(*  Title:      HOL/SPARK/Tools/spark_vcs.ML

    Author:     Stefan Berghofer

    Copyright:  secunet Security Networks AG

Store for verification conditions generated by SPARK/Ada.

*)

signature SPARK_VCS =

sig

  val set_vcs: Fdl_Parser.decls -> Fdl_Parser.rules ->

    (string * string) * Fdl_Parser.vcs -> Path.T -> string -> theory -> theory

  val add_proof_fun: (typ option -> 'a -> term) ->

    string * ((string list * string) option * 'a) ->

    theory -> theory

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

  val lookup_vc: theory -> bool -> string -> (Element.context_i list *

    (string * thm list option * Element.context_i * Element.statement_i)) option

  val get_vcs: theory -> bool ->

    Element.context_i list * (binding * thm) list * (string *

    (string * thm list option * Element.context_i * Element.statement_i)) list

  val mark_proved: string -> thm list -> theory -> theory

  val close: bool -> theory -> theory

  val is_closed: theory -> bool

end;

structure SPARK_VCs: SPARK_VCS =

struct

open Fdl_Parser;

(** theory data **)

fun err_unfinished () = error "An unfinished SPARK environment is still open."

val strip_number = pairself implode o take_suffix Fdl_Lexer.is_digit o raw_explode;

val name_ord = prod_ord string_ord (option_ord int_ord) o

  pairself (strip_number ##> Int.fromString);

structure VCtab = Table(type key = string val ord = name_ord);

structure VCs = Theory_Data

(

  type T =

    {pfuns: ((string list * string) option * term) Symtab.table,

     type_map: (typ * (string * string) list) Symtab.table,

     env:

       {ctxt: Element.context_i list,

        defs: (binding * thm) list,

        types: fdl_type tab,

        funs: (string list * string) tab,

        pfuns: ((string list * string) option * term) Symtab.table,

        ids: (term * string) Symtab.table * Name.context,

        proving: bool,

        vcs: (string * thm list option *

          (string * expr) list * (string * expr) list) VCtab.table,

        path: Path.T,

        prefix: string} option}

  val empty : T = {pfuns = Symtab.empty, type_map = Symtab.empty, env = NONE}

  val extend = I

  fun merge ({pfuns = pfuns1, type_map = type_map1, env = NONE},

        {pfuns = pfuns2, type_map = type_map2, env = NONE}) =

        {pfuns = Symtab.merge (eq_pair (op =) (op aconv)) (pfuns1, pfuns2),

         type_map = Symtab.merge (op =) (type_map1, type_map2),

         env = NONE}

    | merge _ = err_unfinished ()

)

(** utilities **)

val to_lower = raw_explode #> map Symbol.to_ascii_lower #> implode;

val lcase_eq = (op =) o pairself (to_lower o Long_Name.base_name);

fun lookup_prfx "" tab s = Symtab.lookup tab s

  | lookup_prfx prfx tab s = (case Symtab.lookup tab s of

        NONE => Symtab.lookup tab (prfx ^ "__" ^ s)

      | x => x);

fun strip_prfx s =

  let

    fun strip ys [] = ("", implode ys)

      | strip ys ("_" :: "_" :: xs) = (implode (rev xs), implode ys)

      | strip ys (x :: xs) = strip (x :: ys) xs

  in strip [] (rev (raw_explode s)) end;

fun unprefix_pfun "" s = s

  | unprefix_pfun prfx s =

      let val (prfx', s') = strip_prfx s

      in if prfx = prfx' then s' else s end;

fun mk_unop s t =

  let val T = fastype_of t

  in Const (s, T --> T) $ t end;

fun mk_times (t, u) =

  let

    val setT = fastype_of t;

    val T = HOLogic.dest_setT setT;

    val U = HOLogic.dest_setT (fastype_of u)

  in

    Const (@{const_name Sigma}, setT --> (T --> HOLogic.mk_setT U) -->

      HOLogic.mk_setT (HOLogic.mk_prodT (T, U))) $ t $ Abs ("", T, u)

  end;

fun get_type thy prfx ty =

  let val {type_map, ...} = VCs.get thy

  in lookup_prfx prfx type_map ty end;

fun mk_type' _ _ "integer" = (HOLogic.intT, [])

  | mk_type' _ _ "boolean" = (HOLogic.boolT, [])

  | mk_type' thy prfx ty =

      (case get_type thy prfx ty of

         NONE =>

           (Syntax.check_typ (Proof_Context.init_global thy)

              (Type (Sign.full_name thy (Binding.name ty), [])),

            [])

       | SOME p => p);

fun mk_type thy prfx ty = fst (mk_type' thy prfx ty);

val booleanN = "boolean";

val integerN = "integer";

fun define_overloaded (def_name, eq) lthy =

  let

    val ((c, _), rhs) = eq |> Syntax.check_term lthy |>

      Logic.dest_equals |>> dest_Free;

    val ((_, (_, thm)), lthy') = Local_Theory.define

      ((Binding.name c, NoSyn), ((Binding.name def_name, []), rhs)) lthy

    val ctxt_thy = Proof_Context.init_global (Proof_Context.theory_of lthy');

    val thm' = singleton (Proof_Context.export lthy' ctxt_thy) thm

  in (thm', lthy') end;

fun strip_underscores s =

  strip_underscores (unsuffix "_" s) handle Fail _ => s;

fun strip_tilde s =

  unsuffix "~" s ^ "_init" handle Fail _ => s;

val mangle_name = strip_underscores #> strip_tilde;

fun mk_variables thy prfx xs ty (tab, ctxt) =

  let

    val T = mk_type thy prfx ty;

    val (ys, ctxt') = fold_map Name.variant (map mangle_name xs) ctxt;

    val zs = map (Free o rpair T) ys;

  in (zs, (fold (Symtab.update o apsnd (rpair ty)) (xs ~~ zs) tab, ctxt')) end;

fun get_record_info thy T = (case Record.dest_recTs T of

    [(tyname, [@{typ unit}])] =>

      Record.get_info thy (Long_Name.qualifier tyname)

  | _ => NONE);

fun find_field [] fname fields =

      find_first (curry lcase_eq fname o fst) fields

  | find_field cmap fname fields =

      (case AList.lookup (op =) cmap fname of

         NONE => NONE

       | SOME fname' => SOME (fname', the (AList.lookup (op =) fields fname')));

fun find_field' fname = get_first (fn (flds, fldty) =>

  if member (op =) flds fname then SOME fldty else NONE);

fun assoc_ty_err thy T s msg =

  error ("Type " ^ Syntax.string_of_typ_global thy T ^

    " associated with SPARK type " ^ s ^ "\n" ^ msg);

(** generate properties of enumeration types **)

fun add_enum_type tyname tyname' thy =

  let

    val {case_name, ...} = the (Datatype.get_info thy tyname');

    val cs = map Const (the (Datatype.get_constrs thy tyname'));

    val k = length cs;

    val T = Type (tyname', []);

    val p = Const (@{const_name pos}, T --> HOLogic.intT);

    val v = Const (@{const_name val}, HOLogic.intT --> T);

    val card = Const (@{const_name card},

      HOLogic.mk_setT T --> HOLogic.natT) $ HOLogic.mk_UNIV T;

    fun mk_binrel_def s f = Logic.mk_equals

      (Const (s, T --> T --> HOLogic.boolT),

       Abs ("x", T, Abs ("y", T,

         Const (s, HOLogic.intT --> HOLogic.intT --> HOLogic.boolT) $

           (f $ Bound 1) $ (f $ Bound 0))));

    val (((def1, def2), def3), lthy) = thy |>

      Class.instantiation ([tyname'], [], @{sort spark_enum}) |>

      define_overloaded ("pos_" ^ tyname ^ "_def", Logic.mk_equals

        (p,

         list_comb (Const (case_name, replicate k HOLogic.intT @

             [T] ---> HOLogic.intT),

           map (HOLogic.mk_number HOLogic.intT) (0 upto k - 1)))) ||>>

      define_overloaded ("less_eq_" ^ tyname ^ "_def",

        mk_binrel_def @{const_name less_eq} p) ||>>

      define_overloaded ("less_" ^ tyname ^ "_def",

        mk_binrel_def @{const_name less} p);

    val UNIV_eq = Goal.prove lthy [] []

      (HOLogic.mk_Trueprop (HOLogic.mk_eq

         (HOLogic.mk_UNIV T, HOLogic.mk_set T cs)))

      (fn _ =>

         rtac @{thm subset_antisym} 1 THEN

         rtac @{thm subsetI} 1 THEN

         Datatype_Aux.exh_tac (K (#exhaust (Datatype.the_info

           (Proof_Context.theory_of lthy) tyname'))) 1 THEN

         ALLGOALS (asm_full_simp_tac (simpset_of lthy)));

    val finite_UNIV = Goal.prove lthy [] []

      (HOLogic.mk_Trueprop (Const (@{const_name finite},

         HOLogic.mk_setT T --> HOLogic.boolT) $ HOLogic.mk_UNIV T))

      (fn _ => simp_tac (simpset_of lthy addsimps [UNIV_eq]) 1);

    val card_UNIV = Goal.prove lthy [] []

      (HOLogic.mk_Trueprop (HOLogic.mk_eq

         (card, HOLogic.mk_number HOLogic.natT k)))

      (fn _ => simp_tac (simpset_of lthy addsimps [UNIV_eq]) 1);

    val range_pos = Goal.prove lthy [] []

      (HOLogic.mk_Trueprop (HOLogic.mk_eq

         (Const (@{const_name image}, (T --> HOLogic.intT) -->

            HOLogic.mk_setT T --> HOLogic.mk_setT HOLogic.intT) $

              p $ HOLogic.mk_UNIV T,

          Const (@{const_name atLeastLessThan}, HOLogic.intT -->

            HOLogic.intT --> HOLogic.mk_setT HOLogic.intT) $

              HOLogic.mk_number HOLogic.intT 0 $

              (@{term int} $ card))))

      (fn _ =>

         simp_tac (simpset_of lthy addsimps [card_UNIV]) 1 THEN

         simp_tac (simpset_of lthy addsimps [UNIV_eq, def1]) 1 THEN

         rtac @{thm subset_antisym} 1 THEN

         simp_tac (simpset_of lthy) 1 THEN

         rtac @{thm subsetI} 1 THEN

         asm_full_simp_tac (simpset_of lthy addsimps @{thms interval_expand}

           delsimps @{thms atLeastLessThan_iff}) 1);

    val lthy' =

      Class.prove_instantiation_instance (fn _ =>

        Class.intro_classes_tac [] THEN

        rtac finite_UNIV 1 THEN

        rtac range_pos 1 THEN

        simp_tac (HOL_basic_ss addsimps [def3]) 1 THEN

        simp_tac (HOL_basic_ss addsimps [def2]) 1) lthy;

    val (pos_eqs, val_eqs) = split_list (map_index (fn (i, c) =>

      let

        val n = HOLogic.mk_number HOLogic.intT i;

        val th = Goal.prove lthy' [] []

          (HOLogic.mk_Trueprop (HOLogic.mk_eq (p $ c, n)))

          (fn _ => simp_tac (simpset_of lthy' addsimps [def1]) 1);

        val th' = Goal.prove lthy' [] []

          (HOLogic.mk_Trueprop (HOLogic.mk_eq (v $ n, c)))

          (fn _ =>

             rtac (@{thm inj_pos} RS @{thm injD}) 1 THEN

             simp_tac (simpset_of lthy' addsimps

               [@{thm pos_val}, range_pos, card_UNIV, th]) 1)

      in (th, th') end) cs);

    val first_el = Goal.prove lthy' [] []

      (HOLogic.mk_Trueprop (HOLogic.mk_eq

         (Const (@{const_name first_el}, T), hd cs)))

      (fn _ => simp_tac (simpset_of lthy' addsimps

         [@{thm first_el_def}, hd val_eqs]) 1);

    val last_el = Goal.prove lthy' [] []

      (HOLogic.mk_Trueprop (HOLogic.mk_eq

         (Const (@{const_name last_el}, T), List.last cs)))

      (fn _ => simp_tac (simpset_of lthy' addsimps

         [@{thm last_el_def}, List.last val_eqs, card_UNIV]) 1);

  in

    lthy' |>

    Local_Theory.note

      ((Binding.name (tyname ^ "_card"), @{attributes [simp]}), [card_UNIV]) ||>>

    Local_Theory.note

      ((Binding.name (tyname ^ "_pos"), @{attributes [simp]}), pos_eqs) ||>>

    Local_Theory.note

      ((Binding.name (tyname ^ "_val"), @{attributes [simp]}), val_eqs) ||>>

    Local_Theory.note

      ((Binding.name (tyname ^ "_first_el"), @{attributes [simp]}), [first_el]) ||>>

    Local_Theory.note

      ((Binding.name (tyname ^ "_last_el"), @{attributes [simp]}), [last_el]) |> snd |>

    Local_Theory.exit_global

  end;

fun check_no_assoc thy prfx s = case get_type thy prfx s of

    NONE => ()

  | SOME _ => error ("Cannot associate a type with " ^ s ^

      "\nsince it is no record or enumeration type");

fun check_enum [] [] = NONE 

  | check_enum els [] = SOME ("has no element(s) " ^ commas els)

  | check_enum [] cs = SOME ("has extra element(s) " ^

      commas (map (Long_Name.base_name o fst) cs))

  | check_enum (el :: els) ((cname, _) :: cs) =

      if lcase_eq (el, cname) then check_enum els cs

      else SOME ("either has no element " ^ el ^

        " or it is at the wrong position");

fun invert_map [] = I

  | invert_map cmap =

      map (apfst (the o AList.lookup (op =) (map swap cmap)));

fun add_type_def prfx (s, Basic_Type ty) (ids, thy) =

      (check_no_assoc thy prfx s;

       (ids,

        Typedecl.abbrev_global (Binding.name s, [], NoSyn)

          (mk_type thy prfx ty) thy |> snd))

  | add_type_def prfx (s, Enum_Type els) ((tab, ctxt), thy) =

      let

        val (thy', tyname) = (case get_type thy prfx s of

            NONE =>

              let

                val tyb = Binding.name s;

                val tyname = Sign.full_name thy tyb

              in

                (thy |>

                 Datatype.add_datatype {strict = true, quiet = true}

                   [((tyb, [], NoSyn),

                     map (fn s => (Binding.name s, [], NoSyn)) els)] |> snd |>

                 add_enum_type s tyname,

                 tyname)

              end

          | SOME (T as Type (tyname, []), cmap) =>

              (case Datatype.get_constrs thy tyname of

                 NONE => assoc_ty_err thy T s "is not a datatype"

               | SOME cs =>

                   let val (prfx', _) = strip_prfx s

                   in

                     case check_enum (map (unprefix_pfun prfx') els)

                         (invert_map cmap cs) of

                       NONE => (thy, tyname)

                     | SOME msg => assoc_ty_err thy T s msg

                   end)

          | SOME (T, _) => assoc_ty_err thy T s "is not a datatype");

        val cs = map Const (the (Datatype.get_constrs thy' tyname));

      in

        ((fold (Symtab.update_new o apsnd (rpair s)) (els ~~ cs) tab,

          fold Name.declare els ctxt),

         thy')

      end

  | add_type_def prfx (s, Array_Type (argtys, resty)) (ids, thy) =

      (check_no_assoc thy prfx s;

       (ids,

        Typedecl.abbrev_global (Binding.name s, [], NoSyn)

          (foldr1 HOLogic.mk_prodT (map (mk_type thy prfx) argtys) -->

             mk_type thy prfx resty) thy |> snd))

  | add_type_def prfx (s, Record_Type fldtys) (ids, thy) =

      (ids,

       let val fldTs = maps (fn (flds, ty) =>

         map (rpair (mk_type thy prfx ty)) flds) fldtys

       in case get_type thy prfx s of

           NONE =>

             Record.add_record ([], Binding.name s) NONE

               (map (fn (fld, T) => (Binding.name fld, T, NoSyn)) fldTs) thy

         | SOME (rT, cmap) =>

             (case get_record_info thy rT of

                NONE => assoc_ty_err thy rT s "is not a record type"

              | SOME {fields, ...} =>

                  let val fields' = invert_map cmap fields

                  in

                    (case subtract (lcase_eq o pairself fst) fldTs fields' of

                       [] => ()

                     | flds => assoc_ty_err thy rT s ("has extra field(s) " ^

                         commas (map (Long_Name.base_name o fst) flds));

                     map (fn (fld, T) =>

                       case AList.lookup lcase_eq fields' fld of

                         NONE => assoc_ty_err thy rT s ("has no field " ^ fld)

                       | SOME U => T = U orelse assoc_ty_err thy rT s

                           ("has field " ^

                            fld ^ " whose type\n" ^

                            Syntax.string_of_typ_global thy U ^

                            "\ndoes not match declared type\n" ^

                            Syntax.string_of_typ_global thy T)) fldTs;

                     thy)

                  end)

       end)

  | add_type_def prfx (s, Pending_Type) (ids, thy) =

      (ids,

       case get_type thy prfx s of

         SOME _ => thy

       | NONE => Typedecl.typedecl_global

           (Binding.name s, [], NoSyn) thy |> snd);

fun term_of_expr thy prfx types pfuns =

  let

    fun tm_of vs (Funct ("->", [e, e'])) =

          (HOLogic.mk_imp (fst (tm_of vs e), fst (tm_of vs e')), booleanN)

      | tm_of vs (Funct ("<->", [e, e'])) =

          (HOLogic.mk_eq (fst (tm_of vs e), fst (tm_of vs e')), booleanN)

      | tm_of vs (Funct ("or", [e, e'])) =

          (HOLogic.mk_disj (fst (tm_of vs e), fst (tm_of vs e')), booleanN)

      | tm_of vs (Funct ("and", [e, e'])) =

          (HOLogic.mk_conj (fst (tm_of vs e), fst (tm_of vs e')), booleanN)

      | tm_of vs (Funct ("not", [e])) =

          (HOLogic.mk_not (fst (tm_of vs e)), booleanN)

      | tm_of vs (Funct ("=", [e, e'])) =

          (HOLogic.mk_eq (fst (tm_of vs e), fst (tm_of vs e')), booleanN)

      | tm_of vs (Funct ("<>", [e, e'])) = (HOLogic.mk_not

          (HOLogic.mk_eq (fst (tm_of vs e), fst (tm_of vs e'))), booleanN)

      | tm_of vs (Funct ("<", [e, e'])) = (HOLogic.mk_binrel @{const_name less}

          (fst (tm_of vs e), fst (tm_of vs e')), booleanN)

      | tm_of vs (Funct (">", [e, e'])) = (HOLogic.mk_binrel @{const_name less}

          (fst (tm_of vs e'), fst (tm_of vs e)), booleanN)

      | tm_of vs (Funct ("<=", [e, e'])) = (HOLogic.mk_binrel @{const_name less_eq}

          (fst (tm_of vs e), fst (tm_of vs e')), booleanN)

      | tm_of vs (Funct (">=", [e, e'])) = (HOLogic.mk_binrel @{const_name less_eq}

          (fst (tm_of vs e'), fst (tm_of vs e)), booleanN)

      | tm_of vs (Funct ("+", [e, e'])) = (HOLogic.mk_binop @{const_name plus}

          (fst (tm_of vs e), fst (tm_of vs e')), integerN)

      | tm_of vs (Funct ("-", [e, e'])) = (HOLogic.mk_binop @{const_name minus}

          (fst (tm_of vs e), fst (tm_of vs e')), integerN)

      | tm_of vs (Funct ("*", [e, e'])) = (HOLogic.mk_binop @{const_name times}

          (fst (tm_of vs e), fst (tm_of vs e')), integerN)

      | tm_of vs (Funct ("/", [e, e'])) = (HOLogic.mk_binop @{const_name divide}

          (fst (tm_of vs e), fst (tm_of vs e')), integerN)

      | tm_of vs (Funct ("div", [e, e'])) = (HOLogic.mk_binop @{const_name sdiv}

          (fst (tm_of vs e), fst (tm_of vs e')), integerN)

      | tm_of vs (Funct ("mod", [e, e'])) = (HOLogic.mk_binop @{const_name mod}

          (fst (tm_of vs e), fst (tm_of vs e')), integerN)

      | tm_of vs (Funct ("-", [e])) =

          (mk_unop @{const_name uminus} (fst (tm_of vs e)), integerN)

      | tm_of vs (Funct ("**", [e, e'])) =

          (Const (@{const_name power}, HOLogic.intT --> HOLogic.natT -->

             HOLogic.intT) $ fst (tm_of vs e) $

               (@{const nat} $ fst (tm_of vs e')), integerN)

      | tm_of (tab, _) (Ident s) =

          (case Symtab.lookup tab s of

             SOME t_ty => t_ty

           | NONE => (case lookup_prfx prfx pfuns s of

               SOME (SOME ([], resty), t) => (t, resty)

             | _ => error ("Undeclared identifier " ^ s)))

      | tm_of _ (Number i) = (HOLogic.mk_number HOLogic.intT i, integerN)

      | tm_of vs (Quantifier (s, xs, ty, e)) =

          let

            val (ys, vs') = mk_variables thy prfx xs ty vs;

            val q = (case s of

                "for_all" => HOLogic.mk_all

              | "for_some" => HOLogic.mk_exists)

          in

            (fold_rev (fn Free (x, T) => fn t => q (x, T, t))

               ys (fst (tm_of vs' e)),

             booleanN)

          end

      | tm_of vs (Funct (s, es)) =

          (* record field selection *)

          (case try (unprefix "fld_") s of

             SOME fname => (case es of

               [e] =>

                 let

                   val (t, rcdty) = tm_of vs e;

                   val (rT, cmap) = mk_type' thy prfx rcdty

                 in case (get_record_info thy rT, lookup types rcdty) of

                     (SOME {fields, ...}, SOME (Record_Type fldtys)) =>

                       (case (find_field cmap fname fields,

                            find_field' fname fldtys) of

                          (SOME (fname', fT), SOME fldty) =>

                            (Const (fname', rT --> fT) $ t, fldty)

                        | _ => error ("Record " ^ rcdty ^

                            " has no field named " ^ fname))

                   | _ => error (rcdty ^ " is not a record type")

                 end

             | _ => error ("Function " ^ s ^ " expects one argument"))

           | NONE =>

          (* record field update *)

          (case try (unprefix "upf_") s of

             SOME fname => (case es of

               [e, e'] =>

                 let

                   val (t, rcdty) = tm_of vs e;

                   val (rT, cmap) = mk_type' thy prfx rcdty;

                   val (u, fldty) = tm_of vs e';

                   val fT = mk_type thy prfx fldty

                 in case get_record_info thy rT of

                     SOME {fields, ...} =>

                       (case find_field cmap fname fields of

                          SOME (fname', fU) =>

                            if fT = fU then

                              (Const (fname' ^ "_update",

                                 (fT --> fT) --> rT --> rT) $

                                   Abs ("x", fT, u) $ t,

                               rcdty)

                            else error ("Type\n" ^

                              Syntax.string_of_typ_global thy fT ^

                              "\ndoes not match type\n" ^

                              Syntax.string_of_typ_global thy fU ^

                              "\nof field " ^ fname)

                        | NONE => error ("Record " ^ rcdty ^

                            " has no field named " ^ fname))

                   | _ => error (rcdty ^ " is not a record type")

                 end

             | _ => error ("Function " ^ s ^ " expects two arguments"))

           | NONE =>

          (* enumeration type to integer *)

          (case try (unsuffix "__pos") s of

             SOME tyname => (case es of

               [e] => (Const (@{const_name pos},

                   mk_type thy prfx tyname --> HOLogic.intT) $ fst (tm_of vs e),

                 integerN)

             | _ => error ("Function " ^ s ^ " expects one argument"))

           | NONE =>

          (* integer to enumeration type *)

          (case try (unsuffix "__val") s of

             SOME tyname => (case es of

               [e] => (Const (@{const_name val},

                   HOLogic.intT --> mk_type thy prfx tyname) $ fst (tm_of vs e),

                 tyname)

             | _ => error ("Function " ^ s ^ " expects one argument"))

           | NONE =>

          (* successor / predecessor of enumeration type element *)

          if s = "succ" orelse s = "pred" then (case es of

              [e] =>

                let

                  val (t, tyname) = tm_of vs e;

                  val T = mk_type thy prfx tyname

                in (Const

                  (if s = "succ" then @{const_name succ}

                   else @{const_name pred}, T --> T) $ t, tyname)

                end

            | _ => error ("Function " ^ s ^ " expects one argument"))

          (* user-defined proof function *)

          else

            (case lookup_prfx prfx pfuns s of

               SOME (SOME (_, resty), t) =>

                 (list_comb (t, map (fst o tm_of vs) es), resty)

             | _ => error ("Undeclared proof function " ^ s))))))

      | tm_of vs (Element (e, es)) =

          let val (t, ty) = tm_of vs e

          in case lookup types ty of

              SOME (Array_Type (_, elty)) =>

                (t $ foldr1 HOLogic.mk_prod (map (fst o tm_of vs) es), elty)

            | _ => error (ty ^ " is not an array type")

          end

      | tm_of vs (Update (e, es, e')) =

          let val (t, ty) = tm_of vs e

          in case lookup types ty of

              SOME (Array_Type (idxtys, elty)) =>

                let

                  val T = foldr1 HOLogic.mk_prodT

                    (map (mk_type thy prfx) idxtys);

                  val U = mk_type thy prfx elty;

                  val fT = T --> U

                in

                  (Const (@{const_name fun_upd}, fT --> T --> U --> fT) $

                     t $ foldr1 HOLogic.mk_prod (map (fst o tm_of vs) es) $

                       fst (tm_of vs e'),

                   ty)

                end

            | _ => error (ty ^ " is not an array type")

          end

      | tm_of vs (Record (s, flds)) =

          let

            val (T, cmap) = mk_type' thy prfx s;

            val {extension = (ext_name, _), fields, ...} =

              (case get_record_info thy T of

                 NONE => error (s ^ " is not a record type")

               | SOME info => info);

            val flds' = map (apsnd (tm_of vs)) flds;

            val fnames = fields |> invert_map cmap |>

              map (Long_Name.base_name o fst);

            val fnames' = map fst flds;

            val (fvals, ftys) = split_list (map (fn s' =>

              case AList.lookup lcase_eq flds' s' of

                SOME fval_ty => fval_ty

              | NONE => error ("Field " ^ s' ^ " missing in record " ^ s))

                  fnames);

            val _ = (case subtract lcase_eq fnames fnames' of

                [] => ()

              | xs => error ("Extra field(s) " ^ commas xs ^

                  " in record " ^ s));

            val _ = (case duplicates (op =) fnames' of

                [] => ()

              | xs => error ("Duplicate field(s) " ^ commas xs ^

                  " in record " ^ s))

          in

            (list_comb

               (Const (ext_name,

                  map (mk_type thy prfx) ftys @ [HOLogic.unitT] ---> T),

                fvals @ [HOLogic.unit]),

             s)

          end

      | tm_of vs (Array (s, default, assocs)) =

          (case lookup types s of

             SOME (Array_Type (idxtys, elty)) =>

               let

                 val Ts = map (mk_type thy prfx) idxtys;

                 val T = foldr1 HOLogic.mk_prodT Ts;

                 val U = mk_type thy prfx elty;

                 fun mk_idx' T (e, NONE) = HOLogic.mk_set T [fst (tm_of vs e)]

                   | mk_idx' T (e, SOME e') = Const (@{const_name atLeastAtMost},

                       T --> T --> HOLogic.mk_setT T) $

                         fst (tm_of vs e) $ fst (tm_of vs e');

                 fun mk_idx idx =

                   if length Ts <> length idx then

                     error ("Arity mismatch in construction of array " ^ s)

                   else foldr1 mk_times (map2 mk_idx' Ts idx);

                 fun mk_upd (idxs, e) t =

                   if length idxs = 1 andalso forall (is_none o snd) (hd idxs)

                   then

                     Const (@{const_name fun_upd}, (T --> U) -->

                         T --> U --> T --> U) $ t $

                       foldl1 HOLogic.mk_prod

                         (map (fst o tm_of vs o fst) (hd idxs)) $

                       fst (tm_of vs e)

                   else

                     Const (@{const_name fun_upds}, (T --> U) -->

                         HOLogic.mk_setT T --> U --> T --> U) $ t $

                       foldl1 (HOLogic.mk_binop @{const_name sup})

                         (map mk_idx idxs) $

                       fst (tm_of vs e)

               in

                 (fold mk_upd assocs

                    (case default of

                       SOME e => Abs ("x", T, fst (tm_of vs e))

                     | NONE => Const (@{const_name undefined}, T --> U)),

                  s)

               end

           | _ => error (s ^ " is not an array type"))

  in tm_of end;

fun term_of_rule thy prfx types pfuns ids rule =

  let val tm_of = fst o term_of_expr thy prfx types pfuns ids

  in case rule of

      Inference_Rule (es, e) => Logic.list_implies

        (map (HOLogic.mk_Trueprop o tm_of) es, HOLogic.mk_Trueprop (tm_of e))

    | Substitution_Rule (es, e, e') => Logic.list_implies

        (map (HOLogic.mk_Trueprop o tm_of) es,

         HOLogic.mk_Trueprop (HOLogic.mk_eq (tm_of e, tm_of e')))

  end;

val builtin = Symtab.make (map (rpair ())

  ["->", "<->", "or", "and", "not", "=", "<>", "<", ">", "<=", ">=",

   "+", "-", "*", "/", "div", "mod", "**"]);

fun complex_expr (Number _) = false

  | complex_expr (Ident _) = false 

  | complex_expr (Funct (s, es)) =

      not (Symtab.defined builtin s) orelse exists complex_expr es

  | complex_expr (Quantifier (_, _, _, e)) = complex_expr e

  | complex_expr _ = true;

fun complex_rule (Inference_Rule (es, e)) =

      complex_expr e orelse exists complex_expr es

  | complex_rule (Substitution_Rule (es, e, e')) =

      complex_expr e orelse complex_expr e' orelse

      exists complex_expr es;

val is_pfun =

  Symtab.defined builtin orf

  can (unprefix "fld_") orf can (unprefix "upf_") orf

  can (unsuffix "__pos") orf can (unsuffix "__val") orf

  equal "succ" orf equal "pred";

fun fold_opt f = the_default I o Option.map f;

fun fold_pair f g (x, y) = f x #> g y;

fun fold_expr f g (Funct (s, es)) = f s #> fold (fold_expr f g) es

  | fold_expr f g (Ident s) = g s

  | fold_expr f g (Number _) = I

  | fold_expr f g (Quantifier (_, _, _, e)) = fold_expr f g e

  | fold_expr f g (Element (e, es)) =

      fold_expr f g e #> fold (fold_expr f g) es

  | fold_expr f g (Update (e, es, e')) =

      fold_expr f g e #> fold (fold_expr f g) es #> fold_expr f g e'

  | fold_expr f g (Record (_, flds)) = fold (fold_expr f g o snd) flds

  | fold_expr f g (Array (_, default, assocs)) =

      fold_opt (fold_expr f g) default #>

      fold (fold_pair

        (fold (fold (fold_pair

          (fold_expr f g) (fold_opt (fold_expr f g)))))

        (fold_expr f g)) assocs;

fun add_expr_pfuns funs = fold_expr

  (fn s => if is_pfun s then I else insert (op =) s)

  (fn s => if is_none (lookup funs s) then I else insert (op =) s);

val add_expr_idents = fold_expr (K I) (insert (op =));

fun pfun_type thy prfx (argtys, resty) =

  map (mk_type thy prfx) argtys ---> mk_type thy prfx resty;

fun check_pfun_type thy prfx s t optty1 optty2 =

  let

    val T = fastype_of t;

    fun check ty =

      let val U = pfun_type thy prfx ty

      in

        T = U orelse

        error ("Type\n" ^

          Syntax.string_of_typ_global thy T ^

          "\nof function " ^

          Syntax.string_of_term_global thy t ^

          " associated with proof function " ^ s ^

          "\ndoes not match declared type\n" ^

          Syntax.string_of_typ_global thy U)

      end

  in (Option.map check optty1; Option.map check optty2; ()) end;

fun upd_option x y = if is_some x then x else y;

fun check_pfuns_types thy prfx funs =

  Symtab.map (fn s => fn (optty, t) =>

   let val optty' = lookup funs (unprefix_pfun prfx s)

   in

     (check_pfun_type thy prfx s t optty optty';

      (NONE |> upd_option optty |> upd_option optty', t))

   end);

(** the VC store **)

fun pp_vcs msg vcs = Pretty.big_list msg (map (Pretty.str o fst) vcs);

fun pp_open_vcs [] = Pretty.str "All verification conditions have been proved."

  | pp_open_vcs vcs = pp_vcs

      "The following verification conditions remain to be proved:" vcs;

fun partition_vcs vcs = VCtab.fold_rev

  (fn (name, (trace, SOME thms, ps, cs)) =>

        apfst (cons (name, (trace, thms, ps, cs)))

    | (name, (trace, NONE, ps, cs)) =>

        apsnd (cons (name, (trace, ps, cs))))

  vcs ([], []);

fun insert_break prt = Pretty.blk (0, [Pretty.fbrk, prt]);

fun print_open_vcs f vcs =

  (Pretty.writeln (f (pp_open_vcs (snd (partition_vcs vcs)))); vcs);

fun set_env ctxt defs types funs ids vcs path prefix thy = VCs.map (fn

    {pfuns, type_map, env = NONE} =>

      {pfuns = pfuns,

       type_map = type_map,

       env = SOME

         {ctxt = ctxt, defs = defs, types = types, funs = funs,

          pfuns = check_pfuns_types thy prefix funs pfuns,

          ids = ids, proving = false, vcs = print_open_vcs I vcs, path = path,

          prefix = prefix}}

  | _ => err_unfinished ()) thy;

fun mk_pat s = (case Int.fromString s of

    SOME i => [HOLogic.mk_Trueprop (Var (("C", i), HOLogic.boolT))]

  | NONE => error ("Bad conclusion identifier: C" ^ s));

fun mk_vc thy prfx types pfuns ids name_concl (tr, proved, ps, cs) =

  let val prop_of =

    HOLogic.mk_Trueprop o fst o term_of_expr thy prfx types pfuns ids

  in

    (tr, proved,

     Element.Assumes (map (fn (s', e) =>

       ((Binding.name ("H" ^ s'), []), [(prop_of e, [])])) ps),

     Element.Shows (map (fn (s', e) =>

       (if name_concl then (Binding.name ("C" ^ s'), [])

        else Attrib.empty_binding,

        [(prop_of e, mk_pat s')])) cs))

  end;

fun fold_vcs f vcs =

  VCtab.fold (fn (_, (_, _, ps, cs)) => fold f ps #> fold f cs) vcs;

fun pfuns_of_vcs prfx funs pfuns vcs =

  fold_vcs (add_expr_pfuns funs o snd) vcs [] |>

  filter (is_none o lookup_prfx prfx pfuns);

fun declare_missing_pfuns thy prfx funs pfuns vcs (tab, ctxt) =

  let

    val (fs, (tys, Ts)) =

      pfuns_of_vcs prfx funs pfuns vcs |>

      map_filter (fn s => lookup funs s |>

        Option.map (fn ty => (s, (SOME ty, pfun_type thy prfx ty)))) |>

      split_list ||> split_list;

    val (fs', ctxt') = fold_map Name.variant fs ctxt

  in

    (fold Symtab.update_new (fs ~~ (tys ~~ map Free (fs' ~~ Ts))) pfuns,

     Element.Fixes (map2 (fn s => fn T =>

       (Binding.name s, SOME T, NoSyn)) fs' Ts),

     (tab, ctxt'))

  end;

fun map_pfuns f {pfuns, type_map, env} =

  {pfuns = f pfuns, type_map = type_map, env = env}

fun map_pfuns_env f {pfuns, type_map,

      env = SOME {ctxt, defs, types, funs, pfuns = pfuns_env,

        ids, proving, vcs, path, prefix}} =

  if proving then err_unfinished ()

  else

    {pfuns = pfuns, type_map = type_map,

     env = SOME {ctxt = ctxt, defs = defs, types = types, funs = funs,

       pfuns = f pfuns_env, ids = ids, proving = false, vcs = vcs,

       path = path, prefix = prefix}};

fun add_proof_fun prep (s, (optty, raw_t)) thy =

  VCs.map (fn data as {env, ...} =>

    let

      val (optty', prfx, map_pf) = (case env of

          SOME {funs, prefix, ...} =>

            (lookup funs (unprefix_pfun prefix s),

             prefix, map_pfuns_env)

        | NONE => (NONE, "", map_pfuns));

      val optty'' = NONE |> upd_option optty |> upd_option optty';

      val t = prep (Option.map (pfun_type thy prfx) optty'') raw_t;

      val _ = (case fold_aterms (fn u =>

          if is_Var u orelse is_Free u then insert (op =) u else I) t [] of

          [] => ()

        | ts => error ("Term\n" ^ Syntax.string_of_term_global thy t ^

            "\nto be associated with proof function " ^ s ^

            " contains free variable(s) " ^

            commas (map (Syntax.string_of_term_global thy) ts)));

    in

      (check_pfun_type thy prfx s t optty optty';

       if is_some optty'' orelse is_none env then

         map_pf (Symtab.update_new (s, (optty'', t))) data

           handle Symtab.DUP _ => error ("Proof function " ^ s ^

             " already associated with function")

       else error ("Undeclared proof function " ^ s))

    end) thy;

fun check_mapping _ _ [] _ = ()

  | check_mapping err s cmap cs =

      (case duplicates (op = o pairself fst) cmap of

         [] => (case duplicates (op = o pairself snd) cmap of

             [] => (case subtract (op = o apsnd snd) cs cmap of

                 [] => (case subtract (op = o apfst snd) cmap cs of

                     [] => ()

                   | zs => err ("has extra " ^ s ^ "(s) " ^ commas zs))

               | zs => err ("does not have " ^ s ^ "(s) " ^

                   commas (map snd zs)))

           | zs => error ("Several SPARK names are mapped to " ^

               commas (map snd zs)))

       | zs => error ("The SPARK names " ^ commas (map fst zs) ^

           " are mapped to more than one name"));

fun add_type (s, (T as Type (tyname, Ts), cmap)) thy =

      let val cmap' = map (apsnd (Sign.intern_const thy)) cmap

      in

        thy |>

        VCs.map (fn

            {env = SOME _, ...} => err_unfinished ()

          | {pfuns, type_map, env} =>

              {pfuns = pfuns,

               type_map = Symtab.update_new (s, (T, cmap')) type_map,

               env = env}

                handle Symtab.DUP _ => error ("SPARK type " ^ s ^

                  " already associated with type")) |>

        (fn thy' =>

           case Datatype.get_constrs thy' tyname of

             NONE => (case get_record_info thy' T of

               NONE => thy'

             | SOME {fields, ...} =>

                 (check_mapping (assoc_ty_err thy' T s) "field"

                    cmap' (map fst fields);

                  thy'))

           | SOME cs =>

               if null Ts then

                 (map

                    (fn (_, Type (_, [])) => ()

                      | (cname, _) => assoc_ty_err thy' T s

                          ("has illegal constructor " ^

                           Long_Name.base_name cname)) cs;

                  check_mapping (assoc_ty_err thy' T s) "constructor"

                    cmap' (map fst cs);

                  add_enum_type s tyname thy')

               else assoc_ty_err thy' T s "is illegal")

      end

  | add_type (s, (T, _)) thy = assoc_ty_err thy T s "is illegal";

val is_closed = is_none o #env o VCs.get;

fun lookup_vc thy name_concl name =

  (case VCs.get thy of

    {env = SOME {vcs, types, funs, pfuns, ids, ctxt, prefix, ...}, ...} =>

      (case VCtab.lookup vcs name of

         SOME vc =>

           let val (pfuns', ctxt', ids') =

             declare_missing_pfuns thy prefix funs pfuns vcs ids

           in

             SOME (ctxt @ [ctxt'],

               mk_vc thy prefix types pfuns' ids' name_concl vc)

           end

       | NONE => NONE)

  | _ => NONE);

fun get_vcs thy name_concl = (case VCs.get thy of

    {env = SOME {vcs, types, funs, pfuns, ids, ctxt, defs, prefix, ...}, ...} =>

      let val (pfuns', ctxt', ids') =

        declare_missing_pfuns thy prefix funs pfuns vcs ids

      in

        (ctxt @ [ctxt'], defs,

         VCtab.dest vcs |>

         map (apsnd (mk_vc thy prefix types pfuns' ids' name_concl)))

      end

  | _ => ([], [], []));

fun mark_proved name thms = VCs.map (fn

    {pfuns, type_map,

     env = SOME {ctxt, defs, types, funs, pfuns = pfuns_env,

        ids, vcs, path, prefix, ...}} =>

      {pfuns = pfuns,

       type_map = type_map,

       env = SOME {ctxt = ctxt, defs = defs,

         types = types, funs = funs, pfuns = pfuns_env,

         ids = ids,

         proving = true,

         vcs = print_open_vcs insert_break (VCtab.map_entry name

           (fn (trace, _, ps, cs) => (trace, SOME thms, ps, cs)) vcs),

         path = path,

         prefix = prefix}}

  | x => x);

fun close incomplete thy =

  thy |>

  VCs.map (fn

      {pfuns, type_map, env = SOME {vcs, path, ...}} =>

        let

          val (proved, unproved) = partition_vcs vcs;

          val _ = Thm.join_proofs (maps (#2 o snd) proved);

          val (proved', proved'') = List.partition (fn (_, (_, thms, _, _)) =>

            exists (#oracle o Thm.status_of) thms) proved

        in

          (if null unproved then ()

           else (if incomplete then warning else error)

             (Pretty.string_of (pp_open_vcs unproved));

           if null proved' then ()

           else warning (Pretty.string_of (pp_vcs

             "The following VCs are not marked as proved \

             \because their proofs contain oracles:" proved'));

           File.write (Path.ext "prv" path)

             (implode (map (fn (s, _) => snd (strip_number s) ^

                " -- proved by " ^ Distribution.version ^ "\n") proved''));

           {pfuns = pfuns, type_map = type_map, env = NONE})

        end

    | _ => error "No SPARK environment is currently open") |>

  Sign.parent_path;

(** set up verification conditions **)

fun partition_opt f =

  let

    fun part ys zs [] = (rev ys, rev zs)

      | part ys zs (x :: xs) = (case f x of

            SOME y => part (y :: ys) zs xs

          | NONE => part ys (x :: zs) xs)

  in part [] [] end;