# HG changeset patch
# User haftmann
# Date 1352365358 -3600
# Node ID 28f3263d4d1b0907109b7c805c771263915e68c2
# Parent  286dfcab9833a06fc8c571425f50572af8223273
refined stack of library theories implementing int and/or nat by target language numerals

diff -r 286dfcab9833 -r 28f3263d4d1b src/HOL/Library/Code_Binary_Nat.thy
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Library/Code_Binary_Nat.thy	Thu Nov 08 10:02:38 2012 +0100
@@ -0,0 +1,259 @@
+(*  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
+
diff -r 286dfcab9833 -r 28f3263d4d1b src/HOL/Library/Code_Nat.thy
--- a/src/HOL/Library/Code_Nat.thy	Wed Nov 07 20:48:04 2012 +0100
+++ /dev/null	Thu Jan 01 00:00:00 1970 +0000
@@ -1,258 +0,0 @@
-(*  Title:      HOL/Library/Code_Nat.thy
-    Author:     Stefan Berghofer, Florian Haftmann, TU Muenchen
-*)
-
-header {* Implementation of natural numbers as binary numerals *}
-
-theory Code_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 *}
-
-lemma [code_abbrev]:
-  "nat_of_num = numeral"
-  by (fact nat_of_num_numeral)
-
-code_datatype "0::nat" nat_of_num
-
-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)"
-  unfolding Num_def 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_Nat Arith
-
-code_modulename OCaml
-  Code_Nat Arith
-
-code_modulename Haskell
-  Code_Nat Arith
-
-hide_const (open) dup sub
-
-end
diff -r 286dfcab9833 -r 28f3263d4d1b src/HOL/Library/Code_Numeral_Types.thy
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Library/Code_Numeral_Types.thy	Thu Nov 08 10:02:38 2012 +0100
@@ -0,0 +1,870 @@
+(*  Title:      HOL/Library/Code_Numeral_Types.thy
+    Author:     Florian Haftmann, TU Muenchen
+*)
+
+header {* Numeric types for code generation onto target language numerals only *}
+
+theory Code_Numeral_Types
+imports Main Nat_Transfer Divides Lifting
+begin
+
+subsection {* Type of target language integers *}
+
+typedef integer = "UNIV \<Colon> int set"
+  morphisms int_of_integer integer_of_int ..
+
+lemma integer_eq_iff:
+  "k = l \<longleftrightarrow> int_of_integer k = int_of_integer l"
+  using int_of_integer_inject [of k l] ..
+
+lemma integer_eqI:
+  "int_of_integer k = int_of_integer l \<Longrightarrow> k = l"
+  using integer_eq_iff [of k l] by simp
+
+lemma int_of_integer_integer_of_int [simp]:
+  "int_of_integer (integer_of_int k) = k"
+  using integer_of_int_inverse [of k] by simp
+
+lemma integer_of_int_int_of_integer [simp]:
+  "integer_of_int (int_of_integer k) = k"
+  using int_of_integer_inverse [of k] by simp
+
+instantiation integer :: ring_1
+begin
+
+definition
+  "0 = integer_of_int 0"
+
+lemma int_of_integer_zero [simp]:
+  "int_of_integer 0 = 0"
+  by (simp add: zero_integer_def)
+
+definition
+  "1 = integer_of_int 1"
+
+lemma int_of_integer_one [simp]:
+  "int_of_integer 1 = 1"
+  by (simp add: one_integer_def)
+
+definition
+  "k + l = integer_of_int (int_of_integer k + int_of_integer l)"
+
+lemma int_of_integer_plus [simp]:
+  "int_of_integer (k + l) = int_of_integer k + int_of_integer l"
+  by (simp add: plus_integer_def)
+
+definition
+  "- k = integer_of_int (- int_of_integer k)"
+
+lemma int_of_integer_uminus [simp]:
+  "int_of_integer (- k) = - int_of_integer k"
+  by (simp add: uminus_integer_def)
+
+definition
+  "k - l = integer_of_int (int_of_integer k - int_of_integer l)"
+
+lemma int_of_integer_minus [simp]:
+  "int_of_integer (k - l) = int_of_integer k - int_of_integer l"
+  by (simp add: minus_integer_def)
+
+definition
+  "k * l = integer_of_int (int_of_integer k * int_of_integer l)"
+
+lemma int_of_integer_times [simp]:
+  "int_of_integer (k * l) = int_of_integer k * int_of_integer l"
+  by (simp add: times_integer_def)
+
+instance proof
+qed (auto simp add: integer_eq_iff algebra_simps)
+
+end
+
+lemma int_of_integer_of_nat [simp]:
+  "int_of_integer (of_nat n) = of_nat n"
+  by (induct n) simp_all
+
+definition nat_of_integer :: "integer \<Rightarrow> nat"
+where
+  "nat_of_integer k = Int.nat (int_of_integer k)"
+
+lemma nat_of_integer_of_nat [simp]:
+  "nat_of_integer (of_nat n) = n"
+  by (simp add: nat_of_integer_def)
+
+lemma int_of_integer_of_int [simp]:
+  "int_of_integer (of_int k) = k"
+  by (induct k) (simp_all, simp only: neg_numeral_def numeral_One int_of_integer_uminus int_of_integer_one)
+
+lemma integer_integer_of_int_eq_of_integer_integer_of_int [simp, code_abbrev]:
+  "integer_of_int = of_int"
+  by rule (simp add: integer_eq_iff)
+
+lemma of_int_integer_of [simp]:
+  "of_int (int_of_integer k) = (k :: integer)"
+  by (simp add: integer_eq_iff)
+
+lemma int_of_integer_numeral [simp]:
+  "int_of_integer (numeral k) = numeral k"
+  using int_of_integer_of_int [of "numeral k"] by simp
+
+lemma int_of_integer_neg_numeral [simp]:
+  "int_of_integer (neg_numeral k) = neg_numeral k"
+  by (simp only: neg_numeral_def int_of_integer_uminus) simp
+
+lemma int_of_integer_sub [simp]:
+  "int_of_integer (Num.sub k l) = Num.sub k l"
+  by (simp only: Num.sub_def int_of_integer_minus int_of_integer_numeral)
+
+instantiation integer :: "{ring_div, equal, linordered_idom}"
+begin
+
+definition
+  "k div l = of_int (int_of_integer k div int_of_integer l)"
+
+lemma int_of_integer_div [simp]:
+  "int_of_integer (k div l) = int_of_integer k div int_of_integer l"
+  by (simp add: div_integer_def)
+
+definition
+  "k mod l = of_int (int_of_integer k mod int_of_integer l)"
+
+lemma int_of_integer_mod [simp]:
+  "int_of_integer (k mod l) = int_of_integer k mod int_of_integer l"
+  by (simp add: mod_integer_def)
+
+definition
+  "\<bar>k\<bar> = of_int \<bar>int_of_integer k\<bar>"
+
+lemma int_of_integer_abs [simp]:
+  "int_of_integer \<bar>k\<bar> = \<bar>int_of_integer k\<bar>"
+  by (simp add: abs_integer_def)
+
+definition
+  "sgn k = of_int (sgn (int_of_integer k))"
+
+lemma int_of_integer_sgn [simp]:
+  "int_of_integer (sgn k) = sgn (int_of_integer k)"
+  by (simp add: sgn_integer_def)
+
+definition
+  "k \<le> l \<longleftrightarrow> int_of_integer k \<le> int_of_integer l"
+
+definition
+  "k < l \<longleftrightarrow> int_of_integer k < int_of_integer l"
+
+definition
+  "HOL.equal k l \<longleftrightarrow> HOL.equal (int_of_integer k) (int_of_integer l)"
+
+instance proof
+qed (auto simp add: integer_eq_iff algebra_simps
+  less_eq_integer_def less_integer_def equal_integer_def equal
+  intro: mult_strict_right_mono)
+
+end
+
+lemma int_of_integer_min [simp]:
+  "int_of_integer (min k l) = min (int_of_integer k) (int_of_integer l)"
+  by (simp add: min_def less_eq_integer_def)
+
+lemma int_of_integer_max [simp]:
+  "int_of_integer (max k l) = max (int_of_integer k) (int_of_integer l)"
+  by (simp add: max_def less_eq_integer_def)
+
+lemma nat_of_integer_non_positive [simp]:
+  "k \<le> 0 \<Longrightarrow> nat_of_integer k = 0"
+  by (simp add: nat_of_integer_def less_eq_integer_def)
+
+lemma of_nat_of_integer [simp]:
+  "of_nat (nat_of_integer k) = max 0 k"
+  by (simp add: nat_of_integer_def integer_eq_iff less_eq_integer_def max_def)
+
+
+subsection {* Code theorems for target language integers *}
+
+text {* Constructors *}
+
+definition Pos :: "num \<Rightarrow> integer"
+where
+  [simp, code_abbrev]: "Pos = numeral"
+
+definition Neg :: "num \<Rightarrow> integer"
+where
+  [simp, code_abbrev]: "Neg = neg_numeral"
+
+code_datatype "0::integer" Pos Neg
+
+
+text {* Auxiliary operations *}
+
+definition dup :: "integer \<Rightarrow> integer"
+where
+  [simp]: "dup k = k + k"
+
+lemma dup_code [code]:
+  "dup 0 = 0"
+  "dup (Pos n) = Pos (Num.Bit0 n)"
+  "dup (Neg n) = Neg (Num.Bit0 n)"
+  unfolding Pos_def Neg_def neg_numeral_def
+  by (simp_all add: numeral_Bit0)
+
+definition sub :: "num \<Rightarrow> num \<Rightarrow> integer"
+where
+  [simp]: "sub m n = numeral m - numeral n"
+
+lemma sub_code [code]:
+  "sub Num.One Num.One = 0"
+  "sub (Num.Bit0 m) Num.One = Pos (Num.BitM m)"
+  "sub (Num.Bit1 m) Num.One = Pos (Num.Bit0 m)"
+  "sub Num.One (Num.Bit0 n) = Neg (Num.BitM n)"
+  "sub Num.One (Num.Bit1 n) = Neg (Num.Bit0 n)"
+  "sub (Num.Bit0 m) (Num.Bit0 n) = dup (sub m n)"
+  "sub (Num.Bit1 m) (Num.Bit1 n) = dup (sub m n)"
+  "sub (Num.Bit1 m) (Num.Bit0 n) = dup (sub m n) + 1"
+  "sub (Num.Bit0 m) (Num.Bit1 n) = dup (sub m n) - 1"
+  unfolding sub_def dup_def numeral.simps Pos_def Neg_def
+    neg_numeral_def numeral_BitM
+  by (simp_all only: algebra_simps add.comm_neutral)
+
+
+text {* Implementations *}
+
+lemma one_integer_code [code, code_unfold]:
+  "1 = Pos Num.One"
+  by simp
+
+lemma plus_integer_code [code]:
+  "k + 0 = (k::integer)"
+  "0 + l = (l::integer)"
+  "Pos m + Pos n = Pos (m + n)"
+  "Pos m + Neg n = sub m n"
+  "Neg m + Pos n = sub n m"
+  "Neg m + Neg n = Neg (m + n)"
+  by simp_all
+
+lemma uminus_integer_code [code]:
+  "uminus 0 = (0::integer)"
+  "uminus (Pos m) = Neg m"
+  "uminus (Neg m) = Pos m"
+  by simp_all
+
+lemma minus_integer_code [code]:
+  "k - 0 = (k::integer)"
+  "0 - l = uminus (l::integer)"
+  "Pos m - Pos n = sub m n"
+  "Pos m - Neg n = Pos (m + n)"
+  "Neg m - Pos n = Neg (m + n)"
+  "Neg m - Neg n = sub n m"
+  by simp_all
+
+lemma abs_integer_code [code]:
+  "\<bar>k\<bar> = (if (k::integer) < 0 then - k else k)"
+  by simp
+
+lemma sgn_integer_code [code]:
+  "sgn k = (if k = 0 then 0 else if (k::integer) < 0 then - 1 else 1)"
+  by simp
+
+lemma times_integer_code [code]:
+  "k * 0 = (0::integer)"
+  "0 * l = (0::integer)"
+  "Pos m * Pos n = Pos (m * n)"
+  "Pos m * Neg n = Neg (m * n)"
+  "Neg m * Pos n = Neg (m * n)"
+  "Neg m * Neg n = Pos (m * n)"
+  by simp_all
+
+definition divmod_integer :: "integer \<Rightarrow> integer \<Rightarrow> integer \<times> integer"
+where
+  "divmod_integer k l = (k div l, k mod l)"
+
+lemma fst_divmod [simp]:
+  "fst (divmod_integer k l) = k div l"
+  by (simp add: divmod_integer_def)
+
+lemma snd_divmod [simp]:
+  "snd (divmod_integer k l) = k mod l"
+  by (simp add: divmod_integer_def)
+
+definition divmod_abs :: "integer \<Rightarrow> integer \<Rightarrow> integer \<times> integer"
+where
+  "divmod_abs k l = (\<bar>k\<bar> div \<bar>l\<bar>, \<bar>k\<bar> mod \<bar>l\<bar>)"
+
+lemma fst_divmod_abs [simp]:
+  "fst (divmod_abs k l) = \<bar>k\<bar> div \<bar>l\<bar>"
+  by (simp add: divmod_abs_def)
+
+lemma snd_divmod_abs [simp]:
+  "snd (divmod_abs k l) = \<bar>k\<bar> mod \<bar>l\<bar>"
+  by (simp add: divmod_abs_def)
+
+lemma divmod_abs_terminate_code [code]:
+  "divmod_abs (Neg k) (Neg l) = divmod_abs (Pos k) (Pos l)"
+  "divmod_abs (Neg k) (Pos l) = divmod_abs (Pos k) (Pos l)"
+  "divmod_abs (Pos k) (Neg l) = divmod_abs (Pos k) (Pos l)"
+  "divmod_abs j 0 = (0, \<bar>j\<bar>)"
+  "divmod_abs 0 j = (0, 0)"
+  by (simp_all add: prod_eq_iff)
+
+lemma divmod_abs_rec_code [code]:
+  "divmod_abs (Pos k) (Pos l) =
+    (let j = sub k l in
+       if j < 0 then (0, Pos k)
+       else let (q, r) = divmod_abs j (Pos l) in (q + 1, r))"
+  by (auto simp add: prod_eq_iff integer_eq_iff Let_def prod_case_beta
+    sub_non_negative sub_negative div_pos_pos_trivial mod_pos_pos_trivial div_pos_geq mod_pos_geq)
+
+lemma divmod_integer_code [code]: "divmod_integer k l =
+  (if k = 0 then (0, 0) else if l = 0 then (0, k) else
+  (apsnd \<circ> times \<circ> sgn) l (if sgn k = sgn l
+    then divmod_abs k l
+    else (let (r, s) = divmod_abs k l in
+      if s = 0 then (- r, 0) else (- r - 1, \<bar>l\<bar> - s))))"
+proof -
+  have aux1: "\<And>k l::int. sgn k = sgn l \<longleftrightarrow> k = 0 \<and> l = 0 \<or> 0 < l \<and> 0 < k \<or> l < 0 \<and> k < 0"
+    by (auto simp add: sgn_if)
+  have aux2: "\<And>q::int. - int_of_integer k = int_of_integer l * q \<longleftrightarrow> int_of_integer k = int_of_integer l * - q" by auto
+  show ?thesis
+    by (simp add: prod_eq_iff integer_eq_iff prod_case_beta aux1)
+      (auto simp add: zdiv_zminus1_eq_if zmod_zminus1_eq_if div_minus_right mod_minus_right aux2)
+qed
+
+lemma div_integer_code [code]:
+  "k div l = fst (divmod_integer k l)"
+  by simp
+
+lemma mod_integer_code [code]:
+  "k mod l = snd (divmod_integer k l)"
+  by simp
+
+lemma equal_integer_code [code]:
+  "HOL.equal 0 (0::integer) \<longleftrightarrow> True"
+  "HOL.equal 0 (Pos l) \<longleftrightarrow> False"
+  "HOL.equal 0 (Neg l) \<longleftrightarrow> False"
+  "HOL.equal (Pos k) 0 \<longleftrightarrow> False"
+  "HOL.equal (Pos k) (Pos l) \<longleftrightarrow> HOL.equal k l"
+  "HOL.equal (Pos k) (Neg l) \<longleftrightarrow> False"
+  "HOL.equal (Neg k) 0 \<longleftrightarrow> False"
+  "HOL.equal (Neg k) (Pos l) \<longleftrightarrow> False"
+  "HOL.equal (Neg k) (Neg l) \<longleftrightarrow> HOL.equal k l"
+  by (simp_all add: equal integer_eq_iff)
+
+lemma equal_integer_refl [code nbe]:
+  "HOL.equal (k::integer) k \<longleftrightarrow> True"
+  by (fact equal_refl)
+
+lemma less_eq_integer_code [code]:
+  "0 \<le> (0::integer) \<longleftrightarrow> True"
+  "0 \<le> Pos l \<longleftrightarrow> True"
+  "0 \<le> Neg l \<longleftrightarrow> False"
+  "Pos k \<le> 0 \<longleftrightarrow> False"
+  "Pos k \<le> Pos l \<longleftrightarrow> k \<le> l"
+  "Pos k \<le> Neg l \<longleftrightarrow> False"
+  "Neg k \<le> 0 \<longleftrightarrow> True"
+  "Neg k \<le> Pos l \<longleftrightarrow> True"
+  "Neg k \<le> Neg l \<longleftrightarrow> l \<le> k"
+  by (simp_all add: less_eq_integer_def)
+
+lemma less_integer_code [code]:
+  "0 < (0::integer) \<longleftrightarrow> False"
+  "0 < Pos l \<longleftrightarrow> True"
+  "0 < Neg l \<longleftrightarrow> False"
+  "Pos k < 0 \<longleftrightarrow> False"
+  "Pos k < Pos l \<longleftrightarrow> k < l"
+  "Pos k < Neg l \<longleftrightarrow> False"
+  "Neg k < 0 \<longleftrightarrow> True"
+  "Neg k < Pos l \<longleftrightarrow> True"
+  "Neg k < Neg l \<longleftrightarrow> l < k"
+  by (simp_all add: less_integer_def)
+
+definition integer_of_num :: "num \<Rightarrow> integer"
+where
+  "integer_of_num = numeral"
+
+lemma integer_of_num [code]:
+  "integer_of_num num.One = 1"
+  "integer_of_num (num.Bit0 n) = (let k = integer_of_num n in k + k)"
+  "integer_of_num (num.Bit1 n) = (let k = integer_of_num n in k + k + 1)"
+  by (simp_all only: Let_def) (simp_all only: integer_of_num_def numeral.simps)
+
+definition num_of_integer :: "integer \<Rightarrow> num"
+where
+  "num_of_integer = num_of_nat \<circ> nat_of_integer"
+
+lemma num_of_integer_code [code]:
+  "num_of_integer k = (if k \<le> 1 then Num.One
+     else let
+       (l, j) = divmod_integer k 2;
+       l' = num_of_integer l;
+       l'' = l' + l'
+     in if j = 0 then l'' else l'' + Num.One)"
+proof -
+  {
+    assume "int_of_integer k mod 2 = 1"
+    then have "nat (int_of_integer k mod 2) = nat 1" by simp
+    moreover assume *: "1 < int_of_integer k"
+    ultimately have **: "nat (int_of_integer k) mod 2 = 1" by (simp add: nat_mod_distrib)
+    have "num_of_nat (nat (int_of_integer k)) =
+      num_of_nat (2 * (nat (int_of_integer k) div 2) + nat (int_of_integer k) mod 2)"
+      by simp
+    then have "num_of_nat (nat (int_of_integer k)) =
+      num_of_nat (nat (int_of_integer k) div 2 + nat (int_of_integer k) div 2 + nat (int_of_integer k) mod 2)"
+      by (simp add: mult_2)
+    with ** have "num_of_nat (nat (int_of_integer k)) =
+      num_of_nat (nat (int_of_integer k) div 2 + nat (int_of_integer k) div 2 + 1)"
+      by simp
+  }
+  note aux = this
+  show ?thesis
+    by (auto simp add: num_of_integer_def nat_of_integer_def Let_def prod_case_beta
+      not_le integer_eq_iff less_eq_integer_def
+      nat_mult_distrib nat_div_distrib num_of_nat_One num_of_nat_plus_distrib
+       mult_2 [where 'a=nat] aux add_One)
+qed
+
+lemma nat_of_integer_code [code]:
+  "nat_of_integer k = (if k \<le> 0 then 0
+     else let
+       (l, j) = divmod_integer k 2;
+       l' = nat_of_integer l;
+       l'' = l' + l'
+     in if j = 0 then l'' else l'' + 1)"
+proof -
+  obtain j where "k = integer_of_int j"
+  proof
+    show "k = integer_of_int (int_of_integer k)" by simp
+  qed
+  moreover have "2 * (j div 2) = j - j mod 2"
+    by (simp add: zmult_div_cancel mult_commute)
+  ultimately show ?thesis
+    by (auto simp add: split_def Let_def mod_integer_def nat_of_integer_def not_le
+      nat_add_distrib [symmetric] Suc_nat_eq_nat_zadd1)
+qed
+
+lemma int_of_integer_code [code]:
+  "int_of_integer k = (if k < 0 then - (int_of_integer (- k))
+     else if k = 0 then 0
+     else let
+       (l, j) = divmod_integer k 2;
+       l' = 2 * int_of_integer l
+     in if j = 0 then l' else l' + 1)"
+  by (auto simp add: split_def Let_def integer_eq_iff zmult_div_cancel)
+
+lemma integer_of_int_code [code]:
+  "integer_of_int k = (if k < 0 then - (integer_of_int (- k))
+     else if k = 0 then 0
+     else let
+       (l, j) = divmod_int k 2;
+       l' = 2 * integer_of_int l
+     in if j = 0 then l' else l' + 1)"
+  by (auto simp add: split_def Let_def integer_eq_iff zmult_div_cancel)
+
+hide_const (open) Pos Neg sub dup divmod_abs
+
+
+subsection {* Serializer setup for target language integers *}
+
+code_reserved Eval abs
+
+code_type integer
+  (SML "IntInf.int")
+  (OCaml "Big'_int.big'_int")
+  (Haskell "Integer")
+  (Scala "BigInt")
+  (Eval "int")
+
+code_instance integer :: equal
+  (Haskell -)
+
+code_const "0::integer"
+  (SML "0")
+  (OCaml "Big'_int.zero'_big'_int")
+  (Haskell "0")
+  (Scala "BigInt(0)")
+
+setup {*
+  fold (Numeral.add_code @{const_name Code_Numeral_Types.Pos}
+    false Code_Printer.literal_numeral) ["SML", "OCaml", "Haskell", "Scala"]
+*}
+
+setup {*
+  fold (Numeral.add_code @{const_name Code_Numeral_Types.Neg}
+    true Code_Printer.literal_numeral) ["SML", "OCaml", "Haskell", "Scala"]
+*}
+
+code_const "plus :: integer \<Rightarrow> _ \<Rightarrow> _"
+  (SML "IntInf.+ ((_), (_))")
+  (OCaml "Big'_int.add'_big'_int")
+  (Haskell infixl 6 "+")
+  (Scala infixl 7 "+")
+  (Eval infixl 8 "+")
+
+code_const "uminus :: integer \<Rightarrow> _"
+  (SML "IntInf.~")
+  (OCaml "Big'_int.minus'_big'_int")
+  (Haskell "negate")
+  (Scala "!(- _)")
+  (Eval "~/ _")
+
+code_const "minus :: integer \<Rightarrow> _"
+  (SML "IntInf.- ((_), (_))")
+  (OCaml "Big'_int.sub'_big'_int")
+  (Haskell infixl 6 "-")
+  (Scala infixl 7 "-")
+  (Eval infixl 8 "-")
+
+code_const Code_Numeral_Types.dup
+  (SML "IntInf.*/ (2,/ (_))")
+  (OCaml "Big'_int.mult'_big'_int/ (Big'_int.big'_int'_of'_int/ 2)")
+  (Haskell "!(2 * _)")
+  (Scala "!(2 * _)")
+  (Eval "!(2 * _)")
+
+code_const Code_Numeral_Types.sub
+  (SML "!(raise/ Fail/ \"sub\")")
+  (OCaml "failwith/ \"sub\"")
+  (Haskell "error/ \"sub\"")
+  (Scala "!sys.error(\"sub\")")
+
+code_const "times :: integer \<Rightarrow> _ \<Rightarrow> _"
+  (SML "IntInf.* ((_), (_))")
+  (OCaml "Big'_int.mult'_big'_int")
+  (Haskell infixl 7 "*")
+  (Scala infixl 8 "*")
+  (Eval infixl 9 "*")
+
+code_const Code_Numeral_Types.divmod_abs
+  (SML "IntInf.divMod/ (IntInf.abs _,/ IntInf.abs _)")
+  (OCaml "Big'_int.quomod'_big'_int/ (Big'_int.abs'_big'_int _)/ (Big'_int.abs'_big'_int _)")
+  (Haskell "divMod/ (abs _)/ (abs _)")
+  (Scala "!((k: BigInt) => (l: BigInt) =>/ if (l == 0)/ (BigInt(0), k) else/ (k.abs '/% l.abs))")
+  (Eval "Integer.div'_mod/ (abs _)/ (abs _)")
+
+code_const "HOL.equal :: integer \<Rightarrow> _ \<Rightarrow> bool"
+  (SML "!((_ : IntInf.int) = _)")
+  (OCaml "Big'_int.eq'_big'_int")
+  (Haskell infix 4 "==")
+  (Scala infixl 5 "==")
+  (Eval infixl 6 "=")
+
+code_const "less_eq :: integer \<Rightarrow> _ \<Rightarrow> bool"
+  (SML "IntInf.<= ((_), (_))")
+  (OCaml "Big'_int.le'_big'_int")
+  (Haskell infix 4 "<=")
+  (Scala infixl 4 "<=")
+  (Eval infixl 6 "<=")
+
+code_const "less :: integer \<Rightarrow> _ \<Rightarrow> bool"
+  (SML "IntInf.< ((_), (_))")
+  (OCaml "Big'_int.lt'_big'_int")
+  (Haskell infix 4 "<")
+  (Scala infixl 4 "<")
+  (Eval infixl 6 "<")
+
+code_modulename SML
+  Code_Numeral_Types Arith
+
+code_modulename OCaml
+  Code_Numeral_Types Arith
+
+code_modulename Haskell
+  Code_Numeral_Types Arith
+
+
+subsection {* Type of target language naturals *}
+
+typedef natural = "UNIV \<Colon> nat set"
+  morphisms nat_of_natural natural_of_nat ..
+
+lemma natural_eq_iff [termination_simp]:
+  "m = n \<longleftrightarrow> nat_of_natural m = nat_of_natural n"
+  using nat_of_natural_inject [of m n] ..
+
+lemma natural_eqI:
+  "nat_of_natural m = nat_of_natural n \<Longrightarrow> m = n"
+  using natural_eq_iff [of m n] by simp
+
+lemma nat_of_natural_of_nat_inverse [simp]:
+  "nat_of_natural (natural_of_nat n) = n"
+  using natural_of_nat_inverse [of n] by simp
+
+lemma natural_of_nat_of_natural_inverse [simp]:
+  "natural_of_nat (nat_of_natural n) = n"
+  using nat_of_natural_inverse [of n] by simp
+
+instantiation natural :: "{comm_monoid_diff, semiring_1}"
+begin
+
+definition
+  "0 = natural_of_nat 0"
+
+lemma nat_of_natural_zero [simp]:
+  "nat_of_natural 0 = 0"
+  by (simp add: zero_natural_def)
+
+definition
+  "1 = natural_of_nat 1"
+
+lemma nat_of_natural_one [simp]:
+  "nat_of_natural 1 = 1"
+  by (simp add: one_natural_def)
+
+definition
+  "m + n = natural_of_nat (nat_of_natural m + nat_of_natural n)"
+
+lemma nat_of_natural_plus [simp]:
+  "nat_of_natural (m + n) = nat_of_natural m + nat_of_natural n"
+  by (simp add: plus_natural_def)
+
+definition
+  "m - n = natural_of_nat (nat_of_natural m - nat_of_natural n)"
+
+lemma nat_of_natural_minus [simp]:
+  "nat_of_natural (m - n) = nat_of_natural m - nat_of_natural n"
+  by (simp add: minus_natural_def)
+
+definition
+  "m * n = natural_of_nat (nat_of_natural m * nat_of_natural n)"
+
+lemma nat_of_natural_times [simp]:
+  "nat_of_natural (m * n) = nat_of_natural m * nat_of_natural n"
+  by (simp add: times_natural_def)
+
+instance proof
+qed (auto simp add: natural_eq_iff algebra_simps)
+
+end
+
+lemma nat_of_natural_of_nat [simp]:
+  "nat_of_natural (of_nat n) = n"
+  by (induct n) simp_all
+
+lemma natural_of_nat_of_nat [simp, code_abbrev]:
+  "natural_of_nat = of_nat"
+  by rule (simp add: natural_eq_iff)
+
+lemma of_nat_of_natural [simp]:
+  "of_nat (nat_of_natural n) = n"
+  using natural_of_nat_of_natural_inverse [of n] by simp
+
+lemma nat_of_natural_numeral [simp]:
+  "nat_of_natural (numeral k) = numeral k"
+  using nat_of_natural_of_nat [of "numeral k"] by simp
+
+instantiation natural :: "{semiring_div, equal, linordered_semiring}"
+begin
+
+definition
+  "m div n = natural_of_nat (nat_of_natural m div nat_of_natural n)"
+
+lemma nat_of_natural_div [simp]:
+  "nat_of_natural (m div n) = nat_of_natural m div nat_of_natural n"
+  by (simp add: div_natural_def)
+
+definition
+  "m mod n = natural_of_nat (nat_of_natural m mod nat_of_natural n)"
+
+lemma nat_of_natural_mod [simp]:
+  "nat_of_natural (m mod n) = nat_of_natural m mod nat_of_natural n"
+  by (simp add: mod_natural_def)
+
+definition
+  [termination_simp]: "m \<le> n \<longleftrightarrow> nat_of_natural m \<le> nat_of_natural n"
+
+definition
+  [termination_simp]: "m < n \<longleftrightarrow> nat_of_natural m < nat_of_natural n"
+
+definition
+  "HOL.equal m n \<longleftrightarrow> HOL.equal (nat_of_natural m) (nat_of_natural n)"
+
+instance proof
+qed (auto simp add: natural_eq_iff algebra_simps
+  less_eq_natural_def less_natural_def equal_natural_def equal)
+
+end
+
+lemma nat_of_natural_min [simp]:
+  "nat_of_natural (min k l) = min (nat_of_natural k) (nat_of_natural l)"
+  by (simp add: min_def less_eq_natural_def)
+
+lemma nat_of_natural_max [simp]:
+  "nat_of_natural (max k l) = max (nat_of_natural k) (nat_of_natural l)"
+  by (simp add: max_def less_eq_natural_def)
+
+definition natural_of_integer :: "integer \<Rightarrow> natural"
+where
+  "natural_of_integer = of_nat \<circ> nat_of_integer"
+
+definition integer_of_natural :: "natural \<Rightarrow> integer"
+where
+  "integer_of_natural = of_nat \<circ> nat_of_natural"
+
+lemma natural_of_integer_of_natural [simp]:
+  "natural_of_integer (integer_of_natural n) = n"
+  by (simp add: natural_of_integer_def integer_of_natural_def natural_eq_iff)
+
+lemma integer_of_natural_of_integer [simp]:
+  "integer_of_natural (natural_of_integer k) = max 0 k"
+  by (simp add: natural_of_integer_def integer_of_natural_def integer_eq_iff)
+
+lemma int_of_integer_of_natural [simp]:
+  "int_of_integer (integer_of_natural n) = of_nat (nat_of_natural n)"
+  by (simp add: integer_of_natural_def)
+
+lemma integer_of_natural_of_nat [simp]:
+  "integer_of_natural (of_nat n) = of_nat n"
+  by (simp add: integer_eq_iff)
+
+lemma [measure_function]:
+  "is_measure nat_of_natural" by (rule is_measure_trivial)
+
+
+subsection {* Inductive represenation of target language naturals *}
+
+definition Suc :: "natural \<Rightarrow> natural"
+where
+  "Suc = natural_of_nat \<circ> Nat.Suc \<circ> nat_of_natural"
+
+lemma nat_of_natural_Suc [simp]:
+  "nat_of_natural (Suc n) = Nat.Suc (nat_of_natural n)"
+  by (simp add: Suc_def)
+
+rep_datatype "0::natural" Suc
+proof -
+  fix P :: "natural \<Rightarrow> bool"
+  fix n :: natural
+  assume "P 0" then have init: "P (natural_of_nat 0)" by simp
+  assume "\<And>n. P n \<Longrightarrow> P (Suc n)"
+    then have "\<And>n. P (natural_of_nat n) \<Longrightarrow> P (Suc (natural_of_nat n))" .
+    then have step: "\<And>n. P (natural_of_nat n) \<Longrightarrow> P (natural_of_nat (Nat.Suc n))"
+      by (simp add: Suc_def)
+  from init step have "P (natural_of_nat (nat_of_natural n))"
+    by (rule nat.induct)
+  with natural_of_nat_of_natural_inverse show "P n" by simp
+qed (simp_all add: natural_eq_iff)
+
+lemma natural_case [case_names nat, cases type: natural]:
+  fixes m :: natural
+  assumes "\<And>n. m = of_nat n \<Longrightarrow> P"
+  shows P
+  by (rule assms [of "nat_of_natural m"]) simp
+
+lemma [simp, code]:
+  "natural_size = nat_of_natural"
+proof (rule ext)
+  fix n
+  show "natural_size n = nat_of_natural n"
+    by (induct n) simp_all
+qed
+
+lemma [simp, code]:
+  "size = nat_of_natural"
+proof (rule ext)
+  fix n
+  show "size n = nat_of_natural n"
+    by (induct n) simp_all
+qed
+
+lemma natural_decr [termination_simp]:
+  "n \<noteq> 0 \<Longrightarrow> nat_of_natural n - Nat.Suc 0 < nat_of_natural n"
+  by (simp add: natural_eq_iff)
+
+lemma natural_zero_minus_one:
+  "(0::natural) - 1 = 0"
+  by simp
+
+lemma Suc_natural_minus_one:
+  "Suc n - 1 = n"
+  by (simp add: natural_eq_iff)
+
+hide_const (open) Suc
+
+
+subsection {* Code refinement for target language naturals *}
+
+definition Nat :: "integer \<Rightarrow> natural"
+where
+  "Nat = natural_of_integer"
+
+lemma [code abstype]:
+  "Nat (integer_of_natural n) = n"
+  by (unfold Nat_def) (fact natural_of_integer_of_natural)
+
+lemma [code abstract]:
+  "integer_of_natural (natural_of_nat n) = of_nat n"
+  by simp
+
+lemma [code abstract]:
+  "integer_of_natural (natural_of_integer k) = max 0 k"
+  by simp
+
+lemma [code_abbrev]:
+  "natural_of_integer (Code_Numeral_Types.Pos k) = numeral k"
+  by (simp add: nat_of_integer_def natural_of_integer_def)
+
+lemma [code abstract]:
+  "integer_of_natural 0 = 0"
+  by (simp add: integer_eq_iff)
+
+lemma [code abstract]:
+  "integer_of_natural 1 = 1"
+  by (simp add: integer_eq_iff)
+
+lemma [code abstract]:
+  "integer_of_natural (Code_Numeral_Types.Suc n) = integer_of_natural n + 1"
+  by (simp add: integer_eq_iff)
+
+lemma [code]:
+  "nat_of_natural = nat_of_integer \<circ> integer_of_natural"
+  by (simp add: integer_of_natural_def fun_eq_iff)
+
+lemma [code, code_unfold]:
+  "natural_case f g n = (if n = 0 then f else g (n - 1))"
+  by (cases n rule: natural.exhaust) (simp_all, simp add: Suc_def)
+
+declare natural.recs [code del]
+
+lemma [code abstract]:
+  "integer_of_natural (m + n) = integer_of_natural m + integer_of_natural n"
+  by (simp add: integer_eq_iff)
+
+lemma [code abstract]:
+  "integer_of_natural (m - n) = max 0 (integer_of_natural m - integer_of_natural n)"
+  by (simp add: integer_eq_iff)
+
+lemma [code abstract]:
+  "integer_of_natural (m * n) = integer_of_natural m * integer_of_natural n"
+  by (simp add: integer_eq_iff of_nat_mult)
+
+lemma [code abstract]:
+  "integer_of_natural (m div n) = integer_of_natural m div integer_of_natural n"
+  by (simp add: integer_eq_iff zdiv_int)
+
+lemma [code abstract]:
+  "integer_of_natural (m mod n) = integer_of_natural m mod integer_of_natural n"
+  by (simp add: integer_eq_iff zmod_int)
+
+lemma [code]:
+  "HOL.equal m n \<longleftrightarrow> HOL.equal (integer_of_natural m) (integer_of_natural n)"
+  by (simp add: equal natural_eq_iff integer_eq_iff)
+
+lemma [code nbe]:
+  "HOL.equal n (n::natural) \<longleftrightarrow> True"
+  by (simp add: equal)
+
+lemma [code]:
+  "m \<le> n \<longleftrightarrow> (integer_of_natural m) \<le> integer_of_natural n"
+  by (simp add: less_eq_natural_def less_eq_integer_def)
+
+lemma [code]:
+  "m < n \<longleftrightarrow> (integer_of_natural m) < integer_of_natural n"
+  by (simp add: less_natural_def less_integer_def)
+
+hide_const (open) Nat
+
+
+code_reflect Code_Numeral_Types
+  datatypes natural = _
+  functions integer_of_natural natural_of_integer
+
+end
+
diff -r 286dfcab9833 -r 28f3263d4d1b src/HOL/Library/Code_Target_Int.thy
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Library/Code_Target_Int.thy	Thu Nov 08 10:02:38 2012 +0100
@@ -0,0 +1,122 @@
+(*  Title:      HOL/Library/Code_Target_Int.thy
+    Author:     Florian Haftmann, TU Muenchen
+*)
+
+header {* Implementation of integer numbers by target-language integers *}
+
+theory Code_Target_Int
+imports Main "~~/src/HOL/Library/Code_Numeral_Types"
+begin
+
+code_datatype int_of_integer
+
+lemma [code, code del]:
+  "integer_of_int = integer_of_int" ..
+
+lemma [code]:
+  "integer_of_int (int_of_integer k) = k"
+  by (simp add: integer_eq_iff)
+
+lemma [code]:
+  "Int.Pos = int_of_integer \<circ> integer_of_num"
+  by (simp add: integer_of_num_def fun_eq_iff)
+
+lemma [code]:
+  "Int.Neg = int_of_integer \<circ> uminus \<circ> integer_of_num"
+  by (simp add: integer_of_num_def fun_eq_iff)
+
+lemma [code_abbrev]:
+  "int_of_integer (Code_Numeral_Types.Pos k) = Int.Pos k"
+  by simp
+
+lemma [code_abbrev]:
+  "int_of_integer (Code_Numeral_Types.Neg k) = Int.Neg k"
+  by simp
+
+lemma [code]:
+  "0 = int_of_integer 0"
+  by simp
+
+lemma [code]:
+  "1 = int_of_integer 1"
+  by simp
+
+lemma [code]:
+  "k + l = int_of_integer (of_int k + of_int l)"
+  by simp
+
+lemma [code]:
+  "- k = int_of_integer (- of_int k)"
+  by simp
+
+lemma [code]:
+  "k - l = int_of_integer (of_int k - of_int l)"
+  by simp
+
+lemma [code]:
+  "Int.dup k = int_of_integer (Code_Numeral_Types.dup (of_int k))"
+  by simp
+
+lemma [code, code del]:
+  "Int.sub = Int.sub" ..
+
+lemma [code]:
+  "k * l = int_of_integer (of_int k * of_int l)"
+  by simp
+
+lemma [code]:
+  "pdivmod k l = map_pair int_of_integer int_of_integer
+    (Code_Numeral_Types.divmod_abs (of_int k) (of_int l))"
+  by (simp add: prod_eq_iff pdivmod_def)
+
+lemma [code]:
+  "k div l = int_of_integer (of_int k div of_int l)"
+  by simp
+
+lemma [code]:
+  "k mod l = int_of_integer (of_int k mod of_int l)"
+  by simp
+
+lemma [code]:
+  "HOL.equal k l = HOL.equal (of_int k :: integer) (of_int l)"
+  by (simp add: equal integer_eq_iff)
+
+lemma [code]:
+  "k \<le> l \<longleftrightarrow> (of_int k :: integer) \<le> of_int l"
+  by (simp add: less_eq_int_def)
+
+lemma [code]:
+  "k < l \<longleftrightarrow> (of_int k :: integer) < of_int l"
+  by (simp add: less_int_def)
+
+lemma (in ring_1) of_int_code:
+  "of_int k = (if k = 0 then 0
+     else if k < 0 then - of_int (- k)
+     else let
+       (l, j) = divmod_int k 2;
+       l' = 2 * of_int l
+     in if j = 0 then l' else l' + 1)"
+proof -
+  from mod_div_equality have *: "of_int k = of_int (k div 2 * 2 + k mod 2)" by simp
+  show ?thesis
+    by (simp add: Let_def divmod_int_mod_div mod_2_not_eq_zero_eq_one_int
+      of_int_add [symmetric]) (simp add: * mult_commute)
+qed
+
+declare of_int_code [code]
+
+lemma [code]:
+  "nat = nat_of_integer \<circ> of_int"
+  by (simp add: fun_eq_iff nat_of_integer_def)
+
+code_modulename SML
+  Code_Target_Int Arith
+
+code_modulename OCaml
+  Code_Target_Int Arith
+
+code_modulename Haskell
+  Code_Target_Int Arith
+
+end
+
diff -r 286dfcab9833 -r 28f3263d4d1b src/HOL/Library/Code_Target_Nat.thy
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Library/Code_Target_Nat.thy	Thu Nov 08 10:02:38 2012 +0100
@@ -0,0 +1,137 @@
+(*  Title:      HOL/Library/Code_Target_Nat.thy
+    Author:     Stefan Berghofer, Florian Haftmann, TU Muenchen
+*)
+
+header {* Implementation of natural numbers by target-language integers *}
+
+theory Code_Target_Nat
+imports Main Code_Numeral_Types Code_Binary_Nat
+begin
+
+subsection {* Implementation for @{typ nat} *}
+
+definition Nat :: "integer \<Rightarrow> nat"
+where
+  "Nat = nat_of_integer"
+
+definition integer_of_nat :: "nat \<Rightarrow> integer"
+where
+  [code_abbrev]: "integer_of_nat = of_nat"
+
+lemma int_of_integer_integer_of_nat [simp]:
+  "int_of_integer (integer_of_nat n) = of_nat n"
+  by (simp add: integer_of_nat_def)
+
+lemma [code_unfold]:
+  "Int.nat (int_of_integer k) = nat_of_integer k"
+  by (simp add: nat_of_integer_def)
+
+lemma [code abstype]:
+  "Code_Target_Nat.Nat (integer_of_nat n) = n"
+  by (simp add: Nat_def integer_of_nat_def)
+
+lemma [code abstract]:
+  "integer_of_nat (nat_of_integer k) = max 0 k"
+  by (simp add: integer_of_nat_def)
+
+lemma [code_abbrev]:
+  "nat_of_integer (Code_Numeral_Types.Pos k) = nat_of_num k"
+  by (simp add: nat_of_integer_def nat_of_num_numeral)
+
+lemma [code abstract]:
+  "integer_of_nat (nat_of_num n) = integer_of_num n"
+  by (simp add: integer_eq_iff integer_of_num_def nat_of_num_numeral)
+
+lemma [code abstract]:
+  "integer_of_nat 0 = 0"
+  by (simp add: integer_eq_iff integer_of_nat_def)
+
+lemma [code abstract]:
+  "integer_of_nat 1 = 1"
+  by (simp add: integer_eq_iff integer_of_nat_def)
+
+lemma [code abstract]:
+  "integer_of_nat (m + n) = of_nat m + of_nat n"
+  by (simp add: integer_eq_iff integer_of_nat_def)
+
+lemma [code abstract]:
+  "integer_of_nat (Code_Binary_Nat.dup n) = Code_Numeral_Types.dup (of_nat n)"
+  by (simp add: integer_eq_iff Code_Binary_Nat.dup_def integer_of_nat_def)
+
+lemma [code, code del]:
+  "Code_Binary_Nat.sub = Code_Binary_Nat.sub" ..
+
+lemma [code abstract]:
+  "integer_of_nat (m - n) = max 0 (of_nat m - of_nat n)"
+  by (simp add: integer_eq_iff integer_of_nat_def)
+
+lemma [code abstract]:
+  "integer_of_nat (m * n) = of_nat m * of_nat n"
+  by (simp add: integer_eq_iff of_nat_mult integer_of_nat_def)
+
+lemma [code abstract]:
+  "integer_of_nat (m div n) = of_nat m div of_nat n"
+  by (simp add: integer_eq_iff zdiv_int integer_of_nat_def)
+
+lemma [code abstract]:
+  "integer_of_nat (m mod n) = of_nat m mod of_nat n"
+  by (simp add: integer_eq_iff zmod_int integer_of_nat_def)
+
+lemma [code]:
+  "Divides.divmod_nat m n = (m div n, m mod n)"
+  by (simp add: prod_eq_iff)
+
+lemma [code]:
+  "HOL.equal m n = HOL.equal (of_nat m :: integer) (of_nat n)"
+  by (simp add: equal integer_eq_iff)
+
+lemma [code]:
+  "m \<le> n \<longleftrightarrow> (of_nat m :: integer) \<le> of_nat n"
+  by simp
+
+lemma [code]:
+  "m < n \<longleftrightarrow> (of_nat m :: integer) < of_nat n"
+  by simp
+
+lemma num_of_nat_code [code]:
+  "num_of_nat = num_of_integer \<circ> of_nat"
+  by (simp add: fun_eq_iff num_of_integer_def integer_of_nat_def)
+
+lemma (in semiring_1) of_nat_code:
+  "of_nat n = (if n = 0 then 0
+     else let
+       (m, q) = divmod_nat n 2;
+       m' = 2 * of_nat m
+     in if q = 0 then m' else m' + 1)"
+proof -
+  from mod_div_equality have *: "of_nat n = of_nat (n div 2 * 2 + n mod 2)" by simp
+  show ?thesis
+    by (simp add: Let_def divmod_nat_div_mod mod_2_not_eq_zero_eq_one_nat
+      of_nat_add [symmetric])
+      (simp add: * mult_commute of_nat_mult add_commute)
+qed
+
+declare of_nat_code [code]
+
+definition int_of_nat :: "nat \<Rightarrow> int" where
+  [code_abbrev]: "int_of_nat = of_nat"
+
+lemma [code]:
+  "int_of_nat n = int_of_integer (of_nat n)"
+  by (simp add: int_of_nat_def)
+
+lemma [code abstract]:
+  "integer_of_nat (nat k) = max 0 (integer_of_int k)"
+  by (simp add: integer_of_nat_def of_int_of_nat max_def)
+
+code_modulename SML
+  Code_Target_Nat Arith
+
+code_modulename OCaml
+  Code_Target_Nat Arith
+
+code_modulename Haskell
+  Code_Target_Nat Arith
+
+end
+
diff -r 286dfcab9833 -r 28f3263d4d1b src/HOL/Library/Code_Target_Numeral.thy
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Library/Code_Target_Numeral.thy	Thu Nov 08 10:02:38 2012 +0100
@@ -0,0 +1,12 @@
+(*  Title:      HOL/Library/Code_Target_Numeral.thy
+    Author:     Florian Haftmann, TU Muenchen
+*)
+
+header {* Implementation of natural and integer numbers by target-language integers *}
+
+theory Code_Target_Numeral
+imports Code_Target_Int Code_Target_Nat
+begin
+
+end
+
diff -r 286dfcab9833 -r 28f3263d4d1b src/HOL/Library/Efficient_Nat.thy
--- a/src/HOL/Library/Efficient_Nat.thy	Wed Nov 07 20:48:04 2012 +0100
+++ b/src/HOL/Library/Efficient_Nat.thy	Thu Nov 08 10:02:38 2012 +0100
@@ -5,7 +5,7 @@
 header {* Implementation of natural numbers by target-language integers *}
 
 theory Efficient_Nat
-imports Code_Nat Code_Integer Main
+imports Code_Binary_Nat Code_Integer Main
 begin
 
 text {*
@@ -217,14 +217,14 @@
   (Scala infixl 7 "-")
   (Eval "Integer.max/ 0/ (_ -/ _)")
 
-code_const Code_Nat.dup
+code_const Code_Binary_Nat.dup
   (SML "IntInf.*/ (2,/ (_))")
   (OCaml "Big'_int.mult'_big'_int/ 2")
   (Haskell "!(2 * _)")
   (Scala "!(2 * _)")
   (Eval "!(2 * _)")
 
-code_const Code_Nat.sub
+code_const Code_Binary_Nat.sub
   (SML "!(raise/ Fail/ \"sub\")")
   (OCaml "failwith/ \"sub\"")
   (Haskell "error/ \"sub\"")
@@ -302,3 +302,4 @@
 hide_const (open) int
 
 end
+
diff -r 286dfcab9833 -r 28f3263d4d1b src/HOL/Library/Target_Numeral.thy
--- a/src/HOL/Library/Target_Numeral.thy	Wed Nov 07 20:48:04 2012 +0100
+++ /dev/null	Thu Jan 01 00:00:00 1970 +0000
@@ -1,742 +0,0 @@
-theory Target_Numeral
-imports Main Code_Nat
-begin
-
-subsection {* Type of target language numerals *}
-
-typedef int = "UNIV \<Colon> int set"
-  morphisms int_of of_int ..
-
-hide_type (open) int
-hide_const (open) of_int
-
-lemma int_eq_iff:
-  "k = l \<longleftrightarrow> int_of k = int_of l"
-  using int_of_inject [of k l] ..
-
-lemma int_eqI:
-  "int_of k = int_of l \<Longrightarrow> k = l"
-  using int_eq_iff [of k l] by simp
-
-lemma int_of_int [simp]:
-  "int_of (Target_Numeral.of_int k) = k"
-  using of_int_inverse [of k] by simp
-
-lemma of_int_of [simp]:
-  "Target_Numeral.of_int (int_of k) = k"
-  using int_of_inverse [of k] by simp
-
-hide_fact (open) int_eq_iff int_eqI
-
-instantiation Target_Numeral.int :: ring_1
-begin
-
-definition
-  "0 = Target_Numeral.of_int 0"
-
-lemma int_of_zero [simp]:
-  "int_of 0 = 0"
-  by (simp add: zero_int_def)
-
-definition
-  "1 = Target_Numeral.of_int 1"
-
-lemma int_of_one [simp]:
-  "int_of 1 = 1"
-  by (simp add: one_int_def)
-
-definition
-  "k + l = Target_Numeral.of_int (int_of k + int_of l)"
-
-lemma int_of_plus [simp]:
-  "int_of (k + l) = int_of k + int_of l"
-  by (simp add: plus_int_def)
-
-definition
-  "- k = Target_Numeral.of_int (- int_of k)"
-
-lemma int_of_uminus [simp]:
-  "int_of (- k) = - int_of k"
-  by (simp add: uminus_int_def)
-
-definition
-  "k - l = Target_Numeral.of_int (int_of k - int_of l)"
-
-lemma int_of_minus [simp]:
-  "int_of (k - l) = int_of k - int_of l"
-  by (simp add: minus_int_def)
-
-definition
-  "k * l = Target_Numeral.of_int (int_of k * int_of l)"
-
-lemma int_of_times [simp]:
-  "int_of (k * l) = int_of k * int_of l"
-  by (simp add: times_int_def)
-
-instance proof
-qed (auto simp add: Target_Numeral.int_eq_iff algebra_simps)
-
-end
-
-lemma int_of_of_nat [simp]:
-  "int_of (of_nat n) = of_nat n"
-  by (induct n) simp_all
-
-definition nat_of :: "Target_Numeral.int \<Rightarrow> nat" where
-  "nat_of k = Int.nat (int_of k)"
-
-lemma nat_of_of_nat [simp]:
-  "nat_of (of_nat n) = n"
-  by (simp add: nat_of_def)
-
-lemma int_of_of_int [simp]:
-  "int_of (of_int k) = k"
-  by (induct k) (simp_all, simp only: neg_numeral_def numeral_One int_of_uminus int_of_one)
-
-lemma of_int_of_int [simp, code_abbrev]:
-  "Target_Numeral.of_int = of_int"
-  by rule (simp add: Target_Numeral.int_eq_iff)
-
-lemma int_of_numeral [simp]:
-  "int_of (numeral k) = numeral k"
-  using int_of_of_int [of "numeral k"] by simp
-
-lemma int_of_neg_numeral [simp]:
-  "int_of (neg_numeral k) = neg_numeral k"
-  by (simp only: neg_numeral_def int_of_uminus) simp
-
-lemma int_of_sub [simp]:
-  "int_of (Num.sub k l) = Num.sub k l"
-  by (simp only: Num.sub_def int_of_minus int_of_numeral)
-
-instantiation Target_Numeral.int :: "{ring_div, equal, linordered_idom}"
-begin
-
-definition
-  "k div l = of_int (int_of k div int_of l)"
-
-lemma int_of_div [simp]:
-  "int_of (k div l) = int_of k div int_of l"
-  by (simp add: div_int_def)
-
-definition
-  "k mod l = of_int (int_of k mod int_of l)"
-
-lemma int_of_mod [simp]:
-  "int_of (k mod l) = int_of k mod int_of l"
-  by (simp add: mod_int_def)
-
-definition
-  "\<bar>k\<bar> = of_int \<bar>int_of k\<bar>"
-
-lemma int_of_abs [simp]:
-  "int_of \<bar>k\<bar> = \<bar>int_of k\<bar>"
-  by (simp add: abs_int_def)
-
-definition
-  "sgn k = of_int (sgn (int_of k))"
-
-lemma int_of_sgn [simp]:
-  "int_of (sgn k) = sgn (int_of k)"
-  by (simp add: sgn_int_def)
-
-definition
-  "k \<le> l \<longleftrightarrow> int_of k \<le> int_of l"
-
-definition
-  "k < l \<longleftrightarrow> int_of k < int_of l"
-
-definition
-  "HOL.equal k l \<longleftrightarrow> HOL.equal (int_of k) (int_of l)"
-
-instance proof
-qed (auto simp add: Target_Numeral.int_eq_iff algebra_simps
-  less_eq_int_def less_int_def equal_int_def equal)
-
-end
-
-lemma int_of_min [simp]:
-  "int_of (min k l) = min (int_of k) (int_of l)"
-  by (simp add: min_def less_eq_int_def)
-
-lemma int_of_max [simp]:
-  "int_of (max k l) = max (int_of k) (int_of l)"
-  by (simp add: max_def less_eq_int_def)
-
-lemma of_nat_nat_of [simp]:
-  "of_nat (nat_of k) = max 0 k"
-  by (simp add: nat_of_def Target_Numeral.int_eq_iff less_eq_int_def max_def)
-
-
-subsection {* Code theorems for target language numerals *}
-
-text {* Constructors *}
-
-definition Pos :: "num \<Rightarrow> Target_Numeral.int" where
-  [simp, code_abbrev]: "Pos = numeral"
-
-definition Neg :: "num \<Rightarrow> Target_Numeral.int" where
-  [simp, code_abbrev]: "Neg = neg_numeral"
-
-code_datatype "0::Target_Numeral.int" Pos Neg
-
-
-text {* Auxiliary operations *}
-
-definition dup :: "Target_Numeral.int \<Rightarrow> Target_Numeral.int" where
-  [simp]: "dup k = k + k"
-
-lemma dup_code [code]:
-  "dup 0 = 0"
-  "dup (Pos n) = Pos (Num.Bit0 n)"
-  "dup (Neg n) = Neg (Num.Bit0 n)"
-  unfolding Pos_def Neg_def neg_numeral_def
-  by (simp_all add: numeral_Bit0)
-
-definition sub :: "num \<Rightarrow> num \<Rightarrow> Target_Numeral.int" where
-  [simp]: "sub m n = numeral m - numeral n"
-
-lemma sub_code [code]:
-  "sub Num.One Num.One = 0"
-  "sub (Num.Bit0 m) Num.One = Pos (Num.BitM m)"
-  "sub (Num.Bit1 m) Num.One = Pos (Num.Bit0 m)"
-  "sub Num.One (Num.Bit0 n) = Neg (Num.BitM n)"
-  "sub Num.One (Num.Bit1 n) = Neg (Num.Bit0 n)"
-  "sub (Num.Bit0 m) (Num.Bit0 n) = dup (sub m n)"
-  "sub (Num.Bit1 m) (Num.Bit1 n) = dup (sub m n)"
-  "sub (Num.Bit1 m) (Num.Bit0 n) = dup (sub m n) + 1"
-  "sub (Num.Bit0 m) (Num.Bit1 n) = dup (sub m n) - 1"
-  unfolding sub_def dup_def numeral.simps Pos_def Neg_def
-    neg_numeral_def numeral_BitM
-  by (simp_all only: algebra_simps add.comm_neutral)
-
-
-text {* Implementations *}
-
-lemma one_int_code [code, code_unfold]:
-  "1 = Pos Num.One"
-  by simp
-
-lemma plus_int_code [code]:
-  "k + 0 = (k::Target_Numeral.int)"
-  "0 + l = (l::Target_Numeral.int)"
-  "Pos m + Pos n = Pos (m + n)"
-  "Pos m + Neg n = sub m n"
-  "Neg m + Pos n = sub n m"
-  "Neg m + Neg n = Neg (m + n)"
-  by simp_all
-
-lemma uminus_int_code [code]:
-  "uminus 0 = (0::Target_Numeral.int)"
-  "uminus (Pos m) = Neg m"
-  "uminus (Neg m) = Pos m"
-  by simp_all
-
-lemma minus_int_code [code]:
-  "k - 0 = (k::Target_Numeral.int)"
-  "0 - l = uminus (l::Target_Numeral.int)"
-  "Pos m - Pos n = sub m n"
-  "Pos m - Neg n = Pos (m + n)"
-  "Neg m - Pos n = Neg (m + n)"
-  "Neg m - Neg n = sub n m"
-  by simp_all
-
-lemma times_int_code [code]:
-  "k * 0 = (0::Target_Numeral.int)"
-  "0 * l = (0::Target_Numeral.int)"
-  "Pos m * Pos n = Pos (m * n)"
-  "Pos m * Neg n = Neg (m * n)"
-  "Neg m * Pos n = Neg (m * n)"
-  "Neg m * Neg n = Pos (m * n)"
-  by simp_all
-
-definition divmod :: "Target_Numeral.int \<Rightarrow> Target_Numeral.int \<Rightarrow> Target_Numeral.int \<times> Target_Numeral.int" where
-  "divmod k l = (k div l, k mod l)"
-
-lemma fst_divmod [simp]:
-  "fst (divmod k l) = k div l"
-  by (simp add: divmod_def)
-
-lemma snd_divmod [simp]:
-  "snd (divmod k l) = k mod l"
-  by (simp add: divmod_def)
-
-definition divmod_abs :: "Target_Numeral.int \<Rightarrow> Target_Numeral.int \<Rightarrow> Target_Numeral.int \<times> Target_Numeral.int" where
-  "divmod_abs k l = (\<bar>k\<bar> div \<bar>l\<bar>, \<bar>k\<bar> mod \<bar>l\<bar>)"
-
-lemma fst_divmod_abs [simp]:
-  "fst (divmod_abs k l) = \<bar>k\<bar> div \<bar>l\<bar>"
-  by (simp add: divmod_abs_def)
-
-lemma snd_divmod_abs [simp]:
-  "snd (divmod_abs k l) = \<bar>k\<bar> mod \<bar>l\<bar>"
-  by (simp add: divmod_abs_def)
-
-lemma divmod_abs_terminate_code [code]:
-  "divmod_abs (Neg k) (Neg l) = divmod_abs (Pos k) (Pos l)"
-  "divmod_abs (Neg k) (Pos l) = divmod_abs (Pos k) (Pos l)"
-  "divmod_abs (Pos k) (Neg l) = divmod_abs (Pos k) (Pos l)"
-  "divmod_abs j 0 = (0, \<bar>j\<bar>)"
-  "divmod_abs 0 j = (0, 0)"
-  by (simp_all add: prod_eq_iff)
-
-lemma divmod_abs_rec_code [code]:
-  "divmod_abs (Pos k) (Pos l) =
-    (let j = sub k l in
-       if j < 0 then (0, Pos k)
-       else let (q, r) = divmod_abs j (Pos l) in (q + 1, r))"
-  by (auto simp add: prod_eq_iff Target_Numeral.int_eq_iff Let_def prod_case_beta
-    sub_non_negative sub_negative div_pos_pos_trivial mod_pos_pos_trivial div_pos_geq mod_pos_geq)
-
-lemma divmod_code [code]: "divmod k l =
-  (if k = 0 then (0, 0) else if l = 0 then (0, k) else
-  (apsnd \<circ> times \<circ> sgn) l (if sgn k = sgn l
-    then divmod_abs k l
-    else (let (r, s) = divmod_abs k l in
-      if s = 0 then (- r, 0) else (- r - 1, \<bar>l\<bar> - s))))"
-proof -
-  have aux1: "\<And>k l::int. sgn k = sgn l \<longleftrightarrow> k = 0 \<and> l = 0 \<or> 0 < l \<and> 0 < k \<or> l < 0 \<and> k < 0"
-    by (auto simp add: sgn_if)
-  have aux2: "\<And>q::int. - int_of k = int_of l * q \<longleftrightarrow> int_of k = int_of l * - q" by auto
-  show ?thesis
-    by (simp add: prod_eq_iff Target_Numeral.int_eq_iff prod_case_beta aux1)
-      (auto simp add: zdiv_zminus1_eq_if zmod_zminus1_eq_if div_minus_right mod_minus_right aux2)
-qed
-
-lemma div_int_code [code]:
-  "k div l = fst (divmod k l)"
-  by simp
-
-lemma div_mod_code [code]:
-  "k mod l = snd (divmod k l)"
-  by simp
-
-lemma equal_int_code [code]:
-  "HOL.equal 0 (0::Target_Numeral.int) \<longleftrightarrow> True"
-  "HOL.equal 0 (Pos l) \<longleftrightarrow> False"
-  "HOL.equal 0 (Neg l) \<longleftrightarrow> False"
-  "HOL.equal (Pos k) 0 \<longleftrightarrow> False"
-  "HOL.equal (Pos k) (Pos l) \<longleftrightarrow> HOL.equal k l"
-  "HOL.equal (Pos k) (Neg l) \<longleftrightarrow> False"
-  "HOL.equal (Neg k) 0 \<longleftrightarrow> False"
-  "HOL.equal (Neg k) (Pos l) \<longleftrightarrow> False"
-  "HOL.equal (Neg k) (Neg l) \<longleftrightarrow> HOL.equal k l"
-  by (simp_all add: equal Target_Numeral.int_eq_iff)
-
-lemma equal_int_refl [code nbe]:
-  "HOL.equal (k::Target_Numeral.int) k \<longleftrightarrow> True"
-  by (fact equal_refl)
-
-lemma less_eq_int_code [code]:
-  "0 \<le> (0::Target_Numeral.int) \<longleftrightarrow> True"
-  "0 \<le> Pos l \<longleftrightarrow> True"
-  "0 \<le> Neg l \<longleftrightarrow> False"
-  "Pos k \<le> 0 \<longleftrightarrow> False"
-  "Pos k \<le> Pos l \<longleftrightarrow> k \<le> l"
-  "Pos k \<le> Neg l \<longleftrightarrow> False"
-  "Neg k \<le> 0 \<longleftrightarrow> True"
-  "Neg k \<le> Pos l \<longleftrightarrow> True"
-  "Neg k \<le> Neg l \<longleftrightarrow> l \<le> k"
-  by (simp_all add: less_eq_int_def)
-
-lemma less_int_code [code]:
-  "0 < (0::Target_Numeral.int) \<longleftrightarrow> False"
-  "0 < Pos l \<longleftrightarrow> True"
-  "0 < Neg l \<longleftrightarrow> False"
-  "Pos k < 0 \<longleftrightarrow> False"
-  "Pos k < Pos l \<longleftrightarrow> k < l"
-  "Pos k < Neg l \<longleftrightarrow> False"
-  "Neg k < 0 \<longleftrightarrow> True"
-  "Neg k < Pos l \<longleftrightarrow> True"
-  "Neg k < Neg l \<longleftrightarrow> l < k"
-  by (simp_all add: less_int_def)
-
-lemma nat_of_code [code]:
-  "nat_of (Neg k) = 0"
-  "nat_of 0 = 0"
-  "nat_of (Pos k) = nat_of_num k"
-  by (simp_all add: nat_of_def nat_of_num_numeral)
-
-lemma int_of_code [code]:
-  "int_of (Neg k) = neg_numeral k"
-  "int_of 0 = 0"
-  "int_of (Pos k) = numeral k"
-  by simp_all
-
-lemma of_int_code [code]:
-  "Target_Numeral.of_int (Int.Neg k) = neg_numeral k"
-  "Target_Numeral.of_int 0 = 0"
-  "Target_Numeral.of_int (Int.Pos k) = numeral k"
-  by simp_all
-
-definition num_of_int :: "Target_Numeral.int \<Rightarrow> num" where
-  "num_of_int = num_of_nat \<circ> nat_of"
-
-lemma num_of_int_code [code]:
-  "num_of_int k = (if k \<le> 1 then Num.One
-     else let
-       (l, j) = divmod k 2;
-       l' = num_of_int l + num_of_int l
-     in if j = 0 then l' else l' + Num.One)"
-proof -
-  {
-    assume "int_of k mod 2 = 1"
-    then have "nat (int_of k mod 2) = nat 1" by simp
-    moreover assume *: "1 < int_of k"
-    ultimately have **: "nat (int_of k) mod 2 = 1" by (simp add: nat_mod_distrib)
-    have "num_of_nat (nat (int_of k)) =
-      num_of_nat (2 * (nat (int_of k) div 2) + nat (int_of k) mod 2)"
-      by simp
-    then have "num_of_nat (nat (int_of k)) =
-      num_of_nat (nat (int_of k) div 2 + nat (int_of k) div 2 + nat (int_of k) mod 2)"
-      by (simp add: mult_2)
-    with ** have "num_of_nat (nat (int_of k)) =
-      num_of_nat (nat (int_of k) div 2 + nat (int_of k) div 2 + 1)"
-      by simp
-  }
-  note aux = this
-  show ?thesis
-    by (auto simp add: num_of_int_def nat_of_def Let_def prod_case_beta
-      not_le Target_Numeral.int_eq_iff less_eq_int_def
-      nat_mult_distrib nat_div_distrib num_of_nat_One num_of_nat_plus_distrib
-       mult_2 [where 'a=nat] aux add_One)
-qed
-
-hide_const (open) int_of nat_of Pos Neg sub dup divmod_abs num_of_int
-
-
-subsection {* Serializer setup for target language numerals *}
-
-code_type Target_Numeral.int
-  (SML "IntInf.int")
-  (OCaml "Big'_int.big'_int")
-  (Haskell "Integer")
-  (Scala "BigInt")
-  (Eval "int")
-
-code_instance Target_Numeral.int :: equal
-  (Haskell -)
-
-code_const "0::Target_Numeral.int"
-  (SML "0")
-  (OCaml "Big'_int.zero'_big'_int")
-  (Haskell "0")
-  (Scala "BigInt(0)")
-
-setup {*
-  fold (Numeral.add_code @{const_name Target_Numeral.Pos}
-    false Code_Printer.literal_numeral) ["SML", "OCaml", "Haskell", "Scala"]
-*}
-
-setup {*
-  fold (Numeral.add_code @{const_name Target_Numeral.Neg}
-    true Code_Printer.literal_numeral) ["SML", "OCaml", "Haskell", "Scala"]
-*}
-
-code_const "plus :: Target_Numeral.int \<Rightarrow> _ \<Rightarrow> _"
-  (SML "IntInf.+ ((_), (_))")
-  (OCaml "Big'_int.add'_big'_int")
-  (Haskell infixl 6 "+")
-  (Scala infixl 7 "+")
-  (Eval infixl 8 "+")
-
-code_const "uminus :: Target_Numeral.int \<Rightarrow> _"
-  (SML "IntInf.~")
-  (OCaml "Big'_int.minus'_big'_int")
-  (Haskell "negate")
-  (Scala "!(- _)")
-  (Eval "~/ _")
-
-code_const "minus :: Target_Numeral.int \<Rightarrow> _"
-  (SML "IntInf.- ((_), (_))")
-  (OCaml "Big'_int.sub'_big'_int")
-  (Haskell infixl 6 "-")
-  (Scala infixl 7 "-")
-  (Eval infixl 8 "-")
-
-code_const Target_Numeral.dup
-  (SML "IntInf.*/ (2,/ (_))")
-  (OCaml "Big'_int.mult'_big'_int/ 2")
-  (Haskell "!(2 * _)")
-  (Scala "!(2 * _)")
-  (Eval "!(2 * _)")
-
-code_const Target_Numeral.sub
-  (SML "!(raise/ Fail/ \"sub\")")
-  (OCaml "failwith/ \"sub\"")
-  (Haskell "error/ \"sub\"")
-  (Scala "!sys.error(\"sub\")")
-
-code_const "times :: Target_Numeral.int \<Rightarrow> _ \<Rightarrow> _"
-  (SML "IntInf.* ((_), (_))")
-  (OCaml "Big'_int.mult'_big'_int")
-  (Haskell infixl 7 "*")
-  (Scala infixl 8 "*")
-  (Eval infixl 9 "*")
-
-code_const Target_Numeral.divmod_abs
-  (SML "IntInf.divMod/ (IntInf.abs _,/ IntInf.abs _)")
-  (OCaml "Big'_int.quomod'_big'_int/ (Big'_int.abs'_big'_int _)/ (Big'_int.abs'_big'_int _)")
-  (Haskell "divMod/ (abs _)/ (abs _)")
-  (Scala "!((k: BigInt) => (l: BigInt) =>/ if (l == 0)/ (BigInt(0), k) else/ (k.abs '/% l.abs))")
-  (Eval "Integer.div'_mod/ (abs _)/ (abs _)")
-
-code_const "HOL.equal :: Target_Numeral.int \<Rightarrow> _ \<Rightarrow> bool"
-  (SML "!((_ : IntInf.int) = _)")
-  (OCaml "Big'_int.eq'_big'_int")
-  (Haskell infix 4 "==")
-  (Scala infixl 5 "==")
-  (Eval infixl 6 "=")
-
-code_const "less_eq :: Target_Numeral.int \<Rightarrow> _ \<Rightarrow> bool"
-  (SML "IntInf.<= ((_), (_))")
-  (OCaml "Big'_int.le'_big'_int")
-  (Haskell infix 4 "<=")
-  (Scala infixl 4 "<=")
-  (Eval infixl 6 "<=")
-
-code_const "less :: Target_Numeral.int \<Rightarrow> _ \<Rightarrow> bool"
-  (SML "IntInf.< ((_), (_))")
-  (OCaml "Big'_int.lt'_big'_int")
-  (Haskell infix 4 "<")
-  (Scala infixl 4 "<")
-  (Eval infixl 6 "<")
-
-ML {*
-structure Target_Numeral =
-struct
-
-val T = @{typ "Target_Numeral.int"};
-
-end;
-*}
-
-code_reserved Eval Target_Numeral
-
-code_const "Code_Evaluation.term_of \<Colon> Target_Numeral.int \<Rightarrow> term"
-  (Eval "HOLogic.mk'_number/ Target'_Numeral.T")
-
-code_modulename SML
-  Target_Numeral Arith
-
-code_modulename OCaml
-  Target_Numeral Arith
-
-code_modulename Haskell
-  Target_Numeral Arith
-
-
-subsection {* Implementation for @{typ int} *}
-
-code_datatype Target_Numeral.int_of
-
-lemma [code, code del]:
-  "Target_Numeral.of_int = Target_Numeral.of_int" ..
-
-lemma [code]:
-  "Target_Numeral.of_int (Target_Numeral.int_of k) = k"
-  by (simp add: Target_Numeral.int_eq_iff)
-
-declare Int.Pos_def [code]
-
-lemma [code_abbrev]:
-  "Target_Numeral.int_of (Target_Numeral.Pos k) = Int.Pos k"
-  by simp
-
-declare Int.Neg_def [code]
-
-lemma [code_abbrev]:
-  "Target_Numeral.int_of (Target_Numeral.Neg k) = Int.Neg k"
-  by simp
-
-lemma [code]:
-  "0 = Target_Numeral.int_of 0"
-  by simp
-
-lemma [code]:
-  "1 = Target_Numeral.int_of 1"
-  by simp
-
-lemma [code]:
-  "k + l = Target_Numeral.int_of (of_int k + of_int l)"
-  by simp
-
-lemma [code]:
-  "- k = Target_Numeral.int_of (- of_int k)"
-  by simp
-
-lemma [code]:
-  "k - l = Target_Numeral.int_of (of_int k - of_int l)"
-  by simp
-
-lemma [code]:
-  "Int.dup k = Target_Numeral.int_of (Target_Numeral.dup (of_int k))"
-  by simp
-
-lemma [code, code del]:
-  "Int.sub = Int.sub" ..
-
-lemma [code]:
-  "k * l = Target_Numeral.int_of (of_int k * of_int l)"
-  by simp
-
-lemma [code]:
-  "pdivmod k l = map_pair Target_Numeral.int_of Target_Numeral.int_of
-    (Target_Numeral.divmod_abs (of_int k) (of_int l))"
-  by (simp add: prod_eq_iff pdivmod_def)
-
-lemma [code]:
-  "k div l = Target_Numeral.int_of (of_int k div of_int l)"
-  by simp
-
-lemma [code]:
-  "k mod l = Target_Numeral.int_of (of_int k mod of_int l)"
-  by simp
-
-lemma [code]:
-  "HOL.equal k l = HOL.equal (of_int k :: Target_Numeral.int) (of_int l)"
-  by (simp add: equal Target_Numeral.int_eq_iff)
-
-lemma [code]:
-  "k \<le> l \<longleftrightarrow> (of_int k :: Target_Numeral.int) \<le> of_int l"
-  by (simp add: less_eq_int_def)
-
-lemma [code]:
-  "k < l \<longleftrightarrow> (of_int k :: Target_Numeral.int) < of_int l"
-  by (simp add: less_int_def)
-
-lemma (in ring_1) of_int_code:
-  "of_int k = (if k = 0 then 0
-     else if k < 0 then - of_int (- k)
-     else let
-       (l, j) = divmod_int k 2;
-       l' = 2 * of_int l
-     in if j = 0 then l' else l' + 1)"
-proof -
-  from mod_div_equality have *: "of_int k = of_int (k div 2 * 2 + k mod 2)" by simp
-  show ?thesis
-    by (simp add: Let_def divmod_int_mod_div mod_2_not_eq_zero_eq_one_int
-      of_int_add [symmetric]) (simp add: * mult_commute)
-qed
-
-declare of_int_code [code]
-
-
-subsection {* Implementation for @{typ nat} *}
-
-definition Nat :: "Target_Numeral.int \<Rightarrow> nat" where
-  "Nat = Target_Numeral.nat_of"
-
-definition of_nat :: "nat \<Rightarrow> Target_Numeral.int" where
-  [code_abbrev]: "of_nat = Nat.of_nat"
-
-hide_const (open) of_nat Nat
-
-lemma [code_unfold]:
-  "Int.nat (Target_Numeral.int_of k) = Target_Numeral.nat_of k"
-  by (simp add: nat_of_def)
-
-lemma int_of_nat [simp]:
-  "Target_Numeral.int_of (Target_Numeral.of_nat n) = of_nat n"
-  by (simp add: of_nat_def)
-
-lemma [code abstype]:
-  "Target_Numeral.Nat (Target_Numeral.of_nat n) = n"
-  by (simp add: Nat_def nat_of_def)
-
-lemma [code abstract]:
-  "Target_Numeral.of_nat (Target_Numeral.nat_of k) = max 0 k"
-  by (simp add: of_nat_def)
-
-lemma [code_abbrev]:
-  "nat (Int.Pos k) = nat_of_num k"
-  by (simp add: nat_of_num_numeral)
-
-lemma [code abstract]:
-  "Target_Numeral.of_nat 0 = 0"
-  by (simp add: Target_Numeral.int_eq_iff)
-
-lemma [code abstract]:
-  "Target_Numeral.of_nat 1 = 1"
-  by (simp add: Target_Numeral.int_eq_iff)
-
-lemma [code abstract]:
-  "Target_Numeral.of_nat (m + n) = of_nat m + of_nat n"
-  by (simp add: Target_Numeral.int_eq_iff)
-
-lemma [code abstract]:
-  "Target_Numeral.of_nat (Code_Nat.dup n) = Target_Numeral.dup (of_nat n)"
-  by (simp add: Target_Numeral.int_eq_iff Code_Nat.dup_def)
-
-lemma [code, code del]:
-  "Code_Nat.sub = Code_Nat.sub" ..
-
-lemma [code abstract]:
-  "Target_Numeral.of_nat (m - n) = max 0 (of_nat m - of_nat n)"
-  by (simp add: Target_Numeral.int_eq_iff)
-
-lemma [code abstract]:
-  "Target_Numeral.of_nat (m * n) = of_nat m * of_nat n"
-  by (simp add: Target_Numeral.int_eq_iff of_nat_mult)
-
-lemma [code abstract]:
-  "Target_Numeral.of_nat (m div n) = of_nat m div of_nat n"
-  by (simp add: Target_Numeral.int_eq_iff zdiv_int)
-
-lemma [code abstract]:
-  "Target_Numeral.of_nat (m mod n) = of_nat m mod of_nat n"
-  by (simp add: Target_Numeral.int_eq_iff zmod_int)
-
-lemma [code]:
-  "Divides.divmod_nat m n = (m div n, m mod n)"
-  by (simp add: prod_eq_iff)
-
-lemma [code]:
-  "HOL.equal m n = HOL.equal (of_nat m :: Target_Numeral.int) (of_nat n)"
-  by (simp add: equal Target_Numeral.int_eq_iff)
-
-lemma [code]:
-  "m \<le> n \<longleftrightarrow> (of_nat m :: Target_Numeral.int) \<le> of_nat n"
-  by (simp add: less_eq_int_def)
-
-lemma [code]:
-  "m < n \<longleftrightarrow> (of_nat m :: Target_Numeral.int) < of_nat n"
-  by (simp add: less_int_def)
-
-lemma num_of_nat_code [code]:
-  "num_of_nat = Target_Numeral.num_of_int \<circ> of_nat"
-  by (simp add: fun_eq_iff num_of_int_def of_nat_def)
-
-lemma (in semiring_1) of_nat_code:
-  "of_nat n = (if n = 0 then 0
-     else let
-       (m, q) = divmod_nat n 2;
-       m' = 2 * of_nat m
-     in if q = 0 then m' else m' + 1)"
-proof -
-  from mod_div_equality have *: "of_nat n = of_nat (n div 2 * 2 + n mod 2)" by simp
-  show ?thesis
-    by (simp add: Let_def divmod_nat_div_mod mod_2_not_eq_zero_eq_one_nat
-      of_nat_add [symmetric])
-      (simp add: * mult_commute of_nat_mult add_commute)
-qed
-
-declare of_nat_code [code]
-
-text {* Conversions between @{typ nat} and @{typ int} *}
-
-definition int :: "nat \<Rightarrow> int" where
-  [code_abbrev]: "int = of_nat"
-
-hide_const (open) int
-
-lemma [code]:
-  "Target_Numeral.int n = Target_Numeral.int_of (of_nat n)"
-  by (simp add: int_def)
-
-lemma [code abstract]:
-  "Target_Numeral.of_nat (nat k) = max 0 (Target_Numeral.of_int k)"
-  by (simp add: of_nat_def of_int_of_nat max_def)
-
-end
-
diff -r 286dfcab9833 -r 28f3263d4d1b src/HOL/ROOT
--- a/src/HOL/ROOT	Wed Nov 07 20:48:04 2012 +0100
+++ b/src/HOL/ROOT	Thu Nov 08 10:02:38 2012 +0100
@@ -48,7 +48,7 @@
     Efficient_Nat
     (* Code_Prolog  FIXME cf. 76965c356d2a *)
     Code_Real_Approx_By_Float
-    Target_Numeral
+    Code_Target_Numeral
     Refute
   theories [condition = ISABELLE_FULL_TEST]
     Sum_of_Squares_Remote
@@ -415,7 +415,7 @@
   options [timeout = 3600, condition = ISABELLE_POLYML]
   theories [document = false]
     "~~/src/HOL/Library/State_Monad"
-    Code_Nat_examples
+    Code_Binary_Nat_examples
     "~~/src/HOL/Library/FuncSet"
     Eval_Examples
     Normalization_by_Evaluation
diff -r 286dfcab9833 -r 28f3263d4d1b src/HOL/ex/Code_Binary_Nat_examples.thy
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/ex/Code_Binary_Nat_examples.thy	Thu Nov 08 10:02:38 2012 +0100
@@ -0,0 +1,57 @@
+(*  Title:      HOL/ex/Code_Binary_Nat_examples.thy
+    Author:     Florian Haftmann, TU Muenchen
+*)
+
+header {* Simple examples for natural numbers implemented in binary representation theory. *}
+
+theory Code_Binary_Nat_examples
+imports Complex_Main "~~/src/HOL/Library/Efficient_Nat"
+begin
+
+fun to_n :: "nat \<Rightarrow> nat list"
+where
+  "to_n 0 = []"
+| "to_n (Suc 0) = []"
+| "to_n (Suc (Suc 0)) = []"
+| "to_n (Suc n) = n # to_n n"
+
+definition naive_prime :: "nat \<Rightarrow> bool"
+where
+  "naive_prime n \<longleftrightarrow> n \<ge> 2 \<and> filter (\<lambda>m. n mod m = 0) (to_n n) = []"
+
+primrec fac :: "nat \<Rightarrow> nat"
+where
+  "fac 0 = 1"
+| "fac (Suc n) = Suc n * fac n"
+
+primrec harmonic :: "nat \<Rightarrow> rat"
+where
+  "harmonic 0 = 0"
+| "harmonic (Suc n) = 1 / of_nat (Suc n) + harmonic n"
+
+lemma "harmonic 200 \<ge> 5"
+  by eval
+
+lemma "(let (q, r) = quotient_of (harmonic 8) in q div r) \<ge> 2"
+  by normalization
+
+lemma "naive_prime 89"
+  by eval
+
+lemma "naive_prime 89"
+  by normalization
+
+lemma "\<not> naive_prime 87"
+  by eval
+
+lemma "\<not> naive_prime 87"
+  by normalization
+
+lemma "fac 10 > 3000000"
+  by eval
+
+lemma "fac 10 > 3000000"
+  by normalization
+
+end
+
diff -r 286dfcab9833 -r 28f3263d4d1b src/HOL/ex/Code_Nat_examples.thy
--- a/src/HOL/ex/Code_Nat_examples.thy	Wed Nov 07 20:48:04 2012 +0100
+++ /dev/null	Thu Jan 01 00:00:00 1970 +0000
@@ -1,57 +0,0 @@
-(*  Title:      HOL/ex/Code_Nat_examples.thy
-    Author:     Florian Haftmann, TU Muenchen
-*)
-
-header {* Simple examples for Code\_Numeral\_Nat theory. *}
-
-theory Code_Nat_examples
-imports Complex_Main "~~/src/HOL/Library/Efficient_Nat"
-begin
-
-fun to_n :: "nat \<Rightarrow> nat list"
-where
-  "to_n 0 = []"
-| "to_n (Suc 0) = []"
-| "to_n (Suc (Suc 0)) = []"
-| "to_n (Suc n) = n # to_n n"
-
-definition naive_prime :: "nat \<Rightarrow> bool"
-where
-  "naive_prime n \<longleftrightarrow> n \<ge> 2 \<and> filter (\<lambda>m. n mod m = 0) (to_n n) = []"
-
-primrec fac :: "nat \<Rightarrow> nat"
-where
-  "fac 0 = 1"
-| "fac (Suc n) = Suc n * fac n"
-
-primrec harmonic :: "nat \<Rightarrow> rat"
-where
-  "harmonic 0 = 0"
-| "harmonic (Suc n) = 1 / of_nat (Suc n) + harmonic n"
-
-lemma "harmonic 200 \<ge> 5"
-  by eval
-
-lemma "(let (q, r) = quotient_of (harmonic 8) in q div r) \<ge> 2"
-  by normalization
-
-lemma "naive_prime 89"
-  by eval
-
-lemma "naive_prime 89"
-  by normalization
-
-lemma "\<not> naive_prime 87"
-  by eval
-
-lemma "\<not> naive_prime 87"
-  by normalization
-
-lemma "fac 10 > 3000000"
-  by eval
-
-lemma "fac 10 > 3000000"
-  by normalization
-
-end
-