(* Title: HOL/Tools/datatype_realizer.ML
ID: $Id$
Author: Stefan Berghofer, TU Muenchen
Porgram extraction from proofs involving datatypes:
Realizers for induction and case analysis
*)
signature DATATYPE_REALIZER =
sig
val add_dt_realizers: string list -> theory -> theory
val setup: 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 =
Reconstruct.reconstruct_proof (Thm.theory_of_thm thm) (Thm.prop_of thm) (Thm.proof_of thm);
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 tname_of (Type (s, _)) = s
| tname_of _ = "";
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 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 (List.nth (pnames, i), T --> U --> HOLogic.boolT) $ r $ x
else Free (List.nth (pnames, i), 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 (List.concat (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 = Name.variant_list pnames (DatatypeProp.make_tnames Ts);
val recs = List.filter (is_rec_type o fst o fst) (cargs ~~ tnames ~~ Ts);
val frees = tnames ~~ Ts;
fun mk_prems vs [] =
let
val rT = List.nth (rec_result_Ts, i);
val vs' = filter_out is_unit vs;
val f = mk_Free "f" (map fastype_of vs' ---> rT) j;
val f' = Envir.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 = List.nth (rec_result_Ts, k);
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: int) = 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 (List.mapPartial (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 thy;
val inst = map (pairself cert) (map head_of (HOLogic.dest_conj
(HOLogic.dest_Trueprop (concl_of induction))) ~~ ps);
val thm = OldGoals.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_prems_tac,
rewrite_goals_tac (@{thm 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 ind_name = Thm.get_name induction;
val vs = map (fn i => List.nth (pnames, i)) is;
val (thm', thy') = thy
|> Sign.absolute_path
|> PureThy.store_thm (space_implode "_" (ind_name :: vs @ ["correctness"]), thm)
||> Sign.restore_naming thy;
val ivs = rev (Term.add_vars (Logic.varify (DatatypeProp.make_ind [descr] sorts)) []);
val rvs = rev (Thm.fold_terms Term.add_vars thm' []);
val ivs1 = map Var (filter_out (fn (_, T) =>
tname_of (body_type T) mem ["set", "bool"]) ivs);
val ivs2 = map (fn (ixn, _) => Var (ixn, valOf (AList.lookup (op =) rvs ixn))) ivs;
val prf = foldr forall_intr_prf
(foldr (fn ((f, p), prf) =>
(case head_of (strip_abs_body f) of
Free (s, T) =>
let val T' = Logic.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)))
(Proofterm.proof_combP
(prf_of thm', map PBound (length prems - 1 downto 0))) (rec_fns ~~ prems_of thm)) ivs2;
val r' = if null is then r else Logic.varify (foldr (uncurry lambda)
r (map Logic.unvarify ivs1 @ filter_out is_unit
(map (head_of o strip_abs_body) rec_fns)));
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 cert = cterm_of thy;
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 = Name.variant_list ["P", "y"] (DatatypeProp.make_tnames Ts) ~~ 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) = AList.lookup (op =) descr index;
val T = List.nth (get_rec_types descr sorts, index);
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), Logic.legacy_varifyT T);
val y' = Free ("y", T);
val thm = OldGoals.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 exh_name = Thm.get_name exhaustion;
val (thm', thy') = thy
|> Sign.absolute_path
|> PureThy.store_thm (exh_name ^ "_P_correctness", thm)
||> Sign.restore_naming thy;
val P = Var (("P", 0), rT' --> HOLogic.boolT);
val prf = forall_intr_prf (y, forall_intr_prf (P,
foldr (fn ((p, r), prf) =>
forall_intr_prf (Logic.legacy_varify r, AbsP ("H", SOME (Logic.varify p),
prf))) (Proofterm.proof_combP (prf_of thm',
map PBound (length prems - 1 downto 0))) (prems ~~ rs)));
val r' = Logic.legacy_varify (Abs ("y", Logic.legacy_varifyT T,
list_abs (map dest_Free rs, list_comb (r,
map Bound ((length rs - 1 downto 0) @ [length rs])))));
in Extraction.add_realizers_i
[(exh_name, (["P"], r', prf)),
(exh_name, ([], Extraction.nullt, prf_of exhaustion))] thy'
end;
fun add_dt_realizers names thy =
if ! Proofterm.proofs < 2 then thy
else let
val _ = message "Adding realizers for induction and case analysis ..."
val infos = map (DatatypePackage.the_datatype thy) names;
val info :: _ = infos;
in
thy
|> fold_rev (make_ind (#sorts info) info) (subsets 0 (length (#descr info) - 1))
|> fold_rev (make_casedists (#sorts info)) infos
end;
val setup = DatatypePackage.interpretation add_dt_realizers;
end;