Something is wrong with the statement of strong induction for TySch, as the All case is trivial and Fun case unprovable...
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 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 "a |\<notin>| S \<equiv> \<not>(a |\<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 a set -- @{const finsert} *}lemma nil_not_cons: shows "\<not>[] \<approx> x # xs" "\<not>x # xs \<approx> []" by autolemma 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_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: "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: "a \<approx> b" shows "fcard_raw a = fcard_raw b" using a apply(induct a arbitrary: b) apply(auto simp add: memb_def) apply(metis) apply(drule_tac x="[x \<leftarrow> b. x \<noteq> a1]" 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"text {* raw section *}lemma map_rsp_aux: assumes a: "a \<approx> b" shows "map f a \<approx> map f b" using a apply(induct a arbitrary: b) apply(auto) apply(metis rev_image_eqI) donelemma map_rsp[quot_respect]: shows "(op = ===> op \<approx> ===> op \<approx>) map map" by (auto simp add: map_rsp_aux)lemma cons_left_comm: "x # y # A \<approx> y # x # A" by (auto simp add: id_simps)lemma cons_left_idem: "x # x # A \<approx> x # A" by (auto simp add: id_simps)lemma none_mem_nil: "(\<forall>a. a \<notin> set A) = (A \<approx> [])" by simplemma finite_set_raw_strong_cases: "(X = []) \<or> (\<exists>a Y. ((a \<notin> set Y) \<and> (X \<approx> a # Y)))" apply (induct X) apply (simp) apply (rule disjI2) apply (erule disjE) apply (rule_tac x="a" in exI) apply (rule_tac x="[]" in exI) apply (simp) apply (erule exE)+ apply (case_tac "a = aa") apply (rule_tac x="a" in exI) apply (rule_tac x="Y" in exI) apply (simp) apply (rule_tac x="aa" in exI) apply (rule_tac x="a # Y" in exI) apply (auto) 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 mem_delete_raw: "x \<in> set (delete_raw A a) = (x \<in> set A \<and> \<not>(x = a))" by (induct A arbitrary: x a) (auto)lemma mem_delete_raw_ident: "\<not>(a \<in> set (delete_raw A a))" by (induct A) (auto)lemma not_mem_delete_raw_ident: "b \<notin> set A \<Longrightarrow> (delete_raw A b = A)" by (induct A) (auto)lemma finite_set_raw_delete_raw_cases: "X = [] \<or> (\<exists>a. a mem X \<and> X \<approx> a # delete_raw X a)" by (induct X) (auto)lemma list2set_thm: shows "set [] = {}" and "set (h # t) = insert h (set t)" by (auto)lemma list2set_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 fold_raw :: "('a \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> 'b \<Rightarrow> 'a list \<Rightarrow> 'b"where "fold_raw f z [] = z"| "fold_raw f z (a # A) = (if (rsp_fold f) then if a mem A then fold_raw f z A else f a (fold_raw f z A) else z)"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"lemma funion_sym_pre: "a @ b \<approx> b @ a" by autolemma append_rsp[quot_respect]: shows "(op \<approx> ===> op \<approx> ===> op \<approx>) op @ op @" by (auto)lemma set_cong: "(set x = set y) = (x \<approx> y)" apply rule apply simp_all apply (induct x y rule: list_induct2') apply simp_all apply auto donelemma 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)section {* lifted part *}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 a (finsert b S) = finsert b (finsert a S)" by (lifting cons_left_comm)lemma finsert_left_idem: "finsert a (finsert a S) = finsert a 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 {||} = {}" "fset_to_set (finsert (h :: 'b) t) = insert h (fset_to_set t)" by (lifting list2set_thm)lemma in_fset_to_set: "x \<in> fset_to_set xs \<equiv> x |\<in>| xs" by (lifting memb_def[symmetric])lemma none_in_fempty: "(\<forall>a. a \<notin> fset_to_set A) = (A = {||})" by (lifting none_mem_nil)lemma fset_cong: "(fset_to_set x = fset_to_set y) = (x = y)" 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_gt_0: "x \<in> fset_to_set xs \<Longrightarrow> 0 < fcard xs" by (lifting fcard_raw_gt_0)text {* funion *}lemma funion_simps[simp]: "{||} |\<union>| ys = ys" "finsert x xs |\<union>| ys = finsert x (xs |\<union>| ys)" by (lifting append.simps)lemma funion_sym: "a |\<union>| b = b |\<union>| a" by (lifting funion_sym_pre)lemma funion_assoc: "x |\<union>| xa |\<union>| xb = x |\<union>| (xa |\<union>| xb)" by (lifting append_assoc)section {* Induction and Cases rules for finite sets *}lemma fset_strong_cases: "X = {||} \<or> (\<exists>a Y. a \<notin> fset_to_set Y \<and> X = finsert a Y)" by (lifting finite_set_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 done(* fmap *)lemma fmap_simps[simp]: "fmap (f :: 'a \<Rightarrow> 'b) {||} = {||}" "fmap f (finsert x xs) = finsert (f x) (fmap f xs)" by (lifting map.simps)lemma fmap_set_image: "fset_to_set (fmap f fs) = f ` (fset_to_set fs)" apply (induct fs) apply (simp_all)donelemma inj_fmap_eq_iff: "inj f \<Longrightarrow> (fmap f l = fmap f m) = (l = m)" by (lifting inj_map_eq_iff)ML {*fun dest_fsetT (Type ("FSet.fset", [T])) = T | dest_fsetT T = raise TYPE ("dest_fsetT: fset type expected", [T], []);*}end