--- a/src/HOL/Euclidean_Division.thy Thu Sep 29 14:15:01 2022 +0200
+++ b/src/HOL/Euclidean_Division.thy Thu Sep 29 14:03:40 2022 +0000
@@ -10,11 +10,11 @@
begin
subsection \<open>Euclidean (semi)rings with explicit division and remainder\<close>
-
-class euclidean_semiring = semidom_modulo +
+
+class euclidean_semiring = semidom_modulo +
fixes euclidean_size :: "'a \<Rightarrow> nat"
assumes size_0 [simp]: "euclidean_size 0 = 0"
- assumes mod_size_less:
+ assumes mod_size_less:
"b \<noteq> 0 \<Longrightarrow> euclidean_size (a mod b) < euclidean_size b"
assumes size_mult_mono:
"b \<noteq> 0 \<Longrightarrow> euclidean_size a \<le> euclidean_size (a * b)"
@@ -46,7 +46,7 @@
lemma dvd_euclidean_size_eq_imp_dvd:
assumes "a \<noteq> 0" and "euclidean_size a = euclidean_size b"
- and "b dvd a"
+ and "b dvd a"
shows "a dvd b"
proof (rule ccontr)
assume "\<not> a dvd b"
@@ -83,7 +83,7 @@
"is_unit a \<Longrightarrow> euclidean_size a = euclidean_size 1"
using euclidean_size_times_unit [of a 1] by simp
-lemma unit_iff_euclidean_size:
+lemma unit_iff_euclidean_size:
"is_unit a \<longleftrightarrow> euclidean_size a = euclidean_size 1 \<and> a \<noteq> 0"
proof safe
assume A: "a \<noteq> 0" and B: "euclidean_size a = euclidean_size 1"
@@ -96,7 +96,7 @@
shows "euclidean_size b < euclidean_size (a * b)"
proof (rule ccontr)
assume "\<not>euclidean_size b < euclidean_size (a * b)"
- with size_mult_mono'[OF assms(1), of b]
+ with size_mult_mono'[OF assms(1), of b]
have eq: "euclidean_size (a * b) = euclidean_size b" by simp
have "a * b dvd b"
by (rule dvd_euclidean_size_eq_imp_dvd [OF _ eq])
@@ -107,12 +107,12 @@
qed
lemma dvd_imp_size_le:
- assumes "a dvd b" "b \<noteq> 0"
+ assumes "a dvd b" "b \<noteq> 0"
shows "euclidean_size a \<le> euclidean_size b"
using assms by (auto simp: size_mult_mono)
lemma dvd_proper_imp_size_less:
- assumes "a dvd b" "\<not> b dvd a" "b \<noteq> 0"
+ assumes "a dvd b" "\<not> b dvd a" "b \<noteq> 0"
shows "euclidean_size a < euclidean_size b"
proof -
from assms(1) obtain c where "b = a * c" by (erule dvdE)
@@ -160,7 +160,7 @@
then show ?thesis
by simp
qed
-
+
end
@@ -450,7 +450,7 @@
text \<open>Exponentiation respects modular equivalence.\<close>
-lemma power_mod [mod_simps]:
+lemma power_mod [mod_simps]:
"((a mod b) ^ n) mod b = (a ^ n) mod b"
proof (induct n)
case 0
@@ -601,10 +601,10 @@
end
-
+
subsection \<open>Uniquely determined division\<close>
-class unique_euclidean_semiring = euclidean_semiring +
+class unique_euclidean_semiring = euclidean_semiring +
assumes euclidean_size_mult: \<open>euclidean_size (a * b) = euclidean_size a * euclidean_size b\<close>
fixes division_segment :: \<open>'a \<Rightarrow> 'a\<close>
assumes is_unit_division_segment [simp]: \<open>is_unit (division_segment a)\<close>
@@ -838,7 +838,7 @@
class unique_euclidean_ring = euclidean_ring + unique_euclidean_semiring
begin
-
+
subclass euclidean_ring_cancel ..
end
@@ -923,7 +923,7 @@
by (auto simp add: mult_div_unfold ac_simps intro: Max.boundedI)
then show "m div n * n + m mod n = m"
by (simp add: modulo_nat_def)
- assume "n \<noteq> 0"
+ assume "n \<noteq> 0"
show "euclidean_size (m mod n) < euclidean_size n"
proof -
have "m < Suc (m div n) * n"
@@ -1115,7 +1115,7 @@
and mod_less [simp]: "m mod n = m"
if "m < n" for m n :: nat
using that by (auto intro: div_nat_eqI mod_nat_eqI)
-
+
lemma split_div:
\<open>P (m div n) \<longleftrightarrow>
(n = 0 \<longrightarrow> P 0) \<and>
@@ -1229,7 +1229,7 @@
then have "m mod 2 = 0 \<or> m mod 2 = 1"
by arith
then show ?thesis
- by auto
+ by auto
qed
lemma mod_Suc_eq [mod_simps]:
@@ -1501,7 +1501,7 @@
proof
assume P: "P (p - Suc m)"
with \<open>\<not> P 0\<close> have "Suc m < p"
- by (auto intro: ccontr)
+ by (auto intro: ccontr)
then have "Suc (p - Suc m) = p - m"
by arith
moreover from \<open>0 < p\<close> have "p - Suc m < p"
@@ -1540,6 +1540,72 @@
by simp
qed
+lemma mod_eq_dvd_iff_nat:
+ \<open>m mod q = n mod q \<longleftrightarrow> q dvd m - n\<close> (is \<open>?P \<longleftrightarrow> ?Q\<close>)
+ if \<open>m \<ge> n\<close> for m n q :: nat
+proof
+ assume ?Q
+ then obtain s where \<open>m - n = q * s\<close> ..
+ with that have \<open>m = q * s + n\<close>
+ by simp
+ then show ?P
+ by simp
+next
+ assume ?P
+ have \<open>m - n = m div q * q + m mod q - (n div q * q + n mod q)\<close>
+ by simp
+ also have \<open>\<dots> = q * (m div q - n div q)\<close>
+ by (simp only: algebra_simps \<open>?P\<close>)
+ finally show ?Q ..
+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_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 mod_eq_nat1E [OF th xy] obtain q where "y = x + n * q" .
+ then have "x + n * q = y + n * 0"
+ by simp
+ then have "\<exists>q1 q2. x + n * q1 = y + n * q2"
+ by blast
+ }
+ moreover
+ { assume xy: "y \<le> x"
+ from mod_eq_nat1E [OF H xy] obtain q where "x = y + n * q" .
+ then have "x + n * 0 = y + n * q"
+ by simp
+ then have "\<exists>q1 q2. x + n * q1 = y + n * q2"
+ by blast
+ }
+ 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
+
+
subsection \<open>Elementary euclidean division on \<^typ>\<open>int\<close>\<close>
@@ -1577,7 +1643,7 @@
next
fix k l :: int
assume "l \<noteq> 0"
- obtain n m and s t where k: "k = sgn s * int n" and l: "l = sgn t * int m"
+ obtain n m and s t where k: "k = sgn s * int n" and l: "l = sgn t * int m"
by (blast intro: int_sgnE elim: that)
then have "k * l = sgn (s * t) * int (n * m)"
by (simp add: ac_simps sgn_mult)
@@ -1586,7 +1652,7 @@
(auto simp add: algebra_simps sgn_mult sgn_1_pos sgn_0_0)
next
fix k l :: int
- obtain n m and s t where "k = sgn s * int n" and "l = sgn t * int m"
+ obtain n m and s t where "k = sgn s * int n" and "l = sgn t * int m"
by (blast intro: int_sgnE elim: that)
then show "k div l * l + k mod l = k"
by (simp add: divide_int_unfold modulo_int_unfold algebra_simps modulo_nat_def of_nat_diff)
@@ -1736,7 +1802,7 @@
lemma abs_mod_less:
"\<bar>k mod l\<bar> < \<bar>l\<bar>" if "l \<noteq> 0" for k l :: int
proof -
- obtain n m and s t where "k = sgn s * int n" and "l = sgn t * int m"
+ obtain n m and s t where "k = sgn s * int n" and "l = sgn t * int m"
by (blast intro: int_sgnE elim: that)
with that show ?thesis
by (auto simp add: modulo_int_unfold abs_mult mod_greater_zero_iff_not_dvd
@@ -1747,7 +1813,7 @@
lemma sgn_mod:
"sgn (k mod l) = sgn l" if "l \<noteq> 0" "\<not> l dvd k" for k l :: int
proof -
- obtain n m and s t where "k = sgn s * int n" and "l = sgn t * int m"
+ obtain n m and s t where "k = sgn s * int n" and "l = sgn t * int m"
by (blast intro: int_sgnE elim: that)
with that show ?thesis
by (auto simp add: modulo_int_unfold sgn_mult mod_greater_zero_iff_not_dvd
@@ -2143,52 +2209,72 @@
"k mod l = k" if "k \<le> 0" and "l < k" for k l :: int
using that by (simp add: mod_eq_self_iff_div_eq_0)
-lemma div_pos_neg_trivial:
- \<open>k div l = - 1\<close> if \<open>0 < k\<close> and \<open>k + l \<le> 0\<close> for k l :: int
-proof (cases \<open>l dvd k\<close>)
- case True
- then obtain j where \<open>k = l * j\<close> ..
+lemma
+ div_pos_neg_trivial: \<open>k div l = - 1\<close> (is ?Q)
+ and mod_pos_neg_trivial: \<open>k mod l = k + l\<close> (is ?R)
+ if \<open>0 < k\<close> and \<open>k + l \<le> 0\<close> for k l :: int
+proof -
from that have \<open>l < 0\<close>
by simp
- with \<open>k = l * j\<close> \<open>0 < k\<close> have \<open>j \<le> - 1\<close>
- by (simp add: zero_less_mult_iff)
- moreover from \<open>k + l \<le> 0\<close> \<open>k = l * j\<close> have \<open>l * (j + 1) \<le> 0\<close>
- by (simp add: algebra_simps)
- with \<open>l < 0\<close> have \<open>- 1 \<le> j\<close>
- by (simp add: mult_le_0_iff)
- ultimately have \<open>j = - 1\<close>
- by (rule order.antisym)
- with \<open>k = l * j\<close> \<open>l < 0\<close> show ?thesis
- by (simp add: dvd_neg_div)
-next
- case False
- have \<open>k + l < 0\<close>
- proof (rule ccontr)
- assume \<open>\<not> k + l < 0\<close>
- with \<open>k + l \<le> 0\<close> have \<open>k + l = 0\<close>
+ have \<open>(k div l, k mod l) = (- 1, k + l)\<close>
+ proof (cases l \<open>- 1 :: int\<close> \<open>k + l\<close> k rule: euclidean_relation_intI)
+ case by0
+ with \<open>l < 0\<close> show ?case
by simp
- then have \<open>k = - l\<close>
+ next
+ case divides
+ from \<open>l dvd k\<close> obtain j where \<open>k = l * j\<close> ..
+ with \<open>l < 0\<close> \<open>0 < k\<close> have \<open>j < 0\<close>
+ by (simp add: zero_less_mult_iff)
+ moreover from \<open>k + l \<le> 0\<close> \<open>k = l * j\<close> have \<open>l * (j + 1) \<le> 0\<close>
+ by (simp add: algebra_simps)
+ with \<open>l < 0\<close> have \<open>j + 1 \<ge> 0\<close>
+ by (simp add: mult_le_0_iff)
+ with \<open>j < 0\<close> have \<open>j = - 1\<close>
by simp
- then have \<open>l dvd k\<close>
+ with \<open>k = l * j\<close> show ?case
by simp
- with \<open>\<not> l dvd k\<close> show False ..
+ next
+ case euclidean_relation
+ with \<open>k + l \<le> 0\<close> have \<open>k + l < 0\<close>
+ by (auto simp add: less_le add_eq_0_iff)
+ with \<open>0 < k\<close> show ?case
+ by simp
qed
- with that \<open>\<not> l dvd k\<close> show ?thesis
- by (simp add: div_eq_div_abs [of k l])
-qed
-
-lemma mod_pos_neg_trivial:
- \<open>k mod l = k + l\<close> if \<open>0 < k\<close> and \<open>k + l \<le> 0\<close> for k l :: int
-proof -
- from that have \<open>k mod l = k div l * l + k mod l + l\<close>
- by (simp add: div_pos_neg_trivial)
- then show ?thesis by simp
+ then show ?Q and ?R
+ by simp_all
qed
text \<open>There is neither \<open>div_neg_pos_trivial\<close> nor \<open>mod_neg_pos_trivial\<close>
because \<^term>\<open>0 div l = 0\<close> would supersede it.\<close>
+subsubsection \<open>More uniqueness rules\<close>
+
+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>Laws for unary minus\<close>
lemma zmod_zminus1_not_zero:
@@ -2310,7 +2396,7 @@
from * [of \<open>\<lambda>q _. P q\<close>] show ?div .
from * [of \<open>\<lambda>_ r. Q r\<close>] show ?mod .
qed
-
+
text \<open>Enable (lin)arith to deal with \<^const>\<open>divide\<close> and \<^const>\<open>modulo\<close>
when these are applied to some constant that is of the form
\<^term>\<open>numeral k\<close>:\<close>
@@ -2415,6 +2501,421 @@
using that by (simp add: modulo_int_def sgn_if)
+subsubsection \<open>Monotonicity in the First Argument (Dividend)\<close>
+
+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 -
+ from \<open>a \<le> a'\<close> have \<open>b * (a div b) + a mod b \<le> b * (a' div b) + a' mod b\<close>
+ by simp
+ then have \<open>b * (a div b) \<le> (a' mod b - a mod b) + b * (a' div b)\<close>
+ by (simp add: algebra_simps)
+ moreover have \<open>a' mod b < b + a mod b\<close>
+ by (rule less_le_trans [of _ b]) (use \<open>0 < b\<close> in simp_all)
+ ultimately have \<open>b * (a div b) < b * (1 + a' div b)\<close>
+ by (simp add: distrib_left)
+ with \<open>0 < b\<close> have \<open>a div b < 1 + a' div b\<close>
+ by (simp add: mult_less_cancel_left)
+ then show ?thesis
+ by simp
+qed
+
+lemma zdiv_mono1_neg:
+ \<open>a' div b \<le> a div b\<close>
+ if \<open>a \<le> a'\<close> \<open>b < 0\<close>
+ for a a' b :: int
+ using that zdiv_mono1 [of \<open>- a'\<close> \<open>- a\<close> \<open>- b\<close>] by simp
+
+
+subsubsection \<open>Monotonicity in the Second Argument (Divisor)\<close>
+
+lemma zdiv_mono2:
+ \<open>a div b \<le> a div b'\<close> if \<open>0 \<le> a\<close> \<open>0 < b'\<close> \<open>b' \<le> b\<close> for a b b' :: int
+proof -
+ define q q' r r' where **: \<open>q = a div b\<close> \<open>q' = a div b'\<close> \<open>r = a mod b\<close> \<open>r' = a mod b'\<close>
+ then have *: \<open>b * q + r = b' * q' + r'\<close> \<open>0 \<le> b' * q' + r'\<close>
+ \<open>r' < b'\<close> \<open>0 \<le> r\<close> \<open>0 < b'\<close> \<open>b' \<le> b\<close>
+ using that by simp_all
+ have \<open>0 < b' * (q' + 1)\<close>
+ using * by (simp add: distrib_left)
+ with * have \<open>0 \<le> q'\<close>
+ by (simp add: zero_less_mult_iff)
+ moreover have \<open>b * q = r' - r + b' * q'\<close>
+ using * by linarith
+ ultimately have \<open>b * q < b * (q' + 1)\<close>
+ using mult_right_mono * unfolding distrib_left by fastforce
+ with * have \<open>q \<le> q'\<close>
+ by (simp add: mult_less_cancel_left_pos)
+ with ** show ?thesis
+ by simp
+qed
+
+lemma zdiv_mono2_neg:
+ \<open>a div b' \<le> a div b\<close> if \<open>a < 0\<close> \<open>0 < b'\<close> \<open>b' \<le> b\<close> for a b b' :: int
+proof -
+ define q q' r r' where **: \<open>q = a div b\<close> \<open>q' = a div b'\<close> \<open>r = a mod b\<close> \<open>r' = a mod b'\<close>
+ then have *: \<open>b * q + r = b' * q' + r'\<close> \<open>b' * q' + r' < 0\<close>
+ \<open>r < b\<close> \<open>0 \<le> r'\<close> \<open>0 < b'\<close> \<open>b' \<le> b\<close>
+ using that by simp_all
+ have \<open>b' * q' < 0\<close>
+ using * by linarith
+ with * have \<open>q' \<le> 0\<close>
+ by (simp add: mult_less_0_iff)
+ have \<open>b * q' \<le> b' * q'\<close>
+ by (simp add: \<open>q' \<le> 0\<close> * mult_right_mono_neg)
+ then have "b * q' < b * (q + 1)"
+ using * by (simp add: distrib_left)
+ then have \<open>q' \<le> q\<close>
+ using * by (simp add: mult_less_cancel_left)
+ then show ?thesis
+ by (simp add: **)
+qed
+
+
+subsubsection \<open>Quotients of Signs\<close>
+
+lemma div_eq_minus1:
+ \<open>0 < b \<Longrightarrow> - 1 div b = - 1\<close> for b :: int
+ by (simp add: divide_int_def)
+
+lemma zmod_minus1:
+ \<open>0 < b \<Longrightarrow> - 1 mod b = b - 1\<close> 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:
+ \<open>a div b < 0\<close> if \<open>a < 0\<close> \<open>0 < b\<close> for a b :: int
+proof -
+ have "a div b \<le> - 1 div b"
+ using zdiv_mono1 that by auto
+ also have "... \<le> -1"
+ by (simp add: that(2) div_eq_minus1)
+ finally show ?thesis
+ by force
+qed
+
+lemma div_nonneg_neg_le0:
+ \<open>a div b \<le> 0\<close> if \<open>0 \<le> a\<close> \<open>b < 0\<close> for a b :: int
+ using that by (auto dest: zdiv_mono1_neg)
+
+lemma div_nonpos_pos_le0:
+ \<open>a div b \<le> 0\<close> if \<open>a \<le> 0\<close> \<open>0 < b\<close> for a b :: int
+ using that by (auto dest: zdiv_mono1)
+
+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:
+ \<open>0 \<le> a div b \<longleftrightarrow> 0 \<le> a\<close>
+ if \<open>0 < b\<close> for a b :: int
+proof
+ assume \<open>0 \<le> a div b\<close>
+ show \<open>0 \<le> a\<close>
+ proof (rule ccontr)
+ assume \<open>\<not> 0 \<le> a\<close>
+ then have \<open>a < 0\<close>
+ by simp
+ then have \<open>a div b < 0\<close>
+ using that by (rule div_neg_pos_less0)
+ with \<open>0 \<le> a div b\<close> show False
+ by simp
+ qed
+next
+ assume "0 \<le> a"
+ then have "0 div b \<le> a div b"
+ using zdiv_mono1 that by blast
+ then show "0 \<le> a div b"
+ by auto
+qed
+
+lemma neg_imp_zdiv_nonneg_iff:
+ \<open>0 \<le> a div b \<longleftrightarrow> a \<le> 0\<close> if \<open>b < 0\<close> for a b :: int
+ using that pos_imp_zdiv_nonneg_iff [of \<open>- b\<close> \<open>- a\<close>] by simp
+
+lemma pos_imp_zdiv_pos_iff:
+ \<open>0 < (i::int) div k \<longleftrightarrow> k \<le> i\<close> if \<open>0 < k\<close> for i k :: int
+ using that pos_imp_zdiv_nonneg_iff [of k i] zdiv_eq_0_iff [of i k] by arith
+
+lemma pos_imp_zdiv_neg_iff:
+ \<open>a div b < 0 \<longleftrightarrow> a < 0\<close> if \<open>0 < b\<close> for a b :: int
+ \<comment> \<open>But not \<^prop>\<open>a div b \<le> 0 \<longleftrightarrow> a \<le> 0\<close>; consider \<^prop>\<open>a = 1\<close>, \<^prop>\<open>b = 2\<close> when \<^prop>\<open>a div b = 0\<close>.\<close>
+ using that by (simp add: pos_imp_zdiv_nonneg_iff flip: linorder_not_le)
+
+lemma neg_imp_zdiv_neg_iff:
+ \<comment> \<open>But not \<^prop>\<open>a div b \<le> 0 \<longleftrightarrow> 0 \<le> a\<close>; consider \<^prop>\<open>a = - 1\<close>, \<^prop>\<open>b = - 2\<close> when \<^prop>\<open>a div b = 0\<close>.\<close>
+ \<open>a div b < 0 \<longleftrightarrow> 0 < a\<close> if \<open>b < 0\<close> for a b :: int
+ using that by (simp add: neg_imp_zdiv_nonneg_iff flip: linorder_not_le)
+
+lemma nonneg1_imp_zdiv_pos_iff:
+ \<open>a div b > 0 \<longleftrightarrow> a \<ge> b \<and> b > 0\<close> if \<open>0 \<le> a\<close> for a b :: int
+proof -
+ have "0 < a div b \<Longrightarrow> b \<le> a"
+ using div_pos_pos_trivial[of a b] that by arith
+ moreover have "0 < a div b \<Longrightarrow> b > 0"
+ using that 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:
+ \<open>m mod k \<le> m\<close> if \<open>(m::int) \<ge> 0\<close> for m k :: int
+proof -
+ from that have \<open>m > 0 \<or> m = 0\<close>
+ by auto
+ then show ?thesis proof
+ assume \<open>m = 0\<close> then show ?thesis
+ by simp
+ next
+ assume \<open>m > 0\<close> then show ?thesis
+ proof (cases k \<open>0::int\<close> rule: linorder_cases)
+ case less
+ moreover define l where \<open>l = - k\<close>
+ ultimately have \<open>l > 0\<close>
+ by simp
+ with \<open>m > 0\<close> have \<open>int (nat m mod nat l) \<le> m\<close>
+ by (simp flip: le_nat_iff)
+ then have \<open>int (nat m mod nat l) - l \<le> m\<close>
+ using \<open>l > 0\<close> by simp
+ with \<open>m > 0\<close> \<open>l > 0\<close> show ?thesis
+ by (simp add: modulo_int_def l_def flip: le_nat_iff)
+ qed (simp_all add: modulo_int_def flip: le_nat_iff)
+ qed
+qed
+
+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
+
+
+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:
+ \<open>k mod l \<ge> 0 \<longleftrightarrow> l dvd k \<or> l = 0 \<and> k \<ge> 0 \<or> l > 0\<close>
+ 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
+
+lemma abs_div:
+ \<open>\<bar>x div y\<bar> = \<bar>x\<bar> div \<bar>y\<bar>\<close> if \<open>y dvd x\<close> for x y :: int
+ using that by (cases \<open>y = 0\<close>) (auto simp add: abs_mult)
+
+lemma int_power_div_base: \<^marker>\<open>contributor \<open>Matthias Daum\<close>\<close>
+ \<open>k ^ m div k = k ^ (m - Suc 0)\<close> if \<open>0 < m\<close> \<open>0 < k\<close> for k :: int
+ using that by (cases m) simp_all
+
+lemma int_div_less_self: \<^marker>\<open>contributor \<open>Matthias Daum\<close>\<close>
+ \<open>x div k < x\<close> if \<open>0 < x\<close> \<open>1 < k\<close> for x k :: int
+proof -
+ from that have \<open>nat (x div k) = nat x div nat k\<close>
+ by (simp add: nat_div_distrib)
+ also from that have \<open>nat x div nat k < nat x\<close>
+ by simp
+ finally show ?thesis
+ by simp
+qed
+
+
+subsubsection \<open>Computing \<open>div\<close> and \<open>mod\<close> by shifting\<close>
+
+lemma div_pos_geq:
+ \<open>k div l = (k - l) div l + 1\<close> if \<open>0 < l\<close> \<open>l \<le> k\<close> for k l :: int
+proof -
+ have "k = (k - l) + l" by simp
+ then obtain j where k: "k = j + l" ..
+ with that show ?thesis by (simp add: div_add_self2)
+qed
+
+lemma mod_pos_geq:
+ \<open>k mod l = (k - l) mod l\<close> if \<open>0 < l\<close> \<open>l \<le> k\<close> for k l :: int
+proof -
+ have "k = (k - l) + l" by simp
+ then obtain j where k: "k = j + l" ..
+ with that show ?thesis by simp
+qed
+
+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>2 dvd (2 * a)\<close>
+ by simp
+ then have \<open>2 dvd (1 + 2 * b)\<close>
+ using \<open>2 * a dvd 1 + 2 * b\<close> by (rule dvd_trans)
+ then have \<open>2 dvd (1 + b * 2)\<close>
+ by (simp add: ac_simps)
+ then have \<open>is_unit (2 :: int)\<close>
+ by simp
+ 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>2 dvd (2 * a)\<close>
+ by simp
+ then have \<open>2 dvd (1 + 2 * b)\<close>
+ using \<open>2 * a dvd 1 + 2 * b\<close> by (rule dvd_trans)
+ then have \<open>2 dvd (1 + b * 2)\<close>
+ by (simp add: ac_simps)
+ then have \<open>is_unit (2 :: int)\<close>
+ by simp
+ 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]:
+ \<open>numeral (Num.Bit0 v) div numeral (Num.Bit0 w) =
+ numeral v div (numeral w :: int)\<close>
+ unfolding numeral.simps unfolding mult_2 [symmetric]
+ by (rule div_mult_mult1) simp
+
+lemma zdiv_numeral_Bit1 [simp]:
+ \<open>numeral (Num.Bit1 v) div numeral (Num.Bit0 w) =
+ (numeral v div (numeral w :: int))\<close>
+ unfolding numeral.simps
+ unfolding mult_2 [symmetric] add.commute [of _ 1]
+ by (rule pos_zdiv_mult_2) simp
+
+lemma zmod_numeral_Bit0 [simp]:
+ \<open>numeral (Num.Bit0 v) mod numeral (Num.Bit0 w) =
+ (2::int) * (numeral v mod numeral w)\<close>
+ unfolding numeral_Bit0 [of v] numeral_Bit0 [of w]
+ unfolding mult_2 [symmetric] by (rule mod_mult_mult1)
+
+lemma zmod_numeral_Bit1 [simp]:
+ \<open>numeral (Num.Bit1 v) mod numeral (Num.Bit0 w) =
+ 2 * (numeral v mod numeral w) + (1::int)\<close>
+ unfolding numeral_Bit1 [of v] numeral_Bit0 [of w]
+ unfolding mult_2 [symmetric] add.commute [of _ 1]
+ by (rule pos_zmod_mult_2) simp
+
+
subsection \<open>Generic symbolic computations\<close>
text \<open>
@@ -2548,13 +3049,13 @@
"numeral m dvd numeral n \<longleftrightarrow> divides_aux (divmod n m)"
by (simp add: divmod_def mod_eq_0_iff_dvd)
-text \<open>Generic computation of quotient and remainder\<close>
-
-lemma numeral_div_numeral [simp]:
+text \<open>Generic computation of quotient and remainder\<close>
+
+lemma numeral_div_numeral [simp]:
"numeral k div numeral l = fst (divmod k l)"
by (simp add: fst_divmod)
-lemma numeral_mod_numeral [simp]:
+lemma numeral_mod_numeral [simp]:
"numeral k mod numeral l = snd (divmod k l)"
by (simp add: snd_divmod)
@@ -2622,7 +3123,7 @@
context
begin
-
+
qualified definition adjust_div :: "int \<times> int \<Rightarrow> int"
where
"adjust_div qr = (let (q, r) = qr in q + of_bool (r \<noteq> 0))"
@@ -2667,7 +3168,7 @@
then show ?thesis
by (auto simp add: split_def Let_def adjust_div_def divides_aux_def divide_int_def)
qed
-
+
lemma numeral_mod_minus_numeral [simp]:
"numeral m mod - numeral n = - adjust_mod n (snd (divmod m n) :: int)"
proof (cases "snd (divmod m n) = (0::int)")
@@ -2685,7 +3186,7 @@
lemma minus_one_div_numeral [simp]:
"- 1 div numeral n = - (adjust_div (divmod Num.One n) :: int)"
- using minus_numeral_div_numeral [of Num.One n] by simp
+ using minus_numeral_div_numeral [of Num.One n] by simp
lemma minus_one_mod_numeral [simp]:
"- 1 mod numeral n = adjust_mod n (snd (divmod Num.One n) :: int)"
@@ -2694,14 +3195,14 @@
lemma one_div_minus_numeral [simp]:
"1 div - numeral n = - (adjust_div (divmod Num.One n) :: int)"
using numeral_div_minus_numeral [of Num.One n] by simp
-
+
lemma one_mod_minus_numeral [simp]:
"1 mod - numeral n = - adjust_mod n (snd (divmod Num.One n) :: int)"
using numeral_mod_minus_numeral [of Num.One n] by simp
lemma [code]:
fixes k :: int
- shows
+ shows
"k div 0 = 0"
"k mod 0 = k"
"0 div k = 0"
@@ -2788,7 +3289,7 @@
\<close> \<comment> \<open>
There is space for improvement here: the calculation itself
could be carried out outside the logic, and a generic simproc
- (simplifier setup) for generic calculation would be helpful.
+ (simplifier setup) for generic calculation would be helpful.
\<close>