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theory SemilatAlg = Typing_Framework + Product:(* Title: HOL/MicroJava/BV/SemilatAlg.thy
ID: $Id: SemilatAlg.html,v 1.1 2002/11/28 13:16:31 kleing Exp $
Author: Gerwin Klein
Copyright 2002 Technische Universitaet Muenchen
*)
header {* \isaheader{More on Semilattices} *}
theory SemilatAlg = Typing_Framework + Product:
constdefs
lesubstep_type :: "(nat × 's) list \<Rightarrow> 's ord \<Rightarrow> (nat × 's) list \<Rightarrow> bool"
("(_ /<=|_| _)" [50, 0, 51] 50)
"x <=|r| y \<equiv> \<forall>(p,s) \<in> set x. \<exists>s'. (p,s') \<in> set y \<and> s <=_r s'"
consts
"@plusplussub" :: "'a list \<Rightarrow> ('a \<Rightarrow> 'a \<Rightarrow> 'a) \<Rightarrow> 'a \<Rightarrow> 'a" ("(_ /++'__ _)" [65, 1000, 66] 65)
primrec
"[] ++_f y = y"
"(x#xs) ++_f y = xs ++_f (x +_f y)"
constdefs
bounded :: "'s step_type \<Rightarrow> nat \<Rightarrow> bool"
"bounded step n == !p<n. !s. !(q,t):set(step p s). q<n"
pres_type :: "'s step_type \<Rightarrow> nat \<Rightarrow> 's set \<Rightarrow> bool"
"pres_type step n A == \<forall>s\<in>A. \<forall>p<n. \<forall>(q,s')\<in>set (step p s). s' \<in> A"
mono :: "'s ord \<Rightarrow> 's step_type \<Rightarrow> nat \<Rightarrow> 's set \<Rightarrow> bool"
"mono r step n A ==
\<forall>s p t. s \<in> A \<and> p < n \<and> s <=_r t \<longrightarrow> step p s <=|r| step p t"
lemma pres_typeD:
"\<lbrakk> pres_type step n A; s\<in>A; p<n; (q,s')\<in>set (step p s) \<rbrakk> \<Longrightarrow> s' \<in> A"
by (unfold pres_type_def, blast)
lemma monoD:
"\<lbrakk> mono r step n A; p < n; s\<in>A; s <=_r t \<rbrakk> \<Longrightarrow> step p s <=|r| step p t"
by (unfold mono_def, blast)
lemma boundedD:
"\<lbrakk> bounded step n; p < n; (q,t) : set (step p xs) \<rbrakk> \<Longrightarrow> q < n"
by (unfold bounded_def, blast)
lemma lesubstep_type_refl [simp, intro]:
"(\<And>x. x <=_r x) \<Longrightarrow> x <=|r| x"
by (unfold lesubstep_type_def) auto
lemma lesub_step_typeD:
"a <=|r| b \<Longrightarrow> (x,y) \<in> set a \<Longrightarrow> \<exists>y'. (x, y') \<in> set b \<and> y <=_r y'"
by (unfold lesubstep_type_def) blast
lemma list_update_le_listI [rule_format]:
"set xs <= A \<longrightarrow> set ys <= A \<longrightarrow> xs <=[r] ys \<longrightarrow> p < size xs \<longrightarrow>
x <=_r ys!p \<longrightarrow> semilat(A,r,f) \<longrightarrow> x\<in>A \<longrightarrow>
xs[p := x +_f xs!p] <=[r] ys"
apply (unfold Listn.le_def lesub_def semilat_def)
apply (simp add: list_all2_conv_all_nth nth_list_update)
done
lemma plusplus_closed: includes semilat shows
"\<And>y. \<lbrakk> set x \<subseteq> A; y \<in> A\<rbrakk> \<Longrightarrow> x ++_f y \<in> A"
proof (induct x)
show "\<And>y. y \<in> A \<Longrightarrow> [] ++_f y \<in> A" by simp
fix y x xs
assume y: "y \<in> A" and xs: "set (x#xs) \<subseteq> A"
assume IH: "\<And>y. \<lbrakk> set xs \<subseteq> A; y \<in> A\<rbrakk> \<Longrightarrow> xs ++_f y \<in> A"
from xs obtain x: "x \<in> A" and "set xs \<subseteq> A" by simp
from x y have "(x +_f y) \<in> A" ..
with xs have "xs ++_f (x +_f y) \<in> A" by - (rule IH)
thus "(x#xs) ++_f y \<in> A" by simp
qed
lemma (in semilat) pp_ub2:
"\<And>y. \<lbrakk> set x \<subseteq> A; y \<in> A\<rbrakk> \<Longrightarrow> y <=_r x ++_f y"
proof (induct x)
from semilat show "\<And>y. y <=_r [] ++_f y" by simp
fix y a l
assume y: "y \<in> A"
assume "set (a#l) \<subseteq> A"
then obtain a: "a \<in> A" and x: "set l \<subseteq> A" by simp
assume "\<And>y. \<lbrakk>set l \<subseteq> A; y \<in> A\<rbrakk> \<Longrightarrow> y <=_r l ++_f y"
hence IH: "\<And>y. y \<in> A \<Longrightarrow> y <=_r l ++_f y" .
from a y have "y <=_r a +_f y" ..
also from a y have "a +_f y \<in> A" ..
hence "(a +_f y) <=_r l ++_f (a +_f y)" by (rule IH)
finally have "y <=_r l ++_f (a +_f y)" .
thus "y <=_r (a#l) ++_f y" by simp
qed
lemma (in semilat) pp_ub1:
shows "\<And>y. \<lbrakk>set ls \<subseteq> A; y \<in> A; x \<in> set ls\<rbrakk> \<Longrightarrow> x <=_r ls ++_f y"
proof (induct ls)
show "\<And>y. x \<in> set [] \<Longrightarrow> x <=_r [] ++_f y" by simp
fix y s ls
assume "set (s#ls) \<subseteq> A"
then obtain s: "s \<in> A" and ls: "set ls \<subseteq> A" by simp
assume y: "y \<in> A"
assume
"\<And>y. \<lbrakk>set ls \<subseteq> A; y \<in> A; x \<in> set ls\<rbrakk> \<Longrightarrow> x <=_r ls ++_f y"
hence IH: "\<And>y. x \<in> set ls \<Longrightarrow> y \<in> A \<Longrightarrow> x <=_r ls ++_f y" .
assume "x \<in> set (s#ls)"
then obtain xls: "x = s \<or> x \<in> set ls" by simp
moreover {
assume xs: "x = s"
from s y have "s <=_r s +_f y" ..
also from s y have "s +_f y \<in> A" ..
with ls have "(s +_f y) <=_r ls ++_f (s +_f y)" by (rule pp_ub2)
finally have "s <=_r ls ++_f (s +_f y)" .
with xs have "x <=_r ls ++_f (s +_f y)" by simp
}
moreover {
assume "x \<in> set ls"
hence "\<And>y. y \<in> A \<Longrightarrow> x <=_r ls ++_f y" by (rule IH)
moreover from s y have "s +_f y \<in> A" ..
ultimately have "x <=_r ls ++_f (s +_f y)" .
}
ultimately
have "x <=_r ls ++_f (s +_f y)" by blast
thus "x <=_r (s#ls) ++_f y" by simp
qed
lemma (in semilat) pp_lub:
assumes "z \<in> A"
shows
"\<And>y. y \<in> A \<Longrightarrow> set xs \<subseteq> A \<Longrightarrow> \<forall>x \<in> set xs. x <=_r z \<Longrightarrow> y <=_r z \<Longrightarrow> xs ++_f y <=_r z"
proof (induct xs)
fix y assume "y <=_r z" thus "[] ++_f y <=_r z" by simp
next
fix y l ls assume y: "y \<in> A" and "set (l#ls) \<subseteq> A"
then obtain l: "l \<in> A" and ls: "set ls \<subseteq> A" by auto
assume "\<forall>x \<in> set (l#ls). x <=_r z"
then obtain "l <=_r z" and lsz: "\<forall>x \<in> set ls. x <=_r z" by auto
assume "y <=_r z" have "l +_f y <=_r z" ..
moreover
from l y have "l +_f y \<in> A" ..
moreover
assume "\<And>y. y \<in> A \<Longrightarrow> set ls \<subseteq> A \<Longrightarrow> \<forall>x \<in> set ls. x <=_r z \<Longrightarrow> y <=_r z
\<Longrightarrow> ls ++_f y <=_r z"
ultimately
have "ls ++_f (l +_f y) <=_r z" using ls lsz by -
thus "(l#ls) ++_f y <=_r z" by simp
qed
lemma ub1': includes semilat
shows "\<lbrakk>\<forall>(p,s) \<in> set S. s \<in> A; y \<in> A; (a,b) \<in> set S\<rbrakk>
\<Longrightarrow> b <=_r map snd [(p', t')\<in>S. p' = a] ++_f y"
proof -
let "b <=_r ?map ++_f y" = ?thesis
assume "y \<in> A"
moreover
assume "\<forall>(p,s) \<in> set S. s \<in> A"
hence "set ?map \<subseteq> A" by auto
moreover
assume "(a,b) \<in> set S"
hence "b \<in> set ?map" by (induct S, auto)
ultimately
show ?thesis by - (rule pp_ub1)
qed
lemma plusplus_empty:
"\<forall>s'. (q, s') \<in> set S \<longrightarrow> s' +_f ss ! q = ss ! q \<Longrightarrow>
(map snd [(p', t')\<in> S. p' = q] ++_f ss ! q) = ss ! q"
apply (induct S)
apply auto
done
end
lemma pres_typeD:
[| pres_type step n A; s : A; p < n; (q, s') : set (step p s) |] ==> s' : A
lemma monoD:
[| SemilatAlg.mono r step n A; p < n; s : A; s <=_r t |] ==> step p s <=|r| step p t
lemma boundedD:
[| bounded step n; p < n; (q, t) : set (step p xs) |] ==> q < n
lemma lesubstep_type_refl:
(!!x. x <=_r x) ==> x <=|r| x
lemma lesub_step_typeD:
[| a <=|r| b; (x, y) : set a |] ==> EX y'. (x, y') : set b & y <=_r y'
lemma list_update_le_listI:
[| set xs <= A; set ys <= A; xs <=[r] ys; p < length xs; x <=_r ys ! p;
semilat (A, r, f); x : A |]
==> xs[p := x +_f xs ! p] <=[r] ys
lemma
[| semilat (A, r, f); set x <= A; y : A |] ==> x ++_f y : A
lemma pp_ub2:
[| semilat (A, r, f); set x <= A; y : A |] ==> y <=_r x ++_f y
lemma
[| semilat (A, r, f); set ls <= A; y : A; x : set ls |] ==> x <=_r ls ++_f y
lemma
[| semilat (A, r, f); z : A; y : A; set xs <= A; ALL x:set xs. x <=_r z;
y <=_r z |]
==> xs ++_f y <=_r z
lemma
[| semilat (A, r, f); ALL (p, s):set S. s : A; y : A; (a, b) : set S |] ==> b <=_r map snd [(p', t'):S. p' = a] ++_f y
lemma plusplus_empty:
ALL s'. (q, s') : set S --> s' +_f ss ! q = ss ! q ==> map snd [(p', t'):S. p' = q] ++_f ss ! q = ss ! q