header {*
Generic Separation Logic
*}
theory Hoare_gen
imports Main
"../Separation_Algebra/Sep_Tactics"
begin
definition pasrt :: "bool \<Rightarrow> (('a::sep_algebra) \<Rightarrow> bool)" ("<_>" [72] 71)
where "pasrt b = (\<lambda> s . s = 0 \<and> b)"
lemma sep_conj_cond1: "(p \<and>* <cond> \<and>* q) = (<cond> \<and>* p \<and>* q)"
by(simp add: sep_conj_ac)
lemma sep_conj_cond2: "(p \<and>* <cond>) = (<cond> \<and>* p)"
by(simp add: sep_conj_ac)
lemma sep_conj_cond3: "((<cond> \<and>* p) \<and>* r) = (<cond> \<and>* p \<and>* r)"
by (metis sep.mult_assoc)
lemmas sep_conj_cond = sep_conj_cond1 sep_conj_cond2 sep_conj_cond3
lemma cond_true_eq[simp]: "<True> = \<box>"
by(unfold sep_empty_def pasrt_def, auto)
lemma cond_true_eq1: "(<True> \<and>* p) = p"
by(simp)
lemma false_simp [simp]: "<False> = sep_false" (* move *)
by (simp add:pasrt_def)
lemma cond_true_eq2: "(p \<and>* <True>) = p"
by simp
lemma condD: "(<b> ** r) s \<Longrightarrow> b \<and> r s"
by (unfold sep_conj_def pasrt_def, auto)
locale sep_exec =
fixes step :: "'conf \<Rightarrow> 'conf"
and recse:: "'conf \<Rightarrow> 'a::sep_algebra"
begin
definition "run n = step ^^ n"
lemma run_add: "run (n1 + n2) s = run n1 (run n2 s)"
apply (unfold run_def)
by (metis funpow_add o_apply)
definition
Hoare_gen :: "('a \<Rightarrow> bool) \<Rightarrow> ('a \<Rightarrow> bool) \<Rightarrow> ('a \<Rightarrow> bool) \<Rightarrow> bool"
("(\<lbrace>(1_)\<rbrace> / (_)/ \<lbrace>(1_)\<rbrace>)" 50)
where
"\<lbrace> p \<rbrace> c \<lbrace> q \<rbrace> \<equiv>
(\<forall> s r. (p**c**r) (recse s) \<longrightarrow> (\<exists> k. ((q ** c ** r) (recse (run (Suc k) s)))))"
lemma HoareI [case_names Pre]:
assumes h: "\<And> r s. (p**c**r) (recse s) \<Longrightarrow> (\<exists> k. ((q ** c ** r) (recse (run (Suc k) s))))"
shows "\<lbrace> p \<rbrace> c \<lbrace> q \<rbrace>"
using h
by (unfold Hoare_gen_def, auto)
lemma frame_rule:
assumes h: "\<lbrace> p \<rbrace> c \<lbrace> q \<rbrace>"
shows "\<lbrace> p ** r \<rbrace> c \<lbrace> q ** r \<rbrace>"
proof(induct rule: HoareI)
case (Pre r' s')
hence "(p \<and>* c \<and>* r \<and>* r') (recse s')" by (auto simp:sep_conj_ac)
from h[unfolded Hoare_gen_def, rule_format, OF this]
show ?case
by (metis sep_conj_assoc sep_conj_left_commute)
qed
lemma sequencing:
assumes h1: "\<lbrace>p\<rbrace> c \<lbrace>q\<rbrace>"
and h2: "\<lbrace>q\<rbrace> c \<lbrace>r\<rbrace>"
shows "\<lbrace>p\<rbrace> c \<lbrace>r\<rbrace>"
proof(induct rule:HoareI)
case (Pre r' s')
from h1[unfolded Hoare_gen_def, rule_format, OF Pre]
obtain k1 where "(q \<and>* c \<and>* r') (recse (run (Suc k1) s'))" by auto
from h2[unfolded Hoare_gen_def, rule_format, OF this]
obtain k2 where "(r \<and>* c \<and>* r') (recse (run (Suc k2) (run (Suc k1) s')))" by auto
thus ?case
apply (rule_tac x = "Suc (k1 + k2)" in exI)
by (metis add_Suc_right nat_add_commute sep_exec.run_add)
qed
lemma pre_stren:
assumes h1: "\<lbrace>p\<rbrace> c \<lbrace>q\<rbrace>"
and h2: "\<And>s. r s \<Longrightarrow> p s"
shows "\<lbrace>r\<rbrace> c \<lbrace>q\<rbrace>"
proof(induct rule:HoareI)
case (Pre r' s')
with h2
have "(p \<and>* c \<and>* r') (recse s')"
by (metis sep_conj_impl1)
from h1[unfolded Hoare_gen_def, rule_format, OF this]
show ?case .
qed
lemma post_weaken:
assumes h1: "\<lbrace>p\<rbrace> c \<lbrace>q\<rbrace>"
and h2: "\<And> s. q s \<Longrightarrow> r s"
shows "\<lbrace>p\<rbrace> c \<lbrace>r\<rbrace>"
proof(induct rule:HoareI)
case (Pre r' s')
from h1[unfolded Hoare_gen_def, rule_format, OF this]
obtain k where "(q \<and>* c \<and>* r') (recse (run (Suc k) s'))" by blast
with h2
show ?case
by (metis sep_conj_impl1)
qed
lemma hoare_adjust:
assumes h1: "\<lbrace>p1\<rbrace> c \<lbrace>q1\<rbrace>"
and h2: "\<And>s. p s \<Longrightarrow> p1 s"
and h3: "\<And>s. q1 s \<Longrightarrow> q s"
shows "\<lbrace>p\<rbrace> c \<lbrace>q\<rbrace>"
using h1 h2 h3 post_weaken pre_stren
by (metis)
lemma code_exI:
assumes h: "\<And> k. \<lbrace>p\<rbrace> c(k) \<lbrace>q\<rbrace>"
shows "\<lbrace>p\<rbrace> EXS k. c(k) \<lbrace>q\<rbrace>"
proof(unfold pred_ex_def, induct rule:HoareI)
case (Pre r' s')
then obtain k where "(p \<and>* (\<lambda> s. c k s) \<and>* r') (recse s')"
by (auto elim!:sep_conjE intro!:sep_conjI)
from h[unfolded Hoare_gen_def, rule_format, OF this]
show ?case
by (auto elim!:sep_conjE intro!:sep_conjI)
qed
lemma code_extension:
assumes h: "\<lbrace> p \<rbrace> c \<lbrace> q \<rbrace>"
shows "\<lbrace> p \<rbrace> c ** e \<lbrace> q \<rbrace>"
proof(induct rule:HoareI)
case (Pre r' s')
hence "(p \<and>* c \<and>* e \<and>* r') (recse s')" by (auto simp:sep_conj_ac)
from h[unfolded Hoare_gen_def, rule_format, OF this]
show ?case
by (auto elim!:sep_conjE intro!:sep_conjI simp:sep_conj_ac)
qed
lemma code_extension1:
assumes h: "\<lbrace> p \<rbrace> c \<lbrace> q \<rbrace>"
shows "\<lbrace> p \<rbrace> e ** c \<lbrace> q \<rbrace>"
by (metis code_extension h sep.mult_commute)
lemma composition:
assumes h1: "\<lbrace>p\<rbrace> c1 \<lbrace>q\<rbrace>"
and h2: "\<lbrace>q\<rbrace> c2 \<lbrace>r\<rbrace>"
shows "\<lbrace>p\<rbrace> c1 ** c2 \<lbrace>r\<rbrace>"
proof(induct rule:HoareI)
case (Pre r' s')
hence "(p \<and>* c1 \<and>* c2 \<and>* r') (recse s')" by (auto simp:sep_conj_ac)
from h1[unfolded Hoare_gen_def, rule_format, OF this]
obtain k1 where "(q \<and>* c2 \<and>* c1 \<and>* r') (recse (run (Suc k1) s'))"
by (auto simp:sep_conj_ac)
from h2[unfolded Hoare_gen_def, rule_format, OF this, folded run_add]
show ?case
by (auto elim!:sep_conjE intro!:sep_conjI simp:sep_conj_ac)
qed
definition
IHoare :: "('b::sep_algebra \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow>
('b \<Rightarrow> bool) \<Rightarrow> ('a \<Rightarrow> bool) \<Rightarrow> ('b \<Rightarrow> bool) \<Rightarrow> bool"
("(1_).(\<lbrace>(1_)\<rbrace> / (_)/ \<lbrace>(1_)\<rbrace>)" 50)
where
"I. \<lbrace>P\<rbrace> c \<lbrace>Q\<rbrace> = (\<forall>s'. \<lbrace> EXS s. <P s> \<and>* <(s ## s')> \<and>* I(s + s')\<rbrace> c
\<lbrace> EXS s. <Q s> \<and>* <(s ## s')> \<and>* I(s + s')\<rbrace>)"
lemma IHoareI [case_names IPre]:
assumes h: "\<And>s' s r cnf . (<P s> \<and>* <(s ## s')> \<and>* I(s + s') \<and>* c \<and>* r) (recse cnf) \<Longrightarrow>
(\<exists>k t. (<Q t> \<and>* <(t ## s')> \<and>* I(t + s') \<and>* c \<and>* r) (recse (run (Suc k) cnf)))"
shows "I. \<lbrace>P\<rbrace> c \<lbrace>Q\<rbrace>"
unfolding IHoare_def
proof
fix s'
show " \<lbrace>EXS s. <P s> \<and>* <(s ## s')> \<and>* I (s + s')\<rbrace> c
\<lbrace>EXS s. <Q s> \<and>* <(s ## s')> \<and>* I (s + s')\<rbrace>"
proof(unfold pred_ex_def, induct rule:HoareI)
case (Pre r s)
then obtain sa where "(<P sa> \<and>* <(sa ## s')> \<and>* I (sa + s') \<and>* c \<and>* r) (recse s)"
by (auto elim!:sep_conjE intro!:sep_conjI simp:sep_conj_ac)
hence " (\<exists>k t. (<Q t> \<and>* <(t##s')> \<and>* I(t + s') \<and>* c \<and>* r) (recse (run (Suc k) s)))"
by (rule h)
then obtain k t where h2: "(<Q t> \<and>* <(t ## s')> \<and>* I(t + s') \<and>* c \<and>* r) (recse (run (Suc k) s))"
by auto
thus "\<exists>k. ((\<lambda>s. \<exists>sa. (<Q sa> \<and>* <(sa ## s')> \<and>* I (sa + s')) s) \<and>* c \<and>* r) (recse (run (Suc k) s))"
apply (rule_tac x = k in exI)
by (auto intro!:sep_conjI elim!:sep_conjE simp:sep_conj_ac)
qed
qed
lemma I_frame_rule:
assumes h: "I. \<lbrace>P\<rbrace> c \<lbrace>Q\<rbrace>"
shows "I. \<lbrace>P \<and>* R\<rbrace> c \<lbrace>Q \<and>* R\<rbrace>"
proof(induct rule:IHoareI)
case (IPre s' s r cnf)
hence "((<(P \<and>* R) s> \<and>* <(s##s')> \<and>* I (s + s')) \<and>* c \<and>* r) (recse cnf)"
by (auto simp:sep_conj_ac)
then obtain s1 s2
where h1: "((<P s1> \<and>* <((s1 + s2) ## s')> \<and>*I (s1 + s2 + s')) \<and>* c \<and>* r) (recse cnf)"
"s1 ## s2" "s1 + s2 ## s'" "P s1" "R s2"
by (unfold pasrt_def, auto elim!:sep_conjE intro!:sep_conjI)
hence "((EXS s. <P s> \<and>* <(s ## s2 +s')> \<and>*I (s + (s2 + s'))) \<and>* c \<and>* r) (recse cnf)"
apply (sep_cancel, unfold pred_ex_def, auto intro!:sep_conjI sep_disj_addI3 elim!:sep_conjE)
apply (rule_tac x = s1 in exI, unfold pasrt_def,
auto intro!:sep_conjI sep_disj_addI3 elim!:sep_conjE simp:sep_conj_ac)
by (metis sep_add_assoc sep_add_disjD)
from h[unfolded IHoare_def Hoare_gen_def, rule_format, OF this]
obtain k s where
"((<Q s> \<and>* <(s ## s2 + s')> \<and>* I (s + (s2 + s'))) \<and>* c \<and>* r) (recse (run (Suc k) cnf))"
by (unfold pasrt_def pred_ex_def, auto elim!:sep_conjE intro!:sep_conjI)
thus ?case
proof(rule_tac x = k in exI, rule_tac x = "s + s2" in exI, sep_cancel+)
fix h ha
assume hh: "(<Q s> \<and>* <(s ## s2 + s')> \<and>* I (s + (s2 + s'))) ha"
show " (<(Q \<and>* R) (s + s2)> \<and>* <(s + s2 ## s')> \<and>* I (s + s2 + s')) ha"
proof -
from hh have h0: "s ## s2 + s'"
by (metis pasrt_def sep_conjD)
with h1(2, 3)
have h2: "s + s2 ## s'"
by (metis sep_add_disjD sep_disj_addI1)
moreover from h1(2, 3) h2 have h3: "(s + (s2 + s')) = (s + s2 + s')"
by (metis `s ## s2 + s'` sep_add_assoc sep_add_disjD sep_disj_addD1)
moreover from hh have "Q s" by (metis pasrt_def sep_conjD)
moreover from h0 h1(2) h1(3) have "s ## s2"
by (metis sep_add_disjD sep_disj_addD)
moreover note h1(5)
ultimately show ?thesis
by (smt h0 hh sep_conjI)
qed
qed
qed
lemma I_sequencing:
assumes h1: "I. \<lbrace>P\<rbrace> c \<lbrace>Q\<rbrace>"
and h2: "I. \<lbrace>Q\<rbrace> c \<lbrace>R\<rbrace>"
shows "I. \<lbrace>P\<rbrace> c \<lbrace>R\<rbrace>"
using h1 h2 sequencing
by (smt IHoare_def)
lemma I_pre_stren:
assumes h1: "I. \<lbrace>P\<rbrace> c \<lbrace>Q\<rbrace>"
and h2: "\<And>s. R s \<Longrightarrow> P s"
shows "I. \<lbrace>R\<rbrace> c \<lbrace>Q\<rbrace>"
proof(unfold IHoare_def, default)
fix s'
show "\<lbrace>EXS s. <R s> \<and>* <(s ## s')> \<and>* I (s + s')\<rbrace> c
\<lbrace>EXS s. <Q s> \<and>* <(s ## s')> \<and>* I (s + s')\<rbrace>"
proof(rule pre_stren)
from h1[unfolded IHoare_def, rule_format, of s']
show "\<lbrace>EXS s. <P s> \<and>* <(s ## s')> \<and>* I (s + s')\<rbrace> c
\<lbrace>EXS s. <Q s> \<and>* <(s ## s')> \<and>* I (s + s')\<rbrace>" .
next
fix s
show "(EXS s. <R s> \<and>* <(s ## s')> \<and>* I (s + s')) s \<Longrightarrow>
(EXS s. <P s> \<and>* <(s ## s')> \<and>* I (s + s')) s"
apply (unfold pred_ex_def, clarify)
apply (rule_tac x = sa in exI, sep_cancel+)
by (insert h2, auto simp:pasrt_def)
qed
qed
lemma I_post_weaken:
assumes h1: "I. \<lbrace>P\<rbrace> c \<lbrace>Q\<rbrace>"
and h2: "\<And> s. Q s \<Longrightarrow> R s"
shows "I. \<lbrace>P\<rbrace> c \<lbrace>R\<rbrace>"
proof(unfold IHoare_def, default)
fix s'
show "\<lbrace>EXS s. <P s> \<and>* <(s ## s')> \<and>* I (s + s')\<rbrace> c
\<lbrace>EXS s. <R s> \<and>* <(s ## s')> \<and>* I (s + s')\<rbrace>"
proof(rule post_weaken)
from h1[unfolded IHoare_def, rule_format, of s']
show "\<lbrace>EXS s. <P s> \<and>* <(s ## s')> \<and>* I (s + s')\<rbrace> c
\<lbrace>EXS s. <Q s> \<and>* <(s ## s')> \<and>* I (s + s')\<rbrace>" .
next
fix s
show "(EXS s. <Q s> \<and>* <(s ## s')> \<and>* I (s + s')) s \<Longrightarrow>
(EXS s. <R s> \<and>* <(s ## s')> \<and>* I (s + s')) s"
apply (unfold pred_ex_def, clarify)
apply (rule_tac x = sa in exI, sep_cancel+)
by (insert h2, auto simp:pasrt_def)
qed
qed
lemma I_hoare_adjust:
assumes h1: "I. \<lbrace>P1\<rbrace> c \<lbrace>Q1\<rbrace>"
and h2: "\<And>s. P s \<Longrightarrow> P1 s"
and h3: "\<And>s. Q1 s \<Longrightarrow> Q s"
shows "I. \<lbrace>P\<rbrace> c \<lbrace>Q\<rbrace>"
using h1 h2 h3 I_post_weaken I_pre_stren
by (metis)
lemma I_code_exI:
assumes h: "\<And> k. I. \<lbrace>P\<rbrace> c(k) \<lbrace>Q\<rbrace>"
shows "I. \<lbrace>P\<rbrace> EXS k. c(k) \<lbrace>Q\<rbrace>"
using h
by (smt IHoare_def code_exI)
lemma I_code_extension:
assumes h: "I. \<lbrace> P \<rbrace> c \<lbrace> Q \<rbrace>"
shows "I. \<lbrace> P \<rbrace> c ** e \<lbrace> Q \<rbrace>"
using h
by (smt IHoare_def sep_exec.code_extension)
lemma I_composition:
assumes h1: "I. \<lbrace>P\<rbrace> c1 \<lbrace>Q\<rbrace>"
and h2: "I. \<lbrace>Q\<rbrace> c2 \<lbrace>R\<rbrace>"
shows "I. \<lbrace>P\<rbrace> c1 ** c2 \<lbrace>R\<rbrace>"
using h1 h2
by (smt IHoare_def sep_exec.composition)
lemma pre_condI:
assumes h: "cond \<Longrightarrow> \<lbrace>p\<rbrace> c \<lbrace>q\<rbrace>"
shows "\<lbrace><cond> \<and>* p\<rbrace> c \<lbrace>q\<rbrace>"
proof(induct rule:HoareI)
case (Pre r s)
hence "cond" "(p \<and>* c \<and>* r) (recse s)"
apply (metis pasrt_def sep_conjD)
by (smt Pre.hyps pasrt_def sep_add_zero sep_conj_commute sep_conj_def)
from h[OF this(1), unfolded Hoare_gen_def, rule_format, OF this(2)]
show ?case .
qed
lemma I_pre_condI:
assumes h: "cond \<Longrightarrow> I.\<lbrace>P\<rbrace> c \<lbrace>Q\<rbrace>"
shows "I.\<lbrace><cond> \<and>* P\<rbrace> c \<lbrace>Q\<rbrace>"
using h
by (smt IHoareI condD cond_true_eq2 sep.mult_commute)
lemma code_condI:
assumes h: "b \<Longrightarrow> \<lbrace>p\<rbrace> c \<lbrace>q\<rbrace>"
shows "\<lbrace>p\<rbrace> <b>**c \<lbrace>q\<rbrace>"
proof(induct rule: HoareI)
case (Pre r s)
hence h1: "b" "(p \<and>* c \<and>* r) (recse s)"
apply (metis condD sep_conjD sep_conj_assoc)
by (smt Pre.hyps condD sep_conj_impl)
from h[OF h1(1), unfolded Hoare_gen_def, rule_format, OF h1(2)]
and h1(1)
show ?case
by (metis (full_types) cond_true_eq1)
qed
lemma I_code_condI:
assumes h: "b \<Longrightarrow> I. \<lbrace>P\<rbrace> c \<lbrace>Q\<rbrace>"
shows "I.\<lbrace>P\<rbrace> <b>**c \<lbrace>Q\<rbrace>"
using h
by (smt IHoareI condD cond_true_eq2 sep.mult_commute sep_conj_cond1)
lemma precond_exI:
assumes h:"\<And>x. \<lbrace>p x\<rbrace> c \<lbrace>q\<rbrace>"
shows "\<lbrace>EXS x. p x\<rbrace> c \<lbrace>q\<rbrace>"
proof(induct rule:HoareI)
case (Pre r s)
then obtain x where "(p x \<and>* c \<and>* r) (recse s)"
by (unfold pred_ex_def, auto elim!:sep_conjE intro!:sep_conjI)
from h[of x, unfolded Hoare_gen_def, rule_format, OF this]
show ?case .
qed
lemma I_precond_exI:
assumes h:"\<And>x. I. \<lbrace>P x\<rbrace> c \<lbrace>Q\<rbrace>"
shows "I.\<lbrace>EXS x. P x\<rbrace> c \<lbrace>Q\<rbrace>"
proof(induct rule:IHoareI)
case (IPre s' s r cnf)
then obtain x
where "((EXS s. <P x s> \<and>* <(s ## s')> \<and>* I (s + s')) \<and>* c \<and>* r) (recse cnf)"
by ( auto elim!:sep_conjE intro!:sep_conjI simp:pred_ex_def pasrt_def sep_conj_ac)
from h[unfolded IHoare_def Hoare_gen_def, rule_format, OF this]
obtain k t
where "((<Q t> \<and>* <(t ## s')> \<and>* I (t + s')) \<and>* c \<and>* r) (recse (run (Suc k) cnf))"
by (unfold pred_ex_def, auto elim!:sep_conjE intro!:sep_conjI)
thus ?case
by (auto simp:sep_conj_ac)
qed
lemma hoare_sep_false:
"\<lbrace>sep_false\<rbrace> c
\<lbrace>q\<rbrace>"
by(unfold Hoare_gen_def, clarify, simp)
lemma I_hoare_sep_false:
"I. \<lbrace>sep_false\<rbrace> c
\<lbrace>Q\<rbrace>"
by (smt IHoareI condD)
end
instantiation set :: (type)sep_algebra
begin
definition set_zero_def: "0 = {}"
definition plus_set_def: "s1 + s2 = s1 \<union> s2"
definition sep_disj_set_def: "sep_disj s1 s2 = (s1 \<inter> s2 = {})"
lemmas set_ins_def = sep_disj_set_def plus_set_def set_zero_def
instance
apply(default)
apply(simp add:set_ins_def)
apply(simp add:sep_disj_set_def plus_set_def set_zero_def)
apply (metis inf_commute)
apply(simp add:sep_disj_set_def plus_set_def set_zero_def)
apply(simp add:sep_disj_set_def plus_set_def set_zero_def)
apply (metis sup_commute)
apply(simp add:sep_disj_set_def plus_set_def set_zero_def)
apply (metis (lifting) Un_assoc)
apply(simp add:sep_disj_set_def plus_set_def set_zero_def)
apply (metis (lifting) Int_Un_distrib Un_empty inf_sup_distrib1 sup_eq_bot_iff)
apply(simp add:sep_disj_set_def plus_set_def set_zero_def)
by (metis (lifting) Int_Un_distrib Int_Un_distrib2 sup_eq_bot_iff)
end
section {* A big operator of infinite separation conjunction *}
definition "fam_conj I cpt s = (\<exists> p. (s = (\<Union> i \<in> I. p(i))) \<and>
(\<forall> i \<in> I. cpt i (p i)) \<and>
(\<forall> i \<in> I. \<forall> j \<in> I. i \<noteq> j \<longrightarrow> p(i) ## p(j)))"
lemma fam_conj_zero_simp: "fam_conj {} cpt = <True>"
proof
fix s
show "fam_conj {} cpt s = (<True>) s"
proof
assume "fam_conj {} cpt s"
then obtain p where "s = (\<Union> i \<in> {}. p(i))"
by (unfold fam_conj_def, auto)
hence "s = {}" by auto
thus "(<True>) s" by (metis pasrt_def set_zero_def)
next
assume "(<True>) s"
hence eq_s: "s = {}" by (metis pasrt_def set_zero_def)
let ?p = "\<lambda> i. {}"
have "(s = (\<Union> i \<in> {}. ?p(i)))" by (unfold eq_s, auto)
moreover have "(\<forall> i \<in> {}. cpt i (?p i))" by auto
moreover have "(\<forall> i \<in> {}. \<forall> j \<in> {}. i \<noteq> j \<longrightarrow> ?p(i) ## ?p(j))" by auto
ultimately show "fam_conj {} cpt s"
by (unfold eq_s fam_conj_def, auto)
qed
qed
lemma fam_conj_disj_simp_pre:
assumes h1: "I = I1 + I2"
and h2: "I1 ## I2"
shows "fam_conj I cpt = (fam_conj I1 cpt \<and>* fam_conj I2 cpt)"
proof
fix s
let ?fm = "fam_conj I cpt" and ?fm1 = "fam_conj I1 cpt" and ?fm2 = "fam_conj I2 cpt"
show "?fm s = (?fm1 \<and>* ?fm2) s"
proof
assume "?fm s"
then obtain p where pre:
"s = (\<Union> i \<in> I. p(i))"
"(\<forall> i \<in> I. cpt i (p i))"
"(\<forall> i \<in> I. \<forall> j \<in> I. i \<noteq> j \<longrightarrow> p(i) ## p(j))"
unfolding fam_conj_def by metis
from pre(1) h1 h2 have "s = (\<Union> i \<in> I1. p(i)) + (\<Union> i \<in> I2. p(i))"
by (auto simp:set_ins_def)
moreover from pre h1 have "?fm1 (\<Union> i \<in> I1. p(i))"
by (unfold fam_conj_def, rule_tac x = p in exI, auto simp:set_ins_def)
moreover from pre h1 have "?fm2 (\<Union> i \<in> I2. p(i))"
by (unfold fam_conj_def, rule_tac x = p in exI, auto simp:set_ins_def)
moreover have "(\<Union> i \<in> I1. p(i)) ## (\<Union> i \<in> I2. p(i))"
proof -
{ fix x xa xb
assume h: "I1 \<inter> I2 = {}"
"\<forall>i\<in>I1 \<union> I2. \<forall>j\<in>I1 \<union> I2. i \<noteq> j \<longrightarrow> p i \<inter> p j = {}"
"xa \<in> I1" "x \<in> p xa" "xb \<in> I2" "x \<in> p xb"
have "p xa \<inter> p xb = {}"
proof(rule h(2)[rule_format])
from h(1, 3, 5) show "xa \<in> I1 \<union> I2" by auto
next
from h(1, 3, 5) show "xb \<in> I1 \<union> I2" by auto
next
from h(1, 3, 5) show "xa \<noteq> xb" by auto
qed with h(4, 6) have False by auto
} with h1 h2 pre(3) show ?thesis by (auto simp:set_ins_def)
qed
ultimately show "(?fm1 \<and>* ?fm2) s" by (auto intro!:sep_conjI)
next
assume "(?fm1 \<and>* ?fm2) s"
then obtain s1 s2 where pre:
"s = s1 + s2" "s1 ## s2" "?fm1 s1" "?fm2 s2"
by (auto dest!:sep_conjD)
from pre(3) obtain p1 where pre1:
"s1 = (\<Union> i \<in> I1. p1(i))"
"(\<forall> i \<in> I1. cpt i (p1 i))"
"(\<forall> i \<in> I1. \<forall> j \<in> I1. i \<noteq> j \<longrightarrow> p1(i) ## p1(j))"
unfolding fam_conj_def by metis
from pre(4) obtain p2 where pre2:
"s2 = (\<Union> i \<in> I2. p2(i))"
"(\<forall> i \<in> I2. cpt i (p2 i))"
"(\<forall> i \<in> I2. \<forall> j \<in> I2. i \<noteq> j \<longrightarrow> p2(i) ## p2(j))"
unfolding fam_conj_def by metis
let ?p = "\<lambda> i. if i \<in> I1 then p1 i else p2 i"
from h2 pre(2)
have "s = (\<Union> i \<in> I. ?p(i))"
apply (unfold h1 pre(1) pre1(1) pre2(1) set_ins_def, auto split:if_splits)
by (metis disjoint_iff_not_equal)
moreover from h2 pre1(2) pre2(2) have "(\<forall> i \<in> I. cpt i (?p i))"
by (unfold h1 set_ins_def, auto split:if_splits)
moreover from pre1(1, 3) pre2(1, 3) pre(2) h2
have "(\<forall> i \<in> I. \<forall> j \<in> I. i \<noteq> j \<longrightarrow> ?p(i) ## ?p(j))"
apply (unfold h1 set_ins_def, auto split:if_splits)
by (metis IntI empty_iff)
ultimately show "?fm s" by (unfold fam_conj_def, auto)
qed
qed
lemma fam_conj_disj_simp:
assumes h: "I1 ## I2"
shows "fam_conj (I1 + I2) cpt = (fam_conj I1 cpt \<and>* fam_conj I2 cpt)"
proof -
from fam_conj_disj_simp_pre[OF _ h, of "I1 + I2"]
show ?thesis by simp
qed
lemma fam_conj_elm_simp:
assumes h: "i \<in> I"
shows "fam_conj I cpt = (cpt(i) \<and>* fam_conj (I - {i}) cpt)"
proof
fix s
show "fam_conj I cpt s = (cpt i \<and>* fam_conj (I - {i}) cpt) s"
proof
assume pre: "fam_conj I cpt s"
show "(cpt i \<and>* fam_conj (I - {i}) cpt) s"
proof -
from pre obtain p where pres:
"s = (\<Union> i \<in> I. p(i))"
"(\<forall> i \<in> I. cpt i (p i))"
"(\<forall> i \<in> I. \<forall> j \<in> I. i \<noteq> j \<longrightarrow> p(i) ## p(j))"
unfolding fam_conj_def by metis
let ?s = "(\<Union>i\<in>(I - {i}). p i)"
from pres(3) h have disj: "(p i) ## ?s"
by (auto simp:set_ins_def)
moreover from pres(1) h have eq_s: "s = (p i) + ?s"
by (auto simp:set_ins_def)
moreover from pres(2) h have "cpt i (p i)" by auto
moreover from pres have "(fam_conj (I - {i}) cpt) ?s"
by (unfold fam_conj_def, rule_tac x = p in exI, auto)
ultimately show ?thesis by (auto intro!:sep_conjI)
qed
next
let ?fam = "fam_conj (I - {i}) cpt"
assume "(cpt i \<and>* ?fam) s"
then obtain s1 s2 where pre:
"s = s1 + s2" "s1 ## s2" "cpt i s1" "?fam s2"
by (auto dest!:sep_conjD)
from pre(4) obtain p where pres:
"s2 = (\<Union> ia \<in> I - {i}. p(ia))"
"(\<forall> ia \<in> I - {i}. cpt ia (p ia))"
"(\<forall> ia \<in> I - {i}. \<forall> j \<in> I - {i}. ia \<noteq> j \<longrightarrow> p(ia) ## p(j))"
unfolding fam_conj_def by metis
let ?p = "p(i:=s1)"
from h pres(3) pre(2)[unfolded pres(1)]
have " \<forall>ia\<in>I. \<forall>j\<in>I. ia \<noteq> j \<longrightarrow> ?p ia ## ?p j" by (auto simp:set_ins_def)
moreover from pres(1) pre(1) h pre(2)
have "(s = (\<Union> i \<in> I. ?p(i)))" by (auto simp:set_ins_def split:if_splits)
moreover from pre(3) pres(2) h
have "(\<forall> i \<in> I. cpt i (?p i))" by (auto simp:set_ins_def split:if_splits)
ultimately show "fam_conj I cpt s"
by (unfold fam_conj_def, auto)
qed
qed
lemma fam_conj_insert_simp:
assumes h:"i \<notin> I"
shows "fam_conj (insert i I) cpt = (cpt(i) \<and>* fam_conj I cpt)"
proof -
have "i \<in> insert i I" by auto
from fam_conj_elm_simp[OF this] and h
show ?thesis by auto
qed
lemmas fam_conj_simps = fam_conj_insert_simp fam_conj_zero_simp
lemma "fam_conj {0, 2, 3} cpt = (cpt 2 \<and>* cpt (0::int) \<and>* cpt 3)"
by (simp add:fam_conj_simps sep_conj_ac)
lemma fam_conj_ext_eq:
assumes h: "\<And> i . i \<in> I \<Longrightarrow> f i = g i"
shows "fam_conj I f = fam_conj I g"
proof
fix s
show "fam_conj I f s = fam_conj I g s"
by (unfold fam_conj_def, auto simp:h)
qed
end