(* Title: Quotient.thy Author: Cezary Kaliszyk Author: Christian Urban provides a reasoning infrastructure for the type of finite sets*)theory FSetimports Quotient Quotient_List Listbeginfun 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)"lemma list_eq_equivp: shows "equivp list_eq"unfolding equivp_reflp_symp_transp unfolding reflp_def symp_def transp_defby autoquotient_type 'a fset = "'a list" / "list_eq"by (rule list_eq_equivp)section {* empty fset, finsert and membership *}quotient_definition fempty ("{||}")where "fempty :: 'a fset"is "[]::'a list"quotient_definition "finsert :: 'a \<Rightarrow> 'a fset \<Rightarrow> 'a fset" is "op #"syntax "@Finset" :: "args => 'a fset" ("{|(_)|}")translations "{|x, xs|}" == "CONST finsert x {|xs|}" "{|x|}" == "CONST finsert x {||}"definition memb :: "'a \<Rightarrow> 'a list \<Rightarrow> bool"where "memb x xs \<equiv> x \<in> set xs"quotient_definition fin ("_ |\<in>| _" [50, 51] 50)where "fin :: 'a \<Rightarrow> 'a fset \<Rightarrow> bool" is "memb"abbreviation fnotin :: "'a \<Rightarrow> 'a fset \<Rightarrow> bool" ("_ |\<notin>| _" [50, 51] 50)where "x |\<notin>| S \<equiv> \<not>(x |\<in>| S)"lemma memb_rsp[quot_respect]: shows "(op = ===> op \<approx> ===> op =) memb memb"by (auto simp add: memb_def)lemma nil_rsp[quot_respect]: shows "[] \<approx> []"by simplemma cons_rsp[quot_respect]: shows "(op = ===> op \<approx> ===> op \<approx>) op # op #"by simpsection {* Augmenting an fset -- @{const finsert} *}lemma nil_not_cons: shows "\<not> ([] \<approx> x # xs)" "\<not> (x # xs \<approx> [])" by autolemma not_memb_nil: "\<not> memb x []" by (simp add: memb_def)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_consI1: shows "memb x (x # xs)" by (simp add: memb_def)lemma memb_consI2: shows "memb x xs \<Longrightarrow> memb x (y # xs)" by (simp add: memb_def)lemma memb_absorb: shows "memb x xs \<Longrightarrow> x # xs \<approx> xs" by (induct xs) (auto simp add: memb_def id_simps)section {* Singletons *}lemma singleton_list_eq: shows "[x] \<approx> [y] \<longleftrightarrow> x = y" by (simp add: id_simps) autosection {* Union *}quotient_definition funion (infixl "|\<union>|" 65)where "funion :: 'a fset \<Rightarrow> 'a fset \<Rightarrow> 'a fset"is "op @"section {* Cardinality of finite sets *}fun fcard_raw :: "'a list \<Rightarrow> nat"where fcard_raw_nil: "fcard_raw [] = 0"| fcard_raw_cons: "fcard_raw (x # xs) = (if memb x xs then fcard_raw xs else Suc (fcard_raw xs))"quotient_definition "fcard :: 'a fset \<Rightarrow> nat" is "fcard_raw"lemma fcard_raw_0: fixes xs :: "'a list" shows "(fcard_raw xs = 0) = (xs \<approx> [])" by (induct xs) (auto simp add: memb_def)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_not_memb: fixes x :: "'a" shows "\<not>(memb x xs) \<longleftrightarrow> fcard_raw (x # xs) = Suc (fcard_raw xs)" by autolemma fcard_raw_suc: fixes xs :: "'a list" assumes c: "fcard_raw xs = Suc n" shows "\<exists>x ys. \<not>(memb x ys) \<and> xs \<approx> (x # ys) \<and> fcard_raw ys = n" unfolding memb_def using c proof (induct xs) case Nil then show ?case by simp next case (Cons a xs) have f1: "fcard_raw xs = Suc n \<Longrightarrow> \<exists>a ys. a \<notin> set ys \<and> xs \<approx> a # ys \<and> fcard_raw ys = n" by fact have f2: "fcard_raw (a # xs) = Suc n" by fact then show ?case proof (cases "a \<in> set xs") case True then show ?thesis using f1 f2 apply - apply (simp add: memb_def) apply clarify by metis next case False then show ?thesis using f1 f2 apply - apply (rule_tac x="a" in exI) apply (rule_tac x="xs" in exI) apply (simp add: memb_def) done qed qedlemma singleton_fcard_1: shows "set xs = {x} \<Longrightarrow> fcard_raw xs = Suc 0" apply (induct xs) apply simp_all apply auto apply (subgoal_tac "set xs = {x}") apply simp apply (simp add: memb_def) apply auto apply (subgoal_tac "set xs = {}") apply simp by (metis memb_def subset_empty subset_insert)lemma fcard_raw_1: fixes a :: "'a list" 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 (erule singleton_fcard_1) apply auto donelemma fcard_raw_delete_one: "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 apply(induct xs arbitrary: ys) apply(auto simp add: memb_def) apply(metis) apply(drule_tac x="[x \<leftarrow> ys. x \<noteq> a]" in meta_spec) apply(simp add: fcard_raw_delete_one) apply(metis Suc_pred'[OF fcard_raw_gt_0] fcard_raw_delete_one memb_def) donelemma fcard_raw_rsp[quot_respect]: "(op \<approx> ===> op =) fcard_raw fcard_raw" by (simp add: fcard_raw_rsp_aux)section {* fmap and fset comprehension *}quotient_definition "fmap :: ('a \<Rightarrow> 'b) \<Rightarrow> 'a fset \<Rightarrow> 'b fset"is "map"lemma map_append: "map f (xs @ ys) \<approx> (map f xs) @ (map f ys)" by simplemma memb_append: "memb x (xs @ ys) \<longleftrightarrow> memb x xs \<or> memb x ys" by (induct xs) (simp_all add: not_memb_nil memb_cons_iff)text {* raw section *}lemma map_rsp[quot_respect]: shows "(op = ===> op \<approx> ===> op \<approx>) map map" by autolemma cons_left_comm: "x # y # xs \<approx> y # x # xs" by autolemma cons_left_idem: "x # x # xs \<approx> x # xs" by autolemma none_memb_nil: "(\<forall>x. \<not> memb x xs) = (xs \<approx> [])" by (simp add: memb_def)lemma fset_raw_strong_cases: "(xs = []) \<or> (\<exists>x ys. ((\<not> memb x ys) \<and> (xs \<approx> x # ys)))" apply (induct xs) apply (simp) apply (rule disjI2) apply (erule disjE) apply (rule_tac x="a" in exI) apply (rule_tac x="[]" in exI) apply (simp add: memb_def) apply (erule exE)+ apply (case_tac "x = a") apply (rule_tac x="a" in exI) apply (rule_tac x="ys" in exI) apply (simp) apply (rule_tac x="x" in exI) apply (rule_tac x="a # ys" in exI) apply (auto simp add: memb_def) donefun delete_raw :: "'a list \<Rightarrow> 'a \<Rightarrow> 'a list"where "delete_raw [] x = []"| "delete_raw (a # A) x = (if (a = x) then delete_raw A x else a # (delete_raw A x))"lemma memb_delete_raw: "memb x (delete_raw xs y) = (memb x xs \<and> x \<noteq> y)" by (induct xs arbitrary: x y) (auto simp add: memb_def)lemma delete_raw_rsp: "l \<approx> r \<Longrightarrow> delete_raw l x \<approx> delete_raw r x" by (simp add: memb_def[symmetric] memb_delete_raw)lemma [quot_respect]: "(op \<approx> ===> op = ===> op \<approx>) delete_raw delete_raw" by (simp add: memb_def[symmetric] memb_delete_raw)lemma memb_delete_raw_ident: "\<not> memb x (delete_raw xs x)" by (induct xs) (auto simp add: memb_def)lemma not_memb_delete_raw_ident: "\<not> memb x xs \<Longrightarrow> delete_raw xs x = xs" by (induct xs) (auto simp add: memb_def)lemma fset_raw_delete_raw_cases: "xs = [] \<or> (\<exists>x. memb x xs \<and> xs \<approx> x # delete_raw xs x)" by (induct xs) (auto simp add: memb_def)lemma fdelete_raw_filter: "delete_raw xs y = [x \<leftarrow> xs. x \<noteq> y]" by (induct xs) simp_alllemma fcard_raw_delete: "fcard_raw (delete_raw xs y) = (if memb y xs then fcard_raw xs - 1 else fcard_raw xs)" by (simp add: fdelete_raw_filter fcard_raw_delete_one)lemma set_rsp[quot_respect]: "(op \<approx> ===> op =) set set" by autodefinition rsp_foldwhere "rsp_fold f = (\<forall>u v w. (f u (f v w) = f v (f u w)))"primrec 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 # A) = (if (rsp_fold f) then if memb a A then ffold_raw f z A else f a (ffold_raw f z A) else z)"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))" apply (induct b) apply (simp add: not_memb_nil) apply (simp add: ffold_raw.simps) apply (rule conjI) apply (rule_tac [!] impI) apply (rule_tac [!] conjI) apply (rule_tac [!] impI) apply (simp_all add: memb_delete_raw) apply (simp add: memb_cons_iff) apply (simp add: not_memb_delete_raw_ident) apply (simp add: memb_cons_iff rsp_fold_def) donelemma ffold_raw_rsp_pre: "\<forall>e. memb e a = memb e b \<Longrightarrow> ffold_raw f z a = ffold_raw f z b" apply (induct a arbitrary: b) apply (simp add: hd_in_set memb_absorb memb_def none_memb_nil) apply (simp add: ffold_raw.simps) apply (rule conjI) apply (rule_tac [!] impI) apply (rule_tac [!] conjI) apply (rule_tac [!] impI) apply (simp add: in_set_code memb_cons_iff memb_def) apply (metis) apply (metis Nitpick.list_size_simp(2) ffold_raw.simps(2) length_Suc_conv memb_absorb memb_cons_iff nil_not_cons(2)) defer apply (metis Nitpick.list_size_simp(2) ffold_raw.simps(2) length_Suc_conv memb_absorb memb_cons_iff nil_not_cons(2)) apply (subgoal_tac "ffold_raw f z b = f a1 (ffold_raw f z (delete_raw b a1))") apply (simp only:) apply (rule_tac f="f a1" in arg_cong) apply (subgoal_tac "\<forall>e. memb e a2 = memb e (delete_raw b a1)") apply simp apply (simp add: memb_delete_raw) apply (metis memb_cons_iff) apply (erule memb_commute_ffold_raw) apply (drule_tac x="a1" in spec) apply (simp add: memb_cons_iff) donelemma [quot_respect]: "(op = ===> op = ===> op \<approx> ===> op =) ffold_raw ffold_raw" by (simp add: memb_def[symmetric] ffold_raw_rsp_pre)primrec finter_raw :: "'a list \<Rightarrow> 'a list \<Rightarrow> 'a list"where "finter_raw [] l = []"| "finter_raw (h # t) l = (if memb h l then h # (finter_raw t l) else finter_raw t l)"lemma finter_raw_empty: "finter_raw l [] = []" by (induct l) (simp_all add: not_memb_nil)lemma memb_finter_raw: "memb x (finter_raw xs ys) \<longleftrightarrow> memb x xs \<and> memb x ys" apply (induct xs) apply (simp add: not_memb_nil) apply (simp add: finter_raw.simps) apply (simp add: memb_cons_iff) apply auto donelemma [quot_respect]: "(op \<approx> ===> op \<approx> ===> op \<approx>) finter_raw finter_raw" by (simp add: memb_def[symmetric] memb_finter_raw)section {* Constants on the Quotient Type *} quotient_definition "fdelete :: 'a fset \<Rightarrow> 'a \<Rightarrow> 'a fset" is "delete_raw"quotient_definition "fset_to_set :: '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"quotient_definition finter (infix "|\<inter>|" 50)where "finter :: 'a fset \<Rightarrow> 'a fset \<Rightarrow> 'a fset"is "finter_raw"lemma funion_sym_pre: "xs @ ys \<approx> ys @ xs" by autolemma append_rsp[quot_respect]: shows "(op \<approx> ===> op \<approx> ===> op \<approx>) op @ op @" by autolemma set_cong: shows "(set x = set y) = (x \<approx> y)" by autolemma inj_map_eq_iff: "inj f \<Longrightarrow> (map f l \<approx> map f m) = (l \<approx> m)" by (simp add: expand_set_eq[symmetric] inj_image_eq_iff)quotient_definition "fconcat :: ('a fset) fset \<Rightarrow> 'a fset"is "concat"lemma list_equiv_rsp[quot_respect]: shows "(op \<approx> ===> op \<approx> ===> op =) op \<approx> op \<approx>" by autosection {* lifted part *}lemma not_fin_fnil: "x |\<notin>| {||}" by (lifting not_memb_nil)lemma fin_finsert_iff[simp]: "x |\<in>| finsert y S = (x = y \<or> x |\<in>| S)" by (lifting memb_cons_iff)lemma shows finsertI1: "x |\<in>| finsert x S" and finsertI2: "x |\<in>| S \<Longrightarrow> x |\<in>| finsert y S" by (lifting memb_consI1, lifting memb_consI2)lemma finsert_absorb[simp]: shows "x |\<in>| S \<Longrightarrow> finsert x S = S" by (lifting memb_absorb)lemma fempty_not_finsert[simp]: "{||} \<noteq> finsert x S" "finsert x S \<noteq> {||}" by (lifting nil_not_cons)lemma finsert_left_comm: "finsert x (finsert y S) = finsert y (finsert x S)" by (lifting cons_left_comm)lemma finsert_left_idem: "finsert x (finsert x S) = finsert x S" by (lifting cons_left_idem)lemma fsingleton_eq[simp]: shows "{|x|} = {|y|} \<longleftrightarrow> x = y" by (lifting singleton_list_eq)text {* fset_to_set *}lemma fset_to_set_simps[simp]: "fset_to_set {||} = ({} :: 'a set)" "fset_to_set (finsert (h :: 'a) t) = insert h (fset_to_set t)" by (lifting set.simps)lemma in_fset_to_set: "x \<in> fset_to_set 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: "(fset_to_set S = fset_to_set T) = (S = T)" by (lifting set_cong)text {* 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)lemma fcard_0: "(fcard S = 0) = (S = {||})" by (lifting fcard_raw_0)lemma fcard_1: fixes S::"'b fset" shows "(fcard S = 1) = (\<exists>x. S = {|x|})" by (lifting fcard_raw_1)lemma fcard_gt_0: "x \<in> fset_to_set S \<Longrightarrow> 0 < fcard S" by (lifting fcard_raw_gt_0)lemma fcard_not_fin: "(x |\<notin>| S) = (fcard (finsert x S) = Suc (fcard S))" by (lifting fcard_raw_not_memb)lemma fcard_suc: "fcard S = Suc n \<Longrightarrow> \<exists>x T. x |\<notin>| T \<and> S = finsert x T \<and> fcard T = n" by (lifting fcard_raw_suc)lemma fcard_delete: "fcard (fdelete S y) = (if y |\<in>| S then fcard S - 1 else fcard S)" by (lifting fcard_raw_delete)text {* funion *}lemma funion_simps[simp]: "{||} |\<union>| S = S" "finsert x S |\<union>| T = finsert x (S |\<union>| T)" by (lifting append.simps)lemma funion_sym: "S |\<union>| T = T |\<union>| S" by (lifting funion_sym_pre)lemma funion_assoc: "S |\<union>| T |\<union>| U = S |\<union>| (T |\<union>| U)" by (lifting append_assoc)section {* Induction and Cases rules for finite sets *}lemma fset_strong_cases: "S = {||} \<or> (\<exists>x T. x |\<notin>| T \<and> S = finsert x T)" by (lifting fset_raw_strong_cases)lemma fset_exhaust[case_names fempty finsert, cases type: fset]: shows "\<lbrakk>S = {||} \<Longrightarrow> P; \<And>x S'. S = finsert x S' \<Longrightarrow> P\<rbrakk> \<Longrightarrow> P" by (lifting list.exhaust)lemma fset_induct_weak[case_names fempty finsert]: shows "\<lbrakk>P {||}; \<And>x S. P S \<Longrightarrow> P (finsert x S)\<rbrakk> \<Longrightarrow> P S" by (lifting list.induct)lemma fset_induct[case_names fempty finsert, induct type: fset]: assumes prem1: "P {||}" and prem2: "\<And>x S. \<lbrakk>x |\<notin>| S; P S\<rbrakk> \<Longrightarrow> P (finsert x S)" shows "P S"proof(induct S rule: fset_induct_weak) case fempty show "P {||}" by (rule prem1)next case (finsert x S) have asm: "P S" by fact show "P (finsert x S)" proof(cases "x |\<in>| S") case True have "x |\<in>| S" by fact then show "P (finsert x S)" using asm by simp next case False have "x |\<notin>| S" by fact then show "P (finsert x S)" using prem2 asm by simp qedqedlemma fset_induct2: "P {||} {||} \<Longrightarrow> (\<And>x xs. x |\<notin>| xs \<Longrightarrow> P (finsert x xs) {||}) \<Longrightarrow> (\<And>y ys. y |\<notin>| ys \<Longrightarrow> P {||} (finsert y ys)) \<Longrightarrow> (\<And>x xs y ys. \<lbrakk>P xs ys; x |\<notin>| xs; y |\<notin>| ys\<rbrakk> \<Longrightarrow> P (finsert x xs) (finsert y ys)) \<Longrightarrow> P xsa ysa" apply (induct xsa arbitrary: ysa) apply (induct_tac x rule: fset_induct) apply simp_all apply (induct_tac xa rule: fset_induct) apply simp_all donetext {* 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)" 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 fmap_funion: "fmap f (S |\<union>| T) = fmap f S |\<union>| fmap f T" by (lifting map_append)lemma fin_funion: "x |\<in>| S |\<union>| T \<longleftrightarrow> x |\<in>| S \<or> x |\<in>| T" by (lifting memb_append)text {* ffold *}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"])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))" by (lifting memb_commute_ffold_raw)text {* fdelete *}lemma fin_fdelete: shows "x |\<in>| fdelete S y \<longleftrightarrow> x |\<in>| S \<and> x \<noteq> y" by (lifting memb_delete_raw)lemma fin_fdelete_ident: shows "x |\<notin>| fdelete S x" by (lifting memb_delete_raw_ident)lemma not_memb_fdelete_ident: shows "x |\<notin>| S \<Longrightarrow> fdelete S x = S" by (lifting not_memb_delete_raw_ident)lemma fset_fdelete_cases: shows "S = {||} \<or> (\<exists>x. x |\<in>| S \<and> S = finsert x (fdelete S x))" by (lifting fset_raw_delete_raw_cases)text {* inter *}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)" by (lifting finter_raw.simps(2))lemma fin_finter: "x |\<in>| (S |\<inter>| T) \<longleftrightarrow> x |\<in>| S \<and> x |\<in>| T" by (lifting memb_finter_raw)lemma expand_fset_eq: "(S = T) = (\<forall>x. (x |\<in>| S) = (x |\<in>| T))" by (lifting list_eq.simps[simplified memb_def[symmetric]])ML {*fun dest_fsetT (Type ("FSet.fset", [T])) = T | dest_fsetT T = raise TYPE ("dest_fsetT: fset type expected", [T], []);*}no_notation list_eq (infix "\<approx>" 50)end