explicit checks stable_finished_theory/stable_command allow parallel asynchronous command transactions;
tuned;
(* Title: HOL/MicroJava/DFA/Semilat.thy
Author: Tobias Nipkow
Copyright 2000 TUM
*)
header {*
\chapter{Bytecode Verifier}\label{cha:bv}
\isaheader{Semilattices}
*}
theory Semilat
imports Main "~~/src/HOL/Library/While_Combinator"
begin
type_synonym 'a ord = "'a \<Rightarrow> 'a \<Rightarrow> bool"
type_synonym 'a binop = "'a \<Rightarrow> 'a \<Rightarrow> 'a"
type_synonym 'a sl = "'a set \<times> 'a ord \<times> 'a binop"
consts
"lesub" :: "'a \<Rightarrow> 'a ord \<Rightarrow> 'a \<Rightarrow> bool"
"lesssub" :: "'a \<Rightarrow> 'a ord \<Rightarrow> 'a \<Rightarrow> bool"
"plussub" :: "'a \<Rightarrow> ('a \<Rightarrow> 'b \<Rightarrow> 'c) \<Rightarrow> 'b \<Rightarrow> 'c"
(*<*)
notation
"lesub" ("(_ /<='__ _)" [50, 1000, 51] 50) and
"lesssub" ("(_ /<'__ _)" [50, 1000, 51] 50) and
"plussub" ("(_ /+'__ _)" [65, 1000, 66] 65)
(*>*)
notation (xsymbols)
"lesub" ("(_ /\<sqsubseteq>\<^bsub>_\<^esub> _)" [50, 0, 51] 50) and
"lesssub" ("(_ /\<sqsubset>\<^bsub>_\<^esub> _)" [50, 0, 51] 50) and
"plussub" ("(_ /\<squnion>\<^bsub>_\<^esub> _)" [65, 0, 66] 65)
(*<*)
(* allow \<sub> instead of \<bsub>..\<esub> *)
abbreviation (input)
lesub1 :: "'a \<Rightarrow> 'a ord \<Rightarrow> 'a \<Rightarrow> bool" ("(_ /\<sqsubseteq>\<^sub>_ _)" [50, 1000, 51] 50)
where "x \<sqsubseteq>\<^sub>r y == x \<sqsubseteq>\<^bsub>r\<^esub> y"
abbreviation (input)
lesssub1 :: "'a \<Rightarrow> 'a ord \<Rightarrow> 'a \<Rightarrow> bool" ("(_ /\<sqsubset>\<^sub>_ _)" [50, 1000, 51] 50)
where "x \<sqsubset>\<^sub>r y == x \<sqsubset>\<^bsub>r\<^esub> y"
abbreviation (input)
plussub1 :: "'a \<Rightarrow> ('a \<Rightarrow> 'b \<Rightarrow> 'c) \<Rightarrow> 'b \<Rightarrow> 'c" ("(_ /\<squnion>\<^sub>_ _)" [65, 1000, 66] 65)
where "x \<squnion>\<^sub>f y == x \<squnion>\<^bsub>f\<^esub> y"
(*>*)
defs
lesub_def: "x \<sqsubseteq>\<^sub>r y \<equiv> r x y"
lesssub_def: "x \<sqsubset>\<^sub>r y \<equiv> x \<sqsubseteq>\<^sub>r y \<and> x \<noteq> y"
plussub_def: "x \<squnion>\<^sub>f y \<equiv> f x y"
definition ord :: "('a \<times> 'a) set \<Rightarrow> 'a ord" where
"ord r \<equiv> \<lambda>x y. (x,y) \<in> r"
definition order :: "'a ord \<Rightarrow> bool" where
"order r \<equiv> (\<forall>x. x \<sqsubseteq>\<^sub>r x) \<and> (\<forall>x y. x \<sqsubseteq>\<^sub>r y \<and> y \<sqsubseteq>\<^sub>r x \<longrightarrow> x=y) \<and> (\<forall>x y z. x \<sqsubseteq>\<^sub>r y \<and> y \<sqsubseteq>\<^sub>r z \<longrightarrow> x \<sqsubseteq>\<^sub>r z)"
definition top :: "'a ord \<Rightarrow> 'a \<Rightarrow> bool" where
"top r T \<equiv> \<forall>x. x \<sqsubseteq>\<^sub>r T"
definition acc :: "'a ord \<Rightarrow> bool" where
"acc r \<equiv> wf {(y,x). x \<sqsubset>\<^sub>r y}"
definition closed :: "'a set \<Rightarrow> 'a binop \<Rightarrow> bool" where
"closed A f \<equiv> \<forall>x\<in>A. \<forall>y\<in>A. x \<squnion>\<^sub>f y \<in> A"
definition semilat :: "'a sl \<Rightarrow> bool" where
"semilat \<equiv> \<lambda>(A,r,f). order r \<and> closed A f \<and>
(\<forall>x\<in>A. \<forall>y\<in>A. x \<sqsubseteq>\<^sub>r x \<squnion>\<^sub>f y) \<and>
(\<forall>x\<in>A. \<forall>y\<in>A. y \<sqsubseteq>\<^sub>r x \<squnion>\<^sub>f y) \<and>
(\<forall>x\<in>A. \<forall>y\<in>A. \<forall>z\<in>A. x \<sqsubseteq>\<^sub>r z \<and> y \<sqsubseteq>\<^sub>r z \<longrightarrow> x \<squnion>\<^sub>f y \<sqsubseteq>\<^sub>r z)"
definition is_ub :: "('a \<times> 'a) set \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool" where
"is_ub r x y u \<equiv> (x,u)\<in>r \<and> (y,u)\<in>r"
definition is_lub :: "('a \<times> 'a) set \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool" where
"is_lub r x y u \<equiv> is_ub r x y u \<and> (\<forall>z. is_ub r x y z \<longrightarrow> (u,z)\<in>r)"
definition some_lub :: "('a \<times> 'a) set \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a" where
"some_lub r x y \<equiv> SOME z. is_lub r x y z"
locale Semilat =
fixes A :: "'a set"
fixes r :: "'a ord"
fixes f :: "'a binop"
assumes semilat: "semilat (A, r, f)"
lemma order_refl [simp, intro]: "order r \<Longrightarrow> x \<sqsubseteq>\<^sub>r x"
(*<*) by (unfold order_def) (simp (no_asm_simp)) (*>*)
lemma order_antisym: "\<lbrakk> order r; x \<sqsubseteq>\<^sub>r y; y \<sqsubseteq>\<^sub>r x \<rbrakk> \<Longrightarrow> x = y"
(*<*) by (unfold order_def) (simp (no_asm_simp)) (*>*)
lemma order_trans: "\<lbrakk> order r; x \<sqsubseteq>\<^sub>r y; y \<sqsubseteq>\<^sub>r z \<rbrakk> \<Longrightarrow> x \<sqsubseteq>\<^sub>r z"
(*<*) by (unfold order_def) blast (*>*)
lemma order_less_irrefl [intro, simp]: "order r \<Longrightarrow> \<not> x \<sqsubset>\<^sub>r x"
(*<*) by (unfold order_def lesssub_def) blast (*>*)
lemma order_less_trans: "\<lbrakk> order r; x \<sqsubset>\<^sub>r y; y \<sqsubset>\<^sub>r z \<rbrakk> \<Longrightarrow> x \<sqsubset>\<^sub>r z"
(*<*) by (unfold order_def lesssub_def) blast (*>*)
lemma topD [simp, intro]: "top r T \<Longrightarrow> x \<sqsubseteq>\<^sub>r T"
(*<*) by (simp add: top_def) (*>*)
lemma top_le_conv [simp]: "\<lbrakk> order r; top r T \<rbrakk> \<Longrightarrow> (T \<sqsubseteq>\<^sub>r x) = (x = T)"
(*<*) by (blast intro: order_antisym) (*>*)
lemma semilat_Def:
"semilat(A,r,f) \<equiv> order r \<and> closed A f \<and>
(\<forall>x\<in>A. \<forall>y\<in>A. x \<sqsubseteq>\<^sub>r x \<squnion>\<^sub>f y) \<and>
(\<forall>x\<in>A. \<forall>y\<in>A. y \<sqsubseteq>\<^sub>r x \<squnion>\<^sub>f y) \<and>
(\<forall>x\<in>A. \<forall>y\<in>A. \<forall>z\<in>A. x \<sqsubseteq>\<^sub>r z \<and> y \<sqsubseteq>\<^sub>r z \<longrightarrow> x \<squnion>\<^sub>f y \<sqsubseteq>\<^sub>r z)"
(*<*) by (unfold semilat_def) clarsimp (*>*)
lemma (in Semilat) orderI [simp, intro]: "order r"
(*<*) using semilat by (simp add: semilat_Def) (*>*)
lemma (in Semilat) closedI [simp, intro]: "closed A f"
(*<*) using semilat by (simp add: semilat_Def) (*>*)
lemma closedD: "\<lbrakk> closed A f; x\<in>A; y\<in>A \<rbrakk> \<Longrightarrow> x \<squnion>\<^sub>f y \<in> A"
(*<*) by (unfold closed_def) blast (*>*)
lemma closed_UNIV [simp]: "closed UNIV f"
(*<*) by (simp add: closed_def) (*>*)
lemma (in Semilat) closed_f [simp, intro]: "\<lbrakk>x \<in> A; y \<in> A\<rbrakk> \<Longrightarrow> x \<squnion>\<^sub>f y \<in> A"
(*<*) by (simp add: closedD [OF closedI]) (*>*)
lemma (in Semilat) refl_r [intro, simp]: "x \<sqsubseteq>\<^sub>r x" by simp
lemma (in Semilat) antisym_r [intro?]: "\<lbrakk> x \<sqsubseteq>\<^sub>r y; y \<sqsubseteq>\<^sub>r x \<rbrakk> \<Longrightarrow> x = y"
(*<*) by (rule order_antisym) auto (*>*)
lemma (in Semilat) trans_r [trans, intro?]: "\<lbrakk>x \<sqsubseteq>\<^sub>r y; y \<sqsubseteq>\<^sub>r z\<rbrakk> \<Longrightarrow> x \<sqsubseteq>\<^sub>r z"
(*<*) by (auto intro: order_trans) (*>*)
lemma (in Semilat) ub1 [simp, intro?]: "\<lbrakk> x \<in> A; y \<in> A \<rbrakk> \<Longrightarrow> x \<sqsubseteq>\<^sub>r x \<squnion>\<^sub>f y"
(*<*) by (insert semilat) (unfold semilat_Def, simp) (*>*)
lemma (in Semilat) ub2 [simp, intro?]: "\<lbrakk> x \<in> A; y \<in> A \<rbrakk> \<Longrightarrow> y \<sqsubseteq>\<^sub>r x \<squnion>\<^sub>f y"
(*<*) by (insert semilat) (unfold semilat_Def, simp) (*>*)
lemma (in Semilat) lub [simp, intro?]:
"\<lbrakk> x \<sqsubseteq>\<^sub>r z; y \<sqsubseteq>\<^sub>r z; x \<in> A; y \<in> A; z \<in> A \<rbrakk> \<Longrightarrow> x \<squnion>\<^sub>f y \<sqsubseteq>\<^sub>r z";
(*<*) by (insert semilat) (unfold semilat_Def, simp) (*>*)
lemma (in Semilat) plus_le_conv [simp]:
"\<lbrakk> x \<in> A; y \<in> A; z \<in> A \<rbrakk> \<Longrightarrow> (x \<squnion>\<^sub>f y \<sqsubseteq>\<^sub>r z) = (x \<sqsubseteq>\<^sub>r z \<and> y \<sqsubseteq>\<^sub>r z)"
(*<*) by (blast intro: ub1 ub2 lub order_trans) (*>*)
lemma (in Semilat) le_iff_plus_unchanged: "\<lbrakk> x \<in> A; y \<in> A \<rbrakk> \<Longrightarrow> (x \<sqsubseteq>\<^sub>r y) = (x \<squnion>\<^sub>f y = y)"
(*<*)
apply (rule iffI)
apply (blast intro: antisym_r lub ub2)
apply (erule subst)
apply simp
done
(*>*)
lemma (in Semilat) le_iff_plus_unchanged2: "\<lbrakk> x \<in> A; y \<in> A \<rbrakk> \<Longrightarrow> (x \<sqsubseteq>\<^sub>r y) = (y \<squnion>\<^sub>f x = y)"
(*<*)
apply (rule iffI)
apply (blast intro: order_antisym lub ub1)
apply (erule subst)
apply simp
done
(*>*)
lemma (in Semilat) plus_assoc [simp]:
assumes a: "a \<in> A" and b: "b \<in> A" and c: "c \<in> A"
shows "a \<squnion>\<^sub>f (b \<squnion>\<^sub>f c) = a \<squnion>\<^sub>f b \<squnion>\<^sub>f c"
(*<*)
proof -
from a b have ab: "a \<squnion>\<^sub>f b \<in> A" ..
from this c have abc: "(a \<squnion>\<^sub>f b) \<squnion>\<^sub>f c \<in> A" ..
from b c have bc: "b \<squnion>\<^sub>f c \<in> A" ..
from a this have abc': "a \<squnion>\<^sub>f (b \<squnion>\<^sub>f c) \<in> A" ..
show ?thesis
proof
show "a \<squnion>\<^sub>f (b \<squnion>\<^sub>f c) \<sqsubseteq>\<^sub>r (a \<squnion>\<^sub>f b) \<squnion>\<^sub>f c"
proof -
from a b have "a \<sqsubseteq>\<^sub>r a \<squnion>\<^sub>f b" ..
also from ab c have "\<dots> \<sqsubseteq>\<^sub>r \<dots> \<squnion>\<^sub>f c" ..
finally have "a<": "a \<sqsubseteq>\<^sub>r (a \<squnion>\<^sub>f b) \<squnion>\<^sub>f c" .
from a b have "b \<sqsubseteq>\<^sub>r a \<squnion>\<^sub>f b" ..
also from ab c have "\<dots> \<sqsubseteq>\<^sub>r \<dots> \<squnion>\<^sub>f c" ..
finally have "b<": "b \<sqsubseteq>\<^sub>r (a \<squnion>\<^sub>f b) \<squnion>\<^sub>f c" .
from ab c have "c<": "c \<sqsubseteq>\<^sub>r (a \<squnion>\<^sub>f b) \<squnion>\<^sub>f c" ..
from "b<" "c<" b c abc have "b \<squnion>\<^sub>f c \<sqsubseteq>\<^sub>r (a \<squnion>\<^sub>f b) \<squnion>\<^sub>f c" ..
from "a<" this a bc abc show ?thesis ..
qed
show "(a \<squnion>\<^sub>f b) \<squnion>\<^sub>f c \<sqsubseteq>\<^sub>r a \<squnion>\<^sub>f (b \<squnion>\<^sub>f c)"
proof -
from b c have "b \<sqsubseteq>\<^sub>r b \<squnion>\<^sub>f c" ..
also from a bc have "\<dots> \<sqsubseteq>\<^sub>r a \<squnion>\<^sub>f \<dots>" ..
finally have "b<": "b \<sqsubseteq>\<^sub>r a \<squnion>\<^sub>f (b \<squnion>\<^sub>f c)" .
from b c have "c \<sqsubseteq>\<^sub>r b \<squnion>\<^sub>f c" ..
also from a bc have "\<dots> \<sqsubseteq>\<^sub>r a \<squnion>\<^sub>f \<dots>" ..
finally have "c<": "c \<sqsubseteq>\<^sub>r a \<squnion>\<^sub>f (b \<squnion>\<^sub>f c)" .
from a bc have "a<": "a \<sqsubseteq>\<^sub>r a \<squnion>\<^sub>f (b \<squnion>\<^sub>f c)" ..
from "a<" "b<" a b abc' have "a \<squnion>\<^sub>f b \<sqsubseteq>\<^sub>r a \<squnion>\<^sub>f (b \<squnion>\<^sub>f c)" ..
from this "c<" ab c abc' show ?thesis ..
qed
qed
qed
(*>*)
lemma (in Semilat) plus_com_lemma:
"\<lbrakk>a \<in> A; b \<in> A\<rbrakk> \<Longrightarrow> a \<squnion>\<^sub>f b \<sqsubseteq>\<^sub>r b \<squnion>\<^sub>f a"
(*<*)
proof -
assume a: "a \<in> A" and b: "b \<in> A"
from b a have "a \<sqsubseteq>\<^sub>r b \<squnion>\<^sub>f a" ..
moreover from b a have "b \<sqsubseteq>\<^sub>r b \<squnion>\<^sub>f a" ..
moreover note a b
moreover from b a have "b \<squnion>\<^sub>f a \<in> A" ..
ultimately show ?thesis ..
qed
(*>*)
lemma (in Semilat) plus_commutative:
"\<lbrakk>a \<in> A; b \<in> A\<rbrakk> \<Longrightarrow> a \<squnion>\<^sub>f b = b \<squnion>\<^sub>f a"
(*<*) by(blast intro: order_antisym plus_com_lemma) (*>*)
lemma is_lubD:
"is_lub r x y u \<Longrightarrow> is_ub r x y u \<and> (\<forall>z. is_ub r x y z \<longrightarrow> (u,z) \<in> r)"
(*<*) by (simp add: is_lub_def) (*>*)
lemma is_ubI:
"\<lbrakk> (x,u) \<in> r; (y,u) \<in> r \<rbrakk> \<Longrightarrow> is_ub r x y u"
(*<*) by (simp add: is_ub_def) (*>*)
lemma is_ubD:
"is_ub r x y u \<Longrightarrow> (x,u) \<in> r \<and> (y,u) \<in> r"
(*<*) by (simp add: is_ub_def) (*>*)
lemma is_lub_bigger1 [iff]:
"is_lub (r^* ) x y y = ((x,y)\<in>r^* )"
(*<*)
apply (unfold is_lub_def is_ub_def)
apply blast
done
(*>*)
lemma is_lub_bigger2 [iff]:
"is_lub (r^* ) x y x = ((y,x)\<in>r^* )"
(*<*)
apply (unfold is_lub_def is_ub_def)
apply blast
done
(*>*)
lemma extend_lub:
"\<lbrakk> single_valued r; is_lub (r^* ) x y u; (x',x) \<in> r \<rbrakk>
\<Longrightarrow> EX v. is_lub (r^* ) x' y v"
(*<*)
apply (unfold is_lub_def is_ub_def)
apply (case_tac "(y,x) \<in> r^*")
apply (case_tac "(y,x') \<in> r^*")
apply blast
apply (blast elim: converse_rtranclE dest: single_valuedD)
apply (rule exI)
apply (rule conjI)
apply (blast intro: converse_rtrancl_into_rtrancl dest: single_valuedD)
apply (blast intro: rtrancl_into_rtrancl converse_rtrancl_into_rtrancl
elim: converse_rtranclE dest: single_valuedD)
done
(*>*)
lemma single_valued_has_lubs [rule_format]:
"\<lbrakk> single_valued r; (x,u) \<in> r^* \<rbrakk> \<Longrightarrow> (\<forall>y. (y,u) \<in> r^* \<longrightarrow>
(EX z. is_lub (r^* ) x y z))"
(*<*)
apply (erule converse_rtrancl_induct)
apply clarify
apply (erule converse_rtrancl_induct)
apply blast
apply (blast intro: converse_rtrancl_into_rtrancl)
apply (blast intro: extend_lub)
done
(*>*)
lemma some_lub_conv:
"\<lbrakk> acyclic r; is_lub (r^* ) x y u \<rbrakk> \<Longrightarrow> some_lub (r^* ) x y = u"
(*<*)
apply (unfold some_lub_def is_lub_def)
apply (rule someI2)
apply assumption
apply (blast intro: antisymD dest!: acyclic_impl_antisym_rtrancl)
done
(*>*)
lemma is_lub_some_lub:
"\<lbrakk> single_valued r; acyclic r; (x,u)\<in>r^*; (y,u)\<in>r^* \<rbrakk>
\<Longrightarrow> is_lub (r^* ) x y (some_lub (r^* ) x y)";
(*<*) by (fastforce dest: single_valued_has_lubs simp add: some_lub_conv) (*>*)
subsection{*An executable lub-finder*}
definition exec_lub :: "('a * 'a) set \<Rightarrow> ('a \<Rightarrow> 'a) \<Rightarrow> 'a binop" where
"exec_lub r f x y \<equiv> while (\<lambda>z. (x,z) \<notin> r\<^sup>*) f y"
lemma exec_lub_refl: "exec_lub r f T T = T"
by (simp add: exec_lub_def while_unfold)
lemma acyclic_single_valued_finite:
"\<lbrakk>acyclic r; single_valued r; (x,y) \<in> r\<^sup>*\<rbrakk>
\<Longrightarrow> finite (r \<inter> {a. (x, a) \<in> r\<^sup>*} \<times> {b. (b, y) \<in> r\<^sup>*})"
(*<*)
apply(erule converse_rtrancl_induct)
apply(rule_tac B = "{}" in finite_subset)
apply(simp only:acyclic_def)
apply(blast intro:rtrancl_into_trancl2 rtrancl_trancl_trancl)
apply simp
apply(rename_tac x x')
apply(subgoal_tac "r \<inter> {a. (x,a) \<in> r\<^sup>*} \<times> {b. (b,y) \<in> r\<^sup>*} =
insert (x,x') (r \<inter> {a. (x', a) \<in> r\<^sup>*} \<times> {b. (b, y) \<in> r\<^sup>*})")
apply simp
apply(blast intro:converse_rtrancl_into_rtrancl
elim:converse_rtranclE dest:single_valuedD)
done
(*>*)
lemma exec_lub_conv:
"\<lbrakk> acyclic r; \<forall>x y. (x,y) \<in> r \<longrightarrow> f x = y; is_lub (r\<^sup>*) x y u \<rbrakk> \<Longrightarrow>
exec_lub r f x y = u";
(*<*)
apply(unfold exec_lub_def)
apply(rule_tac P = "\<lambda>z. (y,z) \<in> r\<^sup>* \<and> (z,u) \<in> r\<^sup>*" and
r = "(r \<inter> {(a,b). (y,a) \<in> r\<^sup>* \<and> (b,u) \<in> r\<^sup>*})^-1" in while_rule)
apply(blast dest: is_lubD is_ubD)
apply(erule conjE)
apply(erule_tac z = u in converse_rtranclE)
apply(blast dest: is_lubD is_ubD)
apply(blast dest:rtrancl_into_rtrancl)
apply(rename_tac s)
apply(subgoal_tac "is_ub (r\<^sup>*) x y s")
prefer 2; apply(simp add:is_ub_def)
apply(subgoal_tac "(u, s) \<in> r\<^sup>*")
prefer 2; apply(blast dest:is_lubD)
apply(erule converse_rtranclE)
apply blast
apply(simp only:acyclic_def)
apply(blast intro:rtrancl_into_trancl2 rtrancl_trancl_trancl)
apply(rule finite_acyclic_wf)
apply simp
apply(erule acyclic_single_valued_finite)
apply(blast intro:single_valuedI)
apply(simp add:is_lub_def is_ub_def)
apply simp
apply(erule acyclic_subset)
apply blast
apply simp
apply(erule conjE)
apply(erule_tac z = u in converse_rtranclE)
apply(blast dest: is_lubD is_ubD)
apply(blast dest:rtrancl_into_rtrancl)
done
(*>*)
lemma is_lub_exec_lub:
"\<lbrakk> single_valued r; acyclic r; (x,u):r^*; (y,u):r^*; \<forall>x y. (x,y) \<in> r \<longrightarrow> f x = y \<rbrakk>
\<Longrightarrow> is_lub (r^* ) x y (exec_lub r f x y)"
(*<*) by (fastforce dest: single_valued_has_lubs simp add: exec_lub_conv) (*>*)
end