src/HOL/OrderedGroup.thy

     by (simp add: add_sup_inf_distribs sup_ACI
       pprt_def nprt_def diff_minus abs_lattice)
 qed
 
 subclass pordered_ab_group_add_abs
-proof -
+proof
   have abs_ge_zero [simp]: "\<And>a. 0 \<le> \<bar>a\<bar>"
   proof -
     fix a b
     have a: "a \<le> \<bar>a\<bar>" and b: "- a \<le> \<bar>a\<bar>" by (auto simp add: abs_lattice)
     show "0 \<le> \<bar>a\<bar>" by (rule add_mono [OF a b, simplified])
   qed
   have abs_leI: "\<And>a b. a \<le> b \<Longrightarrow> - a \<le> b \<Longrightarrow> \<bar>a\<bar> \<le> b"
     by (simp add: abs_lattice le_supI)
-  show ?thesis
-  proof
-    fix a
-    show "0 \<le> \<bar>a\<bar>" by simp
-  next
-    fix a
-    show "a \<le> \<bar>a\<bar>"
-      by (auto simp add: abs_lattice)
-  next
-    fix a
-    show "\<bar>-a\<bar> = \<bar>a\<bar>"
-      by (simp add: abs_lattice sup_commute)
-  next
-    fix a b
-    show "a \<le> b \<Longrightarrow> - a \<le> b \<Longrightarrow> \<bar>a\<bar> \<le> b" by (erule abs_leI)
-  next
-    fix a b
-    show "\<bar>a + b\<bar> \<le> \<bar>a\<bar> + \<bar>b\<bar>"
-    proof -
-      have g:"abs a + abs b = sup (a+b) (sup (-a-b) (sup (-a+b) (a + (-b))))" (is "_=sup ?m ?n")
-        by (simp add: abs_lattice add_sup_inf_distribs sup_ACI diff_minus)
-      have a:"a+b <= sup ?m ?n" by (simp)
-      have b:"-a-b <= ?n" by (simp) 
-      have c:"?n <= sup ?m ?n" by (simp)
-      from b c have d: "-a-b <= sup ?m ?n" by(rule order_trans)
-      have e:"-a-b = -(a+b)" by (simp add: diff_minus)
-      from a d e have "abs(a+b) <= sup ?m ?n" 
-        by (drule_tac abs_leI, auto)
-      with g[symmetric] show ?thesis by simp
-    qed
-  qed auto
+  fix a b
+  show "0 \<le> \<bar>a\<bar>" by simp
+  show "a \<le> \<bar>a\<bar>"
+    by (auto simp add: abs_lattice)
+  show "\<bar>-a\<bar> = \<bar>a\<bar>"
+    by (simp add: abs_lattice sup_commute)
+  show "a \<le> b \<Longrightarrow> - a \<le> b \<Longrightarrow> \<bar>a\<bar> \<le> b" by (fact abs_leI)
+  show "\<bar>a + b\<bar> \<le> \<bar>a\<bar> + \<bar>b\<bar>"
+  proof -
+    have g:"abs a + abs b = sup (a+b) (sup (-a-b) (sup (-a+b) (a + (-b))))" (is "_=sup ?m ?n")
+      by (simp add: abs_lattice add_sup_inf_distribs sup_ACI diff_minus)
+    have a:"a+b <= sup ?m ?n" by (simp)
+    have b:"-a-b <= ?n" by (simp) 
+    have c:"?n <= sup ?m ?n" by (simp)
+    from b c have d: "-a-b <= sup ?m ?n" by(rule order_trans)
+    have e:"-a-b = -(a+b)" by (simp add: diff_minus)
+    from a d e have "abs(a+b) <= sup ?m ?n" 
+      by (drule_tac abs_leI, auto)
+    with g[symmetric] show ?thesis by simp
+  qed
 qed
 
 end
 
 lemma sup_eq_if: