(* Title: HOL/Isar_Examples/Hoare.thy
Author: Makarius
A formulation of Hoare logic suitable for Isar.
*)
section \<open>Hoare Logic\<close>
theory Hoare
imports Main
begin
subsection \<open>Abstract syntax and semantics\<close>
text \<open>
The following abstract syntax and semantics of Hoare Logic over \<^verbatim>\<open>WHILE\<close>
programs closely follows the existing tradition in Isabelle/HOL of
formalizing the presentation given in @{cite \<open>\S6\<close> "Winskel:1993"}. See also
@{dir "~~/src/HOL/Hoare"} and @{cite "Nipkow:1998:Winskel"}.
\<close>
type_synonym 'a bexp = "'a set"
type_synonym 'a assn = "'a set"
datatype 'a com =
Basic "'a \<Rightarrow> 'a"
| Seq "'a com" "'a com" ("(_;/ _)" [60, 61] 60)
| Cond "'a bexp" "'a com" "'a com"
| While "'a bexp" "'a assn" "'a com"
abbreviation Skip ("SKIP")
where "SKIP \<equiv> Basic id"
type_synonym 'a sem = "'a \<Rightarrow> 'a \<Rightarrow> bool"
primrec iter :: "nat \<Rightarrow> 'a bexp \<Rightarrow> 'a sem \<Rightarrow> 'a sem"
where
"iter 0 b S s s' \<longleftrightarrow> s \<notin> b \<and> s = s'"
| "iter (Suc n) b S s s' \<longleftrightarrow> s \<in> b \<and> (\<exists>s''. S s s'' \<and> iter n b S s'' s')"
primrec Sem :: "'a com \<Rightarrow> 'a sem"
where
"Sem (Basic f) s s' \<longleftrightarrow> s' = f s"
| "Sem (c1; c2) s s' \<longleftrightarrow> (\<exists>s''. Sem c1 s s'' \<and> Sem c2 s'' s')"
| "Sem (Cond b c1 c2) s s' \<longleftrightarrow> (if s \<in> b then Sem c1 s s' else Sem c2 s s')"
| "Sem (While b x c) s s' \<longleftrightarrow> (\<exists>n. iter n b (Sem c) s s')"
definition Valid :: "'a bexp \<Rightarrow> 'a com \<Rightarrow> 'a bexp \<Rightarrow> bool" ("(3\<turnstile> _/ (2_)/ _)" [100, 55, 100] 50)
where "\<turnstile> P c Q \<longleftrightarrow> (\<forall>s s'. Sem c s s' \<longrightarrow> s \<in> P \<longrightarrow> s' \<in> Q)"
lemma ValidI [intro?]: "(\<And>s s'. Sem c s s' \<Longrightarrow> s \<in> P \<Longrightarrow> s' \<in> Q) \<Longrightarrow> \<turnstile> P c Q"
by (simp add: Valid_def)
lemma ValidD [dest?]: "\<turnstile> P c Q \<Longrightarrow> Sem c s s' \<Longrightarrow> s \<in> P \<Longrightarrow> s' \<in> Q"
by (simp add: Valid_def)
subsection \<open>Primitive Hoare rules\<close>
text \<open>
From the semantics defined above, we derive the standard set of primitive
Hoare rules; e.g.\ see @{cite \<open>\S6\<close> "Winskel:1993"}. Usually, variant forms
of these rules are applied in actual proof, see also \S\ref{sec:hoare-isar}
and \S\ref{sec:hoare-vcg}.
\<^medskip>
The \<open>basic\<close> rule represents any kind of atomic access to the state space.
This subsumes the common rules of \<open>skip\<close> and \<open>assign\<close>, as formulated in
\S\ref{sec:hoare-isar}.
\<close>
theorem basic: "\<turnstile> {s. f s \<in> P} (Basic f) P"
proof
fix s s'
assume s: "s \<in> {s. f s \<in> P}"
assume "Sem (Basic f) s s'"
then have "s' = f s" by simp
with s show "s' \<in> P" by simp
qed
text \<open>
The rules for sequential commands and semantic consequences are established
in a straight forward manner as follows.
\<close>
theorem seq: "\<turnstile> P c1 Q \<Longrightarrow> \<turnstile> Q c2 R \<Longrightarrow> \<turnstile> P (c1; c2) R"
proof
assume cmd1: "\<turnstile> P c1 Q" and cmd2: "\<turnstile> Q c2 R"
fix s s'
assume s: "s \<in> P"
assume "Sem (c1; c2) s s'"
then obtain s'' where sem1: "Sem c1 s s''" and sem2: "Sem c2 s'' s'"
by auto
from cmd1 sem1 s have "s'' \<in> Q" ..
with cmd2 sem2 show "s' \<in> R" ..
qed
theorem conseq: "P' \<subseteq> P \<Longrightarrow> \<turnstile> P c Q \<Longrightarrow> Q \<subseteq> Q' \<Longrightarrow> \<turnstile> P' c Q'"
proof
assume P'P: "P' \<subseteq> P" and QQ': "Q \<subseteq> Q'"
assume cmd: "\<turnstile> P c Q"
fix s s' :: 'a
assume sem: "Sem c s s'"
assume "s : P'" with P'P have "s \<in> P" ..
with cmd sem have "s' \<in> Q" ..
with QQ' show "s' \<in> Q'" ..
qed
text \<open>
The rule for conditional commands is directly reflected by the corresponding
semantics; in the proof we just have to look closely which cases apply.
\<close>
theorem cond:
assumes case_b: "\<turnstile> (P \<inter> b) c1 Q"
and case_nb: "\<turnstile> (P \<inter> -b) c2 Q"
shows "\<turnstile> P (Cond b c1 c2) Q"
proof
fix s s'
assume s: "s \<in> P"
assume sem: "Sem (Cond b c1 c2) s s'"
show "s' \<in> Q"
proof cases
assume b: "s \<in> b"
from case_b show ?thesis
proof
from sem b show "Sem c1 s s'" by simp
from s b show "s \<in> P \<inter> b" by simp
qed
next
assume nb: "s \<notin> b"
from case_nb show ?thesis
proof
from sem nb show "Sem c2 s s'" by simp
from s nb show "s : P \<inter> -b" by simp
qed
qed
qed
text \<open>
The \<open>while\<close> rule is slightly less trivial --- it is the only one based on
recursion, which is expressed in the semantics by a Kleene-style least
fixed-point construction. The auxiliary statement below, which is by
induction on the number of iterations is the main point to be proven; the
rest is by routine application of the semantics of \<^verbatim>\<open>WHILE\<close>.
\<close>
theorem while:
assumes body: "\<turnstile> (P \<inter> b) c P"
shows "\<turnstile> P (While b X c) (P \<inter> -b)"
proof
fix s s' assume s: "s \<in> P"
assume "Sem (While b X c) s s'"
then obtain n where "iter n b (Sem c) s s'" by auto
from this and s show "s' \<in> P \<inter> -b"
proof (induct n arbitrary: s)
case 0
then show ?case by auto
next
case (Suc n)
then obtain s'' where b: "s \<in> b" and sem: "Sem c s s''"
and iter: "iter n b (Sem c) s'' s'" by auto
from Suc and b have "s \<in> P \<inter> b" by simp
with body sem have "s'' \<in> P" ..
with iter show ?case by (rule Suc)
qed
qed
subsection \<open>Concrete syntax for assertions\<close>
text \<open>
We now introduce concrete syntax for describing commands (with embedded
expressions) and assertions. The basic technique is that of semantic
``quote-antiquote''. A \<^emph>\<open>quotation\<close> is a syntactic entity delimited by an
implicit abstraction, say over the state space. An \<^emph>\<open>antiquotation\<close> is a
marked expression within a quotation that refers the implicit argument; a
typical antiquotation would select (or even update) components from the
state.
We will see some examples later in the concrete rules and applications.
\<^medskip>
The following specification of syntax and translations is for Isabelle
experts only; feel free to ignore it.
While the first part is still a somewhat intelligible specification of the
concrete syntactic representation of our Hoare language, the actual ``ML
drivers'' is quite involved. Just note that the we re-use the basic
quote/antiquote translations as already defined in Isabelle/Pure (see @{ML
Syntax_Trans.quote_tr}, and @{ML Syntax_Trans.quote_tr'},).
\<close>
syntax
"_quote" :: "'b \<Rightarrow> ('a \<Rightarrow> 'b)"
"_antiquote" :: "('a \<Rightarrow> 'b) \<Rightarrow> 'b" ("\<acute>_" [1000] 1000)
"_Subst" :: "'a bexp \<Rightarrow> 'b \<Rightarrow> idt \<Rightarrow> 'a bexp" ("_[_'/\<acute>_]" [1000] 999)
"_Assert" :: "'a \<Rightarrow> 'a set" ("(\<lbrace>_\<rbrace>)" [0] 1000)
"_Assign" :: "idt \<Rightarrow> 'b \<Rightarrow> 'a com" ("(\<acute>_ :=/ _)" [70, 65] 61)
"_Cond" :: "'a bexp \<Rightarrow> 'a com \<Rightarrow> 'a com \<Rightarrow> 'a com"
("(0IF _/ THEN _/ ELSE _/ FI)" [0, 0, 0] 61)
"_While_inv" :: "'a bexp \<Rightarrow> 'a assn \<Rightarrow> 'a com \<Rightarrow> 'a com"
("(0WHILE _/ INV _ //DO _ /OD)" [0, 0, 0] 61)
"_While" :: "'a bexp \<Rightarrow> 'a com \<Rightarrow> 'a com" ("(0WHILE _ //DO _ /OD)" [0, 0] 61)
translations
"\<lbrace>b\<rbrace>" \<rightharpoonup> "CONST Collect (_quote b)"
"B [a/\<acute>x]" \<rightharpoonup> "\<lbrace>\<acute>(_update_name x (\<lambda>_. a)) \<in> B\<rbrace>"
"\<acute>x := a" \<rightharpoonup> "CONST Basic (_quote (\<acute>(_update_name x (\<lambda>_. a))))"
"IF b THEN c1 ELSE c2 FI" \<rightharpoonup> "CONST Cond \<lbrace>b\<rbrace> c1 c2"
"WHILE b INV i DO c OD" \<rightharpoonup> "CONST While \<lbrace>b\<rbrace> i c"
"WHILE b DO c OD" \<rightleftharpoons> "WHILE b INV CONST undefined DO c OD"
parse_translation \<open>
let
fun quote_tr [t] = Syntax_Trans.quote_tr @{syntax_const "_antiquote"} t
| quote_tr ts = raise TERM ("quote_tr", ts);
in [(@{syntax_const "_quote"}, K quote_tr)] end
\<close>
text \<open>
As usual in Isabelle syntax translations, the part for printing is more
complicated --- we cannot express parts as macro rules as above. Don't look
here, unless you have to do similar things for yourself.
\<close>
print_translation \<open>
let
fun quote_tr' f (t :: ts) =
Term.list_comb (f $ Syntax_Trans.quote_tr' @{syntax_const "_antiquote"} t, ts)
| quote_tr' _ _ = raise Match;
val assert_tr' = quote_tr' (Syntax.const @{syntax_const "_Assert"});
fun bexp_tr' name ((Const (@{const_syntax Collect}, _) $ t) :: ts) =
quote_tr' (Syntax.const name) (t :: ts)
| bexp_tr' _ _ = raise Match;
fun assign_tr' (Abs (x, _, f $ k $ Bound 0) :: ts) =
quote_tr' (Syntax.const @{syntax_const "_Assign"} $ Syntax_Trans.update_name_tr' f)
(Abs (x, dummyT, Syntax_Trans.const_abs_tr' k) :: ts)
| assign_tr' _ = raise Match;
in
[(@{const_syntax Collect}, K assert_tr'),
(@{const_syntax Basic}, K assign_tr'),
(@{const_syntax Cond}, K (bexp_tr' @{syntax_const "_Cond"})),
(@{const_syntax While}, K (bexp_tr' @{syntax_const "_While_inv"}))]
end
\<close>
subsection \<open>Rules for single-step proof \label{sec:hoare-isar}\<close>
text \<open>
We are now ready to introduce a set of Hoare rules to be used in single-step
structured proofs in Isabelle/Isar. We refer to the concrete syntax
introduce above.
\<^medskip>
Assertions of Hoare Logic may be manipulated in calculational proofs, with
the inclusion expressed in terms of sets or predicates. Reversed order is
supported as well.
\<close>
lemma [trans]: "\<turnstile> P c Q \<Longrightarrow> P' \<subseteq> P \<Longrightarrow> \<turnstile> P' c Q"
by (unfold Valid_def) blast
lemma [trans] : "P' \<subseteq> P \<Longrightarrow> \<turnstile> P c Q \<Longrightarrow> \<turnstile> P' c Q"
by (unfold Valid_def) blast
lemma [trans]: "Q \<subseteq> Q' \<Longrightarrow> \<turnstile> P c Q \<Longrightarrow> \<turnstile> P c Q'"
by (unfold Valid_def) blast
lemma [trans]: "\<turnstile> P c Q \<Longrightarrow> Q \<subseteq> Q' \<Longrightarrow> \<turnstile> P c Q'"
by (unfold Valid_def) blast
lemma [trans]:
"\<turnstile> \<lbrace>\<acute>P\<rbrace> c Q \<Longrightarrow> (\<And>s. P' s \<longrightarrow> P s) \<Longrightarrow> \<turnstile> \<lbrace>\<acute>P'\<rbrace> c Q"
by (simp add: Valid_def)
lemma [trans]:
"(\<And>s. P' s \<longrightarrow> P s) \<Longrightarrow> \<turnstile> \<lbrace>\<acute>P\<rbrace> c Q \<Longrightarrow> \<turnstile> \<lbrace>\<acute>P'\<rbrace> c Q"
by (simp add: Valid_def)
lemma [trans]:
"\<turnstile> P c \<lbrace>\<acute>Q\<rbrace> \<Longrightarrow> (\<And>s. Q s \<longrightarrow> Q' s) \<Longrightarrow> \<turnstile> P c \<lbrace>\<acute>Q'\<rbrace>"
by (simp add: Valid_def)
lemma [trans]:
"(\<And>s. Q s \<longrightarrow> Q' s) \<Longrightarrow> \<turnstile> P c \<lbrace>\<acute>Q\<rbrace> \<Longrightarrow> \<turnstile> P c \<lbrace>\<acute>Q'\<rbrace>"
by (simp add: Valid_def)
text \<open>
Identity and basic assignments.\<^footnote>\<open>The \<open>hoare\<close> method introduced in
\S\ref{sec:hoare-vcg} is able to provide proper instances for any number of
basic assignments, without producing additional verification conditions.\<close>
\<close>
lemma skip [intro?]: "\<turnstile> P SKIP P"
proof -
have "\<turnstile> {s. id s \<in> P} SKIP P" by (rule basic)
then show ?thesis by simp
qed
lemma assign: "\<turnstile> P [\<acute>a/\<acute>x::'a] \<acute>x := \<acute>a P"
by (rule basic)
text \<open>
Note that above formulation of assignment corresponds to our preferred way
to model state spaces, using (extensible) record types in HOL @{cite
"Naraschewski-Wenzel:1998:HOOL"}. For any record field \<open>x\<close>, Isabelle/HOL
provides a functions \<open>x\<close> (selector) and \<open>x_update\<close> (update). Above, there is
only a place-holder appearing for the latter kind of function: due to
concrete syntax \<open>\<acute>x := \<acute>a\<close> also contains \<open>x_update\<close>.\<^footnote>\<open>Note that due to the
external nature of HOL record fields, we could not even state a general
theorem relating selector and update functions (if this were required here);
this would only work for any particular instance of record fields introduced
so far.\<close>
\<^medskip>
Sequential composition --- normalizing with associativity achieves proper of
chunks of code verified separately.
\<close>
lemmas [trans, intro?] = seq
lemma seq_assoc [simp]: "\<turnstile> P c1;(c2;c3) Q \<longleftrightarrow> \<turnstile> P (c1;c2);c3 Q"
by (auto simp add: Valid_def)
text \<open>Conditional statements.\<close>
lemmas [trans, intro?] = cond
lemma [trans, intro?]:
"\<turnstile> \<lbrace>\<acute>P \<and> \<acute>b\<rbrace> c1 Q
\<Longrightarrow> \<turnstile> \<lbrace>\<acute>P \<and> \<not> \<acute>b\<rbrace> c2 Q
\<Longrightarrow> \<turnstile> \<lbrace>\<acute>P\<rbrace> IF \<acute>b THEN c1 ELSE c2 FI Q"
by (rule cond) (simp_all add: Valid_def)
text \<open>While statements --- with optional invariant.\<close>
lemma [intro?]: "\<turnstile> (P \<inter> b) c P \<Longrightarrow> \<turnstile> P (While b P c) (P \<inter> -b)"
by (rule while)
lemma [intro?]: "\<turnstile> (P \<inter> b) c P \<Longrightarrow> \<turnstile> P (While b undefined c) (P \<inter> -b)"
by (rule while)
lemma [intro?]:
"\<turnstile> \<lbrace>\<acute>P \<and> \<acute>b\<rbrace> c \<lbrace>\<acute>P\<rbrace>
\<Longrightarrow> \<turnstile> \<lbrace>\<acute>P\<rbrace> WHILE \<acute>b INV \<lbrace>\<acute>P\<rbrace> DO c OD \<lbrace>\<acute>P \<and> \<not> \<acute>b\<rbrace>"
by (simp add: while Collect_conj_eq Collect_neg_eq)
lemma [intro?]:
"\<turnstile> \<lbrace>\<acute>P \<and> \<acute>b\<rbrace> c \<lbrace>\<acute>P\<rbrace>
\<Longrightarrow> \<turnstile> \<lbrace>\<acute>P\<rbrace> WHILE \<acute>b DO c OD \<lbrace>\<acute>P \<and> \<not> \<acute>b\<rbrace>"
by (simp add: while Collect_conj_eq Collect_neg_eq)
subsection \<open>Verification conditions \label{sec:hoare-vcg}\<close>
text \<open>
We now load the \<^emph>\<open>original\<close> ML file for proof scripts and tactic definition
for the Hoare Verification Condition Generator (see @{dir
"~~/src/HOL/Hoare/"}). As far as we are concerned here, the result is a
proof method \<open>hoare\<close>, which may be applied to a Hoare Logic assertion to
extract purely logical verification conditions. It is important to note that
the method requires \<^verbatim>\<open>WHILE\<close> loops to be fully annotated with invariants
beforehand. Furthermore, only \<^emph>\<open>concrete\<close> pieces of code are handled --- the
underlying tactic fails ungracefully if supplied with meta-variables or
parameters, for example.
\<close>
lemma SkipRule: "p \<subseteq> q \<Longrightarrow> Valid p (Basic id) q"
by (auto simp add: 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 iter_aux:
"\<forall>s s'. Sem c s s' \<longrightarrow> s \<in> I \<and> s \<in> b \<longrightarrow> s' \<in> I \<Longrightarrow>
(\<And>s s'. s \<in> I \<Longrightarrow> iter n b (Sem c) s s' \<Longrightarrow> s' \<in> I \<and> s' \<notin> b)"
by (induct n) 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 iter_aux)
prefer 2
apply assumption
apply blast
apply blast
done
lemma Compl_Collect: "- Collect b = {x. \<not> b x}"
by blast
lemmas AbortRule = SkipRule \<comment> "dummy version"
ML_file "~~/src/HOL/Hoare/hoare_tac.ML"
method_setup hoare =
\<open>Scan.succeed (fn ctxt =>
(SIMPLE_METHOD'
(Hoare.hoare_tac ctxt
(simp_tac (put_simpset HOL_basic_ss ctxt addsimps [@{thm "Record.K_record_comp"}] )))))\<close>
"verification condition generator for Hoare logic"
end