--- a/src/HOL/IMP/Abs_Int1.thy Thu Apr 19 12:28:10 2012 +0200
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,358 +0,0 @@
-(* Author: Tobias Nipkow *)
-
-theory Abs_Int1
-imports Abs_Int0 Vars
-begin
-
-instantiation prod :: (preord,preord) preord
-begin
-
-definition "le_prod p1 p2 = (fst p1 \<sqsubseteq> fst p2 \<and> snd p1 \<sqsubseteq> snd p2)"
-
-instance
-proof
- case goal1 show ?case by(simp add: le_prod_def)
-next
- case goal2 thus ?case unfolding le_prod_def by(metis le_trans)
-qed
-
-end
-
-instantiation com :: vars
-begin
-
-fun vars_com :: "com \<Rightarrow> vname set" where
-"vars com.SKIP = {}" |
-"vars (x::=e) = {x} \<union> vars e" |
-"vars (c1;c2) = vars c1 \<union> vars c2" |
-"vars (IF b THEN c1 ELSE c2) = vars b \<union> vars c1 \<union> vars c2" |
-"vars (WHILE b DO c) = vars b \<union> vars c"
-
-instance ..
-
-end
-
-lemma finite_avars: "finite(vars(a::aexp))"
-by(induction a) simp_all
-
-lemma finite_bvars: "finite(vars(b::bexp))"
-by(induction b) (simp_all add: finite_avars)
-
-lemma finite_cvars: "finite(vars(c::com))"
-by(induction c) (simp_all add: finite_avars finite_bvars)
-
-
-subsection "Backward Analysis of Expressions"
-
-hide_const bot
-
-class L_top_bot = SL_top +
-fixes meet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" (infixl "\<sqinter>" 65)
-and bot :: "'a" ("\<bottom>")
-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]: "\<bottom> \<sqsubseteq> x"
-begin
-
-lemma mono_meet: "x \<sqsubseteq> x' \<Longrightarrow> y \<sqsubseteq> y' \<Longrightarrow> x \<sqinter> y \<sqsubseteq> x' \<sqinter> y'"
-by (metis meet_greatest meet_le1 meet_le2 le_trans)
-
-end
-
-locale Val_abs1_gamma =
- Gamma where \<gamma> = \<gamma> for \<gamma> :: "'av::L_top_bot \<Rightarrow> val set" +
-assumes inter_gamma_subset_gamma_meet:
- "\<gamma> a1 \<inter> \<gamma> a2 \<subseteq> \<gamma>(a1 \<sqinter> a2)"
-and gamma_Bot[simp]: "\<gamma> \<bottom> = {}"
-begin
-
-lemma in_gamma_meet: "x : \<gamma> a1 \<Longrightarrow> x : \<gamma> a2 \<Longrightarrow> x : \<gamma>(a1 \<sqinter> a2)"
-by (metis IntI inter_gamma_subset_gamma_meet set_mp)
-
-lemma gamma_meet[simp]: "\<gamma>(a1 \<sqinter> a2) = \<gamma> a1 \<inter> \<gamma> a2"
-by (metis equalityI inter_gamma_subset_gamma_meet le_inf_iff mono_gamma meet_le1 meet_le2)
-
-end
-
-
-locale Val_abs1 =
- Val_abs1_gamma where \<gamma> = \<gamma>
- for \<gamma> :: "'av::L_top_bot \<Rightarrow> val set" +
-fixes test_num' :: "val \<Rightarrow> 'av \<Rightarrow> bool"
-and filter_plus' :: "'av \<Rightarrow> 'av \<Rightarrow> 'av \<Rightarrow> 'av * 'av"
-and filter_less' :: "bool \<Rightarrow> 'av \<Rightarrow> 'av \<Rightarrow> 'av * 'av"
-assumes test_num': "test_num' n a = (n : \<gamma> a)"
-and filter_plus': "filter_plus' a a1 a2 = (b1,b2) \<Longrightarrow>
- n1 : \<gamma> a1 \<Longrightarrow> n2 : \<gamma> a2 \<Longrightarrow> n1+n2 : \<gamma> a \<Longrightarrow> n1 : \<gamma> b1 \<and> n2 : \<gamma> b2"
-and filter_less': "filter_less' (n1<n2) a1 a2 = (b1,b2) \<Longrightarrow>
- n1 : \<gamma> a1 \<Longrightarrow> n2 : \<gamma> a2 \<Longrightarrow> n1 : \<gamma> b1 \<and> n2 : \<gamma> b2"
-
-
-locale Abs_Int1 =
- Val_abs1 where \<gamma> = \<gamma> for \<gamma> :: "'av::L_top_bot \<Rightarrow> val set"
-begin
-
-lemma in_gamma_join_UpI: "s : \<gamma>\<^isub>o S1 \<or> s : \<gamma>\<^isub>o S2 \<Longrightarrow> s : \<gamma>\<^isub>o(S1 \<squnion> S2)"
-by (metis (no_types) join_ge1 join_ge2 mono_gamma_o set_rev_mp)
-
-fun aval'' :: "aexp \<Rightarrow> 'av st option \<Rightarrow> 'av" where
-"aval'' e None = \<bottom>" |
-"aval'' e (Some sa) = aval' e sa"
-
-lemma aval''_sound: "s : \<gamma>\<^isub>o S \<Longrightarrow> aval a s : \<gamma>(aval'' a S)"
-by(cases S)(simp add: aval'_sound)+
-
-subsubsection "Backward analysis"
-
-fun afilter :: "aexp \<Rightarrow> 'av \<Rightarrow> 'av st option \<Rightarrow> 'av st option" where
-"afilter (N n) a S = (if test_num' n 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 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)
- 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 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> 'av st option \<Rightarrow> 'av 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)
- in afilter e1 res1 (afilter e2 res2 S))"
-
-lemma afilter_sound: "s : \<gamma>\<^isub>o S \<Longrightarrow> aval e s : \<gamma> a \<Longrightarrow> s : \<gamma>\<^isub>o (afilter e a S)"
-proof(induction e arbitrary: a S)
- case N thus ?case by simp (metis test_num')
-next
- case (V x)
- obtain S' where "S = Some S'" and "s : \<gamma>\<^isub>f S'" using `s : \<gamma>\<^isub>o S`
- by(auto simp: in_gamma_option_iff)
- moreover hence "s x : \<gamma> (lookup S' x)" by(simp add: \<gamma>_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 \<gamma>_st_def)
- (metis mono_gamma emptyE in_gamma_meet gamma_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 : \<gamma>\<^isub>o S \<Longrightarrow> bv = bval b s \<Longrightarrow> s : \<gamma>\<^isub>o(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(fastforce simp: in_gamma_join_UpI)
-next
- case (Less e1 e2) thus ?case
- by (auto split: prod.split)
- (metis afilter_sound filter_less' aval''_sound Less)
-qed
-
-
-fun step' :: "'av st option \<Rightarrow> 'av st option acom \<Rightarrow> 'av st option acom"
- where
-"step' S (SKIP {P}) = (SKIP {S})" |
-"step' S (x ::= e {P}) =
- 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
- in IF b THEN c1' ELSE c2' {post c1 \<squnion> post c2})" |
-"step' S ({Inv} WHILE b DO c {P}) =
- {S \<squnion> post c}
- WHILE b DO step' (bfilter b True Inv) c
- {bfilter b False Inv}"
-
-definition AI :: "com \<Rightarrow> 'av st option acom option" where
-"AI = lpfp\<^isub>c (step' \<top>)"
-
-lemma strip_step'[simp]: "strip(step' S c) = strip c"
-by(induct c arbitrary: S) (simp_all add: Let_def)
-
-
-subsubsection "Soundness"
-
-lemma in_gamma_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: \<gamma>_st_def lookup_update)
-
-lemma step_preserves_le:
- "\<lbrakk> S \<subseteq> \<gamma>\<^isub>o S'; cs \<le> \<gamma>\<^isub>c ca \<rbrakk> \<Longrightarrow> step S cs \<le> \<gamma>\<^isub>c (step' S' ca)"
-proof(induction cs arbitrary: ca S S')
- case SKIP thus ?case by(auto simp:SKIP_le map_acom_SKIP)
-next
- case Assign thus ?case
- by (fastforce simp: Assign_le map_acom_Assign intro: aval'_sound in_gamma_update
- split: option.splits del:subsetD)
-next
- case Semi thus ?case apply (auto simp: Semi_le map_acom_Semi)
- by (metis le_post post_map_acom)
-next
- case (If b cs1 cs2 P)
- then obtain ca1 ca2 Pa where
- "ca= IF b THEN ca1 ELSE ca2 {Pa}"
- "P \<subseteq> \<gamma>\<^isub>o Pa" "cs1 \<le> \<gamma>\<^isub>c ca1" "cs2 \<le> \<gamma>\<^isub>c ca2"
- by (fastforce simp: If_le map_acom_If)
- moreover have "post cs1 \<subseteq> \<gamma>\<^isub>o(post ca1 \<squnion> post ca2)"
- by (metis (no_types) `cs1 \<le> \<gamma>\<^isub>c ca1` join_ge1 le_post mono_gamma_o order_trans post_map_acom)
- moreover have "post cs2 \<subseteq> \<gamma>\<^isub>o(post ca1 \<squnion> post ca2)"
- by (metis (no_types) `cs2 \<le> \<gamma>\<^isub>c ca2` join_ge2 le_post mono_gamma_o order_trans post_map_acom)
- ultimately show ?case using `S \<subseteq> \<gamma>\<^isub>o S'`
- by (simp add: If.IH subset_iff bfilter_sound)
-next
- case (While I b cs1 P)
- then obtain ca1 Ia Pa where
- "ca = {Ia} WHILE b DO ca1 {Pa}"
- "I \<subseteq> \<gamma>\<^isub>o Ia" "P \<subseteq> \<gamma>\<^isub>o Pa" "cs1 \<le> \<gamma>\<^isub>c ca1"
- by (fastforce simp: map_acom_While While_le)
- moreover have "S \<union> post cs1 \<subseteq> \<gamma>\<^isub>o (S' \<squnion> post ca1)"
- using `S \<subseteq> \<gamma>\<^isub>o S'` le_post[OF `cs1 \<le> \<gamma>\<^isub>c ca1`, simplified]
- by (metis (no_types) join_ge1 join_ge2 le_sup_iff mono_gamma_o order_trans)
- ultimately show ?case by (simp add: While.IH subset_iff bfilter_sound)
-qed
-
-lemma AI_sound: "AI c = Some c' \<Longrightarrow> CS 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 (step UNIV) c \<le> \<gamma>\<^isub>c (step' \<top> c')"
- proof(rule lfp_lowerbound[simplified,OF 3])
- show "step UNIV (\<gamma>\<^isub>c (step' \<top> c')) \<le> \<gamma>\<^isub>c (step' \<top> c')"
- proof(rule step_preserves_le[OF _ _])
- show "UNIV \<subseteq> \<gamma>\<^isub>o \<top>" by simp
- show "\<gamma>\<^isub>c (step' \<top> c') \<le> \<gamma>\<^isub>c c'" by(rule mono_gamma_c[OF 2])
- qed
- qed
- from this 2 show "lfp (step UNIV) c \<le> \<gamma>\<^isub>c c'"
- by (blast intro: mono_gamma_c order_trans)
-qed
-
-
-subsubsection "Commands over a set of variables"
-
-text{* Key invariant: the domains of all abstract states are subsets of the
-set of variables of the program. *}
-
-definition "domo S = (case S of None \<Rightarrow> {} | Some S' \<Rightarrow> set(dom S'))"
-
-definition Com :: "vname set \<Rightarrow> 'a st option acom set" where
-"Com X = {c. \<forall>S \<in> set(annos c). domo S \<subseteq> X}"
-
-lemma domo_Top[simp]: "domo \<top> = {}"
-by(simp add: domo_def Top_st_def Top_option_def)
-
-lemma bot_acom_Com[simp]: "\<bottom>\<^sub>c c \<in> Com X"
-by(simp add: bot_acom_def Com_def domo_def set_annos_anno)
-
-lemma post_in_annos: "post c : set(annos c)"
-by(induction c) simp_all
-
-lemma domo_join: "domo (S \<squnion> T) \<subseteq> domo S \<union> domo T"
-by(auto simp: domo_def join_st_def split: option.split)
-
-lemma domo_afilter: "vars a \<subseteq> X \<Longrightarrow> domo S \<subseteq> X \<Longrightarrow> domo(afilter a i S) \<subseteq> X"
-apply(induction a arbitrary: i S)
-apply(simp add: domo_def)
-apply(simp add: domo_def Let_def update_def lookup_def split: option.splits)
-apply blast
-apply(simp split: prod.split)
-done
-
-lemma domo_bfilter: "vars b \<subseteq> X \<Longrightarrow> domo S \<subseteq> X \<Longrightarrow> domo(bfilter b bv S) \<subseteq> X"
-apply(induction b arbitrary: bv S)
-apply(simp add: domo_def)
-apply(simp)
-apply(simp)
-apply rule
-apply (metis le_sup_iff subset_trans[OF domo_join])
-apply(simp split: prod.split)
-by (metis domo_afilter)
-
-lemma step'_Com:
- "domo S \<subseteq> X \<Longrightarrow> vars(strip c) \<subseteq> X \<Longrightarrow> c : Com X \<Longrightarrow> step' S c : Com X"
-apply(induction c arbitrary: S)
-apply(simp add: Com_def)
-apply(simp add: Com_def domo_def update_def split: option.splits)
-apply(simp (no_asm_use) add: Com_def ball_Un)
-apply (metis post_in_annos)
-apply(simp (no_asm_use) add: Com_def ball_Un)
-apply rule
-apply (metis Un_assoc domo_join order_trans post_in_annos subset_Un_eq)
-apply (metis domo_bfilter)
-apply(simp (no_asm_use) add: Com_def)
-apply rule
-apply (metis domo_join le_sup_iff post_in_annos subset_trans)
-apply rule
-apply (metis domo_bfilter)
-by (metis domo_bfilter)
-
-end
-
-
-subsubsection "Monotonicity"
-
-locale Abs_Int1_mono = Abs_Int1 +
-assumes mono_plus': "a1 \<sqsubseteq> b1 \<Longrightarrow> a2 \<sqsubseteq> b2 \<Longrightarrow> plus' a1 a2 \<sqsubseteq> plus' b1 b2"
-and mono_filter_plus': "a1 \<sqsubseteq> b1 \<Longrightarrow> a2 \<sqsubseteq> b2 \<Longrightarrow> r \<sqsubseteq> r' \<Longrightarrow>
- filter_plus' r a1 a2 \<sqsubseteq> filter_plus' r' b1 b2"
-and mono_filter_less': "a1 \<sqsubseteq> b1 \<Longrightarrow> a2 \<sqsubseteq> b2 \<Longrightarrow>
- filter_less' bv a1 a2 \<sqsubseteq> filter_less' bv b1 b2"
-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
-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: test_num' Let_def split: option.splits prod.splits)
-apply (metis mono_gamma subsetD)
-apply(drule_tac x = "list" in mono_lookup)
-apply (metis mono_meet le_trans)
-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)
-done
-
-lemma mono_step': "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: mono_post mono_bfilter mono_update mono_aval' Let_def le_join_disj
- split: option.split)
-done
-
-lemma mono_step'2: "mono (step' S)"
-by(simp add: mono_def mono_step'[OF le_refl])
-
-end
-
-end