Theory Nominal2_Base

(*  Title:      Nominal2_Base
Authors:    Christian Urban, Brian Huffman, Cezary Kaliszyk

Basic definitions and lemma infrastructure for
Nominal Isabelle.
*)
theory Nominal2_Base
imports "HOL-Library.Infinite_Set"
"HOL-Library.Multiset"
"HOL-Library.FSet"
FinFun.FinFun
keywords
"atom_decl" "equivariance" :: thy_decl
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 ⇒ 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›

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

"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_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_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)

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)

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 ‹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]

lemma permute_swap_cancel2 [simp]:
shows "(a ⇌ b) ∙ (b ⇌ a) ∙ x = x"
unfolding permute_plus [symmetric]

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)"

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
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

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
done

end

lemma permute_self:
shows "p ∙ p = p"
unfolding permute_perm_def

lemma pemute_minus_self:
shows "- p ∙ p = p"
unfolding permute_perm_def

subsection ‹Permutations for functions›

instantiation "fun" :: (pt, pt) pt
begin

definition
"p ∙ f = (λx. p ∙ (f (- p ∙ x)))"

instance
apply standard
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))"

subsection ‹Permutations for booleans›

instantiation bool :: pt
begin

definition "p ∙ (b::bool) = b"

instance
apply standard
done

end

lemma permute_boolE:
fixes P::"bool"
shows "p ∙ P ⟹ P"

lemma permute_boolI:
fixes P::"bool"
shows "P ⟹ p ∙ P"

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

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 ∙ add_mset x M) = add_mset (p ∙ x) (p ∙ M)"
and   "(p ∙ (M + N)) = (p ∙ M) + (p ∙ N)"
unfolding permute_multiset_def
by (simp_all)

subsection ‹Permutations for @{typ "'a fset"}›

instantiation fset :: (pt) pt
begin

context includes fset.lifting begin
lift_definition
"permute_fset" :: "perm ⇒ 'a fset ⇒ 'a fset"
is "permute :: perm ⇒ 'a set ⇒ 'a set" by simp
end

context includes fset.lifting begin
instance
proof
fix x :: "'a fset" and p q :: "perm"
show "0 ∙ x = x" by transfer simp
show "(p + q) ∙ x = p ∙ q ∙ x"  by transfer simp
qed
end

end

context includes fset.lifting
begin
lemma permute_fset [simp]:
fixes S::"('a::pt) fset"
shows "(p ∙ {||}) = ({||} ::('a::pt) fset)"
and   "(p ∙ finsert x S) = finsert (p ∙ x) (p ∙ S)"
done

lemma fset_eqvt:
shows "p ∙ (fset S) = fset (p ∙ S)"
by transfer simp
end

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(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

end

instantiation nat :: pt
begin

definition "p ∙ (n::nat) = n"

instance

end

instantiation int :: pt
begin

definition "p ∙ (i::int) = i"

instance

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››

ML_file ‹nominal_basics.ML›

subsection ‹Eqvt infrastructure›

text ‹Setup of the theorem attributes ‹eqvt› and ‹eqvt_raw›.›

ML_file ‹nominal_thmdecls.ML›

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 ⇒ '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

(* the normal version of this lemma would cause loops *)
lemma permute_eqvt_raw [eqvt_raw]:
shows "p ∙ permute ≡ permute"
apply(subst permute_eqvt)
apply(simp)
done

lemma zero_perm_eqvt [eqvt]:
shows "p ∙ (0::perm) = 0"
unfolding permute_perm_def by simp

fixes p p1 p2 :: perm
shows "p ∙ (p1 + p2) = p ∙ p1 + p ∙ p2"
unfolding permute_perm_def

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

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)"

lemma conj_eqvt [eqvt]:
shows "p ∙ (A ∧ B) ⟷ (p ∙ A) ∧ (p ∙ B)"

lemma imp_eqvt [eqvt]:
shows "p ∙ (A ⟶ B) ⟷ (p ∙ A) ⟶ (p ∙ B)"

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)"

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)"

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(rule unique)
apply(rule_tac p="-p" in permute_boolE)
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(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 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 inter_eqvt [eqvt]:
shows "p ∙ (A ∩ B) = (p ∙ A) ∩ (p ∙ B)"
unfolding Int_def by simp

lemma Inter_eqvt [eqvt]:
shows "p ∙ ⋂S = ⋂(p ∙ S)"
unfolding Inter_eq by simp

lemma union_eqvt [eqvt]:
shows "p ∙ (A ∪ B) = (p ∙ A) ∪ (p ∙ B)"
unfolding Un_def by simp

lemma Union_eqvt [eqvt]:
shows "p ∙ ⋃A = ⋃(p ∙ A)"
unfolding Union_eq
by perm_simp rule

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 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

(* FIXME: eqvt attribute *)
lemma Sigma_eqvt:
shows "(p ∙ (X × Y)) = (p ∙ X) × (p ∙ Y)"
unfolding Sigma_def
by (perm_simp) (rule refl)

text ‹
In order to prove that lfp is equivariant we need two
auxiliary classes which specify that (<=) and
Inf are equivariant. Instances for bool and fun are
given.
›

class le_eqvt = pt +
assumes le_eqvt [eqvt]: "p ∙ (x ≤ y) = ((p ∙ x) ≤ (p ∙ (y :: 'a :: {order, pt})))"

class inf_eqvt = pt +
assumes inf_eqvt [eqvt]: "p ∙ (Inf X) = Inf (p ∙ (X :: 'a :: {complete_lattice, pt} 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
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: 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"}›

context includes fset.lifting begin
lemma in_fset_eqvt [eqvt]:
shows "(p ∙ (x |∈| S)) = ((p ∙ x) |∈| (p ∙ S))"
by transfer 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))"
by transfer simp

lemma subset_fset_eqvt [eqvt]:
shows "(p ∙ (S |⊆| T)) = ((p ∙ S) |⊆| (p ∙ T))"
by transfer simp

lemma map_fset_eqvt [eqvt]:
shows "p ∙ (f || S) = (p ∙ f) || (p ∙ S)"
by transfer simp
end

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

subsection  ‹Type @{typ atom} is finitely-supported.›

lemma supp_atom:
shows "supp a = {a}"
apply (rule finite_supp_unique)
apply simp
apply simp
done

lemma fresh_atom:
shows "a ♯ b ⟷ a ≠ b"
unfolding fresh_def supp_atom by simp

instance atom :: fs

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

lemma supports_perm:
shows "{a. p ∙ a ≠ a} supports p"
unfolding supports_def
unfolding perm_swap_eq

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 (auto simp: perm_eq_iff swap_atom)
done

lemma fresh_perm:
shows "a ♯ p ⟷ p ∙ a = a"
unfolding fresh_def

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

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"
}
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"
}
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"
}
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›
}
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›
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")
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"

lemma supp_Unit:
shows "supp () = {}"

lemma fresh_Unit:
shows "a ♯ ()"

instance prod :: (fs, fs) fs
apply standard
apply (case_tac x)
done

subsection ‹Type @{typ "'a + 'b"} is finitely supported›

lemma supp_Inl:
shows "supp (Inl x) = supp x"

lemma supp_Inr:
shows "supp (Inr x) = supp x"

lemma fresh_Inl:
shows "a ♯ Inl x ⟷ a ♯ x"

lemma fresh_Inr:
shows "a ♯ Inr y ⟷ a ♯ y"

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 = {}"

lemma supp_Some:
shows "supp (Some x) = supp x"

lemma fresh_None:
shows "a ♯ None"

lemma fresh_Some:
shows "a ♯ Some x ⟷ a ♯ x"

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 [] = {}"

lemma fresh_Nil:
shows "a ♯ []"

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"

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)

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 ‹eqvt› and ‹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

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(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
([], []) => 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(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)
using eqvt[simplified eqvt_def]
apply(simp)
apply(rule ex1)
apply(rule THE_defaultI2)
apply(rule_tac p="-p" in permute_boolE)
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(rule assms)
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(rule assms)
apply(subst swap_commute)
done

lemma fresh_finite_atom_set:
fixes S::"atom set"
assumes "finite S"
shows "a ♯ S ⟷ a ∉ S"
unfolding fresh_def

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
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

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

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

lemma supp_set_empty:
shows "supp {} = {}"
unfolding supp_def

lemma fresh_set_empty:
shows "a ♯ {}"

lemma supp_set:
fixes xs :: "('a::fs) list"
shows "supp (set xs) = supp xs"
apply(induct xs)
done

lemma fresh_set:
fixes xs :: "('a::fs) list"
shows "a ♯ (set xs) ⟷ a ♯ xs"
unfolding fresh_def

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 by (induct M) simp_all
have "(⋃x∈set_mset M. supp x) supports M"
by (auto intro!: sw swap_fresh_fresh simp add: fresh_def supports_def)
also have "(⋃x∈set_mset M. supp x) = (⋃{supp x | x. x ∈# M})"
by auto
finally show "(⋃{supp x | x. x ∈# M}) supports M" .
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}) = (⋃x ∈ set_mset M. supp x)" by auto
also have "... = supp (set_mset M)"
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

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_finsert [simp]:
fixes x::"'a::fs"
and   S::"'a fset"
shows "supp (finsert x S) = supp x ∪ supp S"
apply(subst supp_fset[symmetric])
done

lemma fresh_finsert:
fixes x::"'a::fs"
and   S::"'a fset"
shows "a ♯ finsert 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

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"

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)"

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(rule finite_subset)
apply(rule supp_finfun_update)
done

section ‹Freshness and Fresh-Star›

lemma fresh_Unit_elim:
shows "(a ♯ () ⟹ PROP C) ≡ PROP C"

lemma fresh_Pair_elim:
shows "(a ♯ (x, y) ⟹ PROP C) ≡ (a ♯ x ⟹ a ♯ y ⟹ PROP C)"

(* this rule needs to be added before the fresh_prodD is *)
(* added to the simplifier with mksimps                  *)
lemma [simp]:
shows "a ♯ x1 ⟹ a ♯ x2 ⟹ a ♯ (x1, x2)"

lemma fresh_PairD:
shows "a ♯ (x, y) ⟹ a ♯ x"
and   "a ♯ (x, y) ⟹ a ♯ y"

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)
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

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"

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

lemma fresh_star_empty_elim:
"({} ♯* x ⟹ PROP C) ≡ PROP C"

lemma fresh_star_Unit_elim:
shows "(a ♯* () ⟹ PROP C) ≡ PROP C"

lemma fresh_star_Pair_elim:
shows "(a ♯* (x, y) ⟹ PROP C) ≡ (a ♯* x ⟹ a ♯* y ⟹ PROP C)"

lemma fresh_star_zero:
shows "as ♯* (0::perm)"
unfolding fresh_star_def

lemma fresh_star_plus:
fixes p q::perm
shows "⟦a ♯* p;  a ♯* q⟧ ⟹ a ♯* (p + q)"
unfolding fresh_star_def

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)
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›
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›
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)"
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(rule_tac x="p" in exI)
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(rule_tac x="p" in exI)
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 ∙ {}"
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"
define q' where "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(clarify)
apply(rotate_tac 2)
apply(drule subsetD[OF **])
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"
define q' where "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 []"
then show "∃q. (∀b ∈ set []. q ∙ b = p ∙ b) ∧ supp q ⊆ set [] ∪ p ∙ (set [])" by blast
qed

section ‹Concrete Atoms Types›

text ‹
Class ‹at_base› allows types containing multiple sorts of atoms.
Class ‹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(metis)
done

(* solves the freshness only if the inequality can be shown by the
simproc below *)
lemma fresh_ineq_at_base [simp]:
shows "a ≠ atom b ⟹ a ♯ b"

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"

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"

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 ((=)) atms lhs andalso member ((=)) 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 (* 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 ♯ 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 ‹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 ∈ range (λx::unit. 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 ∈ range (λx::unit. 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: 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 ((=))

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  File ‹nominal_basics.ML› (* Title: nominal_basics.ML Author: Christian Urban Author: Tjark Weber Basic functions for nominal. *) infix 1 ||>>> |>>> signature NOMINAL_BASIC = sig val trace: bool Unsynchronized.ref val trace_msg: (unit -> string) -> unit val |>>> : 'a * ('a -> 'b * 'c) -> 'b list * 'c val ||>>> : ('a list * 'b) * ('b -> 'a * 'b) -> 'a list * 'b val last2: 'a list -> 'a * 'a val split_triples: ('a * 'b * 'c) list -> ('a list * 'b list * 'c list) val split_last2: 'a list -> 'a list * 'a * 'a val order: ('a * 'a -> bool) -> 'a list -> ('a * 'b) list -> 'b list val order_default: ('a * 'a -> bool) -> 'b -> 'a list -> ('a * 'b) list -> 'b list val remove_dups: ('a * 'a -> bool) -> 'a list -> 'a list val map4: ('a -> 'b -> 'c -> 'd -> 'e) -> 'a list -> 'b list -> 'c list -> 'd list -> 'e list val split_filter: ('a -> bool) -> 'a list -> 'a list * 'a list val fold_left: ('a * 'a -> 'a) -> 'a list -> 'a -> 'a val is_true: term -> bool val dest_listT: typ -> typ val dest_fsetT: typ -> typ val mk_id: term -> term val mk_all: (string * typ) -> term -> term val mk_All: (string * typ) -> term -> term val mk_exists: (string * typ) -> term -> term val case_sum_const: typ -> typ -> typ -> term val mk_case_sum: term -> term -> term val mk_equiv: thm -> thm val safe_mk_equiv: thm -> thm val mk_minus: term -> term val mk_plus: term -> term -> term val perm_ty: typ -> typ val perm_const: typ -> term val mk_perm_ty: typ -> term -> term -> term val mk_perm: term -> term -> term val dest_perm: term -> term * term (* functions to deal with constants in local contexts *) val long_name: Proof.context -> string -> string val is_fixed: Proof.context -> term -> bool val fixed_nonfixed_args: Proof.context -> term -> term * term list end structure Nominal_Basic: NOMINAL_BASIC = struct val trace = Unsynchronized.ref false fun trace_msg msg = if ! trace then tracing (msg ()) else () infix 1 ||>>> |>>> fun (x |>>> f) = let val (x', y') = f x in ([x'], y') end fun (([], y) ||>>> f) = ([], y) | ((xs, y) ||>>> f) = let val (x', y') = f y in (xs @ [x'], y') end (* orders an AList according to keys - every key needs to be there *) fun order eq keys list = map (the o AList.lookup eq list) keys (* orders an AList according to keys - returns default for non-existing keys *) fun order_default eq default keys list = map (the_default default o AList.lookup eq list) keys (* remove duplicates *) fun remove_dups eq [] = [] | remove_dups eq (x :: xs) = if member eq xs x then remove_dups eq xs else x :: remove_dups eq xs fun split_triples xs = fold (fn (a, b, c) => fn (axs, bxs, cxs) => (a :: axs, b :: bxs, c :: cxs)) xs ([], [], []) fun last2 [] = raise Empty | last2 [_] = raise Empty | last2 [x, y] = (x, y) | last2 (_ :: xs) = last2 xs fun split_last2 xs = let val (xs', x) = split_last xs val (xs'', y) = split_last xs' in (xs'', y, x) end fun map4 _ [] [] [] [] = [] | map4 f (x :: xs) (y :: ys) (z :: zs) (u :: us) = f x y z u :: map4 f xs ys zs us fun split_filter f [] = ([], []) | split_filter f (x :: xs) = let val (r, l) = split_filter f xs in if f x then (x :: r, l) else (r, x :: l) end (* to be used with left-infix binop-operations *) fun fold_left f [] z = z | fold_left f [x] z = x | fold_left f (x :: y :: xs) z = fold_left f (f (x, y) :: xs) z fun is_true @{term "Trueprop True"} = true | is_true _ = false fun dest_listT (Type (@{type_name list}, [T])) = T | dest_listT T = raise TYPE ("dest_listT: list type expected", [T], []) fun dest_fsetT (Type (@{type_name fset}, [T])) = T | dest_fsetT T = raise TYPE ("dest_fsetT: fset type expected", [T], []) fun mk_id trm = HOLogic.id_const (fastype_of trm)$ trm

fun mk_all (a, T) t =  Logic.all_const T $Abs (a, T, t) fun mk_All (a, T) t = HOLogic.all_const T$ Abs (a, T, t)

fun mk_exists (a, T) t =  HOLogic.exists_const T $Abs (a, T, t) fun case_sum_const ty1 ty2 ty3 = Const (@{const_name case_sum}, [ty1 --> ty3, ty2 --> ty3, Type (@{type_name sum}, [ty1, ty2])] ---> ty3) fun mk_case_sum trm1 trm2 = let val ([ty1], ty3) = strip_type (fastype_of trm1) val ty2 = domain_type (fastype_of trm2) in case_sum_const ty1 ty2 ty3$ trm1 $trm2 end fun mk_equiv r = r RS @{thm eq_reflection} fun safe_mk_equiv r = mk_equiv r handle Thm.THM _ => r fun mk_minus p = @{term "uminus::perm => perm"}$ p

fun mk_plus p q = @{term "plus::perm => perm => perm"} $p$ q

fun perm_ty ty = @{typ "perm"} --> ty --> ty
fun perm_const ty  = Const (@{const_name "permute"}, perm_ty ty)
fun mk_perm_ty ty p trm = perm_const ty $p$ trm
fun mk_perm p trm = mk_perm_ty (fastype_of trm) p trm

fun dest_perm (Const (@{const_name "permute"}, _) $p$ t) = (p, t)
| dest_perm t = raise TERM ("dest_perm", [t])

(** functions to deal with constants in local contexts **)

(* returns the fully qualified name of a constant *)
fun long_name ctxt name =
Const (s, _) => s
| _ => error ("Undeclared constant: " ^ quote name)

(* returns true iff the argument term is a fixed Free *)
fun is_fixed_Free ctxt (Free (s, _)) = Variable.is_fixed ctxt s
| is_fixed_Free _ _ = false

(* returns true iff c is a constant or fixed Free applied to
fixed parameters *)
fun is_fixed ctxt c =
let
val (c, args) = strip_comb c
in
(is_Const c orelse is_fixed_Free ctxt c)
andalso List.all (is_fixed_Free ctxt) args
end

(* splits a list into the longest prefix containing only elements
that satisfy p, and the rest of the list *)
fun chop_while p =
let
fun chop_while_aux acc [] =
(rev acc, [])
| chop_while_aux acc (x::xs) =
if p x then chop_while_aux (x::acc) xs else (rev acc, x::xs)
in
chop_while_aux []
end

(* takes a combination "c $fixed1$ ... $fixedN$ not-fixed $..." to the pair ("c$ fixed1 $...$ fixedN", ["not-fixed", ...]). *)
fun fixed_nonfixed_args ctxt c_args =
let
val (c, args)     = strip_comb c_args
val (frees, args) = chop_while (is_fixed_Free ctxt) args
val c_frees       = list_comb (c, frees)
in
(c_frees, args)
end

end (* structure *)

open Nominal_Basic;


File ‹nominal_thmdecls.ML›

(*  Title:      nominal_thmdecls.ML
Author:     Christian Urban
Author:     Tjark Weber

Infrastructure for the lemma collections "eqvts", "eqvts_raw".

Provides the attributes [eqvt] and [eqvt_raw], and the theorem
lists "eqvts" and "eqvts_raw".

The [eqvt] attribute expects a theorem of the form

?p ∙ (c ?x1 ?x2 ...) = c (?p ∙ ?x1) (?p ∙ ?x2) ...    (1)

or, if c is a relation with arity >= 1, of the form

c ?x1 ?x2 ... ==> c (?p ∙ ?x1) (?p ∙ ?x2) ...         (2)

[eqvt] will store this theorem in the form (1) or, if c
is a relation with arity >= 1, in the form

c (?p ∙ ?x1) (?p ∙ ?x2) ... = c ?x1 ?x2 ...           (3)

in "eqvts". (The orientation of (3) was chosen because
Isabelle's simplifier uses equations from left to right.)
[eqvt] will also derive and store the theorem

?p ∙ c == c                                           (4)

in "eqvts_raw".

(1)-(4) are all logically equivalent. We consider (1) and (2)
to be more end-user friendly, i.e., slightly more natural to
understand and prove, while (3) and (4) make the rewriting
system for equivariance more predictable and less prone to
looping in Isabelle.

The [eqvt_raw] attribute expects a theorem of the form (4),
and merely stores it in "eqvts_raw".

[eqvt_raw] is provided because certain equivariance theorems
would lead to looping when used for simplification in the form
(1): notably, equivariance of permute (infix ∙), i.e.,
?p ∙ (?q ∙ ?x) = (?p ∙ ?q) ∙ (?p ∙ ?x).

To support binders such as All/Ex/Ball/Bex etc., which are
typically applied to abstractions, argument terms ?xi (as well
as permuted arguments ?p ∙ ?xi) in (1)-(3) need not be eta-
contracted, i.e., they may be of the form "%z. ?xi z" or
"%z. (?p ∙ ?x) z", respectively.

For convenience, argument terms ?xi (as well as permuted
arguments ?p ∙ ?xi) in (1)-(3) may actually be tuples, e.g.,
"(?xi, ?xj)" or "(?p ∙ ?xi, ?p ∙ ?xj)", respectively.

In (1)-(4), "c" is either a (global) constant or a locally
fixed parameter, e.g., of a locale or type class.
*)

signature NOMINAL_THMDECLS =
sig
val eqvt_del: attribute
val eqvt_raw_del: attribute
val get_eqvts_thms: Proof.context -> thm list
val get_eqvts_raw_thms: Proof.context -> thm list
val eqvt_transform: Proof.context -> thm -> thm
val is_eqvt: Proof.context -> term -> bool
end;

structure Nominal_ThmDecls: NOMINAL_THMDECLS =
struct

structure EqvtData = Generic_Data
( type T = thm Item_Net.T;
val empty = Thm.full_rules;
val extend = I;
val merge = Item_Net.merge);

(* EqvtRawData is implemented with a Termtab (rather than an
Item_Net) so that we can efficiently decide whether a given
constant has a corresponding equivariance theorem stored, cf.
the function is_eqvt. *)
structure EqvtRawData = Generic_Data
( type T = thm Termtab.table;
val empty = Termtab.empty;
val extend = I;
val merge = Termtab.merge (K true));

val eqvts = Item_Net.content o EqvtData.get
val eqvts_raw = map snd o Termtab.dest o EqvtRawData.get

val _ =
Theory.setup

val get_eqvts_thms = eqvts o Context.Proof
val get_eqvts_raw_thms = eqvts_raw o Context.Proof

(** raw equivariance lemmas **)

(* Returns true iff an equivariance lemma exists in "eqvts_raw"
for a given term. *)
val is_eqvt =
Termtab.defined o EqvtRawData.get o Context.Proof

(* Returns c if thm is of the form (4), raises an error
otherwise. *)
fun key_of_raw_thm context thm =
let
fun error_msg () =
error
("Theorem must be of the form \"?p ∙ c ≡ c\", with c a constant or fixed parameter:\n" ^
Syntax.string_of_term (Context.proof_of context) (Thm.prop_of thm))
in
case Thm.prop_of thm of
Const (@{const_name Pure.eq}, _) $(Const (@{const_name "permute"}, _)$ p $c)$ c' =>
if is_Var p andalso is_fixed (Context.proof_of context) c andalso c aconv c' then
c
else
error_msg ()
| _ => error_msg ()
end

let
val c = key_of_raw_thm context thm
in
if Termtab.defined (EqvtRawData.get context) c then
warning ("Replacing existing raw equivariance theorem for \"" ^
Syntax.string_of_term (Context.proof_of context) c ^ "\".")
else ();
EqvtRawData.map (Termtab.update (c, thm)) context
end

fun del_raw_thm thm context =
let
val c = key_of_raw_thm context thm
in
if Termtab.defined (EqvtRawData.get context) c then
EqvtRawData.map (Termtab.delete c) context
else (
warning ("Cannot delete non-existing raw equivariance theorem for \"" ^
Syntax.string_of_term (Context.proof_of context) c ^ "\".");
context
)
end

(** adding/deleting lemmas to/from "eqvts" **)

(
if Item_Net.member (EqvtData.get context) thm then
warning ("Theorem already declared as equivariant:\n" ^
Syntax.string_of_term (Context.proof_of context) (Thm.prop_of thm))
else ();
EqvtData.map (Item_Net.update thm) context
)

fun del_thm thm context =
(
if Item_Net.member (EqvtData.get context) thm then
EqvtData.map (Item_Net.remove thm) context
else (
warning ("Cannot delete non-existing equivariance theorem:\n" ^
Syntax.string_of_term (Context.proof_of context) (Thm.prop_of thm));
context
)
)

(** transformation of equivariance lemmas **)

(* Transforms a theorem of the form (1) into the form (4). *)
local

fun tac ctxt thm =
let
val ss_thms = @{thms "permute_minus_cancel" "permute_prod.simps" "split_paired_all"}
in
REPEAT o FIRST'
[CHANGED o simp_tac (put_simpset HOL_basic_ss ctxt addsimps ss_thms),
resolve_tac ctxt [thm RS @{thm trans}],
resolve_tac ctxt @{thms trans[OF "permute_fun_def"]} THEN'
resolve_tac ctxt @{thms ext}]
end

in

fun thm_4_of_1 ctxt thm =
let
val (p, c) = thm |> Thm.prop_of |> HOLogic.dest_Trueprop
|> HOLogic.dest_eq |> fst |> dest_perm ||> fst o (fixed_nonfixed_args ctxt)
val goal = HOLogic.mk_Trueprop (HOLogic.mk_eq (mk_perm p c, c))
val ([goal', p'], ctxt') = Variable.import_terms false [goal, p] ctxt
in
Goal.prove ctxt [] [] goal' (fn {context, ...} => tac context thm 1)
|> singleton (Proof_Context.export ctxt' ctxt)
|> (fn th => th RS @{thm "eq_reflection"})
|> zero_var_indexes
end
handle TERM _ =>
raise THM ("thm_4_of_1", 0, [thm])

end (* local *)

(* Transforms a theorem of the form (2) into the form (1). *)
local

fun tac ctxt thm thm' =
let
val ss_thms = @{thms "permute_minus_cancel"(2)}
in
EVERY' [resolve_tac ctxt @{thms iffI},
dresolve_tac ctxt @{thms permute_boolE},
resolve_tac ctxt [thm],
assume_tac ctxt,
resolve_tac ctxt @{thms permute_boolI},
dresolve_tac ctxt [thm'],
full_simp_tac (put_simpset HOL_basic_ss ctxt addsimps ss_thms)]
end

in

fun thm_1_of_2 ctxt thm =
let
val (prem, concl) = thm |> Thm.prop_of |> Logic.dest_implies |> apply2 HOLogic.dest_Trueprop
(* since argument terms "?p ∙ ?x1" may actually be eta-expanded
or tuples, we need the following function to find ?p *)
fun find_perm (Const (@{const_name "permute"}, _) $(p as Var _)$ _) = p
| find_perm (Const (@{const_name "Pair"}, _) $x$ _) = find_perm x
| find_perm (Abs (_, _, body)) = find_perm body
| find_perm _ = raise THM ("thm_3_of_2", 0, [thm])
val p = concl |> dest_comb |> snd |> find_perm
val goal = HOLogic.mk_Trueprop (HOLogic.mk_eq (mk_perm p prem, concl))
val ([goal', p'], ctxt') = Variable.import_terms false [goal, p] ctxt
val thm' = infer_instantiate ctxt' [(#1 (dest_Var p), Thm.cterm_of ctxt' (mk_minus p'))] thm
in
Goal.prove ctxt' [] [] goal' (fn {context = ctxt'', ...} => tac ctxt'' thm thm' 1)
|> singleton (Proof_Context.export ctxt' ctxt)
end
handle TERM _ =>
raise THM ("thm_1_of_2", 0, [thm])

end (* local *)

(* Transforms a theorem of the form (1) into the form (3). *)
fun thm_3_of_1 _ thm =
(thm RS (@{thm "permute_bool_def"} RS @{thm "sym"} RS @{thm "trans"}) RS @{thm "sym"})
|> zero_var_indexes

local
val msg = cat_lines
["Equivariance theorem must be of the form",
"  ?p ∙ (c ?x1 ?x2 ...) = c (?p ∙ ?x1) (?p ∙ ?x2) ...",
"or, if c is a relation with arity >= 1, of the form",
"  c ?x1 ?x2 ... ==> c (?p ∙ ?x1) (?p ∙ ?x2) ..."]
in

(* Transforms a theorem of the form (1) or (2) into the form (4). *)
fun eqvt_transform ctxt thm =
(case Thm.prop_of thm of @{const "Trueprop"} $_ => thm_4_of_1 ctxt thm | @{const Pure.imp}$ _ $_ => thm_4_of_1 ctxt (thm_1_of_2 ctxt thm) | _ => error msg) handle THM _ => error msg (* Transforms a theorem of the form (1) into theorems of the form (1) (or, if c is a relation with arity >= 1, of the form (3)) and (4); transforms a theorem of the form (2) into theorems of the form (3) and (4). *) fun eqvt_and_raw_transform ctxt thm = (case Thm.prop_of thm of @{const "Trueprop"}$ (Const (@{const_name "HOL.eq"}, _) $_$ c_args) =>
let
val th' =
if fastype_of c_args = @{typ "bool"}
andalso (not o null) (snd (fixed_nonfixed_args ctxt c_args)) then
thm_3_of_1 ctxt thm
else
thm
in
(th', thm_4_of_1 ctxt thm)
end
| @{const Pure.imp} $_$ _ =>
let
val th1 = thm_1_of_2 ctxt thm
in
(thm_3_of_1 ctxt th1, thm_4_of_1 ctxt th1)
end
| _ =>
error msg)
handle THM _ =>
error msg

end (* local *)

(** attributes **)

val eqvt_raw_del = Thm.declaration_attribute del_raw_thm

Thm.declaration_attribute
(fn thm => fn context =>
let
val (eqvt, raw) = eqvt_and_raw_transform (Context.proof_of context) thm
in
context |> eqvt_fn eqvt |> raw_fn raw
end)

val eqvt_del = eqvt_add_or_del del_thm del_raw_thm

val _ =
Theory.setup
"Declaration of equivariance lemmas - they will automatically be brought into the form ?p ∙ c ≡ c" #>
"Declaration of raw equivariance lemmas - no transformation is performed")

end;


File ‹nominal_permeq.ML›

(*  Title:      nominal_permeq.ML
Author:     Christian Urban
Author:     Brian Huffman
*)

signature NOMINAL_PERMEQ =
sig
datatype eqvt_config =
Eqvt_Config of {strict_mode: bool, pre_thms: thm list, post_thms: thm list, excluded: string list}

val eqvt_relaxed_config: eqvt_config
val eqvt_strict_config: eqvt_config
val addpres : (eqvt_config * thm list) -> eqvt_config
val addposts : (eqvt_config * thm list) -> eqvt_config
val addexcls : (eqvt_config * string list) -> eqvt_config
val delpres : eqvt_config -> eqvt_config
val delposts : eqvt_config -> eqvt_config

val eqvt_conv: Proof.context -> eqvt_config -> conv
val eqvt_rule: Proof.context -> eqvt_config -> thm -> thm
val eqvt_tac: Proof.context -> eqvt_config -> int -> tactic

val perm_simp_meth: thm list * string list -> Proof.context -> Proof.method
val perm_strict_simp_meth: thm list * string list -> Proof.context -> Proof.method
val args_parser: (thm list * string list) context_parser

val trace_eqvt: bool Config.T
end;

(*

- eqvt_tac and eqvt_rule take a  list of theorems which
are first tried to simplify permutations

- the string list contains constants that should not be
analysed (for example there is no raw eqvt-lemma for
the constant The); therefore it should not be analysed

- setting [[trace_eqvt = true]] switches on tracing
information

*)

structure Nominal_Permeq: NOMINAL_PERMEQ =
struct

open Nominal_ThmDecls;

datatype eqvt_config = Eqvt_Config of
{strict_mode: bool, pre_thms: thm list, post_thms: thm list, excluded: string list}

fun (Eqvt_Config {strict_mode, pre_thms, post_thms, excluded}) addpres thms =
Eqvt_Config { strict_mode = strict_mode,
pre_thms = thms @ pre_thms,
post_thms = post_thms,
excluded = excluded }

fun (Eqvt_Config {strict_mode, pre_thms, post_thms, excluded}) addposts thms =
Eqvt_Config { strict_mode = strict_mode,
pre_thms = pre_thms,
post_thms = thms @ post_thms,
excluded = excluded }

fun (Eqvt_Config {strict_mode, pre_thms, post_thms, excluded}) addexcls excls =
Eqvt_Config { strict_mode = strict_mode,
pre_thms = pre_thms,
post_thms = post_thms,
excluded = excls @ excluded }

fun delpres (Eqvt_Config {strict_mode, pre_thms, post_thms, excluded}) =
Eqvt_Config { strict_mode = strict_mode,
pre_thms = [],
post_thms = post_thms,
excluded = excluded }

fun delposts (Eqvt_Config {strict_mode, pre_thms, post_thms, excluded}) =
Eqvt_Config { strict_mode = strict_mode,
pre_thms = pre_thms,
post_thms = [],
excluded = excluded }

val eqvt_relaxed_config =
Eqvt_Config { strict_mode = false,
pre_thms = @{thms eqvt_bound},
post_thms = @{thms permute_pure},
excluded = [] }

val eqvt_strict_config =
Eqvt_Config { strict_mode = true,
pre_thms = @{thms eqvt_bound},
post_thms = @{thms permute_pure},
excluded = [] }

(* tracing infrastructure *)
val trace_eqvt = Attrib.setup_config_bool @{binding "trace_eqvt"} (K false);

fun trace_enabled ctxt = Config.get ctxt trace_eqvt

fun trace_msg ctxt result =
let
val lhs_str = Syntax.string_of_term ctxt (Thm.term_of (Thm.lhs_of result))
val rhs_str = Syntax.string_of_term ctxt (Thm.term_of (Thm.rhs_of result))
in
warning (Pretty.string_of (Pretty.strs ["Rewriting", lhs_str, "to", rhs_str]))
end

fun trace_conv ctxt conv ctrm =
let
val result = conv ctrm
in
if Thm.is_reflexive result
then result
else (trace_msg ctxt result; result)
end

(* this conversion always fails, but prints
out the analysed term  *)
fun trace_info_conv ctxt ctrm =
let
val trm = Thm.term_of ctrm
val _ = case (head_of trm) of
@{const "Trueprop"} => ()
| _ => warning ("Analysing term " ^ Syntax.string_of_term ctxt trm)
in
Conv.no_conv ctrm
end

(* conversion for applications *)
fun eqvt_apply_conv ctrm =
case Thm.term_of ctrm of
Const (@{const_name "permute"}, _) $_$ (_ $_) => let val (perm, t) = Thm.dest_comb ctrm val (_, p) = Thm.dest_comb perm val (f, x) = Thm.dest_comb t val a = Thm.ctyp_of_cterm x; val b = Thm.ctyp_of_cterm t; val ty_insts = map SOME [b, a] val term_insts = map SOME [p, f, x] in Thm.instantiate' ty_insts term_insts @{thm eqvt_apply} end | _ => Conv.no_conv ctrm (* conversion for lambdas *) fun eqvt_lambda_conv ctrm = case Thm.term_of ctrm of Const (@{const_name "permute"}, _)$ _ $(Abs _) => Conv.rewr_conv @{thm eqvt_lambda} ctrm | _ => Conv.no_conv ctrm (* conversion that raises an error or prints a warning message, if a permutation on a constant or application cannot be analysed *) fun is_excluded excluded (Const (a, _)) = member (op=) excluded a | is_excluded _ _ = false fun progress_info_conv ctxt strict_flag excluded ctrm = let fun msg trm = if is_excluded excluded trm then () else (if strict_flag then error else warning) ("Cannot solve equivariance for " ^ (Syntax.string_of_term ctxt trm)) val _ = case Thm.term_of ctrm of Const (@{const_name "permute"}, _)$ _ $(trm as Const _) => msg trm | Const (@{const_name "permute"}, _)$ _ $(trm as _$ _) => msg trm
| _ => ()
in
Conv.all_conv ctrm
end

(* main conversion *)
fun main_eqvt_conv ctxt config ctrm =
let
val Eqvt_Config {strict_mode, pre_thms, post_thms, excluded} = config

val first_conv_wrapper =
if trace_enabled ctxt
then Conv.first_conv o (cons (trace_info_conv ctxt)) o (map (trace_conv ctxt))
else Conv.first_conv

val all_pre_thms = map safe_mk_equiv (pre_thms @ get_eqvts_raw_thms ctxt)
val all_post_thms = map safe_mk_equiv post_thms
in
first_conv_wrapper
[ Conv.rewrs_conv all_pre_thms,
eqvt_apply_conv,
eqvt_lambda_conv,
Conv.rewrs_conv all_post_thms,
progress_info_conv ctxt strict_mode excluded
] ctrm
end

(* the eqvt-conversion first eta-normalises goals in
order to avoid problems with inductions in the
equivariance command. *)
fun eqvt_conv ctxt config =
Conv.top_conv (fn ctxt => Thm.eta_conversion then_conv (main_eqvt_conv ctxt config)) ctxt

(* thms rewriter *)
fun eqvt_rule ctxt config =
Conv.fconv_rule (eqvt_conv ctxt config)

(* tactic *)
fun eqvt_tac ctxt config =
CONVERSION (eqvt_conv ctxt config)

(** methods **)
fun unless_more_args scan = Scan.unless (Scan.lift ((Args.$$"exclude") -- Args.colon)) scan val add_thms_parser = Scan.optional (Scan.lift (Args.add -- Args.colon) |-- Scan.repeat (unless_more_args Attrib.multi_thm) >> flat) []; val exclude_consts_parser = Scan.optional (Scan.lift ((Args.$$$"exclude") -- Args.colon) |-- (Scan.repeat (Args.const {proper = true, strict = true}))) [] val args_parser = add_thms_parser -- exclude_consts_parser fun perm_simp_meth (thms, consts) ctxt = SIMPLE_METHOD (HEADGOAL (eqvt_tac ctxt (eqvt_relaxed_config addpres thms addexcls consts))) fun perm_strict_simp_meth (thms, consts) ctxt = SIMPLE_METHOD (HEADGOAL (eqvt_tac ctxt (eqvt_strict_config addpres thms addexcls consts))) end; (* structure *)  File ‹nominal_library.ML› (* Title: nominal_library.ML Author: Christian Urban Library functions for nominal. *) signature NOMINAL_LIBRARY = sig val mk_sort_of: term -> term val atom_ty: typ -> typ val atom_const: typ -> term val mk_atom_ty: typ -> term -> term val mk_atom: term -> term val mk_atom_set_ty: typ -> term -> term val mk_atom_set: term -> term val mk_atom_fset_ty: typ -> term -> term val mk_atom_fset: term -> term val mk_atom_list_ty: typ -> term -> term val mk_atom_list: term -> term val is_atom: Proof.context -> typ -> bool val is_atom_set: Proof.context -> typ -> bool val is_atom_fset: Proof.context -> typ -> bool val is_atom_list: Proof.context -> typ -> bool val to_set_ty: typ -> term -> term val to_set: term -> term val atomify_ty: Proof.context -> typ -> term -> term val atomify: Proof.context -> term -> term val setify_ty: Proof.context -> typ -> term -> term val setify: Proof.context -> term -> term val listify_ty: Proof.context -> typ -> term -> term val listify: Proof.context -> term -> term val fresh_ty: typ -> typ val fresh_const: typ -> term val mk_fresh_ty: typ -> term -> term -> term val mk_fresh: term -> term -> term val fresh_star_ty: typ -> typ val fresh_star_const: typ -> term val mk_fresh_star_ty: typ -> term -> term -> term val mk_fresh_star: term -> term -> term val supp_ty: typ -> typ val supp_const: typ -> term val mk_supp_ty: typ -> term -> term val mk_supp: term -> term val supp_rel_ty: typ -> typ val supp_rel_const: typ -> term val mk_supp_rel_ty: typ -> term -> term -> term val mk_supp_rel: term -> term -> term val supports_const: typ -> term val mk_supports_ty: typ -> term -> term -> term val mk_supports: term -> term -> term val finite_const: typ -> term val mk_finite_ty: typ -> term -> term val mk_finite: term -> term val mk_diff: term * term -> term val mk_append: term * term -> term val mk_union: term * term -> term val fold_union: term list -> term val fold_append: term list -> term val mk_conj: term * term -> term val fold_conj: term list -> term val fold_conj_balanced: term list -> term (* functions for de-Bruijn open terms *) val mk_binop_env: typ list -> string -> term * term -> term val mk_union_env: typ list -> term * term -> term val fold_union_env: typ list -> term list -> term (* fresh arguments for a term *) val fresh_args: Proof.context -> term -> term list (* some logic operations *) val strip_full_horn: term -> (string * typ) list * term list * term val mk_full_horn: (string * typ) list -> term list -> term -> term (* datatype operations *) type cns_info = (term * typ * typ list * bool list) list val all_dtyp_constrs_types: Old_Datatype_Aux.descr -> cns_info list (* tactics for function package *) val size_ss: simpset val pat_completeness_simp: thm list -> Proof.context -> tactic val prove_termination_ind: Proof.context -> int -> tactic val prove_termination_fun: thm list -> Proof.context -> Function.info * local_theory (* transformations of premises in inductions *) val transform_prem1: Proof.context -> string list -> thm -> thm val transform_prem2: Proof.context -> string list -> thm -> thm (* transformation into the object logic *) val atomize: Proof.context -> thm -> thm val atomize_rule: Proof.context -> int -> thm -> thm val atomize_concl: Proof.context -> thm -> thm (* applies a tactic to a formula composed of conjunctions *) val conj_tac: Proof.context -> (int -> tactic) -> int -> tactic end structure Nominal_Library: NOMINAL_LIBRARY = struct fun mk_sort_of t = @{term "sort_of"}$ t;

fun atom_ty ty = ty --> @{typ "atom"};
fun atom_const ty = Const (@{const_name "atom"}, atom_ty ty)
fun mk_atom_ty ty t = atom_const ty $t; fun mk_atom t = mk_atom_ty (fastype_of t) t; fun mk_atom_set_ty ty t = let val atom_ty = HOLogic.dest_setT ty val img_ty = (atom_ty --> @{typ atom}) --> ty --> @{typ "atom set"}; in Const (@{const_name image}, img_ty)$ atom_const atom_ty $t end fun mk_atom_fset_ty ty t = let val atom_ty = dest_fsetT ty val fmap_ty = (atom_ty --> @{typ atom}) --> ty --> @{typ "atom fset"}; in Const (@{const_name fimage}, fmap_ty)$ atom_const atom_ty $t end fun mk_atom_list_ty ty t = let val atom_ty = dest_listT ty val map_ty = (atom_ty --> @{typ atom}) --> ty --> @{typ "atom list"} in Const (@{const_name map}, map_ty)$ atom_const atom_ty $t end fun mk_atom_set t = mk_atom_set_ty (fastype_of t) t fun mk_atom_fset t = mk_atom_fset_ty (fastype_of t) t fun mk_atom_list t = mk_atom_list_ty (fastype_of t) t (* coerces a list into a set *) fun to_set_ty ty t = case ty of @{typ "atom list"} => @{term "set :: atom list => atom set"}$ t
| @{typ "atom fset"} => @{term "fset :: atom fset => atom set"} $t | _ => t fun to_set t = to_set_ty (fastype_of t) t (* testing for concrete atom types *) fun is_atom ctxt ty = Sign.of_sort (Proof_Context.theory_of ctxt) (ty, @{sort at_base}) fun is_atom_set ctxt (Type ("fun", [ty, @{typ bool}])) = is_atom ctxt ty | is_atom_set _ _ = false; fun is_atom_fset ctxt (Type (@{type_name "fset"}, [ty])) = is_atom ctxt ty | is_atom_fset _ _ = false; fun is_atom_list ctxt (Type (@{type_name "list"}, [ty])) = is_atom ctxt ty | is_atom_list _ _ = false (* functions that coerce singletons, sets, fsets and lists of concrete atoms into general atoms sets / lists *) fun atomify_ty ctxt ty t = if is_atom ctxt ty then mk_atom_ty ty t else if is_atom_set ctxt ty then mk_atom_set_ty ty t else if is_atom_fset ctxt ty then mk_atom_fset_ty ty t else if is_atom_list ctxt ty then mk_atom_list_ty ty t else raise TERM ("atomify: term is not an atom, set or list of atoms", [t]) fun setify_ty ctxt ty t = if is_atom ctxt ty then HOLogic.mk_set @{typ atom} [mk_atom_ty ty t] else if is_atom_set ctxt ty then mk_atom_set_ty ty t else if is_atom_fset ctxt ty then @{term "fset :: atom fset => atom set"}$ mk_atom_fset_ty ty t
else if is_atom_list ctxt ty
then @{term "set :: atom list => atom set"} $mk_atom_list_ty ty t else raise TERM ("setify: term is not an atom, set or list of atoms", [t]) fun listify_ty ctxt ty t = if is_atom ctxt ty then HOLogic.mk_list @{typ atom} [mk_atom_ty ty t] else if is_atom_list ctxt ty then mk_atom_list_ty ty t else raise TERM ("listify: term is not an atom or list of atoms", [t]) fun atomify ctxt t = atomify_ty ctxt (fastype_of t) t fun setify ctxt t = setify_ty ctxt (fastype_of t) t fun listify ctxt t = listify_ty ctxt (fastype_of t) t fun fresh_ty ty = [@{typ atom}, ty] ---> @{typ bool} fun fresh_const ty = Const (@{const_name fresh}, fresh_ty ty) fun mk_fresh_ty ty t1 t2 = fresh_const ty$ t1 $t2 fun mk_fresh t1 t2 = mk_fresh_ty (fastype_of t2) t1 t2 fun fresh_star_ty ty = [@{typ "atom set"}, ty] ---> @{typ bool} fun fresh_star_const ty = Const (@{const_name fresh_star}, fresh_star_ty ty) fun mk_fresh_star_ty ty t1 t2 = fresh_star_const ty$ t1 $t2 fun mk_fresh_star t1 t2 = mk_fresh_star_ty (fastype_of t2) t1 t2 fun supp_ty ty = ty --> @{typ "atom set"}; fun supp_const ty = Const (@{const_name supp}, supp_ty ty) fun mk_supp_ty ty t = supp_const ty$ t
fun mk_supp t = mk_supp_ty (fastype_of t) t

fun supp_rel_ty ty = ([ty, ty] ---> @{typ bool}) --> ty --> @{typ "atom set"};
fun supp_rel_const ty = Const (@{const_name supp_rel}, supp_rel_ty ty)
fun mk_supp_rel_ty ty r t = supp_rel_const ty $r$ t
fun mk_supp_rel r t = mk_supp_rel_ty (fastype_of t) r t

fun supports_const ty = Const (@{const_name supports}, [@{typ "atom set"}, ty] ---> @{typ bool});
fun mk_supports_ty ty t1 t2 = supports_const ty $t1$ t2;
fun mk_supports t1 t2 = mk_supports_ty (fastype_of t2) t1 t2;

fun finite_const ty = Const (@{const_name finite}, ty --> @{typ bool})
fun mk_finite_ty ty t = finite_const ty $t fun mk_finite t = mk_finite_ty (fastype_of t) t (* functions that construct differences, appends and unions but avoid producing empty atom sets or empty atom lists *) fun mk_diff (@{term "{}::atom set"}, _) = @{term "{}::atom set"} | mk_diff (t1, @{term "{}::atom set"}) = t1 | mk_diff (@{term "set ([]::atom list)"}, _) = @{term "set ([]::atom list)"} | mk_diff (t1, @{term "set ([]::atom list)"}) = t1 | mk_diff (t1, t2) = HOLogic.mk_binop @{const_name minus} (t1, t2) fun mk_append (t1, @{term "[]::atom list"}) = t1 | mk_append (@{term "[]::atom list"}, t2) = t2 | mk_append (t1, t2) = HOLogic.mk_binop @{const_name "append"} (t1, t2) fun mk_union (t1, @{term "{}::atom set"}) = t1 | mk_union (@{term "{}::atom set"}, t2) = t2 | mk_union (t1, @{term "set ([]::atom list)"}) = t1 | mk_union (@{term "set ([]::atom list)"}, t2) = t2 | mk_union (t1, t2) = HOLogic.mk_binop @{const_name "sup"} (t1, t2) fun fold_union trms = fold_rev (curry mk_union) trms @{term "{}::atom set"} fun fold_append trms = fold_rev (curry mk_append) trms @{term "[]::atom list"} fun mk_conj (t1, @{term "True"}) = t1 | mk_conj (@{term "True"}, t2) = t2 | mk_conj (t1, t2) = HOLogic.mk_conj (t1, t2) fun fold_conj trms = fold_rev (curry mk_conj) trms @{term "True"} fun fold_conj_balanced ts = Balanced_Tree.make HOLogic.mk_conj ts (* functions for de-Bruijn open terms *) fun mk_binop_env tys c (t, u) = let val ty = fastype_of1 (tys, t) in Const (c, [ty, ty] ---> ty)$ t $u end fun mk_union_env tys (t1, @{term "{}::atom set"}) = t1 | mk_union_env tys (@{term "{}::atom set"}, t2) = t2 | mk_union_env tys (t1, @{term "set ([]::atom list)"}) = t1 | mk_union_env tys (@{term "set ([]::atom list)"}, t2) = t2 | mk_union_env tys (t1, t2) = mk_binop_env tys @{const_name "sup"} (t1, t2) fun fold_union_env tys trms = fold_left (mk_union_env tys) trms @{term "{}::atom set"} (* produces fresh arguments for a term *) fun fresh_args ctxt f = f |> fastype_of |> binder_types |> map (pair "z") |> Variable.variant_frees ctxt [f] |> map Free (** some logic operations **) (* decompses a formula into params, premises and a conclusion *) fun strip_full_horn trm = let fun strip_outer_params (Const (@{const_name Pure.all}, _)$ Abs (a, T, t)) = strip_outer_params t |>> cons (a, T)
| strip_outer_params B = ([], B)

val (params, body) = strip_outer_params trm
val (prems, concl) = Logic.strip_horn body
in
(params, prems, concl)
end

(* composes a formula out of params, premises and a conclusion *)
fun mk_full_horn params prems concl =
Logic.list_implies (prems, concl)
|> fold_rev mk_all params

(** datatypes **)

(* constructor infos *)
type cns_info = (term * typ * typ list * bool list) list

(*  - term for constructor constant
- type of the constructor
- types of the arguments
- flags indicating whether the argument is recursive
*)

(* returns info about constructors in a datatype *)
fun all_dtyp_constrs_info descr =
map (fn (_, (ty, vs, constrs)) => map (pair (ty, vs)) constrs) descr

(* returns the constants of the constructors plus the
corresponding type and types of arguments *)
fun all_dtyp_constrs_types descr =
let
fun aux ((ty_name, vs), (cname, args)) =
let
val vs_tys = map (Old_Datatype_Aux.typ_of_dtyp descr) vs
val ty = Type (ty_name, vs_tys)
val arg_tys = map (Old_Datatype_Aux.typ_of_dtyp descr) args
val is_rec = map Old_Datatype_Aux.is_rec_type args
in
(Const (cname, arg_tys ---> ty), ty, arg_tys, is_rec)
end
in
map (map aux) (all_dtyp_constrs_info descr)
end

(** function package tactics **)

fun pat_completeness_simp simps ctxt =
let
val simpset =
put_simpset HOL_basic_ss ctxt addsimps (@{thms sum.inject sum.distinct} @ simps)
in
Pat_Completeness.pat_completeness_tac ctxt 1
THEN ALLGOALS (asm_full_simp_tac simpset)
end

(* simpset for size goals *)
val size_ss =
simpset_of (put_simpset HOL_ss @{context}
zero_less_Suc prod.size(1) mult_Suc_right})

val natT = @{typ nat}

fun size_prod_const T1 T2 =
let
val T1_fun = T1 --> natT
val T2_fun = T2 --> natT
val prodT = HOLogic.mk_prodT (T1, T2)
in
Const (@{const_name size_prod}, [T1_fun, T2_fun, prodT] ---> natT)
end

fun snd_const T1 T2 =
Const (@{const_name Product_Type.snd}, HOLogic.mk_prodT (T1, T2) --> T2)

fun mk_measure_trm f ctxt T =
HOLogic.dest_setT T
|> fst o HOLogic.dest_prodT
|> f
|> curry (op $) (Const (@{const_name "measure"}, dummyT)) |> Syntax.check_term ctxt (* wf-goal arising in induction_schema *) fun prove_termination_ind ctxt = let fun mk_size_measure T = case T of (Type (@{type_name Sum_Type.sum}, [T1, T2])) => Sum_Tree.mk_sumcase T1 T2 natT (mk_size_measure T1) (mk_size_measure T2) | (Type (@{type_name Product_Type.prod}, [T1, T2])) => HOLogic.mk_comp (mk_size_measure T2, snd_const T1 T2) | _ => HOLogic.size_const T val measure_trm = mk_measure_trm (mk_size_measure) ctxt in Function_Relation.relation_tac ctxt measure_trm end (* wf-goal arising in function definitions *) fun prove_termination_fun size_simps ctxt = let fun mk_size_measure T = case T of (Type (@{type_name Sum_Type.sum}, [T1, T2])) => Sum_Tree.mk_sumcase T1 T2 natT (mk_size_measure T1) (mk_size_measure T2) | (Type (@{type_name Product_Type.prod}, [T1, T2])) => size_prod_const T1 T2$ (mk_size_measure T1) $(mk_size_measure T2) | _ => HOLogic.size_const T val measure_trm = mk_measure_trm (mk_size_measure) ctxt val tac = Function_Relation.relation_tac ctxt measure_trm THEN_ALL_NEW simp_tac (put_simpset size_ss ctxt addsimps size_simps) in Function.prove_termination NONE (HEADGOAL tac) ctxt end (** transformations of premises (in inductive proofs) **) (* given the theorem F[t]; proves the theorem F[f t] - F needs to be monotone - f returns either SOME for a term it fires on and NONE elsewhere *) fun map_term f t = (case f t of NONE => map_term' f t | x => x) and map_term' f (t$ u) =
(case (map_term f t, map_term f u) of
(NONE, NONE) => NONE
| (SOME t'', NONE) => SOME (t'' $u) | (NONE, SOME u'') => SOME (t$ u'')
| (SOME t'', SOME u'') => SOME (t'' $u'')) | map_term' f (Abs (s, T, t)) = (case map_term f t of NONE => NONE | SOME t'' => SOME (Abs (s, T, t''))) | map_term' _ _ = NONE; fun map_thm_tac ctxt tac thm = let val monos = Inductive.get_monos ctxt val simpset = put_simpset HOL_basic_ss ctxt addsimps @{thms split_def} in EVERY [cut_facts_tac [thm] 1, eresolve_tac ctxt [rev_mp] 1, REPEAT_DETERM (FIRSTGOAL (simp_tac simpset THEN' resolve_tac ctxt monos)), REPEAT_DETERM (resolve_tac ctxt [impI] 1 THEN (assume_tac ctxt 1 ORELSE tac))] end fun map_thm ctxt f tac thm = let val opt_goal_trm = map_term f (Thm.prop_of thm) in case opt_goal_trm of NONE => thm | SOME goal => Goal.prove ctxt [] [] goal (fn _ => map_thm_tac ctxt tac thm) end (* inductive premises can be of the form R ... /\ P ...; split_conj_i picks out the part R or P part *) fun split_conj1 names (Const (@{const_name "conj"}, _)$ f1 $_) = (case head_of f1 of Const (name, _) => if member (op =) names name then SOME f1 else NONE | _ => NONE) | split_conj1 _ _ = NONE; fun split_conj2 names (Const (@{const_name "conj"}, _)$ f1 $f2) = (case head_of f1 of Const (name, _) => if member (op =) names name then SOME f2 else NONE | _ => NONE) | split_conj2 _ _ = NONE; fun transform_prem1 ctxt names thm = map_thm ctxt (split_conj1 names) (eresolve_tac ctxt [conjunct1] 1) thm fun transform_prem2 ctxt names thm = map_thm ctxt (split_conj2 names) (eresolve_tac ctxt [conjunct2] 1) thm (* transforms a theorem into one of the object logic *) fun atomize ctxt = Conv.fconv_rule (Object_Logic.atomize ctxt) o forall_intr_vars; fun atomize_rule ctxt i thm = Conv.fconv_rule (Conv.concl_conv i (Object_Logic.atomize ctxt)) thm fun atomize_concl ctxt thm = atomize_rule ctxt (length (Thm.prems_of thm)) thm (* applies a tactic to a formula composed of conjunctions *) fun conj_tac ctxt tac i = let fun select (trm, i) = case trm of @{term "Trueprop"}$ t' => select (t', i)
| @{term "(&)"} $_$ _ =>
EVERY' [resolve_tac ctxt @{thms conjI}, RANGE [conj_tac ctxt tac, conj_tac ctxt tac]] i
| _ => tac i
in
SUBGOAL select i
end

end (* structure *)

open Nominal_Library;


File ‹nominal_atoms.ML›

(*  Title:      nominal_atoms/ML
Authors:    Brian Huffman, Christian Urban

Command for defining concrete atom types.

At the moment, only single-sorted atom types
are supported.
*)

signature ATOM_DECL =
sig
val add_atom_decl: (binding * (binding option)) -> theory -> theory
end;

structure Atom_Decl : ATOM_DECL =
struct

val simp_attr = Attrib.internal (K Simplifier.simp_add)

fun atom_decl_set (str : string) : term =
let
val a = Free ("a", @{typ atom});
val s = Const (@{const_name "Sort"}, @{typ "string => atom_sort list => atom_sort"})
$HOLogic.mk_string str$ HOLogic.nil_const @{typ "atom_sort"};
in
HOLogic.mk_Collect ("a", @{typ atom}, HOLogic.mk_eq (mk_sort_of a, s))
end

fun add_atom_decl (name : binding, arg : binding option) (thy : theory) =
let
val str = Sign.full_name thy name;

(* typedef *)
val set = atom_decl_set str;
fun tac ctxt = resolve_tac ctxt @{thms exists_eq_simple_sort} 1;
val ((full_tname, info as ({Rep_name, Abs_name, ...}, {type_definition, ...})), thy) =
thy
|> Named_Target.theory_map_result (apsnd o Typedef.transform_info)

(* definition of atom and permute *)
val newT = #abs_type (fst info);
val RepC = Const (Rep_name, newT --> @{typ atom});
val AbsC = Const (Abs_name, @{typ atom} --> newT);
val a = Free ("a", newT);
val p = Free ("p", @{typ perm});
val atom_eqn =
HOLogic.mk_Trueprop (HOLogic.mk_eq (mk_atom a, RepC $a)); val permute_eqn = HOLogic.mk_Trueprop (HOLogic.mk_eq (mk_perm p a, AbsC$ (mk_perm p (RepC \$ a))));
val atom_def_name =
Binding.prefix_name "atom_" (Binding.suffix_name "_def" name);
val sort_thm_name =
Binding.prefix_name "atom_" (Binding.suffix_name "_sort" name);
val permute_def_name =
Binding.prefix_name "permute_" (Binding.suffix_name "_def" name);

(* at class instance *)
val lthy =
Class.instantiation ([full_tname], [], @{sort at}) thy;
val ((_, (_, permute_ldef)), lthy) =
Specification.definition NONE [] [] ((permute_def_name, []), permute_eqn) lthy;
val ((_, (_, atom_ldef)), lthy) =
Specification.definition NONE [] [] ((atom_def_name, []), atom_eqn) lthy;
val ctxt_thy = Proof_Context.init_global (Proof_Context.theory_of lthy);
val permute_def = singleton (Proof_Context.export lthy ctxt_thy) permute_ldef;
val atom_def = singleton (Proof_Context.export lthy ctxt_thy) atom_ldef;
val class_thm = @{thm at_class} OF [type_definition, atom_def, permute_def];
val sort_thm = @{thm at_class_sort} OF [type_definition, atom_def]
val thy = lthy
|> snd o (Local_Theory.note ((sort_thm_name, [simp_attr]), [sort_thm]))
|> Class.prove_instantiation_instance (fn ctxt => resolve_tac ctxt [class_thm] 1)
|> Local_Theory.exit_global;
in
thy
end;

(** outer syntax **)
val _ =
Outer_Syntax.command @{command_keyword atom_decl}
"declaration of a concrete atom type"
((Parse.binding -- Scan.option (Args.parens (Parse.binding))) >>

end;


File ‹nominal_eqvt.ML›

(*  Title:      nominal_eqvt.ML
Author:     Stefan Berghofer (original code)
Author:     Christian Urban
Author:     Tjark Weber

Automatic proofs for equivariance of inductive predicates.
*)

signature NOMINAL_EQVT =
sig
val raw_equivariance: Proof.context -> term list -> thm -> thm list -> thm list
val equivariance_cmd: string -> Proof.context -> local_theory
end

structure Nominal_Eqvt : NOMINAL_EQVT =
struct

open Nominal_Permeq;
open Nominal_ThmDecls;

fun atomize_conv ctxt =
Raw_Simplifier.rewrite_cterm (true, false, false) (K (K NONE))
(put_simpset HOL_basic_ss ctxt addsimps @{thms induct_atomize})

fun atomize_intr ctxt = Conv.fconv_rule (Conv.prems_conv ~1 (atomize_conv ctxt))

fun atomize_induct ctxt = Conv.fconv_rule (Conv.prems_conv ~1
(Conv.params_conv ~1 (K (Conv.prems_conv ~1 (atomize_conv ctxt))) ctxt))

(** equivariance tactics **)

fun eqvt_rel_single_case_tac ctxt pred_names pi intro =
let
val cpi = Thm.cterm_of ctxt pi
val pi_intro_rule = Thm.instantiate' [] [NONE, SOME cpi] @{thm permute_boolI}
val eqvt_sconfig = eqvt_strict_config addexcls pred_names
in
eqvt_tac ctxt eqvt_sconfig THEN'
SUBPROOF (fn {prems, context as ctxt, ...} =>
let
val simps1 =
put_simpset HOL_basic_ss ctxt addsimps @{thms permute_fun_def permute_self split_paired_all}
val simps2 =
put_simpset HOL_basic_ss ctxt addsimps @{thms permute_bool_def permute_minus_cancel(2)}
val prems' = map (transform_prem2 ctxt pred_names) prems
val prems'' = map (fn thm => eqvt_rule ctxt eqvt_sconfig (thm RS pi_intro_rule)) prems'
val prems''' = map (simplify simps2 o simplify simps1) prems''
in
resolve_tac ctxt (prems' @ prems'' @ prems'''))
end) ctxt
end

fun eqvt_rel_tac ctxt pred_names pi induct intros =
let
val cases = map (eqvt_rel_single_case_tac ctxt pred_names pi) intros
in
EVERY' ((DETERM o resolve_tac ctxt [induct]) :: cases)
end

(** equivariance procedure **)

fun prepare_goal ctxt pi pred_with_args =
let
val (c, xs) = strip_comb pred_with_args
fun is_nonfixed_Free (Free (s, _)) = not (Variable.is_fixed ctxt s)
| is_nonfixed_Free _ = false
fun mk_perm_nonfixed_Free t =
if is_nonfixed_Free t then mk_perm pi t else t
in
HOLogic.mk_imp (pred_with_args,
list_comb (c, map mk_perm_nonfixed_Free xs))
end

fun name_of (Const (s, _)) = s

fun raw_equivariance ctxt preds raw_induct intrs =
let
(* FIXME: polymorphic predicates should either be rejected or
specialized to arguments of sort pt *)

val is_already_eqvt = filter (is_eqvt ctxt) preds
val _ = if null is_already_eqvt then ()
else error ("Already equivariant: " ^ commas

val pred_names = map (name_of o head_of) preds
val raw_induct' = atomize_induct ctxt raw_induct
val intrs' = map (atomize_intr ctxt) intrs

val (([raw_concl], [raw_pi]), ctxt') =
ctxt
|> Variable.import_terms false [Thm.concl_of raw_induct']
||>> Variable.variant_fixes ["p"]
val pi = Free (raw_pi, @{typ perm})

val preds_with_args = raw_concl
|> HOLogic.dest_Trueprop
|> HOLogic.dest_conj
|> map (fst o HOLogic.dest_imp)

val goal = preds_with_args
|> map (prepare_goal ctxt pi)
|> foldr1 HOLogic.mk_conj
|> HOLogic.mk_Trueprop
in
Goal.prove ctxt' [] [] goal
(fn {context, ...} => eqvt_rel_tac context pred_names pi raw_induct' intrs' 1)
|> Old_Datatype_Aux.split_conj_thm
|> Proof_Context.export ctxt' ctxt
|> map (fn th => th RS mp)
|> map zero_var_indexes
end

(** stores thm under name.eqvt and adds [eqvt]-attribute **)

fun note_named_thm (name, thm) ctxt =
let
val thm_name = Binding.qualified_name
(Long_Name.qualify (Long_Name.base_name name) "eqvt")
val attr = Attrib.internal (K eqvt_add)
val ((_, [thm']), ctxt') = Local_Theory.note ((thm_name, [attr]), [thm]) ctxt
in
(thm', ctxt')
end

(** equivariance command **)

fun equivariance_cmd pred_name ctxt =
let
val ({names, ...}, {preds, raw_induct, intrs, ...}) =
Inductive.the_inductive_global ctxt (long_name ctxt pred_name)
val thms = raw_equivariance ctxt preds raw_induct intrs
in
fold_map note_named_thm (names ~~ thms) ctxt |> snd
end

val _ =
Outer_Syntax.local_theory @{command_keyword equivariance}
"Proves equivariance for inductive predicate involving nominal datatypes."
(Parse.const >> equivariance_cmd)

end (* structure *)


Theory Nominal2_Abs

theory Nominal2_Abs
imports Nominal2_Base
"HOL-Library.Quotient_List"
"HOL-Library.Quotient_Product"
begin

section ‹Abstractions›

fun
alpha_set
where
alpha_set[simp del]:
"alpha_set (bs, x) R f p (cs, y) ⟷
f x - bs = f y - cs ∧
(f x - bs) ♯* p ∧
R (p ∙ x) y ∧
p ∙ bs = cs"

fun
alpha_res
where
alpha_res[simp del]:
"alpha_res (bs, x) R f p (cs, y) ⟷
f x - bs = f y - cs ∧
(f x - bs) ♯* p ∧
R (p ∙ x) y"

fun
alpha_lst
where
alpha_lst[simp del]:
"alpha_lst (bs, x) R f p (cs, y) ⟷
f x - set bs = f y - set cs ∧
(f x - set bs) ♯* p ∧
R (p ∙ x) y ∧
p ∙ bs = cs"

lemmas alphas = alpha_set.simps alpha_res.simps alpha_lst.simps

notation
alpha_set ("_ ≈set _ _ _ _" [100, 100, 100, 100, 100] 100) and
alpha_res ("_ ≈res _ _ _ _" [100, 100, 100, 100, 100] 100) and
alpha_lst ("_ ≈lst _ _ _ _" [100, 100, 100, 100, 100] 100)

section ‹Mono›

lemma [mono]:
shows "R1 ≤ R2 ⟹ alpha_set bs R1 ≤ alpha_set bs R2"
and   "R1 ≤ R2 ⟹ alpha_res bs R1 ≤ alpha_res bs R2"
and   "R1 ≤ R2 ⟹ alpha_lst cs R1 ≤ alpha_lst cs R2"
by (case_tac [!] bs, case_tac [!] cs)
(auto simp: le_fun_def le_bool_def alphas)

section ‹Equivariance›

lemma alpha_eqvt[eqvt]:
shows "(bs, x) ≈set R f q (cs, y) ⟹ (p ∙ bs, p ∙ x) ≈set (p ∙ R) (p ∙ f) (p ∙ q) (p ∙ cs, p ∙ y)"
and   "(bs, x) ≈res R f q (cs, y) ⟹ (p ∙ bs, p ∙ x) ≈res (p ∙ R) (p ∙ f) (p ∙ q) (p ∙ cs, p ∙ y)"
and   "(ds, x) ≈lst R f q (es, y) ⟹ (p ∙ ds, p ∙ x) ≈lst (p ∙ R) (p ∙ f) (p ∙ q) (p ∙ es, p ∙ y)"
unfolding alphas
unfolding permute_eqvt[symmetric]
unfolding set_eqvt[symmetric]
unfolding permute_fun_app_eq[symmetric]
unfolding Diff_eqvt[symmetric]
unfolding eq_eqvt[symmetric]
unfolding fresh_star_eqvt[symmetric]
by (auto simp only: permute_bool_def)

section ‹Equivalence›

lemma alpha_refl:
assumes a: "R x x"
shows "(bs, x) ≈set R f 0 (bs, x)"
and   "(bs, x) ≈res R f 0 (bs, x)"
and   "(cs, x) ≈lst R f 0 (cs, x)"
using a
unfolding alphas
unfolding fresh_star_def

lemma alpha_sym:
assumes a: "R (p ∙ x) y ⟹ R (- p ∙ y) x"
shows "(bs, x) ≈set R f p (cs, y) ⟹ (cs, y) ≈set R f (- p) (bs, x)"
and   "(bs, x) ≈res R f p (cs, y) ⟹ (cs, y) ≈res R f (- p) (bs, x)"
and   "(ds, x) ≈lst R f p (es, y) ⟹ (es, y) ≈lst R f (- p) (ds, x)"
unfolding alphas fresh_star_def
using a
by (auto simp: fresh_minus_perm)

lemma alpha_trans:
assumes a: "⟦R (p ∙ x) y; R (q ∙ y) z⟧ ⟹ R ((q + p) ∙ x) z"
shows "⟦(bs, x) ≈set R f p (cs, y); (cs, y) ≈set R f q (ds, z)⟧ ⟹ (bs, x) ≈set R f (q + p) (ds, z)"
and   "⟦(bs, x) ≈res R f p (cs, y); (cs, y) ≈res R f q (ds, z)⟧ ⟹ (bs, x) ≈res R f (q + p) (ds, z)"
and   "⟦(es, x) ≈lst R f p (gs, y); (gs, y) ≈lst R f q (hs, z)⟧ ⟹ (es, x) ≈lst R f (q + p) (hs, z)"
using a
unfolding alphas fresh_star_def
and   "(ds, x) ≈lst R f p (es, y) ⟹ (es, y) ≈lst R f (- p) (ds,