(* Title: HOL/Tools/datatype_realizer.ML
ID: $Id$
Author: Stefan Berghofer, TU Muenchen
License: GPL (GNU GENERAL PUBLIC LICENSE)
Porgram extraction from proofs involving datatypes:
Realizers for induction and case analysis
*)
signature DATATYPE_REALIZER =
sig
val add_dt_realizers: (string * sort) list ->
DatatypeAux.datatype_info list -> theory -> theory
end;
structure DatatypeRealizer : DATATYPE_REALIZER =
struct
open DatatypeAux;
fun subsets i j = if i <= j then
let val is = subsets (i+1) j
in map (fn ks => i::ks) is @ is end
else [[]];
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 prf_of thm =
let val {sign, prop, der = (_, prf), ...} = rep_thm thm
in Reconstruct.reconstruct_proof sign prop prf end;
fun prf_subst_vars inst =
Proofterm.map_proof_terms (subst_vars ([], inst)) I;
fun is_unit t = snd (strip_type (fastype_of t)) = HOLogic.unitT;
fun mk_realizes T = Const ("realizes", T --> HOLogic.boolT --> HOLogic.boolT);
fun make_ind sorts ({descr, rec_names, rec_rewrites, induction, ...} : datatype_info) (is, thy) =
let
val sg = sign_of thy;
val recTs = get_rec_types descr sorts;
val pnames = if length descr = 1 then ["P"]
else map (fn i => "P" ^ string_of_int i) (1 upto length descr);
val rec_result_Ts = map (fn ((i, _), P) =>
if i mem is then TFree ("'" ^ P, HOLogic.typeS) else HOLogic.unitT)
(descr ~~ pnames);
fun make_pred i T U r x =
if i mem is then
Free (nth_elem (i, pnames), T --> U --> HOLogic.boolT) $ r $ x
else Free (nth_elem (i, pnames), U --> HOLogic.boolT) $ x;
fun mk_all i s T t =
if i mem is then list_all_free ([(s, T)], t) else t;
val (prems, rec_fns) = split_list (flat (snd (foldl_map
(fn (j, ((i, (_, _, constrs)), T)) => foldl_map (fn (j, (cname, cargs)) =>
let
val Ts = map (typ_of_dtyp descr sorts) cargs;
val tnames = variantlist (DatatypeProp.make_tnames Ts, pnames);
val recs = filter (is_rec_type o fst o fst) (cargs ~~ tnames ~~ Ts);
val frees = tnames ~~ Ts;
fun mk_prems vs [] =
let
val rT = nth_elem (i, rec_result_Ts);
val vs' = filter_out is_unit vs;
val f = mk_Free "f" (map fastype_of vs' ---> rT) j;
val f' = Pattern.eta_contract (list_abs_free
(map dest_Free vs, if i mem is then list_comb (f, vs')
else HOLogic.unit));
in (HOLogic.mk_Trueprop (make_pred i rT T (list_comb (f, vs'))
(list_comb (Const (cname, Ts ---> T), map Free frees))), f')
end
| mk_prems vs (((dt, s), T) :: ds) =
let
val k = body_index dt;
val (Us, U) = strip_type T;
val i = length Us;
val rT = nth_elem (k, rec_result_Ts);
val r = Free ("r" ^ s, Us ---> rT);
val (p, f) = mk_prems (vs @ [r]) ds
in (mk_all k ("r" ^ s) (Us ---> rT) (Logic.mk_implies
(list_all (map (pair "x") Us, HOLogic.mk_Trueprop
(make_pred k rT U (app_bnds r i)
(app_bnds (Free (s, T)) i))), p)), f)
end
in (j + 1,
apfst (curry list_all_free frees) (mk_prems (map Free frees) recs))
end) (j, constrs)) (1, descr ~~ recTs))));
fun mk_proj j [] t = t
| mk_proj j (i :: is) t = if null is then t else
if j = i then HOLogic.mk_fst t
else mk_proj j is (HOLogic.mk_snd t);
val tnames = DatatypeProp.make_tnames recTs;
val fTs = map fastype_of rec_fns;
val ps = map (fn ((((i, _), T), U), s) => Abs ("x", T, make_pred i U T
(list_comb (Const (s, fTs ---> T --> U), rec_fns) $ Bound 0) (Bound 0)))
(descr ~~ recTs ~~ rec_result_Ts ~~ rec_names);
val r = if null is then Extraction.nullt else
foldr1 HOLogic.mk_prod (mapfilter (fn (((((i, _), T), U), s), tname) =>
if i mem is then Some
(list_comb (Const (s, fTs ---> T --> U), rec_fns) $ Free (tname, T))
else None) (descr ~~ recTs ~~ rec_result_Ts ~~ rec_names ~~ tnames));
val concl = HOLogic.mk_Trueprop (foldr1 (HOLogic.mk_binop "op &")
(map (fn ((((i, _), T), U), tname) =>
make_pred i U T (mk_proj i is r) (Free (tname, T)))
(descr ~~ recTs ~~ rec_result_Ts ~~ tnames)));
val cert = cterm_of sg;
val inst = map (pairself cert) (map head_of (HOLogic.dest_conj
(HOLogic.dest_Trueprop (concl_of induction))) ~~ ps);
val thm = simple_prove_goal_cterm (cert (Logic.list_implies (prems, concl)))
(fn prems =>
[rewrite_goals_tac (map mk_meta_eq [fst_conv, snd_conv]),
rtac (cterm_instantiate inst induction) 1,
ALLGOALS ObjectLogic.atomize_tac,
rewrite_goals_tac (o_def :: map mk_meta_eq rec_rewrites),
REPEAT ((resolve_tac prems THEN_ALL_NEW (fn i =>
REPEAT (etac allE i) THEN atac i)) 1)]);
val {path, ...} = Sign.rep_sg sg;
val ind_name = Thm.name_of_thm induction;
val vs = map (fn i => nth_elem (i, pnames)) is;
val (thy', thm') = thy
|> Theory.absolute_path
|> PureThy.store_thm
((space_implode "_" (ind_name :: vs @ ["correctness"]), thm), [])
|>> Theory.add_path (NameSpace.pack (if_none path []));
val inst = map (fn ((((i, _), s), T), U) => ((s, 0), if i mem is then
Abs ("r", U, Abs ("x", T, mk_realizes U $ Bound 1 $
(Var ((s, 0), T --> HOLogic.boolT) $ Bound 0)))
else Abs ("x", T, mk_realizes Extraction.nullT $ Extraction.nullt $
(Var ((s, 0), T --> HOLogic.boolT) $
Bound 0)))) (descr ~~ pnames ~~ map Type.varifyT recTs ~~
map Type.varifyT rec_result_Ts);
val ivs = map Var (Drule.vars_of_terms
[Logic.varify (DatatypeProp.make_ind [descr] sorts)]);
val prf = foldr forall_intr_prf (ivs,
prf_subst_vars inst (foldr (fn ((f, p), prf) =>
(case head_of (strip_abs_body f) of
Free (s, T) =>
let val T' = Type.varifyT T
in Abst (s, Some T', Proofterm.prf_abstract_over
(Var ((s, 0), T')) (AbsP ("H", Some p, prf)))
end
| _ => AbsP ("H", Some p, prf)))
(rec_fns ~~ prems_of thm, Proofterm.proof_combP
(prf_of thm', map PBound (length prems - 1 downto 0)))));
val r' = if null is then r else Logic.varify (foldr (uncurry lambda)
(map Logic.unvarify ivs @ filter_out is_unit
(map (head_of o strip_abs_body) rec_fns), r));
in Extraction.add_realizers_i [(ind_name, (vs, r', prf))] thy' end;
fun make_casedists sorts ({index, descr, case_name, case_rewrites, exhaustion, ...} : datatype_info, thy) =
let
val sg = sign_of thy;
val sorts = map (rpair HOLogic.typeS) (distinct (flat (map
(fn (_, (_, ds, _)) => mapfilter (try dest_DtTFree) ds) descr)));
val cert = cterm_of sg;
val rT = TFree ("'P", HOLogic.typeS);
val rT' = TVar (("'P", 0), HOLogic.typeS);
fun make_casedist_prem T (cname, cargs) =
let
val Ts = map (typ_of_dtyp descr sorts) cargs;
val frees = variantlist
(DatatypeProp.make_tnames Ts, ["P", "y"]) ~~ Ts;
val free_ts = map Free frees;
val r = Free ("r" ^ NameSpace.base cname, Ts ---> rT)
in (r, list_all_free (frees, Logic.mk_implies (HOLogic.mk_Trueprop
(HOLogic.mk_eq (Free ("y", T), list_comb (Const (cname, Ts ---> T), free_ts))),
HOLogic.mk_Trueprop (Free ("P", rT --> HOLogic.boolT) $
list_comb (r, free_ts)))))
end;
val Some (_, _, constrs) = assoc (descr, index);
val T = nth_elem (index, get_rec_types descr sorts);
val (rs, prems) = split_list (map (make_casedist_prem T) constrs);
val r = Const (case_name, map fastype_of rs ---> T --> rT);
val y = Var (("y", 0), Type.varifyT T);
val y' = Free ("y", T);
val thm = prove_goalw_cterm [] (cert (Logic.list_implies (prems,
HOLogic.mk_Trueprop (Free ("P", rT --> HOLogic.boolT) $
list_comb (r, rs @ [y'])))))
(fn prems =>
[rtac (cterm_instantiate [(cert y, cert y')] exhaustion) 1,
ALLGOALS (EVERY'
[asm_simp_tac (HOL_basic_ss addsimps case_rewrites),
resolve_tac prems, asm_simp_tac HOL_basic_ss])]);
val {path, ...} = Sign.rep_sg sg;
val exh_name = Thm.name_of_thm exhaustion;
val (thy', thm') = thy
|> Theory.absolute_path
|> PureThy.store_thm ((exh_name ^ "_P_correctness", thm), [])
|>> Theory.add_path (NameSpace.pack (if_none path []));
val P = Var (("P", 0), HOLogic.boolT);
val prf = forall_intr_prf (y, forall_intr_prf (P,
prf_subst_vars [(("P", 0), Abs ("r", rT',
mk_realizes rT' $ Bound 0 $ P))] (foldr (fn ((p, r), prf) =>
forall_intr_prf (Logic.varify r, AbsP ("H", Some (Logic.varify p),
prf))) (prems ~~ rs, Proofterm.proof_combP (prf_of thm',
map PBound (length prems - 1 downto 0))))));
val r' = Logic.varify (Abs ("y", Type.varifyT T,
Abs ("P", HOLogic.boolT, list_abs (map dest_Free rs, list_comb (r,
map Bound ((length rs - 1 downto 0) @ [length rs + 1]))))));
val prf' = forall_intr_prf (y, forall_intr_prf (P, prf_subst_vars
[(("P", 0), mk_realizes Extraction.nullT $ Extraction.nullt $ P)]
(prf_of exhaustion)));
in Extraction.add_realizers_i
[(exh_name, (["P"], r', prf)),
(exh_name, ([], Extraction.nullt, prf'))] thy'
end;
fun add_dt_realizers sorts infos thy = if !proofs < 2 then thy else
(message "Adding realizers for induction and case analysis ..."; thy
|> curry (foldr (make_ind sorts (hd infos)))
(subsets 0 (length (#descr (hd infos)) - 1))
|> curry (foldr (make_casedists sorts)) infos);
end;