(*  Title:      HOL/Hoare/Hoare_Logic_Abort.thy

     Author:     Leonor Prensa Nieto & Tobias Nipkow

     Copyright   2003 TUM

 Like Hoare.thy, but with an Abort statement for modelling run time errors.

 *)

 theory Hoare_Logic_Abort

 imports Main

 begin

 type_synonym 'a bexp = "'a set"

 type_synonym 'a assn = "'a set"

 datatype

  'a com = Basic "'a \<Rightarrow> 'a"

    | Abort

    | Seq "'a com" "'a com"               ("(_;/ _)"      [61,60] 60)

    | Cond "'a bexp" "'a com" "'a com"    ("(1IF _/ THEN _ / ELSE _/ FI)"  [0,0,0] 61)

    | While "'a bexp" "'a assn" "'a com"  ("(1WHILE _/ INV {_} //DO _ /OD)"  [0,0,0] 61)

 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 (c1;c2) s s'"

 | "Sem (IF b THEN c1 ELSE c2 FI) None None"

 | "s \<in> b \<Longrightarrow> Sem c1 (Some s) s' \<Longrightarrow> Sem (IF b THEN c1 ELSE c2 FI) (Some s) s'"

 | "s \<notin> b \<Longrightarrow> Sem c2 (Some s) s' \<Longrightarrow> Sem (IF b THEN c1 ELSE c2 FI) (Some s) s'"

 | "Sem (While b x c) None None"

 | "s \<notin> b \<Longrightarrow> Sem (While b x c) (Some s) (Some s)"

 | "s \<in> b \<Longrightarrow> Sem c (Some s) s'' \<Longrightarrow> Sem (While b x c) s'' s' \<Longrightarrow>

    Sem (While b x c) (Some s) s'"

 inductive_cases [elim!]:

   "Sem (Basic f) s s'" "Sem (c1;c2) s s'"

   "Sem (IF b THEN c1 ELSE c2 FI) s s'"

 definition Valid :: "'a bexp \<Rightarrow> 'a com \<Rightarrow> 'a bexp \<Rightarrow> bool" where

   "Valid p c q == \<forall>s s'. Sem c s s' \<longrightarrow> s : Some ` p \<longrightarrow> s' : Some ` q"

 syntax

   "_assign" :: "idt => 'b => 'a com"  ("(2_ :=/ _)" [70, 65] 61)

 syntax

   "_hoare_abort_vars" :: "[idts, 'a assn,'a com,'a assn] => bool"

                  ("VARS _// {_} // _ // {_}" [0,0,55,0] 50)

 syntax ("" output)

   "_hoare_abort"      :: "['a assn,'a com,'a assn] => bool"

                  ("{_} // _ // {_}" [0,55,0] 50)

 ML_file "hoare_syntax.ML"

 parse_translation {* [(@{syntax_const "_hoare_abort_vars"}, K Hoare_Syntax.hoare_vars_tr)] *}

 print_translation

   {* [(@{const_syntax Valid}, K (Hoare_Syntax.spec_tr' @{syntax_const "_hoare_abort"}))] *}

 (*** The proof rules ***)

 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 (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 INV {i} DO c OD) 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 INV {i} DO c OD" 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 c) q"

 apply(simp add:Valid_def)

 apply(simp (no_asm) add:image_def)

 apply clarify

 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)

 subsection {* Derivation of the proof rules and, most importantly, the VCG tactic *}

 lemma Compl_Collect: "-(Collect b) = {x. ~(b x)}"

   by blast

 ML_file "hoare_tac.ML"

 method_setup vcg = {*

   Scan.succeed (fn ctxt => SIMPLE_METHOD' (hoare_tac ctxt (K all_tac))) *}

   "verification condition generator"

 method_setup vcg_simp = {*

   Scan.succeed (fn ctxt =>

     SIMPLE_METHOD' (hoare_tac ctxt (asm_full_simp_tac ctxt))) *}

   "verification condition generator plus simplification"

 (* Special syntax for guarded statements and guarded array updates: *)

 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" == "IF P THEN c ELSE CONST Abort FI"

   "a[i] := v" => "(i < CONST length a) \<rightarrow> (a := CONST list_update a i v)"

   (* reverse translation not possible because of duplicate "a" *)

 text{* Note: there is no special syntax for guarded array access. Thus

 you must write @{text"j < length a \<rightarrow> a[i] := a!j"}. *}

 end