theory Nominal2_Base
imports "~~/src/HOL/Library/Old_Datatype"
"~~/src/HOL/Library/Infinite_Set"
"~~/src/HOL/Quotient_Examples/FSet"
"~~/src/HOL/Library/FinFun"
keywords
"atom_decl" "equivariance" :: thy_decl
begin
declare [[typedef_overloaded]]
section {* Atoms and Sorts *}
text {* A simple implementation for atom_sorts is strings. *}
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 ⇒ atom_sort"
where
"sort_of (Atom s n) = s"
primrec
nat_of :: "atom ⇒ 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 ∉ X" "sort_of a = s"
proof -
from X have "MOST a. a ∉ X"
unfolding MOST_iff_cofinite by simp
with INFM_sort_of_eq
have "INFM a. sort_of a = s ∧ a ∉ X"
by (rule INFM_conjI)
then obtain a where "a ∉ 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 ⟷ sort_of a = sort_of b ∧ nat_of a = nat_of b"
by (induct a, induct b, simp)
section {* Sort-Respecting Permutations *}
definition
"perm ≡ {f. bij f ∧ finite {a. f a ≠ a} ∧ (∀a. sort_of (f a) = sort_of a)}"
typedef perm = "perm"
proof
show "id ∈ perm" unfolding perm_def by simp
qed
lemma permI:
assumes "bij f" and "MOST x. f x = x" and "⋀a. sort_of (f a) = sort_of a"
shows "f ∈ perm"
using assms unfolding perm_def MOST_iff_cofinite by simp
lemma perm_is_bij: "f ∈ perm ⟹ bij f"
unfolding perm_def by simp
lemma perm_is_finite: "f ∈ perm ⟹ finite {a. f a ≠ a}"
unfolding perm_def by simp
lemma perm_is_sort_respecting: "f ∈ perm ⟹ sort_of (f a) = sort_of a"
unfolding perm_def by simp
lemma perm_MOST: "f ∈ perm ⟹ MOST x. f x = x"
unfolding perm_def MOST_iff_cofinite by simp
lemma perm_id: "id ∈ perm"
unfolding perm_def by simp
lemma perm_comp:
assumes f: "f ∈ perm" and g: "g ∈ perm"
shows "(f ∘ g) ∈ perm"
apply (rule permI)
apply (rule bij_comp)
apply (rule perm_is_bij [OF g])
apply (rule perm_is_bij [OF f])
apply (rule MOST_rev_mp [OF perm_MOST [OF g]])
apply (rule MOST_rev_mp [OF perm_MOST [OF f]])
apply (simp)
apply (simp add: perm_is_sort_respecting [OF f])
apply (simp add: perm_is_sort_respecting [OF g])
done
lemma perm_inv:
assumes f: "f ∈ perm"
shows "(inv f) ∈ perm"
apply (rule permI)
apply (rule bij_imp_bij_inv)
apply (rule perm_is_bij [OF f])
apply (rule MOST_mono [OF perm_MOST [OF f]])
apply (erule subst, rule inv_f_f)
apply (rule bij_is_inj [OF perm_is_bij [OF f]])
apply (rule perm_is_sort_respecting [OF f, THEN sym, THEN trans])
apply (simp add: surj_f_inv_f [OF bij_is_surj [OF perm_is_bij [OF f]]])
done
lemma bij_Rep_perm: "bij (Rep_perm p)"
using Rep_perm [of p] unfolding perm_def by simp
lemma finite_Rep_perm: "finite {a. Rep_perm p a ≠ a}"
using Rep_perm [of p] unfolding perm_def by simp
lemma sort_of_Rep_perm: "sort_of (Rep_perm p a) = sort_of a"
using Rep_perm [of p] unfolding perm_def by simp
lemma Rep_perm_ext:
"Rep_perm p1 = Rep_perm p2 ⟹ p1 = p2"
by (simp add: fun_eq_iff Rep_perm_inject [symmetric])
instance perm :: size ..
subsection {* Permutations form a (multiplicative) group *}
instantiation perm :: group_add
begin
definition
"0 = Abs_perm id"
definition
"- p = Abs_perm (inv (Rep_perm p))"
definition
"p + q = Abs_perm (Rep_perm p ∘ Rep_perm q)"
definition
"(p1::perm) - p2 = p1 + - p2"
lemma Rep_perm_0: "Rep_perm 0 = id"
unfolding zero_perm_def
by (simp add: Abs_perm_inverse perm_id)
lemma Rep_perm_add:
"Rep_perm (p1 + p2) = Rep_perm p1 ∘ Rep_perm p2"
unfolding plus_perm_def
by (simp add: Abs_perm_inverse perm_comp Rep_perm)
lemma Rep_perm_uminus:
"Rep_perm (- p) = inv (Rep_perm p)"
unfolding uminus_perm_def
by (simp add: Abs_perm_inverse perm_inv Rep_perm)
instance
apply standard
unfolding Rep_perm_inject [symmetric]
unfolding minus_perm_def
unfolding Rep_perm_add
unfolding Rep_perm_uminus
unfolding Rep_perm_0
by (simp_all add: o_assoc inv_o_cancel [OF bij_is_inj [OF bij_Rep_perm]])
end
section {* Implementation of swappings *}
definition
swap :: "atom ⇒ atom ⇒ perm" ("'(_ ⇌ _')")
where
"(a ⇌ b) =
Abs_perm (if sort_of a = sort_of b
then (λc. if a = c then b else if b = c then a else c)
else id)"
lemma Rep_perm_swap:
"Rep_perm (a ⇌ b) =
(if sort_of a = sort_of b
then (λc. if a = c then b else if b = c then a else c)
else id)"
unfolding swap_def
apply (rule Abs_perm_inverse)
apply (rule permI)
apply (auto simp: bij_def inj_on_def surj_def)[1]
apply (rule MOST_rev_mp [OF MOST_neq(1) [of a]])
apply (rule MOST_rev_mp [OF MOST_neq(1) [of b]])
apply (simp)
apply (simp)
done
lemmas Rep_perm_simps =
Rep_perm_0
Rep_perm_add
Rep_perm_uminus
Rep_perm_swap
lemma swap_different_sorts [simp]:
"sort_of a ≠ sort_of b ⟹ (a ⇌ b) = 0"
by (rule Rep_perm_ext) (simp add: Rep_perm_simps)
lemma swap_cancel:
shows "(a ⇌ b) + (a ⇌ b) = 0"
and "(a ⇌ b) + (b ⇌ a) = 0"
by (rule_tac [!] Rep_perm_ext)
(simp_all add: Rep_perm_simps fun_eq_iff)
lemma swap_self [simp]:
"(a ⇌ a) = 0"
by (rule Rep_perm_ext, simp add: Rep_perm_simps fun_eq_iff)
lemma minus_swap [simp]:
"- (a ⇌ b) = (a ⇌ b)"
by (rule minus_unique [OF swap_cancel(1)])
lemma swap_commute:
"(a ⇌ b) = (b ⇌ a)"
by (rule Rep_perm_ext)
(simp add: Rep_perm_swap fun_eq_iff)
lemma swap_triple:
assumes "a ≠ b" and "c ≠ b"
assumes "sort_of a = sort_of b" "sort_of b = sort_of c"
shows "(a ⇌ c) + (b ⇌ c) + (a ⇌ c) = (a ⇌ b)"
using assms
by (rule_tac Rep_perm_ext)
(auto simp: Rep_perm_simps fun_eq_iff)
section {* Permutation Types *}
text {*
Infix syntax for @{text permute} has higher precedence than
addition, but lower than unary minus.
*}
class pt =
fixes permute :: "perm ⇒ 'a ⇒ 'a" ("_ ∙ _" [76, 75] 75)
assumes permute_zero [simp]: "0 ∙ x = x"
assumes permute_plus [simp]: "(p + q) ∙ x = p ∙ (q ∙ x)"
begin
lemma permute_diff [simp]:
shows "(p - q) ∙ x = p ∙ - q ∙ x"
using permute_plus [of p "- q" x] by simp
lemma permute_minus_cancel [simp]:
shows "p ∙ - p ∙ x = x"
and "- p ∙ p ∙ x = x"
unfolding permute_plus [symmetric] by simp_all
lemma permute_swap_cancel [simp]:
shows "(a ⇌ b) ∙ (a ⇌ b) ∙ x = x"
unfolding permute_plus [symmetric]
by (simp add: swap_cancel)
lemma permute_swap_cancel2 [simp]:
shows "(a ⇌ b) ∙ (b ⇌ a) ∙ 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 ∙ x = p ∙ y ⟷ x = y"
by (rule inj_permute [THEN inj_eq])
end
subsection {* Permutations for atoms *}
instantiation atom :: pt
begin
definition
"p ∙ a = (Rep_perm p) a"
instance
apply standard
apply(simp_all add: permute_atom_def Rep_perm_simps)
done
end
lemma sort_of_permute [simp]:
shows "sort_of (p ∙ a) = sort_of a"
unfolding permute_atom_def by (rule sort_of_Rep_perm)
lemma swap_atom:
shows "(a ⇌ b) ∙ c =
(if sort_of a = sort_of b
then (if c = a then b else if c = b then a else c) else c)"
unfolding permute_atom_def
by (simp add: Rep_perm_swap)
lemma swap_atom_simps [simp]:
"sort_of a = sort_of b ⟹ (a ⇌ b) ∙ a = b"
"sort_of a = sort_of b ⟹ (a ⇌ b) ∙ b = a"
"c ≠ a ⟹ c ≠ b ⟹ (a ⇌ b) ∙ c = c"
unfolding swap_atom by simp_all
lemma perm_eq_iff:
fixes p q :: "perm"
shows "p = q ⟷ (∀a::atom. p ∙ a = q ∙ a)"
unfolding permute_atom_def
by (metis Rep_perm_ext ext)
subsection {* Permutations for permutations *}
instantiation perm :: pt
begin
definition
"p ∙ q = p + q - p"
instance
apply standard
apply (simp add: permute_perm_def)
apply (simp add: permute_perm_def algebra_simps)
done
end
lemma permute_self:
shows "p ∙ p = p"
unfolding permute_perm_def
by (simp add: add.assoc)
lemma pemute_minus_self:
shows "- p ∙ p = p"
unfolding permute_perm_def
by (simp add: add.assoc)
subsection {* Permutations for functions *}
instantiation "fun" :: (pt, pt) pt
begin
definition
"p ∙ f = (λx. p ∙ (f (- p ∙ x)))"
instance
apply standard
apply (simp add: permute_fun_def)
apply (simp add: permute_fun_def minus_add)
done
end
lemma permute_fun_app_eq:
shows "p ∙ (f x) = (p ∙ f) (p ∙ x)"
unfolding permute_fun_def by simp
lemma permute_fun_comp:
shows "p ∙ f = (permute p) o f o (permute (-p))"
by (simp add: comp_def permute_fun_def)
subsection {* Permutations for booleans *}
instantiation bool :: pt
begin
definition "p ∙ (b::bool) = b"
instance
apply standard
apply(simp_all add: permute_bool_def)
done
end
lemma permute_boolE:
fixes P::"bool"
shows "p ∙ P ⟹ P"
by (simp add: permute_bool_def)
lemma permute_boolI:
fixes P::"bool"
shows "P ⟹ p ∙ P"
by(simp add: permute_bool_def)
subsection {* Permutations for sets *}
instantiation "set" :: (pt) pt
begin
definition
"p ∙ X = {p ∙ x | x. x ∈ X}"
instance
apply standard
apply (auto simp: permute_set_def)
done
end
lemma permute_set_eq:
shows "p ∙ X = {x. - p ∙ x ∈ X}"
unfolding permute_set_def
by (auto) (metis permute_minus_cancel(1))
lemma permute_set_eq_image:
shows "p ∙ X = permute p ` X"
unfolding permute_set_def by auto
lemma permute_set_eq_vimage:
shows "p ∙ X = permute (- p) -` X"
unfolding permute_set_eq vimage_def
by simp
lemma permute_finite [simp]:
shows "finite (p ∙ X) = finite X"
unfolding permute_set_eq_vimage
using bij_permute by (rule finite_vimage_iff)
lemma swap_set_not_in:
assumes a: "a ∉ S" "b ∉ S"
shows "(a ⇌ b) ∙ S = S"
unfolding permute_set_def
using a by (auto simp: swap_atom)
lemma swap_set_in:
assumes a: "a ∈ S" "b ∉ S" "sort_of a = sort_of b"
shows "(a ⇌ b) ∙ S ≠ S"
unfolding permute_set_def
using a by (auto simp: swap_atom)
lemma swap_set_in_eq:
assumes a: "a ∈ S" "b ∉ S" "sort_of a = sort_of b"
shows "(a ⇌ b) ∙ S = (S - {a}) ∪ {b}"
unfolding permute_set_def
using a by (auto simp: swap_atom)
lemma swap_set_both_in:
assumes a: "a ∈ S" "b ∈ S"
shows "(a ⇌ b) ∙ S = S"
unfolding permute_set_def
using a by (auto simp: swap_atom)
lemma mem_permute_iff:
shows "(p ∙ x) ∈ (p ∙ X) ⟷ x ∈ X"
unfolding permute_set_def
by auto
lemma empty_eqvt:
shows "p ∙ {} = {}"
unfolding permute_set_def
by (simp)
lemma insert_eqvt:
shows "p ∙ (insert x A) = insert (p ∙ x) (p ∙ A)"
unfolding permute_set_eq_image image_insert ..
subsection {* Permutations for @{typ unit} *}
instantiation unit :: pt
begin
definition "p ∙ (u::unit) = u"
instance
by standard (simp_all add: permute_unit_def)
end
subsection {* Permutations for products *}
instantiation prod :: (pt, pt) pt
begin
primrec
permute_prod
where
Pair_eqvt: "p ∙ (x, y) = (p ∙ x, p ∙ y)"
instance
by standard auto
end
subsection {* Permutations for sums *}
instantiation sum :: (pt, pt) pt
begin
primrec
permute_sum
where
Inl_eqvt: "p ∙ (Inl x) = Inl (p ∙ x)"
| Inr_eqvt: "p ∙ (Inr y) = Inr (p ∙ y)"
instance
by standard (case_tac [!] x, simp_all)
end
subsection {* Permutations for @{typ "'a list"} *}
instantiation list :: (pt) pt
begin
primrec
permute_list
where
Nil_eqvt: "p ∙ [] = []"
| Cons_eqvt: "p ∙ (x # xs) = p ∙ x # p ∙ xs"
instance
by standard (induct_tac [!] x, simp_all)
end
lemma set_eqvt:
shows "p ∙ (set xs) = set (p ∙ 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 ∙ None = None"
| Some_eqvt: "p ∙ (Some x) = Some (p ∙ x)"
instance
by standard (induct_tac [!] x, simp_all)
end
subsection {* Permutations for @{typ "'a multiset"} *}
instantiation multiset :: (pt) pt
begin
definition
"p ∙ M = {# p ∙ x. x :# M #}"
instance
proof
fix M :: "'a multiset" and p q :: "perm"
show "0 ∙ M = M"
unfolding permute_multiset_def
by (induct_tac M) (simp_all)
show "(p + q) ∙ M = p ∙ q ∙ 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 ∙ {#}) = ({#} ::('a::pt) multiset)"
and "(p ∙ {# x #}) = {# p ∙ x #}"
and "(p ∙ (M + N)) = (p ∙ M) + (p ∙ N)"
unfolding permute_multiset_def
by (simp_all)
subsection {* Permutations for @{typ "'a fset"} *}
lemma permute_fset_rsp[quot_respect]:
shows "(op = ===> list_eq ===> list_eq) permute permute"
unfolding rel_fun_def
by (simp add: set_eqvt[symmetric])
instantiation fset :: (pt) pt
begin
quotient_definition
"permute_fset :: perm ⇒ 'a fset ⇒ 'a fset"
is
"permute :: perm ⇒ 'a list ⇒ 'a list"
by (simp add: set_eqvt[symmetric])
instance
proof
fix x :: "'a fset" and p q :: "perm"
show "0 ∙ x = x" by (descending) (simp)
show "(p + q) ∙ x = p ∙ q ∙ x" by (descending) (simp)
qed
end
lemma permute_fset [simp]:
fixes S::"('a::pt) fset"
shows "(p ∙ {||}) = ({||} ::('a::pt) fset)"
and "(p ∙ insert_fset x S) = insert_fset (p ∙ x) (p ∙ S)"
by (lifting permute_list.simps)
lemma fset_eqvt:
shows "p ∙ (fset S) = fset (p ∙ S)"
by (lifting set_eqvt)
subsection {* Permutations for @{typ "('a, 'b) finfun"} *}
instantiation finfun :: (pt, pt) pt
begin
lift_definition
permute_finfun :: "perm ⇒ ('a, 'b) finfun ⇒ ('a, 'b) finfun"
is
"permute :: perm ⇒ ('a ⇒ 'b) ⇒ ('a ⇒ 'b)"
apply(simp add: permute_fun_comp)
apply(rule finfun_right_compose)
apply(rule finfun_left_compose)
apply(assumption)
apply(simp)
done
instance
apply standard
apply(transfer)
apply(simp)
apply(transfer)
apply(simp)
done
end
subsection {* Permutations for @{typ char}, @{typ nat}, and @{typ int} *}
instantiation char :: pt
begin
definition "p ∙ (c::char) = c"
instance
by standard (simp_all add: permute_char_def)
end
instantiation nat :: pt
begin
definition "p ∙ (n::nat) = n"
instance
by standard (simp_all add: permute_nat_def)
end
instantiation int :: pt
begin
definition "p ∙ (i::int) = i"
instance
by standard (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 ∙ 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 standard (simp add: permute_fun_def permute_pure)
instance set :: (pure) pure
by standard (simp add: permute_set_def permute_pure)
instance prod :: (pure, pure) pure
by standard (induct_tac x, simp add: permute_pure)
instance sum :: (pure, pure) pure
by standard (induct_tac x, simp_all add: permute_pure)
instance list :: (pure) pure
by standard (induct_tac x, simp_all add: permute_pure)
instance option :: (pure) pure
by standard (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 *}
ML_file "nominal_basics.ML"
subsection {* Eqvt infrastructure *}
text {* Setup of the theorem attributes @{text eqvt} and @{text eqvt_raw}. *}
ML_file "nominal_thmdecls.ML"
lemmas [eqvt] =
permute_prod.simps
permute_list.simps
permute_option.simps
permute_sum.simps
empty_eqvt insert_eqvt set_eqvt
permute_fset fset_eqvt
permute_multiset
subsection {* perm_simp infrastructure *}
definition
"unpermute p = permute (- p)"
lemma eqvt_apply:
fixes f :: "'a::pt ⇒ 'b::pt"
and x :: "'a::pt"
shows "p ∙ (f x) ≡ (p ∙ f) (p ∙ x)"
unfolding permute_fun_def by simp
lemma eqvt_lambda:
fixes f :: "'a::pt ⇒ 'b::pt"
shows "p ∙ f ≡ (λx. p ∙ (f (unpermute p x)))"
unfolding permute_fun_def unpermute_def by simp
lemma eqvt_bound:
shows "p ∙ unpermute p x ≡ x"
unfolding unpermute_def by simp
text {* provides perm_simp methods *}
ML_file "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. *}
simproc_setup perm_simproc ("p ∙ t") = {* fn _ => fn ctxt => fn ctrm =>
case Thm.term_of (Thm.dest_arg ctrm) of
Free _ => NONE
| Var _ => NONE
| Const (@{const_name permute}, _) $ _ $ _ => NONE
| _ =>
let
val thm = Nominal_Permeq.eqvt_conv ctxt Nominal_Permeq.eqvt_strict_config ctrm
handle ERROR _ => Thm.reflexive ctrm
in
if Thm.is_reflexive thm then NONE else SOME(thm)
end
*}
subsubsection {* Equivariance for permutations and swapping *}
lemma permute_eqvt:
shows "p ∙ (q ∙ x) = (p ∙ q) ∙ (p ∙ x)"
unfolding permute_perm_def by simp
lemma permute_eqvt_raw [eqvt_raw]:
shows "p ∙ permute ≡ permute"
apply(simp add: fun_eq_iff permute_fun_def)
apply(subst permute_eqvt)
apply(simp)
done
lemma zero_perm_eqvt [eqvt]:
shows "p ∙ (0::perm) = 0"
unfolding permute_perm_def by simp
lemma add_perm_eqvt [eqvt]:
fixes p p1 p2 :: perm
shows "p ∙ (p1 + p2) = p ∙ p1 + p ∙ p2"
unfolding permute_perm_def
by (simp add: perm_eq_iff)
lemma swap_eqvt [eqvt]:
shows "p ∙ (a ⇌ b) = (p ∙ a ⇌ p ∙ b)"
unfolding permute_perm_def
by (auto simp: swap_atom perm_eq_iff)
lemma uminus_eqvt [eqvt]:
fixes p q::"perm"
shows "p ∙ (- q) = - (p ∙ q)"
unfolding permute_perm_def
by (simp add: diff_add_eq_diff_diff_swap)
subsubsection {* Equivariance of Logical Operators *}
lemma eq_eqvt [eqvt]:
shows "p ∙ (x = y) ⟷ (p ∙ x) = (p ∙ y)"
unfolding permute_eq_iff permute_bool_def ..
lemma Not_eqvt [eqvt]:
shows "p ∙ (¬ A) ⟷ ¬ (p ∙ A)"
by (simp add: permute_bool_def)
lemma conj_eqvt [eqvt]:
shows "p ∙ (A ∧ B) ⟷ (p ∙ A) ∧ (p ∙ B)"
by (simp add: permute_bool_def)
lemma imp_eqvt [eqvt]:
shows "p ∙ (A ⟶ B) ⟷ (p ∙ A) ⟶ (p ∙ B)"
by (simp add: permute_bool_def)
declare imp_eqvt[folded HOL.induct_implies_def, eqvt]
lemma all_eqvt [eqvt]:
shows "p ∙ (∀x. P x) = (∀x. (p ∙ P) x)"
unfolding All_def
by (perm_simp) (rule refl)
declare all_eqvt[folded HOL.induct_forall_def, eqvt]
lemma ex_eqvt [eqvt]:
shows "p ∙ (∃x. P x) = (∃x. (p ∙ P) x)"
unfolding Ex_def
by (perm_simp) (rule refl)
lemma ex1_eqvt [eqvt]:
shows "p ∙ (∃!x. P x) = (∃!x. (p ∙ P) x)"
unfolding Ex1_def
by (perm_simp) (rule refl)
lemma if_eqvt [eqvt]:
shows "p ∙ (if b then x else y) = (if p ∙ b then p ∙ x else p ∙ y)"
by (simp add: permute_fun_def permute_bool_def)
lemma True_eqvt [eqvt]:
shows "p ∙ True = True"
unfolding permute_bool_def ..
lemma False_eqvt [eqvt]:
shows "p ∙ False = False"
unfolding permute_bool_def ..
lemma disj_eqvt [eqvt]:
shows "p ∙ (A ∨ B) ⟷ (p ∙ A) ∨ (p ∙ B)"
by (simp add: permute_bool_def)
lemma all_eqvt2:
shows "p ∙ (∀x. P x) = (∀x. p ∙ P (- p ∙ x))"
by (perm_simp add: permute_minus_cancel) (rule refl)
lemma ex_eqvt2:
shows "p ∙ (∃x. P x) = (∃x. p ∙ P (- p ∙ x))"
by (perm_simp add: permute_minus_cancel) (rule refl)
lemma ex1_eqvt2:
shows "p ∙ (∃!x. P x) = (∃!x. p ∙ P (- p ∙ x))"
by (perm_simp add: permute_minus_cancel) (rule refl)
lemma the_eqvt:
assumes unique: "∃!x. P x"
shows "(p ∙ (THE x. P x)) = (THE x. (p ∙ 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: "∃!x. P x"
shows "(p ∙ (THE x. P x)) = (THE x. p ∙ P (- p ∙ x))"
apply(rule the1_equality [symmetric])
apply(simp only: 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 ∙ (x ∈ A) ⟷ (p ∙ x) ∈ (p ∙ A)"
unfolding permute_bool_def permute_set_def
by (auto)
lemma Collect_eqvt [eqvt]:
shows "p ∙ {x. P x} = {x. (p ∙ P) x}"
unfolding permute_set_eq permute_fun_def
by (auto simp: permute_bool_def)
lemma inter_eqvt [eqvt]:
shows "p ∙ (A ∩ B) = (p ∙ A) ∩ (p ∙ B)"
unfolding Int_def by simp
lemma Bex_eqvt [eqvt]:
shows "p ∙ (∃x ∈ S. P x) = (∃x ∈ (p ∙ S). (p ∙ P) x)"
unfolding Bex_def by simp
lemma Ball_eqvt [eqvt]:
shows "p ∙ (∀x ∈ S. P x) = (∀x ∈ (p ∙ S). (p ∙ P) x)"
unfolding Ball_def by simp
lemma image_eqvt [eqvt]:
shows "p ∙ (f ` A) = (p ∙ f) ` (p ∙ A)"
unfolding image_def by simp
lemma Image_eqvt [eqvt]:
shows "p ∙ (R `` A) = (p ∙ R) `` (p ∙ A)"
unfolding Image_def by simp
lemma UNIV_eqvt [eqvt]:
shows "p ∙ UNIV = UNIV"
unfolding UNIV_def
by (perm_simp) (rule refl)
lemma union_eqvt [eqvt]:
shows "p ∙ (A ∪ B) = (p ∙ A) ∪ (p ∙ B)"
unfolding Un_def by simp
lemma UNION_eqvt [eqvt]:
shows "p ∙ (UNION A f) = (UNION (p ∙ A) (p ∙ f))"
unfolding UNION_eq
by (perm_simp) (simp)
lemma Diff_eqvt [eqvt]:
fixes A B :: "'a::pt set"
shows "p ∙ (A - B) = (p ∙ A) - (p ∙ B)"
unfolding set_diff_eq by simp
lemma Compl_eqvt [eqvt]:
fixes A :: "'a::pt set"
shows "p ∙ (- A) = - (p ∙ A)"
unfolding Compl_eq_Diff_UNIV by simp
lemma subset_eqvt [eqvt]:
shows "p ∙ (S ⊆ T) ⟷ (p ∙ S) ⊆ (p ∙ T)"
unfolding subset_eq by simp
lemma psubset_eqvt [eqvt]:
shows "p ∙ (S ⊂ T) ⟷ (p ∙ S) ⊂ (p ∙ T)"
unfolding psubset_eq by simp
lemma vimage_eqvt [eqvt]:
shows "p ∙ (f -` A) = (p ∙ f) -` (p ∙ A)"
unfolding vimage_def by simp
lemma Union_eqvt [eqvt]:
shows "p ∙ (⋃ S) = ⋃ (p ∙ S)"
unfolding Union_eq by simp
lemma Inter_eqvt [eqvt]:
shows "p ∙ (⋂ S) = ⋂ (p ∙ S)"
unfolding Inter_eq by simp
thm foldr.simps
lemma foldr_eqvt[eqvt]:
"p ∙ foldr f xs = foldr (p ∙ f) (p ∙ xs)"
apply(induct xs)
apply(simp_all)
apply(perm_simp exclude: foldr)
apply(simp)
done
lemma Sigma_eqvt:
shows "(p ∙ (X × Y)) = (p ∙ X) × (p ∙ 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 ∙ (x ≤ y) = ((p ∙ x) ≤ (p ∙ (y::('a::{pt, order}))))"
class inf_eqvt = Inf +
assumes inf_eqvt [eqvt]: "p ∙ (Inf X) = Inf (p ∙ (X::('a::{pt, complete_lattice}) set))"
instantiation bool :: le_eqvt
begin
instance
apply standard
unfolding le_bool_def
apply(perm_simp)
apply(rule refl)
done
end
instantiation "fun" :: (pt, le_eqvt) le_eqvt
begin
instance
apply standard
unfolding le_fun_def
apply(perm_simp)
apply(rule refl)
done
end
instantiation bool :: inf_eqvt
begin
instance
apply standard
unfolding Inf_bool_def
apply(perm_simp)
apply(rule refl)
done
end
instantiation "fun" :: (pt, inf_eqvt) inf_eqvt
begin
instance
apply standard
unfolding Inf_fun_def INF_def
apply(perm_simp)
apply(rule refl)
done
end
lemma lfp_eqvt [eqvt]:
fixes F::"('a ⇒ 'b) ⇒ ('a::pt ⇒ 'b::{inf_eqvt, le_eqvt})"
shows "p ∙ (lfp F) = lfp (p ∙ F)"
unfolding lfp_def
by simp
lemma finite_eqvt [eqvt]:
shows "p ∙ finite A = finite (p ∙ A)"
unfolding finite_def
by simp
lemma fun_upd_eqvt[eqvt]:
shows "p ∙ (f(x := y)) = (p ∙ f)((p ∙ x) := (p ∙ y))"
unfolding fun_upd_def
by simp
lemma comp_eqvt [eqvt]:
shows "p ∙ (f ∘ g) = (p ∙ f) ∘ (p ∙ g)"
unfolding comp_def
by simp
subsubsection {* Equivariance for product operations *}
lemma fst_eqvt [eqvt]:
shows "p ∙ (fst x) = fst (p ∙ x)"
by (cases x) simp
lemma snd_eqvt [eqvt]:
shows "p ∙ (snd x) = snd (p ∙ x)"
by (cases x) simp
lemma split_eqvt [eqvt]:
shows "p ∙ (case_prod P x) = case_prod (p ∙ P) (p ∙ x)"
unfolding split_def
by simp
subsubsection {* Equivariance for list operations *}
lemma append_eqvt [eqvt]:
shows "p ∙ (xs @ ys) = (p ∙ xs) @ (p ∙ ys)"
by (induct xs) auto
lemma rev_eqvt [eqvt]:
shows "p ∙ (rev xs) = rev (p ∙ xs)"
by (induct xs) (simp_all add: append_eqvt)
lemma map_eqvt [eqvt]:
shows "p ∙ (map f xs) = map (p ∙ f) (p ∙ xs)"
by (induct xs) (simp_all)
lemma removeAll_eqvt [eqvt]:
shows "p ∙ (removeAll x xs) = removeAll (p ∙ x) (p ∙ xs)"
by (induct xs) (auto)
lemma filter_eqvt [eqvt]:
shows "p ∙ (filter f xs) = filter (p ∙ f) (p ∙ 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 ∙ (distinct xs) = distinct (p ∙ 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 ∙ (length xs) = length (p ∙ xs)"
by (induct xs) (simp_all add: permute_pure)
subsubsection {* Equivariance for @{typ "'a option"} *}
lemma map_option_eqvt[eqvt]:
shows "p ∙ (map_option f x) = map_option (p ∙ f) (p ∙ x)"
by (cases x) (simp_all)
subsubsection {* Equivariance for @{typ "'a fset"} *}
lemma in_fset_eqvt [eqvt]:
shows "(p ∙ (x |∈| S)) = ((p ∙ x) |∈| (p ∙ S))"
unfolding in_fset by simp
lemma union_fset_eqvt [eqvt]:
shows "(p ∙ (S |∪| T)) = ((p ∙ S) |∪| (p ∙ T))"
by (induct S) (simp_all)
lemma inter_fset_eqvt [eqvt]:
shows "(p ∙ (S |∩| T)) = ((p ∙ S) |∩| (p ∙ T))"
apply(descending)
unfolding list_eq_def inter_list_def
apply(simp)
done
lemma subset_fset_eqvt [eqvt]:
shows "(p ∙ (S |⊆| T)) = ((p ∙ S) |⊆| (p ∙ T))"
apply(descending)
unfolding sub_list_def
apply(simp)
done
lemma map_fset_eqvt [eqvt]:
shows "p ∙ (map_fset f S) = map_fset (p ∙ f) (p ∙ S)"
by (lifting map_eqvt)
subsubsection {* Equivariance for @{typ "('a, 'b) finfun"} *}
lemma finfun_update_eqvt [eqvt]:
shows "(p ∙ (finfun_update f a b)) = finfun_update (p ∙ f) (p ∙ a) (p ∙ b)"
by (transfer) (simp)
lemma finfun_const_eqvt [eqvt]:
shows "(p ∙ (finfun_const b)) = finfun_const (p ∙ b)"
by (transfer) (simp)
lemma finfun_apply_eqvt [eqvt]:
shows "(p ∙ (finfun_apply f b)) = finfun_apply (p ∙ f) (p ∙ b)"
by (transfer) (simp)
section {* Supp, Freshness and Supports *}
context pt
begin
definition
supp :: "'a ⇒ atom set"
where
"supp x = {a. infinite {b. (a ⇌ b) ∙ x ≠ x}}"
definition
fresh :: "atom ⇒ 'a ⇒ bool" ("_ ♯ _" [55, 55] 55)
where
"a ♯ x ≡ a ∉ supp x"
end
lemma supp_conv_fresh:
shows "supp x = {a. ¬ a ♯ 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 ⇌ c) ∙ x = x"
assumes "(b ⇌ c) ∙ x = x"
shows "(a ⇌ b) ∙ x = x"
proof (cases)
assume "a = b ∨ c = b"
with assms show "(a ⇌ b) ∙ x = x" by auto
next
assume *: "¬ (a = b ∨ c = b)"
have "((a ⇌ c) + (b ⇌ c) + (a ⇌ c)) ∙ x = x"
using assms by simp
also have "(a ⇌ c) + (b ⇌ c) + (a ⇌ c) = (a ⇌ b)"
using assms * by (simp add: swap_triple)
finally show "(a ⇌ b) ∙ x = x" .
qed
lemma swap_fresh_fresh:
assumes a: "a ♯ x"
and b: "b ♯ x"
shows "(a ⇌ b) ∙ x = x"
proof (cases)
assume asm: "sort_of a = sort_of b"
have "finite {c. (a ⇌ c) ∙ x ≠ x}" "finite {c. (b ⇌ c) ∙ x ≠ x}"
using a b unfolding fresh_def supp_def by simp_all
then have "finite ({c. (a ⇌ c) ∙ x ≠ x} ∪ {c. (b ⇌ c) ∙ x ≠ x})" by simp
then obtain c
where "(a ⇌ c) ∙ x = x" "(b ⇌ c) ∙ x = x" "sort_of c = sort_of b"
by (rule obtain_atom) (auto)
then show "(a ⇌ b) ∙ x = x" using asm by (rule_tac swap_rel_trans) (simp_all)
next
assume "sort_of a ≠ sort_of b"
then show "(a ⇌ b) ∙ x = x" by simp
qed
subsection {* supp and fresh are equivariant *}
lemma supp_eqvt [eqvt]:
shows "p ∙ (supp x) = supp (p ∙ x)"
unfolding supp_def by simp
lemma fresh_eqvt [eqvt]:
shows "p ∙ (a ♯ x) = (p ∙ a) ♯ (p ∙ x)"
unfolding fresh_def by simp
lemma fresh_permute_iff:
shows "(p ∙ a) ♯ (p ∙ x) ⟷ a ♯ x"
by (simp only: fresh_eqvt[symmetric] permute_bool_def)
lemma fresh_permute_left:
shows "a ♯ p ∙ x ⟷ - p ∙ a ♯ x"
proof
assume "a ♯ p ∙ x"
then have "- p ∙ a ♯ - p ∙ p ∙ x" by (simp only: fresh_permute_iff)
then show "- p ∙ a ♯ x" by simp
next
assume "- p ∙ a ♯ x"
then have "p ∙ - p ∙ a ♯ p ∙ x" by (simp only: fresh_permute_iff)
then show "a ♯ p ∙ x" by simp
qed
section {* supports *}
definition
supports :: "atom set ⇒ 'a::pt ⇒ bool" (infixl "supports" 80)
where
"S supports x ≡ ∀a b. (a ∉ S ∧ b ∉ S ⟶ (a ⇌ b) ∙ 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) ⊆ S"
proof (rule ccontr)
assume "¬ (supp x ⊆ S)"
then obtain a where b1: "a ∈ supp x" and b2: "a ∉ S" by auto
from a1 b2 have "∀b. b ∉ S ⟶ (a ⇌ b) ∙ x = x" unfolding supports_def by auto
then have "{b. (a ⇌ b) ∙ x ≠ x} ⊆ S" by auto
with a2 have "finite {b. (a ⇌ b) ∙ x ≠ x}" by (simp add: finite_subset)
then have "a ∉ (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) ⊆ 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 ∉ (supp x) ∧ b ∉ (supp x)"
then have "a ♯ x" and "b ♯ x" by (simp_all add: fresh_def)
then show "(a ⇌ b) ∙ 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 ∉ S"
shows "a ♯ x"
unfolding fresh_def
proof -
have "(supp x) ⊆ S" using a1 a2 by (rule supp_is_subset)
then show "a ∉ (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: "⋀S'. finite S' ⟹ (S' supports x) ⟹ S ⊆ S'"
shows "(supp x) = S"
proof (rule equalityI)
show "(supp x) ⊆ 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 ⊆ supp x" by blast
qed
lemma subsetCI:
shows "(⋀x. x ∈ A ⟹ x ∉ B ⟹ False) ⟹ A ⊆ B"
by auto
lemma finite_supp_unique:
assumes a1: "S supports x"
assumes a2: "finite S"
assumes a3: "⋀a b. ⟦a ∈ S; b ∉ S; sort_of a = sort_of b⟧ ⟹ (a ⇌ b) ∙ x ≠ 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 ⊆ S'"
proof (rule subsetCI)
fix a
assume "a ∈ S" and "a ∉ S'"
have "finite (S ∪ S')"
using `finite S` `finite S'` by simp
then obtain b where "b ∉ S ∪ S'" and "sort_of b = sort_of a"
by (rule obtain_atom)
then have "b ∉ S" and "b ∉ S'" and "sort_of a = sort_of b"
by simp_all
then have "(a ⇌ b) ∙ x = x"
using `a ∉ S'` `S' supports x` by (simp add: supports_def)
moreover have "(a ⇌ b) ∙ x ≠ x"
using `a ∈ S` `b ∉ 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. ¬(R ((a ⇌ b) ∙ 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 ♯ x"
unfolding fresh_def by (simp add: pure_supp)
instance pure < fs
by standard (simp add: pure_supp)
subsection {* Type @{typ atom} is finitely-supported. *}
lemma supp_atom:
shows "supp a = {a}"
apply (rule finite_supp_unique)
apply (clarsimp simp add: supports_def)
apply simp
apply simp
done
lemma fresh_atom:
shows "a ♯ b ⟷ a ≠ b"
unfolding fresh_def supp_atom by simp
instance atom :: fs
by standard (simp add: supp_atom)
section {* Type @{typ perm} is finitely-supported. *}
lemma perm_swap_eq:
shows "(a ⇌ b) ∙ p = p ⟷ (p ∙ (a ⇌ b)) = (a ⇌ b)"
unfolding permute_perm_def
by (metis add_diff_cancel minus_perm_def)
lemma supports_perm:
shows "{a. p ∙ a ≠ a} supports p"
unfolding supports_def
unfolding perm_swap_eq
by (simp add: swap_eqvt)
lemma finite_perm_lemma:
shows "finite {a::atom. p ∙ a ≠ a}"
using finite_Rep_perm [of p]
unfolding permute_atom_def .
lemma supp_perm:
shows "supp p = {a. p ∙ a ≠ a}"
apply (rule finite_supp_unique)
apply (rule supports_perm)
apply (rule finite_perm_lemma)
apply (simp add: perm_swap_eq swap_eqvt)
apply (auto simp: perm_eq_iff swap_atom)
done
lemma fresh_perm:
shows "a ♯ p ⟷ p ∙ a = a"
unfolding fresh_def
by (simp add: supp_perm)
lemma supp_swap:
shows "supp (a ⇌ b) = (if a = b ∨ sort_of a ≠ sort_of b then {} else {a, b})"
by (auto simp: supp_perm swap_atom)
lemma fresh_swap:
shows "a ♯ (b ⇌ c) ⟷ (sort_of b ≠ sort_of c) ∨ b = c ∨ (a ♯ b ∧ a ♯ c)"
by (simp add: fresh_def supp_swap supp_atom)
lemma fresh_zero_perm:
shows "a ♯ (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 ♯ p" "a ♯ q"
shows "a ♯ (p + q)"
using assms
unfolding fresh_def
by (auto simp: supp_perm)
lemma supp_plus_perm:
fixes p q::perm
shows "supp (p + q) ⊆ supp p ∪ supp q"
by (auto simp: supp_perm)
lemma fresh_minus_perm:
fixes p::perm
shows "a ♯ (- p) ⟷ a ♯ p"
unfolding fresh_def
unfolding supp_perm
apply(simp)
apply(metis permute_minus_cancel)
done
lemma supp_minus_perm:
fixes p::perm
shows "supp (- p) = supp p"
unfolding supp_conv_fresh
by (simp add: fresh_minus_perm)
lemma plus_perm_eq:
fixes p q::"perm"
assumes asm: "supp p ∩ supp q = {}"
shows "p + q = q + p"
unfolding perm_eq_iff
proof
fix a::"atom"
show "(p + q) ∙ a = (q + p) ∙ a"
proof -
{ assume "a ∉ supp p" "a ∉ supp q"
then have "(p + q) ∙ a = (q + p) ∙ a"
by (simp add: supp_perm)
}
moreover
{ assume a: "a ∈ supp p" "a ∉ supp q"
then have "p ∙ a ∈ supp p" by (simp add: supp_perm)
then have "p ∙ a ∉ supp q" using asm by auto
with a have "(p + q) ∙ a = (q + p) ∙ a"
by (simp add: supp_perm)
}
moreover
{ assume a: "a ∉ supp p" "a ∈ supp q"
then have "q ∙ a ∈ supp q" by (simp add: supp_perm)
then have "q ∙ a ∉ supp p" using asm by auto
with a have "(p + q) ∙ a = (q + p) ∙ a"
by (simp add: supp_perm)
}
ultimately show "(p + q) ∙ a = (q + p) ∙ a"
using asm by blast
qed
qed
lemma supp_plus_perm_eq:
fixes p q::perm
assumes asm: "supp p ∩ supp q = {}"
shows "supp (p + q) = supp p ∪ supp q"
proof -
{ fix a::"atom"
assume "a ∈ supp p"
then have "a ∉ supp q" using asm by auto
then have "a ∈ supp (p + q)" using `a ∈ supp p`
by (simp add: supp_perm)
}
moreover
{ fix a::"atom"
assume "a ∈ supp q"
then have "a ∉ supp p" using asm by auto
then have "a ∈ supp (q + p)" using `a ∈ supp q`
by (simp add: supp_perm)
then have "a ∈ supp (p + q)" using asm plus_perm_eq
by metis
}
ultimately have "supp p ∪ supp q ⊆ supp (p + q)"
by blast
then show "supp (p + q) = supp p ∪ supp q" using supp_plus_perm
by blast
qed
lemma perm_eq_iff2:
fixes p q :: "perm"
shows "p = q ⟷ (∀a::atom ∈ supp p ∪ supp q. p ∙ a = q ∙ a)"
unfolding perm_eq_iff
apply(auto)
apply(case_tac "a ♯ p ∧ a ♯ q")
apply(simp add: fresh_perm)
apply(simp add: fresh_def)
done
instance perm :: fs
by standard (simp add: supp_perm finite_perm_lemma)
section {* Finite Support instances for other types *}
subsection {* Type @{typ "'a × 'b"} is finitely-supported. *}
lemma supp_Pair:
shows "supp (x, y) = supp x ∪ supp y"
by (simp add: supp_def Collect_imp_eq Collect_neg_eq)
lemma fresh_Pair:
shows "a ♯ (x, y) ⟷ a ♯ x ∧ a ♯ y"
by (simp add: fresh_def supp_Pair)
lemma supp_Unit:
shows "supp () = {}"
by (simp add: supp_def)
lemma fresh_Unit:
shows "a ♯ ()"
by (simp add: fresh_def supp_Unit)
instance prod :: (fs, fs) fs
apply standard
apply (case_tac x)
apply (simp add: supp_Pair finite_supp)
done
subsection {* Type @{typ "'a + 'b"} is finitely supported *}
lemma supp_Inl:
shows "supp (Inl x) = supp x"
by (simp add: supp_def)
lemma supp_Inr:
shows "supp (Inr x) = supp x"
by (simp add: supp_def)
lemma fresh_Inl:
shows "a ♯ Inl x ⟷ a ♯ x"
by (simp add: fresh_def supp_Inl)
lemma fresh_Inr:
shows "a ♯ Inr y ⟷ a ♯ y"
by (simp add: fresh_def supp_Inr)
instance sum :: (fs, fs) fs
apply standard
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 ♯ None"
by (simp add: fresh_def supp_None)
lemma fresh_Some:
shows "a ♯ Some x ⟷ a ♯ x"
by (simp add: fresh_def supp_Some)
instance option :: (fs) fs
apply standard
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 ♯ []"
by (simp add: fresh_def supp_Nil)
lemma supp_Cons:
shows "supp (x # xs) = supp x ∪ supp xs"
by (simp add: supp_def Collect_imp_eq Collect_neg_eq)
lemma fresh_Cons:
shows "a ♯ (x # xs) ⟷ a ♯ x ∧ a ♯ xs"
by (simp add: fresh_def supp_Cons)
lemma supp_append:
shows "supp (xs @ ys) = supp xs ∪ supp ys"
by (induct xs) (auto simp: supp_Nil supp_Cons)
lemma fresh_append:
shows "a ♯ (xs @ ys) ⟷ a ♯ xs ∧ a ♯ 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: supp_append supp_Cons supp_Nil)
lemma fresh_rev:
shows "a ♯ rev xs ⟷ a ♯ xs"
by (induct xs) (auto simp: 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: 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 standard
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 ♯ x ⟷ (MOST b. (a ⇌ b) ∙ x = x)"
unfolding fresh_def supp_def
unfolding MOST_iff_cofinite by simp
lemma fresh_fun_app:
assumes "a ♯ f" and "a ♯ x"
shows "a ♯ 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) ⊆ (supp f) ∪ (supp x)"
using fresh_fun_app
unfolding fresh_def
by auto
subsection {* Equivariance Predicate @{text eqvt} and @{text eqvt_at}*}
definition
"eqvt f ≡ ∀p. p ∙ 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 ≡ ∀p. p ∙ (f x) = f (p ∙ x)"
lemma eqvtI:
shows "(⋀p. p ∙ f ≡ f) ⟹ eqvt f"
unfolding eqvt_def
by simp
lemma eqvt_at_perm:
assumes "eqvt_at f x"
shows "eqvt_at f (q ∙ x)"
proof -
{ fix p::"perm"
have "p ∙ (f (q ∙ x)) = p ∙ q ∙ (f x)"
using assms by (simp add: eqvt_at_def)
also have "… = (p + q) ∙ (f x)" by simp
also have "… = f ((p + q) ∙ x)"
using assms by (simp only: eqvt_at_def)
finally have "p ∙ (f (q ∙ x)) = f (p ∙ q ∙ x)" by simp }
then show "eqvt_at f (q ∙ 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:
assumes a: "eqvt f"
shows "a ♯ f"
using a
unfolding fresh_def
by (simp add: supp_fun_eqvt)
lemma fresh_fun_eqvt_app:
assumes a: "eqvt f"
shows "a ♯ x ⟹ a ♯ f x"
proof -
from a have "supp f = {}" by (simp add: supp_fun_eqvt)
then show "a ♯ x ⟹ a ♯ 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) ⊆ 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) ⊆ supp x"
apply(rule supp_is_subset)
unfolding supports_def
unfolding fresh_def[symmetric]
using asm
apply(simp add: eqvt_at_def)
apply(simp add: swap_fresh_fresh)
apply(rule fin)
done
lemma finite_supp_eqvt_at:
assumes asm: "eqvt_at f x"
and fin: "finite (supp x)"
shows "finite (supp (f x))"
apply(rule finite_subset)
apply(rule supp_eqvt_at[OF asm fin])
apply(rule fin)
done
lemma fresh_eqvt_at:
assumes asm: "eqvt_at f x"
and fin: "finite (supp x)"
and fresh: "a ♯ x"
shows "a ♯ f x"
using fresh
unfolding fresh_def
using supp_eqvt_at[OF asm fin]
by auto
text {* for handling of freshness of functions *}
simproc_setup fresh_fun_simproc ("a ♯ (f::'a::pt ⇒'b::pt)") = {* fn _ => fn ctxt => fn ctrm =>
let
val _ $ _ $ f = Thm.term_of ctrm
in
case (Term.add_frees f [], Term.add_vars f []) of
([], []) => SOME(@{thm fresh_fun_eqvt[simplified eqvt_def, THEN Eq_TrueI]})
| (x::_, []) =>
let
val argx = Free x
val absf = absfree x f
val cty_inst =
[SOME (Thm.ctyp_of ctxt (fastype_of argx)), SOME (Thm.ctyp_of ctxt (fastype_of f))]
val ctrm_inst = [NONE, SOME (Thm.cterm_of ctxt absf), SOME (Thm.cterm_of ctxt argx)]
val thm = Thm.instantiate' cty_inst ctrm_inst @{thm fresh_fun_app}
in
SOME(thm RS @{thm Eq_TrueI})
end
| (_, _) => NONE
end
*}
subsection {* helper functions for nominal_functions *}
lemma THE_defaultI2:
assumes "∃!x. P x" "⋀x. P x ⟹ Q x"
shows "Q (THE_default d P)"
by (iprover intro: assms THE_defaultI')
lemma the_default_eqvt:
assumes unique: "∃!x. P x"
shows "(p ∙ (THE_default d P)) = (THE_default (p ∙ d) (p ∙ 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 == (λx::'a. THE_default (d x) (G x))"
assumes eqvt: "eqvt G"
assumes ex1: "∃!y. G x y"
shows "(p ∙ (f x)) = f (p ∙ 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 == (λx::'a. THE_default (d x) (G x))"
assumes eqvt: "eqvt G"
assumes ex1: "∃!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 == (λx::'a. THE_default (d x) (G x))"
assumes P_all: "⋀x y. G x y ⟹ P x y"
assumes ex1: "∃!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 ♯ S ⟷ a ∉ 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 ♯ S - T ⟷ (a ∉ T ⟶ a ♯ S)"
unfolding fresh_def
by (auto simp: supp_finite_atom_set assms)
lemma Union_supports_set:
shows "(⋃x ∈ S. supp x) supports S"
proof -
{ fix a b
have "∀x ∈ S. (a ⇌ b) ∙ x = x ⟹ (a ⇌ b) ∙ S = S"
unfolding permute_set_def by force
}
then show "(⋃x ∈ 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 (⋃x∈S. supp x)"
using fin by (induct) (auto simp: finite_supp)
lemma Union_included_in_supp:
fixes S::"('a::fs set)"
assumes fin: "finite S"
shows "(⋃x∈S. supp x) ⊆ supp S"
proof -
have eqvt: "eqvt (λS. ⋃x ∈ S. supp x)"
unfolding eqvt_def by simp
have "(⋃x∈S. supp x) = supp (⋃x∈S. supp x)"
by (rule supp_finite_atom_set[symmetric]) (rule Union_of_finite_supp_sets[OF fin])
also have "… ⊆ supp S" using eqvt
by (rule supp_fun_app_eqvt)
finally show "(⋃x∈S. supp x) ⊆ supp S" .
qed
lemma supp_of_finite_sets:
fixes S::"('a::fs set)"
assumes fin: "finite S"
shows "(supp S) = (⋃x∈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 ∪ T) = supp S ∪ supp T"
using fin1 fin2
by (simp add: supp_of_finite_sets)
lemma fresh_finite_union:
fixes S T::"('a::fs) set"
assumes fin1: "finite S"
and fin2: "finite T"
shows "a ♯ (S ∪ T) ⟷ a ♯ S ∧ a ♯ T"
unfolding fresh_def
by (simp add: supp_of_finite_union[OF fin1 fin2])
lemma supp_of_finite_insert:
fixes S::"('a::fs) set"
assumes fin: "finite S"
shows "supp (insert x S) = supp x ∪ 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 ♯ (insert x S) ⟷ a ♯ x ∧ a ♯ 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 ♯ {}"
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 ♯ (set xs) ⟷ a ♯ xs"
unfolding fresh_def
by (simp add: supp_set)
subsection {* Type @{typ "'a multiset"} is finitely supported *}
lemma set_mset_eqvt [eqvt]:
shows "p ∙ (set_mset M) = set_mset (p ∙ M)"
by (induct M) (simp_all add: insert_eqvt empty_eqvt)
lemma supp_set_mset:
shows "supp (set_mset M) ⊆ 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 (⋃{supp x | x. x ∈# M})"
proof -
have "finite (⋃(supp ` {x. x ∈# M}))"
by (induct M) (simp_all add: Collect_imp_eq Collect_neg_eq finite_supp)
then show "finite (⋃{supp x | x. x ∈# M})"
by (simp only: image_Collect)
qed
lemma Union_supports_multiset:
shows "⋃{supp x | x. x :# M} supports M"
proof -
have sw: "⋀a b. ((⋀x. x :# M ⟹ (a ⇌ b) ∙ x = x) ⟹ (a ⇌ b) ∙ M = M)"
unfolding permute_multiset_def
apply(induct M)
apply(simp_all)
done
show "(⋃{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 "(⋃{supp x | x. x ∈# M}) ⊆ supp M"
proof -
have "(⋃{supp x | x. x ∈# M}) = (⋃{supp x | x. x ∈ set_mset M})" by simp
also have "... ⊆ (⋃x ∈ set_mset M. supp x)" by auto
also have "... = supp (set_mset M)"
by (simp add: supp_of_finite_sets)
also have " ... ⊆ supp M" by (rule supp_set_mset)
finally show "(⋃{supp x | x. x ∈# M}) ⊆ supp M" .
qed
lemma supp_of_multisets:
fixes M::"('a::fs multiset)"
shows "(supp M) = (⋃{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 ∪ supp N"
by (auto simp: supp_of_multisets)
lemma supp_empty_mset [simp]:
shows "supp {#} = {}"
unfolding supp_def
by simp
instance multiset :: (fs) fs
by standard (rule multisets_supp_finite)
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 ♯ {||}"
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 ∪ 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 ♯ insert_fset x S ⟷ a ♯ x ∧ a ♯ 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 |∪| T) = supp S ∪ supp T"
by (induct S) (auto)
lemma fresh_union_fset:
fixes S T::"'a::fs fset"
shows "a ♯ S |∪| T ⟷ a ♯ S ∧ a ♯ T"
unfolding fresh_def
by (simp add: supp_union_fset)
instance fset :: (fs) fs
by standard (rule fset_finite_supp)
subsection {* Type @{typ "('a, 'b) finfun"} is finitely supported *}
lemma fresh_finfun_const:
shows "a ♯ (finfun_const b) ⟷ a ♯ b"
by (simp add: fresh_def supp_def)
lemma fresh_finfun_update:
shows "⟦a ♯ f; a ♯ x; a ♯ y⟧ ⟹ a ♯ finfun_update f x y"
unfolding fresh_conv_MOST
unfolding finfun_update_eqvt
by (elim MOST_rev_mp) (simp)
lemma supp_finfun_const:
shows "supp (finfun_const b) = supp(b)"
by (simp add: supp_def)
lemma supp_finfun_update:
shows "supp (finfun_update f x y) ⊆ supp(f, x, y)"
using fresh_finfun_update
by (auto simp: fresh_def supp_Pair)
instance finfun :: (fs, fs) fs
apply standard
apply(induct_tac x rule: finfun_weak_induct)
apply(simp add: supp_finfun_const finite_supp)
apply(rule finite_subset)
apply(rule supp_finfun_update)
apply(simp add: supp_Pair finite_supp)
done
section {* Freshness and Fresh-Star *}
lemma fresh_Unit_elim:
shows "(a ♯ () ⟹ PROP C) ≡ PROP C"
by (simp add: fresh_Unit)
lemma fresh_Pair_elim:
shows "(a ♯ (x, y) ⟹ PROP C) ≡ (a ♯ x ⟹ a ♯ y ⟹ PROP C)"
by rule (simp_all add: fresh_Pair)
lemma [simp]:
shows "a ♯ x1 ⟹ a ♯ x2 ⟹ a ♯ (x1, x2)"
by (simp add: fresh_Pair)
lemma fresh_PairD:
shows "a ♯ (x, y) ⟹ a ♯ x"
and "a ♯ (x, y) ⟹ a ♯ 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 ⇒ 'a::pt ⇒ bool" ("_ ♯* _" [80,80] 80)
where
"as ♯* x ≡ ∀a ∈ as. a ♯ x"
lemma fresh_star_supp_conv:
shows "supp x ♯* y ⟹ supp y ♯* x"
by (auto simp: fresh_star_def fresh_def)
lemma fresh_star_perm_set_conv:
fixes p::"perm"
assumes fresh: "as ♯* p"
and fin: "finite as"
shows "supp p ♯* 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 ♯* bs"
and fin: "finite as" "finite bs"
shows "bs ♯* as"
using fresh
unfolding fresh_star_def fresh_def
by (auto simp: supp_finite_atom_set fin)
lemma atom_fresh_star_disjoint:
assumes fin: "finite bs"
shows "as ♯* bs ⟷ (as ∩ bs = {})"
unfolding fresh_star_def fresh_def
by (auto simp: supp_finite_atom_set fin)
lemma fresh_star_Pair:
shows "as ♯* (x, y) = (as ♯* x ∧ as ♯* y)"
by (auto simp: fresh_star_def fresh_Pair)
lemma fresh_star_list:
shows "as ♯* (xs @ ys) ⟷ as ♯* xs ∧ as ♯* ys"
and "as ♯* (x # xs) ⟷ as ♯* x ∧ as ♯* xs"
and "as ♯* []"
by (auto simp: fresh_star_def fresh_Nil fresh_Cons fresh_append)
lemma fresh_star_set:
fixes xs::"('a::fs) list"
shows "as ♯* set xs ⟷ as ♯* xs"
unfolding fresh_star_def
by (simp add: fresh_set)
lemma fresh_star_singleton:
fixes a::"atom"
shows "as ♯* {a} ⟷ as ♯* a"
by (simp add: fresh_star_def fresh_finite_insert fresh_set_empty)
lemma fresh_star_fset:
fixes xs::"('a::fs) list"
shows "as ♯* fset S ⟷ as ♯* S"
by (simp add: fresh_star_def fresh_def)
lemma fresh_star_Un:
shows "(as ∪ bs) ♯* x = (as ♯* x ∧ bs ♯* x)"
by (auto simp: fresh_star_def)
lemma fresh_star_insert:
shows "(insert a as) ♯* x = (a ♯ x ∧ as ♯* x)"
by (auto simp: fresh_star_def)
lemma fresh_star_Un_elim:
"((as ∪ bs) ♯* x ⟹ PROP C) ≡ (as ♯* x ⟹ bs ♯* x ⟹ PROP C)"
unfolding fresh_star_def
apply(rule)
apply(erule meta_mp)
apply(auto)
done
lemma fresh_star_insert_elim:
"(insert a as ♯* x ⟹ PROP C) ≡ (a ♯ x ⟹ as ♯* x ⟹ PROP C)"
unfolding fresh_star_def
by rule (simp_all add: fresh_star_def)
lemma fresh_star_empty_elim:
"({} ♯* x ⟹ PROP C) ≡ PROP C"
by (simp add: fresh_star_def)
lemma fresh_star_Unit_elim:
shows "(a ♯* () ⟹ PROP C) ≡ PROP C"
by (simp add: fresh_star_def fresh_Unit)
lemma fresh_star_Pair_elim:
shows "(a ♯* (x, y) ⟹ PROP C) ≡ (a ♯* x ⟹ a ♯* y ⟹ PROP C)"
by (rule, simp_all add: fresh_star_Pair)
lemma fresh_star_zero:
shows "as ♯* (0::perm)"
unfolding fresh_star_def
by (simp add: fresh_zero_perm)
lemma fresh_star_plus:
fixes p q::perm
shows "⟦a ♯* p; a ♯* q⟧ ⟹ a ♯* (p + q)"
unfolding fresh_star_def
by (simp add: fresh_plus_perm)
lemma fresh_star_permute_iff:
shows "(p ∙ a) ♯* (p ∙ x) ⟷ a ♯* x"
unfolding fresh_star_def
by (metis mem_permute_iff permute_minus_cancel(1) fresh_permute_iff)
lemma fresh_star_eqvt [eqvt]:
shows "p ∙ (as ♯* x) ⟷ (p ∙ as) ♯* (p ∙ x)"
unfolding fresh_star_def by simp
section {* Induction principle for permutations *}
lemma smaller_supp:
assumes a: "a ∈ supp p"
shows "supp ((p ∙ a ⇌ a) + p) ⊂ supp p"
proof -
have "supp ((p ∙ a ⇌ a) + p) ⊆ supp p"
unfolding supp_perm by (auto simp: swap_atom)
moreover
have "a ∉ supp ((p ∙ a ⇌ a) + p)" by (simp add: supp_perm)
then have "supp ((p ∙ a ⇌ a) + p) ≠ supp p" using a by auto
ultimately
show "supp ((p ∙ a ⇌ a) + p) ⊂ supp p" by auto
qed
lemma perm_struct_induct[consumes 1, case_names zero swap]:
assumes S: "supp p ⊆ S"
and zero: "P 0"
and swap: "⋀p a b. ⟦P p; supp p ⊆ S; a ∈ S; b ∈ S; a ≠ b; sort_of a = sort_of b⟧ ⟹ P ((a ⇌ b) + p)"
shows "P p"
proof -
have "finite (supp p)" by (simp add: finite_supp)
then show "P p" using S
proof(induct A≡"supp p" arbitrary: p rule: finite_psubset_induct)
case (psubset p)
then have ih: "⋀q. supp q ⊂ supp p ⟹ P q" by auto
have as: "supp p ⊆ 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 ≠ {}"
then obtain a where a0: "a ∈ supp p" by blast
then have a1: "p ∙ a ∈ S" "a ∈ S" "sort_of (p ∙ a) = sort_of a" "p ∙ a ≠ a"
using as by (auto simp: supp_atom supp_perm swap_atom)
let ?q = "(p ∙ a ⇌ a) + p"
have a2: "supp ?q ⊂ supp p" using a0 smaller_supp by simp
then have "P ?q" using ih by simp
moreover
have "supp ?q ⊆ S" using as a2 by simp
ultimately have "P ((p ∙ a ⇌ a) + ?q)" using as a1 swap by simp
moreover
have "p = (p ∙ a ⇌ 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: "⋀p a b. ⟦P p; a ≠ b; sort_of a = sort_of b⟧ ⟹ P ((a ⇌ 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 ⊆ S"
assumes zero: "P 0"
assumes swap: "⋀a b. ⟦sort_of a = sort_of b; a ≠ b; a ∈ S; b ∈ S⟧ ⟹ P (a ⇌ b)"
assumes plus: "⋀p1 p2. ⟦P p1; P p2; supp p1 ⊆ S; supp p2 ⊆ S⟧ ⟹ 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: "⋀a b. ⟦sort_of a = sort_of b; a ≠ b⟧ ⟹ P (a ⇌ b)"
assumes plus: "⋀p1 p2. ⟦P p1; P p2⟧ ⟹ 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 ⊆ {b} ⟷ p = 0"
proof -
{ assume "supp p ⊆ {b}"
then have "p = 0"
by (induct p rule: perm_struct_induct) (simp_all)
}
then show "supp p ⊆ {b} ⟷ p = 0" by (auto simp: supp_zero_perm)
qed
lemma supp_perm_pair:
fixes p::"perm"
shows "supp p ⊆ {a, b} ⟷ p = 0 ∨ p = (b ⇌ a)"
proof -
{ assume "supp p ⊆ {a, b}"
then have "p = 0 ∨ p = (b ⇌ a)"
apply (induct p rule: perm_struct_induct)
apply (auto simp: swap_cancel supp_zero_perm supp_swap)
apply (simp add: swap_commute)
done
}
then show "supp p ⊆ {a, b} ⟷ p = 0 ∨ p = (b ⇌ a)"
by (auto simp: supp_zero_perm supp_swap split: if_splits)
qed
lemma supp_perm_eq:
assumes "(supp x) ♯* p"
shows "p ∙ x = x"
proof -
from assms have "supp p ⊆ {a. a ♯ x}"
unfolding supp_perm fresh_star_def fresh_def by auto
then show "p ∙ x = x"
proof (induct p rule: perm_struct_induct)
case zero
show "0 ∙ x = x" by simp
next
case (swap p a b)
then have "a ♯ x" "b ♯ x" "p ∙ x = x" by simp_all
then show "((a ⇌ b) + p) ∙ 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) ♯* p"
shows "p ∙ x = x"
proof -
from assms have "supp p ⊆ {a. a ♯ x}"
unfolding supp_perm fresh_star_def fresh_def by auto
then show "p ∙ x = x"
proof (induct p rule: perm_struct_induct2)
case zero
show "0 ∙ x = x" by simp
next
case (swap a b)
then have "a ♯ x" "b ♯ x" by simp_all
then show "(a ⇌ b) ∙ x = x" by (simp add: swap_fresh_fresh)
next
case (plus p1 p2)
have "p1 ∙ x = x" "p2 ∙ x = x" by fact+
then show "(p1 + p2) ∙ x = x" by simp
qed
qed
lemma perm_supp_eq:
assumes a: "(supp p) ♯* x"
shows "p ∙ x = x"
proof -
from assms have "supp p ⊆ {a. a ♯ x}"
unfolding supp_perm fresh_star_def fresh_def by auto
then show "p ∙ x = x"
proof (induct p rule: perm_struct_induct2)
case zero
show "0 ∙ x = x" by simp
next
case (swap a b)
then have "a ♯ x" "b ♯ x" by simp_all
then show "(a ⇌ b) ∙ x = x" by (simp add: swap_fresh_fresh)
next
case (plus p1 p2)
have "p1 ∙ x = x" "p2 ∙ x = x" by fact+
then show "(p1 + p2) ∙ x = x" by simp
qed
qed
lemma supp_perm_perm_eq:
assumes a: "∀a ∈ supp x. p ∙ a = q ∙ a"
shows "p ∙ x = q ∙ x"
proof -
from a have "∀a ∈ supp x. (-q + p) ∙ a = a" by simp
then have "∀a ∈ supp x. a ∉ supp (-q + p)"
unfolding supp_perm by simp
then have "supp x ♯* (-q + p)"
unfolding fresh_star_def fresh_def by simp
then have "(-q + p) ∙ x = x" by (simp only: supp_perm_eq)
then show "p ∙ x = q ∙ x"
by (metis permute_minus_cancel permute_plus)
qed
text {* disagreement set *}
definition
dset :: "perm ⇒ perm ⇒ atom set"
where
"dset p q = {a::atom. p ∙ a ≠ q ∙ a}"
lemma ds_fresh:
assumes "dset p q ♯* x"
shows "p ∙ x = q ∙ 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 ♯* p"
shows "p ∙ as = as"
proof -
from a have "supp p ⊆ {a. a ∉ as}"
unfolding supp_perm fresh_star_def fresh_def by auto
then show "p ∙ as = as"
proof (induct p rule: perm_struct_induct)
case zero
show "0 ∙ as = as" by simp
next
case (swap p a b)
then have "a ∉ as" "b ∉ as" "p ∙ as = as" by simp_all
then show "((a ⇌ b) + p) ∙ 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 ⊆ As"
and c: "finite As"
shows "∃p. (p ∙ Xs) ∩ As = {} ∧ (supp p) = (Xs ∪ (p ∙ 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 ∙ {} ∩ As = {}" by simp
moreover
have "supp (0::perm) = {} ∪ 0 ∙ {}" by (simp add: supp_zero_perm)
ultimately show ?case by blast
next
case (insert x Xs)
then obtain p where
p1: "(p ∙ Xs) ∩ As = {}" and
p2: "supp p = (Xs ∪ (p ∙ Xs))" by blast
from `x ∈ As` p1 have "x ∉ p ∙ Xs" by fast
with `x ∉ Xs` p2 have "x ∉ supp p" by fast
hence px: "p ∙ x = x" unfolding supp_perm by simp
have "finite (As ∪ p ∙ Xs ∪ supp p)"
using `finite As` `finite Xs`
by (simp add: permute_set_eq_image finite_supp)
then obtain y where "y ∉ (As ∪ p ∙ Xs ∪ supp p)" "sort_of y = sort_of x"
by (rule obtain_atom)
hence y: "y ∉ As" "y ∉ p ∙ Xs" "y ∉ supp p" "sort_of y = sort_of x"
by simp_all
hence py: "p ∙ y = y" "x ≠ y" using `x ∈ As`
by (auto simp: supp_perm)
let ?q = "(x ⇌ y) + p"
have q: "?q ∙ insert x Xs = insert y (p ∙ Xs)"
unfolding insert_eqvt
using `p ∙ x = x` `sort_of y = sort_of x`
using `x ∉ p ∙ Xs` `y ∉ p ∙ Xs`
by (simp add: swap_atom swap_set_not_in)
have "?q ∙ insert x Xs ∩ As = {}"
using `y ∉ As` `p ∙ Xs ∩ As = {}`
unfolding q by simp
moreover
have "supp (x ⇌ y) ∩ 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 ⇌ y) ∪ supp p)"
by (simp add: supp_plus_perm_eq)
then have "supp ?q = insert x Xs ∪ ?q ∙ insert x Xs"
using p2 `sort_of y = sort_of x` `x ≠ 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 ∙ Xs)♯*c" and "(supp p) = (Xs ∪ (p ∙ Xs))"
using a b at_set_avoiding_aux [where Xs="Xs" and As="Xs ∪ supp c"]
unfolding fresh_star_def fresh_def by blast
lemma at_set_avoiding1:
assumes "finite xs"
and "finite (supp c)"
shows "∃p. (p ∙ xs) ♯* 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 ♯* x"
shows "∃p. (p ∙ xs) ♯* c ∧ supp x ♯* 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: fresh_star_def)
done
lemma at_set_avoiding3:
assumes "finite xs"
and "finite (supp c)" "finite (supp x)"
and "xs ♯* x"
shows "∃p. (p ∙ xs) ♯* c ∧ supp x ♯* p ∧ supp p = xs ∪ (p ∙ 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: fresh_star_def)
done
lemma at_set_avoiding2_atom:
assumes "finite (supp c)" "finite (supp x)"
and b: "a ♯ x"
shows "∃p. (p ∙ a) ♯ c ∧ supp x ♯* p"
proof -
have a: "{a} ♯* x" unfolding fresh_star_def by (simp add: b)
obtain p where p1: "(p ∙ {a}) ♯* c" and p2: "supp x ♯* p"
using at_set_avoiding2[of "{a}" "c" "x"] assms a by blast
have c: "(p ∙ a) ♯ c" using p1
unfolding fresh_star_def Ball_def
by(erule_tac x="p ∙ a" in allE) (simp add: permute_set_def)
hence "p ∙ a ♯ c ∧ supp x ♯* p" using p2 by blast
then show "∃p. (p ∙ a) ♯ c ∧ supp x ♯* p" by blast
qed
section {* Renaming permutations *}
lemma set_renaming_perm:
assumes b: "finite bs"
shows "∃q. (∀b ∈ bs. q ∙ b = p ∙ b) ∧ supp q ⊆ bs ∪ (p ∙ bs)"
using b
proof (induct)
case empty
have "(∀b ∈ {}. 0 ∙ b = p ∙ b) ∧ supp (0::perm) ⊆ {} ∪ p ∙ {}"
by (simp add: permute_set_def supp_perm)
then show "∃q. (∀b ∈ {}. q ∙ b = p ∙ b) ∧ supp q ⊆ {} ∪ p ∙ {}" by blast
next
case (insert a bs)
then have " ∃q. (∀b ∈ bs. q ∙ b = p ∙ b) ∧ supp q ⊆ bs ∪ p ∙ bs" by simp
then obtain q where *: "∀b ∈ bs. q ∙ b = p ∙ b" and **: "supp q ⊆ bs ∪ p ∙ bs"
by (metis empty_subsetI insert(3) supp_swap)
{ assume 1: "q ∙ a = p ∙ a"
have "∀b ∈ (insert a bs). q ∙ b = p ∙ b" using 1 * by simp
moreover
have "supp q ⊆ insert a bs ∪ p ∙ insert a bs"
using ** by (auto simp: insert_eqvt)
ultimately
have "∃q. (∀b ∈ insert a bs. q ∙ b = p ∙ b) ∧ supp q ⊆ insert a bs ∪ p ∙ insert a bs" by blast
}
moreover
{ assume 2: "q ∙ a ≠ p ∙ a"
def q' ≡ "((q ∙ a) ⇌ (p ∙ a)) + q"
have "∀b ∈ insert a bs. q' ∙ b = p ∙ b" using 2 * `a ∉ bs` unfolding q'_def
by (auto simp: swap_atom)
moreover
{ have "{q ∙ a, p ∙ a} ⊆ insert a bs ∪ p ∙ insert a bs"
using **
apply (auto simp: supp_perm insert_eqvt)
apply (subgoal_tac "q ∙ a ∈ bs ∪ p ∙ bs")
apply(auto)[1]
apply(subgoal_tac "q ∙ a ∈ {a. q ∙ a ≠ a}")
apply(blast)
apply(simp)
done
then have "supp (q ∙ a ⇌ p ∙ a) ⊆ insert a bs ∪ p ∙ insert a bs"
unfolding supp_swap by auto
moreover
have "supp q ⊆ insert a bs ∪ p ∙ insert a bs"
using ** by (auto simp: insert_eqvt)
ultimately
have "supp q' ⊆ insert a bs ∪ p ∙ insert a bs"
unfolding q'_def using supp_plus_perm by blast
}
ultimately
have "∃q. (∀b ∈ insert a bs. q ∙ b = p ∙ b) ∧ supp q ⊆ insert a bs ∪ p ∙ insert a bs" by blast
}
ultimately show "∃q. (∀b ∈ insert a bs. q ∙ b = p ∙ b) ∧ supp q ⊆ insert a bs ∪ p ∙ insert a bs"
by blast
qed
lemma set_renaming_perm2:
shows "∃q. (∀b ∈ bs. q ∙ b = p ∙ b) ∧ supp q ⊆ bs ∪ (p ∙ bs)"
proof -
have "finite (bs ∩ supp p)" by (simp add: finite_supp)
then obtain q
where *: "∀b ∈ bs ∩ supp p. q ∙ b = p ∙ b" and **: "supp q ⊆ (bs ∩ supp p) ∪ (p ∙ (bs ∩ supp p))"
using set_renaming_perm by blast
from ** have "supp q ⊆ bs ∪ (p ∙ bs)" by (auto simp: inter_eqvt)
moreover
have "∀b ∈ bs - supp p. q ∙ b = p ∙ b"
apply(auto)
apply(subgoal_tac "b ∉ 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 "(∀b ∈ bs. q ∙ b = p ∙ b) ∧ supp q ⊆ bs ∪ (p ∙ bs)" using * by auto
then show "∃q. (∀b ∈ bs. q ∙ b = p ∙ b) ∧ supp q ⊆ bs ∪ (p ∙ bs)" by blast
qed
lemma list_renaming_perm:
shows "∃q. (∀b ∈ set bs. q ∙ b = p ∙ b) ∧ supp q ⊆ set bs ∪ (p ∙ set bs)"
proof (induct bs)
case (Cons a bs)
then have " ∃q. (∀b ∈ set bs. q ∙ b = p ∙ b) ∧ supp q ⊆ set bs ∪ p ∙ (set bs)" by simp
then obtain q where *: "∀b ∈ set bs. q ∙ b = p ∙ b" and **: "supp q ⊆ set bs ∪ p ∙ (set bs)"
by (blast)
{ assume 1: "a ∈ set bs"
have "q ∙ a = p ∙ a" using * 1 by (induct bs) (auto)
then have "∀b ∈ set (a # bs). q ∙ b = p ∙ b" using * by simp
moreover
have "supp q ⊆ set (a # bs) ∪ p ∙ (set (a # bs))" using ** by (auto simp: insert_eqvt)
ultimately
have "∃q. (∀b ∈ set (a # bs). q ∙ b = p ∙ b) ∧ supp q ⊆ set (a # bs) ∪ p ∙ (set (a # bs))" by blast
}
moreover
{ assume 2: "a ∉ set bs"
def q' ≡ "((q ∙ a) ⇌ (p ∙ a)) + q"
have "∀b ∈ set (a # bs). q' ∙ b = p ∙ b"
unfolding q'_def using 2 * `a ∉ set bs` by (auto simp: swap_atom)
moreover
{ have "{q ∙ a, p ∙ a} ⊆ set (a # bs) ∪ p ∙ (set (a # bs))"
using **
apply (auto simp: supp_perm insert_eqvt)
apply (subgoal_tac "q ∙ a ∈ set bs ∪ p ∙ set bs")
apply(auto)[1]
apply(subgoal_tac "q ∙ a ∈ {a. q ∙ a ≠ a}")
apply(blast)
apply(simp)
done
then have "supp (q ∙ a ⇌ p ∙ a) ⊆ set (a # bs) ∪ p ∙ set (a # bs)"
unfolding supp_swap by auto
moreover
have "supp q ⊆ set (a # bs) ∪ p ∙ (set (a # bs))"
using ** by (auto simp: insert_eqvt)
ultimately
have "supp q' ⊆ set (a # bs) ∪ p ∙ (set (a # bs))"
unfolding q'_def using supp_plus_perm by blast
}
ultimately
have "∃q. (∀b ∈ set (a # bs). q ∙ b = p ∙ b) ∧ supp q ⊆ set (a # bs) ∪ p ∙ (set (a # bs))" by blast
}
ultimately show "∃q. (∀b ∈ set (a # bs). q ∙ b = p ∙ b) ∧ supp q ⊆ set (a # bs) ∪ p ∙ (set (a # bs))"
by blast
next
case Nil
have "(∀b ∈ set []. 0 ∙ b = p ∙ b) ∧ supp (0::perm) ⊆ set [] ∪ p ∙ set []"
by (simp add: supp_zero_perm)
then show "∃q. (∀b ∈ set []. q ∙ b = p ∙ b) ∧ supp q ⊆ set [] ∪ p ∙ (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 ⇒ atom"
assumes atom_eq_iff [simp]: "atom a = atom b ⟷ a = b"
assumes atom_eqvt: "p ∙ (atom a) = atom (p ∙ 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) ≠ sort_of (atom b)"
shows "atom a ≠ 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 ≠ sort_of (atom b) ⟹ a ♯ b"
and "a ♯ b ⟷ a ≠ atom b"
unfolding fresh_def
apply(simp_all add: supp_at_base)
apply(metis)
done
lemma fresh_ineq_at_base [simp]:
shows "a ≠ atom b ⟹ a ♯ b"
by (simp add: fresh_at_base)
lemma fresh_atom_at_base [simp]:
fixes b::"'a::at_base"
shows "a ♯ atom b ⟷ a ♯ b"
by (simp add: fresh_def supp_at_base supp_atom)
lemma fresh_star_atom_at_base:
fixes b::"'a::at_base"
shows "as ♯* atom b ⟷ as ♯* b"
by (simp add: fresh_star_def fresh_atom_at_base)
lemma if_fresh_at_base [simp]:
shows "atom a ♯ x ⟹ P (if a = x then t else s) = P s"
and "atom a ♯ x ⟹ P (if x = a then t else s) = P s"
by (simp_all add: fresh_at_base)
simproc_setup fresh_ineq ("x ≠ (y::'a::at_base)") = {* fn _ => fn ctxt => fn ctrm =>
case Thm.term_of ctrm of @{term "HOL.Not"} $ (Const (@{const_name HOL.eq}, _) $ lhs $ rhs) =>
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 (@{const_name 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 first_is_neg lhs rhs thms)
| _ => first_is_neg lhs rhs thms)
val simp_thms = @{thms fresh_Pair fresh_at_base atom_eq_iff}
val prems = Simplifier.prems_of ctxt
|> filter (fn thm => case Thm.prop_of thm of
_ $ (Const (@{const_name fresh}, ty) $ (_ $ a) $ b) =>
(let
val atms = a :: HOLogic.strip_tuple b
in
member (op=) atms lhs andalso member (op=) atms rhs
end)
| _ => false)
|> map (simplify (put_simpset HOL_basic_ss ctxt addsimps simp_thms))
|> map (HOLogic.conj_elims ctxt)
|> flat
in
case first_is_neg lhs rhs prems of
SOME(thm) => SOME(thm RS @{thm Eq_TrueI})
| NONE => NONE
end
| _ => NONE
*}
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 ∉ atom ` ?U" "sort_of b = sort_of (atom a)"
by (rule obtain_atom)
from b(2) have "b = atom ((atom a ⇌ b) ∙ a)"
unfolding atom_eqvt [symmetric]
by (simp add: swap_atom)
hence "b ∈ 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) ⟹ (atom x ⇌ atom y) ∙ x = y"
and "sort_of (atom x) = sort_of (atom y) ⟹ (atom x ⇌ atom y) ∙ y = x"
and "atom x ≠ a ⟹ atom x ≠ b ⟹ (a ⇌ b) ∙ 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 ∉ X"
proof -
have "inj (atom :: 'a ⇒ 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 ≠ (UNIV :: 'a set)"
by auto
then obtain a :: 'a where "atom a ∉ X"
by auto
thus ?thesis ..
qed
lemma obtain_fresh':
assumes fin: "finite (supp x)"
obtains a::"'a::at_base" where "atom a ♯ x"
using obtain_at_base[where X="supp x"]
by (auto simp: fresh_def fin)
lemma obtain_fresh:
fixes x::"'b::fs"
obtains a::"'a::at_base" where "atom a ♯ x"
by (rule obtain_fresh') (auto simp: 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
lemma fresh_finite_set_at_base:
fixes a::"'a::at_base"
assumes a: "finite S"
shows "atom a ♯ S ⟷ a ∉ 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 ∙ a) ♯ p ∙ x ⟷ atom a ♯ x"
unfolding atom_eqvt[symmetric]
by (simp only: fresh_permute_iff)
lemma fresh_at_base_permI:
shows "atom a ♯ p ⟹ p ∙ a = a"
by (simp add: fresh_def supp_perm)
section {* Infrastructure for concrete atom types *}
definition
flip :: "'a::at_base ⇒ 'a ⇒ perm" ("'(_ ↔ _')")
where
"(a ↔ b) = (atom a ⇌ atom b)"
lemma flip_fresh_fresh:
assumes "atom a ♯ x" "atom b ♯ x"
shows "(a ↔ b) ∙ x = x"
using assms
by (simp add: flip_def swap_fresh_fresh)
lemma flip_self [simp]: "(a ↔ a) = 0"
unfolding flip_def by (rule swap_self)
lemma flip_commute: "(a ↔ b) = (b ↔ a)"
unfolding flip_def by (rule swap_commute)
lemma minus_flip [simp]: "- (a ↔ b) = (a ↔ b)"
unfolding flip_def by (rule minus_swap)
lemma add_flip_cancel: "(a ↔ b) + (a ↔ b) = 0"
unfolding flip_def by (rule swap_cancel)
lemma permute_flip_cancel [simp]: "(a ↔ b) ∙ (a ↔ b) ∙ x = x"
unfolding permute_plus [symmetric] add_flip_cancel by simp
lemma permute_flip_cancel2 [simp]: "(a ↔ b) ∙ (b ↔ a) ∙ x = x"
by (simp add: flip_commute)
lemma flip_eqvt [eqvt]:
shows "p ∙ (a ↔ b) = (p ∙ a ↔ p ∙ 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) ⟹ (a ↔ b) ∙ a = b"
and "sort_of (atom a) = sort_of (atom b) ⟹ (a ↔ b) ∙ b = a"
and "⟦a ≠ c; b ≠ c⟧ ⟹ (a ↔ b) ∙ c = c"
and "sort_of (atom a) ≠ sort_of (atom b) ⟹ (a ↔ b) ∙ 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 flip_triple:
fixes a b c::"'a::at"
assumes "a ≠ b" and "c ≠ b"
shows "(a ↔ c) + (b ↔ c) + (a ↔ c) = (a ↔ b)"
unfolding flip_def
by (rule swap_triple) (simp_all add: assms)
lemma permute_flip_at:
fixes a b c::"'a::at"
shows "(a ↔ b) ∙ 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 ↔ b) ∙ a = b"
and "(a ↔ b) ∙ b = a"
unfolding permute_flip_at by simp_all
subsection {* Syntax for coercing at-elements to the atom-type *}
syntax
"_atom_constrain" :: "logic ⇒ type ⇒ 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 "∃a. a ∈ {a. sort_of a = s}"
by (rule_tac x="Atom s 0" in exI, simp)
lemma exists_eq_sort:
shows "∃a. a ∈ {a. sort_of a ∈ range sort_fun}"
by (rule_tac x="Atom (sort_fun x) y" in exI, simp)
lemma at_base_class:
fixes sort_fun :: "'b ⇒ atom_sort"
fixes Rep :: "'a ⇒ atom" and Abs :: "atom ⇒ 'a"
assumes type: "type_definition Rep Abs {a. sort_of a ∈ range sort_fun}"
assumes atom_def: "⋀a. atom a = Rep a"
assumes permute_def: "⋀p a. p ∙ a = Abs (p ∙ Rep a)"
shows "OFCLASS('a, at_base_class)"
proof
interpret type_definition Rep Abs "{a. sort_of a ∈ range sort_fun}" by (rule type)
have sort_of_Rep: "⋀a. sort_of (Rep a) ∈ range sort_fun" using Rep by simp
fix a b :: 'a and p p1 p2 :: perm
show "0 ∙ a = a"
unfolding permute_def by (simp add: Rep_inverse)
show "(p1 + p2) ∙ a = p1 ∙ p2 ∙ a"
unfolding permute_def by (simp add: Abs_inverse sort_of_Rep)
show "atom a = atom b ⟷ a = b"
unfolding atom_def by (simp add: Rep_inject)
show "p ∙ atom a = atom (p ∙ 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 ⇒ atom" and Abs :: "atom ⇒ 'a"
assumes type: "type_definition Rep Abs {a. sort_of a = s}"
assumes atom_def: "⋀a. atom a = Rep a"
assumes permute_def: "⋀p a. p ∙ a = Abs (p ∙ 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: "⋀a. sort_of (Rep a) = s" using Rep by (simp add: image_def)
fix a b :: 'a and p p1 p2 :: perm
show "0 ∙ a = a"
unfolding permute_def by (simp add: Rep_inverse)
show "(p1 + p2) ∙ a = p1 ∙ p2 ∙ 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 ⟷ a = b"
unfolding atom_def by (simp add: Rep_inject)
show "p ∙ atom a = atom (p ∙ 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 ⇒ atom" and Abs :: "atom ⇒ 'a"
fixes a::"'a"
assumes type: "type_definition Rep Abs {a. sort_of a = s}"
assumes atom_def: "⋀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 ⇒ 'a::pt ⇒ 'a"}) *}
setup {* Sign.add_const_constraint
(@{const_name "atom"}, SOME @{typ "'a::at_base ⇒ atom"}) *}
section {* Library functions for the nominal infrastructure *}
ML_file "nominal_library.ML"
section {* The freshness lemma according to Andy Pitts *}
lemma freshness_lemma:
fixes h :: "'a::at ⇒ 'b::pt"
assumes a: "∃a. atom a ♯ (h, h a)"
shows "∃x. ∀a. atom a ♯ h ⟶ h a = x"
proof -
from a obtain b where a1: "atom b ♯ h" and a2: "atom b ♯ h b"
by (auto simp: fresh_Pair)
show "∃x. ∀a. atom a ♯ h ⟶ h a = x"
proof (intro exI allI impI)
fix a :: 'a
assume a3: "atom a ♯ h"
show "h a = h b"
proof (cases "a = b")
assume "a = b"
thus "h a = h b" by simp
next
assume "a ≠ b"
hence "atom a ♯ b" by (simp add: fresh_at_base)
with a3 have "atom a ♯ h b"
by (rule fresh_fun_app)
with a2 have d1: "(atom b ⇌ atom a) ∙ (h b) = (h b)"
by (rule swap_fresh_fresh)
from a1 a3 have d2: "(atom b ⇌ atom a) ∙ h = h"
by (rule swap_fresh_fresh)
from d1 have "h b = (atom b ⇌ atom a) ∙ (h b)" by simp
also have "… = ((atom b ⇌ atom a) ∙ h) ((atom b ⇌ atom a) ∙ b)"
by (rule permute_fun_app_eq)
also have "… = 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 ⇒ 'b::pt"
assumes a: "∃a. atom a ♯ (h, h a)"
shows "∃!x. ∀a. atom a ♯ h ⟶ h a = x"
proof (rule ex_ex1I)
from a show "∃x. ∀a. atom a ♯ h ⟶ h a = x"
by (rule freshness_lemma)
next
fix x y
assume x: "∀a. atom a ♯ h ⟶ h a = x"
assume y: "∀a. atom a ♯ h ⟶ h a = y"
from a x y show "x = y"
by (auto simp: fresh_Pair)
qed
text {* packaging the freshness lemma into a function *}
definition
Fresh :: "('a::at ⇒ 'b::pt) ⇒ 'b"
where
"Fresh h = (THE x. ∀a. atom a ♯ h ⟶ h a = x)"
lemma Fresh_apply:
fixes h :: "'a::at ⇒ 'b::pt"
assumes a: "∃a. atom a ♯ (h, h a)"
assumes b: "atom a ♯ h"
shows "Fresh h = h a"
unfolding Fresh_def
proof (rule the_equality)
show "∀a'. atom a' ♯ h ⟶ h a' = h a"
proof (intro strip)
fix a':: 'a
assume c: "atom a' ♯ h"
from a have "∃x. ∀a. atom a ♯ h ⟶ h a = x" by (rule freshness_lemma)
with b c show "h a' = h a" by auto
qed
next
fix fr :: 'b
assume "∀a. atom a ♯ h ⟶ h a = fr"
with b show "fr = h a" by auto
qed
lemma Fresh_apply':
fixes h :: "'a::at ⇒ 'b::pt"
assumes a: "atom a ♯ h" "atom a ♯ h a"
shows "Fresh h = h a"
apply (rule Fresh_apply)
apply (auto simp: fresh_Pair intro: a)
done
simproc_setup Fresh_simproc ("Fresh (h::'a::at ⇒ 'b::pt)") = {* fn _ => fn ctxt => fn ctrm =>
let
val _ $ h = Thm.term_of ctrm
val cfresh = @{const_name fresh}
val catom = @{const_name atom}
val atoms = Simplifier.prems_of ctxt
|> map_filter (fn thm => case Thm.prop_of thm of
_ $ (Const (cfresh, _) $ (Const (catom, _) $ atm) $ _) => SOME (atm) | _ => NONE)
|> distinct (op=)
fun get_thm atm =
let
val goal1 = HOLogic.mk_Trueprop (mk_fresh (mk_atom atm) h)
val goal2 = HOLogic.mk_Trueprop (mk_fresh (mk_atom atm) (h $ atm))
val thm1 = Goal.prove ctxt [] [] goal1 (K (asm_simp_tac ctxt 1))
val thm2 = Goal.prove ctxt [] [] goal2 (K (asm_simp_tac ctxt 1))
in
SOME (@{thm Fresh_apply'} OF [thm1, thm2] RS eq_reflection)
end handle ERROR _ => NONE
in
get_first get_thm atoms
end
*}
lemma Fresh_eqvt:
fixes h :: "'a::at ⇒ 'b::pt"
assumes a: "∃a. atom a ♯ (h, h a)"
shows "p ∙ (Fresh h) = Fresh (p ∙ h)"
proof -
from a obtain a::"'a::at" where fr: "atom a ♯ h" "atom a ♯ h a"
by (metis fresh_Pair)
then have fr_p: "atom (p ∙ a) ♯ (p ∙ h)" "atom (p ∙ a) ♯ (p ∙ h) (p ∙ a)"
by (metis atom_eqvt fresh_permute_iff eqvt_apply)+
have "p ∙ (Fresh h) = p ∙ (h a)" using fr by simp
also have "... = (p ∙ h) (p ∙ a)" by simp
also have "... = Fresh (p ∙ h)" using fr_p by simp
finally show "p ∙ (Fresh h) = Fresh (p ∙ h)" .
qed
lemma Fresh_supports:
fixes h :: "'a::at ⇒ 'b::pt"
assumes a: "∃a. atom a ♯ (h, h a)"
shows "(supp h) supports (Fresh h)"
apply (simp add: supports_def fresh_def [symmetric])
apply (simp add: Fresh_eqvt [OF a] swap_fresh_fresh)
done
notation Fresh (binder "FRESH " 10)
lemma FRESH_f_iff:
fixes P :: "'a::at ⇒ 'b::pure"
fixes f :: "'b ⇒ '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 ♯ P" using P by (rule obtain_fresh')
then show "(FRESH x. f (P x)) = f (FRESH x. P x)"
by (simp add: pure_fresh)
qed
lemma FRESH_binop_iff:
fixes P :: "'a::at ⇒ 'b::pure"
fixes Q :: "'a::at ⇒ 'c::pure"
fixes binop :: "'b ⇒ 'c ⇒ '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 ♯ (P, Q)" by (rule obtain_fresh')
then show ?thesis
by (simp add: pure_fresh)
qed
lemma FRESH_conj_iff:
fixes P Q :: "'a::at ⇒ bool"
assumes P: "finite (supp P)" and Q: "finite (supp Q)"
shows "(FRESH x. P x ∧ Q x) ⟷ (FRESH x. P x) ∧ (FRESH x. Q x)"
using P Q by (rule FRESH_binop_iff)
lemma FRESH_disj_iff:
fixes P Q :: "'a::at ⇒ bool"
assumes P: "finite (supp P)" and Q: "finite (supp Q)"
shows "(FRESH x. P x ∨ Q x) ⟷ (FRESH x. P x) ∨ (FRESH x. Q x)"
using P Q by (rule FRESH_binop_iff)
section {* Automation for creating concrete atom types *}
text {* At the moment only single-sort concrete atoms are supported. *}
ML_file "nominal_atoms.ML"
section {* Automatic equivariance procedure for inductive definitions *}
ML_file "nominal_eqvt.ML"
end