(*  Title:      Pure/Proof/extraction.ML

     ID:         $Id$

     Author:     Stefan Berghofer, TU Muenchen

 Extraction of programs from proofs.

 *)

 signature EXTRACTION =

 sig

   val set_preprocessor : (theory -> Proofterm.proof -> Proofterm.proof) -> theory -> theory

   val add_realizes_eqns_i : ((term * term) list * (term * term)) list -> theory -> theory

   val add_realizes_eqns : string list -> theory -> theory

   val add_typeof_eqns_i : ((term * term) list * (term * term)) list -> theory -> theory

   val add_typeof_eqns : string list -> theory -> theory

   val add_realizers_i : (string * (string list * term * Proofterm.proof)) list

     -> theory -> theory

   val add_realizers : (thm * (string list * string * string)) list

     -> theory -> theory

   val add_expand_thms : thm list -> theory -> theory

   val add_types : (xstring * ((term -> term option) list *

     (term -> typ -> term -> typ -> term) option)) list -> theory -> theory

   val extract : (thm * string list) list -> theory -> theory

   val nullT : typ

   val nullt : term

   val mk_typ : typ -> term

   val etype_of : theory -> string list -> typ list -> term -> typ

   val realizes_of: theory -> string list -> term -> term -> term

 end;

 structure Extraction : EXTRACTION =

 struct

 open Proofterm;

 (**** tools ****)

 fun add_syntax thy =

   thy

   |> Theory.copy

   |> Sign.root_path

   |> Sign.add_types [("Type", 0, NoSyn), ("Null", 0, NoSyn)]

   |> Sign.add_consts

       [("typeof", "'b::{} => Type", NoSyn),

        ("Type", "'a::{} itself => Type", NoSyn),

        ("Null", "Null", NoSyn),

        ("realizes", "'a::{} => 'b::{} => 'b", NoSyn)];

 val nullT = Type ("Null", []);

 val nullt = Const ("Null", nullT);

 fun mk_typ T =

   Const ("Type", Term.itselfT T --> Type ("Type", [])) $ Logic.mk_type T;

 fun typeof_proc defaultS vs (Const ("typeof", _) $ u) =

       SOME (mk_typ (case strip_comb u of

           (Var ((a, i), _), _) =>

             if member (op =) vs a then TFree ("'" ^ a ^ ":" ^ string_of_int i, defaultS)

             else nullT

         | (Free (a, _), _) =>

             if member (op =) vs a then TFree ("'" ^ a, defaultS) else nullT

         | _ => nullT))

   | typeof_proc _ _ _ = NONE;

 fun rlz_proc (Const ("realizes", Type (_, [Type ("Null", []), _])) $ r $ t) = SOME t

   | rlz_proc (Const ("realizes", Type (_, [T, _])) $ r $ t) =

       (case strip_comb t of

          (Var (ixn, U), ts) => SOME (list_comb (Var (ixn, T --> U), r :: ts))

        | (Free (s, U), ts) => SOME (list_comb (Free (s, T --> U), r :: ts))

        | _ => NONE)

   | rlz_proc _ = NONE;

 val unpack_ixn = apfst implode o apsnd (fst o read_int o tl) o

   take_prefix (not o equal ":") o explode;

 type rules =

   {next: int, rs: ((term * term) list * (term * term)) list,

    net: (int * ((term * term) list * (term * term))) Net.net};

 val empty_rules : rules = {next = 0, rs = [], net = Net.empty};

 fun add_rule (r as (_, (lhs, _)), {next, rs, net} : rules) =

   {next = next - 1, rs = r :: rs, net = Net.insert_term (K false)

      (Envir.eta_contract lhs, (next, r)) net};

 fun merge_rules

   ({next, rs = rs1, net} : rules) ({next = next2, rs = rs2, ...} : rules) =

   List.foldr add_rule {next = next, rs = rs1, net = net} (subtract (op =) rs1 rs2);

 fun condrew thy rules procs =

   let

     fun rew tm =

       Pattern.rewrite_term thy [] (condrew' :: procs) tm

     and condrew' tm =

       let

         val cache = ref ([] : (term * term) list);

         fun lookup f x = (case AList.lookup (op =) (!cache) x of

             NONE =>

               let val y = f x

               in (cache := (x, y) :: !cache; y) end

           | SOME y => y);

       in

         get_first (fn (_, (prems, (tm1, tm2))) =>

         let

           fun ren t = the_default t (Term.rename_abs tm1 tm t);

           val inc = Logic.incr_indexes ([], maxidx_of_term tm + 1);

           val env as (Tenv, tenv) = Pattern.match thy (inc tm1, tm) (Vartab.empty, Vartab.empty);

           val prems' = map (pairself (Envir.subst_vars env o inc o ren)) prems;

           val env' = Envir.Envir

             {maxidx = Library.foldl Int.max

               (~1, map (Int.max o pairself maxidx_of_term) prems'),

              iTs = Tenv, asol = tenv};

           val env'' = fold (Pattern.unify thy o pairself (lookup rew)) prems' env';

         in SOME (Envir.norm_term env'' (inc (ren tm2)))

         end handle Pattern.MATCH => NONE | Pattern.Unif => NONE)

           (sort (int_ord o pairself fst)

             (Net.match_term rules (Envir.eta_contract tm)))

       end;

   in rew end;

 val chtype = change_type o SOME;

 fun extr_name s vs = NameSpace.append "extr" (space_implode "_" (s :: vs));

 fun corr_name s vs = extr_name s vs ^ "_correctness";

 fun msg d s = priority (Symbol.spaces d ^ s);

 fun vars_of t = rev (fold_aterms (fn v as Var _ => insert (op =) v | _ => I) t []);

 fun vfs_of t = vars_of t @ sort Term.term_ord (term_frees t);

 fun forall_intr_prf (t, prf) =

   let val (a, T) = (case t of Var ((a, _), T) => (a, T) | Free p => p)

   in Abst (a, SOME T, prf_abstract_over t prf) end;

 val mkabs = List.foldr (fn (v, t) => Abs ("x", fastype_of v, abstract_over (v, t)));

 fun strip_abs 0 t = t

   | strip_abs n (Abs (_, _, t)) = strip_abs (n-1) t

   | strip_abs _ _ = error "strip_abs: not an abstraction";

 fun prf_subst_TVars tye =

   map_proof_terms (subst_TVars tye) (typ_subst_TVars tye);

 fun relevant_vars types prop = List.foldr (fn

       (Var ((a, i), T), vs) => (case strip_type T of

         (_, Type (s, _)) => if member (op =) types s then a :: vs else vs

       | _ => vs)

     | (_, vs) => vs) [] (vars_of prop);

 fun tname_of (Type (s, _)) = s

   | tname_of _ = "";

 fun get_var_type t =

   let

     val vs = Term.add_vars t [];

     val fs = Term.add_frees t [];

   in fn 

       Var (ixn, _) => (case AList.lookup (op =) vs ixn of

           NONE => error "get_var_type: no such variable in term"

         | SOME T => Var (ixn, T))

     | Free (s, _) => (case AList.lookup (op =) fs s of

           NONE => error "get_var_type: no such variable in term"

         | SOME T => Free (s, T))

     | _ => error "get_var_type: not a variable"

   end;

 (**** theory data ****)

 (* theory data *)

 structure ExtractionData = TheoryDataFun

 (

   type T =

     {realizes_eqns : rules,

      typeof_eqns : rules,

      types : (string * ((term -> term option) list *

        (term -> typ -> term -> typ -> term) option)) list,

      realizers : (string list * (term * proof)) list Symtab.table,

      defs : thm list,

      expand : (string * term) list,

      prep : (theory -> proof -> proof) option}

   val empty =

     {realizes_eqns = empty_rules,

      typeof_eqns = empty_rules,

      types = [],

      realizers = Symtab.empty,

      defs = [],

      expand = [],

      prep = NONE};

   val copy = I;

   val extend = I;

   fun merge _

     (({realizes_eqns = realizes_eqns1, typeof_eqns = typeof_eqns1, types = types1,

        realizers = realizers1, defs = defs1, expand = expand1, prep = prep1},

       {realizes_eqns = realizes_eqns2, typeof_eqns = typeof_eqns2, types = types2,

        realizers = realizers2, defs = defs2, expand = expand2, prep = prep2}) : T * T) =

     {realizes_eqns = merge_rules realizes_eqns1 realizes_eqns2,

      typeof_eqns = merge_rules typeof_eqns1 typeof_eqns2,

      types = AList.merge (op =) (K true) (types1, types2),

      realizers = Symtab.merge_list (gen_eq_set (op =) o pairself #1) (realizers1, realizers2),

      defs = Library.merge Thm.eq_thm (defs1, defs2),

      expand = Library.merge (op =) (expand1, expand2),

      prep = (case prep1 of NONE => prep2 | _ => prep1)};

 );

 fun read_condeq thy =

   let val thy' = add_syntax thy

   in fn s =>

     let val t = Logic.varify (Syntax.read_prop_global thy' s)

     in (map Logic.dest_equals (Logic.strip_imp_prems t),

       Logic.dest_equals (Logic.strip_imp_concl t))

     end handle TERM _ => error ("Not a (conditional) meta equality:\n" ^ s)

   end;

 (** preprocessor **)

 fun set_preprocessor prep thy =

   let val {realizes_eqns, typeof_eqns, types, realizers,

     defs, expand, ...} = ExtractionData.get thy

   in

     ExtractionData.put

       {realizes_eqns = realizes_eqns, typeof_eqns = typeof_eqns, types = types,

        realizers = realizers, defs = defs, expand = expand, prep = SOME prep} thy

   end;

 (** equations characterizing realizability **)

 fun gen_add_realizes_eqns prep_eq eqns thy =

   let val {realizes_eqns, typeof_eqns, types, realizers,

     defs, expand, prep} = ExtractionData.get thy;

   in

     ExtractionData.put

       {realizes_eqns = List.foldr add_rule realizes_eqns (map (prep_eq thy) eqns),

        typeof_eqns = typeof_eqns, types = types, realizers = realizers,

        defs = defs, expand = expand, prep = prep} thy

   end

 val add_realizes_eqns_i = gen_add_realizes_eqns (K I);

 val add_realizes_eqns = gen_add_realizes_eqns read_condeq;

 (** equations characterizing type of extracted program **)

 fun gen_add_typeof_eqns prep_eq eqns thy =

   let

     val {realizes_eqns, typeof_eqns, types, realizers,

       defs, expand, prep} = ExtractionData.get thy;

     val eqns' = map (prep_eq thy) eqns

   in

     ExtractionData.put

       {realizes_eqns = realizes_eqns, realizers = realizers,

        typeof_eqns = List.foldr add_rule typeof_eqns eqns',

        types = types, defs = defs, expand = expand, prep = prep} thy

   end

 val add_typeof_eqns_i = gen_add_typeof_eqns (K I);

 val add_typeof_eqns = gen_add_typeof_eqns read_condeq;

 fun thaw (T as TFree (a, S)) =

       if exists_string (equal ":") a then TVar (unpack_ixn a, S) else T

   | thaw (Type (a, Ts)) = Type (a, map thaw Ts)

   | thaw T = T;

 fun freeze (TVar ((a, i), S)) = TFree (a ^ ":" ^ string_of_int i, S)

   | freeze (Type (a, Ts)) = Type (a, map freeze Ts)

   | freeze T = T;

 fun freeze_thaw f x =

   map_types thaw (f (map_types freeze x));

 fun etype_of thy vs Ts t =

   let

     val {typeof_eqns, ...} = ExtractionData.get thy;

     fun err () = error ("Unable to determine type of extracted program for\n" ^

       Syntax.string_of_term_global thy t)

   in case strip_abs_body (freeze_thaw (condrew thy (#net typeof_eqns)

     [typeof_proc (Sign.defaultS thy) vs]) (list_abs (map (pair "x") (rev Ts),

       Const ("typeof", fastype_of1 (Ts, t) --> Type ("Type", [])) $ t))) of

       Const ("Type", _) $ u => (Logic.dest_type u handle TERM _ => err ())

     | _ => err ()

   end;

 (** realizers for axioms / theorems, together with correctness proofs **)

 fun gen_add_realizers prep_rlz rs thy =

   let val {realizes_eqns, typeof_eqns, types, realizers,

     defs, expand, prep} = ExtractionData.get thy

   in

     ExtractionData.put

       {realizes_eqns = realizes_eqns, typeof_eqns = typeof_eqns, types = types,

        realizers = fold (Symtab.cons_list o prep_rlz thy) rs realizers,

        defs = defs, expand = expand, prep = prep} thy

   end

 fun prep_realizer thy =

   let

     val {realizes_eqns, typeof_eqns, defs, types, ...} =

       ExtractionData.get thy;

     val procs = maps (fst o snd) types;

     val rtypes = map fst types;

     val eqns = Net.merge (K false) (#net realizes_eqns, #net typeof_eqns);

     val thy' = add_syntax thy;

     val rd = ProofSyntax.read_proof thy' false

   in fn (thm, (vs, s1, s2)) =>

     let

       val name = Thm.get_name thm;

       val _ = name <> "" orelse error "add_realizers: unnamed theorem";

       val prop = Pattern.rewrite_term thy'

         (map (Logic.dest_equals o prop_of) defs) [] (prop_of thm);

       val vars = vars_of prop;

       val vars' = filter_out (fn v =>

         member (op =) rtypes (tname_of (body_type (fastype_of v)))) vars;

       val T = etype_of thy' vs [] prop;

       val (T', thw) = Type.freeze_thaw_type

         (if T = nullT then nullT else map fastype_of vars' ---> T);

       val t = map_types thw (OldGoals.simple_read_term thy' T' s1);

       val r' = freeze_thaw (condrew thy' eqns

         (procs @ [typeof_proc (Sign.defaultS thy') vs, rlz_proc]))

           (Const ("realizes", T --> propT --> propT) $

             (if T = nullT then t else list_comb (t, vars')) $ prop);

       val r = fold_rev Logic.all (map (get_var_type r') vars) r';

       val prf = Reconstruct.reconstruct_proof thy' r (rd s2);

     in (name, (vs, (t, prf))) end

   end;

 val add_realizers_i = gen_add_realizers

   (fn _ => fn (name, (vs, t, prf)) => (name, (vs, (t, prf))));

 val add_realizers = gen_add_realizers prep_realizer;

 fun realizes_of thy vs t prop =

   let

     val thy' = add_syntax thy;

     val {realizes_eqns, typeof_eqns, defs, types, ...} =

       ExtractionData.get thy';

     val procs = maps (rev o fst o snd) types;

     val eqns = Net.merge (K false) (#net realizes_eqns, #net typeof_eqns);

     val prop' = Pattern.rewrite_term thy'

       (map (Logic.dest_equals o prop_of) defs) [] prop;

   in freeze_thaw (condrew thy' eqns

     (procs @ [typeof_proc (Sign.defaultS thy') vs, rlz_proc]))

       (Const ("realizes", fastype_of t --> propT --> propT) $ t $ prop')

   end;

 (** expanding theorems / definitions **)

 fun add_expand_thm thm thy =

   let

     val {realizes_eqns, typeof_eqns, types, realizers,

       defs, expand, prep} = ExtractionData.get thy;

     val name = Thm.get_name thm;

     val _ = name <> "" orelse error "add_expand_thms: unnamed theorem";

     val is_def =

       (case strip_comb (fst (Logic.dest_equals (prop_of thm))) of

          (Const _, ts) => forall is_Var ts andalso not (has_duplicates (op =) ts)

            andalso (PureThy.get_kind thm = Thm.definitionK orelse can (Thm.get_axiom_i thy) name)

        | _ => false) handle TERM _ => false;

   in

     (ExtractionData.put (if is_def then

         {realizes_eqns = realizes_eqns,

          typeof_eqns = add_rule (([],

            Logic.dest_equals (prop_of (Drule.abs_def thm))), typeof_eqns),

          types = types,

          realizers = realizers, defs = insert Thm.eq_thm thm defs,

          expand = expand, prep = prep}

       else

         {realizes_eqns = realizes_eqns, typeof_eqns = typeof_eqns, types = types,

          realizers = realizers, defs = defs,

          expand = insert (op =) (name, prop_of thm) expand, prep = prep}) thy)

   end;

 val add_expand_thms = fold add_expand_thm;

 val extraction_expand =

   Attrib.no_args (Thm.declaration_attribute (fn th => Context.mapping (add_expand_thm th) I));

 (** types with computational content **)

 fun add_types tys thy =

   ExtractionData.map

     (fn {realizes_eqns, typeof_eqns, types, realizers, defs, expand, prep} =>

       {realizes_eqns = realizes_eqns, typeof_eqns = typeof_eqns,

        types = fold (AList.update (op =) o apfst (Sign.intern_type thy)) tys types,

        realizers = realizers, defs = defs, expand = expand, prep = prep})

     thy;

 (** Pure setup **)

 val _ = Context.>> (Context.map_theory

   (add_types [("prop", ([], NONE))] #>

    add_typeof_eqns

      ["(typeof (PROP P)) == (Type (TYPE(Null))) ==>  \

     \  (typeof (PROP Q)) == (Type (TYPE('Q))) ==>  \

     \    (typeof (PROP P ==> PROP Q)) == (Type (TYPE('Q)))",

       "(typeof (PROP Q)) == (Type (TYPE(Null))) ==>  \

     \    (typeof (PROP P ==> PROP Q)) == (Type (TYPE(Null)))",

       "(typeof (PROP P)) == (Type (TYPE('P))) ==>  \

     \  (typeof (PROP Q)) == (Type (TYPE('Q))) ==>  \

     \    (typeof (PROP P ==> PROP Q)) == (Type (TYPE('P => 'Q)))",

       "(%x. typeof (PROP P (x))) == (%x. Type (TYPE(Null))) ==>  \

     \    (typeof (!!x. PROP P (x))) == (Type (TYPE(Null)))",

       "(%x. typeof (PROP P (x))) == (%x. Type (TYPE('P))) ==>  \

     \    (typeof (!!x::'a. PROP P (x))) == (Type (TYPE('a => 'P)))",

       "(%x. typeof (f (x))) == (%x. Type (TYPE('f))) ==>  \

     \    (typeof (f)) == (Type (TYPE('f)))"] #>

    add_realizes_eqns

      ["(typeof (PROP P)) == (Type (TYPE(Null))) ==>  \

     \    (realizes (r) (PROP P ==> PROP Q)) ==  \

     \    (PROP realizes (Null) (PROP P) ==> PROP realizes (r) (PROP Q))",

       "(typeof (PROP P)) == (Type (TYPE('P))) ==>  \

     \  (typeof (PROP Q)) == (Type (TYPE(Null))) ==>  \

     \    (realizes (r) (PROP P ==> PROP Q)) ==  \

     \    (!!x::'P. PROP realizes (x) (PROP P) ==> PROP realizes (Null) (PROP Q))",

       "(realizes (r) (PROP P ==> PROP Q)) ==  \

     \  (!!x. PROP realizes (x) (PROP P) ==> PROP realizes (r (x)) (PROP Q))",

       "(%x. typeof (PROP P (x))) == (%x. Type (TYPE(Null))) ==>  \

     \    (realizes (r) (!!x. PROP P (x))) ==  \

     \    (!!x. PROP realizes (Null) (PROP P (x)))",

       "(realizes (r) (!!x. PROP P (x))) ==  \

     \  (!!x. PROP realizes (r (x)) (PROP P (x)))"] #>

    Attrib.add_attributes

      [("extraction_expand", extraction_expand,

        "specify theorems / definitions to be expanded during extraction")]));

 (**** extract program ****)

 val dummyt = Const ("dummy", dummyT);

 fun extract thms thy =

   let

     val thy' = add_syntax thy;

     val {realizes_eqns, typeof_eqns, types, realizers, defs, expand, prep} =

       ExtractionData.get thy;

     val procs = maps (rev o fst o snd) types;

     val rtypes = map fst types;

     val typroc = typeof_proc (Sign.defaultS thy');

     val prep = the_default (K I) prep thy' o ProofRewriteRules.elim_defs thy' false defs o

       Reconstruct.expand_proof thy' (("", NONE) :: map (apsnd SOME) expand);

     val rrews = Net.merge (K false) (#net realizes_eqns, #net typeof_eqns);

     fun find_inst prop Ts ts vs =

       let

         val rvs = relevant_vars rtypes prop;

         val vars = vars_of prop;

         val n = Int.min (length vars, length ts);

         fun add_args ((Var ((a, i), _), t), (vs', tye)) =

           if member (op =) rvs a then

             let val T = etype_of thy' vs Ts t

             in if T = nullT then (vs', tye)

                else (a :: vs', (("'" ^ a, i), T) :: tye)

             end

           else (vs', tye)

       in List.foldr add_args ([], []) (Library.take (n, vars) ~~ Library.take (n, ts)) end;

     fun find (vs: string list) = Option.map snd o find_first (curry (gen_eq_set (op =)) vs o fst);

     fun find' s = map snd o List.filter (equal s o fst)

     fun app_rlz_rews Ts vs t = strip_abs (length Ts) (freeze_thaw

       (condrew thy' rrews (procs @ [typroc vs, rlz_proc])) (list_abs

         (map (pair "x") (rev Ts), t)));

     fun realizes_null vs prop = app_rlz_rews [] vs

       (Const ("realizes", nullT --> propT --> propT) $ nullt $ prop);

     fun corr d defs vs ts Ts hs (PBound i) _ _ = (defs, PBound i)

       | corr d defs vs ts Ts hs (Abst (s, SOME T, prf)) (Abst (_, _, prf')) t =

           let val (defs', corr_prf) = corr d defs vs [] (T :: Ts)

             (dummyt :: hs) prf (incr_pboundvars 1 0 prf')

             (case t of SOME (Abs (_, _, u)) => SOME u | _ => NONE)

           in (defs', Abst (s, SOME T, corr_prf)) end

       | corr d defs vs ts Ts hs (AbsP (s, SOME prop, prf)) (AbsP (_, _, prf')) t =

           let

             val T = etype_of thy' vs Ts prop;

             val u = if T = nullT then 

                 (case t of SOME u => SOME (incr_boundvars 1 u) | NONE => NONE)

               else (case t of SOME (Abs (_, _, u)) => SOME u | _ => NONE);

             val (defs', corr_prf) = corr d defs vs [] (T :: Ts) (prop :: hs)

               (incr_pboundvars 0 1 prf) (incr_pboundvars 0 1 prf') u;

             val rlz = Const ("realizes", T --> propT --> propT)

           in (defs',

             if T = nullT then AbsP ("R",

               SOME (app_rlz_rews Ts vs (rlz $ nullt $ prop)),

                 prf_subst_bounds [nullt] corr_prf)

             else Abst (s, SOME T, AbsP ("R",

               SOME (app_rlz_rews (T :: Ts) vs

                 (rlz $ Bound 0 $ incr_boundvars 1 prop)), corr_prf)))

           end

       | corr d defs vs ts Ts hs (prf % SOME t) (prf' % _) t' =

           let

             val (Us, T) = strip_type (fastype_of1 (Ts, t));

             val (defs', corr_prf) = corr d defs vs (t :: ts) Ts hs prf prf'

               (if member (op =) rtypes (tname_of T) then t'

                else (case t' of SOME (u $ _) => SOME u | _ => NONE));

             val u = if not (member (op =) rtypes (tname_of T)) then t else

               let

                 val eT = etype_of thy' vs Ts t;

                 val (r, Us') = if eT = nullT then (nullt, Us) else

                   (Bound (length Us), eT :: Us);

                 val u = list_comb (incr_boundvars (length Us') t,

                   map Bound (length Us - 1 downto 0));

                 val u' = (case AList.lookup (op =) types (tname_of T) of

                     SOME ((_, SOME f)) => f r eT u T

                   | _ => Const ("realizes", eT --> T --> T) $ r $ u)

               in app_rlz_rews Ts vs (list_abs (map (pair "x") Us', u')) end

           in (defs', corr_prf % SOME u) end

       | corr d defs vs ts Ts hs (prf1 %% prf2) (prf1' %% prf2') t =

           let

             val prop = Reconstruct.prop_of' hs prf2';

             val T = etype_of thy' vs Ts prop;

             val (defs1, f, u) = if T = nullT then (defs, t, NONE) else

               (case t of

                  SOME (f $ u) => (defs, SOME f, SOME u)

                | _ =>

                  let val (defs1, u) = extr d defs vs [] Ts hs prf2'

                  in (defs1, NONE, SOME u) end)

             val (defs2, corr_prf1) = corr d defs1 vs [] Ts hs prf1 prf1' f;

             val (defs3, corr_prf2) = corr d defs2 vs [] Ts hs prf2 prf2' u;

           in

             if T = nullT then (defs3, corr_prf1 %% corr_prf2) else

               (defs3, corr_prf1 % u %% corr_prf2)

           end

       | corr d defs vs ts Ts hs (prf0 as PThm (name, prf, prop, SOME Ts')) _ _ =

           let

             val (vs', tye) = find_inst prop Ts ts vs;

             val tye' = (map fst (term_tvars prop) ~~ Ts') @ tye;

             val T = etype_of thy' vs' [] prop;

             val defs' = if T = nullT then defs

               else fst (extr d defs vs ts Ts hs prf0)

           in

             if T = nullT andalso realizes_null vs' prop aconv prop then (defs, prf0)

             else case Symtab.lookup realizers name of

               NONE => (case find vs' (find' name defs') of

                 NONE =>

                   let

                     val _ = T = nullT orelse error "corr: internal error";

                     val _ = msg d ("Building correctness proof for " ^ quote name ^

                       (if null vs' then ""

                        else " (relevant variables: " ^ commas_quote vs' ^ ")"));

                     val prf' = prep (Reconstruct.reconstruct_proof thy' prop prf);

                     val (defs'', corr_prf) =

                       corr (d + 1) defs' vs' [] [] [] prf' prf' NONE;

                     val corr_prop = Reconstruct.prop_of corr_prf;

                     val corr_prf' = List.foldr forall_intr_prf

                       (proof_combt

                          (PThm (corr_name name vs', corr_prf, corr_prop,

                              SOME (map TVar (term_tvars corr_prop))), vfs_of corr_prop))

                       (map (get_var_type corr_prop) (vfs_of prop))

                   in

                     ((name, (vs', ((nullt, nullt), (corr_prf, corr_prf')))) :: defs'',

                      prf_subst_TVars tye' corr_prf')

                   end

               | SOME (_, (_, prf')) => (defs', prf_subst_TVars tye' prf'))

             | SOME rs => (case find vs' rs of

                 SOME (_, prf') => (defs', prf_subst_TVars tye' prf')

               | NONE => error ("corr: no realizer for instance of theorem " ^

                   quote name ^ ":\n" ^ Syntax.string_of_term_global thy' (Envir.beta_norm

                     (Reconstruct.prop_of (proof_combt (prf0, ts))))))

           end

       | corr d defs vs ts Ts hs (prf0 as PAxm (s, prop, SOME Ts')) _ _ =

           let

             val (vs', tye) = find_inst prop Ts ts vs;

             val tye' = (map fst (term_tvars prop) ~~ Ts') @ tye

           in

             if etype_of thy' vs' [] prop = nullT andalso

               realizes_null vs' prop aconv prop then (defs, prf0)

             else case find vs' (Symtab.lookup_list realizers s) of

               SOME (_, prf) => (defs, prf_subst_TVars tye' prf)

             | NONE => error ("corr: no realizer for instance of axiom " ^

                 quote s ^ ":\n" ^ Syntax.string_of_term_global thy' (Envir.beta_norm

                   (Reconstruct.prop_of (proof_combt (prf0, ts)))))

           end

       | corr d defs vs ts Ts hs _ _ _ = error "corr: bad proof"

     and extr d defs vs ts Ts hs (PBound i) = (defs, Bound i)

       | extr d defs vs ts Ts hs (Abst (s, SOME T, prf)) =

           let val (defs', t) = extr d defs vs []

             (T :: Ts) (dummyt :: hs) (incr_pboundvars 1 0 prf)

           in (defs', Abs (s, T, t)) end

       | extr d defs vs ts Ts hs (AbsP (s, SOME t, prf)) =

           let

             val T = etype_of thy' vs Ts t;

             val (defs', t) = extr d defs vs [] (T :: Ts) (t :: hs)

               (incr_pboundvars 0 1 prf)

           in (defs',

             if T = nullT then subst_bound (nullt, t) else Abs (s, T, t))

           end

       | extr d defs vs ts Ts hs (prf % SOME t) =

           let val (defs', u) = extr d defs vs (t :: ts) Ts hs prf

           in (defs',

             if member (op =) rtypes (tname_of (body_type (fastype_of1 (Ts, t)))) then u

             else u $ t)

           end

       | extr d defs vs ts Ts hs (prf1 %% prf2) =

           let

             val (defs', f) = extr d defs vs [] Ts hs prf1;

             val prop = Reconstruct.prop_of' hs prf2;

             val T = etype_of thy' vs Ts prop

           in

             if T = nullT then (defs', f) else

               let val (defs'', t) = extr d defs' vs [] Ts hs prf2

               in (defs'', f $ t) end

           end

       | extr d defs vs ts Ts hs (prf0 as PThm (s, prf, prop, SOME Ts')) =

           let

             val (vs', tye) = find_inst prop Ts ts vs;

             val tye' = (map fst (term_tvars prop) ~~ Ts') @ tye

           in

             case Symtab.lookup realizers s of

               NONE => (case find vs' (find' s defs) of

                 NONE =>

                   let

                     val _ = msg d ("Extracting " ^ quote s ^

                       (if null vs' then ""

                        else " (relevant variables: " ^ commas_quote vs' ^ ")"));

                     val prf' = prep (Reconstruct.reconstruct_proof thy' prop prf);

                     val (defs', t) = extr (d + 1) defs vs' [] [] [] prf';

                     val (defs'', corr_prf) =

                       corr (d + 1) defs' vs' [] [] [] prf' prf' (SOME t);

                     val nt = Envir.beta_norm t;

                     val args = filter_out (fn v => member (op =) rtypes

                       (tname_of (body_type (fastype_of v)))) (vfs_of prop);

                     val args' = List.filter (fn v => Logic.occs (v, nt)) args;

                     val t' = mkabs nt args';

                     val T = fastype_of t';

                     val cname = extr_name s vs';

                     val c = Const (cname, T);

                     val u = mkabs (list_comb (c, args')) args;

                     val eqn = Logic.mk_equals (c, t');

                     val rlz =

                       Const ("realizes", fastype_of nt --> propT --> propT);

                     val lhs = app_rlz_rews [] vs' (rlz $ nt $ prop);

                     val rhs = app_rlz_rews [] vs' (rlz $ list_comb (c, args') $ prop);

                     val f = app_rlz_rews [] vs'

                       (Abs ("x", T, rlz $ list_comb (Bound 0, args') $ prop));

                     val corr_prf' =

                       chtype [] equal_elim_axm %> lhs %> rhs %%

                        (chtype [propT] symmetric_axm %> rhs %> lhs %%

                          (chtype [propT, T] combination_axm %> f %> f %> c %> t' %%

                            (chtype [T --> propT] reflexive_axm %> f) %%

                            PAxm (cname ^ "_def", eqn,

                              SOME (map TVar (term_tvars eqn))))) %% corr_prf;

                     val corr_prop = Reconstruct.prop_of corr_prf';

                     val corr_prf'' = List.foldr forall_intr_prf

                       (proof_combt

                         (PThm (corr_name s vs', corr_prf', corr_prop,

                           SOME (map TVar (term_tvars corr_prop))),  vfs_of corr_prop))

                       (map (get_var_type corr_prop) (vfs_of prop));

                   in

                     ((s, (vs', ((t', u), (corr_prf', corr_prf'')))) :: defs'',

                      subst_TVars tye' u)

                   end

               | SOME ((_, u), _) => (defs, subst_TVars tye' u))

             | SOME rs => (case find vs' rs of

                 SOME (t, _) => (defs, subst_TVars tye' t)

               | NONE => error ("extr: no realizer for instance of theorem " ^

                   quote s ^ ":\n" ^ Syntax.string_of_term_global thy' (Envir.beta_norm

                     (Reconstruct.prop_of (proof_combt (prf0, ts))))))

           end

       | extr d defs vs ts Ts hs (prf0 as PAxm (s, prop, SOME Ts')) =

           let

             val (vs', tye) = find_inst prop Ts ts vs;

             val tye' = (map fst (term_tvars prop) ~~ Ts') @ tye

           in

             case find vs' (Symtab.lookup_list realizers s) of

               SOME (t, _) => (defs, subst_TVars tye' t)

             | NONE => error ("extr: no realizer for instance of axiom " ^

                 quote s ^ ":\n" ^ Syntax.string_of_term_global thy' (Envir.beta_norm

                   (Reconstruct.prop_of (proof_combt (prf0, ts)))))

           end

       | extr d defs vs ts Ts hs _ = error "extr: bad proof";

     fun prep_thm (thm, vs) =

       let

         val thy = Thm.theory_of_thm thm;

         val prop = Thm.prop_of thm;

         val prf = Thm.proof_of thm;

         val name = Thm.get_name thm;

         val _ = name <> "" orelse error "extraction: unnamed theorem";

         val _ = etype_of thy' vs [] prop <> nullT orelse error ("theorem " ^

           quote name ^ " has no computational content")

       in (Reconstruct.reconstruct_proof thy prop prf, vs) end;

     val defs = Library.foldl (fn (defs, (prf, vs)) =>

       fst (extr 0 defs vs [] [] [] prf)) ([], map prep_thm thms);

     fun add_def (s, (vs, ((t, u), (prf, _)))) thy =

       (case Sign.const_type thy (extr_name s vs) of

          NONE =>

            let

              val corr_prop = Reconstruct.prop_of prf;

              val ft = Type.freeze t;

              val fu = Type.freeze u;

              val (def_thms, thy') = if t = nullt then ([], thy) else

                thy

                |> Sign.add_consts_i [(extr_name s vs, fastype_of ft, NoSyn)]

                |> PureThy.add_defs false [((extr_name s vs ^ "_def",

                     Logic.mk_equals (head_of (strip_abs_body fu), ft)), [])]

            in

              thy'

              |> PureThy.store_thm (corr_name s vs,

                   Thm.varifyT (funpow (length (term_vars corr_prop))

                     (Thm.forall_elim_var 0) (forall_intr_frees

                       (ProofChecker.thm_of_proof thy'

                        (fst (Proofterm.freeze_thaw_prf prf))))))

              |> snd

              |> fold Code.add_default_func def_thms

            end

        | SOME _ => thy);

   in

     thy

     |> Sign.absolute_path

     |> fold_rev add_def defs

     |> Sign.restore_naming thy

   end;

 (**** interface ****)

 structure P = OuterParse and K = OuterKeyword;

 val parse_vars = Scan.optional (P.$$$ "(" |-- P.list1 P.name --| P.$$$ ")") [];

 val _ =

   OuterSyntax.command "realizers"

   "specify realizers for primitive axioms / theorems, together with correctness proof"

   K.thy_decl

     (Scan.repeat1 (P.xname -- parse_vars --| P.$$$ ":" -- P.string -- P.string) >>

      (fn xs => Toplevel.theory (fn thy => add_realizers

        (map (fn (((a, vs), s1), s2) => (PureThy.get_thm thy a, (vs, s1, s2))) xs) thy)));

 val _ =

   OuterSyntax.command "realizability"

   "add equations characterizing realizability" K.thy_decl

   (Scan.repeat1 P.string >> (Toplevel.theory o add_realizes_eqns));

 val _ =

   OuterSyntax.command "extract_type"

   "add equations characterizing type of extracted program" K.thy_decl

   (Scan.repeat1 P.string >> (Toplevel.theory o add_typeof_eqns));

 val _ =

   OuterSyntax.command "extract" "extract terms from proofs" K.thy_decl

     (Scan.repeat1 (P.xname -- parse_vars) >> (fn xs => Toplevel.theory (fn thy =>

       extract (map (apfst (PureThy.get_thm thy)) xs) thy)));

 val etype_of = etype_of o add_syntax;

 end;