theory Equivp

imports "Abs" "Perm" "Tacs" "Rsp"

begin

lemma not_in_union: "c \<notin> a \<union> b \<equiv> (c \<notin> a \<and> c \<notin> b)"

by auto

ML {*

fun supports_tac perm =

  simp_tac (HOL_ss addsimps @{thms supports_def not_in_union} @ perm) THEN_ALL_NEW (

    REPEAT o rtac allI THEN' REPEAT o rtac impI THEN' split_conj_tac THEN'

    asm_full_simp_tac (HOL_ss addsimps @{thms fresh_def[symmetric]

      swap_fresh_fresh fresh_atom swap_at_base_simps(3) swap_atom_image_fresh

      supp_fset_to_set supp_fmap_atom}))

*}

ML {*

fun mk_supp ty x =

  Const (@{const_name supp}, ty --> @{typ "atom set"}) $ x

*}

ML {*

fun mk_supports_eq thy cnstr =

let

  val (tys, ty) = (strip_type o fastype_of) cnstr

  val names = Datatype_Prop.make_tnames tys

  val frees = map Free (names ~~ tys)

  val rhs = list_comb (cnstr, frees)

  fun mk_supp_arg (x, ty) =

    if is_atom thy ty then mk_supp @{typ atom} (mk_atom_ty ty x) else

    if is_atom_set thy ty then mk_supp @{typ "atom set"} (mk_atom_set x) else

    if is_atom_fset thy ty then mk_supp @{typ "atom set"} (mk_atom_fset x)

    else mk_supp ty x

  val lhss = map mk_supp_arg (frees ~~ tys)

  val supports = Const(@{const_name "supports"}, @{typ "atom set"} --> ty --> @{typ bool})

  val eq = HOLogic.mk_Trueprop (supports $ fold_union lhss $ rhs)

in

  (names, eq)

end

*}

ML {*

fun prove_supports ctxt perms cnst =

let

  val (names, eq) = mk_supports_eq ctxt cnst

in

  Goal.prove ctxt names [] eq (fn _ => supports_tac perms 1)

end

*}

ML {*

fun mk_fs tys =

let

  val names = Datatype_Prop.make_tnames tys

  val frees = map Free (names ~~ tys)

  val supps = map2 mk_supp tys frees

  val fin_supps = map (fn x => @{term "finite :: atom set \<Rightarrow> bool"} $ x) supps

in

  (names, HOLogic.mk_Trueprop (mk_conjl fin_supps))

end

*}

ML {*

fun fs_tac induct supports = rtac induct THEN_ALL_NEW (

  rtac @{thm supports_finite} THEN' resolve_tac supports) THEN_ALL_NEW

  asm_full_simp_tac (HOL_ss addsimps @{thms supp_atom supp_atom_image supp_fset_to_set

    supp_fmap_atom finite_insert finite.emptyI finite_Un finite_supp})

*}

ML {*

fun prove_fs ctxt induct supports tys =

let

  val (names, eq) = mk_fs tys

in

  Goal.prove ctxt names [] eq (fn _ => fs_tac induct supports 1)

end

*}

ML {*

fun mk_supp x = Const (@{const_name supp}, fastype_of x --> @{typ "atom set"}) $ x;

fun mk_supp_neq arg (fv, alpha) =

let

  val collect = Const ("Collect", @{typ "(atom \<Rightarrow> bool) \<Rightarrow> atom \<Rightarrow> bool"});

  val ty = fastype_of arg;

  val perm = Const ("Nominal2_Base.pt_class.permute", @{typ perm} --> ty --> ty);

  val finite = @{term "finite :: atom set \<Rightarrow> bool"}

  val rhs = collect $ Abs ("a", @{typ atom},

    HOLogic.mk_not (finite $

      (collect $ Abs ("b", @{typ atom},

        HOLogic.mk_not (alpha $ (perm $ (@{term swap} $ Bound 1 $ Bound 0) $ arg) $ arg)))))

in

  HOLogic.mk_eq (fv $ arg, rhs)

end;

fun supp_eq fv_alphas_lst =

let

  val (fvs_alphas, ls) = split_list fv_alphas_lst;

  val (fv_ts, _) = split_list fvs_alphas;

  val tys = map (domain_type o fastype_of) fv_ts;

  val names = Datatype_Prop.make_tnames tys;

  val args = map Free (names ~~ tys);

  fun supp_eq_arg ((fv, arg), l) =

    mk_conjl

      ((HOLogic.mk_eq (fv $ arg, mk_supp arg)) ::

       (map (mk_supp_neq arg) l))

  val eqs = mk_conjl (map supp_eq_arg ((fv_ts ~~ args) ~~ ls))

in

  (names, HOLogic.mk_Trueprop eqs)

end

*}

ML {*

fun combine_fv_alpha_bns (fv_ts_nobn, fv_ts_bn) (alpha_ts_nobn, alpha_ts_bn) bn_nos =

if length fv_ts_bn < length alpha_ts_bn then

  (fv_ts_nobn ~~ alpha_ts_nobn) ~~ (replicate (length fv_ts_nobn) [])

else let

  val fv_alpha_nos = 0 upto (length fv_ts_nobn - 1);

  fun filter_fn i (x, j) = if j = i then SOME x else NONE;

  val fv_alpha_bn_nos = (fv_ts_bn ~~ alpha_ts_bn) ~~ bn_nos;

  val fv_alpha_bn_all = map (fn i => map_filter (filter_fn i) fv_alpha_bn_nos) fv_alpha_nos;

in

  (fv_ts_nobn ~~ alpha_ts_nobn) ~~ fv_alpha_bn_all

end

*}

(* TODO: this is a hack, it assumes that only one type of Abs's is present

   in the type and chooses this supp_abs. Additionally single atoms are

   treated properly. *)

ML {*

fun choose_alpha_abs eqiff =

let

  fun exists_subterms f ts = member (op =) (map (exists_subterm f) ts) true;

  val terms = map prop_of eqiff;

  fun check cname = exists_subterms (fn x => fst(dest_Const x) = cname handle _ => false) terms

  val no =

    if check @{const_name alpha_lst} then 2 else

    if check @{const_name alpha_res} then 1 else

    if check @{const_name alpha_gen} then 0 else

    error "Failure choosing supp_abs"

in

  nth @{thms supp_abs[symmetric]} no

end

*}

lemma supp_abs_atom: "supp (Abs {atom a} (x :: 'a :: fs)) = supp x - {atom a}"

by (rule supp_abs(1))

lemma supp_abs_sum:

  "supp (Abs x (a :: 'a :: fs)) \<union> supp (Abs x (b :: 'b :: fs)) = supp (Abs x (a, b))"

  "supp (Abs_res x (a :: 'a :: fs)) \<union> supp (Abs_res x (b :: 'b :: fs)) = supp (Abs_res x (a, b))"

  "supp (Abs_lst y (a :: 'a :: fs)) \<union> supp (Abs_lst y (b :: 'b :: fs)) = supp (Abs_lst y (a, b))"

  apply (simp_all add: supp_abs supp_Pair)

  apply blast+

  done

ML {*

fun supp_eq_tac ind fv perm eqiff ctxt =

  rtac ind THEN_ALL_NEW

  asm_full_simp_tac (HOL_basic_ss addsimps fv) THEN_ALL_NEW

  asm_full_simp_tac (HOL_basic_ss addsimps @{thms supp_abs_atom[symmetric]}) THEN_ALL_NEW

  asm_full_simp_tac (HOL_basic_ss addsimps [choose_alpha_abs eqiff]) THEN_ALL_NEW

  simp_tac (HOL_basic_ss addsimps @{thms supp_abs_sum}) THEN_ALL_NEW

  simp_tac (HOL_basic_ss addsimps @{thms supp_def}) THEN_ALL_NEW

  simp_tac (HOL_basic_ss addsimps (@{thms permute_abs} @ perm)) THEN_ALL_NEW

  simp_tac (HOL_basic_ss addsimps (@{thms Abs_eq_iff} @ eqiff)) THEN_ALL_NEW

  simp_tac (HOL_basic_ss addsimps @{thms alphas3 alphas2}) THEN_ALL_NEW

  simp_tac (HOL_basic_ss addsimps @{thms alphas}) THEN_ALL_NEW

  asm_full_simp_tac (HOL_basic_ss addsimps (@{thm supp_Pair} :: sym_eqvts ctxt)) THEN_ALL_NEW

  asm_full_simp_tac (HOL_basic_ss addsimps (@{thm Pair_eq} :: all_eqvts ctxt)) THEN_ALL_NEW

  simp_tac (HOL_basic_ss addsimps @{thms supp_at_base[symmetric,simplified supp_def]}) THEN_ALL_NEW

  simp_tac (HOL_basic_ss addsimps @{thms Collect_disj_eq[symmetric]}) THEN_ALL_NEW

  simp_tac (HOL_basic_ss addsimps @{thms infinite_Un[symmetric]}) THEN_ALL_NEW

  simp_tac (HOL_basic_ss addsimps @{thms Collect_disj_eq[symmetric]}) THEN_ALL_NEW

  simp_tac (HOL_basic_ss addsimps @{thms de_Morgan_conj[symmetric]}) THEN_ALL_NEW

  simp_tac (HOL_basic_ss addsimps @{thms ex_simps(1,2)[symmetric]}) THEN_ALL_NEW

  simp_tac (HOL_ss addsimps @{thms Collect_const finite.emptyI})

*}

end