src/HOL/Library/Code_Abstract_Nat.thy
 author haftmann Wed Feb 26 11:57:52 2014 +0100 (2014-02-26) changeset 55757 9fc71814b8c1 parent 55415 05f5fdb8d093 child 56790 f54097170704 permissions -rw-r--r--
prefer proof context over background theory
```     1 (*  Title:      HOL/Library/Code_Abstract_Nat.thy
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```     2     Author:     Stefan Berghofer, Florian Haftmann, TU Muenchen
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```     3 *)
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```     4
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```     5 header {* Avoidance of pattern matching on natural numbers *}
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```     6
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```     7 theory Code_Abstract_Nat
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```     8 imports Main
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```     9 begin
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```    10
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```    11 text {*
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```    12   When natural numbers are implemented in another than the
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```    13   conventional inductive @{term "0::nat"}/@{term Suc} representation,
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```    14   it is necessary to avoid all pattern matching on natural numbers
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```    15   altogether.  This is accomplished by this theory (up to a certain
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```    16   extent).
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```    17 *}
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```    18
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```    19 subsection {* Case analysis *}
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```    20
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```    21 text {*
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```    22   Case analysis on natural numbers is rephrased using a conditional
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```    23   expression:
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```    24 *}
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```    25
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```    26 lemma [code, code_unfold]:
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```    27   "case_nat = (\<lambda>f g n. if n = 0 then f else g (n - 1))"
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```    28   by (auto simp add: fun_eq_iff dest!: gr0_implies_Suc)
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```    29
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```    30
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```    31 subsection {* Preprocessors *}
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```    32
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```    33 text {*
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```    34   The term @{term "Suc n"} is no longer a valid pattern.  Therefore,
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```    35   all occurrences of this term in a position where a pattern is
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```    36   expected (i.e.~on the left-hand side of a code equation) must be
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```    37   eliminated.  This can be accomplished – as far as possible – by
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```    38   applying the following transformation rule:
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```    39 *}
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```    40
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```    41 lemma Suc_if_eq: "(\<And>n. f (Suc n) \<equiv> h n) \<Longrightarrow> f 0 \<equiv> g \<Longrightarrow>
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```    42   f n \<equiv> if n = 0 then g else h (n - 1)"
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```    43   by (rule eq_reflection) (cases n, simp_all)
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```    44
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```    45 text {*
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```    46   The rule above is built into a preprocessor that is plugged into
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```    47   the code generator.
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```    48 *}
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```    49
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```    50 setup {*
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```    51 let
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```    52
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```    53 fun remove_suc ctxt thms =
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```    54   let
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```    55     val thy = Proof_Context.theory_of ctxt;
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```    56     val vname = singleton (Name.variant_list (map fst
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```    57       (fold (Term.add_var_names o Thm.full_prop_of) thms []))) "n";
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```    58     val cv = cterm_of thy (Var ((vname, 0), HOLogic.natT));
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```    59     fun lhs_of th = snd (Thm.dest_comb
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```    60       (fst (Thm.dest_comb (cprop_of th))));
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```    61     fun rhs_of th = snd (Thm.dest_comb (cprop_of th));
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```    62     fun find_vars ct = (case term_of ct of
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```    63         (Const (@{const_name Suc}, _) \$ Var _) => [(cv, snd (Thm.dest_comb ct))]
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```    64       | _ \$ _ =>
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```    65         let val (ct1, ct2) = Thm.dest_comb ct
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```    66         in
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```    67           map (apfst (fn ct => Thm.apply ct ct2)) (find_vars ct1) @
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```    68           map (apfst (Thm.apply ct1)) (find_vars ct2)
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```    69         end
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```    70       | _ => []);
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```    71     val eqs = maps
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```    72       (fn th => map (pair th) (find_vars (lhs_of th))) thms;
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```    73     fun mk_thms (th, (ct, cv')) =
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```    74       let
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```    75         val th' =
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```    76           Thm.implies_elim
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```    77            (Conv.fconv_rule (Thm.beta_conversion true)
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```    78              (Drule.instantiate'
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```    79                [SOME (ctyp_of_term ct)] [SOME (Thm.lambda cv ct),
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```    80                  SOME (Thm.lambda cv' (rhs_of th)), NONE, SOME cv']
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```    81                @{thm Suc_if_eq})) (Thm.forall_intr cv' th)
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```    82       in
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```    83         case map_filter (fn th'' =>
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```    84             SOME (th'', singleton
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```    85               (Variable.trade (K (fn [th'''] => [th''' RS th']))
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```    86                 (Variable.global_thm_context th'')) th'')
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```    87           handle THM _ => NONE) thms of
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```    88             [] => NONE
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```    89           | thps =>
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```    90               let val (ths1, ths2) = split_list thps
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```    91               in SOME (subtract Thm.eq_thm (th :: ths1) thms @ ths2) end
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```    92       end
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```    93   in get_first mk_thms eqs end;
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```    94
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```    95 fun eqn_suc_base_preproc thy thms =
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```    96   let
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```    97     val dest = fst o Logic.dest_equals o prop_of;
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```    98     val contains_suc = exists_Const (fn (c, _) => c = @{const_name Suc});
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```    99   in
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```   100     if forall (can dest) thms andalso exists (contains_suc o dest) thms
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```   101       then thms |> perhaps_loop (remove_suc thy) |> (Option.map o map) Drule.zero_var_indexes
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```   102        else NONE
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```   103   end;
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```   104
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```   105 val eqn_suc_preproc = Code_Preproc.simple_functrans eqn_suc_base_preproc;
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```   106
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```   107 in
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```   108
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```   109   Code_Preproc.add_functrans ("eqn_Suc", eqn_suc_preproc)
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```   110
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```   111 end;
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```   112 *}
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```   113
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```   114 end
```