(* Title: HOL/Hoare/Hoare_Logic.thy
Author: Leonor Prensa Nieto & Tobias Nipkow
Copyright 1998 TUM
Sugared semantic embedding of Hoare logic.
Strictly speaking a shallow embedding (as implemented by Norbert Galm
following Mike Gordon) would suffice. Maybe the datatype com comes in useful
later.
*)
theory Hoare_Logic
imports Main
begin
type_synonym 'a bexp = "'a set"
type_synonym 'a assn = "'a set"
datatype
'a com = Basic "'a \<Rightarrow> 'a"
| 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 => 'a => bool"
inductive Sem :: "'a com \<Rightarrow> 'a sem"
where
"Sem (Basic f) s (f s)"
| "Sem c1 s s'' \<Longrightarrow> Sem c2 s'' s' \<Longrightarrow> Sem (c1;c2) s s'"
| "s \<in> b \<Longrightarrow> Sem c1 s s' \<Longrightarrow> Sem (IF b THEN c1 ELSE c2 FI) s s'"
| "s \<notin> b \<Longrightarrow> Sem c2 s s' \<Longrightarrow> Sem (IF b THEN c1 ELSE c2 FI) s s'"
| "s \<notin> b \<Longrightarrow> Sem (While b x c) s s"
| "s \<in> b \<Longrightarrow> Sem c s s'' \<Longrightarrow> Sem (While b x c) s'' s' \<Longrightarrow>
Sem (While b x c) 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 \<longleftrightarrow> (!s s'. Sem c s s' --> s : p --> s' : q)"
syntax
"_assign" :: "idt => 'b => 'a com" ("(2_ :=/ _)" [70, 65] 61)
syntax
"_hoare_vars" :: "[idts, 'a assn,'a com,'a assn] => bool"
("VARS _// {_} // _ // {_}" [0,0,55,0] 50)
syntax ("" output)
"_hoare" :: "['a assn,'a com,'a assn] => bool"
("{_} // _ // {_}" [0,55,0] 50)
ML_file "hoare_syntax.ML"
parse_translation {* [(@{syntax_const "_hoare_vars"}, Hoare_Syntax.hoare_vars_tr)] *}
print_translation {* [(@{const_syntax Valid}, Hoare_Syntax.spec_tr' @{syntax_const "_hoare"})] *}
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 (auto simp:Valid_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> I \<and> s \<in> b \<longrightarrow> s' \<in> I \<Longrightarrow>
s \<in> I \<Longrightarrow> s' \<in> I \<and> s' \<notin> 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 (clarsimp simp:Valid_def)
apply(drule While_aux)
apply assumption
apply blast
apply blast
done
lemma Compl_Collect: "-(Collect b) = {x. ~(b x)}"
by blast
lemmas AbortRule = SkipRule -- "dummy version"
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 (simpset_of ctxt)))) *}
"verification condition generator plus simplification"
end