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(* Title: HOL/Library/Efficient_Nat.thy
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Author: Stefan Berghofer, Florian Haftmann, TU Muenchen
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*)
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header {* Implementation of natural numbers by target-language integers *}
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theory Efficient_Nat
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imports Code_Index Code_Integer Main
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begin
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text {*
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When generating code for functions on natural numbers, the
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canonical representation using @{term "0::nat"} and
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@{term "Suc"} is unsuitable for computations involving large
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numbers. The efficiency of the generated code can be improved
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drastically by implementing natural numbers by target-language
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integers. To do this, just include this theory.
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*}
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subsection {* Basic arithmetic *}
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text {*
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Most standard arithmetic functions on natural numbers are implemented
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using their counterparts on the integers:
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*}
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code_datatype number_nat_inst.number_of_nat
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lemma zero_nat_code [code, code inline]:
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"0 = (Numeral0 :: nat)"
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by simp
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lemmas [code post] = zero_nat_code [symmetric]
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lemma one_nat_code [code, code inline]:
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"1 = (Numeral1 :: nat)"
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by simp
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lemmas [code post] = one_nat_code [symmetric]
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lemma Suc_code [code]:
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"Suc n = n + 1"
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by simp
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lemma plus_nat_code [code]:
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"n + m = nat (of_nat n + of_nat m)"
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by simp
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lemma minus_nat_code [code]:
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"n - m = nat (of_nat n - of_nat m)"
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by simp
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lemma times_nat_code [code]:
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"n * m = nat (of_nat n * of_nat m)"
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unfolding of_nat_mult [symmetric] by simp
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text {* Specialized @{term "op div \<Colon> nat \<Rightarrow> nat \<Rightarrow> nat"}
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and @{term "op mod \<Colon> nat \<Rightarrow> nat \<Rightarrow> nat"} operations. *}
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definition divmod_aux :: "nat \<Rightarrow> nat \<Rightarrow> nat \<times> nat" where
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[code del]: "divmod_aux = Divides.divmod"
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lemma [code]:
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"Divides.divmod n m = (if m = 0 then (0, n) else divmod_aux n m)"
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unfolding divmod_aux_def divmod_div_mod by simp
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lemma divmod_aux_code [code]:
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"divmod_aux n m = (nat (of_nat n div of_nat m), nat (of_nat n mod of_nat m))"
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unfolding divmod_aux_def divmod_div_mod zdiv_int [symmetric] zmod_int [symmetric] by simp
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lemma eq_nat_code [code]:
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"eq_class.eq n m \<longleftrightarrow> eq_class.eq (of_nat n \<Colon> int) (of_nat m)"
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by (simp add: eq)
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lemma eq_nat_refl [code nbe]:
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"eq_class.eq (n::nat) n \<longleftrightarrow> True"
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by (rule HOL.eq_refl)
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lemma less_eq_nat_code [code]:
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"n \<le> m \<longleftrightarrow> (of_nat n \<Colon> int) \<le> of_nat m"
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by simp
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lemma less_nat_code [code]:
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"n < m \<longleftrightarrow> (of_nat n \<Colon> int) < of_nat m"
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by simp
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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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"nat_case = (\<lambda>f g n. if n = 0 then f else g (n - 1))"
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by (auto simp add: expand_fun_eq dest!: gr0_implies_Suc)
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subsection {* Preprocessors *}
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text {*
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In contrast to @{term "Suc n"}, the term @{term "n + (1::nat)"} is no longer
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a constructor term. Therefore, all occurrences of this term in a position
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where a pattern is expected (i.e.\ on the left-hand side of a recursion
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equation or in the arguments of an inductive relation in an introduction
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rule) must be eliminated.
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This can be accomplished by applying the following transformation rules:
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*}
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lemma Suc_if_eq': "(\<And>n. f (Suc n) = h n) \<Longrightarrow> f 0 = g \<Longrightarrow>
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f n = (if n = 0 then g else h (n - 1))"
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by (cases n) simp_all
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lemma Suc_if_eq: "(\<And>n. f (Suc n) \<equiv> h n) \<Longrightarrow> f 0 \<equiv> g \<Longrightarrow>
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f n \<equiv> if n = 0 then g else h (n - 1)"
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by (rule eq_reflection, rule Suc_if_eq')
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(rule meta_eq_to_obj_eq, assumption,
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rule meta_eq_to_obj_eq, assumption)
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lemma Suc_clause: "(\<And>n. P n (Suc n)) \<Longrightarrow> n \<noteq> 0 \<Longrightarrow> P (n - 1) n"
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by (cases n) simp_all
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text {*
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The rules above are built into a preprocessor that is plugged into
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the code generator. Since the preprocessor for introduction rules
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does not know anything about modes, some of the modes that worked
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for the canonical representation of natural numbers may no longer work.
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*}
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(*<*)
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setup {*
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let
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fun gen_remove_suc Suc_if_eq dest_judgement thy thms =
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let
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val vname = Name.variant (map fst
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(fold (Term.add_var_names o Thm.full_prop_of) thms [])) "n";
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val cv = cterm_of thy (Var ((vname, 0), HOLogic.natT));
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fun lhs_of th = snd (Thm.dest_comb
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(fst (Thm.dest_comb (dest_judgement (cprop_of th)))));
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fun rhs_of th = snd (Thm.dest_comb (dest_judgement (cprop_of th)));
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fun find_vars ct = (case 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.capply ct ct2)) (find_vars ct1) @
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map (apfst (Thm.capply 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 th => map (pair th) (find_vars (lhs_of th))) thms;
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fun mk_thms (th, (ct, cv')) =
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let
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val th' =
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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 (ctyp_of_term ct)] [SOME (Thm.cabs cv ct),
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SOME (Thm.cabs cv' (rhs_of th)), NONE, SOME cv']
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Suc_if_eq)) (Thm.forall_intr cv' th)
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in
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case map_filter (fn th'' =>
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SOME (th'', singleton
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(Variable.trade (K (fn [th'''] => [th''' RS th'])) (Variable.thm_context th'')) th'')
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handle THM _ => NONE) thms of
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[] => NONE
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| thps =>
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let val (ths1, ths2) = split_list thps
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in SOME (subtract Thm.eq_thm (th :: ths1) thms @ ths2) end
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end
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in get_first mk_thms eqs end;
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fun gen_eqn_suc_preproc Suc_if_eq dest_judgement dest_lhs thy thms =
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let
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val dest = dest_lhs o 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 perhaps_loop (gen_remove_suc Suc_if_eq dest_judgement thy) thms
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else NONE
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end;
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fun eqn_suc_preproc thy = map fst
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#> gen_eqn_suc_preproc
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@{thm Suc_if_eq} I (fst o Logic.dest_equals) thy
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#> (Option.map o map) (Code_Unit.mk_eqn thy);
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fun eqn_suc_preproc' thy thms = gen_eqn_suc_preproc
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@{thm Suc_if_eq'} (snd o Thm.dest_comb) (fst o HOLogic.dest_eq o HOLogic.dest_Trueprop) thy thms
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|> the_default thms;
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fun remove_suc_clause thy thms =
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let
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val vname = Name.variant (map fst
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(fold (Term.add_var_names o Thm.full_prop_of) thms [])) "x";
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fun find_var (t as Const (@{const_name Suc}, _) $ (v as Var _)) = SOME (t, v)
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| find_var (t $ u) = (case find_var t of NONE => find_var u | x => x)
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| find_var _ = NONE;
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fun find_thm th =
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let val th' = Conv.fconv_rule ObjectLogic.atomize th
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in Option.map (pair (th, th')) (find_var (prop_of th')) end
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in
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case get_first find_thm thms of
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NONE => thms
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| SOME ((th, th'), (Sucv, v)) =>
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let
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val cert = cterm_of (Thm.theory_of_thm th);
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val th'' = ObjectLogic.rulify (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 (cert (lambda v (Abs ("x", HOLogic.natT,
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abstract_over (Sucv,
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HOLogic.dest_Trueprop (prop_of th')))))),
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SOME (cert v)] @{thm Suc_clause}))
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(Thm.forall_intr (cert v) th'))
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in
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remove_suc_clause thy (map (fn th''' =>
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if (op = o pairself prop_of) (th''', th) then th'' else th''') thms)
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end
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end;
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fun clause_suc_preproc thy ths =
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let
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val dest = fst o HOLogic.dest_mem o HOLogic.dest_Trueprop
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in
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if forall (can (dest o concl_of)) ths andalso
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exists (fn th => exists (exists_Const (fn (c, _) => c = @{const_name Suc}))
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(map_filter (try dest) (concl_of th :: prems_of th))) ths
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then remove_suc_clause thy ths else ths
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end;
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in
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Codegen.add_preprocessor eqn_suc_preproc'
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#> Codegen.add_preprocessor clause_suc_preproc
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#> Code.add_functrans ("eqn_Suc", eqn_suc_preproc)
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end;
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*}
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(*>*)
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subsection {* Target language setup *}
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text {*
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For ML, we map @{typ nat} to target language integers, where we
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assert that values are always non-negative.
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*}
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code_type nat
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(SML "IntInf.int")
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(OCaml "Big'_int.big'_int")
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types_code
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nat ("int")
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attach (term_of) {*
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val term_of_nat = HOLogic.mk_number HOLogic.natT;
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*}
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attach (test) {*
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fun gen_nat i =
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let val n = random_range 0 i
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in (n, fn () => term_of_nat n) end;
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*}
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text {*
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For Haskell we define our own @{typ nat} type. The reason
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is that we have to distinguish type class instances
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for @{typ nat} and @{typ int}.
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*}
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code_include Haskell "Nat" {*
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newtype Nat = Nat Integer deriving (Show, Eq);
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instance Num Nat where {
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fromInteger k = Nat (if k >= 0 then k else 0);
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Nat n + Nat m = Nat (n + m);
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haftmann@25967
|
275 |
Nat n - Nat m = fromInteger (n - m);
|
haftmann@25967
|
276 |
Nat n * Nat m = Nat (n * m);
|
haftmann@25967
|
277 |
abs n = n;
|
haftmann@25967
|
278 |
signum _ = 1;
|
haftmann@25967
|
279 |
negate n = error "negate Nat";
|
haftmann@25967
|
280 |
};
|
haftmann@25967
|
281 |
|
haftmann@25967
|
282 |
instance Ord Nat where {
|
haftmann@25967
|
283 |
Nat n <= Nat m = n <= m;
|
haftmann@25967
|
284 |
Nat n < Nat m = n < m;
|
haftmann@25967
|
285 |
};
|
haftmann@25967
|
286 |
|
haftmann@25967
|
287 |
instance Real Nat where {
|
haftmann@25967
|
288 |
toRational (Nat n) = toRational n;
|
haftmann@25967
|
289 |
};
|
haftmann@25967
|
290 |
|
haftmann@25967
|
291 |
instance Enum Nat where {
|
haftmann@25967
|
292 |
toEnum k = fromInteger (toEnum k);
|
haftmann@25967
|
293 |
fromEnum (Nat n) = fromEnum n;
|
haftmann@25967
|
294 |
};
|
haftmann@25967
|
295 |
|
haftmann@25967
|
296 |
instance Integral Nat where {
|
haftmann@25967
|
297 |
toInteger (Nat n) = n;
|
haftmann@25967
|
298 |
divMod n m = quotRem n m;
|
haftmann@25967
|
299 |
quotRem (Nat n) (Nat m) = (Nat k, Nat l) where (k, l) = quotRem n m;
|
haftmann@25967
|
300 |
};
|
haftmann@25967
|
301 |
*}
|
haftmann@25967
|
302 |
|
haftmann@25967
|
303 |
code_reserved Haskell Nat
|
haftmann@25967
|
304 |
|
haftmann@25967
|
305 |
code_type nat
|
haftmann@29730
|
306 |
(Haskell "Nat.Nat")
|
haftmann@25967
|
307 |
|
haftmann@25967
|
308 |
code_instance nat :: eq
|
haftmann@25967
|
309 |
(Haskell -)
|
haftmann@25967
|
310 |
|
haftmann@25967
|
311 |
text {*
|
haftmann@25931
|
312 |
Natural numerals.
|
haftmann@25931
|
313 |
*}
|
haftmann@25931
|
314 |
|
haftmann@25967
|
315 |
lemma [code inline, symmetric, code post]:
|
haftmann@25931
|
316 |
"nat (number_of i) = number_nat_inst.number_of_nat i"
|
haftmann@25931
|
317 |
-- {* this interacts as desired with @{thm nat_number_of_def} *}
|
haftmann@25931
|
318 |
by (simp add: number_nat_inst.number_of_nat)
|
haftmann@25931
|
319 |
|
haftmann@25931
|
320 |
setup {*
|
haftmann@25931
|
321 |
fold (Numeral.add_code @{const_name number_nat_inst.number_of_nat}
|
haftmann@25967
|
322 |
true false) ["SML", "OCaml", "Haskell"]
|
haftmann@25931
|
323 |
*}
|
haftmann@25931
|
324 |
|
haftmann@25931
|
325 |
text {*
|
haftmann@25931
|
326 |
Since natural numbers are implemented
|
haftmann@25967
|
327 |
using integers in ML, the coercion function @{const "of_nat"} of type
|
haftmann@25931
|
328 |
@{typ "nat \<Rightarrow> int"} is simply implemented by the identity function.
|
haftmann@25931
|
329 |
For the @{const "nat"} function for converting an integer to a natural
|
haftmann@25931
|
330 |
number, we give a specific implementation using an ML function that
|
haftmann@25931
|
331 |
returns its input value, provided that it is non-negative, and otherwise
|
haftmann@25931
|
332 |
returns @{text "0"}.
|
haftmann@25931
|
333 |
*}
|
haftmann@25931
|
334 |
|
haftmann@25931
|
335 |
definition
|
haftmann@25931
|
336 |
int :: "nat \<Rightarrow> int"
|
haftmann@25931
|
337 |
where
|
haftmann@28562
|
338 |
[code del]: "int = of_nat"
|
haftmann@25931
|
339 |
|
haftmann@28562
|
340 |
lemma int_code' [code]:
|
haftmann@25931
|
341 |
"int (number_of l) = (if neg (number_of l \<Colon> int) then 0 else number_of l)"
|
haftmann@25931
|
342 |
unfolding int_nat_number_of [folded int_def] ..
|
haftmann@25931
|
343 |
|
haftmann@28562
|
344 |
lemma nat_code' [code]:
|
haftmann@25931
|
345 |
"nat (number_of l) = (if neg (number_of l \<Colon> int) then 0 else number_of l)"
|
huffman@28969
|
346 |
unfolding nat_number_of_def number_of_is_id neg_def by simp
|
haftmann@25931
|
347 |
|
haftmann@25931
|
348 |
lemma of_nat_int [code unfold]:
|
haftmann@25931
|
349 |
"of_nat = int" by (simp add: int_def)
|
haftmann@25967
|
350 |
declare of_nat_int [symmetric, code post]
|
haftmann@25931
|
351 |
|
haftmann@25931
|
352 |
code_const int
|
haftmann@25931
|
353 |
(SML "_")
|
haftmann@25931
|
354 |
(OCaml "_")
|
haftmann@25931
|
355 |
|
haftmann@25931
|
356 |
consts_code
|
haftmann@25931
|
357 |
int ("(_)")
|
haftmann@25931
|
358 |
nat ("\<module>nat")
|
haftmann@25931
|
359 |
attach {*
|
haftmann@25931
|
360 |
fun nat i = if i < 0 then 0 else i;
|
haftmann@25931
|
361 |
*}
|
haftmann@25931
|
362 |
|
haftmann@25967
|
363 |
code_const nat
|
haftmann@25967
|
364 |
(SML "IntInf.max/ (/0,/ _)")
|
haftmann@25967
|
365 |
(OCaml "Big'_int.max'_big'_int/ Big'_int.zero'_big'_int")
|
haftmann@25967
|
366 |
|
haftmann@25967
|
367 |
text {* For Haskell, things are slightly different again. *}
|
haftmann@25967
|
368 |
|
haftmann@25967
|
369 |
code_const int and nat
|
haftmann@25967
|
370 |
(Haskell "toInteger" and "fromInteger")
|
haftmann@25931
|
371 |
|
haftmann@25931
|
372 |
text {* Conversion from and to indices. *}
|
haftmann@25931
|
373 |
|
haftmann@29752
|
374 |
code_const Code_Index.of_nat
|
haftmann@25967
|
375 |
(SML "IntInf.toInt")
|
haftmann@25967
|
376 |
(OCaml "Big'_int.int'_of'_big'_int")
|
haftmann@27673
|
377 |
(Haskell "fromEnum")
|
haftmann@25967
|
378 |
|
haftmann@29752
|
379 |
code_const Code_Index.nat_of
|
haftmann@25931
|
380 |
(SML "IntInf.fromInt")
|
haftmann@25931
|
381 |
(OCaml "Big'_int.big'_int'_of'_int")
|
haftmann@27673
|
382 |
(Haskell "toEnum")
|
haftmann@25931
|
383 |
|
haftmann@25931
|
384 |
text {* Using target language arithmetic operations whenever appropriate *}
|
haftmann@25931
|
385 |
|
haftmann@25931
|
386 |
code_const "op + \<Colon> nat \<Rightarrow> nat \<Rightarrow> nat"
|
haftmann@25931
|
387 |
(SML "IntInf.+ ((_), (_))")
|
haftmann@25931
|
388 |
(OCaml "Big'_int.add'_big'_int")
|
haftmann@25931
|
389 |
(Haskell infixl 6 "+")
|
haftmann@25931
|
390 |
|
haftmann@25931
|
391 |
code_const "op * \<Colon> nat \<Rightarrow> nat \<Rightarrow> nat"
|
haftmann@25931
|
392 |
(SML "IntInf.* ((_), (_))")
|
haftmann@25931
|
393 |
(OCaml "Big'_int.mult'_big'_int")
|
haftmann@25931
|
394 |
(Haskell infixl 7 "*")
|
haftmann@25931
|
395 |
|
haftmann@26100
|
396 |
code_const divmod_aux
|
haftmann@26009
|
397 |
(SML "IntInf.divMod/ ((_),/ (_))")
|
haftmann@26009
|
398 |
(OCaml "Big'_int.quomod'_big'_int")
|
haftmann@26009
|
399 |
(Haskell "divMod")
|
haftmann@25931
|
400 |
|
haftmann@28346
|
401 |
code_const "eq_class.eq \<Colon> nat \<Rightarrow> nat \<Rightarrow> bool"
|
haftmann@25931
|
402 |
(SML "!((_ : IntInf.int) = _)")
|
haftmann@25931
|
403 |
(OCaml "Big'_int.eq'_big'_int")
|
haftmann@25931
|
404 |
(Haskell infixl 4 "==")
|
haftmann@25931
|
405 |
|
haftmann@25931
|
406 |
code_const "op \<le> \<Colon> nat \<Rightarrow> nat \<Rightarrow> bool"
|
haftmann@25931
|
407 |
(SML "IntInf.<= ((_), (_))")
|
haftmann@25931
|
408 |
(OCaml "Big'_int.le'_big'_int")
|
haftmann@25931
|
409 |
(Haskell infix 4 "<=")
|
haftmann@25931
|
410 |
|
haftmann@25931
|
411 |
code_const "op < \<Colon> nat \<Rightarrow> nat \<Rightarrow> bool"
|
haftmann@25931
|
412 |
(SML "IntInf.< ((_), (_))")
|
haftmann@25931
|
413 |
(OCaml "Big'_int.lt'_big'_int")
|
haftmann@25931
|
414 |
(Haskell infix 4 "<")
|
haftmann@25931
|
415 |
|
haftmann@25931
|
416 |
consts_code
|
haftmann@28522
|
417 |
"0::nat" ("0")
|
haftmann@28522
|
418 |
"1::nat" ("1")
|
haftmann@25931
|
419 |
Suc ("(_ +/ 1)")
|
haftmann@25931
|
420 |
"op + \<Colon> nat \<Rightarrow> nat \<Rightarrow> nat" ("(_ +/ _)")
|
haftmann@25931
|
421 |
"op * \<Colon> nat \<Rightarrow> nat \<Rightarrow> nat" ("(_ */ _)")
|
haftmann@25931
|
422 |
"op \<le> \<Colon> nat \<Rightarrow> nat \<Rightarrow> bool" ("(_ <=/ _)")
|
haftmann@25931
|
423 |
"op < \<Colon> nat \<Rightarrow> nat \<Rightarrow> bool" ("(_ </ _)")
|
haftmann@25931
|
424 |
|
haftmann@25931
|
425 |
|
haftmann@28228
|
426 |
text {* Evaluation *}
|
haftmann@28228
|
427 |
|
haftmann@28562
|
428 |
lemma [code, code del]:
|
haftmann@28228
|
429 |
"(Code_Eval.term_of \<Colon> nat \<Rightarrow> term) = Code_Eval.term_of" ..
|
haftmann@28228
|
430 |
|
haftmann@28228
|
431 |
code_const "Code_Eval.term_of \<Colon> nat \<Rightarrow> term"
|
haftmann@28228
|
432 |
(SML "HOLogic.mk'_number/ HOLogic.natT")
|
haftmann@28228
|
433 |
|
haftmann@28228
|
434 |
|
haftmann@25931
|
435 |
text {* Module names *}
|
haftmann@23854
|
436 |
|
haftmann@23854
|
437 |
code_modulename SML
|
haftmann@23854
|
438 |
Nat Integer
|
haftmann@23854
|
439 |
Divides Integer
|
haftmann@28683
|
440 |
Ring_and_Field Integer
|
haftmann@23854
|
441 |
Efficient_Nat Integer
|
haftmann@23854
|
442 |
|
haftmann@23854
|
443 |
code_modulename OCaml
|
haftmann@23854
|
444 |
Nat Integer
|
haftmann@23854
|
445 |
Divides Integer
|
haftmann@28683
|
446 |
Ring_and_Field Integer
|
haftmann@23854
|
447 |
Efficient_Nat Integer
|
haftmann@23854
|
448 |
|
haftmann@23854
|
449 |
code_modulename Haskell
|
haftmann@23854
|
450 |
Nat Integer
|
haftmann@24195
|
451 |
Divides Integer
|
haftmann@28683
|
452 |
Ring_and_Field Integer
|
haftmann@23854
|
453 |
Efficient_Nat Integer
|
haftmann@23854
|
454 |
|
haftmann@25931
|
455 |
hide const int
|
haftmann@23854
|
456 |
|
haftmann@23854
|
457 |
end
|