(* Title: Nominal2_Base
Authors: Brian Huffman, Christian Urban
Basic definitions and lemma infrastructure for
Nominal Isabelle.
*)
theory Nominal2_Base
imports Main Infinite_Set
"~~/src/HOL/Quotient_Examples/FSet"
uses ("nominal_library.ML")
("nominal_atoms.ML")
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"
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)
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
lemma atom_components_eq_iff:
fixes a b :: atom
shows "a = b \<longleftrightarrow> sort_of a = sort_of b \<and> nat_of a = nat_of b"
by (induct a, induct b, simp)
section {* Sort-Respecting Permutations *}
typedef perm =
"{f. bij f \<and> finite {a. f a \<noteq> a} \<and> (\<forall>a. sort_of (f a) = sort_of a)}"
proof
show "id \<in> ?perm" by 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: fun_eq_iff Rep_perm_inject [symmetric])
instance perm :: size ..
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 fun_eq_iff)
lemma swap_self [simp]:
"(a \<rightleftharpoons> a) = 0"
by (rule Rep_perm_ext, simp add: Rep_perm_simps fun_eq_iff)
lemma minus_swap [simp]:
"- (a \<rightleftharpoons> b) = (a \<rightleftharpoons> b)"
by (rule minus_unique [OF swap_cancel])
lemma swap_commute:
"(a \<rightleftharpoons> b) = (b \<rightleftharpoons> a)"
by (rule Rep_perm_ext)
(simp add: Rep_perm_swap fun_eq_iff)
lemma swap_triple:
assumes "a \<noteq> b" and "c \<noteq> b"
assumes "sort_of a = sort_of b" "sort_of b = sort_of c"
shows "(a \<rightleftharpoons> c) + (b \<rightleftharpoons> c) + (a \<rightleftharpoons> c) = (a \<rightleftharpoons> b)"
using assms
by (rule_tac Rep_perm_ext)
(auto simp add: Rep_perm_simps fun_eq_iff)
section {* Permutation Types *}
text {*
Infix syntax for @{text permute} has higher precedence than
addition, but lower than unary minus.
*}
class pt =
fixes permute :: "perm \<Rightarrow> 'a \<Rightarrow> 'a" ("_ \<bullet> _" [76, 75] 75)
assumes permute_zero [simp]: "0 \<bullet> x = x"
assumes permute_plus [simp]: "(p + q) \<bullet> x = p \<bullet> (q \<bullet> x)"
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:
shows "p \<bullet> p = p"
unfolding permute_perm_def
by (simp add: diff_minus add_assoc)
lemma permute_eqvt:
shows "p \<bullet> (q \<bullet> x) = (p \<bullet> q) \<bullet> (p \<bullet> x)"
unfolding permute_perm_def by 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)
lemma uminus_eqvt:
fixes p q::"perm"
shows "p \<bullet> (- q) = - (p \<bullet> q)"
unfolding permute_perm_def
by (simp add: diff_minus minus_add add_assoc)
subsection {* Permutations for functions *}
instantiation "fun" :: (pt, pt) 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)
lemma conj_eqvt:
shows "p \<bullet> (A \<and> B) = ((p \<bullet> A) \<and> (p \<bullet> B))"
by (simp add: permute_bool_def)
lemma imp_eqvt:
shows "p \<bullet> (A \<longrightarrow> B) = ((p \<bullet> A) \<longrightarrow> (p \<bullet> B))"
by (simp add: permute_bool_def)
lemma ex_eqvt:
shows "p \<bullet> (\<exists>x. P x) = (\<exists>x. (p \<bullet> P) x)"
unfolding permute_fun_def permute_bool_def
by (auto, rule_tac x="p \<bullet> x" in exI, simp)
lemma all_eqvt:
shows "p \<bullet> (\<forall>x. P x) = (\<forall>x. (p \<bullet> P) x)"
unfolding permute_fun_def permute_bool_def
by (auto, drule_tac x="p \<bullet> x" in spec, simp)
lemma permute_boolE:
fixes P::"bool"
shows "p \<bullet> P \<Longrightarrow> P"
by (simp add: permute_bool_def)
lemma permute_boolI:
fixes P::"bool"
shows "P \<Longrightarrow> p \<bullet> P"
by(simp add: permute_bool_def)
subsection {* Permutations for sets *}
lemma permute_set_eq:
fixes x::"'a::pt"
and p::"perm"
shows "(p \<bullet> X) = {p \<bullet> x | x. x \<in> X}"
unfolding permute_fun_def
unfolding permute_bool_def
apply(auto simp add: mem_def)
apply(rule_tac x="- p \<bullet> x" in exI)
apply(simp)
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 ..
lemma swap_set_not_in:
assumes a: "a \<notin> S" "b \<notin> S"
shows "(a \<rightleftharpoons> b) \<bullet> S = S"
unfolding permute_set_eq
using a by (auto simp add: swap_atom)
lemma swap_set_in:
assumes a: "a \<in> S" "b \<notin> S" "sort_of a = sort_of b"
shows "(a \<rightleftharpoons> b) \<bullet> S \<noteq> S"
unfolding permute_set_eq
using a by (auto simp add: swap_atom)
lemma mem_permute_iff:
shows "(p \<bullet> x) \<in> (p \<bullet> X) \<longleftrightarrow> x \<in> X"
unfolding mem_def permute_fun_def permute_bool_def
by simp
lemma mem_eqvt:
shows "p \<bullet> (x \<in> A) \<longleftrightarrow> (p \<bullet> x) \<in> (p \<bullet> A)"
unfolding mem_def
by (simp add: permute_fun_app_eq)
lemma empty_eqvt:
shows "p \<bullet> {} = {}"
unfolding empty_def Collect_def
by (simp add: permute_fun_def permute_bool_def)
lemma insert_eqvt:
shows "p \<bullet> (insert x A) = insert (p \<bullet> x) (p \<bullet> A)"
unfolding permute_set_eq_image image_insert ..
subsection {* Permutations for units *}
instantiation unit :: pt
begin
definition "p \<bullet> (u::unit) = u"
instance
by (default) (simp_all add: permute_unit_def)
end
subsection {* Permutations for products *}
instantiation prod :: (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 sum :: (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
by (default) (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
by (default) (induct_tac [!] x, simp_all)
end
lemma set_eqvt:
shows "p \<bullet> (set xs) = set (p \<bullet> xs)"
by (induct xs) (simp_all add: empty_eqvt insert_eqvt)
subsection {* Permutations for options *}
instantiation option :: (pt) pt
begin
primrec
permute_option
where
"p \<bullet> None = None"
| "p \<bullet> (Some x) = Some (p \<bullet> x)"
instance
by (default) (induct_tac [!] x, simp_all)
end
subsection {* Permutations for fsets *}
lemma permute_fset_rsp[quot_respect]:
shows "(op = ===> list_eq ===> list_eq) permute permute"
unfolding fun_rel_def
by (simp add: set_eqvt[symmetric])
instantiation fset :: (pt) pt
begin
quotient_definition
"permute_fset :: perm \<Rightarrow> 'a fset \<Rightarrow> 'a fset"
is
"permute :: perm \<Rightarrow> 'a list \<Rightarrow> 'a list"
instance
proof
fix x :: "'a fset" and p q :: "perm"
show "0 \<bullet> x = x" by (descending) (simp)
show "(p + q) \<bullet> x = p \<bullet> q \<bullet> x" by (descending) (simp)
qed
end
lemma permute_fset [simp]:
fixes S::"('a::pt) fset"
shows "(p \<bullet> {||}) = ({||} ::('a::pt) fset)"
and "(p \<bullet> insert_fset x S) = insert_fset (p \<bullet> x) (p \<bullet> S)"
by (lifting permute_list.simps)
subsection {* Permutations for @{typ char}, @{typ nat}, and @{typ int} *}
instantiation char :: pt
begin
definition "p \<bullet> (c::char) = c"
instance
by (default) (simp_all add: permute_char_def)
end
instantiation nat :: pt
begin
definition "p \<bullet> (n::nat) = n"
instance
by (default) (simp_all add: permute_nat_def)
end
instantiation int :: pt
begin
definition "p \<bullet> (i::int) = i"
instance
by (default) (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 prod :: (pure, pure) pure
by default (induct_tac x, simp add: permute_pure)
instance sum :: (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 "(p \<bullet> a) \<sharp> (p \<bullet> x) \<longleftrightarrow> a \<sharp> x" .
qed
lemma fresh_eqvt:
shows "p \<bullet> (a \<sharp> x) = (p \<bullet> a) \<sharp> (p \<bullet> x)"
unfolding permute_bool_def
by (simp add: fresh_permute_iff)
lemma supp_eqvt:
fixes p :: "perm"
and x :: "'a::pt"
shows "p \<bullet> (supp x) = supp (p \<bullet> x)"
unfolding supp_conv_fresh
unfolding Collect_def
unfolding permute_fun_def
by (simp add: Not_eqvt fresh_eqvt)
subsection {* supports *}
definition
supports :: "atom set \<Rightarrow> 'a::pt \<Rightarrow> bool" (infixl "supports" 80)
where
"S supports x \<equiv> \<forall>a b. (a \<notin> S \<and> b \<notin> S \<longrightarrow> (a \<rightleftharpoons> b) \<bullet> x = x)"
lemma supp_is_subset:
fixes S :: "atom set"
and x :: "'a::pt"
assumes a1: "S supports x"
and a2: "finite S"
shows "(supp x) \<subseteq> S"
proof (rule ccontr)
assume "\<not> (supp x \<subseteq> S)"
then obtain a where b1: "a \<in> supp x" and b2: "a \<notin> S" by auto
from a1 b2 have "\<forall>b. b \<notin> S \<longrightarrow> (a \<rightleftharpoons> b) \<bullet> x = x" unfolding supports_def by auto
then have "{b. (a \<rightleftharpoons> b) \<bullet> x \<noteq> x} \<subseteq> S" by auto
with a2 have "finite {b. (a \<rightleftharpoons> b)\<bullet>x \<noteq> x}" by (simp add: finite_subset)
then have "a \<notin> (supp x)" unfolding supp_def by simp
with b1 show False by 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"
unfolding supports_def
proof (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)
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 {* Support w.r.t. relations *}
text {*
This definition is used for unquotient types, where
alpha-equivalence does not coincide with equality.
*}
definition
"supp_rel R x = {a. infinite {b. \<not>(R ((a \<rightleftharpoons> b) \<bullet> x) x)}}"
section {* Finitely-supported types *}
class fs = pt +
assumes finite_supp: "finite (supp x)"
lemma pure_supp:
shows "supp (x::'a::pure) = {}"
unfolding supp_def by (simp add: permute_pure)
lemma pure_fresh:
fixes x::"'a::pure"
shows "a \<sharp> x"
unfolding fresh_def by (simp add: pure_supp)
instance pure < 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 {* Support for finite sets of atoms *}
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)
done
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
unfolding perm_swap_eq
by (simp add: swap_eqvt)
lemma finite_perm_lemma:
shows "finite {a::atom. p \<bullet> a \<noteq> a}"
using finite_Rep_perm [of p]
unfolding permute_atom_def .
lemma supp_perm:
shows "supp p = {a. p \<bullet> a \<noteq> a}"
apply (rule finite_supp_unique)
apply (rule supports_perm)
apply (rule finite_perm_lemma)
apply (simp add: perm_swap_eq swap_eqvt)
apply (auto simp add: expand_perm_eq swap_atom)
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 fresh_plus_perm:
fixes p q::perm
assumes "a \<sharp> p" "a \<sharp> q"
shows "a \<sharp> (p + q)"
using assms
unfolding fresh_def
by (auto simp add: supp_perm)
lemma supp_plus_perm:
fixes p q::perm
shows "supp (p + q) \<subseteq> supp p \<union> supp q"
by (auto simp add: supp_perm)
lemma fresh_minus_perm:
fixes p::perm
shows "a \<sharp> (- p) \<longleftrightarrow> a \<sharp> p"
unfolding fresh_def
unfolding supp_perm
apply(simp)
apply(metis permute_minus_cancel)
done
lemma supp_minus_perm:
fixes p::perm
shows "supp (- p) = supp p"
unfolding supp_conv_fresh
by (simp add: fresh_minus_perm)
instance perm :: fs
by default (simp add: supp_perm finite_perm_lemma)
lemma plus_perm_eq:
fixes p q::"perm"
assumes asm: "supp p \<inter> supp q = {}"
shows "p + q = q + p"
unfolding expand_perm_eq
proof
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
qed
qed
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
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 sum :: (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
section {* Support and Freshness for Applications *}
lemma fresh_conv_MOST:
shows "a \<sharp> x \<longleftrightarrow> (MOST b. (a \<rightleftharpoons> b) \<bullet> x = x)"
unfolding fresh_def supp_def
unfolding MOST_iff_cofinite by simp
lemma supp_subset_fresh:
assumes a: "\<And>a. a \<sharp> x \<Longrightarrow> a \<sharp> y"
shows "supp y \<subseteq> supp x"
using a
unfolding fresh_def
by blast
lemma 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 auto
text {* Support of Equivariant Functions *}
lemma supp_fun_eqvt:
assumes a: "\<And>p. p \<bullet> f = f"
shows "supp f = {}"
unfolding supp_def
using a by simp
lemma fresh_fun_eqvt_app:
assumes a: "\<And>p. p \<bullet> f = 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 auto
qed
section {* Support of Finite Sets of Finitely Supported Elements *}
lemma Union_fresh:
shows "a \<sharp> S \<Longrightarrow> a \<sharp> (\<Union>x \<in> S. supp x)"
unfolding Union_image_eq[symmetric]
apply(rule_tac f="\<lambda>S. \<Union> supp ` S" in fresh_fun_eqvt_app)
apply(simp add: permute_fun_def UNION_def Collect_def Bex_def ex_eqvt mem_def conj_eqvt)
apply(subst permute_fun_app_eq)
back
apply(simp add: supp_eqvt)
apply(assumption)
done
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_eq by force
}
then show "(\<Union>x \<in> S. supp x) supports S"
unfolding supports_def
by (simp add: fresh_def[symmetric] swap_fresh_fresh)
qed
lemma Union_of_finite_supp_sets:
fixes S::"('a::fs set)"
assumes fin: "finite S"
shows "finite (\<Union>x\<in>S. supp x)"
using fin by (induct) (auto simp add: finite_supp)
lemma Union_included_in_supp:
fixes S::"('a::fs set)"
assumes fin: "finite S"
shows "(\<Union>x\<in>S. supp x) \<subseteq> supp S"
proof -
have "(\<Union>x\<in>S. supp x) = supp (\<Union>x\<in>S. supp x)"
by (rule supp_finite_atom_set[symmetric])
(rule Union_of_finite_supp_sets[OF fin])
also have "\<dots> \<subseteq> supp S"
by (rule supp_subset_fresh)
(simp add: Union_fresh)
finally show "(\<Union>x\<in>S. supp x) \<subseteq> supp S" .
qed
lemma 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])
done
lemma finite_sets_supp:
fixes S::"('a::fs set)"
assumes "finite S"
shows "finite (supp S)"
using assms
by (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)
subsection {* Type @{typ "'a fset"} is finitely supported *}
lemma fset_eqvt:
shows "p \<bullet> (fset S) = fset (p \<bullet> S)"
by (lifting set_eqvt)
lemma supp_fset [simp]:
shows "supp (fset S) = supp S"
unfolding supp_def
by (simp add: fset_eqvt fset_cong)
lemma supp_empty_fset [simp]:
shows "supp {||} = {}"
unfolding supp_def
by 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_fset supp_of_finite_insert)
done
lemma fset_finite_supp:
fixes S::"('a::fs) fset"
shows "finite (supp S)"
by (induct S) (simp_all add: finite_supp)
instance fset :: (fs) fs
apply (default)
apply (rule fset_finite_supp)
done
section {* Fresh-Star *}
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_prod:
fixes as::"atom set"
shows "as \<sharp>* (x, y) = (as \<sharp>* x \<and> as \<sharp>* y)"
by (auto simp add: fresh_star_def fresh_Pair)
lemma fresh_star_union:
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)
done
lemma 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_prod_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_prod)
lemma fresh_star_zero:
shows "as \<sharp>* (0::perm)"
unfolding fresh_star_def
by (simp add: fresh_zero_perm)
lemma fresh_star_plus:
fixes p q::perm
shows "\<lbrakk>a \<sharp>* p; a \<sharp>* q\<rbrakk> \<Longrightarrow> a \<sharp>* (p + q)"
unfolding fresh_star_def
by (simp add: fresh_plus_perm)
lemma fresh_star_permute_iff:
shows "(p \<bullet> a) \<sharp>* (p \<bullet> x) \<longleftrightarrow> a \<sharp>* x"
unfolding fresh_star_def
by (metis mem_permute_iff permute_minus_cancel(1) fresh_permute_iff)
lemma fresh_star_eqvt:
shows "(p \<bullet> (as \<sharp>* x)) = (p \<bullet> as) \<sharp>* (p \<bullet> x)"
unfolding fresh_star_def
unfolding Ball_def
apply(simp add: all_eqvt)
apply(subst permute_fun_def)
apply(simp add: imp_eqvt fresh_eqvt mem_eqvt)
done
section {* Induction principle for permutations *}
lemma perm_struct_induct[consumes 1, case_names zero swap]:
assumes S: "supp p \<subseteq> S"
and zero: "P 0"
and swap: "\<And>p a b. \<lbrakk>P p; supp p \<subseteq> S; a \<in> S; b \<in> S; a \<noteq> b; sort_of a = sort_of b\<rbrakk> \<Longrightarrow> P ((a \<rightleftharpoons> b) + p)"
shows "P p"
proof -
have "finite (supp p)" by (simp add: finite_supp)
then show "P p" using S
proof(induct A\<equiv>"supp p" arbitrary: p rule: finite_psubset_induct)
case (psubset p)
then have ih: "\<And>q. supp q \<subset> supp p \<Longrightarrow> P q" by auto
have as: "supp p \<subseteq> S" by fact
{ assume "supp p = {}"
then have "p = 0" by (simp add: supp_perm expand_perm_eq)
then have "P p" using zero by simp
}
moreover
{ assume "supp p \<noteq> {}"
then obtain a where a0: "a \<in> supp p" by blast
then have a1: "p \<bullet> a \<in> S" "a \<in> S" "sort_of (p \<bullet> a) = sort_of a" "p \<bullet> a \<noteq> a"
using as by (auto simp add: supp_atom supp_perm swap_atom)
let ?q = "(p \<bullet> a \<rightleftharpoons> a) + p"
have a2: "supp ?q \<subseteq> supp p" unfolding supp_perm by (auto simp add: swap_atom)
moreover
have "a \<notin> supp ?q" by (simp add: supp_perm)
then have "supp ?q \<noteq> supp p" using a0 by auto
ultimately have "supp ?q \<subset> supp p" using a2 by auto
then have "P ?q" using ih by simp
moreover
have "supp ?q \<subseteq> S" using as a2 by simp
ultimately have "P ((p \<bullet> a \<rightleftharpoons> a) + ?q)" using as a1 swap by simp
moreover
have "p = (p \<bullet> a \<rightleftharpoons> a) + ?q" by (simp add: expand_perm_eq)
ultimately have "P p" by simp
}
ultimately show "P p" by blast
qed
qed
lemma perm_simple_struct_induct[case_names zero swap]:
assumes zero: "P 0"
and swap: "\<And>p a b. \<lbrakk>P p; a \<noteq> b; sort_of a = sort_of b\<rbrakk> \<Longrightarrow> P ((a \<rightleftharpoons> b) + p)"
shows "P p"
by (rule_tac S="supp p" in perm_struct_induct)
(auto intro: zero swap)
lemma perm_subset_induct[consumes 1, case_names zero swap plus]:
assumes S: "supp p \<subseteq> S"
assumes zero: "P 0"
assumes swap: "\<And>a b. \<lbrakk>sort_of a = sort_of b; a \<noteq> b; a \<in> S; b \<in> S\<rbrakk> \<Longrightarrow> P (a \<rightleftharpoons> b)"
assumes plus: "\<And>p1 p2. \<lbrakk>P p1; P p2; supp p1 \<subseteq> S; supp p2 \<subseteq> S\<rbrakk> \<Longrightarrow> P (p1 + p2)"
shows "P p"
using S
by (induct p rule: perm_struct_induct)
(auto intro: zero plus swap simp add: supp_swap)
lemma supp_perm_eq:
assumes "(supp x) \<sharp>* p"
shows "p \<bullet> x = x"
proof -
from assms have "supp p \<subseteq> {a. a \<sharp> x}"
unfolding supp_perm fresh_star_def fresh_def by auto
then show "p \<bullet> x = x"
proof (induct p rule: perm_struct_induct)
case zero
show "0 \<bullet> x = x" by simp
next
case (swap p a b)
then have "a \<sharp> x" "b \<sharp> x" "p \<bullet> x = x" by simp_all
then show "((a \<rightleftharpoons> b) + p) \<bullet> x = x" by (simp add: swap_fresh_fresh)
qed
qed
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_subset_induct)
case zero
show "0 \<bullet> x = x" by simp
next
case (swap a b)
then have "a \<sharp> x" "b \<sharp> x" by simp_all
then show "(a \<rightleftharpoons> b) \<bullet> x = x" by (simp add: swap_fresh_fresh)
next
case (plus p1 p2)
have "p1 \<bullet> x = x" "p2 \<bullet> x = x" by fact+
then show "(p1 + p2) \<bullet> x = x" by simp
qed
qed
section {* 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) \<subseteq> (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) \<subseteq> {} \<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 \<subseteq> (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)"
using `finite As` `finite Xs`
by (simp add: permute_set_eq_image)
then obtain y where "y \<notin> (As \<union> p \<bullet> Xs)" "sort_of y = sort_of x"
by (rule obtain_atom)
hence y: "y \<notin> As" "y \<notin> p \<bullet> Xs" "sort_of y = sort_of x"
by simp_all
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 ?q \<subseteq> insert x Xs \<union> ?q \<bullet> insert x Xs"
using p2 unfolding q
by (intro subset_trans [OF supp_plus_perm])
(auto simp add: supp_swap)
ultimately show ?case by blast
qed
qed
lemma at_set_avoiding:
assumes a: "finite Xs"
and b: "finite (supp c)"
obtains p::"perm" where "(p \<bullet> Xs)\<sharp>*c" and "(supp p) \<subseteq> (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 blast
lemma 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 assms
apply(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_prod)
apply(rule fresh_star_supp_conv)
apply(auto simp add: fresh_star_def)
done
lemma at_set_avoiding2_atom:
assumes "finite (supp c)" "finite (supp x)"
and b: "a \<sharp> x"
shows "\<exists>p. (p \<bullet> a) \<sharp> c \<and> supp x \<sharp>* p"
proof -
have a: "{a} \<sharp>* x" unfolding fresh_star_def by (simp add: b)
obtain p where p1: "(p \<bullet> {a}) \<sharp>* c" and p2: "supp x \<sharp>* p"
using at_set_avoiding2[of "{a}" "c" "x"] assms a by blast
have c: "(p \<bullet> a) \<sharp> c" using p1
unfolding fresh_star_def Ball_def
by(erule_tac x="p \<bullet> a" in allE) (simp add: permute_set_eq)
hence "p \<bullet> a \<sharp> c \<and> supp x \<sharp>* p" using p2 by blast
then show "\<exists>p. (p \<bullet> a) \<sharp> c \<and> supp x \<sharp>* p" by blast
qed
section {* Concrete Atoms Types *}
text {*
Class @{text at_base} allows types containing multiple sorts of atoms.
Class @{text at} only allows types with a single sort.
*}
class at_base = pt +
fixes atom :: "'a \<Rightarrow> atom"
assumes atom_eq_iff [simp]: "atom a = atom b \<longleftrightarrow> a = b"
assumes atom_eqvt: "p \<bullet> (atom a) = atom (p \<bullet> a)"
class at = at_base +
assumes sort_of_atom_eq [simp]: "sort_of (atom a) = sort_of (atom b)"
lemma supp_at_base:
fixes a::"'a::at_base"
shows "supp a = {atom a}"
by (simp add: supp_atom [symmetric] supp_def atom_eqvt)
lemma fresh_at_base:
shows "a \<sharp> b \<longleftrightarrow> a \<noteq> atom b"
unfolding fresh_def by (simp add: supp_at_base)
instance at_base < fs
proof 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 simp
qed
lemma 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_all
lemma 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 ..
qed
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
section {* Infrastructure for concrete atom types *}
section {* A swapping operation for concrete atoms *}
definition
flip :: "'a::at_base \<Rightarrow> 'a \<Rightarrow> perm" ("'(_ \<leftrightarrow> _')")
where
"(a \<leftrightarrow> b) = (atom a \<rightleftharpoons> atom b)"
lemma flip_self [simp]: "(a \<leftrightarrow> a) = 0"
unfolding flip_def by (rule swap_self)
lemma flip_commute: "(a \<leftrightarrow> b) = (b \<leftrightarrow> a)"
unfolding flip_def by (rule swap_commute)
lemma minus_flip [simp]: "- (a \<leftrightarrow> b) = (a \<leftrightarrow> b)"
unfolding flip_def by (rule minus_swap)
lemma add_flip_cancel: "(a \<leftrightarrow> b) + (a \<leftrightarrow> b) = 0"
unfolding flip_def by (rule swap_cancel)
lemma permute_flip_cancel [simp]: "(a \<leftrightarrow> b) \<bullet> (a \<leftrightarrow> b) \<bullet> x = x"
unfolding permute_plus [symmetric] add_flip_cancel by simp
lemma permute_flip_cancel2 [simp]: "(a \<leftrightarrow> b) \<bullet> (b \<leftrightarrow> a) \<bullet> x = x"
by (simp add: flip_commute)
lemma flip_eqvt:
fixes a b c::"'a::at_base"
shows "p \<bullet> (a \<leftrightarrow> b) = (p \<bullet> a \<leftrightarrow> p \<bullet> b)"
unfolding flip_def
by (simp add: swap_eqvt atom_eqvt)
lemma flip_at_base_simps [simp]:
shows "sort_of (atom a) = sort_of (atom b) \<Longrightarrow> (a \<leftrightarrow> b) \<bullet> a = b"
and "sort_of (atom a) = sort_of (atom b) \<Longrightarrow> (a \<leftrightarrow> b) \<bullet> b = a"
and "\<lbrakk>a \<noteq> c; b \<noteq> c\<rbrakk> \<Longrightarrow> (a \<leftrightarrow> b) \<bullet> c = c"
and "sort_of (atom a) \<noteq> sort_of (atom b) \<Longrightarrow> (a \<leftrightarrow> b) \<bullet> x = x"
unfolding flip_def
unfolding atom_eq_iff [symmetric]
unfolding atom_eqvt [symmetric]
by simp_all
text {* 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)
done
lemma 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_all
lemma 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 assms
by (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)
qed
setup {* 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
qed
qed
lemma 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)
qed
text {* packaging the freshness lemma into a function *}
definition
fresh_fun :: "('a::at \<Rightarrow> 'b::pt) \<Rightarrow> 'b"
where
"fresh_fun h = (THE x. \<forall>a. atom a \<sharp> h \<longrightarrow> h a = x)"
lemma fresh_fun_apply:
fixes h :: "'a::at \<Rightarrow> 'b::pt"
assumes a: "\<exists>a. atom a \<sharp> (h, h a)"
assumes b: "atom a \<sharp> h"
shows "fresh_fun h = h a"
unfolding fresh_fun_def
proof (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
qed
next
fix fr :: 'b
assume "\<forall>a. atom a \<sharp> h \<longrightarrow> h a = fr"
with b show "fr = h a" by auto
qed
lemma fresh_fun_apply':
fixes h :: "'a::at \<Rightarrow> 'b::pt"
assumes a: "atom a \<sharp> h" "atom a \<sharp> h a"
shows "fresh_fun h = h a"
apply (rule fresh_fun_apply)
apply (auto simp add: fresh_Pair intro: a)
done
lemma fresh_fun_eqvt:
fixes h :: "'a::at \<Rightarrow> 'b::pt"
assumes a: "\<exists>a. atom a \<sharp> (h, h a)"
shows "p \<bullet> (fresh_fun h) = fresh_fun (p \<bullet> h)"
using a
apply (clarsimp simp add: fresh_Pair)
apply (subst fresh_fun_apply', assumption+)
apply (drule fresh_permute_iff [where p=p, THEN iffD2])
apply (drule fresh_permute_iff [where p=p, THEN iffD2])
apply (simp add: atom_eqvt permute_fun_app_eq [where f=h])
apply (erule (1) fresh_fun_apply' [symmetric])
done
lemma fresh_fun_supports:
fixes h :: "'a::at \<Rightarrow> 'b::pt"
assumes a: "\<exists>a. atom a \<sharp> (h, h a)"
shows "(supp h) supports (fresh_fun h)"
apply (simp add: supports_def fresh_def [symmetric])
apply (simp add: fresh_fun_eqvt [OF a] swap_fresh_fresh)
done
notation fresh_fun (binder "FRESH " 10)
lemma FRESH_f_iff:
fixes P :: "'a::at \<Rightarrow> 'b::pure"
fixes f :: "'b \<Rightarrow> 'c::pure"
assumes P: "finite (supp P)"
shows "(FRESH x. f (P x)) = f (FRESH x. P x)"
proof -
obtain a::'a where "atom a \<notin> supp P"
using P by (rule obtain_at_base)
hence "atom a \<sharp> P"
by (simp add: fresh_def)
show "(FRESH x. f (P x)) = f (FRESH x. P x)"
apply (subst fresh_fun_apply' [where a=a, OF _ pure_fresh])
apply (cut_tac `atom a \<sharp> P`)
apply (simp add: fresh_conv_MOST)
apply (elim MOST_rev_mp, rule MOST_I, clarify)
apply (simp add: permute_fun_def permute_pure fun_eq_iff)
apply (subst fresh_fun_apply' [where a=a, OF `atom a \<sharp> P` pure_fresh])
apply (rule refl)
done
qed
lemma FRESH_binop_iff:
fixes P :: "'a::at \<Rightarrow> 'b::pure"
fixes Q :: "'a::at \<Rightarrow> 'c::pure"
fixes binop :: "'b \<Rightarrow> 'c \<Rightarrow> 'd::pure"
assumes P: "finite (supp P)"
and Q: "finite (supp Q)"
shows "(FRESH x. binop (P x) (Q x)) = binop (FRESH x. P x) (FRESH x. Q x)"
proof -
from assms have "finite (supp P \<union> supp Q)" by simp
then obtain a::'a where "atom a \<notin> (supp P \<union> supp Q)"
by (rule obtain_at_base)
hence "atom a \<sharp> P" and "atom a \<sharp> Q"
by (simp_all add: fresh_def)
show ?thesis
apply (subst fresh_fun_apply' [where a=a, OF _ pure_fresh])
apply (cut_tac `atom a \<sharp> P` `atom a \<sharp> Q`)
apply (simp add: fresh_conv_MOST)
apply (elim MOST_rev_mp, rule MOST_I, clarify)
apply (simp add: permute_fun_def permute_pure fun_eq_iff)
apply (subst fresh_fun_apply' [where a=a, OF `atom a \<sharp> P` pure_fresh])
apply (subst fresh_fun_apply' [where a=a, OF `atom a \<sharp> Q` pure_fresh])
apply (rule refl)
done
qed
lemma FRESH_conj_iff:
fixes P Q :: "'a::at \<Rightarrow> bool"
assumes P: "finite (supp P)" and Q: "finite (supp Q)"
shows "(FRESH x. P x \<and> Q x) \<longleftrightarrow> (FRESH x. P x) \<and> (FRESH x. Q x)"
using P Q by (rule FRESH_binop_iff)
lemma FRESH_disj_iff:
fixes P Q :: "'a::at \<Rightarrow> bool"
assumes P: "finite (supp P)" and Q: "finite (supp Q)"
shows "(FRESH x. P x \<or> Q x) \<longleftrightarrow> (FRESH x. P x) \<or> (FRESH x. Q x)"
using P Q by (rule FRESH_binop_iff)
section {* Library functions for the nominal infrastructure *}
use "nominal_library.ML"
section {* Automation for creating concrete atom types *}
text {* at the moment only single-sort concrete atoms are supported *}
use "nominal_atoms.ML"
end