clarified modules: avoid multiple uses of the same ML file;
clarified concrete vs. abstract syntax;
(* Title: HOL/Hoare/Hoare_Logic_Abort.thy
Author: Leonor Prensa Nieto & Tobias Nipkow
Copyright 2003 TUM
Author: Walter Guttmann (extension to total-correctness proofs)
*)
section \<open>Hoare Logic with an Abort statement for modelling run time errors\<close>
theory Hoare_Logic_Abort
imports Hoare_Syntax Hoare_Tac
begin
type_synonym 'a bexp = "'a set"
type_synonym 'a assn = "'a set"
type_synonym 'a var = "'a \<Rightarrow> nat"
datatype 'a com =
Basic "'a \<Rightarrow> 'a"
| Abort
| Seq "'a com" "'a com"
| Cond "'a bexp" "'a com" "'a com"
| While "'a bexp" "'a assn" "'a var" "'a com"
abbreviation annskip ("SKIP") where "SKIP == Basic id"
type_synonym 'a sem = "'a option => 'a option => bool"
inductive Sem :: "'a com \<Rightarrow> 'a sem"
where
"Sem (Basic f) None None"
| "Sem (Basic f) (Some s) (Some (f s))"
| "Sem Abort s None"
| "Sem c1 s s'' \<Longrightarrow> Sem c2 s'' s' \<Longrightarrow> Sem (Seq c1 c2) s s'"
| "Sem (Cond b c1 c2) None None"
| "s \<in> b \<Longrightarrow> Sem c1 (Some s) s' \<Longrightarrow> Sem (Cond b c1 c2) (Some s) s'"
| "s \<notin> b \<Longrightarrow> Sem c2 (Some s) s' \<Longrightarrow> Sem (Cond b c1 c2) (Some s) s'"
| "Sem (While b x y c) None None"
| "s \<notin> b \<Longrightarrow> Sem (While b x y c) (Some s) (Some s)"
| "s \<in> b \<Longrightarrow> Sem c (Some s) s'' \<Longrightarrow> Sem (While b x y c) s'' s' \<Longrightarrow>
Sem (While b x y c) (Some s) s'"
inductive_cases [elim!]:
"Sem (Basic f) s s'" "Sem (Seq c1 c2) s s'"
"Sem (Cond b c1 c2) s s'"
lemma Sem_deterministic:
assumes "Sem c s s1"
and "Sem c s s2"
shows "s1 = s2"
proof -
have "Sem c s s1 \<Longrightarrow> (\<forall>s2. Sem c s s2 \<longrightarrow> s1 = s2)"
by (induct rule: Sem.induct) (subst Sem.simps, blast)+
thus ?thesis
using assms by simp
qed
definition Valid :: "'a bexp \<Rightarrow> 'a com \<Rightarrow> 'a bexp \<Rightarrow> bool"
where "Valid p c q \<equiv> \<forall>s s'. Sem c s s' \<longrightarrow> s \<in> Some ` p \<longrightarrow> s' \<in> Some ` q"
definition ValidTC :: "'a bexp \<Rightarrow> 'a com \<Rightarrow> 'a bexp \<Rightarrow> bool"
where "ValidTC p c q \<equiv> \<forall>s . s \<in> p \<longrightarrow> (\<exists>t . Sem c (Some s) (Some t) \<and> t \<in> q)"
lemma tc_implies_pc:
"ValidTC p c q \<Longrightarrow> Valid p c q"
by (smt Sem_deterministic ValidTC_def Valid_def image_iff)
lemma tc_extract_function:
"ValidTC p c q \<Longrightarrow> \<exists>f . \<forall>s . s \<in> p \<longrightarrow> f s \<in> q"
by (meson ValidTC_def)
text \<open>The proof rules for partial correctness\<close>
lemma SkipRule: "p \<subseteq> q \<Longrightarrow> Valid p (Basic id) q"
by (auto simp:Valid_def)
lemma BasicRule: "p \<subseteq> {s. f s \<in> q} \<Longrightarrow> Valid p (Basic f) q"
by (auto simp:Valid_def)
lemma SeqRule: "Valid P c1 Q \<Longrightarrow> Valid Q c2 R \<Longrightarrow> Valid P (Seq c1 c2) R"
by (auto simp:Valid_def)
lemma CondRule:
"p \<subseteq> {s. (s \<in> b \<longrightarrow> s \<in> w) \<and> (s \<notin> b \<longrightarrow> s \<in> w')}
\<Longrightarrow> Valid w c1 q \<Longrightarrow> Valid w' c2 q \<Longrightarrow> Valid p (Cond b c1 c2) q"
by (fastforce simp:Valid_def image_def)
lemma While_aux:
assumes "Sem (While b i v c) s s'"
shows "\<forall>s s'. Sem c s s' \<longrightarrow> s \<in> Some ` (I \<inter> b) \<longrightarrow> s' \<in> Some ` I \<Longrightarrow>
s \<in> Some ` I \<Longrightarrow> s' \<in> Some ` (I \<inter> -b)"
using assms
by (induct "While b i v c" s s') auto
lemma WhileRule:
"p \<subseteq> i \<Longrightarrow> Valid (i \<inter> b) c i \<Longrightarrow> i \<inter> (-b) \<subseteq> q \<Longrightarrow> Valid p (While b i v c) q"
apply (clarsimp simp:Valid_def)
apply(drule While_aux)
apply assumption
apply blast
apply blast
done
lemma AbortRule: "p \<subseteq> {s. False} \<Longrightarrow> Valid p Abort q"
by(auto simp:Valid_def)
text \<open>The proof rules for total correctness\<close>
lemma SkipRuleTC:
assumes "p \<subseteq> q"
shows "ValidTC p (Basic id) q"
by (metis Sem.intros(2) ValidTC_def assms id_def subsetD)
lemma BasicRuleTC:
assumes "p \<subseteq> {s. f s \<in> q}"
shows "ValidTC p (Basic f) q"
by (metis Ball_Collect Sem.intros(2) ValidTC_def assms)
lemma SeqRuleTC:
assumes "ValidTC p c1 q"
and "ValidTC q c2 r"
shows "ValidTC p (Seq c1 c2) r"
by (meson assms Sem.intros(4) ValidTC_def)
lemma CondRuleTC:
assumes "p \<subseteq> {s. (s \<in> b \<longrightarrow> s \<in> w) \<and> (s \<notin> b \<longrightarrow> s \<in> w')}"
and "ValidTC w c1 q"
and "ValidTC w' c2 q"
shows "ValidTC p (Cond b c1 c2) q"
proof (unfold ValidTC_def, rule allI)
fix s
show "s \<in> p \<longrightarrow> (\<exists>t . Sem (Cond b c1 c2) (Some s) (Some t) \<and> t \<in> q)"
apply (cases "s \<in> b")
apply (metis (mono_tags, lifting) Ball_Collect Sem.intros(6) ValidTC_def assms(1,2))
by (metis (mono_tags, lifting) Ball_Collect Sem.intros(7) ValidTC_def assms(1,3))
qed
lemma WhileRuleTC:
assumes "p \<subseteq> i"
and "\<And>n::nat . ValidTC (i \<inter> b \<inter> {s . v s = n}) c (i \<inter> {s . v s < n})"
and "i \<inter> uminus b \<subseteq> q"
shows "ValidTC p (While b i v c) q"
proof -
{
fix s n
have "s \<in> i \<and> v s = n \<longrightarrow> (\<exists>t . Sem (While b i v c) (Some s) (Some t) \<and> t \<in> q)"
proof (induction "n" arbitrary: s rule: less_induct)
fix n :: nat
fix s :: 'a
assume 1: "\<And>(m::nat) s::'a . m < n \<Longrightarrow> s \<in> i \<and> v s = m \<longrightarrow> (\<exists>t . Sem (While b i v c) (Some s) (Some t) \<and> t \<in> q)"
show "s \<in> i \<and> v s = n \<longrightarrow> (\<exists>t . Sem (While b i v c) (Some s) (Some t) \<and> t \<in> q)"
proof (rule impI, cases "s \<in> b")
assume 2: "s \<in> b" and "s \<in> i \<and> v s = n"
hence "s \<in> i \<inter> b \<inter> {s . v s = n}"
using assms(1) by auto
hence "\<exists>t . Sem c (Some s) (Some t) \<and> t \<in> i \<inter> {s . v s < n}"
by (metis assms(2) ValidTC_def)
from this obtain t where 3: "Sem c (Some s) (Some t) \<and> t \<in> i \<inter> {s . v s < n}"
by auto
hence "\<exists>u . Sem (While b i v c) (Some t) (Some u) \<and> u \<in> q"
using 1 by auto
thus "\<exists>t . Sem (While b i v c) (Some s) (Some t) \<and> t \<in> q"
using 2 3 Sem.intros(10) by force
next
assume "s \<notin> b" and "s \<in> i \<and> v s = n"
thus "\<exists>t . Sem (While b i v c) (Some s) (Some t) \<and> t \<in> q"
using Sem.intros(9) assms(3) by fastforce
qed
qed
}
thus ?thesis
using assms(1) ValidTC_def by force
qed
subsection \<open>Concrete syntax\<close>
setup \<open>
Hoare_Syntax.setup
{Basic = \<^const_syntax>\<open>Basic\<close>,
Skip = \<^const_syntax>\<open>annskip\<close>,
Seq = \<^const_syntax>\<open>Seq\<close>,
Cond = \<^const_syntax>\<open>Cond\<close>,
While = \<^const_syntax>\<open>While\<close>,
Valid = \<^const_syntax>\<open>Valid\<close>,
ValidTC = \<^const_syntax>\<open>ValidTC\<close>}
\<close>
\<comment> \<open>Special syntax for guarded statements and guarded array updates:\<close>
syntax
"_guarded_com" :: "bool \<Rightarrow> 'a com \<Rightarrow> 'a com" ("(2_ \<rightarrow>/ _)" 71)
"_array_update" :: "'a list \<Rightarrow> nat \<Rightarrow> 'a \<Rightarrow> 'a com" ("(2_[_] :=/ _)" [70, 65] 61)
translations
"P \<rightarrow> c" \<rightleftharpoons> "IF P THEN c ELSE CONST Abort FI"
"a[i] := v" \<rightharpoonup> "(i < CONST length a) \<rightarrow> (a := CONST list_update a i v)"
\<comment> \<open>reverse translation not possible because of duplicate \<open>a\<close>\<close>
text \<open>
Note: there is no special syntax for guarded array access. Thus
you must write \<open>j < length a \<rightarrow> a[i] := a!j\<close>.
\<close>
subsection \<open>Proof methods: VCG\<close>
declare BasicRule [Hoare_Tac.BasicRule]
and SkipRule [Hoare_Tac.SkipRule]
and AbortRule [Hoare_Tac.AbortRule]
and SeqRule [Hoare_Tac.SeqRule]
and CondRule [Hoare_Tac.CondRule]
and WhileRule [Hoare_Tac.WhileRule]
declare BasicRuleTC [Hoare_Tac.BasicRuleTC]
and SkipRuleTC [Hoare_Tac.SkipRuleTC]
and SeqRuleTC [Hoare_Tac.SeqRuleTC]
and CondRuleTC [Hoare_Tac.CondRuleTC]
and WhileRuleTC [Hoare_Tac.WhileRuleTC]
method_setup vcg = \<open>
Scan.succeed (fn ctxt => SIMPLE_METHOD' (Hoare_Tac.hoare_tac ctxt (K all_tac)))\<close>
"verification condition generator"
method_setup vcg_simp = \<open>
Scan.succeed (fn ctxt =>
SIMPLE_METHOD' (Hoare_Tac.hoare_tac ctxt (asm_full_simp_tac ctxt)))\<close>
"verification condition generator plus simplification"
method_setup vcg_tc = \<open>
Scan.succeed (fn ctxt => SIMPLE_METHOD' (Hoare_Tac.hoare_tc_tac ctxt (K all_tac)))\<close>
"verification condition generator"
method_setup vcg_tc_simp = \<open>
Scan.succeed (fn ctxt =>
SIMPLE_METHOD' (Hoare_Tac.hoare_tc_tac ctxt (asm_full_simp_tac ctxt)))\<close>
"verification condition generator plus simplification"
end