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 t =

  case t of

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

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

  | Const (@{const_name insert}, _) $ (Const (@{const_name atom}, _) $ Bound i) $ y =>

      (i, NONE) :: strip_bn_fun y

  | Const (@{const_name Cons}, _) $ (Const (@{const_name atom}, _) $ Bound i) $ y =>

      (i, NONE) :: strip_bn_fun y

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

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

  | (f as Free _) $ Bound i => [(i, SOME f)]

  | _ => error ("Unsupported binding function: " ^ (PolyML.makestring t))

*}

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 {*

let

  fun aux eq = 

  let

    val (lhs, rhs) = eq

      |> strip_qnt_body "all" 

      |> HOLogic.dest_Trueprop

      |> HOLogic.dest_eq

    val (bn_fun, [cnstr]) = strip_comb lhs

    val (_, ty) = dest_Free 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 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_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_names, lthy1) = add_datatype_wrapper raw_dt_names raw_dts lthy

  val morphism_2_0 = ProofContext.export_morphism lthy2 lthy

  fun export_fun f (t, n , l) = (f t, n, map (map (apsnd (Option.map f))) l);

in

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 and raw bn-functions *)

    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 *)

    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;

  val (fv, fvbn, fv_def, lthy3a) = 

    if get_STEPS lthy2 > 3 

    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 (alpha_ts_nobn, alpha_ts_bn) = chop (length fv) alpha_ts

  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 = build_rel_inj alpha_intros (inject_thms @ distinct_thms) alpha_cases 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 (fv_eqvt, lthy6) = prove_eqvt all_tys induct_thm (fv_def @ raw_perm_defs) (fv @ fvbn) lthy5

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

  (* proving alpha equivalence *)

  val _ = warning "Proving equivalence";

  val fv_alpha_all = combine_fv_alpha_bns (fv, fvbn) (alpha_ts_nobn, alpha_ts_bn) bn_nos;

  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;

  val _ = warning "Proving respects";

  val bns_rsp_pre' = build_fvbv_rsps alpha_ts 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 fv_def lthy8 1;

  val fv_rsps = prove_fv_rsp fv_alpha_all alpha_ts 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) (fv @ fvbn) lthy9;

  val fv_rsp = flat (map snd fv_rsp_pre);

    (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_ts alpha_induct (alpha_eq_iff_simp @ rel_dists @ rel_dists_bn) alpha_equivp exhaust_thms alpha_ts_bn 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_ts_bn lthy11

  fun const_rsp_tac _ =

    let val alpha_alphabn = prove_alpha_alphabn alpha_ts alpha_induct alpha_eq_iff_simp alpha_ts_bn lthy11a

      in constr_rsp_tac alpha_eq_iff_simp (fv_rsp @ bns_rsp @ reflps @ 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) (fv @ fvbn)

  val (qfv_ts, qfv_defs, lthy12a) = quotient_lift_consts_export qtys (qfv_names ~~ (fv @ fvbn)) lthy12;

  val qbn_names = map (fn (b, _ , _) => Name.of_binding b) bn_funs

  val (qbn_ts, qbn_defs, lthy12b) = quotient_lift_consts_export qtys (qbn_names ~~ raw_bn_funs) lthy12a;

  val qalpha_bn_names = map (unsuffix "_raw" o Long_Name.base_name o fst o dest_Const) alpha_ts_bn

  val (qalpha_ts_bn, qalphabn_defs, lthy12c) = 

    quotient_lift_consts_export qtys (qalpha_bn_names ~~ alpha_ts_bn) lthy12b;

  val _ = warning "Lifting permutations";

  val thy = Local_Theory.exit_global lthy12c;

  val perm_names = map (fn x => "permute_" ^ x) qty_names

  val lthy13 = Theory_Target.init NONE thy';

  val q_name = space_implode "_" qty_names;

  fun suffix_bind s = Binding.qualify true q_name (Binding.name s);

  val _ = warning "Lifting induction";

  val constr_names = map (Long_Name.base_name o fst o dest_Const) consts;

  val q_induct = Rule_Cases.name constr_names (lift_thm qtys lthy13 induct_thm);

  fun note_suffix s th ctxt =

    snd (Local_Theory.note ((suffix_bind s, []), th) ctxt);

  fun note_simp_suffix s th ctxt =

    snd (Local_Theory.note ((suffix_bind s, [Attrib.internal (K Simplifier.simp_add)]), th) ctxt);

  val (_, lthy14) = Local_Theory.note ((suffix_bind "induct",

    [Attrib.internal (K (Rule_Cases.case_names constr_names))]), 

    [Rule_Cases.name constr_names q_induct]) lthy13;

  val q_inducts = Project_Rule.projects lthy13 (1 upto (length fv)) q_induct

  val (_, lthy14a) = Local_Theory.note ((suffix_bind "inducts", []), q_inducts) lthy14;

  val q_perm = map (lift_thm qtys lthy14) raw_perm_defs;

  val lthy15 = note_simp_suffix "perm" q_perm lthy14a;

  val q_fv = map (lift_thm qtys lthy15) fv_def;

  val lthy16 = note_simp_suffix "fv" q_fv lthy15;

  val q_bn = map (lift_thm qtys lthy16) raw_bn_eqs;

  val lthy17 = note_simp_suffix "bn" q_bn lthy16;

  val _ = warning "Lifting eq-iff";

  (*val _ = map tracing (map PolyML.makestring alpha_eq_iff);*)

  val eq_iff_unfolded0 = map (Local_Defs.unfold lthy17 @{thms alphas}) alpha_eq_iff_simp

  val eq_iff_unfolded1 = map (Local_Defs.unfold lthy17 @{thms Pair_eqvt}) eq_iff_unfolded0

  val q_eq_iff_pre0 = map (lift_thm qtys lthy17) eq_iff_unfolded1;

  val q_eq_iff_pre1 = map (Local_Defs.fold lthy17 @{thms Pair_eqvt}) q_eq_iff_pre0

  val q_eq_iff_pre2 = map (Local_Defs.fold lthy17 @{thms alphas}) q_eq_iff_pre1

  val q_eq_iff = map (Local_Defs.unfold lthy17 (Quotient_Info.id_simps_get lthy17)) q_eq_iff_pre2

  val (_, lthy18) = Local_Theory.note ((suffix_bind "eq_iff", []), q_eq_iff) lthy17;

  val q_dis = map (lift_thm qtys lthy18) rel_dists;

  val lthy19 = note_simp_suffix "distinct" q_dis lthy18;

  val q_eqvt = map (lift_thm qtys lthy19) (bv_eqvt @ fv_eqvt);

  val (_, lthy20) = Local_Theory.note ((Binding.empty,

    [Attrib.internal (fn _ => Nominal_ThmDecls.eqvt_add)]), q_eqvt) lthy19;

  val _ = warning "Supports";

  val supports = map (prove_supports lthy20 q_perm) consts;

  val fin_supp = HOLogic.conj_elims (prove_fs lthy20 q_induct supports qtys);

  val thy3 = Local_Theory.exit_global lthy20;

  val _ = warning "Instantiating FS";

  val lthy21 = Theory_Target.instantiation (qty_full_names, [], @{sort fs}) thy3;

  fun tac _ = Class.intro_classes_tac [] THEN (ALLGOALS (resolve_tac fin_supp))

  val lthy22 = Class.prove_instantiation_instance tac lthy21

  val fv_alpha_all = combine_fv_alpha_bns (qfv_ts_nobn, qfv_ts_bn) (alpha_ts_nobn, qalpha_ts_bn) bn_nos;

  val (names, supp_eq_t) = supp_eq fv_alpha_all;

  val _ = warning "Support Equations";

  fun supp_eq_tac' _ = if !cheat_supp_eq then Skip_Proof.cheat_tac thy else

    supp_eq_tac q_induct q_fv q_perm q_eq_iff lthy22 1;

  val q_supp = HOLogic.conj_elims (Goal.prove lthy22 names [] supp_eq_t supp_eq_tac') handle e =>

    let val _ = warning ("Support eqs failed") in [] end;

  val lthy23 = note_suffix "supp" q_supp lthy22;

in

  (0, lthy23)

end handle TEST ctxt => (0, ctxt)

*}

section {* Preparing and parsing of the specification *}

ML {* 

(* parsing the datatypes and declaring *)

(* constructors in the local theory    *)

fun prepare_dts dt_strs lthy =