(* Title: HOL/Hoare/Hoare_Logic.thy
Author: Leonor Prensa Nieto & Tobias Nipkow
Copyright 1998 TUM
Author: Walter Guttmann (extension to total-correctness proofs)
*)
section \<open>Hoare logic\<close>
theory Hoare_Logic
imports Hoare_Syntax Hoare_Tac
begin
subsection \<open>Sugared semantic embedding of Hoare logic\<close>
text \<open>
Strictly speaking a shallow embedding (as implemented by Norbert Galm
following Mike Gordon) would suffice. Maybe the datatype com comes in useful
later.
\<close>
type_synonym 'a bexp = "'a set"
type_synonym 'a assn = "'a set"
type_synonym 'a var = "'a \<Rightarrow> nat"
datatype 'a com =
Basic "'a \ 'a"
| 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 => 'a => bool"
inductive Sem :: "'a com \ 'a sem"
where
"Sem (Basic f) s (f s)"
| "Sem c1 s s'' \ Sem c2 s'' s' \ Sem (Seq c1 c2) s s'"
| "s \ b \ Sem c1 s s' \ Sem (Cond b c1 c2) s s'"
| "s \ b \ Sem c2 s s' \ Sem (Cond b c1 c2) s s'"
| "s \ b \ Sem (While b x y c) s s"
| "s \ b \ Sem c s s'' \ Sem (While b x y c) s'' s' \
Sem (While b x y c) s s'"
definition Valid :: "'a bexp \ 'a com \ 'a bexp \ bool"
where "Valid p c q \ \s s'. Sem c s s' \ s \ p \ s' \ q"
definition ValidTC :: "'a bexp \ 'a com \ 'a bexp \ bool"
where "ValidTC p c q \ \s. s \ p \ (\t. Sem c s t \ t \ q)"
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 \ (\s2. Sem c s s2 \ s1 = s2)"
by (induct rule: Sem.induct) (subst Sem.simps, blast)+
thus ?thesis
using assms by simp
qed
lemma tc_implies_pc:
"ValidTC p c q \ Valid p c q"
by (metis Sem_deterministic Valid_def ValidTC_def)
lemma tc_extract_function:
"ValidTC p c q \ \f . \s . s \ p \ f s \ q"
by (metis ValidTC_def)
lemma SkipRule: "p \ q \ Valid p (Basic id) q"
by (auto simp:Valid_def)
lemma BasicRule: "p \ {s. f s \ q} \ Valid p (Basic f) q"
by (auto simp:Valid_def)
lemma SeqRule: "Valid P c1 Q \ Valid Q c2 R \ Valid P (Seq c1 c2) R"
by (auto simp:Valid_def)
lemma CondRule:
"p \ {s. (s \ b \ s \ w) \ (s \ b \ s \ 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 i v c) s s'"
shows "\s s'. Sem c s s' \ s \ I \ s \ b \ s' \ I \
s \<in> I \<Longrightarrow> s' \<in> I \<and> s' \<notin> b"
using assms
by (induct "While b i v c" s s') auto
lemma WhileRule:
"p \ i \ Valid (i \ b) c i \ i \ (-b) \ q \ 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 SkipRuleTC:
assumes "p \ q"
shows "ValidTC p (Basic id) q"
by (metis assms Sem.intros(1) ValidTC_def id_apply subsetD)
lemma BasicRuleTC:
assumes "p \ {s. f s \ q}"
shows "ValidTC p (Basic f) q"
by (metis assms Ball_Collect Sem.intros(1) ValidTC_def)
lemma SeqRuleTC:
assumes "ValidTC p c1 q"
and "ValidTC q c2 r"
shows "ValidTC p (Seq c1 c2) r"
by (meson assms Sem.intros(2) ValidTC_def)
lemma CondRuleTC:
assumes "p \ {s. (s \ b \ s \ w) \ (s \ b \ s \ 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 \ p \ (\t . Sem (Cond b c1 c2) s t \ t \ q)"
apply (cases "s \ b")
apply (metis (mono_tags, lifting) assms(1,2) Ball_Collect Sem.intros(3) ValidTC_def)
by (metis (mono_tags, lifting) assms(1,3) Ball_Collect Sem.intros(4) ValidTC_def)
qed
lemma WhileRuleTC:
assumes "p \ i"
and "\n::nat . ValidTC (i \ b \ {s . v s = n}) c (i \ {s . v s < n})"
and "i \ uminus b \ q"
shows "ValidTC p (While b i v c) q"
proof -
{
fix s n
have "s \ i \ v s = n \ (\t . Sem (While b i v c) s t \ t \ q)"
proof (induction "n" arbitrary: s rule: less_induct)
fix n :: nat
fix s :: 'a
assume 1: "\(m::nat) s::'a . m < n \ s \ i \ v s = m \ (\t . Sem (While b i v c) s t \ t \ q)"
show "s \ i \ v s = n \ (\t . Sem (While b i v c) s t \ t \ q)"
proof (rule impI, cases "s \ b")
assume 2: "s \ b" and "s \ i \ v s = n"
hence "s \ i \ b \ {s . v s = n}"
using assms(1) by auto
hence "\t . Sem c s t \ t \ i \ {s . v s < n}"
by (metis assms(2) ValidTC_def)
from this obtain t where 3: "Sem c s t \ t \ i \ {s . v s < n}"
by auto
hence "\u . Sem (While b i v c) t u \ u \ q"
using 1 by auto
thus "\t . Sem (While b i v c) s t \ t \ q"
using 2 3 Sem.intros(6) by force
next
assume "s \ b" and "s \ i \ v s = n"
thus "\t . Sem (While b i v c) s t \ t \ q"
using Sem.intros(5) assms(3) by fastforce
qed
qed
}
thus ?thesis
using assms(1) ValidTC_def by force
qed
subsubsection \<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>
subsubsection \<open>Proof methods: VCG\<close>
declare BasicRule [Hoare_Tac.BasicRule]
and SkipRule [Hoare_Tac.SkipRule]
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)))\
"verification condition generator"
method_setup vcg_simp = \<open>
Scan.succeed (fn ctxt =>
SIMPLE_METHOD' (Hoare_Tac.hoare_tac ctxt (asm_full_simp_tac ctxt)))\
"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)))\
"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)))\
"verification condition generator plus simplification"
end
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