Prove pseudo-inject (eq-iff) on the exported level and rename appropriately.
(* Title: Nominal2_Base+ −
Authors: Brian Huffman, Christian Urban+ −
+ −
Basic definitions and lemma infrastructure for + −
Nominal Isabelle. + −
*)+ −
theory Nominal2_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+ −