src/HOL/Library/Efficient_Nat.thy
 author haftmann Tue May 12 21:17:38 2009 +0200 (2009-05-12) changeset 31128 b3bb28c87409 parent 31090 3be41b271023 child 31203 5c8fb4fd67e0 permissions -rw-r--r--
adapted to changes in module Code
```     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 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) = h n) \<Longrightarrow> f 0 = g \<Longrightarrow>
```
```   109   f n = (if n = 0 then g else h (n - 1))"
```
```   110   by (cases n) simp_all
```
```   111
```
```   112 lemma Suc_if_eq: "(\<And>n. f (Suc n) \<equiv> h n) \<Longrightarrow> f 0 \<equiv> g \<Longrightarrow>
```
```   113   f n \<equiv> if n = 0 then g else h (n - 1)"
```
```   114   by (rule eq_reflection, rule Suc_if_eq')
```
```   115     (rule meta_eq_to_obj_eq, assumption,
```
```   116      rule meta_eq_to_obj_eq, assumption)
```
```   117
```
```   118 lemma Suc_clause: "(\<And>n. P n (Suc n)) \<Longrightarrow> n \<noteq> 0 \<Longrightarrow> P (n - 1) n"
```
```   119   by (cases n) simp_all
```
```   120
```
```   121 text {*
```
```   122   The rules above are built into a preprocessor that is plugged into
```
```   123   the code generator. Since the preprocessor for introduction rules
```
```   124   does not know anything about modes, some of the modes that worked
```
```   125   for the canonical representation of natural numbers may no longer work.
```
```   126 *}
```
```   127
```
```   128 (*<*)
```
```   129 setup {*
```
```   130 let
```
```   131
```
```   132 fun gen_remove_suc Suc_if_eq dest_judgement thy thms =
```
```   133   let
```
```   134     val vname = Name.variant (map fst
```
```   135       (fold (Term.add_var_names o Thm.full_prop_of) thms [])) "n";
```
```   136     val cv = cterm_of thy (Var ((vname, 0), HOLogic.natT));
```
```   137     fun lhs_of th = snd (Thm.dest_comb
```
```   138       (fst (Thm.dest_comb (dest_judgement (cprop_of th)))));
```
```   139     fun rhs_of th = snd (Thm.dest_comb (dest_judgement (cprop_of th)));
```
```   140     fun find_vars ct = (case term_of ct of
```
```   141         (Const (@{const_name Suc}, _) \$ Var _) => [(cv, snd (Thm.dest_comb ct))]
```
```   142       | _ \$ _ =>
```
```   143         let val (ct1, ct2) = Thm.dest_comb ct
```
```   144         in
```
```   145           map (apfst (fn ct => Thm.capply ct ct2)) (find_vars ct1) @
```
```   146           map (apfst (Thm.capply ct1)) (find_vars ct2)
```
```   147         end
```
```   148       | _ => []);
```
```   149     val eqs = maps
```
```   150       (fn th => map (pair th) (find_vars (lhs_of th))) thms;
```
```   151     fun mk_thms (th, (ct, cv')) =
```
```   152       let
```
```   153         val th' =
```
```   154           Thm.implies_elim
```
```   155            (Conv.fconv_rule (Thm.beta_conversion true)
```
```   156              (Drule.instantiate'
```
```   157                [SOME (ctyp_of_term ct)] [SOME (Thm.cabs cv ct),
```
```   158                  SOME (Thm.cabs cv' (rhs_of th)), NONE, SOME cv']
```
```   159                Suc_if_eq)) (Thm.forall_intr cv' th)
```
```   160       in
```
```   161         case map_filter (fn th'' =>
```
```   162             SOME (th'', singleton
```
```   163               (Variable.trade (K (fn [th'''] => [th''' RS th'])) (Variable.thm_context th'')) th'')
```
```   164           handle THM _ => NONE) thms of
```
```   165             [] => NONE
```
```   166           | thps =>
```
```   167               let val (ths1, ths2) = split_list thps
```
```   168               in SOME (subtract Thm.eq_thm (th :: ths1) thms @ ths2) end
```
```   169       end
```
```   170   in get_first mk_thms eqs end;
```
```   171
```
```   172 fun gen_eqn_suc_preproc Suc_if_eq dest_judgement dest_lhs thy thms =
```
```   173   let
```
```   174     val dest = dest_lhs o prop_of;
```
```   175     val contains_suc = exists_Const (fn (c, _) => c = @{const_name Suc});
```
```   176   in
```
```   177     if forall (can dest) thms andalso exists (contains_suc o dest) thms
```
```   178       then perhaps_loop (gen_remove_suc Suc_if_eq dest_judgement thy) thms
```
```   179        else NONE
```
```   180   end;
```
```   181
```
```   182 val eqn_suc_preproc = Code_Preproc.simple_functrans (gen_eqn_suc_preproc
```
```   183   @{thm Suc_if_eq} I (fst o Logic.dest_equals));
```
```   184
```
```   185 fun eqn_suc_preproc' thy thms = gen_eqn_suc_preproc
```
```   186   @{thm Suc_if_eq'} (snd o Thm.dest_comb) (fst o HOLogic.dest_eq o HOLogic.dest_Trueprop) thy thms
```
```   187   |> the_default thms;
```
```   188
```
```   189 fun remove_suc_clause thy thms =
```
```   190   let
```
```   191     val vname = Name.variant (map fst
```
```   192       (fold (Term.add_var_names o Thm.full_prop_of) thms [])) "x";
```
```   193     fun find_var (t as Const (@{const_name Suc}, _) \$ (v as Var _)) = SOME (t, v)
```
```   194       | find_var (t \$ u) = (case find_var t of NONE => find_var u | x => x)
```
```   195       | find_var _ = NONE;
```
```   196     fun find_thm th =
```
```   197       let val th' = Conv.fconv_rule ObjectLogic.atomize th
```
```   198       in Option.map (pair (th, th')) (find_var (prop_of th')) end
```
```   199   in
```
```   200     case get_first find_thm thms of
```
```   201       NONE => thms
```
```   202     | SOME ((th, th'), (Sucv, v)) =>
```
```   203         let
```
```   204           val cert = cterm_of (Thm.theory_of_thm th);
```
```   205           val th'' = ObjectLogic.rulify (Thm.implies_elim
```
```   206             (Conv.fconv_rule (Thm.beta_conversion true)
```
```   207               (Drule.instantiate' []
```
```   208                 [SOME (cert (lambda v (Abs ("x", HOLogic.natT,
```
```   209                    abstract_over (Sucv,
```
```   210                      HOLogic.dest_Trueprop (prop_of th')))))),
```
```   211                  SOME (cert v)] @{thm Suc_clause}))
```
```   212             (Thm.forall_intr (cert v) th'))
```
```   213         in
```
```   214           remove_suc_clause thy (map (fn th''' =>
```
```   215             if (op = o pairself prop_of) (th''', th) then th'' else th''') thms)
```
```   216         end
```
```   217   end;
```
```   218
```
```   219 fun clause_suc_preproc thy ths =
```
```   220   let
```
```   221     val dest = fst o HOLogic.dest_mem o HOLogic.dest_Trueprop
```
```   222   in
```
```   223     if forall (can (dest o concl_of)) ths andalso
```
```   224       exists (fn th => exists (exists_Const (fn (c, _) => c = @{const_name Suc}))
```
```   225         (map_filter (try dest) (concl_of th :: prems_of th))) ths
```
```   226     then remove_suc_clause thy ths else ths
```
```   227   end;
```
```   228 in
```
```   229
```
```   230   Codegen.add_preprocessor eqn_suc_preproc'
```
```   231   #> Codegen.add_preprocessor clause_suc_preproc
```
```   232   #> Code_Preproc.add_functrans ("eqn_Suc", eqn_suc_preproc)
```
```   233
```
```   234 end;
```
```   235 *}
```
```   236 (*>*)
```
```   237
```
```   238
```
```   239 subsection {* Target language setup *}
```
```   240
```
```   241 text {*
```
```   242   For ML, we map @{typ nat} to target language integers, where we
```
```   243   assert that values are always non-negative.
```
```   244 *}
```
```   245
```
```   246 code_type nat
```
```   247   (SML "IntInf.int")
```
```   248   (OCaml "Big'_int.big'_int")
```
```   249
```
```   250 types_code
```
```   251   nat ("int")
```
```   252 attach (term_of) {*
```
```   253 val term_of_nat = HOLogic.mk_number HOLogic.natT;
```
```   254 *}
```
```   255 attach (test) {*
```
```   256 fun gen_nat i =
```
```   257   let val n = random_range 0 i
```
```   258   in (n, fn () => term_of_nat n) end;
```
```   259 *}
```
```   260
```
```   261 text {*
```
```   262   For Haskell we define our own @{typ nat} type.  The reason
```
```   263   is that we have to distinguish type class instances
```
```   264   for @{typ nat} and @{typ int}.
```
```   265 *}
```
```   266
```
```   267 code_include Haskell "Nat" {*
```
```   268 newtype Nat = Nat Integer deriving (Show, Eq);
```
```   269
```
```   270 instance Num Nat where {
```
```   271   fromInteger k = Nat (if k >= 0 then k else 0);
```
```   272   Nat n + Nat m = Nat (n + m);
```
```   273   Nat n - Nat m = fromInteger (n - m);
```
```   274   Nat n * Nat m = Nat (n * m);
```
```   275   abs n = n;
```
```   276   signum _ = 1;
```
```   277   negate n = error "negate Nat";
```
```   278 };
```
```   279
```
```   280 instance Ord Nat where {
```
```   281   Nat n <= Nat m = n <= m;
```
```   282   Nat n < Nat m = n < m;
```
```   283 };
```
```   284
```
```   285 instance Real Nat where {
```
```   286   toRational (Nat n) = toRational n;
```
```   287 };
```
```   288
```
```   289 instance Enum Nat where {
```
```   290   toEnum k = fromInteger (toEnum k);
```
```   291   fromEnum (Nat n) = fromEnum n;
```
```   292 };
```
```   293
```
```   294 instance Integral Nat where {
```
```   295   toInteger (Nat n) = n;
```
```   296   divMod n m = quotRem n m;
```
```   297   quotRem (Nat n) (Nat m) = (Nat k, Nat l) where (k, l) = quotRem n m;
```
```   298 };
```
```   299 *}
```
```   300
```
```   301 code_reserved Haskell Nat
```
```   302
```
```   303 code_type nat
```
```   304   (Haskell "Nat.Nat")
```
```   305
```
```   306 code_instance nat :: eq
```
```   307   (Haskell -)
```
```   308
```
```   309 text {*
```
```   310   Natural numerals.
```
```   311 *}
```
```   312
```
```   313 lemma [code inline, symmetric, code post]:
```
```   314   "nat (number_of i) = number_nat_inst.number_of_nat i"
```
```   315   -- {* this interacts as desired with @{thm nat_number_of_def} *}
```
```   316   by (simp add: number_nat_inst.number_of_nat)
```
```   317
```
```   318 setup {*
```
```   319   fold (Numeral.add_code @{const_name number_nat_inst.number_of_nat}
```
```   320     true false) ["SML", "OCaml", "Haskell"]
```
```   321 *}
```
```   322
```
```   323 text {*
```
```   324   Since natural numbers are implemented
```
```   325   using integers in ML, the coercion function @{const "of_nat"} of type
```
```   326   @{typ "nat \<Rightarrow> int"} is simply implemented by the identity function.
```
```   327   For the @{const "nat"} function for converting an integer to a natural
```
```   328   number, we give a specific implementation using an ML function that
```
```   329   returns its input value, provided that it is non-negative, and otherwise
```
```   330   returns @{text "0"}.
```
```   331 *}
```
```   332
```
```   333 definition
```
```   334   int :: "nat \<Rightarrow> int"
```
```   335 where
```
```   336   [code del]: "int = of_nat"
```
```   337
```
```   338 lemma int_code' [code]:
```
```   339   "int (number_of l) = (if neg (number_of l \<Colon> int) then 0 else number_of l)"
```
```   340   unfolding int_nat_number_of [folded int_def] ..
```
```   341
```
```   342 lemma nat_code' [code]:
```
```   343   "nat (number_of l) = (if neg (number_of l \<Colon> int) then 0 else number_of l)"
```
```   344   unfolding nat_number_of_def number_of_is_id neg_def by simp
```
```   345
```
```   346 lemma of_nat_int [code unfold]:
```
```   347   "of_nat = int" by (simp add: int_def)
```
```   348 declare of_nat_int [symmetric, code post]
```
```   349
```
```   350 code_const int
```
```   351   (SML "_")
```
```   352   (OCaml "_")
```
```   353
```
```   354 consts_code
```
```   355   int ("(_)")
```
```   356   nat ("\<module>nat")
```
```   357 attach {*
```
```   358 fun nat i = if i < 0 then 0 else i;
```
```   359 *}
```
```   360
```
```   361 code_const nat
```
```   362   (SML "IntInf.max/ (/0,/ _)")
```
```   363   (OCaml "Big'_int.max'_big'_int/ Big'_int.zero'_big'_int")
```
```   364
```
```   365 text {* For Haskell, things are slightly different again. *}
```
```   366
```
```   367 code_const int and nat
```
```   368   (Haskell "toInteger" and "fromInteger")
```
```   369
```
```   370 text {* Conversion from and to indices. *}
```
```   371
```
```   372 code_const Code_Index.of_nat
```
```   373   (SML "IntInf.toInt")
```
```   374   (OCaml "Big'_int.int'_of'_big'_int")
```
```   375   (Haskell "fromEnum")
```
```   376
```
```   377 code_const Code_Index.nat_of
```
```   378   (SML "IntInf.fromInt")
```
```   379   (OCaml "Big'_int.big'_int'_of'_int")
```
```   380   (Haskell "toEnum")
```
```   381
```
```   382 text {* Using target language arithmetic operations whenever appropriate *}
```
```   383
```
```   384 code_const "op + \<Colon> nat \<Rightarrow> nat \<Rightarrow> nat"
```
```   385   (SML "IntInf.+ ((_), (_))")
```
```   386   (OCaml "Big'_int.add'_big'_int")
```
```   387   (Haskell infixl 6 "+")
```
```   388
```
```   389 code_const "op * \<Colon> nat \<Rightarrow> nat \<Rightarrow> nat"
```
```   390   (SML "IntInf.* ((_), (_))")
```
```   391   (OCaml "Big'_int.mult'_big'_int")
```
```   392   (Haskell infixl 7 "*")
```
```   393
```
```   394 code_const divmod_aux
```
```   395   (SML "IntInf.divMod/ ((_),/ (_))")
```
```   396   (OCaml "Big'_int.quomod'_big'_int")
```
```   397   (Haskell "divMod")
```
```   398
```
```   399 code_const "eq_class.eq \<Colon> nat \<Rightarrow> nat \<Rightarrow> bool"
```
```   400   (SML "!((_ : IntInf.int) = _)")
```
```   401   (OCaml "Big'_int.eq'_big'_int")
```
```   402   (Haskell infixl 4 "==")
```
```   403
```
```   404 code_const "op \<le> \<Colon> nat \<Rightarrow> nat \<Rightarrow> bool"
```
```   405   (SML "IntInf.<= ((_), (_))")
```
```   406   (OCaml "Big'_int.le'_big'_int")
```
```   407   (Haskell infix 4 "<=")
```
```   408
```
```   409 code_const "op < \<Colon> nat \<Rightarrow> nat \<Rightarrow> bool"
```
```   410   (SML "IntInf.< ((_), (_))")
```
```   411   (OCaml "Big'_int.lt'_big'_int")
```
```   412   (Haskell infix 4 "<")
```
```   413
```
```   414 consts_code
```
```   415   "0::nat"                     ("0")
```
```   416   "1::nat"                     ("1")
```
```   417   Suc                          ("(_ +/ 1)")
```
```   418   "op + \<Colon>  nat \<Rightarrow> nat \<Rightarrow> nat"   ("(_ +/ _)")
```
```   419   "op * \<Colon>  nat \<Rightarrow> nat \<Rightarrow> nat"   ("(_ */ _)")
```
```   420   "op \<le> \<Colon>  nat \<Rightarrow> nat \<Rightarrow> bool"  ("(_ <=/ _)")
```
```   421   "op < \<Colon>  nat \<Rightarrow> nat \<Rightarrow> bool"  ("(_ </ _)")
```
```   422
```
```   423
```
```   424 text {* Evaluation *}
```
```   425
```
```   426 lemma [code, code del]:
```
```   427   "(Code_Eval.term_of \<Colon> nat \<Rightarrow> term) = Code_Eval.term_of" ..
```
```   428
```
```   429 code_const "Code_Eval.term_of \<Colon> nat \<Rightarrow> term"
```
```   430   (SML "HOLogic.mk'_number/ HOLogic.natT")
```
```   431
```
```   432
```
```   433 text {* Module names *}
```
```   434
```
```   435 code_modulename SML
```
```   436   Nat Integer
```
```   437   Divides Integer
```
```   438   Ring_and_Field Integer
```
```   439   Efficient_Nat Integer
```
```   440
```
```   441 code_modulename OCaml
```
```   442   Nat Integer
```
```   443   Divides Integer
```
```   444   Ring_and_Field Integer
```
```   445   Efficient_Nat Integer
```
```   446
```
```   447 code_modulename Haskell
```
```   448   Nat Integer
```
```   449   Divides Integer
```
```   450   Ring_and_Field Integer
```
```   451   Efficient_Nat Integer
```
```   452
```
```   453 hide const int
```
```   454
```
```   455 end
```