--- a/src/HOL/Divides.thy Fri Dec 30 18:02:27 2016 +0100
+++ b/src/HOL/Divides.thy Sat Dec 31 08:12:31 2016 +0100
@@ -1823,6 +1823,103 @@
"is_unit (k::int) \<longleftrightarrow> k = 1 \<or> k = - 1"
by auto
+lemma zdiv_int:
+ "int (a div b) = int a div int b"
+ by (simp add: divide_int_def)
+
+lemma zmod_int:
+ "int (a mod b) = int a mod int b"
+ by (simp add: modulo_int_def int_dvd_iff)
+
+lemma div_abs_eq_div_nat:
+ "\<bar>k\<bar> div \<bar>l\<bar> = int (nat \<bar>k\<bar> div nat \<bar>l\<bar>)"
+ by (simp add: divide_int_def)
+
+lemma mod_abs_eq_div_nat:
+ "\<bar>k\<bar> mod \<bar>l\<bar> = int (nat \<bar>k\<bar> mod nat \<bar>l\<bar>)"
+ by (simp add: modulo_int_def dvd_int_unfold_dvd_nat)
+
+lemma div_sgn_abs_cancel:
+ fixes k l v :: int
+ assumes "v \<noteq> 0"
+ shows "(sgn v * \<bar>k\<bar>) div (sgn v * \<bar>l\<bar>) = \<bar>k\<bar> div \<bar>l\<bar>"
+proof -
+ from assms have "sgn v = - 1 \<or> sgn v = 1"
+ by (cases "v \<ge> 0") auto
+ then show ?thesis
+ using assms unfolding divide_int_def [of "sgn v * \<bar>k\<bar>" "sgn v * \<bar>l\<bar>"]
+ by (auto simp add: not_less div_abs_eq_div_nat)
+qed
+
+lemma div_eq_sgn_abs:
+ fixes k l v :: int
+ assumes "sgn k = sgn l"
+ shows "k div l = \<bar>k\<bar> div \<bar>l\<bar>"
+proof (cases "l = 0")
+ case True
+ then show ?thesis
+ by simp
+next
+ case False
+ with assms have "(sgn k * \<bar>k\<bar>) div (sgn l * \<bar>l\<bar>) = \<bar>k\<bar> div \<bar>l\<bar>"
+ by (simp add: div_sgn_abs_cancel)
+ then show ?thesis
+ by (simp add: sgn_mult_abs)
+qed
+
+lemma div_dvd_sgn_abs:
+ fixes k l :: int
+ assumes "l dvd k"
+ shows "k div l = (sgn k * sgn l) * (\<bar>k\<bar> div \<bar>l\<bar>)"
+proof (cases "k = 0")
+ case True
+ then show ?thesis
+ by simp
+next
+ case False
+ show ?thesis
+ proof (cases "sgn l = sgn k")
+ case True
+ then show ?thesis
+ by (simp add: div_eq_sgn_abs)
+ next
+ case False
+ with \<open>k \<noteq> 0\<close> assms show ?thesis
+ unfolding divide_int_def [of k l]
+ by (auto simp add: zdiv_int)
+ qed
+qed
+
+lemma div_noneq_sgn_abs:
+ fixes k l :: int
+ assumes "l \<noteq> 0"
+ assumes "sgn k \<noteq> sgn l"
+ shows "k div l = - (\<bar>k\<bar> div \<bar>l\<bar>) - of_bool (\<not> l dvd k)"
+ using assms
+ by (simp only: divide_int_def [of k l], auto simp add: not_less zdiv_int)
+
+lemma sgn_mod:
+ fixes k l :: int
+ assumes "l \<noteq> 0" "\<not> l dvd k"
+ shows "sgn (k mod l) = sgn l"
+proof -
+ from \<open>\<not> l dvd k\<close>
+ have "k mod l \<noteq> 0"
+ by (simp add: dvd_eq_mod_eq_0)
+ show ?thesis
+ using \<open>l \<noteq> 0\<close> \<open>\<not> l dvd k\<close>
+ unfolding modulo_int_def [of k l]
+ by (auto simp add: sgn_1_pos sgn_1_neg mod_greater_zero_iff_not_dvd nat_dvd_iff not_less
+ zless_nat_eq_int_zless [symmetric] elim: nonpos_int_cases)
+qed
+
+lemma abs_mod_less:
+ fixes k l :: int
+ assumes "l \<noteq> 0"
+ shows "\<bar>k mod l\<bar> < \<bar>l\<bar>"
+ using assms unfolding modulo_int_def [of k l]
+ by (auto simp add: not_less int_dvd_iff mod_greater_zero_iff_not_dvd elim: pos_int_cases neg_int_cases nonneg_int_cases nonpos_int_cases)
+
instance int :: ring_div
proof
fix k l s :: int
@@ -1870,12 +1967,6 @@
text\<open>Basic laws about division and remainder\<close>
-lemma zdiv_int: "int (a div b) = int a div int b"
- by (simp add: divide_int_def)
-
-lemma zmod_int: "int (a mod b) = int a mod int b"
- by (simp add: modulo_int_def int_dvd_iff)
-
lemma pos_mod_conj: "(0::int) < b \<Longrightarrow> 0 \<le> a mod b \<and> a mod b < b"
using eucl_rel_int [of a b]
by (auto simp add: eucl_rel_int_iff prod_eq_iff)
--- a/src/HOL/Power.thy Fri Dec 30 18:02:27 2016 +0100
+++ b/src/HOL/Power.thy Sat Dec 31 08:12:31 2016 +0100
@@ -582,10 +582,22 @@
context linordered_idom
begin
-lemma power_abs: "\<bar>a ^ n\<bar> = \<bar>a\<bar> ^ n"
- by (induct n) (auto simp add: abs_mult)
+lemma zero_le_power2 [simp]: "0 \<le> a\<^sup>2"
+ by (simp add: power2_eq_square)
+
+lemma zero_less_power2 [simp]: "0 < a\<^sup>2 \<longleftrightarrow> a \<noteq> 0"
+ by (force simp add: power2_eq_square zero_less_mult_iff linorder_neq_iff)
-lemma abs_power_minus [simp]: "\<bar>(-a) ^ n\<bar> = \<bar>a ^ n\<bar>"
+lemma power2_less_0 [simp]: "\<not> a\<^sup>2 < 0"
+ by (force simp add: power2_eq_square mult_less_0_iff)
+
+lemma power_abs: "\<bar>a ^ n\<bar> = \<bar>a\<bar> ^ n" -- \<open>FIXME simp?\<close>
+ by (induct n) (simp_all add: abs_mult)
+
+lemma power_sgn [simp]: "sgn (a ^ n) = sgn a ^ n"
+ by (induct n) (simp_all add: sgn_mult)
+
+lemma abs_power_minus [simp]: "\<bar>(- a) ^ n\<bar> = \<bar>a ^ n\<bar>"
by (simp add: power_abs)
lemma zero_less_power_abs_iff [simp]: "0 < \<bar>a\<bar> ^ n \<longleftrightarrow> a \<noteq> 0 \<or> n = 0"
@@ -600,15 +612,6 @@
lemma zero_le_power_abs [simp]: "0 \<le> \<bar>a\<bar> ^ n"
by (rule zero_le_power [OF abs_ge_zero])
-lemma zero_le_power2 [simp]: "0 \<le> a\<^sup>2"
- by (simp add: power2_eq_square)
-
-lemma zero_less_power2 [simp]: "0 < a\<^sup>2 \<longleftrightarrow> a \<noteq> 0"
- by (force simp add: power2_eq_square zero_less_mult_iff linorder_neq_iff)
-
-lemma power2_less_0 [simp]: "\<not> a\<^sup>2 < 0"
- by (force simp add: power2_eq_square mult_less_0_iff)
-
lemma power2_less_eq_zero_iff [simp]: "a\<^sup>2 \<le> 0 \<longleftrightarrow> a = 0"
by (simp add: le_less)
@@ -618,7 +621,7 @@
lemma power2_abs [simp]: "\<bar>a\<bar>\<^sup>2 = a\<^sup>2"
by (simp add: power2_eq_square)
-lemma odd_power_less_zero: "a < 0 \<Longrightarrow> a ^ Suc (2*n) < 0"
+lemma odd_power_less_zero: "a < 0 \<Longrightarrow> a ^ Suc (2 * n) < 0"
proof (induct n)
case 0
then show ?case by simp
@@ -630,11 +633,11 @@
by (simp del: power_Suc add: Suc mult_less_0_iff mult_neg_neg)
qed
-lemma odd_0_le_power_imp_0_le: "0 \<le> a ^ Suc (2*n) \<Longrightarrow> 0 \<le> a"
+lemma odd_0_le_power_imp_0_le: "0 \<le> a ^ Suc (2 * n) \<Longrightarrow> 0 \<le> a"
using odd_power_less_zero [of a n]
by (force simp add: linorder_not_less [symmetric])
-lemma zero_le_even_power'[simp]: "0 \<le> a ^ (2*n)"
+lemma zero_le_even_power'[simp]: "0 \<le> a ^ (2 * n)"
proof (induct n)
case 0
show ?case by simp