(* Title: HOL/Library/Code_Abstract_Nat.thy
Author: Stefan Berghofer, Florian Haftmann, TU Muenchen
*)
section \<open>Avoidance of pattern matching on natural numbers\<close>
theory Code_Abstract_Nat
imports Main
begin
text \<open>
When natural numbers are implemented in another than the
conventional inductive \<^term>\<open>0::nat\<close>/\<^term>\<open>Suc\<close> representation,
it is necessary to avoid all pattern matching on natural numbers
altogether. This is accomplished by this theory (up to a certain
extent).
\<close>
subsection \<open>Case analysis\<close>
text \<open>
Case analysis on natural numbers is rephrased using a conditional
expression:
\<close>
lemma [code, code_unfold]:
"case_nat = (\f g n. if n = 0 then f else g (n - 1))"
by (auto simp add: fun_eq_iff dest!: gr0_implies_Suc)
subsection \<open>Preprocessors\<close>
text \<open>
The term \<^term>\<open>Suc n\<close> is no longer a valid pattern. Therefore,
all occurrences of this term in a position where a pattern is
expected (i.e.~on the left-hand side of a code equation) must be
eliminated. This can be accomplished -- as far as possible -- by
applying the following transformation rule:
\<close>
lemma Suc_if_eq:
assumes "\n. f (Suc n) \ h n"
assumes "f 0 \ g"
shows "f n \ if n = 0 then g else h (n - 1)"
by (rule eq_reflection) (cases n, insert assms, simp_all)
text \<open>
The rule above is built into a preprocessor that is plugged into
the code generator.
\<close>
setup \<open>
let
val Suc_if_eq = Thm.incr_indexes 1 @{thm Suc_if_eq};
fun remove_suc ctxt thms =
let
val vname = singleton (Name.variant_list (map fst
(fold (Term.add_var_names o Thm.full_prop_of) thms []))) "n";
val cv = Thm.cterm_of ctxt (Var ((vname, 0), HOLogic.natT));
val lhs_of = snd o Thm.dest_comb o fst o Thm.dest_comb o Thm.cprop_of;
val rhs_of = snd o Thm.dest_comb o Thm.cprop_of;
fun find_vars ct = (case Thm.term_of ct of
(Const (\<^const_name>\<open>Suc\<close>, _) $ Var _) => [(cv, snd (Thm.dest_comb ct))]
| _ $ _ =>
let val (ct1, ct2) = Thm.dest_comb ct
in
map (apfst (fn ct => Thm.apply ct ct2)) (find_vars ct1) @
map (apfst (Thm.apply ct1)) (find_vars ct2)
end
| _ => []);
val eqs = maps
(fn thm => map (pair thm) (find_vars (lhs_of thm))) thms;
fun mk_thms (thm, (ct, cv')) =
let
val thm' =
Thm.implies_elim
(Conv.fconv_rule (Thm.beta_conversion true)
(Thm.instantiate'
[SOME (Thm.ctyp_of_cterm ct)] [SOME (Thm.lambda cv ct),
SOME (Thm.lambda cv' (rhs_of thm)), NONE, SOME cv']
Suc_if_eq)) (Thm.forall_intr cv' thm)
in
case map_filter (fn thm'' =>
SOME (thm'', singleton
(Variable.trade (K (fn [thm'''] => [thm''' RS thm']))
(Variable.declare_thm thm'' ctxt)) thm'')
handle THM _ => NONE) thms of
[] => NONE
| thmps =>
let val (thms1, thms2) = split_list thmps
in SOME (subtract Thm.eq_thm (thm :: thms1) thms @ thms2) end
end
in get_first mk_thms eqs end;
fun eqn_suc_base_preproc ctxt thms =
let
val dest = fst o Logic.dest_equals o Thm.prop_of;
val contains_suc = exists_Const (fn (c, _) => c = \<^const_name>\<open>Suc\<close>);
in
if forall (can dest) thms andalso exists (contains_suc o dest) thms
then thms |> perhaps_loop (remove_suc ctxt) |> (Option.map o map) Drule.zero_var_indexes
else NONE
end;
val eqn_suc_preproc = Code_Preproc.simple_functrans eqn_suc_base_preproc;
in
Code_Preproc.add_functrans ("eqn_Suc", eqn_suc_preproc)
end
\<close>
end
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