theory Nominal2

imports 

  Nominal2_Base Nominal2_Abs Nominal2_FCB

uses ("nominal_dt_rawfuns.ML")

     ("nominal_dt_alpha.ML")

     ("nominal_dt_quot.ML")

     ("nominal_induct.ML")

     ("nominal_inductive.ML")

     ("nominal_function_common.ML")

     ("nominal_function_core.ML")

     ("nominal_mutual.ML")

     ("nominal_function.ML")

     ("nominal_dt_data.ML")

begin

use "nominal_dt_data.ML"

ML {* open Nominal_Dt_Data *}

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 *}

(*****************************************)

(* setup for induction principles method *)

use "nominal_induct.ML"

method_setup nominal_induct =

  {* NominalInduct.nominal_induct_method *}

  {* nominal induction *}

(****************************************************)

(* inductive definition involving nominal datatypes *)

use "nominal_inductive.ML" 

(***************************************)

(* forked code of the function package *)

(* for defining nominal functions      *)

use "nominal_function_common.ML"

use "nominal_function_core.ML"

use "nominal_mutual.ML"

use "nominal_function.ML"

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)

val induct_attr = Attrib.internal (K Induct.induct_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, _) =

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

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

    (ty_args, raw_bind bind, NoSyn, 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, _) => 

    (raw_bind bn, SOME (replace_typ dts_env ty), NoSyn)) 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 

*}

ML {* 

fun rawify_bclauses dts_env cnstrs_env bn_fun_env bclauses =

let

  fun rawify_bnds bnds = 

    map (apfst (Option.map (replace_term (cnstrs_env @ bn_fun_env) dts_env))) bnds

  fun rawify_bclause (BC (mode, bnds, bdys)) = BC (mode, rawify_bnds bnds, bdys)

in

  (map o map o map) rawify_bclause bclauses

end

*}

ML {*

(* definition of the raw datatype *)

fun define_raw_dts dts cnstr_names cnstr_tys bn_funs bn_eqs bclauses lthy =

let

  val thy = Local_Theory.exit_global lthy

  val thy_name = Context.theory_name thy

  val dt_names = map (fn (_, s, _, _) => Binding.name_of s) dts

  val dt_full_names = map (Long_Name.qualify thy_name) dt_names 

  val dt_full_names' = add_raws dt_full_names

  val dts_env = dt_full_names ~~ dt_full_names'

  val cnstr_full_names = map (Long_Name.qualify thy_name) cnstr_names

  val cnstr_full_names' = map (fn (x, y) => Long_Name.qualify thy_name 

    (Long_Name.qualify (add_raw x) (add_raw y))) cnstr_tys

  val cnstrs_env = cnstr_full_names ~~ cnstr_full_names'

  val bn_fun_strs = get_bn_fun_strs bn_funs

  val bn_fun_strs' = add_raws bn_fun_strs

  val bn_fun_env = bn_fun_strs ~~ bn_fun_strs'

  val bn_fun_full_env = map (pairself (Long_Name.qualify thy_name)) 

    (bn_fun_strs ~~ bn_fun_strs')

  val (raw_full_dt_names, raw_dts) = rawify_dts dt_names dts dts_env

  val (raw_bn_funs, raw_bn_eqs) = rawify_bn_funs dts_env cnstrs_env bn_fun_env bn_funs bn_eqs 

  val raw_bclauses = rawify_bclauses dts_env cnstrs_env bn_fun_full_env bclauses 

  val (raw_full_dt_names', thy1) = 

    Datatype.add_datatype Datatype.default_config raw_full_dt_names raw_dts thy

  val lthy1 = Named_Target.theory_init thy1

  val dtinfos = map (Datatype.the_info (ProofContext.theory_of lthy1)) raw_full_dt_names' 

  val {descr, sorts, ...} = hd dtinfos

  val raw_tys = Datatype_Aux.get_rec_types descr sorts

  val raw_ty_args = hd raw_tys

    |> snd o dest_Type

    |> map dest_TFree 

  val raw_cns_info = all_dtyp_constrs_types descr sorts

  val raw_all_cns = (map o map) (fn (c, _, _, _) => c) raw_cns_info

  val raw_inject_thms = flat (map #inject dtinfos)

  val raw_distinct_thms = flat (map #distinct dtinfos)

  val raw_induct_thm = #induct (hd dtinfos)

  val raw_induct_thms = #inducts (hd dtinfos)

  val raw_exhaust_thms = map #exhaust dtinfos

  val raw_size_trms = map HOLogic.size_const raw_tys

  val raw_size_thms = Size.size_thms (ProofContext.theory_of lthy1) (hd raw_full_dt_names')

    |> `(fn thms => (length thms) div 2)

    |> uncurry drop

  val raw_result = RawDtInfo 

    {raw_dt_names = raw_full_dt_names',

     raw_dts = raw_dts,

     raw_tys = raw_tys,

     raw_ty_args = raw_ty_args,

     raw_cns_info = raw_cns_info,

     raw_all_cns = raw_all_cns,

     raw_inject_thms = raw_inject_thms,

     raw_distinct_thms = raw_distinct_thms,

     raw_induct_thm = raw_induct_thm,

     raw_induct_thms = raw_induct_thms,

     raw_exhaust_thms = raw_exhaust_thms,

     raw_size_trms = raw_size_trms,

     raw_size_thms = raw_size_thms}

in

  (raw_bclauses, raw_bn_funs, raw_bn_eqs, raw_result, lthy1)

end

*}

ML {*

fun nominal_datatype2 opt_thms_name dts bn_funs bn_eqs bclauses lthy =

let

  val cnstr_names = get_cnstr_strs dts

  val cnstr_tys = get_typed_cnstrs dts

  val _ = trace_msg (K "Defining raw datatypes...")

  val (raw_bclauses, raw_bn_funs, raw_bn_eqs, raw_dt_info, lthy0) =

    define_raw_dts dts cnstr_names cnstr_tys bn_funs bn_eqs bclauses lthy   

  val RawDtInfo 

    {raw_dt_names,

     raw_tys,

     raw_ty_args,

     raw_all_cns,

     raw_inject_thms,

     raw_distinct_thms,

     raw_induct_thm,

     raw_induct_thms,

     raw_exhaust_thms,

     raw_size_trms,

     raw_size_thms, ...} = raw_dt_info

  val _ = trace_msg (K "Defining raw permutations...")

  val ((raw_perm_funs, raw_perm_simps, raw_perm_laws), lthy2a) = define_raw_perms raw_dt_info lthy0

  (* noting the raw permutations as eqvt theorems *)

  val _ = trace_msg (K "Defining raw fv- and bn-functions...")

  val (raw_bns, raw_bn_defs, raw_bn_info, raw_bn_inducts, lthy3a) =

    define_raw_bns raw_dt_info raw_bn_funs raw_bn_eqs lthy3

  (* defining the permute_bn functions *)

  val (raw_perm_bns, raw_perm_bn_simps, lthy3b) = 

    define_raw_bn_perms raw_dt_info raw_bn_info lthy3a

  val (raw_fvs, raw_fv_bns, raw_fv_defs, raw_fv_bns_induct, lthy3c) = 

    define_raw_fvs raw_dt_info raw_bn_info raw_bclauses lthy3b

  val _ = trace_msg (K "Defining alpha relations...")

  val (alpha_result, lthy4) =

    define_raw_alpha raw_dt_info raw_bn_info raw_bclauses raw_fvs lthy3c

  val _ = trace_msg (K "Proving distinct theorems...")

  val alpha_distincts = raw_prove_alpha_distincts lthy4 alpha_result raw_dt_info

  val _ = trace_msg (K "Proving eq-iff theorems...")

  val alpha_eq_iff = raw_prove_alpha_eq_iff lthy4 alpha_result raw_dt_info

  val _ = trace_msg (K "Proving equivariance of bns, fvs, size and alpha...")

  val raw_bn_eqvt = 

    raw_prove_eqvt raw_bns raw_bn_inducts (raw_bn_defs @ raw_perm_simps) lthy4

  (* noting the raw_bn_eqvt lemmas in a temprorary theory *)

  val lthy_tmp = snd (Local_Theory.note ((Binding.empty, [eqvt_attr]), raw_bn_eqvt) lthy4)

  val raw_fv_eqvt = 

    raw_prove_eqvt (raw_fvs @ raw_fv_bns) raw_fv_bns_induct (raw_fv_defs @ raw_perm_simps) 

      (Local_Theory.restore lthy_tmp)

  val raw_size_eqvt =

    let

      val RawDtInfo {raw_size_trms, raw_size_thms, raw_induct_thms, ...} = raw_dt_info

    in

      raw_prove_eqvt raw_size_trms raw_induct_thms (raw_size_thms @ raw_perm_simps) 

        (Local_Theory.restore lthy_tmp)

        |> map (rewrite_rule @{thms permute_nat_def[THEN eq_reflection]})

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

    end 

  val lthy5 = snd (Local_Theory.note ((Binding.empty, [eqvt_attr]), raw_fv_eqvt) lthy_tmp)

  val alpha_eqvt =

    let

      val AlphaResult {alpha_trms, alpha_bn_trms, alpha_raw_induct, alpha_intros, ...} = alpha_result

    in

      Nominal_Eqvt.raw_equivariance (alpha_trms @ alpha_bn_trms) alpha_raw_induct alpha_intros lthy5

    end

  val alpha_eqvt_norm = map (Nominal_ThmDecls.eqvt_transform lthy5) alpha_eqvt

  val _ = trace_msg (K "Proving equivalence of alpha...")

  val alpha_refl_thms = raw_prove_refl lthy5 alpha_result raw_induct_thm  

  val alpha_sym_thms = raw_prove_sym lthy5 alpha_result alpha_eqvt_norm

  val alpha_trans_thms =

    raw_prove_trans lthy5 alpha_result (raw_distinct_thms @ raw_inject_thms) alpha_eqvt_norm

  val (alpha_equivp_thms, alpha_bn_equivp_thms) = 

    raw_prove_equivp lthy5 alpha_result alpha_refl_thms alpha_sym_thms alpha_trans_thms

  val _ = trace_msg (K "Proving alpha implies bn...")

  val alpha_bn_imp_thms = raw_prove_bn_imp lthy5 alpha_result

  val _ = trace_msg (K "Proving respectfulness...")

  val raw_funs_rsp_aux = 

    raw_fv_bn_rsp_aux lthy5 alpha_result raw_fvs raw_bns raw_fv_bns (raw_bn_defs @ raw_fv_defs) 

  val raw_funs_rsp = map mk_funs_rsp raw_funs_rsp_aux

  val raw_size_rsp = 

    raw_size_rsp_aux lthy5 alpha_result (raw_size_thms @ raw_size_eqvt)

      |> map mk_funs_rsp

  val raw_constrs_rsp = 

    raw_constrs_rsp lthy5 alpha_result (flat raw_all_cns) (alpha_bn_imp_thms @ raw_funs_rsp_aux) 

  val alpha_permute_rsp = map mk_alpha_permute_rsp alpha_eqvt

  val alpha_bn_rsp = 

    raw_alpha_bn_rsp alpha_result alpha_bn_equivp_thms alpha_bn_imp_thms

  val raw_perm_bn_rsp =raw_perm_bn_rsp lthy5 alpha_result raw_perm_bns raw_perm_bn_simps

  (* noting the quot_respects lemmas *)

  val (_, lthy6) = 

    Local_Theory.note ((Binding.empty, [rsp_attr]),

      raw_constrs_rsp @ raw_funs_rsp @ raw_size_rsp @ alpha_permute_rsp @ 

      alpha_bn_rsp @ raw_perm_bn_rsp) lthy5

  val _ = trace_msg (K "Defining the quotient types...")

  val qty_descr = map (fn (vs, bind, mx, _) => (vs, bind, mx)) dts

  val (qty_infos, lthy7) = 

    let

      val AlphaResult {alpha_trms, alpha_tys, ...} = alpha_result

    in

      define_qtypes qty_descr alpha_tys alpha_trms alpha_equivp_thms lthy6

    end

  val qtys = map #qtyp qty_infos

  val qty_full_names = map (fst o dest_Type) qtys

  val qty_names = map Long_Name.base_name qty_full_names             

  val _ = trace_msg (K "Defining the quotient constants...")

  val qconstrs_descrs =

    (map2 o map2) (fn (b, _, mx) => fn t => (Variable.check_name b, t, mx)) (get_cnstrs dts) raw_all_cns

  val qbns_descr =

    map2 (fn (b, _, mx) => fn t => (Variable.check_name b, t, mx)) bn_funs raw_bns

  val qfvs_descr = 

    map2 (fn n => fn t => ("fv_" ^ n, t, NoSyn)) qty_names raw_fvs

  val qfv_bns_descr = 

    map2 (fn (b, _, _) => fn t => ("fv_" ^ Variable.check_name b, t, NoSyn)) bn_funs raw_fv_bns

  val qalpha_bns_descr = 

    let

      val AlphaResult {alpha_bn_trms, ...} = alpha_result 

    in

      map2 (fn (b, _, _) => fn t => ("alpha_" ^ Variable.check_name b, t, NoSyn)) bn_funs  alpha_bn_trms

    end