
diff -r 749aaeb40788 -r be6634e99e09 src/HOL/Analysis/Finite_Cartesian_Product.thy
--- a/src/HOL/Analysis/Finite_Cartesian_Product.thy	Fri Dec 14 14:33:26 2018 +0100
+++ b/src/HOL/Analysis/Finite_Cartesian_Product.thy	Tue Jan 22 21:13:23 2019 +0000
@@ -2,7 +2,7 @@
     Author:     Amine Chaieb, University of Cambridge
 *)
 
-section%important \<open>Definition of Finite Cartesian Product Type\<close>
+section \<open>Definition of Finite Cartesian Product Type\<close>
 
 theory Finite_Cartesian_Product
 imports
@@ -104,10 +104,10 @@
     by (auto elim!: countableE)
 qed
 
-lemma%important infinite_UNIV_vec:
+lemma infinite_UNIV_vec:
   assumes "infinite (UNIV :: 'a set)"
   shows   "infinite (UNIV :: ('a^'b) set)"
-proof%unimportant (subst bij_betw_finite)
+proof (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%important CARD_vec [simp]:
+proposition CARD_vec [simp]:
   "CARD('a^'b) = CARD('a) ^ CARD('b)"
-proof%unimportant (cases "finite (UNIV :: 'a set)")
+proof (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%unimportant countable_vector:
+lemma 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}"
@@ -299,7 +299,7 @@
 
 definition%important "scaleR \<equiv> (\<lambda> r x. (\<chi> i. scaleR r (x$i)))"
 
-lemma%important vector_scaleR_component [simp]: "(scaleR r x)$i = scaleR r (x$i)"
+lemma vector_scaleR_component [simp]: "(scaleR r x)$i = scaleR r (x$i)"
   unfolding scaleR_vec_def by simp
 
 instance%unimportant
@@ -318,7 +318,7 @@
     (\<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%unimportant
+instance%unimportant proof
   show "open (UNIV :: ('a ^ 'b) set)"
     unfolding open_vec_def by auto
 next
@@ -346,30 +346,30 @@
 
 end
 
-lemma%unimportant open_vector_box: "\<forall>i. open (S i) \<Longrightarrow> open {x. \<forall>i. x $ i \<in> S i}"
+lemma 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%unimportant open_vimage_vec_nth: "open S \<Longrightarrow> open ((\<lambda>x. x $ i) -` S)"
+lemma 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%unimportant closed_vimage_vec_nth: "closed S \<Longrightarrow> closed ((\<lambda>x. x $ i) -` S)"
+lemma 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%unimportant closed_vector_box: "\<forall>i. closed (S i) \<Longrightarrow> closed {x. \<forall>i. x $ i \<in> S i}"
+lemma 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%important tendsto_vec_nth [tendsto_intros]:
+lemma tendsto_vec_nth [tendsto_intros]:
   assumes "((\<lambda>x. f x) \<longlongrightarrow> a) net"
   shows "((\<lambda>x. f x $ i) \<longlongrightarrow> a $ i) net"
-proof%unimportant (rule topological_tendstoI)
+proof (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)
@@ -379,13 +379,13 @@
     by simp
 qed
 
-lemma%unimportant isCont_vec_nth [simp]: "isCont f a \<Longrightarrow> isCont (\<lambda>x. f x $ i) a"
+lemma isCont_vec_nth [simp]: "isCont f a \<Longrightarrow> isCont (\<lambda>x. f x $ i) a"
   unfolding isCont_def by (rule tendsto_vec_nth)
 
-lemma%important vec_tendstoI:
+lemma vec_tendstoI:
   assumes "\<And>i. ((\<lambda>x. f x $ i) \<longlongrightarrow> a $ i) net"
   shows "((\<lambda>x. f x) \<longlongrightarrow> a) net"
-proof%unimportant (rule topological_tendstoI)
+proof (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"
@@ -398,13 +398,13 @@
     by (rule eventually_mono, simp add: S)
 qed
 
-lemma%unimportant tendsto_vec_lambda [tendsto_intros]:
+lemma 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%important open_image_vec_nth: assumes "open S" shows "open ((\<lambda>x. x $ i) ` S)"
-proof%unimportant (rule openI)
+lemma open_image_vec_nth: assumes "open S" shows "open ((\<lambda>x. x $ i) ` S)"
+proof (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"
@@ -418,7 +418,7 @@
   then show "\<exists>T. open T \<and> a \<in> T \<and> T \<subseteq> (\<lambda>x. x $ i) ` S" by - (rule exI)
 qed
 
-instance vec :: (perfect_space, finite) perfect_space
+instance%unimportant vec :: (perfect_space, finite) perfect_space
 proof
   fix x :: "'a ^ 'b" show "\<not> open {x}"
   proof
@@ -458,10 +458,10 @@
 instantiation%unimportant vec :: (metric_space, finite) metric_space
 begin
 
-lemma%important dist_vec_nth_le: "dist (x $ i) (y $ i) \<le> dist x y"
+proposition 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%unimportant
+instance proof
   fix x y :: "'a ^ 'b"
   show "dist x y = 0 \<longleftrightarrow> x = y"
     unfolding dist_vec_def
@@ -520,15 +520,15 @@
 
 end
 
-lemma%important Cauchy_vec_nth:
+lemma 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%important vec_CauchyI:
+lemma 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%unimportant (rule metric_CauchyI)
+proof (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
@@ -557,7 +557,7 @@
   then show "\<exists>M. \<forall>m\<ge>M. \<forall>n\<ge>M. dist (X m) (X n) < r" ..
 qed
 
-instance vec :: (complete_space, finite) complete_space
+instance%unimportant vec :: (complete_space, finite) complete_space
 proof
   fix X :: "nat \<Rightarrow> 'a ^ 'b" assume "Cauchy X"
   have "\<And>i. (\<lambda>n. X n $ i) \<longlonglongrightarrow> lim (\<lambda>n. X n $ i)"
@@ -579,7 +579,7 @@
 
 definition%important "sgn (x::'a^'b) = scaleR (inverse (norm x)) x"
 
-instance proof%unimportant
+instance%unimportant proof
   fix a :: real and x y :: "'a ^ 'b"
   show "norm x = 0 \<longleftrightarrow> x = 0"
     unfolding norm_vec_def
@@ -601,32 +601,32 @@
 
 end
 
-lemma%unimportant norm_nth_le: "norm (x $ i) \<le> norm x"
+lemma norm_nth_le: "norm (x $ i) \<le> norm x"
 unfolding norm_vec_def
 by (rule member_le_L2_set) simp_all
 
-lemma%important norm_le_componentwise_cart:
+lemma 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%unimportant norm_vec_def
   by%unimportant (rule L2_set_mono) (auto simp: assms)
 
-lemma%unimportant component_le_norm_cart: "\<bar>x$i\<bar> \<le> norm x"
+lemma component_le_norm_cart: "\<bar>x$i\<bar> \<le> norm x"
   apply (simp add: norm_vec_def)
   apply (rule member_le_L2_set, simp_all)
   done
 
-lemma%unimportant norm_bound_component_le_cart: "norm x \<le> e ==> \<bar>x$i\<bar> \<le> e"
+lemma norm_bound_component_le_cart: "norm x \<le> e ==> \<bar>x$i\<bar> \<le> e"
   by (metis component_le_norm_cart order_trans)
 
-lemma%unimportant norm_bound_component_lt_cart: "norm x < e ==> \<bar>x$i\<bar> < e"
+lemma norm_bound_component_lt_cart: "norm x < e ==> \<bar>x$i\<bar> < e"
   by (metis component_le_norm_cart le_less_trans)
 
-lemma%unimportant norm_le_l1_cart: "norm x \<le> sum(\<lambda>i. \<bar>x$i\<bar>) UNIV"
+lemma norm_le_l1_cart: "norm x \<le> sum(\<lambda>i. \<bar>x$i\<bar>) UNIV"
   by (simp add: norm_vec_def L2_set_le_sum)
 
-lemma%unimportant bounded_linear_vec_nth[intro]: "bounded_linear (\<lambda>x. x $ i)"
+lemma bounded_linear_vec_nth[intro]: "bounded_linear (\<lambda>x. x $ i)"
 apply standard
 apply (rule vector_add_component)
 apply (rule vector_scaleR_component)
@@ -643,7 +643,7 @@
 
 definition%important "inner x y = sum (\<lambda>i. inner (x$i) (y$i)) UNIV"
 
-instance proof%unimportant
+instance%unimportant proof
   fix r :: real and x y z :: "'a ^ 'b"
   show "inner x y = inner y x"
     unfolding inner_vec_def
@@ -674,13 +674,13 @@
 
 definition%important "axis k x = (\<chi> i. if i = k then x else 0)"
 
-lemma%unimportant axis_nth [simp]: "axis i x $ i = x"
+lemma axis_nth [simp]: "axis i x $ i = x"
   unfolding axis_def by simp
 
-lemma%unimportant axis_eq_axis: "axis i x = axis j y \<longleftrightarrow> x = y \<and> i = j \<or> x = 0 \<and> y = 0"
+lemma 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%unimportant inner_axis_axis:
+lemma 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")
@@ -690,10 +690,10 @@
   apply (rule sum.neutral, simp add: axis_def)
   done
 
-lemma%unimportant inner_axis: "inner x (axis i y) = inner (x $ i) y"
+lemma 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%unimportant inner_axis': "inner(axis i y) x = inner y (x $ i)"
+lemma inner_axis': "inner(axis i y) x = inner y (x $ i)"
   by (simp add: inner_axis inner_commute)
 
 instantiation%unimportant vec :: (euclidean_space, finite) euclidean_space
@@ -701,7 +701,7 @@
 
 definition%important "Basis = (\<Union>i. \<Union>u\<in>Basis. {axis i u})"
 
-instance proof%unimportant
+instance%unimportant proof
   show "(Basis :: ('a ^ 'b) set) \<noteq> {}"
     unfolding Basis_vec_def by simp
 next
@@ -720,8 +720,8 @@
     by (simp add: inner_axis euclidean_all_zero_iff vec_eq_iff)
 qed
 
-lemma%important DIM_cart [simp]: "DIM('a^'b) = CARD('b) * DIM('a)"
-proof%unimportant -
+proposition DIM_cart [simp]: "DIM('a^'b) = CARD('b) * DIM('a)"
+proof -
   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)"
@@ -732,30 +732,30 @@
 
 end
 
-lemma%unimportant norm_axis_1 [simp]: "norm (axis m (1::real)) = 1"
+lemma norm_axis_1 [simp]: "norm (axis m (1::real)) = 1"
   by (simp add: inner_axis' norm_eq_1)
 
-lemma%important sum_norm_allsubsets_bound_cart:
+lemma 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%unimportant sum_norm_allsubsets_bound[OF assms]
-  by%unimportant simp
+  using sum_norm_allsubsets_bound[OF assms]
+  by simp
 
-lemma%unimportant cart_eq_inner_axis: "a $ i = inner a (axis i 1)"
+lemma cart_eq_inner_axis: "a $ i = inner a (axis i 1)"
   by (simp add: inner_axis)
 
-lemma%unimportant axis_eq_0_iff [simp]:
+lemma axis_eq_0_iff [simp]:
   shows "axis m x = 0 \<longleftrightarrow> x = 0"
   by (simp add: axis_def vec_eq_iff)
 
-lemma%unimportant axis_in_Basis_iff [simp]: "axis i a \<in> Basis \<longleftrightarrow> a \<in> Basis"
+lemma 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%unimportant axis_inverse:
+lemma axis_inverse:
   fixes v :: "real^'n"
   assumes "v \<in> Basis"
   shows "\<exists>i. v = axis i 1"
@@ -766,20 +766,20 @@
     by (force simp add: vec_eq_iff)
 qed
 
-lemma%unimportant axis_index:
+lemma 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%unimportant axis_index_axis [simp]:
+lemma 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%unimportant \<open>A naive proof procedure to lift really trivial arithmetic stuff from the basis of the vector space\<close>
 
-lemma%unimportant sum_cong_aux:
+lemma 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)
 
@@ -811,22 +811,22 @@
 end
 \<close> "lift trivial vector statements to real arith statements"
 
-lemma%unimportant vec_0[simp]: "vec 0 = 0" by vector
-lemma%unimportant vec_1[simp]: "vec 1 = 1" by vector
+lemma vec_0[simp]: "vec 0 = 0" by vector
+lemma vec_1[simp]: "vec 1 = 1" by vector
 
-lemma%unimportant vec_inj[simp]: "vec x = vec y \<longleftrightarrow> x = y" by vector
+lemma vec_inj[simp]: "vec x = vec y \<longleftrightarrow> x = y" by vector
 
-lemma%unimportant vec_in_image_vec: "vec x \<in> (vec ` S) \<longleftrightarrow> x \<in> S" by auto
+lemma vec_in_image_vec: "vec x \<in> (vec ` S) \<longleftrightarrow> x \<in> S" by auto
 
-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_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_scaleR: "vec(c * x) = c *\<^sub>R vec x"
+lemma vec_scaleR: "vec(c * x) = c *\<^sub>R vec x"
   by vector
 
-lemma%unimportant vec_sum:
+lemma vec_sum:
   assumes "finite S"
   shows "vec(sum f S) = sum (vec \<circ> f) S"
   using assms
@@ -840,16 +840,16 @@
 
 text\<open>Obvious "component-pushing".\<close>
 
-lemma%unimportant vec_component [simp]: "vec x $ i = x"
+lemma vec_component [simp]: "vec x $ i = x"
   by vector
 
-lemma%unimportant vector_mult_component [simp]: "(x * y)$i = x$i * y$i"
+lemma vector_mult_component [simp]: "(x * y)$i = x$i * y$i"
   by vector
 
-lemma%unimportant vector_smult_component [simp]: "(c *s y)$i = c * (y$i)"
+lemma vector_smult_component [simp]: "(c *s y)$i = c * (y$i)"
   by vector
 
-lemma%unimportant cond_component: "(if b then x else y)$i = (if b then x$i else y$i)" by vector
+lemma cond_component: "(if b then x else y)$i = (if b then x$i else y$i)" by vector
 
 lemmas%unimportant vector_component =
   vec_component vector_add_component vector_mult_component
@@ -999,7 +999,7 @@
 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%unimportant nth_map_matrix[simp]: "map_matrix f x $ i $ j = f (x $ i $ j)"
+lemma nth_map_matrix[simp]: "map_matrix f x $ i $ j = f (x $ i $ j)"
   by (simp add: map_matrix_def)
 
 definition%important matrix_matrix_mult :: "('a::semiring_1) ^'n^'m \<Rightarrow> 'a ^'p^'n \<Rightarrow> 'a ^ 'p ^'m"
@@ -1022,17 +1022,17 @@
 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%unimportant times0_left [simp]: "(0::'a::semiring_1^'n^'m) ** (A::'a ^'p^'n) = 0" 
+lemma 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%unimportant times0_right [simp]: "(A::'a::semiring_1^'n^'m) ** (0::'a ^'p^'n) = 0" 
+lemma 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%unimportant mat_0[simp]: "mat 0 = 0" by (vector mat_def)
-lemma%unimportant matrix_add_ldistrib: "(A ** (B + C)) = (A ** B) + (A ** C)"
+lemma mat_0[simp]: "mat 0 = 0" by (vector mat_def)
+lemma matrix_add_ldistrib: "(A ** (B + C)) = (A ** B) + (A ** C)"
   by (vector matrix_matrix_mult_def sum.distrib[symmetric] field_simps)
 
-lemma%unimportant matrix_mul_lid [simp]:
+lemma 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)
@@ -1041,7 +1041,7 @@
     mult_1_left mult_zero_left if_True UNIV_I)
   done
 
-lemma%unimportant matrix_mul_rid [simp]:
+lemma 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)
@@ -1050,47 +1050,47 @@
     mult_1_right mult_zero_right if_True UNIV_I cong: if_cong)
   done
 
-lemma%unimportant matrix_mul_assoc: "A ** (B ** C) = (A ** B) ** C"
+proposition 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%unimportant matrix_vector_mul_assoc: "A *v (B *v x) = (A ** B) *v x"
+proposition 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%unimportant scalar_matrix_assoc:
+proposition 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%unimportant matrix_scalar_ac:
+proposition 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%unimportant matrix_vector_mul_lid [simp]: "mat 1 *v x = (x::'a::semiring_1 ^ 'n)"
+lemma 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%unimportant matrix_transpose_mul:
+lemma 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%unimportant matrix_mult_transpose_dot_column:
+lemma 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%unimportant matrix_mult_transpose_dot_row:
+lemma 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)
 
-lemma%unimportant matrix_eq:
+lemma 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
@@ -1103,49 +1103,49 @@
     sum.delta[OF finite] cong del: if_weak_cong)
   done
 
-lemma%unimportant matrix_vector_mul_component: "(A *v x)$k = inner (A$k) x"
+lemma matrix_vector_mul_component: "(A *v x)$k = inner (A$k) x"
   by (simp add: matrix_vector_mult_def inner_vec_def)
 
-lemma%unimportant dot_lmul_matrix: "inner ((x::real ^_) v* A) y = inner x (A *v y)"
+lemma 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%unimportant transpose_mat [simp]: "transpose (mat n) = mat n"
+lemma transpose_mat [simp]: "transpose (mat n) = mat n"
   by (vector transpose_def mat_def)
 
-lemma%unimportant transpose_transpose [simp]: "transpose(transpose A) = A"
+lemma transpose_transpose [simp]: "transpose(transpose A) = A"
   by (vector transpose_def)
 
-lemma%unimportant row_transpose [simp]: "row i (transpose A) = column i A"
+lemma row_transpose [simp]: "row i (transpose A) = column i A"
   by (simp add: row_def column_def transpose_def vec_eq_iff)
 
-lemma%unimportant column_transpose [simp]: "column i (transpose A) = row i A"
+lemma column_transpose [simp]: "column i (transpose A) = row i A"
   by (simp add: row_def column_def transpose_def vec_eq_iff)
 
-lemma%unimportant rows_transpose [simp]: "rows(transpose A) = columns A"
+lemma rows_transpose [simp]: "rows(transpose A) = columns A"
   by (auto simp add: rows_def columns_def intro: set_eqI)
 
-lemma%unimportant columns_transpose [simp]: "columns(transpose A) = rows A"
+lemma columns_transpose [simp]: "columns(transpose A) = rows A"
   by (metis transpose_transpose rows_transpose)
 
-lemma%unimportant transpose_scalar: "transpose (k *\<^sub>R A) = k *\<^sub>R transpose A"
+lemma transpose_scalar: "transpose (k *\<^sub>R A) = k *\<^sub>R transpose A"
   unfolding transpose_def
   by (simp add: vec_eq_iff)
 
-lemma%unimportant transpose_iff [iff]: "transpose A = transpose B \<longleftrightarrow> A = B"
+lemma transpose_iff [iff]: "transpose A = transpose B \<longleftrightarrow> A = B"
   by (metis transpose_transpose)
 
-lemma%unimportant matrix_mult_sum:
+lemma 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%unimportant vector_componentwise:
+lemma 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%unimportant basis_expansion: "sum (\<lambda>i. (x$i) *s axis i 1) UNIV = (x::('a::ring_1) ^'n)"
+lemma basis_expansion: "sum (\<lambda>i. (x$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)
 
 
@@ -1154,51 +1154,51 @@
 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%unimportant matrix_id_mat_1: "matrix id = mat 1"
+lemma matrix_id_mat_1: "matrix id = mat 1"
   by (simp add: mat_def matrix_def axis_def)
 
-lemma%unimportant matrix_scaleR: "(matrix ((*\<^sub>R) r)) = mat r"
+lemma matrix_scaleR: "(matrix ((*\<^sub>R) r)) = mat r"
   by (simp add: mat_def matrix_def axis_def if_distrib cong: if_cong)
 
-lemma%unimportant matrix_vector_mul_linear[intro, simp]: "linear (\<lambda>x. A *v (x::'a::real_algebra_1 ^ _))"
+lemma 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%unimportant vector_matrix_left_distrib [algebra_simps]:
+lemma 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%unimportant matrix_vector_right_distrib [algebra_simps]:
+lemma 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%unimportant matrix_vector_mult_diff_distrib [algebra_simps]:
+lemma 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%unimportant matrix_vector_mult_scaleR[algebra_simps]:
+lemma 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%unimportant matrix_vector_mult_0_right [simp]: "A *v 0 = 0"
+lemma matrix_vector_mult_0_right [simp]: "A *v 0 = 0"
   by (simp add: matrix_vector_mult_def vec_eq_iff)
 
-lemma%unimportant matrix_vector_mult_0 [simp]: "0 *v w = 0"
+lemma matrix_vector_mult_0 [simp]: "0 *v w = 0"
   by (simp add: matrix_vector_mult_def vec_eq_iff)
 
-lemma%unimportant matrix_vector_mult_add_rdistrib [algebra_simps]:
+lemma 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%unimportant matrix_vector_mult_diff_rdistrib [algebra_simps]:
+lemma 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%unimportant matrix_vector_column:
+lemma 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)
 
@@ -1211,17 +1211,17 @@
   "matrix_inv(A:: 'a::semiring_1^'n^'m) =
     (SOME A'::'a^'m^'n. A ** A' = mat 1 \<and> A' ** A = mat 1)"
 
-lemma%unimportant inj_matrix_vector_mult:
+lemma 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%important scalar_invertible:
+lemma scalar_invertible:
   fixes A :: "('a::real_algebra_1)^'m^'n"
   assumes "k \<noteq> 0" and "invertible A"
   shows "invertible (k *\<^sub>R A)"
-proof%unimportant -
+proof -
   obtain A' where "A ** A' = mat 1" and "A' ** A = mat 1"
     using assms unfolding invertible_def by auto
   with \<open>k \<noteq> 0\<close>
@@ -1231,50 +1231,50 @@
     unfolding invertible_def by auto
 qed
 
-lemma%unimportant scalar_invertible_iff:
+proposition 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%unimportant vector_transpose_matrix [simp]: "x v* transpose A = A *v x"
+lemma 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%unimportant transpose_matrix_vector [simp]: "transpose A *v x = x v* A"
+lemma 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%unimportant vector_scalar_commute:
+lemma 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%unimportant scalar_vector_matrix_assoc:
+lemma 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%unimportant vector_matrix_mult_0 [simp]: "0 v* A = 0"
+lemma vector_matrix_mult_0 [simp]: "0 v* A = 0"
   unfolding vector_matrix_mult_def by (simp add: zero_vec_def)
 
-lemma%unimportant vector_matrix_mult_0_right [simp]: "x v* 0 = 0"
+lemma vector_matrix_mult_0_right [simp]: "x v* 0 = 0"
   unfolding vector_matrix_mult_def by (simp add: zero_vec_def)
 
-lemma%unimportant vector_matrix_mul_rid [simp]:
+lemma 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%unimportant scaleR_vector_matrix_assoc:
+lemma 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%important vector_scaleR_matrix_ac:
+proposition 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%unimportant -
+proof -
   have "x v* (k *\<^sub>R A) = (k *\<^sub>R x) v* A"
     unfolding vector_matrix_mult_def
     by (simp add: algebra_simps)




