--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/Quot/Nominal/Nominal2_Base.thy Tue Jan 26 20:07:50 2010 +0100
@@ -0,0 +1,959 @@
+(* Title: Nominal2_Base
+ Authors: Brian Huffman, Christian Urban
+
+ Basic definitions and lemma infrastructure for
+ Nominal Isabelle.
+*)
+theory Nominal2_Base
+imports Main Infinite_Set
+begin
+
+section {* 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 nat
+
+
+text {* Basic projection function. *}
+
+primrec
+ sort_of :: "atom \<Rightarrow> atom_sort"
+where
+ "sort_of (Atom s i) = s"
+
+
+text {* There are infinitely many atoms of each sort. *}
+lemma INFM_sort_of_eq:
+ shows "INFM a. sort_of a = s"
+proof -
+ have "INFM i. sort_of (Atom s i) = s" by simp
+ moreover have "inj (Atom s)" by (simp add: inj_on_def)
+ ultimately show "INFM a. sort_of a = s" by (rule INFM_inj)
+qed
+
+lemma 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 ..
+qed
+
+section {* Sort-Respecting Permutations *}
+
+typedef perm =
+ "{f. bij f \<and> finite {a. f a \<noteq> a} \<and> (\<forall>a. sort_of (f a) = sort_of a)}"
+proof
+ show "id \<in> ?perm" by simp
+qed
+
+lemma permI:
+ assumes "bij f" and "MOST x. f x = x" and "\<And>a. sort_of (f a) = sort_of a"
+ shows "f \<in> perm"
+ using assms unfolding perm_def MOST_iff_cofinite by simp
+
+lemma perm_is_bij: "f \<in> perm \<Longrightarrow> bij f"
+ unfolding perm_def by simp
+
+lemma perm_is_finite: "f \<in> perm \<Longrightarrow> finite {a. f a \<noteq> a}"
+ unfolding perm_def by simp
+
+lemma perm_is_sort_respecting: "f \<in> perm \<Longrightarrow> sort_of (f a) = sort_of a"
+ unfolding perm_def by simp
+
+lemma perm_MOST: "f \<in> perm \<Longrightarrow> MOST x. f x = x"
+ unfolding perm_def MOST_iff_cofinite by simp
+
+lemma perm_id: "id \<in> perm"
+ unfolding perm_def by simp
+
+lemma perm_comp:
+ assumes f: "f \<in> perm" and g: "g \<in> perm"
+ shows "(f \<circ> g) \<in> perm"
+apply (rule permI)
+apply (rule bij_comp)
+apply (rule perm_is_bij [OF g])
+apply (rule perm_is_bij [OF f])
+apply (rule MOST_rev_mp [OF perm_MOST [OF g]])
+apply (rule MOST_rev_mp [OF perm_MOST [OF f]])
+apply (simp)
+apply (simp add: perm_is_sort_respecting [OF f])
+apply (simp add: perm_is_sort_respecting [OF g])
+done
+
+lemma perm_inv:
+ assumes f: "f \<in> perm"
+ shows "(inv f) \<in> perm"
+apply (rule permI)
+apply (rule bij_imp_bij_inv)
+apply (rule perm_is_bij [OF f])
+apply (rule MOST_mono [OF perm_MOST [OF f]])
+apply (erule subst, rule inv_f_f)
+apply (rule bij_is_inj [OF perm_is_bij [OF f]])
+apply (rule perm_is_sort_respecting [OF f, THEN sym, THEN trans])
+apply (simp add: surj_f_inv_f [OF bij_is_surj [OF perm_is_bij [OF f]]])
+done
+
+lemma bij_Rep_perm: "bij (Rep_perm p)"
+ using Rep_perm [of p] unfolding perm_def by simp
+
+lemma finite_Rep_perm: "finite {a. Rep_perm p a \<noteq> a}"
+ using Rep_perm [of p] unfolding perm_def by simp
+
+lemma sort_of_Rep_perm: "sort_of (Rep_perm p a) = sort_of a"
+ using Rep_perm [of p] unfolding perm_def by simp
+
+lemma Rep_perm_ext:
+ "Rep_perm p1 = Rep_perm p2 \<Longrightarrow> p1 = p2"
+ by (simp add: expand_fun_eq Rep_perm_inject [symmetric])
+
+
+subsection {* Permutations form a group *}
+
+instantiation perm :: group_add
+begin
+
+definition
+ "0 = Abs_perm id"
+
+definition
+ "- p = Abs_perm (inv (Rep_perm p))"
+
+definition
+ "p + q = Abs_perm (Rep_perm p \<circ> Rep_perm q)"
+
+definition
+ "(p1::perm) - p2 = p1 + - p2"
+
+lemma Rep_perm_0: "Rep_perm 0 = id"
+ unfolding zero_perm_def
+ by (simp add: Abs_perm_inverse perm_id)
+
+lemma Rep_perm_add:
+ "Rep_perm (p1 + p2) = Rep_perm p1 \<circ> Rep_perm p2"
+ unfolding plus_perm_def
+ by (simp add: Abs_perm_inverse perm_comp Rep_perm)
+
+lemma Rep_perm_uminus:
+ "Rep_perm (- p) = inv (Rep_perm p)"
+ unfolding uminus_perm_def
+ by (simp add: Abs_perm_inverse perm_inv Rep_perm)
+
+instance
+apply default
+unfolding Rep_perm_inject [symmetric]
+unfolding minus_perm_def
+unfolding Rep_perm_add
+unfolding Rep_perm_uminus
+unfolding Rep_perm_0
+by (simp_all add: o_assoc inv_o_cancel [OF bij_is_inj [OF bij_Rep_perm]])
+
+end
+
+
+section {* Implementation of swappings *}
+
+definition
+ swap :: "atom \<Rightarrow> atom \<Rightarrow> perm" ("'(_ \<rightleftharpoons> _')")
+where
+ "(a \<rightleftharpoons> b) =
+ Abs_perm (if sort_of a = sort_of b
+ then (\<lambda>c. if a = c then b else if b = c then a else c)
+ else id)"
+
+lemma Rep_perm_swap:
+ "Rep_perm (a \<rightleftharpoons> b) =
+ (if sort_of a = sort_of b
+ then (\<lambda>c. if a = c then b else if b = c then a else c)
+ else id)"
+unfolding swap_def
+apply (rule Abs_perm_inverse)
+apply (rule permI)
+apply (auto simp add: bij_def inj_on_def surj_def)[1]
+apply (rule MOST_rev_mp [OF MOST_neq(1) [of a]])
+apply (rule MOST_rev_mp [OF MOST_neq(1) [of b]])
+apply (simp)
+apply (simp)
+done
+
+lemmas Rep_perm_simps =
+ Rep_perm_0
+ Rep_perm_add
+ Rep_perm_uminus
+ Rep_perm_swap
+
+lemma swap_different_sorts [simp]:
+ "sort_of a \<noteq> sort_of b \<Longrightarrow> (a \<rightleftharpoons> b) = 0"
+ by (rule Rep_perm_ext) (simp add: Rep_perm_simps)
+
+lemma swap_cancel:
+ "(a \<rightleftharpoons> b) + (a \<rightleftharpoons> b) = 0"
+by (rule Rep_perm_ext)
+ (simp add: Rep_perm_simps expand_fun_eq)
+
+lemma swap_self [simp]:
+ "(a \<rightleftharpoons> a) = 0"
+ by (rule Rep_perm_ext, simp add: Rep_perm_simps expand_fun_eq)
+
+lemma minus_swap [simp]:
+ "- (a \<rightleftharpoons> b) = (a \<rightleftharpoons> b)"
+ by (rule minus_unique [OF swap_cancel])
+
+lemma swap_commute:
+ "(a \<rightleftharpoons> b) = (b \<rightleftharpoons> a)"
+ by (rule Rep_perm_ext)
+ (simp add: Rep_perm_swap expand_fun_eq)
+
+lemma swap_triple:
+ assumes "a \<noteq> b" and "c \<noteq> b"
+ assumes "sort_of a = sort_of b" "sort_of b = sort_of c"
+ shows "(a \<rightleftharpoons> c) + (b \<rightleftharpoons> c) + (a \<rightleftharpoons> c) = (a \<rightleftharpoons> b)"
+ using assms
+ by (rule_tac Rep_perm_ext)
+ (auto simp add: Rep_perm_simps expand_fun_eq)
+
+
+section {* Permutation Types *}
+
+text {*
+ Infix syntax for @{text permute} has higher precedence than
+ addition, but lower than unary minus.
+*}
+
+class pt =
+ fixes permute :: "perm \<Rightarrow> 'a \<Rightarrow> 'a" ("_ \<bullet> _" [76, 75] 75)
+ assumes permute_zero [simp]: "0 \<bullet> x = x"
+ assumes permute_plus [simp]: "(p + q) \<bullet> x = p \<bullet> (q \<bullet> x)"
+begin
+
+lemma permute_diff [simp]:
+ shows "(p - q) \<bullet> x = p \<bullet> - q \<bullet> x"
+ unfolding diff_minus by simp
+
+lemma permute_minus_cancel [simp]:
+ shows "p \<bullet> - p \<bullet> x = x"
+ and "- p \<bullet> p \<bullet> x = x"
+ unfolding permute_plus [symmetric] by simp_all
+
+lemma 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])
+
+end
+
+subsection {* Permutations for atoms *}
+
+instantiation atom :: pt
+begin
+
+definition
+ "p \<bullet> a = Rep_perm p a"
+
+instance
+apply(default)
+apply(simp_all add: permute_atom_def Rep_perm_simps)
+done
+
+end
+
+lemma sort_of_permute [simp]:
+ shows "sort_of (p \<bullet> a) = sort_of a"
+ unfolding permute_atom_def by (rule sort_of_Rep_perm)
+
+lemma swap_atom:
+ shows "(a \<rightleftharpoons> b) \<bullet> c =
+ (if sort_of a = sort_of b
+ then (if c = a then b else if c = b then a else c) else c)"
+ unfolding permute_atom_def
+ by (simp add: Rep_perm_swap)
+
+lemma swap_atom_simps [simp]:
+ "sort_of a = sort_of b \<Longrightarrow> (a \<rightleftharpoons> b) \<bullet> a = b"
+ "sort_of a = sort_of b \<Longrightarrow> (a \<rightleftharpoons> b) \<bullet> b = a"
+ "c \<noteq> a \<Longrightarrow> c \<noteq> b \<Longrightarrow> (a \<rightleftharpoons> b) \<bullet> c = c"
+ unfolding swap_atom by simp_all
+
+lemma expand_perm_eq:
+ fixes p q :: "perm"
+ shows "p = q \<longleftrightarrow> (\<forall>a::atom. p \<bullet> a = q \<bullet> a)"
+ unfolding permute_atom_def
+ by (metis Rep_perm_ext ext)
+
+
+subsection {* Permutations for permutations *}
+
+instantiation perm :: pt
+begin
+
+definition
+ "p \<bullet> q = p + q - p"
+
+instance
+apply default
+apply (simp add: permute_perm_def)
+apply (simp add: permute_perm_def diff_minus minus_add add_assoc)
+done
+
+end
+
+lemma permute_self: "p \<bullet> p = p"
+unfolding permute_perm_def by (simp add: diff_minus add_assoc)
+
+lemma permute_eqvt:
+ shows "p \<bullet> (q \<bullet> x) = (p \<bullet> q) \<bullet> (p \<bullet> x)"
+ unfolding permute_perm_def by simp
+
+lemma zero_perm_eqvt:
+ shows "p \<bullet> (0::perm) = 0"
+ unfolding permute_perm_def by simp
+
+lemma add_perm_eqvt:
+ fixes p p1 p2 :: perm
+ shows "p \<bullet> (p1 + p2) = p \<bullet> p1 + p \<bullet> p2"
+ unfolding permute_perm_def
+ by (simp add: expand_perm_eq)
+
+lemma swap_eqvt:
+ shows "p \<bullet> (a \<rightleftharpoons> b) = (p \<bullet> a \<rightleftharpoons> p \<bullet> b)"
+ unfolding permute_perm_def
+ by (auto simp add: swap_atom expand_perm_eq)
+
+
+subsection {* Permutations for functions *}
+
+instantiation "fun" :: (pt, pt) pt
+begin
+
+definition
+ "p \<bullet> f = (\<lambda>x. p \<bullet> (f (- p \<bullet> x)))"
+
+instance
+apply default
+apply (simp add: permute_fun_def)
+apply (simp add: permute_fun_def minus_add)
+done
+
+end
+
+lemma permute_fun_app_eq:
+ shows "p \<bullet> (f x) = (p \<bullet> f) (p \<bullet> x)"
+unfolding permute_fun_def by simp
+
+
+subsection {* Permutations for booleans *}
+
+instantiation bool :: pt
+begin
+
+definition "p \<bullet> (b::bool) = b"
+
+instance
+apply(default)
+apply(simp_all add: permute_bool_def)
+done
+
+end
+
+lemma Not_eqvt:
+ shows "p \<bullet> (\<not> A) = (\<not> (p \<bullet> A))"
+by (simp add: permute_bool_def)
+
+
+subsection {* Permutations for sets *}
+
+lemma permute_set_eq:
+ fixes x::"'a::pt"
+ and p::"perm"
+ shows "(p \<bullet> X) = {p \<bullet> x | x. x \<in> X}"
+ apply(auto simp add: permute_fun_def permute_bool_def mem_def)
+ apply(rule_tac x="- p \<bullet> x" in exI)
+ apply(simp)
+ done
+
+lemma permute_set_eq_image:
+ shows "p \<bullet> X = permute p ` X"
+unfolding permute_set_eq by auto
+
+lemma permute_set_eq_vimage:
+ shows "p \<bullet> X = permute (- p) -` X"
+unfolding permute_fun_def permute_bool_def
+unfolding vimage_def Collect_def mem_def ..
+
+subsection {* Permutations for units *}
+
+instantiation unit :: pt
+begin
+
+definition "p \<bullet> (u::unit) = u"
+
+instance proof
+qed (simp_all add: permute_unit_def)
+
+end
+
+
+subsection {* Permutations for products *}
+
+instantiation "*" :: (pt, pt) pt
+begin
+
+primrec
+ permute_prod
+where
+ Pair_eqvt: "p \<bullet> (x, y) = (p \<bullet> x, p \<bullet> y)"
+
+instance
+by default auto
+
+end
+
+
+subsection {* Permutations for sums *}
+
+instantiation "+" :: (pt, pt) pt
+begin
+
+primrec
+ permute_sum
+where
+ "p \<bullet> (Inl x) = Inl (p \<bullet> x)"
+| "p \<bullet> (Inr y) = Inr (p \<bullet> y)"
+
+instance proof
+qed (case_tac [!] x, simp_all)
+
+end
+
+subsection {* Permutations for lists *}
+
+instantiation list :: (pt) pt
+begin
+
+primrec
+ permute_list
+where
+ "p \<bullet> [] = []"
+| "p \<bullet> (x # xs) = p \<bullet> x # p \<bullet> xs"
+
+instance proof
+qed (induct_tac [!] x, simp_all)
+
+end
+
+subsection {* Permutations for options *}
+
+instantiation option :: (pt) pt
+begin
+
+primrec
+ permute_option
+where
+ "p \<bullet> None = None"
+| "p \<bullet> (Some x) = Some (p \<bullet> x)"
+
+instance proof
+qed (induct_tac [!] x, simp_all)
+
+end
+
+subsection {* Permutations for @{typ char}, @{typ nat}, and @{typ int} *}
+
+instantiation char :: pt
+begin
+
+definition "p \<bullet> (c::char) = c"
+
+instance proof
+qed (simp_all add: permute_char_def)
+
+end
+
+instantiation nat :: pt
+begin
+
+definition "p \<bullet> (n::nat) = n"
+
+instance proof
+qed (simp_all add: permute_nat_def)
+
+end
+
+instantiation int :: pt
+begin
+
+definition "p \<bullet> (i::int) = i"
+
+instance proof
+qed (simp_all add: permute_int_def)
+
+end
+
+
+section {* 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 :: pure
+proof qed (rule permute_unit_def)
+
+instance bool :: pure
+proof qed (rule permute_bool_def)
+
+text {* Other type constructors preserve purity. *}
+
+instance "fun" :: (pure, pure) pure
+by default (simp add: permute_fun_def permute_pure)
+
+instance "*" :: (pure, pure) pure
+by default (induct_tac x, simp add: permute_pure)
+
+instance "+" :: (pure, pure) pure
+by default (induct_tac x, simp_all add: permute_pure)
+
+instance list :: (pure) pure
+by default (induct_tac x, simp_all add: permute_pure)
+
+instance option :: (pure) pure
+by default (induct_tac x, simp_all add: permute_pure)
+
+
+subsection {* Types @{typ char}, @{typ nat}, and @{typ int} *}
+
+instance char :: pure
+proof qed (rule permute_char_def)
+
+instance nat :: pure
+proof qed (rule permute_nat_def)
+
+instance int :: pure
+proof qed (rule permute_int_def)
+
+
+subsection {* Supp, Freshness and Supports *}
+
+context pt
+begin
+
+definition
+ supp :: "'a \<Rightarrow> atom set"
+where
+ "supp x = {a. infinite {b. (a \<rightleftharpoons> b) \<bullet> x \<noteq> x}}"
+
+end
+
+definition
+ fresh :: "atom \<Rightarrow> 'a::pt \<Rightarrow> bool" ("_ \<sharp> _" [55, 55] 55)
+where
+ "a \<sharp> x \<equiv> a \<notin> supp x"
+
+lemma supp_conv_fresh:
+ shows "supp x = {a. \<not> a \<sharp> x}"
+ unfolding fresh_def by simp
+
+lemma 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 auto
+next
+ 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" .
+qed
+
+lemma 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 simp
+qed
+
+
+subsection {* supp and fresh are equivariant *}
+
+lemma finite_Collect_bij:
+ assumes a: "bij f"
+ shows "finite {x. P (f x)} = finite {x. P x}"
+by (metis a finite_vimage_iff vimage_Collect_eq)
+
+lemma fresh_permute_iff:
+ shows "(p \<bullet> a) \<sharp> (p \<bullet> x) \<longleftrightarrow> a \<sharp> x"
+proof -
+ have "(p \<bullet> a) \<sharp> (p \<bullet> x) \<longleftrightarrow> finite {b. (p \<bullet> a \<rightleftharpoons> b) \<bullet> p \<bullet> x \<noteq> p \<bullet> x}"
+ unfolding fresh_def supp_def by simp
+ also have "\<dots> \<longleftrightarrow> finite {b. (p \<bullet> a \<rightleftharpoons> p \<bullet> b) \<bullet> p \<bullet> x \<noteq> p \<bullet> x}"
+ using bij_permute by (rule finite_Collect_bij [symmetric])
+ also have "\<dots> \<longleftrightarrow> finite {b. p \<bullet> (a \<rightleftharpoons> b) \<bullet> x \<noteq> p \<bullet> x}"
+ by (simp only: permute_eqvt [of p] swap_eqvt)
+ also have "\<dots> \<longleftrightarrow> finite {b. (a \<rightleftharpoons> b) \<bullet> x \<noteq> x}"
+ by (simp only: permute_eq_iff)
+ also have "\<dots> \<longleftrightarrow> a \<sharp> x"
+ unfolding fresh_def supp_def by simp
+ finally show ?thesis .
+qed
+
+lemma fresh_eqvt:
+ shows "p \<bullet> (a \<sharp> x) = (p \<bullet> a) \<sharp> (p \<bullet> x)"
+ by (simp add: permute_bool_def fresh_permute_iff)
+
+lemma supp_eqvt:
+ fixes p :: "perm"
+ and x :: "'a::pt"
+ shows "p \<bullet> (supp x) = supp (p \<bullet> x)"
+ unfolding supp_conv_fresh
+ unfolding permute_fun_def Collect_def
+ by (simp add: Not_eqvt fresh_eqvt)
+
+subsection {* supports *}
+
+definition
+ supports :: "atom set \<Rightarrow> 'a::pt \<Rightarrow> bool" (infixl "supports" 80)
+where
+ "S supports x \<equiv> \<forall>a b. (a \<notin> S \<and> b \<notin> S \<longrightarrow> (a \<rightleftharpoons> b) \<bullet> x = x)"
+
+lemma supp_is_subset:
+ fixes S :: "atom set"
+ and x :: "'a::pt"
+ assumes a1: "S supports x"
+ and a2: "finite S"
+ shows "(supp x) \<subseteq> S"
+proof (rule ccontr)
+ assume "\<not>(supp x \<subseteq> S)"
+ then obtain a where b1: "a \<in> supp x" and b2: "a \<notin> S" by auto
+ from a1 b2 have "\<forall>b. b \<notin> S \<longrightarrow> (a \<rightleftharpoons> b) \<bullet> x = x" by (unfold supports_def) (auto)
+ hence "{b. (a \<rightleftharpoons> b) \<bullet> x \<noteq> x} \<subseteq> S" by auto
+ with a2 have "finite {b. (a \<rightleftharpoons> b)\<bullet>x \<noteq> x}" by (simp add: finite_subset)
+ then have "a \<notin> (supp x)" unfolding supp_def by simp
+ with b1 show False by simp
+qed
+
+lemma 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)
+qed
+
+lemma supp_supports:
+ fixes x :: "'a::pt"
+ shows "(supp x) supports x"
+proof (unfold supports_def, intro strip)
+ fix a b
+ assume "a \<notin> (supp x) \<and> b \<notin> (supp x)"
+ then have "a \<sharp> x" and "b \<sharp> x" by (simp_all add: fresh_def)
+ then show "(a \<rightleftharpoons> b) \<bullet> x = x" by (rule swap_fresh_fresh)
+qed
+
+lemma 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 blast
+qed
+
+lemma subsetCI:
+ shows "(\<And>x. x \<in> A \<Longrightarrow> x \<notin> B \<Longrightarrow> False) \<Longrightarrow> A \<subseteq> B"
+ by auto
+
+lemma 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 a2
+proof (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
+ qed
+qed
+
+section {* Finitely-supported types *}
+
+class fs = pt +
+ assumes finite_supp: "finite (supp x)"
+
+lemma pure_supp:
+ shows "supp (x::'a::pure) = {}"
+ unfolding supp_def by (simp add: permute_pure)
+
+lemma pure_fresh:
+ fixes x::"'a::pure"
+ shows "a \<sharp> x"
+ unfolding fresh_def by (simp add: pure_supp)
+
+instance pure < fs
+by default (simp add: pure_supp)
+
+
+subsection {* Type @{typ atom} is finitely-supported. *}
+
+lemma supp_atom:
+ shows "supp a = {a}"
+apply (rule finite_supp_unique)
+apply (clarsimp simp add: supports_def)
+apply simp
+apply simp
+done
+
+lemma fresh_atom:
+ shows "a \<sharp> b \<longleftrightarrow> a \<noteq> b"
+ unfolding fresh_def supp_atom by simp
+
+instance atom :: fs
+by 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_def
+by (metis add_diff_cancel minus_perm_def)
+
+lemma supports_perm:
+ shows "{a. p \<bullet> a \<noteq> a} supports p"
+ unfolding supports_def
+ by (simp add: perm_swap_eq swap_eqvt)
+
+lemma finite_perm_lemma:
+ shows "finite {a::atom. p \<bullet> a \<noteq> a}"
+ using finite_Rep_perm [of p]
+ unfolding permute_atom_def .
+
+lemma supp_perm:
+ shows "supp p = {a. p \<bullet> a \<noteq> a}"
+apply (rule finite_supp_unique)
+apply (rule supports_perm)
+apply (rule finite_perm_lemma)
+apply (simp add: perm_swap_eq swap_eqvt)
+apply (auto simp add: expand_perm_eq swap_atom)
+done
+
+lemma 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 simp
+
+lemma supp_zero_perm:
+ shows "supp (0::perm) = {}"
+ unfolding supp_perm by simp
+
+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 supp_minus_perm:
+ fixes p::perm
+ shows "supp (- p) = supp p"
+ apply(auto simp add: supp_perm)
+ apply(metis permute_minus_cancel)+
+ done
+
+instance perm :: fs
+by 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)
+
+instance "*" :: (fs, fs) fs
+apply default
+apply (induct_tac x)
+apply (simp add: supp_Pair finite_supp)
+done
+
+
+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 "+" :: (fs, fs) fs
+apply default
+apply (induct_tac x)
+apply (simp_all add: supp_Inl supp_Inr finite_supp)
+done
+
+subsection {* 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) fs
+apply default
+apply (induct_tac x)
+apply (simp_all add: supp_None supp_Some finite_supp)
+done
+
+subsubsection {* Type @{typ "'a list"} is finitely supported *}
+
+lemma supp_Nil:
+ shows "supp [] = {}"
+ by (simp add: supp_def)
+
+lemma supp_Cons:
+ shows "supp (x # xs) = supp x \<union> supp xs"
+by (simp add: supp_def Collect_imp_eq Collect_neg_eq)
+
+lemma fresh_Nil:
+ shows "a \<sharp> []"
+ by (simp add: fresh_def supp_Nil)
+
+lemma fresh_Cons:
+ shows "a \<sharp> (x # xs) \<longleftrightarrow> a \<sharp> x \<and> a \<sharp> xs"
+ by (simp add: fresh_def supp_Cons)
+
+instance list :: (fs) fs
+apply default
+apply (induct_tac x)
+apply (simp_all add: supp_Nil supp_Cons finite_supp)
+done
+
+end