src/HOL/Tools/inductive_realizer.ML

const args: do not store variable names (unused);

(*  Title:      HOL/Tools/inductive_realizer.ML
    ID:         $Id$
    Author:     Stefan Berghofer, TU Muenchen

Porgram extraction from proofs involving inductive predicates:
Realizers for induction and elimination rules
*)

signature INDUCTIVE_REALIZER =
sig
  val add_ind_realizers: string -> string list -> theory -> theory
  val setup: (theory -> theory) list
end;

structure InductiveRealizer : INDUCTIVE_REALIZER =
struct

val all_simps = map (symmetric o mk_meta_eq) (thms "HOL.all_simps");

fun prf_of thm =
  let val {sign, prop, der = (_, prf), ...} = rep_thm thm
  in Reconstruct.reconstruct_proof sign prop prf end;

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, Proofterm.prf_abstract_over t prf) end;

fun subsets [] = [[]]
  | subsets (x::xs) =
      let val ys = subsets xs
      in ys @ map (cons x) ys end;

val set_of = fst o dest_Const o head_of o snd o HOLogic.dest_mem;

fun strip_all t =
  let
    fun strip used (Const ("all", _) $ Abs (s, T, t)) =
          let val s' = variant used s
          in strip (s'::used) (subst_bound (Free (s', T), t)) end
      | strip used ((t as Const ("==>", _) $ P) $ Q) = t $ strip used Q
      | strip _ t = t;
  in strip (add_term_free_names (t, [])) t end;

fun relevant_vars prop = foldr (fn
      (Var ((a, i), T), vs) => (case strip_type T of
        (_, Type (s, _)) => if s mem ["bool", "set"] then (a, T) :: vs else vs
      | _ => vs)
    | (_, vs) => vs) [] (term_vars prop);

fun params_of intr = map (fst o fst o dest_Var) (term_vars
  (snd (HOLogic.dest_mem (HOLogic.dest_Trueprop
    (Logic.strip_imp_concl intr)))));

fun dt_of_intrs thy vs intrs =
  let
    val iTs = term_tvars (prop_of (hd intrs));
    val Tvs = map TVar iTs;
    val (_ $ (_ $ _ $ S)) = Logic.strip_imp_concl (prop_of (hd intrs));
    val (Const (s, _), ts) = strip_comb S;
    val params = map dest_Var ts;
    val tname = space_implode "_" (Sign.base_name s ^ "T" :: vs);
    fun constr_of_intr intr = (Sign.base_name (Thm.name_of_thm intr),
      map (Type.unvarifyT o snd) (rev (Term.add_vars (prop_of intr) []) \\ params) @
        filter_out (equal Extraction.nullT) (map
          (Type.unvarifyT o Extraction.etype_of thy vs []) (prems_of intr)),
            NoSyn);
  in (map (fn a => "'" ^ a) vs @ map (fst o fst) iTs, tname, NoSyn,
    map constr_of_intr intrs)
  end;

fun mk_rlz T = Const ("realizes", [T, HOLogic.boolT] ---> HOLogic.boolT);

(** turn "P" into "%r x. realizes r (P x)" or "%r x. realizes r (x : P)" **)

fun gen_rvar vs (t as Var ((a, 0), T)) =
      let val U = TVar (("'" ^ a, 0), HOLogic.typeS)
      in case try HOLogic.dest_setT T of
          NONE => if body_type T <> HOLogic.boolT then t else
            let
              val Ts = binder_types T;
              val i = length Ts;
              val xs = map (pair "x") Ts;
              val u = list_comb (t, map Bound (i - 1 downto 0))
            in 
              if a mem vs then
                list_abs (("r", U) :: xs, mk_rlz U $ Bound i $ u)
              else list_abs (xs, mk_rlz Extraction.nullT $ Extraction.nullt $ u)
            end
        | SOME T' => if a mem vs then
              Abs ("r", U, Abs ("x", T', mk_rlz U $ Bound 1 $
                (HOLogic.mk_mem (Bound 0, t))))
            else Abs ("x", T', mk_rlz Extraction.nullT $ Extraction.nullt $
              (HOLogic.mk_mem (Bound 0, t)))
      end
  | gen_rvar _ t = t;

fun mk_realizes_eqn n vs intrs =
  let
    val iTs = term_tvars (prop_of (hd intrs));
    val Tvs = map TVar iTs;
    val _ $ (_ $ _ $ S) = concl_of (hd intrs);
    val (Const (s, T), ts') = strip_comb S;
    val setT = body_type T;
    val elT = HOLogic.dest_setT setT;
    val x = Var (("x", 0), elT);
    val rT = if n then Extraction.nullT
      else Type (space_implode "_" (s ^ "T" :: vs),
        map (fn a => TVar (("'" ^ a, 0), HOLogic.typeS)) vs @ Tvs);
    val r = if n then Extraction.nullt else Var ((Sign.base_name s, 0), rT);
    val rvs = relevant_vars S;
    val vs' = map fst rvs \\ vs;
    val rname = space_implode "_" (s ^ "R" :: vs);

    fun mk_Tprem n v =
      let val T = (the o AList.lookup (op =) rvs) v
      in (Const ("typeof", T --> Type ("Type", [])) $ Var ((v, 0), T),
        Extraction.mk_typ (if n then Extraction.nullT
          else TVar (("'" ^ v, 0), HOLogic.typeS)))
      end;

    val prems = map (mk_Tprem true) vs' @ map (mk_Tprem false) vs;
    val ts = map (gen_rvar vs) ts';
    val argTs = map fastype_of ts;

  in ((prems, (Const ("typeof", setT --> Type ("Type", [])) $ S,
       Extraction.mk_typ rT)),
    (prems, (mk_rlz rT $ r $ HOLogic.mk_mem (x, S),
       if n then
         HOLogic.mk_mem (x, list_comb (Const (rname, argTs ---> setT), ts))
       else HOLogic.mk_mem (HOLogic.mk_prod (r, x), list_comb (Const (rname,
         argTs ---> HOLogic.mk_setT (HOLogic.mk_prodT (rT, elT))), ts)))))
  end;

fun fun_of_prem thy rsets vs params rule intr =
  let
    (* add_term_vars and Term.add_vars may return variables in different order *)
    val args = map (Free o apfst fst o dest_Var)
      (add_term_vars (prop_of intr, []) \\ map Var params);
    val args' = map (Free o apfst fst)
      (Term.add_vars (prop_of intr) [] \\ params);
    val rule' = strip_all rule;
    val conclT = Extraction.etype_of thy vs [] (Logic.strip_imp_concl rule');
    val used = map (fst o dest_Free) args;

    fun is_rec t = not (null (term_consts t inter rsets));

    fun is_meta (Const ("all", _) $ Abs (s, _, P)) = is_meta P
      | is_meta (Const ("==>", _) $ _ $ Q) = is_meta Q
      | is_meta (Const ("Trueprop", _) $ (Const ("op :", _) $ _ $ _)) = true
      | is_meta _ = false;

    fun fun_of ts rts args used (prem :: prems) =
          let
            val T = Extraction.etype_of thy vs [] prem;
            val [x, r] = variantlist (["x", "r"], used)
          in if T = Extraction.nullT
            then fun_of ts rts args used prems
            else if is_rec prem then
              if is_meta prem then
                let
                  val prem' :: prems' = prems;
                  val U = Extraction.etype_of thy vs [] prem';
                in if U = Extraction.nullT
                  then fun_of (Free (x, T) :: ts)
                    (Free (r, binder_types T ---> HOLogic.unitT) :: rts)
                    (Free (x, T) :: args) (x :: r :: used) prems'
                  else fun_of (Free (x, T) :: ts) (Free (r, U) :: rts)
                    (Free (r, U) :: Free (x, T) :: args) (x :: r :: used) prems'
                end
              else (case strip_type T of
                  (Ts, Type ("*", [T1, T2])) =>
                    let
                      val fx = Free (x, Ts ---> T1);
                      val fr = Free (r, Ts ---> T2);
                      val bs = map Bound (length Ts - 1 downto 0);
                      val t = list_abs (map (pair "z") Ts,
                        HOLogic.mk_prod (list_comb (fx, bs), list_comb (fr, bs)))
                    in fun_of (fx :: ts) (fr :: rts) (t::args)
                      (x :: r :: used) prems
                    end
                | (Ts, U) => fun_of (Free (x, T) :: ts)
                    (Free (r, binder_types T ---> HOLogic.unitT) :: rts)
                    (Free (x, T) :: args) (x :: r :: used) prems)
            else fun_of (Free (x, T) :: ts) rts (Free (x, T) :: args)
              (x :: used) prems
          end
      | fun_of ts rts args used [] =
          let val xs = rev (rts @ ts)
          in if conclT = Extraction.nullT
            then list_abs_free (map dest_Free xs, HOLogic.unit)
            else list_abs_free (map dest_Free xs, list_comb
              (Free ("r" ^ Sign.base_name (Thm.name_of_thm intr),
                map fastype_of (rev args) ---> conclT), rev args))
          end

  in fun_of args' [] (rev args) used (Logic.strip_imp_prems rule') end;

fun find_first f = Library.find_first f;

fun indrule_realizer thy induct raw_induct rsets params vs rec_names rss intrs dummies =
  let
    val concls = HOLogic.dest_conj (HOLogic.dest_Trueprop (concl_of raw_induct));
    val premss = List.mapPartial (fn (s, rs) => if s mem rsets then
      SOME (map (fn r => List.nth (prems_of raw_induct,
        find_index_eq (prop_of r) (map prop_of intrs))) rs) else NONE) rss;
    val concls' = List.mapPartial (fn (s, _) => if s mem rsets then
        find_first (fn concl => s mem term_consts concl) concls
      else NONE) rss;
    val fs = List.concat (snd (foldl_map (fn (intrs, (prems, dummy)) =>
      let
        val (intrs1, intrs2) = splitAt (length prems, intrs);
        val fs = map (fn (rule, intr) =>
          fun_of_prem thy rsets vs params rule intr) (prems ~~ intrs1)
      in (intrs2, if dummy then Const ("arbitrary",
          HOLogic.unitT --> body_type (fastype_of (hd fs))) :: fs
        else fs)
      end) (intrs, (premss ~~ dummies))));
    val frees = fold Term.add_frees fs [];
    val Ts = map fastype_of fs;
    val rlzs = List.mapPartial (fn (a, concl) =>
      let val T = Extraction.etype_of thy vs [] concl
      in if T = Extraction.nullT then NONE
        else SOME (list_comb (Const (a, Ts ---> T), fs))
      end) (rec_names ~~ concls')
  in if null rlzs then Extraction.nullt else
    let
      val r = foldr1 HOLogic.mk_prod rlzs;
      val x = Free ("x", Extraction.etype_of thy vs [] (hd (prems_of induct)));
      fun name_of_fn intr = "r" ^ Sign.base_name (Thm.name_of_thm intr);
      val r' = list_abs_free (List.mapPartial (fn intr =>
        Option.map (pair (name_of_fn intr)) (AList.lookup (op =) frees (name_of_fn intr))) intrs,
          if length concls = 1 then r $ x else r)
    in
      if length concls = 1 then lambda x r' else r'
    end
  end;

fun add_dummy name dname (x as (_, (vs, s, mfx, cs))) =
  if name = s then (true, (vs, s, mfx, (dname, [HOLogic.unitT], NoSyn) :: cs))
  else x;

fun add_dummies f dts used thy =
  apsnd (pair (map fst dts)) (f (map snd dts) thy)
  handle DatatypeAux.Datatype_Empty name' =>
      let
        val name = Sign.base_name name';
        val dname = variant used "Dummy"
      in add_dummies f (map (add_dummy name dname) dts) (dname :: used) thy
      end;

fun mk_realizer thy vs params ((rule, rrule), rt) =
  let
    val prems = prems_of rule ~~ prems_of rrule;
    val rvs = map fst (relevant_vars (prop_of rule));
    val xs = rev (Term.add_vars (prop_of rule) []);
    val vs1 = map Var (filter_out (fn ((a, _), _) => a mem rvs) xs);
    val rlzvs = rev (Term.add_vars (prop_of rrule) []);
    val vs2 = map (fn (ixn, _) => Var (ixn, (the o AList.lookup (op =) rlzvs) ixn)) xs;
    val rs = gen_rems (op = o pairself fst) (rlzvs, xs);

    fun mk_prf _ [] prf = prf
      | mk_prf rs ((prem, rprem) :: prems) prf =
          if Extraction.etype_of thy vs [] prem = Extraction.nullT
          then AbsP ("H", SOME rprem, mk_prf rs prems prf)
          else forall_intr_prf (Var (hd rs), AbsP ("H", SOME rprem,
            mk_prf (tl rs) prems prf));

  in (Thm.name_of_thm rule, (vs,
    if rt = Extraction.nullt then rt else
      foldr (uncurry lambda) rt vs1,
    foldr forall_intr_prf (mk_prf rs prems (Proofterm.proof_combP
      (prf_of rrule, map PBound (length prems - 1 downto 0)))) vs2))
  end;

fun add_rule r rss =
  let
    val _ $ (_ $ _ $ S) = concl_of r;
    val (Const (s, _), _) = strip_comb S;
  in
    rss
    |> AList.default (op =) (s, [])
    |> AList.map_entry (op =) s (fn rs => rs @ [r])
  end;

fun add_ind_realizer rsets intrs induct raw_induct elims (thy, vs) =
  let
    val iTs = term_tvars (prop_of (hd intrs));
    val ar = length vs + length iTs;
    val (_ $ (_ $ _ $ S)) = Logic.strip_imp_concl (prop_of (hd intrs));
    val (_, params) = strip_comb S;
    val params' = map dest_Var params;
    val rss = [] |> Library.fold add_rule intrs;
    val (prfx, _) = split_last (NameSpace.unpack (fst (hd rss)));
    val tnames = map (fn s => space_implode "_" (s ^ "T" :: vs)) rsets;

    val thy1 = thy |>
      Theory.root_path |>
      Theory.add_path (NameSpace.pack prfx);
    val (ty_eqs, rlz_eqs) = split_list
      (map (fn (s, rs) => mk_realizes_eqn (not (s mem rsets)) vs rs) rss);

    val thy1' = thy1 |>
      Theory.copy |>
      Theory.add_types (map (fn s => (Sign.base_name s, ar, NoSyn)) tnames) |>
      Theory.add_arities_i (map (fn s =>
        (s, replicate ar HOLogic.typeS, HOLogic.typeS)) tnames) |>
        Extraction.add_typeof_eqns_i ty_eqs;
    val dts = List.mapPartial (fn (s, rs) => if s mem rsets then
      SOME (dt_of_intrs thy1' vs rs) else NONE) rss;

    (** datatype representing computational content of inductive set **)

    val (thy2, (dummies, dt_info)) =
      thy1
      |> (if null dts
          then rpair ([], NONE)
          else add_dummies (DatatypePackage.add_datatype_i false false
            (map #2 dts)) (map (pair false) dts) [] #> (fn (v, (b, thy)) => (thy, (b, SOME v))))
      |>> Extraction.add_typeof_eqns_i ty_eqs
      |>> Extraction.add_realizes_eqns_i rlz_eqs;
    fun get f x = getOpt (Option.map f x, []);
    val rec_names = distinct (map (fst o dest_Const o head_of o fst o
      HOLogic.dest_eq o HOLogic.dest_Trueprop o prop_of) (get #rec_thms dt_info));
    val (_, constrss) = foldl_map (fn ((recs, dummies), (s, rs)) =>
      if s mem rsets then
        let
          val (d :: dummies') = dummies;
          val (recs1, recs2) = splitAt (length rs, if d then tl recs else recs)
        in ((recs2, dummies'), map (head_of o hd o rev o snd o strip_comb o
          fst o HOLogic.dest_eq o HOLogic.dest_Trueprop o prop_of) recs1)
        end
      else ((recs, dummies), replicate (length rs) Extraction.nullt))
        ((get #rec_thms dt_info, dummies), rss);
    val rintrs = map (fn (intr, c) => Pattern.eta_contract
      (Extraction.realizes_of thy2 vs
        c (prop_of (forall_intr_list (map (cterm_of (sign_of thy2) o Var)
          (rev (Term.add_vars (prop_of intr) []) \\ params')) intr))))
            (intrs ~~ List.concat constrss);
    val rlzsets = distinct (map (fn rintr => snd (HOLogic.dest_mem
      (HOLogic.dest_Trueprop (Logic.strip_assums_concl rintr)))) rintrs);

    (** realizability predicate **)

    val (thy3', ind_info) = thy2 |>
      InductivePackage.add_inductive_i false true "" false false false
        (map Logic.unvarify rlzsets) (map (fn (rintr, intr) =>
          ((Sign.base_name (Thm.name_of_thm intr), strip_all
            (Logic.unvarify rintr)), [])) (rintrs ~~ intrs)) [] |>>
      Theory.absolute_path;
    val thy3 = PureThy.hide_thms false
      (map Thm.name_of_thm (#intrs ind_info)) thy3';

    (** realizer for induction rule **)

    val Ps = List.mapPartial (fn _ $ M $ P => if set_of M mem rsets then
      SOME (fst (fst (dest_Var (head_of P)))) else NONE)
        (HOLogic.dest_conj (HOLogic.dest_Trueprop (concl_of raw_induct)));

    fun add_ind_realizer (thy, Ps) =
      let
        val r = indrule_realizer thy induct raw_induct rsets params'
          (vs @ Ps) rec_names rss intrs dummies;
        val rlz = strip_all (Logic.unvarify
          (Extraction.realizes_of thy (vs @ Ps) r (prop_of induct)));
        val rews = map mk_meta_eq
          (fst_conv :: snd_conv :: get #rec_thms dt_info);
        val thm = OldGoals.simple_prove_goal_cterm (cterm_of (sign_of thy) rlz) (fn prems =>
          [if length rss = 1 then
             cut_facts_tac [hd prems] 1 THEN etac (#induct ind_info) 1
           else EVERY [rewrite_goals_tac (rews @ all_simps),
             REPEAT (rtac allI 1), rtac (#induct ind_info) 1],
           rewrite_goals_tac rews,
           REPEAT ((resolve_tac prems THEN_ALL_NEW EVERY'
             [K (rewrite_goals_tac rews), ObjectLogic.atomize_tac,
              DEPTH_SOLVE_1 o FIRST' [atac, etac allE, etac impE]]) 1)]);
        val (thy', thm') = PureThy.store_thm ((space_implode "_"
          (Thm.name_of_thm induct :: vs @ Ps @ ["correctness"]), thm), []) thy
      in
        Extraction.add_realizers_i
          [mk_realizer thy' (vs @ Ps) params' ((induct, thm'), r)] thy'
      end;

    (** realizer for elimination rules **)

    val case_names = map (fst o dest_Const o head_of o fst o HOLogic.dest_eq o
      HOLogic.dest_Trueprop o prop_of o hd) (get #case_thms dt_info);

    fun add_elim_realizer Ps
      (((((elim, elimR), intrs), case_thms), case_name), dummy) thy =
      let
        val (prem :: prems) = prems_of elim;
        fun reorder1 (p, intr) =
          Library.foldl (fn (t, ((s, _), T)) => all T $ lambda (Free (s, T)) t)
            (strip_all p, Term.add_vars (prop_of intr) [] \\ params');
        fun reorder2 (intr, i) =
          let
            val fs1 = term_vars (prop_of intr) \\ params;
            val fs2 = Term.add_vars (prop_of intr) [] \\ params'
          in Library.foldl (fn (t, x) => lambda (Var x) t)
            (list_comb (Bound (i + length fs1), fs1), fs2)
          end;
        val p = Logic.list_implies
          (map reorder1 (prems ~~ intrs) @ [prem], concl_of elim);
        val T' = Extraction.etype_of thy (vs @ Ps) [] p;
        val T = if dummy then (HOLogic.unitT --> body_type T') --> T' else T';
        val Ts = map (Extraction.etype_of thy (vs @ Ps) []) (prems_of elim);
        val r = if null Ps then Extraction.nullt
          else list_abs (map (pair "x") Ts, list_comb (Const (case_name, T),
            (if dummy then
               [Abs ("x", HOLogic.unitT, Const ("arbitrary", body_type T))]
             else []) @
            map reorder2 (intrs ~~ (length prems - 1 downto 0)) @
            [Bound (length prems)]));
        val rlz = strip_all (Logic.unvarify
          (Extraction.realizes_of thy (vs @ Ps) r (prop_of elim)));
        val rews = map mk_meta_eq case_thms;
        val thm = OldGoals.simple_prove_goal_cterm (cterm_of (sign_of thy) rlz) (fn prems =>
          [cut_facts_tac [hd prems] 1,
           etac elimR 1,
           ALLGOALS (EVERY' [etac Pair_inject, asm_simp_tac HOL_basic_ss]),
           rewrite_goals_tac rews,
           REPEAT ((resolve_tac prems THEN_ALL_NEW (ObjectLogic.atomize_tac THEN'
             DEPTH_SOLVE_1 o FIRST' [atac, etac allE, etac impE])) 1)]);
        val (thy', thm') = PureThy.store_thm ((space_implode "_"
          (Thm.name_of_thm elim :: vs @ Ps @ ["correctness"]), thm), []) thy
      in
        Extraction.add_realizers_i
          [mk_realizer thy' (vs @ Ps) params' ((elim, thm'), r)] thy'
      end;

    (** add realizers to theory **)

    val rintr_thms = List.concat (map (fn (_, rs) => map (fn r => List.nth
      (#intrs ind_info, find_index_eq r intrs)) rs) rss);
    val thy4 = Library.foldl add_ind_realizer (thy3, subsets Ps);
    val thy5 = Extraction.add_realizers_i
      (map (mk_realizer thy4 vs params')
         (map (fn ((rule, rrule), c) => ((rule, rrule), list_comb (c,
            map Var (rev (Term.add_vars (prop_of rule) []) \\ params')))) 
              (List.concat (map snd rss) ~~ rintr_thms ~~ List.concat constrss))) thy4;
    val elimps = List.mapPartial (fn (s, intrs) => if s mem rsets then
        Option.map (rpair intrs) (find_first (fn (thm, _) =>
          s mem term_consts (hd (prems_of thm))) (elims ~~ #elims ind_info))
      else NONE) rss;
    val thy6 = Library.foldl (fn (thy, p as (((((elim, _), _), _), _), _)) => thy |>
      add_elim_realizer [] p |> add_elim_realizer [fst (fst (dest_Var
        (HOLogic.dest_Trueprop (concl_of elim))))] p) (thy5,
           elimps ~~ get #case_thms dt_info ~~ case_names ~~ dummies)

  in Theory.restore_naming thy thy6 end;

fun add_ind_realizers name rsets thy =
  let
    val (_, {intrs, induct, raw_induct, elims, ...}) =
      (case InductivePackage.get_inductive thy name of
         NONE => error ("Unknown inductive set " ^ quote name)
       | SOME info => info);
    val _ $ (_ $ _ $ S) = concl_of (hd intrs);
    val vss = sort (int_ord o pairself length)
      (subsets (map fst (relevant_vars S)))
  in
    Library.foldl (add_ind_realizer rsets intrs induct raw_induct elims) (thy, vss)
  end

fun rlz_attrib arg (thy, thm) =
  let
    fun err () = error "ind_realizer: bad rule";
    val sets =
      (case HOLogic.dest_conj (HOLogic.dest_Trueprop (concl_of thm)) of
           [_] => [set_of (HOLogic.dest_Trueprop (hd (prems_of thm)))]
         | xs => map (set_of o fst o HOLogic.dest_imp) xs)
         handle TERM _ => err () | Empty => err ();
  in 
    (add_ind_realizers (hd sets) (case arg of
        NONE => sets | SOME NONE => []
      | SOME (SOME sets') => sets \\ sets')
      thy, thm)
  end;

val rlz_attrib_global = Attrib.syntax
 ((Scan.option (Scan.lift (Args.$$$ "irrelevant") |--
    Scan.option (Scan.lift (Args.colon) |--
      Scan.repeat1 Args.global_const))) >> rlz_attrib);

val setup = [Attrib.add_attributes [("ind_realizer",
  (rlz_attrib_global, K Attrib.undef_local_attribute),
  "add realizers for inductive set")]];

end;