src/HOL/Library/Efficient_Nat.thy
 author wenzelm Wed Dec 31 18:53:16 2008 +0100 (2008-12-31) changeset 29270 0eade173f77e parent 29258 bce03c644efb child 29287 5b0bfd63b5da permissions -rw-r--r--
```     1 (*  Title:      HOL/Library/Efficient_Nat.thy
```
```     2     Author:     Stefan Berghofer, Florian Haftmann, TU Muenchen
```
```     3 *)
```
```     4
```
```     5 header {* Implementation of natural numbers by target-language integers *}
```
```     6
```
```     7 theory Efficient_Nat
```
```     8 imports Code_Index Code_Integer
```
```     9 begin
```
```    10
```
```    11 text {*
```
```    12   When generating code for functions on natural numbers, the
```
```    13   canonical representation using @{term "0::nat"} and
```
```    14   @{term "Suc"} is unsuitable for computations involving large
```
```    15   numbers.  The efficiency of the generated code can be improved
```
```    16   drastically by implementing natural numbers by target-language
```
```    17   integers.  To do this, just include this theory.
```
```    18 *}
```
```    19
```
```    20 subsection {* Basic arithmetic *}
```
```    21
```
```    22 text {*
```
```    23   Most standard arithmetic functions on natural numbers are implemented
```
```    24   using their counterparts on the integers:
```
```    25 *}
```
```    26
```
```    27 code_datatype number_nat_inst.number_of_nat
```
```    28
```
```    29 lemma zero_nat_code [code, code inline]:
```
```    30   "0 = (Numeral0 :: nat)"
```
```    31   by simp
```
```    32 lemmas [code post] = zero_nat_code [symmetric]
```
```    33
```
```    34 lemma one_nat_code [code, code inline]:
```
```    35   "1 = (Numeral1 :: nat)"
```
```    36   by simp
```
```    37 lemmas [code post] = one_nat_code [symmetric]
```
```    38
```
```    39 lemma Suc_code [code]:
```
```    40   "Suc n = n + 1"
```
```    41   by simp
```
```    42
```
```    43 lemma plus_nat_code [code]:
```
```    44   "n + m = nat (of_nat n + of_nat m)"
```
```    45   by simp
```
```    46
```
```    47 lemma minus_nat_code [code]:
```
```    48   "n - m = nat (of_nat n - of_nat m)"
```
```    49   by simp
```
```    50
```
```    51 lemma times_nat_code [code]:
```
```    52   "n * m = nat (of_nat n * of_nat m)"
```
```    53   unfolding of_nat_mult [symmetric] by simp
```
```    54
```
```    55 text {* Specialized @{term "op div \<Colon> nat \<Rightarrow> nat \<Rightarrow> nat"}
```
```    56   and @{term "op mod \<Colon> nat \<Rightarrow> nat \<Rightarrow> nat"} operations. *}
```
```    57
```
```    58 definition divmod_aux ::  "nat \<Rightarrow> nat \<Rightarrow> nat \<times> nat" where
```
```    59   [code del]: "divmod_aux = divmod"
```
```    60
```
```    61 lemma [code]:
```
```    62   "divmod n m = (if m = 0 then (0, n) else divmod_aux n m)"
```
```    63   unfolding divmod_aux_def divmod_div_mod by simp
```
```    64
```
```    65 lemma divmod_aux_code [code]:
```
```    66   "divmod_aux n m = (nat (of_nat n div of_nat m), nat (of_nat n mod of_nat m))"
```
```    67   unfolding divmod_aux_def divmod_div_mod zdiv_int [symmetric] zmod_int [symmetric] by simp
```
```    68
```
```    69 lemma eq_nat_code [code]:
```
```    70   "eq_class.eq n m \<longleftrightarrow> eq_class.eq (of_nat n \<Colon> int) (of_nat m)"
```
```    71   by (simp add: eq)
```
```    72
```
```    73 lemma eq_nat_refl [code nbe]:
```
```    74   "eq_class.eq (n::nat) n \<longleftrightarrow> True"
```
```    75   by (rule HOL.eq_refl)
```
```    76
```
```    77 lemma less_eq_nat_code [code]:
```
```    78   "n \<le> m \<longleftrightarrow> (of_nat n \<Colon> int) \<le> of_nat m"
```
```    79   by simp
```
```    80
```
```    81 lemma less_nat_code [code]:
```
```    82   "n < m \<longleftrightarrow> (of_nat n \<Colon> int) < of_nat m"
```
```    83   by simp
```
```    84
```
```    85 subsection {* Case analysis *}
```
```    86
```
```    87 text {*
```
```    88   Case analysis on natural numbers is rephrased using a conditional
```
```    89   expression:
```
```    90 *}
```
```    91
```
```    92 lemma [code, code unfold]:
```
```    93   "nat_case = (\<lambda>f g n. if n = 0 then f else g (n - 1))"
```
```    94   by (auto simp add: expand_fun_eq dest!: gr0_implies_Suc)
```
```    95
```
```    96
```
```    97 subsection {* Preprocessors *}
```
```    98
```
```    99 text {*
```
```   100   In contrast to @{term "Suc n"}, the term @{term "n + (1::nat)"} is no longer
```
```   101   a constructor term. Therefore, all occurrences of this term in a position
```
```   102   where a pattern is expected (i.e.\ on the left-hand side of a recursion
```
```   103   equation or in the arguments of an inductive relation in an introduction
```
```   104   rule) must be eliminated.
```
```   105   This can be accomplished by applying the following transformation rules:
```
```   106 *}
```
```   107
```
```   108 lemma Suc_if_eq: "(\<And>n. f (Suc n) = h n) \<Longrightarrow> f 0 = g \<Longrightarrow>
```
```   109   f n = (if n = 0 then g else h (n - 1))"
```
```   110   by (case_tac n) simp_all
```
```   111
```
```   112 lemma Suc_clause: "(\<And>n. P n (Suc n)) \<Longrightarrow> n \<noteq> 0 \<Longrightarrow> P (n - 1) n"
```
```   113   by (case_tac n) simp_all
```
```   114
```
```   115 text {*
```
```   116   The rules above are built into a preprocessor that is plugged into
```
```   117   the code generator. Since the preprocessor for introduction rules
```
```   118   does not know anything about modes, some of the modes that worked
```
```   119   for the canonical representation of natural numbers may no longer work.
```
```   120 *}
```
```   121
```
```   122 (*<*)
```
```   123 setup {*
```
```   124 let
```
```   125
```
```   126 fun remove_suc thy thms =
```
```   127   let
```
```   128     val vname = Name.variant (map fst
```
```   129       (fold (Term.add_var_names o Thm.full_prop_of) thms [])) "x";
```
```   130     val cv = cterm_of thy (Var ((vname, 0), HOLogic.natT));
```
```   131     fun lhs_of th = snd (Thm.dest_comb
```
```   132       (fst (Thm.dest_comb (snd (Thm.dest_comb (cprop_of th))))));
```
```   133     fun rhs_of th = snd (Thm.dest_comb (snd (Thm.dest_comb (cprop_of th))));
```
```   134     fun find_vars ct = (case term_of ct of
```
```   135         (Const ("Suc", _) \$ Var _) => [(cv, snd (Thm.dest_comb ct))]
```
```   136       | _ \$ _ =>
```
```   137         let val (ct1, ct2) = Thm.dest_comb ct
```
```   138         in
```
```   139           map (apfst (fn ct => Thm.capply ct ct2)) (find_vars ct1) @
```
```   140           map (apfst (Thm.capply ct1)) (find_vars ct2)
```
```   141         end
```
```   142       | _ => []);
```
```   143     val eqs = maps
```
```   144       (fn th => map (pair th) (find_vars (lhs_of th))) thms;
```
```   145     fun mk_thms (th, (ct, cv')) =
```
```   146       let
```
```   147         val th' =
```
```   148           Thm.implies_elim
```
```   149            (Conv.fconv_rule (Thm.beta_conversion true)
```
```   150              (Drule.instantiate'
```
```   151                [SOME (ctyp_of_term ct)] [SOME (Thm.cabs cv ct),
```
```   152                  SOME (Thm.cabs cv' (rhs_of th)), NONE, SOME cv']
```
```   153                @{thm Suc_if_eq})) (Thm.forall_intr cv' th)
```
```   154       in
```
```   155         case map_filter (fn th'' =>
```
```   156             SOME (th'', singleton
```
```   157               (Variable.trade (K (fn [th'''] => [th''' RS th'])) (Variable.thm_context th'')) th'')
```
```   158           handle THM _ => NONE) thms of
```
```   159             [] => NONE
```
```   160           | thps =>
```
```   161               let val (ths1, ths2) = split_list thps
```
```   162               in SOME (subtract Thm.eq_thm (th :: ths1) thms @ ths2) end
```
```   163       end
```
```   164   in case get_first mk_thms eqs of
```
```   165       NONE => thms
```
```   166     | SOME x => remove_suc thy x
```
```   167   end;
```
```   168
```
```   169 fun eqn_suc_preproc thy ths =
```
```   170   let
```
```   171     val dest = fst o HOLogic.dest_eq o HOLogic.dest_Trueprop o prop_of;
```
```   172     fun contains_suc t = member (op =) (OldTerm.term_consts t) @{const_name Suc};
```
```   173   in
```
```   174     if forall (can dest) ths andalso
```
```   175       exists (contains_suc o dest) ths
```
```   176     then remove_suc thy ths else ths
```
```   177   end;
```
```   178
```
```   179 fun remove_suc_clause thy thms =
```
```   180   let
```
```   181     val vname = Name.variant (map fst
```
```   182       (fold (Term.add_var_names o Thm.full_prop_of) thms [])) "x";
```
```   183     fun find_var (t as Const (@{const_name Suc}, _) \$ (v as Var _)) = SOME (t, v)
```
```   184       | find_var (t \$ u) = (case find_var t of NONE => find_var u | x => x)
```
```   185       | find_var _ = NONE;
```
```   186     fun find_thm th =
```
```   187       let val th' = Conv.fconv_rule ObjectLogic.atomize th
```
```   188       in Option.map (pair (th, th')) (find_var (prop_of th')) end
```
```   189   in
```
```   190     case get_first find_thm thms of
```
```   191       NONE => thms
```
```   192     | SOME ((th, th'), (Sucv, v)) =>
```
```   193         let
```
```   194           val cert = cterm_of (Thm.theory_of_thm th);
```
```   195           val th'' = ObjectLogic.rulify (Thm.implies_elim
```
```   196             (Conv.fconv_rule (Thm.beta_conversion true)
```
```   197               (Drule.instantiate' []
```
```   198                 [SOME (cert (lambda v (Abs ("x", HOLogic.natT,
```
```   199                    abstract_over (Sucv,
```
```   200                      HOLogic.dest_Trueprop (prop_of th')))))),
```
```   201                  SOME (cert v)] @{thm Suc_clause}))
```
```   202             (Thm.forall_intr (cert v) th'))
```
```   203         in
```
```   204           remove_suc_clause thy (map (fn th''' =>
```
```   205             if (op = o pairself prop_of) (th''', th) then th'' else th''') thms)
```
```   206         end
```
```   207   end;
```
```   208
```
```   209 fun clause_suc_preproc thy ths =
```
```   210   let
```
```   211     val dest = fst o HOLogic.dest_mem o HOLogic.dest_Trueprop
```
```   212   in
```
```   213     if forall (can (dest o concl_of)) ths andalso
```
```   214       exists (fn th => member (op =) (foldr OldTerm.add_term_consts
```
```   215         [] (map_filter (try dest) (concl_of th :: prems_of th))) "Suc") ths
```
```   216     then remove_suc_clause thy ths else ths
```
```   217   end;
```
```   218
```
```   219 fun lift f thy eqns1 =
```
```   220   let
```
```   221     val eqns2 = burrow_fst Drule.zero_var_indexes_list eqns1;
```
```   222     val thms3 = try (map fst
```
```   223       #> map (fn thm => thm RS @{thm meta_eq_to_obj_eq})
```
```   224       #> f thy
```
```   225       #> map (fn thm => thm RS @{thm eq_reflection})
```
```   226       #> map (Conv.fconv_rule Drule.beta_eta_conversion)) eqns2;
```
```   227     val thms4 = Option.map Drule.zero_var_indexes_list thms3;
```
```   228   in case thms4
```
```   229    of NONE => NONE
```
```   230     | SOME thms4 => if Thm.eq_thms (map fst eqns2, thms4)
```
```   231         then NONE else SOME (map (apfst (AxClass.overload thy) o Code_Unit.mk_eqn thy) thms4)
```
```   232   end
```
```   233
```
```   234 in
```
```   235
```
```   236   Codegen.add_preprocessor eqn_suc_preproc
```
```   237   #> Codegen.add_preprocessor clause_suc_preproc
```
```   238   #> Code.add_functrans ("eqn_Suc", lift eqn_suc_preproc)
```
```   239   #> Code.add_functrans ("clause_Suc", lift clause_suc_preproc)
```
```   240
```
```   241 end;
```
```   242 *}
```
```   243 (*>*)
```
```   244
```
```   245
```
```   246 subsection {* Target language setup *}
```
```   247
```
```   248 text {*
```
```   249   For ML, we map @{typ nat} to target language integers, where we
```
```   250   assert that values are always non-negative.
```
```   251 *}
```
```   252
```
```   253 code_type nat
```
```   254   (SML "IntInf.int")
```
```   255   (OCaml "Big'_int.big'_int")
```
```   256
```
```   257 types_code
```
```   258   nat ("int")
```
```   259 attach (term_of) {*
```
```   260 val term_of_nat = HOLogic.mk_number HOLogic.natT;
```
```   261 *}
```
```   262 attach (test) {*
```
```   263 fun gen_nat i =
```
```   264   let val n = random_range 0 i
```
```   265   in (n, fn () => term_of_nat n) end;
```
```   266 *}
```
```   267
```
```   268 text {*
```
```   269   For Haskell we define our own @{typ nat} type.  The reason
```
```   270   is that we have to distinguish type class instances
```
```   271   for @{typ nat} and @{typ int}.
```
```   272 *}
```
```   273
```
```   274 code_include Haskell "Nat" {*
```
```   275 newtype Nat = Nat Integer deriving (Show, Eq);
```
```   276
```
```   277 instance Num Nat where {
```
```   278   fromInteger k = Nat (if k >= 0 then k else 0);
```
```   279   Nat n + Nat m = Nat (n + m);
```
```   280   Nat n - Nat m = fromInteger (n - m);
```
```   281   Nat n * Nat m = Nat (n * m);
```
```   282   abs n = n;
```
```   283   signum _ = 1;
```
```   284   negate n = error "negate Nat";
```
```   285 };
```
```   286
```
```   287 instance Ord Nat where {
```
```   288   Nat n <= Nat m = n <= m;
```
```   289   Nat n < Nat m = n < m;
```
```   290 };
```
```   291
```
```   292 instance Real Nat where {
```
```   293   toRational (Nat n) = toRational n;
```
```   294 };
```
```   295
```
```   296 instance Enum Nat where {
```
```   297   toEnum k = fromInteger (toEnum k);
```
```   298   fromEnum (Nat n) = fromEnum n;
```
```   299 };
```
```   300
```
```   301 instance Integral Nat where {
```
```   302   toInteger (Nat n) = n;
```
```   303   divMod n m = quotRem n m;
```
```   304   quotRem (Nat n) (Nat m) = (Nat k, Nat l) where (k, l) = quotRem n m;
```
```   305 };
```
```   306 *}
```
```   307
```
```   308 code_reserved Haskell Nat
```
```   309
```
```   310 code_type nat
```
```   311   (Haskell "Nat")
```
```   312
```
```   313 code_instance nat :: eq
```
```   314   (Haskell -)
```
```   315
```
```   316 text {*
```
```   317   Natural numerals.
```
```   318 *}
```
```   319
```
```   320 lemma [code inline, symmetric, code post]:
```
```   321   "nat (number_of i) = number_nat_inst.number_of_nat i"
```
```   322   -- {* this interacts as desired with @{thm nat_number_of_def} *}
```
```   323   by (simp add: number_nat_inst.number_of_nat)
```
```   324
```
```   325 setup {*
```
```   326   fold (Numeral.add_code @{const_name number_nat_inst.number_of_nat}
```
```   327     true false) ["SML", "OCaml", "Haskell"]
```
```   328 *}
```
```   329
```
```   330 text {*
```
```   331   Since natural numbers are implemented
```
```   332   using integers in ML, the coercion function @{const "of_nat"} of type
```
```   333   @{typ "nat \<Rightarrow> int"} is simply implemented by the identity function.
```
```   334   For the @{const "nat"} function for converting an integer to a natural
```
```   335   number, we give a specific implementation using an ML function that
```
```   336   returns its input value, provided that it is non-negative, and otherwise
```
```   337   returns @{text "0"}.
```
```   338 *}
```
```   339
```
```   340 definition
```
```   341   int :: "nat \<Rightarrow> int"
```
```   342 where
```
```   343   [code del]: "int = of_nat"
```
```   344
```
```   345 lemma int_code' [code]:
```
```   346   "int (number_of l) = (if neg (number_of l \<Colon> int) then 0 else number_of l)"
```
```   347   unfolding int_nat_number_of [folded int_def] ..
```
```   348
```
```   349 lemma nat_code' [code]:
```
```   350   "nat (number_of l) = (if neg (number_of l \<Colon> int) then 0 else number_of l)"
```
```   351   unfolding nat_number_of_def number_of_is_id neg_def by simp
```
```   352
```
```   353 lemma of_nat_int [code unfold]:
```
```   354   "of_nat = int" by (simp add: int_def)
```
```   355 declare of_nat_int [symmetric, code post]
```
```   356
```
```   357 code_const int
```
```   358   (SML "_")
```
```   359   (OCaml "_")
```
```   360
```
```   361 consts_code
```
```   362   int ("(_)")
```
```   363   nat ("\<module>nat")
```
```   364 attach {*
```
```   365 fun nat i = if i < 0 then 0 else i;
```
```   366 *}
```
```   367
```
```   368 code_const nat
```
```   369   (SML "IntInf.max/ (/0,/ _)")
```
```   370   (OCaml "Big'_int.max'_big'_int/ Big'_int.zero'_big'_int")
```
```   371
```
```   372 text {* For Haskell, things are slightly different again. *}
```
```   373
```
```   374 code_const int and nat
```
```   375   (Haskell "toInteger" and "fromInteger")
```
```   376
```
```   377 text {* Conversion from and to indices. *}
```
```   378
```
```   379 code_const index_of_nat
```
```   380   (SML "IntInf.toInt")
```
```   381   (OCaml "Big'_int.int'_of'_big'_int")
```
```   382   (Haskell "fromEnum")
```
```   383
```
```   384 code_const nat_of_index
```
```   385   (SML "IntInf.fromInt")
```
```   386   (OCaml "Big'_int.big'_int'_of'_int")
```
```   387   (Haskell "toEnum")
```
```   388
```
```   389 text {* Using target language arithmetic operations whenever appropriate *}
```
```   390
```
```   391 code_const "op + \<Colon> nat \<Rightarrow> nat \<Rightarrow> nat"
```
```   392   (SML "IntInf.+ ((_), (_))")
```
```   393   (OCaml "Big'_int.add'_big'_int")
```
```   394   (Haskell infixl 6 "+")
```
```   395
```
```   396 code_const "op * \<Colon> nat \<Rightarrow> nat \<Rightarrow> nat"
```
```   397   (SML "IntInf.* ((_), (_))")
```
```   398   (OCaml "Big'_int.mult'_big'_int")
```
```   399   (Haskell infixl 7 "*")
```
```   400
```
```   401 code_const divmod_aux
```
```   402   (SML "IntInf.divMod/ ((_),/ (_))")
```
```   403   (OCaml "Big'_int.quomod'_big'_int")
```
```   404   (Haskell "divMod")
```
```   405
```
```   406 code_const "eq_class.eq \<Colon> nat \<Rightarrow> nat \<Rightarrow> bool"
```
```   407   (SML "!((_ : IntInf.int) = _)")
```
```   408   (OCaml "Big'_int.eq'_big'_int")
```
```   409   (Haskell infixl 4 "==")
```
```   410
```
```   411 code_const "op \<le> \<Colon> nat \<Rightarrow> nat \<Rightarrow> bool"
```
```   412   (SML "IntInf.<= ((_), (_))")
```
```   413   (OCaml "Big'_int.le'_big'_int")
```
```   414   (Haskell infix 4 "<=")
```
```   415
```
```   416 code_const "op < \<Colon> nat \<Rightarrow> nat \<Rightarrow> bool"
```
```   417   (SML "IntInf.< ((_), (_))")
```
```   418   (OCaml "Big'_int.lt'_big'_int")
```
```   419   (Haskell infix 4 "<")
```
```   420
```
```   421 consts_code
```
```   422   "0::nat"                     ("0")
```
```   423   "1::nat"                     ("1")
```
```   424   Suc                          ("(_ +/ 1)")
```
```   425   "op + \<Colon>  nat \<Rightarrow> nat \<Rightarrow> nat"   ("(_ +/ _)")
```
```   426   "op * \<Colon>  nat \<Rightarrow> nat \<Rightarrow> nat"   ("(_ */ _)")
```
```   427   "op \<le> \<Colon>  nat \<Rightarrow> nat \<Rightarrow> bool"  ("(_ <=/ _)")
```
```   428   "op < \<Colon>  nat \<Rightarrow> nat \<Rightarrow> bool"  ("(_ </ _)")
```
```   429
```
```   430
```
```   431 text {* Evaluation *}
```
```   432
```
```   433 lemma [code, code del]:
```
```   434   "(Code_Eval.term_of \<Colon> nat \<Rightarrow> term) = Code_Eval.term_of" ..
```
```   435
```
```   436 code_const "Code_Eval.term_of \<Colon> nat \<Rightarrow> term"
```
```   437   (SML "HOLogic.mk'_number/ HOLogic.natT")
```
```   438
```
```   439
```
```   440 text {* Module names *}
```
```   441
```
```   442 code_modulename SML
```
```   443   Nat Integer
```
```   444   Divides Integer
```
```   445   Ring_and_Field Integer
```
```   446   Efficient_Nat Integer
```
```   447
```
```   448 code_modulename OCaml
```
```   449   Nat Integer
```
```   450   Divides Integer
```
```   451   Ring_and_Field Integer
```
```   452   Efficient_Nat Integer
```
```   453
```
```   454 code_modulename Haskell
```
```   455   Nat Integer
```
```   456   Divides Integer
```
```   457   Ring_and_Field Integer
```
```   458   Efficient_Nat Integer
```
```   459
```
```   460 hide const int
```
```   461
```
```   462 end
```