(* Title: Nominal2_Base Authors: Christian Urban, Brian Huffman, Cezary Kaliszyk Basic definitions and lemma infrastructure for Nominal Isabelle. *)theory Nominal2_Baseimports Main "~~/src/HOL/Library/Infinite_Set" "~~/src/HOL/Quotient_Examples/FSet" "GPerm" "~~/src/HOL/Library/List_lexord" "~~/src/HOL/Library/Product_ord" "~~/src/HOL/Library/Efficient_Nat" "~~/src/HOL/Library/Char_ord" "~~/src/HOL/Library/Code_Char_chr" "~~/src/HOL/Library/Code_Char_ord"keywords "atom_decl" "equivariance" :: thy_decluses ("nominal_basics.ML") ("nominal_thmdecls.ML") ("nominal_permeq.ML") ("nominal_library.ML") ("nominal_atoms.ML") ("nominal_eqvt.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 n) = 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 *}definition "sort_respecting p \<longleftrightarrow> (\<forall>a. sort_of (gpermute p a) = sort_of a)"lemma sort_respecting_0[simp]: "sort_respecting (0\<Colon>atom gperm)" by (simp add: sort_respecting_def)typedef (open) perm = "{p::atom gperm. sort_respecting p}" by (auto intro: exI[of _ "0"])setup_lifting type_definition_permlemma perm_eq_rep: "p = q \<longleftrightarrow> Rep_perm p = Rep_perm q" by (simp add: Rep_perm_inject)lift_definition mk_perm :: "atom gperm \<Rightarrow> perm" is "\<lambda>p. if sort_respecting p then p else 0" by simplemma Rep_perm_mk_perm [simp]: "Rep_perm (mk_perm p) = (if sort_respecting p then p else 0)" by (simp add: mk_perm_def Abs_perm_inverse)instance perm :: size ..subsection {* Permutations form a (multiplicative) group *}instantiation perm :: group_addbeginlift_definition zero_perm :: "perm" is "0" by simplift_definition uminus_perm :: "perm \<Rightarrow> perm" is "uminus" unfolding sort_respecting_def by transfer (auto, metis perm_apply_minus)lift_definition plus_perm :: "perm \<Rightarrow> perm \<Rightarrow> perm" is "plus" unfolding sort_respecting_def by transfer (simp add: perm_add_apply)definition "(p :: perm) - q = p + - q"lemma Rep_perm_0 [simp, code abstract]: "Rep_perm 0 = 0" by (metis (mono_tags) zero_perm.rep_eq)lemma Rep_perm_uminus [simp, code abstract]: "Rep_perm (- p) = - (Rep_perm p)" by (metis uminus_perm.rep_eq)lemma Rep_perm_add [simp, code abstract]: "Rep_perm (p + q) = (Rep_perm p) + (Rep_perm q)" by (simp add: plus_perm.rep_eq)instance apply default unfolding minus_perm_def by (transfer, simp add: add_assoc)+endsection {* Implementation of swappings *}lift_definition swap :: "atom \<Rightarrow> atom \<Rightarrow> perm" ("'(_ \<rightleftharpoons> _')") is "(\<lambda>a b. (if sort_of a = sort_of b then mk_perm (gswap a b) else 0))" .lemma sort_respecting_swap [simp]: "sort_of a = sort_of b \<Longrightarrow> sort_respecting (gswap a b)" unfolding sort_respecting_def by transfer autolemma Rep_swap [simp, code abstract]: "Rep_perm (swap a b) = (if sort_of a = sort_of b then gswap a b else 0)" by (simp add: swap_def)lemma swap_different_sorts [simp]: "sort_of a \<noteq> sort_of b \<Longrightarrow> (a \<rightleftharpoons> b) = 0" by (simp add: perm_eq_rep)lemma swap_cancel: shows "(a \<rightleftharpoons> b) + (a \<rightleftharpoons> b) = 0" and "(a \<rightleftharpoons> b) + (b \<rightleftharpoons> a) = 0" by (simp_all add: perm_eq_rep)lemma swap_self [simp]: "(a \<rightleftharpoons> a) = 0" by (simp add: perm_eq_rep)lemma minus_swap [simp]: "- (a \<rightleftharpoons> b) = (a \<rightleftharpoons> b)" by (simp add: perm_eq_rep)lemma swap_commute: "(a \<rightleftharpoons> b) = (b \<rightleftharpoons> a)" by (simp add: perm_eq_rep swap_commute)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 (simp add: perm_eq_rep swap_triple)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 = gpermute (Rep_perm p) a"instance by default (simp_all add: permute_atom_def)endlemma sort_of_permute [simp]: shows "sort_of (p \<bullet> a) = sort_of a" by (metis Rep_perm mem_Collect_eq sort_respecting_def permute_atom_def)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)" by (auto simp add: permute_atom_def)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 perm_eq_iff: fixes p q :: "perm" shows "p = q \<longleftrightarrow> (\<forall>a::atom. p \<bullet> a = q \<bullet> a)" unfolding permute_atom_def perm_eq_rep by (simp add: gperm_eq)subsection {* Permutations for permutations *}instantiation perm :: ptbegindefinition "p \<bullet> q = p + q - p"instance by default (simp_all add: permute_perm_def diff_minus minus_add add_assoc)endlemma permute_self: shows "p \<bullet> p = p" unfolding permute_perm_def by (simp add: diff_minus add_assoc)lemma pemute_minus_self: shows "- p \<bullet> p = p" unfolding permute_perm_def by (simp add: diff_minus add_assoc)subsection {* Permutations for functions *}instantiation "fun" :: (pt, pt) ptbegindefinition "p \<bullet> f = (\<lambda>x. p \<bullet> (f (- p \<bullet> x)))"instance by default (simp_all add: permute_fun_def minus_add)endlemma 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"instance by (default) (simp_all add: permute_bool_def)endlemma 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 *}instantiation "set" :: (pt) ptbegindefinition "p \<bullet> X = {p \<bullet> x | x. x \<in> X}" instanceapply defaultapply (auto simp add: permute_set_def)doneendlemma permute_set_eq: shows "p \<bullet> X = {x. - p \<bullet> x \<in> X}"unfolding permute_set_defby (auto) (metis permute_minus_cancel(1))lemma permute_set_eq_image: shows "p \<bullet> X = permute p ` X" unfolding permute_set_def by autolemma permute_set_eq_vimage: shows "p \<bullet> X = permute (- p) -` X" unfolding permute_set_eq vimage_def by simplemma permute_finite [simp]: shows "finite (p \<bullet> X) = finite X" unfolding permute_set_eq_vimage using bij_permute by (rule finite_vimage_iff)lemma swap_set_not_in: assumes a: "a \<notin> S" "b \<notin> S" shows "(a \<rightleftharpoons> b) \<bullet> S = S" unfolding permute_set_def 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_def using a by (auto simp add: swap_atom)lemma swap_set_in_eq: assumes a: "a \<in> S" "b \<notin> S" "sort_of a = sort_of b" shows "(a \<rightleftharpoons> b) \<bullet> S = (S - {a}) \<union> {b}" unfolding permute_set_def using a by (auto simp add: swap_atom)lemma swap_set_both_in: assumes a: "a \<in> S" "b \<in> S" shows "(a \<rightleftharpoons> b) \<bullet> S = S" unfolding permute_set_def 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 permute_set_def by autolemma empty_eqvt: shows "p \<bullet> {} = {}" unfolding permute_set_def by (simp)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 @{typ unit} *}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 Inl_eqvt: "p \<bullet> (Inl x) = Inl (p \<bullet> x)"| Inr_eqvt: "p \<bullet> (Inr y) = Inr (p \<bullet> y)"instance by (default) (case_tac [!] x, simp_all)endsubsection {* Permutations for @{typ "'a list"} *}instantiation list :: (pt) ptbeginprimrec permute_list where Nil_eqvt: "p \<bullet> [] = []"| Cons_eqvt: "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 @{typ "'a option"} *}instantiation option :: (pt) ptbeginprimrec permute_option where None_eqvt: "p \<bullet> None = None"| Some_eqvt: "p \<bullet> (Some x) = Some (p \<bullet> x)"instance by (default) (induct_tac [!] x, simp_all)endsubsection {* Permutations for @{typ "'a multiset"} *}instantiation multiset :: (pt) ptbegindefinition "p \<bullet> M = {# p \<bullet> x. x :# M #}"instance proof fix M :: "'a multiset" and p q :: "perm" show "0 \<bullet> M = M" unfolding permute_multiset_def by (induct_tac M) (simp_all) show "(p + q) \<bullet> M = p \<bullet> q \<bullet> M" unfolding permute_multiset_def by (induct_tac M) (simp_all)qedendlemma permute_multiset [simp]: fixes M N::"('a::pt) multiset" shows "(p \<bullet> {#}) = ({#} ::('a::pt) multiset)" and "(p \<bullet> {# x #}) = {# p \<bullet> x #}" and "(p \<bullet> (M + N)) = (p \<bullet> M) + (p \<bullet> N)" unfolding permute_multiset_def by (simp_all)subsection {* Permutations for @{typ "'a fset"} *}instantiation fset :: (pt) ptbeginquotient_definition "permute_fset :: perm \<Rightarrow> 'a fset \<Rightarrow> 'a fset"is "permute :: perm \<Rightarrow> 'a list \<Rightarrow> 'a list" by (simp add: set_eqvt[symmetric])instanceproof fix x :: "'a fset" and p q :: "perm" have lst: "\<And>l :: 'a list. 0 \<bullet> l = l" by simp show "0 \<bullet> x = x" by (lifting lst) have lst: "\<And>p q :: perm. \<And>x :: 'a list. (p + q) \<bullet> x = p \<bullet> q \<bullet> x" by simp show "(p + q) \<bullet> x = p \<bullet> q \<bullet> x" by (lifting lst)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)lemma fset_eqvt: shows "p \<bullet> (fset S) = fset (p \<bullet> S)" by (lifting set_eqvt)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 set :: (pure) pureby default (simp add: permute_set_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)section {* Infrastructure for Equivariance and Perm_simp *}subsection {* Basic functions about permutations *}use "nominal_basics.ML"subsection {* Eqvt infrastructure *}text {* Setup of the theorem attributes @{text eqvt} and @{text eqvt_raw} *}use "nominal_thmdecls.ML"setup "Nominal_ThmDecls.setup"lemmas [eqvt] = (* pt types *) permute_prod.simps permute_list.simps permute_option.simps permute_sum.simps (* sets *) empty_eqvt insert_eqvt set_eqvt (* fsets *) permute_fset fset_eqvt (* multisets *) permute_multisetsubsection {* perm_simp infrastructure *}definition "unpermute p = permute (- p)"lemma eqvt_apply: fixes f :: "'a::pt \<Rightarrow> 'b::pt" and x :: "'a::pt" shows "p \<bullet> (f x) \<equiv> (p \<bullet> f) (p \<bullet> x)" unfolding permute_fun_def by simplemma eqvt_lambda: fixes f :: "'a::pt \<Rightarrow> 'b::pt" shows "p \<bullet> f \<equiv> (\<lambda>x. p \<bullet> (f (unpermute p x)))" unfolding permute_fun_def unpermute_def by simplemma eqvt_bound: shows "p \<bullet> unpermute p x \<equiv> x" unfolding unpermute_def by simptext {* provides perm_simp methods *}use "nominal_permeq.ML"method_setup perm_simp = {* Nominal_Permeq.args_parser >> Nominal_Permeq.perm_simp_meth *} {* pushes permutations inside. *}method_setup perm_strict_simp = {* Nominal_Permeq.args_parser >> Nominal_Permeq.perm_strict_simp_meth *} {* pushes permutations inside, raises an error if it cannot solve all permutations. *}subsubsection {* Equivariance for permutations and swapping *}lemma permute_eqvt: shows "p \<bullet> (q \<bullet> x) = (p \<bullet> q) \<bullet> (p \<bullet> x)" unfolding permute_perm_def by simp(* the normal version of this lemma would cause loops *)lemma permute_eqvt_raw [eqvt_raw]: shows "p \<bullet> permute \<equiv> permute"apply(simp add: fun_eq_iff permute_fun_def)apply(subst permute_eqvt)apply(simp)donelemma zero_perm_eqvt [eqvt]: shows "p \<bullet> (0::perm) = 0" unfolding permute_perm_def by simplemma add_perm_eqvt [eqvt]: fixes p p1 p2 :: perm shows "p \<bullet> (p1 + p2) = p \<bullet> p1 + p \<bullet> p2" unfolding permute_perm_def by (simp add: perm_eq_iff)lemma swap_eqvt [eqvt]: shows "p \<bullet> (a \<rightleftharpoons> b) = (p \<bullet> a \<rightleftharpoons> p \<bullet> b)" unfolding permute_perm_def by (auto simp add: swap_atom perm_eq_iff)lemma uminus_eqvt [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)subsubsection {* Equivariance of Logical Operators *}lemma eq_eqvt [eqvt]: shows "p \<bullet> (x = y) \<longleftrightarrow> (p \<bullet> x) = (p \<bullet> y)" unfolding permute_eq_iff permute_bool_def ..lemma Not_eqvt [eqvt]: shows "p \<bullet> (\<not> A) \<longleftrightarrow> \<not> (p \<bullet> A)" by (simp add: permute_bool_def)lemma conj_eqvt [eqvt]: shows "p \<bullet> (A \<and> B) \<longleftrightarrow> (p \<bullet> A) \<and> (p \<bullet> B)" by (simp add: permute_bool_def)lemma imp_eqvt [eqvt]: shows "p \<bullet> (A \<longrightarrow> B) \<longleftrightarrow> (p \<bullet> A) \<longrightarrow> (p \<bullet> B)" by (simp add: permute_bool_def)declare imp_eqvt[folded induct_implies_def, eqvt]lemma all_eqvt [eqvt]: shows "p \<bullet> (\<forall>x. P x) = (\<forall>x. (p \<bullet> P) x)" unfolding All_def by (perm_simp) (rule refl)declare all_eqvt[folded induct_forall_def, eqvt]lemma ex_eqvt [eqvt]: shows "p \<bullet> (\<exists>x. P x) = (\<exists>x. (p \<bullet> P) x)" unfolding Ex_def by (perm_simp) (rule refl)lemma ex1_eqvt [eqvt]: shows "p \<bullet> (\<exists>!x. P x) = (\<exists>!x. (p \<bullet> P) x)" unfolding Ex1_def by (perm_simp) (rule refl)lemma if_eqvt [eqvt]: shows "p \<bullet> (if b then x else y) = (if p \<bullet> b then p \<bullet> x else p \<bullet> y)" by (simp add: permute_fun_def permute_bool_def)lemma Let_eqvt [eqvt]: shows "p \<bullet> Let x y = Let (p \<bullet> x) (p \<bullet> y)" unfolding Let_def permute_fun_app_eq ..lemma True_eqvt [eqvt]: shows "p \<bullet> True = True" unfolding permute_bool_def ..lemma False_eqvt [eqvt]: shows "p \<bullet> False = False" unfolding permute_bool_def ..lemma disj_eqvt [eqvt]: shows "p \<bullet> (A \<or> B) \<longleftrightarrow> (p \<bullet> A) \<or> (p \<bullet> B)" by (simp add: permute_bool_def)lemma all_eqvt2: shows "p \<bullet> (\<forall>x. P x) = (\<forall>x. p \<bullet> P (- p \<bullet> x))" by (perm_simp add: permute_minus_cancel) (rule refl)lemma ex_eqvt2: shows "p \<bullet> (\<exists>x. P x) = (\<exists>x. p \<bullet> P (- p \<bullet> x))" by (perm_simp add: permute_minus_cancel) (rule refl)lemma ex1_eqvt2: shows "p \<bullet> (\<exists>!x. P x) = (\<exists>!x. p \<bullet> P (- p \<bullet> x))" by (perm_simp add: permute_minus_cancel) (rule refl)lemma the_eqvt: assumes unique: "\<exists>!x. P x" shows "(p \<bullet> (THE x. P x)) = (THE x. (p \<bullet> P) x)" apply(rule the1_equality [symmetric]) apply(rule_tac p="-p" in permute_boolE) apply(perm_simp add: permute_minus_cancel) apply(rule unique) apply(rule_tac p="-p" in permute_boolE) apply(perm_simp add: permute_minus_cancel) apply(rule theI'[OF unique]) donelemma the_eqvt2: assumes unique: "\<exists>!x. P x" shows "(p \<bullet> (THE x. P x)) = (THE x. p \<bullet> P (- p \<bullet> x))" apply(rule the1_equality [symmetric]) apply(simp add: ex1_eqvt2[symmetric]) apply(simp add: permute_bool_def unique) apply(simp add: permute_bool_def) apply(rule theI'[OF unique]) donesubsubsection {* Equivariance of Set operators *}lemma mem_eqvt [eqvt]: shows "p \<bullet> (x \<in> A) \<longleftrightarrow> (p \<bullet> x) \<in> (p \<bullet> A)" unfolding permute_bool_def permute_set_def by (auto)lemma Collect_eqvt [eqvt]: shows "p \<bullet> {x. P x} = {x. (p \<bullet> P) x}" unfolding permute_set_eq permute_fun_def by (auto simp add: permute_bool_def)lemma inter_eqvt [eqvt]: shows "p \<bullet> (A \<inter> B) = (p \<bullet> A) \<inter> (p \<bullet> B)" unfolding Int_def by (perm_simp) (rule refl)lemma Bex_eqvt [eqvt]: shows "p \<bullet> (\<exists>x \<in> S. P x) = (\<exists>x \<in> (p \<bullet> S). (p \<bullet> P) x)" unfolding Bex_def by (perm_simp) (rule refl)lemma Ball_eqvt [eqvt]: shows "p \<bullet> (\<forall>x \<in> S. P x) = (\<forall>x \<in> (p \<bullet> S). (p \<bullet> P) x)" unfolding Ball_def by (perm_simp) (rule refl)lemma image_eqvt [eqvt]: shows "p \<bullet> (f ` A) = (p \<bullet> f) ` (p \<bullet> A)" unfolding image_def by (perm_simp) (rule refl)lemma Image_eqvt [eqvt]: shows "p \<bullet> (R `` A) = (p \<bullet> R) `` (p \<bullet> A)" unfolding Image_def by (perm_simp) (rule refl)lemma UNIV_eqvt [eqvt]: shows "p \<bullet> UNIV = UNIV" unfolding UNIV_def by (perm_simp) (rule refl)lemma union_eqvt [eqvt]: shows "p \<bullet> (A \<union> B) = (p \<bullet> A) \<union> (p \<bullet> B)" unfolding Un_def by (perm_simp) (rule refl)lemma Diff_eqvt [eqvt]: fixes A B :: "'a::pt set" shows "p \<bullet> (A - B) = (p \<bullet> A) - (p \<bullet> B)" unfolding set_diff_eq by (perm_simp) (rule refl)lemma Compl_eqvt [eqvt]: fixes A :: "'a::pt set" shows "p \<bullet> (- A) = - (p \<bullet> A)" unfolding Compl_eq_Diff_UNIV by (perm_simp) (rule refl)lemma subset_eqvt [eqvt]: shows "p \<bullet> (S \<subseteq> T) \<longleftrightarrow> (p \<bullet> S) \<subseteq> (p \<bullet> T)" unfolding subset_eq by (perm_simp) (rule refl)lemma psubset_eqvt [eqvt]: shows "p \<bullet> (S \<subset> T) \<longleftrightarrow> (p \<bullet> S) \<subset> (p \<bullet> T)" unfolding psubset_eq by (perm_simp) (rule refl)lemma vimage_eqvt [eqvt]: shows "p \<bullet> (f -` A) = (p \<bullet> f) -` (p \<bullet> A)" unfolding vimage_def by (perm_simp) (rule refl)lemma Union_eqvt [eqvt]: shows "p \<bullet> (\<Union> S) = \<Union> (p \<bullet> S)" unfolding Union_eq by (perm_simp) (rule refl)lemma Inter_eqvt [eqvt]: shows "p \<bullet> (\<Inter> S) = \<Inter> (p \<bullet> S)" unfolding Inter_eq by (perm_simp) (rule refl)lemma foldr_eqvt[eqvt]: "p \<bullet> foldr a b c = foldr (p \<bullet> a) (p \<bullet> b) (p \<bullet> c)" apply (induct b) apply simp_all apply (perm_simp) apply simp done(* FIXME: eqvt attribute *)lemma Sigma_eqvt: shows "(p \<bullet> (X \<times> Y)) = (p \<bullet> X) \<times> (p \<bullet> Y)"unfolding Sigma_defunfolding SUP_defby (perm_simp) (rule refl)text {* In order to prove that lfp is equivariant we need two auxiliary classes which specify that (op <=) and Inf are equivariant. Instances for bool and fun are given.*}class le_eqvt = order + assumes le_eqvt [eqvt]: "p \<bullet> (x \<le> y) = ((p \<bullet> x) \<le> (p \<bullet> (y::('a::{pt, order}))))"class inf_eqvt = complete_lattice + assumes inf_eqvt [eqvt]: "p \<bullet> (Inf X) = Inf (p \<bullet> (X::('a::{pt, Inf}) set))"instantiation bool :: le_eqvtbegininstance apply(default)apply perm_simpapply(rule refl)doneendinstantiation "fun" :: (pt, le_eqvt) le_eqvtbegininstance apply(default)unfolding le_fun_defapply(perm_simp)apply(rule refl)done endinstantiation bool :: inf_eqvtbegininstance apply(default)apply(perm_simp)apply(rule refl)doneendinstantiation "fun" :: (pt, inf_eqvt) inf_eqvtbegininstance apply(default)unfolding Inf_fun_def INF_defapply(perm_simp)apply(rule refl)done endlemma lfp_eqvt [eqvt]: fixes F::"('a \<Rightarrow> 'b) \<Rightarrow> ('a::pt \<Rightarrow> 'b::{inf_eqvt, le_eqvt})" shows "p \<bullet> (lfp F) = lfp (p \<bullet> F)"unfolding lfp_defby (perm_simp) (rule refl)lemma finite_eqvt [eqvt]: shows "p \<bullet> finite A = finite (p \<bullet> A)"unfolding finite_defby (perm_simp) (rule refl)subsubsection {* Equivariance for product operations *}lemma fst_eqvt [eqvt]: shows "p \<bullet> (fst x) = fst (p \<bullet> x)" by (cases x) simplemma snd_eqvt [eqvt]: shows "p \<bullet> (snd x) = snd (p \<bullet> x)" by (cases x) simplemma split_eqvt [eqvt]: shows "p \<bullet> (split P x) = split (p \<bullet> P) (p \<bullet> x)" unfolding split_def by (perm_simp) (rule refl)subsubsection {* Equivariance for list operations *}lemma append_eqvt [eqvt]: shows "p \<bullet> (xs @ ys) = (p \<bullet> xs) @ (p \<bullet> ys)" by (induct xs) autolemma rev_eqvt [eqvt]: shows "p \<bullet> (rev xs) = rev (p \<bullet> xs)" by (induct xs) (simp_all add: append_eqvt)lemma map_eqvt [eqvt]: shows "p \<bullet> (map f xs) = map (p \<bullet> f) (p \<bullet> xs)" by (induct xs) (simp_all, simp only: permute_fun_app_eq)lemma removeAll_eqvt [eqvt]: shows "p \<bullet> (removeAll x xs) = removeAll (p \<bullet> x) (p \<bullet> xs)" by (induct xs) (auto)lemma filter_eqvt [eqvt]: shows "p \<bullet> (filter f xs) = filter (p \<bullet> f) (p \<bullet> xs)"apply(induct xs)apply(simp)apply(simp only: filter.simps permute_list.simps if_eqvt)apply(simp only: permute_fun_app_eq)donelemma distinct_eqvt [eqvt]: shows "p \<bullet> (distinct xs) = distinct (p \<bullet> xs)"apply(induct xs)apply(simp add: permute_bool_def)apply(simp add: conj_eqvt Not_eqvt mem_eqvt set_eqvt)donelemma length_eqvt [eqvt]: shows "p \<bullet> (length xs) = length (p \<bullet> xs)"by (induct xs) (simp_all add: permute_pure)subsubsection {* Equivariance for @{typ "'a option"} *}lemma option_map_eqvt[eqvt]: shows "p \<bullet> (Option.map f x) = Option.map (p \<bullet> f) (p \<bullet> x)" by (cases x) (simp_all, simp add: permute_fun_app_eq)subsubsection {* Equivariance for @{typ "'a fset"} *}lemma in_fset_eqvt [eqvt]: shows "(p \<bullet> (x |\<in>| S)) = ((p \<bullet> x) |\<in>| (p \<bullet> S))"unfolding in_fsetby (perm_simp) (simp)lemma union_fset_eqvt [eqvt]: shows "(p \<bullet> (S |\<union>| T)) = ((p \<bullet> S) |\<union>| (p \<bullet> T))" by (induct S) (simp_all)lemma inter_list_eqvt [eqvt]: shows "p \<bullet> (inter_list S T) = inter_list (p \<bullet> S) (p \<bullet> T)" unfolding list_eq_def inter_list_def by perm_simp simplemma inter_fset_eqvt [eqvt]: shows "(p \<bullet> (S |\<inter>| T)) = ((p \<bullet> S) |\<inter>| (p \<bullet> T))" by (lifting inter_list_eqvt)lemma sub_list_eqvt [eqvt]: shows "p \<bullet> (sub_list S T) = sub_list (p \<bullet> S) (p \<bullet> T)" unfolding sub_list_def by perm_simp simplemma subset_fset_eqvt [eqvt]: shows "(p \<bullet> (S |\<subseteq>| T)) = ((p \<bullet> S) |\<subseteq>| (p \<bullet> T))" by (lifting sub_list_eqvt)lemma map_fset_eqvt [eqvt]: shows "p \<bullet> (map_fset f S) = map_fset (p \<bullet> f) (p \<bullet> S)" by (lifting map_eqvt)section {* Supp, Freshness and Supports *}context ptbegindefinition supp :: "'a \<Rightarrow> atom set"where "supp x = {a. infinite {b. (a \<rightleftharpoons> b) \<bullet> x \<noteq> x}}"definition fresh :: "atom \<Rightarrow> 'a \<Rightarrow> bool" ("_ \<sharp> _" [55, 55] 55)where "a \<sharp> x \<equiv> a \<notin> supp x"endlemma 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 supp_eqvt [eqvt]: shows "p \<bullet> (supp x) = supp (p \<bullet> x)" unfolding supp_def by (perm_simp) (simp only: permute_eqvt[symmetric])lemma fresh_eqvt [eqvt]: shows "p \<bullet> (a \<sharp> x) = (p \<bullet> a) \<sharp> (p \<bullet> x)" unfolding fresh_def by (perm_simp) (rule refl)lemma fresh_permute_iff: shows "(p \<bullet> a) \<sharp> (p \<bullet> x) \<longleftrightarrow> a \<sharp> x" by (simp only: fresh_eqvt[symmetric] permute_bool_def)lemma fresh_permute_left: shows "a \<sharp> p \<bullet> x \<longleftrightarrow> - p \<bullet> a \<sharp> x"proof assume "a \<sharp> p \<bullet> x" then have "- p \<bullet> a \<sharp> - p \<bullet> p \<bullet> x" by (simp only: fresh_permute_iff) then show "- p \<bullet> a \<sharp> x" by simpnext assume "- p \<bullet> a \<sharp> x" then have "p \<bullet> - p \<bullet> a \<sharp> p \<bullet> x" by (simp only: fresh_permute_iff) then show "a \<sharp> p \<bullet> x" by simpqedsection {* 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 supports_fresh: fixes x :: "'a::pt" assumes a1: "S supports x" and a2: "finite S" and a3: "a \<notin> S" shows "a \<sharp> x"unfolding fresh_defproof - have "(supp x) \<subseteq> S" using a1 a2 by (rule supp_is_subset) then show "a \<notin> (supp x)" using a3 by autoqedlemma 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: fixes x::"'a::pure" shows "supp x = {}" 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}" by (rule finite_supp_unique) (auto simp add: supports_def)lemma 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}" unfolding permute_atom_def using finite_gpermute_neq .lemma supp_perm: shows "supp p = {a. p \<bullet> a \<noteq> a}"apply (rule finite_supp_unique)apply (simp_all add: perm_swap_eq swap_eqvt supports_perm finite_perm_lemma)apply (auto simp add: perm_eq_iff swap_atom perm_swap_eq swap_eqvt)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 supp_perm by (simp) (metis permute_minus_cancel(1))lemma supp_minus_perm: fixes p::perm shows "supp (- p) = supp p" unfolding supp_conv_fresh by (simp add: fresh_minus_perm)lemma plus_perm_eq: fixes p q::"perm" assumes asm: "supp p \<inter> supp q = {}" shows "p + q = q + p"unfolding perm_eq_iffproof 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 blastqedinstance perm :: fsby default (simp add: supp_perm finite_perm_lemma)section {* 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) fs by default (auto simp add: supp_Pair finite_supp)subsection {* 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 (case_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 fresh_Nil: shows "a \<sharp> []" by (simp add: fresh_def supp_Nil)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_Cons: shows "a \<sharp> (x # xs) \<longleftrightarrow> a \<sharp> x \<and> a \<sharp> xs" by (simp add: fresh_def supp_Cons)lemma supp_append: shows "supp (xs @ ys) = supp xs \<union> supp ys" by (induct xs) (auto simp add: supp_Nil 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)lemma supp_rev: shows "supp (rev xs) = supp xs" by (induct xs) (auto simp add: supp_append supp_Cons supp_Nil)lemma fresh_rev: shows "a \<sharp> rev xs \<longleftrightarrow> a \<sharp> xs" by (induct xs) (auto simp add: fresh_append fresh_Cons fresh_Nil)lemma supp_removeAll: fixes x::"atom" shows "supp (removeAll x xs) = supp xs - {x}" by (induct xs) (auto simp add: supp_Nil supp_Cons supp_atom)lemma 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)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 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 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 autosubsection {* Equivariance Predicate @{text eqvt} and @{text eqvt_at}*}definition "eqvt f \<equiv> \<forall>p. p \<bullet> f = f"lemma eqvt_boolI: fixes f::"bool" shows "eqvt f" unfolding eqvt_def by (simp add: permute_bool_def)text {* 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 eqvtI: shows "(\<And>p. p \<bullet> f \<equiv> f) \<Longrightarrow> eqvt f"unfolding eqvt_defby simplemma eqvt_at_perm: assumes "eqvt_at f x" shows "eqvt_at f (q \<bullet> x)"proof - { fix p::"perm" have "p \<bullet> (f (q \<bullet> x)) = p \<bullet> q \<bullet> (f x)" using assms by (simp add: eqvt_at_def) also have "\<dots> = (p + q) \<bullet> (f x)" by simp also have "\<dots> = f ((p + q) \<bullet> x)" using assms by (simp add: eqvt_at_def) finally have "p \<bullet> (f (q \<bullet> x)) = f (p \<bullet> q \<bullet> x)" by simp } then show "eqvt_at f (q \<bullet> x)" unfolding eqvt_at_def by simpqedlemma supp_fun_eqvt: assumes a: "eqvt f" shows "supp f = {}" using a unfolding eqvt_def unfolding supp_def by simplemma fresh_fun_eqvt: assumes a: "eqvt f" shows "a \<sharp> f" using a unfolding fresh_def by (simp add: supp_fun_eqvt)lemma 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 autoqedlemma supp_fun_app_eqvt: assumes a: "eqvt f" shows "supp (f x) \<subseteq> supp x" using fresh_fun_eqvt_app[OF a] unfolding fresh_def by autolemma 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 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 {* for handling of freshness of functions *}simproc_setup fresh_fun_simproc ("a \<sharp> (f::'a::pt \<Rightarrow>'b::pt)") = {* fn _ => fn ss => fn ctrm => let val Const(@{const_name fresh}, _) $ _ $ f = term_of ctrm in case (Term.add_frees f [], Term.add_vars f []) of ([], []) => SOME(@{thm fresh_fun_eqvt[simplified eqvt_def, THEN Eq_TrueI]}) | (x::_, []) => let val thy = Proof_Context.theory_of (Simplifier.the_context ss) val argx = Free x val absf = absfree x f val cty_inst = [SOME (ctyp_of thy (fastype_of argx)), SOME (ctyp_of thy (fastype_of f))] val ctrm_inst = [NONE, SOME (cterm_of thy absf), SOME (cterm_of thy argx)] val thm = Drule.instantiate' cty_inst ctrm_inst @{thm fresh_fun_app} in SOME(thm RS @{thm Eq_TrueI}) end | (_, _) => NONE end*}subsection {* helper functions for nominal_functions *}lemma THE_defaultI2: assumes "\<exists>!x. P x" "\<And>x. P x \<Longrightarrow> Q x" shows "Q (THE_default d P)"by (iprover intro: assms THE_defaultI')lemma the_default_eqvt: assumes unique: "\<exists>!x. P x" shows "(p \<bullet> (THE_default d P)) = (THE_default (p \<bullet> 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 x) (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) apply(rule THE_default1_equality [symmetric]) apply(rule_tac p="-p" in permute_boolE) apply(perm_simp add: permute_minus_cancel) using eqvt[simplified eqvt_def] apply(simp) apply(rule ex1) apply(rule THE_defaultI2) apply(rule_tac p="-p" in permute_boolE) apply(perm_simp add: permute_minus_cancel) apply(rule ex1) apply(perm_simp) using eqvt[simplified eqvt_def] apply(simp) donelemma fundef_ex1_eqvt_at: fixes x::"'a::pt" assumes f_def: "f == (\<lambda>x::'a. THE_default (d x) (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)lemma fundef_ex1_prop: fixes x::"'a::pt" assumes f_def: "f \<equiv> (\<lambda>x::'a. THE_default (d x) (G x))" assumes P_all: "\<And>x y. G x y \<Longrightarrow> P x y" assumes ex1: "\<exists>!y. G x y" shows "P x (f x)" unfolding f_def using ex1 apply(erule_tac ex1E) apply(rule THE_defaultI2) apply(blast) apply(rule P_all) apply(assumption) donesection {* 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 supp_cofinite_atom_set: fixes S::"atom set" assumes "finite (UNIV - S)" shows "supp S = (UNIV - S)" apply(rule finite_supp_unique) apply(simp add: supports_def) apply(simp add: swap_set_both_in) apply(rule assms) apply(subst swap_commute) 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 fresh_minus_atom_set: fixes S::"atom set" assumes "finite S" shows "a \<sharp> S - T \<longleftrightarrow> (a \<notin> T \<longrightarrow> a \<sharp> S)" unfolding fresh_def by (auto simp add: supp_finite_atom_set assms)lemma 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_def 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 eqvt: "eqvt (\<lambda>S. \<Union> supp ` S)" unfolding eqvt_def by (perm_simp) (simp) 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> = supp ((\<lambda>S. \<Union> supp ` S) S)" by simp also have "\<dots> \<subseteq> supp S" using eqvt by (rule supp_fun_app_eqvt) 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 multiset"} is finitely supported *}lemma set_of_eqvt[eqvt]: shows "p \<bullet> (set_of M) = set_of (p \<bullet> M)"by (induct M) (simp_all add: insert_eqvt empty_eqvt)lemma supp_set_of: shows "supp (set_of M) \<subseteq> supp M" apply (rule supp_fun_app_eqvt) unfolding eqvt_def apply(perm_simp) apply(simp) donelemma Union_finite_multiset: fixes M::"'a::fs multiset" shows "finite (\<Union>{supp x | x. x \<in># M})"proof - have "finite (\<Union>(supp ` {x. x \<in># M}))" by (induct M) (simp_all add: Collect_imp_eq Collect_neg_eq finite_supp) then show "finite (\<Union>{supp x | x. x \<in># M})" by (simp only: image_Collect)qedlemma Union_supports_multiset: shows "\<Union>{supp x | x. x :# M} supports M"proof - have sw: "\<And>a b. ((\<And>x. x :# M \<Longrightarrow> (a \<rightleftharpoons> b) \<bullet> x = x) \<Longrightarrow> (a \<rightleftharpoons> b) \<bullet> M = M)" unfolding permute_multiset_def apply(induct M) apply(simp_all) done show "(\<Union>{supp x | x. x :# M}) supports M" unfolding supports_def apply(clarify) apply(rule sw) apply(rule swap_fresh_fresh) apply(simp_all only: fresh_def) apply(auto) apply(metis neq0_conv)+ doneqedlemma Union_included_multiset: fixes M::"('a::fs multiset)" shows "(\<Union>{supp x | x. x \<in># M}) \<subseteq> supp M"proof - have "(\<Union>{supp x | x. x \<in># M}) = (\<Union>{supp x | x. x \<in> set_of M})" by simp also have "... \<subseteq> (\<Union>x \<in> set_of M. supp x)" by auto also have "... = supp (set_of M)" by (simp add: subst supp_of_finite_sets) also have " ... \<subseteq> supp M" by (rule supp_set_of) finally show "(\<Union>{supp x | x. x \<in># M}) \<subseteq> supp M" .qedlemma supp_of_multisets: fixes M::"('a::fs multiset)" shows "(supp M) = (\<Union>{supp x | x. x :# M})"apply(rule subset_antisym)apply(rule supp_is_subset)apply(rule Union_supports_multiset)apply(rule Union_finite_multiset)apply(rule Union_included_multiset)donelemma multisets_supp_finite: fixes M::"('a::fs multiset)" shows "finite (supp M)"by (simp only: supp_of_multisets Union_finite_multiset)lemma supp_of_multiset_union: fixes M N::"('a::fs) multiset" shows "supp (M + N) = supp M \<union> supp N" by (auto simp add: supp_of_multisets)lemma supp_empty_mset [simp]: shows "supp {#} = {}" unfolding supp_def by simpinstance multiset :: (fs) fs apply (default) apply (rule multisets_supp_finite) donesubsection {* Type @{typ "'a fset"} is finitely supported *}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)lemma supp_union_fset: fixes S T::"'a::fs fset" shows "supp (S |\<union>| T) = supp S \<union> supp T"by (induct S) (auto)lemma fresh_union_fset: fixes S T::"'a::fs fset" shows "a \<sharp> S |\<union>| T \<longleftrightarrow> a \<sharp> S \<and> a \<sharp> T"unfolding fresh_defby (simp add: supp_union_fset)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)declaration {* fn _ =>let val mksimps_pairs = (@{const_name Nominal2_Base.fresh}, @{thms fresh_PairD}) :: mksimps_pairsin Simplifier.map_ss (fn ss => Simplifier.set_mksimps (mksimps mksimps_pairs) ss)end*}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_perm_set_conv: fixes p::"perm" assumes fresh: "as \<sharp>* p" and fin: "finite as" shows "supp p \<sharp>* as"apply(rule fresh_star_supp_conv)apply(simp add: supp_finite_atom_set fin fresh)donelemma fresh_star_atom_set_conv: assumes fresh: "as \<sharp>* bs" and fin: "finite as" "finite bs" shows "bs \<sharp>* as"using freshunfolding fresh_star_def fresh_defby (auto simp add: supp_finite_atom_set fin)lemma atom_fresh_star_disjoint: assumes fin: "finite bs" shows "as \<sharp>* bs \<longleftrightarrow> (as \<inter> bs = {})"unfolding fresh_star_def fresh_defby (auto simp add: supp_finite_atom_set fin)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 [eqvt]: shows "p \<bullet> (as \<sharp>* x) \<longleftrightarrow> (p \<bullet> as) \<sharp>* (p \<bullet> x)"unfolding fresh_star_defby (perm_simp) (rule refl)section {* Induction principle for permutations *}lemma smaller_supp: assumes a: "a \<in> supp p" shows "supp ((p \<bullet> a \<rightleftharpoons> a) + p) \<subset> supp p"proof - have "supp ((p \<bullet> a \<rightleftharpoons> a) + p) \<subseteq> supp p" unfolding supp_perm by (auto simp add: swap_atom) moreover have "a \<notin> supp ((p \<bullet> a \<rightleftharpoons> a) + p)" by (simp add: supp_perm) then have "supp ((p \<bullet> a \<rightleftharpoons> a) + p) \<noteq> supp p" using a by auto ultimately show "supp ((p \<bullet> a \<rightleftharpoons> a) + p) \<subset> supp p" by autoqedlemma 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 perm_eq_iff) 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 \<subset> supp p" using a0 smaller_supp by simp 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: perm_eq_iff) 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_struct_induct2[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 perm_simple_struct_induct2[case_names zero swap plus]: assumes zero: "P 0" assumes swap: "\<And>a b. \<lbrakk>sort_of a = sort_of b; a \<noteq> b\<rbrakk> \<Longrightarrow> P (a \<rightleftharpoons> b)" assumes plus: "\<And>p1 p2. \<lbrakk>P p1; P p2\<rbrakk> \<Longrightarrow> P (p1 + p2)" shows "P p"by (rule_tac S="supp p" in perm_struct_induct2) (auto intro: zero swap plus)lemma supp_perm_singleton: fixes p::"perm" shows "supp p \<subseteq> {b} \<longleftrightarrow> p = 0"proof - { assume "supp p \<subseteq> {b}" then have "p = 0" by (induct p rule: perm_struct_induct) (simp_all) } then show "supp p \<subseteq> {b} \<longleftrightarrow> p = 0" by (auto simp add: supp_zero_perm)qedlemma supp_perm_pair: fixes p::"perm" shows "supp p \<subseteq> {a, b} \<longleftrightarrow> p = 0 \<or> p = (b \<rightleftharpoons> a)"proof - { assume "supp p \<subseteq> {a, b}" then have "p = 0 \<or> p = (b \<rightleftharpoons> a)" apply (induct p rule: perm_struct_induct) apply (auto simp add: swap_cancel supp_zero_perm supp_swap) apply (simp add: swap_commute) done } then show "supp p \<subseteq> {a, b} \<longleftrightarrow> p = 0 \<or> p = (b \<rightleftharpoons> a)" by (auto simp add: supp_zero_perm supp_swap split: if_splits)qedlemma 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) qedqedtext {* same lemma as above, but proved with a different induction principle *}lemma 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_struct_induct2) 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"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_induct2) 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 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(1) permute_plus)qedtext {* disagreement set *}definition dset :: "perm \<Rightarrow> perm \<Rightarrow> atom set"where "dset p q = {a::atom. p \<bullet> a \<noteq> q \<bullet> a}"lemma ds_fresh: assumes "dset p q \<sharp>* x" shows "p \<bullet> x = q \<bullet> x"using assmsunfolding dset_def fresh_star_def fresh_defby (auto intro: supp_perm_perm_eq)lemma atom_set_perm_eq: assumes a: "as \<sharp>* p" shows "p \<bullet> as = as"proof - from a have "supp p \<subseteq> {a. a \<notin> as}" unfolding supp_perm fresh_star_def fresh_def by auto then show "p \<bullet> as = as" proof (induct p rule: perm_struct_induct) case zero show "0 \<bullet> as = as" by simp next case (swap p a b) then have "a \<notin> as" "b \<notin> as" "p \<bullet> as = as" by simp_all then show "((a \<rightleftharpoons> b) + p) \<bullet> as = as" by (simp add: swap_set_not_in) qedqedsection {* 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_def) 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. (\<forall>b \<in> bs. q \<bullet> b = p \<bullet> b) \<and> supp q \<subseteq> bs \<union> (p \<bullet> bs)"using bproof (induct) case empty have "(\<forall>b \<in> {}. 0 \<bullet> b = p \<bullet> b) \<and> supp (0::perm) \<subseteq> {} \<union> p \<bullet> {}" by (simp add: permute_set_def supp_perm) then show "\<exists>q. (\<forall>b \<in> {}. q \<bullet> b = p \<bullet> b) \<and> supp q \<subseteq> {} \<union> p \<bullet> {}" by blastnext case (insert a bs) then have " \<exists>q. (\<forall>b \<in> bs. q \<bullet> b = p \<bullet> b) \<and> supp q \<subseteq> bs \<union> p \<bullet> bs" by simp then obtain q where *: "\<forall>b \<in> bs. q \<bullet> b = p \<bullet> b" and **: "supp q \<subseteq> bs \<union> p \<bullet> bs" by auto { assume 1: "q \<bullet> a = p \<bullet> a" have "\<forall>b \<in> (insert a bs). q \<bullet> b = p \<bullet> b" using 1 * by simp 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. (\<forall>b \<in> insert a bs. q \<bullet> b = p \<bullet> b) \<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 "\<forall>b \<in> insert a bs. q' \<bullet> b = p \<bullet> b" using 2 * `a \<notin> bs` unfolding q'_def by (auto simp add: swap_atom) 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. (\<forall>b \<in> insert a bs. q \<bullet> b = p \<bullet> b) \<and> supp q \<subseteq> insert a bs \<union> p \<bullet> insert a bs" by blast } ultimately show "\<exists>q. (\<forall>b \<in> insert a bs. q \<bullet> b = p \<bullet> b) \<and> supp q \<subseteq> insert a bs \<union> p \<bullet> insert a bs" by blastqedlemma set_renaming_perm2: shows "\<exists>q. (\<forall>b \<in> bs. q \<bullet> b = p \<bullet> b) \<and> supp q \<subseteq> bs \<union> (p \<bullet> bs)"proof - have "finite (bs \<inter> supp p)" by (simp add: finite_supp) then obtain q where *: "\<forall>b \<in> bs \<inter> supp p. q \<bullet> b = p \<bullet> b" and **: "supp q \<subseteq> (bs \<inter> supp p) \<union> (p \<bullet> (bs \<inter> supp p))" using set_renaming_perm by blast from ** have "supp q \<subseteq> bs \<union> (p \<bullet> bs)" by (auto simp add: inter_eqvt) moreover have "\<forall>b \<in> bs - supp p. q \<bullet> b = p \<bullet> b" apply(auto) apply(subgoal_tac "b \<notin> supp q") apply(simp add: fresh_def[symmetric]) apply(simp add: fresh_perm) apply(clarify) apply(rotate_tac 2) apply(drule subsetD[OF **]) apply(simp add: inter_eqvt supp_eqvt permute_self) done ultimately have "(\<forall>b \<in> bs. q \<bullet> b = p \<bullet> b) \<and> supp q \<subseteq> bs \<union> (p \<bullet> bs)" using * by auto then show "\<exists>q. (\<forall>b \<in> bs. q \<bullet> b = p \<bullet> b) \<and> supp q \<subseteq> bs \<union> (p \<bullet> bs)" by blastqedlemma list_renaming_perm: shows "\<exists>q. (\<forall>b \<in> set bs. q \<bullet> b = p \<bullet> b) \<and> supp q \<subseteq> set bs \<union> (p \<bullet> set bs)"proof (induct bs) case (Cons a bs) then have " \<exists>q. (\<forall>b \<in> set bs. q \<bullet> b = p \<bullet> b) \<and> supp q \<subseteq> set bs \<union> p \<bullet> (set bs)" by simp then obtain q where *: "\<forall>b \<in> set bs. q \<bullet> b = p \<bullet> b" 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 "\<forall>b \<in> set (a # bs). q \<bullet> b = p \<bullet> b" 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. (\<forall>b \<in> set (a # bs). q \<bullet> b = p \<bullet> b) \<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 "\<forall>b \<in> set (a # bs). q' \<bullet> b = p \<bullet> b" unfolding q'_def using 2 * `a \<notin> set bs` by (auto simp add: swap_atom) 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. (\<forall>b \<in> set (a # bs). q \<bullet> b = p \<bullet> b) \<and> supp q \<subseteq> set (a # bs) \<union> p \<bullet> (set (a # bs))" by blast } ultimately show "\<exists>q. (\<forall>b \<in> set (a # bs). q \<bullet> b = p \<bullet> b) \<and> supp q \<subseteq> set (a # bs) \<union> p \<bullet> (set (a # bs))" by blastnext case Nil have "(\<forall>b \<in> set []. 0 \<bullet> b = p \<bullet> b) \<and> supp (0::perm) \<subseteq> set [] \<union> p \<bullet> set []" by (simp add: supp_zero_perm) then show "\<exists>q. (\<forall>b \<in> set []. q \<bullet> b = p \<bullet> b) \<and> supp q \<subseteq> set [] \<union> p \<bullet> (set [])" 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)"declare atom_eqvt[eqvt]class at = at_base + assumes sort_of_atom_eq [simp]: "sort_of (atom a) = sort_of (atom b)"lemma sort_ineq [simp]: assumes "sort_of (atom a) \<noteq> sort_of (atom b)" shows "atom a \<noteq> atom b"using assms by metislemma 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 "sort_of a \<noteq> sort_of (atom b) \<Longrightarrow> a \<sharp> b" and "a \<sharp> b \<longleftrightarrow> a \<noteq> atom b" unfolding fresh_def apply(simp_all add: supp_at_base) apply(metis) donelemma 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)lemma if_fresh_at_base [simp]: shows "atom a \<sharp> x \<Longrightarrow> P (if a = x then t else s) = P s" and "atom a \<sharp> x \<Longrightarrow> P (if x = a then t else s) = P s"by (simp_all add: fresh_at_base)simproc_setup fresh_ineq ("x \<noteq> (y::'a::at_base)") = {* fn _ => fn ss => fn ctrm => let fun first_is_neg lhs rhs [] = NONE | first_is_neg lhs rhs (thm::thms) = (case Thm.prop_of thm of _ $ (@{term "HOL.Not"} $ (Const ("HOL.eq", _) $ l $ r)) => (if l = lhs andalso r = rhs then SOME(thm) else if r = lhs andalso l = rhs then SOME(thm RS @{thm not_sym}) else first_is_neg lhs rhs thms) | _ => first_is_neg lhs rhs thms) val simp_thms = @{thms fresh_Pair fresh_at_base atom_eq_iff} val prems = Simplifier.prems_of ss |> filter (fn thm => case Thm.prop_of thm of _ $ (Const (@{const_name fresh}, _) $ _ $ _) => true | _ => false) |> map (simplify (HOL_basic_ss addsimps simp_thms)) |> map HOLogic.conj_elims |> flat in case term_of ctrm of @{term "HOL.Not"} $ (Const ("HOL.eq", _) $ lhs $ rhs) => (case first_is_neg lhs rhs prems of SOME(thm) => SOME(thm RS @{thm Eq_TrueI}) | NONE => NONE) | _ => NONE end*}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 obtain_fresh': assumes fin: "finite (supp x)" obtains a::"'a::at_base" where "atom a \<sharp> x"using obtain_at_base[where X="supp x"]by (auto simp add: fresh_def fin)lemma obtain_fresh: fixes x::"'b::fs" obtains a::"'a::at_base" where "atom a \<sharp> x" by (rule obtain_fresh') (auto simp add: finite_supp)lemma 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)done(* FIXME lemma supp_cofinite_set_at_base: assumes a: "finite (UNIV - S)" shows "supp S = atom ` (UNIV - S)"apply(rule finite_supp_unique)*)lemma 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) donelemma fresh_at_base_permute_iff [simp]: fixes a::"'a::at_base" shows "atom (p \<bullet> a) \<sharp> p \<bullet> x \<longleftrightarrow> atom a \<sharp> x" unfolding atom_eqvt[symmetric] by (simp add: fresh_permute_iff)section {* Infrastructure for concrete atom types *}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 [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)qedlemma at_class_sort: fixes s :: atom_sort fixes Rep :: "'a \<Rightarrow> atom" and Abs :: "atom \<Rightarrow> 'a" fixes a::"'a" assumes type: "type_definition Rep Abs {a. sort_of a = s}" assumes atom_def: "\<And>a. atom a = Rep a" shows "sort_of (atom a) = s" using atom_def type unfolding type_definition_def by simpsetup {* 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 :: "('a::at \<Rightarrow> 'b::pt) \<Rightarrow> 'b"where "Fresh h = (THE x. \<forall>a. atom a \<sharp> h \<longrightarrow> h a = x)"lemma Fresh_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 h = h a"unfolding Fresh_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_apply': fixes h :: "'a::at \<Rightarrow> 'b::pt" assumes a: "atom a \<sharp> h" "atom a \<sharp> h a" shows "Fresh h = h a" apply (rule Fresh_apply) apply (auto simp add: fresh_Pair intro: a) donelemma Fresh_eqvt: fixes h :: "'a::at \<Rightarrow> 'b::pt" assumes a: "\<exists>a. atom a \<sharp> (h, h a)" shows "p \<bullet> (Fresh h) = Fresh (p \<bullet> h)" using a apply (clarsimp simp add: fresh_Pair) apply (subst Fresh_apply', assumption+) apply (drule fresh_permute_iff [where p=p, THEN iffD2]) apply (drule fresh_permute_iff [where p=p, THEN iffD2]) apply (simp only: atom_eqvt permute_fun_app_eq [where f=h]) apply (erule (1) Fresh_apply' [symmetric]) donelemma Fresh_supports: fixes h :: "'a::at \<Rightarrow> 'b::pt" assumes a: "\<exists>a. atom a \<sharp> (h, h a)" shows "(supp h) supports (Fresh h)" apply (simp add: supports_def fresh_def [symmetric]) apply (simp add: Fresh_eqvt [OF a] swap_fresh_fresh) donenotation Fresh (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 \<sharp> P" using P by (rule obtain_fresh') show "(FRESH x. f (P x)) = f (FRESH x. P x)" apply (subst Fresh_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_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, Q))" by (simp add: supp_Pair) then obtain a::'a where "atom a \<sharp> (P, Q)" by (rule obtain_fresh') then have "atom a \<sharp> P" and "atom a \<sharp> Q" by (simp_all add: fresh_Pair) show ?thesis apply (subst Fresh_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_apply' [where a=a, OF `atom a \<sharp> P` pure_fresh]) apply (subst Fresh_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"section {* automatic equivariance procedure for inductive definitions *}use "nominal_eqvt.ML"instantiation atom_sort :: ord beginfun less_atom_sort where "less_atom_sort (Sort s1 l1) (Sort s2 []) \<longleftrightarrow> s1 < s2"| "less_atom_sort (Sort s1 []) (Sort s2 (h # t)) \<longleftrightarrow> s1 \<le> s2"| "less_atom_sort (Sort s1 (h1 # t1)) (Sort s2 (h2 # t2)) \<longleftrightarrow> s1 < s2 \<or> s1 \<le> s2 \<and> ((less_atom_sort h1 h2) \<or> (h1 = h2 \<and> less_atom_sort (Sort s1 t1) (Sort s2 t2)))"definition less_eq_atom_sort where less_eq_atom_sort_def: "less_eq_atom_sort (x :: atom_sort) y \<longleftrightarrow> x < y \<or> x = y"instance ..endlemma less_st_less: "(Sort s1 l1) < (Sort s2 l2) \<longleftrightarrow> s1 < s2 \<or> s1 \<le> s2 \<and> l1 < l2" by (induct l1 l2 rule: list_induct2') autolemma not_as_le_as: "\<not>((x :: atom_sort) < x)" apply (rule less_atom_sort.induct[of "\<lambda>x y. x = y \<longrightarrow> \<not>x < y" "x" "x", simplified]) ..instance atom_sort :: linorderproof (default, auto simp add: less_eq_atom_sort_def not_as_le_as) fix x y :: atom_sort assume x: "x < y" "y < x" then show False by (induct x y rule: less_atom_sort.induct) (case_tac l1, auto) with x show "x = y" by (induct x y rule: less_atom_sort.induct) (case_tac l1, auto)next fix x y z :: atom_sort assume "x < y" "y < z" then show "x < z" apply (induct x z arbitrary: y rule: less_atom_sort.induct) apply (case_tac [!] y) apply auto apply (case_tac [!] list2) apply auto apply (case_tac l1) apply auto[2] donenext fix x y :: atom_sort assume x: "\<not>x < y" "y \<noteq> x" then show "y < x" apply (induct x y rule: less_atom_sort.induct) apply auto apply (case_tac [!] l1) apply auto doneqedinstantiation atom :: linorder begindefinition less_eq_atom where [simp]: "less_eq_atom x y \<longleftrightarrow> sort_of x < sort_of y \<or> sort_of x \<le> sort_of y \<and> nat_of x \<le> nat_of y"definition less_atom where [simp]: "less_atom x y \<longleftrightarrow> sort_of x < sort_of y \<or> sort_of x \<le> sort_of y \<and> nat_of x < nat_of y"instance apply default apply auto apply (case_tac x, case_tac y) apply auto doneendlemma [code]: "gpermute p = perm_apply (dest_perm p)" apply transfer unfolding Rel_def by (auto, metis perm_eq_def valid_dest_perm_raw_eq(2))instantiation perm :: equal begindefinition "equal_perm a b \<longleftrightarrow> Rep_perm a = Rep_perm b"instance apply default unfolding equal_perm_def perm_eq_rep ..end(* Test: export_code swap in SML *)end