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