|
|
|
|
|
Quellcode-Bibliothek
© Kompilation durch diese Firma
[Weder Korrektheit noch Funktionsfähigkeit der Software werden zugesichert.]
Datei:
Binary_Product_Measure.thy
Sprache: Isabelle
|
|
(* Title: HOL/Quotient_Examples/Quotient_FSet.thy
Author: Cezary Kaliszyk, TU Munich
Author: Christian Urban, TU Munich
Type of finite sets.
*)
(********************************************************************
WARNING: There is a formalization of 'a fset as a subtype of sets in
HOL/Library/FSet.thy using Lifting/Transfer. The user should use
that file rather than this file unless there are some very specific
reasons.
*********************************************************************)
theory Quotient_FSet
imports "HOL-Library.Multiset" "HOL-Library.Quotient_List"
begin
text \<open>
The type of finite sets is created by a quotient construction
over lists. The definition of the equivalence:
\<close>
definition
list_eq :: "'a list \ 'a list \ bool" (infix "\" 50)
where
[simp]: "xs \ ys \ set xs = set ys"
lemma list_eq_reflp:
"reflp list_eq"
by (auto intro: reflpI)
lemma list_eq_symp:
"symp list_eq"
by (auto intro: sympI)
lemma list_eq_transp:
"transp list_eq"
by (auto intro: transpI)
lemma list_eq_equivp:
"equivp list_eq"
by (auto intro: equivpI list_eq_reflp list_eq_symp list_eq_transp)
text \<open>The \<open>fset\<close> type\<close>
quotient_type
'a fset = "'a list" / "list_eq"
by (rule list_eq_equivp)
text \<open>
Definitions for sublist, cardinality,
intersection, difference and respectful fold over
lists.
\<close>
declare List.member_def [simp]
definition
sub_list :: "'a list \ 'a list \ bool"
where
[simp]: "sub_list xs ys \ set xs \ set ys"
definition
card_list :: "'a list \ nat"
where
[simp]: "card_list xs = card (set xs)"
definition
inter_list :: "'a list \ 'a list \ 'a list"
where
[simp]: "inter_list xs ys = [x \ xs. x \ set xs \ x \ set ys]"
definition
diff_list :: "'a list \ 'a list \ 'a list"
where
[simp]: "diff_list xs ys = [x \ xs. x \ set ys]"
definition
rsp_fold :: "('a \ 'b \ 'b) \ bool"
where
"rsp_fold f \ (\u v. f u \ f v = f v \ f u)"
lemma rsp_foldI:
"(\u v. f u \ f v = f v \ f u) \ rsp_fold f"
by (simp add: rsp_fold_def)
lemma rsp_foldE:
assumes "rsp_fold f"
obtains "f u \ f v = f v \ f u"
using assms by (simp add: rsp_fold_def)
definition
fold_once :: "('a \ 'b \ 'b) \ 'a list \ 'b \ 'b"
where
"fold_once f xs = (if rsp_fold f then fold f (remdups xs) else id)"
lemma fold_once_default [simp]:
"\ rsp_fold f \ fold_once f xs = id"
by (simp add: fold_once_def)
lemma fold_once_fold_remdups:
"rsp_fold f \ fold_once f xs = fold f (remdups xs)"
by (simp add: fold_once_def)
section \<open>Quotient composition lemmas\<close>
lemma list_all2_refl':
assumes q: "equivp R"
shows "(list_all2 R) r r"
by (rule list_all2_refl) (metis equivp_def q)
lemma compose_list_refl:
assumes q: "equivp R"
shows "(list_all2 R OOO (\)) r r"
proof
have *: "r \ r" by (rule equivp_reflp[OF fset_equivp])
show "list_all2 R r r" by (rule list_all2_refl'[OF q])
with * show "((\) OO list_all2 R) r r" ..
qed
lemma map_list_eq_cong: "b \ ba \ map f b \ map f ba"
by (simp only: list_eq_def set_map)
lemma quotient_compose_list_g:
assumes q: "Quotient3 R Abs Rep"
and e: "equivp R"
shows "Quotient3 ((list_all2 R) OOO (\))
(abs_fset \<circ> (map Abs)) ((map Rep) \<circ> rep_fset)"
unfolding Quotient3_def comp_def
proof (intro conjI allI)
fix a r s
show "abs_fset (map Abs (map Rep (rep_fset a))) = a"
by (simp add: abs_o_rep[OF q] Quotient3_abs_rep[OF Quotient3_fset] List.map.id)
have b: "list_all2 R (map Rep (rep_fset a)) (map Rep (rep_fset a))"
by (rule list_all2_refl'[OF e])
have c: "((\) OO list_all2 R) (map Rep (rep_fset a)) (map Rep (rep_fset a))"
by (rule, rule equivp_reflp[OF fset_equivp]) (rule b)
show "(list_all2 R OOO (\)) (map Rep (rep_fset a)) (map Rep (rep_fset a))"
by (rule, rule list_all2_refl'[OF e]) (rule c)
show "(list_all2 R OOO (\)) r s = ((list_all2 R OOO (\)) r r \
(list_all2 R OOO (\<approx>)) s s \<and> abs_fset (map Abs r) = abs_fset (map Abs s))"
proof (intro iffI conjI)
show "(list_all2 R OOO (\)) r r" by (rule compose_list_refl[OF e])
show "(list_all2 R OOO (\)) s s" by (rule compose_list_refl[OF e])
next
assume a: "(list_all2 R OOO (\)) r s"
then have b: "map Abs r \ map Abs s"
proof (elim relcomppE)
fix b ba
assume c: "list_all2 R r b"
assume d: "b \ ba"
assume e: "list_all2 R ba s"
have f: "map Abs r = map Abs b"
using Quotient3_rel[OF list_quotient3[OF q]] c by blast
have "map Abs ba = map Abs s"
using Quotient3_rel[OF list_quotient3[OF q]] e by blast
then have g: "map Abs s = map Abs ba" by simp
then show "map Abs r \ map Abs s" using d f map_list_eq_cong by simp
qed
then show "abs_fset (map Abs r) = abs_fset (map Abs s)"
using Quotient3_rel[OF Quotient3_fset] by blast
next
assume a: "(list_all2 R OOO (\)) r r \ (list_all2 R OOO (\)) s s
\<and> abs_fset (map Abs r) = abs_fset (map Abs s)"
then have s: "(list_all2 R OOO (\)) s s" by simp
have d: "map Abs r \ map Abs s"
by (subst Quotient3_rel [OF Quotient3_fset, symmetric]) (simp add: a)
have b: "map Rep (map Abs r) \ map Rep (map Abs s)"
by (rule map_list_eq_cong[OF d])
have y: "list_all2 R (map Rep (map Abs s)) s"
by (fact rep_abs_rsp_left[OF list_quotient3[OF q], OF list_all2_refl'[OF e, of s]])
have c: "((\) OO list_all2 R) (map Rep (map Abs r)) s"
by (rule relcomppI) (rule b, rule y)
have z: "list_all2 R r (map Rep (map Abs r))"
by (fact rep_abs_rsp[OF list_quotient3[OF q], OF list_all2_refl'[OF e, of r]])
then show "(list_all2 R OOO (\)) r s"
using a c relcomppI by simp
qed
qed
lemma quotient_compose_list[quot_thm]:
shows "Quotient3 ((list_all2 (\)) OOO (\))
(abs_fset \<circ> (map abs_fset)) ((map rep_fset) \<circ> rep_fset)"
by (rule quotient_compose_list_g, rule Quotient3_fset, rule list_eq_equivp)
section \<open>Quotient definitions for fsets\<close>
subsection \<open>Finite sets are a bounded, distributive lattice with minus\<close>
instantiation fset :: (type) "{bounded_lattice_bot, distrib_lattice, minus}"
begin
quotient_definition
"bot :: 'a fset"
is "Nil :: 'a list" done
abbreviation
empty_fset ("{||}")
where
"{||} \ bot :: 'a fset"
quotient_definition
"less_eq_fset :: ('a fset \ 'a fset \ bool)"
is "sub_list :: ('a list \ 'a list \ bool)" by simp
abbreviation
subset_fset :: "'a fset \ 'a fset \ bool" (infix "|\|" 50)
where
"xs |\| ys \ xs \ ys"
definition
less_fset :: "'a fset \ 'a fset \ bool"
where
"xs < ys \ xs \ ys \ xs \ (ys::'a fset)"
abbreviation
psubset_fset :: "'a fset \ 'a fset \ bool" (infix "|\|" 50)
where
"xs |\| ys \ xs < ys"
quotient_definition
"sup :: 'a fset \ 'a fset \ 'a fset"
is "append :: 'a list \ 'a list \ 'a list" by simp
abbreviation
union_fset (infixl "|\|" 65)
where
"xs |\| ys \ sup xs (ys::'a fset)"
quotient_definition
"inf :: 'a fset \ 'a fset \ 'a fset"
is "inter_list :: 'a list \ 'a list \ 'a list" by simp
abbreviation
inter_fset (infixl "|\|" 65)
where
"xs |\| ys \ inf xs (ys::'a fset)"
quotient_definition
"minus :: 'a fset \ 'a fset \ 'a fset"
is "diff_list :: 'a list \ 'a list \ 'a list" by fastforce
instance
proof
fix x y z :: "'a fset"
show "x |\| y \ x |\| y \ \ y |\| x"
by (unfold less_fset_def, descending) auto
show "x |\| x" by (descending) (simp)
show "{||} |\| x" by (descending) (simp)
show "x |\| x |\| y" by (descending) (simp)
show "y |\| x |\| y" by (descending) (simp)
show "x |\| y |\| x" by (descending) (auto)
show "x |\| y |\| y" by (descending) (auto)
show "x |\| (y |\| z) = x |\| y |\| (x |\| z)"
by (descending) (auto)
next
fix x y z :: "'a fset"
assume a: "x |\| y"
assume b: "y |\| z"
show "x |\| z" using a b by (descending) (simp)
next
fix x y :: "'a fset"
assume a: "x |\| y"
assume b: "y |\| x"
show "x = y" using a b by (descending) (auto)
next
fix x y z :: "'a fset"
assume a: "y |\| x"
assume b: "z |\| x"
show "y |\| z |\| x" using a b by (descending) (simp)
next
fix x y z :: "'a fset"
assume a: "x |\| y"
assume b: "x |\| z"
show "x |\| y |\| z" using a b by (descending) (auto)
qed
end
subsection \<open>Other constants for fsets\<close>
quotient_definition
"insert_fset :: 'a \ 'a fset \ 'a fset"
is "Cons" by auto
syntax
"_insert_fset" :: "args => 'a fset" ("{|(_)|}")
translations
"{|x, xs|}" == "CONST insert_fset x {|xs|}"
"{|x|}" == "CONST insert_fset x {||}"
quotient_definition
fset_member
where
"fset_member :: 'a fset \ 'a \ bool" is "List.member" by fastforce
abbreviation
in_fset :: "'a \ 'a fset \ bool" (infix "|\|" 50)
where
"x |\| S \ fset_member S x"
abbreviation
notin_fset :: "'a \ 'a fset \ bool" (infix "|\|" 50)
where
"x |\| S \ \ (x |\| S)"
subsection \<open>Other constants on the Quotient Type\<close>
quotient_definition
"card_fset :: 'a fset \ nat"
is card_list by simp
quotient_definition
"map_fset :: ('a \ 'b) \ 'a fset \ 'b fset"
is map by simp
quotient_definition
"remove_fset :: 'a \ 'a fset \ 'a fset"
is removeAll by simp
quotient_definition
"fset :: 'a fset \ 'a set"
is "set" by simp
lemma fold_once_set_equiv:
assumes "xs \ ys"
shows "fold_once f xs = fold_once f ys"
proof (cases "rsp_fold f")
case False then show ?thesis by simp
next
case True
then have "\x y. x \ set (remdups xs) \ y \ set (remdups xs) \ f x \ f y = f y \ f x"
by (rule rsp_foldE)
moreover from assms have "mset (remdups xs) = mset (remdups ys)"
by (simp add: set_eq_iff_mset_remdups_eq)
ultimately have "fold f (remdups xs) = fold f (remdups ys)"
by (rule fold_multiset_equiv)
with True show ?thesis by (simp add: fold_once_fold_remdups)
qed
quotient_definition
"fold_fset :: ('a \ 'b \ 'b) \ 'a fset \ 'b \ 'b"
is fold_once by (rule fold_once_set_equiv)
lemma concat_rsp_pre:
assumes a: "list_all2 (\) x x'"
and b: "x' \ y'"
and c: "list_all2 (\) y' y"
and d: "\x\set x. xa \ set x"
shows "\x\set y. xa \ set x"
proof -
obtain xb where e: "xb \ set x" and f: "xa \ set xb" using d by auto
have "\y. y \ set x' \ xb \ y" by (rule list_all2_find_element[OF e a])
then obtain ya where h: "ya \ set x'" and i: "xb \ ya" by auto
have "ya \ set y'" using b h by simp
then have "\yb. yb \ set y \ ya \ yb" using c by (rule list_all2_find_element)
then show ?thesis using f i by auto
qed
quotient_definition
"concat_fset :: ('a fset) fset \ 'a fset"
is concat
proof (elim relcomppE)
fix a b ba bb
assume a: "list_all2 (\) a ba"
with list_symp [OF list_eq_symp] have a': "list_all2 (\) ba a" by (rule sympE)
assume b: "ba \ bb"
with list_eq_symp have b': "bb \ ba" by (rule sympE)
assume c: "list_all2 (\) bb b"
with list_symp [OF list_eq_symp] have c': "list_all2 (\) b bb" by (rule sympE)
have "\x. (\xa\set a. x \ set xa) = (\xa\set b. x \ set xa)"
proof
fix x
show "(\xa\set a. x \ set xa) = (\xa\set b. x \ set xa)"
proof
assume d: "\xa\set a. x \ set xa"
show "\xa\set b. x \ set xa" by (rule concat_rsp_pre[OF a b c d])
next
assume e: "\xa\set b. x \ set xa"
show "\xa\set a. x \ set xa" by (rule concat_rsp_pre[OF c' b' a' e])
qed
qed
then show "concat a \ concat b" by auto
qed
quotient_definition
"filter_fset :: ('a \ bool) \ 'a fset \ 'a fset"
is filter by force
subsection \<open>Compositional respectfulness and preservation lemmas\<close>
lemma Nil_rsp2 [quot_respect]:
shows "(list_all2 (\) OOO (\)) Nil Nil"
by (rule compose_list_refl, rule list_eq_equivp)
lemma Cons_rsp2 [quot_respect]:
shows "((\) ===> list_all2 (\) OOO (\) ===> list_all2 (\) OOO (\)) Cons Cons"
apply (auto intro!: rel_funI)
apply (rule_tac b="x # b" in relcomppI)
apply auto
apply (rule_tac b="x # ba" in relcomppI)
apply auto
done
lemma Nil_prs2 [quot_preserve]:
assumes "Quotient3 R Abs Rep"
shows "(Abs \ map f) [] = Abs []"
by simp
lemma Cons_prs2 [quot_preserve]:
assumes q: "Quotient3 R1 Abs1 Rep1"
and r: "Quotient3 R2 Abs2 Rep2"
shows "(Rep1 ---> (map Rep1 \ Rep2) ---> (Abs2 \ map Abs1)) (#) = (id ---> Rep2 ---> Abs2) (#)"
by (auto simp add: fun_eq_iff comp_def Quotient3_abs_rep [OF q])
lemma append_prs2 [quot_preserve]:
assumes q: "Quotient3 R1 Abs1 Rep1"
and r: "Quotient3 R2 Abs2 Rep2"
shows "((map Rep1 \ Rep2) ---> (map Rep1 \ Rep2) ---> (Abs2 \ map Abs1)) (@) =
(Rep2 ---> Rep2 ---> Abs2) (@)"
by (simp add: fun_eq_iff abs_o_rep[OF q] List.map.id)
lemma list_all2_app_l:
assumes a: "reflp R"
and b: "list_all2 R l r"
shows "list_all2 R (z @ l) (z @ r)"
using a b by (induct z) (auto elim: reflpE)
lemma append_rsp2_pre0:
assumes a:"list_all2 (\) x x'"
shows "list_all2 (\) (x @ z) (x' @ z)"
using a apply (induct x x' rule: list_induct2')
by simp_all (rule list_all2_refl'[OF list_eq_equivp])
lemma append_rsp2_pre1:
assumes a:"list_all2 (\) x x'"
shows "list_all2 (\) (z @ x) (z @ x')"
using a apply (induct x x' arbitrary: z rule: list_induct2')
apply (rule list_all2_refl'[OF list_eq_equivp])
apply (simp_all del: list_eq_def)
apply (rule list_all2_app_l)
apply (simp_all add: reflpI)
done
lemma append_rsp2_pre:
assumes "list_all2 (\) x x'"
and "list_all2 (\) z z'"
shows "list_all2 (\) (x @ z) (x' @ z')"
using assms by (rule list_all2_appendI)
lemma compositional_rsp3:
assumes "(R1 ===> R2 ===> R3) C C" and "(R4 ===> R5 ===> R6) C C"
shows "(R1 OOO R4 ===> R2 OOO R5 ===> R3 OOO R6) C C"
by (auto intro!: rel_funI)
(metis (full_types) assms rel_funE relcomppI)
lemma append_rsp2 [quot_respect]:
"(list_all2 (\) OOO (\) ===> list_all2 (\) OOO (\) ===> list_all2 (\) OOO (\)) append append"
by (intro compositional_rsp3)
(auto intro!: rel_funI simp add: append_rsp2_pre)
lemma map_rsp2 [quot_respect]:
"(((\) ===> (\)) ===> list_all2 (\) OOO (\) ===> list_all2 (\) OOO (\)) map map"
proof (auto intro!: rel_funI)
fix f f' :: "'a list \<Rightarrow> 'b list"
fix xa ya x y :: "'a list list"
assume fs: "((\) ===> (\)) f f'" and x: "list_all2 (\) xa x" and xy: "set x = set y" and y: "list_all2 (\) y ya"
have a: "(list_all2 (\)) (map f xa) (map f x)"
using x
by (induct xa x rule: list_induct2')
(simp_all, metis fs rel_funE list_eq_def)
have b: "set (map f x) = set (map f y)"
using xy fs
by (induct x y rule: list_induct2')
(simp_all, metis image_insert)
have c: "(list_all2 (\)) (map f y) (map f' ya)"
using y fs
by (induct y ya rule: list_induct2')
(simp_all, metis apply_rsp' list_eq_def)
show "(list_all2 (\) OOO (\)) (map f xa) (map f' ya)"
by (metis a b c list_eq_def relcomppI)
qed
lemma map_prs2 [quot_preserve]:
shows "((abs_fset ---> rep_fset) ---> (map rep_fset \ rep_fset) ---> abs_fset \ map abs_fset) map = (id ---> rep_fset ---> abs_fset) map"
by (auto simp add: fun_eq_iff)
(simp only: map_map[symmetric] map_prs_aux[OF Quotient3_fset Quotient3_fset])
section \<open>Lifted theorems\<close>
subsection \<open>fset\<close>
lemma fset_simps [simp]:
shows "fset {||} = {}"
and "fset (insert_fset x S) = insert x (fset S)"
by (descending, simp)+
lemma finite_fset [simp]:
shows "finite (fset S)"
by (descending) (simp)
lemma fset_cong:
shows "fset S = fset T \ S = T"
by (descending) (simp)
lemma filter_fset [simp]:
shows "fset (filter_fset P xs) = Collect P \ fset xs"
by (descending) (auto)
lemma remove_fset [simp]:
shows "fset (remove_fset x xs) = fset xs - {x}"
by (descending) (simp)
lemma inter_fset [simp]:
shows "fset (xs |\| ys) = fset xs \ fset ys"
by (descending) (auto)
lemma union_fset [simp]:
shows "fset (xs |\| ys) = fset xs \ fset ys"
by (lifting set_append)
lemma minus_fset [simp]:
shows "fset (xs - ys) = fset xs - fset ys"
by (descending) (auto)
subsection \<open>in_fset\<close>
lemma in_fset:
shows "x |\| S \ x \ fset S"
by descending simp
lemma notin_fset:
shows "x |\| S \ x \ fset S"
by (simp add: in_fset)
lemma notin_empty_fset:
shows "x |\| {||}"
by (simp add: in_fset)
lemma fset_eq_iff:
shows "S = T \ (\x. (x |\| S) = (x |\| T))"
by descending auto
lemma none_in_empty_fset:
shows "(\x. x |\| S) \ S = {||}"
by descending simp
subsection \<open>insert_fset\<close>
lemma in_insert_fset_iff [simp]:
shows "x |\| insert_fset y S \ x = y \ x |\| S"
by descending simp
lemma
shows insert_fsetI1: "x |\| insert_fset x S"
and insert_fsetI2: "x |\| S \ x |\| insert_fset y S"
by simp_all
lemma insert_absorb_fset [simp]:
shows "x |\| S \ insert_fset x S = S"
by (descending) (auto)
lemma empty_not_insert_fset[simp]:
shows "{||} \ insert_fset x S"
and "insert_fset x S \ {||}"
by (descending, simp)+
lemma insert_fset_left_comm:
shows "insert_fset x (insert_fset y S) = insert_fset y (insert_fset x S)"
by (descending) (auto)
lemma insert_fset_left_idem:
shows "insert_fset x (insert_fset x S) = insert_fset x S"
by (descending) (auto)
lemma singleton_fset_eq[simp]:
shows "{|x|} = {|y|} \ x = y"
by (descending) (auto)
lemma in_fset_mdef:
shows "x |\| F \ x |\| (F - {|x|}) \ F = insert_fset x (F - {|x|})"
by (descending) (auto)
subsection \<open>union_fset\<close>
lemmas [simp] =
sup_bot_left[where 'a="'a fset"]
sup_bot_right[where 'a="'a fset"]
lemma union_insert_fset [simp]:
shows "insert_fset x S |\| T = insert_fset x (S |\| T)"
by (lifting append.simps(2))
lemma singleton_union_fset_left:
shows "{|a|} |\| S = insert_fset a S"
by simp
lemma singleton_union_fset_right:
shows "S |\| {|a|} = insert_fset a S"
by (subst sup.commute) simp
lemma in_union_fset:
shows "x |\| S |\| T \ x |\| S \ x |\| T"
by (descending) (simp)
subsection \<open>minus_fset\<close>
lemma minus_in_fset:
shows "x |\| (xs - ys) \ x |\| xs \ x |\| ys"
by (descending) (simp)
lemma minus_insert_fset:
shows "insert_fset x xs - ys = (if x |\| ys then xs - ys else insert_fset x (xs - ys))"
by (descending) (auto)
lemma minus_insert_in_fset[simp]:
shows "x |\| ys \ insert_fset x xs - ys = xs - ys"
by (simp add: minus_insert_fset)
lemma minus_insert_notin_fset[simp]:
shows "x |\| ys \ insert_fset x xs - ys = insert_fset x (xs - ys)"
by (simp add: minus_insert_fset)
lemma in_minus_fset:
shows "x |\| F - S \ x |\| S"
unfolding in_fset minus_fset
by blast
lemma notin_minus_fset:
shows "x |\| S \ x |\| F - S"
unfolding in_fset minus_fset
by blast
subsection \<open>remove_fset\<close>
lemma in_remove_fset:
shows "x |\| remove_fset y S \ x |\| S \ x \ y"
by (descending) (simp)
lemma notin_remove_fset:
shows "x |\| remove_fset x S"
by (descending) (simp)
lemma notin_remove_ident_fset:
shows "x |\| S \ remove_fset x S = S"
by (descending) (simp)
lemma remove_fset_cases:
shows "S = {||} \ (\x. x |\| S \ S = insert_fset x (remove_fset x S))"
by (descending) (auto simp add: insert_absorb)
subsection \<open>inter_fset\<close>
lemma inter_empty_fset_l:
shows "{||} |\| S = {||}"
by simp
lemma inter_empty_fset_r:
shows "S |\| {||} = {||}"
by simp
lemma inter_insert_fset:
shows "insert_fset x S |\| T = (if x |\| T then insert_fset x (S |\| T) else S |\| T)"
by (descending) (auto)
lemma in_inter_fset:
shows "x |\| (S |\| T) \ x |\| S \ x |\| T"
by (descending) (simp)
subsection \<open>subset_fset and psubset_fset\<close>
lemma subset_fset:
shows "xs |\| ys \ fset xs \ fset ys"
by (descending) (simp)
lemma psubset_fset:
shows "xs |\| ys \ fset xs \ fset ys"
unfolding less_fset_def
by (descending) (auto)
lemma subset_insert_fset:
shows "(insert_fset x xs) |\| ys \ x |\| ys \ xs |\| ys"
by (descending) (simp)
lemma subset_in_fset:
shows "xs |\| ys = (\x. x |\| xs \ x |\| ys)"
by (descending) (auto)
lemma subset_empty_fset:
shows "xs |\| {||} \ xs = {||}"
by (descending) (simp)
lemma not_psubset_empty_fset:
shows "\ xs |\| {||}"
by (metis fset_simps(1) psubset_fset not_psubset_empty)
subsection \<open>map_fset\<close>
lemma map_fset_simps [simp]:
shows "map_fset f {||} = {||}"
and "map_fset f (insert_fset x S) = insert_fset (f x) (map_fset f S)"
by (descending, simp)+
lemma map_fset_image [simp]:
shows "fset (map_fset f S) = f ` (fset S)"
by (descending) (simp)
lemma inj_map_fset_cong:
shows "inj f \ map_fset f S = map_fset f T \ S = T"
by (descending) (metis inj_vimage_image_eq list_eq_def set_map)
lemma map_union_fset:
shows "map_fset f (S |\| T) = map_fset f S |\| map_fset f T"
by (descending) (simp)
lemma in_fset_map_fset[simp]: "a |\| map_fset f X = (\b. b |\| X \ a = f b)"
by descending auto
subsection \<open>card_fset\<close>
lemma card_fset:
shows "card_fset xs = card (fset xs)"
by (descending) (simp)
lemma card_insert_fset_iff [simp]:
shows "card_fset (insert_fset x S) = (if x |\| S then card_fset S else Suc (card_fset S))"
by (descending) (simp add: insert_absorb)
lemma card_fset_0[simp]:
shows "card_fset S = 0 \ S = {||}"
by (descending) (simp)
lemma card_empty_fset[simp]:
shows "card_fset {||} = 0"
by (simp add: card_fset)
lemma card_fset_1:
shows "card_fset S = 1 \ (\x. S = {|x|})"
by (descending) (auto simp add: card_Suc_eq)
lemma card_fset_gt_0:
shows "x \ fset S \ 0 < card_fset S"
by (descending) (auto simp add: card_gt_0_iff)
lemma card_notin_fset:
shows "(x |\| S) = (card_fset (insert_fset x S) = Suc (card_fset S))"
by simp
lemma card_fset_Suc:
shows "card_fset S = Suc n \ \x T. x |\| T \ S = insert_fset x T \ card_fset T = n"
apply(descending)
apply(auto dest!: card_eq_SucD)
by (metis Diff_insert_absorb set_removeAll)
lemma card_remove_fset_iff [simp]:
shows "card_fset (remove_fset y S) = (if y |\| S then card_fset S - 1 else card_fset S)"
by (descending) (simp)
lemma card_Suc_exists_in_fset:
shows "card_fset S = Suc n \ \a. a |\| S"
by (drule card_fset_Suc) (auto)
lemma in_card_fset_not_0:
shows "a |\| A \ card_fset A \ 0"
by (descending) (auto)
lemma card_fset_mono:
shows "xs |\| ys \ card_fset xs \ card_fset ys"
unfolding card_fset psubset_fset
by (simp add: card_mono subset_fset)
lemma card_subset_fset_eq:
shows "xs |\| ys \ card_fset ys \ card_fset xs \ xs = ys"
unfolding card_fset subset_fset
by (auto dest: card_seteq[OF finite_fset] simp add: fset_cong)
lemma psubset_card_fset_mono:
shows "xs |\| ys \ card_fset xs < card_fset ys"
unfolding card_fset subset_fset
by (metis finite_fset psubset_fset psubset_card_mono)
lemma card_union_inter_fset:
shows "card_fset xs + card_fset ys = card_fset (xs |\| ys) + card_fset (xs |\| ys)"
unfolding card_fset union_fset inter_fset
by (rule card_Un_Int[OF finite_fset finite_fset])
lemma card_union_disjoint_fset:
shows "xs |\| ys = {||} \ card_fset (xs |\| ys) = card_fset xs + card_fset ys"
unfolding card_fset union_fset
apply (rule card_Un_disjoint[OF finite_fset finite_fset])
by (metis inter_fset fset_simps(1))
lemma card_remove_fset_less1:
shows "x |\| xs \ card_fset (remove_fset x xs) < card_fset xs"
unfolding card_fset in_fset remove_fset
by (rule card_Diff1_less[OF finite_fset])
lemma card_remove_fset_less2:
shows "x |\| xs \ y |\| xs \ card_fset (remove_fset y (remove_fset x xs)) < card_fset xs"
unfolding card_fset remove_fset in_fset
by (rule card_Diff2_less[OF finite_fset])
lemma card_remove_fset_le1:
shows "card_fset (remove_fset x xs) \ card_fset xs"
unfolding remove_fset card_fset
by (rule card_Diff1_le[OF finite_fset])
lemma card_psubset_fset:
shows "ys |\| xs \ card_fset ys < card_fset xs \ ys |\| xs"
unfolding card_fset psubset_fset subset_fset
by (rule card_psubset[OF finite_fset])
lemma card_map_fset_le:
shows "card_fset (map_fset f xs) \ card_fset xs"
unfolding card_fset map_fset_image
by (rule card_image_le[OF finite_fset])
lemma card_minus_insert_fset[simp]:
assumes "a |\| A" and "a |\| B"
shows "card_fset (A - insert_fset a B) = card_fset (A - B) - 1"
using assms
unfolding in_fset card_fset minus_fset
by (simp add: card_Diff_insert[OF finite_fset])
lemma card_minus_subset_fset:
assumes "B |\| A"
shows "card_fset (A - B) = card_fset A - card_fset B"
using assms
unfolding subset_fset card_fset minus_fset
by (rule card_Diff_subset[OF finite_fset])
lemma card_minus_fset:
shows "card_fset (A - B) = card_fset A - card_fset (A |\| B)"
unfolding inter_fset card_fset minus_fset
by (rule card_Diff_subset_Int) (simp)
subsection \<open>concat_fset\<close>
lemma concat_empty_fset [simp]:
shows "concat_fset {||} = {||}"
by descending simp
lemma concat_insert_fset [simp]:
shows "concat_fset (insert_fset x S) = x |\| concat_fset S"
by descending simp
lemma concat_union_fset [simp]:
shows "concat_fset (xs |\| ys) = concat_fset xs |\| concat_fset ys"
by descending simp
lemma map_concat_fset:
shows "map_fset f (concat_fset xs) = concat_fset (map_fset (map_fset f) xs)"
by (lifting map_concat)
subsection \<open>filter_fset\<close>
lemma subset_filter_fset:
"filter_fset P xs |\| filter_fset Q xs = (\ x. x |\| xs \ P x \ Q x)"
by descending auto
lemma eq_filter_fset:
"(filter_fset P xs = filter_fset Q xs) = (\x. x |\| xs \ P x = Q x)"
by descending auto
lemma psubset_filter_fset:
"(\x. x |\| xs \ P x \ Q x) \ (x |\| xs & \ P x & Q x) \
filter_fset P xs |\<subset>| filter_fset Q xs"
unfolding less_fset_def by (auto simp add: subset_filter_fset eq_filter_fset)
subsection \<open>fold_fset\<close>
lemma fold_empty_fset:
"fold_fset f {||} = id"
by descending (simp add: fold_once_def)
lemma fold_insert_fset: "fold_fset f (insert_fset a A) =
(if rsp_fold f then if a |\<in>| A then fold_fset f A else fold_fset f A \<circ> f a else id)"
by descending (simp add: fold_once_fold_remdups)
lemma remdups_removeAll:
"remdups (removeAll x xs) = remove1 x (remdups xs)"
by (induct xs) auto
lemma member_commute_fold_once:
assumes "rsp_fold f"
and "x \ set xs"
shows "fold_once f xs = fold_once f (removeAll x xs) \ f x"
proof -
from assms have "fold f (remdups xs) = fold f (remove1 x (remdups xs)) \ f x"
by (auto intro!: fold_remove1_split elim: rsp_foldE)
then show ?thesis using \<open>rsp_fold f\<close> by (simp add: fold_once_fold_remdups remdups_removeAll)
qed
lemma in_commute_fold_fset:
"rsp_fold f \ h |\| b \ fold_fset f b = fold_fset f (remove_fset h b) \ f h"
by descending (simp add: member_commute_fold_once)
subsection \<open>Choice in fsets\<close>
lemma fset_choice:
assumes a: "\x. x |\| A \ (\y. P x y)"
shows "\f. \x. x |\| A \ P x (f x)"
using a
apply(descending)
using finite_set_choice
by (auto simp add: Ball_def)
section \<open>Induction and Cases rules for fsets\<close>
lemma fset_exhaust [case_names empty insert, cases type: fset]:
assumes empty_fset_case: "S = {||} \ P"
and insert_fset_case: "\x S'. S = insert_fset x S' \ P"
shows "P"
using assms by (lifting list.exhaust)
lemma fset_induct [case_names empty insert]:
assumes empty_fset_case: "P {||}"
and insert_fset_case: "\x S. P S \ P (insert_fset x S)"
shows "P S"
using assms
by (descending) (blast intro: list.induct)
lemma fset_induct_stronger [case_names empty insert, induct type: fset]:
assumes empty_fset_case: "P {||}"
and insert_fset_case: "\x S. \x |\| S; P S\ \ P (insert_fset x S)"
shows "P S"
proof(induct S rule: fset_induct)
case empty
show "P {||}" using empty_fset_case by simp
next
case (insert x S)
have "P S" by fact
then show "P (insert_fset x S)" using insert_fset_case
by (cases "x |\| S") (simp_all)
qed
lemma fset_card_induct:
assumes empty_fset_case: "P {||}"
and card_fset_Suc_case: "\S T. Suc (card_fset S) = (card_fset T) \ P S \ P T"
shows "P S"
proof (induct S)
case empty
show "P {||}" by (rule empty_fset_case)
next
case (insert x S)
have h: "P S" by fact
have "x |\| S" by fact
then have "Suc (card_fset S) = card_fset (insert_fset x S)"
using card_fset_Suc by auto
then show "P (insert_fset x S)"
using h card_fset_Suc_case by simp
qed
lemma fset_raw_strong_cases:
obtains "xs = []"
| ys x where "\ List.member ys x" and "xs \ x # ys"
proof (induct xs)
case Nil
then show thesis by simp
next
case (Cons a xs)
have a: "\xs = [] \ thesis; \x ys. \\ List.member ys x; xs \ x # ys\ \ thesis\ \ thesis"
by (rule Cons(1))
have b: "\x' ys'. \\ List.member ys' x'; a # xs \ x' # ys'\ \ thesis" by fact
have c: "xs = [] \ thesis" using b
apply(simp)
by (metis list.set(1) emptyE empty_subsetI)
have "\x ys. \\ List.member ys x; xs \ x # ys\ \ thesis"
proof -
fix x :: 'a
fix ys :: "'a list"
assume d:"\ List.member ys x"
assume e:"xs \ x # ys"
show thesis
proof (cases "x = a")
assume h: "x = a"
then have f: "\ List.member ys a" using d by simp
have g: "a # xs \ a # ys" using e h by auto
show thesis using b f g by simp
next
assume h: "x \ a"
then have f: "\ List.member (a # ys) x" using d by auto
have g: "a # xs \ x # (a # ys)" using e h by auto
show thesis using b f g by (simp del: List.member_def)
qed
qed
then show thesis using a c by blast
qed
lemma fset_strong_cases:
obtains "xs = {||}"
| ys x where "x |\| ys" and "xs = insert_fset x ys"
by (lifting fset_raw_strong_cases)
lemma fset_induct2:
"P {||} {||} \
(\<And>x xs. x |\<notin>| xs \<Longrightarrow> P (insert_fset x xs) {||}) \<Longrightarrow>
(\<And>y ys. y |\<notin>| ys \<Longrightarrow> P {||} (insert_fset y ys)) \<Longrightarrow>
(\<And>x xs y ys. \<lbrakk>P xs ys; x |\<notin>| xs; y |\<notin>| ys\<rbrakk> \<Longrightarrow> P (insert_fset x xs) (insert_fset y ys)) \<Longrightarrow>
P xsa ysa"
apply (induct xsa arbitrary: ysa)
apply (induct_tac x rule: fset_induct_stronger)
apply simp_all
apply (induct_tac xa rule: fset_induct_stronger)
apply simp_all
done
text \<open>Extensionality\<close>
lemma fset_eqI:
assumes "\x. x \ fset A \ x \ fset B"
shows "A = B"
using assms proof (induct A arbitrary: B)
case empty then show ?case
by (auto simp add: in_fset none_in_empty_fset [symmetric] sym)
next
case (insert x A)
from insert.prems insert.hyps(1) have "\z. z \ fset A \ z \ fset (B - {|x|})"
by (auto simp add: in_fset)
then have A: "A = B - {|x|}" by (rule insert.hyps(2))
with insert.prems [symmetric, of x] have "x |\| B" by (simp add: in_fset)
with A show ?case by (metis in_fset_mdef)
qed
subsection \<open>alternate formulation with a different decomposition principle
and a proof of equivalence\<close>
inductive
list_eq2 :: "'a list \ 'a list \ bool" ("_ \2 _")
where
"(a # b # xs) \2 (b # a # xs)"
| "[] \2 []"
| "xs \2 ys \ ys \2 xs"
| "(a # a # xs) \2 (a # xs)"
| "xs \2 ys \ (a # xs) \2 (a # ys)"
| "xs1 \2 xs2 \ xs2 \2 xs3 \ xs1 \2 xs3"
lemma list_eq2_refl:
shows "xs \2 xs"
by (induct xs) (auto intro: list_eq2.intros)
lemma cons_delete_list_eq2:
shows "(a # (removeAll a A)) \2 (if List.member A a then A else a # A)"
apply (induct A)
apply (simp add: list_eq2_refl)
apply (case_tac "List.member (aa # A) a")
apply (simp_all)
apply (case_tac [!] "a = aa")
apply (simp_all)
apply (case_tac "List.member A a")
apply (auto)[2]
apply (metis list_eq2.intros(3) list_eq2.intros(4) list_eq2.intros(5) list_eq2.intros(6))
apply (metis list_eq2.intros(1) list_eq2.intros(5) list_eq2.intros(6))
apply (auto simp add: list_eq2_refl)
done
lemma member_delete_list_eq2:
assumes a: "List.member r e"
shows "(e # removeAll e r) \2 r"
using a cons_delete_list_eq2[of e r]
by simp
lemma list_eq2_equiv:
"(l \ r) \ (list_eq2 l r)"
proof
show "list_eq2 l r \ l \ r" by (induct rule: list_eq2.induct) auto
next
{
fix n
assume a: "card_list l = n" and b: "l \ r"
have "l \2 r"
using a b
proof (induct n arbitrary: l r)
case 0
have "card_list l = 0" by fact
then have "\x. \ List.member l x" by auto
then have z: "l = []" by auto
then have "r = []" using \<open>l \<approx> r\<close> by simp
then show ?case using z list_eq2_refl by simp
next
case (Suc m)
have b: "l \ r" by fact
have d: "card_list l = Suc m" by fact
then have "\a. List.member l a"
apply(simp)
apply(drule card_eq_SucD)
apply(blast)
done
then obtain a where e: "List.member l a" by auto
then have e': "List.member r a" using list_eq_def [simplified List.member_def [symmetric], of l r] b
by auto
have f: "card_list (removeAll a l) = m" using e d by (simp)
have g: "removeAll a l \ removeAll a r" using remove_fset.rsp b by simp
have "(removeAll a l) \2 (removeAll a r)" by (rule Suc.hyps[OF f g])
then have h: "(a # removeAll a l) \2 (a # removeAll a r)" by (rule list_eq2.intros(5))
have i: "l \2 (a # removeAll a l)"
by (rule list_eq2.intros(3)[OF member_delete_list_eq2[OF e]])
have "l \2 (a # removeAll a r)" by (rule list_eq2.intros(6)[OF i h])
then show ?case using list_eq2.intros(6)[OF _ member_delete_list_eq2[OF e']] by simp
qed
}
then show "l \ r \ l \2 r" by blast
qed
(* We cannot write it as "assumes .. shows" since Isabelle changes
the quantifiers to schematic variables and reintroduces them in
a different order *)
lemma fset_eq_cases:
"\a1 = a2;
\<And>a b xs. \<lbrakk>a1 = insert_fset a (insert_fset b xs); a2 = insert_fset b (insert_fset a xs)\<rbrakk> \<Longrightarrow> P;
\<lbrakk>a1 = {||}; a2 = {||}\<rbrakk> \<Longrightarrow> P; \<And>xs ys. \<lbrakk>a1 = ys; a2 = xs; xs = ys\<rbrakk> \<Longrightarrow> P;
\<And>a xs. \<lbrakk>a1 = insert_fset a (insert_fset a xs); a2 = insert_fset a xs\<rbrakk> \<Longrightarrow> P;
\<And>xs ys a. \<lbrakk>a1 = insert_fset a xs; a2 = insert_fset a ys; xs = ys\<rbrakk> \<Longrightarrow> P;
\<And>xs1 xs2 xs3. \<lbrakk>a1 = xs1; a2 = xs3; xs1 = xs2; xs2 = xs3\<rbrakk> \<Longrightarrow> P\<rbrakk>
\<Longrightarrow> P"
by (lifting list_eq2.cases[simplified list_eq2_equiv[symmetric]])
lemma fset_eq_induct:
assumes "x1 = x2"
and "\a b xs. P (insert_fset a (insert_fset b xs)) (insert_fset b (insert_fset a xs))"
and "P {||} {||}"
and "\xs ys. \xs = ys; P xs ys\ \ P ys xs"
and "\a xs. P (insert_fset a (insert_fset a xs)) (insert_fset a xs)"
and "\xs ys a. \xs = ys; P xs ys\ \ P (insert_fset a xs) (insert_fset a ys)"
and "\xs1 xs2 xs3. \xs1 = xs2; P xs1 xs2; xs2 = xs3; P xs2 xs3\ \ P xs1 xs3"
shows "P x1 x2"
using assms
by (lifting list_eq2.induct[simplified list_eq2_equiv[symmetric]])
ML \<open>
fun dest_fsetT (Type (\<^type_name>\<open>fset\<close>, [T])) = T
| dest_fsetT T = raise TYPE ("dest_fsetT: fset type expected", [T], []);
\<close>
no_notation
list_eq (infix "\" 50) and
list_eq2 (infix "\2" 50)
end
¤ Diese beiden folgenden Angebotsgruppen bietet das Unternehmen0.24Angebot
Wie Sie bei der Firma Beratungs- und Dienstleistungen beauftragen können
¤
|
Lebenszyklus
Die hierunter aufgelisteten Ziele sind für diese Firma wichtig
Ziele
Entwicklung einer Software für die statische Quellcodeanalyse
|
|
|
|
|
|
