(* Title: Nominal2_Base+ −
Authors: Brian Huffman, Christian Urban+ −
+ −
Basic definitions and lemma infrastructure for + −
Nominal Isabelle. + −
*)+ −
theory Nominal2_Base+ −
imports Main + −
"~~/src/HOL/Library/Infinite_Set"+ −
"~~/src/HOL/Quotient_Examples/FSet"+ −
uses ("nominal_basics.ML")+ −
("nominal_thmdecls.ML")+ −
("nominal_permeq.ML")+ −
("nominal_library.ML")+ −
("nominal_atoms.ML")+ −
("nominal_eqvt.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 n) = s"+ −
+ −
primrec+ −
nat_of :: "atom \<Rightarrow> nat"+ −
where+ −
"nat_of (Atom s n) = n"+ −
+ −
+ −
text {* There are infinitely many atoms of each sort. *}+ −
lemma INFM_sort_of_eq: + −
shows "INFM a. sort_of a = s"+ −
proof -+ −
have "INFM i. sort_of (Atom s i) = s" by simp+ −
moreover have "inj (Atom s)" by (simp add: inj_on_def)+ −
ultimately show "INFM a. sort_of a = s" by (rule INFM_inj)+ −
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 (multiplicative) 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:+ −
shows "(a \<rightleftharpoons> b) + (a \<rightleftharpoons> b) = 0"+ −
and "(a \<rightleftharpoons> b) + (b \<rightleftharpoons> a) = 0"+ −
by (rule_tac [!] Rep_perm_ext) + −
(simp_all 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(1)])+ −
+ −
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 perm_eq_iff:+ −
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 pemute_minus_self:+ −
shows "- p \<bullet> p = p"+ −
unfolding permute_perm_def + −
by (simp add: diff_minus add_assoc)+ −
+ −
+ −
subsection {* Permutations for functions *}+ −
+ −
instantiation "fun" :: (pt, pt) 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 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"+ −
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 permute_finite [simp]:+ −
shows "finite (p \<bullet> X) = finite X"+ −
unfolding permute_set_eq_vimage+ −
using bij_permute by (rule finite_vimage_iff)+ −
+ −
lemma swap_set_not_in:+ −
assumes a: "a \<notin> S" "b \<notin> S"+ −
shows "(a \<rightleftharpoons> b) \<bullet> S = S"+ −
unfolding permute_set_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 swap_set_in_eq:+ −
assumes a: "a \<in> S" "b \<notin> S" "sort_of a = sort_of b"+ −
shows "(a \<rightleftharpoons> b) \<bullet> S = (S - {a}) \<union> {b}"+ −
unfolding permute_set_eq+ −
using a by (auto simp add: swap_atom)+ −
+ −
lemma swap_set_both_in:+ −
assumes a: "a \<in> S" "b \<in> S"+ −
shows "(a \<rightleftharpoons> b) \<bullet> S = S"+ −
unfolding permute_set_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 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 @{typ unit} *}+ −
+ −
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 @{typ "'a list"} *}+ −
+ −
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 @{typ "'a option"} *}+ −
+ −
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 @{typ "'a fset"} *}+ −
+ −
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)+ −
+ −
lemma fset_eqvt: + −
shows "p \<bullet> (fset S) = fset (p \<bullet> S)"+ −
by (lifting set_eqvt)+ −
+ −
+ −
subsection {* Permutations for @{typ char}, @{typ nat}, and @{typ int} *}+ −
+ −
instantiation char :: 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)+ −
+ −
+ −
section {* Infrastructure for Equivariance and Perm_simp *}+ −
+ −
subsection {* Basic functions about permutations *}+ −
+ −
use "nominal_basics.ML"+ −
+ −
+ −
subsection {* Eqvt infrastructure *}+ −
+ −
text {* Setup of the theorem attributes @{text eqvt} and @{text eqvt_raw} *}+ −
+ −
use "nominal_thmdecls.ML"+ −
setup "Nominal_ThmDecls.setup"+ −
+ −
+ −
lemmas [eqvt] =+ −
(* pt types *)+ −
permute_prod.simps + −
permute_list.simps + −
permute_option.simps + −
permute_sum.simps+ −
+ −
(* sets *)+ −
empty_eqvt insert_eqvt set_eqvt + −
+ −
(* fsets *)+ −
permute_fset fset_eqvt+ −
+ −
+ −
+ −
subsection {* perm_simp infrastructure *}+ −
+ −
definition+ −
"unpermute p = permute (- p)"+ −
+ −
lemma eqvt_apply:+ −
fixes f :: "'a::pt \<Rightarrow> 'b::pt" + −
and x :: "'a::pt"+ −
shows "p \<bullet> (f x) \<equiv> (p \<bullet> f) (p \<bullet> x)"+ −
unfolding permute_fun_def by simp+ −
+ −
lemma eqvt_lambda:+ −
fixes f :: "'a::pt \<Rightarrow> 'b::pt"+ −
shows "p \<bullet> f \<equiv> (\<lambda>x. p \<bullet> (f (unpermute p x)))"+ −
unfolding permute_fun_def unpermute_def by simp+ −
+ −
lemma eqvt_bound:+ −
shows "p \<bullet> unpermute p x \<equiv> x"+ −
unfolding unpermute_def by simp+ −
+ −
text {* provides perm_simp methods *}+ −
+ −
use "nominal_permeq.ML"+ −
+ −
method_setup perm_simp =+ −
{* Nominal_Permeq.args_parser >> Nominal_Permeq.perm_simp_meth *}+ −
{* pushes permutations inside. *}+ −
+ −
method_setup perm_strict_simp =+ −
{* Nominal_Permeq.args_parser >> Nominal_Permeq.perm_strict_simp_meth *}+ −
{* pushes permutations inside, raises an error if it cannot solve all permutations. *}+ −
+ −
+ −
subsubsection {* Equivariance for permutations and swapping *}+ −
+ −
lemma permute_eqvt:+ −
shows "p \<bullet> (q \<bullet> x) = (p \<bullet> q) \<bullet> (p \<bullet> x)"+ −
unfolding permute_perm_def by simp+ −
+ −
(* the normal version of this lemma would cause loops *)+ −
lemma permute_eqvt_raw [eqvt_raw]:+ −
shows "p \<bullet> permute \<equiv> permute"+ −
apply(simp add: fun_eq_iff permute_fun_def)+ −
apply(subst permute_eqvt)+ −
apply(simp)+ −
done+ −
+ −
lemma zero_perm_eqvt [eqvt]:+ −
shows "p \<bullet> (0::perm) = 0"+ −
unfolding permute_perm_def by simp+ −
+ −
lemma add_perm_eqvt [eqvt]:+ −
fixes p p1 p2 :: perm+ −
shows "p \<bullet> (p1 + p2) = p \<bullet> p1 + p \<bullet> p2"+ −
unfolding permute_perm_def+ −
by (simp add: perm_eq_iff)+ −
+ −
lemma swap_eqvt [eqvt]:+ −
shows "p \<bullet> (a \<rightleftharpoons> b) = (p \<bullet> a \<rightleftharpoons> p \<bullet> b)"+ −
unfolding permute_perm_def+ −
by (auto simp add: swap_atom perm_eq_iff)+ −
+ −
lemma uminus_eqvt [eqvt]:+ −
fixes p q::"perm"+ −
shows "p \<bullet> (- q) = - (p \<bullet> q)"+ −
unfolding permute_perm_def+ −
by (simp add: diff_minus minus_add add_assoc)+ −
+ −
subsubsection {* Equivariance of Logical Operators *}+ −
+ −
lemma eq_eqvt [eqvt]:+ −
shows "p \<bullet> (x = y) \<longleftrightarrow> (p \<bullet> x) = (p \<bullet> y)"+ −
unfolding permute_eq_iff permute_bool_def ..+ −
+ −
lemma Not_eqvt [eqvt]:+ −
shows "p \<bullet> (\<not> A) \<longleftrightarrow> \<not> (p \<bullet> A)"+ −
by (simp add: permute_bool_def)+ −
+ −
lemma conj_eqvt [eqvt]:+ −
shows "p \<bullet> (A \<and> B) \<longleftrightarrow> (p \<bullet> A) \<and> (p \<bullet> B)"+ −
by (simp add: permute_bool_def)+ −
+ −
lemma imp_eqvt [eqvt]:+ −
shows "p \<bullet> (A \<longrightarrow> B) \<longleftrightarrow> (p \<bullet> A) \<longrightarrow> (p \<bullet> B)"+ −
by (simp add: permute_bool_def)+ −
+ −
declare imp_eqvt[folded induct_implies_def, eqvt]+ −
+ −
lemma all_eqvt [eqvt]:+ −
shows "p \<bullet> (\<forall>x. P x) = (\<forall>x. (p \<bullet> P) x)"+ −
unfolding All_def+ −
by (perm_simp) (rule refl)+ −
+ −
declare all_eqvt[folded induct_forall_def, eqvt]+ −
+ −
lemma ex_eqvt [eqvt]:+ −
shows "p \<bullet> (\<exists>x. P x) = (\<exists>x. (p \<bullet> P) x)"+ −
unfolding Ex_def+ −
by (perm_simp) (rule refl)+ −
+ −
lemma ex1_eqvt [eqvt]:+ −
shows "p \<bullet> (\<exists>!x. P x) = (\<exists>!x. (p \<bullet> P) x)"+ −
unfolding Ex1_def+ −
by (perm_simp) (rule refl)+ −
+ −
lemma if_eqvt [eqvt]:+ −
shows "p \<bullet> (if b then x else y) = (if p \<bullet> b then p \<bullet> x else p \<bullet> y)"+ −
by (simp add: permute_fun_def permute_bool_def)+ −
+ −
lemma True_eqvt [eqvt]:+ −
shows "p \<bullet> True = True"+ −
unfolding permute_bool_def ..+ −
+ −
lemma False_eqvt [eqvt]:+ −
shows "p \<bullet> False = False"+ −
unfolding permute_bool_def ..+ −
+ −
lemma disj_eqvt [eqvt]:+ −
shows "p \<bullet> (A \<or> B) \<longleftrightarrow> (p \<bullet> A) \<or> (p \<bullet> B)"+ −
by (simp add: permute_bool_def)+ −
+ −
lemma all_eqvt2:+ −
shows "p \<bullet> (\<forall>x. P x) = (\<forall>x. p \<bullet> P (- p \<bullet> x))"+ −
by (perm_simp add: permute_minus_cancel) (rule refl)+ −
+ −
lemma ex_eqvt2:+ −
shows "p \<bullet> (\<exists>x. P x) = (\<exists>x. p \<bullet> P (- p \<bullet> x))"+ −
by (perm_simp add: permute_minus_cancel) (rule refl)+ −
+ −
lemma ex1_eqvt2:+ −
shows "p \<bullet> (\<exists>!x. P x) = (\<exists>!x. p \<bullet> P (- p \<bullet> x))"+ −
by (perm_simp add: permute_minus_cancel) (rule refl)+ −
+ −
lemma the_eqvt:+ −
assumes unique: "\<exists>!x. P x"+ −
shows "(p \<bullet> (THE x. P x)) = (THE x. (p \<bullet> P) x)"+ −
apply(rule the1_equality [symmetric])+ −
apply(rule_tac p="-p" in permute_boolE)+ −
apply(perm_simp add: permute_minus_cancel)+ −
apply(rule unique)+ −
apply(rule_tac p="-p" in permute_boolE)+ −
apply(perm_simp add: permute_minus_cancel)+ −
apply(rule theI'[OF unique])+ −
done+ −
+ −
lemma the_eqvt2:+ −
assumes unique: "\<exists>!x. P x"+ −
shows "(p \<bullet> (THE x. P x)) = (THE x. p \<bullet> P (- p \<bullet> x))"+ −
apply(rule the1_equality [symmetric])+ −
apply(simp add: ex1_eqvt2[symmetric])+ −
apply(simp add: permute_bool_def unique)+ −
apply(simp add: permute_bool_def)+ −
apply(rule theI'[OF unique])+ −
done+ −
+ −
subsubsection {* Equivariance of Set operators *}+ −
+ −
lemma mem_eqvt [eqvt]:+ −
shows "p \<bullet> (x \<in> A) \<longleftrightarrow> (p \<bullet> x) \<in> (p \<bullet> A)"+ −
unfolding mem_def+ −
by (rule permute_fun_app_eq)+ −
+ −
lemma Collect_eqvt [eqvt]:+ −
shows "p \<bullet> {x. P x} = {x. (p \<bullet> P) x}"+ −
unfolding Collect_def permute_fun_def ..+ −
+ −
lemma inter_eqvt [eqvt]:+ −
shows "p \<bullet> (A \<inter> B) = (p \<bullet> A) \<inter> (p \<bullet> B)"+ −
unfolding Int_def+ −
by (perm_simp) (rule refl)+ −
+ −
lemma Bex_eqvt [eqvt]:+ −
shows "p \<bullet> (\<exists>x \<in> S. P x) = (\<exists>x \<in> (p \<bullet> S). (p \<bullet> P) x)"+ −
unfolding Bex_def+ −
by (perm_simp) (rule refl)+ −
+ −
lemma Ball_eqvt [eqvt]:+ −
shows "p \<bullet> (\<forall>x \<in> S. P x) = (\<forall>x \<in> (p \<bullet> S). (p \<bullet> P) x)"+ −
unfolding Ball_def+ −
by (perm_simp) (rule refl)+ −
+ −
lemma image_eqvt [eqvt]:+ −
shows "p \<bullet> (f ` A) = (p \<bullet> f) ` (p \<bullet> A)"+ −
unfolding image_def+ −
by (perm_simp) (rule refl)+ −
+ −
lemma UNIV_eqvt [eqvt]:+ −
shows "p \<bullet> UNIV = UNIV"+ −
unfolding UNIV_def+ −
by (perm_simp) (rule refl)+ −
+ −
lemma union_eqvt [eqvt]:+ −
shows "p \<bullet> (A \<union> B) = (p \<bullet> A) \<union> (p \<bullet> B)"+ −
unfolding Un_def+ −
by (perm_simp) (rule refl)+ −
+ −
lemma Diff_eqvt [eqvt]:+ −
fixes A B :: "'a::pt set"+ −
shows "p \<bullet> (A - B) = (p \<bullet> A) - (p \<bullet> B)"+ −
unfolding set_diff_eq+ −
by (perm_simp) (rule refl)+ −
+ −
lemma Compl_eqvt [eqvt]:+ −
fixes A :: "'a::pt set"+ −
shows "p \<bullet> (- A) = - (p \<bullet> A)"+ −
unfolding Compl_eq_Diff_UNIV+ −
by (perm_simp) (rule refl)+ −
+ −
lemma subset_eqvt [eqvt]:+ −
shows "p \<bullet> (S \<subseteq> T) \<longleftrightarrow> (p \<bullet> S) \<subseteq> (p \<bullet> T)"+ −
unfolding subset_eq+ −
by (perm_simp) (rule refl)+ −
+ −
lemma psubset_eqvt [eqvt]:+ −
shows "p \<bullet> (S \<subset> T) \<longleftrightarrow> (p \<bullet> S) \<subset> (p \<bullet> T)"+ −
unfolding psubset_eq+ −
by (perm_simp) (rule refl)+ −
+ −
lemma vimage_eqvt [eqvt]:+ −
shows "p \<bullet> (f -` A) = (p \<bullet> f) -` (p \<bullet> A)"+ −
unfolding vimage_def+ −
by (perm_simp) (rule refl)+ −
+ −
lemma Union_eqvt [eqvt]:+ −
shows "p \<bullet> (\<Union> S) = \<Union> (p \<bullet> S)"+ −
unfolding Union_eq + −
by (perm_simp) (rule refl)+ −
+ −
lemma Inter_eqvt [eqvt]:+ −
shows "p \<bullet> (\<Inter> S) = \<Inter> (p \<bullet> S)"+ −
unfolding Inter_eq + −
by (perm_simp) (rule refl)+ −
+ −
+ −
(* FIXME: eqvt attribute *)+ −
lemma Sigma_eqvt:+ −
shows "(p \<bullet> (X \<times> Y)) = (p \<bullet> X) \<times> (p \<bullet> Y)"+ −
unfolding Sigma_def+ −
unfolding UNION_eq_Union_image+ −
by (perm_simp) (rule refl)+ −
+ −
text {* + −
In order to prove that lfp is equivariant we need two+ −
auxiliary classes which specify that (op <=) and+ −
Inf are equivariant. Instances for bool and fun are + −
given.+ −
*}+ −
+ −
class le_eqvt = order + + −
assumes le_eqvt [eqvt]: "p \<bullet> (x \<le> y) = ((p \<bullet> x) \<le> (p \<bullet> (y::('a::{pt, order}))))"+ −
+ −
class inf_eqvt = complete_lattice ++ −
assumes inf_eqvt [eqvt]: "p \<bullet> (Inf X) = Inf (p \<bullet> (X::('a::{pt, Inf}) set))"+ −
+ −
instantiation bool :: le_eqvt+ −
begin+ −
+ −
instance + −
apply(default)+ −
unfolding le_bool_def+ −
apply(perm_simp)+ −
apply(rule refl)+ −
done+ −
+ −
end+ −
+ −
instantiation "fun" :: (pt, le_eqvt) le_eqvt+ −
begin+ −
+ −
instance + −
apply(default)+ −
unfolding le_fun_def+ −
apply(perm_simp)+ −
apply(rule refl)+ −
done + −
+ −
end+ −
+ −
instantiation bool :: inf_eqvt+ −
begin+ −
+ −
instance + −
apply(default)+ −
unfolding Inf_bool_def+ −
apply(perm_simp)+ −
apply(rule refl)+ −
done+ −
+ −
end+ −
+ −
instantiation "fun" :: (pt, inf_eqvt) inf_eqvt+ −
begin+ −
+ −
instance + −
apply(default)+ −
unfolding Inf_fun_def+ −
apply(perm_simp)+ −
apply(rule refl)+ −
done + −
+ −
end+ −
+ −
lemma lfp_eqvt [eqvt]:+ −
fixes F::"('a \<Rightarrow> 'b) \<Rightarrow> ('a::pt \<Rightarrow> 'b::{inf_eqvt, le_eqvt})"+ −
shows "p \<bullet> (lfp F) = lfp (p \<bullet> F)"+ −
unfolding lfp_def+ −
by (perm_simp) (rule refl)+ −
+ −
lemma finite_eqvt [eqvt]:+ −
shows "p \<bullet> finite A = finite (p \<bullet> A)"+ −
unfolding finite_def+ −
by (perm_simp) (rule refl)+ −
+ −
+ −
subsubsection {* Equivariance for product operations *}+ −
+ −
lemma fst_eqvt [eqvt]:+ −
shows "p \<bullet> (fst x) = fst (p \<bullet> x)"+ −
by (cases x) simp+ −
+ −
lemma snd_eqvt [eqvt]:+ −
shows "p \<bullet> (snd x) = snd (p \<bullet> x)"+ −
by (cases x) simp+ −
+ −
lemma split_eqvt [eqvt]: + −
shows "p \<bullet> (split P x) = split (p \<bullet> P) (p \<bullet> x)"+ −
unfolding split_def+ −
by (perm_simp) (rule refl)+ −
+ −
+ −
subsubsection {* Equivariance for list operations *}+ −
+ −
lemma append_eqvt [eqvt]:+ −
shows "p \<bullet> (xs @ ys) = (p \<bullet> xs) @ (p \<bullet> ys)"+ −
by (induct xs) auto+ −
+ −
lemma rev_eqvt [eqvt]:+ −
shows "p \<bullet> (rev xs) = rev (p \<bullet> xs)"+ −
by (induct xs) (simp_all add: append_eqvt)+ −
+ −
lemma map_eqvt [eqvt]: + −
shows "p \<bullet> (map f xs) = map (p \<bullet> f) (p \<bullet> xs)"+ −
by (induct xs) (simp_all, simp only: permute_fun_app_eq)+ −
+ −
lemma removeAll_eqvt [eqvt]:+ −
shows "p \<bullet> (removeAll x xs) = removeAll (p \<bullet> x) (p \<bullet> xs)"+ −
by (induct xs) (auto)+ −
+ −
lemma filter_eqvt [eqvt]:+ −
shows "p \<bullet> (filter f xs) = filter (p \<bullet> f) (p \<bullet> xs)"+ −
apply(induct xs)+ −
apply(simp)+ −
apply(simp only: filter.simps permute_list.simps if_eqvt)+ −
apply(simp only: permute_fun_app_eq)+ −
done+ −
+ −
lemma distinct_eqvt [eqvt]:+ −
shows "p \<bullet> (distinct xs) = distinct (p \<bullet> xs)"+ −
apply(induct xs)+ −
apply(simp add: permute_bool_def)+ −
apply(simp add: conj_eqvt Not_eqvt mem_eqvt set_eqvt)+ −
done+ −
+ −
lemma length_eqvt [eqvt]:+ −
shows "p \<bullet> (length xs) = length (p \<bullet> xs)"+ −
by (induct xs) (simp_all add: permute_pure)+ −
+ −
+ −
subsubsection {* Equivariance for @{typ "'a fset"} *}+ −
+ −
lemma in_fset_eqvt [eqvt]:+ −
shows "(p \<bullet> (x |\<in>| S)) = ((p \<bullet> x) |\<in>| (p \<bullet> S))"+ −
unfolding in_fset+ −
by (perm_simp) (simp)+ −
+ −
lemma union_fset_eqvt [eqvt]:+ −
shows "(p \<bullet> (S |\<union>| T)) = ((p \<bullet> S) |\<union>| (p \<bullet> T))"+ −
by (induct S) (simp_all)+ −
+ −
lemma map_fset_eqvt [eqvt]: + −
shows "p \<bullet> (map_fset f S) = map_fset (p \<bullet> f) (p \<bullet> S)"+ −
by (lifting map_eqvt)+ −
+ −
+ −
section {* Supp, Freshness and Supports *}+ −
+ −
context pt+ −
begin+ −
+ −
definition+ −
supp :: "'a \<Rightarrow> atom set"+ −
where+ −
"supp x = {a. infinite {b. (a \<rightleftharpoons> b) \<bullet> x \<noteq> x}}"+ −
+ −
definition+ −
fresh :: "atom \<Rightarrow> 'a \<Rightarrow> bool" ("_ \<sharp> _" [55, 55] 55)+ −
where + −
"a \<sharp> x \<equiv> a \<notin> supp x"+ −
+ −
end+ −
+ −
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 supp_eqvt [eqvt]:+ −
shows "p \<bullet> (supp x) = supp (p \<bullet> x)"+ −
unfolding supp_def+ −
by (perm_simp)+ −
(simp only: permute_eqvt[symmetric])+ −
+ −
lemma fresh_eqvt [eqvt]:+ −
shows "p \<bullet> (a \<sharp> x) = (p \<bullet> a) \<sharp> (p \<bullet> x)"+ −
unfolding fresh_def+ −
by (perm_simp) (rule refl)+ −
+ −
lemma fresh_permute_iff:+ −
shows "(p \<bullet> a) \<sharp> (p \<bullet> x) \<longleftrightarrow> a \<sharp> x"+ −
by (simp only: fresh_eqvt[symmetric] permute_bool_def)+ −
+ −
lemma fresh_permute_left:+ −
shows "a \<sharp> p \<bullet> x \<longleftrightarrow> - p \<bullet> a \<sharp> x"+ −
proof+ −
assume "a \<sharp> p \<bullet> x"+ −
then have "- p \<bullet> a \<sharp> - p \<bullet> p \<bullet> x" by (simp only: fresh_permute_iff)+ −
then show "- p \<bullet> a \<sharp> x" by simp+ −
next+ −
assume "- p \<bullet> a \<sharp> x"+ −
then have "p \<bullet> - p \<bullet> a \<sharp> p \<bullet> x" by (simp only: fresh_permute_iff)+ −
then show "a \<sharp> p \<bullet> x" by simp+ −
qed+ −
+ −
+ −
section {* 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: + −
fixes x::"'a::pure"+ −
shows "supp x = {}"+ −
unfolding supp_def by (simp add: permute_pure)+ −
+ −
lemma pure_fresh:+ −
fixes x::"'a::pure"+ −
shows "a \<sharp> x"+ −
unfolding fresh_def by (simp add: pure_supp)+ −
+ −
instance pure < 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+ −
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: perm_eq_iff 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)+ −
+ −
lemma plus_perm_eq:+ −
fixes p q::"perm"+ −
assumes asm: "supp p \<inter> supp q = {}"+ −
shows "p + q = q + p"+ −
unfolding perm_eq_iff+ −
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+ −
+ −
lemma supp_plus_perm_eq:+ −
fixes p q::perm+ −
assumes asm: "supp p \<inter> supp q = {}"+ −
shows "supp (p + q) = supp p \<union> supp q"+ −
proof -+ −
{ fix a::"atom"+ −
assume "a \<in> supp p"+ −
then have "a \<notin> supp q" using asm by auto+ −
then have "a \<in> supp (p + q)" using `a \<in> supp p` + −
by (simp add: supp_perm)+ −
}+ −
moreover+ −
{ fix a::"atom"+ −
assume "a \<in> supp q"+ −
then have "a \<notin> supp p" using asm by auto+ −
then have "a \<in> supp (q + p)" using `a \<in> supp q` + −
by (simp add: supp_perm)+ −
then have "a \<in> supp (p + q)" using asm plus_perm_eq+ −
by metis+ −
}+ −
ultimately have "supp p \<union> supp q \<subseteq> supp (p + q)"+ −
by blast+ −
then show "supp (p + q) = supp p \<union> supp q" using supp_plus_perm+ −
by blast+ −
qed+ −
+ −
instance perm :: fs+ −
by default (simp add: supp_perm finite_perm_lemma)+ −
+ −
+ −
+ −
section {* Finite Support instances for other types *}+ −
+ −
+ −
subsection {* Type @{typ "'a \<times> 'b"} is finitely-supported. *}+ −
+ −
lemma supp_Pair: + −
shows "supp (x, y) = supp x \<union> supp y"+ −
by (simp add: supp_def Collect_imp_eq Collect_neg_eq)+ −
+ −
lemma fresh_Pair: + −
shows "a \<sharp> (x, y) \<longleftrightarrow> a \<sharp> x \<and> a \<sharp> y"+ −
by (simp add: fresh_def supp_Pair)+ −
+ −
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 (case_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 (case_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 fresh_Nil: + −
shows "a \<sharp> []"+ −
by (simp add: fresh_def supp_Nil)+ −
+ −
lemma supp_Cons: + −
shows "supp (x # xs) = supp x \<union> supp xs"+ −
by (simp add: supp_def Collect_imp_eq Collect_neg_eq)+ −
+ −
lemma fresh_Cons: + −
shows "a \<sharp> (x # xs) \<longleftrightarrow> a \<sharp> x \<and> a \<sharp> xs"+ −
by (simp add: fresh_def supp_Cons)+ −
+ −
lemma supp_append:+ −
shows "supp (xs @ ys) = supp xs \<union> supp ys"+ −
by (induct xs) (auto simp add: supp_Nil supp_Cons)+ −
+ −
lemma fresh_append:+ −
shows "a \<sharp> (xs @ ys) \<longleftrightarrow> a \<sharp> xs \<and> a \<sharp> ys"+ −
by (induct xs) (simp_all add: fresh_Nil fresh_Cons)+ −
+ −
lemma supp_rev:+ −
shows "supp (rev xs) = supp xs"+ −
by (induct xs) (auto simp add: supp_append supp_Cons supp_Nil)+ −
+ −
lemma fresh_rev:+ −
shows "a \<sharp> rev xs \<longleftrightarrow> a \<sharp> xs"+ −
by (induct xs) (auto simp add: fresh_append fresh_Cons fresh_Nil)+ −
+ −
lemma supp_removeAll:+ −
fixes x::"atom"+ −
shows "supp (removeAll x xs) = supp xs - {x}"+ −
by (induct xs)+ −
(auto simp add: supp_Nil supp_Cons supp_atom)+ −
+ −
lemma supp_of_atom_list:+ −
fixes as::"atom list"+ −
shows "supp as = set as"+ −
by (induct as)+ −
(simp_all add: supp_Nil supp_Cons supp_atom)+ −
+ −
instance list :: (fs) 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 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+ −
+ −
+ −
subsection {* Equivariance Predicate @{text eqvt} and @{text eqvt_at}*}+ −
+ −
definition+ −
"eqvt f \<equiv> \<forall>p. p \<bullet> f = f"+ −
+ −
text {* equivariance of a function at a given argument *}+ −
+ −
definition+ −
"eqvt_at f x \<equiv> \<forall>p. p \<bullet> (f x) = f (p \<bullet> x)"+ −
+ −
lemma eqvtI:+ −
shows "(\<And>p. p \<bullet> f \<equiv> f) \<Longrightarrow> eqvt f"+ −
unfolding eqvt_def+ −
by simp+ −
+ −
lemma supp_fun_eqvt:+ −
assumes a: "eqvt f"+ −
shows "supp f = {}"+ −
using a+ −
unfolding eqvt_def+ −
unfolding supp_def + −
by simp+ −
+ −
lemma fresh_fun_eqvt_app:+ −
assumes a: "eqvt f"+ −
shows "a \<sharp> x \<Longrightarrow> a \<sharp> f x"+ −
proof -+ −
from a have "supp f = {}" by (simp add: supp_fun_eqvt)+ −
then show "a \<sharp> x \<Longrightarrow> a \<sharp> f x"+ −
unfolding fresh_def+ −
using supp_fun_app by auto+ −
qed+ −
+ −
lemma supp_fun_app_eqvt:+ −
assumes a: "eqvt f"+ −
shows "supp (f x) \<subseteq> supp x"+ −
using fresh_fun_eqvt_app[OF a]+ −
unfolding fresh_def+ −
by auto+ −
+ −
lemma supp_eqvt_at:+ −
assumes asm: "eqvt_at f x"+ −
and fin: "finite (supp x)"+ −
shows "supp (f x) \<subseteq> supp x"+ −
apply(rule supp_is_subset)+ −
unfolding supports_def+ −
unfolding fresh_def[symmetric]+ −
using asm+ −
apply(simp add: eqvt_at_def)+ −
apply(simp add: swap_fresh_fresh)+ −
apply(rule fin)+ −
done+ −
+ −
lemma finite_supp_eqvt_at:+ −
assumes asm: "eqvt_at f x"+ −
and fin: "finite (supp x)"+ −
shows "finite (supp (f x))"+ −
apply(rule finite_subset)+ −
apply(rule supp_eqvt_at[OF asm fin])+ −
apply(rule fin)+ −
done+ −
+ −
lemma fresh_eqvt_at:+ −
assumes asm: "eqvt_at f x"+ −
and fin: "finite (supp x)"+ −
and fresh: "a \<sharp> x"+ −
shows "a \<sharp> f x"+ −
using fresh+ −
unfolding fresh_def+ −
using supp_eqvt_at[OF asm fin]+ −
by auto+ −
+ −
+ −
subsection {* helper functions for nominal_functions *}+ −
+ −
lemma THE_defaultI2:+ −
assumes "P a" "\<exists>!x. P x" "\<And>x. P x \<Longrightarrow> Q x"+ −
shows "Q (THE_default d P)"+ −
by (iprover intro: assms THE_defaultI')+ −
+ −
lemma the_default_eqvt:+ −
assumes unique: "\<exists>!x. P x"+ −
shows "(p \<bullet> (THE_default d P)) = (THE_default d (p \<bullet> P))"+ −
apply(rule THE_default1_equality [symmetric])+ −
apply(rule_tac p="-p" in permute_boolE)+ −
apply(simp add: ex1_eqvt)+ −
apply(rule unique)+ −
apply(rule_tac p="-p" in permute_boolE)+ −
apply(rule subst[OF permute_fun_app_eq])+ −
apply(simp)+ −
apply(rule THE_defaultI'[OF unique])+ −
done+ −
+ −
lemma fundef_ex1_eqvt:+ −
fixes x::"'a::pt"+ −
assumes f_def: "f == (\<lambda>x::'a. THE_default d (G x))"+ −
assumes eqvt: "eqvt G"+ −
assumes ex1: "\<exists>!y. G x y"+ −
shows "(p \<bullet> (f x)) = f (p \<bullet> x)"+ −
apply(simp only: f_def)+ −
apply(subst the_default_eqvt)+ −
apply(rule ex1)+ −
using eqvt+ −
unfolding eqvt_def+ −
apply(simp add: permute_fun_app_eq)+ −
done+ −
+ −
lemma fundef_ex1_eqvt_at:+ −
fixes x::"'a::pt"+ −
assumes f_def: "f == (\<lambda>x::'a. THE_default d (G x))"+ −
assumes eqvt: "eqvt G"+ −
assumes ex1: "\<exists>!y. G x y"+ −
shows "eqvt_at f x"+ −
unfolding eqvt_at_def+ −
using assms+ −
by (auto intro: fundef_ex1_eqvt)+ −
+ −
(* fixme: polish *)+ −
lemma fundef_ex1_prop:+ −
fixes x::"'a::pt"+ −
assumes f_def: "f == (\<lambda>x::'a. THE_default d (G x))"+ −
assumes P_all: "\<And>y. G x y \<Longrightarrow> P y"+ −
assumes ex1: "\<exists>!y. G x y"+ −
shows "P (f x)"+ −
unfolding f_def+ −
using ex1+ −
apply(erule_tac ex1E)+ −
apply(rule THE_defaultI2)+ −
apply(assumption)+ −
apply(blast)+ −
apply(rule P_all)+ −
apply(assumption)+ −
done+ −
+ −
+ −
section {* Support of Finite Sets of Finitely Supported Elements *}+ −
+ −
text {* support and freshness for atom sets *}+ −
+ −
lemma supp_finite_atom_set:+ −
fixes S::"atom set"+ −
assumes "finite S"+ −
shows "supp S = S"+ −
apply(rule finite_supp_unique)+ −
apply(simp add: supports_def)+ −
apply(simp add: swap_set_not_in)+ −
apply(rule assms)+ −
apply(simp add: swap_set_in)+ −
done+ −
+ −
lemma supp_cofinite_atom_set:+ −
fixes S::"atom set"+ −
assumes "finite (UNIV - S)"+ −
shows "supp S = (UNIV - S)"+ −
apply(rule finite_supp_unique)+ −
apply(simp add: supports_def)+ −
apply(simp add: swap_set_both_in)+ −
apply(rule assms)+ −
apply(subst swap_commute)+ −
apply(simp add: swap_set_in)+ −
done+ −
+ −
lemma fresh_finite_atom_set:+ −
fixes S::"atom set"+ −
assumes "finite S"+ −
shows "a \<sharp> S \<longleftrightarrow> a \<notin> S"+ −
unfolding fresh_def+ −
by (simp add: supp_finite_atom_set[OF assms])+ −
+ −
lemma fresh_minus_atom_set:+ −
fixes S::"atom set"+ −
assumes "finite S"+ −
shows "a \<sharp> S - T \<longleftrightarrow> (a \<notin> T \<longrightarrow> a \<sharp> S)"+ −
unfolding fresh_def+ −
by (auto simp add: supp_finite_atom_set assms)+ −
+ −
lemma Union_supports_set:+ −
shows "(\<Union>x \<in> S. supp x) supports S"+ −
proof -+ −
{ fix a b+ −
have "\<forall>x \<in> S. (a \<rightleftharpoons> b) \<bullet> x = x \<Longrightarrow> (a \<rightleftharpoons> b) \<bullet> S = S"+ −
unfolding permute_set_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 eqvt: "eqvt (\<lambda>S. \<Union> supp ` S)" + −
unfolding eqvt_def+ −
by (perm_simp) (simp)+ −
have "(\<Union>x\<in>S. supp x) = supp (\<Union>x\<in>S. supp x)"+ −
by (rule supp_finite_atom_set[symmetric]) (rule Union_of_finite_supp_sets[OF fin])+ −
also have "\<dots> = supp ((\<lambda>S. \<Union> supp ` S) S)" by simp+ −
also have "\<dots> \<subseteq> supp S" using eqvt+ −
by (rule supp_fun_app_eqvt)+ −
finally show "(\<Union>x\<in>S. supp x) \<subseteq> supp S" .+ −
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)+ −
+ −
lemma fresh_finite_insert:+ −
fixes S::"('a::fs) set"+ −
assumes fin: "finite S"+ −
shows "a \<sharp> (insert x S) \<longleftrightarrow> a \<sharp> x \<and> a \<sharp> S"+ −
using fin unfolding fresh_def+ −
by (simp add: supp_of_finite_insert)+ −
+ −
lemma supp_set_empty:+ −
shows "supp {} = {}"+ −
unfolding supp_def+ −
by (simp add: empty_eqvt)+ −
+ −
lemma fresh_set_empty:+ −
shows "a \<sharp> {}"+ −
by (simp add: fresh_def supp_set_empty)+ −
+ −
lemma supp_set:+ −
fixes xs :: "('a::fs) list"+ −
shows "supp (set xs) = supp xs"+ −
apply(induct xs)+ −
apply(simp add: supp_set_empty supp_Nil)+ −
apply(simp add: supp_Cons supp_of_finite_insert)+ −
done+ −
+ −
lemma fresh_set:+ −
fixes xs :: "('a::fs) list"+ −
shows "a \<sharp> (set xs) \<longleftrightarrow> a \<sharp> xs"+ −
unfolding fresh_def+ −
by (simp add: supp_set)+ −
+ −
+ −
subsection {* Type @{typ "'a fset"} is finitely supported *}+ −
+ −
lemma supp_fset [simp]:+ −
shows "supp (fset S) = supp S"+ −
unfolding supp_def+ −
by (simp add: fset_eqvt fset_cong)+ −
+ −
lemma supp_empty_fset [simp]:+ −
shows "supp {||} = {}"+ −
unfolding supp_def+ −
by simp+ −
+ −
lemma fresh_empty_fset:+ −
shows "a \<sharp> {||}"+ −
unfolding fresh_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_of_finite_insert)+ −
done+ −
+ −
lemma fresh_insert_fset:+ −
fixes x::"'a::fs"+ −
and S::"'a fset"+ −
shows "a \<sharp> insert_fset x S \<longleftrightarrow> a \<sharp> x \<and> a \<sharp> S"+ −
unfolding fresh_def+ −
by (simp)+ −
+ −
lemma fset_finite_supp:+ −
fixes S::"('a::fs) fset"+ −
shows "finite (supp S)"+ −
by (induct S) (simp_all add: finite_supp)+ −
+ −
lemma supp_union_fset:+ −
fixes S T::"'a::fs fset"+ −
shows "supp (S |\<union>| T) = supp S \<union> supp T"+ −
by (induct S) (auto)+ −
+ −
lemma fresh_union_fset:+ −
fixes S T::"'a::fs fset"+ −
shows "a \<sharp> S |\<union>| T \<longleftrightarrow> a \<sharp> S \<and> a \<sharp> T"+ −
unfolding fresh_def+ −
by (simp add: supp_union_fset)+ −
+ −
instance fset :: (fs) fs+ −
apply (default)+ −
apply (rule fset_finite_supp)+ −
done+ −
+ −
+ −
section {* Freshness and Fresh-Star *}+ −
+ −
lemma fresh_Unit_elim: + −
shows "(a \<sharp> () \<Longrightarrow> PROP C) \<equiv> PROP C"+ −
by (simp add: fresh_Unit)+ −
+ −
lemma fresh_Pair_elim: + −
shows "(a \<sharp> (x, y) \<Longrightarrow> PROP C) \<equiv> (a \<sharp> x \<Longrightarrow> a \<sharp> y \<Longrightarrow> PROP C)"+ −
by rule (simp_all add: fresh_Pair)+ −
+ −
(* this rule needs to be added before the fresh_prodD is *)+ −
(* added to the simplifier with mksimps *) + −
lemma [simp]:+ −
shows "a \<sharp> x1 \<Longrightarrow> a \<sharp> x2 \<Longrightarrow> a \<sharp> (x1, x2)"+ −
by (simp add: fresh_Pair)+ −
+ −
lemma fresh_PairD:+ −
shows "a \<sharp> (x, y) \<Longrightarrow> a \<sharp> x"+ −
and "a \<sharp> (x, y) \<Longrightarrow> a \<sharp> y"+ −
by (simp_all add: fresh_Pair)+ −
+ −
ML {*+ −
val mksimps_pairs = (@{const_name Nominal2_Base.fresh}, @{thms fresh_PairD}) :: mksimps_pairs;+ −
*}+ −
+ −
declaration {* fn _ =>+ −
Simplifier.map_ss (fn ss => ss setmksimps (mksimps mksimps_pairs))+ −
*}+ −
+ −
text {* The fresh-star generalisation of fresh is used in strong+ −
induction principles. *}+ −
+ −
definition + −
fresh_star :: "atom set \<Rightarrow> 'a::pt \<Rightarrow> bool" ("_ \<sharp>* _" [80,80] 80)+ −
where + −
"as \<sharp>* x \<equiv> \<forall>a \<in> as. a \<sharp> x"+ −
+ −
lemma fresh_star_supp_conv:+ −
shows "supp x \<sharp>* y \<Longrightarrow> supp y \<sharp>* x"+ −
by (auto simp add: fresh_star_def fresh_def)+ −
+ −
lemma fresh_star_perm_set_conv:+ −
fixes p::"perm"+ −
assumes fresh: "as \<sharp>* p"+ −
and fin: "finite as"+ −
shows "supp p \<sharp>* as"+ −
apply(rule fresh_star_supp_conv)+ −
apply(simp add: supp_finite_atom_set fin fresh)+ −
done+ −
+ −
lemma fresh_star_atom_set_conv:+ −
assumes fresh: "as \<sharp>* bs"+ −
and fin: "finite as" "finite bs"+ −
shows "bs \<sharp>* as"+ −
using fresh+ −
unfolding fresh_star_def fresh_def+ −
by (auto simp add: supp_finite_atom_set fin)+ −
+ −
lemma atom_fresh_star_disjoint:+ −
assumes fin: "finite bs" + −
shows "as \<sharp>* bs \<longleftrightarrow> (as \<inter> bs = {})"+ −
+ −
unfolding fresh_star_def fresh_def+ −
by (auto simp add: supp_finite_atom_set fin)+ −
+ −
+ −
lemma fresh_star_Pair:+ −
shows "as \<sharp>* (x, y) = (as \<sharp>* x \<and> as \<sharp>* y)" + −
by (auto simp add: fresh_star_def fresh_Pair)+ −
+ −
lemma fresh_star_list:+ −
shows "as \<sharp>* (xs @ ys) \<longleftrightarrow> as \<sharp>* xs \<and> as \<sharp>* ys"+ −
and "as \<sharp>* (x # xs) \<longleftrightarrow> as \<sharp>* x \<and> as \<sharp>* xs"+ −
and "as \<sharp>* []"+ −
by (auto simp add: fresh_star_def fresh_Nil fresh_Cons fresh_append)+ −
+ −
lemma fresh_star_set:+ −
fixes xs::"('a::fs) list"+ −
shows "as \<sharp>* set xs \<longleftrightarrow> as \<sharp>* xs"+ −
unfolding fresh_star_def+ −
by (simp add: fresh_set)+ −
+ −
lemma fresh_star_singleton:+ −
fixes a::"atom"+ −
shows "as \<sharp>* {a} \<longleftrightarrow> as \<sharp>* a"+ −
by (simp add: fresh_star_def fresh_finite_insert fresh_set_empty)+ −
+ −
lemma fresh_star_fset:+ −
fixes xs::"('a::fs) list"+ −
shows "as \<sharp>* fset S \<longleftrightarrow> as \<sharp>* S"+ −
by (simp add: fresh_star_def fresh_def) + −
+ −
lemma fresh_star_Un:+ −
shows "(as \<union> bs) \<sharp>* x = (as \<sharp>* x \<and> bs \<sharp>* x)"+ −
by (auto simp add: fresh_star_def)+ −
+ −
lemma fresh_star_insert:+ −
shows "(insert a as) \<sharp>* x = (a \<sharp> x \<and> as \<sharp>* x)"+ −
by (auto simp add: fresh_star_def)+ −
+ −
lemma fresh_star_Un_elim:+ −
"((as \<union> bs) \<sharp>* x \<Longrightarrow> PROP C) \<equiv> (as \<sharp>* x \<Longrightarrow> bs \<sharp>* x \<Longrightarrow> PROP C)"+ −
unfolding fresh_star_def+ −
apply(rule)+ −
apply(erule meta_mp)+ −
apply(auto)+ −
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_Pair_elim: + −
shows "(a \<sharp>* (x, y) \<Longrightarrow> PROP C) \<equiv> (a \<sharp>* x \<Longrightarrow> a \<sharp>* y \<Longrightarrow> PROP C)"+ −
by (rule, simp_all add: fresh_star_Pair)+ −
+ −
lemma fresh_star_zero:+ −
shows "as \<sharp>* (0::perm)"+ −
unfolding fresh_star_def+ −
by (simp add: fresh_zero_perm)+ −
+ −
lemma fresh_star_plus:+ −
fixes p q::perm+ −
shows "\<lbrakk>a \<sharp>* p; a \<sharp>* q\<rbrakk> \<Longrightarrow> a \<sharp>* (p + q)"+ −
unfolding fresh_star_def+ −
by (simp add: fresh_plus_perm)+ −
+ −
lemma fresh_star_permute_iff:+ −
shows "(p \<bullet> a) \<sharp>* (p \<bullet> x) \<longleftrightarrow> a \<sharp>* x"+ −
unfolding fresh_star_def+ −
by (metis mem_permute_iff permute_minus_cancel(1) fresh_permute_iff)+ −
+ −
lemma fresh_star_eqvt [eqvt]:+ −
shows "p \<bullet> (as \<sharp>* x) \<longleftrightarrow> (p \<bullet> as) \<sharp>* (p \<bullet> x)"+ −
unfolding fresh_star_def+ −
by (perm_simp) (rule refl)+ −
+ −
+ −
+ −
section {* Induction principle for permutations *}+ −
+ −
lemma smaller_supp:+ −
assumes a: "a \<in> supp p"+ −
shows "supp ((p \<bullet> a \<rightleftharpoons> a) + p) \<subset> supp p"+ −
proof -+ −
have "supp ((p \<bullet> a \<rightleftharpoons> a) + p) \<subseteq> supp p"+ −
unfolding supp_perm by (auto simp add: swap_atom)+ −
moreover+ −
have "a \<notin> supp ((p \<bullet> a \<rightleftharpoons> a) + p)" by (simp add: supp_perm)+ −
then have "supp ((p \<bullet> a \<rightleftharpoons> a) + p) \<noteq> supp p" using a by auto+ −
ultimately + −
show "supp ((p \<bullet> a \<rightleftharpoons> a) + p) \<subset> supp p" by auto+ −
qed+ −
+ −
+ −
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 perm_eq_iff)+ −
then have "P p" using zero by simp+ −
}+ −
moreover+ −
{ assume "supp p \<noteq> {}"+ −
then obtain a where a0: "a \<in> supp p" by blast+ −
then have a1: "p \<bullet> a \<in> S" "a \<in> S" "sort_of (p \<bullet> a) = sort_of a" "p \<bullet> a \<noteq> a"+ −
using as by (auto simp add: supp_atom supp_perm swap_atom)+ −
let ?q = "(p \<bullet> a \<rightleftharpoons> a) + p"+ −
have a2: "supp ?q \<subset> supp p" using a0 smaller_supp by simp+ −
then have "P ?q" using ih by simp+ −
moreover+ −
have "supp ?q \<subseteq> S" using as a2 by simp+ −
ultimately have "P ((p \<bullet> a \<rightleftharpoons> a) + ?q)" using as a1 swap by simp + −
moreover + −
have "p = (p \<bullet> a \<rightleftharpoons> a) + ?q" by (simp add: perm_eq_iff)+ −
ultimately have "P p" by simp+ −
}+ −
ultimately show "P p" by blast+ −
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_struct_induct2[consumes 1, case_names zero swap plus]:+ −
assumes S: "supp p \<subseteq> S"+ −
assumes zero: "P 0"+ −
assumes swap: "\<And>a b. \<lbrakk>sort_of a = sort_of b; a \<noteq> b; a \<in> S; b \<in> S\<rbrakk> \<Longrightarrow> P (a \<rightleftharpoons> b)"+ −
assumes plus: "\<And>p1 p2. \<lbrakk>P p1; P p2; supp p1 \<subseteq> S; supp p2 \<subseteq> S\<rbrakk> \<Longrightarrow> P (p1 + p2)"+ −
shows "P p"+ −
using S+ −
by (induct p rule: perm_struct_induct)+ −
(auto intro: zero plus swap simp add: supp_swap)+ −
+ −
lemma perm_simple_struct_induct2[case_names zero swap plus]:+ −
assumes zero: "P 0"+ −
assumes swap: "\<And>a b. \<lbrakk>sort_of a = sort_of b; a \<noteq> b\<rbrakk> \<Longrightarrow> P (a \<rightleftharpoons> b)"+ −
assumes plus: "\<And>p1 p2. \<lbrakk>P p1; P p2\<rbrakk> \<Longrightarrow> P (p1 + p2)"+ −
shows "P p"+ −
by (rule_tac S="supp p" in perm_struct_induct2)+ −
(auto intro: zero swap plus)+ −
+ −
lemma supp_perm_singleton:+ −
fixes p::"perm"+ −
shows "supp p \<subseteq> {b} \<longleftrightarrow> p = 0"+ −
proof -+ −
{ assume "supp p \<subseteq> {b}"+ −
then have "p = 0"+ −
by (induct p rule: perm_struct_induct) (simp_all)+ −
}+ −
then show "supp p \<subseteq> {b} \<longleftrightarrow> p = 0" by (auto simp add: supp_zero_perm)+ −
qed+ −
+ −
lemma supp_perm_pair:+ −
fixes p::"perm"+ −
shows "supp p \<subseteq> {a, b} \<longleftrightarrow> p = 0 \<or> p = (b \<rightleftharpoons> a)"+ −
proof -+ −
{ assume "supp p \<subseteq> {a, b}"+ −
then have "p = 0 \<or> p = (b \<rightleftharpoons> a)"+ −
apply (induct p rule: perm_struct_induct) + −
apply (auto simp add: swap_cancel supp_zero_perm supp_swap)+ −
apply (simp add: swap_commute)+ −
done+ −
}+ −
then show "supp p \<subseteq> {a, b} \<longleftrightarrow> p = 0 \<or> p = (b \<rightleftharpoons> a)" + −
by (auto simp add: supp_zero_perm supp_swap split: if_splits)+ −
qed+ −
+ −
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+ −
+ −
text {* same lemma as above, but proved with a different induction principle *}+ −
lemma supp_perm_eq_test:+ −
assumes "(supp x) \<sharp>* p"+ −
shows "p \<bullet> x = x"+ −
proof -+ −
from assms have "supp p \<subseteq> {a. a \<sharp> x}"+ −
unfolding supp_perm fresh_star_def fresh_def by auto+ −
then show "p \<bullet> x = x"+ −
proof (induct p rule: perm_struct_induct2)+ −
case zero+ −
show "0 \<bullet> x = x" by simp+ −
next+ −
case (swap a b)+ −
then have "a \<sharp> x" "b \<sharp> x" by simp_all+ −
then show "(a \<rightleftharpoons> b) \<bullet> x = x" by (simp add: swap_fresh_fresh)+ −
next+ −
case (plus p1 p2)+ −
have "p1 \<bullet> x = x" "p2 \<bullet> x = x" by fact++ −
then show "(p1 + p2) \<bullet> x = x" by simp+ −
qed+ −
qed+ −
+ −
lemma perm_supp_eq:+ −
assumes a: "(supp p) \<sharp>* x"+ −
shows "p \<bullet> x = x"+ −
proof -+ −
from assms have "supp p \<subseteq> {a. a \<sharp> x}"+ −
unfolding supp_perm fresh_star_def fresh_def by auto+ −
then show "p \<bullet> x = x"+ −
proof (induct p rule: perm_struct_induct2)+ −
case zero+ −
show "0 \<bullet> x = x" by simp+ −
next+ −
case (swap a b)+ −
then have "a \<sharp> x" "b \<sharp> x" by simp_all+ −
then show "(a \<rightleftharpoons> b) \<bullet> x = x" by (simp add: swap_fresh_fresh)+ −
next+ −
case (plus p1 p2)+ −
have "p1 \<bullet> x = x" "p2 \<bullet> x = x" by fact++ −
then show "(p1 + p2) \<bullet> x = x" by simp+ −
qed+ −
qed+ −
+ −
+ −
lemma supp_perm_perm_eq:+ −
assumes a: "\<forall>a \<in> supp x. p \<bullet> a = q \<bullet> a"+ −
shows "p \<bullet> x = q \<bullet> x"+ −
proof -+ −
from a have "\<forall>a \<in> supp x. (-q + p) \<bullet> a = a" by simp+ −
then have "\<forall>a \<in> supp x. a \<notin> supp (-q + p)" + −
unfolding supp_perm by simp+ −
then have "supp x \<sharp>* (-q + p)"+ −
unfolding fresh_star_def fresh_def by simp+ −
then have "(-q + p) \<bullet> x = x" by (simp only: supp_perm_eq)+ −
then show "p \<bullet> x = q \<bullet> x"+ −
by (metis permute_minus_cancel permute_plus)+ −
qed+ −
+ −
lemma atom_set_perm_eq:+ −
assumes a: "as \<sharp>* p"+ −
shows "p \<bullet> as = as"+ −
proof -+ −
from a have "supp p \<subseteq> {a. a \<notin> as}"+ −
unfolding supp_perm fresh_star_def fresh_def by auto+ −
then show "p \<bullet> as = as"+ −
proof (induct p rule: perm_struct_induct)+ −
case zero+ −
show "0 \<bullet> as = as" by simp+ −
next+ −
case (swap p a b)+ −
then have "a \<notin> as" "b \<notin> as" "p \<bullet> as = as" by simp_all+ −
then show "((a \<rightleftharpoons> b) + p) \<bullet> as = as" by (simp add: swap_set_not_in)+ −
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) = (Xs \<union> (p \<bullet> Xs))"+ −
proof -+ −
from b c have "finite Xs" by (rule finite_subset)+ −
then show ?thesis using b+ −
proof (induct rule: finite_subset_induct)+ −
case empty+ −
have "0 \<bullet> {} \<inter> As = {}" by simp+ −
moreover+ −
have "supp (0::perm) = {} \<union> 0 \<bullet> {}" by (simp add: supp_zero_perm)+ −
ultimately show ?case by blast+ −
next+ −
case (insert x Xs)+ −
then obtain p where+ −
p1: "(p \<bullet> Xs) \<inter> As = {}" and + −
p2: "supp p = (Xs \<union> (p \<bullet> Xs))" by blast+ −
from `x \<in> As` p1 have "x \<notin> p \<bullet> Xs" by fast+ −
with `x \<notin> Xs` p2 have "x \<notin> supp p" by fast+ −
hence px: "p \<bullet> x = x" unfolding supp_perm by simp+ −
have "finite (As \<union> p \<bullet> Xs \<union> supp p)"+ −
using `finite As` `finite Xs`+ −
by (simp add: permute_set_eq_image finite_supp)+ −
then obtain y where "y \<notin> (As \<union> p \<bullet> Xs \<union> supp p)" "sort_of y = sort_of x"+ −
by (rule obtain_atom)+ −
hence y: "y \<notin> As" "y \<notin> p \<bullet> Xs" "y \<notin> supp p" "sort_of y = sort_of x"+ −
by simp_all+ −
hence py: "p \<bullet> y = y" "x \<noteq> y" using `x \<in> As`+ −
by (auto simp add: supp_perm)+ −
let ?q = "(x \<rightleftharpoons> y) + p"+ −
have q: "?q \<bullet> insert x Xs = insert y (p \<bullet> Xs)"+ −
unfolding insert_eqvt+ −
using `p \<bullet> x = x` `sort_of y = sort_of x`+ −
using `x \<notin> p \<bullet> Xs` `y \<notin> p \<bullet> Xs`+ −
by (simp add: swap_atom swap_set_not_in)+ −
have "?q \<bullet> insert x Xs \<inter> As = {}"+ −
using `y \<notin> As` `p \<bullet> Xs \<inter> As = {}`+ −
unfolding q by simp+ −
moreover+ −
have "supp (x \<rightleftharpoons> y) \<inter> supp p = {}" using px py `sort_of y = sort_of x`+ −
unfolding supp_swap by (simp add: supp_perm)+ −
then have "supp ?q = (supp (x \<rightleftharpoons> y) \<union> supp p)" + −
by (simp add: supp_plus_perm_eq)+ −
then have "supp ?q = insert x Xs \<union> ?q \<bullet> insert x Xs"+ −
using p2 `sort_of y = sort_of x` `x \<noteq> y` unfolding q supp_swap+ −
by auto+ −
ultimately show ?case by blast+ −
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) = (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_avoiding1:+ −
assumes "finite xs"+ −
and "finite (supp c)"+ −
shows "\<exists>p. (p \<bullet> xs) \<sharp>* c"+ −
using assms+ −
apply(erule_tac c="c" in at_set_avoiding)+ −
apply(auto)+ −
done+ −
+ −
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_Pair)+ −
apply(rule fresh_star_supp_conv)+ −
apply(auto simp add: fresh_star_def)+ −
done+ −
+ −
lemma at_set_avoiding3:+ −
assumes "finite xs"+ −
and "finite (supp c)" "finite (supp x)"+ −
and "xs \<sharp>* x"+ −
shows "\<exists>p. (p \<bullet> xs) \<sharp>* c \<and> supp x \<sharp>* p \<and> supp p = xs \<union> (p \<bullet> xs)"+ −
using 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_Pair)+ −
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 {* Renaming permutations *}+ −
+ −
lemma set_renaming_perm:+ −
assumes b: "finite bs"+ −
shows "\<exists>q. (\<forall>b \<in> bs. q \<bullet> b = p \<bullet> b) \<and> supp q \<subseteq> bs \<union> (p \<bullet> bs)"+ −
using b+ −
proof (induct)+ −
case empty+ −
have "(\<forall>b \<in> {}. 0 \<bullet> b = p \<bullet> b) \<and> supp (0::perm) \<subseteq> {} \<union> p \<bullet> {}"+ −
by (simp add: permute_set_eq supp_perm)+ −
then show "\<exists>q. (\<forall>b \<in> {}. q \<bullet> b = p \<bullet> b) \<and> supp q \<subseteq> {} \<union> p \<bullet> {}" by blast+ −
next+ −
case (insert a bs)+ −
then have " \<exists>q. (\<forall>b \<in> bs. q \<bullet> b = p \<bullet> b) \<and> supp q \<subseteq> bs \<union> p \<bullet> bs" by simp + −
then obtain q where *: "\<forall>b \<in> bs. q \<bullet> b = p \<bullet> b" and **: "supp q \<subseteq> bs \<union> p \<bullet> bs"+ −
by (metis empty_subsetI insert(3) supp_swap) + −
{ assume 1: "q \<bullet> a = p \<bullet> a"+ −
have "\<forall>b \<in> (insert a bs). q \<bullet> b = p \<bullet> b" using 1 * by simp+ −
moreover + −
have "supp q \<subseteq> insert a bs \<union> p \<bullet> insert a bs" + −
using ** by (auto simp add: insert_eqvt)+ −
ultimately + −
have "\<exists>q. (\<forall>b \<in> insert a bs. q \<bullet> b = p \<bullet> b) \<and> supp q \<subseteq> insert a bs \<union> p \<bullet> insert a bs" by blast+ −
}+ −
moreover+ −
{ assume 2: "q \<bullet> a \<noteq> p \<bullet> a"+ −
def q' \<equiv> "((q \<bullet> a) \<rightleftharpoons> (p \<bullet> a)) + q"+ −
have "\<forall>b \<in> insert a bs. q' \<bullet> b = p \<bullet> b" using 2 * `a \<notin> bs` unfolding q'_def+ −
by (auto simp add: swap_atom)+ −
moreover + −
{ have "{q \<bullet> a, p \<bullet> a} \<subseteq> insert a bs \<union> p \<bullet> insert a bs"+ −
using ** + −
apply (auto simp add: supp_perm insert_eqvt)+ −
apply (subgoal_tac "q \<bullet> a \<in> bs \<union> p \<bullet> bs")+ −
apply(auto)[1]+ −
apply(subgoal_tac "q \<bullet> a \<in> {a. q \<bullet> a \<noteq> a}")+ −
apply(blast)+ −
apply(simp)+ −
done+ −
then have "supp (q \<bullet> a \<rightleftharpoons> p \<bullet> a) \<subseteq> insert a bs \<union> p \<bullet> insert a bs" by (simp add: supp_swap)+ −
moreover+ −
have "supp q \<subseteq> insert a bs \<union> p \<bullet> insert a bs" + −
using ** by (auto simp add: insert_eqvt)+ −
ultimately + −
have "supp q' \<subseteq> insert a bs \<union> p \<bullet> insert a bs" + −
unfolding q'_def using supp_plus_perm by blast+ −
}+ −
ultimately + −
have "\<exists>q. (\<forall>b \<in> insert a bs. q \<bullet> b = p \<bullet> b) \<and> supp q \<subseteq> insert a bs \<union> p \<bullet> insert a bs" by blast+ −
}+ −
ultimately show "\<exists>q. (\<forall>b \<in> insert a bs. q \<bullet> b = p \<bullet> b) \<and> supp q \<subseteq> insert a bs \<union> p \<bullet> insert a bs"+ −
by blast+ −
qed+ −
+ −
lemma set_renaming_perm2:+ −
shows "\<exists>q. (\<forall>b \<in> bs. q \<bullet> b = p \<bullet> b) \<and> supp q \<subseteq> bs \<union> (p \<bullet> bs)"+ −
proof -+ −
have "finite (bs \<inter> supp p)" by (simp add: finite_supp)+ −
then obtain q + −
where *: "\<forall>b \<in> bs \<inter> supp p. q \<bullet> b = p \<bullet> b" and **: "supp q \<subseteq> (bs \<inter> supp p) \<union> (p \<bullet> (bs \<inter> supp p))"+ −
using set_renaming_perm by blast+ −
from ** have "supp q \<subseteq> bs \<union> (p \<bullet> bs)" by (auto simp add: inter_eqvt)+ −
moreover+ −
have "\<forall>b \<in> bs - supp p. q \<bullet> b = p \<bullet> b" + −
apply(auto)+ −
apply(subgoal_tac "b \<notin> supp q")+ −
apply(simp add: fresh_def[symmetric])+ −
apply(simp add: fresh_perm)+ −
apply(clarify)+ −
apply(rotate_tac 2)+ −
apply(drule subsetD[OF **])+ −
apply(simp add: inter_eqvt supp_eqvt permute_self)+ −
done+ −
ultimately have "(\<forall>b \<in> bs. q \<bullet> b = p \<bullet> b) \<and> supp q \<subseteq> bs \<union> (p \<bullet> bs)" using * by auto+ −
then show "\<exists>q. (\<forall>b \<in> bs. q \<bullet> b = p \<bullet> b) \<and> supp q \<subseteq> bs \<union> (p \<bullet> bs)" by blast+ −
qed+ −
+ −
lemma list_renaming_perm:+ −
shows "\<exists>q. (\<forall>b \<in> set bs. q \<bullet> b = p \<bullet> b) \<and> supp q \<subseteq> set bs \<union> (p \<bullet> set bs)"+ −
proof (induct bs)+ −
case (Cons a bs)+ −
then have " \<exists>q. (\<forall>b \<in> set bs. q \<bullet> b = p \<bullet> b) \<and> supp q \<subseteq> set bs \<union> p \<bullet> (set bs)" by simp+ −
then obtain q where *: "\<forall>b \<in> set bs. q \<bullet> b = p \<bullet> b" and **: "supp q \<subseteq> set bs \<union> p \<bullet> (set bs)"+ −
by (blast)+ −
{ assume 1: "a \<in> set bs"+ −
have "q \<bullet> a = p \<bullet> a" using * 1 by (induct bs) (auto)+ −
then have "\<forall>b \<in> set (a # bs). q \<bullet> b = p \<bullet> b" using * by simp + −
moreover + −
have "supp q \<subseteq> set (a # bs) \<union> p \<bullet> (set (a # bs))" using ** by (auto simp add: insert_eqvt)+ −
ultimately + −
have "\<exists>q. (\<forall>b \<in> set (a # bs). q \<bullet> b = p \<bullet> b) \<and> supp q \<subseteq> set (a # bs) \<union> p \<bullet> (set (a # bs))" by blast+ −
}+ −
moreover+ −
{ assume 2: "a \<notin> set bs"+ −
def q' \<equiv> "((q \<bullet> a) \<rightleftharpoons> (p \<bullet> a)) + q"+ −
have "\<forall>b \<in> set (a # bs). q' \<bullet> b = p \<bullet> b" + −
unfolding q'_def using 2 * `a \<notin> set bs` by (auto simp add: swap_atom)+ −
moreover + −
{ have "{q \<bullet> a, p \<bullet> a} \<subseteq> set (a # bs) \<union> p \<bullet> (set (a # bs))"+ −
using **+ −
apply (auto simp add: supp_perm insert_eqvt)+ −
apply (subgoal_tac "q \<bullet> a \<in> set bs \<union> p \<bullet> set bs")+ −
apply(auto)[1]+ −
apply(subgoal_tac "q \<bullet> a \<in> {a. q \<bullet> a \<noteq> a}")+ −
apply(blast)+ −
apply(simp)+ −
done+ −
then have "supp (q \<bullet> a \<rightleftharpoons> p \<bullet> a) \<subseteq> set (a # bs) \<union> p \<bullet> set (a # bs)" by (simp add: supp_swap)+ −
moreover+ −
have "supp q \<subseteq> set (a # bs) \<union> p \<bullet> (set (a # bs))" + −
using ** by (auto simp add: insert_eqvt)+ −
ultimately + −
have "supp q' \<subseteq> set (a # bs) \<union> p \<bullet> (set (a # bs))" + −
unfolding q'_def using supp_plus_perm by blast+ −
}+ −
ultimately + −
have "\<exists>q. (\<forall>b \<in> set (a # bs). q \<bullet> b = p \<bullet> b) \<and> supp q \<subseteq> set (a # bs) \<union> p \<bullet> (set (a # bs))" by blast+ −
}+ −
ultimately show "\<exists>q. (\<forall>b \<in> set (a # bs). q \<bullet> b = p \<bullet> b) \<and> supp q \<subseteq> set (a # bs) \<union> p \<bullet> (set (a # bs))"+ −
by blast+ −
next+ −
case Nil+ −
have "(\<forall>b \<in> set []. 0 \<bullet> b = p \<bullet> b) \<and> supp (0::perm) \<subseteq> set [] \<union> p \<bullet> set []" + −
by (simp add: supp_zero_perm)+ −
then show "\<exists>q. (\<forall>b \<in> set []. q \<bullet> b = p \<bullet> b) \<and> supp q \<subseteq> set [] \<union> p \<bullet> (set [])" by 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)"+ −
+ −
declare atom_eqvt[eqvt]+ −
+ −
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)+ −
+ −
lemma fresh_atom_at_base: + −
fixes b::"'a::at_base"+ −
shows "a \<sharp> atom b \<longleftrightarrow> a \<sharp> b"+ −
by (simp add: fresh_def supp_at_base supp_atom)+ −
+ −
lemma fresh_star_atom_at_base: + −
fixes b::"'a::at_base"+ −
shows "as \<sharp>* atom b \<longleftrightarrow> as \<sharp>* b"+ −
by (simp add: fresh_star_def fresh_atom_at_base)+ −
+ −
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 obtain_fresh':+ −
assumes fin: "finite (supp x)"+ −
obtains a::"'a::at_base" where "atom a \<sharp> x"+ −
using obtain_at_base[where X="supp x"]+ −
by (auto simp add: fresh_def fin)+ −
+ −
lemma obtain_fresh:+ −
fixes x::"'b::fs"+ −
obtains a::"'a::at_base" where "atom a \<sharp> x"+ −
by (rule obtain_fresh') (auto simp add: finite_supp)+ −
+ −
lemma supp_finite_set_at_base:+ −
assumes a: "finite S"+ −
shows "supp S = atom ` S"+ −
apply(simp add: supp_of_finite_sets[OF a])+ −
apply(simp add: supp_at_base)+ −
apply(auto)+ −
done+ −
+ −
(* FIXME + −
lemma supp_cofinite_set_at_base:+ −
assumes a: "finite (UNIV - S)"+ −
shows "supp S = atom ` (UNIV - S)"+ −
apply(rule finite_supp_unique)+ −
*)+ −
+ −
lemma fresh_finite_set_at_base:+ −
fixes a::"'a::at_base"+ −
assumes a: "finite S"+ −
shows "atom a \<sharp> S \<longleftrightarrow> a \<notin> S"+ −
unfolding fresh_def+ −
apply(simp add: supp_finite_set_at_base[OF a])+ −
apply(subst inj_image_mem_iff)+ −
apply(simp add: inj_on_def)+ −
apply(simp)+ −
done+ −
+ −
lemma fresh_at_base_permute_iff [simp]:+ −
fixes a::"'a::at_base"+ −
shows "atom (p \<bullet> a) \<sharp> p \<bullet> x \<longleftrightarrow> atom a \<sharp> x"+ −
unfolding atom_eqvt[symmetric]+ −
by (simp add: fresh_permute_iff)+ −
+ −
+ −
section {* Infrastructure for concrete atom types *}+ −
+ −
definition+ −
flip :: "'a::at_base \<Rightarrow> 'a \<Rightarrow> perm" ("'(_ \<leftrightarrow> _')")+ −
where+ −
"(a \<leftrightarrow> b) = (atom a \<rightleftharpoons> atom b)"+ −
+ −
+ −
lemma flip_self [simp]: "(a \<leftrightarrow> a) = 0"+ −
unfolding flip_def by (rule swap_self)+ −
+ −
lemma flip_commute: "(a \<leftrightarrow> b) = (b \<leftrightarrow> a)"+ −
unfolding flip_def by (rule swap_commute)+ −
+ −
lemma minus_flip [simp]: "- (a \<leftrightarrow> b) = (a \<leftrightarrow> b)"+ −
unfolding flip_def by (rule minus_swap)+ −
+ −
lemma add_flip_cancel: "(a \<leftrightarrow> b) + (a \<leftrightarrow> b) = 0"+ −
unfolding flip_def by (rule swap_cancel)+ −
+ −
lemma permute_flip_cancel [simp]: "(a \<leftrightarrow> b) \<bullet> (a \<leftrightarrow> b) \<bullet> x = x"+ −
unfolding permute_plus [symmetric] add_flip_cancel by simp+ −
+ −
lemma permute_flip_cancel2 [simp]: "(a \<leftrightarrow> b) \<bullet> (b \<leftrightarrow> a) \<bullet> x = x"+ −
by (simp add: flip_commute)+ −
+ −
lemma flip_eqvt [eqvt]: + −
fixes a b c::"'a::at_base"+ −
shows "p \<bullet> (a \<leftrightarrow> b) = (p \<bullet> a \<leftrightarrow> p \<bullet> b)"+ −
unfolding flip_def+ −
by (simp add: swap_eqvt atom_eqvt)+ −
+ −
lemma flip_at_base_simps [simp]:+ −
shows "sort_of (atom a) = sort_of (atom b) \<Longrightarrow> (a \<leftrightarrow> b) \<bullet> a = b"+ −
and "sort_of (atom a) = sort_of (atom b) \<Longrightarrow> (a \<leftrightarrow> b) \<bullet> b = a"+ −
and "\<lbrakk>a \<noteq> c; b \<noteq> c\<rbrakk> \<Longrightarrow> (a \<leftrightarrow> b) \<bullet> c = c"+ −
and "sort_of (atom a) \<noteq> sort_of (atom b) \<Longrightarrow> (a \<leftrightarrow> b) \<bullet> x = x"+ −
unfolding flip_def+ −
unfolding atom_eq_iff [symmetric]+ −
unfolding atom_eqvt [symmetric]+ −
by simp_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 only: 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 \<sharp> P" using P by (rule obtain_fresh')+ −
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, Q))" by (simp add: supp_Pair)+ −
then obtain a::'a where "atom a \<sharp> (P, Q)" by (rule obtain_fresh') + −
then have "atom a \<sharp> P" and "atom a \<sharp> Q" by (simp_all add: fresh_Pair)+ −
show ?thesis+ −
apply (subst fresh_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"+ −
+ −
+ −
section {* automatic equivariance procedure for inductive definitions *}+ −
+ −
use "nominal_eqvt.ML"+ −
+ −
+ −
+ −
end+ −