src/HOL/Library/Efficient_Nat.thy
 author haftmann Wed Apr 22 19:09:21 2009 +0200 (2009-04-22) changeset 30960 fec1a04b7220 parent 30663 0b6aff7451b2 child 31090 3be41b271023 permissions -rw-r--r--
power operation defined generic
```     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 fun eqn_suc_preproc thy = map fst
```
```   183   #> gen_eqn_suc_preproc
```
```   184       @{thm Suc_if_eq} I (fst o Logic.dest_equals) thy
```
```   185   #> (Option.map o map) (Code_Unit.mk_eqn thy);
```
```   186
```
```   187 fun eqn_suc_preproc' thy thms = gen_eqn_suc_preproc
```
```   188   @{thm Suc_if_eq'} (snd o Thm.dest_comb) (fst o HOLogic.dest_eq o HOLogic.dest_Trueprop) thy thms
```
```   189   |> the_default thms;
```
```   190
```
```   191 fun remove_suc_clause thy thms =
```
```   192   let
```
```   193     val vname = Name.variant (map fst
```
```   194       (fold (Term.add_var_names o Thm.full_prop_of) thms [])) "x";
```
```   195     fun find_var (t as Const (@{const_name Suc}, _) \$ (v as Var _)) = SOME (t, v)
```
```   196       | find_var (t \$ u) = (case find_var t of NONE => find_var u | x => x)
```
```   197       | find_var _ = NONE;
```
```   198     fun find_thm th =
```
```   199       let val th' = Conv.fconv_rule ObjectLogic.atomize th
```
```   200       in Option.map (pair (th, th')) (find_var (prop_of th')) end
```
```   201   in
```
```   202     case get_first find_thm thms of
```
```   203       NONE => thms
```
```   204     | SOME ((th, th'), (Sucv, v)) =>
```
```   205         let
```
```   206           val cert = cterm_of (Thm.theory_of_thm th);
```
```   207           val th'' = ObjectLogic.rulify (Thm.implies_elim
```
```   208             (Conv.fconv_rule (Thm.beta_conversion true)
```
```   209               (Drule.instantiate' []
```
```   210                 [SOME (cert (lambda v (Abs ("x", HOLogic.natT,
```
```   211                    abstract_over (Sucv,
```
```   212                      HOLogic.dest_Trueprop (prop_of th')))))),
```
```   213                  SOME (cert v)] @{thm Suc_clause}))
```
```   214             (Thm.forall_intr (cert v) th'))
```
```   215         in
```
```   216           remove_suc_clause thy (map (fn th''' =>
```
```   217             if (op = o pairself prop_of) (th''', th) then th'' else th''') thms)
```
```   218         end
```
```   219   end;
```
```   220
```
```   221 fun clause_suc_preproc thy ths =
```
```   222   let
```
```   223     val dest = fst o HOLogic.dest_mem o HOLogic.dest_Trueprop
```
```   224   in
```
```   225     if forall (can (dest o concl_of)) ths andalso
```
```   226       exists (fn th => exists (exists_Const (fn (c, _) => c = @{const_name Suc}))
```
```   227         (map_filter (try dest) (concl_of th :: prems_of th))) ths
```
```   228     then remove_suc_clause thy ths else ths
```
```   229   end;
```
```   230 in
```
```   231
```
```   232   Codegen.add_preprocessor eqn_suc_preproc'
```
```   233   #> Codegen.add_preprocessor clause_suc_preproc
```
```   234   #> Code.add_functrans ("eqn_Suc", eqn_suc_preproc)
```
```   235
```
```   236 end;
```
```   237 *}
```
```   238 (*>*)
```
```   239
```
```   240
```
```   241 subsection {* Target language setup *}
```
```   242
```
```   243 text {*
```
```   244   For ML, we map @{typ nat} to target language integers, where we
```
```   245   assert that values are always non-negative.
```
```   246 *}
```
```   247
```
```   248 code_type nat
```
```   249   (SML "IntInf.int")
```
```   250   (OCaml "Big'_int.big'_int")
```
```   251
```
```   252 types_code
```
```   253   nat ("int")
```
```   254 attach (term_of) {*
```
```   255 val term_of_nat = HOLogic.mk_number HOLogic.natT;
```
```   256 *}
```
```   257 attach (test) {*
```
```   258 fun gen_nat i =
```
```   259   let val n = random_range 0 i
```
```   260   in (n, fn () => term_of_nat n) end;
```
```   261 *}
```
```   262
```
```   263 text {*
```
```   264   For Haskell we define our own @{typ nat} type.  The reason
```
```   265   is that we have to distinguish type class instances
```
```   266   for @{typ nat} and @{typ int}.
```
```   267 *}
```
```   268
```
```   269 code_include Haskell "Nat" {*
```
```   270 newtype Nat = Nat Integer deriving (Show, Eq);
```
```   271
```
```   272 instance Num Nat where {
```
```   273   fromInteger k = Nat (if k >= 0 then k else 0);
```
```   274   Nat n + Nat m = Nat (n + m);
```
```   275   Nat n - Nat m = fromInteger (n - m);
```
```   276   Nat n * Nat m = Nat (n * m);
```
```   277   abs n = n;
```
```   278   signum _ = 1;
```
```   279   negate n = error "negate Nat";
```
```   280 };
```
```   281
```
```   282 instance Ord Nat where {
```
```   283   Nat n <= Nat m = n <= m;
```
```   284   Nat n < Nat m = n < m;
```
```   285 };
```
```   286
```
```   287 instance Real Nat where {
```
```   288   toRational (Nat n) = toRational n;
```
```   289 };
```
```   290
```
```   291 instance Enum Nat where {
```
```   292   toEnum k = fromInteger (toEnum k);
```
```   293   fromEnum (Nat n) = fromEnum n;
```
```   294 };
```
```   295
```
```   296 instance Integral Nat where {
```
```   297   toInteger (Nat n) = n;
```
```   298   divMod n m = quotRem n m;
```
```   299   quotRem (Nat n) (Nat m) = (Nat k, Nat l) where (k, l) = quotRem n m;
```
```   300 };
```
```   301 *}
```
```   302
```
```   303 code_reserved Haskell Nat
```
```   304
```
```   305 code_type nat
```
```   306   (Haskell "Nat.Nat")
```
```   307
```
```   308 code_instance nat :: eq
```
```   309   (Haskell -)
```
```   310
```
```   311 text {*
```
```   312   Natural numerals.
```
```   313 *}
```
```   314
```
```   315 lemma [code inline, symmetric, code post]:
```
```   316   "nat (number_of i) = number_nat_inst.number_of_nat i"
```
```   317   -- {* this interacts as desired with @{thm nat_number_of_def} *}
```
```   318   by (simp add: number_nat_inst.number_of_nat)
```
```   319
```
```   320 setup {*
```
```   321   fold (Numeral.add_code @{const_name number_nat_inst.number_of_nat}
```
```   322     true false) ["SML", "OCaml", "Haskell"]
```
```   323 *}
```
```   324
```
```   325 text {*
```
```   326   Since natural numbers are implemented
```
```   327   using integers in ML, the coercion function @{const "of_nat"} of type
```
```   328   @{typ "nat \<Rightarrow> int"} is simply implemented by the identity function.
```
```   329   For the @{const "nat"} function for converting an integer to a natural
```
```   330   number, we give a specific implementation using an ML function that
```
```   331   returns its input value, provided that it is non-negative, and otherwise
```
```   332   returns @{text "0"}.
```
```   333 *}
```
```   334
```
```   335 definition
```
```   336   int :: "nat \<Rightarrow> int"
```
```   337 where
```
```   338   [code del]: "int = of_nat"
```
```   339
```
```   340 lemma int_code' [code]:
```
```   341   "int (number_of l) = (if neg (number_of l \<Colon> int) then 0 else number_of l)"
```
```   342   unfolding int_nat_number_of [folded int_def] ..
```
```   343
```
```   344 lemma nat_code' [code]:
```
```   345   "nat (number_of l) = (if neg (number_of l \<Colon> int) then 0 else number_of l)"
```
```   346   unfolding nat_number_of_def number_of_is_id neg_def by simp
```
```   347
```
```   348 lemma of_nat_int [code unfold]:
```
```   349   "of_nat = int" by (simp add: int_def)
```
```   350 declare of_nat_int [symmetric, code post]
```
```   351
```
```   352 code_const int
```
```   353   (SML "_")
```
```   354   (OCaml "_")
```
```   355
```
```   356 consts_code
```
```   357   int ("(_)")
```
```   358   nat ("\<module>nat")
```
```   359 attach {*
```
```   360 fun nat i = if i < 0 then 0 else i;
```
```   361 *}
```
```   362
```
```   363 code_const nat
```
```   364   (SML "IntInf.max/ (/0,/ _)")
```
```   365   (OCaml "Big'_int.max'_big'_int/ Big'_int.zero'_big'_int")
```
```   366
```
```   367 text {* For Haskell, things are slightly different again. *}
```
```   368
```
```   369 code_const int and nat
```
```   370   (Haskell "toInteger" and "fromInteger")
```
```   371
```
```   372 text {* Conversion from and to indices. *}
```
```   373
```
```   374 code_const Code_Index.of_nat
```
```   375   (SML "IntInf.toInt")
```
```   376   (OCaml "Big'_int.int'_of'_big'_int")
```
```   377   (Haskell "fromEnum")
```
```   378
```
```   379 code_const Code_Index.nat_of
```
```   380   (SML "IntInf.fromInt")
```
```   381   (OCaml "Big'_int.big'_int'_of'_int")
```
```   382   (Haskell "toEnum")
```
```   383
```
```   384 text {* Using target language arithmetic operations whenever appropriate *}
```
```   385
```
```   386 code_const "op + \<Colon> nat \<Rightarrow> nat \<Rightarrow> nat"
```
```   387   (SML "IntInf.+ ((_), (_))")
```
```   388   (OCaml "Big'_int.add'_big'_int")
```
```   389   (Haskell infixl 6 "+")
```
```   390
```
```   391 code_const "op * \<Colon> nat \<Rightarrow> nat \<Rightarrow> nat"
```
```   392   (SML "IntInf.* ((_), (_))")
```
```   393   (OCaml "Big'_int.mult'_big'_int")
```
```   394   (Haskell infixl 7 "*")
```
```   395
```
```   396 code_const divmod_aux
```
```   397   (SML "IntInf.divMod/ ((_),/ (_))")
```
```   398   (OCaml "Big'_int.quomod'_big'_int")
```
```   399   (Haskell "divMod")
```
```   400
```
```   401 code_const "eq_class.eq \<Colon> nat \<Rightarrow> nat \<Rightarrow> bool"
```
```   402   (SML "!((_ : IntInf.int) = _)")
```
```   403   (OCaml "Big'_int.eq'_big'_int")
```
```   404   (Haskell infixl 4 "==")
```
```   405
```
```   406 code_const "op \<le> \<Colon> nat \<Rightarrow> nat \<Rightarrow> bool"
```
```   407   (SML "IntInf.<= ((_), (_))")
```
```   408   (OCaml "Big'_int.le'_big'_int")
```
```   409   (Haskell infix 4 "<=")
```
```   410
```
```   411 code_const "op < \<Colon> nat \<Rightarrow> nat \<Rightarrow> bool"
```
```   412   (SML "IntInf.< ((_), (_))")
```
```   413   (OCaml "Big'_int.lt'_big'_int")
```
```   414   (Haskell infix 4 "<")
```
```   415
```
```   416 consts_code
```
```   417   "0::nat"                     ("0")
```
```   418   "1::nat"                     ("1")
```
```   419   Suc                          ("(_ +/ 1)")
```
```   420   "op + \<Colon>  nat \<Rightarrow> nat \<Rightarrow> nat"   ("(_ +/ _)")
```
```   421   "op * \<Colon>  nat \<Rightarrow> nat \<Rightarrow> nat"   ("(_ */ _)")
```
```   422   "op \<le> \<Colon>  nat \<Rightarrow> nat \<Rightarrow> bool"  ("(_ <=/ _)")
```
```   423   "op < \<Colon>  nat \<Rightarrow> nat \<Rightarrow> bool"  ("(_ </ _)")
```
```   424
```
```   425
```
```   426 text {* Evaluation *}
```
```   427
```
```   428 lemma [code, code del]:
```
```   429   "(Code_Eval.term_of \<Colon> nat \<Rightarrow> term) = Code_Eval.term_of" ..
```
```   430
```
```   431 code_const "Code_Eval.term_of \<Colon> nat \<Rightarrow> term"
```
```   432   (SML "HOLogic.mk'_number/ HOLogic.natT")
```
```   433
```
```   434
```
```   435 text {* Module names *}
```
```   436
```
```   437 code_modulename SML
```
```   438   Nat Integer
```
```   439   Divides Integer
```
```   440   Ring_and_Field Integer
```
```   441   Efficient_Nat Integer
```
```   442
```
```   443 code_modulename OCaml
```
```   444   Nat Integer
```
```   445   Divides Integer
```
```   446   Ring_and_Field Integer
```
```   447   Efficient_Nat Integer
```
```   448
```
```   449 code_modulename Haskell
```
```   450   Nat Integer
```
```   451   Divides Integer
```
```   452   Ring_and_Field Integer
```
```   453   Efficient_Nat Integer
```
```   454
```
```   455 hide const int
```
```   456
```
```   457 end
```