src/HOL/IMP/Abs_Int0.thy

 
 lemma m_o_h: "finite X \<Longrightarrow> m_o X opt \<le> (h*card X + 1)"
 by(cases opt)(auto simp add: m_s_h le_SucI dest: m_s_h)
 
 definition m_c :: "'av st option acom \<Rightarrow> nat" ("m\<^isub>c") where
-"m_c C = (\<Sum>i<size(annos C). m_o (vars C) (annos C ! i))"
+"m_c C = listsum (map (m_o (vars C)) (annos C))"
 
 text{* Upper complexity bound: *}
 lemma m_c_h: "m_c C \<le> size(annos C) * (h * card(vars C) + 1)"
 proof-
   let ?X = "vars C" let ?n = "card ?X" let ?a = "size(annos C)"
-  have "m_c C = (\<Sum>i<?a. m_o ?X (annos C ! i))" by(simp add: m_c_def)
+  have "m_c C = (\<Sum>i<?a. m_o ?X (annos C ! i))"
+    by(simp add: m_c_def listsum_setsum_nth atLeast0LessThan)
   also have "\<dots> \<le> (\<Sum>i<?a. h * ?n + 1)"
     apply(rule setsum_mono) using m_o_h[OF finite_Cvars] by simp
   also have "\<dots> = ?a * (h * ?n + 1)" by simp
   finally show ?thesis .
 qed

 by(auto simp: le_less m_o2)
 
 
 lemma m_c2: "top_on_acom (-vars C1) C1 \<Longrightarrow> top_on_acom (-vars C2) C2 \<Longrightarrow>
   C1 < C2 \<Longrightarrow> m_c C1 > m_c C2"
-proof(auto simp add: le_iff_le_annos m_c_def size_annos_same[of C1 C2] vars_acom_def less_acom_def)
+proof(auto simp add: le_iff_le_annos size_annos_same[of C1 C2] vars_acom_def less_acom_def)
   let ?X = "vars(strip C2)"
   assume top: "top_on_acom (- vars(strip C2)) C1"  "top_on_acom (- vars(strip C2)) C2"
   and strip_eq: "strip C1 = strip C2"
   and 0: "\<forall>i<size(annos C2). annos C1 ! i \<le> annos C2 ! i"
   hence 1: "\<forall>i<size(annos C2). m_o ?X (annos C1 ! i) \<ge> m_o ?X (annos C2 ! i)"

   have topo2: "top_on_opt (- ?X) (annos C2 ! i)"
     using i(1) top(2) by(simp add: top_on_acom_def size_annos_same[OF strip_eq])
   from i have "m_o ?X (annos C1 ! i) > m_o ?X (annos C2 ! i)" (is "?P i")
     by (metis 0 less_option_def m_o2[OF finite_cvars topo1] topo2)
   hence 2: "\<exists>i < size(annos C2). ?P i" using `i < size(annos C2)` by blast
-  show "(\<Sum>i<size(annos C2). m_o ?X (annos C2 ! i))
+  have "(\<Sum>i<size(annos C2). m_o ?X (annos C2 ! i))
          < (\<Sum>i<size(annos C2). m_o ?X (annos C1 ! i))"
     apply(rule setsum_strict_mono_ex1) using 1 2 by (auto)
+  thus ?thesis
+    by(simp add: m_c_def vars_acom_def strip_eq listsum_setsum_nth atLeast0LessThan size_annos_same[OF strip_eq])
 qed
 
 end