theory NewParser

imports "../Nominal-General/Nominal2_Base" 

        "../Nominal-General/Nominal2_Eqvt" 

        "../Nominal-General/Nominal2_Supp" 

begin

(* TODO

  we need to also export a cases-rule for nominal datatypes

  size function

*)

section{* Interface for nominal_datatype *}

ML {* 

(* nominal datatype parser *)

local

  structure P = Parse;

  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 _ = Keyword.keyword "bind"

val _ = Keyword.keyword "bind_set"

val _ = Keyword.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 -- Parse_Spec.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, 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 

*}

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

  in

    (dt_index, (bn_fun, (cnstr_head, rhs_elements)))

  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_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

in

  (dt_full_names', raw_dt_full_names, raw_dts, raw_bclauses, raw_bn_funs, raw_bn_eqs, lthy1)

end

*}

ML {*

fun raw_bn_decls dt_names dts raw_bn_funs raw_bn_eqs constr_thms lthy =

  if null raw_bn_funs 

  then ([], [], [], [], lthy)

  else 

    let

      val (_, lthy1) = Function.add_function raw_bn_funs raw_bn_eqs

        Function_Common.default_config (pat_completeness_simp constr_thms) lthy

      val (info, lthy2) = prove_termination (Local_Theory.restore lthy1)

      val {fs, simps, inducts, ...} = info;

      val raw_bn_induct = (the inducts)

      val raw_bn_eqs = the simps

      val raw_bn_info = 

        prep_bn_info lthy dt_names dts (map prop_of raw_bn_eqs)

    in

      (fs, raw_bn_eqs, raw_bn_info, raw_bn_induct, lthy2)

    end

*}

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 {*

(* 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 _ = warning "Definition of raw datatypes";

  val (dt_names, raw_dt_names, raw_dts, raw_bclauses, raw_bn_funs, raw_bn_eqs, lthy0) =

    if get_STEPS lthy > 0 

    then raw_nominal_decls dts bn_funs bn_eqs bclauses lthy

    else raise TEST lthy

  val dtinfo = Datatype.the_info (ProofContext.theory_of lthy0) (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 lthy0)) all_full_tnames

  val inject_thms = flat (map #inject dtinfos);

  val distinct_thms = flat (map #distinct dtinfos);

  val constr_thms = inject_thms @ distinct_thms

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

  val raw_constrs_distinct = (map #distinct rel_dtinfos); 

  val induct_thm = #induct dtinfo;

  val exhaust_thms = map #exhaust dtinfos;

  (* definitions of raw permutations *)

  val _ = warning "Definition of raw permutations";

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

    if get_STEPS lthy0 > 1 

    then Local_Theory.theory_result (define_raw_perms descr sorts induct_thm (length dts)) lthy0

    else raise TEST lthy0

  (* 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 _ = warning "Definition of raw fv-functions";

  val lthy3 = Theory_Target.init NONE thy;

  val (raw_bn_funs, raw_bn_eqs, raw_bn_info, raw_bn_induct, lthy3a) =

    if get_STEPS lthy3 > 2 

    then raw_bn_decls dt_names raw_dts raw_bn_funs raw_bn_eqs constr_thms lthy3

    else raise TEST lthy3

  val (raw_fvs, raw_fv_bns, raw_fv_defs, raw_fv_bns_induct, lthy3b) = 

    if get_STEPS lthy3a > 3 

    then define_raw_fvs descr sorts raw_bn_info raw_bclauses constr_thms lthy3a

    else raise TEST lthy3a

  (* definition of raw alphas *)

  val _ = warning "Definition of alphas";

  val (alpha_trms, alpha_bn_trms, alpha_intros, alpha_cases, alpha_induct, lthy4) =

    if get_STEPS lthy3b > 4 

    then define_raw_alpha descr sorts raw_bn_info raw_bclauses raw_fvs lthy3b

    else raise TEST lthy3b

  (* definition of alpha-distinct lemmas *)

  val _ = warning "Distinct theorems";

  val (alpha_distincts, alpha_bn_distincts) = 

    mk_alpha_distincts lthy4 alpha_cases raw_constrs_distinct alpha_trms alpha_bn_trms raw_bn_info

  (* definition of raw_alpha_eq_iff  lemmas *)

  val _ = warning "Eq-iff theorems";

  val alpha_eq_iff = 

    if get_STEPS lthy > 5

    then mk_alpha_eq_iff lthy4 alpha_intros distinct_thms inject_thms alpha_cases

    else raise TEST lthy4

  (* proving equivariance lemmas for bns, fvs and alpha *)

  val _ = warning "Proving equivariance";

  val bn_eqvt = 

    if get_STEPS lthy > 6

    then raw_prove_eqvt raw_bn_funs raw_bn_induct (raw_bn_eqs @ raw_perm_defs) lthy4

    else raise TEST lthy4

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

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

  val lthy_tmp = snd (Local_Theory.note ((Binding.empty, [add_eqvt]), bn_eqvt) lthy4)

  val fv_eqvt = 

    if get_STEPS lthy > 7

    then raw_prove_eqvt (raw_fvs @ raw_fv_bns) raw_fv_bns_induct (raw_fv_defs @ raw_perm_defs) 

       (Local_Theory.restore lthy_tmp)

    else raise TEST lthy4

  val lthy_tmp' = snd (Local_Theory.note ((Binding.empty, [add_eqvt]), fv_eqvt) lthy_tmp)

  val (alpha_eqvt, lthy_tmp'') =

    if get_STEPS lthy > 8

    then Nominal_Eqvt.equivariance true (alpha_trms @ alpha_bn_trms) alpha_induct alpha_intros lthy_tmp'

    else raise TEST lthy4

  (* proving alpha equivalence *)

  val _ = warning "Proving equivalence"

  val alpha_refl_thms = 

    if get_STEPS lthy > 9

    then raw_prove_refl alpha_trms alpha_bn_trms alpha_intros induct_thm lthy_tmp'' 

    else raise TEST lthy4

  val alpha_sym_thms = 

    if get_STEPS lthy > 10

    then raw_prove_sym (alpha_trms @ alpha_bn_trms) alpha_intros alpha_induct lthy_tmp'' 

    else raise TEST lthy4

  val alpha_trans_thms = 

    if get_STEPS lthy > 11

    then raw_prove_trans (alpha_trms @ alpha_bn_trms) (distinct_thms @ inject_thms) 

           alpha_intros alpha_induct alpha_cases lthy_tmp'' 

    else raise TEST lthy4

  val alpha_equivp_thms = 

    if get_STEPS lthy > 12

    then raw_prove_equivp alpha_trms alpha_refl_thms alpha_sym_thms alpha_trans_thms lthy_tmp''

    else raise TEST lthy4

  (* proving alpha implies alpha_bn *)

  val _ = warning "Proving alpha implies bn"

  val alpha_bn_imp_thms = 

    if get_STEPS lthy > 13

    then raw_prove_bn_imp alpha_trms alpha_bn_trms alpha_intros alpha_induct lthy_tmp'' 

    else raise TEST lthy4

  val _ = tracing ("alphas " ^ commas (map (Syntax.string_of_term lthy4) alpha_trms))

  val _ = tracing ("alpha_bns " ^ commas (map (Syntax.string_of_term lthy4) alpha_bn_trms))

  val _ = tracing ("alpha_refl\n" ^ 

    cat_lines (map (Syntax.string_of_term lthy4 o prop_of) alpha_refl_thms))

  val _ = tracing ("alpha_bn_imp\n" ^ 

    cat_lines (map (Syntax.string_of_term lthy4 o prop_of) alpha_bn_imp_thms))

  val _ = tracing ("alpha_equivp\n" ^ 

    cat_lines (map (Syntax.string_of_term lthy4 o prop_of) alpha_equivp_thms))

  (* old stuff *)

  val _ = 

    if get_STEPS lthy > 14

    then true else raise TEST lthy4

  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_trms alpha_equivp_thms lthy4;

  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;

  val _ = warning "Proving respects";

  val bn_nos = map (fn (_, i, _) => i) raw_bn_info;

  val bns = raw_bn_funs ~~ bn_nos;

  val bns_rsp_pre' = build_fvbv_rsps alpha_trms alpha_induct raw_bn_eqs (map fst bns) lthy8;

  val (bns_rsp_pre, lthy9) = fold_map (

    fn (bn_t, _) => prove_const_rsp qtys Binding.empty [bn_t] (fn _ =>

       resolve_tac bns_rsp_pre' 1)) bns lthy8;

  val bns_rsp = flat (map snd bns_rsp_pre);

  fun fv_rsp_tac _ = if !cheat_fv_rsp then Skip_Proof.cheat_tac thy

    else fvbv_rsp_tac alpha_induct raw_fv_defs lthy8 1;

  val fv_alpha_all = combine_fv_alpha_bns (raw_fvs, raw_fv_bns) (alpha_trms, alpha_bn_trms) bn_nos

  val fv_rsps = prove_fv_rsp fv_alpha_all alpha_trms fv_rsp_tac lthy9;

  val (fv_rsp_pre, lthy10) = fold_map

    (fn fv => fn ctxt => prove_const_rsp qtys Binding.empty [fv]

    (fn _ => asm_simp_tac (HOL_ss addsimps fv_rsps) 1) ctxt) (raw_fvs @ raw_fv_bns) lthy9;

  val fv_rsp = flat (map snd fv_rsp_pre);

  val (perms_rsp, lthy11) = prove_const_rsp qtys Binding.empty raw_perm_funs

    (fn _ => asm_simp_tac (HOL_ss addsimps alpha_eqvt) 1) lthy10;

  fun alpha_bn_rsp_tac _ = if !cheat_alpha_bn_rsp then Skip_Proof.cheat_tac thy

    else

      let val alpha_bn_rsp_pre = prove_alpha_bn_rsp alpha_trms alpha_induct (alpha_eq_iff @ alpha_distincts @ alpha_bn_distincts) alpha_equivp_thms exhaust_thms alpha_bn_trms lthy11 in asm_simp_tac (HOL_ss addsimps alpha_bn_rsp_pre) 1 end;

  val (alpha_bn_rsps, lthy11a) = fold_map (fn cnst => prove_const_rsp qtys Binding.empty [cnst]

    alpha_bn_rsp_tac) alpha_bn_trms lthy11

  fun const_rsp_tac _ =

    let val alpha_alphabn = prove_alpha_alphabn alpha_trms alpha_induct alpha_eq_iff alpha_bn_trms lthy11a

      in constr_rsp_tac alpha_eq_iff (fv_rsp @ bns_rsp @ alpha_refl_thms @ alpha_alphabn) 1 end

  val (const_rsps, lthy12) = fold_map (fn cnst => prove_const_rsp qtys Binding.empty [cnst]

    const_rsp_tac) raw_consts lthy11a

    val qfv_names = map (unsuffix "_raw" o Long_Name.base_name o fst o dest_Const) (raw_fvs @ raw_fv_bns)

  val (qfv_ts, qfv_defs, lthy12a) = quotient_lift_consts_export qtys (qfv_names ~~ (raw_fvs @ raw_fv_bns)) lthy12;