src/HOL/OrderedGroup.thy

 
 lemma pprt_0[simp]: "pprt 0 = 0" by (simp add: pprt_def)
 lemma nprt_0[simp]: "nprt 0 = 0" by (simp add: nprt_def)
 
 lemma pprt_eq_id [simp, noatp]: "0 \<le> x \<Longrightarrow> pprt x = x"
-by (simp add: pprt_def le_iff_sup sup_ACI)
+by (simp add: pprt_def le_iff_sup sup_aci)
 
 lemma nprt_eq_id [simp, noatp]: "x \<le> 0 \<Longrightarrow> nprt x = x"
-by (simp add: nprt_def le_iff_inf inf_ACI)
+by (simp add: nprt_def le_iff_inf inf_aci)
 
 lemma pprt_eq_0 [simp, noatp]: "x \<le> 0 \<Longrightarrow> pprt x = 0"
-by (simp add: pprt_def le_iff_sup sup_ACI)
+by (simp add: pprt_def le_iff_sup sup_aci)
 
 lemma nprt_eq_0 [simp, noatp]: "0 \<le> x \<Longrightarrow> nprt x = 0"
-by (simp add: nprt_def le_iff_inf inf_ACI)
+by (simp add: nprt_def le_iff_inf inf_aci)
 
 lemma sup_0_imp_0: "sup a (- a) = 0 \<Longrightarrow> a = 0"
 proof -
   {
     fix a::'a

   "0 \<le> a + a \<longleftrightarrow> 0 \<le> a"
 proof
   assume "0 <= a + a"
   hence a:"inf (a+a) 0 = 0" by (simp add: le_iff_inf inf_commute)
   have "(inf a 0)+(inf a 0) = inf (inf (a+a) 0) a" (is "?l=_")
-    by (simp add: add_sup_inf_distribs inf_ACI)
+    by (simp add: add_sup_inf_distribs inf_aci)
   hence "?l = 0 + inf a 0" by (simp add: a, simp add: inf_commute)
   hence "inf a 0 = 0" by (simp only: add_right_cancel)
   then show "0 <= a" by (simp add: le_iff_inf inf_commute)    
 next  
   assume a: "0 <= a"

 
 lemma zero_le_iff_nprt_id: "a \<le> 0 \<longleftrightarrow> nprt a = a"
 by (simp add: le_iff_inf nprt_def inf_commute)
 
 lemma pprt_mono [simp, noatp]: "a \<le> b \<Longrightarrow> pprt a \<le> pprt b"
-by (simp add: le_iff_sup pprt_def sup_ACI sup_assoc [symmetric, of a])
+by (simp add: le_iff_sup pprt_def sup_aci sup_assoc [symmetric, of a])
 
 lemma nprt_mono [simp, noatp]: "a \<le> b \<Longrightarrow> nprt a \<le> nprt b"
-by (simp add: le_iff_inf nprt_def inf_ACI inf_assoc [symmetric, of a])
+by (simp add: le_iff_inf nprt_def inf_aci inf_assoc [symmetric, of a])
 
 end
 
 lemmas add_sup_inf_distribs = add_inf_distrib_right add_inf_distrib_left add_sup_distrib_right add_sup_distrib_left
 

     show ?thesis by (rule add_mono [OF a b, simplified])
   qed
   then have "0 \<le> sup a (- a)" unfolding abs_lattice .
   then have "sup (sup a (- a)) 0 = sup a (- a)" by (rule sup_absorb1)
   then show ?thesis
-    by (simp add: add_sup_inf_distribs sup_ACI
+    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

     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)
+      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)