haftmann@23854: (*  Title:      HOL/Library/Efficient_Nat.thy
haftmann@23854:     ID:         $Id$
haftmann@25931:     Author:     Stefan Berghofer, Florian Haftmann, TU Muenchen
haftmann@23854: *)
haftmann@23854: 
haftmann@25931: header {* Implementation of natural numbers by target-language integers *}
haftmann@23854: 
haftmann@23854: theory Efficient_Nat
haftmann@27368: imports Plain Code_Integer Code_Index
haftmann@23854: begin
haftmann@23854: 
haftmann@23854: text {*
haftmann@25931:   When generating code for functions on natural numbers, the
haftmann@25931:   canonical representation using @{term "0::nat"} and
haftmann@25931:   @{term "Suc"} is unsuitable for computations involving large
haftmann@25931:   numbers.  The efficiency of the generated code can be improved
haftmann@25931:   drastically by implementing natural numbers by target-language
haftmann@25931:   integers.  To do this, just include this theory.
haftmann@23854: *}
haftmann@23854: 
haftmann@25931: subsection {* Basic arithmetic *}
haftmann@23854: 
haftmann@23854: text {*
haftmann@23854:   Most standard arithmetic functions on natural numbers are implemented
haftmann@23854:   using their counterparts on the integers:
haftmann@23854: *}
haftmann@23854: 
haftmann@25931: code_datatype number_nat_inst.number_of_nat
haftmann@24715: 
haftmann@25931: lemma zero_nat_code [code, code unfold]:
haftmann@25931:   "0 = (Numeral0 :: nat)"
haftmann@25931:   by simp
haftmann@25931: lemmas [code post] = zero_nat_code [symmetric]
haftmann@24715: 
haftmann@25931: lemma one_nat_code [code, code unfold]:
haftmann@25931:   "1 = (Numeral1 :: nat)"
haftmann@25931:   by simp
haftmann@25931: lemmas [code post] = one_nat_code [symmetric]
haftmann@24715: 
haftmann@25931: lemma Suc_code [code]:
haftmann@25931:   "Suc n = n + 1"
haftmann@25931:   by simp
haftmann@24715: 
haftmann@25931: lemma plus_nat_code [code]:
haftmann@25931:   "n + m = nat (of_nat n + of_nat m)"
haftmann@25931:   by simp
haftmann@24715: 
haftmann@25931: lemma minus_nat_code [code]:
haftmann@25931:   "n - m = nat (of_nat n - of_nat m)"
haftmann@25931:   by simp
haftmann@24715: 
haftmann@25931: lemma times_nat_code [code]:
haftmann@25931:   "n * m = nat (of_nat n * of_nat m)"
haftmann@25931:   unfolding of_nat_mult [symmetric] by simp
haftmann@24715: 
haftmann@26009: text {* Specialized @{term "op div \<Colon> nat \<Rightarrow> nat \<Rightarrow> nat"} 
haftmann@26009:   and @{term "op mod \<Colon> nat \<Rightarrow> nat \<Rightarrow> nat"} operations. *}
haftmann@26009: 
haftmann@26009: definition
haftmann@26100:   divmod_aux ::  "nat \<Rightarrow> nat \<Rightarrow> nat \<times> nat"
haftmann@26009: where
haftmann@26100:   [code func del]: "divmod_aux = divmod"
haftmann@24715: 
haftmann@26009: lemma [code func]:
haftmann@26100:   "divmod n m = (if m = 0 then (0, n) else divmod_aux n m)"
haftmann@26100:   unfolding divmod_aux_def divmod_div_mod by simp
haftmann@26009: 
haftmann@26100: lemma divmod_aux_code [code]:
haftmann@26100:   "divmod_aux n m = (nat (of_nat n div of_nat m), nat (of_nat n mod of_nat m))"
haftmann@26100:   unfolding divmod_aux_def divmod_div_mod zdiv_int [symmetric] zmod_int [symmetric] by simp
haftmann@24715: 
haftmann@25931: lemma eq_nat_code [code]:
haftmann@25931:   "n = m \<longleftrightarrow> (of_nat n \<Colon> int) = of_nat m"
haftmann@25931:   by simp
haftmann@24715: 
haftmann@25931: lemma less_eq_nat_code [code]:
haftmann@25931:   "n \<le> m \<longleftrightarrow> (of_nat n \<Colon> int) \<le> of_nat m"
haftmann@25931:   by simp
haftmann@23854: 
haftmann@25931: lemma less_nat_code [code]:
haftmann@25931:   "n < m \<longleftrightarrow> (of_nat n \<Colon> int) < of_nat m"
haftmann@25931:   by simp
haftmann@23854: 
haftmann@25931: subsection {* Case analysis *}
haftmann@23854: 
haftmann@23854: text {*
haftmann@25931:   Case analysis on natural numbers is rephrased using a conditional
haftmann@25931:   expression:
haftmann@23854: *}
haftmann@23854: 
haftmann@25931: lemma [code func, code unfold]:
haftmann@25931:   "nat_case = (\<lambda>f g n. if n = 0 then f else g (n - 1))"
haftmann@25931:   by (auto simp add: expand_fun_eq dest!: gr0_implies_Suc)
haftmann@25615: 
haftmann@23854: 
haftmann@23854: subsection {* Preprocessors *}
haftmann@23854: 
haftmann@23854: text {*
haftmann@23854:   In contrast to @{term "Suc n"}, the term @{term "n + (1::nat)"} is no longer
haftmann@23854:   a constructor term. Therefore, all occurrences of this term in a position
haftmann@23854:   where a pattern is expected (i.e.\ on the left-hand side of a recursion
haftmann@23854:   equation or in the arguments of an inductive relation in an introduction
haftmann@23854:   rule) must be eliminated.
haftmann@23854:   This can be accomplished by applying the following transformation rules:
haftmann@23854: *}
haftmann@23854: 
haftmann@25931: lemma Suc_if_eq: "(\<And>n. f (Suc n) = h n) \<Longrightarrow> f 0 = g \<Longrightarrow>
haftmann@23854:   f n = (if n = 0 then g else h (n - 1))"
haftmann@23854:   by (case_tac n) simp_all
haftmann@23854: 
haftmann@25931: lemma Suc_clause: "(\<And>n. P n (Suc n)) \<Longrightarrow> n \<noteq> 0 \<Longrightarrow> P (n - 1) n"
haftmann@23854:   by (case_tac n) simp_all
haftmann@23854: 
haftmann@23854: text {*
haftmann@23854:   The rules above are built into a preprocessor that is plugged into
haftmann@23854:   the code generator. Since the preprocessor for introduction rules
haftmann@23854:   does not know anything about modes, some of the modes that worked
haftmann@23854:   for the canonical representation of natural numbers may no longer work.
haftmann@23854: *}
haftmann@23854: 
haftmann@23854: (*<*)
haftmann@27609: setup {*
haftmann@27609: let
haftmann@23854: 
haftmann@23854: fun remove_suc thy thms =
haftmann@23854:   let
haftmann@23854:     val vname = Name.variant (map fst
haftmann@23854:       (fold (Term.add_varnames o Thm.full_prop_of) thms [])) "x";
haftmann@23854:     val cv = cterm_of thy (Var ((vname, 0), HOLogic.natT));
haftmann@23854:     fun lhs_of th = snd (Thm.dest_comb
haftmann@23854:       (fst (Thm.dest_comb (snd (Thm.dest_comb (cprop_of th))))));
haftmann@23854:     fun rhs_of th = snd (Thm.dest_comb (snd (Thm.dest_comb (cprop_of th))));
haftmann@23854:     fun find_vars ct = (case term_of ct of
haftmann@23854:         (Const ("Suc", _) $ Var _) => [(cv, snd (Thm.dest_comb ct))]
haftmann@23854:       | _ $ _ =>
haftmann@23854:         let val (ct1, ct2) = Thm.dest_comb ct
haftmann@23854:         in 
haftmann@23854:           map (apfst (fn ct => Thm.capply ct ct2)) (find_vars ct1) @
haftmann@23854:           map (apfst (Thm.capply ct1)) (find_vars ct2)
haftmann@23854:         end
haftmann@23854:       | _ => []);
haftmann@23854:     val eqs = maps
haftmann@23854:       (fn th => map (pair th) (find_vars (lhs_of th))) thms;
haftmann@23854:     fun mk_thms (th, (ct, cv')) =
haftmann@23854:       let
haftmann@23854:         val th' =
haftmann@23854:           Thm.implies_elim
haftmann@23854:            (Conv.fconv_rule (Thm.beta_conversion true)
haftmann@23854:              (Drule.instantiate'
haftmann@23854:                [SOME (ctyp_of_term ct)] [SOME (Thm.cabs cv ct),
haftmann@23854:                  SOME (Thm.cabs cv' (rhs_of th)), NONE, SOME cv']
haftmann@24222:                @{thm Suc_if_eq})) (Thm.forall_intr cv' th)
haftmann@23854:       in
haftmann@23854:         case map_filter (fn th'' =>
haftmann@23854:             SOME (th'', singleton
haftmann@23854:               (Variable.trade (K (fn [th'''] => [th''' RS th'])) (Variable.thm_context th'')) th'')
haftmann@23854:           handle THM _ => NONE) thms of
haftmann@23854:             [] => NONE
haftmann@23854:           | thps =>
haftmann@23854:               let val (ths1, ths2) = split_list thps
haftmann@23854:               in SOME (subtract Thm.eq_thm (th :: ths1) thms @ ths2) end
haftmann@23854:       end
haftmann@27609:   in case get_first mk_thms eqs of
haftmann@23854:       NONE => thms
haftmann@23854:     | SOME x => remove_suc thy x
haftmann@23854:   end;
haftmann@23854: 
haftmann@23854: fun eqn_suc_preproc thy ths =
haftmann@23854:   let
haftmann@24222:     val dest = fst o HOLogic.dest_eq o HOLogic.dest_Trueprop o prop_of;
haftmann@24222:     fun contains_suc t = member (op =) (term_consts t) @{const_name Suc};
haftmann@23854:   in
haftmann@23854:     if forall (can dest) ths andalso
haftmann@23854:       exists (contains_suc o dest) ths
haftmann@23854:     then remove_suc thy ths else ths
haftmann@23854:   end;
haftmann@23854: 
haftmann@23854: fun remove_suc_clause thy thms =
haftmann@23854:   let
haftmann@23854:     val vname = Name.variant (map fst
haftmann@23854:       (fold (Term.add_varnames o Thm.full_prop_of) thms [])) "x";
haftmann@24222:     fun find_var (t as Const (@{const_name Suc}, _) $ (v as Var _)) = SOME (t, v)
haftmann@23854:       | find_var (t $ u) = (case find_var t of NONE => find_var u | x => x)
haftmann@23854:       | find_var _ = NONE;
haftmann@23854:     fun find_thm th =
haftmann@23854:       let val th' = Conv.fconv_rule ObjectLogic.atomize th
haftmann@23854:       in Option.map (pair (th, th')) (find_var (prop_of th')) end
haftmann@23854:   in
haftmann@23854:     case get_first find_thm thms of
haftmann@23854:       NONE => thms
haftmann@23854:     | SOME ((th, th'), (Sucv, v)) =>
haftmann@23854:         let
haftmann@23854:           val cert = cterm_of (Thm.theory_of_thm th);
haftmann@23854:           val th'' = ObjectLogic.rulify (Thm.implies_elim
haftmann@23854:             (Conv.fconv_rule (Thm.beta_conversion true)
haftmann@23854:               (Drule.instantiate' []
haftmann@23854:                 [SOME (cert (lambda v (Abs ("x", HOLogic.natT,
haftmann@23854:                    abstract_over (Sucv,
haftmann@23854:                      HOLogic.dest_Trueprop (prop_of th')))))),
haftmann@24222:                  SOME (cert v)] @{thm Suc_clause}))
haftmann@23854:             (Thm.forall_intr (cert v) th'))
haftmann@23854:         in
haftmann@23854:           remove_suc_clause thy (map (fn th''' =>
haftmann@23854:             if (op = o pairself prop_of) (th''', th) then th'' else th''') thms)
haftmann@23854:         end
haftmann@23854:   end;
haftmann@23854: 
haftmann@23854: fun clause_suc_preproc thy ths =
haftmann@23854:   let
haftmann@23854:     val dest = fst o HOLogic.dest_mem o HOLogic.dest_Trueprop
haftmann@23854:   in
haftmann@23854:     if forall (can (dest o concl_of)) ths andalso
haftmann@23854:       exists (fn th => member (op =) (foldr add_term_consts
haftmann@23854:         [] (map_filter (try dest) (concl_of th :: prems_of th))) "Suc") ths
haftmann@23854:     then remove_suc_clause thy ths else ths
haftmann@23854:   end;
haftmann@23854: 
haftmann@27609: fun lift f thy thms1 =
haftmann@27609:   let
haftmann@27609:     val thms2 = Drule.zero_var_indexes_list thms1;
haftmann@27609:     val thms3 = try (map (fn thm => thm RS @{thm meta_eq_to_obj_eq})
haftmann@27609:       #> f thy
haftmann@27609:       #> map (fn thm => thm RS @{thm eq_reflection})
haftmann@27609:       #> map (Conv.fconv_rule Drule.beta_eta_conversion)) thms2;
haftmann@27609:     val thms4 = Option.map Drule.zero_var_indexes_list thms3;
haftmann@27609:   in case thms4
haftmann@27609:    of NONE => NONE
haftmann@27609:     | SOME thms4 => if Thm.eq_thms (thms2, thms4) then NONE else SOME thms4
haftmann@27609:   end
haftmann@23854: 
haftmann@27609: in
haftmann@27609: 
haftmann@23854:   Codegen.add_preprocessor eqn_suc_preproc
haftmann@23854:   #> Codegen.add_preprocessor clause_suc_preproc
haftmann@27609:   #> Code.add_functrans ("eqn_Suc", lift eqn_suc_preproc)
haftmann@27609:   #> Code.add_functrans ("clause_Suc", lift clause_suc_preproc)
haftmann@27609: 
haftmann@27609: end;
haftmann@23854: *}
haftmann@23854: (*>*)
haftmann@23854: 
haftmann@27609: 
haftmann@25931: subsection {* Target language setup *}
haftmann@25931: 
haftmann@25931: text {*
haftmann@25967:   For ML, we map @{typ nat} to target language integers, where we
haftmann@25931:   assert that values are always non-negative.
haftmann@25931: *}
haftmann@25931: 
haftmann@25931: code_type nat
haftmann@27496:   (SML "IntInf.int")
haftmann@25931:   (OCaml "Big'_int.big'_int")
haftmann@25931: 
haftmann@25931: types_code
haftmann@25931:   nat ("int")
haftmann@25931: attach (term_of) {*
haftmann@25931: val term_of_nat = HOLogic.mk_number HOLogic.natT;
haftmann@25931: *}
haftmann@25931: attach (test) {*
haftmann@25931: fun gen_nat i =
haftmann@25931:   let val n = random_range 0 i
haftmann@25931:   in (n, fn () => term_of_nat n) end;
haftmann@25931: *}
haftmann@25931: 
haftmann@25931: text {*
haftmann@25967:   For Haskell we define our own @{typ nat} type.  The reason
haftmann@25967:   is that we have to distinguish type class instances
haftmann@25967:   for @{typ nat} and @{typ int}.
haftmann@25967: *}
haftmann@25967: 
haftmann@25967: code_include Haskell "Nat" {*
haftmann@25967: newtype Nat = Nat Integer deriving (Show, Eq);
haftmann@25967: 
haftmann@25967: instance Num Nat where {
haftmann@25967:   fromInteger k = Nat (if k >= 0 then k else 0);
haftmann@25967:   Nat n + Nat m = Nat (n + m);
haftmann@25967:   Nat n - Nat m = fromInteger (n - m);
haftmann@25967:   Nat n * Nat m = Nat (n * m);
haftmann@25967:   abs n = n;
haftmann@25967:   signum _ = 1;
haftmann@25967:   negate n = error "negate Nat";
haftmann@25967: };
haftmann@25967: 
haftmann@25967: instance Ord Nat where {
haftmann@25967:   Nat n <= Nat m = n <= m;
haftmann@25967:   Nat n < Nat m = n < m;
haftmann@25967: };
haftmann@25967: 
haftmann@25967: instance Real Nat where {
haftmann@25967:   toRational (Nat n) = toRational n;
haftmann@25967: };
haftmann@25967: 
haftmann@25967: instance Enum Nat where {
haftmann@25967:   toEnum k = fromInteger (toEnum k);
haftmann@25967:   fromEnum (Nat n) = fromEnum n;
haftmann@25967: };
haftmann@25967: 
haftmann@25967: instance Integral Nat where {
haftmann@25967:   toInteger (Nat n) = n;
haftmann@25967:   divMod n m = quotRem n m;
haftmann@25967:   quotRem (Nat n) (Nat m) = (Nat k, Nat l) where (k, l) = quotRem n m;
haftmann@25967: };
haftmann@25967: *}
haftmann@25967: 
haftmann@25967: code_reserved Haskell Nat
haftmann@25967: 
haftmann@25967: code_type nat
haftmann@25967:   (Haskell "Nat")
haftmann@25967: 
haftmann@25967: code_instance nat :: eq
haftmann@25967:   (Haskell -)
haftmann@25967: 
haftmann@25967: text {*
haftmann@25931:   Natural numerals.
haftmann@25931: *}
haftmann@25931: 
haftmann@25967: lemma [code inline, symmetric, code post]:
haftmann@25931:   "nat (number_of i) = number_nat_inst.number_of_nat i"
haftmann@25931:   -- {* this interacts as desired with @{thm nat_number_of_def} *}
haftmann@25931:   by (simp add: number_nat_inst.number_of_nat)
haftmann@25931: 
haftmann@25931: setup {*
haftmann@25931:   fold (Numeral.add_code @{const_name number_nat_inst.number_of_nat}
haftmann@25967:     true false) ["SML", "OCaml", "Haskell"]
haftmann@25931: *}
haftmann@25931: 
haftmann@25931: text {*
haftmann@25931:   Since natural numbers are implemented
haftmann@25967:   using integers in ML, the coercion function @{const "of_nat"} of type
haftmann@25931:   @{typ "nat \<Rightarrow> int"} is simply implemented by the identity function.
haftmann@25931:   For the @{const "nat"} function for converting an integer to a natural
haftmann@25931:   number, we give a specific implementation using an ML function that
haftmann@25931:   returns its input value, provided that it is non-negative, and otherwise
haftmann@25931:   returns @{text "0"}.
haftmann@25931: *}
haftmann@25931: 
haftmann@25931: definition
haftmann@25931:   int :: "nat \<Rightarrow> int"
haftmann@25931: where
haftmann@25931:   [code func del]: "int = of_nat"
haftmann@25931: 
haftmann@25931: lemma int_code' [code func]:
haftmann@25931:   "int (number_of l) = (if neg (number_of l \<Colon> int) then 0 else number_of l)"
haftmann@25931:   unfolding int_nat_number_of [folded int_def] ..
haftmann@25931: 
haftmann@25931: lemma nat_code' [code func]:
haftmann@25931:   "nat (number_of l) = (if neg (number_of l \<Colon> int) then 0 else number_of l)"
haftmann@25931:   by auto
haftmann@25931: 
haftmann@25931: lemma of_nat_int [code unfold]:
haftmann@25931:   "of_nat = int" by (simp add: int_def)
haftmann@25967: declare of_nat_int [symmetric, code post]
haftmann@25931: 
haftmann@25931: code_const int
haftmann@25931:   (SML "_")
haftmann@25931:   (OCaml "_")
haftmann@25931: 
haftmann@25931: consts_code
haftmann@25931:   int ("(_)")
haftmann@25931:   nat ("\<module>nat")
haftmann@25931: attach {*
haftmann@25931: fun nat i = if i < 0 then 0 else i;
haftmann@25931: *}
haftmann@25931: 
haftmann@25967: code_const nat
haftmann@25967:   (SML "IntInf.max/ (/0,/ _)")
haftmann@25967:   (OCaml "Big'_int.max'_big'_int/ Big'_int.zero'_big'_int")
haftmann@25967: 
haftmann@25967: text {* For Haskell, things are slightly different again. *}
haftmann@25967: 
haftmann@25967: code_const int and nat
haftmann@25967:   (Haskell "toInteger" and "fromInteger")
haftmann@25931: 
haftmann@25931: text {* Conversion from and to indices. *}
haftmann@25931: 
haftmann@25967: code_const index_of_nat
haftmann@25967:   (SML "IntInf.toInt")
haftmann@25967:   (OCaml "Big'_int.int'_of'_big'_int")
haftmann@25967:   (Haskell "toEnum")
haftmann@25967: 
haftmann@25931: code_const nat_of_index
haftmann@25931:   (SML "IntInf.fromInt")
haftmann@25931:   (OCaml "Big'_int.big'_int'_of'_int")
haftmann@25967:   (Haskell "fromEnum")
haftmann@25931: 
haftmann@25931: text {* Using target language arithmetic operations whenever appropriate *}
haftmann@25931: 
haftmann@25931: code_const "op + \<Colon> nat \<Rightarrow> nat \<Rightarrow> nat"
haftmann@25931:   (SML "IntInf.+ ((_), (_))")
haftmann@25931:   (OCaml "Big'_int.add'_big'_int")
haftmann@25931:   (Haskell infixl 6 "+")
haftmann@25931: 
haftmann@25931: code_const "op * \<Colon> nat \<Rightarrow> nat \<Rightarrow> nat"
haftmann@25931:   (SML "IntInf.* ((_), (_))")
haftmann@25931:   (OCaml "Big'_int.mult'_big'_int")
haftmann@25931:   (Haskell infixl 7 "*")
haftmann@25931: 
haftmann@26100: code_const divmod_aux
haftmann@26009:   (SML "IntInf.divMod/ ((_),/ (_))")
haftmann@26009:   (OCaml "Big'_int.quomod'_big'_int")
haftmann@26009:   (Haskell "divMod")
haftmann@25931: 
haftmann@25931: code_const "op = \<Colon> nat \<Rightarrow> nat \<Rightarrow> bool"
haftmann@25931:   (SML "!((_ : IntInf.int) = _)")
haftmann@25931:   (OCaml "Big'_int.eq'_big'_int")
haftmann@25931:   (Haskell infixl 4 "==")
haftmann@25931: 
haftmann@25931: code_const "op \<le> \<Colon> nat \<Rightarrow> nat \<Rightarrow> bool"
haftmann@25931:   (SML "IntInf.<= ((_), (_))")
haftmann@25931:   (OCaml "Big'_int.le'_big'_int")
haftmann@25931:   (Haskell infix 4 "<=")
haftmann@25931: 
haftmann@25931: code_const "op < \<Colon> nat \<Rightarrow> nat \<Rightarrow> bool"
haftmann@25931:   (SML "IntInf.< ((_), (_))")
haftmann@25931:   (OCaml "Big'_int.lt'_big'_int")
haftmann@25931:   (Haskell infix 4 "<")
haftmann@25931: 
haftmann@25931: consts_code
haftmann@25931:   0                            ("0")
haftmann@25931:   Suc                          ("(_ +/ 1)")
haftmann@25931:   "op + \<Colon>  nat \<Rightarrow> nat \<Rightarrow> nat"   ("(_ +/ _)")
haftmann@25931:   "op * \<Colon>  nat \<Rightarrow> nat \<Rightarrow> nat"   ("(_ */ _)")
haftmann@25931:   "op \<le> \<Colon>  nat \<Rightarrow> nat \<Rightarrow> bool"  ("(_ <=/ _)")
haftmann@25931:   "op < \<Colon>  nat \<Rightarrow> nat \<Rightarrow> bool"  ("(_ </ _)")
haftmann@25931: 
haftmann@25931: 
haftmann@25931: text {* Module names *}
haftmann@23854: 
haftmann@23854: code_modulename SML
haftmann@23854:   Nat Integer
haftmann@23854:   Divides Integer
haftmann@23854:   Efficient_Nat Integer
haftmann@23854: 
haftmann@23854: code_modulename OCaml
haftmann@23854:   Nat Integer
haftmann@23854:   Divides Integer
haftmann@23854:   Efficient_Nat Integer
haftmann@23854: 
haftmann@23854: code_modulename Haskell
haftmann@23854:   Nat Integer
haftmann@24195:   Divides Integer
haftmann@23854:   Efficient_Nat Integer
haftmann@23854: 
haftmann@25931: hide const int
haftmann@23854: 
haftmann@23854: end