
diff -r 4334c91b7405 -r f682f3f7b726 src/HOL/IMP/Abs_Int1.thy
--- a/src/HOL/IMP/Abs_Int1.thy	Wed Nov 23 23:31:32 2011 +0100
+++ b/src/HOL/IMP/Abs_Int1.thy	Thu Nov 24 19:58:37 2011 +0100
@@ -1,7 +1,7 @@
 (* Author: Tobias Nipkow *)
 
 theory Abs_Int1
-imports Abs_Int0_const
+imports Abs_Int0
 begin
 
 instantiation prod :: (preord,preord) preord
@@ -37,20 +37,19 @@
 
 end
 
-
 locale Val_abs1_rep =
   Val_abs rep num' plus'
-  for rep :: "'a::L_top_bot \<Rightarrow> val set" and num' plus' +
+  for rep :: "'a::L_top_bot \<Rightarrow> val set" ("\<gamma>") and num' plus' +
 assumes inter_rep_subset_rep_meet:
   "rep a1 \<inter> rep a2 \<subseteq> rep(a1 \<sqinter> a2)"
-and rep_Bot: "rep \<bottom> = {}"
+and rep_Bot[simp]: "rep \<bottom> = {}"
 begin
 
-lemma in_rep_meet: "x <: a1 \<Longrightarrow> x <: a2 \<Longrightarrow> x <: a1 \<sqinter> a2"
+lemma in_rep_meet: "x : \<gamma> a1 \<Longrightarrow> x : \<gamma> a2 \<Longrightarrow> x : \<gamma>(a1 \<sqinter> a2)"
 by (metis IntI inter_rep_subset_rep_meet set_mp)
 
-lemma rep_meet[simp]: "rep(a1 \<sqinter> a2) = rep a1 \<inter> rep a2"
-by (metis equalityI inter_rep_subset_rep_meet le_inf_iff le_rep meet_le1 meet_le2)
+lemma rep_meet[simp]: "\<gamma>(a1 \<sqinter> a2) = \<gamma> a1 \<inter> \<gamma> a2"
+by (metis equalityI inter_rep_subset_rep_meet le_inf_iff mono_rep meet_le1 meet_le2)
 
 end
 
@@ -59,76 +58,74 @@
 fixes filter_plus' :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a * 'a"
 and filter_less' :: "bool \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a * 'a"
 assumes filter_plus': "filter_plus' a a1 a2 = (a1',a2') \<Longrightarrow>
-  n1 <: a1 \<Longrightarrow> n2 <: a2 \<Longrightarrow> n1+n2 <: a \<Longrightarrow> n1 <: a1' \<and> n2 <: a2'"
+  n1 : \<gamma> a1 \<Longrightarrow> n2 : \<gamma> a2 \<Longrightarrow> n1+n2 : \<gamma> a \<Longrightarrow> n1 : \<gamma> a1' \<and> n2 : \<gamma> a2'"
 and filter_less': "filter_less' (n1<n2) a1 a2 = (a1',a2') \<Longrightarrow>
-  n1 <: a1 \<Longrightarrow> n2 <: a2 \<Longrightarrow> n1 <: a1' \<and> n2 <: a2'"
+  n1 : \<gamma> a1 \<Longrightarrow> n2 : \<gamma> a2 \<Longrightarrow> n1 : \<gamma> a1' \<and> n2 : \<gamma> a2'"
 
 
 locale Abs_Int1 = Val_abs1
 begin
 
-lemma in_rep_join_UpI: "s <:up S1 | s <:up S2 \<Longrightarrow> s <:up S1 \<squnion> S2"
-by (metis join_ge1 join_ge2 up_fun_in_rep_le)
+lemma in_rep_join_UpI: "s : \<gamma>\<^isub>u S1 | s : \<gamma>\<^isub>u S2 \<Longrightarrow> s : \<gamma>\<^isub>u(S1 \<squnion> S2)"
+by (metis (no_types) join_ge1 join_ge2 mono_rep_u set_rev_mp)
 
-fun aval' :: "aexp \<Rightarrow> 'a st up \<Rightarrow> 'a" where
-"aval' _ Bot = \<bottom>" |
-"aval' (N n) _ = num' n" |
-"aval' (V x) (Up S) = lookup S x" |
-"aval' (Plus a1 a2) S = plus' (aval' a1 S) (aval' a2 S)"
+fun aval'' :: "aexp \<Rightarrow> 'a st option \<Rightarrow> 'a" where
+"aval'' e None = \<bottom>" |
+"aval'' e (Some sa) = aval' e sa"
 
-lemma aval'_sound: "s <:up S \<Longrightarrow> aval a s <: aval' a S"
-by(induct a)(auto simp: rep_num' rep_plus' in_rep_up_iff lookup_def rep_st_def)
+lemma aval''_sound: "s : \<gamma>\<^isub>u S \<Longrightarrow> aval a s : \<gamma>(aval'' a S)"
+by(cases S)(simp add: aval'_sound)+
 
 subsubsection "Backward analysis"
 
-fun afilter :: "aexp \<Rightarrow> 'a \<Rightarrow> 'a st up \<Rightarrow> 'a st up" where
-"afilter (N n) a S = (if n <: a then S else Bot)" |
-"afilter (V x) a S = (case S of Bot \<Rightarrow> Bot | Up S \<Rightarrow>
+fun afilter :: "aexp \<Rightarrow> 'a \<Rightarrow> 'a st option \<Rightarrow> 'a st option" where
+"afilter (N n) a S = (if n : \<gamma> a then S else None)" |
+"afilter (V x) a S = (case S of None \<Rightarrow> None | Some S \<Rightarrow>
   let a' = lookup S x \<sqinter> a in
-  if a' \<sqsubseteq> \<bottom> then Bot else Up(update S x a'))" |
+  if a' \<sqsubseteq> \<bottom> then None else Some(update S x a'))" |
 "afilter (Plus e1 e2) a S =
- (let (a1,a2) = filter_plus' a (aval' e1 S) (aval' e2 S)
+ (let (a1,a2) = filter_plus' a (aval'' e1 S) (aval'' e2 S)
   in afilter e1 a1 (afilter e2 a2 S))"
 
-text{* The test for @{const Bot} in the @{const V}-case is important: @{const
-Bot} indicates that a variable has no possible values, i.e.\ that the current
+text{* The test for @{const bot} in the @{const V}-case is important: @{const
+bot} indicates that a variable has no possible values, i.e.\ that the current
 program point is unreachable. But then the abstract state should collapse to
-@{const bot}. Put differently, we maintain the invariant that in an abstract
-state all variables are mapped to non-@{const Bot} values. Otherwise the
-(pointwise) join of two abstract states, one of which contains @{const Bot}
-values, may produce too large a result, thus making the analysis less
-precise. *}
+@{const None}. Put differently, we maintain the invariant that in an abstract
+state of the form @{term"Some s"}, all variables are mapped to non-@{const
+bot} values. Otherwise the (pointwise) join of two abstract states, one of
+which contains @{const bot} values, may produce too large a result, thus
+making the analysis less precise. *}
 
 
-fun bfilter :: "bexp \<Rightarrow> bool \<Rightarrow> 'a st up \<Rightarrow> 'a st up" where
-"bfilter (Bc v) res S = (if v=res then S else Bot)" |
+fun bfilter :: "bexp \<Rightarrow> bool \<Rightarrow> 'a st option \<Rightarrow> 'a st option" where
+"bfilter (Bc v) res S = (if v=res then S else None)" |
 "bfilter (Not b) res S = bfilter b (\<not> res) S" |
 "bfilter (And b1 b2) res S =
   (if res then bfilter b1 True (bfilter b2 True S)
    else bfilter b1 False S \<squnion> bfilter b2 False S)" |
 "bfilter (Less e1 e2) res S =
-  (let (res1,res2) = filter_less' res (aval' e1 S) (aval' e2 S)
+  (let (res1,res2) = filter_less' res (aval'' e1 S) (aval'' e2 S)
    in afilter e1 res1 (afilter e2 res2 S))"
 
-lemma afilter_sound: "s <:up S \<Longrightarrow> aval e s <: a \<Longrightarrow> s <:up afilter e a S"
+lemma afilter_sound: "s : \<gamma>\<^isub>u S \<Longrightarrow> aval e s : \<gamma> a \<Longrightarrow> s : \<gamma>\<^isub>u (afilter e a S)"
 proof(induction e arbitrary: a S)
   case N thus ?case by simp
 next
   case (V x)
-  obtain S' where "S = Up S'" and "s <:f S'" using `s <:up S`
+  obtain S' where "S = Some S'" and "s : \<gamma>\<^isub>f S'" using `s : \<gamma>\<^isub>u S`
     by(auto simp: in_rep_up_iff)
-  moreover hence "s x <: lookup S' x" by(simp add: rep_st_def)
-  moreover have "s x <: a" using V by simp
+  moreover hence "s x : \<gamma> (lookup S' x)" by(simp add: rep_st_def)
+  moreover have "s x : \<gamma> a" using V by simp
   ultimately show ?case using V(1)
     by(simp add: lookup_update Let_def rep_st_def)
-      (metis le_rep emptyE in_rep_meet rep_Bot subset_empty)
+      (metis mono_rep emptyE in_rep_meet rep_Bot subset_empty)
 next
   case (Plus e1 e2) thus ?case
-    using filter_plus'[OF _ aval'_sound[OF Plus(3)] aval'_sound[OF Plus(3)]]
+    using filter_plus'[OF _ aval''_sound[OF Plus(3)] aval''_sound[OF Plus(3)]]
     by (auto split: prod.split)
 qed
 
-lemma bfilter_sound: "s <:up S \<Longrightarrow> bv = bval b s \<Longrightarrow> s <:up bfilter b bv S"
+lemma bfilter_sound: "s : \<gamma>\<^isub>u S \<Longrightarrow> bv = bval b s \<Longrightarrow> s : \<gamma>\<^isub>u(bfilter b bv S)"
 proof(induction b arbitrary: S bv)
   case Bc thus ?case by simp
 next
@@ -138,16 +135,15 @@
 next
   case (Less e1 e2) thus ?case
     by (auto split: prod.split)
-       (metis afilter_sound filter_less' aval'_sound Less)
+       (metis afilter_sound filter_less' aval''_sound Less)
 qed
 
 
-fun step :: "'a st up \<Rightarrow> 'a st up acom \<Rightarrow> 'a st up acom"
+fun step :: "'a st option \<Rightarrow> 'a st option acom \<Rightarrow> 'a st option acom"
  where
 "step S (SKIP {P}) = (SKIP {S})" |
 "step S (x ::= e {P}) =
-  x ::= e {case S of Bot \<Rightarrow> Bot
-           | Up S \<Rightarrow> Up(update S x (aval' e (Up S)))}" |
+  x ::= e {case S of None \<Rightarrow> None | Some S \<Rightarrow> Some(update S x (aval' e S))}" |
 "step S (c1; c2) = step S c1; step (post c1) c2" |
 "step S (IF b THEN c1 ELSE c2 {P}) =
   (let c1' = step (bfilter b True S) c1; c2' = step (bfilter b False S) c2
@@ -157,7 +153,7 @@
    WHILE b DO step (bfilter b True Inv) c
    {bfilter b False Inv}"
 
-definition AI :: "com \<Rightarrow> 'a st up acom option" where
+definition AI :: "com \<Rightarrow> 'a st option acom option" where
 "AI = lpfp\<^isub>c (step \<top>)"
 
 lemma strip_step[simp]: "strip(step S c) = strip c"
@@ -166,68 +162,73 @@
 
 subsubsection "Soundness"
 
-lemma in_rep_update: "\<lbrakk> s <:f S; i <: a \<rbrakk> \<Longrightarrow> s(x := i) <:f update S x a"
+lemma in_rep_update:
+  "\<lbrakk> s : \<gamma>\<^isub>f S; i : \<gamma> a \<rbrakk> \<Longrightarrow> s(x := i) : \<gamma>\<^isub>f(update S x a)"
 by(simp add: rep_st_def lookup_update)
 
-lemma While_final_False: "(WHILE b DO c, s) \<Rightarrow> t \<Longrightarrow> \<not> bval b t"
-by(induct "WHILE b DO c" s t rule: big_step_induct) simp_all
 
-lemma step_sound:
-  "step S c \<sqsubseteq> c \<Longrightarrow> (strip c,s) \<Rightarrow> t \<Longrightarrow> s <:up S \<Longrightarrow> t <:up post c"
-proof(induction c arbitrary: S s t)
+lemma step_preserves_le2:
+  "\<lbrakk> S \<subseteq> \<gamma>\<^isub>u sa; cs \<le> \<gamma>\<^isub>c ca; strip cs = c; strip ca = c \<rbrakk>
+   \<Longrightarrow> step_cs S cs \<le> \<gamma>\<^isub>c (step sa ca)"
+proof(induction c arbitrary: cs ca S sa)
   case SKIP thus ?case
-    by simp (metis skipE up_fun_in_rep_le)
+    by(auto simp:strip_eq_SKIP)
 next
   case Assign thus ?case
-    apply (auto simp del: fun_upd_apply split: up.splits)
-    by (metis aval'_sound fun_in_rep_le in_rep_update rep_up.simps(2))
+    by (fastforce simp: strip_eq_Assign intro: aval'_sound in_rep_update
+      split: option.splits del:subsetD)
 next
-  case Semi thus ?case by simp blast
+  case Semi thus ?case apply (auto simp: strip_eq_Semi)
+    by (metis le_post post_map_acom)
 next
-  case (If b c1 c2 S0)
-  show ?case
-  proof cases
-    assume "bval b s"
-    with If.prems have 1: "step (bfilter b True S) c1 \<sqsubseteq> c1"
-      and 2: "(strip c1, s) \<Rightarrow> t" and 3: "post c1 \<sqsubseteq> S0" by(auto simp: Let_def)
-    from If.IH(1)[OF 1 2 bfilter_sound[OF `s <:up S`]] `bval b s` 3
-    show ?thesis by simp (metis up_fun_in_rep_le)
-  next
-    assume "\<not> bval b s"
-    with If.prems have 1: "step (bfilter b False S) c2 \<sqsubseteq> c2"
-      and 2: "(strip c2, s) \<Rightarrow> t" and 3: "post c2 \<sqsubseteq> S0" by(auto simp: Let_def)
-    from If.IH(2)[OF 1 2 bfilter_sound[OF `s <:up S`]] `\<not> bval b s` 3
-    show ?thesis by simp (metis up_fun_in_rep_le)
-  qed
+  case (If b c1 c2)
+  then obtain cs1 cs2 ca1 ca2 P Pa where
+      "cs= IF b THEN cs1 ELSE cs2 {P}" "ca= IF b THEN ca1 ELSE ca2 {Pa}"
+      "P \<subseteq> \<gamma>\<^isub>u Pa" "cs1 \<le> \<gamma>\<^isub>c ca1" "cs2 \<le> \<gamma>\<^isub>c ca2"
+      "strip cs1 = c1" "strip ca1 = c1" "strip cs2 = c2" "strip ca2 = c2"
+    by (fastforce simp: strip_eq_If)
+  moreover have "post cs1 \<subseteq> \<gamma>\<^isub>u(post ca1 \<squnion> post ca2)"
+    by (metis (no_types) `cs1 \<le> \<gamma>\<^isub>c ca1` join_ge1 le_post mono_rep_u order_trans post_map_acom)
+  moreover have "post cs2 \<subseteq> \<gamma>\<^isub>u(post ca1 \<squnion> post ca2)"
+    by (metis (no_types) `cs2 \<le> \<gamma>\<^isub>c ca2` join_ge2 le_post mono_rep_u order_trans post_map_acom)
+  ultimately show ?case using `S \<subseteq> \<gamma>\<^isub>u sa`
+    by (simp add: If.IH subset_iff bfilter_sound)
 next
-  case (While Inv b c P)
-  from While.prems have inv: "step (bfilter b True Inv) c \<sqsubseteq> c"
-    and "post c \<sqsubseteq> Inv" and "S \<sqsubseteq> Inv" and "bfilter b False Inv \<sqsubseteq> P"
-    by(auto simp: Let_def)
-  { fix s t have "(WHILE b DO strip c,s) \<Rightarrow> t \<Longrightarrow> s <:up Inv \<Longrightarrow> t <:up Inv"
-    proof(induction "WHILE b DO strip c" s t rule: big_step_induct)
-      case WhileFalse thus ?case by simp
-    next
-      case (WhileTrue s1 s2 s3)
-      from WhileTrue.hyps(5)[OF up_fun_in_rep_le[OF While.IH[OF inv `(strip c, s1) \<Rightarrow> s2` bfilter_sound[OF `s1 <:up Inv`]] `post c \<sqsubseteq> Inv`]] `bval b s1`
-      show ?case by simp
-    qed
-  } note Inv = this
-  from  While.prems(2) have "(WHILE b DO strip c, s) \<Rightarrow> t" and "\<not> bval b t"
-    by(auto dest: While_final_False)
-  from Inv[OF this(1) up_fun_in_rep_le[OF `s <:up S` `S \<sqsubseteq> Inv`]]
-  have "t <:up Inv" .
-  from up_fun_in_rep_le[OF bfilter_sound[OF this]  `bfilter b False Inv \<sqsubseteq> P`]
-  show ?case using `\<not> bval b t` by simp
+  case (While b c1)
+  then obtain cs1 ca1 I P Ia Pa where
+    "cs = {I} WHILE b DO cs1 {P}" "ca = {Ia} WHILE b DO ca1 {Pa}"
+    "I \<subseteq> \<gamma>\<^isub>u Ia" "P \<subseteq> \<gamma>\<^isub>u Pa" "cs1 \<le> \<gamma>\<^isub>c ca1"
+    "strip cs1 = c1" "strip ca1 = c1"
+    by (fastforce simp: strip_eq_While)
+  moreover have "S \<union> post cs1 \<subseteq> \<gamma>\<^isub>u (sa \<squnion> post ca1)"
+    using `S \<subseteq> \<gamma>\<^isub>u sa` le_post[OF `cs1 \<le> \<gamma>\<^isub>c ca1`, simplified]
+    by (metis (no_types) join_ge1 join_ge2 le_sup_iff mono_rep_u order_trans)
+  ultimately show ?case by (simp add: While.IH subset_iff bfilter_sound)
 qed
 
-lemma AI_sound: "\<lbrakk> AI c = Some c';  (c,s) \<Rightarrow> t \<rbrakk> \<Longrightarrow> t <:up post c'"
-unfolding AI_def
-by (metis in_rep_Top_up lpfpc_pfp step_sound strip_lpfpc strip_step)
-(*
-by(metis step_sound[of "\<top>" c' s t] strip_lpfp strip_step
-  lpfp_pfp mono_def mono_step[OF le_refl] in_rep_Top_up)
-*)
+lemma step_preserves_le:
+  "\<lbrakk> S \<subseteq> \<gamma>\<^isub>u sa; cs \<le> \<gamma>\<^isub>c ca; strip cs = c \<rbrakk>
+   \<Longrightarrow> step_cs S cs \<le> \<gamma>\<^isub>c(step sa ca)"
+by (metis le_strip step_preserves_le2 strip_acom)
+
+lemma AI_sound: "AI c = Some c' \<Longrightarrow> CS UNIV c \<le> \<gamma>\<^isub>c c'"
+proof(simp add: CS_def AI_def)
+  assume 1: "lpfp\<^isub>c (step \<top>) c = Some c'"
+  have 2: "step \<top> c' \<sqsubseteq> c'" by(rule lpfpc_pfp[OF 1])
+  have 3: "strip (\<gamma>\<^isub>c (step \<top> c')) = c"
+    by(simp add: strip_lpfpc[OF _ 1])
+  have "lfp c (step_cs UNIV) \<le> \<gamma>\<^isub>c (step \<top> c')"
+  proof(rule lfp_lowerbound[OF 3])
+    show "step_cs UNIV (\<gamma>\<^isub>c (step \<top> c')) \<le> \<gamma>\<^isub>c (step \<top> c')"
+    proof(rule step_preserves_le[OF _ _ 3])
+      show "UNIV \<subseteq> \<gamma>\<^isub>u \<top>" by simp
+      show "\<gamma>\<^isub>c (step \<top> c') \<le> \<gamma>\<^isub>c c'" by(rule mono_rep_c[OF 2])
+    qed
+  qed
+  from this 2 show "lfp c (step_cs UNIV) \<le> \<gamma>\<^isub>c c'"
+    by (blast intro: mono_rep_c order_trans)
+qed
+
 end
 
 
@@ -242,39 +243,44 @@
 begin
 
 lemma mono_aval': "S \<sqsubseteq> S' \<Longrightarrow> aval' e S \<sqsubseteq> aval' e S'"
+by(induction e) (auto simp: le_st_def lookup_def mono_plus')
+
+lemma mono_aval'': "S \<sqsubseteq> S' \<Longrightarrow> aval'' e S \<sqsubseteq> aval'' e S'"
 apply(cases S)
  apply simp
 apply(cases S')
  apply simp
-apply simp
-by(induction e) (auto simp: le_st_def lookup_def mono_plus')
+by (simp add: mono_aval')
 
 lemma mono_afilter: "r \<sqsubseteq> r' \<Longrightarrow> S \<sqsubseteq> S' \<Longrightarrow> afilter e r S \<sqsubseteq> afilter e r' S'"
 apply(induction e arbitrary: r r' S S')
-apply(auto simp: Let_def split: up.splits prod.splits)
-apply (metis le_rep subsetD)
+apply(auto simp: Let_def split: option.splits prod.splits)
+apply (metis mono_rep subsetD)
 apply(drule_tac x = "list" in mono_lookup)
 apply (metis mono_meet le_trans)
-apply (metis mono_meet mono_lookup mono_update le_trans)
-apply(metis mono_aval' mono_filter_plus'[simplified le_prod_def] fst_conv snd_conv)
+apply (metis mono_meet mono_lookup mono_update)
+apply(metis mono_aval'' mono_filter_plus'[simplified le_prod_def] fst_conv snd_conv)
 done
 
 lemma mono_bfilter: "S \<sqsubseteq> S' \<Longrightarrow> bfilter b r S \<sqsubseteq> bfilter b r S'"
 apply(induction b arbitrary: r S S')
 apply(auto simp: le_trans[OF _ join_ge1] le_trans[OF _ join_ge2] split: prod.splits)
-apply(metis mono_aval' mono_afilter mono_filter_less'[simplified le_prod_def] fst_conv snd_conv)
+apply(metis mono_aval'' mono_afilter mono_filter_less'[simplified le_prod_def] fst_conv snd_conv)
 done
 
 
 lemma post_le_post: "c \<sqsubseteq> c' \<Longrightarrow> post c \<sqsubseteq> post c'"
 by (induction c c' rule: le_acom.induct) simp_all
 
-lemma mono_step: "S \<sqsubseteq> S' \<Longrightarrow> c \<sqsubseteq> c' \<Longrightarrow> step S c \<sqsubseteq> step S' c'"
+lemma mono_step_aux: "S \<sqsubseteq> S' \<Longrightarrow> c \<sqsubseteq> c' \<Longrightarrow> step S c \<sqsubseteq> step S' c'"
 apply(induction c c' arbitrary: S S' rule: le_acom.induct)
 apply (auto simp: post_le_post Let_def mono_bfilter mono_update mono_aval' le_join_disj
-  split: up.split)
+  split: option.split)
 done
 
+lemma mono_step: "mono (step S)"
+by(simp add: mono_def mono_step_aux[OF le_refl])
+
 end
 
 end




