theory Nominal2

imports 

  Nominal2_Base Nominal2_Eqvt Nominal2_Abs

uses ("nominal_dt_rawfuns.ML")

     ("nominal_dt_alpha.ML")

     ("nominal_dt_quot.ML")

begin

use "nominal_dt_rawfuns.ML"

ML {* open Nominal_Dt_RawFuns *}

use "nominal_dt_alpha.ML"

ML {* open Nominal_Dt_Alpha *}

use "nominal_dt_quot.ML"

ML {* open Nominal_Dt_Quot *}

text {* TEST *}

ML {*

fun strip_outer_params (Const("all", _) $ Abs (a, T, t)) = strip_outer_params t |>> cons (a, T)

  | strip_outer_params B = ([], B)

fun strip_params_prems_concl trm =

  let

    val (params, body) = strip_outer_params trm

    val (prems, concl) = Logic.strip_horn body

  in

    (params, prems, concl)

  end

fun list_params_prems_concl params prems concl =

  Logic.list_implies (prems, concl)

  |> fold_rev mk_all params

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_rev (curry (mk_union_env tys)) trms @{term "{}::atom set"}

*}

ML {*

fun process_ecase lthy c (params, prems, concl) bclauses =

  let

    val tys = map snd params

    val binders = get_all_binders bclauses

    fun prep_binder (opt, i) = 

      let

        val t = Bound (length tys - i - 1)

      in

        case opt of

          NONE => setify_ty lthy (nth tys i) t 

        | SOME bn => to_set_ty (fastype_of1 (tys, bn $ t)) (bn $ t)   

      end 

    val fresh_prem = 

      case binders of

        [] => []                        (* case: no binders *)

      | _ => binders                    (* case: binders *) 

          |> map prep_binder

          |> fold_union_env tys

          |> (fn t => mk_fresh_star t c)

          |> HOLogic.mk_Trueprop

          |> single 

  in

    list_params_prems_concl params (fresh_prem @ prems) concl

  end  

*}

ML {*

(* derives the freshness theorem that there exists a p, such that 

   (p o as) #* (c, t1,\<dots>, tn) *)

fun fresh_thm ctxt c parms binders bn_finite_thms =

  let

    fun prep_binder (opt, i) = 

      case opt of

        NONE => setify ctxt (nth parms i) 

      | SOME bn => to_set (bn $ (nth parms i))  

    fun prep_binder2 (opt, i) = 

      case opt of

        NONE => atomify ctxt (nth parms i)

      | SOME bn => bn $ (nth parms i) 

    val rhs = HOLogic.mk_tuple ([c] @ parms @ (map prep_binder2 binders))

    val lhs = binders

      |> map prep_binder

      |> fold_union

      |> mk_perm (Bound 0)

    val goal = mk_fresh_star lhs rhs

      |> (fn t => HOLogic.mk_exists ("p", @{typ perm}, t))

      |> HOLogic.mk_Trueprop   

    val ss = bn_finite_thms @ @{thms supp_Pair finite_supp finite_sets_supp} 

      @ @{thms finite.intros finite_Un finite_set finite_fset}  

  in

    Goal.prove ctxt [] [] goal

      (K (HEADGOAL (rtac @{thm at_set_avoiding1}

          THEN_ALL_NEW (simp_tac (HOL_ss addsimps ss)))))

  end

*} 

ML {*

(* derives abs_eq theorems of the form Exists s. [as].t = [p o as].s *)

  case binders of

    [] => [] 

  | _ =>

      let

        val binder_trm = comb_binders ctxt bmode parms binders

        val body_trm = foldl1 HOLogic.mk_prod (map (nth parms) bodies)

        val body_ty = fastype_of body_trm

        val (abs_name, binder_ty, abs_ty) = 

          case bmode of

            Lst => (@{const_name "Abs_lst"}, @{typ "atom list"}, @{type_name abs_lst})

          | Set => (@{const_name "Abs_set"}, @{typ "atom set"},  @{type_name abs_set})

          | Res => (@{const_name "Abs_res"}, @{typ "atom set"},  @{type_name abs_res})

        val abs = Const (abs_name, [binder_ty, body_ty] ---> Type (abs_ty, [body_ty]))

        val abs_lhs = abs $ binder_trm $ body_trm

          |> HOLogic.mk_Trueprop  

        val ss = fprops @ bn_finite_thms @ @{thms set.simps set_append union_eqvt}

          @ @{thms fresh_star_Un fresh_star_Pair fresh_star_list fresh_star_singleton fresh_star_fset

     in

     end

*}

ML {*

*}

ML {*

  let

    fun aux_tac prem bclauses =

      case (get_all_binders bclauses) of

        [] => EVERY' [rtac prem, atac]

          let

            val parms = map (term_of o snd) params

            val fthm = fresh_thm ctxt c parms binders bn_finite_thms 

            val ss = @{thms fresh_star_Pair union_eqvt fresh_star_Un}

            val (([(_, fperm)], fprops), ctxt') = Obtain.result

              (K (EVERY1 [etac exE, 

                          full_simp_tac (HOL_basic_ss addsimps ss), 

          in

          end) ctxt

  in

  end

*}

ML {*

  let

    val ([c, a], lthy'') = Variable.variant_fixes ["c", "'a"] lthy'

    val c = Free (c, TFree (a, @{sort fs}))

      |> map prop_of

      |> map Logic.strip_horn

      |> split_list

      |>> (map o map) strip_params_prems_concl     

    val prems = map2 (process_ecase lthy'' c) (flat ecases) (flat bclausesss)

  in

    Goal.prove_multi lthy'' [] prems main_concls

     (fn {prems:thm list, context} =>

        let

           val prems' = partitions prems (map length bclausesss)

        in

          EVERY1 [Goal.conjunction_tac,

        end)

  end

*}

ML {*

val eqvt_attr = Attrib.internal (K Nominal_ThmDecls.eqvt_add)

val rsp_attr = Attrib.internal (K Quotient_Info.rsp_rules_add)

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

*}

section{* Interface for nominal_datatype *}

ML {* print_depth 50 *}

ML {*

fun get_cnstrs dts =

  map (fn (_, _, _, constrs) => constrs) dts

fun get_typed_cnstrs dts =

  flat (map (fn (_, bn, _, constrs) => 

   (map (fn (bn', _, _) => (Binding.name_of bn, Binding.name_of bn')) constrs)) dts)

fun get_cnstr_strs dts =

  map (fn (bn, _, _) => Binding.name_of bn) (flat (get_cnstrs dts))

fun get_bn_fun_strs bn_funs =

  map (fn (bn_fun, _, _) => Binding.name_of bn_fun) bn_funs

*}

text {* Infrastructure for adding "_raw" to types and terms *}

ML {*

fun add_raw s = s ^ "_raw"

fun add_raws ss = map add_raw ss

fun raw_bind bn = Binding.suffix_name "_raw" bn

fun replace_str ss s = 

  case (AList.lookup (op=) ss s) of 

     SOME s' => s'

   | NONE => s

fun replace_typ ty_ss (Type (a, Ts)) = Type (replace_str ty_ss a, map (replace_typ ty_ss) Ts)

  | replace_typ ty_ss T = T  

fun raw_dts ty_ss dts =

let

  fun raw_dts_aux1 (bind, tys, mx) =

    (raw_bind bind, map (replace_typ ty_ss) tys, mx)

  fun raw_dts_aux2 (ty_args, bind, mx, constrs) =

    (ty_args, raw_bind bind, mx, map raw_dts_aux1 constrs)

in

  map raw_dts_aux2 dts

end

fun replace_aterm trm_ss (Const (a, T)) = Const (replace_str trm_ss a, T)

  | replace_aterm trm_ss (Free (a, T)) = Free (replace_str trm_ss a, T)

  | replace_aterm trm_ss trm = trm

fun replace_term trm_ss ty_ss trm =

  trm |> Term.map_aterms (replace_aterm trm_ss) |> map_types (replace_typ ty_ss) 

*}

ML {*

fun rawify_dts dt_names dts dts_env =

let

  val raw_dts = raw_dts dts_env dts

  val raw_dt_names = add_raws dt_names

in

  (raw_dt_names, raw_dts)

end 

*}

ML {*

fun rawify_bn_funs dts_env cnstrs_env bn_fun_env bn_funs bn_eqs =

let

  val bn_funs' = map (fn (bn, ty, mx) => 

    (raw_bind bn, SOME (replace_typ dts_env ty), mx)) bn_funs

  val bn_eqs' = map (fn (attr, trm) => 

    (attr, replace_term (cnstrs_env @ bn_fun_env) dts_env trm)) bn_eqs

in

  (bn_funs', bn_eqs') 

end