(*  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 -> theory -> theory

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

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

   val get_vcs: theory ->

     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: 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 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 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 T => T);

 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 = find_first (curry lcase_eq fname o fst);

 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 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, [])) =>

               (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;

                      val els' =

                        if prfx' = "" then els

                        else map (unprefix (prfx' ^ "__")) els

                          handle Fail _ => error ("Bad enumeration type " ^ s)

                    in

                      case check_enum els' 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 =>

              (case get_record_info thy rT of

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

               | SOME {fields, ...} =>

                   (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)

   | 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 = 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 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 = 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 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 = 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 = map (Long_Name.base_name o fst) fields;

             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 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 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 err_vcs names = error (Pretty.string_of

   (Pretty.big_list "The following verification conditions have not been proved:"

     (map Pretty.str names)))

 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 = 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 (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) =>

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

       thy |>

       VCs.map (fn

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

         | {pfuns, type_map, env} =>

             {pfuns = pfuns,

              type_map = Symtab.update_new (s, T) 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 => 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;

                 add_enum_type s tyname thy')

              else assoc_ty_err thy T s "is illegal")

   | 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 =

   (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' vc) end

        | NONE => NONE)

   | _ => NONE);

 fun get_vcs thy = (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')))

       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 = VCtab.map_entry name (fn (trace, _, ps, cs) =>

            (trace, SOME thms, ps, cs)) vcs,

          path = path,

          prefix = prefix}}

   | x => x);

 fun close thy =

   thy |>

   VCs.map (fn

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

         (case VCtab.fold_rev (fn vc as (_, (_, p, _, _)) =>

              (if is_some p then apfst else apsnd) (cons vc)) vcs ([], []) of

            (proved, []) =>

              (Thm.join_proofs (maps (the o #2 o snd) 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})

          | (_, unproved) => err_vcs (map fst unproved))

     | _ => 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;

 fun dest_def (id, (Substitution_Rule ([], Ident s, rhs))) = SOME (id, (s, rhs))

   | dest_def _ = NONE;

 fun mk_rulename (s, i) = Binding.name (s ^ string_of_int i);

 fun add_const prfx (s, ty) ((tab, ctxt), thy) =

   let

     val T = mk_type thy prfx ty;

     val b = Binding.name s;

     val c = Const (Sign.full_name thy b, T)

   in

     (c,

      ((Symtab.update (s, (c, ty)) tab, Name.declare s ctxt),

       Sign.add_consts_i [(b, T, NoSyn)] thy))

   end;

 fun add_def prfx types pfuns consts (id, (s, e)) (ids as (tab, ctxt), thy) =

   (case lookup consts s of

      SOME ty =>

        let

          val (t, ty') = term_of_expr thy prfx types pfuns ids e;

          val T = mk_type thy prfx ty;

          val T' = mk_type thy prfx ty';

          val _ = T = T' orelse

            error ("Declared type " ^ ty ^ " of " ^ s ^

              "\ndoes not match type " ^ ty' ^ " in definition");

          val id' = mk_rulename id;

          val lthy = Named_Target.theory_init thy;

          val ((t', (_, th)), lthy') = Specification.definition

            (NONE, ((id', []), HOLogic.mk_Trueprop (HOLogic.mk_eq

              (Free (s, T), t)))) lthy;

          val phi = Proof_Context.export_morphism lthy' lthy

        in

          ((id', Morphism.thm phi th),

           ((Symtab.update (s, (Morphism.term phi t', ty)) tab,

             Name.declare s ctxt),

            Local_Theory.exit_global lthy'))

        end

    | NONE => error ("Undeclared constant " ^ s));

 fun add_var prfx (s, ty) (ids, thy) =

   let val ([Free p], ids') = mk_variables thy prfx [s] ty ids

   in (p, (ids', thy)) end;

 fun add_init_vars prfx vcs (ids_thy as ((tab, _), _)) =

   fold_map (add_var prfx)

     (map_filter

        (fn s => case try (unsuffix "~") s of

           SOME s' => (case Symtab.lookup tab s' of

             SOME (_, ty) => SOME (s, ty)

           | NONE => error ("Undeclared identifier " ^ s'))

         | NONE => NONE)

        (fold_vcs (add_expr_idents o snd) vcs []))

     ids_thy;

 fun is_trivial_vc ([], [(_, Ident "true")]) = true

   | is_trivial_vc _ = false;

 fun rulenames rules = commas

   (map (fn ((s, i), _) => s ^ "(" ^ string_of_int i ^ ")") rules);

 (* sort definitions according to their dependency *)

 fun sort_defs _ _ _ _ [] sdefs = rev sdefs

   | sort_defs prfx funs pfuns consts defs sdefs =

       (case find_first (fn (_, (_, e)) =>

          forall (is_some o lookup_prfx prfx pfuns)

            (add_expr_pfuns funs e []) andalso

          forall (fn id =>

            member (fn (s, (_, (s', _))) => s = s') sdefs id orelse

            consts id)

              (add_expr_idents e [])) defs of

          SOME d => sort_defs prfx funs pfuns consts

            (remove (op =) d defs) (d :: sdefs)

        | NONE => error ("Bad definitions: " ^ rulenames defs));

 fun set_vcs ({types, vars, consts, funs} : decls)

       (rules, _) ((_, ident), vcs) path prfx thy =

   let

     val {pfuns, ...} = VCs.get thy;

     val (defs, rules') = partition_opt dest_def rules;

     val consts' =

       subtract (fn ((_, (s, _)), (s', _)) => s = s') defs (items consts);

     (* ignore all complex rules in rls files *)

     val (rules'', other_rules) =

       List.partition (complex_rule o snd) rules';

     val _ = if null rules'' then ()

       else warning ("Ignoring rules: " ^ rulenames rules'');

     val vcs' = VCtab.make (maps (fn (tr, vcs) =>

       map (fn (s, (ps, cs)) => (s, (tr, NONE, ps, cs)))

         (filter_out (is_trivial_vc o snd) vcs)) vcs);

     val _ = (case filter_out (is_some o lookup funs)

         (pfuns_of_vcs prfx funs pfuns vcs') of

         [] => ()

       | fs => error ("Undeclared proof function(s) " ^ commas fs));

     val (((defs', vars''), ivars), (ids, thy')) =

       ((Symtab.empty |>

         Symtab.update ("false", (@{term False}, booleanN)) |>

         Symtab.update ("true", (@{term True}, booleanN)),

         Name.context),

        thy |> Sign.add_path (Long_Name.base_name ident)) |>

       fold (add_type_def prfx) (items types) |>

       fold (snd oo add_const prfx) consts' |> (fn ids_thy as ((tab, _), _) =>

         ids_thy |>

         fold_map (add_def prfx types pfuns consts)

           (sort_defs prfx funs pfuns (Symtab.defined tab) defs []) ||>>

         fold_map (add_var prfx) (items vars) ||>>

         add_init_vars prfx vcs');

     val ctxt =

       [Element.Fixes (map (fn (s, T) =>

          (Binding.name s, SOME T, NoSyn)) (vars'' @ ivars)),

        Element.Assumes (map (fn (id, rl) =>

          ((mk_rulename id, []),

           [(term_of_rule thy' prfx types pfuns ids rl, [])]))

            other_rules),

        Element.Notes ("", [((Binding.name "defns", []), map (rpair [] o single o snd) defs')])]

   in

     set_env ctxt defs' types funs ids vcs' path prfx thy'

   end;

 end;