(*  Title:      Pure/Isar/code.ML

    Author:     Florian Haftmann, TU Muenchen

Abstract executable ingredients of theory.  Management of data

dependent on executable ingredients as synchronized cache; purged

on any change of underlying executable ingredients.

*)

signature CODE =

sig

  (*constants*)

  val check_const: theory -> term -> string

  val read_bare_const: theory -> string -> string * typ

  val read_const: theory -> string -> string

  val string_of_const: theory -> string -> string

  val const_typ: theory -> string -> typ

  val args_number: theory -> string -> int

  (*constructor sets*)

  val constrset_of_consts: theory -> (string * typ) list

    -> string * ((string * sort) list * (string * ((string * sort) list * typ list)) list)

  (*code equations and certificates*)

  val mk_eqn: theory -> thm * bool -> thm * bool

  val mk_eqn_liberal: theory -> thm -> (thm * bool) option

  val assert_eqn: theory -> thm * bool -> thm * bool

  val const_typ_eqn: theory -> thm -> string * typ

  val expand_eta: theory -> int -> thm -> thm

  type cert

  val empty_cert: theory -> string -> cert

  val cert_of_eqns: theory -> string -> (thm * bool) list -> cert

  val constrain_cert: theory -> sort list -> cert -> cert

  val conclude_cert: cert -> cert

  val typargs_deps_of_cert: theory -> cert -> (string * sort) list * (string * typ list) list

  val equations_of_cert: theory -> cert -> ((string * sort) list * typ)

    * (((term * string option) list * (term * string option)) * (thm option * bool)) list

  val bare_thms_of_cert: theory -> cert -> thm list

  val pretty_cert: theory -> cert -> Pretty.T list

  (*executable code*)

  val add_datatype: (string * typ) list -> theory -> theory

  val add_datatype_cmd: string list -> theory -> theory

  val datatype_interpretation:

    (string * ((string * sort) list * (string * ((string * sort) list * typ list)) list)

      -> theory -> theory) -> theory -> theory

  val add_abstype: thm -> theory -> theory

  val abstype_interpretation:

    (string * ((string * sort) list * ((string * ((string * sort) list * typ)) * (string * thm)))

      -> theory -> theory) -> theory -> theory

  val add_eqn: thm -> theory -> theory

  val add_nbe_eqn: thm -> theory -> theory

  val add_abs_eqn: thm -> theory -> theory

  val add_default_eqn: thm -> theory -> theory

  val add_default_eqn_attribute: attribute

  val add_default_eqn_attrib: Attrib.src

  val add_nbe_default_eqn: thm -> theory -> theory

  val add_nbe_default_eqn_attribute: attribute

  val add_nbe_default_eqn_attrib: Attrib.src

  val del_eqn: thm -> theory -> theory

  val del_eqns: string -> theory -> theory

  val add_case: thm -> theory -> theory

  val add_undefined: string -> theory -> theory

  val get_type: theory -> string

    -> ((string * sort) list * (string * ((string * sort) list * typ list)) list) * bool

  val get_type_of_constr_or_abstr: theory -> string -> (string * bool) option

  val is_constr: theory -> string -> bool

  val is_abstr: theory -> string -> bool

  val get_cert: theory -> { functrans: ((thm * bool) list -> (thm * bool) list option) list,

    ss: simpset } -> string -> cert

  val get_case_scheme: theory -> string -> (int * (int * string option list)) option

  val get_case_cong: theory -> string -> thm option

  val undefineds: theory -> string list

  val print_codesetup: theory -> unit

end;

signature CODE_DATA_ARGS =

sig

  type T

  val empty: T

end;

signature CODE_DATA =

sig

  type T

  val change: theory option -> (T -> T) -> T

  val change_yield: theory option -> (T -> 'a * T) -> 'a * T

end;

signature PRIVATE_CODE =

sig

  include CODE

  val declare_data: Any.T -> serial

  val change_yield_data: serial * ('a -> Any.T) * (Any.T -> 'a)

    -> theory -> ('a -> 'b * 'a) -> 'b * 'a

end;

structure Code : PRIVATE_CODE =

struct

(** auxiliary **)

(* printing *)

fun string_of_typ thy =

  Syntax.string_of_typ (Config.put show_sorts true (Syntax.init_pretty_global thy));

fun string_of_const thy c =

  let val ctxt = Proof_Context.init_global thy in

    case Axclass.inst_of_param thy c of

      SOME (c, tyco) =>

        Proof_Context.extern_const ctxt c ^ " " ^ enclose "[" "]"

          (Proof_Context.extern_type ctxt tyco)

    | NONE => Proof_Context.extern_const ctxt c

  end;

(* constants *)

fun const_typ thy = Type.strip_sorts o Sign.the_const_type thy;

fun args_number thy = length o binder_types o const_typ thy;

fun devarify ty =

  let

    val tys = fold_atyps (fn TVar vi_sort => AList.update (op =) vi_sort) ty [];

    val vs = Name.invent Name.context Name.aT (length tys);

    val mapping = map2 (fn v => fn (vi, sort) => (vi, TFree (v, sort))) vs tys;

  in Term.typ_subst_TVars mapping ty end;

fun typscheme thy (c, ty) =

  (map dest_TFree (Sign.const_typargs thy (c, ty)), Type.strip_sorts ty);

fun typscheme_equiv (ty1, ty2) =

  Type.raw_instance (devarify ty1, ty2) andalso Type.raw_instance (devarify ty2, ty1);

fun check_bare_const thy t = case try dest_Const t

 of SOME c_ty => c_ty

  | NONE => error ("Not a constant: " ^ Syntax.string_of_term_global thy t);

fun check_unoverload thy (c, ty) =

  let

    val c' = Axclass.unoverload_const thy (c, ty);

    val ty_decl = Sign.the_const_type thy c';

  in

    if typscheme_equiv (ty_decl, Logic.varifyT_global ty)

    then c'

    else

      error ("Type\n" ^ string_of_typ thy ty ^

        "\nof constant " ^ quote c ^

        "\nis too specific compared to declared type\n" ^

        string_of_typ thy ty_decl)

  end; 

fun check_const thy = check_unoverload thy o check_bare_const thy;

fun read_bare_const thy = check_bare_const thy o Syntax.read_term_global thy;

fun read_const thy = check_unoverload thy o read_bare_const thy;

(** data store **)

(* datatypes *)

datatype typ_spec = Constructors of (string * ((string * sort) list * typ list)) list *

      string list (*references to associated case constructors*)

  | Abstractor of (string * ((string * sort) list * typ)) * (string * thm);

fun constructors_of (Constructors (cos, _)) = (cos, false)

  | constructors_of (Abstractor ((co, (vs, ty)), _)) = ([(co, (vs, [ty]))], true);

fun case_consts_of (Constructors (_, case_consts)) = case_consts

  | case_consts_of (Abstractor _) = [];

(* functions *)

datatype fun_spec = Default of (thm * bool) list * (thm * bool) list lazy

      (* (cache for default equations, lazy computation of default equations)

         -- helps to restore natural order of default equations *)

  | Eqns of (thm * bool) list

  | Proj of term * string

  | Abstr of thm * string;

val empty_fun_spec = Default ([], Lazy.value []);

fun is_default (Default _) = true

  | is_default _ = false;

fun associated_abstype (Abstr (_, tyco)) = SOME tyco

  | associated_abstype _ = NONE;

(* executable code data *)

datatype spec = Spec of {

  history_concluded: bool,

  functions: ((bool * fun_spec) * (serial * fun_spec) list) Symtab.table

    (*with explicit history*),

  types: ((serial * ((string * sort) list * typ_spec)) list) Symtab.table

    (*with explicit history*),

  cases: ((int * (int * string option list)) * thm) Symtab.table * unit Symtab.table

};

fun make_spec (history_concluded, (functions, (types, cases))) =

  Spec { history_concluded = history_concluded, functions = functions, types = types, cases = cases };

fun map_spec f (Spec { history_concluded = history_concluded,

  functions = functions, types = types, cases = cases }) =

  make_spec (f (history_concluded, (functions, (types, cases))));

fun merge_spec (Spec { history_concluded = _, functions = functions1,

    types = types1, cases = (cases1, undefs1) },

  Spec { history_concluded = _, functions = functions2,

    types = types2, cases = (cases2, undefs2) }) =

  let

    val types = Symtab.join (K (AList.merge (op =) (K true))) (types1, types2);

    val case_consts_of' = (maps case_consts_of o map (snd o snd o hd o snd) o Symtab.dest);

    fun merge_functions ((_, history1), (_, history2)) =

      let

        val raw_history = AList.merge (op = : serial * serial -> bool)

          (K true) (history1, history2);

        val filtered_history = filter_out (is_default o snd) raw_history;

        val history = if null filtered_history

          then raw_history else filtered_history;

      in ((false, (snd o hd) history), history) end;

    val all_datatype_specs = map (snd o snd o hd o snd) (Symtab.dest types);

    val all_constructors = maps (map fst o fst o constructors_of) all_datatype_specs;

    val invalidated_case_consts = union (op =) (case_consts_of' types1) (case_consts_of' types2)

      |> subtract (op =) (maps case_consts_of all_datatype_specs)

    val functions = Symtab.join (K merge_functions) (functions1, functions2)

      |> fold (fn c => Symtab.map_entry c (apfst (K (true, empty_fun_spec)))) all_constructors;

    val cases = (Symtab.merge (K true) (cases1, cases2)

      |> fold Symtab.delete invalidated_case_consts, Symtab.merge (K true) (undefs1, undefs2));

  in make_spec (false, (functions, (types, cases))) end;

fun history_concluded (Spec { history_concluded, ... }) = history_concluded;

fun the_functions (Spec { functions, ... }) = functions;

fun the_types (Spec { types, ... }) = types;

fun the_cases (Spec { cases, ... }) = cases;

val map_history_concluded = map_spec o apfst;

val map_functions = map_spec o apsnd o apfst;

val map_typs = map_spec o apsnd o apsnd o apfst;

val map_cases = map_spec o apsnd o apsnd o apsnd;

(* data slots dependent on executable code *)

(*private copy avoids potential conflict of table exceptions*)

structure Datatab = Table(type key = int val ord = int_ord);

local

type kind = { empty: Any.T };

val kinds = Synchronized.var "Code_Data" (Datatab.empty: kind Datatab.table);

fun invoke f k =

  (case Datatab.lookup (Synchronized.value kinds) k of

    SOME kind => f kind

  | NONE => raise Fail "Invalid code data identifier");

in

fun declare_data empty =

  let

    val k = serial ();

    val kind = { empty = empty };

    val _ = Synchronized.change kinds (Datatab.update (k, kind));

  in k end;

fun invoke_init k = invoke (fn kind => #empty kind) k;

end; (*local*)

(* theory store *)

local

type data = Any.T Datatab.table;

fun empty_dataref () = Synchronized.var "code data" (NONE : (data * theory) option);

structure Code_Data = Theory_Data

(

  type T = spec * (data * theory) option Synchronized.var;

  val empty = (make_spec (false, (Symtab.empty,

    (Symtab.empty, (Symtab.empty, Symtab.empty)))), empty_dataref ());

  val extend : T -> T = apsnd (K (empty_dataref ()));

  fun merge ((spec1, _), (spec2, _)) =

    (merge_spec (spec1, spec2), empty_dataref ());

);

in

(* access to executable code *)

val the_exec : theory -> spec = fst o Code_Data.get;

fun map_exec_purge f = Code_Data.map (fn (exec, _) => (f exec, empty_dataref ()));

fun change_fun_spec delete c f = (map_exec_purge o map_functions

  o (if delete then Symtab.map_entry c else Symtab.map_default (c, ((false, empty_fun_spec), [])))

    o apfst) (fn (_, spec) => (true, f spec));

(* tackling equation history *)

fun continue_history thy = if (history_concluded o the_exec) thy

  then thy

    |> (Code_Data.map o apfst o map_history_concluded) (K false)

    |> SOME

  else NONE;

fun conclude_history thy = if (history_concluded o the_exec) thy

  then NONE

  else thy

    |> (Code_Data.map o apfst)

        ((map_functions o Symtab.map) (fn _ => fn ((changed, current), history) =>

          ((false, current),

            if changed then (serial (), current) :: history else history))

        #> map_history_concluded (K true))

    |> SOME;

val _ = Theory.setup

  (Theory.at_begin continue_history #> Theory.at_end conclude_history);

(* access to data dependent on abstract executable code *)

fun change_yield_data (kind, mk, dest) theory f =

  let

    val dataref = (snd o Code_Data.get) theory;

    val (datatab, thy) = case Synchronized.value dataref

     of SOME (datatab, thy) =>

        if Theory.eq_thy (theory, thy)

          then (datatab, thy)

          else (Datatab.empty, theory)

      | NONE => (Datatab.empty, theory)

    val data = case Datatab.lookup datatab kind

     of SOME data => data

      | NONE => invoke_init kind;

    val result as (_, data') = f (dest data);

    val _ = Synchronized.change dataref

      ((K o SOME) (Datatab.update (kind, mk data') datatab, thy));

  in result end;

end; (*local*)

(** foundation **)

(* datatypes *)

fun no_constr thy s (c, ty) = error ("Not a datatype constructor:\n" ^ string_of_const thy c

  ^ " :: " ^ string_of_typ thy ty ^ "\n" ^ enclose "(" ")" s);

fun analyze_constructor thy (c, ty) =

  let

    val _ = Thm.cterm_of thy (Const (c, ty));

    val ty_decl = devarify (const_typ thy c);

    fun last_typ c_ty ty =

      let

        val tfrees = Term.add_tfreesT ty [];

        val (tyco, vs) = (apsnd o map) dest_TFree (dest_Type (body_type ty))

          handle TYPE _ => no_constr thy "bad type" c_ty

        val _ = if tyco = "fun" then no_constr thy "bad type" c_ty else ();

        val _ =

          if has_duplicates (eq_fst (op =)) vs

          then no_constr thy "duplicate type variables in datatype" c_ty else ();

        val _ =

          if length tfrees <> length vs

          then no_constr thy "type variables missing in datatype" c_ty else ();

      in (tyco, vs) end;

    val (tyco, _) = last_typ (c, ty) ty_decl;

    val (_, vs) = last_typ (c, ty) ty;

  in ((tyco, map snd vs), (c, (map fst vs, ty))) end;

fun constrset_of_consts thy consts =

  let

    val _ = map (fn (c, _) => if (is_some o Axclass.class_of_param thy) c

      then error ("Is a class parameter: " ^ string_of_const thy c) else ()) consts;

    val raw_constructors = map (analyze_constructor thy) consts;

    val tyco = case distinct (op =) (map (fst o fst) raw_constructors)

     of [tyco] => tyco

      | [] => error "Empty constructor set"

      | tycos => error ("Different type constructors in constructor set: " ^ commas_quote tycos)

    val vs = Name.invent Name.context Name.aT (Sign.arity_number thy tyco);

    fun inst vs' (c, (vs, ty)) =

      let

        val the_v = the o AList.lookup (op =) (vs ~~ vs');

        val ty' = map_type_tfree (fn (v, _) => TFree (the_v v, [])) ty;

        val (vs'', ty'') = typscheme thy (c, ty');

      in (c, (vs'', binder_types ty'')) end;

    val constructors = map (inst vs o snd) raw_constructors;

  in (tyco, (map (rpair []) vs, constructors)) end;

fun get_type_entry thy tyco = case these (Symtab.lookup ((the_types o the_exec) thy) tyco)

 of (_, entry) :: _ => SOME entry

  | _ => NONE;

fun get_type thy tyco = case get_type_entry thy tyco

 of SOME (vs, spec) => apfst (pair vs) (constructors_of spec)

  | NONE => Sign.arity_number thy tyco

      |> Name.invent Name.context Name.aT

      |> map (rpair [])

      |> rpair []

      |> rpair false;

fun get_abstype_spec thy tyco = case get_type_entry thy tyco

 of SOME (vs, Abstractor spec) => (vs, spec)

  | _ => error ("Not an abstract type: " ^ tyco);

fun get_type_of_constr_or_abstr thy c =

  case (body_type o const_typ thy) c

   of Type (tyco, _) => let val ((_, cos), abstract) = get_type thy tyco

        in if member (op =) (map fst cos) c then SOME (tyco, abstract) else NONE end

    | _ => NONE;

fun is_constr thy c = case get_type_of_constr_or_abstr thy c

 of SOME (_, false) => true

   | _ => false;

fun is_abstr thy c = case get_type_of_constr_or_abstr thy c

 of SOME (_, true) => true

   | _ => false;

(* bare code equations *)

(* convention for variables:

    ?x ?'a   for free-floating theorems (e.g. in the data store)

    ?x  'a   for certificates

     x  'a   for final representation of equations

*)

exception BAD_THM of string;

fun bad_thm msg = raise BAD_THM msg;

fun error_thm f thy (thm, proper) = f (thm, proper)

  handle BAD_THM msg => error (msg ^ ", in theorem:\n" ^ Display.string_of_thm_global thy thm);

fun error_abs_thm f thy thm = f thm

  handle BAD_THM msg => error (msg ^ ", in theorem:\n" ^ Display.string_of_thm_global thy thm);

fun warning_thm f thy (thm, proper) = SOME (f (thm, proper))

  handle BAD_THM msg => (warning (msg ^ ", in theorem:\n" ^ Display.string_of_thm_global thy thm); NONE)

fun try_thm f thm_proper = SOME (f thm_proper)

  handle BAD_THM _ => NONE;

fun is_linear thm =

  let val (_, args) = (strip_comb o fst o Logic.dest_equals o Thm.plain_prop_of) thm

  in not (has_duplicates (op =) ((fold o fold_aterms)

    (fn Var (v, _) => cons v | _ => I) args [])) end;

fun check_decl_ty thy (c, ty) =

  let

    val ty_decl = Sign.the_const_type thy c;

  in if typscheme_equiv (ty_decl, ty) then ()

    else bad_thm ("Type\n" ^ string_of_typ thy ty

      ^ "\nof constant " ^ quote c

      ^ "\nis too specific compared to declared type\n"

      ^ string_of_typ thy ty_decl)

  end; 

fun check_eqn thy { allow_nonlinear, allow_consts, allow_pats } thm (lhs, rhs) =

  let

    fun vars_of t = fold_aterms (fn Var (v, _) => insert (op =) v

      | Free _ => bad_thm "Illegal free variable"

      | _ => I) t [];

    fun tvars_of t = fold_term_types (fn _ =>

      fold_atyps (fn TVar (v, _) => insert (op =) v

        | TFree _ => bad_thm "Illegal free type variable")) t [];

    val lhs_vs = vars_of lhs;

    val rhs_vs = vars_of rhs;

    val lhs_tvs = tvars_of lhs;

    val rhs_tvs = tvars_of rhs;

    val _ = if null (subtract (op =) lhs_vs rhs_vs)

      then ()

      else bad_thm "Free variables on right hand side of equation";

    val _ = if null (subtract (op =) lhs_tvs rhs_tvs)

      then ()

      else bad_thm "Free type variables on right hand side of equation";

    val (head, args) = strip_comb lhs;

    val (c, ty) = case head

     of Const (c_ty as (_, ty)) => (Axclass.unoverload_const thy c_ty, ty)

      | _ => bad_thm "Equation not headed by constant";

    fun check _ (Abs _) = bad_thm "Abstraction on left hand side of equation"

      | check 0 (Var _) = ()

      | check _ (Var _) = bad_thm "Variable with application on left hand side of equation"

      | check n (t1 $ t2) = (check (n+1) t1; check 0 t2)

      | check n (Const (c_ty as (c, ty))) =

          if allow_pats then let

            val c' = Axclass.unoverload_const thy c_ty

          in if n = (length o binder_types) ty

            then if allow_consts orelse is_constr thy c'

              then ()

              else bad_thm (quote c ^ " is not a constructor, on left hand side of equation")

            else bad_thm ("Partially applied constant " ^ quote c ^ " on left hand side of equation")

          end else bad_thm ("Pattern not allowed here, but constant " ^ quote c ^ " encountered on left hand side of equation")

    val _ = map (check 0) args;

    val _ = if allow_nonlinear orelse is_linear thm then ()

      else bad_thm "Duplicate variables on left hand side of equation";

    val _ = if (is_none o Axclass.class_of_param thy) c then ()

      else bad_thm "Overloaded constant as head in equation";

    val _ = if not (is_constr thy c) then ()

      else bad_thm "Constructor as head in equation";

    val _ = if not (is_abstr thy c) then ()

      else bad_thm "Abstractor as head in equation";

    val _ = check_decl_ty thy (c, ty);

    val _ = case strip_type ty

     of (Type (tyco, _) :: _, _) => (case get_type_entry thy tyco

       of SOME (_, Abstractor (_, (proj, _))) => if c = proj

            then bad_thm "Projection as head in equation"

            else ()

        | _ => ())

      | _ => ();

  in () end;

fun gen_assert_eqn thy check_patterns (thm, proper) =

  let

    val (lhs, rhs) = (Logic.dest_equals o Thm.plain_prop_of) thm

      handle TERM _ => bad_thm "Not an equation"

           | THM _ => bad_thm "Not a proper equation";

    val _ = check_eqn thy { allow_nonlinear = not proper,

      allow_consts = not (proper andalso check_patterns), allow_pats = true } thm (lhs, rhs);

  in (thm, proper) end;

fun assert_abs_eqn thy some_tyco thm =

  let

    val (full_lhs, rhs) = (Logic.dest_equals o Thm.plain_prop_of) thm

      handle TERM _ => bad_thm "Not an equation"

           | THM _ => bad_thm "Not a proper equation";

    val (rep, lhs) = dest_comb full_lhs

      handle TERM _ => bad_thm "Not an abstract equation";

    val (rep_const, ty) = dest_Const rep

      handle TERM _ => bad_thm "Not an abstract equation";

    val (tyco, Ts) = (dest_Type o domain_type) ty

      handle TERM _ => bad_thm "Not an abstract equation"

           | TYPE _ => bad_thm "Not an abstract equation";

    val _ = case some_tyco of SOME tyco' => if tyco = tyco' then ()

          else bad_thm ("Abstract type mismatch:" ^ quote tyco ^ " vs. " ^ quote tyco')

      | NONE => ();

    val (vs', (_, (rep', _))) = case try (get_abstype_spec thy) tyco

     of SOME data => data

      | NONE => bad_thm ("Not an abstract type: " ^ tyco);

    val _ = if rep_const = rep' then ()

      else bad_thm ("Projection mismatch: " ^ quote rep_const ^ " vs. " ^ quote rep');

    val _ = check_eqn thy { allow_nonlinear = false,

      allow_consts = false, allow_pats = false } thm (lhs, rhs);

    val _ = if forall2 (fn T => fn (_, sort) => Sign.of_sort thy (T, sort)) Ts vs' then ()

      else error ("Type arguments do not satisfy sort constraints of abstype certificate.");

  in (thm, tyco) end;

fun assert_eqn thy = error_thm (gen_assert_eqn thy true) thy;

fun meta_rewrite thy = Local_Defs.meta_rewrite_rule (Proof_Context.init_global thy);

fun mk_eqn thy = error_thm (gen_assert_eqn thy false) thy o

  apfst (meta_rewrite thy);

fun mk_eqn_liberal thy = Option.map (fn (thm, _) => (thm, is_linear thm))

  o try_thm (gen_assert_eqn thy false) o rpair false o meta_rewrite thy;

fun mk_eqn_maybe_abs thy raw_thm =

  let

    val thm = meta_rewrite thy raw_thm;

    val some_abs_thm = try_thm (assert_abs_eqn thy NONE) thm;

  in case some_abs_thm

   of SOME (thm, tyco) => SOME ((thm, true), SOME tyco)

    | NONE => (Option.map (fn (thm, _) => ((thm, is_linear thm), NONE))

        o warning_thm (gen_assert_eqn thy false) thy) (thm, false)

  end;

fun mk_abs_eqn thy = error_abs_thm (assert_abs_eqn thy NONE) thy o meta_rewrite thy;

val head_eqn = dest_Const o fst o strip_comb o fst o Logic.dest_equals o Thm.plain_prop_of;

fun const_typ_eqn thy thm =

  let

    val (c, ty) = head_eqn thm;

    val c' = Axclass.unoverload_const thy (c, ty);

      (*permissive wrt. to overloaded constants!*)

  in (c', ty) end;

fun const_eqn thy = fst o const_typ_eqn thy;

fun const_abs_eqn thy = Axclass.unoverload_const thy o dest_Const o fst o strip_comb o snd

  o dest_comb o fst o Logic.dest_equals o Thm.plain_prop_of;

fun mk_proj tyco vs ty abs rep =

  let

    val ty_abs = Type (tyco, map TFree vs);

    val xarg = Var (("x", 0), ty);

  in Logic.mk_equals (Const (rep, ty_abs --> ty) $ (Const (abs, ty --> ty_abs) $ xarg), xarg) end;

(* technical transformations of code equations *)

fun expand_eta thy k thm =

  let

    val (lhs, rhs) = (Logic.dest_equals o Thm.plain_prop_of) thm;

    val (_, args) = strip_comb lhs;

    val l = if k = ~1

      then (length o fst o strip_abs) rhs

      else Int.max (0, k - length args);

    val (raw_vars, _) = Term.strip_abs_eta l rhs;

    val vars = burrow_fst (Name.variant_list (map (fst o fst) (Term.add_vars lhs [])))

      raw_vars;

    fun expand (v, ty) thm = Drule.fun_cong_rule thm

      (Thm.cterm_of thy (Var ((v, 0), ty)));

  in

    thm

    |> fold expand vars

    |> Conv.fconv_rule Drule.beta_eta_conversion

  end;

fun same_arity thy thms =

  let

    val num_args_of = length o snd o strip_comb o fst o Logic.dest_equals;

    val k = fold (Integer.max o num_args_of o Thm.prop_of) thms 0;

  in map (expand_eta thy k) thms end;

fun mk_desymbolization pre post mk vs =

  let

    val names = map (pre o fst o fst) vs

      |> map (Name.desymbolize false)

      |> Name.variant_list []

      |> map post;

  in map_filter (fn (((v, i), x), v') =>

    if v = v' andalso i = 0 then NONE

    else SOME (((v, i), x), mk ((v', 0), x))) (vs ~~ names)

  end;

fun desymbolize_tvars thms =

  let

    val tvs = fold (Term.add_tvars o Thm.prop_of) thms [];

    val tvar_subst = mk_desymbolization (unprefix "'") (prefix "'") TVar tvs;

  in map (Thm.certify_instantiate (tvar_subst, [])) thms end;

fun desymbolize_vars thm =

  let

    val vs = Term.add_vars (Thm.prop_of thm) [];

    val var_subst = mk_desymbolization I I Var vs;

  in Thm.certify_instantiate ([], var_subst) thm end;

fun canonize_thms thy = desymbolize_tvars #> same_arity thy #> map desymbolize_vars;

(* abstype certificates *)

fun check_abstype_cert thy proto_thm =

  let

    val thm = (Axclass.unoverload thy o meta_rewrite thy) proto_thm;

    val (lhs, rhs) = Logic.dest_equals (Thm.plain_prop_of thm)

      handle TERM _ => bad_thm "Not an equation"

           | THM _ => bad_thm "Not a proper equation";

    val ((abs, raw_ty), ((rep, rep_ty), param)) = (apsnd (apfst dest_Const o dest_comb)

        o apfst dest_Const o dest_comb) lhs

      handle TERM _ => bad_thm "Not an abstype certificate";

    val _ = pairself (fn c => if (is_some o Axclass.class_of_param thy) c

      then error ("Is a class parameter: " ^ string_of_const thy c) else ()) (abs, rep);

    val _ = check_decl_ty thy (abs, raw_ty);

    val _ = check_decl_ty thy (rep, rep_ty);

    val _ = if length (binder_types raw_ty) = 1

      then ()

      else bad_thm "Bad type for abstract constructor";

    val _ = (fst o dest_Var) param

      handle TERM _ => bad_thm "Not an abstype certificate";

    val _ = if param = rhs then () else bad_thm "Not an abstype certificate";

    val ((tyco, sorts), (abs, (vs, ty'))) =

      analyze_constructor thy (abs, devarify raw_ty);

    val ty = domain_type ty';

    val (vs', _) = typscheme thy (abs, ty');

  in (tyco, (vs ~~ sorts, ((abs, (vs', ty)), (rep, thm)))) end;

(* code equation certificates *)

fun build_head thy (c, ty) =

  Thm.cterm_of thy (Logic.mk_equals (Free ("HEAD", ty), Const (c, ty)));

fun get_head thy cert_thm =

  let

    val [head] = (#hyps o Thm.crep_thm) cert_thm;

    val (_, Const (c, ty)) = (Logic.dest_equals o Thm.term_of) head;

  in (typscheme thy (c, ty), head) end;

fun typscheme_projection thy =

  typscheme thy o dest_Const o fst o dest_comb o fst o Logic.dest_equals;

fun typscheme_abs thy =

  typscheme thy o dest_Const o fst o strip_comb o snd o dest_comb o fst o Logic.dest_equals o Thm.prop_of;

fun constrain_thm thy vs sorts thm =

  let

    val mapping = map2 (fn (v, sort) => fn sort' =>

      (v, Sorts.inter_sort (Sign.classes_of thy) (sort, sort'))) vs sorts;

    val inst = map2 (fn (v, sort) => fn (_, sort') =>

      (((v, 0), sort), TFree (v, sort'))) vs mapping;

    val subst = (map_types o map_type_tfree)

      (fn (v, _) => TFree (v, the (AList.lookup (op =) mapping v)));

  in

    thm

    |> Thm.varifyT_global

    |> Thm.certify_instantiate (inst, [])

    |> pair subst

  end;

fun concretify_abs thy tyco abs_thm =

  let

    val (_, ((c, _), (_, cert))) = get_abstype_spec thy tyco;

    val lhs = (fst o Logic.dest_equals o Thm.prop_of) abs_thm

    val ty = fastype_of lhs;

    val ty_abs = (fastype_of o snd o dest_comb) lhs;

    val abs = Thm.cterm_of thy (Const (c, ty --> ty_abs));

    val raw_concrete_thm = Drule.transitive_thm OF [Thm.symmetric cert, Thm.combination (Thm.reflexive abs) abs_thm];

  in (c, (Thm.varifyT_global o zero_var_indexes) raw_concrete_thm) end;

fun add_rhss_of_eqn thy t =

  let

    val (args, rhs) = (apfst (snd o strip_comb) o Logic.dest_equals) t;

    fun add_const (Const (c, ty)) = insert (op =) (c, Sign.const_typargs thy (c, ty))

      | add_const _ = I

    val add_consts = fold_aterms add_const

  in add_consts rhs o fold add_consts args end;

val dest_eqn = apfst (snd o strip_comb) o Logic.dest_equals o Logic.unvarify_global;

abstype cert = Equations of thm * bool list

  | Projection of term * string

  | Abstract of thm * string

with

fun empty_cert thy c = 

  let

    val raw_ty = devarify (const_typ thy c);

    val (vs, _) = typscheme thy (c, raw_ty);

    val sortargs = case Axclass.class_of_param thy c

     of SOME class => [[class]]

      | NONE => (case get_type_of_constr_or_abstr thy c

         of SOME (tyco, _) => (map snd o fst o the)

              (AList.lookup (op =) ((snd o fst o get_type thy) tyco) c)

          | NONE => replicate (length vs) []);

    val the_sort = the o AList.lookup (op =) (map fst vs ~~ sortargs);

    val ty = map_type_tfree (fn (v, _) => TFree (v, the_sort v)) raw_ty

    val chead = build_head thy (c, ty);

  in Equations (Thm.weaken chead Drule.dummy_thm, []) end;

fun cert_of_eqns thy c [] = empty_cert thy c

  | cert_of_eqns thy c raw_eqns = 

      let

        val eqns = burrow_fst (canonize_thms thy) raw_eqns;

        val _ = map (assert_eqn thy) eqns;

        val (thms, propers) = split_list eqns;

        val _ = map (fn thm => if c = const_eqn thy thm then ()

          else error ("Wrong head of code equation,\nexpected constant "

            ^ string_of_const thy c ^ "\n" ^ Display.string_of_thm_global thy thm)) thms;

        fun tvars_of T = rev (Term.add_tvarsT T []);

        val vss = map (tvars_of o snd o head_eqn) thms;

        fun inter_sorts vs =

          fold (curry (Sorts.inter_sort (Sign.classes_of thy)) o snd) vs [];

        val sorts = map_transpose inter_sorts vss;

        val vts = Name.invent_names Name.context Name.aT sorts;

        val thms' =

          map2 (fn vs => Thm.certify_instantiate (vs ~~ map TFree vts, [])) vss thms;

        val head_thm = Thm.symmetric (Thm.assume (build_head thy (head_eqn (hd thms'))));

        fun head_conv ct = if can Thm.dest_comb ct

          then Conv.fun_conv head_conv ct

          else Conv.rewr_conv head_thm ct;

        val rewrite_head = Conv.fconv_rule (Conv.arg1_conv head_conv);

        val cert_thm = Conjunction.intr_balanced (map rewrite_head thms');

      in Equations (cert_thm, propers) end;

fun cert_of_proj thy c tyco =

  let

    val (vs, ((abs, (_, ty)), (rep, _))) = get_abstype_spec thy tyco;

    val _ = if c = rep then () else

      error ("Wrong head of projection,\nexpected constant " ^ string_of_const thy rep);

  in Projection (mk_proj tyco vs ty abs rep, tyco) end;

fun cert_of_abs thy tyco c raw_abs_thm =

  let

    val abs_thm = singleton (canonize_thms thy) raw_abs_thm;

    val _ = assert_abs_eqn thy (SOME tyco) abs_thm;

    val _ = if c = const_abs_eqn thy abs_thm then ()

      else error ("Wrong head of abstract code equation,\nexpected constant "

        ^ string_of_const thy c ^ "\n" ^ Display.string_of_thm_global thy abs_thm);

  in Abstract (Thm.legacy_freezeT abs_thm, tyco) end;

fun constrain_cert thy sorts (Equations (cert_thm, propers)) =

      let

        val ((vs, _), head) = get_head thy cert_thm;

        val (subst, cert_thm') = cert_thm

          |> Thm.implies_intr head

          |> constrain_thm thy vs sorts;

        val head' = Thm.term_of head

          |> subst

          |> Thm.cterm_of thy;

        val cert_thm'' = cert_thm'

          |> Thm.elim_implies (Thm.assume head');

      in Equations (cert_thm'', propers) end

  | constrain_cert thy _ (cert as Projection _) =

      cert

  | constrain_cert thy sorts (Abstract (abs_thm, tyco)) =

      Abstract (snd (constrain_thm thy (fst (typscheme_abs thy abs_thm)) sorts abs_thm), tyco);

fun conclude_cert (Equations (cert_thm, propers)) =

      (Equations (Thm.close_derivation cert_thm, propers))

  | conclude_cert (cert as Projection _) =

      cert

  | conclude_cert (Abstract (abs_thm, tyco)) =

      (Abstract (Thm.close_derivation abs_thm, tyco));

fun typscheme_of_cert thy (Equations (cert_thm, _)) =

      fst (get_head thy cert_thm)

  | typscheme_of_cert thy (Projection (proj, _)) =

      typscheme_projection thy proj

  | typscheme_of_cert thy (Abstract (abs_thm, _)) =

      typscheme_abs thy abs_thm;

fun typargs_deps_of_cert thy (Equations (cert_thm, propers)) =

      let

        val vs = (fst o fst) (get_head thy cert_thm);

        val equations = if null propers then [] else

          Thm.prop_of cert_thm

          |> Logic.dest_conjunction_balanced (length propers);

      in (vs, fold (add_rhss_of_eqn thy) equations []) end

  | typargs_deps_of_cert thy (Projection (t, _)) =

      (fst (typscheme_projection thy t), add_rhss_of_eqn thy t [])

  | typargs_deps_of_cert thy (Abstract (abs_thm, tyco)) =

      let

        val vs = fst (typscheme_abs thy abs_thm);

        val (_, concrete_thm) = concretify_abs thy tyco abs_thm;

      in (vs, add_rhss_of_eqn thy (Logic.unvarify_types_global (Thm.prop_of concrete_thm)) []) end;

fun equations_of_cert thy (cert as Equations (cert_thm, propers)) =

      let

        val tyscm = typscheme_of_cert thy cert;

        val thms = if null propers then [] else

          cert_thm

          |> Local_Defs.expand [snd (get_head thy cert_thm)]

          |> Thm.varifyT_global

          |> Conjunction.elim_balanced (length propers);

        fun abstractions (args, rhs) = (map (rpair NONE) args, (rhs, NONE));

      in (tyscm, map (abstractions o dest_eqn o Thm.prop_of) thms ~~ (map SOME thms ~~ propers)) end

  | equations_of_cert thy (Projection (t, tyco)) =

      let

        val (_, ((abs, _), _)) = get_abstype_spec thy tyco;

        val tyscm = typscheme_projection thy t;

        val t' = Logic.varify_types_global t;

        fun abstractions (args, rhs) = (map (rpair (SOME abs)) args, (rhs, NONE));

      in (tyscm, [((abstractions o dest_eqn) t', (NONE, true))]) end

  | equations_of_cert thy (Abstract (abs_thm, tyco)) =

      let

        val tyscm = typscheme_abs thy abs_thm;

        val (abs, concrete_thm) = concretify_abs thy tyco abs_thm;

        fun abstractions (args, rhs) = (map (rpair NONE) args, (rhs, (SOME abs)));

      in

        (tyscm, [((abstractions o dest_eqn o Thm.prop_of) concrete_thm,

          (SOME (Thm.varifyT_global abs_thm), true))])

      end;

fun pretty_cert thy (cert as Equations _) =

      (map_filter (Option.map (Display.pretty_thm_global thy o Axclass.overload thy) o fst o snd)

         o snd o equations_of_cert thy) cert

  | pretty_cert thy (Projection (t, _)) =

      [Syntax.pretty_term_global thy (Logic.varify_types_global t)]

  | pretty_cert thy (Abstract (abs_thm, _)) =

      [(Display.pretty_thm_global thy o Axclass.overload thy o Thm.varifyT_global) abs_thm];

fun bare_thms_of_cert thy (cert as Equations _) =

      (map_filter (fn (_, (some_thm, proper)) => if proper then some_thm else NONE)

        o snd o equations_of_cert thy) cert

  | bare_thms_of_cert thy (Projection _) = []

  | bare_thms_of_cert thy (Abstract (abs_thm, tyco)) =

      [Thm.varifyT_global (snd (concretify_abs thy tyco abs_thm))];

end;

(* code certificate access *)

fun retrieve_raw thy c =

  Symtab.lookup ((the_functions o the_exec) thy) c

  |> Option.map (snd o fst)

  |> the_default empty_fun_spec

fun eqn_conv conv ct =

  let

    fun lhs_conv ct = if can Thm.dest_comb ct

      then Conv.combination_conv lhs_conv conv ct

      else Conv.all_conv ct;

  in Conv.combination_conv (Conv.arg_conv lhs_conv) conv ct end;

fun rewrite_eqn thy conv ss =

  let

    val ctxt = Proof_Context.init_global thy;

    val rewrite = Conv.fconv_rule (conv (Simplifier.rewrite (put_simpset ss ctxt)));

  in singleton (Variable.trade (K (map rewrite)) ctxt) end;

fun cert_of_eqns_preprocess thy functrans ss c =

  (map o apfst) (Thm.transfer thy)

  #> perhaps (perhaps_loop (perhaps_apply functrans))

  #> (map o apfst) (rewrite_eqn thy eqn_conv ss) 

  #> (map o apfst) (Axclass.unoverload thy)

  #> cert_of_eqns thy c;

fun get_cert thy { functrans, ss } c =

  case retrieve_raw thy c

   of Default (_, eqns_lazy) => Lazy.force eqns_lazy

        |> cert_of_eqns_preprocess thy functrans ss c

    | Eqns eqns => eqns

        |> cert_of_eqns_preprocess thy functrans ss c

    | Proj (_, tyco) =>

        cert_of_proj thy c tyco

    | Abstr (abs_thm, tyco) => abs_thm

        |> Thm.transfer thy

        |> rewrite_eqn thy Conv.arg_conv ss

        |> Axclass.unoverload thy

        |> cert_of_abs thy tyco c;

(* cases *)

fun case_certificate thm =

  let

    val ((head, raw_case_expr), cases) = (apfst Logic.dest_equals

      o apsnd Logic.dest_conjunctions o Logic.dest_implies o Thm.plain_prop_of) thm;

    val _ = case head of Free _ => true

      | Var _ => true

      | _ => raise TERM ("case_cert", []);

    val ([(case_var, _)], case_expr) = Term.strip_abs_eta 1 raw_case_expr;

    val (Const (case_const, _), raw_params) = strip_comb case_expr;

    val n = find_index (fn Free (v, _) => v = case_var | _ => false) raw_params;

    val _ = if n = ~1 then raise TERM ("case_cert", []) else ();

    val params = map (fst o dest_Var) (nth_drop n raw_params);

    fun dest_case t =

      let

        val (head' $ t_co, rhs) = Logic.dest_equals t;

        val _ = if head' = head then () else raise TERM ("case_cert", []);

        val (Const (co, _), args) = strip_comb t_co;

        val (Var (param, _), args') = strip_comb rhs;

        val _ = if args' = args then () else raise TERM ("case_cert", []);

      in (param, co) end;

    fun analyze_cases cases =

      let

        val co_list = fold (AList.update (op =) o dest_case) cases [];

      in map (AList.lookup (op =) co_list) params end;

    fun analyze_let t =

      let

        val (head' $ arg, Var (param', _) $ arg') = Logic.dest_equals t;

        val _ = if head' = head then () else raise TERM ("case_cert", []);

        val _ = if arg' = arg then () else raise TERM ("case_cert", []);

        val _ = if [param'] = params then () else raise TERM ("case_cert", []);

      in [] end;

    fun analyze (cases as [let_case]) =

          (analyze_cases cases handle Bind => analyze_let let_case)

      | analyze cases = analyze_cases cases;

  in (case_const, (n, analyze cases)) end;

fun case_cert thm = case_certificate thm

  handle Bind => error "bad case certificate"

       | TERM _ => error "bad case certificate";

fun get_case_scheme thy = Option.map fst o Symtab.lookup ((fst o the_cases o the_exec) thy);

fun get_case_cong thy = Option.map snd o Symtab.lookup ((fst o the_cases o the_exec) thy);

val undefineds = Symtab.keys o snd o the_cases o the_exec;

(* diagnostic *)

fun print_codesetup thy =

  let

    val ctxt = Proof_Context.init_global thy;

    val exec = the_exec thy;

    fun pretty_equations const thms =

      (Pretty.block o Pretty.fbreaks)

        (Pretty.str (string_of_const thy const) :: map (Display.pretty_thm_item ctxt) thms);

    fun pretty_function (const, Default (_, eqns_lazy)) =

          pretty_equations const (map fst (Lazy.force eqns_lazy))

      | pretty_function (const, Eqns eqns) = pretty_equations const (map fst eqns)

      | pretty_function (const, Proj (proj, _)) = Pretty.block

          [Pretty.str (string_of_const thy const), Pretty.fbrk, Syntax.pretty_term ctxt proj]

      | pretty_function (const, Abstr (thm, _)) = pretty_equations const [thm];

    fun pretty_typ (tyco, vs) = Pretty.str

      (string_of_typ thy (Type (tyco, map TFree vs)));

    fun pretty_typspec (typ, (cos, abstract)) = if null cos

      then pretty_typ typ

      else (Pretty.block o Pretty.breaks) (

        pretty_typ typ

        :: Pretty.str "="

        :: (if abstract then [Pretty.str "(abstract)"] else [])

        @ separate (Pretty.str "|") (map (fn (c, (_, [])) => Pretty.str (string_of_const thy c)

             | (c, (_, tys)) =>

                 (Pretty.block o Pretty.breaks)

                    (Pretty.str (string_of_const thy c)