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NonDetMonadVCG.thy
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2133 lines (1713 loc) · 91.6 KB
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(*
* Copyright 2014, NICTA
*
* This software may be distributed and modified according to the terms of
* the BSD 2-Clause license. Note that NO WARRANTY is provided.
* See "LICENSE_BSD2.txt" for details.
*
* @TAG(NICTA_BSD)
*)
theory NonDetMonadVCG
imports NonDetMonadLemmas WP WPC
begin
declare K_def [simp]
section "Satisfiability"
text {*
The dual to validity: an existential instead of a universal
quantifier for the post condition. In refinement, it is
often sufficient to know that there is one state that
satisfies a condition.
*}
definition
exs_valid :: "('a \<Rightarrow> bool) \<Rightarrow> ('a, 'b) nondet_monad \<Rightarrow>
('b \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> bool"
("\<lbrace>_\<rbrace> _ \<exists>\<lbrace>_\<rbrace>")
where
"exs_valid P f Q \<equiv> (\<forall>s. P s \<longrightarrow> (\<exists>(rv, s') \<in> fst (f s). Q rv s'))"
text {* The above for the exception monad *}
definition
ex_exs_validE :: "('a \<Rightarrow> bool) \<Rightarrow> ('a, 'e + 'b) nondet_monad \<Rightarrow>
('b \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> ('e \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> bool"
("\<lbrace>_\<rbrace> _ \<exists>\<lbrace>_\<rbrace>, \<lbrace>_\<rbrace>")
where
"ex_exs_validE P f Q E \<equiv>
exs_valid P f (\<lambda>rv. case rv of Inl e \<Rightarrow> E e | Inr v \<Rightarrow> Q v)"
section "Lemmas"
subsection {* Determinism *}
lemma det_set_iff:
"det f \<Longrightarrow> (r \<in> fst (f s)) = (fst (f s) = {r})"
apply (simp add: det_def)
apply (rule iffI)
apply (erule_tac x=s in allE)
apply auto
done
lemma return_det [iff]:
"det (return x)"
by (simp add: det_def return_def)
lemma put_det [iff]:
"det (put s)"
by (simp add: det_def put_def)
lemma get_det [iff]:
"det get"
by (simp add: det_def get_def)
lemma det_gets [iff]:
"det (gets f)"
by (auto simp add: gets_def det_def get_def return_def bind_def)
lemma det_UN:
"det f \<Longrightarrow> (\<Union>x \<in> fst (f s). g x) = (g (THE x. x \<in> fst (f s)))"
unfolding det_def
apply simp
apply (drule spec [of _ s])
apply clarsimp
done
lemma bind_detI [simp, intro!]:
"\<lbrakk> det f; \<forall>x. det (g x) \<rbrakk> \<Longrightarrow> det (f >>= g)"
apply (simp add: bind_def det_def split_def)
apply clarsimp
apply (erule_tac x=s in allE)
apply clarsimp
apply (erule_tac x="a" in allE)
apply (erule_tac x="b" in allE)
apply clarsimp
done
lemma the_run_stateI:
"fst (M s) = {s'} \<Longrightarrow> the_run_state M s = s'"
by (simp add: the_run_state_def)
lemma the_run_state_det:
"\<lbrakk> s' \<in> fst (M s); det M \<rbrakk> \<Longrightarrow> the_run_state M s = s'"
by (simp add: the_run_stateI det_set_iff)
subsection "Lifting and Alternative Basic Definitions"
lemma liftE_liftM: "liftE = liftM Inr"
apply (rule ext)
apply (simp add: liftE_def liftM_def)
done
lemma liftME_liftM: "liftME f = liftM (case_sum Inl (Inr \<circ> f))"
apply (rule ext)
apply (simp add: liftME_def liftM_def bindE_def returnOk_def lift_def)
apply (rule_tac f="bind x" in arg_cong)
apply (rule ext)
apply (case_tac xa)
apply (simp_all add: lift_def throwError_def)
done
lemma liftE_bindE:
"(liftE a) >>=E b = a >>= b"
apply (simp add: liftE_def bindE_def lift_def bind_assoc)
done
lemma liftM_id[simp]: "liftM id = id"
apply (rule ext)
apply (simp add: liftM_def)
done
lemma liftM_bind:
"(liftM t f >>= g) = (f >>= (\<lambda>x. g (t x)))"
by (simp add: liftM_def bind_assoc)
lemma gets_bind_ign: "gets f >>= (\<lambda>x. m) = m"
apply (rule ext)
apply (simp add: bind_def simpler_gets_def)
done
lemma get_bind_apply: "(get >>= f) x = f x x"
by (simp add: get_def bind_def)
lemma exec_gets:
"(gets f >>= m) s = m (f s) s"
by (simp add: simpler_gets_def bind_def)
lemma exec_get:
"(get >>= m) s = m s s"
by (simp add: get_def bind_def)
lemma bind_eqI:
"\<lbrakk> f = f'; \<And>x. g x = g' x \<rbrakk> \<Longrightarrow> f >>= g = f' >>= g'"
apply (rule ext)
apply (simp add: bind_def)
apply (auto simp: split_def)
done
subsection "Simplification Rules for Lifted And/Or"
lemma pred_andE[elim!]: "\<lbrakk> (A and B) x; \<lbrakk> A x; B x \<rbrakk> \<Longrightarrow> R \<rbrakk> \<Longrightarrow> R"
by(simp add:pred_conj_def)
lemma pred_andI[intro!]: "\<lbrakk> A x; B x \<rbrakk> \<Longrightarrow> (A and B) x"
by(simp add:pred_conj_def)
lemma pred_conj_app[simp]: "(P and Q) x = (P x \<and> Q x)"
by(simp add:pred_conj_def)
lemma bipred_andE[elim!]: "\<lbrakk> (A And B) x y; \<lbrakk> A x y; B x y \<rbrakk> \<Longrightarrow> R \<rbrakk> \<Longrightarrow> R"
by(simp add:bipred_conj_def)
lemma bipred_andI[intro!]: "\<lbrakk> A x y; B x y \<rbrakk> \<Longrightarrow> (A And B) x y"
by (simp add:bipred_conj_def)
lemma bipred_conj_app[simp]: "(P And Q) x = (P x and Q x)"
by(simp add:pred_conj_def bipred_conj_def)
lemma pred_disjE[elim!]: "\<lbrakk> (P or Q) x; P x \<Longrightarrow> R; Q x \<Longrightarrow> R \<rbrakk> \<Longrightarrow> R"
by (fastforce simp: pred_disj_def)
lemma pred_disjI1[intro]: "P x \<Longrightarrow> (P or Q) x"
by (simp add: pred_disj_def)
lemma pred_disjI2[intro]: "Q x \<Longrightarrow> (P or Q) x"
by (simp add: pred_disj_def)
lemma pred_disj_app[simp]: "(P or Q) x = (P x \<or> Q x)"
by auto
lemma bipred_disjI1[intro]: "P x y \<Longrightarrow> (P Or Q) x y"
by (simp add: bipred_disj_def)
lemma bipred_disjI2[intro]: "Q x y \<Longrightarrow> (P Or Q) x y"
by (simp add: bipred_disj_def)
lemma bipred_disj_app[simp]: "(P Or Q) x = (P x or Q x)"
by(simp add:pred_disj_def bipred_disj_def)
lemma pred_notnotD[simp]: "(not not P) = P"
by(simp add:pred_neg_def)
lemma pred_and_true[simp]: "(P and \<top>) = P"
by(simp add:pred_conj_def)
lemma pred_and_true_var[simp]: "(\<top> and P) = P"
by(simp add:pred_conj_def)
lemma pred_and_false[simp]: "(P and \<bottom>) = \<bottom>"
by(simp add:pred_conj_def)
lemma pred_and_false_var[simp]: "(\<bottom> and P) = \<bottom>"
by(simp add:pred_conj_def)
subsection "Hoare Logic Rules"
lemma validE_def2:
"\<lbrace>P\<rbrace> f \<lbrace>Q\<rbrace>,\<lbrace>R\<rbrace> \<equiv> \<forall>s. P s \<longrightarrow> (\<forall>(r,s') \<in> fst (f s). case r of Inr b \<Rightarrow> Q b s'
| Inl a \<Rightarrow> R a s')"
by (unfold valid_def validE_def)
lemma seq':
"\<lbrakk> \<lbrace>A\<rbrace> f \<lbrace>B\<rbrace>;
\<forall>x. P x \<longrightarrow> \<lbrace>C\<rbrace> g x \<lbrace>D\<rbrace>;
\<forall>x s. B x s \<longrightarrow> P x \<and> C s \<rbrakk> \<Longrightarrow>
\<lbrace>A\<rbrace> do x \<leftarrow> f; g x od \<lbrace>D\<rbrace>"
apply (clarsimp simp: valid_def bind_def)
apply fastforce
done
lemma seq:
assumes f_valid: "\<lbrace>A\<rbrace> f \<lbrace>B\<rbrace>"
assumes g_valid: "\<And>x. P x \<Longrightarrow> \<lbrace>C\<rbrace> g x \<lbrace>D\<rbrace>"
assumes bind: "\<And>x s. B x s \<Longrightarrow> P x \<and> C s"
shows "\<lbrace>A\<rbrace> do x \<leftarrow> f; g x od \<lbrace>D\<rbrace>"
apply (insert f_valid g_valid bind)
apply (blast intro: seq')
done
lemma seq_ext':
"\<lbrakk> \<lbrace>A\<rbrace> f \<lbrace>B\<rbrace>;
\<forall>x. \<lbrace>B x\<rbrace> g x \<lbrace>C\<rbrace> \<rbrakk> \<Longrightarrow>
\<lbrace>A\<rbrace> do x \<leftarrow> f; g x od \<lbrace>C\<rbrace>"
by (fastforce simp: valid_def bind_def Let_def split_def)
lemma seq_ext:
assumes f_valid: "\<lbrace>A\<rbrace> f \<lbrace>B\<rbrace>"
assumes g_valid: "\<And>x. \<lbrace>B x\<rbrace> g x \<lbrace>C\<rbrace>"
shows "\<lbrace>A\<rbrace> do x \<leftarrow> f; g x od \<lbrace>C\<rbrace>"
apply(insert f_valid g_valid)
apply(blast intro: seq_ext')
done
lemma seqE':
"\<lbrakk> \<lbrace>A\<rbrace> f \<lbrace>B\<rbrace>,\<lbrace>E\<rbrace>;
\<forall>x. \<lbrace>B x\<rbrace> g x \<lbrace>C\<rbrace>,\<lbrace>E\<rbrace> \<rbrakk> \<Longrightarrow>
\<lbrace>A\<rbrace> doE x \<leftarrow> f; g x odE \<lbrace>C\<rbrace>,\<lbrace>E\<rbrace>"
apply(simp add:bindE_def lift_def bind_def Let_def split_def)
apply(clarsimp simp:validE_def2)
apply (fastforce simp add: throwError_def return_def lift_def
split: sum.splits)
done
lemma seqE:
assumes f_valid: "\<lbrace>A\<rbrace> f \<lbrace>B\<rbrace>,\<lbrace>E\<rbrace>"
assumes g_valid: "\<And>x. \<lbrace>B x\<rbrace> g x \<lbrace>C\<rbrace>,\<lbrace>E\<rbrace>"
shows "\<lbrace>A\<rbrace> doE x \<leftarrow> f; g x odE \<lbrace>C\<rbrace>,\<lbrace>E\<rbrace>"
apply(insert f_valid g_valid)
apply(blast intro: seqE')
done
lemma hoare_TrueI: "\<lbrace>P\<rbrace> f \<lbrace>\<lambda>_. \<top>\<rbrace>"
by (simp add: valid_def)
lemma hoareE_TrueI: "\<lbrace>P\<rbrace> f \<lbrace>\<lambda>_. \<top>\<rbrace>, \<lbrace>\<lambda>r. \<top>\<rbrace>"
by (simp add: validE_def valid_def)
lemma hoare_True_E_R [simp]:
"\<lbrace>P\<rbrace> f \<lbrace>\<lambda>r s. True\<rbrace>, -"
by (auto simp add: validE_R_def validE_def valid_def split: sum.splits)
lemma hoare_post_conj [intro!]:
"\<lbrakk> \<lbrace> P \<rbrace> a \<lbrace> Q \<rbrace>; \<lbrace> P \<rbrace> a \<lbrace> R \<rbrace> \<rbrakk> \<Longrightarrow> \<lbrace> P \<rbrace> a \<lbrace> Q And R \<rbrace>"
by (fastforce simp: valid_def split_def bipred_conj_def)
lemma hoare_pre_disj [intro!]:
"\<lbrakk> \<lbrace> P \<rbrace> a \<lbrace> R \<rbrace>; \<lbrace> Q \<rbrace> a \<lbrace> R \<rbrace> \<rbrakk> \<Longrightarrow> \<lbrace> P or Q \<rbrace> a \<lbrace> R \<rbrace>"
by (simp add:valid_def pred_disj_def)
lemma hoare_conj:
"\<lbrakk> \<lbrace>P\<rbrace> f \<lbrace>Q\<rbrace>; \<lbrace>P'\<rbrace> f \<lbrace>Q'\<rbrace> \<rbrakk> \<Longrightarrow> \<lbrace>P and P'\<rbrace> f \<lbrace>Q And Q'\<rbrace>"
unfolding valid_def by auto
lemma hoare_post_taut: "\<lbrace> P \<rbrace> a \<lbrace> \<top>\<top> \<rbrace>"
by (simp add:valid_def)
lemma wp_post_taut: "\<lbrace>\<lambda>r. True\<rbrace> f \<lbrace>\<lambda>r s. True\<rbrace>"
by (rule hoare_post_taut)
lemma wp_post_tautE: "\<lbrace>\<lambda>r. True\<rbrace> f \<lbrace>\<lambda>r s. True\<rbrace>,\<lbrace>\<lambda>f s. True\<rbrace>"
proof -
have P: "\<And>r. (case r of Inl a \<Rightarrow> True | _ \<Rightarrow> True) = True"
by (case_tac r, simp_all)
show ?thesis
by (simp add: validE_def P wp_post_taut)
qed
lemma hoare_pre_cont [simp]: "\<lbrace> \<bottom> \<rbrace> a \<lbrace> P \<rbrace>"
by (simp add:valid_def)
subsection {* Strongest Postcondition Rules *}
lemma get_sp:
"\<lbrace>P\<rbrace> get \<lbrace>\<lambda>a s. s = a \<and> P s\<rbrace>"
by(simp add:get_def valid_def)
lemma put_sp:
"\<lbrace>\<top>\<rbrace> put a \<lbrace>\<lambda>_ s. s = a\<rbrace>"
by(simp add:put_def valid_def)
lemma return_sp:
"\<lbrace>P\<rbrace> return a \<lbrace>\<lambda>b s. b = a \<and> P s\<rbrace>"
by(simp add:return_def valid_def)
lemma assert_sp:
"\<lbrace> P \<rbrace> assert Q \<lbrace> \<lambda>r s. P s \<and> Q \<rbrace>"
by (simp add: assert_def fail_def return_def valid_def)
lemma hoare_gets_sp:
"\<lbrace>P\<rbrace> gets f \<lbrace>\<lambda>rv s. rv = f s \<and> P s\<rbrace>"
by (simp add: valid_def simpler_gets_def)
lemma hoare_return_drop_var [iff]: "\<lbrace> Q \<rbrace> return x \<lbrace> \<lambda>r. Q \<rbrace>"
by (simp add:valid_def return_def)
lemma hoare_gets [intro!]: "\<lbrakk> \<And>s. P s \<Longrightarrow> Q (f s) s \<rbrakk> \<Longrightarrow> \<lbrace> P \<rbrace> gets f \<lbrace> Q \<rbrace>"
by (simp add:valid_def gets_def get_def bind_def return_def)
lemma hoare_modifyE_var [intro!]:
"\<lbrakk> \<And>s. P s \<Longrightarrow> Q (f s) \<rbrakk> \<Longrightarrow> \<lbrace> P \<rbrace> modify f \<lbrace> \<lambda>r s. Q s \<rbrace>"
by(simp add: valid_def modify_def put_def get_def bind_def)
lemma hoare_if [intro!]:
"\<lbrakk> P \<Longrightarrow> \<lbrace> Q \<rbrace> a \<lbrace> R \<rbrace>; \<not> P \<Longrightarrow> \<lbrace> Q \<rbrace> b \<lbrace> R \<rbrace> \<rbrakk> \<Longrightarrow>
\<lbrace> Q \<rbrace> if P then a else b \<lbrace> R \<rbrace>"
by (simp add:valid_def)
lemma hoare_pre_subst: "\<lbrakk> A = B; \<lbrace>A\<rbrace> a \<lbrace>C\<rbrace> \<rbrakk> \<Longrightarrow> \<lbrace>B\<rbrace> a \<lbrace>C\<rbrace>"
by(clarsimp simp:valid_def split_def)
lemma hoare_post_subst: "\<lbrakk> B = C; \<lbrace>A\<rbrace> a \<lbrace>B\<rbrace> \<rbrakk> \<Longrightarrow> \<lbrace>A\<rbrace> a \<lbrace>C\<rbrace>"
by(clarsimp simp:valid_def split_def)
lemma hoare_pre_tautI: "\<lbrakk> \<lbrace>A and P\<rbrace> a \<lbrace>B\<rbrace>; \<lbrace>A and not P\<rbrace> a \<lbrace>B\<rbrace> \<rbrakk> \<Longrightarrow> \<lbrace>A\<rbrace> a \<lbrace>B\<rbrace>"
by(fastforce simp:valid_def split_def pred_conj_def pred_neg_def)
lemma hoare_pre_imp: "\<lbrakk> \<And>s. P s \<Longrightarrow> Q s; \<lbrace>Q\<rbrace> a \<lbrace>R\<rbrace> \<rbrakk> \<Longrightarrow> \<lbrace>P\<rbrace> a \<lbrace>R\<rbrace>"
by (fastforce simp add:valid_def)
lemma hoare_post_imp: "\<lbrakk> \<And>r s. Q r s \<Longrightarrow> R r s; \<lbrace>P\<rbrace> a \<lbrace>Q\<rbrace> \<rbrakk> \<Longrightarrow> \<lbrace>P\<rbrace> a \<lbrace>R\<rbrace>"
by(fastforce simp:valid_def split_def)
lemma hoare_post_impErr': "\<lbrakk> \<lbrace>P\<rbrace> a \<lbrace>Q\<rbrace>,\<lbrace>E\<rbrace>;
\<forall>r s. Q r s \<longrightarrow> R r s;
\<forall>e s. E e s \<longrightarrow> F e s \<rbrakk> \<Longrightarrow>
\<lbrace>P\<rbrace> a \<lbrace>R\<rbrace>,\<lbrace>F\<rbrace>"
apply (simp add: validE_def)
apply (rule_tac Q="\<lambda>r s. case r of Inl a \<Rightarrow> E a s | Inr b \<Rightarrow> Q b s" in hoare_post_imp)
apply (case_tac r)
apply simp_all
done
lemma hoare_post_impErr: "\<lbrakk> \<lbrace>P\<rbrace> a \<lbrace>Q\<rbrace>,\<lbrace>E\<rbrace>;
\<And>r s. Q r s \<Longrightarrow> R r s;
\<And>e s. E e s \<Longrightarrow> F e s \<rbrakk> \<Longrightarrow>
\<lbrace>P\<rbrace> a \<lbrace>R\<rbrace>,\<lbrace>F\<rbrace>"
apply (blast intro: hoare_post_impErr')
done
lemma hoare_validE_cases:
"\<lbrakk> \<lbrace> P \<rbrace> f \<lbrace> Q \<rbrace>, \<lbrace> \<lambda>_ _. True \<rbrace>; \<lbrace> P \<rbrace> f \<lbrace> \<lambda>_ _. True \<rbrace>, \<lbrace> R \<rbrace> \<rbrakk>
\<Longrightarrow> \<lbrace> P \<rbrace> f \<lbrace> Q \<rbrace>, \<lbrace> R \<rbrace>"
by (simp add: validE_def valid_def split: sum.splits) blast
lemma hoare_post_imp_dc:
"\<lbrakk>\<lbrace>P\<rbrace> a \<lbrace>\<lambda>r. Q\<rbrace>; \<And>s. Q s \<Longrightarrow> R s\<rbrakk> \<Longrightarrow> \<lbrace>P\<rbrace> a \<lbrace>\<lambda>r. R\<rbrace>,\<lbrace>\<lambda>r. R\<rbrace>"
by (simp add: validE_def valid_def split: sum.splits) blast
lemma hoare_post_imp_dc2:
"\<lbrakk>\<lbrace>P\<rbrace> a \<lbrace>\<lambda>r. Q\<rbrace>; \<And>s. Q s \<Longrightarrow> R s\<rbrakk> \<Longrightarrow> \<lbrace>P\<rbrace> a \<lbrace>\<lambda>r. R\<rbrace>,\<lbrace>\<lambda>r s. True\<rbrace>"
by (simp add: validE_def valid_def split: sum.splits) blast
lemma hoare_post_imp_dc2E:
"\<lbrakk>\<lbrace>P\<rbrace> a \<lbrace>\<lambda>r. Q\<rbrace>; \<And>s. Q s \<Longrightarrow> R s\<rbrakk> \<Longrightarrow> \<lbrace>P\<rbrace> a \<lbrace>\<lambda>r s. True\<rbrace>, \<lbrace>\<lambda>r. R\<rbrace>"
by (simp add: validE_def valid_def split: sum.splits) fast
lemma hoare_post_imp_dc2E_actual:
"\<lbrakk>\<lbrace>P\<rbrace> a \<lbrace>\<lambda>r. R\<rbrace>\<rbrakk> \<Longrightarrow> \<lbrace>P\<rbrace> a \<lbrace>\<lambda>r s. True\<rbrace>, \<lbrace>\<lambda>r. R\<rbrace>"
by (simp add: validE_def valid_def split: sum.splits) fast
lemma hoare_post_imp_dc2_actual:
"\<lbrakk>\<lbrace>P\<rbrace> a \<lbrace>\<lambda>r. R\<rbrace>\<rbrakk> \<Longrightarrow> \<lbrace>P\<rbrace> a \<lbrace>\<lambda>r. R\<rbrace>, \<lbrace>\<lambda>r s. True\<rbrace>"
by (simp add: validE_def valid_def split: sum.splits) fast
lemma hoare_post_impE: "\<lbrakk> \<And>r s. Q r s \<Longrightarrow> R r s; \<lbrace>P\<rbrace> a \<lbrace>Q\<rbrace> \<rbrakk> \<Longrightarrow> \<lbrace>P\<rbrace> a \<lbrace>R\<rbrace>"
by (fastforce simp:valid_def split_def)
lemma hoare_conjD1:
"\<lbrace>P\<rbrace> f \<lbrace>\<lambda>rv. Q rv and R rv\<rbrace> \<Longrightarrow> \<lbrace>P\<rbrace> f \<lbrace>\<lambda>rv. Q rv\<rbrace>"
unfolding valid_def by auto
lemma hoare_conjD2:
"\<lbrace>P\<rbrace> f \<lbrace>\<lambda>rv. Q rv and R rv\<rbrace> \<Longrightarrow> \<lbrace>P\<rbrace> f \<lbrace>\<lambda>rv. R rv\<rbrace>"
unfolding valid_def by auto
lemma hoare_post_disjI1:
"\<lbrace>P\<rbrace> f \<lbrace>\<lambda>rv. Q rv\<rbrace> \<Longrightarrow> \<lbrace>P\<rbrace> f \<lbrace>\<lambda>rv. Q rv or R rv\<rbrace>"
unfolding valid_def by auto
lemma hoare_post_disjI2:
"\<lbrace>P\<rbrace> f \<lbrace>\<lambda>rv. R rv\<rbrace> \<Longrightarrow> \<lbrace>P\<rbrace> f \<lbrace>\<lambda>rv. Q rv or R rv\<rbrace>"
unfolding valid_def by auto
lemma hoare_weaken_pre:
"\<lbrakk>\<lbrace>Q\<rbrace> a \<lbrace>R\<rbrace>; \<And>s. P s \<Longrightarrow> Q s\<rbrakk> \<Longrightarrow> \<lbrace>P\<rbrace> a \<lbrace>R\<rbrace>"
apply (rule hoare_pre_imp)
prefer 2
apply assumption
apply blast
done
lemma hoare_strengthen_post:
"\<lbrakk>\<lbrace>P\<rbrace> a \<lbrace>Q\<rbrace>; \<And>r s. Q r s \<Longrightarrow> R r s\<rbrakk> \<Longrightarrow> \<lbrace>P\<rbrace> a \<lbrace>R\<rbrace>"
apply (rule hoare_post_imp)
prefer 2
apply assumption
apply blast
done
lemma use_valid: "\<lbrakk>(r, s') \<in> fst (f s); \<lbrace>P\<rbrace> f \<lbrace>Q\<rbrace>; P s \<rbrakk> \<Longrightarrow> Q r s'"
apply (simp add: valid_def)
apply blast
done
lemma use_validE_norm: "\<lbrakk> (Inr r', s') \<in> fst (B s); \<lbrace> P \<rbrace> B \<lbrace> Q \<rbrace>,\<lbrace> E \<rbrace>; P s \<rbrakk> \<Longrightarrow> Q r' s'"
apply (clarsimp simp: validE_def valid_def)
apply force
done
lemma use_validE_except: "\<lbrakk> (Inl r', s') \<in> fst (B s); \<lbrace> P \<rbrace> B \<lbrace> Q \<rbrace>,\<lbrace> E \<rbrace>; P s \<rbrakk> \<Longrightarrow> E r' s'"
apply (clarsimp simp: validE_def valid_def)
apply force
done
lemma in_inv_by_hoareD:
"\<lbrakk> \<And>P. \<lbrace>P\<rbrace> f \<lbrace>\<lambda>_. P\<rbrace>; (x,s') \<in> fst (f s) \<rbrakk> \<Longrightarrow> s' = s"
by (auto simp add: valid_def) blast
subsection "Satisfiability"
lemma exs_hoare_post_imp: "\<lbrakk>\<And>r s. Q r s \<Longrightarrow> R r s; \<lbrace>P\<rbrace> a \<exists>\<lbrace>Q\<rbrace>\<rbrakk> \<Longrightarrow> \<lbrace>P\<rbrace> a \<exists>\<lbrace>R\<rbrace>"
apply (simp add: exs_valid_def)
apply safe
apply (erule_tac x=s in allE, simp)
apply blast
done
lemma use_exs_valid: "\<lbrakk>\<lbrace>P\<rbrace> f \<exists>\<lbrace>Q\<rbrace>; P s \<rbrakk> \<Longrightarrow> \<exists>(r, s') \<in> fst (f s). Q r s'"
by (simp add: exs_valid_def)
definition "exs_postcondition P f \<equiv> (\<lambda>a b. \<exists>(rv, s)\<in> f a b. P rv s)"
lemma exs_valid_is_triple:
"exs_valid P f Q = triple_judgement P f (exs_postcondition Q (\<lambda>s f. fst (f s)))"
by (simp add: triple_judgement_def exs_postcondition_def exs_valid_def)
lemmas [wp_trip] = exs_valid_is_triple
lemma exs_valid_weaken_pre [wp_comb]:
"\<lbrakk> \<lbrace> P' \<rbrace> f \<exists>\<lbrace> Q \<rbrace>; \<And>s. P s \<Longrightarrow> P' s \<rbrakk> \<Longrightarrow> \<lbrace> P \<rbrace> f \<exists>\<lbrace> Q \<rbrace>"
apply atomize
apply (clarsimp simp: exs_valid_def)
done
lemma exs_valid_chain:
"\<lbrakk> \<lbrace> P \<rbrace> f \<exists>\<lbrace> Q \<rbrace>; \<And>s. R s \<Longrightarrow> P s; \<And>r s. Q r s \<Longrightarrow> S r s \<rbrakk> \<Longrightarrow> \<lbrace> R \<rbrace> f \<exists>\<lbrace> S \<rbrace>"
apply atomize
apply (fastforce simp: exs_valid_def Bex_def)
done
lemma exs_valid_assume_pre:
"\<lbrakk> \<And>s. P s \<Longrightarrow> \<lbrace> P \<rbrace> f \<exists>\<lbrace> Q \<rbrace> \<rbrakk> \<Longrightarrow> \<lbrace> P \<rbrace> f \<exists>\<lbrace> Q \<rbrace>"
apply (fastforce simp: exs_valid_def)
done
lemma exs_valid_bind [wp_split]:
"\<lbrakk> \<And>x. \<lbrace>B x\<rbrace> g x \<exists>\<lbrace>C\<rbrace>; \<lbrace>A\<rbrace> f \<exists>\<lbrace>B\<rbrace> \<rbrakk> \<Longrightarrow> \<lbrace> A \<rbrace> f >>= (\<lambda>x. g x) \<exists>\<lbrace> C \<rbrace>"
apply atomize
apply (clarsimp simp: exs_valid_def bind_def')
apply blast
done
lemma exs_valid_return [wp]:
"\<lbrace> Q v \<rbrace> return v \<exists>\<lbrace> Q \<rbrace>"
by (clarsimp simp: exs_valid_def return_def)
lemma exs_valid_select [wp]:
"\<lbrace> \<lambda>s. \<exists>r \<in> S. Q r s \<rbrace> select S \<exists>\<lbrace> Q \<rbrace>"
by (clarsimp simp: exs_valid_def select_def)
lemma exs_valid_get [wp]:
"\<lbrace> \<lambda>s. Q s s \<rbrace> get \<exists>\<lbrace> Q \<rbrace>"
by (clarsimp simp: exs_valid_def get_def)
lemma exs_valid_gets [wp]:
"\<lbrace> \<lambda>s. Q (f s) s \<rbrace> gets f \<exists>\<lbrace> Q \<rbrace>"
by (clarsimp simp: gets_def, wp)
lemma exs_valid_put [wp]:
"\<lbrace> Q v \<rbrace> put v \<exists>\<lbrace> Q \<rbrace>"
by (clarsimp simp: put_def exs_valid_def)
lemma exs_valid_state_assert [wp]:
"\<lbrace> \<lambda>s. Q () s \<and> G s \<rbrace> state_assert G \<exists>\<lbrace> Q \<rbrace>"
by (clarsimp simp: state_assert_def exs_valid_def get_def
assert_def bind_def' return_def)
lemmas exs_valid_guard = exs_valid_state_assert
lemma exs_valid_fail [wp]:
"\<lbrace> \<lambda>_. False \<rbrace> fail \<exists>\<lbrace> Q \<rbrace>"
by (clarsimp simp: fail_def exs_valid_def)
lemma exs_valid_condition [wp]:
"\<lbrakk> \<lbrace> P \<rbrace> L \<exists>\<lbrace> Q \<rbrace>; \<lbrace> P' \<rbrace> R \<exists>\<lbrace> Q \<rbrace> \<rbrakk> \<Longrightarrow>
\<lbrace> \<lambda>s. (C s \<and> P s) \<or> (\<not> C s \<and> P' s) \<rbrace> condition C L R \<exists>\<lbrace> Q \<rbrace>"
by (clarsimp simp: condition_def exs_valid_def split: sum.splits)
subsection MISC
lemma hoare_return_simp:
"\<lbrace>P\<rbrace> return x \<lbrace>Q\<rbrace> = (\<forall>s. P s \<longrightarrow> Q x s)"
by (simp add: valid_def return_def)
lemma hoare_gen_asm:
"(P \<Longrightarrow> \<lbrace>P'\<rbrace> f \<lbrace>Q\<rbrace>) \<Longrightarrow> \<lbrace>P' and K P\<rbrace> f \<lbrace>Q\<rbrace>"
by (fastforce simp add: valid_def)
lemma hoare_when_wp [wp]:
"\<lbrakk> P \<Longrightarrow> \<lbrace>Q\<rbrace> f \<lbrace>R\<rbrace> \<rbrakk> \<Longrightarrow> \<lbrace>if P then Q else R ()\<rbrace> when P f \<lbrace>R\<rbrace>"
by (clarsimp simp: when_def valid_def return_def)
lemma hoare_conjI:
"\<lbrakk> \<lbrace>P\<rbrace> f \<lbrace>Q\<rbrace>; \<lbrace>P\<rbrace> f \<lbrace>R\<rbrace> \<rbrakk> \<Longrightarrow> \<lbrace>P\<rbrace> f \<lbrace>\<lambda>r s. Q r s \<and> R r s\<rbrace>"
unfolding valid_def by blast
lemma hoare_disjI1:
"\<lbrakk> \<lbrace>P\<rbrace> f \<lbrace>Q\<rbrace> \<rbrakk> \<Longrightarrow> \<lbrace>P\<rbrace> f \<lbrace>\<lambda>r s. Q r s \<or> R r s \<rbrace>"
unfolding valid_def by blast
lemma hoare_disjI2:
"\<lbrakk> \<lbrace>P\<rbrace> f \<lbrace>R\<rbrace> \<rbrakk> \<Longrightarrow> \<lbrace>P\<rbrace> f \<lbrace>\<lambda>r s. Q r s \<or> R r s \<rbrace>"
unfolding valid_def by blast
lemma hoare_assume_pre:
"(\<And>s. P s \<Longrightarrow> \<lbrace>P\<rbrace> f \<lbrace>Q\<rbrace>) \<Longrightarrow> \<lbrace>P\<rbrace> f \<lbrace>Q\<rbrace>"
by (auto simp: valid_def)
lemma hoare_returnOk_sp:
"\<lbrace>P\<rbrace> returnOk x \<lbrace>\<lambda>r s. r = x \<and> P s\<rbrace>, \<lbrace>Q\<rbrace>"
by (simp add: valid_def validE_def returnOk_def return_def)
lemma hoare_assume_preE:
"(\<And>s. P s \<Longrightarrow> \<lbrace>P\<rbrace> f \<lbrace>Q\<rbrace>,\<lbrace>R\<rbrace>) \<Longrightarrow> \<lbrace>P\<rbrace> f \<lbrace>Q\<rbrace>,\<lbrace>R\<rbrace>"
by (auto simp: valid_def validE_def)
lemma hoare_allI:
"(\<And>x. \<lbrace>P\<rbrace>f\<lbrace>Q x\<rbrace>) \<Longrightarrow> \<lbrace>P\<rbrace>f\<lbrace>\<lambda>r s. \<forall>x. Q x r s\<rbrace>"
by (simp add: valid_def) blast
lemma validE_allI:
"(\<And>x. \<lbrace>P\<rbrace> f \<lbrace>\<lambda>r s. Q x r s\<rbrace>,\<lbrace>E\<rbrace>) \<Longrightarrow> \<lbrace>P\<rbrace> f \<lbrace>\<lambda>r s. \<forall>x. Q x r s\<rbrace>,\<lbrace>E\<rbrace>"
by (fastforce simp: valid_def validE_def split: sum.splits)
lemma hoare_exI:
"\<lbrace>P\<rbrace> f \<lbrace>Q x\<rbrace> \<Longrightarrow> \<lbrace>P\<rbrace> f \<lbrace>\<lambda>r s. \<exists>x. Q x r s\<rbrace>"
by (simp add: valid_def) blast
lemma hoare_impI:
"(R \<Longrightarrow> \<lbrace>P\<rbrace>f\<lbrace>Q\<rbrace>) \<Longrightarrow> \<lbrace>P\<rbrace>f\<lbrace>\<lambda>r s. R \<longrightarrow> Q r s\<rbrace>"
by (simp add: valid_def) blast
lemma validE_impI:
" \<lbrakk>\<And>E. \<lbrace>P\<rbrace> f \<lbrace>\<lambda>_ _. True\<rbrace>,\<lbrace>E\<rbrace>; (P' \<Longrightarrow> \<lbrace>P\<rbrace> f \<lbrace>Q\<rbrace>,\<lbrace>E\<rbrace>)\<rbrakk> \<Longrightarrow>
\<lbrace>P\<rbrace> f \<lbrace>\<lambda>r s. P' \<longrightarrow> Q r s\<rbrace>, \<lbrace>E\<rbrace>"
by (fastforce simp: validE_def valid_def split: sum.splits)
lemma hoare_case_option_wp:
"\<lbrakk> \<lbrace>P\<rbrace> f None \<lbrace>Q\<rbrace>;
\<And>x. \<lbrace>P' x\<rbrace> f (Some x) \<lbrace>Q' x\<rbrace> \<rbrakk>
\<Longrightarrow> \<lbrace>case_option P P' v\<rbrace> f v \<lbrace>\<lambda>rv. case v of None \<Rightarrow> Q rv | Some x \<Rightarrow> Q' x rv\<rbrace>"
by (cases v) auto
subsection "Reasoning directly about states"
lemma in_throwError:
"((v, s') \<in> fst (throwError e s)) = (v = Inl e \<and> s' = s)"
by (simp add: throwError_def return_def)
lemma in_returnOk:
"((v', s') \<in> fst (returnOk v s)) = (v' = Inr v \<and> s' = s)"
by (simp add: returnOk_def return_def)
lemma in_bind:
"((r,s') \<in> fst ((do x \<leftarrow> f; g x od) s)) =
(\<exists>s'' x. (x, s'') \<in> fst (f s) \<and> (r, s') \<in> fst (g x s''))"
apply (simp add: bind_def split_def)
apply force
done
lemma in_bindE_R:
"((Inr r,s') \<in> fst ((doE x \<leftarrow> f; g x odE) s)) =
(\<exists>s'' x. (Inr x, s'') \<in> fst (f s) \<and> (Inr r, s') \<in> fst (g x s''))"
apply (simp add: bindE_def lift_def split_def bind_def)
apply (clarsimp simp: throwError_def return_def lift_def split: sum.splits)
apply safe
apply (case_tac a)
apply fastforce
apply fastforce
apply force
done
lemma in_bindE_L:
"((Inl r, s') \<in> fst ((doE x \<leftarrow> f; g x odE) s)) \<Longrightarrow>
(\<exists>s'' x. (Inr x, s'') \<in> fst (f s) \<and> (Inl r, s') \<in> fst (g x s'')) \<or> ((Inl r, s') \<in> fst (f s))"
apply (simp add: bindE_def lift_def bind_def)
apply safe
apply (simp add: return_def throwError_def lift_def split_def split: sum.splits split_if_asm)
apply force
done
lemma in_liftE:
"((v, s') \<in> fst (liftE f s)) = (\<exists>v'. v = Inr v' \<and> (v', s') \<in> fst (f s))"
by (force simp add: liftE_def bind_def return_def split_def)
lemma in_whenE: "((v, s') \<in> fst (whenE P f s)) = ((P \<longrightarrow> (v, s') \<in> fst (f s)) \<and>
(\<not>P \<longrightarrow> v = Inr () \<and> s' = s))"
by (simp add: whenE_def in_returnOk)
lemma inl_whenE:
"((Inl x, s') \<in> fst (whenE P f s)) = (P \<and> (Inl x, s') \<in> fst (f s))"
by (auto simp add: in_whenE)
lemma in_fail:
"r \<in> fst (fail s) = False"
by (simp add: fail_def)
lemma in_return:
"(r, s') \<in> fst (return v s) = (r = v \<and> s' = s)"
by (simp add: return_def)
lemma in_assert:
"(r, s') \<in> fst (assert P s) = (P \<and> s' = s)"
by (simp add: assert_def return_def fail_def)
lemma in_assertE:
"(r, s') \<in> fst (assertE P s) = (P \<and> r = Inr () \<and> s' = s)"
by (simp add: assertE_def returnOk_def return_def fail_def)
lemma in_assert_opt:
"(r, s') \<in> fst (assert_opt v s) = (v = Some r \<and> s' = s)"
by (auto simp: assert_opt_def in_fail in_return split: option.splits)
lemma in_get:
"(r, s') \<in> fst (get s) = (r = s \<and> s' = s)"
by (simp add: get_def)
lemma in_gets:
"(r, s') \<in> fst (gets f s) = (r = f s \<and> s' = s)"
by (simp add: simpler_gets_def)
lemma in_put:
"(r, s') \<in> fst (put x s) = (s' = x \<and> r = ())"
by (simp add: put_def)
lemma in_when:
"(v, s') \<in> fst (when P f s) = ((P \<longrightarrow> (v, s') \<in> fst (f s)) \<and> (\<not>P \<longrightarrow> v = () \<and> s' = s))"
by (simp add: when_def in_return)
lemma in_modify:
"(v, s') \<in> fst (modify f s) = (s'=f s \<and> v = ())"
by (simp add: modify_def bind_def get_def put_def)
lemma gets_the_in_monad:
"((v, s') \<in> fst (gets_the f s)) = (s' = s \<and> f s = Some v)"
by (auto simp: gets_the_def in_bind in_gets in_assert_opt split: option.split)
lemma in_alternative:
"(r,s') \<in> fst ((f \<sqinter> g) s) = ((r,s') \<in> fst (f s) \<or> (r,s') \<in> fst (g s))"
by (simp add: alternative_def)
lemmas in_monad = inl_whenE in_whenE in_liftE in_bind in_bindE_L
in_bindE_R in_returnOk in_throwError in_fail
in_assertE in_assert in_return in_assert_opt
in_get in_gets in_put in_when unlessE_whenE
unless_when in_modify gets_the_in_monad
in_alternative
subsection "Non-Failure"
lemma no_failD:
"\<lbrakk> no_fail P m; P s \<rbrakk> \<Longrightarrow> \<not>(snd (m s))"
by (simp add: no_fail_def)
lemma non_fail_modify [wp,simp]:
"no_fail \<top> (modify f)"
by (simp add: no_fail_def modify_def get_def put_def bind_def)
lemma non_fail_gets [simp]:
"no_fail \<top> (gets f)"
by (simp add: no_fail_def simpler_gets_def)
lemma non_fail_select [simp]:
"no_fail \<top> (select S)"
by (simp add: no_fail_def select_def)
lemma no_fail_pre:
"\<lbrakk> no_fail P f; \<And>s. Q s \<Longrightarrow> P s\<rbrakk> \<Longrightarrow> no_fail Q f"
by (simp add: no_fail_def)
lemma no_fail_alt [wp]:
"\<lbrakk> no_fail P f; no_fail Q g \<rbrakk> \<Longrightarrow> no_fail (P and Q) (f OR g)"
by (simp add: no_fail_def alternative_def)
lemma no_fail_return [simp, wp]:
"no_fail \<top> (return x)"
by (simp add: return_def no_fail_def)
lemma no_fail_get [simp, wp]:
"no_fail \<top> get"
by (simp add: get_def no_fail_def)
lemma no_fail_put [simp, wp]:
"no_fail \<top> (put s)"
by (simp add: put_def no_fail_def)
lemma no_fail_when [wp]:
"(P \<Longrightarrow> no_fail Q f) \<Longrightarrow> no_fail (if P then Q else \<top>) (when P f)"
by (simp add: when_def)
lemma no_fail_unless [wp]:
"(\<not>P \<Longrightarrow> no_fail Q f) \<Longrightarrow> no_fail (if P then \<top> else Q) (unless P f)"
by (simp add: unless_def when_def)
lemma no_fail_fail [simp, wp]:
"no_fail \<bottom> fail"
by (simp add: fail_def no_fail_def)
lemmas [wp] = non_fail_gets
lemma no_fail_assert [simp, wp]:
"no_fail (\<lambda>_. P) (assert P)"
by (simp add: assert_def)
lemma no_fail_assert_opt [simp, wp]:
"no_fail (\<lambda>_. P \<noteq> None) (assert_opt P)"
by (simp add: assert_opt_def split: option.splits)
lemma no_fail_case_option [wp]:
assumes f: "no_fail P f"
assumes g: "\<And>x. no_fail (Q x) (g x)"
shows "no_fail (if x = None then P else Q (the x)) (case_option f g x)"
by (clarsimp simp add: f g)
lemma no_fail_if [wp]:
"\<lbrakk> P \<Longrightarrow> no_fail Q f; \<not>P \<Longrightarrow> no_fail R g \<rbrakk> \<Longrightarrow>
no_fail (if P then Q else R) (if P then f else g)"
by simp
lemma no_fail_apply [wp]:
"no_fail P (f (g x)) \<Longrightarrow> no_fail P (f $ g x)"
by simp
lemma no_fail_undefined [simp, wp]:
"no_fail \<bottom> undefined"
by (simp add: no_fail_def)
lemma no_fail_returnOK [simp, wp]:
"no_fail \<top> (returnOk x)"
by (simp add: returnOk_def)
text {* Empty results implies non-failure *}
lemma empty_fail_modify [simp]:
"empty_fail (modify f)"
by (simp add: empty_fail_def simpler_modify_def)
lemma empty_fail_gets [simp]:
"empty_fail (gets f)"
by (simp add: empty_fail_def simpler_gets_def)
lemma empty_failD:
"\<lbrakk> empty_fail m; fst (m s) = {} \<rbrakk> \<Longrightarrow> snd (m s)"
by (simp add: empty_fail_def)
lemma empty_fail_select_f [simp]:
assumes ef: "fst S = {} \<Longrightarrow> snd S"
shows "empty_fail (select_f S)"
by (fastforce simp add: empty_fail_def select_f_def intro: ef)
lemma empty_fail_bind [simp]:
"\<lbrakk> empty_fail a; \<And>x. empty_fail (b x) \<rbrakk> \<Longrightarrow> empty_fail (a >>= b)"
apply (simp add: bind_def empty_fail_def split_def)
apply clarsimp
apply (case_tac "fst (a s) = {}")
apply blast
apply (clarsimp simp: ex_in_conv [symmetric])
done
lemma empty_fail_return [simp]:
"empty_fail (return x)"
by (simp add: empty_fail_def return_def)
lemma empty_fail_mapM [simp]:
assumes m: "\<And>x. empty_fail (m x)"
shows "empty_fail (mapM m xs)"
proof (induct xs)
case Nil
thus ?case by (simp add: mapM_def sequence_def)
next
case Cons
have P: "\<And>m x xs. mapM m (x # xs) = (do y \<leftarrow> m x; ys \<leftarrow> (mapM m xs); return (y # ys) od)"
by (simp add: mapM_def sequence_def Let_def)
from Cons
show ?case by (simp add: P m)
qed
lemma empty_fail [simp]:
"empty_fail fail"
by (simp add: fail_def empty_fail_def)
lemma empty_fail_assert_opt [simp]:
"empty_fail (assert_opt x)"
by (simp add: assert_opt_def split: option.splits)
lemma empty_fail_mk_ef:
"empty_fail (mk_ef o m)"
by (simp add: empty_fail_def mk_ef_def)
subsection "Failure"
lemma fail_wp: "\<lbrace>\<lambda>x. True\<rbrace> fail \<lbrace>Q\<rbrace>"
by (simp add: valid_def fail_def)
lemma failE_wp: "\<lbrace>\<lambda>x. True\<rbrace> fail \<lbrace>Q\<rbrace>,\<lbrace>E\<rbrace>"
by (simp add: validE_def fail_wp)
lemma fail_update [iff]:
"fail (f s) = fail s"
by (simp add: fail_def)
text {* We can prove postconditions using hoare triples *}
lemma post_by_hoare: "\<lbrakk> \<lbrace>P\<rbrace> f \<lbrace>Q\<rbrace>; P s; (r, s') \<in> fst (f s) \<rbrakk> \<Longrightarrow> Q r s'"
apply (simp add: valid_def)
apply blast
done
text {* Weakest Precondition Rules *}
lemma hoare_vcg_prop:
"\<lbrace>\<lambda>s. P\<rbrace> f \<lbrace>\<lambda>rv s. P\<rbrace>"
by (simp add: valid_def)
lemma return_wp:
"\<lbrace>P x\<rbrace> return x \<lbrace>P\<rbrace>"
apply(simp add:valid_def return_def)
done
lemma get_wp:
"\<lbrace>\<lambda>s. P s s\<rbrace> get \<lbrace>P\<rbrace>"
apply(simp add:valid_def split_def get_def)
done
lemma gets_wp:
"\<lbrace>\<lambda>s. P (f s) s\<rbrace> gets f \<lbrace>P\<rbrace>"
apply(simp add:valid_def split_def gets_def return_def get_def bind_def)
done
lemma modify_wp:
"\<lbrace>\<lambda>s. P () (f s)\<rbrace> modify f \<lbrace>P\<rbrace>"
apply(simp add:valid_def split_def modify_def get_def put_def bind_def)
done
lemma put_wp:
"\<lbrace>\<lambda>s. P () x\<rbrace> put x \<lbrace>P\<rbrace>"
apply(simp add:valid_def put_def)
done
lemma returnOk_wp:
"\<lbrace>P x\<rbrace> returnOk x \<lbrace>P\<rbrace>,\<lbrace>E\<rbrace>"
apply(simp add:validE_def2 returnOk_def return_def)
done
lemma throwError_wp:
"\<lbrace>E e\<rbrace> throwError e \<lbrace>P\<rbrace>,\<lbrace>E\<rbrace>"
apply(simp add:validE_def2 throwError_def return_def)
done
lemma returnOKE_R_wp : "\<lbrace>P x\<rbrace> returnOk x \<lbrace>P\<rbrace>, -"
by (simp add: validE_R_def validE_def valid_def
returnOk_def return_def)
lemma liftE_wp:
"\<lbrace>P\<rbrace> f \<lbrace>Q\<rbrace> \<Longrightarrow> \<lbrace>P\<rbrace> liftE f \<lbrace>Q\<rbrace>,\<lbrace>E\<rbrace>"
apply(clarsimp simp:valid_def validE_def2 liftE_def split_def Let_def bind_def return_def)
done
lemma catch_wp:
"\<lbrakk> \<And>x. \<lbrace>E x\<rbrace> handler x \<lbrace>Q\<rbrace>; \<lbrace>P\<rbrace> f \<lbrace>Q\<rbrace>,\<lbrace>E\<rbrace> \<rbrakk> \<Longrightarrow>
\<lbrace>P\<rbrace> catch f handler \<lbrace>Q\<rbrace>"
apply (unfold catch_def valid_def validE_def return_def)
apply (clarsimp simp: bind_def)
apply (fastforce split: sum.splits)
done
lemma handleE'_wp:
"\<lbrakk> \<And>x. \<lbrace>F x\<rbrace> handler x \<lbrace>Q\<rbrace>,\<lbrace>E\<rbrace>; \<lbrace>P\<rbrace> f \<lbrace>Q\<rbrace>,\<lbrace>F\<rbrace> \<rbrakk> \<Longrightarrow>
\<lbrace>P\<rbrace> f <handle2> handler \<lbrace>Q\<rbrace>,\<lbrace>E\<rbrace>"
apply (unfold handleE'_def valid_def validE_def return_def)
apply (clarsimp simp: bind_def)
apply (fastforce split: sum.splits)
done
lemma handleE_wp:
assumes x: "\<And>x. \<lbrace>F x\<rbrace> handler x \<lbrace>Q\<rbrace>,\<lbrace>E\<rbrace>"
assumes y: "\<lbrace>P\<rbrace> f \<lbrace>Q\<rbrace>,\<lbrace>F\<rbrace>"
shows "\<lbrace>P\<rbrace> f <handle> handler \<lbrace>Q\<rbrace>,\<lbrace>E\<rbrace>"
by (simp add: handleE_def handleE'_wp [OF x y])
lemma hoare_vcg_split_if:
"\<lbrakk> P \<Longrightarrow> \<lbrace>Q\<rbrace> f \<lbrace>S\<rbrace>; \<not>P \<Longrightarrow> \<lbrace>R\<rbrace> g \<lbrace>S\<rbrace> \<rbrakk> \<Longrightarrow>
\<lbrace>\<lambda>s. (P \<longrightarrow> Q s) \<and> (\<not>P \<longrightarrow> R s)\<rbrace> if P then f else g \<lbrace>S\<rbrace>"
by simp
lemma hoare_vcg_split_ifE:
"\<lbrakk> P \<Longrightarrow> \<lbrace>Q\<rbrace> f \<lbrace>S\<rbrace>,\<lbrace>E\<rbrace>; \<not>P \<Longrightarrow> \<lbrace>R\<rbrace> g \<lbrace>S\<rbrace>,\<lbrace>E\<rbrace> \<rbrakk> \<Longrightarrow>
\<lbrace>\<lambda>s. (P \<longrightarrow> Q s) \<and> (\<not>P \<longrightarrow> R s)\<rbrace> if P then f else g \<lbrace>S\<rbrace>,\<lbrace>E\<rbrace>"
by simp
lemma hoare_liftM_subst: "\<lbrace>P\<rbrace> liftM f m \<lbrace>Q\<rbrace> = \<lbrace>P\<rbrace> m \<lbrace>Q \<circ> f\<rbrace>"
apply (simp add: liftM_def bind_def return_def split_def)
apply (simp add: valid_def Ball_def)
apply (rule_tac f=All in arg_cong)
apply (rule ext)
apply fastforce
done
lemma liftE_validE[simp]: "\<lbrace>P\<rbrace> liftE f \<lbrace>Q\<rbrace>,\<lbrace>E\<rbrace> = \<lbrace>P\<rbrace> f \<lbrace>Q\<rbrace>"
apply (simp add: liftE_liftM validE_def hoare_liftM_subst o_def)
done
lemma liftM_wp: "\<lbrace>P\<rbrace> m \<lbrace>Q \<circ> f\<rbrace> \<Longrightarrow> \<lbrace>P\<rbrace> liftM f m \<lbrace>Q\<rbrace>"
by (simp add: hoare_liftM_subst)
lemma hoare_liftME_subst: "\<lbrace>P\<rbrace> liftME f m \<lbrace>Q\<rbrace>,\<lbrace>E\<rbrace> = \<lbrace>P\<rbrace> m \<lbrace>Q \<circ> f\<rbrace>,\<lbrace>E\<rbrace>"
apply (simp add: validE_def liftME_liftM hoare_liftM_subst o_def)
apply (rule_tac f="valid P m" in arg_cong)
apply (rule ext)+
apply (case_tac x, simp_all)
done
lemma liftME_wp: "\<lbrace>P\<rbrace> m \<lbrace>Q \<circ> f\<rbrace>,\<lbrace>E\<rbrace> \<Longrightarrow> \<lbrace>P\<rbrace> liftME f m \<lbrace>Q\<rbrace>,\<lbrace>E\<rbrace>"
by (simp add: hoare_liftME_subst)
(* FIXME: Move *)
lemma o_const_simp[simp]: "(\<lambda>x. C) \<circ> f = (\<lambda>x. C)"
by (simp add: o_def)
lemma hoare_vcg_split_case_option:
"\<lbrakk> \<And>x. x = None \<Longrightarrow> \<lbrace>P x\<rbrace> f x \<lbrace>R x\<rbrace>;
\<And>x y. x = Some y \<Longrightarrow> \<lbrace>Q x y\<rbrace> g x y \<lbrace>R x\<rbrace> \<rbrakk> \<Longrightarrow>
\<lbrace>\<lambda>s. (x = None \<longrightarrow> P x s) \<and>
(\<forall>y. x = Some y \<longrightarrow> Q x y s)\<rbrace>
case x of None \<Rightarrow> f x
| Some y \<Rightarrow> g x y
\<lbrace>R x\<rbrace>"
apply(simp add:valid_def split_def)
apply(case_tac x, simp_all)
done
lemma hoare_vcg_split_case_optionE:
assumes none_case: "\<And>x. x = None \<Longrightarrow> \<lbrace>P x\<rbrace> f x \<lbrace>R x\<rbrace>,\<lbrace>E x\<rbrace>"
assumes some_case: "\<And>x y. x = Some y \<Longrightarrow> \<lbrace>Q x y\<rbrace> g x y \<lbrace>R x\<rbrace>,\<lbrace>E x\<rbrace>"
shows "\<lbrace>\<lambda>s. (x = None \<longrightarrow> P x s) \<and>
(\<forall>y. x = Some y \<longrightarrow> Q x y s)\<rbrace>
case x of None \<Rightarrow> f x
| Some y \<Rightarrow> g x y
\<lbrace>R x\<rbrace>,\<lbrace>E x\<rbrace>"
apply(case_tac x, simp_all)
apply(rule none_case, simp)
apply(rule some_case, simp)
done
lemma hoare_vcg_split_case_sum: