--- a/NEWS Mon Sep 19 20:07:39 2016 +0200
+++ b/NEWS Mon Sep 19 23:14:34 2016 +0200
@@ -795,6 +795,9 @@
nn_integral :: 'a measure => ('a => ennreal) => ennreal
INCOMPATIBILITY.
+* Renamed HOL/Quotient_Examples/FSet.thy to
+HOL/Quotient_Examples/Quotient_FSet.thy
+INCOMPATIBILITY.
*** ML ***
--- a/src/HOL/Quotient_Examples/FSet.thy Mon Sep 19 20:07:39 2016 +0200
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,1163 +0,0 @@
-(* Title: HOL/Quotient_Examples/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 FSet
-imports "~~/src/HOL/Library/Multiset" "~~/src/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 \<Rightarrow> 'a list \<Rightarrow> bool" (infix "\<approx>" 50)
-where
- [simp]: "xs \<approx> ys \<longleftrightarrow> 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 \<Rightarrow> 'a list \<Rightarrow> bool"
-where
- [simp]: "sub_list xs ys \<longleftrightarrow> set xs \<subseteq> set ys"
-
-definition
- card_list :: "'a list \<Rightarrow> nat"
-where
- [simp]: "card_list xs = card (set xs)"
-
-definition
- inter_list :: "'a list \<Rightarrow> 'a list \<Rightarrow> 'a list"
-where
- [simp]: "inter_list xs ys = [x \<leftarrow> xs. x \<in> set xs \<and> x \<in> set ys]"
-
-definition
- diff_list :: "'a list \<Rightarrow> 'a list \<Rightarrow> 'a list"
-where
- [simp]: "diff_list xs ys = [x \<leftarrow> xs. x \<notin> set ys]"
-
-definition
- rsp_fold :: "('a \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> bool"
-where
- "rsp_fold f \<longleftrightarrow> (\<forall>u v. f u \<circ> f v = f v \<circ> f u)"
-
-lemma rsp_foldI:
- "(\<And>u v. f u \<circ> f v = f v \<circ> f u) \<Longrightarrow> rsp_fold f"
- by (simp add: rsp_fold_def)
-
-lemma rsp_foldE:
- assumes "rsp_fold f"
- obtains "f u \<circ> f v = f v \<circ> f u"
- using assms by (simp add: rsp_fold_def)
-
-definition
- fold_once :: "('a \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> 'a list \<Rightarrow> 'b \<Rightarrow> 'b"
-where
- "fold_once f xs = (if rsp_fold f then fold f (remdups xs) else id)"
-
-lemma fold_once_default [simp]:
- "\<not> rsp_fold f \<Longrightarrow> fold_once f xs = id"
- by (simp add: fold_once_def)
-
-lemma fold_once_fold_remdups:
- "rsp_fold f \<Longrightarrow> 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 op \<approx>) r r"
-proof
- have *: "r \<approx> r" by (rule equivp_reflp[OF fset_equivp])
- show "list_all2 R r r" by (rule list_all2_refl'[OF q])
- with * show "(op \<approx> OO list_all2 R) r r" ..
-qed
-
-lemma map_list_eq_cong: "b \<approx> ba \<Longrightarrow> map f b \<approx> 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 (op \<approx>))
- (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: "(op \<approx> 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 op \<approx>) (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 op \<approx>) r s = ((list_all2 R OOO op \<approx>) r r \<and>
- (list_all2 R OOO op \<approx>) s s \<and> abs_fset (map Abs r) = abs_fset (map Abs s))"
- proof (intro iffI conjI)
- show "(list_all2 R OOO op \<approx>) r r" by (rule compose_list_refl[OF e])
- show "(list_all2 R OOO op \<approx>) s s" by (rule compose_list_refl[OF e])
- next
- assume a: "(list_all2 R OOO op \<approx>) r s"
- then have b: "map Abs r \<approx> map Abs s"
- proof (elim relcomppE)
- fix b ba
- assume c: "list_all2 R r b"
- assume d: "b \<approx> 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 \<approx> 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 op \<approx>) r r \<and> (list_all2 R OOO op \<approx>) s s
- \<and> abs_fset (map Abs r) = abs_fset (map Abs s)"
- then have s: "(list_all2 R OOO op \<approx>) s s" by simp
- have d: "map Abs r \<approx> map Abs s"
- by (subst Quotient3_rel [OF Quotient3_fset, symmetric]) (simp add: a)
- have b: "map Rep (map Abs r) \<approx> 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: "(op \<approx> 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 op \<approx>) r s"
- using a c relcomppI by simp
- qed
-qed
-
-lemma quotient_compose_list[quot_thm]:
- shows "Quotient3 ((list_all2 op \<approx>) OOO (op \<approx>))
- (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
- "{||} \<equiv> bot :: 'a fset"
-
-quotient_definition
- "less_eq_fset :: ('a fset \<Rightarrow> 'a fset \<Rightarrow> bool)"
- is "sub_list :: ('a list \<Rightarrow> 'a list \<Rightarrow> bool)" by simp
-
-abbreviation
- subset_fset :: "'a fset \<Rightarrow> 'a fset \<Rightarrow> bool" (infix "|\<subseteq>|" 50)
-where
- "xs |\<subseteq>| ys \<equiv> xs \<le> ys"
-
-definition
- less_fset :: "'a fset \<Rightarrow> 'a fset \<Rightarrow> bool"
-where
- "xs < ys \<equiv> xs \<le> ys \<and> xs \<noteq> (ys::'a fset)"
-
-abbreviation
- psubset_fset :: "'a fset \<Rightarrow> 'a fset \<Rightarrow> bool" (infix "|\<subset>|" 50)
-where
- "xs |\<subset>| ys \<equiv> xs < ys"
-
-quotient_definition
- "sup :: 'a fset \<Rightarrow> 'a fset \<Rightarrow> 'a fset"
- is "append :: 'a list \<Rightarrow> 'a list \<Rightarrow> 'a list" by simp
-
-abbreviation
- union_fset (infixl "|\<union>|" 65)
-where
- "xs |\<union>| ys \<equiv> sup xs (ys::'a fset)"
-
-quotient_definition
- "inf :: 'a fset \<Rightarrow> 'a fset \<Rightarrow> 'a fset"
- is "inter_list :: 'a list \<Rightarrow> 'a list \<Rightarrow> 'a list" by simp
-
-abbreviation
- inter_fset (infixl "|\<inter>|" 65)
-where
- "xs |\<inter>| ys \<equiv> inf xs (ys::'a fset)"
-
-quotient_definition
- "minus :: 'a fset \<Rightarrow> 'a fset \<Rightarrow> 'a fset"
- is "diff_list :: 'a list \<Rightarrow> 'a list \<Rightarrow> 'a list" by fastforce
-
-instance
-proof
- fix x y z :: "'a fset"
- show "x |\<subset>| y \<longleftrightarrow> x |\<subseteq>| y \<and> \<not> y |\<subseteq>| x"
- by (unfold less_fset_def, descending) auto
- show "x |\<subseteq>| x" by (descending) (simp)
- show "{||} |\<subseteq>| x" by (descending) (simp)
- show "x |\<subseteq>| x |\<union>| y" by (descending) (simp)
- show "y |\<subseteq>| x |\<union>| y" by (descending) (simp)
- show "x |\<inter>| y |\<subseteq>| x" by (descending) (auto)
- show "x |\<inter>| y |\<subseteq>| y" by (descending) (auto)
- show "x |\<union>| (y |\<inter>| z) = x |\<union>| y |\<inter>| (x |\<union>| z)"
- by (descending) (auto)
-next
- fix x y z :: "'a fset"
- assume a: "x |\<subseteq>| y"
- assume b: "y |\<subseteq>| z"
- show "x |\<subseteq>| z" using a b by (descending) (simp)
-next
- fix x y :: "'a fset"
- assume a: "x |\<subseteq>| y"
- assume b: "y |\<subseteq>| x"
- show "x = y" using a b by (descending) (auto)
-next
- fix x y z :: "'a fset"
- assume a: "y |\<subseteq>| x"
- assume b: "z |\<subseteq>| x"
- show "y |\<union>| z |\<subseteq>| x" using a b by (descending) (simp)
-next
- fix x y z :: "'a fset"
- assume a: "x |\<subseteq>| y"
- assume b: "x |\<subseteq>| z"
- show "x |\<subseteq>| y |\<inter>| z" using a b by (descending) (auto)
-qed
-
-end
-
-
-subsection \<open>Other constants for fsets\<close>
-
-quotient_definition
- "insert_fset :: 'a \<Rightarrow> 'a fset \<Rightarrow> '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 \<Rightarrow> 'a \<Rightarrow> bool" is "List.member" by fastforce
-
-abbreviation
- in_fset :: "'a \<Rightarrow> 'a fset \<Rightarrow> bool" (infix "|\<in>|" 50)
-where
- "x |\<in>| S \<equiv> fset_member S x"
-
-abbreviation
- notin_fset :: "'a \<Rightarrow> 'a fset \<Rightarrow> bool" (infix "|\<notin>|" 50)
-where
- "x |\<notin>| S \<equiv> \<not> (x |\<in>| S)"
-
-
-subsection \<open>Other constants on the Quotient Type\<close>
-
-quotient_definition
- "card_fset :: 'a fset \<Rightarrow> nat"
- is card_list by simp
-
-quotient_definition
- "map_fset :: ('a \<Rightarrow> 'b) \<Rightarrow> 'a fset \<Rightarrow> 'b fset"
- is map by simp
-
-quotient_definition
- "remove_fset :: 'a \<Rightarrow> 'a fset \<Rightarrow> 'a fset"
- is removeAll by simp
-
-quotient_definition
- "fset :: 'a fset \<Rightarrow> 'a set"
- is "set" by simp
-
-lemma fold_once_set_equiv:
- assumes "xs \<approx> 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 "\<And>x y. x \<in> set (remdups xs) \<Longrightarrow> y \<in> set (remdups xs) \<Longrightarrow> f x \<circ> f y = f y \<circ> 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 \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> 'a fset \<Rightarrow> 'b \<Rightarrow> 'b"
- is fold_once by (rule fold_once_set_equiv)
-
-lemma concat_rsp_pre:
- assumes a: "list_all2 op \<approx> x x'"
- and b: "x' \<approx> y'"
- and c: "list_all2 op \<approx> y' y"
- and d: "\<exists>x\<in>set x. xa \<in> set x"
- shows "\<exists>x\<in>set y. xa \<in> set x"
-proof -
- obtain xb where e: "xb \<in> set x" and f: "xa \<in> set xb" using d by auto
- have "\<exists>y. y \<in> set x' \<and> xb \<approx> y" by (rule list_all2_find_element[OF e a])
- then obtain ya where h: "ya \<in> set x'" and i: "xb \<approx> ya" by auto
- have "ya \<in> set y'" using b h by simp
- then have "\<exists>yb. yb \<in> set y \<and> ya \<approx> 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 \<Rightarrow> 'a fset"
- is concat
-proof (elim relcomppE)
-fix a b ba bb
- assume a: "list_all2 op \<approx> a ba"
- with list_symp [OF list_eq_symp] have a': "list_all2 op \<approx> ba a" by (rule sympE)
- assume b: "ba \<approx> bb"
- with list_eq_symp have b': "bb \<approx> ba" by (rule sympE)
- assume c: "list_all2 op \<approx> bb b"
- with list_symp [OF list_eq_symp] have c': "list_all2 op \<approx> b bb" by (rule sympE)
- have "\<forall>x. (\<exists>xa\<in>set a. x \<in> set xa) = (\<exists>xa\<in>set b. x \<in> set xa)"
- proof
- fix x
- show "(\<exists>xa\<in>set a. x \<in> set xa) = (\<exists>xa\<in>set b. x \<in> set xa)"
- proof
- assume d: "\<exists>xa\<in>set a. x \<in> set xa"
- show "\<exists>xa\<in>set b. x \<in> set xa" by (rule concat_rsp_pre[OF a b c d])
- next
- assume e: "\<exists>xa\<in>set b. x \<in> set xa"
- show "\<exists>xa\<in>set a. x \<in> set xa" by (rule concat_rsp_pre[OF c' b' a' e])
- qed
- qed
- then show "concat a \<approx> concat b" by auto
-qed
-
-quotient_definition
- "filter_fset :: ('a \<Rightarrow> bool) \<Rightarrow> 'a fset \<Rightarrow> 'a fset"
- is filter by force
-
-
-subsection \<open>Compositional respectfulness and preservation lemmas\<close>
-
-lemma Nil_rsp2 [quot_respect]:
- shows "(list_all2 op \<approx> OOO op \<approx>) Nil Nil"
- by (rule compose_list_refl, rule list_eq_equivp)
-
-lemma Cons_rsp2 [quot_respect]:
- shows "(op \<approx> ===> list_all2 op \<approx> OOO op \<approx> ===> list_all2 op \<approx> OOO op \<approx>) 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 \<circ> 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 \<circ> Rep2) ---> (Abs2 \<circ> map Abs1)) (op #) = (id ---> Rep2 ---> Abs2) (op #)"
- 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 \<circ> Rep2) ---> (map Rep1 \<circ> Rep2) ---> (Abs2 \<circ> map Abs1)) op @ =
- (Rep2 ---> Rep2 ---> Abs2) op @"
- 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 op \<approx> x x'"
- shows "list_all2 op \<approx> (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 op \<approx> x x'"
- shows "list_all2 op \<approx> (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 op \<approx> x x'"
- and "list_all2 op \<approx> z z'"
- shows "list_all2 op \<approx> (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 op \<approx> OOO op \<approx> ===> list_all2 op \<approx> OOO op \<approx> ===> list_all2 op \<approx> OOO op \<approx>) append append"
- by (intro compositional_rsp3)
- (auto intro!: rel_funI simp add: append_rsp2_pre)
-
-lemma map_rsp2 [quot_respect]:
- "((op \<approx> ===> op \<approx>) ===> list_all2 op \<approx> OOO op \<approx> ===> list_all2 op \<approx> OOO op \<approx>) 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: "(op \<approx> ===> op \<approx>) f f'" and x: "list_all2 op \<approx> xa x" and xy: "set x = set y" and y: "list_all2 op \<approx> y ya"
- have a: "(list_all2 op \<approx>) (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 op \<approx>) (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 op \<approx> OOO op \<approx>) (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 \<circ> rep_fset) ---> abs_fset \<circ> 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 \<longleftrightarrow> S = T"
- by (descending) (simp)
-
-lemma filter_fset [simp]:
- shows "fset (filter_fset P xs) = Collect P \<inter> 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 |\<inter>| ys) = fset xs \<inter> fset ys"
- by (descending) (auto)
-
-lemma union_fset [simp]:
- shows "fset (xs |\<union>| ys) = fset xs \<union> 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 |\<in>| S \<longleftrightarrow> x \<in> fset S"
- by descending simp
-
-lemma notin_fset:
- shows "x |\<notin>| S \<longleftrightarrow> x \<notin> fset S"
- by (simp add: in_fset)
-
-lemma notin_empty_fset:
- shows "x |\<notin>| {||}"
- by (simp add: in_fset)
-
-lemma fset_eq_iff:
- shows "S = T \<longleftrightarrow> (\<forall>x. (x |\<in>| S) = (x |\<in>| T))"
- by descending auto
-
-lemma none_in_empty_fset:
- shows "(\<forall>x. x |\<notin>| S) \<longleftrightarrow> S = {||}"
- by descending simp
-
-
-subsection \<open>insert_fset\<close>
-
-lemma in_insert_fset_iff [simp]:
- shows "x |\<in>| insert_fset y S \<longleftrightarrow> x = y \<or> x |\<in>| S"
- by descending simp
-
-lemma
- shows insert_fsetI1: "x |\<in>| insert_fset x S"
- and insert_fsetI2: "x |\<in>| S \<Longrightarrow> x |\<in>| insert_fset y S"
- by simp_all
-
-lemma insert_absorb_fset [simp]:
- shows "x |\<in>| S \<Longrightarrow> insert_fset x S = S"
- by (descending) (auto)
-
-lemma empty_not_insert_fset[simp]:
- shows "{||} \<noteq> insert_fset x S"
- and "insert_fset x S \<noteq> {||}"
- 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|} \<longleftrightarrow> x = y"
- by (descending) (auto)
-
-lemma in_fset_mdef:
- shows "x |\<in>| F \<longleftrightarrow> x |\<notin>| (F - {|x|}) \<and> 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 |\<union>| T = insert_fset x (S |\<union>| T)"
- by (lifting append.simps(2))
-
-lemma singleton_union_fset_left:
- shows "{|a|} |\<union>| S = insert_fset a S"
- by simp
-
-lemma singleton_union_fset_right:
- shows "S |\<union>| {|a|} = insert_fset a S"
- by (subst sup.commute) simp
-
-lemma in_union_fset:
- shows "x |\<in>| S |\<union>| T \<longleftrightarrow> x |\<in>| S \<or> x |\<in>| T"
- by (descending) (simp)
-
-
-subsection \<open>minus_fset\<close>
-
-lemma minus_in_fset:
- shows "x |\<in>| (xs - ys) \<longleftrightarrow> x |\<in>| xs \<and> x |\<notin>| ys"
- by (descending) (simp)
-
-lemma minus_insert_fset:
- shows "insert_fset x xs - ys = (if x |\<in>| ys then xs - ys else insert_fset x (xs - ys))"
- by (descending) (auto)
-
-lemma minus_insert_in_fset[simp]:
- shows "x |\<in>| ys \<Longrightarrow> insert_fset x xs - ys = xs - ys"
- by (simp add: minus_insert_fset)
-
-lemma minus_insert_notin_fset[simp]:
- shows "x |\<notin>| ys \<Longrightarrow> insert_fset x xs - ys = insert_fset x (xs - ys)"
- by (simp add: minus_insert_fset)
-
-lemma in_minus_fset:
- shows "x |\<in>| F - S \<Longrightarrow> x |\<notin>| S"
- unfolding in_fset minus_fset
- by blast
-
-lemma notin_minus_fset:
- shows "x |\<in>| S \<Longrightarrow> x |\<notin>| F - S"
- unfolding in_fset minus_fset
- by blast
-
-
-subsection \<open>remove_fset\<close>
-
-lemma in_remove_fset:
- shows "x |\<in>| remove_fset y S \<longleftrightarrow> x |\<in>| S \<and> x \<noteq> y"
- by (descending) (simp)
-
-lemma notin_remove_fset:
- shows "x |\<notin>| remove_fset x S"
- by (descending) (simp)
-
-lemma notin_remove_ident_fset:
- shows "x |\<notin>| S \<Longrightarrow> remove_fset x S = S"
- by (descending) (simp)
-
-lemma remove_fset_cases:
- shows "S = {||} \<or> (\<exists>x. x |\<in>| S \<and> 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 "{||} |\<inter>| S = {||}"
- by simp
-
-lemma inter_empty_fset_r:
- shows "S |\<inter>| {||} = {||}"
- by simp
-
-lemma inter_insert_fset:
- shows "insert_fset x S |\<inter>| T = (if x |\<in>| T then insert_fset x (S |\<inter>| T) else S |\<inter>| T)"
- by (descending) (auto)
-
-lemma in_inter_fset:
- shows "x |\<in>| (S |\<inter>| T) \<longleftrightarrow> x |\<in>| S \<and> x |\<in>| T"
- by (descending) (simp)
-
-
-subsection \<open>subset_fset and psubset_fset\<close>
-
-lemma subset_fset:
- shows "xs |\<subseteq>| ys \<longleftrightarrow> fset xs \<subseteq> fset ys"
- by (descending) (simp)
-
-lemma psubset_fset:
- shows "xs |\<subset>| ys \<longleftrightarrow> fset xs \<subset> fset ys"
- unfolding less_fset_def
- by (descending) (auto)
-
-lemma subset_insert_fset:
- shows "(insert_fset x xs) |\<subseteq>| ys \<longleftrightarrow> x |\<in>| ys \<and> xs |\<subseteq>| ys"
- by (descending) (simp)
-
-lemma subset_in_fset:
- shows "xs |\<subseteq>| ys = (\<forall>x. x |\<in>| xs \<longrightarrow> x |\<in>| ys)"
- by (descending) (auto)
-
-lemma subset_empty_fset:
- shows "xs |\<subseteq>| {||} \<longleftrightarrow> xs = {||}"
- by (descending) (simp)
-
-lemma not_psubset_empty_fset:
- shows "\<not> xs |\<subset>| {||}"
- 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 \<Longrightarrow> map_fset f S = map_fset f T \<longleftrightarrow> S = T"
- by (descending) (metis inj_vimage_image_eq list_eq_def set_map)
-
-lemma map_union_fset:
- shows "map_fset f (S |\<union>| T) = map_fset f S |\<union>| map_fset f T"
- by (descending) (simp)
-
-lemma in_fset_map_fset[simp]: "a |\<in>| map_fset f X = (\<exists>b. b |\<in>| X \<and> 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 |\<in>| 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 \<longleftrightarrow> 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 \<longleftrightarrow> (\<exists>x. S = {|x|})"
- by (descending) (auto simp add: card_Suc_eq)
-
-lemma card_fset_gt_0:
- shows "x \<in> fset S \<Longrightarrow> 0 < card_fset S"
- by (descending) (auto simp add: card_gt_0_iff)
-
-lemma card_notin_fset:
- shows "(x |\<notin>| S) = (card_fset (insert_fset x S) = Suc (card_fset S))"
- by simp
-
-lemma card_fset_Suc:
- shows "card_fset S = Suc n \<Longrightarrow> \<exists>x T. x |\<notin>| T \<and> S = insert_fset x T \<and> 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 |\<in>| 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 \<Longrightarrow> \<exists>a. a |\<in>| S"
- by (drule card_fset_Suc) (auto)
-
-lemma in_card_fset_not_0:
- shows "a |\<in>| A \<Longrightarrow> card_fset A \<noteq> 0"
- by (descending) (auto)
-
-lemma card_fset_mono:
- shows "xs |\<subseteq>| ys \<Longrightarrow> card_fset xs \<le> card_fset ys"
- unfolding card_fset psubset_fset
- by (simp add: card_mono subset_fset)
-
-lemma card_subset_fset_eq:
- shows "xs |\<subseteq>| ys \<Longrightarrow> card_fset ys \<le> card_fset xs \<Longrightarrow> 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 |\<subset>| ys \<Longrightarrow> 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 |\<union>| ys) + card_fset (xs |\<inter>| 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 |\<inter>| ys = {||} \<Longrightarrow> card_fset (xs |\<union>| 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 |\<in>| xs \<Longrightarrow> 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 |\<in>| xs \<Longrightarrow> y |\<in>| xs \<Longrightarrow> 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) \<le> card_fset xs"
- unfolding remove_fset card_fset
- by (rule card_Diff1_le[OF finite_fset])
-
-lemma card_psubset_fset:
- shows "ys |\<subseteq>| xs \<Longrightarrow> card_fset ys < card_fset xs \<Longrightarrow> ys |\<subset>| 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) \<le> 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 |\<in>| A" and "a |\<notin>| 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 |\<subseteq>| 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 |\<inter>| 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 |\<union>| concat_fset S"
- by descending simp
-
-lemma concat_union_fset [simp]:
- shows "concat_fset (xs |\<union>| ys) = concat_fset xs |\<union>| 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 |\<subseteq>| filter_fset Q xs = (\<forall> x. x |\<in>| xs \<longrightarrow> P x \<longrightarrow> Q x)"
- by descending auto
-
-lemma eq_filter_fset:
- "(filter_fset P xs = filter_fset Q xs) = (\<forall>x. x |\<in>| xs \<longrightarrow> P x = Q x)"
- by descending auto
-
-lemma psubset_filter_fset:
- "(\<And>x. x |\<in>| xs \<Longrightarrow> P x \<Longrightarrow> Q x) \<Longrightarrow> (x |\<in>| xs & \<not> P x & Q x) \<Longrightarrow>
- 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 \<in> set xs"
- shows "fold_once f xs = fold_once f (removeAll x xs) \<circ> f x"
-proof -
- from assms have "fold f (remdups xs) = fold f (remove1 x (remdups xs)) \<circ> 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 \<Longrightarrow> h |\<in>| b \<Longrightarrow> fold_fset f b = fold_fset f (remove_fset h b) \<circ> f h"
- by descending (simp add: member_commute_fold_once)
-
-
-subsection \<open>Choice in fsets\<close>
-
-lemma fset_choice:
- assumes a: "\<forall>x. x |\<in>| A \<longrightarrow> (\<exists>y. P x y)"
- shows "\<exists>f. \<forall>x. x |\<in>| A \<longrightarrow> 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 = {||} \<Longrightarrow> P"
- and insert_fset_case: "\<And>x S'. S = insert_fset x S' \<Longrightarrow> 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: "\<And>x S. P S \<Longrightarrow> 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: "\<And>x S. \<lbrakk>x |\<notin>| S; P S\<rbrakk> \<Longrightarrow> 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 |\<in>| S") (simp_all)
-qed
-
-lemma fset_card_induct:
- assumes empty_fset_case: "P {||}"
- and card_fset_Suc_case: "\<And>S T. Suc (card_fset S) = (card_fset T) \<Longrightarrow> P S \<Longrightarrow> 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 |\<notin>| 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 "\<not> List.member ys x" and "xs \<approx> x # ys"
-proof (induct xs)
- case Nil
- then show thesis by simp
-next
- case (Cons a xs)
- have a: "\<lbrakk>xs = [] \<Longrightarrow> thesis; \<And>x ys. \<lbrakk>\<not> List.member ys x; xs \<approx> x # ys\<rbrakk> \<Longrightarrow> thesis\<rbrakk> \<Longrightarrow> thesis"
- by (rule Cons(1))
- have b: "\<And>x' ys'. \<lbrakk>\<not> List.member ys' x'; a # xs \<approx> x' # ys'\<rbrakk> \<Longrightarrow> thesis" by fact
- have c: "xs = [] \<Longrightarrow> thesis" using b
- apply(simp)
- by (metis list.set(1) emptyE empty_subsetI)
- have "\<And>x ys. \<lbrakk>\<not> List.member ys x; xs \<approx> x # ys\<rbrakk> \<Longrightarrow> thesis"
- proof -
- fix x :: 'a
- fix ys :: "'a list"
- assume d:"\<not> List.member ys x"
- assume e:"xs \<approx> x # ys"
- show thesis
- proof (cases "x = a")
- assume h: "x = a"
- then have f: "\<not> List.member ys a" using d by simp
- have g: "a # xs \<approx> a # ys" using e h by auto
- show thesis using b f g by simp
- next
- assume h: "x \<noteq> a"
- then have f: "\<not> List.member (a # ys) x" using d by auto
- have g: "a # xs \<approx> 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 |\<notin>| ys" and "xs = insert_fset x ys"
- by (lifting fset_raw_strong_cases)
-
-
-lemma fset_induct2:
- "P {||} {||} \<Longrightarrow>
- (\<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 "\<And>x. x \<in> fset A \<longleftrightarrow> x \<in> 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 "\<And>z. z \<in> fset A \<longleftrightarrow> z \<in> 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 |\<in>| 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 \<Rightarrow> 'a list \<Rightarrow> bool" ("_ \<approx>2 _")
-where
- "(a # b # xs) \<approx>2 (b # a # xs)"
-| "[] \<approx>2 []"
-| "xs \<approx>2 ys \<Longrightarrow> ys \<approx>2 xs"
-| "(a # a # xs) \<approx>2 (a # xs)"
-| "xs \<approx>2 ys \<Longrightarrow> (a # xs) \<approx>2 (a # ys)"
-| "xs1 \<approx>2 xs2 \<Longrightarrow> xs2 \<approx>2 xs3 \<Longrightarrow> xs1 \<approx>2 xs3"
-
-lemma list_eq2_refl:
- shows "xs \<approx>2 xs"
- by (induct xs) (auto intro: list_eq2.intros)
-
-lemma cons_delete_list_eq2:
- shows "(a # (removeAll a A)) \<approx>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) \<approx>2 r"
- using a cons_delete_list_eq2[of e r]
- by simp
-
-lemma list_eq2_equiv:
- "(l \<approx> r) \<longleftrightarrow> (list_eq2 l r)"
-proof
- show "list_eq2 l r \<Longrightarrow> l \<approx> r" by (induct rule: list_eq2.induct) auto
-next
- {
- fix n
- assume a: "card_list l = n" and b: "l \<approx> r"
- have "l \<approx>2 r"
- using a b
- proof (induct n arbitrary: l r)
- case 0
- have "card_list l = 0" by fact
- then have "\<forall>x. \<not> 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 \<approx> r" by fact
- have d: "card_list l = Suc m" by fact
- then have "\<exists>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 \<approx> removeAll a r" using remove_fset.rsp b by simp
- have "(removeAll a l) \<approx>2 (removeAll a r)" by (rule Suc.hyps[OF f g])
- then have h: "(a # removeAll a l) \<approx>2 (a # removeAll a r)" by (rule list_eq2.intros(5))
- have i: "l \<approx>2 (a # removeAll a l)"
- by (rule list_eq2.intros(3)[OF member_delete_list_eq2[OF e]])
- have "l \<approx>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 \<approx> r \<Longrightarrow> l \<approx>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:
- "\<lbrakk>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 "\<And>a b xs. P (insert_fset a (insert_fset b xs)) (insert_fset b (insert_fset a xs))"
- and "P {||} {||}"
- and "\<And>xs ys. \<lbrakk>xs = ys; P xs ys\<rbrakk> \<Longrightarrow> P ys xs"
- and "\<And>a xs. P (insert_fset a (insert_fset a xs)) (insert_fset a xs)"
- and "\<And>xs ys a. \<lbrakk>xs = ys; P xs ys\<rbrakk> \<Longrightarrow> P (insert_fset a xs) (insert_fset a ys)"
- and "\<And>xs1 xs2 xs3. \<lbrakk>xs1 = xs2; P xs1 xs2; xs2 = xs3; P xs2 xs3\<rbrakk> \<Longrightarrow> 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 fset}, [T])) = T
- | dest_fsetT T = raise TYPE ("dest_fsetT: fset type expected", [T], []);
-\<close>
-
-no_notation
- list_eq (infix "\<approx>" 50) and
- list_eq2 (infix "\<approx>2" 50)
-
-end
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Quotient_Examples/Quotient_FSet.thy Mon Sep 19 23:14:34 2016 +0200
@@ -0,0 +1,1163 @@
+(* 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 "~~/src/HOL/Library/Multiset" "~~/src/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 \<Rightarrow> 'a list \<Rightarrow> bool" (infix "\<approx>" 50)
+where
+ [simp]: "xs \<approx> ys \<longleftrightarrow> 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 \<Rightarrow> 'a list \<Rightarrow> bool"
+where
+ [simp]: "sub_list xs ys \<longleftrightarrow> set xs \<subseteq> set ys"
+
+definition
+ card_list :: "'a list \<Rightarrow> nat"
+where
+ [simp]: "card_list xs = card (set xs)"
+
+definition
+ inter_list :: "'a list \<Rightarrow> 'a list \<Rightarrow> 'a list"
+where
+ [simp]: "inter_list xs ys = [x \<leftarrow> xs. x \<in> set xs \<and> x \<in> set ys]"
+
+definition
+ diff_list :: "'a list \<Rightarrow> 'a list \<Rightarrow> 'a list"
+where
+ [simp]: "diff_list xs ys = [x \<leftarrow> xs. x \<notin> set ys]"
+
+definition
+ rsp_fold :: "('a \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> bool"
+where
+ "rsp_fold f \<longleftrightarrow> (\<forall>u v. f u \<circ> f v = f v \<circ> f u)"
+
+lemma rsp_foldI:
+ "(\<And>u v. f u \<circ> f v = f v \<circ> f u) \<Longrightarrow> rsp_fold f"
+ by (simp add: rsp_fold_def)
+
+lemma rsp_foldE:
+ assumes "rsp_fold f"
+ obtains "f u \<circ> f v = f v \<circ> f u"
+ using assms by (simp add: rsp_fold_def)
+
+definition
+ fold_once :: "('a \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> 'a list \<Rightarrow> 'b \<Rightarrow> 'b"
+where
+ "fold_once f xs = (if rsp_fold f then fold f (remdups xs) else id)"
+
+lemma fold_once_default [simp]:
+ "\<not> rsp_fold f \<Longrightarrow> fold_once f xs = id"
+ by (simp add: fold_once_def)
+
+lemma fold_once_fold_remdups:
+ "rsp_fold f \<Longrightarrow> 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 op \<approx>) r r"
+proof
+ have *: "r \<approx> r" by (rule equivp_reflp[OF fset_equivp])
+ show "list_all2 R r r" by (rule list_all2_refl'[OF q])
+ with * show "(op \<approx> OO list_all2 R) r r" ..
+qed
+
+lemma map_list_eq_cong: "b \<approx> ba \<Longrightarrow> map f b \<approx> 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 (op \<approx>))
+ (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: "(op \<approx> 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 op \<approx>) (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 op \<approx>) r s = ((list_all2 R OOO op \<approx>) r r \<and>
+ (list_all2 R OOO op \<approx>) s s \<and> abs_fset (map Abs r) = abs_fset (map Abs s))"
+ proof (intro iffI conjI)
+ show "(list_all2 R OOO op \<approx>) r r" by (rule compose_list_refl[OF e])
+ show "(list_all2 R OOO op \<approx>) s s" by (rule compose_list_refl[OF e])
+ next
+ assume a: "(list_all2 R OOO op \<approx>) r s"
+ then have b: "map Abs r \<approx> map Abs s"
+ proof (elim relcomppE)
+ fix b ba
+ assume c: "list_all2 R r b"
+ assume d: "b \<approx> 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 \<approx> 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 op \<approx>) r r \<and> (list_all2 R OOO op \<approx>) s s
+ \<and> abs_fset (map Abs r) = abs_fset (map Abs s)"
+ then have s: "(list_all2 R OOO op \<approx>) s s" by simp
+ have d: "map Abs r \<approx> map Abs s"
+ by (subst Quotient3_rel [OF Quotient3_fset, symmetric]) (simp add: a)
+ have b: "map Rep (map Abs r) \<approx> 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: "(op \<approx> 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 op \<approx>) r s"
+ using a c relcomppI by simp
+ qed
+qed
+
+lemma quotient_compose_list[quot_thm]:
+ shows "Quotient3 ((list_all2 op \<approx>) OOO (op \<approx>))
+ (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
+ "{||} \<equiv> bot :: 'a fset"
+
+quotient_definition
+ "less_eq_fset :: ('a fset \<Rightarrow> 'a fset \<Rightarrow> bool)"
+ is "sub_list :: ('a list \<Rightarrow> 'a list \<Rightarrow> bool)" by simp
+
+abbreviation
+ subset_fset :: "'a fset \<Rightarrow> 'a fset \<Rightarrow> bool" (infix "|\<subseteq>|" 50)
+where
+ "xs |\<subseteq>| ys \<equiv> xs \<le> ys"
+
+definition
+ less_fset :: "'a fset \<Rightarrow> 'a fset \<Rightarrow> bool"
+where
+ "xs < ys \<equiv> xs \<le> ys \<and> xs \<noteq> (ys::'a fset)"
+
+abbreviation
+ psubset_fset :: "'a fset \<Rightarrow> 'a fset \<Rightarrow> bool" (infix "|\<subset>|" 50)
+where
+ "xs |\<subset>| ys \<equiv> xs < ys"
+
+quotient_definition
+ "sup :: 'a fset \<Rightarrow> 'a fset \<Rightarrow> 'a fset"
+ is "append :: 'a list \<Rightarrow> 'a list \<Rightarrow> 'a list" by simp
+
+abbreviation
+ union_fset (infixl "|\<union>|" 65)
+where
+ "xs |\<union>| ys \<equiv> sup xs (ys::'a fset)"
+
+quotient_definition
+ "inf :: 'a fset \<Rightarrow> 'a fset \<Rightarrow> 'a fset"
+ is "inter_list :: 'a list \<Rightarrow> 'a list \<Rightarrow> 'a list" by simp
+
+abbreviation
+ inter_fset (infixl "|\<inter>|" 65)
+where
+ "xs |\<inter>| ys \<equiv> inf xs (ys::'a fset)"
+
+quotient_definition
+ "minus :: 'a fset \<Rightarrow> 'a fset \<Rightarrow> 'a fset"
+ is "diff_list :: 'a list \<Rightarrow> 'a list \<Rightarrow> 'a list" by fastforce
+
+instance
+proof
+ fix x y z :: "'a fset"
+ show "x |\<subset>| y \<longleftrightarrow> x |\<subseteq>| y \<and> \<not> y |\<subseteq>| x"
+ by (unfold less_fset_def, descending) auto
+ show "x |\<subseteq>| x" by (descending) (simp)
+ show "{||} |\<subseteq>| x" by (descending) (simp)
+ show "x |\<subseteq>| x |\<union>| y" by (descending) (simp)
+ show "y |\<subseteq>| x |\<union>| y" by (descending) (simp)
+ show "x |\<inter>| y |\<subseteq>| x" by (descending) (auto)
+ show "x |\<inter>| y |\<subseteq>| y" by (descending) (auto)
+ show "x |\<union>| (y |\<inter>| z) = x |\<union>| y |\<inter>| (x |\<union>| z)"
+ by (descending) (auto)
+next
+ fix x y z :: "'a fset"
+ assume a: "x |\<subseteq>| y"
+ assume b: "y |\<subseteq>| z"
+ show "x |\<subseteq>| z" using a b by (descending) (simp)
+next
+ fix x y :: "'a fset"
+ assume a: "x |\<subseteq>| y"
+ assume b: "y |\<subseteq>| x"
+ show "x = y" using a b by (descending) (auto)
+next
+ fix x y z :: "'a fset"
+ assume a: "y |\<subseteq>| x"
+ assume b: "z |\<subseteq>| x"
+ show "y |\<union>| z |\<subseteq>| x" using a b by (descending) (simp)
+next
+ fix x y z :: "'a fset"
+ assume a: "x |\<subseteq>| y"
+ assume b: "x |\<subseteq>| z"
+ show "x |\<subseteq>| y |\<inter>| z" using a b by (descending) (auto)
+qed
+
+end
+
+
+subsection \<open>Other constants for fsets\<close>
+
+quotient_definition
+ "insert_fset :: 'a \<Rightarrow> 'a fset \<Rightarrow> '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 \<Rightarrow> 'a \<Rightarrow> bool" is "List.member" by fastforce
+
+abbreviation
+ in_fset :: "'a \<Rightarrow> 'a fset \<Rightarrow> bool" (infix "|\<in>|" 50)
+where
+ "x |\<in>| S \<equiv> fset_member S x"
+
+abbreviation
+ notin_fset :: "'a \<Rightarrow> 'a fset \<Rightarrow> bool" (infix "|\<notin>|" 50)
+where
+ "x |\<notin>| S \<equiv> \<not> (x |\<in>| S)"
+
+
+subsection \<open>Other constants on the Quotient Type\<close>
+
+quotient_definition
+ "card_fset :: 'a fset \<Rightarrow> nat"
+ is card_list by simp
+
+quotient_definition
+ "map_fset :: ('a \<Rightarrow> 'b) \<Rightarrow> 'a fset \<Rightarrow> 'b fset"
+ is map by simp
+
+quotient_definition
+ "remove_fset :: 'a \<Rightarrow> 'a fset \<Rightarrow> 'a fset"
+ is removeAll by simp
+
+quotient_definition
+ "fset :: 'a fset \<Rightarrow> 'a set"
+ is "set" by simp
+
+lemma fold_once_set_equiv:
+ assumes "xs \<approx> 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 "\<And>x y. x \<in> set (remdups xs) \<Longrightarrow> y \<in> set (remdups xs) \<Longrightarrow> f x \<circ> f y = f y \<circ> 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 \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> 'a fset \<Rightarrow> 'b \<Rightarrow> 'b"
+ is fold_once by (rule fold_once_set_equiv)
+
+lemma concat_rsp_pre:
+ assumes a: "list_all2 op \<approx> x x'"
+ and b: "x' \<approx> y'"
+ and c: "list_all2 op \<approx> y' y"
+ and d: "\<exists>x\<in>set x. xa \<in> set x"
+ shows "\<exists>x\<in>set y. xa \<in> set x"
+proof -
+ obtain xb where e: "xb \<in> set x" and f: "xa \<in> set xb" using d by auto
+ have "\<exists>y. y \<in> set x' \<and> xb \<approx> y" by (rule list_all2_find_element[OF e a])
+ then obtain ya where h: "ya \<in> set x'" and i: "xb \<approx> ya" by auto
+ have "ya \<in> set y'" using b h by simp
+ then have "\<exists>yb. yb \<in> set y \<and> ya \<approx> 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 \<Rightarrow> 'a fset"
+ is concat
+proof (elim relcomppE)
+fix a b ba bb
+ assume a: "list_all2 op \<approx> a ba"
+ with list_symp [OF list_eq_symp] have a': "list_all2 op \<approx> ba a" by (rule sympE)
+ assume b: "ba \<approx> bb"
+ with list_eq_symp have b': "bb \<approx> ba" by (rule sympE)
+ assume c: "list_all2 op \<approx> bb b"
+ with list_symp [OF list_eq_symp] have c': "list_all2 op \<approx> b bb" by (rule sympE)
+ have "\<forall>x. (\<exists>xa\<in>set a. x \<in> set xa) = (\<exists>xa\<in>set b. x \<in> set xa)"
+ proof
+ fix x
+ show "(\<exists>xa\<in>set a. x \<in> set xa) = (\<exists>xa\<in>set b. x \<in> set xa)"
+ proof
+ assume d: "\<exists>xa\<in>set a. x \<in> set xa"
+ show "\<exists>xa\<in>set b. x \<in> set xa" by (rule concat_rsp_pre[OF a b c d])
+ next
+ assume e: "\<exists>xa\<in>set b. x \<in> set xa"
+ show "\<exists>xa\<in>set a. x \<in> set xa" by (rule concat_rsp_pre[OF c' b' a' e])
+ qed
+ qed
+ then show "concat a \<approx> concat b" by auto
+qed
+
+quotient_definition
+ "filter_fset :: ('a \<Rightarrow> bool) \<Rightarrow> 'a fset \<Rightarrow> 'a fset"
+ is filter by force
+
+
+subsection \<open>Compositional respectfulness and preservation lemmas\<close>
+
+lemma Nil_rsp2 [quot_respect]:
+ shows "(list_all2 op \<approx> OOO op \<approx>) Nil Nil"
+ by (rule compose_list_refl, rule list_eq_equivp)
+
+lemma Cons_rsp2 [quot_respect]:
+ shows "(op \<approx> ===> list_all2 op \<approx> OOO op \<approx> ===> list_all2 op \<approx> OOO op \<approx>) 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 \<circ> 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 \<circ> Rep2) ---> (Abs2 \<circ> map Abs1)) (op #) = (id ---> Rep2 ---> Abs2) (op #)"
+ 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 \<circ> Rep2) ---> (map Rep1 \<circ> Rep2) ---> (Abs2 \<circ> map Abs1)) op @ =
+ (Rep2 ---> Rep2 ---> Abs2) op @"
+ 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 op \<approx> x x'"
+ shows "list_all2 op \<approx> (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 op \<approx> x x'"
+ shows "list_all2 op \<approx> (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 op \<approx> x x'"
+ and "list_all2 op \<approx> z z'"
+ shows "list_all2 op \<approx> (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 op \<approx> OOO op \<approx> ===> list_all2 op \<approx> OOO op \<approx> ===> list_all2 op \<approx> OOO op \<approx>) append append"
+ by (intro compositional_rsp3)
+ (auto intro!: rel_funI simp add: append_rsp2_pre)
+
+lemma map_rsp2 [quot_respect]:
+ "((op \<approx> ===> op \<approx>) ===> list_all2 op \<approx> OOO op \<approx> ===> list_all2 op \<approx> OOO op \<approx>) 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: "(op \<approx> ===> op \<approx>) f f'" and x: "list_all2 op \<approx> xa x" and xy: "set x = set y" and y: "list_all2 op \<approx> y ya"
+ have a: "(list_all2 op \<approx>) (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 op \<approx>) (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 op \<approx> OOO op \<approx>) (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 \<circ> rep_fset) ---> abs_fset \<circ> 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 \<longleftrightarrow> S = T"
+ by (descending) (simp)
+
+lemma filter_fset [simp]:
+ shows "fset (filter_fset P xs) = Collect P \<inter> 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 |\<inter>| ys) = fset xs \<inter> fset ys"
+ by (descending) (auto)
+
+lemma union_fset [simp]:
+ shows "fset (xs |\<union>| ys) = fset xs \<union> 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 |\<in>| S \<longleftrightarrow> x \<in> fset S"
+ by descending simp
+
+lemma notin_fset:
+ shows "x |\<notin>| S \<longleftrightarrow> x \<notin> fset S"
+ by (simp add: in_fset)
+
+lemma notin_empty_fset:
+ shows "x |\<notin>| {||}"
+ by (simp add: in_fset)
+
+lemma fset_eq_iff:
+ shows "S = T \<longleftrightarrow> (\<forall>x. (x |\<in>| S) = (x |\<in>| T))"
+ by descending auto
+
+lemma none_in_empty_fset:
+ shows "(\<forall>x. x |\<notin>| S) \<longleftrightarrow> S = {||}"
+ by descending simp
+
+
+subsection \<open>insert_fset\<close>
+
+lemma in_insert_fset_iff [simp]:
+ shows "x |\<in>| insert_fset y S \<longleftrightarrow> x = y \<or> x |\<in>| S"
+ by descending simp
+
+lemma
+ shows insert_fsetI1: "x |\<in>| insert_fset x S"
+ and insert_fsetI2: "x |\<in>| S \<Longrightarrow> x |\<in>| insert_fset y S"
+ by simp_all
+
+lemma insert_absorb_fset [simp]:
+ shows "x |\<in>| S \<Longrightarrow> insert_fset x S = S"
+ by (descending) (auto)
+
+lemma empty_not_insert_fset[simp]:
+ shows "{||} \<noteq> insert_fset x S"
+ and "insert_fset x S \<noteq> {||}"
+ 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|} \<longleftrightarrow> x = y"
+ by (descending) (auto)
+
+lemma in_fset_mdef:
+ shows "x |\<in>| F \<longleftrightarrow> x |\<notin>| (F - {|x|}) \<and> 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 |\<union>| T = insert_fset x (S |\<union>| T)"
+ by (lifting append.simps(2))
+
+lemma singleton_union_fset_left:
+ shows "{|a|} |\<union>| S = insert_fset a S"
+ by simp
+
+lemma singleton_union_fset_right:
+ shows "S |\<union>| {|a|} = insert_fset a S"
+ by (subst sup.commute) simp
+
+lemma in_union_fset:
+ shows "x |\<in>| S |\<union>| T \<longleftrightarrow> x |\<in>| S \<or> x |\<in>| T"
+ by (descending) (simp)
+
+
+subsection \<open>minus_fset\<close>
+
+lemma minus_in_fset:
+ shows "x |\<in>| (xs - ys) \<longleftrightarrow> x |\<in>| xs \<and> x |\<notin>| ys"
+ by (descending) (simp)
+
+lemma minus_insert_fset:
+ shows "insert_fset x xs - ys = (if x |\<in>| ys then xs - ys else insert_fset x (xs - ys))"
+ by (descending) (auto)
+
+lemma minus_insert_in_fset[simp]:
+ shows "x |\<in>| ys \<Longrightarrow> insert_fset x xs - ys = xs - ys"
+ by (simp add: minus_insert_fset)
+
+lemma minus_insert_notin_fset[simp]:
+ shows "x |\<notin>| ys \<Longrightarrow> insert_fset x xs - ys = insert_fset x (xs - ys)"
+ by (simp add: minus_insert_fset)
+
+lemma in_minus_fset:
+ shows "x |\<in>| F - S \<Longrightarrow> x |\<notin>| S"
+ unfolding in_fset minus_fset
+ by blast
+
+lemma notin_minus_fset:
+ shows "x |\<in>| S \<Longrightarrow> x |\<notin>| F - S"
+ unfolding in_fset minus_fset
+ by blast
+
+
+subsection \<open>remove_fset\<close>
+
+lemma in_remove_fset:
+ shows "x |\<in>| remove_fset y S \<longleftrightarrow> x |\<in>| S \<and> x \<noteq> y"
+ by (descending) (simp)
+
+lemma notin_remove_fset:
+ shows "x |\<notin>| remove_fset x S"
+ by (descending) (simp)
+
+lemma notin_remove_ident_fset:
+ shows "x |\<notin>| S \<Longrightarrow> remove_fset x S = S"
+ by (descending) (simp)
+
+lemma remove_fset_cases:
+ shows "S = {||} \<or> (\<exists>x. x |\<in>| S \<and> 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 "{||} |\<inter>| S = {||}"
+ by simp
+
+lemma inter_empty_fset_r:
+ shows "S |\<inter>| {||} = {||}"
+ by simp
+
+lemma inter_insert_fset:
+ shows "insert_fset x S |\<inter>| T = (if x |\<in>| T then insert_fset x (S |\<inter>| T) else S |\<inter>| T)"
+ by (descending) (auto)
+
+lemma in_inter_fset:
+ shows "x |\<in>| (S |\<inter>| T) \<longleftrightarrow> x |\<in>| S \<and> x |\<in>| T"
+ by (descending) (simp)
+
+
+subsection \<open>subset_fset and psubset_fset\<close>
+
+lemma subset_fset:
+ shows "xs |\<subseteq>| ys \<longleftrightarrow> fset xs \<subseteq> fset ys"
+ by (descending) (simp)
+
+lemma psubset_fset:
+ shows "xs |\<subset>| ys \<longleftrightarrow> fset xs \<subset> fset ys"
+ unfolding less_fset_def
+ by (descending) (auto)
+
+lemma subset_insert_fset:
+ shows "(insert_fset x xs) |\<subseteq>| ys \<longleftrightarrow> x |\<in>| ys \<and> xs |\<subseteq>| ys"
+ by (descending) (simp)
+
+lemma subset_in_fset:
+ shows "xs |\<subseteq>| ys = (\<forall>x. x |\<in>| xs \<longrightarrow> x |\<in>| ys)"
+ by (descending) (auto)
+
+lemma subset_empty_fset:
+ shows "xs |\<subseteq>| {||} \<longleftrightarrow> xs = {||}"
+ by (descending) (simp)
+
+lemma not_psubset_empty_fset:
+ shows "\<not> xs |\<subset>| {||}"
+ 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 \<Longrightarrow> map_fset f S = map_fset f T \<longleftrightarrow> S = T"
+ by (descending) (metis inj_vimage_image_eq list_eq_def set_map)
+
+lemma map_union_fset:
+ shows "map_fset f (S |\<union>| T) = map_fset f S |\<union>| map_fset f T"
+ by (descending) (simp)
+
+lemma in_fset_map_fset[simp]: "a |\<in>| map_fset f X = (\<exists>b. b |\<in>| X \<and> 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 |\<in>| 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 \<longleftrightarrow> 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 \<longleftrightarrow> (\<exists>x. S = {|x|})"
+ by (descending) (auto simp add: card_Suc_eq)
+
+lemma card_fset_gt_0:
+ shows "x \<in> fset S \<Longrightarrow> 0 < card_fset S"
+ by (descending) (auto simp add: card_gt_0_iff)
+
+lemma card_notin_fset:
+ shows "(x |\<notin>| S) = (card_fset (insert_fset x S) = Suc (card_fset S))"
+ by simp
+
+lemma card_fset_Suc:
+ shows "card_fset S = Suc n \<Longrightarrow> \<exists>x T. x |\<notin>| T \<and> S = insert_fset x T \<and> 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 |\<in>| 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 \<Longrightarrow> \<exists>a. a |\<in>| S"
+ by (drule card_fset_Suc) (auto)
+
+lemma in_card_fset_not_0:
+ shows "a |\<in>| A \<Longrightarrow> card_fset A \<noteq> 0"
+ by (descending) (auto)
+
+lemma card_fset_mono:
+ shows "xs |\<subseteq>| ys \<Longrightarrow> card_fset xs \<le> card_fset ys"
+ unfolding card_fset psubset_fset
+ by (simp add: card_mono subset_fset)
+
+lemma card_subset_fset_eq:
+ shows "xs |\<subseteq>| ys \<Longrightarrow> card_fset ys \<le> card_fset xs \<Longrightarrow> 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 |\<subset>| ys \<Longrightarrow> 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 |\<union>| ys) + card_fset (xs |\<inter>| 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 |\<inter>| ys = {||} \<Longrightarrow> card_fset (xs |\<union>| 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 |\<in>| xs \<Longrightarrow> 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 |\<in>| xs \<Longrightarrow> y |\<in>| xs \<Longrightarrow> 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) \<le> card_fset xs"
+ unfolding remove_fset card_fset
+ by (rule card_Diff1_le[OF finite_fset])
+
+lemma card_psubset_fset:
+ shows "ys |\<subseteq>| xs \<Longrightarrow> card_fset ys < card_fset xs \<Longrightarrow> ys |\<subset>| 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) \<le> 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 |\<in>| A" and "a |\<notin>| 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 |\<subseteq>| 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 |\<inter>| 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 |\<union>| concat_fset S"
+ by descending simp
+
+lemma concat_union_fset [simp]:
+ shows "concat_fset (xs |\<union>| ys) = concat_fset xs |\<union>| 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 |\<subseteq>| filter_fset Q xs = (\<forall> x. x |\<in>| xs \<longrightarrow> P x \<longrightarrow> Q x)"
+ by descending auto
+
+lemma eq_filter_fset:
+ "(filter_fset P xs = filter_fset Q xs) = (\<forall>x. x |\<in>| xs \<longrightarrow> P x = Q x)"
+ by descending auto
+
+lemma psubset_filter_fset:
+ "(\<And>x. x |\<in>| xs \<Longrightarrow> P x \<Longrightarrow> Q x) \<Longrightarrow> (x |\<in>| xs & \<not> P x & Q x) \<Longrightarrow>
+ 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 \<in> set xs"
+ shows "fold_once f xs = fold_once f (removeAll x xs) \<circ> f x"
+proof -
+ from assms have "fold f (remdups xs) = fold f (remove1 x (remdups xs)) \<circ> 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 \<Longrightarrow> h |\<in>| b \<Longrightarrow> fold_fset f b = fold_fset f (remove_fset h b) \<circ> f h"
+ by descending (simp add: member_commute_fold_once)
+
+
+subsection \<open>Choice in fsets\<close>
+
+lemma fset_choice:
+ assumes a: "\<forall>x. x |\<in>| A \<longrightarrow> (\<exists>y. P x y)"
+ shows "\<exists>f. \<forall>x. x |\<in>| A \<longrightarrow> 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 = {||} \<Longrightarrow> P"
+ and insert_fset_case: "\<And>x S'. S = insert_fset x S' \<Longrightarrow> 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: "\<And>x S. P S \<Longrightarrow> 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: "\<And>x S. \<lbrakk>x |\<notin>| S; P S\<rbrakk> \<Longrightarrow> 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 |\<in>| S") (simp_all)
+qed
+
+lemma fset_card_induct:
+ assumes empty_fset_case: "P {||}"
+ and card_fset_Suc_case: "\<And>S T. Suc (card_fset S) = (card_fset T) \<Longrightarrow> P S \<Longrightarrow> 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 |\<notin>| 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 "\<not> List.member ys x" and "xs \<approx> x # ys"
+proof (induct xs)
+ case Nil
+ then show thesis by simp
+next
+ case (Cons a xs)
+ have a: "\<lbrakk>xs = [] \<Longrightarrow> thesis; \<And>x ys. \<lbrakk>\<not> List.member ys x; xs \<approx> x # ys\<rbrakk> \<Longrightarrow> thesis\<rbrakk> \<Longrightarrow> thesis"
+ by (rule Cons(1))
+ have b: "\<And>x' ys'. \<lbrakk>\<not> List.member ys' x'; a # xs \<approx> x' # ys'\<rbrakk> \<Longrightarrow> thesis" by fact
+ have c: "xs = [] \<Longrightarrow> thesis" using b
+ apply(simp)
+ by (metis list.set(1) emptyE empty_subsetI)
+ have "\<And>x ys. \<lbrakk>\<not> List.member ys x; xs \<approx> x # ys\<rbrakk> \<Longrightarrow> thesis"
+ proof -
+ fix x :: 'a
+ fix ys :: "'a list"
+ assume d:"\<not> List.member ys x"
+ assume e:"xs \<approx> x # ys"
+ show thesis
+ proof (cases "x = a")
+ assume h: "x = a"
+ then have f: "\<not> List.member ys a" using d by simp
+ have g: "a # xs \<approx> a # ys" using e h by auto
+ show thesis using b f g by simp
+ next
+ assume h: "x \<noteq> a"
+ then have f: "\<not> List.member (a # ys) x" using d by auto
+ have g: "a # xs \<approx> 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 |\<notin>| ys" and "xs = insert_fset x ys"
+ by (lifting fset_raw_strong_cases)
+
+
+lemma fset_induct2:
+ "P {||} {||} \<Longrightarrow>
+ (\<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 "\<And>x. x \<in> fset A \<longleftrightarrow> x \<in> 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 "\<And>z. z \<in> fset A \<longleftrightarrow> z \<in> 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 |\<in>| 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 \<Rightarrow> 'a list \<Rightarrow> bool" ("_ \<approx>2 _")
+where
+ "(a # b # xs) \<approx>2 (b # a # xs)"
+| "[] \<approx>2 []"
+| "xs \<approx>2 ys \<Longrightarrow> ys \<approx>2 xs"
+| "(a # a # xs) \<approx>2 (a # xs)"
+| "xs \<approx>2 ys \<Longrightarrow> (a # xs) \<approx>2 (a # ys)"
+| "xs1 \<approx>2 xs2 \<Longrightarrow> xs2 \<approx>2 xs3 \<Longrightarrow> xs1 \<approx>2 xs3"
+
+lemma list_eq2_refl:
+ shows "xs \<approx>2 xs"
+ by (induct xs) (auto intro: list_eq2.intros)
+
+lemma cons_delete_list_eq2:
+ shows "(a # (removeAll a A)) \<approx>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) \<approx>2 r"
+ using a cons_delete_list_eq2[of e r]
+ by simp
+
+lemma list_eq2_equiv:
+ "(l \<approx> r) \<longleftrightarrow> (list_eq2 l r)"
+proof
+ show "list_eq2 l r \<Longrightarrow> l \<approx> r" by (induct rule: list_eq2.induct) auto
+next
+ {
+ fix n
+ assume a: "card_list l = n" and b: "l \<approx> r"
+ have "l \<approx>2 r"
+ using a b
+ proof (induct n arbitrary: l r)
+ case 0
+ have "card_list l = 0" by fact
+ then have "\<forall>x. \<not> 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 \<approx> r" by fact
+ have d: "card_list l = Suc m" by fact
+ then have "\<exists>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 \<approx> removeAll a r" using remove_fset.rsp b by simp
+ have "(removeAll a l) \<approx>2 (removeAll a r)" by (rule Suc.hyps[OF f g])
+ then have h: "(a # removeAll a l) \<approx>2 (a # removeAll a r)" by (rule list_eq2.intros(5))
+ have i: "l \<approx>2 (a # removeAll a l)"
+ by (rule list_eq2.intros(3)[OF member_delete_list_eq2[OF e]])
+ have "l \<approx>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 \<approx> r \<Longrightarrow> l \<approx>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:
+ "\<lbrakk>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 "\<And>a b xs. P (insert_fset a (insert_fset b xs)) (insert_fset b (insert_fset a xs))"
+ and "P {||} {||}"
+ and "\<And>xs ys. \<lbrakk>xs = ys; P xs ys\<rbrakk> \<Longrightarrow> P ys xs"
+ and "\<And>a xs. P (insert_fset a (insert_fset a xs)) (insert_fset a xs)"
+ and "\<And>xs ys a. \<lbrakk>xs = ys; P xs ys\<rbrakk> \<Longrightarrow> P (insert_fset a xs) (insert_fset a ys)"
+ and "\<And>xs1 xs2 xs3. \<lbrakk>xs1 = xs2; P xs1 xs2; xs2 = xs3; P xs2 xs3\<rbrakk> \<Longrightarrow> 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 fset}, [T])) = T
+ | dest_fsetT T = raise TYPE ("dest_fsetT: fset type expected", [T], []);
+\<close>
+
+no_notation
+ list_eq (infix "\<approx>" 50) and
+ list_eq2 (infix "\<approx>2" 50)
+
+end
--- a/src/HOL/ROOT Mon Sep 19 20:07:39 2016 +0200
+++ b/src/HOL/ROOT Mon Sep 19 23:14:34 2016 +0200
@@ -996,7 +996,7 @@
options [document = false]
theories
DList
- FSet
+ Quotient_FSet
Quotient_Int
Quotient_Message
Lift_FSet