Sup S - e < x"
 proof -
   have \<open>Sup (ereal ` S) \<noteq> \<infinity>\<close>
     by (metis assms(2) bdd_above_def ereal_less_eq(3) less_SUP_iff less_ereal.simps(4) not_le)
   moreover have \<open>Sup (ereal ` S) \<noteq> -\<infinity>\<close>
     by (simp add: SUP_eq_iff assms(3))
   ultimately have Sup_bdd: \<open>\<bar>Sup (ereal ` S)\<bar> \<noteq> \<infinity>\<close>
     by auto
   then have \<open>\<exists>x'\<in>ereal ` S. Sup (ereal ` S) - ereal e < x'\<close>
     apply (rule_tac Sup_ereal_close)
     using assms by auto
   then obtain x where \<open>x \<in> S\<close> and Sup_x: \<open>Sup (ereal ` S) - ereal e < ereal x\<close>
     by auto
   have \<open>Sup (ereal ` S) = ereal (Sup S)\<close>
     using Sup_bdd by (rule ereal_Sup[symmetric])
   with Sup_x have \<open>ereal (Sup S - e) < ereal x\<close>
     by auto
   then have \<open>Sup S - e < x\<close>
     by auto
   with \<open>x \<in> S\<close> show ?thesis
     by auto
 qed
 
 text \<open>Improved version of @{attribute internalize_sort}: It is not necessary to specify the sort of the type variable.\<close>
 attribute_setup internalize_sort' = \<open>let
 fun find_tvar thm v = let
   val tvars = Term.add_tvars (Thm.prop_of thm) []
   val tv = case find_first (fn (n,sort) => n=v) tvars of
               SOME tv => tv | NONE => raise THM ("Type variable " ^ string_of_indexname v ^ " not found", 0, [thm])
 in 
 TVar tv
 end
 
 fun internalize_sort_attr (tvar:indexname) =
   Thm.rule_attribute [] (fn context => fn thm =>
     (snd (Internalize_Sort.internalize_sort (Thm.ctyp_of (Context.proof_of context) (find_tvar thm tvar)) thm)));
 in
   Scan.lift Args.var >> internalize_sort_attr
 end\<close>
   "internalize a sort"
 
 lemma card_prod_omega: \<open>X *c natLeq =o X\<close> if \<open>Cinfinite X\<close>
   by (simp add: Cinfinite_Cnotzero cprod_infinite1' natLeq_Card_order natLeq_cinfinite natLeq_ordLeq_cinfinite that)
 
 lemma countable_leq_natLeq: \<open>|X| \<le>o natLeq\<close> if \<open>countable X\<close>
   using subset_range_from_nat_into[OF that]
   by (meson card_of_nat ordIso_iff_ordLeq ordLeq_transitive surj_imp_ordLeq)
 
 lemma set_Times_plus_distrib: \<open>(A \<times> B) + (C \<times> D) = (A + C) \<times> (B + D)\<close>
   by (auto simp: Sigma_def set_plus_def)
 
 subsection \<open>Not singleton\<close>
 
 class not_singleton =
   assumes not_singleton_card: "\<exists>x y. x \<noteq> y"
 
 lemma not_singleton_existence[simp]:
   \<open>\<exists> x::('a::not_singleton). x \<noteq> t\<close>
   using not_singleton_card[where ?'a = 'a] by (metis (full_types))
 
 lemma UNIV_not_singleton[simp]: "(UNIV::_::not_singleton set) \<noteq> {x}"
   using not_singleton_existence[of x] by blast
 
 lemma UNIV_not_singleton_converse: 
   assumes"\<And>x::'a. UNIV \<noteq> {x}"
   shows "\<exists>x::'a. \<exists>y. x \<noteq> y"
   using assms
   by fastforce 
 
 subclass (in card2) not_singleton
   apply standard using two_le_card
   by (meson card_2_iff' obtain_subset_with_card_n)
 
 subclass (in perfect_space) not_singleton
   apply intro_classes
   by (metis (mono_tags) Collect_cong Collect_mem_eq UNIV_I local.UNIV_not_singleton local.not_open_singleton local.open_subopen)
 
 lemma class_not_singletonI_monoid_add:
   assumes "(UNIV::'a set) \<noteq> {0}"
   shows "class.not_singleton TYPE('a::monoid_add)"
 proof intro_classes
   let ?univ = "UNIV :: 'a set"
   from assms obtain x::'a where "x \<noteq> 0"
     by auto
   thus "\<exists>x y :: 'a. x \<noteq> y"
     by auto
 qed
 
 lemma not_singleton_vs_CARD_1:
   assumes \<open>\<not> class.not_singleton TYPE('a)\<close>
   shows \<open>class.CARD_1 TYPE('a)\<close>
   using assms unfolding class.not_singleton_def class.CARD_1_def
   by (metis (full_types) One_nat_def UNIV_I card.empty card.insert empty_iff equalityI finite.intros(1) insert_iff subsetI)
 
 subsection \<open>\<^class>\<open>CARD_1\<close>\<close>
 
 context CARD_1 begin
 
 lemma everything_the_same[simp]: "(x::'a)=y"
   by (metis (full_types) UNIV_I card_1_singletonE empty_iff insert_iff local.CARD_1)
 
 lemma CARD_1_UNIV: "UNIV = {x::'a}"
   by (metis (full_types) UNIV_I card_1_singletonE local.CARD_1 singletonD)
 
 lemma CARD_1_ext: "x (a::'a) = y b \<Longrightarrow> x = y"
 proof (rule ext)
   show "x t = y t"
     if "x a = y b"
     for t :: 'a
     using that  apply (subst (asm) everything_the_same[where x=a])
     apply (subst (asm) everything_the_same[where x=b])
     by simp
 qed 
 
 end
 
 instance unit :: CARD_1
   apply standard by auto
 
 instance prod :: (CARD_1, CARD_1) CARD_1
   apply intro_classes
   by (simp add: CARD_1)
 
 instance "fun" :: (CARD_1, CARD_1) CARD_1
   apply intro_classes
   by (auto simp add: card_fun CARD_1)
 
 
 lemma enum_CARD_1: "(Enum.enum :: 'a::{CARD_1,enum} list) = [a]"
 proof -
   let ?enum = "Enum.enum :: 'a::{CARD_1,enum} list"
   have "length ?enum = 1"
     apply (subst card_UNIV_length_enum[symmetric])
     by (rule CARD_1)
   then obtain b where "?enum = [b]"
     apply atomize_elim
     apply (cases ?enum, auto)
     by (metis length_0_conv length_Cons nat.inject)
   thus "?enum = [a]"
     by (subst everything_the_same[of _ b], simp)
 qed
 
 lemma card_not_singleton: \<open>CARD('a::not_singleton) \<noteq> 1\<close>
   by (simp add: card_1_singleton_iff)
 
 
 subsection \<open>Topology\<close>
 
 lemma cauchy_filter_metricI:
   fixes F :: "'a::metric_space filter"
   assumes "\<And>e. e>0 \<Longrightarrow> \<exists>P. eventually P F \<and> (\<forall>x y. P x \<and> P y \<longrightarrow> dist x y < e)"
   shows "cauchy_filter F"
 proof (unfold cauchy_filter_def le_filter_def, auto)
   fix P :: "'a \<times> 'a \<Rightarrow> bool"
   assume "eventually P uniformity"
   then obtain e where e: "e > 0" and P: "dist x y < e \<Longrightarrow> P (x, y)" for x y
     unfolding eventually_uniformity_metric by auto
 
   obtain P' where evP': "eventually P' F" and P'_dist: "P' x \<and> P' y \<Longrightarrow> dist x y < e" for x y
     apply atomize_elim using assms e by auto
 
   from evP' P'_dist P
   show "eventually P (F \<times>\<^sub>F F)"
     unfolding eventually_uniformity_metric eventually_prod_filter eventually_filtermap by metis
 qed
 
 lemma cauchy_filter_metric_filtermapI:
   fixes F :: "'a filter" and f :: "'a\<Rightarrow>'b::metric_space"
   assumes "\<And>e. e>0 \<Longrightarrow> \<exists>P. eventually P F \<and> (\<forall>x y. P x \<and> P y \<longrightarrow> dist (f x) (f y) < e)"
   shows "cauchy_filter (filtermap f F)"
 proof (rule cauchy_filter_metricI)
   fix e :: real assume e: "e > 0"
   with assms obtain P where evP: "eventually P F" and dist: "P x \<and> P y \<Longrightarrow> dist (f x) (f y) < e" for x y
     by atomize_elim auto
   define P' where "P' y = (\<exists>x. P x \<and> y = f x)" for y
   have "eventually P' (filtermap f F)"
     unfolding eventually_filtermap P'_def 
     using evP
     by (smt eventually_mono) 
   moreover have "P' x \<and> P' y \<longrightarrow> dist x y < e" for x y
     unfolding P'_def using dist by metis
   ultimately show "\<exists>P. eventually P (filtermap f F) \<and> (\<forall>x y. P x \<and> P y \<longrightarrow> dist x y < e)"
     by auto
 qed
 
 
 lemma tendsto_add_const_iff:
   \<comment> \<open>This is a generalization of \<open>Limits.tendsto_add_const_iff\<close>, 
       the only difference is that the sort here is more general.\<close>
   "((\<lambda>x. c + f x :: 'a::topological_group_add) \<longlongrightarrow> c + d) F \<longleftrightarrow> (f \<longlongrightarrow> d) F"
   using tendsto_add[OF tendsto_const[of c], of f d]
     and tendsto_add[OF tendsto_const[of "-c"], of "\<lambda>x. c + f x" "c + d"] by auto
 
 lemma finite_subsets_at_top_minus: 
   assumes "A\<subseteq>B"
   shows "finite_subsets_at_top (B - A) \<le> filtermap (\<lambda>F. F - A) (finite_subsets_at_top B)"
 proof (rule filter_leI)
   fix P assume "eventually P (filtermap (\<lambda>F. F - A) (finite_subsets_at_top B))"
   then obtain X where "finite X" and "X \<subseteq> B" 
     and P: "finite Y \<and> X \<subseteq> Y \<and> Y \<subseteq> B \<longrightarrow> P (Y - A)" for Y
     unfolding eventually_filtermap eventually_finite_subsets_at_top by auto
 
   hence "finite (X-A)" and "X-A \<subseteq> B - A"
     by auto
   moreover have "finite Y \<and> X-A \<subseteq> Y \<and> Y \<subseteq> B - A \<longrightarrow> P Y" for Y
     using P[where Y="Y\<union>X"] \<open>finite X\<close> \<open>X \<subseteq> B\<close>
     by (metis Diff_subset Int_Diff Un_Diff finite_Un inf.orderE le_sup_iff sup.orderE sup_ge2)
   ultimately show "eventually P (finite_subsets_at_top (B - A))"
     unfolding eventually_finite_subsets_at_top by meson
 qed
 
 lemma finite_subsets_at_top_inter: 
   assumes "A\<subseteq>B"
   shows "filtermap (\<lambda>F. F \<inter> A) (finite_subsets_at_top B) = finite_subsets_at_top A"
 proof (subst filter_eq_iff, intro allI iffI)
   fix P :: "'a set \<Rightarrow> bool"
   assume "eventually P (finite_subsets_at_top A)"
   then show "eventually P (filtermap (\<lambda>F. F \<inter> A) (finite_subsets_at_top B))"
     unfolding eventually_filtermap
     unfolding eventually_finite_subsets_at_top
     by (metis Int_subset_iff assms finite_Int inf_le2 subset_trans)
 next
   fix P :: "'a set \<Rightarrow> bool"
   assume "eventually P (filtermap (\<lambda>F. F \<inter> A) (finite_subsets_at_top B))"
   then obtain X where \<open>finite X\<close> \<open>X \<subseteq> B\<close> and P: \<open>finite Y \<Longrightarrow> X \<subseteq> Y \<Longrightarrow> Y \<subseteq> B \<Longrightarrow> P (Y \<inter> A)\<close> for Y
     unfolding eventually_filtermap eventually_finite_subsets_at_top by metis
   have *: \<open>finite Y \<Longrightarrow> X \<inter> A \<subseteq> Y \<Longrightarrow> Y \<subseteq> A \<Longrightarrow> P Y\<close> for Y
     using P[where Y=\<open>Y \<union> (B-A)\<close>]
     apply (subgoal_tac \<open>(Y \<union> (B - A)) \<inter> A = Y\<close>)
     apply (smt (verit, best) Int_Un_distrib2 Int_Un_eq(4) P Un_subset_iff \<open>X \<subseteq> B\<close> \<open>finite X\<close> assms finite_UnI inf.orderE sup_ge2)
     by auto
   show "eventually P (finite_subsets_at_top A)"
     unfolding eventually_finite_subsets_at_top
     apply (rule exI[of _ \<open>X\<inter>A\<close>])
     by (auto simp: \<open>finite X\<close> intro!: *)
 qed
 
 lemma tendsto_principal_singleton:
   shows "(f \<longlongrightarrow> f x) (principal {x})"
   unfolding tendsto_def eventually_principal by simp
 
 lemma complete_singleton: 
   "complete {s::'a::uniform_space}"
 proof-
   have "F \<le> principal {s} \<Longrightarrow>
          F \<noteq> bot \<Longrightarrow> cauchy_filter F \<Longrightarrow> F \<le> nhds s" for F
     by (metis eventually_nhds eventually_principal le_filter_def singletonD)
   thus ?thesis
     unfolding complete_uniform
     by simp
 qed
 
 lemma on_closure_eqI:
   fixes f g :: \<open>'a::topological_space \<Rightarrow> 'b::t2_space\<close>
   assumes eq: \<open>\<And>x. x \<in> S \<Longrightarrow> f x = g x\<close>
   assumes xS: \<open>x \<in> closure S\<close>
   assumes cont: \<open>continuous_on UNIV f\<close> \<open>continuous_on UNIV g\<close>
   shows \<open>f x = g x\<close>
 proof -
   define X where \<open>X = {x. f x = g x}\<close>
   have \<open>closed X\<close>
     using cont by (simp add: X_def closed_Collect_eq)
   moreover have \<open>S \<subseteq> X\<close>
     by (simp add: X_def eq subsetI)
   ultimately have \<open>closure S \<subseteq> X\<close>
     using closure_minimal by blast
   with xS have \<open>x \<in> X\<close>
     by auto
   then show ?thesis
     using X_def by blast
 qed
 
 lemma on_closure_leI:
   fixes f g :: \<open>'a::topological_space \<Rightarrow> 'b::linorder_topology\<close>
   assumes eq: \<open>\<And>x. x \<in> S \<Longrightarrow> f x \<le> g x\<close>
   assumes xS: \<open>x \<in> closure S\<close>
   assumes cont: \<open>continuous_on UNIV f\<close> \<open>continuous_on UNIV g\<close> (* Is "isCont f x" "isCont g x" sufficient? *)
   shows \<open>f x \<le> g x\<close>
 proof -
   define X where \<open>X = {x. f x \<le> g x}\<close>
   have \<open>closed X\<close>
     using cont by (simp add: X_def closed_Collect_le)
   moreover have \<open>S \<subseteq> X\<close>
     by (simp add: X_def eq subsetI)
   ultimately have \<open>closure S \<subseteq> X\<close>
     using closure_minimal by blast
   with xS have \<open>x \<in> X\<close>
     by auto
   then show ?thesis
     using X_def by blast
 qed
 
 
 lemma tendsto_compose_at_within:
   assumes f: "(f \<longlongrightarrow> y) F" and g: "(g \<longlongrightarrow> z) (at y within S)" 
     and fg: "eventually (\<lambda>w. f w = y \<longrightarrow> g y = z) F"
     and fS: \<open>\<forall>\<^sub>F w in F. f w \<in> S\<close>
   shows "((g \<circ> f) \<longlongrightarrow> z) F"
 proof (cases \<open>g y = z\<close>)
   case False
   then have 1: "(\<forall>\<^sub>F a in F. f a \<noteq> y)"
     using fg by force
   have 2: "(g \<longlongrightarrow> z) (filtermap f F) \<or> \<not> (\<forall>\<^sub>F a in F. f a \<noteq> y)"
     by (smt (verit, best) eventually_elim2 f fS filterlim_at filterlim_def g tendsto_mono)
   show ?thesis
     using "1" "2" tendsto_compose_filtermap by blast
 next
   case True
   have *: ?thesis if \<open>(g \<longlongrightarrow> z) (filtermap f F)\<close>
     using that by (simp add: tendsto_compose_filtermap)
   from g
   have \<open>(g \<longlongrightarrow> g y) (inf (nhds y) (principal (S-{y})))\<close>
     by (simp add: True at_within_def)
   then have g': \<open>(g \<longlongrightarrow> g y) (inf (nhds y) (principal S))\<close>
     using True g tendsto_at_iff_tendsto_nhds_within by blast
   from f have \<open>filterlim f (nhds y) F\<close>
     by -
   then have f': \<open>filterlim f (inf (nhds y) (principal S)) F\<close>
     using fS
     by (simp add: filterlim_inf filterlim_principal)
   from f' g' show ?thesis
     by (simp add: * True filterlim_compose filterlim_filtermap)
 qed
 
 subsection \<open>Sums\<close>
 
 lemma sum_single: 
   assumes "finite A"
   assumes "\<And>j. j \<noteq> i \<Longrightarrow> j\<in>A \<Longrightarrow> f j = 0"
   shows "sum f A = (if i\<in>A then f i else 0)"
   apply (subst sum.mono_neutral_cong_right[where S=\<open>A \<inter> {i}\<close> and h=f])
   using assms by auto
 
 lemma has_sum_comm_additive_general: 
   \<comment> \<open>This is a strengthening of @{thm [source] has_sum_comm_additive_general}.\<close>
   fixes f :: \<open>'b :: {comm_monoid_add,topological_space} \<Rightarrow> 'c :: {comm_monoid_add,topological_space}\<close>
   assumes f_sum: \<open>\<And>F. finite F \<Longrightarrow> F \<subseteq> S \<Longrightarrow> sum (f o g) F = f (sum g F)\<close>
       \<comment> \<open>Not using \<^const>\<open>additive\<close> because it would add sort constraint \<^class>\<open>ab_group_add\<close>\<close>
   assumes inS: \<open>\<And>F. finite F \<Longrightarrow> sum g F \<in> T\<close>
   assumes cont: \<open>(f \<longlongrightarrow> f x) (at x within T)\<close>
     \<comment> \<open>For \<^class>\<open>t2_space\<close> and \<^term>\<open>T=UNIV\<close>, this is equivalent to \<open>isCont f x\<close> by @{thm [source] isCont_def}.\<close>
   assumes infsum: \<open>has_sum g S x\<close>
   shows \<open>has_sum (f o g) S (f x)\<close> 
 proof -
   have \<open>(sum g \<longlongrightarrow> x) (finite_subsets_at_top S)\<close>
     using infsum has_sum_def by blast
   then have \<open>((f o sum g) \<longlongrightarrow> f x) (finite_subsets_at_top S)\<close>
     apply (rule tendsto_compose_at_within[where S=T])
     using assms by auto
   then have \<open>(sum (f o g) \<longlongrightarrow> f x) (finite_subsets_at_top S)\<close>
     apply (rule tendsto_cong[THEN iffD1, rotated])
     using f_sum by fastforce
   then show \<open>has_sum (f o g) S (f x)\<close>
     using has_sum_def by blast 
 qed
 
 lemma summable_on_comm_additive_general:
   \<comment> \<open>This is a strengthening of @{thm [source] summable_on_comm_additive_general}.\<close>
   fixes g :: \<open>'a \<Rightarrow> 'b :: {comm_monoid_add,topological_space}\<close> and f :: \<open>'b \<Rightarrow> 'c :: {comm_monoid_add,topological_space}\<close>
   assumes \<open>\<And>F. finite F \<Longrightarrow> F \<subseteq> S \<Longrightarrow> sum (f o g) F = f (sum g F)\<close>
     \<comment> \<open>Not using \<^const>\<open>additive\<close> because it would add sort constraint \<^class>\<open>ab_group_add\<close>\<close>
   assumes inS: \<open>\<And>F. finite F \<Longrightarrow> sum g F \<in> T\<close>
   assumes cont: \<open>\<And>x. has_sum g S x \<Longrightarrow> (f \<longlongrightarrow> f x) (at x within T)\<close>
     \<comment> \<open>For \<^class>\<open>t2_space\<close> and \<^term>\<open>T=UNIV\<close>, this is equivalent to \<open>isCont f x\<close> by @{thm [source] isCont_def}.\<close>
   assumes \<open>g summable_on S\<close>
   shows \<open>(f o g) summable_on S\<close>
   by (meson assms summable_on_def has_sum_comm_additive_general has_sum_def infsum_tendsto)
 
 lemma has_sum_metric:
   fixes l :: \<open>'a :: {metric_space, comm_monoid_add}\<close>
   shows \<open>has_sum f A l \<longleftrightarrow> (\<forall>e. e > 0 \<longrightarrow> (\<exists>X. finite X \<and> X \<subseteq> A \<and> (\<forall>Y. finite Y \<and> X \<subseteq> Y \<and> Y \<subseteq> A \<longrightarrow> dist (sum f Y) l < e)))\<close>
   unfolding has_sum_def
   apply (subst tendsto_iff)
   unfolding eventually_finite_subsets_at_top
   by simp
 
 lemma summable_on_product_finite_left:
   fixes f :: \<open>'a\<times>'b \<Rightarrow> 'c::{topological_comm_monoid_add}\<close>
   assumes sum: \<open>\<And>x. x\<in>X \<Longrightarrow> (\<lambda>y. f(x,y)) summable_on Y\<close>
   assumes \<open>finite X\<close>
   shows \<open>f summable_on (X\<times>Y)\<close>
   using \<open>finite X\<close> subset_refl[of X]
 proof (induction rule: finite_subset_induct')
   case empty
   then show ?case
     by simp
 next
   case (insert x F)
   have *: \<open>bij_betw (Pair x) Y ({x} \<times> Y)\<close>
     apply (rule bij_betwI')
     by auto
   from sum[of x]
   have \<open>f summable_on {x} \<times> Y\<close>
     apply (rule summable_on_reindex_bij_betw[THEN iffD1, rotated])
     by (simp_all add: * insert.hyps(2))
   then have \<open>f summable_on {x} \<times> Y \<union> F \<times> Y\<close>
     apply (rule summable_on_Un_disjoint)
     using insert by auto
   then show ?case
     by (metis Sigma_Un_distrib1 insert_is_Un)
 qed
 
 lemma summable_on_product_finite_right:
   fixes f :: \<open>'a\<times>'b \<Rightarrow> 'c::{topological_comm_monoid_add}\<close>
   assumes sum: \<open>\<And>y. y\<in>Y \<Longrightarrow> (\<lambda>x. f(x,y)) summable_on X\<close>
   assumes \<open>finite Y\<close>
   shows \<open>f summable_on (X\<times>Y)\<close>
 proof -
   have \<open>(\<lambda>(y,x). f(x,y)) summable_on (Y\<times>X)\<close>
     apply (rule summable_on_product_finite_left)
     using assms by auto
   then show ?thesis
     apply (subst summable_on_reindex_bij_betw[where g=prod.swap and A=\<open>Y\<times>X\<close>, symmetric])
     apply (simp add: bij_betw_def product_swap)
     by (metis (mono_tags, lifting) case_prod_unfold prod.swap_def summable_on_cong)
 qed
 
 subsection \<open>Complex numbers\<close>
 
 lemma cmod_Re:
   assumes "x \<ge> 0"
   shows "cmod x = Re x"
   using assms unfolding less_eq_complex_def cmod_def
   by auto
 
 lemma abs_complex_real[simp]: "abs x \<in> \<real>" for x :: complex
   by (simp add: abs_complex_def)
 
 lemma Im_abs[simp]: "Im (abs x) = 0"
   using abs_complex_real complex_is_Real_iff by blast
 
 
 lemma cnj_x_x: "cnj x * x = (abs x)\<^sup>2"
 proof (cases x)
   show "cnj x * x = \<bar>x\<bar>\<^sup>2"
     if "x = Complex x1 x2"
     for x1 :: real
       and x2 :: real
     using that
     by (auto simp: complex_cnj complex_mult abs_complex_def
         complex_norm power2_eq_square complex_of_real_def)
 qed
 
 lemma cnj_x_x_geq0[simp]: \<open>cnj x * x \<ge> 0\<close>
   by (simp add: less_eq_complex_def)
 
 lemma complex_of_real_leq_1_iff[iff]: \<open>complex_of_real x \<le> 1 \<longleftrightarrow> x \<le> 1\<close>
   by (simp add: less_eq_complex_def)
 
 lemma x_cnj_x: \<open>x * cnj x = (abs x)\<^sup>2\<close>
   by (metis cnj_x_x mult.commute)
 
 subsection \<open>List indices and enum\<close>
 
 fun index_of where
   "index_of x [] = (0::nat)"
 | "index_of x (y#ys) = (if x=y then 0 else (index_of x ys + 1))"
 
 definition "enum_idx (x::'a::enum) = index_of x (enum_class.enum :: 'a list)"
 
 lemma index_of_length: "index_of x y \<le> length y"
   apply (induction y) by auto
 
 lemma index_of_correct:
   assumes "x \<in> set y"
   shows "y ! index_of x y = x"
   using assms apply (induction y arbitrary: x)
   by auto
 
 lemma enum_idx_correct: 
   "Enum.enum ! enum_idx i = i"
 proof-
   have "i \<in> set enum_class.enum"
     using UNIV_enum by blast 
   thus ?thesis
     unfolding enum_idx_def
     using index_of_correct by metis
 qed
 
 lemma index_of_bound: 
   assumes "y \<noteq> []" and "x \<in> set y"
   shows "index_of x y < length y"
   using assms proof(induction y arbitrary: x)
   case Nil
   thus ?case by auto
 next
   case (Cons a y)
   show ?case 
   proof(cases "a = x")
     case True
     thus ?thesis by auto
   next
     case False
     moreover have "a \<noteq> x \<Longrightarrow> index_of x y < length y"
       using Cons.IH Cons.prems(2) by fastforce      
     ultimately show ?thesis by auto
   qed
 qed
 
 lemma enum_idx_bound: "enum_idx x < length (Enum.enum :: 'a list)" for x :: "'a::enum"
 proof-
   have p1: "False"
     if "(Enum.enum :: 'a list) = []"
   proof-
     have "(UNIV::'a set) = set ([]::'a list)"
       using that UNIV_enum by metis
     also have "\<dots> = {}"
       by blast
     finally have "(UNIV::'a set) = {}".
     thus ?thesis by simp
   qed    
   have p2: "x \<in> set (Enum.enum :: 'a list)"
     using UNIV_enum by auto
   moreover have "(enum_class.enum::'a list) \<noteq> []"
     using p2 by auto
   ultimately show ?thesis
     unfolding enum_idx_def     
     using index_of_bound [where x = x and y = "(Enum.enum :: 'a list)"]
     by auto   
 qed
 
 lemma index_of_nth:
   assumes "distinct xs"
   assumes "i < length xs"
   shows "index_of (xs ! i) xs = i"
   using assms
   by (metis gr_implies_not_zero index_of_bound index_of_correct length_0_conv nth_eq_iff_index_eq nth_mem)
 
 lemma enum_idx_enum: 
   assumes \<open>i < CARD('a::enum)\<close>
   shows \<open>enum_idx (enum_class.enum ! i :: 'a) = i\<close>
   unfolding enum_idx_def apply (rule index_of_nth)
   using assms by (simp_all add: card_UNIV_length_enum enum_distinct)
 
 subsection \<open>Filtering lists/sets\<close>
 
 lemma map_filter_map: "List.map_filter f (map g l) = List.map_filter (f o g) l"
 proof (induction l)
   show "List.map_filter f (map g []) = List.map_filter (f \<circ> g) []"
     by (simp add: map_filter_simps)
   show "List.map_filter f (map g (a # l)) = List.map_filter (f \<circ> g) (a # l)"
     if "List.map_filter f (map g l) = List.map_filter (f \<circ> g) l"
     for a :: 'c
       and l :: "'c list"
     using that  map_filter_simps(1)
     by (metis comp_eq_dest_lhs list.simps(9))
 qed
 
 lemma map_filter_Some[simp]: "List.map_filter (\<lambda>x. Some (f x)) l = map f l"
 proof (induction l)
   show "List.map_filter (\<lambda>x. Some (f x)) [] = map f []"
     by (simp add: map_filter_simps)
   show "List.map_filter (\<lambda>x. Some (f x)) (a # l) = map f (a # l)"
     if "List.map_filter (\<lambda>x. Some (f x)) l = map f l"
     for a :: 'b
       and l :: "'b list"
     using that by (simp add: map_filter_simps(1))
 qed
 
 lemma filter_Un: "Set.filter f (x \<union> y) = Set.filter f x \<union> Set.filter f y"
   unfolding Set.filter_def by auto  
 
 lemma Set_filter_unchanged: "Set.filter P X = X" if "\<And>x. x\<in>X \<Longrightarrow> P x" for P and X :: "'z set"
   using that unfolding Set.filter_def by auto
 
 subsection \<open>Maps\<close>
 
 definition "inj_map \<pi> = (\<forall>x y. \<pi> x = \<pi> y \<and> \<pi> x \<noteq> None \<longrightarrow> x = y)"
 
 definition "inv_map \<pi> = (\<lambda>y. if Some y \<in> range \<pi> then Some (inv \<pi> (Some y)) else None)"
 
 lemma inj_map_total[simp]: "inj_map (Some o \<pi>) = inj \<pi>"
   unfolding inj_map_def inj_def by simp
 
 lemma inj_map_Some[simp]: "inj_map Some"
   by (simp add: inj_map_def)
 
 lemma inv_map_total: 
   assumes "surj \<pi>"
   shows "inv_map (Some o \<pi>) = Some o inv \<pi>"
 proof-
   have "(if Some y \<in> range (\<lambda>x. Some (\<pi> x))
           then Some (SOME x. Some (\<pi> x) = Some y)
           else None) =
          Some (SOME b. \<pi> b = y)"
     if "surj \<pi>"
     for y
     using that by auto
   hence  "surj \<pi> \<Longrightarrow>
     (\<lambda>y. if Some y \<in> range (\<lambda>x. Some (\<pi> x))
          then Some (SOME x. Some (\<pi> x) = Some y) else None) =
     (\<lambda>x. Some (SOME xa. \<pi> xa = x))"
     by (rule ext) 
   thus ?thesis 
     unfolding inv_map_def o_def inv_def
     using assms by linarith
 qed
 
 lemma inj_map_map_comp[simp]: 
   assumes a1: "inj_map f" and a2: "inj_map g" 
   shows "inj_map (f \<circ>\<^sub>m g)"
   using a1 a2
   unfolding inj_map_def
   by (metis (mono_tags, lifting) map_comp_def option.case_eq_if option.expand)
 
 lemma inj_map_inv_map[simp]: "inj_map (inv_map \<pi>)"
 proof (unfold inj_map_def, rule allI, rule allI, rule impI, erule conjE)
   fix x y
   assume same: "inv_map \<pi> x = inv_map \<pi> y"
     and pix_not_None: "inv_map \<pi> x \<noteq> None"
   have x_pi: "Some x \<in> range \<pi>" 
     using pix_not_None unfolding inv_map_def apply auto
     by (meson option.distinct(1))
   have y_pi: "Some y \<in> range \<pi>" 
     using pix_not_None unfolding same unfolding inv_map_def apply auto
     by (meson option.distinct(1))
   have "inv_map \<pi> x = Some (Hilbert_Choice.inv \<pi> (Some x))"
     unfolding inv_map_def using x_pi by simp
   moreover have "inv_map \<pi> y = Some (Hilbert_Choice.inv \<pi> (Some y))"
     unfolding inv_map_def using y_pi by simp
   ultimately have "Hilbert_Choice.inv \<pi> (Some x) = Hilbert_Choice.inv \<pi> (Some y)"
     using same by simp
   thus "x = y"
     by (meson inv_into_injective option.inject x_pi y_pi)
 qed
 
 subsection \<open>Lattices\<close>
 
 unbundle lattice_syntax
 
 text \<open>The following lemma is identical to @{thm [source] Complete_Lattices.uminus_Inf} 
   except for the more general sort.\<close>
 lemma uminus_Inf: "- (\<Sqinter>A) = \<Squnion>(uminus ` A)" for A :: \<open>'a::complete_orthocomplemented_lattice set\<close>
 proof (rule order.antisym)
   show "- \<Sqinter>A \<le> \<Squnion>(uminus ` A)"
     by (rule compl_le_swap2, rule Inf_greatest, rule compl_le_swap2, rule Sup_upper) simp
   show "\<Squnion>(uminus ` A) \<le> - \<Sqinter>A"
     by (rule Sup_least, rule compl_le_swap1, rule Inf_lower) auto
 qed
 
 text \<open>The following lemma is identical to @{thm [source] Complete_Lattices.uminus_INF}
   except for the more general sort.\<close>
 lemma uminus_INF: "- (INF x\<in>A. B x) = (SUP x\<in>A. - B x)" for B :: \<open>'a \<Rightarrow> 'b::complete_orthocomplemented_lattice\<close>
   by (simp add: uminus_Inf image_image)
 
 text \<open>The following lemma is identical to @{thm [source] Complete_Lattices.uminus_Sup}
   except for the more general sort.\<close>
 lemma uminus_Sup: "- (\<Squnion>A) = \<Sqinter>(uminus ` A)" for A :: \<open>'a::complete_orthocomplemented_lattice set\<close>
   by (metis (no_types, lifting) uminus_INF image_cong image_ident ortho_involution)
 
 text \<open>The following lemma is identical to @{thm [source] Complete_Lattices.uminus_SUP}
   except for the more general sort.\<close>
 lemma uminus_SUP: "- (SUP x\<in>A. B x) = (INF x\<in>A. - B x)" for B :: \<open>'a \<Rightarrow> 'b::complete_orthocomplemented_lattice\<close>
   by (simp add: uminus_Sup image_image)
 
 
 end
diff --git a/thys/Complex_Bounded_Operators/extra/Extra_Jordan_Normal_Form.thy b/thys/Complex_Bounded_Operators/extra/Extra_Jordan_Normal_Form.thy
--- a/thys/Complex_Bounded_Operators/extra/Extra_Jordan_Normal_Form.thy
+++ b/thys/Complex_Bounded_Operators/extra/Extra_Jordan_Normal_Form.thy
@@ -1,432 +1,415 @@
 section \<open>\<open>Extra_Jordan_Normal_Form\<close> -- Additional results for \<^session>\<open>Jordan_Normal_Form\<close>\<close>
 (*
 Authors: 
   Dominique Unruh, University of Tartu, unruh@ut.ee      
   Jose Manuel Rodriguez Caballero, University of Tartu, jose.manuel.rodriguez.caballero@ut.ee
 *)                 
 
 theory Extra_Jordan_Normal_Form
   imports
     Jordan_Normal_Form.Matrix Jordan_Normal_Form.Schur_Decomposition
 begin
 
 text \<open>We define bundles to activate/deactivate the notation from \<^session>\<open>Jordan_Normal_Form\<close>.
                                                                          
 Reactivate the notation locally via "@{theory_text \<open>includes jnf_notation\<close>}" in a lemma statement.
 (Or sandwich a declaration using that notation between "@{theory_text \<open>unbundle jnf_notation ... unbundle no_jnf_notation\<close>}.)
 \<close>
 
 bundle jnf_notation begin
 notation transpose_mat ("(_\<^sup>T)" [1000])
 notation cscalar_prod (infix "\<bullet>c" 70)
 notation vec_index (infixl "$" 100)
 notation smult_vec (infixl "\<cdot>\<^sub>v" 70)
 notation scalar_prod (infix "\<bullet>" 70)
 notation index_mat (infixl "$$" 100)
 notation smult_mat (infixl "\<cdot>\<^sub>m" 70)
 notation mult_mat_vec (infixl "*\<^sub>v" 70)
 notation pow_mat (infixr "^\<^sub>m" 75)
 notation append_vec (infixr "@\<^sub>v" 65)
 notation append_rows (infixr "@\<^sub>r" 65)
 end
 
 
 bundle no_jnf_notation begin
 no_notation transpose_mat ("(_\<^sup>T)" [1000])
 no_notation cscalar_prod (infix "\<bullet>c" 70)
 no_notation vec_index (infixl "$" 100)
 no_notation smult_vec (infixl "\<cdot>\<^sub>v" 70)
 no_notation scalar_prod (infix "\<bullet>" 70)
 no_notation index_mat (infixl "$$" 100)
 no_notation smult_mat (infixl "\<cdot>\<^sub>m" 70)
 no_notation mult_mat_vec (infixl "*\<^sub>v" 70)
 no_notation pow_mat (infixr "^\<^sub>m" 75)
 no_notation append_vec (infixr "@\<^sub>v" 65)
 no_notation append_rows (infixr "@\<^sub>r" 65)
 end
 
 unbundle jnf_notation
 
 
 lemma mat_entry_explicit:
   fixes M :: "'a::field mat"
   assumes "M \<in> carrier_mat m n" and "i < m" and "j < n"
   shows   "vec_index (M *\<^sub>v unit_vec n j) i = M $$ (i,j)"
   using assms by auto
 
 
 lemma mat_adjoint_def': "mat_adjoint M = transpose_mat (map_mat conjugate M)"
   apply (rule mat_eq_iff[THEN iffD2])
   apply (auto simp: mat_adjoint_def transpose_mat_def)
   apply (subst mat_of_rows_index)
   by auto
 
 lemma mat_adjoint_swap:
   fixes M ::"complex mat"
   assumes "M \<in> carrier_mat nB nA" and "iA < dim_row M" and "iB < dim_col M"
   shows "(mat_adjoint M)$$(iB,iA) = cnj (M$$(iA,iB))"
   unfolding transpose_mat_def map_mat_def
   by (simp add: assms(2) assms(3) mat_adjoint_def')
 
 lemma cscalar_prod_adjoint:
   fixes M:: "complex mat"
   assumes "M \<in> carrier_mat nB nA" 
     and "dim_vec v = nA"
     and "dim_vec u = nB"
   shows "v \<bullet>c ((mat_adjoint M) *\<^sub>v u) = (M *\<^sub>v v) \<bullet>c u"
   unfolding mat_adjoint_def using assms(1) assms(2,3)[symmetric]
   apply (simp add: scalar_prod_def sum_distrib_left field_simps)
   by (intro sum.swap)
 
 lemma scaleC_minus1_left_vec: "-1 \<cdot>\<^sub>v v = - v" for v :: "_::ring_1 vec"
   unfolding smult_vec_def uminus_vec_def by auto
 
 lemma square_nneg_complex:
   fixes x :: complex
   assumes "x \<in> \<real>" shows "x^2 \<ge> 0"
   apply (cases x) using assms unfolding Reals_def less_eq_complex_def by auto
 
 definition "vec_is_zero n v = (\<forall>i<n. v $ i = 0)"
 
 lemma vec_is_zero: "dim_vec v = n \<Longrightarrow> vec_is_zero n v \<longleftrightarrow> v = 0\<^sub>v n"
   unfolding vec_is_zero_def apply auto
   by (metis index_zero_vec(1))
 
 fun gram_schmidt_sub0
   where "gram_schmidt_sub0 n us [] = us"
   | "gram_schmidt_sub0 n us (w # ws) =
      (let w' = adjuster n w us + w in
       if vec_is_zero n w' then gram_schmidt_sub0 n us ws
                           else gram_schmidt_sub0 n (w' # us) ws)"
 
 lemma (in cof_vec_space) adjuster_already_in_span:
   assumes "w \<in> carrier_vec n"
   assumes us_carrier: "set us \<subseteq> carrier_vec n"
   assumes "corthogonal us"
   assumes "w \<in> span (set us)"
   shows "adjuster n w us + w = 0\<^sub>v n"
 proof -
   define v U where "v = adjuster n w us + w" and "U = set us"
   have span: "v \<in> span U"
     unfolding v_def U_def
     apply (rule adjust_preserves_span[THEN iffD1])
     using assms corthogonal_distinct by simp_all
   have v_carrier: "v \<in> carrier_vec n"
     by (simp add: v_def assms corthogonal_distinct)
   have "v \<bullet>c us!i = 0" if "i < length us" for i
     unfolding v_def
     apply (rule adjust_zero)
     using that assms by simp_all
   hence "v \<bullet>c u = 0" if "u \<in> U" for u
     by (metis assms(3) U_def corthogonal_distinct distinct_Ex1 that)
   hence ortho: "u \<bullet>c v = 0" if "u \<in> U" for u
     apply (subst conjugate_zero_iff[symmetric])
     apply (subst conjugate_vec_sprod_comm)
     using that us_carrier v_carrier apply (auto simp: U_def)[2]
     apply (subst conjugate_conjugate_sprod)
     using that us_carrier v_carrier by (auto simp: U_def)
   from span obtain a where v: "lincomb a U = v"
     apply atomize_elim apply (rule finite_in_span[simplified])
     unfolding U_def using us_carrier by auto
   have "v \<bullet>c v = (\<Sum>u\<in>U. (a u \<cdot>\<^sub>v u) \<bullet>c v)"
     apply (subst v[symmetric])
     unfolding lincomb_def
     apply (subst finsum_scalar_prod_sum)
     using U_def span us_carrier by auto
   also have "\<dots> = (\<Sum>u\<in>U. a u * (u \<bullet>c v))"
     using U_def assms(1) in_mono us_carrier v_def by fastforce
   also have "\<dots> = (\<Sum>u\<in>U. a u * conjugate 0)"
     apply (rule sum.cong, simp)
     using span span_closed U_def us_carrier ortho by auto
   also have "\<dots> = 0"
     by auto
   finally have "v \<bullet>c v = 0"
     by -
   thus "v = 0\<^sub>v n"
     using U_def conjugate_square_eq_0_vec span span_closed us_carrier by blast
 qed
 
 
 lemma (in cof_vec_space) gram_schmidt_sub0_result:
   assumes "gram_schmidt_sub0 n us ws = us'"
     and "set ws \<subseteq> carrier_vec n"
     and "set us \<subseteq> carrier_vec n"
     and "distinct us"
     and "~ lin_dep (set us)"
     and "corthogonal us"
   shows "set us' \<subseteq> carrier_vec n \<and>
          distinct us' \<and>
          corthogonal us' \<and>
          span (set (us @ ws)) = span (set us')"  
   using assms
 proof (induct ws arbitrary: us us')
   case (Cons w ws)
   show ?case
   proof (cases "w \<in> span (set us)")
     case False
     let ?v = "adjuster n w us"
     have wW[simp]: "set (w#ws) \<subseteq> carrier_vec n" using Cons by simp
     hence W[simp]: "set ws \<subseteq> carrier_vec n"
       and w[simp]: "w : carrier_vec n" by auto
     have U[simp]: "set us \<subseteq> carrier_vec n" using Cons by simp
     have UW: "set (us@ws) \<subseteq> carrier_vec n" by simp
     have wU: "set (w#us) \<subseteq> carrier_vec n" by simp
     have dist_U: "distinct us" using Cons by simp
     have w_U: "w \<notin> set us" using False using span_mem by auto
     have ind_U: "~ lin_dep (set us)"
       using Cons by simp
     have ind_wU: "~ lin_dep (insert w (set us))"
       apply (subst lin_dep_iff_in_span[simplified, symmetric])
       using w_U ind_U False by auto
     thm lin_dep_iff_in_span[simplified, symmetric]
     have corth: "corthogonal us" using Cons by simp
     have "?v + w \<noteq> 0\<^sub>v n"
       by (simp add: False adjust_nonzero dist_U)
     hence "\<not> vec_is_zero n (?v + w)"
       by (simp add: vec_is_zero)
     hence U'def: "gram_schmidt_sub0 n ((?v + w)#us) ws = us'" 
       using Cons by simp
     have v: "?v : carrier_vec n" using dist_U by auto
     hence vw: "?v + w : carrier_vec n" by auto
     hence vwU: "set ((?v + w) # us) \<subseteq> carrier_vec n" by auto
     have vsU: "?v : span (set us)" 
       apply (rule adjuster_in_span[OF w])
       using Cons by simp_all
     hence vsUW: "?v : span (set (us @ ws))"
       using span_is_monotone[of "set us" "set (us@ws)"] by auto
     have wsU: "w \<notin> span (set us)"
       using lin_dep_iff_in_span[OF U ind_U w w_U] ind_wU by auto
     hence vwU: "?v + w \<notin> span (set us)" using adjust_not_in_span[OF w U dist_U] by auto
 
     have span: "?v + w \<notin> span (set us)" 
       apply (subst span_add[symmetric])
       by (simp_all add: False vsU)
     hence vwUS: "?v + w \<notin> set us" using span_mem by auto
 
     have vwU: "set ((?v + w) # us) \<subseteq> carrier_vec n" 
       using U w vw by simp
     have dist2: "distinct (((?v + w) # us))" 
       using vwUS
       by (simp add: dist_U)
 
     have orth2: "corthogonal ((adjuster n w us + w) # us)"
       using adjust_orthogonal[OF U corth w wsU].
 
     have ind_vwU: "~ lin_dep (set ((adjuster n w us + w) # us))"
       apply simp
       apply (subst lin_dep_iff_in_span[simplified, symmetric])
       by (simp_all add: ind_U vw vwUS span)
 
     have span_UwW_U': "span (set (us @ w # ws)) = span (set us')"
       using Cons(1)[OF U'def W vwU dist2 ind_vwU orth2] 
       using span_Un[OF vwU wU gram_schmidt_sub_span[OF w U dist_U] W W refl]
       by simp
 
     show ?thesis
       apply (intro conjI)
       using Cons(1)[OF U'def W vwU dist2 ind_vwU orth2] span_UwW_U' by simp_all
   next
     case True
 
     let ?v = "adjuster n w us"
     have "?v + w = 0\<^sub>v n"
       apply (rule adjuster_already_in_span)
       using True Cons by auto
     hence "vec_is_zero n (?v + w)"
       by (simp add: vec_is_zero)
     hence U'_def: "us' = gram_schmidt_sub0 n us ws"
       using Cons by simp
     have span: "span (set (us @ w # ws)) = span (set us')"
     proof -
       have wU_U: "span (set (w # us)) = span (set us)"
         apply (subst already_in_span[OF _ True, simplified])
         using Cons by auto
       have "span (set (us @ w # ws)) = span (set (w # us) \<union> set ws)"
         by simp
       also have "\<dots> = span (set us \<union> set ws)"
         apply (rule span_Un) using wU_U Cons by auto
       also have "\<dots> = local.span (set us')"
         using Cons U'_def by auto
       finally show ?thesis
         by -
     qed
     moreover have "set us' \<subseteq> carrier_vec n \<and> distinct us' \<and> corthogonal us'"
       unfolding U'_def using Cons by simp
     ultimately show ?thesis
       by auto
   qed
 qed simp
 
 text \<open>This is a variant of \<^term>\<open>Gram_Schmidt.gram_schmidt\<close> that does not require the input vectors
   \<^term>\<open>ws\<close> to be distinct or orthogonal. (In comparison to \<^term>\<open>Gram_Schmidt.gram_schmidt\<close>,
   our version also returns the result in reversed order.)\<close>
 definition "gram_schmidt0 n ws = gram_schmidt_sub0 n [] ws"
 
 lemma (in cof_vec_space) gram_schmidt0_result:
   fixes ws
   defines "us' \<equiv> gram_schmidt0 n ws"
   assumes ws: "set ws \<subseteq> carrier_vec n"
   shows "set us' \<subseteq> carrier_vec n"        (is ?thesis1)
     and "distinct us'"                    (is ?thesis2)
     and "corthogonal us'"                 (is ?thesis3)
     and "span (set ws) = span (set us')"  (is ?thesis4)
 proof -
   have carrier_empty: "set [] \<subseteq> carrier_vec n" by auto
   have distinct_empty: "distinct []" by simp
   have indep_empty: "lin_indpt (set [])"
     using basis_def subset_li_is_li unit_vecs_basis by auto
   have ortho_empty: "corthogonal []" by auto
   note gram_schmidt_sub0_result' = gram_schmidt_sub0_result
     [OF us'_def[symmetric, THEN meta_eq_to_obj_eq, unfolded gram_schmidt0_def] ws
       carrier_empty distinct_empty indep_empty ortho_empty]
   thus ?thesis1 ?thesis2 ?thesis3 ?thesis4
     by auto
 qed
 
 locale complex_vec_space = cof_vec_space n "TYPE(complex)" for n :: nat
 
 lemma gram_schmidt0_corthogonal:
   assumes a1: "corthogonal R" 
     and a2: "\<And>x. x \<in> set R \<Longrightarrow> dim_vec x = d"
   shows "gram_schmidt0 d R = rev R"
 proof -
   have "gram_schmidt_sub0 d U R = rev R @ U"
     if "corthogonal ((rev U) @ R)"
       and "\<And>x. x \<in> set U \<union> set R \<Longrightarrow> dim_vec x = d" for U
   proof (insert that, induction R arbitrary: U)
     case Nil
     show ?case 
       by auto
   next
     case (Cons a R)
     have "a \<in> set (rev U @ a # R)"
       by simp      
     moreover have uar: "corthogonal (rev U @ a # R)"
       by (simp add: Cons.prems(1))      
     ultimately have \<open>a \<noteq> 0\<^sub>v d\<close>
       unfolding corthogonal_def
       by (metis conjugate_zero_vec in_set_conv_nth scalar_prod_right_zero zero_carrier_vec)
     then have nonzero_a: "\<not> vec_is_zero d a"
       by (simp add: Cons.prems(2) vec_is_zero)
     define T where "T = rev U @ a # R"
     have "T ! length (rev U) = a"
       unfolding T_def
       by (meson nth_append_length) 
-    moreover have "(T ! i \<bullet>c T ! j = 0) = (i \<noteq> j)"
-      if "i<length T"
-        and "j<length T"
-      for i j
-      using uar 
+    moreover have "(T ! i \<bullet>c T ! j = 0) = (i \<noteq> j)" if "i<length T" and "j<length T" for i j
+      using uar that 
       unfolding corthogonal_def T_def
-      apply auto
-      using T_def that(2) apply auto[1]
-      using T_def that(1) that(2) by auto     
+      by auto
     moreover have "length (rev U) < length T"
       by (simp add: T_def)
-    ultimately have "(T ! (length (rev U)) \<bullet>c T ! j = 0) = (length (rev U) \<noteq> j)"
-      if "j<length T"
-      for j
-      using that by blast    
+    ultimately have "(T ! (length (rev U)) \<bullet>c T ! j = 0) = (length (rev U) \<noteq> j)" if "j<length T" for j
+      using that by blast
     hence "T ! (length (rev U)) \<bullet>c T ! j = 0"
-      if  "j<length T"
-        and "j \<noteq> length (rev U)"
-      for j
-      using that(1) that(2) by blast
-    hence "a \<bullet>c T ! j = 0"
-      if   "j < length (rev U)"
-      for j
+      if  "j<length T" and "j \<noteq> length (rev U)" for j
+      using that by blast
+    hence "a \<bullet>c T ! j = 0" if "j < length (rev U)" for j
       using \<open>T ! length (rev U) = a\<close> that(1)
         \<open>length (rev U) < length T\<close> dual_order.strict_trans by blast
-    moreover have "T ! j = (rev U) ! j"
-      if   "j < length (rev U)"
-      for j
+    moreover have "T ! j = (rev U) ! j" if "j < length (rev U)" for j
       by (smt T_def \<open>length (rev U) < length T\<close> dual_order.strict_trans list_update_append1
           list_update_id nth_list_update_eq that)
-    ultimately have "a \<bullet>c u = 0"
-      if "u \<in> set (rev U)"
-      for u
+    ultimately have "a \<bullet>c u = 0" if "u \<in> set (rev U)" for u
       by (metis in_set_conv_nth that)
-    hence "a \<bullet>c u = 0"
-      if "u \<in> set U"
-      for u
+    hence "a \<bullet>c u = 0" if "u \<in> set U" for u
       by (simp add: that)
     moreover have "\<And>x. x \<in> set U \<Longrightarrow> dim_vec x = d"
       by (simp add: Cons.prems(2))      
     ultimately have "adjuster d a U = 0\<^sub>v d"
     proof(induction U)
       case Nil
       then show ?case by simp
     next
       case (Cons u U)
       moreover have "0 \<cdot>\<^sub>v u + 0\<^sub>v d = 0\<^sub>v d"
       proof-
         have "dim_vec u = d"
           by (simp add: calculation(3))          
         thus ?thesis
           by auto 
       qed
       ultimately show ?case by auto
     qed
     hence adjuster_a: "adjuster d a U + a = a"
       by (simp add: Cons.prems(2) carrier_vecI)      
     have "gram_schmidt_sub0 d U (a # R) = gram_schmidt_sub0 d (a # U) R"
       by (simp add: adjuster_a nonzero_a)
     also have "\<dots> = rev (a # R) @ U"
       apply (subst Cons.IH)
       using Cons.prems by simp_all
     finally show ?case
       by -
   qed
   from this[where U="[]"] show ?thesis
     unfolding gram_schmidt0_def using assms by auto
 qed
 
 lemma adjuster_carrier': (* Like adjuster_carrier but with one assm less *)
   assumes w: "(w :: 'a::conjugatable_field vec) : carrier_vec n"
     and us: "set (us :: 'a vec list) \<subseteq> carrier_vec n"
   shows "adjuster n w us \<in> carrier_vec n"
   by (insert us, induction us, auto)
 
 lemma eq_mat_on_vecI:
   fixes M N :: \<open>'a::field mat\<close>
   assumes eq: \<open>\<And>v. v\<in>carrier_vec nA \<Longrightarrow> M *\<^sub>v v = N *\<^sub>v v\<close>
   assumes [simp]: \<open>M \<in> carrier_mat nB nA\<close> \<open>N \<in> carrier_mat nB nA\<close>
   shows \<open>M = N\<close>
 proof (rule eq_matI)
   show [simp]: \<open>dim_row M = dim_row N\<close> \<open>dim_col M = dim_col N\<close>
     using assms(2) assms(3) by blast+
   fix i j
   assume [simp]: \<open>i < dim_row N\<close> \<open>j < dim_col N\<close>
   show \<open>M $$ (i, j) = N $$ (i, j)\<close>
     thm mat_entry_explicit[where M=M]
     apply (subst mat_entry_explicit[symmetric])
     using assms apply auto[3]
     apply (subst mat_entry_explicit[symmetric])
     using assms apply auto[3]
     apply (subst eq)
      apply auto using assms(3) unit_vec_carrier by blast
 qed
 
 lemma list_of_vec_plus:
   fixes v1 v2 :: \<open>complex vec\<close>
   assumes \<open>dim_vec v1 = dim_vec v2\<close>
   shows \<open>list_of_vec (v1 + v2) = map2 (+) (list_of_vec v1) (list_of_vec v2)\<close>
 proof-
   have \<open>i < dim_vec v1 \<Longrightarrow> (list_of_vec (v1 + v2)) ! i = (map2 (+) (list_of_vec v1) (list_of_vec v2)) ! i\<close>
     for i
     by (simp add: assms)
   thus ?thesis
     by (metis assms index_add_vec(2) length_list_of_vec length_map map_fst_zip nth_equalityI) 
 qed
 
 lemma list_of_vec_mult:
   fixes v :: \<open>complex vec\<close>
   shows \<open>list_of_vec (c \<cdot>\<^sub>v v) = map ((*) c) (list_of_vec v)\<close>
   by (metis (mono_tags, lifting) index_smult_vec(1) index_smult_vec(2) length_list_of_vec length_map nth_equalityI nth_list_of_vec nth_map)
 
 
 
 unbundle no_jnf_notation
 
 
 end
diff --git a/thys/Complex_Bounded_Operators/extra/Extra_Operator_Norm.thy b/thys/Complex_Bounded_Operators/extra/Extra_Operator_Norm.thy
--- a/thys/Complex_Bounded_Operators/extra/Extra_Operator_Norm.thy
+++ b/thys/Complex_Bounded_Operators/extra/Extra_Operator_Norm.thy
@@ -1,392 +1,157 @@
 section \<open>\<open>Extra_Operator_Norm\<close> -- Additional facts bout the operator norm\<close>
 
 theory Extra_Operator_Norm
   imports "HOL-Analysis.Operator_Norm"
     Extra_General
     "HOL-Analysis.Bounded_Linear_Function"
+    Extra_Vector_Spaces
 begin
 
 
 text \<open>This theorem complements \<^theory>\<open>HOL-Analysis.Operator_Norm\<close>
       additional useful facts about operator norms.\<close>
 
-lemma ex_norm1:
-  assumes \<open>(UNIV::'a::real_normed_vector set) \<noteq> {0}\<close>
-  shows \<open>\<exists>x::'a. norm x = 1\<close>
-proof-
-  have \<open>\<exists>x::'a. x \<noteq> 0\<close>
-    using assms by fastforce
-  then obtain x::'a where \<open>x \<noteq> 0\<close>
-    by blast
-  hence \<open>norm x \<noteq> 0\<close>
-    by simp
-  hence \<open>(norm x) / (norm x) = 1\<close>
-    by simp
-  moreover have \<open>(norm x) / (norm x) = norm (x /\<^sub>R (norm x))\<close>
-    by simp
-  ultimately have \<open>norm (x /\<^sub>R (norm x)) = 1\<close>
-    by simp
-  thus ?thesis
-    by blast
-qed
-
-lemma bdd_above_norm_f:
-  assumes "bounded_linear f"
-  shows \<open>bdd_above {norm (f x) |x. norm x = 1}\<close>
-proof-
-  have \<open>\<exists>M. \<forall>x. norm x = 1 \<longrightarrow> norm (f x) \<le> M\<close>
-    using assms
-    by (metis bounded_linear.axioms(2) bounded_linear_axioms_def)
-  thus ?thesis by auto
-qed
-
 lemma onorm_sphere:
   fixes f :: "'a::{real_normed_vector, not_singleton} \<Rightarrow> 'b::real_normed_vector"
   assumes a1: "bounded_linear f"
   shows \<open>onorm f = Sup {norm (f x) | x. norm x = 1}\<close>
 proof(cases \<open>f = (\<lambda> _. 0)\<close>)
   case True
   have \<open>(UNIV::'a set) \<noteq> {0}\<close>
     by simp
   hence \<open>\<exists>x::'a. norm x = 1\<close>
     using  ex_norm1
     by blast
   have \<open>norm (f x) = 0\<close>
     for x
     by (simp add: True)
   hence \<open>{norm (f x) | x. norm x = 1} = {0}\<close>
     using \<open>\<exists>x. norm x = 1\<close> by auto
   hence v1: \<open>Sup {norm (f x) | x. norm x = 1} = 0\<close>
     by simp
   have \<open>onorm f = 0\<close>
     by (simp add: True onorm_eq_0)
   thus ?thesis using v1 by simp
 next
   case False
   have \<open>y \<in> {norm (f x) |x. norm x = 1} \<union> {0}\<close>
     if "y \<in> {norm (f x) / norm x |x. True}"
     for y
   proof(cases \<open>y = 0\<close>)
     case True
     thus ?thesis
       by simp
   next
     case False
     have \<open>\<exists> x. y = norm (f x) / norm x\<close>
       using \<open>y \<in> {norm (f x) / norm x |x. True}\<close> by auto
     then obtain x where \<open>y = norm (f x) / norm x\<close>
       by blast
     hence \<open>y = \<bar>(1/norm x)\<bar> * norm ( f x )\<close>
       by simp
     hence \<open>y = norm ( (1/norm x) *\<^sub>R f x )\<close>
       by simp
     hence \<open>y = norm ( f ((1/norm x) *\<^sub>R x) )\<close>
       apply (subst linear_cmul[of f])
       by (simp_all add: assms bounded_linear.linear)
     moreover have \<open>norm ((1/norm x) *\<^sub>R x) = 1\<close>
       using False \<open>y = norm (f x) / norm x\<close> by auto
     ultimately have \<open>y \<in> {norm (f x) |x. norm x = 1}\<close>
       by blast
     thus ?thesis by blast
   qed
   moreover have "y \<in> {norm (f x) / norm x |x. True}"
     if \<open>y \<in> {norm (f x) |x. norm x = 1} \<union> {0}\<close>
     for y
   proof(cases \<open>y = 0\<close>)
     case True
     thus ?thesis
       by auto
   next
     case False
     hence \<open>y \<notin> {0}\<close>
       by simp
     hence \<open>y \<in> {norm (f x) |x. norm x = 1}\<close>
       using that by auto
     hence \<open>\<exists> x. norm x = 1 \<and> y = norm (f x)\<close>
       by auto
     then obtain x where \<open>norm x = 1\<close> and \<open>y = norm (f x)\<close>
       by auto
     have \<open>y = norm (f x) / norm x\<close> using  \<open>norm x = 1\<close>  \<open>y = norm (f x)\<close>
       by simp
     thus ?thesis
       by auto
   qed
   ultimately have \<open>{norm (f x) / norm x |x. True} = {norm (f x) |x. norm x = 1} \<union> {0}\<close>
     by blast
   hence \<open>Sup {norm (f x) / norm x |x. True} = Sup ({norm (f x) |x. norm x = 1} \<union> {0})\<close>
     by simp
   moreover have \<open>Sup {norm (f x) |x. norm x = 1} \<ge> 0\<close>
   proof-
     have \<open>\<exists> x::'a. norm x = 1\<close>
       by (metis (full_types) False assms linear_simps(3) norm_sgn)
     then obtain x::'a where \<open>norm x = 1\<close>
       by blast
     have \<open>norm (f x) \<ge> 0\<close>
       by simp
     hence \<open>\<exists> x::'a. norm x = 1 \<and> norm (f x) \<ge> 0\<close>
       using \<open>norm x = 1\<close> by blast
     hence \<open>\<exists> y \<in> {norm (f x) |x. norm x = 1}. y \<ge> 0\<close>
       by blast
     then obtain y::real where \<open>y \<in> {norm (f x) |x. norm x = 1}\<close>
       and \<open>y \<ge> 0\<close>
       by auto
     have \<open>{norm (f x) |x. norm x = 1} \<noteq> {}\<close>
       using \<open>y \<in> {norm (f x) |x. norm x = 1}\<close> by blast
     moreover have \<open>bdd_above {norm (f x) |x. norm x = 1}\<close>
       using bdd_above_norm_f
       by (metis (mono_tags, lifting) a1)
     ultimately have \<open>y \<le> Sup {norm (f x) |x. norm x = 1}\<close>
       using \<open>y \<in> {norm (f x) |x. norm x = 1}\<close>
       by (simp add: cSup_upper)
     thus ?thesis using \<open>y \<ge> 0\<close> by simp
   qed
   moreover have \<open>Sup ({norm (f x) |x. norm x = 1} \<union> {0}) = Sup {norm (f x) |x. norm x = 1}\<close>
   proof-
     have \<open>{norm (f x) |x. norm x = 1} \<noteq> {}\<close>
       by (simp add: assms(1) ex_norm1)
     moreover have \<open>bdd_above {norm (f x) |x. norm x = 1}\<close>
       using a1 bdd_above_norm_f by force
     have \<open>{0::real} \<noteq> {}\<close>
       by simp
     moreover have \<open>bdd_above {0::real}\<close>
       by simp
     ultimately have \<open>Sup ({norm (f x) |x. norm x = 1} \<union> {(0::real)})
              = max (Sup {norm (f x) |x. norm x = 1}) (Sup {0::real})\<close>
       by (metis (lifting) \<open>0 \<le> Sup {norm (f x) |x. norm x = 1}\<close> \<open>bdd_above {0}\<close> \<open>bdd_above {norm (f x) |x. norm x = 1}\<close> \<open>{0} \<noteq> {}\<close> \<open>{norm (f x) |x. norm x = 1} \<noteq> {}\<close> cSup_singleton cSup_union_distrib max.absorb_iff1 sup.absorb_iff1)
     moreover have \<open>Sup {(0::real)} = (0::real)\<close>
       by simp
     moreover have \<open>Sup {norm (f x) |x. norm x = 1} \<ge> 0\<close>
       by (simp add: \<open>0 \<le> Sup {norm (f x) |x. norm x = 1}\<close>)
     ultimately show ?thesis
       by simp
   qed
   moreover have \<open>Sup ( {norm (f x) |x. norm x = 1} \<union> {0})
            = max (Sup {norm (f x) |x. norm x = 1}) (Sup {0}) \<close>
     using calculation(2) calculation(3) by auto
   ultimately have w1: "Sup {norm (f x) / norm x | x. True} = Sup {norm (f x) | x. norm x = 1}"
     by simp
 
   have \<open>(SUP x. norm (f x) / (norm x)) = Sup {norm (f x) / norm x | x. True}\<close>
     by (simp add: full_SetCompr_eq)
   also have \<open>... = Sup {norm (f x) | x. norm x = 1}\<close>
     using w1 by auto
   ultimately  have \<open>(SUP x. norm (f x) / (norm x)) = Sup {norm (f x) | x. norm x = 1}\<close>
     by linarith
   thus ?thesis unfolding onorm_def by blast
 qed
 
-
-lemma onorm_Inf_bound:
-  fixes f :: \<open>'a::{real_normed_vector,not_singleton} \<Rightarrow> 'b::real_normed_vector\<close>
-  assumes a1: "bounded_linear f"
-  shows "onorm f = Inf {K. (\<forall>x\<noteq>0. norm (f x) \<le> norm x * K)}"
-proof-
-  have a2: \<open>(UNIV::'a set) \<noteq> {0}\<close>
-    by simp
-
-  define A where \<open>A = {norm (f x) / (norm x) | x. x \<noteq> 0}\<close>
-  have \<open>A \<noteq> {}\<close>
-  proof-
-    have \<open>\<exists> x::'a. x \<noteq> 0\<close>
-      using a2 by auto
-    thus ?thesis using A_def
-      by simp
-  qed
-  moreover have \<open>bdd_above A\<close>
-  proof-
-    have \<open>\<exists> M. \<forall> x.  norm (f x) / (norm x) \<le> M\<close>
-      using \<open>bounded_linear f\<close> le_onorm by auto
-    thus ?thesis using A_def
-      by auto
-  qed
-  ultimately have \<open>Sup A = Inf {b. \<forall>a\<in>A. a \<le> b}\<close>
-    by (simp add: cSup_cInf)
-  moreover have \<open>{b. \<forall>a\<in>A. a \<le> b} = {K. (\<forall>x\<noteq>0. norm (f x)/ norm x \<le>  K)}\<close>
-  proof-
-    have \<open>{b. \<forall>a\<in>A. a \<le> b} = {b. \<forall>a\<in>{norm (f x) / (norm x) | x. x \<noteq> 0}. a \<le> b}\<close>
-      using A_def by blast
-    also have \<open>... = {b. \<forall>x\<in>{x | x. x \<noteq> 0}. norm (f x) / (norm x) \<le> b}\<close>
-      by auto
-    also have \<open>... = {b. \<forall>x\<noteq>0. norm (f x) / (norm x) \<le> b}\<close>
-      by auto
-    finally show ?thesis by blast
-  qed
-  ultimately have \<open>Sup {norm (f x) / (norm x) | x. x \<noteq> 0}
-                    = Inf {K. (\<forall>x\<noteq>0. norm (f x)/ norm x \<le>  K)}\<close>
-    using A_def
-    by simp
-  moreover have \<open>(\<forall>x\<noteq>0. norm (f x) \<le> norm x * K) \<longleftrightarrow> (\<forall>x\<noteq>0. norm (f x)/ norm x \<le> K)\<close>
-    for K
-  proof
-    show "\<forall>x\<noteq>0. norm (f x) / norm x \<le> K"
-      if "\<forall>x\<noteq>0. norm (f x) \<le> norm x * K"
-      using divide_le_eq nonzero_mult_div_cancel_left norm_le_zero_iff that
-      by (simp add: divide_le_eq mult.commute)
-
-    show "\<forall>x\<noteq>0. norm (f x) \<le> norm x * K"
-      if "\<forall>x\<noteq>0. norm (f x) / norm x \<le> K"
-      using divide_le_eq nonzero_mult_div_cancel_left norm_le_zero_iff that
-      by (simp add: divide_le_eq mult.commute)
-  qed
-  ultimately have f1: \<open>Sup {norm (f x) / (norm x) | x. x \<noteq> 0} = Inf {K. (\<forall>x\<noteq>0. norm (f x) \<le> norm x * K)}\<close>
-    by simp
-  moreover
-  have t1: \<open>{norm (f x) / (norm x) | x. x \<noteq> 0} \<union> {norm (f x) / (norm x) | x. x = 0}  = {norm (f x) / (norm x) | x. True}\<close>
-    using Collect_cong by blast
-
-  have \<open>{norm (f x) / (norm x) | x. x \<noteq> 0} \<noteq> {}\<close>
-  proof-
-    have \<open>\<exists> x::'a. x \<noteq> 0\<close>
-      using \<open>UNIV\<noteq>{0}\<close> by auto
-    thus ?thesis
-      by simp
-  qed
-  moreover have \<open>bdd_above {norm (f x) / (norm x) | x. x \<noteq> 0}\<close>
-  proof-
-    have \<open>\<exists> M. \<forall> x.  norm (f x) / (norm x) \<le> M\<close>
-      using \<open>bounded_linear f\<close> bounded_linear.nonneg_bounded
-        mult_divide_mult_cancel_left_if norm_zero real_divide_square_eq
-      using le_onorm by blast
-    thus ?thesis
-      by auto
-  qed
-  moreover have \<open>{norm (f x) / (norm x) | x. x = 0} \<noteq> {}\<close>
-    by simp
-  moreover have \<open>bdd_above {norm (f x) / (norm x) | x. x = 0}\<close>
-    by simp
-  ultimately
-  have d1: \<open>Sup ({norm (f x) / (norm x) | x. x \<noteq> 0} \<union> {norm (f x) / (norm x) | x. x = 0})
-        = max (Sup {norm (f x) / (norm x) | x. x \<noteq> 0}) (Sup {norm (f x) / (norm x) | x. x = 0})\<close>
-    by (metis (no_types, lifting) cSup_union_distrib sup_max)
-  have g1: \<open>Sup {norm (f x) / (norm x) | x. x \<noteq> 0} \<ge> 0\<close>
-  proof-
-    have t2: \<open>{norm (f x) / (norm x) | x. x \<noteq> 0} \<noteq> {}\<close>
-    proof-
-      have \<open>\<exists> x::'a. x \<noteq> 0\<close>
-        using \<open>UNIV\<noteq>{0}\<close> by auto
-      thus ?thesis
-        by auto
-    qed
-    have \<open>\<exists> M. \<forall> x.  norm (f x) / (norm x) \<le> M\<close>
-      using \<open>bounded_linear f\<close>
-      by (metis \<open>\<And>K. (\<forall>x. x \<noteq> 0 \<longrightarrow> norm (f x) \<le> norm x * K) = (\<forall>x. x \<noteq> 0 \<longrightarrow> norm (f x) / norm x \<le> K)\<close> bounded_linear.nonneg_bounded mult_divide_mult_cancel_left_if norm_zero real_divide_square_eq)
-    hence t3: \<open>bdd_above {norm (f x) / (norm x) | x. x \<noteq> 0}\<close>
-      by auto
-    have \<open>norm (f x) / (norm x) \<ge> 0\<close>
-      for x
-      by simp
-    hence \<open>\<forall> y\<in>{norm (f x) / (norm x) | x. x \<noteq> 0}. y \<ge> 0\<close>
-      by blast
-    show ?thesis
-      by (metis (lifting) \<open>\<forall>y\<in>{norm (f x) / norm x |x. x \<noteq> 0}. 0 \<le> y\<close> \<open>bdd_above {norm (f x) / norm x |x. x \<noteq> 0}\<close> \<open>{norm (f x) / norm x |x. x \<noteq> 0} \<noteq> {}\<close> bot.extremum_uniqueI cSup_upper2 subset_emptyI)
-  qed
-  hence r: \<open>Sup ({norm (f x) / (norm x) | x. x \<noteq> 0} \<union> {norm (f x) / (norm x) | x. x = 0})
-         = Sup {norm (f x) / (norm x) | x. True}\<close>
-    using t1 by auto
-  have \<open>{norm (f x) / (norm x) | x. x = 0} = {norm (f 0) / (norm 0)}\<close>
-    by simp
-  hence \<open>Sup {norm (f x) / (norm x) | x. x = 0} = 0\<close>
-    by simp
-  have h1: \<open>Sup {norm (f x) / (norm x) | x. x \<noteq> 0} = Sup {norm (f x) / (norm x) | x. True}\<close>
-    using d1 r g1 by auto
-  have \<open>(SUP x. norm (f x) / (norm x)) = Inf {K. (\<forall>x\<noteq>0. norm (f x) \<le> norm x * K)}\<close>
-    using full_SetCompr_eq
-    by (metis f1 h1)
-  thus ?thesis
-    by (simp add: onorm_def)
-qed
-
-
 lemma onormI:
   assumes "\<And>x. norm (f x) \<le> b * norm x"
     and "x \<noteq> 0" and "norm (f x) = b * norm x"
   shows "onorm f = b"
   apply (unfold onorm_def, rule cSup_eq_maximum)
    apply (smt (verit) UNIV_I assms(2) assms(3) image_iff nonzero_mult_div_cancel_right norm_eq_zero)
   by (smt (verit, del_insts) assms(1) assms(2) divide_nonneg_nonpos norm_ge_zero norm_le_zero_iff pos_divide_le_eq rangeE zero_le_mult_iff)
 
-
-lemma norm_unit_sphere:
-  fixes f::\<open>'a::{real_normed_vector,not_singleton} \<Rightarrow>\<^sub>L 'b::real_normed_vector\<close>
-  assumes a1: "bounded_linear f" and a2: "e > 0"
-  shows \<open>\<exists>x\<in>(sphere 0 1). norm (norm (blinfun_apply f x) - norm f) < e\<close>
-proof-
-  define S::"real set" where \<open>S = { norm (f x)| x. x \<in> sphere 0 1 }\<close>
-  have "\<exists>x::'a. norm x = 1"
-    by (metis (full_types) Collect_empty_eq Extra_General.UNIV_not_singleton UNIV_I equalityI mem_Collect_eq norm_sgn singleton_conv subsetI)
-  hence \<open>\<exists>x::'a. x \<in> sphere 0 1\<close>
-    by simp
-  hence \<open>S\<noteq>{}\<close>unfolding S_def
-    by auto
-  hence t1: \<open>e > 0 \<Longrightarrow> \<exists> y \<in> S. Sup S - e < y\<close>
-    for e
-    by (simp add: less_cSupD)
-  have \<open>onorm f = Sup { norm (f x)| x. norm x = 1 }\<close>
-    using \<open>bounded_linear f\<close> onorm_sphere
-    by auto
-  hence \<open>onorm f = Sup { norm (f x)| x. x \<in> sphere 0 1 }\<close>
-    unfolding sphere_def
-    by simp
-  hence t2: \<open>Sup S = onorm f\<close> unfolding S_def
-    by auto
-  have s1: \<open>\<exists>y\<in>{norm (f x) |x. x \<in> sphere 0 1}. norm (onorm f - y) < e\<close>
-    if "0 < e"
-    for e
-  proof-
-    have \<open>\<exists> y \<in> S. (onorm f) - e < y\<close>
-      using t1 t2 that by auto
-    hence \<open>\<exists> y \<in> S. (onorm f) - y  < e\<close>
-      using that
-      by force
-    have \<open>\<exists> y \<in> S. (onorm f) - y  < e\<close>
-      using \<open>0 < e\<close> \<open>\<exists>y\<in>S. onorm f - y < e\<close> by auto
-    then obtain y where \<open>y \<in> S\<close> and \<open>(onorm f) - y  < e\<close>
-      by blast
-    have \<open>y \<in> {norm (f x) |x. x \<in> sphere 0 1} \<Longrightarrow> y \<le> onorm f\<close>
-    proof-
-      assume \<open>y \<in> {norm (f x) |x. x \<in> sphere 0 1}\<close>
-      hence \<open>\<exists> x \<in> sphere 0 1. y = norm (f x)\<close>
-        by blast
-      then obtain x where \<open>x \<in> sphere 0 1\<close> and \<open>y = norm (f x)\<close>
-        by blast
-      from \<open>y = norm (f x)\<close>
-      have \<open>y \<le> onorm f * norm x\<close>
-        using a1 onorm by auto
-      moreover have \<open>norm x = 1\<close>
-        using  \<open>x \<in> sphere 0 1\<close> unfolding sphere_def by auto
-      ultimately show ?thesis by simp
-    qed
-    hence \<open>bdd_above {norm (f x) |x. x \<in> sphere 0 1}\<close>
-      using a1 bdd_above_norm_f by force
-    hence \<open>bdd_above S\<close> unfolding S_def
-      by blast
-    hence \<open>y \<le> Sup S\<close>
-      using \<open>y \<in> S\<close> \<open>S \<noteq> {}\<close> cSup_upper
-      by blast
-    hence \<open>0 \<le> Sup S - y\<close>
-      by simp
-    hence \<open>0 \<le> onorm f - y\<close>
-      using \<open>Sup S = onorm f\<close>
-      by simp
-    hence \<open>\<bar> (onorm f - y) \<bar> = onorm f - y\<close>
-      by simp
-    hence \<open>norm (onorm f - y)  = onorm f - y\<close>
-      by auto
-    hence \<open>\<exists> y \<in> S. norm ((onorm f) - y)  < e\<close>
-      using \<open>onorm f - y < e\<close> \<open>y \<in> S\<close> by force
-    show ?thesis
-      unfolding S_def
-      using S_def \<open>\<exists>y\<in>S. norm (onorm (blinfun_apply f) - y) < e\<close> by blast
-  qed
-  have f2: "onorm (blinfun_apply f) = Sup S"
-    using S_def \<open>onorm (blinfun_apply f) = Sup {norm (blinfun_apply f x) |x. x \<in> sphere 0 1}\<close> by blast
-  hence "\<exists>a. norm (norm (blinfun_apply f a) - Sup S) < e \<and> a \<in> sphere 0 1"
-    using a1 a2 s1 a2 t2
-    by force
-  thus ?thesis
-    using f2 by (metis (full_types) norm_blinfun.rep_eq)
-qed
-
-
-
 end
diff --git a/thys/Complex_Bounded_Operators/extra/Extra_Ordered_Fields.thy b/thys/Complex_Bounded_Operators/extra/Extra_Ordered_Fields.thy
--- a/thys/Complex_Bounded_Operators/extra/Extra_Ordered_Fields.thy
+++ b/thys/Complex_Bounded_Operators/extra/Extra_Ordered_Fields.thy
@@ -1,956 +1,936 @@
 section \<open>\<open>Extra_Ordered_Fields\<close> -- Additional facts about ordered fields\<close>
 
 theory Extra_Ordered_Fields
   imports Complex_Main "HOL-Library.Complex_Order"
 begin
 
 
 subsection\<open>Ordered Fields\<close>
 text \<open>In this section we introduce some type classes for ordered rings/fields/etc.
 that are weakenings of existing classes. Most theorems in this section are
 copies of the eponymous theorems from Isabelle/HOL, except that they are now proven
 requiring weaker type classes (usually the need for a total order is removed).
 
 Since the lemmas are identical to the originals except for weaker type constraints,
 we use the same names as for the original lemmas. (In fact, the new lemmas could replace
 the original ones in Isabelle/HOL with at most minor incompatibilities.\<close>
 
 subsection \<open>Missing from Orderings.thy\<close>
 
 text \<open>This class is analogous to \<^class>\<open>unbounded_dense_linorder\<close>, except that it does not require a total order\<close>
 
 class unbounded_dense_order = dense_order + no_top + no_bot
 
 instance unbounded_dense_linorder \<subseteq> unbounded_dense_order ..
 
 subsection \<open>Missing from Rings.thy\<close>
 
 text \<open>The existing class \<^class>\<open>abs_if\<close> requires \<^term>\<open>\<bar>a\<bar> = (if a < 0 then - a else a)\<close>.
 However, if \<^term>\<open>(<)\<close> is not a total order, this condition is too strong when \<^term>\<open>a\<close>
 is incomparable with \<^term>\<open>0\<close>. (Namely, it requires the absolute value to be
 the identity on such elements. E.g., the absolute value for complex numbers does not
 satisfy this.) The following class \<open>partial_abs_if\<close> is analogous to \<^class>\<open>abs_if\<close>
 but does not require anything if \<^term>\<open>a\<close> is incomparable with \<^term>\<open>0\<close>.\<close>
 
 
 class partial_abs_if = minus + uminus + ord + zero + abs +
   assumes abs_neg: "a \<le> 0 \<Longrightarrow> abs a = -a"
   assumes abs_pos: "a \<ge> 0 \<Longrightarrow> abs a = a"
 
 class ordered_semiring_1 = ordered_semiring + semiring_1
   \<comment> \<open>missing class analogous to \<^class>\<open>linordered_semiring_1\<close> without requiring a total order\<close>
 begin
 
 lemma convex_bound_le:
   assumes "x \<le> a" and "y \<le> a" and "0 \<le> u" and "0 \<le> v" and "u + v = 1"
   shows "u * x + v * y \<le> a"
 proof-
   from assms have "u * x + v * y \<le> u * a + v * a"
     by (simp add: add_mono mult_left_mono)
   with assms show ?thesis
     unfolding distrib_right[symmetric] by simp
 qed
 
 end
 
 subclass (in linordered_semiring_1) ordered_semiring_1 ..
 
 class ordered_semiring_strict = semiring + comm_monoid_add + ordered_cancel_ab_semigroup_add +
   \<comment> \<open>missing class analogous to \<^class>\<open>linordered_semiring_strict\<close> without requiring a total order\<close>
   assumes mult_strict_left_mono: "a < b \<Longrightarrow> 0 < c \<Longrightarrow> c * a < c * b"
   assumes mult_strict_right_mono: "a < b \<Longrightarrow> 0 < c \<Longrightarrow> a * c < b * c"
 begin
 
 subclass semiring_0_cancel ..
 
 subclass ordered_semiring
 proof
   fix a b c :: 'a
   assume t1: "a \<le> b" and t2: "0 \<le> c"
   thus "c * a \<le> c * b"
     unfolding le_less
     using mult_strict_left_mono by (cases "c = 0") auto
   from t2 show "a * c \<le> b * c"
     unfolding le_less
     by (metis local.antisym_conv2 local.mult_not_zero local.mult_strict_right_mono t1)
 qed
 
 lemma mult_pos_pos[simp]: "0 < a \<Longrightarrow> 0 < b \<Longrightarrow> 0 < a * b"
   using mult_strict_left_mono [of 0 b a] by simp
 
 lemma mult_pos_neg: "0 < a \<Longrightarrow> b < 0 \<Longrightarrow> a * b < 0"
   using mult_strict_left_mono [of b 0 a] by simp
 
 lemma mult_neg_pos: "a < 0 \<Longrightarrow> 0 < b \<Longrightarrow> a * b < 0"
   using mult_strict_right_mono [of a 0 b] by simp
 
 text \<open>Strict monotonicity in both arguments\<close>
 lemma mult_strict_mono:
   assumes t1: "a < b" and t2: "c < d" and t3: "0 < b" and t4: "0 \<le> c"
   shows "a * c < b * d"
 proof-
   have "a * c < b * d"
     by (metis local.dual_order.order_iff_strict local.dual_order.strict_trans2
         local.mult_strict_left_mono local.mult_strict_right_mono local.mult_zero_right t1 t2 t3 t4)
   thus ?thesis
     using assms by blast
 qed
 
 
 text \<open>This weaker variant has more natural premises\<close>
 lemma mult_strict_mono':
   assumes "a < b" and "c < d" and "0 \<le> a" and "0 \<le> c"
   shows "a * c < b * d"
   by (rule mult_strict_mono) (insert assms, auto)
 
 lemma mult_less_le_imp_less:
   assumes t1: "a < b" and t2: "c \<le> d" and t3: "0 \<le> a" and t4: "0 < c"
   shows "a * c < b * d"
   using local.mult_strict_mono' local.mult_strict_right_mono local.order.order_iff_strict
     t1 t2 t3 t4 by auto
 
 lemma mult_le_less_imp_less:
   assumes "a \<le> b" and "c < d" and "0 < a" and "0 \<le> c"
   shows "a * c < b * d"
   by (metis assms(1) assms(2) assms(3) assms(4) local.antisym_conv2 local.dual_order.strict_trans1
       local.mult_strict_left_mono local.mult_strict_mono)
 
 end
 
 subclass (in linordered_semiring_strict) ordered_semiring_strict
-proof
-  show "c * a < c * b"
-    if "a < b"
-      and "0 < c"
-    for a :: 'a
-      and b
-      and c
-    using that
-    by (simp add: local.mult_strict_left_mono)
-  show "a * c < b * c"
-    if "a < b"
-      and "0 < c"
-    for a :: 'a
-      and b
-      and c
-    using that
-    by (simp add: local.mult_strict_right_mono)
-qed
+  apply standard
+  by (auto simp: mult_strict_left_mono mult_strict_right_mono)
 
 class ordered_semiring_1_strict = ordered_semiring_strict + semiring_1
   \<comment> \<open>missing class analogous to \<^class>\<open>linordered_semiring_1_strict\<close> without requiring
   a total order\<close>
 begin
 
 subclass ordered_semiring_1 ..
 
 lemma convex_bound_lt:
   assumes "x < a" and "y < a" and "0 \<le> u" and "0 \<le> v" and "u + v = 1"
   shows "u * x + v * y < a"
 proof -
   from assms have "u * x + v * y < u * a + v * a"
     by (cases "u = 0") (auto intro!: add_less_le_mono mult_strict_left_mono mult_left_mono)
   with assms show ?thesis
     unfolding distrib_right[symmetric] by simp
 qed
 
 end
 
 subclass (in linordered_semiring_1_strict) ordered_semiring_1_strict ..
 
 class ordered_comm_semiring_strict = comm_semiring_0 + ordered_cancel_ab_semigroup_add +
   \<comment> \<open>missing class analogous to \<^class>\<open>linordered_comm_semiring_strict\<close> without requiring a total order\<close>
   assumes comm_mult_strict_left_mono: "a < b \<Longrightarrow> 0 < c \<Longrightarrow> c * a < c * b"
 begin
 
 subclass ordered_semiring_strict
 proof
   fix a b c :: 'a
   assume "a < b" and "0 < c"
   thus "c * a < c * b"
     by (rule comm_mult_strict_left_mono)
   thus "a * c < b * c"
     by (simp only: mult.commute)
 qed
 
 subclass ordered_cancel_comm_semiring
 proof
   fix a b c :: 'a
   assume "a \<le> b" and "0 \<le> c"
   thus "c * a \<le> c * b"
     unfolding le_less
     using mult_strict_left_mono by (cases "c = 0") auto
 qed
 
 end
 
 subclass (in linordered_comm_semiring_strict) ordered_comm_semiring_strict
   apply standard
   by (simp add: local.mult_strict_left_mono)
 
 class ordered_ring_strict = ring + ordered_semiring_strict
   + ordered_ab_group_add + partial_abs_if
   \<comment> \<open>missing class analogous to \<^class>\<open>linordered_ring_strict\<close> without requiring a total order\<close>
 begin
 
 subclass ordered_ring ..
 
 lemma mult_strict_left_mono_neg: "b < a \<Longrightarrow> c < 0 \<Longrightarrow> c * a < c * b"
   using mult_strict_left_mono [of b a "- c"] by simp
 
 lemma mult_strict_right_mono_neg: "b < a \<Longrightarrow> c < 0 \<Longrightarrow> a * c < b * c"
   using mult_strict_right_mono [of b a "- c"] by simp
 
 lemma mult_neg_neg: "a < 0 \<Longrightarrow> b < 0 \<Longrightarrow> 0 < a * b"
   using mult_strict_right_mono_neg [of a 0 b] by simp
 
 end
 
 lemmas mult_sign_intros =
   mult_nonneg_nonneg mult_nonneg_nonpos
   mult_nonpos_nonneg mult_nonpos_nonpos
   mult_pos_pos mult_pos_neg
   mult_neg_pos mult_neg_neg
 
 
 subsection \<open>Ordered fields\<close>
 
 class ordered_field = field + order + ordered_comm_semiring_strict + ordered_ab_group_add
   + partial_abs_if
   \<comment> \<open>missing class analogous to \<^class>\<open>linordered_field\<close> without requiring a total order\<close>
 begin
 
 lemma frac_less_eq:
   "y \<noteq> 0 \<Longrightarrow> z \<noteq> 0 \<Longrightarrow> x / y < w / z \<longleftrightarrow> (x * z - w * y) / (y * z) < 0"
   by (subst less_iff_diff_less_0) (simp add: diff_frac_eq )
 
 lemma frac_le_eq:
   "y \<noteq> 0 \<Longrightarrow> z \<noteq> 0 \<Longrightarrow> x / y \<le> w / z \<longleftrightarrow> (x * z - w * y) / (y * z) \<le> 0"
   by (subst le_iff_diff_le_0) (simp add: diff_frac_eq )
 
 lemmas sign_simps = algebra_simps zero_less_mult_iff mult_less_0_iff
 
 lemmas (in -) sign_simps = algebra_simps zero_less_mult_iff mult_less_0_iff
 
 text\<open>Simplify expressions equated with 1\<close>
 
 lemma zero_eq_1_divide_iff [simp]: "0 = 1 / a \<longleftrightarrow> a = 0"
   by (cases "a = 0") (auto simp: field_simps)
 
 lemma one_divide_eq_0_iff [simp]: "1 / a = 0 \<longleftrightarrow> a = 0"
   using zero_eq_1_divide_iff[of a] by simp
 
 text\<open>Simplify expressions such as \<open>0 < 1/x\<close> to \<open>0 < x\<close>\<close>
 
 text\<open>Simplify quotients that are compared with the value 1.\<close>
 
 text \<open>Conditional Simplification Rules: No Case Splits\<close>
 
 lemma eq_divide_eq_1 [simp]:
   "(1 = b/a) = ((a \<noteq> 0 & a = b))"
   by (auto simp add: eq_divide_eq)
 
 lemma divide_eq_eq_1 [simp]:
   "(b/a = 1) = ((a \<noteq> 0 & a = b))"
   by (auto simp add: divide_eq_eq)
 
 end (* class ordered_field *)
 
 
 text \<open>The following type class intends to capture some important properties
   that are common both to the real and the complex numbers. The purpose is
   to be able to state and prove lemmas that apply both to the real and the complex
   numbers without needing to state the lemma twice.
 \<close>
 
 class nice_ordered_field = ordered_field + zero_less_one + idom_abs_sgn +
   assumes positive_imp_inverse_positive: "0 < a \<Longrightarrow> 0 < inverse a"
     and inverse_le_imp_le: "inverse a \<le> inverse b \<Longrightarrow> 0 < a \<Longrightarrow> b \<le> a"
     and dense_le: "(\<And>x. x < y \<Longrightarrow> x \<le> z) \<Longrightarrow> y \<le> z"
     and nn_comparable: "0 \<le> a \<Longrightarrow> 0 \<le> b \<Longrightarrow> a \<le> b \<or> b \<le> a"
     and abs_nn: "\<bar>x\<bar> \<ge> 0"
 begin
 
 subclass (in linordered_field) nice_ordered_field
 proof
   show "\<bar>a\<bar> = - a"
     if "a \<le> 0"
     for a :: 'a
     using that
     by simp
   show "\<bar>a\<bar> = a"
     if "0 \<le> a"
     for a :: 'a
     using that
     by simp
   show "0 < inverse a"
     if "0 < a"
     for a :: 'a
     using that
     by simp
   show "b \<le> a"
     if "inverse a \<le> inverse b"
       and "0 < a"
     for a :: 'a
       and b
     using that
     using local.inverse_le_imp_le by blast
   show "y \<le> z"
     if "\<And>x::'a. x < y \<Longrightarrow> x \<le> z"
     for y
       and z
     using that
     using local.dense_le by blast
   show "a \<le> b \<or> b \<le> a"
     if "0 \<le> a"
       and "0 \<le> b"
     for a :: 'a
       and b
     using that
     by auto
   show "0 \<le> \<bar>x\<bar>"
     for x :: 'a
     by simp
 qed
 
 lemma comparable:
   assumes h1: "a \<le> c \<or> a \<ge> c"
     and h2: "b \<le> c \<or> b \<ge> c"
   shows "a \<le> b \<or> b \<le> a"
 proof-
   have "a \<le> b"
     if t1: "\<not> b \<le> a" and t2: "a \<le> c" and t3: "b \<le> c"
   proof-
     have "0 \<le> c-a"
       by (simp add: t2)
     moreover have "0 \<le> c-b"
       by (simp add: t3)
     ultimately have "c-a \<le> c-b \<or> c-a \<ge> c-b" by (rule nn_comparable)
     hence "-a \<le> -b \<or> -a \<ge> -b"
       using local.add_le_imp_le_right local.uminus_add_conv_diff by presburger
     thus ?thesis
       by (simp add: t1)
   qed
   moreover have "a \<le> b"
     if t1: "\<not> b \<le> a" and t2: "c \<le> a" and t3: "b \<le> c"
   proof-
     have "b \<le> a"
       using local.dual_order.trans t2 t3 by blast
     thus ?thesis
       using t1 by auto
   qed
   moreover have "a \<le> b"
     if t1: "\<not> b \<le> a" and t2: "c \<le> a" and t3: "c \<le> b"
   proof-
     have "0 \<le> a-c"
       by (simp add: t2)
     moreover have "0 \<le> b-c"
       by (simp add: t3)
     ultimately have "a-c \<le> b-c \<or> a-c \<ge> b-c" by (rule nn_comparable)
     hence "a \<le> b \<or> a \<ge> b"
       by (simp add: local.le_diff_eq)
     thus ?thesis
       by (simp add: t1)
   qed
   ultimately show ?thesis using assms by auto
 qed
 
 lemma negative_imp_inverse_negative:
   "a < 0 \<Longrightarrow> inverse a < 0"
   by (insert positive_imp_inverse_positive [of "-a"],
       simp add: nonzero_inverse_minus_eq less_imp_not_eq)
 
 lemma inverse_positive_imp_positive:
   assumes inv_gt_0: "0 < inverse a" and nz: "a \<noteq> 0"
   shows "0 < a"
 proof -
   have "0 < inverse (inverse a)"
     using inv_gt_0 by (rule positive_imp_inverse_positive)
   thus "0 < a"
     using nz by (simp add: nonzero_inverse_inverse_eq)
 qed
 
 lemma inverse_negative_imp_negative:
   assumes inv_less_0: "inverse a < 0" and nz: "a \<noteq> 0"
   shows "a < 0"
 proof-
   have "inverse (inverse a) < 0"
     using inv_less_0 by (rule negative_imp_inverse_negative)
   thus "a < 0" using nz by (simp add: nonzero_inverse_inverse_eq)
 qed
 
 lemma linordered_field_no_lb:
   "\<forall>x. \<exists>y. y < x"
 proof
   fix x::'a
   have m1: "- (1::'a) < 0" by simp
   from add_strict_right_mono[OF m1, where c=x]
   have "(- 1) + x < x" by simp
   thus "\<exists>y. y < x" by blast
 qed
 
 lemma linordered_field_no_ub:
   "\<forall>x. \<exists>y. y > x"
 proof
   fix x::'a
   have m1: " (1::'a) > 0" by simp
   from add_strict_right_mono[OF m1, where c=x]
   have "1 + x > x" by simp
   thus "\<exists>y. y > x" by blast
 qed
 
 lemma less_imp_inverse_less:
   assumes less: "a < b" and apos:  "0 < a"
   shows "inverse b < inverse a"
   using assms by (metis local.dual_order.strict_iff_order
       local.inverse_inverse_eq local.inverse_le_imp_le local.positive_imp_inverse_positive)
 
 lemma inverse_less_imp_less:
   "inverse a < inverse b \<Longrightarrow> 0 < a \<Longrightarrow> b < a"
   using local.inverse_le_imp_le local.order.strict_iff_order by blast
 
 text\<open>Both premises are essential. Consider -1 and 1.\<close>
 lemma inverse_less_iff_less [simp]:
   "0 < a \<Longrightarrow> 0 < b \<Longrightarrow> inverse a < inverse b \<longleftrightarrow> b < a"
   by (blast intro: less_imp_inverse_less dest: inverse_less_imp_less)
 
 lemma le_imp_inverse_le:
   "a \<le> b \<Longrightarrow> 0 < a \<Longrightarrow> inverse b \<le> inverse a"
   by (force simp add: le_less less_imp_inverse_less)
 
 lemma inverse_le_iff_le [simp]:
   "0 < a \<Longrightarrow> 0 < b \<Longrightarrow> inverse a \<le> inverse b \<longleftrightarrow> b \<le> a"
   by (blast intro: le_imp_inverse_le dest: inverse_le_imp_le)
 
 
 text\<open>These results refer to both operands being negative.  The opposite-sign
 case is trivial, since inverse preserves signs.\<close>
 lemma inverse_le_imp_le_neg:
   "inverse a \<le> inverse b \<Longrightarrow> b < 0 \<Longrightarrow> b \<le> a"
   by (metis local.inverse_le_imp_le local.inverse_minus_eq local.neg_0_less_iff_less
       local.neg_le_iff_le)
 
 lemma inverse_less_imp_less_neg:
   "inverse a < inverse b \<Longrightarrow> b < 0 \<Longrightarrow> b < a"
   using local.dual_order.strict_iff_order local.inverse_le_imp_le_neg by blast
 
 lemma inverse_less_iff_less_neg [simp]:
   "a < 0 \<Longrightarrow> b < 0 \<Longrightarrow> inverse a < inverse b \<longleftrightarrow> b < a"
   by (metis local.antisym_conv2 local.inverse_less_imp_less_neg local.negative_imp_inverse_negative
       local.nonzero_inverse_inverse_eq local.order.strict_implies_order)
 
 lemma le_imp_inverse_le_neg:
   "a \<le> b \<Longrightarrow> b < 0 \<Longrightarrow> inverse b \<le> inverse a"
   by (force simp add: le_less less_imp_inverse_less_neg)
 
 lemma inverse_le_iff_le_neg [simp]:
   "a < 0 \<Longrightarrow> b < 0 \<Longrightarrow> inverse a \<le> inverse b \<longleftrightarrow> b \<le> a"
   by (blast intro: le_imp_inverse_le_neg dest: inverse_le_imp_le_neg)
 
 lemma one_less_inverse:
   "0 < a \<Longrightarrow> a < 1 \<Longrightarrow> 1 < inverse a"
   using less_imp_inverse_less [of a 1, unfolded inverse_1] .
 
 lemma one_le_inverse:
   "0 < a \<Longrightarrow> a \<le> 1 \<Longrightarrow> 1 \<le> inverse a"
   using le_imp_inverse_le [of a 1, unfolded inverse_1] .
 
 lemma pos_le_divide_eq [field_simps]:
   assumes "0 < c"
   shows "a \<le> b / c \<longleftrightarrow> a * c \<le> b"
   using assms by (metis local.divide_eq_imp local.divide_inverse_commute
       local.dual_order.order_iff_strict local.dual_order.strict_iff_order
       local.mult_right_mono local.mult_strict_left_mono local.nonzero_divide_eq_eq
       local.order.strict_implies_order local.positive_imp_inverse_positive)
 
 lemma pos_less_divide_eq [field_simps]:
   assumes "0 < c"
   shows "a < b / c \<longleftrightarrow> a * c < b"
   using assms local.dual_order.strict_iff_order local.nonzero_divide_eq_eq local.pos_le_divide_eq
   by auto
 
 lemma neg_less_divide_eq [field_simps]:
   assumes "c < 0"
   shows "a < b / c \<longleftrightarrow> b < a * c"
   by (metis assms local.minus_divide_divide local.mult_minus_right local.neg_0_less_iff_less
       local.neg_less_iff_less local.pos_less_divide_eq)
 
 lemma neg_le_divide_eq [field_simps]:
   assumes "c < 0"
   shows "a \<le> b / c \<longleftrightarrow> b \<le> a * c"
   by (metis assms local.dual_order.order_iff_strict local.dual_order.strict_iff_order
       local.neg_less_divide_eq local.nonzero_divide_eq_eq)
 
 lemma pos_divide_le_eq [field_simps]:
   assumes "0 < c"
   shows "b / c \<le> a \<longleftrightarrow> b \<le> a * c"
   by (metis assms local.dual_order.strict_iff_order local.nonzero_eq_divide_eq
       local.pos_le_divide_eq)
 
 lemma pos_divide_less_eq [field_simps]:
   assumes "0 < c"
   shows "b / c < a \<longleftrightarrow> b < a * c"
   by (metis assms local.minus_divide_left local.mult_minus_left local.neg_less_iff_less
       local.pos_less_divide_eq)
 
 lemma neg_divide_le_eq [field_simps]:
   assumes "c < 0"
   shows "b / c \<le> a \<longleftrightarrow> a * c \<le> b"
   by (metis assms local.minus_divide_left local.mult_minus_left local.neg_le_divide_eq
       local.neg_le_iff_le)
 
 lemma neg_divide_less_eq [field_simps]:
   assumes "c < 0"
   shows "b / c < a \<longleftrightarrow> a * c < b"
   using assms local.dual_order.strict_iff_order local.neg_divide_le_eq by auto
 
 text\<open>The following \<open>field_simps\<close> rules are necessary, as minus is always moved atop of
 division but we want to get rid of division.\<close>
 
 lemma pos_le_minus_divide_eq [field_simps]: "0 < c \<Longrightarrow> a \<le> - (b / c) \<longleftrightarrow> a * c \<le> - b"
   unfolding minus_divide_left by (rule pos_le_divide_eq)
 
 lemma neg_le_minus_divide_eq [field_simps]: "c < 0 \<Longrightarrow> a \<le> - (b / c) \<longleftrightarrow> - b \<le> a * c"
   unfolding minus_divide_left by (rule neg_le_divide_eq)
 
 lemma pos_less_minus_divide_eq [field_simps]: "0 < c \<Longrightarrow> a < - (b / c) \<longleftrightarrow> a * c < - b"
   unfolding minus_divide_left by (rule pos_less_divide_eq)
 
 lemma neg_less_minus_divide_eq [field_simps]: "c < 0 \<Longrightarrow> a < - (b / c) \<longleftrightarrow> - b < a * c"
   unfolding minus_divide_left by (rule neg_less_divide_eq)
 
 lemma pos_minus_divide_less_eq [field_simps]: "0 < c \<Longrightarrow> - (b / c) < a \<longleftrightarrow> - b < a * c"
   unfolding minus_divide_left by (rule pos_divide_less_eq)
 
 lemma neg_minus_divide_less_eq [field_simps]: "c < 0 \<Longrightarrow> - (b / c) < a \<longleftrightarrow> a * c < - b"
   unfolding minus_divide_left by (rule neg_divide_less_eq)
 
 lemma pos_minus_divide_le_eq [field_simps]: "0 < c \<Longrightarrow> - (b / c) \<le> a \<longleftrightarrow> - b \<le> a * c"
   unfolding minus_divide_left by (rule pos_divide_le_eq)
 
 lemma neg_minus_divide_le_eq [field_simps]: "c < 0 \<Longrightarrow> - (b / c) \<le> a \<longleftrightarrow> a * c \<le> - b"
   unfolding minus_divide_left by (rule neg_divide_le_eq)
 
 lemma frac_less_eq:
   "y \<noteq> 0 \<Longrightarrow> z \<noteq> 0 \<Longrightarrow> x / y < w / z \<longleftrightarrow> (x * z - w * y) / (y * z) < 0"
   by (subst less_iff_diff_less_0) (simp add: diff_frac_eq )
 
 lemma frac_le_eq:
   "y \<noteq> 0 \<Longrightarrow> z \<noteq> 0 \<Longrightarrow> x / y \<le> w / z \<longleftrightarrow> (x * z - w * y) / (y * z) \<le> 0"
   by (subst le_iff_diff_le_0) (simp add: diff_frac_eq )
 
 
 text\<open>Lemmas \<open>sign_simps\<close> is a first attempt to automate proofs
 of positivity/negativity needed for \<open>field_simps\<close>. Have not added \<open>sign_simps\<close> to \<open>field_simps\<close>
   because the former can lead to case explosions.\<close>
 
 lemma divide_pos_pos[simp]:
   "0 < x \<Longrightarrow> 0 < y \<Longrightarrow> 0 < x / y"
   by(simp add:field_simps)
 
 lemma divide_nonneg_pos:
   "0 \<le> x \<Longrightarrow> 0 < y \<Longrightarrow> 0 \<le> x / y"
   by(simp add:field_simps)
 
 lemma divide_neg_pos:
   "x < 0 \<Longrightarrow> 0 < y \<Longrightarrow> x / y < 0"
   by(simp add:field_simps)
 
 lemma divide_nonpos_pos:
   "x \<le> 0 \<Longrightarrow> 0 < y \<Longrightarrow> x / y \<le> 0"
   by(simp add:field_simps)
 
 lemma divide_pos_neg:
   "0 < x \<Longrightarrow> y < 0 \<Longrightarrow> x / y < 0"
   by(simp add:field_simps)
 
 lemma divide_nonneg_neg:
   "0 \<le> x \<Longrightarrow> y < 0 \<Longrightarrow> x / y \<le> 0"
   by(simp add:field_simps)
 
 lemma divide_neg_neg:
   "x < 0 \<Longrightarrow> y < 0 \<Longrightarrow> 0 < x / y"
   by(simp add:field_simps)
 
 lemma divide_nonpos_neg:
   "x \<le> 0 \<Longrightarrow> y < 0 \<Longrightarrow> 0 \<le> x / y"
   by(simp add:field_simps)
 
 lemma divide_strict_right_mono:
   "a < b \<Longrightarrow> 0 < c \<Longrightarrow> a / c < b / c"
   by (simp add: less_imp_not_eq2 divide_inverse mult_strict_right_mono
       positive_imp_inverse_positive)
 
 
 lemma divide_strict_right_mono_neg:
   "b < a \<Longrightarrow> c < 0 \<Longrightarrow> a / c < b / c"
   by (simp add: local.neg_less_divide_eq)
 
 text\<open>The last premise ensures that \<^term>\<open>a\<close> and \<^term>\<open>b\<close>
       have the same sign\<close>
 lemma divide_strict_left_mono:
   "b < a \<Longrightarrow> 0 < c \<Longrightarrow> 0 < a*b \<Longrightarrow> c / a < c / b"
   by (metis local.divide_neg_pos local.dual_order.strict_iff_order local.frac_less_eq local.less_iff_diff_less_0 local.mult_not_zero local.mult_strict_left_mono)
 
 lemma divide_left_mono:
   "b \<le> a \<Longrightarrow> 0 \<le> c \<Longrightarrow> 0 < a*b \<Longrightarrow> c / a \<le> c / b"
   using local.divide_cancel_left local.divide_strict_left_mono local.dual_order.order_iff_strict by auto
 
 lemma divide_strict_left_mono_neg:
   "a < b \<Longrightarrow> c < 0 \<Longrightarrow> 0 < a*b \<Longrightarrow> c / a < c / b"
   by (metis local.divide_strict_left_mono local.minus_divide_left local.neg_0_less_iff_less local.neg_less_iff_less mult_commute)
 
 lemma mult_imp_div_pos_le: "0 < y \<Longrightarrow> x \<le> z * y \<Longrightarrow> x / y \<le> z"
   by (subst pos_divide_le_eq, assumption+)
 
 lemma mult_imp_le_div_pos: "0 < y \<Longrightarrow> z * y \<le> x \<Longrightarrow> z \<le> x / y"
   by(simp add:field_simps)
 
 lemma mult_imp_div_pos_less: "0 < y \<Longrightarrow> x < z * y \<Longrightarrow> x / y < z"
   by(simp add:field_simps)
 
 lemma mult_imp_less_div_pos: "0 < y \<Longrightarrow> z * y < x \<Longrightarrow> z < x / y"
   by(simp add:field_simps)
 
 lemma frac_le: "0 \<le> x \<Longrightarrow> x \<le> y \<Longrightarrow> 0 < w \<Longrightarrow> w \<le> z  \<Longrightarrow> x / z \<le> y / w"
   using local.mult_imp_div_pos_le local.mult_imp_le_div_pos local.mult_mono by auto
 
 lemma frac_less: "0 \<le> x \<Longrightarrow> x < y \<Longrightarrow> 0 < w \<Longrightarrow> w \<le> z \<Longrightarrow> x / z < y / w"
 proof-
   assume a1: "w \<le> z"
   assume a2: "0 < w"
   assume a3: "0 \<le> x"
   assume a4: "x < y"
   have f5: "a = 0 \<or> (b = c / a) = (b * a = c)"
     for a b c::'a
     by (meson local.nonzero_eq_divide_eq)
   have f6: "0 < z"
     using a2 a1 less_le_trans by blast
   have "z \<noteq> 0"
     using a2 a1 by (meson local.leD)
   moreover have "x / z \<noteq> y / w"
     using a1 a2 a3 a4 local.frac_eq_eq local.mult_less_le_imp_less by fastforce
   ultimately have "x / z \<noteq> y / w"
     using f5 by (metis (no_types))
   thus ?thesis
     using a4 a3 a2 a1 by (meson local.frac_le local.order.not_eq_order_implies_strict
         local.order.strict_implies_order)
 qed
 
 
 lemma frac_less2: "0 < x \<Longrightarrow> x \<le> y \<Longrightarrow> 0 < w \<Longrightarrow> w < z  \<Longrightarrow> x / z < y / w"
   by (metis local.antisym_conv2 local.divide_cancel_left local.dual_order.strict_implies_order
       local.frac_le local.frac_less)
 
 lemma less_half_sum: "a < b \<Longrightarrow> a < (a+b) / (1+1)"
   by (metis local.add_pos_pos local.add_strict_left_mono local.mult_imp_less_div_pos local.semiring_normalization_rules(4) local.zero_less_one mult_commute)
 
 lemma gt_half_sum: "a < b \<Longrightarrow> (a+b)/(1+1) < b"
   by (metis local.add_pos_pos local.add_strict_left_mono local.mult_imp_div_pos_less local.semiring_normalization_rules(24) local.semiring_normalization_rules(4) local.zero_less_one mult_commute)
 
 subclass unbounded_dense_order
 proof
   fix x y :: 'a
   have less_add_one: "a < a + 1" for a::'a by auto
   from less_add_one show "\<exists>y. x < y"
     by blast
 
   from less_add_one have "x + (- 1) < (x + 1) + (- 1)"
     by (rule add_strict_right_mono)
   hence "x - 1 < x + 1 - 1" by simp
   hence "x - 1 < x" by (simp add: algebra_simps)
   thus "\<exists>y. y < x" ..
   show "x < y \<Longrightarrow> \<exists>z>x. z < y" by (blast intro!: less_half_sum gt_half_sum)
 qed
 
 
 lemma dense_le_bounded:
   fixes x y z :: 'a
   assumes "x < y"
     and *: "\<And>w. \<lbrakk> x < w ; w < y \<rbrakk> \<Longrightarrow> w \<le> z"
   shows "y \<le> z"
 proof (rule dense_le)
   fix w assume "w < y"
   from dense[OF \<open>x < y\<close>] obtain u where "x < u" "u < y" by safe
   have "u \<le> w \<or> w \<le> u"
     using \<open>u < y\<close> \<open>w < y\<close> comparable local.order.strict_implies_order by blast
   thus "w \<le> z"
     using "*" \<open>u < y\<close> \<open>w < y\<close> \<open>x < u\<close> local.dual_order.trans local.order.strict_trans2 by blast
 qed
 
 subclass field_abs_sgn ..
 
 
 lemma nonzero_abs_inverse:
   "a \<noteq> 0 \<Longrightarrow> \<bar>inverse a\<bar> = inverse \<bar>a\<bar>"
   by (rule abs_inverse)
 
 lemma nonzero_abs_divide:
   "b \<noteq> 0 \<Longrightarrow> \<bar>a / b\<bar> = \<bar>a\<bar> / \<bar>b\<bar>"
   by (rule abs_divide)
 
 lemma field_le_epsilon:
   assumes e: "\<And>e. 0 < e \<Longrightarrow> x \<le> y + e"
   shows "x \<le> y"
 proof (rule dense_le)
   fix t assume "t < x"
   hence "0 < x - t" by (simp add: less_diff_eq)
   from e [OF this] have "x + 0 \<le> x + (y - t)" by (simp add: algebra_simps)
   hence "0 \<le> y - t" by (simp only: add_le_cancel_left)
   thus "t \<le> y" by (simp add: algebra_simps)
 qed
 
 lemma inverse_positive_iff_positive [simp]:
   "(0 < inverse a) = (0 < a)"
   using local.positive_imp_inverse_positive by fastforce
 
 lemma inverse_negative_iff_negative [simp]:
   "(inverse a < 0) = (a < 0)"
   using local.negative_imp_inverse_negative by fastforce
 
 lemma inverse_nonnegative_iff_nonnegative [simp]:
   "0 \<le> inverse a \<longleftrightarrow> 0 \<le> a"
   by (simp add: local.dual_order.order_iff_strict)
 
 lemma inverse_nonpositive_iff_nonpositive [simp]:
   "inverse a \<le> 0 \<longleftrightarrow> a \<le> 0"
   using local.inverse_nonnegative_iff_nonnegative local.neg_0_le_iff_le by fastforce
 
 lemma one_less_inverse_iff: "1 < inverse x \<longleftrightarrow> 0 < x \<and> x < 1"
   using less_trans[of 1 x 0 for x]
   by (metis local.dual_order.strict_trans local.inverse_1 local.inverse_less_imp_less local.inverse_positive_iff_positive local.one_less_inverse local.zero_less_one)
 
 lemma one_le_inverse_iff: "1 \<le> inverse x \<longleftrightarrow> 0 < x \<and> x \<le> 1"
   by (metis local.dual_order.strict_trans1 local.inverse_1 local.inverse_le_imp_le local.inverse_positive_iff_positive local.one_le_inverse local.zero_less_one)
 
 lemma inverse_less_1_iff: "inverse x < 1 \<longleftrightarrow> x \<le> 0 \<or> 1 < x"
 proof (rule)
   assume invx1: "inverse x < 1"
   have "inverse x \<le> 0 \<or> inverse x \<ge> 0"
     using comparable invx1 local.order.strict_implies_order local.zero_less_one by blast
   then consider (leq0) "inverse x \<le> 0" | (pos) "inverse x > 0" | (zero) "inverse x = 0"
     using local.antisym_conv1 by blast
   thus "x \<le> 0 \<or> 1 < x"
     by (metis invx1 local.eq_refl local.inverse_1 inverse_less_imp_less
               inverse_nonpositive_iff_nonpositive inverse_positive_iff_positive)
 next
   assume "x \<le> 0 \<or> 1 < x"
   then consider (neg) "x \<le> 0" | (g1) "1 < x" by auto
   thus "inverse x < 1"
     by (metis local.dual_order.not_eq_order_implies_strict local.dual_order.strict_trans
         local.inverse_1 local.inverse_negative_iff_negative local.inverse_zero
         local.less_imp_inverse_less local.zero_less_one)
 qed
 
 lemma inverse_le_1_iff: "inverse x \<le> 1 \<longleftrightarrow> x \<le> 0 \<or> 1 \<le> x"
   by (metis local.dual_order.order_iff_strict local.inverse_1 local.inverse_le_iff_le
       local.inverse_less_1_iff local.one_le_inverse_iff)
 
 text\<open>Simplify expressions such as \<open>0 < 1/x\<close> to \<open>0 < x\<close>\<close>
 
 lemma zero_le_divide_1_iff [simp]:
   "0 \<le> 1 / a \<longleftrightarrow> 0 \<le> a"
   using local.dual_order.order_iff_strict local.inverse_eq_divide
     local.inverse_positive_iff_positive by auto
 
 lemma zero_less_divide_1_iff [simp]:
   "0 < 1 / a \<longleftrightarrow> 0 < a"
   by (simp add: local.dual_order.strict_iff_order)
 
 lemma divide_le_0_1_iff [simp]:
   "1 / a \<le> 0 \<longleftrightarrow> a \<le> 0"
   by (smt local.abs_0 local.abs_1 local.abs_divide local.abs_neg local.abs_nn
       local.divide_cancel_left local.le_minus_iff local.minus_divide_right local.zero_neq_one)
 
 lemma divide_less_0_1_iff [simp]:
   "1 / a < 0 \<longleftrightarrow> a < 0"
   using local.dual_order.strict_iff_order by auto
 
 lemma divide_right_mono:
   "a \<le> b \<Longrightarrow> 0 \<le> c \<Longrightarrow> a/c \<le> b/c"
   using local.divide_cancel_right local.divide_strict_right_mono local.dual_order.order_iff_strict by blast
 
 lemma divide_right_mono_neg: "a \<le> b
     \<Longrightarrow> c \<le> 0 \<Longrightarrow> b / c \<le> a / c"
   by (metis local.divide_cancel_right local.divide_strict_right_mono_neg local.dual_order.strict_implies_order local.eq_refl local.le_imp_less_or_eq)
 
 lemma divide_left_mono_neg: "a \<le> b
     \<Longrightarrow> c \<le> 0 \<Longrightarrow> 0 < a * b \<Longrightarrow> c / a \<le> c / b"
   by (metis local.divide_left_mono local.minus_divide_left local.neg_0_le_iff_le local.neg_le_iff_le mult_commute)
 
 lemma divide_nonneg_nonneg [simp]:
   "0 \<le> x \<Longrightarrow> 0 \<le> y \<Longrightarrow> 0 \<le> x / y"
   using local.divide_eq_0_iff local.divide_nonneg_pos local.dual_order.order_iff_strict by blast
 
 lemma divide_nonpos_nonpos:
   "x \<le> 0 \<Longrightarrow> y \<le> 0 \<Longrightarrow> 0 \<le> x / y"
   using local.divide_nonpos_neg local.dual_order.order_iff_strict by auto
 
 lemma divide_nonneg_nonpos:
   "0 \<le> x \<Longrightarrow> y \<le> 0 \<Longrightarrow> x / y \<le> 0"
   by (metis local.divide_eq_0_iff local.divide_nonneg_neg local.dual_order.order_iff_strict)
 
 lemma divide_nonpos_nonneg:
   "x \<le> 0 \<Longrightarrow> 0 \<le> y \<Longrightarrow> x / y \<le> 0"
   using local.divide_nonpos_pos local.dual_order.order_iff_strict by auto
 
 text \<open>Conditional Simplification Rules: No Case Splits\<close>
 
 lemma le_divide_eq_1_pos [simp]:
   "0 < a \<Longrightarrow> (1 \<le> b/a) = (a \<le> b)"
   by (simp add: local.pos_le_divide_eq)
 
 lemma le_divide_eq_1_neg [simp]:
   "a < 0 \<Longrightarrow> (1 \<le> b/a) = (b \<le> a)"
   by (metis local.le_divide_eq_1_pos local.minus_divide_divide local.neg_0_less_iff_less local.neg_le_iff_le)
 
 lemma divide_le_eq_1_pos [simp]:
   "0 < a \<Longrightarrow> (b/a \<le> 1) = (b \<le> a)"
   using local.pos_divide_le_eq by auto
 
 lemma divide_le_eq_1_neg [simp]:
   "a < 0 \<Longrightarrow> (b/a \<le> 1) = (a \<le> b)"
   by (metis local.divide_le_eq_1_pos local.minus_divide_divide local.neg_0_less_iff_less
       local.neg_le_iff_le)
 
 lemma less_divide_eq_1_pos [simp]:
   "0 < a \<Longrightarrow> (1 < b/a) = (a < b)"
   by (simp add: local.dual_order.strict_iff_order)
 
 lemma less_divide_eq_1_neg [simp]:
   "a < 0 \<Longrightarrow> (1 < b/a) = (b < a)"
   using local.dual_order.strict_iff_order by auto
 
 lemma divide_less_eq_1_pos [simp]:
   "0 < a \<Longrightarrow> (b/a < 1) = (b < a)"
   using local.divide_le_eq_1_pos local.dual_order.strict_iff_order by auto
 
 lemma divide_less_eq_1_neg [simp]:
   "a < 0 \<Longrightarrow> b/a < 1 \<longleftrightarrow> a < b"
   using local.dual_order.strict_iff_order by auto
 
 lemma abs_div_pos: "0 < y \<Longrightarrow>
     \<bar>x\<bar> / y = \<bar>x / y\<bar>"
   by (simp add: local.abs_pos)
 
 lemma zero_le_divide_abs_iff [simp]: "(0 \<le> a / \<bar>b\<bar>) = (0 \<le> a | b = 0)"
 proof
   assume assm: "0 \<le> a / \<bar>b\<bar>"
   have absb: "abs b \<ge> 0" by (fact abs_nn)
   thus "0 \<le> a \<or> b = 0"
     using absb assm local.abs_eq_0_iff local.mult_nonneg_nonneg by fastforce
 next
   assume "0 \<le> a \<or> b = 0"
   then consider (a) "0 \<le> a" | (b) "b = 0" by atomize_elim auto
   thus "0 \<le> a / \<bar>b\<bar>"
     by (metis local.abs_eq_0_iff local.abs_nn local.divide_eq_0_iff local.divide_nonneg_nonneg)
 qed
 
 
 lemma divide_le_0_abs_iff [simp]: "(a / \<bar>b\<bar> \<le> 0) = (a \<le> 0 | b = 0)"
   by (metis local.minus_divide_left local.neg_0_le_iff_le local.zero_le_divide_abs_iff)
 
 text\<open>For creating values between \<^term>\<open>u\<close> and \<^term>\<open>v\<close>.\<close>
 lemma scaling_mono:
   assumes "u \<le> v" and "0 \<le> r" and "r \<le> s"
   shows "u + r * (v - u) / s \<le> v"
 proof -
   have "r/s \<le> 1" using assms
     by (metis local.divide_le_eq_1_pos local.division_ring_divide_zero
         local.dual_order.order_iff_strict local.dual_order.trans local.zero_less_one)
   hence "(r/s) * (v - u) \<le> 1 * (v - u)"
     using assms(1) local.diff_ge_0_iff_ge local.mult_right_mono by blast
   thus ?thesis
     by (simp add: field_simps)
 qed
 
 end (* class nice_ordered_field *)
 
 
 code_identifier
   code_module Ordered_Fields \<rightharpoonup> (SML) Arith and (OCaml) Arith and (Haskell) Arith
 
-subsection\<open>Ordered Complex\<close>
-
-subsection\<open>Ordered Complex\<close>
-
 subsection \<open>Ordering on complex numbers\<close>
 
 instantiation complex :: nice_ordered_field begin
 instance
 proof intro_classes
   note defs = less_eq_complex_def less_complex_def abs_complex_def
   fix x y z a b c :: complex
   show "a \<le> 0 \<Longrightarrow> \<bar>a\<bar> = - a" unfolding defs
     by (simp add: cmod_eq_Re complex_is_Real_iff)
   show "0 \<le> a \<Longrightarrow> \<bar>a\<bar> = a"
     unfolding defs
     by (metis abs_of_nonneg cmod_eq_Re comp_apply complex.exhaust_sel complex_of_real_def zero_complex.simps(1) zero_complex.simps(2))
   show "a < b \<Longrightarrow> 0 < c \<Longrightarrow> c * a < c * b" unfolding defs by auto
   show "0 < (1::complex)" unfolding defs by simp
   show "0 < a \<Longrightarrow> 0 < inverse a" unfolding defs by auto
   define ra ia rb ib rc ic where "ra = Re a" "ia = Im a" "rb = Re b" "ib = Im b" "rc = Re c" "ic = Im c"
   note ri = this[symmetric]
   hence "a = Complex ra ia" "b = Complex rb ib" "c = Complex rc ic" by auto
   note ri = this ri
   have "rb \<le> ra"
     if "1 / ra \<le> (if rb = 0 then 0 else 1 / rb)"
       and "ia = 0" and "0 < ra" and "ib = 0"
   proof(cases "rb = 0")
     case True
     thus ?thesis
       using that(3) by auto
   next
     case False
     thus ?thesis
       by (smt nice_ordered_field_class.frac_less2 that(1) that(3))
   qed
   thus "inverse a \<le> inverse b \<Longrightarrow> 0 < a \<Longrightarrow> b \<le> a" unfolding defs ri
     by (auto simp: power2_eq_square)
   show "(\<And>a. a < b \<Longrightarrow> a \<le> c) \<Longrightarrow> b \<le> c" unfolding defs ri
     by (metis complex.sel(1) complex.sel(2) dense less_le_not_le
         nice_ordered_field_class.linordered_field_no_lb not_le_imp_less)
   show "0 \<le> a \<Longrightarrow> 0 \<le> b \<Longrightarrow> a \<le> b \<or> b \<le> a" unfolding defs by auto
   show "0 \<le> \<bar>x\<bar>" unfolding defs by auto
 qed
 end
 
 lemma less_eq_complexI: "Re x \<le> Re y \<Longrightarrow> Im x = Im y \<Longrightarrow> x\<le>y" unfolding less_eq_complex_def
   by simp
 lemma less_complexI: "Re x < Re y \<Longrightarrow> Im x = Im y \<Longrightarrow> x<y" unfolding less_complex_def
   by simp
 
 lemma complex_of_real_mono:
   "x \<le> y \<Longrightarrow> complex_of_real x \<le> complex_of_real y"
   unfolding less_eq_complex_def by auto
 
 lemma complex_of_real_mono_iff[simp]:
   "complex_of_real x \<le> complex_of_real y \<longleftrightarrow> x \<le> y"
   unfolding less_eq_complex_def by auto
 
 lemma complex_of_real_strict_mono_iff[simp]:
   "complex_of_real x < complex_of_real y \<longleftrightarrow> x < y"
   unfolding less_complex_def by auto
 
 lemma complex_of_real_nn_iff[simp]:
   "0 \<le> complex_of_real y \<longleftrightarrow> 0 \<le> y"
   unfolding less_eq_complex_def by auto
 
 lemma complex_of_real_pos_iff[simp]:
   "0 < complex_of_real y \<longleftrightarrow> 0 < y"
   unfolding less_complex_def by auto
 
 lemma Re_mono: "x \<le> y \<Longrightarrow> Re x \<le> Re y"
   unfolding less_eq_complex_def by simp
 
 lemma comp_Im_same: "x \<le> y \<Longrightarrow> Im x = Im y"
   unfolding less_eq_complex_def by simp
 
 lemma Re_strict_mono: "x < y \<Longrightarrow> Re x < Re y"
   unfolding less_complex_def by simp
 
 lemma complex_of_real_cmod: assumes "x \<ge> 0" shows "complex_of_real (cmod x) = x"
   by (metis Reals_cases abs_of_nonneg assms comp_Im_same complex_is_Real_iff complex_of_real_nn_iff norm_of_real zero_complex.simps(2))
 
 
 end
diff --git a/thys/Complex_Bounded_Operators/extra/Extra_Vector_Spaces.thy b/thys/Complex_Bounded_Operators/extra/Extra_Vector_Spaces.thy
--- a/thys/Complex_Bounded_Operators/extra/Extra_Vector_Spaces.thy
+++ b/thys/Complex_Bounded_Operators/extra/Extra_Vector_Spaces.thy
@@ -1,178 +1,209 @@
 section \<open>\<open>Extra_Vector_Spaces\<close> -- Additional facts about vector spaces\<close>
 
 theory Extra_Vector_Spaces
   imports
     "HOL-Analysis.Inner_Product"
     "HOL-Analysis.Euclidean_Space"
     "HOL-Library.Indicator_Function"
     "HOL-Analysis.Topology_Euclidean_Space"
     "HOL-Analysis.Line_Segment"
     "HOL-Analysis.Bounded_Linear_Function"
     Extra_General
 begin
 
 subsection \<open>Euclidean spaces\<close>
 
 typedef 'a euclidean_space = "UNIV :: ('a \<Rightarrow> real) set" ..
 setup_lifting type_definition_euclidean_space
 
 instantiation euclidean_space :: (type) real_vector begin
 lift_definition plus_euclidean_space ::
   "'a euclidean_space \<Rightarrow> 'a euclidean_space \<Rightarrow> 'a euclidean_space"
   is "\<lambda>f g x. f x + g x" .
 lift_definition zero_euclidean_space :: "'a euclidean_space" is "\<lambda>_. 0" .
 lift_definition uminus_euclidean_space :: 
   "'a euclidean_space \<Rightarrow> 'a euclidean_space" 
   is "\<lambda>f x. - f x" .
 lift_definition minus_euclidean_space :: 
   "'a euclidean_space \<Rightarrow> 'a euclidean_space \<Rightarrow> 'a euclidean_space" 
   is "\<lambda>f g x. f x - g x".
 lift_definition scaleR_euclidean_space :: 
   "real \<Rightarrow> 'a euclidean_space \<Rightarrow> 'a euclidean_space" 
   is "\<lambda>c f x. c * f x" .
 instance
   apply intro_classes
   by (transfer; auto intro: distrib_left distrib_right)+
 end
 
 instantiation euclidean_space :: (finite) real_inner begin
 lift_definition inner_euclidean_space :: "'a euclidean_space \<Rightarrow> 'a euclidean_space \<Rightarrow> real"
   is "\<lambda>f g. \<Sum>x\<in>UNIV. f x * g x :: real" .
 definition "norm_euclidean_space (x::'a euclidean_space) = sqrt (inner x x)"
 definition "dist_euclidean_space (x::'a euclidean_space) y = norm (x-y)"
 definition "sgn x = x /\<^sub>R norm x" for x::"'a euclidean_space"
 definition "uniformity = (INF e\<in>{0<..}. principal {(x::'a euclidean_space, y). dist x y < e})"
 definition "open U = (\<forall>x\<in>U. \<forall>\<^sub>F (x'::'a euclidean_space, y) in uniformity. x' = x \<longrightarrow> y \<in> U)"
 instance
 proof intro_classes
   fix x :: "'a euclidean_space"
     and y :: "'a euclidean_space"
     and z :: "'a euclidean_space"
   show "dist (x::'a euclidean_space) y = norm (x - y)"
     and "sgn (x::'a euclidean_space) = x /\<^sub>R norm x"
     and "uniformity = (INF e\<in>{0<..}. principal {(x, y). dist (x::'a euclidean_space) y < e})"
     and "open U = (\<forall>x\<in>U. \<forall>\<^sub>F (x', y) in uniformity. (x'::'a euclidean_space) = x \<longrightarrow> y \<in> U)"
     and "norm x = sqrt (inner x x)" for U
     unfolding dist_euclidean_space_def norm_euclidean_space_def sgn_euclidean_space_def
       uniformity_euclidean_space_def open_euclidean_space_def
     by simp_all
 
   show "inner x y = inner y x"
     apply transfer
     by (simp add: mult.commute)
   show "inner (x + y) z = inner x z + inner y z"
   proof transfer
     fix x y z::"'a \<Rightarrow> real"
     have "(\<Sum>i\<in>UNIV. (x i + y i) * z i) = (\<Sum>i\<in>UNIV. x i * z i + y i * z i)"
       by (simp add: distrib_left mult.commute)
     thus "(\<Sum>i\<in>UNIV. (x i + y i) * z i) = (\<Sum>j\<in>UNIV. x j * z j) + (\<Sum>k\<in>UNIV. y k * z k)"
       by (subst sum.distrib[symmetric])      
   qed
 
   show "inner (r *\<^sub>R x) y = r * (inner x y)" for r
   proof transfer
     fix r and x y::"'a\<Rightarrow>real"
     have "(\<Sum>i\<in>UNIV. r * x i * y i) = (\<Sum>i\<in>UNIV. r * (x i * y i))"
       by (simp add: mult.assoc)
     thus "(\<Sum>i\<in>UNIV. r * x i * y i) = r * (\<Sum>j\<in>UNIV. x j * y j)"
       by (subst sum_distrib_left)
   qed
   show "0 \<le> inner x x"
     apply transfer
     by (simp add: sum_nonneg)
   show "(inner x x = 0) = (x = 0)"
   proof (transfer, rule)
     fix f :: "'a \<Rightarrow> real"
     assume "(\<Sum>i\<in>UNIV. f i * f i) = 0"
     hence "f x * f x = 0" for x
       apply (rule_tac sum_nonneg_eq_0_iff[THEN iffD1, rule_format, where A=UNIV and x=x])
       by auto
     thus "f = (\<lambda>_. 0)"
       by auto
   qed auto
 qed
 end
 
 instantiation euclidean_space :: (finite) euclidean_space begin
 lift_definition euclidean_space_basis_vector :: "'a \<Rightarrow> 'a euclidean_space" is
   "\<lambda>x. indicator {x}" .
 definition "Basis_euclidean_space == (euclidean_space_basis_vector ` UNIV)"
 instance
 proof intro_classes
   fix u :: "'a euclidean_space"
     and v :: "'a euclidean_space"
   show "(Basis::'a euclidean_space set) \<noteq> {}"
     unfolding Basis_euclidean_space_def by simp
   show "finite (Basis::'a euclidean_space set)"
     unfolding Basis_euclidean_space_def by simp
   show "inner u v = (if u = v then 1 else 0)"
     if "u \<in> Basis" and "v \<in> Basis"
     using that unfolding Basis_euclidean_space_def
     apply transfer apply auto
     by (auto simp: indicator_def)
   show "(\<forall>v\<in>Basis. inner u v = 0) = (u = 0)"
     unfolding Basis_euclidean_space_def
     apply transfer
     by auto
 qed
 end (* euclidean_space :: (finite) euclidean_space *)
 
 subsection \<open>Misc\<close>
 
 lemma closure_bounded_linear_image_subset_eq:
   assumes f: "bounded_linear f"
   shows "closure (f ` closure S) = closure (f ` S)"
   by (meson closed_closure closure_bounded_linear_image_subset closure_minimal closure_mono closure_subset f image_mono subset_antisym)
 
 lemma not_singleton_real_normed_is_perfect_space[simp]: \<open>class.perfect_space (open :: 'a::{not_singleton,real_normed_vector} set \<Rightarrow> bool)\<close>
   apply standard
   by (metis UNIV_not_singleton clopen closed_singleton empty_not_insert)
 
 lemma infsum_bounded_linear:
   assumes \<open>bounded_linear f\<close>
   assumes \<open>g summable_on S\<close>
   shows \<open>infsum (f \<circ> g) S = f (infsum g S)\<close>
   apply (rule infsum_comm_additive)
   using assms blinfun_apply_induct blinfun.additive_right
   by (auto simp: linear_continuous_within)
 
 lemma has_sum_bounded_linear: 
   assumes \<open>bounded_linear f\<close>
   assumes \<open>has_sum g S x\<close>
   shows \<open>has_sum (f o g) S (f x)\<close>
   apply (rule has_sum_comm_additive)
   using assms blinfun_apply_induct blinfun.additive_right apply auto
   using isCont_def linear_continuous_at by fastforce
 
 lemma abs_summable_on_bounded_linear:
   assumes \<open>bounded_linear f\<close>
   assumes \<open>g abs_summable_on S\<close>
   shows \<open>(f o g) abs_summable_on S\<close>
 proof -
   have bound: \<open>norm (f (g x)) \<le> onorm f * norm (g x)\<close> for x
     apply (rule onorm)
     by (simp add: assms(1))
 
   from assms(2) have \<open>(\<lambda>x. onorm f *\<^sub>R g x) abs_summable_on S\<close>
     by (auto intro!: summable_on_cmult_right)
   then have \<open>(\<lambda>x. f (g x)) abs_summable_on S\<close>
     apply (rule abs_summable_on_comparison_test)
     using bound by (auto simp: assms(1) onorm_pos_le)
   then show ?thesis
     by auto
 qed
 
 lemma norm_plus_leq_norm_prod: \<open>norm (a + b) \<le> sqrt 2 * norm (a, b)\<close>
 proof -
   have \<open>(norm (a + b))\<^sup>2 \<le> (norm a + norm b)\<^sup>2\<close>
     using norm_triangle_ineq by auto
   also have \<open>\<dots> \<le> 2 * ((norm a)\<^sup>2 + (norm b)\<^sup>2)\<close>
     by (smt (verit, best) power2_sum sum_squares_bound)
   also have \<open>\<dots> \<le> (sqrt 2 * norm (a, b))\<^sup>2\<close>
     by (auto simp: power_mult_distrib norm_prod_def simp del: power_mono_iff)
   finally show ?thesis
     by auto
 qed
 
+lemma ex_norm1:
+  assumes \<open>(UNIV::'a::real_normed_vector set) \<noteq> {0}\<close>
+  shows \<open>\<exists>x::'a. norm x = 1\<close>
+proof-
+  have \<open>\<exists>x::'a. x \<noteq> 0\<close>
+    using assms by fastforce
+  then obtain x::'a where \<open>x \<noteq> 0\<close>
+    by blast
+  hence \<open>norm x \<noteq> 0\<close>
+    by simp
+  hence \<open>(norm x) / (norm x) = 1\<close>
+    by simp
+  moreover have \<open>(norm x) / (norm x) = norm (x /\<^sub>R (norm x))\<close>
+    by simp
+  ultimately have \<open>norm (x /\<^sub>R (norm x)) = 1\<close>
+    by simp
+  thus ?thesis
+    by blast
+qed
+
+lemma bdd_above_norm_f:
+  assumes "bounded_linear f"
+  shows \<open>bdd_above {norm (f x) |x. norm x = 1}\<close>
+proof-
+  have \<open>\<exists>M. \<forall>x. norm x = 1 \<longrightarrow> norm (f x) \<le> M\<close>
+    using assms
+    by (metis bounded_linear.axioms(2) bounded_linear_axioms_def)
+  thus ?thesis by auto
+qed
+
+
 
 end
diff --git a/thys/Registers/Finite_Tensor_Product.thy b/thys/Registers/Finite_Tensor_Product.thy
--- a/thys/Registers/Finite_Tensor_Product.thy
+++ b/thys/Registers/Finite_Tensor_Product.thy
@@ -1,790 +1,790 @@
 section \<open>Tensor products (finite dimensional)\<close>
 
 theory Finite_Tensor_Product
   imports Complex_Bounded_Operators.Complex_L2 Misc
 begin
 
 declare cblinfun.scaleC_right[simp]
 
 unbundle cblinfun_notation
 no_notation m_inv ("inv\<index> _" [81] 80)
 
 lift_definition tensor_ell2 :: \<open>'a::finite ell2 \<Rightarrow> 'b::finite ell2 \<Rightarrow> ('a\<times>'b) ell2\<close> (infixr "\<otimes>\<^sub>s" 70) is
   \<open>\<lambda>\<psi> \<phi> (i,j). \<psi> i * \<phi> j\<close>
   by simp
 
 lemma tensor_ell2_add2: \<open>tensor_ell2 a (b + c) = tensor_ell2 a b + tensor_ell2 a c\<close>
   apply transfer apply (rule ext) apply (auto simp: case_prod_beta)
   by (meson algebra_simps)
 
 lemma tensor_ell2_add1: \<open>tensor_ell2 (a + b) c = tensor_ell2 a c + tensor_ell2 b c\<close>
   apply transfer apply (rule ext) apply (auto simp: case_prod_beta)
   by (simp add: vector_space_over_itself.scale_left_distrib)
 
 lemma tensor_ell2_scaleC2: \<open>tensor_ell2 a (c *\<^sub>C b) = c *\<^sub>C tensor_ell2 a b\<close>
   apply transfer apply (rule ext) by (auto simp: case_prod_beta)
 
 lemma tensor_ell2_scaleC1: \<open>tensor_ell2 (c *\<^sub>C a) b = c *\<^sub>C tensor_ell2 a b\<close>
   apply transfer apply (rule ext) by (auto simp: case_prod_beta)
 
-lemma tensor_ell2_inner_prod[simp]: \<open>\<langle>tensor_ell2 a b, tensor_ell2 c d\<rangle> = \<langle>a,c\<rangle> * \<langle>b,d\<rangle>\<close>
+lemma tensor_ell2_inner_prod[simp]: \<open>tensor_ell2 a b \<bullet>\<^sub>C tensor_ell2 c d = (a \<bullet>\<^sub>C c) * (b \<bullet>\<^sub>C d)\<close>
   apply transfer
   by (auto simp: case_prod_beta sum_product sum.cartesian_product mult.assoc mult.left_commute)
 
 lemma clinear_tensor_ell21: "clinear (\<lambda>b. tensor_ell2 a b)"
   apply (rule clinearI; transfer)
    apply (auto simp: case_prod_beta)
   by (simp add: cond_case_prod_eta algebra_simps)
 
 lemma clinear_tensor_ell22: "clinear (\<lambda>a. tensor_ell2 a b)"
   apply (rule clinearI; transfer)
    apply (auto simp: case_prod_beta)
   by (simp add: case_prod_beta' algebra_simps)
 
 lemma tensor_ell2_ket[simp]: "tensor_ell2 (ket i) (ket j) = ket (i,j)"
   apply transfer by auto
 
 
 definition tensor_op :: \<open>('a ell2, 'b::finite ell2) cblinfun \<Rightarrow> ('c ell2, 'd::finite ell2) cblinfun
       \<Rightarrow> (('a\<times>'c) ell2, ('b\<times>'d) ell2) cblinfun\<close> (infixr "\<otimes>\<^sub>o" 70) where
   \<open>tensor_op M N = (SOME P. \<forall>a c. P *\<^sub>V (ket (a,c))
       = tensor_ell2 (M *\<^sub>V ket a) (N *\<^sub>V ket c))\<close>
 
 lemma tensor_op_ket: 
   fixes a :: \<open>'a::finite\<close> and b :: \<open>'b\<close> and c :: \<open>'c::finite\<close> and d :: \<open>'d\<close>
   shows \<open>tensor_op M N *\<^sub>V (ket (a,c)) = tensor_ell2 (M *\<^sub>V ket a) (N *\<^sub>V ket c)\<close>
 proof -
   define S :: \<open>('a\<times>'c) ell2 set\<close> where "S = ket ` UNIV"
   define \<phi> where \<open>\<phi> = (\<lambda>(a,c). tensor_ell2 (M *\<^sub>V ket a) (N *\<^sub>V ket c))\<close>
   define \<phi>' where \<open>\<phi>' = \<phi> \<circ> inv ket\<close>
 
   have def: \<open>tensor_op M N = (SOME P. \<forall>a c. P *\<^sub>V (ket (a,c)) = \<phi> (a,c))\<close>
     unfolding tensor_op_def \<phi>_def by auto
 
   have \<open>cindependent S\<close>
     using S_def cindependent_ket by blast
   moreover have \<open>cspan S = UNIV\<close>
     using S_def cspan_range_ket_finite by blast
   ultimately have "cblinfun_extension_exists S \<phi>'"
     by (rule cblinfun_extension_exists_finite_dim)
   then have "\<exists>P. \<forall>x\<in>S. P *\<^sub>V x = \<phi>' x"
     unfolding cblinfun_extension_exists_def by auto
   then have ex: \<open>\<exists>P. \<forall>a c. P *\<^sub>V ket (a,c) = \<phi> (a,c)\<close>
     by (metis S_def \<phi>'_def comp_eq_dest_lhs inj_ket inv_f_f rangeI)
 
 
   then have \<open>tensor_op M N *\<^sub>V (ket (a,c)) = \<phi> (a,c)\<close>
     unfolding def apply (rule someI2_ex[where P=\<open>\<lambda>P. \<forall>a c. P *\<^sub>V (ket (a,c)) = \<phi> (a,c)\<close>])
     by auto
   then show ?thesis
     unfolding \<phi>_def by auto
 qed
 
 
 lemma tensor_op_ell2: "tensor_op A B *\<^sub>V tensor_ell2 \<psi> \<phi> = tensor_ell2 (A *\<^sub>V \<psi>) (B *\<^sub>V \<phi>)"
 proof -
   have 1: \<open>clinear (\<lambda>a. tensor_op A B *\<^sub>V tensor_ell2 a (ket b))\<close> for b
     by (auto intro!: clinearI simp: tensor_ell2_add1 tensor_ell2_scaleC1 cblinfun.add_right)
   have 2: \<open>clinear (\<lambda>a. tensor_ell2 (A *\<^sub>V a) (B *\<^sub>V ket b))\<close> for b
     by (auto intro!: clinearI simp: tensor_ell2_add1 tensor_ell2_scaleC1 cblinfun.add_right)
   have 3: \<open>clinear (\<lambda>a. tensor_op A B *\<^sub>V tensor_ell2 \<psi> a)\<close>
     by (auto intro!: clinearI simp: tensor_ell2_add2 tensor_ell2_scaleC2 cblinfun.add_right)
   have 4: \<open>clinear (\<lambda>a. tensor_ell2 (A *\<^sub>V \<psi>) (B *\<^sub>V a))\<close>
     by (auto intro!: clinearI simp: tensor_ell2_add2 tensor_ell2_scaleC2 cblinfun.add_right)
 
   have eq_ket_ket: \<open>tensor_op A B *\<^sub>V tensor_ell2 (ket a) (ket b) = tensor_ell2 (A *\<^sub>V ket a) (B *\<^sub>V ket b)\<close> for a b
     by (simp add: tensor_op_ket)
   have eq_ket: \<open>tensor_op A B *\<^sub>V tensor_ell2 \<psi> (ket b) = tensor_ell2 (A *\<^sub>V \<psi>) (B *\<^sub>V ket b)\<close> for b
     apply (rule fun_cong[where x=\<psi>])
     using 1 2 eq_ket_ket by (rule clinear_equal_ket)
   show ?thesis 
     apply (rule fun_cong[where x=\<phi>])
     using 3 4 eq_ket by (rule clinear_equal_ket)
 qed
 
 lemma comp_tensor_op: "(tensor_op a b) o\<^sub>C\<^sub>L (tensor_op c d) = tensor_op (a o\<^sub>C\<^sub>L c) (b o\<^sub>C\<^sub>L d)"
   for a :: "'e::finite ell2 \<Rightarrow>\<^sub>C\<^sub>L 'c::finite ell2" and b :: "'f::finite ell2 \<Rightarrow>\<^sub>C\<^sub>L 'd::finite ell2" and
       c :: "'a::finite ell2 \<Rightarrow>\<^sub>C\<^sub>L 'e ell2" and d :: "'b::finite ell2 \<Rightarrow>\<^sub>C\<^sub>L 'f ell2"
   apply (rule equal_ket)
   apply (rename_tac ij, case_tac ij, rename_tac i j, hypsubst_thin)
   by (simp flip: tensor_ell2_ket add: tensor_op_ell2 cblinfun_apply_cblinfun_compose)
 
 
 lemma tensor_op_cbilinear: \<open>cbilinear (tensor_op :: 'a::finite ell2 \<Rightarrow>\<^sub>C\<^sub>L 'b::finite ell2
                                                  \<Rightarrow> 'c::finite ell2 \<Rightarrow>\<^sub>C\<^sub>L 'd::finite ell2 \<Rightarrow> _)\<close>
 proof -
   have \<open>clinear (\<lambda>b::'c ell2 \<Rightarrow>\<^sub>C\<^sub>L 'd ell2. tensor_op a b)\<close> for a :: \<open>'a ell2 \<Rightarrow>\<^sub>C\<^sub>L 'b ell2\<close>
     apply (rule clinearI)
      apply (rule equal_ket, rename_tac ij, case_tac ij, rename_tac i j, hypsubst_thin)
      apply (simp flip: tensor_ell2_ket add: tensor_op_ell2 cblinfun.add_left tensor_ell2_add2)
     apply (rule equal_ket, rename_tac ij, case_tac ij, rename_tac i j, hypsubst_thin)
     by (simp add: scaleC_cblinfun.rep_eq tensor_ell2_scaleC2 tensor_op_ket)
 
   moreover have \<open>clinear (\<lambda>a::'a::finite ell2 \<Rightarrow>\<^sub>C\<^sub>L 'b::finite ell2. tensor_op a b)\<close> for b :: \<open>'c ell2 \<Rightarrow>\<^sub>C\<^sub>L 'd ell2\<close>
     apply (rule clinearI)
      apply (rule equal_ket, rename_tac ij, case_tac ij, rename_tac i j, hypsubst_thin)
      apply (simp flip: tensor_ell2_ket add: tensor_op_ell2 cblinfun.add_left tensor_ell2_add1)
     apply (rule equal_ket, rename_tac ij, case_tac ij, rename_tac i j, hypsubst_thin)
     by (simp add: scaleC_cblinfun.rep_eq tensor_ell2_scaleC1 tensor_op_ket)
 
   ultimately show ?thesis
     unfolding cbilinear_def by auto
 qed
 
 
 lemma tensor_butter: \<open>tensor_op (butterket i j) (butterket k l) = butterket (i,k) (j,l)\<close>
   for i :: "_" and j :: "_::finite" and k :: "_" and l :: "_::finite"
   apply (rule equal_ket, rename_tac x, case_tac x)
   apply (auto simp flip: tensor_ell2_ket simp: cblinfun_apply_cblinfun_compose tensor_op_ell2 butterfly_def)
   by (auto simp: tensor_ell2_scaleC1 tensor_ell2_scaleC2)
 
 lemma cspan_tensor_op: \<open>cspan {tensor_op (butterket i j) (butterket k l)| i (j::_::finite) k (l::_::finite). True} = UNIV\<close>
   unfolding tensor_butter
   apply (subst cspan_butterfly_ket[symmetric])
   by (metis surj_pair)
 
 lemma cindependent_tensor_op: \<open>cindependent {tensor_op (butterket i j) (butterket k l)| i (j::_::finite) k (l::_::finite). True}\<close>
   unfolding tensor_butter
   using cindependent_butterfly_ket
   by (smt (z3) Collect_mono_iff complex_vector.independent_mono)
 
 
 lemma tensor_extensionality:
   fixes F G :: \<open>((('a::finite \<times> 'b::finite) ell2) \<Rightarrow>\<^sub>C\<^sub>L (('c::finite \<times> 'd::finite) ell2)) \<Rightarrow> 'e::complex_vector\<close>
   assumes [simp]: "clinear F" "clinear G"
   assumes tensor_eq: "(\<And>a b. F (tensor_op a b) = G (tensor_op a b))"
   shows "F = G"
 proof (rule ext, rule complex_vector.linear_eq_on_span[where f=F and g=G])
   show \<open>clinear F\<close> and \<open>clinear G\<close>
     using assms by (simp_all add: cbilinear_def)
   show \<open>x \<in> cspan  {tensor_op (butterket i j) (butterket k l)| i j k l. True}\<close> 
     for x :: \<open>('a \<times> 'b) ell2 \<Rightarrow>\<^sub>C\<^sub>L ('c \<times> 'd) ell2\<close>
     using cspan_tensor_op by auto
   show \<open>F x = G x\<close> if \<open>x \<in> {tensor_op (butterket i j) (butterket k l) |i j k l. True}\<close> for x
     using that by (auto simp: tensor_eq)
 qed
 
 lemma tensor_id[simp]: \<open>tensor_op id_cblinfun id_cblinfun = id_cblinfun\<close>
   apply (rule equal_ket, rename_tac x, case_tac x)
   by (simp flip: tensor_ell2_ket add: tensor_op_ell2)
 
 lemma tensor_op_adjoint: \<open>(tensor_op a b)* = tensor_op (a*) (b*)\<close>
   apply (rule cinner_ket_adjointI[symmetric])
   apply (auto simp flip: tensor_ell2_ket simp: tensor_op_ell2)
   by (simp add: cinner_adj_left)
 
 lemma tensor_butterfly[simp]: "tensor_op (butterfly \<psi> \<psi>') (butterfly \<phi> \<phi>') = butterfly (tensor_ell2 \<psi> \<phi>) (tensor_ell2 \<psi>' \<phi>')"
   apply (rule equal_ket, rename_tac x, case_tac x)
   by (simp flip: tensor_ell2_ket add: tensor_op_ell2 butterfly_def
       cblinfun_apply_cblinfun_compose tensor_ell2_scaleC1 tensor_ell2_scaleC2)
 
 definition tensor_lift :: \<open>(('a1::finite ell2 \<Rightarrow>\<^sub>C\<^sub>L 'a2::finite ell2) \<Rightarrow> ('b1::finite ell2 \<Rightarrow>\<^sub>C\<^sub>L 'b2::finite ell2) \<Rightarrow> 'c)
                         \<Rightarrow> ((('a1\<times>'b1) ell2 \<Rightarrow>\<^sub>C\<^sub>L ('a2\<times>'b2) ell2) \<Rightarrow> 'c::complex_vector)\<close> where
   "tensor_lift F2 = (SOME G. clinear G \<and> (\<forall>a b. G (tensor_op a b) = F2 a b))"
 
 lemma 
   fixes F2 :: "'a::finite ell2 \<Rightarrow>\<^sub>C\<^sub>L 'b::finite ell2
             \<Rightarrow> 'c::finite ell2 \<Rightarrow>\<^sub>C\<^sub>L 'd::finite ell2
             \<Rightarrow> 'e::complex_normed_vector"
   assumes "cbilinear F2"
   shows tensor_lift_clinear: "clinear (tensor_lift F2)"
     and tensor_lift_correct:  \<open>(\<lambda>a b. tensor_lift F2 (tensor_op a b)) = F2\<close>
 proof -
   define F2' t4 \<phi> where
     \<open>F2' = tensor_lift F2\<close> and
     \<open>t4 = (\<lambda>(i,j,k,l). tensor_op (butterket i j) (butterket k l))\<close> and
     \<open>\<phi> m = (let (i,j,k,l) = inv t4 m in F2 (butterket i j) (butterket k l))\<close> for m
   have t4inj: "x = y" if "t4 x = t4 y" for x y
   proof (rule ccontr)
     obtain i  j  k  l  where x: "x = (i,j,k,l)" by (meson prod_cases4) 
     obtain i' j' k' l' where y: "y = (i',j',k',l')" by (meson prod_cases4) 
     have 1: "bra (i,k) *\<^sub>V t4 x *\<^sub>V ket (j,l) = 1"
       by (auto simp: t4_def x tensor_op_ell2 butterfly_def cinner_ket simp flip: tensor_ell2_ket)
     assume \<open>x \<noteq> y\<close>
     then have 2: "bra (i,k) *\<^sub>V t4 y *\<^sub>V ket (j,l) = 0"
       by (auto simp: t4_def x y tensor_op_ell2 butterfly_def cblinfun_apply_cblinfun_compose cinner_ket
                simp flip: tensor_ell2_ket)
     from 1 2 that
     show False
       by auto
   qed
   have \<open>\<phi> (tensor_op (butterket i j) (butterket k l)) = F2 (butterket i j) (butterket k l)\<close> for i j k l
     apply (subst asm_rl[of \<open>tensor_op (butterket i j) (butterket k l) = t4 (i,j,k,l)\<close>])
      apply (simp add: t4_def)
     by (auto simp add: injI t4inj inv_f_f \<phi>_def)
 
   have *: \<open>range t4 = {tensor_op (butterket i j) (butterket k l) |i j k l. True}\<close>
     apply (auto simp: case_prod_beta t4_def)
     using image_iff by fastforce
 
   have "cblinfun_extension_exists (range t4) \<phi>"
     thm cblinfun_extension_exists_finite_dim[where \<phi>=\<phi>]
     apply (rule cblinfun_extension_exists_finite_dim)
      apply auto unfolding * 
     using cindependent_tensor_op
     using cspan_tensor_op
     by auto
 
   then obtain G where G: \<open>G *\<^sub>V (t4 (i,j,k,l)) = F2 (butterket i j) (butterket k l)\<close> for i j k l
     apply atomize_elim
     unfolding cblinfun_extension_exists_def
     apply auto
     by (metis (no_types, lifting) t4inj \<phi>_def f_inv_into_f rangeI split_conv)
 
   have *: \<open>G *\<^sub>V tensor_op (butterket i j) (butterket k l) = F2 (butterket i j) (butterket k l)\<close> for i j k l
     using G by (auto simp: t4_def)
   have *: \<open>G *\<^sub>V tensor_op a (butterket k l) = F2 a (butterket k l)\<close> for a k l
     apply (rule complex_vector.linear_eq_on_span[where g=\<open>\<lambda>a. F2 a _\<close> and B=\<open>{butterket k l|k l. True}\<close>])
     unfolding cspan_butterfly_ket
     using * apply (auto intro!: clinear_compose[unfolded o_def, where f=\<open>\<lambda>a. tensor_op a _\<close> and g=\<open>(*\<^sub>V) G\<close>])
      apply (metis cbilinear_def tensor_op_cbilinear)
     using assms unfolding cbilinear_def by blast
   have G_F2: \<open>G *\<^sub>V tensor_op a b = F2 a b\<close> for a b
     apply (rule complex_vector.linear_eq_on_span[where g=\<open>F2 a\<close> and B=\<open>{butterket k l|k l. True}\<close>])
     unfolding cspan_butterfly_ket
     using * apply (auto simp: cblinfun.add_right clinearI
                         intro!: clinear_compose[unfolded o_def, where f=\<open>tensor_op a\<close> and g=\<open>(*\<^sub>V) G\<close>])
     apply (meson cbilinear_def tensor_op_cbilinear)
     using assms unfolding cbilinear_def by blast
 
   have \<open>clinear F2' \<and> (\<forall>a b. F2' (tensor_op a b) = F2 a b)\<close>
     unfolding F2'_def tensor_lift_def 
     apply (rule someI[where x=\<open>(*\<^sub>V) G\<close> and P=\<open>\<lambda>G. clinear G \<and> (\<forall>a b. G (tensor_op a b) = F2 a b)\<close>])
     using G_F2 by (simp add: cblinfun.add_right clinearI)
 
   then show \<open>clinear F2'\<close> and \<open>(\<lambda>a b. tensor_lift F2 (tensor_op a b)) = F2\<close>
     unfolding F2'_def by auto
 qed
 
 lift_definition assoc_ell20 :: \<open>(('a::finite\<times>'b::finite)\<times>'c::finite) ell2 \<Rightarrow> ('a\<times>('b\<times>'c)) ell2\<close> is
   \<open>\<lambda>f (a,(b,c)). f ((a,b),c)\<close>
   by auto
 
 lift_definition assoc_ell20' :: \<open>('a::finite\<times>('b::finite\<times>'c::finite)) ell2 \<Rightarrow> (('a\<times>'b)\<times>'c) ell2\<close> is
   \<open>\<lambda>f ((a,b),c). f (a,(b,c))\<close>
   by auto
 
 lift_definition assoc_ell2 :: \<open>(('a::finite\<times>'b::finite)\<times>'c::finite) ell2 \<Rightarrow>\<^sub>C\<^sub>L ('a\<times>('b\<times>'c)) ell2\<close>
   is assoc_ell20
   apply (subst bounded_clinear_finite_dim)
    apply (rule clinearI; transfer)
   by auto
 
 lift_definition assoc_ell2' :: \<open>('a::finite\<times>('b::finite\<times>'c::finite)) ell2 \<Rightarrow>\<^sub>C\<^sub>L (('a\<times>'b)\<times>'c) ell2\<close> is
   assoc_ell20'
   apply (subst bounded_clinear_finite_dim)
    apply (rule clinearI; transfer)
   by auto
 
 lemma assoc_ell2_tensor: \<open>assoc_ell2 *\<^sub>V tensor_ell2 (tensor_ell2 a b) c = tensor_ell2 a (tensor_ell2 b c)\<close>
   apply (rule clinear_equal_ket[THEN fun_cong, where x=a])
     apply (simp add: cblinfun.add_right clinearI tensor_ell2_add1 tensor_ell2_scaleC1)
    apply (simp add: clinear_tensor_ell22)
   apply (rule clinear_equal_ket[THEN fun_cong, where x=b])
     apply (simp add: cblinfun.add_right clinearI tensor_ell2_add1 tensor_ell2_add2 tensor_ell2_scaleC1 tensor_ell2_scaleC2)
    apply (simp add: clinearI tensor_ell2_add1 tensor_ell2_add2 tensor_ell2_scaleC1 tensor_ell2_scaleC2)
   apply (rule clinear_equal_ket[THEN fun_cong, where x=c])
     apply (simp add: cblinfun.add_right clinearI tensor_ell2_add2 tensor_ell2_scaleC2)
    apply (simp add: clinearI tensor_ell2_add2 tensor_ell2_scaleC2)
   unfolding assoc_ell2.rep_eq
   apply transfer
   by auto
 
 lemma assoc_ell2'_tensor: \<open>assoc_ell2' *\<^sub>V tensor_ell2 a (tensor_ell2 b c) = tensor_ell2 (tensor_ell2 a b) c\<close>
   apply (rule clinear_equal_ket[THEN fun_cong, where x=a])
     apply (simp add: cblinfun.add_right clinearI tensor_ell2_add1 tensor_ell2_scaleC1)
    apply (simp add: clinearI tensor_ell2_add1 tensor_ell2_scaleC1)
   apply (rule clinear_equal_ket[THEN fun_cong, where x=b])
     apply (simp add: cblinfun.add_right clinearI tensor_ell2_add1 tensor_ell2_add2 tensor_ell2_scaleC1 tensor_ell2_scaleC2)
    apply (simp add: clinearI tensor_ell2_add1 tensor_ell2_add2 tensor_ell2_scaleC1 tensor_ell2_scaleC2)
   apply (rule clinear_equal_ket[THEN fun_cong, where x=c])
     apply (simp add: cblinfun.add_right clinearI tensor_ell2_add2 tensor_ell2_scaleC2)
    apply (simp add: clinearI tensor_ell2_add2 tensor_ell2_scaleC2)
   unfolding assoc_ell2'.rep_eq
   apply transfer
   by auto
 
 lemma adjoint_assoc_ell2[simp]: \<open>assoc_ell2* = assoc_ell2'\<close>
 proof (rule adjoint_eqI[symmetric])
   have [simp]: \<open>clinear (cinner (assoc_ell2' *\<^sub>V x))\<close> for x :: \<open>('a \<times> 'b \<times> 'c) ell2\<close>
     by (metis (no_types, lifting) cblinfun.add_right cinner_scaleC_right clinearI complex_scaleC_def mult.comm_neutral of_complex_def vector_to_cblinfun_adj_apply)
-  have [simp]: \<open>clinear (\<lambda>a. \<langle>x, assoc_ell2 *\<^sub>V a\<rangle>)\<close> for x :: \<open>('a \<times> 'b \<times> 'c) ell2\<close>
+  have [simp]: \<open>clinear (\<lambda>a. x \<bullet>\<^sub>C (assoc_ell2 *\<^sub>V a))\<close> for x :: \<open>('a \<times> 'b \<times> 'c) ell2\<close>
     by (simp add: cblinfun.add_right cinner_add_right clinearI)
-  have [simp]: \<open>antilinear (\<lambda>a. \<langle>a, y\<rangle>)\<close> for y :: \<open>('a \<times> 'b \<times> 'c) ell2\<close>
+  have [simp]: \<open>antilinear (\<lambda>a. a \<bullet>\<^sub>C y)\<close> for y :: \<open>('a \<times> 'b \<times> 'c) ell2\<close>
     using bounded_antilinear_cinner_left bounded_antilinear_def by blast
-  have [simp]: \<open>antilinear (\<lambda>a. \<langle>assoc_ell2' *\<^sub>V a, y\<rangle>)\<close> for y :: \<open>(('a \<times> 'b) \<times> 'c) ell2\<close>
+  have [simp]: \<open>antilinear (\<lambda>a. (assoc_ell2' *\<^sub>V a) \<bullet>\<^sub>C y)\<close> for y :: \<open>(('a \<times> 'b) \<times> 'c) ell2\<close>
     by (simp add: cblinfun.add_right cinner_add_left antilinearI)
-  have \<open>\<langle>assoc_ell2' *\<^sub>V (ket x), ket y\<rangle> = \<langle>ket x, assoc_ell2 *\<^sub>V ket y\<rangle>\<close> for x :: \<open>'a \<times> 'b \<times> 'c\<close> and y
+  have \<open>(assoc_ell2' *\<^sub>V ket x) \<bullet>\<^sub>C ket y = ket x \<bullet>\<^sub>C (assoc_ell2 *\<^sub>V ket y)\<close> for x :: \<open>'a \<times> 'b \<times> 'c\<close> and y
     apply (cases x, cases y)
     by (simp flip: tensor_ell2_ket add: assoc_ell2'_tensor assoc_ell2_tensor)
-  then have \<open>\<langle>assoc_ell2' *\<^sub>V (ket x), y\<rangle> = \<langle>ket x, assoc_ell2 *\<^sub>V y\<rangle>\<close> for x :: \<open>'a \<times> 'b \<times> 'c\<close> and y
+  then have \<open>(assoc_ell2' *\<^sub>V ket x) \<bullet>\<^sub>C y = ket x \<bullet>\<^sub>C (assoc_ell2 *\<^sub>V y)\<close> for x :: \<open>'a \<times> 'b \<times> 'c\<close> and y
     by (rule clinear_equal_ket[THEN fun_cong, rotated 2], simp_all)
-  then show \<open>\<langle>assoc_ell2' *\<^sub>V x, y\<rangle> = \<langle>x, assoc_ell2 *\<^sub>V y\<rangle>\<close> for x :: \<open>('a \<times> 'b \<times> 'c) ell2\<close> and y
+  then show \<open>(assoc_ell2' *\<^sub>V x) \<bullet>\<^sub>C y = x \<bullet>\<^sub>C (assoc_ell2 *\<^sub>V y)\<close> for x :: \<open>('a \<times> 'b \<times> 'c) ell2\<close> and y
     by (rule antilinear_equal_ket[THEN fun_cong, rotated 2], simp_all)
 qed
 
 lemma adjoint_assoc_ell2'[simp]: \<open>assoc_ell2'* = assoc_ell2\<close>
   by (simp flip: adjoint_assoc_ell2)
 
 
 lift_definition swap_ell20 :: \<open>('a::finite\<times>'b::finite) ell2 \<Rightarrow> ('b\<times>'a) ell2\<close> is
   \<open>\<lambda>f (a,b). f (b,a)\<close>
   by auto
 
 lift_definition swap_ell2 :: \<open>('a::finite\<times>'b::finite) ell2 \<Rightarrow>\<^sub>C\<^sub>L ('b\<times>'a) ell2\<close>
   is swap_ell20
   apply (subst bounded_clinear_finite_dim)
    apply (rule clinearI; transfer)
   by auto
 
 lemma swap_ell2_tensor[simp]: \<open>swap_ell2 *\<^sub>V tensor_ell2 a b = tensor_ell2 b a\<close>
   apply (rule clinear_equal_ket[THEN fun_cong, where x=a])
     apply (simp add: cblinfun.add_right clinearI tensor_ell2_add1 tensor_ell2_scaleC1)
    apply (simp add: clinear_tensor_ell21)
   apply (rule clinear_equal_ket[THEN fun_cong, where x=b])
     apply (simp add: cblinfun.add_right clinearI tensor_ell2_add1 tensor_ell2_add2 tensor_ell2_scaleC1 tensor_ell2_scaleC2)
    apply (simp add: clinearI tensor_ell2_add1 tensor_ell2_add2 tensor_ell2_scaleC1 tensor_ell2_scaleC2)
   unfolding swap_ell2.rep_eq
   apply transfer
   by auto
 
 lemma adjoint_swap_ell2[simp]: \<open>swap_ell2* = swap_ell2\<close>
 proof (rule adjoint_eqI[symmetric])
   have [simp]: \<open>clinear (cinner (swap_ell2 *\<^sub>V x))\<close> for x :: \<open>('a \<times> 'b) ell2\<close>
     by (metis (no_types, lifting) cblinfun.add_right cinner_scaleC_right clinearI complex_scaleC_def mult.comm_neutral of_complex_def vector_to_cblinfun_adj_apply)
-  have [simp]: \<open>clinear (\<lambda>a. \<langle>x, swap_ell2 *\<^sub>V a\<rangle>)\<close> for x :: \<open>('a \<times> 'b) ell2\<close>
+  have [simp]: \<open>clinear (\<lambda>a. x \<bullet>\<^sub>C (swap_ell2 *\<^sub>V a))\<close> for x :: \<open>('a \<times> 'b) ell2\<close>
     by (simp add: cblinfun.add_right cinner_add_right clinearI)
-  have [simp]: \<open>antilinear (\<lambda>a. \<langle>a, y\<rangle>)\<close> for y :: \<open>('a \<times> 'b) ell2\<close>
+  have [simp]: \<open>antilinear (\<lambda>a. a \<bullet>\<^sub>C y)\<close> for y :: \<open>('a \<times> 'b) ell2\<close>
     using bounded_antilinear_cinner_left bounded_antilinear_def by blast
-  have [simp]: \<open>antilinear (\<lambda>a. \<langle>swap_ell2 *\<^sub>V a, y\<rangle>)\<close> for y :: \<open>('b \<times> 'a) ell2\<close>
+  have [simp]: \<open>antilinear (\<lambda>a. (swap_ell2 *\<^sub>V a) \<bullet>\<^sub>C y)\<close> for y :: \<open>('b \<times> 'a) ell2\<close>
     by (simp add: cblinfun.add_right cinner_add_left antilinearI)
-  have \<open>\<langle>swap_ell2 *\<^sub>V (ket x), ket y\<rangle> = \<langle>ket x, swap_ell2 *\<^sub>V ket y\<rangle>\<close> for x :: \<open>'a \<times> 'b\<close> and y
+  have \<open>(swap_ell2 *\<^sub>V ket x) \<bullet>\<^sub>C ket y = ket x \<bullet>\<^sub>C (swap_ell2 *\<^sub>V ket y)\<close> for x :: \<open>'a \<times> 'b\<close> and y
     apply (cases x, cases y)
     by (simp flip: tensor_ell2_ket add: swap_ell2_tensor)
-  then have \<open>\<langle>swap_ell2 *\<^sub>V (ket x), y\<rangle> = \<langle>ket x, swap_ell2 *\<^sub>V y\<rangle>\<close> for x :: \<open>'a \<times> 'b\<close> and y
+  then have \<open>(swap_ell2 *\<^sub>V ket x) \<bullet>\<^sub>C y = ket x \<bullet>\<^sub>C (swap_ell2 *\<^sub>V y)\<close> for x :: \<open>'a \<times> 'b\<close> and y
     by (rule clinear_equal_ket[THEN fun_cong, rotated 2], simp_all)
-  then show \<open>\<langle>swap_ell2 *\<^sub>V x, y\<rangle> = \<langle>x, swap_ell2 *\<^sub>V y\<rangle>\<close> for x :: \<open>('a \<times> 'b) ell2\<close> and y
+  then show \<open>(swap_ell2 *\<^sub>V x) \<bullet>\<^sub>C y = x \<bullet>\<^sub>C (swap_ell2 *\<^sub>V y)\<close> for x :: \<open>('a \<times> 'b) ell2\<close> and y
     apply (rule antilinear_equal_ket[THEN fun_cong, rotated 2])
     by simp_all
 qed
 
 
 lemma tensor_ell2_extensionality:
   assumes "(\<And>s t. a *\<^sub>V (s \<otimes>\<^sub>s t) = b *\<^sub>V (s \<otimes>\<^sub>s t))"
   shows "a = b"
   apply (rule equal_ket, case_tac x, hypsubst_thin)
   by (simp add: assms flip: tensor_ell2_ket)
 
 lemma assoc_ell2'_assoc_ell2[simp]: \<open>assoc_ell2' o\<^sub>C\<^sub>L assoc_ell2 = id_cblinfun\<close>
   by (auto intro!: equal_ket simp: cblinfun_apply_cblinfun_compose assoc_ell2'_tensor assoc_ell2_tensor simp flip: tensor_ell2_ket)
 
 lemma assoc_ell2_assoc_ell2'[simp]: \<open>assoc_ell2 o\<^sub>C\<^sub>L assoc_ell2' = id_cblinfun\<close>
   by (auto intro!: equal_ket simp: cblinfun_apply_cblinfun_compose assoc_ell2'_tensor assoc_ell2_tensor simp flip: tensor_ell2_ket)
 
 lemma unitary_assoc_ell2[simp]: "unitary assoc_ell2"
   unfolding unitary_def by auto
 
 lemma unitary_assoc_ell2'[simp]: "unitary assoc_ell2'"
   unfolding unitary_def by auto
 
 lemma tensor_op_left_add: \<open>(x + y) \<otimes>\<^sub>o b = x \<otimes>\<^sub>o b + y \<otimes>\<^sub>o b\<close>
   for x y :: \<open>'a::finite ell2 \<Rightarrow>\<^sub>C\<^sub>L 'c::finite ell2\<close> and b :: \<open>'b::finite ell2 \<Rightarrow>\<^sub>C\<^sub>L 'd::finite ell2\<close>
   apply (auto intro!: equal_ket simp: tensor_op_ket)
   by (simp add: plus_cblinfun.rep_eq tensor_ell2_add1 tensor_op_ket)
 
 lemma tensor_op_right_add: \<open>b \<otimes>\<^sub>o (x + y) = b \<otimes>\<^sub>o x + b \<otimes>\<^sub>o y\<close>
   for x y :: \<open>'a::finite ell2 \<Rightarrow>\<^sub>C\<^sub>L 'c::finite ell2\<close> and b :: \<open>'b::finite ell2 \<Rightarrow>\<^sub>C\<^sub>L 'd::finite ell2\<close>
   apply (auto intro!: equal_ket simp: tensor_op_ket)
   by (simp add: plus_cblinfun.rep_eq tensor_ell2_add2 tensor_op_ket)
 
 lemma tensor_op_scaleC_left: \<open>(c *\<^sub>C x) \<otimes>\<^sub>o b = c *\<^sub>C (x \<otimes>\<^sub>o b)\<close>
   for x :: \<open>'a::finite ell2 \<Rightarrow>\<^sub>C\<^sub>L 'c::finite ell2\<close> and b :: \<open>'b::finite ell2 \<Rightarrow>\<^sub>C\<^sub>L 'd::finite ell2\<close>
   apply (auto intro!: equal_ket simp: tensor_op_ket)
   by (metis scaleC_cblinfun.rep_eq tensor_ell2_ket tensor_ell2_scaleC1 tensor_op_ell2)
 
 lemma tensor_op_scaleC_right: \<open>b \<otimes>\<^sub>o (c *\<^sub>C x) = c *\<^sub>C (b \<otimes>\<^sub>o x)\<close>
   for x :: \<open>'a::finite ell2 \<Rightarrow>\<^sub>C\<^sub>L 'c::finite ell2\<close> and b :: \<open>'b::finite ell2 \<Rightarrow>\<^sub>C\<^sub>L 'd::finite ell2\<close>
   apply (auto intro!: equal_ket simp: tensor_op_ket)
   by (metis scaleC_cblinfun.rep_eq tensor_ell2_ket tensor_ell2_scaleC2 tensor_op_ell2)
 
 lemma clinear_tensor_left[simp]: \<open>clinear (\<lambda>a. a \<otimes>\<^sub>o b :: _::finite ell2 \<Rightarrow>\<^sub>C\<^sub>L _::finite ell2)\<close>
   apply (rule clinearI)
    apply (rule tensor_op_left_add)
   by (rule tensor_op_scaleC_left)
 
 lemma clinear_tensor_right[simp]: \<open>clinear (\<lambda>b. a \<otimes>\<^sub>o b :: _::finite ell2 \<Rightarrow>\<^sub>C\<^sub>L _::finite ell2)\<close>
   apply (rule clinearI)
    apply (rule tensor_op_right_add)
   by (rule tensor_op_scaleC_right)
 
 lemma tensor_ell2_nonzero: \<open>a \<otimes>\<^sub>s b \<noteq> 0\<close> if \<open>a \<noteq> 0\<close> and \<open>b \<noteq> 0\<close>
   apply (use that in transfer)
   apply auto
   by (metis mult_eq_0_iff old.prod.case)
 
 lemma tensor_op_nonzero:
   fixes a :: \<open>'a::finite ell2 \<Rightarrow>\<^sub>C\<^sub>L 'c::finite ell2\<close> and b :: \<open>'b::finite ell2 \<Rightarrow>\<^sub>C\<^sub>L 'd::finite ell2\<close>
   assumes \<open>a \<noteq> 0\<close> and \<open>b \<noteq> 0\<close>
   shows \<open>a \<otimes>\<^sub>o b \<noteq> 0\<close>
 proof -
   from \<open>a \<noteq> 0\<close> obtain i where i: \<open>a *\<^sub>V ket i \<noteq> 0\<close>
     by (metis cblinfun.zero_left equal_ket)
   from \<open>b \<noteq> 0\<close> obtain j where j: \<open>b *\<^sub>V ket j \<noteq> 0\<close>
     by (metis cblinfun.zero_left equal_ket)
   from i j have ijneq0: \<open>(a *\<^sub>V ket i) \<otimes>\<^sub>s (b *\<^sub>V ket j) \<noteq> 0\<close>
     by (simp add: tensor_ell2_nonzero)
   have \<open>(a *\<^sub>V ket i) \<otimes>\<^sub>s (b *\<^sub>V ket j) = (a \<otimes>\<^sub>o b) *\<^sub>V ket (i,j)\<close>
     by (simp add: tensor_op_ket)
   with ijneq0 show \<open>a \<otimes>\<^sub>o b \<noteq> 0\<close>
     by force
 qed
 
 lemma inj_tensor_ell2_left: \<open>inj (\<lambda>a::'a::finite ell2. a \<otimes>\<^sub>s b)\<close> if \<open>b \<noteq> 0\<close> for b :: \<open>'b::finite ell2\<close>
 proof (rule injI, rule ccontr)
   fix x y :: \<open>'a ell2\<close>
   assume eq: \<open>x \<otimes>\<^sub>s b = y \<otimes>\<^sub>s b\<close>
   assume neq: \<open>x \<noteq> y\<close>
   define a where \<open>a = x - y\<close>
   from neq a_def have neq0: \<open>a \<noteq> 0\<close>
     by auto
   with \<open>b \<noteq> 0\<close> have \<open>a \<otimes>\<^sub>s b \<noteq> 0\<close>
     by (simp add: tensor_ell2_nonzero)
   then have \<open>x \<otimes>\<^sub>s b \<noteq> y \<otimes>\<^sub>s b\<close>
     unfolding a_def
     by (metis add_cancel_left_left diff_add_cancel tensor_ell2_add1)
   with eq show False
     by auto
 qed
 
 lemma inj_tensor_ell2_right: \<open>inj (\<lambda>b::'b::finite ell2. a \<otimes>\<^sub>s b)\<close> if \<open>a \<noteq> 0\<close> for a :: \<open>'a::finite ell2\<close>
 proof (rule injI, rule ccontr)
   fix x y :: \<open>'b ell2\<close>
   assume eq: \<open>a \<otimes>\<^sub>s x = a \<otimes>\<^sub>s y\<close>
   assume neq: \<open>x \<noteq> y\<close>
   define b where \<open>b = x - y\<close>
   from neq b_def have neq0: \<open>b \<noteq> 0\<close>
     by auto
   with \<open>a \<noteq> 0\<close> have \<open>a \<otimes>\<^sub>s b \<noteq> 0\<close>
     by (simp add: tensor_ell2_nonzero)
   then have \<open>a \<otimes>\<^sub>s x \<noteq> a \<otimes>\<^sub>s y\<close>
     unfolding b_def
     by (metis add_cancel_left_left diff_add_cancel tensor_ell2_add2)
   with eq show False
     by auto
 qed
 
 
 
 lemma inj_tensor_left: \<open>inj (\<lambda>a::'a::finite ell2 \<Rightarrow>\<^sub>C\<^sub>L 'c::finite ell2. a \<otimes>\<^sub>o b)\<close> if \<open>b \<noteq> 0\<close> for b :: \<open>'b::finite ell2 \<Rightarrow>\<^sub>C\<^sub>L 'd::finite ell2\<close>
 proof (rule injI, rule ccontr)
   fix x y :: \<open>'a ell2 \<Rightarrow>\<^sub>C\<^sub>L 'c ell2\<close>
   assume eq: \<open>x \<otimes>\<^sub>o b = y \<otimes>\<^sub>o b\<close>
   assume neq: \<open>x \<noteq> y\<close>
   define a where \<open>a = x - y\<close>
   from neq a_def have neq0: \<open>a \<noteq> 0\<close>
     by auto
   with \<open>b \<noteq> 0\<close> have \<open>a \<otimes>\<^sub>o b \<noteq> 0\<close>
     by (simp add: tensor_op_nonzero)
   then have \<open>x \<otimes>\<^sub>o b \<noteq> y \<otimes>\<^sub>o b\<close>
     unfolding a_def
     by (metis add_cancel_left_left diff_add_cancel tensor_op_left_add) 
   with eq show False
     by auto
 qed
 
 lemma inj_tensor_right: \<open>inj (\<lambda>b::'b::finite ell2 \<Rightarrow>\<^sub>C\<^sub>L 'c::finite ell2. a \<otimes>\<^sub>o b)\<close> if \<open>a \<noteq> 0\<close> for a :: \<open>'a::finite ell2 \<Rightarrow>\<^sub>C\<^sub>L 'd::finite ell2\<close>
 proof (rule injI, rule ccontr)
   fix x y :: \<open>'b ell2 \<Rightarrow>\<^sub>C\<^sub>L 'c ell2\<close>
   assume eq: \<open>a \<otimes>\<^sub>o x = a \<otimes>\<^sub>o y\<close>
   assume neq: \<open>x \<noteq> y\<close>
   define b where \<open>b = x - y\<close>
   from neq b_def have neq0: \<open>b \<noteq> 0\<close>
     by auto
   with \<open>a \<noteq> 0\<close> have \<open>a \<otimes>\<^sub>o b \<noteq> 0\<close>
     by (simp add: tensor_op_nonzero)
   then have \<open>a \<otimes>\<^sub>o x \<noteq> a \<otimes>\<^sub>o y\<close>
     unfolding b_def
     by (metis add_cancel_left_left diff_add_cancel tensor_op_right_add) 
   with eq show False
     by auto
 qed
 
 lemma tensor_ell2_almost_injective:
   assumes \<open>tensor_ell2 a b = tensor_ell2 c d\<close>
   assumes \<open>a \<noteq> 0\<close>
   shows \<open>\<exists>\<gamma>. b = \<gamma> *\<^sub>C d\<close>
 proof -
   from \<open>a \<noteq> 0\<close> obtain i where i: \<open>cinner (ket i) a \<noteq> 0\<close>
     by (metis cinner_eq_zero_iff cinner_ket_left ell2_pointwise_ortho)
   have \<open>cinner (ket i \<otimes>\<^sub>s ket j) (a \<otimes>\<^sub>s b) = cinner (ket i \<otimes>\<^sub>s ket j) (c \<otimes>\<^sub>s d)\<close> for j
     using assms by simp
   then have eq2: \<open>(cinner (ket i) a) * (cinner (ket j) b) = (cinner (ket i) c) * (cinner (ket j) d)\<close> for j
     by (metis tensor_ell2_inner_prod)
   then obtain \<gamma> where \<open>cinner (ket i) c = \<gamma> * cinner (ket i) a\<close>
     by (metis i eq_divide_eq)
   with eq2 have \<open>(cinner (ket i) a) * (cinner (ket j) b) = (cinner (ket i) a) * (\<gamma> * cinner (ket j) d)\<close> for j
     by simp
   then have \<open>cinner (ket j) b = cinner (ket j) (\<gamma> *\<^sub>C d)\<close> for j
     using i by force
   then have \<open>b = \<gamma> *\<^sub>C d\<close>
     by (simp add: cinner_ket_eqI)
   then show ?thesis
     by auto
 qed
 
 
 lemma tensor_op_almost_injective:
   fixes a c :: \<open>'a::finite ell2 \<Rightarrow>\<^sub>C\<^sub>L 'b::finite ell2\<close>
     and b d :: \<open>'c::finite ell2 \<Rightarrow>\<^sub>C\<^sub>L 'd::finite ell2\<close>
   assumes \<open>tensor_op a b = tensor_op c d\<close>
   assumes \<open>a \<noteq> 0\<close>
   shows \<open>\<exists>\<gamma>. b = \<gamma> *\<^sub>C d\<close>
 proof (cases \<open>d = 0\<close>)
   case False
   from \<open>a \<noteq> 0\<close> obtain \<psi> where \<psi>: \<open>a *\<^sub>V \<psi> \<noteq> 0\<close>
     by (metis cblinfun.zero_left cblinfun_eqI)
   have \<open>(a \<otimes>\<^sub>o b) (\<psi> \<otimes>\<^sub>s \<phi>) = (c \<otimes>\<^sub>o d) (\<psi> \<otimes>\<^sub>s \<phi>)\<close> for \<phi>
     using assms by simp
   then have eq2: \<open>(a \<psi>) \<otimes>\<^sub>s (b \<phi>) = (c \<psi>) \<otimes>\<^sub>s (d \<phi>)\<close> for \<phi>
     by (simp add: tensor_op_ell2)
   then have eq2': \<open>(d \<phi>) \<otimes>\<^sub>s (c \<psi>) = (b \<phi>) \<otimes>\<^sub>s (a \<psi>)\<close> for \<phi>
     by (metis swap_ell2_tensor)
   from False obtain \<phi>0 where \<phi>0: \<open>d \<phi>0 \<noteq> 0\<close>
     by (metis cblinfun.zero_left cblinfun_eqI)
   obtain \<gamma> where \<open>c \<psi> = \<gamma> *\<^sub>C a \<psi>\<close>
     apply atomize_elim
     using eq2' \<phi>0 by (rule tensor_ell2_almost_injective)
   with eq2 have \<open>(a \<psi>) \<otimes>\<^sub>s (b \<phi>) = (a \<psi>) \<otimes>\<^sub>s (\<gamma> *\<^sub>C d \<phi>)\<close> for \<phi>
     by (simp add: tensor_ell2_scaleC1 tensor_ell2_scaleC2)
   then have \<open>b \<phi> = \<gamma> *\<^sub>C d \<phi>\<close> for \<phi>
     by (smt (verit, best) \<psi> complex_vector.scale_cancel_right tensor_ell2_almost_injective tensor_ell2_nonzero tensor_ell2_scaleC2)
   then have \<open>b = \<gamma> *\<^sub>C d\<close>
     by (simp add: cblinfun_eqI)
   then show ?thesis
     by auto
 next
   case True
   then have \<open>c \<otimes>\<^sub>o d = 0\<close>
     by (metis add_cancel_right_left tensor_op_right_add)
   then have \<open>a \<otimes>\<^sub>o b = 0\<close>
     using assms(1) by presburger
   with \<open>a \<noteq> 0\<close> have \<open>b = 0\<close>
     by (meson tensor_op_nonzero)
   then show ?thesis
     by auto
 qed
 
 
 lemma tensor_ell2_0_left[simp]: \<open>tensor_ell2 0 x = 0\<close>
   apply transfer by auto
 
 lemma tensor_ell2_0_right[simp]: \<open>tensor_ell2 x 0 = 0\<close>
   apply transfer by auto
 
 lemma tensor_op_0_left[simp]: \<open>tensor_op 0 x = (0 :: ('a::finite*'b::finite) ell2 \<Rightarrow>\<^sub>C\<^sub>L ('c::finite*'d::finite) ell2)\<close>
   apply (rule equal_ket)
   by (auto simp flip: tensor_ell2_ket simp: tensor_op_ell2)
 
 lemma tensor_op_0_right[simp]: \<open>tensor_op x 0 = (0 :: ('a::finite*'b::finite) ell2 \<Rightarrow>\<^sub>C\<^sub>L ('c::finite*'d::finite) ell2)\<close>
   apply (rule equal_ket)
   by (auto simp flip: tensor_ell2_ket simp: tensor_op_ell2)
 
 lemma bij_tensor_ell2_one_dim_left:
   assumes \<open>\<psi> \<noteq> 0\<close>
   shows \<open>bij (\<lambda>x::'b::finite ell2. (\<psi> :: 'a::CARD_1 ell2) \<otimes>\<^sub>s x)\<close>
 proof (rule bijI)
   show \<open>inj (\<lambda>x::'b::finite ell2. (\<psi> :: 'a::CARD_1 ell2) \<otimes>\<^sub>s x)\<close>
     using assms by (rule inj_tensor_ell2_right)
   have \<open>\<exists>x. \<psi> \<otimes>\<^sub>s x = \<phi>\<close> for \<phi> :: \<open>('a*'b) ell2\<close>
   proof (use assms in transfer)
     fix \<psi> :: \<open>'a \<Rightarrow> complex\<close> and \<phi> :: \<open>'a*'b \<Rightarrow> complex\<close>
     assume \<open>has_ell2_norm \<phi>\<close> and \<open>\<psi> \<noteq> (\<lambda>_. 0)\<close>
     define c where \<open>c = \<psi> undefined\<close>
     then have \<open>\<psi> a = c\<close> for a 
       apply (subst everything_the_same[of _ undefined])
       by simp
     with \<open>\<psi> \<noteq> (\<lambda>_. 0)\<close> have \<open>c \<noteq> 0\<close>
       by auto
 
     define x where \<open>x j = \<phi> (undefined, j) / c\<close> for j
     have \<open>(\<lambda>(i, j). \<psi> i * x j) = \<phi>\<close>
       apply (auto intro!: ext simp: x_def \<open>\<psi> _ = c\<close> \<open>c \<noteq> 0\<close>)
       apply (subst (2) everything_the_same[of _ undefined])
       by simp
     then show \<open>\<exists>x\<in>Collect has_ell2_norm. (\<lambda>(i, j). \<psi> i * x j) = \<phi>\<close>
       apply (rule bexI[where x=x])
       by simp
   qed
 
   then show \<open>surj (\<lambda>x::'b::finite ell2. (\<psi> :: 'a::CARD_1 ell2) \<otimes>\<^sub>s x)\<close>
     by (metis surj_def)
 qed
 
 lemma bij_tensor_op_one_dim_left:
   assumes \<open>a \<noteq> 0\<close>
   shows \<open>bij (\<lambda>x::'c::finite ell2 \<Rightarrow>\<^sub>C\<^sub>L 'd::finite ell2. (a :: 'a::{CARD_1,enum} ell2 \<Rightarrow>\<^sub>C\<^sub>L 'b::{CARD_1,enum} ell2) \<otimes>\<^sub>o x)\<close>
 proof (rule bijI)
   define t where \<open>t = (\<lambda>x::'c ell2 \<Rightarrow>\<^sub>C\<^sub>L 'd ell2. (a :: 'a ell2 \<Rightarrow>\<^sub>C\<^sub>L 'b ell2) \<otimes>\<^sub>o x)\<close>
   define i where
     \<open>i = tensor_lift (\<lambda>(x::'a ell2 \<Rightarrow>\<^sub>C\<^sub>L 'b ell2) (y::'c ell2 \<Rightarrow>\<^sub>C\<^sub>L 'd ell2). (one_dim_iso x / one_dim_iso a) *\<^sub>C y)\<close>
 
   have [simp]: \<open>clinear i\<close>
     by (auto intro!: tensor_lift_clinear simp: i_def cbilinear_def clinearI scaleC_add_left add_divide_distrib)
   have [simp]: \<open>clinear t\<close>
     by (simp add: t_def)
   have \<open>i (x \<otimes>\<^sub>o y) = (one_dim_iso x / one_dim_iso a) *\<^sub>C y\<close> for x y
     by (auto intro!: clinearI tensor_lift_correct[THEN fun_cong, THEN fun_cong] simp: t_def i_def cbilinear_def  scaleC_add_left add_divide_distrib)
   then have \<open>t (i (x \<otimes>\<^sub>o y)) = x \<otimes>\<^sub>o y\<close> for x y
     apply (simp add: t_def)
     by (smt (z3) assms complex_vector.scale_eq_0_iff nonzero_mult_div_cancel_right one_dim_scaleC_1 scaleC_scaleC tensor_op_scaleC_left tensor_op_scaleC_right times_divide_eq_left)
   then have \<open>t (i x) = x\<close> for x
     apply (rule_tac fun_cong[where x=x])
     apply (rule tensor_extensionality)
     by (auto intro: clinear_compose complex_vector.module_hom_ident simp flip: o_def[of t i])
   then show \<open>surj t\<close> 
     by (rule surjI)
 
   show \<open>inj t\<close>
     unfolding t_def using assms by (rule inj_tensor_right)
 qed
 
 lemma swap_ell2_selfinv[simp]: \<open>swap_ell2 o\<^sub>C\<^sub>L swap_ell2 = id_cblinfun\<close>
   apply (rule tensor_ell2_extensionality)
   by auto
 
 lemma bij_tensor_op_one_dim_right:
   assumes \<open>b \<noteq> 0\<close>
   shows \<open>bij (\<lambda>x::'c::finite ell2 \<Rightarrow>\<^sub>C\<^sub>L 'd::finite ell2. x \<otimes>\<^sub>o (b :: 'a::{CARD_1,enum} ell2 \<Rightarrow>\<^sub>C\<^sub>L 'b::{CARD_1,enum} ell2))\<close>
     (is \<open>bij ?f\<close>)
 proof -
   let ?sf = \<open>(\<lambda>x. swap_ell2 o\<^sub>C\<^sub>L (?f x) o\<^sub>C\<^sub>L swap_ell2)\<close>
   let ?s = \<open>(\<lambda>x. swap_ell2 o\<^sub>C\<^sub>L x o\<^sub>C\<^sub>L swap_ell2)\<close>
   let ?g = \<open>(\<lambda>x::'c::finite ell2 \<Rightarrow>\<^sub>C\<^sub>L 'd::finite ell2. (b :: 'a::{CARD_1,enum} ell2 \<Rightarrow>\<^sub>C\<^sub>L 'b::{CARD_1,enum} ell2) \<otimes>\<^sub>o x)\<close>
   have \<open>?sf = ?g\<close>
     by (auto intro!: ext tensor_ell2_extensionality simp add: swap_ell2_tensor tensor_op_ell2)
   have \<open>bij ?g\<close>
     using assms by (rule bij_tensor_op_one_dim_left)
   have \<open>?s o ?sf = ?f\<close>
     apply (auto intro!: ext simp: cblinfun_assoc_left)
     by (auto simp: cblinfun_assoc_right)
   also have \<open>bij ?s\<close>
     apply (rule o_bij[where g=\<open>(\<lambda>x. swap_ell2 o\<^sub>C\<^sub>L x o\<^sub>C\<^sub>L swap_ell2)\<close>])
      apply (auto intro!: ext simp: cblinfun_assoc_left)
     by (auto simp: cblinfun_assoc_right)
   show \<open>bij ?f\<close>
     apply (subst \<open>?s o ?sf = ?f\<close>[symmetric], subst \<open>?sf = ?g\<close>)
     using \<open>bij ?g\<close> \<open>bij ?s\<close> by (rule bij_comp)
 qed
 
 lemma overlapping_tensor:
   fixes a23 :: \<open>('a2::finite*'a3::finite) ell2 \<Rightarrow>\<^sub>C\<^sub>L ('b2::finite*'b3::finite) ell2\<close>
     and b12 :: \<open>('a1::finite*'a2) ell2 \<Rightarrow>\<^sub>C\<^sub>L ('b1::finite*'b2) ell2\<close>
   assumes eq: \<open>butterfly \<psi> \<psi>' \<otimes>\<^sub>o a23 = assoc_ell2 o\<^sub>C\<^sub>L (b12 \<otimes>\<^sub>o butterfly \<phi> \<phi>') o\<^sub>C\<^sub>L assoc_ell2'\<close>
   assumes \<open>\<psi> \<noteq> 0\<close> \<open>\<psi>' \<noteq> 0\<close> \<open>\<phi> \<noteq> 0\<close> \<open>\<phi>' \<noteq> 0\<close>
   shows \<open>\<exists>c. butterfly \<psi> \<psi>' \<otimes>\<^sub>o a23 = butterfly \<psi> \<psi>' \<otimes>\<^sub>o c \<otimes>\<^sub>o butterfly \<phi> \<phi>'\<close>
 proof -
   note [[show_types]]
   let ?id1 = \<open>id_cblinfun :: unit ell2 \<Rightarrow>\<^sub>C\<^sub>L unit ell2\<close>
   note id_cblinfun_eq_1[simp del]
   define d where \<open>d = butterfly \<psi> \<psi>' \<otimes>\<^sub>o a23\<close>
 
   define \<psi>\<^sub>n \<psi>\<^sub>n' a23\<^sub>n where \<open>\<psi>\<^sub>n = \<psi> /\<^sub>C norm \<psi>\<close> and \<open>\<psi>\<^sub>n' = \<psi>' /\<^sub>C norm \<psi>'\<close> and \<open>a23\<^sub>n = norm \<psi> *\<^sub>C norm \<psi>' *\<^sub>C a23\<close>
   have [simp]: \<open>norm \<psi>\<^sub>n = 1\<close> \<open>norm \<psi>\<^sub>n' = 1\<close>
     using \<open>\<psi> \<noteq> 0\<close> \<open>\<psi>' \<noteq> 0\<close> by (auto simp: \<psi>\<^sub>n_def \<psi>\<^sub>n'_def norm_inverse)
   have n1: \<open>butterfly \<psi>\<^sub>n \<psi>\<^sub>n' \<otimes>\<^sub>o a23\<^sub>n = butterfly \<psi> \<psi>' \<otimes>\<^sub>o a23\<close>
     apply (auto simp: \<psi>\<^sub>n_def \<psi>\<^sub>n'_def a23\<^sub>n_def tensor_op_scaleC_left tensor_op_scaleC_right)
     by (metis (no_types, lifting) assms(2) assms(3) inverse_mult_distrib mult.commute no_zero_divisors norm_eq_zero of_real_eq_0_iff right_inverse scaleC_one)
 
   define \<phi>\<^sub>n \<phi>\<^sub>n' b12\<^sub>n where \<open>\<phi>\<^sub>n = \<phi> /\<^sub>C norm \<phi>\<close> and \<open>\<phi>\<^sub>n' = \<phi>' /\<^sub>C norm \<phi>'\<close> and \<open>b12\<^sub>n = norm \<phi> *\<^sub>C norm \<phi>' *\<^sub>C b12\<close>
   have [simp]: \<open>norm \<phi>\<^sub>n = 1\<close> \<open>norm \<phi>\<^sub>n' = 1\<close>
     using \<open>\<phi> \<noteq> 0\<close> \<open>\<phi>' \<noteq> 0\<close> by (auto simp: \<phi>\<^sub>n_def \<phi>\<^sub>n'_def norm_inverse)
   have n2: \<open>b12\<^sub>n \<otimes>\<^sub>o butterfly \<phi>\<^sub>n \<phi>\<^sub>n' = b12 \<otimes>\<^sub>o butterfly \<phi> \<phi>'\<close>
     apply (auto simp: \<phi>\<^sub>n_def \<phi>\<^sub>n'_def b12\<^sub>n_def tensor_op_scaleC_left tensor_op_scaleC_right)
     by (metis (no_types, lifting) assms(4) assms(5) field_class.field_inverse inverse_mult_distrib mult.commute no_zero_divisors norm_eq_zero of_real_hom.hom_0 scaleC_one)
 
   define c' :: \<open>(unit*'a2*unit) ell2 \<Rightarrow>\<^sub>C\<^sub>L (unit*'b2*unit) ell2\<close> 
     where \<open>c' = (vector_to_cblinfun \<psi>\<^sub>n \<otimes>\<^sub>o id_cblinfun \<otimes>\<^sub>o vector_to_cblinfun \<phi>\<^sub>n)* o\<^sub>C\<^sub>L d
             o\<^sub>C\<^sub>L (vector_to_cblinfun \<psi>\<^sub>n' \<otimes>\<^sub>o id_cblinfun \<otimes>\<^sub>o vector_to_cblinfun \<phi>\<^sub>n')\<close>
 
   define c'' :: \<open>'a2 ell2 \<Rightarrow>\<^sub>C\<^sub>L 'b2 ell2\<close>
     where \<open>c'' = inv (\<lambda>c''. id_cblinfun \<otimes>\<^sub>o c'' \<otimes>\<^sub>o id_cblinfun) c'\<close>
 
   have *: \<open>bij (\<lambda>c''::'a2 ell2 \<Rightarrow>\<^sub>C\<^sub>L 'b2 ell2. ?id1 \<otimes>\<^sub>o c'' \<otimes>\<^sub>o ?id1)\<close>
     apply (subst asm_rl[of \<open>_ = (\<lambda>x. id_cblinfun \<otimes>\<^sub>o x) o (\<lambda>c''. c'' \<otimes>\<^sub>o id_cblinfun)\<close>])
     using [[show_consts]]
     by (auto intro!: bij_comp bij_tensor_op_one_dim_left bij_tensor_op_one_dim_right)
 
   have c'_c'': \<open>c' = ?id1 \<otimes>\<^sub>o c'' \<otimes>\<^sub>o ?id1\<close>
     unfolding c''_def 
     apply (rule surj_f_inv_f[where y=c', symmetric])
     using * by (rule bij_is_surj)
 
   define c :: \<open>'a2 ell2 \<Rightarrow>\<^sub>C\<^sub>L 'b2 ell2\<close>
     where \<open>c = c'' /\<^sub>C norm \<psi> /\<^sub>C norm \<psi>' /\<^sub>C norm \<phi> /\<^sub>C norm \<phi>'\<close>
 
   have aux: \<open>assoc_ell2' o\<^sub>C\<^sub>L (assoc_ell2 o\<^sub>C\<^sub>L x o\<^sub>C\<^sub>L assoc_ell2') o\<^sub>C\<^sub>L assoc_ell2 = x\<close> for x
     apply (simp add: cblinfun_assoc_left)
     by (simp add: cblinfun_assoc_right)
   have aux2: \<open>(assoc_ell2 o\<^sub>C\<^sub>L ((x \<otimes>\<^sub>o y) \<otimes>\<^sub>o z) o\<^sub>C\<^sub>L assoc_ell2') = x \<otimes>\<^sub>o (y \<otimes>\<^sub>o z)\<close> for x y z
     apply (rule equal_ket, rename_tac xyz)
     apply (case_tac xyz, hypsubst_thin)
     by (simp flip: tensor_ell2_ket add: assoc_ell2'_tensor assoc_ell2_tensor tensor_op_ell2)
 
   have \<open>d = (butterfly \<psi>\<^sub>n \<psi>\<^sub>n \<otimes>\<^sub>o id_cblinfun) o\<^sub>C\<^sub>L d o\<^sub>C\<^sub>L (butterfly \<psi>\<^sub>n' \<psi>\<^sub>n' \<otimes>\<^sub>o id_cblinfun)\<close>
     by (auto simp: d_def n1[symmetric] comp_tensor_op cnorm_eq_1[THEN iffD1])
   also have \<open>\<dots> = (butterfly \<psi>\<^sub>n \<psi>\<^sub>n \<otimes>\<^sub>o id_cblinfun) o\<^sub>C\<^sub>L assoc_ell2 o\<^sub>C\<^sub>L (b12\<^sub>n \<otimes>\<^sub>o butterfly \<phi>\<^sub>n \<phi>\<^sub>n')
                   o\<^sub>C\<^sub>L assoc_ell2' o\<^sub>C\<^sub>L (butterfly \<psi>\<^sub>n' \<psi>\<^sub>n' \<otimes>\<^sub>o id_cblinfun)\<close>
     by (auto simp: d_def eq n2 cblinfun_assoc_left)
   also have \<open>\<dots> = (butterfly \<psi>\<^sub>n \<psi>\<^sub>n \<otimes>\<^sub>o id_cblinfun) o\<^sub>C\<^sub>L assoc_ell2 o\<^sub>C\<^sub>L 
                ((id_cblinfun \<otimes>\<^sub>o butterfly \<phi>\<^sub>n \<phi>\<^sub>n) o\<^sub>C\<^sub>L (b12\<^sub>n \<otimes>\<^sub>o butterfly \<phi>\<^sub>n \<phi>\<^sub>n') o\<^sub>C\<^sub>L (id_cblinfun \<otimes>\<^sub>o butterfly \<phi>\<^sub>n' \<phi>\<^sub>n'))
                o\<^sub>C\<^sub>L assoc_ell2' o\<^sub>C\<^sub>L (butterfly \<psi>\<^sub>n' \<psi>\<^sub>n' \<otimes>\<^sub>o id_cblinfun)\<close>
     by (auto simp: comp_tensor_op cnorm_eq_1[THEN iffD1])
   also have \<open>\<dots> = (butterfly \<psi>\<^sub>n \<psi>\<^sub>n \<otimes>\<^sub>o id_cblinfun) o\<^sub>C\<^sub>L assoc_ell2 o\<^sub>C\<^sub>L 
                ((id_cblinfun \<otimes>\<^sub>o butterfly \<phi>\<^sub>n \<phi>\<^sub>n) o\<^sub>C\<^sub>L (assoc_ell2' o\<^sub>C\<^sub>L d o\<^sub>C\<^sub>L assoc_ell2) o\<^sub>C\<^sub>L (id_cblinfun \<otimes>\<^sub>o butterfly \<phi>\<^sub>n' \<phi>\<^sub>n'))
                o\<^sub>C\<^sub>L assoc_ell2' o\<^sub>C\<^sub>L (butterfly \<psi>\<^sub>n' \<psi>\<^sub>n' \<otimes>\<^sub>o id_cblinfun)\<close>
     by (auto simp: d_def n2 eq aux)
   also have \<open>\<dots> = ((butterfly \<psi>\<^sub>n \<psi>\<^sub>n \<otimes>\<^sub>o id_cblinfun) o\<^sub>C\<^sub>L (assoc_ell2 o\<^sub>C\<^sub>L (id_cblinfun \<otimes>\<^sub>o butterfly \<phi>\<^sub>n \<phi>\<^sub>n) o\<^sub>C\<^sub>L assoc_ell2'))
                o\<^sub>C\<^sub>L d o\<^sub>C\<^sub>L ((assoc_ell2 o\<^sub>C\<^sub>L (id_cblinfun \<otimes>\<^sub>o butterfly \<phi>\<^sub>n' \<phi>\<^sub>n') o\<^sub>C\<^sub>L assoc_ell2') o\<^sub>C\<^sub>L (butterfly \<psi>\<^sub>n' \<psi>\<^sub>n' \<otimes>\<^sub>o id_cblinfun))\<close>
     by (auto simp: sandwich_def cblinfun_assoc_left)
   also have \<open>\<dots> = (butterfly \<psi>\<^sub>n \<psi>\<^sub>n \<otimes>\<^sub>o id_cblinfun \<otimes>\<^sub>o butterfly \<phi>\<^sub>n \<phi>\<^sub>n)
                o\<^sub>C\<^sub>L d o\<^sub>C\<^sub>L (butterfly \<psi>\<^sub>n' \<psi>\<^sub>n' \<otimes>\<^sub>o id_cblinfun \<otimes>\<^sub>o butterfly \<phi>\<^sub>n' \<phi>\<^sub>n')\<close>
     apply (simp only: tensor_id[symmetric] comp_tensor_op aux2)
     by (simp add: cnorm_eq_1[THEN iffD1])
   also have \<open>\<dots> = (vector_to_cblinfun \<psi>\<^sub>n \<otimes>\<^sub>o id_cblinfun \<otimes>\<^sub>o vector_to_cblinfun \<phi>\<^sub>n)
                o\<^sub>C\<^sub>L c' o\<^sub>C\<^sub>L (vector_to_cblinfun \<psi>\<^sub>n' \<otimes>\<^sub>o id_cblinfun \<otimes>\<^sub>o vector_to_cblinfun \<phi>\<^sub>n')*\<close>
     apply (simp add: c'_def butterfly_def_one_dim[where 'c="unit ell2"] cblinfun_assoc_left comp_tensor_op
                       tensor_op_adjoint cnorm_eq_1[THEN iffD1])
     by (simp add: cblinfun_assoc_right comp_tensor_op)
   also have \<open>\<dots> = butterfly \<psi>\<^sub>n \<psi>\<^sub>n' \<otimes>\<^sub>o c'' \<otimes>\<^sub>o butterfly \<phi>\<^sub>n \<phi>\<^sub>n'\<close>
     by (simp add: c'_c'' comp_tensor_op tensor_op_adjoint butterfly_def_one_dim[symmetric])
   also have \<open>\<dots> = butterfly \<psi> \<psi>' \<otimes>\<^sub>o c \<otimes>\<^sub>o butterfly \<phi> \<phi>'\<close>
     by (simp add: \<psi>\<^sub>n_def \<psi>\<^sub>n'_def \<phi>\<^sub>n_def \<phi>\<^sub>n'_def c_def tensor_op_scaleC_left tensor_op_scaleC_right)
   finally have d_c: \<open>d = butterfly \<psi> \<psi>' \<otimes>\<^sub>o c \<otimes>\<^sub>o butterfly \<phi> \<phi>'\<close>
     by -
   then show ?thesis
     by (auto simp: d_def)
 qed
 
 lemma norm_tensor_ell2: \<open>norm (a \<otimes>\<^sub>s b) = norm a * norm b\<close>
   apply transfer
   by (simp add: ell2_norm_finite sum_product sum.cartesian_product case_prod_beta
       norm_mult power_mult_distrib flip: real_sqrt_mult)
 
 lemma bounded_cbilinear_tensor_ell2[bounded_cbilinear]: \<open>bounded_cbilinear (\<otimes>\<^sub>s)\<close>
 proof standard
   fix a a' :: "'a ell2" and b b' :: "'b ell2" and r :: complex
   show \<open>tensor_ell2 (a + a') b = tensor_ell2 a b + tensor_ell2 a' b\<close>
     by (meson tensor_ell2_add1)
   show \<open>tensor_ell2 a (b + b') = tensor_ell2 a b + tensor_ell2 a b'\<close>
     by (simp add: tensor_ell2_add2)  
   show \<open>tensor_ell2 (r *\<^sub>C a) b = r *\<^sub>C tensor_ell2 a b\<close>
     by (simp add: tensor_ell2_scaleC1)
   show \<open>tensor_ell2 a (r *\<^sub>C b) = r *\<^sub>C tensor_ell2 a b\<close>
     by (simp add: tensor_ell2_scaleC2)
   show \<open>\<exists>K. \<forall>a b. norm (tensor_ell2 a b) \<le> norm a * norm b * K \<close>
     apply (rule exI[of _ 1])
     by (simp add: norm_tensor_ell2)
 qed
 
 
 end