(* Title: HOL/Tools/Datatype/datatype_realizer.ML
Author: Stefan Berghofer, TU Muenchen
Program extraction from proofs involving datatypes:
realizers for induction and case analysis.
*)
signature DATATYPE_REALIZER =
sig
val add_dt_realizers: Datatype.config -> string list -> theory -> theory
val setup: theory -> theory
end;
structure Datatype_Realizer : DATATYPE_REALIZER =
struct
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 prf_of thm =
Reconstruct.reconstruct_proof (Thm.theory_of_thm thm) (Thm.prop_of thm) (Thm.proof_of thm);
fun is_unit t = body_type (fastype_of t) = HOLogic.unitT;
fun tname_of (Type (s, _)) = s
| tname_of _ = "";
fun make_ind sorts ({descr, rec_names, rec_rewrites, induct, ...} : Datatype_Aux.info) is thy =
let
val recTs = Datatype_Aux.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 member (op =) is i then TFree ("'" ^ P, HOLogic.typeS) else HOLogic.unitT)
(descr ~~ pnames);
fun make_pred i T U r x =
if member (op =) is i then
Free (nth pnames i, T --> U --> HOLogic.boolT) $ r $ x
else Free (nth pnames i, U --> HOLogic.boolT) $ x;
fun mk_all i s T t =
if member (op =) is i then list_all_free ([(s, T)], t) else t;
val (prems, rec_fns) = split_list (flat (fst (fold_map
(fn ((i, (_, _, constrs)), T) => fold_map (fn (cname, cargs) => fn j =>
let
val Ts = map (Datatype_Aux.typ_of_dtyp descr sorts) cargs;
val tnames = Name.variant_list pnames (Datatype_Prop.make_tnames Ts);
val recs = filter (Datatype_Aux.is_rec_type o fst o fst) (cargs ~~ tnames ~~ Ts);
val frees = tnames ~~ Ts;
fun mk_prems vs [] =
let
val rT = nth (rec_result_Ts) i;
val vs' = filter_out is_unit vs;
val f = Datatype_Aux.mk_Free "f" (map fastype_of vs' ---> rT) j;
val f' = Envir.eta_contract (list_abs_free
(map dest_Free vs, if member (op =) is i 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 = Datatype_Aux.body_index dt;
val (Us, U) = strip_type T;
val i = length Us;
val rT = 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 (Datatype_Aux.app_bnds r i)
(Datatype_Aux.app_bnds (Free (s, T)) i))), p)), f)
end
in (apfst (curry list_all_free frees) (mk_prems (map Free frees) recs), j + 1) end)
constrs) (descr ~~ recTs) 1)));
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 = Datatype_Prop.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 (map_filter (fn (((((i, _), T), U), s), tname) =>
if member (op =) is i 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 @{const_name HOL.conj})
(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 induct))) ~~ ps);
val thm = Goal.prove_internal (map cert prems) (cert concl)
(fn prems =>
EVERY [
rewrite_goals_tac (map mk_meta_eq [@{thm fst_conv}, @{thm snd_conv}]),
rtac (cterm_instantiate inst induct) 1,
ALLGOALS Object_Logic.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)])
|> Drule.export_without_context;
val ind_name = Thm.derivation_name induct;
val vs = map (nth pnames) is;
val (thm', thy') = thy
|> Sign.root_path
|> Global_Theory.store_thm
(Binding.qualified_name (space_implode "_" (ind_name :: vs @ ["correctness"])), thm)
||> Sign.restore_naming thy;
val ivs = rev (Term.add_vars (Logic.varify_global (Datatype_Prop.make_ind [descr] sorts)) []);
val rvs = rev (Thm.fold_terms Term.add_vars thm' []);
val ivs1 = map Var (filter_out (fn (_, T) =>
@{type_name bool} = tname_of (body_type T)) ivs);
val ivs2 = map (fn (ixn, _) => Var (ixn, the (AList.lookup (op =) rvs ixn))) ivs;
val prf =
Extraction.abs_corr_shyps thy' induct vs ivs2
(fold_rev (fn (f, p) => fn prf =>
(case head_of (strip_abs_body f) of
Free (s, T) =>
let val T' = Logic.varifyT_global 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_global (fold_rev lambda
(map Logic.unvarify_global ivs1 @ 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, exhaust, ...} : Datatype_Aux.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 (Datatype_Aux.typ_of_dtyp descr sorts) cargs;
val frees = Name.variant_list ["P", "y"] (Datatype_Prop.make_tnames Ts) ~~ Ts;
val free_ts = map Free frees;
val r = Free ("r" ^ Long_Name.base_name 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 = nth (Datatype_Aux.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.varifyT_global T);
val y' = Free ("y", T);
val thm = Goal.prove_internal (map cert prems)
(cert (HOLogic.mk_Trueprop (Free ("P", rT --> HOLogic.boolT) $ list_comb (r, rs @ [y']))))
(fn prems =>
EVERY [
rtac (cterm_instantiate [(cert y, cert y')] exhaust) 1,
ALLGOALS (EVERY'
[asm_simp_tac (HOL_basic_ss addsimps case_rewrites),
resolve_tac prems, asm_simp_tac HOL_basic_ss])])
|> Drule.export_without_context;
val exh_name = Thm.derivation_name exhaust;
val (thm', thy') = thy
|> Sign.root_path
|> Global_Theory.store_thm (Binding.qualified_name (exh_name ^ "_P_correctness"), thm)
||> Sign.restore_naming thy;
val P = Var (("P", 0), rT' --> HOLogic.boolT);
val prf = Extraction.abs_corr_shyps thy' exhaust ["P"] [y, P]
(fold_rev (fn (p, r) => fn prf =>
Proofterm.forall_intr_proof' (Logic.varify_global r)
(AbsP ("H", SOME (Logic.varify_global p), prf)))
(prems ~~ rs)
(Proofterm.proof_combP (prf_of thm', map PBound (length prems - 1 downto 0))));
val prf' = Extraction.abs_corr_shyps thy' exhaust []
(map Var (Term.add_vars (prop_of exhaust) [])) (prf_of exhaust);
val r' = Logic.varify_global (Abs ("y", 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'))] thy'
end;
fun add_dt_realizers config names thy =
if not (Proofterm.proofs_enabled ()) then thy
else
let
val _ = Datatype_Aux.message config "Adding realizers for induction and case analysis ...";
val infos = map (Datatype.the_info 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 = Datatype.interpretation add_dt_realizers;
end;