Synchronize Nominal2_Base_Exec with Nominal2_Base, equivariance for Let, avoid overloading approx twice and changes for new isabelle
(* Title: Nominal2_Base
Authors: Christian Urban, Brian Huffman, Cezary Kaliszyk
Basic definitions and lemma infrastructure for
Nominal Isabelle.
*)
theory Nominal2_Base
imports Main
"~~/src/HOL/Library/Infinite_Set"
"~~/src/HOL/Quotient_Examples/FSet"
"GPerm"
"~~/src/HOL/Library/List_lexord"
"~~/src/HOL/Library/Product_ord"
"~~/src/HOL/Library/Efficient_Nat"
"~~/src/HOL/Library/Char_ord"
"~~/src/HOL/Library/Code_Char_chr"
"~~/src/HOL/Library/Code_Char_ord"
keywords
"atom_decl" "equivariance" :: thy_decl
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 *}
definition "sort_respecting p \<longleftrightarrow> (\<forall>a. sort_of (gpermute p a) = sort_of a)"
lemma sort_respecting_0[simp]:
"sort_respecting (0\<Colon>atom gperm)"
by (simp add: sort_respecting_def)
typedef (open) perm = "{p::atom gperm. sort_respecting p}"
by (auto intro: exI[of _ "0"])
setup_lifting type_definition_perm
lemma perm_eq_rep:
"p = q \<longleftrightarrow> Rep_perm p = Rep_perm q"
by (simp add: Rep_perm_inject)
lift_definition mk_perm :: "atom gperm \<Rightarrow> perm" is
"\<lambda>p. if sort_respecting p then p else 0" by simp
lemma Rep_perm_mk_perm [simp]:
"Rep_perm (mk_perm p) = (if sort_respecting p then p else 0)"
by (simp add: mk_perm_def Abs_perm_inverse)
instance perm :: size ..
subsection {* Permutations form a (multiplicative) group *}
instantiation perm :: group_add
begin
lift_definition zero_perm :: "perm" is "0" by simp
lift_definition uminus_perm :: "perm \<Rightarrow> perm" is "uminus"
unfolding sort_respecting_def
by transfer (auto, metis perm_apply_minus)
lift_definition plus_perm :: "perm \<Rightarrow> perm \<Rightarrow> perm" is "plus"
unfolding sort_respecting_def
by transfer (simp add: perm_add_apply)
definition "(p :: perm) - q = p + - q"
lemma Rep_perm_0 [simp, code abstract]:
"Rep_perm 0 = 0"
by (metis (mono_tags) zero_perm.rep_eq)
lemma Rep_perm_uminus [simp, code abstract]:
"Rep_perm (- p) = - (Rep_perm p)"
by (metis uminus_perm.rep_eq)
lemma Rep_perm_add [simp, code abstract]:
"Rep_perm (p + q) = (Rep_perm p) + (Rep_perm q)"
by (simp add: plus_perm.rep_eq)
instance
apply default
unfolding minus_perm_def
by (transfer, simp add: add_assoc)+
end
section {* Implementation of swappings *}
lift_definition swap :: "atom \<Rightarrow> atom \<Rightarrow> perm" ("'(_ \<rightleftharpoons> _')")
is "(\<lambda>a b. (if sort_of a = sort_of b then mk_perm (gswap a b) else 0))" .
lemma sort_respecting_swap [simp]:
"sort_of a = sort_of b \<Longrightarrow> sort_respecting (gswap a b)"
unfolding sort_respecting_def
by transfer auto
lemma Rep_swap [simp, code abstract]:
"Rep_perm (swap a b) = (if sort_of a = sort_of b then gswap a b else 0)"
by (simp add: swap_def)
lemma swap_different_sorts [simp]:
"sort_of a \<noteq> sort_of b \<Longrightarrow> (a \<rightleftharpoons> b) = 0"
by (simp add: perm_eq_rep)
lemma swap_cancel:
shows "(a \<rightleftharpoons> b) + (a \<rightleftharpoons> b) = 0"
and "(a \<rightleftharpoons> b) + (b \<rightleftharpoons> a) = 0"
by (simp_all add: perm_eq_rep)
lemma swap_self [simp]:
"(a \<rightleftharpoons> a) = 0"
by (simp add: perm_eq_rep)
lemma minus_swap [simp]:
"- (a \<rightleftharpoons> b) = (a \<rightleftharpoons> b)"
by (simp add: perm_eq_rep)
lemma swap_commute:
"(a \<rightleftharpoons> b) = (b \<rightleftharpoons> a)"
by (simp add: perm_eq_rep swap_commute)
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 (simp add: perm_eq_rep swap_triple)
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 = gpermute (Rep_perm p) a"
instance
by default (simp_all add: permute_atom_def)
end
lemma sort_of_permute [simp]:
shows "sort_of (p \<bullet> a) = sort_of a"
by (metis Rep_perm mem_Collect_eq sort_respecting_def permute_atom_def)
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)"
by (auto simp add: permute_atom_def)
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 perm_eq_rep
by (simp add: gperm_eq)
subsection {* Permutations for permutations *}
instantiation perm :: pt
begin
definition
"p \<bullet> q = p + q - p"
instance
by default
(simp_all add: permute_perm_def diff_minus minus_add add_assoc)
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
by default
(simp_all add: permute_fun_def minus_add)
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
by (default)
(simp_all add: permute_bool_def)
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 *}
instantiation "set" :: (pt) pt
begin
definition
"p \<bullet> X = {p \<bullet> x | x. x \<in> X}"
instance
apply default
apply (auto simp add: permute_set_def)
done
end
lemma permute_set_eq:
shows "p \<bullet> X = {x. - p \<bullet> x \<in> X}"
unfolding permute_set_def
by (auto) (metis permute_minus_cancel(1))
lemma permute_set_eq_image:
shows "p \<bullet> X = permute p ` X"
unfolding permute_set_def by auto
lemma permute_set_eq_vimage:
shows "p \<bullet> X = permute (- p) -` X"
unfolding permute_set_eq vimage_def
by simp
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_def
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_def
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_def
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_def
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 permute_set_def
by auto
lemma empty_eqvt:
shows "p \<bullet> {} = {}"
unfolding permute_set_def
by (simp)
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
Inl_eqvt: "p \<bullet> (Inl x) = Inl (p \<bullet> x)"
| Inr_eqvt: "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
Nil_eqvt: "p \<bullet> [] = []"
| Cons_eqvt: "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
None_eqvt: "p \<bullet> None = None"
| Some_eqvt: "p \<bullet> (Some x) = Some (p \<bullet> x)"
instance
by (default) (induct_tac [!] x, simp_all)
end
subsection {* Permutations for @{typ "'a multiset"} *}
instantiation multiset :: (pt) pt
begin
definition
"p \<bullet> M = {# p \<bullet> x. x :# M #}"
instance
proof
fix M :: "'a multiset" and p q :: "perm"
show "0 \<bullet> M = M"
unfolding permute_multiset_def
by (induct_tac M) (simp_all)
show "(p + q) \<bullet> M = p \<bullet> q \<bullet> M"
unfolding permute_multiset_def
by (induct_tac M) (simp_all)
qed
end
lemma permute_multiset [simp]:
fixes M N::"('a::pt) multiset"
shows "(p \<bullet> {#}) = ({#} ::('a::pt) multiset)"
and "(p \<bullet> {# x #}) = {# p \<bullet> x #}"
and "(p \<bullet> (M + N)) = (p \<bullet> M) + (p \<bullet> N)"
unfolding permute_multiset_def
by (simp_all)
subsection {* Permutations for @{typ "'a fset"} *}
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"
by (simp add: set_eqvt[symmetric])
instance
proof
fix x :: "'a fset" and p q :: "perm"
have lst: "\<And>l :: 'a list. 0 \<bullet> l = l" by simp
show "0 \<bullet> x = x" by (lifting lst)
have lst: "\<And>p q :: perm. \<And>x :: 'a list. (p + q) \<bullet> x = p \<bullet> q \<bullet> x" by simp
show "(p + q) \<bullet> x = p \<bullet> q \<bullet> x" by (lifting lst)
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 set :: (pure) pure
by default (simp add: permute_set_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
(* multisets *)
permute_multiset
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 permute_bool_def permute_set_def
by (auto)
lemma Collect_eqvt [eqvt]:
shows "p \<bullet> {x. P x} = {x. (p \<bullet> P) x}"
unfolding permute_set_eq permute_fun_def
by (auto simp add: permute_bool_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 Image_eqvt [eqvt]:
shows "p \<bullet> (R `` A) = (p \<bullet> R) `` (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)
lemma foldr_eqvt[eqvt]:
"p \<bullet> foldr a b c = foldr (p \<bullet> a) (p \<bullet> b) (p \<bullet> c)"
apply (induct b)
apply simp_all
apply (perm_simp)
apply simp
done
(* FIXME: eqvt attribute *)
lemma Sigma_eqvt:
shows "(p \<bullet> (X \<times> Y)) = (p \<bullet> X) \<times> (p \<bullet> Y)"
unfolding Sigma_def
unfolding SUP_def
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)
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)
apply(perm_simp)
apply(rule refl)
done
end
instantiation "fun" :: (pt, inf_eqvt) inf_eqvt
begin
instance
apply(default)
unfolding Inf_fun_def INF_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)
lemma Let_eqvt [eqvt]:
"p \<bullet> Let x y = Let (p \<bullet> x) (p \<bullet> y)"
unfolding Let_def permute_fun_app_eq ..
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 option"} *}
lemma option_map_eqvt[eqvt]:
shows "p \<bullet> (Option.map f x) = Option.map (p \<bullet> f) (p \<bullet> x)"
by (cases x) (simp_all, simp add: permute_fun_app_eq)
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 inter_list_eqvt [eqvt]:
shows "p \<bullet> (inter_list S T) = inter_list (p \<bullet> S) (p \<bullet> T)"
unfolding list_eq_def inter_list_def
by perm_simp simp
lemma inter_fset_eqvt [eqvt]:
shows "(p \<bullet> (S |\<inter>| T)) = ((p \<bullet> S) |\<inter>| (p \<bullet> T))"
by (lifting inter_list_eqvt)
lemma sub_list_eqvt [eqvt]:
shows "p \<bullet> (sub_list S T) = sub_list (p \<bullet> S) (p \<bullet> T)"
unfolding sub_list_def
by perm_simp simp
lemma subset_fset_eqvt [eqvt]:
shows "(p \<bullet> (S |\<subseteq>| T)) = ((p \<bullet> S) |\<subseteq>| (p \<bullet> T))"
by (lifting sub_list_eqvt)
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 supports_fresh:
fixes x :: "'a::pt"
assumes a1: "S supports x"
and a2: "finite S"
and a3: "a \<notin> S"
shows "a \<sharp> x"
unfolding fresh_def
proof -
have "(supp x) \<subseteq> S" using a1 a2 by (rule supp_is_subset)
then show "a \<notin> (supp x)" using a3 by auto
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}"
by (rule finite_supp_unique)
(auto simp add: supports_def)
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}"
unfolding permute_atom_def
using finite_gpermute_neq .
lemma supp_perm:
shows "supp p = {a. p \<bullet> a \<noteq> a}"
apply (rule finite_supp_unique)
apply (simp_all add: perm_swap_eq swap_eqvt supports_perm finite_perm_lemma)
apply (auto simp add: perm_eq_iff swap_atom perm_swap_eq swap_eqvt)
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 supp_perm
by (simp) (metis permute_minus_cancel(1))
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
by default (auto simp add: supp_Pair finite_supp)
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"
lemma eqvt_boolI:
fixes f::"bool"
shows "eqvt f"
unfolding eqvt_def
by (simp add: permute_bool_def)
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 eqvt_at_perm:
assumes "eqvt_at f x"
shows "eqvt_at f (q \<bullet> x)"
proof -
{ fix p::"perm"
have "p \<bullet> (f (q \<bullet> x)) = p \<bullet> q \<bullet> (f x)"
using assms by (simp add: eqvt_at_def)
also have "\<dots> = (p + q) \<bullet> (f x)" by simp
also have "\<dots> = f ((p + q) \<bullet> x)"
using assms by (simp add: eqvt_at_def)
finally have "p \<bullet> (f (q \<bullet> x)) = f (p \<bullet> q \<bullet> x)" by simp }
then show "eqvt_at f (q \<bullet> x)" unfolding eqvt_at_def
by simp
qed
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 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 "\<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 (p \<bullet> 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 x) (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)
apply(rule THE_default1_equality [symmetric])
apply(rule_tac p="-p" in permute_boolE)
apply(perm_simp add: permute_minus_cancel)
using eqvt[simplified eqvt_def]
apply(simp)
apply(rule ex1)
apply(rule THE_defaultI2)
apply(rule_tac p="-p" in permute_boolE)
apply(perm_simp add: permute_minus_cancel)
apply(rule ex1)
apply(perm_simp)
using eqvt[simplified eqvt_def]
apply(simp)
done
lemma fundef_ex1_eqvt_at:
fixes x::"'a::pt"
assumes f_def: "f == (\<lambda>x::'a. THE_default (d x) (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)
lemma fundef_ex1_prop:
fixes x::"'a::pt"
assumes f_def: "f \<equiv> (\<lambda>x::'a. THE_default (d x) (G x))"
assumes P_all: "\<And>x y. G x y \<Longrightarrow> P x y"
assumes ex1: "\<exists>!y. G x y"
shows "P x (f x)"
unfolding f_def
using ex1
apply(erule_tac ex1E)
apply(rule THE_defaultI2)
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_def 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 multiset"} is finitely supported *}
lemma set_of_eqvt[eqvt]:
shows "p \<bullet> (set_of M) = set_of (p \<bullet> M)"
by (induct M) (simp_all add: insert_eqvt empty_eqvt)
lemma supp_set_of:
shows "supp (set_of M) \<subseteq> supp M"
apply (rule supp_fun_app_eqvt)
unfolding eqvt_def
apply(perm_simp)
apply(simp)
done
lemma Union_finite_multiset:
fixes M::"'a::fs multiset"
shows "finite (\<Union>{supp x | x. x \<in># M})"
proof -
have "finite (\<Union>(supp ` {x. x \<in># M}))"
by (induct M) (simp_all add: Collect_imp_eq Collect_neg_eq finite_supp)
then show "finite (\<Union>{supp x | x. x \<in># M})"
by (simp only: image_Collect)
qed
lemma Union_supports_multiset:
shows "\<Union>{supp x | x. x :# M} supports M"
proof -
have sw: "\<And>a b. ((\<And>x. x :# M \<Longrightarrow> (a \<rightleftharpoons> b) \<bullet> x = x) \<Longrightarrow> (a \<rightleftharpoons> b) \<bullet> M = M)"
unfolding permute_multiset_def
apply(induct M)
apply(simp_all)
done
show "(\<Union>{supp x | x. x :# M}) supports M"
unfolding supports_def
apply(clarify)
apply(rule sw)
apply(rule swap_fresh_fresh)
apply(simp_all only: fresh_def)
apply(auto)
apply(metis neq0_conv)+
done
qed
lemma Union_included_multiset:
fixes M::"('a::fs multiset)"
shows "(\<Union>{supp x | x. x \<in># M}) \<subseteq> supp M"
proof -
have "(\<Union>{supp x | x. x \<in># M}) = (\<Union>{supp x | x. x \<in> set_of M})" by simp
also have "... \<subseteq> (\<Union>x \<in> set_of M. supp x)" by auto
also have "... = supp (set_of M)" by (simp add: subst supp_of_finite_sets)
also have " ... \<subseteq> supp M" by (rule supp_set_of)
finally show "(\<Union>{supp x | x. x \<in># M}) \<subseteq> supp M" .
qed
lemma supp_of_multisets:
fixes M::"('a::fs multiset)"
shows "(supp M) = (\<Union>{supp x | x. x :# M})"
apply(rule subset_antisym)
apply(rule supp_is_subset)
apply(rule Union_supports_multiset)
apply(rule Union_finite_multiset)
apply(rule Union_included_multiset)
done
lemma multisets_supp_finite:
fixes M::"('a::fs multiset)"
shows "finite (supp M)"
by (simp only: supp_of_multisets Union_finite_multiset)
lemma supp_of_multiset_union:
fixes M N::"('a::fs) multiset"
shows "supp (M + N) = supp M \<union> supp N"
by (auto simp add: supp_of_multisets)
lemma supp_empty_mset [simp]:
shows "supp {#} = {}"
unfolding supp_def
by simp
instance multiset :: (fs) fs
apply (default)
apply (rule multisets_supp_finite)
done
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)
declaration {* fn _ =>
let
val mksimps_pairs = (@{const_name Nominal2_Base.fresh}, @{thms fresh_PairD}) :: mksimps_pairs
in
Simplifier.map_ss (fn ss => Simplifier.set_mksimps (mksimps mksimps_pairs) ss)
end
*}
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(1) permute_plus)
qed
text {* disagreement set *}
definition
dset :: "perm \<Rightarrow> perm \<Rightarrow> atom set"
where
"dset p q = {a::atom. p \<bullet> a \<noteq> q \<bullet> a}"
lemma ds_fresh:
assumes "dset p q \<sharp>* x"
shows "p \<bullet> x = q \<bullet> x"
using assms
unfolding dset_def fresh_star_def fresh_def
by (auto intro: supp_perm_perm_eq)
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_def)
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_def 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 auto
{ 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 sort_ineq [simp]:
assumes "sort_of (atom a) \<noteq> sort_of (atom b)"
shows "atom a \<noteq> atom b"
using assms by metis
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 "sort_of a \<noteq> sort_of (atom b) \<Longrightarrow> a \<sharp> b"
and "a \<sharp> b \<longleftrightarrow> a \<noteq> atom b"
unfolding fresh_def
apply(simp_all add: supp_at_base)
apply(metis)
done
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)
lemma if_fresh_at_base [simp]:
shows "atom a \<sharp> x \<Longrightarrow> P (if a = x then t else s) = P s"
and "atom a \<sharp> x \<Longrightarrow> P (if x = a then t else s) = P s"
by (simp_all add: fresh_at_base)
simproc_setup fresh_ineq ("x \<noteq> (y::'a::at_base)") = {* fn _ => fn ss => fn ctrm =>
let
fun first_is_neg lhs rhs [] = NONE
| first_is_neg lhs rhs (thm::thms) =
(case Thm.prop_of thm of
_ $ (@{term "HOL.Not"} $ (Const ("HOL.eq", _) $ l $ r)) =>
(if l = lhs andalso r = rhs then SOME(thm)
else if r = lhs andalso l = rhs then SOME(thm RS @{thm not_sym})
else NONE)
| _ => first_is_neg lhs rhs thms)
val simp_thms = @{thms fresh_Pair fresh_at_base atom_eq_iff}
val prems = Simplifier.prems_of ss
|> filter (fn thm => case Thm.prop_of thm of
_ $ (Const (@{const_name fresh}, _) $ _ $ _) => true | _ => false)
|> map (simplify (HOL_basic_ss addsimps simp_thms))
|> map HOLogic.conj_elims
|> flat
in
case term_of ctrm of
@{term "HOL.Not"} $ (Const ("HOL.eq", _) $ lhs $ rhs) =>
(case first_is_neg lhs rhs prems of
SOME(thm) => SOME(thm RS @{thm Eq_TrueI})
| NONE => NONE)
| _ => NONE
end
*}
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
lemma at_class_sort:
fixes s :: atom_sort
fixes Rep :: "'a \<Rightarrow> atom" and Abs :: "atom \<Rightarrow> 'a"
fixes a::"'a"
assumes type: "type_definition Rep Abs {a. sort_of a = s}"
assumes atom_def: "\<And>a. atom a = Rep a"
shows "sort_of (atom a) = s"
using atom_def type
unfolding type_definition_def by simp
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"
instantiation atom_sort :: ord begin
fun less_atom_sort where
"less_atom_sort (Sort s1 l1) (Sort s2 []) \<longleftrightarrow> s1 < s2"
| "less_atom_sort (Sort s1 []) (Sort s2 (h # t)) \<longleftrightarrow> s1 \<le> s2"
| "less_atom_sort (Sort s1 (h1 # t1)) (Sort s2 (h2 # t2)) \<longleftrightarrow> s1 < s2 \<or> s1 \<le> s2 \<and> ((less_atom_sort h1 h2) \<or> (h1 = h2 \<and> less_atom_sort (Sort s1 t1) (Sort s2 t2)))"
definition less_eq_atom_sort where
less_eq_atom_sort_def: "less_eq_atom_sort (x :: atom_sort) y \<longleftrightarrow> x < y \<or> x = y"
instance ..
end
lemma less_st_less: "(Sort s1 l1) < (Sort s2 l2) \<longleftrightarrow> s1 < s2 \<or> s1 \<le> s2 \<and> l1 < l2"
by (induct l1 l2 rule: list_induct2') auto
lemma not_as_le_as: "\<not>((x :: atom_sort) < x)"
apply (rule less_atom_sort.induct[of "\<lambda>x y. x = y \<longrightarrow> \<not>x < y" "x" "x", simplified]) ..
instance atom_sort :: linorder
proof (default, auto simp add: less_eq_atom_sort_def not_as_le_as)
fix x y :: atom_sort
assume x: "x < y" "y < x"
then show False
by (induct x y rule: less_atom_sort.induct) (case_tac l1, auto)
with x show "x = y"
by (induct x y rule: less_atom_sort.induct) (case_tac l1, auto)
next
fix x y z :: atom_sort
assume "x < y" "y < z"
then show "x < z"
apply (induct x z arbitrary: y rule: less_atom_sort.induct)
apply (case_tac [!] y) apply auto
apply (case_tac [!] list2) apply auto
apply (case_tac l1) apply auto[2]
done
next
fix x y :: atom_sort
assume x: "\<not>x < y" "y \<noteq> x"
then show "y < x"
apply (induct x y rule: less_atom_sort.induct)
apply auto
apply (case_tac [!] l1)
apply auto
done
qed
instantiation atom :: linorder begin
definition less_eq_atom where
[simp]: "less_eq_atom x y \<longleftrightarrow> sort_of x < sort_of y \<or> sort_of x \<le> sort_of y \<and> nat_of x \<le> nat_of y"
definition less_atom where
[simp]: "less_atom x y \<longleftrightarrow> sort_of x < sort_of y \<or> sort_of x \<le> sort_of y \<and> nat_of x < nat_of y"
instance apply default
apply auto
apply (case_tac x, case_tac y)
apply auto
done
end
lemma [code]:
"gpermute p = perm_apply (dest_perm p)"
apply transfer
unfolding Rel_def
by (auto, metis perm_eq_def valid_dest_perm_raw_eq(2))
instantiation perm :: equal begin
definition "equal_perm a b \<longleftrightarrow> Rep_perm a = Rep_perm b"
instance
apply default
unfolding equal_perm_def perm_eq_rep ..
end
(* Test: export_code swap in SML *)
end