src/HOL/Library/Efficient_Nat.thy
 author haftmann Tue Jul 07 17:37:00 2009 +0200 (2009-07-07) changeset 31954 8db19c99b00a parent 31377 a48f9ef9de15 child 31998 2c7a24f74db9 permissions -rw-r--r--
Stefan Berghofer's code generator uses Pure equality instead of HOL equality now
```     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_Integer Main
```
```     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 = Divides.divmod"
```
```    60
```
```    61 lemma [code]:
```
```    62   "Divides.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) \<equiv> h n) \<Longrightarrow> f 0 \<equiv> g \<Longrightarrow>
```
```   109   f n \<equiv> if n = 0 then g else h (n - 1)"
```
```   110   by (rule eq_reflection) (cases 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 (cases 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 [])) "n";
```
```   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 (cprop_of th))));
```
```   133     fun rhs_of th = snd (Thm.dest_comb (cprop_of th));
```
```   134     fun find_vars ct = (case term_of ct of
```
```   135         (Const (@{const_name 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 get_first mk_thms eqs end;
```
```   165
```
```   166 fun eqn_suc_preproc thy thms =
```
```   167   let
```
```   168     val dest = fst o Logic.dest_equals o prop_of;
```
```   169     val contains_suc = exists_Const (fn (c, _) => c = @{const_name Suc});
```
```   170   in
```
```   171     if forall (can dest) thms andalso exists (contains_suc o dest) thms
```
```   172       then perhaps_loop (remove_suc thy) thms
```
```   173        else NONE
```
```   174   end;
```
```   175
```
```   176 val eqn_suc_preproc1 = Code_Preproc.simple_functrans eqn_suc_preproc;
```
```   177
```
```   178 fun eqn_suc_preproc2 thy thms = eqn_suc_preproc thy thms
```
```   179   |> the_default thms;
```
```   180
```
```   181 fun remove_suc_clause thy thms =
```
```   182   let
```
```   183     val vname = Name.variant (map fst
```
```   184       (fold (Term.add_var_names o Thm.full_prop_of) thms [])) "x";
```
```   185     fun find_var (t as Const (@{const_name Suc}, _) \$ (v as Var _)) = SOME (t, v)
```
```   186       | find_var (t \$ u) = (case find_var t of NONE => find_var u | x => x)
```
```   187       | find_var _ = NONE;
```
```   188     fun find_thm th =
```
```   189       let val th' = Conv.fconv_rule ObjectLogic.atomize th
```
```   190       in Option.map (pair (th, th')) (find_var (prop_of th')) end
```
```   191   in
```
```   192     case get_first find_thm thms of
```
```   193       NONE => thms
```
```   194     | SOME ((th, th'), (Sucv, v)) =>
```
```   195         let
```
```   196           val cert = cterm_of (Thm.theory_of_thm th);
```
```   197           val th'' = ObjectLogic.rulify (Thm.implies_elim
```
```   198             (Conv.fconv_rule (Thm.beta_conversion true)
```
```   199               (Drule.instantiate' []
```
```   200                 [SOME (cert (lambda v (Abs ("x", HOLogic.natT,
```
```   201                    abstract_over (Sucv,
```
```   202                      HOLogic.dest_Trueprop (prop_of th')))))),
```
```   203                  SOME (cert v)] @{thm Suc_clause}))
```
```   204             (Thm.forall_intr (cert v) th'))
```
```   205         in
```
```   206           remove_suc_clause thy (map (fn th''' =>
```
```   207             if (op = o pairself prop_of) (th''', th) then th'' else th''') thms)
```
```   208         end
```
```   209   end;
```
```   210
```
```   211 fun clause_suc_preproc thy ths =
```
```   212   let
```
```   213     val dest = fst o HOLogic.dest_mem o HOLogic.dest_Trueprop
```
```   214   in
```
```   215     if forall (can (dest o concl_of)) ths andalso
```
```   216       exists (fn th => exists (exists_Const (fn (c, _) => c = @{const_name Suc}))
```
```   217         (map_filter (try dest) (concl_of th :: prems_of th))) ths
```
```   218     then remove_suc_clause thy ths else ths
```
```   219   end;
```
```   220 in
```
```   221
```
```   222   Codegen.add_preprocessor eqn_suc_preproc2
```
```   223   #> Codegen.add_preprocessor clause_suc_preproc
```
```   224   #> Code_Preproc.add_functrans ("eqn_Suc", eqn_suc_preproc1)
```
```   225
```
```   226 end;
```
```   227 *}
```
```   228 (*>*)
```
```   229
```
```   230
```
```   231 subsection {* Target language setup *}
```
```   232
```
```   233 text {*
```
```   234   For ML, we map @{typ nat} to target language integers, where we
```
```   235   assert that values are always non-negative.
```
```   236 *}
```
```   237
```
```   238 code_type nat
```
```   239   (SML "IntInf.int")
```
```   240   (OCaml "Big'_int.big'_int")
```
```   241
```
```   242 types_code
```
```   243   nat ("int")
```
```   244 attach (term_of) {*
```
```   245 val term_of_nat = HOLogic.mk_number HOLogic.natT;
```
```   246 *}
```
```   247 attach (test) {*
```
```   248 fun gen_nat i =
```
```   249   let val n = random_range 0 i
```
```   250   in (n, fn () => term_of_nat n) end;
```
```   251 *}
```
```   252
```
```   253 text {*
```
```   254   For Haskell we define our own @{typ nat} type.  The reason
```
```   255   is that we have to distinguish type class instances
```
```   256   for @{typ nat} and @{typ int}.
```
```   257 *}
```
```   258
```
```   259 code_include Haskell "Nat" {*
```
```   260 newtype Nat = Nat Integer deriving (Show, Eq);
```
```   261
```
```   262 instance Num Nat where {
```
```   263   fromInteger k = Nat (if k >= 0 then k else 0);
```
```   264   Nat n + Nat m = Nat (n + m);
```
```   265   Nat n - Nat m = fromInteger (n - m);
```
```   266   Nat n * Nat m = Nat (n * m);
```
```   267   abs n = n;
```
```   268   signum _ = 1;
```
```   269   negate n = error "negate Nat";
```
```   270 };
```
```   271
```
```   272 instance Ord Nat where {
```
```   273   Nat n <= Nat m = n <= m;
```
```   274   Nat n < Nat m = n < m;
```
```   275 };
```
```   276
```
```   277 instance Real Nat where {
```
```   278   toRational (Nat n) = toRational n;
```
```   279 };
```
```   280
```
```   281 instance Enum Nat where {
```
```   282   toEnum k = fromInteger (toEnum k);
```
```   283   fromEnum (Nat n) = fromEnum n;
```
```   284 };
```
```   285
```
```   286 instance Integral Nat where {
```
```   287   toInteger (Nat n) = n;
```
```   288   divMod n m = quotRem n m;
```
```   289   quotRem (Nat n) (Nat m) = (Nat k, Nat l) where (k, l) = quotRem n m;
```
```   290 };
```
```   291 *}
```
```   292
```
```   293 code_reserved Haskell Nat
```
```   294
```
```   295 code_type nat
```
```   296   (Haskell "Nat.Nat")
```
```   297
```
```   298 code_instance nat :: eq
```
```   299   (Haskell -)
```
```   300
```
```   301 text {*
```
```   302   Natural numerals.
```
```   303 *}
```
```   304
```
```   305 lemma [code inline, symmetric, code post]:
```
```   306   "nat (number_of i) = number_nat_inst.number_of_nat i"
```
```   307   -- {* this interacts as desired with @{thm nat_number_of_def} *}
```
```   308   by (simp add: number_nat_inst.number_of_nat)
```
```   309
```
```   310 setup {*
```
```   311   fold (Numeral.add_code @{const_name number_nat_inst.number_of_nat}
```
```   312     false true) ["SML", "OCaml", "Haskell"]
```
```   313 *}
```
```   314
```
```   315 text {*
```
```   316   Since natural numbers are implemented
```
```   317   using integers in ML, the coercion function @{const "of_nat"} of type
```
```   318   @{typ "nat \<Rightarrow> int"} is simply implemented by the identity function.
```
```   319   For the @{const "nat"} function for converting an integer to a natural
```
```   320   number, we give a specific implementation using an ML function that
```
```   321   returns its input value, provided that it is non-negative, and otherwise
```
```   322   returns @{text "0"}.
```
```   323 *}
```
```   324
```
```   325 definition
```
```   326   int :: "nat \<Rightarrow> int"
```
```   327 where
```
```   328   [code del]: "int = of_nat"
```
```   329
```
```   330 lemma int_code' [code]:
```
```   331   "int (number_of l) = (if neg (number_of l \<Colon> int) then 0 else number_of l)"
```
```   332   unfolding int_nat_number_of [folded int_def] ..
```
```   333
```
```   334 lemma nat_code' [code]:
```
```   335   "nat (number_of l) = (if neg (number_of l \<Colon> int) then 0 else number_of l)"
```
```   336   unfolding nat_number_of_def number_of_is_id neg_def by simp
```
```   337
```
```   338 lemma of_nat_int [code unfold]:
```
```   339   "of_nat = int" by (simp add: int_def)
```
```   340 declare of_nat_int [symmetric, code post]
```
```   341
```
```   342 code_const int
```
```   343   (SML "_")
```
```   344   (OCaml "_")
```
```   345
```
```   346 consts_code
```
```   347   int ("(_)")
```
```   348   nat ("\<module>nat")
```
```   349 attach {*
```
```   350 fun nat i = if i < 0 then 0 else i;
```
```   351 *}
```
```   352
```
```   353 code_const nat
```
```   354   (SML "IntInf.max/ (/0,/ _)")
```
```   355   (OCaml "Big'_int.max'_big'_int/ Big'_int.zero'_big'_int")
```
```   356
```
```   357 text {* For Haskell, things are slightly different again. *}
```
```   358
```
```   359 code_const int and nat
```
```   360   (Haskell "toInteger" and "fromInteger")
```
```   361
```
```   362 text {* Conversion from and to indices. *}
```
```   363
```
```   364 code_const Code_Numeral.of_nat
```
```   365   (SML "IntInf.toInt")
```
```   366   (OCaml "_")
```
```   367   (Haskell "fromEnum")
```
```   368
```
```   369 code_const Code_Numeral.nat_of
```
```   370   (SML "IntInf.fromInt")
```
```   371   (OCaml "_")
```
```   372   (Haskell "toEnum")
```
```   373
```
```   374 text {* Using target language arithmetic operations whenever appropriate *}
```
```   375
```
```   376 code_const "op + \<Colon> nat \<Rightarrow> nat \<Rightarrow> nat"
```
```   377   (SML "IntInf.+ ((_), (_))")
```
```   378   (OCaml "Big'_int.add'_big'_int")
```
```   379   (Haskell infixl 6 "+")
```
```   380
```
```   381 code_const "op * \<Colon> nat \<Rightarrow> nat \<Rightarrow> nat"
```
```   382   (SML "IntInf.* ((_), (_))")
```
```   383   (OCaml "Big'_int.mult'_big'_int")
```
```   384   (Haskell infixl 7 "*")
```
```   385
```
```   386 code_const divmod_aux
```
```   387   (SML "IntInf.divMod/ ((_),/ (_))")
```
```   388   (OCaml "Big'_int.quomod'_big'_int")
```
```   389   (Haskell "divMod")
```
```   390
```
```   391 code_const "eq_class.eq \<Colon> nat \<Rightarrow> nat \<Rightarrow> bool"
```
```   392   (SML "!((_ : IntInf.int) = _)")
```
```   393   (OCaml "Big'_int.eq'_big'_int")
```
```   394   (Haskell infixl 4 "==")
```
```   395
```
```   396 code_const "op \<le> \<Colon> nat \<Rightarrow> nat \<Rightarrow> bool"
```
```   397   (SML "IntInf.<= ((_), (_))")
```
```   398   (OCaml "Big'_int.le'_big'_int")
```
```   399   (Haskell infix 4 "<=")
```
```   400
```
```   401 code_const "op < \<Colon> nat \<Rightarrow> nat \<Rightarrow> bool"
```
```   402   (SML "IntInf.< ((_), (_))")
```
```   403   (OCaml "Big'_int.lt'_big'_int")
```
```   404   (Haskell infix 4 "<")
```
```   405
```
```   406 consts_code
```
```   407   "0::nat"                     ("0")
```
```   408   "1::nat"                     ("1")
```
```   409   Suc                          ("(_ +/ 1)")
```
```   410   "op + \<Colon>  nat \<Rightarrow> nat \<Rightarrow> nat"   ("(_ +/ _)")
```
```   411   "op * \<Colon>  nat \<Rightarrow> nat \<Rightarrow> nat"   ("(_ */ _)")
```
```   412   "op \<le> \<Colon>  nat \<Rightarrow> nat \<Rightarrow> bool"  ("(_ <=/ _)")
```
```   413   "op < \<Colon>  nat \<Rightarrow> nat \<Rightarrow> bool"  ("(_ </ _)")
```
```   414
```
```   415
```
```   416 text {* Evaluation *}
```
```   417
```
```   418 lemma [code, code del]:
```
```   419   "(Code_Eval.term_of \<Colon> nat \<Rightarrow> term) = Code_Eval.term_of" ..
```
```   420
```
```   421 code_const "Code_Eval.term_of \<Colon> nat \<Rightarrow> term"
```
```   422   (SML "HOLogic.mk'_number/ HOLogic.natT")
```
```   423
```
```   424
```
```   425 text {* Module names *}
```
```   426
```
```   427 code_modulename SML
```
```   428   Nat Integer
```
```   429   Divides Integer
```
```   430   Ring_and_Field Integer
```
```   431   Efficient_Nat Integer
```
```   432
```
```   433 code_modulename OCaml
```
```   434   Nat Integer
```
```   435   Divides Integer
```
```   436   Ring_and_Field Integer
```
```   437   Efficient_Nat Integer
```
```   438
```
```   439 code_modulename Haskell
```
```   440   Nat Integer
```
```   441   Divides Integer
```
```   442   Ring_and_Field Integer
```
```   443   Efficient_Nat Integer
```
```   444
```
```   445 hide const int
```
```   446
```
```   447 end
```