theory NewParser

imports "../Nominal-General/Nominal2_Base" 

        "../Nominal-General/Nominal2_Eqvt" 

        "../Nominal-General/Nominal2_Supp" 

        "Perm" "NewAlpha" "Tacs" "Equivp" "Lift"

begin

section{* Interface for nominal_datatype *}

ML {* 

(* nominal datatype parser *)

local

  structure P = OuterParse;

  structure S = Scan

  fun triple1 ((x, y), z) = (x, y, z)

  fun triple2 (x, (y, z)) = (x, y, z)

  fun tuple ((x, y, z), u) = (x, y, z, u)

  fun tswap (((x, y), z), u) = (x, y, u, z)

in

val _ = OuterKeyword.keyword "bind"

val _ = OuterKeyword.keyword "bind_set"

val _ = OuterKeyword.keyword "bind_res"

val anno_typ = S.option (P.name --| P.$$$ "::") -- P.typ

val bind_mode = P.$$$ "bind" || P.$$$ "bind_set" || P.$$$ "bind_res"

val bind_clauses = 

  P.enum "," (bind_mode -- S.repeat1 P.term -- (P.$$$ "in" |-- S.repeat1 P.name) >> triple1)

val cnstr_parser =

  P.binding -- S.repeat anno_typ -- bind_clauses -- P.opt_mixfix >> tswap

(* datatype parser *)

val dt_parser =

  (P.type_args -- P.binding -- P.opt_mixfix >> triple1) -- 

    (P.$$$ "=" |-- P.enum1 "|" cnstr_parser) >> tuple

(* binding function parser *)

val bnfun_parser = 

  S.optional (P.$$$ "binder" |-- P.fixes -- SpecParse.where_alt_specs) ([], [])

(* main parser *)

val main_parser =

  P.and_list1 dt_parser -- bnfun_parser >> triple2

end

*}

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

*}

ML {* 

(* exports back the results *)

fun add_primrec_wrapper funs eqs lthy = 

  if null funs then ([], [], lthy)

  else 

   let 

     val eqs' = map (fn (_, eq) => (Attrib.empty_binding, eq)) eqs

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

     val ((funs'', eqs''), lthy') = Primrec.add_primrec funs' eqs' lthy

     val phi = ProofContext.export_morphism lthy' lthy

   in 

     (map (Morphism.term phi) funs'', map (Morphism.thm phi) eqs'', lthy')

   end

*}

ML {*

fun add_datatype_wrapper dt_names dts =

let

  val conf = Datatype.default_config

in

  Local_Theory.theory_result (Datatype.add_datatype conf dt_names dts)

end

*}

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, 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 

*}

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 (map (map rawify_bclause)) bclauses

end

*}

(* strip_bn_fun takes a rhs of a bn function: this can only contain unions or

   appends of elements; in case of recursive calls it retruns also the applied

   bn function *)

ML {*

fun strip_bn_fun lthy args t =

let 

  fun aux t =

    case t of

      Const (@{const_name sup}, _) $ l $ r => aux l @ aux r

    | Const (@{const_name append}, _) $ l $ r => aux l @ aux r

    | Const (@{const_name insert}, _) $ (Const (@{const_name atom}, _) $ (x as Var _)) $ y =>

        (find_index (equal x) args, NONE) :: aux y

    | Const (@{const_name Cons}, _) $ (Const (@{const_name atom}, _) $ (x as Var _)) $ y =>

        (find_index (equal x) args, NONE) :: aux y

    | Const (@{const_name bot}, _) => []

    | Const (@{const_name Nil}, _) => []

    | (f as Const _) $ (x as Var _) => [(find_index (equal x) args, SOME f)]

    | _ => error ("Unsupported binding function: " ^ (Syntax.string_of_term lthy t))

in

  aux t

end  

*}

ML {*

fun find [] _ = error ("cannot find element")

  | find ((x, z)::xs) y = if (Long_Name.base_name x) = y then z else find xs y

*}

ML {*

fun prep_bn_info lthy dt_names dts eqs = 

let

  fun aux eq = 

  let

    val (lhs, rhs) = eq

      |> HOLogic.dest_Trueprop

      |> HOLogic.dest_eq

    val (bn_fun, [cnstr]) = strip_comb lhs

    val (_, ty) = dest_Const bn_fun

    val (ty_name, _) = dest_Type (domain_type ty)

    val dt_index = find_index (fn x => x = ty_name) dt_names

    val (cnstr_head, cnstr_args) = strip_comb cnstr    

    val rhs_elements = strip_bn_fun lthy cnstr_args rhs

    val included = map (apfst (fn i => length (cnstr_args) - i - 1)) rhs_elements

  in

    (dt_index, (bn_fun, (cnstr_head, included)))

  end

  fun order dts i ts = 

  let

    val dt = nth dts i

    val cts = map (fn (x, _, _) => Binding.name_of x) ((fn (_, _, _, x) => x) dt)

    val ts' = map (fn (x, y) => (fst (dest_Const x), y)) ts

  in

    map (find ts') cts

  end

  val unordered = AList.group (op=) (map aux eqs)

  val unordered' = map (fn (x, y) =>  (x, AList.group (op=) y)) unordered

  val ordered = map (fn (x, y) => (x, map (fn (v, z) => (v, order dts x z)) y)) unordered' 

  val ordered' = flat (map (fn (ith, l) => map (fn (bn, data) => (bn, ith, data)) l) ordered)

  (*val _ = tracing ("eqs\n" ^ cat_lines (map (Syntax.string_of_term lthy) eqs))*)

  (*val _ = tracing ("map eqs\n" ^ @{make_string} (map aux2 eqs))*)

  (*val _ = tracing ("ordered'\n" ^ @{make_string} ordered')*)

in

  ordered'

end

*}

ML {*

fun raw_nominal_decls dts bn_funs bn_eqs binds lthy =

let

  val thy = ProofContext.theory_of 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 cnstrs = get_cnstr_strs dts

  val cnstrs_ty = get_typed_cnstrs dts

  val cnstrs_full_names = map (Long_Name.qualify thy_name) cnstrs

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

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

  val cnstrs_env = cnstrs_full_names ~~ cnstrs_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_dt_names, raw_dts) = rawify_dts dt_names dts dts_env

  val raw_dt_names' =  map (Long_Name.qualify thy_name) raw_dt_names

  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 binds 

  val (raw_dt_full_names, lthy1) = add_datatype_wrapper raw_dt_names raw_dts lthy

  val (raw_bn_funs', raw_bn_eqs', lthy2) = add_primrec_wrapper raw_bn_funs raw_bn_eqs lthy1

  val raw_bn_info = 

    prep_bn_info lthy dt_full_names' raw_dts (map prop_of raw_bn_eqs')

in

  (raw_dt_full_names, raw_bclauses, raw_bn_funs', raw_bn_eqs', raw_bn_info, lthy2)

end

*}

lemma equivp_hack: "equivp x"

sorry

ML {*

fun equivp_hack ctxt rel =

let

  val thy = ProofContext.theory_of ctxt

  val ty = domain_type (fastype_of rel)

  val cty = ctyp_of thy ty

  val ct = cterm_of thy rel

in

  Drule.instantiate' [SOME cty] [SOME ct] @{thm equivp_hack}

end

*}

ML {* val cheat_equivp = Unsynchronized.ref false *}

ML {* val cheat_fv_rsp = Unsynchronized.ref false *}

ML {* val cheat_alpha_bn_rsp = Unsynchronized.ref false *}

ML {* val cheat_supp_eq = Unsynchronized.ref false *}

ML {*

fun remove_loop t =

  let val _ = HOLogic.dest_eq (HOLogic.dest_Trueprop (prop_of t)) in t end

  handle TERM _ => @{thm eqTrueI} OF [t]

*}

ML {*

(* for testing porposes - to exit the procedure early *)

exception TEST of Proof.context

val (STEPS, STEPS_setup) = Attrib.config_int "STEPS" (K 10);

fun get_STEPS ctxt = Config.get ctxt STEPS

*}

setup STEPS_setup

ML {*

fun nominal_datatype2 dts bn_funs bn_eqs bclauses lthy =

let

  (* definition of the raw datatypes *)

  val (raw_dt_names, raw_bclauses, raw_bn_funs, raw_bn_eqs, raw_bn_info, lthy1) =

    if get_STEPS lthy > 1 

    then raw_nominal_decls dts bn_funs bn_eqs bclauses lthy

    else raise TEST lthy

  val dtinfo = Datatype.the_info (ProofContext.theory_of lthy1) (hd raw_dt_names)

  val {descr, sorts, ...} = dtinfo

  val all_tys = map (fn (i, _) => nth_dtyp descr sorts i) descr

  val all_full_tnames = map (fn (_, (n, _, _)) => n) descr

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

  val inject_thms = flat (map #inject dtinfos);

  val distinct_thms = flat (map #distinct dtinfos);

  val rel_dtinfos = List.take (dtinfos, (length dts)); 

  val rel_distinct = map #distinct rel_dtinfos;  (* thm list list *)

  val induct_thm = #induct dtinfo;

  val exhaust_thms = map #exhaust dtinfos;

  (* definitions of raw permutations *)

  val ((raw_perm_funs, raw_perm_defs, raw_perm_simps), lthy2) =

    if get_STEPS lthy1 > 2 

    then Local_Theory.theory_result (define_raw_perms descr sorts induct_thm (length dts)) lthy1

    else raise TEST lthy1

  (* noting the raw permutations as eqvt theorems *)

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

  val (_, lthy2a) = Local_Theory.note ((Binding.empty, [eqvt_attrib]), raw_perm_defs) lthy2

  val thy = Local_Theory.exit_global lthy2a;

  val thy_name = Context.theory_name thy

  (* definition of raw fv_functions *)

  val lthy3 = Theory_Target.init NONE thy;

    if get_STEPS lthy2 > 3 

    then define_raw_fvs descr sorts raw_bn_info raw_bclauses lthy3

    else raise TEST lthy3

  (* definition of raw alphas *)

  val (alpha_ts, alpha_intros, alpha_cases, alpha_induct, lthy4) =

    if get_STEPS lthy > 4 

    else raise TEST lthy3a

  val dts_names = map (fn (i, (s, _, _)) => (s, i)) descr;

  val bn_tys = map (domain_type o fastype_of) raw_bn_funs;

  val bn_nos = map (dtyp_no_of_typ dts_names) bn_tys;

  val bns = raw_bn_funs ~~ bn_nos;

  val rel_dists = flat (map (distinct_rel lthy4 alpha_cases)

    (rel_distinct ~~ alpha_ts_nobn));

  val rel_dists_bn = flat (map (distinct_rel lthy4 alpha_cases)

    ((map (fn i => nth rel_distinct i) bn_nos) ~~ alpha_ts_bn))

  (* definition of raw_alpha_eq_iff  lemmas *)

  val alpha_eq_iff = 

    if get_STEPS lthy > 5

    then build_rel_inj alpha_intros (inject_thms @ distinct_thms) alpha_cases lthy4

    else raise TEST lthy4

  val alpha_eq_iff_simp = map remove_loop alpha_eq_iff;

  (* proving equivariance lemmas *)

  val _ = warning "Proving equivariance";

  val (bv_eqvt, lthy5) = prove_eqvt all_tys induct_thm ((*raw_bn_eqs @*) raw_perm_defs) (map fst bns) lthy4

  val (alpha_eqvt, lthy6a) = Nominal_Eqvt.equivariance alpha_ts alpha_induct alpha_intros lthy6;

  (* proving alpha equivalence *)

  val _ = warning "Proving equivalence";

  val reflps = build_alpha_refl fv_alpha_all alpha_ts induct_thm alpha_eq_iff_simp lthy6a;

  val alpha_equivp =

    if !cheat_equivp then map (equivp_hack lthy6a) alpha_ts

    else build_equivps alpha_ts reflps alpha_induct

      inject_thms alpha_eq_iff_simp distinct_thms alpha_cases alpha_eqvt lthy6a;

  val qty_binds = map (fn (_, b, _, _) => b) dts;

  val qty_names = map Name.of_binding qty_binds;

  val qty_full_names = map (Long_Name.qualify thy_name) qty_names

  val (qtys, lthy7) = define_quotient_types qty_binds all_tys alpha_ts_nobn alpha_equivp lthy6a;

  val const_names = map Name.of_binding (flat (map (fn (_, _, _, t) => map (fn (b, _, _) => b) t) dts));

  val raw_consts =

    flat (map (fn (i, (_, _, l)) =>

      map (fn (cname, dts) =>

        Const (cname, map (Datatype_Aux.typ_of_dtyp descr sorts) dts --->

          Datatype_Aux.typ_of_dtyp descr sorts (Datatype_Aux.DtRec i))) l) descr);

  val (consts, const_defs, lthy8) = quotient_lift_consts_export qtys (const_names ~~ raw_consts) lthy7;