# HG changeset patch
# User paulson <lp15@cam.ac.uk>
# Date 1433886491 -3600
# Node ID f0bd2a6a31850326b6fc7145bd000b729beb32cf
# Parent  a58c48befade61a038186af3402bf128d2155413
more tidying up of proofs

diff -r a58c48befade -r f0bd2a6a3185 src/HOL/Multivariate_Analysis/Integration.thy
--- a/src/HOL/Multivariate_Analysis/Integration.thy	Mon Jun 08 23:52:21 2015 +0100
+++ b/src/HOL/Multivariate_Analysis/Integration.thy	Tue Jun 09 22:48:11 2015 +0100
@@ -2089,10 +2089,7 @@
     obtain n where n: "(\<Sum>i\<in>Basis. b \<bullet> i - a \<bullet> i) / e < 2 ^ n"
       using real_arch_pow2[of "(setsum (\<lambda>i. b\<bullet>i - a\<bullet>i) Basis) / e"] ..
     show ?case
-      apply (rule_tac x=n in exI)
-      apply rule
-      apply rule
-    proof -
+    proof (rule exI [where x=n], clarify)
       fix x y
       assume xy: "x\<in>cbox (A n) (B n)" "y\<in>cbox (A n) (B n)"
       have "dist x y \<le> setsum (\<lambda>i. abs((x - y)\<bullet>i)) Basis"
@@ -2150,25 +2147,20 @@
     assume "e > 0"
     from interv[OF this] obtain n
       where n: "\<forall>x\<in>cbox (A n) (B n). \<forall>y\<in>cbox (A n) (B n). dist x y < e" ..
-    show "\<exists>c d. x0 \<in> cbox c d \<and> cbox c d \<subseteq> ball x0 e \<and> cbox c d \<subseteq> cbox a b \<and> \<not> P (cbox c d)"
+    have "\<not> P (cbox (A n) (B n))"
+      apply (cases "0 < n")
+      using AB(3)[of "n - 1"] assms(3) AB(1-2)
+      apply auto
+      done
+    moreover have "cbox (A n) (B n) \<subseteq> ball x0 e"
+      using n using x0[of n] by auto
+    moreover have "cbox (A n) (B n) \<subseteq> cbox a b"
+      unfolding AB(1-2)[symmetric] by (rule ABsubset) auto
+    ultimately show "\<exists>c d. x0 \<in> cbox c d \<and> cbox c d \<subseteq> ball x0 e \<and> cbox c d \<subseteq> cbox a b \<and> \<not> P (cbox c d)"
       apply (rule_tac x="A n" in exI)
       apply (rule_tac x="B n" in exI)
-      apply rule
-      apply (rule x0)
-      apply rule
-      defer
-      apply rule
-    proof -
-      show "\<not> P (cbox (A n) (B n))"
-        apply (cases "0 < n")
-        using AB(3)[of "n - 1"] assms(3) AB(1-2)
-        apply auto
-        done
-      show "cbox (A n) (B n) \<subseteq> ball x0 e"
-        using n using x0[of n] by auto
-      show "cbox (A n) (B n) \<subseteq> cbox a b"
-        unfolding AB(1-2)[symmetric] by (rule ABsubset) auto
-    qed
+      apply (auto simp: x0)
+      done
   qed
 qed
 
@@ -2205,22 +2197,17 @@
     assume "p tagged_division_of s" "g fine p" "p' tagged_division_of t" "g fine p'"
       "interior s \<inter> interior t = {}"
     then show "\<exists>p. p tagged_division_of s \<union> t \<and> g fine p"
-      apply -
       apply (rule_tac x="p \<union> p'" in exI)
-      apply rule
-      apply (rule tagged_division_union)
-      prefer 4
-      apply (rule fine_union)
-      apply auto
+      apply (simp add: tagged_division_union fine_union)
       done
   qed blast
   obtain e where e: "e > 0" "ball x e \<subseteq> g x"
     using gaugeD[OF assms, of x] unfolding open_contains_ball by auto
-  from x(2)[OF e(1)] obtain c d where c_d:
-    "x \<in> cbox c d"
-    "cbox c d \<subseteq> ball x e"
-    "cbox c d \<subseteq> cbox a b"
-    "\<not> (\<exists>p. p tagged_division_of cbox c d \<and> g fine p)"
+  from x(2)[OF e(1)] 
+  obtain c d where c_d: "x \<in> cbox c d"
+                        "cbox c d \<subseteq> ball x e"
+                        "cbox c d \<subseteq> cbox a b"
+                        "\<not> (\<exists>p. p tagged_division_of cbox c d \<and> g fine p)"
     by blast
   have "g fine {(x, cbox c d)}"
     unfolding fine_def using e using c_d(2) by auto
@@ -2281,11 +2268,7 @@
   {
     presume "\<not> (\<exists>a b. i = cbox a b) \<Longrightarrow> False"
     then show False
-      apply -
-      apply (cases "\<exists>a b. i = cbox a b")
-      using assms
-      apply (auto simp add:has_integral intro:lem[OF _ _ as])
-      done
+      using as assms lem by blast
   }
   assume as: "\<not> (\<exists>a b. i = cbox a b)"
   obtain B1 where B1:
@@ -2339,23 +2322,11 @@
   have lem: "\<And>a b. \<And>f::'n \<Rightarrow> 'a.
     (\<forall>x\<in>cbox a b. f(x) = 0) \<Longrightarrow> (f has_integral 0) (cbox a b)"
     unfolding has_integral
-    apply rule
-    apply rule
-  proof -
+  proof clarify
     fix a b e
     fix f :: "'n \<Rightarrow> 'a"
     assume as: "\<forall>x\<in>cbox a b. f x = 0" "0 < (e::real)"
-    show "\<exists>d. gauge d \<and>
-      (\<forall>p. p tagged_division_of (cbox a b) \<and> d fine p \<longrightarrow> norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R f x) - 0) < e)"
-      apply (rule_tac x="\<lambda>x. ball x 1" in exI)
-      apply rule
-      apply (rule gaugeI)
-      unfolding centre_in_ball
-      defer
-      apply (rule open_ball)
-      apply rule
-      apply rule
-      apply (erule conjE)
+    have "\<And>p. p tagged_division_of cbox a b \<Longrightarrow> (\<lambda>x. ball x 1) fine p \<Longrightarrow> norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R f x) - 0) < e"
     proof -
       case goal1
       have "(\<Sum>(x, k)\<in>p. content k *\<^sub>R f x) = 0"
@@ -2372,16 +2343,15 @@
       qed
       then show ?case
         using as by auto
-    qed auto
+    qed
+    then show "\<exists>d. gauge d \<and>
+                   (\<forall>p. p tagged_division_of (cbox a b) \<and> d fine p \<longrightarrow> norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R f x) - 0) < e)"
+      by auto
   qed
   {
     presume "\<not> (\<exists>a b. s = cbox a b) \<Longrightarrow> ?thesis"
-    then show ?thesis
-      apply -
-      apply (cases "\<exists>a b. s = cbox a b")
-      using assms
-      apply (auto simp add:has_integral intro: lem)
-      done
+    with assms lem show ?thesis
+      by blast
   }
   have *: "(\<lambda>x. if x \<in> s then f x else 0) = (\<lambda>x. 0)"
     apply (rule ext)
@@ -2390,26 +2360,8 @@
     done
   assume "\<not> (\<exists>a b. s = cbox a b)"
   then show ?thesis
-    apply (subst has_integral_alt)
-    unfolding if_not_P *
-    apply rule
-    apply rule
-    apply (rule_tac x=1 in exI)
-    apply rule
-    defer
-    apply rule
-    apply rule
-    apply rule
-  proof -
-    fix e :: real
-    fix a b
-    assume "e > 0"
-    then show "\<exists>z. ((\<lambda>x::'n. 0::'a) has_integral z) (cbox a b) \<and> norm (z - 0) < e"
-      apply (rule_tac x=0 in exI)
-      apply(rule,rule lem)
-      apply auto
-      done
-  qed auto
+    using lem
+    by (subst has_integral_alt) (force simp add: *)
 qed
 
 lemma has_integral_0[simp]: "((\<lambda>x::'n::euclidean_space. 0) has_integral 0) s"
@@ -2430,20 +2382,17 @@
     by blast
   have lem: "\<And>(f :: 'n \<Rightarrow> 'a) y a b.
     (f has_integral y) (cbox a b) \<Longrightarrow> ((h o f) has_integral h y) (cbox a b)"
-    apply (subst has_integral)
-    apply rule
-    apply rule
-  proof -
+  unfolding has_integral
+  proof clarify
     case goal1
     from pos_bounded
     obtain B where B: "0 < B" "\<And>x. norm (h x) \<le> norm x * B"
       by blast
     have *: "e / B > 0" using goal1(2) B by simp
-    obtain g where g:
-      "gauge g"
-      "\<And>p. p tagged_division_of (cbox a b) \<Longrightarrow> g fine p \<Longrightarrow>
-        norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R f x) - y) < e / B"
-      by (rule has_integralD[OF goal1(1) *]) blast
+    obtain g where g: "gauge g"
+              "\<And>p. p tagged_division_of (cbox a b) \<Longrightarrow> g fine p \<Longrightarrow>
+                norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R f x) - y) < e / B"
+        using "*" goal1(1) by auto
     show ?case
       apply (rule_tac x=g in exI)
       apply rule
@@ -2472,19 +2421,11 @@
   {
     presume "\<not> (\<exists>a b. s = cbox a b) \<Longrightarrow> ?thesis"
     then show ?thesis
-      apply -
-      apply (cases "\<exists>a b. s = cbox a b")
-      using assms
-      apply (auto simp add:has_integral intro!:lem)
-      done
+      using assms(1) lem by blast
   }
   assume as: "\<not> (\<exists>a b. s = cbox a b)"
   then show ?thesis
-    apply (subst has_integral_alt)
-    unfolding if_not_P
-    apply rule
-    apply rule
-  proof -
+  proof (subst has_integral_alt, clarsimp)
     fix e :: real
     assume e: "e > 0"
     have *: "0 < e/B" using e B(1) by simp
@@ -2495,34 +2436,18 @@
       using has_integral_altD[OF assms(1) as *] by blast
     show "\<exists>B>0. \<forall>a b. ball 0 B \<subseteq> cbox a b \<longrightarrow>
       (\<exists>z. ((\<lambda>x. if x \<in> s then (h \<circ> f) x else 0) has_integral z) (cbox a b) \<and> norm (z - h y) < e)"
-      apply (rule_tac x=M in exI)
-      apply rule
-      apply (rule M(1))
-      apply rule
-      apply rule
-      apply rule
-    proof -
+    proof (rule_tac x=M in exI, clarsimp simp add: M)
       case goal1
       obtain z where z:
         "((\<lambda>x. if x \<in> s then f x else 0) has_integral z) (cbox a b)"
         "norm (z - y) < e / B"
         using M(2)[OF goal1(1)] by blast
       have *: "(\<lambda>x. if x \<in> s then (h \<circ> f) x else 0) = h \<circ> (\<lambda>x. if x \<in> s then f x else 0)"
-        unfolding o_def
-        apply (rule ext)
-        using zero
-        apply auto
-        done
+        using zero by auto
       show ?case
         apply (rule_tac x="h z" in exI)
-        apply rule
-        unfolding *
-        apply (rule lem[OF z(1)])
-        unfolding diff[symmetric]
-        apply (rule le_less_trans[OF B(2)])
-        using B(1) z(2)
-        apply (auto simp add: field_simps)
-        done
+        apply (simp add: "*" lem z(1))
+        by (metis B diff le_less_trans pos_less_divide_eq z(2))
     qed
   qed
 qed
@@ -2538,9 +2463,7 @@
 
 lemma has_integral_cmul: "(f has_integral k) s \<Longrightarrow> ((\<lambda>x. c *\<^sub>R f x) has_integral (c *\<^sub>R k)) s"
   unfolding o_def[symmetric]
-  apply (rule has_integral_linear,assumption)
-  apply (rule bounded_linear_scaleR_right)
-  done
+  by (metis has_integral_linear bounded_linear_scaleR_right)
 
 lemma has_integral_cmult_real:
   fixes c :: real
@@ -2556,9 +2479,7 @@
 qed
 
 lemma has_integral_neg: "(f has_integral k) s \<Longrightarrow> ((\<lambda>x. -(f x)) has_integral (-k)) s"
-  apply (drule_tac c="-1" in has_integral_cmul)
-  apply auto
-  done
+  by (drule_tac c="-1" in has_integral_cmul) auto
 
 lemma has_integral_add:
   fixes f :: "'n::euclidean_space \<Rightarrow> 'a::real_normed_vector"
@@ -2574,9 +2495,7 @@
     case goal1
     show ?case
       unfolding has_integral
-      apply rule
-      apply rule
-    proof -
+    proof clarify
       fix e :: real
       assume e: "e > 0"
       then have *: "e/2 > 0"
@@ -2592,30 +2511,23 @@
           norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R g x) - l) < e / 2"
         using has_integralD[OF goal1(2) *] by blast
       show "\<exists>d. gauge d \<and> (\<forall>p. p tagged_division_of (cbox a b) \<and> d fine p \<longrightarrow>
-        norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R (f x + g x)) - (k + l)) < e)"
-        apply (rule_tac x="\<lambda>x. (d1 x) \<inter> (d2 x)" in exI)
-        apply rule
-        apply (rule gauge_inter[OF d1(1) d2(1)])
-        apply (rule,rule,erule conjE)
-      proof -
+                norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R (f x + g x)) - (k + l)) < e)"
+      proof (rule exI [where x="\<lambda>x. (d1 x) \<inter> (d2 x)"], clarsimp simp add: gauge_inter[OF d1(1) d2(1)])
         fix p
         assume as: "p tagged_division_of (cbox a b)" "(\<lambda>x. d1 x \<inter> d2 x) fine p"
         have *: "(\<Sum>(x, k)\<in>p. content k *\<^sub>R (f x + g x)) =
           (\<Sum>(x, k)\<in>p. content k *\<^sub>R f x) + (\<Sum>(x, k)\<in>p. content k *\<^sub>R g x)"
           unfolding scaleR_right_distrib setsum.distrib[of "\<lambda>(x,k). content k *\<^sub>R f x" "\<lambda>(x,k). content k *\<^sub>R g x" p,symmetric]
           by (rule setsum.cong) auto
+        from as have fine: "d1 fine p" "d2 fine p"
+          unfolding fine_inter by auto
         have "norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R (f x + g x)) - (k + l)) =
-          norm (((\<Sum>(x, k)\<in>p. content k *\<^sub>R f x) - k) + ((\<Sum>(x, k)\<in>p. content k *\<^sub>R g x) - l))"
+              norm (((\<Sum>(x, k)\<in>p. content k *\<^sub>R f x) - k) + ((\<Sum>(x, k)\<in>p. content k *\<^sub>R g x) - l))"
           unfolding * by (auto simp add: algebra_simps)
-        also
-        let ?res = "\<dots>"
-        from as have *: "d1 fine p" "d2 fine p"
-          unfolding fine_inter by auto
-        have "?res < e/2 + e/2"
+        also have "\<dots> < e/2 + e/2"
           apply (rule le_less_trans[OF norm_triangle_ineq])
-          apply (rule add_strict_mono)
-          using d1(2)[OF as(1) *(1)] and d2(2)[OF as(1) *(2)]
-          apply auto
+          using as d1 d2 fine 
+          apply (blast intro: add_strict_mono)
           done
         finally show "norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R (f x + g x)) - (k + l)) < e"
           by auto
@@ -2625,19 +2537,11 @@
   {
     presume "\<not> (\<exists>a b. s = cbox a b) \<Longrightarrow> ?thesis"
     then show ?thesis
-      apply -
-      apply (cases "\<exists>a b. s = cbox a b")
-      using assms
-      apply (auto simp add:has_integral intro!:lem)
-      done
+      using assms lem by force
   }
   assume as: "\<not> (\<exists>a b. s = cbox a b)"
   then show ?thesis
-    apply (subst has_integral_alt)
-    unfolding if_not_P
-    apply rule
-    apply rule
-  proof -
+  proof (subst has_integral_alt, clarsimp)
     case goal1
     then have *: "e/2 > 0"
       by auto
@@ -2654,14 +2558,7 @@
           \<exists>z. ((\<lambda>x. if x \<in> s then g x else 0) has_integral z) (cbox a b) \<and> norm (z - l) < e / 2"
       by blast
     show ?case
-      apply (rule_tac x="max B1 B2" in exI)
-      apply rule
-      apply (rule max.strict_coboundedI1)
-      apply (rule B1)
-      apply rule
-      apply rule
-      apply rule
-    proof -
+    proof (rule_tac x="max B1 B2" in exI, clarsimp simp add: max.strict_coboundedI1 B1)
       fix a b
       assume "ball 0 (max B1 B2) \<subseteq> cbox a (b::'n)"
       then have *: "ball 0 B1 \<subseteq> cbox a (b::'n)" "ball 0 B2 \<subseteq> cbox a (b::'n)"
@@ -2679,8 +2576,7 @@
         by auto
       show "\<exists>z. ((\<lambda>x. if x \<in> s then f x + g x else 0) has_integral z) (cbox a b) \<and> norm (z - (k + l)) < e"
         apply (rule_tac x="w + z" in exI)
-        apply rule
-        apply (rule lem[OF w(1) z(1), unfolded *[symmetric]])
+        apply (simp add: lem[OF w(1) z(1), unfolded *[symmetric]])
         using norm_triangle_ineq[of "w - k" "z - l"] w(2) z(2)
         apply (auto simp add: field_simps)
         done
@@ -2701,33 +2597,17 @@
 
 lemma integral_add: "f integrable_on s \<Longrightarrow> g integrable_on s \<Longrightarrow>
     integral s (\<lambda>x. f x + g x) = integral s f + integral s g"
-  apply (rule integral_unique)
-  apply (drule integrable_integral)+
-  apply (rule has_integral_add)
-  apply assumption+
-  done
+  by (rule integral_unique) (metis integrable_integral has_integral_add)
 
 lemma integral_cmul: "f integrable_on s \<Longrightarrow> integral s (\<lambda>x. c *\<^sub>R f x) = c *\<^sub>R integral s f"
-  apply (rule integral_unique)
-  apply (drule integrable_integral)+
-  apply (rule has_integral_cmul)
-  apply assumption+
-  done
+  by (rule integral_unique) (metis integrable_integral has_integral_cmul)
 
 lemma integral_neg: "f integrable_on s \<Longrightarrow> integral s (\<lambda>x. - f x) = - integral s f"
-  apply (rule integral_unique)
-  apply (drule integrable_integral)+
-  apply (rule has_integral_neg)
-  apply assumption+
-  done
+  by (rule integral_unique) (metis integrable_integral has_integral_neg)
 
 lemma integral_sub: "f integrable_on s \<Longrightarrow> g integrable_on s \<Longrightarrow>
     integral s (\<lambda>x. f x - g x) = integral s f - integral s g"
-  apply (rule integral_unique)
-  apply (drule integrable_integral)+
-  apply (rule has_integral_sub)
-  apply assumption+
-  done
+  by (rule integral_unique) (metis integrable_integral has_integral_sub)
 
 lemma integrable_0: "(\<lambda>x. 0) integrable_on s"
   unfolding integrable_on_def using has_integral_0 by auto
@@ -2758,13 +2638,8 @@
 
 lemma integral_linear:
   "f integrable_on s \<Longrightarrow> bounded_linear h \<Longrightarrow> integral s (h \<circ> f) = h (integral s f)"
-  apply (rule has_integral_unique)
-  defer
-  unfolding has_integral_integral
-  apply (drule (2) has_integral_linear)
-  unfolding has_integral_integral[symmetric]
-  apply (rule integrable_linear)
-  apply assumption+
+  apply (rule has_integral_unique [where i=s and f = "h \<circ> f"])
+  apply (simp_all add: integrable_integral integrable_linear has_integral_linear )
   done
 
 lemma integral_component_eq[simp]:
@@ -2783,24 +2658,17 @@
   then show ?case by auto
 next
   case (insert x F)
-  show ?case
-    unfolding setsum.insert[OF insert(1,3)]
-    apply (rule has_integral_add)
-    using insert assms
-    apply auto
-    done
-qed
-
-lemma integral_setsum: "finite t \<Longrightarrow> \<forall>a\<in>t. (f a) integrable_on s \<Longrightarrow>
-  integral s (\<lambda>x. setsum (\<lambda>a. f a x) t) = setsum (\<lambda>a. integral s (f a)) t"
-  apply (rule integral_unique)
-  apply (rule has_integral_setsum)
-  using integrable_integral
-  apply auto
-  done
+  with assms show ?case
+    by (simp add: has_integral_add)
+qed
+
+lemma integral_setsum:
+  "\<lbrakk>finite t;  \<forall>a\<in>t. (f a) integrable_on s\<rbrakk> \<Longrightarrow>
+   integral s (\<lambda>x. setsum (\<lambda>a. f a x) t) = setsum (\<lambda>a. integral s (f a)) t"
+  by (auto intro: has_integral_setsum integrable_integral)
 
 lemma integrable_setsum:
-  "finite t \<Longrightarrow> \<forall>a \<in> t. (f a) integrable_on s \<Longrightarrow> (\<lambda>x. setsum (\<lambda>a. f a x) t) integrable_on s"
+  "\<lbrakk>finite t;  \<forall>a\<in>t. (f a) integrable_on s\<rbrakk> \<Longrightarrow> (\<lambda>x. setsum (\<lambda>a. f a x) t) integrable_on s"
   unfolding integrable_on_def
   apply (drule bchoice)
   using has_integral_setsum[of t]
@@ -2857,19 +2725,10 @@
   by (metis assms box_real(2) has_integral_null)
 
 lemma has_integral_null_eq[simp]: "content (cbox a b) = 0 \<Longrightarrow> (f has_integral i) (cbox a b) \<longleftrightarrow> i = 0"
-  apply rule
-  apply (rule has_integral_unique)
-  apply assumption
-  apply (drule (1) has_integral_null)
-  apply (drule has_integral_null)
-  apply auto
-  done
+  by (auto simp add: has_integral_null dest!: integral_unique)
 
 lemma integral_null[dest]: "content (cbox a b) = 0 \<Longrightarrow> integral (cbox a b) f = 0"
-  apply (rule integral_unique)
-  apply (drule has_integral_null)
-  apply assumption
-  done
+  by (metis has_integral_null integral_unique)
 
 lemma integrable_on_null[dest]: "content (cbox a b) = 0 \<Longrightarrow> f integrable_on (cbox a b)"
   unfolding integrable_on_def
@@ -2878,19 +2737,10 @@
   done
 
 lemma has_integral_empty[intro]: "(f has_integral 0) {}"
-  unfolding empty_as_interval
-  apply (rule has_integral_null)
-  using content_empty
-  unfolding empty_as_interval
-  apply assumption
-  done
+  by (simp add: has_integral_is_0)
 
 lemma has_integral_empty_eq[simp]: "(f has_integral i) {} \<longleftrightarrow> i = 0"
-  apply rule
-  apply (rule has_integral_unique)
-  apply assumption
-  apply auto
-  done
+  by (auto simp add: has_integral_empty has_integral_unique)
 
 lemma integrable_on_empty[intro]: "f integrable_on {}"
   unfolding integrable_on_def by auto
@@ -2954,17 +2804,10 @@
       done
     note d=this[rule_format]
     show ?case
-      apply (rule_tac x=d in exI)
-      apply rule
-      apply (rule d)
-      apply rule
-      apply rule
-      apply rule
-      apply (erule conjE)+
-    proof -
+    proof (rule_tac x=d in exI, clarsimp simp: d)
       fix p1 p2
       assume as: "p1 tagged_division_of (cbox a b)" "d fine p1"
-        "p2 tagged_division_of (cbox a b)" "d fine p2"
+                 "p2 tagged_division_of (cbox a b)" "d fine p2"
       show "norm ((\<Sum>(x, k)\<in>p1. content k *\<^sub>R f x) - (\<Sum>(x, k)\<in>p2. content k *\<^sub>R f x)) < e"
         apply (rule dist_triangle_half_l[where y=y,unfolded dist_norm])
         using d(2)[OF conjI[OF as(1-2)]] d(2)[OF conjI[OF as(3-4)]] .