(* Title: HOL/LList.thy
ID: $Id$
Author: Lawrence C Paulson, Cambridge University Computer Laboratory
Copyright 1997 University of Cambridge
*)
header {*The "filter" functional for coinductive lists
--defined by a combination of induction and coinduction*}
theory LFilter = LList:
consts
findRel :: "('a => bool) => ('a llist * 'a llist)set"
inductive "findRel p"
intros
found: "p x ==> (LCons x l, LCons x l) \<in> findRel p"
seek: "[| ~p x; (l,l') \<in> findRel p |] ==> (LCons x l, l') \<in> findRel p"
declare findRel.intros [intro]
constdefs
find :: "['a => bool, 'a llist] => 'a llist"
"find p l == @l'. (l,l'): findRel p | (l' = LNil & l ~: Domain(findRel p))"
lfilter :: "['a => bool, 'a llist] => 'a llist"
"lfilter p l == llist_corec l (%l. case find p l of
LNil => None
| LCons y z => Some(y,z))"
subsection {* @{text findRel}: basic laws *}
inductive_cases
findRel_LConsE [elim!]: "(LCons x l, l'') \<in> findRel p"
lemma findRel_functional [rule_format]:
"(l,l'): findRel p ==> (l,l''): findRel p --> l'' = l'"
by (erule findRel.induct, auto)
lemma findRel_imp_LCons [rule_format]:
"(l,l'): findRel p ==> \<exists>x l''. l' = LCons x l'' & p x"
by (erule findRel.induct, auto)
lemma findRel_LNil [elim!]: "(LNil,l): findRel p ==> R"
by (blast elim: findRel.cases)
subsection {* Properties of @{text "Domain (findRel p)"} *}
lemma LCons_Domain_findRel [simp]:
"LCons x l \<in> Domain(findRel p) = (p x | l \<in> Domain(findRel p))"
by auto
lemma Domain_findRel_iff:
"(l \<in> Domain (findRel p)) = (\<exists>x l'. (l, LCons x l') \<in> findRel p & p x)"
by (blast dest: findRel_imp_LCons)
lemma Domain_findRel_mono:
"[| !!x. p x ==> q x |] ==> Domain (findRel p) <= Domain (findRel q)"
apply clarify
apply (erule findRel.induct, blast+)
done
subsection {* @{text find}: basic equations *}
lemma find_LNil [simp]: "find p LNil = LNil"
by (unfold find_def, blast)
lemma findRel_imp_find [simp]: "(l,l') \<in> findRel p ==> find p l = l'"
apply (unfold find_def)
apply (blast dest: findRel_functional)
done
lemma find_LCons_found: "p x ==> find p (LCons x l) = LCons x l"
by (blast intro: findRel_imp_find)
lemma diverge_find_LNil [simp]: "l ~: Domain(findRel p) ==> find p l = LNil"
by (unfold find_def, blast)
lemma find_LCons_seek: "~ (p x) ==> find p (LCons x l) = find p l"
apply (case_tac "LCons x l \<in> Domain (findRel p) ")
apply auto
apply (blast intro: findRel_imp_find)
done
lemma find_LCons [simp]:
"find p (LCons x l) = (if p x then LCons x l else find p l)"
by (simp add: find_LCons_seek find_LCons_found)
subsection {* @{text lfilter}: basic equations *}
lemma lfilter_LNil [simp]: "lfilter p LNil = LNil"
by (rule lfilter_def [THEN def_llist_corec, THEN trans], simp)
lemma diverge_lfilter_LNil [simp]:
"l ~: Domain(findRel p) ==> lfilter p l = LNil"
by (rule lfilter_def [THEN def_llist_corec, THEN trans], simp)
lemma lfilter_LCons_found:
"p x ==> lfilter p (LCons x l) = LCons x (lfilter p l)"
by (rule lfilter_def [THEN def_llist_corec, THEN trans], simp)
lemma findRel_imp_lfilter [simp]:
"(l, LCons x l') \<in> findRel p ==> lfilter p l = LCons x (lfilter p l')"
by (rule lfilter_def [THEN def_llist_corec, THEN trans], simp)
lemma lfilter_LCons_seek: "~ (p x) ==> lfilter p (LCons x l) = lfilter p l"
apply (rule lfilter_def [THEN def_llist_corec, THEN trans], simp)
apply (case_tac "LCons x l \<in> Domain (findRel p) ")
apply (simp add: Domain_findRel_iff, auto)
done
lemma lfilter_LCons [simp]:
"lfilter p (LCons x l) =
(if p x then LCons x (lfilter p l) else lfilter p l)"
by (simp add: lfilter_LCons_found lfilter_LCons_seek)
declare llistD_Fun_LNil_I [intro!] llistD_Fun_LCons_I [intro!]
lemma lfilter_eq_LNil: "lfilter p l = LNil ==> l ~: Domain(findRel p)"
apply (auto iff: Domain_findRel_iff)
done
lemma lfilter_eq_LCons [rule_format]:
"lfilter p l = LCons x l' -->
(\<exists>l''. l' = lfilter p l'' & (l, LCons x l'') \<in> findRel p)"
apply (subst lfilter_def [THEN def_llist_corec])
apply (case_tac "l \<in> Domain (findRel p) ")
apply (auto iff: Domain_findRel_iff)
done
lemma lfilter_cases: "lfilter p l = LNil |
(\<exists>y l'. lfilter p l = LCons y (lfilter p l') & p y)"
apply (case_tac "l \<in> Domain (findRel p) ")
apply (auto iff: Domain_findRel_iff)
done
subsection {* @{text lfilter}: simple facts by coinduction *}
lemma lfilter_K_True: "lfilter (%x. True) l = l"
by (rule_tac l = "l" in llist_fun_equalityI, simp_all)
lemma lfilter_idem: "lfilter p (lfilter p l) = lfilter p l"
apply (rule_tac l = "l" in llist_fun_equalityI, simp_all)
apply safe
txt{*Cases: @{text "p x"} is true or false*}
apply (rule lfilter_cases [THEN disjE])
apply (erule ssubst, auto)
done
subsection {* Numerous lemmas required to prove @{text lfilter_conj} *}
lemma findRel_conj_lemma [rule_format]:
"(l,l') \<in> findRel q
==> l' = LCons x l'' --> p x --> (l,l') \<in> findRel (%x. p x & q x)"
by (erule findRel.induct, auto)
lemmas findRel_conj = findRel_conj_lemma [OF _ refl]
lemma findRel_not_conj_Domain [rule_format]:
"(l,l'') \<in> findRel (%x. p x & q x)
==> (l, LCons x l') \<in> findRel q --> ~ p x -->
l' \<in> Domain (findRel (%x. p x & q x))"
by (erule findRel.induct, auto)
lemma findRel_conj2 [rule_format]:
"(l,lxx) \<in> findRel q
==> lxx = LCons x lx --> (lx,lz) \<in> findRel(%x. p x & q x) --> ~ p x
--> (l,lz) \<in> findRel (%x. p x & q x)"
by (erule findRel.induct, auto)
lemma findRel_lfilter_Domain_conj [rule_format]:
"(lx,ly) \<in> findRel p
==> \<forall>l. lx = lfilter q l --> l \<in> Domain (findRel(%x. p x & q x))"
apply (erule findRel.induct)
apply (blast dest!: sym [THEN lfilter_eq_LCons] intro: findRel_conj, auto)
apply (drule sym [THEN lfilter_eq_LCons], auto)
apply (drule spec)
apply (drule refl [THEN rev_mp])
apply (blast intro: findRel_conj2)
done
lemma findRel_conj_lfilter [rule_format]:
"(l,l'') \<in> findRel(%x. p x & q x)
==> l'' = LCons y l' -->
(lfilter q l, LCons y (lfilter q l')) \<in> findRel p"
by (erule findRel.induct, auto)
lemma lfilter_conj_lemma:
"(lfilter p (lfilter q l), lfilter (%x. p x & q x) l)
\<in> llistD_Fun (range (%u. (lfilter p (lfilter q u),
lfilter (%x. p x & q x) u)))"
apply (case_tac "l \<in> Domain (findRel q)")
apply (subgoal_tac [2] "l ~: Domain (findRel (%x. p x & q x))")
prefer 3 apply (blast intro: rev_subsetD [OF _ Domain_findRel_mono])
txt{*There are no @{text qs} in @{text l}: both lists are @{text LNil}*}
apply (simp_all add: Domain_findRel_iff, clarify)
txt{*case @{text "q x"}*}
apply (case_tac "p x")
apply (simp_all add: findRel_conj [THEN findRel_imp_lfilter])
txt{*case @{text "q x"} and @{text "~(p x)"} *}
apply (case_tac "l' \<in> Domain (findRel (%x. p x & q x))")
txt{*subcase: there is no @{text "p & q"} in @{text l'} and therefore none in @{text l}*}
apply (subgoal_tac [2] "l ~: Domain (findRel (%x. p x & q x))")
prefer 3 apply (blast intro: findRel_not_conj_Domain)
apply (subgoal_tac [2] "lfilter q l' ~: Domain (findRel p) ")
prefer 3 apply (blast intro: findRel_lfilter_Domain_conj)
txt{* {\dots} and therefore too, no @{text p} in @{text "lfilter q l'"}.
Both results are @{text LNil}*}
apply (simp_all add: Domain_findRel_iff, clarify)
txt{*subcase: there is a @{text "p & q"} in @{text l'} and therefore also one in @{text l} *}
apply (subgoal_tac " (l, LCons xa l'a) \<in> findRel (%x. p x & q x) ")
prefer 2 apply (blast intro: findRel_conj2)
apply (subgoal_tac " (lfilter q l', LCons xa (lfilter q l'a)) \<in> findRel p")
apply simp
apply (blast intro: findRel_conj_lfilter)
done
lemma lfilter_conj: "lfilter p (lfilter q l) = lfilter (%x. p x & q x) l"
apply (rule_tac l = "l" in llist_fun_equalityI, simp_all)
apply (blast intro: lfilter_conj_lemma rev_subsetD [OF _ llistD_Fun_mono])
done
subsection {* Numerous lemmas required to prove ??:
@{text "lfilter p (lmap f l) = lmap f (lfilter (%x. p(f x)) l)"}
*}
lemma findRel_lmap_Domain:
"(l,l') \<in> findRel(%x. p (f x)) ==> lmap f l \<in> Domain(findRel p)"
by (erule findRel.induct, auto)
lemma lmap_eq_LCons [rule_format]: "lmap f l = LCons x l' -->
(\<exists>y l''. x = f y & l' = lmap f l'' & l = LCons y l'')"
apply (subst lmap_def [THEN def_llist_corec])
apply (rule_tac l = "l" in llistE, auto)
done
lemma lmap_LCons_findRel_lemma [rule_format]:
"(lx,ly) \<in> findRel p
==> \<forall>l. lmap f l = lx --> ly = LCons x l' -->
(\<exists>y l''. x = f y & l' = lmap f l'' &
(l, LCons y l'') \<in> findRel(%x. p(f x)))"
apply (erule findRel.induct, simp_all)
apply (blast dest!: lmap_eq_LCons)+
done
lemmas lmap_LCons_findRel = lmap_LCons_findRel_lemma [OF _ refl refl]
lemma lfilter_lmap: "lfilter p (lmap f l) = lmap f (lfilter (p o f) l)"
apply (rule_tac l = "l" in llist_fun_equalityI, simp_all)
apply safe
apply (case_tac "lmap f l \<in> Domain (findRel p)")
apply (simp add: Domain_findRel_iff, clarify)
apply (frule lmap_LCons_findRel, force)
apply (subgoal_tac "l ~: Domain (findRel (%x. p (f x)))", simp)
apply (blast intro: findRel_lmap_Domain)
done
end