# HG changeset patch
# User wenzelm
# Date 1329824700 -3600
# Node ID 26977429b784d36da5d8745c47c10fd99082362f
# Parent  0133d31f9ab4ca122025f98794d7cc8440a17f58# Parent  092f4eca98485add0d670d01abbde4ca76dd1bf3
merged;

diff -r 0133d31f9ab4 -r 26977429b784 src/HOL/Algebra/Group.thy
--- a/src/HOL/Algebra/Group.thy	Mon Feb 20 22:35:32 2012 +0100
+++ b/src/HOL/Algebra/Group.thy	Tue Feb 21 12:45:00 2012 +0100
@@ -41,7 +41,7 @@
 begin
   definition "int_pow G a z =
    (let p = nat_rec \<one>\<^bsub>G\<^esub> (%u b. b \<otimes>\<^bsub>G\<^esub> a)
-    in if neg z then inv\<^bsub>G\<^esub> (p (nat (-z))) else p (nat z))"
+    in if z < 0 then inv\<^bsub>G\<^esub> (p (nat (-z))) else p (nat z))"
 end
 
 locale monoid =
@@ -391,7 +391,7 @@
 text {* Power *}
 
 lemma (in group) int_pow_def2:
-  "a (^) (z::int) = (if neg z then inv (a (^) (nat (-z))) else a (^) (nat z))"
+  "a (^) (z::int) = (if z < 0 then inv (a (^) (nat (-z))) else a (^) (nat z))"
   by (simp add: int_pow_def nat_pow_def Let_def)
 
 lemma (in group) int_pow_0 [simp]:
diff -r 0133d31f9ab4 -r 26977429b784 src/HOL/Big_Operators.thy
diff -r 0133d31f9ab4 -r 26977429b784 src/HOL/Complete_Lattices.thy
diff -r 0133d31f9ab4 -r 26977429b784 src/HOL/Divides.thy
--- a/src/HOL/Divides.thy	Mon Feb 20 22:35:32 2012 +0100
+++ b/src/HOL/Divides.thy	Tue Feb 21 12:45:00 2012 +0100
@@ -523,9 +523,6 @@
   means of @{const divmod_nat_rel}:
 *}
 
-instantiation nat :: semiring_div
-begin
-
 definition divmod_nat :: "nat \<Rightarrow> nat \<Rightarrow> nat \<times> nat" where
   "divmod_nat m n = (THE qr. divmod_nat_rel m n qr)"
 
@@ -543,28 +540,39 @@
   shows "divmod_nat m n = qr"
   using assms by (auto intro: divmod_nat_rel_unique divmod_nat_rel_divmod_nat)
 
+instantiation nat :: semiring_div
+begin
+
 definition div_nat where
   "m div n = fst (divmod_nat m n)"
 
+lemma fst_divmod_nat [simp]:
+  "fst (divmod_nat m n) = m div n"
+  by (simp add: div_nat_def)
+
 definition mod_nat where
   "m mod n = snd (divmod_nat m n)"
 
+lemma snd_divmod_nat [simp]:
+  "snd (divmod_nat m n) = m mod n"
+  by (simp add: mod_nat_def)
+
 lemma divmod_nat_div_mod:
   "divmod_nat m n = (m div n, m mod n)"
-  unfolding div_nat_def mod_nat_def by simp
+  by (simp add: prod_eq_iff)
 
 lemma div_eq:
   assumes "divmod_nat_rel m n (q, r)" 
   shows "m div n = q"
-  using assms by (auto dest: divmod_nat_eq simp add: divmod_nat_div_mod)
+  using assms by (auto dest!: divmod_nat_eq simp add: prod_eq_iff)
 
 lemma mod_eq:
   assumes "divmod_nat_rel m n (q, r)" 
   shows "m mod n = r"
-  using assms by (auto dest: divmod_nat_eq simp add: divmod_nat_div_mod)
+  using assms by (auto dest!: divmod_nat_eq simp add: prod_eq_iff)
 
 lemma divmod_nat_rel: "divmod_nat_rel m n (m div n, m mod n)"
-  by (simp add: div_nat_def mod_nat_def divmod_nat_rel_divmod_nat)
+  using divmod_nat_rel_divmod_nat by (simp add: divmod_nat_div_mod)
 
 lemma divmod_nat_zero:
   "divmod_nat m 0 = (0, m)"
@@ -613,25 +621,25 @@
   fixes m n :: nat
   assumes "m < n"
   shows "m div n = 0"
-  using assms divmod_nat_base divmod_nat_div_mod by simp
+  using assms divmod_nat_base by (simp add: prod_eq_iff)
 
 lemma le_div_geq:
   fixes m n :: nat
   assumes "0 < n" and "n \<le> m"
   shows "m div n = Suc ((m - n) div n)"
-  using assms divmod_nat_step divmod_nat_div_mod by simp
+  using assms divmod_nat_step by (simp add: prod_eq_iff)
 
 lemma mod_less [simp]:
   fixes m n :: nat
   assumes "m < n"
   shows "m mod n = m"
-  using assms divmod_nat_base divmod_nat_div_mod by simp
+  using assms divmod_nat_base by (simp add: prod_eq_iff)
 
 lemma le_mod_geq:
   fixes m n :: nat
   assumes "n \<le> m"
   shows "m mod n = (m - n) mod n"
-  using assms divmod_nat_step divmod_nat_div_mod by (cases "n = 0") simp_all
+  using assms divmod_nat_step by (cases "n = 0") (simp_all add: prod_eq_iff)
 
 instance proof -
   have [simp]: "\<And>n::nat. n div 0 = 0"
@@ -673,8 +681,7 @@
 
 lemma divmod_nat_if [code]: "divmod_nat m n = (if n = 0 \<or> m < n then (0, m) else
   let (q, r) = divmod_nat (m - n) n in (Suc q, r))"
-by (simp add: divmod_nat_zero divmod_nat_base divmod_nat_step)
-    (simp add: divmod_nat_div_mod)
+  by (simp add: prod_eq_iff prod_case_beta not_less le_div_geq le_mod_geq)
 
 text {* Simproc for cancelling @{const div} and @{const mod} *}
 
@@ -760,27 +767,23 @@
 subsubsection {* Quotient and Remainder *}
 
 lemma divmod_nat_rel_mult1_eq:
-  "divmod_nat_rel b c (q, r) \<Longrightarrow> c > 0
+  "divmod_nat_rel b c (q, r)
    \<Longrightarrow> divmod_nat_rel (a * b) c (a * q + a * r div c, a * r mod c)"
 by (auto simp add: split_ifs divmod_nat_rel_def algebra_simps)
 
 lemma div_mult1_eq:
   "(a * b) div c = a * (b div c) + a * (b mod c) div (c::nat)"
-apply (cases "c = 0", simp)
-apply (blast intro: divmod_nat_rel [THEN divmod_nat_rel_mult1_eq, THEN div_eq])
-done
+by (blast intro: divmod_nat_rel [THEN divmod_nat_rel_mult1_eq, THEN div_eq])
 
 lemma divmod_nat_rel_add1_eq:
-  "divmod_nat_rel a c (aq, ar) \<Longrightarrow> divmod_nat_rel b c (bq, br) \<Longrightarrow>  c > 0
+  "divmod_nat_rel a c (aq, ar) \<Longrightarrow> divmod_nat_rel b c (bq, br)
    \<Longrightarrow> divmod_nat_rel (a + b) c (aq + bq + (ar + br) div c, (ar + br) mod c)"
 by (auto simp add: split_ifs divmod_nat_rel_def algebra_simps)
 
 (*NOT suitable for rewriting: the RHS has an instance of the LHS*)
 lemma div_add1_eq:
   "(a+b) div (c::nat) = a div c + b div c + ((a mod c + b mod c) div c)"
-apply (cases "c = 0", simp)
-apply (blast intro: divmod_nat_rel_add1_eq [THEN div_eq] divmod_nat_rel)
-done
+by (blast intro: divmod_nat_rel_add1_eq [THEN div_eq] divmod_nat_rel)
 
 lemma mod_lemma: "[| (0::nat) < c; r < b |] ==> b * (q mod c) + r < b * c"
   apply (cut_tac m = q and n = c in mod_less_divisor)
@@ -790,29 +793,22 @@
   done
 
 lemma divmod_nat_rel_mult2_eq:
-  "divmod_nat_rel a b (q, r) \<Longrightarrow> 0 < b \<Longrightarrow> 0 < c
+  "divmod_nat_rel a b (q, r)
    \<Longrightarrow> divmod_nat_rel a (b * c) (q div c, b *(q mod c) + r)"
 by (auto simp add: mult_ac divmod_nat_rel_def add_mult_distrib2 [symmetric] mod_lemma)
 
 lemma div_mult2_eq: "a div (b*c) = (a div b) div (c::nat)"
-  apply (cases "b = 0", simp)
-  apply (cases "c = 0", simp)
-  apply (force simp add: divmod_nat_rel [THEN divmod_nat_rel_mult2_eq, THEN div_eq])
-  done
+by (force simp add: divmod_nat_rel [THEN divmod_nat_rel_mult2_eq, THEN div_eq])
 
 lemma mod_mult2_eq: "a mod (b*c) = b*(a div b mod c) + a mod (b::nat)"
-  apply (cases "b = 0", simp)
-  apply (cases "c = 0", simp)
-  apply (auto simp add: mult_commute divmod_nat_rel [THEN divmod_nat_rel_mult2_eq, THEN mod_eq])
-  done
-
-
-subsubsection{*Further Facts about Quotient and Remainder*}
+by (auto simp add: mult_commute divmod_nat_rel [THEN divmod_nat_rel_mult2_eq, THEN mod_eq])
+
+
+subsubsection {* Further Facts about Quotient and Remainder *}
 
 lemma div_1 [simp]: "m div Suc 0 = m"
 by (induct m) (simp_all add: div_geq)
 
-
 (* Monotonicity of div in first argument *)
 lemma div_le_mono [rule_format (no_asm)]:
     "\<forall>m::nat. m \<le> n --> (m div k) \<le> (n div k)"
@@ -1018,7 +1014,7 @@
 qed
 
 
-subsubsection {*An ``induction'' law for modulus arithmetic.*}
+subsubsection {* An ``induction'' law for modulus arithmetic. *}
 
 lemma mod_induct_0:
   assumes step: "\<forall>i<p. P i \<longrightarrow> P ((Suc i) mod p)"
@@ -1211,8 +1207,6 @@
   (auto simp add: mult_2)
 
 text{*algorithm for the general case @{term "b\<noteq>0"}*}
-definition negateSnd :: "int \<times> int \<Rightarrow> int \<times> int" where
-  [code_unfold]: "negateSnd = apsnd uminus"
 
 definition divmod_int :: "int \<Rightarrow> int \<Rightarrow> int \<times> int" where
     --{*The full division algorithm considers all possible signs for a, b
@@ -1220,19 +1214,27 @@
        @{term negDivAlg} requires @{term "a<0"}.*}
   "divmod_int a b = (if 0 \<le> a then if 0 \<le> b then posDivAlg a b
                   else if a = 0 then (0, 0)
-                       else negateSnd (negDivAlg (-a) (-b))
+                       else apsnd uminus (negDivAlg (-a) (-b))
                else 
                   if 0 < b then negDivAlg a b
-                  else negateSnd (posDivAlg (-a) (-b)))"
+                  else apsnd uminus (posDivAlg (-a) (-b)))"
 
 instantiation int :: Divides.div
 begin
 
-definition
+definition div_int where
   "a div b = fst (divmod_int a b)"
 
-definition
- "a mod b = snd (divmod_int a b)"
+lemma fst_divmod_int [simp]:
+  "fst (divmod_int a b) = a div b"
+  by (simp add: div_int_def)
+
+definition mod_int where
+  "a mod b = snd (divmod_int a b)"
+
+lemma snd_divmod_int [simp]:
+  "snd (divmod_int a b) = a mod b"
+  by (simp add: mod_int_def)
 
 instance ..
 
@@ -1240,7 +1242,7 @@
 
 lemma divmod_int_mod_div:
   "divmod_int p q = (p div q, p mod q)"
-  by (auto simp add: div_int_def mod_int_def)
+  by (simp add: prod_eq_iff)
 
 text{*
 Here is the division algorithm in ML:
@@ -1271,8 +1273,7 @@
 *}
 
 
-
-subsubsection{*Uniqueness and Monotonicity of Quotients and Remainders*}
+subsubsection {* Uniqueness and Monotonicity of Quotients and Remainders *}
 
 lemma unique_quotient_lemma:
      "[| b*q' + r'  \<le> b*q + r;  0 \<le> r';  r' < b;  r < b |]  
@@ -1294,7 +1295,7 @@
     auto)
 
 lemma unique_quotient:
-     "[| divmod_int_rel a b (q, r); divmod_int_rel a b (q', r');  b \<noteq> 0 |]  
+     "[| divmod_int_rel a b (q, r); divmod_int_rel a b (q', r') |]  
       ==> q = q'"
 apply (simp add: divmod_int_rel_def linorder_neq_iff split: split_if_asm)
 apply (blast intro: order_antisym
@@ -1304,7 +1305,7 @@
 
 
 lemma unique_remainder:
-     "[| divmod_int_rel a b (q, r); divmod_int_rel a b (q', r');  b \<noteq> 0 |]  
+     "[| divmod_int_rel a b (q, r); divmod_int_rel a b (q', r') |]  
       ==> r = r'"
 apply (subgoal_tac "q = q'")
  apply (simp add: divmod_int_rel_def)
@@ -1312,7 +1313,7 @@
 done
 
 
-subsubsection{*Correctness of @{term posDivAlg}, the Algorithm for Non-Negative Dividends*}
+subsubsection {* Correctness of @{term posDivAlg}, the Algorithm for Non-Negative Dividends *}
 
 text{*And positive divisors*}
 
@@ -1347,7 +1348,7 @@
   done
 
 
-subsubsection{*Correctness of @{term negDivAlg}, the Algorithm for Negative Dividends*}
+subsubsection {* Correctness of @{term negDivAlg}, the Algorithm for Negative Dividends *}
 
 text{*And positive divisors*}
 
@@ -1377,7 +1378,7 @@
   done
 
 
-subsubsection{*Existence Shown by Proving the Division Algorithm to be Correct*}
+subsubsection {* Existence Shown by Proving the Division Algorithm to be Correct *}
 
 (*the case a=0*)
 lemma divmod_int_rel_0: "b \<noteq> 0 ==> divmod_int_rel 0 b (0, 0)"
@@ -1389,10 +1390,7 @@
 lemma negDivAlg_minus1 [simp]: "negDivAlg -1 b = (-1, b - 1)"
 by (subst negDivAlg.simps, auto)
 
-lemma negateSnd_eq [simp]: "negateSnd(q,r) = (q,-r)"
-by (simp add: negateSnd_def)
-
-lemma divmod_int_rel_neg: "divmod_int_rel (-a) (-b) qr ==> divmod_int_rel a b (negateSnd qr)"
+lemma divmod_int_rel_neg: "divmod_int_rel (-a) (-b) qr ==> divmod_int_rel a b (apsnd uminus qr)"
 by (auto simp add: split_ifs divmod_int_rel_def)
 
 lemma divmod_int_correct: "b \<noteq> 0 ==> divmod_int_rel a b (divmod_int a b)"
@@ -1411,7 +1409,7 @@
 lemma zmod_zdiv_equality: "(a::int) = b * (a div b) + (a mod b)"
 apply (case_tac "b = 0", simp)
 apply (cut_tac a = a and b = b in divmod_int_correct)
-apply (auto simp add: divmod_int_rel_def div_int_def mod_int_def)
+apply (auto simp add: divmod_int_rel_def prod_eq_iff)
 done
 
 lemma zdiv_zmod_equality: "(b * (a div b) + (a mod b)) + k = (a::int)+k"
@@ -1442,7 +1440,7 @@
 
 lemma pos_mod_conj : "(0::int) < b ==> 0 \<le> a mod b & a mod b < b"
 apply (cut_tac a = a and b = b in divmod_int_correct)
-apply (auto simp add: divmod_int_rel_def mod_int_def)
+apply (auto simp add: divmod_int_rel_def prod_eq_iff)
 done
 
 lemmas pos_mod_sign [simp] = pos_mod_conj [THEN conjunct1]
@@ -1450,24 +1448,28 @@
 
 lemma neg_mod_conj : "b < (0::int) ==> a mod b \<le> 0 & b < a mod b"
 apply (cut_tac a = a and b = b in divmod_int_correct)
-apply (auto simp add: divmod_int_rel_def div_int_def mod_int_def)
+apply (auto simp add: divmod_int_rel_def prod_eq_iff)
 done
 
 lemmas neg_mod_sign [simp] = neg_mod_conj [THEN conjunct1]
    and neg_mod_bound [simp] = neg_mod_conj [THEN conjunct2]
 
 
-subsubsection{*General Properties of div and mod*}
+subsubsection {* General Properties of div and mod *}
 
 lemma divmod_int_rel_div_mod: "b \<noteq> 0 ==> divmod_int_rel a b (a div b, a mod b)"
 apply (cut_tac a = a and b = b in zmod_zdiv_equality)
 apply (force simp add: divmod_int_rel_def linorder_neq_iff)
 done
 
-lemma divmod_int_rel_div: "[| divmod_int_rel a b (q, r);  b \<noteq> 0 |] ==> a div b = q"
+lemma divmod_int_rel_div: "[| divmod_int_rel a b (q, r) |] ==> a div b = q"
+apply (cases "b = 0")
+apply (simp add: divmod_int_rel_def)
 by (simp add: divmod_int_rel_div_mod [THEN unique_quotient])
 
-lemma divmod_int_rel_mod: "[| divmod_int_rel a b (q, r);  b \<noteq> 0 |] ==> a mod b = r"
+lemma divmod_int_rel_mod: "[| divmod_int_rel a b (q, r) |] ==> a mod b = r"
+apply (cases "b = 0")
+apply (simp add: divmod_int_rel_def)
 by (simp add: divmod_int_rel_div_mod [THEN unique_remainder])
 
 lemma div_pos_pos_trivial: "[| (0::int) \<le> a;  a < b |] ==> a div b = 0"
@@ -1521,7 +1523,7 @@
 done
 
 
-subsubsection{*Laws for div and mod with Unary Minus*}
+subsubsection {* Laws for div and mod with Unary Minus *}
 
 lemma zminus1_lemma:
      "divmod_int_rel a b (q, r)
@@ -1569,7 +1571,7 @@
   unfolding zmod_zminus2_eq_if by auto 
 
 
-subsubsection{*Division of a Number by Itself*}
+subsubsection {* Division of a Number by Itself *}
 
 lemma self_quotient_aux1: "[| (0::int) < a; a = r + a*q; r < a |] ==> 1 \<le> q"
 apply (subgoal_tac "0 < a*q")
@@ -1582,7 +1584,7 @@
 apply (simp add: right_diff_distrib)
 done
 
-lemma self_quotient: "[| divmod_int_rel a a (q, r);  a \<noteq> (0::int) |] ==> q = 1"
+lemma self_quotient: "[| divmod_int_rel a a (q, r) |] ==> q = 1"
 apply (simp add: split_ifs divmod_int_rel_def linorder_neq_iff)
 apply (rule order_antisym, safe, simp_all)
 apply (rule_tac [3] a = "-a" and r = "-r" in self_quotient_aux1)
@@ -1590,8 +1592,8 @@
 apply (force intro: self_quotient_aux1 self_quotient_aux2 simp add: add_commute)+
 done
 
-lemma self_remainder: "[| divmod_int_rel a a (q, r);  a \<noteq> (0::int) |] ==> r = 0"
-apply (frule self_quotient, assumption)
+lemma self_remainder: "[| divmod_int_rel a a (q, r) |] ==> r = 0"
+apply (frule self_quotient)
 apply (simp add: divmod_int_rel_def)
 done
 
@@ -1605,7 +1607,7 @@
 done
 
 
-subsubsection{*Computation of Division and Remainder*}
+subsubsection {* Computation of Division and Remainder *}
 
 lemma zdiv_zero [simp]: "(0::int) div b = 0"
 by (simp add: div_int_def divmod_int_def)
@@ -1638,40 +1640,39 @@
 text{*a positive, b negative *}
 
 lemma div_pos_neg:
-     "[| 0 < a;  b < 0 |] ==> a div b = fst (negateSnd (negDivAlg (-a) (-b)))"
+     "[| 0 < a;  b < 0 |] ==> a div b = fst (apsnd uminus (negDivAlg (-a) (-b)))"
 by (simp add: div_int_def divmod_int_def)
 
 lemma mod_pos_neg:
-     "[| 0 < a;  b < 0 |] ==> a mod b = snd (negateSnd (negDivAlg (-a) (-b)))"
+     "[| 0 < a;  b < 0 |] ==> a mod b = snd (apsnd uminus (negDivAlg (-a) (-b)))"
 by (simp add: mod_int_def divmod_int_def)
 
 text{*a negative, b negative *}
 
 lemma div_neg_neg:
-     "[| a < 0;  b \<le> 0 |] ==> a div b = fst (negateSnd (posDivAlg (-a) (-b)))"
+     "[| a < 0;  b \<le> 0 |] ==> a div b = fst (apsnd uminus (posDivAlg (-a) (-b)))"
 by (simp add: div_int_def divmod_int_def)
 
 lemma mod_neg_neg:
-     "[| a < 0;  b \<le> 0 |] ==> a mod b = snd (negateSnd (posDivAlg (-a) (-b)))"
+     "[| a < 0;  b \<le> 0 |] ==> a mod b = snd (apsnd uminus (posDivAlg (-a) (-b)))"
 by (simp add: mod_int_def divmod_int_def)
 
 text {*Simplify expresions in which div and mod combine numerical constants*}
 
 lemma int_div_pos_eq: "\<lbrakk>(a::int) = b * q + r; 0 \<le> r; r < b\<rbrakk> \<Longrightarrow> a div b = q"
-  by (rule divmod_int_rel_div [of a b q r],
-    simp add: divmod_int_rel_def, simp)
+  by (rule divmod_int_rel_div [of a b q r]) (simp add: divmod_int_rel_def)
 
 lemma int_div_neg_eq: "\<lbrakk>(a::int) = b * q + r; r \<le> 0; b < r\<rbrakk> \<Longrightarrow> a div b = q"
   by (rule divmod_int_rel_div [of a b q r],
-    simp add: divmod_int_rel_def, simp)
+    simp add: divmod_int_rel_def)
 
 lemma int_mod_pos_eq: "\<lbrakk>(a::int) = b * q + r; 0 \<le> r; r < b\<rbrakk> \<Longrightarrow> a mod b = r"
   by (rule divmod_int_rel_mod [of a b q r],
-    simp add: divmod_int_rel_def, simp)
+    simp add: divmod_int_rel_def)
 
 lemma int_mod_neg_eq: "\<lbrakk>(a::int) = b * q + r; r \<le> 0; b < r\<rbrakk> \<Longrightarrow> a mod b = r"
   by (rule divmod_int_rel_mod [of a b q r],
-    simp add: divmod_int_rel_def, simp)
+    simp add: divmod_int_rel_def)
 
 lemmas arithmetic_simps =
   arith_simps
@@ -1748,7 +1749,7 @@
 lemmas negDivAlg_eqn_1_number_of [simp] = negDivAlg_eqn [of concl: 1 "number_of w"] for w
 
 
-subsubsection{*Monotonicity in the First Argument (Dividend)*}
+subsubsection {* Monotonicity in the First Argument (Dividend) *}
 
 lemma zdiv_mono1: "[| a \<le> a';  0 < (b::int) |] ==> a div b \<le> a' div b"
 apply (cut_tac a = a and b = b in zmod_zdiv_equality)
@@ -1767,7 +1768,7 @@
 done
 
 
-subsubsection{*Monotonicity in the Second Argument (Divisor)*}
+subsubsection {* Monotonicity in the Second Argument (Divisor) *}
 
 lemma q_pos_lemma:
      "[| 0 \<le> b'*q' + r'; r' < b';  0 < b' |] ==> 0 \<le> (q'::int)"
@@ -1829,12 +1830,12 @@
 done
 
 
-subsubsection{*More Algebraic Laws for div and mod*}
+subsubsection {* More Algebraic Laws for div and mod *}
 
 text{*proving (a*b) div c = a * (b div c) + a * (b mod c) *}
 
 lemma zmult1_lemma:
-     "[| divmod_int_rel b c (q, r);  c \<noteq> 0 |]  
+     "[| divmod_int_rel b c (q, r) |]  
       ==> divmod_int_rel (a * b) c (a*q + a*r div c, a*r mod c)"
 by (auto simp add: split_ifs divmod_int_rel_def linorder_neq_iff right_distrib mult_ac)
 
@@ -1856,7 +1857,7 @@
 text{*proving (a+b) div c = a div c + b div c + ((a mod c + b mod c) div c) *}
 
 lemma zadd1_lemma:
-     "[| divmod_int_rel a c (aq, ar);  divmod_int_rel b c (bq, br);  c \<noteq> 0 |]  
+     "[| divmod_int_rel a c (aq, ar);  divmod_int_rel b c (bq, br) |]  
       ==> divmod_int_rel (a+b) c (aq + bq + (ar+br) div c, (ar+br) mod c)"
 by (force simp add: split_ifs divmod_int_rel_def linorder_neq_iff right_distrib)
 
@@ -1890,8 +1891,7 @@
       done
     moreover with `c \<noteq> 0` divmod_int_rel_div_mod have "divmod_int_rel b c (b div c, b mod c)" by auto
     ultimately have "divmod_int_rel (a * b) (a * c) (b div c, a * (b mod c))" .
-    moreover from  `a \<noteq> 0` `c \<noteq> 0` have "a * c \<noteq> 0" by simp
-    ultimately show ?thesis by (rule divmod_int_rel_div)
+    from this show ?thesis by (rule divmod_int_rel_div)
   qed
 qed auto
 
@@ -1905,7 +1905,7 @@
   case False with assms posDivAlg_correct
     have "divmod_int_rel k l (fst (posDivAlg k l), snd (posDivAlg k l))"
     by simp
-  from divmod_int_rel_div [OF this `l \<noteq> 0`] divmod_int_rel_mod [OF this `l \<noteq> 0`]
+  from divmod_int_rel_div [OF this] divmod_int_rel_mod [OF this]
   show ?thesis by simp
 qed
 
@@ -1918,7 +1918,7 @@
   from assms negDivAlg_correct
     have "divmod_int_rel k l (fst (negDivAlg k l), snd (negDivAlg k l))"
     by simp
-  from divmod_int_rel_div [OF this `l \<noteq> 0`] divmod_int_rel_mod [OF this `l \<noteq> 0`]
+  from divmod_int_rel_div [OF this] divmod_int_rel_mod [OF this]
   show ?thesis by simp
 qed
 
@@ -1929,7 +1929,7 @@
 lemmas zmod_eq_0D [dest!] = zmod_eq_0_iff [THEN iffD1]
 
 
-subsubsection{*Proving  @{term "a div (b*c) = (a div b) div c"} *}
+subsubsection {* Proving  @{term "a div (b*c) = (a div b) div c"} *}
 
 (*The condition c>0 seems necessary.  Consider that 7 div ~6 = ~2 but
   7 div 2 div ~3 = 3 div ~3 = ~1.  The subcase (a div b) mod c = 0 seems
@@ -1972,7 +1972,7 @@
 apply simp
 done
 
-lemma zmult2_lemma: "[| divmod_int_rel a b (q, r);  b \<noteq> 0;  0 < c |]  
+lemma zmult2_lemma: "[| divmod_int_rel a b (q, r); 0 < c |]  
       ==> divmod_int_rel a (b * c) (q div c, b*(q mod c) + r)"
 by (auto simp add: mult_ac divmod_int_rel_def linorder_neq_iff
                    zero_less_mult_iff right_distrib [symmetric] 
@@ -1990,7 +1990,7 @@
 done
 
 
-subsubsection {*Splitting Rules for div and mod*}
+subsubsection {* Splitting Rules for div and mod *}
 
 text{*The proofs of the two lemmas below are essentially identical*}
 
@@ -2051,7 +2051,7 @@
 declare split_zmod [of _ _ "number_of k", arith_split] for k
 
 
-subsubsection{*Speeding up the Division Algorithm with Shifting*}
+subsubsection {* Speeding up the Division Algorithm with Shifting *}
 
 text{*computing div by shifting *}
 
@@ -2105,7 +2105,7 @@
 done
 
 
-subsubsection{*Computing mod by Shifting (proofs resemble those for div)*}
+subsubsection {* Computing mod by Shifting (proofs resemble those for div) *}
 
 lemma pos_zmod_mult_2:
   fixes a b :: int
@@ -2169,7 +2169,7 @@
 qed
 
 
-subsubsection{*Quotients of Signs*}
+subsubsection {* Quotients of Signs *}
 
 lemma div_neg_pos_less0: "[| a < (0::int);  0 < b |] ==> a div b < 0"
 apply (subgoal_tac "a div b \<le> -1", force)
@@ -2225,7 +2225,6 @@
 using int_one_le_iff_zero_less[of "a div b"] zdiv_mono1[of b a b]apply simp
 done
 
-
 lemma zmod_le_nonneg_dividend: "(m::int) \<ge> 0 ==> m mod k \<le> m"
 apply (rule split_zmod[THEN iffD2])
 apply(fastforce dest: q_pos_lemma intro: split_mult_pos_le)
@@ -2384,7 +2383,7 @@
 proof-
   from xy have th: "int x - int y = int (x - y)" by simp 
   from xyn have "int x mod int n = int y mod int n" 
-    by (simp add: zmod_int[symmetric])
+    by (simp add: zmod_int [symmetric])
   hence "int n dvd int x - int y" by (simp only: zmod_eq_dvd_iff[symmetric]) 
   hence "n dvd x - y" by (simp add: th zdvd_int)
   then show ?thesis using xy unfolding dvd_def apply clarsimp apply (rule_tac x="k" in exI) by arith
diff -r 0133d31f9ab4 -r 26977429b784 src/HOL/IsaMakefile
--- a/src/HOL/IsaMakefile	Mon Feb 20 22:35:32 2012 +0100
+++ b/src/HOL/IsaMakefile	Tue Feb 21 12:45:00 2012 +0100
@@ -1059,9 +1059,9 @@
   ex/Lagrange.thy ex/List_to_Set_Comprehension_Examples.thy		\
   ex/LocaleTest2.thy ex/MT.thy ex/MergeSort.thy ex/Meson_Test.thy	\
   ex/MonoidGroup.thy ex/Multiquote.thy ex/NatSum.thy			\
-  ex/Normalization_by_Evaluation.thy ex/Numeral.thy ex/PER.thy		\
-  ex/PresburgerEx.thy ex/Primrec.thy ex/Quickcheck_Examples.thy		\
-  ex/Quickcheck_Lattice_Examples.thy					\
+  ex/Normalization_by_Evaluation.thy ex/Numeral_Representation.thy	\
+  ex/PER.thy ex/PresburgerEx.thy ex/Primrec.thy				\
+  ex/Quickcheck_Examples.thy ex/Quickcheck_Lattice_Examples.thy		\
   ex/Quickcheck_Narrowing_Examples.thy ex/Quicksort.thy ex/ROOT.ML	\
   ex/Records.thy ex/ReflectionEx.thy ex/Refute_Examples.thy		\
   ex/SAT_Examples.thy ex/Serbian.thy ex/Set_Theory.thy			\
diff -r 0133d31f9ab4 -r 26977429b784 src/HOL/Lattices.thy
diff -r 0133d31f9ab4 -r 26977429b784 src/HOL/Matrix/ComputeNumeral.thy
--- a/src/HOL/Matrix/ComputeNumeral.thy	Mon Feb 20 22:35:32 2012 +0100
+++ b/src/HOL/Matrix/ComputeNumeral.thy	Tue Feb 21 12:45:00 2012 +0100
@@ -147,19 +147,16 @@
 lemma adjust: "adjust b (q, r) = (if 0 \<le> r - b then (2 * q + 1, r - b) else (2 * q, r))"
   by (auto simp only: adjust_def)
 
-lemma negateSnd: "negateSnd (q, r) = (q, -r)" 
-  by (simp add: negateSnd_def)
-
 lemma divmod: "divmod_int a b = (if 0\<le>a then
                   if 0\<le>b then posDivAlg a b
                   else if a=0 then (0, 0)
-                       else negateSnd (negDivAlg (-a) (-b))
+                       else apsnd uminus (negDivAlg (-a) (-b))
                else 
                   if 0<b then negDivAlg a b
-                  else negateSnd (posDivAlg (-a) (-b)))"
+                  else apsnd uminus (posDivAlg (-a) (-b)))"
   by (auto simp only: divmod_int_def)
 
-lemmas compute_div_mod = div_int_def mod_int_def divmod adjust negateSnd posDivAlg.simps negDivAlg.simps
+lemmas compute_div_mod = div_int_def mod_int_def divmod adjust apsnd_def map_pair_def posDivAlg.simps negDivAlg.simps
 
 
 
diff -r 0133d31f9ab4 -r 26977429b784 src/HOL/Orderings.thy
diff -r 0133d31f9ab4 -r 26977429b784 src/HOL/Predicate.thy
diff -r 0133d31f9ab4 -r 26977429b784 src/HOL/Product_Type.thy
diff -r 0133d31f9ab4 -r 26977429b784 src/HOL/ZF/Games.thy
diff -r 0133d31f9ab4 -r 26977429b784 src/HOL/ex/Numeral.thy
--- a/src/HOL/ex/Numeral.thy	Mon Feb 20 22:35:32 2012 +0100
+++ /dev/null	Thu Jan 01 00:00:00 1970 +0000
@@ -1,1116 +0,0 @@
-(*  Title:      HOL/ex/Numeral.thy
-    Author:     Florian Haftmann
-*)
-
-header {* An experimental alternative numeral representation. *}
-
-theory Numeral
-imports Main
-begin
-
-subsection {* The @{text num} type *}
-
-datatype num = One | Dig0 num | Dig1 num
-
-text {* Increment function for type @{typ num} *}
-
-primrec inc :: "num \<Rightarrow> num" where
-  "inc One = Dig0 One"
-| "inc (Dig0 x) = Dig1 x"
-| "inc (Dig1 x) = Dig0 (inc x)"
-
-text {* Converting between type @{typ num} and type @{typ nat} *}
-
-primrec nat_of_num :: "num \<Rightarrow> nat" where
-  "nat_of_num One = Suc 0"
-| "nat_of_num (Dig0 x) = nat_of_num x + nat_of_num x"
-| "nat_of_num (Dig1 x) = Suc (nat_of_num x + nat_of_num x)"
-
-primrec num_of_nat :: "nat \<Rightarrow> num" where
-  "num_of_nat 0 = One"
-| "num_of_nat (Suc n) = (if 0 < n then inc (num_of_nat n) else One)"
-
-lemma nat_of_num_pos: "0 < nat_of_num x"
-  by (induct x) simp_all
-
-lemma nat_of_num_neq_0: "nat_of_num x \<noteq> 0"
-  by (induct x) simp_all
-
-lemma nat_of_num_inc: "nat_of_num (inc x) = Suc (nat_of_num x)"
-  by (induct x) simp_all
-
-lemma num_of_nat_double:
-  "0 < n \<Longrightarrow> num_of_nat (n + n) = Dig0 (num_of_nat n)"
-  by (induct n) simp_all
-
-text {*
-  Type @{typ num} is isomorphic to the strictly positive
-  natural numbers.
-*}
-
-lemma nat_of_num_inverse: "num_of_nat (nat_of_num x) = x"
-  by (induct x) (simp_all add: num_of_nat_double nat_of_num_pos)
-
-lemma num_of_nat_inverse: "0 < n \<Longrightarrow> nat_of_num (num_of_nat n) = n"
-  by (induct n) (simp_all add: nat_of_num_inc)
-
-lemma num_eq_iff: "x = y \<longleftrightarrow> nat_of_num x = nat_of_num y"
-proof
-  assume "nat_of_num x = nat_of_num y"
-  then have "num_of_nat (nat_of_num x) = num_of_nat (nat_of_num y)" by simp
-  then show "x = y" by (simp add: nat_of_num_inverse)
-qed simp
-
-lemma num_induct [case_names One inc]:
-  fixes P :: "num \<Rightarrow> bool"
-  assumes One: "P One"
-    and inc: "\<And>x. P x \<Longrightarrow> P (inc x)"
-  shows "P x"
-proof -
-  obtain n where n: "Suc n = nat_of_num x"
-    by (cases "nat_of_num x", simp_all add: nat_of_num_neq_0)
-  have "P (num_of_nat (Suc n))"
-  proof (induct n)
-    case 0 show ?case using One by simp
-  next
-    case (Suc n)
-    then have "P (inc (num_of_nat (Suc n)))" by (rule inc)
-    then show "P (num_of_nat (Suc (Suc n)))" by simp
-  qed
-  with n show "P x"
-    by (simp add: nat_of_num_inverse)
-qed
-
-text {*
-  From now on, there are two possible models for @{typ num}: as
-  positive naturals (rule @{text "num_induct"}) and as digit
-  representation (rules @{text "num.induct"}, @{text "num.cases"}).
-
-  It is not entirely clear in which context it is better to use the
-  one or the other, or whether the construction should be reversed.
-*}
-
-
-subsection {* Numeral operations *}
-
-ML {*
-structure Dig_Simps = Named_Thms
-(
-  val name = @{binding numeral}
-  val description = "simplification rules for numerals"
-)
-*}
-
-setup Dig_Simps.setup
-
-instantiation num :: "{plus,times,ord}"
-begin
-
-definition plus_num :: "num \<Rightarrow> num \<Rightarrow> num" where
-  "m + n = num_of_nat (nat_of_num m + nat_of_num n)"
-
-definition times_num :: "num \<Rightarrow> num \<Rightarrow> num" where
-  "m * n = num_of_nat (nat_of_num m * nat_of_num n)"
-
-definition less_eq_num :: "num \<Rightarrow> num \<Rightarrow> bool" where
-  "m \<le> n \<longleftrightarrow> nat_of_num m \<le> nat_of_num n"
-
-definition less_num :: "num \<Rightarrow> num \<Rightarrow> bool" where
-  "m < n \<longleftrightarrow> nat_of_num m < nat_of_num n"
-
-instance ..
-
-end
-
-lemma nat_of_num_add: "nat_of_num (x + y) = nat_of_num x + nat_of_num y"
-  unfolding plus_num_def
-  by (intro num_of_nat_inverse add_pos_pos nat_of_num_pos)
-
-lemma nat_of_num_mult: "nat_of_num (x * y) = nat_of_num x * nat_of_num y"
-  unfolding times_num_def
-  by (intro num_of_nat_inverse mult_pos_pos nat_of_num_pos)
-
-lemma Dig_plus [numeral, simp, code]:
-  "One + One = Dig0 One"
-  "One + Dig0 m = Dig1 m"
-  "One + Dig1 m = Dig0 (m + One)"
-  "Dig0 n + One = Dig1 n"
-  "Dig0 n + Dig0 m = Dig0 (n + m)"
-  "Dig0 n + Dig1 m = Dig1 (n + m)"
-  "Dig1 n + One = Dig0 (n + One)"
-  "Dig1 n + Dig0 m = Dig1 (n + m)"
-  "Dig1 n + Dig1 m = Dig0 (n + m + One)"
-  by (simp_all add: num_eq_iff nat_of_num_add)
-
-lemma Dig_times [numeral, simp, code]:
-  "One * One = One"
-  "One * Dig0 n = Dig0 n"
-  "One * Dig1 n = Dig1 n"
-  "Dig0 n * One = Dig0 n"
-  "Dig0 n * Dig0 m = Dig0 (n * Dig0 m)"
-  "Dig0 n * Dig1 m = Dig0 (n * Dig1 m)"
-  "Dig1 n * One = Dig1 n"
-  "Dig1 n * Dig0 m = Dig0 (n * Dig0 m + m)"
-  "Dig1 n * Dig1 m = Dig1 (n * Dig1 m + m)"
-  by (simp_all add: num_eq_iff nat_of_num_add nat_of_num_mult
-                    left_distrib right_distrib)
-
-lemma less_eq_num_code [numeral, simp, code]:
-  "One \<le> n \<longleftrightarrow> True"
-  "Dig0 m \<le> One \<longleftrightarrow> False"
-  "Dig1 m \<le> One \<longleftrightarrow> False"
-  "Dig0 m \<le> Dig0 n \<longleftrightarrow> m \<le> n"
-  "Dig0 m \<le> Dig1 n \<longleftrightarrow> m \<le> n"
-  "Dig1 m \<le> Dig1 n \<longleftrightarrow> m \<le> n"
-  "Dig1 m \<le> Dig0 n \<longleftrightarrow> m < n"
-  using nat_of_num_pos [of n] nat_of_num_pos [of m]
-  by (auto simp add: less_eq_num_def less_num_def)
-
-lemma less_num_code [numeral, simp, code]:
-  "m < One \<longleftrightarrow> False"
-  "One < One \<longleftrightarrow> False"
-  "One < Dig0 n \<longleftrightarrow> True"
-  "One < Dig1 n \<longleftrightarrow> True"
-  "Dig0 m < Dig0 n \<longleftrightarrow> m < n"
-  "Dig0 m < Dig1 n \<longleftrightarrow> m \<le> n"
-  "Dig1 m < Dig1 n \<longleftrightarrow> m < n"
-  "Dig1 m < Dig0 n \<longleftrightarrow> m < n"
-  using nat_of_num_pos [of n] nat_of_num_pos [of m]
-  by (auto simp add: less_eq_num_def less_num_def)
-
-text {* Rules using @{text One} and @{text inc} as constructors *}
-
-lemma add_One: "x + One = inc x"
-  by (simp add: num_eq_iff nat_of_num_add nat_of_num_inc)
-
-lemma add_inc: "x + inc y = inc (x + y)"
-  by (simp add: num_eq_iff nat_of_num_add nat_of_num_inc)
-
-lemma mult_One: "x * One = x"
-  by (simp add: num_eq_iff nat_of_num_mult)
-
-lemma mult_inc: "x * inc y = x * y + x"
-  by (simp add: num_eq_iff nat_of_num_mult nat_of_num_add nat_of_num_inc)
-
-text {* A double-and-decrement function *}
-
-primrec DigM :: "num \<Rightarrow> num" where
-  "DigM One = One"
-| "DigM (Dig0 n) = Dig1 (DigM n)"
-| "DigM (Dig1 n) = Dig1 (Dig0 n)"
-
-lemma DigM_plus_one: "DigM n + One = Dig0 n"
-  by (induct n) simp_all
-
-lemma add_One_commute: "One + n = n + One"
-  by (induct n) simp_all
-
-lemma one_plus_DigM: "One + DigM n = Dig0 n"
-  by (simp add: add_One_commute DigM_plus_one)
-
-text {* Squaring and exponentiation *}
-
-primrec square :: "num \<Rightarrow> num" where
-  "square One = One"
-| "square (Dig0 n) = Dig0 (Dig0 (square n))"
-| "square (Dig1 n) = Dig1 (Dig0 (square n + n))"
-
-primrec pow :: "num \<Rightarrow> num \<Rightarrow> num" where
-  "pow x One = x"
-| "pow x (Dig0 y) = square (pow x y)"
-| "pow x (Dig1 y) = x * square (pow x y)"
-
-
-subsection {* Binary numerals *}
-
-text {*
-  We embed binary representations into a generic algebraic
-  structure using @{text of_num}.
-*}
-
-class semiring_numeral = semiring + monoid_mult
-begin
-
-primrec of_num :: "num \<Rightarrow> 'a" where
-  of_num_One [numeral]: "of_num One = 1"
-| "of_num (Dig0 n) = of_num n + of_num n"
-| "of_num (Dig1 n) = of_num n + of_num n + 1"
-
-lemma of_num_inc: "of_num (inc n) = of_num n + 1"
-  by (induct n) (simp_all add: add_ac)
-
-lemma of_num_add: "of_num (m + n) = of_num m + of_num n"
-  by (induct n rule: num_induct) (simp_all add: add_One add_inc of_num_inc add_ac)
-
-lemma of_num_mult: "of_num (m * n) = of_num m * of_num n"
-  by (induct n rule: num_induct) (simp_all add: mult_One mult_inc of_num_add of_num_inc right_distrib)
-
-declare of_num.simps [simp del]
-
-end
-
-ML {*
-fun mk_num k =
-  if k > 1 then
-    let
-      val (l, b) = Integer.div_mod k 2;
-      val bit = (if b = 0 then @{term Dig0} else @{term Dig1});
-    in bit $ (mk_num l) end
-  else if k = 1 then @{term One}
-  else error ("mk_num " ^ string_of_int k);
-
-fun dest_num @{term One} = 1
-  | dest_num (@{term Dig0} $ n) = 2 * dest_num n
-  | dest_num (@{term Dig1} $ n) = 2 * dest_num n + 1
-  | dest_num t = raise TERM ("dest_num", [t]);
-
-fun mk_numeral phi T k = Morphism.term phi (Const (@{const_name of_num}, @{typ num} --> T))
-  $ mk_num k
-
-fun dest_numeral phi (u $ t) =
-  if Term.aconv_untyped (u, Morphism.term phi (Const (@{const_name of_num}, dummyT)))
-  then (range_type (fastype_of u), dest_num t)
-  else raise TERM ("dest_numeral", [u, t]);
-*}
-
-syntax
-  "_Numerals" :: "xnum_token \<Rightarrow> 'a"    ("_")
-
-parse_translation {*
-let
-  fun num_of_int n = if n > 0 then case IntInf.quotRem (n, 2)
-     of (0, 1) => Const (@{const_name One}, dummyT)
-      | (n, 0) => Const (@{const_name Dig0}, dummyT) $ num_of_int n
-      | (n, 1) => Const (@{const_name Dig1}, dummyT) $ num_of_int n
-    else raise Match;
-  fun numeral_tr [Free (num, _)] =
-        let
-          val {leading_zeros, value, ...} = Lexicon.read_xnum num;
-          val _ = leading_zeros = 0 andalso value > 0
-            orelse error ("Bad numeral: " ^ num);
-        in Const (@{const_name of_num}, @{typ num} --> dummyT) $ num_of_int value end
-    | numeral_tr ts = raise TERM ("numeral_tr", ts);
-in [(@{syntax_const "_Numerals"}, numeral_tr)] end
-*}
-
-typed_print_translation (advanced) {*
-let
-  fun dig b n = b + 2 * n; 
-  fun int_of_num' (Const (@{const_syntax Dig0}, _) $ n) =
-        dig 0 (int_of_num' n)
-    | int_of_num' (Const (@{const_syntax Dig1}, _) $ n) =
-        dig 1 (int_of_num' n)
-    | int_of_num' (Const (@{const_syntax One}, _)) = 1;
-  fun num_tr' ctxt T [n] =
-    let
-      val k = int_of_num' n;
-      val t' = Syntax.const @{syntax_const "_Numerals"} $ Syntax.free ("#" ^ string_of_int k);
-    in
-      case T of
-        Type (@{type_name fun}, [_, T']) =>
-          if not (Config.get ctxt show_types) andalso can Term.dest_Type T' then t'
-          else Syntax.const @{syntax_const "_constrain"} $ t' $ Syntax_Phases.term_of_typ ctxt T'
-      | T' => if T' = dummyT then t' else raise Match
-    end;
-in [(@{const_syntax of_num}, num_tr')] end
-*}
-
-
-subsection {* Class-specific numeral rules *}
-
-subsubsection {* Class @{text semiring_numeral} *}
-
-context semiring_numeral
-begin
-
-abbreviation "Num1 \<equiv> of_num One"
-
-text {*
-  Alas, there is still the duplication of @{term 1}, although the
-  duplicated @{term 0} has disappeared.  We could get rid of it by
-  replacing the constructor @{term 1} in @{typ num} by two
-  constructors @{text two} and @{text three}, resulting in a further
-  blow-up.  But it could be worth the effort.
-*}
-
-lemma of_num_plus_one [numeral]:
-  "of_num n + 1 = of_num (n + One)"
-  by (simp only: of_num_add of_num_One)
-
-lemma of_num_one_plus [numeral]:
-  "1 + of_num n = of_num (One + n)"
-  by (simp only: of_num_add of_num_One)
-
-lemma of_num_plus [numeral]:
-  "of_num m + of_num n = of_num (m + n)"
-  by (simp only: of_num_add)
-
-lemma of_num_times_one [numeral]:
-  "of_num n * 1 = of_num n"
-  by simp
-
-lemma of_num_one_times [numeral]:
-  "1 * of_num n = of_num n"
-  by simp
-
-lemma of_num_times [numeral]:
-  "of_num m * of_num n = of_num (m * n)"
-  unfolding of_num_mult ..
-
-end
-
-
-subsubsection {* Structures with a zero: class @{text semiring_1} *}
-
-context semiring_1
-begin
-
-subclass semiring_numeral ..
-
-lemma of_nat_of_num [numeral]: "of_nat (of_num n) = of_num n"
-  by (induct n)
-    (simp_all add: semiring_numeral_class.of_num.simps of_num.simps add_ac)
-
-declare of_nat_1 [numeral]
-
-lemma Dig_plus_zero [numeral]:
-  "0 + 1 = 1"
-  "0 + of_num n = of_num n"
-  "1 + 0 = 1"
-  "of_num n + 0 = of_num n"
-  by simp_all
-
-lemma Dig_times_zero [numeral]:
-  "0 * 1 = 0"
-  "0 * of_num n = 0"
-  "1 * 0 = 0"
-  "of_num n * 0 = 0"
-  by simp_all
-
-end
-
-lemma nat_of_num_of_num: "nat_of_num = of_num"
-proof
-  fix n
-  have "of_num n = nat_of_num n"
-    by (induct n) (simp_all add: of_num.simps)
-  then show "nat_of_num n = of_num n" by simp
-qed
-
-
-subsubsection {* Equality: class @{text semiring_char_0} *}
-
-context semiring_char_0
-begin
-
-lemma of_num_eq_iff [numeral]: "of_num m = of_num n \<longleftrightarrow> m = n"
-  unfolding of_nat_of_num [symmetric] nat_of_num_of_num [symmetric]
-    of_nat_eq_iff num_eq_iff ..
-
-lemma of_num_eq_one_iff [numeral]: "of_num n = 1 \<longleftrightarrow> n = One"
-  using of_num_eq_iff [of n One] by (simp add: of_num_One)
-
-lemma one_eq_of_num_iff [numeral]: "1 = of_num n \<longleftrightarrow> One = n"
-  using of_num_eq_iff [of One n] by (simp add: of_num_One)
-
-end
-
-
-subsubsection {* Comparisons: class @{text linordered_semidom} *}
-
-text {*
-  Perhaps the underlying structure could even 
-  be more general than @{text linordered_semidom}.
-*}
-
-context linordered_semidom
-begin
-
-lemma of_num_pos [numeral]: "0 < of_num n"
-  by (induct n) (simp_all add: of_num.simps add_pos_pos)
-
-lemma of_num_not_zero [numeral]: "of_num n \<noteq> 0"
-  using of_num_pos [of n] by simp
-
-lemma of_num_less_eq_iff [numeral]: "of_num m \<le> of_num n \<longleftrightarrow> m \<le> n"
-proof -
-  have "of_nat (of_num m) \<le> of_nat (of_num n) \<longleftrightarrow> m \<le> n"
-    unfolding less_eq_num_def nat_of_num_of_num of_nat_le_iff ..
-  then show ?thesis by (simp add: of_nat_of_num)
-qed
-
-lemma of_num_less_eq_one_iff [numeral]: "of_num n \<le> 1 \<longleftrightarrow> n \<le> One"
-  using of_num_less_eq_iff [of n One] by (simp add: of_num_One)
-
-lemma one_less_eq_of_num_iff [numeral]: "1 \<le> of_num n"
-  using of_num_less_eq_iff [of One n] by (simp add: of_num_One)
-
-lemma of_num_less_iff [numeral]: "of_num m < of_num n \<longleftrightarrow> m < n"
-proof -
-  have "of_nat (of_num m) < of_nat (of_num n) \<longleftrightarrow> m < n"
-    unfolding less_num_def nat_of_num_of_num of_nat_less_iff ..
-  then show ?thesis by (simp add: of_nat_of_num)
-qed
-
-lemma of_num_less_one_iff [numeral]: "\<not> of_num n < 1"
-  using of_num_less_iff [of n One] by (simp add: of_num_One)
-
-lemma one_less_of_num_iff [numeral]: "1 < of_num n \<longleftrightarrow> One < n"
-  using of_num_less_iff [of One n] by (simp add: of_num_One)
-
-lemma of_num_nonneg [numeral]: "0 \<le> of_num n"
-  by (induct n) (simp_all add: of_num.simps add_nonneg_nonneg)
-
-lemma of_num_less_zero_iff [numeral]: "\<not> of_num n < 0"
-  by (simp add: not_less of_num_nonneg)
-
-lemma of_num_le_zero_iff [numeral]: "\<not> of_num n \<le> 0"
-  by (simp add: not_le of_num_pos)
-
-end
-
-context linordered_idom
-begin
-
-lemma minus_of_num_less_of_num_iff: "- of_num m < of_num n"
-proof -
-  have "- of_num m < 0" by (simp add: of_num_pos)
-  also have "0 < of_num n" by (simp add: of_num_pos)
-  finally show ?thesis .
-qed
-
-lemma minus_of_num_not_equal_of_num: "- of_num m \<noteq> of_num n"
-  using minus_of_num_less_of_num_iff [of m n] by simp
-
-lemma minus_of_num_less_one_iff: "- of_num n < 1"
-  using minus_of_num_less_of_num_iff [of n One] by (simp add: of_num_One)
-
-lemma minus_one_less_of_num_iff: "- 1 < of_num n"
-  using minus_of_num_less_of_num_iff [of One n] by (simp add: of_num_One)
-
-lemma minus_one_less_one_iff: "- 1 < 1"
-  using minus_of_num_less_of_num_iff [of One One] by (simp add: of_num_One)
-
-lemma minus_of_num_le_of_num_iff: "- of_num m \<le> of_num n"
-  by (simp add: less_imp_le minus_of_num_less_of_num_iff)
-
-lemma minus_of_num_le_one_iff: "- of_num n \<le> 1"
-  by (simp add: less_imp_le minus_of_num_less_one_iff)
-
-lemma minus_one_le_of_num_iff: "- 1 \<le> of_num n"
-  by (simp add: less_imp_le minus_one_less_of_num_iff)
-
-lemma minus_one_le_one_iff: "- 1 \<le> 1"
-  by (simp add: less_imp_le minus_one_less_one_iff)
-
-lemma of_num_le_minus_of_num_iff: "\<not> of_num m \<le> - of_num n"
-  by (simp add: not_le minus_of_num_less_of_num_iff)
-
-lemma one_le_minus_of_num_iff: "\<not> 1 \<le> - of_num n"
-  by (simp add: not_le minus_of_num_less_one_iff)
-
-lemma of_num_le_minus_one_iff: "\<not> of_num n \<le> - 1"
-  by (simp add: not_le minus_one_less_of_num_iff)
-
-lemma one_le_minus_one_iff: "\<not> 1 \<le> - 1"
-  by (simp add: not_le minus_one_less_one_iff)
-
-lemma of_num_less_minus_of_num_iff: "\<not> of_num m < - of_num n"
-  by (simp add: not_less minus_of_num_le_of_num_iff)
-
-lemma one_less_minus_of_num_iff: "\<not> 1 < - of_num n"
-  by (simp add: not_less minus_of_num_le_one_iff)
-
-lemma of_num_less_minus_one_iff: "\<not> of_num n < - 1"
-  by (simp add: not_less minus_one_le_of_num_iff)
-
-lemma one_less_minus_one_iff: "\<not> 1 < - 1"
-  by (simp add: not_less minus_one_le_one_iff)
-
-lemmas le_signed_numeral_special [numeral] =
-  minus_of_num_le_of_num_iff
-  minus_of_num_le_one_iff
-  minus_one_le_of_num_iff
-  minus_one_le_one_iff
-  of_num_le_minus_of_num_iff
-  one_le_minus_of_num_iff
-  of_num_le_minus_one_iff
-  one_le_minus_one_iff
-
-lemmas less_signed_numeral_special [numeral] =
-  minus_of_num_less_of_num_iff
-  minus_of_num_not_equal_of_num
-  minus_of_num_less_one_iff
-  minus_one_less_of_num_iff
-  minus_one_less_one_iff
-  of_num_less_minus_of_num_iff
-  one_less_minus_of_num_iff
-  of_num_less_minus_one_iff
-  one_less_minus_one_iff
-
-end
-
-subsubsection {* Structures with subtraction: class @{text semiring_1_minus} *}
-
-class semiring_minus = semiring + minus + zero +
-  assumes minus_inverts_plus1: "a + b = c \<Longrightarrow> c - b = a"
-  assumes minus_minus_zero_inverts_plus1: "a + b = c \<Longrightarrow> b - c = 0 - a"
-begin
-
-lemma minus_inverts_plus2: "a + b = c \<Longrightarrow> c - a = b"
-  by (simp add: add_ac minus_inverts_plus1 [of b a])
-
-lemma minus_minus_zero_inverts_plus2: "a + b = c \<Longrightarrow> a - c = 0 - b"
-  by (simp add: add_ac minus_minus_zero_inverts_plus1 [of b a])
-
-end
-
-class semiring_1_minus = semiring_1 + semiring_minus
-begin
-
-lemma Dig_of_num_pos:
-  assumes "k + n = m"
-  shows "of_num m - of_num n = of_num k"
-  using assms by (simp add: of_num_plus minus_inverts_plus1)
-
-lemma Dig_of_num_zero:
-  shows "of_num n - of_num n = 0"
-  by (rule minus_inverts_plus1) simp
-
-lemma Dig_of_num_neg:
-  assumes "k + m = n"
-  shows "of_num m - of_num n = 0 - of_num k"
-  by (rule minus_minus_zero_inverts_plus1) (simp add: of_num_plus assms)
-
-lemmas Dig_plus_eval =
-  of_num_plus of_num_eq_iff Dig_plus refl [of One, THEN eqTrueI] num.inject
-
-simproc_setup numeral_minus ("of_num m - of_num n") = {*
-  let
-    (*TODO proper implicit use of morphism via pattern antiquotations*)
-    fun cdest_of_num ct = (List.last o snd o Drule.strip_comb) ct;
-    fun cdest_minus ct = case (rev o snd o Drule.strip_comb) ct of [n, m] => (m, n);
-    fun attach_num ct = (dest_num (Thm.term_of ct), ct);
-    fun cdifference t = (pairself (attach_num o cdest_of_num) o cdest_minus) t;
-    val simplify = Raw_Simplifier.rewrite false (map mk_meta_eq @{thms Dig_plus_eval});
-    fun cert ck cl cj = @{thm eqTrueE} OF [@{thm meta_eq_to_obj_eq}
-      OF [simplify (Drule.list_comb (@{cterm "op = :: num \<Rightarrow> _"},
-        [Drule.list_comb (@{cterm "op + :: num \<Rightarrow> _"}, [ck, cl]), cj]))]];
-  in fn phi => fn _ => fn ct => case try cdifference ct
-   of NONE => (NONE)
-    | SOME ((k, ck), (l, cl)) => SOME (let val j = k - l in if j = 0
-        then Raw_Simplifier.rewrite false [mk_meta_eq (Morphism.thm phi @{thm Dig_of_num_zero})] ct
-        else mk_meta_eq (let
-          val cj = Thm.cterm_of (Thm.theory_of_cterm ct) (mk_num (abs j));
-        in
-          (if j > 0 then (Morphism.thm phi @{thm Dig_of_num_pos}) OF [cert cj cl ck]
-          else (Morphism.thm phi @{thm Dig_of_num_neg}) OF [cert cj ck cl])
-        end) end)
-  end
-*}
-
-lemma Dig_of_num_minus_zero [numeral]:
-  "of_num n - 0 = of_num n"
-  by (simp add: minus_inverts_plus1)
-
-lemma Dig_one_minus_zero [numeral]:
-  "1 - 0 = 1"
-  by (simp add: minus_inverts_plus1)
-
-lemma Dig_one_minus_one [numeral]:
-  "1 - 1 = 0"
-  by (simp add: minus_inverts_plus1)
-
-lemma Dig_of_num_minus_one [numeral]:
-  "of_num (Dig0 n) - 1 = of_num (DigM n)"
-  "of_num (Dig1 n) - 1 = of_num (Dig0 n)"
-  by (auto intro: minus_inverts_plus1 simp add: DigM_plus_one of_num.simps of_num_plus_one)
-
-lemma Dig_one_minus_of_num [numeral]:
-  "1 - of_num (Dig0 n) = 0 - of_num (DigM n)"
-  "1 - of_num (Dig1 n) = 0 - of_num (Dig0 n)"
-  by (auto intro: minus_minus_zero_inverts_plus1 simp add: DigM_plus_one of_num.simps of_num_plus_one)
-
-end
-
-
-subsubsection {* Structures with negation: class @{text ring_1} *}
-
-context ring_1
-begin
-
-subclass semiring_1_minus proof
-qed (simp_all add: algebra_simps)
-
-lemma Dig_zero_minus_of_num [numeral]:
-  "0 - of_num n = - of_num n"
-  by simp
-
-lemma Dig_zero_minus_one [numeral]:
-  "0 - 1 = - 1"
-  by simp
-
-lemma Dig_uminus_uminus [numeral]:
-  "- (- of_num n) = of_num n"
-  by simp
-
-lemma Dig_plus_uminus [numeral]:
-  "of_num m + - of_num n = of_num m - of_num n"
-  "- of_num m + of_num n = of_num n - of_num m"
-  "- of_num m + - of_num n = - (of_num m + of_num n)"
-  "of_num m - - of_num n = of_num m + of_num n"
-  "- of_num m - of_num n = - (of_num m + of_num n)"
-  "- of_num m - - of_num n = of_num n - of_num m"
-  by (simp_all add: diff_minus add_commute)
-
-lemma Dig_times_uminus [numeral]:
-  "- of_num n * of_num m = - (of_num n * of_num m)"
-  "of_num n * - of_num m = - (of_num n * of_num m)"
-  "- of_num n * - of_num m = of_num n * of_num m"
-  by simp_all
-
-lemma of_int_of_num [numeral]: "of_int (of_num n) = of_num n"
-by (induct n)
-  (simp_all only: of_num.simps semiring_numeral_class.of_num.simps of_int_add, simp_all)
-
-declare of_int_1 [numeral]
-
-end
-
-
-subsubsection {* Structures with exponentiation *}
-
-lemma of_num_square: "of_num (square x) = of_num x * of_num x"
-by (induct x)
-   (simp_all add: of_num.simps of_num_add algebra_simps)
-
-lemma of_num_pow: "of_num (pow x y) = of_num x ^ of_num y"
-by (induct y)
-   (simp_all add: of_num.simps of_num_square of_num_mult power_add)
-
-lemma power_of_num [numeral]: "of_num x ^ of_num y = of_num (pow x y)"
-  unfolding of_num_pow ..
-
-lemma power_zero_of_num [numeral]:
-  "0 ^ of_num n = (0::'a::semiring_1)"
-  using of_num_pos [where n=n and ?'a=nat]
-  by (simp add: power_0_left)
-
-lemma power_minus_Dig0 [numeral]:
-  fixes x :: "'a::ring_1"
-  shows "(- x) ^ of_num (Dig0 n) = x ^ of_num (Dig0 n)"
-  by (induct n rule: num_induct) (simp_all add: of_num.simps of_num_inc)
-
-lemma power_minus_Dig1 [numeral]:
-  fixes x :: "'a::ring_1"
-  shows "(- x) ^ of_num (Dig1 n) = - (x ^ of_num (Dig1 n))"
-  by (induct n rule: num_induct) (simp_all add: of_num.simps of_num_inc)
-
-declare power_one [numeral]
-
-
-subsubsection {* Greetings to @{typ nat}. *}
-
-instance nat :: semiring_1_minus proof
-qed simp_all
-
-lemma Suc_of_num [numeral]: "Suc (of_num n) = of_num (n + One)"
-  unfolding of_num_plus_one [symmetric] by simp
-
-lemma nat_number:
-  "1 = Suc 0"
-  "of_num One = Suc 0"
-  "of_num (Dig0 n) = Suc (of_num (DigM n))"
-  "of_num (Dig1 n) = Suc (of_num (Dig0 n))"
-  by (simp_all add: of_num.simps DigM_plus_one Suc_of_num)
-
-declare diff_0_eq_0 [numeral]
-
-
-subsection {* Proof tools setup *}
-
-subsubsection {* Numeral equations as default simplification rules *}
-
-declare (in semiring_numeral) of_num_One [simp]
-declare (in semiring_numeral) of_num_plus_one [simp]
-declare (in semiring_numeral) of_num_one_plus [simp]
-declare (in semiring_numeral) of_num_plus [simp]
-declare (in semiring_numeral) of_num_times [simp]
-
-declare (in semiring_1) of_nat_of_num [simp]
-
-declare (in semiring_char_0) of_num_eq_iff [simp]
-declare (in semiring_char_0) of_num_eq_one_iff [simp]
-declare (in semiring_char_0) one_eq_of_num_iff [simp]
-
-declare (in linordered_semidom) of_num_pos [simp]
-declare (in linordered_semidom) of_num_not_zero [simp]
-declare (in linordered_semidom) of_num_less_eq_iff [simp]
-declare (in linordered_semidom) of_num_less_eq_one_iff [simp]
-declare (in linordered_semidom) one_less_eq_of_num_iff [simp]
-declare (in linordered_semidom) of_num_less_iff [simp]
-declare (in linordered_semidom) of_num_less_one_iff [simp]
-declare (in linordered_semidom) one_less_of_num_iff [simp]
-declare (in linordered_semidom) of_num_nonneg [simp]
-declare (in linordered_semidom) of_num_less_zero_iff [simp]
-declare (in linordered_semidom) of_num_le_zero_iff [simp]
-
-declare (in linordered_idom) le_signed_numeral_special [simp]
-declare (in linordered_idom) less_signed_numeral_special [simp]
-
-declare (in semiring_1_minus) Dig_of_num_minus_one [simp]
-declare (in semiring_1_minus) Dig_one_minus_of_num [simp]
-
-declare (in ring_1) Dig_plus_uminus [simp]
-declare (in ring_1) of_int_of_num [simp]
-
-declare power_of_num [simp]
-declare power_zero_of_num [simp]
-declare power_minus_Dig0 [simp]
-declare power_minus_Dig1 [simp]
-
-declare Suc_of_num [simp]
-
-
-subsubsection {* Reorientation of equalities *}
-
-setup {*
-  Reorient_Proc.add
-    (fn Const(@{const_name of_num}, _) $ _ => true
-      | Const(@{const_name uminus}, _) $
-          (Const(@{const_name of_num}, _) $ _) => true
-      | _ => false)
-*}
-
-simproc_setup reorient_num ("of_num n = x" | "- of_num m = y") = Reorient_Proc.proc
-
-
-subsubsection {* Constant folding for multiplication in semirings *}
-
-context semiring_numeral
-begin
-
-lemma mult_of_num_commute: "x * of_num n = of_num n * x"
-by (induct n)
-  (simp_all only: of_num.simps left_distrib right_distrib mult_1_left mult_1_right)
-
-definition
-  "commutes_with a b \<longleftrightarrow> a * b = b * a"
-
-lemma commutes_with_commute: "commutes_with a b \<Longrightarrow> a * b = b * a"
-unfolding commutes_with_def .
-
-lemma commutes_with_left_commute: "commutes_with a b \<Longrightarrow> a * (b * c) = b * (a * c)"
-unfolding commutes_with_def by (simp only: mult_assoc [symmetric])
-
-lemma commutes_with_numeral: "commutes_with x (of_num n)" "commutes_with (of_num n) x"
-unfolding commutes_with_def by (simp_all add: mult_of_num_commute)
-
-lemmas mult_ac_numeral =
-  mult_assoc
-  commutes_with_commute
-  commutes_with_left_commute
-  commutes_with_numeral
-
-end
-
-ML {*
-structure Semiring_Times_Assoc_Data : ASSOC_FOLD_DATA =
-struct
-  val assoc_ss = HOL_ss addsimps @{thms mult_ac_numeral}
-  val eq_reflection = eq_reflection
-  fun is_numeral (Const(@{const_name of_num}, _) $ _) = true
-    | is_numeral _ = false;
-end;
-
-structure Semiring_Times_Assoc = Assoc_Fold (Semiring_Times_Assoc_Data);
-*}
-
-simproc_setup semiring_assoc_fold' ("(a::'a::semiring_numeral) * b") =
-  {* fn phi => fn ss => fn ct =>
-    Semiring_Times_Assoc.proc ss (Thm.term_of ct) *}
-
-
-subsection {* Code generator setup for @{typ int} *}
-
-text {* Reversing standard setup *}
-
-lemma [code_unfold del]: "(0::int) \<equiv> Numeral0" by simp
-lemma [code_unfold del]: "(1::int) \<equiv> Numeral1" by simp
-declare zero_is_num_zero [code_unfold del]
-declare one_is_num_one [code_unfold del]
-  
-lemma [code, code del]:
-  "(1 :: int) = 1"
-  "(op + :: int \<Rightarrow> int \<Rightarrow> int) = op +"
-  "(uminus :: int \<Rightarrow> int) = uminus"
-  "(op - :: int \<Rightarrow> int \<Rightarrow> int) = op -"
-  "(op * :: int \<Rightarrow> int \<Rightarrow> int) = op *"
-  "(HOL.equal :: int \<Rightarrow> int \<Rightarrow> bool) = HOL.equal"
-  "(op \<le> :: int \<Rightarrow> int \<Rightarrow> bool) = op \<le>"
-  "(op < :: int \<Rightarrow> int \<Rightarrow> bool) = op <"
-  by rule+
-
-text {* Constructors *}
-
-definition Pls :: "num \<Rightarrow> int" where
-  [simp, code_post]: "Pls n = of_num n"
-
-definition Mns :: "num \<Rightarrow> int" where
-  [simp, code_post]: "Mns n = - of_num n"
-
-code_datatype "0::int" Pls Mns
-
-lemmas [code_unfold] = Pls_def [symmetric] Mns_def [symmetric]
-
-text {* Auxiliary operations *}
-
-definition dup :: "int \<Rightarrow> int" where
-  [simp]: "dup k = k + k"
-
-lemma Dig_dup [code]:
-  "dup 0 = 0"
-  "dup (Pls n) = Pls (Dig0 n)"
-  "dup (Mns n) = Mns (Dig0 n)"
-  by (simp_all add: of_num.simps)
-
-definition sub :: "num \<Rightarrow> num \<Rightarrow> int" where
-  [simp]: "sub m n = (of_num m - of_num n)"
-
-lemma Dig_sub [code]:
-  "sub One One = 0"
-  "sub (Dig0 m) One = of_num (DigM m)"
-  "sub (Dig1 m) One = of_num (Dig0 m)"
-  "sub One (Dig0 n) = - of_num (DigM n)"
-  "sub One (Dig1 n) = - of_num (Dig0 n)"
-  "sub (Dig0 m) (Dig0 n) = dup (sub m n)"
-  "sub (Dig1 m) (Dig1 n) = dup (sub m n)"
-  "sub (Dig1 m) (Dig0 n) = dup (sub m n) + 1"
-  "sub (Dig0 m) (Dig1 n) = dup (sub m n) - 1"
-  by (simp_all add: algebra_simps num_eq_iff nat_of_num_add)
-
-text {* Implementations *}
-
-lemma one_int_code [code]:
-  "1 = Pls One"
-  by simp
-
-lemma plus_int_code [code]:
-  "k + 0 = (k::int)"
-  "0 + l = (l::int)"
-  "Pls m + Pls n = Pls (m + n)"
-  "Pls m + Mns n = sub m n"
-  "Mns m + Pls n = sub n m"
-  "Mns m + Mns n = Mns (m + n)"
-  by simp_all
-
-lemma uminus_int_code [code]:
-  "uminus 0 = (0::int)"
-  "uminus (Pls m) = Mns m"
-  "uminus (Mns m) = Pls m"
-  by simp_all
-
-lemma minus_int_code [code]:
-  "k - 0 = (k::int)"
-  "0 - l = uminus (l::int)"
-  "Pls m - Pls n = sub m n"
-  "Pls m - Mns n = Pls (m + n)"
-  "Mns m - Pls n = Mns (m + n)"
-  "Mns m - Mns n = sub n m"
-  by simp_all
-
-lemma times_int_code [code]:
-  "k * 0 = (0::int)"
-  "0 * l = (0::int)"
-  "Pls m * Pls n = Pls (m * n)"
-  "Pls m * Mns n = Mns (m * n)"
-  "Mns m * Pls n = Mns (m * n)"
-  "Mns m * Mns n = Pls (m * n)"
-  by simp_all
-
-lemma eq_int_code [code]:
-  "HOL.equal 0 (0::int) \<longleftrightarrow> True"
-  "HOL.equal 0 (Pls l) \<longleftrightarrow> False"
-  "HOL.equal 0 (Mns l) \<longleftrightarrow> False"
-  "HOL.equal (Pls k) 0 \<longleftrightarrow> False"
-  "HOL.equal (Pls k) (Pls l) \<longleftrightarrow> HOL.equal k l"
-  "HOL.equal (Pls k) (Mns l) \<longleftrightarrow> False"
-  "HOL.equal (Mns k) 0 \<longleftrightarrow> False"
-  "HOL.equal (Mns k) (Pls l) \<longleftrightarrow> False"
-  "HOL.equal (Mns k) (Mns l) \<longleftrightarrow> HOL.equal k l"
-  by (auto simp add: equal dest: sym)
-
-lemma [code nbe]:
-  "HOL.equal (k::int) k \<longleftrightarrow> True"
-  by (fact equal_refl)
-
-lemma less_eq_int_code [code]:
-  "0 \<le> (0::int) \<longleftrightarrow> True"
-  "0 \<le> Pls l \<longleftrightarrow> True"
-  "0 \<le> Mns l \<longleftrightarrow> False"
-  "Pls k \<le> 0 \<longleftrightarrow> False"
-  "Pls k \<le> Pls l \<longleftrightarrow> k \<le> l"
-  "Pls k \<le> Mns l \<longleftrightarrow> False"
-  "Mns k \<le> 0 \<longleftrightarrow> True"
-  "Mns k \<le> Pls l \<longleftrightarrow> True"
-  "Mns k \<le> Mns l \<longleftrightarrow> l \<le> k"
-  by simp_all
-
-lemma less_int_code [code]:
-  "0 < (0::int) \<longleftrightarrow> False"
-  "0 < Pls l \<longleftrightarrow> True"
-  "0 < Mns l \<longleftrightarrow> False"
-  "Pls k < 0 \<longleftrightarrow> False"
-  "Pls k < Pls l \<longleftrightarrow> k < l"
-  "Pls k < Mns l \<longleftrightarrow> False"
-  "Mns k < 0 \<longleftrightarrow> True"
-  "Mns k < Pls l \<longleftrightarrow> True"
-  "Mns k < Mns l \<longleftrightarrow> l < k"
-  by simp_all
-
-hide_const (open) sub dup
-
-text {* Pretty literals *}
-
-ML {*
-local open Code_Thingol in
-
-fun add_code print target =
-  let
-    fun dest_num one' dig0' dig1' thm =
-      let
-        fun dest_dig (IConst (c, _)) = if c = dig0' then 0
-              else if c = dig1' then 1
-              else Code_Printer.eqn_error thm "Illegal numeral expression: illegal dig"
-          | dest_dig _ = Code_Printer.eqn_error thm "Illegal numeral expression: illegal digit";
-        fun dest_num (IConst (c, _)) = if c = one' then 1
-              else Code_Printer.eqn_error thm "Illegal numeral expression: illegal leading digit"
-          | dest_num (t1 `$ t2) = 2 * dest_num t2 + dest_dig t1
-          | dest_num _ = Code_Printer.eqn_error thm "Illegal numeral expression: illegal term";
-      in dest_num end;
-    fun pretty sgn literals [one', dig0', dig1'] _ thm _ _ [(t, _)] =
-      (Code_Printer.str o print literals o sgn o dest_num one' dig0' dig1' thm) t
-    fun add_syntax (c, sgn) = Code_Target.add_const_syntax target c
-      (SOME (Code_Printer.complex_const_syntax
-        (1, ([@{const_name One}, @{const_name Dig0}, @{const_name Dig1}],
-          pretty sgn))));
-  in
-    add_syntax (@{const_name Pls}, I)
-    #> add_syntax (@{const_name Mns}, (fn k => ~ k))
-  end;
-
-end
-*}
-
-hide_const (open) One Dig0 Dig1
-
-
-subsection {* Toy examples *}
-
-definition "foo \<longleftrightarrow> #4 * #2 + #7 = (#8 :: nat)"
-definition "bar \<longleftrightarrow> #4 * #2 + #7 \<ge> (#8 :: int) - #3"
-
-code_thms foo bar
-export_code foo bar checking SML OCaml? Haskell? Scala?
-
-text {* This is an ad-hoc @{text Code_Integer} setup. *}
-
-setup {*
-  fold (add_code Code_Printer.literal_numeral)
-    [Code_ML.target_SML, Code_ML.target_OCaml, Code_Haskell.target, Code_Scala.target]
-*}
-
-code_type int
-  (SML "IntInf.int")
-  (OCaml "Big'_int.big'_int")
-  (Haskell "Integer")
-  (Scala "BigInt")
-  (Eval "int")
-
-code_const "0::int"
-  (SML "0/ :/ IntInf.int")
-  (OCaml "Big'_int.zero")
-  (Haskell "0")
-  (Scala "BigInt(0)")
-  (Eval "0/ :/ int")
-
-code_const Int.pred
-  (SML "IntInf.- ((_), 1)")
-  (OCaml "Big'_int.pred'_big'_int")
-  (Haskell "!(_/ -/ 1)")
-  (Scala "!(_ -/ 1)")
-  (Eval "!(_/ -/ 1)")
-
-code_const Int.succ
-  (SML "IntInf.+ ((_), 1)")
-  (OCaml "Big'_int.succ'_big'_int")
-  (Haskell "!(_/ +/ 1)")
-  (Scala "!(_ +/ 1)")
-  (Eval "!(_/ +/ 1)")
-
-code_const "op + \<Colon> int \<Rightarrow> int \<Rightarrow> int"
-  (SML "IntInf.+ ((_), (_))")
-  (OCaml "Big'_int.add'_big'_int")
-  (Haskell infixl 6 "+")
-  (Scala infixl 7 "+")
-  (Eval infixl 8 "+")
-
-code_const "uminus \<Colon> int \<Rightarrow> int"
-  (SML "IntInf.~")
-  (OCaml "Big'_int.minus'_big'_int")
-  (Haskell "negate")
-  (Scala "!(- _)")
-  (Eval "~/ _")
-
-code_const "op - \<Colon> int \<Rightarrow> int \<Rightarrow> int"
-  (SML "IntInf.- ((_), (_))")
-  (OCaml "Big'_int.sub'_big'_int")
-  (Haskell infixl 6 "-")
-  (Scala infixl 7 "-")
-  (Eval infixl 8 "-")
-
-code_const "op * \<Colon> int \<Rightarrow> int \<Rightarrow> int"
-  (SML "IntInf.* ((_), (_))")
-  (OCaml "Big'_int.mult'_big'_int")
-  (Haskell infixl 7 "*")
-  (Scala infixl 8 "*")
-  (Eval infixl 9 "*")
-
-code_const pdivmod
-  (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 \<Colon> int \<Rightarrow> int \<Rightarrow> bool"
-  (SML "!((_ : IntInf.int) = _)")
-  (OCaml "Big'_int.eq'_big'_int")
-  (Haskell infix 4 "==")
-  (Scala infixl 5 "==")
-  (Eval infixl 6 "=")
-
-code_const "op \<le> \<Colon> int \<Rightarrow> int \<Rightarrow> bool"
-  (SML "IntInf.<= ((_), (_))")
-  (OCaml "Big'_int.le'_big'_int")
-  (Haskell infix 4 "<=")
-  (Scala infixl 4 "<=")
-  (Eval infixl 6 "<=")
-
-code_const "op < \<Colon> int \<Rightarrow> int \<Rightarrow> bool"
-  (SML "IntInf.< ((_), (_))")
-  (OCaml "Big'_int.lt'_big'_int")
-  (Haskell infix 4 "<")
-  (Scala infixl 4 "<")
-  (Eval infixl 6 "<")
-
-code_const Code_Numeral.int_of
-  (SML "IntInf.fromInt")
-  (OCaml "_")
-  (Haskell "toInteger")
-  (Scala "!_.as'_BigInt")
-  (Eval "_")
-
-export_code foo bar checking SML OCaml? Haskell? Scala?
-
-end
diff -r 0133d31f9ab4 -r 26977429b784 src/HOL/ex/Numeral_Representation.thy
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/ex/Numeral_Representation.thy	Tue Feb 21 12:45:00 2012 +0100
@@ -0,0 +1,1116 @@
+(*  Title:      HOL/ex/Numeral_Representation.thy
+    Author:     Florian Haftmann
+*)
+
+header {* An experimental alternative numeral representation. *}
+
+theory Numeral_Representation
+imports Main
+begin
+
+subsection {* The @{text num} type *}
+
+datatype num = One | Dig0 num | Dig1 num
+
+text {* Increment function for type @{typ num} *}
+
+primrec inc :: "num \<Rightarrow> num" where
+  "inc One = Dig0 One"
+| "inc (Dig0 x) = Dig1 x"
+| "inc (Dig1 x) = Dig0 (inc x)"
+
+text {* Converting between type @{typ num} and type @{typ nat} *}
+
+primrec nat_of_num :: "num \<Rightarrow> nat" where
+  "nat_of_num One = Suc 0"
+| "nat_of_num (Dig0 x) = nat_of_num x + nat_of_num x"
+| "nat_of_num (Dig1 x) = Suc (nat_of_num x + nat_of_num x)"
+
+primrec num_of_nat :: "nat \<Rightarrow> num" where
+  "num_of_nat 0 = One"
+| "num_of_nat (Suc n) = (if 0 < n then inc (num_of_nat n) else One)"
+
+lemma nat_of_num_pos: "0 < nat_of_num x"
+  by (induct x) simp_all
+
+lemma nat_of_num_neq_0: "nat_of_num x \<noteq> 0"
+  by (induct x) simp_all
+
+lemma nat_of_num_inc: "nat_of_num (inc x) = Suc (nat_of_num x)"
+  by (induct x) simp_all
+
+lemma num_of_nat_double:
+  "0 < n \<Longrightarrow> num_of_nat (n + n) = Dig0 (num_of_nat n)"
+  by (induct n) simp_all
+
+text {*
+  Type @{typ num} is isomorphic to the strictly positive
+  natural numbers.
+*}
+
+lemma nat_of_num_inverse: "num_of_nat (nat_of_num x) = x"
+  by (induct x) (simp_all add: num_of_nat_double nat_of_num_pos)
+
+lemma num_of_nat_inverse: "0 < n \<Longrightarrow> nat_of_num (num_of_nat n) = n"
+  by (induct n) (simp_all add: nat_of_num_inc)
+
+lemma num_eq_iff: "x = y \<longleftrightarrow> nat_of_num x = nat_of_num y"
+proof
+  assume "nat_of_num x = nat_of_num y"
+  then have "num_of_nat (nat_of_num x) = num_of_nat (nat_of_num y)" by simp
+  then show "x = y" by (simp add: nat_of_num_inverse)
+qed simp
+
+lemma num_induct [case_names One inc]:
+  fixes P :: "num \<Rightarrow> bool"
+  assumes One: "P One"
+    and inc: "\<And>x. P x \<Longrightarrow> P (inc x)"
+  shows "P x"
+proof -
+  obtain n where n: "Suc n = nat_of_num x"
+    by (cases "nat_of_num x", simp_all add: nat_of_num_neq_0)
+  have "P (num_of_nat (Suc n))"
+  proof (induct n)
+    case 0 show ?case using One by simp
+  next
+    case (Suc n)
+    then have "P (inc (num_of_nat (Suc n)))" by (rule inc)
+    then show "P (num_of_nat (Suc (Suc n)))" by simp
+  qed
+  with n show "P x"
+    by (simp add: nat_of_num_inverse)
+qed
+
+text {*
+  From now on, there are two possible models for @{typ num}: as
+  positive naturals (rule @{text "num_induct"}) and as digit
+  representation (rules @{text "num.induct"}, @{text "num.cases"}).
+
+  It is not entirely clear in which context it is better to use the
+  one or the other, or whether the construction should be reversed.
+*}
+
+
+subsection {* Numeral operations *}
+
+ML {*
+structure Dig_Simps = Named_Thms
+(
+  val name = @{binding numeral}
+  val description = "simplification rules for numerals"
+)
+*}
+
+setup Dig_Simps.setup
+
+instantiation num :: "{plus,times,ord}"
+begin
+
+definition plus_num :: "num \<Rightarrow> num \<Rightarrow> num" where
+  "m + n = num_of_nat (nat_of_num m + nat_of_num n)"
+
+definition times_num :: "num \<Rightarrow> num \<Rightarrow> num" where
+  "m * n = num_of_nat (nat_of_num m * nat_of_num n)"
+
+definition less_eq_num :: "num \<Rightarrow> num \<Rightarrow> bool" where
+  "m \<le> n \<longleftrightarrow> nat_of_num m \<le> nat_of_num n"
+
+definition less_num :: "num \<Rightarrow> num \<Rightarrow> bool" where
+  "m < n \<longleftrightarrow> nat_of_num m < nat_of_num n"
+
+instance ..
+
+end
+
+lemma nat_of_num_add: "nat_of_num (x + y) = nat_of_num x + nat_of_num y"
+  unfolding plus_num_def
+  by (intro num_of_nat_inverse add_pos_pos nat_of_num_pos)
+
+lemma nat_of_num_mult: "nat_of_num (x * y) = nat_of_num x * nat_of_num y"
+  unfolding times_num_def
+  by (intro num_of_nat_inverse mult_pos_pos nat_of_num_pos)
+
+lemma Dig_plus [numeral, simp, code]:
+  "One + One = Dig0 One"
+  "One + Dig0 m = Dig1 m"
+  "One + Dig1 m = Dig0 (m + One)"
+  "Dig0 n + One = Dig1 n"
+  "Dig0 n + Dig0 m = Dig0 (n + m)"
+  "Dig0 n + Dig1 m = Dig1 (n + m)"
+  "Dig1 n + One = Dig0 (n + One)"
+  "Dig1 n + Dig0 m = Dig1 (n + m)"
+  "Dig1 n + Dig1 m = Dig0 (n + m + One)"
+  by (simp_all add: num_eq_iff nat_of_num_add)
+
+lemma Dig_times [numeral, simp, code]:
+  "One * One = One"
+  "One * Dig0 n = Dig0 n"
+  "One * Dig1 n = Dig1 n"
+  "Dig0 n * One = Dig0 n"
+  "Dig0 n * Dig0 m = Dig0 (n * Dig0 m)"
+  "Dig0 n * Dig1 m = Dig0 (n * Dig1 m)"
+  "Dig1 n * One = Dig1 n"
+  "Dig1 n * Dig0 m = Dig0 (n * Dig0 m + m)"
+  "Dig1 n * Dig1 m = Dig1 (n * Dig1 m + m)"
+  by (simp_all add: num_eq_iff nat_of_num_add nat_of_num_mult
+                    left_distrib right_distrib)
+
+lemma less_eq_num_code [numeral, simp, code]:
+  "One \<le> n \<longleftrightarrow> True"
+  "Dig0 m \<le> One \<longleftrightarrow> False"
+  "Dig1 m \<le> One \<longleftrightarrow> False"
+  "Dig0 m \<le> Dig0 n \<longleftrightarrow> m \<le> n"
+  "Dig0 m \<le> Dig1 n \<longleftrightarrow> m \<le> n"
+  "Dig1 m \<le> Dig1 n \<longleftrightarrow> m \<le> n"
+  "Dig1 m \<le> Dig0 n \<longleftrightarrow> m < n"
+  using nat_of_num_pos [of n] nat_of_num_pos [of m]
+  by (auto simp add: less_eq_num_def less_num_def)
+
+lemma less_num_code [numeral, simp, code]:
+  "m < One \<longleftrightarrow> False"
+  "One < One \<longleftrightarrow> False"
+  "One < Dig0 n \<longleftrightarrow> True"
+  "One < Dig1 n \<longleftrightarrow> True"
+  "Dig0 m < Dig0 n \<longleftrightarrow> m < n"
+  "Dig0 m < Dig1 n \<longleftrightarrow> m \<le> n"
+  "Dig1 m < Dig1 n \<longleftrightarrow> m < n"
+  "Dig1 m < Dig0 n \<longleftrightarrow> m < n"
+  using nat_of_num_pos [of n] nat_of_num_pos [of m]
+  by (auto simp add: less_eq_num_def less_num_def)
+
+text {* Rules using @{text One} and @{text inc} as constructors *}
+
+lemma add_One: "x + One = inc x"
+  by (simp add: num_eq_iff nat_of_num_add nat_of_num_inc)
+
+lemma add_inc: "x + inc y = inc (x + y)"
+  by (simp add: num_eq_iff nat_of_num_add nat_of_num_inc)
+
+lemma mult_One: "x * One = x"
+  by (simp add: num_eq_iff nat_of_num_mult)
+
+lemma mult_inc: "x * inc y = x * y + x"
+  by (simp add: num_eq_iff nat_of_num_mult nat_of_num_add nat_of_num_inc)
+
+text {* A double-and-decrement function *}
+
+primrec DigM :: "num \<Rightarrow> num" where
+  "DigM One = One"
+| "DigM (Dig0 n) = Dig1 (DigM n)"
+| "DigM (Dig1 n) = Dig1 (Dig0 n)"
+
+lemma DigM_plus_one: "DigM n + One = Dig0 n"
+  by (induct n) simp_all
+
+lemma add_One_commute: "One + n = n + One"
+  by (induct n) simp_all
+
+lemma one_plus_DigM: "One + DigM n = Dig0 n"
+  by (simp add: add_One_commute DigM_plus_one)
+
+text {* Squaring and exponentiation *}
+
+primrec square :: "num \<Rightarrow> num" where
+  "square One = One"
+| "square (Dig0 n) = Dig0 (Dig0 (square n))"
+| "square (Dig1 n) = Dig1 (Dig0 (square n + n))"
+
+primrec pow :: "num \<Rightarrow> num \<Rightarrow> num" where
+  "pow x One = x"
+| "pow x (Dig0 y) = square (pow x y)"
+| "pow x (Dig1 y) = x * square (pow x y)"
+
+
+subsection {* Binary numerals *}
+
+text {*
+  We embed binary representations into a generic algebraic
+  structure using @{text of_num}.
+*}
+
+class semiring_numeral = semiring + monoid_mult
+begin
+
+primrec of_num :: "num \<Rightarrow> 'a" where
+  of_num_One [numeral]: "of_num One = 1"
+| "of_num (Dig0 n) = of_num n + of_num n"
+| "of_num (Dig1 n) = of_num n + of_num n + 1"
+
+lemma of_num_inc: "of_num (inc n) = of_num n + 1"
+  by (induct n) (simp_all add: add_ac)
+
+lemma of_num_add: "of_num (m + n) = of_num m + of_num n"
+  by (induct n rule: num_induct) (simp_all add: add_One add_inc of_num_inc add_ac)
+
+lemma of_num_mult: "of_num (m * n) = of_num m * of_num n"
+  by (induct n rule: num_induct) (simp_all add: mult_One mult_inc of_num_add of_num_inc right_distrib)
+
+declare of_num.simps [simp del]
+
+end
+
+ML {*
+fun mk_num k =
+  if k > 1 then
+    let
+      val (l, b) = Integer.div_mod k 2;
+      val bit = (if b = 0 then @{term Dig0} else @{term Dig1});
+    in bit $ (mk_num l) end
+  else if k = 1 then @{term One}
+  else error ("mk_num " ^ string_of_int k);
+
+fun dest_num @{term One} = 1
+  | dest_num (@{term Dig0} $ n) = 2 * dest_num n
+  | dest_num (@{term Dig1} $ n) = 2 * dest_num n + 1
+  | dest_num t = raise TERM ("dest_num", [t]);
+
+fun mk_numeral phi T k = Morphism.term phi (Const (@{const_name of_num}, @{typ num} --> T))
+  $ mk_num k
+
+fun dest_numeral phi (u $ t) =
+  if Term.aconv_untyped (u, Morphism.term phi (Const (@{const_name of_num}, dummyT)))
+  then (range_type (fastype_of u), dest_num t)
+  else raise TERM ("dest_numeral", [u, t]);
+*}
+
+syntax
+  "_Numerals" :: "xnum_token \<Rightarrow> 'a"    ("_")
+
+parse_translation {*
+let
+  fun num_of_int n = if n > 0 then case IntInf.quotRem (n, 2)
+     of (0, 1) => Const (@{const_name One}, dummyT)
+      | (n, 0) => Const (@{const_name Dig0}, dummyT) $ num_of_int n
+      | (n, 1) => Const (@{const_name Dig1}, dummyT) $ num_of_int n
+    else raise Match;
+  fun numeral_tr [Free (num, _)] =
+        let
+          val {leading_zeros, value, ...} = Lexicon.read_xnum num;
+          val _ = leading_zeros = 0 andalso value > 0
+            orelse error ("Bad numeral: " ^ num);
+        in Const (@{const_name of_num}, @{typ num} --> dummyT) $ num_of_int value end
+    | numeral_tr ts = raise TERM ("numeral_tr", ts);
+in [(@{syntax_const "_Numerals"}, numeral_tr)] end
+*}
+
+typed_print_translation (advanced) {*
+let
+  fun dig b n = b + 2 * n; 
+  fun int_of_num' (Const (@{const_syntax Dig0}, _) $ n) =
+        dig 0 (int_of_num' n)
+    | int_of_num' (Const (@{const_syntax Dig1}, _) $ n) =
+        dig 1 (int_of_num' n)
+    | int_of_num' (Const (@{const_syntax One}, _)) = 1;
+  fun num_tr' ctxt T [n] =
+    let
+      val k = int_of_num' n;
+      val t' = Syntax.const @{syntax_const "_Numerals"} $ Syntax.free ("#" ^ string_of_int k);
+    in
+      case T of
+        Type (@{type_name fun}, [_, T']) =>
+          if not (Config.get ctxt show_types) andalso can Term.dest_Type T' then t'
+          else Syntax.const @{syntax_const "_constrain"} $ t' $ Syntax_Phases.term_of_typ ctxt T'
+      | T' => if T' = dummyT then t' else raise Match
+    end;
+in [(@{const_syntax of_num}, num_tr')] end
+*}
+
+
+subsection {* Class-specific numeral rules *}
+
+subsubsection {* Class @{text semiring_numeral} *}
+
+context semiring_numeral
+begin
+
+abbreviation "Num1 \<equiv> of_num One"
+
+text {*
+  Alas, there is still the duplication of @{term 1}, although the
+  duplicated @{term 0} has disappeared.  We could get rid of it by
+  replacing the constructor @{term 1} in @{typ num} by two
+  constructors @{text two} and @{text three}, resulting in a further
+  blow-up.  But it could be worth the effort.
+*}
+
+lemma of_num_plus_one [numeral]:
+  "of_num n + 1 = of_num (n + One)"
+  by (simp only: of_num_add of_num_One)
+
+lemma of_num_one_plus [numeral]:
+  "1 + of_num n = of_num (One + n)"
+  by (simp only: of_num_add of_num_One)
+
+lemma of_num_plus [numeral]:
+  "of_num m + of_num n = of_num (m + n)"
+  by (simp only: of_num_add)
+
+lemma of_num_times_one [numeral]:
+  "of_num n * 1 = of_num n"
+  by simp
+
+lemma of_num_one_times [numeral]:
+  "1 * of_num n = of_num n"
+  by simp
+
+lemma of_num_times [numeral]:
+  "of_num m * of_num n = of_num (m * n)"
+  unfolding of_num_mult ..
+
+end
+
+
+subsubsection {* Structures with a zero: class @{text semiring_1} *}
+
+context semiring_1
+begin
+
+subclass semiring_numeral ..
+
+lemma of_nat_of_num [numeral]: "of_nat (of_num n) = of_num n"
+  by (induct n)
+    (simp_all add: semiring_numeral_class.of_num.simps of_num.simps add_ac)
+
+declare of_nat_1 [numeral]
+
+lemma Dig_plus_zero [numeral]:
+  "0 + 1 = 1"
+  "0 + of_num n = of_num n"
+  "1 + 0 = 1"
+  "of_num n + 0 = of_num n"
+  by simp_all
+
+lemma Dig_times_zero [numeral]:
+  "0 * 1 = 0"
+  "0 * of_num n = 0"
+  "1 * 0 = 0"
+  "of_num n * 0 = 0"
+  by simp_all
+
+end
+
+lemma nat_of_num_of_num: "nat_of_num = of_num"
+proof
+  fix n
+  have "of_num n = nat_of_num n"
+    by (induct n) (simp_all add: of_num.simps)
+  then show "nat_of_num n = of_num n" by simp
+qed
+
+
+subsubsection {* Equality: class @{text semiring_char_0} *}
+
+context semiring_char_0
+begin
+
+lemma of_num_eq_iff [numeral]: "of_num m = of_num n \<longleftrightarrow> m = n"
+  unfolding of_nat_of_num [symmetric] nat_of_num_of_num [symmetric]
+    of_nat_eq_iff num_eq_iff ..
+
+lemma of_num_eq_one_iff [numeral]: "of_num n = 1 \<longleftrightarrow> n = One"
+  using of_num_eq_iff [of n One] by (simp add: of_num_One)
+
+lemma one_eq_of_num_iff [numeral]: "1 = of_num n \<longleftrightarrow> One = n"
+  using of_num_eq_iff [of One n] by (simp add: of_num_One)
+
+end
+
+
+subsubsection {* Comparisons: class @{text linordered_semidom} *}
+
+text {*
+  Perhaps the underlying structure could even 
+  be more general than @{text linordered_semidom}.
+*}
+
+context linordered_semidom
+begin
+
+lemma of_num_pos [numeral]: "0 < of_num n"
+  by (induct n) (simp_all add: of_num.simps add_pos_pos)
+
+lemma of_num_not_zero [numeral]: "of_num n \<noteq> 0"
+  using of_num_pos [of n] by simp
+
+lemma of_num_less_eq_iff [numeral]: "of_num m \<le> of_num n \<longleftrightarrow> m \<le> n"
+proof -
+  have "of_nat (of_num m) \<le> of_nat (of_num n) \<longleftrightarrow> m \<le> n"
+    unfolding less_eq_num_def nat_of_num_of_num of_nat_le_iff ..
+  then show ?thesis by (simp add: of_nat_of_num)
+qed
+
+lemma of_num_less_eq_one_iff [numeral]: "of_num n \<le> 1 \<longleftrightarrow> n \<le> One"
+  using of_num_less_eq_iff [of n One] by (simp add: of_num_One)
+
+lemma one_less_eq_of_num_iff [numeral]: "1 \<le> of_num n"
+  using of_num_less_eq_iff [of One n] by (simp add: of_num_One)
+
+lemma of_num_less_iff [numeral]: "of_num m < of_num n \<longleftrightarrow> m < n"
+proof -
+  have "of_nat (of_num m) < of_nat (of_num n) \<longleftrightarrow> m < n"
+    unfolding less_num_def nat_of_num_of_num of_nat_less_iff ..
+  then show ?thesis by (simp add: of_nat_of_num)
+qed
+
+lemma of_num_less_one_iff [numeral]: "\<not> of_num n < 1"
+  using of_num_less_iff [of n One] by (simp add: of_num_One)
+
+lemma one_less_of_num_iff [numeral]: "1 < of_num n \<longleftrightarrow> One < n"
+  using of_num_less_iff [of One n] by (simp add: of_num_One)
+
+lemma of_num_nonneg [numeral]: "0 \<le> of_num n"
+  by (induct n) (simp_all add: of_num.simps add_nonneg_nonneg)
+
+lemma of_num_less_zero_iff [numeral]: "\<not> of_num n < 0"
+  by (simp add: not_less of_num_nonneg)
+
+lemma of_num_le_zero_iff [numeral]: "\<not> of_num n \<le> 0"
+  by (simp add: not_le of_num_pos)
+
+end
+
+context linordered_idom
+begin
+
+lemma minus_of_num_less_of_num_iff: "- of_num m < of_num n"
+proof -
+  have "- of_num m < 0" by (simp add: of_num_pos)
+  also have "0 < of_num n" by (simp add: of_num_pos)
+  finally show ?thesis .
+qed
+
+lemma minus_of_num_not_equal_of_num: "- of_num m \<noteq> of_num n"
+  using minus_of_num_less_of_num_iff [of m n] by simp
+
+lemma minus_of_num_less_one_iff: "- of_num n < 1"
+  using minus_of_num_less_of_num_iff [of n One] by (simp add: of_num_One)
+
+lemma minus_one_less_of_num_iff: "- 1 < of_num n"
+  using minus_of_num_less_of_num_iff [of One n] by (simp add: of_num_One)
+
+lemma minus_one_less_one_iff: "- 1 < 1"
+  using minus_of_num_less_of_num_iff [of One One] by (simp add: of_num_One)
+
+lemma minus_of_num_le_of_num_iff: "- of_num m \<le> of_num n"
+  by (simp add: less_imp_le minus_of_num_less_of_num_iff)
+
+lemma minus_of_num_le_one_iff: "- of_num n \<le> 1"
+  by (simp add: less_imp_le minus_of_num_less_one_iff)
+
+lemma minus_one_le_of_num_iff: "- 1 \<le> of_num n"
+  by (simp add: less_imp_le minus_one_less_of_num_iff)
+
+lemma minus_one_le_one_iff: "- 1 \<le> 1"
+  by (simp add: less_imp_le minus_one_less_one_iff)
+
+lemma of_num_le_minus_of_num_iff: "\<not> of_num m \<le> - of_num n"
+  by (simp add: not_le minus_of_num_less_of_num_iff)
+
+lemma one_le_minus_of_num_iff: "\<not> 1 \<le> - of_num n"
+  by (simp add: not_le minus_of_num_less_one_iff)
+
+lemma of_num_le_minus_one_iff: "\<not> of_num n \<le> - 1"
+  by (simp add: not_le minus_one_less_of_num_iff)
+
+lemma one_le_minus_one_iff: "\<not> 1 \<le> - 1"
+  by (simp add: not_le minus_one_less_one_iff)
+
+lemma of_num_less_minus_of_num_iff: "\<not> of_num m < - of_num n"
+  by (simp add: not_less minus_of_num_le_of_num_iff)
+
+lemma one_less_minus_of_num_iff: "\<not> 1 < - of_num n"
+  by (simp add: not_less minus_of_num_le_one_iff)
+
+lemma of_num_less_minus_one_iff: "\<not> of_num n < - 1"
+  by (simp add: not_less minus_one_le_of_num_iff)
+
+lemma one_less_minus_one_iff: "\<not> 1 < - 1"
+  by (simp add: not_less minus_one_le_one_iff)
+
+lemmas le_signed_numeral_special [numeral] =
+  minus_of_num_le_of_num_iff
+  minus_of_num_le_one_iff
+  minus_one_le_of_num_iff
+  minus_one_le_one_iff
+  of_num_le_minus_of_num_iff
+  one_le_minus_of_num_iff
+  of_num_le_minus_one_iff
+  one_le_minus_one_iff
+
+lemmas less_signed_numeral_special [numeral] =
+  minus_of_num_less_of_num_iff
+  minus_of_num_not_equal_of_num
+  minus_of_num_less_one_iff
+  minus_one_less_of_num_iff
+  minus_one_less_one_iff
+  of_num_less_minus_of_num_iff
+  one_less_minus_of_num_iff
+  of_num_less_minus_one_iff
+  one_less_minus_one_iff
+
+end
+
+subsubsection {* Structures with subtraction: class @{text semiring_1_minus} *}
+
+class semiring_minus = semiring + minus + zero +
+  assumes minus_inverts_plus1: "a + b = c \<Longrightarrow> c - b = a"
+  assumes minus_minus_zero_inverts_plus1: "a + b = c \<Longrightarrow> b - c = 0 - a"
+begin
+
+lemma minus_inverts_plus2: "a + b = c \<Longrightarrow> c - a = b"
+  by (simp add: add_ac minus_inverts_plus1 [of b a])
+
+lemma minus_minus_zero_inverts_plus2: "a + b = c \<Longrightarrow> a - c = 0 - b"
+  by (simp add: add_ac minus_minus_zero_inverts_plus1 [of b a])
+
+end
+
+class semiring_1_minus = semiring_1 + semiring_minus
+begin
+
+lemma Dig_of_num_pos:
+  assumes "k + n = m"
+  shows "of_num m - of_num n = of_num k"
+  using assms by (simp add: of_num_plus minus_inverts_plus1)
+
+lemma Dig_of_num_zero:
+  shows "of_num n - of_num n = 0"
+  by (rule minus_inverts_plus1) simp
+
+lemma Dig_of_num_neg:
+  assumes "k + m = n"
+  shows "of_num m - of_num n = 0 - of_num k"
+  by (rule minus_minus_zero_inverts_plus1) (simp add: of_num_plus assms)
+
+lemmas Dig_plus_eval =
+  of_num_plus of_num_eq_iff Dig_plus refl [of One, THEN eqTrueI] num.inject
+
+simproc_setup numeral_minus ("of_num m - of_num n") = {*
+  let
+    (*TODO proper implicit use of morphism via pattern antiquotations*)
+    fun cdest_of_num ct = (List.last o snd o Drule.strip_comb) ct;
+    fun cdest_minus ct = case (rev o snd o Drule.strip_comb) ct of [n, m] => (m, n);
+    fun attach_num ct = (dest_num (Thm.term_of ct), ct);
+    fun cdifference t = (pairself (attach_num o cdest_of_num) o cdest_minus) t;
+    val simplify = Raw_Simplifier.rewrite false (map mk_meta_eq @{thms Dig_plus_eval});
+    fun cert ck cl cj = @{thm eqTrueE} OF [@{thm meta_eq_to_obj_eq}
+      OF [simplify (Drule.list_comb (@{cterm "op = :: num \<Rightarrow> _"},
+        [Drule.list_comb (@{cterm "op + :: num \<Rightarrow> _"}, [ck, cl]), cj]))]];
+  in fn phi => fn _ => fn ct => case try cdifference ct
+   of NONE => (NONE)
+    | SOME ((k, ck), (l, cl)) => SOME (let val j = k - l in if j = 0
+        then Raw_Simplifier.rewrite false [mk_meta_eq (Morphism.thm phi @{thm Dig_of_num_zero})] ct
+        else mk_meta_eq (let
+          val cj = Thm.cterm_of (Thm.theory_of_cterm ct) (mk_num (abs j));
+        in
+          (if j > 0 then (Morphism.thm phi @{thm Dig_of_num_pos}) OF [cert cj cl ck]
+          else (Morphism.thm phi @{thm Dig_of_num_neg}) OF [cert cj ck cl])
+        end) end)
+  end
+*}
+
+lemma Dig_of_num_minus_zero [numeral]:
+  "of_num n - 0 = of_num n"
+  by (simp add: minus_inverts_plus1)
+
+lemma Dig_one_minus_zero [numeral]:
+  "1 - 0 = 1"
+  by (simp add: minus_inverts_plus1)
+
+lemma Dig_one_minus_one [numeral]:
+  "1 - 1 = 0"
+  by (simp add: minus_inverts_plus1)
+
+lemma Dig_of_num_minus_one [numeral]:
+  "of_num (Dig0 n) - 1 = of_num (DigM n)"
+  "of_num (Dig1 n) - 1 = of_num (Dig0 n)"
+  by (auto intro: minus_inverts_plus1 simp add: DigM_plus_one of_num.simps of_num_plus_one)
+
+lemma Dig_one_minus_of_num [numeral]:
+  "1 - of_num (Dig0 n) = 0 - of_num (DigM n)"
+  "1 - of_num (Dig1 n) = 0 - of_num (Dig0 n)"
+  by (auto intro: minus_minus_zero_inverts_plus1 simp add: DigM_plus_one of_num.simps of_num_plus_one)
+
+end
+
+
+subsubsection {* Structures with negation: class @{text ring_1} *}
+
+context ring_1
+begin
+
+subclass semiring_1_minus proof
+qed (simp_all add: algebra_simps)
+
+lemma Dig_zero_minus_of_num [numeral]:
+  "0 - of_num n = - of_num n"
+  by simp
+
+lemma Dig_zero_minus_one [numeral]:
+  "0 - 1 = - 1"
+  by simp
+
+lemma Dig_uminus_uminus [numeral]:
+  "- (- of_num n) = of_num n"
+  by simp
+
+lemma Dig_plus_uminus [numeral]:
+  "of_num m + - of_num n = of_num m - of_num n"
+  "- of_num m + of_num n = of_num n - of_num m"
+  "- of_num m + - of_num n = - (of_num m + of_num n)"
+  "of_num m - - of_num n = of_num m + of_num n"
+  "- of_num m - of_num n = - (of_num m + of_num n)"
+  "- of_num m - - of_num n = of_num n - of_num m"
+  by (simp_all add: diff_minus add_commute)
+
+lemma Dig_times_uminus [numeral]:
+  "- of_num n * of_num m = - (of_num n * of_num m)"
+  "of_num n * - of_num m = - (of_num n * of_num m)"
+  "- of_num n * - of_num m = of_num n * of_num m"
+  by simp_all
+
+lemma of_int_of_num [numeral]: "of_int (of_num n) = of_num n"
+by (induct n)
+  (simp_all only: of_num.simps semiring_numeral_class.of_num.simps of_int_add, simp_all)
+
+declare of_int_1 [numeral]
+
+end
+
+
+subsubsection {* Structures with exponentiation *}
+
+lemma of_num_square: "of_num (square x) = of_num x * of_num x"
+by (induct x)
+   (simp_all add: of_num.simps of_num_add algebra_simps)
+
+lemma of_num_pow: "of_num (pow x y) = of_num x ^ of_num y"
+by (induct y)
+   (simp_all add: of_num.simps of_num_square of_num_mult power_add)
+
+lemma power_of_num [numeral]: "of_num x ^ of_num y = of_num (pow x y)"
+  unfolding of_num_pow ..
+
+lemma power_zero_of_num [numeral]:
+  "0 ^ of_num n = (0::'a::semiring_1)"
+  using of_num_pos [where n=n and ?'a=nat]
+  by (simp add: power_0_left)
+
+lemma power_minus_Dig0 [numeral]:
+  fixes x :: "'a::ring_1"
+  shows "(- x) ^ of_num (Dig0 n) = x ^ of_num (Dig0 n)"
+  by (induct n rule: num_induct) (simp_all add: of_num.simps of_num_inc)
+
+lemma power_minus_Dig1 [numeral]:
+  fixes x :: "'a::ring_1"
+  shows "(- x) ^ of_num (Dig1 n) = - (x ^ of_num (Dig1 n))"
+  by (induct n rule: num_induct) (simp_all add: of_num.simps of_num_inc)
+
+declare power_one [numeral]
+
+
+subsubsection {* Greetings to @{typ nat}. *}
+
+instance nat :: semiring_1_minus proof
+qed simp_all
+
+lemma Suc_of_num [numeral]: "Suc (of_num n) = of_num (n + One)"
+  unfolding of_num_plus_one [symmetric] by simp
+
+lemma nat_number:
+  "1 = Suc 0"
+  "of_num One = Suc 0"
+  "of_num (Dig0 n) = Suc (of_num (DigM n))"
+  "of_num (Dig1 n) = Suc (of_num (Dig0 n))"
+  by (simp_all add: of_num.simps DigM_plus_one Suc_of_num)
+
+declare diff_0_eq_0 [numeral]
+
+
+subsection {* Proof tools setup *}
+
+subsubsection {* Numeral equations as default simplification rules *}
+
+declare (in semiring_numeral) of_num_One [simp]
+declare (in semiring_numeral) of_num_plus_one [simp]
+declare (in semiring_numeral) of_num_one_plus [simp]
+declare (in semiring_numeral) of_num_plus [simp]
+declare (in semiring_numeral) of_num_times [simp]
+
+declare (in semiring_1) of_nat_of_num [simp]
+
+declare (in semiring_char_0) of_num_eq_iff [simp]
+declare (in semiring_char_0) of_num_eq_one_iff [simp]
+declare (in semiring_char_0) one_eq_of_num_iff [simp]
+
+declare (in linordered_semidom) of_num_pos [simp]
+declare (in linordered_semidom) of_num_not_zero [simp]
+declare (in linordered_semidom) of_num_less_eq_iff [simp]
+declare (in linordered_semidom) of_num_less_eq_one_iff [simp]
+declare (in linordered_semidom) one_less_eq_of_num_iff [simp]
+declare (in linordered_semidom) of_num_less_iff [simp]
+declare (in linordered_semidom) of_num_less_one_iff [simp]
+declare (in linordered_semidom) one_less_of_num_iff [simp]
+declare (in linordered_semidom) of_num_nonneg [simp]
+declare (in linordered_semidom) of_num_less_zero_iff [simp]
+declare (in linordered_semidom) of_num_le_zero_iff [simp]
+
+declare (in linordered_idom) le_signed_numeral_special [simp]
+declare (in linordered_idom) less_signed_numeral_special [simp]
+
+declare (in semiring_1_minus) Dig_of_num_minus_one [simp]
+declare (in semiring_1_minus) Dig_one_minus_of_num [simp]
+
+declare (in ring_1) Dig_plus_uminus [simp]
+declare (in ring_1) of_int_of_num [simp]
+
+declare power_of_num [simp]
+declare power_zero_of_num [simp]
+declare power_minus_Dig0 [simp]
+declare power_minus_Dig1 [simp]
+
+declare Suc_of_num [simp]
+
+
+subsubsection {* Reorientation of equalities *}
+
+setup {*
+  Reorient_Proc.add
+    (fn Const(@{const_name of_num}, _) $ _ => true
+      | Const(@{const_name uminus}, _) $
+          (Const(@{const_name of_num}, _) $ _) => true
+      | _ => false)
+*}
+
+simproc_setup reorient_num ("of_num n = x" | "- of_num m = y") = Reorient_Proc.proc
+
+
+subsubsection {* Constant folding for multiplication in semirings *}
+
+context semiring_numeral
+begin
+
+lemma mult_of_num_commute: "x * of_num n = of_num n * x"
+by (induct n)
+  (simp_all only: of_num.simps left_distrib right_distrib mult_1_left mult_1_right)
+
+definition
+  "commutes_with a b \<longleftrightarrow> a * b = b * a"
+
+lemma commutes_with_commute: "commutes_with a b \<Longrightarrow> a * b = b * a"
+unfolding commutes_with_def .
+
+lemma commutes_with_left_commute: "commutes_with a b \<Longrightarrow> a * (b * c) = b * (a * c)"
+unfolding commutes_with_def by (simp only: mult_assoc [symmetric])
+
+lemma commutes_with_numeral: "commutes_with x (of_num n)" "commutes_with (of_num n) x"
+unfolding commutes_with_def by (simp_all add: mult_of_num_commute)
+
+lemmas mult_ac_numeral =
+  mult_assoc
+  commutes_with_commute
+  commutes_with_left_commute
+  commutes_with_numeral
+
+end
+
+ML {*
+structure Semiring_Times_Assoc_Data : ASSOC_FOLD_DATA =
+struct
+  val assoc_ss = HOL_ss addsimps @{thms mult_ac_numeral}
+  val eq_reflection = eq_reflection
+  fun is_numeral (Const(@{const_name of_num}, _) $ _) = true
+    | is_numeral _ = false;
+end;
+
+structure Semiring_Times_Assoc = Assoc_Fold (Semiring_Times_Assoc_Data);
+*}
+
+simproc_setup semiring_assoc_fold' ("(a::'a::semiring_numeral) * b") =
+  {* fn phi => fn ss => fn ct =>
+    Semiring_Times_Assoc.proc ss (Thm.term_of ct) *}
+
+
+subsection {* Code generator setup for @{typ int} *}
+
+text {* Reversing standard setup *}
+
+lemma [code_unfold del]: "(0::int) \<equiv> Numeral0" by simp
+lemma [code_unfold del]: "(1::int) \<equiv> Numeral1" by simp
+declare zero_is_num_zero [code_unfold del]
+declare one_is_num_one [code_unfold del]
+  
+lemma [code, code del]:
+  "(1 :: int) = 1"
+  "(op + :: int \<Rightarrow> int \<Rightarrow> int) = op +"
+  "(uminus :: int \<Rightarrow> int) = uminus"
+  "(op - :: int \<Rightarrow> int \<Rightarrow> int) = op -"
+  "(op * :: int \<Rightarrow> int \<Rightarrow> int) = op *"
+  "(HOL.equal :: int \<Rightarrow> int \<Rightarrow> bool) = HOL.equal"
+  "(op \<le> :: int \<Rightarrow> int \<Rightarrow> bool) = op \<le>"
+  "(op < :: int \<Rightarrow> int \<Rightarrow> bool) = op <"
+  by rule+
+
+text {* Constructors *}
+
+definition Pls :: "num \<Rightarrow> int" where
+  [simp, code_post]: "Pls n = of_num n"
+
+definition Mns :: "num \<Rightarrow> int" where
+  [simp, code_post]: "Mns n = - of_num n"
+
+code_datatype "0::int" Pls Mns
+
+lemmas [code_unfold] = Pls_def [symmetric] Mns_def [symmetric]
+
+text {* Auxiliary operations *}
+
+definition dup :: "int \<Rightarrow> int" where
+  [simp]: "dup k = k + k"
+
+lemma Dig_dup [code]:
+  "dup 0 = 0"
+  "dup (Pls n) = Pls (Dig0 n)"
+  "dup (Mns n) = Mns (Dig0 n)"
+  by (simp_all add: of_num.simps)
+
+definition sub :: "num \<Rightarrow> num \<Rightarrow> int" where
+  [simp]: "sub m n = (of_num m - of_num n)"
+
+lemma Dig_sub [code]:
+  "sub One One = 0"
+  "sub (Dig0 m) One = of_num (DigM m)"
+  "sub (Dig1 m) One = of_num (Dig0 m)"
+  "sub One (Dig0 n) = - of_num (DigM n)"
+  "sub One (Dig1 n) = - of_num (Dig0 n)"
+  "sub (Dig0 m) (Dig0 n) = dup (sub m n)"
+  "sub (Dig1 m) (Dig1 n) = dup (sub m n)"
+  "sub (Dig1 m) (Dig0 n) = dup (sub m n) + 1"
+  "sub (Dig0 m) (Dig1 n) = dup (sub m n) - 1"
+  by (simp_all add: algebra_simps num_eq_iff nat_of_num_add)
+
+text {* Implementations *}
+
+lemma one_int_code [code]:
+  "1 = Pls One"
+  by simp
+
+lemma plus_int_code [code]:
+  "k + 0 = (k::int)"
+  "0 + l = (l::int)"
+  "Pls m + Pls n = Pls (m + n)"
+  "Pls m + Mns n = sub m n"
+  "Mns m + Pls n = sub n m"
+  "Mns m + Mns n = Mns (m + n)"
+  by simp_all
+
+lemma uminus_int_code [code]:
+  "uminus 0 = (0::int)"
+  "uminus (Pls m) = Mns m"
+  "uminus (Mns m) = Pls m"
+  by simp_all
+
+lemma minus_int_code [code]:
+  "k - 0 = (k::int)"
+  "0 - l = uminus (l::int)"
+  "Pls m - Pls n = sub m n"
+  "Pls m - Mns n = Pls (m + n)"
+  "Mns m - Pls n = Mns (m + n)"
+  "Mns m - Mns n = sub n m"
+  by simp_all
+
+lemma times_int_code [code]:
+  "k * 0 = (0::int)"
+  "0 * l = (0::int)"
+  "Pls m * Pls n = Pls (m * n)"
+  "Pls m * Mns n = Mns (m * n)"
+  "Mns m * Pls n = Mns (m * n)"
+  "Mns m * Mns n = Pls (m * n)"
+  by simp_all
+
+lemma eq_int_code [code]:
+  "HOL.equal 0 (0::int) \<longleftrightarrow> True"
+  "HOL.equal 0 (Pls l) \<longleftrightarrow> False"
+  "HOL.equal 0 (Mns l) \<longleftrightarrow> False"
+  "HOL.equal (Pls k) 0 \<longleftrightarrow> False"
+  "HOL.equal (Pls k) (Pls l) \<longleftrightarrow> HOL.equal k l"
+  "HOL.equal (Pls k) (Mns l) \<longleftrightarrow> False"
+  "HOL.equal (Mns k) 0 \<longleftrightarrow> False"
+  "HOL.equal (Mns k) (Pls l) \<longleftrightarrow> False"
+  "HOL.equal (Mns k) (Mns l) \<longleftrightarrow> HOL.equal k l"
+  by (auto simp add: equal dest: sym)
+
+lemma [code nbe]:
+  "HOL.equal (k::int) k \<longleftrightarrow> True"
+  by (fact equal_refl)
+
+lemma less_eq_int_code [code]:
+  "0 \<le> (0::int) \<longleftrightarrow> True"
+  "0 \<le> Pls l \<longleftrightarrow> True"
+  "0 \<le> Mns l \<longleftrightarrow> False"
+  "Pls k \<le> 0 \<longleftrightarrow> False"
+  "Pls k \<le> Pls l \<longleftrightarrow> k \<le> l"
+  "Pls k \<le> Mns l \<longleftrightarrow> False"
+  "Mns k \<le> 0 \<longleftrightarrow> True"
+  "Mns k \<le> Pls l \<longleftrightarrow> True"
+  "Mns k \<le> Mns l \<longleftrightarrow> l \<le> k"
+  by simp_all
+
+lemma less_int_code [code]:
+  "0 < (0::int) \<longleftrightarrow> False"
+  "0 < Pls l \<longleftrightarrow> True"
+  "0 < Mns l \<longleftrightarrow> False"
+  "Pls k < 0 \<longleftrightarrow> False"
+  "Pls k < Pls l \<longleftrightarrow> k < l"
+  "Pls k < Mns l \<longleftrightarrow> False"
+  "Mns k < 0 \<longleftrightarrow> True"
+  "Mns k < Pls l \<longleftrightarrow> True"
+  "Mns k < Mns l \<longleftrightarrow> l < k"
+  by simp_all
+
+hide_const (open) sub dup
+
+text {* Pretty literals *}
+
+ML {*
+local open Code_Thingol in
+
+fun add_code print target =
+  let
+    fun dest_num one' dig0' dig1' thm =
+      let
+        fun dest_dig (IConst (c, _)) = if c = dig0' then 0
+              else if c = dig1' then 1
+              else Code_Printer.eqn_error thm "Illegal numeral expression: illegal dig"
+          | dest_dig _ = Code_Printer.eqn_error thm "Illegal numeral expression: illegal digit";
+        fun dest_num (IConst (c, _)) = if c = one' then 1
+              else Code_Printer.eqn_error thm "Illegal numeral expression: illegal leading digit"
+          | dest_num (t1 `$ t2) = 2 * dest_num t2 + dest_dig t1
+          | dest_num _ = Code_Printer.eqn_error thm "Illegal numeral expression: illegal term";
+      in dest_num end;
+    fun pretty sgn literals [one', dig0', dig1'] _ thm _ _ [(t, _)] =
+      (Code_Printer.str o print literals o sgn o dest_num one' dig0' dig1' thm) t
+    fun add_syntax (c, sgn) = Code_Target.add_const_syntax target c
+      (SOME (Code_Printer.complex_const_syntax
+        (1, ([@{const_name One}, @{const_name Dig0}, @{const_name Dig1}],
+          pretty sgn))));
+  in
+    add_syntax (@{const_name Pls}, I)
+    #> add_syntax (@{const_name Mns}, (fn k => ~ k))
+  end;
+
+end
+*}
+
+hide_const (open) One Dig0 Dig1
+
+
+subsection {* Toy examples *}
+
+definition "foo \<longleftrightarrow> #4 * #2 + #7 = (#8 :: nat)"
+definition "bar \<longleftrightarrow> #4 * #2 + #7 \<ge> (#8 :: int) - #3"
+
+code_thms foo bar
+export_code foo bar checking SML OCaml? Haskell? Scala?
+
+text {* This is an ad-hoc @{text Code_Integer} setup. *}
+
+setup {*
+  fold (add_code Code_Printer.literal_numeral)
+    [Code_ML.target_SML, Code_ML.target_OCaml, Code_Haskell.target, Code_Scala.target]
+*}
+
+code_type int
+  (SML "IntInf.int")
+  (OCaml "Big'_int.big'_int")
+  (Haskell "Integer")
+  (Scala "BigInt")
+  (Eval "int")
+
+code_const "0::int"
+  (SML "0/ :/ IntInf.int")
+  (OCaml "Big'_int.zero")
+  (Haskell "0")
+  (Scala "BigInt(0)")
+  (Eval "0/ :/ int")
+
+code_const Int.pred
+  (SML "IntInf.- ((_), 1)")
+  (OCaml "Big'_int.pred'_big'_int")
+  (Haskell "!(_/ -/ 1)")
+  (Scala "!(_ -/ 1)")
+  (Eval "!(_/ -/ 1)")
+
+code_const Int.succ
+  (SML "IntInf.+ ((_), 1)")
+  (OCaml "Big'_int.succ'_big'_int")
+  (Haskell "!(_/ +/ 1)")
+  (Scala "!(_ +/ 1)")
+  (Eval "!(_/ +/ 1)")
+
+code_const "op + \<Colon> int \<Rightarrow> int \<Rightarrow> int"
+  (SML "IntInf.+ ((_), (_))")
+  (OCaml "Big'_int.add'_big'_int")
+  (Haskell infixl 6 "+")
+  (Scala infixl 7 "+")
+  (Eval infixl 8 "+")
+
+code_const "uminus \<Colon> int \<Rightarrow> int"
+  (SML "IntInf.~")
+  (OCaml "Big'_int.minus'_big'_int")
+  (Haskell "negate")
+  (Scala "!(- _)")
+  (Eval "~/ _")
+
+code_const "op - \<Colon> int \<Rightarrow> int \<Rightarrow> int"
+  (SML "IntInf.- ((_), (_))")
+  (OCaml "Big'_int.sub'_big'_int")
+  (Haskell infixl 6 "-")
+  (Scala infixl 7 "-")
+  (Eval infixl 8 "-")
+
+code_const "op * \<Colon> int \<Rightarrow> int \<Rightarrow> int"
+  (SML "IntInf.* ((_), (_))")
+  (OCaml "Big'_int.mult'_big'_int")
+  (Haskell infixl 7 "*")
+  (Scala infixl 8 "*")
+  (Eval infixl 9 "*")
+
+code_const pdivmod
+  (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 \<Colon> int \<Rightarrow> int \<Rightarrow> bool"
+  (SML "!((_ : IntInf.int) = _)")
+  (OCaml "Big'_int.eq'_big'_int")
+  (Haskell infix 4 "==")
+  (Scala infixl 5 "==")
+  (Eval infixl 6 "=")
+
+code_const "op \<le> \<Colon> int \<Rightarrow> int \<Rightarrow> bool"
+  (SML "IntInf.<= ((_), (_))")
+  (OCaml "Big'_int.le'_big'_int")
+  (Haskell infix 4 "<=")
+  (Scala infixl 4 "<=")
+  (Eval infixl 6 "<=")
+
+code_const "op < \<Colon> int \<Rightarrow> int \<Rightarrow> bool"
+  (SML "IntInf.< ((_), (_))")
+  (OCaml "Big'_int.lt'_big'_int")
+  (Haskell infix 4 "<")
+  (Scala infixl 4 "<")
+  (Eval infixl 6 "<")
+
+code_const Code_Numeral.int_of
+  (SML "IntInf.fromInt")
+  (OCaml "_")
+  (Haskell "toInteger")
+  (Scala "!_.as'_BigInt")
+  (Eval "_")
+
+export_code foo bar checking SML OCaml? Haskell? Scala?
+
+end
diff -r 0133d31f9ab4 -r 26977429b784 src/HOL/ex/ROOT.ML
--- a/src/HOL/ex/ROOT.ML	Mon Feb 20 22:35:32 2012 +0100
+++ b/src/HOL/ex/ROOT.ML	Tue Feb 21 12:45:00 2012 +0100
@@ -17,7 +17,7 @@
 use_thys [
   "Iff_Oracle",
   "Coercion_Examples",
-  "Numeral",
+  "Numeral_Representation",
   "Higher_Order_Logic",
   "Abstract_NAT",
   "Guess",