val prove_fv_supp: typ list -> term list -> term list -> term list -> thm list -> thm list -> 

    thm list -> thm list -> thm list -> thm list -> bclause list list -> Proof.context -> thm list 

fun lookup xs x = the (AList.lookup (op=) xs x)

  let  

    val vs = fresh_args ctxt qtrm

    val rhs = list_comb (qtrm, vs)

    val lhs = fold (curry HOLogic.mk_prod) vs @{term "()"}

      |> mk_supp

  in

    mk_supports lhs rhs

    |> HOLogic.mk_Trueprop

  end

  let

    val ss1 = HOL_basic_ss addsimps @{thms supports_def fresh_def[symmetric]}

    val ss2 = HOL_ss addsimps @{thms swap_fresh_fresh fresh_Pair}

  in

    EVERY' [ simp_tac ss1,

             Nominal_Permeq.eqvt_strict_tac ctxt perm_simps [],

             simp_tac ss2 ]

  end

  let

    val goal = mk_supports_goal ctxt qtrm 

    val ctxt' = Variable.auto_fixes goal ctxt

  in

    Goal.prove ctxt' [] [] goal

      (K (HEADGOAL (supports_tac ctxt perm_simps)))

    |> singleton (ProofContext.export ctxt' ctxt)

  end

  let

    val (vs, ctxt') = Variable.variant_fixes (replicate (length qtys) "x") ctxt

    val goals = vs ~~ qtys

      |> map Free

      |> map (mk_finite o mk_supp)

      |> foldr1 (HOLogic.mk_conj)

      |> HOLogic.mk_Trueprop

    val tac = 

      EVERY' [ rtac @{thm supports_finite},

               resolve_tac qsupports_thms,

               asm_simp_tac (HOL_ss addsimps @{thms finite_supp supp_Pair finite_Un}) ]

  in

    Goal.prove ctxt' [] [] goals

      (K (HEADGOAL (rtac qinduct THEN_ALL_NEW tac)))

    |> singleton (ProofContext.export ctxt' ctxt)

    |> Datatype_Aux.split_conj_thm

    |> map zero_var_indexes

  end

  let

    val lthy1 = 

      lthy

      |> Local_Theory.exit_global

      |> Class.instantiation (qfull_ty_names, tvs, @{sort fs}) 

    fun tac _ =

      Class.intro_classes_tac [] THEN

        (ALLGOALS (resolve_tac qfsupp_thms))

  in

    lthy1

    |> Class.prove_instantiation_exit tac 

    |> Named_Target.theory_init

  end

(* proves that fv and fv_bn equals supp *)

fun mk_fvs_goals ty_arg_map fv =

  let

    val arg = fastype_of fv

      |> domain_type

      |> lookup ty_arg_map

  in

    (fv $ arg, mk_supp arg)

      |> HOLogic.mk_eq

      |> HOLogic.mk_Trueprop 

  end

fun mk_fv_bns_goals ty_arg_map fv_bn alpha_bn =

  let

    val arg = fastype_of fv_bn

      |> domain_type

      |> lookup ty_arg_map

  in

    (fv_bn $ arg, mk_supp_rel alpha_bn arg)

      |> HOLogic.mk_eq

      |> HOLogic.mk_Trueprop 

  end

fun add_ss thms =

  HOL_basic_ss addsimps thms

fun symmetric thms = 

  map (fn thm => thm RS @{thm sym}) thms

val supp_abs_set = @{thms supp_abs(1)[symmetric]}

val supp_abs_res = @{thms supp_abs(2)[symmetric]}

val supp_abs_lst = @{thms supp_abs(3)[symmetric]}

fun mk_supp_abs ctxt (BC (Set, _, _)) = EqSubst.eqsubst_tac ctxt [1] supp_abs_set 

  | mk_supp_abs ctxt (BC (Res, _, _)) = EqSubst.eqsubst_tac ctxt [1] supp_abs_res

  | mk_supp_abs ctxt (BC (Lst, _, _)) = EqSubst.eqsubst_tac ctxt [1] supp_abs_lst

fun mk_supp_abs_tac ctxt [] = []

  | mk_supp_abs_tac ctxt (BC (_, [], _)::xs) = mk_supp_abs_tac ctxt xs

  | mk_supp_abs_tac ctxt (bc::xs) = (DETERM o mk_supp_abs ctxt bc)::mk_supp_abs_tac ctxt xs

fun mk_bn_supp_abs_tac thm =

  (prop_of thm)

  |> HOLogic.dest_Trueprop

  |> snd o HOLogic.dest_eq

  |> fastype_of

  |> (fn ty => case ty of

        @{typ "atom set"}  => simp_tac (add_ss supp_abs_set)

      | @{typ "atom list"} => simp_tac (add_ss supp_abs_lst)

      | _ => raise TERM ("mk_bn_supp_abs_tac", [prop_of thm]))

val thms1 = @{thms supp_Pair supp_eqvt[symmetric] Un_assoc conj_assoc}

val thms2 = @{thms de_Morgan_conj Collect_disj_eq finite_Un}

val thms3 = @{thms alphas prod_alpha_def prod_fv.simps prod_rel.simps permute_prod_def 

  prod.recs prod.cases prod.inject not_True_eq_False empty_def[symmetric] Finite_Set.finite.emptyI}

fun ind_tac ctxt qinducts = 

  let

    fun CASES_TAC_TO_TAC cases_tac st = Seq.map snd (cases_tac st)

  in

    DETERM o (CASES_TAC_TO_TAC o (Induct.induct_tac ctxt false [] [] [] (SOME qinducts) []))

  end

fun p_tac msg i = 

  if false then print_tac ("ptest: " ^ msg) else all_tac

fun q_tac msg i = 

  if true then print_tac ("qtest: " ^ msg) else all_tac

fun prove_fv_supp qtys fvs fv_bns alpha_bns bn_simps fv_simps eq_iffs perm_simps 

  fv_bn_eqvts qinducts bclausess ctxt =

  let

    val (arg_names, ctxt') = 

      Variable.variant_fixes (replicate (length qtys) "x") ctxt 

    val args = map2 (curry Free) arg_names qtys 

    val ty_arg_map = qtys ~~ args

    val ind_args = map SOME arg_names

    val goals1 = map (mk_fvs_goals ty_arg_map) fvs

    val goals2 = map2 (mk_fv_bns_goals ty_arg_map) fv_bns alpha_bns

    fun fv_tac ctxt bclauses =

      SOLVED' (EVERY' [

        (fn i => print_tac ("FV Goal: " ^ string_of_int i ^ "  with  " ^ (@{make_string} bclauses))),

        TRY o asm_full_simp_tac (add_ss (@{thm supp_Pair[symmetric]}::fv_simps)), 

        p_tac "A",

        TRY o EVERY' (mk_supp_abs_tac ctxt bclauses),

        p_tac "B",

        TRY o simp_tac (add_ss @{thms supp_def supp_rel_def}),

        p_tac "C",

        TRY o Nominal_Permeq.eqvt_tac ctxt (perm_simps @ fv_bn_eqvts) [], 

        p_tac "D",

        TRY o simp_tac (add_ss (@{thms Abs_eq_iff} @ eq_iffs)),

        p_tac "E",

        TRY o asm_full_simp_tac (add_ss thms3),

        p_tac "F",

        TRY o simp_tac (add_ss thms2),

        p_tac "G",

        TRY o asm_full_simp_tac (add_ss (thms1 @ (symmetric fv_bn_eqvts))),

        p_tac "H",

        (fn i => print_tac ("finished with FV Goal: " ^ string_of_int i))

        ])

    fun bn_tac ctxt bn_simp =

      SOLVED' (EVERY' [

        (fn i => print_tac ("BN Goal: " ^ string_of_int i)),

        TRY o asm_full_simp_tac (add_ss (@{thm supp_Pair[symmetric]}::fv_simps)),

        q_tac "A",

        TRY o mk_bn_supp_abs_tac bn_simp,

        q_tac "B",

        TRY o simp_tac (add_ss @{thms supp_def supp_rel_def}),

        q_tac "C",

        TRY o Nominal_Permeq.eqvt_tac ctxt (perm_simps @ fv_bn_eqvts) [], 

        q_tac "D",

        TRY o simp_tac (add_ss (@{thms Abs_eq_iff} @ eq_iffs)),

        q_tac "E",

        TRY o asm_full_simp_tac (add_ss thms3),

        q_tac "F",

        TRY o simp_tac (add_ss thms2),

        q_tac "G",

        TRY o asm_full_simp_tac (add_ss (thms1 @ (symmetric fv_bn_eqvts))),

        (fn i => print_tac ("finished with BN Goal: " ^ string_of_int i))

        ])

    fun fv_tacs ctxt = map (fv_tac ctxt) bclausess

    fun bn_tacs ctxt = map (bn_tac ctxt) bn_simps

  in

    Goal.prove_multi ctxt' [] [] (goals1 @ goals2)

     (fn {context as ctxt, ...} => HEADGOAL 

       (ind_tac ctxt qinducts THEN' RANGE  (fv_tacs ctxt @ bn_tacs ctxt)))

    |> ProofContext.export ctxt' ctxt

  end