--- a/Nominal/FSet.thy Tue Oct 19 15:08:24 2010 +0100
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,1164 +0,0 @@
-(* Title: HOL/Quotient_Examples/FSet.thy
- Author: Cezary Kaliszyk, TU Munich
- Author: Christian Urban, TU Munich
-
- Type of finite sets.
-*)
-
-theory FSet
-imports Quotient_List
-begin
-
-text {*
- The type of finite sets is created by a quotient construction
- over lists. The definition of the equivalence:
-*}
-
-fun
- list_eq :: "'a list \<Rightarrow> 'a list \<Rightarrow> bool" (infix "\<approx>" 50)
-where
- "list_eq xs ys \<longleftrightarrow> set xs = set ys"
-
-lemma list_eq_equivp:
- shows "equivp list_eq"
- unfolding equivp_reflp_symp_transp
- unfolding reflp_def symp_def transp_def
- by auto
-
-text {* Fset type *}
-
-quotient_type
- 'a fset = "'a list" / "list_eq"
- by (rule list_eq_equivp)
-
-text {*
- Definitions for membership, sublist, cardinality,
- intersection, difference and respectful fold over
- lists.
-*}
-
-definition
- memb :: "'a \<Rightarrow> 'a list \<Rightarrow> bool"
-where
- [simp]: "memb x xs \<longleftrightarrow> x \<in> set xs"
-
-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
-where
- "rsp_fold f \<equiv> \<forall>u v w. (f u (f v w) = f v (f u w))"
-
-primrec
- fold_list :: "('a \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> 'b \<Rightarrow> 'a list \<Rightarrow> 'b"
-where
- "fold_list f z [] = z"
-| "fold_list f z (a # xs) =
- (if (rsp_fold f) then
- if a \<in> set xs then fold_list f z xs
- else f a (fold_list f z xs)
- else z)"
-
-
-
-section {* Quotient composition lemmas *}
-
-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"
- unfolding list_eq.simps
- by (simp only: set_map)
-
-lemma quotient_compose_list_g:
- assumes q: "Quotient R Abs Rep"
- and e: "equivp R"
- shows "Quotient ((list_all2 R) OOO (op \<approx>))
- (abs_fset \<circ> (map Abs)) ((map Rep) \<circ> rep_fset)"
- unfolding Quotient_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] Quotient_abs_rep[OF Quotient_fset] 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 pred_compE)
- 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 Quotient_rel[OF list_quotient[OF q]] c by blast
- have "map Abs ba = map Abs s"
- using Quotient_rel[OF list_quotient[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 Quotient_rel[OF Quotient_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 Quotient_rel[OF Quotient_fset]) (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_quotient[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 pred_compI) (rule b, rule y)
- have z: "list_all2 R r (map Rep (map Abs r))"
- by (fact rep_abs_rsp[OF list_quotient[OF q], OF list_all2_refl'[OF e, of r]])
- then show "(list_all2 R OOO op \<approx>) r s"
- using a c pred_compI by simp
- qed
-qed
-
-lemma quotient_compose_list[quot_thm]:
- shows "Quotient ((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 Quotient_fset, rule list_eq_equivp)
-
-
-
-subsection {* Respectfulness lemmas for list operations *}
-
-lemma list_equiv_rsp [quot_respect]:
- shows "(op \<approx> ===> op \<approx> ===> op =) op \<approx> op \<approx>"
- by auto
-
-lemma append_rsp [quot_respect]:
- shows "(op \<approx> ===> op \<approx> ===> op \<approx>) append append"
- by simp
-
-lemma sub_list_rsp [quot_respect]:
- shows "(op \<approx> ===> op \<approx> ===> op =) sub_list sub_list"
- by simp
-
-lemma memb_rsp [quot_respect]:
- shows "(op = ===> op \<approx> ===> op =) memb memb"
- by simp
-
-lemma nil_rsp [quot_respect]:
- shows "(op \<approx>) Nil Nil"
- by simp
-
-lemma cons_rsp [quot_respect]:
- shows "(op = ===> op \<approx> ===> op \<approx>) Cons Cons"
- by simp
-
-lemma map_rsp [quot_respect]:
- shows "(op = ===> op \<approx> ===> op \<approx>) map map"
- by auto
-
-lemma set_rsp [quot_respect]:
- "(op \<approx> ===> op =) set set"
- by auto
-
-lemma inter_list_rsp [quot_respect]:
- shows "(op \<approx> ===> op \<approx> ===> op \<approx>) inter_list inter_list"
- by simp
-
-lemma removeAll_rsp [quot_respect]:
- shows "(op = ===> op \<approx> ===> op \<approx>) removeAll removeAll"
- by simp
-
-lemma diff_list_rsp [quot_respect]:
- shows "(op \<approx> ===> op \<approx> ===> op \<approx>) diff_list diff_list"
- by simp
-
-lemma card_list_rsp [quot_respect]:
- shows "(op \<approx> ===> op =) card_list card_list"
- by simp
-
-lemma filter_rsp [quot_respect]:
- shows "(op = ===> op \<approx> ===> op \<approx>) filter filter"
- by simp
-
-lemma memb_commute_fold_list:
- assumes a: "rsp_fold f"
- and b: "x \<in> set xs"
- shows "fold_list f y xs = f x (fold_list f y (removeAll x xs))"
- using a b by (induct xs) (auto simp add: rsp_fold_def)
-
-lemma fold_list_rsp_pre:
- assumes a: "set xs = set ys"
- shows "fold_list f z xs = fold_list f z ys"
- using a
- apply (induct xs arbitrary: ys)
- apply (simp)
- apply (simp (no_asm_use))
- apply (rule conjI)
- apply (rule_tac [!] impI)
- apply (rule_tac [!] conjI)
- apply (rule_tac [!] impI)
- apply (metis insert_absorb)
- apply (metis List.insert_def List.set.simps(2) List.set_insert fold_list.simps(2))
- apply (metis Diff_insert_absorb insertI1 memb_commute_fold_list set_removeAll)
- apply(drule_tac x="removeAll a ys" in meta_spec)
- apply(auto)
- apply(drule meta_mp)
- apply(blast)
- by (metis List.set.simps(2) emptyE fold_list.simps(2) in_listsp_conv_set listsp.simps mem_def)
-
-lemma fold_list_rsp [quot_respect]:
- shows "(op = ===> op = ===> op \<approx> ===> op =) fold_list fold_list"
- unfolding fun_rel_def
- by(auto intro: fold_list_rsp_pre)
-
-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
-
-lemma concat_rsp [quot_respect]:
- shows "(list_all2 op \<approx> OOO op \<approx> ===> op \<approx>) concat concat"
-proof (rule fun_relI, elim pred_compE)
- fix a b ba bb
- assume a: "list_all2 op \<approx> a ba"
- assume b: "ba \<approx> bb"
- assume c: "list_all2 op \<approx> bb b"
- 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"
- have a': "list_all2 op \<approx> ba a" by (rule list_all2_symp[OF list_eq_equivp, OF a])
- have b': "bb \<approx> ba" by (rule equivp_symp[OF list_eq_equivp, OF b])
- have c': "list_all2 op \<approx> b bb" by (rule list_all2_symp[OF list_eq_equivp, OF c])
- 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
-
-
-
-section {* Quotient definitions for fsets *}
-
-
-subsection {* Finite sets are a bounded, distributive lattice with minus *}
-
-instantiation fset :: (type) "{bounded_lattice_bot, distrib_lattice, minus}"
-begin
-
-quotient_definition
- "bot :: 'a fset"
- is "Nil :: 'a list"
-
-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)"
-
-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"
-
-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"
-
-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"
-
-
-instance
-proof
- fix x y z :: "'a fset"
- show "x |\<subset>| y \<longleftrightarrow> x |\<subseteq>| y \<and> \<not> y |\<subseteq>| x"
- unfolding less_fset_def
- by (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 {* Other constants for fsets *}
-
-quotient_definition
- "insert_fset :: 'a \<Rightarrow> 'a fset \<Rightarrow> 'a fset"
- is "Cons"
-
-syntax
- "@Insert_fset" :: "args => 'a fset" ("{|(_)|}")
-
-translations
- "{|x, xs|}" == "CONST insert_fset x {|xs|}"
- "{|x|}" == "CONST insert_fset x {||}"
-
-quotient_definition
- in_fset (infix "|\<in>|" 50)
-where
- "in_fset :: 'a \<Rightarrow> 'a fset \<Rightarrow> bool" is "memb"
-
-abbreviation
- notin_fset :: "'a \<Rightarrow> 'a fset \<Rightarrow> bool" (infix "|\<notin>|" 50)
-where
- "x |\<notin>| S \<equiv> \<not> (x |\<in>| S)"
-
-
-subsection {* Other constants on the Quotient Type *}
-
-quotient_definition
- "card_fset :: 'a fset \<Rightarrow> nat"
- is card_list
-
-quotient_definition
- "map_fset :: ('a \<Rightarrow> 'b) \<Rightarrow> 'a fset \<Rightarrow> 'b fset"
- is map
-
-quotient_definition
- "remove_fset :: 'a \<Rightarrow> 'a fset \<Rightarrow> 'a fset"
- is removeAll
-
-quotient_definition
- "fset :: 'a fset \<Rightarrow> 'a set"
- is "set"
-
-quotient_definition
- "fold_fset :: ('a \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> 'b \<Rightarrow> 'a fset \<Rightarrow> 'b"
- is fold_list
-
-quotient_definition
- "concat_fset :: ('a fset) fset \<Rightarrow> 'a fset"
- is concat
-
-quotient_definition
- "filter_fset :: ('a \<Rightarrow> bool) \<Rightarrow> 'a fset \<Rightarrow> 'a fset"
- is filter
-
-
-subsection {* Compositional respectfulness and preservation lemmas *}
-
-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
- apply (rule_tac b="x # b" in pred_compI)
- apply auto
- apply (rule_tac b="x # ba" in pred_compI)
- apply auto
- done
-
-lemma map_prs [quot_preserve]:
- shows "(abs_fset \<circ> map f) [] = abs_fset []"
- by simp
-
-lemma insert_fset_rsp [quot_preserve]:
- "(rep_fset ---> (map rep_fset \<circ> rep_fset) ---> (abs_fset \<circ> map abs_fset)) Cons = insert_fset"
- by (simp add: fun_eq_iff Quotient_abs_rep[OF Quotient_fset]
- abs_o_rep[OF Quotient_fset] map_id insert_fset_def)
-
-lemma union_fset_rsp [quot_preserve]:
- "((map rep_fset \<circ> rep_fset) ---> (map rep_fset \<circ> rep_fset) ---> (abs_fset \<circ> map abs_fset))
- append = union_fset"
- by (simp add: fun_eq_iff Quotient_abs_rep[OF Quotient_fset]
- abs_o_rep[OF Quotient_fset] map_id sup_fset_def)
-
-lemma list_all2_app_l:
- assumes a: "reflp R"
- and b: "list_all2 R l r"
- shows "list_all2 R (z @ l) (z @ r)"
- by (induct z) (simp_all add: b rev_iffD1[OF a meta_eq_to_obj_eq[OF reflp_def]])
-
-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.simps)
- apply (rule list_all2_app_l)
- apply (simp_all add: reflp_def)
- done
-
-lemma append_rsp2_pre:
- assumes a:"list_all2 op \<approx> x x'"
- and b: "list_all2 op \<approx> z z'"
- shows "list_all2 op \<approx> (x @ z) (x' @ z')"
- apply (rule list_all2_transp[OF fset_equivp])
- apply (rule append_rsp2_pre0)
- apply (rule a)
- using b apply (induct z z' rule: list_induct2')
- apply (simp_all only: append_Nil2)
- apply (rule list_all2_refl'[OF list_eq_equivp])
- apply simp_all
- apply (rule append_rsp2_pre1)
- apply simp
- done
-
-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"
-proof (intro fun_relI, elim pred_compE)
- fix x y z w x' z' y' w' :: "'a list list"
- assume a:"list_all2 op \<approx> x x'"
- and b: "x' \<approx> y'"
- and c: "list_all2 op \<approx> y' y"
- assume aa: "list_all2 op \<approx> z z'"
- and bb: "z' \<approx> w'"
- and cc: "list_all2 op \<approx> w' w"
- have a': "list_all2 op \<approx> (x @ z) (x' @ z')" using a aa append_rsp2_pre by auto
- have b': "x' @ z' \<approx> y' @ w'" using b bb by simp
- have c': "list_all2 op \<approx> (y' @ w') (y @ w)" using c cc append_rsp2_pre by auto
- have d': "(op \<approx> OO list_all2 op \<approx>) (x' @ z') (y @ w)"
- by (rule pred_compI) (rule b', rule c')
- show "(list_all2 op \<approx> OOO op \<approx>) (x @ z) (y @ w)"
- by (rule pred_compI) (rule a', rule d')
-qed
-
-
-
-section {* Lifted theorems *}
-
-subsection {* fset *}
-
-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) = P \<inter> fset xs"
- by (descending) (auto simp add: mem_def)
-
-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 {* in_fset *}
-
-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 {* insert_fset *}
-
-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 {* union_fset *}
-
-lemmas [simp] =
- sup_bot_left[where 'a="'a fset", standard]
- sup_bot_right[where 'a="'a fset", standard]
-
-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 {* minus_fset *}
-
-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 {* remove_fset *}
-
-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 {* inter_fset *}
-
-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 {* subset_fset and psubset_fset *}
-
-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 {* map_fset *}
-
-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.simps 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)
-
-
-subsection {* card_fset *}
-
-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 {* concat_fset *}
-
-lemma concat_empty_fset [simp]:
- shows "concat_fset {||} = {||}"
- by (lifting concat.simps(1))
-
-lemma concat_insert_fset [simp]:
- shows "concat_fset (insert_fset x S) = x |\<union>| concat_fset S"
- by (lifting concat.simps(2))
-
-lemma concat_inter_fset [simp]:
- shows "concat_fset (xs |\<union>| ys) = concat_fset xs |\<union>| concat_fset ys"
- by (lifting concat_append)
-
-
-subsection {* filter_fset *}
-
-lemma subset_filter_fset:
- shows "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:
- shows "(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:
- shows "(\<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 {* fold_fset *}
-
-lemma fold_empty_fset:
- shows "fold_fset f z {||} = z"
- by (descending) (simp)
-
-lemma fold_insert_fset: "fold_fset f z (insert_fset a A) =
- (if rsp_fold f then if a |\<in>| A then fold_fset f z A else f a (fold_fset f z A) else z)"
- by (descending) (simp)
-
-lemma in_commute_fold_fset:
- "\<lbrakk>rsp_fold f; h |\<in>| b\<rbrakk> \<Longrightarrow> fold_fset f z b = f h (fold_fset f z (remove_fset h b))"
- by (descending) (simp add: memb_commute_fold_list)
-
-
-subsection {* Choice in fsets *}
-
-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 {* Induction and Cases rules for fsets *}
-
-lemma fset_exhaust [case_names empty_fset insert_fset, 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_fset insert_fset]:
- 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_fset insert_fset, 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_fset
- show "P {||}" using empty_fset_case by simp
-next
- case (insert_fset 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_fset
- show "P {||}" by (rule empty_fset_case)
-next
- case (insert_fset 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 = []"
- | x ys where "\<not> memb x ys" and "xs \<approx> x # ys"
-proof (induct xs arbitrary: x ys)
- case Nil
- then show thesis by simp
-next
- case (Cons a xs)
- have a: "\<lbrakk>xs = [] \<Longrightarrow> thesis; \<And>x ys. \<lbrakk>\<not> memb x ys; xs \<approx> x # ys\<rbrakk> \<Longrightarrow> thesis\<rbrakk> \<Longrightarrow> thesis" by fact
- have b: "\<And>x' ys'. \<lbrakk>\<not> memb x' ys'; a # xs \<approx> x' # ys'\<rbrakk> \<Longrightarrow> thesis" by fact
- have c: "xs = [] \<Longrightarrow> thesis" using b
- apply(simp)
- by (metis List.set.simps(1) emptyE empty_subsetI)
- have "\<And>x ys. \<lbrakk>\<not> memb x ys; xs \<approx> x # ys\<rbrakk> \<Longrightarrow> thesis"
- proof -
- fix x :: 'a
- fix ys :: "'a list"
- assume d:"\<not> memb x ys"
- assume e:"xs \<approx> x # ys"
- show thesis
- proof (cases "x = a")
- assume h: "x = a"
- then have f: "\<not> memb a ys" 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> memb x (a # ys)" 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: memb_def)
- qed
- qed
- then show thesis using a c by blast
-qed
-
-
-lemma fset_strong_cases:
- obtains "xs = {||}"
- | x ys 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
-
-
-
-subsection {* alternate formulation with a different decomposition principle
- and a proof of equivalence *}
-
-inductive
- list_eq2 ("_ \<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)"
-| "\<lbrakk>xs1 \<approx>2 xs2; xs2 \<approx>2 xs3\<rbrakk> \<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 memb a A then A else a # A)"
- apply (induct A)
- apply (simp add: list_eq2_refl)
- apply (case_tac "memb a (aa # A)")
- apply (simp_all)
- apply (case_tac [!] "a = aa")
- apply (simp_all)
- apply (case_tac "memb 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 memb_def)
- done
-
-lemma memb_delete_list_eq2:
- assumes a: "memb e r"
- 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> memb x l" by auto
- then have z: "l = []" by auto
- then have "r = []" using `l \<approx> r` 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. memb a l"
- apply(simp)
- apply(drule card_eq_SucD)
- apply(blast)
- done
- then obtain a where e: "memb a l" by auto
- then have e': "memb a r" using list_eq.simps[simplified memb_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 removeAll_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 memb_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 _ memb_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 {*
-fun dest_fsetT (Type (@{type_name fset}, [T])) = T
- | dest_fsetT T = raise TYPE ("dest_fsetT: fset type expected", [T], []);
-*}
-
-no_notation
- list_eq (infix "\<approx>" 50) and
- list_eq2 (infix "\<approx>2" 50)
-
-end