(*  Title:      src/Tools/isac/BridgeJEdit/Calculation.thy

    Author:     Walther Neuper, JKU Linz

    (c) due to copyright terms

Outer syntax for Isabelle/Isac. Compare src/Pure/Pure.thy

*)

theory Calculation

  imports "~~/src/Tools/isac/MathEngine/MathEngine" "HOL-SPARK.SPARK"

keywords

  "Example" :: thy_load ("str") and (* "spark_vc" :: thy_goal *)

  "Problem" and (* root-problem + recursively in Solution *)

  "Specification" "Model" "References" "Solution" and (* structure only *)

  "Given" "Find" "Relate" "Where" and (* await input of term *)

  "RTheory" (* await input of string; "R" distingues "Problem".."RProblem" *)

  "RProblem" "RMethod" and (* await input of string list *)

  "Tactic" (* optional for each step 0..end of calculation *)

begin

ML_file parseC.sml

section \<open>code for spark_open: cp.to cp_spark_vcs.ML, cp_spark_vcs.ML\<close>

text \<open>

  We minimally change code for Example, until all works.

  The code is separated from Calculation.thy into files cp_spark_vcs.ML and cp_spark_commands.ML, 

  because text\<open>...\<close> and (*...*) for const_name>\<open>...\<close> etc and @ {thms ...} causes errors.

  Thus for Test_Isac (**)ML_file "cp_spark_vcs.ML", ""(*cp_spark_commands.ML*), "HOL-SPARK.SPARK"

  must be outcommented.

\<close>

subsection \<open>cp. HOL/SPARK/Tools/spark_vcs.ML\<close>

 ML \<open>

\<close> ML \<open>

(*  Title:      src/Tools/isac/BridgeJEdit/cp_spark_vcs.ML

    Author:     Walther Neuper, JKU Linz

    (c) due to copyright terms

Preliminary code for developing Outer_Syntax.command..Example from spark_open as a model.

*)

(** theory data **)

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

val strip_number = apply2 implode o chop_suffix Fdl_Lexer.is_digit o raw_explode;

val name_ord = prod_ord string_ord (option_ord int_ord) o

  apply2 (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_Parser.fdl_type Fdl_Parser.tab,

        funs: (string list * string) Fdl_Parser.tab,

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

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

        proving: bool,

        vcs: (string * thm list option *

          (string * Fdl_Parser.expr) list * (string * Fdl_Parser.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;

(** 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, (Fdl_Parser.Substitution_Rule ([], Fdl_Parser.Ident s, rhs))) = SOME (id, (s, rhs))

  | dest_def _ = NONE;

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

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

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

fun complex_expr (Fdl_Parser.Number _) = false

  | complex_expr (Fdl_Parser.Ident _) = false

  | complex_expr (Fdl_Parser.Funct (s, es)) =

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

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

  | complex_expr _ = true;

fun complex_rule (Fdl_Parser.Inference_Rule (es, e)) =

      complex_expr e orelse exists complex_expr es

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

      complex_expr e orelse complex_expr e' orelse

      exists complex_expr es;

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

  | is_trivial_vc _ = false;

fun rulenames rules = commas

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

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

  | fold_expr f g (Fdl_Parser.Ident s) = g s

  | fold_expr f g (Fdl_Parser.Number _) = I

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

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

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

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

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

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

  | fold_expr f g (Fdl_Parser.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 (Fdl_Parser.lookup funs s) then I else insert (op =) s);

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 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);

val booleanN = "boolean";

val integerN = "integer";

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);

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 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;

(** generate properties of enumeration types **)

fun add_enum_type tyname tyname' thy =

  let

    val {case_name, ...} = the (BNF_LFP_Compat.get_info thy [BNF_LFP_Compat.Keep_Nesting] tyname');

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

    val k = length cs;

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

    val p = Const (\<^const_name>\<open>pos\<close>, T --> HOLogic.intT);

    val v = Const (\<^const_name>\<open>val\<close>, HOLogic.intT --> T);

    val card = Const (\<^const_name>\<open>card\<close>,

      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>\<open>spark_enum\<close>) |>

      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>\<open>less_eq\<close> p) ||>>

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

        mk_binrel_def \<^const_name>\<open>less\<close> p);

    val UNIV_eq = Goal.prove lthy [] []

      (HOLogic.mk_Trueprop (HOLogic.mk_eq

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

      (fn {context = ctxt, ...} =>

         resolve_tac ctxt @{thms subset_antisym} 1 THEN

         resolve_tac ctxt @{thms subsetI} 1 THEN

         Old_Datatype_Aux.exh_tac ctxt (K (#exhaust (BNF_LFP_Compat.the_info

           (Proof_Context.theory_of ctxt) [BNF_LFP_Compat.Keep_Nesting] tyname'))) 1 THEN

         ALLGOALS (asm_full_simp_tac ctxt));

    val finite_UNIV = Goal.prove lthy [] []

      (HOLogic.mk_Trueprop (Const (\<^const_name>\<open>finite\<close>,

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

      (fn {context = ctxt, ...} => simp_tac (ctxt addsimps [UNIV_eq]) 1);

    val card_UNIV = Goal.prove lthy [] []

      (HOLogic.mk_Trueprop (HOLogic.mk_eq

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

      (fn {context = ctxt, ...} => simp_tac (ctxt addsimps [UNIV_eq]) 1);

    val range_pos = Goal.prove lthy [] []

      (HOLogic.mk_Trueprop (HOLogic.mk_eq

         (Const (\<^const_name>\<open>image\<close>, (T --> HOLogic.intT) -->

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

              p $ HOLogic.mk_UNIV T,

          Const (\<^const_name>\<open>atLeastLessThan\<close>, HOLogic.intT -->

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

              HOLogic.mk_number HOLogic.intT 0 $

              (\<^term>\<open>int\<close> $ card))))

      (fn {context = ctxt, ...} =>

         simp_tac (ctxt addsimps [card_UNIV]) 1 THEN

         simp_tac (ctxt addsimps [UNIV_eq, def1]) 1 THEN

         resolve_tac ctxt @{thms subset_antisym} 1 THEN

         simp_tac ctxt 1 THEN

         resolve_tac ctxt @{thms subsetI} 1 THEN

         asm_full_simp_tac (ctxt addsimps @{thms interval_expand}

           delsimps @{thms atLeastLessThan_iff}) 1);

    val lthy' =

      Class.prove_instantiation_instance (fn ctxt =>

        Class.intro_classes_tac ctxt [] THEN

        resolve_tac ctxt [finite_UNIV] 1 THEN

        resolve_tac ctxt [range_pos] 1 THEN

        simp_tac (put_simpset HOL_basic_ss ctxt addsimps [def3]) 1 THEN

        simp_tac (put_simpset HOL_basic_ss ctxt 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 {context = ctxt, ...} => simp_tac (ctxt addsimps [def1]) 1);

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

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

          (fn {context = ctxt, ...} =>

             resolve_tac ctxt [@{thm inj_pos} RS @{thm injD}] 1 THEN

             simp_tac (ctxt 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>\<open>first_el\<close>, T), hd cs)))

      (fn {context = ctxt, ...} => simp_tac (ctxt addsimps [@{thm first_el_def}, hd val_eqs]) 1);

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

      (HOLogic.mk_Trueprop (HOLogic.mk_eq

         (Const (\<^const_name>\<open>last_el\<close>, T), List.last cs)))

      (fn {context = ctxt, ...} =>

        simp_tac (ctxt 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;

val lcase_eq = (op =) o apply2 (to_lower o Long_Name.base_name);

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 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 assoc_ty_err thy T s msg =

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

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

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 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 invert_map [] = I

  | invert_map cmap =

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

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

    [(tyname, [\<^typ>\<open>unit\<close>])] =>

      Record.get_info thy (Long_Name.qualifier tyname)

  | _ => NONE);

fun add_type_def prfx (s, Fdl_Parser.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, Fdl_Parser.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 |>

                 BNF_LFP_Compat.add_datatype [BNF_LFP_Compat.Keep_Nesting]

                   [((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 BNF_LFP_Compat.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 (BNF_LFP_Compat.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, Fdl_Parser.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, Fdl_Parser.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 {overloaded = false} ([], 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 apply2 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, Fdl_Parser.Pending_Type) (ids, thy) =

      (ids,

       case get_type thy prfx s of

         SOME _ => thy

       | NONE => Typedecl.typedecl_global {final = true} (Binding.name s, [], NoSyn) thy |> snd);

fun add_type_def prfx (s, Fdl_Parser.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, Fdl_Parser.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 |>

                 BNF_LFP_Compat.add_datatype [BNF_LFP_Compat.Keep_Nesting]

                   [((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 BNF_LFP_Compat.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 (BNF_LFP_Compat.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, Fdl_Parser.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, Fdl_Parser.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 {overloaded = false} ([], 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 apply2 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, Fdl_Parser.Pending_Type) (ids, thy) =

      (ids,

       case get_type thy prfx s of

         SOME _ => thy

       | NONE => Typedecl.typedecl_global {final = true} (Binding.name s, [], NoSyn) thy |> snd);

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 [(b, T, NoSyn)] thy))

  end;

fun mk_unop s t =

  let val T = fastype_of t

  in Const (s, T --> T) $ t 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 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 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>\<open>Sigma\<close>, setT --> (T --> HOLogic.mk_setT U) -->

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

  end;

fun term_of_expr thy prfx types pfuns =

  let

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      | tm_of vs (Fdl_Parser.Funct ("<", [e, e'])) = (HOLogic.mk_binrel \<^const_name>\<open>less\<close>

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

      | tm_of vs (Fdl_Parser.Funct (">", [e, e'])) = (HOLogic.mk_binrel \<^const_name>\<open>less\<close>

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

      | tm_of vs (Fdl_Parser.Funct ("<=", [e, e'])) = (HOLogic.mk_binrel \<^const_name>\<open>less_eq\<close>

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

      | tm_of vs (Fdl_Parser.Funct (">=", [e, e'])) = (HOLogic.mk_binrel \<^const_name>\<open>less_eq\<close>

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

      | tm_of vs (Fdl_Parser.Funct ("+", [e, e'])) = (HOLogic.mk_binop \<^const_name>\<open>plus\<close>

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

      | tm_of vs (Fdl_Parser.Funct ("-", [e, e'])) = (HOLogic.mk_binop \<^const_name>\<open>minus\<close>

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

      | tm_of vs (Fdl_Parser.Funct ("*", [e, e'])) = (HOLogic.mk_binop \<^const_name>\<open>times\<close>

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

      | tm_of vs (Fdl_Parser.Funct ("/", [e, e'])) = (HOLogic.mk_binop \<^const_name>\<open>divide\<close>

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

      | tm_of vs (Fdl_Parser.Funct ("div", [e, e'])) = (HOLogic.mk_binop \<^const_name>\<open>sdiv\<close>

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

      | tm_of vs (Fdl_Parser.Funct ("mod", [e, e'])) = (HOLogic.mk_binop \<^const_name>\<open>modulo\<close>

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

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

          (mk_unop \<^const_name>\<open>uminus\<close> (fst (tm_of vs e)), integerN)

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

          (Const (\<^const_name>\<open>power\<close>, HOLogic.intT --> HOLogic.natT -->

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

               (\<^const>\<open>nat\<close> $ fst (tm_of vs e')), integerN)

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

      | tm_of vs (Fdl_Parser.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 (Fdl_Parser.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, Fdl_Parser.lookup types rcdty) of

                     (SOME {fields, ...}, SOME (Fdl_Parser.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>\<open>pos\<close>,

                   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>\<open>val\<close>,

                   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>\<open>succ\<close>

                   else \<^const_name>\<open>pred\<close>, 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 (Fdl_Parser.Element (e, es)) =

          let val (t, ty) = tm_of vs e

          in case Fdl_Parser.lookup types ty of

              SOME (Fdl_Parser.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 (Fdl_Parser.Update (e, es, e')) =

          let val (t, ty) = tm_of vs e

          in case Fdl_Parser.lookup types ty of

              SOME (Fdl_Parser.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>\<open>fun_upd\<close>, 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 (Fdl_Parser.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 (Fdl_Parser.Array (s, default, assocs)) =

          (case Fdl_Parser.lookup types s of

             SOME (Fdl_Parser.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>\<open>atLeastAtMost\<close>,

                       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>\<open>fun_upd\<close>, (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>\<open>fun_upds\<close>, (T --> U) -->

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

                       foldl1 (HOLogic.mk_binop \<^const_name>\<open>sup\<close>)

                         (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>\<open>undefined\<close>, T --> U)),

                  s)

               end

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

  in tm_of end;

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

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

  (case Fdl_Parser.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 ((t', (_, th)), lthy') = Named_Target.theory_init thy

          |> Specification.definition NONE [] []

            ((id', []), HOLogic.mk_Trueprop (HOLogic.mk_eq (Free (s, T), t)));

        val phi =

          Proof_Context.export_morphism lthy'

            (Proof_Context.init_global (Proof_Context.theory_of 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));

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

(* 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 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 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

      Fdl_Parser.Inference_Rule (es, e) => Logic.list_implies

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

    | Fdl_Parser.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;

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' = Fdl_Parser.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);

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 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;

\<close> ML \<open>

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

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

  let

    val prfx' =