tagged 21 theories in the Analysis library for the manual

--- a/src/HOL/Analysis/Arcwise_Connected.thy	Mon Aug 27 22:58:36 2018 +0200
+++ b/src/HOL/Analysis/Arcwise_Connected.thy	Tue Aug 28 13:28:39 2018 +0100
@@ -9,14 +9,14 @@
 
 begin
 
-subsection\<open>The Brouwer reduction theorem\<close>
+subsection%important \<open>The Brouwer reduction theorem\<close>
 
-theorem Brouwer_reduction_theorem_gen:
+theorem%important Brouwer_reduction_theorem_gen:
   fixes S :: "'a::euclidean_space set"
   assumes "closed S" "\<phi> S"
       and \<phi>: "\<And>F. \<lbrakk>\<And>n. closed(F n); \<And>n. \<phi>(F n); \<And>n. F(Suc n) \<subseteq> F n\<rbrakk> \<Longrightarrow> \<phi>(\<Inter>range F)"
   obtains T where "T \<subseteq> S" "closed T" "\<phi> T" "\<And>U. \<lbrakk>U \<subseteq> S; closed U; \<phi> U\<rbrakk> \<Longrightarrow> \<not> (U \<subset> T)"
-proof -
+proof%unimportant -
   obtain B :: "nat \<Rightarrow> 'a set"
     where "inj B" "\<And>n. open(B n)" and open_cov: "\<And>S. open S \<Longrightarrow> \<exists>K. S = \<Union>(B ` K)"
       by (metis Setcompr_eq_image that univ_second_countable_sequence)
@@ -79,13 +79,13 @@
   qed
 qed
 
-corollary Brouwer_reduction_theorem:
+corollary%important Brouwer_reduction_theorem:
   fixes S :: "'a::euclidean_space set"
   assumes "compact S" "\<phi> S" "S \<noteq> {}"
       and \<phi>: "\<And>F. \<lbrakk>\<And>n. compact(F n); \<And>n. F n \<noteq> {}; \<And>n. \<phi>(F n); \<And>n. F(Suc n) \<subseteq> F n\<rbrakk> \<Longrightarrow> \<phi>(\<Inter>range F)"
   obtains T where "T \<subseteq> S" "compact T" "T \<noteq> {}" "\<phi> T"
                   "\<And>U. \<lbrakk>U \<subseteq> S; closed U; U \<noteq> {}; \<phi> U\<rbrakk> \<Longrightarrow> \<not> (U \<subset> T)"
-proof (rule Brouwer_reduction_theorem_gen [of S "\<lambda>T. T \<noteq> {} \<and> T \<subseteq> S \<and> \<phi> T"])
+proof%unimportant (rule Brouwer_reduction_theorem_gen [of S "\<lambda>T. T \<noteq> {} \<and> T \<subseteq> S \<and> \<phi> T"])
   fix F
   assume cloF: "\<And>n. closed (F n)"
      and F: "\<And>n. F n \<noteq> {} \<and> F n \<subseteq> S \<and> \<phi> (F n)" and Fsub: "\<And>n. F (Suc n) \<subseteq> F n"
@@ -107,14 +107,14 @@
 qed (meson assms compact_imp_closed seq_compact_closed_subset seq_compact_eq_compact)+
 
 
-subsection\<open>Arcwise Connections\<close>
+subsection%important\<open>Arcwise Connections\<close>(*FIX ME this subsection is empty(?) *)
 
-subsection\<open>Density of points with dyadic rational coordinates\<close>
+subsection%important\<open>Density of points with dyadic rational coordinates\<close>
 
-proposition closure_dyadic_rationals:
+proposition%important closure_dyadic_rationals:
     "closure (\<Union>k. \<Union>f \<in> Basis \<rightarrow> \<int>.
                    { \<Sum>i :: 'a :: euclidean_space \<in> Basis. (f i / 2^k) *\<^sub>R i }) = UNIV"
-proof -
+proof%unimportant -
   have "x \<in> closure (\<Union>k. \<Union>f \<in> Basis \<rightarrow> \<int>. {\<Sum>i \<in> Basis. (f i / 2^k) *\<^sub>R i})" for x::'a
   proof (clarsimp simp: closure_approachable)
     fix e::real
@@ -152,9 +152,9 @@
   then show ?thesis by auto
 qed
 
-corollary closure_rational_coordinates:
+corollary%important closure_rational_coordinates:
     "closure (\<Union>f \<in> Basis \<rightarrow> \<rat>. { \<Sum>i :: 'a :: euclidean_space \<in> Basis. f i *\<^sub>R i }) = UNIV"
-proof -
+proof%unimportant -
   have *: "(\<Union>k. \<Union>f \<in> Basis \<rightarrow> \<int>. { \<Sum>i::'a \<in> Basis. (f i / 2^k) *\<^sub>R i })
            \<subseteq> (\<Union>f \<in> Basis \<rightarrow> \<rat>. { \<Sum>i \<in> Basis. f i *\<^sub>R i })"
   proof clarsimp
@@ -168,7 +168,7 @@
     using closure_dyadic_rationals closure_mono [OF *] by blast
 qed
 
-lemma closure_dyadic_rationals_in_convex_set:
+lemma%unimportant closure_dyadic_rationals_in_convex_set:
    "\<lbrakk>convex S; interior S \<noteq> {}\<rbrakk>
         \<Longrightarrow> closure(S \<inter>
                     (\<Union>k. \<Union>f \<in> Basis \<rightarrow> \<int>.
@@ -176,7 +176,7 @@
             closure S"
   by (simp add: closure_dyadic_rationals closure_convex_Int_superset)
 
-lemma closure_rationals_in_convex_set:
+lemma%unimportant closure_rationals_in_convex_set:
    "\<lbrakk>convex S; interior S \<noteq> {}\<rbrakk>
     \<Longrightarrow> closure(S \<inter> (\<Union>f \<in> Basis \<rightarrow> \<rat>. { \<Sum>i :: 'a :: euclidean_space \<in> Basis. f i *\<^sub>R i })) =
         closure S"
@@ -187,11 +187,11 @@
 path connection is equivalent to arcwise connection for distinct points.
 The proof is based on Whyburn's "Topological Analysis".\<close>
 
-lemma closure_dyadic_rationals_in_convex_set_pos_1:
+lemma%important closure_dyadic_rationals_in_convex_set_pos_1:
   fixes S :: "real set"
   assumes "convex S" and intnz: "interior S \<noteq> {}" and pos: "\<And>x. x \<in> S \<Longrightarrow> 0 \<le> x"
     shows "closure(S \<inter> (\<Union>k m. {of_nat m / 2^k})) = closure S"
-proof -
+proof%unimportant -
   have "\<exists>m. f 1/2^k = real m / 2^k" if "(f 1) / 2^k \<in> S" "f 1 \<in> \<int>" for k and f :: "real \<Rightarrow> real"
     using that by (force simp: Ints_def zero_le_divide_iff power_le_zero_eq dest: pos zero_le_imp_eq_int)
   then have "S \<inter> (\<Union>k m. {real m / 2^k}) = S \<inter>
@@ -202,13 +202,13 @@
 qed
 
 
-definition dyadics :: "'a::field_char_0 set" where "dyadics \<equiv> \<Union>k m. {of_nat m / 2^k}"
+definition%important dyadics :: "'a::field_char_0 set" where "dyadics \<equiv> \<Union>k m. {of_nat m / 2^k}"
 
-lemma real_in_dyadics [simp]: "real m \<in> dyadics"
+lemma%unimportant real_in_dyadics [simp]: "real m \<in> dyadics"
   apply (simp add: dyadics_def)
   by (metis divide_numeral_1 numeral_One power_0)
 
-lemma nat_neq_4k1: "of_nat m \<noteq> (4 * of_nat k + 1) / (2 * 2^n :: 'a::field_char_0)"
+lemma%unimportant nat_neq_4k1: "of_nat m \<noteq> (4 * of_nat k + 1) / (2 * 2^n :: 'a::field_char_0)"
 proof
   assume "of_nat m = (4 * of_nat k + 1) / (2 * 2^n :: 'a)"
   then have "of_nat (m * (2 * 2^n)) = (of_nat (Suc (4 * k)) :: 'a)"
@@ -221,8 +221,8 @@
     by simp
 qed
 
-lemma nat_neq_4k3: "of_nat m \<noteq> (4 * of_nat k + 3) / (2 * 2^n :: 'a::field_char_0)"
-proof
+lemma%important nat_neq_4k3: "of_nat m \<noteq> (4 * of_nat k + 3) / (2 * 2^n :: 'a::field_char_0)"
+proof%unimportant
   assume "of_nat m = (4 * of_nat k + 3) / (2 * 2^n :: 'a)"
   then have "of_nat (m * (2 * 2^n)) = (of_nat (4 * k + 3) :: 'a)"
     by (simp add: divide_simps)
@@ -234,10 +234,10 @@
     by simp
 qed
 
-lemma iff_4k:
+lemma%important iff_4k:
   assumes "r = real k" "odd k"
     shows "(4 * real m + r) / (2 * 2^n) = (4 * real m' + r) / (2 * 2 ^ n') \<longleftrightarrow> m=m' \<and> n=n'"
-proof -
+proof%unimportant -
   { assume "(4 * real m + r) / (2 * 2^n) = (4 * real m' + r) / (2 * 2 ^ n')"
     then have "real ((4 * m + k) * (2 * 2 ^ n')) = real ((4 * m' + k) * (2 * 2^n))"
       using assms by (auto simp: field_simps)
@@ -254,8 +254,8 @@
   then show ?thesis by blast
 qed
 
-lemma neq_4k1_k43: "(4 * real m + 1) / (2 * 2^n) \<noteq> (4 * real m' + 3) / (2 * 2 ^ n')"
-proof
+lemma%important neq_4k1_k43: "(4 * real m + 1) / (2 * 2^n) \<noteq> (4 * real m' + 3) / (2 * 2 ^ n')"
+proof%unimportant
   assume "(4 * real m + 1) / (2 * 2^n) = (4 * real m' + 3) / (2 * 2 ^ n')"
   then have "real (Suc (4 * m) * (2 * 2 ^ n')) = real ((4 * m' + 3) * (2 * 2^n))"
     by (auto simp: field_simps)
@@ -271,11 +271,11 @@
     using even_Suc by presburger
 qed
 
-lemma dyadic_413_cases:
+lemma%important dyadic_413_cases:
   obtains "(of_nat m::'a::field_char_0) / 2^k \<in> Nats"
   | m' k' where "k' < k" "(of_nat m:: 'a) / 2^k = of_nat (4*m' + 1) / 2^Suc k'"
   | m' k' where "k' < k" "(of_nat m:: 'a) / 2^k = of_nat (4*m' + 3) / 2^Suc k'"
-proof (cases "m>0")
+proof%unimportant (cases "m>0")
   case False
   then have "m=0" by simp
   with that show ?thesis by auto
@@ -324,11 +324,11 @@
 qed
 
 
-lemma dyadics_iff:
+lemma%important dyadics_iff:
    "(dyadics :: 'a::field_char_0 set) =
     Nats \<union> (\<Union>k m. {of_nat (4*m + 1) / 2^Suc k}) \<union> (\<Union>k m. {of_nat (4*m + 3) / 2^Suc k})"
            (is "_ = ?rhs")
-proof
+proof%unimportant
   show "dyadics \<subseteq> ?rhs"
     unfolding dyadics_def
     apply clarify
@@ -355,14 +355,14 @@
      apply (fastforce simp add: dyadics_iff Nats_def field_simps)+
   done
 
-lemma dyadics_levels: "dyadics = (\<Union>K. \<Union>k<K. \<Union> m. {of_nat m / 2^k})"
+lemma%unimportant dyadics_levels: "dyadics = (\<Union>K. \<Union>k<K. \<Union> m. {of_nat m / 2^k})"
   unfolding dyadics_def by auto
 
-lemma dyad_rec_level_termination:
+lemma%important dyad_rec_level_termination:
   assumes "k < K"
   shows "dyad_rec_dom(b, l, r, real m / 2^k)"
   using assms
-proof (induction K arbitrary: k m)
+proof%unimportant (induction K arbitrary: k m)
   case 0
   then show ?case by auto
 next
@@ -427,22 +427,22 @@
 qed
 
 
-lemma dyad_rec_termination: "x \<in> dyadics \<Longrightarrow> dyad_rec_dom(b,l,r,x)"
+lemma%unimportant dyad_rec_termination: "x \<in> dyadics \<Longrightarrow> dyad_rec_dom(b,l,r,x)"
   by (auto simp: dyadics_levels intro: dyad_rec_level_termination)
 
-lemma dyad_rec_of_nat [simp]: "dyad_rec b l r (real m) = b m"
+lemma%unimportant dyad_rec_of_nat [simp]: "dyad_rec b l r (real m) = b m"
   by (simp add: dyad_rec.psimps dyad_rec_termination)
 
-lemma dyad_rec_41 [simp]: "dyad_rec b l r ((4 * real m + 1) / 2 ^ (Suc n)) = l (dyad_rec b l r ((2*m + 1) / 2^n))"
+lemma%unimportant dyad_rec_41 [simp]: "dyad_rec b l r ((4 * real m + 1) / 2 ^ (Suc n)) = l (dyad_rec b l r ((2*m + 1) / 2^n))"
   apply (rule dyad_rec.psimps)
   by (metis dyad_rec_level_termination lessI add.commute of_nat_Suc of_nat_mult of_nat_numeral)
 
 
-lemma dyad_rec_43 [simp]: "dyad_rec b l r ((4 * real m + 3) / 2 ^ (Suc n)) = r (dyad_rec b l r ((2*m + 1) / 2^n))"
+lemma%unimportant dyad_rec_43 [simp]: "dyad_rec b l r ((4 * real m + 3) / 2 ^ (Suc n)) = r (dyad_rec b l r ((2*m + 1) / 2^n))"
   apply (rule dyad_rec.psimps)
   by (metis dyad_rec_level_termination lessI of_nat_add of_nat_mult of_nat_numeral)
 
-lemma dyad_rec_41_times2:
+lemma%unimportant dyad_rec_41_times2:
   assumes "n > 0"
     shows "dyad_rec b l r (2 * ((4 * real m + 1) / 2^Suc n)) = l (dyad_rec b l r (2 * (2 * real m + 1) / 2^n))"
 proof -
@@ -461,10 +461,10 @@
   finally show ?thesis .
 qed
 
-lemma dyad_rec_43_times2:
+lemma%important dyad_rec_43_times2:
   assumes "n > 0"
     shows "dyad_rec b l r (2 * ((4 * real m + 3) / 2^Suc n)) = r (dyad_rec b l r (2 * (2 * real m + 1) / 2^n))"
-proof -
+proof%unimportant -
   obtain n' where n': "n = Suc n'"
     using assms not0_implies_Suc by blast
   have "dyad_rec b l r (2 * ((4 * real m + 3) / 2^Suc n)) = dyad_rec b l r ((2 * (4 * real m + 3)) / (2 * 2^n))"
@@ -480,22 +480,22 @@
   finally show ?thesis .
 qed
 
-definition dyad_rec2
+definition%important dyad_rec2
     where "dyad_rec2 u v lc rc x =
              dyad_rec (\<lambda>z. (u,v)) (\<lambda>(a,b). (a, lc a b (midpoint a b))) (\<lambda>(a,b). (rc a b (midpoint a b), b)) (2*x)"
 
 abbreviation leftrec where "leftrec u v lc rc x \<equiv> fst (dyad_rec2 u v lc rc x)"
 abbreviation rightrec where "rightrec u v lc rc x \<equiv> snd (dyad_rec2 u v lc rc x)"
 
-lemma leftrec_base: "leftrec u v lc rc (real m / 2) = u"
+lemma%unimportant leftrec_base: "leftrec u v lc rc (real m / 2) = u"
   by (simp add: dyad_rec2_def)
 
-lemma leftrec_41: "n > 0 \<Longrightarrow> leftrec u v lc rc ((4 * real m + 1) / 2 ^ (Suc n)) = leftrec u v lc rc ((2 * real m + 1) / 2^n)"
+lemma%unimportant leftrec_41: "n > 0 \<Longrightarrow> leftrec u v lc rc ((4 * real m + 1) / 2 ^ (Suc n)) = leftrec u v lc rc ((2 * real m + 1) / 2^n)"
   apply (simp only: dyad_rec2_def dyad_rec_41_times2)
   apply (simp add: case_prod_beta)
   done
 
-lemma leftrec_43: "n > 0 \<Longrightarrow>
+lemma%unimportant leftrec_43: "n > 0 \<Longrightarrow>
              leftrec u v lc rc ((4 * real m + 3) / 2 ^ (Suc n)) =
                  rc (leftrec u v lc rc ((2 * real m + 1) / 2^n)) (rightrec u v lc rc ((2 * real m + 1) / 2^n))
                  (midpoint (leftrec u v lc rc ((2 * real m + 1) / 2^n)) (rightrec u v lc rc ((2 * real m + 1) / 2^n)))"
@@ -503,10 +503,10 @@
   apply (simp add: case_prod_beta)
   done
 
-lemma rightrec_base: "rightrec u v lc rc (real m / 2) = v"
+lemma%unimportant rightrec_base: "rightrec u v lc rc (real m / 2) = v"
   by (simp add: dyad_rec2_def)
 
-lemma rightrec_41: "n > 0 \<Longrightarrow>
+lemma%unimportant rightrec_41: "n > 0 \<Longrightarrow>
              rightrec u v lc rc ((4 * real m + 1) / 2 ^ (Suc n)) =
                  lc (leftrec u v lc rc ((2 * real m + 1) / 2^n)) (rightrec u v lc rc ((2 * real m + 1) / 2^n))
                  (midpoint (leftrec u v lc rc ((2 * real m + 1) / 2^n)) (rightrec u v lc rc ((2 * real m + 1) / 2^n)))"
@@ -514,24 +514,24 @@
   apply (simp add: case_prod_beta)
   done
 
-lemma rightrec_43: "n > 0 \<Longrightarrow> rightrec u v lc rc ((4 * real m + 3) / 2 ^ (Suc n)) = rightrec u v lc rc ((2 * real m + 1) / 2^n)"
+lemma%unimportant rightrec_43: "n > 0 \<Longrightarrow> rightrec u v lc rc ((4 * real m + 3) / 2 ^ (Suc n)) = rightrec u v lc rc ((2 * real m + 1) / 2^n)"
   apply (simp only: dyad_rec2_def dyad_rec_43_times2)
   apply (simp add: case_prod_beta)
   done
 
-lemma dyadics_in_open_unit_interval:
+lemma%unimportant dyadics_in_open_unit_interval:
    "{0<..<1} \<inter> (\<Union>k m. {real m / 2^k}) = (\<Union>k. \<Union>m \<in> {0<..<2^k}. {real m / 2^k})"
   by (auto simp: divide_simps)
 
 
 
-theorem homeomorphic_monotone_image_interval:
+theorem%important homeomorphic_monotone_image_interval:
   fixes f :: "real \<Rightarrow> 'a::{real_normed_vector,complete_space}"
   assumes cont_f: "continuous_on {0..1} f"
       and conn: "\<And>y. connected ({0..1} \<inter> f -` {y})"
       and f_1not0: "f 1 \<noteq> f 0"
     shows "(f ` {0..1}) homeomorphic {0..1::real}"
-proof -
+proof%unimportant -
   have "\<exists>c d. a \<le> c \<and> c \<le> m \<and> m \<le> d \<and> d \<le> b \<and>
               (\<forall>x \<in> {c..d}. f x = f m) \<and>
               (\<forall>x \<in> {a..<c}. (f x \<noteq> f m)) \<and>
@@ -1664,11 +1664,11 @@
 qed
 
 
-theorem path_contains_arc:
+theorem%important path_contains_arc:
   fixes p :: "real \<Rightarrow> 'a::{complete_space,real_normed_vector}"
   assumes "path p" and a: "pathstart p = a" and b: "pathfinish p = b" and "a \<noteq> b"
   obtains q where "arc q" "path_image q \<subseteq> path_image p" "pathstart q = a" "pathfinish q = b"
-proof -
+proof%unimportant -
   have ucont_p: "uniformly_continuous_on {0..1} p"
     using \<open>path p\<close> unfolding path_def
     by (metis compact_Icc compact_uniformly_continuous)
@@ -1961,12 +1961,12 @@
 qed
 
 
-corollary path_connected_arcwise:
+corollary%important path_connected_arcwise:
   fixes S :: "'a::{complete_space,real_normed_vector} set"
   shows "path_connected S \<longleftrightarrow>
          (\<forall>x \<in> S. \<forall>y \<in> S. x \<noteq> y \<longrightarrow> (\<exists>g. arc g \<and> path_image g \<subseteq> S \<and> pathstart g = x \<and> pathfinish g = y))"
         (is "?lhs = ?rhs")
-proof (intro iffI impI ballI)
+proof%unimportant (intro iffI impI ballI)
   fix x y
   assume "path_connected S" "x \<in> S" "y \<in> S" "x \<noteq> y"
   then obtain p where p: "path p" "path_image p \<subseteq> S" "pathstart p = x" "pathfinish p = y"
@@ -1992,23 +1992,23 @@
 qed
 
 
-corollary arc_connected_trans:
+corollary%important arc_connected_trans:
   fixes g :: "real \<Rightarrow> 'a::{complete_space,real_normed_vector}"
   assumes "arc g" "arc h" "pathfinish g = pathstart h" "pathstart g \<noteq> pathfinish h"
   obtains i where "arc i" "path_image i \<subseteq> path_image g \<union> path_image h"
                   "pathstart i = pathstart g" "pathfinish i = pathfinish h"
-  by (metis (no_types, hide_lams) arc_imp_path assms path_contains_arc path_image_join path_join pathfinish_join pathstart_join)
+  by%unimportant (metis (no_types, hide_lams) arc_imp_path assms path_contains_arc path_image_join path_join pathfinish_join pathstart_join)
 
 
 
 
-subsection\<open>Accessibility of frontier points\<close>
+subsection%important\<open>Accessibility of frontier points\<close>
 
-lemma dense_accessible_frontier_points:
+lemma%important dense_accessible_frontier_points:
   fixes S :: "'a::{complete_space,real_normed_vector} set"
   assumes "open S" and opeSV: "openin (subtopology euclidean (frontier S)) V" and "V \<noteq> {}"
   obtains g where "arc g" "g ` {0..<1} \<subseteq> S" "pathstart g \<in> S" "pathfinish g \<in> V"
-proof -
+proof%unimportant -
   obtain z where "z \<in> V"
     using \<open>V \<noteq> {}\<close> by auto
   then obtain r where "r > 0" and r: "ball z r \<inter> frontier S \<subseteq> V"
@@ -2099,12 +2099,12 @@
 qed
 
 
-lemma dense_accessible_frontier_points_connected:
+lemma%important dense_accessible_frontier_points_connected:
   fixes S :: "'a::{complete_space,real_normed_vector} set"
   assumes "open S" "connected S" "x \<in> S" "V \<noteq> {}"
       and ope: "openin (subtopology euclidean (frontier S)) V"
   obtains g where "arc g" "g ` {0..<1} \<subseteq> S" "pathstart g = x" "pathfinish g \<in> V"
-proof -
+proof%unimportant -
   have "V \<subseteq> frontier S"
     using ope openin_imp_subset by blast
   with \<open>open S\<close> \<open>x \<in> S\<close> have "x \<notin> V"
@@ -2134,14 +2134,14 @@
     using h \<open>pathfinish g \<in> V\<close> by auto
 qed
 
-lemma dense_access_fp_aux:
+lemma%important dense_access_fp_aux:
   fixes S :: "'a::{complete_space,real_normed_vector} set"
   assumes S: "open S" "connected S"
       and opeSU: "openin (subtopology euclidean (frontier S)) U"
       and opeSV: "openin (subtopology euclidean (frontier S)) V"
       and "V \<noteq> {}" "\<not> U \<subseteq> V"
   obtains g where "arc g" "pathstart g \<in> U" "pathfinish g \<in> V" "g ` {0<..<1} \<subseteq> S"
-proof -
+proof%unimportant -
   have "S \<noteq> {}"
     using opeSV \<open>V \<noteq> {}\<close> by (metis frontier_empty openin_subtopology_empty)
   then obtain x where "x \<in> S" by auto
@@ -2181,14 +2181,14 @@
   qed
 qed
 
-lemma dense_accessible_frontier_point_pairs:
+lemma%important dense_accessible_frontier_point_pairs:
   fixes S :: "'a::{complete_space,real_normed_vector} set"
   assumes S: "open S" "connected S"
       and opeSU: "openin (subtopology euclidean (frontier S)) U"
       and opeSV: "openin (subtopology euclidean (frontier S)) V"
       and "U \<noteq> {}" "V \<noteq> {}" "U \<noteq> V"
     obtains g where "arc g" "pathstart g \<in> U" "pathfinish g \<in> V" "g ` {0<..<1} \<subseteq> S"
-proof -
+proof%unimportant -
   consider "\<not> U \<subseteq> V" | "\<not> V \<subseteq> U"
     using \<open>U \<noteq> V\<close> by blast
   then show ?thesis

--- a/src/HOL/Analysis/Binary_Product_Measure.thy	Mon Aug 27 22:58:36 2018 +0200
+++ b/src/HOL/Analysis/Binary_Product_Measure.thy	Tue Aug 28 13:28:39 2018 +0100
@@ -2,67 +2,67 @@
     Author:     Johannes Hölzl, TU München
 *)
 
-section \<open>Binary product measures\<close>
+section%important \<open>Binary product measures\<close>
 
 theory Binary_Product_Measure
 imports Nonnegative_Lebesgue_Integration
 begin
 
-lemma Pair_vimage_times[simp]: "Pair x -` (A \<times> B) = (if x \<in> A then B else {})"
+lemma%unimportant Pair_vimage_times[simp]: "Pair x -` (A \<times> B) = (if x \<in> A then B else {})"
   by auto
 
-lemma rev_Pair_vimage_times[simp]: "(\<lambda>x. (x, y)) -` (A \<times> B) = (if y \<in> B then A else {})"
+lemma%unimportant rev_Pair_vimage_times[simp]: "(\<lambda>x. (x, y)) -` (A \<times> B) = (if y \<in> B then A else {})"
   by auto
 
-subsection "Binary products"
+subsection%important "Binary products"
 
-definition pair_measure (infixr "\<Otimes>\<^sub>M" 80) where
+definition%important pair_measure (infixr "\<Otimes>\<^sub>M" 80) where
   "A \<Otimes>\<^sub>M B = measure_of (space A \<times> space B)
       {a \<times> b | a b. a \<in> sets A \<and> b \<in> sets B}
       (\<lambda>X. \<integral>\<^sup>+x. (\<integral>\<^sup>+y. indicator X (x,y) \<partial>B) \<partial>A)"
 
-lemma pair_measure_closed: "{a \<times> b | a b. a \<in> sets A \<and> b \<in> sets B} \<subseteq> Pow (space A \<times> space B)"
-  using sets.space_closed[of A] sets.space_closed[of B] by auto
+lemma%important pair_measure_closed: "{a \<times> b | a b. a \<in> sets A \<and> b \<in> sets B} \<subseteq> Pow (space A \<times> space B)"
+  using%unimportant sets.space_closed[of A] sets.space_closed[of B] by auto
 
-lemma space_pair_measure:
+lemma%important space_pair_measure:
   "space (A \<Otimes>\<^sub>M B) = space A \<times> space B"
   unfolding pair_measure_def using pair_measure_closed[of A B]
-  by (rule space_measure_of)
+  by%unimportant (rule space_measure_of)
 
-lemma SIGMA_Collect_eq: "(SIGMA x:space M. {y\<in>space N. P x y}) = {x\<in>space (M \<Otimes>\<^sub>M N). P (fst x) (snd x)}"
+lemma%unimportant SIGMA_Collect_eq: "(SIGMA x:space M. {y\<in>space N. P x y}) = {x\<in>space (M \<Otimes>\<^sub>M N). P (fst x) (snd x)}"
   by (auto simp: space_pair_measure)
 
-lemma sets_pair_measure:
+lemma%unimportant sets_pair_measure:
   "sets (A \<Otimes>\<^sub>M B) = sigma_sets (space A \<times> space B) {a \<times> b | a b. a \<in> sets A \<and> b \<in> sets B}"
   unfolding pair_measure_def using pair_measure_closed[of A B]
   by (rule sets_measure_of)
 
-lemma sets_pair_measure_cong[measurable_cong, cong]:
+lemma%unimportant sets_pair_measure_cong[measurable_cong, cong]:
   "sets M1 = sets M1' \<Longrightarrow> sets M2 = sets M2' \<Longrightarrow> sets (M1 \<Otimes>\<^sub>M M2) = sets (M1' \<Otimes>\<^sub>M M2')"
   unfolding sets_pair_measure by (simp cong: sets_eq_imp_space_eq)
 
-lemma pair_measureI[intro, simp, measurable]:
+lemma%unimportant pair_measureI[intro, simp, measurable]:
   "x \<in> sets A \<Longrightarrow> y \<in> sets B \<Longrightarrow> x \<times> y \<in> sets (A \<Otimes>\<^sub>M B)"
   by (auto simp: sets_pair_measure)
 
-lemma sets_Pair: "{x} \<in> sets M1 \<Longrightarrow> {y} \<in> sets M2 \<Longrightarrow> {(x, y)} \<in> sets (M1 \<Otimes>\<^sub>M M2)"
+lemma%unimportant sets_Pair: "{x} \<in> sets M1 \<Longrightarrow> {y} \<in> sets M2 \<Longrightarrow> {(x, y)} \<in> sets (M1 \<Otimes>\<^sub>M M2)"
   using pair_measureI[of "{x}" M1 "{y}" M2] by simp
 
-lemma measurable_pair_measureI:
+lemma%unimportant measurable_pair_measureI:
   assumes 1: "f \<in> space M \<rightarrow> space M1 \<times> space M2"
   assumes 2: "\<And>A B. A \<in> sets M1 \<Longrightarrow> B \<in> sets M2 \<Longrightarrow> f -` (A \<times> B) \<inter> space M \<in> sets M"
   shows "f \<in> measurable M (M1 \<Otimes>\<^sub>M M2)"
   unfolding pair_measure_def using 1 2
   by (intro measurable_measure_of) (auto dest: sets.sets_into_space)
 
-lemma measurable_split_replace[measurable (raw)]:
+lemma%unimportant measurable_split_replace[measurable (raw)]:
   "(\<lambda>x. f x (fst (g x)) (snd (g x))) \<in> measurable M N \<Longrightarrow> (\<lambda>x. case_prod (f x) (g x)) \<in> measurable M N"
   unfolding split_beta' .
 
-lemma measurable_Pair[measurable (raw)]:
+lemma%important measurable_Pair[measurable (raw)]:
   assumes f: "f \<in> measurable M M1" and g: "g \<in> measurable M M2"
   shows "(\<lambda>x. (f x, g x)) \<in> measurable M (M1 \<Otimes>\<^sub>M M2)"
-proof (rule measurable_pair_measureI)
+proof%unimportant (rule measurable_pair_measureI)
   show "(\<lambda>x. (f x, g x)) \<in> space M \<rightarrow> space M1 \<times> space M2"
     using f g by (auto simp: measurable_def)
   fix A B assume *: "A \<in> sets M1" "B \<in> sets M2"
@@ -73,59 +73,59 @@
   finally show "(\<lambda>x. (f x, g x)) -` (A \<times> B) \<inter> space M \<in> sets M" .
 qed
 
-lemma measurable_fst[intro!, simp, measurable]: "fst \<in> measurable (M1 \<Otimes>\<^sub>M M2) M1"
+lemma%unimportant measurable_fst[intro!, simp, measurable]: "fst \<in> measurable (M1 \<Otimes>\<^sub>M M2) M1"
   by (auto simp: fst_vimage_eq_Times space_pair_measure sets.sets_into_space times_Int_times
     measurable_def)
 
-lemma measurable_snd[intro!, simp, measurable]: "snd \<in> measurable (M1 \<Otimes>\<^sub>M M2) M2"
+lemma%unimportant measurable_snd[intro!, simp, measurable]: "snd \<in> measurable (M1 \<Otimes>\<^sub>M M2) M2"
   by (auto simp: snd_vimage_eq_Times space_pair_measure sets.sets_into_space times_Int_times
     measurable_def)
 
-lemma measurable_Pair_compose_split[measurable_dest]:
+lemma%unimportant measurable_Pair_compose_split[measurable_dest]:
   assumes f: "case_prod f \<in> measurable (M1 \<Otimes>\<^sub>M M2) N"
   assumes g: "g \<in> measurable M M1" and h: "h \<in> measurable M M2"
   shows "(\<lambda>x. f (g x) (h x)) \<in> measurable M N"
   using measurable_compose[OF measurable_Pair f, OF g h] by simp
 
-lemma measurable_Pair1_compose[measurable_dest]:
+lemma%unimportant measurable_Pair1_compose[measurable_dest]:
   assumes f: "(\<lambda>x. (f x, g x)) \<in> measurable M (M1 \<Otimes>\<^sub>M M2)"
   assumes [measurable]: "h \<in> measurable N M"
   shows "(\<lambda>x. f (h x)) \<in> measurable N M1"
   using measurable_compose[OF f measurable_fst] by simp
 
-lemma measurable_Pair2_compose[measurable_dest]:
+lemma%unimportant measurable_Pair2_compose[measurable_dest]:
   assumes f: "(\<lambda>x. (f x, g x)) \<in> measurable M (M1 \<Otimes>\<^sub>M M2)"
   assumes [measurable]: "h \<in> measurable N M"
   shows "(\<lambda>x. g (h x)) \<in> measurable N M2"
   using measurable_compose[OF f measurable_snd] by simp
 
-lemma measurable_pair:
+lemma%unimportant measurable_pair:
   assumes "(fst \<circ> f) \<in> measurable M M1" "(snd \<circ> f) \<in> measurable M M2"
   shows "f \<in> measurable M (M1 \<Otimes>\<^sub>M M2)"
   using measurable_Pair[OF assms] by simp
 
-lemma
+lemma%unimportant (*FIX ME needs a name *)
   assumes f[measurable]: "f \<in> measurable M (N \<Otimes>\<^sub>M P)"
   shows measurable_fst': "(\<lambda>x. fst (f x)) \<in> measurable M N"
     and measurable_snd': "(\<lambda>x. snd (f x)) \<in> measurable M P"
   by simp_all
 
-lemma
+lemma%unimportant (*FIX ME needs a name *)
   assumes f[measurable]: "f \<in> measurable M N"
   shows measurable_fst'': "(\<lambda>x. f (fst x)) \<in> measurable (M \<Otimes>\<^sub>M P) N"
     and measurable_snd'': "(\<lambda>x. f (snd x)) \<in> measurable (P \<Otimes>\<^sub>M M) N"
   by simp_all
 
-lemma sets_pair_in_sets:
+lemma%unimportant sets_pair_in_sets:
   assumes "\<And>a b. a \<in> sets A \<Longrightarrow> b \<in> sets B \<Longrightarrow> a \<times> b \<in> sets N"
   shows "sets (A \<Otimes>\<^sub>M B) \<subseteq> sets N"
   unfolding sets_pair_measure
   by (intro sets.sigma_sets_subset') (auto intro!: assms)
 
-lemma sets_pair_eq_sets_fst_snd:
+lemma%important  sets_pair_eq_sets_fst_snd:
   "sets (A \<Otimes>\<^sub>M B) = sets (Sup {vimage_algebra (space A \<times> space B) fst A, vimage_algebra (space A \<times> space B) snd B})"
     (is "?P = sets (Sup {?fst, ?snd})")
-proof -
+proof%unimportant -
   { fix a b assume ab: "a \<in> sets A" "b \<in> sets B"
     then have "a \<times> b = (fst -` a \<inter> (space A \<times> space B)) \<inter> (snd -` b \<inter> (space A \<times> space B))"
       by (auto dest: sets.sets_into_space)
@@ -157,29 +157,29 @@
     done
 qed
 
-lemma measurable_pair_iff:
+lemma%unimportant measurable_pair_iff:
   "f \<in> measurable M (M1 \<Otimes>\<^sub>M M2) \<longleftrightarrow> (fst \<circ> f) \<in> measurable M M1 \<and> (snd \<circ> f) \<in> measurable M M2"
   by (auto intro: measurable_pair[of f M M1 M2])
 
-lemma measurable_split_conv:
+lemma%unimportant  measurable_split_conv:
   "(\<lambda>(x, y). f x y) \<in> measurable A B \<longleftrightarrow> (\<lambda>x. f (fst x) (snd x)) \<in> measurable A B"
   by (intro arg_cong2[where f="(\<in>)"]) auto
 
-lemma measurable_pair_swap': "(\<lambda>(x,y). (y, x)) \<in> measurable (M1 \<Otimes>\<^sub>M M2) (M2 \<Otimes>\<^sub>M M1)"
+lemma%unimportant measurable_pair_swap': "(\<lambda>(x,y). (y, x)) \<in> measurable (M1 \<Otimes>\<^sub>M M2) (M2 \<Otimes>\<^sub>M M1)"
   by (auto intro!: measurable_Pair simp: measurable_split_conv)
 
-lemma measurable_pair_swap:
+lemma%unimportant  measurable_pair_swap:
   assumes f: "f \<in> measurable (M1 \<Otimes>\<^sub>M M2) M" shows "(\<lambda>(x,y). f (y, x)) \<in> measurable (M2 \<Otimes>\<^sub>M M1) M"
   using measurable_comp[OF measurable_Pair f] by (auto simp: measurable_split_conv comp_def)
 
-lemma measurable_pair_swap_iff:
+lemma%unimportant measurable_pair_swap_iff:
   "f \<in> measurable (M2 \<Otimes>\<^sub>M M1) M \<longleftrightarrow> (\<lambda>(x,y). f (y,x)) \<in> measurable (M1 \<Otimes>\<^sub>M M2) M"
   by (auto dest: measurable_pair_swap)
 
-lemma measurable_Pair1': "x \<in> space M1 \<Longrightarrow> Pair x \<in> measurable M2 (M1 \<Otimes>\<^sub>M M2)"
+lemma%unimportant measurable_Pair1': "x \<in> space M1 \<Longrightarrow> Pair x \<in> measurable M2 (M1 \<Otimes>\<^sub>M M2)"
   by simp
 
-lemma sets_Pair1[measurable (raw)]:
+lemma%unimportant sets_Pair1[measurable (raw)]:
   assumes A: "A \<in> sets (M1 \<Otimes>\<^sub>M M2)" shows "Pair x -` A \<in> sets M2"
 proof -
   have "Pair x -` A = (if x \<in> space M1 then Pair x -` A \<inter> space M2 else {})"
@@ -189,10 +189,10 @@
   finally show ?thesis .
 qed
 
-lemma measurable_Pair2': "y \<in> space M2 \<Longrightarrow> (\<lambda>x. (x, y)) \<in> measurable M1 (M1 \<Otimes>\<^sub>M M2)"
+lemma%unimportant measurable_Pair2': "y \<in> space M2 \<Longrightarrow> (\<lambda>x. (x, y)) \<in> measurable M1 (M1 \<Otimes>\<^sub>M M2)"
   by (auto intro!: measurable_Pair)
 
-lemma sets_Pair2: assumes A: "A \<in> sets (M1 \<Otimes>\<^sub>M M2)" shows "(\<lambda>x. (x, y)) -` A \<in> sets M1"
+lemma%unimportant sets_Pair2: assumes A: "A \<in> sets (M1 \<Otimes>\<^sub>M M2)" shows "(\<lambda>x. (x, y)) -` A \<in> sets M1"
 proof -
   have "(\<lambda>x. (x, y)) -` A = (if y \<in> space M2 then (\<lambda>x. (x, y)) -` A \<inter> space M1 else {})"
     using A[THEN sets.sets_into_space] by (auto simp: space_pair_measure)
@@ -201,23 +201,23 @@
   finally show ?thesis .
 qed
 
-lemma measurable_Pair2:
+lemma%unimportant measurable_Pair2:
   assumes f: "f \<in> measurable (M1 \<Otimes>\<^sub>M M2) M" and x: "x \<in> space M1"
   shows "(\<lambda>y. f (x, y)) \<in> measurable M2 M"
   using measurable_comp[OF measurable_Pair1' f, OF x]
   by (simp add: comp_def)
 
-lemma measurable_Pair1:
+lemma%unimportant measurable_Pair1:
   assumes f: "f \<in> measurable (M1 \<Otimes>\<^sub>M M2) M" and y: "y \<in> space M2"
   shows "(\<lambda>x. f (x, y)) \<in> measurable M1 M"
   using measurable_comp[OF measurable_Pair2' f, OF y]
   by (simp add: comp_def)
 
-lemma Int_stable_pair_measure_generator: "Int_stable {a \<times> b | a b. a \<in> sets A \<and> b \<in> sets B}"
+lemma%unimportant Int_stable_pair_measure_generator: "Int_stable {a \<times> b | a b. a \<in> sets A \<and> b \<in> sets B}"
   unfolding Int_stable_def
   by safe (auto simp add: times_Int_times)
 
-lemma (in finite_measure) finite_measure_cut_measurable:
+lemma%unimportant (in finite_measure) finite_measure_cut_measurable:
   assumes [measurable]: "Q \<in> sets (N \<Otimes>\<^sub>M M)"
   shows "(\<lambda>x. emeasure M (Pair x -` Q)) \<in> borel_measurable N"
     (is "?s Q \<in> _")
@@ -239,7 +239,7 @@
     unfolding sets_pair_measure[symmetric] by simp
 qed (auto simp add: if_distrib Int_def[symmetric] intro!: measurable_If)
 
-lemma (in sigma_finite_measure) measurable_emeasure_Pair:
+lemma%unimportant (in sigma_finite_measure) measurable_emeasure_Pair:
   assumes Q: "Q \<in> sets (N \<Otimes>\<^sub>M M)" shows "(\<lambda>x. emeasure M (Pair x -` Q)) \<in> borel_measurable N" (is "?s Q \<in> _")
 proof -
   from sigma_finite_disjoint guess F . note F = this
@@ -279,7 +279,7 @@
     by auto
 qed
 
-lemma (in sigma_finite_measure) measurable_emeasure[measurable (raw)]:
+lemma%unimportant (in sigma_finite_measure) measurable_emeasure[measurable (raw)]:
   assumes space: "\<And>x. x \<in> space N \<Longrightarrow> A x \<subseteq> space M"
   assumes A: "{x\<in>space (N \<Otimes>\<^sub>M M). snd x \<in> A (fst x)} \<in> sets (N \<Otimes>\<^sub>M M)"
   shows "(\<lambda>x. emeasure M (A x)) \<in> borel_measurable N"
@@ -290,7 +290,7 @@
     by (auto cong: measurable_cong)
 qed
 
-lemma (in sigma_finite_measure) emeasure_pair_measure:
+lemma%unimportant (in sigma_finite_measure) emeasure_pair_measure:
   assumes "X \<in> sets (N \<Otimes>\<^sub>M M)"
   shows "emeasure (N \<Otimes>\<^sub>M M) X = (\<integral>\<^sup>+ x. \<integral>\<^sup>+ y. indicator X (x, y) \<partial>M \<partial>N)" (is "_ = ?\<mu> X")
 proof (rule emeasure_measure_of[OF pair_measure_def])
@@ -314,7 +314,7 @@
     using sets.space_closed[of N] sets.space_closed[of M] by auto
 qed fact
 
-lemma (in sigma_finite_measure) emeasure_pair_measure_alt:
+lemma%unimportant (in sigma_finite_measure) emeasure_pair_measure_alt:
   assumes X: "X \<in> sets (N \<Otimes>\<^sub>M M)"
   shows "emeasure (N  \<Otimes>\<^sub>M M) X = (\<integral>\<^sup>+x. emeasure M (Pair x -` X) \<partial>N)"
 proof -
@@ -324,10 +324,10 @@
     using X by (auto intro!: nn_integral_cong simp: emeasure_pair_measure sets_Pair1)
 qed
 
-lemma (in sigma_finite_measure) emeasure_pair_measure_Times:
+lemma%important (in sigma_finite_measure) emeasure_pair_measure_Times:
   assumes A: "A \<in> sets N" and B: "B \<in> sets M"
   shows "emeasure (N \<Otimes>\<^sub>M M) (A \<times> B) = emeasure N A * emeasure M B"
-proof -
+proof%unimportant -
   have "emeasure (N \<Otimes>\<^sub>M M) (A \<times> B) = (\<integral>\<^sup>+x. emeasure M B * indicator A x \<partial>N)"
     using A B by (auto intro!: nn_integral_cong simp: emeasure_pair_measure_alt)
   also have "\<dots> = emeasure M B * emeasure N A"
@@ -336,18 +336,18 @@
     by (simp add: ac_simps)
 qed
 
-subsection \<open>Binary products of $\sigma$-finite emeasure spaces\<close>
+subsection%important \<open>Binary products of $\sigma$-finite emeasure spaces\<close>
 
-locale pair_sigma_finite = M1?: sigma_finite_measure M1 + M2?: sigma_finite_measure M2
+locale%important pair_sigma_finite = M1?: sigma_finite_measure M1 + M2?: sigma_finite_measure M2
   for M1 :: "'a measure" and M2 :: "'b measure"
 
-lemma (in pair_sigma_finite) measurable_emeasure_Pair1:
+lemma%unimportant (in pair_sigma_finite) measurable_emeasure_Pair1:
   "Q \<in> sets (M1 \<Otimes>\<^sub>M M2) \<Longrightarrow> (\<lambda>x. emeasure M2 (Pair x -` Q)) \<in> borel_measurable M1"
   using M2.measurable_emeasure_Pair .
 
-lemma (in pair_sigma_finite) measurable_emeasure_Pair2:
+lemma%important (in pair_sigma_finite) measurable_emeasure_Pair2:
   assumes Q: "Q \<in> sets (M1 \<Otimes>\<^sub>M M2)" shows "(\<lambda>y. emeasure M1 ((\<lambda>x. (x, y)) -` Q)) \<in> borel_measurable M2"
-proof -
+proof%unimportant -
   have "(\<lambda>(x, y). (y, x)) -` Q \<inter> space (M2 \<Otimes>\<^sub>M M1) \<in> sets (M2 \<Otimes>\<^sub>M M1)"
     using Q measurable_pair_swap' by (auto intro: measurable_sets)
   note M1.measurable_emeasure_Pair[OF this]
@@ -356,11 +356,11 @@
   ultimately show ?thesis by simp
 qed
 
-lemma (in pair_sigma_finite) sigma_finite_up_in_pair_measure_generator:
+lemma%important (in pair_sigma_finite) sigma_finite_up_in_pair_measure_generator:
   defines "E \<equiv> {A \<times> B | A B. A \<in> sets M1 \<and> B \<in> sets M2}"
   shows "\<exists>F::nat \<Rightarrow> ('a \<times> 'b) set. range F \<subseteq> E \<and> incseq F \<and> (\<Union>i. F i) = space M1 \<times> space M2 \<and>
     (\<forall>i. emeasure (M1 \<Otimes>\<^sub>M M2) (F i) \<noteq> \<infinity>)"
-proof -
+proof%unimportant -
   from M1.sigma_finite_incseq guess F1 . note F1 = this
   from M2.sigma_finite_incseq guess F2 . note F2 = this
   from F1 F2 have space: "space M1 = (\<Union>i. F1 i)" "space M2 = (\<Union>i. F2 i)" by auto
@@ -394,7 +394,7 @@
   qed
 qed
 
-sublocale pair_sigma_finite \<subseteq> P?: sigma_finite_measure "M1 \<Otimes>\<^sub>M M2"
+sublocale%important pair_sigma_finite \<subseteq> P?: sigma_finite_measure "M1 \<Otimes>\<^sub>M M2"
 proof
   from M1.sigma_finite_countable guess F1 ..
   moreover from M2.sigma_finite_countable guess F2 ..
@@ -404,7 +404,7 @@
        (auto simp: M2.emeasure_pair_measure_Times space_pair_measure set_eq_iff subset_eq ennreal_mult_eq_top_iff)
 qed
 
-lemma sigma_finite_pair_measure:
+lemma%unimportant sigma_finite_pair_measure:
   assumes A: "sigma_finite_measure A" and B: "sigma_finite_measure B"
   shows "sigma_finite_measure (A \<Otimes>\<^sub>M B)"
 proof -
@@ -414,14 +414,14 @@
   show ?thesis ..
 qed
 
-lemma sets_pair_swap:
+lemma%unimportant sets_pair_swap:
   assumes "A \<in> sets (M1 \<Otimes>\<^sub>M M2)"
   shows "(\<lambda>(x, y). (y, x)) -` A \<inter> space (M2 \<Otimes>\<^sub>M M1) \<in> sets (M2 \<Otimes>\<^sub>M M1)"
   using measurable_pair_swap' assms by (rule measurable_sets)
 
-lemma (in pair_sigma_finite) distr_pair_swap:
+lemma%important (in pair_sigma_finite) distr_pair_swap:
   "M1 \<Otimes>\<^sub>M M2 = distr (M2 \<Otimes>\<^sub>M M1) (M1 \<Otimes>\<^sub>M M2) (\<lambda>(x, y). (y, x))" (is "?P = ?D")
-proof -
+proof%unimportant -
   from sigma_finite_up_in_pair_measure_generator guess F :: "nat \<Rightarrow> ('a \<times> 'b) set" .. note F = this
   let ?E = "{a \<times> b |a b. a \<in> sets M1 \<and> b \<in> sets M2}"
   show ?thesis
@@ -446,11 +446,11 @@
   qed
 qed
 
-lemma (in pair_sigma_finite) emeasure_pair_measure_alt2:
+lemma%unimportant (in pair_sigma_finite) emeasure_pair_measure_alt2:
   assumes A: "A \<in> sets (M1 \<Otimes>\<^sub>M M2)"
   shows "emeasure (M1 \<Otimes>\<^sub>M M2) A = (\<integral>\<^sup>+y. emeasure M1 ((\<lambda>x. (x, y)) -` A) \<partial>M2)"
     (is "_ = ?\<nu> A")
-proof -
+proof%unimportant -
   have [simp]: "\<And>y. (Pair y -` ((\<lambda>(x, y). (y, x)) -` A \<inter> space (M2 \<Otimes>\<^sub>M M1))) = (\<lambda>x. (x, y)) -` A"
     using sets.sets_into_space[OF A] by (auto simp: space_pair_measure)
   show ?thesis using A
@@ -459,7 +459,7 @@
                  M1.emeasure_pair_measure_alt emeasure_distr[OF measurable_pair_swap' A])
 qed
 
-lemma (in pair_sigma_finite) AE_pair:
+lemma%unimportant (in pair_sigma_finite) AE_pair:
   assumes "AE x in (M1 \<Otimes>\<^sub>M M2). Q x"
   shows "AE x in M1. (AE y in M2. Q (x, y))"
 proof -
@@ -485,11 +485,11 @@
   qed
 qed
 
-lemma (in pair_sigma_finite) AE_pair_measure:
+lemma%important (in pair_sigma_finite) AE_pair_measure:
   assumes "{x\<in>space (M1 \<Otimes>\<^sub>M M2). P x} \<in> sets (M1 \<Otimes>\<^sub>M M2)"
   assumes ae: "AE x in M1. AE y in M2. P (x, y)"
   shows "AE x in M1 \<Otimes>\<^sub>M M2. P x"
-proof (subst AE_iff_measurable[OF _ refl])
+proof%unimportant (subst AE_iff_measurable[OF _ refl])
   show "{x\<in>space (M1 \<Otimes>\<^sub>M M2). \<not> P x} \<in> sets (M1 \<Otimes>\<^sub>M M2)"
     by (rule sets.sets_Collect) fact
   then have "emeasure (M1 \<Otimes>\<^sub>M M2) {x \<in> space (M1 \<Otimes>\<^sub>M M2). \<not> P x} =
@@ -505,12 +505,12 @@
   finally show "emeasure (M1 \<Otimes>\<^sub>M M2) {x \<in> space (M1 \<Otimes>\<^sub>M M2). \<not> P x} = 0" by simp
 qed
 
-lemma (in pair_sigma_finite) AE_pair_iff:
+lemma%unimportant (in pair_sigma_finite) AE_pair_iff:
   "{x\<in>space (M1 \<Otimes>\<^sub>M M2). P (fst x) (snd x)} \<in> sets (M1 \<Otimes>\<^sub>M M2) \<Longrightarrow>
     (AE x in M1. AE y in M2. P x y) \<longleftrightarrow> (AE x in (M1 \<Otimes>\<^sub>M M2). P (fst x) (snd x))"
   using AE_pair[of "\<lambda>x. P (fst x) (snd x)"] AE_pair_measure[of "\<lambda>x. P (fst x) (snd x)"] by auto
 
-lemma (in pair_sigma_finite) AE_commute:
+lemma%unimportant (in pair_sigma_finite) AE_commute:
   assumes P: "{x\<in>space (M1 \<Otimes>\<^sub>M M2). P (fst x) (snd x)} \<in> sets (M1 \<Otimes>\<^sub>M M2)"
   shows "(AE x in M1. AE y in M2. P x y) \<longleftrightarrow> (AE y in M2. AE x in M1. P x y)"
 proof -
@@ -531,16 +531,16 @@
     done
 qed
 
-subsection "Fubinis theorem"
+subsection%important "Fubinis theorem"
 
-lemma measurable_compose_Pair1:
+lemma%unimportant measurable_compose_Pair1:
   "x \<in> space M1 \<Longrightarrow> g \<in> measurable (M1 \<Otimes>\<^sub>M M2) L \<Longrightarrow> (\<lambda>y. g (x, y)) \<in> measurable M2 L"
   by simp
 
-lemma (in sigma_finite_measure) borel_measurable_nn_integral_fst:
+lemma%unimportant (in sigma_finite_measure) borel_measurable_nn_integral_fst:
   assumes f: "f \<in> borel_measurable (M1 \<Otimes>\<^sub>M M)"
   shows "(\<lambda>x. \<integral>\<^sup>+ y. f (x, y) \<partial>M) \<in> borel_measurable M1"
-using f proof induct
+using f proof%unimportant induct
   case (cong u v)
   then have "\<And>w x. w \<in> space M1 \<Longrightarrow> x \<in> space M \<Longrightarrow> u (w, x) = v (w, x)"
     by (auto simp: space_pair_measure)
@@ -561,10 +561,10 @@
                    nn_integral_monotone_convergence_SUP incseq_def le_fun_def
               cong: measurable_cong)
 
-lemma (in sigma_finite_measure) nn_integral_fst:
+lemma%unimportant (in sigma_finite_measure) nn_integral_fst:
   assumes f: "f \<in> borel_measurable (M1 \<Otimes>\<^sub>M M)"
   shows "(\<integral>\<^sup>+ x. \<integral>\<^sup>+ y. f (x, y) \<partial>M \<partial>M1) = integral\<^sup>N (M1 \<Otimes>\<^sub>M M) f" (is "?I f = _")
-using f proof induct
+  using f proof induct
   case (cong u v)
   then have "?I u = ?I v"
     by (intro nn_integral_cong) (auto simp: space_pair_measure)
@@ -575,14 +575,14 @@
                    borel_measurable_nn_integral_fst nn_integral_mono incseq_def le_fun_def
               cong: nn_integral_cong)
 
-lemma (in sigma_finite_measure) borel_measurable_nn_integral[measurable (raw)]:
+lemma%unimportant (in sigma_finite_measure) borel_measurable_nn_integral[measurable (raw)]:
   "case_prod f \<in> borel_measurable (N \<Otimes>\<^sub>M M) \<Longrightarrow> (\<lambda>x. \<integral>\<^sup>+ y. f x y \<partial>M) \<in> borel_measurable N"
   using borel_measurable_nn_integral_fst[of "case_prod f" N] by simp
 
-lemma (in pair_sigma_finite) nn_integral_snd:
+lemma%important (in pair_sigma_finite) nn_integral_snd:
   assumes f[measurable]: "f \<in> borel_measurable (M1 \<Otimes>\<^sub>M M2)"
   shows "(\<integral>\<^sup>+ y. (\<integral>\<^sup>+ x. f (x, y) \<partial>M1) \<partial>M2) = integral\<^sup>N (M1 \<Otimes>\<^sub>M M2) f"
-proof -
+proof%unimportant -
   note measurable_pair_swap[OF f]
   from M1.nn_integral_fst[OF this]
   have "(\<integral>\<^sup>+ y. (\<integral>\<^sup>+ x. f (x, y) \<partial>M1) \<partial>M2) = (\<integral>\<^sup>+ (x, y). f (y, x) \<partial>(M2 \<Otimes>\<^sub>M M1))"
@@ -592,24 +592,24 @@
   finally show ?thesis .
 qed
 
-lemma (in pair_sigma_finite) Fubini:
+lemma%important (in pair_sigma_finite) Fubini:
   assumes f: "f \<in> borel_measurable (M1 \<Otimes>\<^sub>M M2)"
   shows "(\<integral>\<^sup>+ y. (\<integral>\<^sup>+ x. f (x, y) \<partial>M1) \<partial>M2) = (\<integral>\<^sup>+ x. (\<integral>\<^sup>+ y. f (x, y) \<partial>M2) \<partial>M1)"
   unfolding nn_integral_snd[OF assms] M2.nn_integral_fst[OF assms] ..
 
-lemma (in pair_sigma_finite) Fubini':
+lemma%important (in pair_sigma_finite) Fubini':
   assumes f: "case_prod f \<in> borel_measurable (M1 \<Otimes>\<^sub>M M2)"
   shows "(\<integral>\<^sup>+ y. (\<integral>\<^sup>+ x. f x y \<partial>M1) \<partial>M2) = (\<integral>\<^sup>+ x. (\<integral>\<^sup>+ y. f x y \<partial>M2) \<partial>M1)"
   using Fubini[OF f] by simp
 
-subsection \<open>Products on counting spaces, densities and distributions\<close>
+subsection%important \<open>Products on counting spaces, densities and distributions\<close>
 
-lemma sigma_prod:
+lemma%important sigma_prod:
   assumes X_cover: "\<exists>E\<subseteq>A. countable E \<and> X = \<Union>E" and A: "A \<subseteq> Pow X"
   assumes Y_cover: "\<exists>E\<subseteq>B. countable E \<and> Y = \<Union>E" and B: "B \<subseteq> Pow Y"
   shows "sigma X A \<Otimes>\<^sub>M sigma Y B = sigma (X \<times> Y) {a \<times> b | a b. a \<in> A \<and> b \<in> B}"
     (is "?P = ?S")
-proof (rule measure_eqI)
+proof%unimportant (rule measure_eqI)
   have [simp]: "snd \<in> X \<times> Y \<rightarrow> Y" "fst \<in> X \<times> Y \<rightarrow> X"
     by auto
   let ?XY = "{{fst -` a \<inter> X \<times> Y | a. a \<in> A}, {snd -` b \<inter> X \<times> Y | b. b \<in> B}}"
@@ -662,7 +662,7 @@
     by (simp add: emeasure_pair_measure_alt emeasure_sigma)
 qed
 
-lemma sigma_sets_pair_measure_generator_finite:
+lemma%unimportant sigma_sets_pair_measure_generator_finite:
   assumes "finite A" and "finite B"
   shows "sigma_sets (A \<times> B) { a \<times> b | a b. a \<subseteq> A \<and> b \<subseteq> B} = Pow (A \<times> B)"
   (is "sigma_sets ?prod ?sets = _")
@@ -686,14 +686,14 @@
   show "a \<in> A" and "b \<in> B" by auto
 qed
 
-lemma sets_pair_eq:
+lemma%important sets_pair_eq:
   assumes Ea: "Ea \<subseteq> Pow (space A)" "sets A = sigma_sets (space A) Ea"
     and Ca: "countable Ca" "Ca \<subseteq> Ea" "\<Union>Ca = space A"
     and Eb: "Eb \<subseteq> Pow (space B)" "sets B = sigma_sets (space B) Eb"
     and Cb: "countable Cb" "Cb \<subseteq> Eb" "\<Union>Cb = space B"
   shows "sets (A \<Otimes>\<^sub>M B) = sets (sigma (space A \<times> space B) { a \<times> b | a b. a \<in> Ea \<and> b \<in> Eb })"
     (is "_ = sets (sigma ?\<Omega> ?E)")
-proof
+proof%unimportant
   show "sets (sigma ?\<Omega> ?E) \<subseteq> sets (A \<Otimes>\<^sub>M B)"
     using Ea(1) Eb(1) by (subst sigma_le_sets) (auto simp: Ea(2) Eb(2))
   have "?E \<subseteq> Pow ?\<Omega>"
@@ -733,10 +733,10 @@
   finally show "sets (A \<Otimes>\<^sub>M B) \<subseteq> sets (sigma ?\<Omega> ?E)" .
 qed
 
-lemma borel_prod:
+lemma%important borel_prod:
   "(borel \<Otimes>\<^sub>M borel) = (borel :: ('a::second_countable_topology \<times> 'b::second_countable_topology) measure)"
   (is "?P = ?B")
-proof -
+proof%unimportant -
   have "?B = sigma UNIV {A \<times> B | A B. open A \<and> open B}"
     by (rule second_countable_borel_measurable[OF open_prod_generated])
   also have "\<dots> = ?P"
@@ -745,10 +745,10 @@
   finally show ?thesis ..
 qed
 
-lemma pair_measure_count_space:
+lemma%important pair_measure_count_space:
   assumes A: "finite A" and B: "finite B"
   shows "count_space A \<Otimes>\<^sub>M count_space B = count_space (A \<times> B)" (is "?P = ?C")
-proof (rule measure_eqI)
+proof%unimportant (rule measure_eqI)
   interpret A: finite_measure "count_space A" by (rule finite_measure_count_space) fact
   interpret B: finite_measure "count_space B" by (rule finite_measure_count_space) fact
   interpret P: pair_sigma_finite "count_space A" "count_space B" ..
@@ -776,14 +776,14 @@
 qed
 
 
-lemma emeasure_prod_count_space:
+lemma%unimportant emeasure_prod_count_space:
   assumes A: "A \<in> sets (count_space UNIV \<Otimes>\<^sub>M M)" (is "A \<in> sets (?A \<Otimes>\<^sub>M ?B)")
   shows "emeasure (?A \<Otimes>\<^sub>M ?B) A = (\<integral>\<^sup>+ x. \<integral>\<^sup>+ y. indicator A (x, y) \<partial>?B \<partial>?A)"
   by (rule emeasure_measure_of[OF pair_measure_def])
      (auto simp: countably_additive_def positive_def suminf_indicator A
                  nn_integral_suminf[symmetric] dest: sets.sets_into_space)
 
-lemma emeasure_prod_count_space_single[simp]: "emeasure (count_space UNIV \<Otimes>\<^sub>M count_space UNIV) {x} = 1"
+lemma%unimportant emeasure_prod_count_space_single[simp]: "emeasure (count_space UNIV \<Otimes>\<^sub>M count_space UNIV) {x} = 1"
 proof -
   have [simp]: "\<And>a b x y. indicator {(a, b)} (x, y) = (indicator {a} x * indicator {b} y::ennreal)"
     by (auto split: split_indicator)
@@ -791,11 +791,11 @@
     by (cases x) (auto simp: emeasure_prod_count_space nn_integral_cmult sets_Pair)
 qed
 
-lemma emeasure_count_space_prod_eq:
+lemma%important emeasure_count_space_prod_eq:
   fixes A :: "('a \<times> 'b) set"
   assumes A: "A \<in> sets (count_space UNIV \<Otimes>\<^sub>M count_space UNIV)" (is "A \<in> sets (?A \<Otimes>\<^sub>M ?B)")
   shows "emeasure (?A \<Otimes>\<^sub>M ?B) A = emeasure (count_space UNIV) A"
-proof -
+proof%unimportant -
   { fix A :: "('a \<times> 'b) set" assume "countable A"
     then have "emeasure (?A \<Otimes>\<^sub>M ?B) (\<Union>a\<in>A. {a}) = (\<integral>\<^sup>+a. emeasure (?A \<Otimes>\<^sub>M ?B) {a} \<partial>count_space A)"
       by (intro emeasure_UN_countable) (auto simp: sets_Pair disjoint_family_on_def)
@@ -822,7 +822,7 @@
   qed
 qed
 
-lemma nn_integral_count_space_prod_eq:
+lemma%unimportant nn_integral_count_space_prod_eq:
   "nn_integral (count_space UNIV \<Otimes>\<^sub>M count_space UNIV) f = nn_integral (count_space UNIV) f"
     (is "nn_integral ?P f = _")
 proof cases
@@ -874,12 +874,12 @@
     by (simp add: top_unique)
 qed
 
-lemma pair_measure_density:
+lemma%important pair_measure_density:
   assumes f: "f \<in> borel_measurable M1"
   assumes g: "g \<in> borel_measurable M2"
   assumes "sigma_finite_measure M2" "sigma_finite_measure (density M2 g)"
   shows "density M1 f \<Otimes>\<^sub>M density M2 g = density (M1 \<Otimes>\<^sub>M M2) (\<lambda>(x,y). f x * g y)" (is "?L = ?R")
-proof (rule measure_eqI)
+proof%unimportant (rule measure_eqI)
   interpret M2: sigma_finite_measure M2 by fact
   interpret D2: sigma_finite_measure "density M2 g" by fact
 
@@ -894,7 +894,7 @@
              cong: nn_integral_cong)
 qed simp
 
-lemma sigma_finite_measure_distr:
+lemma%unimportant sigma_finite_measure_distr:
   assumes "sigma_finite_measure (distr M N f)" and f: "f \<in> measurable M N"
   shows "sigma_finite_measure M"
 proof -
@@ -909,7 +909,7 @@
   qed
 qed
 
-lemma pair_measure_distr:
+lemma%unimportant pair_measure_distr:
   assumes f: "f \<in> measurable M S" and g: "g \<in> measurable N T"
   assumes "sigma_finite_measure (distr N T g)"
   shows "distr M S f \<Otimes>\<^sub>M distr N T g = distr (M \<Otimes>\<^sub>M N) (S \<Otimes>\<^sub>M T) (\<lambda>(x, y). (f x, g y))" (is "?P = ?D")
@@ -924,12 +924,12 @@
              intro!: nn_integral_cong arg_cong[where f="emeasure N"])
 qed simp
 
-lemma pair_measure_eqI:
+lemma%important pair_measure_eqI:
   assumes "sigma_finite_measure M1" "sigma_finite_measure M2"
   assumes sets: "sets (M1 \<Otimes>\<^sub>M M2) = sets M"
   assumes emeasure: "\<And>A B. A \<in> sets M1 \<Longrightarrow> B \<in> sets M2 \<Longrightarrow> emeasure M1 A * emeasure M2 B = emeasure M (A \<times> B)"
   shows "M1 \<Otimes>\<^sub>M M2 = M"
-proof -
+proof%unimportant -
   interpret M1: sigma_finite_measure M1 by fact
   interpret M2: sigma_finite_measure M2 by fact
   interpret pair_sigma_finite M1 M2 ..
@@ -959,11 +959,11 @@
   qed
 qed
 
-lemma sets_pair_countable:
+lemma%important sets_pair_countable:
   assumes "countable S1" "countable S2"
   assumes M: "sets M = Pow S1" and N: "sets N = Pow S2"
   shows "sets (M \<Otimes>\<^sub>M N) = Pow (S1 \<times> S2)"
-proof auto
+proof%unimportant auto
   fix x a b assume x: "x \<in> sets (M \<Otimes>\<^sub>M N)" "(a, b) \<in> x"
   from sets.sets_into_space[OF x(1)] x(2)
     sets_eq_imp_space_eq[of N "count_space S2"] sets_eq_imp_space_eq[of M "count_space S1"] M N
@@ -980,10 +980,10 @@
   finally show "X \<in> sets (M \<Otimes>\<^sub>M N)" .
 qed
 
-lemma pair_measure_countable:
+lemma%important pair_measure_countable:
   assumes "countable S1" "countable S2"
   shows "count_space S1 \<Otimes>\<^sub>M count_space S2 = count_space (S1 \<times> S2)"
-proof (rule pair_measure_eqI)
+proof%unimportant (rule pair_measure_eqI)
   show "sigma_finite_measure (count_space S1)" "sigma_finite_measure (count_space S2)"
     using assms by (auto intro!: sigma_finite_measure_count_space_countable)
   show "sets (count_space S1 \<Otimes>\<^sub>M count_space S2) = sets (count_space (S1 \<times> S2))"
@@ -995,10 +995,10 @@
     by (subst (1 2 3) emeasure_count_space) (auto simp: finite_cartesian_product_iff ennreal_mult_top ennreal_top_mult)
 qed
 
-lemma nn_integral_fst_count_space:
+lemma%important nn_integral_fst_count_space:
   "(\<integral>\<^sup>+ x. \<integral>\<^sup>+ y. f (x, y) \<partial>count_space UNIV \<partial>count_space UNIV) = integral\<^sup>N (count_space UNIV) f"
   (is "?lhs = ?rhs")
-proof(cases)
+proof%unimportant(cases)
   assume *: "countable {xy. f xy \<noteq> 0}"
   let ?A = "fst ` {xy. f xy \<noteq> 0}"
   let ?B = "snd ` {xy. f xy \<noteq> 0}"
@@ -1088,20 +1088,20 @@
   finally show ?thesis .
 qed
 
-lemma measurable_pair_measure_countable1:
+lemma%unimportant measurable_pair_measure_countable1:
   assumes "countable A"
   and [measurable]: "\<And>x. x \<in> A \<Longrightarrow> (\<lambda>y. f (x, y)) \<in> measurable N K"
   shows "f \<in> measurable (count_space A \<Otimes>\<^sub>M N) K"
 using _ _ assms(1)
 by(rule measurable_compose_countable'[where f="\<lambda>a b. f (a, snd b)" and g=fst and I=A, simplified])simp_all
 
-subsection \<open>Product of Borel spaces\<close>
+subsection%important \<open>Product of Borel spaces\<close>
 
-lemma borel_Times:
+lemma%important borel_Times:
   fixes A :: "'a::topological_space set" and B :: "'b::topological_space set"
   assumes A: "A \<in> sets borel" and B: "B \<in> sets borel"
   shows "A \<times> B \<in> sets borel"
-proof -
+proof%unimportant -
   have "A \<times> B = (A\<times>UNIV) \<inter> (UNIV \<times> B)"
     by auto
   moreover
@@ -1146,7 +1146,7 @@
     by auto
 qed
 
-lemma finite_measure_pair_measure:
+lemma%unimportant finite_measure_pair_measure:
   assumes "finite_measure M" "finite_measure N"
   shows "finite_measure (N  \<Otimes>\<^sub>M M)"
 proof (rule finite_measureI)

--- a/src/HOL/Analysis/Bochner_Integration.thy	Mon Aug 27 22:58:36 2018 +0200
+++ b/src/HOL/Analysis/Bochner_Integration.thy	Tue Aug 28 13:28:39 2018 +0100
@@ -2,7 +2,7 @@
     Author:     Johannes Hölzl, TU München
 *)
 
-section \<open>Bochner Integration for Vector-Valued Functions\<close>
+section%important \<open>Bochner Integration for Vector-Valued Functions\<close>
 
 theory Bochner_Integration
   imports Finite_Product_Measure
@@ -15,12 +15,12 @@
 
 \<close>
 
-lemma borel_measurable_implies_sequence_metric:
+lemma%important borel_measurable_implies_sequence_metric:
   fixes f :: "'a \<Rightarrow> 'b :: {metric_space, second_countable_topology}"
   assumes [measurable]: "f \<in> borel_measurable M"
   shows "\<exists>F. (\<forall>i. simple_function M (F i)) \<and> (\<forall>x\<in>space M. (\<lambda>i. F i x) \<longlonglongrightarrow> f x) \<and>
     (\<forall>i. \<forall>x\<in>space M. dist (F i x) z \<le> 2 * dist (f x) z)"
-proof -
+proof%unimportant -
   obtain D :: "'b set" where "countable D" and D: "\<And>X. open X \<Longrightarrow> X \<noteq> {} \<Longrightarrow> \<exists>d\<in>D. d \<in> X"
     by (erule countable_dense_setE)
 
@@ -155,14 +155,14 @@
   qed
 qed
 
-lemma
+lemma%unimportant
   fixes f :: "'a \<Rightarrow> 'b::semiring_1" assumes "finite A"
   shows sum_mult_indicator[simp]: "(\<Sum>x \<in> A. f x * indicator (B x) (g x)) = (\<Sum>x\<in>{x\<in>A. g x \<in> B x}. f x)"
   and sum_indicator_mult[simp]: "(\<Sum>x \<in> A. indicator (B x) (g x) * f x) = (\<Sum>x\<in>{x\<in>A. g x \<in> B x}. f x)"
   unfolding indicator_def
   using assms by (auto intro!: sum.mono_neutral_cong_right split: if_split_asm)
 
-lemma borel_measurable_induct_real[consumes 2, case_names set mult add seq]:
+lemma%unimportant borel_measurable_induct_real[consumes 2, case_names set mult add seq]:
   fixes P :: "('a \<Rightarrow> real) \<Rightarrow> bool"
   assumes u: "u \<in> borel_measurable M" "\<And>x. 0 \<le> u x"
   assumes set: "\<And>A. A \<in> sets M \<Longrightarrow> P (indicator A)"
@@ -227,7 +227,7 @@
   qed
 qed
 
-lemma scaleR_cong_right:
+lemma%unimportant scaleR_cong_right:
   fixes x :: "'a :: real_vector"
   shows "(x \<noteq> 0 \<Longrightarrow> r = p) \<Longrightarrow> r *\<^sub>R x = p *\<^sub>R x"
   by (cases "x = 0") auto
@@ -236,7 +236,7 @@
   "simple_function M f \<Longrightarrow> emeasure M {y\<in>space M. f y \<noteq> 0} \<noteq> \<infinity> \<Longrightarrow>
     simple_bochner_integrable M f"
 
-lemma simple_bochner_integrable_compose2:
+lemma%unimportant simple_bochner_integrable_compose2:
   assumes p_0: "p 0 0 = 0"
   shows "simple_bochner_integrable M f \<Longrightarrow> simple_bochner_integrable M g \<Longrightarrow>
     simple_bochner_integrable M (\<lambda>x. p (f x) (g x))"
@@ -261,7 +261,7 @@
     using fin by (auto simp: top_unique)
 qed
 
-lemma simple_function_finite_support:
+lemma%unimportant simple_function_finite_support:
   assumes f: "simple_function M f" and fin: "(\<integral>\<^sup>+x. f x \<partial>M) < \<infinity>" and nn: "\<And>x. 0 \<le> f x"
   shows "emeasure M {x\<in>space M. f x \<noteq> 0} \<noteq> \<infinity>"
 proof cases
@@ -296,7 +296,7 @@
   show ?thesis unfolding * by simp
 qed
 
-lemma simple_bochner_integrableI_bounded:
+lemma%unimportant simple_bochner_integrableI_bounded:
   assumes f: "simple_function M f" and fin: "(\<integral>\<^sup>+x. norm (f x) \<partial>M) < \<infinity>"
   shows "simple_bochner_integrable M f"
 proof
@@ -309,16 +309,16 @@
   then show "emeasure M {y \<in> space M. f y \<noteq> 0} \<noteq> \<infinity>" by simp
 qed fact
 
-definition simple_bochner_integral :: "'a measure \<Rightarrow> ('a \<Rightarrow> 'b::real_vector) \<Rightarrow> 'b" where
+definition%important simple_bochner_integral :: "'a measure \<Rightarrow> ('a \<Rightarrow> 'b::real_vector) \<Rightarrow> 'b" where
   "simple_bochner_integral M f = (\<Sum>y\<in>f`space M. measure M {x\<in>space M. f x = y} *\<^sub>R y)"
 
-lemma simple_bochner_integral_partition:
+lemma%important simple_bochner_integral_partition:
   assumes f: "simple_bochner_integrable M f" and g: "simple_function M g"
   assumes sub: "\<And>x y. x \<in> space M \<Longrightarrow> y \<in> space M \<Longrightarrow> g x = g y \<Longrightarrow> f x = f y"
   assumes v: "\<And>x. x \<in> space M \<Longrightarrow> f x = v (g x)"
   shows "simple_bochner_integral M f = (\<Sum>y\<in>g ` space M. measure M {x\<in>space M. g x = y} *\<^sub>R v y)"
     (is "_ = ?r")
-proof -
+proof%unimportant -
   from f g have [simp]: "finite (f`space M)" "finite (g`space M)"
     by (auto simp: simple_function_def elim: simple_bochner_integrable.cases)
 
@@ -394,7 +394,7 @@
     by (simp add: sum.distrib[symmetric] scaleR_add_right)
 qed
 
-lemma simple_bochner_integral_linear:
+lemma%unimportant simple_bochner_integral_linear:
   assumes "linear f"
   assumes g: "simple_bochner_integrable M g"
   shows "simple_bochner_integral M (\<lambda>x. f (g x)) = f (simple_bochner_integral M g)"
@@ -484,14 +484,14 @@
   finally show ?thesis .
 qed
 
-lemma simple_bochner_integral_bounded:
+lemma%important simple_bochner_integral_bounded:
   fixes f :: "'a \<Rightarrow> 'b::{real_normed_vector, second_countable_topology}"
   assumes f[measurable]: "f \<in> borel_measurable M"
   assumes s: "simple_bochner_integrable M s" and t: "simple_bochner_integrable M t"
   shows "ennreal (norm (simple_bochner_integral M s - simple_bochner_integral M t)) \<le>
     (\<integral>\<^sup>+ x. norm (f x - s x) \<partial>M) + (\<integral>\<^sup>+ x. norm (f x - t x) \<partial>M)"
     (is "ennreal (norm (?s - ?t)) \<le> ?S + ?T")
-proof -
+proof%unimportant -
   have [measurable]: "s \<in> borel_measurable M" "t \<in> borel_measurable M"
     using s t by (auto intro: borel_measurable_simple_function elim: simple_bochner_integrable.cases)
 
@@ -520,13 +520,13 @@
     (\<lambda>i. simple_bochner_integral M (s i)) \<longlonglongrightarrow> x \<Longrightarrow>
     has_bochner_integral M f x"
 
-lemma has_bochner_integral_cong:
+lemma%unimportant has_bochner_integral_cong:
   assumes "M = N" "\<And>x. x \<in> space N \<Longrightarrow> f x = g x" "x = y"
   shows "has_bochner_integral M f x \<longleftrightarrow> has_bochner_integral N g y"
   unfolding has_bochner_integral.simps assms(1,3)
   using assms(2) by (simp cong: measurable_cong_strong nn_integral_cong_strong)
 
-lemma has_bochner_integral_cong_AE:
+lemma%unimportant has_bochner_integral_cong_AE:
   "f \<in> borel_measurable M \<Longrightarrow> g \<in> borel_measurable M \<Longrightarrow> (AE x in M. f x = g x) \<Longrightarrow>
     has_bochner_integral M f x \<longleftrightarrow> has_bochner_integral M g x"
   unfolding has_bochner_integral.simps
@@ -534,22 +534,22 @@
             nn_integral_cong_AE)
      auto
 
-lemma borel_measurable_has_bochner_integral:
+lemma%unimportant borel_measurable_has_bochner_integral:
   "has_bochner_integral M f x \<Longrightarrow> f \<in> borel_measurable M"
   by (rule has_bochner_integral.cases)
 
-lemma borel_measurable_has_bochner_integral'[measurable_dest]:
+lemma%unimportant borel_measurable_has_bochner_integral'[measurable_dest]:
   "has_bochner_integral M f x \<Longrightarrow> g \<in> measurable N M \<Longrightarrow> (\<lambda>x. f (g x)) \<in> borel_measurable N"
   using borel_measurable_has_bochner_integral[measurable] by measurable
 
-lemma has_bochner_integral_simple_bochner_integrable:
+lemma%unimportant has_bochner_integral_simple_bochner_integrable:
   "simple_bochner_integrable M f \<Longrightarrow> has_bochner_integral M f (simple_bochner_integral M f)"
   by (rule has_bochner_integral.intros[where s="\<lambda>_. f"])
      (auto intro: borel_measurable_simple_function
            elim: simple_bochner_integrable.cases
            simp: zero_ennreal_def[symmetric])
 
-lemma has_bochner_integral_real_indicator:
+lemma%unimportant has_bochner_integral_real_indicator:
   assumes [measurable]: "A \<in> sets M" and A: "emeasure M A < \<infinity>"
   shows "has_bochner_integral M (indicator A) (measure M A)"
 proof -
@@ -567,7 +567,7 @@
     by (metis has_bochner_integral_simple_bochner_integrable)
 qed
 
-lemma has_bochner_integral_add[intro]:
+lemma%unimportant has_bochner_integral_add[intro]:
   "has_bochner_integral M f x \<Longrightarrow> has_bochner_integral M g y \<Longrightarrow>
     has_bochner_integral M (\<lambda>x. f x + g x) (x + y)"
 proof (safe intro!: has_bochner_integral.intros elim!: has_bochner_integral.cases)
@@ -604,7 +604,7 @@
   qed
 qed (auto simp: simple_bochner_integral_add tendsto_add)
 
-lemma has_bochner_integral_bounded_linear:
+lemma%unimportant has_bochner_integral_bounded_linear:
   assumes "bounded_linear T"
   shows "has_bochner_integral M f x \<Longrightarrow> has_bochner_integral M (\<lambda>x. T (f x)) (T x)"
 proof (safe intro!: has_bochner_integral.intros elim!: has_bochner_integral.cases)
@@ -650,79 +650,79 @@
     by (auto intro!: T.tendsto simp: simple_bochner_integral_linear T.linear_axioms)
 qed
 
-lemma has_bochner_integral_zero[intro]: "has_bochner_integral M (\<lambda>x. 0) 0"
+lemma%unimportant has_bochner_integral_zero[intro]: "has_bochner_integral M (\<lambda>x. 0) 0"
   by (auto intro!: has_bochner_integral.intros[where s="\<lambda>_ _. 0"]
            simp: zero_ennreal_def[symmetric] simple_bochner_integrable.simps
                  simple_bochner_integral_def image_constant_conv)
 
-lemma has_bochner_integral_scaleR_left[intro]:
+lemma%unimportant has_bochner_integral_scaleR_left[intro]:
   "(c \<noteq> 0 \<Longrightarrow> has_bochner_integral M f x) \<Longrightarrow> has_bochner_integral M (\<lambda>x. f x *\<^sub>R c) (x *\<^sub>R c)"
   by (cases "c = 0") (auto simp add: has_bochner_integral_bounded_linear[OF bounded_linear_scaleR_left])
 
-lemma has_bochner_integral_scaleR_right[intro]:
+lemma%unimportant has_bochner_integral_scaleR_right[intro]:
   "(c \<noteq> 0 \<Longrightarrow> has_bochner_integral M f x) \<Longrightarrow> has_bochner_integral M (\<lambda>x. c *\<^sub>R f x) (c *\<^sub>R x)"
   by (cases "c = 0") (auto simp add: has_bochner_integral_bounded_linear[OF bounded_linear_scaleR_right])
 
-lemma has_bochner_integral_mult_left[intro]:
+lemma%unimportant has_bochner_integral_mult_left[intro]:
   fixes c :: "_::{real_normed_algebra,second_countable_topology}"
   shows "(c \<noteq> 0 \<Longrightarrow> has_bochner_integral M f x) \<Longrightarrow> has_bochner_integral M (\<lambda>x. f x * c) (x * c)"
   by (cases "c = 0") (auto simp add: has_bochner_integral_bounded_linear[OF bounded_linear_mult_left])
 
-lemma has_bochner_integral_mult_right[intro]:
+lemma%unimportant has_bochner_integral_mult_right[intro]:
   fixes c :: "_::{real_normed_algebra,second_countable_topology}"
   shows "(c \<noteq> 0 \<Longrightarrow> has_bochner_integral M f x) \<Longrightarrow> has_bochner_integral M (\<lambda>x. c * f x) (c * x)"
   by (cases "c = 0") (auto simp add: has_bochner_integral_bounded_linear[OF bounded_linear_mult_right])
 
-lemmas has_bochner_integral_divide =
+lemmas%unimportant has_bochner_integral_divide =
   has_bochner_integral_bounded_linear[OF bounded_linear_divide]
 
-lemma has_bochner_integral_divide_zero[intro]:
+lemma%unimportant has_bochner_integral_divide_zero[intro]:
   fixes c :: "_::{real_normed_field, field, second_countable_topology}"
   shows "(c \<noteq> 0 \<Longrightarrow> has_bochner_integral M f x) \<Longrightarrow> has_bochner_integral M (\<lambda>x. f x / c) (x / c)"
   using has_bochner_integral_divide by (cases "c = 0") auto
 
-lemma has_bochner_integral_inner_left[intro]:
+lemma%unimportant has_bochner_integral_inner_left[intro]:
   "(c \<noteq> 0 \<Longrightarrow> has_bochner_integral M f x) \<Longrightarrow> has_bochner_integral M (\<lambda>x. f x \<bullet> c) (x \<bullet> c)"
   by (cases "c = 0") (auto simp add: has_bochner_integral_bounded_linear[OF bounded_linear_inner_left])
 
-lemma has_bochner_integral_inner_right[intro]:
+lemma%unimportant has_bochner_integral_inner_right[intro]:
   "(c \<noteq> 0 \<Longrightarrow> has_bochner_integral M f x) \<Longrightarrow> has_bochner_integral M (\<lambda>x. c \<bullet> f x) (c \<bullet> x)"
   by (cases "c = 0") (auto simp add: has_bochner_integral_bounded_linear[OF bounded_linear_inner_right])
 
-lemmas has_bochner_integral_minus =
+lemmas%unimportant has_bochner_integral_minus =
   has_bochner_integral_bounded_linear[OF bounded_linear_minus[OF bounded_linear_ident]]
-lemmas has_bochner_integral_Re =
+lemmas%unimportant has_bochner_integral_Re =
   has_bochner_integral_bounded_linear[OF bounded_linear_Re]
-lemmas has_bochner_integral_Im =
+lemmas%unimportant has_bochner_integral_Im =
   has_bochner_integral_bounded_linear[OF bounded_linear_Im]
-lemmas has_bochner_integral_cnj =
+lemmas%unimportant has_bochner_integral_cnj =
   has_bochner_integral_bounded_linear[OF bounded_linear_cnj]
-lemmas has_bochner_integral_of_real =
+lemmas%unimportant has_bochner_integral_of_real =
   has_bochner_integral_bounded_linear[OF bounded_linear_of_real]
-lemmas has_bochner_integral_fst =
+lemmas%unimportant has_bochner_integral_fst =
   has_bochner_integral_bounded_linear[OF bounded_linear_fst]
-lemmas has_bochner_integral_snd =
+lemmas%unimportant has_bochner_integral_snd =
   has_bochner_integral_bounded_linear[OF bounded_linear_snd]
 
-lemma has_bochner_integral_indicator:
+lemma%unimportant has_bochner_integral_indicator:
   "A \<in> sets M \<Longrightarrow> emeasure M A < \<infinity> \<Longrightarrow>
     has_bochner_integral M (\<lambda>x. indicator A x *\<^sub>R c) (measure M A *\<^sub>R c)"
   by (intro has_bochner_integral_scaleR_left has_bochner_integral_real_indicator)
 
-lemma has_bochner_integral_diff:
+lemma%unimportant has_bochner_integral_diff:
   "has_bochner_integral M f x \<Longrightarrow> has_bochner_integral M g y \<Longrightarrow>
     has_bochner_integral M (\<lambda>x. f x - g x) (x - y)"
   unfolding diff_conv_add_uminus
   by (intro has_bochner_integral_add has_bochner_integral_minus)
 
-lemma has_bochner_integral_sum:
+lemma%unimportant has_bochner_integral_sum:
   "(\<And>i. i \<in> I \<Longrightarrow> has_bochner_integral M (f i) (x i)) \<Longrightarrow>
     has_bochner_integral M (\<lambda>x. \<Sum>i\<in>I. f i x) (\<Sum>i\<in>I. x i)"
   by (induct I rule: infinite_finite_induct) auto
 
-lemma has_bochner_integral_implies_finite_norm:
+lemma%important has_bochner_integral_implies_finite_norm:
   "has_bochner_integral M f x \<Longrightarrow> (\<integral>\<^sup>+x. norm (f x) \<partial>M) < \<infinity>"
-proof (elim has_bochner_integral.cases)
+proof%unimportant (elim has_bochner_integral.cases)
   fix s v
   assume [measurable]: "f \<in> borel_measurable M" and s: "\<And>i. simple_bochner_integrable M (s i)" and
     lim_0: "(\<lambda>i. \<integral>\<^sup>+ x. ennreal (norm (f x - s i x)) \<partial>M) \<longlonglongrightarrow> 0"
@@ -756,10 +756,10 @@
   finally show "(\<integral>\<^sup>+ x. ennreal (norm (f x)) \<partial>M) < \<infinity>" .
 qed
 
-lemma has_bochner_integral_norm_bound:
+lemma%important has_bochner_integral_norm_bound:
   assumes i: "has_bochner_integral M f x"
   shows "norm x \<le> (\<integral>\<^sup>+x. norm (f x) \<partial>M)"
-using assms proof
+using assms proof%unimportant
   fix s assume
     x: "(\<lambda>i. simple_bochner_integral M (s i)) \<longlonglongrightarrow> x" (is "?s \<longlonglongrightarrow> x") and
     s[simp]: "\<And>i. simple_bochner_integrable M (s i)" and
@@ -797,9 +797,9 @@
   qed
 qed
 
-lemma has_bochner_integral_eq:
+lemma%important has_bochner_integral_eq:
   "has_bochner_integral M f x \<Longrightarrow> has_bochner_integral M f y \<Longrightarrow> x = y"
-proof (elim has_bochner_integral.cases)
+proof%unimportant (elim has_bochner_integral.cases)
   assume f[measurable]: "f \<in> borel_measurable M"
 
   fix s t
@@ -834,7 +834,7 @@
   then show "x = y" by simp
 qed
 
-lemma has_bochner_integralI_AE:
+lemma%unimportant has_bochner_integralI_AE:
   assumes f: "has_bochner_integral M f x"
     and g: "g \<in> borel_measurable M"
     and ae: "AE x in M. f x = g x"
@@ -848,7 +848,7 @@
   finally show "(\<lambda>i. \<integral>\<^sup>+ x. ennreal (norm (g x - s i x)) \<partial>M) \<longlonglongrightarrow> 0" .
 qed (auto intro: g)
 
-lemma has_bochner_integral_eq_AE:
+lemma%unimportant has_bochner_integral_eq_AE:
   assumes f: "has_bochner_integral M f x"
     and g: "has_bochner_integral M g y"
     and ae: "AE x in M. f x = g x"
@@ -860,7 +860,7 @@
     by (rule has_bochner_integral_eq)
 qed
 
-lemma simple_bochner_integrable_restrict_space:
+lemma%unimportant simple_bochner_integrable_restrict_space:
   fixes f :: "_ \<Rightarrow> 'b::real_normed_vector"
   assumes \<Omega>: "\<Omega> \<inter> space M \<in> sets M"
   shows "simple_bochner_integrable (restrict_space M \<Omega>) f \<longleftrightarrow>
@@ -869,13 +869,13 @@
     simple_function_restrict_space[OF \<Omega>] emeasure_restrict_space[OF \<Omega>] Collect_restrict
     indicator_eq_0_iff conj_left_commute)
 
-lemma simple_bochner_integral_restrict_space:
+lemma%important simple_bochner_integral_restrict_space:
   fixes f :: "_ \<Rightarrow> 'b::real_normed_vector"
   assumes \<Omega>: "\<Omega> \<inter> space M \<in> sets M"
   assumes f: "simple_bochner_integrable (restrict_space M \<Omega>) f"
   shows "simple_bochner_integral (restrict_space M \<Omega>) f =
     simple_bochner_integral M (\<lambda>x. indicator \<Omega> x *\<^sub>R f x)"
-proof -
+proof%unimportant -
   have "finite ((\<lambda>x. indicator \<Omega> x *\<^sub>R f x)`space M)"
     using f simple_bochner_integrable_restrict_space[OF \<Omega>, of f]
     by (simp add: simple_bochner_integrable.simps simple_function_def)
@@ -895,7 +895,7 @@
 
 end
 
-definition lebesgue_integral ("integral\<^sup>L") where
+definition%important lebesgue_integral ("integral\<^sup>L") where
   "integral\<^sup>L M f = (if \<exists>x. has_bochner_integral M f x then THE x. has_bochner_integral M f x else 0)"
 
 syntax
@@ -910,155 +910,155 @@
 translations
   "LINT x|M. f" == "CONST lebesgue_integral M (\<lambda>x. f)"
 
-lemma has_bochner_integral_integral_eq: "has_bochner_integral M f x \<Longrightarrow> integral\<^sup>L M f = x"
+lemma%unimportant has_bochner_integral_integral_eq: "has_bochner_integral M f x \<Longrightarrow> integral\<^sup>L M f = x"
   by (metis the_equality has_bochner_integral_eq lebesgue_integral_def)
 
-lemma has_bochner_integral_integrable:
+lemma%unimportant has_bochner_integral_integrable:
   "integrable M f \<Longrightarrow> has_bochner_integral M f (integral\<^sup>L M f)"
   by (auto simp: has_bochner_integral_integral_eq integrable.simps)
 
-lemma has_bochner_integral_iff:
+lemma%unimportant has_bochner_integral_iff:
   "has_bochner_integral M f x \<longleftrightarrow> integrable M f \<and> integral\<^sup>L M f = x"
   by (metis has_bochner_integral_integrable has_bochner_integral_integral_eq integrable.intros)
 
-lemma simple_bochner_integrable_eq_integral:
+lemma%unimportant simple_bochner_integrable_eq_integral:
   "simple_bochner_integrable M f \<Longrightarrow> simple_bochner_integral M f = integral\<^sup>L M f"
   using has_bochner_integral_simple_bochner_integrable[of M f]
   by (simp add: has_bochner_integral_integral_eq)
 
-lemma not_integrable_integral_eq: "\<not> integrable M f \<Longrightarrow> integral\<^sup>L M f = 0"
+lemma%unimportant not_integrable_integral_eq: "\<not> integrable M f \<Longrightarrow> integral\<^sup>L M f = 0"
   unfolding integrable.simps lebesgue_integral_def by (auto intro!: arg_cong[where f=The])
 
-lemma integral_eq_cases:
+lemma%unimportant integral_eq_cases:
   "integrable M f \<longleftrightarrow> integrable N g \<Longrightarrow>
     (integrable M f \<Longrightarrow> integrable N g \<Longrightarrow> integral\<^sup>L M f = integral\<^sup>L N g) \<Longrightarrow>
     integral\<^sup>L M f = integral\<^sup>L N g"
   by (metis not_integrable_integral_eq)
 
-lemma borel_measurable_integrable[measurable_dest]: "integrable M f \<Longrightarrow> f \<in> borel_measurable M"
+lemma%unimportant borel_measurable_integrable[measurable_dest]: "integrable M f \<Longrightarrow> f \<in> borel_measurable M"
   by (auto elim: integrable.cases has_bochner_integral.cases)
 
-lemma borel_measurable_integrable'[measurable_dest]:
+lemma%unimportant borel_measurable_integrable'[measurable_dest]:
   "integrable M f \<Longrightarrow> g \<in> measurable N M \<Longrightarrow> (\<lambda>x. f (g x)) \<in> borel_measurable N"
   using borel_measurable_integrable[measurable] by measurable
 
-lemma integrable_cong:
+lemma%unimportant integrable_cong:
   "M = N \<Longrightarrow> (\<And>x. x \<in> space N \<Longrightarrow> f x = g x) \<Longrightarrow> integrable M f \<longleftrightarrow> integrable N g"
   by (simp cong: has_bochner_integral_cong add: integrable.simps)
 
-lemma integrable_cong_AE:
+lemma%unimportant integrable_cong_AE:
   "f \<in> borel_measurable M \<Longrightarrow> g \<in> borel_measurable M \<Longrightarrow> AE x in M. f x = g x \<Longrightarrow>
     integrable M f \<longleftrightarrow> integrable M g"
   unfolding integrable.simps
   by (intro has_bochner_integral_cong_AE arg_cong[where f=Ex] ext)
 
-lemma integrable_cong_AE_imp:
+lemma%unimportant integrable_cong_AE_imp:
   "integrable M g \<Longrightarrow> f \<in> borel_measurable M \<Longrightarrow> (AE x in M. g x = f x) \<Longrightarrow> integrable M f"
   using integrable_cong_AE[of f M g] by (auto simp: eq_commute)
 
-lemma integral_cong:
+lemma%unimportant integral_cong:
   "M = N \<Longrightarrow> (\<And>x. x \<in> space N \<Longrightarrow> f x = g x) \<Longrightarrow> integral\<^sup>L M f = integral\<^sup>L N g"
   by (simp cong: has_bochner_integral_cong cong del: if_weak_cong add: lebesgue_integral_def)
 
-lemma integral_cong_AE:
+lemma%unimportant integral_cong_AE:
   "f \<in> borel_measurable M \<Longrightarrow> g \<in> borel_measurable M \<Longrightarrow> AE x in M. f x = g x \<Longrightarrow>
     integral\<^sup>L M f = integral\<^sup>L M g"
   unfolding lebesgue_integral_def
   by (rule arg_cong[where x="has_bochner_integral M f"]) (intro has_bochner_integral_cong_AE ext)
 
-lemma integrable_add[simp, intro]: "integrable M f \<Longrightarrow> integrable M g \<Longrightarrow> integrable M (\<lambda>x. f x + g x)"
+lemma%unimportant integrable_add[simp, intro]: "integrable M f \<Longrightarrow> integrable M g \<Longrightarrow> integrable M (\<lambda>x. f x + g x)"
   by (auto simp: integrable.simps)
 
-lemma integrable_zero[simp, intro]: "integrable M (\<lambda>x. 0)"
+lemma%unimportant integrable_zero[simp, intro]: "integrable M (\<lambda>x. 0)"
   by (metis has_bochner_integral_zero integrable.simps)
 
-lemma integrable_sum[simp, intro]: "(\<And>i. i \<in> I \<Longrightarrow> integrable M (f i)) \<Longrightarrow> integrable M (\<lambda>x. \<Sum>i\<in>I. f i x)"
+lemma%unimportant integrable_sum[simp, intro]: "(\<And>i. i \<in> I \<Longrightarrow> integrable M (f i)) \<Longrightarrow> integrable M (\<lambda>x. \<Sum>i\<in>I. f i x)"
   by (metis has_bochner_integral_sum integrable.simps)
 
-lemma integrable_indicator[simp, intro]: "A \<in> sets M \<Longrightarrow> emeasure M A < \<infinity> \<Longrightarrow>
+lemma%unimportant integrable_indicator[simp, intro]: "A \<in> sets M \<Longrightarrow> emeasure M A < \<infinity> \<Longrightarrow>
   integrable M (\<lambda>x. indicator A x *\<^sub>R c)"
   by (metis has_bochner_integral_indicator integrable.simps)
 
-lemma integrable_real_indicator[simp, intro]: "A \<in> sets M \<Longrightarrow> emeasure M A < \<infinity> \<Longrightarrow>
+lemma%unimportant integrable_real_indicator[simp, intro]: "A \<in> sets M \<Longrightarrow> emeasure M A < \<infinity> \<Longrightarrow>
   integrable M (indicator A :: 'a \<Rightarrow> real)"
   by (metis has_bochner_integral_real_indicator integrable.simps)
 
-lemma integrable_diff[simp, intro]: "integrable M f \<Longrightarrow> integrable M g \<Longrightarrow> integrable M (\<lambda>x. f x - g x)"
+lemma%unimportant integrable_diff[simp, intro]: "integrable M f \<Longrightarrow> integrable M g \<Longrightarrow> integrable M (\<lambda>x. f x - g x)"
   by (auto simp: integrable.simps intro: has_bochner_integral_diff)
 
-lemma integrable_bounded_linear: "bounded_linear T \<Longrightarrow> integrable M f \<Longrightarrow> integrable M (\<lambda>x. T (f x))"
+lemma%unimportant integrable_bounded_linear: "bounded_linear T \<Longrightarrow> integrable M f \<Longrightarrow> integrable M (\<lambda>x. T (f x))"
   by (auto simp: integrable.simps intro: has_bochner_integral_bounded_linear)
 
-lemma integrable_scaleR_left[simp, intro]: "(c \<noteq> 0 \<Longrightarrow> integrable M f) \<Longrightarrow> integrable M (\<lambda>x. f x *\<^sub>R c)"
+lemma%unimportant integrable_scaleR_left[simp, intro]: "(c \<noteq> 0 \<Longrightarrow> integrable M f) \<Longrightarrow> integrable M (\<lambda>x. f x *\<^sub>R c)"
   unfolding integrable.simps by fastforce
 
-lemma integrable_scaleR_right[simp, intro]: "(c \<noteq> 0 \<Longrightarrow> integrable M f) \<Longrightarrow> integrable M (\<lambda>x. c *\<^sub>R f x)"
+lemma%unimportant integrable_scaleR_right[simp, intro]: "(c \<noteq> 0 \<Longrightarrow> integrable M f) \<Longrightarrow> integrable M (\<lambda>x. c *\<^sub>R f x)"
   unfolding integrable.simps by fastforce
 
-lemma integrable_mult_left[simp, intro]:
+lemma%unimportant integrable_mult_left[simp, intro]:
   fixes c :: "_::{real_normed_algebra,second_countable_topology}"
   shows "(c \<noteq> 0 \<Longrightarrow> integrable M f) \<Longrightarrow> integrable M (\<lambda>x. f x * c)"
   unfolding integrable.simps by fastforce
 
-lemma integrable_mult_right[simp, intro]:
+lemma%unimportant integrable_mult_right[simp, intro]:
   fixes c :: "_::{real_normed_algebra,second_countable_topology}"
   shows "(c \<noteq> 0 \<Longrightarrow> integrable M f) \<Longrightarrow> integrable M (\<lambda>x. c * f x)"
   unfolding integrable.simps by fastforce
 
-lemma integrable_divide_zero[simp, intro]:
+lemma%unimportant integrable_divide_zero[simp, intro]:
   fixes c :: "_::{real_normed_field, field, second_countable_topology}"
   shows "(c \<noteq> 0 \<Longrightarrow> integrable M f) \<Longrightarrow> integrable M (\<lambda>x. f x / c)"
   unfolding integrable.simps by fastforce
 
-lemma integrable_inner_left[simp, intro]:
+lemma%unimportant integrable_inner_left[simp, intro]:
   "(c \<noteq> 0 \<Longrightarrow> integrable M f) \<Longrightarrow> integrable M (\<lambda>x. f x \<bullet> c)"
   unfolding integrable.simps by fastforce
 
-lemma integrable_inner_right[simp, intro]:
+lemma%unimportant integrable_inner_right[simp, intro]:
   "(c \<noteq> 0 \<Longrightarrow> integrable M f) \<Longrightarrow> integrable M (\<lambda>x. c \<bullet> f x)"
   unfolding integrable.simps by fastforce
 
-lemmas integrable_minus[simp, intro] =
+lemmas%unimportant integrable_minus[simp, intro] =
   integrable_bounded_linear[OF bounded_linear_minus[OF bounded_linear_ident]]
-lemmas integrable_divide[simp, intro] =
+lemmas%unimportant integrable_divide[simp, intro] =
   integrable_bounded_linear[OF bounded_linear_divide]
-lemmas integrable_Re[simp, intro] =
+lemmas%unimportant integrable_Re[simp, intro] =
   integrable_bounded_linear[OF bounded_linear_Re]
-lemmas integrable_Im[simp, intro] =
+lemmas%unimportant integrable_Im[simp, intro] =
   integrable_bounded_linear[OF bounded_linear_Im]
-lemmas integrable_cnj[simp, intro] =
+lemmas%unimportant integrable_cnj[simp, intro] =
   integrable_bounded_linear[OF bounded_linear_cnj]
-lemmas integrable_of_real[simp, intro] =
+lemmas%unimportant integrable_of_real[simp, intro] =
   integrable_bounded_linear[OF bounded_linear_of_real]
-lemmas integrable_fst[simp, intro] =
+lemmas%unimportant integrable_fst[simp, intro] =
   integrable_bounded_linear[OF bounded_linear_fst]
-lemmas integrable_snd[simp, intro] =
+lemmas%unimportant integrable_snd[simp, intro] =
   integrable_bounded_linear[OF bounded_linear_snd]
 
-lemma integral_zero[simp]: "integral\<^sup>L M (\<lambda>x. 0) = 0"
+lemma%unimportant integral_zero[simp]: "integral\<^sup>L M (\<lambda>x. 0) = 0"
   by (intro has_bochner_integral_integral_eq has_bochner_integral_zero)
 
-lemma integral_add[simp]: "integrable M f \<Longrightarrow> integrable M g \<Longrightarrow>
+lemma%unimportant integral_add[simp]: "integrable M f \<Longrightarrow> integrable M g \<Longrightarrow>
     integral\<^sup>L M (\<lambda>x. f x + g x) = integral\<^sup>L M f + integral\<^sup>L M g"
   by (intro has_bochner_integral_integral_eq has_bochner_integral_add has_bochner_integral_integrable)
 
-lemma integral_diff[simp]: "integrable M f \<Longrightarrow> integrable M g \<Longrightarrow>
+lemma%unimportant integral_diff[simp]: "integrable M f \<Longrightarrow> integrable M g \<Longrightarrow>
     integral\<^sup>L M (\<lambda>x. f x - g x) = integral\<^sup>L M f - integral\<^sup>L M g"
   by (intro has_bochner_integral_integral_eq has_bochner_integral_diff has_bochner_integral_integrable)
 
-lemma integral_sum: "(\<And>i. i \<in> I \<Longrightarrow> integrable M (f i)) \<Longrightarrow>
+lemma%unimportant integral_sum: "(\<And>i. i \<in> I \<Longrightarrow> integrable M (f i)) \<Longrightarrow>
   integral\<^sup>L M (\<lambda>x. \<Sum>i\<in>I. f i x) = (\<Sum>i\<in>I. integral\<^sup>L M (f i))"
   by (intro has_bochner_integral_integral_eq has_bochner_integral_sum has_bochner_integral_integrable)
 
-lemma integral_sum'[simp]: "(\<And>i. i \<in> I =simp=> integrable M (f i)) \<Longrightarrow>
+lemma%unimportant integral_sum'[simp]: "(\<And>i. i \<in> I =simp=> integrable M (f i)) \<Longrightarrow>
   integral\<^sup>L M (\<lambda>x. \<Sum>i\<in>I. f i x) = (\<Sum>i\<in>I. integral\<^sup>L M (f i))"
   unfolding simp_implies_def by (rule integral_sum)
 
-lemma integral_bounded_linear: "bounded_linear T \<Longrightarrow> integrable M f \<Longrightarrow>
+lemma%unimportant integral_bounded_linear: "bounded_linear T \<Longrightarrow> integrable M f \<Longrightarrow>
     integral\<^sup>L M (\<lambda>x. T (f x)) = T (integral\<^sup>L M f)"
   by (metis has_bochner_integral_bounded_linear has_bochner_integral_integrable has_bochner_integral_integral_eq)
 
-lemma integral_bounded_linear':
+lemma%unimportant integral_bounded_linear':
   assumes T: "bounded_linear T" and T': "bounded_linear T'"
   assumes *: "\<not> (\<forall>x. T x = 0) \<Longrightarrow> (\<forall>x. T' (T x) = x)"
   shows "integral\<^sup>L M (\<lambda>x. T (f x)) = T (integral\<^sup>L M f)"
@@ -1085,76 +1085,76 @@
   qed
 qed
 
-lemma integral_scaleR_left[simp]: "(c \<noteq> 0 \<Longrightarrow> integrable M f) \<Longrightarrow> (\<integral> x. f x *\<^sub>R c \<partial>M) = integral\<^sup>L M f *\<^sub>R c"
+lemma%unimportant integral_scaleR_left[simp]: "(c \<noteq> 0 \<Longrightarrow> integrable M f) \<Longrightarrow> (\<integral> x. f x *\<^sub>R c \<partial>M) = integral\<^sup>L M f *\<^sub>R c"
   by (intro has_bochner_integral_integral_eq has_bochner_integral_integrable has_bochner_integral_scaleR_left)
 
-lemma integral_scaleR_right[simp]: "(\<integral> x. c *\<^sub>R f x \<partial>M) = c *\<^sub>R integral\<^sup>L M f"
+lemma%unimportant integral_scaleR_right[simp]: "(\<integral> x. c *\<^sub>R f x \<partial>M) = c *\<^sub>R integral\<^sup>L M f"
   by (rule integral_bounded_linear'[OF bounded_linear_scaleR_right bounded_linear_scaleR_right[of "1 / c"]]) simp
 
-lemma integral_mult_left[simp]:
+lemma%unimportant integral_mult_left[simp]:
   fixes c :: "_::{real_normed_algebra,second_countable_topology}"
   shows "(c \<noteq> 0 \<Longrightarrow> integrable M f) \<Longrightarrow> (\<integral> x. f x * c \<partial>M) = integral\<^sup>L M f * c"
   by (intro has_bochner_integral_integral_eq has_bochner_integral_integrable has_bochner_integral_mult_left)
 
-lemma integral_mult_right[simp]:
+lemma%unimportant integral_mult_right[simp]:
   fixes c :: "_::{real_normed_algebra,second_countable_topology}"
   shows "(c \<noteq> 0 \<Longrightarrow> integrable M f) \<Longrightarrow> (\<integral> x. c * f x \<partial>M) = c * integral\<^sup>L M f"
   by (intro has_bochner_integral_integral_eq has_bochner_integral_integrable has_bochner_integral_mult_right)
 
-lemma integral_mult_left_zero[simp]:
+lemma%unimportant integral_mult_left_zero[simp]:
   fixes c :: "_::{real_normed_field,second_countable_topology}"
   shows "(\<integral> x. f x * c \<partial>M) = integral\<^sup>L M f * c"
   by (rule integral_bounded_linear'[OF bounded_linear_mult_left bounded_linear_mult_left[of "1 / c"]]) simp
 
-lemma integral_mult_right_zero[simp]:
+lemma%unimportant integral_mult_right_zero[simp]:
   fixes c :: "_::{real_normed_field,second_countable_topology}"
   shows "(\<integral> x. c * f x \<partial>M) = c * integral\<^sup>L M f"
   by (rule integral_bounded_linear'[OF bounded_linear_mult_right bounded_linear_mult_right[of "1 / c"]]) simp
 
-lemma integral_inner_left[simp]: "(c \<noteq> 0 \<Longrightarrow> integrable M f) \<Longrightarrow> (\<integral> x. f x \<bullet> c \<partial>M) = integral\<^sup>L M f \<bullet> c"
+lemma%unimportant integral_inner_left[simp]: "(c \<noteq> 0 \<Longrightarrow> integrable M f) \<Longrightarrow> (\<integral> x. f x \<bullet> c \<partial>M) = integral\<^sup>L M f \<bullet> c"
   by (intro has_bochner_integral_integral_eq has_bochner_integral_integrable has_bochner_integral_inner_left)
 
-lemma integral_inner_right[simp]: "(c \<noteq> 0 \<Longrightarrow> integrable M f) \<Longrightarrow> (\<integral> x. c \<bullet> f x \<partial>M) = c \<bullet> integral\<^sup>L M f"
+lemma%unimportant integral_inner_right[simp]: "(c \<noteq> 0 \<Longrightarrow> integrable M f) \<Longrightarrow> (\<integral> x. c \<bullet> f x \<partial>M) = c \<bullet> integral\<^sup>L M f"
   by (intro has_bochner_integral_integral_eq has_bochner_integral_integrable has_bochner_integral_inner_right)
 
-lemma integral_divide_zero[simp]:
+lemma%unimportant integral_divide_zero[simp]:
   fixes c :: "_::{real_normed_field, field, second_countable_topology}"
   shows "integral\<^sup>L M (\<lambda>x. f x / c) = integral\<^sup>L M f / c"
   by (rule integral_bounded_linear'[OF bounded_linear_divide bounded_linear_mult_left[of c]]) simp
 
-lemma integral_minus[simp]: "integral\<^sup>L M (\<lambda>x. - f x) = - integral\<^sup>L M f"
+lemma%unimportant integral_minus[simp]: "integral\<^sup>L M (\<lambda>x. - f x) = - integral\<^sup>L M f"
   by (rule integral_bounded_linear'[OF bounded_linear_minus[OF bounded_linear_ident] bounded_linear_minus[OF bounded_linear_ident]]) simp
 
-lemma integral_complex_of_real[simp]: "integral\<^sup>L M (\<lambda>x. complex_of_real (f x)) = of_real (integral\<^sup>L M f)"
+lemma%unimportant integral_complex_of_real[simp]: "integral\<^sup>L M (\<lambda>x. complex_of_real (f x)) = of_real (integral\<^sup>L M f)"
   by (rule integral_bounded_linear'[OF bounded_linear_of_real bounded_linear_Re]) simp
 
-lemma integral_cnj[simp]: "integral\<^sup>L M (\<lambda>x. cnj (f x)) = cnj (integral\<^sup>L M f)"
+lemma%unimportant integral_cnj[simp]: "integral\<^sup>L M (\<lambda>x. cnj (f x)) = cnj (integral\<^sup>L M f)"
   by (rule integral_bounded_linear'[OF bounded_linear_cnj bounded_linear_cnj]) simp
 
-lemmas integral_divide[simp] =
+lemmas%unimportant integral_divide[simp] =
   integral_bounded_linear[OF bounded_linear_divide]
-lemmas integral_Re[simp] =
+lemmas%unimportant integral_Re[simp] =
   integral_bounded_linear[OF bounded_linear_Re]
-lemmas integral_Im[simp] =
+lemmas%unimportant integral_Im[simp] =
   integral_bounded_linear[OF bounded_linear_Im]
-lemmas integral_of_real[simp] =
+lemmas%unimportant integral_of_real[simp] =
   integral_bounded_linear[OF bounded_linear_of_real]
-lemmas integral_fst[simp] =
+lemmas%unimportant integral_fst[simp] =
   integral_bounded_linear[OF bounded_linear_fst]
-lemmas integral_snd[simp] =
+lemmas%unimportant integral_snd[simp] =
   integral_bounded_linear[OF bounded_linear_snd]
 
-lemma integral_norm_bound_ennreal:
+lemma%unimportant integral_norm_bound_ennreal:
   "integrable M f \<Longrightarrow> norm (integral\<^sup>L M f) \<le> (\<integral>\<^sup>+x. norm (f x) \<partial>M)"
   by (metis has_bochner_integral_integrable has_bochner_integral_norm_bound)
 
-lemma integrableI_sequence:
+lemma%important integrableI_sequence:
   fixes f :: "'a \<Rightarrow> 'b::{banach, second_countable_topology}"
   assumes f[measurable]: "f \<in> borel_measurable M"
   assumes s: "\<And>i. simple_bochner_integrable M (s i)"
   assumes lim: "(\<lambda>i. \<integral>\<^sup>+x. norm (f x - s i x) \<partial>M) \<longlonglongrightarrow> 0" (is "?S \<longlonglongrightarrow> 0")
   shows "integrable M f"
-proof -
+proof%unimportant -
   let ?s = "\<lambda>n. simple_bochner_integral M (s n)"
 
   have "\<exists>x. ?s \<longlonglongrightarrow> x"
@@ -1184,7 +1184,7 @@
     by (rule, rule) fact+
 qed
 
-lemma nn_integral_dominated_convergence_norm:
+lemma%important nn_integral_dominated_convergence_norm:
   fixes u' :: "_ \<Rightarrow> _::{real_normed_vector, second_countable_topology}"
   assumes [measurable]:
        "\<And>i. u i \<in> borel_measurable M" "u' \<in> borel_measurable M" "w \<in> borel_measurable M"
@@ -1192,7 +1192,7 @@
     and w: "(\<integral>\<^sup>+x. w x \<partial>M) < \<infinity>"
     and u': "AE x in M. (\<lambda>i. u i x) \<longlonglongrightarrow> u' x"
   shows "(\<lambda>i. (\<integral>\<^sup>+x. norm (u' x - u i x) \<partial>M)) \<longlonglongrightarrow> 0"
-proof -
+proof%unimportant -
   have "AE x in M. \<forall>j. norm (u j x) \<le> w x"
     unfolding AE_all_countable by rule fact
   with u' have bnd: "AE x in M. \<forall>j. norm (u' x - u j x) \<le> 2 * w x"
@@ -1225,11 +1225,11 @@
   then show ?thesis by simp
 qed
 
-lemma integrableI_bounded:
+lemma%important integrableI_bounded:
   fixes f :: "'a \<Rightarrow> 'b::{banach, second_countable_topology}"
   assumes f[measurable]: "f \<in> borel_measurable M" and fin: "(\<integral>\<^sup>+x. norm (f x) \<partial>M) < \<infinity>"
   shows "integrable M f"
-proof -
+proof%unimportant -
   from borel_measurable_implies_sequence_metric[OF f, of 0] obtain s where
     s: "\<And>i. simple_function M (s i)" and
     pointwise: "\<And>x. x \<in> space M \<Longrightarrow> (\<lambda>i. s i x) \<longlonglongrightarrow> f x" and
@@ -1266,14 +1266,14 @@
   qed fact
 qed
 
-lemma integrableI_bounded_set:
+lemma%important integrableI_bounded_set:
   fixes f :: "'a \<Rightarrow> 'b::{banach, second_countable_topology}"
   assumes [measurable]: "A \<in> sets M" "f \<in> borel_measurable M"
   assumes finite: "emeasure M A < \<infinity>"
     and bnd: "AE x in M. x \<in> A \<longrightarrow> norm (f x) \<le> B"
     and null: "AE x in M. x \<notin> A \<longrightarrow> f x = 0"
   shows "integrable M f"
-proof (rule integrableI_bounded)
+proof%unimportant (rule integrableI_bounded)
   { fix x :: 'b have "norm x \<le> B \<Longrightarrow> 0 \<le> B"
       using norm_ge_zero[of x] by arith }
   with bnd null have "(\<integral>\<^sup>+ x. ennreal (norm (f x)) \<partial>M) \<le> (\<integral>\<^sup>+ x. ennreal (max 0 B) * indicator A x \<partial>M)"
@@ -1283,37 +1283,37 @@
   finally show "(\<integral>\<^sup>+ x. ennreal (norm (f x)) \<partial>M) < \<infinity>" .
 qed simp
 
-lemma integrableI_bounded_set_indicator:
+lemma%unimportant integrableI_bounded_set_indicator:
   fixes f :: "'a \<Rightarrow> 'b::{banach, second_countable_topology}"
   shows "A \<in> sets M \<Longrightarrow> f \<in> borel_measurable M \<Longrightarrow>
     emeasure M A < \<infinity> \<Longrightarrow> (AE x in M. x \<in> A \<longrightarrow> norm (f x) \<le> B) \<Longrightarrow>
     integrable M (\<lambda>x. indicator A x *\<^sub>R f x)"
   by (rule integrableI_bounded_set[where A=A]) auto
 
-lemma integrableI_nonneg:
+lemma%important integrableI_nonneg:
   fixes f :: "'a \<Rightarrow> real"
   assumes "f \<in> borel_measurable M" "AE x in M. 0 \<le> f x" "(\<integral>\<^sup>+x. f x \<partial>M) < \<infinity>"
   shows "integrable M f"
-proof -
+proof%unimportant -
   have "(\<integral>\<^sup>+x. norm (f x) \<partial>M) = (\<integral>\<^sup>+x. f x \<partial>M)"
     using assms by (intro nn_integral_cong_AE) auto
   then show ?thesis
     using assms by (intro integrableI_bounded) auto
 qed
 
-lemma integrable_iff_bounded:
+lemma%important integrable_iff_bounded:
   fixes f :: "'a \<Rightarrow> 'b::{banach, second_countable_topology}"
   shows "integrable M f \<longleftrightarrow> f \<in> borel_measurable M \<and> (\<integral>\<^sup>+x. norm (f x) \<partial>M) < \<infinity>"
   using integrableI_bounded[of f M] has_bochner_integral_implies_finite_norm[of M f]
   unfolding integrable.simps has_bochner_integral.simps[abs_def] by auto
 
-lemma integrable_bound:
+lemma%important integrable_bound:
   fixes f :: "'a \<Rightarrow> 'b::{banach, second_countable_topology}"
     and g :: "'a \<Rightarrow> 'c::{banach, second_countable_topology}"
   shows "integrable M f \<Longrightarrow> g \<in> borel_measurable M \<Longrightarrow> (AE x in M. norm (g x) \<le> norm (f x)) \<Longrightarrow>
     integrable M g"
   unfolding integrable_iff_bounded
-proof safe
+proof%unimportant safe
   assume "f \<in> borel_measurable M" "g \<in> borel_measurable M"
   assume "AE x in M. norm (g x) \<le> norm (f x)"
   then have "(\<integral>\<^sup>+ x. ennreal (norm (g x)) \<partial>M) \<le> (\<integral>\<^sup>+ x. ennreal (norm (f x)) \<partial>M)"
@@ -1322,71 +1322,71 @@
   finally show "(\<integral>\<^sup>+ x. ennreal (norm (g x)) \<partial>M) < \<infinity>" .
 qed
 
-lemma integrable_mult_indicator:
+lemma%unimportant integrable_mult_indicator:
   fixes f :: "'a \<Rightarrow> 'b::{banach, second_countable_topology}"
   shows "A \<in> sets M \<Longrightarrow> integrable M f \<Longrightarrow> integrable M (\<lambda>x. indicator A x *\<^sub>R f x)"
   by (rule integrable_bound[of M f]) (auto split: split_indicator)
 
-lemma integrable_real_mult_indicator:
+lemma%unimportant integrable_real_mult_indicator:
   fixes f :: "'a \<Rightarrow> real"
   shows "A \<in> sets M \<Longrightarrow> integrable M f \<Longrightarrow> integrable M (\<lambda>x. f x * indicator A x)"
   using integrable_mult_indicator[of A M f] by (simp add: mult_ac)
 
-lemma integrable_abs[simp, intro]:
+lemma%unimportant integrable_abs[simp, intro]:
   fixes f :: "'a \<Rightarrow> real"
   assumes [measurable]: "integrable M f" shows "integrable M (\<lambda>x. \<bar>f x\<bar>)"
   using assms by (rule integrable_bound) auto
 
-lemma integrable_norm[simp, intro]:
+lemma%unimportant integrable_norm[simp, intro]:
   fixes f :: "'a \<Rightarrow> 'b::{banach, second_countable_topology}"
   assumes [measurable]: "integrable M f" shows "integrable M (\<lambda>x. norm (f x))"
   using assms by (rule integrable_bound) auto
 
-lemma integrable_norm_cancel:
+lemma%unimportant integrable_norm_cancel:
   fixes f :: "'a \<Rightarrow> 'b::{banach, second_countable_topology}"
   assumes [measurable]: "integrable M (\<lambda>x. norm (f x))" "f \<in> borel_measurable M" shows "integrable M f"
   using assms by (rule integrable_bound) auto
 
-lemma integrable_norm_iff:
+lemma%unimportant integrable_norm_iff:
   fixes f :: "'a \<Rightarrow> 'b::{banach, second_countable_topology}"
   shows "f \<in> borel_measurable M \<Longrightarrow> integrable M (\<lambda>x. norm (f x)) \<longleftrightarrow> integrable M f"
   by (auto intro: integrable_norm_cancel)
 
-lemma integrable_abs_cancel:
+lemma%unimportant integrable_abs_cancel:
   fixes f :: "'a \<Rightarrow> real"
   assumes [measurable]: "integrable M (\<lambda>x. \<bar>f x\<bar>)" "f \<in> borel_measurable M" shows "integrable M f"
   using assms by (rule integrable_bound) auto
 
-lemma integrable_abs_iff:
+lemma%unimportant integrable_abs_iff:
   fixes f :: "'a \<Rightarrow> real"
   shows "f \<in> borel_measurable M \<Longrightarrow> integrable M (\<lambda>x. \<bar>f x\<bar>) \<longleftrightarrow> integrable M f"
   by (auto intro: integrable_abs_cancel)
 
-lemma integrable_max[simp, intro]:
+lemma%unimportant integrable_max[simp, intro]:
   fixes f :: "'a \<Rightarrow> real"
   assumes fg[measurable]: "integrable M f" "integrable M g"
   shows "integrable M (\<lambda>x. max (f x) (g x))"
   using integrable_add[OF integrable_norm[OF fg(1)] integrable_norm[OF fg(2)]]
   by (rule integrable_bound) auto
 
-lemma integrable_min[simp, intro]:
+lemma%unimportant integrable_min[simp, intro]:
   fixes f :: "'a \<Rightarrow> real"
   assumes fg[measurable]: "integrable M f" "integrable M g"
   shows "integrable M (\<lambda>x. min (f x) (g x))"
   using integrable_add[OF integrable_norm[OF fg(1)] integrable_norm[OF fg(2)]]
   by (rule integrable_bound) auto
 
-lemma integral_minus_iff[simp]:
+lemma%unimportant integral_minus_iff[simp]:
   "integrable M (\<lambda>x. - f x ::'a::{banach, second_countable_topology}) \<longleftrightarrow> integrable M f"
   unfolding integrable_iff_bounded
   by (auto intro: borel_measurable_uminus[of "\<lambda>x. - f x" M, simplified])
 
-lemma integrable_indicator_iff:
+lemma%unimportant integrable_indicator_iff:
   "integrable M (indicator A::_ \<Rightarrow> real) \<longleftrightarrow> A \<inter> space M \<in> sets M \<and> emeasure M (A \<inter> space M) < \<infinity>"
   by (simp add: integrable_iff_bounded borel_measurable_indicator_iff ennreal_indicator nn_integral_indicator'
            cong: conj_cong)
 
-lemma integral_indicator[simp]: "integral\<^sup>L M (indicator A) = measure M (A \<inter> space M)"
+lemma%unimportant integral_indicator[simp]: "integral\<^sup>L M (indicator A) = measure M (A \<inter> space M)"
 proof cases
   assume *: "A \<inter> space M \<in> sets M \<and> emeasure M (A \<inter> space M) < \<infinity>"
   have "integral\<^sup>L M (indicator A) = integral\<^sup>L M (indicator (A \<inter> space M))"
@@ -1405,7 +1405,7 @@
   finally show ?thesis .
 qed
 
-lemma integrable_discrete_difference:
+lemma%unimportant integrable_discrete_difference:
   fixes f :: "'a \<Rightarrow> 'b::{banach, second_countable_topology}"
   assumes X: "countable X"
   assumes null: "\<And>x. x \<in> X \<Longrightarrow> emeasure M {x} = 0"
@@ -1429,7 +1429,7 @@
     by simp
 qed
 
-lemma integral_discrete_difference:
+lemma%unimportant integral_discrete_difference:
   fixes f :: "'a \<Rightarrow> 'b::{banach, second_countable_topology}"
   assumes X: "countable X"
   assumes null: "\<And>x. x \<in> X \<Longrightarrow> emeasure M {x} = 0"
@@ -1453,7 +1453,7 @@
   qed
 qed
 
-lemma has_bochner_integral_discrete_difference:
+lemma%unimportant has_bochner_integral_discrete_difference:
   fixes f :: "'a \<Rightarrow> 'b::{banach, second_countable_topology}"
   assumes X: "countable X"
   assumes null: "\<And>x. x \<in> X \<Longrightarrow> emeasure M {x} = 0"
@@ -1464,7 +1464,7 @@
   using integral_discrete_difference[of X M f g, OF assms]
   by (metis has_bochner_integral_iff)
 
-lemma
+lemma%important (*FIX ME needs name *)
   fixes f :: "'a \<Rightarrow> 'b::{banach, second_countable_topology}" and w :: "'a \<Rightarrow> real"
   assumes "f \<in> borel_measurable M" "\<And>i. s i \<in> borel_measurable M" "integrable M w"
   assumes lim: "AE x in M. (\<lambda>i. s i x) \<longlonglongrightarrow> f x"
@@ -1472,7 +1472,7 @@
   shows integrable_dominated_convergence: "integrable M f"
     and integrable_dominated_convergence2: "\<And>i. integrable M (s i)"
     and integral_dominated_convergence: "(\<lambda>i. integral\<^sup>L M (s i)) \<longlonglongrightarrow> integral\<^sup>L M f"
-proof -
+proof%unimportant -
   have w_nonneg: "AE x in M. 0 \<le> w x"
     using bound[of 0] by eventually_elim (auto intro: norm_ge_zero order_trans)
   then have "(\<integral>\<^sup>+x. w x \<partial>M) = (\<integral>\<^sup>+x. norm (w x) \<partial>M)"
@@ -1539,7 +1539,7 @@
   assumes bound: "\<forall>\<^sub>F i in at_top. AE x in M. norm (s i x) \<le> w x"
 begin
 
-lemma integral_dominated_convergence_at_top: "((\<lambda>t. integral\<^sup>L M (s t)) \<longlongrightarrow> integral\<^sup>L M f) at_top"
+lemma%unimportant integral_dominated_convergence_at_top: "((\<lambda>t. integral\<^sup>L M (s t)) \<longlongrightarrow> integral\<^sup>L M f) at_top"
 proof (rule tendsto_at_topI_sequentially)
   fix X :: "nat \<Rightarrow> real" assume X: "filterlim X at_top sequentially"
   from filterlim_iff[THEN iffD1, OF this, rule_format, OF bound]
@@ -1560,7 +1560,7 @@
   qed fact+
 qed
 
-lemma integrable_dominated_convergence_at_top: "integrable M f"
+lemma%unimportant integrable_dominated_convergence_at_top: "integrable M f"
 proof -
   from bound obtain N where w: "\<And>n. N \<le> n \<Longrightarrow> AE x in M. norm (s n x) \<le> w x"
     by (auto simp: eventually_at_top_linorder)
@@ -1581,13 +1581,13 @@
 
 end
 
-lemma integrable_mult_left_iff:
+lemma%unimportant integrable_mult_left_iff:
   fixes f :: "'a \<Rightarrow> real"
   shows "integrable M (\<lambda>x. c * f x) \<longleftrightarrow> c = 0 \<or> integrable M f"
   using integrable_mult_left[of c M f] integrable_mult_left[of "1 / c" M "\<lambda>x. c * f x"]
   by (cases "c = 0") auto
 
-lemma integrableI_nn_integral_finite:
+lemma%unimportant integrableI_nn_integral_finite:
   assumes [measurable]: "f \<in> borel_measurable M"
     and nonneg: "AE x in M. 0 \<le> f x"
     and finite: "(\<integral>\<^sup>+x. f x \<partial>M) = ennreal x"
@@ -1599,7 +1599,7 @@
     by auto
 qed simp
 
-lemma integral_nonneg_AE:
+lemma%unimportant integral_nonneg_AE:
   fixes f :: "'a \<Rightarrow> real"
   assumes nonneg: "AE x in M. 0 \<le> f x"
   shows "0 \<le> integral\<^sup>L M f"
@@ -1635,16 +1635,16 @@
   finally show ?thesis .
 qed (simp add: not_integrable_integral_eq)
 
-lemma integral_nonneg[simp]:
+lemma%unimportant integral_nonneg[simp]:
   fixes f :: "'a \<Rightarrow> real"
   shows "(\<And>x. x \<in> space M \<Longrightarrow> 0 \<le> f x) \<Longrightarrow> 0 \<le> integral\<^sup>L M f"
   by (intro integral_nonneg_AE) auto
 
-lemma nn_integral_eq_integral:
+lemma%important nn_integral_eq_integral:
   assumes f: "integrable M f"
   assumes nonneg: "AE x in M. 0 \<le> f x"
   shows "(\<integral>\<^sup>+ x. f x \<partial>M) = integral\<^sup>L M f"
-proof -
+proof%unimportant -
   { fix f :: "'a \<Rightarrow> real" assume f: "f \<in> borel_measurable M" "\<And>x. 0 \<le> f x" "integrable M f"
     then have "(\<integral>\<^sup>+ x. f x \<partial>M) = integral\<^sup>L M f"
     proof (induct rule: borel_measurable_induct_real)
@@ -1687,7 +1687,7 @@
   finally show ?thesis .
 qed
 
-lemma nn_integral_eq_integrable:
+lemma%unimportant nn_integral_eq_integrable:
   assumes f: "f \<in> M \<rightarrow>\<^sub>M borel" "AE x in M. 0 \<le> f x" and "0 \<le> x"
   shows "(\<integral>\<^sup>+x. f x \<partial>M) = ennreal x \<longleftrightarrow> (integrable M f \<and> integral\<^sup>L M f = x)"
 proof (safe intro!: nn_integral_eq_integral assms)
@@ -1697,7 +1697,7 @@
     by (simp_all add: * assms integral_nonneg_AE)
 qed
 
-lemma
+lemma%unimportant (* FIX ME needs name*)
   fixes f :: "_ \<Rightarrow> _ \<Rightarrow> 'a :: {banach, second_countable_topology}"
   assumes integrable[measurable]: "\<And>i. integrable M (f i)"
   and summable: "AE x in M. summable (\<lambda>i. norm (f i x))"
@@ -1748,10 +1748,10 @@
     unfolding sums_iff by auto
 qed
 
-lemma integral_norm_bound [simp]:
+lemma%important integral_norm_bound [simp]:
   fixes f :: "_ \<Rightarrow> 'a :: {banach, second_countable_topology}"
   shows "norm (integral\<^sup>L M f) \<le> (\<integral>x. norm (f x) \<partial>M)"
-proof (cases "integrable M f")
+proof%unimportant (cases "integrable M f")
   case True then show ?thesis
     using nn_integral_eq_integral[of M "\<lambda>x. norm (f x)"] integral_norm_bound_ennreal[of M f]
     by (simp add: integral_nonneg_AE)
@@ -1762,11 +1762,11 @@
   ultimately show ?thesis by simp
 qed
 
-lemma integral_abs_bound [simp]:
+lemma%unimportant integral_abs_bound [simp]:
   fixes f :: "'a \<Rightarrow> real" shows "abs (\<integral>x. f x \<partial>M) \<le> (\<integral>x. \<bar>f x\<bar> \<partial>M)"
 using integral_norm_bound[of M f] by auto
 
-lemma integral_eq_nn_integral:
+lemma%unimportant integral_eq_nn_integral:
   assumes [measurable]: "f \<in> borel_measurable M"
   assumes nonneg: "AE x in M. 0 \<le> f x"
   shows "integral\<^sup>L M f = enn2real (\<integral>\<^sup>+ x. ennreal (f x) \<partial>M)"
@@ -1787,7 +1787,7 @@
     by (simp add: integral_nonneg_AE)
 qed
 
-lemma enn2real_nn_integral_eq_integral:
+lemma%unimportant enn2real_nn_integral_eq_integral:
   assumes eq: "AE x in M. f x = ennreal (g x)" and nn: "AE x in M. 0 \<le> g x"
     and fin: "(\<integral>\<^sup>+x. f x \<partial>M) < top"
     and [measurable]: "g \<in> M \<rightarrow>\<^sub>M borel"
@@ -1812,20 +1812,20 @@
     using nn by (simp add: integral_nonneg_AE)
 qed
 
-lemma has_bochner_integral_nn_integral:
+lemma%unimportant has_bochner_integral_nn_integral:
   assumes "f \<in> borel_measurable M" "AE x in M. 0 \<le> f x" "0 \<le> x"
   assumes "(\<integral>\<^sup>+x. f x \<partial>M) = ennreal x"
   shows "has_bochner_integral M f x"
   unfolding has_bochner_integral_iff
   using assms by (auto simp: assms integral_eq_nn_integral intro: integrableI_nn_integral_finite)
 
-lemma integrableI_simple_bochner_integrable:
+lemma%unimportant integrableI_simple_bochner_integrable:
   fixes f :: "'a \<Rightarrow> 'b::{banach, second_countable_topology}"
   shows "simple_bochner_integrable M f \<Longrightarrow> integrable M f"
   by (intro integrableI_sequence[where s="\<lambda>_. f"] borel_measurable_simple_function)
      (auto simp: zero_ennreal_def[symmetric] simple_bochner_integrable.simps)
 
-lemma integrable_induct[consumes 1, case_names base add lim, induct pred: integrable]:
+lemma%important integrable_induct[consumes 1, case_names base add lim, induct pred: integrable]:
   fixes f :: "'a \<Rightarrow> 'b::{banach, second_countable_topology}"
   assumes "integrable M f"
   assumes base: "\<And>A c. A \<in> sets M \<Longrightarrow> emeasure M A < \<infinity> \<Longrightarrow> P (\<lambda>x. indicator A x *\<^sub>R c)"
@@ -1834,7 +1834,7 @@
    (\<And>x. x \<in> space M \<Longrightarrow> (\<lambda>i. s i x) \<longlonglongrightarrow> f x) \<Longrightarrow>
    (\<And>i x. x \<in> space M \<Longrightarrow> norm (s i x) \<le> 2 * norm (f x)) \<Longrightarrow> integrable M f \<Longrightarrow> P f"
   shows "P f"
-proof -
+proof%unimportant -
   from \<open>integrable M f\<close> have f: "f \<in> borel_measurable M" "(\<integral>\<^sup>+x. norm (f x) \<partial>M) < \<infinity>"
     unfolding integrable_iff_bounded by auto
   from borel_measurable_implies_sequence_metric[OF f(1)]
@@ -1895,12 +1895,12 @@
   qed fact
 qed
 
-lemma integral_eq_zero_AE:
+lemma%unimportant integral_eq_zero_AE:
   "(AE x in M. f x = 0) \<Longrightarrow> integral\<^sup>L M f = 0"
   using integral_cong_AE[of f M "\<lambda>_. 0"]
   by (cases "integrable M f") (simp_all add: not_integrable_integral_eq)
 
-lemma integral_nonneg_eq_0_iff_AE:
+lemma%unimportant integral_nonneg_eq_0_iff_AE:
   fixes f :: "_ \<Rightarrow> real"
   assumes f[measurable]: "integrable M f" and nonneg: "AE x in M. 0 \<le> f x"
   shows "integral\<^sup>L M f = 0 \<longleftrightarrow> (AE x in M. f x = 0)"
@@ -1914,7 +1914,7 @@
     by auto
 qed (auto simp add: integral_eq_zero_AE)
 
-lemma integral_mono_AE:
+lemma%unimportant integral_mono_AE:
   fixes f :: "'a \<Rightarrow> real"
   assumes "integrable M f" "integrable M g" "AE x in M. f x \<le> g x"
   shows "integral\<^sup>L M f \<le> integral\<^sup>L M g"
@@ -1926,7 +1926,7 @@
   finally show ?thesis by simp
 qed
 
-lemma integral_mono:
+lemma%unimportant integral_mono:
   fixes f :: "'a \<Rightarrow> real"
   shows "integrable M f \<Longrightarrow> integrable M g \<Longrightarrow> (\<And>x. x \<in> space M \<Longrightarrow> f x \<le> g x) \<Longrightarrow>
     integral\<^sup>L M f \<le> integral\<^sup>L M g"
@@ -1936,11 +1936,11 @@
 integrability assumption. The price to pay is that the upper function has to be nonnegative,
 but this is often true and easy to check in computations.\<close>
 
-lemma integral_mono_AE':
+lemma%important integral_mono_AE':
   fixes f::"_ \<Rightarrow> real"
   assumes "integrable M f" "AE x in M. g x \<le> f x" "AE x in M. 0 \<le> f x"
   shows "(\<integral>x. g x \<partial>M) \<le> (\<integral>x. f x \<partial>M)"
-proof (cases "integrable M g")
+proof%unimportant (cases "integrable M g")
   case True
   show ?thesis by (rule integral_mono_AE, auto simp add: assms True)
 next
@@ -1950,16 +1950,16 @@
   finally show ?thesis by simp
 qed
 
-lemma integral_mono':
+lemma%important integral_mono':
   fixes f::"_ \<Rightarrow> real"
   assumes "integrable M f" "\<And>x. x \<in> space M \<Longrightarrow> g x \<le> f x" "\<And>x. x \<in> space M \<Longrightarrow> 0 \<le> f x"
   shows "(\<integral>x. g x \<partial>M) \<le> (\<integral>x. f x \<partial>M)"
 by (rule integral_mono_AE', insert assms, auto)
 
-lemma (in finite_measure) integrable_measure:
+lemma%important (in finite_measure) integrable_measure:
   assumes I: "disjoint_family_on X I" "countable I"
   shows "integrable (count_space I) (\<lambda>i. measure M (X i))"
-proof -
+proof%unimportant -
   have "(\<integral>\<^sup>+i. measure M (X i) \<partial>count_space I) = (\<integral>\<^sup>+i. measure M (if X i \<in> sets M then X i else {}) \<partial>count_space I)"
     by (auto intro!: nn_integral_cong measure_notin_sets)
   also have "\<dots> = measure M (\<Union>i\<in>I. if X i \<in> sets M then X i else {})"
@@ -1969,7 +1969,7 @@
     by (auto intro!: integrableI_bounded)
 qed
 
-lemma integrableI_real_bounded:
+lemma%unimportant integrableI_real_bounded:
   assumes f: "f \<in> borel_measurable M" and ae: "AE x in M. 0 \<le> f x" and fin: "integral\<^sup>N M f < \<infinity>"
   shows "integrable M f"
 proof (rule integrableI_bounded)
@@ -1979,13 +1979,13 @@
   finally show "(\<integral>\<^sup>+ x. ennreal (norm (f x)) \<partial>M) < \<infinity>" .
 qed fact
 
-lemma nn_integral_nonneg_infinite:
+lemma%unimportant nn_integral_nonneg_infinite:
   fixes f::"'a \<Rightarrow> real"
   assumes "f \<in> borel_measurable M" "\<not> integrable M f" "AE x in M. f x \<ge> 0"
   shows "(\<integral>\<^sup>+x. f x \<partial>M) = \<infinity>"
 using assms integrableI_real_bounded less_top by auto
 
-lemma integral_real_bounded:
+lemma%unimportant integral_real_bounded:
   assumes "0 \<le> r" "integral\<^sup>N M f \<le> ennreal r"
   shows "integral\<^sup>L M f \<le> r"
 proof cases
@@ -2009,22 +2009,22 @@
     using \<open>0 \<le> r\<close> by (simp add: not_integrable_integral_eq)
 qed
 
-lemma integrable_MIN:
+lemma%unimportant integrable_MIN:
   fixes f:: "_ \<Rightarrow> _ \<Rightarrow> real"
   shows "\<lbrakk> finite I;  I \<noteq> {}; \<And>i. i \<in> I \<Longrightarrow> integrable M (f i) \<rbrakk>
    \<Longrightarrow> integrable M (\<lambda>x. MIN i\<in>I. f i x)"
 by (induct rule: finite_ne_induct) simp+
 
-lemma integrable_MAX:
+lemma%unimportant integrable_MAX:
   fixes f:: "_ \<Rightarrow> _ \<Rightarrow> real"
   shows "\<lbrakk> finite I;  I \<noteq> {};  \<And>i. i \<in> I \<Longrightarrow> integrable M (f i) \<rbrakk>
   \<Longrightarrow> integrable M (\<lambda>x. MAX i\<in>I. f i x)"
 by (induct rule: finite_ne_induct) simp+
 
-lemma integral_Markov_inequality:
+lemma%important integral_Markov_inequality:
   assumes [measurable]: "integrable M u" and "AE x in M. 0 \<le> u x" "0 < (c::real)"
   shows "(emeasure M) {x\<in>space M. u x \<ge> c} \<le> (1/c) * (\<integral>x. u x \<partial>M)"
-proof -
+proof%unimportant -
   have "(\<integral>\<^sup>+ x. ennreal(u x) * indicator (space M) x \<partial>M) \<le> (\<integral>\<^sup>+ x. u x \<partial>M)"
     by (rule nn_integral_mono_AE, auto simp add: \<open>c>0\<close> less_eq_real_def)
   also have "... = (\<integral>x. u x \<partial>M)"
@@ -2044,7 +2044,7 @@
     using \<open>0<c\<close> by (simp add: ennreal_mult'[symmetric])
 qed
 
-lemma integral_ineq_eq_0_then_AE:
+lemma%unimportant integral_ineq_eq_0_then_AE:
   fixes f::"_ \<Rightarrow> real"
   assumes "AE x in M. f x \<le> g x" "integrable M f" "integrable M g"
           "(\<integral>x. f x \<partial>M) = (\<integral>x. g x \<partial>M)"
@@ -2057,7 +2057,7 @@
   then show ?thesis unfolding h_def by auto
 qed
 
-lemma not_AE_zero_int_E:
+lemma%unimportant not_AE_zero_int_E:
   fixes f::"'a \<Rightarrow> real"
   assumes "AE x in M. f x \<ge> 0" "(\<integral>x. f x \<partial>M) > 0"
       and [measurable]: "f \<in> borel_measurable M"
@@ -2069,12 +2069,12 @@
   then show False using assms(2) by simp
 qed
 
-lemma tendsto_L1_int:
+lemma%important tendsto_L1_int:
   fixes u :: "_ \<Rightarrow> _ \<Rightarrow> 'b::{banach, second_countable_topology}"
   assumes [measurable]: "\<And>n. integrable M (u n)" "integrable M f"
           and "((\<lambda>n. (\<integral>\<^sup>+x. norm(u n x - f x) \<partial>M)) \<longlongrightarrow> 0) F"
   shows "((\<lambda>n. (\<integral>x. u n x \<partial>M)) \<longlongrightarrow> (\<integral>x. f x \<partial>M)) F"
-proof -
+proof%unimportant -
   have "((\<lambda>n. norm((\<integral>x. u n x \<partial>M) - (\<integral>x. f x \<partial>M))) \<longlongrightarrow> (0::ennreal)) F"
   proof (rule tendsto_sandwich[of "\<lambda>_. 0", where ?h = "\<lambda>n. (\<integral>\<^sup>+x. norm(u n x - f x) \<partial>M)"], auto simp add: assms)
     {
@@ -2103,12 +2103,12 @@
 text \<open>The next lemma asserts that, if a sequence of functions converges in $L^1$, then
 it admits a subsequence that converges almost everywhere.\<close>
 
-lemma tendsto_L1_AE_subseq:
+lemma%important tendsto_L1_AE_subseq:
   fixes u :: "nat \<Rightarrow> 'a \<Rightarrow> 'b::{banach, second_countable_topology}"
   assumes [measurable]: "\<And>n. integrable M (u n)"
       and "(\<lambda>n. (\<integral>x. norm(u n x) \<partial>M)) \<longlonglongrightarrow> 0"
   shows "\<exists>r::nat\<Rightarrow>nat. strict_mono r \<and> (AE x in M. (\<lambda>n. u (r n) x) \<longlonglongrightarrow> 0)"
-proof -
+proof%unimportant -
   {
     fix k
     have "eventually (\<lambda>n. (\<integral>x. norm(u n x) \<partial>M) < (1/2)^k) sequentially"
@@ -2201,9 +2201,9 @@
   then show ?thesis using \<open>strict_mono r\<close> by auto
 qed
 
-subsection \<open>Restricted measure spaces\<close>
-
-lemma integrable_restrict_space:
+subsection%important \<open>Restricted measure spaces\<close>
+
+lemma%unimportant integrable_restrict_space:
   fixes f :: "'a \<Rightarrow> 'b::{banach, second_countable_topology}"
   assumes \<Omega>[simp]: "\<Omega> \<inter> space M \<in> sets M"
   shows "integrable (restrict_space M \<Omega>) f \<longleftrightarrow> integrable M (\<lambda>x. indicator \<Omega> x *\<^sub>R f x)"
@@ -2212,11 +2212,11 @@
     nn_integral_restrict_space[OF \<Omega>]
   by (simp add: ac_simps ennreal_indicator ennreal_mult)
 
-lemma integral_restrict_space:
+lemma%important integral_restrict_space:
   fixes f :: "'a \<Rightarrow> 'b::{banach, second_countable_topology}"
   assumes \<Omega>[simp]: "\<Omega> \<inter> space M \<in> sets M"
   shows "integral\<^sup>L (restrict_space M \<Omega>) f = integral\<^sup>L M (\<lambda>x. indicator \<Omega> x *\<^sub>R f x)"
-proof (rule integral_eq_cases)
+proof%unimportant (rule integral_eq_cases)
   assume "integrable (restrict_space M \<Omega>) f"
   then show ?thesis
   proof induct
@@ -2243,7 +2243,7 @@
   qed
 qed (simp add: integrable_restrict_space)
 
-lemma integral_empty:
+lemma%unimportant integral_empty:
   assumes "space M = {}"
   shows "integral\<^sup>L M f = 0"
 proof -
@@ -2252,9 +2252,9 @@
   thus ?thesis by simp
 qed
 
-subsection \<open>Measure spaces with an associated density\<close>
-
-lemma integrable_density:
+subsection%important \<open>Measure spaces with an associated density\<close>
+
+lemma%unimportant integrable_density:
   fixes f :: "'a \<Rightarrow> 'b::{banach, second_countable_topology}" and g :: "'a \<Rightarrow> real"
   assumes [measurable]: "f \<in> borel_measurable M" "g \<in> borel_measurable M"
     and nn: "AE x in M. 0 \<le> g x"
@@ -2265,12 +2265,12 @@
   apply (auto simp: ennreal_mult)
   done
 
-lemma integral_density:
+lemma%important integral_density:
   fixes f :: "'a \<Rightarrow> 'b::{banach, second_countable_topology}" and g :: "'a \<Rightarrow> real"
   assumes f: "f \<in> borel_measurable M"
     and g[measurable]: "g \<in> borel_measurable M" "AE x in M. 0 \<le> g x"
   shows "integral\<^sup>L (density M g) f = integral\<^sup>L M (\<lambda>x. g x *\<^sub>R f x)"
-proof (rule integral_eq_cases)
+proof%unimportant (rule integral_eq_cases)
   assume "integrable (density M g) f"
   then show ?thesis
   proof induct
@@ -2325,38 +2325,38 @@
   qed
 qed (simp add: f g integrable_density)
 
-lemma
+lemma%unimportant (*FIX ME needs name *)
   fixes g :: "'a \<Rightarrow> real"
   assumes "f \<in> borel_measurable M" "AE x in M. 0 \<le> f x" "g \<in> borel_measurable M"
   shows integral_real_density: "integral\<^sup>L (density M f) g = (\<integral> x. f x * g x \<partial>M)"
     and integrable_real_density: "integrable (density M f) g \<longleftrightarrow> integrable M (\<lambda>x. f x * g x)"
   using assms integral_density[of g M f] integrable_density[of g M f] by auto
 
-lemma has_bochner_integral_density:
+lemma%important has_bochner_integral_density:
   fixes f :: "'a \<Rightarrow> 'b::{banach, second_countable_topology}" and g :: "'a \<Rightarrow> real"
   shows "f \<in> borel_measurable M \<Longrightarrow> g \<in> borel_measurable M \<Longrightarrow> (AE x in M. 0 \<le> g x) \<Longrightarrow>
     has_bochner_integral M (\<lambda>x. g x *\<^sub>R f x) x \<Longrightarrow> has_bochner_integral (density M g) f x"
   by (simp add: has_bochner_integral_iff integrable_density integral_density)
 
-subsection \<open>Distributions\<close>
-
-lemma integrable_distr_eq:
+subsection%important \<open>Distributions\<close>
+
+lemma%unimportant integrable_distr_eq:
   fixes f :: "'a \<Rightarrow> 'b::{banach, second_countable_topology}"
   assumes [measurable]: "g \<in> measurable M N" "f \<in> borel_measurable N"
   shows "integrable (distr M N g) f \<longleftrightarrow> integrable M (\<lambda>x. f (g x))"
   unfolding integrable_iff_bounded by (simp_all add: nn_integral_distr)
 
-lemma integrable_distr:
+lemma%unimportant integrable_distr:
   fixes f :: "'a \<Rightarrow> 'b::{banach, second_countable_topology}"
   shows "T \<in> measurable M M' \<Longrightarrow> integrable (distr M M' T) f \<Longrightarrow> integrable M (\<lambda>x. f (T x))"
   by (subst integrable_distr_eq[symmetric, where g=T])
      (auto dest: borel_measurable_integrable)
 
-lemma integral_distr:
+lemma%important integral_distr:
   fixes f :: "'a \<Rightarrow> 'b::{banach, second_countable_topology}"
   assumes g[measurable]: "g \<in> measurable M N" and f: "f \<in> borel_measurable N"
   shows "integral\<^sup>L (distr M N g) f = integral\<^sup>L M (\<lambda>x. f (g x))"
-proof (rule integral_eq_cases)
+proof%unimportant (rule integral_eq_cases)
   assume "integrable (distr M N g) f"
   then show ?thesis
   proof induct
@@ -2404,27 +2404,27 @@
   qed
 qed (simp add: f g integrable_distr_eq)
 
-lemma has_bochner_integral_distr:
+lemma%important has_bochner_integral_distr:
   fixes f :: "'a \<Rightarrow> 'b::{banach, second_countable_topology}"
   shows "f \<in> borel_measurable N \<Longrightarrow> g \<in> measurable M N \<Longrightarrow>
     has_bochner_integral M (\<lambda>x. f (g x)) x \<Longrightarrow> has_bochner_integral (distr M N g) f x"
-  by (simp add: has_bochner_integral_iff integrable_distr_eq integral_distr)
-
-subsection \<open>Lebesgue integration on @{const count_space}\<close>
-
-lemma integrable_count_space:
+  by%unimportant (simp add: has_bochner_integral_iff integrable_distr_eq integral_distr)
+
+subsection%important \<open>Lebesgue integration on @{const count_space}\<close>
+
+lemma%unimportant integrable_count_space:
   fixes f :: "'a \<Rightarrow> 'b::{banach,second_countable_topology}"
   shows "finite X \<Longrightarrow> integrable (count_space X) f"
   by (auto simp: nn_integral_count_space integrable_iff_bounded)
 
-lemma measure_count_space[simp]:
+lemma%unimportant measure_count_space[simp]:
   "B \<subseteq> A \<Longrightarrow> finite B \<Longrightarrow> measure (count_space A) B = card B"
   unfolding measure_def by (subst emeasure_count_space ) auto
 
-lemma lebesgue_integral_count_space_finite_support:
+lemma%important lebesgue_integral_count_space_finite_support:
   assumes f: "finite {a\<in>A. f a \<noteq> 0}"
   shows "(\<integral>x. f x \<partial>count_space A) = (\<Sum>a | a \<in> A \<and> f a \<noteq> 0. f a)"
-proof -
+proof%unimportant -
   have eq: "\<And>x. x \<in> A \<Longrightarrow> (\<Sum>a | x = a \<and> a \<in> A \<and> f a \<noteq> 0. f a) = (\<Sum>x\<in>{x}. f x)"
     by (intro sum.mono_neutral_cong_left) auto
 
@@ -2436,17 +2436,17 @@
     by auto
 qed
 
-lemma lebesgue_integral_count_space_finite: "finite A \<Longrightarrow> (\<integral>x. f x \<partial>count_space A) = (\<Sum>a\<in>A. f a)"
+lemma%unimportant lebesgue_integral_count_space_finite: "finite A \<Longrightarrow> (\<integral>x. f x \<partial>count_space A) = (\<Sum>a\<in>A. f a)"
   by (subst lebesgue_integral_count_space_finite_support)
      (auto intro!: sum.mono_neutral_cong_left)
 
-lemma integrable_count_space_nat_iff:
+lemma%unimportant integrable_count_space_nat_iff:
   fixes f :: "nat \<Rightarrow> _::{banach,second_countable_topology}"
   shows "integrable (count_space UNIV) f \<longleftrightarrow> summable (\<lambda>x. norm (f x))"
   by (auto simp add: integrable_iff_bounded nn_integral_count_space_nat ennreal_suminf_neq_top
            intro:  summable_suminf_not_top)
 
-lemma sums_integral_count_space_nat:
+lemma%unimportant sums_integral_count_space_nat:
   fixes f :: "nat \<Rightarrow> _::{banach,second_countable_topology}"
   assumes *: "integrable (count_space UNIV) f"
   shows "f sums (integral\<^sup>L (count_space UNIV) f)"
@@ -2471,18 +2471,18 @@
   finally show ?thesis .
 qed
 
-lemma integral_count_space_nat:
+lemma%unimportant integral_count_space_nat:
   fixes f :: "nat \<Rightarrow> _::{banach,second_countable_topology}"
   shows "integrable (count_space UNIV) f \<Longrightarrow> integral\<^sup>L (count_space UNIV) f = (\<Sum>x. f x)"
   using sums_integral_count_space_nat by (rule sums_unique)
 
-lemma integrable_bij_count_space:
+lemma%unimportant integrable_bij_count_space:
   fixes f :: "'a \<Rightarrow> 'b::{banach, second_countable_topology}"
   assumes g: "bij_betw g A B"
   shows "integrable (count_space A) (\<lambda>x. f (g x)) \<longleftrightarrow> integrable (count_space B) f"
   unfolding integrable_iff_bounded by (subst nn_integral_bij_count_space[OF g]) auto
 
-lemma integral_bij_count_space:
+lemma%important integral_bij_count_space:
   fixes f :: "'a \<Rightarrow> 'b::{banach, second_countable_topology}"
   assumes g: "bij_betw g A B"
   shows "integral\<^sup>L (count_space A) (\<lambda>x. f (g x)) = integral\<^sup>L (count_space B) f"
@@ -2492,49 +2492,49 @@
   apply auto
   done
 
-lemma has_bochner_integral_count_space_nat:
+lemma%important has_bochner_integral_count_space_nat:
   fixes f :: "nat \<Rightarrow> _::{banach,second_countable_topology}"
   shows "has_bochner_integral (count_space UNIV) f x \<Longrightarrow> f sums x"
   unfolding has_bochner_integral_iff by (auto intro!: sums_integral_count_space_nat)
 
-subsection \<open>Point measure\<close>
-
-lemma lebesgue_integral_point_measure_finite:
+subsection%important \<open>Point measure\<close>
+
+lemma%unimportant lebesgue_integral_point_measure_finite:
   fixes g :: "'a \<Rightarrow> 'b::{banach, second_countable_topology}"
   shows "finite A \<Longrightarrow> (\<And>a. a \<in> A \<Longrightarrow> 0 \<le> f a) \<Longrightarrow>
     integral\<^sup>L (point_measure A f) g = (\<Sum>a\<in>A. f a *\<^sub>R g a)"
   by (simp add: lebesgue_integral_count_space_finite AE_count_space integral_density point_measure_def)
 
-lemma integrable_point_measure_finite:
+lemma%important integrable_point_measure_finite:
   fixes g :: "'a \<Rightarrow> 'b::{banach, second_countable_topology}" and f :: "'a \<Rightarrow> real"
   shows "finite A \<Longrightarrow> integrable (point_measure A f) g"
-  unfolding point_measure_def
+  unfolding%unimportant point_measure_def
   apply (subst density_cong[where f'="\<lambda>x. ennreal (max 0 (f x))"])
   apply (auto split: split_max simp: ennreal_neg)
   apply (subst integrable_density)
   apply (auto simp: AE_count_space integrable_count_space)
   done
 
-subsection \<open>Lebesgue integration on @{const null_measure}\<close>
-
-lemma has_bochner_integral_null_measure_iff[iff]:
+subsection%important \<open>Lebesgue integration on @{const null_measure}\<close>
+
+lemma%unimportant has_bochner_integral_null_measure_iff[iff]:
   "has_bochner_integral (null_measure M) f 0 \<longleftrightarrow> f \<in> borel_measurable M"
   by (auto simp add: has_bochner_integral.simps simple_bochner_integral_def[abs_def]
            intro!: exI[of _ "\<lambda>n x. 0"] simple_bochner_integrable.intros)
 
-lemma integrable_null_measure_iff[iff]: "integrable (null_measure M) f \<longleftrightarrow> f \<in> borel_measurable M"
-  by (auto simp add: integrable.simps)
-
-lemma integral_null_measure[simp]: "integral\<^sup>L (null_measure M) f = 0"
+lemma%important integrable_null_measure_iff[iff]: "integrable (null_measure M) f \<longleftrightarrow> f \<in> borel_measurable M"
+  by%unimportant (auto simp add: integrable.simps)
+
+lemma%unimportant integral_null_measure[simp]: "integral\<^sup>L (null_measure M) f = 0"
   by (cases "integrable (null_measure M) f")
      (auto simp add: not_integrable_integral_eq has_bochner_integral_integral_eq)
 
-subsection \<open>Legacy lemmas for the real-valued Lebesgue integral\<close>
-
-lemma real_lebesgue_integral_def:
+subsection%important \<open>Legacy lemmas for the real-valued Lebesgue integral\<close>
+
+lemma%important real_lebesgue_integral_def:
   assumes f[measurable]: "integrable M f"
   shows "integral\<^sup>L M f = enn2real (\<integral>\<^sup>+x. f x \<partial>M) - enn2real (\<integral>\<^sup>+x. ennreal (- f x) \<partial>M)"
-proof -
+proof%unimportant -
   have "integral\<^sup>L M f = integral\<^sup>L M (\<lambda>x. max 0 (f x) - max 0 (- f x))"
     by (auto intro!: arg_cong[where f="integral\<^sup>L M"])
   also have "\<dots> = integral\<^sup>L M (\<lambda>x. max 0 (f x)) - integral\<^sup>L M (\<lambda>x. max 0 (- f x))"
@@ -2546,11 +2546,11 @@
   finally show ?thesis .
 qed
 
-lemma real_integrable_def:
+lemma%important real_integrable_def:
   "integrable M f \<longleftrightarrow> f \<in> borel_measurable M \<and>
     (\<integral>\<^sup>+ x. ennreal (f x) \<partial>M) \<noteq> \<infinity> \<and> (\<integral>\<^sup>+ x. ennreal (- f x) \<partial>M) \<noteq> \<infinity>"
   unfolding integrable_iff_bounded
-proof (safe del: notI)
+proof%unimportant (safe del: notI)
   assume *: "(\<integral>\<^sup>+ x. ennreal (norm (f x)) \<partial>M) < \<infinity>"
   have "(\<integral>\<^sup>+ x. ennreal (f x) \<partial>M) \<le> (\<integral>\<^sup>+ x. ennreal (norm (f x)) \<partial>M)"
     by (intro nn_integral_mono) auto
@@ -2574,12 +2574,12 @@
   finally show "(\<integral>\<^sup>+ x. norm (f x) \<partial>M) < \<infinity>" .
 qed
 
-lemma integrableD[dest]:
+lemma%unimportant integrableD[dest]:
   assumes "integrable M f"
   shows "f \<in> borel_measurable M" "(\<integral>\<^sup>+ x. ennreal (f x) \<partial>M) \<noteq> \<infinity>" "(\<integral>\<^sup>+ x. ennreal (- f x) \<partial>M) \<noteq> \<infinity>"
   using assms unfolding real_integrable_def by auto
 
-lemma integrableE:
+lemma%unimportant integrableE:
   assumes "integrable M f"
   obtains r q where "0 \<le> r" "0 \<le> q"
     "(\<integral>\<^sup>+x. ennreal (f x)\<partial>M) = ennreal r"
@@ -2588,7 +2588,7 @@
   using assms unfolding real_integrable_def real_lebesgue_integral_def[OF assms]
   by (cases rule: ennreal2_cases[of "(\<integral>\<^sup>+x. ennreal (-f x)\<partial>M)" "(\<integral>\<^sup>+x. ennreal (f x)\<partial>M)"]) auto
 
-lemma integral_monotone_convergence_nonneg:
+lemma%important integral_monotone_convergence_nonneg:
   fixes f :: "nat \<Rightarrow> 'a \<Rightarrow> real"
   assumes i: "\<And>i. integrable M (f i)" and mono: "AE x in M. mono (\<lambda>n. f n x)"
     and pos: "\<And>i. AE x in M. 0 \<le> f i x"
@@ -2597,7 +2597,7 @@
     and u: "u \<in> borel_measurable M"
   shows "integrable M u"
   and "integral\<^sup>L M u = x"
-proof -
+proof%unimportant -
   have nn: "AE x in M. \<forall>i. 0 \<le> f i x"
     using pos unfolding AE_all_countable by auto
   with lim have u_nn: "AE x in M. 0 \<le> u x"
@@ -2629,7 +2629,7 @@
     by (auto simp: real_integrable_def real_lebesgue_integral_def u)
 qed
 
-lemma
+lemma%unimportant (*FIX ME needs name *)
   fixes f :: "nat \<Rightarrow> 'a \<Rightarrow> real"
   assumes f: "\<And>i. integrable M (f i)" and mono: "AE x in M. mono (\<lambda>n. f n x)"
   and lim: "AE x in M. (\<lambda>i. f i x) \<longlonglongrightarrow> u x"
@@ -2660,7 +2660,7 @@
     by (metis has_bochner_integral_integrable)
 qed
 
-lemma integral_norm_eq_0_iff:
+lemma%unimportant integral_norm_eq_0_iff:
   fixes f :: "'a \<Rightarrow> 'b::{banach, second_countable_topology}"
   assumes f[measurable]: "integrable M f"
   shows "(\<integral>x. norm (f x) \<partial>M) = 0 \<longleftrightarrow> emeasure M {x\<in>space M. f x \<noteq> 0} = 0"
@@ -2675,18 +2675,18 @@
     by simp
 qed
 
-lemma integral_0_iff:
+lemma%unimportant integral_0_iff:
   fixes f :: "'a \<Rightarrow> real"
   shows "integrable M f \<Longrightarrow> (\<integral>x. \<bar>f x\<bar> \<partial>M) = 0 \<longleftrightarrow> emeasure M {x\<in>space M. f x \<noteq> 0} = 0"
   using integral_norm_eq_0_iff[of M f] by simp
 
-lemma (in finite_measure) integrable_const[intro!, simp]: "integrable M (\<lambda>x. a)"
+lemma%unimportant (in finite_measure) integrable_const[intro!, simp]: "integrable M (\<lambda>x. a)"
   using integrable_indicator[of "space M" M a] by (simp cong: integrable_cong add: less_top[symmetric])
 
-lemma lebesgue_integral_const[simp]:
+lemma%important lebesgue_integral_const[simp]:
   fixes a :: "'a :: {banach, second_countable_topology}"
   shows "(\<integral>x. a \<partial>M) = measure M (space M) *\<^sub>R a"
-proof -
+proof%unimportant -
   { assume "emeasure M (space M) = \<infinity>" "a \<noteq> 0"
     then have ?thesis
       by (auto simp add: not_integrable_integral_eq ennreal_mult_less_top measure_def integrable_iff_bounded) }
@@ -2704,7 +2704,7 @@
   ultimately show ?thesis by blast
 qed
 
-lemma (in finite_measure) integrable_const_bound:
+lemma%unimportant (in finite_measure) integrable_const_bound:
   fixes f :: "'a \<Rightarrow> 'b::{banach,second_countable_topology}"
   shows "AE x in M. norm (f x) \<le> B \<Longrightarrow> f \<in> borel_measurable M \<Longrightarrow> integrable M f"
   apply (rule integrable_bound[OF integrable_const[of B], of f])
@@ -2713,19 +2713,19 @@
   apply auto
   done
 
-lemma (in finite_measure) integral_bounded_eq_bound_then_AE:
+lemma%unimportant (in finite_measure) integral_bounded_eq_bound_then_AE:
   assumes "AE x in M. f x \<le> (c::real)"
     "integrable M f" "(\<integral>x. f x \<partial>M) = c * measure M (space M)"
   shows "AE x in M. f x = c"
   apply (rule integral_ineq_eq_0_then_AE) using assms by auto
 
-lemma integral_indicator_finite_real:
+lemma%important integral_indicator_finite_real:
   fixes f :: "'a \<Rightarrow> real"
   assumes [simp]: "finite A"
   assumes [measurable]: "\<And>a. a \<in> A \<Longrightarrow> {a} \<in> sets M"
   assumes finite: "\<And>a. a \<in> A \<Longrightarrow> emeasure M {a} < \<infinity>"
   shows "(\<integral>x. f x * indicator A x \<partial>M) = (\<Sum>a\<in>A. f a * measure M {a})"
-proof -
+proof%unimportant -
   have "(\<integral>x. f x * indicator A x \<partial>M) = (\<integral>x. (\<Sum>a\<in>A. f a * indicator {a} x) \<partial>M)"
   proof (intro integral_cong refl)
     fix x show "f x * indicator A x = (\<Sum>a\<in>A. f a * indicator {a} x)"
@@ -2736,7 +2736,7 @@
   finally show ?thesis .
 qed
 
-lemma (in finite_measure) ennreal_integral_real:
+lemma%unimportant (in finite_measure) ennreal_integral_real:
   assumes [measurable]: "f \<in> borel_measurable M"
   assumes ae: "AE x in M. f x \<le> ennreal B" "0 \<le> B"
   shows "ennreal (\<integral>x. enn2real (f x) \<partial>M) = (\<integral>\<^sup>+x. f x \<partial>M)"
@@ -2747,7 +2747,7 @@
     using ae by (intro nn_integral_cong_AE) (auto simp: le_less_trans[OF _ ennreal_less_top])
 qed auto
 
-lemma (in finite_measure) integral_less_AE:
+lemma%unimportant (in finite_measure) integral_less_AE:
   fixes X Y :: "'a \<Rightarrow> real"
   assumes int: "integrable M X" "integrable M Y"
   assumes A: "(emeasure M) A \<noteq> 0" "A \<in> sets M" "AE x in M. x \<in> A \<longrightarrow> X x \<noteq> Y x"
@@ -2778,14 +2778,14 @@
   ultimately show ?thesis by auto
 qed
 
-lemma (in finite_measure) integral_less_AE_space:
+lemma%unimportant (in finite_measure) integral_less_AE_space:
   fixes X Y :: "'a \<Rightarrow> real"
   assumes int: "integrable M X" "integrable M Y"
   assumes gt: "AE x in M. X x < Y x" "emeasure M (space M) \<noteq> 0"
   shows "integral\<^sup>L M X < integral\<^sup>L M Y"
   using gt by (intro integral_less_AE[OF int, where A="space M"]) auto
 
-lemma tendsto_integral_at_top:
+lemma%unimportant tendsto_integral_at_top:
   fixes f :: "real \<Rightarrow> 'a::{banach, second_countable_topology}"
   assumes [measurable_cong]: "sets M = sets borel" and f[measurable]: "integrable M f"
   shows "((\<lambda>y. \<integral> x. indicator {.. y} x *\<^sub>R f x \<partial>M) \<longlongrightarrow> \<integral> x. f x \<partial>M) at_top"
@@ -2809,7 +2809,7 @@
   qed auto
 qed
 
-lemma
+lemma%unimportant (*FIX ME needs name *)
   fixes f :: "real \<Rightarrow> real"
   assumes M: "sets M = sets borel"
   assumes nonneg: "AE x in M. 0 \<le> f x"
@@ -2840,32 +2840,32 @@
     by (auto simp: _has_bochner_integral_iff)
 qed
 
-subsection \<open>Product measure\<close>
-
-lemma (in sigma_finite_measure) borel_measurable_lebesgue_integrable[measurable (raw)]:
+subsection%important \<open>Product measure\<close>
+
+lemma%important (in sigma_finite_measure) borel_measurable_lebesgue_integrable[measurable (raw)]:
   fixes f :: "_ \<Rightarrow> _ \<Rightarrow> _::{banach, second_countable_topology}"
   assumes [measurable]: "case_prod f \<in> borel_measurable (N \<Otimes>\<^sub>M M)"
   shows "Measurable.pred N (\<lambda>x. integrable M (f x))"
-proof -
+proof%unimportant -
   have [simp]: "\<And>x. x \<in> space N \<Longrightarrow> integrable M (f x) \<longleftrightarrow> (\<integral>\<^sup>+y. norm (f x y) \<partial>M) < \<infinity>"
     unfolding integrable_iff_bounded by simp
   show ?thesis
     by (simp cong: measurable_cong)
 qed
 
-lemma Collect_subset [simp]: "{x\<in>A. P x} \<subseteq> A" by auto
-
-lemma (in sigma_finite_measure) measurable_measure[measurable (raw)]:
+lemma%unimportant Collect_subset [simp]: "{x\<in>A. P x} \<subseteq> A" by auto
+
+lemma%unimportant (in sigma_finite_measure) measurable_measure[measurable (raw)]:
   "(\<And>x. x \<in> space N \<Longrightarrow> A x \<subseteq> space M) \<Longrightarrow>
     {x \<in> space (N \<Otimes>\<^sub>M M). snd x \<in> A (fst x)} \<in> sets (N \<Otimes>\<^sub>M M) \<Longrightarrow>
     (\<lambda>x. measure M (A x)) \<in> borel_measurable N"
   unfolding measure_def by (intro measurable_emeasure borel_measurable_enn2real) auto
 
-lemma (in sigma_finite_measure) borel_measurable_lebesgue_integral[measurable (raw)]:
+lemma%important (in sigma_finite_measure) borel_measurable_lebesgue_integral[measurable (raw)]:
   fixes f :: "_ \<Rightarrow> _ \<Rightarrow> _::{banach, second_countable_topology}"
   assumes f[measurable]: "case_prod f \<in> borel_measurable (N \<Otimes>\<^sub>M M)"
   shows "(\<lambda>x. \<integral>y. f x y \<partial>M) \<in> borel_measurable N"
-proof -
+proof%unimportant -
   from borel_measurable_implies_sequence_metric[OF f, of 0] guess s ..
   then have s: "\<And>i. simple_function (N \<Otimes>\<^sub>M M) (s i)"
     "\<And>x y. x \<in> space N \<Longrightarrow> y \<in> space M \<Longrightarrow> (\<lambda>i. s i (x, y)) \<longlonglongrightarrow> f x y"
@@ -2936,7 +2936,7 @@
   qed
 qed
 
-lemma (in pair_sigma_finite) integrable_product_swap:
+lemma%unimportant (in pair_sigma_finite) integrable_product_swap:
   fixes f :: "_ \<Rightarrow> _::{banach, second_countable_topology}"
   assumes "integrable (M1 \<Otimes>\<^sub>M M2) f"
   shows "integrable (M2 \<Otimes>\<^sub>M M1) (\<lambda>(x,y). f (y,x))"
@@ -2948,7 +2948,7 @@
        (simp add: distr_pair_swap[symmetric] assms)
 qed
 
-lemma (in pair_sigma_finite) integrable_product_swap_iff:
+lemma%unimportant (in pair_sigma_finite) integrable_product_swap_iff:
   fixes f :: "_ \<Rightarrow> _::{banach, second_countable_topology}"
   shows "integrable (M2 \<Otimes>\<^sub>M M1) (\<lambda>(x,y). f (y,x)) \<longleftrightarrow> integrable (M1 \<Otimes>\<^sub>M M2) f"
 proof -
@@ -2957,7 +2957,7 @@
   show ?thesis by auto
 qed
 
-lemma (in pair_sigma_finite) integral_product_swap:
+lemma%unimportant (in pair_sigma_finite) integral_product_swap:
   fixes f :: "_ \<Rightarrow> _::{banach, second_countable_topology}"
   assumes f: "f \<in> borel_measurable (M1 \<Otimes>\<^sub>M M2)"
   shows "(\<integral>(x,y). f (y,x) \<partial>(M2 \<Otimes>\<^sub>M M1)) = integral\<^sup>L (M1 \<Otimes>\<^sub>M M2) f"
@@ -2967,13 +2967,13 @@
     by (simp add: integral_distr[symmetric, OF measurable_pair_swap' f] distr_pair_swap[symmetric])
 qed
 
-lemma (in pair_sigma_finite) Fubini_integrable:
+lemma%important (in pair_sigma_finite) Fubini_integrable:
   fixes f :: "_ \<Rightarrow> _::{banach, second_countable_topology}"
   assumes f[measurable]: "f \<in> borel_measurable (M1 \<Otimes>\<^sub>M M2)"
     and integ1: "integrable M1 (\<lambda>x. \<integral> y. norm (f (x, y)) \<partial>M2)"
     and integ2: "AE x in M1. integrable M2 (\<lambda>y. f (x, y))"
   shows "integrable (M1 \<Otimes>\<^sub>M M2) f"
-proof (rule integrableI_bounded)
+proof%unimportant (rule integrableI_bounded)
   have "(\<integral>\<^sup>+ p. norm (f p) \<partial>(M1 \<Otimes>\<^sub>M M2)) = (\<integral>\<^sup>+ x. (\<integral>\<^sup>+ y. norm (f (x, y)) \<partial>M2) \<partial>M1)"
     by (simp add: M2.nn_integral_fst [symmetric])
   also have "\<dots> = (\<integral>\<^sup>+ x. \<bar>\<integral>y. norm (f (x, y)) \<partial>M2\<bar> \<partial>M1)"
@@ -2994,7 +2994,7 @@
   finally show "(\<integral>\<^sup>+ p. norm (f p) \<partial>(M1 \<Otimes>\<^sub>M M2)) < \<infinity>" .
 qed fact
 
-lemma (in pair_sigma_finite) emeasure_pair_measure_finite:
+lemma%unimportant (in pair_sigma_finite) emeasure_pair_measure_finite:
   assumes A: "A \<in> sets (M1 \<Otimes>\<^sub>M M2)" and finite: "emeasure (M1 \<Otimes>\<^sub>M M2) A < \<infinity>"
   shows "AE x in M1. emeasure M2 {y\<in>space M2. (x, y) \<in> A} < \<infinity>"
 proof -
@@ -3008,7 +3008,7 @@
   ultimately show ?thesis by (auto simp: less_top)
 qed
 
-lemma (in pair_sigma_finite) AE_integrable_fst':
+lemma%unimportant (in pair_sigma_finite) AE_integrable_fst':
   fixes f :: "_ \<Rightarrow> _::{banach, second_countable_topology}"
   assumes f[measurable]: "integrable (M1 \<Otimes>\<^sub>M M2) f"
   shows "AE x in M1. integrable M2 (\<lambda>y. f (x, y))"
@@ -3025,7 +3025,7 @@
        (auto simp: integrable_iff_bounded measurable_compose[OF _ borel_measurable_integrable[OF f]] less_top)
 qed
 
-lemma (in pair_sigma_finite) integrable_fst':
+lemma%unimportant (in pair_sigma_finite) integrable_fst':
   fixes f :: "_ \<Rightarrow> _::{banach, second_countable_topology}"
   assumes f[measurable]: "integrable (M1 \<Otimes>\<^sub>M M2) f"
   shows "integrable M1 (\<lambda>x. \<integral>y. f (x, y) \<partial>M2)"
@@ -3042,11 +3042,11 @@
   finally show "(\<integral>\<^sup>+ x. ennreal (norm (\<integral> y. f (x, y) \<partial>M2)) \<partial>M1) < \<infinity>" .
 qed
 
-lemma (in pair_sigma_finite) integral_fst':
+lemma%important (in pair_sigma_finite) integral_fst':
   fixes f :: "_ \<Rightarrow> _::{banach, second_countable_topology}"
   assumes f: "integrable (M1 \<Otimes>\<^sub>M M2) f"
   shows "(\<integral>x. (\<integral>y. f (x, y) \<partial>M2) \<partial>M1) = integral\<^sup>L (M1 \<Otimes>\<^sub>M M2) f"
-using f proof induct
+using f proof%unimportant induct
   case (base A c)
   have A[measurable]: "A \<in> sets (M1 \<Otimes>\<^sub>M M2)" by fact
 
@@ -3155,7 +3155,7 @@
   qed
 qed
 
-lemma (in pair_sigma_finite)
+lemma%unimportant (in pair_sigma_finite) (* FIX ME needs name *)
   fixes f :: "_ \<Rightarrow> _ \<Rightarrow> _::{banach, second_countable_topology}"
   assumes f: "integrable (M1 \<Otimes>\<^sub>M M2) (case_prod f)"
   shows AE_integrable_fst: "AE x in M1. integrable M2 (\<lambda>y. f x y)" (is "?AE")
@@ -3163,7 +3163,7 @@
     and integral_fst: "(\<integral>x. (\<integral>y. f x y \<partial>M2) \<partial>M1) = integral\<^sup>L (M1 \<Otimes>\<^sub>M M2) (\<lambda>(x, y). f x y)" (is "?EQ")
   using AE_integrable_fst'[OF f] integrable_fst'[OF f] integral_fst'[OF f] by auto
 
-lemma (in pair_sigma_finite)
+lemma%unimportant (in pair_sigma_finite) (* FIX ME needs name *)
   fixes f :: "_ \<Rightarrow> _ \<Rightarrow> _::{banach, second_countable_topology}"
   assumes f[measurable]: "integrable (M1 \<Otimes>\<^sub>M M2) (case_prod f)"
   shows AE_integrable_snd: "AE y in M2. integrable M1 (\<lambda>x. f x y)" (is "?AE")
@@ -3179,13 +3179,13 @@
     using integral_product_swap[of "case_prod f"] by simp
 qed
 
-lemma (in pair_sigma_finite) Fubini_integral:
+lemma%unimportant (in pair_sigma_finite) Fubini_integral:
   fixes f :: "_ \<Rightarrow> _ \<Rightarrow> _ :: {banach, second_countable_topology}"
   assumes f: "integrable (M1 \<Otimes>\<^sub>M M2) (case_prod f)"
   shows "(\<integral>y. (\<integral>x. f x y \<partial>M1) \<partial>M2) = (\<integral>x. (\<integral>y. f x y \<partial>M2) \<partial>M1)"
   unfolding integral_snd[OF assms] integral_fst[OF assms] ..
 
-lemma (in product_sigma_finite) product_integral_singleton:
+lemma%unimportant (in product_sigma_finite) product_integral_singleton:
   fixes f :: "_ \<Rightarrow> _::{banach, second_countable_topology}"
   shows "f \<in> borel_measurable (M i) \<Longrightarrow> (\<integral>x. f (x i) \<partial>Pi\<^sub>M {i} M) = integral\<^sup>L (M i) f"
   apply (subst distr_singleton[symmetric])
@@ -3193,7 +3193,7 @@
   apply simp_all
   done
 
-lemma (in product_sigma_finite) product_integral_fold:
+lemma%unimportant (in product_sigma_finite) product_integral_fold:
   fixes f :: "_ \<Rightarrow> _::{banach, second_countable_topology}"
   assumes IJ[simp]: "I \<inter> J = {}" and fin: "finite I" "finite J"
   and f: "integrable (Pi\<^sub>M (I \<union> J) M) f"
@@ -3220,12 +3220,12 @@
     done
 qed
 
-lemma (in product_sigma_finite) product_integral_insert:
+lemma%important (in product_sigma_finite) product_integral_insert:
   fixes f :: "_ \<Rightarrow> _::{banach, second_countable_topology}"
   assumes I: "finite I" "i \<notin> I"
     and f: "integrable (Pi\<^sub>M (insert i I) M) f"
   shows "integral\<^sup>L (Pi\<^sub>M (insert i I) M) f = (\<integral>x. (\<integral>y. f (x(i:=y)) \<partial>M i) \<partial>Pi\<^sub>M I M)"
-proof -
+proof%unimportant -
   have "integral\<^sup>L (Pi\<^sub>M (insert i I) M) f = integral\<^sup>L (Pi\<^sub>M (I \<union> {i}) M) f"
     by simp
   also have "\<dots> = (\<integral>x. (\<integral>y. f (merge I {i} (x,y)) \<partial>Pi\<^sub>M {i} M) \<partial>Pi\<^sub>M I M)"
@@ -3244,11 +3244,11 @@
   finally show ?thesis .
 qed
 
-lemma (in product_sigma_finite) product_integrable_prod:
+lemma%important (in product_sigma_finite) product_integrable_prod:
   fixes f :: "'i \<Rightarrow> 'a \<Rightarrow> _::{real_normed_field,banach,second_countable_topology}"
   assumes [simp]: "finite I" and integrable: "\<And>i. i \<in> I \<Longrightarrow> integrable (M i) (f i)"
   shows "integrable (Pi\<^sub>M I M) (\<lambda>x. (\<Prod>i\<in>I. f i (x i)))" (is "integrable _ ?f")
-proof (unfold integrable_iff_bounded, intro conjI)
+proof%unimportant (unfold integrable_iff_bounded, intro conjI)
   interpret finite_product_sigma_finite M I by standard fact
 
   show "?f \<in> borel_measurable (Pi\<^sub>M I M)"
@@ -3263,11 +3263,11 @@
   finally show "(\<integral>\<^sup>+ x. ennreal (norm (\<Prod>i\<in>I. f i (x i))) \<partial>Pi\<^sub>M I M) < \<infinity>" .
 qed
 
-lemma (in product_sigma_finite) product_integral_prod:
+lemma%important (in product_sigma_finite) product_integral_prod:
   fixes f :: "'i \<Rightarrow> 'a \<Rightarrow> _::{real_normed_field,banach,second_countable_topology}"
   assumes "finite I" and integrable: "\<And>i. i \<in> I \<Longrightarrow> integrable (M i) (f i)"
   shows "(\<integral>x. (\<Prod>i\<in>I. f i (x i)) \<partial>Pi\<^sub>M I M) = (\<Prod>i\<in>I. integral\<^sup>L (M i) (f i))"
-using assms proof induct
+using assms proof%unimportant induct
   case empty
   interpret finite_measure "Pi\<^sub>M {} M"
     by rule (simp add: space_PiM)
@@ -3286,7 +3286,7 @@
     by (simp add: * insert prod subset_insertI)
 qed
 
-lemma integrable_subalgebra:
+lemma%unimportant integrable_subalgebra:
   fixes f :: "'a \<Rightarrow> 'b::{banach, second_countable_topology}"
   assumes borel: "f \<in> borel_measurable N"
   and N: "sets N \<subseteq> sets M" "space N = space M" "\<And>A. A \<in> sets N \<Longrightarrow> emeasure N A = emeasure M A"
@@ -3298,12 +3298,12 @@
     using assms by (auto simp: integrable_iff_bounded nn_integral_subalgebra)
 qed
 
-lemma integral_subalgebra:
+lemma%important integral_subalgebra:
   fixes f :: "'a \<Rightarrow> 'b::{banach, second_countable_topology}"
   assumes borel: "f \<in> borel_measurable N"
   and N: "sets N \<subseteq> sets M" "space N = space M" "\<And>A. A \<in> sets N \<Longrightarrow> emeasure N A = emeasure M A"
   shows "integral\<^sup>L N f = integral\<^sup>L M f"
-proof cases
+proof%unimportant cases
   assume "integrable N f"
   then show ?thesis
   proof induct

--- a/src/HOL/Analysis/Borel_Space.thy	Mon Aug 27 22:58:36 2018 +0200
+++ b/src/HOL/Analysis/Borel_Space.thy	Tue Aug 28 13:28:39 2018 +0100
@@ -3,55 +3,55 @@
     Author:     Armin Heller, TU München
 *)
 
-section \<open>Borel spaces\<close>
+section%important \<open>Borel spaces\<close>
 
 theory Borel_Space
 imports
   Measurable Derivative Ordered_Euclidean_Space Extended_Real_Limits
 begin
 
-lemma sets_Collect_eventually_sequentially[measurable]:
+lemma%unimportant sets_Collect_eventually_sequentially[measurable]:
   "(\<And>i. {x\<in>space M. P x i} \<in> sets M) \<Longrightarrow> {x\<in>space M. eventually (P x) sequentially} \<in> sets M"
   unfolding eventually_sequentially by simp
 
-lemma topological_basis_trivial: "topological_basis {A. open A}"
+lemma%unimportant topological_basis_trivial: "topological_basis {A. open A}"
   by (auto simp: topological_basis_def)
 
-lemma open_prod_generated: "open = generate_topology {A \<times> B | A B. open A \<and> open B}"
-proof -
+lemma%important open_prod_generated: "open = generate_topology {A \<times> B | A B. open A \<and> open B}"
+proof%unimportant -
   have "{A \<times> B :: ('a \<times> 'b) set | A B. open A \<and> open B} = ((\<lambda>(a, b). a \<times> b) ` ({A. open A} \<times> {A. open A}))"
     by auto
   then show ?thesis
     by (auto intro: topological_basis_prod topological_basis_trivial topological_basis_imp_subbasis)
 qed
 
-definition "mono_on f A \<equiv> \<forall>r s. r \<in> A \<and> s \<in> A \<and> r \<le> s \<longrightarrow> f r \<le> f s"
+definition%important "mono_on f A \<equiv> \<forall>r s. r \<in> A \<and> s \<in> A \<and> r \<le> s \<longrightarrow> f r \<le> f s"
 
-lemma mono_onI:
+lemma%unimportant mono_onI:
   "(\<And>r s. r \<in> A \<Longrightarrow> s \<in> A \<Longrightarrow> r \<le> s \<Longrightarrow> f r \<le> f s) \<Longrightarrow> mono_on f A"
   unfolding mono_on_def by simp
 
-lemma mono_onD:
+lemma%unimportant mono_onD:
   "\<lbrakk>mono_on f A; r \<in> A; s \<in> A; r \<le> s\<rbrakk> \<Longrightarrow> f r \<le> f s"
   unfolding mono_on_def by simp
 
-lemma mono_imp_mono_on: "mono f \<Longrightarrow> mono_on f A"
+lemma%unimportant mono_imp_mono_on: "mono f \<Longrightarrow> mono_on f A"
   unfolding mono_def mono_on_def by auto
 
-lemma mono_on_subset: "mono_on f A \<Longrightarrow> B \<subseteq> A \<Longrightarrow> mono_on f B"
+lemma%unimportant mono_on_subset: "mono_on f A \<Longrightarrow> B \<subseteq> A \<Longrightarrow> mono_on f B"
   unfolding mono_on_def by auto
 
-definition "strict_mono_on f A \<equiv> \<forall>r s. r \<in> A \<and> s \<in> A \<and> r < s \<longrightarrow> f r < f s"
+definition%important "strict_mono_on f A \<equiv> \<forall>r s. r \<in> A \<and> s \<in> A \<and> r < s \<longrightarrow> f r < f s"
 
-lemma strict_mono_onI:
+lemma%unimportant strict_mono_onI:
   "(\<And>r s. r \<in> A \<Longrightarrow> s \<in> A \<Longrightarrow> r < s \<Longrightarrow> f r < f s) \<Longrightarrow> strict_mono_on f A"
   unfolding strict_mono_on_def by simp
 
-lemma strict_mono_onD:
+lemma%unimportant strict_mono_onD:
   "\<lbrakk>strict_mono_on f A; r \<in> A; s \<in> A; r < s\<rbrakk> \<Longrightarrow> f r < f s"
   unfolding strict_mono_on_def by simp
 
-lemma mono_on_greaterD:
+lemma%unimportant mono_on_greaterD:
   assumes "mono_on g A" "x \<in> A" "y \<in> A" "g x > (g (y::_::linorder) :: _ :: linorder)"
   shows "x > y"
 proof (rule ccontr)
@@ -61,7 +61,7 @@
   with assms(4) show False by simp
 qed
 
-lemma strict_mono_inv:
+lemma%unimportant strict_mono_inv:
   fixes f :: "('a::linorder) \<Rightarrow> ('b::linorder)"
   assumes "strict_mono f" and "surj f" and inv: "\<And>x. g (f x) = x"
   shows "strict_mono g"
@@ -72,7 +72,7 @@
   with inv show "g x < g y" by simp
 qed
 
-lemma strict_mono_on_imp_inj_on:
+lemma%unimportant strict_mono_on_imp_inj_on:
   assumes "strict_mono_on (f :: (_ :: linorder) \<Rightarrow> (_ :: preorder)) A"
   shows "inj_on f A"
 proof (rule inj_onI)
@@ -82,7 +82,7 @@
        (auto dest: strict_mono_onD[OF assms, of x y] strict_mono_onD[OF assms, of y x])
 qed
 
-lemma strict_mono_on_leD:
+lemma%unimportant strict_mono_on_leD:
   assumes "strict_mono_on (f :: (_ :: linorder) \<Rightarrow> _ :: preorder) A" "x \<in> A" "y \<in> A" "x \<le> y"
   shows "f x \<le> f y"
 proof (insert le_less_linear[of y x], elim disjE)
@@ -91,17 +91,17 @@
   thus ?thesis by (rule less_imp_le)
 qed (insert assms, simp)
 
-lemma strict_mono_on_eqD:
+lemma%unimportant strict_mono_on_eqD:
   fixes f :: "(_ :: linorder) \<Rightarrow> (_ :: preorder)"
   assumes "strict_mono_on f A" "f x = f y" "x \<in> A" "y \<in> A"
   shows "y = x"
   using assms by (rule_tac linorder_cases[of x y]) (auto dest: strict_mono_onD)
 
-lemma mono_on_imp_deriv_nonneg:
+lemma%important mono_on_imp_deriv_nonneg:
   assumes mono: "mono_on f A" and deriv: "(f has_real_derivative D) (at x)"
   assumes "x \<in> interior A"
   shows "D \<ge> 0"
-proof (rule tendsto_lowerbound)
+proof%unimportant (rule tendsto_lowerbound)
   let ?A' = "(\<lambda>y. y - x) ` interior A"
   from deriv show "((\<lambda>h. (f (x + h) - f x) / h) \<longlongrightarrow> D) (at 0)"
       by (simp add: field_has_derivative_at has_field_derivative_def)
@@ -124,16 +124,16 @@
   qed
 qed simp
 
-lemma strict_mono_on_imp_mono_on:
+lemma%unimportant strict_mono_on_imp_mono_on:
   "strict_mono_on (f :: (_ :: linorder) \<Rightarrow> _ :: preorder) A \<Longrightarrow> mono_on f A"
   by (rule mono_onI, rule strict_mono_on_leD)
 
-lemma mono_on_ctble_discont:
+lemma%important mono_on_ctble_discont:
   fixes f :: "real \<Rightarrow> real"
   fixes A :: "real set"
   assumes "mono_on f A"
   shows "countable {a\<in>A. \<not> continuous (at a within A) f}"
-proof -
+proof%unimportant -
   have mono: "\<And>x y. x \<in> A \<Longrightarrow> y \<in> A \<Longrightarrow> x \<le> y \<Longrightarrow> f x \<le> f y"
     using \<open>mono_on f A\<close> by (simp add: mono_on_def)
   have "\<forall>a \<in> {a\<in>A. \<not> continuous (at a within A) f}. \<exists>q :: nat \<times> rat.
@@ -196,12 +196,12 @@
     by (rule countableI')
 qed
 
-lemma mono_on_ctble_discont_open:
+lemma%important mono_on_ctble_discont_open:
   fixes f :: "real \<Rightarrow> real"
   fixes A :: "real set"
   assumes "open A" "mono_on f A"
   shows "countable {a\<in>A. \<not>isCont f a}"
-proof -
+proof%unimportant -
   have "{a\<in>A. \<not>isCont f a} = {a\<in>A. \<not>(continuous (at a within A) f)}"
     by (auto simp add: continuous_within_open [OF _ \<open>open A\<close>])
   thus ?thesis
@@ -209,13 +209,13 @@
     by (rule mono_on_ctble_discont, rule assms)
 qed
 
-lemma mono_ctble_discont:
+lemma%important mono_ctble_discont:
   fixes f :: "real \<Rightarrow> real"
   assumes "mono f"
   shows "countable {a. \<not> isCont f a}"
-using assms mono_on_ctble_discont [of f UNIV] unfolding mono_on_def mono_def by auto
+using%unimportant assms mono_on_ctble_discont [of f UNIV] unfolding mono_on_def mono_def by auto
 
-lemma has_real_derivative_imp_continuous_on:
+lemma%important has_real_derivative_imp_continuous_on:
   assumes "\<And>x. x \<in> A \<Longrightarrow> (f has_real_derivative f' x) (at x)"
   shows "continuous_on A f"
   apply (intro differentiable_imp_continuous_on, unfold differentiable_on_def)
@@ -224,11 +224,11 @@
   apply simp_all
   done
 
-lemma closure_contains_Sup:
+lemma%important closure_contains_Sup:
   fixes S :: "real set"
   assumes "S \<noteq> {}" "bdd_above S"
   shows "Sup S \<in> closure S"
-proof-
+proof%unimportant -
   have "Inf (uminus ` S) \<in> closure (uminus ` S)"
       using assms by (intro closure_contains_Inf) auto
   also have "Inf (uminus ` S) = -Sup S" by (simp add: Inf_real_def)
@@ -237,22 +237,22 @@
   finally show ?thesis by auto
 qed
 
-lemma closed_contains_Sup:
+lemma%unimportant closed_contains_Sup:
   fixes S :: "real set"
   shows "S \<noteq> {} \<Longrightarrow> bdd_above S \<Longrightarrow> closed S \<Longrightarrow> Sup S \<in> S"
   by (subst closure_closed[symmetric], assumption, rule closure_contains_Sup)
 
-lemma closed_subset_contains_Sup:
+lemma%unimportant closed_subset_contains_Sup:
   fixes A C :: "real set"
   shows "closed C \<Longrightarrow> A \<subseteq> C \<Longrightarrow> A \<noteq> {} \<Longrightarrow> bdd_above A \<Longrightarrow> Sup A \<in> C"
   by (metis closure_contains_Sup closure_minimal subset_eq)
 
-lemma deriv_nonneg_imp_mono:
+lemma%important deriv_nonneg_imp_mono:
   assumes deriv: "\<And>x. x \<in> {a..b} \<Longrightarrow> (g has_real_derivative g' x) (at x)"
   assumes nonneg: "\<And>x. x \<in> {a..b} \<Longrightarrow> g' x \<ge> 0"
   assumes ab: "a \<le> b"
   shows "g a \<le> g b"
-proof (cases "a < b")
+proof%unimportant (cases "a < b")
   assume "a < b"
   from deriv have "\<And>x. \<lbrakk>x \<ge> a; x \<le> b\<rbrakk> \<Longrightarrow> (g has_real_derivative g' x) (at x)" by simp
   with MVT2[OF \<open>a < b\<close>] and deriv
@@ -261,11 +261,11 @@
   with g_ab show ?thesis by simp
 qed (insert ab, simp)
 
-lemma continuous_interval_vimage_Int:
+lemma%important continuous_interval_vimage_Int:
   assumes "continuous_on {a::real..b} g" and mono: "\<And>x y. a \<le> x \<Longrightarrow> x \<le> y \<Longrightarrow> y \<le> b \<Longrightarrow> g x \<le> g y"
   assumes "a \<le> b" "(c::real) \<le> d" "{c..d} \<subseteq> {g a..g b}"
   obtains c' d' where "{a..b} \<inter> g -` {c..d} = {c'..d'}" "c' \<le> d'" "g c' = c" "g d' = d"
-proof-
+proof%unimportant-
   let ?A = "{a..b} \<inter> g -` {c..d}"
   from IVT'[of g a c b, OF _ _ \<open>a \<le> b\<close> assms(1)] assms(4,5)
   obtain c'' where c'': "c'' \<in> ?A" "g c'' = c" by auto
@@ -297,48 +297,48 @@
   ultimately show ?thesis using that by blast
 qed
 
-subsection \<open>Generic Borel spaces\<close>
+subsection%important \<open>Generic Borel spaces\<close>
 
-definition (in topological_space) borel :: "'a measure" where
+definition%important (in topological_space) borel :: "'a measure" where
   "borel = sigma UNIV {S. open S}"
 
 abbreviation "borel_measurable M \<equiv> measurable M borel"
 
-lemma in_borel_measurable:
+lemma%important in_borel_measurable:
    "f \<in> borel_measurable M \<longleftrightarrow>
     (\<forall>S \<in> sigma_sets UNIV {S. open S}. f -` S \<inter> space M \<in> sets M)"
-  by (auto simp add: measurable_def borel_def)
+  by%unimportant (auto simp add: measurable_def borel_def)
 
-lemma in_borel_measurable_borel:
+lemma%important in_borel_measurable_borel:
    "f \<in> borel_measurable M \<longleftrightarrow>
     (\<forall>S \<in> sets borel.
       f -` S \<inter> space M \<in> sets M)"
-  by (auto simp add: measurable_def borel_def)
+  by%unimportant (auto simp add: measurable_def borel_def)
 
-lemma space_borel[simp]: "space borel = UNIV"
+lemma%unimportant space_borel[simp]: "space borel = UNIV"
   unfolding borel_def by auto
 
-lemma space_in_borel[measurable]: "UNIV \<in> sets borel"
+lemma%unimportant space_in_borel[measurable]: "UNIV \<in> sets borel"
   unfolding borel_def by auto
 
-lemma sets_borel: "sets borel = sigma_sets UNIV {S. open S}"
+lemma%unimportant sets_borel: "sets borel = sigma_sets UNIV {S. open S}"
   unfolding borel_def by (rule sets_measure_of) simp
 
-lemma measurable_sets_borel:
+lemma%unimportant measurable_sets_borel:
     "\<lbrakk>f \<in> measurable borel M; A \<in> sets M\<rbrakk> \<Longrightarrow> f -` A \<in> sets borel"
   by (drule (1) measurable_sets) simp
 
-lemma pred_Collect_borel[measurable (raw)]: "Measurable.pred borel P \<Longrightarrow> {x. P x} \<in> sets borel"
+lemma%unimportant pred_Collect_borel[measurable (raw)]: "Measurable.pred borel P \<Longrightarrow> {x. P x} \<in> sets borel"
   unfolding borel_def pred_def by auto
 
-lemma borel_open[measurable (raw generic)]:
+lemma%unimportant borel_open[measurable (raw generic)]:
   assumes "open A" shows "A \<in> sets borel"
 proof -
   have "A \<in> {S. open S}" unfolding mem_Collect_eq using assms .
   thus ?thesis unfolding borel_def by auto
 qed
 
-lemma borel_closed[measurable (raw generic)]:
+lemma%unimportant borel_closed[measurable (raw generic)]:
   assumes "closed A" shows "A \<in> sets borel"
 proof -
   have "space borel - (- A) \<in> sets borel"
@@ -346,11 +346,11 @@
   thus ?thesis by simp
 qed
 
-lemma borel_singleton[measurable]:
+lemma%unimportant borel_singleton[measurable]:
   "A \<in> sets borel \<Longrightarrow> insert x A \<in> sets (borel :: 'a::t1_space measure)"
   unfolding insert_def by (rule sets.Un) auto
 
-lemma sets_borel_eq_count_space: "sets (borel :: 'a::{countable, t2_space} measure) = count_space UNIV"
+lemma%unimportant sets_borel_eq_count_space: "sets (borel :: 'a::{countable, t2_space} measure) = count_space UNIV"
 proof -
   have "(\<Union>a\<in>A. {a}) \<in> sets borel" for A :: "'a set"
     by (intro sets.countable_UN') auto
@@ -358,16 +358,16 @@
     by auto
 qed
 
-lemma borel_comp[measurable]: "A \<in> sets borel \<Longrightarrow> - A \<in> sets borel"
+lemma%unimportant borel_comp[measurable]: "A \<in> sets borel \<Longrightarrow> - A \<in> sets borel"
   unfolding Compl_eq_Diff_UNIV by simp
 
-lemma borel_measurable_vimage:
+lemma%unimportant borel_measurable_vimage:
   fixes f :: "'a \<Rightarrow> 'x::t2_space"
   assumes borel[measurable]: "f \<in> borel_measurable M"
   shows "f -` {x} \<inter> space M \<in> sets M"
   by simp
 
-lemma borel_measurableI:
+lemma%unimportant borel_measurableI:
   fixes f :: "'a \<Rightarrow> 'x::topological_space"
   assumes "\<And>S. open S \<Longrightarrow> f -` S \<inter> space M \<in> sets M"
   shows "f \<in> borel_measurable M"
@@ -377,30 +377,30 @@
     using assms[of S] by simp
 qed
 
-lemma borel_measurable_const:
+lemma%unimportant borel_measurable_const:
   "(\<lambda>x. c) \<in> borel_measurable M"
   by auto
 
-lemma borel_measurable_indicator:
+lemma%unimportant borel_measurable_indicator:
   assumes A: "A \<in> sets M"
   shows "indicator A \<in> borel_measurable M"
   unfolding indicator_def [abs_def] using A
   by (auto intro!: measurable_If_set)
 
-lemma borel_measurable_count_space[measurable (raw)]:
+lemma%unimportant borel_measurable_count_space[measurable (raw)]:
   "f \<in> borel_measurable (count_space S)"
   unfolding measurable_def by auto
 
-lemma borel_measurable_indicator'[measurable (raw)]:
+lemma%unimportant borel_measurable_indicator'[measurable (raw)]:
   assumes [measurable]: "{x\<in>space M. f x \<in> A x} \<in> sets M"
   shows "(\<lambda>x. indicator (A x) (f x)) \<in> borel_measurable M"
   unfolding indicator_def[abs_def]
   by (auto intro!: measurable_If)
 
-lemma borel_measurable_indicator_iff:
+lemma%important borel_measurable_indicator_iff:
   "(indicator A :: 'a \<Rightarrow> 'x::{t1_space, zero_neq_one}) \<in> borel_measurable M \<longleftrightarrow> A \<inter> space M \<in> sets M"
     (is "?I \<in> borel_measurable M \<longleftrightarrow> _")
-proof
+proof%unimportant
   assume "?I \<in> borel_measurable M"
   then have "?I -` {1} \<inter> space M \<in> sets M"
     unfolding measurable_def by auto
@@ -415,12 +415,12 @@
   ultimately show "?I \<in> borel_measurable M" by auto
 qed
 
-lemma borel_measurable_subalgebra:
+lemma%unimportant borel_measurable_subalgebra:
   assumes "sets N \<subseteq> sets M" "space N = space M" "f \<in> borel_measurable N"
   shows "f \<in> borel_measurable M"
   using assms unfolding measurable_def by auto
 
-lemma borel_measurable_restrict_space_iff_ereal:
+lemma%unimportant borel_measurable_restrict_space_iff_ereal:
   fixes f :: "'a \<Rightarrow> ereal"
   assumes \<Omega>[measurable, simp]: "\<Omega> \<inter> space M \<in> sets M"
   shows "f \<in> borel_measurable (restrict_space M \<Omega>) \<longleftrightarrow>
@@ -428,7 +428,7 @@
   by (subst measurable_restrict_space_iff)
      (auto simp: indicator_def if_distrib[where f="\<lambda>x. a * x" for a] cong del: if_weak_cong)
 
-lemma borel_measurable_restrict_space_iff_ennreal:
+lemma%unimportant borel_measurable_restrict_space_iff_ennreal:
   fixes f :: "'a \<Rightarrow> ennreal"
   assumes \<Omega>[measurable, simp]: "\<Omega> \<inter> space M \<in> sets M"
   shows "f \<in> borel_measurable (restrict_space M \<Omega>) \<longleftrightarrow>
@@ -436,7 +436,7 @@
   by (subst measurable_restrict_space_iff)
      (auto simp: indicator_def if_distrib[where f="\<lambda>x. a * x" for a] cong del: if_weak_cong)
 
-lemma borel_measurable_restrict_space_iff:
+lemma%unimportant borel_measurable_restrict_space_iff:
   fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"
   assumes \<Omega>[measurable, simp]: "\<Omega> \<inter> space M \<in> sets M"
   shows "f \<in> borel_measurable (restrict_space M \<Omega>) \<longleftrightarrow>
@@ -445,27 +445,27 @@
      (auto simp: indicator_def if_distrib[where f="\<lambda>x. x *\<^sub>R a" for a] ac_simps
        cong del: if_weak_cong)
 
-lemma cbox_borel[measurable]: "cbox a b \<in> sets borel"
+lemma%unimportant cbox_borel[measurable]: "cbox a b \<in> sets borel"
   by (auto intro: borel_closed)
 
-lemma box_borel[measurable]: "box a b \<in> sets borel"
+lemma%unimportant box_borel[measurable]: "box a b \<in> sets borel"
   by (auto intro: borel_open)
 
-lemma borel_compact: "compact (A::'a::t2_space set) \<Longrightarrow> A \<in> sets borel"
+lemma%unimportant borel_compact: "compact (A::'a::t2_space set) \<Longrightarrow> A \<in> sets borel"
   by (auto intro: borel_closed dest!: compact_imp_closed)
 
-lemma borel_sigma_sets_subset:
+lemma%unimportant borel_sigma_sets_subset:
   "A \<subseteq> sets borel \<Longrightarrow> sigma_sets UNIV A \<subseteq> sets borel"
   using sets.sigma_sets_subset[of A borel] by simp
 
-lemma borel_eq_sigmaI1:
+lemma%important borel_eq_sigmaI1:
   fixes F :: "'i \<Rightarrow> 'a::topological_space set" and X :: "'a::topological_space set set"
   assumes borel_eq: "borel = sigma UNIV X"
   assumes X: "\<And>x. x \<in> X \<Longrightarrow> x \<in> sets (sigma UNIV (F ` A))"
   assumes F: "\<And>i. i \<in> A \<Longrightarrow> F i \<in> sets borel"
   shows "borel = sigma UNIV (F ` A)"
   unfolding borel_def
-proof (intro sigma_eqI antisym)
+proof%unimportant (intro sigma_eqI antisym)
   have borel_rev_eq: "sigma_sets UNIV {S::'a set. open S} = sets borel"
     unfolding borel_def by simp
   also have "\<dots> = sigma_sets UNIV X"
@@ -477,7 +477,7 @@
     unfolding borel_rev_eq using F by (intro borel_sigma_sets_subset) auto
 qed auto
 
-lemma borel_eq_sigmaI2:
+lemma%unimportant borel_eq_sigmaI2:
   fixes F :: "'i \<Rightarrow> 'j \<Rightarrow> 'a::topological_space set"
     and G :: "'l \<Rightarrow> 'k \<Rightarrow> 'a::topological_space set"
   assumes borel_eq: "borel = sigma UNIV ((\<lambda>(i, j). G i j)`B)"
@@ -487,7 +487,7 @@
   using assms
   by (intro borel_eq_sigmaI1[where X="(\<lambda>(i, j). G i j) ` B" and F="(\<lambda>(i, j). F i j)"]) auto
 
-lemma borel_eq_sigmaI3:
+lemma%unimportant borel_eq_sigmaI3:
   fixes F :: "'i \<Rightarrow> 'j \<Rightarrow> 'a::topological_space set" and X :: "'a::topological_space set set"
   assumes borel_eq: "borel = sigma UNIV X"
   assumes X: "\<And>x. x \<in> X \<Longrightarrow> x \<in> sets (sigma UNIV ((\<lambda>(i, j). F i j) ` A))"
@@ -495,7 +495,7 @@
   shows "borel = sigma UNIV ((\<lambda>(i, j). F i j) ` A)"
   using assms by (intro borel_eq_sigmaI1[where X=X and F="(\<lambda>(i, j). F i j)"]) auto
 
-lemma borel_eq_sigmaI4:
+lemma%unimportant borel_eq_sigmaI4:
   fixes F :: "'i \<Rightarrow> 'a::topological_space set"
     and G :: "'l \<Rightarrow> 'k \<Rightarrow> 'a::topological_space set"
   assumes borel_eq: "borel = sigma UNIV ((\<lambda>(i, j). G i j)`A)"
@@ -504,7 +504,7 @@
   shows "borel = sigma UNIV (range F)"
   using assms by (intro borel_eq_sigmaI1[where X="(\<lambda>(i, j). G i j) ` A" and F=F]) auto
 
-lemma borel_eq_sigmaI5:
+lemma%unimportant borel_eq_sigmaI5:
   fixes F :: "'i \<Rightarrow> 'j \<Rightarrow> 'a::topological_space set" and G :: "'l \<Rightarrow> 'a::topological_space set"
   assumes borel_eq: "borel = sigma UNIV (range G)"
   assumes X: "\<And>i. G i \<in> sets (sigma UNIV (range (\<lambda>(i, j). F i j)))"
@@ -512,12 +512,12 @@
   shows "borel = sigma UNIV (range (\<lambda>(i, j). F i j))"
   using assms by (intro borel_eq_sigmaI1[where X="range G" and F="(\<lambda>(i, j). F i j)"]) auto
 
-lemma second_countable_borel_measurable:
+lemma%important second_countable_borel_measurable:
   fixes X :: "'a::second_countable_topology set set"
   assumes eq: "open = generate_topology X"
   shows "borel = sigma UNIV X"
   unfolding borel_def
-proof (intro sigma_eqI sigma_sets_eqI)
+proof%unimportant (intro sigma_eqI sigma_sets_eqI)
   interpret X: sigma_algebra UNIV "sigma_sets UNIV X"
     by (rule sigma_algebra_sigma_sets) simp
 
@@ -556,7 +556,7 @@
   qed auto
 qed (auto simp: eq intro: generate_topology.Basis)
 
-lemma borel_eq_closed: "borel = sigma UNIV (Collect closed)"
+lemma%unimportant borel_eq_closed: "borel = sigma UNIV (Collect closed)"
   unfolding borel_def
 proof (intro sigma_eqI sigma_sets_eqI, safe)
   fix x :: "'a set" assume "open x"
@@ -572,13 +572,13 @@
   finally show "x \<in> sigma_sets UNIV (Collect open)" by simp
 qed simp_all
 
-lemma borel_eq_countable_basis:
+lemma%important borel_eq_countable_basis:
   fixes B::"'a::topological_space set set"
   assumes "countable B"
   assumes "topological_basis B"
   shows "borel = sigma UNIV B"
   unfolding borel_def
-proof (intro sigma_eqI sigma_sets_eqI, safe)
+proof%unimportant (intro sigma_eqI sigma_sets_eqI, safe)
   interpret countable_basis using assms by unfold_locales
   fix X::"'a set" assume "open X"
   from open_countable_basisE[OF this] guess B' . note B' = this
@@ -590,7 +590,7 @@
   thus "b \<in> sigma_sets UNIV (Collect open)" by auto
 qed simp_all
 
-lemma borel_measurable_continuous_on_restrict:
+lemma%unimportant borel_measurable_continuous_on_restrict:
   fixes f :: "'a::topological_space \<Rightarrow> 'b::topological_space"
   assumes f: "continuous_on A f"
   shows "f \<in> borel_measurable (restrict_space borel A)"
@@ -602,16 +602,16 @@
     by (force simp add: sets_restrict_space space_restrict_space)
 qed
 
-lemma borel_measurable_continuous_on1: "continuous_on UNIV f \<Longrightarrow> f \<in> borel_measurable borel"
+lemma%unimportant borel_measurable_continuous_on1: "continuous_on UNIV f \<Longrightarrow> f \<in> borel_measurable borel"
   by (drule borel_measurable_continuous_on_restrict) simp
 
-lemma borel_measurable_continuous_on_if:
+lemma%unimportant borel_measurable_continuous_on_if:
   "A \<in> sets borel \<Longrightarrow> continuous_on A f \<Longrightarrow> continuous_on (- A) g \<Longrightarrow>
     (\<lambda>x. if x \<in> A then f x else g x) \<in> borel_measurable borel"
   by (auto simp add: measurable_If_restrict_space_iff Collect_neg_eq
            intro!: borel_measurable_continuous_on_restrict)
 
-lemma borel_measurable_continuous_countable_exceptions:
+lemma%unimportant borel_measurable_continuous_countable_exceptions:
   fixes f :: "'a::t1_space \<Rightarrow> 'b::topological_space"
   assumes X: "countable X"
   assumes "continuous_on (- X) f"
@@ -623,23 +623,23 @@
     by (intro borel_measurable_continuous_on_if assms continuous_intros)
 qed auto
 
-lemma borel_measurable_continuous_on:
+lemma%unimportant borel_measurable_continuous_on:
   assumes f: "continuous_on UNIV f" and g: "g \<in> borel_measurable M"
   shows "(\<lambda>x. f (g x)) \<in> borel_measurable M"
   using measurable_comp[OF g borel_measurable_continuous_on1[OF f]] by (simp add: comp_def)
 
-lemma borel_measurable_continuous_on_indicator:
+lemma%unimportant borel_measurable_continuous_on_indicator:
   fixes f g :: "'a::topological_space \<Rightarrow> 'b::real_normed_vector"
   shows "A \<in> sets borel \<Longrightarrow> continuous_on A f \<Longrightarrow> (\<lambda>x. indicator A x *\<^sub>R f x) \<in> borel_measurable borel"
   by (subst borel_measurable_restrict_space_iff[symmetric])
      (auto intro: borel_measurable_continuous_on_restrict)
 
-lemma borel_measurable_Pair[measurable (raw)]:
+lemma%important borel_measurable_Pair[measurable (raw)]:
   fixes f :: "'a \<Rightarrow> 'b::second_countable_topology" and g :: "'a \<Rightarrow> 'c::second_countable_topology"
   assumes f[measurable]: "f \<in> borel_measurable M"
   assumes g[measurable]: "g \<in> borel_measurable M"
   shows "(\<lambda>x. (f x, g x)) \<in> borel_measurable M"
-proof (subst borel_eq_countable_basis)
+proof%unimportant (subst borel_eq_countable_basis)
   let ?B = "SOME B::'b set set. countable B \<and> topological_basis B"
   let ?C = "SOME B::'c set set. countable B \<and> topological_basis B"
   let ?P = "(\<lambda>(b, c). b \<times> c) ` (?B \<times> ?C)"
@@ -660,21 +660,21 @@
   qed auto
 qed
 
-lemma borel_measurable_continuous_Pair:
+lemma%important borel_measurable_continuous_Pair:
   fixes f :: "'a \<Rightarrow> 'b::second_countable_topology" and g :: "'a \<Rightarrow> 'c::second_countable_topology"
   assumes [measurable]: "f \<in> borel_measurable M"
   assumes [measurable]: "g \<in> borel_measurable M"
   assumes H: "continuous_on UNIV (\<lambda>x. H (fst x) (snd x))"
   shows "(\<lambda>x. H (f x) (g x)) \<in> borel_measurable M"
-proof -
+proof%unimportant -
   have eq: "(\<lambda>x. H (f x) (g x)) = (\<lambda>x. (\<lambda>x. H (fst x) (snd x)) (f x, g x))" by auto
   show ?thesis
     unfolding eq by (rule borel_measurable_continuous_on[OF H]) auto
 qed
 
-subsection \<open>Borel spaces on order topologies\<close>
+subsection%important \<open>Borel spaces on order topologies\<close>
 
-lemma [measurable]:
+lemma%unimportant [measurable]:
   fixes a b :: "'a::linorder_topology"
   shows lessThan_borel: "{..< a} \<in> sets borel"
     and greaterThan_borel: "{a <..} \<in> sets borel"
@@ -688,7 +688,7 @@
   by (blast intro: borel_open borel_closed open_lessThan open_greaterThan open_greaterThanLessThan
                    closed_atMost closed_atLeast closed_atLeastAtMost)+
 
-lemma borel_Iio:
+lemma%unimportant borel_Iio:
   "borel = sigma UNIV (range lessThan :: 'a::{linorder_topology, second_countable_topology} set set)"
   unfolding second_countable_borel_measurable[OF open_generated_order]
 proof (intro sigma_eqI sigma_sets_eqI)
@@ -726,7 +726,7 @@
   qed auto
 qed auto
 
-lemma borel_Ioi:
+lemma%unimportant borel_Ioi:
   "borel = sigma UNIV (range greaterThan :: 'a::{linorder_topology, second_countable_topology} set set)"
   unfolding second_countable_borel_measurable[OF open_generated_order]
 proof (intro sigma_eqI sigma_sets_eqI)
@@ -764,29 +764,29 @@
   qed auto
 qed auto
 
-lemma borel_measurableI_less:
+lemma%unimportant borel_measurableI_less:
   fixes f :: "'a \<Rightarrow> 'b::{linorder_topology, second_countable_topology}"
   shows "(\<And>y. {x\<in>space M. f x < y} \<in> sets M) \<Longrightarrow> f \<in> borel_measurable M"
   unfolding borel_Iio
   by (rule measurable_measure_of) (auto simp: Int_def conj_commute)
 
-lemma borel_measurableI_greater:
+lemma%important borel_measurableI_greater:
   fixes f :: "'a \<Rightarrow> 'b::{linorder_topology, second_countable_topology}"
   shows "(\<And>y. {x\<in>space M. y < f x} \<in> sets M) \<Longrightarrow> f \<in> borel_measurable M"
   unfolding borel_Ioi
-  by (rule measurable_measure_of) (auto simp: Int_def conj_commute)
+  by%unimportant (rule measurable_measure_of) (auto simp: Int_def conj_commute)
 
-lemma borel_measurableI_le:
+lemma%unimportant borel_measurableI_le:
   fixes f :: "'a \<Rightarrow> 'b::{linorder_topology, second_countable_topology}"
   shows "(\<And>y. {x\<in>space M. f x \<le> y} \<in> sets M) \<Longrightarrow> f \<in> borel_measurable M"
   by (rule borel_measurableI_greater) (auto simp: not_le[symmetric])
 
-lemma borel_measurableI_ge:
+lemma%unimportant borel_measurableI_ge:
   fixes f :: "'a \<Rightarrow> 'b::{linorder_topology, second_countable_topology}"
   shows "(\<And>y. {x\<in>space M. y \<le> f x} \<in> sets M) \<Longrightarrow> f \<in> borel_measurable M"
   by (rule borel_measurableI_less) (auto simp: not_le[symmetric])
 
-lemma borel_measurable_less[measurable]:
+lemma%unimportant borel_measurable_less[measurable]:
   fixes f :: "'a \<Rightarrow> 'b::{second_countable_topology, linorder_topology}"
   assumes "f \<in> borel_measurable M"
   assumes "g \<in> borel_measurable M"
@@ -800,7 +800,7 @@
   finally show ?thesis .
 qed
 
-lemma
+lemma%important
   fixes f :: "'a \<Rightarrow> 'b::{second_countable_topology, linorder_topology}"
   assumes f[measurable]: "f \<in> borel_measurable M"
   assumes g[measurable]: "g \<in> borel_measurable M"
@@ -808,23 +808,23 @@
     and borel_measurable_eq[measurable]: "{w \<in> space M. f w = g w} \<in> sets M"
     and borel_measurable_neq: "{w \<in> space M. f w \<noteq> g w} \<in> sets M"
   unfolding eq_iff not_less[symmetric]
-  by measurable
+  by%unimportant measurable
 
-lemma borel_measurable_SUP[measurable (raw)]:
+lemma%important borel_measurable_SUP[measurable (raw)]:
   fixes F :: "_ \<Rightarrow> _ \<Rightarrow> _::{complete_linorder, linorder_topology, second_countable_topology}"
   assumes [simp]: "countable I"
   assumes [measurable]: "\<And>i. i \<in> I \<Longrightarrow> F i \<in> borel_measurable M"
   shows "(\<lambda>x. SUP i:I. F i x) \<in> borel_measurable M"
-  by (rule borel_measurableI_greater) (simp add: less_SUP_iff)
+  by%unimportant (rule borel_measurableI_greater) (simp add: less_SUP_iff)
 
-lemma borel_measurable_INF[measurable (raw)]:
+lemma%unimportant borel_measurable_INF[measurable (raw)]:
   fixes F :: "_ \<Rightarrow> _ \<Rightarrow> _::{complete_linorder, linorder_topology, second_countable_topology}"
   assumes [simp]: "countable I"
   assumes [measurable]: "\<And>i. i \<in> I \<Longrightarrow> F i \<in> borel_measurable M"
   shows "(\<lambda>x. INF i:I. F i x) \<in> borel_measurable M"
   by (rule borel_measurableI_less) (simp add: INF_less_iff)
 
-lemma borel_measurable_cSUP[measurable (raw)]:
+lemma%unimportant borel_measurable_cSUP[measurable (raw)]:
   fixes F :: "_ \<Rightarrow> _ \<Rightarrow> 'a::{conditionally_complete_linorder, linorder_topology, second_countable_topology}"
   assumes [simp]: "countable I"
   assumes [measurable]: "\<And>i. i \<in> I \<Longrightarrow> F i \<in> borel_measurable M"
@@ -846,13 +846,13 @@
   qed
 qed
 
-lemma borel_measurable_cINF[measurable (raw)]:
+lemma%important borel_measurable_cINF[measurable (raw)]:
   fixes F :: "_ \<Rightarrow> _ \<Rightarrow> 'a::{conditionally_complete_linorder, linorder_topology, second_countable_topology}"
   assumes [simp]: "countable I"
   assumes [measurable]: "\<And>i. i \<in> I \<Longrightarrow> F i \<in> borel_measurable M"
   assumes bdd: "\<And>x. x \<in> space M \<Longrightarrow> bdd_below ((\<lambda>i. F i x) ` I)"
   shows "(\<lambda>x. INF i:I. F i x) \<in> borel_measurable M"
-proof cases
+proof%unimportant cases
   assume "I = {}" then show ?thesis
     unfolding \<open>I = {}\<close> image_empty by simp
 next
@@ -868,7 +868,7 @@
   qed
 qed
 
-lemma borel_measurable_lfp[consumes 1, case_names continuity step]:
+lemma%unimportant borel_measurable_lfp[consumes 1, case_names continuity step]:
   fixes F :: "('a \<Rightarrow> 'b) \<Rightarrow> ('a \<Rightarrow> 'b::{complete_linorder, linorder_topology, second_countable_topology})"
   assumes "sup_continuous F"
   assumes *: "\<And>f. f \<in> borel_measurable M \<Longrightarrow> F f \<in> borel_measurable M"
@@ -885,7 +885,7 @@
   finally show ?thesis .
 qed
 
-lemma borel_measurable_gfp[consumes 1, case_names continuity step]:
+lemma%unimportant borel_measurable_gfp[consumes 1, case_names continuity step]:
   fixes F :: "('a \<Rightarrow> 'b) \<Rightarrow> ('a \<Rightarrow> 'b::{complete_linorder, linorder_topology, second_countable_topology})"
   assumes "inf_continuous F"
   assumes *: "\<And>f. f \<in> borel_measurable M \<Longrightarrow> F f \<in> borel_measurable M"
@@ -902,56 +902,56 @@
   finally show ?thesis .
 qed
 
-lemma borel_measurable_max[measurable (raw)]:
+lemma%unimportant borel_measurable_max[measurable (raw)]:
   "f \<in> borel_measurable M \<Longrightarrow> g \<in> borel_measurable M \<Longrightarrow> (\<lambda>x. max (g x) (f x) :: 'b::{second_countable_topology, linorder_topology}) \<in> borel_measurable M"
   by (rule borel_measurableI_less) simp
 
-lemma borel_measurable_min[measurable (raw)]:
+lemma%unimportant borel_measurable_min[measurable (raw)]:
   "f \<in> borel_measurable M \<Longrightarrow> g \<in> borel_measurable M \<Longrightarrow> (\<lambda>x. min (g x) (f x) :: 'b::{second_countable_topology, linorder_topology}) \<in> borel_measurable M"
   by (rule borel_measurableI_greater) simp
 
-lemma borel_measurable_Min[measurable (raw)]:
+lemma%unimportant borel_measurable_Min[measurable (raw)]:
   "finite I \<Longrightarrow> (\<And>i. i \<in> I \<Longrightarrow> f i \<in> borel_measurable M) \<Longrightarrow> (\<lambda>x. Min ((\<lambda>i. f i x)`I) :: 'b::{second_countable_topology, linorder_topology}) \<in> borel_measurable M"
 proof (induct I rule: finite_induct)
   case (insert i I) then show ?case
     by (cases "I = {}") auto
 qed auto
 
-lemma borel_measurable_Max[measurable (raw)]:
+lemma%unimportant borel_measurable_Max[measurable (raw)]:
   "finite I \<Longrightarrow> (\<And>i. i \<in> I \<Longrightarrow> f i \<in> borel_measurable M) \<Longrightarrow> (\<lambda>x. Max ((\<lambda>i. f i x)`I) :: 'b::{second_countable_topology, linorder_topology}) \<in> borel_measurable M"
 proof (induct I rule: finite_induct)
   case (insert i I) then show ?case
     by (cases "I = {}") auto
 qed auto
 
-lemma borel_measurable_sup[measurable (raw)]:
+lemma%unimportant borel_measurable_sup[measurable (raw)]:
   "f \<in> borel_measurable M \<Longrightarrow> g \<in> borel_measurable M \<Longrightarrow> (\<lambda>x. sup (g x) (f x) :: 'b::{lattice, second_countable_topology, linorder_topology}) \<in> borel_measurable M"
   unfolding sup_max by measurable
 
-lemma borel_measurable_inf[measurable (raw)]:
+lemma%unimportant borel_measurable_inf[measurable (raw)]:
   "f \<in> borel_measurable M \<Longrightarrow> g \<in> borel_measurable M \<Longrightarrow> (\<lambda>x. inf (g x) (f x) :: 'b::{lattice, second_countable_topology, linorder_topology}) \<in> borel_measurable M"
   unfolding inf_min by measurable
 
-lemma [measurable (raw)]:
+lemma%unimportant [measurable (raw)]:
   fixes f :: "nat \<Rightarrow> 'a \<Rightarrow> 'b::{complete_linorder, second_countable_topology, linorder_topology}"
   assumes "\<And>i. f i \<in> borel_measurable M"
   shows borel_measurable_liminf: "(\<lambda>x. liminf (\<lambda>i. f i x)) \<in> borel_measurable M"
     and borel_measurable_limsup: "(\<lambda>x. limsup (\<lambda>i. f i x)) \<in> borel_measurable M"
   unfolding liminf_SUP_INF limsup_INF_SUP using assms by auto
 
-lemma measurable_convergent[measurable (raw)]:
+lemma%unimportant measurable_convergent[measurable (raw)]:
   fixes f :: "nat \<Rightarrow> 'a \<Rightarrow> 'b::{complete_linorder, second_countable_topology, linorder_topology}"
   assumes [measurable]: "\<And>i. f i \<in> borel_measurable M"
   shows "Measurable.pred M (\<lambda>x. convergent (\<lambda>i. f i x))"
   unfolding convergent_ereal by measurable
 
-lemma sets_Collect_convergent[measurable]:
+lemma%unimportant sets_Collect_convergent[measurable]:
   fixes f :: "nat \<Rightarrow> 'a \<Rightarrow> 'b::{complete_linorder, second_countable_topology, linorder_topology}"
   assumes f[measurable]: "\<And>i. f i \<in> borel_measurable M"
   shows "{x\<in>space M. convergent (\<lambda>i. f i x)} \<in> sets M"
   by measurable
 
-lemma borel_measurable_lim[measurable (raw)]:
+lemma%unimportant borel_measurable_lim[measurable (raw)]:
   fixes f :: "nat \<Rightarrow> 'a \<Rightarrow> 'b::{complete_linorder, second_countable_topology, linorder_topology}"
   assumes [measurable]: "\<And>i. f i \<in> borel_measurable M"
   shows "(\<lambda>x. lim (\<lambda>i. f i x)) \<in> borel_measurable M"
@@ -962,7 +962,7 @@
     by simp
 qed
 
-lemma borel_measurable_LIMSEQ_order:
+lemma%unimportant borel_measurable_LIMSEQ_order:
   fixes u :: "nat \<Rightarrow> 'a \<Rightarrow> 'b::{complete_linorder, second_countable_topology, linorder_topology}"
   assumes u': "\<And>x. x \<in> space M \<Longrightarrow> (\<lambda>i. u i x) \<longlonglongrightarrow> u' x"
   and u: "\<And>i. u i \<in> borel_measurable M"
@@ -973,16 +973,16 @@
   with u show ?thesis by (simp cong: measurable_cong)
 qed
 
-subsection \<open>Borel spaces on topological monoids\<close>
+subsection%important \<open>Borel spaces on topological monoids\<close>
 
-lemma borel_measurable_add[measurable (raw)]:
+lemma%unimportant borel_measurable_add[measurable (raw)]:
   fixes f g :: "'a \<Rightarrow> 'b::{second_countable_topology, topological_monoid_add}"
   assumes f: "f \<in> borel_measurable M"
   assumes g: "g \<in> borel_measurable M"
   shows "(\<lambda>x. f x + g x) \<in> borel_measurable M"
   using f g by (rule borel_measurable_continuous_Pair) (intro continuous_intros)
 
-lemma borel_measurable_sum[measurable (raw)]:
+lemma%unimportant borel_measurable_sum[measurable (raw)]:
   fixes f :: "'c \<Rightarrow> 'a \<Rightarrow> 'b::{second_countable_topology, topological_comm_monoid_add}"
   assumes "\<And>i. i \<in> S \<Longrightarrow> f i \<in> borel_measurable M"
   shows "(\<lambda>x. \<Sum>i\<in>S. f i x) \<in> borel_measurable M"
@@ -991,30 +991,30 @@
   thus ?thesis using assms by induct auto
 qed simp
 
-lemma borel_measurable_suminf_order[measurable (raw)]:
+lemma%important borel_measurable_suminf_order[measurable (raw)]:
   fixes f :: "nat \<Rightarrow> 'a \<Rightarrow> 'b::{complete_linorder, second_countable_topology, linorder_topology, topological_comm_monoid_add}"
   assumes f[measurable]: "\<And>i. f i \<in> borel_measurable M"
   shows "(\<lambda>x. suminf (\<lambda>i. f i x)) \<in> borel_measurable M"
   unfolding suminf_def sums_def[abs_def] lim_def[symmetric] by simp
 
-subsection \<open>Borel spaces on Euclidean spaces\<close>
+subsection%important \<open>Borel spaces on Euclidean spaces\<close>
 
-lemma borel_measurable_inner[measurable (raw)]:
+lemma%important borel_measurable_inner[measurable (raw)]:
   fixes f g :: "'a \<Rightarrow> 'b::{second_countable_topology, real_inner}"
   assumes "f \<in> borel_measurable M"
   assumes "g \<in> borel_measurable M"
   shows "(\<lambda>x. f x \<bullet> g x) \<in> borel_measurable M"
   using assms
-  by (rule borel_measurable_continuous_Pair) (intro continuous_intros)
+  by%unimportant (rule borel_measurable_continuous_Pair) (intro continuous_intros)
 
 notation
   eucl_less (infix "<e" 50)
 
-lemma box_oc: "{x. a <e x \<and> x \<le> b} = {x. a <e x} \<inter> {..b}"
+lemma%important box_oc: "{x. a <e x \<and> x \<le> b} = {x. a <e x} \<inter> {..b}"
   and box_co: "{x. a \<le> x \<and> x <e b} = {a..} \<inter> {x. x <e b}"
   by auto
 
-lemma eucl_ivals[measurable]:
+lemma%important eucl_ivals[measurable]:
   fixes a b :: "'a::ordered_euclidean_space"
   shows "{x. x <e a} \<in> sets borel"
     and "{x. a <e x} \<in> sets borel"
@@ -1026,7 +1026,7 @@
   unfolding box_oc box_co
   by (auto intro: borel_open borel_closed)
 
-lemma
+lemma%unimportant (*FIX ME this has no name *)
   fixes i :: "'a::{second_countable_topology, real_inner}"
   shows hafspace_less_borel: "{x. a < x \<bullet> i} \<in> sets borel"
     and hafspace_greater_borel: "{x. x \<bullet> i < a} \<in> sets borel"
@@ -1034,7 +1034,7 @@
     and hafspace_greater_eq_borel: "{x. x \<bullet> i \<le> a} \<in> sets borel"
   by simp_all
 
-lemma borel_eq_box:
+lemma%unimportant borel_eq_box:
   "borel = sigma UNIV (range (\<lambda> (a, b). box a b :: 'a :: euclidean_space set))"
     (is "_ = ?SIGMA")
 proof (rule borel_eq_sigmaI1[OF borel_def])
@@ -1047,7 +1047,7 @@
     done
 qed (auto simp: box_def)
 
-lemma halfspace_gt_in_halfspace:
+lemma%unimportant halfspace_gt_in_halfspace:
   assumes i: "i \<in> A"
   shows "{x::'a. a < x \<bullet> i} \<in>
     sigma_sets UNIV ((\<lambda> (a, i). {x::'a::euclidean_space. x \<bullet> i < a}) ` (UNIV \<times> A))"
@@ -1073,7 +1073,7 @@
     by (auto intro!: Diff sigma_sets_Inter i)
 qed
 
-lemma borel_eq_halfspace_less:
+lemma%unimportant borel_eq_halfspace_less:
   "borel = sigma UNIV ((\<lambda>(a, i). {x::'a::euclidean_space. x \<bullet> i < a}) ` (UNIV \<times> Basis))"
   (is "_ = ?SIGMA")
 proof (rule borel_eq_sigmaI2[OF borel_eq_box])
@@ -1086,7 +1086,7 @@
   finally show "box a b \<in> sets ?SIGMA" .
 qed auto
 
-lemma borel_eq_halfspace_le:
+lemma%unimportant borel_eq_halfspace_le:
   "borel = sigma UNIV ((\<lambda> (a, i). {x::'a::euclidean_space. x \<bullet> i \<le> a}) ` (UNIV \<times> Basis))"
   (is "_ = ?SIGMA")
 proof (rule borel_eq_sigmaI2[OF borel_eq_halfspace_less])
@@ -1110,7 +1110,7 @@
     by (intro sets.countable_UN) (auto intro: i)
 qed auto
 
-lemma borel_eq_halfspace_ge:
+lemma%unimportant borel_eq_halfspace_ge:
   "borel = sigma UNIV ((\<lambda> (a, i). {x::'a::euclidean_space. a \<le> x \<bullet> i}) ` (UNIV \<times> Basis))"
   (is "_ = ?SIGMA")
 proof (rule borel_eq_sigmaI2[OF borel_eq_halfspace_less])
@@ -1120,10 +1120,10 @@
     using i by (intro sets.compl_sets) auto
 qed auto
 
-lemma borel_eq_halfspace_greater:
+lemma%important borel_eq_halfspace_greater:
   "borel = sigma UNIV ((\<lambda> (a, i). {x::'a::euclidean_space. a < x \<bullet> i}) ` (UNIV \<times> Basis))"
   (is "_ = ?SIGMA")
-proof (rule borel_eq_sigmaI2[OF borel_eq_halfspace_le])
+proof%unimportant (rule borel_eq_sigmaI2[OF borel_eq_halfspace_le])
   fix a :: real and i :: 'a assume "(a, i) \<in> (UNIV \<times> Basis)"
   then have i: "i \<in> Basis" by auto
   have *: "{x::'a. x\<bullet>i \<le> a} = space ?SIGMA - {x::'a. a < x\<bullet>i}" by auto
@@ -1131,7 +1131,7 @@
     by (intro sets.compl_sets) (auto intro: i)
 qed auto
 
-lemma borel_eq_atMost:
+lemma%unimportant borel_eq_atMost:
   "borel = sigma UNIV (range (\<lambda>a. {..a::'a::ordered_euclidean_space}))"
   (is "_ = ?SIGMA")
 proof (rule borel_eq_sigmaI4[OF borel_eq_halfspace_le])
@@ -1150,7 +1150,7 @@
     by (intro sets.countable_UN) auto
 qed auto
 
-lemma borel_eq_greaterThan:
+lemma%unimportant borel_eq_greaterThan:
   "borel = sigma UNIV (range (\<lambda>a::'a::ordered_euclidean_space. {x. a <e x}))"
   (is "_ = ?SIGMA")
 proof (rule borel_eq_sigmaI4[OF borel_eq_halfspace_le])
@@ -1177,7 +1177,7 @@
     done
 qed auto
 
-lemma borel_eq_lessThan:
+lemma%unimportant borel_eq_lessThan:
   "borel = sigma UNIV (range (\<lambda>a::'a::ordered_euclidean_space. {x. x <e a}))"
   (is "_ = ?SIGMA")
 proof (rule borel_eq_sigmaI4[OF borel_eq_halfspace_ge])
@@ -1203,10 +1203,10 @@
     done
 qed auto
 
-lemma borel_eq_atLeastAtMost:
+lemma%important borel_eq_atLeastAtMost:
   "borel = sigma UNIV (range (\<lambda>(a,b). {a..b} ::'a::ordered_euclidean_space set))"
   (is "_ = ?SIGMA")
-proof (rule borel_eq_sigmaI5[OF borel_eq_atMost])
+proof%unimportant (rule borel_eq_sigmaI5[OF borel_eq_atMost])
   fix a::'a
   have *: "{..a} = (\<Union>n::nat. {- real n *\<^sub>R One .. a})"
   proof (safe, simp_all add: eucl_le[where 'a='a])
@@ -1225,12 +1225,12 @@
        (auto intro!: sigma_sets_top)
 qed auto
 
-lemma borel_set_induct[consumes 1, case_names empty interval compl union]:
+lemma%important borel_set_induct[consumes 1, case_names empty interval compl union]:
   assumes "A \<in> sets borel"
   assumes empty: "P {}" and int: "\<And>a b. a \<le> b \<Longrightarrow> P {a..b}" and compl: "\<And>A. A \<in> sets borel \<Longrightarrow> P A \<Longrightarrow> P (-A)" and
           un: "\<And>f. disjoint_family f \<Longrightarrow> (\<And>i. f i \<in> sets borel) \<Longrightarrow>  (\<And>i. P (f i)) \<Longrightarrow> P (\<Union>i::nat. f i)"
   shows "P (A::real set)"
-proof-
+proof%unimportant -
   let ?G = "range (\<lambda>(a,b). {a..b::real})"
   have "Int_stable ?G" "?G \<subseteq> Pow UNIV" "A \<in> sigma_sets UNIV ?G"
       using assms(1) by (auto simp add: borel_eq_atLeastAtMost Int_stable_def)
@@ -1247,7 +1247,7 @@
   qed (auto intro: empty compl simp: Compl_eq_Diff_UNIV[symmetric] borel_eq_atLeastAtMost)
 qed
 
-lemma borel_sigma_sets_Ioc: "borel = sigma UNIV (range (\<lambda>(a, b). {a <.. b::real}))"
+lemma%unimportant borel_sigma_sets_Ioc: "borel = sigma UNIV (range (\<lambda>(a, b). {a <.. b::real}))"
 proof (rule borel_eq_sigmaI5[OF borel_eq_atMost])
   fix i :: real
   have "{..i} = (\<Union>j::nat. {-j <.. i})"
@@ -1257,10 +1257,10 @@
   finally show "{..i} \<in> sets (sigma UNIV (range (\<lambda>(i, j). {i<..j})))" .
 qed simp
 
-lemma eucl_lessThan: "{x::real. x <e a} = lessThan a"
+lemma%unimportant eucl_lessThan: "{x::real. x <e a} = lessThan a"
   by (simp add: eucl_less_def lessThan_def)
 
-lemma borel_eq_atLeastLessThan:
+lemma%unimportant borel_eq_atLeastLessThan:
   "borel = sigma UNIV (range (\<lambda>(a, b). {a ..< b :: real}))" (is "_ = ?SIGMA")
 proof (rule borel_eq_sigmaI5[OF borel_eq_lessThan])
   have move_uminus: "\<And>x y::real. -x \<le> y \<longleftrightarrow> -y \<le> x" by auto
@@ -1271,7 +1271,7 @@
     by (auto intro: sigma_sets.intros(2-) simp: eucl_lessThan)
 qed auto
 
-lemma borel_measurable_halfspacesI:
+lemma%unimportant borel_measurable_halfspacesI:
   fixes f :: "'a \<Rightarrow> 'c::euclidean_space"
   assumes F: "borel = sigma UNIV (F ` (UNIV \<times> Basis))"
   and S_eq: "\<And>a i. S a i = f -` F (a,i) \<inter> space M"
@@ -1286,112 +1286,112 @@
     by (auto intro!: measurable_measure_of simp: S_eq F)
 qed
 
-lemma borel_measurable_iff_halfspace_le:
+lemma%unimportant borel_measurable_iff_halfspace_le:
   fixes f :: "'a \<Rightarrow> 'c::euclidean_space"
   shows "f \<in> borel_measurable M = (\<forall>i\<in>Basis. \<forall>a. {w \<in> space M. f w \<bullet> i \<le> a} \<in> sets M)"
   by (rule borel_measurable_halfspacesI[OF borel_eq_halfspace_le]) auto
 
-lemma borel_measurable_iff_halfspace_less:
+lemma%unimportant borel_measurable_iff_halfspace_less:
   fixes f :: "'a \<Rightarrow> 'c::euclidean_space"
   shows "f \<in> borel_measurable M \<longleftrightarrow> (\<forall>i\<in>Basis. \<forall>a. {w \<in> space M. f w \<bullet> i < a} \<in> sets M)"
   by (rule borel_measurable_halfspacesI[OF borel_eq_halfspace_less]) auto
 
-lemma borel_measurable_iff_halfspace_ge:
+lemma%unimportant borel_measurable_iff_halfspace_ge:
   fixes f :: "'a \<Rightarrow> 'c::euclidean_space"
   shows "f \<in> borel_measurable M = (\<forall>i\<in>Basis. \<forall>a. {w \<in> space M. a \<le> f w \<bullet> i} \<in> sets M)"
   by (rule borel_measurable_halfspacesI[OF borel_eq_halfspace_ge]) auto
 
-lemma borel_measurable_iff_halfspace_greater:
+lemma%unimportant borel_measurable_iff_halfspace_greater:
   fixes f :: "'a \<Rightarrow> 'c::euclidean_space"
   shows "f \<in> borel_measurable M \<longleftrightarrow> (\<forall>i\<in>Basis. \<forall>a. {w \<in> space M. a < f w \<bullet> i} \<in> sets M)"
   by (rule borel_measurable_halfspacesI[OF borel_eq_halfspace_greater]) auto
 
-lemma borel_measurable_iff_le:
+lemma%unimportant borel_measurable_iff_le:
   "(f::'a \<Rightarrow> real) \<in> borel_measurable M = (\<forall>a. {w \<in> space M. f w \<le> a} \<in> sets M)"
   using borel_measurable_iff_halfspace_le[where 'c=real] by simp
 
-lemma borel_measurable_iff_less:
+lemma%unimportant borel_measurable_iff_less:
   "(f::'a \<Rightarrow> real) \<in> borel_measurable M = (\<forall>a. {w \<in> space M. f w < a} \<in> sets M)"
   using borel_measurable_iff_halfspace_less[where 'c=real] by simp
 
-lemma borel_measurable_iff_ge:
+lemma%unimportant borel_measurable_iff_ge:
   "(f::'a \<Rightarrow> real) \<in> borel_measurable M = (\<forall>a. {w \<in> space M. a \<le> f w} \<in> sets M)"
   using borel_measurable_iff_halfspace_ge[where 'c=real]
   by simp
 
-lemma borel_measurable_iff_greater:
+lemma%unimportant borel_measurable_iff_greater:
   "(f::'a \<Rightarrow> real) \<in> borel_measurable M = (\<forall>a. {w \<in> space M. a < f w} \<in> sets M)"
   using borel_measurable_iff_halfspace_greater[where 'c=real] by simp
 
-lemma borel_measurable_euclidean_space:
+lemma%important borel_measurable_euclidean_space:
   fixes f :: "'a \<Rightarrow> 'c::euclidean_space"
   shows "f \<in> borel_measurable M \<longleftrightarrow> (\<forall>i\<in>Basis. (\<lambda>x. f x \<bullet> i) \<in> borel_measurable M)"
-proof safe
+proof%unimportant safe
   assume f: "\<forall>i\<in>Basis. (\<lambda>x. f x \<bullet> i) \<in> borel_measurable M"
   then show "f \<in> borel_measurable M"
     by (subst borel_measurable_iff_halfspace_le) auto
 qed auto
 
-subsection "Borel measurable operators"
+subsection%important "Borel measurable operators"
 
-lemma borel_measurable_norm[measurable]: "norm \<in> borel_measurable borel"
-  by (intro borel_measurable_continuous_on1 continuous_intros)
+lemma%important borel_measurable_norm[measurable]: "norm \<in> borel_measurable borel"
+  by%unimportant (intro borel_measurable_continuous_on1 continuous_intros)
 
-lemma borel_measurable_sgn [measurable]: "(sgn::'a::real_normed_vector \<Rightarrow> 'a) \<in> borel_measurable borel"
+lemma%unimportant borel_measurable_sgn [measurable]: "(sgn::'a::real_normed_vector \<Rightarrow> 'a) \<in> borel_measurable borel"
   by (rule borel_measurable_continuous_countable_exceptions[where X="{0}"])
      (auto intro!: continuous_on_sgn continuous_on_id)
 
-lemma borel_measurable_uminus[measurable (raw)]:
+lemma%important borel_measurable_uminus[measurable (raw)]:
   fixes g :: "'a \<Rightarrow> 'b::{second_countable_topology, real_normed_vector}"
   assumes g: "g \<in> borel_measurable M"
   shows "(\<lambda>x. - g x) \<in> borel_measurable M"
-  by (rule borel_measurable_continuous_on[OF _ g]) (intro continuous_intros)
+  by%unimportant (rule borel_measurable_continuous_on[OF _ g]) (intro continuous_intros)
 
-lemma borel_measurable_diff[measurable (raw)]:
+lemma%important borel_measurable_diff[measurable (raw)]:
   fixes f :: "'a \<Rightarrow> 'b::{second_countable_topology, real_normed_vector}"
   assumes f: "f \<in> borel_measurable M"
   assumes g: "g \<in> borel_measurable M"
   shows "(\<lambda>x. f x - g x) \<in> borel_measurable M"
-  using borel_measurable_add [of f M "- g"] assms by (simp add: fun_Compl_def)
+  using%unimportant borel_measurable_add [of f M "- g"] assms by (simp add: fun_Compl_def)
 
-lemma borel_measurable_times[measurable (raw)]:
+lemma%unimportant borel_measurable_times[measurable (raw)]:
   fixes f :: "'a \<Rightarrow> 'b::{second_countable_topology, real_normed_algebra}"
   assumes f: "f \<in> borel_measurable M"
   assumes g: "g \<in> borel_measurable M"
   shows "(\<lambda>x. f x * g x) \<in> borel_measurable M"
   using f g by (rule borel_measurable_continuous_Pair) (intro continuous_intros)
 
-lemma borel_measurable_prod[measurable (raw)]:
+lemma%important borel_measurable_prod[measurable (raw)]:
   fixes f :: "'c \<Rightarrow> 'a \<Rightarrow> 'b::{second_countable_topology, real_normed_field}"
   assumes "\<And>i. i \<in> S \<Longrightarrow> f i \<in> borel_measurable M"
   shows "(\<lambda>x. \<Prod>i\<in>S. f i x) \<in> borel_measurable M"
-proof cases
+proof%unimportant cases
   assume "finite S"
   thus ?thesis using assms by induct auto
 qed simp
 
-lemma borel_measurable_dist[measurable (raw)]:
+lemma%important borel_measurable_dist[measurable (raw)]:
   fixes g f :: "'a \<Rightarrow> 'b::{second_countable_topology, metric_space}"
   assumes f: "f \<in> borel_measurable M"
   assumes g: "g \<in> borel_measurable M"
   shows "(\<lambda>x. dist (f x) (g x)) \<in> borel_measurable M"
-  using f g by (rule borel_measurable_continuous_Pair) (intro continuous_intros)
+  using%unimportant f g by (rule borel_measurable_continuous_Pair) (intro continuous_intros)
 
-lemma borel_measurable_scaleR[measurable (raw)]:
+lemma%unimportant borel_measurable_scaleR[measurable (raw)]:
   fixes g :: "'a \<Rightarrow> 'b::{second_countable_topology, real_normed_vector}"
   assumes f: "f \<in> borel_measurable M"
   assumes g: "g \<in> borel_measurable M"
   shows "(\<lambda>x. f x *\<^sub>R g x) \<in> borel_measurable M"
   using f g by (rule borel_measurable_continuous_Pair) (intro continuous_intros)
 
-lemma borel_measurable_uminus_eq [simp]:
+lemma%unimportant borel_measurable_uminus_eq [simp]:
   fixes f :: "'a \<Rightarrow> 'b::{second_countable_topology, real_normed_vector}"
   shows "(\<lambda>x. - f x) \<in> borel_measurable M \<longleftrightarrow> f \<in> borel_measurable M" (is "?l = ?r")
 proof
   assume ?l from borel_measurable_uminus[OF this] show ?r by simp
 qed auto
 
-lemma affine_borel_measurable_vector:
+lemma%unimportant affine_borel_measurable_vector:
   fixes f :: "'a \<Rightarrow> 'x::real_normed_vector"
   assumes "f \<in> borel_measurable M"
   shows "(\<lambda>x. a + b *\<^sub>R f x) \<in> borel_measurable M"
@@ -1412,15 +1412,15 @@
   qed simp
 qed
 
-lemma borel_measurable_const_scaleR[measurable (raw)]:
+lemma%unimportant borel_measurable_const_scaleR[measurable (raw)]:
   "f \<in> borel_measurable M \<Longrightarrow> (\<lambda>x. b *\<^sub>R f x ::'a::real_normed_vector) \<in> borel_measurable M"
   using affine_borel_measurable_vector[of f M 0 b] by simp
 
-lemma borel_measurable_const_add[measurable (raw)]:
+lemma%unimportant borel_measurable_const_add[measurable (raw)]:
   "f \<in> borel_measurable M \<Longrightarrow> (\<lambda>x. a + f x ::'a::real_normed_vector) \<in> borel_measurable M"
   using affine_borel_measurable_vector[of f M a 1] by simp
 
-lemma borel_measurable_inverse[measurable (raw)]:
+lemma%unimportant borel_measurable_inverse[measurable (raw)]:
   fixes f :: "'a \<Rightarrow> 'b::real_normed_div_algebra"
   assumes f: "f \<in> borel_measurable M"
   shows "(\<lambda>x. inverse (f x)) \<in> borel_measurable M"
@@ -1429,27 +1429,27 @@
   apply (auto intro!: continuous_on_inverse continuous_on_id)
   done
 
-lemma borel_measurable_divide[measurable (raw)]:
+lemma%unimportant borel_measurable_divide[measurable (raw)]:
   "f \<in> borel_measurable M \<Longrightarrow> g \<in> borel_measurable M \<Longrightarrow>
     (\<lambda>x. f x / g x::'b::{second_countable_topology, real_normed_div_algebra}) \<in> borel_measurable M"
   by (simp add: divide_inverse)
 
-lemma borel_measurable_abs[measurable (raw)]:
+lemma%unimportant borel_measurable_abs[measurable (raw)]:
   "f \<in> borel_measurable M \<Longrightarrow> (\<lambda>x. \<bar>f x :: real\<bar>) \<in> borel_measurable M"
   unfolding abs_real_def by simp
 
-lemma borel_measurable_nth[measurable (raw)]:
+lemma%unimportant borel_measurable_nth[measurable (raw)]:
   "(\<lambda>x::real^'n. x $ i) \<in> borel_measurable borel"
   by (simp add: cart_eq_inner_axis)
 
-lemma convex_measurable:
+lemma%important convex_measurable:
   fixes A :: "'a :: euclidean_space set"
   shows "X \<in> borel_measurable M \<Longrightarrow> X ` space M \<subseteq> A \<Longrightarrow> open A \<Longrightarrow> convex_on A q \<Longrightarrow>
     (\<lambda>x. q (X x)) \<in> borel_measurable M"
-  by (rule measurable_compose[where f=X and N="restrict_space borel A"])
+  by%unimportant (rule measurable_compose[where f=X and N="restrict_space borel A"])
      (auto intro!: borel_measurable_continuous_on_restrict convex_on_continuous measurable_restrict_space2)
 
-lemma borel_measurable_ln[measurable (raw)]:
+lemma%unimportant borel_measurable_ln[measurable (raw)]:
   assumes f: "f \<in> borel_measurable M"
   shows "(\<lambda>x. ln (f x :: real)) \<in> borel_measurable M"
   apply (rule measurable_compose[OF f])
@@ -1457,15 +1457,15 @@
   apply (auto intro!: continuous_on_ln continuous_on_id)
   done
 
-lemma borel_measurable_log[measurable (raw)]:
+lemma%unimportant borel_measurable_log[measurable (raw)]:
   "f \<in> borel_measurable M \<Longrightarrow> g \<in> borel_measurable M \<Longrightarrow> (\<lambda>x. log (g x) (f x)) \<in> borel_measurable M"
   unfolding log_def by auto
 
-lemma borel_measurable_exp[measurable]:
+lemma%unimportant borel_measurable_exp[measurable]:
   "(exp::'a::{real_normed_field,banach}\<Rightarrow>'a) \<in> borel_measurable borel"
   by (intro borel_measurable_continuous_on1 continuous_at_imp_continuous_on ballI isCont_exp)
 
-lemma measurable_real_floor[measurable]:
+lemma%unimportant measurable_real_floor[measurable]:
   "(floor :: real \<Rightarrow> int) \<in> measurable borel (count_space UNIV)"
 proof -
   have "\<And>a x. \<lfloor>x\<rfloor> = a \<longleftrightarrow> (real_of_int a \<le> x \<and> x < real_of_int (a + 1))"
@@ -1474,44 +1474,44 @@
     by (auto simp: vimage_def measurable_count_space_eq2_countable)
 qed
 
-lemma measurable_real_ceiling[measurable]:
+lemma%unimportant measurable_real_ceiling[measurable]:
   "(ceiling :: real \<Rightarrow> int) \<in> measurable borel (count_space UNIV)"
   unfolding ceiling_def[abs_def] by simp
 
-lemma borel_measurable_real_floor: "(\<lambda>x::real. real_of_int \<lfloor>x\<rfloor>) \<in> borel_measurable borel"
+lemma%unimportant borel_measurable_real_floor: "(\<lambda>x::real. real_of_int \<lfloor>x\<rfloor>) \<in> borel_measurable borel"
   by simp
 
-lemma borel_measurable_root [measurable]: "root n \<in> borel_measurable borel"
+lemma%unimportant borel_measurable_root [measurable]: "root n \<in> borel_measurable borel"
   by (intro borel_measurable_continuous_on1 continuous_intros)
 
-lemma borel_measurable_sqrt [measurable]: "sqrt \<in> borel_measurable borel"
+lemma%unimportant borel_measurable_sqrt [measurable]: "sqrt \<in> borel_measurable borel"
   by (intro borel_measurable_continuous_on1 continuous_intros)
 
-lemma borel_measurable_power [measurable (raw)]:
+lemma%unimportant borel_measurable_power [measurable (raw)]:
   fixes f :: "_ \<Rightarrow> 'b::{power,real_normed_algebra}"
   assumes f: "f \<in> borel_measurable M"
   shows "(\<lambda>x. (f x) ^ n) \<in> borel_measurable M"
   by (intro borel_measurable_continuous_on [OF _ f] continuous_intros)
 
-lemma borel_measurable_Re [measurable]: "Re \<in> borel_measurable borel"
+lemma%unimportant borel_measurable_Re [measurable]: "Re \<in> borel_measurable borel"
   by (intro borel_measurable_continuous_on1 continuous_intros)
 
-lemma borel_measurable_Im [measurable]: "Im \<in> borel_measurable borel"
+lemma%unimportant borel_measurable_Im [measurable]: "Im \<in> borel_measurable borel"
   by (intro borel_measurable_continuous_on1 continuous_intros)
 
-lemma borel_measurable_of_real [measurable]: "(of_real :: _ \<Rightarrow> (_::real_normed_algebra)) \<in> borel_measurable borel"
+lemma%unimportant borel_measurable_of_real [measurable]: "(of_real :: _ \<Rightarrow> (_::real_normed_algebra)) \<in> borel_measurable borel"
   by (intro borel_measurable_continuous_on1 continuous_intros)
 
-lemma borel_measurable_sin [measurable]: "(sin :: _ \<Rightarrow> (_::{real_normed_field,banach})) \<in> borel_measurable borel"
+lemma%unimportant borel_measurable_sin [measurable]: "(sin :: _ \<Rightarrow> (_::{real_normed_field,banach})) \<in> borel_measurable borel"
   by (intro borel_measurable_continuous_on1 continuous_intros)
 
-lemma borel_measurable_cos [measurable]: "(cos :: _ \<Rightarrow> (_::{real_normed_field,banach})) \<in> borel_measurable borel"
+lemma%unimportant borel_measurable_cos [measurable]: "(cos :: _ \<Rightarrow> (_::{real_normed_field,banach})) \<in> borel_measurable borel"
   by (intro borel_measurable_continuous_on1 continuous_intros)
 
-lemma borel_measurable_arctan [measurable]: "arctan \<in> borel_measurable borel"
+lemma%unimportant borel_measurable_arctan [measurable]: "arctan \<in> borel_measurable borel"
   by (intro borel_measurable_continuous_on1 continuous_intros)
 
-lemma borel_measurable_complex_iff:
+lemma%important borel_measurable_complex_iff:
   "f \<in> borel_measurable M \<longleftrightarrow>
     (\<lambda>x. Re (f x)) \<in> borel_measurable M \<and> (\<lambda>x. Im (f x)) \<in> borel_measurable M"
   apply auto
@@ -1520,21 +1520,21 @@
   apply auto
   done
 
-lemma powr_real_measurable [measurable]:
+lemma%important powr_real_measurable [measurable]:
   assumes "f \<in> measurable M borel" "g \<in> measurable M borel"
   shows   "(\<lambda>x. f x powr g x :: real) \<in> measurable M borel"
-  using assms by (simp_all add: powr_def)
+  using%unimportant assms by (simp_all add: powr_def)
 
-lemma measurable_of_bool[measurable]: "of_bool \<in> count_space UNIV \<rightarrow>\<^sub>M borel"
+lemma%unimportant measurable_of_bool[measurable]: "of_bool \<in> count_space UNIV \<rightarrow>\<^sub>M borel"
   by simp
 
-subsection "Borel space on the extended reals"
+subsection%important "Borel space on the extended reals"
 
-lemma borel_measurable_ereal[measurable (raw)]:
+lemma%unimportant borel_measurable_ereal[measurable (raw)]:
   assumes f: "f \<in> borel_measurable M" shows "(\<lambda>x. ereal (f x)) \<in> borel_measurable M"
   using continuous_on_ereal f by (rule borel_measurable_continuous_on) (rule continuous_on_id)
 
-lemma borel_measurable_real_of_ereal[measurable (raw)]:
+lemma%unimportant borel_measurable_real_of_ereal[measurable (raw)]:
   fixes f :: "'a \<Rightarrow> ereal"
   assumes f: "f \<in> borel_measurable M"
   shows "(\<lambda>x. real_of_ereal (f x)) \<in> borel_measurable M"
@@ -1543,7 +1543,7 @@
   apply (auto intro: continuous_on_real simp: Compl_eq_Diff_UNIV)
   done
 
-lemma borel_measurable_ereal_cases:
+lemma%unimportant borel_measurable_ereal_cases:
   fixes f :: "'a \<Rightarrow> ereal"
   assumes f: "f \<in> borel_measurable M"
   assumes H: "(\<lambda>x. H (ereal (real_of_ereal (f x)))) \<in> borel_measurable M"
@@ -1554,20 +1554,20 @@
   with f H show ?thesis by simp
 qed
 
-lemma
+lemma%unimportant (*FIX ME needs a name *)
   fixes f :: "'a \<Rightarrow> ereal" assumes f[measurable]: "f \<in> borel_measurable M"
   shows borel_measurable_ereal_abs[measurable(raw)]: "(\<lambda>x. \<bar>f x\<bar>) \<in> borel_measurable M"
     and borel_measurable_ereal_inverse[measurable(raw)]: "(\<lambda>x. inverse (f x) :: ereal) \<in> borel_measurable M"
     and borel_measurable_uminus_ereal[measurable(raw)]: "(\<lambda>x. - f x :: ereal) \<in> borel_measurable M"
   by (auto simp del: abs_real_of_ereal simp: borel_measurable_ereal_cases[OF f] measurable_If)
 
-lemma borel_measurable_uminus_eq_ereal[simp]:
+lemma%unimportant borel_measurable_uminus_eq_ereal[simp]:
   "(\<lambda>x. - f x :: ereal) \<in> borel_measurable M \<longleftrightarrow> f \<in> borel_measurable M" (is "?l = ?r")
 proof
   assume ?l from borel_measurable_uminus_ereal[OF this] show ?r by simp
 qed auto
 
-lemma set_Collect_ereal2:
+lemma%important set_Collect_ereal2:
   fixes f g :: "'a \<Rightarrow> ereal"
   assumes f: "f \<in> borel_measurable M"
   assumes g: "g \<in> borel_measurable M"
@@ -1577,7 +1577,7 @@
     "{x \<in> space borel. H (ereal x) (-\<infinity>)} \<in> sets borel"
     "{x \<in> space borel. H (ereal x) (\<infinity>)} \<in> sets borel"
   shows "{x \<in> space M. H (f x) (g x)} \<in> sets M"
-proof -
+proof%unimportant -
   let ?G = "\<lambda>y x. if g x = \<infinity> then H y \<infinity> else if g x = -\<infinity> then H y (-\<infinity>) else H y (ereal (real_of_ereal (g x)))"
   let ?F = "\<lambda>x. if f x = \<infinity> then ?G \<infinity> x else if f x = -\<infinity> then ?G (-\<infinity>) x else ?G (ereal (real_of_ereal (f x))) x"
   { fix x have "H (f x) (g x) = ?F x" by (cases "f x" "g x" rule: ereal2_cases) auto }
@@ -1586,7 +1586,7 @@
     by (subst *) (simp del: space_borel split del: if_split)
 qed
 
-lemma borel_measurable_ereal_iff:
+lemma%unimportant borel_measurable_ereal_iff:
   shows "(\<lambda>x. ereal (f x)) \<in> borel_measurable M \<longleftrightarrow> f \<in> borel_measurable M"
 proof
   assume "(\<lambda>x. ereal (f x)) \<in> borel_measurable M"
@@ -1594,15 +1594,15 @@
   show "f \<in> borel_measurable M" by auto
 qed auto
 
-lemma borel_measurable_erealD[measurable_dest]:
+lemma%unimportant borel_measurable_erealD[measurable_dest]:
   "(\<lambda>x. ereal (f x)) \<in> borel_measurable M \<Longrightarrow> g \<in> measurable N M \<Longrightarrow> (\<lambda>x. f (g x)) \<in> borel_measurable N"
   unfolding borel_measurable_ereal_iff by simp
 
-lemma borel_measurable_ereal_iff_real:
+lemma%important borel_measurable_ereal_iff_real:
   fixes f :: "'a \<Rightarrow> ereal"
   shows "f \<in> borel_measurable M \<longleftrightarrow>
     ((\<lambda>x. real_of_ereal (f x)) \<in> borel_measurable M \<and> f -` {\<infinity>} \<inter> space M \<in> sets M \<and> f -` {-\<infinity>} \<inter> space M \<in> sets M)"
-proof safe
+proof%unimportant safe
   assume *: "(\<lambda>x. real_of_ereal (f x)) \<in> borel_measurable M" "f -` {\<infinity>} \<inter> space M \<in> sets M" "f -` {-\<infinity>} \<inter> space M \<in> sets M"
   have "f -` {\<infinity>} \<inter> space M = {x\<in>space M. f x = \<infinity>}" "f -` {-\<infinity>} \<inter> space M = {x\<in>space M. f x = -\<infinity>}" by auto
   with * have **: "{x\<in>space M. f x = \<infinity>} \<in> sets M" "{x\<in>space M. f x = -\<infinity>} \<in> sets M" by simp_all
@@ -1612,15 +1612,15 @@
   finally show "f \<in> borel_measurable M" .
 qed simp_all
 
-lemma borel_measurable_ereal_iff_Iio:
+lemma%unimportant borel_measurable_ereal_iff_Iio:
   "(f::'a \<Rightarrow> ereal) \<in> borel_measurable M \<longleftrightarrow> (\<forall>a. f -` {..< a} \<inter> space M \<in> sets M)"
   by (auto simp: borel_Iio measurable_iff_measure_of)
 
-lemma borel_measurable_ereal_iff_Ioi:
+lemma%unimportant borel_measurable_ereal_iff_Ioi:
   "(f::'a \<Rightarrow> ereal) \<in> borel_measurable M \<longleftrightarrow> (\<forall>a. f -` {a <..} \<inter> space M \<in> sets M)"
   by (auto simp: borel_Ioi measurable_iff_measure_of)
 
-lemma vimage_sets_compl_iff:
+lemma%unimportant vimage_sets_compl_iff:
   "f -` A \<inter> space M \<in> sets M \<longleftrightarrow> f -` (- A) \<inter> space M \<in> sets M"
 proof -
   { fix A assume "f -` A \<inter> space M \<in> sets M"
@@ -1630,15 +1630,15 @@
     by (metis double_complement)
 qed
 
-lemma borel_measurable_iff_Iic_ereal:
+lemma%unimportant borel_measurable_iff_Iic_ereal:
   "(f::'a\<Rightarrow>ereal) \<in> borel_measurable M \<longleftrightarrow> (\<forall>a. f -` {..a} \<inter> space M \<in> sets M)"
   unfolding borel_measurable_ereal_iff_Ioi vimage_sets_compl_iff[where A="{a <..}" for a] by simp
 
-lemma borel_measurable_iff_Ici_ereal:
+lemma%unimportant borel_measurable_iff_Ici_ereal:
   "(f::'a \<Rightarrow> ereal) \<in> borel_measurable M \<longleftrightarrow> (\<forall>a. f -` {a..} \<inter> space M \<in> sets M)"
   unfolding borel_measurable_ereal_iff_Iio vimage_sets_compl_iff[where A="{..< a}" for a] by simp
 
-lemma borel_measurable_ereal2:
+lemma%important borel_measurable_ereal2:
   fixes f g :: "'a \<Rightarrow> ereal"
   assumes f: "f \<in> borel_measurable M"
   assumes g: "g \<in> borel_measurable M"
@@ -1648,7 +1648,7 @@
     "(\<lambda>x. H (ereal (real_of_ereal (f x))) (-\<infinity>)) \<in> borel_measurable M"
     "(\<lambda>x. H (ereal (real_of_ereal (f x))) (\<infinity>)) \<in> borel_measurable M"
   shows "(\<lambda>x. H (f x) (g x)) \<in> borel_measurable M"
-proof -
+proof%unimportant -
   let ?G = "\<lambda>y x. if g x = \<infinity> then H y \<infinity> else if g x = - \<infinity> then H y (-\<infinity>) else H y (ereal (real_of_ereal (g x)))"
   let ?F = "\<lambda>x. if f x = \<infinity> then ?G \<infinity> x else if f x = - \<infinity> then ?G (-\<infinity>) x else ?G (ereal (real_of_ereal (f x))) x"
   { fix x have "H (f x) (g x) = ?F x" by (cases "f x" "g x" rule: ereal2_cases) auto }
@@ -1656,14 +1656,14 @@
   from assms show ?thesis unfolding * by simp
 qed
 
-lemma [measurable(raw)]:
+lemma%unimportant [measurable(raw)]:
   fixes f :: "'a \<Rightarrow> ereal"
   assumes [measurable]: "f \<in> borel_measurable M" "g \<in> borel_measurable M"
   shows borel_measurable_ereal_add: "(\<lambda>x. f x + g x) \<in> borel_measurable M"
     and borel_measurable_ereal_times: "(\<lambda>x. f x * g x) \<in> borel_measurable M"
   by (simp_all add: borel_measurable_ereal2)
 
-lemma [measurable(raw)]:
+lemma%unimportant [measurable(raw)]:
   fixes f g :: "'a \<Rightarrow> ereal"
   assumes "f \<in> borel_measurable M"
   assumes "g \<in> borel_measurable M"
@@ -1671,42 +1671,42 @@
     and borel_measurable_ereal_divide: "(\<lambda>x. f x / g x) \<in> borel_measurable M"
   using assms by (simp_all add: minus_ereal_def divide_ereal_def)
 
-lemma borel_measurable_ereal_sum[measurable (raw)]:
+lemma%unimportant borel_measurable_ereal_sum[measurable (raw)]:
   fixes f :: "'c \<Rightarrow> 'a \<Rightarrow> ereal"
   assumes "\<And>i. i \<in> S \<Longrightarrow> f i \<in> borel_measurable M"
   shows "(\<lambda>x. \<Sum>i\<in>S. f i x) \<in> borel_measurable M"
   using assms by (induction S rule: infinite_finite_induct) auto
 
-lemma borel_measurable_ereal_prod[measurable (raw)]:
+lemma%unimportant borel_measurable_ereal_prod[measurable (raw)]:
   fixes f :: "'c \<Rightarrow> 'a \<Rightarrow> ereal"
   assumes "\<And>i. i \<in> S \<Longrightarrow> f i \<in> borel_measurable M"
   shows "(\<lambda>x. \<Prod>i\<in>S. f i x) \<in> borel_measurable M"
   using assms by (induction S rule: infinite_finite_induct) auto
 
-lemma borel_measurable_extreal_suminf[measurable (raw)]:
+lemma%unimportant borel_measurable_extreal_suminf[measurable (raw)]:
   fixes f :: "nat \<Rightarrow> 'a \<Rightarrow> ereal"
   assumes [measurable]: "\<And>i. f i \<in> borel_measurable M"
   shows "(\<lambda>x. (\<Sum>i. f i x)) \<in> borel_measurable M"
   unfolding suminf_def sums_def[abs_def] lim_def[symmetric] by simp
 
-subsection "Borel space on the extended non-negative reals"
+subsection%important "Borel space on the extended non-negative reals"
 
 text \<open> @{type ennreal} is a topological monoid, so no rules for plus are required, also all order
   statements are usually done on type classes. \<close>
 
-lemma measurable_enn2ereal[measurable]: "enn2ereal \<in> borel \<rightarrow>\<^sub>M borel"
+lemma%unimportant measurable_enn2ereal[measurable]: "enn2ereal \<in> borel \<rightarrow>\<^sub>M borel"
   by (intro borel_measurable_continuous_on1 continuous_on_enn2ereal)
 
-lemma measurable_e2ennreal[measurable]: "e2ennreal \<in> borel \<rightarrow>\<^sub>M borel"
+lemma%unimportant measurable_e2ennreal[measurable]: "e2ennreal \<in> borel \<rightarrow>\<^sub>M borel"
   by (intro borel_measurable_continuous_on1 continuous_on_e2ennreal)
 
-lemma borel_measurable_enn2real[measurable (raw)]:
+lemma%unimportant borel_measurable_enn2real[measurable (raw)]:
   "f \<in> M \<rightarrow>\<^sub>M borel \<Longrightarrow> (\<lambda>x. enn2real (f x)) \<in> M \<rightarrow>\<^sub>M borel"
   unfolding enn2real_def[abs_def] by measurable
 
-definition [simp]: "is_borel f M \<longleftrightarrow> f \<in> borel_measurable M"
+definition%important [simp]: "is_borel f M \<longleftrightarrow> f \<in> borel_measurable M"
 
-lemma is_borel_transfer[transfer_rule]: "rel_fun (rel_fun (=) pcr_ennreal) (=) is_borel is_borel"
+lemma%unimportant is_borel_transfer[transfer_rule]: "rel_fun (rel_fun (=) pcr_ennreal) (=) is_borel is_borel"
   unfolding is_borel_def[abs_def]
 proof (safe intro!: rel_funI ext dest!: rel_fun_eq_pcr_ennreal[THEN iffD1])
   fix f and M :: "'a measure"
@@ -1718,14 +1718,14 @@
   includes ennreal.lifting
 begin
 
-lemma measurable_ennreal[measurable]: "ennreal \<in> borel \<rightarrow>\<^sub>M borel"
+lemma%unimportant measurable_ennreal[measurable]: "ennreal \<in> borel \<rightarrow>\<^sub>M borel"
   unfolding is_borel_def[symmetric]
   by transfer simp
 
-lemma borel_measurable_ennreal_iff[simp]:
+lemma%important borel_measurable_ennreal_iff[simp]:
   assumes [simp]: "\<And>x. x \<in> space M \<Longrightarrow> 0 \<le> f x"
   shows "(\<lambda>x. ennreal (f x)) \<in> M \<rightarrow>\<^sub>M borel \<longleftrightarrow> f \<in> M \<rightarrow>\<^sub>M borel"
-proof safe
+proof%unimportant safe
   assume "(\<lambda>x. ennreal (f x)) \<in> M \<rightarrow>\<^sub>M borel"
   then have "(\<lambda>x. enn2real (ennreal (f x))) \<in> M \<rightarrow>\<^sub>M borel"
     by measurable
@@ -1733,44 +1733,44 @@
     by (rule measurable_cong[THEN iffD1, rotated]) auto
 qed measurable
 
-lemma borel_measurable_times_ennreal[measurable (raw)]:
+lemma%unimportant borel_measurable_times_ennreal[measurable (raw)]:
   fixes f g :: "'a \<Rightarrow> ennreal"
   shows "f \<in> M \<rightarrow>\<^sub>M borel \<Longrightarrow> g \<in> M \<rightarrow>\<^sub>M borel \<Longrightarrow> (\<lambda>x. f x * g x) \<in> M \<rightarrow>\<^sub>M borel"
   unfolding is_borel_def[symmetric] by transfer simp
 
-lemma borel_measurable_inverse_ennreal[measurable (raw)]:
+lemma%unimportant borel_measurable_inverse_ennreal[measurable (raw)]:
   fixes f :: "'a \<Rightarrow> ennreal"
   shows "f \<in> M \<rightarrow>\<^sub>M borel \<Longrightarrow> (\<lambda>x. inverse (f x)) \<in> M \<rightarrow>\<^sub>M borel"
   unfolding is_borel_def[symmetric] by transfer simp
 
-lemma borel_measurable_divide_ennreal[measurable (raw)]:
+lemma%unimportant borel_measurable_divide_ennreal[measurable (raw)]:
   fixes f :: "'a \<Rightarrow> ennreal"
   shows "f \<in> M \<rightarrow>\<^sub>M borel \<Longrightarrow> g \<in> M \<rightarrow>\<^sub>M borel \<Longrightarrow> (\<lambda>x. f x / g x) \<in> M \<rightarrow>\<^sub>M borel"
   unfolding divide_ennreal_def by simp
 
-lemma borel_measurable_minus_ennreal[measurable (raw)]:
+lemma%unimportant borel_measurable_minus_ennreal[measurable (raw)]:
   fixes f :: "'a \<Rightarrow> ennreal"
   shows "f \<in> M \<rightarrow>\<^sub>M borel \<Longrightarrow> g \<in> M \<rightarrow>\<^sub>M borel \<Longrightarrow> (\<lambda>x. f x - g x) \<in> M \<rightarrow>\<^sub>M borel"
   unfolding is_borel_def[symmetric] by transfer simp
 
-lemma borel_measurable_prod_ennreal[measurable (raw)]:
+lemma%important borel_measurable_prod_ennreal[measurable (raw)]:
   fixes f :: "'c \<Rightarrow> 'a \<Rightarrow> ennreal"
   assumes "\<And>i. i \<in> S \<Longrightarrow> f i \<in> borel_measurable M"
   shows "(\<lambda>x. \<Prod>i\<in>S. f i x) \<in> borel_measurable M"
-  using assms by (induction S rule: infinite_finite_induct) auto
+  using%unimportant assms by (induction S rule: infinite_finite_induct) auto
 
 end
 
 hide_const (open) is_borel
 
-subsection \<open>LIMSEQ is borel measurable\<close>
+subsection%important \<open>LIMSEQ is borel measurable\<close>
 
-lemma borel_measurable_LIMSEQ_real:
+lemma%important borel_measurable_LIMSEQ_real:
   fixes u :: "nat \<Rightarrow> 'a \<Rightarrow> real"
   assumes u': "\<And>x. x \<in> space M \<Longrightarrow> (\<lambda>i. u i x) \<longlonglongrightarrow> u' x"
   and u: "\<And>i. u i \<in> borel_measurable M"
   shows "u' \<in> borel_measurable M"
-proof -
+proof%unimportant -
   have "\<And>x. x \<in> space M \<Longrightarrow> liminf (\<lambda>n. ereal (u n x)) = ereal (u' x)"
     using u' by (simp add: lim_imp_Liminf)
   moreover from u have "(\<lambda>x. liminf (\<lambda>n. ereal (u n x))) \<in> borel_measurable M"
@@ -1778,13 +1778,13 @@
   ultimately show ?thesis by (simp cong: measurable_cong add: borel_measurable_ereal_iff)
 qed
 
-lemma borel_measurable_LIMSEQ_metric:
+lemma%important borel_measurable_LIMSEQ_metric:
   fixes f :: "nat \<Rightarrow> 'a \<Rightarrow> 'b :: metric_space"
   assumes [measurable]: "\<And>i. f i \<in> borel_measurable M"
   assumes lim: "\<And>x. x \<in> space M \<Longrightarrow> (\<lambda>i. f i x) \<longlonglongrightarrow> g x"
   shows "g \<in> borel_measurable M"
   unfolding borel_eq_closed
-proof (safe intro!: measurable_measure_of)
+proof%unimportant (safe intro!: measurable_measure_of)
   fix A :: "'b set" assume "closed A"
 
   have [measurable]: "(\<lambda>x. infdist (g x) A) \<in> borel_measurable M"
@@ -1809,13 +1809,13 @@
   qed simp
 qed auto
 
-lemma sets_Collect_Cauchy[measurable]:
+lemma%important sets_Collect_Cauchy[measurable]:
   fixes f :: "nat \<Rightarrow> 'a => 'b::{metric_space, second_countable_topology}"
   assumes f[measurable]: "\<And>i. f i \<in> borel_measurable M"
   shows "{x\<in>space M. Cauchy (\<lambda>i. f i x)} \<in> sets M"
   unfolding metric_Cauchy_iff2 using f by auto
 
-lemma borel_measurable_lim_metric[measurable (raw)]:
+lemma%unimportant borel_measurable_lim_metric[measurable (raw)]:
   fixes f :: "nat \<Rightarrow> 'a \<Rightarrow> 'b::{banach, second_countable_topology}"
   assumes f[measurable]: "\<And>i. f i \<in> borel_measurable M"
   shows "(\<lambda>x. lim (\<lambda>i. f i x)) \<in> borel_measurable M"
@@ -1837,17 +1837,17 @@
     unfolding * by measurable
 qed
 
-lemma borel_measurable_suminf[measurable (raw)]:
+lemma%unimportant borel_measurable_suminf[measurable (raw)]:
   fixes f :: "nat \<Rightarrow> 'a \<Rightarrow> 'b::{banach, second_countable_topology}"
   assumes f[measurable]: "\<And>i. f i \<in> borel_measurable M"
   shows "(\<lambda>x. suminf (\<lambda>i. f i x)) \<in> borel_measurable M"
   unfolding suminf_def sums_def[abs_def] lim_def[symmetric] by simp
 
-lemma Collect_closed_imp_pred_borel: "closed {x. P x} \<Longrightarrow> Measurable.pred borel P"
+lemma%unimportant Collect_closed_imp_pred_borel: "closed {x. P x} \<Longrightarrow> Measurable.pred borel P"
   by (simp add: pred_def)
 
 (* Proof by Jeremy Avigad and Luke Serafin *)
-lemma isCont_borel_pred[measurable]:
+lemma%unimportant isCont_borel_pred[measurable]:
   fixes f :: "'b::metric_space \<Rightarrow> 'a::metric_space"
   shows "Measurable.pred borel (isCont f)"
 proof (subst measurable_cong)
@@ -1898,21 +1898,21 @@
 qed (intro pred_intros_countable closed_Collect_all closed_Collect_le open_Collect_less
            Collect_closed_imp_pred_borel closed_Collect_imp open_Collect_conj continuous_intros)
 
-lemma isCont_borel:
+lemma%unimportant isCont_borel:
   fixes f :: "'b::metric_space \<Rightarrow> 'a::metric_space"
   shows "{x. isCont f x} \<in> sets borel"
   by simp
 
-lemma is_real_interval:
+lemma%important is_real_interval:
   assumes S: "is_interval S"
   shows "\<exists>a b::real. S = {} \<or> S = UNIV \<or> S = {..<b} \<or> S = {..b} \<or> S = {a<..} \<or> S = {a..} \<or>
     S = {a<..<b} \<or> S = {a<..b} \<or> S = {a..<b} \<or> S = {a..b}"
   using S unfolding is_interval_1 by (blast intro: interval_cases)
 
-lemma real_interval_borel_measurable:
+lemma%important real_interval_borel_measurable:
   assumes "is_interval (S::real set)"
   shows "S \<in> sets borel"
-proof -
+proof%unimportant -
   from assms is_real_interval have "\<exists>a b::real. S = {} \<or> S = UNIV \<or> S = {..<b} \<or> S = {..b} \<or>
     S = {a<..} \<or> S = {a..} \<or> S = {a<..<b} \<or> S = {a<..b} \<or> S = {a..<b} \<or> S = {a..b}" by auto
   then guess a ..
@@ -1924,7 +1924,7 @@
 text \<open>The next lemmas hold in any second countable linorder (including ennreal or ereal for instance),
 but in the current state they are restricted to reals.\<close>
 
-lemma borel_measurable_mono_on_fnc:
+lemma%important borel_measurable_mono_on_fnc:
   fixes f :: "real \<Rightarrow> real" and A :: "real set"
   assumes "mono_on f A"
   shows "f \<in> borel_measurable (restrict_space borel A)"
@@ -1935,18 +1935,18 @@
               intro!: borel_measurable_continuous_on_restrict intro: continuous_within_subset)
   done
 
-lemma borel_measurable_piecewise_mono:
+lemma%unimportant borel_measurable_piecewise_mono:
   fixes f::"real \<Rightarrow> real" and C::"real set set"
   assumes "countable C" "\<And>c. c \<in> C \<Longrightarrow> c \<in> sets borel" "\<And>c. c \<in> C \<Longrightarrow> mono_on f c" "(\<Union>C) = UNIV"
   shows "f \<in> borel_measurable borel"
-by (rule measurable_piecewise_restrict[of C], auto intro: borel_measurable_mono_on_fnc simp: assms)
+  by (rule measurable_piecewise_restrict[of C], auto intro: borel_measurable_mono_on_fnc simp: assms)
 
-lemma borel_measurable_mono:
+lemma%unimportant borel_measurable_mono:
   fixes f :: "real \<Rightarrow> real"
   shows "mono f \<Longrightarrow> f \<in> borel_measurable borel"
   using borel_measurable_mono_on_fnc[of f UNIV] by (simp add: mono_def mono_on_def)
 
-lemma measurable_bdd_below_real[measurable (raw)]:
+lemma%unimportant measurable_bdd_below_real[measurable (raw)]:
   fixes F :: "'a \<Rightarrow> 'i \<Rightarrow> real"
   assumes [simp]: "countable I" and [measurable]: "\<And>i. i \<in> I \<Longrightarrow> F i \<in> M \<rightarrow>\<^sub>M borel"
   shows "Measurable.pred M (\<lambda>x. bdd_below ((\<lambda>i. F i x)`I))"
@@ -1957,12 +1957,12 @@
     using countable_int by measurable
 qed
 
-lemma borel_measurable_cINF_real[measurable (raw)]:
+lemma%important borel_measurable_cINF_real[measurable (raw)]:
   fixes F :: "_ \<Rightarrow> _ \<Rightarrow> real"
   assumes [simp]: "countable I"
   assumes F[measurable]: "\<And>i. i \<in> I \<Longrightarrow> F i \<in> borel_measurable M"
   shows "(\<lambda>x. INF i:I. F i x) \<in> borel_measurable M"
-proof (rule measurable_piecewise_restrict)
+proof%unimportant (rule measurable_piecewise_restrict)
   let ?\<Omega> = "{x\<in>space M. bdd_below ((\<lambda>i. F i x)`I)}"
   show "countable {?\<Omega>, - ?\<Omega>}" "space M \<subseteq> \<Union>{?\<Omega>, - ?\<Omega>}" "\<And>X. X \<in> {?\<Omega>, - ?\<Omega>} \<Longrightarrow> X \<inter> space M \<in> sets M"
     by auto
@@ -1982,7 +1982,7 @@
   qed
 qed
 
-lemma borel_Ici: "borel = sigma UNIV (range (\<lambda>x::real. {x ..}))"
+lemma%unimportant borel_Ici: "borel = sigma UNIV (range (\<lambda>x::real. {x ..}))"
 proof (safe intro!: borel_eq_sigmaI1[OF borel_Iio])
   fix x :: real
   have eq: "{..<x} = space (sigma UNIV (range atLeast)) - {x ..}"
@@ -1991,7 +1991,7 @@
     unfolding eq by (intro sets.compl_sets) auto
 qed auto
 
-lemma borel_measurable_pred_less[measurable (raw)]:
+lemma%unimportant borel_measurable_pred_less[measurable (raw)]:
   fixes f :: "'a \<Rightarrow> 'b::{second_countable_topology, linorder_topology}"
   shows "f \<in> borel_measurable M \<Longrightarrow> g \<in> borel_measurable M \<Longrightarrow> Measurable.pred M (\<lambda>w. f w < g w)"
   unfolding Measurable.pred_def by (rule borel_measurable_less)
@@ -1999,19 +1999,19 @@
 no_notation
   eucl_less (infix "<e" 50)
 
-lemma borel_measurable_Max2[measurable (raw)]:
+lemma%important borel_measurable_Max2[measurable (raw)]:
   fixes f::"_ \<Rightarrow> _ \<Rightarrow> 'a::{second_countable_topology, dense_linorder, linorder_topology}"
   assumes "finite I"
     and [measurable]: "\<And>i. f i \<in> borel_measurable M"
   shows "(\<lambda>x. Max{f i x |i. i \<in> I}) \<in> borel_measurable M"
-by (simp add: borel_measurable_Max[OF assms(1), where ?f=f and ?M=M] Setcompr_eq_image)
+by%unimportant (simp add: borel_measurable_Max[OF assms(1), where ?f=f and ?M=M] Setcompr_eq_image)
 
-lemma measurable_compose_n [measurable (raw)]:
+lemma%unimportant measurable_compose_n [measurable (raw)]:
   assumes "T \<in> measurable M M"
   shows "(T^^n) \<in> measurable M M"
 by (induction n, auto simp add: measurable_compose[OF _ assms])
 
-lemma measurable_real_imp_nat:
+lemma%unimportant measurable_real_imp_nat:
   fixes f::"'a \<Rightarrow> nat"
   assumes [measurable]: "(\<lambda>x. real(f x)) \<in> borel_measurable M"
   shows "f \<in> measurable M (count_space UNIV)"
@@ -2023,7 +2023,7 @@
   then show ?thesis using measurable_count_space_eq2_countable by blast
 qed
 
-lemma measurable_equality_set [measurable]:
+lemma%unimportant measurable_equality_set [measurable]:
   fixes f g::"_\<Rightarrow> 'a::{second_countable_topology, t2_space}"
   assumes [measurable]: "f \<in> borel_measurable M" "g \<in> borel_measurable M"
   shows "{x \<in> space M. f x = g x} \<in> sets M"
@@ -2038,7 +2038,7 @@
   then show ?thesis unfolding A_def by simp
 qed
 
-lemma measurable_inequality_set [measurable]:
+lemma%unimportant measurable_inequality_set [measurable]:
   fixes f g::"_ \<Rightarrow> 'a::{second_countable_topology, linorder_topology}"
   assumes [measurable]: "f \<in> borel_measurable M" "g \<in> borel_measurable M"
   shows "{x \<in> space M. f x \<le> g x} \<in> sets M"
@@ -2066,7 +2066,7 @@
   ultimately show "{x \<in> space M. f x > g x} \<in> sets M" using * by (metis (mono_tags, lifting) measurable_sets)
 qed
 
-lemma measurable_limit [measurable]:
+lemma%unimportant measurable_limit [measurable]:
   fixes f::"nat \<Rightarrow> 'a \<Rightarrow> 'b::first_countable_topology"
   assumes [measurable]: "\<And>n::nat. f n \<in> borel_measurable M"
   shows "Measurable.pred M (\<lambda>x. (\<lambda>n. f n x) \<longlonglongrightarrow> c)"
@@ -2106,11 +2106,11 @@
   then show ?thesis by auto
 qed
 
-lemma measurable_limit2 [measurable]:
+lemma%important measurable_limit2 [measurable]:
   fixes u::"nat \<Rightarrow> 'a \<Rightarrow> real"
   assumes [measurable]: "\<And>n. u n \<in> borel_measurable M" "v \<in> borel_measurable M"
   shows "Measurable.pred M (\<lambda>x. (\<lambda>n. u n x) \<longlonglongrightarrow> v x)"
-proof -
+proof%unimportant -
   define w where "w = (\<lambda>n x. u n x - v x)"
   have [measurable]: "w n \<in> borel_measurable M" for n unfolding w_def by auto
   have "((\<lambda>n. u n x) \<longlonglongrightarrow> v x) \<longleftrightarrow> ((\<lambda>n. w n x) \<longlonglongrightarrow> 0)" for x
@@ -2118,7 +2118,7 @@
   then show ?thesis using measurable_limit by auto
 qed
 
-lemma measurable_P_restriction [measurable (raw)]:
+lemma%unimportant measurable_P_restriction [measurable (raw)]:
   assumes [measurable]: "Measurable.pred M P" "A \<in> sets M"
   shows "{x \<in> A. P x} \<in> sets M"
 proof -
@@ -2127,7 +2127,7 @@
   then show ?thesis by auto
 qed
 
-lemma measurable_sum_nat [measurable (raw)]:
+lemma%unimportant measurable_sum_nat [measurable (raw)]:
   fixes f :: "'c \<Rightarrow> 'a \<Rightarrow> nat"
   assumes "\<And>i. i \<in> S \<Longrightarrow> f i \<in> measurable M (count_space UNIV)"
   shows "(\<lambda>x. \<Sum>i\<in>S. f i x) \<in> measurable M (count_space UNIV)"
@@ -2137,7 +2137,7 @@
 qed simp
 
 
-lemma measurable_abs_powr [measurable]:
+lemma%unimportant measurable_abs_powr [measurable]:
   fixes p::real
   assumes [measurable]: "f \<in> borel_measurable M"
   shows "(\<lambda>x. \<bar>f x\<bar> powr p) \<in> borel_measurable M"
@@ -2145,7 +2145,7 @@
 
 text \<open>The next one is a variation around \verb+measurable_restrict_space+.\<close>
 
-lemma measurable_restrict_space3:
+lemma%unimportant measurable_restrict_space3:
   assumes "f \<in> measurable M N" and
           "f \<in> A \<rightarrow> B"
   shows "f \<in> measurable (restrict_space M A) (restrict_space N B)"
@@ -2157,12 +2157,12 @@
 
 text \<open>The next one is a variation around \verb+measurable_piecewise_restrict+.\<close>
 
-lemma measurable_piecewise_restrict2:
+lemma%important measurable_piecewise_restrict2:
   assumes [measurable]: "\<And>n. A n \<in> sets M"
       and "space M = (\<Union>(n::nat). A n)"
           "\<And>n. \<exists>h \<in> measurable M N. (\<forall>x \<in> A n. f x = h x)"
   shows "f \<in> measurable M N"
-proof (rule measurableI)
+proof%unimportant (rule measurableI)
   fix B assume [measurable]: "B \<in> sets N"
   {
     fix n::nat

--- a/src/HOL/Analysis/Caratheodory.thy	Mon Aug 27 22:58:36 2018 +0200
+++ b/src/HOL/Analysis/Caratheodory.thy	Tue Aug 28 13:28:39 2018 +0100
@@ -3,7 +3,7 @@
     Author:     Johannes Hölzl, TU München
 *)
 
-section \<open>Caratheodory Extension Theorem\<close>
+section%important \<open>Caratheodory Extension Theorem\<close>
 
 theory Caratheodory
   imports Measure_Space
@@ -13,7 +13,7 @@
   Originally from the Hurd/Coble measure theory development, translated by Lawrence Paulson.
 \<close>
 
-lemma suminf_ennreal_2dimen:
+lemma%unimportant suminf_ennreal_2dimen:
   fixes f:: "nat \<times> nat \<Rightarrow> ennreal"
   assumes "\<And>m. g m = (\<Sum>n. f (m,n))"
   shows "(\<Sum>i. f (prod_decode i)) = suminf g"
@@ -49,18 +49,18 @@
     by (simp add: suminf_eq_SUP)
 qed
 
-subsection \<open>Characterizations of Measures\<close>
+subsection%important \<open>Characterizations of Measures\<close>
 
-definition outer_measure_space where
+definition%important outer_measure_space where
   "outer_measure_space M f \<longleftrightarrow> positive M f \<and> increasing M f \<and> countably_subadditive M f"
 
-subsubsection \<open>Lambda Systems\<close>
+subsubsection%important \<open>Lambda Systems\<close>
 
-definition lambda_system :: "'a set \<Rightarrow> 'a set set \<Rightarrow> ('a set \<Rightarrow> ennreal) \<Rightarrow> 'a set set"
+definition%important lambda_system :: "'a set \<Rightarrow> 'a set set \<Rightarrow> ('a set \<Rightarrow> ennreal) \<Rightarrow> 'a set set"
 where
   "lambda_system \<Omega> M f = {l \<in> M. \<forall>x \<in> M. f (l \<inter> x) + f ((\<Omega> - l) \<inter> x) = f x}"
 
-lemma (in algebra) lambda_system_eq:
+lemma%unimportant (in algebra) lambda_system_eq:
   "lambda_system \<Omega> M f = {l \<in> M. \<forall>x \<in> M. f (x \<inter> l) + f (x - l) = f x}"
 proof -
   have [simp]: "\<And>l x. l \<in> M \<Longrightarrow> x \<in> M \<Longrightarrow> (\<Omega> - l) \<inter> x = x - l"
@@ -69,13 +69,13 @@
     by (auto simp add: lambda_system_def) (metis Int_commute)+
 qed
 
-lemma (in algebra) lambda_system_empty: "positive M f \<Longrightarrow> {} \<in> lambda_system \<Omega> M f"
+lemma%unimportant (in algebra) lambda_system_empty: "positive M f \<Longrightarrow> {} \<in> lambda_system \<Omega> M f"
   by (auto simp add: positive_def lambda_system_eq)
 
-lemma lambda_system_sets: "x \<in> lambda_system \<Omega> M f \<Longrightarrow> x \<in> M"
+lemma%unimportant lambda_system_sets: "x \<in> lambda_system \<Omega> M f \<Longrightarrow> x \<in> M"
   by (simp add: lambda_system_def)
 
-lemma (in algebra) lambda_system_Compl:
+lemma%unimportant (in algebra) lambda_system_Compl:
   fixes f:: "'a set \<Rightarrow> ennreal"
   assumes x: "x \<in> lambda_system \<Omega> M f"
   shows "\<Omega> - x \<in> lambda_system \<Omega> M f"
@@ -88,7 +88,7 @@
     by (force simp add: lambda_system_def ac_simps)
 qed
 
-lemma (in algebra) lambda_system_Int:
+lemma%unimportant (in algebra) lambda_system_Int:
   fixes f:: "'a set \<Rightarrow> ennreal"
   assumes xl: "x \<in> lambda_system \<Omega> M f" and yl: "y \<in> lambda_system \<Omega> M f"
   shows "x \<inter> y \<in> lambda_system \<Omega> M f"
@@ -122,7 +122,7 @@
   qed
 qed
 
-lemma (in algebra) lambda_system_Un:
+lemma%unimportant (in algebra) lambda_system_Un:
   fixes f:: "'a set \<Rightarrow> ennreal"
   assumes xl: "x \<in> lambda_system \<Omega> M f" and yl: "y \<in> lambda_system \<Omega> M f"
   shows "x \<union> y \<in> lambda_system \<Omega> M f"
@@ -136,7 +136,7 @@
     by (metis lambda_system_Compl lambda_system_Int xl yl)
 qed
 
-lemma (in algebra) lambda_system_algebra:
+lemma%unimportant (in algebra) lambda_system_algebra:
   "positive M f \<Longrightarrow> algebra \<Omega> (lambda_system \<Omega> M f)"
   apply (auto simp add: algebra_iff_Un)
   apply (metis lambda_system_sets set_mp sets_into_space)
@@ -145,7 +145,7 @@
   apply (metis lambda_system_Un)
   done
 
-lemma (in algebra) lambda_system_strong_additive:
+lemma%unimportant (in algebra) lambda_system_strong_additive:
   assumes z: "z \<in> M" and disj: "x \<inter> y = {}"
       and xl: "x \<in> lambda_system \<Omega> M f" and yl: "y \<in> lambda_system \<Omega> M f"
   shows "f (z \<inter> (x \<union> y)) = f (z \<inter> x) + f (z \<inter> y)"
@@ -160,7 +160,7 @@
     by (simp add: lambda_system_eq)
 qed
 
-lemma (in algebra) lambda_system_additive: "additive (lambda_system \<Omega> M f) f"
+lemma%unimportant (in algebra) lambda_system_additive: "additive (lambda_system \<Omega> M f) f"
 proof (auto simp add: additive_def)
   fix x and y
   assume disj: "x \<inter> y = {}"
@@ -171,13 +171,13 @@
     by (simp add: Un)
 qed
 
-lemma lambda_system_increasing: "increasing M f \<Longrightarrow> increasing (lambda_system \<Omega> M f) f"
+lemma%unimportant lambda_system_increasing: "increasing M f \<Longrightarrow> increasing (lambda_system \<Omega> M f) f"
   by (simp add: increasing_def lambda_system_def)
 
-lemma lambda_system_positive: "positive M f \<Longrightarrow> positive (lambda_system \<Omega> M f) f"
+lemma%unimportant lambda_system_positive: "positive M f \<Longrightarrow> positive (lambda_system \<Omega> M f) f"
   by (simp add: positive_def lambda_system_def)
 
-lemma (in algebra) lambda_system_strong_sum:
+lemma%unimportant (in algebra) lambda_system_strong_sum:
   fixes A:: "nat \<Rightarrow> 'a set" and f :: "'a set \<Rightarrow> ennreal"
   assumes f: "positive M f" and a: "a \<in> M"
       and A: "range A \<subseteq> lambda_system \<Omega> M f"
@@ -199,12 +199,12 @@
     by (simp add: atLeastLessThanSuc lambda_system_strong_additive [OF a 2 3 4])
 qed
 
-lemma (in sigma_algebra) lambda_system_caratheodory:
+lemma%important (in sigma_algebra) lambda_system_caratheodory:
   assumes oms: "outer_measure_space M f"
       and A: "range A \<subseteq> lambda_system \<Omega> M f"
       and disj: "disjoint_family A"
   shows  "(\<Union>i. A i) \<in> lambda_system \<Omega> M f \<and> (\<Sum>i. f (A i)) = f (\<Union>i. A i)"
-proof -
+proof%unimportant -
   have pos: "positive M f" and inc: "increasing M f"
    and csa: "countably_subadditive M f"
     by (metis oms outer_measure_space_def)+
@@ -274,11 +274,11 @@
     by (simp add: lambda_system_eq sums_iff U_eq U_in)
 qed
 
-lemma (in sigma_algebra) caratheodory_lemma:
+lemma%important (in sigma_algebra) caratheodory_lemma:
   assumes oms: "outer_measure_space M f"
   defines "L \<equiv> lambda_system \<Omega> M f"
   shows "measure_space \<Omega> L f"
-proof -
+proof%unimportant -
   have pos: "positive M f"
     by (metis oms outer_measure_space_def)
   have alg: "algebra \<Omega> L"
@@ -297,11 +297,11 @@
     using pos by (simp add: measure_space_def)
 qed
 
-definition outer_measure :: "'a set set \<Rightarrow> ('a set \<Rightarrow> ennreal) \<Rightarrow> 'a set \<Rightarrow> ennreal" where
+definition%important outer_measure :: "'a set set \<Rightarrow> ('a set \<Rightarrow> ennreal) \<Rightarrow> 'a set \<Rightarrow> ennreal" where
    "outer_measure M f X =
      (INF A:{A. range A \<subseteq> M \<and> disjoint_family A \<and> X \<subseteq> (\<Union>i. A i)}. \<Sum>i. f (A i))"
 
-lemma (in ring_of_sets) outer_measure_agrees:
+lemma%unimportant (in ring_of_sets) outer_measure_agrees:
   assumes posf: "positive M f" and ca: "countably_additive M f" and s: "s \<in> M"
   shows "outer_measure M f s = f s"
   unfolding outer_measure_def
@@ -326,19 +326,19 @@
        (auto simp: disjoint_family_on_def)
 qed
 
-lemma outer_measure_empty:
+lemma%unimportant outer_measure_empty:
   "positive M f \<Longrightarrow> {} \<in> M \<Longrightarrow> outer_measure M f {} = 0"
   unfolding outer_measure_def
   by (intro antisym INF_lower2[of  "\<lambda>_. {}"]) (auto simp: disjoint_family_on_def positive_def)
 
-lemma (in ring_of_sets) positive_outer_measure:
+lemma%unimportant (in ring_of_sets) positive_outer_measure:
   assumes "positive M f" shows "positive (Pow \<Omega>) (outer_measure M f)"
   unfolding positive_def by (auto simp: assms outer_measure_empty)
 
-lemma (in ring_of_sets) increasing_outer_measure: "increasing (Pow \<Omega>) (outer_measure M f)"
+lemma%unimportant (in ring_of_sets) increasing_outer_measure: "increasing (Pow \<Omega>) (outer_measure M f)"
   by (force simp: increasing_def outer_measure_def intro!: INF_greatest intro: INF_lower)
 
-lemma (in ring_of_sets) outer_measure_le:
+lemma%unimportant (in ring_of_sets) outer_measure_le:
   assumes pos: "positive M f" and inc: "increasing M f" and A: "range A \<subseteq> M" and X: "X \<subseteq> (\<Union>i. A i)"
   shows "outer_measure M f X \<le> (\<Sum>i. f (A i))"
   unfolding outer_measure_def
@@ -351,11 +351,11 @@
     by (blast intro!: suminf_le)
 qed (auto simp: X UN_disjointed_eq disjoint_family_disjointed)
 
-lemma (in ring_of_sets) outer_measure_close:
+lemma%unimportant (in ring_of_sets) outer_measure_close:
   "outer_measure M f X < e \<Longrightarrow> \<exists>A. range A \<subseteq> M \<and> disjoint_family A \<and> X \<subseteq> (\<Union>i. A i) \<and> (\<Sum>i. f (A i)) < e"
   unfolding outer_measure_def INF_less_iff by auto
 
-lemma (in ring_of_sets) countably_subadditive_outer_measure:
+lemma%unimportant (in ring_of_sets) countably_subadditive_outer_measure:
   assumes posf: "positive M f" and inc: "increasing M f"
   shows "countably_subadditive (Pow \<Omega>) (outer_measure M f)"
 proof (simp add: countably_subadditive_def, safe)
@@ -398,12 +398,12 @@
   qed
 qed
 
-lemma (in ring_of_sets) outer_measure_space_outer_measure:
+lemma%unimportant (in ring_of_sets) outer_measure_space_outer_measure:
   "positive M f \<Longrightarrow> increasing M f \<Longrightarrow> outer_measure_space (Pow \<Omega>) (outer_measure M f)"
   by (simp add: outer_measure_space_def
     positive_outer_measure increasing_outer_measure countably_subadditive_outer_measure)
 
-lemma (in ring_of_sets) algebra_subset_lambda_system:
+lemma%unimportant (in ring_of_sets) algebra_subset_lambda_system:
   assumes posf: "positive M f" and inc: "increasing M f"
       and add: "additive M f"
   shows "M \<subseteq> lambda_system \<Omega> (Pow \<Omega>) (outer_measure M f)"
@@ -457,15 +457,15 @@
     by (rule order_antisym)
 qed
 
-lemma measure_down: "measure_space \<Omega> N \<mu> \<Longrightarrow> sigma_algebra \<Omega> M \<Longrightarrow> M \<subseteq> N \<Longrightarrow> measure_space \<Omega> M \<mu>"
+lemma%unimportant measure_down: "measure_space \<Omega> N \<mu> \<Longrightarrow> sigma_algebra \<Omega> M \<Longrightarrow> M \<subseteq> N \<Longrightarrow> measure_space \<Omega> M \<mu>"
   by (auto simp add: measure_space_def positive_def countably_additive_def subset_eq)
 
-subsection \<open>Caratheodory's theorem\<close>
+subsection%important \<open>Caratheodory's theorem\<close>
 
-theorem (in ring_of_sets) caratheodory':
+theorem%important (in ring_of_sets) caratheodory':
   assumes posf: "positive M f" and ca: "countably_additive M f"
   shows "\<exists>\<mu> :: 'a set \<Rightarrow> ennreal. (\<forall>s \<in> M. \<mu> s = f s) \<and> measure_space \<Omega> (sigma_sets \<Omega> M) \<mu>"
-proof -
+proof%unimportant -
   have inc: "increasing M f"
     by (metis additive_increasing ca countably_additive_additive posf)
   let ?O = "outer_measure M f"
@@ -489,37 +489,37 @@
     by (intro exI[of _ ?O]) auto
 qed
 
-lemma (in ring_of_sets) caratheodory_empty_continuous:
+lemma%important (in ring_of_sets) caratheodory_empty_continuous:
   assumes f: "positive M f" "additive M f" and fin: "\<And>A. A \<in> M \<Longrightarrow> f A \<noteq> \<infinity>"
   assumes cont: "\<And>A. range A \<subseteq> M \<Longrightarrow> decseq A \<Longrightarrow> (\<Inter>i. A i) = {} \<Longrightarrow> (\<lambda>i. f (A i)) \<longlonglongrightarrow> 0"
   shows "\<exists>\<mu> :: 'a set \<Rightarrow> ennreal. (\<forall>s \<in> M. \<mu> s = f s) \<and> measure_space \<Omega> (sigma_sets \<Omega> M) \<mu>"
-proof (intro caratheodory' empty_continuous_imp_countably_additive f)
+proof%unimportant (intro caratheodory' empty_continuous_imp_countably_additive f)
   show "\<forall>A\<in>M. f A \<noteq> \<infinity>" using fin by auto
 qed (rule cont)
 
-subsection \<open>Volumes\<close>
+subsection%important \<open>Volumes\<close>
 
-definition volume :: "'a set set \<Rightarrow> ('a set \<Rightarrow> ennreal) \<Rightarrow> bool" where
+definition%important volume :: "'a set set \<Rightarrow> ('a set \<Rightarrow> ennreal) \<Rightarrow> bool" where
   "volume M f \<longleftrightarrow>
   (f {} = 0) \<and> (\<forall>a\<in>M. 0 \<le> f a) \<and>
   (\<forall>C\<subseteq>M. disjoint C \<longrightarrow> finite C \<longrightarrow> \<Union>C \<in> M \<longrightarrow> f (\<Union>C) = (\<Sum>c\<in>C. f c))"
 
-lemma volumeI:
+lemma%unimportant volumeI:
   assumes "f {} = 0"
   assumes "\<And>a. a \<in> M \<Longrightarrow> 0 \<le> f a"
   assumes "\<And>C. C \<subseteq> M \<Longrightarrow> disjoint C \<Longrightarrow> finite C \<Longrightarrow> \<Union>C \<in> M \<Longrightarrow> f (\<Union>C) = (\<Sum>c\<in>C. f c)"
   shows "volume M f"
   using assms by (auto simp: volume_def)
 
-lemma volume_positive:
+lemma%unimportant volume_positive:
   "volume M f \<Longrightarrow> a \<in> M \<Longrightarrow> 0 \<le> f a"
   by (auto simp: volume_def)
 
-lemma volume_empty:
+lemma%unimportant volume_empty:
   "volume M f \<Longrightarrow> f {} = 0"
   by (auto simp: volume_def)
 
-lemma volume_finite_additive:
+lemma%unimportant volume_finite_additive:
   assumes "volume M f"
   assumes A: "\<And>i. i \<in> I \<Longrightarrow> A i \<in> M" "disjoint_family_on A I" "finite I" "UNION I A \<in> M"
   shows "f (UNION I A) = (\<Sum>i\<in>I. f (A i))"
@@ -540,7 +540,7 @@
     by simp
 qed
 
-lemma (in ring_of_sets) volume_additiveI:
+lemma%unimportant (in ring_of_sets) volume_additiveI:
   assumes pos: "\<And>a. a \<in> M \<Longrightarrow> 0 \<le> \<mu> a"
   assumes [simp]: "\<mu> {} = 0"
   assumes add: "\<And>a b. a \<in> M \<Longrightarrow> b \<in> M \<Longrightarrow> a \<inter> b = {} \<Longrightarrow> \<mu> (a \<union> b) = \<mu> a + \<mu> b"
@@ -557,10 +557,10 @@
   qed simp
 qed fact+
 
-lemma (in semiring_of_sets) extend_volume:
+lemma%important (in semiring_of_sets) extend_volume:
   assumes "volume M \<mu>"
   shows "\<exists>\<mu>'. volume generated_ring \<mu>' \<and> (\<forall>a\<in>M. \<mu>' a = \<mu> a)"
-proof -
+proof%unimportant -
   let ?R = generated_ring
   have "\<forall>a\<in>?R. \<exists>m. \<exists>C\<subseteq>M. a = \<Union>C \<and> finite C \<and> disjoint C \<and> m = (\<Sum>c\<in>C. \<mu> c)"
     by (auto simp: generated_ring_def)
@@ -635,12 +635,12 @@
   qed
 qed
 
-subsubsection \<open>Caratheodory on semirings\<close>
+subsubsection%important \<open>Caratheodory on semirings\<close>
 
-theorem (in semiring_of_sets) caratheodory:
+theorem%important (in semiring_of_sets) caratheodory:
   assumes pos: "positive M \<mu>" and ca: "countably_additive M \<mu>"
   shows "\<exists>\<mu>' :: 'a set \<Rightarrow> ennreal. (\<forall>s \<in> M. \<mu>' s = \<mu> s) \<and> measure_space \<Omega> (sigma_sets \<Omega> M) \<mu>'"
-proof -
+proof%unimportant -
   have "volume M \<mu>"
   proof (rule volumeI)
     { fix a assume "a \<in> M" then show "0 \<le> \<mu> a"
@@ -816,7 +816,7 @@
     by (intro exI[of _ \<mu>']) (auto intro: generated_ringI_Basic)
 qed
 
-lemma extend_measure_caratheodory:
+lemma%important extend_measure_caratheodory:
   fixes G :: "'i \<Rightarrow> 'a set"
   assumes M: "M = extend_measure \<Omega> I G \<mu>"
   assumes "i \<in> I"
@@ -827,7 +827,8 @@
   assumes add: "\<And>A::nat \<Rightarrow> 'i. \<And>j. A \<in> UNIV \<rightarrow> I \<Longrightarrow> j \<in> I \<Longrightarrow> disjoint_family (G \<circ> A) \<Longrightarrow>
     (\<Union>i. G (A i)) = G j \<Longrightarrow> (\<Sum>n. \<mu> (A n)) = \<mu> j"
   shows "emeasure M (G i) = \<mu> i"
-proof -
+
+proof%unimportant -
   interpret semiring_of_sets \<Omega> "G ` I"
     by fact
   have "\<forall>g\<in>G`I. \<exists>i\<in>I. g = G i"
@@ -860,7 +861,7 @@
   qed fact
 qed
 
-lemma extend_measure_caratheodory_pair:
+lemma%important extend_measure_caratheodory_pair:
   fixes G :: "'i \<Rightarrow> 'j \<Rightarrow> 'a set"
   assumes M: "M = extend_measure \<Omega> {(a, b). P a b} (\<lambda>(a, b). G a b) (\<lambda>(a, b). \<mu> a b)"
   assumes "P i j"
@@ -872,7 +873,7 @@
     (\<And>n. P (A n) (B n)) \<Longrightarrow> P j k \<Longrightarrow> disjoint_family (\<lambda>n. G (A n) (B n)) \<Longrightarrow>
     (\<Union>i. G (A i) (B i)) = G j k \<Longrightarrow> (\<Sum>n. \<mu> (A n) (B n)) = \<mu> j k"
   shows "emeasure M (G i j) = \<mu> i j"
-proof -
+proof%unimportant -
   have "emeasure M ((\<lambda>(a, b). G a b) (i, j)) = (\<lambda>(a, b). \<mu> a b) (i, j)"
   proof (rule extend_measure_caratheodory[OF M])
     show "semiring_of_sets \<Omega> ((\<lambda>(a, b). G a b) ` {(a, b). P a b})"

--- a/src/HOL/Analysis/Cartesian_Euclidean_Space.thy	Mon Aug 27 22:58:36 2018 +0200
+++ b/src/HOL/Analysis/Cartesian_Euclidean_Space.thy	Tue Aug 28 13:28:39 2018 +0100
@@ -2,19 +2,19 @@
    Some material by Jose Divasón, Tim Makarios and L C Paulson
 *)
 
-section \<open>Instantiates the finite Cartesian product of Euclidean spaces as a Euclidean space\<close>
+section%important \<open>Instantiates the finite Cartesian product of Euclidean spaces as a Euclidean space\<close>
 
 theory Cartesian_Euclidean_Space
 imports Cartesian_Space Derivative
 begin
 
-lemma subspace_special_hyperplane: "subspace {x. x $ k = 0}"
+lemma%unimportant subspace_special_hyperplane: "subspace {x. x $ k = 0}"
   by (simp add: subspace_def)
 
-lemma sum_mult_product:
+lemma%important sum_mult_product:
   "sum h {..<A * B :: nat} = (\<Sum>i\<in>{..<A}. \<Sum>j\<in>{..<B}. h (j + i * B))"
   unfolding sum_nat_group[of h B A, unfolded atLeast0LessThan, symmetric]
-proof (rule sum.cong, simp, rule sum.reindex_cong)
+proof%unimportant (rule sum.cong, simp, rule sum.reindex_cong)
   fix i
   show "inj_on (\<lambda>j. j + i * B) {..<B}" by (auto intro!: inj_onI)
   show "{i * B..<i * B + B} = (\<lambda>j. j + i * B) ` {..<B}"
@@ -25,32 +25,32 @@
   qed simp
 qed simp
 
-lemma interval_cbox_cart: "{a::real^'n..b} = cbox a b"
+lemma%unimportant interval_cbox_cart: "{a::real^'n..b} = cbox a b"
   by (auto simp add: less_eq_vec_def mem_box Basis_vec_def inner_axis)
 
-lemma differentiable_vec:
+lemma%unimportant differentiable_vec:
   fixes S :: "'a::euclidean_space set"
   shows "vec differentiable_on S"
   by (simp add: linear_linear bounded_linear_imp_differentiable_on)
 
-lemma continuous_vec [continuous_intros]:
+lemma%unimportant continuous_vec [continuous_intros]:
   fixes x :: "'a::euclidean_space"
   shows "isCont vec x"
   apply (clarsimp simp add: continuous_def LIM_def dist_vec_def L2_set_def)
   apply (rule_tac x="r / sqrt (real CARD('b))" in exI)
   by (simp add: mult.commute pos_less_divide_eq real_sqrt_mult)
 
-lemma box_vec_eq_empty [simp]:
+lemma%unimportant box_vec_eq_empty [simp]:
   shows "cbox (vec a) (vec b) = {} \<longleftrightarrow> cbox a b = {}"
         "box (vec a) (vec b) = {} \<longleftrightarrow> box a b = {}"
   by (auto simp: Basis_vec_def mem_box box_eq_empty inner_axis)
 
-subsection\<open>Closures and interiors of halfspaces\<close>
+subsection%important\<open>Closures and interiors of halfspaces\<close>
 
-lemma interior_halfspace_le [simp]:
+lemma%important interior_halfspace_le [simp]:
   assumes "a \<noteq> 0"
     shows "interior {x. a \<bullet> x \<le> b} = {x. a \<bullet> x < b}"
-proof -
+proof%unimportant -
   have *: "a \<bullet> x < b" if x: "x \<in> S" and S: "S \<subseteq> {x. a \<bullet> x \<le> b}" and "open S" for S x
   proof -
     obtain e where "e>0" and e: "cball x e \<subseteq> S"
@@ -70,15 +70,15 @@
     by (rule interior_unique) (auto simp: open_halfspace_lt *)
 qed
 
-lemma interior_halfspace_ge [simp]:
+lemma%unimportant interior_halfspace_ge [simp]:
    "a \<noteq> 0 \<Longrightarrow> interior {x. a \<bullet> x \<ge> b} = {x. a \<bullet> x > b}"
 using interior_halfspace_le [of "-a" "-b"] by simp
 
-lemma interior_halfspace_component_le [simp]:
+lemma%important interior_halfspace_component_le [simp]:
      "interior {x. x$k \<le> a} = {x :: (real^'n). x$k < a}" (is "?LE")
   and interior_halfspace_component_ge [simp]:
      "interior {x. x$k \<ge> a} = {x :: (real^'n). x$k > a}" (is "?GE")
-proof -
+proof%unimportant -
   have "axis k (1::real) \<noteq> 0"
     by (simp add: axis_def vec_eq_iff)
   moreover have "axis k (1::real) \<bullet> x = x$k" for x
@@ -88,7 +88,7 @@
           interior_halfspace_ge [of "axis k (1::real)" a] by auto
 qed
 
-lemma closure_halfspace_lt [simp]:
+lemma%unimportant closure_halfspace_lt [simp]:
   assumes "a \<noteq> 0"
     shows "closure {x. a \<bullet> x < b} = {x. a \<bullet> x \<le> b}"
 proof -
@@ -99,15 +99,15 @@
     by (force simp: closure_interior)
 qed
 
-lemma closure_halfspace_gt [simp]:
+lemma%unimportant closure_halfspace_gt [simp]:
    "a \<noteq> 0 \<Longrightarrow> closure {x. a \<bullet> x > b} = {x. a \<bullet> x \<ge> b}"
 using closure_halfspace_lt [of "-a" "-b"] by simp
 
-lemma closure_halfspace_component_lt [simp]:
+lemma%important closure_halfspace_component_lt [simp]:
      "closure {x. x$k < a} = {x :: (real^'n). x$k \<le> a}" (is "?LE")
   and closure_halfspace_component_gt [simp]:
      "closure {x. x$k > a} = {x :: (real^'n). x$k \<ge> a}" (is "?GE")
-proof -
+proof%unimportant -
   have "axis k (1::real) \<noteq> 0"
     by (simp add: axis_def vec_eq_iff)
   moreover have "axis k (1::real) \<bullet> x = x$k" for x
@@ -117,17 +117,17 @@
           closure_halfspace_gt [of "axis k (1::real)" a] by auto
 qed
 
-lemma interior_hyperplane [simp]:
+lemma%unimportant interior_hyperplane [simp]:
   assumes "a \<noteq> 0"
     shows "interior {x. a \<bullet> x = b} = {}"
-proof -
+proof%unimportant -
   have [simp]: "{x. a \<bullet> x = b} = {x. a \<bullet> x \<le> b} \<inter> {x. a \<bullet> x \<ge> b}"
     by (force simp:)
   then show ?thesis
     by (auto simp: assms)
 qed
 
-lemma frontier_halfspace_le:
+lemma%unimportant frontier_halfspace_le:
   assumes "a \<noteq> 0 \<or> b \<noteq> 0"
     shows "frontier {x. a \<bullet> x \<le> b} = {x. a \<bullet> x = b}"
 proof (cases "a = 0")
@@ -137,7 +137,7 @@
     by (force simp: frontier_def closed_halfspace_le)
 qed
 
-lemma frontier_halfspace_ge:
+lemma%unimportant frontier_halfspace_ge:
   assumes "a \<noteq> 0 \<or> b \<noteq> 0"
     shows "frontier {x. a \<bullet> x \<ge> b} = {x. a \<bullet> x = b}"
 proof (cases "a = 0")
@@ -147,7 +147,7 @@
     by (force simp: frontier_def closed_halfspace_ge)
 qed
 
-lemma frontier_halfspace_lt:
+lemma%unimportant frontier_halfspace_lt:
   assumes "a \<noteq> 0 \<or> b \<noteq> 0"
     shows "frontier {x. a \<bullet> x < b} = {x. a \<bullet> x = b}"
 proof (cases "a = 0")
@@ -157,19 +157,19 @@
     by (force simp: frontier_def interior_open open_halfspace_lt)
 qed
 
-lemma frontier_halfspace_gt:
+lemma%important frontier_halfspace_gt:
   assumes "a \<noteq> 0 \<or> b \<noteq> 0"
     shows "frontier {x. a \<bullet> x > b} = {x. a \<bullet> x = b}"
-proof (cases "a = 0")
+proof%unimportant (cases "a = 0")
   case True with assms show ?thesis by simp
 next
   case False then show ?thesis
     by (force simp: frontier_def interior_open open_halfspace_gt)
 qed
 
-lemma interior_standard_hyperplane:
+lemma%important interior_standard_hyperplane:
    "interior {x :: (real^'n). x$k = a} = {}"
-proof -
+proof%unimportant -
   have "axis k (1::real) \<noteq> 0"
     by (simp add: axis_def vec_eq_iff)
   moreover have "axis k (1::real) \<bullet> x = x$k" for x
@@ -179,20 +179,20 @@
     by force
 qed
 
-lemma matrix_mult_transpose_dot_column:
+lemma%unimportant matrix_mult_transpose_dot_column:
   shows "transpose A ** A = (\<chi> i j. inner (column i A) (column j A))"
   by (simp add: matrix_matrix_mult_def vec_eq_iff transpose_def column_def inner_vec_def)
 
-lemma matrix_mult_transpose_dot_row:
+lemma%unimportant matrix_mult_transpose_dot_row:
   shows "A ** transpose A = (\<chi> i j. inner (row i A) (row j A))"
   by (simp add: matrix_matrix_mult_def vec_eq_iff transpose_def row_def inner_vec_def)
 
 text\<open>Two sometimes fruitful ways of looking at matrix-vector multiplication.\<close>
 
-lemma matrix_mult_dot: "A *v x = (\<chi> i. inner (A$i) x)"
+lemma%important matrix_mult_dot: "A *v x = (\<chi> i. inner (A$i) x)"
   by (simp add: matrix_vector_mult_def inner_vec_def)
 
-lemma adjoint_matrix: "adjoint(\<lambda>x. (A::real^'n^'m) *v x) = (\<lambda>x. transpose A *v x)"
+lemma%unimportant adjoint_matrix: "adjoint(\<lambda>x. (A::real^'n^'m) *v x) = (\<lambda>x. transpose A *v x)"
   apply (rule adjoint_unique)
   apply (simp add: transpose_def inner_vec_def matrix_vector_mult_def
     sum_distrib_right sum_distrib_left)
@@ -200,9 +200,9 @@
   apply (simp add:  ac_simps)
   done
 
-lemma matrix_adjoint: assumes lf: "linear (f :: real^'n \<Rightarrow> real ^'m)"
+lemma%important matrix_adjoint: assumes lf: "linear (f :: real^'n \<Rightarrow> real ^'m)"
   shows "matrix(adjoint f) = transpose(matrix f)"
-proof -
+proof%unimportant -
   have "matrix(adjoint f) = matrix(adjoint (( *v) (matrix f)))"
     by (simp add: lf)
   also have "\<dots> = transpose(matrix f)"
@@ -212,17 +212,17 @@
   finally show ?thesis .
 qed
 
-lemma matrix_vector_mul_bounded_linear[intro, simp]: "bounded_linear (( *v) A)" for A :: "'a::{euclidean_space,real_algebra_1}^'n^'m"
+lemma%unimportant matrix_vector_mul_bounded_linear[intro, simp]: "bounded_linear (( *v) A)" for A :: "'a::{euclidean_space,real_algebra_1}^'n^'m"
   using matrix_vector_mul_linear[of A]
   by (simp add: linear_conv_bounded_linear linear_matrix_vector_mul_eq)
 
-lemma
+lemma%unimportant (* FIX ME needs name*)
   fixes A :: "'a::{euclidean_space,real_algebra_1}^'n^'m"
   shows matrix_vector_mult_linear_continuous_at [continuous_intros]: "isCont (( *v) A) z"
     and matrix_vector_mult_linear_continuous_on [continuous_intros]: "continuous_on S (( *v) A)"
   by (simp_all add: linear_continuous_at linear_continuous_on)
 
-lemma scalar_invertible:
+lemma%unimportant scalar_invertible:
   fixes A :: "('a::real_algebra_1)^'m^'n"
   assumes "k \<noteq> 0" and "invertible A"
   shows "invertible (k *\<^sub>R A)"
@@ -236,50 +236,50 @@
     unfolding invertible_def by auto
 qed
 
-lemma scalar_invertible_iff:
+lemma%unimportant scalar_invertible_iff:
   fixes A :: "('a::real_algebra_1)^'m^'n"
   assumes "k \<noteq> 0" and "invertible A"
   shows "invertible (k *\<^sub>R A) \<longleftrightarrow> k \<noteq> 0 \<and> invertible A"
   by (simp add: assms scalar_invertible)
 
-lemma vector_transpose_matrix [simp]: "x v* transpose A = A *v x"
+lemma%unimportant vector_transpose_matrix [simp]: "x v* transpose A = A *v x"
   unfolding transpose_def vector_matrix_mult_def matrix_vector_mult_def
   by simp
 
-lemma transpose_matrix_vector [simp]: "transpose A *v x = x v* A"
+lemma%unimportant transpose_matrix_vector [simp]: "transpose A *v x = x v* A"
   unfolding transpose_def vector_matrix_mult_def matrix_vector_mult_def
   by simp
 
-lemma vector_scalar_commute:
+lemma%unimportant vector_scalar_commute:
   fixes A :: "'a::{field}^'m^'n"
   shows "A *v (c *s x) = c *s (A *v x)"
   by (simp add: vector_scalar_mult_def matrix_vector_mult_def mult_ac sum_distrib_left)
 
-lemma scalar_vector_matrix_assoc:
+lemma%unimportant scalar_vector_matrix_assoc:
   fixes k :: "'a::{field}" and x :: "'a::{field}^'n" and A :: "'a^'m^'n"
   shows "(k *s x) v* A = k *s (x v* A)"
   by (metis transpose_matrix_vector vector_scalar_commute)
  
-lemma vector_matrix_mult_0 [simp]: "0 v* A = 0"
+lemma%unimportant vector_matrix_mult_0 [simp]: "0 v* A = 0"
   unfolding vector_matrix_mult_def by (simp add: zero_vec_def)
 
-lemma vector_matrix_mult_0_right [simp]: "x v* 0 = 0"
+lemma%unimportant vector_matrix_mult_0_right [simp]: "x v* 0 = 0"
   unfolding vector_matrix_mult_def by (simp add: zero_vec_def)
 
-lemma vector_matrix_mul_rid [simp]:
+lemma%unimportant vector_matrix_mul_rid [simp]:
   fixes v :: "('a::semiring_1)^'n"
   shows "v v* mat 1 = v"
   by (metis matrix_vector_mul_lid transpose_mat vector_transpose_matrix)
 
-lemma scaleR_vector_matrix_assoc:
+lemma%unimportant scaleR_vector_matrix_assoc:
   fixes k :: real and x :: "real^'n" and A :: "real^'m^'n"
   shows "(k *\<^sub>R x) v* A = k *\<^sub>R (x v* A)"
   by (metis matrix_vector_mult_scaleR transpose_matrix_vector)
 
-lemma vector_scaleR_matrix_ac:
+lemma%important vector_scaleR_matrix_ac:
   fixes k :: real and x :: "real^'n" and A :: "real^'m^'n"
   shows "x v* (k *\<^sub>R A) = k *\<^sub>R (x v* A)"
-proof -
+proof%unimportant -
   have "x v* (k *\<^sub>R A) = (k *\<^sub>R x) v* A"
     unfolding vector_matrix_mult_def
     by (simp add: algebra_simps)
@@ -289,12 +289,12 @@
 qed
 
 
-subsection\<open>Some bounds on components etc. relative to operator norm\<close>
+subsection%important\<open>Some bounds on components etc. relative to operator norm\<close>
 
-lemma norm_column_le_onorm:
+lemma%important norm_column_le_onorm:
   fixes A :: "real^'n^'m"
   shows "norm(column i A) \<le> onorm(( *v) A)"
-proof -
+proof%unimportant -
   have "norm (\<chi> j. A $ j $ i) \<le> norm (A *v axis i 1)"
     by (simp add: matrix_mult_dot cart_eq_inner_axis)
   also have "\<dots> \<le> onorm (( *v) A)"
@@ -304,10 +304,10 @@
     unfolding column_def .
 qed
 
-lemma matrix_component_le_onorm:
+lemma%important matrix_component_le_onorm:
   fixes A :: "real^'n^'m"
   shows "\<bar>A $ i $ j\<bar> \<le> onorm(( *v) A)"
-proof -
+proof%unimportant -
   have "\<bar>A $ i $ j\<bar> \<le> norm (\<chi> n. (A $ n $ j))"
     by (metis (full_types, lifting) component_le_norm_cart vec_lambda_beta)
   also have "\<dots> \<le> onorm (( *v) A)"
@@ -315,15 +315,15 @@
   finally show ?thesis .
 qed
 
-lemma component_le_onorm:
+lemma%unimportant component_le_onorm:
   fixes f :: "real^'m \<Rightarrow> real^'n"
   shows "linear f \<Longrightarrow> \<bar>matrix f $ i $ j\<bar> \<le> onorm f"
   by (metis linear_matrix_vector_mul_eq matrix_component_le_onorm matrix_vector_mul)
 
-lemma onorm_le_matrix_component_sum:
+lemma%important onorm_le_matrix_component_sum:
   fixes A :: "real^'n^'m"
   shows "onorm(( *v) A) \<le> (\<Sum>i\<in>UNIV. \<Sum>j\<in>UNIV. \<bar>A $ i $ j\<bar>)"
-proof (rule onorm_le)
+proof%unimportant (rule onorm_le)
   fix x
   have "norm (A *v x) \<le> (\<Sum>i\<in>UNIV. \<bar>(A *v x) $ i\<bar>)"
     by (rule norm_le_l1_cart)
@@ -342,11 +342,11 @@
     by (simp add: sum_distrib_right)
 qed
 
-lemma onorm_le_matrix_component:
+lemma%important onorm_le_matrix_component:
   fixes A :: "real^'n^'m"
   assumes "\<And>i j. abs(A$i$j) \<le> B"
   shows "onorm(( *v) A) \<le> real (CARD('m)) * real (CARD('n)) * B"
-proof (rule onorm_le)
+proof%unimportant (rule onorm_le)
   fix x :: "real^'n::_"
   have "norm (A *v x) \<le> (\<Sum>i\<in>UNIV. \<bar>(A *v x) $ i\<bar>)"
     by (rule norm_le_l1_cart)
@@ -366,11 +366,11 @@
   finally show "norm (A *v x) \<le> CARD('m) * real (CARD('n)) * B * norm x" .
 qed
 
-subsection \<open>lambda skolemization on cartesian products\<close>
+subsection%important \<open>lambda skolemization on cartesian products\<close>
 
-lemma lambda_skolem: "(\<forall>i. \<exists>x. P i x) \<longleftrightarrow>
+lemma%important lambda_skolem: "(\<forall>i. \<exists>x. P i x) \<longleftrightarrow>
    (\<exists>x::'a ^ 'n. \<forall>i. P i (x $ i))" (is "?lhs \<longleftrightarrow> ?rhs")
-proof -
+proof%unimportant -
   let ?S = "(UNIV :: 'n set)"
   { assume H: "?rhs"
     then have ?lhs by auto }
@@ -387,16 +387,16 @@
   ultimately show ?thesis by metis
 qed
 
-lemma rational_approximation:
+lemma%unimportant rational_approximation:
   assumes "e > 0"
   obtains r::real where "r \<in> \<rat>" "\<bar>r - x\<bar> < e"
   using Rats_dense_in_real [of "x - e/2" "x + e/2"] assms by auto
 
-lemma matrix_rational_approximation:
+lemma%important matrix_rational_approximation:
   fixes A :: "real^'n^'m"
   assumes "e > 0"
   obtains B where "\<And>i j. B$i$j \<in> \<rat>" "onorm(\<lambda>x. (A - B) *v x) < e"
-proof -
+proof%unimportant -
   have "\<forall>i j. \<exists>q \<in> \<rat>. \<bar>q - A $ i $ j\<bar> < e / (2 * CARD('m) * CARD('n))"
     using assms by (force intro: rational_approximation [of "e / (2 * CARD('m) * CARD('n))"])
   then obtain B where B: "\<And>i j. B$i$j \<in> \<rat>" and Bclo: "\<And>i j. \<bar>B$i$j - A $ i $ j\<bar> < e / (2 * CARD('m) * CARD('n))"
@@ -413,7 +413,7 @@
   qed (use B in auto)
 qed
 
-lemma vector_sub_project_orthogonal_cart: "(b::real^'n) \<bullet> (x - ((b \<bullet> x) / (b \<bullet> b)) *s b) = 0"
+lemma%unimportant vector_sub_project_orthogonal_cart: "(b::real^'n) \<bullet> (x - ((b \<bullet> x) / (b \<bullet> b)) *s b) = 0"
   unfolding inner_simps scalar_mult_eq_scaleR by auto
 
 
@@ -422,51 +422,51 @@
 
 text \<open>Considering an n-element vector as an n-by-1 or 1-by-n matrix.\<close>
 
-definition "rowvector v = (\<chi> i j. (v$j))"
+definition%unimportant "rowvector v = (\<chi> i j. (v$j))"
 
-definition "columnvector v = (\<chi> i j. (v$i))"
+definition%unimportant "columnvector v = (\<chi> i j. (v$i))"
 
-lemma transpose_columnvector: "transpose(columnvector v) = rowvector v"
+lemma%unimportant transpose_columnvector: "transpose(columnvector v) = rowvector v"
   by (simp add: transpose_def rowvector_def columnvector_def vec_eq_iff)
 
-lemma transpose_rowvector: "transpose(rowvector v) = columnvector v"
+lemma%unimportant transpose_rowvector: "transpose(rowvector v) = columnvector v"
   by (simp add: transpose_def columnvector_def rowvector_def vec_eq_iff)
 
-lemma dot_rowvector_columnvector: "columnvector (A *v v) = A ** columnvector v"
+lemma%unimportant dot_rowvector_columnvector: "columnvector (A *v v) = A ** columnvector v"
   by (vector columnvector_def matrix_matrix_mult_def matrix_vector_mult_def)
 
-lemma dot_matrix_product:
+lemma%unimportant dot_matrix_product:
   "(x::real^'n) \<bullet> y = (((rowvector x ::real^'n^1) ** (columnvector y :: real^1^'n))$1)$1"
   by (vector matrix_matrix_mult_def rowvector_def columnvector_def inner_vec_def)
 
-lemma dot_matrix_vector_mul:
+lemma%unimportant dot_matrix_vector_mul:
   fixes A B :: "real ^'n ^'n" and x y :: "real ^'n"
   shows "(A *v x) \<bullet> (B *v y) =
       (((rowvector x :: real^'n^1) ** ((transpose A ** B) ** (columnvector y :: real ^1^'n)))$1)$1"
   unfolding dot_matrix_product transpose_columnvector[symmetric]
     dot_rowvector_columnvector matrix_transpose_mul matrix_mul_assoc ..
 
-lemma infnorm_cart:"infnorm (x::real^'n) = Sup {\<bar>x$i\<bar> |i. i\<in>UNIV}"
+lemma%unimportant infnorm_cart:"infnorm (x::real^'n) = Sup {\<bar>x$i\<bar> |i. i\<in>UNIV}"
   by (simp add: infnorm_def inner_axis Basis_vec_def) (metis (lifting) inner_axis real_inner_1_right)
 
-lemma component_le_infnorm_cart: "\<bar>x$i\<bar> \<le> infnorm (x::real^'n)"
+lemma%unimportant component_le_infnorm_cart: "\<bar>x$i\<bar> \<le> infnorm (x::real^'n)"
   using Basis_le_infnorm[of "axis i 1" x]
   by (simp add: Basis_vec_def axis_eq_axis inner_axis)
 
-lemma continuous_component[continuous_intros]: "continuous F f \<Longrightarrow> continuous F (\<lambda>x. f x $ i)"
+lemma%unimportant continuous_component[continuous_intros]: "continuous F f \<Longrightarrow> continuous F (\<lambda>x. f x $ i)"
   unfolding continuous_def by (rule tendsto_vec_nth)
 
-lemma continuous_on_component[continuous_intros]: "continuous_on s f \<Longrightarrow> continuous_on s (\<lambda>x. f x $ i)"
+lemma%unimportant continuous_on_component[continuous_intros]: "continuous_on s f \<Longrightarrow> continuous_on s (\<lambda>x. f x $ i)"
   unfolding continuous_on_def by (fast intro: tendsto_vec_nth)
 
-lemma continuous_on_vec_lambda[continuous_intros]:
+lemma%unimportant continuous_on_vec_lambda[continuous_intros]:
   "(\<And>i. continuous_on S (f i)) \<Longrightarrow> continuous_on S (\<lambda>x. \<chi> i. f i x)"
   unfolding continuous_on_def by (auto intro: tendsto_vec_lambda)
 
-lemma closed_positive_orthant: "closed {x::real^'n. \<forall>i. 0 \<le>x$i}"
+lemma%unimportant closed_positive_orthant: "closed {x::real^'n. \<forall>i. 0 \<le>x$i}"
   by (simp add: Collect_all_eq closed_INT closed_Collect_le continuous_on_const continuous_on_id continuous_on_component)
 
-lemma bounded_component_cart: "bounded s \<Longrightarrow> bounded ((\<lambda>x. x $ i) ` s)"
+lemma%unimportant bounded_component_cart: "bounded s \<Longrightarrow> bounded ((\<lambda>x. x $ i) ` s)"
   unfolding bounded_def
   apply clarify
   apply (rule_tac x="x $ i" in exI)
@@ -475,13 +475,13 @@
   apply (rule order_trans [OF dist_vec_nth_le], simp)
   done
 
-lemma compact_lemma_cart:
+lemma%important compact_lemma_cart:
   fixes f :: "nat \<Rightarrow> 'a::heine_borel ^ 'n"
   assumes f: "bounded (range f)"
   shows "\<exists>l r. strict_mono r \<and>
         (\<forall>e>0. eventually (\<lambda>n. \<forall>i\<in>d. dist (f (r n) $ i) (l $ i) < e) sequentially)"
     (is "?th d")
-proof -
+proof%unimportant -
   have "\<forall>d' \<subseteq> d. ?th d'"
     by (rule compact_lemma_general[where unproj=vec_lambda])
       (auto intro!: f bounded_component_cart simp: vec_lambda_eta)
@@ -517,19 +517,19 @@
   with r show "\<exists>l r. strict_mono r \<and> ((f \<circ> r) \<longlongrightarrow> l) sequentially" by auto
 qed
 
-lemma interval_cart:
+lemma%unimportant interval_cart:
   fixes a :: "real^'n"
   shows "box a b = {x::real^'n. \<forall>i. a$i < x$i \<and> x$i < b$i}"
     and "cbox a b = {x::real^'n. \<forall>i. a$i \<le> x$i \<and> x$i \<le> b$i}"
   by (auto simp add: set_eq_iff less_vec_def less_eq_vec_def mem_box Basis_vec_def inner_axis)
 
-lemma mem_box_cart:
+lemma%unimportant mem_box_cart:
   fixes a :: "real^'n"
   shows "x \<in> box a b \<longleftrightarrow> (\<forall>i. a$i < x$i \<and> x$i < b$i)"
     and "x \<in> cbox a b \<longleftrightarrow> (\<forall>i. a$i \<le> x$i \<and> x$i \<le> b$i)"
   using interval_cart[of a b] by (auto simp add: set_eq_iff less_vec_def less_eq_vec_def)
 
-lemma interval_eq_empty_cart:
+lemma%unimportant interval_eq_empty_cart:
   fixes a :: "real^'n"
   shows "(box a b = {} \<longleftrightarrow> (\<exists>i. b$i \<le> a$i))" (is ?th1)
     and "(cbox a b = {} \<longleftrightarrow> (\<exists>i. b$i < a$i))" (is ?th2)
@@ -565,14 +565,14 @@
   ultimately show ?th2 by blast
 qed
 
-lemma interval_ne_empty_cart:
+lemma%unimportant interval_ne_empty_cart:
   fixes a :: "real^'n"
   shows "cbox a b \<noteq> {} \<longleftrightarrow> (\<forall>i. a$i \<le> b$i)"
     and "box a b \<noteq> {} \<longleftrightarrow> (\<forall>i. a$i < b$i)"
   unfolding interval_eq_empty_cart[of a b] by (auto simp add: not_less not_le)
     (* BH: Why doesn't just "auto" work here? *)
 
-lemma subset_interval_imp_cart:
+lemma%unimportant subset_interval_imp_cart:
   fixes a :: "real^'n"
   shows "(\<forall>i. a$i \<le> c$i \<and> d$i \<le> b$i) \<Longrightarrow> cbox c d \<subseteq> cbox a b"
     and "(\<forall>i. a$i < c$i \<and> d$i < b$i) \<Longrightarrow> cbox c d \<subseteq> box a b"
@@ -581,13 +581,13 @@
   unfolding subset_eq[unfolded Ball_def] unfolding mem_box_cart
   by (auto intro: order_trans less_le_trans le_less_trans less_imp_le) (* BH: Why doesn't just "auto" work here? *)
 
-lemma interval_sing:
+lemma%unimportant interval_sing:
   fixes a :: "'a::linorder^'n"
   shows "{a .. a} = {a} \<and> {a<..<a} = {}"
   apply (auto simp add: set_eq_iff less_vec_def less_eq_vec_def vec_eq_iff)
   done
 
-lemma subset_interval_cart:
+lemma%unimportant subset_interval_cart:
   fixes a :: "real^'n"
   shows "cbox c d \<subseteq> cbox a b \<longleftrightarrow> (\<forall>i. c$i \<le> d$i) --> (\<forall>i. a$i \<le> c$i \<and> d$i \<le> b$i)" (is ?th1)
     and "cbox c d \<subseteq> box a b \<longleftrightarrow> (\<forall>i. c$i \<le> d$i) --> (\<forall>i. a$i < c$i \<and> d$i < b$i)" (is ?th2)
@@ -595,7 +595,7 @@
     and "box c d \<subseteq> box a b \<longleftrightarrow> (\<forall>i. c$i < d$i) --> (\<forall>i. a$i \<le> c$i \<and> d$i \<le> b$i)" (is ?th4)
   using subset_box[of c d a b] by (simp_all add: Basis_vec_def inner_axis)
 
-lemma disjoint_interval_cart:
+lemma%unimportant disjoint_interval_cart:
   fixes a::"real^'n"
   shows "cbox a b \<inter> cbox c d = {} \<longleftrightarrow> (\<exists>i. (b$i < a$i \<or> d$i < c$i \<or> b$i < c$i \<or> d$i < a$i))" (is ?th1)
     and "cbox a b \<inter> box c d = {} \<longleftrightarrow> (\<exists>i. (b$i < a$i \<or> d$i \<le> c$i \<or> b$i \<le> c$i \<or> d$i \<le> a$i))" (is ?th2)
@@ -603,53 +603,53 @@
     and "box a b \<inter> box c d = {} \<longleftrightarrow> (\<exists>i. (b$i \<le> a$i \<or> d$i \<le> c$i \<or> b$i \<le> c$i \<or> d$i \<le> a$i))" (is ?th4)
   using disjoint_interval[of a b c d] by (simp_all add: Basis_vec_def inner_axis)
 
-lemma Int_interval_cart:
+lemma%unimportant Int_interval_cart:
   fixes a :: "real^'n"
   shows "cbox a b \<inter> cbox c d =  {(\<chi> i. max (a$i) (c$i)) .. (\<chi> i. min (b$i) (d$i))}"
   unfolding Int_interval
   by (auto simp: mem_box less_eq_vec_def)
     (auto simp: Basis_vec_def inner_axis)
 
-lemma closed_interval_left_cart:
+lemma%unimportant closed_interval_left_cart:
   fixes b :: "real^'n"
   shows "closed {x::real^'n. \<forall>i. x$i \<le> b$i}"
   by (simp add: Collect_all_eq closed_INT closed_Collect_le continuous_on_const continuous_on_id continuous_on_component)
 
-lemma closed_interval_right_cart:
+lemma%unimportant closed_interval_right_cart:
   fixes a::"real^'n"
   shows "closed {x::real^'n. \<forall>i. a$i \<le> x$i}"
   by (simp add: Collect_all_eq closed_INT closed_Collect_le continuous_on_const continuous_on_id continuous_on_component)
 
-lemma is_interval_cart:
+lemma%unimportant is_interval_cart:
   "is_interval (s::(real^'n) set) \<longleftrightarrow>
     (\<forall>a\<in>s. \<forall>b\<in>s. \<forall>x. (\<forall>i. ((a$i \<le> x$i \<and> x$i \<le> b$i) \<or> (b$i \<le> x$i \<and> x$i \<le> a$i))) \<longrightarrow> x \<in> s)"
   by (simp add: is_interval_def Ball_def Basis_vec_def inner_axis imp_ex)
 
-lemma closed_halfspace_component_le_cart: "closed {x::real^'n. x$i \<le> a}"
+lemma%unimportant closed_halfspace_component_le_cart: "closed {x::real^'n. x$i \<le> a}"
   by (simp add: closed_Collect_le continuous_on_const continuous_on_id continuous_on_component)
 
-lemma closed_halfspace_component_ge_cart: "closed {x::real^'n. x$i \<ge> a}"
+lemma%unimportant closed_halfspace_component_ge_cart: "closed {x::real^'n. x$i \<ge> a}"
   by (simp add: closed_Collect_le continuous_on_const continuous_on_id continuous_on_component)
 
-lemma open_halfspace_component_lt_cart: "open {x::real^'n. x$i < a}"
+lemma%unimportant open_halfspace_component_lt_cart: "open {x::real^'n. x$i < a}"
   by (simp add: open_Collect_less continuous_on_const continuous_on_id continuous_on_component)
 
-lemma open_halfspace_component_gt_cart: "open {x::real^'n. x$i  > a}"
+lemma%unimportant open_halfspace_component_gt_cart: "open {x::real^'n. x$i  > a}"
   by (simp add: open_Collect_less continuous_on_const continuous_on_id continuous_on_component)
 
-lemma Lim_component_le_cart:
+lemma%unimportant Lim_component_le_cart:
   fixes f :: "'a \<Rightarrow> real^'n"
   assumes "(f \<longlongrightarrow> l) net" "\<not> (trivial_limit net)"  "eventually (\<lambda>x. f x $i \<le> b) net"
   shows "l$i \<le> b"
   by (rule tendsto_le[OF assms(2) tendsto_const tendsto_vec_nth, OF assms(1, 3)])
 
-lemma Lim_component_ge_cart:
+lemma%unimportant Lim_component_ge_cart:
   fixes f :: "'a \<Rightarrow> real^'n"
   assumes "(f \<longlongrightarrow> l) net"  "\<not> (trivial_limit net)"  "eventually (\<lambda>x. b \<le> (f x)$i) net"
   shows "b \<le> l$i"
   by (rule tendsto_le[OF assms(2) tendsto_vec_nth tendsto_const, OF assms(1, 3)])
 
-lemma Lim_component_eq_cart:
+lemma%unimportant Lim_component_eq_cart:
   fixes f :: "'a \<Rightarrow> real^'n"
   assumes net: "(f \<longlongrightarrow> l) net" "~(trivial_limit net)" and ev:"eventually (\<lambda>x. f(x)$i = b) net"
   shows "l$i = b"
@@ -657,18 +657,18 @@
     Lim_component_ge_cart[OF net, of b i] and
     Lim_component_le_cart[OF net, of i b] by auto
 
-lemma connected_ivt_component_cart:
+lemma%unimportant connected_ivt_component_cart:
   fixes x :: "real^'n"
   shows "connected s \<Longrightarrow> x \<in> s \<Longrightarrow> y \<in> s \<Longrightarrow> x$k \<le> a \<Longrightarrow> a \<le> y$k \<Longrightarrow> (\<exists>z\<in>s.  z$k = a)"
   using connected_ivt_hyperplane[of s x y "axis k 1" a]
   by (auto simp add: inner_axis inner_commute)
 
-lemma subspace_substandard_cart: "vec.subspace {x. (\<forall>i. P i \<longrightarrow> x$i = 0)}"
+lemma%unimportant subspace_substandard_cart: "vec.subspace {x. (\<forall>i. P i \<longrightarrow> x$i = 0)}"
   unfolding vec.subspace_def by auto
 
-lemma closed_substandard_cart:
+lemma%important closed_substandard_cart:
   "closed {x::'a::real_normed_vector ^ 'n. \<forall>i. P i \<longrightarrow> x$i = 0}"
-proof -
+proof%unimportant -
   { fix i::'n
     have "closed {x::'a ^ 'n. P i \<longrightarrow> x$i = 0}"
       by (cases "P i") (simp_all add: closed_Collect_eq continuous_on_const continuous_on_id continuous_on_component) }
@@ -676,9 +676,9 @@
     unfolding Collect_all_eq by (simp add: closed_INT)
 qed
 
-lemma dim_substandard_cart: "vec.dim {x::'a::field^'n. \<forall>i. i \<notin> d \<longrightarrow> x$i = 0} = card d"
+lemma%important dim_substandard_cart: "vec.dim {x::'a::field^'n. \<forall>i. i \<notin> d \<longrightarrow> x$i = 0} = card d"
   (is "vec.dim ?A = _")
-proof (rule vec.dim_unique)
+proof%unimportant (rule vec.dim_unique)
   let ?B = "((\<lambda>x. axis x 1) ` d)"
   have subset_basis: "?B \<subseteq> cart_basis"
     by (auto simp: cart_basis_def)
@@ -703,27 +703,27 @@
   then show "?A \<subseteq> vec.span ?B" by auto
 qed (simp add: card_image inj_on_def axis_eq_axis)
 
-lemma dim_subset_UNIV_cart_gen:
+lemma%unimportant dim_subset_UNIV_cart_gen:
   fixes S :: "('a::field^'n) set"
   shows "vec.dim S \<le> CARD('n)"
   by (metis vec.dim_eq_full vec.dim_subset_UNIV vec.span_UNIV vec_dim_card)
 
-lemma dim_subset_UNIV_cart:
+lemma%unimportant dim_subset_UNIV_cart:
   fixes S :: "(real^'n) set"
   shows "dim S \<le> CARD('n)"
   using dim_subset_UNIV_cart_gen[of S] by (simp add: dim_vec_eq)
 
-lemma affinity_inverses:
+lemma%unimportant affinity_inverses:
   assumes m0: "m \<noteq> (0::'a::field)"
   shows "(\<lambda>x. m *s x + c) \<circ> (\<lambda>x. inverse(m) *s x + (-(inverse(m) *s c))) = id"
   "(\<lambda>x. inverse(m) *s x + (-(inverse(m) *s c))) \<circ> (\<lambda>x. m *s x + c) = id"
   using m0
   by (auto simp add: fun_eq_iff vector_add_ldistrib diff_conv_add_uminus simp del: add_uminus_conv_diff)
 
-lemma vector_affinity_eq:
+lemma%important vector_affinity_eq:
   assumes m0: "(m::'a::field) \<noteq> 0"
   shows "m *s x + c = y \<longleftrightarrow> x = inverse m *s y + -(inverse m *s c)"
-proof
+proof%unimportant
   assume h: "m *s x + c = y"
   hence "m *s x = y - c" by (simp add: field_simps)
   hence "inverse m *s (m *s x) = inverse m *s (y - c)" by simp
@@ -735,48 +735,48 @@
     using m0 by (simp add: vector_smult_assoc vector_ssub_ldistrib)
 qed
 
-lemma vector_eq_affinity:
+lemma%unimportant vector_eq_affinity:
     "(m::'a::field) \<noteq> 0 ==> (y = m *s x + c \<longleftrightarrow> inverse(m) *s y + -(inverse(m) *s c) = x)"
   using vector_affinity_eq[where m=m and x=x and y=y and c=c]
   by metis
 
-lemma vector_cart:
+lemma%unimportant vector_cart:
   fixes f :: "real^'n \<Rightarrow> real"
   shows "(\<chi> i. f (axis i 1)) = (\<Sum>i\<in>Basis. f i *\<^sub>R i)"
   unfolding euclidean_eq_iff[where 'a="real^'n"]
   by simp (simp add: Basis_vec_def inner_axis)
 
-lemma const_vector_cart:"((\<chi> i. d)::real^'n) = (\<Sum>i\<in>Basis. d *\<^sub>R i)"
+lemma%unimportant const_vector_cart:"((\<chi> i. d)::real^'n) = (\<Sum>i\<in>Basis. d *\<^sub>R i)"
   by (rule vector_cart)
 
-subsection "Convex Euclidean Space"
+subsection%important "Convex Euclidean Space"
 
-lemma Cart_1:"(1::real^'n) = \<Sum>Basis"
+lemma%unimportant Cart_1:"(1::real^'n) = \<Sum>Basis"
   using const_vector_cart[of 1] by (simp add: one_vec_def)
 
 declare vector_add_ldistrib[simp] vector_ssub_ldistrib[simp] vector_smult_assoc[simp] vector_smult_rneg[simp]
 declare vector_sadd_rdistrib[simp] vector_sub_rdistrib[simp]
 
-lemmas vector_component_simps = vector_minus_component vector_smult_component vector_add_component less_eq_vec_def vec_lambda_beta vector_uminus_component
+lemmas%unimportant vector_component_simps = vector_minus_component vector_smult_component vector_add_component less_eq_vec_def vec_lambda_beta vector_uminus_component
 
-lemma convex_box_cart:
+lemma%unimportant convex_box_cart:
   assumes "\<And>i. convex {x. P i x}"
   shows "convex {x. \<forall>i. P i (x$i)}"
   using assms unfolding convex_def by auto
 
-lemma convex_positive_orthant_cart: "convex {x::real^'n. (\<forall>i. 0 \<le> x$i)}"
+lemma%unimportant convex_positive_orthant_cart: "convex {x::real^'n. (\<forall>i. 0 \<le> x$i)}"
   by (rule convex_box_cart) (simp add: atLeast_def[symmetric])
 
-lemma unit_interval_convex_hull_cart:
+lemma%unimportant unit_interval_convex_hull_cart:
   "cbox (0::real^'n) 1 = convex hull {x. \<forall>i. (x$i = 0) \<or> (x$i = 1)}"
   unfolding Cart_1 unit_interval_convex_hull[where 'a="real^'n"] box_real[symmetric]
   by (rule arg_cong[where f="\<lambda>x. convex hull x"]) (simp add: Basis_vec_def inner_axis)
 
-lemma cube_convex_hull_cart:
+lemma%important cube_convex_hull_cart:
   assumes "0 < d"
   obtains s::"(real^'n) set"
     where "finite s" "cbox (x - (\<chi> i. d)) (x + (\<chi> i. d)) = convex hull s"
-proof -
+proof%unimportant -
   from assms obtain s where "finite s"
     and "cbox (x - sum (( *\<^sub>R) d) Basis) (x + sum (( *\<^sub>R) d) Basis) = convex hull s"
     by (rule cube_convex_hull)
@@ -785,14 +785,14 @@
 qed
 
 
-subsection "Derivative"
+subsection%important "Derivative"
 
-definition "jacobian f net = matrix(frechet_derivative f net)"
+definition%important "jacobian f net = matrix(frechet_derivative f net)"
 
-lemma jacobian_works:
+lemma%important jacobian_works:
   "(f::(real^'a) \<Rightarrow> (real^'b)) differentiable net \<longleftrightarrow>
     (f has_derivative (\<lambda>h. (jacobian f net) *v h)) net" (is "?lhs = ?rhs")
-proof
+proof%unimportant
   assume ?lhs then show ?rhs
     by (simp add: frechet_derivative_works has_derivative_linear jacobian_def)
 next
@@ -801,10 +801,10 @@
 qed
 
 
-subsection \<open>Component of the differential must be zero if it exists at a local
+subsection%important \<open>Component of the differential must be zero if it exists at a local
   maximum or minimum for that corresponding component\<close>
 
-lemma differential_zero_maxmin_cart:
+lemma%important differential_zero_maxmin_cart:
   fixes f::"real^'a \<Rightarrow> real^'b"
   assumes "0 < e" "((\<forall>y \<in> ball x e. (f y)$k \<le> (f x)$k) \<or> (\<forall>y\<in>ball x e. (f x)$k \<le> (f y)$k))"
     "f differentiable (at x)"
@@ -813,7 +813,7 @@
     vector_cart[of "\<lambda>j. frechet_derivative f (at x) j $ k"]
   by (simp add: Basis_vec_def axis_eq_axis inner_axis jacobian_def matrix_def)
 
-subsection \<open>Lemmas for working on @{typ "real^1"}\<close>
+subsection%unimportant \<open>Lemmas for working on @{typ "real^1"}\<close>
 
 lemma forall_1[simp]: "(\<forall>i::1. P i) \<longleftrightarrow> P 1"
   by (metis (full_types) num1_eq_iff)
@@ -893,7 +893,7 @@
 instance num1 :: wellorder
   by intro_classes (auto simp: less_eq_num1_def less_num1_def)
 
-subsection\<open>The collapse of the general concepts to dimension one\<close>
+subsection%unimportant\<open>The collapse of the general concepts to dimension one\<close>
 
 lemma vector_one: "(x::'a ^1) = (\<chi> i. (x$1))"
   by (simp add: vec_eq_iff)
@@ -918,11 +918,11 @@
 lemma dist_real: "dist(x::real ^ 1) y = \<bar>(x$1) - (y$1)\<bar>"
   by (auto simp add: norm_real dist_norm)
 
-subsection\<open> Rank of a matrix\<close>
+subsection%important\<open> Rank of a matrix\<close>
 
 text\<open>Equivalence of row and column rank is taken from George Mackiw's paper, Mathematics Magazine 1995, p. 285.\<close>
 
-lemma matrix_vector_mult_in_columnspace_gen:
+lemma%unimportant matrix_vector_mult_in_columnspace_gen:
   fixes A :: "'a::field^'n^'m"
   shows "(A *v x) \<in> vec.span(columns A)"
   apply (simp add: matrix_vector_column columns_def transpose_def column_def)
@@ -930,17 +930,17 @@
   apply (force intro: vec.span_base)
   done
 
-lemma matrix_vector_mult_in_columnspace:
+lemma%unimportant matrix_vector_mult_in_columnspace:
   fixes A :: "real^'n^'m"
   shows "(A *v x) \<in> span(columns A)"
   using matrix_vector_mult_in_columnspace_gen[of A x] by (simp add: span_vec_eq)
 
-lemma orthogonal_nullspace_rowspace:
+lemma%important orthogonal_nullspace_rowspace:
   fixes A :: "real^'n^'m"
   assumes 0: "A *v x = 0" and y: "y \<in> span(rows A)"
   shows "orthogonal x y"
   using y
-proof (induction rule: span_induct)
+proof%unimportant (induction rule: span_induct)
   case base
   then show ?case
     by (simp add: subspace_orthogonal_to_vector)
@@ -953,28 +953,28 @@
     by (simp add: mult.commute) (metis (no_types) vec_lambda_beta zero_index)
 qed
 
-lemma nullspace_inter_rowspace:
+lemma%unimportant nullspace_inter_rowspace:
   fixes A :: "real^'n^'m"
   shows "A *v x = 0 \<and> x \<in> span(rows A) \<longleftrightarrow> x = 0"
   using orthogonal_nullspace_rowspace orthogonal_self span_zero matrix_vector_mult_0_right
   by blast
 
-lemma matrix_vector_mul_injective_on_rowspace:
+lemma%unimportant matrix_vector_mul_injective_on_rowspace:
   fixes A :: "real^'n^'m"
   shows "\<lbrakk>A *v x = A *v y; x \<in> span(rows A); y \<in> span(rows A)\<rbrakk> \<Longrightarrow> x = y"
   using nullspace_inter_rowspace [of A "x-y"]
   by (metis diff_eq_diff_eq diff_self matrix_vector_mult_diff_distrib span_diff)
 
-definition rank :: "'a::field^'n^'m=>nat"
+definition%important rank :: "'a::field^'n^'m=>nat"
   where row_rank_def_gen: "rank A \<equiv> vec.dim(rows A)"
 
-lemma row_rank_def: "rank A = dim (rows A)" for A::"real^'n^'m"
-  by (auto simp: row_rank_def_gen dim_vec_eq)
+lemma%important row_rank_def: "rank A = dim (rows A)" for A::"real^'n^'m"
+  by%unimportant (auto simp: row_rank_def_gen dim_vec_eq)
 
-lemma dim_rows_le_dim_columns:
+lemma%important dim_rows_le_dim_columns:
   fixes A :: "real^'n^'m"
   shows "dim(rows A) \<le> dim(columns A)"
-proof -
+proof%unimportant -
   have "dim (span (rows A)) \<le> dim (span (columns A))"
   proof -
     obtain B where "independent B" "span(rows A) \<subseteq> span B"
@@ -999,32 +999,32 @@
     by (simp add: dim_span)
 qed
 
-lemma column_rank_def:
+lemma%unimportant column_rank_def:
   fixes A :: "real^'n^'m"
   shows "rank A = dim(columns A)"
   unfolding row_rank_def
   by (metis columns_transpose dim_rows_le_dim_columns le_antisym rows_transpose)
 
-lemma rank_transpose:
+lemma%unimportant rank_transpose:
   fixes A :: "real^'n^'m"
   shows "rank(transpose A) = rank A"
   by (metis column_rank_def row_rank_def rows_transpose)
 
-lemma matrix_vector_mult_basis:
+lemma%unimportant matrix_vector_mult_basis:
   fixes A :: "real^'n^'m"
   shows "A *v (axis k 1) = column k A"
   by (simp add: cart_eq_inner_axis column_def matrix_mult_dot)
 
-lemma columns_image_basis:
+lemma%unimportant columns_image_basis:
   fixes A :: "real^'n^'m"
   shows "columns A = ( *v) A ` (range (\<lambda>i. axis i 1))"
   by (force simp: columns_def matrix_vector_mult_basis [symmetric])
 
-lemma rank_dim_range:
+lemma%important rank_dim_range:
   fixes A :: "real^'n^'m"
   shows "rank A = dim(range (\<lambda>x. A *v x))"
   unfolding column_rank_def
-proof (rule span_eq_dim)
+proof%unimportant (rule span_eq_dim)
   have "span (columns A) \<subseteq> span (range (( *v) A))" (is "?l \<subseteq> ?r")
     by (simp add: columns_image_basis image_subsetI span_mono)
   then show "?l = ?r"
@@ -1032,45 +1032,45 @@
         span_eq span_span)
 qed
 
-lemma rank_bound:
+lemma%unimportant rank_bound:
   fixes A :: "real^'n^'m"
   shows "rank A \<le> min CARD('m) (CARD('n))"
   by (metis (mono_tags, lifting) dim_subset_UNIV_cart min.bounded_iff
       column_rank_def row_rank_def)
 
-lemma full_rank_injective:
+lemma%unimportant full_rank_injective:
   fixes A :: "real^'n^'m"
   shows "rank A = CARD('n) \<longleftrightarrow> inj (( *v) A)"
   by (simp add: matrix_left_invertible_injective [symmetric] matrix_left_invertible_span_rows row_rank_def
       dim_eq_full [symmetric] card_cart_basis vec.dimension_def)
 
-lemma full_rank_surjective:
+lemma%unimportant full_rank_surjective:
   fixes A :: "real^'n^'m"
   shows "rank A = CARD('m) \<longleftrightarrow> surj (( *v) A)"
   by (simp add: matrix_right_invertible_surjective [symmetric] left_invertible_transpose [symmetric]
                 matrix_left_invertible_injective full_rank_injective [symmetric] rank_transpose)
 
-lemma rank_I: "rank(mat 1::real^'n^'n) = CARD('n)"
+lemma%unimportant rank_I: "rank(mat 1::real^'n^'n) = CARD('n)"
   by (simp add: full_rank_injective inj_on_def)
 
-lemma less_rank_noninjective:
+lemma%unimportant less_rank_noninjective:
   fixes A :: "real^'n^'m"
   shows "rank A < CARD('n) \<longleftrightarrow> \<not> inj (( *v) A)"
 using less_le rank_bound by (auto simp: full_rank_injective [symmetric])
 
-lemma matrix_nonfull_linear_equations_eq:
+lemma%unimportant matrix_nonfull_linear_equations_eq:
   fixes A :: "real^'n^'m"
   shows "(\<exists>x. (x \<noteq> 0) \<and> A *v x = 0) \<longleftrightarrow> ~(rank A = CARD('n))"
   by (meson matrix_left_invertible_injective full_rank_injective matrix_left_invertible_ker)
 
-lemma rank_eq_0: "rank A = 0 \<longleftrightarrow> A = 0" and rank_0 [simp]: "rank (0::real^'n^'m) = 0"
+lemma%unimportant rank_eq_0: "rank A = 0 \<longleftrightarrow> A = 0" and rank_0 [simp]: "rank (0::real^'n^'m) = 0"
   for A :: "real^'n^'m"
   by (auto simp: rank_dim_range matrix_eq)
 
-lemma rank_mul_le_right:
+lemma%important rank_mul_le_right:
   fixes A :: "real^'n^'m" and B :: "real^'p^'n"
   shows "rank(A ** B) \<le> rank B"
-proof -
+proof%unimportant -
   have "rank(A ** B) \<le> dim (( *v) A ` range (( *v) B))"
     by (auto simp: rank_dim_range image_comp o_def matrix_vector_mul_assoc)
   also have "\<dots> \<le> rank B"
@@ -1078,12 +1078,12 @@
   finally show ?thesis .
 qed
 
-lemma rank_mul_le_left:
+lemma%unimportant rank_mul_le_left:
   fixes A :: "real^'n^'m" and B :: "real^'p^'n"
   shows "rank(A ** B) \<le> rank A"
   by (metis matrix_transpose_mul rank_mul_le_right rank_transpose)
 
-subsection\<open>Routine results connecting the types @{typ "real^1"} and @{typ real}\<close>
+subsection%unimportant\<open>Routine results connecting the types @{typ "real^1"} and @{typ real}\<close>
 
 lemma vector_one_nth [simp]:
   fixes x :: "'a^1" shows "vec (x $ 1) = x"
@@ -1146,7 +1146,7 @@
     done
 
 
-subsection\<open>Explicit vector construction from lists\<close>
+subsection%unimportant\<open>Explicit vector construction from lists\<close>
 
 definition "vector l = (\<chi> i. foldr (\<lambda>x f n. fun_upd (f (n+1)) n x) l (\<lambda>n x. 0) 1 i)"
 

--- a/src/HOL/Analysis/Cartesian_Space.thy	Mon Aug 27 22:58:36 2018 +0200
+++ b/src/HOL/Analysis/Cartesian_Space.thy	Tue Aug 28 13:28:39 2018 +0100
@@ -11,19 +11,19 @@
     Finite_Cartesian_Product Linear_Algebra
 begin
 
-definition "cart_basis = {axis i 1 | i. i\<in>UNIV}"
+definition%unimportant "cart_basis = {axis i 1 | i. i\<in>UNIV}"
 
-lemma finite_cart_basis: "finite (cart_basis)" unfolding cart_basis_def
+lemma%unimportant finite_cart_basis: "finite (cart_basis)" unfolding cart_basis_def
   using finite_Atleast_Atmost_nat by fastforce
 
-lemma card_cart_basis: "card (cart_basis::('a::zero_neq_one^'i) set) = CARD('i)"
+lemma%unimportant card_cart_basis: "card (cart_basis::('a::zero_neq_one^'i) set) = CARD('i)"
   unfolding cart_basis_def Setcompr_eq_image
   by (rule card_image) (auto simp: inj_on_def axis_eq_axis)
 
 interpretation vec: vector_space "( *s) "
   by unfold_locales (vector algebra_simps)+
 
-lemma independent_cart_basis:
+lemma%unimportant independent_cart_basis:
   "vec.independent (cart_basis)"
 proof (rule vec.independent_if_scalars_zero)
   show "finite (cart_basis)" using finite_cart_basis .
@@ -48,7 +48,7 @@
   finally show "f x = 0" ..
 qed
 
-lemma span_cart_basis:
+lemma%unimportant span_cart_basis:
   "vec.span (cart_basis) = UNIV"
 proof (auto)
   fix x::"('a, 'b) vec"
@@ -93,12 +93,12 @@
 interpretation vec: finite_dimensional_vector_space "( *s)" "cart_basis"
   by (unfold_locales, auto simp add: finite_cart_basis independent_cart_basis span_cart_basis)
 
-lemma matrix_vector_mul_linear_gen[intro, simp]:
+lemma%unimportant matrix_vector_mul_linear_gen[intro, simp]:
   "Vector_Spaces.linear ( *s) ( *s) (( *v) A)"
   by unfold_locales
     (vector matrix_vector_mult_def sum.distrib algebra_simps)+
 
-lemma span_vec_eq: "vec.span X = span X"
+lemma%important span_vec_eq: "vec.span X = span X"
   and dim_vec_eq: "vec.dim X = dim X"
   and dependent_vec_eq: "vec.dependent X = dependent X"
   and subspace_vec_eq: "vec.subspace X = subspace X"
@@ -106,11 +106,11 @@
   unfolding span_raw_def dim_raw_def dependent_raw_def subspace_raw_def
   by (auto simp: scalar_mult_eq_scaleR)
 
-lemma linear_componentwise:
+lemma%important linear_componentwise:
   fixes f:: "'a::field ^'m \<Rightarrow> 'a ^ 'n"
   assumes lf: "Vector_Spaces.linear ( *s) ( *s) f"
   shows "(f x)$j = sum (\<lambda>i. (x$i) * (f (axis i 1)$j)) (UNIV :: 'm set)" (is "?lhs = ?rhs")
-proof -
+proof%unimportant -
   interpret lf: Vector_Spaces.linear "( *s)" "( *s)" f
     using lf .
   let ?M = "(UNIV :: 'm set)"
@@ -128,7 +128,7 @@
 
 interpretation vec: finite_dimensional_vector_space_pair "( *s)" cart_basis "( *s)" cart_basis ..
 
-lemma matrix_works:
+lemma%unimportant matrix_works:
   assumes lf: "Vector_Spaces.linear ( *s) ( *s) f"
   shows "matrix f *v x = f (x::'a::field ^ 'n)"
   apply (simp add: matrix_def matrix_vector_mult_def vec_eq_iff mult.commute)
@@ -136,45 +136,45 @@
   apply (rule linear_componentwise[OF lf, symmetric])
   done
 
-lemma matrix_of_matrix_vector_mul[simp]: "matrix(\<lambda>x. A *v (x :: 'a::field ^ 'n)) = A"
+lemma%unimportant matrix_of_matrix_vector_mul[simp]: "matrix(\<lambda>x. A *v (x :: 'a::field ^ 'n)) = A"
   by (simp add: matrix_eq matrix_works)
 
-lemma matrix_compose_gen:
+lemma%unimportant matrix_compose_gen:
   assumes lf: "Vector_Spaces.linear ( *s) ( *s) (f::'a::{field}^'n \<Rightarrow> 'a^'m)"
     and lg: "Vector_Spaces.linear ( *s) ( *s) (g::'a^'m \<Rightarrow> 'a^_)"
   shows "matrix (g o f) = matrix g ** matrix f"
   using lf lg Vector_Spaces.linear_compose[OF lf lg] matrix_works[OF Vector_Spaces.linear_compose[OF lf lg]]
   by (simp add: matrix_eq matrix_works matrix_vector_mul_assoc[symmetric] o_def)
 
-lemma matrix_compose:
+lemma%unimportant matrix_compose:
   assumes "linear (f::real^'n \<Rightarrow> real^'m)" "linear (g::real^'m \<Rightarrow> real^_)"
   shows "matrix (g o f) = matrix g ** matrix f"
   using matrix_compose_gen[of f g] assms
   by (simp add: linear_def scalar_mult_eq_scaleR)
 
-lemma left_invertible_transpose:
+lemma%unimportant left_invertible_transpose:
   "(\<exists>(B). B ** transpose (A) = mat (1::'a::comm_semiring_1)) \<longleftrightarrow> (\<exists>(B). A ** B = mat 1)"
   by (metis matrix_transpose_mul transpose_mat transpose_transpose)
 
-lemma right_invertible_transpose:
+lemma%unimportant right_invertible_transpose:
   "(\<exists>(B). transpose (A) ** B = mat (1::'a::comm_semiring_1)) \<longleftrightarrow> (\<exists>(B). B ** A = mat 1)"
   by (metis matrix_transpose_mul transpose_mat transpose_transpose)
 
-lemma linear_matrix_vector_mul_eq:
+lemma%unimportant linear_matrix_vector_mul_eq:
   "Vector_Spaces.linear ( *s) ( *s) f \<longleftrightarrow> linear (f :: real^'n \<Rightarrow> real ^'m)"
   by (simp add: scalar_mult_eq_scaleR linear_def)
 
-lemma matrix_vector_mul[simp]:
+lemma%unimportant matrix_vector_mul[simp]:
   "Vector_Spaces.linear ( *s) ( *s) g \<Longrightarrow> (\<lambda>y. matrix g *v y) = g"
   "linear f \<Longrightarrow> (\<lambda>x. matrix f *v x) = f"
   "bounded_linear f \<Longrightarrow> (\<lambda>x. matrix f *v x) = f"
   for f :: "real^'n \<Rightarrow> real ^'m"
   by (simp_all add: ext matrix_works linear_matrix_vector_mul_eq linear_linear)
 
-lemma matrix_left_invertible_injective:
+lemma%important matrix_left_invertible_injective:
   fixes A :: "'a::field^'n^'m"
   shows "(\<exists>B. B ** A = mat 1) \<longleftrightarrow> inj (( *v) A)"
-proof safe
+proof%unimportant safe
   fix B
   assume B: "B ** A = mat 1"
   show "inj (( *v) A)"
@@ -192,15 +192,15 @@
     by metis
 qed
 
-lemma matrix_left_invertible_ker:
+lemma%unimportant matrix_left_invertible_ker:
   "(\<exists>B. (B::'a::{field} ^'m^'n) ** (A::'a::{field}^'n^'m) = mat 1) \<longleftrightarrow> (\<forall>x. A *v x = 0 \<longrightarrow> x = 0)"
   unfolding matrix_left_invertible_injective
   using vec.inj_on_iff_eq_0[OF vec.subspace_UNIV, of A]
   by (simp add: inj_on_def)
 
-lemma matrix_right_invertible_surjective:
+lemma%important matrix_right_invertible_surjective:
   "(\<exists>B. (A::'a::field^'n^'m) ** (B::'a::field^'m^'n) = mat 1) \<longleftrightarrow> surj (\<lambda>x. A *v x)"
-proof -
+proof%unimportant -
   { fix B :: "'a ^'m^'n"
     assume AB: "A ** B = mat 1"
     { fix x :: "'a ^ 'm"
@@ -223,12 +223,12 @@
   ultimately show ?thesis unfolding surj_def by blast
 qed
 
-lemma matrix_left_invertible_independent_columns:
+lemma%important matrix_left_invertible_independent_columns:
   fixes A :: "'a::{field}^'n^'m"
   shows "(\<exists>(B::'a ^'m^'n). B ** A = mat 1) \<longleftrightarrow>
       (\<forall>c. sum (\<lambda>i. c i *s column i A) (UNIV :: 'n set) = 0 \<longrightarrow> (\<forall>i. c i = 0))"
     (is "?lhs \<longleftrightarrow> ?rhs")
-proof -
+proof%unimportant -
   let ?U = "UNIV :: 'n set"
   { assume k: "\<forall>x. A *v x = 0 \<longrightarrow> x = 0"
     { fix c i
@@ -250,7 +250,7 @@
   ultimately show ?thesis unfolding matrix_left_invertible_ker by auto
 qed
 
-lemma matrix_right_invertible_independent_rows:
+lemma%unimportant matrix_right_invertible_independent_rows:
   fixes A :: "'a::{field}^'n^'m"
   shows "(\<exists>(B::'a^'m^'n). A ** B = mat 1) \<longleftrightarrow>
     (\<forall>c. sum (\<lambda>i. c i *s row i A) (UNIV :: 'm set) = 0 \<longrightarrow> (\<forall>i. c i = 0))"
@@ -258,10 +258,10 @@
     matrix_left_invertible_independent_columns
   by (simp add:)
 
-lemma matrix_right_invertible_span_columns:
+lemma%important matrix_right_invertible_span_columns:
   "(\<exists>(B::'a::field ^'n^'m). (A::'a ^'m^'n) ** B = mat 1) \<longleftrightarrow>
     vec.span (columns A) = UNIV" (is "?lhs = ?rhs")
-proof -
+proof%unimportant -
   let ?U = "UNIV :: 'm set"
   have fU: "finite ?U" by simp
   have lhseq: "?lhs \<longleftrightarrow> (\<forall>y. \<exists>(x::'a^'m). sum (\<lambda>i. (x$i) *s column i A) ?U = y)"
@@ -322,21 +322,21 @@
   ultimately show ?thesis by blast
 qed
 
-lemma matrix_left_invertible_span_rows_gen:
+lemma%unimportant matrix_left_invertible_span_rows_gen:
   "(\<exists>(B::'a^'m^'n). B ** (A::'a::field^'n^'m) = mat 1) \<longleftrightarrow> vec.span (rows A) = UNIV"
   unfolding right_invertible_transpose[symmetric]
   unfolding columns_transpose[symmetric]
   unfolding matrix_right_invertible_span_columns
   ..
 
-lemma matrix_left_invertible_span_rows:
+lemma%unimportant matrix_left_invertible_span_rows:
   "(\<exists>(B::real^'m^'n). B ** (A::real^'n^'m) = mat 1) \<longleftrightarrow> span (rows A) = UNIV"
   using matrix_left_invertible_span_rows_gen[of A] by (simp add: span_vec_eq)
 
-lemma matrix_left_right_inverse:
+lemma%important matrix_left_right_inverse:
   fixes A A' :: "'a::{field}^'n^'n"
   shows "A ** A' = mat 1 \<longleftrightarrow> A' ** A = mat 1"
-proof -
+proof%unimportant -
   { fix A A' :: "'a ^'n^'n"
     assume AA': "A ** A' = mat 1"
     have sA: "surj (( *v) A)"
@@ -356,21 +356,21 @@
   then show ?thesis by blast
 qed
 
-lemma invertible_left_inverse:
+lemma%unimportant invertible_left_inverse:
   fixes A :: "'a::{field}^'n^'n"
   shows "invertible A \<longleftrightarrow> (\<exists>(B::'a^'n^'n). B ** A = mat 1)"
   by (metis invertible_def matrix_left_right_inverse)
 
-lemma invertible_right_inverse:
+lemma%unimportant invertible_right_inverse:
   fixes A :: "'a::{field}^'n^'n"
   shows "invertible A \<longleftrightarrow> (\<exists>(B::'a^'n^'n). A** B = mat 1)"
   by (metis invertible_def matrix_left_right_inverse)
 
-lemma invertible_mult:
+lemma%important invertible_mult:
   assumes inv_A: "invertible A"
   and inv_B: "invertible B"
   shows "invertible (A**B)"
-proof -
+proof%unimportant -
   obtain A' where AA': "A ** A' = mat 1" and A'A: "A' ** A = mat 1" 
     using inv_A unfolding invertible_def by blast
   obtain B' where BB': "B ** B' = mat 1" and B'B: "B' ** B = mat 1" 
@@ -393,28 +393,28 @@
   qed
 qed
 
-lemma transpose_invertible:
+lemma%unimportant transpose_invertible:
   fixes A :: "real^'n^'n"
   assumes "invertible A"
   shows "invertible (transpose A)"
   by (meson assms invertible_def matrix_left_right_inverse right_invertible_transpose)
 
-lemma vector_matrix_mul_assoc:
+lemma%important vector_matrix_mul_assoc:
   fixes v :: "('a::comm_semiring_1)^'n"
   shows "(v v* M) v* N = v v* (M ** N)"
-proof -
+proof%unimportant -
   from matrix_vector_mul_assoc
   have "transpose N *v (transpose M *v v) = (transpose N ** transpose M) *v v" by fast
   thus "(v v* M) v* N = v v* (M ** N)"
     by (simp add: matrix_transpose_mul [symmetric])
 qed
 
-lemma matrix_scaleR_vector_ac:
+lemma%unimportant matrix_scaleR_vector_ac:
   fixes A :: "real^('m::finite)^'n"
   shows "A *v (k *\<^sub>R v) = k *\<^sub>R A *v v"
   by (metis matrix_vector_mult_scaleR transpose_scalar vector_scaleR_matrix_ac vector_transpose_matrix)
 
-lemma scaleR_matrix_vector_assoc:
+lemma%unimportant scaleR_matrix_vector_assoc:
   fixes A :: "real^('m::finite)^'n"
   shows "k *\<^sub>R (A *v v) = k *\<^sub>R A *v v"
   by (metis matrix_scaleR_vector_ac matrix_vector_mult_scaleR)
@@ -430,8 +430,8 @@
   and BasisB :: "('b set)"
   and f :: "('b=>'c)"
 
-lemma vec_dim_card: "vec.dim (UNIV::('a::{field}^'n) set) = CARD ('n)"
-proof -
+lemma%important vec_dim_card: "vec.dim (UNIV::('a::{field}^'n) set) = CARD ('n)"
+proof%unimportant -
   let ?f="\<lambda>i::'n. axis i (1::'a)"
   have "vec.dim (UNIV::('a::{field}^'n) set) = card (cart_basis::('a^'n) set)"
     unfolding vec.dim_UNIV ..

--- a/src/HOL/Analysis/Change_Of_Vars.thy	Mon Aug 27 22:58:36 2018 +0200
+++ b/src/HOL/Analysis/Change_Of_Vars.thy	Tue Aug 28 13:28:39 2018 +0100
@@ -9,9 +9,9 @@
 
 begin
 
-subsection\<open>Induction on matrix row operations\<close>
+subsection%important\<open>Induction on matrix row operations\<close>
 
-lemma induct_matrix_row_operations:
+lemma%unimportant induct_matrix_row_operations:
   fixes P :: "real^'n^'n \<Rightarrow> bool"
   assumes zero_row: "\<And>A i. row i A = 0 \<Longrightarrow> P A"
     and diagonal: "\<And>A. (\<And>i j. i \<noteq> j \<Longrightarrow> A$i$j = 0) \<Longrightarrow> P A"
@@ -120,7 +120,7 @@
     by blast
 qed
 
-lemma induct_matrix_elementary:
+lemma%unimportant induct_matrix_elementary:
   fixes P :: "real^'n^'n \<Rightarrow> bool"
   assumes mult: "\<And>A B. \<lbrakk>P A; P B\<rbrakk> \<Longrightarrow> P(A ** B)"
     and zero_row: "\<And>A i. row i A = 0 \<Longrightarrow> P A"
@@ -151,7 +151,7 @@
     by (rule induct_matrix_row_operations [OF zero_row diagonal swap row])
 qed
 
-lemma induct_matrix_elementary_alt:
+lemma%unimportant induct_matrix_elementary_alt:
   fixes P :: "real^'n^'n \<Rightarrow> bool"
   assumes mult: "\<And>A B. \<lbrakk>P A; P B\<rbrakk> \<Longrightarrow> P(A ** B)"
     and zero_row: "\<And>A i. row i A = 0 \<Longrightarrow> P A"
@@ -185,11 +185,11 @@
     by (rule induct_matrix_elementary) (auto intro: assms *)
 qed
 
-lemma matrix_vector_mult_matrix_matrix_mult_compose:
+lemma%unimportant matrix_vector_mult_matrix_matrix_mult_compose:
   "( *v) (A ** B) = ( *v) A \<circ> ( *v) B"
   by (auto simp: matrix_vector_mul_assoc)
 
-lemma induct_linear_elementary:
+lemma%unimportant induct_linear_elementary:
   fixes f :: "real^'n \<Rightarrow> real^'n"
   assumes "linear f"
     and comp: "\<And>f g. \<lbrakk>linear f; linear g; P f; P g\<rbrakk> \<Longrightarrow> P(f \<circ> g)"
@@ -250,14 +250,14 @@
 qed
 
 
-proposition
+proposition%important
   fixes a :: "real^'n"
   assumes "m \<noteq> n" and ab_ne: "cbox a b \<noteq> {}" and an: "0 \<le> a$n"
   shows measurable_shear_interval: "(\<lambda>x. \<chi> i. if i = m then x$m + x$n else x$i) ` (cbox a b) \<in> lmeasurable"
        (is  "?f ` _ \<in> _")
    and measure_shear_interval: "measure lebesgue ((\<lambda>x. \<chi> i. if i = m then x$m + x$n else x$i) ` cbox a b)
                = measure lebesgue (cbox a b)" (is "?Q")
-proof -
+proof%unimportant -
   have lin: "linear ?f"
     by (rule linearI) (auto simp: plus_vec_def scaleR_vec_def algebra_simps)
   show fab: "?f ` cbox a b \<in> lmeasurable"
@@ -359,13 +359,13 @@
 qed
 
 
-proposition
+proposition%important
   fixes S :: "(real^'n) set"
   assumes "S \<in> lmeasurable"
   shows measurable_stretch: "((\<lambda>x. \<chi> k. m k * x$k) ` S) \<in> lmeasurable" (is  "?f ` S \<in> _")
     and measure_stretch: "measure lebesgue ((\<lambda>x. \<chi> k. m k * x$k) ` S) = \<bar>prod m UNIV\<bar> * measure lebesgue S"
     (is "?MEQ")
-proof -
+proof%unimportant -
   have "(?f ` S) \<in> lmeasurable \<and> ?MEQ"
   proof (cases "\<forall>k. m k \<noteq> 0")
     case True
@@ -463,12 +463,12 @@
 qed
 
 
-proposition
+proposition%important
  fixes f :: "real^'n::{finite,wellorder} \<Rightarrow> real^'n::_"
   assumes "linear f" "S \<in> lmeasurable"
   shows measurable_linear_image: "(f ` S) \<in> lmeasurable"
     and measure_linear_image: "measure lebesgue (f ` S) = \<bar>det (matrix f)\<bar> * measure lebesgue S" (is "?Q f S")
-proof -
+proof%unimportant -
   have "\<forall>S \<in> lmeasurable. (f ` S) \<in> lmeasurable \<and> ?Q f S"
   proof (rule induct_linear_elementary [OF \<open>linear f\<close>]; intro ballI)
     fix f g and S :: "(real,'n) vec set"
@@ -611,7 +611,7 @@
 qed
 
 
-lemma
+lemma%unimportant
  fixes f :: "real^'n::{finite,wellorder} \<Rightarrow> real^'n::_"
   assumes f: "orthogonal_transformation f" and S: "S \<in> lmeasurable"
   shows measurable_orthogonal_image: "f ` S \<in> lmeasurable"
@@ -625,19 +625,19 @@
     by (simp add: measure_linear_image \<open>linear f\<close> S f)
 qed
 
-subsection\<open>@{text F_sigma} and @{text G_delta} sets.\<close>
+subsection%important\<open>@{text F_sigma} and @{text G_delta} sets.\<close>(*FIX ME mv *)
 
 (*https://en.wikipedia.org/wiki/F\<sigma>_set*)
-inductive fsigma :: "'a::topological_space set \<Rightarrow> bool" where
+inductive%important fsigma :: "'a::topological_space set \<Rightarrow> bool" where
   "(\<And>n::nat. closed (F n)) \<Longrightarrow> fsigma (UNION UNIV F)"
 
-inductive gdelta :: "'a::topological_space set \<Rightarrow> bool" where
+inductive%important gdelta :: "'a::topological_space set \<Rightarrow> bool" where
   "(\<And>n::nat. open (F n)) \<Longrightarrow> gdelta (INTER UNIV F)"
 
-lemma fsigma_Union_compact:
+lemma%important fsigma_Union_compact:
   fixes S :: "'a::{real_normed_vector,heine_borel} set"
   shows "fsigma S \<longleftrightarrow> (\<exists>F::nat \<Rightarrow> 'a set. range F \<subseteq> Collect compact \<and> S = UNION UNIV F)"
-proof safe
+proof%unimportant safe
   assume "fsigma S"
   then obtain F :: "nat \<Rightarrow> 'a set" where F: "range F \<subseteq> Collect closed" "S = UNION UNIV F"
     by (meson fsigma.cases image_subsetI mem_Collect_eq)
@@ -668,7 +668,7 @@
     by (simp add: compact_imp_closed fsigma.intros image_subset_iff)
 qed
 
-lemma gdelta_imp_fsigma: "gdelta S \<Longrightarrow> fsigma (- S)"
+lemma%unimportant gdelta_imp_fsigma: "gdelta S \<Longrightarrow> fsigma (- S)"
 proof (induction rule: gdelta.induct)
   case (1 F)
   have "- INTER UNIV F = (\<Union>i. -(F i))"
@@ -677,7 +677,7 @@
     by (simp add: fsigma.intros closed_Compl 1)
 qed
 
-lemma fsigma_imp_gdelta: "fsigma S \<Longrightarrow> gdelta (- S)"
+lemma%unimportant fsigma_imp_gdelta: "fsigma S \<Longrightarrow> gdelta (- S)"
 proof (induction rule: fsigma.induct)
   case (1 F)
   have "- UNION UNIV F = (\<Inter>i. -(F i))"
@@ -686,11 +686,11 @@
     by (simp add: 1 gdelta.intros open_closed)
 qed
 
-lemma gdelta_complement: "gdelta(- S) \<longleftrightarrow> fsigma S"
+lemma%unimportant gdelta_complement: "gdelta(- S) \<longleftrightarrow> fsigma S"
   using fsigma_imp_gdelta gdelta_imp_fsigma by force
 
 text\<open>A Lebesgue set is almost an @{text F_sigma} or @{text G_delta}.\<close>
-lemma lebesgue_set_almost_fsigma:
+lemma%unimportant lebesgue_set_almost_fsigma:
   assumes "S \<in> sets lebesgue"
   obtains C T where "fsigma C" "negligible T" "C \<union> T = S" "disjnt C T"
 proof -
@@ -725,7 +725,7 @@
   qed
 qed
 
-lemma lebesgue_set_almost_gdelta:
+lemma%unimportant lebesgue_set_almost_gdelta:
   assumes "S \<in> sets lebesgue"
   obtains C T where "gdelta C" "negligible T" "S \<union> T = C" "disjnt S T"
 proof -
@@ -743,12 +743,12 @@
 qed
 
 
-proposition measure_semicontinuous_with_hausdist_explicit:
+proposition%important measure_semicontinuous_with_hausdist_explicit:
   assumes "bounded S" and neg: "negligible(frontier S)" and "e > 0"
   obtains d where "d > 0"
                   "\<And>T. \<lbrakk>T \<in> lmeasurable; \<And>y. y \<in> T \<Longrightarrow> \<exists>x. x \<in> S \<and> dist x y < d\<rbrakk>
                         \<Longrightarrow> measure lebesgue T < measure lebesgue S + e"
-proof (cases "S = {}")
+proof%unimportant (cases "S = {}")
   case True
   with that \<open>e > 0\<close> show ?thesis by force
 next
@@ -812,10 +812,10 @@
   qed
 qed
 
-proposition lebesgue_regular_inner:
+proposition%important lebesgue_regular_inner:
  assumes "S \<in> sets lebesgue"
  obtains K C where "negligible K" "\<And>n::nat. compact(C n)" "S = (\<Union>n. C n) \<union> K"
-proof -
+proof%unimportant -
   have "\<exists>T. closed T \<and> T \<subseteq> S \<and> (S - T) \<in> lmeasurable \<and> measure lebesgue (S - T) < (1/2)^n" for n
     using sets_lebesgue_inner_closed assms
     by (metis sets_lebesgue_inner_closed zero_less_divide_1_iff zero_less_numeral zero_less_power)
@@ -865,7 +865,7 @@
 qed
 
 
-lemma sets_lebesgue_continuous_image:
+lemma%unimportant sets_lebesgue_continuous_image:
   assumes T: "T \<in> sets lebesgue" and contf: "continuous_on S f"
     and negim: "\<And>T. \<lbrakk>negligible T; T \<subseteq> S\<rbrakk> \<Longrightarrow> negligible(f ` T)" and "T \<subseteq> S"
  shows "f ` T \<in> sets lebesgue"
@@ -890,7 +890,7 @@
     by (simp add: Teq image_Un image_Union)
 qed
 
-lemma differentiable_image_in_sets_lebesgue:
+lemma%unimportant differentiable_image_in_sets_lebesgue:
   fixes f :: "'m::euclidean_space \<Rightarrow> 'n::euclidean_space"
   assumes S: "S \<in> sets lebesgue" and dim: "DIM('m) \<le> DIM('n)" and f: "f differentiable_on S"
   shows "f`S \<in> sets lebesgue"
@@ -902,7 +902,7 @@
     by (auto simp: intro!: negligible_differentiable_image_negligible [OF dim])
 qed auto
 
-lemma sets_lebesgue_on_continuous_image:
+lemma%unimportant sets_lebesgue_on_continuous_image:
   assumes S: "S \<in> sets lebesgue" and X: "X \<in> sets (lebesgue_on S)" and contf: "continuous_on S f"
     and negim: "\<And>T. \<lbrakk>negligible T; T \<subseteq> S\<rbrakk> \<Longrightarrow> negligible(f ` T)"
   shows "f ` X \<in> sets (lebesgue_on (f ` S))"
@@ -917,7 +917,7 @@
     by (auto simp: sets_restrict_space_iff)
 qed
 
-lemma differentiable_image_in_sets_lebesgue_on:
+lemma%unimportant differentiable_image_in_sets_lebesgue_on:
   fixes f :: "'m::euclidean_space \<Rightarrow> 'n::euclidean_space"
   assumes S: "S \<in> sets lebesgue" and X: "X \<in> sets (lebesgue_on S)" and dim: "DIM('m) \<le> DIM('n)"
        and f: "f differentiable_on S"
@@ -931,7 +931,7 @@
 qed
 
 
-proposition
+proposition%important
  fixes f :: "real^'n::{finite,wellorder} \<Rightarrow> real^'n::_"
   assumes S: "S \<in> lmeasurable"
   and deriv: "\<And>x. x \<in> S \<Longrightarrow> (f has_derivative f' x) (at x within S)"
@@ -941,7 +941,7 @@
        "f ` S \<in> lmeasurable"
     and measure_bounded_differentiable_image:
        "measure lebesgue (f ` S) \<le> B * measure lebesgue S" (is "?M")
-proof -
+proof%unimportant -
   have "f ` S \<in> lmeasurable \<and> measure lebesgue (f ` S) \<le> B * measure lebesgue S"
   proof (cases "B < 0")
     case True
@@ -1166,13 +1166,13 @@
   then show "f ` S \<in> lmeasurable" ?M by blast+
 qed
 
-lemma
+lemma%important
  fixes f :: "real^'n::{finite,wellorder} \<Rightarrow> real^'n::_"
   assumes S: "S \<in> lmeasurable"
     and deriv: "\<And>x. x \<in> S \<Longrightarrow> (f has_derivative f' x) (at x within S)"
     and int: "(\<lambda>x. \<bar>det (matrix (f' x))\<bar>) integrable_on S"
   shows m_diff_image_weak: "f ` S \<in> lmeasurable \<and> measure lebesgue (f ` S) \<le> integral S (\<lambda>x. \<bar>det (matrix (f' x))\<bar>)"
-proof -
+proof%unimportant -
   let ?\<mu> = "measure lebesgue"
   have aint_S: "(\<lambda>x. \<bar>det (matrix (f' x))\<bar>) absolutely_integrable_on S"
     using int unfolding absolutely_integrable_on_def by auto
@@ -1364,7 +1364,7 @@
 qed
 
 
-theorem
+theorem%important
  fixes f :: "real^'n::{finite,wellorder} \<Rightarrow> real^'n::_"
   assumes S: "S \<in> sets lebesgue"
     and deriv: "\<And>x. x \<in> S \<Longrightarrow> (f has_derivative f' x) (at x within S)"
@@ -1372,7 +1372,7 @@
   shows measurable_differentiable_image: "f ` S \<in> lmeasurable"
     and measure_differentiable_image:
        "measure lebesgue (f ` S) \<le> integral S (\<lambda>x. \<bar>det (matrix (f' x))\<bar>)" (is "?M")
-proof -
+proof%unimportant -
   let ?I = "\<lambda>n::nat. cbox (vec (-n)) (vec n) \<inter> S"
   let ?\<mu> = "measure lebesgue"
   have "x \<in> cbox (vec (- real (nat \<lceil>norm x\<rceil>))) (vec (real (nat \<lceil>norm x\<rceil>)))" for x :: "real^'n::_"
@@ -1417,7 +1417,7 @@
 qed
 
 
-lemma borel_measurable_simple_function_limit_increasing:
+lemma%unimportant borel_measurable_simple_function_limit_increasing:
   fixes f :: "'a::euclidean_space \<Rightarrow> real"
   shows "(f \<in> borel_measurable lebesgue \<and> (\<forall>x. 0 \<le> f x)) \<longleftrightarrow>
          (\<exists>g. (\<forall>n x. 0 \<le> g n x \<and> g n x \<le> f x) \<and> (\<forall>n x. g n x \<le> (g(Suc n) x)) \<and>
@@ -1614,9 +1614,9 @@
     by (blast intro: order_trans)
 qed
 
-subsection\<open>Borel measurable Jacobian determinant\<close>
+subsection%important\<open>Borel measurable Jacobian determinant\<close>
 
-lemma lemma_partial_derivatives0:
+lemma%unimportant lemma_partial_derivatives0:
   fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
   assumes "linear f" and lim0: "((\<lambda>x. f x /\<^sub>R norm x) \<longlongrightarrow> 0) (at 0 within S)"
     and lb: "\<And>v. v \<noteq> 0 \<Longrightarrow> (\<exists>k>0. \<forall>e>0. \<exists>x. x \<in> S - {0} \<and> norm x < e \<and> k * norm x \<le> \<bar>v \<bullet> x\<bar>)"
@@ -1689,7 +1689,7 @@
         mem_Collect_eq module_hom_zero span_base span_raw_def)
 qed
 
-lemma lemma_partial_derivatives:
+lemma%unimportant lemma_partial_derivatives:
   fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
   assumes "linear f" and lim: "((\<lambda>x. f (x - a) /\<^sub>R norm (x - a)) \<longlongrightarrow> 0) (at a within S)"
     and lb: "\<And>v. v \<noteq> 0 \<Longrightarrow> (\<exists>k>0.  \<forall>e>0. \<exists>x \<in> S - {a}. norm(a - x) < e \<and> k * norm(a - x) \<le> \<bar>v \<bullet> (x - a)\<bar>)"
@@ -1707,12 +1707,12 @@
 qed
 
 
-proposition borel_measurable_partial_derivatives:
+proposition%important borel_measurable_partial_derivatives:
   fixes f :: "real^'m::{finite,wellorder} \<Rightarrow> real^'n"
   assumes S: "S \<in> sets lebesgue"
     and f: "\<And>x. x \<in> S \<Longrightarrow> (f has_derivative f' x) (at x within S)"
   shows "(\<lambda>x. (matrix(f' x)$m$n)) \<in> borel_measurable (lebesgue_on S)"
-proof -
+proof%unimportant -
   have contf: "continuous_on S f"
     using continuous_on_eq_continuous_within f has_derivative_continuous by blast
   have "{x \<in> S.  (matrix (f' x)$m$n) \<le> b} \<in> sets lebesgue" for b
@@ -2298,21 +2298,21 @@
 qed
 
 
-theorem borel_measurable_det_Jacobian:
+theorem%important borel_measurable_det_Jacobian:
  fixes f :: "real^'n::{finite,wellorder} \<Rightarrow> real^'n::_"
   assumes S: "S \<in> sets lebesgue" and f: "\<And>x. x \<in> S \<Longrightarrow> (f has_derivative f' x) (at x within S)"
   shows "(\<lambda>x. det(matrix(f' x))) \<in> borel_measurable (lebesgue_on S)"
   unfolding det_def
-  by (intro measurable) (auto intro: f borel_measurable_partial_derivatives [OF S])
+  by%unimportant (intro measurable) (auto intro: f borel_measurable_partial_derivatives [OF S])
 
 text\<open>The localisation wrt S uses the same argument for many similar results.\<close>
 (*See HOL Light's MEASURABLE_ON_LEBESGUE_MEASURABLE_PREIMAGE_BOREL, etc.*)
-lemma borel_measurable_lebesgue_on_preimage_borel:
+lemma%important borel_measurable_lebesgue_on_preimage_borel:
   fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
   assumes "S \<in> sets lebesgue"
   shows "f \<in> borel_measurable (lebesgue_on S) \<longleftrightarrow>
          (\<forall>T. T \<in> sets borel \<longrightarrow> {x \<in> S. f x \<in> T} \<in> sets lebesgue)"
-proof -
+proof%unimportant -
   have "{x. (if x \<in> S then f x else 0) \<in> T} \<in> sets lebesgue \<longleftrightarrow> {x \<in> S. f x \<in> T} \<in> sets lebesgue"
          if "T \<in> sets borel" for T
     proof (cases "0 \<in> T")
@@ -2334,7 +2334,7 @@
       by blast
 qed
 
-lemma sets_lebesgue_almost_borel:
+lemma%unimportant sets_lebesgue_almost_borel:
   assumes "S \<in> sets lebesgue"
   obtains B N where "B \<in> sets borel" "negligible N" "B \<union> N = S"
 proof -
@@ -2344,7 +2344,7 @@
     by (metis negligible_iff_null_sets negligible_subset null_sets_completionI that)
 qed
 
-lemma double_lebesgue_sets:
+lemma%unimportant double_lebesgue_sets:
  assumes S: "S \<in> sets lebesgue" and T: "T \<in> sets lebesgue" and fim: "f ` S \<subseteq> T"
  shows "(\<forall>U. U \<in> sets lebesgue \<and> U \<subseteq> T \<longrightarrow> {x \<in> S. f x \<in> U} \<in> sets lebesgue) \<longleftrightarrow>
           f \<in> borel_measurable (lebesgue_on S) \<and>
@@ -2387,9 +2387,9 @@
 qed
 
 
-subsection\<open>Simplest case of Sard's theorem (we don't need continuity of derivative)\<close>
+subsection%important\<open>Simplest case of Sard's theorem (we don't need continuity of derivative)\<close>
 
-lemma Sard_lemma00:
+lemma%important Sard_lemma00:
   fixes P :: "'b::euclidean_space set"
   assumes "a \<ge> 0" and a: "a *\<^sub>R i \<noteq> 0" and i: "i \<in> Basis"
     and P: "P \<subseteq> {x. a *\<^sub>R i \<bullet> x = 0}"
@@ -2397,7 +2397,7 @@
  obtains S where "S \<in> lmeasurable"
             and "{z. norm z \<le> m \<and> (\<exists>t \<in> P. norm(z - t) \<le> e)} \<subseteq> S"
             and "measure lebesgue S \<le> (2 * e) * (2 * m) ^ (DIM('b) - 1)"
-proof -
+proof%unimportant -
   have "a > 0"
     using assms by simp
   let ?v = "(\<Sum>j\<in>Basis. (if j = i then e else m) *\<^sub>R j)"
@@ -2425,7 +2425,7 @@
 qed
 
 text\<open>As above, but reorienting the vector (HOL Light's @text{GEOM\_BASIS\_MULTIPLE\_TAC})\<close>
-lemma Sard_lemma0:
+lemma%unimportant Sard_lemma0:
   fixes P :: "(real^'n::{finite,wellorder}) set"
   assumes "a \<noteq> 0"
     and P: "P \<subseteq> {x. a \<bullet> x = 0}" and "0 \<le> m" "0 \<le> e"
@@ -2485,13 +2485,13 @@
 qed
 
 text\<open>As above, but translating the sets (HOL Light's @text{GEN\_GEOM\_ORIGIN\_TAC})\<close>
-lemma Sard_lemma1:
+lemma%important Sard_lemma1:
   fixes P :: "(real^'n::{finite,wellorder}) set"
    assumes P: "dim P < CARD('n)" and "0 \<le> m" "0 \<le> e"
  obtains S where "S \<in> lmeasurable"
             and "{z. norm(z - w) \<le> m \<and> (\<exists>t \<in> P. norm(z - w - t) \<le> e)} \<subseteq> S"
             and "measure lebesgue S \<le> (2 * e) * (2 * m) ^ (CARD('n) - 1)"
-proof -
+proof%unimportant -
   obtain a where "a \<noteq> 0" "P \<subseteq> {x. a \<bullet> x = 0}"
     using lowdim_subset_hyperplane [of P] P span_base by auto
   then obtain S where S: "S \<in> lmeasurable"
@@ -2509,7 +2509,7 @@
   qed
 qed
 
-lemma Sard_lemma2:
+lemma%important Sard_lemma2:
   fixes f :: "real^'m::{finite,wellorder} \<Rightarrow> real^'n::{finite,wellorder}"
   assumes mlen: "CARD('m) \<le> CARD('n)" (is "?m \<le> ?n")
     and "B > 0" "bounded S"
@@ -2517,7 +2517,7 @@
     and rank: "\<And>x. x \<in> S \<Longrightarrow> rank(matrix(f' x)) < CARD('n)"
     and B: "\<And>x. x \<in> S \<Longrightarrow> onorm(f' x) \<le> B"
   shows "negligible(f ` S)"
-proof -
+proof%unimportant -
   have lin_f': "\<And>x. x \<in> S \<Longrightarrow> linear(f' x)"
     using derS has_derivative_linear by blast
   show ?thesis
@@ -2724,13 +2724,13 @@
 qed
 
 
-theorem baby_Sard:
+theorem%important baby_Sard:
   fixes f :: "real^'m::{finite,wellorder} \<Rightarrow> real^'n::{finite,wellorder}"
   assumes mlen: "CARD('m) \<le> CARD('n)"
     and der: "\<And>x. x \<in> S \<Longrightarrow> (f has_derivative f' x) (at x within S)"
     and rank: "\<And>x. x \<in> S \<Longrightarrow> rank(matrix(f' x)) < CARD('n)"
   shows "negligible(f ` S)"
-proof -
+proof%unimportant -
   let ?U = "\<lambda>n. {x \<in> S. norm(x) \<le> n \<and> onorm(f' x) \<le> real n}"
   have "\<And>x. x \<in> S \<Longrightarrow> \<exists>n. norm x \<le> real n \<and> onorm (f' x) \<le> real n"
     by (meson linear order_trans real_arch_simple)
@@ -2750,9 +2750,9 @@
 qed
 
 
-subsection\<open>A one-way version of change-of-variables not assuming injectivity. \<close>
+subsection%important\<open>A one-way version of change-of-variables not assuming injectivity. \<close>
 
-lemma integral_on_image_ubound_weak:
+lemma%important integral_on_image_ubound_weak:
   fixes f :: "real^'n::{finite,wellorder} \<Rightarrow> real"
   assumes S: "S \<in> sets lebesgue"
       and f: "f \<in> borel_measurable (lebesgue_on (g ` S))"
@@ -2763,7 +2763,7 @@
     shows "f integrable_on (g ` S) \<and>
            integral (g ` S) f \<le> integral S (\<lambda>x. \<bar>det (matrix (g' x))\<bar> * f(g x))"
          (is "_ \<and> _ \<le> ?b")
-proof -
+proof%unimportant -
   let ?D = "\<lambda>x. \<bar>det (matrix (g' x))\<bar>"
   have cont_g: "continuous_on S g"
     using der_g has_derivative_continuous_on by blast
@@ -2941,14 +2941,14 @@
 qed
 
 
-lemma integral_on_image_ubound_nonneg:
+lemma%important integral_on_image_ubound_nonneg:
   fixes f :: "real^'n::{finite,wellorder} \<Rightarrow> real"
   assumes nonneg_fg: "\<And>x. x \<in> S \<Longrightarrow> 0 \<le> f(g x)"
       and der_g:   "\<And>x. x \<in> S \<Longrightarrow> (g has_derivative g' x) (at x within S)"
       and intS: "(\<lambda>x. \<bar>det (matrix (g' x))\<bar> * f(g x)) integrable_on S"
   shows "f integrable_on (g ` S) \<and> integral (g ` S) f \<le> integral S (\<lambda>x. \<bar>det (matrix (g' x))\<bar> * f(g x))"
          (is "_ \<and> _ \<le> ?b")
-proof -
+proof%unimportant -
   let ?D = "\<lambda>x. det (matrix (g' x))"
   define S' where "S' \<equiv> {x \<in> S. ?D x * f(g x) \<noteq> 0}"
   then have der_gS': "\<And>x. x \<in> S' \<Longrightarrow> (g has_derivative g' x) (at x within S')"
@@ -3071,7 +3071,7 @@
 qed
 
 
-lemma absolutely_integrable_on_image_real:
+lemma%unimportant absolutely_integrable_on_image_real:
   fixes f :: "real^'n::{finite,wellorder} \<Rightarrow> real" and g :: "real^'n::_ \<Rightarrow> real^'n::_"
   assumes der_g: "\<And>x. x \<in> S \<Longrightarrow> (g has_derivative g' x) (at x within S)"
     and intS: "(\<lambda>x. \<bar>det (matrix (g' x))\<bar> * f(g x)) absolutely_integrable_on S"
@@ -3146,29 +3146,29 @@
 qed
 
 
-proposition absolutely_integrable_on_image:
+proposition%important absolutely_integrable_on_image:
   fixes f :: "real^'m::{finite,wellorder} \<Rightarrow> real^'n" and g :: "real^'m::_ \<Rightarrow> real^'m::_"
   assumes der_g: "\<And>x. x \<in> S \<Longrightarrow> (g has_derivative g' x) (at x within S)"
     and intS: "(\<lambda>x. \<bar>det (matrix (g' x))\<bar> *\<^sub>R f(g x)) absolutely_integrable_on S"
   shows "f absolutely_integrable_on (g ` S)"
   apply (rule absolutely_integrable_componentwise [OF absolutely_integrable_on_image_real [OF der_g]])
-  using absolutely_integrable_component [OF intS]  by auto
+  using%unimportant absolutely_integrable_component [OF intS]  by%unimportant auto
 
-proposition integral_on_image_ubound:
+proposition%important integral_on_image_ubound:
   fixes f :: "real^'n::{finite,wellorder} \<Rightarrow> real" and g :: "real^'n::_ \<Rightarrow> real^'n::_"
   assumes"\<And>x. x \<in> S \<Longrightarrow> 0 \<le> f(g x)"
     and "\<And>x. x \<in> S \<Longrightarrow> (g has_derivative g' x) (at x within S)"
     and "(\<lambda>x. \<bar>det (matrix (g' x))\<bar> * f(g x)) integrable_on S"
   shows "integral (g ` S) f \<le> integral S (\<lambda>x. \<bar>det (matrix (g' x))\<bar> * f(g x))"
-  using integral_on_image_ubound_nonneg [OF assms] by simp
+  using%unimportant integral_on_image_ubound_nonneg [OF assms] by%unimportant simp
 
 
-subsection\<open>Change-of-variables theorem\<close>
+subsection%important\<open>Change-of-variables theorem\<close>
 
 text\<open>The classic change-of-variables theorem. We have two versions with quite general hypotheses,
 the first that the transforming function has a continuous inverse, the second that the base set is
 Lebesgue measurable.\<close>
-lemma cov_invertible_nonneg_le:
+lemma%unimportant cov_invertible_nonneg_le:
   fixes f :: "real^'n::{finite,wellorder} \<Rightarrow> real" and g :: "real^'n::_ \<Rightarrow> real^'n::_"
   assumes der_g: "\<And>x. x \<in> S \<Longrightarrow> (g has_derivative g' x) (at x within S)"
     and der_h: "\<And>y. y \<in> T \<Longrightarrow> (h has_derivative h' y) (at y within T)"
@@ -3239,7 +3239,7 @@
 qed
 
 
-lemma cov_invertible_nonneg_eq:
+lemma%unimportant cov_invertible_nonneg_eq:
   fixes f :: "real^'n::{finite,wellorder} \<Rightarrow> real" and g :: "real^'n::_ \<Rightarrow> real^'n::_"
   assumes "\<And>x. x \<in> S \<Longrightarrow> (g has_derivative g' x) (at x within S)"
       and "\<And>y. y \<in> T \<Longrightarrow> (h has_derivative h' y) (at y within T)"
@@ -3252,7 +3252,7 @@
   by (simp add: has_integral_iff) (meson eq_iff)
 
 
-lemma cov_invertible_real:
+lemma%unimportant cov_invertible_real:
   fixes f :: "real^'n::{finite,wellorder} \<Rightarrow> real" and g :: "real^'n::_ \<Rightarrow> real^'n::_"
   assumes der_g: "\<And>x. x \<in> S \<Longrightarrow> (g has_derivative g' x) (at x within S)"
       and der_h: "\<And>y. y \<in> T \<Longrightarrow> (h has_derivative h' y) (at y within T)"
@@ -3418,7 +3418,7 @@
 qed
 
 
-lemma cv_inv_version3:
+lemma%unimportant cv_inv_version3:
   fixes f :: "real^'m::{finite,wellorder} \<Rightarrow> real^'n" and g :: "real^'m::_ \<Rightarrow> real^'m::_"
   assumes der_g: "\<And>x. x \<in> S \<Longrightarrow> (g has_derivative g' x) (at x within S)"
     and der_h: "\<And>y. y \<in> T \<Longrightarrow> (h has_derivative h' y) (at y within T)"
@@ -3444,7 +3444,7 @@
 qed
 
 
-lemma cv_inv_version4:
+lemma%unimportant cv_inv_version4:
   fixes f :: "real^'m::{finite,wellorder} \<Rightarrow> real^'n" and g :: "real^'m::_ \<Rightarrow> real^'m::_"
   assumes der_g: "\<And>x. x \<in> S \<Longrightarrow> (g has_derivative g' x) (at x within S) \<and> invertible(matrix(g' x))"
     and hg: "\<And>x. x \<in> S \<Longrightarrow> continuous_on (g ` S) h \<and> h(g x) = x"
@@ -3469,7 +3469,7 @@
 qed
 
 
-proposition has_absolute_integral_change_of_variables_invertible:
+proposition%important has_absolute_integral_change_of_variables_invertible:
   fixes f :: "real^'m::{finite,wellorder} \<Rightarrow> real^'n" and g :: "real^'m::_ \<Rightarrow> real^'m::_"
   assumes der_g: "\<And>x. x \<in> S \<Longrightarrow> (g has_derivative g' x) (at x within S)"
       and hg: "\<And>x. x \<in> S \<Longrightarrow> h(g x) = x"
@@ -3477,7 +3477,7 @@
   shows "(\<lambda>x. \<bar>det (matrix (g' x))\<bar> *\<^sub>R f(g x)) absolutely_integrable_on S \<and> integral S (\<lambda>x. \<bar>det (matrix (g' x))\<bar> *\<^sub>R f(g x)) = b \<longleftrightarrow>
          f absolutely_integrable_on (g ` S) \<and> integral (g ` S) f = b"
     (is "?lhs = ?rhs")
-proof -
+proof%unimportant -
   let ?S = "{x \<in> S. invertible (matrix (g' x))}" and ?D = "\<lambda>x. \<bar>det (matrix (g' x))\<bar> *\<^sub>R f(g x)"
   have *: "?D absolutely_integrable_on ?S \<and> integral ?S ?D = b
            \<longleftrightarrow> f absolutely_integrable_on (g ` ?S) \<and> integral (g ` ?S) f = b"
@@ -3513,7 +3513,7 @@
 
 
 
-lemma has_absolute_integral_change_of_variables_compact:
+lemma%unimportant has_absolute_integral_change_of_variables_compact:
   fixes f :: "real^'m::{finite,wellorder} \<Rightarrow> real^'n" and g :: "real^'m::_ \<Rightarrow> real^'m::_"
   assumes "compact S"
       and der_g: "\<And>x. x \<in> S \<Longrightarrow> (g has_derivative g' x) (at x within S)"
@@ -3531,7 +3531,7 @@
 qed
 
 
-lemma has_absolute_integral_change_of_variables_compact_family:
+lemma%unimportant has_absolute_integral_change_of_variables_compact_family:
   fixes f :: "real^'m::{finite,wellorder} \<Rightarrow> real^'n" and g :: "real^'m::_ \<Rightarrow> real^'m::_"
   assumes compact: "\<And>n::nat. compact (F n)"
       and der_g: "\<And>x. x \<in> (\<Union>n. F n) \<Longrightarrow> (g has_derivative g' x) (at x within (\<Union>n. F n))"
@@ -3710,7 +3710,7 @@
 qed
 
 
-proposition has_absolute_integral_change_of_variables:
+proposition%important has_absolute_integral_change_of_variables:
   fixes f :: "real^'m::{finite,wellorder} \<Rightarrow> real^'n" and g :: "real^'m::_ \<Rightarrow> real^'m::_"
   assumes S: "S \<in> sets lebesgue"
     and der_g: "\<And>x. x \<in> S \<Longrightarrow> (g has_derivative g' x) (at x within S)"
@@ -3718,7 +3718,7 @@
   shows "(\<lambda>x. \<bar>det (matrix (g' x))\<bar> *\<^sub>R f(g x)) absolutely_integrable_on S \<and>
            integral S (\<lambda>x. \<bar>det (matrix (g' x))\<bar> *\<^sub>R f(g x)) = b
      \<longleftrightarrow> f absolutely_integrable_on (g ` S) \<and> integral (g ` S) f = b"
-proof -
+proof%unimportant -
   obtain C N where "fsigma C" "negligible N" and CNS: "C \<union> N = S" and "disjnt C N"
     using lebesgue_set_almost_fsigma [OF S] .
   then obtain F :: "nat \<Rightarrow> (real^'m::_) set"
@@ -3772,16 +3772,16 @@
 qed
 
 
-corollary absolutely_integrable_change_of_variables:
+corollary%important absolutely_integrable_change_of_variables:
   fixes f :: "real^'m::{finite,wellorder} \<Rightarrow> real^'n" and g :: "real^'m::_ \<Rightarrow> real^'m::_"
   assumes "S \<in> sets lebesgue"
     and "\<And>x. x \<in> S \<Longrightarrow> (g has_derivative g' x) (at x within S)"
     and "inj_on g S"
   shows "f absolutely_integrable_on (g ` S)
      \<longleftrightarrow> (\<lambda>x. \<bar>det (matrix (g' x))\<bar> *\<^sub>R f(g x)) absolutely_integrable_on S"
-  using assms has_absolute_integral_change_of_variables by blast
+  using%unimportant assms has_absolute_integral_change_of_variables by%unimportant blast
 
-corollary integral_change_of_variables:
+corollary%important integral_change_of_variables:
   fixes f :: "real^'m::{finite,wellorder} \<Rightarrow> real^'n" and g :: "real^'m::_ \<Rightarrow> real^'m::_"
   assumes S: "S \<in> sets lebesgue"
     and der_g: "\<And>x. x \<in> S \<Longrightarrow> (g has_derivative g' x) (at x within S)"
@@ -3789,10 +3789,10 @@
     and disj: "(f absolutely_integrable_on (g ` S) \<or>
         (\<lambda>x. \<bar>det (matrix (g' x))\<bar> *\<^sub>R f(g x)) absolutely_integrable_on S)"
   shows "integral (g ` S) f = integral S (\<lambda>x. \<bar>det (matrix (g' x))\<bar> *\<^sub>R f(g x))"
-  using has_absolute_integral_change_of_variables [OF S der_g inj] disj
-  by blast
+  using%unimportant has_absolute_integral_change_of_variables [OF S der_g inj] disj
+  by%unimportant blast
 
-lemma has_absolute_integral_change_of_variables_1:
+lemma%unimportant has_absolute_integral_change_of_variables_1:
   fixes f :: "real \<Rightarrow> real^'n::{finite,wellorder}" and g :: "real \<Rightarrow> real"
   assumes S: "S \<in> sets lebesgue"
     and der_g: "\<And>x. x \<in> S \<Longrightarrow> (g has_vector_derivative g' x) (at x within S)"
@@ -3831,19 +3831,19 @@
 qed
 
 
-corollary absolutely_integrable_change_of_variables_1:
+corollary%important absolutely_integrable_change_of_variables_1:
   fixes f :: "real \<Rightarrow> real^'n::{finite,wellorder}" and g :: "real \<Rightarrow> real"
   assumes S: "S \<in> sets lebesgue"
     and der_g: "\<And>x. x \<in> S \<Longrightarrow> (g has_vector_derivative g' x) (at x within S)"
     and inj: "inj_on g S"
   shows "(f absolutely_integrable_on g ` S \<longleftrightarrow>
              (\<lambda>x. \<bar>g' x\<bar> *\<^sub>R f(g x)) absolutely_integrable_on S)"
-  using has_absolute_integral_change_of_variables_1 [OF assms] by auto
+  using%unimportant has_absolute_integral_change_of_variables_1 [OF assms] by%unimportant auto
 
 
-subsection\<open>Change of variables for integrals: special case of linear function\<close>
+subsection%important\<open>Change of variables for integrals: special case of linear function\<close>
 
-lemma has_absolute_integral_change_of_variables_linear:
+lemma%unimportant has_absolute_integral_change_of_variables_linear:
   fixes f :: "real^'m::{finite,wellorder} \<Rightarrow> real^'n" and g :: "real^'m::_ \<Rightarrow> real^'m::_"
   assumes "linear g"
   shows "(\<lambda>x. \<bar>det (matrix g)\<bar> *\<^sub>R f(g x)) absolutely_integrable_on S \<and>
@@ -3868,14 +3868,14 @@
   qed (use h in auto)
 qed
 
-lemma absolutely_integrable_change_of_variables_linear:
+lemma%unimportant absolutely_integrable_change_of_variables_linear:
   fixes f :: "real^'m::{finite,wellorder} \<Rightarrow> real^'n" and g :: "real^'m::_ \<Rightarrow> real^'m::_"
   assumes "linear g"
   shows "(\<lambda>x. \<bar>det (matrix g)\<bar> *\<^sub>R f(g x)) absolutely_integrable_on S
      \<longleftrightarrow> f absolutely_integrable_on (g ` S)"
   using assms has_absolute_integral_change_of_variables_linear by blast
 
-lemma absolutely_integrable_on_linear_image:
+lemma%unimportant absolutely_integrable_on_linear_image:
   fixes f :: "real^'m::{finite,wellorder} \<Rightarrow> real^'n" and g :: "real^'m::_ \<Rightarrow> real^'m::_"
   assumes "linear g"
   shows "f absolutely_integrable_on (g ` S)
@@ -3883,12 +3883,12 @@
   unfolding assms absolutely_integrable_change_of_variables_linear [OF assms, symmetric] absolutely_integrable_on_scaleR_iff
   by (auto simp: set_integrable_def)
 
-lemma integral_change_of_variables_linear:
+lemma%important integral_change_of_variables_linear:
   fixes f :: "real^'m::{finite,wellorder} \<Rightarrow> real^'n" and g :: "real^'m::_ \<Rightarrow> real^'m::_"
   assumes "linear g"
       and "f absolutely_integrable_on (g ` S) \<or> (f \<circ> g) absolutely_integrable_on S"
     shows "integral (g ` S) f = \<bar>det (matrix g)\<bar> *\<^sub>R integral S (f \<circ> g)"
-proof -
+proof%unimportant -
   have "((\<lambda>x. \<bar>det (matrix g)\<bar> *\<^sub>R f (g x)) absolutely_integrable_on S) \<or> (f absolutely_integrable_on g ` S)"
     using absolutely_integrable_on_linear_image assms by blast
   moreover
@@ -3900,20 +3900,20 @@
     by blast
 qed
 
-subsection\<open>Change of variable for measure\<close>
+subsection%important\<open>Change of variable for measure\<close>
 
-lemma has_measure_differentiable_image:
+lemma%unimportant has_measure_differentiable_image:
   fixes f :: "real^'n::{finite,wellorder} \<Rightarrow> real^'n::_"
   assumes "S \<in> sets lebesgue"
       and "\<And>x. x \<in> S \<Longrightarrow> (f has_derivative f' x) (at x within S)"
       and "inj_on f S"
   shows "f ` S \<in> lmeasurable \<and> measure lebesgue (f ` S) = m
      \<longleftrightarrow> ((\<lambda>x. \<bar>det (matrix (f' x))\<bar>) has_integral m) S"
-  using has_absolute_integral_change_of_variables [OF assms, of "\<lambda>x. (1::real^1)" "vec m"]
-  unfolding absolutely_integrable_on_1_iff integral_on_1_eq integrable_on_1_iff absolutely_integrable_on_def
-  by (auto simp: has_integral_iff lmeasurable_iff_integrable_on lmeasure_integral)
+  using%unimportant has_absolute_integral_change_of_variables [OF assms, of "\<lambda>x. (1::real^1)" "vec m"]
+  unfolding%unimportant absolutely_integrable_on_1_iff integral_on_1_eq integrable_on_1_iff absolutely_integrable_on_def
+  by%unimportant (auto simp: has_integral_iff lmeasurable_iff_integrable_on lmeasure_integral)
 
-lemma measurable_differentiable_image_eq:
+lemma%unimportant measurable_differentiable_image_eq:
   fixes f :: "real^'n::{finite,wellorder} \<Rightarrow> real^'n::_"
   assumes "S \<in> sets lebesgue"
       and "\<And>x. x \<in> S \<Longrightarrow> (f has_derivative f' x) (at x within S)"
@@ -3922,23 +3922,23 @@
   using has_measure_differentiable_image [OF assms]
   by blast
 
-lemma measurable_differentiable_image_alt:
+lemma%important measurable_differentiable_image_alt:
   fixes f :: "real^'n::{finite,wellorder} \<Rightarrow> real^'n::_"
   assumes "S \<in> sets lebesgue"
     and "\<And>x. x \<in> S \<Longrightarrow> (f has_derivative f' x) (at x within S)"
     and "inj_on f S"
   shows "f ` S \<in> lmeasurable \<longleftrightarrow> (\<lambda>x. \<bar>det (matrix (f' x))\<bar>) absolutely_integrable_on S"
-  using measurable_differentiable_image_eq [OF assms]
-  by (simp only: absolutely_integrable_on_iff_nonneg)
+  using%unimportant measurable_differentiable_image_eq [OF assms]
+  by%unimportant (simp only: absolutely_integrable_on_iff_nonneg)
 
-lemma measure_differentiable_image_eq:
+lemma%important measure_differentiable_image_eq:
   fixes f :: "real^'n::{finite,wellorder} \<Rightarrow> real^'n::_"
   assumes S: "S \<in> sets lebesgue"
     and der_f: "\<And>x. x \<in> S \<Longrightarrow> (f has_derivative f' x) (at x within S)"
     and inj: "inj_on f S"
     and intS: "(\<lambda>x. \<bar>det (matrix (f' x))\<bar>) integrable_on S"
   shows "measure lebesgue (f ` S) = integral S (\<lambda>x. \<bar>det (matrix (f' x))\<bar>)"
-  using measurable_differentiable_image_eq [OF S der_f inj]
-        assms has_measure_differentiable_image by blast
+  using%unimportant measurable_differentiable_image_eq [OF S der_f inj]
+        assms has_measure_differentiable_image by%unimportant blast
 
 end

--- a/src/HOL/Analysis/Cross3.thy	Mon Aug 27 22:58:36 2018 +0200
+++ b/src/HOL/Analysis/Cross3.thy	Tue Aug 28 13:28:39 2018 +0100
@@ -13,7 +13,7 @@
 context includes no_Set_Product_syntax 
 begin \<comment>\<open>locally disable syntax for set product, to avoid warnings\<close>
 
-definition cross3 :: "[real^3, real^3] \<Rightarrow> real^3"  (infixr "\<times>" 80)
+definition%important cross3 :: "[real^3, real^3] \<Rightarrow> real^3"  (infixr "\<times>" 80)
   where "a \<times> b \<equiv>
     vector [a$2 * b$3 - a$3 * b$2,
             a$3 * b$1 - a$1 * b$3,
@@ -33,76 +33,76 @@
 
 unbundle cross3_syntax
 
-subsection\<open> Basic lemmas.\<close>
+subsection%important\<open> Basic lemmas.\<close>
 
 lemmas cross3_simps = cross3_def inner_vec_def sum_3 det_3 vec_eq_iff vector_def algebra_simps
 
-lemma dot_cross_self: "x \<bullet> (x \<times> y) = 0" "x \<bullet> (y \<times> x) = 0" "(x \<times> y) \<bullet> y = 0" "(y \<times> x) \<bullet> y = 0"
+lemma%unimportant dot_cross_self: "x \<bullet> (x \<times> y) = 0" "x \<bullet> (y \<times> x) = 0" "(x \<times> y) \<bullet> y = 0" "(y \<times> x) \<bullet> y = 0"
   by (simp_all add: orthogonal_def cross3_simps)
 
-lemma orthogonal_cross: "orthogonal (x \<times> y) x" "orthogonal (x \<times> y) y"  
+lemma%unimportant  orthogonal_cross: "orthogonal (x \<times> y) x" "orthogonal (x \<times> y) y"  
                         "orthogonal y (x \<times> y)" "orthogonal (x \<times> y) x"
   by (simp_all add: orthogonal_def dot_cross_self)
 
-lemma cross_zero_left [simp]: "0 \<times> x = 0" and cross_zero_right [simp]: "x \<times> 0 = 0" for x::"real^3"
+lemma%unimportant  cross_zero_left [simp]: "0 \<times> x = 0" and cross_zero_right [simp]: "x \<times> 0 = 0" for x::"real^3"
   by (simp_all add: cross3_simps)
 
-lemma cross_skew: "(x \<times> y) = -(y \<times> x)" for x::"real^3"
+lemma%unimportant  cross_skew: "(x \<times> y) = -(y \<times> x)" for x::"real^3"
   by (simp add: cross3_simps)
 
-lemma cross_refl [simp]: "x \<times> x = 0" for x::"real^3"
+lemma%unimportant  cross_refl [simp]: "x \<times> x = 0" for x::"real^3"
   by (simp add: cross3_simps)
 
-lemma cross_add_left: "(x + y) \<times> z = (x \<times> z) + (y \<times> z)" for x::"real^3"
+lemma%unimportant  cross_add_left: "(x + y) \<times> z = (x \<times> z) + (y \<times> z)" for x::"real^3"
   by (simp add: cross3_simps)
 
-lemma cross_add_right: "x \<times> (y + z) = (x \<times> y) + (x \<times> z)" for x::"real^3"
+lemma%unimportant  cross_add_right: "x \<times> (y + z) = (x \<times> y) + (x \<times> z)" for x::"real^3"
   by (simp add: cross3_simps)
 
-lemma cross_mult_left: "(c *\<^sub>R x) \<times> y = c *\<^sub>R (x \<times> y)" for x::"real^3"
+lemma%unimportant  cross_mult_left: "(c *\<^sub>R x) \<times> y = c *\<^sub>R (x \<times> y)" for x::"real^3"
   by (simp add: cross3_simps)
 
-lemma cross_mult_right: "x \<times> (c *\<^sub>R y) = c *\<^sub>R (x \<times> y)" for x::"real^3"
+lemma%unimportant  cross_mult_right: "x \<times> (c *\<^sub>R y) = c *\<^sub>R (x \<times> y)" for x::"real^3"
   by (simp add: cross3_simps)
 
-lemma cross_minus_left [simp]: "(-x) \<times> y = - (x \<times> y)" for x::"real^3"
+lemma%unimportant  cross_minus_left [simp]: "(-x) \<times> y = - (x \<times> y)" for x::"real^3"
   by (simp add: cross3_simps)
 
-lemma cross_minus_right [simp]: "x \<times> -y = - (x \<times> y)" for x::"real^3"
+lemma%unimportant  cross_minus_right [simp]: "x \<times> -y = - (x \<times> y)" for x::"real^3"
   by (simp add: cross3_simps)
 
-lemma left_diff_distrib: "(x - y) \<times> z = x \<times> z - y \<times> z" for x::"real^3"
+lemma%unimportant  left_diff_distrib: "(x - y) \<times> z = x \<times> z - y \<times> z" for x::"real^3"
   by (simp add: cross3_simps)
 
-lemma right_diff_distrib: "x \<times> (y - z) = x \<times> y - x \<times> z" for x::"real^3"
+lemma%unimportant  right_diff_distrib: "x \<times> (y - z) = x \<times> y - x \<times> z" for x::"real^3"
   by (simp add: cross3_simps)
 
 hide_fact (open) left_diff_distrib right_diff_distrib
 
-lemma Jacobi: "x \<times> (y \<times> z) + y \<times> (z \<times> x) + z \<times> (x \<times> y) = 0" for x::"real^3"
-  by (simp add: cross3_simps)
+lemma%important  Jacobi: "x \<times> (y \<times> z) + y \<times> (z \<times> x) + z \<times> (x \<times> y) = 0" for x::"real^3"
+  by%unimportant (simp add: cross3_simps)
 
-lemma Lagrange: "x \<times> (y \<times> z) = (x \<bullet> z) *\<^sub>R y - (x \<bullet> y) *\<^sub>R z"
-  by (simp add: cross3_simps) (metis (full_types) exhaust_3)
+lemma%important  Lagrange: "x \<times> (y \<times> z) = (x \<bullet> z) *\<^sub>R y - (x \<bullet> y) *\<^sub>R z"
+  by%unimportant (simp add: cross3_simps) (metis (full_types) exhaust_3)
 
-lemma cross_triple: "(x \<times> y) \<bullet> z = (y \<times> z) \<bullet> x"
+lemma%unimportant  cross_triple: "(x \<times> y) \<bullet> z = (y \<times> z) \<bullet> x"
   by (simp add: cross3_def inner_vec_def sum_3 vec_eq_iff algebra_simps)
 
-lemma cross_components:
+lemma%unimportant  cross_components:
    "(x \<times> y)$1 = x$2 * y$3 - y$2 * x$3" "(x \<times> y)$2 = x$3 * y$1 - y$3 * x$1" "(x \<times> y)$3 = x$1 * y$2 - y$1 * x$2"
   by (simp_all add: cross3_def inner_vec_def sum_3 vec_eq_iff algebra_simps)
 
-lemma cross_basis: "(axis 1 1) \<times> (axis 2 1) = axis 3 1" "(axis 2 1) \<times> (axis 1 1) = -(axis 3 1)" 
+lemma%unimportant  cross_basis: "(axis 1 1) \<times> (axis 2 1) = axis 3 1" "(axis 2 1) \<times> (axis 1 1) = -(axis 3 1)" 
                    "(axis 2 1) \<times> (axis 3 1) = axis 1 1" "(axis 3 1) \<times> (axis 2 1) = -(axis 1 1)" 
                    "(axis 3 1) \<times> (axis 1 1) = axis 2 1" "(axis 1 1) \<times> (axis 3 1) = -(axis 2 1)"
   using exhaust_3
   by (force simp add: axis_def cross3_simps)+
 
-lemma cross_basis_nonzero:
+lemma%unimportant  cross_basis_nonzero:
   "u \<noteq> 0 \<Longrightarrow> ~(u \<times> axis 1 1 = 0) \<or> ~(u \<times> axis 2 1 = 0) \<or> ~(u \<times> axis 3 1 = 0)"
   by (clarsimp simp add: axis_def cross3_simps) (metis vector_3 exhaust_3)
 
-lemma cross_dot_cancel:
+lemma%unimportant  cross_dot_cancel:
   fixes x::"real^3"
   assumes deq: "x \<bullet> y = x \<bullet> z" and veq: "x \<times> y = x \<times> z" and x: "x \<noteq> 0"
   shows "y = z" 
@@ -116,25 +116,25 @@
     using eq_iff_diff_eq_0 by blast
 qed
 
-lemma norm_cross_dot: "(norm (x \<times> y))\<^sup>2 + (x \<bullet> y)\<^sup>2 = (norm x * norm y)\<^sup>2"
+lemma%unimportant  norm_cross_dot: "(norm (x \<times> y))\<^sup>2 + (x \<bullet> y)\<^sup>2 = (norm x * norm y)\<^sup>2"
   unfolding power2_norm_eq_inner power_mult_distrib
   by (simp add: cross3_simps power2_eq_square)
 
-lemma dot_cross_det: "x \<bullet> (y \<times> z) = det(vector[x,y,z])"
+lemma%unimportant  dot_cross_det: "x \<bullet> (y \<times> z) = det(vector[x,y,z])"
   by (simp add: cross3_simps) 
 
-lemma cross_cross_det: "(w \<times> x) \<times> (y \<times> z) = det(vector[w,x,z]) *\<^sub>R y - det(vector[w,x,y]) *\<^sub>R z"
+lemma%unimportant  cross_cross_det: "(w \<times> x) \<times> (y \<times> z) = det(vector[w,x,z]) *\<^sub>R y - det(vector[w,x,y]) *\<^sub>R z"
   using exhaust_3 by (force simp add: cross3_simps) 
 
-lemma dot_cross: "(w \<times> x) \<bullet> (y \<times> z) = (w \<bullet> y) * (x \<bullet> z) - (w \<bullet> z) * (x \<bullet> y)"
-  by (force simp add: cross3_simps) 
+lemma%important  dot_cross: "(w \<times> x) \<bullet> (y \<times> z) = (w \<bullet> y) * (x \<bullet> z) - (w \<bullet> z) * (x \<bullet> y)"
+  by%unimportant (force simp add: cross3_simps) 
 
-lemma norm_cross: "(norm (x \<times> y))\<^sup>2 = (norm x)\<^sup>2 * (norm y)\<^sup>2 - (x \<bullet> y)\<^sup>2"
+lemma%unimportant  norm_cross: "(norm (x \<times> y))\<^sup>2 = (norm x)\<^sup>2 * (norm y)\<^sup>2 - (x \<bullet> y)\<^sup>2"
   unfolding power2_norm_eq_inner power_mult_distrib
   by (simp add: cross3_simps power2_eq_square)
 
-lemma cross_eq_0: "x \<times> y = 0 \<longleftrightarrow> collinear{0,x,y}"
-proof -
+lemma%important  cross_eq_0: "x \<times> y = 0 \<longleftrightarrow> collinear{0,x,y}"
+proof%unimportant -
   have "x \<times> y = 0 \<longleftrightarrow> norm (x \<times> y) = 0"
     by simp
   also have "... \<longleftrightarrow> (norm x * norm y)\<^sup>2 = (x \<bullet> y)\<^sup>2"
@@ -147,11 +147,11 @@
   finally show ?thesis .
 qed
 
-lemma cross_eq_self: "x \<times> y = x \<longleftrightarrow> x = 0" "x \<times> y = y \<longleftrightarrow> y = 0"
+lemma%unimportant  cross_eq_self: "x \<times> y = x \<longleftrightarrow> x = 0" "x \<times> y = y \<longleftrightarrow> y = 0"
   apply (metis cross_zero_left dot_cross_self(1) inner_eq_zero_iff)
   by (metis cross_zero_right dot_cross_self(2) inner_eq_zero_iff)
 
-lemma norm_and_cross_eq_0:
+lemma%unimportant  norm_and_cross_eq_0:
    "x \<bullet> y = 0 \<and> x \<times> y = 0 \<longleftrightarrow> x = 0 \<or> y = 0" (is "?lhs = ?rhs")
 proof 
   assume ?lhs
@@ -159,7 +159,7 @@
     by (metis cross_dot_cancel cross_zero_right inner_zero_right)
 qed auto
 
-lemma bilinear_cross: "bilinear(\<times>)"
+lemma%unimportant  bilinear_cross: "bilinear(\<times>)"
   apply (auto simp add: bilinear_def linear_def)
   apply unfold_locales
   apply (simp add: cross_add_right)
@@ -168,33 +168,33 @@
   apply (simp add: cross_mult_left)
   done
 
-subsection\<open>Preservation by rotation, or other orthogonal transformation up to sign.\<close>
+subsection%important   \<open>Preservation by rotation, or other orthogonal transformation up to sign.\<close>
 
-lemma cross_matrix_mult: "transpose A *v ((A *v x) \<times> (A *v y)) = det A *\<^sub>R (x \<times> y)"
+lemma%unimportant  cross_matrix_mult: "transpose A *v ((A *v x) \<times> (A *v y)) = det A *\<^sub>R (x \<times> y)"
   apply (simp add: vec_eq_iff   )
   apply (simp add: vector_matrix_mult_def matrix_vector_mult_def forall_3 cross3_simps)
   done
 
-lemma cross_orthogonal_matrix:
+lemma%important  cross_orthogonal_matrix:
   assumes "orthogonal_matrix A"
   shows "(A *v x) \<times> (A *v y) = det A *\<^sub>R (A *v (x \<times> y))"
-proof -
+proof%unimportant -
   have "mat 1 = transpose (A ** transpose A)"
     by (metis (no_types) assms orthogonal_matrix_def transpose_mat)
   then show ?thesis
     by (metis (no_types) vector_matrix_mul_rid vector_transpose_matrix cross_matrix_mult matrix_vector_mul_assoc matrix_vector_mult_scaleR)
 qed
 
-lemma cross_rotation_matrix: "rotation_matrix A \<Longrightarrow> (A *v x) \<times> (A *v y) =  A *v (x \<times> y)"
+lemma%unimportant  cross_rotation_matrix: "rotation_matrix A \<Longrightarrow> (A *v x) \<times> (A *v y) =  A *v (x \<times> y)"
   by (simp add: rotation_matrix_def cross_orthogonal_matrix)
 
-lemma cross_rotoinversion_matrix: "rotoinversion_matrix A \<Longrightarrow> (A *v x) \<times> (A *v y) = - A *v (x \<times> y)"
+lemma%unimportant  cross_rotoinversion_matrix: "rotoinversion_matrix A \<Longrightarrow> (A *v x) \<times> (A *v y) = - A *v (x \<times> y)"
   by (simp add: rotoinversion_matrix_def cross_orthogonal_matrix scaleR_matrix_vector_assoc)
 
-lemma cross_orthogonal_transformation:
+lemma%important  cross_orthogonal_transformation:
   assumes "orthogonal_transformation f"
   shows   "(f x) \<times> (f y) = det(matrix f) *\<^sub>R f(x \<times> y)"
-proof -
+proof%unimportant -
   have orth: "orthogonal_matrix (matrix f)"
     using assms orthogonal_transformation_matrix by blast
   have "matrix f *v z = f z" for z
@@ -203,20 +203,20 @@
     by simp
 qed
 
-lemma cross_linear_image:
+lemma%unimportant  cross_linear_image:
    "\<lbrakk>linear f; \<And>x. norm(f x) = norm x; det(matrix f) = 1\<rbrakk>
            \<Longrightarrow> (f x) \<times> (f y) = f(x \<times> y)"
   by (simp add: cross_orthogonal_transformation orthogonal_transformation)
 
-subsection\<open>Continuity\<close>
+subsection%unimportant \<open>Continuity\<close>
 
-lemma continuous_cross: "\<lbrakk>continuous F f; continuous F g\<rbrakk> \<Longrightarrow> continuous F (\<lambda>x. (f x) \<times> (g x))"
+lemma%unimportant  continuous_cross: "\<lbrakk>continuous F f; continuous F g\<rbrakk> \<Longrightarrow> continuous F (\<lambda>x. (f x) \<times> (g x))"
   apply (subst continuous_componentwise)
   apply (clarsimp simp add: cross3_simps)
   apply (intro continuous_intros; simp)
   done
 
-lemma continuous_on_cross:
+lemma%unimportant  continuous_on_cross:
   fixes f :: "'a::t2_space \<Rightarrow> real^3"
   shows "\<lbrakk>continuous_on S f; continuous_on S g\<rbrakk> \<Longrightarrow> continuous_on S (\<lambda>x. (f x) \<times> (g x))"
   by (simp add: continuous_on_eq_continuous_within continuous_cross)

--- a/src/HOL/Analysis/Determinants.thy	Mon Aug 27 22:58:36 2018 +0200
+++ b/src/HOL/Analysis/Determinants.thy	Tue Aug 28 13:28:39 2018 +0100
@@ -10,40 +10,40 @@
   "HOL-Library.Permutations"
 begin
 
-subsection \<open>Trace\<close>
+subsection%important  \<open>Trace\<close>
 
-definition trace :: "'a::semiring_1^'n^'n \<Rightarrow> 'a"
+definition%important  trace :: "'a::semiring_1^'n^'n \<Rightarrow> 'a"
   where "trace A = sum (\<lambda>i. ((A$i)$i)) (UNIV::'n set)"
 
-lemma trace_0: "trace (mat 0) = 0"
+lemma%unimportant  trace_0: "trace (mat 0) = 0"
   by (simp add: trace_def mat_def)
 
-lemma trace_I: "trace (mat 1 :: 'a::semiring_1^'n^'n) = of_nat(CARD('n))"
+lemma%unimportant  trace_I: "trace (mat 1 :: 'a::semiring_1^'n^'n) = of_nat(CARD('n))"
   by (simp add: trace_def mat_def)
 
-lemma trace_add: "trace ((A::'a::comm_semiring_1^'n^'n) + B) = trace A + trace B"
+lemma%unimportant  trace_add: "trace ((A::'a::comm_semiring_1^'n^'n) + B) = trace A + trace B"
   by (simp add: trace_def sum.distrib)
 
-lemma trace_sub: "trace ((A::'a::comm_ring_1^'n^'n) - B) = trace A - trace B"
+lemma%unimportant  trace_sub: "trace ((A::'a::comm_ring_1^'n^'n) - B) = trace A - trace B"
   by (simp add: trace_def sum_subtractf)
 
-lemma trace_mul_sym: "trace ((A::'a::comm_semiring_1^'n^'m) ** B) = trace (B**A)"
+lemma%important  trace_mul_sym: "trace ((A::'a::comm_semiring_1^'n^'m) ** B) = trace (B**A)"
   apply (simp add: trace_def matrix_matrix_mult_def)
   apply (subst sum.swap)
   apply (simp add: mult.commute)
   done
 
-subsubsection \<open>Definition of determinant\<close>
+subsubsection%important  \<open>Definition of determinant\<close>
 
-definition det:: "'a::comm_ring_1^'n^'n \<Rightarrow> 'a" where
+definition%important  det:: "'a::comm_ring_1^'n^'n \<Rightarrow> 'a" where
   "det A =
     sum (\<lambda>p. of_int (sign p) * prod (\<lambda>i. A$i$p i) (UNIV :: 'n set))
       {p. p permutes (UNIV :: 'n set)}"
 
 text \<open>Basic determinant properties\<close>
 
-lemma det_transpose [simp]: "det (transpose A) = det (A::'a::comm_ring_1 ^'n^'n)"
-proof -
+lemma%important  det_transpose [simp]: "det (transpose A) = det (A::'a::comm_ring_1 ^'n^'n)"
+proof%unimportant -
   let ?di = "\<lambda>A i j. A$i$j"
   let ?U = "(UNIV :: 'n set)"
   have fU: "finite ?U" by simp
@@ -80,7 +80,7 @@
     by (subst sum_permutations_inverse) (blast intro: sum.cong)
 qed
 
-lemma det_lowerdiagonal:
+lemma%unimportant  det_lowerdiagonal:
   fixes A :: "'a::comm_ring_1^('n::{finite,wellorder})^('n::{finite,wellorder})"
   assumes ld: "\<And>i j. i < j \<Longrightarrow> A$i$j = 0"
   shows "det A = prod (\<lambda>i. A$i$i) (UNIV:: 'n set)"
@@ -107,11 +107,11 @@
     unfolding det_def by (simp add: sign_id)
 qed
 
-lemma det_upperdiagonal:
+lemma%important  det_upperdiagonal:
   fixes A :: "'a::comm_ring_1^'n::{finite,wellorder}^'n::{finite,wellorder}"
   assumes ld: "\<And>i j. i > j \<Longrightarrow> A$i$j = 0"
   shows "det A = prod (\<lambda>i. A$i$i) (UNIV:: 'n set)"
-proof -
+proof%unimportant -
   let ?U = "UNIV:: 'n set"
   let ?PU = "{p. p permutes ?U}"
   let ?pp = "(\<lambda>p. of_int (sign p) * prod (\<lambda>i. A$i$p i) (UNIV :: 'n set))"
@@ -134,11 +134,11 @@
     unfolding det_def by (simp add: sign_id)
 qed
 
-lemma det_diagonal:
+lemma%important  det_diagonal:
   fixes A :: "'a::comm_ring_1^'n^'n"
   assumes ld: "\<And>i j. i \<noteq> j \<Longrightarrow> A$i$j = 0"
   shows "det A = prod (\<lambda>i. A$i$i) (UNIV::'n set)"
-proof -
+proof%unimportant -
   let ?U = "UNIV:: 'n set"
   let ?PU = "{p. p permutes ?U}"
   let ?pp = "\<lambda>p. of_int (sign p) * prod (\<lambda>i. A$i$p i) (UNIV :: 'n set)"
@@ -161,13 +161,13 @@
     unfolding det_def by (simp add: sign_id)
 qed
 
-lemma det_I [simp]: "det (mat 1 :: 'a::comm_ring_1^'n^'n) = 1"
+lemma%unimportant  det_I [simp]: "det (mat 1 :: 'a::comm_ring_1^'n^'n) = 1"
   by (simp add: det_diagonal mat_def)
 
-lemma det_0 [simp]: "det (mat 0 :: 'a::comm_ring_1^'n^'n) = 0"
+lemma%unimportant  det_0 [simp]: "det (mat 0 :: 'a::comm_ring_1^'n^'n) = 0"
   by (simp add: det_def prod_zero power_0_left)
 
-lemma det_permute_rows:
+lemma%unimportant  det_permute_rows:
   fixes A :: "'a::comm_ring_1^'n^'n"
   assumes p: "p permutes (UNIV :: 'n::finite set)"
   shows "det (\<chi> i. A$p i :: 'a^'n^'n) = of_int (sign p) * det A"
@@ -204,11 +204,11 @@
     done
 qed
 
-lemma det_permute_columns:
+lemma%important  det_permute_columns:
   fixes A :: "'a::comm_ring_1^'n^'n"
   assumes p: "p permutes (UNIV :: 'n set)"
   shows "det(\<chi> i j. A$i$ p j :: 'a^'n^'n) = of_int (sign p) * det A"
-proof -
+proof%unimportant -
   let ?Ap = "\<chi> i j. A$i$ p j :: 'a^'n^'n"
   let ?At = "transpose A"
   have "of_int (sign p) * det A = det (transpose (\<chi> i. transpose A $ p i))"
@@ -220,7 +220,7 @@
     by simp
 qed
 
-lemma det_identical_columns:
+lemma%unimportant  det_identical_columns:
   fixes A :: "'a::comm_ring_1^'n^'n"
   assumes jk: "j \<noteq> k"
     and r: "column j A = column k A"
@@ -300,24 +300,24 @@
   finally show "det A = 0" by simp
 qed
 
-lemma det_identical_rows:
+lemma%unimportant  det_identical_rows:
   fixes A :: "'a::comm_ring_1^'n^'n"
   assumes ij: "i \<noteq> j" and r: "row i A = row j A"
   shows "det A = 0"
   by (metis column_transpose det_identical_columns det_transpose ij r)
 
-lemma det_zero_row:
+lemma%unimportant  det_zero_row:
   fixes A :: "'a::{idom, ring_char_0}^'n^'n" and F :: "'b::{field}^'m^'m"
   shows "row i A = 0 \<Longrightarrow> det A = 0" and "row j F = 0 \<Longrightarrow> det F = 0"
   by (force simp: row_def det_def vec_eq_iff sign_nz intro!: sum.neutral)+
 
-lemma det_zero_column:
+lemma%unimportant  det_zero_column:
   fixes A :: "'a::{idom, ring_char_0}^'n^'n" and F :: "'b::{field}^'m^'m"
   shows "column i A = 0 \<Longrightarrow> det A = 0" and "column j F = 0 \<Longrightarrow> det F = 0"
   unfolding atomize_conj atomize_imp
   by (metis det_transpose det_zero_row row_transpose)
 
-lemma det_row_add:
+lemma%unimportant  det_row_add:
   fixes a b c :: "'n::finite \<Rightarrow> _ ^ 'n"
   shows "det((\<chi> i. if i = k then a i + b i else c i)::'a::comm_ring_1^'n^'n) =
     det((\<chi> i. if i = k then a i else c i)::'a::comm_ring_1^'n^'n) +
@@ -358,7 +358,7 @@
     by (simp add: field_simps)
 qed auto
 
-lemma det_row_mul:
+lemma%unimportant  det_row_mul:
   fixes a b :: "'n::finite \<Rightarrow> _ ^ 'n"
   shows "det((\<chi> i. if i = k then c *s a i else b i)::'a::comm_ring_1^'n^'n) =
     c * det((\<chi> i. if i = k then a i else b i)::'a::comm_ring_1^'n^'n)"
@@ -395,7 +395,7 @@
     by (simp add: field_simps)
 qed auto
 
-lemma det_row_0:
+lemma%unimportant  det_row_0:
   fixes b :: "'n::finite \<Rightarrow> _ ^ 'n"
   shows "det((\<chi> i. if i = k then 0 else b i)::'a::comm_ring_1^'n^'n) = 0"
   using det_row_mul[of k 0 "\<lambda>i. 1" b]
@@ -403,11 +403,11 @@
   apply (simp only: vector_smult_lzero)
   done
 
-lemma det_row_operation:
+lemma%important  det_row_operation:
   fixes A :: "'a::{comm_ring_1}^'n^'n"
   assumes ij: "i \<noteq> j"
   shows "det (\<chi> k. if k = i then row i A + c *s row j A else row k A) = det A"
-proof -
+proof%unimportant -
   let ?Z = "(\<chi> k. if k = i then row j A else row k A) :: 'a ^'n^'n"
   have th: "row i ?Z = row j ?Z" by (vector row_def)
   have th2: "((\<chi> k. if k = i then row i A else row k A) :: 'a^'n^'n) = A"
@@ -417,7 +417,7 @@
     by simp
 qed
 
-lemma det_row_span:
+lemma%unimportant  det_row_span:
   fixes A :: "'a::{field}^'n^'n"
   assumes x: "x \<in> vec.span {row j A |j. j \<noteq> i}"
   shows "det (\<chi> k. if k = i then row i A + x else row k A) = det A"
@@ -449,10 +449,10 @@
     unfolding scalar_mult_eq_scaleR .
 qed
 
-lemma matrix_id [simp]: "det (matrix id) = 1"
+lemma%unimportant  matrix_id [simp]: "det (matrix id) = 1"
   by (simp add: matrix_id_mat_1)
 
-lemma det_matrix_scaleR [simp]: "det (matrix ((( *\<^sub>R) r)) :: real^'n^'n) = r ^ CARD('n::finite)"
+lemma%important  det_matrix_scaleR [simp]: "det (matrix ((( *\<^sub>R) r)) :: real^'n^'n) = r ^ CARD('n::finite)"
   apply (subst det_diagonal)
    apply (auto simp: matrix_def mat_def)
   apply (simp add: cart_eq_inner_axis inner_axis_axis)
@@ -463,7 +463,7 @@
   exact duplicates by considering the rows/columns as a set.
 \<close>
 
-lemma det_dependent_rows:
+lemma%unimportant  det_dependent_rows:
   fixes A:: "'a::{field}^'n^'n"
   assumes d: "vec.dependent (rows A)"
   shows "det A = 0"
@@ -491,23 +491,23 @@
   qed
 qed
 
-lemma det_dependent_columns:
+lemma%unimportant  det_dependent_columns:
   assumes d: "vec.dependent (columns (A::real^'n^'n))"
   shows "det A = 0"
   by (metis d det_dependent_rows rows_transpose det_transpose)
 
 text \<open>Multilinearity and the multiplication formula\<close>
 
-lemma Cart_lambda_cong: "(\<And>x. f x = g x) \<Longrightarrow> (vec_lambda f::'a^'n) = (vec_lambda g :: 'a^'n)"
+lemma%unimportant  Cart_lambda_cong: "(\<And>x. f x = g x) \<Longrightarrow> (vec_lambda f::'a^'n) = (vec_lambda g :: 'a^'n)"
   by auto
 
-lemma det_linear_row_sum:
+lemma%unimportant  det_linear_row_sum:
   assumes fS: "finite S"
   shows "det ((\<chi> i. if i = k then sum (a i) S else c i)::'a::comm_ring_1^'n^'n) =
     sum (\<lambda>j. det ((\<chi> i. if i = k then a  i j else c i)::'a^'n^'n)) S"
   using fS  by (induct rule: finite_induct; simp add: det_row_0 det_row_add cong: if_cong)
 
-lemma finite_bounded_functions:
+lemma%unimportant  finite_bounded_functions:
   assumes fS: "finite S"
   shows "finite {f. (\<forall>i \<in> {1.. (k::nat)}. f i \<in> S) \<and> (\<forall>i. i \<notin> {1 .. k} \<longrightarrow> f i = i)}"
 proof (induct k)
@@ -532,14 +532,14 @@
 qed
 
 
-lemma det_linear_rows_sum_lemma:
+lemma%important  det_linear_rows_sum_lemma:
   assumes fS: "finite S"
     and fT: "finite T"
   shows "det ((\<chi> i. if i \<in> T then sum (a i) S else c i):: 'a::comm_ring_1^'n^'n) =
     sum (\<lambda>f. det((\<chi> i. if i \<in> T then a i (f i) else c i)::'a^'n^'n))
       {f. (\<forall>i \<in> T. f i \<in> S) \<and> (\<forall>i. i \<notin> T \<longrightarrow> f i = i)}"
   using fT
-proof (induct T arbitrary: a c set: finite)
+proof%unimportant (induct T arbitrary: a c set: finite)
   case empty
   have th0: "\<And>x y. (\<chi> i. if i \<in> {} then x i else y i) = (\<chi> i. y i)"
     by vector
@@ -577,7 +577,7 @@
        (auto intro!: cong[OF refl[of det]] simp: vec_eq_iff)
 qed
 
-lemma det_linear_rows_sum:
+lemma%unimportant  det_linear_rows_sum:
   fixes S :: "'n::finite set"
   assumes fS: "finite S"
   shows "det (\<chi> i. sum (a i) S) =
@@ -589,12 +589,12 @@
   show ?thesis by simp
 qed
 
-lemma matrix_mul_sum_alt:
+lemma%unimportant  matrix_mul_sum_alt:
   fixes A B :: "'a::comm_ring_1^'n^'n"
   shows "A ** B = (\<chi> i. sum (\<lambda>k. A$i$k *s B $ k) (UNIV :: 'n set))"
   by (vector matrix_matrix_mult_def sum_component)
 
-lemma det_rows_mul:
+lemma%unimportant  det_rows_mul:
   "det((\<chi> i. c i *s a i)::'a::comm_ring_1^'n^'n) =
     prod (\<lambda>i. c i) (UNIV:: 'n set) * det((\<chi> i. a i)::'a^'n^'n)"
 proof (simp add: det_def sum_distrib_left cong add: prod.cong, rule sum.cong)
@@ -612,10 +612,10 @@
     by (simp add: field_simps)
 qed rule
 
-lemma det_mul:
+lemma%important  det_mul:
   fixes A B :: "'a::comm_ring_1^'n^'n"
   shows "det (A ** B) = det A * det B"
-proof -
+proof%unimportant -
   let ?U = "UNIV :: 'n set"
   let ?F = "{f. (\<forall>i \<in> ?U. f i \<in> ?U) \<and> (\<forall>i. i \<notin> ?U \<longrightarrow> f i = i)}"
   let ?PU = "{p. p permutes ?U}"
@@ -712,12 +712,12 @@
 qed
 
 
-subsection \<open>Relation to invertibility\<close>
+subsection%important \<open>Relation to invertibility\<close>
 
-lemma invertible_det_nz:
+lemma%important  invertible_det_nz:
   fixes A::"'a::{field}^'n^'n"
   shows "invertible A \<longleftrightarrow> det A \<noteq> 0"
-proof (cases "invertible A")
+proof%unimportant (cases "invertible A")
   case True
   then obtain B :: "'a^'n^'n" where B: "A ** B = mat 1"
     unfolding invertible_right_inverse by blast
@@ -748,7 +748,7 @@
 qed
 
 
-lemma det_nz_iff_inj_gen:
+lemma%unimportant  det_nz_iff_inj_gen:
   fixes f :: "'a::field^'n \<Rightarrow> 'a::field^'n"
   assumes "Vector_Spaces.linear ( *s) ( *s) f"
   shows "det (matrix f) \<noteq> 0 \<longleftrightarrow> inj f"
@@ -763,26 +763,26 @@
     by (metis assms invertible_det_nz invertible_left_inverse matrix_compose_gen matrix_id_mat_1)
 qed
 
-lemma det_nz_iff_inj:
+lemma%unimportant  det_nz_iff_inj:
   fixes f :: "real^'n \<Rightarrow> real^'n"
   assumes "linear f"
   shows "det (matrix f) \<noteq> 0 \<longleftrightarrow> inj f"
   using det_nz_iff_inj_gen[of f] assms
   unfolding linear_matrix_vector_mul_eq .
 
-lemma det_eq_0_rank:
+lemma%unimportant  det_eq_0_rank:
   fixes A :: "real^'n^'n"
   shows "det A = 0 \<longleftrightarrow> rank A < CARD('n)"
   using invertible_det_nz [of A]
   by (auto simp: matrix_left_invertible_injective invertible_left_inverse less_rank_noninjective)
 
-subsubsection\<open>Invertibility of matrices and corresponding linear functions\<close>
+subsubsection%important  \<open>Invertibility of matrices and corresponding linear functions\<close>
 
-lemma matrix_left_invertible_gen:
+lemma%important  matrix_left_invertible_gen:
   fixes f :: "'a::field^'m \<Rightarrow> 'a::field^'n"
   assumes "Vector_Spaces.linear ( *s) ( *s) f"
   shows "((\<exists>B. B ** matrix f = mat 1) \<longleftrightarrow> (\<exists>g. Vector_Spaces.linear ( *s) ( *s) g \<and> g \<circ> f = id))"
-proof safe
+proof%unimportant safe
   fix B
   assume 1: "B ** matrix f = mat 1"
   show "\<exists>g. Vector_Spaces.linear ( *s) ( *s) g \<and> g \<circ> f = id"
@@ -801,12 +801,12 @@
   then show "\<exists>B. B ** matrix f = mat 1" ..
 qed
 
-lemma matrix_left_invertible:
+lemma%unimportant  matrix_left_invertible:
   "linear f \<Longrightarrow> ((\<exists>B. B ** matrix f = mat 1) \<longleftrightarrow> (\<exists>g. linear g \<and> g \<circ> f = id))" for f::"real^'m \<Rightarrow> real^'n"
   using matrix_left_invertible_gen[of f]
   by (auto simp: linear_matrix_vector_mul_eq)
 
-lemma matrix_right_invertible_gen:
+lemma%unimportant  matrix_right_invertible_gen:
   fixes f :: "'a::field^'m \<Rightarrow> 'a^'n"
   assumes "Vector_Spaces.linear ( *s) ( *s) f"
   shows "((\<exists>B. matrix f ** B = mat 1) \<longleftrightarrow> (\<exists>g. Vector_Spaces.linear ( *s) ( *s) g \<and> f \<circ> g = id))"
@@ -829,12 +829,12 @@
   then show "\<exists>B. matrix f ** B = mat 1" ..
 qed
 
-lemma matrix_right_invertible:
+lemma%unimportant  matrix_right_invertible:
   "linear f \<Longrightarrow> ((\<exists>B. matrix f ** B = mat 1) \<longleftrightarrow> (\<exists>g. linear g \<and> f \<circ> g = id))" for f::"real^'m \<Rightarrow> real^'n"
   using matrix_right_invertible_gen[of f]
   by (auto simp: linear_matrix_vector_mul_eq)
 
-lemma matrix_invertible_gen:
+lemma%unimportant  matrix_invertible_gen:
   fixes f :: "'a::field^'m \<Rightarrow> 'a::field^'n"
   assumes "Vector_Spaces.linear ( *s) ( *s) f"
   shows  "invertible (matrix f) \<longleftrightarrow> (\<exists>g. Vector_Spaces.linear ( *s) ( *s) g \<and> f \<circ> g = id \<and> g \<circ> f = id)"
@@ -847,13 +847,13 @@
     by (metis assms invertible_def matrix_compose_gen matrix_id_mat_1)
 qed
 
-lemma matrix_invertible:
+lemma%unimportant  matrix_invertible:
   "linear f \<Longrightarrow> invertible (matrix f) \<longleftrightarrow> (\<exists>g. linear g \<and> f \<circ> g = id \<and> g \<circ> f = id)"
   for f::"real^'m \<Rightarrow> real^'n"
   using matrix_invertible_gen[of f]
   by (auto simp: linear_matrix_vector_mul_eq)
 
-lemma invertible_eq_bij:
+lemma%unimportant  invertible_eq_bij:
   fixes m :: "'a::field^'m^'n"
   shows "invertible m \<longleftrightarrow> bij (( *v) m)"
   using matrix_invertible_gen[OF matrix_vector_mul_linear_gen, of m, simplified matrix_of_matrix_vector_mul]
@@ -861,15 +861,15 @@
       vec.linear_injective_left_inverse vec.linear_surjective_right_inverse)
 
 
-subsection \<open>Cramer's rule\<close>
+subsection%important \<open>Cramer's rule\<close>
 
-lemma cramer_lemma_transpose:
+lemma%important  cramer_lemma_transpose:
   fixes A:: "'a::{field}^'n^'n"
     and x :: "'a::{field}^'n"
   shows "det ((\<chi> i. if i = k then sum (\<lambda>i. x$i *s row i A) (UNIV::'n set)
                              else row i A)::'a::{field}^'n^'n) = x$k * det A"
   (is "?lhs = ?rhs")
-proof -
+proof%unimportant -
   let ?U = "UNIV :: 'n set"
   let ?Uk = "?U - {k}"
   have U: "?U = insert k ?Uk"
@@ -899,10 +899,10 @@
     done
 qed
 
-lemma cramer_lemma:
+lemma%important  cramer_lemma:
   fixes A :: "'a::{field}^'n^'n"
   shows "det((\<chi> i j. if j = k then (A *v x)$i else A$i$j):: 'a::{field}^'n^'n) = x$k * det A"
-proof -
+proof%unimportant -
   let ?U = "UNIV :: 'n set"
   have *: "\<And>c. sum (\<lambda>i. c i *s row i (transpose A)) ?U = sum (\<lambda>i. c i *s column i A) ?U"
     by (auto intro: sum.cong)
@@ -916,11 +916,11 @@
     done
 qed
 
-lemma cramer:
+lemma%important  cramer:
   fixes A ::"'a::{field}^'n^'n"
   assumes d0: "det A \<noteq> 0"
   shows "A *v x = b \<longleftrightarrow> x = (\<chi> k. det(\<chi> i j. if j=k then b$i else A$i$j) / det A)"
-proof -
+proof%unimportant -
   from d0 obtain B where B: "A ** B = mat 1" "B ** A = mat 1"
     unfolding invertible_det_nz[symmetric] invertible_def
     by blast
@@ -941,14 +941,14 @@
     by auto
 qed
 
-subsection \<open>Orthogonality of a transformation and matrix\<close>
+subsection%important  \<open>Orthogonality of a transformation and matrix\<close>
 
-definition "orthogonal_transformation f \<longleftrightarrow> linear f \<and> (\<forall>v w. f v \<bullet> f w = v \<bullet> w)"
+definition%important  "orthogonal_transformation f \<longleftrightarrow> linear f \<and> (\<forall>v w. f v \<bullet> f w = v \<bullet> w)"
 
-definition "orthogonal_matrix (Q::'a::semiring_1^'n^'n) \<longleftrightarrow>
+definition%important  "orthogonal_matrix (Q::'a::semiring_1^'n^'n) \<longleftrightarrow>
   transpose Q ** Q = mat 1 \<and> Q ** transpose Q = mat 1"
 
-lemma orthogonal_transformation:
+lemma%unimportant  orthogonal_transformation:
   "orthogonal_transformation f \<longleftrightarrow> linear f \<and> (\<forall>v. norm (f v) = norm v)"
   unfolding orthogonal_transformation_def
   apply auto
@@ -957,70 +957,70 @@
   apply (simp add: dot_norm linear_add[symmetric])
   done
 
-lemma orthogonal_transformation_id [simp]: "orthogonal_transformation (\<lambda>x. x)"
+lemma%unimportant  orthogonal_transformation_id [simp]: "orthogonal_transformation (\<lambda>x. x)"
   by (simp add: linear_iff orthogonal_transformation_def)
 
-lemma orthogonal_orthogonal_transformation:
+lemma%unimportant  orthogonal_orthogonal_transformation:
     "orthogonal_transformation f \<Longrightarrow> orthogonal (f x) (f y) \<longleftrightarrow> orthogonal x y"
   by (simp add: orthogonal_def orthogonal_transformation_def)
 
-lemma orthogonal_transformation_compose:
+lemma%unimportant  orthogonal_transformation_compose:
    "\<lbrakk>orthogonal_transformation f; orthogonal_transformation g\<rbrakk> \<Longrightarrow> orthogonal_transformation(f \<circ> g)"
   by (auto simp: orthogonal_transformation_def linear_compose)
 
-lemma orthogonal_transformation_neg:
+lemma%unimportant  orthogonal_transformation_neg:
   "orthogonal_transformation(\<lambda>x. -(f x)) \<longleftrightarrow> orthogonal_transformation f"
   by (auto simp: orthogonal_transformation_def dest: linear_compose_neg)
 
-lemma orthogonal_transformation_scaleR: "orthogonal_transformation f \<Longrightarrow> f (c *\<^sub>R v) = c *\<^sub>R f v"
+lemma%unimportant  orthogonal_transformation_scaleR: "orthogonal_transformation f \<Longrightarrow> f (c *\<^sub>R v) = c *\<^sub>R f v"
   by (simp add: linear_iff orthogonal_transformation_def)
 
-lemma orthogonal_transformation_linear:
+lemma%unimportant  orthogonal_transformation_linear:
    "orthogonal_transformation f \<Longrightarrow> linear f"
   by (simp add: orthogonal_transformation_def)
 
-lemma orthogonal_transformation_inj:
+lemma%unimportant  orthogonal_transformation_inj:
   "orthogonal_transformation f \<Longrightarrow> inj f"
   unfolding orthogonal_transformation_def inj_on_def
   by (metis vector_eq)
 
-lemma orthogonal_transformation_surj:
+lemma%unimportant  orthogonal_transformation_surj:
   "orthogonal_transformation f \<Longrightarrow> surj f"
   for f :: "'a::euclidean_space \<Rightarrow> 'a::euclidean_space"
   by (simp add: linear_injective_imp_surjective orthogonal_transformation_inj orthogonal_transformation_linear)
 
-lemma orthogonal_transformation_bij:
+lemma%unimportant  orthogonal_transformation_bij:
   "orthogonal_transformation f \<Longrightarrow> bij f"
   for f :: "'a::euclidean_space \<Rightarrow> 'a::euclidean_space"
   by (simp add: bij_def orthogonal_transformation_inj orthogonal_transformation_surj)
 
-lemma orthogonal_transformation_inv:
+lemma%unimportant  orthogonal_transformation_inv:
   "orthogonal_transformation f \<Longrightarrow> orthogonal_transformation (inv f)"
   for f :: "'a::euclidean_space \<Rightarrow> 'a::euclidean_space"
   by (metis (no_types, hide_lams) bijection.inv_right bijection_def inj_linear_imp_inv_linear orthogonal_transformation orthogonal_transformation_bij orthogonal_transformation_inj)
 
-lemma orthogonal_transformation_norm:
+lemma%unimportant  orthogonal_transformation_norm:
   "orthogonal_transformation f \<Longrightarrow> norm (f x) = norm x"
   by (metis orthogonal_transformation)
 
-lemma orthogonal_matrix: "orthogonal_matrix (Q:: real ^'n^'n) \<longleftrightarrow> transpose Q ** Q = mat 1"
+lemma%unimportant  orthogonal_matrix: "orthogonal_matrix (Q:: real ^'n^'n) \<longleftrightarrow> transpose Q ** Q = mat 1"
   by (metis matrix_left_right_inverse orthogonal_matrix_def)
 
-lemma orthogonal_matrix_id: "orthogonal_matrix (mat 1 :: _^'n^'n)"
+lemma%unimportant  orthogonal_matrix_id: "orthogonal_matrix (mat 1 :: _^'n^'n)"
   by (simp add: orthogonal_matrix_def)
 
-lemma orthogonal_matrix_mul:
+lemma%unimportant  orthogonal_matrix_mul:
   fixes A :: "real ^'n^'n"
   assumes  "orthogonal_matrix A" "orthogonal_matrix B"
   shows "orthogonal_matrix(A ** B)"
   using assms
   by (simp add: orthogonal_matrix matrix_transpose_mul matrix_left_right_inverse matrix_mul_assoc)
 
-lemma orthogonal_transformation_matrix:
+lemma%important  orthogonal_transformation_matrix:
   fixes f:: "real^'n \<Rightarrow> real^'n"
   shows "orthogonal_transformation f \<longleftrightarrow> linear f \<and> orthogonal_matrix(matrix f)"
   (is "?lhs \<longleftrightarrow> ?rhs")
-proof -
+proof%unimportant -
   let ?mf = "matrix f"
   let ?ot = "orthogonal_transformation f"
   let ?U = "UNIV :: 'n set"
@@ -1063,11 +1063,11 @@
     by (auto simp: linear_def scalar_mult_eq_scaleR)
 qed
 
-lemma det_orthogonal_matrix:
+lemma%important  det_orthogonal_matrix:
   fixes Q:: "'a::linordered_idom^'n^'n"
   assumes oQ: "orthogonal_matrix Q"
   shows "det Q = 1 \<or> det Q = - 1"
-proof -
+proof%unimportant -
   have "Q ** transpose Q = mat 1"
     by (metis oQ orthogonal_matrix_def)
   then have "det (Q ** transpose Q) = det (mat 1:: 'a^'n^'n)"
@@ -1078,20 +1078,20 @@
     by (simp add: square_eq_1_iff)
 qed
 
-lemma orthogonal_transformation_det [simp]:
+lemma%important  orthogonal_transformation_det [simp]:
   fixes f :: "real^'n \<Rightarrow> real^'n"
   shows "orthogonal_transformation f \<Longrightarrow> \<bar>det (matrix f)\<bar> = 1"
-  using det_orthogonal_matrix orthogonal_transformation_matrix by fastforce
+  using%unimportant det_orthogonal_matrix orthogonal_transformation_matrix by fastforce
 
 
-subsection \<open>Linearity of scaling, and hence isometry, that preserves origin\<close>
+subsection%important  \<open>Linearity of scaling, and hence isometry, that preserves origin\<close>
 
-lemma scaling_linear:
+lemma%important  scaling_linear:
   fixes f :: "'a::real_inner \<Rightarrow> 'a::real_inner"
   assumes f0: "f 0 = 0"
     and fd: "\<forall>x y. dist (f x) (f y) = c * dist x y"
   shows "linear f"
-proof -
+proof%unimportant -
   {
     fix v w
     have "norm (f x) = c * norm x" for x
@@ -1105,13 +1105,13 @@
     by (simp add: inner_add field_simps)
 qed
 
-lemma isometry_linear:
+lemma%unimportant  isometry_linear:
   "f (0::'a::real_inner) = (0::'a) \<Longrightarrow> \<forall>x y. dist(f x) (f y) = dist x y \<Longrightarrow> linear f"
   by (rule scaling_linear[where c=1]) simp_all
 
 text \<open>Hence another formulation of orthogonal transformation\<close>
 
-lemma orthogonal_transformation_isometry:
+lemma%important  orthogonal_transformation_isometry:
   "orthogonal_transformation f \<longleftrightarrow> f(0::'a::real_inner) = (0::'a) \<and> (\<forall>x y. dist(f x) (f y) = dist x y)"
   unfolding orthogonal_transformation
   apply (auto simp: linear_0 isometry_linear)
@@ -1119,7 +1119,7 @@
   by (metis dist_0_norm)
 
 
-lemma image_orthogonal_transformation_ball:
+lemma%unimportant  image_orthogonal_transformation_ball:
   fixes f :: "'a::euclidean_space \<Rightarrow> 'a"
   assumes "orthogonal_transformation f"
   shows "f ` ball x r = ball (f x) r"
@@ -1135,7 +1135,7 @@
     by (auto simp: orthogonal_transformation_isometry)
 qed
 
-lemma image_orthogonal_transformation_cball:
+lemma%unimportant  image_orthogonal_transformation_cball:
   fixes f :: "'a::euclidean_space \<Rightarrow> 'a"
   assumes "orthogonal_transformation f"
   shows "f ` cball x r = cball (f x) r"
@@ -1151,31 +1151,31 @@
     by (auto simp: orthogonal_transformation_isometry)
 qed
 
-subsection\<open> We can find an orthogonal matrix taking any unit vector to any other\<close>
+subsection%important \<open> We can find an orthogonal matrix taking any unit vector to any other\<close>
 
-lemma orthogonal_matrix_transpose [simp]:
+lemma%unimportant  orthogonal_matrix_transpose [simp]:
      "orthogonal_matrix(transpose A) \<longleftrightarrow> orthogonal_matrix A"
   by (auto simp: orthogonal_matrix_def)
 
-lemma orthogonal_matrix_orthonormal_columns:
+lemma%unimportant  orthogonal_matrix_orthonormal_columns:
   fixes A :: "real^'n^'n"
   shows "orthogonal_matrix A \<longleftrightarrow>
           (\<forall>i. norm(column i A) = 1) \<and>
           (\<forall>i j. i \<noteq> j \<longrightarrow> orthogonal (column i A) (column j A))"
   by (auto simp: orthogonal_matrix matrix_mult_transpose_dot_column vec_eq_iff mat_def norm_eq_1 orthogonal_def)
 
-lemma orthogonal_matrix_orthonormal_rows:
+lemma%unimportant  orthogonal_matrix_orthonormal_rows:
   fixes A :: "real^'n^'n"
   shows "orthogonal_matrix A \<longleftrightarrow>
           (\<forall>i. norm(row i A) = 1) \<and>
           (\<forall>i j. i \<noteq> j \<longrightarrow> orthogonal (row i A) (row j A))"
   using orthogonal_matrix_orthonormal_columns [of "transpose A"] by simp
 
-lemma orthogonal_matrix_exists_basis:
+lemma%important  orthogonal_matrix_exists_basis:
   fixes a :: "real^'n"
   assumes "norm a = 1"
   obtains A where "orthogonal_matrix A" "A *v (axis k 1) = a"
-proof -
+proof%unimportant -
   obtain S where "a \<in> S" "pairwise orthogonal S" and noS: "\<And>x. x \<in> S \<Longrightarrow> norm x = 1"
    and "independent S" "card S = CARD('n)" "span S = UNIV"
     using vector_in_orthonormal_basis assms by force
@@ -1198,11 +1198,11 @@
   qed
 qed
 
-lemma orthogonal_transformation_exists_1:
+lemma%unimportant  orthogonal_transformation_exists_1:
   fixes a b :: "real^'n"
   assumes "norm a = 1" "norm b = 1"
   obtains f where "orthogonal_transformation f" "f a = b"
-proof -
+proof%unimportant -
   obtain k::'n where True
     by simp
   obtain A B where AB: "orthogonal_matrix A" "orthogonal_matrix B" and eq: "A *v (axis k 1) = a" "B *v (axis k 1) = b"
@@ -1220,11 +1220,11 @@
   qed
 qed
 
-lemma orthogonal_transformation_exists:
+lemma%important  orthogonal_transformation_exists:
   fixes a b :: "real^'n"
   assumes "norm a = norm b"
   obtains f where "orthogonal_transformation f" "f a = b"
-proof (cases "a = 0 \<or> b = 0")
+proof%unimportant (cases "a = 0 \<or> b = 0")
   case True
   with assms show ?thesis
     using that by force
@@ -1246,14 +1246,14 @@
 qed
 
 
-subsection \<open>Can extend an isometry from unit sphere\<close>
+subsection%important  \<open>Can extend an isometry from unit sphere\<close>
 
-lemma isometry_sphere_extend:
+lemma%important  isometry_sphere_extend:
   fixes f:: "'a::real_inner \<Rightarrow> 'a"
   assumes f1: "\<And>x. norm x = 1 \<Longrightarrow> norm (f x) = 1"
     and fd1: "\<And>x y. \<lbrakk>norm x = 1; norm y = 1\<rbrakk> \<Longrightarrow> dist (f x) (f y) = dist x y"
   shows "\<exists>g. orthogonal_transformation g \<and> (\<forall>x. norm x = 1 \<longrightarrow> g x = f x)"
-proof -
+proof%unimportant -
   {
     fix x y x' y' u v u' v' :: "'a"
     assume H: "x = norm x *\<^sub>R u" "y = norm y *\<^sub>R v"
@@ -1286,31 +1286,31 @@
     by (rule exI[where x= ?g]) (metis thfg thd)
 qed
 
-subsection \<open>Rotation, reflection, rotoinversion\<close>
+subsection%important  \<open>Rotation, reflection, rotoinversion\<close>
 
-definition "rotation_matrix Q \<longleftrightarrow> orthogonal_matrix Q \<and> det Q = 1"
-definition "rotoinversion_matrix Q \<longleftrightarrow> orthogonal_matrix Q \<and> det Q = - 1"
+definition%important  "rotation_matrix Q \<longleftrightarrow> orthogonal_matrix Q \<and> det Q = 1"
+definition%important  "rotoinversion_matrix Q \<longleftrightarrow> orthogonal_matrix Q \<and> det Q = - 1"
 
-lemma orthogonal_rotation_or_rotoinversion:
+lemma%important  orthogonal_rotation_or_rotoinversion:
   fixes Q :: "'a::linordered_idom^'n^'n"
   shows " orthogonal_matrix Q \<longleftrightarrow> rotation_matrix Q \<or> rotoinversion_matrix Q"
-  by (metis rotoinversion_matrix_def rotation_matrix_def det_orthogonal_matrix)
+  by%unimportant (metis rotoinversion_matrix_def rotation_matrix_def det_orthogonal_matrix)
 
 text \<open>Explicit formulas for low dimensions\<close>
 
-lemma prod_neutral_const: "prod f {(1::nat)..1} = f 1"
+lemma%unimportant  prod_neutral_const: "prod f {(1::nat)..1} = f 1"
   by simp
 
-lemma prod_2: "prod f {(1::nat)..2} = f 1 * f 2"
+lemma%unimportant  prod_2: "prod f {(1::nat)..2} = f 1 * f 2"
   by (simp add: eval_nat_numeral atLeastAtMostSuc_conv mult.commute)
 
-lemma prod_3: "prod f {(1::nat)..3} = f 1 * f 2 * f 3"
+lemma%unimportant  prod_3: "prod f {(1::nat)..3} = f 1 * f 2 * f 3"
   by (simp add: eval_nat_numeral atLeastAtMostSuc_conv mult.commute)
 
-lemma det_1: "det (A::'a::comm_ring_1^1^1) = A$1$1"
+lemma%unimportant  det_1: "det (A::'a::comm_ring_1^1^1) = A$1$1"
   by (simp add: det_def sign_id)
 
-lemma det_2: "det (A::'a::comm_ring_1^2^2) = A$1$1 * A$2$2 - A$1$2 * A$2$1"
+lemma%unimportant  det_2: "det (A::'a::comm_ring_1^2^2) = A$1$1 * A$2$2 - A$1$2 * A$2$1"
 proof -
   have f12: "finite {2::2}" "1 \<notin> {2::2}" by auto
   show ?thesis
@@ -1320,7 +1320,7 @@
     by (simp add: sign_swap_id sign_id swap_id_eq)
 qed
 
-lemma det_3:
+lemma%unimportant  det_3:
   "det (A::'a::comm_ring_1^3^3) =
     A$1$1 * A$2$2 * A$3$3 +
     A$1$2 * A$2$3 * A$3$1 +
@@ -1344,7 +1344,7 @@
 
 text\<open> Slightly stronger results giving rotation, but only in two or more dimensions\<close>
 
-lemma rotation_matrix_exists_basis:
+lemma%unimportant  rotation_matrix_exists_basis:
   fixes a :: "real^'n"
   assumes 2: "2 \<le> CARD('n)" and "norm a = 1"
   obtains A where "rotation_matrix A" "A *v (axis k 1) = a"
@@ -1383,7 +1383,7 @@
   qed
 qed
 
-lemma rotation_exists_1:
+lemma%unimportant  rotation_exists_1:
   fixes a :: "real^'n"
   assumes "2 \<le> CARD('n)" "norm a = 1" "norm b = 1"
   obtains f where "orthogonal_transformation f" "det(matrix f) = 1" "f a = b"
@@ -1406,7 +1406,7 @@
   qed
 qed
 
-lemma rotation_exists:
+lemma%unimportant  rotation_exists:
   fixes a :: "real^'n"
   assumes 2: "2 \<le> CARD('n)" and eq: "norm a = norm b"
   obtains f where "orthogonal_transformation f" "det(matrix f) = 1" "f a = b"
@@ -1428,7 +1428,7 @@
   with f show thesis ..
 qed
 
-lemma rotation_rightward_line:
+lemma%unimportant  rotation_rightward_line:
   fixes a :: "real^'n"
   obtains f where "orthogonal_transformation f" "2 \<le> CARD('n) \<Longrightarrow> det(matrix f) = 1"
                   "f(norm a *\<^sub>R axis k 1) = a"

--- a/src/HOL/Analysis/Finite_Cartesian_Product.thy	Mon Aug 27 22:58:36 2018 +0200
+++ b/src/HOL/Analysis/Finite_Cartesian_Product.thy	Tue Aug 28 13:28:39 2018 +0100
@@ -2,7 +2,7 @@
     Author:     Amine Chaieb, University of Cambridge
 *)
 
-section \<open>Definition of finite Cartesian product types\<close>
+section%important \<open>Definition of finite Cartesian product types\<close>
 
 theory Finite_Cartesian_Product
 imports
@@ -13,7 +13,7 @@
   "HOL-Library.FuncSet"
 begin
 
-subsection \<open>Finite Cartesian products, with indexing and lambdas\<close>
+subsection%unimportant \<open>Finite Cartesian products, with indexing and lambdas\<close>
 
 typedef ('a, 'b) vec = "UNIV :: ('b::finite \<Rightarrow> 'a) set"
   morphisms vec_nth vec_lambda ..
@@ -68,7 +68,7 @@
 lemma vec_lambda_eta [simp]: "(\<chi> i. (g$i)) = g"
   by (simp add: vec_eq_iff)
 
-subsection \<open>Cardinality of vectors\<close>
+subsection%important \<open>Cardinality of vectors\<close>
 
 instance vec :: (finite, finite) finite
 proof
@@ -104,10 +104,10 @@
     by (auto elim!: countableE)
 qed
 
-lemma infinite_UNIV_vec:
+lemma%important infinite_UNIV_vec:
   assumes "infinite (UNIV :: 'a set)"
   shows   "infinite (UNIV :: ('a^'b) set)"
-proof (subst bij_betw_finite)
+proof%unimportant (subst bij_betw_finite)
   show "bij_betw vec_nth UNIV (Pi (UNIV :: 'b set) (\<lambda>_. UNIV :: 'a set))"
     by (intro bij_betwI[of _ _ _ vec_lambda]) (auto simp: vec_eq_iff)
   have "infinite (PiE (UNIV :: 'b set) (\<lambda>_. UNIV :: 'a set))" (is "infinite ?A")
@@ -125,9 +125,9 @@
   finally show "infinite (Pi (UNIV :: 'b set) (\<lambda>_. UNIV :: 'a set))" .
 qed
 
-lemma CARD_vec [simp]:
+lemma%important CARD_vec [simp]:
   "CARD('a^'b) = CARD('a) ^ CARD('b)"
-proof (cases "finite (UNIV :: 'a set)")
+proof%unimportant (cases "finite (UNIV :: 'a set)")
   case True
   show ?thesis
   proof (subst bij_betw_same_card)
@@ -143,7 +143,7 @@
   qed
 qed (simp_all add: infinite_UNIV_vec)
 
-lemma countable_vector:
+lemma%unimportant countable_vector:
   fixes B:: "'n::finite \<Rightarrow> 'a set"
   assumes "\<And>i. countable (B i)"
   shows "countable {V. \<forall>i::'n::finite. V $ i \<in> B i}"
@@ -165,7 +165,7 @@
     by (simp add: image_comp o_def vec_nth_inverse)
 qed
 
-subsection \<open>Group operations and class instances\<close>
+subsection%unimportant \<open>Group operations and class instances\<close>
 
 instantiation vec :: (zero, finite) zero
 begin
@@ -230,7 +230,7 @@
   by standard (simp_all add: vec_eq_iff)
 
 
-subsection\<open>Basic componentwise operations on vectors\<close>
+subsection%unimportant\<open>Basic componentwise operations on vectors\<close>
 
 instantiation vec :: (times, finite) times
 begin
@@ -304,33 +304,33 @@
   "scalar_product v w = (\<Sum> i \<in> UNIV. v $ i * w $ i)"
 
 
-subsection \<open>Real vector space\<close>
+subsection%important \<open>Real vector space\<close>
 
-instantiation vec :: (real_vector, finite) real_vector
+instantiation%unimportant vec :: (real_vector, finite) real_vector
 begin
 
-definition "scaleR \<equiv> (\<lambda> r x. (\<chi> i. scaleR r (x$i)))"
+definition%important "scaleR \<equiv> (\<lambda> r x. (\<chi> i. scaleR r (x$i)))"
 
-lemma vector_scaleR_component [simp]: "(scaleR r x)$i = scaleR r (x$i)"
+lemma%important vector_scaleR_component [simp]: "(scaleR r x)$i = scaleR r (x$i)"
   unfolding scaleR_vec_def by simp
 
-instance
+instance%unimportant
   by standard (simp_all add: vec_eq_iff scaleR_left_distrib scaleR_right_distrib)
 
 end
 
 
-subsection \<open>Topological space\<close>
+subsection%important \<open>Topological space\<close>
 
-instantiation vec :: (topological_space, finite) topological_space
+instantiation%unimportant vec :: (topological_space, finite) topological_space
 begin
 
-definition [code del]:
+definition%important [code del]:
   "open (S :: ('a ^ 'b) set) \<longleftrightarrow>
     (\<forall>x\<in>S. \<exists>A. (\<forall>i. open (A i) \<and> x$i \<in> A i) \<and>
       (\<forall>y. (\<forall>i. y$i \<in> A i) \<longrightarrow> y \<in> S))"
 
-instance proof
+instance proof%unimportant
   show "open (UNIV :: ('a ^ 'b) set)"
     unfolding open_vec_def by auto
 next
@@ -358,30 +358,30 @@
 
 end
 
-lemma open_vector_box: "\<forall>i. open (S i) \<Longrightarrow> open {x. \<forall>i. x $ i \<in> S i}"
+lemma%unimportant open_vector_box: "\<forall>i. open (S i) \<Longrightarrow> open {x. \<forall>i. x $ i \<in> S i}"
   unfolding open_vec_def by auto
 
-lemma open_vimage_vec_nth: "open S \<Longrightarrow> open ((\<lambda>x. x $ i) -` S)"
+lemma%unimportant open_vimage_vec_nth: "open S \<Longrightarrow> open ((\<lambda>x. x $ i) -` S)"
   unfolding open_vec_def
   apply clarify
   apply (rule_tac x="\<lambda>k. if k = i then S else UNIV" in exI, simp)
   done
 
-lemma closed_vimage_vec_nth: "closed S \<Longrightarrow> closed ((\<lambda>x. x $ i) -` S)"
+lemma%unimportant closed_vimage_vec_nth: "closed S \<Longrightarrow> closed ((\<lambda>x. x $ i) -` S)"
   unfolding closed_open vimage_Compl [symmetric]
   by (rule open_vimage_vec_nth)
 
-lemma closed_vector_box: "\<forall>i. closed (S i) \<Longrightarrow> closed {x. \<forall>i. x $ i \<in> S i}"
+lemma%unimportant closed_vector_box: "\<forall>i. closed (S i) \<Longrightarrow> closed {x. \<forall>i. x $ i \<in> S i}"
 proof -
   have "{x. \<forall>i. x $ i \<in> S i} = (\<Inter>i. (\<lambda>x. x $ i) -` S i)" by auto
   thus "\<forall>i. closed (S i) \<Longrightarrow> closed {x. \<forall>i. x $ i \<in> S i}"
     by (simp add: closed_INT closed_vimage_vec_nth)
 qed
 
-lemma tendsto_vec_nth [tendsto_intros]:
+lemma%important tendsto_vec_nth [tendsto_intros]:
   assumes "((\<lambda>x. f x) \<longlongrightarrow> a) net"
   shows "((\<lambda>x. f x $ i) \<longlongrightarrow> a $ i) net"
-proof (rule topological_tendstoI)
+proof%unimportant (rule topological_tendstoI)
   fix S assume "open S" "a $ i \<in> S"
   then have "open ((\<lambda>y. y $ i) -` S)" "a \<in> ((\<lambda>y. y $ i) -` S)"
     by (simp_all add: open_vimage_vec_nth)
@@ -391,13 +391,13 @@
     by simp
 qed
 
-lemma isCont_vec_nth [simp]: "isCont f a \<Longrightarrow> isCont (\<lambda>x. f x $ i) a"
+lemma%unimportant isCont_vec_nth [simp]: "isCont f a \<Longrightarrow> isCont (\<lambda>x. f x $ i) a"
   unfolding isCont_def by (rule tendsto_vec_nth)
 
-lemma vec_tendstoI:
+lemma%important vec_tendstoI:
   assumes "\<And>i. ((\<lambda>x. f x $ i) \<longlongrightarrow> a $ i) net"
   shows "((\<lambda>x. f x) \<longlongrightarrow> a) net"
-proof (rule topological_tendstoI)
+proof%unimportant (rule topological_tendstoI)
   fix S assume "open S" and "a \<in> S"
   then obtain A where A: "\<And>i. open (A i)" "\<And>i. a $ i \<in> A i"
     and S: "\<And>y. \<forall>i. y $ i \<in> A i \<Longrightarrow> y \<in> S"
@@ -410,13 +410,13 @@
     by (rule eventually_mono, simp add: S)
 qed
 
-lemma tendsto_vec_lambda [tendsto_intros]:
+lemma%unimportant tendsto_vec_lambda [tendsto_intros]:
   assumes "\<And>i. ((\<lambda>x. f x i) \<longlongrightarrow> a i) net"
   shows "((\<lambda>x. \<chi> i. f x i) \<longlongrightarrow> (\<chi> i. a i)) net"
   using assms by (simp add: vec_tendstoI)
 
-lemma open_image_vec_nth: assumes "open S" shows "open ((\<lambda>x. x $ i) ` S)"
-proof (rule openI)
+lemma%important open_image_vec_nth: assumes "open S" shows "open ((\<lambda>x. x $ i) ` S)"
+proof%unimportant (rule openI)
   fix a assume "a \<in> (\<lambda>x. x $ i) ` S"
   then obtain z where "a = z $ i" and "z \<in> S" ..
   then obtain A where A: "\<forall>i. open (A i) \<and> z $ i \<in> A i"
@@ -442,38 +442,38 @@
 qed
 
 
-subsection \<open>Metric space\<close>
+subsection%important \<open>Metric space\<close>
 (* TODO: Product of uniform spaces and compatibility with metric_spaces! *)
 
-instantiation vec :: (metric_space, finite) dist
+instantiation%unimportant vec :: (metric_space, finite) dist
 begin
 
-definition
+definition%important
   "dist x y = L2_set (\<lambda>i. dist (x$i) (y$i)) UNIV"
 
 instance ..
 end
 
-instantiation vec :: (metric_space, finite) uniformity_dist
+instantiation%unimportant vec :: (metric_space, finite) uniformity_dist
 begin
 
-definition [code del]:
+definition%important [code del]:
   "(uniformity :: (('a^'b::_) \<times> ('a^'b::_)) filter) =
     (INF e:{0 <..}. principal {(x, y). dist x y < e})"
 
-instance
+instance%unimportant
   by standard (rule uniformity_vec_def)
 end
 
 declare uniformity_Abort[where 'a="'a :: metric_space ^ 'b :: finite", code]
 
-instantiation vec :: (metric_space, finite) metric_space
+instantiation%unimportant vec :: (metric_space, finite) metric_space
 begin
 
-lemma dist_vec_nth_le: "dist (x $ i) (y $ i) \<le> dist x y"
+lemma%important dist_vec_nth_le: "dist (x $ i) (y $ i) \<le> dist x y"
   unfolding dist_vec_def by (rule member_le_L2_set) simp_all
 
-instance proof
+instance proof%unimportant
   fix x y :: "'a ^ 'b"
   show "dist x y = 0 \<longleftrightarrow> x = y"
     unfolding dist_vec_def
@@ -532,15 +532,15 @@
 
 end
 
-lemma Cauchy_vec_nth:
+lemma%important Cauchy_vec_nth:
   "Cauchy (\<lambda>n. X n) \<Longrightarrow> Cauchy (\<lambda>n. X n $ i)"
   unfolding Cauchy_def by (fast intro: le_less_trans [OF dist_vec_nth_le])
 
-lemma vec_CauchyI:
+lemma%important vec_CauchyI:
   fixes X :: "nat \<Rightarrow> 'a::metric_space ^ 'n"
   assumes X: "\<And>i. Cauchy (\<lambda>n. X n $ i)"
   shows "Cauchy (\<lambda>n. X n)"
-proof (rule metric_CauchyI)
+proof%unimportant (rule metric_CauchyI)
   fix r :: real assume "0 < r"
   hence "0 < r / of_nat CARD('n)" (is "0 < ?s") by simp
   define N where "N i = (LEAST N. \<forall>m\<ge>N. \<forall>n\<ge>N. dist (X m $ i) (X n $ i) < ?s)" for i
@@ -582,16 +582,16 @@
 qed
 
 
-subsection \<open>Normed vector space\<close>
+subsection%important \<open>Normed vector space\<close>
 
-instantiation vec :: (real_normed_vector, finite) real_normed_vector
+instantiation%unimportant vec :: (real_normed_vector, finite) real_normed_vector
 begin
 
-definition "norm x = L2_set (\<lambda>i. norm (x$i)) UNIV"
+definition%important "norm x = L2_set (\<lambda>i. norm (x$i)) UNIV"
 
-definition "sgn (x::'a^'b) = scaleR (inverse (norm x)) x"
+definition%important "sgn (x::'a^'b) = scaleR (inverse (norm x)) x"
 
-instance proof
+instance proof%unimportant
   fix a :: real and x y :: "'a ^ 'b"
   show "norm x = 0 \<longleftrightarrow> x = 0"
     unfolding norm_vec_def
@@ -613,32 +613,32 @@
 
 end
 
-lemma norm_nth_le: "norm (x $ i) \<le> norm x"
+lemma%unimportant norm_nth_le: "norm (x $ i) \<le> norm x"
 unfolding norm_vec_def
 by (rule member_le_L2_set) simp_all
 
-lemma norm_le_componentwise_cart:
+lemma%important norm_le_componentwise_cart:
   fixes x :: "'a::real_normed_vector^'n"
   assumes "\<And>i. norm(x$i) \<le> norm(y$i)"
   shows "norm x \<le> norm y"
-  unfolding norm_vec_def
-  by (rule L2_set_mono) (auto simp: assms)
+  unfolding%unimportant norm_vec_def
+  by%unimportant (rule L2_set_mono) (auto simp: assms)
 
-lemma component_le_norm_cart: "\<bar>x$i\<bar> \<le> norm x"
+lemma%unimportant component_le_norm_cart: "\<bar>x$i\<bar> \<le> norm x"
   apply (simp add: norm_vec_def)
   apply (rule member_le_L2_set, simp_all)
   done
 
-lemma norm_bound_component_le_cart: "norm x \<le> e ==> \<bar>x$i\<bar> \<le> e"
+lemma%unimportant norm_bound_component_le_cart: "norm x \<le> e ==> \<bar>x$i\<bar> \<le> e"
   by (metis component_le_norm_cart order_trans)
 
-lemma norm_bound_component_lt_cart: "norm x < e ==> \<bar>x$i\<bar> < e"
+lemma%unimportant norm_bound_component_lt_cart: "norm x < e ==> \<bar>x$i\<bar> < e"
   by (metis component_le_norm_cart le_less_trans)
 
-lemma norm_le_l1_cart: "norm x \<le> sum(\<lambda>i. \<bar>x$i\<bar>) UNIV"
+lemma%unimportant norm_le_l1_cart: "norm x \<le> sum(\<lambda>i. \<bar>x$i\<bar>) UNIV"
   by (simp add: norm_vec_def L2_set_le_sum)
 
-lemma bounded_linear_vec_nth: "bounded_linear (\<lambda>x. x $ i)"
+lemma%unimportant bounded_linear_vec_nth: "bounded_linear (\<lambda>x. x $ i)"
 apply standard
 apply (rule vector_add_component)
 apply (rule vector_scaleR_component)
@@ -648,14 +648,14 @@
 instance vec :: (banach, finite) banach ..
 
 
-subsection \<open>Inner product space\<close>
+subsection%important \<open>Inner product space\<close>
 
-instantiation vec :: (real_inner, finite) real_inner
+instantiation%unimportant vec :: (real_inner, finite) real_inner
 begin
 
-definition "inner x y = sum (\<lambda>i. inner (x$i) (y$i)) UNIV"
+definition%important "inner x y = sum (\<lambda>i. inner (x$i) (y$i)) UNIV"
 
-instance proof
+instance proof%unimportant
   fix r :: real and x y z :: "'a ^ 'b"
   show "inner x y = inner y x"
     unfolding inner_vec_def
@@ -680,19 +680,19 @@
 end
 
 
-subsection \<open>Euclidean space\<close>
+subsection%important \<open>Euclidean space\<close>
 
 text \<open>Vectors pointing along a single axis.\<close>
 
-definition "axis k x = (\<chi> i. if i = k then x else 0)"
+definition%important "axis k x = (\<chi> i. if i = k then x else 0)"
 
-lemma axis_nth [simp]: "axis i x $ i = x"
+lemma%unimportant axis_nth [simp]: "axis i x $ i = x"
   unfolding axis_def by simp
 
-lemma axis_eq_axis: "axis i x = axis j y \<longleftrightarrow> x = y \<and> i = j \<or> x = 0 \<and> y = 0"
+lemma%unimportant axis_eq_axis: "axis i x = axis j y \<longleftrightarrow> x = y \<and> i = j \<or> x = 0 \<and> y = 0"
   unfolding axis_def vec_eq_iff by auto
 
-lemma inner_axis_axis:
+lemma%unimportant inner_axis_axis:
   "inner (axis i x) (axis j y) = (if i = j then inner x y else 0)"
   unfolding inner_vec_def
   apply (cases "i = j")
@@ -702,18 +702,18 @@
   apply (rule sum.neutral, simp add: axis_def)
   done
 
-lemma inner_axis: "inner x (axis i y) = inner (x $ i) y"
+lemma%unimportant inner_axis: "inner x (axis i y) = inner (x $ i) y"
   by (simp add: inner_vec_def axis_def sum.remove [where x=i])
 
-lemma inner_axis': "inner(axis i y) x = inner y (x $ i)"
+lemma%unimportant inner_axis': "inner(axis i y) x = inner y (x $ i)"
   by (simp add: inner_axis inner_commute)
 
-instantiation vec :: (euclidean_space, finite) euclidean_space
+instantiation%unimportant vec :: (euclidean_space, finite) euclidean_space
 begin
 
-definition "Basis = (\<Union>i. \<Union>u\<in>Basis. {axis i u})"
+definition%important "Basis = (\<Union>i. \<Union>u\<in>Basis. {axis i u})"
 
-instance proof
+instance proof%unimportant
   show "(Basis :: ('a ^ 'b) set) \<noteq> {}"
     unfolding Basis_vec_def by simp
 next
@@ -732,8 +732,8 @@
     by (simp add: inner_axis euclidean_all_zero_iff vec_eq_iff)
 qed
 
-lemma DIM_cart [simp]: "DIM('a^'b) = CARD('b) * DIM('a)"
-proof -
+lemma%important DIM_cart [simp]: "DIM('a^'b) = CARD('b) * DIM('a)"
+proof%unimportant -
   have "card (\<Union>i::'b. \<Union>u::'a\<in>Basis. {axis i u}) = (\<Sum>i::'b\<in>UNIV. card (\<Union>u::'a\<in>Basis. {axis i u}))"
     by (rule card_UN_disjoint) (auto simp: axis_eq_axis) 
   also have "... = CARD('b) * DIM('a)"
@@ -744,30 +744,30 @@
 
 end
 
-lemma norm_axis_1 [simp]: "norm (axis m (1::real)) = 1"
+lemma%unimportant norm_axis_1 [simp]: "norm (axis m (1::real)) = 1"
   by (simp add: inner_axis' norm_eq_1)
 
-lemma sum_norm_allsubsets_bound_cart:
+lemma%important sum_norm_allsubsets_bound_cart:
   fixes f:: "'a \<Rightarrow> real ^'n"
   assumes fP: "finite P" and fPs: "\<And>Q. Q \<subseteq> P \<Longrightarrow> norm (sum f Q) \<le> e"
   shows "sum (\<lambda>x. norm (f x)) P \<le> 2 * real CARD('n) *  e"
-  using sum_norm_allsubsets_bound[OF assms]
-  by simp
+  using%unimportant sum_norm_allsubsets_bound[OF assms]
+  by%unimportant simp
 
-lemma cart_eq_inner_axis: "a $ i = inner a (axis i 1)"
+lemma%unimportant cart_eq_inner_axis: "a $ i = inner a (axis i 1)"
   by (simp add: inner_axis)
 
-lemma axis_eq_0_iff [simp]:
+lemma%unimportant axis_eq_0_iff [simp]:
   shows "axis m x = 0 \<longleftrightarrow> x = 0"
   by (simp add: axis_def vec_eq_iff)
 
-lemma axis_in_Basis_iff [simp]: "axis i a \<in> Basis \<longleftrightarrow> a \<in> Basis"
+lemma%unimportant axis_in_Basis_iff [simp]: "axis i a \<in> Basis \<longleftrightarrow> a \<in> Basis"
   by (auto simp: Basis_vec_def axis_eq_axis)
 
 text\<open>Mapping each basis element to the corresponding finite index\<close>
 definition axis_index :: "('a::comm_ring_1)^'n \<Rightarrow> 'n" where "axis_index v \<equiv> SOME i. v = axis i 1"
 
-lemma axis_inverse:
+lemma%unimportant axis_inverse:
   fixes v :: "real^'n"
   assumes "v \<in> Basis"
   shows "\<exists>i. v = axis i 1"
@@ -778,20 +778,20 @@
     by (force simp add: vec_eq_iff)
 qed
 
-lemma axis_index:
+lemma%unimportant axis_index:
   fixes v :: "real^'n"
   assumes "v \<in> Basis"
   shows "v = axis (axis_index v) 1"
   by (metis (mono_tags) assms axis_inverse axis_index_def someI_ex)
 
-lemma axis_index_axis [simp]:
+lemma%unimportant axis_index_axis [simp]:
   fixes UU :: "real^'n"
   shows "(axis_index (axis u 1 :: real^'n)) = (u::'n)"
   by (simp add: axis_eq_axis axis_index_def)
 
-subsection \<open>A naive proof procedure to lift really trivial arithmetic stuff from the basis of the vector space\<close>
+subsection%unimportant \<open>A naive proof procedure to lift really trivial arithmetic stuff from the basis of the vector space\<close>
 
-lemma sum_cong_aux:
+lemma%unimportant sum_cong_aux:
   "(\<And>x. x \<in> A \<Longrightarrow> f x = g x) \<Longrightarrow> sum f A = sum g A"
   by (auto intro: sum.cong)
 
@@ -823,22 +823,22 @@
 end
 \<close> "lift trivial vector statements to real arith statements"
 
-lemma vec_0[simp]: "vec 0 = 0" by vector
-lemma vec_1[simp]: "vec 1 = 1" by vector
+lemma%unimportant vec_0[simp]: "vec 0 = 0" by vector
+lemma%unimportant vec_1[simp]: "vec 1 = 1" by vector
 
-lemma vec_inj[simp]: "vec x = vec y \<longleftrightarrow> x = y" by vector
+lemma%unimportant vec_inj[simp]: "vec x = vec y \<longleftrightarrow> x = y" by vector
 
-lemma vec_in_image_vec: "vec x \<in> (vec ` S) \<longleftrightarrow> x \<in> S" by auto
+lemma%unimportant vec_in_image_vec: "vec x \<in> (vec ` S) \<longleftrightarrow> x \<in> S" by auto
 
-lemma vec_add: "vec(x + y) = vec x + vec y"  by vector
-lemma vec_sub: "vec(x - y) = vec x - vec y" by vector
-lemma vec_cmul: "vec(c * x) = c *s vec x " by vector
-lemma vec_neg: "vec(- x) = - vec x " by vector
+lemma%unimportant vec_add: "vec(x + y) = vec x + vec y"  by vector
+lemma%unimportant vec_sub: "vec(x - y) = vec x - vec y" by vector
+lemma%unimportant vec_cmul: "vec(c * x) = c *s vec x " by vector
+lemma%unimportant vec_neg: "vec(- x) = - vec x " by vector
 
-lemma vec_scaleR: "vec(c * x) = c *\<^sub>R vec x"
+lemma%unimportant vec_scaleR: "vec(c * x) = c *\<^sub>R vec x"
   by vector
 
-lemma vec_sum:
+lemma%unimportant vec_sum:
   assumes "finite S"
   shows "vec(sum f S) = sum (vec \<circ> f) S"
   using assms
@@ -852,24 +852,24 @@
 
 text\<open>Obvious "component-pushing".\<close>
 
-lemma vec_component [simp]: "vec x $ i = x"
+lemma%unimportant vec_component [simp]: "vec x $ i = x"
   by vector
 
-lemma vector_mult_component [simp]: "(x * y)$i = x$i * y$i"
+lemma%unimportant vector_mult_component [simp]: "(x * y)$i = x$i * y$i"
   by vector
 
-lemma vector_smult_component [simp]: "(c *s y)$i = c * (y$i)"
+lemma%unimportant vector_smult_component [simp]: "(c *s y)$i = c * (y$i)"
   by vector
 
-lemma cond_component: "(if b then x else y)$i = (if b then x$i else y$i)" by vector
+lemma%unimportant cond_component: "(if b then x else y)$i = (if b then x$i else y$i)" by vector
 
-lemmas vector_component =
+lemmas%unimportant vector_component =
   vec_component vector_add_component vector_mult_component
   vector_smult_component vector_minus_component vector_uminus_component
   vector_scaleR_component cond_component
 
 
-subsection \<open>Some frequently useful arithmetic lemmas over vectors\<close>
+subsection%unimportant \<open>Some frequently useful arithmetic lemmas over vectors\<close>
 
 instance vec :: (semigroup_mult, finite) semigroup_mult
   by standard (vector mult.assoc)
@@ -1004,47 +1004,47 @@
   apply (auto )
   by unfold_locales (vector algebra_simps)+
 
-subsection \<open>Matrix operations\<close>
+subsection%important \<open>Matrix operations\<close>
 
 text\<open>Matrix notation. NB: an MxN matrix is of type @{typ "'a^'n^'m"}, not @{typ "'a^'m^'n"}\<close>
 
-definition map_matrix::"('a \<Rightarrow> 'b) \<Rightarrow> (('a, 'i::finite)vec, 'j::finite) vec \<Rightarrow> (('b, 'i)vec, 'j) vec" where
+definition%important map_matrix::"('a \<Rightarrow> 'b) \<Rightarrow> (('a, 'i::finite)vec, 'j::finite) vec \<Rightarrow> (('b, 'i)vec, 'j) vec" where
   "map_matrix f x = (\<chi> i j. f (x $ i $ j))"
 
-lemma nth_map_matrix[simp]: "map_matrix f x $ i $ j = f (x $ i $ j)"
+lemma%unimportant nth_map_matrix[simp]: "map_matrix f x $ i $ j = f (x $ i $ j)"
   by (simp add: map_matrix_def)
 
-definition matrix_matrix_mult :: "('a::semiring_1) ^'n^'m \<Rightarrow> 'a ^'p^'n \<Rightarrow> 'a ^ 'p ^'m"
+definition%important matrix_matrix_mult :: "('a::semiring_1) ^'n^'m \<Rightarrow> 'a ^'p^'n \<Rightarrow> 'a ^ 'p ^'m"
     (infixl "**" 70)
   where "m ** m' == (\<chi> i j. sum (\<lambda>k. ((m$i)$k) * ((m'$k)$j)) (UNIV :: 'n set)) ::'a ^ 'p ^'m"
 
-definition matrix_vector_mult :: "('a::semiring_1) ^'n^'m \<Rightarrow> 'a ^'n \<Rightarrow> 'a ^ 'm"
+definition%important matrix_vector_mult :: "('a::semiring_1) ^'n^'m \<Rightarrow> 'a ^'n \<Rightarrow> 'a ^ 'm"
     (infixl "*v" 70)
   where "m *v x \<equiv> (\<chi> i. sum (\<lambda>j. ((m$i)$j) * (x$j)) (UNIV ::'n set)) :: 'a^'m"
 
-definition vector_matrix_mult :: "'a ^ 'm \<Rightarrow> ('a::semiring_1) ^'n^'m \<Rightarrow> 'a ^'n "
+definition%important vector_matrix_mult :: "'a ^ 'm \<Rightarrow> ('a::semiring_1) ^'n^'m \<Rightarrow> 'a ^'n "
     (infixl "v*" 70)
   where "v v* m == (\<chi> j. sum (\<lambda>i. ((m$i)$j) * (v$i)) (UNIV :: 'm set)) :: 'a^'n"
 
-definition "(mat::'a::zero => 'a ^'n^'n) k = (\<chi> i j. if i = j then k else 0)"
-definition transpose where
+definition%unimportant "(mat::'a::zero => 'a ^'n^'n) k = (\<chi> i j. if i = j then k else 0)"
+definition%unimportant transpose where
   "(transpose::'a^'n^'m \<Rightarrow> 'a^'m^'n) A = (\<chi> i j. ((A$j)$i))"
-definition "(row::'m => 'a ^'n^'m \<Rightarrow> 'a ^'n) i A = (\<chi> j. ((A$i)$j))"
-definition "(column::'n =>'a^'n^'m =>'a^'m) j A = (\<chi> i. ((A$i)$j))"
-definition "rows(A::'a^'n^'m) = { row i A | i. i \<in> (UNIV :: 'm set)}"
-definition "columns(A::'a^'n^'m) = { column i A | i. i \<in> (UNIV :: 'n set)}"
+definition%unimportant "(row::'m => 'a ^'n^'m \<Rightarrow> 'a ^'n) i A = (\<chi> j. ((A$i)$j))"
+definition%unimportant "(column::'n =>'a^'n^'m =>'a^'m) j A = (\<chi> i. ((A$i)$j))"
+definition%unimportant "rows(A::'a^'n^'m) = { row i A | i. i \<in> (UNIV :: 'm set)}"
+definition%unimportant "columns(A::'a^'n^'m) = { column i A | i. i \<in> (UNIV :: 'n set)}"
 
-lemma times0_left [simp]: "(0::'a::semiring_1^'n^'m) ** (A::'a ^'p^'n) = 0" 
+lemma%unimportant times0_left [simp]: "(0::'a::semiring_1^'n^'m) ** (A::'a ^'p^'n) = 0" 
   by (simp add: matrix_matrix_mult_def zero_vec_def)
 
-lemma times0_right [simp]: "(A::'a::semiring_1^'n^'m) ** (0::'a ^'p^'n) = 0" 
+lemma%unimportant times0_right [simp]: "(A::'a::semiring_1^'n^'m) ** (0::'a ^'p^'n) = 0" 
   by (simp add: matrix_matrix_mult_def zero_vec_def)
 
-lemma mat_0[simp]: "mat 0 = 0" by (vector mat_def)
-lemma matrix_add_ldistrib: "(A ** (B + C)) = (A ** B) + (A ** C)"
+lemma%unimportant mat_0[simp]: "mat 0 = 0" by (vector mat_def)
+lemma%unimportant matrix_add_ldistrib: "(A ** (B + C)) = (A ** B) + (A ** C)"
   by (vector matrix_matrix_mult_def sum.distrib[symmetric] field_simps)
 
-lemma matrix_mul_lid [simp]:
+lemma%unimportant matrix_mul_lid [simp]:
   fixes A :: "'a::semiring_1 ^ 'm ^ 'n"
   shows "mat 1 ** A = A"
   apply (simp add: matrix_matrix_mult_def mat_def)
@@ -1053,7 +1053,7 @@
     mult_1_left mult_zero_left if_True UNIV_I)
   done
 
-lemma matrix_mul_rid [simp]:
+lemma%unimportant matrix_mul_rid [simp]:
   fixes A :: "'a::semiring_1 ^ 'm ^ 'n"
   shows "A ** mat 1 = A"
   apply (simp add: matrix_matrix_mult_def mat_def)
@@ -1062,39 +1062,39 @@
     mult_1_right mult_zero_right if_True UNIV_I cong: if_cong)
   done
 
-lemma matrix_mul_assoc: "A ** (B ** C) = (A ** B) ** C"
+lemma%unimportant matrix_mul_assoc: "A ** (B ** C) = (A ** B) ** C"
   apply (vector matrix_matrix_mult_def sum_distrib_left sum_distrib_right mult.assoc)
   apply (subst sum.swap)
   apply simp
   done
 
-lemma matrix_vector_mul_assoc: "A *v (B *v x) = (A ** B) *v x"
+lemma%unimportant matrix_vector_mul_assoc: "A *v (B *v x) = (A ** B) *v x"
   apply (vector matrix_matrix_mult_def matrix_vector_mult_def
     sum_distrib_left sum_distrib_right mult.assoc)
   apply (subst sum.swap)
   apply simp
   done
 
-lemma scalar_matrix_assoc:
+lemma%unimportant scalar_matrix_assoc:
   fixes A :: "('a::real_algebra_1)^'m^'n"
   shows "k *\<^sub>R (A ** B) = (k *\<^sub>R A) ** B"
   by (simp add: matrix_matrix_mult_def sum_distrib_left mult_ac vec_eq_iff scaleR_sum_right)
 
-lemma matrix_scalar_ac:
+lemma%unimportant matrix_scalar_ac:
   fixes A :: "('a::real_algebra_1)^'m^'n"
   shows "A ** (k *\<^sub>R B) = k *\<^sub>R A ** B"
   by (simp add: matrix_matrix_mult_def sum_distrib_left mult_ac vec_eq_iff)
 
-lemma matrix_vector_mul_lid [simp]: "mat 1 *v x = (x::'a::semiring_1 ^ 'n)"
+lemma%unimportant matrix_vector_mul_lid [simp]: "mat 1 *v x = (x::'a::semiring_1 ^ 'n)"
   apply (vector matrix_vector_mult_def mat_def)
   apply (simp add: if_distrib if_distribR sum.delta' cong del: if_weak_cong)
   done
 
-lemma matrix_transpose_mul:
+lemma%unimportant matrix_transpose_mul:
     "transpose(A ** B) = transpose B ** transpose (A::'a::comm_semiring_1^_^_)"
   by (simp add: matrix_matrix_mult_def transpose_def vec_eq_iff mult.commute)
 
-lemma matrix_eq:
+lemma%unimportant matrix_eq:
   fixes A B :: "'a::semiring_1 ^ 'n ^ 'm"
   shows "A = B \<longleftrightarrow>  (\<forall>x. A *v x = B *v x)" (is "?lhs \<longleftrightarrow> ?rhs")
   apply auto
@@ -1107,125 +1107,125 @@
     sum.delta[OF finite] cong del: if_weak_cong)
   done
 
-lemma matrix_vector_mul_component: "(A *v x)$k = inner (A$k) x"
+lemma%unimportant matrix_vector_mul_component: "(A *v x)$k = inner (A$k) x"
   by (simp add: matrix_vector_mult_def inner_vec_def)
 
-lemma dot_lmul_matrix: "inner ((x::real ^_) v* A) y = inner x (A *v y)"
+lemma%unimportant dot_lmul_matrix: "inner ((x::real ^_) v* A) y = inner x (A *v y)"
   apply (simp add: inner_vec_def matrix_vector_mult_def vector_matrix_mult_def sum_distrib_right sum_distrib_left ac_simps)
   apply (subst sum.swap)
   apply simp
   done
 
-lemma transpose_mat [simp]: "transpose (mat n) = mat n"
+lemma%unimportant transpose_mat [simp]: "transpose (mat n) = mat n"
   by (vector transpose_def mat_def)
 
-lemma transpose_transpose [simp]: "transpose(transpose A) = A"
+lemma%unimportant transpose_transpose [simp]: "transpose(transpose A) = A"
   by (vector transpose_def)
 
-lemma row_transpose [simp]: "row i (transpose A) = column i A"
+lemma%unimportant row_transpose [simp]: "row i (transpose A) = column i A"
   by (simp add: row_def column_def transpose_def vec_eq_iff)
 
-lemma column_transpose [simp]: "column i (transpose A) = row i A"
+lemma%unimportant column_transpose [simp]: "column i (transpose A) = row i A"
   by (simp add: row_def column_def transpose_def vec_eq_iff)
 
-lemma rows_transpose [simp]: "rows(transpose A) = columns A"
+lemma%unimportant rows_transpose [simp]: "rows(transpose A) = columns A"
   by (auto simp add: rows_def columns_def intro: set_eqI)
 
-lemma columns_transpose [simp]: "columns(transpose A) = rows A"
+lemma%unimportant columns_transpose [simp]: "columns(transpose A) = rows A"
   by (metis transpose_transpose rows_transpose)
 
-lemma transpose_scalar: "transpose (k *\<^sub>R A) = k *\<^sub>R transpose A"
+lemma%unimportant transpose_scalar: "transpose (k *\<^sub>R A) = k *\<^sub>R transpose A"
   unfolding transpose_def
   by (simp add: vec_eq_iff)
 
-lemma transpose_iff [iff]: "transpose A = transpose B \<longleftrightarrow> A = B"
+lemma%unimportant transpose_iff [iff]: "transpose A = transpose B \<longleftrightarrow> A = B"
   by (metis transpose_transpose)
 
-lemma matrix_mult_sum:
+lemma%unimportant matrix_mult_sum:
   "(A::'a::comm_semiring_1^'n^'m) *v x = sum (\<lambda>i. (x$i) *s column i A) (UNIV:: 'n set)"
   by (simp add: matrix_vector_mult_def vec_eq_iff column_def mult.commute)
 
-lemma vector_componentwise:
+lemma%unimportant vector_componentwise:
   "(x::'a::ring_1^'n) = (\<chi> j. \<Sum>i\<in>UNIV. (x$i) * (axis i 1 :: 'a^'n) $ j)"
   by (simp add: axis_def if_distrib sum.If_cases vec_eq_iff)
 
-lemma basis_expansion: "sum (\<lambda>i. (x$i) *s axis i 1) UNIV = (x::('a::ring_1) ^'n)"
+lemma%unimportant basis_expansion: "sum (\<lambda>i. (x$i) *s axis i 1) UNIV = (x::('a::ring_1) ^'n)"
   by (auto simp add: axis_def vec_eq_iff if_distrib sum.If_cases cong del: if_weak_cong)
 
 
 text\<open>Correspondence between matrices and linear operators.\<close>
 
-definition matrix :: "('a::{plus,times, one, zero}^'m \<Rightarrow> 'a ^ 'n) \<Rightarrow> 'a^'m^'n"
+definition%important matrix :: "('a::{plus,times, one, zero}^'m \<Rightarrow> 'a ^ 'n) \<Rightarrow> 'a^'m^'n"
   where "matrix f = (\<chi> i j. (f(axis j 1))$i)"
 
-lemma matrix_id_mat_1: "matrix id = mat 1"
+lemma%unimportant matrix_id_mat_1: "matrix id = mat 1"
   by (simp add: mat_def matrix_def axis_def)
 
-lemma matrix_scaleR: "(matrix (( *\<^sub>R) r)) = mat r"
+lemma%unimportant matrix_scaleR: "(matrix (( *\<^sub>R) r)) = mat r"
   by (simp add: mat_def matrix_def axis_def if_distrib cong: if_cong)
 
-lemma matrix_vector_mul_linear[intro, simp]: "linear (\<lambda>x. A *v (x::'a::real_algebra_1 ^ _))"
+lemma%unimportant matrix_vector_mul_linear[intro, simp]: "linear (\<lambda>x. A *v (x::'a::real_algebra_1 ^ _))"
   by (simp add: linear_iff matrix_vector_mult_def vec_eq_iff field_simps sum_distrib_left
       sum.distrib scaleR_right.sum)
 
-lemma vector_matrix_left_distrib [algebra_simps]:
+lemma%unimportant vector_matrix_left_distrib [algebra_simps]:
   shows "(x + y) v* A = x v* A + y v* A"
   unfolding vector_matrix_mult_def
   by (simp add: algebra_simps sum.distrib vec_eq_iff)
 
-lemma matrix_vector_right_distrib [algebra_simps]:
+lemma%unimportant matrix_vector_right_distrib [algebra_simps]:
   "A *v (x + y) = A *v x + A *v y"
   by (vector matrix_vector_mult_def sum.distrib distrib_left)
 
-lemma matrix_vector_mult_diff_distrib [algebra_simps]:
+lemma%unimportant matrix_vector_mult_diff_distrib [algebra_simps]:
   fixes A :: "'a::ring_1^'n^'m"
   shows "A *v (x - y) = A *v x - A *v y"
   by (vector matrix_vector_mult_def sum_subtractf right_diff_distrib)
 
-lemma matrix_vector_mult_scaleR[algebra_simps]:
+lemma%unimportant matrix_vector_mult_scaleR[algebra_simps]:
   fixes A :: "real^'n^'m"
   shows "A *v (c *\<^sub>R x) = c *\<^sub>R (A *v x)"
   using linear_iff matrix_vector_mul_linear by blast
 
-lemma matrix_vector_mult_0_right [simp]: "A *v 0 = 0"
+lemma%unimportant matrix_vector_mult_0_right [simp]: "A *v 0 = 0"
   by (simp add: matrix_vector_mult_def vec_eq_iff)
 
-lemma matrix_vector_mult_0 [simp]: "0 *v w = 0"
+lemma%unimportant matrix_vector_mult_0 [simp]: "0 *v w = 0"
   by (simp add: matrix_vector_mult_def vec_eq_iff)
 
-lemma matrix_vector_mult_add_rdistrib [algebra_simps]:
+lemma%unimportant matrix_vector_mult_add_rdistrib [algebra_simps]:
   "(A + B) *v x = (A *v x) + (B *v x)"
   by (vector matrix_vector_mult_def sum.distrib distrib_right)
 
-lemma matrix_vector_mult_diff_rdistrib [algebra_simps]:
+lemma%unimportant matrix_vector_mult_diff_rdistrib [algebra_simps]:
   fixes A :: "'a :: ring_1^'n^'m"
   shows "(A - B) *v x = (A *v x) - (B *v x)"
   by (vector matrix_vector_mult_def sum_subtractf left_diff_distrib)
 
-lemma matrix_vector_column:
+lemma%unimportant matrix_vector_column:
   "(A::'a::comm_semiring_1^'n^_) *v x = sum (\<lambda>i. (x$i) *s ((transpose A)$i)) (UNIV:: 'n set)"
   by (simp add: matrix_vector_mult_def transpose_def vec_eq_iff mult.commute)
 
-subsection\<open>Inverse matrices  (not necessarily square)\<close>
+subsection%important\<open>Inverse matrices  (not necessarily square)\<close>
 
-definition
+definition%important
   "invertible(A::'a::semiring_1^'n^'m) \<longleftrightarrow> (\<exists>A'::'a^'m^'n. A ** A' = mat 1 \<and> A' ** A = mat 1)"
 
-definition
+definition%important
   "matrix_inv(A:: 'a::semiring_1^'n^'m) =
     (SOME A'::'a^'m^'n. A ** A' = mat 1 \<and> A' ** A = mat 1)"
 
-lemma inj_matrix_vector_mult:
+lemma%unimportant inj_matrix_vector_mult:
   fixes A::"'a::field^'n^'m"
   assumes "invertible A"
   shows "inj (( *v) A)"
   by (metis assms inj_on_inverseI invertible_def matrix_vector_mul_assoc matrix_vector_mul_lid)
 
-lemma scalar_invertible:
+lemma%important scalar_invertible:
   fixes A :: "('a::real_algebra_1)^'m^'n"
   assumes "k \<noteq> 0" and "invertible A"
   shows "invertible (k *\<^sub>R A)"
-proof -
+proof%unimportant -
   obtain A' where "A ** A' = mat 1" and "A' ** A = mat 1"
     using assms unfolding invertible_def by auto
   with `k \<noteq> 0`
@@ -1235,50 +1235,50 @@
     unfolding invertible_def by auto
 qed
 
-lemma scalar_invertible_iff:
+lemma%unimportant scalar_invertible_iff:
   fixes A :: "('a::real_algebra_1)^'m^'n"
   assumes "k \<noteq> 0" and "invertible A"
   shows "invertible (k *\<^sub>R A) \<longleftrightarrow> k \<noteq> 0 \<and> invertible A"
   by (simp add: assms scalar_invertible)
 
-lemma vector_transpose_matrix [simp]: "x v* transpose A = A *v x"
+lemma%unimportant vector_transpose_matrix [simp]: "x v* transpose A = A *v x"
   unfolding transpose_def vector_matrix_mult_def matrix_vector_mult_def
   by simp
 
-lemma transpose_matrix_vector [simp]: "transpose A *v x = x v* A"
+lemma%unimportant transpose_matrix_vector [simp]: "transpose A *v x = x v* A"
   unfolding transpose_def vector_matrix_mult_def matrix_vector_mult_def
   by simp
 
-lemma vector_scalar_commute:
+lemma%unimportant vector_scalar_commute:
   fixes A :: "'a::{field}^'m^'n"
   shows "A *v (c *s x) = c *s (A *v x)"
   by (simp add: vector_scalar_mult_def matrix_vector_mult_def mult_ac sum_distrib_left)
 
-lemma scalar_vector_matrix_assoc:
+lemma%unimportant scalar_vector_matrix_assoc:
   fixes k :: "'a::{field}" and x :: "'a::{field}^'n" and A :: "'a^'m^'n"
   shows "(k *s x) v* A = k *s (x v* A)"
   by (metis transpose_matrix_vector vector_scalar_commute)
  
-lemma vector_matrix_mult_0 [simp]: "0 v* A = 0"
+lemma%unimportant vector_matrix_mult_0 [simp]: "0 v* A = 0"
   unfolding vector_matrix_mult_def by (simp add: zero_vec_def)
 
-lemma vector_matrix_mult_0_right [simp]: "x v* 0 = 0"
+lemma%unimportant vector_matrix_mult_0_right [simp]: "x v* 0 = 0"
   unfolding vector_matrix_mult_def by (simp add: zero_vec_def)
 
-lemma vector_matrix_mul_rid [simp]:
+lemma%unimportant vector_matrix_mul_rid [simp]:
   fixes v :: "('a::semiring_1)^'n"
   shows "v v* mat 1 = v"
   by (metis matrix_vector_mul_lid transpose_mat vector_transpose_matrix)
 
-lemma scaleR_vector_matrix_assoc:
+lemma%unimportant scaleR_vector_matrix_assoc:
   fixes k :: real and x :: "real^'n" and A :: "real^'m^'n"
   shows "(k *\<^sub>R x) v* A = k *\<^sub>R (x v* A)"
   by (metis matrix_vector_mult_scaleR transpose_matrix_vector)
 
-lemma vector_scaleR_matrix_ac:
+lemma%important vector_scaleR_matrix_ac:
   fixes k :: real and x :: "real^'n" and A :: "real^'m^'n"
   shows "x v* (k *\<^sub>R A) = k *\<^sub>R (x v* A)"
-proof -
+proof%unimportant -
   have "x v* (k *\<^sub>R A) = (k *\<^sub>R x) v* A"
     unfolding vector_matrix_mult_def
     by (simp add: algebra_simps)

--- a/src/HOL/Analysis/Finite_Product_Measure.thy	Mon Aug 27 22:58:36 2018 +0200
+++ b/src/HOL/Analysis/Finite_Product_Measure.thy	Tue Aug 28 13:28:39 2018 +0100
@@ -2,20 +2,20 @@
     Author:     Johannes Hölzl, TU München
 *)
 
-section \<open>Finite product measures\<close>
+section%important \<open>Finite product measures\<close>
 
 theory Finite_Product_Measure
 imports Binary_Product_Measure
 begin
 
-lemma PiE_choice: "(\<exists>f\<in>Pi\<^sub>E I F. \<forall>i\<in>I. P i (f i)) \<longleftrightarrow> (\<forall>i\<in>I. \<exists>x\<in>F i. P i x)"
+lemma%unimportant PiE_choice: "(\<exists>f\<in>Pi\<^sub>E I F. \<forall>i\<in>I. P i (f i)) \<longleftrightarrow> (\<forall>i\<in>I. \<exists>x\<in>F i. P i x)"
   by (auto simp: Bex_def PiE_iff Ball_def dest!: choice_iff'[THEN iffD1])
      (force intro: exI[of _ "restrict f I" for f])
 
-lemma case_prod_const: "(\<lambda>(i, j). c) = (\<lambda>_. c)"
+lemma%unimportant case_prod_const: "(\<lambda>(i, j). c) = (\<lambda>_. c)"
   by auto
 
-subsubsection \<open>More about Function restricted by @{const extensional}\<close>
+subsubsection%unimportant \<open>More about Function restricted by @{const extensional}\<close>
 
 definition
   "merge I J = (\<lambda>(x, y) i. if i \<in> I then x i else if i \<in> J then y i else undefined)"
@@ -109,59 +109,59 @@
   "I \<inter> J = {} \<Longrightarrow> merge I J -` Pi\<^sub>E (I \<union> J) E = Pi I E \<times> Pi J E"
   by (auto simp: restrict_Pi_cancel PiE_def)
 
-subsection \<open>Finite product spaces\<close>
+subsection%important \<open>Finite product spaces\<close>
 
-subsubsection \<open>Products\<close>
+subsubsection%important \<open>Products\<close>
 
-definition prod_emb where
+definition%important prod_emb where
   "prod_emb I M K X = (\<lambda>x. restrict x K) -` X \<inter> (\<Pi>\<^sub>E i\<in>I. space (M i))"
 
-lemma prod_emb_iff:
+lemma%important prod_emb_iff:
   "f \<in> prod_emb I M K X \<longleftrightarrow> f \<in> extensional I \<and> (restrict f K \<in> X) \<and> (\<forall>i\<in>I. f i \<in> space (M i))"
-  unfolding prod_emb_def PiE_def by auto
+  unfolding%unimportant prod_emb_def PiE_def by auto
 
-lemma
+lemma%unimportant (*FIX ME needs a name *)
   shows prod_emb_empty[simp]: "prod_emb M L K {} = {}"
     and prod_emb_Un[simp]: "prod_emb M L K (A \<union> B) = prod_emb M L K A \<union> prod_emb M L K B"
     and prod_emb_Int: "prod_emb M L K (A \<inter> B) = prod_emb M L K A \<inter> prod_emb M L K B"
     and prod_emb_UN[simp]: "prod_emb M L K (\<Union>i\<in>I. F i) = (\<Union>i\<in>I. prod_emb M L K (F i))"
     and prod_emb_INT[simp]: "I \<noteq> {} \<Longrightarrow> prod_emb M L K (\<Inter>i\<in>I. F i) = (\<Inter>i\<in>I. prod_emb M L K (F i))"
     and prod_emb_Diff[simp]: "prod_emb M L K (A - B) = prod_emb M L K A - prod_emb M L K B"
-  by (auto simp: prod_emb_def)
+  by%unimportant (auto simp: prod_emb_def)
 
-lemma prod_emb_PiE: "J \<subseteq> I \<Longrightarrow> (\<And>i. i \<in> J \<Longrightarrow> E i \<subseteq> space (M i)) \<Longrightarrow>
+lemma%unimportant prod_emb_PiE: "J \<subseteq> I \<Longrightarrow> (\<And>i. i \<in> J \<Longrightarrow> E i \<subseteq> space (M i)) \<Longrightarrow>
     prod_emb I M J (\<Pi>\<^sub>E i\<in>J. E i) = (\<Pi>\<^sub>E i\<in>I. if i \<in> J then E i else space (M i))"
   by (force simp: prod_emb_def PiE_iff if_split_mem2)
 
-lemma prod_emb_PiE_same_index[simp]:
+lemma%unimportant prod_emb_PiE_same_index[simp]:
     "(\<And>i. i \<in> I \<Longrightarrow> E i \<subseteq> space (M i)) \<Longrightarrow> prod_emb I M I (Pi\<^sub>E I E) = Pi\<^sub>E I E"
   by (auto simp: prod_emb_def PiE_iff)
 
-lemma prod_emb_trans[simp]:
+lemma%unimportant prod_emb_trans[simp]:
   "J \<subseteq> K \<Longrightarrow> K \<subseteq> L \<Longrightarrow> prod_emb L M K (prod_emb K M J X) = prod_emb L M J X"
   by (auto simp add: Int_absorb1 prod_emb_def PiE_def)
 
-lemma prod_emb_Pi:
+lemma%unimportant prod_emb_Pi:
   assumes "X \<in> (\<Pi> j\<in>J. sets (M j))" "J \<subseteq> K"
   shows "prod_emb K M J (Pi\<^sub>E J X) = (\<Pi>\<^sub>E i\<in>K. if i \<in> J then X i else space (M i))"
   using assms sets.space_closed
   by (auto simp: prod_emb_def PiE_iff split: if_split_asm) blast+
 
-lemma prod_emb_id:
+lemma%unimportant prod_emb_id:
   "B \<subseteq> (\<Pi>\<^sub>E i\<in>L. space (M i)) \<Longrightarrow> prod_emb L M L B = B"
   by (auto simp: prod_emb_def subset_eq extensional_restrict)
 
-lemma prod_emb_mono:
+lemma%unimportant prod_emb_mono:
   "F \<subseteq> G \<Longrightarrow> prod_emb A M B F \<subseteq> prod_emb A M B G"
   by (auto simp: prod_emb_def)
 
-definition PiM :: "'i set \<Rightarrow> ('i \<Rightarrow> 'a measure) \<Rightarrow> ('i \<Rightarrow> 'a) measure" where
+definition%important PiM :: "'i set \<Rightarrow> ('i \<Rightarrow> 'a measure) \<Rightarrow> ('i \<Rightarrow> 'a) measure" where
   "PiM I M = extend_measure (\<Pi>\<^sub>E i\<in>I. space (M i))
     {(J, X). (J \<noteq> {} \<or> I = {}) \<and> finite J \<and> J \<subseteq> I \<and> X \<in> (\<Pi> j\<in>J. sets (M j))}
     (\<lambda>(J, X). prod_emb I M J (\<Pi>\<^sub>E j\<in>J. X j))
     (\<lambda>(J, X). \<Prod>j\<in>J \<union> {i\<in>I. emeasure (M i) (space (M i)) \<noteq> 1}. if j \<in> J then emeasure (M j) (X j) else emeasure (M j) (space (M j)))"
 
-definition prod_algebra :: "'i set \<Rightarrow> ('i \<Rightarrow> 'a measure) \<Rightarrow> ('i \<Rightarrow> 'a) set set" where
+definition%important prod_algebra :: "'i set \<Rightarrow> ('i \<Rightarrow> 'a measure) \<Rightarrow> ('i \<Rightarrow> 'a) set set" where
   "prod_algebra I M = (\<lambda>(J, X). prod_emb I M J (\<Pi>\<^sub>E j\<in>J. X j)) `
     {(J, X). (J \<noteq> {} \<or> I = {}) \<and> finite J \<and> J \<subseteq> I \<and> X \<in> (\<Pi> j\<in>J. sets (M j))}"
 
@@ -173,20 +173,20 @@
 translations
   "\<Pi>\<^sub>M x\<in>I. M" == "CONST PiM I (%x. M)"
 
-lemma extend_measure_cong:
+lemma%important extend_measure_cong:
   assumes "\<Omega> = \<Omega>'" "I = I'" "G = G'" "\<And>i. i \<in> I' \<Longrightarrow> \<mu> i = \<mu>' i"
   shows "extend_measure \<Omega> I G \<mu> = extend_measure \<Omega>' I' G' \<mu>'"
-  unfolding extend_measure_def by (auto simp add: assms)
+  unfolding%unimportant extend_measure_def by (auto simp add: assms)
 
-lemma Pi_cong_sets:
+lemma%unimportant Pi_cong_sets:
     "\<lbrakk>I = J; \<And>x. x \<in> I \<Longrightarrow> M x = N x\<rbrakk> \<Longrightarrow> Pi I M = Pi J N"
   unfolding Pi_def by auto
 
-lemma PiM_cong:
+lemma%important PiM_cong:
   assumes "I = J" "\<And>x. x \<in> I \<Longrightarrow> M x = N x"
   shows "PiM I M = PiM J N"
   unfolding PiM_def
-proof (rule extend_measure_cong, goal_cases)
+proof%unimportant (rule extend_measure_cong, goal_cases)
   case 1
   show ?case using assms
     by (subst assms(1), intro PiE_cong[of J "\<lambda>i. space (M i)" "\<lambda>i. space (N i)"]) simp_all
@@ -206,14 +206,14 @@
 qed
 
 
-lemma prod_algebra_sets_into_space:
+lemma%unimportant prod_algebra_sets_into_space:
   "prod_algebra I M \<subseteq> Pow (\<Pi>\<^sub>E i\<in>I. space (M i))"
   by (auto simp: prod_emb_def prod_algebra_def)
 
-lemma prod_algebra_eq_finite:
+lemma%important prod_algebra_eq_finite:
   assumes I: "finite I"
   shows "prod_algebra I M = {(\<Pi>\<^sub>E i\<in>I. X i) |X. X \<in> (\<Pi> j\<in>I. sets (M j))}" (is "?L = ?R")
-proof (intro iffI set_eqI)
+proof%unimportant (intro iffI set_eqI)
   fix A assume "A \<in> ?L"
   then obtain J E where J: "J \<noteq> {} \<or> I = {}" "finite J" "J \<subseteq> I" "\<forall>i\<in>J. E i \<in> sets (M i)"
     and A: "A = prod_emb I M J (\<Pi>\<^sub>E j\<in>J. E j)"
@@ -232,32 +232,32 @@
     by (auto simp: prod_algebra_def)
 qed
 
-lemma prod_algebraI:
+lemma%unimportant prod_algebraI:
   "finite J \<Longrightarrow> (J \<noteq> {} \<or> I = {}) \<Longrightarrow> J \<subseteq> I \<Longrightarrow> (\<And>i. i \<in> J \<Longrightarrow> E i \<in> sets (M i))
     \<Longrightarrow> prod_emb I M J (\<Pi>\<^sub>E j\<in>J. E j) \<in> prod_algebra I M"
   by (auto simp: prod_algebra_def)
 
-lemma prod_algebraI_finite:
+lemma%unimportant prod_algebraI_finite:
   "finite I \<Longrightarrow> (\<forall>i\<in>I. E i \<in> sets (M i)) \<Longrightarrow> (Pi\<^sub>E I E) \<in> prod_algebra I M"
   using prod_algebraI[of I I E M] prod_emb_PiE_same_index[of I E M, OF sets.sets_into_space] by simp
 
-lemma Int_stable_PiE: "Int_stable {Pi\<^sub>E J E | E. \<forall>i\<in>I. E i \<in> sets (M i)}"
+lemma%unimportant Int_stable_PiE: "Int_stable {Pi\<^sub>E J E | E. \<forall>i\<in>I. E i \<in> sets (M i)}"
 proof (safe intro!: Int_stableI)
   fix E F assume "\<forall>i\<in>I. E i \<in> sets (M i)" "\<forall>i\<in>I. F i \<in> sets (M i)"
   then show "\<exists>G. Pi\<^sub>E J E \<inter> Pi\<^sub>E J F = Pi\<^sub>E J G \<and> (\<forall>i\<in>I. G i \<in> sets (M i))"
     by (auto intro!: exI[of _ "\<lambda>i. E i \<inter> F i"] simp: PiE_Int)
 qed
 
-lemma prod_algebraE:
+lemma%unimportant prod_algebraE:
   assumes A: "A \<in> prod_algebra I M"
   obtains J E where "A = prod_emb I M J (\<Pi>\<^sub>E j\<in>J. E j)"
     "finite J" "J \<noteq> {} \<or> I = {}" "J \<subseteq> I" "\<And>i. i \<in> J \<Longrightarrow> E i \<in> sets (M i)"
   using A by (auto simp: prod_algebra_def)
 
-lemma prod_algebraE_all:
+lemma%important prod_algebraE_all:
   assumes A: "A \<in> prod_algebra I M"
   obtains E where "A = Pi\<^sub>E I E" "E \<in> (\<Pi> i\<in>I. sets (M i))"
-proof -
+proof%unimportant -
   from A obtain E J where A: "A = prod_emb I M J (Pi\<^sub>E J E)"
     and J: "J \<subseteq> I" and E: "E \<in> (\<Pi> i\<in>J. sets (M i))"
     by (auto simp: prod_algebra_def)
@@ -270,7 +270,7 @@
   ultimately show ?thesis using that by auto
 qed
 
-lemma Int_stable_prod_algebra: "Int_stable (prod_algebra I M)"
+lemma%unimportant Int_stable_prod_algebra: "Int_stable (prod_algebra I M)"
 proof (unfold Int_stable_def, safe)
   fix A assume "A \<in> prod_algebra I M"
   from prod_algebraE[OF this] guess J E . note A = this
@@ -291,11 +291,11 @@
   finally show "A \<inter> B \<in> prod_algebra I M" .
 qed
 
-lemma prod_algebra_mono:
+lemma%unimportant prod_algebra_mono:
   assumes space: "\<And>i. i \<in> I \<Longrightarrow> space (E i) = space (F i)"
   assumes sets: "\<And>i. i \<in> I \<Longrightarrow> sets (E i) \<subseteq> sets (F i)"
   shows "prod_algebra I E \<subseteq> prod_algebra I F"
-proof
+proof%unimportant
   fix A assume "A \<in> prod_algebra I E"
   then obtain J G where J: "J \<noteq> {} \<or> I = {}" "finite J" "J \<subseteq> I"
     and A: "A = prod_emb I E J (\<Pi>\<^sub>E i\<in>J. G i)"
@@ -316,17 +316,17 @@
     done
 qed
 
-lemma prod_algebra_cong:
+lemma%unimportant prod_algebra_cong:
   assumes "I = J" and sets: "(\<And>i. i \<in> I \<Longrightarrow> sets (M i) = sets (N i))"
   shows "prod_algebra I M = prod_algebra J N"
-proof -
+proof%unimportant -
   have space: "\<And>i. i \<in> I \<Longrightarrow> space (M i) = space (N i)"
     using sets_eq_imp_space_eq[OF sets] by auto
   with sets show ?thesis unfolding \<open>I = J\<close>
     by (intro antisym prod_algebra_mono) auto
 qed
 
-lemma space_in_prod_algebra:
+lemma%unimportant space_in_prod_algebra:
   "(\<Pi>\<^sub>E i\<in>I. space (M i)) \<in> prod_algebra I M"
 proof cases
   assume "I = {}" then show ?thesis
@@ -341,26 +341,26 @@
   finally show ?thesis .
 qed
 
-lemma space_PiM: "space (\<Pi>\<^sub>M i\<in>I. M i) = (\<Pi>\<^sub>E i\<in>I. space (M i))"
+lemma%unimportant space_PiM: "space (\<Pi>\<^sub>M i\<in>I. M i) = (\<Pi>\<^sub>E i\<in>I. space (M i))"
   using prod_algebra_sets_into_space unfolding PiM_def prod_algebra_def by (intro space_extend_measure) simp
 
-lemma prod_emb_subset_PiM[simp]: "prod_emb I M K X \<subseteq> space (PiM I M)"
+lemma%unimportant prod_emb_subset_PiM[simp]: "prod_emb I M K X \<subseteq> space (PiM I M)"
   by (auto simp: prod_emb_def space_PiM)
 
-lemma space_PiM_empty_iff[simp]: "space (PiM I M) = {} \<longleftrightarrow>  (\<exists>i\<in>I. space (M i) = {})"
+lemma%unimportant space_PiM_empty_iff[simp]: "space (PiM I M) = {} \<longleftrightarrow>  (\<exists>i\<in>I. space (M i) = {})"
   by (auto simp: space_PiM PiE_eq_empty_iff)
 
-lemma undefined_in_PiM_empty[simp]: "(\<lambda>x. undefined) \<in> space (PiM {} M)"
+lemma%unimportant undefined_in_PiM_empty[simp]: "(\<lambda>x. undefined) \<in> space (PiM {} M)"
   by (auto simp: space_PiM)
 
-lemma sets_PiM: "sets (\<Pi>\<^sub>M i\<in>I. M i) = sigma_sets (\<Pi>\<^sub>E i\<in>I. space (M i)) (prod_algebra I M)"
+lemma%unimportant sets_PiM: "sets (\<Pi>\<^sub>M i\<in>I. M i) = sigma_sets (\<Pi>\<^sub>E i\<in>I. space (M i)) (prod_algebra I M)"
   using prod_algebra_sets_into_space unfolding PiM_def prod_algebra_def by (intro sets_extend_measure) simp
 
-lemma sets_PiM_single: "sets (PiM I M) =
+lemma%important sets_PiM_single: "sets (PiM I M) =
     sigma_sets (\<Pi>\<^sub>E i\<in>I. space (M i)) {{f\<in>\<Pi>\<^sub>E i\<in>I. space (M i). f i \<in> A} | i A. i \<in> I \<and> A \<in> sets (M i)}"
     (is "_ = sigma_sets ?\<Omega> ?R")
   unfolding sets_PiM
-proof (rule sigma_sets_eqI)
+proof%unimportant (rule sigma_sets_eqI)
   interpret R: sigma_algebra ?\<Omega> "sigma_sets ?\<Omega> ?R" by (rule sigma_algebra_sigma_sets) auto
   fix A assume "A \<in> prod_algebra I M"
   from prod_algebraE[OF this] guess J X . note X = this
@@ -388,7 +388,7 @@
   finally show "A \<in> sigma_sets ?\<Omega> (prod_algebra I M)" .
 qed
 
-lemma sets_PiM_eq_proj:
+lemma%unimportant sets_PiM_eq_proj:
   "I \<noteq> {} \<Longrightarrow> sets (PiM I M) = sets (SUP i:I. vimage_algebra (\<Pi>\<^sub>E i\<in>I. space (M i)) (\<lambda>x. x i) (M i))"
   apply (simp add: sets_PiM_single)
   apply (subst sets_Sup_eq[where X="\<Pi>\<^sub>E i\<in>I. space (M i)"])
@@ -401,18 +401,18 @@
   apply (auto intro!: arg_cong2[where f=sigma_sets])
   done
 
-lemma
+lemma%unimportant (*FIX ME needs name *)
   shows space_PiM_empty: "space (Pi\<^sub>M {} M) = {\<lambda>k. undefined}"
     and sets_PiM_empty: "sets (Pi\<^sub>M {} M) = { {}, {\<lambda>k. undefined} }"
   by (simp_all add: space_PiM sets_PiM_single image_constant sigma_sets_empty_eq)
 
-lemma sets_PiM_sigma:
+lemma%important sets_PiM_sigma:
   assumes \<Omega>_cover: "\<And>i. i \<in> I \<Longrightarrow> \<exists>S\<subseteq>E i. countable S \<and> \<Omega> i = \<Union>S"
   assumes E: "\<And>i. i \<in> I \<Longrightarrow> E i \<subseteq> Pow (\<Omega> i)"
   assumes J: "\<And>j. j \<in> J \<Longrightarrow> finite j" "\<Union>J = I"
   defines "P \<equiv> {{f\<in>(\<Pi>\<^sub>E i\<in>I. \<Omega> i). \<forall>i\<in>j. f i \<in> A i} | A j. j \<in> J \<and> A \<in> Pi j E}"
   shows "sets (\<Pi>\<^sub>M i\<in>I. sigma (\<Omega> i) (E i)) = sets (sigma (\<Pi>\<^sub>E i\<in>I. \<Omega> i) P)"
-proof cases
+proof%unimportant cases
   assume "I = {}"
   with \<open>\<Union>J = I\<close> have "P = {{\<lambda>_. undefined}} \<or> P = {}"
     by (auto simp: P_def)
@@ -496,21 +496,21 @@
   finally show "?thesis" .
 qed
 
-lemma sets_PiM_in_sets:
+lemma%unimportant sets_PiM_in_sets:
   assumes space: "space N = (\<Pi>\<^sub>E i\<in>I. space (M i))"
   assumes sets: "\<And>i A. i \<in> I \<Longrightarrow> A \<in> sets (M i) \<Longrightarrow> {x\<in>space N. x i \<in> A} \<in> sets N"
   shows "sets (\<Pi>\<^sub>M i \<in> I. M i) \<subseteq> sets N"
   unfolding sets_PiM_single space[symmetric]
   by (intro sets.sigma_sets_subset subsetI) (auto intro: sets)
 
-lemma sets_PiM_cong[measurable_cong]:
+lemma%unimportant sets_PiM_cong[measurable_cong]:
   assumes "I = J" "\<And>i. i \<in> J \<Longrightarrow> sets (M i) = sets (N i)" shows "sets (PiM I M) = sets (PiM J N)"
   using assms sets_eq_imp_space_eq[OF assms(2)] by (simp add: sets_PiM_single cong: PiE_cong conj_cong)
 
-lemma sets_PiM_I:
+lemma%important sets_PiM_I:
   assumes "finite J" "J \<subseteq> I" "\<forall>i\<in>J. E i \<in> sets (M i)"
   shows "prod_emb I M J (\<Pi>\<^sub>E j\<in>J. E j) \<in> sets (\<Pi>\<^sub>M i\<in>I. M i)"
-proof cases
+proof%unimportant cases
   assume "J = {}"
   then have "prod_emb I M J (\<Pi>\<^sub>E j\<in>J. E j) = (\<Pi>\<^sub>E j\<in>I. space (M j))"
     by (auto simp: prod_emb_def)
@@ -521,7 +521,7 @@
     by (force simp add: sets_PiM prod_algebra_def)
 qed
 
-lemma measurable_PiM:
+lemma%unimportant measurable_PiM:
   assumes space: "f \<in> space N \<rightarrow> (\<Pi>\<^sub>E i\<in>I. space (M i))"
   assumes sets: "\<And>X J. J \<noteq> {} \<or> I = {} \<Longrightarrow> finite J \<Longrightarrow> J \<subseteq> I \<Longrightarrow> (\<And>i. i \<in> J \<Longrightarrow> X i \<in> sets (M i)) \<Longrightarrow>
     f -` prod_emb I M J (Pi\<^sub>E J X) \<inter> space N \<in> sets N"
@@ -533,13 +533,13 @@
   with sets[of J X] show "f -` A \<inter> space N \<in> sets N" by auto
 qed
 
-lemma measurable_PiM_Collect:
+lemma%important measurable_PiM_Collect:
   assumes space: "f \<in> space N \<rightarrow> (\<Pi>\<^sub>E i\<in>I. space (M i))"
   assumes sets: "\<And>X J. J \<noteq> {} \<or> I = {} \<Longrightarrow> finite J \<Longrightarrow> J \<subseteq> I \<Longrightarrow> (\<And>i. i \<in> J \<Longrightarrow> X i \<in> sets (M i)) \<Longrightarrow>
     {\<omega>\<in>space N. \<forall>i\<in>J. f \<omega> i \<in> X i} \<in> sets N"
   shows "f \<in> measurable N (PiM I M)"
   using sets_PiM prod_algebra_sets_into_space space
-proof (rule measurable_sigma_sets)
+proof%unimportant (rule measurable_sigma_sets)
   fix A assume "A \<in> prod_algebra I M"
   from prod_algebraE[OF this] guess J X . note X = this
   then have "f -` A \<inter> space N = {\<omega> \<in> space N. \<forall>i\<in>J. f \<omega> i \<in> X i}"
@@ -548,7 +548,7 @@
   finally show "f -` A \<inter> space N \<in> sets N" .
 qed
 
-lemma measurable_PiM_single:
+lemma%unimportant measurable_PiM_single:
   assumes space: "f \<in> space N \<rightarrow> (\<Pi>\<^sub>E i\<in>I. space (M i))"
   assumes sets: "\<And>A i. i \<in> I \<Longrightarrow> A \<in> sets (M i) \<Longrightarrow> {\<omega> \<in> space N. f \<omega> i \<in> A} \<in> sets N"
   shows "f \<in> measurable N (PiM I M)"
@@ -562,11 +562,11 @@
   finally show "f -` A \<inter> space N \<in> sets N" .
 qed (auto simp: space)
 
-lemma measurable_PiM_single':
+lemma%important measurable_PiM_single':
   assumes f: "\<And>i. i \<in> I \<Longrightarrow> f i \<in> measurable N (M i)"
     and "(\<lambda>\<omega> i. f i \<omega>) \<in> space N \<rightarrow> (\<Pi>\<^sub>E i\<in>I. space (M i))"
   shows "(\<lambda>\<omega> i. f i \<omega>) \<in> measurable N (Pi\<^sub>M I M)"
-proof (rule measurable_PiM_single)
+proof%unimportant (rule measurable_PiM_single)
   fix A i assume A: "i \<in> I" "A \<in> sets (M i)"
   then have "{\<omega> \<in> space N. f i \<omega> \<in> A} = f i -` A \<inter> space N"
     by auto
@@ -574,12 +574,12 @@
     using A f by (auto intro!: measurable_sets)
 qed fact
 
-lemma sets_PiM_I_finite[measurable]:
+lemma%unimportant sets_PiM_I_finite[measurable]:
   assumes "finite I" and sets: "(\<And>i. i \<in> I \<Longrightarrow> E i \<in> sets (M i))"
   shows "(\<Pi>\<^sub>E j\<in>I. E j) \<in> sets (\<Pi>\<^sub>M i\<in>I. M i)"
   using sets_PiM_I[of I I E M] sets.sets_into_space[OF sets] \<open>finite I\<close> sets by auto
 
-lemma measurable_component_singleton[measurable (raw)]:
+lemma%unimportant measurable_component_singleton[measurable (raw)]:
   assumes "i \<in> I" shows "(\<lambda>x. x i) \<in> measurable (Pi\<^sub>M I M) (M i)"
 proof (unfold measurable_def, intro CollectI conjI ballI)
   fix A assume "A \<in> sets (M i)"
@@ -590,30 +590,30 @@
     using \<open>A \<in> sets (M i)\<close> \<open>i \<in> I\<close> by (auto intro!: sets_PiM_I)
 qed (insert \<open>i \<in> I\<close>, auto simp: space_PiM)
 
-lemma measurable_component_singleton'[measurable_dest]:
+lemma%unimportant measurable_component_singleton'[measurable_dest]:
   assumes f: "f \<in> measurable N (Pi\<^sub>M I M)"
   assumes g: "g \<in> measurable L N"
   assumes i: "i \<in> I"
   shows "(\<lambda>x. (f (g x)) i) \<in> measurable L (M i)"
   using measurable_compose[OF measurable_compose[OF g f] measurable_component_singleton, OF i] .
 
-lemma measurable_PiM_component_rev:
+lemma%unimportant measurable_PiM_component_rev:
   "i \<in> I \<Longrightarrow> f \<in> measurable (M i) N \<Longrightarrow> (\<lambda>x. f (x i)) \<in> measurable (PiM I M) N"
   by simp
 
-lemma measurable_case_nat[measurable (raw)]:
+lemma%unimportant measurable_case_nat[measurable (raw)]:
   assumes [measurable (raw)]: "i = 0 \<Longrightarrow> f \<in> measurable M N"
     "\<And>j. i = Suc j \<Longrightarrow> (\<lambda>x. g x j) \<in> measurable M N"
   shows "(\<lambda>x. case_nat (f x) (g x) i) \<in> measurable M N"
   by (cases i) simp_all
 
-lemma measurable_case_nat'[measurable (raw)]:
+lemma%unimportant measurable_case_nat'[measurable (raw)]:
   assumes fg[measurable]: "f \<in> measurable N M" "g \<in> measurable N (\<Pi>\<^sub>M i\<in>UNIV. M)"
   shows "(\<lambda>x. case_nat (f x) (g x)) \<in> measurable N (\<Pi>\<^sub>M i\<in>UNIV. M)"
   using fg[THEN measurable_space]
   by (auto intro!: measurable_PiM_single' simp add: space_PiM PiE_iff split: nat.split)
 
-lemma measurable_add_dim[measurable]:
+lemma%unimportant measurable_add_dim[measurable]:
   "(\<lambda>(f, y). f(i := y)) \<in> measurable (Pi\<^sub>M I M \<Otimes>\<^sub>M M i) (Pi\<^sub>M (insert i I) M)"
     (is "?f \<in> measurable ?P ?I")
 proof (rule measurable_PiM_single)
@@ -627,12 +627,12 @@
   finally show "{\<omega> \<in> space ?P. case_prod (\<lambda>f. fun_upd f i) \<omega> j \<in> A} \<in> sets ?P" .
 qed (auto simp: space_pair_measure space_PiM PiE_def)
 
-lemma measurable_fun_upd:
+lemma%important measurable_fun_upd:
   assumes I: "I = J \<union> {i}"
   assumes f[measurable]: "f \<in> measurable N (PiM J M)"
   assumes h[measurable]: "h \<in> measurable N (M i)"
   shows "(\<lambda>x. (f x) (i := h x)) \<in> measurable N (PiM I M)"
-proof (intro measurable_PiM_single')
+proof%unimportant (intro measurable_PiM_single')
   fix j assume "j \<in> I" then show "(\<lambda>\<omega>. ((f \<omega>)(i := h \<omega>)) j) \<in> measurable N (M j)"
     unfolding I by (cases "j = i") auto
 next
@@ -641,14 +641,14 @@
     by (auto simp: space_PiM PiE_iff extensional_def)
 qed
 
-lemma measurable_component_update:
+lemma%unimportant measurable_component_update:
   "x \<in> space (Pi\<^sub>M I M) \<Longrightarrow> i \<notin> I \<Longrightarrow> (\<lambda>v. x(i := v)) \<in> measurable (M i) (Pi\<^sub>M (insert i I) M)"
   by simp
 
-lemma measurable_merge[measurable]:
+lemma%important measurable_merge[measurable]:
   "merge I J \<in> measurable (Pi\<^sub>M I M \<Otimes>\<^sub>M Pi\<^sub>M J M) (Pi\<^sub>M (I \<union> J) M)"
     (is "?f \<in> measurable ?P ?U")
-proof (rule measurable_PiM_single)
+proof%unimportant (rule measurable_PiM_single)
   fix i A assume A: "A \<in> sets (M i)" "i \<in> I \<union> J"
   then have "{\<omega> \<in> space ?P. merge I J \<omega> i \<in> A} =
     (if i \<in> I then ((\<lambda>x. x i) \<circ> fst) -` A \<inter> space ?P else ((\<lambda>x. x i) \<circ> snd) -` A \<inter> space ?P)"
@@ -659,7 +659,7 @@
   finally show "{\<omega> \<in> space ?P. merge I J \<omega> i \<in> A} \<in> sets ?P" .
 qed (auto simp: space_pair_measure space_PiM PiE_iff merge_def extensional_def)
 
-lemma measurable_restrict[measurable (raw)]:
+lemma%unimportant measurable_restrict[measurable (raw)]:
   assumes X: "\<And>i. i \<in> I \<Longrightarrow> X i \<in> measurable N (M i)"
   shows "(\<lambda>x. \<lambda>i\<in>I. X i x) \<in> measurable N (Pi\<^sub>M I M)"
 proof (rule measurable_PiM_single)
@@ -670,14 +670,14 @@
     using A X by (auto intro!: measurable_sets)
 qed (insert X, auto simp add: PiE_def dest: measurable_space)
 
-lemma measurable_abs_UNIV:
+lemma%unimportant measurable_abs_UNIV:
   "(\<And>n. (\<lambda>\<omega>. f n \<omega>) \<in> measurable M (N n)) \<Longrightarrow> (\<lambda>\<omega> n. f n \<omega>) \<in> measurable M (PiM UNIV N)"
   by (intro measurable_PiM_single) (auto dest: measurable_space)
 
-lemma measurable_restrict_subset: "J \<subseteq> L \<Longrightarrow> (\<lambda>f. restrict f J) \<in> measurable (Pi\<^sub>M L M) (Pi\<^sub>M J M)"
+lemma%unimportant measurable_restrict_subset: "J \<subseteq> L \<Longrightarrow> (\<lambda>f. restrict f J) \<in> measurable (Pi\<^sub>M L M) (Pi\<^sub>M J M)"
   by (intro measurable_restrict measurable_component_singleton) auto
 
-lemma measurable_restrict_subset':
+lemma%unimportant measurable_restrict_subset':
   assumes "J \<subseteq> L" "\<And>x. x \<in> J \<Longrightarrow> sets (M x) = sets (N x)"
   shows "(\<lambda>f. restrict f J) \<in> measurable (Pi\<^sub>M L M) (Pi\<^sub>M J N)"
 proof-
@@ -688,12 +688,12 @@
   finally show ?thesis .
 qed
 
-lemma measurable_prod_emb[intro, simp]:
+lemma%unimportant measurable_prod_emb[intro, simp]:
   "J \<subseteq> L \<Longrightarrow> X \<in> sets (Pi\<^sub>M J M) \<Longrightarrow> prod_emb L M J X \<in> sets (Pi\<^sub>M L M)"
   unfolding prod_emb_def space_PiM[symmetric]
   by (auto intro!: measurable_sets measurable_restrict measurable_component_singleton)
 
-lemma merge_in_prod_emb:
+lemma%unimportant merge_in_prod_emb:
   assumes "y \<in> space (PiM I M)" "x \<in> X" and X: "X \<in> sets (Pi\<^sub>M J M)" and "J \<subseteq> I"
   shows "merge J I (x, y) \<in> prod_emb I M J X"
   using assms sets.sets_into_space[OF X]
@@ -701,7 +701,7 @@
            cong: if_cong restrict_cong)
      (simp add: extensional_def)
 
-lemma prod_emb_eq_emptyD:
+lemma%unimportant prod_emb_eq_emptyD:
   assumes J: "J \<subseteq> I" and ne: "space (PiM I M) \<noteq> {}" and X: "X \<in> sets (Pi\<^sub>M J M)"
     and *: "prod_emb I M J X = {}"
   shows "X = {}"
@@ -712,19 +712,19 @@
   from merge_in_prod_emb[OF this \<open>x\<in>X\<close> X J] * show "x \<in> {}" by auto
 qed
 
-lemma sets_in_Pi_aux:
+lemma%unimportant sets_in_Pi_aux:
   "finite I \<Longrightarrow> (\<And>j. j \<in> I \<Longrightarrow> {x\<in>space (M j). x \<in> F j} \<in> sets (M j)) \<Longrightarrow>
   {x\<in>space (PiM I M). x \<in> Pi I F} \<in> sets (PiM I M)"
   by (simp add: subset_eq Pi_iff)
 
-lemma sets_in_Pi[measurable (raw)]:
+lemma%unimportant sets_in_Pi[measurable (raw)]:
   "finite I \<Longrightarrow> f \<in> measurable N (PiM I M) \<Longrightarrow>
   (\<And>j. j \<in> I \<Longrightarrow> {x\<in>space (M j). x \<in> F j} \<in> sets (M j)) \<Longrightarrow>
   Measurable.pred N (\<lambda>x. f x \<in> Pi I F)"
   unfolding pred_def
   by (rule measurable_sets_Collect[of f N "PiM I M", OF _ sets_in_Pi_aux]) auto
 
-lemma sets_in_extensional_aux:
+lemma%unimportant sets_in_extensional_aux:
   "{x\<in>space (PiM I M). x \<in> extensional I} \<in> sets (PiM I M)"
 proof -
   have "{x\<in>space (PiM I M). x \<in> extensional I} = space (PiM I M)"
@@ -732,12 +732,12 @@
   then show ?thesis by simp
 qed
 
-lemma sets_in_extensional[measurable (raw)]:
+lemma%unimportant sets_in_extensional[measurable (raw)]:
   "f \<in> measurable N (PiM I M) \<Longrightarrow> Measurable.pred N (\<lambda>x. f x \<in> extensional I)"
   unfolding pred_def
   by (rule measurable_sets_Collect[of f N "PiM I M", OF _ sets_in_extensional_aux]) auto
 
-lemma sets_PiM_I_countable:
+lemma%unimportant sets_PiM_I_countable:
   assumes I: "countable I" and E: "\<And>i. i \<in> I \<Longrightarrow> E i \<in> sets (M i)" shows "Pi\<^sub>E I E \<in> sets (Pi\<^sub>M I M)"
 proof cases
   assume "I \<noteq> {}"
@@ -748,11 +748,11 @@
   finally show ?thesis .
 qed (simp add: sets_PiM_empty)
 
-lemma sets_PiM_D_countable:
+lemma%important sets_PiM_D_countable:
   assumes A: "A \<in> PiM I M"
   shows "\<exists>J\<subseteq>I. \<exists>X\<in>PiM J M. countable J \<and> A = prod_emb I M J X"
   using A[unfolded sets_PiM_single]
-proof induction
+proof%unimportant induction
   case (Basic A)
   then obtain i X where *: "i \<in> I" "X \<in> sets (M i)" and "A = {f \<in> \<Pi>\<^sub>E i\<in>I. space (M i). f i \<in> X}"
     by auto
@@ -784,12 +784,12 @@
   qed(auto intro: X)
 qed
 
-lemma measure_eqI_PiM_finite:
+lemma%important measure_eqI_PiM_finite:
   assumes [simp]: "finite I" "sets P = PiM I M" "sets Q = PiM I M"
   assumes eq: "\<And>A. (\<And>i. i \<in> I \<Longrightarrow> A i \<in> sets (M i)) \<Longrightarrow> P (Pi\<^sub>E I A) = Q (Pi\<^sub>E I A)"
   assumes A: "range A \<subseteq> prod_algebra I M" "(\<Union>i. A i) = space (PiM I M)" "\<And>i::nat. P (A i) \<noteq> \<infinity>"
   shows "P = Q"
-proof (rule measure_eqI_generator_eq[OF Int_stable_prod_algebra prod_algebra_sets_into_space])
+proof%unimportant (rule measure_eqI_generator_eq[OF Int_stable_prod_algebra prod_algebra_sets_into_space])
   show "range A \<subseteq> prod_algebra I M" "(\<Union>i. A i) = (\<Pi>\<^sub>E i\<in>I. space (M i))" "\<And>i. P (A i) \<noteq> \<infinity>"
     unfolding space_PiM[symmetric] by fact+
   fix X assume "X \<in> prod_algebra I M"
@@ -800,13 +800,13 @@
     unfolding X by (subst (1 2) prod_emb_Pi) (auto simp: eq)
 qed (simp_all add: sets_PiM)
 
-lemma measure_eqI_PiM_infinite:
+lemma%important measure_eqI_PiM_infinite:
   assumes [simp]: "sets P = PiM I M" "sets Q = PiM I M"
   assumes eq: "\<And>A J. finite J \<Longrightarrow> J \<subseteq> I \<Longrightarrow> (\<And>i. i \<in> J \<Longrightarrow> A i \<in> sets (M i)) \<Longrightarrow>
     P (prod_emb I M J (Pi\<^sub>E J A)) = Q (prod_emb I M J (Pi\<^sub>E J A))"
   assumes A: "finite_measure P"
   shows "P = Q"
-proof (rule measure_eqI_generator_eq[OF Int_stable_prod_algebra prod_algebra_sets_into_space])
+proof%unimportant (rule measure_eqI_generator_eq[OF Int_stable_prod_algebra prod_algebra_sets_into_space])
   interpret finite_measure P by fact
   define i where "i = (SOME i. i \<in> I)"
   have i: "I \<noteq> {} \<Longrightarrow> i \<in> I"
@@ -829,23 +829,23 @@
     by (auto intro!: eq)
 qed (auto simp: sets_PiM)
 
-locale product_sigma_finite =
+locale%unimportant product_sigma_finite =
   fixes M :: "'i \<Rightarrow> 'a measure"
   assumes sigma_finite_measures: "\<And>i. sigma_finite_measure (M i)"
 
-sublocale product_sigma_finite \<subseteq> M?: sigma_finite_measure "M i" for i
+sublocale%unimportant product_sigma_finite \<subseteq> M?: sigma_finite_measure "M i" for i
   by (rule sigma_finite_measures)
 
-locale finite_product_sigma_finite = product_sigma_finite M for M :: "'i \<Rightarrow> 'a measure" +
+locale%unimportant finite_product_sigma_finite = product_sigma_finite M for M :: "'i \<Rightarrow> 'a measure" +
   fixes I :: "'i set"
   assumes finite_index: "finite I"
 
-lemma (in finite_product_sigma_finite) sigma_finite_pairs:
+lemma%important (in finite_product_sigma_finite) sigma_finite_pairs:
   "\<exists>F::'i \<Rightarrow> nat \<Rightarrow> 'a set.
     (\<forall>i\<in>I. range (F i) \<subseteq> sets (M i)) \<and>
     (\<forall>k. \<forall>i\<in>I. emeasure (M i) (F i k) \<noteq> \<infinity>) \<and> incseq (\<lambda>k. \<Pi>\<^sub>E i\<in>I. F i k) \<and>
     (\<Union>k. \<Pi>\<^sub>E i\<in>I. F i k) = space (PiM I M)"
-proof -
+proof%unimportant -
   have "\<forall>i::'i. \<exists>F::nat \<Rightarrow> 'a set. range F \<subseteq> sets (M i) \<and> incseq F \<and> (\<Union>i. F i) = space (M i) \<and> (\<forall>k. emeasure (M i) (F k) \<noteq> \<infinity>)"
     using M.sigma_finite_incseq by metis
   from choice[OF this] guess F :: "'i \<Rightarrow> nat \<Rightarrow> 'a set" ..
@@ -871,7 +871,7 @@
   qed
 qed
 
-lemma emeasure_PiM_empty[simp]: "emeasure (PiM {} M) {\<lambda>_. undefined} = 1"
+lemma%unimportant emeasure_PiM_empty[simp]: "emeasure (PiM {} M) {\<lambda>_. undefined} = 1"
 proof -
   let ?\<mu> = "\<lambda>A. if A = {} then 0 else (1::ennreal)"
   have "emeasure (Pi\<^sub>M {} M) (prod_emb {} M {} (\<Pi>\<^sub>E i\<in>{}. {})) = 1"
@@ -888,12 +888,12 @@
     by simp
 qed
 
-lemma PiM_empty: "PiM {} M = count_space {\<lambda>_. undefined}"
+lemma%unimportant PiM_empty: "PiM {} M = count_space {\<lambda>_. undefined}"
   by (rule measure_eqI) (auto simp add: sets_PiM_empty)
 
-lemma (in product_sigma_finite) emeasure_PiM:
+lemma%important (in product_sigma_finite) emeasure_PiM:
   "finite I \<Longrightarrow> (\<And>i. i\<in>I \<Longrightarrow> A i \<in> sets (M i)) \<Longrightarrow> emeasure (PiM I M) (Pi\<^sub>E I A) = (\<Prod>i\<in>I. emeasure (M i) (A i))"
-proof (induct I arbitrary: A rule: finite_induct)
+proof%unimportant (induct I arbitrary: A rule: finite_induct)
   case (insert i I)
   interpret finite_product_sigma_finite M I by standard fact
   have "finite (insert i I)" using \<open>finite I\<close> by auto
@@ -948,7 +948,7 @@
     by (subst (asm) prod_emb_PiE_same_index) (auto intro!: sets.sets_into_space)
 qed simp
 
-lemma (in product_sigma_finite) PiM_eqI:
+lemma%unimportant (in product_sigma_finite) PiM_eqI:
   assumes I[simp]: "finite I" and P: "sets P = PiM I M"
   assumes eq: "\<And>A. (\<And>i. i \<in> I \<Longrightarrow> A i \<in> sets (M i)) \<Longrightarrow> P (Pi\<^sub>E I A) = (\<Prod>i\<in>I. emeasure (M i) (A i))"
   shows "P = PiM I M"
@@ -966,7 +966,7 @@
   qed
 qed
 
-lemma (in product_sigma_finite) sigma_finite:
+lemma%unimportant (in product_sigma_finite) sigma_finite:
   assumes "finite I"
   shows "sigma_finite_measure (PiM I M)"
 proof
@@ -987,19 +987,19 @@
 sublocale finite_product_sigma_finite \<subseteq> sigma_finite_measure "Pi\<^sub>M I M"
   using sigma_finite[OF finite_index] .
 
-lemma (in finite_product_sigma_finite) measure_times:
+lemma%unimportant (in finite_product_sigma_finite) measure_times:
   "(\<And>i. i \<in> I \<Longrightarrow> A i \<in> sets (M i)) \<Longrightarrow> emeasure (Pi\<^sub>M I M) (Pi\<^sub>E I A) = (\<Prod>i\<in>I. emeasure (M i) (A i))"
   using emeasure_PiM[OF finite_index] by auto
 
-lemma (in product_sigma_finite) nn_integral_empty:
+lemma%unimportant (in product_sigma_finite) nn_integral_empty:
   "0 \<le> f (\<lambda>k. undefined) \<Longrightarrow> integral\<^sup>N (Pi\<^sub>M {} M) f = f (\<lambda>k. undefined)"
   by (simp add: PiM_empty nn_integral_count_space_finite max.absorb2)
 
-lemma (in product_sigma_finite) distr_merge:
+lemma%important (in product_sigma_finite) distr_merge:
   assumes IJ[simp]: "I \<inter> J = {}" and fin: "finite I" "finite J"
   shows "distr (Pi\<^sub>M I M \<Otimes>\<^sub>M Pi\<^sub>M J M) (Pi\<^sub>M (I \<union> J) M) (merge I J) = Pi\<^sub>M (I \<union> J) M"
    (is "?D = ?P")
-proof (rule PiM_eqI)
+proof%unimportant (rule PiM_eqI)
   interpret I: finite_product_sigma_finite M I by standard fact
   interpret J: finite_product_sigma_finite M J by standard fact
   fix A assume A: "\<And>i. i \<in> I \<union> J \<Longrightarrow> A i \<in> sets (M i)"
@@ -1011,12 +1011,12 @@
        (auto simp: * J.emeasure_pair_measure_Times I.measure_times J.measure_times prod.union_disjoint)
 qed (insert fin, simp_all)
 
-lemma (in product_sigma_finite) product_nn_integral_fold:
+lemma%important (in product_sigma_finite) product_nn_integral_fold:
   assumes IJ: "I \<inter> J = {}" "finite I" "finite J"
   and f[measurable]: "f \<in> borel_measurable (Pi\<^sub>M (I \<union> J) M)"
   shows "integral\<^sup>N (Pi\<^sub>M (I \<union> J) M) f =
     (\<integral>\<^sup>+ x. (\<integral>\<^sup>+ y. f (merge I J (x, y)) \<partial>(Pi\<^sub>M J M)) \<partial>(Pi\<^sub>M I M))"
-proof -
+proof%unimportant -
   interpret I: finite_product_sigma_finite M I by standard fact
   interpret J: finite_product_sigma_finite M J by standard fact
   interpret P: pair_sigma_finite "Pi\<^sub>M I M" "Pi\<^sub>M J M" by standard
@@ -1031,7 +1031,7 @@
     done
 qed
 
-lemma (in product_sigma_finite) distr_singleton:
+lemma%unimportant (in product_sigma_finite) distr_singleton:
   "distr (Pi\<^sub>M {i} M) (M i) (\<lambda>x. x i) = M i" (is "?D = _")
 proof (intro measure_eqI[symmetric])
   interpret I: finite_product_sigma_finite M "{i}" by standard simp
@@ -1043,7 +1043,7 @@
     by (simp add: emeasure_distr measurable_component_singleton)
 qed simp
 
-lemma (in product_sigma_finite) product_nn_integral_singleton:
+lemma%unimportant (in product_sigma_finite) product_nn_integral_singleton:
   assumes f: "f \<in> borel_measurable (M i)"
   shows "integral\<^sup>N (Pi\<^sub>M {i} M) (\<lambda>x. f (x i)) = integral\<^sup>N (M i) f"
 proof -
@@ -1055,11 +1055,11 @@
     done
 qed
 
-lemma (in product_sigma_finite) product_nn_integral_insert:
+lemma%important (in product_sigma_finite) product_nn_integral_insert:
   assumes I[simp]: "finite I" "i \<notin> I"
     and f: "f \<in> borel_measurable (Pi\<^sub>M (insert i I) M)"
   shows "integral\<^sup>N (Pi\<^sub>M (insert i I) M) f = (\<integral>\<^sup>+ x. (\<integral>\<^sup>+ y. f (x(i := y)) \<partial>(M i)) \<partial>(Pi\<^sub>M I M))"
-proof -
+proof%unimportant -
   interpret I: finite_product_sigma_finite M I by standard auto
   interpret i: finite_product_sigma_finite M "{i}" by standard auto
   have IJ: "I \<inter> {i} = {}" and insert: "I \<union> {i} = insert i I"
@@ -1079,7 +1079,7 @@
   qed
 qed
 
-lemma (in product_sigma_finite) product_nn_integral_insert_rev:
+lemma%unimportant (in product_sigma_finite) product_nn_integral_insert_rev:
   assumes I[simp]: "finite I" "i \<notin> I"
     and [measurable]: "f \<in> borel_measurable (Pi\<^sub>M (insert i I) M)"
   shows "integral\<^sup>N (Pi\<^sub>M (insert i I) M) f = (\<integral>\<^sup>+ y. (\<integral>\<^sup>+ x. f (x(i := y)) \<partial>(Pi\<^sub>M I M)) \<partial>(M i))"
@@ -1090,7 +1090,7 @@
   apply measurable
   done
 
-lemma (in product_sigma_finite) product_nn_integral_prod:
+lemma%unimportant (in product_sigma_finite) product_nn_integral_prod:
   assumes "finite I" "\<And>i. i \<in> I \<Longrightarrow> f i \<in> borel_measurable (M i)"
   shows "(\<integral>\<^sup>+ x. (\<Prod>i\<in>I. f i (x i)) \<partial>Pi\<^sub>M I M) = (\<Prod>i\<in>I. integral\<^sup>N (M i) (f i))"
 using assms proof (induction I)
@@ -1113,11 +1113,11 @@
     done
 qed (simp add: space_PiM)
 
-lemma (in product_sigma_finite) product_nn_integral_pair:
+lemma%important (in product_sigma_finite) product_nn_integral_pair:
   assumes [measurable]: "case_prod f \<in> borel_measurable (M x \<Otimes>\<^sub>M M y)"
   assumes xy: "x \<noteq> y"
   shows "(\<integral>\<^sup>+\<sigma>. f (\<sigma> x) (\<sigma> y) \<partial>PiM {x, y} M) = (\<integral>\<^sup>+z. f (fst z) (snd z) \<partial>(M x \<Otimes>\<^sub>M M y))"
-proof-
+proof%unimportant -
   interpret psm: pair_sigma_finite "M x" "M y"
     unfolding pair_sigma_finite_def using sigma_finite_measures by simp_all
   have "{x, y} = {y, x}" by auto
@@ -1130,7 +1130,7 @@
   finally show ?thesis .
 qed
 
-lemma (in product_sigma_finite) distr_component:
+lemma%unimportant (in product_sigma_finite) distr_component:
   "distr (M i) (Pi\<^sub>M {i} M) (\<lambda>x. \<lambda>i\<in>{i}. x) = Pi\<^sub>M {i} M" (is "?D = ?P")
 proof (intro PiM_eqI)
   fix A assume A: "\<And>ia. ia \<in> {i} \<Longrightarrow> A ia \<in> sets (M ia)"
@@ -1140,7 +1140,7 @@
     by (subst emeasure_distr) (auto intro!: sets_PiM_I_finite measurable_restrict)
 qed simp_all
 
-lemma (in product_sigma_finite)
+lemma%unimportant (in product_sigma_finite)
   assumes IJ: "I \<inter> J = {}" "finite I" "finite J" and A: "A \<in> sets (Pi\<^sub>M (I \<union> J) M)"
   shows emeasure_fold_integral:
     "emeasure (Pi\<^sub>M (I \<union> J) M) A = (\<integral>\<^sup>+x. emeasure (Pi\<^sub>M J M) ((\<lambda>y. merge I J (x, y)) -` A \<inter> space (Pi\<^sub>M J M)) \<partial>Pi\<^sub>M I M)" (is ?I)
@@ -1165,11 +1165,11 @@
     by (simp add: vimage_comp comp_def space_pair_measure cong: measurable_cong)
 qed
 
-lemma sets_Collect_single:
+lemma%unimportant sets_Collect_single:
   "i \<in> I \<Longrightarrow> A \<in> sets (M i) \<Longrightarrow> { x \<in> space (Pi\<^sub>M I M). x i \<in> A } \<in> sets (Pi\<^sub>M I M)"
   by simp
 
-lemma pair_measure_eq_distr_PiM:
+lemma%unimportant pair_measure_eq_distr_PiM:
   fixes M1 :: "'a measure" and M2 :: "'a measure"
   assumes "sigma_finite_measure M1" "sigma_finite_measure M2"
   shows "(M1 \<Otimes>\<^sub>M M2) = distr (Pi\<^sub>M UNIV (case_bool M1 M2)) (M1 \<Otimes>\<^sub>M M2) (\<lambda>x. (x True, x False))"
@@ -1196,7 +1196,7 @@
     by (subst emeasure_distr) (auto simp: measurable_pair_iff)
 qed simp
 
-lemma infprod_in_sets[intro]:
+lemma%unimportant infprod_in_sets[intro]:
   fixes E :: "nat \<Rightarrow> 'a set" assumes E: "\<And>i. E i \<in> sets (M i)"
   shows "Pi UNIV E \<in> sets (\<Pi>\<^sub>M i\<in>UNIV::nat set. M i)"
 proof -

--- a/src/HOL/Analysis/Function_Topology.thy	Mon Aug 27 22:58:36 2018 +0200
+++ b/src/HOL/Analysis/Function_Topology.thy	Tue Aug 28 13:28:39 2018 +0100
@@ -7,7 +7,7 @@
 begin
 
 
-section \<open>Product topology\<close>
+section%important \<open>Product topology\<close>
 
 text \<open>We want to define the product topology.
 
@@ -62,9 +62,9 @@
 I realized afterwards that this theory has a lot in common with \verb+Fin_Map.thy+.
 \<close>
 
-subsection \<open>Topology without type classes\<close>
+subsection%important \<open>Topology without type classes\<close>
 
-subsubsection \<open>The topology generated by some (open) subsets\<close>
+subsubsection%important \<open>The topology generated by some (open) subsets\<close>
 
 text \<open>In the definition below of a generated topology, the \<open>Empty\<close> case is not necessary,
 as it follows from \<open>UN\<close> taking for $K$ the empty set. However, it is convenient to have,
@@ -79,14 +79,14 @@
 | UN: "(\<And>k. k \<in> K \<Longrightarrow> generate_topology_on S k) \<Longrightarrow> generate_topology_on S (\<Union>K)"
 | Basis: "s \<in> S \<Longrightarrow> generate_topology_on S s"
 
-lemma istopology_generate_topology_on:
+lemma%unimportant istopology_generate_topology_on:
   "istopology (generate_topology_on S)"
 unfolding istopology_def by (auto intro: generate_topology_on.intros)
 
 text \<open>The basic property of the topology generated by a set $S$ is that it is the
 smallest topology containing all the elements of $S$:\<close>
 
-lemma generate_topology_on_coarsest:
+lemma%unimportant generate_topology_on_coarsest:
   assumes "istopology T"
           "\<And>s. s \<in> S \<Longrightarrow> T s"
           "generate_topology_on S s0"
@@ -94,21 +94,21 @@
 using assms(3) apply (induct rule: generate_topology_on.induct)
 using assms(1) assms(2) unfolding istopology_def by auto
 
-definition topology_generated_by::"('a set set) \<Rightarrow> ('a topology)"
+definition%unimportant topology_generated_by::"('a set set) \<Rightarrow> ('a topology)"
   where "topology_generated_by S = topology (generate_topology_on S)"
 
-lemma openin_topology_generated_by_iff:
+lemma%unimportant openin_topology_generated_by_iff:
   "openin (topology_generated_by S) s \<longleftrightarrow> generate_topology_on S s"
 using topology_inverse'[OF istopology_generate_topology_on[of S]]
 unfolding topology_generated_by_def by simp
 
-lemma openin_topology_generated_by:
+lemma%unimportant openin_topology_generated_by:
   "openin (topology_generated_by S) s \<Longrightarrow> generate_topology_on S s"
 using openin_topology_generated_by_iff by auto
 
-lemma topology_generated_by_topspace:
+lemma%important topology_generated_by_topspace:
   "topspace (topology_generated_by S) = (\<Union>S)"
-proof
+proof%unimportant
   {
     fix s assume "openin (topology_generated_by S) s"
     then have "generate_topology_on S s" by (rule openin_topology_generated_by)
@@ -123,41 +123,41 @@
     unfolding topspace_def using openin_topology_generated_by_iff by auto
 qed
 
-lemma topology_generated_by_Basis:
+lemma%important topology_generated_by_Basis:
   "s \<in> S \<Longrightarrow> openin (topology_generated_by S) s"
-by (simp only: openin_topology_generated_by_iff, auto simp: generate_topology_on.Basis)
+by%unimportant (simp only: openin_topology_generated_by_iff, auto simp: generate_topology_on.Basis)
 
-subsubsection \<open>Continuity\<close>
+subsubsection%important \<open>Continuity\<close>
 
 text \<open>We will need to deal with continuous maps in terms of topologies and not in terms
 of type classes, as defined below.\<close>
 
-definition continuous_on_topo::"'a topology \<Rightarrow> 'b topology \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> bool"
+definition%important continuous_on_topo::"'a topology \<Rightarrow> 'b topology \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> bool"
   where "continuous_on_topo T1 T2 f = ((\<forall> U. openin T2 U \<longrightarrow> openin T1 (f-`U \<inter> topspace(T1)))
                                       \<and> (f`(topspace T1) \<subseteq> (topspace T2)))"
 
-lemma continuous_on_continuous_on_topo:
+lemma%important continuous_on_continuous_on_topo:
   "continuous_on s f \<longleftrightarrow> continuous_on_topo (subtopology euclidean s) euclidean f"
-unfolding continuous_on_open_invariant openin_open vimage_def continuous_on_topo_def
+unfolding%unimportant continuous_on_open_invariant openin_open vimage_def continuous_on_topo_def
 topspace_euclidean_subtopology open_openin topspace_euclidean by fast
 
-lemma continuous_on_topo_UNIV:
+lemma%unimportant continuous_on_topo_UNIV:
   "continuous_on UNIV f \<longleftrightarrow> continuous_on_topo euclidean euclidean f"
 using continuous_on_continuous_on_topo[of UNIV f] subtopology_UNIV[of euclidean] by auto
 
-lemma continuous_on_topo_open [intro]:
+lemma%important continuous_on_topo_open [intro]:
   "continuous_on_topo T1 T2 f \<Longrightarrow> openin T2 U \<Longrightarrow> openin T1 (f-`U \<inter> topspace(T1))"
-unfolding continuous_on_topo_def by auto
+unfolding%unimportant continuous_on_topo_def by auto
 
-lemma continuous_on_topo_topspace [intro]:
+lemma%unimportant continuous_on_topo_topspace [intro]:
   "continuous_on_topo T1 T2 f \<Longrightarrow> f`(topspace T1) \<subseteq> (topspace T2)"
 unfolding continuous_on_topo_def by auto
 
-lemma continuous_on_generated_topo_iff:
+lemma%important continuous_on_generated_topo_iff:
   "continuous_on_topo T1 (topology_generated_by S) f \<longleftrightarrow>
       ((\<forall>U. U \<in> S \<longrightarrow> openin T1 (f-`U \<inter> topspace(T1))) \<and> (f`(topspace T1) \<subseteq> (\<Union> S)))"
 unfolding continuous_on_topo_def topology_generated_by_topspace
-proof (auto simp add: topology_generated_by_Basis)
+proof%unimportant (auto simp add: topology_generated_by_Basis)
   assume H: "\<forall>U. U \<in> S \<longrightarrow> openin T1 (f -` U \<inter> topspace T1)"
   fix U assume "openin (topology_generated_by S) U"
   then have "generate_topology_on S U" by (rule openin_topology_generated_by)
@@ -179,17 +179,17 @@
   qed (auto simp add: H)
 qed
 
-lemma continuous_on_generated_topo:
+lemma%important continuous_on_generated_topo:
   assumes "\<And>U. U \<in>S \<Longrightarrow> openin T1 (f-`U \<inter> topspace(T1))"
           "f`(topspace T1) \<subseteq> (\<Union> S)"
   shows "continuous_on_topo T1 (topology_generated_by S) f"
-using assms continuous_on_generated_topo_iff by blast
+using%unimportant assms continuous_on_generated_topo_iff by blast
 
-lemma continuous_on_topo_compose:
+lemma%important continuous_on_topo_compose:
   assumes "continuous_on_topo T1 T2 f" "continuous_on_topo T2 T3 g"
   shows "continuous_on_topo T1 T3 (g o f)"
-using assms unfolding continuous_on_topo_def
-proof (auto)
+using%unimportant assms unfolding continuous_on_topo_def
+proof%unimportant (auto)
   fix U :: "'c set"
   assume H: "openin T3 U"
   have "openin T1 (f -` (g -` U \<inter> topspace T2) \<inter> topspace T1)"
@@ -200,13 +200,13 @@
     by simp
 qed (blast)
 
-lemma continuous_on_topo_preimage_topspace [intro]:
+lemma%unimportant continuous_on_topo_preimage_topspace [intro]:
   assumes "continuous_on_topo T1 T2 f"
   shows "f-`(topspace T2) \<inter> topspace T1 = topspace T1"
 using assms unfolding continuous_on_topo_def by auto
 
 
-subsubsection \<open>Pullback topology\<close>
+subsubsection%important \<open>Pullback topology\<close>
 
 text \<open>Pulling back a topology by map gives again a topology. \<open>subtopology\<close> is
 a special case of this notion, pulling back by the identity. We introduce the general notion as
@@ -216,12 +216,12 @@
 text \<open>\verb+pullback_topology A f T+ is the pullback of the topology $T$ by the map $f$ on
 the set $A$.\<close>
 
-definition pullback_topology::"('a set) \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> ('b topology) \<Rightarrow> ('a topology)"
+definition%important pullback_topology::"('a set) \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> ('b topology) \<Rightarrow> ('a topology)"
   where "pullback_topology A f T = topology (\<lambda>S. \<exists>U. openin T U \<and> S = f-`U \<inter> A)"
 
-lemma istopology_pullback_topology:
+lemma%important istopology_pullback_topology:
   "istopology (\<lambda>S. \<exists>U. openin T U \<and> S = f-`U \<inter> A)"
-unfolding istopology_def proof (auto)
+unfolding%unimportant istopology_def proof (auto)
   fix K assume "\<forall>S\<in>K. \<exists>U. openin T U \<and> S = f -` U \<inter> A"
   then have "\<exists>U. \<forall>S\<in>K. openin T (U S) \<and> S = f-`(U S) \<inter> A"
     by (rule bchoice)
@@ -232,19 +232,19 @@
   then show "\<exists>V. openin T V \<and> \<Union>K = f -` V \<inter> A" by auto
 qed
 
-lemma openin_pullback_topology:
+lemma%unimportant openin_pullback_topology:
   "openin (pullback_topology A f T) S \<longleftrightarrow> (\<exists>U. openin T U \<and> S = f-`U \<inter> A)"
 unfolding pullback_topology_def topology_inverse'[OF istopology_pullback_topology] by auto
 
-lemma topspace_pullback_topology:
+lemma%unimportant topspace_pullback_topology:
   "topspace (pullback_topology A f T) = f-`(topspace T) \<inter> A"
 by (auto simp add: topspace_def openin_pullback_topology)
 
-lemma continuous_on_topo_pullback [intro]:
+lemma%important continuous_on_topo_pullback [intro]:
   assumes "continuous_on_topo T1 T2 g"
   shows "continuous_on_topo (pullback_topology A f T1) T2 (g o f)"
 unfolding continuous_on_topo_def
-proof (auto)
+proof%unimportant (auto)
   fix U::"'b set" assume "openin T2 U"
   then have "openin T1 (g-`U \<inter> topspace T1)"
     using assms unfolding continuous_on_topo_def by auto
@@ -264,11 +264,11 @@
     using assms unfolding continuous_on_topo_def by auto
 qed
 
-lemma continuous_on_topo_pullback' [intro]:
+lemma%important continuous_on_topo_pullback' [intro]:
   assumes "continuous_on_topo T1 T2 (f o g)" "topspace T1 \<subseteq> g-`A"
   shows "continuous_on_topo T1 (pullback_topology A f T2) g"
 unfolding continuous_on_topo_def
-proof (auto)
+proof%unimportant (auto)
   fix U assume "openin (pullback_topology A f T2) U"
   then have "\<exists>V. openin T2 V \<and> U = f-`V \<inter> A"
     unfolding openin_pullback_topology by auto
@@ -295,7 +295,7 @@
 qed
 
 
-subsection \<open>The product topology\<close>
+subsection%important \<open>The product topology\<close>
 
 text \<open>We can now define the product topology, as generated by
 the sets which are products of open sets along finitely many coordinates, and the whole
@@ -308,16 +308,16 @@
 So, we use the first formulation, which moreover seems to give rise to more straightforward proofs.
 \<close>
 
-definition product_topology::"('i \<Rightarrow> ('a topology)) \<Rightarrow> ('i set) \<Rightarrow> (('i \<Rightarrow> 'a) topology)"
+definition%important product_topology::"('i \<Rightarrow> ('a topology)) \<Rightarrow> ('i set) \<Rightarrow> (('i \<Rightarrow> 'a) topology)"
   where "product_topology T I =
     topology_generated_by {(\<Pi>\<^sub>E i\<in>I. X i) |X. (\<forall>i. openin (T i) (X i)) \<and> finite {i. X i \<noteq> topspace (T i)}}"
 
 text \<open>The total set of the product topology is the product of the total sets
 along each coordinate.\<close>
 
-lemma product_topology_topspace:
+lemma%important product_topology_topspace:
   "topspace (product_topology T I) = (\<Pi>\<^sub>E i\<in>I. topspace(T i))"
-proof
+proof%unimportant
   show "topspace (product_topology T I) \<subseteq> (\<Pi>\<^sub>E i\<in>I. topspace (T i))"
     unfolding product_topology_def apply (simp only: topology_generated_by_topspace)
     unfolding topspace_def by auto
@@ -327,16 +327,16 @@
     unfolding product_topology_def using PiE_def by (auto simp add: topology_generated_by_topspace)
 qed
 
-lemma product_topology_basis:
+lemma%unimportant product_topology_basis:
   assumes "\<And>i. openin (T i) (X i)" "finite {i. X i \<noteq> topspace (T i)}"
   shows "openin (product_topology T I) (\<Pi>\<^sub>E i\<in>I. X i)"
 unfolding product_topology_def apply (rule topology_generated_by_Basis) using assms by auto
 
-lemma product_topology_open_contains_basis:
+lemma%important product_topology_open_contains_basis:
   assumes "openin (product_topology T I) U"
           "x \<in> U"
   shows "\<exists>X. x \<in> (\<Pi>\<^sub>E i\<in>I. X i) \<and> (\<forall>i. openin (T i) (X i)) \<and> finite {i. X i \<noteq> topspace (T i)} \<and> (\<Pi>\<^sub>E i\<in>I. X i) \<subseteq> U"
-proof -
+proof%unimportant -
   have "generate_topology_on {(\<Pi>\<^sub>E i\<in>I. X i) |X. (\<forall>i. openin (T i) (X i)) \<and> finite {i. X i \<noteq> topspace (T i)}} U"
     using assms unfolding product_topology_def by (intro openin_topology_generated_by) auto
   then have "\<And>x. x\<in>U \<Longrightarrow> \<exists>X. x \<in> (\<Pi>\<^sub>E i\<in>I. X i) \<and> (\<forall>i. openin (T i) (X i)) \<and> finite {i. X i \<noteq> topspace (T i)} \<and> (\<Pi>\<^sub>E i\<in>I. X i) \<subseteq> U"
@@ -377,7 +377,7 @@
 
 text \<open>The basic property of the product topology is the continuity of projections:\<close>
 
-lemma continuous_on_topo_product_coordinates [simp]:
+lemma%unimportant continuous_on_topo_product_coordinates [simp]:
   assumes "i \<in> I"
   shows "continuous_on_topo (product_topology T I) (T i) (\<lambda>x. x i)"
 proof -
@@ -399,12 +399,12 @@
     by (auto simp add: assms product_topology_topspace PiE_iff)
 qed
 
-lemma continuous_on_topo_coordinatewise_then_product [intro]:
+lemma%important continuous_on_topo_coordinatewise_then_product [intro]:
   assumes "\<And>i. i \<in> I \<Longrightarrow> continuous_on_topo T1 (T i) (\<lambda>x. f x i)"
           "\<And>i x. i \<notin> I \<Longrightarrow> x \<in> topspace T1 \<Longrightarrow> f x i = undefined"
   shows "continuous_on_topo T1 (product_topology T I) f"
 unfolding product_topology_def
-proof (rule continuous_on_generated_topo)
+proof%unimportant (rule continuous_on_generated_topo)
   fix U assume "U \<in> {Pi\<^sub>E I X |X. (\<forall>i. openin (T i) (X i)) \<and> finite {i. X i \<noteq> topspace (T i)}}"
   then obtain X where H: "U = Pi\<^sub>E I X" "\<And>i. openin (T i) (X i)" "finite {i. X i \<noteq> topspace (T i)}"
     by blast
@@ -432,7 +432,7 @@
     using assms unfolding continuous_on_topo_def by auto
 qed
 
-lemma continuous_on_topo_product_then_coordinatewise [intro]:
+lemma%unimportant continuous_on_topo_product_then_coordinatewise [intro]:
   assumes "continuous_on_topo T1 (product_topology T I) f"
   shows "\<And>i. i \<in> I \<Longrightarrow> continuous_on_topo T1 (T i) (\<lambda>x. f x i)"
         "\<And>i x. i \<notin> I \<Longrightarrow> x \<in> topspace T1 \<Longrightarrow> f x i = undefined"
@@ -454,7 +454,7 @@
     using \<open>i \<notin> I\<close> by (auto simp add: PiE_iff extensional_def)
 qed
 
-lemma continuous_on_restrict:
+lemma%unimportant continuous_on_restrict:
   assumes "J \<subseteq> I"
   shows "continuous_on_topo (product_topology T I) (product_topology T J) (\<lambda>x. restrict x J)"
 proof (rule continuous_on_topo_coordinatewise_then_product)
@@ -469,15 +469,15 @@
 qed
 
 
-subsubsection \<open>Powers of a single topological space as a topological space, using type classes\<close>
+subsubsection%important \<open>Powers of a single topological space as a topological space, using type classes\<close>
 
 instantiation "fun" :: (type, topological_space) topological_space
 begin
 
-definition open_fun_def:
+definition%important open_fun_def:
   "open U = openin (product_topology (\<lambda>i. euclidean) UNIV) U"
 
-instance proof
+instance proof%unimportant
   have "topspace (product_topology (\<lambda>(i::'a). euclidean::('b topology)) UNIV) = UNIV"
     unfolding product_topology_topspace topspace_euclidean by auto
   then show "open (UNIV::('a \<Rightarrow> 'b) set)"
@@ -486,15 +486,15 @@
 
 end
 
-lemma euclidean_product_topology:
+lemma%unimportant euclidean_product_topology:
   "euclidean = product_topology (\<lambda>i. euclidean::('b::topological_space) topology) UNIV"
 by (metis open_openin topology_eq open_fun_def)
 
-lemma product_topology_basis':
+lemma%important product_topology_basis':
   fixes x::"'i \<Rightarrow> 'a" and U::"'i \<Rightarrow> ('b::topological_space) set"
   assumes "finite I" "\<And>i. i \<in> I \<Longrightarrow> open (U i)"
   shows "open {f. \<forall>i\<in>I. f (x i) \<in> U i}"
-proof -
+proof%unimportant -
   define J where "J = x`I"
   define V where "V = (\<lambda>y. if y \<in> J then \<Inter>{U i|i. i\<in>I \<and> x i = y} else UNIV)"
   define X where "X = (\<lambda>y. if y \<in> J then V y else UNIV)"
@@ -524,34 +524,34 @@
 text \<open>The results proved in the general situation of products of possibly different
 spaces have their counterparts in this simpler setting.\<close>
 
-lemma continuous_on_product_coordinates [simp]:
+lemma%unimportant continuous_on_product_coordinates [simp]:
   "continuous_on UNIV (\<lambda>x. x i::('b::topological_space))"
 unfolding continuous_on_topo_UNIV euclidean_product_topology
 by (rule continuous_on_topo_product_coordinates, simp)
 
-lemma continuous_on_coordinatewise_then_product [intro, continuous_intros]:
+lemma%unimportant continuous_on_coordinatewise_then_product [intro, continuous_intros]:
   assumes "\<And>i. continuous_on UNIV (\<lambda>x. f x i)"
   shows "continuous_on UNIV f"
 using assms unfolding continuous_on_topo_UNIV euclidean_product_topology
 by (rule continuous_on_topo_coordinatewise_then_product, simp)
 
-lemma continuous_on_product_then_coordinatewise:
+lemma%unimportant continuous_on_product_then_coordinatewise:
   assumes "continuous_on UNIV f"
   shows "continuous_on UNIV (\<lambda>x. f x i)"
 using assms unfolding continuous_on_topo_UNIV euclidean_product_topology
 by (rule continuous_on_topo_product_then_coordinatewise(1), simp)
 
-lemma continuous_on_product_coordinatewise_iff:
+lemma%unimportant continuous_on_product_coordinatewise_iff:
   "continuous_on UNIV f \<longleftrightarrow> (\<forall>i. continuous_on UNIV (\<lambda>x. f x i))"
 by (auto intro: continuous_on_product_then_coordinatewise)
 
-subsubsection \<open>Topological countability for product spaces\<close>
+subsubsection%important \<open>Topological countability for product spaces\<close>
 
 text \<open>The next two lemmas are useful to prove first or second countability
 of product spaces, but they have more to do with countability and could
 be put in the corresponding theory.\<close>
 
-lemma countable_nat_product_event_const:
+lemma%unimportant countable_nat_product_event_const:
   fixes F::"'a set" and a::'a
   assumes "a \<in> F" "countable F"
   shows "countable {x::(nat \<Rightarrow> 'a). (\<forall>i. x i \<in> F) \<and> finite {i. x i \<noteq> a}}"
@@ -588,7 +588,7 @@
   then show ?thesis using countable_subset[OF *] by auto
 qed
 
-lemma countable_product_event_const:
+lemma%unimportant countable_product_event_const:
   fixes F::"('a::countable) \<Rightarrow> 'b set" and b::'b
   assumes "\<And>i. countable (F i)"
   shows "countable {f::('a \<Rightarrow> 'b). (\<forall>i. f i \<in> F i) \<and> (finite {i. f i \<noteq> b})}"
@@ -621,7 +621,7 @@
 qed
 
 instance "fun" :: (countable, first_countable_topology) first_countable_topology
-proof
+proof%unimportant
   fix x::"'a \<Rightarrow> 'b"
   have "\<exists>A::('b \<Rightarrow> nat \<Rightarrow> 'b set). \<forall>x. (\<forall>i. x \<in> A x i \<and> open (A x i)) \<and> (\<forall>S. open S \<and> x \<in> S \<longrightarrow> (\<exists>i. A x i \<subseteq> S))"
     apply (rule choice) using first_countable_basis by auto
@@ -693,11 +693,11 @@
     using \<open>countable K\<close> \<open>\<And>k. k \<in> K \<Longrightarrow> x \<in> k\<close> \<open>\<And>k. k \<in> K \<Longrightarrow> open k\<close> Inc by auto
 qed
 
-lemma product_topology_countable_basis:
+lemma%important product_topology_countable_basis:
   shows "\<exists>K::(('a::countable \<Rightarrow> 'b::second_countable_topology) set set).
           topological_basis K \<and> countable K \<and>
           (\<forall>k\<in>K. \<exists>X. (k = Pi\<^sub>E UNIV X) \<and> (\<forall>i. open (X i)) \<and> finite {i. X i \<noteq> UNIV})"
-proof -
+proof%unimportant -
   obtain B::"'b set set" where B: "countable B \<and> topological_basis B"
     using ex_countable_basis by auto
   then have "B \<noteq> {}" by (meson UNIV_I empty_iff open_UNIV topological_basisE)
@@ -776,7 +776,7 @@
   using product_topology_countable_basis topological_basis_imp_subbasis by auto
 
 
-subsection \<open>The strong operator topology on continuous linear operators\<close>
+subsection%important \<open>The strong operator topology on continuous linear operators\<close> (* FIX ME mv*)
 
 text \<open>Let 'a and 'b be two normed real vector spaces. Then the space of linear continuous
 operators from 'a to 'b has a canonical norm, and therefore a canonical corresponding topology
@@ -797,18 +797,18 @@
 defined analogously.
 \<close>
 
-definition strong_operator_topology::"('a::real_normed_vector \<Rightarrow>\<^sub>L'b::real_normed_vector) topology"
+definition%important strong_operator_topology::"('a::real_normed_vector \<Rightarrow>\<^sub>L'b::real_normed_vector) topology"
 where "strong_operator_topology = pullback_topology UNIV blinfun_apply euclidean"
 
-lemma strong_operator_topology_topspace:
+lemma%unimportant strong_operator_topology_topspace:
   "topspace strong_operator_topology = UNIV"
 unfolding strong_operator_topology_def topspace_pullback_topology topspace_euclidean by auto
 
-lemma strong_operator_topology_basis:
+lemma%important strong_operator_topology_basis:
   fixes f::"('a::real_normed_vector \<Rightarrow>\<^sub>L'b::real_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 strong_operator_topology {f. \<forall>i\<in>I. blinfun_apply f (x i) \<in> U i}"
-proof -
+proof%unimportant -
   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. blinfun_apply f (x i) \<in> U i}
@@ -818,16 +818,16 @@
     unfolding strong_operator_topology_def open_openin apply (subst openin_pullback_topology) by auto
 qed
 
-lemma strong_operator_topology_continuous_evaluation:
+lemma%important strong_operator_topology_continuous_evaluation:
   "continuous_on_topo strong_operator_topology euclidean (\<lambda>f. blinfun_apply f x)"
-proof -
+proof%unimportant -
   have "continuous_on_topo strong_operator_topology euclidean ((\<lambda>f. f x) o blinfun_apply)"
     unfolding strong_operator_topology_def apply (rule continuous_on_topo_pullback)
     using continuous_on_topo_UNIV continuous_on_product_coordinates by fastforce
   then show ?thesis unfolding comp_def by simp
 qed
 
-lemma continuous_on_strong_operator_topo_iff_coordinatewise:
+lemma%unimportant continuous_on_strong_operator_topo_iff_coordinatewise:
   "continuous_on_topo T strong_operator_topology f
     \<longleftrightarrow> (\<forall>x. continuous_on_topo T euclidean (\<lambda>y. blinfun_apply (f y) x))"
 proof (auto)
@@ -850,13 +850,13 @@
     by (auto simp add: \<open>continuous_on_topo T euclidean (blinfun_apply o f)\<close>)
 qed
 
-lemma strong_operator_topology_weaker_than_euclidean:
+lemma%important strong_operator_topology_weaker_than_euclidean:
   "continuous_on_topo euclidean strong_operator_topology (\<lambda>f. f)"
-by (subst continuous_on_strong_operator_topo_iff_coordinatewise,
+by%unimportant (subst continuous_on_strong_operator_topo_iff_coordinatewise,
     auto simp add: continuous_on_topo_UNIV[symmetric] linear_continuous_on)
 
 
-subsection \<open>Metrics on product spaces\<close>
+subsection%important \<open>Metrics on product spaces\<close>
 
 
 text \<open>In general, the product topology is not metrizable, unless the index set is countable.
@@ -875,10 +875,10 @@
 instantiation "fun" :: (countable, metric_space) metric_space
 begin
 
-definition dist_fun_def:
+definition%important dist_fun_def:
   "dist x y = (\<Sum>n. (1/2)^n * min (dist (x (from_nat n)) (y (from_nat n))) 1)"
 
-definition uniformity_fun_def:
+definition%important uniformity_fun_def:
   "(uniformity::(('a \<Rightarrow> 'b) \<times> ('a \<Rightarrow> 'b)) filter) = (INF e:{0<..}. principal {(x, y). dist (x::('a\<Rightarrow>'b)) y < e})"
 
 text \<open>Except for the first one, the auxiliary lemmas below are only useful when proving the
@@ -886,9 +886,9 @@
 spaces. It would thus be desirable to hide them once the instance is proved, but I do not know how
 to do this.\<close>
 
-lemma dist_fun_le_dist_first_terms:
+lemma%important dist_fun_le_dist_first_terms:
   "dist x y \<le> 2 * Max {dist (x (from_nat n)) (y (from_nat n))|n. n \<le> N} + (1/2)^N"
-proof -
+proof%unimportant -
   have "(\<Sum>n. (1 / 2) ^ (n+Suc N) * min (dist (x (from_nat (n+Suc N))) (y (from_nat (n+Suc N)))) 1)
           = (\<Sum>n. (1 / 2) ^ (Suc N) * ((1/2) ^ n * min (dist (x (from_nat (n+Suc N))) (y (from_nat (n+Suc N)))) 1))"
     by (rule suminf_cong, simp add: power_add)
@@ -936,7 +936,7 @@
   finally show ?thesis unfolding M_def by simp
 qed
 
-lemma open_fun_contains_ball_aux:
+lemma%unimportant open_fun_contains_ball_aux:
   assumes "open (U::(('a \<Rightarrow> 'b) set))"
           "x \<in> U"
   shows "\<exists>e>0. {y. dist x y < e} \<subseteq> U"
@@ -999,7 +999,7 @@
   then show "\<exists>m>0. {y. dist x y < m} \<subseteq> U" using * by blast
 qed
 
-lemma ball_fun_contains_open_aux:
+lemma%unimportant ball_fun_contains_open_aux:
   fixes x::"('a \<Rightarrow> 'b)" and e::real
   assumes "e>0"
   shows "\<exists>U. open U \<and> x \<in> U \<and> U \<subseteq> {y. dist x y < e}"
@@ -1046,7 +1046,7 @@
   ultimately show ?thesis by auto
 qed
 
-lemma fun_open_ball_aux:
+lemma%unimportant fun_open_ball_aux:
   fixes U::"('a \<Rightarrow> 'b) set"
   shows "open U \<longleftrightarrow> (\<forall>x\<in>U. \<exists>e>0. \<forall>y. dist x y < e \<longrightarrow> y \<in> U)"
 proof (auto)
@@ -1233,7 +1233,7 @@
 by standard
 
 
-subsection \<open>Measurability\<close>
+subsection%important \<open>Measurability\<close> (*FIX ME mv *)
 
 text \<open>There are two natural sigma-algebras on a product space: the borel sigma algebra,
 generated by open sets in the product, and the product sigma algebra, countably generated by
@@ -1251,11 +1251,11 @@
 compare it with the product sigma algebra as explained above.
 \<close>
 
-lemma measurable_product_coordinates [measurable (raw)]:
+lemma%unimportant measurable_product_coordinates [measurable (raw)]:
   "(\<lambda>x. x i) \<in> measurable borel borel"
 by (rule borel_measurable_continuous_on1[OF continuous_on_product_coordinates])
 
-lemma measurable_product_then_coordinatewise:
+lemma%unimportant measurable_product_then_coordinatewise:
   fixes f::"'a \<Rightarrow> 'b \<Rightarrow> ('c::topological_space)"
   assumes [measurable]: "f \<in> borel_measurable M"
   shows "(\<lambda>x. f x i) \<in> borel_measurable M"
@@ -1269,10 +1269,10 @@
 of the product sigma algebra that is more similar to the one we used above for the product
 topology.\<close>
 
-lemma sets_PiM_finite:
+lemma%important sets_PiM_finite:
   "sets (Pi\<^sub>M I M) = sigma_sets (\<Pi>\<^sub>E i\<in>I. space (M i))
         {(\<Pi>\<^sub>E i\<in>I. X i) |X. (\<forall>i. X i \<in> sets (M i)) \<and> finite {i. X i \<noteq> space (M i)}}"
-proof
+proof%unimportant
   have "{(\<Pi>\<^sub>E i\<in>I. X i) |X. (\<forall>i. X i \<in> sets (M i)) \<and> finite {i. X i \<noteq> space (M i)}} \<subseteq> sets (Pi\<^sub>M I M)"
   proof (auto)
     fix X assume H: "\<forall>i. X i \<in> sets (M i)" "finite {i. X i \<noteq> space (M i)}"
@@ -1312,9 +1312,9 @@
     done
 qed
 
-lemma sets_PiM_subset_borel:
+lemma%important sets_PiM_subset_borel:
   "sets (Pi\<^sub>M UNIV (\<lambda>_. borel)) \<subseteq> sets borel"
-proof -
+proof%unimportant -
   have *: "Pi\<^sub>E UNIV X \<in> sets borel" if [measurable]: "\<And>i. X i \<in> sets borel" "finite {i. X i \<noteq> UNIV}" for X::"'a \<Rightarrow> 'b set"
   proof -
     define I where "I = {i. X i \<noteq> UNIV}"
@@ -1331,9 +1331,9 @@
     by (simp add: * sets.sigma_sets_subset')
 qed
 
-lemma sets_PiM_equal_borel:
+lemma%important sets_PiM_equal_borel:
   "sets (Pi\<^sub>M UNIV (\<lambda>i::('a::countable). borel::('b::second_countable_topology measure))) = sets borel"
-proof
+proof%unimportant
   obtain K::"('a \<Rightarrow> 'b) set set" where K: "topological_basis K" "countable K"
             "\<And>k. k \<in> K \<Longrightarrow> \<exists>X. (k = Pi\<^sub>E UNIV X) \<and> (\<forall>i. open (X i)) \<and> finite {i. X i \<noteq> UNIV}"
     using product_topology_countable_basis by fast
@@ -1358,11 +1358,11 @@
     unfolding borel_def by auto
 qed (simp add: sets_PiM_subset_borel)
 
-lemma measurable_coordinatewise_then_product:
+lemma%important measurable_coordinatewise_then_product:
   fixes f::"'a \<Rightarrow> ('b::countable) \<Rightarrow> ('c::second_countable_topology)"
   assumes [measurable]: "\<And>i. (\<lambda>x. f x i) \<in> borel_measurable M"
   shows "f \<in> borel_measurable M"
-proof -
+proof%unimportant -
   have "f \<in> measurable M (Pi\<^sub>M UNIV (\<lambda>_. borel))"
     by (rule measurable_PiM_single', auto simp add: assms)
   then show ?thesis using sets_PiM_equal_borel measurable_cong_sets by blast

--- a/src/HOL/Analysis/Further_Topology.thy	Mon Aug 27 22:58:36 2018 +0200
+++ b/src/HOL/Analysis/Further_Topology.thy	Tue Aug 28 13:28:39 2018 +0100
@@ -1,4 +1,4 @@
-section \<open>Extending Continous Maps, Invariance of Domain, etc\<close>
+section%important \<open>Extending Continous Maps, Invariance of Domain, etc\<close>
 
 text\<open>Ported from HOL Light (moretop.ml) by L C Paulson\<close>
 
@@ -6,15 +6,15 @@
   imports Equivalence_Lebesgue_Henstock_Integration Weierstrass_Theorems Polytope Complex_Transcendental
 begin
 
-subsection\<open>A map from a sphere to a higher dimensional sphere is nullhomotopic\<close>
-
-lemma spheremap_lemma1:
+subsection%important\<open>A map from a sphere to a higher dimensional sphere is nullhomotopic\<close>
+
+lemma%important spheremap_lemma1:
   fixes f :: "'a::euclidean_space \<Rightarrow> 'a::euclidean_space"
   assumes "subspace S" "subspace T" and dimST: "dim S < dim T"
       and "S \<subseteq> T"
       and diff_f: "f differentiable_on sphere 0 1 \<inter> S"
     shows "f ` (sphere 0 1 \<inter> S) \<noteq> sphere 0 1 \<inter> T"
-proof
+proof%unimportant
   assume fim: "f ` (sphere 0 1 \<inter> S) = sphere 0 1 \<inter> T"
   have inS: "\<And>x. \<lbrakk>x \<in> S; x \<noteq> 0\<rbrakk> \<Longrightarrow> (x /\<^sub>R norm x) \<in> S"
     using subspace_mul \<open>subspace S\<close> by blast
@@ -158,14 +158,14 @@
 qed
 
 
-lemma spheremap_lemma2:
+lemma%important spheremap_lemma2:
   fixes f :: "'a::euclidean_space \<Rightarrow> 'a::euclidean_space"
   assumes ST: "subspace S" "subspace T" "dim S < dim T"
       and "S \<subseteq> T"
       and contf: "continuous_on (sphere 0 1 \<inter> S) f"
       and fim: "f ` (sphere 0 1 \<inter> S) \<subseteq> sphere 0 1 \<inter> T"
     shows "\<exists>c. homotopic_with (\<lambda>x. True) (sphere 0 1 \<inter> S) (sphere 0 1 \<inter> T) f (\<lambda>x. c)"
-proof -
+proof%unimportant -
   have [simp]: "\<And>x. \<lbrakk>norm x = 1; x \<in> S\<rbrakk> \<Longrightarrow> norm (f x) = 1"
     using fim by (simp add: image_subset_iff)
   have "compact (sphere 0 1 \<inter> S)"
@@ -252,11 +252,11 @@
 qed
 
 
-lemma spheremap_lemma3:
+lemma%important spheremap_lemma3:
   assumes "bounded S" "convex S" "subspace U" and affSU: "aff_dim S \<le> dim U"
   obtains T where "subspace T" "T \<subseteq> U" "S \<noteq> {} \<Longrightarrow> aff_dim T = aff_dim S"
                   "(rel_frontier S) homeomorphic (sphere 0 1 \<inter> T)"
-proof (cases "S = {}")
+proof%unimportant (cases "S = {}")
   case True
   with \<open>subspace U\<close> subspace_0 show ?thesis
     by (rule_tac T = "{0}" in that) auto
@@ -300,14 +300,14 @@
 qed
 
 
-proposition inessential_spheremap_lowdim_gen:
+proposition%important inessential_spheremap_lowdim_gen:
   fixes f :: "'M::euclidean_space \<Rightarrow> 'a::euclidean_space"
   assumes "convex S" "bounded S" "convex T" "bounded T"
       and affST: "aff_dim S < aff_dim T"
       and contf: "continuous_on (rel_frontier S) f"
       and fim: "f ` (rel_frontier S) \<subseteq> rel_frontier T"
   obtains c where "homotopic_with (\<lambda>z. True) (rel_frontier S) (rel_frontier T) f (\<lambda>x. c)"
-proof (cases "S = {}")
+proof%unimportant (cases "S = {}")
   case True
   then show ?thesis
     by (simp add: that)
@@ -350,7 +350,7 @@
   qed
 qed
 
-lemma inessential_spheremap_lowdim:
+lemma%unimportant inessential_spheremap_lowdim:
   fixes f :: "'M::euclidean_space \<Rightarrow> 'a::euclidean_space"
   assumes
    "DIM('M) < DIM('a)" and f: "continuous_on (sphere a r) f" "f ` (sphere a r) \<subseteq> (sphere b s)"
@@ -377,9 +377,9 @@
 
 
 
-subsection\<open> Some technical lemmas about extending maps from cell complexes\<close>
-
-lemma extending_maps_Union_aux:
+subsection%important\<open> Some technical lemmas about extending maps from cell complexes\<close>
+
+lemma%unimportant extending_maps_Union_aux:
   assumes fin: "finite \<F>"
       and "\<And>S. S \<in> \<F> \<Longrightarrow> closed S"
       and "\<And>S T. \<lbrakk>S \<in> \<F>; T \<in> \<F>; S \<noteq> T\<rbrakk> \<Longrightarrow> S \<inter> T \<subseteq> K"
@@ -410,7 +410,7 @@
     done
 qed
 
-lemma extending_maps_Union:
+lemma%unimportant extending_maps_Union:
   assumes fin: "finite \<F>"
       and "\<And>S. S \<in> \<F> \<Longrightarrow> \<exists>g. continuous_on S g \<and> g ` S \<subseteq> T \<and> (\<forall>x \<in> S \<inter> K. g x = h x)"
       and "\<And>S. S \<in> \<F> \<Longrightarrow> closed S"
@@ -422,14 +422,14 @@
 by (metis K psubsetI)
 
 
-lemma extend_map_lemma:
+lemma%important extend_map_lemma:
   assumes "finite \<F>" "\<G> \<subseteq> \<F>" "convex T" "bounded T"
       and poly: "\<And>X. X \<in> \<F> \<Longrightarrow> polytope X"
       and aff: "\<And>X. X \<in> \<F> - \<G> \<Longrightarrow> aff_dim X < aff_dim T"
       and face: "\<And>S T. \<lbrakk>S \<in> \<F>; T \<in> \<F>\<rbrakk> \<Longrightarrow> (S \<inter> T) face_of S \<and> (S \<inter> T) face_of T"
       and contf: "continuous_on (\<Union>\<G>) f" and fim: "f ` (\<Union>\<G>) \<subseteq> rel_frontier T"
   obtains g where "continuous_on (\<Union>\<F>) g" "g ` (\<Union>\<F>) \<subseteq> rel_frontier T" "\<And>x. x \<in> \<Union>\<G> \<Longrightarrow> g x = f x"
-proof (cases "\<F> - \<G> = {}")
+proof%unimportant (cases "\<F> - \<G> = {}")
   case True
   then have "\<Union>\<F> \<subseteq> \<Union>\<G>"
     by (simp add: Union_mono)
@@ -608,7 +608,7 @@
     using extendf [of i] unfolding eq by (metis that)
 qed
 
-lemma extend_map_lemma_cofinite0:
+lemma%unimportant extend_map_lemma_cofinite0:
   assumes "finite \<F>"
       and "pairwise (\<lambda>S T. S \<inter> T \<subseteq> K) \<F>"
       and "\<And>S. S \<in> \<F> \<Longrightarrow> \<exists>a g. a \<notin> U \<and> continuous_on (S - {a}) g \<and> g ` (S - {a}) \<subseteq> T \<and> (\<forall>x \<in> S \<inter> K. g x = h x)"
@@ -665,7 +665,7 @@
 qed
 
 
-lemma extend_map_lemma_cofinite1:
+lemma%unimportant extend_map_lemma_cofinite1:
 assumes "finite \<F>"
     and \<F>: "\<And>X. X \<in> \<F> \<Longrightarrow> \<exists>a g. a \<notin> U \<and> continuous_on (X - {a}) g \<and> g ` (X - {a}) \<subseteq> T \<and> (\<forall>x \<in> X \<inter> K. g x = h x)"
     and clo: "\<And>X. X \<in> \<F> \<Longrightarrow> closed X"
@@ -695,7 +695,7 @@
 qed
 
 
-lemma extend_map_lemma_cofinite:
+lemma%important extend_map_lemma_cofinite:
   assumes "finite \<F>" "\<G> \<subseteq> \<F>" and T: "convex T" "bounded T"
       and poly: "\<And>X. X \<in> \<F> \<Longrightarrow> polytope X"
       and contf: "continuous_on (\<Union>\<G>) f" and fim: "f ` (\<Union>\<G>) \<subseteq> rel_frontier T"
@@ -704,7 +704,7 @@
   obtains C g where
      "finite C" "disjnt C (\<Union>\<G>)" "card C \<le> card \<F>" "continuous_on (\<Union>\<F> - C) g"
      "g ` (\<Union> \<F> - C) \<subseteq> rel_frontier T" "\<And>x. x \<in> \<Union>\<G> \<Longrightarrow> g x = f x"
-proof -
+proof%unimportant -
   define \<H> where "\<H> \<equiv> \<G> \<union> {D. \<exists>C \<in> \<F> - \<G>. D face_of C \<and> aff_dim D < aff_dim T}"
   have "finite \<G>"
     using assms finite_subset by blast
@@ -869,7 +869,7 @@
 qed
 
 text\<open>The next two proofs are similar\<close>
-theorem extend_map_cell_complex_to_sphere:
+theorem%important extend_map_cell_complex_to_sphere:
   assumes "finite \<F>" and S: "S \<subseteq> \<Union>\<F>" "closed S" and T: "convex T" "bounded T"
       and poly: "\<And>X. X \<in> \<F> \<Longrightarrow> polytope X"
       and aff: "\<And>X. X \<in> \<F> \<Longrightarrow> aff_dim X < aff_dim T"
@@ -877,7 +877,7 @@
       and contf: "continuous_on S f" and fim: "f ` S \<subseteq> rel_frontier T"
   obtains g where "continuous_on (\<Union>\<F>) g"
      "g ` (\<Union>\<F>) \<subseteq> rel_frontier T" "\<And>x. x \<in> S \<Longrightarrow> g x = f x"
-proof -
+proof%unimportant -
   obtain V g where "S \<subseteq> V" "open V" "continuous_on V g" and gim: "g ` V \<subseteq> rel_frontier T" and gf: "\<And>x. x \<in> S \<Longrightarrow> g x = f x"
     using neighbourhood_extension_into_ANR [OF contf fim _ \<open>closed S\<close>] ANR_rel_frontier_convex T by blast
   have "compact S"
@@ -924,7 +924,7 @@
 qed
 
 
-theorem extend_map_cell_complex_to_sphere_cofinite:
+theorem%important extend_map_cell_complex_to_sphere_cofinite:
   assumes "finite \<F>" and S: "S \<subseteq> \<Union>\<F>" "closed S" and T: "convex T" "bounded T"
       and poly: "\<And>X. X \<in> \<F> \<Longrightarrow> polytope X"
       and aff: "\<And>X. X \<in> \<F> \<Longrightarrow> aff_dim X \<le> aff_dim T"
@@ -932,7 +932,7 @@
       and contf: "continuous_on S f" and fim: "f ` S \<subseteq> rel_frontier T"
   obtains C g where "finite C" "disjnt C S" "continuous_on (\<Union>\<F> - C) g"
      "g ` (\<Union>\<F> - C) \<subseteq> rel_frontier T" "\<And>x. x \<in> S \<Longrightarrow> g x = f x"
-proof -
+proof%unimportant -
   obtain V g where "S \<subseteq> V" "open V" "continuous_on V g" and gim: "g ` V \<subseteq> rel_frontier T" and gf: "\<And>x. x \<in> S \<Longrightarrow> g x = f x"
     using neighbourhood_extension_into_ANR [OF contf fim _ \<open>closed S\<close>] ANR_rel_frontier_convex T by blast
   have "compact S"
@@ -991,12 +991,12 @@
 
 
 
-subsection\<open> Special cases and corollaries involving spheres\<close>
-
-lemma disjnt_Diff1: "X \<subseteq> Y' \<Longrightarrow> disjnt (X - Y) (X' - Y')"
+subsection%important\<open> Special cases and corollaries involving spheres\<close>
+
+lemma%unimportant disjnt_Diff1: "X \<subseteq> Y' \<Longrightarrow> disjnt (X - Y) (X' - Y')"
   by (auto simp: disjnt_def)
 
-proposition extend_map_affine_to_sphere_cofinite_simple:
+proposition%important extend_map_affine_to_sphere_cofinite_simple:
   fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
   assumes "compact S" "convex U" "bounded U"
       and aff: "aff_dim T \<le> aff_dim U"
@@ -1005,7 +1005,7 @@
  obtains K g where "finite K" "K \<subseteq> T" "disjnt K S" "continuous_on (T - K) g"
                    "g ` (T - K) \<subseteq> rel_frontier U"
                    "\<And>x. x \<in> S \<Longrightarrow> g x = f x"
-proof -
+proof%unimportant -
   have "\<exists>K g. finite K \<and> disjnt K S \<and> continuous_on (T - K) g \<and>
               g ` (T - K) \<subseteq> rel_frontier U \<and> (\<forall>x \<in> S. g x = f x)"
        if "affine T" "S \<subseteq> T" and aff: "aff_dim T \<le> aff_dim U"  for T
@@ -1140,18 +1140,18 @@
     by (rule_tac K="K \<inter> T" and g=g in that) (auto simp: disjnt_iff Diff_Int contg)
 qed
 
-subsection\<open>Extending maps to spheres\<close>
+subsection%important\<open>Extending maps to spheres\<close>
 
 (*Up to extend_map_affine_to_sphere_cofinite_gen*)
 
-lemma extend_map_affine_to_sphere1:
+lemma%important extend_map_affine_to_sphere1:
   fixes f :: "'a::euclidean_space \<Rightarrow> 'b::topological_space"
   assumes "finite K" "affine U" and contf: "continuous_on (U - K) f"
       and fim: "f ` (U - K) \<subseteq> T"
       and comps: "\<And>C. \<lbrakk>C \<in> components(U - S); C \<inter> K \<noteq> {}\<rbrakk> \<Longrightarrow> C \<inter> L \<noteq> {}"
       and clo: "closedin (subtopology euclidean U) S" and K: "disjnt K S" "K \<subseteq> U"
   obtains g where "continuous_on (U - L) g" "g ` (U - L) \<subseteq> T" "\<And>x. x \<in> S \<Longrightarrow> g x = f x"
-proof (cases "K = {}")
+proof%unimportant (cases "K = {}")
   case True
   then show ?thesis
     by (metis Diff_empty Diff_subset contf fim continuous_on_subset image_subsetI rev_image_eqI subset_iff that)
@@ -1436,7 +1436,7 @@
 qed
 
 
-lemma extend_map_affine_to_sphere2:
+lemma%important extend_map_affine_to_sphere2:
   fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
   assumes "compact S" "convex U" "bounded U" "affine T" "S \<subseteq> T"
       and affTU: "aff_dim T \<le> aff_dim U"
@@ -1446,7 +1446,7 @@
     obtains K g where "finite K" "K \<subseteq> L" "K \<subseteq> T" "disjnt K S"
                       "continuous_on (T - K) g" "g ` (T - K) \<subseteq> rel_frontier U"
                       "\<And>x. x \<in> S \<Longrightarrow> g x = f x"
-proof -
+proof%unimportant -
   obtain K g where K: "finite K" "K \<subseteq> T" "disjnt K S"
                and contg: "continuous_on (T - K) g"
                and gim: "g ` (T - K) \<subseteq> rel_frontier U"
@@ -1490,7 +1490,7 @@
 qed
 
 
-proposition extend_map_affine_to_sphere_cofinite_gen:
+proposition%important extend_map_affine_to_sphere_cofinite_gen:
   fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
   assumes SUT: "compact S" "convex U" "bounded U" "affine T" "S \<subseteq> T"
       and aff: "aff_dim T \<le> aff_dim U"
@@ -1500,7 +1500,7 @@
  obtains K g where "finite K" "K \<subseteq> L" "K \<subseteq> T" "disjnt K S" "continuous_on (T - K) g"
                    "g ` (T - K) \<subseteq> rel_frontier U"
                    "\<And>x. x \<in> S \<Longrightarrow> g x = f x"
-proof (cases "S = {}")
+proof%unimportant (cases "S = {}")
   case True
   show ?thesis
   proof (cases "rel_frontier U = {}")
@@ -1645,7 +1645,7 @@
 qed
 
 
-corollary extend_map_affine_to_sphere_cofinite:
+corollary%important extend_map_affine_to_sphere_cofinite:
   fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
   assumes SUT: "compact S" "affine T" "S \<subseteq> T"
       and aff: "aff_dim T \<le> DIM('b)" and "0 \<le> r"
@@ -1654,7 +1654,7 @@
       and dis: "\<And>C. \<lbrakk>C \<in> components(T - S); bounded C\<rbrakk> \<Longrightarrow> C \<inter> L \<noteq> {}"
   obtains K g where "finite K" "K \<subseteq> L" "K \<subseteq> T" "disjnt K S" "continuous_on (T - K) g"
                     "g ` (T - K) \<subseteq> sphere a r" "\<And>x. x \<in> S \<Longrightarrow> g x = f x"
-proof (cases "r = 0")
+proof%unimportant (cases "r = 0")
   case True
   with fim show ?thesis
     by (rule_tac K="{}" and g = "\<lambda>x. a" in that) (auto simp: continuous_on_const)
@@ -1670,7 +1670,7 @@
     done
 qed
 
-corollary extend_map_UNIV_to_sphere_cofinite:
+corollary%important extend_map_UNIV_to_sphere_cofinite:
   fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
   assumes aff: "DIM('a) \<le> DIM('b)" and "0 \<le> r"
       and SUT: "compact S"
@@ -1684,7 +1684,7 @@
  apply (auto simp: assms that Compl_eq_Diff_UNIV [symmetric])
 done
 
-corollary extend_map_UNIV_to_sphere_no_bounded_component:
+corollary%important extend_map_UNIV_to_sphere_no_bounded_component:
   fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
   assumes aff: "DIM('a) \<le> DIM('b)" and "0 \<le> r"
       and SUT: "compact S"
@@ -1696,14 +1696,14 @@
    apply (auto simp: that dest: dis)
 done
 
-theorem Borsuk_separation_theorem_gen:
+theorem%important Borsuk_separation_theorem_gen:
   fixes S :: "'a::euclidean_space set"
   assumes "compact S"
     shows "(\<forall>c \<in> components(- S). ~bounded c) \<longleftrightarrow>
            (\<forall>f. continuous_on S f \<and> f ` S \<subseteq> sphere (0::'a) 1
                 \<longrightarrow> (\<exists>c. homotopic_with (\<lambda>x. True) S (sphere 0 1) f (\<lambda>x. c)))"
        (is "?lhs = ?rhs")
-proof
+proof%unimportant
   assume L [rule_format]: ?lhs
   show ?rhs
   proof clarify
@@ -1734,14 +1734,14 @@
 qed
 
 
-corollary Borsuk_separation_theorem:
+corollary%important Borsuk_separation_theorem:
   fixes S :: "'a::euclidean_space set"
   assumes "compact S" and 2: "2 \<le> DIM('a)"
     shows "connected(- S) \<longleftrightarrow>
            (\<forall>f. continuous_on S f \<and> f ` S \<subseteq> sphere (0::'a) 1
                 \<longrightarrow> (\<exists>c. homotopic_with (\<lambda>x. True) S (sphere 0 1) f (\<lambda>x. c)))"
        (is "?lhs = ?rhs")
-proof
+proof%unimportant
   assume L: ?lhs
   show ?rhs
   proof (cases "S = {}")
@@ -1764,7 +1764,7 @@
 qed
 
 
-lemma homotopy_eqv_separation:
+lemma%unimportant homotopy_eqv_separation:
   fixes S :: "'a::euclidean_space set" and T :: "'a set"
   assumes "S homotopy_eqv T" and "compact S" and "compact T"
   shows "connected(- S) \<longleftrightarrow> connected(- T)"
@@ -1783,11 +1783,11 @@
   qed
 qed
 
-lemma Jordan_Brouwer_separation:
+lemma%important Jordan_Brouwer_separation:
   fixes S :: "'a::euclidean_space set" and a::'a
   assumes hom: "S homeomorphic sphere a r" and "0 < r"
     shows "\<not> connected(- S)"
-proof -
+proof%unimportant -
   have "- sphere a r \<inter> ball a r \<noteq> {}"
     using \<open>0 < r\<close> by (simp add: Int_absorb1 subset_eq)
   moreover
@@ -1817,11 +1817,11 @@
 qed
 
 
-lemma Jordan_Brouwer_frontier:
+lemma%important Jordan_Brouwer_frontier:
   fixes S :: "'a::euclidean_space set" and a::'a
   assumes S: "S homeomorphic sphere a r" and T: "T \<in> components(- S)" and 2: "2 \<le> DIM('a)"
     shows "frontier T = S"
-proof (cases r rule: linorder_cases)
+proof%unimportant (cases r rule: linorder_cases)
   assume "r < 0"
   with S T show ?thesis by auto
 next
@@ -1866,11 +1866,11 @@
   qed (rule T)
 qed
 
-lemma Jordan_Brouwer_nonseparation:
+lemma%important Jordan_Brouwer_nonseparation:
   fixes S :: "'a::euclidean_space set" and a::'a
   assumes S: "S homeomorphic sphere a r" and "T \<subset> S" and 2: "2 \<le> DIM('a)"
     shows "connected(- T)"
-proof -
+proof%unimportant -
   have *: "connected(C \<union> (S - T))" if "C \<in> components(- S)" for C
   proof (rule connected_intermediate_closure)
     show "connected C"
@@ -1892,9 +1892,9 @@
     done
 qed
 
-subsection\<open> Invariance of domain and corollaries\<close>
-
-lemma invariance_of_domain_ball:
+subsection%important\<open> Invariance of domain and corollaries\<close>
+
+lemma%unimportant invariance_of_domain_ball:
   fixes f :: "'a \<Rightarrow> 'a::euclidean_space"
   assumes contf: "continuous_on (cball a r) f" and "0 < r"
      and inj: "inj_on f (cball a r)"
@@ -1982,12 +1982,12 @@
 
 
 text\<open>Proved by L. E. J. Brouwer (1912)\<close>
-theorem invariance_of_domain:
+theorem%important invariance_of_domain:
   fixes f :: "'a \<Rightarrow> 'a::euclidean_space"
   assumes "continuous_on S f" "open S" "inj_on f S"
     shows "open(f ` S)"
   unfolding open_subopen [of "f`S"]
-proof clarify
+proof%unimportant clarify
   fix a
   assume "a \<in> S"
   obtain \<delta> where "\<delta> > 0" and \<delta>: "cball a \<delta> \<subseteq> S"
@@ -2003,7 +2003,7 @@
   qed
 qed
 
-lemma inv_of_domain_ss0:
+lemma%unimportant inv_of_domain_ss0:
   fixes f :: "'a \<Rightarrow> 'a::euclidean_space"
   assumes contf: "continuous_on U f" and injf: "inj_on f U" and fim: "f ` U \<subseteq> S"
       and "subspace S" and dimS: "dim S = DIM('b::euclidean_space)"
@@ -2049,7 +2049,7 @@
     by (metis (no_types, hide_lams) homeomorphism_imp_open_map homhk image_comp open_openin subtopology_UNIV)
 qed
 
-lemma inv_of_domain_ss1:
+lemma%unimportant inv_of_domain_ss1:
   fixes f :: "'a \<Rightarrow> 'a::euclidean_space"
   assumes contf: "continuous_on U f" and injf: "inj_on f U" and fim: "f ` U \<subseteq> S"
       and "subspace S"
@@ -2090,14 +2090,14 @@
 qed
 
 
-corollary invariance_of_domain_subspaces:
+corollary%important invariance_of_domain_subspaces:
   fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
   assumes ope: "openin (subtopology euclidean U) S"
       and "subspace U" "subspace V" and VU: "dim V \<le> dim U"
       and contf: "continuous_on S f" and fim: "f ` S \<subseteq> V"
       and injf: "inj_on f S"
     shows "openin (subtopology euclidean V) (f ` S)"
-proof -
+proof%unimportant -
   obtain V' where "subspace V'" "V' \<subseteq> U" "dim V' = dim V"
     using choose_subspace_of_subspace [OF VU]
     by (metis span_eq_iff \<open>subspace U\<close>)
@@ -2134,14 +2134,14 @@
   qed
 qed
 
-corollary invariance_of_dimension_subspaces:
+corollary%important invariance_of_dimension_subspaces:
   fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
   assumes ope: "openin (subtopology euclidean U) S"
       and "subspace U" "subspace V"
       and contf: "continuous_on S f" and fim: "f ` S \<subseteq> V"
       and injf: "inj_on f S" and "S \<noteq> {}"
     shows "dim U \<le> dim V"
-proof -
+proof%unimportant -
   have "False" if "dim V < dim U"
   proof -
     obtain T where "subspace T" "T \<subseteq> U" "dim T = dim V"
@@ -2169,14 +2169,14 @@
     using not_less by blast
 qed
 
-corollary invariance_of_domain_affine_sets:
+corollary%important invariance_of_domain_affine_sets:
   fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
   assumes ope: "openin (subtopology euclidean U) S"
       and aff: "affine U" "affine V" "aff_dim V \<le> aff_dim U"
       and contf: "continuous_on S f" and fim: "f ` S \<subseteq> V"
       and injf: "inj_on f S"
     shows "openin (subtopology euclidean V) (f ` S)"
-proof (cases "S = {}")
+proof%unimportant (cases "S = {}")
   case True
   then show ?thesis by auto
 next
@@ -2204,14 +2204,14 @@
     by (metis (no_types, lifting) homeomorphism_imp_open_map homeomorphism_translation image_comp translation_galois)
 qed
 
-corollary invariance_of_dimension_affine_sets:
+corollary%important invariance_of_dimension_affine_sets:
   fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
   assumes ope: "openin (subtopology euclidean U) S"
       and aff: "affine U" "affine V"
       and contf: "continuous_on S f" and fim: "f ` S \<subseteq> V"
       and injf: "inj_on f S" and "S \<noteq> {}"
     shows "aff_dim U \<le> aff_dim V"
-proof -
+proof%unimportant -
   obtain a b where "a \<in> S" "a \<in> U" "b \<in> V"
     using \<open>S \<noteq> {}\<close> fim ope openin_contains_cball by fastforce
   have "dim ((+) (- a) ` U) \<le> dim ((+) (- b) ` V)"
@@ -2233,25 +2233,25 @@
     by (metis \<open>a \<in> U\<close> \<open>b \<in> V\<close> aff_dim_eq_dim affine_hull_eq aff of_nat_le_iff)
 qed
 
-corollary invariance_of_dimension:
+corollary%important invariance_of_dimension:
   fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
   assumes contf: "continuous_on S f" and "open S"
       and injf: "inj_on f S" and "S \<noteq> {}"
     shows "DIM('a) \<le> DIM('b)"
-  using invariance_of_dimension_subspaces [of UNIV S UNIV f] assms
+  using%unimportant invariance_of_dimension_subspaces [of UNIV S UNIV f] assms
   by auto
 
 
-corollary continuous_injective_image_subspace_dim_le:
+corollary%important continuous_injective_image_subspace_dim_le:
   fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
   assumes "subspace S" "subspace T"
       and contf: "continuous_on S f" and fim: "f ` S \<subseteq> T"
       and injf: "inj_on f S"
     shows "dim S \<le> dim T"
   apply (rule invariance_of_dimension_subspaces [of S S _ f])
-  using assms by (auto simp: subspace_affine)
-
-lemma invariance_of_dimension_convex_domain:
+  using%unimportant assms by (auto simp: subspace_affine)
+
+lemma%unimportant invariance_of_dimension_convex_domain:
   fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
   assumes "convex S"
       and contf: "continuous_on S f" and fim: "f ` S \<subseteq> affine hull T"
@@ -2280,7 +2280,7 @@
 qed
 
 
-lemma homeomorphic_convex_sets_le:
+lemma%unimportant homeomorphic_convex_sets_le:
   assumes "convex S" "S homeomorphic T"
   shows "aff_dim S \<le> aff_dim T"
 proof -
@@ -2297,23 +2297,23 @@
   qed
 qed
 
-lemma homeomorphic_convex_sets:
+lemma%unimportant homeomorphic_convex_sets:
   assumes "convex S" "convex T" "S homeomorphic T"
   shows "aff_dim S = aff_dim T"
   by (meson assms dual_order.antisym homeomorphic_convex_sets_le homeomorphic_sym)
 
-lemma homeomorphic_convex_compact_sets_eq:
+lemma%unimportant homeomorphic_convex_compact_sets_eq:
   assumes "convex S" "compact S" "convex T" "compact T"
   shows "S homeomorphic T \<longleftrightarrow> aff_dim S = aff_dim T"
   by (meson assms homeomorphic_convex_compact_sets homeomorphic_convex_sets)
 
-lemma invariance_of_domain_gen:
+lemma%unimportant invariance_of_domain_gen:
   fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
   assumes "open S" "continuous_on S f" "inj_on f S" "DIM('b) \<le> DIM('a)"
     shows "open(f ` S)"
   using invariance_of_domain_subspaces [of UNIV S UNIV f] assms by auto
 
-lemma injective_into_1d_imp_open_map_UNIV:
+lemma%unimportant injective_into_1d_imp_open_map_UNIV:
   fixes f :: "'a::euclidean_space \<Rightarrow> real"
   assumes "open T" "continuous_on S f" "inj_on f S" "T \<subseteq> S"
     shows "open (f ` T)"
@@ -2321,7 +2321,7 @@
   using assms apply (auto simp: elim: continuous_on_subset subset_inj_on)
   done
 
-lemma continuous_on_inverse_open:
+lemma%unimportant continuous_on_inverse_open:
   fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
   assumes "open S" "continuous_on S f" "DIM('b) \<le> DIM('a)" and gf: "\<And>x. x \<in> S \<Longrightarrow> g(f x) = x"
     shows "continuous_on (f ` S) g"
@@ -2340,7 +2340,7 @@
     by (metis \<open>open T\<close> continuous_on_subset inj_onI inj_on_subset invariance_of_domain_gen openin_open openin_open_eq)
 qed
 
-lemma invariance_of_domain_homeomorphism:
+lemma%unimportant invariance_of_domain_homeomorphism:
   fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
   assumes "open S" "continuous_on S f" "DIM('b) \<le> DIM('a)" "inj_on f S"
   obtains g where "homeomorphism S (f ` S) f g"
@@ -2349,14 +2349,14 @@
     by (simp add: assms continuous_on_inverse_open homeomorphism_def)
 qed
 
-corollary invariance_of_domain_homeomorphic:
+corollary%important invariance_of_domain_homeomorphic:
   fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
   assumes "open S" "continuous_on S f" "DIM('b) \<le> DIM('a)" "inj_on f S"
   shows "S homeomorphic (f ` S)"
-  using invariance_of_domain_homeomorphism [OF assms]
-  by (meson homeomorphic_def)
-
-lemma continuous_image_subset_interior:
+  using%unimportant invariance_of_domain_homeomorphism [OF assms]
+  by%unimportant (meson homeomorphic_def)
+
+lemma%unimportant continuous_image_subset_interior:
   fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
   assumes "continuous_on S f" "inj_on f S" "DIM('b) \<le> DIM('a)"
   shows "f ` (interior S) \<subseteq> interior(f ` S)"
@@ -2367,13 +2367,13 @@
      apply (auto simp: subset_inj_on interior_subset continuous_on_subset)
   done
 
-lemma homeomorphic_interiors_same_dimension:
+lemma%important homeomorphic_interiors_same_dimension:
   fixes S :: "'a::euclidean_space set" and T :: "'b::euclidean_space set"
   assumes "S homeomorphic T" and dimeq: "DIM('a) = DIM('b)"
   shows "(interior S) homeomorphic (interior T)"
   using assms [unfolded homeomorphic_minimal]
   unfolding homeomorphic_def
-proof (clarify elim!: ex_forward)
+proof%unimportant (clarify elim!: ex_forward)
   fix f g
   assume S: "\<forall>x\<in>S. f x \<in> T \<and> g (f x) = x" and T: "\<forall>y\<in>T. g y \<in> S \<and> f (g y) = y"
      and contf: "continuous_on S f" and contg: "continuous_on T g"
@@ -2417,7 +2417,7 @@
   qed
 qed
 
-lemma homeomorphic_open_imp_same_dimension:
+lemma%unimportant homeomorphic_open_imp_same_dimension:
   fixes S :: "'a::euclidean_space set" and T :: "'b::euclidean_space set"
   assumes "S homeomorphic T" "open S" "S \<noteq> {}" "open T" "T \<noteq> {}"
   shows "DIM('a) = DIM('b)"
@@ -2426,7 +2426,7 @@
     apply (rule order_antisym; metis inj_onI invariance_of_dimension)
     done
 
-lemma homeomorphic_interiors:
+lemma%unimportant homeomorphic_interiors:
   fixes S :: "'a::euclidean_space set" and T :: "'b::euclidean_space set"
   assumes "S homeomorphic T" "interior S = {} \<longleftrightarrow> interior T = {}"
     shows "(interior S) homeomorphic (interior T)"
@@ -2444,7 +2444,7 @@
     by (rule homeomorphic_interiors_same_dimension [OF \<open>S homeomorphic T\<close>])
 qed
 
-lemma homeomorphic_frontiers_same_dimension:
+lemma%unimportant homeomorphic_frontiers_same_dimension:
   fixes S :: "'a::euclidean_space set" and T :: "'b::euclidean_space set"
   assumes "S homeomorphic T" "closed S" "closed T" and dimeq: "DIM('a) = DIM('b)"
   shows "(frontier S) homeomorphic (frontier T)"
@@ -2500,7 +2500,7 @@
   qed
 qed
 
-lemma homeomorphic_frontiers:
+lemma%unimportant homeomorphic_frontiers:
   fixes S :: "'a::euclidean_space set" and T :: "'b::euclidean_space set"
   assumes "S homeomorphic T" "closed S" "closed T"
           "interior S = {} \<longleftrightarrow> interior T = {}"
@@ -2517,7 +2517,7 @@
     using False assms homeomorphic_interiors homeomorphic_open_imp_same_dimension by blast
 qed
 
-lemma continuous_image_subset_rel_interior:
+lemma%unimportant continuous_image_subset_rel_interior:
   fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
   assumes contf: "continuous_on S f" and injf: "inj_on f S" and fim: "f ` S \<subseteq> T"
       and TS: "aff_dim T \<le> aff_dim S"
@@ -2540,7 +2540,7 @@
   qed auto
 qed
 
-lemma homeomorphic_rel_interiors_same_dimension:
+lemma%unimportant homeomorphic_rel_interiors_same_dimension:
   fixes S :: "'a::euclidean_space set" and T :: "'b::euclidean_space set"
   assumes "S homeomorphic T" and aff: "aff_dim S = aff_dim T"
   shows "(rel_interior S) homeomorphic (rel_interior T)"
@@ -2592,11 +2592,11 @@
   qed
 qed
 
-lemma homeomorphic_rel_interiors:
+lemma%important homeomorphic_rel_interiors:
   fixes S :: "'a::euclidean_space set" and T :: "'b::euclidean_space set"
   assumes "S homeomorphic T" "rel_interior S = {} \<longleftrightarrow> rel_interior T = {}"
     shows "(rel_interior S) homeomorphic (rel_interior T)"
-proof (cases "rel_interior T = {}")
+proof%unimportant (cases "rel_interior T = {}")
   case True
   with assms show ?thesis by auto
 next
@@ -2625,7 +2625,7 @@
 qed
 
 
-lemma homeomorphic_rel_boundaries_same_dimension:
+lemma%unimportant homeomorphic_rel_boundaries_same_dimension:
   fixes S :: "'a::euclidean_space set" and T :: "'b::euclidean_space set"
   assumes "S homeomorphic T" and aff: "aff_dim S = aff_dim T"
   shows "(S - rel_interior S) homeomorphic (T - rel_interior T)"
@@ -2659,7 +2659,7 @@
   qed
 qed
 
-lemma homeomorphic_rel_boundaries:
+lemma%unimportant homeomorphic_rel_boundaries:
   fixes S :: "'a::euclidean_space set" and T :: "'b::euclidean_space set"
   assumes "S homeomorphic T" "rel_interior S = {} \<longleftrightarrow> rel_interior T = {}"
     shows "(S - rel_interior S) homeomorphic (T - rel_interior T)"
@@ -2691,11 +2691,11 @@
     by (rule homeomorphic_rel_boundaries_same_dimension [OF \<open>S homeomorphic T\<close>])
 qed
 
-proposition uniformly_continuous_homeomorphism_UNIV_trivial:
+proposition%important uniformly_continuous_homeomorphism_UNIV_trivial:
   fixes f :: "'a::euclidean_space \<Rightarrow> 'a"
   assumes contf: "uniformly_continuous_on S f" and hom: "homeomorphism S UNIV f g"
   shows "S = UNIV"
-proof (cases "S = {}")
+proof%unimportant (cases "S = {}")
   case True
   then show ?thesis
     by (metis UNIV_I hom empty_iff homeomorphism_def image_eqI)
@@ -2737,9 +2737,9 @@
     using clopen [of S] False  by simp
 qed
 
-subsection\<open>Dimension-based conditions for various homeomorphisms\<close>
-
-lemma homeomorphic_subspaces_eq:
+subsection%important\<open>Dimension-based conditions for various homeomorphisms\<close>
+
+lemma%unimportant homeomorphic_subspaces_eq:
   fixes S :: "'a::euclidean_space set" and T :: "'b::euclidean_space set"
   assumes "subspace S" "subspace T"
   shows "S homeomorphic T \<longleftrightarrow> dim S = dim T"
@@ -2760,7 +2760,7 @@
     by (simp add: assms homeomorphic_subspaces)
 qed
 
-lemma homeomorphic_affine_sets_eq:
+lemma%unimportant homeomorphic_affine_sets_eq:
   fixes S :: "'a::euclidean_space set" and T :: "'b::euclidean_space set"
   assumes "affine S" "affine T"
   shows "S homeomorphic T \<longleftrightarrow> aff_dim S = aff_dim T"
@@ -2778,19 +2778,19 @@
     by (metis affine_imp_convex assms homeomorphic_affine_sets homeomorphic_convex_sets)
 qed
 
-lemma homeomorphic_hyperplanes_eq:
+lemma%unimportant homeomorphic_hyperplanes_eq:
   fixes a :: "'a::euclidean_space" and c :: "'b::euclidean_space"
   assumes "a \<noteq> 0" "c \<noteq> 0"
   shows "({x. a \<bullet> x = b} homeomorphic {x. c \<bullet> x = d} \<longleftrightarrow> DIM('a) = DIM('b))"
   apply (auto simp: homeomorphic_affine_sets_eq affine_hyperplane assms)
   by (metis DIM_positive Suc_pred)
 
-lemma homeomorphic_UNIV_UNIV:
+lemma%unimportant homeomorphic_UNIV_UNIV:
   shows "(UNIV::'a set) homeomorphic (UNIV::'b set) \<longleftrightarrow>
     DIM('a::euclidean_space) = DIM('b::euclidean_space)"
   by (simp add: homeomorphic_subspaces_eq)
 
-lemma simply_connected_sphere_gen:
+lemma%unimportant simply_connected_sphere_gen:
    assumes "convex S" "bounded S" and 3: "3 \<le> aff_dim S"
    shows "simply_connected(rel_frontier S)"
 proof -
@@ -2814,16 +2814,16 @@
   qed
 qed
 
-subsection\<open>more invariance of domain\<close>
-
-proposition invariance_of_domain_sphere_affine_set_gen:
+subsection%important\<open>more invariance of domain\<close>
+
+proposition%important invariance_of_domain_sphere_affine_set_gen:
   fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
   assumes contf: "continuous_on S f" and injf: "inj_on f S" and fim: "f ` S \<subseteq> T"
       and U: "bounded U" "convex U"
       and "affine T" and affTU: "aff_dim T < aff_dim U"
       and ope: "openin (subtopology euclidean (rel_frontier U)) S"
    shows "openin (subtopology euclidean T) (f ` S)"
-proof (cases "rel_frontier U = {}")
+proof%unimportant (cases "rel_frontier U = {}")
   case True
   then show ?thesis
     using ope openin_subset by force
@@ -2922,7 +2922,7 @@
 qed
 
 
-lemma invariance_of_domain_sphere_affine_set:
+lemma%unimportant invariance_of_domain_sphere_affine_set:
   fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
   assumes contf: "continuous_on S f" and injf: "inj_on f S" and fim: "f ` S \<subseteq> T"
       and "r \<noteq> 0" "affine T" and affTU: "aff_dim T < DIM('a)"
@@ -2943,7 +2943,7 @@
   qed
 qed
 
-lemma no_embedding_sphere_lowdim:
+lemma%unimportant no_embedding_sphere_lowdim:
   fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
   assumes contf: "continuous_on (sphere a r) f" and injf: "inj_on f (sphere a r)" and "r > 0"
    shows "DIM('a) \<le> DIM('b)"
@@ -2964,7 +2964,7 @@
     using not_less by blast
 qed
 
-lemma simply_connected_sphere:
+lemma%unimportant simply_connected_sphere:
   fixes a :: "'a::euclidean_space"
   assumes "3 \<le> DIM('a)"
     shows "simply_connected(sphere a r)"
@@ -2981,7 +2981,7 @@
     by (simp add: aff_dim_cball)
 qed
 
-lemma simply_connected_sphere_eq:
+lemma%unimportant simply_connected_sphere_eq:
   fixes a :: "'a::euclidean_space"
   shows "simply_connected(sphere a r) \<longleftrightarrow> 3 \<le> DIM('a) \<or> r \<le> 0"  (is "?lhs = ?rhs")
 proof (cases "r \<le> 0")
@@ -3024,7 +3024,7 @@
 qed
 
 
-lemma simply_connected_punctured_universe_eq:
+lemma%unimportant simply_connected_punctured_universe_eq:
   fixes a :: "'a::euclidean_space"
   shows "simply_connected(- {a}) \<longleftrightarrow> 3 \<le> DIM('a)"
 proof -
@@ -3042,17 +3042,17 @@
   finally show ?thesis .
 qed
 
-lemma not_simply_connected_circle:
+lemma%unimportant not_simply_connected_circle:
   fixes a :: complex
   shows "0 < r \<Longrightarrow> \<not> simply_connected(sphere a r)"
 by (simp add: simply_connected_sphere_eq)
 
 
-proposition simply_connected_punctured_convex:
+proposition%important simply_connected_punctured_convex:
   fixes a :: "'a::euclidean_space"
   assumes "convex S" and 3: "3 \<le> aff_dim S"
     shows "simply_connected(S - {a})"
-proof (cases "a \<in> rel_interior S")
+proof%unimportant (cases "a \<in> rel_interior S")
   case True
   then obtain e where "a \<in> S" "0 < e" and e: "cball a e \<inter> affine hull S \<subseteq> S"
     by (auto simp: rel_interior_cball)
@@ -3091,11 +3091,11 @@
     by (meson Diff_subset closure_subset subset_trans)
 qed
 
-corollary simply_connected_punctured_universe:
+corollary%important simply_connected_punctured_universe:
   fixes a :: "'a::euclidean_space"
   assumes "3 \<le> DIM('a)"
   shows "simply_connected(- {a})"
-proof -
+proof%unimportant -
   have [simp]: "affine hull cball a 1 = UNIV"
     apply auto
     by (metis UNIV_I aff_dim_cball aff_dim_lt_full zero_less_one not_less_iff_gr_or_eq)
@@ -3109,12 +3109,12 @@
 qed
 
 
-subsection\<open>The power, squaring and exponential functions as covering maps\<close>
-
-proposition covering_space_power_punctured_plane:
+subsection%important\<open>The power, squaring and exponential functions as covering maps\<close>
+
+proposition%important covering_space_power_punctured_plane:
   assumes "0 < n"
     shows "covering_space (- {0}) (\<lambda>z::complex. z^n) (- {0})"
-proof -
+proof%unimportant -
   consider "n = 1" | "2 \<le> n" using assms by linarith
   then obtain e where "0 < e"
                 and e: "\<And>w z. cmod(w - z) < e * cmod z \<Longrightarrow> (w^n = z^n \<longleftrightarrow> w = z)"
@@ -3362,14 +3362,14 @@
     done
 qed
 
-corollary covering_space_square_punctured_plane:
+corollary%important covering_space_square_punctured_plane:
   "covering_space (- {0}) (\<lambda>z::complex. z^2) (- {0})"
-  by (simp add: covering_space_power_punctured_plane)
-
-
-proposition covering_space_exp_punctured_plane:
+  by%unimportant (simp add: covering_space_power_punctured_plane)
+
+
+proposition%important covering_space_exp_punctured_plane:
   "covering_space UNIV (\<lambda>z::complex. exp z) (- {0})"
-proof (simp add: covering_space_def, intro conjI ballI)
+proof%unimportant (simp add: covering_space_def, intro conjI ballI)
   show "continuous_on UNIV (\<lambda>z::complex. exp z)"
     by (rule continuous_on_exp [OF continuous_on_id])
   show "range exp = - {0::complex}"
@@ -3483,9 +3483,9 @@
 qed
 
 
-subsection\<open>Hence the Borsukian results about mappings into circles\<close>
-
-lemma inessential_eq_continuous_logarithm:
+subsection%important\<open>Hence the Borsukian results about mappings into circles\<close>
+
+lemma%unimportant inessential_eq_continuous_logarithm:
   fixes f :: "'a::real_normed_vector \<Rightarrow> complex"
   shows "(\<exists>a. homotopic_with (\<lambda>h. True) S (-{0}) f (\<lambda>t. a)) \<longleftrightarrow>
          (\<exists>g. continuous_on S g \<and> (\<forall>x \<in> S. f x = exp(g x)))"
@@ -3509,7 +3509,7 @@
     by (simp add: f homotopic_with_eq)
 qed
 
-corollary inessential_imp_continuous_logarithm_circle:
+corollary%important inessential_imp_continuous_logarithm_circle:
   fixes f :: "'a::real_normed_vector \<Rightarrow> complex"
   assumes "homotopic_with (\<lambda>h. True) S (sphere 0 1) f (\<lambda>t. a)"
   obtains g where "continuous_on S g" and "\<And>x. x \<in> S \<Longrightarrow> f x = exp(g x)"
@@ -3521,7 +3521,7 @@
 qed
 
 
-lemma inessential_eq_continuous_logarithm_circle:
+lemma%unimportant inessential_eq_continuous_logarithm_circle:
   fixes f :: "'a::real_normed_vector \<Rightarrow> complex"
   shows "(\<exists>a. homotopic_with (\<lambda>h. True) S (sphere 0 1) f (\<lambda>t. a)) \<longleftrightarrow>
          (\<exists>g. continuous_on S g \<and> (\<forall>x \<in> S. f x = exp(\<i> * of_real(g x))))"
@@ -3561,12 +3561,12 @@
     by (simp add: homotopic_with_eq)
 qed
 
-lemma homotopic_with_sphere_times:
+lemma%important homotopic_with_sphere_times:
   fixes f :: "'a::real_normed_vector \<Rightarrow> complex"
   assumes hom: "homotopic_with (\<lambda>x. True) S (sphere 0 1) f g" and conth: "continuous_on S h"
       and hin: "\<And>x. x \<in> S \<Longrightarrow> h x \<in> sphere 0 1"
     shows "homotopic_with (\<lambda>x. True) S (sphere 0 1) (\<lambda>x. f x * h x) (\<lambda>x. g x * h x)"
-proof -
+proof%unimportant -
   obtain k where contk: "continuous_on ({0..1::real} \<times> S) k"
              and kim: "k ` ({0..1} \<times> S) \<subseteq> sphere 0 1"
              and k0:  "\<And>x. k(0, x) = f x"
@@ -3582,13 +3582,13 @@
 qed
 
 
-lemma homotopic_circlemaps_divide:
+lemma%important homotopic_circlemaps_divide:
   fixes f :: "'a::real_normed_vector \<Rightarrow> complex"
     shows "homotopic_with (\<lambda>x. True) S (sphere 0 1) f g \<longleftrightarrow>
            continuous_on S f \<and> f ` S \<subseteq> sphere 0 1 \<and>
            continuous_on S g \<and> g ` S \<subseteq> sphere 0 1 \<and>
            (\<exists>c. homotopic_with (\<lambda>x. True) S (sphere 0 1) (\<lambda>x. f x / g x) (\<lambda>x. c))"
-proof -
+proof%unimportant -
   have "homotopic_with (\<lambda>x. True) S (sphere 0 1) (\<lambda>x. f x / g x) (\<lambda>x. 1)"
        if "homotopic_with (\<lambda>x. True) S (sphere 0 1) (\<lambda>x. f x / g x) (\<lambda>x. c)" for c
   proof -
@@ -3634,7 +3634,7 @@
     by (simp add: *)
 qed
 
-subsection\<open>Upper and lower hemicontinuous functions\<close>
+subsection%important\<open>Upper and lower hemicontinuous functions\<close>
 
 text\<open>And relation in the case of preimage map to open and closed maps, and fact that upper and lower
 hemicontinuity together imply continuity in the sense of the Hausdorff metric (at points where the
@@ -3642,7 +3642,7 @@
 
 
 text\<open>Many similar proofs below.\<close>
-lemma upper_hemicontinuous:
+lemma%unimportant upper_hemicontinuous:
   assumes "\<And>x. x \<in> S \<Longrightarrow> f x \<subseteq> T"
     shows "((\<forall>U. openin (subtopology euclidean T) U
                  \<longrightarrow> openin (subtopology euclidean S) {x \<in> S. f x \<subseteq> U}) \<longleftrightarrow>
@@ -3673,7 +3673,7 @@
     by (simp add: openin_closedin_eq)
 qed
 
-lemma lower_hemicontinuous:
+lemma%unimportant lower_hemicontinuous:
   assumes "\<And>x. x \<in> S \<Longrightarrow> f x \<subseteq> T"
     shows "((\<forall>U. closedin (subtopology euclidean T) U
                  \<longrightarrow> closedin (subtopology euclidean S) {x \<in> S. f x \<subseteq> U}) \<longleftrightarrow>
@@ -3704,7 +3704,7 @@
     by (simp add: openin_closedin_eq)
 qed
 
-lemma open_map_iff_lower_hemicontinuous_preimage:
+lemma%unimportant open_map_iff_lower_hemicontinuous_preimage:
   assumes "f ` S \<subseteq> T"
     shows "((\<forall>U. openin (subtopology euclidean S) U
                  \<longrightarrow> openin (subtopology euclidean T) (f ` U)) \<longleftrightarrow>
@@ -3735,7 +3735,7 @@
     using assms openin_imp_subset [OF opeSU] by (force simp: openin_closedin_eq)
 qed
 
-lemma closed_map_iff_upper_hemicontinuous_preimage:
+lemma%unimportant closed_map_iff_upper_hemicontinuous_preimage:
   assumes "f ` S \<subseteq> T"
     shows "((\<forall>U. closedin (subtopology euclidean S) U
                  \<longrightarrow> closedin (subtopology euclidean T) (f ` U)) \<longleftrightarrow>
@@ -3766,7 +3766,7 @@
     by (simp add: openin_closedin_eq)
 qed
 
-proposition upper_lower_hemicontinuous_explicit:
+proposition%important upper_lower_hemicontinuous_explicit:
   fixes T :: "('b::{real_normed_vector,heine_borel}) set"
   assumes fST: "\<And>x. x \<in> S \<Longrightarrow> f x \<subseteq> T"
       and ope: "\<And>U. openin (subtopology euclidean T) U
@@ -3778,7 +3778,7 @@
              "\<And>x'. \<lbrakk>x' \<in> S; dist x x' < d\<rbrakk>
                            \<Longrightarrow> (\<forall>y \<in> f x. \<exists>y'. y' \<in> f x' \<and> dist y y' < e) \<and>
                                (\<forall>y' \<in> f x'. \<exists>y. y \<in> f x \<and> dist y' y < e)"
-proof -
+proof%unimportant -
   have "openin (subtopology euclidean T) (T \<inter> (\<Union>a\<in>f x. \<Union>b\<in>ball 0 e. {a + b}))"
     by (auto simp: open_sums openin_open_Int)
   with ope have "openin (subtopology euclidean S)
@@ -3837,13 +3837,13 @@
 qed
 
 
-subsection\<open>Complex logs exist on various "well-behaved" sets\<close>
-
-lemma continuous_logarithm_on_contractible:
+subsection%important\<open>Complex logs exist on various "well-behaved" sets\<close>
+
+lemma%important continuous_logarithm_on_contractible:
   fixes f :: "'a::real_normed_vector \<Rightarrow> complex"
   assumes "continuous_on S f" "contractible S" "\<And>z. z \<in> S \<Longrightarrow> f z \<noteq> 0"
   obtains g where "continuous_on S g" "\<And>x. x \<in> S \<Longrightarrow> f x = exp(g x)"
-proof -
+proof%unimportant -
   obtain c where hom: "homotopic_with (\<lambda>h. True) S (-{0}) f (\<lambda>x. c)"
     using nullhomotopic_from_contractible assms
     by (metis imageE subset_Compl_singleton)
@@ -3851,27 +3851,27 @@
     by (metis inessential_eq_continuous_logarithm that)
 qed
 
-lemma continuous_logarithm_on_simply_connected:
+lemma%important continuous_logarithm_on_simply_connected:
   fixes f :: "'a::real_normed_vector \<Rightarrow> complex"
   assumes contf: "continuous_on S f" and S: "simply_connected S" "locally path_connected S"
       and f: "\<And>z. z \<in> S \<Longrightarrow> f z \<noteq> 0"
   obtains g where "continuous_on S g" "\<And>x. x \<in> S \<Longrightarrow> f x = exp(g x)"
-  using covering_space_lift [OF covering_space_exp_punctured_plane S contf]
-  by (metis (full_types) f imageE subset_Compl_singleton)
-
-lemma continuous_logarithm_on_cball:
+  using%unimportant covering_space_lift [OF covering_space_exp_punctured_plane S contf]
+  by%unimportant (metis (full_types) f imageE subset_Compl_singleton)
+
+lemma%unimportant continuous_logarithm_on_cball:
   fixes f :: "'a::real_normed_vector \<Rightarrow> complex"
   assumes "continuous_on (cball a r) f" and "\<And>z. z \<in> cball a r \<Longrightarrow> f z \<noteq> 0"
     obtains h where "continuous_on (cball a r) h" "\<And>z. z \<in> cball a r \<Longrightarrow> f z = exp(h z)"
   using assms continuous_logarithm_on_contractible convex_imp_contractible by blast
 
-lemma continuous_logarithm_on_ball:
+lemma%unimportant continuous_logarithm_on_ball:
   fixes f :: "'a::real_normed_vector \<Rightarrow> complex"
   assumes "continuous_on (ball a r) f" and "\<And>z. z \<in> ball a r \<Longrightarrow> f z \<noteq> 0"
   obtains h where "continuous_on (ball a r) h" "\<And>z. z \<in> ball a r \<Longrightarrow> f z = exp(h z)"
   using assms continuous_logarithm_on_contractible convex_imp_contractible by blast
 
-lemma continuous_sqrt_on_contractible:
+lemma%unimportant continuous_sqrt_on_contractible:
   fixes f :: "'a::real_normed_vector \<Rightarrow> complex"
   assumes "continuous_on S f" "contractible S"
       and "\<And>z. z \<in> S \<Longrightarrow> f z \<noteq> 0"
@@ -3888,7 +3888,7 @@
   qed
 qed
 
-lemma continuous_sqrt_on_simply_connected:
+lemma%unimportant continuous_sqrt_on_simply_connected:
   fixes f :: "'a::real_normed_vector \<Rightarrow> complex"
   assumes contf: "continuous_on S f" and S: "simply_connected S" "locally path_connected S"
       and f: "\<And>z. z \<in> S \<Longrightarrow> f z \<noteq> 0"
@@ -3906,14 +3906,14 @@
 qed
 
 
-subsection\<open>Another simple case where sphere maps are nullhomotopic\<close>
-
-lemma inessential_spheremap_2_aux:
+subsection%important\<open>Another simple case where sphere maps are nullhomotopic\<close>
+
+lemma%important inessential_spheremap_2_aux:
   fixes f :: "'a::euclidean_space \<Rightarrow> complex"
   assumes 2: "2 < DIM('a)" and contf: "continuous_on (sphere a r) f" 
       and fim: "f `(sphere a r) \<subseteq> (sphere 0 1)" 
   obtains c where "homotopic_with (\<lambda>z. True) (sphere a r) (sphere 0 1) f (\<lambda>x. c)"
-proof -
+proof%unimportant -
   obtain g where contg: "continuous_on (sphere a r) g" 
              and feq: "\<And>x. x \<in> sphere a r \<Longrightarrow> f x = exp(g x)"
   proof (rule continuous_logarithm_on_simply_connected [OF contf])
@@ -3935,12 +3935,12 @@
     by metis
 qed
 
-lemma inessential_spheremap_2:
+lemma%important inessential_spheremap_2:
   fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
   assumes a2: "2 < DIM('a)" and b2: "DIM('b) = 2" 
       and contf: "continuous_on (sphere a r) f" and fim: "f `(sphere a r) \<subseteq> (sphere b s)"
   obtains c where "homotopic_with (\<lambda>z. True) (sphere a r) (sphere b s) f (\<lambda>x. c)"
-proof (cases "s \<le> 0")
+proof%unimportant (cases "s \<le> 0")
   case True
   then show ?thesis
     using contf contractible_sphere fim nullhomotopic_into_contractible that by blast
@@ -3972,14 +3972,14 @@
 qed
 
 
-subsection\<open>Holomorphic logarithms and square roots\<close>
-
-lemma contractible_imp_holomorphic_log:
+subsection%important\<open>Holomorphic logarithms and square roots\<close>
+
+lemma%important contractible_imp_holomorphic_log:
   assumes holf: "f holomorphic_on S"
       and S: "contractible S"
       and fnz: "\<And>z. z \<in> S \<Longrightarrow> f z \<noteq> 0"
   obtains g where "g holomorphic_on S" "\<And>z. z \<in> S \<Longrightarrow> f z = exp(g z)"
-proof -
+proof%unimportant -
   have contf: "continuous_on S f"
     by (simp add: holf holomorphic_on_imp_continuous_on)
   obtain g where contg: "continuous_on S g" and feq: "\<And>x. x \<in> S \<Longrightarrow> f x = exp (g x)"
@@ -4022,12 +4022,12 @@
 qed
 
 (*Identical proofs*)
-lemma simply_connected_imp_holomorphic_log:
+lemma%important simply_connected_imp_holomorphic_log:
   assumes holf: "f holomorphic_on S"
       and S: "simply_connected S" "locally path_connected S"
       and fnz: "\<And>z. z \<in> S \<Longrightarrow> f z \<noteq> 0"
   obtains g where "g holomorphic_on S" "\<And>z. z \<in> S \<Longrightarrow> f z = exp(g z)"
-proof -
+proof%unimportant -
   have contf: "continuous_on S f"
     by (simp add: holf holomorphic_on_imp_continuous_on)
   obtain g where contg: "continuous_on S g" and feq: "\<And>x. x \<in> S \<Longrightarrow> f x = exp (g x)"
@@ -4070,7 +4070,7 @@
 qed
 
 
-lemma contractible_imp_holomorphic_sqrt:
+lemma%unimportant contractible_imp_holomorphic_sqrt:
   assumes holf: "f holomorphic_on S"
       and S: "contractible S"
       and fnz: "\<And>z. z \<in> S \<Longrightarrow> f z \<noteq> 0"
@@ -4088,7 +4088,7 @@
   qed
 qed
 
-lemma simply_connected_imp_holomorphic_sqrt:
+lemma%unimportant simply_connected_imp_holomorphic_sqrt:
   assumes holf: "f holomorphic_on S"
       and S: "simply_connected S" "locally path_connected S"
       and fnz: "\<And>z. z \<in> S \<Longrightarrow> f z \<noteq> 0"
@@ -4108,11 +4108,11 @@
 
 text\<open> Related theorems about holomorphic inverse cosines.\<close>
 
-lemma contractible_imp_holomorphic_arccos:
+lemma%important contractible_imp_holomorphic_arccos:
   assumes holf: "f holomorphic_on S" and S: "contractible S"
       and non1: "\<And>z. z \<in> S \<Longrightarrow> f z \<noteq> 1 \<and> f z \<noteq> -1"
   obtains g where "g holomorphic_on S" "\<And>z. z \<in> S \<Longrightarrow> f z = cos(g z)"
-proof -
+proof%unimportant -
   have hol1f: "(\<lambda>z. 1 - f z ^ 2) holomorphic_on S"
     by (intro holomorphic_intros holf)
   obtain g where holg: "g holomorphic_on S" and eq: "\<And>z. z \<in> S \<Longrightarrow> 1 - (f z)\<^sup>2 = (g z)\<^sup>2"
@@ -4146,11 +4146,11 @@
 qed
 
 
-lemma contractible_imp_holomorphic_arccos_bounded:
+lemma%important contractible_imp_holomorphic_arccos_bounded:
   assumes holf: "f holomorphic_on S" and S: "contractible S" and "a \<in> S"
       and non1: "\<And>z. z \<in> S \<Longrightarrow> f z \<noteq> 1 \<and> f z \<noteq> -1"
   obtains g where "g holomorphic_on S" "norm(g a) \<le> pi + norm(f a)" "\<And>z. z \<in> S \<Longrightarrow> f z = cos(g z)"
-proof -
+proof%unimportant -
   obtain g where holg: "g holomorphic_on S" and feq: "\<And>z. z \<in> S \<Longrightarrow> f z = cos (g z)"
     using contractible_imp_holomorphic_arccos [OF holf S non1] by blast
   obtain b where "cos b = f a" "norm b \<le> pi + norm (f a)"
@@ -4186,21 +4186,21 @@
 qed
 
 
-subsection\<open>The "Borsukian" property of sets\<close>
+subsection%important\<open>The "Borsukian" property of sets\<close>
 
 text\<open>This doesn't have a standard name. Kuratowski uses ``contractible with respect to $[S^1]$''
  while Whyburn uses ``property b''. It's closely related to unicoherence.\<close>
 
-definition Borsukian where
+definition%important Borsukian where
     "Borsukian S \<equiv>
         \<forall>f. continuous_on S f \<and> f ` S \<subseteq> (- {0::complex})
             \<longrightarrow> (\<exists>a. homotopic_with (\<lambda>h. True) S (- {0}) f (\<lambda>x. a))"
 
-lemma Borsukian_retraction_gen:
+lemma%important Borsukian_retraction_gen:
   assumes "Borsukian S" "continuous_on S h" "h ` S = T"
           "continuous_on T k"  "k ` T \<subseteq> S"  "\<And>y. y \<in> T \<Longrightarrow> h(k y) = y"
     shows "Borsukian T"
-proof -
+proof%unimportant -
   interpret R: Retracts S h T k
     using assms by (simp add: Retracts.intro)
   show ?thesis
@@ -4210,42 +4210,42 @@
     done
 qed
 
-lemma retract_of_Borsukian: "\<lbrakk>Borsukian T; S retract_of T\<rbrakk> \<Longrightarrow> Borsukian S"
+lemma%unimportant retract_of_Borsukian: "\<lbrakk>Borsukian T; S retract_of T\<rbrakk> \<Longrightarrow> Borsukian S"
   apply (auto simp: retract_of_def retraction_def)
   apply (erule (1) Borsukian_retraction_gen)
   apply (meson retraction retraction_def)
     apply (auto simp: continuous_on_id)
     done
 
-lemma homeomorphic_Borsukian: "\<lbrakk>Borsukian S; S homeomorphic T\<rbrakk> \<Longrightarrow> Borsukian T"
+lemma%unimportant homeomorphic_Borsukian: "\<lbrakk>Borsukian S; S homeomorphic T\<rbrakk> \<Longrightarrow> Borsukian T"
   using Borsukian_retraction_gen order_refl
   by (fastforce simp add: homeomorphism_def homeomorphic_def)
 
-lemma homeomorphic_Borsukian_eq:
+lemma%unimportant homeomorphic_Borsukian_eq:
    "S homeomorphic T \<Longrightarrow> Borsukian S \<longleftrightarrow> Borsukian T"
   by (meson homeomorphic_Borsukian homeomorphic_sym)
 
-lemma Borsukian_translation:
+lemma%unimportant Borsukian_translation:
   fixes S :: "'a::real_normed_vector set"
   shows "Borsukian (image (\<lambda>x. a + x) S) \<longleftrightarrow> Borsukian S"
   apply (rule homeomorphic_Borsukian_eq)
     using homeomorphic_translation homeomorphic_sym by blast
 
-lemma Borsukian_injective_linear_image:
+lemma%unimportant Borsukian_injective_linear_image:
   fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
   assumes "linear f" "inj f"
     shows "Borsukian(f ` S) \<longleftrightarrow> Borsukian S"
   apply (rule homeomorphic_Borsukian_eq)
   using assms homeomorphic_sym linear_homeomorphic_image by blast
 
-lemma homotopy_eqv_Borsukianness:
+lemma%unimportant homotopy_eqv_Borsukianness:
   fixes S :: "'a::real_normed_vector set"
     and T :: "'b::real_normed_vector set"
    assumes "S homotopy_eqv T"
      shows "(Borsukian S \<longleftrightarrow> Borsukian T)"
   by (meson Borsukian_def assms homotopy_eqv_cohomotopic_triviality_null)
 
-lemma Borsukian_alt:
+lemma%unimportant Borsukian_alt:
   fixes S :: "'a::real_normed_vector set"
   shows
    "Borsukian S \<longleftrightarrow>
@@ -4255,20 +4255,20 @@
   unfolding Borsukian_def homotopic_triviality
   by (simp add: path_connected_punctured_universe)
 
-lemma Borsukian_continuous_logarithm:
+lemma%unimportant Borsukian_continuous_logarithm:
   fixes S :: "'a::real_normed_vector set"
   shows "Borsukian S \<longleftrightarrow>
             (\<forall>f. continuous_on S f \<and> f ` S \<subseteq> (- {0::complex})
                  \<longrightarrow> (\<exists>g. continuous_on S g \<and> (\<forall>x \<in> S. f x = exp(g x))))"
   by (simp add: Borsukian_def inessential_eq_continuous_logarithm)
 
-lemma Borsukian_continuous_logarithm_circle:
+lemma%important Borsukian_continuous_logarithm_circle:
   fixes S :: "'a::real_normed_vector set"
   shows "Borsukian S \<longleftrightarrow>
              (\<forall>f. continuous_on S f \<and> f ` S \<subseteq> sphere (0::complex) 1
                   \<longrightarrow> (\<exists>g. continuous_on S g \<and> (\<forall>x \<in> S. f x = exp(g x))))"
    (is "?lhs = ?rhs")
-proof
+proof%unimportant
   assume ?lhs then show ?rhs
     by (force simp: Borsukian_continuous_logarithm)
 next
@@ -4298,13 +4298,13 @@
 qed
 
 
-lemma Borsukian_continuous_logarithm_circle_real:
+lemma%important Borsukian_continuous_logarithm_circle_real:
   fixes S :: "'a::real_normed_vector set"
   shows "Borsukian S \<longleftrightarrow>
          (\<forall>f. continuous_on S f \<and> f ` S \<subseteq> sphere (0::complex) 1
               \<longrightarrow> (\<exists>g. continuous_on S (complex_of_real \<circ> g) \<and> (\<forall>x \<in> S. f x = exp(\<i> * of_real(g x)))))"
    (is "?lhs = ?rhs")
-proof
+proof%unimportant
   assume LHS: ?lhs
   show ?rhs
   proof (clarify)
@@ -4331,52 +4331,52 @@
   qed
 qed
 
-lemma Borsukian_circle:
+lemma%unimportant Borsukian_circle:
   fixes S :: "'a::real_normed_vector set"
   shows "Borsukian S \<longleftrightarrow>
          (\<forall>f. continuous_on S f \<and> f ` S \<subseteq> sphere (0::complex) 1
               \<longrightarrow> (\<exists>a. homotopic_with (\<lambda>h. True) S (sphere (0::complex) 1) f (\<lambda>x. a)))"
 by (simp add: inessential_eq_continuous_logarithm_circle Borsukian_continuous_logarithm_circle_real)
 
-lemma contractible_imp_Borsukian: "contractible S \<Longrightarrow> Borsukian S"
+lemma%unimportant contractible_imp_Borsukian: "contractible S \<Longrightarrow> Borsukian S"
   by (meson Borsukian_def nullhomotopic_from_contractible)
 
-lemma simply_connected_imp_Borsukian:
+lemma%unimportant simply_connected_imp_Borsukian:
   fixes S :: "'a::real_normed_vector set"
   shows  "\<lbrakk>simply_connected S; locally path_connected S\<rbrakk> \<Longrightarrow> Borsukian S"
   apply (simp add: Borsukian_continuous_logarithm)
   by (metis (no_types, lifting) continuous_logarithm_on_simply_connected image_iff)
 
-lemma starlike_imp_Borsukian:
+lemma%unimportant starlike_imp_Borsukian:
   fixes S :: "'a::real_normed_vector set"
   shows "starlike S \<Longrightarrow> Borsukian S"
   by (simp add: contractible_imp_Borsukian starlike_imp_contractible)
 
-lemma Borsukian_empty: "Borsukian {}"
+lemma%unimportant Borsukian_empty: "Borsukian {}"
   by (auto simp: contractible_imp_Borsukian)
 
-lemma Borsukian_UNIV: "Borsukian (UNIV :: 'a::real_normed_vector set)"
+lemma%unimportant Borsukian_UNIV: "Borsukian (UNIV :: 'a::real_normed_vector set)"
   by (auto simp: contractible_imp_Borsukian)
 
-lemma convex_imp_Borsukian:
+lemma%unimportant convex_imp_Borsukian:
   fixes S :: "'a::real_normed_vector set"
   shows "convex S \<Longrightarrow> Borsukian S"
   by (meson Borsukian_def convex_imp_contractible nullhomotopic_from_contractible)
 
-lemma Borsukian_sphere:
+lemma%unimportant Borsukian_sphere:
   fixes a :: "'a::euclidean_space"
   shows "3 \<le> DIM('a) \<Longrightarrow> Borsukian (sphere a r)"
   apply (rule simply_connected_imp_Borsukian)
   using simply_connected_sphere apply blast
   using ENR_imp_locally_path_connected ENR_sphere by blast
 
-lemma Borsukian_open_Un:
+lemma%important Borsukian_open_Un:
   fixes S :: "'a::real_normed_vector set"
   assumes opeS: "openin (subtopology euclidean (S \<union> T)) S"
       and opeT: "openin (subtopology euclidean (S \<union> T)) T"
       and BS: "Borsukian S" and BT: "Borsukian T" and ST: "connected(S \<inter> T)"
     shows "Borsukian(S \<union> T)"
-proof (clarsimp simp add: Borsukian_continuous_logarithm)
+proof%unimportant (clarsimp simp add: Borsukian_continuous_logarithm)
   fix f :: "'a \<Rightarrow> complex"
   assume contf: "continuous_on (S \<union> T) f" and 0: "0 \<notin> f ` (S \<union> T)"
   then have contfS: "continuous_on S f" and contfT: "continuous_on T f"
@@ -4440,13 +4440,13 @@
 qed
 
 text\<open>The proof is a duplicate of that of \<open>Borsukian_open_Un\<close>.\<close>
-lemma Borsukian_closed_Un:
+lemma%important Borsukian_closed_Un:
   fixes S :: "'a::real_normed_vector set"
   assumes cloS: "closedin (subtopology euclidean (S \<union> T)) S"
       and cloT: "closedin (subtopology euclidean (S \<union> T)) T"
       and BS: "Borsukian S" and BT: "Borsukian T" and ST: "connected(S \<inter> T)"
     shows "Borsukian(S \<union> T)"
-proof (clarsimp simp add: Borsukian_continuous_logarithm)
+proof%unimportant (clarsimp simp add: Borsukian_continuous_logarithm)
   fix f :: "'a \<Rightarrow> complex"
   assume contf: "continuous_on (S \<union> T) f" and 0: "0 \<notin> f ` (S \<union> T)"
   then have contfS: "continuous_on S f" and contfT: "continuous_on T f"
@@ -4509,18 +4509,18 @@
   qed
 qed
 
-lemma Borsukian_separation_compact:
+lemma%unimportant Borsukian_separation_compact:
   fixes S :: "complex set"
   assumes "compact S"
     shows "Borsukian S \<longleftrightarrow> connected(- S)"
   by (simp add: Borsuk_separation_theorem Borsukian_circle assms)
 
-lemma Borsukian_monotone_image_compact:
+lemma%important Borsukian_monotone_image_compact:
   fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
   assumes "Borsukian S" and contf: "continuous_on S f" and fim: "f ` S = T"
       and "compact S" and conn: "\<And>y. y \<in> T \<Longrightarrow> connected {x. x \<in> S \<and> f x = y}"
     shows "Borsukian T"
-proof (clarsimp simp add: Borsukian_continuous_logarithm)
+proof%unimportant (clarsimp simp add: Borsukian_continuous_logarithm)
   fix g :: "'b \<Rightarrow> complex"
   assume contg: "continuous_on T g" and 0: "0 \<notin> g ` T"
   have "continuous_on S (g \<circ> f)"
@@ -4580,13 +4580,13 @@
 qed
 
 
-lemma Borsukian_open_map_image_compact:
+lemma%important Borsukian_open_map_image_compact:
   fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
   assumes "Borsukian S" and contf: "continuous_on S f" and fim: "f ` S = T" and "compact S"
       and ope: "\<And>U. openin (subtopology euclidean S) U
                      \<Longrightarrow> openin (subtopology euclidean T) (f ` U)"
     shows "Borsukian T"
-proof (clarsimp simp add: Borsukian_continuous_logarithm_circle_real)
+proof%unimportant (clarsimp simp add: Borsukian_continuous_logarithm_circle_real)
   fix g :: "'b \<Rightarrow> complex"
   assume contg: "continuous_on T g" and gim: "g ` T \<subseteq> sphere 0 1"
   have "continuous_on S (g \<circ> f)"
@@ -4667,12 +4667,12 @@
 
 
 text\<open>If two points are separated by a closed set, there's a minimal one.\<close>
-proposition closed_irreducible_separator:
+proposition%important closed_irreducible_separator:
   fixes a :: "'a::real_normed_vector"
   assumes "closed S" and ab: "\<not> connected_component (- S) a b"
   obtains T where "T \<subseteq> S" "closed T" "T \<noteq> {}" "\<not> connected_component (- T) a b"
                   "\<And>U. U \<subset> T \<Longrightarrow> connected_component (- U) a b"
-proof (cases "a \<in> S \<or> b \<in> S")
+proof%unimportant (cases "a \<in> S \<or> b \<in> S")
   case True
   then show ?thesis
   proof
@@ -4775,7 +4775,7 @@
   qed
 qed
 
-lemma frontier_minimal_separating_closed_pointwise:
+lemma%unimportant frontier_minimal_separating_closed_pointwise:
   fixes S :: "'a::real_normed_vector set"
   assumes S: "closed S" "a \<notin> S" and nconn: "\<not> connected_component (- S) a b"
       and conn: "\<And>T. \<lbrakk>closed T; T \<subset> S\<rbrakk> \<Longrightarrow> connected_component (- T) a b"
@@ -4804,29 +4804,29 @@
 qed
 
 
-subsection\<open>Unicoherence (closed)\<close>
-
-definition unicoherent where
+subsection%important\<open>Unicoherence (closed)\<close>
+
+definition%important unicoherent where
   "unicoherent U \<equiv>
   \<forall>S T. connected S \<and> connected T \<and> S \<union> T = U \<and>
         closedin (subtopology euclidean U) S \<and> closedin (subtopology euclidean U) T
         \<longrightarrow> connected (S \<inter> T)"
 
-lemma unicoherentI [intro?]:
+lemma%unimportant unicoherentI [intro?]:
   assumes "\<And>S T. \<lbrakk>connected S; connected T; U = S \<union> T; closedin (subtopology euclidean U) S; closedin (subtopology euclidean U) T\<rbrakk>
           \<Longrightarrow> connected (S \<inter> T)"
   shows "unicoherent U"
   using assms unfolding unicoherent_def by blast
 
-lemma unicoherentD:
+lemma%unimportant unicoherentD:
   assumes "unicoherent U" "connected S" "connected T" "U = S \<union> T" "closedin (subtopology euclidean U) S" "closedin (subtopology euclidean U) T"
   shows "connected (S \<inter> T)"
   using assms unfolding unicoherent_def by blast
 
-lemma homeomorphic_unicoherent:
+lemma%important homeomorphic_unicoherent:
   assumes ST: "S homeomorphic T" and S: "unicoherent S"
   shows "unicoherent T"
-proof -
+proof%unimportant -
   obtain f g where gf: "\<And>x. x \<in> S \<Longrightarrow> g (f x) = x" and fim: "T = f ` S" and gfim: "g ` f ` S = S"
     and contf: "continuous_on S f" and contg: "continuous_on (f ` S) g"
     using ST by (auto simp: homeomorphic_def homeomorphism_def)
@@ -4867,24 +4867,24 @@
 qed
 
 
-lemma homeomorphic_unicoherent_eq:
+lemma%unimportant homeomorphic_unicoherent_eq:
    "S homeomorphic T \<Longrightarrow> (unicoherent S \<longleftrightarrow> unicoherent T)"
   by (meson homeomorphic_sym homeomorphic_unicoherent)
 
-lemma unicoherent_translation:
+lemma%unimportant unicoherent_translation:
   fixes S :: "'a::real_normed_vector set"
   shows
    "unicoherent (image (\<lambda>x. a + x) S) \<longleftrightarrow> unicoherent S"
   using homeomorphic_translation homeomorphic_unicoherent_eq by blast
 
-lemma unicoherent_injective_linear_image:
+lemma%unimportant unicoherent_injective_linear_image:
   fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
   assumes "linear f" "inj f"
   shows "(unicoherent(f ` S) \<longleftrightarrow> unicoherent S)"
   using assms homeomorphic_unicoherent_eq linear_homeomorphic_image by blast
 
 
-lemma Borsukian_imp_unicoherent:
+lemma%unimportant Borsukian_imp_unicoherent:
   fixes U :: "'a::euclidean_space set"
   assumes "Borsukian U"  shows "unicoherent U"
   unfolding unicoherent_def
@@ -4994,27 +4994,27 @@
 qed
 
 
-corollary contractible_imp_unicoherent:
+corollary%important contractible_imp_unicoherent:
   fixes U :: "'a::euclidean_space set"
   assumes "contractible U"  shows "unicoherent U"
-  by (simp add: Borsukian_imp_unicoherent assms contractible_imp_Borsukian)
-
-corollary convex_imp_unicoherent:
+  by%unimportant (simp add: Borsukian_imp_unicoherent assms contractible_imp_Borsukian)
+
+corollary%important convex_imp_unicoherent:
   fixes U :: "'a::euclidean_space set"
   assumes "convex U"  shows "unicoherent U"
-  by (simp add: Borsukian_imp_unicoherent assms convex_imp_Borsukian)
+  by%unimportant (simp add: Borsukian_imp_unicoherent assms convex_imp_Borsukian)
 
 text\<open>If the type class constraint can be relaxed, I don't know how!\<close>
-corollary unicoherent_UNIV: "unicoherent (UNIV :: 'a :: euclidean_space set)"
-  by (simp add: convex_imp_unicoherent)
-
-
-lemma unicoherent_monotone_image_compact:
+corollary%important unicoherent_UNIV: "unicoherent (UNIV :: 'a :: euclidean_space set)"
+  by%unimportant (simp add: convex_imp_unicoherent)
+
+
+lemma%important unicoherent_monotone_image_compact:
   fixes T :: "'b :: t2_space set"
   assumes S: "unicoherent S" "compact S" and contf: "continuous_on S f" and fim: "f ` S = T"
   and conn: "\<And>y. y \<in> T \<Longrightarrow> connected (S \<inter> f -` {y})"
   shows "unicoherent T"
-proof
+proof%unimportant
   fix U V
   assume UV: "connected U" "connected V" "T = U \<union> V"
      and cloU: "closedin (subtopology euclidean T) U"
@@ -5048,9 +5048,9 @@
 qed
 
 
-subsection\<open>Several common variants of unicoherence\<close>
-
-lemma connected_frontier_simple:
+subsection%important\<open>Several common variants of unicoherence\<close>
+
+lemma%unimportant connected_frontier_simple:
   fixes S :: "'a :: euclidean_space set"
   assumes "connected S" "connected(- S)" shows "connected(frontier S)"
   unfolding frontier_closures
@@ -5058,18 +5058,18 @@
       apply (simp_all add: assms connected_imp_connected_closure)
   by (simp add: closure_def)
 
-lemma connected_frontier_component_complement:
+lemma%unimportant connected_frontier_component_complement:
   fixes S :: "'a :: euclidean_space set"
   assumes "connected S" and C: "C \<in> components(- S)" shows "connected(frontier C)"
   apply (rule connected_frontier_simple)
   using C in_components_connected apply blast
   by (metis Compl_eq_Diff_UNIV connected_UNIV assms top_greatest component_complement_connected)
 
-lemma connected_frontier_disjoint:
+lemma%important connected_frontier_disjoint:
   fixes S :: "'a :: euclidean_space set"
   assumes "connected S" "connected T" "disjnt S T" and ST: "frontier S \<subseteq> frontier T"
   shows "connected(frontier S)"
-proof (cases "S = UNIV")
+proof%unimportant (cases "S = UNIV")
   case True then show ?thesis
     by simp
 next
@@ -5101,13 +5101,13 @@
 qed
 
 
-subsection\<open>Some separation results\<close>
-
-lemma separation_by_component_closed_pointwise:
+subsection%important\<open>Some separation results\<close>
+
+lemma%important separation_by_component_closed_pointwise:
   fixes S :: "'a :: euclidean_space set"
   assumes "closed S" "\<not> connected_component (- S) a b"
   obtains C where "C \<in> components S" "\<not> connected_component(- C) a b"
-proof (cases "a \<in> S \<or> b \<in> S")
+proof%unimportant (cases "a \<in> S \<or> b \<in> S")
   case True
   then show ?thesis
     using connected_component_in componentsI that by fastforce
@@ -5140,11 +5140,11 @@
 qed
 
 
-lemma separation_by_component_closed:
+lemma%important separation_by_component_closed:
   fixes S :: "'a :: euclidean_space set"
   assumes "closed S" "\<not> connected(- S)"
   obtains C where "C \<in> components S" "\<not> connected(- C)"
-proof -
+proof%unimportant -
   obtain x y where "closed S" "x \<notin> S" "y \<notin> S" and "\<not> connected_component (- S) x y"
     using assms by (auto simp: connected_iff_connected_component)
   then obtain C where "C \<in> components S" "\<not> connected_component(- C) x y"
@@ -5154,12 +5154,12 @@
     by (metis Compl_iff \<open>C \<in> components S\<close> \<open>x \<notin> S\<close> \<open>y \<notin> S\<close> connected_component_eq connected_component_eq_eq connected_iff_connected_component that)
 qed
 
-lemma separation_by_Un_closed_pointwise:
+lemma%important separation_by_Un_closed_pointwise:
   fixes S :: "'a :: euclidean_space set"
   assumes ST: "closed S" "closed T" "S \<inter> T = {}"
       and conS: "connected_component (- S) a b" and conT: "connected_component (- T) a b"
     shows "connected_component (- (S \<union> T)) a b"
-proof (rule ccontr)
+proof%unimportant (rule ccontr)
   have "a \<notin> S" "b \<notin> S" "a \<notin> T" "b \<notin> T"
     using conS conT connected_component_in by auto
   assume "\<not> connected_component (- (S \<union> T)) a b"
@@ -5178,14 +5178,14 @@
     by (meson Compl_anti_mono C conS conT connected_component_of_subset)
 qed
 
-lemma separation_by_Un_closed:
+lemma%unimportant separation_by_Un_closed:
   fixes S :: "'a :: euclidean_space set"
   assumes ST: "closed S" "closed T" "S \<inter> T = {}" and conS: "connected(- S)" and conT: "connected(- T)"
   shows "connected(- (S \<union> T))"
   using assms separation_by_Un_closed_pointwise
   by (fastforce simp add: connected_iff_connected_component)
 
-lemma open_unicoherent_UNIV:
+lemma%unimportant open_unicoherent_UNIV:
   fixes S :: "'a :: euclidean_space set"
   assumes "open S" "open T" "connected S" "connected T" "S \<union> T = UNIV"
   shows "connected(S \<inter> T)"
@@ -5196,7 +5196,7 @@
     by simp
 qed
 
-lemma separation_by_component_open_aux:
+lemma%unimportant separation_by_component_open_aux:
   fixes S :: "'a :: euclidean_space set"
   assumes ST: "closed S" "closed T" "S \<inter> T = {}"
       and "S \<noteq> {}" "T \<noteq> {}"
@@ -5276,11 +5276,11 @@
 qed
 
 
-lemma separation_by_component_open:
+lemma%important separation_by_component_open:
   fixes S :: "'a :: euclidean_space set"
   assumes "open S" and non: "\<not> connected(- S)"
   obtains C where "C \<in> components S" "\<not> connected(- C)"
-proof -
+proof%unimportant -
   obtain T U
     where "closed T" "closed U" and TU: "T \<union> U = - S" "T \<inter> U = {}" "T \<noteq> {}" "U \<noteq> {}"
     using assms by (auto simp: connected_closed_set closed_def)
@@ -5303,18 +5303,18 @@
   qed
 qed
 
-lemma separation_by_Un_open:
+lemma%unimportant separation_by_Un_open:
   fixes S :: "'a :: euclidean_space set"
   assumes "open S" "open T" "S \<inter> T = {}" and cS: "connected(-S)" and cT: "connected(-T)"
     shows "connected(- (S \<union> T))"
   using assms unicoherent_UNIV unfolding unicoherent_def by force
 
 
-lemma nonseparation_by_component_eq:
+lemma%important nonseparation_by_component_eq:
   fixes S :: "'a :: euclidean_space set"
   assumes "open S \<or> closed S"
   shows "((\<forall>C \<in> components S. connected(-C)) \<longleftrightarrow> connected(- S))" (is "?lhs = ?rhs")
-proof
+proof%unimportant
   assume ?lhs with assms show ?rhs
     by (meson separation_by_component_closed separation_by_component_open)
 next
@@ -5324,13 +5324,13 @@
 
 
 text\<open>Another interesting equivalent of an inessential mapping into C-{0}\<close>
-proposition inessential_eq_extensible:
+proposition%important inessential_eq_extensible:
   fixes f :: "'a::euclidean_space \<Rightarrow> complex"
   assumes "closed S"
   shows "(\<exists>a. homotopic_with (\<lambda>h. True) S (-{0}) f (\<lambda>t. a)) \<longleftrightarrow>
          (\<exists>g. continuous_on UNIV g \<and> (\<forall>x \<in> S. g x = f x) \<and> (\<forall>x. g x \<noteq> 0))"
      (is "?lhs = ?rhs")
-proof
+proof%unimportant
   assume ?lhs
   then obtain a where a: "homotopic_with (\<lambda>h. True) S (-{0}) f (\<lambda>t. a)" ..
   show ?rhs
@@ -5390,14 +5390,14 @@
     by (simp add: inessential_eq_continuous_logarithm)
 qed
 
-lemma inessential_on_clopen_Union:
+lemma%important inessential_on_clopen_Union:
   fixes \<F> :: "'a::euclidean_space set set"
   assumes T: "path_connected T"
       and "\<And>S. S \<in> \<F> \<Longrightarrow> closedin (subtopology euclidean (\<Union>\<F>)) S"
       and "\<And>S. S \<in> \<F> \<Longrightarrow> openin (subtopology euclidean (\<Union>\<F>)) S"
       and hom: "\<And>S. S \<in> \<F> \<Longrightarrow> \<exists>a. homotopic_with (\<lambda>x. True) S T f (\<lambda>x. a)"
   obtains a where "homotopic_with (\<lambda>x. True) (\<Union>\<F>) T f (\<lambda>x. a)"
-proof (cases "\<Union>\<F> = {}")
+proof%unimportant (cases "\<Union>\<F> = {}")
   case True
   with that show ?thesis
     by force
@@ -5438,12 +5438,12 @@
   then show ?thesis ..
 qed
 
-lemma Janiszewski_dual:
+lemma%important Janiszewski_dual:
   fixes S :: "complex set"
   assumes
    "compact S" "compact T" "connected S" "connected T" "connected(- (S \<union> T))"
  shows "connected(S \<inter> T)"
-proof -
+proof%unimportant -
   have ST: "compact (S \<union> T)"
     by (simp add: assms compact_Un)
   with Borsukian_imp_unicoherent [of "S \<union> T"] ST assms

--- a/src/HOL/Analysis/Homeomorphism.thy	Mon Aug 27 22:58:36 2018 +0200
+++ b/src/HOL/Analysis/Homeomorphism.thy	Tue Aug 28 13:28:39 2018 +0100
@@ -2,13 +2,13 @@
     Author: LC Paulson (ported from HOL Light)
 *)
 
-section \<open>Homeomorphism Theorems\<close>
+section%important \<open>Homeomorphism Theorems\<close>
 
 theory Homeomorphism
 imports Path_Connected
 begin
 
-lemma homeomorphic_spheres':
+lemma%unimportant homeomorphic_spheres':
   fixes a ::"'a::euclidean_space" and b ::"'b::euclidean_space"
   assumes "0 < \<delta>" and dimeq: "DIM('a) = DIM('b)"
   shows "(sphere a \<delta>) homeomorphic (sphere b \<delta>)"
@@ -28,7 +28,7 @@
     done
 qed
 
-lemma homeomorphic_spheres_gen:
+lemma%unimportant homeomorphic_spheres_gen:
     fixes a :: "'a::euclidean_space" and b :: "'b::euclidean_space"
   assumes "0 < r" "0 < s" "DIM('a::euclidean_space) = DIM('b::euclidean_space)"
   shows "(sphere a r homeomorphic sphere b s)"
@@ -36,9 +36,9 @@
   using assms  apply auto
   done
 
-subsection \<open>Homeomorphism of all convex compact sets with nonempty interior\<close>
+subsection%important \<open>Homeomorphism of all convex compact sets with nonempty interior\<close>
 
-proposition ray_to_rel_frontier:
+proposition%important ray_to_rel_frontier:
   fixes a :: "'a::real_inner"
   assumes "bounded S"
       and a: "a \<in> rel_interior S"
@@ -46,7 +46,7 @@
       and "l \<noteq> 0"
   obtains d where "0 < d" "(a + d *\<^sub>R l) \<in> rel_frontier S"
            "\<And>e. \<lbrakk>0 \<le> e; e < d\<rbrakk> \<Longrightarrow> (a + e *\<^sub>R l) \<in> rel_interior S"
-proof -
+proof%unimportant -
   have aaff: "a \<in> affine hull S"
     by (meson a hull_subset rel_interior_subset rev_subsetD)
   let ?D = "{d. 0 < d \<and> a + d *\<^sub>R l \<notin> rel_interior S}"
@@ -139,14 +139,14 @@
     by (rule that [OF \<open>0 < d\<close> infront inint])
 qed
 
-corollary ray_to_frontier:
+corollary%important ray_to_frontier:
   fixes a :: "'a::euclidean_space"
   assumes "bounded S"
       and a: "a \<in> interior S"
       and "l \<noteq> 0"
   obtains d where "0 < d" "(a + d *\<^sub>R l) \<in> frontier S"
            "\<And>e. \<lbrakk>0 \<le> e; e < d\<rbrakk> \<Longrightarrow> (a + e *\<^sub>R l) \<in> interior S"
-proof -
+proof%unimportant -
   have "interior S = rel_interior S"
     using a rel_interior_nonempty_interior by auto
   then have "a \<in> rel_interior S"
@@ -158,7 +158,7 @@
 qed
 
 
-lemma segment_to_rel_frontier_aux:
+lemma%unimportant segment_to_rel_frontier_aux:
   fixes x :: "'a::euclidean_space"
   assumes "convex S" "bounded S" and x: "x \<in> rel_interior S" and y: "y \<in> S" and xy: "x \<noteq> y"
   obtains z where "z \<in> rel_frontier S" "y \<in> closed_segment x z"
@@ -198,7 +198,7 @@
   qed
 qed
 
-lemma segment_to_rel_frontier:
+lemma%unimportant segment_to_rel_frontier:
   fixes x :: "'a::euclidean_space"
   assumes S: "convex S" "bounded S" and x: "x \<in> rel_interior S"
       and y: "y \<in> S" and xy: "~(x = y \<and> S = {x})"
@@ -216,11 +216,11 @@
     using segment_to_rel_frontier_aux [OF S x y] that by blast
 qed
 
-proposition rel_frontier_not_sing:
+proposition%important rel_frontier_not_sing:
   fixes a :: "'a::euclidean_space"
   assumes "bounded S"
     shows "rel_frontier S \<noteq> {a}"
-proof (cases "S = {}")
+proof%unimportant (cases "S = {}")
   case True  then show ?thesis  by simp
 next
   case False
@@ -278,7 +278,7 @@
   qed
 qed
 
-proposition
+proposition%important
   fixes S :: "'a::euclidean_space set"
   assumes "compact S" and 0: "0 \<in> rel_interior S"
       and star: "\<And>x. x \<in> S \<Longrightarrow> open_segment 0 x \<subseteq> rel_interior S"
@@ -288,7 +288,7 @@
       and starlike_compact_projective2_0:
             "S homeomorphic cball 0 1 \<inter> affine hull S"
             (is "S homeomorphic ?CBALL")
-proof -
+proof%unimportant -
   have starI: "(u *\<^sub>R x) \<in> rel_interior S" if "x \<in> S" "0 \<le> u" "u < 1" for x u
   proof (cases "x=0 \<or> u=0")
     case True with 0 show ?thesis by force
@@ -540,7 +540,7 @@
     done
 qed
 
-corollary
+corollary%important
   fixes S :: "'a::euclidean_space set"
   assumes "compact S" and a: "a \<in> rel_interior S"
       and star: "\<And>x. x \<in> S \<Longrightarrow> open_segment a x \<subseteq> rel_interior S"
@@ -548,7 +548,7 @@
             "S - rel_interior S homeomorphic sphere a 1 \<inter> affine hull S"
       and starlike_compact_projective2:
             "S homeomorphic cball a 1 \<inter> affine hull S"
-proof -
+proof%unimportant -
   have 1: "compact ((+) (-a) ` S)" by (meson assms compact_translation)
   have 2: "0 \<in> rel_interior ((+) (-a) ` S)"
     by (simp add: a rel_interior_translation)
@@ -580,12 +580,12 @@
   finally show "S homeomorphic cball a 1 \<inter> affine hull S" .
 qed
 
-corollary starlike_compact_projective_special:
+corollary%important starlike_compact_projective_special:
   assumes "compact S"
     and cb01: "cball (0::'a::euclidean_space) 1 \<subseteq> S"
     and scale: "\<And>x u. \<lbrakk>x \<in> S; 0 \<le> u; u < 1\<rbrakk> \<Longrightarrow> u *\<^sub>R x \<in> S - frontier S"
   shows "S homeomorphic (cball (0::'a::euclidean_space) 1)"
-proof -
+proof%unimportant -
   have "ball 0 1 \<subseteq> interior S"
     using cb01 interior_cball interior_mono by blast
   then have 0: "0 \<in> rel_interior S"
@@ -604,13 +604,13 @@
     using starlike_compact_projective2_0 [OF \<open>compact S\<close> 0 star] by simp
 qed
 
-lemma homeomorphic_convex_lemma:
+lemma%important homeomorphic_convex_lemma:
   fixes S :: "'a::euclidean_space set" and T :: "'b::euclidean_space set"
   assumes "convex S" "compact S" "convex T" "compact T"
       and affeq: "aff_dim S = aff_dim T"
     shows "(S - rel_interior S) homeomorphic (T - rel_interior T) \<and>
            S homeomorphic T"
-proof (cases "rel_interior S = {} \<or> rel_interior T = {}")
+proof%unimportant (cases "rel_interior S = {} \<or> rel_interior T = {}")
   case True
     then show ?thesis
       by (metis Diff_empty affeq \<open>convex S\<close> \<open>convex T\<close> aff_dim_empty homeomorphic_empty rel_interior_eq_empty aff_dim_empty)
@@ -708,7 +708,7 @@
     using 1 2 by blast
 qed
 
-lemma homeomorphic_convex_compact_sets:
+lemma%unimportant homeomorphic_convex_compact_sets:
   fixes S :: "'a::euclidean_space set" and T :: "'b::euclidean_space set"
   assumes "convex S" "compact S" "convex T" "compact T"
       and affeq: "aff_dim S = aff_dim T"
@@ -716,7 +716,7 @@
 using homeomorphic_convex_lemma [OF assms] assms
 by (auto simp: rel_frontier_def)
 
-lemma homeomorphic_rel_frontiers_convex_bounded_sets:
+lemma%unimportant homeomorphic_rel_frontiers_convex_bounded_sets:
   fixes S :: "'a::euclidean_space set" and T :: "'b::euclidean_space set"
   assumes "convex S" "bounded S" "convex T" "bounded T"
       and affeq: "aff_dim S = aff_dim T"
@@ -725,11 +725,11 @@
 by (simp add: rel_frontier_def convex_rel_interior_closure)
 
 
-subsection\<open>Homeomorphisms between punctured spheres and affine sets\<close>
+subsection%important\<open>Homeomorphisms between punctured spheres and affine sets\<close>
 text\<open>Including the famous stereoscopic projection of the 3-D sphere to the complex plane\<close>
 
 text\<open>The special case with centre 0 and radius 1\<close>
-lemma homeomorphic_punctured_affine_sphere_affine_01:
+lemma%unimportant homeomorphic_punctured_affine_sphere_affine_01:
   assumes "b \<in> sphere 0 1" "affine T" "0 \<in> T" "b \<in> T" "affine p"
       and affT: "aff_dim T = aff_dim p + 1"
     shows "(sphere 0 1 \<inter> T) - {b} homeomorphic p"
@@ -823,12 +823,12 @@
   finally show ?thesis .
 qed
 
-theorem homeomorphic_punctured_affine_sphere_affine:
+theorem%important homeomorphic_punctured_affine_sphere_affine:
   fixes a :: "'a :: euclidean_space"
   assumes "0 < r" "b \<in> sphere a r" "affine T" "a \<in> T" "b \<in> T" "affine p"
       and aff: "aff_dim T = aff_dim p + 1"
     shows "(sphere a r \<inter> T) - {b} homeomorphic p"
-proof -
+proof%unimportant -
   have "a \<noteq> b" using assms by auto
   then have inj: "inj (\<lambda>x::'a. x /\<^sub>R norm (a - b))"
     by (simp add: inj_on_def)
@@ -847,14 +847,14 @@
   finally show ?thesis .
 qed
 
-corollary homeomorphic_punctured_sphere_affine:
+corollary%important homeomorphic_punctured_sphere_affine:
   fixes a :: "'a :: euclidean_space"
   assumes "0 < r" and b: "b \<in> sphere a r"
       and "affine T" and affS: "aff_dim T + 1 = DIM('a)"
     shows "(sphere a r - {b}) homeomorphic T"
-  using homeomorphic_punctured_affine_sphere_affine [of r b a UNIV T] assms by auto
+  using%unimportant homeomorphic_punctured_affine_sphere_affine [of r b a UNIV T] assms by%unimportant auto
 
-corollary homeomorphic_punctured_sphere_hyperplane:
+corollary%important homeomorphic_punctured_sphere_hyperplane:
   fixes a :: "'a :: euclidean_space"
   assumes "0 < r" and b: "b \<in> sphere a r"
       and "c \<noteq> 0"
@@ -864,12 +864,12 @@
 apply (auto simp: affine_hyperplane of_nat_diff)
 done
 
-proposition homeomorphic_punctured_sphere_affine_gen:
+proposition%important homeomorphic_punctured_sphere_affine_gen:
   fixes a :: "'a :: euclidean_space"
   assumes "convex S" "bounded S" and a: "a \<in> rel_frontier S"
       and "affine T" and affS: "aff_dim S = aff_dim T + 1"
     shows "rel_frontier S - {a} homeomorphic T"
-proof -
+proof%unimportant -
   obtain U :: "'a set" where "affine U" "convex U" and affdS: "aff_dim U = aff_dim S"
     using choose_affine_subset [OF affine_UNIV aff_dim_geq]
     by (meson aff_dim_affine_hull affine_affine_hull affine_imp_convex)
@@ -912,13 +912,13 @@
   is homeomorphic to a closed subset of a convex set, and
   if the set is locally compact we can take the convex set to be the universe.\<close>
 
-proposition homeomorphic_closedin_convex:
+proposition%important homeomorphic_closedin_convex:
   fixes S :: "'m::euclidean_space set"
   assumes "aff_dim S < DIM('n)"
   obtains U and T :: "'n::euclidean_space set"
      where "convex U" "U \<noteq> {}" "closedin (subtopology euclidean U) T"
            "S homeomorphic T"
-proof (cases "S = {}")
+proof%unimportant (cases "S = {}")
   case True then show ?thesis
     by (rule_tac U=UNIV and T="{}" in that) auto
 next
@@ -1004,16 +1004,16 @@
     done
 qed
 
-subsection\<open>Locally compact sets in an open set\<close>
+subsection%important\<open>Locally compact sets in an open set\<close>
 
 text\<open> Locally compact sets are closed in an open set and are homeomorphic
   to an absolutely closed set if we have one more dimension to play with.\<close>
 
-lemma locally_compact_open_Int_closure:
+lemma%important locally_compact_open_Int_closure:
   fixes S :: "'a :: metric_space set"
   assumes "locally compact S"
   obtains T where "open T" "S = T \<inter> closure S"
-proof -
+proof%unimportant -
   have "\<forall>x\<in>S. \<exists>T v u. u = S \<inter> T \<and> x \<in> u \<and> u \<subseteq> v \<and> v \<subseteq> S \<and> open T \<and> compact v"
     by (metis assms locally_compact openin_open)
   then obtain t v where
@@ -1048,14 +1048,14 @@
 qed
 
 
-lemma locally_compact_closedin_open:
+lemma%unimportant locally_compact_closedin_open:
     fixes S :: "'a :: metric_space set"
     assumes "locally compact S"
     obtains T where "open T" "closedin (subtopology euclidean T) S"
   by (metis locally_compact_open_Int_closure [OF assms] closed_closure closedin_closed_Int)
 
 
-lemma locally_compact_homeomorphism_projection_closed:
+lemma%unimportant locally_compact_homeomorphism_projection_closed:
   assumes "locally compact S"
   obtains T and f :: "'a \<Rightarrow> 'a :: euclidean_space \<times> 'b :: euclidean_space"
     where "closed T" "homeomorphism S T f fst"
@@ -1108,14 +1108,14 @@
       done
 qed
 
-lemma locally_compact_closed_Int_open:
+lemma%unimportant locally_compact_closed_Int_open:
   fixes S :: "'a :: euclidean_space set"
   shows
     "locally compact S \<longleftrightarrow> (\<exists>U u. closed U \<and> open u \<and> S = U \<inter> u)"
 by (metis closed_closure closed_imp_locally_compact inf_commute locally_compact_Int locally_compact_open_Int_closure open_imp_locally_compact)
 
 
-lemma lowerdim_embeddings:
+lemma%unimportant lowerdim_embeddings:
   assumes  "DIM('a) < DIM('b)"
   obtains f :: "'a::euclidean_space*real \<Rightarrow> 'b::euclidean_space" 
       and g :: "'b \<Rightarrow> 'a*real"
@@ -1161,11 +1161,11 @@
   qed
 qed
 
-proposition locally_compact_homeomorphic_closed:
+proposition%important locally_compact_homeomorphic_closed:
   fixes S :: "'a::euclidean_space set"
   assumes "locally compact S" and dimlt: "DIM('a) < DIM('b)"
   obtains T :: "'b::euclidean_space set" where "closed T" "S homeomorphic T"
-proof -
+proof%unimportant -
   obtain U:: "('a*real)set" and h
     where "closed U" and homU: "homeomorphism S U h fst"
     using locally_compact_homeomorphism_projection_closed assms by metis
@@ -1191,13 +1191,13 @@
 qed
 
 
-lemma homeomorphic_convex_compact_lemma:
+lemma%important homeomorphic_convex_compact_lemma:
   fixes S :: "'a::euclidean_space set"
   assumes "convex S"
     and "compact S"
     and "cball 0 1 \<subseteq> S"
   shows "S homeomorphic (cball (0::'a) 1)"
-proof (rule starlike_compact_projective_special[OF assms(2-3)])
+proof%unimportant (rule starlike_compact_projective_special[OF assms(2-3)])
   fix x u
   assume "x \<in> S" and "0 \<le> u" and "u < (1::real)"
   have "open (ball (u *\<^sub>R x) (1 - u))"
@@ -1223,7 +1223,7 @@
     using frontier_def and interior_subset by auto
 qed
 
-proposition homeomorphic_convex_compact_cball:
+proposition%important homeomorphic_convex_compact_cball:
   fixes e :: real
     and S :: "'a::euclidean_space set"
   assumes "convex S"
@@ -1231,7 +1231,7 @@
     and "interior S \<noteq> {}"
     and "e > 0"
   shows "S homeomorphic (cball (b::'a) e)"
-proof -
+proof%unimportant -
   obtain a where "a \<in> interior S"
     using assms(3) by auto
   then obtain d where "d > 0" and d: "cball a d \<subseteq> S"
@@ -1259,7 +1259,7 @@
     done
 qed
 
-corollary homeomorphic_convex_compact:
+corollary%important homeomorphic_convex_compact:
   fixes S :: "'a::euclidean_space set"
     and T :: "'a set"
   assumes "convex S" "compact S" "interior S \<noteq> {}"
@@ -1268,9 +1268,9 @@
   using assms
   by (meson zero_less_one homeomorphic_trans homeomorphic_convex_compact_cball homeomorphic_sym)
 
-subsection\<open>Covering spaces and lifting results for them\<close>
+subsection%important\<open>Covering spaces and lifting results for them\<close>
 
-definition covering_space
+definition%important covering_space
            :: "'a::topological_space set \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'b::topological_space set \<Rightarrow> bool"
   where
   "covering_space c p S \<equiv>
@@ -1281,19 +1281,19 @@
                         pairwise disjnt v \<and>
                         (\<forall>u \<in> v. \<exists>q. homeomorphism u T p q)))"
 
-lemma covering_space_imp_continuous: "covering_space c p S \<Longrightarrow> continuous_on c p"
+lemma%unimportant covering_space_imp_continuous: "covering_space c p S \<Longrightarrow> continuous_on c p"
   by (simp add: covering_space_def)
 
-lemma covering_space_imp_surjective: "covering_space c p S \<Longrightarrow> p ` c = S"
+lemma%unimportant covering_space_imp_surjective: "covering_space c p S \<Longrightarrow> p ` c = S"
   by (simp add: covering_space_def)
 
-lemma homeomorphism_imp_covering_space: "homeomorphism S T f g \<Longrightarrow> covering_space S f T"
+lemma%unimportant homeomorphism_imp_covering_space: "homeomorphism S T f g \<Longrightarrow> covering_space S f T"
   apply (simp add: homeomorphism_def covering_space_def, clarify)
   apply (rule_tac x=T in exI, simp)
   apply (rule_tac x="{S}" in exI, auto)
   done
 
-lemma covering_space_local_homeomorphism:
+lemma%unimportant covering_space_local_homeomorphism:
   assumes "covering_space c p S" "x \<in> c"
   obtains T u q where "x \<in> T" "openin (subtopology euclidean c) T"
                       "p x \<in> u" "openin (subtopology euclidean S) u"
@@ -1304,13 +1304,13 @@
   by (metis IntI UnionE vimage_eq) 
 
 
-lemma covering_space_local_homeomorphism_alt:
+lemma%important covering_space_local_homeomorphism_alt:
   assumes p: "covering_space c p S" and "y \<in> S"
   obtains x T U q where "p x = y"
                         "x \<in> T" "openin (subtopology euclidean c) T"
                         "y \<in> U" "openin (subtopology euclidean S) U"
                           "homeomorphism T U p q"
-proof -
+proof%unimportant -
   obtain x where "p x = y" "x \<in> c"
     using assms covering_space_imp_surjective by blast
   show ?thesis
@@ -1318,11 +1318,11 @@
     using that \<open>p x = y\<close> by blast
 qed
 
-proposition covering_space_open_map:
+proposition%important covering_space_open_map:
   fixes S :: "'a :: metric_space set" and T :: "'b :: metric_space set"
   assumes p: "covering_space c p S" and T: "openin (subtopology euclidean c) T"
     shows "openin (subtopology euclidean S) (p ` T)"
-proof -
+proof%unimportant -
   have pce: "p ` c = S"
    and covs:
        "\<And>x. x \<in> S \<Longrightarrow>
@@ -1368,7 +1368,7 @@
   with openin_subopen show ?thesis by blast
 qed
 
-lemma covering_space_lift_unique_gen:
+lemma%important covering_space_lift_unique_gen:
   fixes f :: "'a::topological_space \<Rightarrow> 'b::topological_space"
   fixes g1 :: "'a \<Rightarrow> 'c::real_normed_vector"
   assumes cov: "covering_space c p S"
@@ -1380,7 +1380,7 @@
       and fg2: "\<And>x. x \<in> T \<Longrightarrow> f x = p(g2 x)"
       and u_compt: "U \<in> components T" and "a \<in> U" "x \<in> U"
     shows "g1 x = g2 x"
-proof -
+proof%unimportant -
   have "U \<subseteq> T" by (rule in_components_subset [OF u_compt])
   define G12 where "G12 \<equiv> {x \<in> U. g1 x - g2 x = 0}"
   have "connected U" by (rule in_components_connected [OF u_compt])
@@ -1427,7 +1427,7 @@
     using \<open>x \<in> U\<close> by force
 qed
 
-proposition covering_space_lift_unique:
+proposition%important covering_space_lift_unique:
   fixes f :: "'a::topological_space \<Rightarrow> 'b::topological_space"
   fixes g1 :: "'a \<Rightarrow> 'c::real_normed_vector"
   assumes "covering_space c p S"
@@ -1437,9 +1437,10 @@
           "continuous_on T g2"  "g2 ` T \<subseteq> c"  "\<And>x. x \<in> T \<Longrightarrow> f x = p(g2 x)"
           "connected T"  "a \<in> T"   "x \<in> T"
    shows "g1 x = g2 x"
-using covering_space_lift_unique_gen [of c p S] in_components_self assms ex_in_conv by blast
+  using%unimportant covering_space_lift_unique_gen [of c p S] in_components_self assms ex_in_conv
+  by%unimportant blast
 
-lemma covering_space_locally:
+lemma%unimportant covering_space_locally:
   fixes p :: "'a::real_normed_vector \<Rightarrow> 'b::real_normed_vector"
   assumes loc: "locally \<phi> C" and cov: "covering_space C p S"
       and pim: "\<And>T. \<lbrakk>T \<subseteq> C; \<phi> T\<rbrakk> \<Longrightarrow> \<psi>(p ` T)"
@@ -1455,14 +1456,14 @@
 qed
 
 
-proposition covering_space_locally_eq:
+proposition%important covering_space_locally_eq:
   fixes p :: "'a::real_normed_vector \<Rightarrow> 'b::real_normed_vector"
   assumes cov: "covering_space C p S"
       and pim: "\<And>T. \<lbrakk>T \<subseteq> C; \<phi> T\<rbrakk> \<Longrightarrow> \<psi>(p ` T)"
       and qim: "\<And>q U. \<lbrakk>U \<subseteq> S; continuous_on U q; \<psi> U\<rbrakk> \<Longrightarrow> \<phi>(q ` U)"
     shows "locally \<psi> S \<longleftrightarrow> locally \<phi> C"
          (is "?lhs = ?rhs")
-proof
+proof%unimportant
   assume L: ?lhs
   show ?rhs
   proof (rule locallyI)
@@ -1517,7 +1518,7 @@
     using cov covering_space_locally pim by blast
 qed
 
-lemma covering_space_locally_compact_eq:
+lemma%unimportant covering_space_locally_compact_eq:
   fixes p :: "'a::real_normed_vector \<Rightarrow> 'b::real_normed_vector"
   assumes "covering_space C p S"
   shows "locally compact S \<longleftrightarrow> locally compact C"
@@ -1525,7 +1526,7 @@
    apply (meson assms compact_continuous_image continuous_on_subset covering_space_imp_continuous)
   using compact_continuous_image by blast
 
-lemma covering_space_locally_connected_eq:
+lemma%unimportant covering_space_locally_connected_eq:
   fixes p :: "'a::real_normed_vector \<Rightarrow> 'b::real_normed_vector"
   assumes "covering_space C p S"
     shows "locally connected S \<longleftrightarrow> locally connected C"
@@ -1533,7 +1534,7 @@
    apply (meson connected_continuous_image assms continuous_on_subset covering_space_imp_continuous)
   using connected_continuous_image by blast
 
-lemma covering_space_locally_path_connected_eq:
+lemma%unimportant covering_space_locally_path_connected_eq:
   fixes p :: "'a::real_normed_vector \<Rightarrow> 'b::real_normed_vector"
   assumes "covering_space C p S"
     shows "locally path_connected S \<longleftrightarrow> locally path_connected C"
@@ -1542,26 +1543,26 @@
   using path_connected_continuous_image by blast
 
 
-lemma covering_space_locally_compact:
+lemma%unimportant covering_space_locally_compact:
   fixes p :: "'a::real_normed_vector \<Rightarrow> 'b::real_normed_vector"
   assumes "locally compact C" "covering_space C p S"
   shows "locally compact S"
   using assms covering_space_locally_compact_eq by blast
 
 
-lemma covering_space_locally_connected:
+lemma%unimportant covering_space_locally_connected:
   fixes p :: "'a::real_normed_vector \<Rightarrow> 'b::real_normed_vector"
   assumes "locally connected C" "covering_space C p S"
   shows "locally connected S"
   using assms covering_space_locally_connected_eq by blast
 
-lemma covering_space_locally_path_connected:
+lemma%unimportant covering_space_locally_path_connected:
   fixes p :: "'a::real_normed_vector \<Rightarrow> 'b::real_normed_vector"
   assumes "locally path_connected C" "covering_space C p S"
   shows "locally path_connected S"
   using assms covering_space_locally_path_connected_eq by blast
 
-proposition covering_space_lift_homotopy:
+proposition%important covering_space_lift_homotopy:
   fixes p :: "'a::real_normed_vector \<Rightarrow> 'b::real_normed_vector"
     and h :: "real \<times> 'c::real_normed_vector \<Rightarrow> 'b"
   assumes cov: "covering_space C p S"
@@ -1573,7 +1574,7 @@
                     "k ` ({0..1} \<times> U) \<subseteq> C"
                     "\<And>y. y \<in> U \<Longrightarrow> k(0, y) = f y"
                     "\<And>z. z \<in> {0..1} \<times> U \<Longrightarrow> h z = p(k z)"
-proof -
+proof%unimportant -
   have "\<exists>V k. openin (subtopology euclidean U) V \<and> y \<in> V \<and>
               continuous_on ({0..1} \<times> V) k \<and> k ` ({0..1} \<times> V) \<subseteq> C \<and>
               (\<forall>z \<in> V. k(0, z) = f z) \<and> (\<forall>z \<in> {0..1} \<times> V. h z = p(k z))"
@@ -1911,7 +1912,7 @@
   qed (auto simp: contk)
 qed
 
-corollary covering_space_lift_homotopy_alt:
+corollary%important covering_space_lift_homotopy_alt:
   fixes p :: "'a::real_normed_vector \<Rightarrow> 'b::real_normed_vector"
     and h :: "'c::real_normed_vector \<times> real \<Rightarrow> 'b"
   assumes cov: "covering_space C p S"
@@ -1923,7 +1924,7 @@
                   "k ` (U \<times> {0..1}) \<subseteq> C"
                   "\<And>y. y \<in> U \<Longrightarrow> k(y, 0) = f y"
                   "\<And>z. z \<in> U \<times> {0..1} \<Longrightarrow> h z = p(k z)"
-proof -
+proof%unimportant -
   have "continuous_on ({0..1} \<times> U) (h \<circ> (\<lambda>z. (snd z, fst z)))"
     by (intro continuous_intros continuous_on_subset [OF conth]) auto
   then obtain k where contk: "continuous_on ({0..1} \<times> U) k"
@@ -1939,7 +1940,7 @@
     done
 qed
 
-corollary covering_space_lift_homotopic_function:
+corollary%important covering_space_lift_homotopic_function:
   fixes p :: "'a::real_normed_vector \<Rightarrow> 'b::real_normed_vector" and g:: "'c::real_normed_vector \<Rightarrow> 'a"
   assumes cov: "covering_space C p S"
       and contg: "continuous_on U g"
@@ -1947,7 +1948,7 @@
       and pgeq: "\<And>y. y \<in> U \<Longrightarrow> p(g y) = f y"
       and hom: "homotopic_with (\<lambda>x. True) U S f f'"
     obtains g' where "continuous_on U g'" "image g' U \<subseteq> C" "\<And>y. y \<in> U \<Longrightarrow> p(g' y) = f' y"
-proof -
+proof%unimportant -
   obtain h where conth: "continuous_on ({0..1::real} \<times> U) h"
              and him: "h ` ({0..1} \<times> U) \<subseteq> S"
              and h0:  "\<And>x. h(0, x) = f x"
@@ -1971,12 +1972,12 @@
   qed
 qed
 
-corollary covering_space_lift_inessential_function:
+corollary%important covering_space_lift_inessential_function:
   fixes p :: "'a::real_normed_vector \<Rightarrow> 'b::real_normed_vector" and U :: "'c::real_normed_vector set"
   assumes cov: "covering_space C p S"
       and hom: "homotopic_with (\<lambda>x. True) U S f (\<lambda>x. a)"
   obtains g where "continuous_on U g" "g ` U \<subseteq> C" "\<And>y. y \<in> U \<Longrightarrow> p(g y) = f y"
-proof (cases "U = {}")
+proof%unimportant (cases "U = {}")
   case True
   then show ?thesis
     using that continuous_on_empty by blast
@@ -1994,16 +1995,16 @@
     done
 qed
 
-subsection\<open> Lifting of general functions to covering space\<close>
+subsection%important\<open> Lifting of general functions to covering space\<close>
 
-proposition covering_space_lift_path_strong:
+proposition%important covering_space_lift_path_strong:
   fixes p :: "'a::real_normed_vector \<Rightarrow> 'b::real_normed_vector"
     and f :: "'c::real_normed_vector \<Rightarrow> 'b"
   assumes cov: "covering_space C p S" and "a \<in> C"
       and "path g" and pag: "path_image g \<subseteq> S" and pas: "pathstart g = p a"
     obtains h where "path h" "path_image h \<subseteq> C" "pathstart h = a"
                 and "\<And>t. t \<in> {0..1} \<Longrightarrow> p(h t) = g t"
-proof -
+proof%unimportant -
   obtain k:: "real \<times> 'c \<Rightarrow> 'a"
     where contk: "continuous_on ({0..1} \<times> {undefined}) k"
       and kim: "k ` ({0..1} \<times> {undefined}) \<subseteq> C"
@@ -2032,11 +2033,11 @@
   qed
 qed
 
-corollary covering_space_lift_path:
+corollary%important covering_space_lift_path:
   fixes p :: "'a::real_normed_vector \<Rightarrow> 'b::real_normed_vector"
   assumes cov: "covering_space C p S" and "path g" and pig: "path_image g \<subseteq> S"
   obtains h where "path h" "path_image h \<subseteq> C" "\<And>t. t \<in> {0..1} \<Longrightarrow> p(h t) = g t"
-proof -
+proof%unimportant -
   obtain a where "a \<in> C" "pathstart g = p a"
     by (metis pig cov covering_space_imp_surjective imageE pathstart_in_path_image subsetCE)
   show ?thesis
@@ -2045,7 +2046,7 @@
 qed
 
   
-proposition covering_space_lift_homotopic_paths:
+proposition%important covering_space_lift_homotopic_paths:
   fixes p :: "'a::real_normed_vector \<Rightarrow> 'b::real_normed_vector"
   assumes cov: "covering_space C p S"
       and "path g1" and pig1: "path_image g1 \<subseteq> S"
@@ -2055,7 +2056,7 @@
       and "path h2" and pih2: "path_image h2 \<subseteq> C" and ph2: "\<And>t. t \<in> {0..1} \<Longrightarrow> p(h2 t) = g2 t"
       and h1h2: "pathstart h1 = pathstart h2"
     shows "homotopic_paths C h1 h2"
-proof -
+proof%unimportant -
   obtain h :: "real \<times> real \<Rightarrow> 'b"
      where conth: "continuous_on ({0..1} \<times> {0..1}) h"
        and him: "h ` ({0..1} \<times> {0..1}) \<subseteq> S"
@@ -2116,7 +2117,7 @@
 qed
 
 
-corollary covering_space_monodromy:
+corollary%important covering_space_monodromy:
   fixes p :: "'a::real_normed_vector \<Rightarrow> 'b::real_normed_vector"
   assumes cov: "covering_space C p S"
       and "path g1" and pig1: "path_image g1 \<subseteq> S"
@@ -2126,10 +2127,11 @@
       and "path h2" and pih2: "path_image h2 \<subseteq> C" and ph2: "\<And>t. t \<in> {0..1} \<Longrightarrow> p(h2 t) = g2 t"
       and h1h2: "pathstart h1 = pathstart h2"
     shows "pathfinish h1 = pathfinish h2"
-  using covering_space_lift_homotopic_paths [OF assms] homotopic_paths_imp_pathfinish by blast
+  using%unimportant covering_space_lift_homotopic_paths [OF assms] homotopic_paths_imp_pathfinish
+  by%unimportant blast
 
 
-corollary covering_space_lift_homotopic_path:
+corollary%important covering_space_lift_homotopic_path:
   fixes p :: "'a::real_normed_vector \<Rightarrow> 'b::real_normed_vector"
   assumes cov: "covering_space C p S"
       and hom: "homotopic_paths S f f'"
@@ -2138,7 +2140,7 @@
       and pgeq: "\<And>t. t \<in> {0..1} \<Longrightarrow> p(g t) = f t"
   obtains g' where "path g'" "path_image g' \<subseteq> C"
                    "pathstart g' = a" "pathfinish g' = b" "\<And>t. t \<in> {0..1} \<Longrightarrow> p(g' t) = f' t"
-proof (rule covering_space_lift_path_strong [OF cov, of a f'])
+proof%unimportant (rule covering_space_lift_path_strong [OF cov, of a f'])
   show "a \<in> C"
     using a pig by auto
   show "path f'" "path_image f' \<subseteq> S"
@@ -2148,7 +2150,7 @@
 qed (metis (mono_tags, lifting) assms cov covering_space_monodromy hom homotopic_paths_imp_path homotopic_paths_imp_subset pgeq pig)
 
 
-proposition covering_space_lift_general:
+proposition%important covering_space_lift_general:
   fixes p :: "'a::real_normed_vector \<Rightarrow> 'b::real_normed_vector"
     and f :: "'c::real_normed_vector \<Rightarrow> 'b"
   assumes cov: "covering_space C p S" and "a \<in> C" "z \<in> U"
@@ -2160,7 +2162,7 @@
                              pathstart q = a \<and> pathfinish q = a \<and>
                              homotopic_paths S (f \<circ> r) (p \<circ> q)"
   obtains g where "continuous_on U g" "g ` U \<subseteq> C" "g z = a" "\<And>y. y \<in> U \<Longrightarrow> p(g y) = f y"
-proof -
+proof%unimportant -
   have *: "\<exists>g h. path g \<and> path_image g \<subseteq> U \<and>
                  pathstart g = z \<and> pathfinish g = y \<and>
                  path h \<and> path_image h \<subseteq> C \<and> pathstart h = a \<and>
@@ -2403,7 +2405,7 @@
 qed
 
 
-corollary covering_space_lift_stronger:
+corollary%important covering_space_lift_stronger:
   fixes p :: "'a::real_normed_vector \<Rightarrow> 'b::real_normed_vector"
     and f :: "'c::real_normed_vector \<Rightarrow> 'b"
   assumes cov: "covering_space C p S" "a \<in> C" "z \<in> U"
@@ -2413,7 +2415,7 @@
       and hom: "\<And>r. \<lbrakk>path r; path_image r \<subseteq> U; pathstart r = z; pathfinish r = z\<rbrakk>
                      \<Longrightarrow> \<exists>b. homotopic_paths S (f \<circ> r) (linepath b b)"
   obtains g where "continuous_on U g" "g ` U \<subseteq> C" "g z = a" "\<And>y. y \<in> U \<Longrightarrow> p(g y) = f y"
-proof (rule covering_space_lift_general [OF cov U contf fim feq])
+proof%unimportant (rule covering_space_lift_general [OF cov U contf fim feq])
   fix r
   assume "path r" "path_image r \<subseteq> U" "pathstart r = z" "pathfinish r = z"
   then obtain b where b: "homotopic_paths S (f \<circ> r) (linepath b b)"
@@ -2430,7 +2432,7 @@
     by (force simp: \<open>a \<in> C\<close>)
 qed auto
 
-corollary covering_space_lift_strong:
+corollary%important covering_space_lift_strong:
   fixes p :: "'a::real_normed_vector \<Rightarrow> 'b::real_normed_vector"
     and f :: "'c::real_normed_vector \<Rightarrow> 'b"
   assumes cov: "covering_space C p S" "a \<in> C" "z \<in> U"
@@ -2438,7 +2440,7 @@
       and contf: "continuous_on U f" and fim: "f ` U \<subseteq> S"
       and feq: "f z = p a"
   obtains g where "continuous_on U g" "g ` U \<subseteq> C" "g z = a" "\<And>y. y \<in> U \<Longrightarrow> p(g y) = f y"
-proof (rule covering_space_lift_stronger [OF cov _ lpcU contf fim feq])
+proof%unimportant (rule covering_space_lift_stronger [OF cov _ lpcU contf fim feq])
   show "path_connected U"
     using scU simply_connected_eq_contractible_loop_some by blast
   fix r
@@ -2451,14 +2453,14 @@
     by blast
 qed blast
 
-corollary covering_space_lift:
+corollary%important covering_space_lift:
   fixes p :: "'a::real_normed_vector \<Rightarrow> 'b::real_normed_vector"
     and f :: "'c::real_normed_vector \<Rightarrow> 'b"
   assumes cov: "covering_space C p S"
       and U: "simply_connected U"  "locally path_connected U"
       and contf: "continuous_on U f" and fim: "f ` U \<subseteq> S"
     obtains g where "continuous_on U g" "g ` U \<subseteq> C" "\<And>y. y \<in> U \<Longrightarrow> p(g y) = f y"
-proof (cases "U = {}")
+proof%unimportant (cases "U = {}")
   case True
   with that show ?thesis by auto
 next

--- a/src/HOL/Analysis/Operator_Norm.thy	Mon Aug 27 22:58:36 2018 +0200
+++ b/src/HOL/Analysis/Operator_Norm.thy	Tue Aug 28 13:28:39 2018 +0100
@@ -3,7 +3,7 @@
     Author:     Brian Huffman
 *)
 
-section \<open>Operator Norm\<close>
+section%important \<open>Operator Norm\<close>
 
 theory Operator_Norm
 imports Complex_Main
@@ -11,14 +11,14 @@
 
 text \<open>This formulation yields zero if \<open>'a\<close> is the trivial vector space.\<close>
 
-definition onorm :: "('a::real_normed_vector \<Rightarrow> 'b::real_normed_vector) \<Rightarrow> real"
+definition%important onorm :: "('a::real_normed_vector \<Rightarrow> 'b::real_normed_vector) \<Rightarrow> real"
   where "onorm f = (SUP x. norm (f x) / norm x)"
 
-lemma onorm_bound:
+lemma%important onorm_bound:
   assumes "0 \<le> b" and "\<And>x. norm (f x) \<le> b * norm x"
   shows "onorm f \<le> b"
   unfolding onorm_def
-proof (rule cSUP_least)
+proof%unimportant (rule cSUP_least)
   fix x
   show "norm (f x) / norm x \<le> b"
     using assms by (cases "x = 0") (simp_all add: pos_divide_le_eq)
@@ -26,11 +26,11 @@
 
 text \<open>In non-trivial vector spaces, the first assumption is redundant.\<close>
 
-lemma onorm_le:
+lemma%important onorm_le:
   fixes f :: "'a::{real_normed_vector, perfect_space} \<Rightarrow> 'b::real_normed_vector"
   assumes "\<And>x. norm (f x) \<le> b * norm x"
   shows "onorm f \<le> b"
-proof (rule onorm_bound [OF _ assms])
+proof%unimportant (rule onorm_bound [OF _ assms])
   have "{0::'a} \<noteq> UNIV" by (metis not_open_singleton open_UNIV)
   then obtain a :: 'a where "a \<noteq> 0" by fast
   have "0 \<le> b * norm a"
@@ -39,10 +39,10 @@
     by (simp add: zero_le_mult_iff)
 qed
 
-lemma le_onorm:
+lemma%important le_onorm:
   assumes "bounded_linear f"
   shows "norm (f x) / norm x \<le> onorm f"
-proof -
+proof%unimportant -
   interpret f: bounded_linear f by fact
   obtain b where "0 \<le> b" and "\<forall>x. norm (f x) \<le> norm x * b"
     using f.nonneg_bounded by auto
@@ -55,10 +55,10 @@
     unfolding onorm_def by (rule cSUP_upper)
 qed
 
-lemma onorm:
+lemma%important onorm:
   assumes "bounded_linear f"
   shows "norm (f x) \<le> onorm f * norm x"
-proof -
+proof%unimportant -
   interpret f: bounded_linear f by fact
   show ?thesis
   proof (cases)
@@ -73,12 +73,12 @@
   qed
 qed
 
-lemma onorm_pos_le:
+lemma%unimportant onorm_pos_le:
   assumes f: "bounded_linear f"
   shows "0 \<le> onorm f"
   using le_onorm [OF f, where x=0] by simp
 
-lemma onorm_zero: "onorm (\<lambda>x. 0) = 0"
+lemma%unimportant onorm_zero: "onorm (\<lambda>x. 0) = 0"
 proof (rule order_antisym)
   show "onorm (\<lambda>x. 0) \<le> 0"
     by (simp add: onorm_bound)
@@ -86,20 +86,20 @@
     using bounded_linear_zero by (rule onorm_pos_le)
 qed
 
-lemma onorm_eq_0:
+lemma%unimportant onorm_eq_0:
   assumes f: "bounded_linear f"
   shows "onorm f = 0 \<longleftrightarrow> (\<forall>x. f x = 0)"
   using onorm [OF f] by (auto simp: fun_eq_iff [symmetric] onorm_zero)
 
-lemma onorm_pos_lt:
+lemma%unimportant onorm_pos_lt:
   assumes f: "bounded_linear f"
   shows "0 < onorm f \<longleftrightarrow> \<not> (\<forall>x. f x = 0)"
   by (simp add: less_le onorm_pos_le [OF f] onorm_eq_0 [OF f])
 
-lemma onorm_id_le: "onorm (\<lambda>x. x) \<le> 1"
+lemma%unimportant onorm_id_le: "onorm (\<lambda>x. x) \<le> 1"
   by (rule onorm_bound) simp_all
 
-lemma onorm_id: "onorm (\<lambda>x. x::'a::{real_normed_vector, perfect_space}) = 1"
+lemma%unimportant onorm_id: "onorm (\<lambda>x. x::'a::{real_normed_vector, perfect_space}) = 1"
 proof (rule antisym[OF onorm_id_le])
   have "{0::'a} \<noteq> UNIV" by (metis not_open_singleton open_UNIV)
   then obtain x :: 'a where "x \<noteq> 0" by fast
@@ -110,11 +110,11 @@
   finally show "1 \<le> onorm (\<lambda>x::'a. x)" .
 qed
 
-lemma onorm_compose:
+lemma%important onorm_compose:
   assumes f: "bounded_linear f"
   assumes g: "bounded_linear g"
   shows "onorm (f \<circ> g) \<le> onorm f * onorm g"
-proof (rule onorm_bound)
+proof%unimportant (rule onorm_bound)
   show "0 \<le> onorm f * onorm g"
     by (intro mult_nonneg_nonneg onorm_pos_le f g)
 next
@@ -127,7 +127,7 @@
     by (simp add: mult.assoc)
 qed
 
-lemma onorm_scaleR_lemma:
+lemma%unimportant onorm_scaleR_lemma:
   assumes f: "bounded_linear f"
   shows "onorm (\<lambda>x. r *\<^sub>R f x) \<le> \<bar>r\<bar> * onorm f"
 proof (rule onorm_bound)
@@ -141,10 +141,10 @@
     by (simp only: norm_scaleR mult.assoc)
 qed
 
-lemma onorm_scaleR:
+lemma%important onorm_scaleR:
   assumes f: "bounded_linear f"
   shows "onorm (\<lambda>x. r *\<^sub>R f x) = \<bar>r\<bar> * onorm f"
-proof (cases "r = 0")
+proof%unimportant (cases "r = 0")
   assume "r \<noteq> 0"
   show ?thesis
   proof (rule order_antisym)
@@ -160,7 +160,7 @@
   qed
 qed (simp add: onorm_zero)
 
-lemma onorm_scaleR_left_lemma:
+lemma%unimportant onorm_scaleR_left_lemma:
   assumes r: "bounded_linear r"
   shows "onorm (\<lambda>x. r x *\<^sub>R f) \<le> onorm r * norm f"
 proof (rule onorm_bound)
@@ -173,10 +173,10 @@
     by (simp add: ac_simps)
 qed (intro mult_nonneg_nonneg norm_ge_zero onorm_pos_le r)
 
-lemma onorm_scaleR_left:
+lemma%important onorm_scaleR_left:
   assumes f: "bounded_linear r"
   shows "onorm (\<lambda>x. r x *\<^sub>R f) = onorm r * norm f"
-proof (cases "f = 0")
+proof%unimportant (cases "f = 0")
   assume "f \<noteq> 0"
   show ?thesis
   proof (rule order_antisym)
@@ -200,15 +200,15 @@
   qed
 qed (simp add: onorm_zero)
 
-lemma onorm_neg:
+lemma%unimportant onorm_neg:
   shows "onorm (\<lambda>x. - f x) = onorm f"
   unfolding onorm_def by simp
 
-lemma onorm_triangle:
+lemma%important onorm_triangle:
   assumes f: "bounded_linear f"
   assumes g: "bounded_linear g"
   shows "onorm (\<lambda>x. f x + g x) \<le> onorm f + onorm g"
-proof (rule onorm_bound)
+proof%unimportant (rule onorm_bound)
   show "0 \<le> onorm f + onorm g"
     by (intro add_nonneg_nonneg onorm_pos_le f g)
 next
@@ -221,26 +221,26 @@
     by (simp only: distrib_right)
 qed
 
-lemma onorm_triangle_le:
+lemma%important onorm_triangle_le:
   assumes "bounded_linear f"
   assumes "bounded_linear g"
   assumes "onorm f + onorm g \<le> e"
   shows "onorm (\<lambda>x. f x + g x) \<le> e"
-  using assms by (rule onorm_triangle [THEN order_trans])
+  using%unimportant assms by%unimportant (rule onorm_triangle [THEN order_trans])
 
-lemma onorm_triangle_lt:
+lemma%unimportant onorm_triangle_lt:
   assumes "bounded_linear f"
   assumes "bounded_linear g"
   assumes "onorm f + onorm g < e"
   shows "onorm (\<lambda>x. f x + g x) < e"
   using assms by (rule onorm_triangle [THEN order_le_less_trans])
 
-lemma onorm_sum:
+lemma%important onorm_sum:
   assumes "finite S"
   assumes "\<And>s. s \<in> S \<Longrightarrow> bounded_linear (f s)"
   shows "onorm (\<lambda>x. sum (\<lambda>s. f s x) S) \<le> sum (\<lambda>s. onorm (f s)) S"
-  using assms
-  by (induction) (auto simp: onorm_zero intro!: onorm_triangle_le bounded_linear_sum)
+  using%unimportant assms
+  by%unimportant (induction) (auto simp: onorm_zero intro!: onorm_triangle_le bounded_linear_sum)
 
 lemmas onorm_sum_le = onorm_sum[THEN order_trans]
 

--- a/src/HOL/Analysis/Ordered_Euclidean_Space.thy	Mon Aug 27 22:58:36 2018 +0200
+++ b/src/HOL/Analysis/Ordered_Euclidean_Space.thy	Tue Aug 28 13:28:39 2018 +0100
@@ -4,7 +4,7 @@
   "HOL-Library.Product_Order"
 begin
 
-subsection \<open>An ordering on euclidean spaces that will allow us to talk about intervals\<close>
+subsection%important \<open>An ordering on euclidean spaces that will allow us to talk about intervals\<close>
 
 class ordered_euclidean_space = ord + inf + sup + abs + Inf + Sup + euclidean_space +
   assumes eucl_le: "x \<le> y \<longleftrightarrow> (\<forall>i\<in>Basis. x \<bullet> i \<le> y \<bullet> i)"
@@ -33,7 +33,7 @@
   by standard (auto simp: eucl_inf eucl_sup sup_inf_distrib1 intro!: euclidean_eqI)
 
 subclass conditionally_complete_lattice
-proof
+proof%unimportant
   fix z::'a and X::"'a set"
   assume "X \<noteq> {}"
   hence "\<And>i. (\<lambda>x. x \<bullet> i) ` X \<noteq> {}" by simp
@@ -44,42 +44,42 @@
       simp: bdd_below_def bdd_above_def preorder_class.bdd_below_def preorder_class.bdd_above_def
         eucl_Inf eucl_Sup eucl_le)+
 
-lemma inner_Basis_inf_left: "i \<in> Basis \<Longrightarrow> inf x y \<bullet> i = inf (x \<bullet> i) (y \<bullet> i)"
+lemma%unimportant inner_Basis_inf_left: "i \<in> Basis \<Longrightarrow> inf x y \<bullet> i = inf (x \<bullet> i) (y \<bullet> i)"
   and inner_Basis_sup_left: "i \<in> Basis \<Longrightarrow> sup x y \<bullet> i = sup (x \<bullet> i) (y \<bullet> i)"
   by (simp_all add: eucl_inf eucl_sup inner_sum_left inner_Basis if_distrib comm_monoid_add_class.sum.delta
       cong: if_cong)
 
-lemma inner_Basis_INF_left: "i \<in> Basis \<Longrightarrow> (INF x:X. f x) \<bullet> i = (INF x:X. f x \<bullet> i)"
+lemma%unimportant inner_Basis_INF_left: "i \<in> Basis \<Longrightarrow> (INF x:X. f x) \<bullet> i = (INF x:X. f x \<bullet> i)"
   and inner_Basis_SUP_left: "i \<in> Basis \<Longrightarrow> (SUP x:X. f x) \<bullet> i = (SUP x:X. f x \<bullet> i)"
   using eucl_Sup [of "f ` X"] eucl_Inf [of "f ` X"] by (simp_all add: comp_def)
 
-lemma abs_inner: "i \<in> Basis \<Longrightarrow> \<bar>x\<bar> \<bullet> i = \<bar>x \<bullet> i\<bar>"
+lemma%unimportant abs_inner: "i \<in> Basis \<Longrightarrow> \<bar>x\<bar> \<bullet> i = \<bar>x \<bullet> i\<bar>"
   by (auto simp: eucl_abs)
 
-lemma
+lemma%unimportant
   abs_scaleR: "\<bar>a *\<^sub>R b\<bar> = \<bar>a\<bar> *\<^sub>R \<bar>b\<bar>"
   by (auto simp: eucl_abs abs_mult intro!: euclidean_eqI)
 
-lemma interval_inner_leI:
+lemma%unimportant interval_inner_leI:
   assumes "x \<in> {a .. b}" "0 \<le> i"
   shows "a\<bullet>i \<le> x\<bullet>i" "x\<bullet>i \<le> b\<bullet>i"
   using assms
   unfolding euclidean_inner[of a i] euclidean_inner[of x i] euclidean_inner[of b i]
   by (auto intro!: ordered_comm_monoid_add_class.sum_mono mult_right_mono simp: eucl_le)
 
-lemma inner_nonneg_nonneg:
+lemma%unimportant inner_nonneg_nonneg:
   shows "0 \<le> a \<Longrightarrow> 0 \<le> b \<Longrightarrow> 0 \<le> a \<bullet> b"
   using interval_inner_leI[of a 0 a b]
   by auto
 
-lemma inner_Basis_mono:
+lemma%unimportant inner_Basis_mono:
   shows "a \<le> b \<Longrightarrow> c \<in> Basis  \<Longrightarrow> a \<bullet> c \<le> b \<bullet> c"
   by (simp add: eucl_le)
 
-lemma Basis_nonneg[intro, simp]: "i \<in> Basis \<Longrightarrow> 0 \<le> i"
+lemma%unimportant Basis_nonneg[intro, simp]: "i \<in> Basis \<Longrightarrow> 0 \<le> i"
   by (auto simp: eucl_le inner_Basis)
 
-lemma Sup_eq_maximum_componentwise:
+lemma%unimportant Sup_eq_maximum_componentwise:
   fixes s::"'a set"
   assumes i: "\<And>b. b \<in> Basis \<Longrightarrow> X \<bullet> b = i b \<bullet> b"
   assumes sup: "\<And>b x. b \<in> Basis \<Longrightarrow> x \<in> s \<Longrightarrow> x \<bullet> b \<le> X \<bullet> b"
@@ -89,7 +89,7 @@
   unfolding eucl_Sup euclidean_representation_sum
   by (auto intro!: conditionally_complete_lattice_class.cSup_eq_maximum)
 
-lemma Inf_eq_minimum_componentwise:
+lemma%unimportant Inf_eq_minimum_componentwise:
   assumes i: "\<And>b. b \<in> Basis \<Longrightarrow> X \<bullet> b = i b \<bullet> b"
   assumes sup: "\<And>b x. b \<in> Basis \<Longrightarrow> x \<in> s \<Longrightarrow> X \<bullet> b \<le> x \<bullet> b"
   assumes i_s: "\<And>b. b \<in> Basis \<Longrightarrow> (i b \<bullet> b) \<in> (\<lambda>x. x \<bullet> b) ` s"
@@ -100,12 +100,12 @@
 
 end
 
-lemma
+lemma%important
   compact_attains_Inf_componentwise:
   fixes b::"'a::ordered_euclidean_space"
   assumes "b \<in> Basis" assumes "X \<noteq> {}" "compact X"
   obtains x where "x \<in> X" "x \<bullet> b = Inf X \<bullet> b" "\<And>y. y \<in> X \<Longrightarrow> x \<bullet> b \<le> y \<bullet> b"
-proof atomize_elim
+proof%unimportant atomize_elim
   let ?proj = "(\<lambda>x. x \<bullet> b) ` X"
   from assms have "compact ?proj" "?proj \<noteq> {}"
     by (auto intro!: compact_continuous_image continuous_intros)
@@ -122,12 +122,12 @@
   with x show "\<exists>x. x \<in> X \<and> x \<bullet> b = Inf X \<bullet> b \<and> (\<forall>y. y \<in> X \<longrightarrow> x \<bullet> b \<le> y \<bullet> b)" by blast
 qed
 
-lemma
+lemma%important
   compact_attains_Sup_componentwise:
   fixes b::"'a::ordered_euclidean_space"
   assumes "b \<in> Basis" assumes "X \<noteq> {}" "compact X"
   obtains x where "x \<in> X" "x \<bullet> b = Sup X \<bullet> b" "\<And>y. y \<in> X \<Longrightarrow> y \<bullet> b \<le> x \<bullet> b"
-proof atomize_elim
+proof%unimportant atomize_elim
   let ?proj = "(\<lambda>x. x \<bullet> b) ` X"
   from assms have "compact ?proj" "?proj \<noteq> {}"
     by (auto intro!: compact_continuous_image continuous_intros)
@@ -144,14 +144,14 @@
   with x show "\<exists>x. x \<in> X \<and> x \<bullet> b = Sup X \<bullet> b \<and> (\<forall>y. y \<in> X \<longrightarrow> y \<bullet> b \<le> x \<bullet> b)" by blast
 qed
 
-lemma (in order) atLeastatMost_empty'[simp]:
+lemma%unimportant (in order) atLeastatMost_empty'[simp]:
   "(~ a <= b) \<Longrightarrow> {a..b} = {}"
   by (auto)
 
 instance real :: ordered_euclidean_space
   by standard auto
 
-lemma in_Basis_prod_iff:
+lemma%unimportant in_Basis_prod_iff:
   fixes i::"'a::euclidean_space*'b::euclidean_space"
   shows "i \<in> Basis \<longleftrightarrow> fst i = 0 \<and> snd i \<in> Basis \<or> snd i = 0 \<and> fst i \<in> Basis"
   by (cases i) (auto simp: Basis_prod_def)
@@ -174,52 +174,52 @@
 
 text\<open>Instantiation for intervals on \<open>ordered_euclidean_space\<close>\<close>
 
-lemma
+lemma%important
   fixes a :: "'a::ordered_euclidean_space"
   shows cbox_interval: "cbox a b = {a..b}"
     and interval_cbox: "{a..b} = cbox a b"
     and eucl_le_atMost: "{x. \<forall>i\<in>Basis. x \<bullet> i <= a \<bullet> i} = {..a}"
     and eucl_le_atLeast: "{x. \<forall>i\<in>Basis. a \<bullet> i <= x \<bullet> i} = {a..}"
-    by (auto simp: eucl_le[where 'a='a] eucl_less_def box_def cbox_def)
+    by%unimportant (auto simp: eucl_le[where 'a='a] eucl_less_def box_def cbox_def)
 
-lemma vec_nth_real_1_iff_cbox [simp]:
+lemma%unimportant vec_nth_real_1_iff_cbox [simp]:
   fixes a b :: real
   shows "(\<lambda>x::real^1. x $ 1) ` S = {a..b} \<longleftrightarrow> S = cbox (vec a) (vec b)"
   by (metis interval_cbox vec_nth_1_iff_cbox)
 
-lemma closed_eucl_atLeastAtMost[simp, intro]:
+lemma%unimportant closed_eucl_atLeastAtMost[simp, intro]:
   fixes a :: "'a::ordered_euclidean_space"
   shows "closed {a..b}"
   by (simp add: cbox_interval[symmetric] closed_cbox)
 
-lemma closed_eucl_atMost[simp, intro]:
+lemma%unimportant closed_eucl_atMost[simp, intro]:
   fixes a :: "'a::ordered_euclidean_space"
   shows "closed {..a}"
   by (simp add: closed_interval_left eucl_le_atMost[symmetric])
 
-lemma closed_eucl_atLeast[simp, intro]:
+lemma%unimportant closed_eucl_atLeast[simp, intro]:
   fixes a :: "'a::ordered_euclidean_space"
   shows "closed {a..}"
   by (simp add: closed_interval_right eucl_le_atLeast[symmetric])
 
-lemma bounded_closed_interval [simp]:
+lemma%unimportant bounded_closed_interval [simp]:
   fixes a :: "'a::ordered_euclidean_space"
   shows "bounded {a .. b}"
   using bounded_cbox[of a b]
   by (metis interval_cbox)
 
-lemma convex_closed_interval [simp]:
+lemma%unimportant convex_closed_interval [simp]:
   fixes a :: "'a::ordered_euclidean_space"
   shows "convex {a .. b}"
   using convex_box[of a b]
   by (metis interval_cbox)
 
-lemma image_smult_interval:"(\<lambda>x. m *\<^sub>R (x::_::ordered_euclidean_space)) ` {a .. b} =
+lemma%unimportant image_smult_interval:"(\<lambda>x. m *\<^sub>R (x::_::ordered_euclidean_space)) ` {a .. b} =
   (if {a .. b} = {} then {} else if 0 \<le> m then {m *\<^sub>R a .. m *\<^sub>R b} else {m *\<^sub>R b .. m *\<^sub>R a})"
   using image_smult_cbox[of m a b]
   by (simp add: cbox_interval)
 
-lemma [simp]:
+lemma%unimportant [simp]:
   fixes a b::"'a::ordered_euclidean_space" and r s::real
   shows is_interval_io: "is_interval {..<r}"
     and is_interval_ic: "is_interval {..a}"
@@ -231,15 +231,15 @@
     and is_interval_cc: "is_interval {b..a}"
   by (force simp: is_interval_def eucl_le[where 'a='a])+
 
-lemma is_interval_real_ereal_oo: "is_interval (real_of_ereal ` {N<..<M::ereal})"
+lemma%unimportant is_interval_real_ereal_oo: "is_interval (real_of_ereal ` {N<..<M::ereal})"
   by (auto simp: real_atLeastGreaterThan_eq)
 
-lemma compact_interval [simp]:
+lemma%unimportant compact_interval [simp]:
   fixes a b::"'a::ordered_euclidean_space"
   shows "compact {a .. b}"
   by (metis compact_cbox interval_cbox)
 
-lemma homeomorphic_closed_intervals:
+lemma%unimportant homeomorphic_closed_intervals:
   fixes a :: "'a::euclidean_space" and b and c :: "'a::euclidean_space" and d
   assumes "box a b \<noteq> {}" and "box c d \<noteq> {}"
     shows "(cbox a b) homeomorphic (cbox c d)"
@@ -247,7 +247,7 @@
 using assms apply auto
 done
 
-lemma homeomorphic_closed_intervals_real:
+lemma%unimportant homeomorphic_closed_intervals_real:
   fixes a::real and b and c::real and d
   assumes "a<b" and "c<d"
   shows "{a..b} homeomorphic {c..d}"
@@ -256,10 +256,10 @@
 no_notation
   eucl_less (infix "<e" 50)
 
-lemma One_nonneg: "0 \<le> (\<Sum>Basis::'a::ordered_euclidean_space)"
+lemma%unimportant One_nonneg: "0 \<le> (\<Sum>Basis::'a::ordered_euclidean_space)"
   by (auto intro: sum_nonneg)
 
-lemma
+lemma%unimportant
   fixes a b::"'a::ordered_euclidean_space"
   shows bdd_above_cbox[intro, simp]: "bdd_above (cbox a b)"
     and bdd_below_cbox[intro, simp]: "bdd_below (cbox a b)"
@@ -272,11 +272,11 @@
 instantiation vec :: (ordered_euclidean_space, finite) ordered_euclidean_space
 begin
 
-definition "inf x y = (\<chi> i. inf (x $ i) (y $ i))"
-definition "sup x y = (\<chi> i. sup (x $ i) (y $ i))"
-definition "Inf X = (\<chi> i. (INF x:X. x $ i))"
-definition "Sup X = (\<chi> i. (SUP x:X. x $ i))"
-definition "\<bar>x\<bar> = (\<chi> i. \<bar>x $ i\<bar>)"
+definition%important "inf x y = (\<chi> i. inf (x $ i) (y $ i))"
+definition%important "sup x y = (\<chi> i. sup (x $ i) (y $ i))"
+definition%important "Inf X = (\<chi> i. (INF x:X. x $ i))"
+definition%important "Sup X = (\<chi> i. (SUP x:X. x $ i))"
+definition%important "\<bar>x\<bar> = (\<chi> i. \<bar>x $ i\<bar>)"
 
 instance
   apply standard
@@ -288,12 +288,12 @@
 
 end
 
-lemma ANR_interval [iff]:
+lemma%unimportant ANR_interval [iff]:
     fixes a :: "'a::ordered_euclidean_space"
     shows "ANR{a..b}"
 by (simp add: interval_cbox)
 
-lemma ENR_interval [iff]:
+lemma%unimportant ENR_interval [iff]:
     fixes a :: "'a::ordered_euclidean_space"
     shows "ENR{a..b}"
   by (auto simp: interval_cbox)

--- a/src/HOL/Analysis/Poly_Roots.thy	Mon Aug 27 22:58:36 2018 +0200
+++ b/src/HOL/Analysis/Poly_Roots.thy	Tue Aug 28 13:28:39 2018 +0100
@@ -2,19 +2,19 @@
     Ported from "hol_light/Multivariate/complexes.ml" by L C Paulson
 *)
 
-section \<open>polynomial functions: extremal behaviour and root counts\<close>
+section%important \<open>polynomial functions: extremal behaviour and root counts\<close>
 
 theory Poly_Roots
 imports Complex_Main
 begin
 
-subsection\<open>Basics about polynomial functions: extremal behaviour and root counts\<close>
+subsection%important\<open>Basics about polynomial functions: extremal behaviour and root counts\<close>
 
-lemma sub_polyfun:
+lemma%important sub_polyfun:
   fixes x :: "'a::{comm_ring,monoid_mult}"
   shows   "(\<Sum>i\<le>n. a i * x^i) - (\<Sum>i\<le>n. a i * y^i) =
            (x - y) * (\<Sum>j<n. \<Sum>k= Suc j..n. a k * y^(k - Suc j) * x^j)"
-proof -
+proof%unimportant -
   have "(\<Sum>i\<le>n. a i * x^i) - (\<Sum>i\<le>n. a i * y^i) =
         (\<Sum>i\<le>n. a i * (x^i - y^i))"
     by (simp add: algebra_simps sum_subtractf [symmetric])
@@ -27,7 +27,7 @@
   finally show ?thesis .
 qed
 
-lemma sub_polyfun_alt:
+lemma%unimportant sub_polyfun_alt:
   fixes x :: "'a::{comm_ring,monoid_mult}"
   shows   "(\<Sum>i\<le>n. a i * x^i) - (\<Sum>i\<le>n. a i * y^i) =
            (x - y) * (\<Sum>j<n. \<Sum>k<n-j. a (j+k+1) * y^k * x^j)"
@@ -40,7 +40,7 @@
     by (simp add: sub_polyfun)
 qed
 
-lemma polyfun_linear_factor:
+lemma%unimportant polyfun_linear_factor:
   fixes a :: "'a::{comm_ring,monoid_mult}"
   shows  "\<exists>b. \<forall>z. (\<Sum>i\<le>n. c i * z^i) =
                   (z-a) * (\<Sum>i<n. b i * z^i) + (\<Sum>i\<le>n. c i * a^i)"
@@ -57,21 +57,21 @@
     by (intro exI allI)
 qed
 
-lemma polyfun_linear_factor_root:
+lemma%important polyfun_linear_factor_root:
   fixes a :: "'a::{comm_ring,monoid_mult}"
   assumes "(\<Sum>i\<le>n. c i * a^i) = 0"
   shows  "\<exists>b. \<forall>z. (\<Sum>i\<le>n. c i * z^i) = (z-a) * (\<Sum>i<n. b i * z^i)"
-  using polyfun_linear_factor [of c n a] assms
-  by simp
+  using%unimportant polyfun_linear_factor [of c n a] assms
+  by%unimportant simp
 
-lemma adhoc_norm_triangle: "a + norm(y) \<le> b ==> norm(x) \<le> a ==> norm(x + y) \<le> b"
+lemma%unimportant adhoc_norm_triangle: "a + norm(y) \<le> b ==> norm(x) \<le> a ==> norm(x + y) \<le> b"
   by (metis norm_triangle_mono order.trans order_refl)
 
-lemma polyfun_extremal_lemma:
+lemma%important polyfun_extremal_lemma:
   fixes c :: "nat \<Rightarrow> 'a::real_normed_div_algebra"
   assumes "e > 0"
     shows "\<exists>M. \<forall>z. M \<le> norm z \<longrightarrow> norm(\<Sum>i\<le>n. c i * z^i) \<le> e * norm(z) ^ Suc n"
-proof (induction n)
+proof%unimportant (induction n)
   case 0
   show ?case
     by (rule exI [where x="norm (c 0) / e"]) (auto simp: mult.commute pos_divide_le_eq assms)
@@ -102,7 +102,7 @@
     qed
 qed
 
-lemma norm_lemma_xy: assumes "\<bar>b\<bar> + 1 \<le> norm(y) - a" "norm(x) \<le> a" shows "b \<le> norm(x + y)"
+lemma%unimportant norm_lemma_xy: assumes "\<bar>b\<bar> + 1 \<le> norm(y) - a" "norm(x) \<le> a" shows "b \<le> norm(x + y)"
 proof -
   have "b \<le> norm y - norm x"
     using assms by linarith
@@ -110,12 +110,12 @@
     by (metis (no_types) add.commute norm_diff_ineq order_trans)
 qed
 
-lemma polyfun_extremal:
+lemma%important polyfun_extremal:
   fixes c :: "nat \<Rightarrow> 'a::real_normed_div_algebra"
   assumes "\<exists>k. k \<noteq> 0 \<and> k \<le> n \<and> c k \<noteq> 0"
     shows "eventually (\<lambda>z. norm(\<Sum>i\<le>n. c i * z^i) \<ge> B) at_infinity"
 using assms
-proof (induction n)
+proof%unimportant (induction n)
   case 0 then show ?case
     by simp
 next
@@ -149,12 +149,12 @@
   qed
 qed
 
-lemma polyfun_rootbound:
+lemma%important polyfun_rootbound:
  fixes c :: "nat \<Rightarrow> 'a::{comm_ring,real_normed_div_algebra}"
  assumes "\<exists>k. k \<le> n \<and> c k \<noteq> 0"
    shows "finite {z. (\<Sum>i\<le>n. c i * z^i) = 0} \<and> card {z. (\<Sum>i\<le>n. c i * z^i) = 0} \<le> n"
 using assms
-proof (induction n arbitrary: c)
+proof%unimportant (induction n arbitrary: c)
  case (Suc n) show ?case
  proof (cases "{z. (\<Sum>i\<le>Suc n. c i * z^i) = 0} = {}")
    case False
@@ -182,17 +182,17 @@
    qed simp
 qed simp
 
-corollary
+corollary%important (*FIX ME needs name *)
   fixes c :: "nat \<Rightarrow> 'a::{comm_ring,real_normed_div_algebra}"
   assumes "\<exists>k. k \<le> n \<and> c k \<noteq> 0"
     shows polyfun_rootbound_finite: "finite {z. (\<Sum>i\<le>n. c i * z^i) = 0}"
       and polyfun_rootbound_card:   "card {z. (\<Sum>i\<le>n. c i * z^i) = 0} \<le> n"
 using polyfun_rootbound [OF assms] by auto
 
-lemma polyfun_finite_roots:
+lemma%important polyfun_finite_roots:
   fixes c :: "nat \<Rightarrow> 'a::{comm_ring,real_normed_div_algebra}"
     shows  "finite {z. (\<Sum>i\<le>n. c i * z^i) = 0} \<longleftrightarrow> (\<exists>k. k \<le> n \<and> c k \<noteq> 0)"
-proof (cases " \<exists>k\<le>n. c k \<noteq> 0")
+proof%unimportant (cases " \<exists>k\<le>n. c k \<noteq> 0")
   case True then show ?thesis
     by (blast intro: polyfun_rootbound_finite)
 next
@@ -200,7 +200,7 @@
     by (auto simp: infinite_UNIV_char_0)
 qed
 
-lemma polyfun_eq_0:
+lemma%unimportant polyfun_eq_0:
   fixes c :: "nat \<Rightarrow> 'a::{comm_ring,real_normed_div_algebra}"
     shows  "(\<forall>z. (\<Sum>i\<le>n. c i * z^i) = 0) \<longleftrightarrow> (\<forall>k. k \<le> n \<longrightarrow> c k = 0)"
 proof (cases "(\<forall>z. (\<Sum>i\<le>n. c i * z^i) = 0)")
@@ -215,10 +215,10 @@
     by auto
 qed
 
-lemma polyfun_eq_const:
+lemma%important polyfun_eq_const:
   fixes c :: "nat \<Rightarrow> 'a::{comm_ring,real_normed_div_algebra}"
     shows  "(\<forall>z. (\<Sum>i\<le>n. c i * z^i) = k) \<longleftrightarrow> c 0 = k \<and> (\<forall>k. k \<noteq> 0 \<and> k \<le> n \<longrightarrow> c k = 0)"
-proof -
+proof%unimportant -
   {fix z
     have "(\<Sum>i\<le>n. c i * z^i) = (\<Sum>i\<le>n. (if i = 0 then c 0 - k else c i) * z^i) + k"
       by (induct n) auto

--- a/src/HOL/Analysis/Polytope.thy	Mon Aug 27 22:58:36 2018 +0200
+++ b/src/HOL/Analysis/Polytope.thy	Tue Aug 28 13:28:39 2018 +0100
@@ -6,18 +6,18 @@
 imports Cartesian_Euclidean_Space
 begin
 
-subsection \<open>Faces of a (usually convex) set\<close>
-
-definition face_of :: "['a::real_vector set, 'a set] \<Rightarrow> bool" (infixr "(face'_of)" 50)
+subsection%important \<open>Faces of a (usually convex) set\<close>
+
+definition%important face_of :: "['a::real_vector set, 'a set] \<Rightarrow> bool" (infixr "(face'_of)" 50)
   where
   "T face_of S \<longleftrightarrow>
         T \<subseteq> S \<and> convex T \<and>
         (\<forall>a \<in> S. \<forall>b \<in> S. \<forall>x \<in> T. x \<in> open_segment a b \<longrightarrow> a \<in> T \<and> b \<in> T)"
 
-lemma face_ofD: "\<lbrakk>T face_of S; x \<in> open_segment a b; a \<in> S; b \<in> S; x \<in> T\<rbrakk> \<Longrightarrow> a \<in> T \<and> b \<in> T"
+lemma%unimportant face_ofD: "\<lbrakk>T face_of S; x \<in> open_segment a b; a \<in> S; b \<in> S; x \<in> T\<rbrakk> \<Longrightarrow> a \<in> T \<and> b \<in> T"
   unfolding face_of_def by blast
 
-lemma face_of_translation_eq [simp]:
+lemma%unimportant face_of_translation_eq [simp]:
     "((+) a ` T face_of (+) a ` S) \<longleftrightarrow> T face_of S"
 proof -
   have *: "\<And>a T S. T face_of S \<Longrightarrow> ((+) a ` T face_of (+) a ` S)"
@@ -32,52 +32,52 @@
     done
 qed
 
-lemma face_of_linear_image:
+lemma%unimportant face_of_linear_image:
   assumes "linear f" "inj f"
     shows "(f ` c face_of f ` S) \<longleftrightarrow> c face_of S"
 by (simp add: face_of_def inj_image_subset_iff inj_image_mem_iff open_segment_linear_image assms)
 
-lemma face_of_refl: "convex S \<Longrightarrow> S face_of S"
+lemma%unimportant face_of_refl: "convex S \<Longrightarrow> S face_of S"
   by (auto simp: face_of_def)
 
-lemma face_of_refl_eq: "S face_of S \<longleftrightarrow> convex S"
+lemma%unimportant face_of_refl_eq: "S face_of S \<longleftrightarrow> convex S"
   by (auto simp: face_of_def)
 
-lemma empty_face_of [iff]: "{} face_of S"
+lemma%unimportant empty_face_of [iff]: "{} face_of S"
   by (simp add: face_of_def)
 
-lemma face_of_empty [simp]: "S face_of {} \<longleftrightarrow> S = {}"
+lemma%unimportant face_of_empty [simp]: "S face_of {} \<longleftrightarrow> S = {}"
   by (meson empty_face_of face_of_def subset_empty)
 
-lemma face_of_trans [trans]: "\<lbrakk>S face_of T; T face_of u\<rbrakk> \<Longrightarrow> S face_of u"
+lemma%unimportant face_of_trans [trans]: "\<lbrakk>S face_of T; T face_of u\<rbrakk> \<Longrightarrow> S face_of u"
   unfolding face_of_def by (safe; blast)
 
-lemma face_of_face: "T face_of S \<Longrightarrow> (f face_of T \<longleftrightarrow> f face_of S \<and> f \<subseteq> T)"
+lemma%unimportant face_of_face: "T face_of S \<Longrightarrow> (f face_of T \<longleftrightarrow> f face_of S \<and> f \<subseteq> T)"
   unfolding face_of_def by (safe; blast)
 
-lemma face_of_subset: "\<lbrakk>F face_of S; F \<subseteq> T; T \<subseteq> S\<rbrakk> \<Longrightarrow> F face_of T"
+lemma%unimportant face_of_subset: "\<lbrakk>F face_of S; F \<subseteq> T; T \<subseteq> S\<rbrakk> \<Longrightarrow> F face_of T"
   unfolding face_of_def by (safe; blast)
 
-lemma face_of_slice: "\<lbrakk>F face_of S; convex T\<rbrakk> \<Longrightarrow> (F \<inter> T) face_of (S \<inter> T)"
+lemma%unimportant face_of_slice: "\<lbrakk>F face_of S; convex T\<rbrakk> \<Longrightarrow> (F \<inter> T) face_of (S \<inter> T)"
   unfolding face_of_def by (blast intro: convex_Int)
 
-lemma face_of_Int: "\<lbrakk>t1 face_of S; t2 face_of S\<rbrakk> \<Longrightarrow> (t1 \<inter> t2) face_of S"
+lemma%unimportant face_of_Int: "\<lbrakk>t1 face_of S; t2 face_of S\<rbrakk> \<Longrightarrow> (t1 \<inter> t2) face_of S"
   unfolding face_of_def by (blast intro: convex_Int)
 
-lemma face_of_Inter: "\<lbrakk>A \<noteq> {}; \<And>T. T \<in> A \<Longrightarrow> T face_of S\<rbrakk> \<Longrightarrow> (\<Inter> A) face_of S"
+lemma%unimportant face_of_Inter: "\<lbrakk>A \<noteq> {}; \<And>T. T \<in> A \<Longrightarrow> T face_of S\<rbrakk> \<Longrightarrow> (\<Inter> A) face_of S"
   unfolding face_of_def by (blast intro: convex_Inter)
 
-lemma face_of_Int_Int: "\<lbrakk>F face_of T; F' face_of t'\<rbrakk> \<Longrightarrow> (F \<inter> F') face_of (T \<inter> t')"
+lemma%unimportant face_of_Int_Int: "\<lbrakk>F face_of T; F' face_of t'\<rbrakk> \<Longrightarrow> (F \<inter> F') face_of (T \<inter> t')"
   unfolding face_of_def by (blast intro: convex_Int)
 
-lemma face_of_imp_subset: "T face_of S \<Longrightarrow> T \<subseteq> S"
+lemma%unimportant face_of_imp_subset: "T face_of S \<Longrightarrow> T \<subseteq> S"
   unfolding face_of_def by blast
 
-lemma face_of_imp_eq_affine_Int:
+lemma%important face_of_imp_eq_affine_Int:
   fixes S :: "'a::euclidean_space set"
   assumes S: "convex S"  and T: "T face_of S"
   shows "T = (affine hull T) \<inter> S"
-proof -
+proof%unimportant -
   have "convex T" using T by (simp add: face_of_def)
   have *: False if x: "x \<in> affine hull T" and "x \<in> S" "x \<notin> T" and y: "y \<in> rel_interior T" for x y
   proof -
@@ -114,15 +114,15 @@
     done
 qed
 
-lemma face_of_imp_closed:
+lemma%unimportant face_of_imp_closed:
      fixes S :: "'a::euclidean_space set"
      assumes "convex S" "closed S" "T face_of S" shows "closed T"
   by (metis affine_affine_hull affine_closed closed_Int face_of_imp_eq_affine_Int assms)
 
-lemma face_of_Int_supporting_hyperplane_le_strong:
+lemma%important face_of_Int_supporting_hyperplane_le_strong:
     assumes "convex(S \<inter> {x. a \<bullet> x = b})" and aleb: "\<And>x. x \<in> S \<Longrightarrow> a \<bullet> x \<le> b"
       shows "(S \<inter> {x. a \<bullet> x = b}) face_of S"
-proof -
+proof%unimportant -
   have *: "a \<bullet> u = a \<bullet> x" if "x \<in> open_segment u v" "u \<in> S" "v \<in> S" and b: "b = a \<bullet> x"
           for u v x
   proof (rule antisym)
@@ -145,33 +145,33 @@
     using "*" open_segment_commute by blast
 qed
 
-lemma face_of_Int_supporting_hyperplane_ge_strong:
+lemma%unimportant face_of_Int_supporting_hyperplane_ge_strong:
    "\<lbrakk>convex(S \<inter> {x. a \<bullet> x = b}); \<And>x. x \<in> S \<Longrightarrow> a \<bullet> x \<ge> b\<rbrakk>
     \<Longrightarrow> (S \<inter> {x. a \<bullet> x = b}) face_of S"
   using face_of_Int_supporting_hyperplane_le_strong [of S "-a" "-b"] by simp
 
-lemma face_of_Int_supporting_hyperplane_le:
+lemma%unimportant face_of_Int_supporting_hyperplane_le:
     "\<lbrakk>convex S; \<And>x. x \<in> S \<Longrightarrow> a \<bullet> x \<le> b\<rbrakk> \<Longrightarrow> (S \<inter> {x. a \<bullet> x = b}) face_of S"
   by (simp add: convex_Int convex_hyperplane face_of_Int_supporting_hyperplane_le_strong)
 
-lemma face_of_Int_supporting_hyperplane_ge:
+lemma%unimportant face_of_Int_supporting_hyperplane_ge:
     "\<lbrakk>convex S; \<And>x. x \<in> S \<Longrightarrow> a \<bullet> x \<ge> b\<rbrakk> \<Longrightarrow> (S \<inter> {x. a \<bullet> x = b}) face_of S"
   by (simp add: convex_Int convex_hyperplane face_of_Int_supporting_hyperplane_ge_strong)
 
-lemma face_of_imp_convex: "T face_of S \<Longrightarrow> convex T"
+lemma%unimportant face_of_imp_convex: "T face_of S \<Longrightarrow> convex T"
   using face_of_def by blast
 
-lemma face_of_imp_compact:
+lemma%unimportant face_of_imp_compact:
     fixes S :: "'a::euclidean_space set"
     shows "\<lbrakk>convex S; compact S; T face_of S\<rbrakk> \<Longrightarrow> compact T"
   by (meson bounded_subset compact_eq_bounded_closed face_of_imp_closed face_of_imp_subset)
 
-lemma face_of_Int_subface:
+lemma%unimportant face_of_Int_subface:
      "\<lbrakk>A \<inter> B face_of A; A \<inter> B face_of B; C face_of A; D face_of B\<rbrakk>
       \<Longrightarrow> (C \<inter> D) face_of C \<and> (C \<inter> D) face_of D"
   by (meson face_of_Int_Int face_of_face inf_le1 inf_le2)
 
-lemma subset_of_face_of:
+lemma%unimportant subset_of_face_of:
     fixes S :: "'a::real_normed_vector set"
     assumes "T face_of S" "u \<subseteq> S" "T \<inter> (rel_interior u) \<noteq> {}"
       shows "u \<subseteq> T"
@@ -213,7 +213,7 @@
   qed
 qed
 
-lemma face_of_eq:
+lemma%unimportant face_of_eq:
     fixes S :: "'a::real_normed_vector set"
     assumes "T face_of S" "u face_of S" "(rel_interior T) \<inter> (rel_interior u) \<noteq> {}"
       shows "T = u"
@@ -221,13 +221,13 @@
   apply (metis assms disjoint_iff_not_equal face_of_imp_subset rel_interior_subset subsetCE subset_of_face_of)
   by (metis assms disjoint_iff_not_equal face_of_imp_subset rel_interior_subset subset_iff subset_of_face_of)
 
-lemma face_of_disjoint_rel_interior:
+lemma%unimportant face_of_disjoint_rel_interior:
       fixes S :: "'a::real_normed_vector set"
       assumes "T face_of S" "T \<noteq> S"
         shows "T \<inter> rel_interior S = {}"
   by (meson assms subset_of_face_of face_of_imp_subset order_refl subset_antisym)
 
-lemma face_of_disjoint_interior:
+lemma%unimportant face_of_disjoint_interior:
       fixes S :: "'a::real_normed_vector set"
       assumes "T face_of S" "T \<noteq> S"
         shows "T \<inter> interior S = {}"
@@ -238,19 +238,19 @@
     by (metis (no_types) Int_greatest assms face_of_disjoint_rel_interior inf_sup_ord(1) subset_empty)
 qed
 
-lemma face_of_subset_rel_boundary:
+lemma%unimportant face_of_subset_rel_boundary:
   fixes S :: "'a::real_normed_vector set"
   assumes "T face_of S" "T \<noteq> S"
     shows "T \<subseteq> (S - rel_interior S)"
 by (meson DiffI assms disjoint_iff_not_equal face_of_disjoint_rel_interior face_of_imp_subset rev_subsetD subsetI)
 
-lemma face_of_subset_rel_frontier:
+lemma%unimportant face_of_subset_rel_frontier:
     fixes S :: "'a::real_normed_vector set"
     assumes "T face_of S" "T \<noteq> S"
       shows "T \<subseteq> rel_frontier S"
   using assms closure_subset face_of_disjoint_rel_interior face_of_imp_subset rel_frontier_def by fastforce
 
-lemma face_of_aff_dim_lt:
+lemma%unimportant face_of_aff_dim_lt:
   fixes S :: "'a::euclidean_space set"
   assumes "convex S" "T face_of S" "T \<noteq> S"
     shows "aff_dim T < aff_dim S"
@@ -268,7 +268,7 @@
     by simp
 qed
 
-lemma subset_of_face_of_affine_hull:
+lemma%unimportant subset_of_face_of_affine_hull:
     fixes S :: "'a::euclidean_space set"
   assumes T: "T face_of S" and "convex S" "U \<subseteq> S" and dis: "~disjnt (affine hull T) (rel_interior U)"
   shows "U \<subseteq> T"
@@ -277,13 +277,13 @@
   using rel_interior_subset [of U] dis
   using \<open>U \<subseteq> S\<close> disjnt_def by fastforce
 
-lemma affine_hull_face_of_disjoint_rel_interior:
+lemma%unimportant affine_hull_face_of_disjoint_rel_interior:
     fixes S :: "'a::euclidean_space set"
   assumes "convex S" "F face_of S" "F \<noteq> S"
   shows "affine hull F \<inter> rel_interior S = {}"
   by (metis assms disjnt_def face_of_imp_subset order_refl subset_antisym subset_of_face_of_affine_hull)
 
-lemma affine_diff_divide:
+lemma%unimportant affine_diff_divide:
     assumes "affine S" "k \<noteq> 0" "k \<noteq> 1" and xy: "x \<in> S" "y /\<^sub>R (1 - k) \<in> S"
       shows "(x - y) /\<^sub>R k \<in> S"
 proof -
@@ -294,10 +294,10 @@
     using \<open>affine S\<close> xy by (auto simp: affine_alt)
 qed
 
-lemma face_of_convex_hulls:
+lemma%important face_of_convex_hulls:
       assumes S: "finite S" "T \<subseteq> S" and disj: "affine hull T \<inter> convex hull (S - T) = {}"
       shows  "(convex hull T) face_of (convex hull S)"
-proof -
+proof%unimportant -
   have fin: "finite T" "finite (S - T)" using assms
     by (auto simp: finite_subset)
   have *: "x \<in> convex hull T"
@@ -391,16 +391,16 @@
     using open_segment_commute by (auto simp: face_of_def intro: *)
 qed
 
-proposition face_of_convex_hull_insert:
+proposition%important face_of_convex_hull_insert:
    "\<lbrakk>finite S; a \<notin> affine hull S; T face_of convex hull S\<rbrakk> \<Longrightarrow> T face_of convex hull insert a S"
   apply (rule face_of_trans, blast)
   apply (rule face_of_convex_hulls; force simp: insert_Diff_if)
   done
 
-proposition face_of_affine_trivial:
+proposition%important face_of_affine_trivial:
     assumes "affine S" "T face_of S"
     shows "T = {} \<or> T = S"
-proof (rule ccontr, clarsimp)
+proof%unimportant (rule ccontr, clarsimp)
   assume "T \<noteq> {}" "T \<noteq> S"
   then obtain a where "a \<in> T" by auto
   then have "a \<in> S"
@@ -430,16 +430,16 @@
 qed
 
 
-lemma face_of_affine_eq:
+lemma%unimportant face_of_affine_eq:
    "affine S \<Longrightarrow> (T face_of S \<longleftrightarrow> T = {} \<or> T = S)"
 using affine_imp_convex face_of_affine_trivial face_of_refl by auto
 
 
-lemma Inter_faces_finite_altbound:
+lemma%important Inter_faces_finite_altbound:
     fixes T :: "'a::euclidean_space set set"
     assumes cfaI: "\<And>c. c \<in> T \<Longrightarrow> c face_of S"
     shows "\<exists>F'. finite F' \<and> F' \<subseteq> T \<and> card F' \<le> DIM('a) + 2 \<and> \<Inter>F' = \<Inter>T"
-proof (cases "\<forall>F'. finite F' \<and> F' \<subseteq> T \<and> card F' \<le> DIM('a) + 2 \<longrightarrow> (\<exists>c. c \<in> T \<and> c \<inter> (\<Inter>F') \<subset> (\<Inter>F'))")
+proof%unimportant (cases "\<forall>F'. finite F' \<and> F' \<subseteq> T \<and> card F' \<le> DIM('a) + 2 \<longrightarrow> (\<exists>c. c \<in> T \<and> c \<inter> (\<Inter>F') \<subset> (\<Inter>F'))")
   case True
   then obtain c where c:
        "\<And>F'. \<lbrakk>finite F'; F' \<subseteq> T; card F' \<le> DIM('a) + 2\<rbrakk> \<Longrightarrow> c F' \<in> T \<and> c F' \<inter> (\<Inter>F') \<subset> (\<Inter>F')"
@@ -499,17 +499,17 @@
     by blast
 qed
 
-lemma faces_of_translation:
+lemma%unimportant faces_of_translation:
    "{F. F face_of image (\<lambda>x. a + x) S} = image (image (\<lambda>x. a + x)) {F. F face_of S}"
 apply (rule subset_antisym, clarify)
 apply (auto simp: image_iff)
 apply (metis face_of_imp_subset face_of_translation_eq subset_imageE)
 done
 
-proposition face_of_Times:
+proposition%important face_of_Times:
   assumes "F face_of S" and "F' face_of S'"
     shows "(F \<times> F') face_of (S \<times> S')"
-proof -
+proof%unimportant -
   have "F \<times> F' \<subseteq> S \<times> S'"
     using assms [unfolded face_of_def] by blast
   moreover
@@ -531,11 +531,11 @@
     unfolding face_of_def by blast
 qed
 
-corollary face_of_Times_decomp:
+corollary%important face_of_Times_decomp:
     fixes S :: "'a::euclidean_space set" and S' :: "'b::euclidean_space set"
     shows "c face_of (S \<times> S') \<longleftrightarrow> (\<exists>F F'. F face_of S \<and> F' face_of S' \<and> c = F \<times> F')"
      (is "?lhs = ?rhs")
-proof
+proof%unimportant
   assume c: ?lhs
   show ?rhs
   proof (cases "c = {}")
@@ -582,13 +582,13 @@
   assume ?rhs with face_of_Times show ?lhs by auto
 qed
 
-lemma face_of_Times_eq:
+lemma%unimportant face_of_Times_eq:
     fixes S :: "'a::euclidean_space set" and S' :: "'b::euclidean_space set"
     shows "(F \<times> F') face_of (S \<times> S') \<longleftrightarrow>
            F = {} \<or> F' = {} \<or> F face_of S \<and> F' face_of S'"
 by (auto simp: face_of_Times_decomp times_eq_iff)
 
-lemma hyperplane_face_of_halfspace_le: "{x. a \<bullet> x = b} face_of {x. a \<bullet> x \<le> b}"
+lemma%unimportant hyperplane_face_of_halfspace_le: "{x. a \<bullet> x = b} face_of {x. a \<bullet> x \<le> b}"
 proof -
   have "{x. a \<bullet> x \<le> b} \<inter> {x. a \<bullet> x = b} = {x. a \<bullet> x = b}"
     by auto
@@ -596,7 +596,7 @@
   show ?thesis by auto
 qed
 
-lemma hyperplane_face_of_halfspace_ge: "{x. a \<bullet> x = b} face_of {x. a \<bullet> x \<ge> b}"
+lemma%unimportant hyperplane_face_of_halfspace_ge: "{x. a \<bullet> x = b} face_of {x. a \<bullet> x \<ge> b}"
 proof -
   have "{x. a \<bullet> x \<ge> b} \<inter> {x. a \<bullet> x = b} = {x. a \<bullet> x = b}"
     by auto
@@ -604,12 +604,12 @@
   show ?thesis by auto
 qed
 
-lemma face_of_halfspace_le:
+lemma%important face_of_halfspace_le:
   fixes a :: "'n::euclidean_space"
   shows "F face_of {x. a \<bullet> x \<le> b} \<longleftrightarrow>
          F = {} \<or> F = {x. a \<bullet> x = b} \<or> F = {x. a \<bullet> x \<le> b}"
      (is "?lhs = ?rhs")
-proof (cases "a = 0")
+proof%unimportant (cases "a = 0")
   case True then show ?thesis
     using face_of_affine_eq affine_UNIV by auto
 next
@@ -635,37 +635,37 @@
   qed
 qed
 
-lemma face_of_halfspace_ge:
+lemma%unimportant face_of_halfspace_ge:
   fixes a :: "'n::euclidean_space"
   shows "F face_of {x. a \<bullet> x \<ge> b} \<longleftrightarrow>
          F = {} \<or> F = {x. a \<bullet> x = b} \<or> F = {x. a \<bullet> x \<ge> b}"
 using face_of_halfspace_le [of F "-a" "-b"] by simp
 
-subsection\<open>Exposed faces\<close>
+subsection%important\<open>Exposed faces\<close>
 
 text\<open>That is, faces that are intersection with supporting hyperplane\<close>
 
-definition exposed_face_of :: "['a::euclidean_space set, 'a set] \<Rightarrow> bool"
+definition%important exposed_face_of :: "['a::euclidean_space set, 'a set] \<Rightarrow> bool"
                                (infixr "(exposed'_face'_of)" 50)
   where "T exposed_face_of S \<longleftrightarrow>
          T face_of S \<and> (\<exists>a b. S \<subseteq> {x. a \<bullet> x \<le> b} \<and> T = S \<inter> {x. a \<bullet> x = b})"
 
-lemma empty_exposed_face_of [iff]: "{} exposed_face_of S"
+lemma%unimportant empty_exposed_face_of [iff]: "{} exposed_face_of S"
   apply (simp add: exposed_face_of_def)
   apply (rule_tac x=0 in exI)
   apply (rule_tac x=1 in exI, force)
   done
 
-lemma exposed_face_of_refl_eq [simp]: "S exposed_face_of S \<longleftrightarrow> convex S"
+lemma%unimportant exposed_face_of_refl_eq [simp]: "S exposed_face_of S \<longleftrightarrow> convex S"
   apply (simp add: exposed_face_of_def face_of_refl_eq, auto)
   apply (rule_tac x=0 in exI)+
   apply force
   done
 
-lemma exposed_face_of_refl: "convex S \<Longrightarrow> S exposed_face_of S"
+lemma%unimportant exposed_face_of_refl: "convex S \<Longrightarrow> S exposed_face_of S"
   by simp
 
-lemma exposed_face_of:
+lemma%unimportant exposed_face_of:
     "T exposed_face_of S \<longleftrightarrow>
      T face_of S \<and>
      (T = {} \<or> T = S \<or>
@@ -688,19 +688,19 @@
   qed
 qed
 
-lemma exposed_face_of_Int_supporting_hyperplane_le:
+lemma%unimportant exposed_face_of_Int_supporting_hyperplane_le:
    "\<lbrakk>convex S; \<And>x. x \<in> S \<Longrightarrow> a \<bullet> x \<le> b\<rbrakk> \<Longrightarrow> (S \<inter> {x. a \<bullet> x = b}) exposed_face_of S"
 by (force simp: exposed_face_of_def face_of_Int_supporting_hyperplane_le)
 
-lemma exposed_face_of_Int_supporting_hyperplane_ge:
+lemma%unimportant exposed_face_of_Int_supporting_hyperplane_ge:
    "\<lbrakk>convex S; \<And>x. x \<in> S \<Longrightarrow> a \<bullet> x \<ge> b\<rbrakk> \<Longrightarrow> (S \<inter> {x. a \<bullet> x = b}) exposed_face_of S"
 using exposed_face_of_Int_supporting_hyperplane_le [of S "-a" "-b"] by simp
 
-proposition exposed_face_of_Int:
+proposition%important exposed_face_of_Int:
   assumes "T exposed_face_of S"
       and "u exposed_face_of S"
     shows "(T \<inter> u) exposed_face_of S"
-proof -
+proof%unimportant -
   obtain a b where T: "S \<inter> {x. a \<bullet> x = b} face_of S"
                and S: "S \<subseteq> {x. a \<bullet> x \<le> b}"
                and teq: "T = S \<inter> {x. a \<bullet> x = b}"
@@ -722,12 +722,12 @@
     done
 qed
 
-proposition exposed_face_of_Inter:
+proposition%important exposed_face_of_Inter:
     fixes P :: "'a::euclidean_space set set"
   assumes "P \<noteq> {}"
       and "\<And>T. T \<in> P \<Longrightarrow> T exposed_face_of S"
     shows "\<Inter>P exposed_face_of S"
-proof -
+proof%unimportant -
   obtain Q where "finite Q" and QsubP: "Q \<subseteq> P" "card Q \<le> DIM('a) + 2" and IntQ: "\<Inter>Q = \<Inter>P"
     using Inter_faces_finite_altbound [of P S] assms [unfolded exposed_face_of]
     by force
@@ -749,14 +749,14 @@
   qed
 qed
 
-proposition exposed_face_of_sums:
+proposition%important exposed_face_of_sums:
   assumes "convex S" and "convex T"
       and "F exposed_face_of {x + y | x y. x \<in> S \<and> y \<in> T}"
           (is "F exposed_face_of ?ST")
   obtains k l
     where "k exposed_face_of S" "l exposed_face_of T"
           "F = {x + y | x y. x \<in> k \<and> y \<in> l}"
-proof (cases "F = {}")
+proof%unimportant (cases "F = {}")
   case True then show ?thesis
     using that by blast
 next
@@ -805,14 +805,14 @@
   qed
 qed
 
-lemma exposed_face_of_parallel:
+lemma%important exposed_face_of_parallel:
    "T exposed_face_of S \<longleftrightarrow>
          T face_of S \<and>
          (\<exists>a b. S \<subseteq> {x. a \<bullet> x \<le> b} \<and> T = S \<inter> {x. a \<bullet> x = b} \<and>
                 (T \<noteq> {} \<longrightarrow> T \<noteq> S \<longrightarrow> a \<noteq> 0) \<and>
                 (T \<noteq> S \<longrightarrow> (\<forall>w \<in> affine hull S. (w + a) \<in> affine hull S)))"
   (is "?lhs = ?rhs")
-proof
+proof%unimportant
   assume ?lhs then show ?rhs
   proof (clarsimp simp: exposed_face_of_def)
     fix a b
@@ -874,40 +874,40 @@
     unfolding exposed_face_of_def by blast
 qed
 
-subsection\<open>Extreme points of a set: its singleton faces\<close>
-
-definition extreme_point_of :: "['a::real_vector, 'a set] \<Rightarrow> bool"
+subsection%important\<open>Extreme points of a set: its singleton faces\<close>
+
+definition%important extreme_point_of :: "['a::real_vector, 'a set] \<Rightarrow> bool"
                                (infixr "(extreme'_point'_of)" 50)
   where "x extreme_point_of S \<longleftrightarrow>
          x \<in> S \<and> (\<forall>a \<in> S. \<forall>b \<in> S. x \<notin> open_segment a b)"
 
-lemma extreme_point_of_stillconvex:
+lemma%unimportant extreme_point_of_stillconvex:
    "convex S \<Longrightarrow> (x extreme_point_of S \<longleftrightarrow> x \<in> S \<and> convex(S - {x}))"
   by (fastforce simp add: convex_contains_segment extreme_point_of_def open_segment_def)
 
-lemma face_of_singleton:
+lemma%unimportant face_of_singleton:
    "{x} face_of S \<longleftrightarrow> x extreme_point_of S"
 by (fastforce simp add: extreme_point_of_def face_of_def)
 
-lemma extreme_point_not_in_REL_INTERIOR:
+lemma%unimportant extreme_point_not_in_REL_INTERIOR:
     fixes S :: "'a::real_normed_vector set"
     shows "\<lbrakk>x extreme_point_of S; S \<noteq> {x}\<rbrakk> \<Longrightarrow> x \<notin> rel_interior S"
 apply (simp add: face_of_singleton [symmetric])
 apply (blast dest: face_of_disjoint_rel_interior)
 done
 
-lemma extreme_point_not_in_interior:
+lemma%important extreme_point_not_in_interior:
     fixes S :: "'a::{real_normed_vector, perfect_space} set"
     shows "x extreme_point_of S \<Longrightarrow> x \<notin> interior S"
 apply (case_tac "S = {x}")
 apply (simp add: empty_interior_finite)
 by (meson contra_subsetD extreme_point_not_in_REL_INTERIOR interior_subset_rel_interior)
 
-lemma extreme_point_of_face:
+lemma%unimportant extreme_point_of_face:
      "F face_of S \<Longrightarrow> v extreme_point_of F \<longleftrightarrow> v extreme_point_of S \<and> v \<in> F"
   by (meson empty_subsetI face_of_face face_of_singleton insert_subset)
 
-lemma extreme_point_of_convex_hull:
+lemma%unimportant extreme_point_of_convex_hull:
    "x extreme_point_of (convex hull S) \<Longrightarrow> x \<in> S"
 apply (simp add: extreme_point_of_stillconvex)
 using hull_minimal [of S "(convex hull S) - {x}" convex]
@@ -915,52 +915,52 @@
 apply blast
 done
 
-lemma extreme_points_of_convex_hull:
+lemma%important extreme_points_of_convex_hull:
    "{x. x extreme_point_of (convex hull S)} \<subseteq> S"
-using extreme_point_of_convex_hull by auto
-
-lemma extreme_point_of_empty [simp]: "~ (x extreme_point_of {})"
+using%unimportant extreme_point_of_convex_hull by auto
+
+lemma%unimportant extreme_point_of_empty [simp]: "~ (x extreme_point_of {})"
   by (simp add: extreme_point_of_def)
 
-lemma extreme_point_of_singleton [iff]: "x extreme_point_of {a} \<longleftrightarrow> x = a"
+lemma%unimportant extreme_point_of_singleton [iff]: "x extreme_point_of {a} \<longleftrightarrow> x = a"
   using extreme_point_of_stillconvex by auto
 
-lemma extreme_point_of_translation_eq:
+lemma%unimportant extreme_point_of_translation_eq:
    "(a + x) extreme_point_of (image (\<lambda>x. a + x) S) \<longleftrightarrow> x extreme_point_of S"
 by (auto simp: extreme_point_of_def)
 
-lemma extreme_points_of_translation:
+lemma%important extreme_points_of_translation:
    "{x. x extreme_point_of (image (\<lambda>x. a + x) S)} =
     (\<lambda>x. a + x) ` {x. x extreme_point_of S}"
-using extreme_point_of_translation_eq
-by auto (metis (no_types, lifting) image_iff mem_Collect_eq minus_add_cancel)
-
-lemma extreme_point_of_Int:
+using%unimportant extreme_point_of_translation_eq
+by%unimportant auto (metis (no_types, lifting) image_iff mem_Collect_eq minus_add_cancel)
+
+lemma%unimportant extreme_point_of_Int:
    "\<lbrakk>x extreme_point_of S; x extreme_point_of T\<rbrakk> \<Longrightarrow> x extreme_point_of (S \<inter> T)"
 by (simp add: extreme_point_of_def)
 
-lemma extreme_point_of_Int_supporting_hyperplane_le:
+lemma%important extreme_point_of_Int_supporting_hyperplane_le:
    "\<lbrakk>S \<inter> {x. a \<bullet> x = b} = {c}; \<And>x. x \<in> S \<Longrightarrow> a \<bullet> x \<le> b\<rbrakk> \<Longrightarrow> c extreme_point_of S"
 apply (simp add: face_of_singleton [symmetric])
 by (metis face_of_Int_supporting_hyperplane_le_strong convex_singleton)
 
-lemma extreme_point_of_Int_supporting_hyperplane_ge:
+lemma%unimportant extreme_point_of_Int_supporting_hyperplane_ge:
    "\<lbrakk>S \<inter> {x. a \<bullet> x = b} = {c}; \<And>x. x \<in> S \<Longrightarrow> a \<bullet> x \<ge> b\<rbrakk> \<Longrightarrow> c extreme_point_of S"
 apply (simp add: face_of_singleton [symmetric])
 by (metis face_of_Int_supporting_hyperplane_ge_strong convex_singleton)
 
-lemma exposed_point_of_Int_supporting_hyperplane_le:
+lemma%unimportant exposed_point_of_Int_supporting_hyperplane_le:
    "\<lbrakk>S \<inter> {x. a \<bullet> x = b} = {c}; \<And>x. x \<in> S \<Longrightarrow> a \<bullet> x \<le> b\<rbrakk> \<Longrightarrow> {c} exposed_face_of S"
 apply (simp add: exposed_face_of_def face_of_singleton)
 apply (force simp: extreme_point_of_Int_supporting_hyperplane_le)
 done
 
-lemma exposed_point_of_Int_supporting_hyperplane_ge:
+lemma%unimportant exposed_point_of_Int_supporting_hyperplane_ge:
     "\<lbrakk>S \<inter> {x. a \<bullet> x = b} = {c}; \<And>x. x \<in> S \<Longrightarrow> a \<bullet> x \<ge> b\<rbrakk> \<Longrightarrow> {c} exposed_face_of S"
 using exposed_point_of_Int_supporting_hyperplane_le [of S "-a" "-b" c]
 by simp
 
-lemma extreme_point_of_convex_hull_insert:
+lemma%unimportant extreme_point_of_convex_hull_insert:
    "\<lbrakk>finite S; a \<notin> convex hull S\<rbrakk> \<Longrightarrow> a extreme_point_of (convex hull (insert a S))"
 apply (case_tac "a \<in> S")
 apply (simp add: hull_inc)
@@ -968,40 +968,40 @@
 apply (auto simp: face_of_singleton hull_same)
 done
 
-subsection\<open>Facets\<close>
-
-definition facet_of :: "['a::euclidean_space set, 'a set] \<Rightarrow> bool"
+subsection%important\<open>Facets\<close>
+
+definition%important facet_of :: "['a::euclidean_space set, 'a set] \<Rightarrow> bool"
                     (infixr "(facet'_of)" 50)
   where "F facet_of S \<longleftrightarrow> F face_of S \<and> F \<noteq> {} \<and> aff_dim F = aff_dim S - 1"
 
-lemma facet_of_empty [simp]: "~ S facet_of {}"
+lemma%unimportant facet_of_empty [simp]: "~ S facet_of {}"
   by (simp add: facet_of_def)
 
-lemma facet_of_irrefl [simp]: "~ S facet_of S "
+lemma%unimportant facet_of_irrefl [simp]: "~ S facet_of S "
   by (simp add: facet_of_def)
 
-lemma facet_of_imp_face_of: "F facet_of S \<Longrightarrow> F face_of S"
+lemma%unimportant facet_of_imp_face_of: "F facet_of S \<Longrightarrow> F face_of S"
   by (simp add: facet_of_def)
 
-lemma facet_of_imp_subset: "F facet_of S \<Longrightarrow> F \<subseteq> S"
+lemma%unimportant facet_of_imp_subset: "F facet_of S \<Longrightarrow> F \<subseteq> S"
   by (simp add: face_of_imp_subset facet_of_def)
 
-lemma hyperplane_facet_of_halfspace_le:
+lemma%unimportant hyperplane_facet_of_halfspace_le:
    "a \<noteq> 0 \<Longrightarrow> {x. a \<bullet> x = b} facet_of {x. a \<bullet> x \<le> b}"
 unfolding facet_of_def hyperplane_eq_empty
 by (auto simp: hyperplane_face_of_halfspace_ge hyperplane_face_of_halfspace_le
            DIM_positive Suc_leI of_nat_diff aff_dim_halfspace_le)
 
-lemma hyperplane_facet_of_halfspace_ge:
+lemma%unimportant hyperplane_facet_of_halfspace_ge:
     "a \<noteq> 0 \<Longrightarrow> {x. a \<bullet> x = b} facet_of {x. a \<bullet> x \<ge> b}"
 unfolding facet_of_def hyperplane_eq_empty
 by (auto simp: hyperplane_face_of_halfspace_le hyperplane_face_of_halfspace_ge
            DIM_positive Suc_leI of_nat_diff aff_dim_halfspace_ge)
 
-lemma facet_of_halfspace_le:
+lemma%important facet_of_halfspace_le:
     "F facet_of {x. a \<bullet> x \<le> b} \<longleftrightarrow> a \<noteq> 0 \<and> F = {x. a \<bullet> x = b}"
     (is "?lhs = ?rhs")
-proof
+proof%unimportant
   assume c: ?lhs
   with c facet_of_irrefl show ?rhs
     by (force simp: aff_dim_halfspace_le facet_of_def face_of_halfspace_le cong: conj_cong split: if_split_asm)
@@ -1010,26 +1010,26 @@
     by (simp add: hyperplane_facet_of_halfspace_le)
 qed
 
-lemma facet_of_halfspace_ge:
+lemma%unimportant facet_of_halfspace_ge:
     "F facet_of {x. a \<bullet> x \<ge> b} \<longleftrightarrow> a \<noteq> 0 \<and> F = {x. a \<bullet> x = b}"
 using facet_of_halfspace_le [of F "-a" "-b"] by simp
 
-subsection \<open>Edges: faces of affine dimension 1\<close>
-
-definition edge_of :: "['a::euclidean_space set, 'a set] \<Rightarrow> bool"  (infixr "(edge'_of)" 50)
+subsection%important \<open>Edges: faces of affine dimension 1\<close>
+
+definition%important edge_of :: "['a::euclidean_space set, 'a set] \<Rightarrow> bool"  (infixr "(edge'_of)" 50)
   where "e edge_of S \<longleftrightarrow> e face_of S \<and> aff_dim e = 1"
 
-lemma edge_of_imp_subset:
+lemma%unimportant edge_of_imp_subset:
    "S edge_of T \<Longrightarrow> S \<subseteq> T"
 by (simp add: edge_of_def face_of_imp_subset)
 
-subsection\<open>Existence of extreme points\<close>
-
-lemma different_norm_3_collinear_points:
+subsection%important\<open>Existence of extreme points\<close>
+
+lemma%important different_norm_3_collinear_points:
   fixes a :: "'a::euclidean_space"
   assumes "x \<in> open_segment a b" "norm(a) = norm(b)" "norm(x) = norm(b)"
   shows False
-proof -
+proof%unimportant -
   obtain u where "norm ((1 - u) *\<^sub>R a + u *\<^sub>R b) = norm b"
              and "a \<noteq> b"
              and u01: "0 < u" "u < 1"
@@ -1050,11 +1050,11 @@
     using \<open>a \<noteq> b\<close> by force
 qed
 
-proposition extreme_point_exists_convex:
+proposition%important extreme_point_exists_convex:
   fixes S :: "'a::euclidean_space set"
   assumes "compact S" "convex S" "S \<noteq> {}"
   obtains x where "x extreme_point_of S"
-proof -
+proof%unimportant -
   obtain x where "x \<in> S" and xsup: "\<And>y. y \<in> S \<Longrightarrow> norm y \<le> norm x"
     using distance_attains_sup [of S 0] assms by auto
   have False if "a \<in> S" "b \<in> S" and x: "x \<in> open_segment a b" for a b
@@ -1099,13 +1099,13 @@
     done
 qed
 
-subsection\<open>Krein-Milman, the weaker form\<close>
-
-proposition Krein_Milman:
+subsection%important\<open>Krein-Milman, the weaker form\<close>
+
+proposition%important Krein_Milman:
   fixes S :: "'a::euclidean_space set"
   assumes "compact S" "convex S"
     shows "S = closure(convex hull {x. x extreme_point_of S})"
-proof (cases "S = {}")
+proof%unimportant (cases "S = {}")
   case True then show ?thesis   by simp
 next
   case False
@@ -1155,12 +1155,12 @@
 
 text\<open>Now the sharper form.\<close>
 
-lemma Krein_Milman_Minkowski_aux:
+lemma%important Krein_Milman_Minkowski_aux:
   fixes S :: "'a::euclidean_space set"
   assumes n: "dim S = n" and S: "compact S" "convex S" "0 \<in> S"
     shows "0 \<in> convex hull {x. x extreme_point_of S}"
 using n S
-proof (induction n arbitrary: S rule: less_induct)
+proof%unimportant (induction n arbitrary: S rule: less_induct)
   case (less n S) show ?case
   proof (cases "0 \<in> rel_interior S")
     case True with Krein_Milman show ?thesis
@@ -1197,11 +1197,11 @@
 qed
 
 
-theorem Krein_Milman_Minkowski:
+theorem%important Krein_Milman_Minkowski:
   fixes S :: "'a::euclidean_space set"
   assumes "compact S" "convex S"
     shows "S = convex hull {x. x extreme_point_of S}"
-proof
+proof%unimportant
   show "S \<subseteq> convex hull {x. x extreme_point_of S}"
   proof
     fix a assume [simp]: "a \<in> S"
@@ -1225,17 +1225,17 @@
 qed
 
 
-subsection\<open>Applying it to convex hulls of explicitly indicated finite sets\<close>
-
-lemma Krein_Milman_polytope:
+subsection%important\<open>Applying it to convex hulls of explicitly indicated finite sets\<close>
+
+lemma%important Krein_Milman_polytope:
   fixes S :: "'a::euclidean_space set"
   shows
    "finite S
        \<Longrightarrow> convex hull S =
            convex hull {x. x extreme_point_of (convex hull S)}"
-by (simp add: Krein_Milman_Minkowski finite_imp_compact_convex_hull)
-
-lemma extreme_points_of_convex_hull_eq:
+by%unimportant (simp add: Krein_Milman_Minkowski finite_imp_compact_convex_hull)
+
+lemma%unimportant extreme_points_of_convex_hull_eq:
   fixes S :: "'a::euclidean_space set"
   shows
    "\<lbrakk>compact S; \<And>T. T \<subset> S \<Longrightarrow> convex hull T \<noteq> convex hull S\<rbrakk>
@@ -1243,18 +1243,18 @@
 by (metis (full_types) Krein_Milman_Minkowski compact_convex_hull convex_convex_hull extreme_points_of_convex_hull psubsetI)
 
 
-lemma extreme_point_of_convex_hull_eq:
+lemma%unimportant extreme_point_of_convex_hull_eq:
   fixes S :: "'a::euclidean_space set"
   shows
    "\<lbrakk>compact S; \<And>T. T \<subset> S \<Longrightarrow> convex hull T \<noteq> convex hull S\<rbrakk>
     \<Longrightarrow> (x extreme_point_of (convex hull S) \<longleftrightarrow> x \<in> S)"
 using extreme_points_of_convex_hull_eq by auto
 
-lemma extreme_point_of_convex_hull_convex_independent:
+lemma%important extreme_point_of_convex_hull_convex_independent:
   fixes S :: "'a::euclidean_space set"
   assumes "compact S" and S: "\<And>a. a \<in> S \<Longrightarrow> a \<notin> convex hull (S - {a})"
   shows "(x extreme_point_of (convex hull S) \<longleftrightarrow> x \<in> S)"
-proof -
+proof%unimportant -
   have "convex hull T \<noteq> convex hull S" if "T \<subset> S" for T
   proof -
     obtain a where  "T \<subseteq> S" "a \<in> S" "a \<notin> T" using \<open>T \<subset> S\<close> by blast
@@ -1265,7 +1265,7 @@
     by (rule extreme_point_of_convex_hull_eq [OF \<open>compact S\<close>])
 qed
 
-lemma extreme_point_of_convex_hull_affine_independent:
+lemma%unimportant extreme_point_of_convex_hull_affine_independent:
   fixes S :: "'a::euclidean_space set"
   shows
    "~ affine_dependent S
@@ -1273,7 +1273,7 @@
 by (metis aff_independent_finite affine_dependent_def affine_hull_convex_hull extreme_point_of_convex_hull_convex_independent finite_imp_compact hull_inc)
 
 text\<open>Elementary proofs exist, not requiring Euclidean spaces and all this development\<close>
-lemma extreme_point_of_convex_hull_2:
+lemma%unimportant extreme_point_of_convex_hull_2:
   fixes x :: "'a::euclidean_space"
   shows "x extreme_point_of (convex hull {a,b}) \<longleftrightarrow> x = a \<or> x = b"
 proof -
@@ -1283,13 +1283,13 @@
     by simp
 qed
 
-lemma extreme_point_of_segment:
+lemma%unimportant extreme_point_of_segment:
   fixes x :: "'a::euclidean_space"
   shows
    "x extreme_point_of closed_segment a b \<longleftrightarrow> x = a \<or> x = b"
 by (simp add: extreme_point_of_convex_hull_2 segment_convex_hull)
 
-lemma face_of_convex_hull_subset:
+lemma%unimportant face_of_convex_hull_subset:
   fixes S :: "'a::euclidean_space set"
   assumes "compact S" and T: "T face_of (convex hull S)"
   obtains s' where "s' \<subseteq> S" "T = convex hull s'"
@@ -1298,11 +1298,11 @@
 by (metis (no_types) Krein_Milman_Minkowski assms compact_convex_hull convex_convex_hull face_of_imp_compact face_of_imp_convex)
 
 
-lemma face_of_convex_hull_aux:
+lemma%important face_of_convex_hull_aux:
   assumes eq: "x *\<^sub>R p = u *\<^sub>R a + v *\<^sub>R b + w *\<^sub>R c"
     and x: "u + v + w = x" "x \<noteq> 0" and S: "affine S" "a \<in> S" "b \<in> S" "c \<in> S"
   shows "p \<in> S"
-proof -
+proof%unimportant -
   have "p = (u *\<^sub>R a + v *\<^sub>R b + w *\<^sub>R c) /\<^sub>R x"
     by (metis \<open>x \<noteq> 0\<close> eq mult.commute right_inverse scaleR_one scaleR_scaleR)
   moreover have "affine hull {a,b,c} \<subseteq> S"
@@ -1317,14 +1317,14 @@
   ultimately show ?thesis by force
 qed
 
-proposition face_of_convex_hull_insert_eq:
+proposition%important face_of_convex_hull_insert_eq:
   fixes a :: "'a :: euclidean_space"
   assumes "finite S" and a: "a \<notin> affine hull S"
   shows "(F face_of (convex hull (insert a S)) \<longleftrightarrow>
           F face_of (convex hull S) \<or>
           (\<exists>F'. F' face_of (convex hull S) \<and> F = convex hull (insert a F')))"
          (is "F face_of ?CAS \<longleftrightarrow> _")
-proof safe
+proof%unimportant safe
   assume F: "F face_of ?CAS"
     and *: "\<nexists>F'. F' face_of convex hull S \<and> F = convex hull insert a F'"
   obtain T where T: "T \<subseteq> insert a S" and FeqT: "F = convex hull T"
@@ -1465,18 +1465,18 @@
   qed
 qed
 
-lemma face_of_convex_hull_insert2:
+lemma%unimportant face_of_convex_hull_insert2:
   fixes a :: "'a :: euclidean_space"
   assumes S: "finite S" and a: "a \<notin> affine hull S" and F: "F face_of convex hull S"
   shows "convex hull (insert a F) face_of convex hull (insert a S)"
   by (metis F face_of_convex_hull_insert_eq [OF S a])
 
-proposition face_of_convex_hull_affine_independent:
+proposition%important face_of_convex_hull_affine_independent:
   fixes S :: "'a::euclidean_space set"
   assumes "~ affine_dependent S"
     shows "(T face_of (convex hull S) \<longleftrightarrow> (\<exists>c. c \<subseteq> S \<and> T = convex hull c))"
           (is "?lhs = ?rhs")
-proof
+proof%unimportant
   assume ?lhs
   then show ?rhs
     by (meson \<open>T face_of convex hull S\<close> aff_independent_finite assms face_of_convex_hull_subset finite_imp_compact)
@@ -1493,7 +1493,7 @@
     by (metis face_of_convex_hulls \<open>c \<subseteq> S\<close> aff_independent_finite assms T)
 qed
 
-lemma facet_of_convex_hull_affine_independent:
+lemma%unimportant facet_of_convex_hull_affine_independent:
   fixes S :: "'a::euclidean_space set"
   assumes "~ affine_dependent S"
     shows "T facet_of (convex hull S) \<longleftrightarrow>
@@ -1540,7 +1540,7 @@
     done
 qed
 
-lemma facet_of_convex_hull_affine_independent_alt:
+lemma%unimportant facet_of_convex_hull_affine_independent_alt:
   fixes S :: "'a::euclidean_space set"
   shows
    "~affine_dependent S
@@ -1551,7 +1551,7 @@
 apply (metis Diff_cancel Int_empty_right Int_insert_right_if1  aff_independent_finite card_eq_0_iff card_insert_if card_mono card_subset_eq convex_hull_eq_empty eq_iff equals0D finite_insert finite_subset inf.absorb_iff2 insert_absorb insert_not_empty  not_less_eq_eq numeral_2_eq_2)
 done
 
-lemma segment_face_of:
+lemma%unimportant segment_face_of:
   assumes "(closed_segment a b) face_of S"
   shows "a extreme_point_of S" "b extreme_point_of S"
 proof -
@@ -1571,12 +1571,12 @@
 qed
 
 
-lemma Krein_Milman_frontier:
+lemma%important Krein_Milman_frontier:
   fixes S :: "'a::euclidean_space set"
   assumes "convex S" "compact S"
     shows "S = convex hull (frontier S)"
           (is "?lhs = ?rhs")
-proof
+proof%unimportant
   have "?lhs \<subseteq> convex hull {x. x extreme_point_of S}"
     using Krein_Milman_Minkowski assms by blast
   also have "... \<subseteq> ?rhs"
@@ -1593,36 +1593,36 @@
   finally show "?rhs \<subseteq> ?lhs" .
 qed
 
-subsection\<open>Polytopes\<close>
-
-definition polytope where
+subsection%important\<open>Polytopes\<close>
+
+definition%important polytope where
  "polytope S \<equiv> \<exists>v. finite v \<and> S = convex hull v"
 
-lemma polytope_translation_eq: "polytope (image (\<lambda>x. a + x) S) \<longleftrightarrow> polytope S"
+lemma%unimportant polytope_translation_eq: "polytope (image (\<lambda>x. a + x) S) \<longleftrightarrow> polytope S"
 apply (simp add: polytope_def, safe)
 apply (metis convex_hull_translation finite_imageI translation_galois)
 by (metis convex_hull_translation finite_imageI)
 
-lemma polytope_linear_image: "\<lbrakk>linear f; polytope p\<rbrakk> \<Longrightarrow> polytope(image f p)"
+lemma%unimportant polytope_linear_image: "\<lbrakk>linear f; polytope p\<rbrakk> \<Longrightarrow> polytope(image f p)"
   unfolding polytope_def using convex_hull_linear_image by blast
 
-lemma polytope_empty: "polytope {}"
+lemma%unimportant polytope_empty: "polytope {}"
   using convex_hull_empty polytope_def by blast
 
-lemma polytope_convex_hull: "finite S \<Longrightarrow> polytope(convex hull S)"
+lemma%unimportant polytope_convex_hull: "finite S \<Longrightarrow> polytope(convex hull S)"
   using polytope_def by auto
 
-lemma polytope_Times: "\<lbrakk>polytope S; polytope T\<rbrakk> \<Longrightarrow> polytope(S \<times> T)"
+lemma%unimportant polytope_Times: "\<lbrakk>polytope S; polytope T\<rbrakk> \<Longrightarrow> polytope(S \<times> T)"
   unfolding polytope_def
   by (metis finite_cartesian_product convex_hull_Times)
 
-lemma face_of_polytope_polytope:
+lemma%unimportant face_of_polytope_polytope:
   fixes S :: "'a::euclidean_space set"
   shows "\<lbrakk>polytope S; F face_of S\<rbrakk> \<Longrightarrow> polytope F"
 unfolding polytope_def
 by (meson face_of_convex_hull_subset finite_imp_compact finite_subset)
 
-lemma finite_polytope_faces:
+lemma%unimportant finite_polytope_faces:
   fixes S :: "'a::euclidean_space set"
   assumes "polytope S"
   shows "finite {F. F face_of S}"
@@ -1637,48 +1637,48 @@
     by (blast intro: finite_subset)
 qed
 
-lemma finite_polytope_facets:
+lemma%unimportant finite_polytope_facets:
   assumes "polytope S"
   shows "finite {T. T facet_of S}"
 by (simp add: assms facet_of_def finite_polytope_faces)
 
-lemma polytope_scaling:
+lemma%unimportant polytope_scaling:
   assumes "polytope S"  shows "polytope (image (\<lambda>x. c *\<^sub>R x) S)"
 by (simp add: assms polytope_linear_image)
 
-lemma polytope_imp_compact:
+lemma%unimportant polytope_imp_compact:
   fixes S :: "'a::real_normed_vector set"
   shows "polytope S \<Longrightarrow> compact S"
 by (metis finite_imp_compact_convex_hull polytope_def)
 
-lemma polytope_imp_convex: "polytope S \<Longrightarrow> convex S"
+lemma%unimportant polytope_imp_convex: "polytope S \<Longrightarrow> convex S"
   by (metis convex_convex_hull polytope_def)
 
-lemma polytope_imp_closed:
+lemma%unimportant polytope_imp_closed:
   fixes S :: "'a::real_normed_vector set"
   shows "polytope S \<Longrightarrow> closed S"
 by (simp add: compact_imp_closed polytope_imp_compact)
 
-lemma polytope_imp_bounded:
+lemma%unimportant polytope_imp_bounded:
   fixes S :: "'a::real_normed_vector set"
   shows "polytope S \<Longrightarrow> bounded S"
 by (simp add: compact_imp_bounded polytope_imp_compact)
 
-lemma polytope_interval: "polytope(cbox a b)"
+lemma%unimportant polytope_interval: "polytope(cbox a b)"
   unfolding polytope_def by (meson closed_interval_as_convex_hull)
 
-lemma polytope_sing: "polytope {a}"
+lemma%unimportant polytope_sing: "polytope {a}"
   using polytope_def by force
 
-lemma face_of_polytope_insert:
+lemma%unimportant face_of_polytope_insert:
      "\<lbrakk>polytope S; a \<notin> affine hull S; F face_of S\<rbrakk> \<Longrightarrow> F face_of convex hull (insert a S)"
   by (metis (no_types, lifting) affine_hull_convex_hull face_of_convex_hull_insert hull_insert polytope_def)
 
-lemma face_of_polytope_insert2:
+lemma%important face_of_polytope_insert2:
   fixes a :: "'a :: euclidean_space"
   assumes "polytope S" "a \<notin> affine hull S" "F face_of S"
   shows "convex hull (insert a F) face_of convex hull (insert a S)"
-proof -
+proof%unimportant -
   obtain V where "finite V" "S = convex hull V"
     using assms by (auto simp: polytope_def)
   then have "convex hull (insert a F) face_of convex hull (insert a V)"
@@ -1688,31 +1688,31 @@
 qed
 
 
-subsection\<open>Polyhedra\<close>
-
-definition polyhedron where
+subsection%important\<open>Polyhedra\<close>
+
+definition%important polyhedron where
  "polyhedron S \<equiv>
         \<exists>F. finite F \<and>
             S = \<Inter> F \<and>
             (\<forall>h \<in> F. \<exists>a b. a \<noteq> 0 \<and> h = {x. a \<bullet> x \<le> b})"
 
-lemma polyhedron_Int [intro,simp]:
+lemma%unimportant polyhedron_Int [intro,simp]:
    "\<lbrakk>polyhedron S; polyhedron T\<rbrakk> \<Longrightarrow> polyhedron (S \<inter> T)"
   apply (simp add: polyhedron_def, clarify)
   apply (rename_tac F G)
   apply (rule_tac x="F \<union> G" in exI, auto)
   done
 
-lemma polyhedron_UNIV [iff]: "polyhedron UNIV"
+lemma%unimportant polyhedron_UNIV [iff]: "polyhedron UNIV"
   unfolding polyhedron_def
   by (rule_tac x="{}" in exI) auto
 
-lemma polyhedron_Inter [intro,simp]:
+lemma%unimportant polyhedron_Inter [intro,simp]:
    "\<lbrakk>finite F; \<And>S. S \<in> F \<Longrightarrow> polyhedron S\<rbrakk> \<Longrightarrow> polyhedron(\<Inter>F)"
 by (induction F rule: finite_induct) auto
 
 
-lemma polyhedron_empty [iff]: "polyhedron ({} :: 'a :: euclidean_space set)"
+lemma%unimportant polyhedron_empty [iff]: "polyhedron ({} :: 'a :: euclidean_space set)"
 proof -
   have "\<exists>a. a \<noteq> 0 \<and>
              (\<exists>b. {x. (SOME i. i \<in> Basis) \<bullet> x \<le> - 1} = {x. a \<bullet> x \<le> b})"
@@ -1731,7 +1731,7 @@
     done
 qed
 
-lemma polyhedron_halfspace_le:
+lemma%unimportant polyhedron_halfspace_le:
   fixes a :: "'a :: euclidean_space"
   shows "polyhedron {x. a \<bullet> x \<le> b}"
 proof (cases "a = 0")
@@ -1743,53 +1743,53 @@
     by (rule_tac x="{{x. a \<bullet> x \<le> b}}" in exI) auto
 qed
 
-lemma polyhedron_halfspace_ge:
+lemma%unimportant polyhedron_halfspace_ge:
   fixes a :: "'a :: euclidean_space"
   shows "polyhedron {x. a \<bullet> x \<ge> b}"
 using polyhedron_halfspace_le [of "-a" "-b"] by simp
 
-lemma polyhedron_hyperplane:
+lemma%important polyhedron_hyperplane:
   fixes a :: "'a :: euclidean_space"
   shows "polyhedron {x. a \<bullet> x = b}"
-proof -
+proof%unimportant -
   have "{x. a \<bullet> x = b} = {x. a \<bullet> x \<le> b} \<inter> {x. a \<bullet> x \<ge> b}"
     by force
   then show ?thesis
     by (simp add: polyhedron_halfspace_ge polyhedron_halfspace_le)
 qed
 
-lemma affine_imp_polyhedron:
+lemma%unimportant affine_imp_polyhedron:
   fixes S :: "'a :: euclidean_space set"
   shows "affine S \<Longrightarrow> polyhedron S"
 by (metis affine_hull_eq polyhedron_Inter polyhedron_hyperplane affine_hull_finite_intersection_hyperplanes [of S])
 
-lemma polyhedron_imp_closed:
+lemma%unimportant polyhedron_imp_closed:
   fixes S :: "'a :: euclidean_space set"
   shows "polyhedron S \<Longrightarrow> closed S"
 apply (simp add: polyhedron_def)
 using closed_halfspace_le by fastforce
 
-lemma polyhedron_imp_convex:
+lemma%unimportant polyhedron_imp_convex:
   fixes S :: "'a :: euclidean_space set"
   shows "polyhedron S \<Longrightarrow> convex S"
 apply (simp add: polyhedron_def)
 using convex_Inter convex_halfspace_le by fastforce
 
-lemma polyhedron_affine_hull:
+lemma%unimportant polyhedron_affine_hull:
   fixes S :: "'a :: euclidean_space set"
   shows "polyhedron(affine hull S)"
 by (simp add: affine_imp_polyhedron)
 
 
-subsection\<open>Canonical polyhedron representation making facial structure explicit\<close>
-
-lemma polyhedron_Int_affine:
+subsection%important\<open>Canonical polyhedron representation making facial structure explicit\<close>
+
+lemma%important polyhedron_Int_affine:
   fixes S :: "'a :: euclidean_space set"
   shows "polyhedron S \<longleftrightarrow>
            (\<exists>F. finite F \<and> S = (affine hull S) \<inter> \<Inter>F \<and>
                 (\<forall>h \<in> F. \<exists>a b. a \<noteq> 0 \<and> h = {x. a \<bullet> x \<le> b}))"
         (is "?lhs = ?rhs")
-proof
+proof%unimportant
   assume ?lhs then show ?rhs
     apply (simp add: polyhedron_def)
     apply (erule ex_forward)
@@ -1803,13 +1803,13 @@
     done
 qed
 
-proposition rel_interior_polyhedron_explicit:
+proposition%important rel_interior_polyhedron_explicit:
   assumes "finite F"
       and seq: "S = affine hull S \<inter> \<Inter>F"
       and faceq: "\<And>h. h \<in> F \<Longrightarrow> a h \<noteq> 0 \<and> h = {x. a h \<bullet> x \<le> b h}"
       and psub: "\<And>F'. F' \<subset> F \<Longrightarrow> S \<subset> affine hull S \<inter> \<Inter>F'"
     shows "rel_interior S = {x \<in> S. \<forall>h \<in> F. a h \<bullet> x < b h}"
-proof -
+proof%unimportant -
   have rels: "\<And>x. x \<in> rel_interior S \<Longrightarrow> x \<in> S"
     by (meson IntE mem_rel_interior)
   moreover have "a i \<bullet> x < b i" if x: "x \<in> rel_interior S" and "i \<in> F" for x i
@@ -1877,7 +1877,7 @@
 qed
 
 
-lemma polyhedron_Int_affine_parallel:
+lemma%important polyhedron_Int_affine_parallel:
   fixes S :: "'a :: euclidean_space set"
   shows "polyhedron S \<longleftrightarrow>
          (\<exists>F. finite F \<and>
@@ -1885,7 +1885,7 @@
               (\<forall>h \<in> F. \<exists>a b. a \<noteq> 0 \<and> h = {x. a \<bullet> x \<le> b} \<and>
                              (\<forall>x \<in> affine hull S. (x + a) \<in> affine hull S)))"
     (is "?lhs = ?rhs")
-proof
+proof%unimportant
   assume ?lhs
   then obtain F where "finite F" and seq: "S = (affine hull S) \<inter> \<Inter>F"
                   and faces: "\<And>h. h \<in> F \<Longrightarrow> \<exists>a b. a \<noteq> 0 \<and> h = {x. a \<bullet> x \<le> b}"
@@ -1931,7 +1931,7 @@
 qed
 
 
-proposition polyhedron_Int_affine_parallel_minimal:
+proposition%important polyhedron_Int_affine_parallel_minimal:
   fixes S :: "'a :: euclidean_space set"
   shows "polyhedron S \<longleftrightarrow>
          (\<exists>F. finite F \<and>
@@ -1940,7 +1940,7 @@
                              (\<forall>x \<in> affine hull S. (x + a) \<in> affine hull S)) \<and>
               (\<forall>F'. F' \<subset> F \<longrightarrow> S \<subset> (affine hull S) \<inter> (\<Inter>F')))"
     (is "?lhs = ?rhs")
-proof
+proof%unimportant
   assume ?lhs
   then obtain f0
            where f0: "finite f0"
@@ -1976,7 +1976,7 @@
 qed
 
 
-lemma polyhedron_Int_affine_minimal:
+lemma%unimportant polyhedron_Int_affine_minimal:
   fixes S :: "'a :: euclidean_space set"
   shows "polyhedron S \<longleftrightarrow>
          (\<exists>F. finite F \<and> S = (affine hull S) \<inter> \<Inter>F \<and>
@@ -1987,13 +1987,13 @@
 apply (auto simp: polyhedron_Int_affine elim!: ex_forward)
 done
 
-proposition facet_of_polyhedron_explicit:
+proposition%important facet_of_polyhedron_explicit:
   assumes "finite F"
       and seq: "S = affine hull S \<inter> \<Inter>F"
       and faceq: "\<And>h. h \<in> F \<Longrightarrow> a h \<noteq> 0 \<and> h = {x. a h \<bullet> x \<le> b h}"
       and psub: "\<And>F'. F' \<subset> F \<Longrightarrow> S \<subset> affine hull S \<inter> \<Inter>F'"
     shows "c facet_of S \<longleftrightarrow> (\<exists>h. h \<in> F \<and> c = S \<inter> {x. a h \<bullet> x = b h})"
-proof (cases "S = {}")
+proof%unimportant (cases "S = {}")
   case True with psub show ?thesis by force
 next
   case False
@@ -2198,7 +2198,7 @@
 qed
 
 
-lemma face_of_polyhedron_subset_explicit:
+lemma%important face_of_polyhedron_subset_explicit:
   fixes S :: "'a :: euclidean_space set"
   assumes "finite F"
       and seq: "S = affine hull S \<inter> \<Inter>F"
@@ -2206,7 +2206,7 @@
       and psub: "\<And>F'. F' \<subset> F \<Longrightarrow> S \<subset> affine hull S \<inter> \<Inter>F'"
       and c: "c face_of S" and "c \<noteq> {}" "c \<noteq> S"
    obtains h where "h \<in> F" "c \<subseteq> S \<inter> {x. a h \<bullet> x = b h}"
-proof -
+proof%unimportant -
   have "c \<subseteq> S" using \<open>c face_of S\<close>
     by (simp add: face_of_imp_subset)
   have "polyhedron S"
@@ -2245,7 +2245,7 @@
 qed
 
 text\<open>Initial part of proof duplicates that above\<close>
-proposition face_of_polyhedron_explicit:
+proposition%important face_of_polyhedron_explicit:
   fixes S :: "'a :: euclidean_space set"
   assumes "finite F"
       and seq: "S = affine hull S \<inter> \<Inter>F"
@@ -2253,7 +2253,7 @@
       and psub: "\<And>F'. F' \<subset> F \<Longrightarrow> S \<subset> affine hull S \<inter> \<Inter>F'"
       and c: "c face_of S" and "c \<noteq> {}" "c \<noteq> S"
     shows "c = \<Inter>{S \<inter> {x. a h \<bullet> x = b h} | h. h \<in> F \<and> c \<subseteq> S \<inter> {x. a h \<bullet> x = b h}}"
-proof -
+proof%unimportant -
   let ?ab = "\<lambda>h. {x. a h \<bullet> x = b h}"
   have "c \<subseteq> S" using \<open>c face_of S\<close>
     by (simp add: face_of_imp_subset)
@@ -2359,10 +2359,10 @@
 
 subsection\<open>More general corollaries from the explicit representation\<close>
 
-corollary facet_of_polyhedron:
+corollary%important facet_of_polyhedron:
   assumes "polyhedron S" and "c facet_of S"
   obtains a b where "a \<noteq> 0" "S \<subseteq> {x. a \<bullet> x \<le> b}" "c = S \<inter> {x. a \<bullet> x = b}"
-proof -
+proof%unimportant -
   obtain F where "finite F" and seq: "S = affine hull S \<inter> \<Inter>F"
              and faces: "\<And>h. h \<in> F \<Longrightarrow> \<exists>a b. a \<noteq> 0 \<and> h = {x. a \<bullet> x \<le> b}"
              and min: "\<And>F'. F' \<subset> F \<Longrightarrow> S \<subset> (affine hull S) \<inter> \<Inter>F'"
@@ -2379,10 +2379,10 @@
     by (rule_tac a = "a i" and b = "b i" in that) (simp_all add: ab)
 qed
 
-corollary face_of_polyhedron:
+corollary%important face_of_polyhedron:
   assumes "polyhedron S" and "c face_of S" and "c \<noteq> {}" and "c \<noteq> S"
     shows "c = \<Inter>{F. F facet_of S \<and> c \<subseteq> F}"
-proof -
+proof%unimportant -
   obtain F where "finite F" and seq: "S = affine hull S \<inter> \<Inter>F"
              and faces: "\<And>h. h \<in> F \<Longrightarrow> \<exists>a b. a \<noteq> 0 \<and> h = {x. a \<bullet> x \<le> b}"
              and min: "\<And>F'. F' \<subset> F \<Longrightarrow> S \<subset> (affine hull S) \<inter> \<Inter>F'"
@@ -2395,14 +2395,14 @@
     done
 qed
 
-lemma face_of_polyhedron_subset_facet:
+lemma%unimportant face_of_polyhedron_subset_facet:
   assumes "polyhedron S" and "c face_of S" and "c \<noteq> {}" and "c \<noteq> S"
   obtains F where "F facet_of S" "c \<subseteq> F"
 using face_of_polyhedron assms
 by (metis (no_types, lifting) Inf_greatest antisym_conv face_of_imp_subset mem_Collect_eq)
 
 
-lemma exposed_face_of_polyhedron:
+lemma%unimportant exposed_face_of_polyhedron:
   assumes "polyhedron S"
     shows "F exposed_face_of S \<longleftrightarrow> F face_of S"
 proof
@@ -2430,12 +2430,12 @@
   qed
 qed
 
-lemma face_of_polyhedron_polyhedron:
+lemma%unimportant face_of_polyhedron_polyhedron:
   fixes S :: "'a :: euclidean_space set"
   assumes "polyhedron S" "c face_of S" shows "polyhedron c"
 by (metis assms face_of_imp_eq_affine_Int polyhedron_Int polyhedron_affine_hull polyhedron_imp_convex)
 
-lemma finite_polyhedron_faces:
+lemma%unimportant finite_polyhedron_faces:
   fixes S :: "'a :: euclidean_space set"
   assumes "polyhedron S"
     shows "finite {F. F face_of S}"
@@ -2459,29 +2459,29 @@
     by (meson finite.emptyI finite.insertI finite_Diff2 finite_subset)
 qed
 
-lemma finite_polyhedron_exposed_faces:
+lemma%unimportant finite_polyhedron_exposed_faces:
    "polyhedron S \<Longrightarrow> finite {F. F exposed_face_of S}"
 using exposed_face_of_polyhedron finite_polyhedron_faces by fastforce
 
-lemma finite_polyhedron_extreme_points:
+lemma%unimportant finite_polyhedron_extreme_points:
   fixes S :: "'a :: euclidean_space set"
   shows "polyhedron S \<Longrightarrow> finite {v. v extreme_point_of S}"
 apply (simp add: face_of_singleton [symmetric])
 apply (rule finite_subset [OF _ finite_vimageI [OF finite_polyhedron_faces]], auto)
 done
 
-lemma finite_polyhedron_facets:
+lemma%unimportant finite_polyhedron_facets:
   fixes S :: "'a :: euclidean_space set"
   shows "polyhedron S \<Longrightarrow> finite {F. F facet_of S}"
 unfolding facet_of_def
 by (blast intro: finite_subset [OF _ finite_polyhedron_faces])
 
 
-proposition rel_interior_of_polyhedron:
+proposition%important rel_interior_of_polyhedron:
   fixes S :: "'a :: euclidean_space set"
   assumes "polyhedron S"
     shows "rel_interior S = S - \<Union>{F. F facet_of S}"
-proof -
+proof%unimportant -
   obtain F where "finite F" and seq: "S = affine hull S \<inter> \<Inter>F"
              and faces: "\<And>h. h \<in> F \<Longrightarrow> \<exists>a b. a \<noteq> 0 \<and> h = {x. a \<bullet> x \<le> b}"
              and min: "\<And>F'. F' \<subset> F \<Longrightarrow> S \<subset> (affine hull S) \<inter> \<Inter>F'"
@@ -2521,19 +2521,19 @@
     by (force simp: rel)
 qed
 
-lemma rel_boundary_of_polyhedron:
+lemma%unimportant rel_boundary_of_polyhedron:
   fixes S :: "'a :: euclidean_space set"
   assumes "polyhedron S"
     shows "S - rel_interior S = \<Union> {F. F facet_of S}"
 using facet_of_imp_subset by (fastforce simp add: rel_interior_of_polyhedron assms)
 
-lemma rel_frontier_of_polyhedron:
+lemma%unimportant rel_frontier_of_polyhedron:
   fixes S :: "'a :: euclidean_space set"
   assumes "polyhedron S"
     shows "rel_frontier S = \<Union> {F. F facet_of S}"
 by (simp add: assms rel_frontier_def polyhedron_imp_closed rel_boundary_of_polyhedron)
 
-lemma rel_frontier_of_polyhedron_alt:
+lemma%unimportant rel_frontier_of_polyhedron_alt:
   fixes S :: "'a :: euclidean_space set"
   assumes "polyhedron S"
     shows "rel_frontier S = \<Union> {F. F face_of S \<and> (F \<noteq> S)}"
@@ -2544,11 +2544,11 @@
 
 text\<open>A characterization of polyhedra as having finitely many faces\<close>
 
-proposition polyhedron_eq_finite_exposed_faces:
+proposition%important polyhedron_eq_finite_exposed_faces:
   fixes S :: "'a :: euclidean_space set"
   shows "polyhedron S \<longleftrightarrow> closed S \<and> convex S \<and> finite {F. F exposed_face_of S}"
          (is "?lhs = ?rhs")
-proof
+proof%unimportant
   assume ?lhs
   then show ?rhs
     by (auto simp: polyhedron_imp_closed polyhedron_imp_convex finite_polyhedron_exposed_faces)
@@ -2635,11 +2635,11 @@
   qed
 qed
 
-corollary polyhedron_eq_finite_faces:
+corollary%important polyhedron_eq_finite_faces:
   fixes S :: "'a :: euclidean_space set"
   shows "polyhedron S \<longleftrightarrow> closed S \<and> convex S \<and> finite {F. F face_of S}"
          (is "?lhs = ?rhs")
-proof
+proof%unimportant
   assume ?lhs
   then show ?rhs
     by (simp add: finite_polyhedron_faces polyhedron_imp_closed polyhedron_imp_convex)
@@ -2649,7 +2649,7 @@
     by (force simp: polyhedron_eq_finite_exposed_faces exposed_face_of intro: finite_subset)
 qed
 
-lemma polyhedron_linear_image_eq:
+lemma%unimportant polyhedron_linear_image_eq:
   fixes h :: "'a :: euclidean_space \<Rightarrow> 'b :: euclidean_space"
   assumes "linear h" "bij h"
     shows "polyhedron (h ` S) \<longleftrightarrow> polyhedron S"
@@ -2669,19 +2669,19 @@
     done
 qed
 
-lemma polyhedron_negations:
+lemma%unimportant polyhedron_negations:
   fixes S :: "'a :: euclidean_space set"
   shows   "polyhedron S \<Longrightarrow> polyhedron(image uminus S)"
   by (subst polyhedron_linear_image_eq)
     (auto simp: bij_uminus intro!: linear_uminus)
 
-subsection\<open>Relation between polytopes and polyhedra\<close>
-
-lemma polytope_eq_bounded_polyhedron:
+subsection%important\<open>Relation between polytopes and polyhedra\<close>
+
+lemma%important polytope_eq_bounded_polyhedron:
   fixes S :: "'a :: euclidean_space set"
   shows "polytope S \<longleftrightarrow> polyhedron S \<and> bounded S"
          (is "?lhs = ?rhs")
-proof
+proof%unimportant
   assume ?lhs
   then show ?rhs
     by (simp add: finite_polytope_faces polyhedron_eq_finite_faces
@@ -2694,28 +2694,28 @@
     done
 qed
 
-lemma polytope_Int:
+lemma%unimportant polytope_Int:
   fixes S :: "'a :: euclidean_space set"
   shows "\<lbrakk>polytope S; polytope T\<rbrakk> \<Longrightarrow> polytope(S \<inter> T)"
 by (simp add: polytope_eq_bounded_polyhedron bounded_Int)
 
 
-lemma polytope_Int_polyhedron:
+lemma%important polytope_Int_polyhedron:
   fixes S :: "'a :: euclidean_space set"
   shows "\<lbrakk>polytope S; polyhedron T\<rbrakk> \<Longrightarrow> polytope(S \<inter> T)"
-by (simp add: bounded_Int polytope_eq_bounded_polyhedron)
-
-lemma polyhedron_Int_polytope:
+by%unimportant (simp add: bounded_Int polytope_eq_bounded_polyhedron)
+
+lemma%important polyhedron_Int_polytope:
   fixes S :: "'a :: euclidean_space set"
   shows "\<lbrakk>polyhedron S; polytope T\<rbrakk> \<Longrightarrow> polytope(S \<inter> T)"
-by (simp add: bounded_Int polytope_eq_bounded_polyhedron)
-
-lemma polytope_imp_polyhedron:
+by%unimportant (simp add: bounded_Int polytope_eq_bounded_polyhedron)
+
+lemma%important polytope_imp_polyhedron:
   fixes S :: "'a :: euclidean_space set"
   shows "polytope S \<Longrightarrow> polyhedron S"
-by (simp add: polytope_eq_bounded_polyhedron)
-
-lemma polytope_facet_exists:
+by%unimportant (simp add: polytope_eq_bounded_polyhedron)
+
+lemma%unimportant polytope_facet_exists:
   fixes p :: "'a :: euclidean_space set"
   assumes "polytope p" "0 < aff_dim p"
   obtains F where "F facet_of p"
@@ -2732,16 +2732,16 @@
        all_not_in_conv assms face_of_singleton less_irrefl singletonI that)
 qed
 
-lemma polyhedron_interval [iff]: "polyhedron(cbox a b)"
+lemma%unimportant polyhedron_interval [iff]: "polyhedron(cbox a b)"
 by (metis polytope_imp_polyhedron polytope_interval)
 
-lemma polyhedron_convex_hull:
+lemma%unimportant polyhedron_convex_hull:
   fixes S :: "'a :: euclidean_space set"
   shows "finite S \<Longrightarrow> polyhedron(convex hull S)"
 by (simp add: polytope_convex_hull polytope_imp_polyhedron)
 
 
-subsection\<open>Relative and absolute frontier of a polytope\<close>
+subsection%important\<open>Relative and absolute frontier of a polytope\<close>
 
 lemma rel_boundary_of_convex_hull:
     fixes S :: "'a::euclidean_space set"
@@ -3194,9 +3194,9 @@
   qed (use C in auto)
 qed
 
-subsection\<open>Simplicial complexes and triangulations\<close>
-
-definition simplicial_complex where
+subsection%important \<open>Simplicial complexes and triangulations\<close>
+
+definition%important simplicial_complex where
  "simplicial_complex \<C> \<equiv>
         finite \<C> \<and>
         (\<forall>S \<in> \<C>. \<exists>n. n simplex S) \<and>
@@ -3204,7 +3204,7 @@
         (\<forall>S S'. S \<in> \<C> \<and> S' \<in> \<C>
                 \<longrightarrow> (S \<inter> S') face_of S \<and> (S \<inter> S') face_of S')"
 
-definition triangulation where
+definition%important triangulation where
  "triangulation \<T> \<equiv>
         finite \<T> \<and>
         (\<forall>T \<in> \<T>. \<exists>n. n simplex T) \<and>
@@ -3212,9 +3212,9 @@
                 \<longrightarrow> (T \<inter> T') face_of T \<and> (T \<inter> T') face_of T')"
 
 
-subsection\<open>Refining a cell complex to a simplicial complex\<close>
-
-lemma convex_hull_insert_Int_eq:
+subsection%important\<open>Refining a cell complex to a simplicial complex\<close>
+
+lemma%important convex_hull_insert_Int_eq:
   fixes z :: "'a :: euclidean_space"
   assumes z: "z \<in> rel_interior S"
       and T: "T \<subseteq> rel_frontier S"
@@ -3222,7 +3222,7 @@
       and "convex S" "convex T" "convex U"
   shows "convex hull (insert z T) \<inter> convex hull (insert z U) = convex hull (insert z (T \<inter> U))"
     (is "?lhs = ?rhs")
-proof
+proof%unimportant
   show "?lhs \<subseteq> ?rhs"
   proof (cases "T={} \<or> U={}")
     case True then show ?thesis by auto
@@ -3294,7 +3294,7 @@
     by (metis inf_greatest hull_mono inf.cobounded1 inf.cobounded2 insert_mono)
 qed
 
-lemma simplicial_subdivision_aux:
+lemma%important simplicial_subdivision_aux:
   assumes "finite \<M>"
       and "\<And>C. C \<in> \<M> \<Longrightarrow> polytope C"
       and "\<And>C. C \<in> \<M> \<Longrightarrow> aff_dim C \<le> of_nat n"
@@ -3306,7 +3306,7 @@
                 (\<forall>C \<in> \<M>. \<exists>F. finite F \<and> F \<subseteq> \<T> \<and> C = \<Union>F) \<and>
                 (\<forall>K \<in> \<T>. \<exists>C. C \<in> \<M> \<and> K \<subseteq> C)"
   using assms
-proof (induction n arbitrary: \<M> rule: less_induct)
+proof%unimportant (induction n arbitrary: \<M> rule: less_induct)
   case (less n)
   then have poly\<M>: "\<And>C. C \<in> \<M> \<Longrightarrow> polytope C"
       and aff\<M>:    "\<And>C. C \<in> \<M> \<Longrightarrow> aff_dim C \<le> of_nat n"
@@ -3756,7 +3756,7 @@
 qed
 
 
-lemma simplicial_subdivision_of_cell_complex_lowdim:
+lemma%important simplicial_subdivision_of_cell_complex_lowdim:
   assumes "finite \<M>"
       and poly: "\<And>C. C \<in> \<M> \<Longrightarrow> polytope C"
       and face: "\<And>C1 C2. \<lbrakk>C1 \<in> \<M>; C2 \<in> \<M>\<rbrakk> \<Longrightarrow> C1 \<inter> C2 face_of C1 \<and> C1 \<inter> C2 face_of C2"
@@ -3765,7 +3765,7 @@
                   "\<Union>\<T> = \<Union>\<M>"
                   "\<And>C. C \<in> \<M> \<Longrightarrow> \<exists>F. finite F \<and> F \<subseteq> \<T> \<and> C = \<Union>F"
                   "\<And>K. K \<in> \<T> \<Longrightarrow> \<exists>C. C \<in> \<M> \<and> K \<subseteq> C"
-proof (cases "d \<ge> 0")
+proof%unimportant (cases "d \<ge> 0")
   case True
   then obtain n where n: "d = of_nat n"
     using zero_le_imp_eq_int by blast
@@ -3822,7 +3822,7 @@
   qed auto
 qed
 
-proposition simplicial_subdivision_of_cell_complex:
+proposition%important simplicial_subdivision_of_cell_complex:
   assumes "finite \<M>"
       and poly: "\<And>C. C \<in> \<M> \<Longrightarrow> polytope C"
       and face: "\<And>C1 C2. \<lbrakk>C1 \<in> \<M>; C2 \<in> \<M>\<rbrakk> \<Longrightarrow> C1 \<inter> C2 face_of C1 \<and> C1 \<inter> C2 face_of C2"
@@ -3830,9 +3830,9 @@
                   "\<Union>\<T> = \<Union>\<M>"
                   "\<And>C. C \<in> \<M> \<Longrightarrow> \<exists>F. finite F \<and> F \<subseteq> \<T> \<and> C = \<Union>F"
                   "\<And>K. K \<in> \<T> \<Longrightarrow> \<exists>C. C \<in> \<M> \<and> K \<subseteq> C"
-  by (blast intro: simplicial_subdivision_of_cell_complex_lowdim [OF assms aff_dim_le_DIM])
-
-corollary fine_simplicial_subdivision_of_cell_complex:
+  by%unimportant (blast intro: simplicial_subdivision_of_cell_complex_lowdim [OF assms aff_dim_le_DIM])
+
+corollary%important fine_simplicial_subdivision_of_cell_complex:
   assumes "0 < e" "finite \<M>"
       and poly: "\<And>C. C \<in> \<M> \<Longrightarrow> polytope C"
       and face: "\<And>C1 C2. \<lbrakk>C1 \<in> \<M>; C2 \<in> \<M>\<rbrakk> \<Longrightarrow> C1 \<inter> C2 face_of C1 \<and> C1 \<inter> C2 face_of C2"
@@ -3841,7 +3841,7 @@
                   "\<Union>\<T> = \<Union>\<M>"
                   "\<And>C. C \<in> \<M> \<Longrightarrow> \<exists>F. finite F \<and> F \<subseteq> \<T> \<and> C = \<Union>F"
                   "\<And>K. K \<in> \<T> \<Longrightarrow> \<exists>C. C \<in> \<M> \<and> K \<subseteq> C"
-proof -
+proof%unimportant -
   obtain \<N> where \<N>: "finite \<N>" "\<Union>\<N> = \<Union>\<M>" 
               and diapoly: "\<And>X. X \<in> \<N> \<Longrightarrow> diameter X < e" "\<And>X. X \<in> \<N> \<Longrightarrow> polytope X"
                and      "\<And>X Y. \<lbrakk>X \<in> \<N>; Y \<in> \<N>\<rbrakk> \<Longrightarrow> X \<inter> Y face_of X \<and> X \<inter> Y face_of Y"
@@ -3880,13 +3880,13 @@
   qed
 qed
 
-subsection\<open>Some results on cell division with full-dimensional cells only\<close>
-
-lemma convex_Union_fulldim_cells:
+subsection%important\<open>Some results on cell division with full-dimensional cells only\<close>
+
+lemma%important convex_Union_fulldim_cells:
   assumes "finite \<S>" and clo: "\<And>C. C \<in> \<S> \<Longrightarrow> closed C" and con: "\<And>C. C \<in> \<S> \<Longrightarrow> convex C"
       and eq: "\<Union>\<S> = U"and  "convex U"
  shows "\<Union>{C \<in> \<S>. aff_dim C = aff_dim U} = U"  (is "?lhs = U")
-proof -
+proof%unimportant -
   have "closed U"
     using \<open>finite \<S>\<close> clo eq by blast
   have "?lhs \<subseteq> U"
@@ -3935,7 +3935,7 @@
   ultimately show ?thesis by blast
 qed
 
-proposition fine_triangular_subdivision_of_cell_complex:
+proposition%important fine_triangular_subdivision_of_cell_complex:
   assumes "0 < e" "finite \<M>"
       and poly: "\<And>C. C \<in> \<M> \<Longrightarrow> polytope C"
       and aff: "\<And>C. C \<in> \<M> \<Longrightarrow> aff_dim C = d"
@@ -3944,7 +3944,7 @@
                  "\<And>k. k \<in> \<T> \<Longrightarrow> aff_dim k = d" "\<Union>\<T> = \<Union>\<M>"
                  "\<And>C. C \<in> \<M> \<Longrightarrow> \<exists>f. finite f \<and> f \<subseteq> \<T> \<and> C = \<Union>f"
                  "\<And>k. k \<in> \<T> \<Longrightarrow> \<exists>C. C \<in> \<M> \<and> k \<subseteq> C"
-proof -
+proof%unimportant -
   obtain \<T> where "simplicial_complex \<T>"
              and dia\<T>: "\<And>K. K \<in> \<T> \<Longrightarrow> diameter K < e"
              and "\<Union>\<T> = \<Union>\<M>"

--- a/src/HOL/Analysis/Vitali_Covering_Theorem.thy	Mon Aug 27 22:58:36 2018 +0200
+++ b/src/HOL/Analysis/Vitali_Covering_Theorem.thy	Tue Aug 28 13:28:39 2018 +0100
@@ -2,29 +2,29 @@
     Authors:    LC Paulson, based on material from HOL Light
 *)
 
-section\<open>Vitali Covering Theorem and an Application to Negligibility\<close>
+section%important  \<open>Vitali Covering Theorem and an Application to Negligibility\<close>
 
 theory Vitali_Covering_Theorem
   imports Ball_Volume "HOL-Library.Permutations"
 
 begin
 
-lemma stretch_Galois:
+lemma%important stretch_Galois:
   fixes x :: "real^'n"
   shows "(\<And>k. m k \<noteq> 0) \<Longrightarrow> ((y = (\<chi> k. m k * x$k)) \<longleftrightarrow> (\<chi> k. y$k / m k) = x)"
-  by auto
+  by%unimportant auto
 
-lemma lambda_swap_Galois:
+lemma%important lambda_swap_Galois:
    "(x = (\<chi> i. y $ Fun.swap m n id i) \<longleftrightarrow> (\<chi> i. x $ Fun.swap m n id i) = y)"
-  by (auto; simp add: pointfree_idE vec_eq_iff)
+  by%unimportant (auto; simp add: pointfree_idE vec_eq_iff)
 
-lemma lambda_add_Galois:
+lemma%important lambda_add_Galois:
   fixes x :: "real^'n"
   shows "m \<noteq> n \<Longrightarrow> (x = (\<chi> i. if i = m then y$m + y$n else y$i) \<longleftrightarrow> (\<chi> i. if i = m then x$m - x$n else x$i) = y)"
-  by (safe; simp add: vec_eq_iff)
+  by%unimportant (safe; simp add: vec_eq_iff)
 
 
-lemma Vitali_covering_lemma_cballs_balls:
+lemma%important Vitali_covering_lemma_cballs_balls:
   fixes a :: "'a \<Rightarrow> 'b::euclidean_space"
   assumes "\<And>i. i \<in> K \<Longrightarrow> 0 < r i \<and> r i \<le> B"
   obtains C where "countable C" "C \<subseteq> K"
@@ -32,7 +32,7 @@
      "\<And>i. i \<in> K \<Longrightarrow> \<exists>j. j \<in> C \<and>
                         \<not> disjnt (cball (a i) (r i)) (cball (a j) (r j)) \<and>
                         cball (a i) (r i) \<subseteq> ball (a j) (5 * r j)"
-proof (cases "K = {}")
+proof%unimportant (cases "K = {}")
   case True
   with that show ?thesis
     by auto
@@ -234,16 +234,16 @@
   qed
 qed
 
-subsection\<open>Vitali covering theorem\<close>
+subsection%important\<open>Vitali covering theorem\<close>
 
-lemma Vitali_covering_lemma_cballs:
+lemma%unimportant Vitali_covering_lemma_cballs:
   fixes a :: "'a \<Rightarrow> 'b::euclidean_space"
   assumes S: "S \<subseteq> (\<Union>i\<in>K. cball (a i) (r i))"
       and r: "\<And>i. i \<in> K \<Longrightarrow> 0 < r i \<and> r i \<le> B"
   obtains C where "countable C" "C \<subseteq> K"
      "pairwise (\<lambda>i j. disjnt (cball (a i) (r i)) (cball (a j) (r j))) C"
      "S \<subseteq> (\<Union>i\<in>C. cball (a i) (5 * r i))"
-proof -
+proof%unimportant -
   obtain C where C: "countable C" "C \<subseteq> K"
                     "pairwise (\<lambda>i j. disjnt (cball (a i) (r i)) (cball (a j) (r j))) C"
            and cov: "\<And>i. i \<in> K \<Longrightarrow> \<exists>j. j \<in> C \<and> \<not> disjnt (cball (a i) (r i)) (cball (a j) (r j)) \<and>
@@ -258,14 +258,14 @@
   qed (use C in auto)
 qed
 
-lemma Vitali_covering_lemma_balls:
+lemma%important Vitali_covering_lemma_balls:
   fixes a :: "'a \<Rightarrow> 'b::euclidean_space"
   assumes S: "S \<subseteq> (\<Union>i\<in>K. ball (a i) (r i))"
       and r: "\<And>i. i \<in> K \<Longrightarrow> 0 < r i \<and> r i \<le> B"
   obtains C where "countable C" "C \<subseteq> K"
      "pairwise (\<lambda>i j. disjnt (ball (a i) (r i)) (ball (a j) (r j))) C"
      "S \<subseteq> (\<Union>i\<in>C. ball (a i) (5 * r i))"
-proof -
+proof%unimportant -
   obtain C where C: "countable C" "C \<subseteq> K"
            and pw:  "pairwise (\<lambda>i j. disjnt (cball (a i) (r i)) (cball (a j) (r j))) C"
            and cov: "\<And>i. i \<in> K \<Longrightarrow> \<exists>j. j \<in> C \<and> \<not> disjnt (cball (a i) (r i)) (cball (a j) (r j)) \<and>
@@ -285,7 +285,7 @@
 qed
 
 
-proposition Vitali_covering_theorem_cballs:
+proposition%important Vitali_covering_theorem_cballs:
   fixes a :: "'a \<Rightarrow> 'n::euclidean_space"
   assumes r: "\<And>i. i \<in> K \<Longrightarrow> 0 < r i"
       and S: "\<And>x d. \<lbrakk>x \<in> S; 0 < d\<rbrakk>
@@ -293,7 +293,7 @@
   obtains C where "countable C" "C \<subseteq> K"
      "pairwise (\<lambda>i j. disjnt (cball (a i) (r i)) (cball (a j) (r j))) C"
      "negligible(S - (\<Union>i \<in> C. cball (a i) (r i)))"
-proof -
+proof%unimportant -
   let ?\<mu> = "measure lebesgue"
   have *: "\<exists>C. countable C \<and> C \<subseteq> K \<and>
             pairwise (\<lambda>i j. disjnt (cball (a i) (r i)) (cball (a j) (r j))) C \<and>
@@ -493,13 +493,13 @@
 qed
 
 
-proposition Vitali_covering_theorem_balls:
+proposition%important Vitali_covering_theorem_balls:
   fixes a :: "'a \<Rightarrow> 'b::euclidean_space"
   assumes S: "\<And>x d. \<lbrakk>x \<in> S; 0 < d\<rbrakk> \<Longrightarrow> \<exists>i. i \<in> K \<and> x \<in> ball (a i) (r i) \<and> r i < d"
   obtains C where "countable C" "C \<subseteq> K"
      "pairwise (\<lambda>i j. disjnt (ball (a i) (r i)) (ball (a j) (r j))) C"
      "negligible(S - (\<Union>i \<in> C. ball (a i) (r i)))"
-proof -
+proof%unimportant -
   have 1: "\<exists>i. i \<in> {i \<in> K. 0 < r i} \<and> x \<in> cball (a i) (r i) \<and> r i < d"
          if xd: "x \<in> S" "d > 0" for x d
     by (metis (mono_tags, lifting) assms ball_eq_empty less_eq_real_def mem_Collect_eq mem_ball mem_cball not_le xd(1) xd(2))
@@ -523,7 +523,7 @@
 qed
 
 
-lemma negligible_eq_zero_density_alt:
+lemma%unimportant negligible_eq_zero_density_alt:
      "negligible S \<longleftrightarrow>
       (\<forall>x \<in> S. \<forall>e > 0.
         \<exists>d U. 0 < d \<and> d \<le> e \<and> S \<inter> ball x d \<subseteq> U \<and>
@@ -635,11 +635,11 @@
     by metis
 qed
 
-proposition negligible_eq_zero_density:
+proposition%important negligible_eq_zero_density:
    "negligible S \<longleftrightarrow>
     (\<forall>x\<in>S. \<forall>r>0. \<forall>e>0. \<exists>d. 0 < d \<and> d \<le> r \<and>
                    (\<exists>U. S \<inter> ball x d \<subseteq> U \<and> U \<in> lmeasurable \<and> measure lebesgue U < e * measure lebesgue (ball x d)))"
-proof -
+proof%unimportant -
   let ?Q = "\<lambda>x d e. \<exists>U. S \<inter> ball x d \<subseteq> U \<and> U \<in> lmeasurable \<and> measure lebesgue U < e * content (ball x d)"
   have "(\<forall>e>0. \<exists>d>0. d \<le> e \<and> ?Q x d e) = (\<forall>r>0. \<forall>e>0. \<exists>d>0. d \<le> r \<and> ?Q x d e)"
     if "x \<in> S" for x

--- a/src/HOL/Analysis/Weierstrass_Theorems.thy	Mon Aug 27 22:58:36 2018 +0200
+++ b/src/HOL/Analysis/Weierstrass_Theorems.thy	Tue Aug 28 13:28:39 2018 +0100
@@ -6,29 +6,29 @@
 imports Uniform_Limit Path_Connected Derivative
 begin
 
-subsection \<open>Bernstein polynomials\<close>
+subsection%important \<open>Bernstein polynomials\<close>
 
-definition Bernstein :: "[nat,nat,real] \<Rightarrow> real" where
+definition%important Bernstein :: "[nat,nat,real] \<Rightarrow> real" where
   "Bernstein n k x \<equiv> of_nat (n choose k) * x ^ k * (1 - x) ^ (n - k)"
 
-lemma Bernstein_nonneg: "\<lbrakk>0 \<le> x; x \<le> 1\<rbrakk> \<Longrightarrow> 0 \<le> Bernstein n k x"
+lemma%unimportant Bernstein_nonneg: "\<lbrakk>0 \<le> x; x \<le> 1\<rbrakk> \<Longrightarrow> 0 \<le> Bernstein n k x"
   by (simp add: Bernstein_def)
 
-lemma Bernstein_pos: "\<lbrakk>0 < x; x < 1; k \<le> n\<rbrakk> \<Longrightarrow> 0 < Bernstein n k x"
+lemma%unimportant Bernstein_pos: "\<lbrakk>0 < x; x < 1; k \<le> n\<rbrakk> \<Longrightarrow> 0 < Bernstein n k x"
   by (simp add: Bernstein_def)
 
-lemma sum_Bernstein [simp]: "(\<Sum>k\<le>n. Bernstein n k x) = 1"
+lemma%unimportant sum_Bernstein [simp]: "(\<Sum>k\<le>n. Bernstein n k x) = 1"
   using binomial_ring [of x "1-x" n]
   by (simp add: Bernstein_def)
 
-lemma binomial_deriv1:
+lemma%unimportant binomial_deriv1:
     "(\<Sum>k\<le>n. (of_nat k * of_nat (n choose k)) * a^(k-1) * b^(n-k)) = real_of_nat n * (a+b) ^ (n-1)"
   apply (rule DERIV_unique [where f = "\<lambda>a. (a+b)^n" and x=a])
   apply (subst binomial_ring)
   apply (rule derivative_eq_intros sum.cong | simp add: atMost_atLeast0)+
   done
 
-lemma binomial_deriv2:
+lemma%unimportant binomial_deriv2:
     "(\<Sum>k\<le>n. (of_nat k * of_nat (k-1) * of_nat (n choose k)) * a^(k-2) * b^(n-k)) =
      of_nat n * of_nat (n-1) * (a+b::real) ^ (n-2)"
   apply (rule DERIV_unique [where f = "\<lambda>a. of_nat n * (a+b::real) ^ (n-1)" and x=a])
@@ -36,14 +36,14 @@
   apply (rule derivative_eq_intros sum.cong | simp add: Num.numeral_2_eq_2)+
   done
 
-lemma sum_k_Bernstein [simp]: "(\<Sum>k\<le>n. real k * Bernstein n k x) = of_nat n * x"
+lemma%important sum_k_Bernstein [simp]: "(\<Sum>k\<le>n. real k * Bernstein n k x) = of_nat n * x"
   apply (subst binomial_deriv1 [of n x "1-x", simplified, symmetric])
   apply (simp add: sum_distrib_right)
   apply (auto simp: Bernstein_def algebra_simps power_eq_if intro!: sum.cong)
   done
 
-lemma sum_kk_Bernstein [simp]: "(\<Sum>k\<le>n. real k * (real k - 1) * Bernstein n k x) = real n * (real n - 1) * x\<^sup>2"
-proof -
+lemma%important sum_kk_Bernstein [simp]: "(\<Sum>k\<le>n. real k * (real k - 1) * Bernstein n k x) = real n * (real n - 1) * x\<^sup>2"
+proof%unimportant -
   have "(\<Sum>k\<le>n. real k * (real k - 1) * Bernstein n k x) =
         (\<Sum>k\<le>n. real k * real (k - Suc 0) * real (n choose k) * x ^ (k - 2) * (1 - x) ^ (n - k) * x\<^sup>2)"
   proof (rule sum.cong [OF refl], simp)
@@ -65,14 +65,14 @@
     by auto
 qed
 
-subsection \<open>Explicit Bernstein version of the 1D Weierstrass approximation theorem\<close>
+subsection%important \<open>Explicit Bernstein version of the 1D Weierstrass approximation theorem\<close>
 
-lemma Bernstein_Weierstrass:
+lemma%important Bernstein_Weierstrass:
   fixes f :: "real \<Rightarrow> real"
   assumes contf: "continuous_on {0..1} f" and e: "0 < e"
     shows "\<exists>N. \<forall>n x. N \<le> n \<and> x \<in> {0..1}
                     \<longrightarrow> \<bar>f x - (\<Sum>k\<le>n. f(k/n) * Bernstein n k x)\<bar> < e"
-proof -
+proof%unimportant -
   have "bounded (f ` {0..1})"
     using compact_continuous_image compact_imp_bounded contf by blast
   then obtain M where M: "\<And>x. 0 \<le> x \<Longrightarrow> x \<le> 1 \<Longrightarrow> \<bar>f x\<bar> \<le> M"
@@ -167,7 +167,7 @@
 qed
 
 
-subsection \<open>General Stone-Weierstrass theorem\<close>
+subsection%important \<open>General Stone-Weierstrass theorem\<close>
 
 text\<open>Source:
 Bruno Brosowski and Frank Deutsch.
@@ -176,7 +176,7 @@
 Volume 81, Number 1, January 1981.
 DOI: 10.2307/2043993  https://www.jstor.org/stable/2043993\<close>
 
-locale function_ring_on =
+locale%important function_ring_on =
   fixes R :: "('a::t2_space \<Rightarrow> real) set" and S :: "'a set"
   assumes compact: "compact S"
   assumes continuous: "f \<in> R \<Longrightarrow> continuous_on S f"
@@ -186,39 +186,39 @@
   assumes separable: "x \<in> S \<Longrightarrow> y \<in> S \<Longrightarrow> x \<noteq> y \<Longrightarrow> \<exists>f\<in>R. f x \<noteq> f y"
 
 begin
-  lemma minus: "f \<in> R \<Longrightarrow> (\<lambda>x. - f x) \<in> R"
+  lemma%unimportant minus: "f \<in> R \<Longrightarrow> (\<lambda>x. - f x) \<in> R"
     by (frule mult [OF const [of "-1"]]) simp
 
-  lemma diff: "f \<in> R \<Longrightarrow> g \<in> R \<Longrightarrow> (\<lambda>x. f x - g x) \<in> R"
+  lemma%unimportant diff: "f \<in> R \<Longrightarrow> g \<in> R \<Longrightarrow> (\<lambda>x. f x - g x) \<in> R"
     unfolding diff_conv_add_uminus by (metis add minus)
 
-  lemma power: "f \<in> R \<Longrightarrow> (\<lambda>x. f x ^ n) \<in> R"
+  lemma%unimportant power: "f \<in> R \<Longrightarrow> (\<lambda>x. f x ^ n) \<in> R"
     by (induct n) (auto simp: const mult)
 
-  lemma sum: "\<lbrakk>finite I; \<And>i. i \<in> I \<Longrightarrow> f i \<in> R\<rbrakk> \<Longrightarrow> (\<lambda>x. \<Sum>i \<in> I. f i x) \<in> R"
+  lemma%unimportant sum: "\<lbrakk>finite I; \<And>i. i \<in> I \<Longrightarrow> f i \<in> R\<rbrakk> \<Longrightarrow> (\<lambda>x. \<Sum>i \<in> I. f i x) \<in> R"
     by (induct I rule: finite_induct; simp add: const add)
 
-  lemma prod: "\<lbrakk>finite I; \<And>i. i \<in> I \<Longrightarrow> f i \<in> R\<rbrakk> \<Longrightarrow> (\<lambda>x. \<Prod>i \<in> I. f i x) \<in> R"
+  lemma%unimportant prod: "\<lbrakk>finite I; \<And>i. i \<in> I \<Longrightarrow> f i \<in> R\<rbrakk> \<Longrightarrow> (\<lambda>x. \<Prod>i \<in> I. f i x) \<in> R"
     by (induct I rule: finite_induct; simp add: const mult)
 
-  definition normf :: "('a::t2_space \<Rightarrow> real) \<Rightarrow> real"
+  definition%important normf :: "('a::t2_space \<Rightarrow> real) \<Rightarrow> real"
     where "normf f \<equiv> SUP x:S. \<bar>f x\<bar>"
 
-  lemma normf_upper: "\<lbrakk>continuous_on S f; x \<in> S\<rbrakk> \<Longrightarrow> \<bar>f x\<bar> \<le> normf f"
+  lemma%unimportant normf_upper: "\<lbrakk>continuous_on S f; x \<in> S\<rbrakk> \<Longrightarrow> \<bar>f x\<bar> \<le> normf f"
     apply (simp add: normf_def)
     apply (rule cSUP_upper, assumption)
     by (simp add: bounded_imp_bdd_above compact compact_continuous_image compact_imp_bounded continuous_on_rabs)
 
-  lemma normf_least: "S \<noteq> {} \<Longrightarrow> (\<And>x. x \<in> S \<Longrightarrow> \<bar>f x\<bar> \<le> M) \<Longrightarrow> normf f \<le> M"
+  lemma%unimportant normf_least: "S \<noteq> {} \<Longrightarrow> (\<And>x. x \<in> S \<Longrightarrow> \<bar>f x\<bar> \<le> M) \<Longrightarrow> normf f \<le> M"
     by (simp add: normf_def cSUP_least)
 
 end
 
-lemma (in function_ring_on) one:
+lemma%important (in function_ring_on) one:
   assumes U: "open U" and t0: "t0 \<in> S" "t0 \<in> U" and t1: "t1 \<in> S-U"
     shows "\<exists>V. open V \<and> t0 \<in> V \<and> S \<inter> V \<subseteq> U \<and>
                (\<forall>e>0. \<exists>f \<in> R. f ` S \<subseteq> {0..1} \<and> (\<forall>t \<in> S \<inter> V. f t < e) \<and> (\<forall>t \<in> S - U. f t > 1 - e))"
-proof -
+proof%unimportant -
   have "\<exists>pt \<in> R. pt t0 = 0 \<and> pt t > 0 \<and> pt ` S \<subseteq> {0..1}" if t: "t \<in> S - U" for t
   proof -
     have "t \<noteq> t0" using t t0 by auto
@@ -437,7 +437,7 @@
 qed
 
 text\<open>Non-trivial case, with @{term A} and @{term B} both non-empty\<close>
-lemma (in function_ring_on) two_special:
+lemma%unimportant (in function_ring_on) two_special:
   assumes A: "closed A" "A \<subseteq> S" "a \<in> A"
       and B: "closed B" "B \<subseteq> S" "b \<in> B"
       and disj: "A \<inter> B = {}"
@@ -542,7 +542,7 @@
   show ?thesis by blast
 qed
 
-lemma (in function_ring_on) two:
+lemma%unimportant (in function_ring_on) two:
   assumes A: "closed A" "A \<subseteq> S"
       and B: "closed B" "B \<subseteq> S"
       and disj: "A \<inter> B = {}"
@@ -566,11 +566,11 @@
 qed
 
 text\<open>The special case where @{term f} is non-negative and @{term"e<1/3"}\<close>
-lemma (in function_ring_on) Stone_Weierstrass_special:
+lemma%important (in function_ring_on) Stone_Weierstrass_special:
   assumes f: "continuous_on S f" and fpos: "\<And>x. x \<in> S \<Longrightarrow> f x \<ge> 0"
       and e: "0 < e" "e < 1/3"
   shows "\<exists>g \<in> R. \<forall>x\<in>S. \<bar>f x - g x\<bar> < 2*e"
-proof -
+proof%unimportant -
   define n where "n = 1 + nat \<lceil>normf f / e\<rceil>"
   define A where "A j = {x \<in> S. f x \<le> (j - 1/3)*e}" for j :: nat
   define B where "B j = {x \<in> S. f x \<ge> (j + 1/3)*e}" for j :: nat
@@ -711,10 +711,10 @@
 qed
 
 text\<open>The ``unpretentious'' formulation\<close>
-lemma (in function_ring_on) Stone_Weierstrass_basic:
+lemma%important (in function_ring_on) Stone_Weierstrass_basic:
   assumes f: "continuous_on S f" and e: "e > 0"
   shows "\<exists>g \<in> R. \<forall>x\<in>S. \<bar>f x - g x\<bar> < e"
-proof -
+proof%unimportant -
   have "\<exists>g \<in> R. \<forall>x\<in>S. \<bar>(f x + normf f) - g x\<bar> < 2 * min (e/2) (1/4)"
     apply (rule Stone_Weierstrass_special)
     apply (rule Limits.continuous_on_add [OF f Topological_Spaces.continuous_on_const])
@@ -730,10 +730,10 @@
 qed
 
 
-theorem (in function_ring_on) Stone_Weierstrass:
+theorem%important (in function_ring_on) Stone_Weierstrass:
   assumes f: "continuous_on S f"
   shows "\<exists>F\<in>UNIV \<rightarrow> R. LIM n sequentially. F n :> uniformly_on S f"
-proof -
+proof%unimportant -
   { fix e::real
     assume e: "0 < e"
     then obtain N::nat where N: "0 < N" "0 < inverse N" "inverse N < e"
@@ -762,7 +762,7 @@
 qed
 
 text\<open>A HOL Light formulation\<close>
-corollary Stone_Weierstrass_HOL:
+corollary%important Stone_Weierstrass_HOL:
   fixes R :: "('a::t2_space \<Rightarrow> real) set" and S :: "'a set"
   assumes "compact S"  "\<And>c. P(\<lambda>x. c::real)"
           "\<And>f. P f \<Longrightarrow> continuous_on S f"
@@ -771,7 +771,7 @@
           "continuous_on S f"
        "0 < e"
     shows "\<exists>g. P(g) \<and> (\<forall>x \<in> S. \<bar>f x - g x\<bar> < e)"
-proof -
+proof%unimportant -
   interpret PR: function_ring_on "Collect P"
     apply unfold_locales
     using assms
@@ -782,7 +782,7 @@
 qed
 
 
-subsection \<open>Polynomial functions\<close>
+subsection%important \<open>Polynomial functions\<close>
 
 inductive real_polynomial_function :: "('a::real_normed_vector \<Rightarrow> real) \<Rightarrow> bool" where
     linear: "bounded_linear f \<Longrightarrow> real_polynomial_function f"
@@ -792,11 +792,11 @@
 
 declare real_polynomial_function.intros [intro]
 
-definition polynomial_function :: "('a::real_normed_vector \<Rightarrow> 'b::real_normed_vector) \<Rightarrow> bool"
+definition%important polynomial_function :: "('a::real_normed_vector \<Rightarrow> 'b::real_normed_vector) \<Rightarrow> bool"
   where
    "polynomial_function p \<equiv> (\<forall>f. bounded_linear f \<longrightarrow> real_polynomial_function (f o p))"
 
-lemma real_polynomial_function_eq: "real_polynomial_function p = polynomial_function p"
+lemma%unimportant real_polynomial_function_eq: "real_polynomial_function p = polynomial_function p"
 unfolding polynomial_function_def
 proof
   assume "real_polynomial_function p"
@@ -820,21 +820,21 @@
     by (simp add: o_def)
 qed
 
-lemma polynomial_function_const [iff]: "polynomial_function (\<lambda>x. c)"
+lemma%unimportant polynomial_function_const [iff]: "polynomial_function (\<lambda>x. c)"
   by (simp add: polynomial_function_def o_def const)
 
-lemma polynomial_function_bounded_linear:
+lemma%unimportant polynomial_function_bounded_linear:
   "bounded_linear f \<Longrightarrow> polynomial_function f"
   by (simp add: polynomial_function_def o_def bounded_linear_compose real_polynomial_function.linear)
 
-lemma polynomial_function_id [iff]: "polynomial_function(\<lambda>x. x)"
+lemma%unimportant polynomial_function_id [iff]: "polynomial_function(\<lambda>x. x)"
   by (simp add: polynomial_function_bounded_linear)
 
-lemma polynomial_function_add [intro]:
+lemma%unimportant polynomial_function_add [intro]:
     "\<lbrakk>polynomial_function f; polynomial_function g\<rbrakk> \<Longrightarrow> polynomial_function (\<lambda>x. f x + g x)"
   by (auto simp: polynomial_function_def bounded_linear_def linear_add real_polynomial_function.add o_def)
 
-lemma polynomial_function_mult [intro]:
+lemma%unimportant polynomial_function_mult [intro]:
   assumes f: "polynomial_function f" and g: "polynomial_function g"
     shows "polynomial_function (\<lambda>x. f x *\<^sub>R g x)"
   using g
@@ -844,45 +844,45 @@
   apply (auto simp: real_polynomial_function_eq)
   done
 
-lemma polynomial_function_cmul [intro]:
+lemma%unimportant polynomial_function_cmul [intro]:
   assumes f: "polynomial_function f"
     shows "polynomial_function (\<lambda>x. c *\<^sub>R f x)"
   by (rule polynomial_function_mult [OF polynomial_function_const f])
 
-lemma polynomial_function_minus [intro]:
+lemma%unimportant polynomial_function_minus [intro]:
   assumes f: "polynomial_function f"
     shows "polynomial_function (\<lambda>x. - f x)"
   using polynomial_function_cmul [OF f, of "-1"] by simp
 
-lemma polynomial_function_diff [intro]:
+lemma%unimportant polynomial_function_diff [intro]:
     "\<lbrakk>polynomial_function f; polynomial_function g\<rbrakk> \<Longrightarrow> polynomial_function (\<lambda>x. f x - g x)"
   unfolding add_uminus_conv_diff [symmetric]
   by (metis polynomial_function_add polynomial_function_minus)
 
-lemma polynomial_function_sum [intro]:
+lemma%unimportant polynomial_function_sum [intro]:
     "\<lbrakk>finite I; \<And>i. i \<in> I \<Longrightarrow> polynomial_function (\<lambda>x. f x i)\<rbrakk> \<Longrightarrow> polynomial_function (\<lambda>x. sum (f x) I)"
 by (induct I rule: finite_induct) auto
 
-lemma real_polynomial_function_minus [intro]:
+lemma%unimportant real_polynomial_function_minus [intro]:
     "real_polynomial_function f \<Longrightarrow> real_polynomial_function (\<lambda>x. - f x)"
   using polynomial_function_minus [of f]
   by (simp add: real_polynomial_function_eq)
 
-lemma real_polynomial_function_diff [intro]:
+lemma%unimportant real_polynomial_function_diff [intro]:
     "\<lbrakk>real_polynomial_function f; real_polynomial_function g\<rbrakk> \<Longrightarrow> real_polynomial_function (\<lambda>x. f x - g x)"
   using polynomial_function_diff [of f]
   by (simp add: real_polynomial_function_eq)
 
-lemma real_polynomial_function_sum [intro]:
+lemma%unimportant real_polynomial_function_sum [intro]:
     "\<lbrakk>finite I; \<And>i. i \<in> I \<Longrightarrow> real_polynomial_function (\<lambda>x. f x i)\<rbrakk> \<Longrightarrow> real_polynomial_function (\<lambda>x. sum (f x) I)"
   using polynomial_function_sum [of I f]
   by (simp add: real_polynomial_function_eq)
 
-lemma real_polynomial_function_power [intro]:
+lemma%unimportant real_polynomial_function_power [intro]:
     "real_polynomial_function f \<Longrightarrow> real_polynomial_function (\<lambda>x. f x ^ n)"
   by (induct n) (simp_all add: const mult)
 
-lemma real_polynomial_function_compose [intro]:
+lemma%unimportant real_polynomial_function_compose [intro]:
   assumes f: "polynomial_function f" and g: "real_polynomial_function g"
     shows "real_polynomial_function (g o f)"
   using g
@@ -891,13 +891,13 @@
   apply (simp_all add: polynomial_function_def o_def const add mult)
   done
 
-lemma polynomial_function_compose [intro]:
+lemma%unimportant polynomial_function_compose [intro]:
   assumes f: "polynomial_function f" and g: "polynomial_function g"
     shows "polynomial_function (g o f)"
   using g real_polynomial_function_compose [OF f]
   by (auto simp: polynomial_function_def o_def)
 
-lemma sum_max_0:
+lemma%unimportant sum_max_0:
   fixes x::real (*in fact "'a::comm_ring_1"*)
   shows "(\<Sum>i\<le>max m n. x^i * (if i \<le> m then a i else 0)) = (\<Sum>i\<le>m. x^i * a i)"
 proof -
@@ -910,7 +910,7 @@
   finally show ?thesis .
 qed
 
-lemma real_polynomial_function_imp_sum:
+lemma%unimportant real_polynomial_function_imp_sum:
   assumes "real_polynomial_function f"
     shows "\<exists>a n::nat. f = (\<lambda>x. \<Sum>i\<le>n. a i * x ^ i)"
 using assms
@@ -955,19 +955,19 @@
     done
 qed
 
-lemma real_polynomial_function_iff_sum:
+lemma%unimportant real_polynomial_function_iff_sum:
      "real_polynomial_function f \<longleftrightarrow> (\<exists>a n::nat. f = (\<lambda>x. \<Sum>i\<le>n. a i * x ^ i))"
   apply (rule iffI)
   apply (erule real_polynomial_function_imp_sum)
   apply (auto simp: linear mult const real_polynomial_function_power real_polynomial_function_sum)
   done
 
-lemma polynomial_function_iff_Basis_inner:
+lemma%important polynomial_function_iff_Basis_inner:
   fixes f :: "'a::real_normed_vector \<Rightarrow> 'b::euclidean_space"
   shows "polynomial_function f \<longleftrightarrow> (\<forall>b\<in>Basis. real_polynomial_function (\<lambda>x. inner (f x) b))"
         (is "?lhs = ?rhs")
 unfolding polynomial_function_def
-proof (intro iffI allI impI)
+proof%unimportant (intro iffI allI impI)
   assume "\<forall>h. bounded_linear h \<longrightarrow> real_polynomial_function (h \<circ> f)"
   then show ?rhs
     by (force simp add: bounded_linear_inner_left o_def)
@@ -983,17 +983,17 @@
     by (simp add: euclidean_representation_sum_fun)
 qed
 
-subsection \<open>Stone-Weierstrass theorem for polynomial functions\<close>
+subsection%important \<open>Stone-Weierstrass theorem for polynomial functions\<close>
 
 text\<open>First, we need to show that they are continous, differentiable and separable.\<close>
 
-lemma continuous_real_polymonial_function:
+lemma%unimportant continuous_real_polymonial_function:
   assumes "real_polynomial_function f"
     shows "continuous (at x) f"
 using assms
 by (induct f) (auto simp: linear_continuous_at)
 
-lemma continuous_polymonial_function:
+lemma%unimportant continuous_polymonial_function:
   fixes f :: "'a::real_normed_vector \<Rightarrow> 'b::euclidean_space"
   assumes "polynomial_function f"
     shows "continuous (at x) f"
@@ -1002,14 +1002,14 @@
   apply (force dest: continuous_real_polymonial_function intro: isCont_scaleR)
   done
 
-lemma continuous_on_polymonial_function:
+lemma%unimportant continuous_on_polymonial_function:
   fixes f :: "'a::real_normed_vector \<Rightarrow> 'b::euclidean_space"
   assumes "polynomial_function f"
     shows "continuous_on S f"
   using continuous_polymonial_function [OF assms] continuous_at_imp_continuous_on
   by blast
 
-lemma has_real_derivative_polynomial_function:
+lemma%unimportant has_real_derivative_polynomial_function:
   assumes "real_polynomial_function p"
     shows "\<exists>p'. real_polynomial_function p' \<and>
                  (\<forall>x. (p has_real_derivative (p' x)) (at x))"
@@ -1043,7 +1043,7 @@
     done
 qed
 
-lemma has_vector_derivative_polynomial_function:
+lemma%unimportant has_vector_derivative_polynomial_function:
   fixes p :: "real \<Rightarrow> 'a::euclidean_space"
   assumes "polynomial_function p"
   obtains p' where "polynomial_function p'" "\<And>x. (p has_vector_derivative (p' x)) (at x)"
@@ -1074,7 +1074,7 @@
     done
 qed
 
-lemma real_polynomial_function_separable:
+lemma%unimportant real_polynomial_function_separable:
   fixes x :: "'a::euclidean_space"
   assumes "x \<noteq> y" shows "\<exists>f. real_polynomial_function f \<and> f x \<noteq> f y"
 proof -
@@ -1090,11 +1090,11 @@
     done
 qed
 
-lemma Stone_Weierstrass_real_polynomial_function:
+lemma%important Stone_Weierstrass_real_polynomial_function:
   fixes f :: "'a::euclidean_space \<Rightarrow> real"
   assumes "compact S" "continuous_on S f" "0 < e"
   obtains g where "real_polynomial_function g" "\<And>x. x \<in> S \<Longrightarrow> \<bar>f x - g x\<bar> < e"
-proof -
+proof%unimportant -
   interpret PR: function_ring_on "Collect real_polynomial_function"
     apply unfold_locales
     using assms continuous_on_polymonial_function real_polynomial_function_eq
@@ -1105,13 +1105,13 @@
     by blast
 qed
 
-lemma Stone_Weierstrass_polynomial_function:
+lemma%important Stone_Weierstrass_polynomial_function:
   fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
   assumes S: "compact S"
       and f: "continuous_on S f"
       and e: "0 < e"
     shows "\<exists>g. polynomial_function g \<and> (\<forall>x \<in> S. norm(f x - g x) < e)"
-proof -
+proof%unimportant -
   { fix b :: 'b
     assume "b \<in> Basis"
     have "\<exists>p. real_polynomial_function p \<and> (\<forall>x \<in> S. \<bar>f x \<bullet> b - p x\<bar> < e / DIM('b))"
@@ -1151,12 +1151,12 @@
     done
 qed
 
-lemma Stone_Weierstrass_uniform_limit:
+lemma%important Stone_Weierstrass_uniform_limit:
   fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
   assumes S: "compact S"
     and f: "continuous_on S f"
   obtains g where "uniform_limit S g f sequentially" "\<And>n. polynomial_function (g n)"
-proof -
+proof%unimportant -
   have pos: "inverse (Suc n) > 0" for n by auto
   obtain g where g: "\<And>n. polynomial_function (g n)" "\<And>x n. x \<in> S \<Longrightarrow> norm(f x - g n x) < inverse (Suc n)"
     using Stone_Weierstrass_polynomial_function[OF S f pos]
@@ -1175,17 +1175,17 @@
 qed
 
 
-subsection\<open>Polynomial functions as paths\<close>
+subsection%important\<open>Polynomial functions as paths\<close>
 
 text\<open>One application is to pick a smooth approximation to a path,
 or just pick a smooth path anyway in an open connected set\<close>
 
-lemma path_polynomial_function:
+lemma%unimportant path_polynomial_function:
     fixes g  :: "real \<Rightarrow> 'b::euclidean_space"
     shows "polynomial_function g \<Longrightarrow> path g"
   by (simp add: path_def continuous_on_polymonial_function)
 
-lemma path_approx_polynomial_function:
+lemma%unimportant path_approx_polynomial_function:
     fixes g :: "real \<Rightarrow> 'b::euclidean_space"
     assumes "path g" "0 < e"
     shows "\<exists>p. polynomial_function p \<and>
@@ -1211,13 +1211,13 @@
     done
 qed
 
-lemma connected_open_polynomial_connected:
+lemma%important connected_open_polynomial_connected:
   fixes S :: "'a::euclidean_space set"
   assumes S: "open S" "connected S"
       and "x \<in> S" "y \<in> S"
     shows "\<exists>g. polynomial_function g \<and> path_image g \<subseteq> S \<and>
                pathstart g = x \<and> pathfinish g = y"
-proof -
+proof%unimportant -
   have "path_connected S" using assms
     by (simp add: connected_open_path_connected)
   with \<open>x \<in> S\<close> \<open>y \<in> S\<close> obtain p where p: "path p" "path_image p \<subseteq> S" "pathstart p = x" "pathfinish p = y"
@@ -1247,7 +1247,7 @@
     done
 qed
 
-lemma has_derivative_componentwise_within:
+lemma%unimportant has_derivative_componentwise_within:
    "(f has_derivative f') (at a within S) \<longleftrightarrow>
     (\<forall>i \<in> Basis. ((\<lambda>x. f x \<bullet> i) has_derivative (\<lambda>x. f' x \<bullet> i)) (at a within S))"
   apply (simp add: has_derivative_within)
@@ -1256,7 +1256,7 @@
   apply (simp add: algebra_simps)
   done
 
-lemma differentiable_componentwise_within:
+lemma%unimportant differentiable_componentwise_within:
    "f differentiable (at a within S) \<longleftrightarrow>
     (\<forall>i \<in> Basis. (\<lambda>x. f x \<bullet> i) differentiable at a within S)"
 proof -
@@ -1277,7 +1277,7 @@
     by blast
 qed
 
-lemma polynomial_function_inner [intro]:
+lemma%unimportant polynomial_function_inner [intro]:
   fixes i :: "'a::euclidean_space"
   shows "polynomial_function g \<Longrightarrow> polynomial_function (\<lambda>x. g x \<bullet> i)"
   apply (subst euclidean_representation [where x=i, symmetric])
@@ -1286,26 +1286,26 @@
 
 text\<open> Differentiability of real and vector polynomial functions.\<close>
 
-lemma differentiable_at_real_polynomial_function:
+lemma%unimportant differentiable_at_real_polynomial_function:
    "real_polynomial_function f \<Longrightarrow> f differentiable (at a within S)"
   by (induction f rule: real_polynomial_function.induct)
      (simp_all add: bounded_linear_imp_differentiable)
 
-lemma differentiable_on_real_polynomial_function:
+lemma%unimportant differentiable_on_real_polynomial_function:
    "real_polynomial_function p \<Longrightarrow> p differentiable_on S"
 by (simp add: differentiable_at_imp_differentiable_on differentiable_at_real_polynomial_function)
 
-lemma differentiable_at_polynomial_function:
+lemma%unimportant differentiable_at_polynomial_function:
   fixes f :: "_ \<Rightarrow> 'a::euclidean_space"
   shows "polynomial_function f \<Longrightarrow> f differentiable (at a within S)"
   by (metis differentiable_at_real_polynomial_function polynomial_function_iff_Basis_inner differentiable_componentwise_within)
 
-lemma differentiable_on_polynomial_function:
+lemma%unimportant differentiable_on_polynomial_function:
   fixes f :: "_ \<Rightarrow> 'a::euclidean_space"
   shows "polynomial_function f \<Longrightarrow> f differentiable_on S"
 by (simp add: differentiable_at_polynomial_function differentiable_on_def)
 
-lemma vector_eq_dot_span:
+lemma%unimportant vector_eq_dot_span:
   assumes "x \<in> span B" "y \<in> span B" and i: "\<And>i. i \<in> B \<Longrightarrow> i \<bullet> x = i \<bullet> y"
   shows "x = y"
 proof -
@@ -1318,7 +1318,7 @@
     then show ?thesis by simp
 qed
 
-lemma orthonormal_basis_expand:
+lemma%unimportant orthonormal_basis_expand:
   assumes B: "pairwise orthogonal B"
       and 1: "\<And>i. i \<in> B \<Longrightarrow> norm i = 1"
       and "x \<in> span B"
@@ -1343,7 +1343,7 @@
 qed
 
 
-lemma Stone_Weierstrass_polynomial_function_subspace:
+lemma%important Stone_Weierstrass_polynomial_function_subspace:
   fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
   assumes "compact S"
       and contf: "continuous_on S f"
@@ -1351,7 +1351,7 @@
       and "subspace T" "f ` S \<subseteq> T"
     obtains g where "polynomial_function g" "g ` S \<subseteq> T"
                     "\<And>x. x \<in> S \<Longrightarrow> norm(f x - g x) < e"
-proof -
+proof%unimportant -
   obtain B where "B \<subseteq> T" and orthB: "pairwise orthogonal B"
              and B1: "\<And>x. x \<in> B \<Longrightarrow> norm x = 1"
              and "independent B" and cardB: "card B = dim T"