src/HOL/Library/Efficient_Nat.thy
 author haftmann Tue Jul 15 16:02:07 2008 +0200 (2008-07-15) changeset 27609 b23c9ad0fe7d parent 27557 151731493264 child 27673 52056ddac194 permissions -rw-r--r--
tuned code theorem bookkeeping
```     1 (*  Title:      HOL/Library/Efficient_Nat.thy
```
```     2     ID:         \$Id\$
```
```     3     Author:     Stefan Berghofer, Florian Haftmann, TU Muenchen
```
```     4 *)
```
```     5
```
```     6 header {* Implementation of natural numbers by target-language integers *}
```
```     7
```
```     8 theory Efficient_Nat
```
```     9 imports Plain Code_Integer Code_Index
```
```    10 begin
```
```    11
```
```    12 text {*
```
```    13   When generating code for functions on natural numbers, the
```
```    14   canonical representation using @{term "0::nat"} and
```
```    15   @{term "Suc"} is unsuitable for computations involving large
```
```    16   numbers.  The efficiency of the generated code can be improved
```
```    17   drastically by implementing natural numbers by target-language
```
```    18   integers.  To do this, just include this theory.
```
```    19 *}
```
```    20
```
```    21 subsection {* Basic arithmetic *}
```
```    22
```
```    23 text {*
```
```    24   Most standard arithmetic functions on natural numbers are implemented
```
```    25   using their counterparts on the integers:
```
```    26 *}
```
```    27
```
```    28 code_datatype number_nat_inst.number_of_nat
```
```    29
```
```    30 lemma zero_nat_code [code, code unfold]:
```
```    31   "0 = (Numeral0 :: nat)"
```
```    32   by simp
```
```    33 lemmas [code post] = zero_nat_code [symmetric]
```
```    34
```
```    35 lemma one_nat_code [code, code unfold]:
```
```    36   "1 = (Numeral1 :: nat)"
```
```    37   by simp
```
```    38 lemmas [code post] = one_nat_code [symmetric]
```
```    39
```
```    40 lemma Suc_code [code]:
```
```    41   "Suc n = n + 1"
```
```    42   by simp
```
```    43
```
```    44 lemma plus_nat_code [code]:
```
```    45   "n + m = nat (of_nat n + of_nat m)"
```
```    46   by simp
```
```    47
```
```    48 lemma minus_nat_code [code]:
```
```    49   "n - m = nat (of_nat n - of_nat m)"
```
```    50   by simp
```
```    51
```
```    52 lemma times_nat_code [code]:
```
```    53   "n * m = nat (of_nat n * of_nat m)"
```
```    54   unfolding of_nat_mult [symmetric] by simp
```
```    55
```
```    56 text {* Specialized @{term "op div \<Colon> nat \<Rightarrow> nat \<Rightarrow> nat"}
```
```    57   and @{term "op mod \<Colon> nat \<Rightarrow> nat \<Rightarrow> nat"} operations. *}
```
```    58
```
```    59 definition
```
```    60   divmod_aux ::  "nat \<Rightarrow> nat \<Rightarrow> nat \<times> nat"
```
```    61 where
```
```    62   [code func del]: "divmod_aux = divmod"
```
```    63
```
```    64 lemma [code func]:
```
```    65   "divmod n m = (if m = 0 then (0, n) else divmod_aux n m)"
```
```    66   unfolding divmod_aux_def divmod_div_mod by simp
```
```    67
```
```    68 lemma divmod_aux_code [code]:
```
```    69   "divmod_aux n m = (nat (of_nat n div of_nat m), nat (of_nat n mod of_nat m))"
```
```    70   unfolding divmod_aux_def divmod_div_mod zdiv_int [symmetric] zmod_int [symmetric] by simp
```
```    71
```
```    72 lemma eq_nat_code [code]:
```
```    73   "n = m \<longleftrightarrow> (of_nat n \<Colon> int) = of_nat m"
```
```    74   by simp
```
```    75
```
```    76 lemma less_eq_nat_code [code]:
```
```    77   "n \<le> m \<longleftrightarrow> (of_nat n \<Colon> int) \<le> of_nat m"
```
```    78   by simp
```
```    79
```
```    80 lemma less_nat_code [code]:
```
```    81   "n < m \<longleftrightarrow> (of_nat n \<Colon> int) < of_nat m"
```
```    82   by simp
```
```    83
```
```    84 subsection {* Case analysis *}
```
```    85
```
```    86 text {*
```
```    87   Case analysis on natural numbers is rephrased using a conditional
```
```    88   expression:
```
```    89 *}
```
```    90
```
```    91 lemma [code func, code unfold]:
```
```    92   "nat_case = (\<lambda>f g n. if n = 0 then f else g (n - 1))"
```
```    93   by (auto simp add: expand_fun_eq dest!: gr0_implies_Suc)
```
```    94
```
```    95
```
```    96 subsection {* Preprocessors *}
```
```    97
```
```    98 text {*
```
```    99   In contrast to @{term "Suc n"}, the term @{term "n + (1::nat)"} is no longer
```
```   100   a constructor term. Therefore, all occurrences of this term in a position
```
```   101   where a pattern is expected (i.e.\ on the left-hand side of a recursion
```
```   102   equation or in the arguments of an inductive relation in an introduction
```
```   103   rule) must be eliminated.
```
```   104   This can be accomplished by applying the following transformation rules:
```
```   105 *}
```
```   106
```
```   107 lemma Suc_if_eq: "(\<And>n. f (Suc n) = h n) \<Longrightarrow> f 0 = g \<Longrightarrow>
```
```   108   f n = (if n = 0 then g else h (n - 1))"
```
```   109   by (case_tac n) simp_all
```
```   110
```
```   111 lemma Suc_clause: "(\<And>n. P n (Suc n)) \<Longrightarrow> n \<noteq> 0 \<Longrightarrow> P (n - 1) n"
```
```   112   by (case_tac n) simp_all
```
```   113
```
```   114 text {*
```
```   115   The rules above are built into a preprocessor that is plugged into
```
```   116   the code generator. Since the preprocessor for introduction rules
```
```   117   does not know anything about modes, some of the modes that worked
```
```   118   for the canonical representation of natural numbers may no longer work.
```
```   119 *}
```
```   120
```
```   121 (*<*)
```
```   122 setup {*
```
```   123 let
```
```   124
```
```   125 fun remove_suc thy thms =
```
```   126   let
```
```   127     val vname = Name.variant (map fst
```
```   128       (fold (Term.add_varnames o Thm.full_prop_of) thms [])) "x";
```
```   129     val cv = cterm_of thy (Var ((vname, 0), HOLogic.natT));
```
```   130     fun lhs_of th = snd (Thm.dest_comb
```
```   131       (fst (Thm.dest_comb (snd (Thm.dest_comb (cprop_of th))))));
```
```   132     fun rhs_of th = snd (Thm.dest_comb (snd (Thm.dest_comb (cprop_of th))));
```
```   133     fun find_vars ct = (case term_of ct of
```
```   134         (Const ("Suc", _) \$ Var _) => [(cv, snd (Thm.dest_comb ct))]
```
```   135       | _ \$ _ =>
```
```   136         let val (ct1, ct2) = Thm.dest_comb ct
```
```   137         in
```
```   138           map (apfst (fn ct => Thm.capply ct ct2)) (find_vars ct1) @
```
```   139           map (apfst (Thm.capply ct1)) (find_vars ct2)
```
```   140         end
```
```   141       | _ => []);
```
```   142     val eqs = maps
```
```   143       (fn th => map (pair th) (find_vars (lhs_of th))) thms;
```
```   144     fun mk_thms (th, (ct, cv')) =
```
```   145       let
```
```   146         val th' =
```
```   147           Thm.implies_elim
```
```   148            (Conv.fconv_rule (Thm.beta_conversion true)
```
```   149              (Drule.instantiate'
```
```   150                [SOME (ctyp_of_term ct)] [SOME (Thm.cabs cv ct),
```
```   151                  SOME (Thm.cabs cv' (rhs_of th)), NONE, SOME cv']
```
```   152                @{thm Suc_if_eq})) (Thm.forall_intr cv' th)
```
```   153       in
```
```   154         case map_filter (fn th'' =>
```
```   155             SOME (th'', singleton
```
```   156               (Variable.trade (K (fn [th'''] => [th''' RS th'])) (Variable.thm_context th'')) th'')
```
```   157           handle THM _ => NONE) thms of
```
```   158             [] => NONE
```
```   159           | thps =>
```
```   160               let val (ths1, ths2) = split_list thps
```
```   161               in SOME (subtract Thm.eq_thm (th :: ths1) thms @ ths2) end
```
```   162       end
```
```   163   in case get_first mk_thms eqs of
```
```   164       NONE => thms
```
```   165     | SOME x => remove_suc thy x
```
```   166   end;
```
```   167
```
```   168 fun eqn_suc_preproc thy ths =
```
```   169   let
```
```   170     val dest = fst o HOLogic.dest_eq o HOLogic.dest_Trueprop o prop_of;
```
```   171     fun contains_suc t = member (op =) (term_consts t) @{const_name Suc};
```
```   172   in
```
```   173     if forall (can dest) ths andalso
```
```   174       exists (contains_suc o dest) ths
```
```   175     then remove_suc thy ths else ths
```
```   176   end;
```
```   177
```
```   178 fun remove_suc_clause thy thms =
```
```   179   let
```
```   180     val vname = Name.variant (map fst
```
```   181       (fold (Term.add_varnames o Thm.full_prop_of) thms [])) "x";
```
```   182     fun find_var (t as Const (@{const_name Suc}, _) \$ (v as Var _)) = SOME (t, v)
```
```   183       | find_var (t \$ u) = (case find_var t of NONE => find_var u | x => x)
```
```   184       | find_var _ = NONE;
```
```   185     fun find_thm th =
```
```   186       let val th' = Conv.fconv_rule ObjectLogic.atomize th
```
```   187       in Option.map (pair (th, th')) (find_var (prop_of th')) end
```
```   188   in
```
```   189     case get_first find_thm thms of
```
```   190       NONE => thms
```
```   191     | SOME ((th, th'), (Sucv, v)) =>
```
```   192         let
```
```   193           val cert = cterm_of (Thm.theory_of_thm th);
```
```   194           val th'' = ObjectLogic.rulify (Thm.implies_elim
```
```   195             (Conv.fconv_rule (Thm.beta_conversion true)
```
```   196               (Drule.instantiate' []
```
```   197                 [SOME (cert (lambda v (Abs ("x", HOLogic.natT,
```
```   198                    abstract_over (Sucv,
```
```   199                      HOLogic.dest_Trueprop (prop_of th')))))),
```
```   200                  SOME (cert v)] @{thm Suc_clause}))
```
```   201             (Thm.forall_intr (cert v) th'))
```
```   202         in
```
```   203           remove_suc_clause thy (map (fn th''' =>
```
```   204             if (op = o pairself prop_of) (th''', th) then th'' else th''') thms)
```
```   205         end
```
```   206   end;
```
```   207
```
```   208 fun clause_suc_preproc thy ths =
```
```   209   let
```
```   210     val dest = fst o HOLogic.dest_mem o HOLogic.dest_Trueprop
```
```   211   in
```
```   212     if forall (can (dest o concl_of)) ths andalso
```
```   213       exists (fn th => member (op =) (foldr add_term_consts
```
```   214         [] (map_filter (try dest) (concl_of th :: prems_of th))) "Suc") ths
```
```   215     then remove_suc_clause thy ths else ths
```
```   216   end;
```
```   217
```
```   218 fun lift f thy thms1 =
```
```   219   let
```
```   220     val thms2 = Drule.zero_var_indexes_list thms1;
```
```   221     val thms3 = try (map (fn thm => thm RS @{thm meta_eq_to_obj_eq})
```
```   222       #> f thy
```
```   223       #> map (fn thm => thm RS @{thm eq_reflection})
```
```   224       #> map (Conv.fconv_rule Drule.beta_eta_conversion)) thms2;
```
```   225     val thms4 = Option.map Drule.zero_var_indexes_list thms3;
```
```   226   in case thms4
```
```   227    of NONE => NONE
```
```   228     | SOME thms4 => if Thm.eq_thms (thms2, thms4) then NONE else SOME thms4
```
```   229   end
```
```   230
```
```   231 in
```
```   232
```
```   233   Codegen.add_preprocessor eqn_suc_preproc
```
```   234   #> Codegen.add_preprocessor clause_suc_preproc
```
```   235   #> Code.add_functrans ("eqn_Suc", lift eqn_suc_preproc)
```
```   236   #> Code.add_functrans ("clause_Suc", lift clause_suc_preproc)
```
```   237
```
```   238 end;
```
```   239 *}
```
```   240 (*>*)
```
```   241
```
```   242
```
```   243 subsection {* Target language setup *}
```
```   244
```
```   245 text {*
```
```   246   For ML, we map @{typ nat} to target language integers, where we
```
```   247   assert that values are always non-negative.
```
```   248 *}
```
```   249
```
```   250 code_type nat
```
```   251   (SML "IntInf.int")
```
```   252   (OCaml "Big'_int.big'_int")
```
```   253
```
```   254 types_code
```
```   255   nat ("int")
```
```   256 attach (term_of) {*
```
```   257 val term_of_nat = HOLogic.mk_number HOLogic.natT;
```
```   258 *}
```
```   259 attach (test) {*
```
```   260 fun gen_nat i =
```
```   261   let val n = random_range 0 i
```
```   262   in (n, fn () => term_of_nat n) end;
```
```   263 *}
```
```   264
```
```   265 text {*
```
```   266   For Haskell we define our own @{typ nat} type.  The reason
```
```   267   is that we have to distinguish type class instances
```
```   268   for @{typ nat} and @{typ int}.
```
```   269 *}
```
```   270
```
```   271 code_include Haskell "Nat" {*
```
```   272 newtype Nat = Nat Integer deriving (Show, Eq);
```
```   273
```
```   274 instance Num Nat where {
```
```   275   fromInteger k = Nat (if k >= 0 then k else 0);
```
```   276   Nat n + Nat m = Nat (n + m);
```
```   277   Nat n - Nat m = fromInteger (n - m);
```
```   278   Nat n * Nat m = Nat (n * m);
```
```   279   abs n = n;
```
```   280   signum _ = 1;
```
```   281   negate n = error "negate Nat";
```
```   282 };
```
```   283
```
```   284 instance Ord Nat where {
```
```   285   Nat n <= Nat m = n <= m;
```
```   286   Nat n < Nat m = n < m;
```
```   287 };
```
```   288
```
```   289 instance Real Nat where {
```
```   290   toRational (Nat n) = toRational n;
```
```   291 };
```
```   292
```
```   293 instance Enum Nat where {
```
```   294   toEnum k = fromInteger (toEnum k);
```
```   295   fromEnum (Nat n) = fromEnum n;
```
```   296 };
```
```   297
```
```   298 instance Integral Nat where {
```
```   299   toInteger (Nat n) = n;
```
```   300   divMod n m = quotRem n m;
```
```   301   quotRem (Nat n) (Nat m) = (Nat k, Nat l) where (k, l) = quotRem n m;
```
```   302 };
```
```   303 *}
```
```   304
```
```   305 code_reserved Haskell Nat
```
```   306
```
```   307 code_type nat
```
```   308   (Haskell "Nat")
```
```   309
```
```   310 code_instance nat :: eq
```
```   311   (Haskell -)
```
```   312
```
```   313 text {*
```
```   314   Natural numerals.
```
```   315 *}
```
```   316
```
```   317 lemma [code inline, symmetric, code post]:
```
```   318   "nat (number_of i) = number_nat_inst.number_of_nat i"
```
```   319   -- {* this interacts as desired with @{thm nat_number_of_def} *}
```
```   320   by (simp add: number_nat_inst.number_of_nat)
```
```   321
```
```   322 setup {*
```
```   323   fold (Numeral.add_code @{const_name number_nat_inst.number_of_nat}
```
```   324     true false) ["SML", "OCaml", "Haskell"]
```
```   325 *}
```
```   326
```
```   327 text {*
```
```   328   Since natural numbers are implemented
```
```   329   using integers in ML, the coercion function @{const "of_nat"} of type
```
```   330   @{typ "nat \<Rightarrow> int"} is simply implemented by the identity function.
```
```   331   For the @{const "nat"} function for converting an integer to a natural
```
```   332   number, we give a specific implementation using an ML function that
```
```   333   returns its input value, provided that it is non-negative, and otherwise
```
```   334   returns @{text "0"}.
```
```   335 *}
```
```   336
```
```   337 definition
```
```   338   int :: "nat \<Rightarrow> int"
```
```   339 where
```
```   340   [code func del]: "int = of_nat"
```
```   341
```
```   342 lemma int_code' [code func]:
```
```   343   "int (number_of l) = (if neg (number_of l \<Colon> int) then 0 else number_of l)"
```
```   344   unfolding int_nat_number_of [folded int_def] ..
```
```   345
```
```   346 lemma nat_code' [code func]:
```
```   347   "nat (number_of l) = (if neg (number_of l \<Colon> int) then 0 else number_of l)"
```
```   348   by auto
```
```   349
```
```   350 lemma of_nat_int [code unfold]:
```
```   351   "of_nat = int" by (simp add: int_def)
```
```   352 declare of_nat_int [symmetric, code post]
```
```   353
```
```   354 code_const int
```
```   355   (SML "_")
```
```   356   (OCaml "_")
```
```   357
```
```   358 consts_code
```
```   359   int ("(_)")
```
```   360   nat ("\<module>nat")
```
```   361 attach {*
```
```   362 fun nat i = if i < 0 then 0 else i;
```
```   363 *}
```
```   364
```
```   365 code_const nat
```
```   366   (SML "IntInf.max/ (/0,/ _)")
```
```   367   (OCaml "Big'_int.max'_big'_int/ Big'_int.zero'_big'_int")
```
```   368
```
```   369 text {* For Haskell, things are slightly different again. *}
```
```   370
```
```   371 code_const int and nat
```
```   372   (Haskell "toInteger" and "fromInteger")
```
```   373
```
```   374 text {* Conversion from and to indices. *}
```
```   375
```
```   376 code_const index_of_nat
```
```   377   (SML "IntInf.toInt")
```
```   378   (OCaml "Big'_int.int'_of'_big'_int")
```
```   379   (Haskell "toEnum")
```
```   380
```
```   381 code_const nat_of_index
```
```   382   (SML "IntInf.fromInt")
```
```   383   (OCaml "Big'_int.big'_int'_of'_int")
```
```   384   (Haskell "fromEnum")
```
```   385
```
```   386 text {* Using target language arithmetic operations whenever appropriate *}
```
```   387
```
```   388 code_const "op + \<Colon> nat \<Rightarrow> nat \<Rightarrow> nat"
```
```   389   (SML "IntInf.+ ((_), (_))")
```
```   390   (OCaml "Big'_int.add'_big'_int")
```
```   391   (Haskell infixl 6 "+")
```
```   392
```
```   393 code_const "op * \<Colon> nat \<Rightarrow> nat \<Rightarrow> nat"
```
```   394   (SML "IntInf.* ((_), (_))")
```
```   395   (OCaml "Big'_int.mult'_big'_int")
```
```   396   (Haskell infixl 7 "*")
```
```   397
```
```   398 code_const divmod_aux
```
```   399   (SML "IntInf.divMod/ ((_),/ (_))")
```
```   400   (OCaml "Big'_int.quomod'_big'_int")
```
```   401   (Haskell "divMod")
```
```   402
```
```   403 code_const "op = \<Colon> nat \<Rightarrow> nat \<Rightarrow> bool"
```
```   404   (SML "!((_ : IntInf.int) = _)")
```
```   405   (OCaml "Big'_int.eq'_big'_int")
```
```   406   (Haskell infixl 4 "==")
```
```   407
```
```   408 code_const "op \<le> \<Colon> nat \<Rightarrow> nat \<Rightarrow> bool"
```
```   409   (SML "IntInf.<= ((_), (_))")
```
```   410   (OCaml "Big'_int.le'_big'_int")
```
```   411   (Haskell infix 4 "<=")
```
```   412
```
```   413 code_const "op < \<Colon> nat \<Rightarrow> nat \<Rightarrow> bool"
```
```   414   (SML "IntInf.< ((_), (_))")
```
```   415   (OCaml "Big'_int.lt'_big'_int")
```
```   416   (Haskell infix 4 "<")
```
```   417
```
```   418 consts_code
```
```   419   0                            ("0")
```
```   420   Suc                          ("(_ +/ 1)")
```
```   421   "op + \<Colon>  nat \<Rightarrow> nat \<Rightarrow> nat"   ("(_ +/ _)")
```
```   422   "op * \<Colon>  nat \<Rightarrow> nat \<Rightarrow> nat"   ("(_ */ _)")
```
```   423   "op \<le> \<Colon>  nat \<Rightarrow> nat \<Rightarrow> bool"  ("(_ <=/ _)")
```
```   424   "op < \<Colon>  nat \<Rightarrow> nat \<Rightarrow> bool"  ("(_ </ _)")
```
```   425
```
```   426
```
```   427 text {* Module names *}
```
```   428
```
```   429 code_modulename SML
```
```   430   Nat Integer
```
```   431   Divides Integer
```
```   432   Efficient_Nat Integer
```
```   433
```
```   434 code_modulename OCaml
```
```   435   Nat Integer
```
```   436   Divides Integer
```
```   437   Efficient_Nat Integer
```
```   438
```
```   439 code_modulename Haskell
```
```   440   Nat Integer
```
```   441   Divides Integer
```
```   442   Efficient_Nat Integer
```
```   443
```
```   444 hide const int
```
```   445
```
```   446 end
```