# HG changeset patch
# User nipkow
# Date 1326113294 -3600
# Node ID 8b5f1f91ef383c625e94baff804c047a57f81cc2
# Parent  3d518b508bbb10552140d3ca85aa1f03cf6f9e3d
Added termination to IMP Abs_Int

diff -r 3d518b508bbb -r 8b5f1f91ef38 src/HOL/IMP/Abs_Int0.thy
--- a/src/HOL/IMP/Abs_Int0.thy	Mon Jan 09 11:41:38 2012 +0100
+++ b/src/HOL/IMP/Abs_Int0.thy	Mon Jan 09 13:48:14 2012 +0100
@@ -132,4 +132,260 @@
 
 end
 
+
+subsubsection "Ascending Chain Condition"
+
+hide_const acc
+
+abbreviation "strict r == r \<inter> -(r^-1)"
+abbreviation "acc r == wf((strict r)^-1)"
+
+lemma strict_inv_image: "strict(inv_image r f) = inv_image (strict r) f"
+by(auto simp: inv_image_def)
+
+lemma acc_inv_image:
+  "acc r \<Longrightarrow> acc (inv_image r f)"
+by (metis converse_inv_image strict_inv_image wf_inv_image)
+
+text{* ACC for option type: *}
+
+lemma acc_option: assumes "acc {(x,y::'a::preord). x \<sqsubseteq> y}"
+shows "acc {(x,y::'a option). x \<sqsubseteq> y}"
+proof(auto simp: wf_eq_minimal)
+  fix xo :: "'a option" and Qo assume "xo : Qo"
+  let ?Q = "{x. Some x \<in> Qo}"
+  show "\<exists>yo\<in>Qo. \<forall>zo. yo \<sqsubseteq> zo \<and> ~ zo \<sqsubseteq> yo \<longrightarrow> zo \<notin> Qo" (is "\<exists>zo\<in>Qo. ?P zo")
+  proof cases
+    assume "?Q = {}"
+    hence "?P None" by auto
+    moreover have "None \<in> Qo" using `?Q = {}` `xo : Qo`
+      by auto (metis not_Some_eq)
+    ultimately show ?thesis by blast
+  next
+    assume "?Q \<noteq> {}"
+    with assms show ?thesis
+      apply(auto simp: wf_eq_minimal)
+      apply(erule_tac x="?Q" in allE)
+      apply auto
+      apply(rule_tac x = "Some z" in bexI)
+      by auto
+  qed
+qed
+
+text{* ACC for abstract states, via measure functions. *}
+
+(*FIXME mv*)
+lemma setsum_strict_mono1:
+fixes f :: "'a \<Rightarrow> 'b::{comm_monoid_add, ordered_cancel_ab_semigroup_add}"
+assumes "finite A" and "ALL x:A. f x \<le> g x" and "EX a:A. f a < g a"
+shows "setsum f A < setsum g A"
+proof-
+  from assms(3) obtain a where a: "a:A" "f a < g a" by blast
+  have "setsum f A = setsum f ((A-{a}) \<union> {a})"
+    by(simp add:insert_absorb[OF `a:A`])
+  also have "\<dots> = setsum f (A-{a}) + setsum f {a}"
+    using `finite A` by(subst setsum_Un_disjoint) auto
+  also have "setsum f (A-{a}) \<le> setsum g (A-{a})"
+    by(rule setsum_mono)(simp add: assms(2))
+  also have "setsum f {a} < setsum g {a}" using a by simp
+  also have "setsum g (A - {a}) + setsum g {a} = setsum g((A-{a}) \<union> {a})"
+    using `finite A` by(subst setsum_Un_disjoint[symmetric]) auto
+  also have "\<dots> = setsum g A" by(simp add:insert_absorb[OF `a:A`])
+  finally show ?thesis by (metis add_right_mono add_strict_left_mono)
+qed
+
+lemma measure_st: assumes "(strict{(x,y::'a::SL_top). x \<sqsubseteq> y})^-1 <= measure m"
+and "\<forall>x y::'a. x \<sqsubseteq> y \<and> y \<sqsubseteq> x \<longrightarrow> m x = m y"
+shows "(strict{(S,S'::'a st). S \<sqsubseteq> S'})^-1 \<subseteq>
+  measure(%fd. \<Sum>x| x\<in>set(dom fd) \<and> ~ \<top> \<sqsubseteq> fun fd x. m(fun fd x)+1)"
+proof-
+  { fix S S' :: "'a st" assume "S \<sqsubseteq> S'" "~ S' \<sqsubseteq> S"
+    let ?X = "set(dom S)" let ?Y = "set(dom S')"
+    let ?f = "fun S" let ?g = "fun S'"
+    let ?X' = "{x:?X. ~ \<top> \<sqsubseteq> ?f x}" let ?Y' = "{y:?Y. ~ \<top> \<sqsubseteq> ?g y}"
+    from `S \<sqsubseteq> S'` have "ALL y:?Y'\<inter>?X. ?f y \<sqsubseteq> ?g y"
+      by(auto simp: le_st_def lookup_def)
+    hence 1: "ALL y:?Y'\<inter>?X. m(?g y)+1 \<le> m(?f y)+1"
+      using assms(1,2) by(fastforce)
+    from `~ S' \<sqsubseteq> S` obtain u where u: "u : ?X" "~ lookup S' u \<sqsubseteq> ?f u"
+      by(auto simp: le_st_def)
+    hence "u : ?X'" by simp (metis preord_class.le_trans top)
+    have "?Y'-?X = {}" using `S \<sqsubseteq> S'` by(fastforce simp: le_st_def lookup_def)
+    have "?Y'\<inter>?X <= ?X'" apply auto
+      apply (metis `S \<sqsubseteq> S'` le_st_def lookup_def preord_class.le_trans)
+      done
+    have "(\<Sum>y\<in>?Y'. m(?g y)+1) = (\<Sum>y\<in>(?Y'-?X) \<union> (?Y'\<inter>?X). m(?g y)+1)"
+      by (metis Un_Diff_Int)
+    also have "\<dots> = (\<Sum>y\<in>?Y'\<inter>?X. m(?g y)+1)"
+      using `?Y'-?X = {}` by (metis Un_empty_left)
+    also have "\<dots> < (\<Sum>x\<in>?X'. m(?f x)+1)"
+    proof cases
+      assume "u \<in> ?Y'"
+      hence "m(?g u) < m(?f u)" using assms(1) `S \<sqsubseteq> S'` u
+        by (fastforce simp: le_st_def lookup_def)
+      have "(\<Sum>y\<in>?Y'\<inter>?X. m(?g y)+1) < (\<Sum>y\<in>?Y'\<inter>?X. m(?f y)+1)"
+        using `u:?X` `u:?Y'` `m(?g u) < m(?f u)`
+        by(fastforce intro!: setsum_strict_mono1[OF _ 1])
+      also have "\<dots> \<le> (\<Sum>y\<in>?X'. m(?f y)+1)"
+        by(simp add: setsum_mono3[OF _ `?Y'\<inter>?X <= ?X'`])
+      finally show ?thesis .
+    next
+      assume "u \<notin> ?Y'"
+      with `?Y'\<inter>?X <= ?X'` have "?Y'\<inter>?X - {u} <= ?X' - {u}" by blast
+      have "(\<Sum>y\<in>?Y'\<inter>?X. m(?g y)+1) = (\<Sum>y\<in>?Y'\<inter>?X - {u}. m(?g y)+1)"
+      proof-
+        have "?Y'\<inter>?X = ?Y'\<inter>?X - {u}" using `u \<notin> ?Y'` by auto
+        thus ?thesis by metis
+      qed
+      also have "\<dots> < (\<Sum>y\<in>?Y'\<inter>?X-{u}. m(?g y)+1) + (\<Sum>y\<in>{u}. m(?f y)+1)" by simp
+      also have "(\<Sum>y\<in>?Y'\<inter>?X-{u}. m(?g y)+1) \<le> (\<Sum>y\<in>?Y'\<inter>?X-{u}. m(?f y)+1)"
+        using 1 by(blast intro: setsum_mono)
+      also have "\<dots> \<le> (\<Sum>y\<in>?X'-{u}. m(?f y)+1)"
+        by(simp add: setsum_mono3[OF _ `?Y'\<inter>?X-{u} <= ?X'-{u}`])
+      also have "\<dots> + (\<Sum>y\<in>{u}. m(?f y)+1)= (\<Sum>y\<in>(?X'-{u}) \<union> {u}. m(?f y)+1)"
+        using `u:?X'` by(subst setsum_Un_disjoint[symmetric]) auto
+      also have "\<dots> = (\<Sum>x\<in>?X'. m(?f x)+1)"
+        using `u : ?X'` by(simp add:insert_absorb)
+      finally show ?thesis by (blast intro: add_right_mono)
+    qed
+    finally have "(\<Sum>y\<in>?Y'. m(?g y)+1) < (\<Sum>x\<in>?X'. m(?f x)+1)" .
+  } thus ?thesis by(auto simp add: measure_def inv_image_def)
+qed
+
+text{* ACC for acom. First the ordering on acom is related to an ordering on
+lists of annotations. *}
+
+(* FIXME mv and add [simp] *)
+lemma listrel_Cons_iff:
+  "(x#xs, y#ys) : listrel r \<longleftrightarrow> (x,y) \<in> r \<and> (xs,ys) \<in> listrel r"
+by (blast intro:listrel.Cons)
+
+lemma listrel_app: "(xs1,ys1) : listrel r \<Longrightarrow> (xs2,ys2) : listrel r
+  \<Longrightarrow> (xs1@xs2, ys1@ys2) : listrel r"
+by(auto simp add: listrel_iff_zip)
+
+lemma listrel_app_same_size: "size xs1 = size ys1 \<Longrightarrow> size xs2 = size ys2 \<Longrightarrow>
+  (xs1@xs2, ys1@ys2) : listrel r \<longleftrightarrow>
+  (xs1,ys1) : listrel r \<and> (xs2,ys2) : listrel r"
+by(auto simp add: listrel_iff_zip)
+
+lemma listrel_converse: "listrel(r^-1) = (listrel r)^-1"
+proof-
+  { fix xs ys
+    have "(xs,ys) : listrel(r^-1) \<longleftrightarrow> (ys,xs) : listrel r"
+      apply(induct xs arbitrary: ys)
+      apply (fastforce simp: listrel.Nil)
+      apply (fastforce simp: listrel_Cons_iff)
+      done
+  } thus ?thesis by auto
+qed
+
+(* It would be nice to get rid of refl & trans and build them into the proof *)
+lemma acc_listrel: fixes r :: "('a*'a)set" assumes "refl r" and "trans r"
+and "acc r" shows "acc (listrel r - {([],[])})"
+proof-
+  have refl: "!!x. (x,x) : r" using `refl r` unfolding refl_on_def by blast
+  have trans: "!!x y z. (x,y) : r \<Longrightarrow> (y,z) : r \<Longrightarrow> (x,z) : r"
+    using `trans r` unfolding trans_def by blast
+  from assms(3) obtain mx :: "'a set \<Rightarrow> 'a" where
+    mx: "!!S x. x:S \<Longrightarrow> mx S : S \<and> (\<forall>y. (mx S,y) : strict r \<longrightarrow> y \<notin> S)"
+    by(simp add: wf_eq_minimal) metis
+  let ?R = "listrel r - {([], [])}"
+  { fix Q and xs :: "'a list"
+    have "xs \<in> Q \<Longrightarrow> \<exists>ys. ys\<in>Q \<and> (\<forall>zs. (ys, zs) \<in> strict ?R \<longrightarrow> zs \<notin> Q)"
+      (is "_ \<Longrightarrow> \<exists>ys. ?P Q ys")
+    proof(induction xs arbitrary: Q rule: length_induct)
+      case (1 xs)
+      { have "!!ys Q. size ys < size xs \<Longrightarrow> ys : Q \<Longrightarrow> EX ms. ?P Q ms"
+          using "1.IH" by blast
+      } note IH = this
+      show ?case
+      proof(cases xs)
+        case Nil with `xs : Q` have "?P Q []" by auto
+        thus ?thesis by blast
+      next
+        case (Cons x ys)
+        let ?Q1 = "{a. \<exists>bs. size bs = size ys \<and> a#bs : Q}"
+        have "x : ?Q1" using `xs : Q` Cons by auto
+        from mx[OF this] obtain m1 where
+          1: "m1 \<in> ?Q1 \<and> (\<forall>y. (m1,y) \<in> strict r \<longrightarrow> y \<notin> ?Q1)" by blast
+        then obtain ms1 where "size ms1 = size ys" "m1#ms1 : Q" by blast+
+        hence "size ms1 < size xs" using Cons by auto
+        let ?Q2 = "{bs. \<exists>m1'. (m1',m1):r \<and> (m1,m1'):r \<and> m1'#bs : Q \<and> size bs = size ms1}"
+        have "ms1 : ?Q2" using `m1#ms1 : Q` by(blast intro: refl)
+        from IH[OF `size ms1 < size xs` this]
+        obtain ms where 2: "?P ?Q2 ms" by auto
+        then obtain m1' where m1': "(m1',m1) : r \<and> (m1,m1') : r \<and> m1'#ms : Q"
+          by blast
+        hence "\<forall>ab. (m1'#ms,ab) : strict ?R \<longrightarrow> ab \<notin> Q" using 1 2
+          apply (auto simp: listrel_Cons_iff)
+          apply (metis `length ms1 = length ys` listrel_eq_len trans)
+          by (metis `length ms1 = length ys` listrel_eq_len trans)
+        with m1' show ?thesis by blast
+      qed
+    qed
+  }
+  thus ?thesis unfolding wf_eq_minimal by (metis converse_iff)
+qed
+
+
+fun annos :: "'a acom \<Rightarrow> 'a list" where
+"annos (SKIP {a}) = [a]" |
+"annos (x::=e {a}) = [a]" |
+"annos (c1;c2) = annos c1 @ annos c2" |
+"annos (IF b THEN c1 ELSE c2 {a}) = a #  annos c1 @ annos c2" |
+"annos ({i} WHILE b DO c {a}) = i # a # annos c"
+
+lemma size_annos_same: "strip c1 = strip c2 \<Longrightarrow> size(annos c1) = size(annos c2)"
+apply(induct c2 arbitrary: c1)
+apply (auto simp: strip_eq_SKIP strip_eq_Assign strip_eq_Semi strip_eq_If strip_eq_While)
+done
+
+lemma le_iff_le_annos: "c1 \<sqsubseteq> c2 \<longleftrightarrow>
+ (annos c1, annos c2) : listrel{(x,y). x \<sqsubseteq> y} \<and> strip c1 = strip c2"
+apply(induct c1 c2 rule: le_acom.induct)
+apply (auto simp: listrel.Nil listrel_Cons_iff listrel_app size_annos_same)
+apply (metis listrel_app_same_size size_annos_same)+
+done
+
+lemma le_acom_subset_same_annos:
+ "(strict{(c,c'::'a::preord acom). c \<sqsubseteq> c'})^-1 \<subseteq>
+  (strict(inv_image (listrel{(a,a'::'a). a \<sqsubseteq> a'} - {([],[])}) annos))^-1"
+by(auto simp: le_iff_le_annos)
+
+lemma acc_acom: "acc {(a,a'::'a::preord). a \<sqsubseteq> a'} \<Longrightarrow>
+  acc {(c,c'::'a acom). c \<sqsubseteq> c'}"
+apply(rule wf_subset[OF _ le_acom_subset_same_annos])
+apply(rule acc_inv_image[OF acc_listrel])
+apply(auto simp: refl_on_def trans_def intro: le_trans)
+done
+
+text{* Termination of the fixed-point finders, assuming monotone functions: *}
+
+lemma pfp_termination:
+fixes x0 :: "'a::preord"
+assumes mono: "\<And>x y. x \<sqsubseteq> y \<Longrightarrow> f x \<sqsubseteq> f y" and "acc {(x::'a,y). x \<sqsubseteq> y}"
+and "x0 \<sqsubseteq> f x0" shows "EX x. pfp f x0 = Some x"
+proof(simp add: pfp_def, rule wf_while_option_Some[where P = "%x. x \<sqsubseteq> f x"])
+  show "wf {(x, s). (s \<sqsubseteq> f s \<and> \<not> f s \<sqsubseteq> s) \<and> x = f s}"
+    by(rule wf_subset[OF assms(2)]) auto
+next
+  show "x0 \<sqsubseteq> f x0" by(rule assms)
+next
+  fix x assume "x \<sqsubseteq> f x" thus "f x \<sqsubseteq> f(f x)" by(rule mono)
+qed
+
+lemma lpfpc_termination:
+  fixes f :: "(('a::SL_top)option acom \<Rightarrow> 'a option acom)"
+  assumes "acc {(x::'a,y). x \<sqsubseteq> y}" and "\<And>x y. x \<sqsubseteq> y \<Longrightarrow> f x \<sqsubseteq> f y"
+  and "\<And>c. strip(f c) = strip c"
+  shows "\<exists>c'. lpfp\<^isub>c f c = Some c'"
+unfolding lpfp\<^isub>c_def
+apply(rule pfp_termination)
+  apply(erule assms(2))
+ apply(rule acc_acom[OF acc_option[OF assms(1)]])
+apply(simp add: bot_acom assms(3))
+done
+
+
 end
diff -r 3d518b508bbb -r 8b5f1f91ef38 src/HOL/IMP/Abs_Int0_const.thy
--- a/src/HOL/IMP/Abs_Int0_const.thy	Mon Jan 09 11:41:38 2012 +0100
+++ b/src/HOL/IMP/Abs_Int0_const.thy	Mon Jan 09 13:48:14 2012 +0100
@@ -6,29 +6,29 @@
 
 subsection "Constant Propagation"
 
-datatype cval = Const val | Any
+datatype const = Const val | Any
 
-fun \<gamma>_cval where
-"\<gamma>_cval (Const n) = {n}" |
-"\<gamma>_cval (Any) = UNIV"
+fun \<gamma>_const where
+"\<gamma>_const (Const n) = {n}" |
+"\<gamma>_const (Any) = UNIV"
 
-fun plus_cval where
-"plus_cval (Const m) (Const n) = Const(m+n)" |
-"plus_cval _ _ = Any"
+fun plus_const where
+"plus_const (Const m) (Const n) = Const(m+n)" |
+"plus_const _ _ = Any"
 
-lemma plus_cval_cases: "plus_cval a1 a2 =
+lemma plus_const_cases: "plus_const a1 a2 =
   (case (a1,a2) of (Const m, Const n) \<Rightarrow> Const(m+n) | _ \<Rightarrow> Any)"
-by(auto split: prod.split cval.split)
+by(auto split: prod.split const.split)
 
-instantiation cval :: SL_top
+instantiation const :: SL_top
 begin
 
-fun le_cval where
+fun le_const where
 "_ \<sqsubseteq> Any = True" |
 "Const n \<sqsubseteq> Const m = (n=m)" |
 "Any \<sqsubseteq> Const _ = False"
 
-fun join_cval where
+fun join_const where
 "Const m \<squnion> Const n = (if n=m then Const m else Any)" |
 "_ \<squnion> _ = Any"
 
@@ -46,29 +46,29 @@
 next
   case goal5 thus ?case by(cases z, cases y, cases x, simp_all)
 next
-  case goal6 thus ?case by(simp add: Top_cval_def)
+  case goal6 thus ?case by(simp add: Top_const_def)
 qed
 
 end
 
 
 interpretation Val_abs
-where \<gamma> = \<gamma>_cval and num' = Const and plus' = plus_cval
+where \<gamma> = \<gamma>_const and num' = Const and plus' = plus_const
 defines aval'_const is aval'
 proof
   case goal1 thus ?case
     by(cases a, cases b, simp, simp, cases b, simp, simp)
 next
-  case goal2 show ?case by(simp add: Top_cval_def)
+  case goal2 show ?case by(simp add: Top_const_def)
 next
   case goal3 show ?case by simp
 next
   case goal4 thus ?case
-    by(auto simp: plus_cval_cases split: cval.split)
+    by(auto simp: plus_const_cases split: const.split)
 qed
 
 interpretation Abs_Int
-where \<gamma> = \<gamma>_cval and num' = Const and plus' = plus_cval
+where \<gamma> = \<gamma>_const and num' = Const and plus' = plus_const
 defines AI_const is AI
 and step_const is step'
 proof qed
@@ -77,12 +77,30 @@
 text{* Monotonicity: *}
 
 interpretation Abs_Int_mono
-where \<gamma> = \<gamma>_cval and num' = Const and plus' = plus_cval
+where \<gamma> = \<gamma>_const and num' = Const and plus' = plus_const
 proof
   case goal1 thus ?case
-    by(auto simp: plus_cval_cases split: cval.split)
+    by(auto simp: plus_const_cases split: const.split)
 qed
 
+text{* Termination: *}
+
+lemma measure_const:
+  "(strict{(x::const,y). x \<sqsubseteq> y})^-1 \<subseteq>
+  measure(%x. case x of Const _ \<Rightarrow> 1 | Any \<Rightarrow> 0)"
+by(auto split: const.splits)
+
+lemma measure_const_eq:
+  "\<forall> x y::const. x \<sqsubseteq> y \<and> y \<sqsubseteq> x \<longrightarrow> (%x. case x of Const _ \<Rightarrow> 1 | Any \<Rightarrow> 0) x = (%x. case x of Const _ \<Rightarrow> 1 | Any \<Rightarrow> 0) y"
+by(auto split: const.splits)
+
+lemma acc_const_st: "Abs_Int0.acc{(x::const st,y). x \<sqsubseteq> y}"
+by(rule wf_subset[OF wf_measure measure_st[OF measure_const measure_const_eq]])
+
+lemma "EX c'. AI_const c = Some c'"
+by(metis AI_def lpfpc_termination[OF acc_const_st, where f = "step_const \<top>",
+  OF mono_step'[OF le_refl] strip_step'])
+
 
 subsubsection "Tests"