src/Pure/Proof/extraction.ML
author wenzelm
Mon Aug 01 19:20:37 2005 +0200 (2005-08-01)
changeset 16983 c895701d55ea
parent 16865 fb39dcfc1c24
child 17057 0934ac31985f
permissions -rw-r--r--
replaced atless by term_ord;
     1 (*  Title:      Pure/Proof/extraction.ML
     2     ID:         $Id$
     3     Author:     Stefan Berghofer, TU Muenchen
     4 
     5 Extraction of programs from proofs.
     6 *)
     7 
     8 signature EXTRACTION =
     9 sig
    10   val set_preprocessor : (theory -> Proofterm.proof -> Proofterm.proof) -> theory -> theory
    11   val add_realizes_eqns_i : ((term * term) list * (term * term)) list -> theory -> theory
    12   val add_realizes_eqns : string list -> theory -> theory
    13   val add_typeof_eqns_i : ((term * term) list * (term * term)) list -> theory -> theory
    14   val add_typeof_eqns : string list -> theory -> theory
    15   val add_realizers_i : (string * (string list * term * Proofterm.proof)) list
    16     -> theory -> theory
    17   val add_realizers : (thm * (string list * string * string)) list
    18     -> theory -> theory
    19   val add_expand_thms : thm list -> theory -> theory
    20   val add_types : (xstring * ((term -> term option) list *
    21     (term -> typ -> term -> typ -> term) option)) list -> theory -> theory
    22   val extract : (thm * string list) list -> theory -> theory
    23   val nullT : typ
    24   val nullt : term
    25   val mk_typ : typ -> term
    26   val etype_of : theory -> string list -> typ list -> term -> typ
    27   val realizes_of: theory -> string list -> term -> term -> term
    28 end;
    29 
    30 structure Extraction : EXTRACTION =
    31 struct
    32 
    33 open Proofterm;
    34 
    35 (**** tools ****)
    36 
    37 fun add_syntax thy =
    38   thy
    39   |> Theory.copy
    40   |> Theory.root_path
    41   |> Theory.add_types [("Type", 0, NoSyn), ("Null", 0, NoSyn)]
    42   |> Theory.add_consts
    43       [("typeof", "'b::{} => Type", NoSyn),
    44        ("Type", "'a::{} itself => Type", NoSyn),
    45        ("Null", "Null", NoSyn),
    46        ("realizes", "'a::{} => 'b::{} => 'b", NoSyn)];
    47 
    48 val nullT = Type ("Null", []);
    49 val nullt = Const ("Null", nullT);
    50 
    51 fun mk_typ T =
    52   Const ("Type", itselfT T --> Type ("Type", [])) $ Logic.mk_type T;
    53 
    54 fun typeof_proc defaultS vs (Const ("typeof", _) $ u) =
    55       SOME (mk_typ (case strip_comb u of
    56           (Var ((a, i), _), _) =>
    57             if a mem vs then TFree ("'" ^ a ^ ":" ^ string_of_int i, defaultS)
    58             else nullT
    59         | (Free (a, _), _) =>
    60             if a mem vs then TFree ("'" ^ a, defaultS) else nullT
    61         | _ => nullT))
    62   | typeof_proc _ _ _ = NONE;
    63 
    64 fun rlz_proc (Const ("realizes", Type (_, [Type ("Null", []), _])) $ r $ t) = SOME t
    65   | rlz_proc (Const ("realizes", Type (_, [T, _])) $ r $ t) =
    66       (case strip_comb t of
    67          (Var (ixn, U), ts) => SOME (list_comb (Var (ixn, T --> U), r :: ts))
    68        | (Free (s, U), ts) => SOME (list_comb (Free (s, T --> U), r :: ts))
    69        | _ => NONE)
    70   | rlz_proc _ = NONE;
    71 
    72 val unpack_ixn = apfst implode o apsnd (fst o read_int o tl) o
    73   take_prefix (not o equal ":") o explode;
    74 
    75 type rules =
    76   {next: int, rs: ((term * term) list * (term * term)) list,
    77    net: (int * ((term * term) list * (term * term))) Net.net};
    78 
    79 val empty_rules : rules = {next = 0, rs = [], net = Net.empty};
    80 
    81 fun add_rule (r as (_, (lhs, _)), {next, rs, net} : rules) =
    82   {next = next - 1, rs = r :: rs, net = Net.insert_term (K false)
    83      (Pattern.eta_contract lhs, (next, r)) net};
    84 
    85 fun merge_rules
    86   ({next, rs = rs1, net} : rules) ({next = next2, rs = rs2, ...} : rules) =
    87   foldr add_rule {next = next, rs = rs1, net = net} (rs2 \\ rs1);
    88 
    89 fun condrew thy rules procs =
    90   let
    91     val tsig = Sign.tsig_of thy;
    92 
    93     fun rew tm =
    94       Pattern.rewrite_term tsig [] (condrew' :: procs) tm
    95     and condrew' tm =
    96       let
    97         val cache = ref ([] : (term * term) list);
    98         fun lookup f x = (case gen_assoc (op =) (!cache, x) of
    99             NONE =>
   100               let val y = f x
   101               in (cache := (x, y) :: !cache; y) end
   102           | SOME y => y);
   103       in
   104         get_first (fn (_, (prems, (tm1, tm2))) =>
   105         let
   106           fun ren t = getOpt (Term.rename_abs tm1 tm t, t);
   107           val inc = Logic.incr_indexes ([], maxidx_of_term tm + 1);
   108           val env as (Tenv, tenv) = Pattern.match tsig (inc tm1, tm);
   109           val prems' = map (pairself (Envir.subst_vars env o inc o ren)) prems;
   110           val env' = Envir.Envir
   111             {maxidx = Library.foldl Int.max
   112               (~1, map (Int.max o pairself maxidx_of_term) prems'),
   113              iTs = Tenv, asol = tenv};
   114           val env'' = Library.foldl (fn (env, p) =>
   115             Pattern.unify (thy, env, [pairself (lookup rew) p])) (env', prems')
   116         in SOME (Envir.norm_term env'' (inc (ren tm2)))
   117         end handle Pattern.MATCH => NONE | Pattern.Unif => NONE)
   118           (sort (int_ord o pairself fst)
   119             (Net.match_term rules (Pattern.eta_contract tm)))
   120       end;
   121 
   122   in rew end;
   123 
   124 val chtype = change_type o SOME;
   125 
   126 fun extr_name s vs = NameSpace.append "extr" (space_implode "_" (s :: vs));
   127 fun corr_name s vs = extr_name s vs ^ "_correctness";
   128 
   129 fun msg d s = priority (Symbol.spaces d ^ s);
   130 
   131 fun vars_of t = rev (fold_aterms (fn v as Var _ => insert (op =) v | _ => I) t []);
   132 fun vfs_of t = vars_of t @ sort Term.term_ord (term_frees t);
   133 
   134 fun forall_intr (t, prop) =
   135   let val (a, T) = (case t of Var ((a, _), T) => (a, T) | Free p => p)
   136   in all T $ Abs (a, T, abstract_over (t, prop)) end;
   137 
   138 fun forall_intr_prf (t, prf) =
   139   let val (a, T) = (case t of Var ((a, _), T) => (a, T) | Free p => p)
   140   in Abst (a, SOME T, prf_abstract_over t prf) end;
   141 
   142 val mkabs = foldr (fn (v, t) => Abs ("x", fastype_of v, abstract_over (v, t)));
   143 
   144 fun strip_abs 0 t = t
   145   | strip_abs n (Abs (_, _, t)) = strip_abs (n-1) t
   146   | strip_abs _ _ = error "strip_abs: not an abstraction";
   147 
   148 fun prf_subst_TVars tye =
   149   map_proof_terms (subst_TVars tye) (typ_subst_TVars tye);
   150 
   151 fun relevant_vars types prop = foldr (fn
   152       (Var ((a, i), T), vs) => (case strip_type T of
   153         (_, Type (s, _)) => if s mem types then a :: vs else vs
   154       | _ => vs)
   155     | (_, vs) => vs) [] (vars_of prop);
   156 
   157 fun tname_of (Type (s, _)) = s
   158   | tname_of _ = "";
   159 
   160 fun get_var_type t =
   161   let
   162     val vs = Term.add_vars t [];
   163     val fs = Term.add_frees t [];
   164   in fn 
   165       Var (ixn, _) => (case assoc_string_int (vs, ixn) of
   166           NONE => error "get_var_type: no such variable in term"
   167         | SOME T => Var (ixn, T))
   168     | Free (s, _) => (case assoc_string (fs, s) of
   169           NONE => error "get_var_type: no such variable in term"
   170         | SOME T => Free (s, T))
   171     | _ => error "get_var_type: not a variable"
   172   end;
   173 
   174 
   175 (**** theory data ****)
   176 
   177 (* data kind 'Pure/extraction' *)
   178 
   179 structure ExtractionData = TheoryDataFun
   180 (struct
   181   val name = "Pure/extraction";
   182   type T =
   183     {realizes_eqns : rules,
   184      typeof_eqns : rules,
   185      types : (string * ((term -> term option) list *
   186        (term -> typ -> term -> typ -> term) option)) list,
   187      realizers : (string list * (term * proof)) list Symtab.table,
   188      defs : thm list,
   189      expand : (string * term) list,
   190      prep : (theory -> proof -> proof) option}
   191 
   192   val empty =
   193     {realizes_eqns = empty_rules,
   194      typeof_eqns = empty_rules,
   195      types = [],
   196      realizers = Symtab.empty,
   197      defs = [],
   198      expand = [],
   199      prep = NONE};
   200   val copy = I;
   201   val extend = I;
   202 
   203   fun merge _
   204     (({realizes_eqns = realizes_eqns1, typeof_eqns = typeof_eqns1, types = types1,
   205        realizers = realizers1, defs = defs1, expand = expand1, prep = prep1},
   206       {realizes_eqns = realizes_eqns2, typeof_eqns = typeof_eqns2, types = types2,
   207        realizers = realizers2, defs = defs2, expand = expand2, prep = prep2}) : T * T) =
   208     {realizes_eqns = merge_rules realizes_eqns1 realizes_eqns2,
   209      typeof_eqns = merge_rules typeof_eqns1 typeof_eqns2,
   210      types = merge_alists types1 types2,
   211      realizers = Symtab.merge_multi' (eq_set o pairself #1)
   212        (realizers1, realizers2),
   213      defs = gen_merge_lists eq_thm defs1 defs2,
   214      expand = merge_lists expand1 expand2,
   215      prep = (case prep1 of NONE => prep2 | _ => prep1)};
   216 
   217   fun print _ _ = ();
   218 end);
   219 
   220 val _ = Context.add_setup [ExtractionData.init];
   221 
   222 fun read_condeq thy =
   223   let val thy' = add_syntax thy
   224   in fn s =>
   225     let val t = Logic.varify (term_of (read_cterm thy' (s, propT)))
   226     in (map Logic.dest_equals (Logic.strip_imp_prems t),
   227       Logic.dest_equals (Logic.strip_imp_concl t))
   228     end handle TERM _ => error ("Not a (conditional) meta equality:\n" ^ s)
   229   end;
   230 
   231 (** preprocessor **)
   232 
   233 fun set_preprocessor prep thy =
   234   let val {realizes_eqns, typeof_eqns, types, realizers,
   235     defs, expand, ...} = ExtractionData.get thy
   236   in
   237     ExtractionData.put
   238       {realizes_eqns = realizes_eqns, typeof_eqns = typeof_eqns, types = types,
   239        realizers = realizers, defs = defs, expand = expand, prep = SOME prep} thy
   240   end;
   241 
   242 (** equations characterizing realizability **)
   243 
   244 fun gen_add_realizes_eqns prep_eq eqns thy =
   245   let val {realizes_eqns, typeof_eqns, types, realizers,
   246     defs, expand, prep} = ExtractionData.get thy;
   247   in
   248     ExtractionData.put
   249       {realizes_eqns = foldr add_rule realizes_eqns (map (prep_eq thy) eqns),
   250        typeof_eqns = typeof_eqns, types = types, realizers = realizers,
   251        defs = defs, expand = expand, prep = prep} thy
   252   end
   253 
   254 val add_realizes_eqns_i = gen_add_realizes_eqns (K I);
   255 val add_realizes_eqns = gen_add_realizes_eqns read_condeq;
   256 
   257 (** equations characterizing type of extracted program **)
   258 
   259 fun gen_add_typeof_eqns prep_eq eqns thy =
   260   let
   261     val {realizes_eqns, typeof_eqns, types, realizers,
   262       defs, expand, prep} = ExtractionData.get thy;
   263     val eqns' = map (prep_eq thy) eqns
   264   in
   265     ExtractionData.put
   266       {realizes_eqns = realizes_eqns, realizers = realizers,
   267        typeof_eqns = foldr add_rule typeof_eqns eqns',
   268        types = types, defs = defs, expand = expand, prep = prep} thy
   269   end
   270 
   271 val add_typeof_eqns_i = gen_add_typeof_eqns (K I);
   272 val add_typeof_eqns = gen_add_typeof_eqns read_condeq;
   273 
   274 fun thaw (T as TFree (a, S)) =
   275       if exists_string (equal ":") a then TVar (unpack_ixn a, S) else T
   276   | thaw (Type (a, Ts)) = Type (a, map thaw Ts)
   277   | thaw T = T;
   278 
   279 fun freeze (TVar ((a, i), S)) = TFree (a ^ ":" ^ string_of_int i, S)
   280   | freeze (Type (a, Ts)) = Type (a, map freeze Ts)
   281   | freeze T = T;
   282 
   283 fun freeze_thaw f x =
   284   map_term_types thaw (f (map_term_types freeze x));
   285 
   286 fun etype_of thy vs Ts t =
   287   let
   288     val {typeof_eqns, ...} = ExtractionData.get thy;
   289     fun err () = error ("Unable to determine type of extracted program for\n" ^
   290       Sign.string_of_term thy t)
   291   in case strip_abs_body (freeze_thaw (condrew thy (#net typeof_eqns)
   292     [typeof_proc (Sign.defaultS thy) vs]) (list_abs (map (pair "x") (rev Ts),
   293       Const ("typeof", fastype_of1 (Ts, t) --> Type ("Type", [])) $ t))) of
   294       Const ("Type", _) $ u => (Logic.dest_type u handle TERM _ => err ())
   295     | _ => err ()
   296   end;
   297 
   298 (** realizers for axioms / theorems, together with correctness proofs **)
   299 
   300 fun gen_add_realizers prep_rlz rs thy =
   301   let val {realizes_eqns, typeof_eqns, types, realizers,
   302     defs, expand, prep} = ExtractionData.get thy
   303   in
   304     ExtractionData.put
   305       {realizes_eqns = realizes_eqns, typeof_eqns = typeof_eqns, types = types,
   306        realizers = foldr Symtab.update_multi
   307          realizers (map (prep_rlz thy) (rev rs)),
   308        defs = defs, expand = expand, prep = prep} thy
   309   end
   310 
   311 fun prep_realizer thy =
   312   let
   313     val {realizes_eqns, typeof_eqns, defs, types, ...} =
   314       ExtractionData.get thy;
   315     val procs = List.concat (map (fst o snd) types);
   316     val rtypes = map fst types;
   317     val eqns = Net.merge (K false) (#net realizes_eqns, #net typeof_eqns);
   318     val thy' = add_syntax thy;
   319     val rd = ProofSyntax.read_proof thy' false
   320   in fn (thm, (vs, s1, s2)) =>
   321     let
   322       val name = Thm.name_of_thm thm;
   323       val _ = assert (name <> "") "add_realizers: unnamed theorem";
   324       val prop = Pattern.rewrite_term (Sign.tsig_of thy')
   325         (map (Logic.dest_equals o prop_of) defs) [] (prop_of thm);
   326       val vars = vars_of prop;
   327       val vars' = filter_out (fn v =>
   328         tname_of (body_type (fastype_of v)) mem rtypes) vars;
   329       val T = etype_of thy' vs [] prop;
   330       val (T', thw) = Type.freeze_thaw_type
   331         (if T = nullT then nullT else map fastype_of vars' ---> T);
   332       val t = map_term_types thw (term_of (read_cterm thy' (s1, T')));
   333       val r' = freeze_thaw (condrew thy' eqns
   334         (procs @ [typeof_proc (Sign.defaultS thy') vs, rlz_proc]))
   335           (Const ("realizes", T --> propT --> propT) $
   336             (if T = nullT then t else list_comb (t, vars')) $ prop);
   337       val r = foldr forall_intr r' (map (get_var_type r') vars);
   338       val prf = Reconstruct.reconstruct_proof thy' r (rd s2);
   339     in (name, (vs, (t, prf))) end
   340   end;
   341 
   342 val add_realizers_i = gen_add_realizers
   343   (fn _ => fn (name, (vs, t, prf)) => (name, (vs, (t, prf))));
   344 val add_realizers = gen_add_realizers prep_realizer;
   345 
   346 fun realizes_of thy vs t prop =
   347   let
   348     val thy' = add_syntax thy;
   349     val {realizes_eqns, typeof_eqns, defs, types, ...} =
   350       ExtractionData.get thy';
   351     val procs = List.concat (map (fst o snd) types);
   352     val eqns = Net.merge (K false) (#net realizes_eqns, #net typeof_eqns);
   353     val prop' = Pattern.rewrite_term (Sign.tsig_of thy')
   354       (map (Logic.dest_equals o prop_of) defs) [] prop;
   355   in freeze_thaw (condrew thy' eqns
   356     (procs @ [typeof_proc (Sign.defaultS thy') vs, rlz_proc]))
   357       (Const ("realizes", fastype_of t --> propT --> propT) $ t $ prop')
   358   end;
   359 
   360 (** expanding theorems / definitions **)
   361 
   362 fun add_expand_thm (thy, thm) =
   363   let
   364     val {realizes_eqns, typeof_eqns, types, realizers,
   365       defs, expand, prep} = ExtractionData.get thy;
   366 
   367     val name = Thm.name_of_thm thm;
   368     val _ = assert (name <> "") "add_expand_thms: unnamed theorem";
   369 
   370     val is_def =
   371       (case strip_comb (fst (Logic.dest_equals (prop_of thm))) of
   372          (Const _, ts) => forall is_Var ts andalso null (duplicates ts)
   373            andalso can (Thm.get_axiom_i thy) name
   374        | _ => false) handle TERM _ => false;
   375   in
   376     (ExtractionData.put (if is_def then
   377         {realizes_eqns = realizes_eqns,
   378          typeof_eqns = add_rule (([],
   379            Logic.dest_equals (prop_of (Drule.abs_def thm))), typeof_eqns),
   380          types = types,
   381          realizers = realizers, defs = gen_ins eq_thm (thm, defs),
   382          expand = expand, prep = prep}
   383       else
   384         {realizes_eqns = realizes_eqns, typeof_eqns = typeof_eqns, types = types,
   385          realizers = realizers, defs = defs,
   386          expand = (name, prop_of thm) ins expand, prep = prep}) thy, thm)
   387   end;
   388 
   389 fun add_expand_thms thms thy = Library.foldl (fst o add_expand_thm) (thy, thms);
   390 
   391 
   392 (** types with computational content **)
   393 
   394 fun add_types tys thy =
   395   let val {realizes_eqns, typeof_eqns, types, realizers,
   396     defs, expand, prep} = ExtractionData.get thy;
   397   in
   398     ExtractionData.put
   399       {realizes_eqns = realizes_eqns, typeof_eqns = typeof_eqns,
   400        types = map (apfst (Sign.intern_type thy)) tys @ types,
   401        realizers = realizers, defs = defs, expand = expand, prep = prep} thy
   402   end;
   403 
   404 
   405 (** Pure setup **)
   406 
   407 val _ = Context.add_setup
   408   [add_types [("prop", ([], NONE))],
   409 
   410    add_typeof_eqns
   411      ["(typeof (PROP P)) == (Type (TYPE(Null))) ==>  \
   412     \  (typeof (PROP Q)) == (Type (TYPE('Q))) ==>  \
   413     \    (typeof (PROP P ==> PROP Q)) == (Type (TYPE('Q)))",
   414 
   415       "(typeof (PROP Q)) == (Type (TYPE(Null))) ==>  \
   416     \    (typeof (PROP P ==> PROP Q)) == (Type (TYPE(Null)))",
   417 
   418       "(typeof (PROP P)) == (Type (TYPE('P))) ==>  \
   419     \  (typeof (PROP Q)) == (Type (TYPE('Q))) ==>  \
   420     \    (typeof (PROP P ==> PROP Q)) == (Type (TYPE('P => 'Q)))",
   421 
   422       "(%x. typeof (PROP P (x))) == (%x. Type (TYPE(Null))) ==>  \
   423     \    (typeof (!!x. PROP P (x))) == (Type (TYPE(Null)))",
   424 
   425       "(%x. typeof (PROP P (x))) == (%x. Type (TYPE('P))) ==>  \
   426     \    (typeof (!!x::'a. PROP P (x))) == (Type (TYPE('a => 'P)))",
   427 
   428       "(%x. typeof (f (x))) == (%x. Type (TYPE('f))) ==>  \
   429     \    (typeof (f)) == (Type (TYPE('f)))"],
   430 
   431    add_realizes_eqns
   432      ["(typeof (PROP P)) == (Type (TYPE(Null))) ==>  \
   433     \    (realizes (r) (PROP P ==> PROP Q)) ==  \
   434     \    (PROP realizes (Null) (PROP P) ==> PROP realizes (r) (PROP Q))",
   435 
   436       "(typeof (PROP P)) == (Type (TYPE('P))) ==>  \
   437     \  (typeof (PROP Q)) == (Type (TYPE(Null))) ==>  \
   438     \    (realizes (r) (PROP P ==> PROP Q)) ==  \
   439     \    (!!x::'P. PROP realizes (x) (PROP P) ==> PROP realizes (Null) (PROP Q))",
   440 
   441       "(realizes (r) (PROP P ==> PROP Q)) ==  \
   442     \  (!!x. PROP realizes (x) (PROP P) ==> PROP realizes (r (x)) (PROP Q))",
   443 
   444       "(%x. typeof (PROP P (x))) == (%x. Type (TYPE(Null))) ==>  \
   445     \    (realizes (r) (!!x. PROP P (x))) ==  \
   446     \    (!!x. PROP realizes (Null) (PROP P (x)))",
   447 
   448       "(realizes (r) (!!x. PROP P (x))) ==  \
   449     \  (!!x. PROP realizes (r (x)) (PROP P (x)))"],
   450 
   451    Attrib.add_attributes
   452      [("extraction_expand",
   453        (Attrib.no_args add_expand_thm, K Attrib.undef_local_attribute),
   454        "specify theorems / definitions to be expanded during extraction")]];
   455 
   456 
   457 (**** extract program ****)
   458 
   459 val dummyt = Const ("dummy", dummyT);
   460 
   461 fun extract thms thy =
   462   let
   463     val thy' = add_syntax thy;
   464     val {realizes_eqns, typeof_eqns, types, realizers, defs, expand, prep} =
   465       ExtractionData.get thy;
   466     val procs = List.concat (map (fst o snd) types);
   467     val rtypes = map fst types;
   468     val typroc = typeof_proc (Sign.defaultS thy');
   469     val prep = getOpt (prep, K I) thy' o ProofRewriteRules.elim_defs thy' false defs o
   470       Reconstruct.expand_proof thy' (("", NONE) :: map (apsnd SOME) expand);
   471     val rrews = Net.merge (K false) (#net realizes_eqns, #net typeof_eqns);
   472 
   473     fun find_inst prop Ts ts vs =
   474       let
   475         val rvs = relevant_vars rtypes prop;
   476         val vars = vars_of prop;
   477         val n = Int.min (length vars, length ts);
   478 
   479         fun add_args ((Var ((a, i), _), t), (vs', tye)) =
   480           if a mem rvs then
   481             let val T = etype_of thy' vs Ts t
   482             in if T = nullT then (vs', tye)
   483                else (a :: vs', (("'" ^ a, i), T) :: tye)
   484             end
   485           else (vs', tye)
   486 
   487       in foldr add_args ([], []) (Library.take (n, vars) ~~ Library.take (n, ts)) end;
   488 
   489     fun find vs = Option.map snd o find_first (curry eq_set vs o fst);
   490     fun find' s = map snd o List.filter (equal s o fst)
   491 
   492     fun app_rlz_rews Ts vs t = strip_abs (length Ts) (freeze_thaw
   493       (condrew thy' rrews (procs @ [typroc vs, rlz_proc])) (list_abs
   494         (map (pair "x") (rev Ts), t)));
   495 
   496     fun realizes_null vs prop = app_rlz_rews [] vs
   497       (Const ("realizes", nullT --> propT --> propT) $ nullt $ prop);
   498 
   499     fun corr d defs vs ts Ts hs (PBound i) _ _ = (defs, PBound i)
   500 
   501       | corr d defs vs ts Ts hs (Abst (s, SOME T, prf)) (Abst (_, _, prf')) t =
   502           let val (defs', corr_prf) = corr d defs vs [] (T :: Ts)
   503             (dummyt :: hs) prf (incr_pboundvars 1 0 prf')
   504             (case t of SOME (Abs (_, _, u)) => SOME u | _ => NONE)
   505           in (defs', Abst (s, SOME T, corr_prf)) end
   506 
   507       | corr d defs vs ts Ts hs (AbsP (s, SOME prop, prf)) (AbsP (_, _, prf')) t =
   508           let
   509             val T = etype_of thy' vs Ts prop;
   510             val u = if T = nullT then 
   511                 (case t of SOME u => SOME (incr_boundvars 1 u) | NONE => NONE)
   512               else (case t of SOME (Abs (_, _, u)) => SOME u | _ => NONE);
   513             val (defs', corr_prf) = corr d defs vs [] (T :: Ts) (prop :: hs)
   514               (incr_pboundvars 0 1 prf) (incr_pboundvars 0 1 prf') u;
   515             val rlz = Const ("realizes", T --> propT --> propT)
   516           in (defs',
   517             if T = nullT then AbsP ("R",
   518               SOME (app_rlz_rews Ts vs (rlz $ nullt $ prop)),
   519                 prf_subst_bounds [nullt] corr_prf)
   520             else Abst (s, SOME T, AbsP ("R",
   521               SOME (app_rlz_rews (T :: Ts) vs
   522                 (rlz $ Bound 0 $ incr_boundvars 1 prop)), corr_prf)))
   523           end
   524 
   525       | corr d defs vs ts Ts hs (prf % SOME t) (prf' % _) t' =
   526           let
   527             val (Us, T) = strip_type (fastype_of1 (Ts, t));
   528             val (defs', corr_prf) = corr d defs vs (t :: ts) Ts hs prf prf'
   529               (if tname_of T mem rtypes then t'
   530                else (case t' of SOME (u $ _) => SOME u | _ => NONE));
   531             val u = if not (tname_of T mem rtypes) then t else
   532               let
   533                 val eT = etype_of thy' vs Ts t;
   534                 val (r, Us') = if eT = nullT then (nullt, Us) else
   535                   (Bound (length Us), eT :: Us);
   536                 val u = list_comb (incr_boundvars (length Us') t,
   537                   map Bound (length Us - 1 downto 0));
   538                 val u' = (case assoc_string (types, tname_of T) of
   539                     SOME ((_, SOME f)) => f r eT u T
   540                   | _ => Const ("realizes", eT --> T --> T) $ r $ u)
   541               in app_rlz_rews Ts vs (list_abs (map (pair "x") Us', u')) end
   542           in (defs', corr_prf % SOME u) end
   543 
   544       | corr d defs vs ts Ts hs (prf1 %% prf2) (prf1' %% prf2') t =
   545           let
   546             val prop = Reconstruct.prop_of' hs prf2';
   547             val T = etype_of thy' vs Ts prop;
   548             val (defs1, f, u) = if T = nullT then (defs, t, NONE) else
   549               (case t of
   550                  SOME (f $ u) => (defs, SOME f, SOME u)
   551                | _ =>
   552                  let val (defs1, u) = extr d defs vs [] Ts hs prf2'
   553                  in (defs1, NONE, SOME u) end)
   554             val (defs2, corr_prf1) = corr d defs1 vs [] Ts hs prf1 prf1' f;
   555             val (defs3, corr_prf2) = corr d defs2 vs [] Ts hs prf2 prf2' u;
   556           in
   557             if T = nullT then (defs3, corr_prf1 %% corr_prf2) else
   558               (defs3, corr_prf1 % u %% corr_prf2)
   559           end
   560 
   561       | corr d defs vs ts Ts hs (prf0 as PThm ((name, _), prf, prop, SOME Ts')) _ _ =
   562           let
   563             val (vs', tye) = find_inst prop Ts ts vs;
   564             val tye' = (map fst (term_tvars prop) ~~ Ts') @ tye;
   565             val T = etype_of thy' vs' [] prop;
   566             val defs' = if T = nullT then defs
   567               else fst (extr d defs vs ts Ts hs prf0)
   568           in
   569             if T = nullT andalso realizes_null vs' prop aconv prop then (defs, prf0)
   570             else case Symtab.lookup (realizers, name) of
   571               NONE => (case find vs' (find' name defs') of
   572                 NONE =>
   573                   let
   574                     val _ = assert (T = nullT) "corr: internal error";
   575                     val _ = msg d ("Building correctness proof for " ^ quote name ^
   576                       (if null vs' then ""
   577                        else " (relevant variables: " ^ commas_quote vs' ^ ")"));
   578                     val prf' = prep (Reconstruct.reconstruct_proof thy' prop prf);
   579                     val (defs'', corr_prf) =
   580                       corr (d + 1) defs' vs' [] [] [] prf' prf' NONE;
   581                     val corr_prop = Reconstruct.prop_of corr_prf;
   582                     val corr_prf' = foldr forall_intr_prf
   583                       (proof_combt
   584                          (PThm ((corr_name name vs', []), corr_prf, corr_prop,
   585                              SOME (map TVar (term_tvars corr_prop))), vfs_of corr_prop))
   586 		      (map (get_var_type corr_prop) (vfs_of prop))
   587                   in
   588                     ((name, (vs', ((nullt, nullt), (corr_prf, corr_prf')))) :: defs'',
   589                      prf_subst_TVars tye' corr_prf')
   590                   end
   591               | SOME (_, (_, prf')) => (defs', prf_subst_TVars tye' prf'))
   592             | SOME rs => (case find vs' rs of
   593                 SOME (_, prf') => (defs', prf_subst_TVars tye' prf')
   594               | NONE => error ("corr: no realizer for instance of theorem " ^
   595                   quote name ^ ":\n" ^ Sign.string_of_term thy' (Envir.beta_norm
   596                     (Reconstruct.prop_of (proof_combt (prf0, ts))))))
   597           end
   598 
   599       | corr d defs vs ts Ts hs (prf0 as PAxm (s, prop, SOME Ts')) _ _ =
   600           let
   601             val (vs', tye) = find_inst prop Ts ts vs;
   602             val tye' = (map fst (term_tvars prop) ~~ Ts') @ tye
   603           in
   604             if etype_of thy' vs' [] prop = nullT andalso
   605               realizes_null vs' prop aconv prop then (defs, prf0)
   606             else case find vs' (Symtab.lookup_multi (realizers, s)) of
   607               SOME (_, prf) => (defs, prf_subst_TVars tye' prf)
   608             | NONE => error ("corr: no realizer for instance of axiom " ^
   609                 quote s ^ ":\n" ^ Sign.string_of_term thy' (Envir.beta_norm
   610                   (Reconstruct.prop_of (proof_combt (prf0, ts)))))
   611           end
   612 
   613       | corr d defs vs ts Ts hs _ _ _ = error "corr: bad proof"
   614 
   615     and extr d defs vs ts Ts hs (PBound i) = (defs, Bound i)
   616 
   617       | extr d defs vs ts Ts hs (Abst (s, SOME T, prf)) =
   618           let val (defs', t) = extr d defs vs []
   619             (T :: Ts) (dummyt :: hs) (incr_pboundvars 1 0 prf)
   620           in (defs', Abs (s, T, t)) end
   621 
   622       | extr d defs vs ts Ts hs (AbsP (s, SOME t, prf)) =
   623           let
   624             val T = etype_of thy' vs Ts t;
   625             val (defs', t) = extr d defs vs [] (T :: Ts) (t :: hs)
   626               (incr_pboundvars 0 1 prf)
   627           in (defs',
   628             if T = nullT then subst_bound (nullt, t) else Abs (s, T, t))
   629           end
   630 
   631       | extr d defs vs ts Ts hs (prf % SOME t) =
   632           let val (defs', u) = extr d defs vs (t :: ts) Ts hs prf
   633           in (defs',
   634             if tname_of (body_type (fastype_of1 (Ts, t))) mem rtypes then u
   635             else u $ t)
   636           end
   637 
   638       | extr d defs vs ts Ts hs (prf1 %% prf2) =
   639           let
   640             val (defs', f) = extr d defs vs [] Ts hs prf1;
   641             val prop = Reconstruct.prop_of' hs prf2;
   642             val T = etype_of thy' vs Ts prop
   643           in
   644             if T = nullT then (defs', f) else
   645               let val (defs'', t) = extr d defs' vs [] Ts hs prf2
   646               in (defs'', f $ t) end
   647           end
   648 
   649       | extr d defs vs ts Ts hs (prf0 as PThm ((s, _), prf, prop, SOME Ts')) =
   650           let
   651             val (vs', tye) = find_inst prop Ts ts vs;
   652             val tye' = (map fst (term_tvars prop) ~~ Ts') @ tye
   653           in
   654             case Symtab.lookup (realizers, s) of
   655               NONE => (case find vs' (find' s defs) of
   656                 NONE =>
   657                   let
   658                     val _ = msg d ("Extracting " ^ quote s ^
   659                       (if null vs' then ""
   660                        else " (relevant variables: " ^ commas_quote vs' ^ ")"));
   661                     val prf' = prep (Reconstruct.reconstruct_proof thy' prop prf);
   662                     val (defs', t) = extr (d + 1) defs vs' [] [] [] prf';
   663                     val (defs'', corr_prf) =
   664                       corr (d + 1) defs' vs' [] [] [] prf' prf' (SOME t);
   665 
   666                     val nt = Envir.beta_norm t;
   667                     val args = filter_out (fn v => tname_of (body_type
   668                       (fastype_of v)) mem rtypes) (vfs_of prop);
   669                     val args' = List.filter (fn v => Logic.occs (v, nt)) args;
   670                     val t' = mkabs nt args';
   671                     val T = fastype_of t';
   672                     val cname = extr_name s vs';
   673                     val c = Const (cname, T);
   674                     val u = mkabs (list_comb (c, args')) args;
   675                     val eqn = Logic.mk_equals (c, t');
   676                     val rlz =
   677                       Const ("realizes", fastype_of nt --> propT --> propT);
   678                     val lhs = app_rlz_rews [] vs' (rlz $ nt $ prop);
   679                     val rhs = app_rlz_rews [] vs' (rlz $ list_comb (c, args') $ prop);
   680                     val f = app_rlz_rews [] vs'
   681                       (Abs ("x", T, rlz $ list_comb (Bound 0, args') $ prop));
   682 
   683                     val corr_prf' =
   684                       chtype [] equal_elim_axm %> lhs %> rhs %%
   685                        (chtype [propT] symmetric_axm %> rhs %> lhs %%
   686                          (chtype [propT, T] combination_axm %> f %> f %> c %> t' %%
   687                            (chtype [T --> propT] reflexive_axm %> f) %%
   688                            PAxm (cname ^ "_def", eqn,
   689                              SOME (map TVar (term_tvars eqn))))) %% corr_prf;
   690                     val corr_prop = Reconstruct.prop_of corr_prf';
   691                     val corr_prf'' = foldr forall_intr_prf
   692                       (proof_combt
   693                         (PThm ((corr_name s vs', []), corr_prf', corr_prop,
   694                           SOME (map TVar (term_tvars corr_prop))),  vfs_of corr_prop))
   695 		      (map (get_var_type corr_prop) (vfs_of prop));
   696                   in
   697                     ((s, (vs', ((t', u), (corr_prf', corr_prf'')))) :: defs'',
   698                      subst_TVars tye' u)
   699                   end
   700               | SOME ((_, u), _) => (defs, subst_TVars tye' u))
   701             | SOME rs => (case find vs' rs of
   702                 SOME (t, _) => (defs, subst_TVars tye' t)
   703               | NONE => error ("extr: no realizer for instance of theorem " ^
   704                   quote s ^ ":\n" ^ Sign.string_of_term thy' (Envir.beta_norm
   705                     (Reconstruct.prop_of (proof_combt (prf0, ts))))))
   706           end
   707 
   708       | extr d defs vs ts Ts hs (prf0 as PAxm (s, prop, SOME Ts')) =
   709           let
   710             val (vs', tye) = find_inst prop Ts ts vs;
   711             val tye' = (map fst (term_tvars prop) ~~ Ts') @ tye
   712           in
   713             case find vs' (Symtab.lookup_multi (realizers, s)) of
   714               SOME (t, _) => (defs, subst_TVars tye' t)
   715             | NONE => error ("extr: no realizer for instance of axiom " ^
   716                 quote s ^ ":\n" ^ Sign.string_of_term thy' (Envir.beta_norm
   717                   (Reconstruct.prop_of (proof_combt (prf0, ts)))))
   718           end
   719 
   720       | extr d defs vs ts Ts hs _ = error "extr: bad proof";
   721 
   722     fun prep_thm (thm, vs) =
   723       let
   724         val {prop, der = (_, prf), sign, ...} = rep_thm thm;
   725         val name = Thm.name_of_thm thm;
   726         val _ = assert (name <> "") "extraction: unnamed theorem";
   727         val _ = assert (etype_of thy' vs [] prop <> nullT) ("theorem " ^
   728           quote name ^ " has no computational content")
   729       in (Reconstruct.reconstruct_proof sign prop prf, vs) end;
   730 
   731     val defs = Library.foldl (fn (defs, (prf, vs)) =>
   732       fst (extr 0 defs vs [] [] [] prf)) ([], map prep_thm thms);
   733 
   734     fun add_def (s, (vs, ((t, u), (prf, _)))) thy =
   735       (case Sign.const_type thy (extr_name s vs) of
   736          NONE =>
   737            let
   738              val corr_prop = Reconstruct.prop_of prf;
   739              val ft = Type.freeze t;
   740              val fu = Type.freeze u;
   741              val thy' = if t = nullt then thy else thy |>
   742                Theory.add_consts_i [(extr_name s vs, fastype_of ft, NoSyn)] |>
   743                fst o PureThy.add_defs_i false [((extr_name s vs ^ "_def",
   744                  Logic.mk_equals (head_of (strip_abs_body fu), ft)), [])];
   745            in
   746              fst (PureThy.store_thm ((corr_name s vs,
   747                Thm.varifyT (funpow (length (term_vars corr_prop))
   748                  (forall_elim_var 0) (forall_intr_frees
   749                    (ProofChecker.thm_of_proof thy'
   750                      (fst (Proofterm.freeze_thaw_prf prf)))))), []) thy')
   751            end
   752        | SOME _ => thy);
   753 
   754   in
   755     thy
   756     |> Theory.absolute_path
   757     |> fold_rev add_def defs
   758     |> Theory.restore_naming thy
   759   end;
   760 
   761 
   762 (**** interface ****)
   763 
   764 structure P = OuterParse and K = OuterSyntax.Keyword;
   765 
   766 val parse_vars = Scan.optional (P.$$$ "(" |-- P.list1 P.name --| P.$$$ ")") [];
   767 
   768 val realizersP =
   769   OuterSyntax.command "realizers"
   770   "specify realizers for primitive axioms / theorems, together with correctness proof"
   771   K.thy_decl
   772     (Scan.repeat1 (P.xname -- parse_vars --| P.$$$ ":" -- P.string -- P.string) >>
   773      (fn xs => Toplevel.theory (fn thy => add_realizers
   774        (map (fn (((a, vs), s1), s2) =>
   775          (PureThy.get_thm thy (Name a), (vs, s1, s2))) xs) thy)));
   776 
   777 val realizabilityP =
   778   OuterSyntax.command "realizability"
   779   "add equations characterizing realizability" K.thy_decl
   780   (Scan.repeat1 P.string >> (Toplevel.theory o add_realizes_eqns));
   781 
   782 val typeofP =
   783   OuterSyntax.command "extract_type"
   784   "add equations characterizing type of extracted program" K.thy_decl
   785   (Scan.repeat1 P.string >> (Toplevel.theory o add_typeof_eqns));
   786 
   787 val extractP =
   788   OuterSyntax.command "extract" "extract terms from proofs" K.thy_decl
   789     (Scan.repeat1 (P.xname -- parse_vars) >> (fn xs => Toplevel.theory
   790       (fn thy => extract (map (apfst (PureThy.get_thm thy o Name)) xs) thy)));
   791 
   792 val _ = OuterSyntax.add_parsers [realizersP, realizabilityP, typeofP, extractP];
   793 
   794 val etype_of = etype_of o add_syntax;
   795 
   796 end;