Prove pseudo-inject (eq-iff) on the exported level and rename appropriately.
(* Title: Nominal2_Base Authors: Brian Huffman, Christian Urban Basic definitions and lemma infrastructure for Nominal Isabelle. *)theory Nominal2_Baseimports Main Infinite_Setbeginsection {* 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"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 ..qedsection {* 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: expand_fun_eq Rep_perm_inject [symmetric])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 expand_fun_eq)lemma swap_self [simp]: "(a \<rightleftharpoons> a) = 0" by (rule Rep_perm_ext, simp add: Rep_perm_simps expand_fun_eq)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 expand_fun_eq)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 expand_fun_eq)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: "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)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)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}" apply(auto simp add: permute_fun_def permute_bool_def 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_defunfolding vimage_def Collect_def mem_def ..lemma swap_set_not_in: assumes a: "a \<notin> S" "b \<notin> S" shows "(a \<rightleftharpoons> b) \<bullet> S = S" using a by (auto simp add: permute_set_eq 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" using a by (auto simp add: permute_set_eq swap_atom)subsection {* Permutations for units *}instantiation unit :: ptbegindefinition "p \<bullet> (u::unit) = u"instance proofqed (simp_all add: permute_unit_def)endsubsection {* Permutations for products *}instantiation "*" :: (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 "+" :: (pt, pt) ptbeginprimrec permute_sum where "p \<bullet> (Inl x) = Inl (p \<bullet> x)"| "p \<bullet> (Inr y) = Inr (p \<bullet> y)"instance proofqed (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 proofqed (induct_tac [!] x, simp_all)endsubsection {* Permutations for options *}instantiation option :: (pt) ptbeginprimrec permute_option where "p \<bullet> None = None"| "p \<bullet> (Some x) = Some (p \<bullet> x)"instance proofqed (induct_tac [!] x, simp_all)endsubsection {* Permutations for @{typ char}, @{typ nat}, and @{typ int} *}instantiation char :: ptbegindefinition "p \<bullet> (c::char) = c"instance proofqed (simp_all add: permute_char_def)endinstantiation nat :: ptbegindefinition "p \<bullet> (n::nat) = n"instance proofqed (simp_all add: permute_nat_def)endinstantiation int :: ptbegindefinition "p \<bullet> (i::int) = i"instance proofqed (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 "*" :: (pure, pure) pureby default (induct_tac x, simp add: permute_pure)instance "+" :: (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 ?thesis .qedlemma fresh_eqvt: shows "p \<bullet> (a \<sharp> x) = (p \<bullet> a) \<sharp> (p \<bullet> x)" by (simp add: permute_bool_def 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 permute_fun_def Collect_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" by (unfold supports_def) (auto) hence "{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"proof (unfold supports_def, 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 (rule 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 {* 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 by (simp add: perm_swap_eq 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 apply(auto simp add: supp_perm) 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 qedqedsection {* 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)instance "*" :: (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 "+" :: (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 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)instance list :: (fs) fsapply defaultapply (induct_tac x)apply (simp_all add: supp_Nil supp_Cons finite_supp)donesection {* Support and freshness for applications *}lemma supp_fun_app: shows "supp (f x) \<subseteq> (supp f) \<union> (supp x)"proof (rule subsetCI) fix a::"atom" assume a: "a \<in> supp (f x)" assume b: "a \<notin> supp f \<union> supp x" then have "finite {b. (a \<rightleftharpoons> b) \<bullet> f \<noteq> f}" "finite {b. (a \<rightleftharpoons> b) \<bullet> x \<noteq> x}" unfolding supp_def by auto then have "finite ({b. (a \<rightleftharpoons> b) \<bullet> f \<noteq> f} \<union> {b. (a \<rightleftharpoons> b) \<bullet> x \<noteq> x})" by simp moreover have "{b. ((a \<rightleftharpoons> b) \<bullet> f) ((a \<rightleftharpoons> b) \<bullet> x) \<noteq> f x} \<subseteq> ({b. (a \<rightleftharpoons> b) \<bullet> f \<noteq> f} \<union> {b. (a \<rightleftharpoons> b) \<bullet> x \<noteq> x})" by auto ultimately have "finite {b. ((a \<rightleftharpoons> b) \<bullet> f) ((a \<rightleftharpoons> b) \<bullet> x) \<noteq> f x}" using finite_subset by auto then have "a \<notin> supp (f x)" unfolding supp_def by (simp add: permute_fun_app_eq) with a show "False" by simpqedlemma fresh_fun_app: shows "a \<sharp> (f, x) \<Longrightarrow> a \<sharp> f x"unfolding fresh_defusing supp_fun_appby (auto simp add: supp_Pair)lemma fresh_fun_eqvt_app: assumes a: "\<forall>p. p \<bullet> f = f" shows "a \<sharp> x \<Longrightarrow> a \<sharp> f x"proof - from a have b: "supp f = {}" unfolding supp_def by simp show "a \<sharp> x \<Longrightarrow> a \<sharp> f x" unfolding fresh_def using supp_fun_app b by autoqedend