(* 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 SOME T = assoc (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 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)) (assoc (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, valOf (assoc (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 (rss, r) =
let
val _ $ (_ $ _ $ S) = concl_of r;
val (Const (s, _), _) = strip_comb S;
val rs = getOpt (assoc (rss, s), []);
in AList.update (op =) (s, rs @ [r]) rss 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.foldl 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
apsnd (apsnd SOME) o add_dummies (DatatypePackage.add_datatype_i false false
(map #2 dts)) (map (pair false) dts) []) |>>
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 = 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 = 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;