
diff -r be91db94e526 -r 64e8d4afcf10 src/HOL/Divides.thy
--- a/src/HOL/Divides.thy	Thu Sep 29 14:15:01 2022 +0200
+++ b/src/HOL/Divides.thy	Thu Sep 29 14:03:40 2022 +0000
@@ -1,537 +1,12 @@
 (*  Title:      HOL/Divides.thy
-    Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
-    Copyright   1999  University of Cambridge
 *)
 
-section \<open>More on quotient and remainder\<close>
+section \<open>Lemmas of doubtful value\<close>
 
 theory Divides
 imports Parity
 begin
 
-subsection \<open>More on division\<close>
-
-subsubsection \<open>Monotonicity in the First Argument (Dividend)\<close>
-
-lemma unique_quotient_lemma:
-  assumes "b * q' + r' \<le> b * q + r" "0 \<le> r'" "r' < b" "r < b" shows "q' \<le> (q::int)"
-proof -
-  have "r' + b * (q'-q) \<le> r"
-    using assms by (simp add: right_diff_distrib)
-  moreover have "0 < b * (1 + q - q') "
-    using assms by (simp add: right_diff_distrib distrib_left)
-  moreover have "b * q' < b * (1 + q)"
-    using assms by (simp add: right_diff_distrib distrib_left)
-  ultimately show ?thesis
-    using assms by (simp add: mult_less_cancel_left)
-qed
-
-lemma unique_quotient_lemma_neg:
-  "b * q' + r' \<le> b*q + r \<Longrightarrow> r \<le> 0 \<Longrightarrow> b < r \<Longrightarrow> b < r' \<Longrightarrow> q \<le> (q'::int)"
-  using unique_quotient_lemma[where b = "-b" and r = "-r'" and r'="-r"] by auto
-
-lemma zdiv_mono1:
-  \<open>a div b \<le> a' div b\<close>
-  if \<open>a \<le> a'\<close> \<open>0 < b\<close>
-  for a b b' :: int
-proof (rule unique_quotient_lemma)
-  show "b * (a div b) + a mod b \<le> b * (a' div b) + a' mod b"
-    using \<open>a \<le> a'\<close> by auto
-qed (use that in auto)
-
-lemma zdiv_mono1_neg:
-  fixes b::int
-  assumes "a \<le> a'" "b < 0" shows "a' div b \<le> a div b"
-proof (rule unique_quotient_lemma_neg)
-  show "b * (a div b) + a mod b \<le> b * (a' div b) + a' mod b"
-    using assms(1) by auto
-qed (use assms in auto)
-
-
-subsubsection \<open>Monotonicity in the Second Argument (Divisor)\<close>
-
-lemma q_pos_lemma:
-  fixes q'::int
-  assumes "0 \<le> b'*q' + r'" "r' < b'" "0 < b'"
-  shows "0 \<le> q'"
-proof -
-  have "0 < b'* (q' + 1)"
-    using assms by (simp add: distrib_left)
-  with assms show ?thesis
-    by (simp add: zero_less_mult_iff)
-qed
-
-lemma zdiv_mono2_lemma:
-  fixes q'::int
-  assumes eq: "b*q + r = b'*q' + r'" and le: "0 \<le> b'*q' + r'" and "r' < b'" "0 \<le> r" "0 < b'" "b' \<le> b"
-  shows "q \<le> q'"
-proof -
-  have "0 \<le> q'"
-    using q_pos_lemma le \<open>r' < b'\<close> \<open>0 < b'\<close> by blast
-  moreover have "b*q = r' - r + b'*q'"
-    using eq by linarith
-  ultimately have "b*q < b* (q' + 1)"
-    using mult_right_mono assms unfolding distrib_left by fastforce
-  with assms show ?thesis
-    by (simp add: mult_less_cancel_left_pos)
-qed
-
-lemma zdiv_mono2:
-  fixes a::int
-  assumes "0 \<le> a" "0 < b'" "b' \<le> b" shows "a div b \<le> a div b'"
-proof (rule zdiv_mono2_lemma)
-  have "b \<noteq> 0"
-    using assms by linarith
-  show "b * (a div b) + a mod b = b' * (a div b') + a mod b'"
-    by simp
-qed (use assms in auto)
-
-lemma zdiv_mono2_neg_lemma:
-    fixes q'::int
-    assumes "b*q + r = b'*q' + r'" "b'*q' + r' < 0" "r < b" "0 \<le> r'" "0 < b'" "b' \<le> b"
-    shows "q' \<le> q"
-proof -
-  have "b'*q' < 0"
-    using assms by linarith
-  with assms have "q' \<le> 0"
-    by (simp add: mult_less_0_iff)
-  have "b*q' \<le> b'*q'"
-    by (simp add: \<open>q' \<le> 0\<close> assms(6) mult_right_mono_neg)
-  then have "b*q' < b* (q + 1)"
-    using assms by (simp add: distrib_left)
-  then show ?thesis
-    using assms by (simp add: mult_less_cancel_left)
-qed
-
-lemma zdiv_mono2_neg:
-  fixes a::int
-  assumes "a < 0" "0 < b'" "b' \<le> b" shows "a div b' \<le> a div b"
-proof (rule zdiv_mono2_neg_lemma)
-  have "b \<noteq> 0"
-    using assms by linarith
-  show "b * (a div b) + a mod b = b' * (a div b') + a mod b'"
-    by simp
-qed (use assms in auto)
-
-
-subsubsection \<open>Quotients of Signs\<close>
-
-lemma div_eq_minus1: "0 < b \<Longrightarrow> - 1 div b = - 1" for b :: int
-  by (simp add: divide_int_def)
-
-lemma zmod_minus1: "0 < b \<Longrightarrow> - 1 mod b = b - 1" for b :: int
-  by (auto simp add: modulo_int_def)
-
-lemma minus_mod_int_eq:
-  \<open>- k mod l = l - 1 - (k - 1) mod l\<close> if \<open>l \<ge> 0\<close> for k l :: int
-proof (cases \<open>l = 0\<close>)
-  case True
-  then show ?thesis
-    by simp
-next
-  case False
-  with that have \<open>l > 0\<close>
-    by simp
-  then show ?thesis
-  proof (cases \<open>l dvd k\<close>)
-    case True
-    then obtain j where \<open>k = l * j\<close> ..
-    moreover have \<open>(l * j mod l - 1) mod l = l - 1\<close>
-      using \<open>l > 0\<close> by (simp add: zmod_minus1)
-    then have \<open>(l * j - 1) mod l = l - 1\<close>
-      by (simp only: mod_simps)
-    ultimately show ?thesis
-      by simp
-  next
-    case False
-    moreover have 1: \<open>0 < k mod l\<close>
-      using \<open>0 < l\<close> False le_less by fastforce
-    moreover have 2: \<open>k mod l < 1 + l\<close>
-      using \<open>0 < l\<close> pos_mod_bound[of l k] by linarith
-    from 1 2 \<open>l > 0\<close> have \<open>(k mod l - 1) mod l = k mod l - 1\<close>
-      by (simp add: zmod_trivial_iff)
-    ultimately show ?thesis
-      by (simp only: zmod_zminus1_eq_if)
-         (simp add: mod_eq_0_iff_dvd algebra_simps mod_simps)
-  qed
-qed
-
-lemma div_neg_pos_less0:
-  fixes a::int
-  assumes "a < 0" "0 < b" 
-  shows "a div b < 0"
-proof -
-  have "a div b \<le> - 1 div b"
-    using zdiv_mono1 assms by auto
-  also have "... \<le> -1"
-    by (simp add: assms(2) div_eq_minus1)
-  finally show ?thesis 
-    by force
-qed
-
-lemma div_nonneg_neg_le0: "[| (0::int) \<le> a; b < 0 |] ==> a div b \<le> 0"
-  by (drule zdiv_mono1_neg, auto)
-
-lemma div_nonpos_pos_le0: "[| (a::int) \<le> 0; b > 0 |] ==> a div b \<le> 0"
-  by (drule zdiv_mono1, auto)
-
-text\<open>Now for some equivalences of the form \<open>a div b >=< 0 \<longleftrightarrow> \<dots>\<close>
-conditional upon the sign of \<open>a\<close> or \<open>b\<close>. There are many more.
-They should all be simp rules unless that causes too much search.\<close>
-
-lemma pos_imp_zdiv_nonneg_iff:
-      fixes a::int
-      assumes "0 < b" 
-      shows "(0 \<le> a div b) = (0 \<le> a)"
-proof
-  show "0 \<le> a div b \<Longrightarrow> 0 \<le> a"
-    using assms
-    by (simp add: linorder_not_less [symmetric]) (blast intro: div_neg_pos_less0)
-next
-  assume "0 \<le> a"
-  then have "0 div b \<le> a div b"
-    using zdiv_mono1 assms by blast
-  then show "0 \<le> a div b"
-    by auto
-qed
-
-lemma pos_imp_zdiv_pos_iff:
-  "0<k \<Longrightarrow> 0 < (i::int) div k \<longleftrightarrow> k \<le> i"
-  using pos_imp_zdiv_nonneg_iff[of k i] zdiv_eq_0_iff[of i k] by arith
-
-lemma neg_imp_zdiv_nonneg_iff:
-  fixes a::int
-  assumes "b < 0" 
-  shows "(0 \<le> a div b) = (a \<le> 0)"
-  using assms by (simp add: div_minus_minus [of a, symmetric] pos_imp_zdiv_nonneg_iff del: div_minus_minus)
-
-(*But not (a div b \<le> 0 iff a\<le>0); consider a=1, b=2 when a div b = 0.*)
-lemma pos_imp_zdiv_neg_iff: "(0::int) < b ==> (a div b < 0) = (a < 0)"
-  by (simp add: linorder_not_le [symmetric] pos_imp_zdiv_nonneg_iff)
-
-(*Again the law fails for \<le>: consider a = -1, b = -2 when a div b = 0*)
-lemma neg_imp_zdiv_neg_iff: "b < (0::int) ==> (a div b < 0) = (0 < a)"
-  by (simp add: linorder_not_le [symmetric] neg_imp_zdiv_nonneg_iff)
-
-lemma nonneg1_imp_zdiv_pos_iff:
-  fixes a::int
-  assumes "0 \<le> a" 
-  shows "a div b > 0 \<longleftrightarrow> a \<ge> b \<and> b>0"
-proof -
-  have "0 < a div b \<Longrightarrow> b \<le> a"
-    using div_pos_pos_trivial[of a b] assms by arith
-  moreover have "0 < a div b \<Longrightarrow> b > 0"
-    using assms div_nonneg_neg_le0[of a b]  by(cases "b=0"; force)
-  moreover have "b \<le> a \<and> 0 < b \<Longrightarrow> 0 < a div b"
-    using int_one_le_iff_zero_less[of "a div b"] zdiv_mono1[of b a b] by simp
-  ultimately show ?thesis
-    by blast
-qed
-
-lemma zmod_le_nonneg_dividend: "(m::int) \<ge> 0 \<Longrightarrow> m mod k \<le> m"
-  by (rule split_zmod[THEN iffD2]) (fastforce dest: q_pos_lemma intro: split_mult_pos_le)
-
-lemma sgn_div_eq_sgn_mult:
-  \<open>sgn (k div l) = of_bool (k div l \<noteq> 0) * sgn (k * l)\<close>
-  for k l :: int
-proof (cases \<open>k div l = 0\<close>)
-  case True
-  then show ?thesis
-    by simp
-next
-  case False
-  have \<open>0 \<le> \<bar>k\<bar> div \<bar>l\<bar>\<close>
-    by (cases \<open>l = 0\<close>) (simp_all add: pos_imp_zdiv_nonneg_iff)
-  then have \<open>\<bar>k\<bar> div \<bar>l\<bar> \<noteq> 0 \<longleftrightarrow> 0 < \<bar>k\<bar> div \<bar>l\<bar>\<close>
-    by (simp add: less_le)
-  also have \<open>\<dots> \<longleftrightarrow> \<bar>k\<bar> \<ge> \<bar>l\<bar>\<close>
-    using False nonneg1_imp_zdiv_pos_iff by auto
-  finally have *: \<open>\<bar>k\<bar> div \<bar>l\<bar> \<noteq> 0 \<longleftrightarrow> \<bar>l\<bar> \<le> \<bar>k\<bar>\<close> .
-  show ?thesis
-    using \<open>0 \<le> \<bar>k\<bar> div \<bar>l\<bar>\<close> False
-  by (auto simp add: div_eq_div_abs [of k l] div_eq_sgn_abs [of k l]
-    sgn_mult sgn_1_pos sgn_1_neg sgn_eq_0_iff nonneg1_imp_zdiv_pos_iff * dest: sgn_not_eq_imp)
-qed
-
-lemma
-  fixes a b q r :: int
-  assumes \<open>a = b * q + r\<close> \<open>0 \<le> r\<close> \<open>r < b\<close>
-  shows int_div_pos_eq:
-      \<open>a div b = q\<close> (is ?Q)
-    and int_mod_pos_eq:
-      \<open>a mod b = r\<close> (is ?R)
-proof -
-  from assms have \<open>(a div b, a mod b) = (q, r)\<close>
-    by (cases b q r a rule: euclidean_relation_intI)
-      (auto simp add: ac_simps dvd_add_left_iff sgn_1_pos le_less dest: zdvd_imp_le)
-  then show ?Q and ?R
-    by simp_all
-qed
-
-lemma int_div_neg_eq:
-  \<open>a div b = q\<close> if \<open>a = b * q + r\<close> \<open>r \<le> 0\<close> \<open>b < r\<close> for a b q r :: int
-  using that int_div_pos_eq [of a \<open>- b\<close> \<open>- q\<close> \<open>- r\<close>] by simp_all
-
-lemma int_mod_neg_eq:
-  \<open>a mod b = r\<close> if \<open>a = b * q + r\<close> \<open>r \<le> 0\<close> \<open>b < r\<close> for a b q r :: int
-  using that int_div_neg_eq [of a b q r] by simp
-
-
-subsubsection \<open>Further properties\<close>
-
-lemma div_int_pos_iff:
-  "k div l \<ge> 0 \<longleftrightarrow> k = 0 \<or> l = 0 \<or> k \<ge> 0 \<and> l \<ge> 0
-    \<or> k < 0 \<and> l < 0"
-  for k l :: int
-proof (cases "k = 0 \<or> l = 0")
-  case False
-  then have *: "k \<noteq> 0" "l \<noteq> 0"
-    by auto
-  then have "0 \<le> k div l \<Longrightarrow> \<not> k < 0 \<Longrightarrow> 0 \<le> l"
-    by (meson neg_imp_zdiv_neg_iff not_le not_less_iff_gr_or_eq)
-  then show ?thesis
-   using * by (auto simp add: pos_imp_zdiv_nonneg_iff neg_imp_zdiv_nonneg_iff)
-qed auto
-
-lemma mod_int_pos_iff:
-  "k mod l \<ge> 0 \<longleftrightarrow> l dvd k \<or> l = 0 \<and> k \<ge> 0 \<or> l > 0"
-  for k l :: int
-proof (cases "l > 0")
-  case False
-  then show ?thesis 
-    by (simp add: dvd_eq_mod_eq_0) (use neg_mod_sign [of l k] in \<open>auto simp add: le_less not_less\<close>)
-qed auto
-
-text \<open>Simplify expressions in which div and mod combine numerical constants\<close>
-
-lemma abs_div: "(y::int) dvd x \<Longrightarrow> \<bar>x div y\<bar> = \<bar>x\<bar> div \<bar>y\<bar>"
-  unfolding dvd_def by (cases "y=0") (auto simp add: abs_mult)
-
-text\<open>Suggested by Matthias Daum\<close>
-lemma int_power_div_base:
-  fixes k :: int
-  assumes "0 < m" "0 < k"
-  shows "k ^ m div k = (k::int) ^ (m - Suc 0)"
-proof -
-  have eq: "k ^ m = k ^ ((m - Suc 0) + Suc 0)"
-    by (simp add: assms)
-  show ?thesis
-    using assms by (simp only: power_add eq) auto
-qed
-
-text\<open>Suggested by Matthias Daum\<close>
-lemma int_div_less_self:
-  fixes x::int
-  assumes "0 < x" "1 < k"
-  shows  "x div k < x"
-proof -
-  have "nat x div nat k < nat x"
-    by (simp add: assms)
-  with assms show ?thesis
-    by (simp add: nat_div_distrib [symmetric])
-qed
-
-lemma mod_eq_dvd_iff_nat:
-  "m mod q = n mod q \<longleftrightarrow> q dvd m - n" if "m \<ge> n" for m n q :: nat
-proof -
-  have "int m mod int q = int n mod int q \<longleftrightarrow> int q dvd int m - int n"
-    by (simp add: mod_eq_dvd_iff)
-  with that have "int (m mod q) = int (n mod q) \<longleftrightarrow> int q dvd int (m - n)"
-    by (simp only: of_nat_mod of_nat_diff)
-  then show ?thesis
-    by simp
-qed
-
-lemma mod_eq_nat1E:
-  fixes m n q :: nat
-  assumes "m mod q = n mod q" and "m \<ge> n"
-  obtains s where "m = n + q * s"
-proof -
-  from assms have "q dvd m - n"
-    by (simp add: mod_eq_dvd_iff_nat)
-  then obtain s where "m - n = q * s" ..
-  with \<open>m \<ge> n\<close> have "m = n + q * s"
-    by simp
-  with that show thesis .
-qed
-
-lemma mod_eq_nat2E:
-  fixes m n q :: nat
-  assumes "m mod q = n mod q" and "n \<ge> m"
-  obtains s where "n = m + q * s"
-  using assms mod_eq_nat1E [of n q m] by (auto simp add: ac_simps)
-
-lemma nat_mod_eq_lemma:
-  assumes "(x::nat) mod n = y mod n" and "y \<le> x"
-  shows "\<exists>q. x = y + n * q"
-  using assms by (rule mod_eq_nat1E) (rule exI)
-
-lemma nat_mod_eq_iff: "(x::nat) mod n = y mod n \<longleftrightarrow> (\<exists>q1 q2. x + n * q1 = y + n * q2)"
-  (is "?lhs = ?rhs")
-proof
-  assume H: "x mod n = y mod n"
-  {assume xy: "x \<le> y"
-    from H have th: "y mod n = x mod n" by simp
-    from nat_mod_eq_lemma[OF th xy] have ?rhs
-    proof
-      fix q
-      assume "y = x + n * q"
-      then have "x + n * q = y + n * 0"
-        by simp
-      then show "\<exists>q1 q2. x + n * q1 = y + n * q2"
-        by blast
-    qed}
-  moreover
-  {assume xy: "y \<le> x"
-    from nat_mod_eq_lemma[OF H xy] have ?rhs
-    proof
-      fix q
-      assume "x = y + n * q"
-      then have "x + n * 0 = y + n * q"
-        by simp
-      then show "\<exists>q1 q2. x + n * q1 = y + n * q2"
-        by blast
-    qed}
-  ultimately  show ?rhs using linear[of x y] by blast
-next
-  assume ?rhs then obtain q1 q2 where q12: "x + n * q1 = y + n * q2" by blast
-  hence "(x + n * q1) mod n = (y + n * q2) mod n" by simp
-  thus  ?lhs by simp
-qed
-
-
-subsubsection \<open>Computing \<open>div\<close> and \<open>mod\<close> with shifting\<close>
-
-lemma div_pos_geq:
-  fixes k l :: int
-  assumes "0 < l" and "l \<le> k"
-  shows "k div l = (k - l) div l + 1"
-proof -
-  have "k = (k - l) + l" by simp
-  then obtain j where k: "k = j + l" ..
-  with assms show ?thesis by (simp add: div_add_self2)
-qed
-
-lemma mod_pos_geq:
-  fixes k l :: int
-  assumes "0 < l" and "l \<le> k"
-  shows "k mod l = (k - l) mod l"
-proof -
-  have "k = (k - l) + l" by simp
-  then obtain j where k: "k = j + l" ..
-  with assms show ?thesis by simp
-qed
-
-text\<open>computing div by shifting\<close>
-
-lemma pos_zdiv_mult_2: \<open>(1 + 2 * b) div (2 * a) = b div a\<close> (is ?Q)
-  and pos_zmod_mult_2: \<open>(1 + 2 * b) mod (2 * a) = 1 + 2 * (b mod a)\<close> (is ?R)
-  if \<open>0 \<le> a\<close> for a b :: int
-proof -
-  have \<open>((1 + 2 * b) div (2 * a), (1 + 2 * b) mod (2 * a)) = (b div a, 1 + 2 * (b mod a))\<close>
-  proof (cases \<open>2 * a\<close> \<open>b div a\<close> \<open>1 + 2 * (b mod a)\<close> \<open>1 + 2 * b\<close> rule: euclidean_relation_intI)
-    case by0
-    then show ?case
-      by simp
-  next
-    case divides
-    have \<open>even (2 * a)\<close>
-      by simp
-    then have \<open>even (1 + 2 * b)\<close>
-      using \<open>2 * a dvd 1 + 2 * b\<close> by (rule dvd_trans)
-    then show ?case
-      by simp
-  next
-    case euclidean_relation
-    with that have \<open>a > 0\<close>
-      by simp
-    moreover have \<open>b mod a < a\<close>
-      using \<open>a > 0\<close> by simp
-    then have \<open>1 + 2 * (b mod a) < 2 * a\<close>
-      by simp
-    moreover have \<open>2 * (b mod a) + a * (2 * (b div a)) = 2 * (b div a * a + b mod a)\<close>
-      by (simp only: algebra_simps)
-    moreover have \<open>0 \<le> 2 * (b mod a)\<close>
-      using \<open>a > 0\<close> by simp
-    ultimately show ?case
-      by (simp add: algebra_simps)
-  qed
-  then show ?Q and ?R
-    by simp_all
-qed
-
-lemma neg_zdiv_mult_2: \<open>(1 + 2 * b) div (2 * a) = (b + 1) div a\<close> (is ?Q)
-  and neg_zmod_mult_2: \<open>(1 + 2 * b) mod (2 * a) = 2 * ((b + 1) mod a) - 1\<close> (is ?R)
-  if \<open>a \<le> 0\<close> for a b :: int
-proof -
-  have \<open>((1 + 2 * b) div (2 * a), (1 + 2 * b) mod (2 * a)) = ((b + 1) div a, 2 * ((b + 1) mod a) - 1)\<close>
-  proof (cases \<open>2 * a\<close> \<open>(b + 1) div a\<close> \<open>2 * ((b + 1) mod a) - 1\<close> \<open>1 + 2 * b\<close> rule: euclidean_relation_intI)
-    case by0
-    then show ?case
-      by simp
-  next
-    case divides
-    have \<open>even (2 * a)\<close>
-      by simp
-    then have \<open>even (1 + 2 * b)\<close>
-      using \<open>2 * a dvd 1 + 2 * b\<close> by (rule dvd_trans)
-    then show ?case
-      by simp
-  next
-    case euclidean_relation
-    with that have \<open>a < 0\<close>
-      by simp
-    moreover have \<open>(b + 1) mod a > a\<close>
-      using \<open>a < 0\<close> by simp
-    then have \<open>2 * ((b + 1) mod a) > 1 + 2 * a\<close>
-      by simp
-    moreover have \<open>((1 + b) mod a) \<le> 0\<close>
-      using \<open>a < 0\<close> by simp
-    then have \<open>2 * ((1 + b) mod a) \<le> 0\<close>
-      by simp
-    moreover have \<open>2 * ((1 + b) mod a) + a * (2 * ((1 + b) div a)) =
-      2 * ((1 + b) div a * a + (1 + b) mod a)\<close>
-      by (simp only: algebra_simps)
-    ultimately show ?case
-      by (simp add: algebra_simps sgn_mult abs_mult)
-  qed
-  then show ?Q and ?R
-    by simp_all
-qed
-
-lemma zdiv_numeral_Bit0 [simp]:
-  "numeral (Num.Bit0 v) div numeral (Num.Bit0 w) =
-    numeral v div (numeral w :: int)"
-  unfolding numeral.simps unfolding mult_2 [symmetric]
-  by (rule div_mult_mult1, simp)
-
-lemma zdiv_numeral_Bit1 [simp]:
-  "numeral (Num.Bit1 v) div numeral (Num.Bit0 w) =
-    (numeral v div (numeral w :: int))"
-  unfolding numeral.simps
-  unfolding mult_2 [symmetric] add.commute [of _ 1]
-  by (rule pos_zdiv_mult_2, simp)
-
-lemma zmod_numeral_Bit0 [simp]:
-  "numeral (Num.Bit0 v) mod numeral (Num.Bit0 w) =
-    (2::int) * (numeral v mod numeral w)"
-  unfolding numeral_Bit0 [of v] numeral_Bit0 [of w]
-  unfolding mult_2 [symmetric] by (rule mod_mult_mult1)
-
-lemma zmod_numeral_Bit1 [simp]:
-  "numeral (Num.Bit1 v) mod numeral (Num.Bit0 w) =
-    2 * (numeral v mod numeral w) + (1::int)"
-  unfolding numeral_Bit1 [of v] numeral_Bit0 [of w]
-  unfolding mult_2 [symmetric] add.commute [of _ 1]
-  by (rule pos_zmod_mult_2, simp)
-
-  
-code_identifier
-  code_module Divides \<rightharpoonup> (SML) Arith and (OCaml) Arith and (Haskell) Arith
-
-
-subsection \<open>Lemmas of doubtful value\<close>
-
 class unique_euclidean_semiring_numeral = unique_euclidean_semiring_with_nat + linordered_semidom +
   assumes div_less: "0 \<le> a \<Longrightarrow> a < b \<Longrightarrow> a div b = 0"
     and mod_less: " 0 \<le> a \<Longrightarrow> a < b \<Longrightarrow> a mod b = a"
@@ -646,4 +121,7 @@
   "k div l > 0" if "k \<ge> l" and "l > 0" for k l :: int
   using that by (simp add: nonneg1_imp_zdiv_pos_iff)
 
+code_identifier
+  code_module Divides \<rightharpoonup> (SML) Arith and (OCaml) Arith and (Haskell) Arith
+
 end




