theory NewParser

imports "../Nominal-General/Nominal2_Base" 

        "../Nominal-General/Nominal2_Eqvt" 

        "../Nominal-General/Nominal2_Supp" 

        "Perm" "NewFv" "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 {* 

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 (BEmy n) = BEmy n

    | rawify_bclause (BLst (bnds, bdys)) = BLst (rawify_bnds bnds, bdys)

    | rawify_bclause (BSet (bnds, bdys)) = BSet (rawify_bnds bnds, bdys)

    | rawify_bclause (BRes (bnds, bdys)) = BRes (rawify_bnds bnds, bdys)

in

  map (map (map rawify_bclause)) bclauses

end

*}

(* strip_bn_fun takes a bn function defined by the user as a union or

   append of elements and returns those elements together with bn functions

   applied *)

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 {* print_depth 50 *}

ML {*

fun prep_bn lthy dt_names dts eqs = 

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 aux2 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

    (bn_fun, dt_index, (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' (*map aux2 eqs*)

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_bns = prep_bn lthy dt_full_names' raw_dts (map snd raw_bn_eqs)

in

  lthy 

  |> add_datatype_wrapper raw_dt_names raw_dts 

  ||>> add_primrec_wrapper raw_bn_funs raw_bn_eqs

  ||>> pair raw_bclauses

  ||>> pair raw_bns

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]

*}

text {* 

  nominal_datatype2 does the following things in order:

Parser.thy/raw_nominal_decls

  1) define the raw datatype

  2) define the raw binding functions 

Perm.thy/define_raw_perms

  3) define permutations of the raw datatype and show that the raw type is 

     in the pt typeclass

Lift.thy/define_fv_alpha_export, Fv.thy/define_fv & define_alpha

  4) define fv and fv_bn

  5) define alpha and alpha_bn

Perm.thy/distinct_rel

  6) prove alpha_distincts (C1 x \<notsimeq> C2 y ...)             (Proof by cases; simp)

Tacs.thy/build_rel_inj

  6) prove alpha_eq_iff    (C1 x = C2 y \<leftrightarrow> P x y ...)

     (left-to-right by intro rule, right-to-left by cases; simp)

Equivp.thy/prove_eqvt

  7) prove bn_eqvt (common induction on the raw datatype)

  8) prove fv_eqvt (common induction on the raw datatype with help of above)

Rsp.thy/build_alpha_eqvts

  9) prove alpha_eqvt and alpha_bn_eqvt

     (common alpha-induction, unfolding alpha_gen, permute of #* and =)

Equivp.thy/build_alpha_refl & Equivp.thy/build_equivps

 10) prove that alpha and alpha_bn are equivalence relations

     (common induction and application of 'compose' lemmas)

Lift.thy/define_quotient_types

 11) define quotient types

Rsp.thy/build_fvbv_rsps

 12) prove bn respects     (common induction and simp with alpha_gen)

Rsp.thy/prove_const_rsp

 13) prove fv respects     (common induction and simp with alpha_gen)

 14) prove permute respects    (unfolds to alpha_eqvt)

Rsp.thy/prove_alpha_bn_rsp

 15) prove alpha_bn respects

     (alpha_induct then cases then sym and trans of the relations)

Rsp.thy/prove_alpha_alphabn

 16) show that alpha implies alpha_bn (by unduction, needed in following step)

Rsp.thy/prove_const_rsp

 17) prove respects for all datatype constructors

     (unfold eq_iff and alpha_gen; introduce zero permutations; simp)

Perm.thy/quotient_lift_consts_export

 18) define lifted constructors, fv, bn, alpha_bn, permutations

Perm.thy/define_lifted_perms

 19) lift permutation zero and add properties to show that quotient type is in the pt typeclass

Lift.thy/lift_thm

 20) lift permutation simplifications

 21) lift induction

 22) lift fv

 23) lift bn

 24) lift eq_iff

 25) lift alpha_distincts

 26) lift fv and bn eqvts

Equivp.thy/prove_supports

 27) prove that union of arguments supports constructors

Equivp.thy/prove_fs

 28) show that the lifted type is in fs typeclass     (* by q_induct, supports *)

Equivp.thy/supp_eq

 29) prove supp = fv

*}

ML {*

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

exception TEST of Proof.context

val STEPS = 10

*}

ML {*

fun nominal_datatype2 dts bn_funs bn_eqs bclauses lthy =

let

  (* definition of the raw datatype and raw bn-functions *)

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

    if STEPS > 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 raw_tys = map (fn (i, _) => nth_dtyp descr sorts i) descr;

  val all_typs = map (fn i => Datatype_Aux.typ_of_dtyp descr sorts (Datatype_Aux.DtRec i)) (map fst 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_def, raw_perm_simps, perms), lthy2) =

    if STEPS > 2 then Local_Theory.theory_result (define_raw_perms descr sorts induct_thm (length dts)) lthy1

    else raise TEST lthy1

  (* definition of raw fv_functions *)

  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);

  val bn_funs_decls = map (export_fun (Morphism.term morphism_2_0)) raw_bns;

  val (_, lthy2a) = Local_Theory.note ((Binding.empty,

    [Attrib.internal (fn _ => Nominal_ThmDecls.eqvt_add)]), raw_perm_def) lthy2;

  val thy = Local_Theory.exit_global lthy2a;

  val thy_name = Context.theory_name thy

  val lthy3 = Theory_Target.init NONE thy;

  val raw_bn_funs = map (fn (f, _, _) => f) bn_funs_decls;

  val _ = tracing ("raw_bns\n" ^ @{make_string} raw_bns)

  val _ = tracing ("bn_funs\n" ^ @{make_string} bn_funs_decls)

  val ((fv, fvbn), fv_def, lthy3a) = 

    if STEPS > 3 then define_raw_fv descr sorts bn_funs_decls raw_bclauses lthy3

    else raise TEST lthy3

  (* definition of raw alphas *)

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

    if STEPS > 4 then define_raw_alpha dtinfo bn_funs_decls raw_bclauses fv lthy3a

    else raise TEST lthy3a

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

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

  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;

  val _ = map tracing (map PolyML.makestring alpha_eq_iff_simp);

  (* proving equivariance lemmas *)

  val _ = warning "Proving equivariance";

  val (bv_eqvt, lthy5) = prove_eqvt raw_tys induct_thm (raw_bn_eqs @ raw_perm_def) (map fst bns) lthy4

  val (fv_eqvt, lthy6) = prove_eqvt raw_tys induct_thm (fv_def @ raw_perm_def) (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_typs 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);

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

    (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]