--- a/src/HOL/IMP/Abs_Int1.thy Thu Mar 07 18:14:30 2013 +0100
+++ b/src/HOL/IMP/Abs_Int1.thy Fri Mar 08 11:28:04 2013 +0100
@@ -197,7 +197,6 @@
locale Measure1 =
fixes m :: "'av::order \<Rightarrow> nat"
fixes h :: "nat"
-assumes m1: "x \<le> y \<Longrightarrow> m x \<ge> m y"
assumes h: "m x \<le> h"
begin
@@ -208,31 +207,12 @@
by(simp add: L_st_def m_s_def)
(metis nat_mult_commute of_nat_id setsum_bounded[OF h])
-lemma m_s1: "S1 \<le> S2 \<Longrightarrow> m_s S1 \<ge> m_s S2"
-proof(auto simp add: less_eq_st_def m_s_def)
- assume "\<forall>x\<in>dom S2. fun S1 x \<le> fun S2 x"
- hence "\<forall>x\<in>dom S2. m(fun S1 x) \<ge> m(fun S2 x)" by (metis m1)
- thus "(\<Sum>x\<in>dom S2. m (fun S2 x)) \<le> (\<Sum>x\<in>dom S2. m (fun S1 x))"
- by (metis setsum_mono)
-qed
-
definition m_o :: "nat \<Rightarrow> 'av st option \<Rightarrow> nat" ("m\<^isub>o") where
"m_o d opt = (case opt of None \<Rightarrow> h*d+1 | Some S \<Rightarrow> m_s S)"
lemma m_o_h: "ost \<in> L X \<Longrightarrow> finite X \<Longrightarrow> m_o (card X) ost \<le> (h*card X + 1)"
by(auto simp add: m_o_def m_s_h split: option.split dest!:m_s_h)
-lemma m_o1: "finite X \<Longrightarrow> o1 \<in> L X \<Longrightarrow> o2 \<in> L X \<Longrightarrow>
- o1 \<le> o2 \<Longrightarrow> m_o (card X) o1 \<ge> m_o (card X) o2"
-proof(induction o1 o2 rule: less_eq_option.induct)
- case 1 thus ?case by (simp add: m_o_def)(metis m_s1)
-next
- case 2 thus ?case
- by(simp add: L_option_def m_o_def le_SucI m_s_h split: option.splits)
-next
- case 3 thus ?case by simp
-qed
-
definition m_c :: "'av st option acom \<Rightarrow> nat" ("m\<^isub>c") where
"m_c C = (\<Sum>i<size(annos C). m_o (card(vars(strip C))) (annos C ! i))"
@@ -257,6 +237,9 @@
assumes m2: "x < y \<Longrightarrow> m x > m y"
begin
+lemma m1: "x \<le> y \<Longrightarrow> m x \<ge> m y"
+by(auto simp: le_less m2)
+
lemma m_s2: "finite(dom S1) \<Longrightarrow> S1 < S2 \<Longrightarrow> m_s S1 > m_s S2"
proof(auto simp add: less_st_def less_eq_st_def m_s_def)
assume "finite(dom S2)" and 0: "\<forall>x\<in>dom S2. fun S1 x \<le> fun S2 x"
@@ -277,7 +260,10 @@
case 3 thus ?case by (auto simp: less_option_def)
qed
-find_theorems "(_<_) = _"
+lemma m_o1: "finite X \<Longrightarrow> o1 \<in> L X \<Longrightarrow> o2 \<in> L X \<Longrightarrow>
+ o1 \<le> o2 \<Longrightarrow> m_o (card X) o1 \<ge> m_o (card X) o2"
+by(auto simp: le_less m_o2)
+
lemma m_c2: "C1 \<in> L(vars(strip C1)) \<Longrightarrow> C2 \<in> L(vars(strip 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] less_acom_def L_acom_def)
--- a/src/HOL/IMP/Abs_Int1_const.thy Thu Mar 07 18:14:30 2013 +0100
+++ b/src/HOL/IMP/Abs_Int1_const.thy Fri Mar 08 11:28:04 2013 +0100
@@ -143,9 +143,7 @@
proof
case goal1 thus ?case by(auto simp: m_const_def split: const.splits)
next
- case goal2 thus ?case by(auto simp: m_const_def split: const.splits)
-next
- case goal3 thus ?case by(auto simp: m_const_def less_const_def split: const.splits)
+ case goal2 thus ?case by(auto simp: m_const_def less_const_def split: const.splits)
qed
thm AI_Some_measure
--- a/src/HOL/IMP/Abs_Int1_parity.thy Thu Mar 07 18:14:30 2013 +0100
+++ b/src/HOL/IMP/Abs_Int1_parity.thy Fri Mar 08 11:28:04 2013 +0100
@@ -173,9 +173,7 @@
proof
case goal1 thus ?case by(auto simp add: m_parity_def less_eq_parity_def)
next
- case goal2 thus ?case by(auto simp add: m_parity_def less_eq_parity_def)
-next
- case goal3 thus ?case by(auto simp add: m_parity_def less_eq_parity_def less_parity_def)
+ case goal2 thus ?case by(auto simp add: m_parity_def less_eq_parity_def less_parity_def)
qed
thm AI_Some_measure
--- a/src/HOL/IMP/Abs_Int3.thy Thu Mar 07 18:14:30 2013 +0100
+++ b/src/HOL/IMP/Abs_Int3.thy Fri Mar 08 11:28:04 2013 +0100
@@ -426,18 +426,27 @@
locale Measure_WN = Measure1 where m=m for m :: "'av::WN \<Rightarrow> nat" +
fixes n :: "'av \<Rightarrow> nat"
+assumes m_anti_mono: "x \<le> y \<Longrightarrow> m x \<ge> m y"
assumes m_widen: "~ y \<le> x \<Longrightarrow> m(x \<nabla> y) < m x"
assumes n_mono: "x \<le> y \<Longrightarrow> n x \<le> n y"
assumes n_narrow: "y \<le> x \<Longrightarrow> ~ x \<le> x \<triangle> y \<Longrightarrow> n(x \<triangle> y) < n x"
begin
+lemma m_s_anti_mono: "S1 \<le> S2 \<Longrightarrow> m_s S1 \<ge> m_s S2"
+proof(auto simp add: less_eq_st_def m_s_def)
+ assume "\<forall>x\<in>dom S2. fun S1 x \<le> fun S2 x"
+ hence "\<forall>x\<in>dom S2. m(fun S1 x) \<ge> m(fun S2 x)" by (metis m_anti_mono)
+ thus "(\<Sum>x\<in>dom S2. m (fun S2 x)) \<le> (\<Sum>x\<in>dom S2. m (fun S1 x))"
+ by (metis setsum_mono)
+qed
+
lemma m_s_widen: "S1 \<in> L X \<Longrightarrow> S2 \<in> L X \<Longrightarrow> finite X \<Longrightarrow>
~ S2 \<le> S1 \<Longrightarrow> m_s(S1 \<nabla> S2) < m_s S1"
proof(auto simp add: less_eq_st_def m_s_def L_st_def widen_st_def)
assume "finite(dom S1)"
have 1: "\<forall>x\<in>dom S1. m(fun S1 x) \<ge> m(fun S1 x \<nabla> fun S2 x)"
- by (metis m1 WN_class.widen1)
+ by (metis m_anti_mono WN_class.widen1)
fix x assume "x \<in> dom S1" "\<not> fun S2 x \<le> fun S1 x"
hence 2: "EX x : dom S1. m(fun S1 x) > m(fun S1 x \<nabla> fun S2 x)"
using m_widen by blast
@@ -445,6 +454,17 @@
show "(\<Sum>x\<in>dom S1. m (fun S1 x \<nabla> fun S2 x)) < (\<Sum>x\<in>dom S1. m (fun S1 x))" .
qed
+lemma m_o_anti_mono: "finite X \<Longrightarrow> o1 \<in> L X \<Longrightarrow> o2 \<in> L X \<Longrightarrow>
+ o1 \<le> o2 \<Longrightarrow> m_o (card X) o1 \<ge> m_o (card X) o2"
+proof(induction o1 o2 rule: less_eq_option.induct)
+ case 1 thus ?case by (simp add: m_o_def)(metis m_s_anti_mono)
+next
+ case 2 thus ?case
+ by(simp add: L_option_def m_o_def le_SucI m_s_h split: option.splits)
+next
+ case 3 thus ?case by simp
+qed
+
lemma m_o_widen: "\<lbrakk> S1 \<in> L X; S2 \<in> L X; finite X; \<not> S2 \<le> S1 \<rbrakk> \<Longrightarrow>
m_o (card X) (S1 \<nabla> S2) < m_o (card X) S1"
by(auto simp: m_o_def L_st_def m_s_h less_Suc_eq_le m_s_widen split: option.split)
@@ -458,7 +478,7 @@
apply(rule setsum_strict_mono_ex1)
apply simp
apply (clarsimp)
-apply(simp add: m_o1 finite_cvars widen1[where c = "strip C2"])
+apply(simp add: m_o_anti_mono finite_cvars widen1[where c = "strip C2"])
apply(auto simp: le_iff_le_annos listrel_iff_nth)
apply(rule_tac x=i in bexI)
prefer 2 apply simp
@@ -622,9 +642,9 @@
and filter_plus' = filter_plus_ivl and filter_less' = filter_less_ivl
and m = m_ivl and n = n_ivl and h = 3
proof
- case goal1 thus ?case by(rule m_ivl_anti_mono)
+ case goal2 thus ?case by(rule m_ivl_anti_mono)
next
- case goal2 thus ?case by(rule m_ivl_height)
+ case goal1 thus ?case by(rule m_ivl_height)
next
case goal3 thus ?case by(rule m_ivl_widen)
next