(* Title: Nominal2_Base Authors: Brian Huffman, Christian Urban Basic definitions and lemma infrastructure for Nominal Isabelle. *)theory Nominal2_Baseimports Main "~~/src/HOL/Library/Infinite_Set" "~~/src/HOL/Quotient_Examples/FSet"uses ("nominal_library.ML") ("nominal_atoms.ML")beginsection {* Atoms and Sorts *}text {* A simple implementation for atom_sorts is strings. *}(* types atom_sort = string *)text {* To deal with Church-like binding we use trees of strings as sorts. *}datatype atom_sort = Sort "string" "atom_sort list"datatype atom = Atom atom_sort nattext {* Basic projection function. *}primrec sort_of :: "atom \<Rightarrow> atom_sort"where "sort_of (Atom s i) = s"primrec nat_of :: "atom \<Rightarrow> nat"where "nat_of (Atom s n) = n"text {* There are infinitely many atoms of each sort. *}lemma INFM_sort_of_eq: shows "INFM a. sort_of a = s"proof - have "INFM i. sort_of (Atom s i) = s" by simp moreover have "inj (Atom s)" by (simp add: inj_on_def) ultimately show "INFM a. sort_of a = s" by (rule INFM_inj)qedlemma infinite_sort_of_eq: shows "infinite {a. sort_of a = s}" using INFM_sort_of_eq unfolding INFM_iff_infinite .lemma atom_infinite [simp]: shows "infinite (UNIV :: atom set)" using subset_UNIV infinite_sort_of_eq by (rule infinite_super)lemma obtain_atom: fixes X :: "atom set" assumes X: "finite X" obtains a where "a \<notin> X" "sort_of a = s"proof - from X have "MOST a. a \<notin> X" unfolding MOST_iff_cofinite by simp with INFM_sort_of_eq have "INFM a. sort_of a = s \<and> a \<notin> X" by (rule INFM_conjI) then obtain a where "a \<notin> X" "sort_of a = s" by (auto elim: INFM_E) then show ?thesis ..qedlemma atom_components_eq_iff: fixes a b :: atom shows "a = b \<longleftrightarrow> sort_of a = sort_of b \<and> nat_of a = nat_of b" by (induct a, induct b, simp)section {* Sort-Respecting Permutations *}typedef perm = "{f. bij f \<and> finite {a. f a \<noteq> a} \<and> (\<forall>a. sort_of (f a) = sort_of a)}"proof show "id \<in> ?perm" by simpqedlemma permI: assumes "bij f" and "MOST x. f x = x" and "\<And>a. sort_of (f a) = sort_of a" shows "f \<in> perm" using assms unfolding perm_def MOST_iff_cofinite by simplemma perm_is_bij: "f \<in> perm \<Longrightarrow> bij f" unfolding perm_def by simplemma perm_is_finite: "f \<in> perm \<Longrightarrow> finite {a. f a \<noteq> a}" unfolding perm_def by simplemma perm_is_sort_respecting: "f \<in> perm \<Longrightarrow> sort_of (f a) = sort_of a" unfolding perm_def by simplemma perm_MOST: "f \<in> perm \<Longrightarrow> MOST x. f x = x" unfolding perm_def MOST_iff_cofinite by simplemma perm_id: "id \<in> perm" unfolding perm_def by simplemma perm_comp: assumes f: "f \<in> perm" and g: "g \<in> perm" shows "(f \<circ> g) \<in> perm"apply (rule permI)apply (rule bij_comp)apply (rule perm_is_bij [OF g])apply (rule perm_is_bij [OF f])apply (rule MOST_rev_mp [OF perm_MOST [OF g]])apply (rule MOST_rev_mp [OF perm_MOST [OF f]])apply (simp)apply (simp add: perm_is_sort_respecting [OF f])apply (simp add: perm_is_sort_respecting [OF g])donelemma perm_inv: assumes f: "f \<in> perm" shows "(inv f) \<in> perm"apply (rule permI)apply (rule bij_imp_bij_inv)apply (rule perm_is_bij [OF f])apply (rule MOST_mono [OF perm_MOST [OF f]])apply (erule subst, rule inv_f_f)apply (rule bij_is_inj [OF perm_is_bij [OF f]])apply (rule perm_is_sort_respecting [OF f, THEN sym, THEN trans])apply (simp add: surj_f_inv_f [OF bij_is_surj [OF perm_is_bij [OF f]]])donelemma bij_Rep_perm: "bij (Rep_perm p)" using Rep_perm [of p] unfolding perm_def by simplemma finite_Rep_perm: "finite {a. Rep_perm p a \<noteq> a}" using Rep_perm [of p] unfolding perm_def by simplemma sort_of_Rep_perm: "sort_of (Rep_perm p a) = sort_of a" using Rep_perm [of p] unfolding perm_def by simplemma Rep_perm_ext: "Rep_perm p1 = Rep_perm p2 \<Longrightarrow> p1 = p2" by (simp add: fun_eq_iff Rep_perm_inject [symmetric])instance perm :: size ..subsection {* Permutations form a group *}instantiation perm :: group_addbegindefinition "0 = Abs_perm id"definition "- p = Abs_perm (inv (Rep_perm p))"definition "p + q = Abs_perm (Rep_perm p \<circ> Rep_perm q)"definition "(p1::perm) - p2 = p1 + - p2"lemma Rep_perm_0: "Rep_perm 0 = id" unfolding zero_perm_def by (simp add: Abs_perm_inverse perm_id)lemma Rep_perm_add: "Rep_perm (p1 + p2) = Rep_perm p1 \<circ> Rep_perm p2" unfolding plus_perm_def by (simp add: Abs_perm_inverse perm_comp Rep_perm)lemma Rep_perm_uminus: "Rep_perm (- p) = inv (Rep_perm p)" unfolding uminus_perm_def by (simp add: Abs_perm_inverse perm_inv Rep_perm)instanceapply defaultunfolding Rep_perm_inject [symmetric]unfolding minus_perm_defunfolding Rep_perm_addunfolding Rep_perm_uminusunfolding Rep_perm_0by (simp_all add: o_assoc inv_o_cancel [OF bij_is_inj [OF bij_Rep_perm]])endsection {* Implementation of swappings *}definition swap :: "atom \<Rightarrow> atom \<Rightarrow> perm" ("'(_ \<rightleftharpoons> _')")where "(a \<rightleftharpoons> b) = Abs_perm (if sort_of a = sort_of b then (\<lambda>c. if a = c then b else if b = c then a else c) else id)"lemma Rep_perm_swap: "Rep_perm (a \<rightleftharpoons> b) = (if sort_of a = sort_of b then (\<lambda>c. if a = c then b else if b = c then a else c) else id)"unfolding swap_defapply (rule Abs_perm_inverse)apply (rule permI)apply (auto simp add: bij_def inj_on_def surj_def)[1]apply (rule MOST_rev_mp [OF MOST_neq(1) [of a]])apply (rule MOST_rev_mp [OF MOST_neq(1) [of b]])apply (simp)apply (simp)donelemmas Rep_perm_simps = Rep_perm_0 Rep_perm_add Rep_perm_uminus Rep_perm_swaplemma swap_different_sorts [simp]: "sort_of a \<noteq> sort_of b \<Longrightarrow> (a \<rightleftharpoons> b) = 0" by (rule Rep_perm_ext) (simp add: Rep_perm_simps)lemma swap_cancel: "(a \<rightleftharpoons> b) + (a \<rightleftharpoons> b) = 0" by (rule Rep_perm_ext) (simp add: Rep_perm_simps fun_eq_iff)lemma swap_self [simp]: "(a \<rightleftharpoons> a) = 0" by (rule Rep_perm_ext, simp add: Rep_perm_simps fun_eq_iff)lemma minus_swap [simp]: "- (a \<rightleftharpoons> b) = (a \<rightleftharpoons> b)" by (rule minus_unique [OF swap_cancel])lemma swap_commute: "(a \<rightleftharpoons> b) = (b \<rightleftharpoons> a)" by (rule Rep_perm_ext) (simp add: Rep_perm_swap fun_eq_iff)lemma swap_triple: assumes "a \<noteq> b" and "c \<noteq> b" assumes "sort_of a = sort_of b" "sort_of b = sort_of c" shows "(a \<rightleftharpoons> c) + (b \<rightleftharpoons> c) + (a \<rightleftharpoons> c) = (a \<rightleftharpoons> b)" using assms by (rule_tac Rep_perm_ext) (auto simp add: Rep_perm_simps fun_eq_iff)section {* Permutation Types *}text {* Infix syntax for @{text permute} has higher precedence than addition, but lower than unary minus.*}class pt = fixes permute :: "perm \<Rightarrow> 'a \<Rightarrow> 'a" ("_ \<bullet> _" [76, 75] 75) assumes permute_zero [simp]: "0 \<bullet> x = x" assumes permute_plus [simp]: "(p + q) \<bullet> x = p \<bullet> (q \<bullet> x)"beginlemma permute_diff [simp]: shows "(p - q) \<bullet> x = p \<bullet> - q \<bullet> x" unfolding diff_minus by simplemma permute_minus_cancel [simp]: shows "p \<bullet> - p \<bullet> x = x" and "- p \<bullet> p \<bullet> x = x" unfolding permute_plus [symmetric] by simp_alllemma permute_swap_cancel [simp]: shows "(a \<rightleftharpoons> b) \<bullet> (a \<rightleftharpoons> b) \<bullet> x = x" unfolding permute_plus [symmetric] by (simp add: swap_cancel)lemma permute_swap_cancel2 [simp]: shows "(a \<rightleftharpoons> b) \<bullet> (b \<rightleftharpoons> a) \<bullet> x = x" unfolding permute_plus [symmetric] by (simp add: swap_commute)lemma inj_permute [simp]: shows "inj (permute p)" by (rule inj_on_inverseI) (rule permute_minus_cancel)lemma surj_permute [simp]: shows "surj (permute p)" by (rule surjI, rule permute_minus_cancel)lemma bij_permute [simp]: shows "bij (permute p)" by (rule bijI [OF inj_permute surj_permute])lemma inv_permute: shows "inv (permute p) = permute (- p)" by (rule inv_equality) (simp_all)lemma permute_minus: shows "permute (- p) = inv (permute p)" by (simp add: inv_permute)lemma permute_eq_iff [simp]: shows "p \<bullet> x = p \<bullet> y \<longleftrightarrow> x = y" by (rule inj_permute [THEN inj_eq])endsubsection {* Permutations for atoms *}instantiation atom :: ptbegindefinition "p \<bullet> a = (Rep_perm p) a"instance apply(default)apply(simp_all add: permute_atom_def Rep_perm_simps)doneendlemma sort_of_permute [simp]: shows "sort_of (p \<bullet> a) = sort_of a" unfolding permute_atom_def by (rule sort_of_Rep_perm)lemma swap_atom: shows "(a \<rightleftharpoons> b) \<bullet> c = (if sort_of a = sort_of b then (if c = a then b else if c = b then a else c) else c)" unfolding permute_atom_def by (simp add: Rep_perm_swap)lemma swap_atom_simps [simp]: "sort_of a = sort_of b \<Longrightarrow> (a \<rightleftharpoons> b) \<bullet> a = b" "sort_of a = sort_of b \<Longrightarrow> (a \<rightleftharpoons> b) \<bullet> b = a" "c \<noteq> a \<Longrightarrow> c \<noteq> b \<Longrightarrow> (a \<rightleftharpoons> b) \<bullet> c = c" unfolding swap_atom by simp_alllemma expand_perm_eq: fixes p q :: "perm" shows "p = q \<longleftrightarrow> (\<forall>a::atom. p \<bullet> a = q \<bullet> a)" unfolding permute_atom_def by (metis Rep_perm_ext ext)subsection {* Permutations for permutations *}instantiation perm :: ptbegindefinition "p \<bullet> q = p + q - p"instanceapply defaultapply (simp add: permute_perm_def)apply (simp add: permute_perm_def diff_minus minus_add add_assoc)doneendlemma permute_self: shows "p \<bullet> p = p" unfolding permute_perm_def by (simp add: diff_minus add_assoc)lemma permute_eqvt: shows "p \<bullet> (q \<bullet> x) = (p \<bullet> q) \<bullet> (p \<bullet> x)" unfolding permute_perm_def by simplemma zero_perm_eqvt: shows "p \<bullet> (0::perm) = 0" unfolding permute_perm_def by simplemma add_perm_eqvt: fixes p p1 p2 :: perm shows "p \<bullet> (p1 + p2) = p \<bullet> p1 + p \<bullet> p2" unfolding permute_perm_def by (simp add: expand_perm_eq)lemma swap_eqvt: shows "p \<bullet> (a \<rightleftharpoons> b) = (p \<bullet> a \<rightleftharpoons> p \<bullet> b)" unfolding permute_perm_def by (auto simp add: swap_atom expand_perm_eq)lemma uminus_eqvt: fixes p q::"perm" shows "p \<bullet> (- q) = - (p \<bullet> q)" unfolding permute_perm_def by (simp add: diff_minus minus_add add_assoc)subsection {* Permutations for functions *}instantiation "fun" :: (pt, pt) ptbegindefinition "p \<bullet> f = (\<lambda>x. p \<bullet> (f (- p \<bullet> x)))"instanceapply defaultapply (simp add: permute_fun_def)apply (simp add: permute_fun_def minus_add)doneendlemma permute_fun_app_eq: shows "p \<bullet> (f x) = (p \<bullet> f) (p \<bullet> x)" unfolding permute_fun_def by simpsubsection {* Permutations for booleans *}instantiation bool :: ptbegindefinition "p \<bullet> (b::bool) = b"instanceapply(default) apply(simp_all add: permute_bool_def)doneendlemma Not_eqvt: shows "p \<bullet> (\<not> A) = (\<not> (p \<bullet> A))"by (simp add: permute_bool_def)lemma conj_eqvt: shows "p \<bullet> (A \<and> B) = ((p \<bullet> A) \<and> (p \<bullet> B))" by (simp add: permute_bool_def)lemma imp_eqvt: shows "p \<bullet> (A \<longrightarrow> B) = ((p \<bullet> A) \<longrightarrow> (p \<bullet> B))" by (simp add: permute_bool_def)lemma ex_eqvt: shows "p \<bullet> (\<exists>x. P x) = (\<exists>x. (p \<bullet> P) x)" unfolding permute_fun_def permute_bool_def by (auto, rule_tac x="p \<bullet> x" in exI, simp)lemma all_eqvt: shows "p \<bullet> (\<forall>x. P x) = (\<forall>x. (p \<bullet> P) x)" unfolding permute_fun_def permute_bool_def by (auto, drule_tac x="p \<bullet> x" in spec, simp)lemma ex1_eqvt: shows "p \<bullet> (\<exists>!x. P x) = (\<exists>!x. (p \<bullet> P) x)" unfolding Ex1_def apply(simp add: ex_eqvt) unfolding permute_fun_def apply(simp add: conj_eqvt all_eqvt) unfolding permute_fun_def apply(simp add: imp_eqvt permute_bool_def) donelemma permute_boolE: fixes P::"bool" shows "p \<bullet> P \<Longrightarrow> P" by (simp add: permute_bool_def)lemma permute_boolI: fixes P::"bool" shows "P \<Longrightarrow> p \<bullet> P" by(simp add: permute_bool_def)subsection {* Permutations for sets *}lemma permute_set_eq: fixes x::"'a::pt" and p::"perm" shows "(p \<bullet> X) = {p \<bullet> x | x. x \<in> X}" unfolding permute_fun_def unfolding permute_bool_def apply(auto simp add: mem_def) apply(rule_tac x="- p \<bullet> x" in exI) apply(simp) donelemma permute_set_eq_image: shows "p \<bullet> X = permute p ` X" unfolding permute_set_eq by autolemma permute_set_eq_vimage: shows "p \<bullet> X = permute (- p) -` X" unfolding permute_fun_def permute_bool_def unfolding vimage_def Collect_def mem_def ..lemma permute_finite [simp]: shows "finite (p \<bullet> X) = finite X"apply(simp add: permute_set_eq_image)apply(rule iffI)apply(drule finite_imageD)using inj_permute[where p="p"]apply(simp add: inj_on_def)apply(assumption)apply(rule finite_imageI)apply(assumption)donelemma swap_set_not_in: assumes a: "a \<notin> S" "b \<notin> S" shows "(a \<rightleftharpoons> b) \<bullet> S = S" unfolding permute_set_eq using a by (auto simp add: swap_atom)lemma swap_set_in: assumes a: "a \<in> S" "b \<notin> S" "sort_of a = sort_of b" shows "(a \<rightleftharpoons> b) \<bullet> S \<noteq> S" unfolding permute_set_eq using a by (auto simp add: swap_atom)lemma mem_permute_iff: shows "(p \<bullet> x) \<in> (p \<bullet> X) \<longleftrightarrow> x \<in> X" unfolding mem_def permute_fun_def permute_bool_def by simplemma mem_eqvt: shows "p \<bullet> (x \<in> A) \<longleftrightarrow> (p \<bullet> x) \<in> (p \<bullet> A)" unfolding mem_def by (simp add: permute_fun_app_eq)lemma empty_eqvt: shows "p \<bullet> {} = {}" unfolding empty_def Collect_def by (simp add: permute_fun_def permute_bool_def)lemma insert_eqvt: shows "p \<bullet> (insert x A) = insert (p \<bullet> x) (p \<bullet> A)" unfolding permute_set_eq_image image_insert ..subsection {* Permutations for units *}instantiation unit :: ptbegindefinition "p \<bullet> (u::unit) = u"instance by (default) (simp_all add: permute_unit_def)endsubsection {* Permutations for products *}instantiation prod :: (pt, pt) ptbeginprimrec permute_prod where Pair_eqvt: "p \<bullet> (x, y) = (p \<bullet> x, p \<bullet> y)"instanceby default autoendsubsection {* Permutations for sums *}instantiation sum :: (pt, pt) ptbeginprimrec permute_sum where "p \<bullet> (Inl x) = Inl (p \<bullet> x)"| "p \<bullet> (Inr y) = Inr (p \<bullet> y)"instance by (default) (case_tac [!] x, simp_all)endsubsection {* Permutations for lists *}instantiation list :: (pt) ptbeginprimrec permute_list where "p \<bullet> [] = []"| "p \<bullet> (x # xs) = p \<bullet> x # p \<bullet> xs"instance by (default) (induct_tac [!] x, simp_all)endlemma set_eqvt: shows "p \<bullet> (set xs) = set (p \<bullet> xs)" by (induct xs) (simp_all add: empty_eqvt insert_eqvt)subsection {* Permutations for options *}instantiation option :: (pt) ptbeginprimrec permute_option where "p \<bullet> None = None"| "p \<bullet> (Some x) = Some (p \<bullet> x)"instance by (default) (induct_tac [!] x, simp_all)endsubsection {* Permutations for fsets *}lemma permute_fset_rsp[quot_respect]: shows "(op = ===> list_eq ===> list_eq) permute permute" unfolding fun_rel_def by (simp add: set_eqvt[symmetric])instantiation fset :: (pt) ptbeginquotient_definition "permute_fset :: perm \<Rightarrow> 'a fset \<Rightarrow> 'a fset"is "permute :: perm \<Rightarrow> 'a list \<Rightarrow> 'a list"instance proof fix x :: "'a fset" and p q :: "perm" show "0 \<bullet> x = x" by (descending) (simp) show "(p + q) \<bullet> x = p \<bullet> q \<bullet> x" by (descending) (simp)qedendlemma permute_fset [simp]: fixes S::"('a::pt) fset" shows "(p \<bullet> {||}) = ({||} ::('a::pt) fset)" and "(p \<bullet> insert_fset x S) = insert_fset (p \<bullet> x) (p \<bullet> S)" by (lifting permute_list.simps)subsection {* Permutations for @{typ char}, @{typ nat}, and @{typ int} *}instantiation char :: ptbegindefinition "p \<bullet> (c::char) = c"instance by (default) (simp_all add: permute_char_def)endinstantiation nat :: ptbegindefinition "p \<bullet> (n::nat) = n"instance by (default) (simp_all add: permute_nat_def)endinstantiation int :: ptbegindefinition "p \<bullet> (i::int) = i"instance by (default) (simp_all add: permute_int_def)endsection {* Pure types *}text {* Pure types will have always empty support. *}class pure = pt + assumes permute_pure: "p \<bullet> x = x"text {* Types @{typ unit} and @{typ bool} are pure. *}instance unit :: pureproof qed (rule permute_unit_def)instance bool :: pureproof qed (rule permute_bool_def)text {* Other type constructors preserve purity. *}instance "fun" :: (pure, pure) pureby default (simp add: permute_fun_def permute_pure)instance prod :: (pure, pure) pureby default (induct_tac x, simp add: permute_pure)instance sum :: (pure, pure) pureby default (induct_tac x, simp_all add: permute_pure)instance list :: (pure) pureby default (induct_tac x, simp_all add: permute_pure)instance option :: (pure) pureby default (induct_tac x, simp_all add: permute_pure)subsection {* Types @{typ char}, @{typ nat}, and @{typ int} *}instance char :: pureproof qed (rule permute_char_def)instance nat :: pureproof qed (rule permute_nat_def)instance int :: pureproof qed (rule permute_int_def)subsection {* Supp, Freshness and Supports *}context ptbegindefinition supp :: "'a \<Rightarrow> atom set"where "supp x = {a. infinite {b. (a \<rightleftharpoons> b) \<bullet> x \<noteq> x}}"enddefinition fresh :: "atom \<Rightarrow> 'a::pt \<Rightarrow> bool" ("_ \<sharp> _" [55, 55] 55)where "a \<sharp> x \<equiv> a \<notin> supp x"lemma supp_conv_fresh: shows "supp x = {a. \<not> a \<sharp> x}" unfolding fresh_def by simplemma swap_rel_trans: assumes "sort_of a = sort_of b" assumes "sort_of b = sort_of c" assumes "(a \<rightleftharpoons> c) \<bullet> x = x" assumes "(b \<rightleftharpoons> c) \<bullet> x = x" shows "(a \<rightleftharpoons> b) \<bullet> x = x"proof (cases) assume "a = b \<or> c = b" with assms show "(a \<rightleftharpoons> b) \<bullet> x = x" by autonext assume *: "\<not> (a = b \<or> c = b)" have "((a \<rightleftharpoons> c) + (b \<rightleftharpoons> c) + (a \<rightleftharpoons> c)) \<bullet> x = x" using assms by simp also have "(a \<rightleftharpoons> c) + (b \<rightleftharpoons> c) + (a \<rightleftharpoons> c) = (a \<rightleftharpoons> b)" using assms * by (simp add: swap_triple) finally show "(a \<rightleftharpoons> b) \<bullet> x = x" .qedlemma swap_fresh_fresh: assumes a: "a \<sharp> x" and b: "b \<sharp> x" shows "(a \<rightleftharpoons> b) \<bullet> x = x"proof (cases) assume asm: "sort_of a = sort_of b" have "finite {c. (a \<rightleftharpoons> c) \<bullet> x \<noteq> x}" "finite {c. (b \<rightleftharpoons> c) \<bullet> x \<noteq> x}" using a b unfolding fresh_def supp_def by simp_all then have "finite ({c. (a \<rightleftharpoons> c) \<bullet> x \<noteq> x} \<union> {c. (b \<rightleftharpoons> c) \<bullet> x \<noteq> x})" by simp then obtain c where "(a \<rightleftharpoons> c) \<bullet> x = x" "(b \<rightleftharpoons> c) \<bullet> x = x" "sort_of c = sort_of b" by (rule obtain_atom) (auto) then show "(a \<rightleftharpoons> b) \<bullet> x = x" using asm by (rule_tac swap_rel_trans) (simp_all)next assume "sort_of a \<noteq> sort_of b" then show "(a \<rightleftharpoons> b) \<bullet> x = x" by simpqedsubsection {* supp and fresh are equivariant *}lemma finite_Collect_bij: assumes a: "bij f" shows "finite {x. P (f x)} = finite {x. P x}"by (metis a finite_vimage_iff vimage_Collect_eq)lemma fresh_permute_iff: shows "(p \<bullet> a) \<sharp> (p \<bullet> x) \<longleftrightarrow> a \<sharp> x"proof - have "(p \<bullet> a) \<sharp> (p \<bullet> x) \<longleftrightarrow> finite {b. (p \<bullet> a \<rightleftharpoons> b) \<bullet> p \<bullet> x \<noteq> p \<bullet> x}" unfolding fresh_def supp_def by simp also have "\<dots> \<longleftrightarrow> finite {b. (p \<bullet> a \<rightleftharpoons> p \<bullet> b) \<bullet> p \<bullet> x \<noteq> p \<bullet> x}" using bij_permute by (rule finite_Collect_bij[symmetric]) also have "\<dots> \<longleftrightarrow> finite {b. p \<bullet> (a \<rightleftharpoons> b) \<bullet> x \<noteq> p \<bullet> x}" by (simp only: permute_eqvt [of p] swap_eqvt) also have "\<dots> \<longleftrightarrow> finite {b. (a \<rightleftharpoons> b) \<bullet> x \<noteq> x}" by (simp only: permute_eq_iff) also have "\<dots> \<longleftrightarrow> a \<sharp> x" unfolding fresh_def supp_def by simp finally show "(p \<bullet> a) \<sharp> (p \<bullet> x) \<longleftrightarrow> a \<sharp> x" .qedlemma fresh_eqvt: shows "p \<bullet> (a \<sharp> x) = (p \<bullet> a) \<sharp> (p \<bullet> x)" unfolding permute_bool_def by (simp add: fresh_permute_iff)lemma supp_eqvt: fixes p :: "perm" and x :: "'a::pt" shows "p \<bullet> (supp x) = supp (p \<bullet> x)" unfolding supp_conv_fresh unfolding Collect_def unfolding permute_fun_def by (simp add: Not_eqvt fresh_eqvt)subsection {* supports *}definition supports :: "atom set \<Rightarrow> 'a::pt \<Rightarrow> bool" (infixl "supports" 80)where "S supports x \<equiv> \<forall>a b. (a \<notin> S \<and> b \<notin> S \<longrightarrow> (a \<rightleftharpoons> b) \<bullet> x = x)"lemma supp_is_subset: fixes S :: "atom set" and x :: "'a::pt" assumes a1: "S supports x" and a2: "finite S" shows "(supp x) \<subseteq> S"proof (rule ccontr) assume "\<not> (supp x \<subseteq> S)" then obtain a where b1: "a \<in> supp x" and b2: "a \<notin> S" by auto from a1 b2 have "\<forall>b. b \<notin> S \<longrightarrow> (a \<rightleftharpoons> b) \<bullet> x = x" unfolding supports_def by auto then have "{b. (a \<rightleftharpoons> b) \<bullet> x \<noteq> x} \<subseteq> S" by auto with a2 have "finite {b. (a \<rightleftharpoons> b)\<bullet>x \<noteq> x}" by (simp add: finite_subset) then have "a \<notin> (supp x)" unfolding supp_def by simp with b1 show False by simpqedlemma supports_finite: fixes S :: "atom set" and x :: "'a::pt" assumes a1: "S supports x" and a2: "finite S" shows "finite (supp x)"proof - have "(supp x) \<subseteq> S" using a1 a2 by (rule supp_is_subset) then show "finite (supp x)" using a2 by (simp add: finite_subset)qedlemma supp_supports: fixes x :: "'a::pt" shows "(supp x) supports x"unfolding supports_defproof (intro strip) fix a b assume "a \<notin> (supp x) \<and> b \<notin> (supp x)" then have "a \<sharp> x" and "b \<sharp> x" by (simp_all add: fresh_def) then show "(a \<rightleftharpoons> b) \<bullet> x = x" by (simp add: swap_fresh_fresh)qedlemma supp_is_least_supports: fixes S :: "atom set" and x :: "'a::pt" assumes a1: "S supports x" and a2: "finite S" and a3: "\<And>S'. finite S' \<Longrightarrow> (S' supports x) \<Longrightarrow> S \<subseteq> S'" shows "(supp x) = S"proof (rule equalityI) show "(supp x) \<subseteq> S" using a1 a2 by (rule supp_is_subset) with a2 have fin: "finite (supp x)" by (rule rev_finite_subset) have "(supp x) supports x" by (rule supp_supports) with fin a3 show "S \<subseteq> supp x" by blastqedlemma subsetCI: shows "(\<And>x. x \<in> A \<Longrightarrow> x \<notin> B \<Longrightarrow> False) \<Longrightarrow> A \<subseteq> B" by autolemma finite_supp_unique: assumes a1: "S supports x" assumes a2: "finite S" assumes a3: "\<And>a b. \<lbrakk>a \<in> S; b \<notin> S; sort_of a = sort_of b\<rbrakk> \<Longrightarrow> (a \<rightleftharpoons> b) \<bullet> x \<noteq> x" shows "(supp x) = S" using a1 a2proof (rule supp_is_least_supports) fix S' assume "finite S'" and "S' supports x" show "S \<subseteq> S'" proof (rule subsetCI) fix a assume "a \<in> S" and "a \<notin> S'" have "finite (S \<union> S')" using `finite S` `finite S'` by simp then obtain b where "b \<notin> S \<union> S'" and "sort_of b = sort_of a" by (rule obtain_atom) then have "b \<notin> S" and "b \<notin> S'" and "sort_of a = sort_of b" by simp_all then have "(a \<rightleftharpoons> b) \<bullet> x = x" using `a \<notin> S'` `S' supports x` by (simp add: supports_def) moreover have "(a \<rightleftharpoons> b) \<bullet> x \<noteq> x" using `a \<in> S` `b \<notin> S` `sort_of a = sort_of b` by (rule a3) ultimately show "False" by simp qedqedsection {* Support w.r.t. relations *}text {* This definition is used for unquotient types, where alpha-equivalence does not coincide with equality.*}definition "supp_rel R x = {a. infinite {b. \<not>(R ((a \<rightleftharpoons> b) \<bullet> x) x)}}"section {* Finitely-supported types *}class fs = pt + assumes finite_supp: "finite (supp x)"lemma pure_supp: shows "supp (x::'a::pure) = {}" unfolding supp_def by (simp add: permute_pure)lemma pure_fresh: fixes x::"'a::pure" shows "a \<sharp> x" unfolding fresh_def by (simp add: pure_supp)instance pure < fsby default (simp add: pure_supp)subsection {* Type @{typ atom} is finitely-supported. *}lemma supp_atom: shows "supp a = {a}"apply (rule finite_supp_unique)apply (clarsimp simp add: supports_def)apply simpapply simpdonelemma fresh_atom: shows "a \<sharp> b \<longleftrightarrow> a \<noteq> b" unfolding fresh_def supp_atom by simpinstance atom :: fsby default (simp add: supp_atom)section {* Type @{typ perm} is finitely-supported. *}lemma perm_swap_eq: shows "(a \<rightleftharpoons> b) \<bullet> p = p \<longleftrightarrow> (p \<bullet> (a \<rightleftharpoons> b)) = (a \<rightleftharpoons> b)"unfolding permute_perm_defby (metis add_diff_cancel minus_perm_def)lemma supports_perm: shows "{a. p \<bullet> a \<noteq> a} supports p" unfolding supports_def unfolding perm_swap_eq by (simp add: swap_eqvt)lemma finite_perm_lemma: shows "finite {a::atom. p \<bullet> a \<noteq> a}" using finite_Rep_perm [of p] unfolding permute_atom_def .lemma supp_perm: shows "supp p = {a. p \<bullet> a \<noteq> a}"apply (rule finite_supp_unique)apply (rule supports_perm)apply (rule finite_perm_lemma)apply (simp add: perm_swap_eq swap_eqvt)apply (auto simp add: expand_perm_eq swap_atom)donelemma fresh_perm: shows "a \<sharp> p \<longleftrightarrow> p \<bullet> a = a" unfolding fresh_def by (simp add: supp_perm)lemma supp_swap: shows "supp (a \<rightleftharpoons> b) = (if a = b \<or> sort_of a \<noteq> sort_of b then {} else {a, b})" by (auto simp add: supp_perm swap_atom)lemma fresh_zero_perm: shows "a \<sharp> (0::perm)" unfolding fresh_perm by simplemma supp_zero_perm: shows "supp (0::perm) = {}" unfolding supp_perm by simplemma fresh_plus_perm: fixes p q::perm assumes "a \<sharp> p" "a \<sharp> q" shows "a \<sharp> (p + q)" using assms unfolding fresh_def by (auto simp add: supp_perm)lemma supp_plus_perm: fixes p q::perm shows "supp (p + q) \<subseteq> supp p \<union> supp q" by (auto simp add: supp_perm)lemma fresh_minus_perm: fixes p::perm shows "a \<sharp> (- p) \<longleftrightarrow> a \<sharp> p" unfolding fresh_def unfolding supp_perm apply(simp) apply(metis permute_minus_cancel) donelemma supp_minus_perm: fixes p::perm shows "supp (- p) = supp p" unfolding supp_conv_fresh by (simp add: fresh_minus_perm)instance perm :: fsby default (simp add: supp_perm finite_perm_lemma)lemma plus_perm_eq: fixes p q::"perm" assumes asm: "supp p \<inter> supp q = {}" shows "p + q = q + p"unfolding expand_perm_eqproof fix a::"atom" show "(p + q) \<bullet> a = (q + p) \<bullet> a" proof - { assume "a \<notin> supp p" "a \<notin> supp q" then have "(p + q) \<bullet> a = (q + p) \<bullet> a" by (simp add: supp_perm) } moreover { assume a: "a \<in> supp p" "a \<notin> supp q" then have "p \<bullet> a \<in> supp p" by (simp add: supp_perm) then have "p \<bullet> a \<notin> supp q" using asm by auto with a have "(p + q) \<bullet> a = (q + p) \<bullet> a" by (simp add: supp_perm) } moreover { assume a: "a \<notin> supp p" "a \<in> supp q" then have "q \<bullet> a \<in> supp q" by (simp add: supp_perm) then have "q \<bullet> a \<notin> supp p" using asm by auto with a have "(p + q) \<bullet> a = (q + p) \<bullet> a" by (simp add: supp_perm) } ultimately show "(p + q) \<bullet> a = (q + p) \<bullet> a" using asm by blast qedqedlemma supp_plus_perm_eq: fixes p q::perm assumes asm: "supp p \<inter> supp q = {}" shows "supp (p + q) = supp p \<union> supp q"proof - { fix a::"atom" assume "a \<in> supp p" then have "a \<notin> supp q" using asm by auto then have "a \<in> supp (p + q)" using `a \<in> supp p` by (simp add: supp_perm) } moreover { fix a::"atom" assume "a \<in> supp q" then have "a \<notin> supp p" using asm by auto then have "a \<in> supp (q + p)" using `a \<in> supp q` by (simp add: supp_perm) then have "a \<in> supp (p + q)" using asm plus_perm_eq by metis } ultimately have "supp p \<union> supp q \<subseteq> supp (p + q)" by blast then show "supp (p + q) = supp p \<union> supp q" using supp_plus_perm by blastqedsection {* Finite Support instances for other types *}subsection {* Type @{typ "'a \<times> 'b"} is finitely-supported. *}lemma supp_Pair: shows "supp (x, y) = supp x \<union> supp y" by (simp add: supp_def Collect_imp_eq Collect_neg_eq)lemma fresh_Pair: shows "a \<sharp> (x, y) \<longleftrightarrow> a \<sharp> x \<and> a \<sharp> y" by (simp add: fresh_def supp_Pair)lemma supp_Unit: shows "supp () = {}" by (simp add: supp_def)lemma fresh_Unit: shows "a \<sharp> ()" by (simp add: fresh_def supp_Unit)instance prod :: (fs, fs) fsapply defaultapply (induct_tac x)apply (simp add: supp_Pair finite_supp)donesubsection {* Type @{typ "'a + 'b"} is finitely supported *}lemma supp_Inl: shows "supp (Inl x) = supp x" by (simp add: supp_def)lemma supp_Inr: shows "supp (Inr x) = supp x" by (simp add: supp_def)lemma fresh_Inl: shows "a \<sharp> Inl x \<longleftrightarrow> a \<sharp> x" by (simp add: fresh_def supp_Inl)lemma fresh_Inr: shows "a \<sharp> Inr y \<longleftrightarrow> a \<sharp> y" by (simp add: fresh_def supp_Inr)instance sum :: (fs, fs) fsapply defaultapply (induct_tac x)apply (simp_all add: supp_Inl supp_Inr finite_supp)donesubsection {* Type @{typ "'a option"} is finitely supported *}lemma supp_None: shows "supp None = {}"by (simp add: supp_def)lemma supp_Some: shows "supp (Some x) = supp x" by (simp add: supp_def)lemma fresh_None: shows "a \<sharp> None" by (simp add: fresh_def supp_None)lemma fresh_Some: shows "a \<sharp> Some x \<longleftrightarrow> a \<sharp> x" by (simp add: fresh_def supp_Some)instance option :: (fs) fsapply defaultapply (induct_tac x)apply (simp_all add: supp_None supp_Some finite_supp)donesubsubsection {* Type @{typ "'a list"} is finitely supported *}lemma supp_Nil: shows "supp [] = {}" by (simp add: supp_def)lemma supp_Cons: shows "supp (x # xs) = supp x \<union> supp xs"by (simp add: supp_def Collect_imp_eq Collect_neg_eq)lemma supp_append: shows "supp (xs @ ys) = supp xs \<union> supp ys" by (induct xs) (auto simp add: supp_Nil supp_Cons)lemma fresh_Nil: shows "a \<sharp> []" by (simp add: fresh_def supp_Nil)lemma fresh_Cons: shows "a \<sharp> (x # xs) \<longleftrightarrow> a \<sharp> x \<and> a \<sharp> xs" by (simp add: fresh_def supp_Cons)lemma fresh_append: shows "a \<sharp> (xs @ ys) \<longleftrightarrow> a \<sharp> xs \<and> a \<sharp> ys" by (induct xs) (simp_all add: fresh_Nil fresh_Cons)instance list :: (fs) fsapply defaultapply (induct_tac x)apply (simp_all add: supp_Nil supp_Cons finite_supp)donelemma supp_of_atom_list: fixes as::"atom list" shows "supp as = set as"by (induct as) (simp_all add: supp_Nil supp_Cons supp_atom)section {* Support and Freshness for Applications *}lemma fresh_conv_MOST: shows "a \<sharp> x \<longleftrightarrow> (MOST b. (a \<rightleftharpoons> b) \<bullet> x = x)" unfolding fresh_def supp_def unfolding MOST_iff_cofinite by simplemma supp_subset_fresh: assumes a: "\<And>a. a \<sharp> x \<Longrightarrow> a \<sharp> y" shows "supp y \<subseteq> supp x" using a unfolding fresh_def by blastlemma fresh_fun_app: assumes "a \<sharp> f" and "a \<sharp> x" shows "a \<sharp> f x" using assms unfolding fresh_conv_MOST unfolding permute_fun_app_eq by (elim MOST_rev_mp, simp)lemma supp_fun_app: shows "supp (f x) \<subseteq> (supp f) \<union> (supp x)" using fresh_fun_app unfolding fresh_def by autotext {* Equivariant Functions *}definition "eqvt f \<equiv> \<forall>p. p \<bullet> f = f"lemma eqvtI: shows "(\<And>p. p \<bullet> f \<equiv> f) \<Longrightarrow> eqvt f"unfolding eqvt_defby simplemma supp_fun_eqvt: assumes a: "eqvt f" shows "supp f = {}" using a unfolding eqvt_def unfolding supp_def by simplemma fresh_fun_eqvt_app: assumes a: "eqvt f" shows "a \<sharp> x \<Longrightarrow> a \<sharp> f x"proof - from a have "supp f = {}" by (simp add: supp_fun_eqvt) then show "a \<sharp> x \<Longrightarrow> a \<sharp> f x" unfolding fresh_def using supp_fun_app by autoqedtext {* equivariance of a function at a given argument *}definition "eqvt_at f x \<equiv> \<forall>p. p \<bullet> (f x) = f (p \<bullet> x)"lemma supp_eqvt_at: assumes asm: "eqvt_at f x" and fin: "finite (supp x)" shows "supp (f x) \<subseteq> supp x"apply(rule supp_is_subset)unfolding supports_defunfolding fresh_def[symmetric]using asmapply(simp add: eqvt_at_def)apply(simp add: swap_fresh_fresh)apply(rule fin)donelemma finite_supp_eqvt_at: assumes asm: "eqvt_at f x" and fin: "finite (supp x)" shows "finite (supp (f x))"apply(rule finite_subset)apply(rule supp_eqvt_at[OF asm fin])apply(rule fin)donelemma fresh_eqvt_at: assumes asm: "eqvt_at f x" and fin: "finite (supp x)" and fresh: "a \<sharp> x" shows "a \<sharp> f x"using freshunfolding fresh_defusing supp_eqvt_at[OF asm fin]by autotext {* helper functions for nominal_functions *}lemma the_default_eqvt: assumes unique: "\<exists>!x. P x" shows "(p \<bullet> (THE_default d P)) = (THE_default d (p \<bullet> P))" apply(rule THE_default1_equality [symmetric]) apply(rule_tac p="-p" in permute_boolE) apply(simp add: ex1_eqvt) apply(rule unique) apply(rule_tac p="-p" in permute_boolE) apply(rule subst[OF permute_fun_app_eq]) apply(simp) apply(rule THE_defaultI'[OF unique]) donelemma fundef_ex1_eqvt: fixes x::"'a::pt" assumes f_def: "f == (\<lambda>x::'a. THE_default d (G x))" assumes eqvt: "eqvt G" assumes ex1: "\<exists>!y. G x y" shows "(p \<bullet> (f x)) = f (p \<bullet> x)" apply(simp only: f_def) apply(subst the_default_eqvt) apply(rule ex1) using eqvt unfolding eqvt_def apply(simp add: permute_fun_app_eq) donelemma fundef_ex1_eqvt_at: fixes x::"'a::pt" assumes f_def: "f == (\<lambda>x::'a. THE_default d (G x))" assumes eqvt: "eqvt G" assumes ex1: "\<exists>!y. G x y" shows "eqvt_at f x" unfolding eqvt_at_def using assms by (auto intro: fundef_ex1_eqvt)section {* Support of Finite Sets of Finitely Supported Elements *}text {* support and freshness for atom sets *}lemma supp_finite_atom_set: fixes S::"atom set" assumes "finite S" shows "supp S = S" apply(rule finite_supp_unique) apply(simp add: supports_def) apply(simp add: swap_set_not_in) apply(rule assms) apply(simp add: swap_set_in)donelemma fresh_finite_atom_set: fixes S::"atom set" assumes "finite S" shows "a \<sharp> S \<longleftrightarrow> a \<notin> S" unfolding fresh_def by (simp add: supp_finite_atom_set[OF assms])lemma Union_fresh: shows "a \<sharp> S \<Longrightarrow> a \<sharp> (\<Union>x \<in> S. supp x)" unfolding Union_image_eq[symmetric] apply(rule_tac f="\<lambda>S. \<Union> supp ` S" in fresh_fun_eqvt_app) unfolding eqvt_def unfolding permute_fun_def apply(simp) unfolding UNION_def unfolding Collect_def unfolding Bex_def apply(simp add: ex_eqvt) unfolding permute_fun_def apply(simp add: conj_eqvt) apply(simp add: mem_eqvt) apply(simp add: supp_eqvt) unfolding permute_fun_def apply(simp) apply(assumption) donelemma Union_supports_set: shows "(\<Union>x \<in> S. supp x) supports S"proof - { fix a b have "\<forall>x \<in> S. (a \<rightleftharpoons> b) \<bullet> x = x \<Longrightarrow> (a \<rightleftharpoons> b) \<bullet> S = S" unfolding permute_set_eq by force } then show "(\<Union>x \<in> S. supp x) supports S" unfolding supports_def by (simp add: fresh_def[symmetric] swap_fresh_fresh)qedlemma Union_of_finite_supp_sets: fixes S::"('a::fs set)" assumes fin: "finite S" shows "finite (\<Union>x\<in>S. supp x)" using fin by (induct) (auto simp add: finite_supp)lemma Union_included_in_supp: fixes S::"('a::fs set)" assumes fin: "finite S" shows "(\<Union>x\<in>S. supp x) \<subseteq> supp S"proof - have "(\<Union>x\<in>S. supp x) = supp (\<Union>x\<in>S. supp x)" by (rule supp_finite_atom_set[symmetric]) (rule Union_of_finite_supp_sets[OF fin]) also have "\<dots> \<subseteq> supp S" by (rule supp_subset_fresh) (simp add: Union_fresh) finally show "(\<Union>x\<in>S. supp x) \<subseteq> supp S" .qedlemma supp_of_finite_sets: fixes S::"('a::fs set)" assumes fin: "finite S" shows "(supp S) = (\<Union>x\<in>S. supp x)"apply(rule subset_antisym)apply(rule supp_is_subset)apply(rule Union_supports_set)apply(rule Union_of_finite_supp_sets[OF fin])apply(rule Union_included_in_supp[OF fin])donelemma finite_sets_supp: fixes S::"('a::fs set)" assumes "finite S" shows "finite (supp S)"using assmsby (simp only: supp_of_finite_sets Union_of_finite_supp_sets)lemma supp_of_finite_union: fixes S T::"('a::fs) set" assumes fin1: "finite S" and fin2: "finite T" shows "supp (S \<union> T) = supp S \<union> supp T" using fin1 fin2 by (simp add: supp_of_finite_sets)lemma supp_of_finite_insert: fixes S::"('a::fs) set" assumes fin: "finite S" shows "supp (insert x S) = supp x \<union> supp S" using fin by (simp add: supp_of_finite_sets)lemma fresh_finite_insert: fixes S::"('a::fs) set" assumes fin: "finite S" shows "a \<sharp> (insert x S) \<longleftrightarrow> a \<sharp> x \<and> a \<sharp> S" using fin unfolding fresh_def by (simp add: supp_of_finite_insert)lemma supp_set_empty: shows "supp {} = {}" unfolding supp_def by (simp add: empty_eqvt)lemma fresh_set_empty: shows "a \<sharp> {}" by (simp add: fresh_def supp_set_empty)lemma supp_set: fixes xs :: "('a::fs) list" shows "supp (set xs) = supp xs"apply(induct xs)apply(simp add: supp_set_empty supp_Nil)apply(simp add: supp_Cons supp_of_finite_insert)donelemma fresh_set: fixes xs :: "('a::fs) list" shows "a \<sharp> (set xs) \<longleftrightarrow> a \<sharp> xs"unfolding fresh_defby (simp add: supp_set)subsection {* Type @{typ "'a fset"} is finitely supported *}lemma fset_eqvt: shows "p \<bullet> (fset S) = fset (p \<bullet> S)" by (lifting set_eqvt)lemma supp_fset [simp]: shows "supp (fset S) = supp S" unfolding supp_def by (simp add: fset_eqvt fset_cong)lemma supp_empty_fset [simp]: shows "supp {||} = {}" unfolding supp_def by simplemma fresh_empty_fset: shows "a \<sharp> {||}"unfolding fresh_defby (simp)lemma supp_insert_fset [simp]: fixes x::"'a::fs" and S::"'a fset" shows "supp (insert_fset x S) = supp x \<union> supp S" apply(subst supp_fset[symmetric]) apply(simp add: supp_of_finite_insert) donelemma fresh_insert_fset: fixes x::"'a::fs" and S::"'a fset" shows "a \<sharp> insert_fset x S \<longleftrightarrow> a \<sharp> x \<and> a \<sharp> S" unfolding fresh_def by (simp)lemma fset_finite_supp: fixes S::"('a::fs) fset" shows "finite (supp S)" by (induct S) (simp_all add: finite_supp)instance fset :: (fs) fs apply (default) apply (rule fset_finite_supp) donesection {* Freshness and Fresh-Star *}lemma fresh_Unit_elim: shows "(a \<sharp> () \<Longrightarrow> PROP C) \<equiv> PROP C" by (simp add: fresh_Unit)lemma fresh_Pair_elim: shows "(a \<sharp> (x, y) \<Longrightarrow> PROP C) \<equiv> (a \<sharp> x \<Longrightarrow> a \<sharp> y \<Longrightarrow> PROP C)" by rule (simp_all add: fresh_Pair)(* this rule needs to be added before the fresh_prodD is *)(* added to the simplifier with mksimps *) lemma [simp]: shows "a \<sharp> x1 \<Longrightarrow> a \<sharp> x2 \<Longrightarrow> a \<sharp> (x1, x2)" by (simp add: fresh_Pair)lemma fresh_PairD: shows "a \<sharp> (x, y) \<Longrightarrow> a \<sharp> x" and "a \<sharp> (x, y) \<Longrightarrow> a \<sharp> y" by (simp_all add: fresh_Pair)ML {* val mksimps_pairs = (@{const_name Nominal2_Base.fresh}, @{thms fresh_PairD}) :: mksimps_pairs;*}declaration {* fn _ => Simplifier.map_ss (fn ss => ss setmksimps (mksimps mksimps_pairs))*}text {* The fresh-star generalisation of fresh is used in strong induction principles. *}definition fresh_star :: "atom set \<Rightarrow> 'a::pt \<Rightarrow> bool" ("_ \<sharp>* _" [80,80] 80)where "as \<sharp>* x \<equiv> \<forall>a \<in> as. a \<sharp> x"lemma fresh_star_supp_conv: shows "supp x \<sharp>* y \<Longrightarrow> supp y \<sharp>* x"by (auto simp add: fresh_star_def fresh_def)lemma fresh_star_Pair: shows "as \<sharp>* (x, y) = (as \<sharp>* x \<and> as \<sharp>* y)" by (auto simp add: fresh_star_def fresh_Pair)lemma fresh_star_list: shows "as \<sharp>* (xs @ ys) \<longleftrightarrow> as \<sharp>* xs \<and> as \<sharp>* ys" and "as \<sharp>* (x # xs) \<longleftrightarrow> as \<sharp>* x \<and> as \<sharp>* xs" and "as \<sharp>* []"by (auto simp add: fresh_star_def fresh_Nil fresh_Cons fresh_append)lemma fresh_star_set: fixes xs::"('a::fs) list" shows "as \<sharp>* set xs \<longleftrightarrow> as \<sharp>* xs"unfolding fresh_star_defby (simp add: fresh_set)lemma fresh_star_singleton: fixes a::"atom" shows "as \<sharp>* {a} \<longleftrightarrow> as \<sharp>* a" by (simp add: fresh_star_def fresh_finite_insert fresh_set_empty)lemma fresh_star_fset: fixes xs::"('a::fs) list" shows "as \<sharp>* fset S \<longleftrightarrow> as \<sharp>* S"by (simp add: fresh_star_def fresh_def) lemma fresh_star_Un: shows "(as \<union> bs) \<sharp>* x = (as \<sharp>* x \<and> bs \<sharp>* x)" by (auto simp add: fresh_star_def)lemma fresh_star_insert: shows "(insert a as) \<sharp>* x = (a \<sharp> x \<and> as \<sharp>* x)" by (auto simp add: fresh_star_def)lemma fresh_star_Un_elim: "((as \<union> bs) \<sharp>* x \<Longrightarrow> PROP C) \<equiv> (as \<sharp>* x \<Longrightarrow> bs \<sharp>* x \<Longrightarrow> PROP C)" unfolding fresh_star_def apply(rule) apply(erule meta_mp) apply(auto) donelemma fresh_star_insert_elim: "(insert a as \<sharp>* x \<Longrightarrow> PROP C) \<equiv> (a \<sharp> x \<Longrightarrow> as \<sharp>* x \<Longrightarrow> PROP C)" unfolding fresh_star_def by rule (simp_all add: fresh_star_def)lemma fresh_star_empty_elim: "({} \<sharp>* x \<Longrightarrow> PROP C) \<equiv> PROP C" by (simp add: fresh_star_def)lemma fresh_star_Unit_elim: shows "(a \<sharp>* () \<Longrightarrow> PROP C) \<equiv> PROP C" by (simp add: fresh_star_def fresh_Unit) lemma fresh_star_Pair_elim: shows "(a \<sharp>* (x, y) \<Longrightarrow> PROP C) \<equiv> (a \<sharp>* x \<Longrightarrow> a \<sharp>* y \<Longrightarrow> PROP C)" by (rule, simp_all add: fresh_star_Pair)lemma fresh_star_zero: shows "as \<sharp>* (0::perm)" unfolding fresh_star_def by (simp add: fresh_zero_perm)lemma fresh_star_plus: fixes p q::perm shows "\<lbrakk>a \<sharp>* p; a \<sharp>* q\<rbrakk> \<Longrightarrow> a \<sharp>* (p + q)" unfolding fresh_star_def by (simp add: fresh_plus_perm)lemma fresh_star_permute_iff: shows "(p \<bullet> a) \<sharp>* (p \<bullet> x) \<longleftrightarrow> a \<sharp>* x" unfolding fresh_star_def by (metis mem_permute_iff permute_minus_cancel(1) fresh_permute_iff)lemma fresh_star_eqvt: shows "p \<bullet> (as \<sharp>* x) \<longleftrightarrow> (p \<bullet> as) \<sharp>* (p \<bullet> x)"unfolding fresh_star_defunfolding Ball_defapply(simp add: all_eqvt)apply(subst permute_fun_def)apply(simp add: imp_eqvt fresh_eqvt mem_eqvt)donelemma at_fresh_star_inter: assumes a: "(p \<bullet> bs) \<sharp>* bs" and b: "finite bs" shows "p \<bullet> bs \<inter> bs = {}"using a bunfolding fresh_star_defunfolding fresh_defby (auto simp add: supp_finite_atom_set)section {* Induction principle for permutations *}lemma perm_struct_induct[consumes 1, case_names zero swap]: assumes S: "supp p \<subseteq> S" and zero: "P 0" and swap: "\<And>p a b. \<lbrakk>P p; supp p \<subseteq> S; a \<in> S; b \<in> S; a \<noteq> b; sort_of a = sort_of b\<rbrakk> \<Longrightarrow> P ((a \<rightleftharpoons> b) + p)" shows "P p"proof - have "finite (supp p)" by (simp add: finite_supp) then show "P p" using S proof(induct A\<equiv>"supp p" arbitrary: p rule: finite_psubset_induct) case (psubset p) then have ih: "\<And>q. supp q \<subset> supp p \<Longrightarrow> P q" by auto have as: "supp p \<subseteq> S" by fact { assume "supp p = {}" then have "p = 0" by (simp add: supp_perm expand_perm_eq) then have "P p" using zero by simp } moreover { assume "supp p \<noteq> {}" then obtain a where a0: "a \<in> supp p" by blast then have a1: "p \<bullet> a \<in> S" "a \<in> S" "sort_of (p \<bullet> a) = sort_of a" "p \<bullet> a \<noteq> a" using as by (auto simp add: supp_atom supp_perm swap_atom) let ?q = "(p \<bullet> a \<rightleftharpoons> a) + p" have a2: "supp ?q \<subseteq> supp p" unfolding supp_perm by (auto simp add: swap_atom) moreover have "a \<notin> supp ?q" by (simp add: supp_perm) then have "supp ?q \<noteq> supp p" using a0 by auto ultimately have "supp ?q \<subset> supp p" using a2 by auto then have "P ?q" using ih by simp moreover have "supp ?q \<subseteq> S" using as a2 by simp ultimately have "P ((p \<bullet> a \<rightleftharpoons> a) + ?q)" using as a1 swap by simp moreover have "p = (p \<bullet> a \<rightleftharpoons> a) + ?q" by (simp add: expand_perm_eq) ultimately have "P p" by simp } ultimately show "P p" by blast qedqedlemma perm_simple_struct_induct[case_names zero swap]: assumes zero: "P 0" and swap: "\<And>p a b. \<lbrakk>P p; a \<noteq> b; sort_of a = sort_of b\<rbrakk> \<Longrightarrow> P ((a \<rightleftharpoons> b) + p)" shows "P p"by (rule_tac S="supp p" in perm_struct_induct) (auto intro: zero swap)lemma perm_subset_induct[consumes 1, case_names zero swap plus]: assumes S: "supp p \<subseteq> S" assumes zero: "P 0" assumes swap: "\<And>a b. \<lbrakk>sort_of a = sort_of b; a \<noteq> b; a \<in> S; b \<in> S\<rbrakk> \<Longrightarrow> P (a \<rightleftharpoons> b)" assumes plus: "\<And>p1 p2. \<lbrakk>P p1; P p2; supp p1 \<subseteq> S; supp p2 \<subseteq> S\<rbrakk> \<Longrightarrow> P (p1 + p2)" shows "P p"using Sby (induct p rule: perm_struct_induct) (auto intro: zero plus swap simp add: supp_swap)lemma supp_perm_eq: assumes "(supp x) \<sharp>* p" shows "p \<bullet> x = x"proof - from assms have "supp p \<subseteq> {a. a \<sharp> x}" unfolding supp_perm fresh_star_def fresh_def by auto then show "p \<bullet> x = x" proof (induct p rule: perm_struct_induct) case zero show "0 \<bullet> x = x" by simp next case (swap p a b) then have "a \<sharp> x" "b \<sharp> x" "p \<bullet> x = x" by simp_all then show "((a \<rightleftharpoons> b) + p) \<bullet> x = x" by (simp add: swap_fresh_fresh) qedqedlemma supp_perm_eq_test: assumes "(supp x) \<sharp>* p" shows "p \<bullet> x = x"proof - from assms have "supp p \<subseteq> {a. a \<sharp> x}" unfolding supp_perm fresh_star_def fresh_def by auto then show "p \<bullet> x = x" proof (induct p rule: perm_subset_induct) case zero show "0 \<bullet> x = x" by simp next case (swap a b) then have "a \<sharp> x" "b \<sharp> x" by simp_all then show "(a \<rightleftharpoons> b) \<bullet> x = x" by (simp add: swap_fresh_fresh) next case (plus p1 p2) have "p1 \<bullet> x = x" "p2 \<bullet> x = x" by fact+ then show "(p1 + p2) \<bullet> x = x" by simp qedqedlemma perm_supp_eq: assumes a: "(supp p) \<sharp>* x" shows "p \<bullet> x = x"by (rule supp_perm_eq) (simp add: fresh_star_supp_conv a)lemma supp_perm_perm_eq: assumes a: "\<forall>a \<in> supp x. p \<bullet> a = q \<bullet> a" shows "p \<bullet> x = q \<bullet> x"proof - from a have "\<forall>a \<in> supp x. (-q + p) \<bullet> a = a" by simp then have "\<forall>a \<in> supp x. a \<notin> supp (-q + p)" unfolding supp_perm by simp then have "supp x \<sharp>* (-q + p)" unfolding fresh_star_def fresh_def by simp then have "(-q + p) \<bullet> x = x" by (simp only: supp_perm_eq) then show "p \<bullet> x = q \<bullet> x" by (metis permute_minus_cancel permute_plus)qedsection {* Avoiding of atom sets *}text {* For every set of atoms, there is another set of atoms avoiding a finitely supported c and there is a permutation which 'translates' between both sets.*}lemma at_set_avoiding_aux: fixes Xs::"atom set" and As::"atom set" assumes b: "Xs \<subseteq> As" and c: "finite As" shows "\<exists>p. (p \<bullet> Xs) \<inter> As = {} \<and> (supp p) = (Xs \<union> (p \<bullet> Xs))"proof - from b c have "finite Xs" by (rule finite_subset) then show ?thesis using b proof (induct rule: finite_subset_induct) case empty have "0 \<bullet> {} \<inter> As = {}" by simp moreover have "supp (0::perm) = {} \<union> 0 \<bullet> {}" by (simp add: supp_zero_perm) ultimately show ?case by blast next case (insert x Xs) then obtain p where p1: "(p \<bullet> Xs) \<inter> As = {}" and p2: "supp p = (Xs \<union> (p \<bullet> Xs))" by blast from `x \<in> As` p1 have "x \<notin> p \<bullet> Xs" by fast with `x \<notin> Xs` p2 have "x \<notin> supp p" by fast hence px: "p \<bullet> x = x" unfolding supp_perm by simp have "finite (As \<union> p \<bullet> Xs \<union> supp p)" using `finite As` `finite Xs` by (simp add: permute_set_eq_image finite_supp) then obtain y where "y \<notin> (As \<union> p \<bullet> Xs \<union> supp p)" "sort_of y = sort_of x" by (rule obtain_atom) hence y: "y \<notin> As" "y \<notin> p \<bullet> Xs" "y \<notin> supp p" "sort_of y = sort_of x" by simp_all hence py: "p \<bullet> y = y" "x \<noteq> y" using `x \<in> As` by (auto simp add: supp_perm) let ?q = "(x \<rightleftharpoons> y) + p" have q: "?q \<bullet> insert x Xs = insert y (p \<bullet> Xs)" unfolding insert_eqvt using `p \<bullet> x = x` `sort_of y = sort_of x` using `x \<notin> p \<bullet> Xs` `y \<notin> p \<bullet> Xs` by (simp add: swap_atom swap_set_not_in) have "?q \<bullet> insert x Xs \<inter> As = {}" using `y \<notin> As` `p \<bullet> Xs \<inter> As = {}` unfolding q by simp moreover have "supp (x \<rightleftharpoons> y) \<inter> supp p = {}" using px py `sort_of y = sort_of x` unfolding supp_swap by (simp add: supp_perm) then have "supp ?q = (supp (x \<rightleftharpoons> y) \<union> supp p)" by (simp add: supp_plus_perm_eq) then have "supp ?q = insert x Xs \<union> ?q \<bullet> insert x Xs" using p2 `sort_of y = sort_of x` `x \<noteq> y` unfolding q supp_swap by auto ultimately show ?case by blast qedqedlemma at_set_avoiding: assumes a: "finite Xs" and b: "finite (supp c)" obtains p::"perm" where "(p \<bullet> Xs)\<sharp>*c" and "(supp p) = (Xs \<union> (p \<bullet> Xs))" using a b at_set_avoiding_aux [where Xs="Xs" and As="Xs \<union> supp c"] unfolding fresh_star_def fresh_def by blastlemma at_set_avoiding1: assumes "finite xs" and "finite (supp c)" shows "\<exists>p. (p \<bullet> xs) \<sharp>* c"using assmsapply(erule_tac c="c" in at_set_avoiding)apply(auto)donelemma at_set_avoiding2: assumes "finite xs" and "finite (supp c)" "finite (supp x)" and "xs \<sharp>* x" shows "\<exists>p. (p \<bullet> xs) \<sharp>* c \<and> supp x \<sharp>* p"using assmsapply(erule_tac c="(c, x)" in at_set_avoiding)apply(simp add: supp_Pair)apply(rule_tac x="p" in exI)apply(simp add: fresh_star_Pair)apply(rule fresh_star_supp_conv)apply(auto simp add: fresh_star_def)donelemma at_set_avoiding3: assumes "finite xs" and "finite (supp c)" "finite (supp x)" and "xs \<sharp>* x" shows "\<exists>p. (p \<bullet> xs) \<sharp>* c \<and> supp x \<sharp>* p \<and> supp p = xs \<union> (p \<bullet> xs)"using assmsapply(erule_tac c="(c, x)" in at_set_avoiding)apply(simp add: supp_Pair)apply(rule_tac x="p" in exI)apply(simp add: fresh_star_Pair)apply(rule fresh_star_supp_conv)apply(auto simp add: fresh_star_def)donelemma at_set_avoiding2_atom: assumes "finite (supp c)" "finite (supp x)" and b: "a \<sharp> x" shows "\<exists>p. (p \<bullet> a) \<sharp> c \<and> supp x \<sharp>* p"proof - have a: "{a} \<sharp>* x" unfolding fresh_star_def by (simp add: b) obtain p where p1: "(p \<bullet> {a}) \<sharp>* c" and p2: "supp x \<sharp>* p" using at_set_avoiding2[of "{a}" "c" "x"] assms a by blast have c: "(p \<bullet> a) \<sharp> c" using p1 unfolding fresh_star_def Ball_def by(erule_tac x="p \<bullet> a" in allE) (simp add: permute_set_eq) hence "p \<bullet> a \<sharp> c \<and> supp x \<sharp>* p" using p2 by blast then show "\<exists>p. (p \<bullet> a) \<sharp> c \<and> supp x \<sharp>* p" by blastqedsection {* Renaming permutations *}lemma set_renaming_perm: assumes b: "finite bs" shows "\<exists>q. q \<bullet> bs = p \<bullet> bs \<and> supp q \<subseteq> bs \<union> (p \<bullet> bs)"using bproof (induct) case empty have "0 \<bullet> {} = p \<bullet> {} \<and> supp (0::perm) \<subseteq> {} \<union> p \<bullet> {}" by (simp add: permute_set_eq supp_perm) then show "\<exists>q. q \<bullet> {} = p \<bullet> {} \<and> supp q \<subseteq> {} \<union> p \<bullet> {}" by blastnext case (insert a bs) then have " \<exists>q. q \<bullet> bs = p \<bullet> bs \<and> supp q \<subseteq> bs \<union> p \<bullet> bs" by (simp add: insert_eqvt) then obtain q where *: "q \<bullet> bs = p \<bullet> bs" and **: "supp q \<subseteq> bs \<union> p \<bullet> bs" by blast { assume 1: "q \<bullet> a = p \<bullet> a" have "q \<bullet> insert a bs = p \<bullet> insert a bs" using 1 * by (simp add: insert_eqvt) moreover have "supp q \<subseteq> insert a bs \<union> p \<bullet> insert a bs" using ** by (auto simp add: insert_eqvt) ultimately have "\<exists>q. q \<bullet> insert a bs = p \<bullet> insert a bs \<and> supp q \<subseteq> insert a bs \<union> p \<bullet> insert a bs" by blast } moreover { assume 2: "q \<bullet> a \<noteq> p \<bullet> a" def q' \<equiv> "((q \<bullet> a) \<rightleftharpoons> (p \<bullet> a)) + q" { have "(q \<bullet> a) \<notin> (p \<bullet> bs)" using `a \<notin> bs` *[symmetric] by (simp add: mem_permute_iff) moreover have "(p \<bullet> a) \<notin> (p \<bullet> bs)" using `a \<notin> bs` by (simp add: mem_permute_iff) ultimately have "q' \<bullet> insert a bs = p \<bullet> insert a bs" using 2 * unfolding q'_def by (simp add: insert_eqvt swap_set_not_in) } moreover { have "{q \<bullet> a, p \<bullet> a} \<subseteq> insert a bs \<union> p \<bullet> insert a bs" using ** apply (auto simp add: supp_perm insert_eqvt) apply (subgoal_tac "q \<bullet> a \<in> bs \<union> p \<bullet> bs") apply(auto)[1] apply(subgoal_tac "q \<bullet> a \<in> {a. q \<bullet> a \<noteq> a}") apply(blast) apply(simp) done then have "supp (q \<bullet> a \<rightleftharpoons> p \<bullet> a) \<subseteq> insert a bs \<union> p \<bullet> insert a bs" by (simp add: supp_swap) moreover have "supp q \<subseteq> insert a bs \<union> p \<bullet> insert a bs" using ** by (auto simp add: insert_eqvt) ultimately have "supp q' \<subseteq> insert a bs \<union> p \<bullet> insert a bs" unfolding q'_def using supp_plus_perm by blast } ultimately have "\<exists>q. q \<bullet> insert a bs = p \<bullet> insert a bs \<and> supp q \<subseteq> insert a bs \<union> p \<bullet> insert a bs" by blast } ultimately show "\<exists>q. q \<bullet> insert a bs = p \<bullet> insert a bs \<and> supp q \<subseteq> insert a bs \<union> p \<bullet> insert a bs" by blastqedlemma list_renaming_perm: fixes bs::"atom list" shows "\<exists>q. q \<bullet> bs = p \<bullet> bs \<and> supp q \<subseteq> (set bs) \<union> (p \<bullet> (set bs))"proof (induct bs) case Nil have "0 \<bullet> [] = p \<bullet> [] \<and> supp (0::perm) \<subseteq> set [] \<union> p \<bullet> set []" by (simp add: permute_set_eq supp_perm) then show "\<exists>q. q \<bullet> [] = p \<bullet> [] \<and> supp q \<subseteq> set [] \<union> p \<bullet> (set [])" by blastnext case (Cons a bs) then have " \<exists>q. q \<bullet> bs = p \<bullet> bs \<and> supp q \<subseteq> set bs \<union> p \<bullet> (set bs)" by (simp add: insert_eqvt) then obtain q where *: "q \<bullet> bs = p \<bullet> bs" and **: "supp q \<subseteq> set bs \<union> p \<bullet> (set bs)" by blast { assume 1: "a \<in> set bs" have "q \<bullet> a = p \<bullet> a" using * 1 by (induct bs) (auto) then have "q \<bullet> (a # bs) = p \<bullet> (a # bs)" using * by simp moreover have "supp q \<subseteq> set (a # bs) \<union> p \<bullet> (set (a # bs))" using ** by (auto simp add: insert_eqvt) ultimately have "\<exists>q. q \<bullet> (a # bs) = p \<bullet> (a # bs) \<and> supp q \<subseteq> set (a # bs) \<union> p \<bullet> (set (a # bs))" by blast } moreover { assume 2: "a \<notin> set bs" def q' \<equiv> "((q \<bullet> a) \<rightleftharpoons> (p \<bullet> a)) + q" { have "(q \<bullet> a) \<sharp> (p \<bullet> bs)" using `a \<notin> set bs` *[symmetric] by (simp add: fresh_permute_iff) (simp add: fresh_def supp_of_atom_list) moreover have "(p \<bullet> a) \<sharp> (p \<bullet> bs)" using `a \<notin> set bs` by (simp add: fresh_permute_iff) (simp add: fresh_def supp_of_atom_list) ultimately have "q' \<bullet> (a # bs) = p \<bullet> (a # bs)" using 2 * unfolding q'_def by (simp add: swap_fresh_fresh) } moreover { have "{q \<bullet> a, p \<bullet> a} \<subseteq> set (a # bs) \<union> p \<bullet> (set (a # bs))" using ** apply (auto simp add: supp_perm insert_eqvt) apply (subgoal_tac "q \<bullet> a \<in> set bs \<union> p \<bullet> set bs") apply(auto)[1] apply(subgoal_tac "q \<bullet> a \<in> {a. q \<bullet> a \<noteq> a}") apply(blast) apply(simp) done then have "supp (q \<bullet> a \<rightleftharpoons> p \<bullet> a) \<subseteq> set (a # bs) \<union> p \<bullet> set (a # bs)" by (simp add: supp_swap) moreover have "supp q \<subseteq> set (a # bs) \<union> p \<bullet> (set (a # bs))" using ** by (auto simp add: insert_eqvt) ultimately have "supp q' \<subseteq> set (a # bs) \<union> p \<bullet> (set (a # bs))" unfolding q'_def using supp_plus_perm by blast } ultimately have "\<exists>q. q \<bullet> (a # bs) = p \<bullet> (a # bs) \<and> supp q \<subseteq> set (a # bs) \<union> p \<bullet> (set (a # bs))" by blast } ultimately show "\<exists>q. q \<bullet> (a # bs) = p \<bullet> (a # bs) \<and> supp q \<subseteq> set (a # bs) \<union> p \<bullet> (set (a # bs))" by blastqedsection {* Concrete Atoms Types *}text {* Class @{text at_base} allows types containing multiple sorts of atoms. Class @{text at} only allows types with a single sort.*}class at_base = pt + fixes atom :: "'a \<Rightarrow> atom" assumes atom_eq_iff [simp]: "atom a = atom b \<longleftrightarrow> a = b" assumes atom_eqvt: "p \<bullet> (atom a) = atom (p \<bullet> a)"class at = at_base + assumes sort_of_atom_eq [simp]: "sort_of (atom a) = sort_of (atom b)"lemma supp_at_base: fixes a::"'a::at_base" shows "supp a = {atom a}" by (simp add: supp_atom [symmetric] supp_def atom_eqvt)lemma fresh_at_base: shows "a \<sharp> b \<longleftrightarrow> a \<noteq> atom b" unfolding fresh_def by (simp add: supp_at_base)lemma fresh_atom_at_base: fixes b::"'a::at_base" shows "a \<sharp> atom b \<longleftrightarrow> a \<sharp> b" by (simp add: fresh_def supp_at_base supp_atom)lemma fresh_star_atom_at_base: fixes b::"'a::at_base" shows "as \<sharp>* atom b \<longleftrightarrow> as \<sharp>* b" by (simp add: fresh_star_def fresh_atom_at_base)instance at_base < fsproof qed (simp add: supp_at_base)lemma at_base_infinite [simp]: shows "infinite (UNIV :: 'a::at_base set)" (is "infinite ?U")proof obtain a :: 'a where "True" by auto assume "finite ?U" hence "finite (atom ` ?U)" by (rule finite_imageI) then obtain b where b: "b \<notin> atom ` ?U" "sort_of b = sort_of (atom a)" by (rule obtain_atom) from b(2) have "b = atom ((atom a \<rightleftharpoons> b) \<bullet> a)" unfolding atom_eqvt [symmetric] by (simp add: swap_atom) hence "b \<in> atom ` ?U" by simp with b(1) show "False" by simpqedlemma swap_at_base_simps [simp]: fixes x y::"'a::at_base" shows "sort_of (atom x) = sort_of (atom y) \<Longrightarrow> (atom x \<rightleftharpoons> atom y) \<bullet> x = y" and "sort_of (atom x) = sort_of (atom y) \<Longrightarrow> (atom x \<rightleftharpoons> atom y) \<bullet> y = x" and "atom x \<noteq> a \<Longrightarrow> atom x \<noteq> b \<Longrightarrow> (a \<rightleftharpoons> b) \<bullet> x = x" unfolding atom_eq_iff [symmetric] unfolding atom_eqvt [symmetric] by simp_alllemma obtain_at_base: assumes X: "finite X" obtains a::"'a::at_base" where "atom a \<notin> X"proof - have "inj (atom :: 'a \<Rightarrow> atom)" by (simp add: inj_on_def) with X have "finite (atom -` X :: 'a set)" by (rule finite_vimageI) with at_base_infinite have "atom -` X \<noteq> (UNIV :: 'a set)" by auto then obtain a :: 'a where "atom a \<notin> X" by auto thus ?thesis ..qedlemma supp_finite_set_at_base: assumes a: "finite S" shows "supp S = atom ` S"apply(simp add: supp_of_finite_sets[OF a])apply(simp add: supp_at_base)apply(auto)donelemma fresh_finite_set_at_base: fixes a::"'a::at_base" assumes a: "finite S" shows "atom a \<sharp> S \<longleftrightarrow> a \<notin> S" unfolding fresh_def apply(simp add: supp_finite_set_at_base[OF a]) apply(subst inj_image_mem_iff) apply(simp add: inj_on_def) apply(simp) donesection {* Infrastructure for concrete atom types *}section {* A swapping operation for concrete atoms *}definition flip :: "'a::at_base \<Rightarrow> 'a \<Rightarrow> perm" ("'(_ \<leftrightarrow> _')")where "(a \<leftrightarrow> b) = (atom a \<rightleftharpoons> atom b)"lemma flip_self [simp]: "(a \<leftrightarrow> a) = 0" unfolding flip_def by (rule swap_self)lemma flip_commute: "(a \<leftrightarrow> b) = (b \<leftrightarrow> a)" unfolding flip_def by (rule swap_commute)lemma minus_flip [simp]: "- (a \<leftrightarrow> b) = (a \<leftrightarrow> b)" unfolding flip_def by (rule minus_swap)lemma add_flip_cancel: "(a \<leftrightarrow> b) + (a \<leftrightarrow> b) = 0" unfolding flip_def by (rule swap_cancel)lemma permute_flip_cancel [simp]: "(a \<leftrightarrow> b) \<bullet> (a \<leftrightarrow> b) \<bullet> x = x" unfolding permute_plus [symmetric] add_flip_cancel by simplemma permute_flip_cancel2 [simp]: "(a \<leftrightarrow> b) \<bullet> (b \<leftrightarrow> a) \<bullet> x = x" by (simp add: flip_commute)lemma flip_eqvt: fixes a b c::"'a::at_base" shows "p \<bullet> (a \<leftrightarrow> b) = (p \<bullet> a \<leftrightarrow> p \<bullet> b)" unfolding flip_def by (simp add: swap_eqvt atom_eqvt)lemma flip_at_base_simps [simp]: shows "sort_of (atom a) = sort_of (atom b) \<Longrightarrow> (a \<leftrightarrow> b) \<bullet> a = b" and "sort_of (atom a) = sort_of (atom b) \<Longrightarrow> (a \<leftrightarrow> b) \<bullet> b = a" and "\<lbrakk>a \<noteq> c; b \<noteq> c\<rbrakk> \<Longrightarrow> (a \<leftrightarrow> b) \<bullet> c = c" and "sort_of (atom a) \<noteq> sort_of (atom b) \<Longrightarrow> (a \<leftrightarrow> b) \<bullet> x = x" unfolding flip_def unfolding atom_eq_iff [symmetric] unfolding atom_eqvt [symmetric] by simp_alltext {* the following two lemmas do not hold for at_base, only for single sort atoms from at *}lemma permute_flip_at: fixes a b c::"'a::at" shows "(a \<leftrightarrow> b) \<bullet> c = (if c = a then b else if c = b then a else c)" unfolding flip_def apply (rule atom_eq_iff [THEN iffD1]) apply (subst atom_eqvt [symmetric]) apply (simp add: swap_atom) donelemma flip_at_simps [simp]: fixes a b::"'a::at" shows "(a \<leftrightarrow> b) \<bullet> a = b" and "(a \<leftrightarrow> b) \<bullet> b = a" unfolding permute_flip_at by simp_alllemma flip_fresh_fresh: fixes a b::"'a::at_base" assumes "atom a \<sharp> x" "atom b \<sharp> x" shows "(a \<leftrightarrow> b) \<bullet> x = x"using assmsby (simp add: flip_def swap_fresh_fresh)subsection {* Syntax for coercing at-elements to the atom-type *}syntax "_atom_constrain" :: "logic \<Rightarrow> type \<Rightarrow> logic" ("_:::_" [4, 0] 3)translations "_atom_constrain a t" => "CONST atom (_constrain a t)"subsection {* A lemma for proving instances of class @{text at}. *}setup {* Sign.add_const_constraint (@{const_name "permute"}, NONE) *}setup {* Sign.add_const_constraint (@{const_name "atom"}, NONE) *}text {* New atom types are defined as subtypes of @{typ atom}.*}lemma exists_eq_simple_sort: shows "\<exists>a. a \<in> {a. sort_of a = s}" by (rule_tac x="Atom s 0" in exI, simp)lemma exists_eq_sort: shows "\<exists>a. a \<in> {a. sort_of a \<in> range sort_fun}" by (rule_tac x="Atom (sort_fun x) y" in exI, simp)lemma at_base_class: fixes sort_fun :: "'b \<Rightarrow>atom_sort" fixes Rep :: "'a \<Rightarrow> atom" and Abs :: "atom \<Rightarrow> 'a" assumes type: "type_definition Rep Abs {a. sort_of a \<in> range sort_fun}" assumes atom_def: "\<And>a. atom a = Rep a" assumes permute_def: "\<And>p a. p \<bullet> a = Abs (p \<bullet> Rep a)" shows "OFCLASS('a, at_base_class)"proof interpret type_definition Rep Abs "{a. sort_of a \<in> range sort_fun}" by (rule type) have sort_of_Rep: "\<And>a. sort_of (Rep a) \<in> range sort_fun" using Rep by simp fix a b :: 'a and p p1 p2 :: perm show "0 \<bullet> a = a" unfolding permute_def by (simp add: Rep_inverse) show "(p1 + p2) \<bullet> a = p1 \<bullet> p2 \<bullet> a" unfolding permute_def by (simp add: Abs_inverse sort_of_Rep) show "atom a = atom b \<longleftrightarrow> a = b" unfolding atom_def by (simp add: Rep_inject) show "p \<bullet> atom a = atom (p \<bullet> a)" unfolding permute_def atom_def by (simp add: Abs_inverse sort_of_Rep)qed(*lemma at_class: fixes s :: atom_sort fixes Rep :: "'a \<Rightarrow> atom" and Abs :: "atom \<Rightarrow> 'a" assumes type: "type_definition Rep Abs {a. sort_of a \<in> range (\<lambda>x::unit. s)}" assumes atom_def: "\<And>a. atom a = Rep a" assumes permute_def: "\<And>p a. p \<bullet> a = Abs (p \<bullet> Rep a)" shows "OFCLASS('a, at_class)"proof interpret type_definition Rep Abs "{a. sort_of a \<in> range (\<lambda>x::unit. s)}" by (rule type) have sort_of_Rep: "\<And>a. sort_of (Rep a) = s" using Rep by (simp add: image_def) fix a b :: 'a and p p1 p2 :: perm show "0 \<bullet> a = a" unfolding permute_def by (simp add: Rep_inverse) show "(p1 + p2) \<bullet> a = p1 \<bullet> p2 \<bullet> a" unfolding permute_def by (simp add: Abs_inverse sort_of_Rep) show "sort_of (atom a) = sort_of (atom b)" unfolding atom_def by (simp add: sort_of_Rep) show "atom a = atom b \<longleftrightarrow> a = b" unfolding atom_def by (simp add: Rep_inject) show "p \<bullet> atom a = atom (p \<bullet> a)" unfolding permute_def atom_def by (simp add: Abs_inverse sort_of_Rep)qed*)lemma at_class: fixes s :: atom_sort fixes Rep :: "'a \<Rightarrow> atom" and Abs :: "atom \<Rightarrow> 'a" assumes type: "type_definition Rep Abs {a. sort_of a = s}" assumes atom_def: "\<And>a. atom a = Rep a" assumes permute_def: "\<And>p a. p \<bullet> a = Abs (p \<bullet> Rep a)" shows "OFCLASS('a, at_class)"proof interpret type_definition Rep Abs "{a. sort_of a = s}" by (rule type) have sort_of_Rep: "\<And>a. sort_of (Rep a) = s" using Rep by (simp add: image_def) fix a b :: 'a and p p1 p2 :: perm show "0 \<bullet> a = a" unfolding permute_def by (simp add: Rep_inverse) show "(p1 + p2) \<bullet> a = p1 \<bullet> p2 \<bullet> a" unfolding permute_def by (simp add: Abs_inverse sort_of_Rep) show "sort_of (atom a) = sort_of (atom b)" unfolding atom_def by (simp add: sort_of_Rep) show "atom a = atom b \<longleftrightarrow> a = b" unfolding atom_def by (simp add: Rep_inject) show "p \<bullet> atom a = atom (p \<bullet> a)" unfolding permute_def atom_def by (simp add: Abs_inverse sort_of_Rep)qedsetup {* Sign.add_const_constraint (@{const_name "permute"}, SOME @{typ "perm \<Rightarrow> 'a::pt \<Rightarrow> 'a"}) *}setup {* Sign.add_const_constraint (@{const_name "atom"}, SOME @{typ "'a::at_base \<Rightarrow> atom"}) *}section {* The freshness lemma according to Andy Pitts *}lemma freshness_lemma: fixes h :: "'a::at \<Rightarrow> 'b::pt" assumes a: "\<exists>a. atom a \<sharp> (h, h a)" shows "\<exists>x. \<forall>a. atom a \<sharp> h \<longrightarrow> h a = x"proof - from a obtain b where a1: "atom b \<sharp> h" and a2: "atom b \<sharp> h b" by (auto simp add: fresh_Pair) show "\<exists>x. \<forall>a. atom a \<sharp> h \<longrightarrow> h a = x" proof (intro exI allI impI) fix a :: 'a assume a3: "atom a \<sharp> h" show "h a = h b" proof (cases "a = b") assume "a = b" thus "h a = h b" by simp next assume "a \<noteq> b" hence "atom a \<sharp> b" by (simp add: fresh_at_base) with a3 have "atom a \<sharp> h b" by (rule fresh_fun_app) with a2 have d1: "(atom b \<rightleftharpoons> atom a) \<bullet> (h b) = (h b)" by (rule swap_fresh_fresh) from a1 a3 have d2: "(atom b \<rightleftharpoons> atom a) \<bullet> h = h" by (rule swap_fresh_fresh) from d1 have "h b = (atom b \<rightleftharpoons> atom a) \<bullet> (h b)" by simp also have "\<dots> = ((atom b \<rightleftharpoons> atom a) \<bullet> h) ((atom b \<rightleftharpoons> atom a) \<bullet> b)" by (rule permute_fun_app_eq) also have "\<dots> = h a" using d2 by simp finally show "h a = h b" by simp qed qedqedlemma freshness_lemma_unique: fixes h :: "'a::at \<Rightarrow> 'b::pt" assumes a: "\<exists>a. atom a \<sharp> (h, h a)" shows "\<exists>!x. \<forall>a. atom a \<sharp> h \<longrightarrow> h a = x"proof (rule ex_ex1I) from a show "\<exists>x. \<forall>a. atom a \<sharp> h \<longrightarrow> h a = x" by (rule freshness_lemma)next fix x y assume x: "\<forall>a. atom a \<sharp> h \<longrightarrow> h a = x" assume y: "\<forall>a. atom a \<sharp> h \<longrightarrow> h a = y" from a x y show "x = y" by (auto simp add: fresh_Pair)qedtext {* packaging the freshness lemma into a function *}definition fresh_fun :: "('a::at \<Rightarrow> 'b::pt) \<Rightarrow> 'b"where "fresh_fun h = (THE x. \<forall>a. atom a \<sharp> h \<longrightarrow> h a = x)"lemma fresh_fun_apply: fixes h :: "'a::at \<Rightarrow> 'b::pt" assumes a: "\<exists>a. atom a \<sharp> (h, h a)" assumes b: "atom a \<sharp> h" shows "fresh_fun h = h a"unfolding fresh_fun_defproof (rule the_equality) show "\<forall>a'. atom a' \<sharp> h \<longrightarrow> h a' = h a" proof (intro strip) fix a':: 'a assume c: "atom a' \<sharp> h" from a have "\<exists>x. \<forall>a. atom a \<sharp> h \<longrightarrow> h a = x" by (rule freshness_lemma) with b c show "h a' = h a" by auto qednext fix fr :: 'b assume "\<forall>a. atom a \<sharp> h \<longrightarrow> h a = fr" with b show "fr = h a" by autoqedlemma fresh_fun_apply': fixes h :: "'a::at \<Rightarrow> 'b::pt" assumes a: "atom a \<sharp> h" "atom a \<sharp> h a" shows "fresh_fun h = h a" apply (rule fresh_fun_apply) apply (auto simp add: fresh_Pair intro: a) donelemma fresh_fun_eqvt: fixes h :: "'a::at \<Rightarrow> 'b::pt" assumes a: "\<exists>a. atom a \<sharp> (h, h a)" shows "p \<bullet> (fresh_fun h) = fresh_fun (p \<bullet> h)" using a apply (clarsimp simp add: fresh_Pair) apply (subst fresh_fun_apply', assumption+) apply (drule fresh_permute_iff [where p=p, THEN iffD2]) apply (drule fresh_permute_iff [where p=p, THEN iffD2]) apply (simp add: atom_eqvt permute_fun_app_eq [where f=h]) apply (erule (1) fresh_fun_apply' [symmetric]) donelemma fresh_fun_supports: fixes h :: "'a::at \<Rightarrow> 'b::pt" assumes a: "\<exists>a. atom a \<sharp> (h, h a)" shows "(supp h) supports (fresh_fun h)" apply (simp add: supports_def fresh_def [symmetric]) apply (simp add: fresh_fun_eqvt [OF a] swap_fresh_fresh) donenotation fresh_fun (binder "FRESH " 10)lemma FRESH_f_iff: fixes P :: "'a::at \<Rightarrow> 'b::pure" fixes f :: "'b \<Rightarrow> 'c::pure" assumes P: "finite (supp P)" shows "(FRESH x. f (P x)) = f (FRESH x. P x)"proof - obtain a::'a where "atom a \<notin> supp P" using P by (rule obtain_at_base) hence "atom a \<sharp> P" by (simp add: fresh_def) show "(FRESH x. f (P x)) = f (FRESH x. P x)" apply (subst fresh_fun_apply' [where a=a, OF _ pure_fresh]) apply (cut_tac `atom a \<sharp> P`) apply (simp add: fresh_conv_MOST) apply (elim MOST_rev_mp, rule MOST_I, clarify) apply (simp add: permute_fun_def permute_pure fun_eq_iff) apply (subst fresh_fun_apply' [where a=a, OF `atom a \<sharp> P` pure_fresh]) apply (rule refl) doneqedlemma FRESH_binop_iff: fixes P :: "'a::at \<Rightarrow> 'b::pure" fixes Q :: "'a::at \<Rightarrow> 'c::pure" fixes binop :: "'b \<Rightarrow> 'c \<Rightarrow> 'd::pure" assumes P: "finite (supp P)" and Q: "finite (supp Q)" shows "(FRESH x. binop (P x) (Q x)) = binop (FRESH x. P x) (FRESH x. Q x)"proof - from assms have "finite (supp P \<union> supp Q)" by simp then obtain a::'a where "atom a \<notin> (supp P \<union> supp Q)" by (rule obtain_at_base) hence "atom a \<sharp> P" and "atom a \<sharp> Q" by (simp_all add: fresh_def) show ?thesis apply (subst fresh_fun_apply' [where a=a, OF _ pure_fresh]) apply (cut_tac `atom a \<sharp> P` `atom a \<sharp> Q`) apply (simp add: fresh_conv_MOST) apply (elim MOST_rev_mp, rule MOST_I, clarify) apply (simp add: permute_fun_def permute_pure fun_eq_iff) apply (subst fresh_fun_apply' [where a=a, OF `atom a \<sharp> P` pure_fresh]) apply (subst fresh_fun_apply' [where a=a, OF `atom a \<sharp> Q` pure_fresh]) apply (rule refl) doneqedlemma FRESH_conj_iff: fixes P Q :: "'a::at \<Rightarrow> bool" assumes P: "finite (supp P)" and Q: "finite (supp Q)" shows "(FRESH x. P x \<and> Q x) \<longleftrightarrow> (FRESH x. P x) \<and> (FRESH x. Q x)"using P Q by (rule FRESH_binop_iff)lemma FRESH_disj_iff: fixes P Q :: "'a::at \<Rightarrow> bool" assumes P: "finite (supp P)" and Q: "finite (supp Q)" shows "(FRESH x. P x \<or> Q x) \<longleftrightarrow> (FRESH x. P x) \<or> (FRESH x. Q x)"using P Q by (rule FRESH_binop_iff)section {* Library functions for the nominal infrastructure *}use "nominal_library.ML"section {* Automation for creating concrete atom types *}text {* at the moment only single-sort concrete atoms are supported *}use "nominal_atoms.ML"end