Add the supp intersection conditions.
(* 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 ..
lemma swap_set_not_in:
assumes a: "a \<notin> S" "b \<notin> S"
shows "(a \<rightleftharpoons> b) \<bullet> S = S"
using a by (auto simp add: permute_set_eq swap_atom)
lemma swap_set_in:
assumes a: "a \<in> S" "b \<notin> S" "sort_of a = sort_of b"
shows "(a \<rightleftharpoons> b) \<bullet> S \<noteq> S"
using a by (auto simp add: permute_set_eq swap_atom)
subsection {* Permutations for units *}
instantiation unit :: 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 fresh_plus_perm:
fixes p q::perm
assumes "a \<sharp> p" "a \<sharp> q"
shows "a \<sharp> (p + q)"
using assms
unfolding fresh_def
by (auto simp add: supp_perm)
lemma supp_plus_perm:
fixes p q::perm
shows "supp (p + q) \<subseteq> supp p \<union> supp q"
by (auto simp add: supp_perm)
lemma fresh_minus_perm:
fixes p::perm
shows "a \<sharp> (- p) \<longleftrightarrow> a \<sharp> p"
unfolding fresh_def
apply(auto simp add: supp_perm)
apply(metis permute_minus_cancel)+
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)
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
section {* Support and freshness for applications *}
lemma supp_fun_app:
shows "supp (f x) \<subseteq> (supp f) \<union> (supp x)"
proof (rule subsetCI)
fix a::"atom"
assume a: "a \<in> supp (f x)"
assume b: "a \<notin> supp f \<union> supp x"
then have "finite {b. (a \<rightleftharpoons> b) \<bullet> f \<noteq> f}" "finite {b. (a \<rightleftharpoons> b) \<bullet> x \<noteq> x}"
unfolding supp_def by auto
then have "finite ({b. (a \<rightleftharpoons> b) \<bullet> f \<noteq> f} \<union> {b. (a \<rightleftharpoons> b) \<bullet> x \<noteq> x})" by simp
moreover
have "{b. ((a \<rightleftharpoons> b) \<bullet> f) ((a \<rightleftharpoons> b) \<bullet> x) \<noteq> f x} \<subseteq> ({b. (a \<rightleftharpoons> b) \<bullet> f \<noteq> f} \<union> {b. (a \<rightleftharpoons> b) \<bullet> x \<noteq> x})"
by auto
ultimately have "finite {b. ((a \<rightleftharpoons> b) \<bullet> f) ((a \<rightleftharpoons> b) \<bullet> x) \<noteq> f x}"
using finite_subset by auto
then have "a \<notin> supp (f x)" unfolding supp_def
by (simp add: permute_fun_app_eq)
with a show "False" by simp
qed
lemma fresh_fun_app:
shows "a \<sharp> (f, x) \<Longrightarrow> a \<sharp> f x"
unfolding fresh_def
using supp_fun_app
by (auto simp add: supp_Pair)
lemma fresh_fun_eqvt_app:
assumes a: "\<forall>p. p \<bullet> f = f"
shows "a \<sharp> x \<Longrightarrow> a \<sharp> f x"
proof -
from a have b: "supp f = {}"
unfolding supp_def by simp
show "a \<sharp> x \<Longrightarrow> a \<sharp> f x"
unfolding fresh_def
using supp_fun_app b
by auto
qed
end