nominal2: comparison Nominal/FSet.thy

equal deleted inserted replaced

-:e903c32ec24f
+:693562f03eee
 text {* Definiton of List relation and the quotient type *}
 fun
 list_eq :: "'a list \<Rightarrow> 'a list \<Rightarrow> bool" (infix "\<approx>" 50)
 where
-"list_eq xs ys = (\<forall>x. x \<in> set xs \<longleftrightarrow> x \<in> set ys)"
+"list_eq xs ys = (set xs = set ys)"
 lemma list_eq_equivp:
 shows "equivp list_eq"
 unfolding equivp_reflp_symp_transp
 unfolding reflp_def symp_def transp_def
 "memb x xs \<equiv> x \<in> set xs"
 definition
 sub_list :: "'a list \<Rightarrow> 'a list \<Rightarrow> bool"
 where
-"sub_list xs ys \<equiv> (\<forall>x. x \<in> set xs \<longrightarrow> x \<in> set ys)"
+"sub_list xs ys \<equiv> set xs \<subseteq> set ys"
-fun
+definition
 fcard_raw :: "'a list \<Rightarrow> nat"
 where
-fcard_raw_nil:  "fcard_raw [] = 0"
+"fcard_raw xs = card (set xs)"
-| fcard_raw_cons: "fcard_raw (x # xs) =
-(if memb x xs then fcard_raw xs else Suc (fcard_raw xs))"
 primrec
 finter_raw :: "'a list \<Rightarrow> 'a list \<Rightarrow> 'a list"
 where
-"finter_raw [] l = []"
+"finter_raw [] ys = []"
-| "finter_raw (h # t) l =
+| "finter_raw (x # xs) ys =
-(if memb h l then h # (finter_raw t l) else finter_raw t l)"
+(if x \<in> set ys then x # (finter_raw xs ys) else finter_raw xs ys)"
-primrec
-delete_raw :: "'a list \<Rightarrow> 'a \<Rightarrow> 'a list"
-where
-"delete_raw [] x = []"
-| "delete_raw (a # xs) x =
-(if (a = x) then delete_raw xs x else a # (delete_raw xs x))"
 primrec
 fminus_raw :: "'a list \<Rightarrow> 'a list \<Rightarrow> 'a list"
 where
-"fminus_raw l [] = l"
+"fminus_raw ys [] = ys"
-| "fminus_raw l (h # t) = fminus_raw (delete_raw l h) t"
+| "fminus_raw ys (x # xs) = fminus_raw (removeAll x ys) xs"
 definition
 rsp_fold
 where
 "rsp_fold f = (\<forall>u v w. (f u (f v w) = f v (f u w)))"
 ffold_raw :: "('a \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> 'b \<Rightarrow> 'a list \<Rightarrow> 'b"
 where
 "ffold_raw f z [] = z"
 | "ffold_raw f z (a # xs) =
 (if (rsp_fold f) then
-if memb a xs then ffold_raw f z xs
+if a \<in> set xs then ffold_raw f z xs
 else f a (ffold_raw f z xs)
 else z)"
 text {* Composition Quotient *}
 lemma Quotient_fset_list:
 shows "Quotient (list_all2 op \<approx>) (map abs_fset) (map rep_fset)"
 by (fact list_quotient[OF Quotient_fset])
+(*
 lemma set_in_eq: "(\<forall>e. ((e \<in> xs) \<longleftrightarrow> (e \<in> ys))) \<equiv> xs = ys"
 by (rule eq_reflection) auto
+*)
 lemma map_rel_cong: "b \<approx> ba \<Longrightarrow> map f b \<approx> map f ba"
 unfolding list_eq.simps
-by (simp only: set_map set_in_eq)
+by (simp only: set_map)
 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)"
 unfolding Quotient_def comp_def
 qed
 text {* Respectfullness *}
 lemma append_rsp[quot_respect]:
-shows "(op \<approx> ===> op \<approx> ===> op \<approx>) op @ op @"
+shows "(op \<approx> ===> op \<approx> ===> op \<approx>) append append"
-apply(simp del: list_eq.simps)
+by (simp)
-by auto
+(*
 lemma append_rsp_unfolded:
 "\<lbrakk>x \<approx> y; v \<approx> w\<rbrakk> \<Longrightarrow> x @ v \<approx> y @ w"
 by auto
+*)
 lemma [quot_respect]:
 shows "(op \<approx> ===> op \<approx> ===> op =) sub_list sub_list"
 by (auto simp add: sub_list_def)
 lemma nil_rsp[quot_respect]:
 shows "[] \<approx> []"
 by simp
 lemma cons_rsp[quot_respect]:
-shows "(op = ===> op \<approx> ===> op \<approx>) op # op #"
+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 memb_cons_iff:
 shows "memb x (y # xs) = (x = y \<or> memb x xs)"
 by (induct xs) (auto simp add: memb_def)
-lemma memb_finter_raw:
+lemma set_finter_raw[simp]:
-"memb x (finter_raw xs ys) \<longleftrightarrow> memb x xs \<and> memb x ys"
+"set (finter_raw xs ys) = set xs \<inter> set ys"
-by (induct xs) (auto simp add: not_memb_nil memb_cons_iff)
+by (induct xs) (auto simp add: memb_def)
 lemma [quot_respect]:
-"(op \<approx> ===> op \<approx> ===> op \<approx>) finter_raw finter_raw"
+shows "(op \<approx> ===> op \<approx> ===> op \<approx>) finter_raw finter_raw"
-by (simp add: memb_def[symmetric] memb_finter_raw)
+by (simp)
-lemma memb_delete_raw:
+(*
-"memb x (delete_raw xs y) = (memb x xs \<and> x \<noteq> y)"
+lemma memb_removeAll:
+"memb x (removeAll y xs) \<longleftrightarrow> memb x xs \<and> x \<noteq> y"
 by (induct xs arbitrary: x y) (auto simp add: memb_def)
+*)
 lemma [quot_respect]:
-"(op \<approx> ===> op = ===> op \<approx>) delete_raw delete_raw"
+shows "(op = ===> op \<approx> ===> op \<approx>) removeAll removeAll"
-by (simp add: memb_def[symmetric] memb_delete_raw)
+by (simp)
-lemma fminus_raw_memb: "memb x (fminus_raw xs ys) = (memb x xs \<and> \<not> memb x ys)"
+lemma set_fminus_raw[simp]:
-by (induct ys arbitrary: xs)
+"set (fminus_raw xs ys) = (set xs - set ys)"
-(simp_all add: not_memb_nil memb_delete_raw memb_cons_iff)
+by (induct ys arbitrary: xs) (auto)
 lemma [quot_respect]:
-"(op \<approx> ===> op \<approx> ===> op \<approx>) fminus_raw fminus_raw"
+shows "(op \<approx> ===> op \<approx> ===> op \<approx>) fminus_raw fminus_raw"
-by (simp add: memb_def[symmetric] fminus_raw_memb)
+by simp
-lemma fcard_raw_gt_0:
-assumes a: "x \<in> set xs"
-shows "0 < fcard_raw xs"
-using a by (induct xs) (auto simp add: memb_def)
-lemma fcard_raw_delete_one:
-shows "fcard_raw ([x \<leftarrow> xs. x \<noteq> y]) = (if memb y xs then fcard_raw xs - 1 else fcard_raw xs)"
-by (induct xs) (auto dest: fcard_raw_gt_0 simp add: memb_def)
-lemma fcard_raw_rsp_aux:
-assumes a: "xs \<approx> ys"
-shows "fcard_raw xs = fcard_raw ys"
-using a
-proof (induct xs arbitrary: ys)
-case Nil
-show ?case using Nil.prems by simp
-next
-case (Cons a xs)
-have a: "a # xs \<approx> ys" by fact
-have b: "\<And>ys. xs \<approx> ys \<Longrightarrow> fcard_raw xs = fcard_raw ys" by fact
-show ?case proof (cases "a \<in> set xs")
-assume c: "a \<in> set xs"
-have "\<forall>x. (x \<in> set xs) = (x \<in> set ys)"
-proof (intro allI iffI)
-fix x
-assume "x \<in> set xs"
-then show "x \<in> set ys" using a by auto
-next
-fix x
-assume d: "x \<in> set ys"
-have e: "(x \<in> set (a # xs)) = (x \<in> set ys)" using a by simp
-show "x \<in> set xs" using c d e unfolding list_eq.simps by simp blast
-qed
-then show ?thesis using b c by (simp add: memb_def)
-next
-assume c: "a \<notin> set xs"
-have d: "xs \<approx> [x\<leftarrow>ys . x \<noteq> a] \<Longrightarrow> fcard_raw xs = fcard_raw [x\<leftarrow>ys . x \<noteq> a]" using b by simp
-have "Suc (fcard_raw xs) = fcard_raw ys"
-proof (cases "a \<in> set ys")
-assume e: "a \<in> set ys"
-have f: "\<forall>x. (x \<in> set xs) = (x \<in> set ys \<and> x \<noteq> a)" using a c
-by (auto simp add: fcard_raw_delete_one)
-have "fcard_raw ys = Suc (fcard_raw ys - 1)" by (rule Suc_pred'[OF fcard_raw_gt_0]) (rule e)
-then show ?thesis using d e f by (simp_all add: fcard_raw_delete_one memb_def)
-next
-case False then show ?thesis using a c d by auto
-qed
-then show ?thesis using a c d by (simp add: memb_def)
-qed
-qed
 lemma fcard_raw_rsp[quot_respect]:
 shows "(op \<approx> ===> op =) fcard_raw fcard_raw"
-by (simp add: fcard_raw_rsp_aux)
+by (simp add: fcard_raw_def)
 lemma memb_absorb:
 shows "memb x xs \<Longrightarrow> x # xs \<approx> xs"
 by (induct xs) (auto simp add: memb_def)
 lemma none_memb_nil:
 "(\<forall>x. \<not> memb x xs) = (xs \<approx> [])"
 by (simp add: memb_def)
-lemma not_memb_delete_raw_ident:
+lemma not_memb_removeAll_ident:
-shows "\<not> memb x xs \<Longrightarrow> delete_raw xs x = xs"
+shows "\<not> memb x xs \<Longrightarrow> removeAll x xs = xs"
 by (induct xs) (auto simp add: memb_def)
 lemma memb_commute_ffold_raw:
-"rsp_fold f \<Longrightarrow> memb h b \<Longrightarrow> ffold_raw f z b = f h (ffold_raw f z (delete_raw b h))"
+"rsp_fold f \<Longrightarrow> h \<in> set b \<Longrightarrow> ffold_raw f z b = f h (ffold_raw f z (removeAll h b))"
 apply (induct b)
-apply (simp_all add: not_memb_nil)
+apply (auto simp add: rsp_fold_def)
-apply (auto)
-apply (simp_all add: memb_delete_raw not_memb_delete_raw_ident rsp_fold_def  memb_cons_iff)
 done
 lemma ffold_raw_rsp_pre:
-"\<forall>e. memb e a = memb e b \<Longrightarrow> ffold_raw f z a = ffold_raw f z b"
+"set a = set b \<Longrightarrow> ffold_raw f z a = ffold_raw f z b"
 apply (induct a arbitrary: b)
-apply (simp add: memb_absorb memb_def none_memb_nil)
 apply (simp)
+apply (simp (no_asm_use))
 apply (rule conjI)
 apply (rule_tac [!] impI)
 apply (rule_tac [!] conjI)
 apply (rule_tac [!] impI)
-apply (subgoal_tac "\<forall>e. memb e a2 = memb e b")
+apply (metis insert_absorb)
-apply (simp)
+apply (metis List.insert_def List.set.simps(2) List.set_insert ffold_raw.simps(2))
-apply (simp add: memb_cons_iff memb_def)
+apply (metis Diff_insert_absorb insertI1 memb_commute_ffold_raw set_removeAll)
-apply (auto)[1]
+apply(drule_tac x="removeAll a1 b" in meta_spec)
-apply (drule_tac x="e" in spec)
+apply(auto)
-apply (blast)
+apply(drule meta_mp)
-apply (case_tac b)
+apply(blast)
-apply (simp_all)
+by (metis List.set.simps(2) emptyE ffold_raw.simps(2) in_listsp_conv_set listsp.simps mem_def)
-apply (subgoal_tac "ffold_raw f z b = f a1 (ffold_raw f z (delete_raw b a1))")
-apply (simp only:)
+lemma [quot_respect]:
-apply (rule_tac f="f a1" in arg_cong)
+shows "(op = ===> op = ===> op \<approx> ===> op =) ffold_raw ffold_raw"
-apply (subgoal_tac "\<forall>e. memb e a2 = memb e (delete_raw b a1)")
+unfolding fun_rel_def
-apply (simp)
+apply(auto intro: ffold_raw_rsp_pre)
-apply (simp add: memb_delete_raw)
-apply (auto simp add: memb_cons_iff)[1]
-apply (erule memb_commute_ffold_raw)
-apply (drule_tac x="a1" in spec)
-apply (simp add: memb_cons_iff)
-apply (simp add: memb_cons_iff)
-apply (case_tac b)
-apply (simp_all)
 done
-lemma [quot_respect]:
-"(op = ===> op = ===> op \<approx> ===> op =) ffold_raw ffold_raw"
-by (simp add: memb_def[symmetric] ffold_raw_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"
 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
+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
+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 simp
+then show "concat a \<approx> concat b" by auto
 qed
 lemma concat_rsp_unfolded:
 "\<lbrakk>list_all2 op \<approx> a ba; ba \<approx> bb; list_all2 op \<approx> bb b\<rbrakk> \<Longrightarrow> concat a \<approx> concat b"
 proof -
 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 simp
+then show "concat a \<approx> concat b" by auto
 qed
 lemma [quot_respect]:
-"((op =) ===> op \<approx> ===> op \<approx>) filter filter"
+shows "((op =) ===> op \<approx> ===> op \<approx>) filter filter"
 by auto
 text {* Distributive lattice with bot *}
 lemma sub_list_not_eq:
 "sub_list y (x @ y)"
 by (simp add: sub_list_def)
 lemma sub_list_inter_left:
 shows "sub_list (finter_raw x y) x"
-by (simp add: sub_list_def memb_def[symmetric] memb_finter_raw)
+by (simp add: sub_list_def)
 lemma sub_list_inter_right:
 shows "sub_list (finter_raw x y) y"
-by (simp add: sub_list_def memb_def[symmetric] memb_finter_raw)
+by (simp add: sub_list_def)
 lemma sub_list_list_eq:
 "sub_list x y \<Longrightarrow> sub_list y x \<Longrightarrow> list_eq x y"
 unfolding sub_list_def list_eq.simps by blast
 "sub_list y x \<Longrightarrow> sub_list z x \<Longrightarrow> sub_list (y @ z) x"
 unfolding sub_list_def by auto
 lemma sub_list_inter:
 "sub_list x y \<Longrightarrow> sub_list x z \<Longrightarrow> sub_list x (finter_raw y z)"
-by (simp add: sub_list_def memb_def[symmetric] memb_finter_raw)
+by (simp add: sub_list_def)
 lemma append_inter_distrib:
 "x @ (finter_raw y z) \<approx> finter_raw (x @ y) (x @ z)"
 apply (induct x)
-apply (simp_all add: memb_def)
+apply (auto)
-apply (simp add: memb_def[symmetric] memb_finter_raw)
-apply (auto simp add: memb_def)
 done
 instantiation fset :: (type) "{bounded_lattice_bot,distrib_lattice,minus}"
 begin
 f_subset_eq :: "'a fset \<Rightarrow> 'a fset \<Rightarrow> bool" (infix "|\<subseteq>|" 50)
 where
 "xs |\<subseteq>| ys \<equiv> xs \<le> ys"
 definition
-less_fset:
+less_fset :: "'a fset \<Rightarrow> 'a fset \<Rightarrow> bool"
-"(xs :: 'a fset) < ys \<equiv> xs \<le> ys \<and> xs \<noteq> ys"
+where
+"xs < ys \<equiv> xs \<le> ys \<and> xs \<noteq> (ys::'a fset)"
 abbreviation
-f_subset :: "'a fset \<Rightarrow> 'a fset \<Rightarrow> bool" (infix "|\<subset>|" 50)
+fsubset :: "'a fset \<Rightarrow> 'a fset \<Rightarrow> bool" (infix "|\<subset>|" 50)
 where
 "xs |\<subset>| ys \<equiv> xs < ys"
 quotient_definition
-"sup  \<Colon> ('a fset \<Rightarrow> 'a fset \<Rightarrow> 'a fset)"
+"sup :: 'a fset \<Rightarrow> 'a fset \<Rightarrow> 'a fset"
 is
-"(op @) \<Colon> ('a list \<Rightarrow> 'a list \<Rightarrow> 'a list)"
+"append :: 'a list \<Rightarrow> 'a list \<Rightarrow> 'a list"
 abbreviation
 funion (infixl "|\<union>|" 65)
 where
-"xs |\<union>| ys \<equiv> sup (xs :: 'a fset) ys"
+"xs |\<union>| ys \<equiv> sup xs (ys::'a fset)"
 quotient_definition
-"inf \<Colon> ('a fset \<Rightarrow> 'a fset \<Rightarrow> 'a fset)"
+"inf :: 'a fset \<Rightarrow> 'a fset \<Rightarrow> 'a fset"
 is
-"finter_raw \<Colon> ('a list \<Rightarrow> 'a list \<Rightarrow> 'a list)"
+"finter_raw :: 'a list \<Rightarrow> 'a list \<Rightarrow> 'a list"
 abbreviation
 finter (infixl "|\<inter>|" 65)
 where
-"xs |\<inter>| ys \<equiv> inf (xs :: 'a fset) ys"
+"xs |\<inter>| ys \<equiv> inf xs (ys::'a fset)"
 quotient_definition
-"minus \<Colon> ('a fset \<Rightarrow> 'a fset \<Rightarrow> 'a fset)"
+"minus :: 'a fset \<Rightarrow> 'a fset \<Rightarrow> 'a fset"
 is
-"fminus_raw \<Colon> ('a list \<Rightarrow> 'a list \<Rightarrow> 'a list)"
+"fminus_raw :: 'a list \<Rightarrow> 'a list \<Rightarrow> 'a list"
 instance
 proof
 fix x y z :: "'a fset"
 show "(x |\<subset>| y) = (x |\<subseteq>| y \<and> \<not> y |\<subseteq>| x)"
-unfolding less_fset by (lifting sub_list_not_eq)
+unfolding less_fset_def by (lifting sub_list_not_eq)
 show "x |\<subseteq>| x" by (lifting sub_list_refl)
 show "{||} |\<subseteq>| x" by (lifting sub_list_empty)
 show "x |\<subseteq>| x |\<union>| y" by (lifting sub_list_append_left)
 show "y |\<subseteq>| x |\<union>| y" by (lifting sub_list_append_right)
 show "x |\<inter>| y |\<subseteq>| x" by (lifting sub_list_inter_left)
 section {* Finsert and Membership *}
 quotient_definition
 "finsert :: 'a \<Rightarrow> 'a fset \<Rightarrow> 'a fset"
-is "op #"
+is "Cons"
 syntax
 "@Finset"     :: "args => 'a fset"  ("{|(_)|}")
 translations
 section {* Other constants on the Quotient Type *}
 quotient_definition
 "fcard :: 'a fset \<Rightarrow> nat"
-is
+is fcard_raw
-"fcard_raw"
 quotient_definition
 "fmap :: ('a \<Rightarrow> 'b) \<Rightarrow> 'a fset \<Rightarrow> 'b fset"
-is
+is map
-"map"
 quotient_definition
-"fdelete :: 'a fset \<Rightarrow> 'a \<Rightarrow> 'a fset"
+"fdelete :: 'a \<Rightarrow> 'a fset \<Rightarrow> 'a fset"
-is "delete_raw"
+is removeAll
 quotient_definition
-"fset_to_set :: 'a fset \<Rightarrow> 'a set"
+"fset :: 'a fset \<Rightarrow> 'a set"
 is "set"
 quotient_definition
 "ffold :: ('a \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> 'b \<Rightarrow> 'a fset \<Rightarrow> 'b"
 is "ffold_raw"
 lemma [quot_preserve]: "(abs_fset \<circ> map f) [] = abs_fset []"
 by simp
 lemma [quot_respect]:
-"(op \<approx> ===> list_all2 op \<approx> OOO op \<approx> ===> list_all2 op \<approx> OOO op \<approx>) op # op #"
+shows "(op \<approx> ===> list_all2 op \<approx> OOO op \<approx> ===> list_all2 op \<approx> OOO op \<approx>) Cons Cons"
 apply auto
-apply (simp add: set_in_eq)
 apply (rule_tac b="x # b" in pred_compI)
 apply auto
 apply (rule_tac b="x # ba" in pred_compI)
 apply auto
 done
 shows "memb x xs \<Longrightarrow> memb x (y # xs)"
 by (simp add: memb_def)
 lemma singleton_list_eq:
 shows "[x] \<approx> [y] \<longleftrightarrow> x = y"
-by (simp add:) auto
+by (simp)
 lemma sub_list_cons:
 "sub_list (x # xs) ys = (memb x ys \<and> sub_list xs ys)"
 by (auto simp add: memb_def sub_list_def)
-lemma fminus_raw_red: "fminus_raw (x # xs) ys = (if memb x ys then fminus_raw xs ys else x # (fminus_raw xs ys))"
+lemma fminus_raw_red:
-by (induct ys arbitrary: xs x)
+"fminus_raw (x # xs) ys = (if x \<in> set ys then fminus_raw xs ys else x # (fminus_raw xs ys))"
-(simp_all add: not_memb_nil memb_delete_raw memb_cons_iff)
+by (induct ys arbitrary: xs x) (simp_all)
 text {* Cardinality of finite sets *}
+(* used in memb_card_not_0 *)
 lemma fcard_raw_0:
 shows "fcard_raw xs = 0 \<longleftrightarrow> xs \<approx> []"
-by (induct xs) (auto simp add: memb_def)
+unfolding fcard_raw_def
+by (induct xs) (auto)
-lemma fcard_raw_not_memb:
-shows "\<not> memb x xs \<longleftrightarrow> fcard_raw (x # xs) = Suc (fcard_raw xs)"
+(* used in list_eq2_equiv *)
-by auto
-lemma fcard_raw_suc:
-assumes a: "fcard_raw xs = Suc n"
-shows "\<exists>x ys. \<not> (memb x ys) \<and> xs \<approx> (x # ys) \<and> fcard_raw ys = n"
-using a
-by (induct xs) (auto simp add: memb_def split_ifs)
-lemma singleton_fcard_1:
-shows "set xs = {x} \<Longrightarrow> fcard_raw xs = 1"
-by (induct xs) (auto simp add: memb_def subset_insert)
-lemma fcard_raw_1:
-shows "fcard_raw xs = 1 \<longleftrightarrow> (\<exists>x. xs \<approx> [x])"
-apply (auto dest!: fcard_raw_suc)
-apply (simp add: fcard_raw_0)
-apply (rule_tac x="x" in exI)
-apply simp
-apply (subgoal_tac "set xs = {x}")
-apply (drule singleton_fcard_1)
-apply auto
-done
-lemma fcard_raw_suc_memb:
-assumes a: "fcard_raw A = Suc n"
-shows "\<exists>a. memb a A"
-using a
-by (induct A) (auto simp add: memb_def)
 lemma memb_card_not_0:
 assumes a: "memb a A"
 shows "\<not>(fcard_raw A = 0)"
 proof -
 have "\<not>(\<forall>x. \<not> memb x A)" using a by auto
 then have "\<not>A \<approx> []" using none_memb_nil[of A] by simp
 then show ?thesis using fcard_raw_0[of A] by simp
 qed
-text {* fmap *}
+section {* fmap *}
+(* there is another fmap section below *)
 lemma map_append:
 "map f (xs @ ys) \<approx> (map f xs) @ (map f ys)"
 by simp
 qed
 qed
 then show thesis using a c by blast
 qed
 section {* deletion *}
-lemma memb_delete_raw_ident:
+lemma memb_removeAll_ident:
-shows "\<not> memb x (delete_raw xs x)"
+shows "\<not> memb x (removeAll x xs)"
 by (induct xs) (auto simp add: memb_def)
-lemma fset_raw_delete_raw_cases:
+lemma fset_raw_removeAll_cases:
-"xs = [] \<or> (\<exists>x. memb x xs \<and> xs \<approx> x # delete_raw xs x)"
+"xs = [] \<or> (\<exists>x. memb x xs \<and> xs \<approx> x # removeAll x xs)"
 by (induct xs) (auto simp add: memb_def)
-lemma fdelete_raw_filter:
+lemma fremoveAll_filter:
-"delete_raw xs y = [x \<leftarrow> xs. x \<noteq> y]"
+"removeAll y xs = [x \<leftarrow> xs. x \<noteq> y]"
 by (induct xs) simp_all
 lemma fcard_raw_delete:
-"fcard_raw (delete_raw xs y) = (if memb y xs then fcard_raw xs - 1 else fcard_raw xs)"
+"fcard_raw (removeAll y xs) = (if memb y xs then fcard_raw xs - 1 else fcard_raw xs)"
-by (simp add: fdelete_raw_filter fcard_raw_delete_one)
+by (auto simp add: fcard_raw_def memb_def)
 lemma finter_raw_empty:
 "finter_raw l [] = []"
 by (induct l) (simp_all add: not_memb_nil)
-lemma set_cong:
-shows "(x \<approx> y) = (set x = set y)"
-by auto
 lemma inj_map_eq_iff:
 "inj f \<Longrightarrow> (map f l \<approx> map f m) = (l \<approx> m)"
 by (simp add: set_eq_iff[symmetric] inj_image_eq_iff)
 lemma list_eq2_refl:
 shows "list_eq2 xs xs"
 by (induct xs) (auto intro: list_eq2.intros)
 lemma cons_delete_list_eq2:
-shows "list_eq2 (a # (delete_raw A a)) (if memb a A then A else a # A)"
+shows "list_eq2 (a # (removeAll a A)) (if memb a A then A else a # A)"
 apply (induct A)
 apply (simp add: memb_def list_eq2_refl)
 apply (case_tac "memb a (aa # A)")
 apply (simp_all only: memb_cons_iff)
 apply (case_tac [!] "a = aa")
 apply (simp_all)
 apply (case_tac "memb a A")
 apply (auto simp add: memb_def)[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 not_memb_delete_raw_ident)
+apply (auto simp add: list_eq2_refl not_memb_removeAll_ident)
 done
 lemma memb_delete_list_eq2:
 assumes a: "memb e r"
-shows "list_eq2 (e # delete_raw r e) r"
+shows "list_eq2 (e # removeAll e r) r"
 using a cons_delete_list_eq2[of e r]
 by simp
-lemma delete_raw_rsp:
+lemma removeAll_rsp:
-"xs \<approx> ys \<Longrightarrow> delete_raw xs x \<approx> delete_raw ys x"
+"xs \<approx> ys \<Longrightarrow> removeAll x xs \<approx> removeAll x ys"
-by (simp add: memb_def[symmetric] memb_delete_raw)
+by (simp add: memb_def[symmetric])
 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
 then show ?case using z list_eq2_refl by simp
 next
 case (Suc m)
 have b: "l \<approx> r" by fact
 have d: "fcard_raw l = Suc m" by fact
-then have "\<exists>a. memb a l" by (rule fcard_raw_suc_memb)
+then have "\<exists>a. memb a l"
+	apply(simp add: fcard_raw_def memb_def)
+	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
+then have e': "memb a r" using list_eq.simps[simplified memb_def[symmetric], of l r] b
-have f: "fcard_raw (delete_raw l a) = m" using fcard_raw_delete[of l a] e d by simp
+	unfolding memb_def by auto
-have g: "delete_raw l a \<approx> delete_raw r a" using delete_raw_rsp[OF b] by simp
+have f: "fcard_raw (removeAll a l) = m" using fcard_raw_delete[of a l] e d by simp
-have "list_eq2 (delete_raw l a) (delete_raw r a)" by (rule Suc.hyps[OF f g])
+have g: "removeAll a l \<approx> removeAll a r" using removeAll_rsp[OF b] by simp
-then have h: "list_eq2 (a # delete_raw l a) (a # delete_raw r a)" by (rule list_eq2.intros(5))
+have "list_eq2 (removeAll a l) (removeAll a r)" by (rule Suc.hyps[OF f g])
-have i: "list_eq2 l (a # delete_raw l a)"
+then have h: "list_eq2 (a # removeAll a l) (a # removeAll a r)" by (rule list_eq2.intros(5))
+have i: "list_eq2 l (a # removeAll a l)"
 by (rule list_eq2.intros(3)[OF memb_delete_list_eq2[OF e]])
-have "list_eq2 l (a # delete_raw r a)" by (rule list_eq2.intros(6)[OF i h])
+have "list_eq2 l (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> list_eq2 l r" by blast
 qed
-text {* Set *}
-lemma sub_list_set: "sub_list xs ys = (set xs \<subseteq> set ys)"
-by (metis rev_append set_append set_cong set_rev sub_list_append sub_list_append_left sub_list_def sub_list_not_eq subset_Un_eq)
-lemma sub_list_neq_set: "(sub_list xs ys \<and> \<not> list_eq xs ys) = (set xs \<subset> set ys)"
-by (auto simp add: sub_list_set)
-lemma fcard_raw_set: "fcard_raw xs = card (set xs)"
-by (induct xs) (auto simp add: insert_absorb memb_def card_insert_disjoint)
-lemma memb_set: "memb x xs = (x \<in> set xs)"
-by (simp only: memb_def)
-lemma filter_set: "set (filter P xs) = P \<inter> (set xs)"
-by (induct xs, simp)
-(metis Int_insert_right_if0 Int_insert_right_if1 List.set.simps(2) filter.simps(2) mem_def)
-lemma delete_raw_set: "set (delete_raw xs x) = set xs - {x}"
-by (induct xs) auto
-lemma inter_raw_set: "set (finter_raw xs ys) = set xs \<inter> set ys"
-by (induct xs) (simp_all add: memb_def)
-lemma fminus_raw_set: "set (fminus_raw xs ys) = set xs - set ys"
-by (induct ys arbitrary: xs)
-(simp_all add: delete_raw_set, blast)
-text {* Raw theorems of ffilter *}
-lemma sub_list_filter: "sub_list (filter P xs) (filter Q xs) = (\<forall> x. memb x xs \<longrightarrow> P x \<longrightarrow> Q x)"
-unfolding sub_list_def memb_def by auto
-lemma list_eq_filter: "list_eq (filter P xs) (filter Q xs) = (\<forall>x. memb x xs \<longrightarrow> P x = Q x)"
-unfolding memb_def by auto
 text {* Lifted theorems *}
 lemma not_fin_fnil: "x |\<notin>| {||}"
 by (lifting not_memb_nil)
 lemma fsingleton_eq[simp]:
 shows "{|x|} = {|y|} \<longleftrightarrow> x = y"
 by (lifting singleton_list_eq)
-text {* fset_to_set *}
+text {* fset *}
-lemma fset_to_set_simps[simp]:
-"fset_to_set {||} = ({} :: 'a set)"
+lemma fset_simps[simp]:
-"fset_to_set (finsert (h :: 'a) t) = insert h (fset_to_set t)"
+"fset {||} = ({} :: 'a set)"
+"fset (finsert (h :: 'a) t) = insert h (fset t)"
 by (lifting set.simps)
-lemma in_fset_to_set:
+lemma in_fset:
-"x \<in> fset_to_set S \<equiv> x |\<in>| S"
+"x \<in> fset S \<equiv> x |\<in>| S"
 by (lifting memb_def[symmetric])
 lemma none_fin_fempty:
 "(\<forall>x. x |\<notin>| S) = (S = {||})"
 by (lifting none_memb_nil)
 lemma fset_cong:
-"(S = T) = (fset_to_set S = fset_to_set T)"
+"(S = T) = (fset S = fset T)"
-by (lifting set_cong)
+by (lifting list_eq.simps)
-text {* fcard *}
+section {* fcard *}
-lemma fcard_fempty [simp]:
-shows "fcard {||} = 0"
-by (lifting fcard_raw_nil)
 lemma fcard_finsert_if [simp]:
 shows "fcard (finsert x S) = (if x |\<in>| S then fcard S else Suc (fcard S))"
-by (lifting fcard_raw_cons)
+by (descending) (auto simp add: fcard_raw_def memb_def insert_absorb)
-lemma fcard_0: "(fcard S = 0) = (S = {||})"
+lemma fcard_0[simp]:
-by (lifting fcard_raw_0)
+shows "fcard S = 0 \<longleftrightarrow> S = {||}"
+by (descending) (simp add: fcard_raw_def)
+lemma fcard_fempty[simp]:
+shows "fcard {||} = 0"
+by (simp add: fcard_0)
 lemma fcard_1:
-shows "(fcard S = 1) = (\<exists>x. S = {|x|})"
+shows "fcard S = 1 \<longleftrightarrow> (\<exists>x. S = {|x|})"
-by (lifting fcard_raw_1)
+by (descending) (auto simp add: fcard_raw_def card_Suc_eq)
 lemma fcard_gt_0:
-shows "x \<in> fset_to_set S \<Longrightarrow> 0 < fcard S"
+shows "x \<in> fset S \<Longrightarrow> 0 < fcard S"
-by (lifting fcard_raw_gt_0)
+by (descending) (auto simp add: fcard_raw_def card_gt_0_iff)
 lemma fcard_not_fin:
-shows "(x |\<notin>| S) = (fcard (finsert x S) = Suc (fcard S))"
+assumes a: "x |\<notin>| S"
-by (lifting fcard_raw_not_memb)
+shows "fcard (finsert x S) = Suc (fcard S)"
+using a
-lemma fcard_suc: "fcard S = Suc n \<Longrightarrow> \<exists>x T. x |\<notin>| T \<and> S = finsert x T \<and> fcard T = n"
+by (descending) (simp add: memb_def fcard_raw_def)
-by (lifting fcard_raw_suc)
+lemma fcard_suc:
+shows "fcard S = Suc n \<Longrightarrow> \<exists>x T. x |\<notin>| T \<and> S = finsert x T \<and> fcard T = n"
+apply(descending)
+apply(simp add: fcard_raw_def memb_def)
+apply(drule card_eq_SucD)
+apply(auto)
+apply(rule_tac x="b" in exI)
+apply(rule_tac x="removeAll b S" in exI)
+apply(auto)
+done
 lemma fcard_delete:
-"fcard (fdelete S y) = (if y |\<in>| S then fcard S - 1 else fcard S)"
+"fcard (fdelete y S) = (if y |\<in>| S then fcard S - 1 else fcard S)"
 by (lifting fcard_raw_delete)
-lemma fcard_suc_memb: "fcard A = Suc n \<Longrightarrow> \<exists>a. a |\<in>| A"
+lemma fcard_suc_memb:
-by (lifting fcard_raw_suc_memb)
+shows "fcard A = Suc n \<Longrightarrow> \<exists>a. a |\<in>| A"
+apply(descending)
-lemma fin_fcard_not_0: "a |\<in>| A \<Longrightarrow> fcard A \<noteq> 0"
+apply(simp add: fcard_raw_def memb_def)
-by (lifting memb_card_not_0)
+apply(drule card_eq_SucD)
+apply(auto)
-text {* funion *}
+done
+lemma fin_fcard_not_0:
+shows "a |\<in>| A \<Longrightarrow> fcard A \<noteq> 0"
+by (descending) (auto simp add: fcard_raw_def memb_def)
+section {* funion *}
 lemmas [simp] =
 sup_bot_left[where 'a="'a fset", standard]
 sup_bot_right[where 'a="'a fset", standard]
 lemma funion_finsert[simp]:
 shows "finsert x S |\<union>| T = finsert x (S |\<union>| T)"
 by (lifting append.simps(2))
 lemma singleton_union_left:
-"{|a|} |\<union>| S = finsert a S"
+shows "{|a|} |\<union>| S = finsert a S"
 by simp
 lemma singleton_union_right:
-"S |\<union>| {|a|} = finsert a S"
+shows "S |\<union>| {|a|} = finsert a S"
 by (subst sup.commute) simp
-section {* Induction and Cases rules for finite sets *}
+section {* Induction and Cases rules for fsets *}
 lemma fset_strong_cases:
 obtains "xs = {||}"
 | x ys where "x |\<notin>| ys" and "xs = finsert x ys"
 by (lifting fset_raw_strong_cases)
 assume "x |\<notin>| zs"
 then have H1: "Suc (fcard zs) = fcard (finsert x zs)" using fcard_suc by auto
 then show "P (finsert x zs)" using b h by simp
 qed
-text {* fmap *}
+section {* fmap *}
 lemma fmap_simps[simp]:
 "fmap (f :: 'a \<Rightarrow> 'b) {||} = {||}"
 "fmap f (finsert x S) = finsert (f x) (fmap f S)"
 by (lifting map.simps)
 lemma fmap_set_image:
-"fset_to_set (fmap f S) = f ` (fset_to_set S)"
+"fset (fmap f S) = f ` (fset S)"
 by (induct S) simp_all
 lemma inj_fmap_eq_iff:
 "inj f \<Longrightarrow> (fmap f S = fmap f T) = (S = T)"
 by (lifting inj_map_eq_iff)
 lemma fin_funion:
 "x |\<in>| S |\<union>| T \<longleftrightarrow> x |\<in>| S \<or> x |\<in>| T"
 by (lifting memb_append)
-text {* to_set *}
+section {* fset *}
-lemma fin_set: "(x |\<in>| xs) = (x \<in> fset_to_set xs)"
-by (lifting memb_set)
+lemma fin_set:
+shows "x |\<in>| xs \<longleftrightarrow> x \<in> fset xs"
-lemma fnotin_set: "(x |\<notin>| xs) = (x \<notin> fset_to_set xs)"
+by (lifting memb_def)
+lemma fnotin_set:
+shows "x |\<notin>| xs \<longleftrightarrow> x \<notin> fset xs"
 by (simp add: fin_set)
-lemma fcard_set: "fcard xs = card (fset_to_set xs)"
+lemma fcard_set:
-by (lifting fcard_raw_set)
+shows "fcard xs = card (fset xs)"
+by (lifting fcard_raw_def)
-lemma fsubseteq_set: "(xs |\<subseteq>| ys) = (fset_to_set xs \<subseteq> fset_to_set ys)"
-by (lifting sub_list_set)
+lemma fsubseteq_set:
+shows "xs |\<subseteq>| ys \<longleftrightarrow> fset xs \<subseteq> fset ys"
-lemma fsubset_set: "(xs |\<subset>| ys) = (fset_to_set xs \<subset> fset_to_set ys)"
+by (lifting sub_list_def)
-unfolding less_fset by (lifting sub_list_neq_set)
+lemma fsubset_set:
-lemma ffilter_set: "fset_to_set (ffilter P xs) = P \<inter> fset_to_set xs"
+shows "xs |\<subset>| ys \<longleftrightarrow> fset xs \<subset> fset ys"
-by (lifting filter_set)
+unfolding less_fset_def
+by (descending) (auto simp add: sub_list_def)
-lemma fdelete_set: "fset_to_set (fdelete xs x) = fset_to_set xs - {x}"
-by (lifting delete_raw_set)
+lemma ffilter_set [simp]:
+shows "fset (ffilter P xs) = P \<inter> fset xs"
-lemma inter_set: "fset_to_set (xs |\<inter>| ys) = fset_to_set xs \<inter> fset_to_set ys"
+by (descending) (auto simp add: mem_def)
-by (lifting inter_raw_set)
+lemma fdelete_set [simp]:
-lemma union_set: "fset_to_set (xs |\<union>| ys) = fset_to_set xs \<union> fset_to_set ys"
+shows "fset (fdelete x xs) = fset xs - {x}"
+by (lifting set_removeAll)
+lemma finter_set [simp]:
+shows "fset (xs |\<inter>| ys) = fset xs \<inter> fset ys"
+by (lifting set_finter_raw)
+lemma funion_set [simp]:
+shows "fset (xs |\<union>| ys) = fset xs \<union> fset ys"
 by (lifting set_append)
-lemma fminus_set: "fset_to_set (xs - ys) = fset_to_set xs - fset_to_set ys"
+lemma fminus_set [simp]:
-by (lifting fminus_raw_set)
+shows "fset (xs - ys) = fset xs - fset ys"
+by (lifting set_fminus_raw)
-lemmas fset_to_set_trans =
-fin_set fnotin_set fcard_set fsubseteq_set fsubset_set
-inter_set union_set ffilter_set fset_to_set_simps
-fset_cong fdelete_set fmap_set_image fminus_set
+section {* ffold *}
+lemma ffold_nil:
-text {* ffold *}
+shows "ffold f z {||} = z"
-lemma ffold_nil: "ffold f z {||} = z"
 by (lifting ffold_raw.simps(1)[where 'a="'b" and 'b="'a"])
 lemma ffold_finsert: "ffold f z (finsert a A) =
 (if rsp_fold f then if a |\<in>| A then ffold f z A else f a (ffold f z A) else z)"
-by (lifting ffold_raw.simps(2)[where 'a="'b" and 'b="'a"])
+by (descending) (simp add: memb_def)
 lemma fin_commute_ffold:
-"\<lbrakk>rsp_fold f; h |\<in>| b\<rbrakk> \<Longrightarrow> ffold f z b = f h (ffold f z (fdelete b h))"
+"\<lbrakk>rsp_fold f; h |\<in>| b\<rbrakk> \<Longrightarrow> ffold f z b = f h (ffold f z (fdelete h b))"
-by (lifting memb_commute_ffold_raw)
+by (descending) (simp add: memb_def memb_commute_ffold_raw)
-text {* fdelete *}
+section {* fdelete *}
 lemma fin_fdelete:
-shows "x |\<in>| fdelete S y \<longleftrightarrow> x |\<in>| S \<and> x \<noteq> y"
+shows "x |\<in>| fdelete y S \<longleftrightarrow> x |\<in>| S \<and> x \<noteq> y"
-by (lifting memb_delete_raw)
+by (descending) (simp add: memb_def)
 lemma fin_fdelete_ident:
-shows "x |\<notin>| fdelete S x"
+shows "x |\<notin>| fdelete x S"
-by (lifting memb_delete_raw_ident)
+by (lifting memb_removeAll_ident)
 lemma not_memb_fdelete_ident:
-shows "x |\<notin>| S \<Longrightarrow> fdelete S x = S"
+shows "x |\<notin>| S \<Longrightarrow> fdelete x S = S"
-by (lifting not_memb_delete_raw_ident)
+by (lifting not_memb_removeAll_ident)
 lemma fset_fdelete_cases:
-shows "S = {||} \<or> (\<exists>x. x |\<in>| S \<and> S = finsert x (fdelete S x))"
+shows "S = {||} \<or> (\<exists>x. x |\<in>| S \<and> S = finsert x (fdelete x S))"
-by (lifting fset_raw_delete_raw_cases)
+by (lifting fset_raw_removeAll_cases)
-text {* inter *}
+section {* finter *}
 lemma finter_empty_l: "({||} |\<inter>| S) = {||}"
 by (lifting finter_raw.simps(1))
 lemma finter_empty_r: "(S |\<inter>| {||}) = {||}"
 by (lifting finter_raw_empty)
 lemma finter_finsert:
-"finsert x S |\<inter>| T = (if x |\<in>| T then finsert x (S |\<inter>| T) else S |\<inter>| T)"
+shows "finsert x S |\<inter>| T = (if x |\<in>| T then finsert x (S |\<inter>| T) else S |\<inter>| T)"
-by (lifting finter_raw.simps(2))
+by (descending) (simp add: memb_def)
 lemma fin_finter:
-"x |\<in>| (S |\<inter>| T) \<longleftrightarrow> x |\<in>| S \<and> x |\<in>| T"
+shows "x |\<in>| (S |\<inter>| T) \<longleftrightarrow> x |\<in>| S \<and> x |\<in>| T"
-by (lifting memb_finter_raw)
+by (descending) (simp add: memb_def)
 lemma fsubset_finsert:
-"(finsert x xs |\<subseteq>| ys) = (x |\<in>| ys \<and> xs |\<subseteq>| ys)"
+shows "(finsert x xs |\<subseteq>| ys) = (x |\<in>| ys \<and> xs |\<subseteq>| ys)"
 by (lifting sub_list_cons)
-lemma "xs |\<subseteq>| ys \<equiv> \<forall>x. x |\<in>| xs \<longrightarrow> x |\<in>| ys"
+lemma
-by (lifting sub_list_def[simplified memb_def[symmetric]])
+shows "xs |\<subseteq>| ys \<equiv> \<forall>x. x |\<in>| xs \<longrightarrow> x |\<in>| ys"
+by (descending) (auto simp add: sub_list_def memb_def)
-lemma fsubset_fin: "xs |\<subseteq>| ys = (\<forall>x. x |\<in>| xs \<longrightarrow> x |\<in>| ys)"
-by (rule meta_eq_to_obj_eq)
+lemma fsubset_fin:
-(lifting sub_list_def[simplified memb_def[symmetric]])
+shows "xs |\<subseteq>| ys = (\<forall>x. x |\<in>| xs \<longrightarrow> x |\<in>| ys)"
+by (descending) (auto simp add: sub_list_def memb_def)
-lemma fminus_fin: "(x |\<in>| xs - ys) = (x |\<in>| xs \<and> x |\<notin>| ys)"
-by (lifting fminus_raw_memb)
+lemma fminus_fin:
+shows "(x |\<in>| xs - ys) = (x |\<in>| xs \<and> x |\<notin>| ys)"
-lemma fminus_red: "finsert x xs - ys = (if x |\<in>| ys then xs - ys else finsert x (xs - ys))"
+by (descending) (simp add: memb_def)
-by (lifting fminus_raw_red)
+lemma fminus_red:
-lemma fminus_red_fin[simp]: "x |\<in>| ys \<Longrightarrow> finsert x xs - ys = xs - ys"
+shows "finsert x xs - ys = (if x |\<in>| ys then xs - ys else finsert x (xs - ys))"
+by (descending) (auto simp add: memb_def)
+lemma fminus_red_fin[simp]:
+shows "x |\<in>| ys \<Longrightarrow> finsert x xs - ys = xs - ys"
 by (simp add: fminus_red)
-lemma fminus_red_fnotin[simp]: "x |\<notin>| ys \<Longrightarrow> finsert x xs - ys = finsert x (xs - ys)"
+lemma fminus_red_fnotin[simp]:
+shows "x |\<notin>| ys \<Longrightarrow> finsert x xs - ys = finsert x (xs - ys)"
 by (simp add: fminus_red)
 lemma fset_eq_iff:
 "(S = T) = (\<forall>x. (x |\<in>| S) = (x |\<in>| T))"
-by (lifting list_eq.simps[simplified memb_def[symmetric]])
+by (descending) (auto simp add: memb_def)
 (* 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:
 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]])
-text {* concat *}
+section {* fconcat *}
 lemma fconcat_empty:
 shows "fconcat {||} = {||}"
 by (lifting concat.simps(1))
 lemma fconcat_insert:
 shows "fconcat (finsert x S) = x |\<union>| fconcat S"
 by (lifting concat.simps(2))
-text {* ffilter *}
+section {* ffilter *}
-lemma subseteq_filter: "ffilter P xs <= ffilter Q xs = (\<forall> x. x |\<in>| xs \<longrightarrow> P x \<longrightarrow> Q x)"
-by (lifting sub_list_filter)
+lemma subseteq_filter:
+shows "ffilter P xs <= ffilter Q xs = (\<forall> x. x |\<in>| xs \<longrightarrow> P x \<longrightarrow> Q x)"
-lemma eq_ffilter: "(ffilter P xs = ffilter Q xs) = (\<forall>x. x |\<in>| xs \<longrightarrow> P x = Q x)"
+by  (descending) (auto simp add: memb_def sub_list_def)
-by (lifting list_eq_filter)
+lemma eq_ffilter:
+shows "(ffilter P xs = ffilter Q xs) = (\<forall>x. x |\<in>| xs \<longrightarrow> P x = Q x)"
+by (descending) (auto simp add: memb_def)
 lemma subset_ffilter:
 "(\<And>x. x |\<in>| xs \<Longrightarrow> P x \<Longrightarrow> Q x) \<Longrightarrow> (x |\<in>| xs & \<not> P x & Q x) \<Longrightarrow> ffilter P xs < ffilter Q xs"
-unfolding less_fset by (auto simp add: subseteq_filter eq_ffilter)
+unfolding less_fset_def by (auto simp add: subseteq_filter eq_ffilter)
 section {* lemmas transferred from Finite_Set theory *}
 text {* finiteness for finite sets holds *}
-lemma finite_fset: "finite (fset_to_set S)"
+lemma finite_fset [simp]:
+shows "finite (fset S)"
 by (induct S) auto
-lemma fset_choice: "\<forall>x. x |\<in>| A \<longrightarrow> (\<exists>y. P x y) \<Longrightarrow> \<exists>f. \<forall>x. x |\<in>| A \<longrightarrow> P x (f x)"
+lemma fset_choice:
-unfolding fset_to_set_trans
+shows "\<forall>x. x |\<in>| A \<longrightarrow> (\<exists>y. P x y) \<Longrightarrow> \<exists>f. \<forall>x. x |\<in>| A \<longrightarrow> P x (f x)"
-by (rule finite_set_choice[simplified Ball_def, OF finite_fset])
+apply(descending)
+apply(simp add: memb_def)
-lemma fsubseteq_fnil: "xs |\<subseteq>| {||} = (xs = {||})"
+apply(rule finite_set_choice[simplified Ball_def])
-unfolding fset_to_set_trans
+apply(simp_all)
-by (rule subset_empty)
+done
-lemma not_fsubset_fnil: "\<not> xs |\<subset>| {||}"
+lemma fsubseteq_fempty:
-unfolding fset_to_set_trans
+shows "xs |\<subseteq>| {||} = (xs = {||})"
-by (rule not_psubset_empty)
+by (metis finter_empty_r le_iff_inf)
-lemma fcard_mono: "xs |\<subseteq>| ys \<Longrightarrow> fcard xs \<le> fcard ys"
+lemma not_fsubset_fnil:
-unfolding fset_to_set_trans
+shows "\<not> xs |\<subset>| {||}"
+by (metis fset_simps(1) fsubset_set not_psubset_empty)
+lemma fcard_mono:
+shows "xs |\<subseteq>| ys \<Longrightarrow> fcard xs \<le> fcard ys"
+unfolding fcard_set fsubseteq_set
 by (rule card_mono[OF finite_fset])
-lemma fcard_fseteq: "xs |\<subseteq>| ys \<Longrightarrow> fcard ys \<le> fcard xs \<Longrightarrow> xs = ys"
+lemma fcard_fseteq:
-unfolding fset_to_set_trans
+shows "xs |\<subseteq>| ys \<Longrightarrow> fcard ys \<le> fcard xs \<Longrightarrow> xs = ys"
-by (rule card_seteq[OF finite_fset])
+unfolding fcard_set fsubseteq_set
+by (simp add: card_seteq[OF finite_fset] fset_cong)
-lemma psubset_fcard_mono: "xs |\<subset>| ys \<Longrightarrow> fcard xs < fcard ys"
-unfolding fset_to_set_trans
+lemma psubset_fcard_mono:
+shows "xs |\<subset>| ys \<Longrightarrow> fcard xs < fcard ys"
+unfolding fcard_set fsubset_set
 by (rule psubset_card_mono[OF finite_fset])
-lemma fcard_funion_finter: "fcard xs + fcard ys = fcard (xs |\<union>| ys) + fcard (xs |\<inter>| ys)"
+lemma fcard_funion_finter:
-unfolding fset_to_set_trans
+shows "fcard xs + fcard ys = fcard (xs |\<union>| ys) + fcard (xs |\<inter>| ys)"
+unfolding fcard_set funion_set finter_set
 by (rule card_Un_Int[OF finite_fset finite_fset])
-lemma fcard_funion_disjoint: "xs |\<inter>| ys = {||} \<Longrightarrow> fcard (xs |\<union>| ys) = fcard xs + fcard ys"
+lemma fcard_funion_disjoint:
-unfolding fset_to_set_trans
+shows "xs |\<inter>| ys = {||} \<Longrightarrow> fcard (xs |\<union>| ys) = fcard xs + fcard ys"
-by (rule card_Un_disjoint[OF finite_fset finite_fset])
+unfolding fcard_set funion_set
+apply (rule card_Un_disjoint[OF finite_fset finite_fset])
-lemma fcard_delete1_less: "x |\<in>| xs \<Longrightarrow> fcard (fdelete xs x) < fcard xs"
+by (metis finter_set fset_simps(1))
-unfolding fset_to_set_trans
+lemma fcard_delete1_less:
+shows "x |\<in>| xs \<Longrightarrow> fcard (fdelete x xs) < fcard xs"
+unfolding fcard_set fin_set fdelete_set
 by (rule card_Diff1_less[OF finite_fset])
-lemma fcard_delete2_less: "x |\<in>| xs \<Longrightarrow> y |\<in>| xs \<Longrightarrow> fcard (fdelete (fdelete xs x) y) < fcard xs"
+lemma fcard_delete2_less:
-unfolding fset_to_set_trans
+shows "x |\<in>| xs \<Longrightarrow> y |\<in>| xs \<Longrightarrow> fcard (fdelete y (fdelete x xs)) < fcard xs"
+unfolding fcard_set fdelete_set fin_set
 by (rule card_Diff2_less[OF finite_fset])
-lemma fcard_delete1_le: "fcard (fdelete xs x) <= fcard xs"
+lemma fcard_delete1_le:
-unfolding fset_to_set_trans
+shows "fcard (fdelete x xs) <= fcard xs"
+unfolding fdelete_set fcard_set
 by (rule card_Diff1_le[OF finite_fset])
-lemma fcard_psubset: "ys |\<subseteq>| xs \<Longrightarrow> fcard ys < fcard xs \<Longrightarrow> ys |\<subset>| xs"
+lemma fcard_psubset:
-unfolding fset_to_set_trans
+shows "ys |\<subseteq>| xs \<Longrightarrow> fcard ys < fcard xs \<Longrightarrow> ys |\<subset>| xs"
+unfolding fcard_set fsubseteq_set fsubset_set
 by (rule card_psubset[OF finite_fset])
-lemma fcard_fmap_le: "fcard (fmap f xs) \<le> fcard xs"
+lemma fcard_fmap_le:
-unfolding fset_to_set_trans
+shows "fcard (fmap f xs) \<le> fcard xs"
+unfolding fcard_set  fmap_set_image
 by (rule card_image_le[OF finite_fset])
-lemma fin_fminus_fnotin: "x |\<in>| F - S \<Longrightarrow> x |\<notin>| S"
+lemma fin_fminus_fnotin:
-unfolding fset_to_set_trans
+shows "x |\<in>| F - S \<Longrightarrow> x |\<notin>| S"
+unfolding fin_set fminus_set
 by blast
-lemma fin_fnotin_fminus: "x |\<in>| S \<Longrightarrow> x |\<notin>| F - S"
+lemma fin_fnotin_fminus:
-unfolding fset_to_set_trans
+shows "x |\<in>| S \<Longrightarrow> x |\<notin>| F - S"
+unfolding fin_set fminus_set
 by blast
-lemma fin_mdef: "x |\<in>| F = ((x |\<notin>| (F - {|x|})) & (F = finsert x (F - {|x|})))"
+lemma fin_mdef:
-unfolding fset_to_set_trans
+shows "x |\<in>| F = ((x |\<notin>| (F - {|x|})) & (F = finsert x (F - {|x|})))"
+unfolding fin_set fset_simps fset_cong fminus_set
 by blast
 lemma fcard_fminus_finsert[simp]:
 assumes "a |\<in>| A" and "a |\<notin>| B"
-shows "fcard(A - finsert a B) = fcard(A - B) - 1"
+shows "fcard (A - finsert a B) = fcard (A - B) - 1"
-using assms unfolding fset_to_set_trans
+using assms
-by (rule card_Diff_insert[OF finite_fset])
+unfolding fin_set fcard_set fminus_set
+by (simp add: card_Diff_insert[OF finite_fset])
 lemma fcard_fminus_fsubset:
 assumes "B |\<subseteq>| A"
 shows "fcard (A - B) = fcard A - fcard B"
-using assms unfolding fset_to_set_trans
+using assms
+unfolding fsubseteq_set fcard_set fminus_set
 by (rule card_Diff_subset[OF finite_fset])
 lemma fcard_fminus_subset_finter:
-"fcard (A - B) = fcard A - fcard (A |\<inter>| B)"
+shows "fcard (A - B) = fcard A - fcard (A |\<inter>| B)"
-unfolding fset_to_set_trans
+using assms
-by (rule card_Diff_subset_Int) (fold inter_set, rule finite_fset)
+unfolding finter_set fcard_set fminus_set
+by (rule card_Diff_subset_Int) (simp)
-lemma ball_reg_right_unfolded: "(\<forall>x. R x \<longrightarrow> P x \<longrightarrow> Q x) \<longrightarrow> (All P \<longrightarrow> Ball R Q)"
-apply rule
-apply (rule ball_reg_right)
-apply auto
-done
 lemma list_all2_refl:
 assumes q: "equivp R"
 shows "(list_all2 R) r r"
 by (rule list_all2_refl) (metis equivp_def q)

changeset 2524	693562f03eee
parent 2479	a9b6a00b1ba0
child 2525	c848f93807b9