src/HOL/Library/Code_Binary_Nat.thy

+(*  Title:      HOL/Library/Code_Binary_Nat.thy
+    Author:     Stefan Berghofer, Florian Haftmann, TU Muenchen
+*)
+
+header {* Implementation of natural numbers as binary numerals *}
+
+theory Code_Binary_Nat
+imports Main
+begin
+
+text {*
+  When generating code for functions on natural numbers, the
+  canonical representation using @{term "0::nat"} and
+  @{term Suc} is unsuitable for computations involving large
+  numbers.  This theory refines the representation of
+  natural numbers for code generation to use binary
+  numerals, which do not grow linear in size but logarithmic.
+*}
+
+subsection {* Representation *}
+
+code_datatype "0::nat" nat_of_num
+
+lemma [code_abbrev]:
+  "nat_of_num = numeral"
+  by (fact nat_of_num_numeral)
+
+lemma [code]:
+  "num_of_nat 0 = Num.One"
+  "num_of_nat (nat_of_num k) = k"
+  by (simp_all add: nat_of_num_inverse)
+
+lemma [code]:
+  "(1\<Colon>nat) = Numeral1"
+  by simp
+
+lemma [code_abbrev]: "Numeral1 = (1\<Colon>nat)"
+  by simp
+
+lemma [code]:
+  "Suc n = n + 1"
+  by simp
+
+
+subsection {* Basic arithmetic *}
+
+lemma [code, code del]:
+  "(plus :: nat \<Rightarrow> _) = plus" ..
+
+lemma plus_nat_code [code]:
+  "nat_of_num k + nat_of_num l = nat_of_num (k + l)"
+  "m + 0 = (m::nat)"
+  "0 + n = (n::nat)"
+  by (simp_all add: nat_of_num_numeral)
+
+text {* Bounded subtraction needs some auxiliary *}
+
+definition dup :: "nat \<Rightarrow> nat" where
+  "dup n = n + n"
+
+lemma dup_code [code]:
+  "dup 0 = 0"
+  "dup (nat_of_num k) = nat_of_num (Num.Bit0 k)"
+  by (simp_all add: dup_def numeral_Bit0)
+
+definition sub :: "num \<Rightarrow> num \<Rightarrow> nat option" where
+  "sub k l = (if k \<ge> l then Some (numeral k - numeral l) else None)"
+
+lemma sub_code [code]:
+  "sub Num.One Num.One = Some 0"
+  "sub (Num.Bit0 m) Num.One = Some (nat_of_num (Num.BitM m))"
+  "sub (Num.Bit1 m) Num.One = Some (nat_of_num (Num.Bit0 m))"
+  "sub Num.One (Num.Bit0 n) = None"
+  "sub Num.One (Num.Bit1 n) = None"
+  "sub (Num.Bit0 m) (Num.Bit0 n) = Option.map dup (sub m n)"
+  "sub (Num.Bit1 m) (Num.Bit1 n) = Option.map dup (sub m n)"
+  "sub (Num.Bit1 m) (Num.Bit0 n) = Option.map (\<lambda>q. dup q + 1) (sub m n)"
+  "sub (Num.Bit0 m) (Num.Bit1 n) = (case sub m n of None \<Rightarrow> None
+     | Some q \<Rightarrow> if q = 0 then None else Some (dup q - 1))"
+  apply (auto simp add: nat_of_num_numeral
+    Num.dbl_def Num.dbl_inc_def Num.dbl_dec_def
+    Let_def le_imp_diff_is_add BitM_plus_one sub_def dup_def)
+  apply (simp_all add: sub_non_positive)
+  apply (simp_all add: sub_non_negative [symmetric, where ?'a = int])
+  done
+
+lemma [code, code del]:
+  "(minus :: nat \<Rightarrow> _) = minus" ..
+
+lemma minus_nat_code [code]:
+  "nat_of_num k - nat_of_num l = (case sub k l of None \<Rightarrow> 0 | Some j \<Rightarrow> j)"
+  "m - 0 = (m::nat)"
+  "0 - n = (0::nat)"
+  by (simp_all add: nat_of_num_numeral sub_non_positive sub_def)
+
+lemma [code, code del]:
+  "(times :: nat \<Rightarrow> _) = times" ..
+
+lemma times_nat_code [code]:
+  "nat_of_num k * nat_of_num l = nat_of_num (k * l)"
+  "m * 0 = (0::nat)"
+  "0 * n = (0::nat)"
+  by (simp_all add: nat_of_num_numeral)
+
+lemma [code, code del]:
+  "(HOL.equal :: nat \<Rightarrow> _) = HOL.equal" ..
+
+lemma equal_nat_code [code]:
+  "HOL.equal 0 (0::nat) \<longleftrightarrow> True"
+  "HOL.equal 0 (nat_of_num l) \<longleftrightarrow> False"
+  "HOL.equal (nat_of_num k) 0 \<longleftrightarrow> False"
+  "HOL.equal (nat_of_num k) (nat_of_num l) \<longleftrightarrow> HOL.equal k l"
+  by (simp_all add: nat_of_num_numeral equal)
+
+lemma equal_nat_refl [code nbe]:
+  "HOL.equal (n::nat) n \<longleftrightarrow> True"
+  by (rule equal_refl)
+
+lemma [code, code del]:
+  "(less_eq :: nat \<Rightarrow> _) = less_eq" ..
+
+lemma less_eq_nat_code [code]:
+  "0 \<le> (n::nat) \<longleftrightarrow> True"
+  "nat_of_num k \<le> 0 \<longleftrightarrow> False"
+  "nat_of_num k \<le> nat_of_num l \<longleftrightarrow> k \<le> l"
+  by (simp_all add: nat_of_num_numeral)
+
+lemma [code, code del]:
+  "(less :: nat \<Rightarrow> _) = less" ..
+
+lemma less_nat_code [code]:
+  "(m::nat) < 0 \<longleftrightarrow> False"
+  "0 < nat_of_num l \<longleftrightarrow> True"
+  "nat_of_num k < nat_of_num l \<longleftrightarrow> k < l"
+  by (simp_all add: nat_of_num_numeral)
+
+
+subsection {* Conversions *}
+
+lemma [code, code del]:
+  "of_nat = of_nat" ..
+
+lemma of_nat_code [code]:
+  "of_nat 0 = 0"
+  "of_nat (nat_of_num k) = numeral k"
+  by (simp_all add: nat_of_num_numeral)
+
+
+subsection {* Case analysis *}
+
+text {*
+  Case analysis on natural numbers is rephrased using a conditional
+  expression:
+*}
+
+lemma [code, code_unfold]:
+  "nat_case = (\<lambda>f g n. if n = 0 then f else g (n - 1))"
+  by (auto simp add: fun_eq_iff dest!: gr0_implies_Suc)
+
+
+subsection {* Preprocessors *}
+
+text {*
+  The term @{term "Suc n"} is no longer a valid pattern.
+  Therefore, all occurrences of this term in a position
+  where a pattern is expected (i.e.~on the left-hand side of a recursion
+  equation) must be eliminated.
+  This can be accomplished by applying the following transformation rules:
+*}
+
+lemma Suc_if_eq: "(\<And>n. f (Suc n) \<equiv> h n) \<Longrightarrow> f 0 \<equiv> g \<Longrightarrow>
+  f n \<equiv> if n = 0 then g else h (n - 1)"
+  by (rule eq_reflection) (cases n, simp_all)
+
+text {*
+  The rules above are built into a preprocessor that is plugged into
+  the code generator. Since the preprocessor for introduction rules
+  does not know anything about modes, some of the modes that worked
+  for the canonical representation of natural numbers may no longer work.
+*}
+
+(*<*)
+setup {*
+let
+
+fun remove_suc thy thms =
+  let
+    val vname = singleton (Name.variant_list (map fst
+      (fold (Term.add_var_names o Thm.full_prop_of) thms []))) "n";
+    val cv = cterm_of thy (Var ((vname, 0), HOLogic.natT));
+    fun lhs_of th = snd (Thm.dest_comb
+      (fst (Thm.dest_comb (cprop_of th))));
+    fun rhs_of th = snd (Thm.dest_comb (cprop_of th));
+    fun find_vars ct = (case term_of ct of
+        (Const (@{const_name Suc}, _) $ Var _) => [(cv, snd (Thm.dest_comb ct))]
+      | _ $ _ =>
+        let val (ct1, ct2) = Thm.dest_comb ct
+        in 
+          map (apfst (fn ct => Thm.apply ct ct2)) (find_vars ct1) @
+          map (apfst (Thm.apply ct1)) (find_vars ct2)
+        end
+      | _ => []);
+    val eqs = maps
+      (fn th => map (pair th) (find_vars (lhs_of th))) thms;
+    fun mk_thms (th, (ct, cv')) =
+      let
+        val th' =
+          Thm.implies_elim
+           (Conv.fconv_rule (Thm.beta_conversion true)
+             (Drule.instantiate'
+               [SOME (ctyp_of_term ct)] [SOME (Thm.lambda cv ct),
+                 SOME (Thm.lambda cv' (rhs_of th)), NONE, SOME cv']
+               @{thm Suc_if_eq})) (Thm.forall_intr cv' th)
+      in
+        case map_filter (fn th'' =>
+            SOME (th'', singleton
+              (Variable.trade (K (fn [th'''] => [th''' RS th']))
+                (Variable.global_thm_context th'')) th'')
+          handle THM _ => NONE) thms of
+            [] => NONE
+          | thps =>
+              let val (ths1, ths2) = split_list thps
+              in SOME (subtract Thm.eq_thm (th :: ths1) thms @ ths2) end
+      end
+  in get_first mk_thms eqs end;
+
+fun eqn_suc_base_preproc thy thms =
+  let
+    val dest = fst o Logic.dest_equals o prop_of;
+    val contains_suc = exists_Const (fn (c, _) => c = @{const_name Suc});
+  in
+    if forall (can dest) thms andalso exists (contains_suc o dest) thms
+      then thms |> perhaps_loop (remove_suc thy) |> (Option.map o map) Drule.zero_var_indexes
+       else NONE
+  end;
+
+val eqn_suc_preproc = Code_Preproc.simple_functrans eqn_suc_base_preproc;
+
+in
+
+  Code_Preproc.add_functrans ("eqn_Suc", eqn_suc_preproc)
+
+end;
+*}
+(*>*)
+
+code_modulename SML
+  Code_Binary_Nat Arith
+
+code_modulename OCaml
+  Code_Binary_Nat Arith
+
+code_modulename Haskell
+  Code_Binary_Nat Arith
+
+hide_const (open) dup sub
+
+end
+