src/HOL/Library/Code_Abstract_Nat.thy
 author Andreas Lochbihler Fri Sep 20 10:09:16 2013 +0200 (2013-09-20) changeset 53745 788730ab7da4 parent 51113 222fb6cb2c3e child 55415 05f5fdb8d093 permissions -rw-r--r--
prefer Code.abort over code_abort
```     1 (*  Title:      HOL/Library/Code_Abstract_Nat.thy
```
```     2     Author:     Stefan Berghofer, Florian Haftmann, TU Muenchen
```
```     3 *)
```
```     4
```
```     5 header {* Avoidance of pattern matching on natural numbers *}
```
```     6
```
```     7 theory Code_Abstract_Nat
```
```     8 imports Main
```
```     9 begin
```
```    10
```
```    11 text {*
```
```    12   When natural numbers are implemented in another than the
```
```    13   conventional inductive @{term "0::nat"}/@{term Suc} representation,
```
```    14   it is necessary to avoid all pattern matching on natural numbers
```
```    15   altogether.  This is accomplished by this theory (up to a certain
```
```    16   extent).
```
```    17 *}
```
```    18
```
```    19 subsection {* Case analysis *}
```
```    20
```
```    21 text {*
```
```    22   Case analysis on natural numbers is rephrased using a conditional
```
```    23   expression:
```
```    24 *}
```
```    25
```
```    26 lemma [code, code_unfold]:
```
```    27   "nat_case = (\<lambda>f g n. if n = 0 then f else g (n - 1))"
```
```    28   by (auto simp add: fun_eq_iff dest!: gr0_implies_Suc)
```
```    29
```
```    30
```
```    31 subsection {* Preprocessors *}
```
```    32
```
```    33 text {*
```
```    34   The term @{term "Suc n"} is no longer a valid pattern.  Therefore,
```
```    35   all occurrences of this term in a position where a pattern is
```
```    36   expected (i.e.~on the left-hand side of a code equation) must be
```
```    37   eliminated.  This can be accomplished – as far as possible – by
```
```    38   applying the following transformation rule:
```
```    39 *}
```
```    40
```
```    41 lemma Suc_if_eq: "(\<And>n. f (Suc n) \<equiv> h n) \<Longrightarrow> f 0 \<equiv> g \<Longrightarrow>
```
```    42   f n \<equiv> if n = 0 then g else h (n - 1)"
```
```    43   by (rule eq_reflection) (cases n, simp_all)
```
```    44
```
```    45 text {*
```
```    46   The rule above is built into a preprocessor that is plugged into
```
```    47   the code generator.
```
```    48 *}
```
```    49
```
```    50 setup {*
```
```    51 let
```
```    52
```
```    53 fun remove_suc thy thms =
```
```    54   let
```
```    55     val vname = singleton (Name.variant_list (map fst
```
```    56       (fold (Term.add_var_names o Thm.full_prop_of) thms []))) "n";
```
```    57     val cv = cterm_of thy (Var ((vname, 0), HOLogic.natT));
```
```    58     fun lhs_of th = snd (Thm.dest_comb
```
```    59       (fst (Thm.dest_comb (cprop_of th))));
```
```    60     fun rhs_of th = snd (Thm.dest_comb (cprop_of th));
```
```    61     fun find_vars ct = (case term_of ct of
```
```    62         (Const (@{const_name Suc}, _) \$ Var _) => [(cv, snd (Thm.dest_comb ct))]
```
```    63       | _ \$ _ =>
```
```    64         let val (ct1, ct2) = Thm.dest_comb ct
```
```    65         in
```
```    66           map (apfst (fn ct => Thm.apply ct ct2)) (find_vars ct1) @
```
```    67           map (apfst (Thm.apply ct1)) (find_vars ct2)
```
```    68         end
```
```    69       | _ => []);
```
```    70     val eqs = maps
```
```    71       (fn th => map (pair th) (find_vars (lhs_of th))) thms;
```
```    72     fun mk_thms (th, (ct, cv')) =
```
```    73       let
```
```    74         val th' =
```
```    75           Thm.implies_elim
```
```    76            (Conv.fconv_rule (Thm.beta_conversion true)
```
```    77              (Drule.instantiate'
```
```    78                [SOME (ctyp_of_term ct)] [SOME (Thm.lambda cv ct),
```
```    79                  SOME (Thm.lambda cv' (rhs_of th)), NONE, SOME cv']
```
```    80                @{thm Suc_if_eq})) (Thm.forall_intr cv' th)
```
```    81       in
```
```    82         case map_filter (fn th'' =>
```
```    83             SOME (th'', singleton
```
```    84               (Variable.trade (K (fn [th'''] => [th''' RS th']))
```
```    85                 (Variable.global_thm_context th'')) th'')
```
```    86           handle THM _ => NONE) thms of
```
```    87             [] => NONE
```
```    88           | thps =>
```
```    89               let val (ths1, ths2) = split_list thps
```
```    90               in SOME (subtract Thm.eq_thm (th :: ths1) thms @ ths2) end
```
```    91       end
```
```    92   in get_first mk_thms eqs end;
```
```    93
```
```    94 fun eqn_suc_base_preproc thy thms =
```
```    95   let
```
```    96     val dest = fst o Logic.dest_equals o prop_of;
```
```    97     val contains_suc = exists_Const (fn (c, _) => c = @{const_name Suc});
```
```    98   in
```
```    99     if forall (can dest) thms andalso exists (contains_suc o dest) thms
```
```   100       then thms |> perhaps_loop (remove_suc thy) |> (Option.map o map) Drule.zero_var_indexes
```
```   101        else NONE
```
```   102   end;
```
```   103
```
```   104 val eqn_suc_preproc = Code_Preproc.simple_functrans eqn_suc_base_preproc;
```
```   105
```
```   106 in
```
```   107
```
```   108   Code_Preproc.add_functrans ("eqn_Suc", eqn_suc_preproc)
```
```   109
```
```   110 end;
```
```   111 *}
```
```   112
```
```   113 end
```