haftmann@23854: (*  Title:      HOL/Library/Efficient_Nat.thy
haftmann@25931:     Author:     Stefan Berghofer, Florian Haftmann, TU Muenchen
haftmann@23854: *)
haftmann@23854: 
haftmann@25931: header {* Implementation of natural numbers by target-language integers *}
haftmann@23854: 
haftmann@23854: theory Efficient_Nat
haftmann@31203: imports Code_Integer Main
haftmann@23854: begin
haftmann@23854: 
haftmann@23854: text {*
haftmann@25931:   When generating code for functions on natural numbers, the
haftmann@25931:   canonical representation using @{term "0::nat"} and
haftmann@37388:   @{term Suc} is unsuitable for computations involving large
haftmann@25931:   numbers.  The efficiency of the generated code can be improved
haftmann@25931:   drastically by implementing natural numbers by target-language
haftmann@25931:   integers.  To do this, just include this theory.
haftmann@23854: *}
haftmann@23854: 
haftmann@25931: subsection {* Basic arithmetic *}
haftmann@23854: 
haftmann@23854: text {*
haftmann@23854:   Most standard arithmetic functions on natural numbers are implemented
haftmann@23854:   using their counterparts on the integers:
haftmann@23854: *}
haftmann@23854: 
haftmann@25931: code_datatype number_nat_inst.number_of_nat
haftmann@24715: 
haftmann@46028: lemma zero_nat_code [code, code_unfold]:
haftmann@25931:   "0 = (Numeral0 :: nat)"
haftmann@25931:   by simp
haftmann@24715: 
haftmann@46028: lemma one_nat_code [code, code_unfold]:
haftmann@25931:   "1 = (Numeral1 :: nat)"
haftmann@25931:   by simp
haftmann@24715: 
haftmann@25931: lemma Suc_code [code]:
haftmann@25931:   "Suc n = n + 1"
haftmann@25931:   by simp
haftmann@24715: 
haftmann@25931: lemma plus_nat_code [code]:
haftmann@25931:   "n + m = nat (of_nat n + of_nat m)"
haftmann@25931:   by simp
haftmann@24715: 
haftmann@25931: lemma minus_nat_code [code]:
haftmann@25931:   "n - m = nat (of_nat n - of_nat m)"
haftmann@25931:   by simp
haftmann@24715: 
haftmann@25931: lemma times_nat_code [code]:
haftmann@25931:   "n * m = nat (of_nat n * of_nat m)"
haftmann@25931:   unfolding of_nat_mult [symmetric] by simp
haftmann@24715: 
haftmann@37958: lemma divmod_nat_code [code]:
haftmann@40607:   "divmod_nat n m = map_pair nat nat (pdivmod (of_nat n) (of_nat m))"
haftmann@40607:   by (simp add: map_pair_def split_def pdivmod_def nat_div_distrib nat_mod_distrib divmod_nat_div_mod)
haftmann@24715: 
haftmann@25931: lemma eq_nat_code [code]:
haftmann@38857:   "HOL.equal n m \<longleftrightarrow> HOL.equal (of_nat n \<Colon> int) (of_nat m)"
haftmann@38857:   by (simp add: equal)
haftmann@28351: 
haftmann@28351: lemma eq_nat_refl [code nbe]:
haftmann@38857:   "HOL.equal (n::nat) n \<longleftrightarrow> True"
haftmann@38857:   by (rule equal_refl)
haftmann@24715: 
haftmann@25931: lemma less_eq_nat_code [code]:
haftmann@25931:   "n \<le> m \<longleftrightarrow> (of_nat n \<Colon> int) \<le> of_nat m"
haftmann@25931:   by simp
haftmann@23854: 
haftmann@25931: lemma less_nat_code [code]:
haftmann@25931:   "n < m \<longleftrightarrow> (of_nat n \<Colon> int) < of_nat m"
haftmann@25931:   by simp
haftmann@23854: 
haftmann@25931: subsection {* Case analysis *}
haftmann@23854: 
haftmann@23854: text {*
haftmann@25931:   Case analysis on natural numbers is rephrased using a conditional
haftmann@25931:   expression:
haftmann@23854: *}
haftmann@23854: 
haftmann@31998: lemma [code, code_unfold]:
haftmann@25931:   "nat_case = (\<lambda>f g n. if n = 0 then f else g (n - 1))"
nipkow@39302:   by (auto simp add: fun_eq_iff dest!: gr0_implies_Suc)
haftmann@25615: 
haftmann@23854: 
haftmann@23854: subsection {* Preprocessors *}
haftmann@23854: 
haftmann@23854: text {*
haftmann@23854:   In contrast to @{term "Suc n"}, the term @{term "n + (1::nat)"} is no longer
haftmann@23854:   a constructor term. Therefore, all occurrences of this term in a position
haftmann@23854:   where a pattern is expected (i.e.\ on the left-hand side of a recursion
haftmann@23854:   equation or in the arguments of an inductive relation in an introduction
haftmann@23854:   rule) must be eliminated.
haftmann@23854:   This can be accomplished by applying the following transformation rules:
haftmann@23854: *}
haftmann@23854: 
haftmann@29937: lemma Suc_if_eq: "(\<And>n. f (Suc n) \<equiv> h n) \<Longrightarrow> f 0 \<equiv> g \<Longrightarrow>
haftmann@29937:   f n \<equiv> if n = 0 then g else h (n - 1)"
haftmann@31954:   by (rule eq_reflection) (cases n, simp_all)
haftmann@29937: 
haftmann@25931: lemma Suc_clause: "(\<And>n. P n (Suc n)) \<Longrightarrow> n \<noteq> 0 \<Longrightarrow> P (n - 1) n"
haftmann@29932:   by (cases n) simp_all
haftmann@23854: 
haftmann@23854: text {*
haftmann@23854:   The rules above are built into a preprocessor that is plugged into
haftmann@23854:   the code generator. Since the preprocessor for introduction rules
haftmann@23854:   does not know anything about modes, some of the modes that worked
haftmann@23854:   for the canonical representation of natural numbers may no longer work.
haftmann@23854: *}
haftmann@23854: 
haftmann@23854: (*<*)
haftmann@27609: setup {*
haftmann@27609: let
haftmann@23854: 
haftmann@31954: fun remove_suc thy thms =
haftmann@23854:   let
wenzelm@43324:     val vname = singleton (Name.variant_list (map fst
wenzelm@43324:       (fold (Term.add_var_names o Thm.full_prop_of) thms []))) "n";
haftmann@23854:     val cv = cterm_of thy (Var ((vname, 0), HOLogic.natT));
haftmann@23854:     fun lhs_of th = snd (Thm.dest_comb
haftmann@31954:       (fst (Thm.dest_comb (cprop_of th))));
haftmann@31954:     fun rhs_of th = snd (Thm.dest_comb (cprop_of th));
haftmann@23854:     fun find_vars ct = (case term_of ct of
haftmann@29932:         (Const (@{const_name Suc}, _) $ Var _) => [(cv, snd (Thm.dest_comb ct))]
haftmann@23854:       | _ $ _ =>
haftmann@23854:         let val (ct1, ct2) = Thm.dest_comb ct
haftmann@23854:         in 
wenzelm@46497:           map (apfst (fn ct => Thm.apply ct ct2)) (find_vars ct1) @
wenzelm@46497:           map (apfst (Thm.apply ct1)) (find_vars ct2)
haftmann@23854:         end
haftmann@23854:       | _ => []);
haftmann@23854:     val eqs = maps
haftmann@23854:       (fn th => map (pair th) (find_vars (lhs_of th))) thms;
haftmann@23854:     fun mk_thms (th, (ct, cv')) =
haftmann@23854:       let
haftmann@23854:         val th' =
haftmann@23854:           Thm.implies_elim
haftmann@23854:            (Conv.fconv_rule (Thm.beta_conversion true)
haftmann@23854:              (Drule.instantiate'
wenzelm@46497:                [SOME (ctyp_of_term ct)] [SOME (Thm.lambda cv ct),
wenzelm@46497:                  SOME (Thm.lambda cv' (rhs_of th)), NONE, SOME cv']
haftmann@31954:                @{thm Suc_if_eq})) (Thm.forall_intr cv' th)
haftmann@23854:       in
haftmann@23854:         case map_filter (fn th'' =>
haftmann@23854:             SOME (th'', singleton
wenzelm@36603:               (Variable.trade (K (fn [th'''] => [th''' RS th']))
wenzelm@36603:                 (Variable.global_thm_context th'')) th'')
haftmann@23854:           handle THM _ => NONE) thms of
haftmann@23854:             [] => NONE
haftmann@23854:           | thps =>
haftmann@23854:               let val (ths1, ths2) = split_list thps
haftmann@23854:               in SOME (subtract Thm.eq_thm (th :: ths1) thms @ ths2) end
haftmann@23854:       end
haftmann@29937:   in get_first mk_thms eqs end;
haftmann@29937: 
haftmann@34893: fun eqn_suc_base_preproc thy thms =
haftmann@29937:   let
haftmann@31954:     val dest = fst o Logic.dest_equals o prop_of;
haftmann@29937:     val contains_suc = exists_Const (fn (c, _) => c = @{const_name Suc});
haftmann@29937:   in
haftmann@29937:     if forall (can dest) thms andalso exists (contains_suc o dest) thms
haftmann@32348:       then thms |> perhaps_loop (remove_suc thy) |> (Option.map o map) Drule.zero_var_indexes
haftmann@29937:        else NONE
haftmann@23854:   end;
haftmann@23854: 
haftmann@34893: val eqn_suc_preproc = Code_Preproc.simple_functrans eqn_suc_base_preproc;
haftmann@23854: 
haftmann@27609: in
haftmann@27609: 
haftmann@34893:   Code_Preproc.add_functrans ("eqn_Suc", eqn_suc_preproc)
haftmann@27609: 
haftmann@27609: end;
haftmann@23854: *}
haftmann@23854: (*>*)
haftmann@23854: 
haftmann@27609: 
haftmann@25931: subsection {* Target language setup *}
haftmann@25931: 
haftmann@25931: text {*
haftmann@25967:   For ML, we map @{typ nat} to target language integers, where we
haftmann@34899:   ensure that values are always non-negative.
haftmann@25931: *}
haftmann@25931: 
haftmann@25931: code_type nat
haftmann@27496:   (SML "IntInf.int")
haftmann@25931:   (OCaml "Big'_int.big'_int")
haftmann@37958:   (Eval "int")
haftmann@25931: 
haftmann@25931: text {*
bulwahn@45185:   For Haskell and Scala we define our own @{typ nat} type.  The reason
haftmann@34899:   is that we have to distinguish type class instances for @{typ nat}
haftmann@34899:   and @{typ int}.
haftmann@25967: *}
haftmann@25967: 
haftmann@38774: code_include Haskell "Nat"
haftmann@38774: {*newtype Nat = Nat Integer deriving (Eq, Show, Read);
haftmann@25967: 
haftmann@25967: instance Num Nat where {
haftmann@25967:   fromInteger k = Nat (if k >= 0 then k else 0);
haftmann@25967:   Nat n + Nat m = Nat (n + m);
haftmann@25967:   Nat n - Nat m = fromInteger (n - m);
haftmann@25967:   Nat n * Nat m = Nat (n * m);
haftmann@25967:   abs n = n;
haftmann@25967:   signum _ = 1;
haftmann@25967:   negate n = error "negate Nat";
haftmann@25967: };
haftmann@25967: 
haftmann@25967: instance Ord Nat where {
haftmann@25967:   Nat n <= Nat m = n <= m;
haftmann@25967:   Nat n < Nat m = n < m;
haftmann@25967: };
haftmann@25967: 
haftmann@25967: instance Real Nat where {
haftmann@25967:   toRational (Nat n) = toRational n;
haftmann@25967: };
haftmann@25967: 
haftmann@25967: instance Enum Nat where {
haftmann@25967:   toEnum k = fromInteger (toEnum k);
haftmann@25967:   fromEnum (Nat n) = fromEnum n;
haftmann@25967: };
haftmann@25967: 
haftmann@25967: instance Integral Nat where {
haftmann@25967:   toInteger (Nat n) = n;
haftmann@25967:   divMod n m = quotRem n m;
haftmann@37958:   quotRem (Nat n) (Nat m)
haftmann@37958:     | (m == 0) = (0, Nat n)
haftmann@37958:     | otherwise = (Nat k, Nat l) where (k, l) = quotRem n m;
haftmann@25967: };
haftmann@25967: *}
haftmann@25967: 
haftmann@25967: code_reserved Haskell Nat
haftmann@25967: 
haftmann@38774: code_include Scala "Nat"
haftmann@38968: {*object Nat {
haftmann@34899: 
haftmann@34899:   def apply(numeral: BigInt): Nat = new Nat(numeral max 0)
haftmann@34899:   def apply(numeral: Int): Nat = Nat(BigInt(numeral))
haftmann@34899:   def apply(numeral: String): Nat = Nat(BigInt(numeral))
haftmann@34899: 
haftmann@34899: }
haftmann@34899: 
haftmann@34899: class Nat private(private val value: BigInt) {
haftmann@34899: 
haftmann@34899:   override def hashCode(): Int = this.value.hashCode()
haftmann@34899: 
haftmann@34899:   override def equals(that: Any): Boolean = that match {
haftmann@34899:     case that: Nat => this equals that
haftmann@34899:     case _ => false
haftmann@34899:   }
haftmann@34899: 
haftmann@34899:   override def toString(): String = this.value.toString
haftmann@34899: 
haftmann@34899:   def equals(that: Nat): Boolean = this.value == that.value
haftmann@34899: 
haftmann@34899:   def as_BigInt: BigInt = this.value
haftmann@39781:   def as_Int: Int = if (this.value >= scala.Int.MinValue && this.value <= scala.Int.MaxValue)
haftmann@34944:       this.value.intValue
haftmann@37969:     else error("Int value out of range: " + this.value.toString)
haftmann@34899: 
haftmann@34899:   def +(that: Nat): Nat = new Nat(this.value + that.value)
haftmann@37223:   def -(that: Nat): Nat = Nat(this.value - that.value)
haftmann@34899:   def *(that: Nat): Nat = new Nat(this.value * that.value)
haftmann@34899: 
haftmann@34899:   def /%(that: Nat): (Nat, Nat) = if (that.value == 0) (new Nat(0), this)
haftmann@34899:     else {
haftmann@34899:       val (k, l) = this.value /% that.value
haftmann@34899:       (new Nat(k), new Nat(l))
haftmann@34899:     }
haftmann@34899: 
haftmann@34899:   def <=(that: Nat): Boolean = this.value <= that.value
haftmann@34899: 
haftmann@34899:   def <(that: Nat): Boolean = this.value < that.value
haftmann@34899: 
haftmann@34899: }
haftmann@34899: *}
haftmann@34899: 
haftmann@34899: code_reserved Scala Nat
haftmann@34899: 
haftmann@25967: code_type nat
haftmann@29793:   (Haskell "Nat.Nat")
haftmann@38968:   (Scala "Nat")
haftmann@25967: 
haftmann@38857: code_instance nat :: equal
haftmann@25967:   (Haskell -)
haftmann@25967: 
haftmann@25967: text {*
haftmann@25931:   Natural numerals.
haftmann@25931: *}
haftmann@25931: 
haftmann@46028: lemma [code_abbrev]:
haftmann@46028:   "number_nat_inst.number_of_nat i = nat (number_of i)"
haftmann@25931:   -- {* this interacts as desired with @{thm nat_number_of_def} *}
haftmann@25931:   by (simp add: number_nat_inst.number_of_nat)
haftmann@25931: 
haftmann@25931: setup {*
haftmann@25931:   fold (Numeral.add_code @{const_name number_nat_inst.number_of_nat}
haftmann@37958:     false Code_Printer.literal_positive_numeral) ["SML", "OCaml", "Haskell", "Scala"]
haftmann@25931: *}
haftmann@25931: 
haftmann@25931: text {*
haftmann@25931:   Since natural numbers are implemented
haftmann@25967:   using integers in ML, the coercion function @{const "of_nat"} of type
haftmann@25931:   @{typ "nat \<Rightarrow> int"} is simply implemented by the identity function.
haftmann@37388:   For the @{const nat} function for converting an integer to a natural
haftmann@25931:   number, we give a specific implementation using an ML function that
haftmann@25931:   returns its input value, provided that it is non-negative, and otherwise
haftmann@25931:   returns @{text "0"}.
haftmann@25931: *}
haftmann@25931: 
haftmann@32073: definition int :: "nat \<Rightarrow> int" where
haftmann@46028:   [code del, code_abbrev]: "int = of_nat"
haftmann@25931: 
haftmann@28562: lemma int_code' [code]:
haftmann@25931:   "int (number_of l) = (if neg (number_of l \<Colon> int) then 0 else number_of l)"
haftmann@25931:   unfolding int_nat_number_of [folded int_def] ..
haftmann@25931: 
haftmann@28562: lemma nat_code' [code]:
haftmann@25931:   "nat (number_of l) = (if neg (number_of l \<Colon> int) then 0 else number_of l)"
huffman@28969:   unfolding nat_number_of_def number_of_is_id neg_def by simp
haftmann@25931: 
haftmann@46028: lemma of_nat_int: (* FIXME delete candidate *)
haftmann@25931:   "of_nat = int" by (simp add: int_def)
haftmann@25931: 
haftmann@32073: lemma of_nat_aux_int [code_unfold]:
haftmann@32073:   "of_nat_aux (\<lambda>i. i + 1) k 0 = int k"
haftmann@32073:   by (simp add: int_def Nat.of_nat_code)
haftmann@32073: 
haftmann@25931: code_const int
haftmann@25931:   (SML "_")
haftmann@25931:   (OCaml "_")
haftmann@25931: 
haftmann@25967: code_const nat
haftmann@43653:   (SML "IntInf.max/ (0,/ _)")
haftmann@25967:   (OCaml "Big'_int.max'_big'_int/ Big'_int.zero'_big'_int")
haftmann@37958:   (Eval "Integer.max/ _/ 0")
haftmann@25967: 
haftmann@35689: text {* For Haskell and Scala, things are slightly different again. *}
haftmann@25967: 
haftmann@25967: code_const int and nat
haftmann@25967:   (Haskell "toInteger" and "fromInteger")
haftmann@38968:   (Scala "!_.as'_BigInt" and "Nat")
haftmann@25931: 
haftmann@37958: text {* Conversion from and to code numerals. *}
haftmann@25931: 
haftmann@31205: code_const Code_Numeral.of_nat
haftmann@25967:   (SML "IntInf.toInt")
haftmann@31377:   (OCaml "_")
haftmann@37958:   (Haskell "!(fromInteger/ ./ toInteger)")
haftmann@38968:   (Scala "!Natural(_.as'_BigInt)")
haftmann@37958:   (Eval "_")
haftmann@25967: 
haftmann@31205: code_const Code_Numeral.nat_of
haftmann@25931:   (SML "IntInf.fromInt")
haftmann@31377:   (OCaml "_")
haftmann@37958:   (Haskell "!(fromInteger/ ./ toInteger)")
haftmann@38968:   (Scala "!Nat(_.as'_BigInt)")
haftmann@37958:   (Eval "_")
haftmann@25931: 
haftmann@25931: text {* Using target language arithmetic operations whenever appropriate *}
haftmann@25931: 
haftmann@25931: code_const "op + \<Colon> nat \<Rightarrow> nat \<Rightarrow> nat"
haftmann@25931:   (SML "IntInf.+ ((_), (_))")
haftmann@25931:   (OCaml "Big'_int.add'_big'_int")
haftmann@25931:   (Haskell infixl 6 "+")
haftmann@34899:   (Scala infixl 7 "+")
haftmann@37958:   (Eval infixl 8 "+")
haftmann@34899: 
haftmann@34899: code_const "op - \<Colon> nat \<Rightarrow> nat \<Rightarrow> nat"
haftmann@34899:   (Haskell infixl 6 "-")
haftmann@34899:   (Scala infixl 7 "-")
haftmann@25931: 
haftmann@25931: code_const "op * \<Colon> nat \<Rightarrow> nat \<Rightarrow> nat"
haftmann@25931:   (SML "IntInf.* ((_), (_))")
haftmann@25931:   (OCaml "Big'_int.mult'_big'_int")
haftmann@25931:   (Haskell infixl 7 "*")
haftmann@34899:   (Scala infixl 8 "*")
haftmann@37958:   (Eval infixl 9 "*")
haftmann@25931: 
haftmann@37958: code_const divmod_nat
haftmann@26009:   (SML "IntInf.divMod/ ((_),/ (_))")
haftmann@26009:   (OCaml "Big'_int.quomod'_big'_int")
haftmann@26009:   (Haskell "divMod")
haftmann@34899:   (Scala infixl 8 "/%")
haftmann@37958:   (Eval "Integer.div'_mod")
haftmann@25931: 
haftmann@38857: code_const "HOL.equal \<Colon> nat \<Rightarrow> nat \<Rightarrow> bool"
haftmann@25931:   (SML "!((_ : IntInf.int) = _)")
haftmann@25931:   (OCaml "Big'_int.eq'_big'_int")
haftmann@39272:   (Haskell infix 4 "==")
haftmann@34899:   (Scala infixl 5 "==")
haftmann@37958:   (Eval infixl 6 "=")
haftmann@25931: 
haftmann@25931: code_const "op \<le> \<Colon> nat \<Rightarrow> nat \<Rightarrow> bool"
haftmann@25931:   (SML "IntInf.<= ((_), (_))")
haftmann@25931:   (OCaml "Big'_int.le'_big'_int")
haftmann@25931:   (Haskell infix 4 "<=")
haftmann@34899:   (Scala infixl 4 "<=")
haftmann@37958:   (Eval infixl 6 "<=")
haftmann@25931: 
haftmann@25931: code_const "op < \<Colon> nat \<Rightarrow> nat \<Rightarrow> bool"
haftmann@25931:   (SML "IntInf.< ((_), (_))")
haftmann@25931:   (OCaml "Big'_int.lt'_big'_int")
haftmann@25931:   (Haskell infix 4 "<")
haftmann@34899:   (Scala infixl 4 "<")
haftmann@37958:   (Eval infixl 6 "<")
haftmann@25931: 
haftmann@25931: 
haftmann@28228: text {* Evaluation *}
haftmann@28228: 
haftmann@28562: lemma [code, code del]:
haftmann@32657:   "(Code_Evaluation.term_of \<Colon> nat \<Rightarrow> term) = Code_Evaluation.term_of" ..
haftmann@28228: 
haftmann@32657: code_const "Code_Evaluation.term_of \<Colon> nat \<Rightarrow> term"
haftmann@28228:   (SML "HOLogic.mk'_number/ HOLogic.natT")
haftmann@28228: 
bulwahn@45793: text {* Evaluation with @{text "Quickcheck_Narrowing"} does not work, as
bulwahn@45793:   @{text "code_module"} is very aggressive leading to bad Haskell code.
bulwahn@45793:   Therefore, we simply deactivate the narrowing-based quickcheck from here on.
bulwahn@45793: *}
bulwahn@45793: 
bulwahn@45793: declare [[quickcheck_narrowing_active = false]] 
haftmann@28228: 
haftmann@25931: text {* Module names *}
haftmann@23854: 
haftmann@23854: code_modulename SML
haftmann@33364:   Efficient_Nat Arith
haftmann@23854: 
haftmann@23854: code_modulename OCaml
haftmann@33364:   Efficient_Nat Arith
haftmann@23854: 
haftmann@23854: code_modulename Haskell
haftmann@33364:   Efficient_Nat Arith
haftmann@23854: 
wenzelm@36176: hide_const int
haftmann@23854: 
haftmann@23854: end