author berghofe Wed Aug 07 16:46:15 2002 +0200 (2002-08-07) changeset 13467 d66b526192bf parent 13466 42766aa25460 child 13468 71118807d303
Module for defining realizers for induction and case analysis theorems
for datatypes.
```     1.1 --- /dev/null	Thu Jan 01 00:00:00 1970 +0000
1.2 +++ b/src/HOL/Tools/datatype_realizer.ML	Wed Aug 07 16:46:15 2002 +0200
1.3 @@ -0,0 +1,258 @@
1.4 +(*  Title:      HOL/Tools/datatype_realizer.ML
1.5 +    ID:         \$Id\$
1.6 +    Author:     Stefan Berghofer, TU Muenchen
1.8 +
1.9 +Porgram extraction from proofs involving datatypes:
1.10 +Realizers for induction and case analysis
1.11 +*)
1.12 +
1.13 +signature DATATYPE_REALIZER =
1.14 +sig
1.15 +  val add_dt_realizers: (string * sort) list ->
1.16 +    DatatypeAux.datatype_info list -> theory -> theory
1.17 +end;
1.18 +
1.19 +structure DatatypeRealizer : DATATYPE_REALIZER =
1.20 +struct
1.21 +
1.22 +open DatatypeAux;
1.23 +
1.24 +fun subsets i j = if i <= j then
1.25 +       let val is = subsets (i+1) j
1.26 +       in map (fn ks => i::ks) is @ is end
1.27 +     else [[]];
1.28 +
1.29 +fun forall_intr_prf (t, prf) =
1.30 +  let val (a, T) = (case t of Var ((a, _), T) => (a, T) | Free p => p)
1.31 +  in Abst (a, Some T, Proofterm.prf_abstract_over t prf) end;
1.32 +
1.33 +fun prove_goal' sg p f =
1.34 +  let
1.35 +    val (_, As, B) = Logic.strip_horn p;
1.36 +    val cAs = map (cterm_of sg) As;
1.37 +    val asms = map (norm_hhf_rule o assume) cAs;
1.38 +    fun check thm = if nprems_of thm > 0 then
1.39 +      error "prove_goal': unsolved goals" else thm
1.40 +  in
1.41 +    standard (implies_intr_list cAs
1.42 +      (check (Seq.hd (EVERY (f asms) (trivial (cterm_of sg B))))))
1.43 +  end;
1.44 +
1.45 +fun prf_of thm =
1.46 +  let val {sign, prop, der = (_, prf), ...} = rep_thm thm
1.47 +  in Reconstruct.reconstruct_proof sign prop prf end;
1.48 +
1.49 +fun prf_subst_vars inst =
1.50 +  Proofterm.map_proof_terms (subst_vars ([], inst)) I;
1.51 +
1.52 +fun is_unit t = snd (strip_type (fastype_of t)) = HOLogic.unitT;
1.53 +
1.54 +fun mk_realizes T = Const ("realizes", T --> HOLogic.boolT --> HOLogic.boolT);
1.55 +
1.56 +fun make_ind sorts ({descr, rec_names, rec_rewrites, induction, ...} : datatype_info) (is, thy) =
1.57 +  let
1.58 +    val sg = sign_of thy;
1.59 +    val recTs = get_rec_types descr sorts;
1.60 +    val pnames = if length descr = 1 then ["P"]
1.61 +      else map (fn i => "P" ^ string_of_int i) (1 upto length descr);
1.62 +
1.63 +    val rec_result_Ts = map (fn ((i, _), P) =>
1.64 +      if i mem is then TFree ("'" ^ P, HOLogic.typeS) else HOLogic.unitT)
1.65 +        (descr ~~ pnames);
1.66 +
1.67 +    fun make_pred i T U r x =
1.68 +      if i mem is then
1.69 +        Free (nth_elem (i, pnames), T --> U --> HOLogic.boolT) \$ r \$ x
1.70 +      else Free (nth_elem (i, pnames), U --> HOLogic.boolT) \$ x;
1.71 +
1.72 +    fun mk_all i s T t =
1.73 +      if i mem is then list_all_free ([(s, T)], t) else t;
1.74 +
1.75 +    val (prems, rec_fns) = split_list (flat (snd (foldl_map
1.76 +      (fn (j, ((i, (_, _, constrs)), T)) => foldl_map (fn (j, (cname, cargs)) =>
1.77 +        let
1.78 +          val Ts = map (typ_of_dtyp descr sorts) cargs;
1.79 +          val tnames = variantlist (DatatypeProp.make_tnames Ts, pnames);
1.80 +          val recs = filter (is_rec_type o fst o fst) (cargs ~~ tnames ~~ Ts);
1.81 +          val frees = tnames ~~ Ts;
1.82 +
1.83 +          fun mk_prems vs [] =
1.84 +                let
1.85 +                  val rT = nth_elem (i, rec_result_Ts);
1.86 +                  val vs' = filter_out is_unit vs;
1.87 +                  val f = mk_Free "f" (map fastype_of vs' ---> rT) j;
1.88 +                  val f' = Pattern.eta_contract (list_abs_free
1.89 +                    (map dest_Free vs, if i mem is then list_comb (f, vs')
1.90 +                      else HOLogic.unit));
1.91 +                in (HOLogic.mk_Trueprop (make_pred i rT T (list_comb (f, vs'))
1.92 +                  (list_comb (Const (cname, Ts ---> T), map Free frees))), f')
1.93 +                end
1.94 +            | mk_prems vs (((DtRec k, s), T) :: ds) =
1.95 +                let
1.96 +                  val rT = nth_elem (k, rec_result_Ts);
1.97 +                  val r = Free ("r" ^ s, rT);
1.98 +                  val (p, f) = mk_prems (vs @ [r]) ds
1.99 +                in (mk_all k ("r" ^ s) rT (Logic.mk_implies
1.100 +                  (HOLogic.mk_Trueprop (make_pred k rT T r (Free (s, T))), p)), f)
1.101 +                end
1.102 +            | mk_prems vs (((DtType ("fun", [_, DtRec k]), s),
1.103 +                  T' as Type ("fun", [T, U])) :: ds) =
1.104 +                let
1.105 +                  val rT = nth_elem (k, rec_result_Ts);
1.106 +                  val r = Free ("r" ^ s, T --> rT);
1.107 +                  val (p, f) = mk_prems (vs @ [r]) ds
1.108 +                in (mk_all k ("r" ^ s) (T --> rT) (Logic.mk_implies
1.109 +                  (all T \$ Abs ("x", T, HOLogic.mk_Trueprop (make_pred k rT U
1.110 +                    (r \$ Bound 0) (Free (s, T') \$ Bound 0))), p)), f)
1.111 +                end
1.112 +
1.113 +        in (j + 1,
1.114 +          apfst (curry list_all_free frees) (mk_prems (map Free frees) recs))
1.115 +        end) (j, constrs)) (1, descr ~~ recTs))));
1.116 +
1.117 +    fun mk_proj j [] t = t
1.118 +      | mk_proj j (i :: is) t = if null is then t else
1.119 +          if j = i then HOLogic.mk_fst t
1.120 +          else mk_proj j is (HOLogic.mk_snd t);
1.121 +
1.122 +    val tnames = DatatypeProp.make_tnames recTs;
1.123 +    val fTs = map fastype_of rec_fns;
1.124 +    val ps = map (fn ((((i, _), T), U), s) => Abs ("x", T, make_pred i U T
1.125 +      (list_comb (Const (s, fTs ---> T --> U), rec_fns) \$ Bound 0) (Bound 0)))
1.126 +        (descr ~~ recTs ~~ rec_result_Ts ~~ rec_names);
1.127 +    val r = if null is then Extraction.nullt else
1.128 +      foldr1 HOLogic.mk_prod (mapfilter (fn (((((i, _), T), U), s), tname) =>
1.129 +        if i mem is then Some
1.130 +          (list_comb (Const (s, fTs ---> T --> U), rec_fns) \$ Free (tname, T))
1.131 +        else None) (descr ~~ recTs ~~ rec_result_Ts ~~ rec_names ~~ tnames));
1.132 +    val concl = HOLogic.mk_Trueprop (foldr1 (HOLogic.mk_binop "op &")
1.133 +      (map (fn ((((i, _), T), U), tname) =>
1.134 +        make_pred i U T (mk_proj i is r) (Free (tname, T)))
1.135 +          (descr ~~ recTs ~~ rec_result_Ts ~~ tnames)));
1.136 +    val cert = cterm_of sg;
1.137 +    val inst = map (pairself cert) (map head_of (HOLogic.dest_conj
1.138 +      (HOLogic.dest_Trueprop (concl_of induction))) ~~ ps);
1.139 +
1.140 +    val thm = prove_goal' sg (Logic.list_implies (prems, concl))
1.141 +      (fn prems =>
1.142 +         [rewrite_goals_tac (map mk_meta_eq [fst_conv, snd_conv]),
1.143 +          rtac (cterm_instantiate inst induction) 1,
1.144 +          ALLGOALS ObjectLogic.atomize_tac,
1.145 +          rewrite_goals_tac (o_def :: map mk_meta_eq rec_rewrites),
1.146 +          REPEAT ((resolve_tac prems THEN_ALL_NEW (fn i =>
1.147 +            REPEAT (etac allE i) THEN atac i)) 1)]);
1.148 +
1.149 +    val {path, ...} = Sign.rep_sg sg;
1.150 +    val ind_name = Thm.name_of_thm induction;
1.151 +    val vs = map (fn i => nth_elem (i, pnames)) is;
1.152 +    val (thy', thm') = thy
1.153 +      |> Theory.absolute_path
1.154 +      |> PureThy.store_thm
1.155 +        ((space_implode "_" (ind_name :: vs @ ["correctness"]), thm), [])
1.156 +      |>> Theory.add_path (NameSpace.pack (if_none path []));
1.157 +
1.158 +    val inst = map (fn ((((i, _), s), T), U) => ((s, 0), if i mem is then
1.159 +        Abs ("r", U, Abs ("x", T, mk_realizes U \$ Bound 1 \$
1.160 +          (Var ((s, 0), T --> HOLogic.boolT) \$ Bound 0)))
1.161 +      else Abs ("x", T, mk_realizes Extraction.nullT \$ Extraction.nullt \$
1.162 +        (Var ((s, 0), T --> HOLogic.boolT) \$
1.163 +          Bound 0)))) (descr ~~ pnames ~~ map Type.varifyT recTs ~~
1.164 +            map Type.varifyT rec_result_Ts);
1.165 +
1.166 +    val ivs = map Var (Drule.vars_of_terms
1.167 +      [Logic.varify (DatatypeProp.make_ind [descr] sorts)]);
1.168 +
1.169 +    val prf = foldr forall_intr_prf (ivs,
1.170 +      prf_subst_vars inst (foldr (fn ((f, p), prf) =>
1.171 +        (case head_of (strip_abs_body f) of
1.172 +           Free (s, T) =>
1.173 +             let val T' = Type.varifyT T
1.174 +             in Abst (s, Some T', Proofterm.prf_abstract_over
1.175 +               (Var ((s, 0), T')) (AbsP ("H", Some p, prf)))
1.176 +             end
1.177 +         | _ => AbsP ("H", Some p, prf)))
1.178 +           (rec_fns ~~ prems_of thm, Proofterm.proof_combP
1.179 +             (prf_of thm', map PBound (length prems - 1 downto 0)))));
1.180 +
1.181 +    val r' = if null is then r else Logic.varify (foldr (uncurry lambda)
1.182 +      (map Logic.unvarify ivs @ filter_out is_unit
1.183 +        (map (head_of o strip_abs_body) rec_fns), r));
1.184 +
1.185 +  in Extraction.add_realizers_i [(ind_name, (vs, r', prf))] thy' end;
1.186 +
1.187 +
1.188 +fun make_casedists sorts ({index, descr, case_name, case_rewrites, exhaustion, ...} : datatype_info, thy) =
1.189 +  let
1.190 +    val sg = sign_of thy;
1.191 +    val sorts = map (rpair HOLogic.typeS) (distinct (flat (map
1.192 +      (fn (_, (_, ds, _)) => mapfilter (try dest_DtTFree) ds) descr)));
1.193 +    val cert = cterm_of sg;
1.194 +    val rT = TFree ("'P", HOLogic.typeS);
1.195 +    val rT' = TVar (("'P", 0), HOLogic.typeS);
1.196 +
1.197 +    fun make_casedist_prem T (cname, cargs) =
1.198 +      let
1.199 +        val Ts = map (typ_of_dtyp descr sorts) cargs;
1.200 +        val frees = variantlist
1.201 +          (DatatypeProp.make_tnames Ts, ["P", "y"]) ~~ Ts;
1.202 +        val free_ts = map Free frees;
1.203 +        val r = Free ("r" ^ NameSpace.base cname, Ts ---> rT)
1.204 +      in (r, list_all_free (frees, Logic.mk_implies (HOLogic.mk_Trueprop
1.205 +        (HOLogic.mk_eq (Free ("y", T), list_comb (Const (cname, Ts ---> T), free_ts))),
1.206 +          HOLogic.mk_Trueprop (Free ("P", rT --> HOLogic.boolT) \$
1.207 +            list_comb (r, free_ts)))))
1.208 +      end;
1.209 +
1.210 +    val Some (_, _, constrs) = assoc (descr, index);
1.211 +    val T = nth_elem (index, get_rec_types descr sorts);
1.212 +    val (rs, prems) = split_list (map (make_casedist_prem T) constrs);
1.213 +    val r = Const (case_name, map fastype_of rs ---> T --> rT);
1.214 +
1.215 +    val y = Var (("y", 0), Type.varifyT T);
1.216 +    val y' = Free ("y", T);
1.217 +
1.218 +    val thm = prove_goalw_cterm [] (cert (Logic.list_implies (prems,
1.219 +      HOLogic.mk_Trueprop (Free ("P", rT --> HOLogic.boolT) \$
1.220 +        list_comb (r, rs @ [y'])))))
1.221 +      (fn prems =>
1.222 +         [rtac (cterm_instantiate [(cert y, cert y')] exhaustion) 1,
1.223 +          ALLGOALS (EVERY'
1.224 +            [asm_simp_tac (HOL_basic_ss addsimps case_rewrites),
1.225 +             resolve_tac prems, asm_simp_tac HOL_basic_ss])]);
1.226 +
1.227 +    val {path, ...} = Sign.rep_sg sg;
1.228 +    val exh_name = Thm.name_of_thm exhaustion;
1.229 +    val (thy', thm') = thy
1.230 +      |> Theory.absolute_path
1.231 +      |> PureThy.store_thm ((exh_name ^ "_P_correctness", thm), [])
1.232 +      |>> Theory.add_path (NameSpace.pack (if_none path []));
1.233 +
1.234 +    val P = Var (("P", 0), HOLogic.boolT);
1.235 +    val prf = forall_intr_prf (y, forall_intr_prf (P,
1.236 +      prf_subst_vars [(("P", 0), Abs ("r", rT',
1.237 +        mk_realizes rT' \$ Bound 0 \$ P))] (foldr (fn ((p, r), prf) =>
1.238 +          forall_intr_prf (Logic.varify r, AbsP ("H", Some (Logic.varify p),
1.239 +            prf))) (prems ~~ rs, Proofterm.proof_combP (prf_of thm',
1.240 +              map PBound (length prems - 1 downto 0))))));
1.241 +    val r' = Logic.varify (Abs ("y", Type.varifyT T,
1.242 +      Abs ("P", HOLogic.boolT, list_abs (map dest_Free rs, list_comb (r,
1.243 +        map Bound ((length rs - 1 downto 0) @ [length rs + 1]))))));
1.244 +
1.245 +    val prf' = forall_intr_prf (y, forall_intr_prf (P, prf_subst_vars
1.246 +      [(("P", 0), mk_realizes Extraction.nullT \$ Extraction.nullt \$ P)]
1.247 +        (prf_of exhaustion)));
1.248 +