theory Nominal2

imports 

  "../Nominal-General/Nominal2_Base" 

  "../Nominal-General/Nominal2_Eqvt" 

  "Nominal2_FSet"

  "Abs"

uses ("nominal_dt_rawperm.ML")

     ("nominal_dt_rawfuns.ML")

     ("nominal_dt_alpha.ML")

     ("nominal_dt_quot.ML")

     ("nominal_dt_supp.ML")

begin

use "nominal_dt_rawperm.ML"

ML {* open Nominal_Dt_RawPerm *}

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 *}

use "nominal_dt_supp.ML"

ML {* open Nominal_Dt_Supp *}

section{* Interface for nominal_datatype *}

ML {*

(* attributes *)

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)

*}

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, 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 {*

(** definition of the raw binding functions **)

(* TODO: needs cleaning *)

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

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

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

fun define_raw_bns dt_names dts raw_bn_funs raw_bn_eqs constr_thms size_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 size_thms (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 {*

fun define_raw_dts dts bn_funs bn_eqs binds 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 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, thy1) = 

    Datatype.add_datatype Datatype.default_config raw_dt_names raw_dts thy

  val lthy1 = Named_Target.theory_init thy1

in

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

end

*}

ML {*

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

exception TEST of Proof.context

val (STEPS, STEPS_setup) = Attrib.config_int "STEPS" (K 100);

fun get_STEPS ctxt = Config.get ctxt STEPS

*}

setup STEPS_setup

ML {*

fun nominal_datatype2 opt_thms_name dts bn_funs bn_eqs bclauses lthy =

let

  (* definition of the raw datatypes *)

  val _ = warning "Definition of raw datatypes";

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

    if get_STEPS lthy > 0 

    then define_raw_dts 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 raw_tys = all_dtyps descr sorts

  val raw_full_ty_names = map (fst o dest_Type) raw_tys

  val tvs = hd raw_tys

    |> snd o dest_Type

    |> map dest_TFree  

  val dtinfos = map (Datatype.the_info (ProofContext.theory_of lthy0)) raw_full_ty_names  

  val raw_cns_info = all_dtyp_constrs_types descr sorts

  val raw_constrs = flat (map (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 dtinfo

  val raw_induct_thms = #inducts dtinfo

  val raw_exhaust_thms = map #exhaust dtinfos

  val raw_size_trms = map size_const raw_tys

  val raw_size_thms = Size.size_thms (ProofContext.theory_of lthy0) (hd raw_dt_names)

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

    |> uncurry drop

  (* definitions of raw permutations by primitive recursion *)

  val _ = warning "Definition of raw permutations";

  val ((raw_perm_funs, raw_perm_simps, raw_perm_laws), lthy2a) =

    if get_STEPS lthy0 > 1 

    then define_raw_perms raw_full_ty_names raw_tys tvs raw_constrs raw_induct_thm lthy0

    else raise TEST lthy0

  (* noting the raw permutations as eqvt theorems *)

  val (_, lthy3) = Local_Theory.note ((Binding.empty, [eqvt_attr]), raw_perm_simps) lthy2a

  (* definition of raw fv_functions *)

  val _ = warning "Definition of raw fv-functions";

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

    if get_STEPS lthy3 > 2 

    then define_raw_bns raw_full_ty_names raw_dts raw_bn_funs raw_bn_eqs 

      (raw_inject_thms @ raw_distinct_thms) raw_size_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 raw_full_ty_names raw_tys raw_cns_info raw_bn_info raw_bclauses 

      (raw_inject_thms @ raw_distinct_thms) raw_size_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 raw_full_ty_names raw_tys raw_cns_info raw_bn_info raw_bclauses raw_fvs lthy3b

    else raise TEST lthy3b

  val alpha_tys = map (domain_type o fastype_of) alpha_trms  

  (* definition of alpha-distinct lemmas *)

  val _ = warning "Distinct theorems";

  val alpha_distincts = 

    mk_alpha_distincts lthy4 alpha_cases raw_distinct_thms alpha_trms raw_tys

  (* definition of 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 raw_distinct_thms raw_inject_thms alpha_cases

    else raise TEST lthy4

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

  val _ = warning "Proving equivariance";

  val raw_bn_eqvt = 

    if get_STEPS lthy > 6

    then raw_prove_eqvt raw_bns raw_bn_induct (raw_bn_defs @ raw_perm_simps) lthy4

    else raise TEST 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 = 

    if get_STEPS lthy > 7

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

      (Local_Theory.restore lthy_tmp)

    else raise TEST lthy4

  val raw_size_eqvt = 

    if get_STEPS lthy > 8

    then 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})

    else raise TEST lthy4

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

  val (alpha_eqvt, lthy6) =

    if get_STEPS lthy > 9

    then Nominal_Eqvt.equivariance true (alpha_trms @ alpha_bn_trms) alpha_induct alpha_intros lthy5

    else raise TEST lthy4

  (* proving alpha equivalence *)

  val _ = warning "Proving equivalence"

  val alpha_refl_thms = 

    if get_STEPS lthy > 10

    then raw_prove_refl alpha_trms alpha_bn_trms alpha_intros raw_induct_thm lthy6 

    else raise TEST lthy6

  val alpha_sym_thms = 

    if get_STEPS lthy > 11

    then raw_prove_sym (alpha_trms @ alpha_bn_trms) alpha_intros alpha_induct lthy6 

    else raise TEST lthy6

  val alpha_trans_thms = 

    if get_STEPS lthy > 12

    then raw_prove_trans (alpha_trms @ alpha_bn_trms) (raw_distinct_thms @ raw_inject_thms) 

           alpha_intros alpha_induct alpha_cases lthy6

    else raise TEST lthy6

  val (alpha_equivp_thms, alpha_bn_equivp_thms) = 

    if get_STEPS lthy > 13

    then raw_prove_equivp alpha_trms alpha_bn_trms alpha_refl_thms alpha_sym_thms 

       alpha_trans_thms lthy6

    else raise TEST lthy6

  (* proving alpha implies alpha_bn *)

  val _ = warning "Proving alpha implies bn"

  val alpha_bn_imp_thms = 

    if get_STEPS lthy > 14

    then raw_prove_bn_imp alpha_trms alpha_bn_trms alpha_intros alpha_induct lthy6 

    else raise TEST lthy6

  (* respectfulness proofs *)

  val raw_funs_rsp_aux = 

    if get_STEPS lthy > 15

    then raw_fv_bn_rsp_aux alpha_trms alpha_bn_trms raw_fvs 

      raw_bns raw_fv_bns alpha_induct (raw_bn_defs @ raw_fv_defs) lthy6

    else raise TEST lthy6 

  val raw_funs_rsp = 

    if get_STEPS lthy > 16

    then map mk_funs_rsp raw_funs_rsp_aux

    else raise TEST lthy6 

  val raw_size_rsp = 

    if get_STEPS lthy > 17

    then

      raw_size_rsp_aux (alpha_trms @ alpha_bn_trms) alpha_induct 

      (raw_size_thms @ raw_size_eqvt) lthy6

      |> map mk_funs_rsp

    else raise TEST lthy6 

  val raw_constrs_rsp = 

    if get_STEPS lthy > 18

    then raw_constrs_rsp raw_constrs alpha_trms alpha_intros

      (alpha_bn_imp_thms @ raw_funs_rsp_aux) lthy6 

    else raise TEST lthy6 

  val alpha_permute_rsp = 

    if get_STEPS lthy > 19

    then map mk_alpha_permute_rsp alpha_eqvt

    else raise TEST lthy6 

  val alpha_bn_rsp = 

    if get_STEPS lthy > 20

    then raw_alpha_bn_rsp alpha_bn_trms alpha_bn_equivp_thms alpha_bn_imp_thms

    else raise TEST lthy6 

  (* noting the quot_respects lemmas *)

  val (_, lthy6a) = 

    if get_STEPS lthy > 21

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

      raw_constrs_rsp @ raw_funs_rsp @ raw_size_rsp @ alpha_permute_rsp @ alpha_bn_rsp) lthy6

    else raise TEST lthy6

  (* defining the quotient type *)

  val _ = warning "Declaring the quotient types"

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

  val (qty_infos, lthy7) = 

    if get_STEPS lthy > 22

    then define_qtypes qty_descr alpha_tys alpha_trms alpha_equivp_thms lthy6a

    else raise TEST lthy6a

  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             

  (* defining of quotient term-constructors, binding functions, free vars functions *)

  val _ = warning "Defining the quotient constants"

  val qconstrs_descr = 

    flat (map (fn (_, _, _, cs) => map (fn (b, _, mx) => (Name.of_binding b, mx)) cs) dts)

    |> map2 (fn t => fn (b, mx) => (b, t, mx)) raw_constrs

  val qbns_descr =

    map2 (fn (b, _, mx) => fn t => (Name.of_binding 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_" ^ Name.of_binding b, t, NoSyn)) bn_funs raw_fv_bns

  val qalpha_bns_descr = 

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

  val qperm_descr =

    map2 (fn n => fn t => ("permute_" ^ n, Type.legacy_freeze t, NoSyn)) qty_names raw_perm_funs

  val qsize_descr =

    map2 (fn n => fn t => ("size_" ^ n, t, NoSyn)) qty_names raw_size_trms

  val (((((qconstrs_info, qbns_info), qfvs_info), qfv_bns_info), qalpha_bns_info), lthy8) = 

    if get_STEPS lthy > 23

    then 

      lthy7

      |> define_qconsts qtys qconstrs_descr 

      ||>> define_qconsts qtys qbns_descr