src/HOL/Tools/datatype_realizer.ML
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Sat, 15 Nov 2008 21:31:17 +0100
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Thm.proof_of returns proof_body;
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(*  Title:      HOL/Tools/datatype_realizer.ML
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    ID:         $Id$
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    Author:     Stefan Berghofer, TU Muenchen
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Porgram extraction from proofs involving datatypes:
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Realizers for induction and case analysis
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*)
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signature DATATYPE_REALIZER =
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sig
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  val add_dt_realizers: string list -> theory -> theory
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  val setup: theory -> theory
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end;
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structure DatatypeRealizer : DATATYPE_REALIZER =
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struct
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open DatatypeAux;
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fun subsets i j = if i <= j then
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       let val is = subsets (i+1) j
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       in map (fn ks => i::ks) is @ is end
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     else [[]];
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fun forall_intr_prf (t, prf) =
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  let val (a, T) = (case t of Var ((a, _), T) => (a, T) | Free p => p)
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  in Abst (a, SOME T, Proofterm.prf_abstract_over t prf) end;
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fun prf_of thm =
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  Reconstruct.reconstruct_proof (Thm.theory_of_thm thm) (Thm.prop_of thm)
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    (Proofterm.proof_of (Thm.proof_of thm));
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fun prf_subst_vars inst =
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  Proofterm.map_proof_terms (subst_vars ([], inst)) I;
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fun is_unit t = snd (strip_type (fastype_of t)) = HOLogic.unitT;
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fun tname_of (Type (s, _)) = s
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  | tname_of _ = "";
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fun mk_realizes T = Const ("realizes", T --> HOLogic.boolT --> HOLogic.boolT);
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fun make_ind sorts ({descr, rec_names, rec_rewrites, induction, ...} : datatype_info) is thy =
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  let
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    val recTs = get_rec_types descr sorts;
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    val pnames = if length descr = 1 then ["P"]
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      else map (fn i => "P" ^ string_of_int i) (1 upto length descr);
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    val rec_result_Ts = map (fn ((i, _), P) =>
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      if i mem is then TFree ("'" ^ P, HOLogic.typeS) else HOLogic.unitT)
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        (descr ~~ pnames);
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    fun make_pred i T U r x =
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      if i mem is then
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        Free (List.nth (pnames, i), T --> U --> HOLogic.boolT) $ r $ x
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      else Free (List.nth (pnames, i), U --> HOLogic.boolT) $ x;
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    fun mk_all i s T t =
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      if i mem is then list_all_free ([(s, T)], t) else t;
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    val (prems, rec_fns) = split_list (List.concat (snd (foldl_map
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      (fn (j, ((i, (_, _, constrs)), T)) => foldl_map (fn (j, (cname, cargs)) =>
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        let
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          val Ts = map (typ_of_dtyp descr sorts) cargs;
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          val tnames = Name.variant_list pnames (DatatypeProp.make_tnames Ts);
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          val recs = List.filter (is_rec_type o fst o fst) (cargs ~~ tnames ~~ Ts);
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          val frees = tnames ~~ Ts;
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          fun mk_prems vs [] = 
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                let
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                  val rT = List.nth (rec_result_Ts, i);
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                  val vs' = filter_out is_unit vs;
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                  val f = mk_Free "f" (map fastype_of vs' ---> rT) j;
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                  val f' = Envir.eta_contract (list_abs_free
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                    (map dest_Free vs, if i mem is then list_comb (f, vs')
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                      else HOLogic.unit));
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                in (HOLogic.mk_Trueprop (make_pred i rT T (list_comb (f, vs'))
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                  (list_comb (Const (cname, Ts ---> T), map Free frees))), f')
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                end
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            | mk_prems vs (((dt, s), T) :: ds) = 
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                let
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                  val k = body_index dt;
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                  val (Us, U) = strip_type T;
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                  val i = length Us;
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                  val rT = List.nth (rec_result_Ts, k);
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                  val r = Free ("r" ^ s, Us ---> rT);
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                  val (p, f) = mk_prems (vs @ [r]) ds
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                in (mk_all k ("r" ^ s) (Us ---> rT) (Logic.mk_implies
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                  (list_all (map (pair "x") Us, HOLogic.mk_Trueprop
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                    (make_pred k rT U (app_bnds r i)
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                      (app_bnds (Free (s, T)) i))), p)), f)
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                end
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        in (j + 1,
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          apfst (curry list_all_free frees) (mk_prems (map Free frees) recs))
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        end) (j, constrs)) (1, descr ~~ recTs))));
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    fun mk_proj j [] t = t
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      | mk_proj j (i :: is) t = if null is then t else
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          if (j: int) = i then HOLogic.mk_fst t
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          else mk_proj j is (HOLogic.mk_snd t);
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    val tnames = DatatypeProp.make_tnames recTs;
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    val fTs = map fastype_of rec_fns;
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    val ps = map (fn ((((i, _), T), U), s) => Abs ("x", T, make_pred i U T
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      (list_comb (Const (s, fTs ---> T --> U), rec_fns) $ Bound 0) (Bound 0)))
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        (descr ~~ recTs ~~ rec_result_Ts ~~ rec_names);
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    val r = if null is then Extraction.nullt else
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      foldr1 HOLogic.mk_prod (List.mapPartial (fn (((((i, _), T), U), s), tname) =>
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        if i mem is then SOME
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          (list_comb (Const (s, fTs ---> T --> U), rec_fns) $ Free (tname, T))
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        else NONE) (descr ~~ recTs ~~ rec_result_Ts ~~ rec_names ~~ tnames));
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    val concl = HOLogic.mk_Trueprop (foldr1 (HOLogic.mk_binop "op &")
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      (map (fn ((((i, _), T), U), tname) =>
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        make_pred i U T (mk_proj i is r) (Free (tname, T)))
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          (descr ~~ recTs ~~ rec_result_Ts ~~ tnames)));
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    val cert = cterm_of thy;
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    val inst = map (pairself cert) (map head_of (HOLogic.dest_conj
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      (HOLogic.dest_Trueprop (concl_of induction))) ~~ ps);
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    val thm = OldGoals.simple_prove_goal_cterm (cert (Logic.list_implies (prems, concl)))
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      (fn prems =>
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         [rewrite_goals_tac (map mk_meta_eq [fst_conv, snd_conv]),
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          rtac (cterm_instantiate inst induction) 1,
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          ALLGOALS ObjectLogic.atomize_prems_tac,
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          rewrite_goals_tac (@{thm o_def} :: map mk_meta_eq rec_rewrites),
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          REPEAT ((resolve_tac prems THEN_ALL_NEW (fn i =>
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            REPEAT (etac allE i) THEN atac i)) 1)]);
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    val ind_name = Thm.get_name induction;
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    val vs = map (fn i => List.nth (pnames, i)) is;
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    val (thm', thy') = thy
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      |> Sign.absolute_path
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      |> PureThy.store_thm (space_implode "_" (ind_name :: vs @ ["correctness"]), thm)
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      ||> Sign.restore_naming thy;
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    val ivs = rev (Term.add_vars (Logic.varify (DatatypeProp.make_ind [descr] sorts)) []);
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    val rvs = rev (Thm.fold_terms Term.add_vars thm' []);
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    val ivs1 = map Var (filter_out (fn (_, T) =>
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      tname_of (body_type T) mem ["set", "bool"]) ivs);
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    val ivs2 = map (fn (ixn, _) => Var (ixn, valOf (AList.lookup (op =) rvs ixn))) ivs;
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    val prf = foldr forall_intr_prf
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     (foldr (fn ((f, p), prf) =>
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        (case head_of (strip_abs_body f) of
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           Free (s, T) =>
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             let val T' = Logic.varifyT T
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             in Abst (s, SOME T', Proofterm.prf_abstract_over
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               (Var ((s, 0), T')) (AbsP ("H", SOME p, prf)))
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             end
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         | _ => AbsP ("H", SOME p, prf)))
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           (Proofterm.proof_combP
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             (prf_of thm', map PBound (length prems - 1 downto 0))) (rec_fns ~~ prems_of thm)) ivs2;
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    val r' = if null is then r else Logic.varify (foldr (uncurry lambda)
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      r (map Logic.unvarify ivs1 @ filter_out is_unit
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          (map (head_of o strip_abs_body) rec_fns)));
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  in Extraction.add_realizers_i [(ind_name, (vs, r', prf))] thy' end;
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fun make_casedists sorts ({index, descr, case_name, case_rewrites, exhaustion, ...} : datatype_info) thy =
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  let
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    val cert = cterm_of thy;
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    val rT = TFree ("'P", HOLogic.typeS);
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    val rT' = TVar (("'P", 0), HOLogic.typeS);
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    fun make_casedist_prem T (cname, cargs) =
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      let
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        val Ts = map (typ_of_dtyp descr sorts) cargs;
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        val frees = Name.variant_list ["P", "y"] (DatatypeProp.make_tnames Ts) ~~ Ts;
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        val free_ts = map Free frees;
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        val r = Free ("r" ^ NameSpace.base cname, Ts ---> rT)
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      in (r, list_all_free (frees, Logic.mk_implies (HOLogic.mk_Trueprop
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        (HOLogic.mk_eq (Free ("y", T), list_comb (Const (cname, Ts ---> T), free_ts))),
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          HOLogic.mk_Trueprop (Free ("P", rT --> HOLogic.boolT) $
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            list_comb (r, free_ts)))))
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      end;
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    val SOME (_, _, constrs) = AList.lookup (op =) descr index;
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    val T = List.nth (get_rec_types descr sorts, index);
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    val (rs, prems) = split_list (map (make_casedist_prem T) constrs);
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    val r = Const (case_name, map fastype_of rs ---> T --> rT);
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    val y = Var (("y", 0), Logic.legacy_varifyT T);
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    val y' = Free ("y", T);
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    val thm = OldGoals.prove_goalw_cterm [] (cert (Logic.list_implies (prems,
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      HOLogic.mk_Trueprop (Free ("P", rT --> HOLogic.boolT) $
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        list_comb (r, rs @ [y'])))))
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      (fn prems =>
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         [rtac (cterm_instantiate [(cert y, cert y')] exhaustion) 1,
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          ALLGOALS (EVERY'
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            [asm_simp_tac (HOL_basic_ss addsimps case_rewrites),
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             resolve_tac prems, asm_simp_tac HOL_basic_ss])]);
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    val exh_name = Thm.get_name exhaustion;
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    val (thm', thy') = thy
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      |> Sign.absolute_path
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      |> PureThy.store_thm (exh_name ^ "_P_correctness", thm)
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      ||> Sign.restore_naming thy;
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    val P = Var (("P", 0), rT' --> HOLogic.boolT);
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    val prf = forall_intr_prf (y, forall_intr_prf (P,
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      foldr (fn ((p, r), prf) =>
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        forall_intr_prf (Logic.legacy_varify r, AbsP ("H", SOME (Logic.varify p),
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          prf))) (Proofterm.proof_combP (prf_of thm',
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            map PBound (length prems - 1 downto 0))) (prems ~~ rs)));
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    val r' = Logic.legacy_varify (Abs ("y", Logic.legacy_varifyT T,
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      list_abs (map dest_Free rs, list_comb (r,
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        map Bound ((length rs - 1 downto 0) @ [length rs])))));
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  in Extraction.add_realizers_i
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    [(exh_name, (["P"], r', prf)),
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     (exh_name, ([], Extraction.nullt, prf_of exhaustion))] thy'
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  end;
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fun add_dt_realizers names thy =
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  if ! Proofterm.proofs < 2 then thy
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  else let
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    val _ = message "Adding realizers for induction and case analysis ..."
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    val infos = map (DatatypePackage.the_datatype thy) names;
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    val info :: _ = infos;
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  in
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    thy
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    |> fold_rev (make_ind (#sorts info) info) (subsets 0 (length (#descr info) - 1))
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    |> fold_rev (make_casedists (#sorts info)) infos
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  end;
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val setup = DatatypePackage.interpretation add_dt_realizers;
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end;