--- a/src/HOL/IMP/Abs_Int_Den/Abs_Int_den1.thy Tue Feb 23 15:47:39 2016 +0000
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,216 +0,0 @@
-(* Author: Tobias Nipkow *)
-
-theory Abs_Int_den1
-imports Abs_Int_den0_const
-begin
-
-subsection "Backward Analysis of Expressions"
-
-class L_top_bot = SL_top +
-fixes meet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" (infixl "\<sqinter>" 65)
-and Bot :: "'a"
-assumes meet_le1 [simp]: "x \<sqinter> y \<sqsubseteq> x"
-and meet_le2 [simp]: "x \<sqinter> y \<sqsubseteq> y"
-and meet_greatest: "x \<sqsubseteq> y \<Longrightarrow> x \<sqsubseteq> z \<Longrightarrow> x \<sqsubseteq> y \<sqinter> z"
-assumes bot[simp]: "Bot \<sqsubseteq> x"
-
-locale Rep1 = Rep rep for rep :: "'a::L_top_bot \<Rightarrow> 'b set" +
-assumes inter_rep_subset_rep_meet: "rep a1 \<inter> rep a2 \<subseteq> rep(a1 \<sqinter> a2)"
-and rep_Bot: "rep Bot = {}"
-begin
-
-lemma in_rep_meet: "x <: a1 \<Longrightarrow> x <: a2 \<Longrightarrow> x <: 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)
-
-end
-
-
-locale Val_abs1 = Val_abs rep num' plus' + Rep1 rep
- for rep :: "'a::L_top_bot \<Rightarrow> int set" and num' plus' +
-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'"
-and filter_less': "filter_less' (n1<n2) a1 a2 = (a1',a2') \<Longrightarrow>
- n1 <: a1 \<Longrightarrow> n2 <: a2 \<Longrightarrow> n1 <: a1' \<and> n2 <: a2'"
-
-datatype 'a up = bot | Up 'a
-
-instantiation up :: (SL_top)SL_top
-begin
-
-fun le_up where
-"Up x \<sqsubseteq> Up y = (x \<sqsubseteq> y)" |
-"bot \<sqsubseteq> y = True" |
-"Up _ \<sqsubseteq> bot = False"
-
-lemma [simp]: "(x \<sqsubseteq> bot) = (x = bot)"
-by (cases x) simp_all
-
-lemma [simp]: "(Up x \<sqsubseteq> u) = (EX y. u = Up y & x \<sqsubseteq> y)"
-by (cases u) auto
-
-fun join_up where
-"Up x \<squnion> Up y = Up(x \<squnion> y)" |
-"bot \<squnion> y = y" |
-"x \<squnion> bot = x"
-
-lemma [simp]: "x \<squnion> bot = x"
-by (cases x) simp_all
-
-
-definition "Top = Up Top"
-
-instance proof
- case goal1 show ?case by(cases x, simp_all)
-next
- case goal2 thus ?case
- by(cases z, simp, cases y, simp, cases x, auto intro: le_trans)
-next
- case goal3 thus ?case by(cases x, simp, cases y, simp_all)
-next
- case goal4 thus ?case by(cases y, simp, cases x, simp_all)
-next
- case goal5 thus ?case by(cases z, simp, cases y, simp, cases x, simp_all)
-next
- case goal6 thus ?case by(cases x, simp_all add: Top_up_def)
-qed
-
-end
-
-
-locale Abs_Int1 = Val_abs1 +
-fixes pfp :: "('a astate up \<Rightarrow> 'a astate up) \<Rightarrow> 'a astate up \<Rightarrow> 'a astate up"
-assumes pfp: "f(pfp f x0) \<sqsubseteq> pfp f x0"
-assumes above: "x0 \<sqsubseteq> pfp f x0"
-begin
-
-(* FIXME avoid duplicating this defn *)
-abbreviation astate_in_rep (infix "<:" 50) where
-"s <: S == ALL x. s x <: lookup S x"
-
-abbreviation in_rep_up :: "state \<Rightarrow> 'a astate up \<Rightarrow> bool" (infix "<::" 50) where
-"s <:: S == EX S0. S = Up S0 \<and> s <: S0"
-
-lemma in_rep_up_trans: "(s::state) <:: S \<Longrightarrow> S \<sqsubseteq> T \<Longrightarrow> s <:: T"
-apply auto
-by (metis in_mono le_astate_def le_rep lookup_def top)
-
-lemma in_rep_join_UpI: "s <:: S1 | s <:: S2 \<Longrightarrow> s <:: S1 \<squnion> S2"
-by (metis in_rep_up_trans SL_top_class.join_ge1 SL_top_class.join_ge2)
-
-fun aval' :: "aexp \<Rightarrow> 'a astate up \<Rightarrow> 'a" ("aval\<^sup>#") where
-"aval' _ bot = Bot" |
-"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)"
-
-lemma aval'_sound: "s <:: S \<Longrightarrow> aval a s <: aval' a S"
-by (induct a) (auto simp: rep_num' rep_plus')
-
-fun afilter :: "aexp \<Rightarrow> 'a \<Rightarrow> 'a astate up \<Rightarrow> 'a astate 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>
- let a' = lookup S x \<sqinter> a in
- if a' \<sqsubseteq> Bot then bot else Up(update S x a'))" |
-"afilter (Plus e1 e2) a 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
-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. *}
-
-
-fun bfilter :: "bexp \<Rightarrow> bool \<Rightarrow> 'a astate up \<Rightarrow> 'a astate up" where
-"bfilter (Bc v) res S = (if v=res then S else bot)" |
-"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)
- in afilter e1 res1 (afilter e2 res2 S))"
-
-lemma afilter_sound: "s <:: S \<Longrightarrow> aval e s <: a \<Longrightarrow> s <:: 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 <: S'" using `s <:: S` by auto
- moreover hence "s x <: lookup S' x" by(simp)
- moreover have "s x <: a" using V by simp
- ultimately show ?case using V(1)
- by(simp add: lookup_update Let_def)
- (metis le_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)]]
- by (auto split: prod.split)
-qed
-
-lemma bfilter_sound: "s <:: S \<Longrightarrow> bv = bval b s \<Longrightarrow> s <:: bfilter b bv S"
-proof(induction b arbitrary: S bv)
- case Bc thus ?case by simp
-next
- case (Not b) thus ?case by simp
-next
- case (And b1 b2) thus ?case by (auto simp: in_rep_join_UpI)
-next
- case (Less e1 e2) thus ?case
- apply hypsubst_thin
- apply (auto split: prod.split)
- apply (metis afilter_sound filter_less' aval'_sound Less)
- done
-qed
-
-fun AI :: "com \<Rightarrow> 'a astate up \<Rightarrow> 'a astate up" where
-"AI SKIP S = S" |
-"AI (x ::= a) S =
- (case S of bot \<Rightarrow> bot | Up S \<Rightarrow> Up(update S x (aval' a (Up S))))" |
-"AI (c1;;c2) S = AI c2 (AI c1 S)" |
-"AI (IF b THEN c1 ELSE c2) S =
- AI c1 (bfilter b True S) \<squnion> AI c2 (bfilter b False S)" |
-"AI (WHILE b DO c) S =
- bfilter b False (pfp (\<lambda>S. AI c (bfilter b True S)) S)"
-
-lemma AI_sound: "(c,s) \<Rightarrow> t \<Longrightarrow> s <:: S \<Longrightarrow> t <:: AI c S"
-proof(induction c arbitrary: s t S)
- case SKIP thus ?case by fastforce
-next
- case Assign thus ?case
- by (auto simp: lookup_update aval'_sound)
-next
- case Seq thus ?case by fastforce
-next
- case If thus ?case by (auto simp: in_rep_join_UpI bfilter_sound)
-next
- case (While b c)
- let ?P = "pfp (\<lambda>S. AI c (bfilter b True S)) S"
- { fix s t
- have "(WHILE b DO c,s) \<Rightarrow> t \<Longrightarrow> s <:: ?P \<Longrightarrow>
- t <:: bfilter b False ?P"
- proof(induction "WHILE b DO c" s t rule: big_step_induct)
- case WhileFalse thus ?case by(metis bfilter_sound)
- next
- case WhileTrue show ?case
- by(rule WhileTrue, rule in_rep_up_trans[OF _ pfp],
- rule While.IH[OF WhileTrue(2)],
- rule bfilter_sound[OF WhileTrue.prems], simp add: WhileTrue(1))
- qed
- }
- with in_rep_up_trans[OF `s <:: S` above] While(2,3) AI.simps(5)
- show ?case by simp
-qed
-
-end
-
-end