Session Nominal2

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

declare [[typedef_overloaded]]


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›

instantiation perm :: group_add
begin

definition
  "0 = Abs_perm id"

definition
  "- p = Abs_perm (inv (Rep_perm p))"

definition
  "p + q = Abs_perm (Rep_perm p  Rep_perm q)"

definition
  "(p1::perm) - p2 = p1 + - p2"

lemma Rep_perm_0: "Rep_perm 0 = id"
  unfolding zero_perm_def
  by (simp add: Abs_perm_inverse perm_id)

lemma Rep_perm_add:
  "Rep_perm (p1 + p2) = Rep_perm p1  Rep_perm p2"
  unfolding plus_perm_def
  by (simp add: Abs_perm_inverse perm_comp Rep_perm)

lemma Rep_perm_uminus:
  "Rep_perm (- p) = inv (Rep_perm p)"
  unfolding uminus_perm_def
  by (simp add: Abs_perm_inverse perm_inv Rep_perm)

instance
apply standard
unfolding Rep_perm_inject [symmetric]
unfolding minus_perm_def
unfolding Rep_perm_add
unfolding Rep_perm_uminus
unfolding Rep_perm_0
by (simp_all add: o_assoc inv_o_cancel [OF bij_is_inj [OF bij_Rep_perm]])

end


section ‹Implementation of swappings›

definition
  swap :: "atom  atom  perm" ("'(_  _')")
where
  "(a  b) =
    Abs_perm (if sort_of a = sort_of b
              then (λc. if a = c then b else if b = c then a else c)
              else id)"

lemma Rep_perm_swap:
  "Rep_perm (a  b) =
    (if sort_of a = sort_of b
     then (λc. if a = c then b else if b = c then a else c)
     else id)"
unfolding swap_def
apply (rule Abs_perm_inverse)
apply (rule permI)
apply (auto simp: bij_def inj_on_def surj_def)[1]
apply (rule MOST_rev_mp [OF MOST_neq(1) [of a]])
apply (rule MOST_rev_mp [OF MOST_neq(1) [of b]])
apply (simp)
apply (simp)
done

lemmas Rep_perm_simps =
  Rep_perm_0
  Rep_perm_add
  Rep_perm_uminus
  Rep_perm_swap

lemma swap_different_sorts [simp]:
  "sort_of a  sort_of b  (a  b) = 0"
  by (rule Rep_perm_ext) (simp add: Rep_perm_simps)

lemma swap_cancel:
  shows "(a  b) + (a  b) = 0"
  and   "(a  b) + (b  a) = 0"
  by (rule_tac [!] Rep_perm_ext)
     (simp_all add: Rep_perm_simps fun_eq_iff)

lemma swap_self [simp]:
  "(a  a) = 0"
  by (rule Rep_perm_ext, simp add: Rep_perm_simps fun_eq_iff)

lemma minus_swap [simp]:
  "- (a  b) = (a  b)"
  by (rule minus_unique [OF swap_cancel(1)])

lemma swap_commute:
  "(a  b) = (b  a)"
  by (rule Rep_perm_ext)
     (simp add: Rep_perm_swap fun_eq_iff)

lemma swap_triple:
  assumes "a  b" and "c  b"
  assumes "sort_of a = sort_of b" "sort_of b = sort_of c"
  shows "(a  c) + (b  c) + (a  c) = (a  b)"
  using assms
  by (rule_tac Rep_perm_ext)
     (auto simp: Rep_perm_simps fun_eq_iff)


section ‹Permutation Types›

text ‹
  Infix syntax for permute› has higher precedence than
  addition, but lower than unary minus.
›

class pt =
  fixes permute :: "perm  'a  'a" ("_  _" [76, 75] 75)
  assumes permute_zero [simp]: "0  x = x"
  assumes permute_plus [simp]: "(p + q)  x = p  (q  x)"
begin

lemma permute_diff [simp]:
  shows "(p - q)  x = p  - q  x"
  using permute_plus [of p "- q" x] by simp

lemma permute_minus_cancel [simp]:
  shows "p  - p  x = x"
  and   "- p  p  x = x"
  unfolding permute_plus [symmetric] by simp_all

lemma permute_swap_cancel [simp]:
  shows "(a  b)  (a  b)  x = x"
  unfolding permute_plus [symmetric]
  by (simp add: swap_cancel)

lemma permute_swap_cancel2 [simp]:
  shows "(a  b)  (b  a)  x = x"
  unfolding permute_plus [symmetric]
  by (simp add: swap_commute)

lemma inj_permute [simp]:
  shows "inj (permute p)"
  by (rule inj_on_inverseI)
     (rule permute_minus_cancel)

lemma surj_permute [simp]:
  shows "surj (permute p)"
  by (rule surjI, rule permute_minus_cancel)

lemma bij_permute [simp]:
  shows "bij (permute p)"
  by (rule bijI [OF inj_permute surj_permute])

lemma inv_permute:
  shows "inv (permute p) = permute (- p)"
  by (rule inv_equality) (simp_all)

lemma permute_minus:
  shows "permute (- p) = inv (permute p)"
  by (simp add: inv_permute)

lemma permute_eq_iff [simp]:
  shows "p  x = p  y  x = y"
  by (rule inj_permute [THEN inj_eq])

end

subsection ‹Permutations for atoms›

instantiation atom :: pt
begin

definition
  "p  a = (Rep_perm p) a"

instance
apply standard
apply(simp_all add: permute_atom_def Rep_perm_simps)
done

end

lemma sort_of_permute [simp]:
  shows "sort_of (p  a) = sort_of a"
  unfolding permute_atom_def by (rule sort_of_Rep_perm)

lemma swap_atom:
  shows "(a  b)  c =
           (if sort_of a = sort_of b
            then (if c = a then b else if c = b then a else c) else c)"
  unfolding permute_atom_def
  by (simp add: Rep_perm_swap)

lemma swap_atom_simps [simp]:
  "sort_of a = sort_of b  (a  b)  a = b"
  "sort_of a = sort_of b  (a  b)  b = a"
  "c  a  c  b  (a  b)  c = c"
  unfolding swap_atom by simp_all

lemma perm_eq_iff:
  fixes p q :: "perm"
  shows "p = q  (a::atom. p  a = q  a)"
  unfolding permute_atom_def
  by (metis Rep_perm_ext ext)

subsection ‹Permutations for permutations›

instantiation perm :: pt
begin

definition
  "p  q = p + q - p"

instance
apply standard
apply (simp add: permute_perm_def)
apply (simp add: permute_perm_def algebra_simps)
done

end

lemma permute_self:
  shows "p  p = p"
  unfolding permute_perm_def
  by (simp add: add.assoc)

lemma pemute_minus_self:
  shows "- p  p = p"
  unfolding permute_perm_def
  by (simp add: add.assoc)


subsection ‹Permutations for functions›

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

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

instance
apply standard
apply (simp add: permute_fun_def)
apply (simp add: permute_fun_def minus_add)
done

end

lemma permute_fun_app_eq:
  shows "p  (f x) = (p  f) (p  x)"
  unfolding permute_fun_def by simp

lemma permute_fun_comp:
  shows "p  f  = (permute p) o f o (permute (-p))"
by (simp add: comp_def permute_fun_def)

subsection ‹Permutations for booleans›

instantiation bool :: pt
begin

definition "p  (b::bool) = b"

instance
apply standard
apply(simp_all add: permute_bool_def)
done

end

lemma permute_boolE:
  fixes P::"bool"
  shows "p  P  P"
  by (simp add: permute_bool_def)

lemma permute_boolI:
  fixes P::"bool"
  shows "P  p  P"
  by(simp add: permute_bool_def)

subsection ‹Permutations for sets›

instantiation "set" :: (pt) pt
begin

definition
  "p  X = {p  x | x. x  X}"

instance
apply standard
apply (auto simp: permute_set_def)
done

end

lemma permute_set_eq:
 shows "p  X = {x. - p  x  X}"
unfolding permute_set_def
by (auto) (metis permute_minus_cancel(1))

lemma permute_set_eq_image:
  shows "p  X = permute p ` X"
  unfolding permute_set_def by auto

lemma permute_set_eq_vimage:
  shows "p  X = permute (- p) -` X"
  unfolding permute_set_eq vimage_def
  by simp

lemma permute_finite [simp]:
  shows "finite (p  X) = finite X"
  unfolding permute_set_eq_vimage
  using bij_permute by (rule finite_vimage_iff)

lemma swap_set_not_in:
  assumes a: "a  S" "b  S"
  shows "(a  b)  S = S"
  unfolding permute_set_def
  using a by (auto simp: swap_atom)

lemma swap_set_in:
  assumes a: "a  S" "b  S" "sort_of a = sort_of b"
  shows "(a  b)  S  S"
  unfolding permute_set_def
  using a by (auto simp: swap_atom)

lemma swap_set_in_eq:
  assumes a: "a  S" "b  S" "sort_of a = sort_of b"
  shows "(a  b)  S = (S - {a})  {b}"
  unfolding permute_set_def
  using a by (auto simp: swap_atom)

lemma swap_set_both_in:
  assumes a: "a  S" "b  S"
  shows "(a  b)  S = S"
  unfolding permute_set_def
  using a by (auto simp: swap_atom)

lemma mem_permute_iff:
  shows "(p  x)  (p  X)  x  X"
  unfolding permute_set_def
  by auto

lemma empty_eqvt:
  shows "p  {} = {}"
  unfolding permute_set_def
  by (simp)

lemma insert_eqvt:
  shows "p  (insert x A) = insert (p  x) (p  A)"
  unfolding permute_set_eq_image image_insert ..


subsection ‹Permutations for @{typ unit}

instantiation unit :: pt
begin

definition "p  (u::unit) = u"

instance
  by standard (simp_all add: permute_unit_def)

end


subsection ‹Permutations for products›

instantiation prod :: (pt, pt) pt
begin

primrec
  permute_prod
where
  Pair_eqvt: "p  (x, y) = (p  x, p  y)"

instance
  by standard auto

end

subsection ‹Permutations for sums›

instantiation sum :: (pt, pt) pt
begin

primrec
  permute_sum
where
  Inl_eqvt: "p  (Inl x) = Inl (p  x)"
| Inr_eqvt: "p  (Inr y) = Inr (p  y)"

instance
  by standard (case_tac [!] x, simp_all)

end

subsection ‹Permutations for @{typ "'a list"}

instantiation list :: (pt) pt
begin

primrec
  permute_list
where
  Nil_eqvt:  "p  [] = []"
| Cons_eqvt: "p  (x # xs) = p  x # p  xs"

instance
  by standard (induct_tac [!] x, simp_all)

end

lemma set_eqvt:
  shows "p  (set xs) = set (p  xs)"
  by (induct xs) (simp_all add: empty_eqvt insert_eqvt)



subsection ‹Permutations for @{typ "'a option"}

instantiation option :: (pt) pt
begin

primrec
  permute_option
where
  None_eqvt: "p  None = None"
| Some_eqvt: "p  (Some x) = Some (p  x)"

instance
  by standard (induct_tac [!] x, simp_all)

end

subsection ‹Permutations for @{typ "'a multiset"}

instantiation multiset :: (pt) pt
begin

definition
  "p  M = {# p  x. x :# M #}"

instance
proof
  fix M :: "'a multiset" and p q :: "perm"
  show "0  M = M"
    unfolding permute_multiset_def
    by (induct_tac M) (simp_all)
  show "(p + q)  M = p  q  M"
    unfolding permute_multiset_def
    by (induct_tac M) (simp_all)
qed

end

lemma permute_multiset [simp]:
  fixes M N::"('a::pt) multiset"
  shows "(p  {#}) = ({#} ::('a::pt) multiset)"
  and   "(p  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)"
  apply (transfer, simp add: empty_eqvt)
  apply (transfer, simp add: insert_eqvt)
  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(simp add: permute_fun_comp)
  apply(rule finfun_right_compose)
  apply(rule finfun_left_compose)
  apply(assumption)
  apply(simp)
  done

instance
apply standard
apply(transfer)
apply(simp)
apply(transfer)
apply(simp)
done

end


subsection ‹Permutations for @{typ char}, @{typ nat}, and @{typ int}

instantiation char :: pt
begin

definition "p  (c::char) = c"

instance
  by standard (simp_all add: permute_char_def)

end

instantiation nat :: pt
begin

definition "p  (n::nat) = n"

instance
  by standard (simp_all add: permute_nat_def)

end

instantiation int :: pt
begin

definition "p  (i::int) = i"

instance
  by standard (simp_all add: permute_int_def)

end


section ‹Pure types›

text ‹Pure types will have always empty support.›

class pure = pt +
  assumes permute_pure: "p  x = x"

text ‹Types @{typ unit} and @{typ bool} are pure.›

instance unit :: pure
proof qed (rule permute_unit_def)

instance bool :: pure
proof qed (rule permute_bool_def)


text ‹Other type constructors preserve purity.›

instance "fun" :: (pure, pure) pure
  by standard (simp add: permute_fun_def permute_pure)

instance set :: (pure) pure
  by standard (simp add: permute_set_def permute_pure)

instance prod :: (pure, pure) pure
  by standard (induct_tac x, simp add: permute_pure)

instance sum :: (pure, pure) pure
  by standard (induct_tac x, simp_all add: permute_pure)

instance list :: (pure) pure
  by standard (induct_tac x, simp_all add: permute_pure)

instance option :: (pure) pure
  by standard (induct_tac x, simp_all add: permute_pure)


subsection ‹Types @{typ char}, @{typ nat}, and @{typ int}

instance char :: pure
proof qed (rule permute_char_def)

instance nat :: pure
proof qed (rule permute_nat_def)

instance int :: pure
proof qed (rule permute_int_def)


section ‹Infrastructure for Equivariance and Perm_simp›

subsection ‹Basic functions about permutations›

ML_file ‹nominal_basics.ML›


subsection ‹Eqvt infrastructure›

text ‹Setup of the theorem attributes 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(simp add: fun_eq_iff permute_fun_def)
apply(subst permute_eqvt)
apply(simp)
done

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

lemma add_perm_eqvt [eqvt]:
  fixes p p1 p2 :: perm
  shows "p  (p1 + p2) = p  p1 + p  p2"
  unfolding permute_perm_def
  by (simp add: perm_eq_iff)

lemma swap_eqvt [eqvt]:
  shows "p  (a  b) = (p  a  p  b)"
  unfolding permute_perm_def
  by (auto simp: swap_atom perm_eq_iff)

lemma uminus_eqvt [eqvt]:
  fixes p q::"perm"
  shows "p  (- q) = - (p  q)"
  unfolding permute_perm_def
  by (simp add: diff_add_eq_diff_diff_swap)


subsubsection ‹Equivariance of Logical Operators›

lemma eq_eqvt [eqvt]:
  shows "p  (x = y)  (p  x) = (p  y)"
  unfolding permute_eq_iff permute_bool_def ..

lemma Not_eqvt [eqvt]:
  shows "p  (¬ A)  ¬ (p  A)"
  by (simp add: permute_bool_def)

lemma conj_eqvt [eqvt]:
  shows "p  (A  B)  (p  A)  (p  B)"
  by (simp add: permute_bool_def)

lemma imp_eqvt [eqvt]:
  shows "p  (A  B)  (p  A)  (p  B)"
  by (simp add: permute_bool_def)

declare imp_eqvt[folded HOL.induct_implies_def, eqvt]

lemma all_eqvt [eqvt]:
  shows "p  (x. P x) = (x. (p  P) x)"
  unfolding All_def
  by (perm_simp) (rule refl)

declare all_eqvt[folded HOL.induct_forall_def, eqvt]

lemma ex_eqvt [eqvt]:
  shows "p  (x. P x) = (x. (p  P) x)"
  unfolding Ex_def
  by (perm_simp) (rule refl)

lemma ex1_eqvt [eqvt]:
  shows "p  (∃!x. P x) = (∃!x. (p  P) x)"
  unfolding Ex1_def
  by (perm_simp) (rule refl)

lemma if_eqvt [eqvt]:
  shows "p  (if b then x else y) = (if p  b then p  x else p  y)"
  by (simp add: permute_fun_def permute_bool_def)

lemma True_eqvt [eqvt]:
  shows "p  True = True"
  unfolding permute_bool_def ..

lemma False_eqvt [eqvt]:
  shows "p  False = False"
  unfolding permute_bool_def ..

lemma disj_eqvt [eqvt]:
  shows "p  (A  B)  (p  A)  (p  B)"
  by (simp add: permute_bool_def)

lemma all_eqvt2:
  shows "p  (x. P x) = (x. p  P (- p  x))"
  by (perm_simp add: permute_minus_cancel) (rule refl)

lemma ex_eqvt2:
  shows "p  (x. P x) = (x. p  P (- p  x))"
  by (perm_simp add: permute_minus_cancel) (rule refl)

lemma ex1_eqvt2:
  shows "p  (∃!x. P x) = (∃!x. p  P (- p  x))"
  by (perm_simp add: permute_minus_cancel) (rule refl)

lemma the_eqvt:
  assumes unique: "∃!x. P x"
  shows "(p  (THE x. P x)) = (THE x. (p  P) x)"
  apply(rule the1_equality [symmetric])
  apply(rule_tac p="-p" in permute_boolE)
  apply(perm_simp add: permute_minus_cancel)
  apply(rule unique)
  apply(rule_tac p="-p" in permute_boolE)
  apply(perm_simp add: permute_minus_cancel)
  apply(rule theI'[OF unique])
  done

lemma the_eqvt2:
  assumes unique: "∃!x. P x"
  shows "(p  (THE x. P x)) = (THE x. p  P (- p  x))"
  apply(rule the1_equality [symmetric])
  apply(simp only: ex1_eqvt2[symmetric])
  apply(simp add: permute_bool_def unique)
  apply(simp add: permute_bool_def)
  apply(rule theI'[OF unique])
  done

subsubsection ‹Equivariance of Set operators›

lemma mem_eqvt [eqvt]:
  shows "p  (x  A)  (p  x)  (p  A)"
  unfolding permute_bool_def permute_set_def
  by (auto)

lemma Collect_eqvt [eqvt]:
  shows "p  {x. P x} = {x. (p  P) x}"
  unfolding permute_set_eq permute_fun_def
  by (auto simp: permute_bool_def)

lemma 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: permute_bool_def)
apply(simp add: conj_eqvt Not_eqvt mem_eqvt set_eqvt)
done

lemma length_eqvt [eqvt]:
  shows "p  (length xs) = length (p  xs)"
by (induct xs) (simp_all add: permute_pure)


subsubsection ‹Equivariance for @{typ "'a option"}

lemma map_option_eqvt[eqvt]:
  shows "p  (map_option f x) = map_option (p  f) (p  x)"
  by (cases x) (simp_all)


subsubsection ‹Equivariance for @{typ "'a fset"}

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
  by standard (simp add: pure_supp)


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

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

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

instance atom :: fs
  by standard (simp add: supp_atom)


section ‹Type @{typ perm} is finitely-supported.›

lemma perm_swap_eq:
  shows "(a  b)  p = p  (p  (a  b)) = (a  b)"
unfolding permute_perm_def
by (metis add_diff_cancel minus_perm_def)

lemma supports_perm:
  shows "{a. p  a  a} supports p"
  unfolding supports_def
  unfolding perm_swap_eq
  by (simp add: swap_eqvt)

lemma finite_perm_lemma:
  shows "finite {a::atom. p  a  a}"
  using finite_Rep_perm [of p]
  unfolding permute_atom_def .

lemma supp_perm:
  shows "supp p = {a. p  a  a}"
apply (rule finite_supp_unique)
apply (rule supports_perm)
apply (rule finite_perm_lemma)
apply (simp add: perm_swap_eq swap_eqvt)
apply (auto simp: perm_eq_iff swap_atom)
done

lemma fresh_perm:
  shows "a  p  p  a = a"
  unfolding fresh_def
  by (simp add: supp_perm)

lemma supp_swap:
  shows "supp (a  b) = (if a = b  sort_of a  sort_of b then {} else {a, b})"
  by (auto simp: supp_perm swap_atom)

lemma fresh_swap:
  shows "a  (b  c)  (sort_of b  sort_of c)  b = c  (a  b  a  c)"
  by (simp add: fresh_def supp_swap supp_atom)

lemma fresh_zero_perm:
  shows "a  (0::perm)"
  unfolding fresh_perm by simp

lemma supp_zero_perm:
  shows "supp (0::perm) = {}"
  unfolding supp_perm by simp

lemma fresh_plus_perm:
  fixes p q::perm
  assumes "a  p" "a  q"
  shows "a  (p + q)"
  using assms
  unfolding fresh_def
  by (auto simp: supp_perm)

lemma supp_plus_perm:
  fixes p q::perm
  shows "supp (p + q)  supp p  supp q"
  by (auto simp: supp_perm)

lemma fresh_minus_perm:
  fixes p::perm
  shows "a  (- p)  a  p"
  unfolding fresh_def
  unfolding supp_perm
  apply(simp)
  apply(metis permute_minus_cancel)
  done

lemma supp_minus_perm:
  fixes p::perm
  shows "supp (- p) = supp p"
  unfolding supp_conv_fresh
  by (simp add: fresh_minus_perm)

lemma plus_perm_eq:
  fixes p q::"perm"
  assumes asm: "supp p  supp q = {}"
  shows "p + q = q + p"
unfolding perm_eq_iff
proof
  fix a::"atom"
  show "(p + q)  a = (q + p)  a"
  proof -
    { assume "a  supp p" "a  supp q"
      then have "(p + q)  a = (q + p)  a"
        by (simp add: supp_perm)
    }
    moreover
    { assume a: "a  supp p" "a  supp q"
      then have "p  a  supp p" by (simp add: supp_perm)
      then have "p  a  supp q" using asm by auto
      with a have "(p + q)  a = (q + p)  a"
        by (simp add: supp_perm)
    }
    moreover
    { assume a: "a  supp p" "a  supp q"
      then have "q  a  supp q" by (simp add: supp_perm)
      then have "q  a  supp p" using asm by auto
      with a have "(p + q)  a = (q + p)  a"
        by (simp add: supp_perm)
    }
    ultimately show "(p + q)  a = (q + p)  a"
      using asm by blast
  qed
qed

lemma supp_plus_perm_eq:
  fixes p q::perm
  assumes asm: "supp p  supp q = {}"
  shows "supp (p + q) = supp p  supp q"
proof -
  { fix a::"atom"
    assume "a  supp p"
    then have "a  supp q" using asm by auto
    then have "a  supp (p + q)" using a  supp p
      by (simp add: supp_perm)
  }
  moreover
  { fix a::"atom"
    assume "a  supp q"
    then have "a  supp p" using asm by auto
    then have "a  supp (q + p)" using a  supp q
      by (simp add: supp_perm)
    then have "a  supp (p + q)" using asm plus_perm_eq
      by metis
  }
  ultimately have "supp p  supp q  supp (p + q)"
    by blast
  then show "supp (p + q) = supp p  supp q" using supp_plus_perm
    by blast
qed

lemma perm_eq_iff2:
  fixes p q :: "perm"
  shows "p = q  (a::atom  supp p  supp q. p  a = q  a)"
  unfolding perm_eq_iff
  apply(auto)
  apply(case_tac "a  p  a  q")
  apply(simp add: fresh_perm)
  apply(simp add: fresh_def)
  done


instance perm :: fs
  by standard (simp add: supp_perm finite_perm_lemma)



section ‹Finite Support instances for other types›


subsection ‹Type @{typ "'a × 'b"} is finitely-supported.›

lemma supp_Pair:
  shows "supp (x, y) = supp x  supp y"
  by (simp add: supp_def Collect_imp_eq Collect_neg_eq)

lemma fresh_Pair:
  shows "a  (x, y)  a  x  a  y"
  by (simp add: fresh_def supp_Pair)

lemma supp_Unit:
  shows "supp () = {}"
  by (simp add: supp_def)

lemma fresh_Unit:
  shows "a  ()"
  by (simp add: fresh_def supp_Unit)

instance prod :: (fs, fs) fs
apply standard
apply (case_tac x)
apply (simp add: supp_Pair finite_supp)
done


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

lemma supp_Inl:
  shows "supp (Inl x) = supp x"
  by (simp add: supp_def)

lemma supp_Inr:
  shows "supp (Inr x) = supp x"
  by (simp add: supp_def)

lemma fresh_Inl:
  shows "a  Inl x  a  x"
  by (simp add: fresh_def supp_Inl)

lemma fresh_Inr:
  shows "a  Inr y  a  y"
  by (simp add: fresh_def supp_Inr)

instance sum :: (fs, fs) fs
apply standard
apply (case_tac x)
apply (simp_all add: supp_Inl supp_Inr finite_supp)
done


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

lemma supp_None:
  shows "supp None = {}"
by (simp add: supp_def)

lemma supp_Some:
  shows "supp (Some x) = supp x"
  by (simp add: supp_def)

lemma fresh_None:
  shows "a  None"
  by (simp add: fresh_def supp_None)

lemma fresh_Some:
  shows "a  Some x  a  x"
  by (simp add: fresh_def supp_Some)

instance option :: (fs) fs
apply standard
apply (induct_tac x)
apply (simp_all add: supp_None supp_Some finite_supp)
done


subsubsection ‹Type @{typ "'a list"} is finitely supported›

lemma supp_Nil:
  shows "supp [] = {}"
  by (simp add: supp_def)

lemma fresh_Nil:
  shows "a  []"
  by (simp add: fresh_def supp_Nil)

lemma supp_Cons:
  shows "supp (x # xs) = supp x  supp xs"
by (simp add: supp_def Collect_imp_eq Collect_neg_eq)

lemma fresh_Cons:
  shows "a  (x # xs)  a  x  a  xs"
  by (simp add: fresh_def supp_Cons)

lemma supp_append:
  shows "supp (xs @ ys) = supp xs  supp ys"
  by (induct xs) (auto simp: supp_Nil supp_Cons)

lemma fresh_append:
  shows "a  (xs @ ys)  a  xs  a  ys"
  by (induct xs) (simp_all add: fresh_Nil fresh_Cons)

lemma supp_rev:
  shows "supp (rev xs) = supp xs"
  by (induct xs) (auto simp: supp_append supp_Cons supp_Nil)

lemma fresh_rev:
  shows "a  rev xs  a  xs"
  by (induct xs) (auto simp: fresh_append fresh_Cons fresh_Nil)

lemma supp_removeAll:
  fixes x::"atom"
  shows "supp (removeAll x xs) = supp xs - {x}"
  by (induct xs)
     (auto simp: supp_Nil supp_Cons supp_atom)

lemma supp_of_atom_list:
  fixes as::"atom list"
  shows "supp as = set as"
by (induct as)
   (simp_all add: supp_Nil supp_Cons supp_atom)

instance list :: (fs) fs
apply standard
apply (induct_tac x)
apply (simp_all add: supp_Nil supp_Cons finite_supp)
done


section ‹Support and Freshness for Applications›

lemma fresh_conv_MOST:
  shows "a  x  (MOST b. (a  b)  x = x)"
  unfolding fresh_def supp_def
  unfolding MOST_iff_cofinite by simp

lemma fresh_fun_app:
  assumes "a  f" and "a  x"
  shows "a  f x"
  using assms
  unfolding fresh_conv_MOST
  unfolding permute_fun_app_eq
  by (elim MOST_rev_mp) (simp)

lemma supp_fun_app:
  shows "supp (f x)  (supp f)  (supp x)"
  using fresh_fun_app
  unfolding fresh_def
  by auto


subsection ‹Equivariance Predicate 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
  by (simp add: supp_fun_eqvt)

lemma fresh_fun_eqvt_app:
  assumes a: "eqvt f"
  shows "a  x  a  f x"
proof -
  from a have "supp f = {}" by (simp add: supp_fun_eqvt)
  then show "a  x  a  f x"
    unfolding fresh_def
    using supp_fun_app by auto
qed

lemma supp_fun_app_eqvt:
  assumes a: "eqvt f"
  shows "supp (f x)  supp x"
  using fresh_fun_eqvt_app[OF a]
  unfolding fresh_def
  by auto

lemma supp_eqvt_at:
  assumes asm: "eqvt_at f x"
  and     fin: "finite (supp x)"
  shows "supp (f x)  supp x"
apply(rule supp_is_subset)
unfolding supports_def
unfolding fresh_def[symmetric]
using asm
apply(simp add: eqvt_at_def)
apply(simp add: swap_fresh_fresh)
apply(rule fin)
done

lemma finite_supp_eqvt_at:
  assumes asm: "eqvt_at f x"
  and     fin: "finite (supp x)"
  shows "finite (supp (f x))"
apply(rule finite_subset)
apply(rule supp_eqvt_at[OF asm fin])
apply(rule fin)
done

lemma fresh_eqvt_at:
  assumes asm: "eqvt_at f x"
  and     fin: "finite (supp x)"
  and     fresh: "a  x"
  shows "a  f x"
using fresh
unfolding fresh_def
using supp_eqvt_at[OF asm fin]
by auto

text ‹for handling of freshness of functions›

simproc_setup fresh_fun_simproc ("a  (f::'a::pt 'b::pt)") = fn _ => fn ctxt => fn ctrm =>
  let
    val _ $ _ $ f = Thm.term_of ctrm
  in
    case (Term.add_frees f [], Term.add_vars f []) of
      ([], []) => SOME(@{thm fresh_fun_eqvt[simplified eqvt_def, THEN Eq_TrueI]})
    | (x::_, []) =>
      let
        val argx = Free x
        val absf = absfree x f
        val cty_inst =
          [SOME (Thm.ctyp_of ctxt (fastype_of argx)), SOME (Thm.ctyp_of ctxt (fastype_of f))]
        val ctrm_inst = [NONE, SOME (Thm.cterm_of ctxt absf), SOME (Thm.cterm_of ctxt argx)]
        val thm = Thm.instantiate' cty_inst ctrm_inst @{thm fresh_fun_app}
      in
        SOME(thm RS @{thm Eq_TrueI})
      end
    | (_, _) => NONE
  end

subsection ‹helper functions for nominal_functions›

lemma THE_defaultI2:
  assumes "∃!x. P x" "x. P x  Q x"
  shows "Q (THE_default d P)"
by (iprover intro: assms THE_defaultI')

lemma the_default_eqvt:
  assumes unique: "∃!x. P x"
  shows "(p  (THE_default d P)) = (THE_default (p  d) (p  P))"
  apply(rule THE_default1_equality [symmetric])
  apply(rule_tac p="-p" in permute_boolE)
  apply(simp add: ex1_eqvt)
  apply(rule unique)
  apply(rule_tac p="-p" in permute_boolE)
  apply(rule subst[OF permute_fun_app_eq])
  apply(simp)
  apply(rule THE_defaultI'[OF unique])
  done

lemma fundef_ex1_eqvt:
  fixes x::"'a::pt"
  assumes f_def: "f == (λx::'a. THE_default (d x) (G x))"
  assumes eqvt: "eqvt G"
  assumes ex1: "∃!y. G x y"
  shows "(p  (f x)) = f (p  x)"
  apply(simp only: f_def)
  apply(subst the_default_eqvt)
  apply(rule ex1)
  apply(rule THE_default1_equality [symmetric])
  apply(rule_tac p="-p" in permute_boolE)
  apply(perm_simp add: permute_minus_cancel)
  using eqvt[simplified eqvt_def]
  apply(simp)
  apply(rule ex1)
  apply(rule THE_defaultI2)
  apply(rule_tac p="-p" in permute_boolE)
  apply(perm_simp add: permute_minus_cancel)
  apply(rule ex1)
  apply(perm_simp)
  using eqvt[simplified eqvt_def]
  apply(simp)
  done

lemma fundef_ex1_eqvt_at:
  fixes x::"'a::pt"
  assumes f_def: "f == (λx::'a. THE_default (d x) (G x))"
  assumes eqvt: "eqvt G"
  assumes ex1: "∃!y. G x y"
  shows "eqvt_at f x"
  unfolding eqvt_at_def
  using assms
  by (auto intro: fundef_ex1_eqvt)

lemma fundef_ex1_prop:
  fixes x::"'a::pt"
  assumes f_def: "f == (λx::'a. THE_default (d x) (G x))"
  assumes P_all: "x y. G x y  P x y"
  assumes ex1: "∃!y. G x y"
  shows "P x (f x)"
  unfolding f_def
  using ex1
  apply(erule_tac ex1E)
  apply(rule THE_defaultI2)
  apply(blast)
  apply(rule P_all)
  apply(assumption)
  done


section ‹Support of Finite Sets of Finitely Supported Elements›

text ‹support and freshness for atom sets›

lemma supp_finite_atom_set:
  fixes S::"atom set"
  assumes "finite S"
  shows "supp S = S"
  apply(rule finite_supp_unique)
  apply(simp add: supports_def)
  apply(simp add: swap_set_not_in)
  apply(rule assms)
  apply(simp add: swap_set_in)
done

lemma supp_cofinite_atom_set:
  fixes S::"atom set"
  assumes "finite (UNIV - S)"
  shows "supp S = (UNIV - S)"
  apply(rule finite_supp_unique)
  apply(simp add: supports_def)
  apply(simp add: swap_set_both_in)
  apply(rule assms)
  apply(subst swap_commute)
  apply(simp add: swap_set_in)
done

lemma fresh_finite_atom_set:
  fixes S::"atom set"
  assumes "finite S"
  shows "a  S  a  S"
  unfolding fresh_def
  by (simp add: supp_finite_atom_set[OF assms])

lemma fresh_minus_atom_set:
  fixes S::"atom set"
  assumes "finite S"
  shows "a  S - T  (a  T  a  S)"
  unfolding fresh_def
  by (auto simp: supp_finite_atom_set assms)

lemma Union_supports_set:
  shows "(x  S. supp x) supports S"
proof -
  { fix a b
    have "x  S. (a  b)  x = x  (a  b)  S = S"
      unfolding permute_set_def by force
  }
  then show "(x  S. supp x) supports S"
    unfolding supports_def
    by (simp add: fresh_def[symmetric] swap_fresh_fresh)
qed

lemma Union_of_finite_supp_sets:
  fixes S::"('a::fs set)"
  assumes fin: "finite S"
  shows "finite (xS. 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 "(xS. supp x)  supp S"
proof -
  have eqvt: "eqvt (λS. x  S. supp x)"
    unfolding eqvt_def by simp
  have "(xS. supp x) = supp (xS. 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 "(xS. supp x)  supp S" .
qed

lemma supp_of_finite_sets:
  fixes S::"('a::fs set)"
  assumes fin: "finite S"
  shows "(supp S) = (xS. supp x)"
apply(rule subset_antisym)
apply(rule supp_is_subset)
apply(rule Union_supports_set)
apply(rule Union_of_finite_supp_sets[OF fin])
apply(rule Union_included_in_supp[OF fin])
done

lemma finite_sets_supp:
  fixes S::"('a::fs set)"
  assumes "finite S"
  shows "finite (supp S)"
using assms
by (simp only: supp_of_finite_sets Union_of_finite_supp_sets)

lemma supp_of_finite_union:
  fixes S T::"('a::fs) set"
  assumes fin1: "finite S"
  and     fin2: "finite T"
  shows "supp (S  T) = supp S  supp T"
  using fin1 fin2
  by (simp add: supp_of_finite_sets)

lemma fresh_finite_union:
  fixes S T::"('a::fs) set"
  assumes fin1: "finite S"
  and     fin2: "finite T"
  shows "a  (S  T)  a  S  a  T"
  unfolding fresh_def
  by (simp add: supp_of_finite_union[OF fin1 fin2])

lemma supp_of_finite_insert:
  fixes S::"('a::fs) set"
  assumes fin:  "finite S"
  shows "supp (insert x S) = supp x  supp S"
  using fin
  by (simp add: supp_of_finite_sets)

lemma fresh_finite_insert:
  fixes S::"('a::fs) set"
  assumes fin:  "finite S"
  shows "a  (insert x S)  a  x  a  S"
  using fin unfolding fresh_def
  by (simp add: supp_of_finite_insert)

lemma supp_set_empty:
  shows "supp {} = {}"
  unfolding supp_def
  by (simp add: empty_eqvt)

lemma fresh_set_empty:
  shows "a  {}"
  by (simp add: fresh_def supp_set_empty)

lemma supp_set:
  fixes xs :: "('a::fs) list"
  shows "supp (set xs) = supp xs"
apply(induct xs)
apply(simp add: supp_set_empty supp_Nil)
apply(simp add: supp_Cons supp_of_finite_insert)
done

lemma fresh_set:
  fixes xs :: "('a::fs) list"
  shows "a  (set xs)  a  xs"
unfolding fresh_def
by (simp add: supp_set)


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

lemma set_mset_eqvt [eqvt]:
  shows "p  (set_mset M) = set_mset (p  M)"
by (induct M) (simp_all add: insert_eqvt empty_eqvt)

lemma supp_set_mset:
  shows "supp (set_mset M)  supp M"
  apply (rule supp_fun_app_eqvt)
  unfolding eqvt_def
  apply(perm_simp)
  apply(simp)
  done

lemma Union_finite_multiset:
  fixes M::"'a::fs multiset"
  shows "finite ({supp x | x. x ∈# M})"
proof -
  have "finite ((supp ` {x. x ∈# M}))"
    by (induct M) (simp_all add: Collect_imp_eq Collect_neg_eq finite_supp)
  then show "finite ({supp x | x. x ∈# M})"
    by (simp only: image_Collect)
qed

lemma Union_supports_multiset:
  shows "{supp x | x. x ∈# M} supports M"
proof -
  have sw: "a b. ((x. x ∈# M  (a  b)  x = x)  (a  b)  M = M)"
    unfolding permute_multiset_def by (induct M) simp_all
  have "(xset_mset M. supp x) supports M"
    by (auto intro!: sw swap_fresh_fresh simp add: fresh_def supports_def)
  also have "(xset_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)"
    by (simp add: supp_of_finite_sets)
  also have " ...  supp M" by (rule supp_set_mset)
  finally show "({supp x | x. x ∈# M})  supp M" .
qed

lemma supp_of_multisets:
  fixes M::"('a::fs multiset)"
  shows "(supp M) = ({supp x | x. x ∈# M})"
apply(rule subset_antisym)
apply(rule supp_is_subset)
apply(rule Union_supports_multiset)
apply(rule Union_finite_multiset)
apply(rule Union_included_multiset)
done

lemma multisets_supp_finite:
  fixes M::"('a::fs multiset)"
  shows "finite (supp M)"
by (simp only: supp_of_multisets Union_finite_multiset)

lemma supp_of_multiset_union:
  fixes M N::"('a::fs) multiset"
  shows "supp (M + N) = supp M  supp N"
  by (auto simp: supp_of_multisets)

lemma supp_empty_mset [simp]:
  shows "supp {#} = {}"
  unfolding supp_def
  by simp

instance multiset :: (fs) fs
  by standard (rule multisets_supp_finite)

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

lemma supp_fset [simp]:
  shows "supp (fset S) = supp S"
  unfolding supp_def
  by (simp add: fset_eqvt fset_cong)

lemma supp_empty_fset [simp]:
  shows "supp {||} = {}"
  unfolding supp_def
  by simp

lemma fresh_empty_fset:
  shows "a  {||}"
unfolding fresh_def
by (simp)

lemma supp_finsert [simp]:
  fixes x::"'a::fs"
  and   S::"'a fset"
  shows "supp (finsert x S) = supp x  supp S"
  apply(subst supp_fset[symmetric])
  apply(simp add: supp_of_finite_insert)
  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
by (simp add: supp_union_fset)

instance fset :: (fs) fs
  by standard (rule fset_finite_supp)


subsection ‹Type @{typ "('a, 'b) finfun"} is finitely supported›

lemma fresh_finfun_const:
  shows "a  (finfun_const b)  a  b"
  by (simp add: fresh_def supp_def)

lemma fresh_finfun_update:
  shows "a  f; a  x; a  y  a  finfun_update f x y"
  unfolding fresh_conv_MOST
  unfolding finfun_update_eqvt
  by (elim MOST_rev_mp) (simp)

lemma supp_finfun_const:
  shows "supp (finfun_const b) = supp(b)"
  by (simp add: supp_def)

lemma supp_finfun_update:
  shows "supp (finfun_update f x y)  supp(f, x, y)"
using fresh_finfun_update
by (auto simp: fresh_def supp_Pair)

instance finfun :: (fs, fs) fs
  apply standard
  apply(induct_tac x rule: finfun_weak_induct)
  apply(simp add: supp_finfun_const finite_supp)
  apply(rule finite_subset)
  apply(rule supp_finfun_update)
  apply(simp add: supp_Pair finite_supp)
  done


section ‹Freshness and Fresh-Star›

lemma fresh_Unit_elim:
  shows "(a  ()  PROP C)  PROP C"
  by (simp add: fresh_Unit)

lemma fresh_Pair_elim:
  shows "(a  (x, y)  PROP C)  (a  x  a  y  PROP C)"
  by rule (simp_all add: fresh_Pair)

(* 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)"
  by (simp add: fresh_Pair)

lemma fresh_PairD:
  shows "a  (x, y)  a  x"
  and   "a  (x, y)  a  y"
  by (simp_all add: fresh_Pair)

declaration fn _ =>
let
  val mksimps_pairs = (@{const_name Nominal2_Base.fresh}, @{thms fresh_PairD}) :: mksimps_pairs
in
  Simplifier.map_ss (fn ss => Simplifier.set_mksimps (mksimps mksimps_pairs) ss)
end


text ‹The fresh-star generalisation of fresh is used in strong
  induction principles.›

definition
  fresh_star :: "atom set  'a::pt  bool" ("_ ♯* _" [80,80] 80)
where
  "as ♯* x  a  as. a  x"

lemma fresh_star_supp_conv:
  shows "supp x ♯* y  supp y ♯* x"
by (auto simp: fresh_star_def fresh_def)

lemma fresh_star_perm_set_conv:
  fixes p::"perm"
  assumes fresh: "as ♯* p"
  and     fin: "finite as"
  shows "supp p ♯* as"
apply(rule fresh_star_supp_conv)
apply(simp add: supp_finite_atom_set fin fresh)
done

lemma fresh_star_atom_set_conv:
  assumes fresh: "as ♯* bs"
  and     fin: "finite as" "finite bs"
  shows "bs ♯* as"
using fresh
unfolding fresh_star_def fresh_def
by (auto simp: supp_finite_atom_set fin)

lemma atom_fresh_star_disjoint:
  assumes fin: "finite bs"
  shows "as ♯* bs  (as  bs = {})"

unfolding fresh_star_def fresh_def
by (auto simp: supp_finite_atom_set fin)


lemma fresh_star_Pair:
  shows "as ♯* (x, y) = (as ♯* x  as ♯* y)"
  by (auto simp: fresh_star_def fresh_Pair)

lemma fresh_star_list:
  shows "as ♯* (xs @ ys)  as ♯* xs  as ♯* ys"
  and   "as ♯* (x # xs)  as ♯* x  as ♯* xs"
  and   "as ♯* []"
by (auto simp: fresh_star_def fresh_Nil fresh_Cons fresh_append)

lemma fresh_star_set:
  fixes xs::"('a::fs) list"
  shows "as ♯* set xs  as ♯* xs"
unfolding fresh_star_def
by (simp add: fresh_set)

lemma fresh_star_singleton:
  fixes a::"atom"
  shows "as ♯* {a}  as ♯* a"
  by (simp add: fresh_star_def fresh_finite_insert fresh_set_empty)

lemma fresh_star_fset:
  fixes xs::"('a::fs) list"
  shows "as ♯* fset S  as ♯* S"
by (simp add: fresh_star_def fresh_def)

lemma fresh_star_Un:
  shows "(as  bs) ♯* x = (as ♯* x  bs ♯* x)"
  by (auto simp: fresh_star_def)

lemma fresh_star_insert:
  shows "(insert a as) ♯* x = (a  x  as ♯* x)"
  by (auto simp: fresh_star_def)

lemma fresh_star_Un_elim:
  "((as  bs) ♯* x  PROP C)  (as ♯* x  bs ♯* x  PROP C)"
  unfolding fresh_star_def
  apply(rule)
  apply(erule meta_mp)
  apply(auto)
  done

lemma fresh_star_insert_elim:
  "(insert a as ♯* x  PROP C)  (a  x  as ♯* x  PROP C)"
  unfolding fresh_star_def
  by rule (simp_all add: fresh_star_def)

lemma fresh_star_empty_elim:
  "({} ♯* x  PROP C)  PROP C"
  by (simp add: fresh_star_def)

lemma fresh_star_Unit_elim:
  shows "(a ♯* ()  PROP C)  PROP C"
  by (simp add: fresh_star_def fresh_Unit)

lemma fresh_star_Pair_elim:
  shows "(a ♯* (x, y)  PROP C)  (a ♯* x  a ♯* y  PROP C)"
  by (rule, simp_all add: fresh_star_Pair)

lemma fresh_star_zero:
  shows "as ♯* (0::perm)"
  unfolding fresh_star_def
  by (simp add: fresh_zero_perm)

lemma fresh_star_plus:
  fixes p q::perm
  shows "a ♯* p;  a ♯* q  a ♯* (p + q)"
  unfolding fresh_star_def
  by (simp add: fresh_plus_perm)

lemma fresh_star_permute_iff:
  shows "(p  a) ♯* (p  x)  a ♯* x"
  unfolding fresh_star_def
  by (metis mem_permute_iff permute_minus_cancel(1) fresh_permute_iff)

lemma fresh_star_eqvt [eqvt]:
  shows "p  (as ♯* x)  (p  as) ♯* (p  x)"
unfolding fresh_star_def by simp


section ‹Induction principle for permutations›

lemma smaller_supp:
  assumes a: "a  supp p"
  shows "supp ((p  a  a) + p)  supp p"
proof -
  have "supp ((p  a  a) + p)  supp p"
    unfolding supp_perm by (auto simp: swap_atom)
  moreover
  have "a  supp ((p  a  a) + p)" by (simp add: supp_perm)
  then have "supp ((p  a  a) + p)  supp p" using a by auto
  ultimately
  show "supp ((p  a  a) + p)  supp p" by auto
qed


lemma perm_struct_induct[consumes 1, case_names zero swap]:
  assumes S: "supp p  S"
  and zero: "P 0"
  and swap: "p a b. P p; supp p  S; a  S; b  S; a  b; sort_of a = sort_of b  P ((a  b) + p)"
  shows "P p"
proof -
  have "finite (supp p)" by (simp add: finite_supp)
  then show "P p" using S
  proof(induct A"supp p" arbitrary: p rule: finite_psubset_induct)
    case (psubset p)
    then have ih: "q. supp q  supp p  P q" by auto
    have as: "supp p  S" by fact
    { assume "supp p = {}"
      then have "p = 0" by (simp add: supp_perm perm_eq_iff)
      then have "P p" using zero by simp
    }
    moreover
    { assume "supp p  {}"
      then obtain a where a0: "a  supp p" by blast
      then have a1: "p  a  S" "a  S" "sort_of (p  a) = sort_of a" "p  a  a"
        using as by (auto simp: supp_atom supp_perm swap_atom)
      let ?q = "(p  a  a) + p"
      have a2: "supp ?q  supp p" using a0 smaller_supp by simp
      then have "P ?q" using ih by simp
      moreover
      have "supp ?q  S" using as a2 by simp
      ultimately  have "P ((p  a  a) + ?q)" using as a1 swap by simp
      moreover
      have "p = (p  a  a) + ?q" by (simp add: perm_eq_iff)
      ultimately have "P p" by simp
    }
    ultimately show "P p" by blast
  qed
qed

lemma perm_simple_struct_induct[case_names zero swap]:
  assumes zero: "P 0"
  and     swap: "p a b. P p; a  b; sort_of a = sort_of b  P ((a  b) + p)"
  shows "P p"
by (rule_tac S="supp p" in perm_struct_induct)
   (auto intro: zero swap)

lemma perm_struct_induct2[consumes 1, case_names zero swap plus]:
  assumes S: "supp p  S"
  assumes zero: "P 0"
  assumes swap: "a b. sort_of a = sort_of b; a  b; a  S; b  S  P (a  b)"
  assumes plus: "p1 p2. P p1; P p2; supp p1  S; supp p2  S  P (p1 + p2)"
  shows "P p"
using S
by (induct p rule: perm_struct_induct)
   (auto intro: zero plus swap simp add: supp_swap)

lemma perm_simple_struct_induct2[case_names zero swap plus]:
  assumes zero: "P 0"
  assumes swap: "a b. sort_of a = sort_of b; a  b  P (a  b)"
  assumes plus: "p1 p2. P p1; P p2  P (p1 + p2)"
  shows "P p"
by (rule_tac S="supp p" in perm_struct_induct2)
   (auto intro: zero swap plus)

lemma supp_perm_singleton:
  fixes p::"perm"
  shows "supp p  {b}  p = 0"
proof -
  { assume "supp p  {b}"
    then have "p = 0"
      by (induct p rule: perm_struct_induct) (simp_all)
  }
  then show "supp p  {b}  p = 0" by (auto simp: supp_zero_perm)
qed

lemma supp_perm_pair:
  fixes p::"perm"
  shows "supp p  {a, b}  p = 0  p = (b  a)"
proof -
  { assume "supp p  {a, b}"
    then have "p = 0  p = (b  a)"
      apply (induct p rule: perm_struct_induct)
      apply (auto simp: swap_cancel supp_zero_perm supp_swap)
      apply (simp add: swap_commute)
      done
  }
  then show "supp p  {a, b}  p = 0  p = (b  a)"
    by (auto simp: supp_zero_perm supp_swap split: if_splits)
qed

lemma supp_perm_eq:
  assumes "(supp x) ♯* p"
  shows "p  x = x"
proof -
  from assms have "supp p  {a. a  x}"
    unfolding supp_perm fresh_star_def fresh_def by auto
  then show "p  x = x"
  proof (induct p rule: perm_struct_induct)
    case zero
    show "0  x = x" by simp
  next
    case (swap p a b)
    then have "a  x" "b  x" "p  x = x" by simp_all
    then show "((a  b) + p)  x = x" by (simp add: swap_fresh_fresh)
  qed
qed

text ‹same lemma as above, but proved with a different induction principle›
lemma supp_perm_eq_test:
  assumes "(supp x) ♯* p"
  shows "p  x = x"
proof -
  from assms have "supp p  {a. a  x}"
    unfolding supp_perm fresh_star_def fresh_def by auto
  then show "p  x = x"
  proof (induct p rule: perm_struct_induct2)
    case zero
    show "0  x = x" by simp
  next
    case (swap a b)
    then have "a  x" "b  x" by simp_all
    then show "(a  b)  x = x" by (simp add: swap_fresh_fresh)
  next
    case (plus p1 p2)
    have "p1  x = x" "p2  x = x" by fact+
    then show "(p1 + p2)  x = x" by simp
  qed
qed

lemma perm_supp_eq:
  assumes a: "(supp p) ♯* x"
  shows "p  x = x"
proof -
  from assms have "supp p  {a. a  x}"
    unfolding supp_perm fresh_star_def fresh_def by auto
  then show "p  x = x"
  proof (induct p rule: perm_struct_induct2)
    case zero
    show "0  x = x" by simp
  next
    case (swap a b)
    then have "a  x" "b  x" by simp_all
    then show "(a  b)  x = x" by (simp add: swap_fresh_fresh)
  next
    case (plus p1 p2)
    have "p1  x = x" "p2  x = x" by fact+
    then show "(p1 + p2)  x = x" by simp
  qed
qed

lemma supp_perm_perm_eq:
  assumes a: "a  supp x. p  a = q  a"
  shows "p  x = q  x"
proof -
  from a have "a  supp x. (-q + p)  a = a" by simp
  then have "a  supp x. a  supp (-q + p)"
    unfolding supp_perm by simp
  then have "supp x ♯* (-q + p)"
    unfolding fresh_star_def fresh_def by simp
  then have "(-q + p)  x = x" by (simp only: supp_perm_eq)
  then show "p  x = q  x"
    by (metis permute_minus_cancel permute_plus)
qed

text ‹disagreement set›

definition
  dset :: "perm  perm  atom set"
where
  "dset p q = {a::atom. p  a  q  a}"

lemma ds_fresh:
  assumes "dset p q ♯* x"
  shows "p  x = q  x"
using assms
unfolding dset_def fresh_star_def fresh_def
by (auto intro: supp_perm_perm_eq)

lemma atom_set_perm_eq:
  assumes a: "as ♯* p"
  shows "p  as = as"
proof -
  from a have "supp p  {a. a  as}"
    unfolding supp_perm fresh_star_def fresh_def by auto
  then show "p  as = as"
  proof (induct p rule: perm_struct_induct)
    case zero
    show "0  as = as" by simp
  next
    case (swap p a b)
    then have "a  as" "b  as" "p  as = as" by simp_all
    then show "((a  b) + p)  as = as" by (simp add: swap_set_not_in)
  qed
qed

section ‹Avoiding of atom sets›

text ‹
  For every set of atoms, there is another set of atoms
  avoiding a finitely supported c and there is a permutation
  which 'translates' between both sets.
›

lemma at_set_avoiding_aux:
  fixes Xs::"atom set"
  and   As::"atom set"
  assumes b: "Xs  As"
  and     c: "finite As"
  shows "p. (p  Xs)  As = {}  (supp p) = (Xs  (p  Xs))"
proof -
  from b c have "finite Xs" by (rule finite_subset)
  then show ?thesis using b
  proof (induct rule: finite_subset_induct)
    case empty
    have "0  {}  As = {}" by simp
    moreover
    have "supp (0::perm) = {}  0  {}" by (simp add: supp_zero_perm)
    ultimately show ?case by blast
  next
    case (insert x Xs)
    then obtain p where
      p1: "(p  Xs)  As = {}" and
      p2: "supp p = (Xs  (p  Xs))" by blast
    from x  As p1 have "x  p  Xs" by fast
    with x  Xs p2 have "x  supp p" by fast
    hence px: "p  x = x" unfolding supp_perm by simp
    have "finite (As  p  Xs  supp p)"
      using ‹finite As ‹finite Xs
      by (simp add: permute_set_eq_image finite_supp)
    then obtain y where "y  (As  p  Xs  supp p)" "sort_of y = sort_of x"
      by (rule obtain_atom)
    hence y: "y  As" "y  p  Xs" "y  supp p" "sort_of y = sort_of x"
      by simp_all
    hence py: "p  y = y" "x  y" using x  As
      by (auto simp: supp_perm)
    let ?q = "(x  y) + p"
    have q: "?q  insert x Xs = insert y (p  Xs)"
      unfolding insert_eqvt
      using p  x = x ‹sort_of y = sort_of x
      using x  p  Xs y  p  Xs
      by (simp add: swap_atom swap_set_not_in)
    have "?q  insert x Xs  As = {}"
      using y  As p  Xs  As = {}
      unfolding q by simp
    moreover
    have "supp (x  y)  supp p = {}" using px py ‹sort_of y = sort_of x
      unfolding supp_swap by (simp add: supp_perm)
    then have "supp ?q = (supp (x  y)  supp p)"
      by (simp add: supp_plus_perm_eq)
    then have "supp ?q = insert x Xs  ?q  insert x Xs"
      using p2 ‹sort_of y = sort_of x x  y unfolding q supp_swap
      by auto
    ultimately show ?case by blast
  qed
qed

lemma at_set_avoiding:
  assumes a: "finite Xs"
  and     b: "finite (supp c)"
  obtains p::"perm" where "(p  Xs)♯*c" and "(supp p) = (Xs  (p  Xs))"
  using a b at_set_avoiding_aux [where Xs="Xs" and As="Xs  supp c"]
  unfolding fresh_star_def fresh_def by blast

lemma at_set_avoiding1:
  assumes "finite xs"
  and     "finite (supp c)"
  shows "p. (p  xs) ♯* c"
using assms
apply(erule_tac c="c" in at_set_avoiding)
apply(auto)
done

lemma at_set_avoiding2:
  assumes "finite xs"
  and     "finite (supp c)" "finite (supp x)"
  and     "xs ♯* x"
  shows "p. (p  xs) ♯* c  supp x ♯* p"
using assms
apply(erule_tac c="(c, x)" in at_set_avoiding)
apply(simp add: supp_Pair)
apply(rule_tac x="p" in exI)
apply(simp add: fresh_star_Pair)
apply(rule fresh_star_supp_conv)
apply(auto simp: fresh_star_def)
done

lemma at_set_avoiding3:
  assumes "finite xs"
  and     "finite (supp c)" "finite (supp x)"
  and     "xs ♯* x"
  shows "p. (p  xs) ♯* c  supp x ♯* p  supp p = xs  (p  xs)"
using assms
apply(erule_tac c="(c, x)" in at_set_avoiding)
apply(simp add: supp_Pair)
apply(rule_tac x="p" in exI)
apply(simp add: fresh_star_Pair)
apply(rule fresh_star_supp_conv)
apply(auto simp: fresh_star_def)
done

lemma at_set_avoiding2_atom:
  assumes "finite (supp c)" "finite (supp x)"
  and     b: "a  x"
  shows "p. (p  a)  c  supp x ♯* p"
proof -
  have a: "{a} ♯* x" unfolding fresh_star_def by (simp add: b)
  obtain p where p1: "(p  {a}) ♯* c" and p2: "supp x ♯* p"
    using at_set_avoiding2[of "{a}" "c" "x"] assms a by blast
  have c: "(p  a)  c" using p1
    unfolding fresh_star_def Ball_def
    by(erule_tac x="p  a" in allE) (simp add: permute_set_def)
  hence "p  a  c  supp x ♯* p" using p2 by blast
  then show "p. (p  a)  c  supp x ♯* p" by blast
qed


section ‹Renaming permutations›

lemma set_renaming_perm:
  assumes b: "finite bs"
  shows "q. (b  bs. q  b = p  b)  supp q  bs  (p  bs)"
using b
proof (induct)
  case empty
  have "(b  {}. 0  b = p  b)  supp (0::perm)  {}  p  {}"
    by (simp add: permute_set_def supp_perm)
  then show "q. (b  {}. q  b = p  b)  supp q  {}  p  {}" by blast
next
  case (insert a bs)
  then have " q. (b  bs. q  b = p  b)  supp q  bs  p  bs" by simp
  then obtain q where *: "b  bs. q  b = p  b" and **: "supp q  bs  p  bs"
    by (metis empty_subsetI insert(3) supp_swap)
  { assume 1: "q  a = p  a"
    have "b  (insert a bs). q  b = p  b" using 1 * by simp
    moreover
    have "supp q  insert a bs  p  insert a bs"
      using ** by (auto simp: insert_eqvt)
    ultimately
    have "q. (b  insert a bs. q  b = p  b)  supp q  insert a bs  p  insert a bs" by blast
  }
  moreover
  { assume 2: "q  a  p  a"
    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(simp add: fresh_def[symmetric])
    apply(simp add: fresh_perm)
    apply(clarify)
    apply(rotate_tac 2)
    apply(drule subsetD[OF **])
    apply(simp add: inter_eqvt supp_eqvt permute_self)
    done
  ultimately have "(b  bs. q  b = p  b)  supp q  bs  (p  bs)" using * by auto
  then show "q. (b  bs. q  b = p  b)  supp q  bs  (p  bs)" by blast
qed

lemma list_renaming_perm:
  shows "q. (b  set bs. q  b = p  b)  supp q  set bs  (p  set bs)"
proof (induct bs)
  case (Cons a bs)
  then have " q. (b  set bs. q  b = p  b)  supp q  set bs  p  (set bs)"  by simp
  then obtain q where *: "b  set bs. q  b = p  b" and **: "supp q  set bs  p  (set bs)"
    by (blast)
  { assume 1: "a  set bs"
    have "q  a = p  a" using * 1 by (induct bs) (auto)
    then have "b  set (a # bs). q  b = p  b" using * by simp
    moreover
    have "supp q  set (a # bs)  p  (set (a # bs))" using ** by (auto simp: insert_eqvt)
    ultimately
    have "q. (b  set (a # bs). q  b = p  b)  supp q  set (a # bs)  p  (set (a # bs))" by blast
  }
  moreover
  { assume 2: "a  set bs"
    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 []"
    by (simp add: supp_zero_perm)
  then show "q. (b  set []. q  b = p  b)  supp q  set []  p  (set [])" by blast
qed


section ‹Concrete Atoms Types›

text ‹
  Class 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(simp_all add: supp_at_base)
  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"
  by (simp add: fresh_at_base)


lemma fresh_atom_at_base [simp]:
  fixes b::"'a::at_base"
  shows "a  atom b  a  b"
  by (simp add: fresh_def supp_at_base supp_atom)

lemma fresh_star_atom_at_base:
  fixes b::"'a::at_base"
  shows "as ♯* atom b  as ♯* b"
  by (simp add: fresh_star_def fresh_atom_at_base)

lemma if_fresh_at_base [simp]:
  shows "atom a  x  P (if a = x then t else s) = P s"
  and   "atom a  x  P (if x = a then t else s) = P s"
by (simp_all add: fresh_at_base)


simproc_setup fresh_ineq ("x  (y::'a::at_base)") = fn _ => fn ctxt => fn ctrm =>
  case Thm.term_of ctrm of @{term "HOL.Not"} $ (Const (@{const_name HOL.eq}, _) $ lhs $ rhs) =>
    let
      fun first_is_neg lhs rhs [] = NONE
        | first_is_neg lhs rhs (thm::thms) =
          (case Thm.prop_of thm of
             _ $ (@{term "HOL.Not"} $ (Const (@{const_name HOL.eq}, _) $ l $ r)) =>
               (if l = lhs andalso r = rhs then SOME(thm)
                else if r = lhs andalso l = rhs then SOME(thm RS @{thm not_sym})
                else first_is_neg lhs rhs thms)
        | _ => first_is_neg lhs rhs thms)

      val simp_thms = @{thms fresh_Pair fresh_at_base atom_eq_iff}
      val prems = Simplifier.prems_of ctxt
         |> filter (fn thm => case Thm.prop_of thm of
            _ $ (Const (@{const_name fresh}, ty) $ (_ $ a) $ b) =>
            (let
               val atms = a :: HOLogic.strip_tuple b
             in
               member ((=)) 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 =
  case head_of (Syntax.read_term ctxt name) of
    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_add: attribute
  val eqvt_del: attribute
  val eqvt_raw_add: 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
   (Global_Theory.add_thms_dynamic (@{binding "eqvts"}, eqvts) #>
    Global_Theory.add_thms_dynamic (@{binding "eqvts_raw"}, eqvts_raw))

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

fun add_raw_thm thm context =
  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" **)

fun add_thm thm context =
  (
    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_add = Thm.declaration_attribute add_raw_thm
val eqvt_raw_del = Thm.declaration_attribute del_raw_thm

fun eqvt_add_or_del eqvt_fn raw_fn =
  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_add = eqvt_add_or_del add_thm add_raw_thm
val eqvt_del = eqvt_add_or_del del_thm del_raw_thm

val _ =
  Theory.setup
   (Attrib.setup @{binding "eqvt"} (Attrib.add_del eqvt_add eqvt_del)
      "Declaration of equivariance lemmas - they will automatically be brought into the form ?p ∙ c ≡ c" #>
    Attrib.setup @{binding "eqvt_raw"} (Attrib.add_del eqvt_raw_add eqvt_raw_del)
      "Declaration of raw equivariance lemmas - no transformation is performed")

end;

File ‹nominal_permeq.ML›

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

infix 4 addpres addposts addexcls

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}
   addsimprocs [@{simproc natless_cancel_numerals}]
   addsimps @{thms in_measure wf_measure sum.case add_Suc_right add.right_neutral
     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)
        (Typedef.add_typedef {overloaded = false} (name, [], NoSyn) set NONE tac);

    (* 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))) >>
        (Toplevel.theory o add_atom_decl))

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
        HEADGOAL (resolve_tac ctxt [intro] THEN_ALL_NEW
          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
        (map (Syntax.string_of_term ctxt) is_already_eqvt))

    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
  by (simp_all add: fresh_zero_perm)

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
  by (simp_all add: fresh_plus_perm)

lemma alpha_sym_eqvt:
  assumes a: "R (p  x) y  R y (p  x)"
  and     b: "p  R = R"
  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,